(a) Suppose the indirect utility function v(p, m) is differentiable. Prove that dv(p, m) av (p, m) 0. Σ am + m Pi api (b) Consider a differentiable quasi-linear utility function u(x) = x₁ + p(x2, X3), where x₁ can take on any value (including a negative value). Use the first-order conditions of the UMP and EMP for an interior solution to prove that ox, (p, m)/ap, = h,(p, u)/ap, for i = 2 and 3. (c) Prove that the profit function 7 (p, w) is convex in input prices w.

(a) Suppose the indirect utility function v(p, m) is differentiable. Prove that dv(p, m) av (p, m) 0. Σ am + m Pi api (b) Consider a differentiable quasi-linear utility function u(x) = x₁ + p(x2, X3), where x₁ can take on any value (including a negative value). Use the first-order conditions of the UMP and EMP for an interior solution to prove that ox, (p, m)/ap, = h,(p, u)/ap, for i = 2 and 3. (c) Prove that the profit function 7 (p, w) is convex in input prices w.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.13P

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