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Consumer and Firm Behavior: The Work-Leisure Decision and Profit Maximization Chapter 4 ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 1 Static vs Dynamic Decision Making • In this and the next chapters, we are considering static decision making, i.e., planning over a single period. • From chapter 6 to 9, we are going to discuss dynamic decision making, i.e., planning over more one period. • Chapter 4 first recalls what you’ve learnt in the last semester: the micro behavior of a representative consumer and a representative firm. • Chapter 5 then assembles these in a macro model in order to address some important macro issues. The role of government is also introduced. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 2 Objectives of the Representative Consumer & the Representative Firm • Representative Consumer: To maximize utility subject to budget (and time) constraint by allocating time between work and leisure; • Representative Firm: To maximize profits subject to technological constraint by deciding how much labor to be hired. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 3 Assumptions of the Model 1) Two Goods: – Consumption good, which is an aggregation of all consumer goods in the economy. – Leisure, which is any time spent other than working in the market. e.g. Recreational activities, sleep and household work. 2) One Consumer: – All consumers are identical in terms of preferences, ability, time constraint and budget constraint. Then, the economy will behave as if there were only one consumer, one that we refer to as the representative consumer. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 4 Assumptions of the Model 3) Price-Taking: – The representative consumer is a price-taker, i.e., he takes all market prices as given, and acts as if his actions had no effect on those prices. 4) No Money: – The economy we’re considering is a barter economy, i.e., all trade involves barter exchanges of goods for goods in the absence of money. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 5 The Representative Consumer’s Optimization Problem • Objective: to make himself as well off as possible given the constraints he faces. • Two Ingredients in this problem: – Consumer’s preferences – Consumer’s budget constraint ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 6 Preferences • The preferences of the representative consumer is captured by the utility function, U(C , l) where C is the quantity of consumption, l is the quantity of leisure • Any particular pair of consumption and leisure (C , l) is called a consumption bundle. • For each consumption bundle, the utility function U assigns a real number so that different bundles can be ranked. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 7 Preferences • Consider two distinct bundles (C1 , l1) and (C2 , l2) – (C1 , l1) is strictly preferred to (C2 , l2) if U(C1 , l1) > U(C2 , l2) – Consumer is indifferent between the two bundles if U(C1 , l1) = U(C2 , l2) • Assumptions on Preferences: 1) More is always preferred to less – A consumer always prefers a consumption bundle that contains more consumption, more leisure, or both. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 8 Preferences 2) Consumer prefers a more diversified consumption bundle. – If the consumer is indifferent between (C1 , l1) and (C2 , l2), then some mixture of the two will be preferable to either one. – Example: Consider a new bundle (C3 , l3), where C3 = lC1 + (1 – l)C2, l3 = ll1 + (1 – l)l2 and l lies between 0 and 1 (a fraction), then U(C3 , l3) > U(C1 , l1) = U(C2 , l2) 3) Consumption and leisure are normal goods. – A good is normal (inferior) for a consumer if the quantity of the good that he/she purchases increases (decreases) when income increases. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 9 Graphical Representation of Preferences • An indifference curve connects a set of points, with these points representing consumption bundles among which the consumer is indifferent. • A family of indifference curves is called indifference map. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 10 Properties of Indifference Curves • Consider a consumption bundle B. Since a consumer prefers Consumption more to less, any bundle that is indifferent to B must lie within quadrant II and IV. IV I • Implication: An indifference curve B slopes downward. III II Leisure ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 11 Properties of Indifference Curves • Consider any two bundles A and B, since a consumer prefers a Consumption more diversified bundle C to either A or B, the set of bundles A that are indifferent to A and B must lie below the straight line C AB. • Implication: An indifference curve is B convex, that is bowed-in toward the origin. Leisure ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 12 Graphical Representation of Preferences Consumption, C A C1 B D I2 C2 I1 l1 l2 Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 13 Marginal Rate of Substitution • Marginal rate of substitution of leisure for consumption (MRSl,C) is Consumption, C the rate at which the consumer is Slope = MRSl,C just willing to substitute leisure for consumption good. A C1 • It is also minus the slope of the indifference curve. C2 D B • Convexity of indifference curve is I2 equivalent to I1 l1 l2 – Diminishing marginal rate of substitution. Leisure, l (compared slope at A and slope at B) ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 14 Marginal Rate of Substitution • The MRS at A is larger (in terms of absolute magnitude) than the MRS Consumption, C at B. Slope = MRSl,C • As we increase l and reduce C, i.e. moving from A to B along I1, the A C1 consumer needs to be compensated more in terms of l to C2 D B give up another unit of C. I2 • The consumer requires this extra I1 consumption because of a l1 l2 Leisure, l preference for diversity. