Recall that the Cobb-Douglas Utility function U(X,Y) = XaY1-a has the

Recall that the Cobb-Douglas Utility function U(X,Y) = XaY1-a has the unusual property that the demand for each good depends only on its own price.

Therefore, a consumer will always allocate the same proportion of income to each good. Specifically, the demand for X is
X* = aI/px
where I is income and px is the price of X.
a. What is the price elasticity of demand for X?
b. What is the direction of the income effect on X of an increase in px?

ANSWER

a. Using calculus,
∂X/∂px = -aI/px2
Then E = -1. Because a constant proportion of income is devoted to each good, an increase in price will be matched by an equal offsetting proportional decrease in quantity.
b. X is a normal good from the demand equation. Therefore, the income effect from a price increase will be to reduce consumption of X.