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f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 1 CHAPTER 3 version 29-9-09 intl 18pt INTRODUCTION TO THE MARGINALIST APPROACH 3.1. Introduction 3.1.1. We proceed to contrast the classical with the marginalist, or supply-and-demand, or neoclassical, approach to value and distribution, and to highlight the central analytical differences between them. This chapter presents the basic structure of the marginalist approach (in particular, the factor substitution mechanisms motivating the thesis that the „real‟ forces operating in a competitive market economy push toward a full-employment equilibrium) assuming on the part of the reader only the knowledge of the microeconomics taught in a good introductory economics course[1]; the more rigorous study of consumer theory, producer theory, and general equilibrium theory will come afterwards. The advantage of this non-conventional order of topics is threefold. First, it is better to be clear as soon as possible on the main differences between classical and marginalist approach: this will allow a better focus on the issues central to a comparison of the scientific robustness of the two approaches. Second, when studying the marginalist approach to the theory of the consumer and of the producer, the reader will have a clearer idea of the analytical motivation of the several assumptions. Third, when studying the important shift undergone in recent decades by the approach, from long-period formulations to the modern very-short-period (or neo- Walrasian) formulations, the reader will find it easier to understand the basic theses that the shift was trying to preserve in the face of the difficulties encountered by the earlier formulations. 3.1.2. The birth of the marginalist approach to value and distribution can be located around 1870, when almost simultaneously and largely independently Stanley Jevons in Great Britan, Carl Menger in Austria, and Léon Walras in France published books which present the first explicit formulations of that approach. By the end of the century the approach was the dominant one, as testified by the contributions – to name just the more important ones – of Marshall, Edgeworth and Wicksteed in Great Britain, Wieser and Böhm-Bawerk in Austria, Pareto and Barone in Italy, J. B. Clark and I. Fisher in the United States of America, Wicksell and Cassel in Sweden. We do not stop 1 More specifically, this chapter assumes that the reader knows the basic calculus of functions of several variables, and the following notions at the level of a good introductory economics course: utility function, marginal utility, indifference curve, marginal rate of substitution (i.e. slope of an indifference curve), budget line, tangency betwen budget line and (convex) indifference curve as condition for utility maximization; production function, marginal product, isoquant, technical rate of substitution (i.e. slope of an isoquant), isocost, tangency between isocost and (convex) isoquant as condition for cost minimization, average cost, marginal cost, marginal revenue, equality between marginal cost and marginal revenue as condition for maximum profit, coincidence between marginal revenue and product price under price taking, elasticity, externality, public goods. These notions are re-examined more rigorously in subsequent chapters. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 2 on the differences among these authors. The fact, that the marginalist approach is what students are introduced to from the very beginning of their studies, makes it possible and convenient to go straight to how its basic insights are reached; on the contrary for the classical or surplus approach, still often misunderstood, probably unfamiliar to a majority of students, and whose resumption is still in an early stage, a gradual introduction to its historical development seemed the best way to lead students into its structure, problems, and mode of approaching economic issues. Jevons, Walras, Menger were all acutely conscious of the novelty of their approach vis-à-vis the Ricardian one. Alfred Marshall on the contrary tried to minimize the extent of the change, and this probably helped the new doctrine to be rapidly accepted; but nowadays it seems clear that the other three authors were correct on the radicality of the change in approach. The surplus approach views the capitalist economy as conflictual, generally unable to realize the full employment of labour, and, at least according to Marx and some other classical authors, prone to crises; the distribution of income between wages and profits is the result of the relative bargaining power of social classes with conflicting interests, bargaining power which depends on unemployment but also on institutions and on other social and political elements; the income of capitalists is not the result of some productive contribution of theirs but only of their superior bargaining strength, that allows them to keep wages below their potential maximum. The marginalist approach, as we will see, views the capitalist economy as essentially efficient (the economy tends to the full employment of resources if the working of free competition is not obstructed, and to a composition of production that reflects consumers‟ desires), views production as characterized by co-operation between the efforts and sacrifices of the suppliers of productive services, and views income distribution as obeying a criterion, "to each according to his/her contribution to social welfare", which may need corrections but can hardly be found basically unjust. The differences could not be greater. The purpose of this chapter is to point out the analytical roots of these differences as simply as possible. Contrary to widespread views that locate the fundamental novelty of the marginalist approach in the notion of marginal utility, it will be argued that the really important differences from the classical approach derive from a central analytical novelty: the claim that the Ricardian theory of differential rent can and must be extended, from the determination of how the social product is divided between land rent on one side and the ensemble of wages and profits (interest) on the other side (the role of rent theory in Ricardo), to the determination of how this second portion is divided between wages and profits (interest) too: the implication being that wages too, as well as income from the property of capital, are determined in the same way as the rent of land; the theory of rent applies to all factor incomes. It will be shown that it is from this difference that the other important differences derive, e.g. the existence in the marginalist approach, and absence in the classical approach, of decreasing demand curves for factors; the much broader role of deductive reasoning in the marginalist approach; or the different conclusions as to whether labour is exploited. In subsequent chapters we will have to discuss the difficulties that this extension has encountered, difficulties whose overcoming has been attempted in different ways; this will oblige us f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 3 to examine several versions of the marginalist approach to value and distribution. 3.2. Equilibrium and gravitation. 3.2.1. The marginalist approach argues that in a market economy prices, quantities produced, and the distribution of the social product between wages, rents, interest[2] are determined by the interaction between supply and demand (viewed as functions of product prices and of income distribution); this interaction – it is argued – causes the economy to gravitate toward a state of equality, or equilibrium, between supply and demand for all goods and all [services of] productive factors[3]. Why is equality between supply and demand described as an equilibrium? Because in such a state the force (on the demand side) pulling toward a rise in price, and the force (on the supply side) pushing toward a fall in price neutralize each other, and are therefore in equilibrium. A physical analogy is an object hanging by an elastic string: the weight of the object pushes it downwards, but the lower the object, the greater the upward pull exercised by the string, so the object will oscillate up and down until it comes to rest in a position where the downward push and the upward pull are in equilibrium. An equilibrium such as the one just described indicates the situation toward which the physical system comes back if perturbed (it is then called a stable equilibrium); other equilibria, e.g. an egg standing in equilibrium on its tip, do not share this characteristic, if disturbed the egg rolls on a side i.e. further away from the original equilibrium. In order to be useful as an indication of the average arond which day-by-day prices and quantities gravitate, the equilibrium between supply and demand must be of the first type, i.e. stable: when demand is different from supply, this must cause the price to tend toward its equilibrium value. The meaning of stability of equilibrium is easy to grasp in the analysis of a single market: when demand and supply are not equal, the higgling and bargaining – with a lower price being asked by sellers who are unable to find purchasers, and a higher price being offered by demanders unable to get the good – will cause the average price[4] to increase if demand is greater than supply, and to decrease in the opposite case; for the equilibrium to be stable, the average market price must increase when it 2 In the marginalist approach, the rate of interest takes the place that the rate of profit has in the classical approach. The meaning itself of 'profit' is altered, to indicate what is left of the firm's revenue after paying all costs including interest on the capital employed (and also the addition to interest necessary to cover risk, which here we neglect for simplicity); for the classical authors, on the contrary, profits were what was left before paying interest; thus the marginalist approach describes a situation, where the revenue of a firm is just sufficient to cover costs and interest at the normal rate, as a situation of zero profits, while a classical author would describe it as a situation of profits equal to interests, i.e. of rate of profit equal to the rate of interest. 3 In the marginalist approach, what is actually offered and demanded is the services of land, labour, and capital; but for simplicity one usually speaks of supply and demand of land, labour, and capital. 4 Reference to some average price is made necessary by the fact that, in disequilibrium, price will not generally be uniform for all the units exchanged of a commodity. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 4 is below the equilibrium price, must decrease when it is above the equilibrium price. We find here again the notion of a gravitation toward a 'normal' position that we have found in the classical authors' conception of a gravitation of market prices toward natural prices or prices of production. But in the marginalist approach the determination of the 'normal' positions is different from the classical one. The fundamental difference concerns the determination of income distribution, and to this we now turn. 3.3. The labour-land-corn economy: the role of direct (or technological) factor substitutability. 3.3.1. The marginalist approach argues that, in a freely competitive economy, income distribution is determined by the operation of mechanisms that act in the same way for all incomes from the property of „factors of production‟, and ensure a tendency toward equilibrium (equality between supply and demand) on all factor markets. The use of the general name „factor of production‟ for each kind of labour, for each kind of land, and for capital, is justified precisely by the fundamental symmetry of the mechanisms acting in the markets for their services. In order clearly to grasp these mechanisms, in this chapter we restrict the analysis to very simple cases, whose study will only require notions familiar to anyone who has read a good introductory economics textbook. Let us first consider an economy where a single consumption good, corn, is produced by employing labour and land. (We neglect the need for seed-corn for the time being.) For simplicity we assume that there is only one kind of labour, and only one quality of land. There is free competition which tends to establish the same price for all units of the same good or service; we choose corn as the unit in which relative prices are expressed, i.e., as the numéraire commodity, hence the price of corn is 1, and there are only two prices to be determined: the rate of wages, w, and the rate of rent, . The production of corn can be carried out according to a variety of different methods, which we assume can be represented through a production function q=f(L,T), a function of two variables which indicates the maximum quantity of corn producible in a year with the utilization of amounts L and T of services of labour and of land. The inputs to a production function are amounts of services of factors of production, but for brevity they are often called amounts of factors of production. We assume that this production function is continuously differentiable. Then the marginal product MPi of the i-th factor, which, approximately speaking, is the increase in output obtainable if the amount employed of that factor is increased by one (small) unit while the amounts employed of the other factors remain unchanged, can be rigorously defined as the partial derivative of the production function: e.g. MPL(L,T):=∂f(L,T)/∂L. In order initially to leave consumer choices f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 5 out of the picture, we assume that the aggregate supply of labour services to firms, L*, is rigid, and that the same holds for the supply of land T*[5], and we also assume that there are no savings: the income of a period is entirely spent in that period. In this economy there are three markets. A general equilibrium of such an economy is a situation in which the supply of labour equals the demand for labour, the supply of land equals the demand for land, and the supply of the product, corn, equals the demand for it. Our assumption that the income of a period is entirely spent in that period guarantees equilibrium on the product market: by definition, the aggregate income of the economy is the value of aggregate production, which is distributed to some consumer or other as wages, as rents, and as residual entrepreneurial profits[ 6 ] (possibly negative for some firms); since this income is by assumption all spent in the purchase of the product, supply of corn and demand for corn are necessarily equal[7]. What remains to study is the forces acting on the labour market and on the land market. 3.3.2. It is convenient initially to assume that landlords (the owners of land) and workers are two separate groups of people, and further to assume that the landlords act as entrepreneurs, each one setting up a firm which produces corn with the landlord's own land and with hired wage labour. Then the equality between supply and demand for land is guaranteed; the sole market on which the equilibrium between supply and demand must be reached is the labour market. Each firm is assumed to have a production function q=f(L,T) which for the moment we assume specific to the firm: the production function reflects the technical knowledge of the entrepreneur. For each firm, given our assumption that the entrepreneur uses her own land (supplied in a given amount T°), labour is the sole variable factor. The microeconomic theory of the firm teaches that the optimal decision for a firm is to employ such a quantity of each variable factor as will make the „value marginal product‟ of the factor equal to its 'price' (or rather to its rental, i.e. price of its services). In our case, since the firm is competitive and the price of the product is 1, the firm must employ the amount of labour that, with MPL the marginal product of labour and w the real wage, establishes the following equality[8]: 5 To such an end, it is not necessary to assume that there is no direct demand for labour or land services by consumers (e.g. demand for domestic labour, or for pleasure gardens or hunting grounds); one can assume that such demands are given, and are subtracted from the given total aggregate supplies of labour and of land; what is left is the supply to firms. We will relax these assumptions later. 6 In this chapter we use „profits‟ in the marginalist sense, cf. fn. 2. 7 If some reader familiar with Walras‟ Law finds this statement surprising, he is asked to wait for §3.4.2. 8 We recall the demonstration: let MR(q(L,T)) be the marginal revenue of the firm i.e. the increase in revenue obtainable from one more (small) unit of output when total output is q; let MPL(L,T) be the marginal product of labour; then the increase in revenue obtainable with one more (small) unit of labour, i.e. the value marginal product of labour, is MR∙MPL; as long as this is greater than the wage rate, the utilization of one more unit of labour increases revenue more than cost i.e. it increases profit, so the firm finds it convenient to → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 6 MPL = w . Let us draw the curve representing MP L (on the vertical axis) as a function of the amount of labour employed. As usual we draw it as a decreasing curve (the horizontal portion will be discussed later). If we also measure w on the vertical axis, the curve indicates the labour demanded by the firm as a function of w, cf. Fig. 3.1. When e.g. w=w1, the optimal labour employment is the one such that MPL(L) = w1, i.e. L1. It is clear that as w decreases, the demand for labour by the firm increases. The curve of the marginal product of labour coincides with the (inverse) demand curve for labour; it is the same curve, only with the independent variable (the wage rate) measured on the vertical axis. Given w, the optimal labour employment L(w,T°)=MPL–1(w,T°) determines how much corn is produced; labour gets MPL∙L, and the landlord gets the residual, f(L(w,T°),T°)–wL(w) . intl 1 line MPL, w MPL labour supply w1 we aggr. labour demand L1 L' L* L (a) (b) Fig. 3.1 By assuming that landlords are the entrepreneurs, each one with his own land and his own technical knowledge, we have implicitly also assumed that there is a finite, given number of firms, each one with its demand curve for labour. We can determine the aggregate demand for labour for go on increasing the utilization of the factor; but since MP L is decreasing and MR is non-increasing, the increase will stop being convenient when an equality is reached between value marginal product and factor rental; thus optimal labour employment requires w=MR∙MPL. With perfect competition, as we are here assuming, the entrepreneur is price-taker, i.e. treats the product price as given and esteems that additional units of output will sell at the same price as previous units, hence MR=p; here p=1, hence optimal labour employment requires w=MPL. In this book we will generally use the term 'factor rental' to indicate the price of the services of a factor, i.e. the rental price, the price of hiring the factor for one period. Most other textbooks use the term 'factor price' in the same sense, but this may create confusion when one can purchase the factor itself: for example in common parlance 'price of land' means the purchase price, not the rental price, of land. The same ambiguity arises with capital goods, e.g. a tractor can be hired, or it can be purchased. The 'factor price' that must be equalized with the value marginal product is the price of the services i.e. the rental, not the purchase price, of land or of a tractor. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 7 each level of w by simple summation of the demands of each firm. Since the demand curve of each firm is decreasing, so will be the aggregate demand curve for labour. We can proceed to determine the equilibrium wage by drawing in the same diagram the aggregate labour demand curve and the aggregate labour supply curve. By assumption the latter is vertical, reflecting a rigid labour supply L*. Therefore there cannot be more than one intersection of the two curves(9): the equilibrium wage we is unique, cf. Fig. 3.1b. Furthermore, when w > we labour demand is smaller than labour supply, i.e. there is unemployment, and if the unemployed offer to work for a lower wage rather than go without a job, then w will tend to fall; conversely when w < we demand is greater than supply, and if the entrepreneurs who are unable to get all the workers they desire compete for workers by raising the wage they offer, then w will tend to rise. Thus the equilibrium is not only unique, it is also stable i.e. there is a tendency toward it when the market is in disequilibrium. 3.3.3. But the assumption that production functions are specific to each firm, as well as the assumption that only the landlords are entrepreneurs, with each landlord only using his own land (then the number of firms is given), are very restrictive, and must be removed. Let us remove the first one. Learning and imitative processes do exist in actual economies, so it is plausible that, given time, the knowledge of the most efficient production methods will become general, and firms will tend to converge to a similar level of efficiency. This process of technological convergence is slow when production uses very durable capital goods, which often go on being used even when technologically obsolete; but for the moment we neglet this complication (we are discussing an economy without capital goods), and we assume that the process of technological convergence has a velocity of the same order of magnitude as the velocity of the processes that cause wages to tend to uniformity in different firms, and of the processes that cause the average wage to change in response to differences between supply and demand for labour; note that both the latter processes take considerable time (in the absence of centralized bargaining), involving the hiring and firing of workers, the search for new jobs, the collection of information, etc., thus the assumption of a similar velocity is not so implausible. Thus we assume now that the labour market equilibrium is associated with a uniform technical knowledge of firms. We keep assuming that land is fully utilized. The assumption that technical knowledge is equally shared implies that all firms will utilize the same production function. We assume that the common production function exhibits constant technological returns to 9 It is possible that there be no intersection (in the positive quadrant), labour demand might be less than supply even for w=0; but even in this case the equilibrium wage exists and is uniquely determined as equal to zero, its lowest possible value. A market in which supply is greater than demand and the price is zero is also considered in equilibrium, in that the price has no incentive to change (it cannot become negative!). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 8 scale (to be abbreviated as CRS), which means that an equal proportional increase of all factors determines an increase of the product in the same proportion: e.g. doubling all inputs doubles the output. Mathematically, the production function is homogeneous of degree one i.e. satisfies the condition (with c a positive scalar) f(cL,cT) = cf(L,T). 22pt w SL MPL = DL L° L* L Fig. 3.2 A property of differentiable functions homogeneous of degree k is that their partial derivatives are themselves homogeneous functions but of degree k−1 (cf. the Mathematical Appendix); applied to a CRS production function this means that the marginal product of a factor is a homogeneous function of degree zero, i.e. does not change if all factors change in the same percentage, it depends only on the proportions among factors: in our case, MPL and MPT (the marginal product of land) depend only on the amount of labour per unit of land L/T, i.e. do not change if L and T change in the same proportion. It can also be shown (cf. chapter 5) that in the two-factors case the relationship is invertible: a given marginal product of a factor uniquely determines L/T [10]. This also means that a given MPL uniquely determines the associated MPT and vice-versa. In this case, since firms converge toward a situation of identical production functions, then a given w implies that all firms will want to reach the same MP L=w and therefore all firms will tend to adopt the same L/T ratio. Let us indicate this ratio as[11] Λ(w); it is the ratio that renders MPL=w. 10 When the marginal product of a factor is initially increasing, there may be two L/T ratios determining the same MPL. But, as argued in §3.3.4, only the higher of the two L/T ratios is economically relevant. 11 The symbol Λ is capital λ (the Greek letter „lambda‟). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 9 Then if the total amount of land employed is T*, the aggregate demand for labour is T*∙Λ(w). Example: if firms desire to employ 7 units of labour per unit of land, and in the aggregate they employ 100 units of land, the demand for labour is 700. Note (this will be relevant when we remove the assumption that each entrepreneur only uses his own land) that since Λ is the same in all firms, firm size is irrelevant to the determination of labour demand: it is irrelevant whether the 100 units of land are distributed among 100 identical firms, each one demanding 7 units of labour, or among 10 firms each one demanding 70 units of labour, or in any other way. So we can also determine the demand for labour as if it came from a hypothetical single firm, employing the entire supply of land and having the same CRS production function as each firm. The curve of the marginal product of labour of this mega-firm can be called the economy-wide marginal-product-of-labour curve, and it coincides with the (inverse) demand curve for labour, cf. Fig. 3.2. (The horizontal section of the labour demand curve in Fig. 3.2 is explained in §3.3.6.) 3.3.4. In our simple competitive economy the labour demand curve coincides with the curve of the economy-wide marginal product of labour, which, at least from a certain point onwards, is necessarily downward sloping (it is inconceivable that additional units of labour on a given amount of land produce the same addition to corn output indefinitely, unless land is not actually necessary for the production of corn). What about the income of landlords? It is determined residually, as what is left of the total product Q*=F(L*,T*) after labour gets its marginal product; it is therefore given by [3.1] landlords' equilibrium total income = Q* – MPL*∙L* = F(L*,T*) – MPL*∙L*. The rate of land rent β thus residually determined is given by the landlords‟ income divided by T*. Now we use another important mathematical property of constant-returns-to-scale (CRS) production functions, the so-called product exhaustion theorem, that states: if a differentiable production function has constant returns to scale (i.e. is homogeneous of degree one), then the payment to each factor of its marginal product exhausts (i.e. adds up to) the product(12). In our case: [3.2] F(L,T) = MPL∙L + MPT∙T. This result means that if one factor is paid its marginal product, what is left of the product is just enough to pay the other factor its marginal product. Applied to full-employment production, 12 This result is an immediate corollary of the so-called Euler Theorem on homogeneous functions, that states that if f(cx1,...cxn)=ckf(x1,...,xn), then ∑i(xi∙∂f/∂xi)=kf(x1,...,xn), cf. the Mathematical Appendix. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 10 this means Q* = MPL*∙L* + MPT* ∙T*. But then [3.3] β ≡ (Q* – MPL*∙L*)/T* = MPT*. The residual income per unit of land accruing to landlords once labour receives its marginal product is just what their land would receive if it were paid its marginal product! But then the equilibrium would not be affected if the role of entrepreneurs were taken by someone else, the labourers for example, or a third party hiring both labour and land. If labourers are the entrepreneurs (e.g. grouped in co-operatives), and hire land, then labour supply is automatically equal to labour demand, and the land market is the sole market where equilibrium needs to be brought about by a price, now the rate of rent . Assuming technology is the same as in the other case, one obtains a demand curve for land coinciding with the marginal product of land in the economy as a whole, and equilibrium obtains when the rent rate equals the full-employment marginal product of land, i.e. = ∂F(L*,T*)/∂T =MPT*. Landlords earn the same as when they are entrepreneurs; and workers too, because their residual income is Q* – MPT*∙T* = MPL*∙L*. This perfect symmetry shows that according to this theory all factors tend to earn their marginal product independently of who acts as entrepreneur. Land earns its full-employment marginal product not only when it is hired by labourers-entrepreneurs but also when it is the landlords who act as entrepreneurs. And if the entrepreneurs are a third party, hiring labour and land, profit maximization requires paying each factor its marginal product, so in equilibrium land still earns its full-employment marginal product (and the same is true for labour); then the product exhaustion theorem implies that in equilibrium the entrepreneur as such makes neither profit nor loss. 3.3.5. In this approach the role of entrepreneurs yields nothing, in equilibrium. This is not surprising in view of the fact that, as shown by the production function, entrepreneurial activity does not contribute to production. Smart entrepreneurs can earn profit in disequilibrium by adjusting faster than the others and exploiting opportunities for profit before the market adjustments wipe those opportunities out; but in equilibrium they only have the legal role of owners of the firm, and this role, according to marginalist theory, does not give them any right to an income[13]. (If, besides being the owners, they also work as managers/supervisors, then they are contributing some labour to the production process and then will receive a salary for that labour, determined by the marginal product of their labour. We have not considered this possibility for simplicity, owing to 13 As Wicksell noted, if the entrepreneur “could obtain a share of the product merely in his capacity of entrepreneur (a share not based on either labour nor land) then it might be thought that everybody would rush to obtain such an easily earned income.” (Wicksell, Lectures vol. I p. 126) f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 11 our assumption of homogeneous labour.) That entrepreneurs as such earn nothing in equilibrium is indeed lucky for the theory. Suppose the product exhaustion theorem not to hold. Suppose that it is the case that F(L,T) > MPL∙L + MPT∙T . Then the residual income of landlords-entrepreneurs is greater than the marginal product of their land, F(L,T) – MPL∙L > MPT∙T, and landlords will prefer to act as entrepreneurs rather than to rent their land to others and earn its marginal product; but the same preference for acting as entrepreneurs holds for labourers, who can earn a residual income F(L,T) – MPT∙T > MPL∙L; so everybody wants to act as entrepreneur, nobody wants to supply their factors to others, and production cannot start. Now suppose that F(L,T) < MPL∙L + MPT∙T . Then no factor owner wants to act as entrepreneur because she would earn less than by supplying her factor to other firms and earning its marginal product; and no third party wants to act as entrepreneur because she would incur losses; so again production cannot start. 3.3.6. The above reasonings show that the assumption that all the entrepreneurs are landlords was motivated only by pedagogical usefulness, and can be abandoned without altering the theory's conclusions. Of course we must also abandon the assumption that each entrepreneur only utilizes the land he owns[14]; and thus also the corollary that the number of firms is given. But we have seen that, with CRS, the number and size of firms is irrelevant for the determination of the demand for labour. What remains to clarify is what is happening to the rent of land along the demand curve for labour, now that the possibility to hire land means that land earns a rent, and thus firms must include β in their costs (as an actual payment, or as an opportunity cost since the land owned by the firm might be rented out). The answer is that, for each level of labour employment, land rent will equal the marginal product of land because, if it is lower, entrepreneurs will wish to increase their use of land and demand for land will exceed supply, and if it is higher, demand for land will be less than supply: so equality of land rent and marginal product of the fully utilized land is necessary for the equality between supply and demand for land. We now briefly discuss the implications of not assuming CRS for individual firms. If the production function is not CRS, then it can be such as to yield a U-shaped average cost curve (once factor rentals are given), implying that initially there are increasing returns to scale, which at a certain point become constant and then decreasing[15]; or the average cost curve can be initially 14 The tendency of the exploitation of technical knowledge (and hence efficiency) toward uniformity is reinforced by the fact that land can be hired, and less efficient landowners/entrepreneurs will be glad to rent out their land to more efficient entrepreneurs for a rent higher than the residual income they would be able to get on their own. 15 This can proved as follows. With CRS, at given factor rentals the average cost curve is horizontal, → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 12 decreasing and then horizontal; or it can be indefinitely decreasing. The last case is considered incompatible with competition, because the bigger the firm, the lower the average cost, hence a possibility arises for the bigger firms to undercut the smaller rivals and grow, until very few firms survive and the market is no longer competitive. In the other two cases, as explained in the standard first-year introduction to the long-period supply curve of an industry, as long as the minimum optimal dimension of firms is small relative to total production (a condition presumably required for the industry to be competitive), owing to competition and free entry the maximum displacement of the dimension of each firm from the optimal dimension will be negligible( 16), so it is an acceptable approximation to assume that all firms adopt the minimum-average-cost input uses, which (since firms tend to adopt the same optimal technology) will be the same in all firms( 17). Thus here too we can aggregate firms into a single firm with a CRS production function, where the proportionality between output variation and total input variation is now obtained by variation in the number of identical firms rather than in the dimension of firms(18). In chapter 5 it will be shown that, at the point of minimum average cost, a differentiable production function has CRS locally, thus the product exhaustion theorem applies and payment to each factor of its marginal product does not create problems. The demand curve for labour can then be derived as before, as coinciding with the curve of the economy-wide marginal product of labour; therefore it will be decreasing, and the equilibrium of the labour market will be stable. In conclusion, a variable number and size of firms does not alter the result: the real wage will tend toward an equilibrium level which, if positive, guarantees the full employment of labour and reflects the full-employment marginal product of labour. If L* is the supply of labour and T* is the supply of land, we obtain that in equilibrium Q=F(L*,T*) and w = ∂f(L*,T*)/∂L = MP L*. 3.3.7. Now we explain the horizontal section of the labour demand curve in Fig. 3.2. Textbooks generally prefer to assume (sometimes only implicitly) that marginal products are always reflecting the fact that optimal factor/output ratios are independent of output: for example, an increase of output implies an increase of all inputs, and therefore of total cost, in the same proportion as output. A decreasing average cost curve is only possible if an increase of all inputs by the same percentage causes an increase of output by a greater percentage: this means increasing returns to scale. Analogously one proves that an increasing average cost curve implies decreasing returns to scale. The issue will be formally examined in chapter 5. 16 For example if there is room for 100 firms of optimal dimension, a 1% increase in the industry's output is sufficient to make it convenient for one more firm to enter the industry, so each firm will at most produce 1% more than its optimal output, which (remembering the U shape of the average cost curve) means a negligible deviation of its average cost from the minimum. 17 We leave aside here the possibility that the same minimum average cost be achievable with different input proportions. 18 If the firm's production function is CRS or is so at least from a certain minimum dimension onwards, then one can have both variations in the dimension of firms, and variations in the number of firms, without variation in the economy-wide Λ corresponding to a given real wage. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 13 positive. This is a very unrealistic assumption. Imagine a corn field on which more and more labour is used: a point will be reached when the ground will be so squashed by the workers' shoes that the product will be lower than with fewer workers. The same would happen with excessive quantities of fertilizer or of irrigation. Or imagine a given number of workers who are asked to till a greater and greater amount of land: after a point, the attempt to till a greater and greater surface will result in shabby tilling which will produce a lower product than if workers had concentrated on a smaller area, tilling it optimally. Clearly the general case is that the marginal product of a factor becomes zero and then negative if the proportion in which it is combined with another factor increases too much. But it is generally possible to leave some part of a factor idle, and in this way to avoid negative marginal products. Let us then assume that the marginal product of land becomes negative if T/L increases beyond a value 1/λ or, equivalently, if L/T goes below λ. Assume that the employment of land is given and equal to T*, and let the real wage rise; the employment of labour decreases more and more in order to maintain the equality between w and MP L; as MPL rises, MPT decreases[19], and at the labour employment L° such that L°/T*=λ the marginal product of land becomes zero. At that point, the land rent rate is zero and the product exhaustion theorem shows that wages absorb the entire product, i.e. the wage rate equals the average product of labour F(L°,T*)/L°, which is in fact the maximum average product of labour[20] and therefore the maximum possible wage rate. From that point on, if L is further decreased, it is convenient to decrease the utilization of land in the same proportion: if the entire supply of land were utilized, the marginal product of land would be negative, and production could be increased by reducing land utilization until the marginal product of land stopped being negative, i.e. down to the point where L/T=λ. Thus, if L continues to decrease, since labour and land utilization decrease in the same proportion, Q/L does not vary, and the wage remains equal to the maximum average product of labour, while land rent is zero. This explains the horizontal stretch of the labour demand curve in Fig. 3.2: at a wage equal to the maximum possible wage, labour demand can be any amount between zero and L°. The reader will wonder what the dotted curve is in Fig. 3.2. It is the marginal product of labour if, for L<L°, land is fully utilized, against economic convenience. Its behaviour is easily grasped once one draws the total product curve of labour, to be illustrated presently. 19 We are assuming that the marginal product of a factor is a decreasing function of the proportion between that factor and the other one, hence MPL rises and MPT decreases as L/T decreases. 