A consumer of two goods has utility u(x1, x2) = x1x2. 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 consume except insofar as it reduces the price she pays for good 1. (a) () Find this consumer's Marshallian demand for each c > 0. Solution: If w 0. Solution: From part (a), this consumer's indirect utility function is e(p, u) = W W P1' P2 w² w- { AP1P2 ² ²P+72) Using the identity v(p, e(p, u)) = u, this corresponds to n{2√/p₁p2u, √2p1p2u+ = min if √2-1 otherwise. = max
A consumer of two goods has utility u(x1, x2) = x1x2. 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 consume except insofar as it reduces the price she pays for good 1. (a) () Find this consumer's Marshallian demand for each c > 0. Solution: If w 0. Solution: From part (a), this consumer's indirect utility function is e(p, u) = W W P1' P2 w² w- { AP1P2 ² ²P+72) Using the identity v(p, e(p, u)) = u, this corresponds to n{2√/p₁p2u, √2p1p2u+ = min if √2-1 otherwise. = max
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![1. A consumer of two goods has utility u(x1, x2) = x12. 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.
(a) () Find this consumer's Marshallian demand for each c > 0.
-
Solution: If w<c, then she cannot purchase the card. If w≥c, 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), w c) = (w c)²/2p1p2, where v(p, w) is the usual indirect
utility function associated with u(x) = x1x2. Hence she prefers to purchase the card
whenever (w - c)2/2p1P2 2 w²/4p1p2, or equivalently, w 2 √2c/(√2-1). Therefore,
using the expression from class for the Marshallian demand with Cobb-Douglas utility,
we have
x(p, w)
[(
2(w-c) wic
P2
P1
W W
P1 P2
w>√2c
√2-1
otherwise.
if
(b) () Find this consumer's expenditure function for each c > 0.
Solution: From part (a), this consumer's indirect utility function is
(p, w) = max.
w-c)
{4P1P2² 2P1P²}
Using the identity v(p, e(p, u)) = u, this corresponds to
e(p, u) = min {2√/P1P2u, √√20₁2 + +c}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7d1c47f-ebeb-4433-a23d-a68418d8e04e%2F5a2633f1-a71d-47cd-92fb-ba1d85c9e1d9%2F79l645f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. A consumer of two goods has utility u(x1, x2) = x12. 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.
(a) () Find this consumer's Marshallian demand for each c > 0.
-
Solution: If w<c, then she cannot purchase the card. If w≥c, 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), w c) = (w c)²/2p1p2, where v(p, w) is the usual indirect
utility function associated with u(x) = x1x2. Hence she prefers to purchase the card
whenever (w - c)2/2p1P2 2 w²/4p1p2, or equivalently, w 2 √2c/(√2-1). Therefore,
using the expression from class for the Marshallian demand with Cobb-Douglas utility,
we have
x(p, w)
[(
2(w-c) wic
P2
P1
W W
P1 P2
w>√2c
√2-1
otherwise.
if
(b) () Find this consumer's expenditure function for each c > 0.
Solution: From part (a), this consumer's indirect utility function is
(p, w) = max.
w-c)
{4P1P2² 2P1P²}
Using the identity v(p, e(p, u)) = u, this corresponds to
e(p, u) = min {2√/P1P2u, √√20₁2 + +c}.
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