1. State whether each of the following statements is true or false. Explain your answers clearly. A counter-example or a diagram may be useful in some situations.

(a) Consider a monopolist facing a market demand given by p(q) = 15 2q Statement: Without any additional information (on costs etc.), we can deduce that the monopolist will produce some quantity

q M > 15 /4

(b) Statement: The no free lunch assumption on production sets states that the firm can always shut down without having to incur any cost.

2. A researchers production function for papers is given by f(K, L) = min {2L, 2K}, where L is the number of research assistants and K is the number of computers.

(a) Find the conditional factor demand for computers and research assistants (tht is, Solve the firms cost minimization problem).

(b) What is the minimum cost that the researcher must incur in terms of a research assistants wage w and and the rent for computers r if she wants to write 5 papers? Hint: First, find the cost function, then plug in what you know about factor demand.

3. Depict a production set that is convex, closed, and allows for inaction. Can you say anything about the returns to scale that it exhibits?

4. Suppose there are two firms in an economy, and that the production set of firm 2 is a subset of the production set of firm 1 (Y2 Y1). Show that at any equilibrium, profit of firm 2 cannot exceed profit of firm 1.

5. Consider the inverse demand function p(q) = 15 2q.

a. Express the price elasticity of demand in terms of q. How does it change with q?

b. Consider a monopolist facing its market demand given by the linear function above. Without any additional information (on costs etc.), can you deduce the range of outputs the monopolist will possibly produce?

c. Consider a market consisting of 100 individuals, the demand of each is given by the linear function above. Derive the market demand function. Sketch the individual and market demand curves on the same diagram. [Hint: Given a certain price p, suppose that there are two individuals in the market. What is the total demand from both individuals? The sum of their demands.

6. A monopolist faces a demand function given by q = 15 p , where p denotes the price of the good while q denotes the quantity demanded. The monopolist has access to two different technologies to produce output, the first one is called A and the second one is called B. Producing q units of the good using technology A costs 3q + 2q 2 . Producing the same using B costs 6q + q 2 2 .

(a) How much of the good would the monopolist produce if forced to use technology A?

(b) If forced to use technology B, how much of the good would the monopolist produce?

(c) If the monopolist could choose which technology to use, which would be chosen?

7. Suppose Keshon starts a firm making high-end basketball shoes. He has a production function q(L, K) = AL1 2 K 1 2

(a) State and solve his cost minimization problem to find his factor demands.

(b) Suppose the firm hires L = 16 and rents capital K = 121. Let A = 1. How many units will it produce?

(c) How does the marginal product of each input behave as L, K, and q increase?

(d) Suppose the wage w = 7 and rental rate of capital r = 10. For L = 16 and K = 121, what is the value of the marginal product of labor? Assume w = selling price of good (constant MR).

(e) Now, as a result of an employee training program hosted by the Chicago Bulls training facility, the overall technological level becomes A = 3. For the same L = 16 and level of capital K = 121, calculate how many units will it produce?

(f) Plot units of labor L and q(L, K), keeping K constant. Show the production function from parts b and e in your figure. What does the function appear to do when the technological level A increases from 1 to 3?

(g) What does the Law of Diminishing Returns say? What aspect of the production function suggests there are diminishing returns to the inputs?