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 Uðx, yÞ¼ðxd þ yd

Þ

1=d

[in this function the elasticity of substitution s ¼ 1/(1 – d)].

a. Show that the indirect utility function for the utility function just given is V ¼ Ið pr x þ pr yÞ

#1=r

, where r ¼ d/(d – 1) ¼ 1 – s.

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 ¼ Vð pr x þ pr yÞ

1=r

: