Intermediate Microeconomics Assignment
Q1. Write down the equation of a utility function and draw an indifference curve that corresponds to each of the following cases.
(a) Carol is always willing to give up 2 pizzas in exchange for 1 additional beer no matter how many pizzas and beers she has in her consumption bundle.
(b) Bob always consumes beer and pizza in fixed proportions. Specifically, if Bob has 1 pizza, he consumes precisely 2 beers and additional beers represent no additional utility for Bob. Similarly, if Bob has 2 beers, he consumes precisely 1 pizza and additional pizzas represent no additional utility for Bob. In addition, when Bob has 4 pizzas and 10 beers, his utility is 32.
(c) Kathy loves pizza and is neutral toward beer (i.e., she neither likes nor dislikes beer). She derives 4 units of satisfaction from each pizza she consumes.
Q2. Suppose the indifference curves of a consumer are given by x2(x1) = c - 5x1. Give an example of a utility function that represents the underlying preferences.
Q3. Suppose the utility function is given by U(x1, x2) = (x12x25)5/4. Calculate the demand function, that is, calculate the optimal consumption bundle as a function of m, p1, and p2.
Q4. Suppose the utility function is given by U(x1, x2) = 14 min{2x, 3y}. Calculate the demand function, that is, calculate the optimal consumption bundle as a function of m, p1, and p2.
Q5. Jeremy's favorite flowers are tulips x1 and daffodils x2. Suppose p1 = 2, p2 = 4 and m = 40.
(a) Write down Jeremy's budget constraint (a formula) and plot all Jeremy's affordable bundles in the graph (his budget set). Find the slope of the budget line (number). Give an economic interpretation for the slope of the budget line (one sentence).
(b) Jeremy's utility function is given by U(x1, x2) = √((lnx1 + lnx2)2 + 7). Propose a simpler utility function that represents the same preferences (give a formula). Explain why your utility represents the same preferences (one sentence).
(c) Plot Jeremy's indifference curve map (graph), find the MRS analytically (give a formula) and find its value at bundle (2, 4) (one number). Give an economic interpretation of this number (one sentence). Mark its value in the graph.
(d) Find the optimal bundle (x1, x2) (two numbers). Is your solution interior?
(e) Find the optimal bundle given p1 = 2, p2 = 4 and m = 40 assuming U(x1, x2) = 2x1 + 3x2. Is your solution interior?
Q6. Suppose prices are given by p1 = 2 and p2 = 4. Which of the following consumption patterns are consistent with the assumption that the consumer chooses the consumption bundle that he prefers the most among those affordable if his preferences are monotone ("more-is-better") and strictly convex (the strict version of "averages-are-better-than-extremes"?
(a) If m = 10, he consumes (2, 1). If m = 14, he consumes (3, 2).
(b) If m = 12, he consumes (2, 2). If m = 18, he consumes (2, 4).
(c) If m = 12, he consumes (2, 2). If m = 18, he consumes (1, 4).
(d) If m = 12, he consumes (2, 2). If m = 18, he consumes (1, 1).
(e) If m = 10, he consumes (1, 2). If m = 16, he consumes (2, 3).
(f) If m = 10, he consumes (1, 2). If m = 12, he consumes (0, 3).