A consumer has preferences represented by the utility function U(x1, x2) = ln x1 + x2 . Denote prices and income by p1, p2 and I respectively.

a) Write the utility maximization problem and get the first order conditions.

b) Solve the system of equations and find the demand functions (i.e. the optimal choice of x1 and x2 as a function of prices and income).

c) Suppose initially p1 = 10, p2 = 50 and I = 100. What is the optimal choice of x1 and x2? Compute, then draw a graph with the budget constraint and the indifference curve showing the optimal point.

d) Suppose now that the price of good 2 increases, becoming p2 = 120 (still p1 = 10 and I = 100). How does the optimal choice of x1 and x2 change? [Be careful: do you still have an interior solution? can you still use the conditions as in the previous case? how do you compute the optimal x1 now?] How does the MRS compare to the relative price? Compute and show graphically what happens, explaining briefly the economic reason of what you find.