VIEWS: 134 PAGES: 38 CATEGORY: Medications & Treatments POSTED ON: 6/4/2010
Welfare economics Lecture 3 Welfare Economics Welfare economics is essentially about judging the desirability of social outcomes. In this lecture we will introduce the normative concept of Pareto efficiency and two positive concepts; the core and general competitive equilibrium. Much of the lecture will be devoted to analyzing the relationships among these concepts. General equilibrium in pure exchange To begin our discussion of welfare economics we will consider a general equilibrium model of a pure (barter) exchange economy. In such an economy we will analyze n consumers trading in m markets. For now, there is no production, no prices, and no money. Define N = (1, 2, ... , n) -- the set of traders (consumers) in the economy. M = (1, 2, ... , m) -- the set of goods available in the economy. Assume that each good is of homogeneous quality. Two goods that are the same except for quality should be thought of as different goods. Furthermore, assume that each good is perfectly divisible. This last is not really necessary, but it makes the analysis easier. Endowments Instead of consumers purchasing a bundle of goods out of money income, consumers are endowed with a bundle of the goods in the economy. Let, wij -- i's endowment of good j. wi = (wi1, wi2, ... , wim) -- i's endowment bundle. W = (w1, w2, ... , wn) -- the economy's endowment. Allocations -- consumers take their endowments to market and trade with others to obtain another bundle of goods which they consume. Let, xij -- i's final demand (consumption ) of good j. xi = (xi1, xi2, ... , xim) -- i's consumption bundle. X = (x1, x2, ... , xn) -- an allocation. Feasibility Note that, ∑i∈N wij = w1 j + w2 j + . .. + wnj , is the aggregate amount of good j available in the economy. Furthermore, 3.1 Welfare economics ∑i∈N xij = x1 j + x 2 j + . .. + x nj is the aggregate consumption of good j. Definition An allocation X is a feasible allocation if and only if ∑i∈N wij ≥ ∑i∈N xij , ∀j ∈ M . Note that a feasible allocation is one in which the final consumption of each good does not exceed the amount available at the start of trade. Preferences Each individual is fully described by their preferences and endowments. Assume that each individual trader's preferences are fully described by a utility function. For person i, ui(xi) = ui(xi1, xi2, ... , xim). Most of the time we will assume that each utility function is strongly monotonic and strictly quasi-concave. Edgeworth Box Let's restrict our economy to two individuals (A, B) trading two goods (1, 2). To illustrate barter trade we construct what is known as an Edgeworth Box. To construct this box first consider our description (preferences and endowment points) of the two individuals. We do not need to draw budget lines since there are no prices in this economy. 2 2 wB2 1 u1 A uB wA2 0 0 uA uB Α wA1 1 Β wB1 1 Now, take B's indifference map, rotate it 180 degree and place it on top of A's indifference map so the endowment points coincide. 3.2 Welfare economics 2 * x B1 wB1 Β 1 0 uB P x* * xB2 wA2 + wB2 A2 wA2 wB2 W u0 A 1 Α wA1 x* A1 2 wA1 + wB1 Remarks 1) The economy's endowment is W = [wA, wB] = [(wA1, wA2), (wB1, wB2)]. It is also called the no-trade allocation. The indifference curves that pass through W gives 0 us the utility levels in the absence of trade ( u0 , uB ) . If these individuals are to A trade with each other they have to do at least as well as this. 2) The height of the box is the amount of good 2 available in the economy, while the width of the box is the amount of good 1 available. 3) Any point in the box or on the boundary represents a feasible allocation. To see ∗ ∗ ∗ this take point P = [ x ∗ , x B ] = [( x ∗ , x ∗ ), ( x B1 , x B2 )], and note that A A1 A2 ∗ x ∗ + x B1 = wA1 + wB1 A1 (supply is equal to demand for good 1) and ∗ x ∗ + x B2 = wA2 + wB2. A2 (supply is equal to demand for good 2) Trade in the Edgeworth Box Consider the following: 2 Β 1 Q u2 A P u1 W A Α u0 A 1 0 uB B 2 u1 u B 2 3.3 Welfare economics W is the endowment point again. Trade between the two individuals will take place when both are made better off. For example, they might trade to allocation P. However, they won't stop trading at P because there are still gains from trade to be had. At allocation Q, all gains from trade are exhausted. Notes about Q 1) It is a feasible allocation. 2) There is no other feasible allocation that will make one of them better off without harming the other. 3) An indifference curve of A's is tangent to an indifference curve of B's. 4) Both individuals prefer allocation Q to allocation W. Pareto Efficient Allocations Definition 1: A Pareto efficient allocation is a feasible allocation from which there is no way to make at least one individual better off without harming another. ( 0 0 ) Definition 2: A feasible allocation X0 = x10 , x2 ,..., xn is Pareto efficient if and only 1 ( 1 1 ) if there is no other feasible allocation X1 = x1 , x 2 , ..., x n such that ui(X1) ≥ ui(X0) ∀ i ∈ N [no one is harmed by moving to X1] and ui(X1) > ui(X0) for at least one i ∈ N. [at least one person is better off] Remarks 1) The two definitions are equivalent if one accepts the assumption that preferences can be represented by a monotonic utility function. 2) In the previous graph, allocation Q is Pareto efficient, while P and the endowment W are not. 3) These definitions require nothing about tangent indifference curves. Thus, tangent indifference curves is sufficient for Pareto efficiency, but not necessary. More on this later. 4) Notice that there is nothing about fairness or equity in these definitions. More later. Pareto Efficiency as a Constrained Optimization Problem Consider a pure-exchange economy with two people (A, B) and two goods (1, 2), and the constrained maximization problem of choosing an allocation X = [xA, xB] = [(xA1, xA2), (xB1, xB2)] to solve max uA(xA1, xA2) 0 s.t. uB(xB1, xB2) = uB i) xA1 + xB1 = wA1 + wB1 ii) xA2 + xB2 = wA2 + wB2. iii) 1) 3.4 Welfare economics What we are doing here is trying to find a feasible allocation [constraints ii) and iii)] to make A as well off as possible (the objective) without harming B [constraint i)]. The solution to this problem will be a Pareto efficient allocation. The Lagrange equation for 1) is L = uA(xA1, xA2) + µ[uB(xB1, xB2) - uB ] 0 + λ1[wA1 + wB1 - xA1 - xB1] + λ2[wA2 + wB2 - xA2 - xB2]. The first-order conditions for an interior solution to 1) are ∂L ∂ uA = − λ1 = 0 2) ∂ x A1 ∂ x A1 ∂L ∂ uA = − λ2 = 0 3) ∂ x A2 ∂ x A2 ∂L ∂ uB = µ − λ1 = 0 4) ∂ x B1 ∂ x B1 ∂L ∂ uB = µ − λ2 = 0 5) ∂ xB2 ∂ x B2 ∂L ∂L ∂L = = = 0 6) ∂µ ∂ λ1 ∂ λ2 ∂ u A ∂ x A1 λ The first-order conditions 2) and 3) imply = 1. 7) ∂ u A ∂ x A2 λ2 ∂ uB ∂ x B1 λ The first-order conditions 4) and 5) imply = 1. 8) ∂ uB ∂ x B 2 λ2 ∂ u A ∂ x A1 ∂ uB ∂ x B1 λ 7) and 8) imply = = 1. 