(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.

ENGR.ECONOMIC ANALYSIS
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Author:NEWNAN
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Chapter1: Making Economics Decisions
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(a) Suppose the indirect utility function v(p,m) is differentiable. Prove that
dv(p, m)
dv(p,m)
Pi
+ m-
дт
= 0.
(b) Consider a differentiable quasi-lincar utility function u(x) = x +(x2,X3), where x,
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 dx,(p, m)/ap; = ah;(p, u)/dp; for
i = 2 and 3.
(c) Prove that the profit function (p, w) is convex in input prices w.
ide
Transcribed Image Text:(a) Suppose the indirect utility function v(p,m) is differentiable. Prove that dv(p, m) dv(p,m) Pi + m- дт = 0. (b) Consider a differentiable quasi-lincar utility function u(x) = x +(x2,X3), where x, 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 dx,(p, m)/ap; = ah;(p, u)/dp; for i = 2 and 3. (c) Prove that the profit function (p, w) is convex in input prices w. ide
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