2.9. V = u(x,y) + a(x-f(l x,t x )) + b(y-g(l y,t y )) + c(l o -L x -L y ) + d(t o -T x -T y ) (1) (a) Suggested Answer: V u a 0 (2) u x = -a b 0 (3)

Transcription

1 2.9 V = u(,) + a(-f(, )) + b(-g(, )) + c( o - - ) + d( o - - ) (1) (a) Suggested Answer: V u a 0 (2) u = -a V u b 0 (3) u = -b V f a d 0 (4) V V V b g d 0 (5) a f c 0 (6) b g c 0 (7) g c/ b c c/a f (b) g d/b d d/a f f (c) is the additional available from a marginal reallocation of to f g divided b the additional available from the same marginal reallocation to g. Hence the marginal rate of transformation. (d) u a d / b g u b d / a f u a c/ b g u b c/ a f 1

2 (e) here is a tpo in the problem. he correct statement is w=p f =p g ; r=p f =p g. From this we get a d / b g b d / a f a c/ b g b c/a f 2

3 Problem Suggested Answer 3.5 Consider an Edgeworth bo (two households, A and B, two goods, and ). Household A is characterized as: (a) endowment = (10, 0), ten units of and zero of ; (b) U A ( A, A ) = A + 4 A ; A likes four times as much as A likes. Household B is characterized as: (a) endowment = (0, 10), ten units of and zero of ; (b) U B ( B, B ) = 5 B + B ; B likes five times as much as B likes. For both households, the two goods are perfect substitutes with MRS s respectivel of (1/4) and 5. (i) Draw an Edgeworth bo for this econom. Show the endowment point, contract curve, competitive equilibrium (a) and the set of Pareto efficient points. Because of the linear preferences, the Pareto efficient set will not be a locus of smooth tangencies don t bother differentiating anthing. Show that ( A, A ) = (0, 10), ( B, B ) = (10, 0) is a competitive equilibrium. Suggested Answer: he set {( A, A ) = (10, C), ( B, B ) = (0, 10 C); ( A, A ) = (10 C, 0), ( B, B ) = (C, 10) 0 C 10} is the Pareto efficient set. he subset with A 2.5, B 2 is the contract curve. Set (p, p ) = (1/2, 1/2). hen ( A, A ) = (0, 10), ( B, B ) = (10, 0) fulfills budget constraint, market clearing, is maimal subject to budget constraint and nonnegativit for each household. So ( A, A ) = (0, 10), ( B, B ) = (10, 0) is a competitive equilibrium allocation. (ii) Some writers would argue that: the contract curve for this econom is equivalent to the set of competitive equilibria. hat is, an individuall rational Pareto efficient point in this Edgeworth bo can be supported as a competitive equilibrium. hese competitive equilibrium allocations would include those of the form ( A, A ), 2.5 < A 10, A = 0; ( B, B ), B = 10, B = 10 A. Eplain the reasoning for this argument (hint: think inside the bo). Suggested Answer: Supporting prices are (p, p ) = ( A 10, 1 A 10 ). A budget line from the endowment point to the suggested allocation neatl separates the upper contour sets of the two households. For each household the suggested allocation is maimal subject to (1) budget constraint, (2) nonnegativit of own consumption, (3) nonnegativit of other household s consumption. he assertion is false. Eplain wh it is mistaken (hint: think outside the bo). Suggested Answer: Point (3) in the argument above should not enter into the household s optimization. It should onl optimize subject to points (1) and (2). At the posted prices, household B would like more and less, creating an ecess demand for, a disequilibrium. 1

4 CB046/Starr December 30, :10 Problems 4.7 and 4.8 are based on the following model. Consider the production of goods and in a competitive econom with two factors of production, land denoted, and labor denoted. Assume all functions are differentiable. Assume interior solutions (no boundar solutions). he available suppl of labor is 0. he available suppl of land is 0. Good is produced in a single firm, called firm, b the production function f(, ) =, where is used to produce, is used to produce. f(, ) 0 for 0, 0; f(0, 0) = 0. Good is produced in a single firm b the production function g(, ) = where is used to produce, is used to produce. g(, ) 0 for 0, 0; g(0, 0) = 0. he resource constraints of the econom are + = 0 + = 0. 1 he allocation of and is said to be technicall efficient if there is no reallocation of and across firms that would increase the output of without reducing the output of. echnical efficienc is a necessar condition for Pareto efficienc. We ll characterize technical efficienc as maimizing the output of for a given level of output of. hat is, choose, to maimize g(, ) subject to f(, ) = X 0 + = 0 + = 0. Restating the problem as choosing, to maimize g(, ) subject to f( 0, 0 ) = X 0. he agrangian for this problem can be stated as M = g(, ) λ[f( 0, 0 ) X 0 ]. Differentiating M with respect to and (letting subscripts denote partial derivatives) and setting the result equal to 0, we have M = g λf = 0 (4.10) M = g λf = 0.(corrected) (4.11) hese are first-order conditions for technical efficienc in this model.

