1. Katie spends her income m on two goods, r and y. Prices of r and y are given by p. and Py, respectively. Her preferences are represented by the following utility function: U(x, y) = - Inz + -Iny (a) What kind of preferences are represented by this utility function? Roughly sketch two indifference curves with a on the horizontal axis and y on the vertical axis. (3 points) (b) What do these indifference curves indicate for Katie's preferences for z and y? In partic- ular: are these goods perfect substitutes, perfect complements or imperfect substitutes? Explain why. (3 points) (c) Write down Katie's Utility Maximisation Problem and the Lagrangian associated with that problem. (2 points) (d) Using the Lagrange method, find the demand functions z(Pz, py, m) and y(P.. pu, m) that solve Katie's maximisation problem. (9 points) (e) Draw the Engel curves for goods r and y. Make sure you label your graph carefully, including giving the intercept and the slope of the Engel curves. Are goods z and y normal or inferior goods? Why? (6 points) (f) Given p2 = p. = 1 and m = 100, how much of goods a and y does Katie consume at the optimum? (2 points) (g) Now, assume that pz changes to p. = 2. Compute the change in demand for good z. What is the proportion of this total effect that is due to the Slutsky substitution effect? (8 points) (Please turn over) (h) Using your answer in part (g) and considering the same change in prices, what would have been the proportion of the total effect on the demand for good a due to the Slutsky substitution effect if Katie's preferences have been represented by the utility function ! Ing? (2 points)