U(x, y) = (xª + y®)!/a in this function the elasticity of substitution o = 1/(1 – ò)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p;,)¯/", where r = d/(d - 1) = 1 - 6. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and in c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by
U(x, y) = (xª + y®)!/a in this function the elasticity of substitution o = 1/(1 – ò)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p;,)¯/", where r = d/(d - 1) = 1 - 6. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and in c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by
Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.13P
Related questions
Question
![4.13 CES indirect utility and expenditure functions
In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure
functions. Suppose utility is given by
U(x, y) = (x° +y®)'/8
[in this function the elasticity of substitution o = 1/(1 – 6)].
a. Show that the indirect utility function for the utility function just given is
V = I(p, + p,)¬/",
where r = 8/(ò – 1) = 1 – 0.
b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income.
c. Show that this function is strictly increasing in income.
d. Show that this function is strictly decreasing in any price.
e. Show that the expenditure function for this case of CES utility is given by
E = V(p', + p,)''".
f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices.
g. Show that this expenditure function is increasing in each of the prices.
h. Show that the function is concave in each price.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb1776ef4-eb5a-42b4-8bd9-b41536cc546d%2Ff84ddcb8-4ffa-46eb-8c17-4bc7949e49bc%2Fc83u82m_processed.png&w=3840&q=75)
Transcribed Image Text:4.13 CES indirect utility and expenditure functions
In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure
functions. Suppose utility is given by
U(x, y) = (x° +y®)'/8
[in this function the elasticity of substitution o = 1/(1 – 6)].
a. Show that the indirect utility function for the utility function just given is
V = I(p, + p,)¬/",
where r = 8/(ò – 1) = 1 – 0.
b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income.
c. Show that this function is strictly increasing in income.
d. Show that this function is strictly decreasing in any price.
e. Show that the expenditure function for this case of CES utility is given by
E = V(p', + p,)''".
f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices.
g. Show that this expenditure function is increasing in each of the prices.
h. Show that the function is concave in each price.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Principles of Economics 2e
Economics
ISBN:
9781947172364
Author:
Steven A. Greenlaw; David Shapiro
Publisher:
OpenStax