9) ∂ u A ∂ x A2 ∂ uB ∂ x B 2 λ2 ∂ ui ∂ xij Recall that is the marginal rate of substitution between goods j and k for ∂ ui ∂ xik person i. It is the slope of an indifference curve of i's. In terms of valuation: The marginal rate of substitution is the subjective marginal value i places on consumption of good j in terms of good k. Equation 9) states that at an interior solution to the optimization problem 1), the slope of an indifference curve for A is equal to the slope of an indifference curve for B. In terms of value 9) says that at an interior solution, A's subjective marginal valuation of good 1 in terms of good 2 must be equal to B's subjective marginal valuation of good 1 in terms of good 2. 3.5 Welfare economics "Equal slopes" and feasibility require a tangency. One way to think about the solution to 0 1) is to choose the highest indifference curve for A that satisfies the constraint uB . 2 B 1 (x * A1 , x* 2 , x*1, x* 2 A B B ) u2 A u1 A A u0 A 1 0 2 uB The Set of Efficient Allocations Definition: A contract curve is the set of Pareto efficient allocations. Actually, the "contract curve" may be a space or a point or something other than a continuous function. In the optimization problem above, the contract curve is found by 0 varying the uB constraint. 2 B 1 u2 A u1 A u0 A A 1 2 uB u1 B 0 uB 2 3.6 Welfare economics Core Allocations Definition (blocking): A coalition S ⊆ N can block an allocation X0 if there is some other allocation X1 such that ∑ i ∈S xij = 1 ∑ i ∈S wij , ∀j ∈ M i) ui(X1) ≥ ui(X0) ∀ i ∈ S ii) ui(X1) > ui(X0) for at least one i ∈ S. iii) The first requirement for blocking i) is that the allocation X1 is feasible for the coalition S. That is, the members of S can 'afford' xi1 ( ) . The requirements ii) and iii) are that at i ∈S least one member of S is better off and no member is harmed. Definition (core): A feasible allocation X is a core allocation if it cannot be blocked by any coalition S ⊆ N. In a core allocation no subgroup can break away from the rest of the economy and trade only among themselves and be better off. Notice that the definition for blocking is like the definition for a Pareto efficient allocation. In fact, the similarity is very real. Proposition: Every core allocation is also a Pareto efficient allocation. Proof: Toward a contradiction assume that an allocation X0 is a core allocation but is not Pareto efficient. If X0 is not Pareto efficient there exists another feasible allocation X1 such that ui(X1) ≥ ui(X0) ∀ i ∈ N and ui(X1) > ui(X0) for at least one i ∈ N. But then, X0 can be blocked by S = N. Hence, X0 could not have been a core allocation. This contradiction proves the proposition. The following graph illustrates the set of core allocations in our two-person, two-good economy. 3.7 Welfare economics 2 Β 1 P contract curve Q core allocations W u0 A Α 1 0 uB 2 Notice that the allocation P is efficient but is not in the core because it can be blocked by person B (he can choose to consume his own endowment). Thus, the reverse of the proposition is not true. That is, not every Pareto efficient allocation is a core allocation. Allocation Q is a core allocation because neither A or B, or both together can block it. A would be worse off consuming her own endowment as would B. Together they cannot block Q because they can't move from it without harming one or the other or both. Notice that core allocations depend on the initial endowments, while Pareto efficient allocations do not. Positive and normative concepts The concept of Pareto efficiency is a normative concept. That is, it is a notion of 'what ought to be'. This definition of efficiency is not derivable from some objective theory of economic behavior, so, like all normative concepts, there is a subjective value system underlying its use. Be aware of this. The core on the other hand is an equilibrium concept. It is reasonable to require that an economic equilibrium be stable in the sense that no coalition can block it using their own resources. As an equilibrium concept, the definition of the core is also not derivable from objective economic theorizing. However, the notion itself is devoid of value statements. 3.8 Welfare economics Competitive exchange So far we have considered barter exchange to define efficient and core allocations. Now we want to look at trade governed by competitive pricing. Once we have characterized the outcomes of competitive trading (i.e., competitive general equilibrium) we will analyze these outcomes in terms of efficiency and the core. Assume: 1) A general equilibrium model as before 2) Again there is no production, no storage, and no money. 3) Each good has a price per unit which traders take as given (the basic assumption of competitive behavior). 4) Each trader has a strongly monotonic and strictly quasi-concave utility function. Let: N = (1, 2, ... , n) -- the set of traders (consumers) in the economy. M = (1, 2, ... , m) -- the set of goods available in the economy. pj -- the competitive price of the jth good The budget constraint for any trader i is p1xi1 + p2xi2 + ... + pmxim ≤ p1wi1 + p2wi2 + ... + pmwim or ∑p x j ∈M j ij ≤ ∑p w j ∈M j ij . Remarks 1) pjxij is the market value of i's consumption of good j. Therefore, ∑Mpjxij is the market value of i's consumption bundle, or i's expenditure on consumption. 2) pjwij is the market value of i's endowment of good j. Therefore, ∑Mpjwij is the market value of i's endowment bundle. 3) Since we assume that utility functions are strongly monotonic we can replace '≤' with '='. We will do this from now on. The assumption of competitive behavior also requires that traders maximize their utility subject to their budget constraint. Thus, each i ∈ N chooses a consumption bundle (xi1, xi2, ... , xim) to solve max ui(xi1, xi2, ... , xim). s.t. ∑p x j ∈M j ij = ∑p w j ∈M j ij 10) 3.9 Welfare economics The Lagrange equation for 10) is Li = ui(xi1, xi2, ... , xim) + λi[∑Mpjwij - ∑Mpjxij]. 11) The first-order conditions for an interior optimum are ∂ Li ∂ ui = − λi pj = 0, ∀ j ∈ M 12) ∂ xij ∂ xij ∂ Li = 0. 13) ∂ λi 12) implies that for any two goods h and k from M, ∂ ui ∂ xih p = h , ∀ h and k ∈ M, h ≠ k. 14) ∂ ui ∂ xik pk This is the result that at an individual, interior optimum, the marginal rate of substitution between any two goods must be equal to the price ratio. Note that since i was chosen arbitrarily, 14) must be true for each i ∈ N. The optimal consumption bundle for i is illustrated for the case M = (1, 2) in the first graph below. The second graph illustrates the comparative static of an increase in the price of good 1. 