Problem 1. (7 points)
Suppose Philip’s utility function over two goods, 1 and 2, is given by U(q1, q2) = 2q10.5 + q2.
Let p1, p2, and Y denote prices and income.
(a) Derive his demand functions for the two goods. (3 points)
(b) Derive his Engel curves for each of the two goods. Briefly discuss the interesting results. (4 points)
Problem 2. (7 points)
Pat eats eggs and toast for breakfast and insists on having three pieces of toast for every two eggs he eats.
(a) Write down a utility function that would represent his preferences over bundles of eggs and toast. (1 point)
(b) Suppose the price of eggs increases. Show in a graph how Pat’s optimal consumption bundle will change. Be sure to label all of the lines, curves, and bundles, and explain your graph. (4 points)
(c) How much of the total effect is driven by the income effect, and how much by the substitution effect? (2 points)
Problem 3. (6 points)
Suppose Don spends his money only on food and operas. Furthermore, Don considers food to be an inferior good.
(a) Does he view an opera performance as an inferior good or a normal good? Explain your answer carefully. (3points)
(b) In a graph with opera performances on the horizontal axis and food on the vertical axis, draw a possible income-consumption curve for Don. (3 points)
Problem 4. (10 points)
Marvin has a Cobb-Douglas utility function, U(q1, q2) = q10.5 q20.5, his income is Y = $100, and, initially, he faces prices of p1 = 1 and p2 = 2.
(a) Derive Marvin’s uncompensated demand functions for the two goods, D1(Y, p1, p2) and D2(Y, p1, p2). (2 points)
(b) Derive Marvin’s compensated demand functions for each good, H1(U, p1, p2) and H2(U, p1, p2). (2 points)
(c) Suppose p1 increases to 2. Calculate the change in his consumer surplus. (Hint: You should work carefully through Solved Problem 5.1 in the textbook) (2 points)
(d) For the same change in p1 (i.e. increase from 1 to 2), use the expenditure function to calculate Marvin’s compensating variation (CV) and equivalent variation (EV). (4 points)