Problem 1. Alice and Barbra sell used CDs at music festivals. Each is deciding whether or not to set up their booth at the last festival of the summer. The festival is scheduled to take place in Alton, very near where Alice lives. It will cost her only $30 to travel the festival. Barbra is farther away, and it will cost her $100 to travel to Alton. Both Alice and Barbra would prefer to be only CD sellers at the festival, since they would avoid competition. If only one seller is at the festival, she will make $150 during the day (not counting travel costs). If both Barbra and Alice sell CDs at the festival, they will lower their prices and each make $50 during the day. Both Alice and Barbra receive $0 for not attending the festival.
1. Draw the Normal Form of the game Alice and Bob are playing, be sure to label the game completely.
2. Does either player have a dominant strategy? If so, what is it?
3. List all pure strategy Nash equilibria for this game. Remember that a Nash equilibrium is a strategy profile.
Problem 2. Let market demand in the cement industry be given by Q(P) = 200 P. There are only two firms in the industry and the total cost function for each firm is C(qi) = 20qi + 400, where i = 1; 2.
1. What is the marginal cost for both firms?
2. Represent the market price as a function of each firm's output, q1 and q2.
3. Find the best-response function of each firm.
4. Suppose firm 2 produces nothing, that is, q2 = 0. What is firm 1's optimal output level? Compare this quantity with the monopoly output level (if there was only one firm with the same cost curve).
5. Using the Cournot model, find each firm's output, profit, and price in equilibrium.
6. Now suppose technical innovation makes each firm's fixed cost is zero, that is, C(qi) = 20qi, where i = 1; 2. Find each firm's output, profit, and price.
Compare with them before the innovation.
7. What is consumer surplus in equilibrium? How does it compare with consumer surplus under monopoly?
8. Is the equilibrium outcome efficient? Why or why not?
Problem 3. The market for a new carbohydrate-free strand of rice is served by three firms, AsiaRice, Ben's Rice, and CountryRice (or A, B, and C). These firms are engaged in (simultaneous) Cournot competition. Total demand for a ton of carb-free rice is described by P = 4000 Q, where Q is the total quantity produced by all firms in the market. The three firms have an identical total cost function Ci(qi) = 400qi for i = fa; b; cg.
1. Write AsiaRice's profit as a function of the quantities produced by each firm qa; qb; and qc?
2. What is the best response function of AsiaRice?
3. Suppose (for this part only) that Ben's Rice faces a lower marginal cost of 200. So it's total cost function is Cb(qb) = 200qb. Would your answer to part 2 change? Why or why not?
4. What is the equilibrium strategy profile for this game (Hint: note that firms are symmetric)?
5. What is the equilibrium price of a ton of carb-free rice?
6. How much profit does each firm earn in equilibrium?
7. Suppose Ben's and CountryRice propose a merger|reducing the number of firms in the industry to two. Would you expect the total quantity produced in the industry to rise or fall following the merger? Explain your answer, but you do not need to calculate the post-merger quantity produced.
Problem 4. Two different companies have discovered a cure for the common cold. Theraquick has the exact same effectiveness as Cureall, but is more expensive to produce. Whereas Cureall's cost curve is C(qc) = 60qc. The cost curve of Theraquick is C(qt) = 140qt. Demand for a cure to the common cold is P = 700 4Q. Both Cureall and Theraquick are owned by profit maximizing firms who compete under Cournot competition.
1. What is the profit maximization problem for Cureall?
2. What is Cureall's best response function?
3. What is the profit maximization problem for Theraquick?
4. What is Theraquick's best response function?
5. What is the Nash equilibrium strategy profile of this game?
6. What are the equilibrium profits of each firm?
7. If Theraquick were to exit the market, would consumers be better off, worse off, or neither better nor worse off? Explain your answer.