2 p1 ' 2 slope = − , p1′ > p1 p2 ∂ ui ∂ xi1 p1 = ∂ ui ∂ xi 2 p2 xi∗∗ ( xi∗ = xi∗1 , xi∗2 ) xi∗ wi2 ui0 wi2 ui1 i wi1 1 i wi1 1 3.10 Welfare economics How are prices determined? -- The Walrasian Auctioneer To mimic actual market operations we add a player whose role is as follows: It calls out a set of prices. Each trader tells the auctioneer its optimal consumption bundle at those prices. If quantity demanded is not equal to the amount available for each good, the auctioneer adjusts prices until all markets clear. When all the markets clear, the traders consume their final consumption bundles. That all markets clear is another requirement for a competitive equilibrium. That is, we require that final consumption in a competitive equilibrium be feasible: ∑x i ∈N ij = ∑w i ∈N ij , ∀ j ∈ M. 15) An illustration Consider the graph of a two-person (A, B), two-good (1, 2) economy below. Suppose 0 that the auctioneer calls out initial prices ( p10 , p2 ) which results in the allocation 0 0 0 [( x 0 1 , x 0 2 ), ( x B1 , x B 2 )] . At ( p10 , p2 ) neither market clears. In fact, A A x 0 1 + x B1 < w A1 + w B1 A 0 (excess supply of good 1) and x 0 2 + x B 2 > w A2 + w B 2 A 0 (excess demand for good 2). In this situation, to move toward market clearing, the auctioneer should decrease the relative price of good 1 and increase the relative price of good 2. That is, in the next round the auctioneer should call out ( p1 , p2 ) such that p1 p2 < p10 p2 . 1 1 1 1 0 At prices (p1*, p2*) and allocation X*, each consumer is optimizing on their budget sets and both markets clear. This set of prices and allocation is a competitive equilibrium. 2 u0 A u1 A 0 xB1 Β 1 x0 A2 0 xB 2 X* W ∗ ∗ slope = − p1 p2 0 0 slope = − p1 p2 Α x01 A 1 u1 B 0 uB 2 3.11 Welfare economics Definition: A competitive (Walrasian) equilibrium in a pure exchange economy is a set of prices p = (p1, p2, ... , pm) and an allocation X* = (x1*, x2*, ... , xn*) such that A) For each i ∈ N, xi* = (xi1*, xi2*, ... , xim*) is the solution to max ui(xi1, xi2, ... , xim). s.t. ∑p x j ∈M j ij = ∑p w j ∈M j ij (utility maximization on a budget set) 10) B) For each j ∈ M, ∑x i ∈N ∗ ij = ∑w i ∈N ij . (feasibility) 15) In the graph above, you noticed that the competitive equilibrium allocation X* is also a Pareto efficient allocation. It is also a core allocation. It turns out that these are general results. The First Theorem of Welfare Economics ∗ ∗ ∗ If [( x1 , x 2 , . .. , x n ), ( p1 , p2 , . .. , pm )] ≡ [X*, p] is a competitive equilibrium, then X* is a core allocation. By implication it is also a Pareto efficient allocation. Proof: To prove the theorem we use the following facts: Fact 1 Let [X*, p] be a competitive equilibrium. If ui(xi') > ui(xi*) for some xi', then ∑ p j xij ' > ∑ p j wij = ∑ p j xij . ∗ j ∈M j ∈M j ∈M That is, if xi' is strictly preferred to xi* by i, it must not be affordable for i. To show this assume that i can afford xi'. Then, since he prefers xi' to xi*, he would have chosen xi' instead of xi*. But then, X* could not have been a competitive equilibrium allocation. This contradiction establishes the result. Fact 2 Let [X*, p] be a competitive equilibrium. If ui(xi') ≥ ui(xi*) for some xi', then ∑ p j xij ' ≥ ∑ p j wij = ∑ p j xij ∗ . j ∈M j ∈M j ∈M That is, if xi' is weakly preferred to xi* by i, it cannot cost less than xi*. 3.12 Welfare economics Toward a contradiction of the welfare theorem, suppose that [X*, p] is a competitive equilibrium but X* is not a core allocation. From the definition of the core, if X* is not a core allocation there exists another allocation X1 and a blocking coalition S ⊆ N for which i) ∑ i ∈S xij = 1 ∑ i ∈S wij , ∀j ∈ M . -- the members of S must be able to achieve their part of X1 with their own resources. ii) ui(xi1) ≥ ui(xi*), ∀ i ∈ S. -- no member of S strictly prefers X* to X1. iii) ui(xi1) > ui(xi*), for at least one i ∈ S. -- at least one person in S strictly prefers X1 to X*. Fact 2 and ii) imply that iv) ∑p x j ∈M j 1 ij ≥ ∑p w j ∈M j ij , ∀ i ∈ S. Fact 1 and iii) v) ∑p x j ∈M 1 j ij > ∑p w j ∈M j ij , for at least one i ∈ S. Summing iv) and v) over the members of S yields ∑ ∑p x i ∈S j ∈M j 1 ij > ∑ ∑p w i ∈S j ∈M j ij , which can be written as p1 ∑ x i11 + p2 ∑ x i12 +...+ pm ∑ x im > p1 ∑ wi1 + p 2 ∑ wi 2 + ...+ p m ∑ wim . 1 i ∈S i ∈S i ∈S i ∈S i ∈S i ∈S This can rewritten again as ∑ p ∑ x − ∑ w j ∈M j i ∈S 1 ij i ∈S ij > 0. But, since prices are positive, ∑ xij − ∑ wij 1 ≠ 0 for some j ∈ M. i ∈S i ∈S This implies that X1 cannot be a feasible allocation for S -- it violates i). We conclude that an allocation like X1 cannot exist. But this contradicts our assertion that X* is not a 3.13 Welfare economics core allocation. Therefore, X* must be a core allocation. Furthermore, since all core allocations are Pareto efficient, X* must be Pareto efficient. Q.E.D. Remarks a) The theorem implies that competitive behavior will lead to a desirable (in the Pareto sense) social outcome. b) Unfortunately the theorem does not hold if the assumptions of competitive trading are not met (i.e, no government intervention, no externalities, no market power, etc.). c) Still we haven't said anything about fairness. There should be no presumption that competitive trading will lead to a fair allocation. However, the Second Welfare Theorem reveals that we can induce a fair (by some criteria) and efficient allocation. The Second Welfare Theorem Suppose that all traders have strongly monotonic and strictly quasi-concave utility functions. Let X* be an efficient allocation such that xij* > 0, ∀ i ∈ N and ∀ j ∈ M. Then there exists a set of prices p = (p1, p2, ... , pm) and an assignment of endowments W = (w1, w2, ... , wn) such that (X*, p) is a competitive equilibrium. Notes a) The assumption that xij* > 0, ∀ i ∈ N and ∀ j ∈ M can be relaxed. b) In a sense, the theorem says that if you let me choose prices and endowments I can guarantee that any efficient allocation of your choice will be a competitive equilibrium allocation. To illustrate the Second Welfare Theorem, consider trade in an Edgeworth box. 2 Β 1 u1 A u1 B X* W0 u0 A 0 uB slope = − p1 p 2 W1 Α 1 2 3.14 Welfare economics Suppose that W0 is the initial allocation of endowments. Suppose we think that trade between the two individuals will lead to an unfair allocation, and we prefer to see them trade to the 'fair' and efficient allocation X*. The theorem guarantees that if we pick the appropriate prices and rearrangement of endowments, X* will result from competitive trading. The appropriate prices here are (p1, p2) so that p1/p2 = MRSA(X*) = MRSB(X*). Now pick an endowment W1 so that the budget line in the Edgeworth box is the common tangent line at X*. Now if A and B start at W1 and trade at prices (p1, p2) they will trade to X*. Remarks a) The welfare theorems are important because they let us conclude that if we believe that people trade in competitive situations, any complaints about the price system can be reduced to issues of equity. Furthermore, issues of equity can be addressed by rearranging endowments. b) In the real-world, we can rearrange endowments by what are called lump-sum transfers. Lump-sum transfers are tax/subsidy policies that don't distort competitive