Assignment

This assignment covers consumer behavior, choice under uncertainty, and production decisions. Please answer the questions below, which are related to Chapters 4 through 8 of P&R. Beyond reviewing the material from class, the problem sets are intended to develop the ability to apply the concepts from class to new situations. You are encouraged to collaborate with others, to research sources outside of class, and to ask questions during office hours. However, you must write up your responses individually.

You should explain all of your answers in detail. Your score will depend on both the correctness of your solutions and the completeness of your explanations. Please write the question number next to your answer for each question.

1. Olive has the utility function U(X,Y) = X^1/3Y^2/3, 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 P_X^a = 2 and the price of oranges is P_Y = 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 P_X^b = 1 and the price of oranges stays constant at P_Y = 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)?]

2. Penny has the utility function U(I) =√I, where I is the level of income. Penny's ?income is 16 with probability 1/2 and 36 with probability 1/2.

(a)  Is Penny risk averse or risk loving?

(b)  What is Penny's expected utility?

(c)  What is the certainty equivalent for the lottery above? [Hint: what is the fixed amount of money that generates the same expected utility as in part (b)?]

(d)  If Penny were to buy insurance, then her income after paying the insurance premium would be 22 with probability 1/2 and 26 with probability 1/2. Would Penny choose to purchase insurance?

3.  A consumer has the utility function U(Y) = Y^1/2, where Y denotes income. There are two states A and B each occurring with probability 1/2. If the consumer obtains S units of insurance, then her income in state A is Y = 16 - S, and her income in state B isY =4+2S/3.

(a)  What is the certainty equivalent of the income distribution when S = 0?

(b)  What is the expected utility of the consumer when S > 0?

(c)  How many units of insurance should the consumer obtain to maximize her expected utility?

(d)  Explain whether the insurance is priced at an actuarially fair level.

4. Suppose that a firm has the production function F(K,L) = K^1/4L^1/4, where K and L respectively denote capital and labor. Let r = 2 be the price of K and w = 2 be the price of L. Assume that both K and L are variable.

(a) Find the greatest output that the firm can produce at a total cost of C.

(b) What are the total, marginal, and average costs as a function of output Q?

(c) What is the elasticity of the total cost with respect to output?

(d) Explain whether the cost structure displays economies or diseconomies of scale.

5. Consider a competitive firm in the short run. The total cost of production is given by TC = q³/3 - q² + 3q + 1. Let P denote the output price.

(a) What is the output level that maximizes the profits of the firm if P = 6?

(b) What is the maximum profit level for the firm when P = 6?

(c) What is the producer surplus of the firm if P = 6?

(d) What is the lowest price P at which the firm stays open in the short run?

6. Consider a competitive industry composed of identical firms. Assume that the market is in a long-run equilibrium. The long-run total cost of each firm is given by T C = 0 ifq=0andbyTC=q²+1ifq>0. LettheaggregatedemandbegivenbyQ=100-10P. Assume that this is a decreasing-cost industry.

(a) What is the long-run equilibrium output of each firm?

(b) What is the long-run equilibrium price?

(c) What is the long-run equilibrium number of firms?

(d) Explain how a rise in output demand would affect the long-run equilibrium price.

7. This question is for fun. Imagine that you are in the following real life situation. You are interviewing for a job at a retail store. The interviewer asks you what classes you are taking and what you are learning in class. You tell her that you are studying microeconomic theory and that you are learning about profit maximization. The interviewer becomes excited. She asks you for advice on increasing the profits of the store. What do you tell the interviewer?