Intermediate Microeconomic Theory Problem Set
1. The frog and the scorpion (from Varian (2010)). We begin with the fable of the frog and the scorpion. They were standing on the bank of the river, trying to figure out a way across. I know, said the scorpion I will climb on your back and you can swim across the river. The frog said, But what if you sting me with your stinger? The scorpion said, Why would I do that? Then we would both die. The frog found this convincing, so the scorpion climbed on his back and they started across the river. Halfway across, at the deepest point, the scorpion stung the frog. Writhing in pain, the frog cried out, Why did you do that? Now we are both doomed! Alas, said the scorpion, as he sank into the river, it is my nature. Lets look at the frog and the scorpion from the viewpoint of game theory.
(a) Depict the sequential game with payoffs consistent with the story. Start at the bottom of the game tree. If the frog refuses the scorpion, both get nothing. Looking up one line, we see that if the frog carries the scorpion, he receives utility 5, for doing a good deed, and the scorpion receives a payoff of 3, for getting across the river. In line where the frog is stung, he receives a payoff of -10, and the scorpion gets a payoff of 5, representing the satisfaction from fulfilling his natural instincts.
(b) Solve the game by backwards induction. What is the SPNE of the game?
2. Consider a monopolist that faces a linear inverse demand of: P(q) = 304 - 4q The firm has the cost function: C(q) = 100 + 4q + 2q 2 What are the monopolistic market price (pM), quantity(qM), and profits (πM)? Show derivations (neatly!) below. pM = qM = πM =
3. (Final Fall 2005) Suppose you become a successful entrepreneur of computers and other electronic devices, with your firm called PineApple. You are a monopolist in the market, with a cost function c(Q) = 2Q2 + 25000. There are two groups of consumers, college students and everyone else. The demand curve of your devices for non-students is Q1(p) = 250 - 1 5 p. The demand curve for students is Q2(p) = 600 - 1 2 p.
(a) If you could segment the market into students and non-students. What quantity is sold in each market? What is total quantity sold?
(b) What price does the monopolist charge in each market?
(c) How much profit does the firm make?
(d) In equilibrium, what is the monopolist's elasticity of demand in each of the two markets?
(e) Circle the correct choice: Comparing your answers to (b) and (d), this firm charges a higher price in the market in which its demand curve is more (elastic / inelastic).
4. Consider a Cournot duopoly with both firms producing a homogeneous good and facing linear demand of the form: P(Q) = 124 - 30Q, where Q = q1 + q2. Assume both firms have the same costs: C(q) = 4q.
(a) For each firm, derive the best response function below. Show all work.
(b) Graph the best response functions for both firms on the axes below. As in class, q1 should be on the y - axis and q2 should be on the x - axis.
(c) Solve for the Nash Equilibrium output of each firm.
(d) Solve for the equilibrium market price.
(e) Solve for the equilibrium profits of each firm.
5. Stackelberg Duopoly with Heterogeneous Costs. Now consider a Stackelberg duopoly with both firms producing a homogeneous good and facing a linear demand of the form P(Q) = 138 - 6Q where Q = q1 + q2. Assume the firms have different costs. Firm 1 has costs C1(q) = 3q + 3 and Firm 2 has costs of C2(q) = 6q + 12. Firm 1 will be the first mover in this case.
(a) What is the Equilibrium output of each firm under Stackelberg Competition?
(b) What are the equilibrium profits of each firm?