prices. Unfortunately, there aren't many types of transfers that don't distort prices. c) Though the welfare theorems are quite powerful, they do depend heavily on the assumptions of competitive behavior. d) There is another problem that we can't address. What criteria will we use to determine what is and what is not fair? Furthermore, what rule do we use to choose among fairness criteria? Shadow prices and competitive prices The purpose of this section is to show that the Lagrange multipliers from the constrained optimization problem that characterizes efficient allocations coincide with competitive market prices. Proposition 1: Suppose all traders have strongly monotonic and strictly quasi-concave utility functions. Then, if (X*, p) is a competitive equilibrium with xij* > 0, ∀ i ∈ N and ∀ j ∈ M, ∂ ui ∂ xih p = h , ∀ h and k ∈ M, and ∀ i ∈ N. ∂ ui ∂ xik pk Proof: Recall that if (X*, p) is a competitive equilibrium each i ∈ N will choose a consumption bundle (xi1, xi2, ... , xim) to solve max ui(xi1, xi2, ... , xim). s.t. ∑p x j ∈M j ij = ∑p w j ∈M j ij 10) 3.15 Welfare economics Recall that the necessary conditions for an interior solution to this problem include ∂ ui ∂ xih p = h , ∀ h and j ∈ M. 14) ∂ ui ∂ xik pk Note that 5) must be true for each i ∈ N. Q.E.D. Proposition 2: Continue to assume that all traders have strongly monotonic and strictly quasi-concave utility functions. Then, if X* is a Pareto efficient allocation with xij* > 0 ∀ i ∈ N and ∀ j ∈ M, ∂ ui ∂ xih λ = h , ∀ h and j ∈ M, and ∀ i ∈ N, ∂ ui ∂ xij λj where λk is the Lagrange multiplier for the feasibility constraint on the kth good. Proof: If X* is an efficient allocation, it solves the following optimization problem for each i ∈ N: max ui(xi1, xi2, ... , xim) ( xij , ∀j ∈M , ∀i ∈N ) s.t. uk(xk1, xk2, ... , xkm) = uk , ∀ k ∈ N, k ≠ i 0 ∑x i ∈N ij = ∑w i ∈N ij , ∀ j ∈ M. For an arbitrary i ∈ N, the Lagrange equation is Li = ui(xi1, xi2, ... , xim) + ∑ k ∈N , k ≠ i [ µ k uk ( xk ) − uk + 0 ] ∑ λ ∑ w − ∑ x . j ∈M j i ∈N ij i ∈N ij The first-order conditions are ∂ Li ∂ ui i) = − λ j = 0 , ∀ j ∈ M. ∂ xij ∂ xij ∂ Li ∂ uk ii) = µk − λ j = 0 , ∀ j ∈ M, ∀ k ∈ N, k ≠ i. ∂ x kj ∂ x kj ∂ Li ∂ Li iii) = = 0 , ∀ j ∈ M, ∀ k ∈ N, k ≠ i. ∂λj ∂ µk From i) and ii) 3.16 Welfare economics ∂ ui ∂ xih λ = h , ∀ h and j ∈ M and ∀ i ∈ N. Q.E.D. 16) ∂ ui ∂ xij λj Now, Propositions 1 and 2 imply that ph λ = h , ∀ h and j ∈ M. pj λj Thus, we can interpret the Lagrange multipliers from the problem of finding efficient allocations as the market prices that would emerge from competitive trading. Welfare maximization Assume the existence of a social welfare function. This is a mapping U: Rn→R such that U(u1, u2, ... , un) gives us the collective welfare of N = (1, 2, ..., n) for any distribution of private utility levels (u1, u2, ... , un). Typically we assume that the social welfare function is increasing in each private utility, That is, ∂ U/∂ ui > 0, for all i ∈ N. Now suppose we have the worthwhile goal of maximizing social welfare. How does the solution to this optimization relate to Pareto efficiency? Proposition: If an allocation X* maximizes U, X* is efficient. Proof: Toward a contradiction of the proposition, suppose that X* maximizes social welfare but is not efficient. If X* is not efficient, there exists a feasible allocation X0, such that i) ui(xi0) ≥ ui(xi*), ∀ i ∈ N and ii) ui(xi0) > ui(xi*), for at least one i ∈ N. But, since U is monotonically increasing in each ui, i) and ii) imply U[u1(x10), u2(x20), ... , un(xn0)] > U[u1(x1*), u2(x2*), ... , un(xn*)]. Therefore, X* could not have maximized U. This contradiction proves the proposition. Q.E.D. Now, consider the problem of maximizing social welfare subject to the feasibility constraints: 3.17 Welfare economics max U[u1(x1), u2(x2), ... , un(xn)] ( xij , ∀j ∈M , ∀i ∈N ) s.t. ∑x i ∈N ij = ∑w i ∈N ij , ∀ j ∈ M. The Lagrange equation for this problem is L = U[u1(x1), u2(x2), ... , un(xn)] + ∑ λ ∑ w − ∑ x j ∈M j i ∈N ij i ∈N ij . Assuming an interior solution, the first-order conditions are ∂L ∂ U ∂ ui i) = ∗ − λ j = 0, ∀ j ∈ M, and ∀ i ∈ N. ∂ xij ∂ ui ∂ xij ∂L ii) = 0 , ∀ j ∈ M. ∂ λj From i) we have ∂ ui ∂ xih λ = h , ∀ h and j ∈ M, and ∀ i ∈ N. ∂ ui ∂ xij λj Since this holds for every i, ∂ ui ∂ xih ∂ uk ∂ xkh = , ∀ h and j ∈ M, and ∀ i and k ∈ N. ∂ ui ∂ xij ∂ uk ∂ xkj These marginal conditions are the same as those for Pareto efficient allocations. Remarks a) Though an allocation that maximizes social welfare is efficient, an efficient allocation does not necessarily maximize a particular social welfare function. This implies that though we may have an efficient allocation, there might be another that gives us greater social welfare. In such a case, we would be able to make society better off in aggregate, but doing so would harm someone. b) However, under certain conditions, it can be shown that an efficient allocation always maximizes some social welfare function. c) There are real problems with assuming that a social welfare function exists. But, at times they are convenient to use. 3.18 Welfare economics General equilibrium and the welfare theorems with production We have examined the relationships among Pareto efficient allocations, core allocations, and competitive equilibria in pure exchange economies. Now we introduce production into the economy. Let H = (1, 2, ... , h) -- the set of firms in the economy. M = (1, 2, ... , m) -- the set of goods available in the economy. p = (p1, p2 , ... , pm) -- constant (competitive) prices. A production plan for the kth firm is yk = (yk1, yk2 , ... , ykm). If ykj > 0, firm k produces good j as an output ykj < 0, firm k uses good j as an input Note that the goods set M includes outputs for consumption and inputs to production. An aggregate production plan for the entire economy is y = (y1, y2 , ... , yh). A production possibilities set for the kth firm is a collection of all production plans that are technically feasible. Denote the production possibilities set of the kth firm as Yk. Assume that firms are competitive, and that they choose a production plan (yk) to maximize profit (πk), taking the vector of prices (p) and the production possibilities set (Yk) as given. Here, πk = p1yk1 + p2yk2 + ... + pmykm = ∑p y j ∈M j kj Note that if good j is an input pjykj < 0 (a cost to the firm), and if good j is an output pjykj > 0 (a source of revenue for the firm). The kth firm's optimization problem is to choose a feasible production plan yk to solve max πk = ∑p y j ∈M j kj s.t. yk ∈ Yk 17) 3.19 Welfare economics The solution to 17) is a production plan (y ∗ k1 ) , y k 2 , . .. , y km = ( y k 1 ( p), y k 2 ( p), . .. , y km ( p)) . ∗ ∗ ∗ In vector notation yk = yk(p). Note that if ykj(p) < 0, it is an input demand function for good j, and if ykj(p) > 0, it is a supply function for good j. An aggregate production plan in which each firm chooses inputs and outputs to maximize profit is y(p) = [y1(p), y2(p) , ... , yh(p)]. Proposition: An aggregate production plan y(p) maximizes aggregate profit ∑k∈Hπk if and only if each firm's production plan yk(p) maximizes its individual profit πk. [For this proposition and its proof see Varian, pg. 339]. Note: For a competitive equilibrium we are going to require that each firm maximizes profit. Sometimes it is more convenient to maximize aggregate profit. The proposition tells us that there is no difference between the two operations. Consumers Recall that in a competitive exchange economy we required that an equilibrium allocation X* = (x1*, x2*, ... , xn*) satisfy max ui(xi*) s.t. ∑ p j xij = ∗ ∑ p j wij , ∀ i ∈ N. j ∈M j ∈M In an economy with production there is a complication. What do we do with profit? Assume that each firm is owned by consumers (not necessarily all consumers). Suppose that if i is an owner of firm k, she is entitled to a share sik of its profit. Assume 1) Each firm is completely owned by individuals so that ∑s i ∈N ik = 1. 