(4) Consider a Cobb-Douglas utility function U(1,2) = ria, where U is the utility level, x, and 2 are the amounts of two commodities, respectively, and a > 0, b> 0, a+b=1. Suppose that the commodity prices for 1 and 2 are p and P2, respectively. (a) Set up the utility maximization problem subject to the budget constraint such that Pix+P2x2 = I, where I is the income level. (b) Suppose that P1, P2, and I are given exogenously. Solve the utility-maximizing levels of r and r2. [Hint: Derive the demands for 1 and 2, respectively, as a function of pi. P2, and I. (c) Compute the indirect utility function V (P1, P2, I). [Hint: Write the utility as a func- tion of P1, P2, and I.] (d) Verify the Roy's identity such that av (pi PJ) др OV (p1.p2.1) ar for i = 1,2.
(4) Consider a Cobb-Douglas utility function U(1,2) = ria, where U is the utility level, x, and 2 are the amounts of two commodities, respectively, and a > 0, b> 0, a+b=1. Suppose that the commodity prices for 1 and 2 are p and P2, respectively. (a) Set up the utility maximization problem subject to the budget constraint such that Pix+P2x2 = I, where I is the income level. (b) Suppose that P1, P2, and I are given exogenously. Solve the utility-maximizing levels of r and r2. [Hint: Derive the demands for 1 and 2, respectively, as a function of pi. P2, and I. (c) Compute the indirect utility function V (P1, P2, I). [Hint: Write the utility as a func- tion of P1, P2, and I.] (d) Verify the Roy's identity such that av (pi PJ) др OV (p1.p2.1) ar for i = 1,2.
Microeconomics A Contemporary Intro
10th Edition
ISBN:9781285635101
Author:MCEACHERN
Publisher:MCEACHERN
Chapter6: Consumer Choice And Demand
Section: Chapter Questions
Problem 6QFR
Related questions
Question
![(4) Consider a Cobb-Douglas utility function U(1,2) = ria, where U is the utility
level, x, and 2 are the amounts of two commodities, respectively, and a > 0, b> 0, a+b=1.
Suppose that the commodity prices for 1 and 2 are p and
P2, respectively.
(a) Set up the utility maximization problem subject to the budget constraint such that
Pix+P2x2 = I, where I is the income level.
(b) Suppose that P1, P2, and I are given exogenously. Solve the utility-maximizing levels
of r and r2. [Hint: Derive the demands for 1 and 2, respectively, as a function of pi. P2,
and I.
(c) Compute the indirect utility function V (P1, P2, I). [Hint: Write the utility as a func-
tion of P1, P2, and I.]
(d) Verify the Roy's identity such that
av (pi PJ)
др
OV (p1.p2.1)
ar
for i = 1,2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21e37180-5079-46cf-8bca-5e4b3f7ff116%2F816cabaa-dffb-4fd2-b465-ee3fecc4f2c4%2Fa70m59_processed.png&w=3840&q=75)
Transcribed Image Text:(4) Consider a Cobb-Douglas utility function U(1,2) = ria, where U is the utility
level, x, and 2 are the amounts of two commodities, respectively, and a > 0, b> 0, a+b=1.
Suppose that the commodity prices for 1 and 2 are p and
P2, respectively.
(a) Set up the utility maximization problem subject to the budget constraint such that
Pix+P2x2 = I, where I is the income level.
(b) Suppose that P1, P2, and I are given exogenously. Solve the utility-maximizing levels
of r and r2. [Hint: Derive the demands for 1 and 2, respectively, as a function of pi. P2,
and I.
(c) Compute the indirect utility function V (P1, P2, I). [Hint: Write the utility as a func-
tion of P1, P2, and I.]
(d) Verify the Roy's identity such that
av (pi PJ)
др
OV (p1.p2.1)
ar
for i = 1,2.
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