5 CB046/Starr December 30, : Firm s marginal rate of technical substitution of for is defined as MRS = f f. Show that technical efficienc requires that the firms respective MRS s be equated. hat is, show that at a technicall efficient allocation of and, MRS = f f = g g = MRS.It is a well established result that at a competitive equilibrium (r/w) = f f = g g, where w is the wage rate on and r is the rental rate on. hus ou have just shown that a competitive equilibrium allocation is (or fulfills a necessar condition for being) technicall efficient. Suggested Answer: Restate 4.10 and 4.11 as g = λf g = λf and divide through to get g g = f f. 4.8 et a tpical household utilit function be u(, ). u and u denote marginal utilities, partial derivatives of u with respect to and. he marginal cost of at a competitive equilibrium is (w/f ) = (r/f ). As usual in competitive equilibrium price equals marginal cost. et p be the price of, p be the price of. We have p = (w/f ) = (r/f ), p = (w/g ) = (r/g ). he marginal rate of transformation of for (also known as the rate of product transformation of for ) is (g /f ) = (g /f ). It represents the (absolute value of the) slope of the production frontier the additional volume of that can be achieved b sacrificing a unit of. From chapter 3 we have (u /u ) = (p /p ) in competitive equilibrium. We established in chapter 2 (in the special case where f = 1; ou ma assume that it generalizes) that a necessar condition for Pareto efficienc is (g /f ) = (u /u ) (4.12) marginal rate of substitution equals marginal rate of transformation. Show that (4.12) is fulfilled in the competitive equilibrium of this model. hus ou ve shown that competitive equilibrium in a two-good econom fulfills a necessar condition for Pareto efficienc. Suggested Answer: (u /u ) = (p /p ) = (w/f ) (w/g ) = g /f.

6 Suggested Answer, September 2004, Part 2 1. he first order condition is u / u = p /p = 1. So that is fulfilled at (50, 50). Yes the allocation is locall at marginal cost. Pareto efficienc is a bit trick since the production conditions are not concave (there is increasing marginal product; concavit requires diminishing marginal product). We can do a quick check for efficienc b looking for a utilit improvement at nearb points. (35, 75) is possible, where u(35, 75) = 2625 > 2500 = u(50, 50). So the allocation is not Pareto efficient. 2. Since the firms have diminishing marginal costs, as price takers, the firms will find increasing production in the region of diminishing costs, above 55 units, profitable. 3. No. he Second Fundamental heorem requires conveit, and the scale econom depicted here is a non-conveit. 4. Yes. he first order condition in question 1 is still fulfilled. 5. B allocating all of to producing good, we can achieve an allocation of (145, 0) with v(145,0) = 145 > v(50,50) = 100. But as noted in 2, the allocation (50, 50) is not a competitive equilibrium, so the First Fundamental heorem does not appl. here is no countereample.

7 Suggested Answer, June 2014, Micro Qual, #3 Homotheticit and strict concavit make the problem particularl simple. here is just one set of preferences to deal with --- this could be Robinson Crusoe without production and his identical twin. (a) Hard to tell if the question purposefull or carelessl omits the assumptions of continuit and nonnegativit. So assume u is a continuous function, that its domain is R m +, and that e i 0 (co-ordinatewise). hat s definitel sufficient to ensure eistence of equilibrium. (b) No trade requires that the two households have the same MRS s at endowment. But we know the have identical homothetic preferences, so it is sufficient that their endowments be linear multiples of each other. hat is e 1 = ke 2 some k > 0. (c) his is just a redistribution of endowment. he conditions in (a) are sufficient for eistence of equilibrium and the First Fundamental heorem of Welfare Economics applies, so the allocation is Pareto efficient.