5.13 The almost ideal demand system The general form for the expenditure function of the almost ideal demand system (AIDS) is given by ln Eð p1,..., pn, UÞ ¼ a0 þXn i¼1 ai ln pi þ

1 2

Xn i¼1 Xn j¼1 gij ln pi ln pj þ Ub0 Y

k i¼1 p

bk k , For analytical ease, assume that the following restrictions apply:

gij ¼ gji, Xn i¼1 ai ¼ 1, and Xn j¼1 gij ¼ Xn k¼1 bk ¼ 0:

a. Derive the AIDS functional form for a two-goods case.

b. Given the previous restrictions, show that this expenditure function is homogeneous of degree 1 in all prices. This, along with the fact that this function resembles closely the actual data, makes it an ‘‘ideal’’ function.

c. Using the fact that sx ¼ d ln E d ln px (see Problem 5.8), calculate the income share of each of the two goods.