Question

Your utility function over x and y is U ( x , y ) = l n ( x ) + 0.25 y. Your income is $20. You don’t know the prices of x or y so leave them as variables (p x and p y).

a) (8 points) Find x*, your demand function for x. Find y*, your demand function for y.

b) (10 points) Find the cross-price elasticity of demand for x (E x ∗ , p y: the elasticity of x* with respect to p y ). Is x a substitute or a complement for y, or neither? Find the cross-price elasticity of demand for y (E y ∗ , p x: the elasticity of y* with respect to p x).   Is y a substitute or a complement for x, or neither? HINT: your two answers here should be different!

c) (10 points) Assume that p y = 1. In a graph with x* on the horizontal axis and p x on the vertical axis, plot the points corresponding to p x = 1, p x = 2, and p x = 4 (hint: leave yourself some extra space on this graph for the next part of this problem). Draw the demand curve for x through those points. It does not have to be accurate beyond having a slope that is the right sign. Label this curve x*. Once you have done that, imagine that there are two identical consumers in the market. Add market quantity demanded when p x = 1, p x = 2, and p x = 4, and draw the market demand curve through these points. Label this curve Q D.

Answer 1