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.
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.
Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.13P
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Questions g) and h)
![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%2Fe519f3f8-6201-4da0-bbaf-b27202e9c84e%2Frch34tg_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.
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