2) The shares sik are fixed, and hence, are not traded. In this model there is no stock market although we could have included one. 3.20 Welfare economics Individual i's share of the profit from firm k is sikπk = sik ∑ p j y kj ( p) . j ∈M Individual i's income from owning shares in a number of firms is ∑s k ∈H ik πk = ∑s ∑ p y k ∈H ik j ∈M j kj ( p) . Thus, i's budget constraint in this economy with production is ∑p j ∈M j x ij = ∑p w j ∈M j ij + ∑s ∑ p k ∈H ik j ∈M j y kj ( p) . 18) We will require that in a competitive equilibrium with production, each i maximizes utility subject to 18). Efficient allocations and competitive equilibria Recall: X denotes a consumption allocation. y denotes an aggregate production plan. The pair (X, y) will now be called an allocation. An allocation (X, y) is feasible if and only if ∑x i ∈N ij = ∑w i ∈N ij + ∑y k ∈H kj , ∀ j ∈ M. An allocation (X, y) is Pareto efficient if and only if there is no other allocation (X0, y0) such that i) ∑x i ∈N 0 ij = ∑w i ∈N ij + ∑y k ∈H 0 ij ,, ∀ j ∈ M. [(X0, y0) is feasible] ii) ui(xi0) ≥ ui(xi), ∀ i ∈ N. [no one is harmed by moving to (X0, y0)] iii) ui(xi0) > ui(xi), for some i ∈ N. [at least one person is better off at (X0, y0)] 3.21 Welfare economics A competitive equilibrium is a triple (X, y, p) such that i) Each production plan yk ∈ y = (y1, y2 , ... , yh) is the solution to max πk = ∑p y j ∈M j kj s.t. yk ∈ Yk, ii) Each consumption bundle xi* ∈ X is the solution to max ui(xi*) s.t. ∑p j ∈M j x ij = ∑p w j ∈M j ij + ∑s ∑ p k ∈H ik j ∈M j y kj ( p) iii) The consumption allocation X is feasible, so that ∑x i ∈N ij = ∑w i ∈N ij + ∑y k ∈H kj ∀ j ∈ M. The First Welfare Theorem If (X, y, p) is a competitive equilibrium, then (X, y) is a core allocation. It is also a Pareto efficient allocation. [For the proof, see Varian, pp. 345-346]. The Second Welfare Theorem Suppose that (X, y) is a Pareto efficient allocation with xij > 0, ∀ i ∈ N and ∀ j ∈ M. Assume further that each consumer has a strongly monotonic and strictly quasi-concave utility function, and each firm has a closed and convex production possibilities set. Then with an appropriate choice of endowments and profit shares, there exists a set of prices p such that (X, y, p) is a competitive equilibrium. Note: For the second theorem to hold we need each firm's production possibilities set Yk to be closed and convex. A set is convex if every point on a line segment joining two points in the set is also in the set. A set is closed if the boundaries of the set are included in the set. A concave production function will imply a closed and convex production possibilities set. However, a quasi-concave production function may not. If there is a region of increasing returns to scale, the production possibilities set will not be convex. 3.22 Welfare economics General equilibrium and efficiency with production: The calculus approach Now we are going to characterize efficient allocations and competitive equilibria with the marginal conditions from a series of optimization problems. We will derive the marginal conditions for 1) technical efficiency, 2) Pareto efficiency, and 3) competitive equilibria. In order to keep things simple we will not fully specify the economy in as much detail as we did above Assume Two consumers, A and B. Two consumption goods, 1 and 2. Two inputs into production, L and K. Technical efficiency Assume production functions that are strictly concave: x1 = f(L1, K1), x2 = g(L2, K2). The resource constraints are L = L1 + L2, K = K1 + K2, where L and K are the aggregate amounts available in the economy. We will ignore the question of where they come from and who owns them. To characterize technical efficiency we choose (L1, L2, K1, K2) to solve the following: max x1 = f(L1, K1) 0 s.t. x2 = g(L2, K2) L = L1 + L2 K = K1 + K2. The Lagrange equation is Φ = f(L1, K1) + λ[g(L2, K2) - x2 ] + λL[L - L1 - L2] + λK[K - K1 - K2]. 0 The necessary conditions are ∂ Φ ∂ L1 = f L − λ L = 0 19) ∂ Φ ∂ K1 = f K − λ K = 0 20) ∂ Φ ∂ L2 = λg L − λ L = 0 21) ∂ Φ ∂ K 2 = λg K − λ K = 0 22) 3.23 Welfare economics ∂ Φ ∂ λ = ∂ Φ ∂ λL = ∂ Φ ∂ λK = 0 23) The first-order conditions 19) through 22) imply fL g λ = L = L. 24) fK gK λK That is, the ratio of the marginal products must be equal for all goods. Recall that the ratio of marginal products is called the marginal rate of technical substitution. Any production plan (x1, x2, L1, L2, K1, K2) that satisfies 24) and the resource constraints L = L1 + L2 and K = K1 + K2, is technically efficient. Production possibilities frontier Note that the first-order conditions 19) through 23) implicitly define L1 = L∗ (x2, L, K) 1 L2 = L∗ (x2, L, K) 2 K1 = K1∗ (x2, L, K) ∗ K2 = K2 (x2, L, K) λ = λ∗ (x2, L, K) λL = λ∗L (x2, L, K) and λK = λ∗K (x2, L, K). The indirect objective function is ∗ x1 = f( L∗ , K1∗ ) = x1 (x2, L, K). 1 This is the production possibilities frontier (PPF). It gives us the maximum possible production of x1 for each level of x2, given the availability of L and K. From the Envelope Theorem, we have the slope of the PPF ∂ x1∗ ∂ Φ∗ = = − λ∗ . ∂ x2 ∂ x2 And from the first-order conditions we have fL f − λ∗ = − = − K < 0. gL gK 3.24 Welfare economics Since ∂ x1 ∂ x 2 = − λ∗ < 0, the PPF is downward sloping. The slope of the PPF is ∗ sometimes called the marginal rate of transformation. If you check the second derivative ∗ of x1 (x2, L, K).you will realize that it is strictly concave if f(L1, K1) and g(L2, K2) are both strictly concave. x1 ∗ ∂ x1 f f = − L = − K ∂ x2 gL gK PPF x2 Remarks 1) The production possibilities frontier collects all the combinations of the production of the two goods that are technically efficient. 