20 Take labour employment as constant and increase the employment of land: output, and hence output per unit of labour (i.e. the average product of labour) too, increases as long as the marginal product of land is positive, reaching a maximum when the latter becomes zero; if now labour and land are reduced in the same proportion until land employment goes back to the initial level, output too is reduced in the same proportion, hence the average product of labour does not vary, remaining at the maximum level achievable. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 14 intl1 q(x1, x2”) A x2 ‟ x2 " a b x1* x 1” x1 ‟ x1 * x1 ” Fig. 3.3(a) Fig. 3.3(b) Isoquant when marginal products can become negative Total product curve of factor 1 If the marginal product of a factor becomes negative for a sufficient increase of the proportion in which it is combined with other factors, then isoquants too have a special shape, and one must distinguish the technological from the economic isoquants. Let us remember what an isoquant is. If the production function is q=f(x1,x2) where x1 and x2 are the amounts of factors, and if the amount of output is kept fixed at a level q*, the condition q*=f(x1,x2) locates all couples (x1,x2) such that the output is q*; the set of all such couples is called the isoquant associated with the given q*; if one represents it graphically in a Cartesian diagram with x1 on the abscissa and x2 on the ordinate, one obtains a curve which is the graphical representation of the isoquant, actually the graphical representation of the implicit function x2=x2(x1,q*) determined by the condition q*=f(x1,x2). An isoquant x2=x2(x1,q*) indicates the minimum amount of factor 2 necessary to produce q* if the quantity of the other factor is x1. If both marginal products are positive, an isoquant has negative slope, i.e. if x1 increases, x2 decreases: if MP1 is the marginal product of factor 1 in the initial situation, and we increase its employment by a very small amount dx 1, q becomes q*+dx1∙MP1; to bring it back to q* one must then vary the employment of factor 2 by an amount dx 2 such that dx2∙MP2= –MP1; in other words the two variations must be such that dx1∙MP1+dx2∙MP2 = 0; hence the slope of the isoquant curve, called Technical Rate of Substitution, TRS, is given by TRS ≡ dx2/dx1 = – MP1/MP2. If both marginal products are positive, then dx2 and dx1 must be of opposite sign, i.e. the slope of the isoquant curve is negative. What happens if marginal products can become negative? We restrict the discussion to a CRS differentiable production function. In this case we know that marginal products depend only on factor proportions; all isoquants are radial expansions of any one of them, i.e. along a ray from the origin both marginal products are constant and hence all isoquants have the same slope, given by the TRS. Let us assume that the marginal product of either factor becomes negative if its ratio to the other factor is sufficiently increased. Suppose more specifically (cf. Fig. 3.3(a)) that MP1 is zero if x2/x1=a and negative if x2/x1<a, that MP2 is zero if x2/x1=b and negative if x2/x1>b where b>a, and that between these two proportions the isoquants are strictly f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 15 convex[ 21 ] (the marginal product of factor 1 increases, and the marginal product of factor 2 decreases, if x2/x1 increases). Then if the amounts of inputs indicated on the axes are all employed in production, we obtain what we can call technological isoquants, which have zero slope along the ray with slope a, infinite slope along the ray with slope b, and are upward-sloping outside this cone because one of the two marginal products is negative. A representative technological isoquant of this type is the heavy continuous line in Fig. 3.3(a). But firms can leave part of the available amount of a factor idle. Part of the available land can be left untilled; or some of the labourers can be ordered not to enter the field at all. The firm will find it convenient not to employ the quantity of an input in excess of what yields a zero marginal product; for example, if it must produce the quantity associated with the isoquant shown in Fig. 3.3(a), and if it has at its disposal the quantity x1‟ of input 1, it will choose not to employ the whole of it (it would then be obliged to demand the quantity x2‟ of factor 2): it will prefer to employ only the amount x 1” of factor 1 and thus demand only x2” of factor 2. If the firm must maximize output on the basis of an employment x1=x1* of factor 1, it will find it convenient to leave idle any amount of factor 2 in excess of x 2", which is the amount for which x1/x2=b i.e. MP2=0. Thus the economic isoquant, the one truly reflecting the economically optimal choices, coincides with the technological isoquant only in the cone where marginal products are not negative, but outside that cone it becomes a straight line, horizontal to the right of the cone, vertical to the left. For an example cf. Exercise 7 in chapter 5. One must then also distinguish the technological from the economic total product curve. The total product curve of a factor indicates output as a function of the quantity employed of that factor, for given amounts employed of all other factors[22]. Its slope measures the marginal product of that factor. The technological total product curve of factor 1 corresponding to Fig. 3.3(a) and to x 2=x2" is the heavy continuous line in Fig. 3.3(b). It is easy to understand its negative slope for x1>x1": in that range the marginal product of factor 1 is negative (x1/x2>a), an increase in its employment decreases output. Then the firm will find it convenient to leave the amount of factor 1 in excess of x1" idle: the economic total product curve will continue horizontal after x1". It is also important to understand the reason for the initial convexity of the technological total product curve. Let AP1 stand for the output per unit of factor 1, or average product of factor 1. Graphically, it is measured by the slope that connects a point on the total product curve with the origin; with CRS, it only depends on factor proportions. Assume factor 2 is fully employed and x1 is decreased from an initial level such that MP2>0; as demonstrated in fn. 16??, AP1 increases and the maximum AP1 is reached 21A curve is convex (respectively, concave) if in between any two of its points it does not lie above (below) the segment connecting those two points; it is strictly convex (strictly concave) if it lies entirely below (above) that segment, except of course at the two extremes of the segment. The curve ⌣ is strictly convex, the curve ⌢ is strictly concave; a straight line is both convex and concave, but not strictly so. Cf. chapter 4 for a more formal definition. 22 Therefore the total product curve shifts if the given amount employed of some other factor is changed. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 16 when MP2=0, i.e. at x1* in Fig. 3.3(b). If x1 is further decreased to a level below x1* and x2 is decreased in the same proportion, output decreases in the same proportion and AP1 does not change; if now, with x1 fixed, x2 is increased to bring it back to full employment, its marginal product becomes negative and output decreases, hence the technological total product curve is below the economic total product curve, and the average product of factor 1 is below the maximum achieved at x1*. As x1 decreases further, its technological average product decreases too, because, relative to the economical average product, the utilization of excess land with a negative marginal product increases. Hence the slope of the technological total product curve (the continuous line) shown in Fig. 3.3(b), which exhibits a lower and lower average product of factor 1 as its employment decreases. The economic total product curve for x1<x1* is on the contrary a straight line with AP1 constant and equal to the maximum AP1 (the broken line in Fig. 3.3(b)) because the part of factor 2 actually utilized is varied in step with x1, maintaining MP2=0. The economic marginal product of factor 1, given by the slope of the economic total product curve, is therefore initially constant and equal to the maximum AP1 up to x1*, because until then factor 2 is only partially utilized, its marginal product is zero, and its use increases in proportion with the increase of factor 1. The reader will see that this explains the initially horizontal section of the marginal product curve and of the demand curve for a factor in Fig. 3.2, as well as the shape of the dotted curve in that same diagram, that indicates the slope of the technological total product curve for labour employment levels corresponding to x1<x1* . The importance of all this is that proportions in factor supplies can determine whether a factor earns a positive or a zero rental. The theory cannot exclude that income may go entirely to only one factor. (This is empirically not observed. We do not discuss here whether this constitutes a serious criticism of the theory, but the reader should keep this question in her mind.) It is clear that the approach can be extended to several types of labour, and several types of land. For each factor, once the quantities employed of all the other factors are given, one can derive the curve of its economy-wide marginal product, which is also the demand curve for that factor; and in equilibrium the factor will earn its full-employment marginal product. 3.3.8. When the entrepreneurs are a third party, hiring labour and land, both labour employment and land employment are variable. This possibility raises new problems. I do not know of a discussion by marginalist authors of these problems, but it seems possible to derive from their theories how they would have tackled them. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 17 intl1 MPL, w B A w1 w2 B' A' L2 L1 L Fig. 3.4. A shift of the labour marginal product curve from AA' to BB' due to a decrease in the amount of land employed may entail that a lower w is associated with a lower demand for labour. First, it becomes possible that at a certain moment there be some unemployment both of labour and of land. Suppose that then both the money wage and the money land rent decrease; and suppose that the ratio w/β remains unaffected; then firms have no incentive to change the L/T ratio; furthermore, remembering that the income spent on the product market equals the income actually earned by factors, we see that firms have no incentive to increase production, because money demand for the product is decreasing in the same proportion as their costs. Thus there might be an indefinite decrease in money prices with no tendency toward the full employment of factors. The reason why this possibility did not worry traditional marginalist economists would appear to be their monetary theory. In various forms, at least before the Keynesian revolution[23] they adhered to the quantity theory of money, which argues that the price level will tend to be established at the level that renders the total demand for money − which is the sum of the average money holdings desired by the economic agents, and depends on money prices, on the transactions to be performed and their time structure, and on the velocity of circulation of money − equal to the total (and given) supply of money. Economic agents want to hold on average a certain amount of money because of the payments they know they must effect (transactions motive), and as reserve in case of unforeseen contingencies requiring special money disbursements (precautionary motive). (Keynes would later add a desire to have money available in order to be able to exploit possibilities of speculation on financial markets, but this motive for holding money cannot exist in our acapitalistic economy.) When on average consumers or firms have more money than needed, it will 23 With Keynes everything changed, but the new theories were concerned with capitalistic production, where part of the demand for the product comes from investment, which is not present in the simple land- labour economy. For something more on the quantity theory of money and its role in the determination of equilibrium prices cf. §3.?? below. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 18 be expended, raising demand for goods and for factors; if factors are fully employed, the result will be a rise in money prices; and in the opposite case of a supply of money less than the demand for money, the result will be a reduction of demand and hence of money prices; thus one has a process that brings about the equilibrium level of money prices, a process illustrated by Wicksell (for the case of a demand for money greater than the supply of money) as follows: Let us suppose that for some reason or other commodity prices rise while the stock of money remains unchanged, or that the stock of money is diminished while prices remain temporarily unchanged. The cash balances will gradually appear to be too small ... I can rely on a higher level of receipts in the future. But meanwhile I run the risk of being unable to meet my obligations punctually, and at best I may easily be forced by shortage of ready money to forgo some purchase that would otherwise have been profitable. I therefore seek to enlarge my balance ... through a reduction in my demand for goods and services, or through an increase in the supply of my own commodity ... the universal reduction in demand and increase in supply of commodities will necessarily bring about a continuous fall in all prices. This can only cease when prices have fallen to the level at which the cash balances are regarded as adequate. (Wicksell, 1936, pp. 39–40, italics in the original). Wicksell‟s reasoning implicitly assumes given quantities i.e. full employment; if we apply the same reasoning to an excess supply of money created by money price decreases but with factors partly unemployed, the increase in demand will plausibly resolve at least partially in an increase in production, thus bringing about a tendency toward an increased employment of both factors. Second, even with one factor fully employed, when one studies what happens on the market of the second factor one can no longer assume that the utilization of the first factor is given. It is no longer guaranteed that a lower wage will increase the demand for labour: if, for some reason, the lower wage causes a reduction in the entrepreneurs‟ utilization of land, this causes the curve of the marginal product of labour to shift downwards, and the result might be a reduced demand for labour in spite of the lower wage, cf. Fig. 3.4. Here the answer is that there is no reason why such a decrease in labour demand should happen: it seems legitimate to assume β initially unchanged; then the decrease of w/β will induce entrepreneurs to raise the L/T ratio, and this might mean a decrease of land utilization, but not to the point of lowering labour utilization too: if the entrepreneur provisionally treats land utilization as given, the demand for labour rises; if the entrepreneur considers the amount to be produced as given she will move along the labour-land isoquant in the direction of utilizing more labour and less land, and thus, again, demand for labour increases (although less than in the other case): furthermore, since in either case (with β given) average cost has decreased, unless the product price decreases by the same amount (and at least initially this is unlikely, because adjustments take time) the entrepreneur has an incentive to expand production. In either instance, an increase in production is the likely outcome, with the entrepreneurs counting on being able to sell the greater production at f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 19 the same or at a slightly lower money price(24); nor will their expectations be totally disappointed, because if all income is spent (as we are assuming), then aggregate demand equals the value of distributed incomes, so aggregate earnings are sufficient to cover costs. Also, in either case the increased marginal product of land, due to the increased labour-land ratio, creates an incentive for entrepreneurs to increase their demand for land; so the employment of land, even if initially reduced, will then increase again(25). And when the employment of land increases, this raises the marginal product of labour, thus the decrease of w has an even greater effect on the demand for labour. It seems therefore impossible to find a reason why the demand for labour should decrease. The conclusion is that the reasons in favour of the tendency toward the full employment of a factor whose rental decreases are not weakened by the fact that other factors are or may become not fully employed. But then, a marginalist economist will argue, on each factor market there is active a tendency toward a full-employment equilibrium, whatever the employment level of other factors, and therefore a full-employment equilibrium will be reached sooner or later on all factor markets. Then when one studies the demand curve for a factor, to assume that the other factors are fully employed only means to go directly to what will be in any case the final result. 3.3.9. The existence of a tendency toward equilibrium, in this simple economy, can in the last analysis be attributed to the fact that the proportion L/T in which the two factors are demanded is a decreasing function of the ratio between factor rentals w/; if a factor is in excess supply, a decrease of its rental brings about an increase in the proportion in which firms wish to combine it with the other factor, and thus an increase in its demand; this generates the tendency of demand to adapt to supply. This is called the mechanism of direct, or technological, factor substitution because it derives from the fact that, in order to produce a given quantity of product, firms move along the corresponding isoquant in the direction of the factor which has become relatively cheaper; so firms change the technology they use, substituting to some extent the factor which has become cheaper to the other factor(s). This mechanism requires the existence of some factor substitutability; it would 24 We have so far spoken as if all transactions were directly in corn, but it was only a way of stressing the exclusive importance of relative prices for our reasonings; in any concrete economy prices are in money terms, and entrepreneurs reason in terms of money prices. 25 The argument is reinforced by the consideration that, in more realistic economies where production also uses capital goods, firms usually own durable capital goods and will go on using them, so an assumption that the stock of durable capital goods does not decrease when the real wage decreases has considerable plausibility. However, the overall argument suffers from a weakness: a decrease in the rental of a factor, with other things initially given (including technology and product prices, which take some time to adapt), decreases the income of that factor, so, during the time required for entrepreneurs to increase their employment of that factor and for product prices to decrease, aggregate income decreases; hence demand decreases and discourages production. This problem is anyway overshadowed by the problems with aggregate demand caused by investment, which will be considered in chapter 13, so we only mention it and pass on. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 20 be unable to operate if labour and land were to be associated in rigid proportions, i.e. if isoquants were L-shaped. It is useful to stop on the latter case in order to clarify the meaning of the demand curve for a factor. Suppose only one fixed-coefficients method is known for the production of corn, which requires the use of λ units of labour and τ units of land per unit of output. Both labour and land are supplied in rigid quantities L* and T*, independent of the rates of wages and of rent. If land is fully employed, then the maximum amount of labour employable is T*∙λ/τ=T*Λ, because λ/τ is the labour/land ratio, the amount of labour to be employed per unit of land[ 26], which we can still indicate with the symbol Λ, cf. §3.3.3. If L*>T*Λ, some labour is inevitably unemployed and if the wage rate decreases as long as there is labour unemployment, the equilibrium wage rate can only be zero; then the rate of rent will rise, owing to competition among entrepreneurs, up to absorbing the entire product, i.e. β=1/τ. If L*<T*Λ, labour is insufficient to permit to full utilization of land, land is necessarily partly unemployed and β falls to zero while wages rise to absorb the entire product i.e. w=1/λ. If L*=T*Λ, the full employment of one factor entails the full employment of the other one too, and income distribution is indeterminate. The (inverse) demand curve for labour is then the heavy line in Fig. 3.4b. intl1.5 w L1 L2 L3 L4 1/λ w+ O T*Λ L Fig. 3.4b It is a strange demand curve; it is not, rigorously speaking, a demand function: for w=1/λ labour demand is indeterminate between zero and the amount employable with the given supply of land. However it is the labour demand „curve‟ needed to represent the results of the theory, because it is the „curve‟ whose intersections with the labour supply curve yield the correct equilibrium wage rates: thus the labour supply curves L1 and L2 yield a wage rate equal to the maximum one because land is in excess supply, the supply curve L4 yields w=0 because labour is in excess supply, and the upward-sloping dotted labour supply curve L3 shows the only possibility of avoiding those totally 26If you are unclear about this, think of it this way: 1/τ is the maximum amount of output producible with one unit of land; and the amount of labour required to produce 1/τ units of output is λ/τ. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 21 implausible results. This shows that the demand curve for a factor must show the locus of combinations of factor employment levels and factor rental levels representing equilibria corresponding to hypothetical different supplies of the factor (for a given supply or supply function[27] of the other factor). 3.4. Comparative statics 3.4.2. Let us suppose that migration flows alter the supply of labour. The variation of the average wage level due to this fact will be explained by the variation in the equilibrium wage, due to the new intersection of the shifted supply curve with the unchanged demand curve. Thus, if in Fig. 3.5(a) the supply of labour was initially L1 and then owing to emigration it decreases to L2, the real wage will gravitate toward the new higher equilibrium level w2. intl1 Demand Supply w2 w* w1 L2 L1 L^ L* (a) (b) Fig. 3.5 Another important example of comparative statics is the study of the effects of a real wage imposed by non-competitive forces, e.g. by legislation or by the bargaining strength of trade unions. If the real wage is given, firms will only demand the amount of labour which renders the marginal product of labour equal to that real wage. Thus if, again in Fig. 3.5(a), the supply of labour is L1 and the real wage is rigid at the level w2, the demand for labour will be L2 and the amount of labour (L1– L2) will remain unemployed. If the real wage is left free to decrease, and the unemployed workers really prefer working to remaining unemployed (as implied by the excess supply of labour at w=w2), then they will also be ready to work for a slightly lower wage – marginalist economists argue –, and then free competition will lower the wage until the full employment of labour is 27 Exercise. The reader is invited to re-draw Fig. 3.4b if land supply is not rigid but instead a function of β, say a+bβ. The assumption as to land employment must be that land is fully utilized if possible. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 22 reached. Hence two conclusions, typical of traditional marginalist economic theory and still now quite popular: – if there is labour unemployment in excess of the ineliminable frictional unemployment(28), the cure consists in letting competition operate freely, bringing about a decrease of the real wage; – unemployment might be only apparent, in that the workers registered as unemployed might in fact be waiting for job offers at a higher wage than the ruling one, and might not be ready to work at the ruling wage (nor, of course, at a slightly lower one): in this case they are voluntarily unemployed, they are freely deciding not to supply their labour at the current or at a slightly lower real wage, and the labour market is then actually in equilibrium[29]; if unemployment is real and persistent, the responsibility must lie with the causes of the insufficient decrease in real wages; these causes are either trade unions (and therefore ultimately the workers' support of trade unions), or minimum-wage legislation (and therefore ultimately the workers' support for the political parties that have implemented that legislation). Hence the frequent conclusion that it is the workers themselves who are responsible for labour unemployment (in excess of frictional unemployment). 3.4.2. What is assumed about the other markets when one analyzes in this way the effects of a given real wage? When the wage level w2 causes a labour demand L2 inferior to labour supply, are the land market and the product market in equilibrium? The labour demand curve is derived precisely under the assumption that this is the case. For the product market, we have already argued that our assumption of zero savings (all income spent in the period in which it is earned) renders the demand for the product equal to its supply, because in equilibrium the value of the product equals 28 In any concrete economy, there are continuous changes in the composition of demand, in technology, etc., and these entail closures of firms, geographical dislocations of firms, etc., and thus firings. Once some workers are fired, some time is necessary for them to find a new job; hence the inevitability of some frictional unemployment. Full employment must therefore be realistically interpreted to mean that unemployment is only frictional. The determination of the rate of unemployment that may be considered frictional is hotly debated by economists; estimates vary from 1% to 5% or even higher, and involve also another hot topic, how to measure unemployment. Anyway historical experience shows that some countries (e.g. Great Britain or France in the years after World War II) have been able to achieve unemployment rates as low as 1.5% for several years in a row, so I would side with the lower estimates of frictional unemployment. 29 A borderline case arises if the unemployed workers, or many of them, are ready to work at the ruling wage but their supply of labour jumps to zero if the wage decreases, even only infinitesimally (the supply curve of labour is then horizontal at the ruing wage). In this case, there may be no wage level capable of ensuring the equality between supply and demand for labour. In Fig. 3.5(b) a wage level w* causes a demand for labour L^, smaller than the supply L*, but any decrease of the wage causes labour supply to jump to zero. However, at least according to Keynes in this case unemployment should be considered voluntary: unemployment, he writes, is involuntary when labour supply is greater than demand at a real wage rate slightly lower than the ruling one (GeneralTheory p. ??). The idea appears to be, that unemployment is voluntary if the unemployed workers are not ready to renounce something in order to get a job. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 23 the incomes of the factors which produced it, and by assumption this income is all spent in the purchase of the product. As to the land market, by the product exhaustion theorem what is left after paying L2 its marginal product is precisely what must go to land according to its marginal product, so the demand for land equals the amount of land utilized − in other words, our assumption that land is fully utilized is acceptable, because firms are satisfied with the land employment assumed; supply and demand for land are equal. Only one market is in disequilibrium, the labour market. Some readers, the ones familiar with Walras‟s Law, will protest that there must be a mistake somewhere in the argument, because Walras‟s Law implies that if all markets but one are in equilibrium then the last market must be in equilibrium too (as long as prices are positive). For these readers (the other ones will have to wait for the discussion of this Law in chapters 4 and 5) I anticipate that there was no mistake: the conclusion that the labour market may well be the sole market in disequilibrium derives from the assumption that the income of factor owners corresponds, not to the value of the factor services they would like to supply, but to the value of the factor services that are demanded by firms, that is, the factor services that find employment (assuming supply to be greater than demand[30]). If there is labour unemployment, the unemployed have no income and therefore cannot demand the product. In this economy, if the unemployed workers were to disappear, the markets would not realize it (as long as the wage rate did not change). The economy behaves as if the supply of labour were equal to the amount of labour that finds employment. Therefore, formally, when we determine the demand curve for labour what we are doing is as follows. The starting point is the equations determining the full-employment equilibrium in this economy, which are the following (remember that Λ=L/T and that the product is the numéraire): Q = F(L,T) w = MPL(Λ) [3.4] = MPT(Λ) T = T* L = L* The data are the production function and the supplies of labour L* and of land T* (which for simplicity are assumed rigid). L and T are the amounts employed of labour and land. The variables are Q, L, T, w, , as many as the equations. If, when Λ=L*/T*, both marginal products are positive, 30When the demand for a factor is greater than the supply, the equations indicate what the factor employment, and the production level, would be if there were available an amount of the factor equal to the demand for it. Thus it is a hypothetical construction. It determines what the full-employment equilibrium would be, if the rigid labour supply were such as to cause the equilibrium wage level to be at the level one is assuming. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 24 then a solution exists with no need for inequalities, i.e. the last two equations can be solved as equalities (the full-employment of both factors is possible). To dervie from these equations the demand curve for labour, we drop the last equation and take w as a parameter. Then we have four equations and four variables; for each level of w this system of equations endogenously determines L. The curve L(w) traced by L as we vary w is the demand curve for labour: for L(w)≤L*, it indicates the labour employment that will indeed be realized, with L*−L(w) the amount of unemployment[31]. It can also be interpreted as showing what the supply of labour would have to be in order to be equal to the demand for labour at that level of w, if land employment is T*. 3.5. The role of consumer choice: the indirect factor substitution mechanism. 3.5.1. What difference does it make to introduce consumer choice into the analysis? If one assumes, as we have done up to now, no savings, only one product, and a given (i.e. rigid) supply of land, consumer choices have no role in the above derivation of the decreasing demand curve for labour; consumer choice can only influence the supply curve of labour. If one also assumes a rigid supply of labour, consumer choice has no role at all in the determination of the equilibrium real wage, nor in the arguments supporting a tendency toward (i.e. the stability of) equilibrium. The mechanism causing the gravitation toward equilibrium only relies on the technical substitutability between factors. The general properties of consumer choice have assumed great relevance in the marginalist/neoclassical approach above all because consumer choice among different consumption goods was considered to supply a reason, for decreasing demand curves for factors, alternative or additional to the technical substitutability among factors in each industry, because activating an indirect factor substitution mechanism, which was argued to provide further support to the plausibility of the marginalist explanation of income distribution(32). In order for this second factor substitution mechanism to operate there must be more than one consumption good. So let us now assume that there are two consumption goods, corn (good 1) and iron (good 2), produced by separate industries (the factors are still labour and land). In order to examine the working of this factor substitution mechanism in isolation, let us assume zero technical 31 The above is valid for a demand for labour not greater than the supply. If the given w is such that L(w)>L*, then the corresponding solution is unattainable and it only indicates the tendency of the economy – namely, a pressure on the wage to rise because the amount of labour that firms would like to employ, L, is greater than the available labour supply. Exercise: in such a case, will there be disequilibrium only on the labour market? 32 On the contrary, consumer choice concerning factor supplies was immediately perceived to be a possible cause of trouble for the plausibility of the marginalist approach, owing to the possibility of „backward-bending‟ factor supply curves which could cause multiple and unstable equilibria, cf. below, §3.6.3. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 25 factor substitutability: in the production of each good only one method of production is known, characterized by given technical coefficients and therefore by a fixed L/T ratio, which is different as between industries(33). Let these technical coefficients be L1, T1 in the corn industry and L2, T2 in the iron industry: L1 units of labour * T1 units of land → 1 unit of corn L2 units of labour * T2 units of land → 1 unit of iron. Assume L1/T1 > L2/T2 , i.e. the corn industry employs more labour per unit of land than the iron industry, or, for brevity, is the more labour-intensive one. In a competitive economy, product prices generally vary more rapidly than factor rentals; therefore we can treat w and as given when we determine the prices of corn and iron. The tendency of prices toward costs of production means a tendency of prices toward the levels determined by p1 = L1w + T1, p2 = L2w + T2. It is easy to show that p1/p2 depends on w/ and that, given our assumption that corn is more labour intensive, p1/p2 decreases when w/ decreases(34). Thus, the more labour-intensive good becomes relatively cheaper when the real wage decreases. The argument then proceeds as follows. Assume that land is fully utilized and there is labour unemployment. The real wage goes down(35), and w/ decreases. Corn becomes cheaper relative to iron and then, it is argued, plausibly(36) the composition of the demand for consumption goods will 33 In the microeconomic jargon, labour and land are perfect complements in both industries; the isoquants are L-shaped with the vertexes aligned on a straight line through the origin. w T1 L1 p1 L1w T1 34 By dividing the first equation by the second we obtain . The sign of p 2 L 2 w T2 L w T 2 2 the derivative of p1/p2 with respect to w/ is the same as the sign of L1T2–L2T1 i.e. is ≶ 0 according as L1/T1 ≶ L2/T2 . 35 As long as all technical coefficients are positive, it does not matter in terms of which good the real wage is measured, because when w/p1 changes, w/p2 changes in the same direction. To prove it, choose either good as numéraire, e.g. put p1=1. When w increases, p2 increases by a lesser percentage than w, because, from the first price equation, decreases; hence w/p2 increases too. 36 But not necessarily; and we will have to return on this issue in chapter 6. For example, assume that labourers and landowners are two distinct groups of consumers, each one with a rigid composition of demand for corn and iron, and that labourers demand corn in a greater ratio to iron than landowners. The aggregate ratio of corn to iron in consumer demand is a weighted average of the ratios of the two groups of consumers; a decrease of the real wage means a redistribution of income from wages to rents, hence a smaller share of demand coming from wages, and therefore a lower corn/iron ratio in aggregate consumer demand. Now assume that the composition of demand is not rigid and each group shifts somewhat the composition of its demand in favour of corn: this may not be enough to counter the effect of the shift in → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 26 shift in favour of corn. The adaptation of the composition of production to the composition of demand will then require shifting some units of land from the production of iron to the production of corn; since in the corn industry each unit of land is combined with more units of labour than in the iron industry, this shift will increase the demand for labour. (Exercise: by how much does labour employment increase for each unit of land shifted from the iron to the corn industry?) The increase in the demand for labour is brought about, in this case, by variations in the relative size of different industries; the more labour intensive industry expands, and the other one shrinks, when labour becomes relatively cheaper; this causes an increase in the average proportion in which labour is demanded relative to land, and thus an increase in the demand for labour since land employment is assumed fixed. By now it should be clear why this is called the indirect factor substitution mechanism. By substituting a consumption good for another one, consumers indirectly substitute a factor for another one in the economy-wide employment of factors. Substitution in consumption means movement along an indifference curve; now, when a consumer moves along an indifference curve e.g. demanding more corn and less iron, she is actually indirectly demanding more labour services and less land services; therefore she is indirectly substituting labour for land in the „production‟ of her utility[37]. Now that there is more than one product, equilibrium implies a simultaneous determination of income distribution, relative product prices, and quantities produced. The equilibrium composition of the demand for consumption goods must be such as to ensure the full employment of both factors; the relative price p1/p2 of consumption goods must therefore be such as to ensure the appropriate composition of demand; and since p1/p2 depends on w/, income distribution must be such as to ensure the appropriate relative price of consumption goods. 3.5.2. We confirm the verbal reasoning by formulating the general equilibrium system of equations for this economy. The data are factor endowments, consumer preferences, and available production methods; in short: endowments, tastes, and technology. We assume no savings: all income of a period is spent in that period. Each consumer has a given endowment of labour and/or land (which we assume are entirely supplied), and given income from labourers to landowners. Analogous phenomena can result from one of the goods being an inferior good for one of the two groups. Thus income effects might cause factor demand curves to be upward-sloping, as in Fig. 3.6 below, with a high likelihood of the implausible outcomes to be noted there. Our presentation here of the indirect substitution mechanism assumes the absence of such „damaging‟ income effects. 37 As will be shown later in this chapter, one can actually derive factor (indirect) indifference curves which show the combinations of factor uses in production among which a consumer is indifferent because the resulting consumption products yield her the same utility. