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A consumer of two goods has utility u(₁,2)=12. She can purchase each good i = 1,2 at a price of p, dollars per unit. Alternatively, she can buy a discount card for fixed fee of e dollars that allows her to purchase good 1 at a price of p₁/2 dollars per unit. (The discount card has no effect on the price of good 2.) The discount card is of no value to the consumer except insofar as it reduces the price she pays for good 1. Find this consumer's Marshallian demand for each e > 0. Solution: If w 0. Solution: From part (a), this consumer's indirect utility function is 5 (p, u) = max {4P1P2 2P1P2) w² w-cl Using the identity v(p, e(p, u)) = u, this corresponds to e(p, u) = min {2√/P₁P2u, √/2p₁p₂u+c}.

A consumer of two goods has utility u(₁,2)=12. She can purchase each good i = 1,2 at a price of p, dollars per unit. Alternatively, she can buy a discount card for fixed fee of e dollars that allows her to purchase good 1 at a price of p₁/2 dollars per unit. (The discount card has no effect on the price of good 2.) The discount card is of no value to the consumer except insofar as it reduces the price she pays for good 1. Find this consumer's Marshallian demand for each e > 0. Solution: If w 0. Solution: From part (a), this consumer's indirect utility function is 5 (p, u) = max {4P1P2 2P1P2) w² w-cl Using the identity v(p, e(p, u)) = u, this corresponds to e(p, u) = min {2√/P₁P2u, √/2p₁p₂u+c}.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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A consumer of two goods has utility u(x₁, 72) = 2₁2. She can purchase each good i = 1,2 at a price of p, dollars per unit. Alternatively, she can buy a discount card for fixed fee of c dollars that allows her to purchase good 1 at a price of p₁/2 dollars per unit. (The discount card has no effect on the price of good 2.) The discount card is of no value to the consumer except insofar as it reduces the price she pays for good 1. Find this consumer's Marshallian demand for each e > 0. Solution: If w<c, then she cannot purchase the card. If w2c, if she does not purchase the card, she obtains utility v(p, w) = w²/4p₁p2, whereas if she purchases the card, she obtains utility v((p₁/2, p₂), wc) = (w - c)²/2p1p2, where v(p, w) is the usual indirect utility function associated with u(x) = 12. Hence she prefers to purchase the card whenever (w - c)²/2p1p2 ≥ w²/4p1p2, or equivalently, w≥ √2c/(√2-1). Therefore, using the expression from class for the Marshallian demand with Cobb-Douglas utility, we have x(p, w) = (2(x-c) c PI 1 (6) P₁ P2, w2√že √2-1 otherwise. Find this consumer's expenditure function for each c > 0. Solution: From part (a), this consumer's indirect utility function is w² w-cl {4PAP2' ²PLP2}· Using the identity v(p, e(p, u)) = u, this corresponds to e(p, u) = min {2√Pip2u, √2₁2+ (p, w) = max +c}.
Transcribed Image Text:A consumer of two goods has utility u(x₁, 72) = 2₁2. She can purchase each good i = 1,2 at a price of p, dollars per unit. Alternatively, she can buy a discount card for fixed fee of c dollars that allows her to purchase good 1 at a price of p₁/2 dollars per unit. (The discount card has no effect on the price of good 2.) The discount card is of no value to the consumer except insofar as it reduces the price she pays for good 1. Find this consumer's Marshallian demand for each e > 0. Solution: If w<c, then she cannot purchase the card. If w2c, if she does not purchase the card, she obtains utility v(p, w) = w²/4p₁p2, whereas if she purchases the card, she obtains utility v((p₁/2, p₂), wc) = (w - c)²/2p1p2, where v(p, w) is the usual indirect utility function associated with u(x) = 12. Hence she prefers to purchase the card whenever (w - c)²/2p1p2 ≥ w²/4p1p2, or equivalently, w≥ √2c/(√2-1). Therefore, using the expression from class for the Marshallian demand with Cobb-Douglas utility, we have x(p, w) = (2(x-c) c PI 1 (6) P₁ P2, w2√že √2-1 otherwise. Find this consumer's expenditure function for each c > 0. Solution: From part (a), this consumer's indirect utility function is w² w-cl {4PAP2' ²PLP2}· Using the identity v(p, e(p, u)) = u, this corresponds to e(p, u) = min {2√Pip2u, √2₁2+ (p, w) = max +c}.
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