2) Technical efficiency is a necessary condition for Pareto efficiency Pareto efficiency As before, to find the set of Pareto efficient allocations we choose an allocation (xA1, xA2, xB1, xB2) to solve max uA(xA1, xA2) 0 s.t. uB(xB1, xB2) = uB i) xA1 + xB1 = x1 ii) xA2 + xB2 = x2 iii) ∗ x1 (x2, L, K) = x1 iv) Constraints ii), iii) and iv) are 'feasibility' constraints. Constraints ii) and iii) state the supply of each good must be equal to aggregate demand, and iv) states that a production plan (x1, x2, L1, L2, K1, K2) must be technically efficient. Combine the last three constraints into one to make the problem a little simpler: ∗ xA1 + xB1 = x1 (xA2 + xB2, L, K) v) 3.25 Welfare economics The Lagrange equation is then ∗ L = uA(xA1, xA2) + µ[uB(xB1, xB2) - uB ] + θ [ x1 (xA2 + xB2, L, K) - xA1 - xB1] 0 The first-order conditions are ∂L ∂ uA = − θ = 0 ∂ x A1 ∂ x A1 ∂ u A ∂ x A2 ∂ x∗ ⇒ = − 1 , 25) ∗ ∂ u A ∂ x A1 ∂ x2 ∂L ∂ uA ∂ x1 = + θ = 0 ∂ x A2 ∂ x A2 ∂ x2 ∂L ∂ uB = −µ − θ = 0 ∂ x B1 ∂ x B1 ∂ u B ∂ x B2 ∂ x∗ ⇒ = − 1 , 26) ∗ ∂ u B ∂ x B1 ∂ x2 ∂L ∂ uB ∂ x1 = −µ + θ = 0 ∂ x B2 ∂ x B2 ∂ x2 and Lθ = Lµ = 0. Equations 25) and 26) imply ∂ u A ∂ x A2 ∂ uB ∂ x B 2 ∂ x∗ = = − 1. 27) ∂ u A ∂ x A1 ∂ u B ∂ x B1 ∂ x2 Equation 27) states that at a Pareto efficient allocation, the marginal rates of substitution between any two goods is equal for every consumer, and in turn equal to the slope of the production possibilities frontier. x1 ∗ ∂ x1 slope = ∂ x2 ∂ u A ∂ x A2 ∂ uB ∂ xB 2 u0 = A ∂ uA ∂ x A1 ∂ uB ∂ xB1 0 uB PPF x2 3.26 Welfare economics Competitive behavior Profit maximization: Let prices in the economy be (p1, p2, pL, pK). For the production of good 1 we choose (L1, K1) to solve max p1f(L1, K1) - pLL1 - pKK1. The necessary conditions are p1 f L − p L = 0 fL p ⇒ = L 28) p1 f K − p K = 0 fK pK Similarly, the necessary conditions for a profit maximizing plan to produce good 2 imply gL p = L 29) gK pK 28) and 29) imply fL g p = L = L 30) fK gK pK Compare 30) and 24) to note that competitive, profit maximizing behavior induces a technically efficient allocation of L and K to the production of x1 and x2. Utility maximization: Each consumer chooses (xi1, xi2) to solve max ui(xi1, xi2) s.t. a budget constraint, i = A, B. The necessary conditions imply ∂ u A ∂ x A2 ∂ uB ∂ x B 2 p = = 2 31) ∂ u A ∂ x A1 ∂ u B ∂ x B1 p1 Compare 31) and 27) to verify that competitive behavior by consumers induces a Pareto efficient consumption allocation. Also note that the ratio of the goods prices is equal to the slope of the production possibilities frontier. Notes 1) Obviously, all the marginal conditions are not enough to make the statements we have been making. We also need the resource constraints. 2) These marginal relationships are not the only ones that can be inferred. See Silberberg for more. 3.27 Welfare economics The compensation criterion As a criterion for evaluating public policy proposals, the concept of Pareto efficiency is considered by most to be too restrictive. Specifically, the criterion is said to be incomplete in the sense that it does not allow us to rank every possible allocation (or, more generally, social outcome). For example, consider the following graph. According to the Pareto criterion, a social planner will not be able to rank bundles R and T. Even worse, the Pareto criterion does not tell us that society prefers R and T to S even though S is inefficient. 2 contract curve Β 1 T S R 0 uB u0 A Α 1 2 In this lecture we will consider a modification of the Pareto criterion called the compensation criterion, which is commonly used in applied welfare economics. It is similar to the Pareto criterion but it allows us to compare more outcomes. That is, it is more complete than the Pareto criterion. Unfortunately, as you will see, it is also incomplete and it can provide inconsistent comparisons. The Pareto frontier It will be useful to use Pareto (sometimes called utility possibility) frontiers. A Pareto frontier collects all bundles of utility levels that are generated by efficient allocations. Consider a society with two people (A, B) and two goods (1, 2) and the problem of finding the Pareto efficient allocations: max uA(xA1, xA2) 0 s.t. uB(xB1, xB2) = uB xA1 + xB1 = wA1 + wB1 xA2 + xB2 = wA2 + wB2. 32) 3.28 Welfare economics The Lagrange equation for 32) is L = uA(xA1, xA2) + µ[uB(xB1, xB2) - uB ] 0 + λ1[wA1 + wB1 - xA1 - xB1] + λ2[wA2 + wB2 - xA2 - xB2]. The necessary conditions for an interior solution to 32) are ∂L ∂ uA ∂ x A1 = ∂ x A1 − λ 1 = 0 33) ∂L ∂ uA ∂ x A2 = ∂ x A2 − λ 2 = 0 34) ∂L ∂ uB ∂ x B1 = µ ∂ x B1 − λ 1 = 0 35) ∂L ∂ uB ∂ x B2 = µ ∂ xB2 − λ 2 = 0 36) ∂L ∂L ∂L ∂µ = ∂ λ1 = ∂ λ2 = 0 37) As usual, these first-order conditions imply that at an efficient allocation with xij > 0 for i ∈ (A, B) and j ∈ (1, 2), ∂ u A ∂ x A1 ∂ uB ∂ x B1 = ∂ u A ∂ x A2 ∂ uB ∂ x B 2 0 uB(xB1, xB2) = uB xA1 + xB1 = wA1 + wB1 xA2 + xB2 = wA2 + wB2 38) [Note that the collection of conditions 38) are identical to 33) through 37)]. Assuming that a solution to 32) exists and is unique, conditions 38) implicitly define xij = x ij (u B , w A1 + w B1 , w A2 + w B 2 ) , i ∈ (A, B) and j ∈ (1, 2) ∗ λ j = λ∗j ( u B , w A1 + w B1 , w A2 + w B 2 ) , j ∈ (1, 2) µ = µ ∗ (u B , w A1 + w B1 , w A 2 + w B 2 ) . The indirect objective function is u A = u ∗ (u B , w A1 + w B1 , w A 2 + w B 2 ) . A 3.29 Welfare economics So that there is no confusion later on, let u ∗ (⋅) = v( u B , w A1 + w B1 , w A 2 + w B 2 ) . This is A the Pareto frontier. From the Envelope Theorem ∂ u∗ ∂v A = = − µ ∗ < 0. ∂ uB ∂ uB Hence, the Pareto frontier is downward sloping. In the graph below, I have drawn the Pareto frontier as concave, although we cannot guarantee this. Instead of thinking about the Pareto frontier as a function , it will be convenient sometimes to describe utility possibilities as a set: [ [ ] ] U = u( X ) = u A ( x A ), u B ( x B ) such that X = ( x A , x B ) is an efficient allocation . In the graph below, each point in this space is a utility vector (uA, uB). Points on the frontier are utility vectors generated by efficient allocations. Points under the frontier are feasible utility vectors, but they are generated by inefficient allocations. 2 contract curve Β 1 T R S 0 uB u0 A Α 1 2 uA u(T) Pareto frontier u(S) u(R) uB 3.30 Welfare economics The compensation criterion In order to define the compensation criterion, we first define the Pareto criterion. Definition: A feasible allocation X0 is said to Pareto dominate another feasible allocation X1 if ui(X0) ≥ ui(X1) ∀ i ∈ N and ui(X0) > ui(X1) for some i ∈ N. In the graph below X0 Pareto dominates only those allocations that induce utility vectors in the shaded area. For example, X0 dominates X1. However, X0 and X2 are not comparable by the Pareto criterion. That is, X0 does not dominate X2 and X2 does not dominate X0. This is what we mean by incompleteness. The Pareto criterion does not give us a basis to judge the desirability of all allocations relative to all others. uA u (X 2 ) u (X 0 ) u( X 1) uB Definition: A feasible allocation X0 is said to be potentially Pareto preferred to another feasible allocation X if there is some reallocation of X0, say X1, such that ∑x =∑x i ∈N 1 ij i ∈N 0 ij ∀j∈M (X1 is a reallocation of X0) and ( ) ui X 1 ≥ ui ( X ) ∀i ∈ N . (X1 Pareto dominates X) u (X ) i 1 > ui ( X ) for some i ∈ N 3.31 Welfare economics This is the essence of the compensation criterion. A social outcome X0 is preferred to X if there is a third outcome X1 attainable from X0 that Pareto dominates X. Note that the definition does not require that we actually move to X1, it only requires its existence. Put another way: Almost any public policy change is likely to produce winners and losers (i.e., some will benefit by the change and others will lose). The change is potentially Pareto preferred if the winners could compensate the losers (a reallocation) so that if the compensation takes place, no one is harmed by the change and some are strictly better off. The catch is that the compensation need not take place. To illustrate, consider two mutually exclusive outcomes: R ≡ (reduce emissions of some industrial pollutant) and NR ≡ (no environmental regulation). Relative to NR, option R will hurt the owners of some firms but will provide a cleaner environment for the enjoyment of others. Consider our two-person society, and suppose that the Pareto frontiers in the graph below correspond to the two policy options. Suppose further that a policy option to achieve R results in allocation X0 and utility vector u(X0), while policy option NR will result in allocation X and utility vector u(X). Note that X0 does not Pareto dominate X. However, there is a third allocation X1 that is attainable with policy option R that does Pareto dominate X. Therefore, X0 is potentially Pareto preferred to X. So, according to the compensation criterion emissions of the industrial pollutant should be reduced even though person A is harmed. uA u ( X 1) u( X ) u ( X 0) NR R uB 3.32 Welfare economics Incompleteness The graph above illustrates the incompleteness of the compensation criterion. Here X0 is not potentially preferred to X1, and X1 is not potentially preferred to X0. However, relative to the Pareto criterion we can compare more outcomes using the compensation criterion. Inconsistency A major shortcoming of the compensation criterion is that it will generate inconsistent results in some situations. Suppose that the Pareto frontiers for the policy options we considered above are as in the graph below. Here, allocation X0 is potentially Pareto preferred to X through the reallocation X1. So, according to the compensation criterion, emissions should be regulated. However, allocation X is potentially Pareto preferred to X0 through the reallocation X2. So, according to the compensation criterion, emissions should not be regulated. In such a case, the compensation criterion gives us an inconsistent ranking. It tells us to regulate emissions, but also do not regulate emissions. uA u( X 2) u ( X 0) u (X 1 ) u( X ) NR R uB The compensation criterion and transferable utility It is often assumed in economics and game theory that utility can be transferred from one person to others. Utility is transferable only if there exists some commodity that enters each individual's utility function linearly and separately from all other commodities. For example, utility is transferable in our two-person, two-good society only if the utility functions have the form ui(xi1, xi2) = vi(xi1) + xi2, i ∈ (A, B). 39) 3.33 Welfare economics Utility functions of this form are called quasi-linear utility functions. Given quasi-linear utility functions, utility is transferable if individuals can make payments to each other using good 2. In the language of game theory, we say that side-payments are allowed. (Sometimes we assume that good 2 is money). Note that the cost to an individual of increasing the utility of the other by one unit is one unit of utility. Now suppose that we want to choose an allocation that maximizes the sum of individual utilities subject to the resource constraints: max V = uA(xA1, xA2) + uB(xB1, xB2) = vA(xA1) + xA2 + vB(xB1) + xB2, s.t. xA1 + xB1 = wA1 + wB1 xA2 + xB2 = wA2 + wB2, 40) [Note that V is a social welfare function]. If vA(xA1) and vB(xB1) are both monotonically increasing and strictly concave then a solution to 40) exists and it is unique. Given a solution to 40), we can define the indirect objective function V* = V ( w A1 + w B1 , w A2 + w B 2 ) . This function gives us the maximal attainable utility for the two-person society. Now, if side-payments are allowed (utility is transferable), V* can be allocated in any way so that uA + uB = V*. 41) Recall that an allocation that maximizes a social welfare function is Pareto efficient. Define the set of utility vectors [( ) US = u V ∗ = (u A , u B ) such that u A + u B = V ∗ . ] 42) This set describes the Pareto frontier when utility is transferable. The function describing the frontier in the (uB, uA) space is uA = V* - uB 42') Note that this is a linearly decreasing function with slope equal to -1 and horizontal and vertical intercepts equal to V*. Recall that before we described the Pareto frontier as the set [ [ ] ] U = u( X ) = u A ( x A ), u B ( x B ) such that X = ( x A , x B ) is an efficient allocation , 43) 3.34 Welfare economics and the function u A = v( u B , w A1 + w B1 , w A 2 + w B 2 ) . 43') Now, we did not allow side-payments when we generated 43) and 43'). In the case of the quasi-linear utility functions 39) with vA(xA1) and vB(xB1) monotonically increasing and strictly concave, the function 43') is decreasing and strictly concave in uB. The graph below is of the two frontiers. Note that both frontiers are derived from the same individual utility functions and society's endowments of the two goods, wAj + wBj, j ∈ (1, 2). The only difference between them is that side-payments are allowed along US and are not allowed along U. uA V* u (X* ) u1 u ( X 0) U US uB V* Suppose that X* is the allocation that maximizes social welfare as defined by 40). Without a transfer of utility between the individuals, the utility vector at this allocation u(X*) is the point of tangency between US and U. Note that even though X0 efficient in the absence of side-payments, X* does not Pareto dominate X0 and vice-versa. However, suppose that the two players agree to the allocation X* and a transfer of utility from A to B so that the utility vector u1 is achieved. [Actually, the players can agree to this scheme or it can be imposed by a social planner]. Since both players are better off with X* and the transfer than they would be at X0, we can say that X* is potentially Pareto preferred to X0. Note the difference between what we have just done and our definition of "potentially Pareto preferred". There an allocation X* was preferred to another X0 if there was a reallocation of consumption bundles that Pareto dominated X0. With transferable utility, we do not reallocate the consumption allocation to reach a dominant outcome -- we choose the social welfare maximizing consumption allocation and then reallocate utility. 