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 27 preferences; each vector of prices (p1,p2,w,) generates a certain income for each consumer(38) and, on the basis of this income, the consumer chooses to demand the amounts of corn and of iron that maximize her utility, in the way learned in the introductory economics course (tangency of budget line and indifference curve), which, as we will study in chapter 4, can be generally assumed to yield demand functions for corn and for iron, depending on those four prices; we omit the specification of how these demand functions are obtained, and simply note that the sum of the demands of the several consumers yield aggregate demand functions for corn and for iron, which we indicate as Q1(p1,p2,w,) and Q2(p1,p2,w,). Factor supplies are assumed rigid for simplicity. Let the quantities produced be q1 and q2. In equilibrium they must equal the quantities demanded. Now, the quantities produced cannot be considered functions of prices, because firms have constant returns to scale, so, given the factor rentals, the supply of each good is horizontal at the price equal to average cost: if price is less than average cost, supply is zero; if price is more than average cost, supply is infinite and therefore factor demand is infinite too; at the price equal to average cost, supply is indeterminate. But since adjustments take time and we are interested in determining a long-period equilibrium, we can assume that in disequilibrium the variation of the quantity produced through variation in the number and/or dimension of firms is gradual, not instantaneous (in a partial-equilibrium framework it can be described as operating through shifts of the short-period supply curve as shown in any introductory textbook), and therefore supply is able to adapt to demand at a price equal to average cost. Hence: [3.5] p1 = L1w + T1, [3.6] p2 = L2w + T2 [3.7] q1 = Q1(p1,p2,w,) [3.8] q2 = Q2(p1,p2,w,). There remains to formalize the condition of equilibrium on factor markets. The production of the quantity qi , i=1,2, entails the employment of Liqi units of labour and Tiqi units of land; thus the demands for labour and for land are given by DL=L1q1+L2q2, DT=T1q1+T2q2. Then equality between supply and demand on factor markets requires (assuming that the full 38 The consumer's income is determined by assuming that the consumer is able to sell her supplies of factors; this assumption is legitimate for the determination of equilibrium. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 28 employment of both factors can be reached – this is not necessarily the case, see further down), with L*, T* the factor supplies (assumed given): [3.09] L1q1+L2q2 = L* [3.10] T1q1+T2q2 = T*. These equations satisfy four groups of conditions: a) product prices equal to average costs; b) supplies of produced goods equal to the quantities demanded, derived from consumer choice; c) determination of demands for factors, from the quantities produced, and from the cost- minimizing technical choices of firms (that depend on relative factor prices – here there was no choice but in general there is[39]); d) conditions of equilibrium on factor markets. Consumer choices, as well as firm choices, only depend on relative prices(40), and therefore the equilibrium can only determine relative prices; so we must arbitrarily fix the scale of prices by choosing a numéraire commodity, e.g. by putting a price equal to 1, or by putting the price of a bundle of goods equal to 1; to avoid complications in case a price were zero in equilibrium, the second solution is often preferred; thus we can for example write: [3.13] p1+p2=1 The equilibrium equations, [3.5] to [3.11], are 7 equations in the 6 variables q1, q2, p1, p2, w, . Is the system overdetermined? No, one of the equations supply=demand can be derived from the other ones(41) and therefore is not an independent equation. This is because we are assuming that all 39 If there is technical choice, technical coefficients become functions of relative factor prices, so they do not appear as further variables; cf. §3.6.1, and ch. 5 for a more general treatment. 40 We remind the reader that, when the income of a consumer derives from her endowments, then a proportional variation of all product prices and factor rentals does not shift the budget constraint and therefore the optimal choices are not affected. This issue will be re-examined in the next chapter. As to firms, we are assuming constant returns to scale and free entry, which implies that prices must equal average costs; if all prices and factor rentals vary by the same percentage, prices remain equal to average costs. 41 We are actually applying here a corollary of a general property of general equilibrium systems of equations, known as Walras' Law, that states that, whichever the relative prices, if we consider the intended demands and supplies of agents at those prices, and if the consumers‟ intended budgets are balanced and firms either make zero profits or distribute them to consumers, then the aggregate exchange value of all demands must equal the aggregate exchange value of all supplies. The corollary is that if the equality supply=demand is satisfied on all markets but one, since this implies that the aggregate exchange value of demand on those markets equals the aggregate exchange value of supply on those same markets, then on the last market too the exchange value of demand equals the exchange value of supply, and hence, if the price on → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 29 consumers spend their entire income, so the total value of demands must equal the total value of incomes: p1Q1(p1,p2,w,) + p2Q2(p1,p2,w,) = wL* + T*. Now, the conditions price = average cost imply that the total value of production equals the total value of incomes( 42): p1q1 + p2q2 = wL* + T*. Thus we obtain that the total value of demands must equal the total value of productions: [3.12] p1Q1(p1,p2,w,) + p2Q2(p1,p2,w,) = p1q1 + p2q2. This equality implies that if p1Q1(p1,p2,w,)=p1q1, then p2Q2(p1,p2,w,)=p2q2; the latter equality, if p2>0, implies equation [3.8]. Thus we can eliminate equation [3.8] from the system(43). The independent equations are 6, as many as the variables to be determined. Let us now admit the possibility that the full employment of both factors is impossible to reach. Equations [3.9-3.10] are two linear equations in two variables, q1 and q2; being linear, they can be assumed to have a solution(44), and furthermore a unique one, but nothing guarantees that the solution is non-negative. This possibility can be given an economic interpretation. Suppose that L1/T1>L2/T2; the proportion L/T in which labour and land are demanded cannot become greater than L1/T1 (when only good 1 is produced) nor smaller than L2/T2 (when only good 2 is produced). If the ratio L*/T* lies outside this interval, the sole way to reach a solution of equations [3.9-3.10] is to produce a negative quantity of at least one of the products, which makes no economic sense. What this means is that one of the two factor supplies cannot be fully employed. Actually, this can happen even when the solution to those two equations is positive, because there may be no price vector that, introduced into equations [3.5-3.8], determines the necessary ratio q1/q2: relative product prices cannot go outside the range determined by w=0 and =0, and the ratio q1/q2 that solves equations [3.9-3.10] might require a p1/p2 outside that range. What will happen if a factor cannot be fully employed? It seems reasonable to admit that, since no price or rental can become negative, if the factor‟s rental decreases as long as the supply of a factor is greater than demand, then if full employment of that factor is impossible to achieve, the rental of that factor will go to zero and stay there. Now, a market must be considered in equilibrium if the price is zero and supply is still greater than demand: the price cannot get any lower, so it no that market is positive (a condition sometimes forgotten), the equality supply=demand is satisfied on that market too. We will rigorously prove this law and examine its meaning in chapters 4 and 5. 42 This is quite intuitive, but as an Exercise: obtain this equality rigorously from the equilibrium equations. 43 If p =0, equation [3.12] no longer implies equations [3.8], but it implies equation [3.7], so there always 2 is one equation supply=demand that can be derived from the other ones and from the assumption of balanced budgets of consumers. 44 Linearity of the equations does not guarantee the existence of solutions: y=x, y=x+1 is a system of two linear equations with no solution. When a solution exists, it need not be unique: y=x, 2y=2x have infinite solutions, because the two equations are not linearly independent. But these cases can be considered flukes of zero probability. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 30 longer changes – which is the real definition of equilibrium. In order to admit this case, after specifying that the solution cannot include negative variables, we must replace equations [3.9-3.10] with inequalities, accompanied by „side conditions‟ (called complementary slackness conditions[45]) specifying that the factor rental must be zero if supply is greater than demand: Conditions that no variable can be negative: q1, q2, p1, p2, w, ≥ 0. [3.12] L1q1 + L2q2 ≤ L* , and w = 0 if the inequality is strict [3.13] T1q1 + T2q2 ≤ T* , and = 0 if the inequality is strict. Even this re-writing may be insufficient to guarantee the existence of solutions, as we will see in subsequent chapters. But for many decades the difficulties on this last issue were not perceived, and in this chapter we intend to present the marginalist approach in the positive light in which it was seen in the first decades after its birth, so as to convey to the reader a feeling of the persuasiveness of the approach at the time of its rise to dominance. The price level 3.5.3. The considerations and equations advanced so far only determine relative prices; the price level needs additional considerations in order to be determined. The key is in the notion of equilibrium as a state where no one has an incentive to alter her/his behaviour. Then in equilibrium, assuming relative prices to be the equilibrium ones, money prices must be such, and the supply of money must be so distributed among economic agents (and continually circulating among them), that each agent is satisfied with the amount of money she holds on average, and wishes neither to increase nor to decrease it. The amount of money, that an agent (consumer or firm) needs to hold on average in order not to wish to alter it, depends on the price level and on the volume and time structure of the transactions of the agent; thus we might derive, for each agent, the amount of money she needs to hold on average as a function of the price level, if her actions are the equilibrium ones determined by the equations illustrated. If, somewhat incorrectly, we call this amount of money needed for equilibrium behaviour the „demand for money‟ of the agent, then by summing up the demands for money of all agents we obtain a function that tells us, for each price level, what the total amount of money should be in order for equilibrium behaviour of all agents to be possible (this will also require that that amount of money be distributed in a certain way among agents). Given the supply of money, the equilibrium price level is determined by the equality of this supply with the aggregate „demand for money‟. Unless the price level is the equilibrium one and the supply of money is distributed among agents in the 45 So called because they indicate a condition that must hold if a constraint is 'slack' i.e. does not hold with equality. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 31 necessary way, equilibrium behaviour is impossible; but the process illustrated by Wicksell in the lines quoted in §3.3.8 above gives reason to believe that the adjustments performed by agents when they do not hold on average the „correct‟ amount of money will set up a simultaneous tendency toward the price level. and toward the distribution of the supply of money among agents, required by the intended transactions. Thus the tendency toward the equilibrium relative prices operates simultaneously with a tendency toward the equilibrium price level, and money does not create permanent difficulties to the tendency toward the equilibrium productions and exchanges. In order to determine the nominal (i.e. monetary) equilibrium prices, traditional marginalist theorists added to the „real‟ equations written so far (so called because determining relative i.e. „real‟ prices – that is, in terms of some physical numéraire – rather than nominal prices) an equation establishing the equality between supply and demand for money where the demand for money[46] depends on the price level. This could be for example the standard Fisherine equation MV = PT. Here M is the supply of money, V its average velocity of circulation (the number of transactions performed on average by each money unit in the unit time period), and PT is the total money value of the transactions to be performed in one period, traditionally represented in very simple terms as the product of an index of the price level and of an index of the amount of real transactions performed (a more disaggregated representation would also be possible). This equation, if re-written as M = PT ∙ 1/V, has the (given) supply of money on the left-hand side, and the demand for money on the right-hand side. It determines the equilibrium price level if the supply of money, the velocity of circulation and the „real‟ transactions to be performed are given. The system of equations resulting from the addition of this equation to equations [3.5 - 3.11] exhibits the so-called Classical Dichotomy: the „real‟ magnitudes (quantities and relative prices) are determined by equations where the quantity of money has no role and which are mathematically independent of the last equation, which determines the price level. It is important to stress a point about this theory, which is not always correctly grasped. The determination of consumer and of firm decisions, that lies behind the equations we have written so far, implicitly assumes that the consumer or the firm has at its disposal the necessary money 46 "Demand for money" means here the average money holding desired by consumers and firms in the long period in order to carry through the intended (equilibrium) transactions; thus it is the demand for an average stock. It depends on the price level, on the agent's transactions and their time sequence, on the advantages and opportunity cost of holding money balances in excess of those strictly necessary for transaction purposes, on the cost of short-period borrowing, and so on. Cf. Petri (2004, ch. 5, Appendix 1) for a discussion of different meanings of demand for money; and Ackley (1978, pp. 88-99) for a discussion of traditional analyses of the demand for money. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 32 balances, neither more nor less; therefore it implicitly assumes that not only relative prices, but also the price level and the distribution of money among the several agents, are the equilibrium ones. Because of the implicit assumption that the amount of money held by an agent is appropriate to her transactions, money balances are not an additional constraint on the agent‟s decisions: they are determined endogenously by the process described by Wicksell[47]. This is why they do not appear in the agents‟ maximization problems from which demand and supply functions are derived. Since a consideration of monetary problems falls outside the purposes of this book, on this issue we stop here. 3.6. The simultaneous operation of both substitution mechanisms, the importance of highly elastic factor demand curves, and the role of consumer demand in the determination of relative product prices. 3.6.1. We have seen how technical substitutability among factors, or psychological substitutability among consumption goods, can each be the basis for an argument that the demand curve for a factor is decreasing. The mechanism inducing a higher demand for a factor whose rental decreases is called the direct substitution mechanism in the former case, the indirect substitution mechanism in the latter case. We have examined these mechanisms separately so as to highlight their functioning as clearly as possible. But generally in a real economy there will be both the possibility to choose among different ways of producing a product, and some flexibility of consumer choices in response to changes in relative prices, hence both mechanisms will be simultaneously at work. This simultaneous presence can be introduced into the general equilibrium equations of the economy of the previous Section, where two consumption goods are produced by labour and land, by assuming that in each industry factor proportions are variable. We can for example assume that in each industry there is a CRS differentiable production function. This requires very little modification in the equations we have written down: we only have to interpret the technical coefficients L1, T1, L2, T2 as, not given, but instead functions of w/ because chosen by entrepreneurs depending on the factor rental ratio (as the reader no doubt already knows, but anyway we will rigorously study in chapter 5, with constant-returns-to-scale production functions the technical coefficients only depend on relative factor rentals). Indeed, by choosing the point of tangency between isocost and isoquant, the entrepreneur also chooses the amount of each input per 47 This endogenous determination requires time, but this is perfectly compatible with the aim of traditional marginalist value theory, which was the determination of long-period equilibria. A long controversy has debated the correctness of the Classical Dichotomy, but only because, owing to transformations undergone by marginalist theory (and to be discussed here in chapter 7), that aim was lost sight of, cf. Petri (2004, ch. 5, Appendix 1). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 33 unit of output, i.e. the technical coefficients[48]. Apart from this reinterpretation of the technical coefficients not as data but as functions (of w/), and therefore endogenously determined, the equations remain exactly the same[49]. But the possibility to vary factor proportions in each industry renders the demand for each factor more responsive to changes in its relative rental; in the economic jargon, the factor demand curve is more elastic. 3.6.2.??Mettere nell‟App. Mat. tutto questo par.? Let us remember the meaning of the elasticity of a function. Let y=f(x) be a continuous function. If x changes from x‟ to x”=x‟+Δx, y changes from y‟=f(x‟) to y”=f(x”)=y‟+Δy. The interval elasticity of f(x) is defined as the proportional, or percentage, variation of y divided by the percentage variation of x that causes it, i.e. y y % y , sometimes represented as . If we consider smaller and smaller variations of x, tending in x % x x the limit to an infinitesimal variation dx, the variation of y tends in the limit to dy=f‟(x)∙dx and the value of the interval elasticity tends in the limit to the point elasticity ε = (dy/y)/(dx/x) = dy/dx ∙ x/y = f‟(x) ∙ x/f(x). Let us clarify the meaning of this notion. Suppose x is positive and increases by 1%; generally for such a small percentage increase of x, the point elasticity is a good approximation to the interval elasticity; if x>0 and f(x)>0 as usually in economics, the sign of ε is the same as the sign of dy/dx; then |ε|>1, respectively =1, or <1, means that y varies by a greater percentage, respectively the same percentage, or a smaller percentage, than x; e.g. if dy/dx is negative and |ε|>1, a 1% increase of x causes a decrease of y by more than 1%[50]. In approximate terms, the elasticity tells us the percent variation of the dependent variable caused by a 1% increase in the independent variable. The difference between derivative and elasticity is illustrated by a decreasing linear function 48 To make this point evident, we present an example, at the cost of anticipating some notions to be studied again in chapter 5. Suppose the production function is q=xαy1−α; if the factor rentals are vx and vy, technical coefficients can be derived as follows: since in this case isoquants can be shown to be strictly convex, cost minimization requires that the slope of the isoquant equals the slope of the isocost i.e. y (∂q/∂x)/(∂q/∂y) ≡ = vx/vy; this implies y=x∙(vx/vy)∙(1−α)/α from which one derives q/x = 1 x xα∙(vx/vy)∙(1−α)/α, whose reciprocal is the technical coefficient of factor x, a function of vx/vy. We leave the derivation of the technical coefficient of factor y to the reader. 49 The fact that technical coefficients are now variable does not mean that they become additional unknowns, because they become functions that express their dependence from w/β; e.g. the given number L1 is replaced by the function L1(w/β). The derivation of the precise form of these functions would require knowledge of the production functions, but for a general analysis of the theory this derivation is unnecessary. 50 The value and sign of ε is the same for increases and for decreases of x because if dx is negative then dy also changes sign. If x is a negative number, it is useful to remember that the sign of dx/x is the same as the sign of d|x|/|x| (make sure you are convinced of this fact). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 34 q=f(p)=a–bp, with a,b>0. This might represent a demand function, with p the price, and q the quantity demanded. The derivative is constant and equal to –b, while the elasticity of demand, ε = dq/dp ∙ p/q = –bp/q, changes continuously, and if we restrict attention to the non-negative quadrant we obtain that as p varies from zero to the level where q becomes zero, |ε| varies from zero to +∞, reaching value 1 exactly halfway, cf. the continuous black demand curve in Fig. 3.5bis (where, following the common practice in economics, the unit price p is measured on the vertical axis although it is the independent variable). Another difference emerges by comparing this demand curve with the dotted demand curve in red: the slope is the same, but at the price at which |ε|=1 with the first demand curve, elasticity is almost infinite with the second demand curve. An outward shift of a demand curve without change in slope decreases the elasticity associated with each price and increases the elasticity associated with each quantity. intl1.5 p |ε|=+∞ |ε|>1 |ε|=1 |ε|<1 ε=0 O qmax/2 qmax Fig. 3.5bis Differently from the slope or derivative, the elasticity of a function is independent of the units in which the independent and the dependent variable are measured. In the previous demand function, suppose that the good is weeks of vacation while the price is per day of vacation; the measurement of the quantity of vacation in days rather than weeks alters the slope (and the horizontal intercept, which becomes seven times bigger), but not the elasticity. Furthermore, even with given units, elasticity gives a measure of the strength of the influence of the independent variable on the dependent variable, which for many purposes is more useful than the slope. For example, suppose that there is 10% unemployment and the aggregate demand for labour, measured in number of labourers, is a linear decreasing function of the monthly real wage, measured in dollars, with a derivative equal to –100. Thus for each decrease of the real wage rate by one dollar, labour employment increases by 100 unit. This means an irrelevant influence of the real wage on unemployment if the real wage is 100 dollars and employment is 1 million (halving the real wage only increases employment by half a percentage point, there is little hope that, if there is unemploment, a wage decrease may significantly alleviate it); it means a big influence if the real f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 35 wage is 1000 dollars and employment is 50,000, because then halving the wage doubles employment. This is grasped by the elasticity of labour demand with respect to the real wage, which is –0.01 in the first case, and –2 in the second case. A useful trick with elasticities is to remember that if we have a differentiable function y=f(x) with y,x>0 then the point elasticity (dy/y)/(dx/x) is identical with the logarithmic derivative d(log y(x))/d(log x).[51] Thus if one graphs log y as a function of log x, the slope measures the elasticity of y=f(x). 3.6.3. Back to the main argument, suppose that a certain reduction in real wages would increase the demand for labour by 2% if the indirect substitution mechanism were the sole one allowed to operate (i.e. if changes in technical coefficients were forbidden); and suppose that the same reduction in real wages would increase the demand for labour by 3% if only the direct substitution were allowed to operate (i.e. if changes in the composition of demand by consumers were forbidden); then if both substitution mechanisms are allowed to operate, the increase in the demand for labour can be assumed to be approximately 5%(52). This is important because the theory, in order to be a plausible explanation of how income distribution is determined in market economies, needs not only decreasing, but also significantly elastic demand curves for factors. We proceed to motivate this statement. 15 pt w supply wmax demand w* 0 L* L Fig. 3.6 51 Let f‟(x) stand for the derivative of y=f(x). By the definition of differential, in x° it is df(x)=f‟(x°)∙(x– x°)=f‟(x°)∙dx. Therefore d log f(x) = [(1/f(x)) f‟(x)]dx, and d log x = (1/x)dx, and their ratio is [x/f(x)]f‟(x). 52 The technical substitution brought about by the lower real wage may reduce or increase the difference in factor proportions between the production processes of the several industries. With many industries, it is not impossible that the changes roughly compensate one another, and that therefore the strength of the indirect substitution mechanism be not altered. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 36 The need for decreasing factor demand curves is easily grasped by noting that otherwise the theory would completely lose credibility: in Fig. 3.6, the increasing demand curve for labour entails that the sole equilibrium with positive rentals for both factors is unstable: if w>w* then wages keep increasing up to absorbing the entire output, if w<w* then wages keep decreasing down to zero; the economy therefore ends up with the entire income going to only one factor, a result in contradiction with experience. The need for a considerable elasticity of the decreasing demand curve for labour (but analogous considerations apply to any other factor which is observed to have positive rental) arises from the need to minimize the likelihood of four types of occurrences which, unless extremely unlikely, would raise serious doubts (for reasons to be explained presently) on the correctness of this theory as an explanation of observed income distribution. The four types are: – a zero or implausibly low equilibrium real wage; – enormous changes in income distribution owing to only small changes in relative factor supplies; – multiple equilibria; – practically indeterminate equilibria. These possibilities are illustrated in Figg. 3.7 and 3.8 with reference to the labour market; it is also shown how a higher elasticity of the demand for labour would render them less likely. Fig. 3.7(a) illustrates how a decreasing but highly inelastic demand curve for labour (the heavy continuous line) may easily imply even a zero real wage, besides implausible changes in income distribution owing to small changes in labour supply. The small increase in labour supply from L1 to L2 causes the wage rate to fall from a high level to zero(53); the small decrease from L1 to L3 causes the real wage to rise up to absorbing the entire product (in the horizontal part of the labour demand curve, the supply of land is not fully utilized because if it were fully utilized its marginal product would be negative; the marginal product of the utilized land is zero, and the rent of land is zero). The more elastic demand curve DD' would avoid these outcomes which have no correspondence with empirical observation. Fig. 3.7(b) illustrates the possibility of multiple equilibria due to a backward-bending labour supply curve, i.e a supply of labour with a downward-sloping section[54]. We have up to now left aside this problem by assuming a rigid labour supply. But this was only a simplifying assumption. As no doubt the reader knows from her introductory economics course, a rise in wages makes labour suppliers wealthier, and the result may well be that they decide to work less (think of labour supply as decided on a household basis: a rise in the husband‟s wage may well induce the 53 Even if for very low levels of the real wage the supply of labour fell to zero, and this prevented the real wage from reaching zero, still the equilibrium real wage might be so low as to result implausible anyway. 54 As will be explained in detail in chapter 4, the term „backward-bending‟ derives from the thesis that initially (i.e. for very low wage levels) the supply of labour is necessarily an increasing function of the wage. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 37 household to send children to school for more years, or the wife may renounce working when she has small children, or grandpa may retire earlier); indeed, most economists in the history of marginalist economics have admitted that, for high levels of the wage, such an outcome is the more probable one; and many would also admit that poor people often try to work more if their real wage decreases. Then there may easily be three (or even more) equilibria in spite of the decreasing demand curve. This means that, if any shock pushes the labour market into a state of disequilibrium, the real wage toward which the economy will converge cannot be univocally predicted; in Fig. 3.7(b) the highest and lowest equilibrium wage are 'locally stable' equilibria, i.e. the economy tends to one of them only if w is sufficiently close to the corresponding equilibrium value; the intermediate equilibrium is unstable: on either side of it, the relationship between supply and demand is such that the wage tends to move away from the equilibrium. (Figure with spacing 20 pt) w demand D supply A D' A' L3 L1 L2 L Fig. 3.7(a) Fig. 3.7(b) It might also happen that a transitory disturbance has permanent effects: suppose that the economy has settled in one of the locally stable equilibria and then a transitory shock, which does not permanently shift either the demand or the supply or the demand curve, causes the wage to pass on the other side of the unstable equilibrium; after the disturbance has disappeared, the economy does not gravitate back to the old equilibrium but rather toward the other locally stable equilibrium. This will happen the easier, the closer the equilibria are to each other, and the theory offers no reason to consider the case of multiple equilibria close to each other less likely than the case of multiple equilibria ver far from each other. Now, a theoretical prediction that accidental and transitory divergences from an equilibrium may easily cause income distribution to jump to another, considerably different equilibrium would be hardly reconcilable with the historical evidence: the distribution of income among wages, rents and interest shows a considerable persistence through f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 38 time, undergoing only very gradual changes. To be credible, the theory must argue that such a jump is extremely unlikely; it can do so, if it can argue that factor demand curves are highly elastic, because here too, the more elastic the demand curve, the less likely the multiplicity of equilibria, as shown in Fig. 3.7(b) by the more elastic demand curve AA' that ensures that equilibrium is unique. intl1 w SL w1 we w2 DL O L Fig. 3.8 Practically indeterminate equilibrium: between w1 and w2 the discrepancy between supply and demand is so weak that the tendency toward we is practically non-existent. Finally, the possibility of what we will call practically indeterminate equilibria is much less discussed than multiple equilibria, but would appear to be potentially as damaging to the theory. When the labour supply curve is backward-bending, it cannot be excluded that the labour demand curve and the labour supply curve stay very close to each other for an ample interval of values of the real wage, cf. the interval between w1 and w2 in Fig. 3.8. In this case, even when the intersection between the two curves is unique and, strictly speaking, stable (as in Fig. 3.8), since the forces pushing toward equilibrium cannot but be the weaker the smaller the discrepancy between supply and demand, in an ample interval of values of the real wage around the unique equilibrium value the tendency toward equilibrium is extremely weak, and therefore in all likelihood unable to bring the real wage close to the equilibrium value in a reasonable time, and also easily blocked by even very weak rigidities: so the real wage would have to be considered indeterminate in such an interval of values, and one would have to have recourse to other forces in order to explain what determines it and renders it usually quite persistent. Here too, the likelihood of such a phenomenon, like the likelihood of multiple equilibria, decreases as the elasticity of the labour demand curve increases. 3.6.4. It is easier now to understand the great space allotted in marginalist writings to a general analysis of the influence of changes in relative prices on consumer choices. Psychological substitutability is important in the marginalist approach above all because it is thought to provide a f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 39 reason, for highly elastic factor demand curves, additional to the reason derivable from technical substitutability. This role of consumer choice explains why in this approach great space is given to examining the general qualitative properties of consumer choices and in particular how these choices change with prices, while comparatively little space is given to examining the contents of these choices, their determinants, and their changes over time. What the approach is really interested in(55) is, to what extent the general properties of consumer choice can supply reasons in support of decreasing and sufficiently elastic demand curves for factors; the study of consumer choice is important above all for its implications for the uniqueness and stability of equilibrium in factor markets. Still, the indubitable existence of both technical choice and consumer choice does not tell us much on the extent of technical and psychological substitutability. Some authors (e.g. Joan Robinson) have argued that this extent is exaggerated by marginalist writers: when income distribution changes (within plausible limits – too great changes are unfeasible because they would disrupt social life and hence the functioning of the economy), according to these authors optimal production techniques are often unaffected, and substitution in consumption is also very weak, the dominant influence on consumption habits being the income level (and income effects can change the composition of consumption in any direction). Other authors disagree, and argue that for example the household‟s choice between performing itself most of the domestic labour it needs, or supplying more salaried labour and purchasing domestic labour services, is considerably affected by the real wage of servants and maids. It seems impossible to decide on the scientific robustness of the marginalist approach on the sole basis of these arguments. We must continue our exploration of the approach. 3.6.5. Now the way consumer preferences affect relative product prices is easily understood. The above equations of the equilibrium of production and exchange show that, once factor prices are given, product prices are given as well. (Chapter 5 will confirm this conclusion, reached here for a simplified two-goods two-factors economy.) So preferences have no direct effect on product prices; they can influence relative product prices only through their influence on relative factor prices. Let us first illustrate this influence for the case of a fixed-coefficients economy (with a unique equilibrium and no „damaging‟(56) income effects). Consider the economy where labour and land 55 Anyway the search for explanations of the contents of consumer choices, i.e. for explanations of preferences, does not sit easily with the structure of marginalist theory, because it risks revealing that preferences are themselves to a large extent determined by the functioning of the economy (e.g. by advertising), but this would question the legitimacy of taking preferences as data of the equilibrium (as well as of basing welfare judgments on them). 56 . „Damaging‟ in the sense of damaging for the likelihood of uniqueness and stability of neoclassical equilibria, and therefore possibly damaging for the plausibility of the approach. Here the absence of → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 40 (inelastically supplied) produce consumption goods 1 and 2 in fixed-coefficients industries, good 1 being the labour-intensive one. Suppose that starting from an initial full-employment equilibrium, tastes shift in favour of good 1, i.e. at the initial factor and product prices the share of consumption expenditure going to good1 increases. The rise in demand for good 1 causes an excess demand for labour if land remains fully employed, or possibly an excess demand for labour and excess supply of land. Either way the wage-rent ratio rises, and this causes a rise in the price of good 1 relative to good 2, which goes on until consumers go back to the previous equilibrium composition of demand, which in this economy (owing to fixed coefficients, rigid factor supplies, and as many consumption goods as factors) is the sole one compatible with the full employment of both factors. In the new equilibrium the quantities produced are the same as in the old, but the increased preference for good 1 has caused a rise in the relative price of the factor more intensively used in industry 1. Let us extend the example to three consumption goods 1, 2 and 3, again produced in fixed- coefficients industries with rigidly supplied labour and land. Suppose that, starting from an initial equilibrium, tastes shift in favour of consumption good 1 while leaving relative preference for the other two goods unchanged, i.e. at the initial factor and product prices the share of consumption expenditure going to good 1 increases, while the composition of the remaining consumption demand remains unchanged. Assume that good 1 is more labour-intensive than the average, i.e. the labour-land ratio in its production is higher than the ratio of aggregate labour supply to aggregate land supply. The increase in the demand for good 1 again tends to increase the demand for labour, so the wage/rent ratio tends to increase. Suppose that good 2 is more labour-intensive than good 3; then p2/p3 rises, and then the argument is that the q2/q3 ratio in demand will generally tend to decrease. So the average labour-land ratio in the industries 2 and 3 jointly considered decreases. If this decrease is not enough to counterbalance the tendency to an increase of the average economy- wide labour-land ratio caused by the expansion of production of good 1, there will still be excess demand for labour, and the wage/rent ratio will keep increasing, until the demand for good 1 is sufficiently discouraged and the q2/q3 ratio has sufficiently decreased for the simultaneous full employment of labour and land to be again possible. Now the shift of tastes in favour of good 1 is able to cause an expansion of its production because by raising the relative price of labour it induces a lesser utilization of labour in the ensemble of the other industries.(57) „damaging‟ income effects means 1) that when a consumption good increases in price relative to other consumption goods, the composition of demand shifts against it and 2) that factor supplies are not backward- bending. That things should work like this is far from guaranteed, but here we are interested in pointing out how the theory was traditionally believed to work: the existence of some difficulties was admitted but these difficulties were estimated to be of minor importance and hence negligible. Later (chapter 6 and elsewhere) we will discuss whether this neglect was legitimate. 57 Exercise: discuss the effects on distribution and on the composition of demand in this three-goods economy if there is a shift of tastes in favour of good 1 but this good is produced with exactly the average labour-land proportion. