Pam has the utility function U(X,Y)=X^(1/3)*Y(2/3) , where X is the quantity of apples consumed, and Y is the quantity is the quantity of oranges consumed. Let income be I=90.

a) suppose the price of apples is Px=2 and the price of oranges is Py=2. What are the quantities of apples and oranges demanded when Pam maximizes her utility subject to the budget constraint?

b) Suppose that the price of apples decreases to Px=1 and the price of oranges stays constant at Py=2. What are the quantities of apples and oranges demanded by Pam after the price change?

c) What is the substitution effect from the price change above? [Hint: what is the expenditure minimizing way of achieving the utility level in part (a) at the prices in part (b)?]

d) What is the income effect from the price change above? [Hint: what is the difference between the total effect in part (b) and the substitution effect in part (c)?]

Problem set 1: Preferences and Utility Exercise 1: Suppose the utility function for two goods, X and Y, has the Cobb-Douglas form utility= U(X, Y) = VX . Y. a. Graph the U = 10 indifference curve associated with this utility function. b. If X = 5, what must Y equal to be on the U = 10 indifference curve? What is the MRS at this point? c. In general, develop an expression for the MRS for this utility function. Show how this can be interpreted as the ratio of the marginal utilities for X and Y. d. Consider a logarithmic transformation of this utility function: U' = log U where log is the logarithmic function to base 10. Show that for this transformation the U' = 1 indifference curve has the same properties as the U' = 10 curve calculated in part (a) and (b). What is the general expression for the MRS of this transformed utility function? Exercise 2: Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether they obey the assumption of diminishing MRS): a. U = 3X + Y. b. U = VX . Y. c. U = VX2 + Yz. d. U = vx2 _ yz e. U = X2/3p1/3 f. U = log* + logY. Exercise 3: Consider the following utility functions: a. U(X, Y) = XY. b. U(X, Y ) = x2yz c. U(X, Y) = InX + InY. Show that each of these has a diminishing MRS, but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?1. (16 points) Olive has the utility function U(X, Y) = X1/3y2/3, where X is the quantity of apples consumed, and Y is the quantity of oranges consumed. Let income be I = 90. (a) Suppose that the price of apples is Po = 2 and the price of oranges is Py = 2. What are the quantities of apples and oranges demanded when Olive maximizes her utility subject to her budget constraint? (b) Suppose that the price of apples decreases to Py = 1 and the price of oranges stays constant at Py = 2. What are the quantities of apples and oranges demanded by Olive after this price change? (c) What is the substitution effect from the price change above? Hint: what is the ex- penditure minimizing way of achieving the utility level in part (a) at the prices in part (b)? ] (d) What is the income effect from the price change above? [Hint: what is the difference between the total effect in part (b) and the substitution effect in part (c)?]