Intermediate Microeconomic Theory Fall 2013 Homework #1 – Part B Choice: Exercise 7: Assume preferences can be represented by the following utility function: u(x1; x2) = 3 ln (x1) + ln (x2) a. Is the utility function monotonic? Justify. b. How can this person be better o造than by consuming the bundle (x1; x2) = (10; 10)? c. Set up the utility maximization problem for the consumer, when facing prices p1 = 6; p2 = 2 and income m = 32: d. Solve the problem by nding (x1; x2) : e. Graph the budget set, a couple of indi造erence curves and the optimal choice. Exercise 8: Assume preferences can be represented by the following utility function: u(x1; x2) = ??x12 + 150×1 ?? 2×22 + 100×2 + x1 x2 a. Is the utility function monotonic? Justify. b. How can this person be better o造than by consuming the bundle (x1; x2) = (100; 100)? c. Set up the utility maximization problem for the consumer, when facing: prices p1 = 2; p2 = 1 and income m = 30: d. Solve the problem by nding (x1; x2) : Exercise 9: Assume preferences can be represented by the following utility function: u(x1; x2) = 4 ln (x1) + x2 a. Is the utility function monotonic? Justify. b. Set up the consumers utility maximization problem for prices p1; p2 and income m (the general case) c. Solve the problem. You will obtain solutions x1(p1; p2;m) ; x2(p1; p2;m) in terms of the parameters of the model (p1; p2;m) : 1
Demand: Exercise 10: You are the owner of a supermarket that wants to understand your clients preferences so that you can optimally price your products. You record a clients purchases of two products x1 and x2 in 8 di造erent occasions. The following table summarizes the results (for a similar exercise see Varian, ch.5, sec.4): Obs. p1 p2 m x1 x2 1 3 2 94 12 29 2 2 2 106 15 38 3 2 3 95 10 25 4 1 3 170 20 50 5 3 1 83 15 38 6 2 1 135 30 75 7 4 1 143 22 55 8 1 2 168 28 70 a. Notice observations number 2 and 5. Quantities purchased are the same but prices are not. What does this mean in terms of the marginal rate of…