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 41 If there is technical substitutability, then the argument is that the above process will be accompanied by a process of technical factor substitution within each industry, again induced by the rise in the relative price of the factor more intensively used in the production of the good whose demand has increased. The shift in tastes in favour of good 1 has again the initial effect of causing a rise in the wage/rent ratio. But now this rise, besides inducing psychological substitution among the other goods in favour of land-intensive goods, also induces technical substitution in favour of land in all industries; the change in relative factor prices necessary to reconcile a given shift in tastes with full factor utilization is therefore smaller than in the fixed-coefficients case. The possibility to vary the labour-land ratio in each industry means that even when the number of consumption goods is equal to the number of factors the full-employment composition of output is not rigidly determined: for example with two factors and two goods, when the production of good 1, the labour-intensive good, is increased, the average demand for labour per unit of land initially increases, but if the consequent increase in the wage-rent ratio brings about a decrease in the labour- land ratio in at least one industry, this allows re-establishing the full employment of both factors without having to go back to the original composition of production. As in the case of fixed coefficients, the factor used more intensively in the expanding industry rises in relative rental; and this causes a rise in the relative price of the product of the expanding industry. Let us supply a graphical proof of the last statement. Assume that consumption goods A and B are produced by inelastically supplied labour and land, with CRS production functions that yield smooth, strictly convex isoquants. Assume initially that good A is more labour-intensive than good B at all wage-rent ratios (then isoquants can only cross once, see Fig. 3.8bis(a)??). We want to study the effect on pA/pB of a change in general equilibrium due to a shift of preferences in favour of good A. Choose units for the two goods, such that in the initial equilibrium their costs of production are the same, pA/pB=1. This is represented in Fig. 3.8bis(a)?? as the same continuous- line isocost being tangent to both the unit isoquant of good A and the unit isoquant of good B; the tangency points are H and K. The reader can check that the isoquants are such that for all slopes of isocosts the optimal labour-land ratio is always higher in the production of good A. Assume now that the wage-rent ratio increases: the new tangency between unit isoquant and isocost is represented by points M and N and the isocost is farther from the origin for good A, indicating a higher cost of production and hence pA/pB>1. intl1.5 T T T unit isoquant of A unit isoquant of B unit isoquant of B K H unit isoquant of A ▪X f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 42 1/ L L L (a) (b) (c) Fig. 3.8bis The analysis is somewhat less simple if there is factor intensity reversal, that is, if the labour- land ratio can be higher for good A or for good B depending on the wage-rent ratio. This will be the case if the isoquants can cross twice, as shown in Fig. 3.8bis(b). [??Make it an Exercise] Then, assuming smooth isoquants, there is a wage-rent ratio such that factor intensity is the same in both industries; let us assume that there is only one such wage-rent ratio. Let us choose units for the consumption goods such that when the wage-rent ratio induces the same factor intensity, the cost of production is the same. Then the unit isoquants are tangent to each other, as in Fig. 3.8bis(c), at a labour-land intensity . Let us suppose that good A is the more labour-intensive good if the wage- rent ratio is higher than the equal-intensities level, and the less labour-intensive good if the wage- rent ratio is lower than that level, i.e. that the elasticity of substitution of good A's isoquant is the higher of the two; then A's unit isoquant is below B's unit isoquant except at the point of equal factor intensity. It is then apparent that if the wage-rent ratio moves away from the equal-intensities level, pA/pB decreases both when the wage-rent ratio increases and when it decreases (cf. the dotted isocosts in Fig. 3.8bis(c)??). If in the initial equilibrium the wage-rent ratio is precisely the one yielding equal factor intensity (which means that the ratio of labour supply to land supply L/T is ), then changes in the composition of demand cause no change in distribution. Suppose now that the ratio between factor supplies is L/T> (e.g. the endowment point is X in Fig. 3.8bis(c)??), then in the initial equilibrium the wage-rent ratio is lower than the equal-intensities level, pA<pB, and good A is the more labour-intensive one; a shift of preferences in favour of good A causes an increase in the labour-land ratio in total demand, hence the wage-rent ratio increases, the isocosts become steeper and the distance between them decreases (also relative to their distance from the origin), indicating that pA/pB increases. The reader will have no difficulty in applying a similar reasoning to the other possibilities (L/T<, and/or a shift of preferences away from good A). Thus here again an increase in the supply of a good is associated with an increase in the relative price of the good, as long at least as factor intensities are different, because the change in the composition of output requires a change in relative factor prices. 3.7. Efficiency, the forest exchange economy, choice curves, and equilibrium in the Edgeworth box. 3.7.1. The marginalist picture of the working of market economies implies that these tend to the full employment of resources and to reflect consumers' desires: all supplies find purchasers, the composition of production adapts to the composition of demand, accumulation reflects the desire of f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 43 consumers to renounce current consumption in favour of future consumption. The efficiency implications of this picture are highlighted by the so-called First Theorem of Welfare Economics. This states: a competitive equilibrium is Pareto-efficient. Here too, we postpone a rigorous examination of this theorem and of its (very restrictive) assumptions to a later chapter, and only present a simple discussion of this result. Pareto efficiency is a notion intended to apply to situations where there is more than one consumer. A situation is Pareto-efficient if no improvement in the utility level of one consumer is possible without a decrease of at least one of the utility levels of the other consumers[58]. The notion can be illustrated with the help of a diagram known as the Edgeworth box. We explain it in some detail, for the readers who are not familiar with it. Assume two consumers A and B who have given endowments of two consumption goods, good 1 and good 2, and can exchange these goods. The vector of endowments of consumer A is ωA=(ω1A,ω2A) and that of consumer B is ωB=(ω1B,ω2B). Draw the indifference curves of consumer A and those of consumer B in two separate diagrams; then rotate the diagram of consumer B by 180° and superpose it to the diagram of consumer A in such a way that the intersection of the axes forms a rectangle of length equal to ω1A+ω1B and height equal to ω2A+ω2B. In this rectangle, called the Edgeworth box, the indifference curves of consumer A have the lower left-hand corner as their origin, and the indifference curves of consumer B appear upside down, with the upper right-hand corner as their origin. The length of the sides of the box represents the total amounts of goods available, i.e. the total endowment (ω1≡ω1A+ω1B, ω2≡ω2A+ω2B) of the 'economy' consisting of these two consumers. We interpret any point of the box as indicating simultaneously two consumption bundles: its coordinates relative to the lower left-hand corner indicate a consumption bundle of consumer A; its coordinates relative to the upper right-hand corner indicate a consumption bundle of consumer B; since these two bundles sum up to the total endowment, any point x=(x1,x2) of the box (we use the coordinates relative to the lower left-hand corner to indicate a point in the box; the coordinates relative to the upper right-hand corner can be obtained by subtraction from the length of the sides of the box) represents a possible allocation (x1A, x2A; x1B, x2B) of the total endowment between the two consumers, where x1A=x1, x2A=x2, x1B=ω1–x1A, x2B=ω2–x2A. For example the point x=(ω1A,ω2A) indicates the endowment of consumer A, but also the endowment of consumer B because its coordinates relative to the upper right-hand corner are (ω1B,ω2B). Consider then the Edgeworth box of Fig. 3.9, where point Ω=(ω1A,ω2A) is the initial allocation, i.e. the endowments. We assume that consumers prefer to have more of a good to having 58 When there is only one consumer and her utility cannot be further increased, the situation is certainly efficient but it is not generally named Pareto-efficient. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 44 less, so indifference curves are downward-sloping( 59 ); we furthermore assume that indifference curves are strictly convex and 'smooth' (i.e. without kinks). In the case we have drawn, the initial allocation Ω as well as an allocation such as X are not Pareto-efficient, because there exist allocations where one of the two consumers is better off than in either of the first two, while the other consumer is not worse off. For example allocation K is associated for both consumers with a higher utility level than in Ω. Allocation K is Pareto-efficient because any movement away from it causes the utility level of at least one consumer to decrease. Allocations Z, H, Y are Pareto-efficient too, for the same reason. Pareto-efficient allocations are therefore many, differing in the utility levels of the two consumers. In the case of Fig. 3.9, clearly the Pareto-efficient allocations are all those where the indifference curves of the two consumers are tangent to each other. Any internal point of the Edgeworth box where the indifference curves of the two consumers are not tangent to each other is not Pareto-efficient (show it graphically!). intl1 ω2A+ω2B B's origin good 2 ● Z H■ ▪K Y● X● ω2 A Ω● A's origin good 1 ω1 A ω1A+ω1B Fig. 3.9. Edgeworth box. The continuous heavy curves are indifference curves of consumer A, the broken heavy curves are indifference curves of consumer B. They are drawn so as to make it clear that they extend outside the box. The Pareto set (actually, part of it) is the curve through Y, H, K and Z. Ω is the endowment point. intl1.5 ●Ω 59 Prove it as an Exercise. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 45 Ω● (a) (b): perfect complements Fig. 3.10. The indifference curves of consumer A are drawn as continuous lines, those of consumer B as broken lines. The locus of Pareto-efficient allocations in an Edgeworth box is called the Pareto set[60]. Exercise: find the Pareto set in the cases of Fig. 3.10(a),(b). Now we shall use the Edgeworth box to show graphically that, for an economy representable in an Edgeworth box (an exchange economy), any equilibrium allocation is in the Pareto set, and hence the First Theorem of Welfare Economics is valid. 3.7.2. We start by characterizing the equilibrium of an exchange economy. As a help to intuition, let us give concreteness to our description of the exchange economy. Let us imagine an economy consisting of households who live scattered in a forest and who once a month gather in a market place for the monthly fair, in which goods are exchanged. Only two goods are exchanged, corn and meat. Strict social customary rules impose to each household always to bring to the fair the same given quantities (specific to the household) of each good, but leave to the household to decide how much of these quantities to offer for sale and how much to retain. This allows us to neglect the processes of production of corn and meat, which go on in the forest and do not affect the quantities brought to the fair. At the market fair there is competition and a free determination of prices on the basis of higgling and bargaining. We treat each household as a single decision unit, and we call it a 'consumer'. Let us assume there are H consumers distinguished by the index h=1,...,H. We assume that each consumer h has a well-defined utility function uh(x1,x2), which generates smooth, decreasing, convex indifference curves. Corn is good 1, meat is good 2. Let us call 'endowment' of consumer h the vector ωh=(ωh1,ωh2)T of quantities of corn and meat that she brings each month to the fair. The consumer's 60 Sometimes it is also called the contract curve, but other economists restrict the latter term to mean only a portion of the Pareto set, the portion such that at all its points neither consumer is worse off than with her initial endowments; with convex indifference curves, it is the portion of the Pareto set included in the „lens‟ formed by the indifference curves through the initial endowment point, the portion between points H and Z in Fig. 3.9. The argument is that only points in this portion can be reached by efficient contracting; cf. Ch. 8 on the core. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 46 problem is how much of these to sell and how much to retain at the market fair. It will depend on the relative price of the two goods. We intend to determine the relative price p2/p1 that will tend to be established over a sufficient repetition of fairs. We analyze the problem as follows. We assume that in this economy people are very wary of getting into debt, so consumers want a balanced budget i.e. want to exit each fair having spent on the purchase of other consumers‟ goods only and all what they earn by selling all or part of their supplies in that fair. Let xh1, xh2 be the quantities of the two goods that the h-th consumer intends to take back home from the fair; these are called the demands[61] of the consumer. The quantity zhj=xhj–ωhj indicates, if xhj>ωhj, the quantity of good j that the consumer intends to purchase from other consumers; if it is negative i.e. if xhj<ωhj, then |zhj| is the quantity that the consumer intends to sell to other consumers; zhj is called the (individual) excess demand or net demand for good j. The opposites of excess demands, –zhj, are called (individual) excess supplies. The balanced budget assumption means that the value of the consumer's planned sales must equal the value of the planned purchases, i.e. her plans must satisfy the budget constraint: p1zh1+p2zh2 = 0. The algebraic sum of the values of excess demands must be zero. E.g. if p1zh1>0, then it must be p2zh2<0: if both prices are positive, the consumer must be a net supplier of one good in order to be a net demander of the other good. Note that this budget constraint is equivalent to (replacing z h1 and zh2 with their definitions) p1(xh1–ωh1)+p2(xh2–ωh2) = 0, i.e., rearranging: p1xh1+p2xh2 = p1ωh1+p2ωh2. This formulation states that the value of demands must equal the income of the consumer, where the income is defined here as the value of the endowments. Graphically, if we measure the consumer's demand for good 1 on the horizontal axis, and of good 2 on the vertical axis, the last formulation shows that the budget line necessarily passes through the endowment point ω=(ω1,ω2) – for simplicity we now drop the superscript h – , because the consumption bundle x=(x1=ω1, x2=ω2) satisfies the budget constraint whatever the prices. So if p1/p2, the (absolute value of the) slope of the budget line, changes, the budget line rotates around the endowment point. The position of the budget line is not altered by changes of all prices in the same proportion: it only depends on the endowment point and on relative prices. intl1.5 x2 61 We assume here that they are single-valued functions of endowments and of the relative price. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 47 ω2 ω1 x1 Fig. 3.10bis. Two budget lines when income derives from given endowments: the budget line necessarily goes through the endowment point and therefore only depends on relative prices. The consumer, whose indifference curves we assume are of the usual strictly convex smooth type, maximizes her utility by choosing the point on the budget line corresponding the the indifference curve „farthest‟ from the origin, and therefore − if this point is internal − the point where the budget line is tangent to an indifference curve. Given the slope of the budget line, the tangency condition determines the optimal choice (x1,x2) and, by difference from the endowment point, also the vector of net or excess demands (z1,z2), cf. Fig. 3.11(a). Now let p1/p2 change; the consumer's choice changes; the curve which indicates all the optimal consumption bundles corresponding to different relative prices is usually called the offer curve of the consumer. I find this term inappropriate, the curve does not indicate 'offer' decisions (the meaning of 'offer' is anyway unclear), it is the locus of optimal consumption choices of the consumer, and therefore I propose to call it choice curve; it is the heavy blue line in Fig. 3.11(b). Note that a choice curve can „turn back on itself‟, horseshoe-like; in other words, it can have upward-sloping sections even when indifference curves cannot. With smooth, strictly convex indifference curves, the choice curve is continuous and necessarily passes through the endowment point, where it is tangent to (i.e. has the same slope as) the indifference curve passing through that point and hence has the same slope as the budget line that makes the consumer satisfied with her endowment. The choice curve shows the demand for each good as a function of relative prices. The straight line connecting a point x=(x1,x2) of the choice curve with the endowment point ω has slope equal to the price ratio –p1/p2 that induces the consumer to choose x. As p1/p2 changes, the intersection of the budget line with the choice curve moves, tracing how x 1 and x2 change with relative prices. intl1.5 x2 x2 x f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 48 z2 ω ω● -z1 x1 x1 Fig. 3.11(a) Fig. 3.11(b): indifference curves are red, budget lines are black, the choice curve is the blue heavy line p h x1 ( 1 , ) The vector function xh(p1/p2,ωh)= that indicates how the quantities demanded x1 p2 p1 h x2(p , ) 2 and x2 depend on (p1/p2) and on the endowment is called the Walrasian(62) demand function of the consumer. 3.7.3. Suppose now that the economy comprises 10000 consumers identical to A and 10000 consumers identical to B. Then the Edgeworth box of Fig. 3.8 can be used to study the equilibrium of such an economy, because it reproduces in scale 1:10000 what happens in the economy. The high number of consumers makes it plausible that each consumer will be price-taker, i.e. will treat prices as uninfluenced by her own decisions. Equilibrium requires that relative prices be such that for each good i the algebraic sum of the excess demands of all consumers, called the aggregate excess demand or simply the excess demand for good i, be nonpositive, and associated with a zero price if negative (indicating an excess supply). intl1 ω2A+ω2B choice curve of A B‟s origin x2=x2A K● ●X choice curve H● of B ω2 A ●Ω 62We shall use the term Walrasian demand function for the demand function derived from given endowments (then the consumer‟s income derives from the value of her endowments), and the term Marshallian demand function for the demand function derived from a given income or wealth (a value sum) whose origin is left unspecified. For the moment we assume that to each relative price there corresponds a unique choice, hence we obtain demand functions rather than correspondences. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 49 A‟ origin x1=x1A ω1 A ω1A+ω1B Fig. 3.12 The introduction of choice curves in the Edgeworth box allows us quickly to determine the equilibrium or equilibria without the explicit representation of indifference curves. Consider Fig. 3.12. In it the choice curves of the two representative consumers cross in the endowment point Ω and again in point X=(x1,x2). Point X is then an equilibrium, associated with the price ratio that causes the budget line to go through it. In fact, by construction of the choice curves, when the budget line goes through X consumer A chooses X and consumer B chooses X; thus the excess demands of the two representative consumers cancel out. Let us measure quantities on A‟s axes. Consumer A demands x1A=x1<ω1A of good 1 and therefore has a net supply of good 1 equal to ω1A–x1; consumer B demands x1B=(ω1A+ω1B)–x1 and therefore has a net demand for good 1 equal to ω1A–x1; net supply and net demand balance out. The reader can check that the same is true for good 2. Since by assumption the economy is only a replica, 10000 times larger, of what is happening in this Edgeworth box, the economy is in equilibrium. (One can also show that in the case of Fig. 3.12 the equilibrium is stable, i.e. that if p1/p2 is not the equilibrium ratio, then the differences between net demands and net supplies will cause p1/p2 to tend toward the equilibrium value. This is left as an Exercise for the reader: for example the reader can check the sign of excess demands when the budget line is such as to cause consumer A to choose point H, and consumer B to choose point K.) We can now show that the equilibrium is Pareto-efficient. When the budget line goes through X, consumer A chooses X because that's the point of tangency between indifference curve and budget line. The same is true of consumer B. Hence the indifference curves through X of the two consumers, being both tangent to the same line, are also tangent to each other, which means that the point is in the Pareto set. The choice curves also cross in the endowment point Ω but there is no equilibrium there, because the indifference curves of the two consumers through that point have the same slopes as the respective choice curves and therefore have different slopes, hence no relative price can induce both consumers to choose their endowment point. An interior endowment point can be an equilibrium only if the choice curves are tangent in that point. intl1 B f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 50 2A A 1A Fig. 3.12 bis. Three equilibria in an Edgeworth box. Consumer A's choice curve is the heavy continuous line, consumer B's choice curve is the heavy broken line. If the two choice curves cross more than once in points other than the endowment point (this is possible, cf. Fig. 3.12bis), then all these points are equilibria, and in all of them, assuming smooth indifference curves, the indifference curves of the two consumers are tangent to each other because both tangent to the budget line; so all these points are Pareto-efficient. Exercise. Determine the equilibrium or equilibria in the two cases of Fig. 3.10 and show that all of them are Pareto-efficient. 3.8. Pareto efficiency in the production economy. 3.8.1. Let us now extend the notion of Pareto efficiency and the First Theorem of Welfare Economics to a production economy. We consider an economy where labour and land produce a variety of consumption goods in single-product industries; we assume insatiable consumers (more is always better). We also assume that there are no externalities, no relevant government activities, no public goods (cf. chapter 14). Initially we assume that: factor supplies are rigid; indifference curves and isoquants are smooth and strictly convex; all consumers demand all consumption goods in positive quantities; all firms employ both factors in positive quantities. Then it is necessary and sufficient for Pareto efficiency that three conditions be satisfied (the new terms that appear in these conditions will be explained in the course of the proof): a) Efficiency in production: it must not be possible to produce more of a good without decreasing the production of any other good. In other words, the economy must be on its production possibility frontier, PPF. This requires a1) the full employment of all supplies of factors, a2) efficiency within each enterprise (for each vector of amounts of factors employed by the firm, it must not be possible to produce more with the production possibilities known in the economy), and a3) equality among firms of the marginal technical rate of substitution (TRS) between factors. b) Efficiency in the allocation among consumers of any given total production vector of consumer goods: all goods must be allocated to some consumer (no pure waste) and it must not be possible, by reallocating among consumers the given quantities available, to increase the utility of one consumer without decreasing the utility of some other consumer. This requires the equality among consumers of the marginal rate of (psychological) substitution between goods, MRS. c) Efficiency in the composition of production: it must not be possible to increase the utility f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 51 level of one consumer without decreasing the utility level of any other consumer by changing the composition of production (i.e. the point on the PPF). This requires that the marginal rate of transformation MRT between any two produced goods (i.e. the slope of the production possibility frontier restricted to those two goods) be equal to the marginal rate of psychological substitution MRS between them, for all consumers. Proof. The necessity of the first condition can be shown as follows. The need for the full employment of factor supplies and for the efficient use of the best available technology is obvious, if waste is to be avoided. The need for the equality of the TRS of different firms can be shown through the analogous of an Edgeworth box. The TRS is the slope of the isoquant, i.e. the (negative of the) ratio between the marginal products of the two factors, TRSLT = – MPL/MPT (labour is measured on the horizontal axis). Take any two firms A and B which are employing the quantities LA, TA, LB, TB of labour and land. Draw an Edgeworth box with length equal to LA+LB and height equal to TA+TB. In this box draw the isoquants of firm A with the lower left-hand corner as their origin, and (upside down) the isoquants of firm B with the upper right-hand corner as their origin. The picture is similar to the Edgeworth box of two consumers, so we can refer to Fig. 3.9 by simply reinterpreting it; each point in it indicates a possible allocation between the firms of the total labour and of the total land employed by the two of them. We can analogousy determine the Pareto set, i.e. the efficiency locus, as the locus of allocations such that it is not possible, by reallocating the factors, for a firm to produce more without the other firm producing less; this set, if interior, will be the locus of points of tangency between isoquants if these are convex. From any other point it is possible to produce more of both goods by moving toward the Pareto set. Efficiency requires that for any couple of firms the allocation between them of the factors they employ be Pareto-efficient; given our assumptions, the allocation is interior, hence if it is Pareto-efficient the isoquants through it are tangent to each other and hence satisfy the condition of equal TRS. Let MPLi, MPTi be the marginal product of labour and of land in the production of good i, then the condition of equal TRS, MPLi/MPTi=MPLj/MPTj, can be re-written MPLj/MPLi=MPTj/MPTi, a ratio that it will be useful to indicate with a symbol, μ. When condition (a) is satisfied, the quantities produced are some point of the economy‟s production possibility frontier, PPF, which is the locus (in Rn if there are n goods and factors) of the combinations of quantities produced, such that it is impossible to produce more of a good without producing less of another good[63]. The PPF can be represented graphically as a curve if one takes as given all factor employments, and all quantities produced but two. For example, if the sole 63 If factor supplies are not rigid and assumed fully employed as here, then the PPF is a surface in R n+m if there are n goods and m factors, and it indicates the combinations of quantities produced and factors employed, such that it is not possible to produce more of a good or employ less of a factor without producing less of some other good or employing more of some other factor. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 52 consumption goods produced are corn and iron, the PPF between corn and iron can be drawn as a curve once the total amounts employed of labour and of land are given. The slope of the PPF restricted to two goods indicates, if units are chosen very small, how many units of the second good (measured on the vertical axis) can be obtained by giving up one unit of the first good; it is called the (economy-wide) marginal rate of transformation of the first good into the second, and indicated as MRT1,2.[64] We now show that if condition (a) is satisfied then MRT1,2= –μ. Let (x1, x2) be a point on the PPF. Suppose you reallocate a very small quantity of labour dL>0 from producing good 1 to producing good 2; the (positive) change in the quantity produced of good 2 is dx2 = dL·MPL2, the (negative) change in x1 is dx1 = –dL·MPL1, hence dx2/dx1 = −MPL2/MPL1 = −μ. If a very small amount of land were reallocated from producing good 1 to producing good 2 we would reach dx2/dx1 = −MPT2/MPT1 = −μ; it is then easy (and left to the reader as an Exercise) to show that dx2/dx1 = – μ also when the reallocation is both of labour and of land. Hence the slope of the PPF is MRT1,2 = −μ. The value of μ indicates how many units of the second good must be given up if one wants to produce one more unit of the first good, and therefore it indicates the opportunity cost for society of good 1 in terms of good 2. As we show later, in a competitive economy this opportunity cost equals the ratio between marginal cost of the first and of the second good, i.e. the ratio between price of the first and price of the second good. Under the assumption of constant-returns-to-scale industries the PPF is concave, and generally strictly concave if the several consumption goods are produced by different production functions at least one of which is differentiable. We do not prove this result here but give some intuition for it through an example. Suppose the two consumption goods are cannons and butter; cannons are produced by labour alone with fixed coefficients, one unit of labour produces one cannon; butter is produced by labour and land according to a differentiable production function with decreasing marginal products. The given supply of land is entirely used in the production of butter, hence if the production of butter is increased by transferring labour from the production of cannons, further units of labour have a lower and lower marginal product owing to the fixed supply of land, hence to a constant decrease in the production of cannons there corresponds a smaller and smaller increase in the production of butter, and the PPF is strictly concave.[65] Note that efficiency in production does not determine the point of the production possibility 64 The MRT between two products is a negative number, but economists often use the term to refer to its absolute value. The qualification "economy-wide" distinguishes this notion of marginal rate of transformation from the analogous notion defined for a single firm that produces jointly more than one product. 65 Exercise: change the assumption on how cannons are produced: by labour and land, with fixed coefficients; butter is still produced by a differentiable production function with constant returns to scale; prove that there is a case in which the PPF is linear. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 53 frontier on which the economy locates itself. The proof of the need for condition (b) is similar. The marginal rate of (psychological) substitution of a consumer is the slope of her indifference curve, MRS1,2 = – MU1/MU2. Take any two consumers A and B and let us suppose that they are allocated given quantities x1A, x2A, x1B, x2B of two consumption goods. Draw in the corresponding Edgeworth box the indifference curves of the two consumers. Unless the point X corresponding to the given allocation is in the Pareto set, it is possible to raise the utility level of both consumers, therefore there is Pareto inefficiency. In our hypotheses, X is interior so if it is in the Pareto set it satisfies the tangency condition between indifference curves i.e. condition (b). Let us now show that (a) and (b) do not suffice for Pareto efficiency, condition (c) is necessary too. Assume that production is on a point of the PPF (condition (a) is satisfied) and that the resulting quantities are allocated efficiently among consumers so that condition (b) is satisfied, but that the slope of the PPF between goods 1 and 2, i.e. MRT1,2, is – 4, while for consumers it is MRS1,2 = – 3. If we measure the goods in very small units, by giving up one unit of good 1 the economy can obtain 4 more units of good 2; suppose that this change in production is implemented, and one consumer receives 1 unit less of good 1 and 4 more units of good 2, while everybody else consumes the same as before; the consumer's utility increases because only 3 more units of good 2 were necessary to compensate her for the loss of 1 unit of good 1. A Pareto improvement has been achieved: the initial situation was not Pareto-efficient. intl1.5 Good 2 PPF H ■ F■ P Y K X O Good 1 Fig. 3.13. If point Y indicates the quantities produced, and X their allocation among the two consumers, Pareto efficiency is not achieved because curve HK indicates the locus of quantities produced which would make it possible to leave the utilities of both consumers unchanged, and the slope of this curve f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 54 in Y (equal to the common slope of the indifference curves in X) is different from the slope of the PPF in Y, so curve HK goes below the PPF, indicating that it is possible to make both consumers better off. A graphical representation of this last condition can be obtained as follows for the case of two goods and two consumers (cf. Fig. 3.13). To each point of the PPF, e.g. to point Y, there corresponds an Edgeworth box with the upper-right corner in that point and the lower-left corner at the origin O; each point in this Edgeworth box describes a possible allocation of the quantities produced between the two consumers A (with origin in the origin), and B with origin in Y; the Pareto set of this Edgeworth box describes the efficient allocations of the given quantities produced. If we are given a point on the PPF and an allocation of the quantities produced that is efficient relative to these quantities, e.g. points Y and X, we can determine how the total quantities produced should change so as to make it possible to leave utility unchanged for both consumers, by moving the upper-right corner of the Edgeworth box (and hence shifting B‟s indifference curves) so that the indifference curve through X of consumer B slides along the indifference curve of consumer A while remaining tangent to it (cf. the red and the green indifference curves corresponding to the red and green upper-right corners); the upper-right corner of the Edgeworth box will then move along the convex curve HK[ 66 ]. The initial slope of this movement of the upper-right corner of the Edgeworth box is given by the MRS in X, and if the latter differs from the MRT in Y, the HK curve goes below the PPF; that is, in order to guarantee both consumers the same utility as in X it suffices to produce quantities below the PPF, e.g. corresponding to point F; but then by producing the quantities of a point on the PPF between Y and H, e.g. point P, it is possible to improve the utility of both consumers. This is impossible only if the HK curve does not go below the PPF, which requires that it be tangent to the PPF in Y, i.e. that its slope in Y, equal – remember – to the MRS in X, be also equal to the slope of the PPF in Y. The necessity of the three conditions has been shown. Their sufficiency for Pareto efficiency is easily proved (under our assumptions): the assumed strict convexity of isoquants and of indifference curves imply that both in a production and in a consumption Edgeworth box the interior points in the Pareto set and only those points exhibit tangency respectively of isoquants, and of indifference curves; hence common TRS plus interiority (plus full factor employment and technological efficiency, of course) implies that the given factor allocation is in the production Pareto set, and common MRS plus interiority (plus no pure waste, of course) implies that the given goods allocation is in the consumption Pareto set; hence the quantities produced are on the PPF and are efficiently allocated. The addition of condition c) guarantees that the convex curve HK in Fig. 3.13 is tangent to the concave production possibility frontier, hence the HK curve does not go below the PPF and it is impossible to increase the utility of one consumer without decreasing the utility of 66 This curve is called a Scitovsky community indifference curve. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 55 the other. ■ 3.8.2. Let us now show that these three conditions are satisfied by a competitive general equilibrium and therefore the First Welfare Theorem holds. For simplicity we continue to refer to the labour-land economy, w is the real wage, β the land rent. In competitive equilibrium factor supplies are fully utilized, and in order to minimize cost all firms adopt TRS = –w/ (tangency between isoquant and isocost), i.e. they all adopt the same TRS because they face the same factor costs; so condition (a) is satisfied. All consumers maximize utility by setting MRSi,j = – pi/pj, and since they face the same price ratios, they all choose the same MRS for the same couple of goods, and condition (b) is satisfied. There remains prove condition (c). Profit maximization in a competitive economy requires that product price equals marginal cost, pi=MCi. Hence the ratio pi/pj between product prices equals the ratio between their marginal costs. Since for all consumers MRSi,j = – pi/pj, we obtain MRSi,j = –MCi/MCj. So if we prove that –MCi/MCj = MRTi,j, this implies MRSi,j = MRTi,j, and condition (c) is satisfied. To prove it, suppose we decrease the production of good i and increase the production of good j, by transferring factors from one industry to the other. If the production of good i varies by a small amount dxi (a negative number), the value of the factors employed in its production, i.e. the industry's total cost C i, decreases by dCi = MCi∙dxi, where MCi is the marginal cost of good i. If the factors thus left free in the i-th industry are employed in the j-th industry, they determine an increase dxj of its production. The slope of the PPF restricted to goods i, j is dxj/dxi. For small variations in the composition of production, factor prices can be treated as constant, hence the increase of Cj, the total cost in the j-th industry, must equal the decrease of Ci, i.e. dCj = MCj∙dxj = – dCi = – MCi∙dxi . This implies MRTi,j ≡ dxj/dxi = – MCi/MCj = – pi/pj = MRSi,j. ■ (Note that through a simple reinterpretation this demonstration is also valid if we admit that the supply of labour and the supply of land are not rigid but depend on prices and on income distribution. We must only admit that among the goods that consumers demand there are also leisure, and 'land for self-enjoyment', goods that are 'produced' respectively by „labour‟ alone and by „land‟ alone, on a one-to-one basis, the total supply of „labour‟ being treated as equal to the sum of leisure and of actual labour, and the same for „land‟. For example, a consumer‟s decision to offer 6 hours of labour and enjoy 18 hours of leisure per day is formalized as a supply of 24 hours of „labour‟, but 18 of these hours are employed by the consumer himself to produce 18 hours of leisure, and only the remaining 6 hours are offered to other firms. The marginal cost and price of leisure is then w, the marginal cost and price of 'land for self-enjoyment' is , and the total supplies of „labour‟ and of „land‟ can be treated as given. This treatment continues implicitly to assume that factor owners are indifferent as to the activities in which their factor supplies are utilized; but whether admitting that workers are not indifferent between different types of labour would destroy the First Welfare Theorem is a question we must postpone to a further chapter. Also postponed is a discussion of the welfare implications of a dependence of labour efficiency on income distribution, whether it be for physical health reasons, or for psychological motivational reasons.) f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 56 The above demonstration assumed no 'corner solutions': all firms employ both factors and all consumers demand all goods. To the contrary, in real economies with very many factors and goods it is normal that not all factors are employed in a firm, and that not all goods are demanded by a consumer. So now we briefly discuss why these instances do not disturb the First Theorem of Welfare Economics. Let us start by noting that the consumer‟s utility maximization condition MU1/MU2 = p1/p2 can be re-written as MU1/p1 = MU2/p2. This equality can be interpreted as equality of the marginal utility of money spent on either good, where the marginal utility of money spent on good i is the increment of utility obtained by the last small unit of money spent on that good, i.e. MUi/pi.