A consumer has preferences over goods x and y that can be represented by the utility function (,)=+() where ln is the (natural) log function. The consumer has income I (all to be spent on x and y) and the price of x and y are px and py respectively. (You may assume the "at least as good as x" set B(x) is a convex set, so the solution to the consumer's problem will be a maximum).

a. Solve the consumer's problem (maximise utility subject to the budget constraint) to derive this consumer's demand function for x: =(,,).

b. Find the consumer's income elasticity of demand for good x (as a function of income and prices). What kind of good is x for this consumer?

c. Suppose I = 20 and py = 1. Originally, =1 but the price of x then increases to the new price =2. Find the substitution and income effect.

d. Calculate the change in Consumer Surplus from the price change.