3.35 Welfare economics Excercises [1] We know that a Pareto efficient allocation is one from which there is no move that can make at least one person better off without harming another. Define a Pareto improving move as one that makes one person better off without harming another. Note that a Pareto improving move does not necessarily result in a Pareto efficient allocation. Define a Pareto superior allocation as one that results from a Pareto improving move. Note that this definition does not require a superior allocation to be an efficient one. In the following graph, the point W is the endowment point, while R, S and T are alternative allocations. Consider moves to and from these points to illustrate the concepts of Pareto efficiency, Pareto improving moves and Pareto superior allocations. 2 Β 1 R contract curve T S Α W u0 A 1 0 uB 2 [2] Consider a two-person pure exchange economy with two goods. Suppose that A does not value good 2 at all. That is, only increases in the consumption of good 1 will increase A's utility. Person B's indifference curves have the usual shape. In an Edgeworth Box locate the contract curve and the core. [3] "Jack Sprat can eat no fat, his wife can eat no lean." In an Edgeworth Box find the contract 'curve'. What about the core? [4] Discuss the relationship between: [a] efficient and core allocations [b] efficient allocations and endowments [c] core allocations and endowment [5] In a pure exchange economy with two people (A and B) and two goods (1 and 2), suppose that uA(xA1, xA2) = 2xA1 + xA2 uB(xB1, xB2) = xB1xB2 wA = (10, 0) wB = (0, 10) [a] Draw an Edgeworth box to illustrate the economy. Draw a few indifference curves and the point W = (wA, wB). [b] Solve for the Pareto efficient allocations. Illustrate them graphically. Illustrate the core allocations graphically. 3.36 Welfare economics [6] Explain how the Pareto efficient allocations in a two-person, two-good exchange economy can be characterized with the appropriate constrained optimization problem. You don't have to go as far as deriving first-order conditions. Just set the problem up to fit the definition of Pareto efficient allocations. [7] Consider a two-person (A and B), two-good (1 and 2), pure exchange economy. The consumers have identical utility functions, ui(xi1, xi2) = ln(xi1) + xi2, i = A, B. Their endowments are (wA1, wA2) = (2, 0) and (wB1, wB2) = (0, 2). [a] Find the contract curve and draw it in an Edgeworth box. [b] Can the allocation [(xA1, xA2), (xB1, xB2)] = [(1, 1), (1, 1)] be a competitive equilibrium allocation? [8] In a two-person, two-good, exchange economy, the consumers have utility functions ui(xi1, xi2) = xi1xi2, i = A, B, and endowments (wA1, wA2) = (10, 0) and (wB1, wB2) = (0, 10). Find the competitive equilibrium allocation for prices (p1, p2) = (1, 1). Is this allocation Pareto efficient? [9] In a pure exchange economy with two people (A and B) and two goods (1 and 2), suppose that uA(xA1, xA2) = 2xA1 + xA2 uB(xB1, xB2) = xB1xB2 wA = (10, 0) wB = (0, 10). Denote the prices of the two goods as p1 and p2. [a] Find the competitive equilibrium (or equilibria). Verify the first welfare theorem. [b] Assume that (p1, p2) = (4, 1). Show that these prices cannot support a competitive equilibrium. Do the same for (p1, p2) = (4, 3). [c] Verify that the allocation X = [(xA1, xA2) = (25/3, 20/3), (xB1, xB2) = (5/3, 10/3)] is an efficient allocation. Find the set of prices and endowments such that X is a competitive equilibrium allocation. Use this exercise to illustrate the second welfare theorem. [10] Waldo and Penelope have preferences over liver and onions. In fact, both view liver and onions as perfect complements to be consumed in one-to-one proportions -- one onion for every pound of liver. Assume that Waldo is endowed with 5 onions and no liver, and Penelope is endowed with 5 onions and 20 pounds of liver. [a] Find the contract curve (or space) and the core. [b] How does the contract curve and the core change if 5 pounds of liver is transferred from Penelope to Waldo. 3.37 Welfare economics [c] Find the set of competitive equilibria for the original endowment. How does this set change if the endowments change as in b)? [d] Comment on the necessity of equating marginal rates of substitution to find efficient allocations and competitive equilibria in this case and in general. [11] Use the welfare theorems as applied to exchange economies to discuss the relationships among core allocations, efficient allocations, and competitive equilibria. Use words, not math. [12] Ken and Barbie have preferences over quiche and Perrier. Ken only consumes quiche and Perrier in one-to-one proportions -- one bottle of Perrier for every slice of quiche. Barbie has indifference curves that have the normal convex shape. Ken is endowed with two slices of quiche and two bottles of Perrier, while Barbie is endowed with four slices of quiche and 4 bottles of Perrier. [a] Find the contract curve and the core. How does the contract curve and the core change if we transfer one slice of quiche from Ken to Barbie. [b] For the original endowments, find the competitive equilibrium. [c] Verify that the allocation in which each person has three units of each good is an efficient allocation. Use this allocation to illustrate the second welfare theorem. [13] Any complaints about the operation of a competitive market system can be reduced to complaints about equity, and such complaints can be addressed by lump-sum transfers. State the welfare theorems in the context of an exchange economy and use them to evaluate this statement. [14] Bennie and Joon have preference over Macadamia nuts and Listerine. Each considers Macadamia nuts and Listerine as complements to be consumed in one-to-one proportions -- one pound of Macadamia nuts for every gallon of Listerine. Bennie is endowed with two pounds of Macadamia nuts and one gallon of Listerine, while Joon is endowed with one pound of Macadamia nuts and two gallons of Listerine. [a] Find the contract curve and the core. [b] Use the allocation in which both individuals consume 1.5 pounds of Macadamia nuts and 1.5 gallons of Listerine to illustrate the second welfare theorem. [15] If two allocations are not comparable by the Pareto criterion, they are not comparable by the compensation criterion. State the Pareto criterion and the compensation criterion, then tell me whether the statement is true or false and prove your conclusion. [16] Show that every Pareto efficient allocation is potentially Pareto preferred to every Pareto inefficient allocation. [17] The compensation criterion and the Pareto criterion are normative concepts that are used to examine the 'desirability' of social states. Compare and contrast these two criteria. 3.38