[67] That such an equality is required for utility maximization is intuitive: if MU1/p1 < MU2/p2, utility can be increased by transferring one unit of money from buying good 1 to buying good 2. But suppose that the consumer goes on transferring money to good 2, without eliminating the inequality so that in the end her demand for good 1 is zero. It means that the marginal utility of even the first small unit of money spent on good 1 is not higher (and generally smaller) than the marginal utility of money spent on good 2, i.e. MU1/p1≤MU2/p2 (where j stands for any good in positive demand) even when x1=0. If the strict inequality holds, then the MRS1,2 of this consumer is not the same as for consumers who demand both goods in positive amounts; condition (b) does not hold; but this does not mean an inefficiency, because it is efficient not to allocate any amount of good 1 to this consumer: if a small unit of good 1 were allocated to our consumer, she could obtain a Paretian improvement by exchanging it for good j at the market price ratio with consumers for whom MRS1,2= –p1/p2; the utility of the other consumers would not vary, but the utility of our consumer would increase. Analogously, if a firm only employs one factor, e.g. only labour, it must mean that the marginal product of even the first small money unit spent on land, MP T/, is not greater (and generally smaller) than for labour: MPT/ ≤ MPL/w; if the inequality is strict, condition (a) does not hold, but it is efficient not to allocate land to this firm, for the same reason as in the case of the consumer who does not purchase good i. Finally, if good 1 might be produced but is not produced because at a price equal to its marginal cost no one demands it, then for that good condition (c) (as well as condition (b)) may not hold, but here too there is no inefficiency, because it would be inefficient to employ factors to produce that good: they would 'produce' no more, and generally less, utility if so employed, than when employed for the production of goods in positive demand. 3.8.3. The last statement can be made even more intuitive with the following choice of units 67The increment of utility obtained by a small addition dqi of good i is given by MUi∙dqi, and dqi=1/pi is how much additional good i can be purchased with one additional unit of money spent on it. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 57 in which to measure utility. The choice of units for the measurement of a magnitude is always arbitrary (e.g. length can be measured in metres, centimetres, yards, inches, etc.). Let us choose as unit for the utility of each consumer her equilibrium marginal utility of money (or of income), i.e. the common marginal utility of the last unit of money spent on any one of the goods she demands in positive amounts. Then for all consumers who demand good i it is MUi/pi = 1, that is, MUi = pi : the price of a good measures its marginal utility for all consumers who consume it; it is greater than the marginal utility of the (first unit of the) good, for the consumers who choose not to buy it[68]. This allows a useful interpretation of factor rentals. Readers know from introductory microeconomics that under competition the optimal employment of a factor requires that its rental be equal to its marginal product times the product price, e.g. for labour employed in the production of good i it must be w = MPLi∙pi.[69] This equality can be written, owing to MUi=pi, as w = MPLi∙MUi , where the product on the right-hand side can be interpreted as the marginal indirect utility of labour through production of good i(70 ), because it indicates by how much the quantity of that good, produced by the last small unit of labour employed to produce that good, increases the utility of the consumer who gets it. The equilibrium conditions plus our choice of units for utility imply then that the marginal indirect utility of a factor is the same in all productions where it is utilized (as long as one considers consumers who demand those production). Then the reason why a good, say good 1, is not produced can be expressed as follows: the marginal indirect utility of factors utilized to produce even the first unit of that good would be less than the marginal indirect utility they have elsewhere. Therefore it is efficient not to employ them in the production of good 1. It is then easy to understand why monopolistic markets violate Pareto efficiency. Suppose good i is produced in monopoly conditions. From first-year economics the reader knows that, in a monopolistic market, marginal revenue is less than price (also cf. below, chapter 11); since the value marginal product of a factor is its physical marginal product times marginal revenue, one 68 It is just possible that a consumer does not buy a good and yet there is uniformity of the marginal utility of money: it is the case of tangency between indifference curve and budget line precisely on one of the axes. We will neglect this unlikely fluke. 69 This is just another way to say that product price equals marginal cost, because producing one more unit of output requires the additional use of 1/MPL units of labour, hence an additional cost w/MPL, or an additional use of a combination of factors of the same cost (owing to the equality between TRS and factor rental ratios). 70 This marginal indirect utility of a factor must not be confused with the partial derivative of the so- called indirect utility function (a function of prices and income) that will be defined in chapter 4; it is the increase in the utility of the consumer, brought about by one more small unit of the quantity of a factor utilized to produce goods consumed by the consumer. Cf., in the enclosed CD, Appendix 1 to this chapter for a definition of the marginal (indirect) utility of a factor when there is no technical substitutability. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 58 obtains, e.g. for labour, that w < MP Li∙pi , hence w < MPLi∙MUi and the marginal (indirect) utility of labour in the production of good i is greater than in the production of goods whose price equals marginal revenue: the utility of at least some consumer could be raised by allocating more labour to producing the monopolized good. 3.9. Robinson Crusoe, and market valuation as reflecting 'natural' laws. Let us now compare the results of the working of markets according to this theory, with the decisions of a single consumer-producer, Robinson Crusoe, who, alone on a desert island, must decide how much of his time he should dedicate to producing his consumption goods. Assume that Robinson produces a single consumption good via a differentiable production function q=f(L,T) where L is labour time and T land. Since the amount of land available to Robinson is given we can omit it and write q=f(L). Robinson's utility depends on how much he works (labour causes disutility because it means less leisure time and possibly because intrinsically unpleasant) and on how much he consumes. Let MPL be the marginal product of labour in the production of the consumption good. Let us follow the usual habit of representing the supply of labour as the giving up of some potential leisure time: let R (mnemonic for „rest‟) stand for leisure time, and let R* stand for the given maximum leisure time (e.g. 1440 minutes (=24 hours), if the day is the length of the period in the measurement of consumption[71] and minutes are the time unit in which labour and leisure per day are measured); then it is R=R*−L; utility can be represented as a function of consumption and of leisure, u=u(q,R), with marginal utilities MUR≡∂u/∂R and MUq≡∂u/∂q. Labour affects utility directly i.e. through its amount (represented here by the amount of leisure) and indirectly through its effect on consumption. The direct effect can be determined by writing u=u(q,R*−L), which yields MUR= −MUL; if the marginal utility of leisure is positive, the direct marginal utility of labour is negative (its absolute value is then called the marginal disutility of labour). If Robinson increases labour time per day by one minute, production of the consumption good increases by MPL, and this increases Robinson's utility by an amount MUq·MPL; the decrease in leisure time decreases his utility by MUR; so the total derivative of utility with respect to labour is du/dL = MUq·MPL–MUR. As long as du/dL is positive, Robinson finds it convenient to increase his labour time. Since land is limited, and since the marginal disutility of labour increases with the amount of labour, du/dL decreases as L increases; utility is maximized when du/dL=0 i.e. when the last unit of labour adds to utility through the increase in consumption as much as it subtracts from utility due to its 71 Consumption too is measured per unit of time; it is a flow. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 59 unpleasantness and to the decrease of leisure time, a condition that can be written as: MPq·MPL = MUR . Let us now consider a consumer in a competitive economy who supplies labour and consumes a single consumption good q. Let u = u(q,R) be his utility function, and wL his nominal income, with w the money wage. L obeys L=R*–R, thus the budget constraint, pq = wL, can also be written pq = wR*–wR. This is just a special case of the consumer with income deriving from endowments, here the endowment is a zero amount of consumption, and the amount R* of leisure. If as usual we measure leisure on the horizontal axis and consumption on the vertical axis (cf. Fig. 3.13bis), the budget line starts at R* on the horizontal axis and rises with a negative slope equal to –w/p (make sure you understand why!). Utility maximization, assuming an interior maximum, requires the tangency between budget line and indifference curve i.e. MUR/MUq = w/p , that can be re-written MUR = MUq∙w/p. In competitive equilibrium the real wage w/p equals the marginal product of labour, w/p = MP L, so we obtain MUR = MUq∙MPL , the same condition as in Robinson Crusoe's case. intl1.5 q −w/p R R* R L Fig. 3.13bis This can be interpreted to mean that Robinson the planner, when deciding how much labour to employ in production, obeys the same conditions as a competitive economy, implicitly attributing to labour the same price (or shadow price, as it is called, because not an actual exchange value but only an imputed value) as a competitive economy would. The analogy can be easily extended to a case with several consumption goods. Thus suppose that Robinson can allocate his labour time to either of two consumption goods. Clearly he will find it optimal to allocate it in such a way that the marginal indirect utility of labour is the same in both productions (otherwise it would be convenient for him to reallocate some labour to the production where its marginal indirect utility is higher). We have seen earlier that this condition is also realized by a competitive equilibrium. The analogy can be further extended, but the intuition should by now be clear. The conclusion f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 60 marginalist economists draw from these observations is that efficient planning requires the same valuation of resources and goods as is brought about by a competitive market[72]. On this basis, marginalist economists have argued that the market solves problems of allocative efficiency that arise in any society, and therefore equilibrium prices reflect the value that should be attributed to goods, and to factors, in any efficient type of social organization, i.e. reflect 'natural', a-historical, laws of evaluation. Thus we find John Bates Clark starting the Preface to his treatise The Distribution of Wealth with these lines: "It is the purpose of this work to show that the distribution of the income of society is controlled by a natural law". 3.10. Introduction of the rate of interest and of capital in the marginalist theory of distribution 3.10.1. So far the rate of interest has not appeared. In order to determine it, the marginalist approach once again refers to an equilibrium between supply and demand, this time supply and demand for savings. Let us first examine the logic of this explanation in a pure exchange economy. Let us consider time as divided into „periods‟, with only one big fair in each period, for example (if the period is a week) only on monday mornings. We now assume that in this economy there is the possibility to transfer purchasing power from one period to another one through loans and debts. A consumer can consume more than the value of her endowments by borrowing purchasing power from other consumers, i.e. by selling a bond − a promise of repayment of a certain amount of some good in some future period − to some other consumer. Suppose that good 1 is the numéraire and that there is only one type of bond, a promise to give one unit of good 1 to the bond holder one period later. Purchasing such a bond in period t means to lend an amount of good 1 to the issuer of the bond, against a promise of repayment of 1 unit of good 1 in period t+1. If the price of this bond is, e.g., 0.909, the loan of 0.909 units of good 1 yields 1 unit of the same good after one period, with a rate of interest r=10% determined by 0.909(1+r)=1. 0.909 is the discounted value of 1 unit of good 1 obtainable one period later. Let us assume that one can also buy fractions of such a bond. 72 “... no matter how technocratic the bias of the planner and how abhorrent to him are the unplanned workings of the free market, every optimal planning decision which he makes must have implicit in it the rationale of the pricing mechanism and the allocation of resources produced by the profit system” (W. Baumol, Economic Theory and Operations Analysis, II ed., 1965 (1968 Prentice-Hall India reprint), p. 114) f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 61 To simplify as much as possible, let us assume that good 1 is the sole good. A consumer‟s utility in period 0 depends on her consumption of the good in that period, to be indicated as c 0, and on the amount of the good available in period 1, to be indicated as c1. Let the consumer‟s endowment of the good in periods 0 and 1 be ω0 and ω1; the amount of the good available in period 0 is ω0 plus the number B0 of net bonds owned by the consumer and coming due that period, a positive or negative number depending on whether the consumer issued or bought bonds the period before. If in period 0 the consumer consumes c0, the difference ω0+B0−c0 is her savings (or dissavings, if negative), which she uses to buy bonds, earning the market rate of interest r on her savings. The number of bonds B1 that she can buy is such that its discounted value equals her savings, B1/(1+r)=(ω0+B0−c0). The amount of the good available in period 1 is then c1 = ω1+B1 = ω1 + (ω0+B0−c0)(1+r). This is the consumer‟s intertemporal budget constraint over two periods. (It is customary in introductory economics textbooks to assume that the economy extends over only two periods, with no past and no further future, which entails that B0=0 and that c1 is entirely consumed; but this unrealistic assumption can be dispensed with, and a past and a future can be admitted, by admitting a non zero B0 and by assuming that utility depends on consumption in period 0 and, not on consumption in period 1, but rather on the amount of the good available in period 1, which the consumer can then consume or save.) Utility maximization can then be analyzed graphically with the help of indifference curves between c0 and c1 and of the choice curve derived from them. In the graph with c0 on the abscissa and c1 on the ordinate, the budget line passes through the point (ω0+B0,ω1) and has slope −(1+r). The slope of an indifference curve is the marginal rate of substitution between consumption of the good to-day and availability of the good next period; its absolute value minus 1 is also called the marginal rate of time preference: if a consumer needs 1.1 more units of the good next period to compensate for the loss of 1 unit of consumption to-day, her marginal rate of time preference is 10%. A portion of the choice curve is shown in thick black in Fig. 3.13??. Note that, as shown in the Figure, a rise of the interest rate may well cause an increase of c0, which means a decrease of savings. intl1 c1 ω1 f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 62 ω0+B0 −(1+r) c0 Fig. 3.13 ter?? In this simple economy the equilibrium between supply and demand for savings (i.e. between supply and demand for bonds) requires some consumers to dissave so as to compensate the savings of other consumers. The price of savings is the interest rate. If we graph the algebraic sum of savings and dissavings − the net or excess supply of savings − as a function of the rate of interest (with, as usual, the price − here the interest rate − on the ordinate, and the quantity − here the net supply of savings − on the abscissa), equilibrium will be unique and stable if this function is upward-sloping and cuts the ordinate, cf. Fig. 3.13 quater??. (But multiple equilibria are perfectly possible, owing to the possibility that, as pointed out, savings may be a decreasing function of the interest rate at least over a range of values of the latter.) The equilibrium interest rate − the one at which the net supply of savings is zero − may be positive or negative; the latter will be the case if the willingness to dissave must be stimulated by a negative interest rate in order to rise to absorb the savings supplied by positive savers. The interest rate reflects intertemporal preferences: in equilibrium, 1+r will be equal to the marginal rate of substitution between consumption to-day and availability of consumption tomorrow. Very little changes if we admit production but without capital goods. The sole difference is that in each period the income of consumers, besides depending on the loans and debts contracted in the previous periods, depends on the rentals of the productive factors they own; but here too it will be possible to consume it all, or to save part of it, or to contract debt in order to spend more than one‟s income. Again it will be possible for some consumers to save only if other consumers dissave; as in the pure exchange economy, the total consumption of each period cannot be altered, it can only be redistributed among consumers. And again, the interest rate will be determined (assuming a unique and stable equilibrium) by the condition that the net supply of savings must be zero, and will reflect intertemporal preferences alone. interl1 interest rate net supply of savings f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 63 Fig. 3.13quater?? 3.10.2. Things change once one admits production with capital goods, i.e. with produced means of production. Then the interest rate becomes the price − the rental − of the factor capital, reflecting the marginal product of capital just like wage and land rent reflect the marginal products of labour and of land; the influence of intertemporal preferences on the rate of interest operates essentially by influencing the supply − the endowment − of capital; and it becomes possible for the economy as a whole to perform net savings, and to transfer consumption from one period to subsequent periods, by allocating the productive resources freed by the decreased consumption to the production of capital goods, which will increase future production. We proceed to illustrate these statements. But the reader must be warned that the introduction of capital goods in the marginalist framework raises problems that will require extensive discussion in later chapters; now we limit ourselves to the case which raises the fewest problems: only one capital good. Let us initially assume that the single capital good is homogeneous with the product. The economy produces only corn, via the use of labour, and of capital consisting of seed-corn advanced at the beginning of the year; the product comes out at the end of the year. (Of course land is also necessary, but we assume now that it is overabundant and hence a free good, so we can neglect it.) Wages are paid in arrears. The corn-capital employed is entirely used-up in a single production cycle; therefore capital is entirely circulating capital, i.e. consumed in a single production cycle. The fact that capital is worn out by use obliges us to distinguish the physical (or gross) from the net marginal product of capital. The net marginal product is the increase in corn production due to one more unit of corn-capital, minus the reintegration of the capital consumed. The rate of interest r (leaving risk aside for simplicity) will tend to equal the net marginal product of corn-capital, i.e. its gross marginal product minus one. Example: if by employing one more unit of corn-capital a farmer obtains 1.2 more units of corn, i.e. if the gross marginal product of capital is 1.2, she will find it convenient to borrow one more unit of capital if the real rate of interest does not exceed 20%, which is the net marginal product of capital. MPKG can indicate the gross or physical marginal product of capital, MPKN the net marginal product, and the condition of equality between net marginal product of corn-capital and rate of interest can be written MPKG=1+r or equivalently MPKN=r. Similarly, one must distinguish the gross from the net product; e.g. if the gross product is 1000 and it has required the employment of 400 units of corn-capital, the net product is 600[73]. 73 Note that the gross product as defined here does not correspond to the gross product of the dominant national accounting conventions: the latter is defined net of the replacement of intermediate goods and only gross with respect to durable capital; thus in this economy, where no capital is durable, the usual national → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 64 Apart from this need to distinguish gross from net magnitudes, the reasoning applied when the factors were labour and land applies identically to labour and corn-capital. In particular, we can again assume initially that the capitalists (the owners of corn-capital) and the labourers are two different groups of people, and we can first assume that the capitalists act as entrepreneurs, and then we can demonstrate that the equilibrium is the same as if it were the labourers to act as entrepreneurs. Competition will ensure a tendency toward the same, most efficient production function; let Q = F(L,K) indicate the production function of the gross product; the net product is simply Y = Q–K = F(L,K)–K. The gross marginal product of corn-capital, MPKG, is the partial derivative of the gross production function i.e. ∂F/∂K; the net marginal product is ∂[F(L,K)–K]/∂K = MPKG–1. If F(L,K) is a CRS production function, so is F(L,K)–K (Exercise: prove it!). The same reasoning applied to labour or land in the labour-land economy can be applied here to corn-capital to derive a decreasing demand curve for capital; if on the vertical axis we measure the rate of interest, the demand curve will coincide with the curve of the net marginal product of corn-capital; if we want to refer to the curve of the gross marginal product of corn- capital, then this will indicate the demand for K as a function of (1+r). The product exhaustion theorem becomes Q = F(K,L) = MPL∙L + MPK∙K = wL + (1+r)K, and it can be used here too to demonstrate that who acts as entrepreneur makes no difference to the equilibrium, and that in equilibrium the entrepreneur makes neither profit nor loss. 3.10.3. The assumption of a given supply of corn-capital can be relaxed. In this economy, at the end of each production cycle what is given is the harvest, not the stock of seed-corn available for the next production cycle: the latter will be determined by the consumers' decisions as to how much of the harvest to consume and how much to save. In order to reflect this fact, we can depict the economy as holding a big market fair and reaching equilibrium just after the harvest has been distributed to the owners of the factors of production according to the latters‟ marginal products; then each consumer has an endowment of corn, and the equilibrium must also determine whether the consumer consumes the whole of it, or saves part of it, or dis-saves. The supply of corn-capital to firms for employment in the next production cycle is the aggregate (gross) savings of consumers, the aggregate non-consumed corn endowments of those consumers who perform positive gross savings, minus the aggregate dissavings (negative numbers) of those consumers who consume more than their endowments. The supply of corn-capital to firms is then a variable, that depends on income distribution. Now the total consumption of consumers is no longer given; the less consumers consume in the aggregate, the greater the supply of corn-capital, and thus the greater accounting definitions would make gross and net product coincide. Here the gross product is defined gross also of the replacement of circulating capital goods. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 65 next period‟s harvest[74]. In this way, society as a whole can transfer consumption − increased by the net marginal product of corn-capital − from a period to the next; it is no longer necessary that someone else dis-saves in order for a consumer to be able to save and thus transfer consumption intertemporally from a period to the next. The rate of interest is at the same time equal to the net marginal product of corn-capital, and equal to the marginal rate of time preference. But if the marginalist economist wants this imaginary economy to mirror – albeit in simplified form – the working of real economies, then she will note that in real economies capital goods and consumption goods are different; that at each moment the stock of capital goods cannot be quickly transformed into consumption goods; and that the change in the stock of capital due to accumulation (net savings) is slow relative to the speed with which factor prices can change; so she will conclude that, when one wants to use this simple economy to understand how income distribution is determined, the assumption that the stock of corn-capital is given is the assumption that more closely mirrors the situation in more complex economies. 3.10.4. It is left to the reader to demonstrate that there is no difficulty in adding land as a third production factor in this economy. In equilibrium each factor will receive its full-employment marginal product (net marginal product, for capital). Note then the symmmetry: all factors are on an equal footing, the same mechanism determines all their rentals. This explains why labour, land, capital are all called 'factors of production': they are seen as having a similar role in the economy. This economy, where capital and product are the same good, and distribution is determined by the technical substitution mechanism, presents the marginalist explanation of the origin of an income from the property of capital in the simplest way. The rate of interest is explained as the reward of the net productive contribution of the factor 'capital', a factor whose supply derives from a decision to save part of the product of the preceding productive cycle and to employ it as input in the next production cycle. 3.10.5. Let us now study the indirect substitution mechanism in an economy where corn and iron are produced by corn-capital and labour and there are fixed technical coefficients. Let K1 and K2 be the technical coefficients of corn-capital in the production, respectively, of corn (good 1) and of iron (good 2). The technical conditions of production are thus: L1 units of labour * K1 units of seed-corn → 1 unit of corn L2 units of labour * K2 units of seed-corn → 1 unit of iron. (Clearly for the economy to be viable it must be K1<1.) The price = cost equations are now: p1 = wL1 + (1+r)p1K1 74 For simplicity I am still assuming that the supply of labour is given. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 66 p2 = wL2 + (1+r)p1K2. The cost of capital in the production of either good is given by the value of the capital employed multiplied by (1+r), because at the end of the year the firm must repay the borrowed capital and the interest rate on it, so, for each unit of corn-capital borrowed, the firm must repay (1+r) units of corn. (If the firm owns the capital, it must anyway impute an opportunity cost[75] equal to (1+r) to each unit of capital, because this is what it would earn by lending it to other firms.) Note an important difference relative to equations [3.5]-[3.6] of the labour-land economy: here p1 appears also on the right-hand side of the equations. But in this simple economy the difference can be made to disappear by choosing corn as the numèraire good i.e. by putting p 1=1. Then the price equations become [3.13] 1 = wL1 + (1+r)K1 [3.14] p2 = wL2 + (1+r)K2, formally indistinguishable (with 1+r having the role of , and K the role of T) from the equations of the labour-land case if there too corn is chosen as numéraire. Here too, in order to derive decreasing demand curves for labour and for corn-capital the marginalist approach relies on the fact that p1/p2 changes as w/r changes, and the more capital- intensive good becomes relatively cheaper if the rate of interest decreases, which is argued generally[76] to entail a shift in the composition of consumption demand in its favour. However, the determination of the demand curve for corn-capital presents a complication, which is that the capital desired by firms in order to produce the goods demanded by consumers requires the determination of the vertically integrated industries producing respectively the corn demanded by consumers, and the iron, as net products[77]. 75 The opportunity cost of the employment of a factor in a certain use is what the factor could have earned in the best alternative use. 76 Cf. fn. 34?? and fn. 51?? 77 A vertically integrated industry is a generalization of the notion of subsystem (ch. 2): the vertically integrated industry producing a vector y as its net product is an ensemble of industries producing as total gross product a vector x such that y equals x minus the means of production used up in the production of x. The reason for the term „vertically integrated‟ is the following. There is a widespread visual image in economics, of production flowing downwards from firms producing a good to other firms that use that good as input; if a firm also produces some of its inputs, it is called „vertically integrated‟. An example of vertical integration in a multi-product economy could be a firm that produces bread and owns a plant producing the flour it needs (vertical integration would be pushed further if the same firm also owned a farm producing the wheat it needs). Complete vertical integration means that a firm produces itself all the produced inputs it utilizes. This is never the case with firms, but it is conceivable as a logical construction for industries: a (completely) vertically integrated industry is an industry that produces itself all the inputs it utilizes in order to produce its net product. If the vertically integrated industry were on an island, it would need to import no produced inputs from the mainland. In the simple economy considered in this paragraph, where the sole produced input is corn, the vertically integrated corn industry is the ensemble of corn-producing firms that → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 67 Suppose that consumers demand 1 unit of corn per period. What demand for corn-capital is caused by such a demand for corn for consumption purposes? In order for a repeated delivery of 1 unit of corn per period to consumers to be possible, the economy must replace every period the capital used-up in the production of that unit; which means that a gross production of corn per period is needed, that besides replacing the seed-corn it has consumed, produces 1 unit of corn as net product per period. The gross production of corn that satisfies such a condition is the quantity Q1 that solves Q1 − K1Q1 = 1 i.e. Q1 = 1/(1−K1). An ensemble of firms producing Q1 is called the vertically integrated industry producing a net product of 1 unit of corn. This ensemble of firms employs (i.e. demands) K1Q1 units of seed-corn; thus K1Q1 = K1/(1−K1) can be called the vertically integrated technical coefficient of capital, that is the total employment of capital in the vertically integrated industry producing one unit of corn as net output; this is the demand for capital implied by a demand for 1 unit of corn by consumers, in that if this economy produced a net product consisting of only 1 unit of corn, one would observe that the corn-capital employed in this economy would be that amount. Let χ1 indicate this coefficient: then χ1y1 is the demand for corn-capital associated with a demand of y1 units of corn by consumers. The demand for labour λ1 associated with a consumer demand for 1 unit of corn is determined in an analogous manner, as the labour needed by the vertically integrated industry producing a net output of 1 unit of corn and hence a gross output of Q1 units of corn, so λ1 = L1Q1 = L1/(1−K1). We can analogously build the vertically integrated industry producing one unit of iron as net product. In order to produce one unit of iron K2 units of corn are used-up, so in order repeatedly to produce one unit of iron this quantity of corn must be produced as net product of an ensemble of firms producing corn. Therefore a vertically integrated industry with one unit of iron as net product will include firms producing iron, and firms producing corn, and it must produce every period a gross product consisting of one unit of iron plus such a quantity of corn Q2 as will satisfy Q2−K1Q2=K2 i.e. Q2 = K2/(1−K1). The corn-capital such a vertically integrated industry employs (and produces) is therefore given by the corn-capital employed in the production of one unit of iron, K2, plus the corn-capital employed in the production of Q2 units of corn which is K1Q2=K1K2/(1−K1). Thus the vertically integrated technical coefficient of corn-capital in the production of iron (that is, the total amount of corn employed per unit of net product of iron) is χ2 = K2 + K1K2/(1−K1). Analogously, the vertically integrated technical coefficient of labour in the production of iron produce enough corn for their net product to be the economy‟s net output of corn; it is smaller than the entire corn industry because the vertically integrated iron industry includes both firms producing iron, and firms producing corn: the corn needed for the production of iron plus the seed necessary to produce it again. The text gives the mathematical determination of these magnitudes. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 68 can be determined as λ2 = L2 + L1Q2 = L2 + L1K2/(1−K1). We can therefore specify the factor uses relevant for the determination of factor demands as: λ1 units of labour * χ1 units of seed-corn → 1 net unit of corn λ2 units of labour * χ2 units of seed-corn → 1 net unit of iron. On this basis we can determine the demand for corn-capital, assuming the supply of labour to be fully employed (e.g. because the labourers are the entrepreneurs, organized in teams), in a way strictly analogous to the labour-land case. Let L* be the supply of labour, and let y1 and y2 be the consumption demands respectively for corn and for iron. The full employment of labour requires λ1y1+λ2y2=L*. The demand for capital will be χ1y1+χ2y2. What we must now demonstrate is that the demand curve for capital is downward-sloping. Assume e.g. that χ1/λ1>χ2/λ2. This will be so if and only if K1/L1>K2/L2, because χ1/λ1=K1/L1, i.e. the capital-labour ratio in the vertically integrated corn industry is the same as in the non-integrated one, while the vertically integrated iron industry includes a production of corn where the capital-labour ratio is again χ1/λ1=K1/L1, plus a non- integrated iron industry which renders the overall capital-labour ratio in the vertically integrated iron industry intermediate between K1/L1 and K2/L2.[78] Therefore we can apply the same reasoning as for the labour-land economy: if the rate of interest decreases, making corn cheaper relative to iron, and if the composition of consumer demand shifts in favour of corn, some units of labour must be shifted from the vertically integrated iron production to the vertically integrated corn production, where they are associated with a greater employment of corn-capital, and the overall demand for corn-capital increases. The decreasing demand curve for capital, combined with the given supply, determines the equilibrium interest rate and thus also the equilibrium real wage and product prices. 3.10.6. We discuss now a fine theoretical point that requires attention. The reader might object that in the previous paragraph we − following traditional marginalist analyses − have implicitly assumed that the net product of the economy consists entirely of consumption goods, excluding any capital accumulation, while more often than not in real economies the net product also includes capital goods in excess of the reintegration of the used-up capital, that is, there is net capital accumulation (net savings). In the economy just described, this would mean that the demand for a net product of corn includes some corn demanded not for consumption purposes but in order to use it as additional capital. The composition of demand would then depend not only on the composition of consumer demand, but also on net savings. It might then be argued that the demand curve for corn-capital as determined in the previous paragraph only applies to stationary economies 78 Note therefore that the difference in capital-labour ratio between two vertically integrated industries must be expected to be, in general, less than the one between non-integrated industries, suggesting a more limited capacity of the indirect substitution mechanism to affect factor demands than in the labour-land economy. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 69 (where the net product consists entirely of consumption goods), and not to economies where there is also a demand for corn for accumulation purposes. The objection then is that, in so far as equilibrium income distribution depends on the composition of demand, it will also depend on the share of net savings in net output; hence the demand curve for corn-capital determined by assuming no net savings cannot yield the correct equilibrium income distribution. To this objection one finds in traditional marginalist economists an explicit answer and an implicit one. The explicit answer is that the net demand for new capital goods due to net savings only marginally alters the composition of demand relative to a stationary economy, because it is small compared with the replacement demand, and therefore for the determination of income distribution the stationary-state assumption is a generally legitimate simplification[79]. The implicit answer is that the demand for capital as determined above, that is, the persistent demand for capital motivated by a net product consisting only of consumption goods – let us call it the standard demand for capital –, is the notion necessary to understand what motivates capital accumulation in the marginalist approach. The point is that capital accumulation would come out to be unjustified, and would then give rise to capital disaccumulation, if the standard demand for capital did not increase. To see why in simple terms, suppose (in order to leave aside Keynesian problems) that when there are net savings, these translate without difficulty into net investment because the corresponding production of new capital is demanded and bought by the consumers/savers themselves. Suppose now that at a certain moment there is equilibrium (both the supply of labour and the given stock of capital are fully employed) with the net product including some net investment (corresponding to some net savings), so the stock of capital is increasing through time; but suppose that the determinants of the standard demand for capital (i.e. labour employment – equal to labour supply –, preferences, technology, and rate of interest – and hence real wage) are not changing through time. Then relative prices are constant, thus the composition of consumer demand is constant and therefore, since all relative demands and technical coefficients remain unchanged, the desired aggregate capital-labour ratio does not change and, since labour employment does not change, the demand for capital does not change either: so the extra capital accumulated finds no employment. This shows that in order for increases in the quantity of capital 79 “For the general fund of capital is the product of labour and waiting; and the extra work, and the extra waiting, to which a rise in the rate of interest would act as an incentive, would not quickly amount to much as compared with the work and waiting, of which the total existing stock of capital is the result.” (Marshall, 1920 [1970], VI, ii, 4, p. 443). “In the case of capital, the normal case is a growing total stock; but this is so large that the net production in a short period of time will not make an appreciable difference in the demand price” (Knight, 1946, p. 400). Note furthermore that, in order to make an appreciable difference, the factor proportions in the production of the goods comprising net investment should be considerably different from average factor proportions in consumption demand. