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EC9D3 Advanced Microeconomics, Part I: Lecture 2

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EC9D3 Advanced Microeconomics, Part I: Lecture 2 EC9D3 Advanced Microeconomics, Part I: Lecture 2 Francesco Squintani August, 2020 Budget Set Up to now we focused on how to represent the consumer's preferences. We shall now consider the sour note of the constraint that is imposed on such preferences. Definition (Budget Set) The consumer's budget set is: B(p; m) = fx j (p x) ≤ m; x 2 X g Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 2 / 49 Budget Set (2) 6 x1 2 L = 2 X = R+ c c c c c c c c c c c c B(p; m) c c c c c c c - x2 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 3 / 49 Income and Prices The two exogenous variables that characterize the consumer's budget set are: the level of income m the vector of prices p = (p1;:::; pL). Often the budget set is characterized by a level of income represented by the value of the consumer's endowment x0 (labour supply): m = (p x0) Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 4 / 49 Utility Maximization The basic consumer's problem (with rational, continuous and monotonic preferences): max u(x) fxg s:t: x 2 B(p; m) Result If p > 0 and u(·) is continuous, then the utility maximization problem has a solution. Proof: If p > 0 (i.e. pl > 0, 8l = 1;:::; L) the budget set is compact (closed, bounded) hence by Weierstrass theorem the maximization of a continuous function on a compact set has a solution. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 5 / 49 First Order Condition Result If u(·) is continuously differentiable, the solution x∗ = x(p; m) to the consumer's problem is characterized by the following necessary conditions. There exists a Lagrange multiplier λ such that: ru(x∗) ≤ λ p x∗ [ru(x∗) − λ p] = 0 p x∗ ≤ m λ [p x∗ − m] = 0: where ∗ ∗ ∗ ru(x ) = [u1(x );:::; uL(x )]: Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 6 / 49 First Order Condition (2) Meaning that 8l = 1;:::; L: ∗ ul (x ) ≤ λpl and ∗ ∗ xl [ul (x ) − λpl ] = 0 ∗ ∗ ∗ ∗ That is if xl > 0 then ul (x ) = λpl while if ul (x ) < λpl then xl = 0. Moreover L " L # X ∗ X ∗ pl xl ≤ m, and λ pl xl − m = 0 l=1 l=1 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 7 / 49 First Order Condition (3) In other words: if λ > 0 then L X ∗ pl xl = m: l=1 if L X ∗ pl xl < m: l=1 then λ = 0 If preferences are strongly monotonic (or locally non-satiated) then L X ∗ pl xl = m: l=1 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 8 / 49 First Order Condition (4) 2 In the case L = 2 and X = R+ these conditions are: ∗ ∗ u1 p1 if x1 > 0 and x2 > 0 then = u2 p2 u1 p1 ∗ ∗ if < then x1 = 0 and x2 > 0 u2 p2 u1 p1 ∗ ∗ if > then x1 > 0 and x2 = 0 u2 p2 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 9 / 49 Interior Solution L = 2 x2 6 ∗ ∗ (x1 ; x2 ) u(x1; x2) =u ¯ p1x1 + p2x2 = m - x1 0 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 10 / 49 r Corner Solution L = 2 u(x1; x2) =u ¯ x2 6 p1x1 + p2x2 = m (x∗; x∗) 1 2 - x1 0 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 11 / 49 Sufficient Conditions The conditions we stated are merely necessary. What about sufficient conditions? Result If u(·) is quasi-concave and monotone, ru(x) 6= 0 for all x 2 X , then the Kuhn-Tucker first order conditions are sufficient. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 12 / 49 Sufficient Conditions (2) Result If u(·) is not quasi-concave then a u(·) locally quasi-concave at x∗, where x∗ satisfies FOC, will suffice for a local maximum. Local (strict) quasi-concavity can be verified by checking whether the determinants of the bordered leading principal minors of order r = 2;:::; L of the Hessian matrix of u(·) at x∗ have the sign of (−1)r : Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 13 / 49 Sufficient Conditions (3) The Hessian is: 0 @2u @2u 1 2 ··· @x @x @x1 1 L B . C H = B . .. C @ A @2u @2u @x @x ··· 2 1 L @xL The bordered leading principal minor of order r of the Hessian is: ∗ T Hr [ru(x )]r ∗ [ru(x )]r 0 Hr is the leading principal minor of order r of the Hessian matrix H ∗ ∗ and [ru(x )]r is the vector of the first r elements of ru(x ): Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 14 / 49 Sufficient Conditions (4) In the case L = 2 the second order conditions hold whenever the determinant @2u @2u @u 2 @x @x @x @x1 1 2 1 @2u @2u @u > 0: @x @x 2 @x 1 2 @x2 2 @u @u 0 @x1 @x2 That is equivalent to: d @u=@x 2 < 0 dx1 @u=@x1 @u=@x2 p1 provided that = and p1x1 + p2x2 = m. @u=@x1 p2 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 15 / 49 Marshallian Demands Definition (Marshallian Demands) The Marshallian or uncompensated demand functions are the solution to the utility maximization problem: 0 1 x1(p1;:::; pL; m) B . C x = x(p; m) = @ . A xL(p1;:::; pL; m) Notice that strong monotonicity of preferences implies that the budget constraint will be binding when computed at the value of the Marshallian demands. (building block of Walras Law) Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 16 / 49 Indirect Utility Function Definition The function obtained by substituting the Marshallian demands in the consumer's utility function is the indirect utility function: V (p; m) = u(x∗(p; m)) We derive next the properties of the indirect utility function and of the Marshallian demands. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 17 / 49 Properties of the Indirect Utility Function @V @V 1 ≥ 0 and ≤ 0 for every i = 1;:::; L. @m @pi 2 V (p; m) continuous in (p; m). It rules out situations in which the consumption feasible set is non-convex (e.g. indivisibility). Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 18 / 49 Properties of the Indirect Utility Function (2) 3 V (p; m) homogeneous of degree 0 in (p; m). Definition r F (x) is homogeneous of degree r iff F (k x) = k F (x) 8k 2 R+ Proof: Multiply both the vector of prices p and the level of income m by the same positive scalar α 2 R+ we obtain the budget set: B(α p; α m) = fx 2 X j α p x ≤ α mg = B(p; m) hence the indirect utility (and Marshallian demands) are the same. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 19 / 49 Properties of the Indirect Utility Function (3) 4 V (p; m) is quasi-convex in p, that is: fp j V (p; m) ≤ kg is a convex set for every k 2 R. Proof: let p, m and p0, be such that: V (p; m) ≤ k V (p0; m) ≤ k: and p00 = tp + (1 − t)p0 for some 0 < t < 1. We need to show that also V (p00; m) ≤ k. Define: B = fx j (p x) ≤ mg B0 = x j (p0 x) ≤ m B00 = x j (p00 x) ≤ m Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 20 / 49 Properties of the Indirect Utility Function (4) Claim It is the case that: B00 ⊆ B [ B0 Proof: Consider x 2 B00, then p00x = [tp + (1 − t)p0] x = t (p x) + (1 − t)(p0 x) ≤ m which implies either p x ≤ m or/and p0 x ≤ m, or x 2 B [ B0. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 21 / 49 Properties of the Indirect Utility Function (5) Now V (p00; m)= max u(x) s:t: x 2 B00 fxg ≤ max u(x) s:t: x 2 B [ B0 fxg = max V (p; m); V (p0; m) ≤ k Since by assumption: V (p; m) ≤ k and V (p0; m0) ≤ k. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 22 / 49 Properties of the Marshallian Demand x(p; m) 1 x(p; m) is continuous in (p; m), (consequence of the convexity of preferences). 2 xi (p; m) homogeneous of degree 0 in (p; m). Proof: Once again if we multiply (p; m) by α > 0: B(α p; α m) = fx 2 X j α p x ≤ α mg = B(p; m) the solution to the utility maximization problem is the same. Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 23 / 49 Constrained Envelope Theorem Consider the problem: max f (x) x s.t. g(x; a) = 0 The Lagrangian is: L(x; λ, a) = f (x) − λ g(x; a) The necessary FOC are: @g(x∗; a) f 0(x∗) − λ∗ = 0 @x g(x∗(a); a) = 0 Francesco Squintani EC9D3 Advanced Microeconomics, Part I August, 2020 24 / 49 Constrained Envelope Theorem (2) Substituting x∗(a) and λ∗(a) in the Lagrangian we get: L(a) = f (x∗(a)) − λ∗(a) g(x∗(a); a) Differentiating, by the necessary FOC, we get: dL(a) @g(x∗; a) d x∗(a) = f 0(x∗) − λ∗ − d a @x d a dλ∗(a) @g(x∗; a) −g(x∗(a); a) − λ∗(a) da @a @g(x∗; a) = −λ∗(a) @a In other words: to the first order only the direct effect of a on the Lagrangian function matters.
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