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 15 Marginal Rate of Substitution Mathematical Derivations: • Suppose indifference curve I1 Consumption, C represents the utility level , Slope = MRSl,C A • Totally differentiate this with C1 respect to C and l gives C2 D B I2 I1 l1 l2 Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 16 Constraints faced by The Representative Consumer • Two constraints: – Time constraint for l – Budget constraint for C • The time constraint for the consumer is given by s l+N =h where h is the total number of hours available (e.g., 24 hours a day), l is s the leisure time and N is the time spent working (or labor supply). ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 17 Budget Constraint • Sources of income: s 1) Real wage income, wN – w is the real wage, i.e., the price of one unit of labor time in terms of consumption goods (the numeraire). 2) Real dividend income, p – Since the firms are owned by the representative consumer, any profits made by firms are distributed to him as dividends. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 18 Budget Constraint • Taxation T: A lump-sum tax, i.e. a tax that does not depend on the actions of the economic agent who is being taxed. s • Real Disposable Income = wN + p – T • The consumer first receives income and pays taxes in terms of consumption goods, and then decides on how much to consume out of the disposable income. • All disposable income is consumed, i.e. s C = wN + p – T = w(h – l) + p – T ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 19 Budget Constraint • Reasons: – Since the consumer only lives for one period, there is no incentive to save anything. – Since more is preferred to less, any wastage is not optimal. • The consumer’s budget constraint can be written as C + wl = wh + p – T • RHS = Total implicit real disposable income • LHS = Implicit real expenditure on consumption goods and leisure • Note: w can also be interpreted as the market price, or the opportunity cost, of leisure time. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 20 Graphical Representation of the Budget Constraint • Budget constraint: C = –wl + (wh + p – T) Consumption, C thus slope = –w. wh + p – T A • The vertical intercept, wh + p – T, is the maximum consumption that C = –wl + wh + p – T can be achieved when the consumer consumes no leisure. • Case 1: p < T B h + (p – T)/w h Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 21 Graphical Representation of the Budget Constraint • Case 2: p > T Consumption, C – The consumer can still enjoy C = p – T > 0 even if he A chooses not to work. – When C = 0, l = h + (p – T)/w, C = –wl + wh + p – T but it is not feasible as the maximum time can only be h p–T B – When l = h, C = p – T D h Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 22 Graphical Representation of the Budget Constraint • The budget constraint tells us what Consumption, C consumption bundles are feasible to consume given the market real A wage (w), dividend income (p) and taxes (T). Not Feasible • The consumption bundles within the shaded regions and on the Feasible B budget constraint, are feasible. • Thus the shaded region together D with the budget constraint is called h the feasible set. Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 23 Consumer Optimization • The representative consumer is assumed to be rational, i.e. he always chooses the best feasible consumption bundle, or the optimal consumption bundle. • “Best” in the sense that it lies on the highest possible indifference curve. • “Feasible” in the sense that it lies within the feasible set. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 24 Graphical Solution • Suppose p > T. • Claim: H is the optimal consumption Consumption, C bundle. A Reasons: F • Any bundle inside the budget H constraint is not optimal J I2 (compare J to F). E I1 p–T B • B is preferred to any point on BD. D • For any point on AB, the h consumer can always improve by Leisure, l moving closer to H. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 25 Mathematical Solution • The consumer tries to solve the following constrained optimization problem max U(C , l) C,l subject to C = w(h – l) + p – T and C ³ 0, h ³ l ³ 0. • Lagrangian L = U(C , l) + l[w(h – l) + p – T – C] where l is the Lagrangian multiplier. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 26 Mathematical Solution • We assume that an interior solution can be obtained. This means choosing C = 0, l = h or l = 0 are not optimal (so that we can ignore the last two constraints). • Formally, we can impose the restrictions: Uc(0 , l) = ∞ and Ul(C , 0) = ∞ For any C and l, to guarantee an interior solution. • First-order (Necessary) conditions (FOCs): – Obtained by differentiating the Lagrangian with respect to C, l and l. (Recall: Lagrangian equation L = U(C , l) + l[w(h – l) + p – T – C] Uc(C , l) = l, Ul(C , l) = lw, w(h – l) + p – T – C = 0. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 27 Mathematical Solution • From the FOCs, we obtain Consumption, C A • At H, where an indifference curve F is just tangent to the budget H constraint, the above equality holds. J I2 • If MRS > w (e.g. at F), the E I1 p–T B consumer would be better off by increasing l and reducing C, thus D moving closer to H. h Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 28 Comparative Statics • To determine how C and l changes when any of p, T and w changes. • Recall the FOC of the consumer’s problem, which can be written as Ul(C , l) – wUc(C , l) = 0. (1) • From the budget constraint, w(h – l) + p – T – C = 0. (2) • The two form a system of equations in terms of C and l (endogenous variables). ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 29 Comparative Statics • Totally differentiate the two equations –dC – wdl + (h – l)dw + dp – dT = 0 from (2) [Ucl – wUcc]dC + [Ull – wUcl]dl – Ucdw = 0 from (1) In matrix form, A • Determinant of the bordered Hessian matrix A is Ñ = –Ull + 2wUcl – w2Ucc • Strict quasiconcavity of U è Ñ > 0. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 30 1) Changes in p and/or T • Using Cramer’s Rule, we get • The assumption that consumption and leisure are normal goods is equivalent to the conditions –Ull + wUcl > 0 and Ucl – wUcc > 0. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 31 1) Changes in p and/or T Graphical Illustration • Consider a net increase in p – T. Consumption, C • Since w (slope) remains the same, F the budget constraint makes a parallel shift (from AB to FJ). A • Since disposable income while , C2 K prices remain the same, there is C1 H I2 J only a pure income effect on the I1 consumer’s choices. B • The new optimal consumption bundle is K, where both C and l D (normal goods). l1 l2 h Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 32 1) Changes in p and/or T Graphical Illustration Remark: Consumption, C • The increase in consumption F (C2 – C1) is less than the increase in nonwage income (distance A AF). C2 K • Since the consumer is working C1 H J I2 less (leisure wage income ¯. ), I1 • This will offset part of the B consumption increase. D l1 l2 h Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 33 2) Changes in w • Using Cramer’s rule, • C is normal good è –Ull + wUcl > 0, together with Ñ > 0 and Uc > 0 è ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 34 2) Changes in w • However, we cannot determine the effect of a change in w on l. • Reason: It depends on the relative magnitude of the opposing income and substitution effects. • Substitution effect: w Opportunity cost of leisure è (l becomes more expensive relative to C) è Demand for leisure ¯ • Income effect: w Wage income è è Demand for leisure (normal good) ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 35 2) Changes in w Graphical Illustration • Suppose p > T and w . Consumption, C • The budget constraint shifts from I1 I ABD to EBD (with a steeper E 2 slope). J • This shows a special case in which A H leisure remains unaffected. C2 O • Pure substitution effect: Movement C1 F from F to O (on the same B indifferent curve). K • Pure income effect: Movement from O to H. D • Both income and substitution l1 h effects act to C. Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 36 2) Changes in w Graphical Illustration • Labor supply curve which specifies how much labor the consumer wishes Real Wage, w to supply given any real wage. s N • Algebraically, the labor supply curve is s N (w) = h – l(w), where l(w) is the demand function for leisure. • Substitution effect > Income effect è Upward sloping labor supply curve • Net (p – T) in Employment, N è Upward shift in labor supply curve ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 37 Example: C and l are perfect complements • Suppose the consumer’s utility Consumption, C function can be represented by U(C , l) = min{C , al}. C = al (Leontief Function) A where a is a positive constant. I2 E • Note that more is not always I1 F preferred to less. The consumer can B be better off only if he receives more of both goods. • Thus, it is always optimal to choose D h C = al. Leisure, l ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 38 Example: C and l are perfect complements • Combining C = al and the budget constraint gives • In this case, • This is because with perfect complements, there are no substitution effects. Thus leisure real wages as . ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 39 The Representative Firm • The firm owns productive capital and hires labor to produce consumption goods. • Production technology is captured by the production function, which describes the technological possibilities for converting factor inputs d (capital K and labor N ) into outputs Y. d Y = zF(K , N ) where z is total factor productivity. d • z both K and N will be more productive. è ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 40 Assumptions on Production Function • Production function exhibits constant returns to scale (or homogenous of degree one). d – For any x > 0, xY = zF(xK , xN ). – If all factor inputs are changed by a factor x, then output changes by the same factor x. – In this case, a perfectly competitive economy with numerous small firms will behave in exactly the same way as one with a single representative firm (same level of efficiency). d d – Increasing return to scale: zF(xK , xN ) > xzF(K , N ). d d – Decreasing return to scale: zF(xK , xN ) < xzF(K , N ). ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 41 Assumptions on Production Function • Positive marginal product of capital (MPK) and marginal product of labor Output, Y (MPN). Slope = MPK – MPK (MPN) is the additional output * F(K , N ) that can be produced with one additional unit of capital (labor), A holding constant the quantities of labor (capital). * – Fix the quantity of labor at N , then * the MPK at K is the slope of the production function at point A. * K Capital Input, K ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 42 Assumptions on Production Function – Algebraically, we assume that, d d FK(K , N ) > 0 and FNd(K , N ) > 0. Marginal Product of Labor, MPN – Conceptually, this simply means: more inputs yield more output. • Diminishing Marginal Product – The declining MPK and MPN is equivalent to the concavity of the production function. MPN – Algebraically, this means d FKK(K , N ) < 0, d d and FNdNd(K , N ) < 0. Labor Input, N – Implicitly, we assume that F(. , .) is twice differentiable. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 43 Assumptions on Production Function • MPN K as – Algebraically, this means Marginal Product of Labor, MPN – Increase in the quantity of machinery and equipment enhances the productivity of the 2 MPN workers. 1 MPN • F(. , .) is quasiconcave. d Labor Input, N ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 44 Cobb-Douglas Production Function • Probably the most commonly used form of production function which satisfies all the above properties a d b Y = zK (N ) where 0 < a, b < 1. • a + b = 1 è Constant return to scale. a + b > (<) 1 è Increasing (decreasing) return to scale. • If there are profit-maximizing price-taking firms and a + b = 1, then a will be the share that capital receives of national income. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 45 Changes in Total Factor Productivity (z) • Changes in z is critical to our Output, Y understanding of the causes of * Z2F(K , N ) d economic growth and business cycles * Z1F(K , N ) d (real business cycles theory). • Effects of z : 1)dOutput given values of K and for Labour Input, N N. Marginal Product of Labor, MP 2) MPN given value of K. N for • Factors that would affect z: – Technological innovation – Weather – Government regulations MPN 2 1 – Price of energy MPN d Labor Input, N ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 46 Profit Maximization Problem • Assume that capital K is fixed. Then the firm’s problem is to choose a d quantity of N in order to maximize its profits. • The representative firm is assumed to behave competitively, i.e. taking the real wage w as given. d • The problem can be stated as (choosing N ) d d max p = zF(K , N ) – wN • Similar to the consumer’s problem, we assume FNd(K , 0) = ∞ and FNd(K , ∞) = 0 to ensure interior solution in the firm’s profit maximization problem. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 47 Profit Maximization Problem • The optimal condition (FOC) is d d z[∂F(K , N )/∂N ] = w Revenue, Variable Costs d wN • This states that it is optimal for the d zF(K , N ) firm to hire workers up to a level in which the MPN equals the real wage. A • Graphically, the optimal quantity of labor N* is at A, where the slope of total revenue function is equal to B the slope of the total variable cost function. * N Labor Input, N d • The maximized profits p* is given by the distance AB. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 48 Profit Maximization Problem • The FOC of the profit- maximization can also be Real Wage, w interpreted as the firm’s demand curve for labor, for given values of z and K . • The optimal condition (FOC) is MPN or Labor Demand Curve MPN(K , N) = w. • Diminishing MPN implies w and N are inversely related. d Quantity of Labor Demanded, N ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 49 Comparative Statics • Recall the FOC of the firm’s problem d zFNd(K , N ) = w • Totally differentiate this gives d zFNdNddN – dw + FNddz + zFKNddK = 0. Thus, we obtain ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 50 Quasiconcavity • A function f(x) is quasiconcave if f(x1) ³ f(x2) è f[ax1 + (1 – a)x2] ³ f(x2) for any 1 ³ a ³ 0. • f is strictly quasiconcave if f(x1) ³ f(x2) è f[ax1 + (1 – a)x2] > f(x2) for any 1 > a > 0. • Consider a strictly quasiconcave utility function U(C , l). Suppose x1 = (C1 , l1), x2 = (C2 , l2), then U(x1) = U(x2) è U[ax1 + (1 – a)x2] > U(x1) = U(x2) for any 1 > a > 0. Thus the indifference curves are strictly convex. ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 51 Quasiconcavity • Strict quasiconcavity also implies that the bordered Hessian matrix of the utility function is negative definite, i.e., –Ull + 2wUcl – w2Ucc > 0 ECO 2021 Intermediate Macroeconomic Theory 7/12/2013 Professor C. K. Yip 52

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