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 70 to find employment, the standard demand for capital must increase. This can be due to a decrease of the interest rate that causes a downward movement along the standard demand-for-capital curve[80], or to a rightward shift of the curve itself, due to an increase of labour supply (and hence of labour employment) or to a kind of technical progress that raises the desired capital-labour ratio for any given level of the interest rate. The above shows that unless the standard demand for capital increases, capital accumulation comes out to find no employment and therefore will be discontinued and reversed[81]. It is the changes over time of the standard demand for capital that motivate persistent increases in the demand for capital, and hence capital accumulation. Including the demand for newly produced capital goods corresponding to net savings among the elements influencing the demand for the existing capital stock would only obscure this fact. Accordingly, in the marginalist approach a non-accidental presence of net savings is due to a gradual increase over time of the standard demand for capital, in turn due (if we leave technical change aside) either to increases in the supply of labour, or to decreases of the rate of interest. There need be no decrease of the rate of interest if labour supply is increasing in step with the stock of capital, so that when the newly produced capital goods come into operation there is no need to change the K/L proportion and the marginal product of capital does not decrease; there will be a decrease of the rate of interest if entrepreneurs expect the increased supply of capital to have a lower marginal product because it has increased more than the supply of labour, so that a rise in the K/L ratio is necessary in order to maintain the full employment of factors, and the lower marginal product of capital requires then a lower rate of interest. In this second case the rate of interest does not correspond to the marginal product of the already existing capital goods; so one can characterize the situation as one of disequilibrium; but the disequilibrium will be nearly unnoticeable, because of 80 When technical choice is admitted, the decrease of the interest rate can cause an increase in the desired capital-labour ratio because of technical substitution in addition to the indirect (psychological) substitution that, strictly speaking, is the sole one we should be considering in the fixed-coefficients economy we are discussing. 81 If now we drop the unrealistic assumption that savings translate directly into demand for new capital goods because the latter are bought by the consumers/savers, and if we admit more realistically that in order for savings to translate into investment the supply of monetary net savings must be absorbed by a demand for them motivated by a desire for net investment by firms, then the absence of an increase in the standard demand for capital would mean that the attempt to perform net savings would be frustrated by a lack of demand for net investment that, as Keynes has shown, would cause a lack of sufficient aggregate demand and hence an aggregate output and income below the full-employment level: in this case, instead of having an initial net accumulation that then comes out to be a mistake, the net savings and connected net capital accumulation would be unable to come into being (because income would decrease until net savings disappeared: with further problems, due to the decreased demand for capital caused by the lower level of aggregate production, that we cannot discuss here). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 71 the slowness with which the K/L ratio changes; the quasi-rents earned by the already existing capital goods will be very nearly what they would have earned in the absence of net savings. 3.11. Money and the rate of interest The rate of interest has been determined so far by the equilibrium between supply and demand for corn-capital. In real economies, on the contrary, the rate of interest is the 'price' of credit, and loans are normally in money terms. Can the above theory apply to an economy where goods are exchanged against money and the rate of interest is the 'price' of monetary loans? The basic idea of the marginalist approach on this issue is that (if financial intermediaries do not disturb the process and if, for simplicity, we assume no inflation) the rate of interest, in bringing into equality the supply and demand for money loans, also brings into equality the supply and demand for 'real' capital (capital goods). The supply of money loans corresponds to the part of income which is not spent on consumption, and it is demanded by firms in order to buy capital goods; now, firms find it profitable to accept loans as long as the rate of interest they must pay on a loan is not higher than the rate of return on the capital goods they can buy with the loans (for simplicity we neglect here the difference between rate of interest and rate of return on capital goods required to cover entrepreneurial risk). Therefore, given the rate of interest, firms push their demand for loans up to the point where the marginal product of the capital they can employ with those loans equals the rate of interest. The lower the rate of interest, the greater the demand for loans because firms want to employ more capital goods; therefore the demand for money loans is a decreasing function of the rate of interest because it reflects the demand for capital goods. In the corn economy, since the supply of loans is equal to the value of total production (= total income) minus the value of the demand for consumption, it is equal to the value of the production of corn which must be absorbed by the demand for corn-capital; when the demand for loans is equal to the supply of loans, it is utilized to buy exactly that amount of corn; so the equilibrium between supply and demand for loans also means an equilibrium between the supply and the demand for corn-capital. In the corn-iron economy, this double equality also requires that the composition of production be the correct one, and mistaken expectations of entrepreneurs as to the composition of demand might cause some difficulties to the adjustment process toward equilibrium; but the marginalist idea is that since mistaken expectations are corrected by experience, the composition of production tends to adapt to the composition of demand and these difficulties can only be transitory. Thus the equilibrium rate of interest, although directly determined by the equilibrium between supply and demand for money loans, is in fact determined by the full-employment marginal product of corn- capital. 3.12. Accumulation f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 72 At this point we can ask: why does the quantity of capital tend to grow as time passes? In the marginalist approach, the answer is: Because consumers decide not to consume the entire net product, and what is left of it is added to the capital stock. For example, in the economy producing corn with corn-capital and labour, if 1100 units of corn are produced in a certain year with the employment of 400 units of corn-capital (hence the net product is 700), and if consumers decide to consume only 600 units of corn, the supply of corn to firms is 500: there are net savings amounting to 100 units of corn, and the stock of corn-capital grows from 400 to 500. The increased supply of money savings causes a decrease of the interest rate until it is absorbed by the firms‟ demand for credit: since the value of production equals the value of incomes, the value of savings equals the value of that part of production that is not bought by consumers, hence if all savings are used by firms to buy corn-capital, the demand for corn- capital absorbs its supply. (This reasoning confirms the arguments in §3.10.6 because it is the lower interest rate that induces firms to desire more capital per unit of labour.) In such a perspective, the capital stock of an economy is the result of the sum of past decisions to perform net savings out of the full-employment production. Evidently – marginalist economists conclude – since there has been and there continues to be capital accumulation, consumers have been having such preferences that, at the rate of interest corresponding to the full-employment marginal product of capital, in the aggregate they prefer to consume in each period less than the net product of that period, in order to consume more in the future. Consumers might well wish to consume more than the net product, in which case there would be decumulation of capital; but this has not happened. Evidently the rate of interest is such and preferences are such as to induce consumers to perform net savings, preferring to trade off some present consumption for more consumption in the future. Were it not for the increase in labour supply and for technical progress that raises the marginal product of capital, this continuous capital accumulation would have caused a continuous decrease of the rate of interest, down probably to zero. 3.13. A comparison between the classical and the marginalist approaches to income distribution: the basic analytical difference and some implications. 3.13.1. We are now able to grasp the radical difference between the classical or surplus approach, and the marginalist or neoclassical approach. The central difference is in the explanation of income distribution. The rent of land is explained on the basis of similar principles in the two approaches (we will see this in detail later). The difference lies in the explanation of the division between wages, and income from the property of capital (‘profits’ in the classical authors, ‘interest’ in marginalist authors), of what is left of the net product after paying land rents. As we have seen, in the classical authors the real wage is determined by complex socio-political forces that reflect the balance of bargaining power between f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 73 social groups with different interests (so that Marx, for example, summarized the elements influencing the real wage under the umbrella term 'class struggle'), and the rate of profit is residually determined; in the marginalist approach the real wage rate and the rate of profit (or rather rate of interest) have symmetrical roles, and are simultaneously determined by the tendency toward an equilibrium between supply and demand. It might then seem that the difference between classical and marginalist authors lies in whether the labour market is seen as competitive or not competitive: it might seem that the marginalists view the labour market as basically competitive so that the interplay of supply and demand is able to push the market toward its equilibrium level, while according to the classicals two coalitions confront each other on the labour market, the real wage is determined by the relative bargaining power of the two coalitions, and it is this non-competitive nature of the labour market that authorizes the treatment of the real wage as already given when determining the rate of profit. But this way of characterizing the roots of the difference would be mistaken. The root of the difference is analytical: it lies in the presence in the marginalist authors, and absence in the Classical authors, of the conception of technical choices and consumer choices as the basis of factor substitution mechanisms which engender a decreasing demand curve for each factor. In the Classical authors there is no notion of demand curves for factors. This is confirmed by the presence in the marginalist authors, and absence in the Classical authors, of a necessary connection between real wage and level of employment. Marginalist authors sometimes admit that the labour market may not be fully competitive, and that trade unions or other forms of labour coalitions may be able to fix the real wage before the rate of interest, and at a level higher than the full-employment level. But the presence in the marginalist approach of a decreasing demand curve for labour means that the given real wage renders the level of labour employment endogenously determined; there is a necessary connection between real wage and labour employment. This is precisely why the marginalist approach tends to view increases in real wages as having a negative effect on employment, and to consider persistent unemployment as due to too high a real wage. In the classical authors there is no necessary, univocal connection between real wages and labour employment: the same level of labour employment can correspond to different real wages depending on relative bargaining power, which will depend for example on the extent of unemployment[82]; and the sole certain effect of a change in real wages is a change of the rate of 82 Let us be clear about this difference. In the marginalist approach, if we consider two economies that differ in the supply of labour but not in the utilization of other factors nor in technology nor in the composition of demand, so that the labour demand curve is the same, the observation of an equal level of labour employment in the two economies (and hence of different unemployment) would imply that the real wage is fixed at the same level in both economies. In the classical approach, if in two economies utilization of other factors and technology and composition of demand and level of labour employment are the same but unemployment differs, there is no reason why one should expect an equal real wage, there is rather an → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 74 profits in the opposite direction, and its further effects, in particular its effects on employment, are not univocally determined, depending on the circumstances, and possibly on the extent of the variation. E.g. in Marx an increase in wages may sometimes increase employment by stimulating demand for consumption goods, in other occasions – especially if it is a big increase accompanied by much social unrest – it may cause a crisis by discouraging investment. In Ricardo, who accepted Say‟s Law, a decrease in wages only causes a faster accumulation, and it is only this growth in the stock of capital (plus possibly a slowdown in the growth of population) that may cause over the years a reduction of unemployment; for marginalist economists on the contrary the increase in the demand for labour is brought about, without any need to increase the stock of capital, by the shift in methods of production and in the composition of consumption, engendered by the lower real wage. The absence of a univocal connection between income distribution and quantities produced also explains the Classical separate determination of the quantities produced, and their treatment as given when one determines the rate of profits (cf. chapter 1): if a regular univocal connection existed between income distribution and quantities produced, a separate determination of quantities would make little sense, a simultaneous determination would be the natural approach. 3.13.2. The presence in the marginalist authors, and absence in the classical authors, of the notion of a negative elasticity of labour employment with respect to real wages entails a different role of social and political elements in the two approaches, and a different conception of how competition in the labour market operates. In the marginalist approach a wage fixed by socio-political elements is an impediment to the free working of competition, which if left free to operate would produce a fully determinate level of the real wages. The marginalist conception itself of the "free working of competition" in labour markets relies on the marginalist analysis of its results: without a significant positive effect of decreases of real wages upon employment, the marginalist conception of competition in the labour market as entailing an indefinite downward flexibility of wages as long as there is unemployment would clearly be unacceptable: its operation would bring wages to zero (or anyway to implausibly low levels) whenever unemployment arose, with a clear contradiction with observation. In the classical approach the absence of the notion of a decreasing demand curve for labour would have entailed precisely such implausible results, if competition had been thought to operate as the marginalists conceive it: the little response of labour employment to changes in wages and the constant presence of unemployment, sometimes very considerable unemployment, in capitalist economies would have entailed a prediction of a tendency of wages to zero, clearly incompatible with observation. It is understandable then that in the classical approach one does not find the view expectation of lower wages where unemployment is greater, unless institutional elements counterbalance this greater weakness of workers. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 75 that wages will decrease as long as the demand for labour is less than its supply, and one finds instead a stress on custom, on the need to defend social status, on balance of relative bargaining power, on fear of popular revolts, and the like, as the elements capable of determining real wages and their variations. The social customs and political elements that in the marginalist approach are, if at all, superposed upon a self-sufficient mechanism and impede its autonomous working are, to the contrary, indispensable in a classical approach and are accordingly viewed as an obvious component of an economy based on wage labour; hence for a classical economist social, political, institutional and historical considerations are an integral part of economic analysis[83]. Accordingly, the operation of competition in the labour markets is conceived as embodying those same social forces that establish the level of wages. Thus a skilled worker will not, only because he has been unemployed for a few months, accept a lower wage than the one that social custom normally grants him. Unemployment will not necessarily mean a tendency of wages to decrease, and even when it does (generally because unusually high and furthermore combined with other socio-political factors, e.g. a right-wing government), the decrease will be slow, and seldom will go below what Adam Smith called the (historically specific) lowest level "compatible with common humanity". 3.13.3. Indeed, if one abandons the presumption of a significant negative elasticity of employment with respect to the real wage, then a stickiness of real wages in the face of unemployment becomes not only necessary in order to avoid absurd conclusions, but also easily understandable. If the level of employment is not significantly improved by real wage decreases, then it is only to be expected that historical experience will have taught workers that wage undercutting must be avoided. If the unemployed workers offer to work for less than the current wage, it suffices that the employed workers themselves accept the lower wage, and they will not be replaced by the unemployed, since labour turnover does imply at least some minimal costs. This is implicitly admitted also by the marginalist approach, where it is the increased demand for labour that ensures that the lower wage gets the unemployed a job (in addition to the previously employed workers), not their ability to get hired in place of previously employed workers. But if the resulting lower real wage does not increase employment, the unemployed workers have gained nothing by offering themselves at a lower wage – they are still unemployed, and have only made the employed workers worse off (and themselves too, in so far as they receive support from the income of their employed relatives). No wonder, then, that popular culture should have developed a variety of ways ('fair wage' notions, a culture of solidarity, sanctions against strike breakers etc.) to spare new entrants into the „reserve army of the unemployed‟ the need to learn through experience – a learning process which would greatly damage their fellow workers in the meanwhile – that wage 83 A modern example of such a classical approach is M. Kalecki, "Political aspects of full employment", 1943. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 76 undercutting brings no advantage to the unemployed even from a strictly selfish viewpoint. The resulting habits and social conventions are so strong that usually the thought does not occur at all to unemployed people that they might try to replace already employed people by offering to accept a lower wage. The strength in this respect of social conventions and customs is often admitted even by neoclassical economists, e.g. by Marshall, Pigou and more recently by Solow (“On theories of unemployment” Am.Ec.Rev. 1980, and The labor market as a social institution, 1990) and Truman Bewley (in several writings, e.g. in European Economic Review, May 1998, 459-490)[84]. On the contrary the marginalist approach tends naturally to result in a separation of economic analysis from those historical and socio-political elements which were an integral part of economics in the classical authors. In the marginalist approach supply and demand, derived from consumer choices and firm choices, are capable of determining prices, quantities and income distribution, on the basis of a very restricted set of institutions: essentially, competition and a general respect of contracts and of private property. Apart from these institutional pre-requisites, the market is a self- sufficient sphere of social life, capable of endogenously determining prices, quantities and income distribution. 3.14. Exploitation? 3.14.1. The analytical differences just illustrated entail radically different answers to the question of the nature of income from the property of capital. In the classical approach the answer that implicitly emerges to the question "what is the origin of profits?"(85) is: profits originate in the fact that, because of the capitalists' collective monopoly of means of production, workers can only work if capitalists give them permission, and the permission 84 Solow (in the abovementioned 1990 book), Hahn and Solow (A critical essay on modern macroeconomic theory, 1997), and De Francesco (“Norme sociali, rigidità dei salari e disoccupazione involontaria”, Economia Politica, Aprile 1993) attempt to explain such a behaviour as an optimal strategy in a repeated game (the basic idea is that, if in each period the workers who remain unemployed are randomly chosen among all workers, then workers will choose to co-operate, i.e. not to lower the wage down to the full-employment level, if the expected decrease in the probability of being unemployed the following period is so small as not to compensate for the decrease in wage; this is the more likely to happen, the lower the elasticity of the demand for labour with respect to the real wage), but the way the strategic situation is formalized appears to be in all three cases not very convincing; for example the possibility is neglected, that the employed workers may themselves accept the lower wage asked for by the unemployed. Also, the existence of a full-employment level of the real wage assumes the correctness of the marginalist approach, which is on the contrary questionable. We return on these issues in chapter 12. 85 Remember that in classical terminology „profit‟ is the income from the property of capital, it includes interest, differently from the marginalist definition of profits, which are what is left after also paying interest (and also the normal addition to interest to cover risk). This terminological difference reflects the marginalist view of interest as the reward for the productive contribution of the factor of production 'capital' and therefore analogous to wages or land rents, hence a cost. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 77 will only be given if the workers‟ wages leave a surplus for the capitalists. Marx – re-expressing in more precise terms what had been already argued by other critics of capitalism, e.g. the so-called Ricardian Socialists – concludes that, behind the veil of the apparent freedom and formal (i.e. juridical) equality of the two parties to the wage labour contract, there is a substantial inequality of bargaining power, that determines an exploitation of wage labour (which produces everything but is expropriated of the surplus product going to profits and rents – as made clear by the fact that labour performed is greater than the labour necessary to produce as net product what workers get) analogous to the exploitation of feudal serfs through corvées. In feudal society, however, the thing is more evident than under capitalism, where the apparent equality and freedom of the parties to the wage contract makes it more difficult to perceive the reality of the situation. 3.14.2. A marginalist author would reply that such a characterization of the origin of profits or interest derives from an imperfect comprehension of the forces that determine income distribution. Each unit of each factor receives a reward equal to the value of its marginal product, i.e. equal to its contribution to social production and to the utility of consumers (where this contribution is measured as what society would lose if that unit of factor were withdrawn from production). The thing is particularly clear in the economy producing corn with labour and corn-capital. In such an economy a worker, if she decided to give up working, would cause society to lose the contribution of her labour, equal to her marginal product; and her wage is just equal to that marginal product; therefore she receives as much as she contributes[86]. Analogously, the owner of a unit of capital receives as interest the (net) marginal product of her capital, i.e. as much as she contributes to society; therefore it does not seem legitimate to argue that she is subtracting something from labourers. Factors co-operate in production and each one receives in exchange for the value of its contribution goods of the same value. The lines from J. B. Clark at the end of Section 3.12 can now be more fully quoted: It is the purpose of this work to show that the distribution of the income of society is controlled by a natural law, and that this law, if it worked without friction, would give to every agent of production the amount of wealth which that agent creates. 86 Actually this is only approximately true, because the equality between marginal product and rental is true only at the margin; if a worker supplies 2000 hours of labour a year, the real hourly wage equals the marginal product of the last hour; if that worker stopped working, production would decrease by more that 2000 times the wage, because the inframarginal labour units have a higher marginal product; thus the total income of the worker is less than the „total contribution of the worker to production‟ measured as what society would lose if the worker stopped working. But it can be plausibly argued that in most situations the difference in marginal product between the last and the first hour of labour supplied by a single worker (if other factor supplies are unaltered) is negligible, and therefore the expression in the text is fundamentally correct. Also, all factor suppliers are in the same situation, see Exercise ?? f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 78 The co-operation is also highlighted by the fact that an increase in the amount employed of one factor increases the marginal product of other factors( 87 ); this shows that factors help one another to be more productive. Now, the marginalist author might continue, it might be argued that, differently from the labourer who contributes her/his labour, the contribution of capital goods to production is not a contribution of their owners, so it is unclear why it should be the owners who appropriate that contribution. But the existence of capital, he will then continue, is due to the past sacrifices of those who have renounced consuming a part of their gross income and have saved it, so behind the existence of each unit of capital there is the sacrifice of abstinence from immediate consumption. In the same way as the wage repays a sacrifice (labour's unpleasantness), so the rate of interest repays a sacrifice (abstinence from immediate consumption – this must be unpleasant too, at least at the margin, otherwise more gross savings would be forthcoming), and both rewards are equal to the contribution of the sacrifice to social welfare. Thus the picture of capitalism as a conflictual society prone to unemployment and crises and where labour is exploited, developed by the classical critics of capitalism and motivating the left- wing parties' aspiration to a different society, is replaced by the picture of a harmonious and efficient co-operation among sacrifices, with each unit of sacrifice rewarded according to its usefulness to society. The existence of a positive income from the property of capital is viewed as reflecting the contribution to production of the sacrifice consisting in abstinence from immediate consumption, and obeying therefore, fundamentally, a criterion of justice[88], rather than reflecting an exploitation of labour. The contributions of these sacrifices are efficiently utilized, there is the tendency to the full employment of factors and in non-wasteful ways as shown by the tendency toward Pareto efficiency. Finally, the attempt on the part of labourers to obtain more than their full- employment marginal product only causes a reduction of employment and a damage to society because total production decreases (in the classical approach, on the contrary, the sole certain effect of wage increases is that they reduce the rate of profits, but their effects on employment vary 87 With three factors, an increase in the employment of factor 1 may reduce the marginal product of factor 2 (this will happen if factor 1 in order to be optimally utilized must be combined with a large amount of the services of factor 3 so that, when the employment of factor 1 increases, it is convenient to leave a smaller amount of the services of factor 3 to co-operate with factor 2), but the marginal product of the composite factor (2+3) increases (as long as all marginal products are positive). See Exercise 5.?? 88 The remaining weak spot of this defence of income from the property of capital is the right to enjoy the fruits of sacrifices performed by one‟s parents or ancestors: it is unclear why one should be rich only because the child or grandchild of a successful businessman or artist. This is one of the great taboos of modern society: that one has the right to inherit one‟s parents‟ wealth is very seldom questioned, although it is clearly not easily justifiable on grounds of equity or meritoriousness. A society, where wealth could be accumulated but not bequeathed, and where the state would appropriate bequests and redistribute them by giving a sum to those who exited minor age, is certainly conceivable and it could work in interesting ways. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 79 according to the specific situation: according to Marx, for example, as already noted in many cases non-excessive wage increases may be a stimulus to employment, because of the incentive to production and investment coming from the increased sales of consumption goods due to the greater purchasing power of workers). 3.15. Technical progress; relative wages; the origins of the marginalist approach. 3.15.1. One might point out many other differences between the two approaches. We will only briefly mention technical progress, and relative wages. According to the marginalist approach, technical progress has no persistent negative effect on employment. Innovations may cause the bankruptcy of some firms and hence an increase in frictional unemployment, but only during the transition to the general adoption of the new techniques and of the new equilibrium real wage. As to the effect of an innovation on wages, this will generally be positive, owing to a generally positive effect of technical progress on the marginal product of labour. It is not inconceivable that technical progress may raise the product per person while at the same time reducing the full-employment marginal product of labour; this possibility is shown in Fig. 3.14, which illustrates two „total product curves‟ of labour. Given a production function q=f(x,y), the total product curve of factor x is simply q as a function of x, for a given amount of y; its slope measures the marginal product of x. Fig. 3.14 shows a case in which technical progress alters the production function such that the total product of the amount of labour L* rises, but its marginal product decreases. But this occurrence is in all likelihood (it is argued) unlikely, and anyway soon to be reversed by the general tendency – confirmed by historical experience – toward a rise in the marginal product of labour. Thus technical progress can be assumed to be generally beneficial to workers. intl1.5 total product L* L Fig. 3.14. A case in which technical progress causes an upward shift of the total product curve of labour (whose slope measures the marginal product of labour) and therefore an increase of production, but together with such a change of its slope that the full-employment marginal product of labour falls. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 80 In the classical approach, on the contrary, technical progress makes a rise in real wages without a decrease of the rate of profits possible, but its immediate effect is generally to cause firings because it allows the production of the same quantities with less labour, and the increase in unemployment will probably tend to decrease wages. This effect might be countered by increases in the quantities produced, but these increases need not happen, and especially so when technical progress takes the form of process innovation i.e. of ways of producing the same final products in a cheaper way and therefore, most of the time, with less labour. The overall effect of technical progress on employment and wages can therefore be negative even for very long periods, and particularly so in periods of slow growth of aggregate production. 3.15.2. As to relative wages or wage differentials, the marginalist approach explains the average real wage of each different type of labour as determined by its full-employment value marginal product, and so if the wage of a type of labour is twice the wage of another type, the marginalist explanation is that the first value marginal product is twice the second. Wage differences are justified by the different contribution to production or to social welfare. According to the classical approach, the explanation must on the contrary often be looked for in the difference in bargaining strength of different types of labourers; certain categories of labourers are highly paid because they occupy positions in the production process, such that any strike causes great damages to production, which gives them great bargaining power; other categories are well paid because it is convenient for the dominant groups to have them as allies in the class conflict. Foremen whose main task is to make sure that other workers work must be paid more than the other workers, otherwise they will side with them rather than with the firm‟s owners. Marx, and later Lenin, argued that the so-called labour aristocracy (the more qualified and better paid strata of the working class) was paid much better than the rest of the working class in order to soothe its opposition to capitalism and make it more conservative[89]. Wage differences are as much the product of conflict 89 A version of the theory of the labour aristocracy is that nearly all labour in advanced economies is nowadays a labour aristocracy whose high wages benefit from the fact that the labour of less developed countries is paid much less. The analytical basis of such a thesis is very simple. Consider a two-good economy such as the one described by equations [1.32], [1.33] of chapter 1, §1.9.3; but suppose now that the labour employed in industry 1 receives a different real wage from the labour employed in industry 2. Assuming wages paid in arrears, choosing good 1 as the numéraire, and measuring labour in units such that aL2=1, we obtain: (*) p1 = (1+r)(a11p1 + a21p2)+aL1w1 = 1 (**) p2 = (1+r)(a12p1 + a22p2)+w2. For a given level of the rate of profit, there is a decreasing relationship between the two wage rates: a rise of w1 in equation (*) will not affect r if p2 decreases sufficiently, which can be obtained (of course within limits) by a sufficient decrease of w2 in equation (**); utilizing the fact that for the economy to be viable it must be (1+r)a22<1, it can be proved (try it as an Exercise) that the elasticity of w2 with respect to p2 is >1 i.e. → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 81 and alliances as the general average wage, and may be very unjust. 3.15.3. The effect on politics, on ideology, and on other disciplines such as sociology and political science, of the dominance in scientific circles of the marginalist approach since the end of the 19th century, can therefore easily be suspected to have been enormous, and so far underestimated. The implications for policy will also be generally very different, according to which approach one considers to be the correct one. It appears therefore of the utmost importance to try to answer the question: which approach is scientifically more solid? Head counting of the opinions of economists on this issue does not help to reach a solid answer. Nowadays the marginalist approach is the more widely accepted one, but this is no guarantee of its scientific superiority: at the time of Copernicus and Galileo, Ptolemaic astronomy was the dominant school, and yet it was mistaken. Nor can one deduce more than suspicions on the scientific robustness of an approach from the support it may have derived because of its politico- ideological implications. In economics, the political implications of scientific theories no doubt have an influence on their diffusion, for example they affect the funding obtained by supporters of different approaches (big business, for example, will not easily fund thinktanks dedicated to showing that big business should be fought); but even if it were concluded that one main reason for the rise to dominance of the marginalist approach was extra-scientific[90] – the favourable picture of capitalism emerging from it –, this would be far from a sufficient reason to dismiss it as unscientific or incorrect: a scientific theory might well be correct, and at the same time have conservative implications. No, the sole way is to proceed scientifically, examining the logical consistency, and the correspondence with empirical evidence, of the two approaches. Now, prima facie the marginalist approach may appear to be the more solid one; its argument is based on the simple existence of technological choices and consumer choices, and both these choices do exist in real economies. Accordingly, it might be argued that the marginalist approach draws out some implications of universally admitted facts – the existence of alternatives in production technology and in consumption –, implications that the classical authors were unable to notice; and that therefore the that w2/p2 decreases when p2 decreases: thus when w1 increases, the real wage of industry-2 labour decreases also in terms of commodity 2. There is therefore a potential conflict of interests between industry-1 workers and industry-2 workers. Suppose now that industry 2 is located in less developed countries, industry 1 in advanced countries. The high real wage in advanced countries is then at least partly due to the low real wage of less developed countries. If the rate of profit cannot be decreased, raising the real wage in less developed countries would entail lowering the real wage in advanced countries. 