Microeconomics II. MOSEC, LUISS Guido Carli Problem Set n 3

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1 Microeconomics II MOSEC, LUISS Guido Carli Problem Set n 3 Problem 1 Consider an economy 1 1, with one firm (or technology and one consumer (firm owner, as in the textbook (MWG section 15.C. The set of feasible consumption allocations is F := { x R 2 + : x i2 f ( L x1, x1 L } This set is nonempty (it contains x = (ω, 0, convex (it is the intersection of two convex sets, R 2 + and the lower contour set of a concave function over the domain [0, L], compact (0 x ( L, f( L, f is a continuous function. Define P x = { x R 2I + : u(x > u(x } where x is a Pareto efficient allocation relative to F, hence it is on the boundary of F (otherwise, it would contradict production efficiency, namely it would be that x has been obtained by picking a production pair in the interior of the production possibility set Y of this economy. P x, containing consumption bundles strictly preferred to x, is nonempty and convex; moreover, it does not intersect F. By the separating hyperplane theorem, there exists a c R and a vector p R 2 such that p x c for all x P x, and p x c for all x F. Since x belongs to the boundary of F, p x c = p x, for all x P x. Letting individual wealth be w := p x, the latest implies that x is expenditure minimizing at (p, w. Notice that if x = (x 1, x 2 =: ( L z, q are in P x, they are not in F; implying ( z, q is not an element of Y. Hence, given x = (x 1, x 2 =: ( L z, q, 1

2 2 p x p x for all x P x, is equivalent to p ( L z, q ( L z, q, or to p ( z, q ( z, q whenever ( z, q is not in Y. The latest establishes that ( z, q is profit maximizing at p: π(p = p ( z, q. Finally, we can check that indeed w is the sum of the endowment value and profits evaluated at p, w := p x = p 1 ( L z + p 2 q = p 1 L + p ( z, q = p 1 L + π(p Problem 2 Economy 2 by 1 1. Define the set of feasible allocations, A, the set of consumption-feasible allocations, F. For I = 2, ω is the aggregate endowments vector. (z, y > 0, f( = { A := (x, (z, y R 2I + R 2 + : i x i2 y + ω 2, y f(z, i x i1 ω 1 z } F := { x R 2I + : i x i2 f ( ω 1 i x i1 + ω 2, i x i1 ω 1 } 2. The picture of the set F is drawn using X to denote aggregate consumption, and ω to denote aggregate endowments. 3. The set of Pareto efficient allocations is, O := {x F : u i (x u i (x all i, u h (x > u h (x some h, only if x not in F} At an efficient allocation, the corresponding individual MRS s equalize and are equal to the MRT. By contradiction, suppose that at an efficient allocation, there exists a commodity pari (x 1, x 2 and some i such that the corresponding marginal rats differ, say MRS i > MRT. Then, two are the interesting cases: i agents have equalized MRSs but these are above MRT, ii for some agents i, j, MRS i > MRS j = MRT. Case ii we know is incompatible with

3 3 X 2 f(ω 1 + ω 2 f(ω 1 -X 1 F ω 1 X 1 Figure 1: The upper envelope of F identifies production efficient bundles of aggregate consumption. Pareto efficiency; this would define a failure of efficiency in the distribution of the resources available for consumption. Case i denotes an inefficiency in the way production is organized; precisely (on a plane (x 1, x 2, MRS > MRT with MRS i = MRS for all i, detects an overproduction of 2 (and an underproduction of 1 relative to the efficient level. Indeed, even if the allocation proposed does not waste resources (i.e. it is on the frontier of the economy production possibility set, a different choice is feasible and welfare improving. To see this, notice that to have an extra marginal unit of good 1 each consumer is willing to give up MRS units of good 2; to produce an extra unit of good 1 (in an productive efficient way it is required to give up only MRT of good 2. Thus we can take out from a consumer only MRT < MRS of 2 and give him an extra marginal unit of 1, making him better-off, without affecting the others. This contradicts the assert that the initial allocation was Pareto efficient. Clearly, in an economy with J > 1, the above changes of output may also correspond to a reallocation of factors of productions among different technologies/firms j, not only to different sectors. 4. An economy is an element of Ξ := I = L = 2, J = 1. Let p 1 = 1, and p := p 2. ( {u i, X i, ω i, θ i } I i=1, {Y j} J j=1, where

4 4 A competitive equilibrium of an economy ξ in Ξ is a price p > 0 such that (x i1 (p ω i1 z(p = 0 i=1,2 This is a Cobb-Douglas economy. Individual demand is x i1 (p, w i = αw i, where w i = w i (p := p ω i + π(pθ i. x i2 (p, w i = (1 α w i p To recover individual wealth at p, solve the firm s optimal problem. π(p = max z>0 p z z to find y(p = p/2, z(p = (p/2 2 and π(p = (p/2 2. Compute (w i i : w 1 (p = 1 + π(p/2 = 1 + (1/8p 2, w 1 (p = p + π(p/2 = p + (1/8p 2. Compute the aggregate demand of good 1: i x i1(p, w i = α i w i x i1 (p, w i = α (1 + p + 14 p2 Impose market clearing: Yielding, i i=1,2 ( p 2 (x i1 (p 1 = 0 2 ( α p 2 + αp + α 1 = 0 4 For α = 2/3, the latter polynomial has a single positive root, p =.4. By substitution in the above demand and supply functions, you can recover the equilibrium allocation: x 1 = (0.68, 0.28, x 2 = (0.85, 0.35, y = 1.2, z = The preference parameter α = 2/3 reveals both consumers have a stronger taste for good 1. In equilibrium, the technology is used to reduce the aggregate consumption of good 2 and increase that of good 1, with respect to the initial aggregate endowment of (1, 1. This is supported by a price p = p 2 /p 1 =.4 < 1.

5 5 Problem 3 1. U is convex if utilities are strictly-quasi concave. Let X i be convex, u i : X i R, the function u i is concave iff for any x, x X, u i (x θ θu i (x+ (1 θu i (x for any θ [0, 1], where x θ := θx+(1 θx. Suppose this holds for all individuals i, and assume (x, y, (x, y are two feasible allocations, in A. For every i define x i,θ := θx i + (1 θx i, and define y θ in the same way. The new allocation (x θ, y θ is in A, by the convexity of the latest set. U(x θ = (..., u i (x i,θ,... (..., θu i (x i + (1 θu i (x i,... = θ (..., u i (x i,... + (1 θ (..., u i (x i,... = θu(x + (1 θu(x 2. Take utilities which are quasi-concave but not strictly-quasi concave. For example, X i = R + for all i = 1, 2 { x 0.5 u i (x i = i : if x i > 1 x i : if 0 x i 1 Letting ω = 2 be the total resources of x, the Pareto frontier is a curve intersecting the vertical axis (u 2 in 2, with graph { 2 u1 for 0 u 1 1 u 2 = 2 (u 1 2 for u 1 > 1 The upper envelope of the following graph is a non-concave Pareto frontier.

6 Figure 2: Non-concave Pareto frontier 6