90 We do not take sides on this issue. Especially in the first decades after its birth the marginalist approach looked very persuasive, and even Marxist thinkers found it nearly impossible not to be influenced at least to some degree by its outlook. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 82 marginalist approach represents a clear analytical progress relative to the classical approach (which only arrived at grasping the tendency of prices to converge to minimum costs of production, the fact that if real wages increase the rate of return on capital decreases, and – as we will explain – that land rent is determined according to marginalist principles). This opinion, long dominant, has been questioned in recent decades. A number of economists, basing themselves on results due to Piero Sraffa and others, have argued that the absence of decreasing factor demand curves in the classical approach, far from being a weakness of this approach, is a strength, because the marginalist derivation of these curves is logically faulty, and that the inconsistencies are so radical that the entire marginalist approach to income distribution must be rejected; the classical approach, which is free of such logical faults, should therefore be reconsidered – these economists continue – because it was prematurely discarded, and it looks like being a promising basis for a reconstruction of the theory of value and income distribution. 3.15.4. It will be possible to present and evaluate these claims only after the more detailed analysis of the marginalist approach that will occupy us for some chapters. But before we embark upon this task, we tackle a question that may have puzzled our readers. New theories do not come out of the blue; usually they have originate from the development of elements that were present in the theories that preceded them. Given that the classical approach was the dominant one before the marginalist one, the origins of the latter approach should be looked for, at least up to a point, in the classical approach itself. But how could the classical approach be the origin of the marginalist one? The answer that emerges from the present state of understanding of the history of economic theory is of great interest because it helps us better to understand the difference between the two approaches. This answer is that the marginalist approach came out of a generalization of Ricardo's theory of intensive differential rent. The issue is explained in the remainder of this chapter. (In order to make the issue clear, we also explain the analysis of production functions and isoquants deriving from a finite number of alternative fixed-coefficients production methods, rather than from smooth factor substitutability.) The implication is that the central analytical novelty of the marginalist approach is the thesis, that the theory of intensive differential rent can be applied, not only (as in Ricardo) to the division of the net social product between land rents on one side, and the sum of wages and profits (interest) on the other side, but also to the division between wages and profits (interest) of this second portion, an issue that the classical economists had analyzed on the basis of totally different principles. The theory of rent, according to this argument, explains all incomes[91]. 91 This was very clearly stated by John Bates Clark in an article named precisely “Distribution as Determined by a Law of Rent”: “The principle that has been made to govern the income derived from land actually governs those derived from capital and from labor. Interest as a whole is rent; and even wages as a whole are so. Both of these incomes are "differential gains", and are gauged in amount by the Ricardian formula.” Quarterly Journal of Economics, volume 5, (1890-91). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 83 We present now this thesis in the context of a simple model, and we clarify further aspects of the marginalist approach on its basis. 3.16. Premises to a comparison of marginalist theory and Ricardian differential rent: kinked isoquants. 3.16.1. Sometimes the assumption of continuous substituability among inputs is not realistic; it may be a closer approximation to the available technological alternatives to assume that there is a finite number of possible ways of combining the several factors in fixed proportions. Each of these ways will be called a (fixed-coefficients) method of production (or an activity, an older terminology now little used). Technical choice then consists in the choice among a number of different methods. Remember that a technical coefficient is the quantity of input per unit of output; note that if technical coeffcients are fixed, this implies CRS. We explain now how isoquants and total productivity functions look in such a case. In order to use graphical illustrations, we restrict the discussion to two factors. Suppose that corn can be produced by land and labour in 3 different constant-returns-to-scale ways, representable as follows (* stands for 'together with', → stands for 'produce', the subscripts to the symbols of the technical coefficients of labour and land indicate the method): l1 * t1 → 1 unit of corn l2 * t2 → 1 unit of corn l3 * t3 → 1 unit of corn. (Note carefully that, differently from earlier in this chapter, the subscripts now refer to different methods to produce the same product, and not to different products.) In Fig. 3. 15, a diagram with L and T on the axes, these technical coefficients are indicated by points A, B, C. According to the usual terminology, in each production method labour and land are 'perfect complements', so the isoquant associated with a single method is L-shaped (one such isoquant, associated with point B, is shown with red broken lines in Fig. 3.15). But by divisibility and CRS, one can utilize each method on a smaller or greater scale and thus one can produce 1 unit of corn with any linear combination of the three methods; let represent the quantity produced with the first method, and γ the quantity produced with the second method, with 0≤≤1, 0≤γ≤1, and +γ≤1; we can write (l1+γl2+(1––γ)l3) * (t1+γt2+(1––γ)t3) → 1 unit of corn . These linear combinations include all points of the triangle ABC. But only the points along the segments AB and BC (representing linear combinations of methods 1 and 2, or 2 and 3) can be cost-minimizing; from any other point in the triangle ABC it is possible to move to a point on one of these segments, which employs less of both factors and therefore costs less. Therefore the unit isoquant, i.e the locus of cost-minimizing combinations of factors corresponding to 1 unit of product, is the kinked line ABC, plus the vertical halfline starting from A and the horizontal halfline starting from C; these two halflines must be included in the isoquant because they indicate f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 84 combinations of factors which, in spite of using more of one factor than the minimum necessary, are anyway cost-minimizing if that factor has rental equal to zero. This isoquant is shown as a heavy line in Fig. 3.15. Owing to CRS, all other isoquants are radial expansions or contractions of this one. Kinked isoquants of this type are called activity-analysis isoquants because first systematically studied in the 1950s in the course of studies classified at the time as 'activity analysis'. The firm minimizes the cost of producing a given level of output in the usual way, by looking on the corresponding isoquant for the point that touches the isocost closest to the origin. Given the shape of the isoquant, this will usually mean that the optimal choice will be at one of the kinks of the isoquant, e.g. at point B in Fig. 3.15; the firm can choose an internal point of one of the segments of the isoquant only if the slope of the isocost happens to coincide with the slope of that segment; in this case the firm is indifferent among all points of the segment. (The occurrence of such a case may appear highly improbable from this picture, but when we come to the entire economy and introduce the condition of full employment of both factors we will see that it is the normal case.) intl1 land method 1 method 2 isocost A • B' method 3 B C O labour Fig. 3.15. Activity-analysis isoquant with three alternative methods. Note that it is possible that a method may be inefficient relative to other methods or linear combinations of other methods, and therefore it may not appear on the isoquant. For example if method 2 had been less productive than assumed so far, and such that the quantities of factors needed to produce 1 unit of corn with that method were indicated by point B', then it would never be efficient to use it, and the isoquant would consist of the line from A to C, plus, as before, the f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 85 vertical halfline starting from A and the horizontal halfline starting from C.(92) The more numerous the different methods appearing on the isoquant, the more kinks it will have and the shorter will each straight segment be; the smooth isoquant of introductory textbooks is simply the limit of the activity-analysis isoquant, as one assumes more and more alternative methods and closer and closer kinks. 3.16.2. Let us now consider the total product curve of labour. Assume the availability of the three methods depicted in Fig. 3.14. Note that Fig. 3.15 implies that we have ordered the methods such that l1<l2<l3 and then necessarily(93) t1>t2>t3. Assume that the firm has at its disposal a fixed quantity T* of land, which can be utilized entirely or in part, with no variation in its cost to the firm; the firm must decide how much labour to employ and according to which method, for each given quantity to be produced. As long as the wage is positive, the firm wants to economize on labour. Therefore for very low levels of production, such that not all the land is needed, the firm chooses method 1 because this method employs the least amount of labour per unit of product; up to the maximum amount producible with method 1 and with the given quantity of land, method 1 remains the most convenient one, and production is proportional to labour employment; as production increases, a greater and greater portion of the available land is utilized. The maximum amount producible with method 1 is q = T*/t1, because 1/t1 is the product per unit of land when method 1 is employed, and there are T* units of land. The associated labour employment is T*∙(l1/t1), because li/ti is the labour employed per unit of land with method i. Therefore up to the level of labour employment L = T*∙(l1/t1), labour's total product curve is a straight line starting from the origin, and with slope (T*/t1)/( T*∙l1/t1) = 1/l1 , cf. Fig. 3.13. 92 Thus isoquants cannot be concave. 93 Necessarily, otherwise a method would be inefficient. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 86 21 pt q T*/t3 q(L) T*/t2 T*/t1 O T*∙l1/t1 T*∙l2/t2 T*∙l3/t3 L MPL L Fig. 3.16 f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 87 In order to increase production beyond this level, the firm must use, on at least part of the land, a method that allows a greater production per unit of land. Method 2 uses less labour per unit of product than method 3, so the firm will initially switch units of land from utilization with method 1, to utilization with method 2. For each unit of land thus switched, labour employment increases by a constant amount, equal to the labour per unit of land with method 2, minus the labour per unit of land with method 1, i.e. l2/t2 – l1/t1 , and production increases by a constant amount, equal to the product per unit of land with method 2, minus the product per unit of land with method 1, i.e. 1/t2 – 1/t1 . When all land has been switched to method 2, it is q = T*/t2 and L = T*∙l2/t2 . Hence as L rises from L = T*∙l1/t1 to L = T*∙l2/t2 labour's total product function rises as a straight line from q = T*/t1 to q = T*/t2 , so its slope is (T*/t2 – T*/t1) / (T*∙l2/t2 – T*∙l1/t1) = (t1 – t2) / (l2t1 – l1t2). Along the first segment of labour's total product function, land is only partly utilized, and method 1 is the sole method in use. Along the second segment, land is fully utilized, and except at the extreme points both method 1 and method 2 are in use; method 2 is used on a greater and greater portion of land as production increases, and finally it is used on the entire land at L = T*∙l2/t2 . In order to increase production beyond T*/t2, the firm must replace method 2 with method 3 on at least some of the land. We apply the same reasoning as before and conclude that the third segment of labour's total product function goes from L = T*∙l2/t2 to L = T*∙l3/t3 and from q = T*/t2 to q = T*/t3 , with slope (t2 – t3) / (l3t2 – l2t3) . Along this segment, except at the extreme points, both methods 2 and 3 are in use; at L = T*∙l3/t3 only method 3 is in use. Unless further methods are available with a still higher product per unit of land, no further increase in production is possible; to the right of L = T*∙l3/t3 labour's total product curve is a horizontal straight halfline. Exercise: show the movement in the isoquant diagram of Fig. 3.15, corresponding to the increase of production and labour employment described by labour's total product curve. Exercise: derive the shape of the total product function if, in the isoquant diagram, method 2 corresponded to point B‟ instead of point B, and show that this means that there is no intrinsic technological necessity of a decreasing marginal product of a factor, the decrease derives from the entrepreneur‟s maximizing choices. The slope of this kinked total product curve of labour, where defined, can again be interpreted as the marginal product of labour; it indicates the increase in production made possible by the employment of one more unit of labour when the employment of land is given. The marginal product of labour is therefore a step function; as always the price-taking firm facing given factor prices will find it convenient to increase the employment of labour as long as the marginal product of labour is less than the real wage; when the real wage coincides with one of the values where the marginal product curve is horizontal, the firm is indifferent among the labour employments f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 88 corresponding to that horizontal segment. A perfectly symmetrical analysis can be applied to land, if the roles of labour and land are reversed, i.e. if it is the amount of labour that is given, while land employment is variable. Thus suppose that the amount of labour is given and equal to L* and a cause of a given cost to the firm; the firm must decide how much land to employ, and according to which method, to produce each given amount of product. The previous reasoning can be applied with labour and land exchanging their roles, and accordingly with a reversal in the order of adoption of the three methods. If the rent of land is positive, initially method 3 is adopted because labour is not fully utilized and method 3 is the one that uses less land per unit of product; land's total product curve is a straight line starting from the origin with slope equal to 1/t3. At the employment of land T = L*∙t3/l3 the total land productivity curve has a kink, after which it has slope (l3 – l2) / (l3t2 – l2t3) and methods 2 and 3 are both in use. At T = L*∙t2/l2 there is another kink, after which land's total product curve has slope (l2 – l1) / (l2t1 – l1t2) and methods 1 and 2 are both in use. At T = L*∙t1/l1 there is the last kink, after which land's total product curve becomes horizontal, only method 1 is in use. Here too, the slope of land's total product curve measures the marginal product of land, constant along each segment and changing discontinuously at the kinks. An important consequence is the following. Suppose that the real wage equals the slope of one of the segments of labour's total product curve, and that therefore the firm employs an amount of labour in between the extremes of that segment, thus ensuring the equality between wage and marginal product of labour: if competition ensures that the entrepreneur makes neither profit nor loss, so that what does not go to labour goes to land, then land too is earning its marginal product. Thus assume that the real wage equals (t2 – t3) / (l3t2 – l2t3) and the firm employs an amount of labour L* such that T*∙l2/t2 < L* < T*∙l3/t3 , i.e. methods 2 and 3 are both in use. Total production is T*/t2 plus the marginal product of labour times the excess of L* over T*∙l2/t2 : q = T*/t2 + [(t2 – t3) / (l3t2 – l2t3)]∙( L* – T*∙l2/t2) and each unit of land earns a rate of rent given by = (q–wL*)/T* = {T*/t2 + [(t2–t3) / (l3t2–l2t3)]∙( L*–T*∙l2/t2) – L*[(t2–t3) / (l3t2–l2t3)]} / T* = = 1/t2 – [(t2 – t3) / (l3t2 – l2t3)]∙l2/t2] = (l3 – l2) / (l3t2 – l2t3) . This has been shown to be the marginal product of land when methods 2 and 3 are both in use. █ 3.16.3. This result shows that, when − assuming that only one good is produced, and land and labour are the only factors − one passes from the single firm to the entire economy, where each factor will tend to earn its marginal product, then for points other than the kinks, the product exhaustion theorem (the payment to each factor of its marginal product exhausts the product) holds also in economies where technical choice is among a finite number of fixed-coefficients methods. If the full employment of factors corresponded to a kink, marginal products would not be defined, and factor rentals would have some freedom within an interval; the tendency of factor payments to sum up to the entire value of the product might still be argued, on the basis of the tendency of f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 89 competition to wipe out profits and reduce prices to average costs; but the product exhaustion theorem (whose importance was that it reconciled such a result of competition in product markets with the tendency of firms to pay to each factor its value marginal product) would not apply because of the absence of well-defined marginal products. But it would be a total fluke if the full employment of all factors corresponded precisely to a kink; in our example, assuming that corn is the sole good produced and that the total product curve of labour of Fig. 3.16 refers to the entire economy rather than to a single firm, if the supply of land is T*, there are only three levels of labour supply corresponding to kinks, out of possible labour supplies that form a continuum: the probability that labour supply takes exactly one of those three values is nil(94). We can conclude that the existence of technical substitutability limited to a finite number of alternative fixed- coefficients methods does not disturb the validity of the product exhaustion theorem. 3.17. Ricardian differential rent and the marginalist approach. 3.17.1. We are now ready to discuss the role of Ricardian differential rent in the birth of the marginalist approach. Already in Ricardo‟s time there had been attempts to explain the income from the property of capital as due to a contribution of capitalists to production, described as abstinence (Senior). But these attempts did not have an analytical base, until the discovery of the fact that Ricardo‟s theory of intensive differential rent could be also used to explain what was left to capital-cum-labour as in its turn a differential rent. What is capital-cum-labour? Ricardo and the Ricardian school conceive the wage as consisting essentially of corn; the corn wage is given and advanced, and it is often identified with the entire capital advanced; even when the distinction is made between advanced wages and seed- corn, the proportion between seed-corn and wage corn is treated as fixed, so one can treat the labour employed and the capital advanced as proportional to each other, and one can treat them as a single composite factor, capital-cum-labour. Thus one can conceive corn as produced by capital-cum- labour and land. Since the wage is included in the capital advanced, which is homogeneous with the product, the rate of profit is very simply determined: e.g. if production of 1000 units of corn on a certain land requires advancing 400 units of corn-capital, and if land rent (paid at the end of the year) takes away 200 units of the product, profits are 400 and the rate of profits is 100%. In Ricardo's theory of extensive differential rent, land is of different qualities. Imagine production is gradually increased; at first it will use only (part of) the land which, when rents are zero, yields the highest rate of profits. As production increases, a point is reached when the first kind of land is fully utilized and a second kind of land starts being utilized, the one which yields the 94 This is a first example of cases so-called of 'measure zero' in the space of possible cases. We will meet other ones later, and the notion of 'measure zero' will be then explained. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 90 second highest r when rents are zero; but since it yields a lower r, competition among capitalists for the first type of land causes a rent to be born on it, which will rise so as to equalize r on the two types of land (at the level attainable on the second land). When the further increase of production causes a third type of land to be utilized, rent arises on the second land, and rises on the first one. The Ricardian theory of extensive differential rent opened the way to the marginalist notion of decreasing marginal utility. In Ricardo's theory, further doses of capital-cum-labour, when one must pass to lower-quality land, yield a lower increase in product; one obtains an extensive decreasing marginal product, so to speak. This idea was imported into consumption theory through the intermediary of the Benthamite conception of utility as something measurable, plus the idea of the several needs satisfied by a commodity as capable of being ordered in succession, from the most important to the less important. One must only treat needs as analogous to different qualities of land, and consumption as analogous to capital-cum-labour in that it 'produces' different quantities of utility(95) according to the need that it satisfies; then, on the basis of the reasonable principle that one will first satisfy the most urgent needs, one obtains an extensive decreasing marginal 'product', in terms of utility, of successive doses of consumption of a good. Thus one finds in Wicksell (1934, p. 31) the example (taken from Böhm-Bawerk and Carl Menger) of "a colonist living alone in the virgin forest by agriculture" who harvests corn and who allocates the first sack of corn to "the maintenance of life", the second sack to "eat his fill and preserve his health and bodily strength", the third sack "to keep fowl and thus procure a necessary change in an otherwise purely cereal diet", the fourth sack "to making spirits", the fifth one to "providing for a few parrots". The successive sacks of corn yield smaller and smaller amounts of additional utility because allocated to the satisfaction of less and less important needs, just like the successive units of capital-cum-labour yield smaller and smaller amounts of additional production because utilized on less and less fertile land. The extension of this insight to the case of several consumption goods was made easier by the initial conception of utility as additive, i.e. as the sum of the utilities caused by each consumption good, u(x1,x2) = v(1)(x1) + v(2)(x2) if there are two goods. It was then easy to conceive consumers as allocating further units of income to whichever use yields higher marginal utility among the remaining uses, in analogy with the farmer who allocates further units of capital-cum-labour to the land where they yield a greater amount of additional product. The next step was the realization that one will be ready to exchange only if the increase of utility from what one obtains is greater than the loss of utility from what one gives up; and thus, at 95. “Utility must be considered as measured by, or even as actually identical with, the addition made to a person‟s happiness. It is a convenient name for the aggregate of the favourable balance of feeling produced – the sum of the pleasure created and the pain prevented” (S. W. Jevons, The theory of political economy (1871), Pelican ed., 1980, p. 103.) The whole section from which this passage is taken is very clear on the cardinal conception of utility and of marginal utility. On p. 152 it is very clear that Jevons treats the marginal utility of each good as independent of the quantity consumed of other goods. The same holds true for Walras. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 91 given exchange rates, one will stop at the point where the last dose of the good given away would have yielded the same increase in utility as the last dose of the good acquired. This realization made theorists believe they had finally found the explanation for the „paradox of value‟ (i.e. why indispensable goods like water cost so little and superfluous goods like diamonds cost so much) in the distinction between total and marginal utility, and in the dependence of the latter on the scarcity of goods. The less there is of a good, the higher its marginal utility( 96). The relative prices of goods would end up reflecting their relative marginal utilities. 3.17.2. But such a theory must then explain the abundance or scarcity of consumption goods, since most of them are produced. So the discovery of an explanation of exchange value of given quantities of consumption goods on the basis of marginal utility would not have generated a new, anti-Ricardian approach to income distribution, if it had not been sensed that one could extend the insight to „factors of production‟ because the scarcity of utility-producing goods could be seen to derive from the scarcity of the factors producing them, and conversely the factors, by producing consumption goods, could be seen as indirectly producing utility – so that one could also speak of a marginal (indirect) utility of factors, and one could say that ultimately, by exchanging products, people are really exchanging factor services. The extension to factors was carried out by Menger and Walras initially on the basis of the mechanism of indirect factor substitution. The first step is to imagine that each consumption good is produced by a specific factor. Suppose that a consumer produces meat with labour, and another one produces strawberries with his own land: in exchanging meat against strawberries, they indirectly exchange (the services of) labour against (the services of) land; given an exchange ratio between meat and strawberries, since meat represents services of labour, and strawberries services of land, there is implicit, in the exchange ratio between products, an exchange ratio between the services of labour and of land, and when exchange stops this implicit exchange ratio will equal the ratio of their marginal (indirect) utilities. Once this equilibrium is reached, the utility of each consumer can be seen as „produced‟ by (the services of) the labour and the land which have gone to produce the goods she consumes, and the real wage and rent will reflect the marginal (indirect) utilities of labour and land. The successive step – the extension of the theory to the case when labour and land are employed in the production both of strawberries and of meat – is achieved through the type of analysis presented in the Appendix "Marginal (indirect) utility of factors with fixed coefficients". But it is doubtful that the theory would have gained as much following as it did, if it had only been based on consumer substitution and the indirect factor substitution mechanism. Substitutability in consumption is limited; and it only permits the full employment of factors if the proportion in 96The concept of scarcity was sometimes (e.g. by Walras) considered as in fact synonymous with marginal utility. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 92 which factors are supplied falls within fairly restricted limits. Consider again the corn-iron economy with fixed coefficients, with corn the more labour-intensive good, i.e. L1/T1 > L2/T2, and fixed factor supplies. In this economy the full employment of both labour and land is only achievable if the ratio L*/T* in which labour and land are supplied is intermediate between L1/T1 and L2/T2, which are the proportions in the demand for factors if only corn, respectively only iron, is produced; and the actual range of variability of the composition of factor demand must be considered generally narrower, perhaps much narrower, because those extreme proportions can only be approached if the demand for corn or for iron approaches zero, and it is highly improbable that such extreme compositions of the demand for consumption goods may ever be approached; not the least reason being that the variability in the relative prices of consumption goods is in turn limited: p1/p2 can only vary between the values associated with w=0 and =0, and if all factor coefficients are positive, these values will be both positive, and the closer to each other the closer is L1/T1 to L2/T2. Therefore it is highly likely that the discovery of the direct factor substitution mechanism was the really decisive element in the success of the marginalist approach. This was developed on the basis of a reinterpretation of Ricardo's theory of intensive differential rent (cf. chapter 1, §1.5). 3.17.3. We remind the reader of the gist of that theory. The capital/labour ratio is fixed and the given corn wages are included in the advanced capital. Suppose there is only one type of land, but several known methods of cultivating it. As long as land is not fully utilized, there is no rent and the method adopted is the one yielding the highest rate of profits. When land becomes insufficient to satisfy demand with that method, the rising price of corn causes a competition for land which causes a rent to be born on it. If there is another production method which, when rent was zero, was less convenient but which produces more corn per acre of land, then the rise of rent decreases the rate of profits obtainable with this second method less rapidly than with the first, so it is possible that a rate of rent exists which leaves a positive rate of profits and which makes the two methods equally profitable. The numerical example used in chapter 1 was the following, where the numbers indicate the corn-capital advanced, and the product obtained, per acre of land: A:5g 9g B : 12 g 18 g Method A when rent is zero yields a rate of profits of 80%, method B of 50%, but B is less land-intensive so when rent starts to rise, the rate of profit with method B decreases slower than with method A. Let rA and rB be the two rates of profit, and the rent per acre, paid at the end of the year. Then we have: 5(1+rA)+ = 9 12(1+rB)+ = 18 The two methods are equally profitable for the value of which solves the above system of two equations if we put rA = rB: the solution is β = 18/7, rA = rB = 2/7 28.6%. When the two methods are equally profitable, an increase in the production of corn is obtained by extending the use of method B and restricting that of method A. Thus corn production f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 93 can vary, without any further change in rent, up to a maximum obtained when only method B is used. If rent became greater than 18/7 before demand had reached this limit, r B would become greater than rA, method B would be universally preferred, but the increase of production beyond demand would make some land redundant, so the rate of rent would start decreasing back toward the value 18/7. When the increase of demand has caused the whole of land to be cultivated with method B, if demand keeps increasing there will be again a tendency of rent to increase; this may make a third process C equally convenient with B, which was not convenient before, e.g.: C : 16 g 23 g B and C are equally profitable when = 3 , r = 25% ; before, with =18/7, method C yielded a rate of profit less than 2/7. So now production is expanded via a gradual replacement of method B with method C. Except at switchpoints, there are always two methods on the same land. The rent thus determined is called intensive because it derives from the need to introduce methods of more intensive cultivation of the same land. 3.17.4. Let us now examine more analytically this theory so as to bring out the symmetry between land and capital-cum-labour. Suppose a single type of land whose surface is T=1. There are several known methods of producing corn, with land technical coefficient ti and capital-cum-labour technical coefficient (a quantity of corn) i: ti * i 1 unit of corn As long as rent is zero, the most convenient method is the one with the smallest i ; let it be method 1. When the need arises to increase production beyond 1/t1, rent arises and if another method has i>1 but ti<t1, it is possible that there is a positive rate of rent which makes the two methods equiprofitable. The thing can be studied as follows: let p1 = 1 be the price of corn produced with method 1 ; let pj be the unit cost (inclusive of rent and profits) of corn produced with other methods when the rate of rent is given and the rate of profits r is residually determined by method 1. Then: 1 p1 1 r 1 t 1 p j 1 r j t j j1 Since j>1 by assumption, when = 0 it is certainly p j 1 , but if tj<t1, as increases pj decreases and it becomes less than 1 before r becomes zero (Exercise: prove it!). Let us assume this happens for j=2 and for j=3 and that as β increases p2 becomes equal to 1 before p3. At that point is determined by: (1+r)1 + t1 = 1 = (1+r)2 + t2 Let us call 1,2 the rate of rent which renders methods 1 and 2 equiprofitable. For each unit of 1 1 land where the method passes from 1 to 2, production increases by the amount , and the t 2 t1 surplus of corn (production minus capital consumption including wages) increases by f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 94 1 2 1 1 0. By altering the fraction of land tilled with method 2 total production can t2 t1 vary from q1 = 1/t1 to q2= 1/t2. If q must increase beyond q2, the sole possibility is that there be a third method with 3>2 and t3<t2 which becomes equiprofitable with the second method for a higher rent and a non-negative rate of profit. Let us assume this is the case and let us represent graphically the amount of corn produced as a function of the corn-capital employed. As long as only method 1 is employed, production increases with slope equal to 1/1, which is equal to 1+r because there is no rent so (1+r)1=1. Let us indicate with L the employment of capital-cum-labour, which for brevity we also indicate simply as capital. The maximum amount of capital employable with method 1 is L1=1/t1. When method 2 is gradually replacing method 1, the slope is (q1–q2)/(L2–L1) = (q1–q2)/(2q2–1q1).. This slope again equals 1 + r, as can be shown with just a bit of manipulation. From q1=1/t1 and q2=1/t2 it follows that q 2 q1 1 t 1 t1 t t 2 t 1 1 t 2 t1 t 2 2 1 2 . 2 q 2 1q1 2 1 t1 t 2 t1 t 2 2 t 1 1 t 2 t 2 t1 1 1 r 1 On the other hand from (1+r)1+t1 = (1+r)2+t2 = 1 it follows that and t1 t1 t 2 substituting into (1+r)2+t2 = 1 one obtains 1 r . And the same holds true when 2 t 1 1 t 2 the second and third method co-exist. So 1+r equals the marginal product of capital-cum-labour. (fig.interlinea 21 pt) q q(L) q2 q1 L1 L2 L3 L1 = 1q1 = 1/t1, L2 = 2q2 = 2/t2, L3 = 3q3 = 3/t3. Fig. 3.17 But note now that the analysis might be inverted, and instead of taking the amount of land as f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 95 given and the employment of corn-capital as variable, one might take the amount of corn-capital as given and the employment of land as variable. Let us do it, assuming the same three methods. The expansion of production from zero will initially not need the entire stock of corn-capital, 1+r will equal zero (i.e. r = –1)(97), and method 3 will be preferred as yielding the highest rent per acre; as production is increased, a point will be reached where the use of method 3 will fully utilize the given stock of corn-capital, then r will rise until method 2 becomes equally convenient with method 3, then production is increased with r and β constant by gradually raising the fraction of corn-capital used with method 2 until only method 2 is used, then r rises again until method 2 and 3 become equally convenient, with a reasoning perfectly symmetrical to the case with given land and variable corn-cum-labour. Therefore the situation, for example, where methods 1 and 2 co-exist can be seen as a point in the process of extension of method 2 and contraction of method 1 on a given amount of land, or as a point in the process of extension of method 1 and contraction of method 2 with a given amount of corn-capital. Given the complete symmetry of the reasoning, the implication must be that it is simultaneously true that 1+r equals the (gross) marginal product of capital, and that equals the marginal product of land. Indeed we have shown that this is the case; graphically, by replacing L with T, i with ti etc. (and vice-versa) in Figure 3.17 and in the reasoning it depicts, one obtains that the slope of q(T) is (1–2)/(1t2–2t1) and that it equals . 3.17.5. No doubt the reader has noticed the striking similarity between Fig. 3.17 and Fig. 3.16[98]. Let us then suppose that, in this Ricardian economy, corn is the sole commodity produced, that land rent is intensive differential rent, and that there is a given stock of corn-capital which is fully utilized. A marginalist economist familiar with the q(L) curve (the total product curve of labour) obtained earlier for the case of 'activity analysis' production possibilities of a firm with land and labour as factors (cf. Fig. 3.16 and §3.16.3) would find that, in this Ricardian economy, income distribution between land on the one hand, and capital-cum-labour on the other, is determined in exactly the same way as he would have determined it for the land-labour economy, with the sole difference that the role of labour is taken here by capital-cum-labour (that includes advanced wages at a given rate), and therefore w is replaced by (1+r). He would therefore point out that it would be one-sided to consider what goes to capital-cum-labour as residually determined, after first 97 Obviously for r to go to –1 the supply of seed-corn must not be storable, nor desired for consumption: unrealistic hypotheses, implied by our decision to treat the supply of seed-corn as fixed; but here we need not discuss the consequences of relaxing them, the basic point is clear anyway. 98 The analogous of Fig. 3.15 might also be obtained in the Ricardian economy. Indeed if, given the utilization of land, corn production can be increased by increasing the corn-capital utilized through a change in the proportion between two methods, then it would also be possible to increase the corn-capital utilized while keeping corn production unchanged, via an analogous change in the proportion between the two methods, but associated now with a decrease in the utilization of land. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 96 determining the income going to land as differential rent; with exactly the same legitimacy, he would insist, one could first determine the income going to capital-cum-labour as differential rent and one could consider land rent as residually determined. The truth, he would argue, is that both incomes have exactly the same origin, they are determined by the same mechanism, they are both determined by the full-utilization marginal products of the respective factors (land, and capital-cum- labour). He would then draw two important implications. The first one is that the division of the net product between land rent on one side, and profits-plus-wages on the other, can be seen as determined by the tendency toward the full utilization both of land, and of capital-cum-labour, brought about by changes in their rates of remuneration. In Ricardo, such a tendency is assumed to work for land, once the amount of corn to be produced is determined by the demand for corn (that depends on the stage reached by accumulation plus possibly other things such as a prohibition on corn imports): it is the appropriate rise of land rent that, given the demand for corn, brings land utilization to coincide with the availability of land by influencing the choice of (and proportion between) production methods; the rate of profit, being residually determined, is not conceived as having an analogous role, the available stock of corn-capital is fully utilized simply because savings always translate into investment (Ricardo‟s simplistic view of Say‟s Law). But, the marginalist economist would point out, since it has been found that a symmetrical mechanism can be seen to apply also to capital-cum-labour, then the rate of profit too must be seen as having the role of ensuring that, given the full utilization of land, the appropriate production methods are chosen so as to ensure the full utilization of the stock of capital. On this basis, one can conceive of the demand for either factor as being a decreasing function of its rate of remuneration, once the utilization of the other factor(s) is given. The second implication is that there seems to be no reason why the same insight should not be applied also to the division between wages and profits of what is left after land rent has been paid. Indeed, if one supposes initially land to be overabundant and hence free, and that there are several methods utilizing different proportions of labour and seed corn (now for greater clarity corn-capital can be assumed to be only seed-capital and not to include wages, by assuming that wages are paid in arrears), one can apply the same kind of analysis to production with labour and corn-capital. Corn-capital can be treated like land, and it will earn a rate of profit determined as intensive differential rent, i.e. as marginal product, depending on how much labour is combined with it; but then labour‟s residual income can be shown − in exactly the same way as for the land&labour or land&ccapital-cum-labour cases − to equal its marginal product too. Actually, the marginalist economist would continue, since a continuously decreasing marginal product can be seen as the limit of assuming more and more (and more and more similar) alternative production methods, then as long as there is a high number of alternative different proportions in which to combine labour f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 97 and seed-capital the situation is well approximated by the analysis of §3.7.1[99]. And, he would conclude, the analysis can be extended to admitting land rent too, since the same intensive-rent logic can be extended to the simultaneous use of more than two factors (this is intuitive but is formally shown in Appendix 3 to the chapter). With this we have succinctly summarized a process of generalization of Ricardian intensive rent theory, based on a gradual realization of its symmetric implications, that started with Von Thünen and Jevons, achieved completion around the end of the 19th century, and represents the central analytical novelty of the marginalist approach. More than in marginal utility, the real root of the differences between the marginalist/neoclassical and the classical approach lies here. The discovery of the symmetry between the roles of whichever two factors are considered, and the realization that the reasoning was also applicable to more than two factors, induced the founders of the marginalist approach to believe that, since the analysis was in principle extensible to any number of factors, it had to be applicable to all incomes from property of factors: therefore it could explain, not only (as in the Classical authors) the division of the net product between land rent on one side, and the joint income of labour and capital on the other side, but also the division, which the Classical authors had tackled in an entirely different way, of the latter part between real wages on one side, and profits or interest on the other side; the reason being that the same rent-type explanation could be applied also to the explanation of the rate of wages alone, and to the explanation of the rate of interest or rate of profit alone. So Jevons in the 1860s developed an analysis strictly parallel to the one we have developed for land and corn-capital, but with labour in place of corn-capital, and then he went on to try and apply the same rent-principle to capital; and not much later we find J. B. Clark (1891) stating, as already observed, that the correct theory of distribution explains all kinds of income as determined by a law of rent. The further development of the theory brought about the joint consideration of the direct and indirect substitution mechanisms, which at first had been analyzed by different authors (mainly the English economists the first one, mainly Walras the second one). It was then realized that the two mechanisms shared a fundamental similarity: both relied on the thesis that a decrease in the relative rental of a factor increases the proportion in which that factor is utilized relative to other factors in the economy as a whole. Less clarity existed on another fundamental similarity of the two mechanisms: both, in order to be used to replace the classical approach to wages and to the rate of 99 Actually the existence of „true‟ marginal products (partial derivatives of production functions) can entail significant analytical differences relative to the case where marginal products are only defined by changes in the proportions between co-existing methods and require therefore that a good be produced by at least two different methods (this is not the case with „true‟ marginal products, each good will be produced by a single method). But these differences only become relevant with heterogeneous capital goods and therefore their discussion is postponed to ch. 7. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 98 profits with a generalization of rent theory, had to argue that capital can be treated as analogous to labour or land. The correctness of this last thesis will occupy us for several later chapters. Now let us underline the common core of the two mechanisms. Both mechanisms rely on the possibility of 'producing' something (a quantity of product, or a quantity of utility) with different proportions of factors. It is this possibility of substitution in the production of a given quantity of the final product that allows one to draw decreasing marginal product curves. The direct substitution mechanism comes out to supply the general picture, which also includes the indirect substitution mechanism, because utility can be seen as 'produced' by factors in perfect analogy with the production of physical products. Imagine a firm where labour and land produce the final product, not directly, but indirectly, by first producing some intermediate products A, B, and C, which then produce the final product. An example might be a firm where labour and land produce wine and also all that is necessary for the production of wine, by first producing the must and barrels and bottles and corks and labels, which after some time become bottles of wine ready to be sold. The production function of this firm can be specified in terms of labour and land producing wine bottles( 100 ) (indeed we do not generally worry as to which intermediate products appear and disappear during the production process of a firm, we only ask about the expenses caused by paid inputs, and about the revenue caused by the final product). So factor prices in terms of wine bottles will correspond to the (indirect) marginal products of labour and land. Now re-interpret wine bottles as the utility of a consumer, and A, B and C as consumption goods, and you have the picture of labour and land indirectly 'producing' utility, and you also obtain the intuition of why factor prices measure the (indirect) marginal utilities of land and of labour. This picture, of the utility of a single consumer as indirectly 'produced' by the factor services that produce the goods consumed by the consumer, allows us to grasp an implication, that will be rigorously examined in chapter 5: that by supplying factor services and purchasing consumption goods, consumers are in fact exchanging factor services. This is very easy to visualize, in the case in which factors are fully specialized: if strawberries are produced by land alone, and meat by labour alone, then by purchasing strawberries one in fact purchases land services, and by purchasing meat one purchases labour services; so if a landowner sells strawberries and buys meat, he exchanges land services against labour services. But the same picture can be reached when factors co-operate in production. Suppose you supply to the market 100 units of labour and that the goods you purchase with your wage income require (assuming CRS) 60 units of labour and 60 units of land: then it is as if you consumed yourself 60 units of your labour and you exchanged the other 40 units of labour against 60 units of land(101). 100 Attention must be given to the element of time in a fully fledged analysis, but here we can neglect this issue. If w is the wage rate and the land rent rate, your consumption goods must cost in the aggregate 101 60w+60, while your income is 100w. Assuming a balanced budged, it must be 60w + 60 = 100w i.e. w/ → f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 99 3.17.6. With this, the general „vision‟ of the functioning of market economies emerging from the marginalist approach should be clear. Now we must enquire into its robustness. To such an end, we shall have to study in detail the rigorous formalizations of this theory, i.e. the theory of general equilibrium (GE). In many modern introductions to GE theory, most of the space if not all of it is dedicated to consumption and to the general equilibrium of "pure-exchange economies", where agents exchange given endowments of consumption goods, endowments whose origin is not analyzed. However, it is important to be clear that GE theory intends above all to be a theory of income distribution, i.e. a theory of what determines wages, rents, the rate of return on capital (i.e. – if we leave aside the reward for the “risk and trouble” of entrepreneurship – the rate of interest) in a capitalist economy; and that it is the formalization of an approach explicitly born as an alternative to the Classical approach, in explicit opposition to the Classical view of the origin of income from property of capital. In the last analysis, the central thesis of this anti-Classical alternative is: the utility-maximizing choices of consumers and the cost-minimizing choices of firms cause the average proportions in which factors are demanded to be inversely related to relative factor rentals; therefore, from the indisputable empirical facts of alternative consumption possibilities for consumers, and alternative production methods for firms, one can derive decreasing demand curves for factors, which give plausibility to a tendency toward a simultaneous equilibrium between supply and demand on all factor markets. This is the theory that we pass rigorously to examine in the next chapters. It will be useful, in reading these chapters, to bear two points in mind. The first is that the theory‟s plausibility rests on its ability to confirm (i) that the theory is not underdetermined (i.e. that it is not the case that one has fewer equations than variables to be determined), and that at least one equilibrium exists, (ii) that the equilibrium is unique, (iii) that the economy tends to it. If the theory is unable to prove the existence of a well-defined equilibrium then it is clearly a failure; but if it surmounts this hurdle but predicts that there are many equilibria to which the economy may converge, then it is still hardly persuasive, because reality does not seem to exhibit the indeterminacy of outcomes that the theory would predict. Some modern presentations of general equilibrium theory completely omit a discussion of issue (iii); but without a convincing argument that there are forces pushing the economy toward the marginalist equilibrium, the latter cannot aim at being an explanation of what determines distribution, quantities and employment in real economies. Indeed, if the tendency toward a marginalist equilibrium cannot be convincingly argued, then, since prices, income distribution, quantities and employment are anyway determined in real economies, the forces determining them must be other from the ones postulated by the marginalist = 3/2 , which is exactly the exchange ratio implicit in the exchange of 40 units of labour against 60 units of land. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 100 approach. Evidently − one would have to conclude − marginalist equilibrium theory, in its formalization of the decisions of economic agents and/or of their consequences, misses or misrepresents some important element. The second point is that it is imperative to arrive at how general equilibrium theory treats capital and determines the rate of interest in cases less simple than the ones we have discussed so far, in which there is only one capital good. The discussion of the equilibrium of pure exchange economies and of production economies without capital or with only one capital good must be understood to be only preliminary to the discussion of the equilibrium of economies with heterogeneous capital goods, which raise difficult problems. It is nonetheless useful to start from the pure exchange economy, which allows the presentation of the supply-and-demand determination of prices and quantities in its simplest form. Actually, we will see in chapter 5 that the general equilibrium of „acapitalistic‟ production economies can be „reduced‟ to a general equilibrium of pure exchange; and modern general equilibrium theory (differing, on this, from earlier marginalist authors) argues that all that is needed to include capital goods into general equilibrium theory is a reinterpretation of the acapitalistic production economy model, and therefore ultimately of the pure exchange model; according to this position, therefore, the pure exchange model really includes all the essentials of the theory. We shall criticize this view, but in order to be able to do so we must first be clear on the pure exchange model and the „acapitalistic‟ production model. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 101 3.18. APPENDIX 1. Concavity of the production possibility frontier. We first need to study the Pareto set in an Edgeworth production box. Assume two factors, labour L and land T, rigidly supplied, and used in two consumption good industries with production functions x1=F(L1,T1), x2=G(L2,T2), with smooth and strictly convex isoquants. Draw an Edgeworth production box with side lengths equal to the factor endowments and draw in it the isoquants of industry 1 (with the lower-left corner A as origin) and of industry 2 (with the upper-right corner B as origin); let us look for the Pareto set, where isoquants are tangent. Because of constant returns to scale, for each industry the slope of isoquants is constant along a ray from the origin. Thus if a point of the Pareto set is on the diagonal connecting the origins, then the Pareto set coincides with that diagonal, and this means that the two goods are produced (with an opportune choice of units) by the same production function (then the PPF is a straight line); if point P on the Pareto set is below the diagonal, then the production functions are different and (apart from points A, P, B) the Pareto set must be entirely below the segments AP and PB (along AP the slope of good 1‟s isoquant is the same as in P but the one of good 2‟s isoquant is not, and the reader can easily complete the proof). An analogous reasoning holding if P is above the diagonal allows us to conclude that when the production functions are different, the Pareto set is either strictly convex, or strictly concave. intl1.5 B D land P A labour C Fig. 3.18 Back to the PPF, the set of points on or inside the PPF is called the production possibility set; it represents the set of feasible vectors of productions of the two goods, given the factor endowments (which we assume are rigidly supplied) and the available technologies. Consider two points Y=(x1,x2) and Z=(x1‟,x2‟) on the production possibility frontier. We assume that the two goods are produced by different production functions. Then factor proportions in the production of x1 may be different from the ones in the production of x1‟, and the same may be true for x2 and x2‟. The production of either vector employs the full supplies of labour and of land; because of constant returns to scale, the production, with the same factor proportions as in Y, of αY=α(x 1,x2), 0<α<1, f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 102 will employ the fraction α of factor supplies, and the production, with the same factor proportions as in Z, of (1−α)Z=(1−α)(x1‟,x2‟) will employ the remainder of factor supplies, hence producing α(x1,x2)+(1−α)(x1‟,x2‟) is feasible. Therefore the segment connecting points Y and Z (cf. Fig. 3.19) belongs to the production possibility set, which is thus shown to be a convex set. This means that the PPF is concave. intl1.5 Y Z Fig. 3.19 What about strict concavity of the PPF? In the Edgeworth production box with the sides measuring the endowments of labour and of land, production vectors Y and Z correspond to points C and D which indicate the allocations of labour and of land between industries 1 and 2 which produce those two vectors; since in Y and Z there is efficiency, points C and D are on the Pareto set of the Edgeworth box. As we vary α from 1 to 0, the corresponding factor allocations move along the segment from C to D (cf. the red segment in Fig. 3.18); since C and D are not on the diagonal of the box because the production functions of the two goods are not the same, the factor allocations on this segment are not Pareto optimal, so it is possible to produce more, hence the PPF between Y and Z is not on the segment joining them but rather farther out, hence it is strictly concave . ■ The above discussion relies on CRS production functions. If there are increasing returns to an industry, the PPF may be at least in part strictly convex. 3.19. APPENDIX 2 - Intensive differential rent with three factors. Let us suppose corn is produced by three distinct inputs: homogeneous land, homogeneous labour, and corn-capital consisting only of seed-capital (wages are paid in arrears). Several productive methods are known, each one with fixed technical coefficients tj, l j, kj where kj is the corn-capital technical coefficient. Let the total amounts utilized e.g. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 103 of land T* and of labour L* be given, and let us see how intensive rent is determined for them as a function of the amount produced or, equivalently, of the amount of corn-capital utilized. Imagine a gradual increase of production. At first both land and labour are not fully utilized, w=β=0, and entrepreneurs choose the production method that maximizes the rate of profit i.e. the one with the smallest ki, let it be method 1. At a certain point either labour or land become fully employed; assume it is labour. Any further attempted increase in production creates an excess demand for labour, then w rises and r decreases until − we suppose − another method, let it be method 2, becomes as profitable as method 1. It must be k2>k1 and l 2< l 1. Land rent is still zero. Now suppose that as method 2 gradually replaces method 1, at a certain point land too becomes fully utilized (it must be t2>t1). Then any further attempted increase in production creates an excess demand for land, and land rent too starts rising, until − we suppose − a third method becomes as profitable as the other two. Suppose it is method 3. It must be: i) less profitable than methods 1 and 2 as long as land rent is zero, and ii) such that its cost increases less rapidly than for either of the other two methods when land rent rises. Condition ii) requires t3<t2 and t3<t1, which means that the introduction and extension of method 3 permits an increase in production from a given land utilization[ 102]. Condition i) can be visualized as follows. If land rent is zero, the equations price=cost for the three methods are (with corn the numéraire, and putting for brevity 1+r = ρ): 1=ρk1+w l 1 1=ρk2+w l 2 1=ρk3+w l 3 These generate three linear wage functions wj(ρ). With ρ in abscissa and w in ordinate, the horizontal intercepts are 1/kj and the vertical ones are 1/l j. Method 1 must have the greatest ρ when w=0, hence the least kj; w2(ρ) must cross w1(ρ) in the positive orthant, hence it must be k2>k1 and l2<l1; w3(ρ) must be below w1(ρ) for ρ greater than the value at which w1(ρ) and w2(ρ) intersect, otherwise method 3 would be adopted before method 2; hence k3>k1 and w3(ρ1,2)<w1(ρ1,2) where ρ1,2 is the value of 1+r where w1(ρ)=w2(ρ), thus in Fig. 3.20 w3(ρ) might e.g. be one of the broken straight lines. Because of condition ii), as land rent rises the three straight lines (now of the form 1=ρki+w l i+ti with β>0 given) move in toward the origin, but the one of method 3 moves in slower than the other two until the three lines cross in the same point. 102 . The introduction and extension of method 3 must leave labour fully utilized, and this will require a different change in the activity levels of the other two methods, see below. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 104 21 pt w w2(ρ) B w1(ρ) ρ C A Fig. 3.20 When (and if) such a point is reached, in addition to the previous conditions the following equations must hold, where q is the total quantity of corn produced, and qi is the quantity produced with method i: 1=(1+r)k1+w l 1+t1 1=(1+r)k2+w l 2+t2 1=(1+r)k3+w l 3+t3 l 1q1+ l 2q2+ l 3q3=L* t1q1+t2q2+t3q3=T* q1+q2+q3=q For given q, these are 6 equations in 6 variables: r, w, , q1, q2, q3 (103). If the solution is positive for a given q, it remains positive in a neighbourhood of q, which means that q can be increased within limits. The increase will require a change of all three q1, q2 and q3: q3 must certainly increase, and at least one of q1 or q2 must decrease, but whether only one and which one depends on the technical coefficients[104]. What is certain is that the increase in q obtained via changes in the activity levels of the three methods raises the utilization of corn- 103 . Cf. chapter 10 for a more general formalization that endogenously determines the methods utilized. 104 . If in the initial situation for each unit of land utilized by method 1 there are h units of land utilized by method 2, then an equiproportional contraction of the activity levels of methods 1 and 2, that leaves free one unit of land for use with method 3, causes a reduction of labour employment equal to (l1/t1+hl2/t2)/(1+h): if this is e.g. greater than l3/t3, labour unemployment would result, and q1/q2 must change in the direction of an expansion of the one, of methods 1 and 2, that has the higher labour-land ratio: which is precisely what economic incentives can be assumed to cause, because that is the method made relatively more convenient by the tendency of the wage rate to decrease owing to unemployment. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 105 capital. This is because, if from initial utilizations T*, L*, K* producing q it were possible to produce q+Δq>q while leaving the utilization of land and of labour unchanged and with a utilization of corn-capital K≤K*, then a method 4 would exist (the method with technical coefficients T*/(q+Δq), L*/(q+Δq), K/(q+Δq)) with lower unit costs than either of the three methods 1, 2, 3, and this is impossible because this method 4 could only be a combination of methods 1, 2, 3 in different proportions, but then it cannot have lower unit cost than the methods forming it. Thus now the q(K) curve (for given L and T) will have a first segment where only corn-capital earns a positive rental; along the second segment either labour or land earns a positive rental too; from the third segment on, all three factors earn positive rentals, and increases of the production of corn require that, while the last method introduced is gradually extended and the utilization of corn-capital increases, the proportion between the other two methods changes so as to keep both labour and land fully utilized. From the third segment on, the reader can check that if, as corn production is increased along a segment, the proportion between the other two methods does not change as required to keep both labour and land fully utilized, the excess demands thus created for labour and land will make it convenient to go to the correct proportion[ 105]. Since along a segment the joint cost of labour and land is unchanged, and since the value of the product equals the total cost, it is clear that the slope (where defined) of the q(K) curve equals (1+r), or, to look at the thing from the perspective of the marginalist economist, if (1+r) were less than the slope of the q(K) curve at the K equalling the supply of corn-capital, then there would be entrepreneurial profits and an incentive to expand production. 105 . Cf. the previous footnote. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 106 3.20. APPENDIX 3 Marginal indirect utility of factors with fixed technical coefficients [To be re-written] In the discussion of Pareto efficiency we introduced the notion of marginal indirect utility of a factor, for the case in which physical marginal products of factors are well defined. Can the marginal indirect utility of a factor be defined for cases where production requires fixed factor proportions? Will it equal the factor rental? Let us refer again for simplicity to an economy where labour and land, both in fixed supply, produced a number of consumption goods; but let us no longer restrict ourselves to the case of only two consumption goods. Thus let q1, ... , qn be the quantities of n different consumption goods produced in this economy, with good i produced with technical coefficients Li, Ti. Let us assume the economy is in equilibrium, with labour and land both fully utilized, and all the n consumption goods produced in the positive quantities in which they are demanded by consumers; the rate of wages and the rate of rent are such as to ensure such a composition of demand as will fully employ both factors. Suppose now that labour supply decreases by one very small unit. The composition of production must change so as to utilize one less unit of labour with the unchanged supply of land. With more than two consumption goods, the change in the composition of production is not uniquely determined, it will depend on how consumer choices change after the rise in the wage rate brought about by the decreased supply of labour. But now let us temporarily assume that income distribution and prices do not change. Then the total value of the produced goods necessarily decreases by one wage, because the value of products equals their cost. Let us attribute all the change in production to a single consumer who buys all consumption goods in positive quantity. Let us choose as unit for this consumer‟s utility her marginal utility of money. We know that in this way the price of each good equals its marginal utility. We can conclude that for this consumer the decrease in labour supply by one unit brings about a decrease in utility equal to the wage. composition, of this economy will be characterized by the full employment of labour and land, utilized in each industry in such proportions as to guarantee the tangency between isoquant and isocost, with the quantities produced of the several consumption goods determined by consumer choices. The logic is the same as for the corn-iron economy Let us refer again for simplicity to the economy where labour and land, both in fixed supply, produce corn and iron in separate industries, with fixed coefficients in each industry. In this economy, as long as L1/T1≠L2/T2, the vector (q1,q2) of quantities produced of corn (good 1) and iron (good 2) uniquely determines the vector (L,T) of quantities employed of labour and land, owing to the following system of two equations: [3.15] L = L1q1 + L2q2 [3.16] T = T1q1 + T2q2. (If both q1 and q2 are multiplied by the same number, the solution (L,T) is multiplied by the f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 107 same number, which shows that the ratio L/T depends only on the ratio q1/q2.) Conversely, if the vector (L,T) is given, the system univocally determines the quantities (q1,q2) that must be produced in order fully to employ (L,T)[106]. In the latter case q1 or q2 might come out negative; this will happen if the full employment of both factors is impossible, i.e. if the amounts of factors to be employed are such that their ratio L/T is not included in the interval between the factor employment ratios determined by producing only good 1 or only good 2, i.e. between L1/T1 and L2/T2; Assuming for simplicity that the full employment of both factors is possible, we can define an indirect factor utility function for a consumer, as follows. Let the consumer's utility function be u(q1,q2); the indirect factor utility function U(L,T) is that function such that U(L,T) = u(q1(L,T),q2(L,T)) where (q1,q2) are the quantities to be produced in order to employ (L,T). The marginal indirect utility of labour is then defined as ∂U/∂L = (∂u/∂q1)(∂q1/∂L)+(∂u/∂q2)(∂q2/∂L). From the expressions for q1(L,T) and q2(L,T) in footnote 86?? one obtains ∂q1/∂L = T2/(L1T2 – L2T1) ∂q2/∂L = – T1/(L1T2 – L2T1) Thus, indicating the partial derivatives of u(q1,q2) as MU1 and MU2: [3.17] ∂U/∂L = (MU1T2 – MU2T1)/(L1T2 – L2T1) . Let us now choose as utility unit, for each consumer, her marginal utility of income; then (§3.11.3) the price of each consumption good equals the marginal utility of that good for each consumer who demands it in positive quantity. Then we can re-write equation [3.17] as [3.18] ∂U/∂L = (p1T2 – p2T1)/(L1T2 – L2T1) . Suppose now the economy to be in equilibrium. Product prices obey, with w the wage rate and β the land rate: p1 = L1w + T1, p2 = L2w + T2 From these equations we obtain [3.19] w = (p1T2 – p2T1)/(L1T2 – L2T1) . So ∂U/∂L = w . (The same reasoning proves that ∂U/∂T = .) Thus the answer is 'yes' to both questions at the beginning of this Appendix. The reason we reach this result is because w equals the value, at the given prices, of the vector of variations in quantities produced associated with the employment of one more small unit of labour. It is necessarily so, because the value of production equals costs, and if one more unit of labour is employed, costs increase by w. (For marginal variations in the quantities produced, relative product prices can be taken as unchanging.) Having adopted the marginal utility of income as unit for utility, as long as we consider small changes these variations in quantities produced 106 By substitution one obtains q1(L,T)=(T2L-L2T)/(T2L1-T1L2) and q2(L,T)=(L1T-T1L)/(T2L1-T1L2). f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 108 cause variations in the utilities of consumers which can be aggregated and whose total value equals the change in the value of total production i.e. equals w (respectively, ). The increase in total utility accrues therefore to the consumer who supplies one more small unit of a factor, since she will increase the value of her purchases by exactly the increase in value due to her supply of one more unit of the factor. What she contributes (at the margin), she receives back, just like in the corn economy. Therefore even with no technological factor substitutability, substitutability in consumption allows the marginalist approach to argue that in equilibrium each factor receives its 'marginal product', to be intended here as a psychological marginal product, its marginal (indirect) utility. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 109 EXERCISES Exercise 1 In the neoclassical economy where corn is produced by labour and land according to a 'well- behaved' differentiable production function, assume that the supply of labour is backward-bending, and show that in this case there is no guarantee that the demand curve for land (assuming equilibrium on the labour market) is downward-sloping. Exercise 2 Derive graphically the form of the labour demand curve if labour and land produce only corn, when a) only a single fixed-coefficients method is known, b) two fixed-coefficient methods are known, which generate activity-analysis-type isoquants. Land is in fixed supply and fully employed. Exercise 3 Suppose labour and land produce two consumption goods, call them corn and iron, in two separate single-product industries, each one with fixed technical coefficients. Labour and land yield no direct utility. Show how one may derive indirect indifference curves for labour and land from the indifference curves for corn and iron of a consumer, and show that the wage-rent ratio will equal the marginal rate of substitution along the indirect indifference curve. Exercise 4. How would a marginalist economy analyze the determination of equilibrium in the following economy? Corn is the sole product and can be produced with labour and either one of two types of land, T and Z, according to different fixed technical coefficients: Lt*t→1 unit of corn Lz*z→1 unit of corn Factor supplies are rigid and neither land is sufficient fully to employ the labour supply. Write the system of competitive general equilibrium of this economy, discuss the possibility that not all three factors earn a positive rental, and conclude whether the difference between the rates of rent of the two types of land is or not a case of Ricardo‟s extensive differential rent. Exercise 5. Suppose that labour and land, inelastically supplied, produce corn and iron in two single- product fixed-coefficients industries. The labour-land ratio is higher in the production of iron. According to the marginalist approach, if there is a shift in tastes in favour of iron, what will be the effect on quantities produced and on income distribution? Exercise 6. Try to formalize the budget constraint behind the construction of the demand curve for labour f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 110 in the labour-land-corn economy (assume rigid factor supplies for simplicity), so as to obtain that when the real wage is higher than its equilibrium level, it is possible that there be disequilibrium only on the labour market. Exercise. In the Edgeworth box, are points of tangency of the indifference curves of the two consumers necessarily in the Pareto set? (Hint: consider non-convex indifference curves) Exercise 8. Suppose that labour and land produce corn, iron and silk in three separate industries, each industry having fixed coefficients, the labour-land ratio being lowest in the production of corn and highest in the production of silk. Explain the effects on income distribution, on the quantities produced and on the relative price of silk of a shift in preferences in favour of silk, according to the marginalist approach (assume that the composition of consumer demand shifts in favour of a good whose relative price decreases). Exercise: in the fixed-coefficients economy producing corn and iron with labour and land, by how much does labour employment vary for each unit of land shifted from the iron to the corn industry? Exercise: Prove that, with constant returns to scale, production is insufficient to pay all factors their total contribution to production as defined in fn. 64??. Exercise [to be moved to chapter 5]: Remembering that the marginal product of a factor is the partial derivative of the production function, prove that not all marginal products of the other factors necessarily increase when the amount of factor 1 increases in the following production function: q=2∙min(x1,x2)+x2αx31−α. Hint: you must first determine how factor 2 is allocated to the other two factors. Exercise. Prove that if G=F(L,K) has constant returns to scale, so does Y=F(L,K)−K. Exercise. Check the correctness of equation [3.3], §3.3.4, for the so-called Cobb-Douglas production function q=x1αx21−α, 0<α<1. Exercise 7 [to be moved to chapter 5]. Check that the production function q = x1αx21−α −βx1−γx2, with α, β, γ positive and less than 1, has constant returns to scale; show that it makes economic sense only if β+γ<1; then find the interval of factor proportions outside which one marginal product is negative. f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 111 Control questions: check that you are able to answer, orally or in writing, the following questions. ??TO BE TRANSLATED INTO ENGLISH 1. Mostrare, nel modello capital-cum-labour, che la teoria Ricardiana della rendita intensiva dà luogo a curve del prodotto di grano, come funzioni dell‟impiego di capital- cum-labour, indistinguibili da quelle neoclassiche della produttività totale di un fattore, per cui quanto va al capital-cum-labour è interpretabile come determinato dalla sua produttività marginale. 2. Spiegare perché è una fortuna per la teoria della concorrenza perfetta che valga il teorema dell‟esaustione del prodotto o teor. di Eulero. (suggerimento: chi vorrebbe fare l‟imprenditore, in concorrenza perfetta, se la somma dei pagamenti ai fattori secondo le loro produttività marginali in valore eccedesse il valore del prodotto? Chi accetterebbe di cedere i suoi fattori ad altri invece di fare lui stesso l‟imprenditore, se fosse vero l‟opposto?) 3. Nel derivare la curva di domanda di un fattore, quale legittimità ha il supporre pienamente occupati gli altri fattori? 4. Quando si deriva la curva di domanda di un fattore, se il prezzo di questo fattore è maggiore di quello di equilibrio, si ha disequilibrio sul solo mercato di questo fattore; non si ha disequilibrio su nessun altro mercato, contro - almeno in apparenza - la legge di Walras. Come mai? 5. Spiegare il meccanismo di sostituzione indiretta tra fattori, il suo ruolo nell‟impostazione marginalista, e perché può funzionare „male‟. 6. Individuare la differenza analitica di fondo tra impostazione classica e impostazione marginalista. 7. Dimostrare che se la funzione di produzione ha rendimenti di scala decrescenti, l‟ottimo impiego dei fattori implica profitti positivi per l‟imprenditore. 8. Cosa si intende per rendimenti di scala locali? 9. Dimostrare i due risultati importanti sulle funzioni omogenee: derivate parziali omogenee di grado k-1, e teorema di Eulero. 10. La legge di Walras implica o no che se su tutti i mercati meno uno vi è equilibrio, vi è necessariamente equilibrio anche sull‟ultimo mercato? 11. Discutere qualche caso in cui non esiste equilibrio. 12. Mostrare la possibilità di equilibri multipli, di cui alcuni instabili, nell‟economia neoclassica terra-lavoro-grano. 13. Dimostrare la riducibilità dell‟equilibrio generale di produzione e scambio a un equilibrio di puro scambio (sia pure indiretto) di fattori. 14. Spiegare perché nell‟impostazione marginalista l‟aumento della domanda di un bene ne fa in generale aumentare il prezzo normale anche nel lungo periodo. 15. Se nell‟economia neoclassica lavoro-terra-grano invece di assumere offerte fisse dei f petri Adv Micro ch 3 introduction to marginalism 18/04/2011 p. 112 fattori si assumono offerte dei fattori dipendenti dalle scelte dei consumatori, è possibile che la curva di domanda di un fattore (ricavata supponendo pienamente impiegato l‟altro) sia crescente? 16. Spiegare il meccanismo di sostituzione indiretta tra fattori, il suo ruolo nell‟impostazione marginalista, e perché può funzionare „male‟. 17. Dimostrare che se la funzione di produzione ha rendimenti di scala decrescenti, l‟ottimo impiego dei fattori implica profitti positivi per l‟imprenditore. 18. Cosa si intende per rendimenti di scala locali? 19. Mostrare la possibilità di equilibri multipli, di cui alcuni instabili, nell‟economia neoclassica terra-lavoro-grano. 20. Mostrare la riducibilità dell‟equilibrio di produzione e scambio a un equilibrio di puro scambio indiretto di fattori, in assenza di produzione congiunta. Parte II 21. Quali condizioni deve soddisfare un ottimo Paretiano non di frontiera in un‟economia con produzione e con utilità marginali e prodotti marginali ben definiti? 22. 23. Quando si deriva la curva di domanda di un fattore, se il prezzo di questo fattore è maggiore di quello di equilibrio, si ha disequilibrio sul solo mercato di questo fattore; non si ha disequilibrio su nessun altro mercato, contro - almeno in apparenza - la legge di Walras. Come mai? 24. Individuare la differenza analitica di fondo tra impostazione classica e impostazione marginalista.