Econ 431: David Reiley
Due Monday, 12 February 2007

## Problem Set #3

Remember to get an early start on this problem set, so that I can help you with questions and problems you may run into. As before, you may discuss these questions with your classmates as much as you like. However, you are responsible for preparing your own independent writeup, in your own words. Merely copying someone else's work constitutes academic dishonesty, and will be punished.

The first problem asks you to rewrite the text of Problem Set #1, based on the feedback you have received. The rest of the problem set comes from the material in chapters 5 and 6.

2. (10 points) DS, question 5.2.
3. (15 points) DS, question 5.5. I have two clarifications on this question. First, the problem says that the consumers are uniformly distributed along the beach, so x is the fraction of customers who buy from cart 0 and 1-x is the fraction of customers who buy from cart 1, but it doesn't say how many total customers there are on the beach. Let's suppose that there are a total of 1000 customers on the beach, so that x represents the number of customers (in thousands) who buy from cart 0. Second, in part (a) you are being asked to write down each firm's demand as a function of the prices p0 and p1.
4. (5 points) DS, question 5.7.
5. (5 points) DS, question 5.8. Note that in the first printing of the second edition of this textbook, there is a typo in this problem and in the original discussion on page 150. In the third round of reasoning, we should restrict X to the range between 9 and 12.75. The correct number should be 12.75 instead of 13.75 in both places.
6. (20 points) DS, question 6.1. In addition to problem 3.2(a), please also do this for problem 3.2(c). Remember from our discussion in class that 3.2(c) had eight possible strategies per player, but when we eliminated redundant strategies we only got four strategies per player. You can save yourself a lot of work if you only consider four strategies per player.
7. (15 points) DS, question 6.3. For help with part (c), see the last three paragraphs of Section 6.4. That discussion applies just as well to two-player games as it does to three-player games.
8. (15 points) DS, question 6.4.
9. (20 points) DS, question 6.5. In part (d), you may find that you get the same equilibria both ways. If you do, discuss how coordination on a single equilibrium may be easier in one version of the game versus the other.
10. (20 points) DS, question 6.7. For help with part (c), see the last three paragraphs of Section 6.4. For part (d), you don't have to put the matrix in strategic form: just use the tree to find the subgame-perfect equilibrium for the new order of moves, and describe how and why it is different from the answer in part (c). Consider differences in strategies, not just payoff outcomes.
11. (15 points) DS, question 6.9.
12. (40 points) This is a two-stage game involving continuous strategies, combining the concepts in Chapters 5 and 6. Two firms, Bilge and Chem, compete in the soft-drink market in the town of Saint James. They sell identical products, and since their good is a liquid, they can easily choose to produce fractions of units. Since they are the only two firms in this market, the price of the good (in dollars), P, is determined by P = (30 - QB - QC), where QB is the quantity produced by Bilge and QC is the quantity produced by Chem (each measured in liters).

At this time both firms are considering whether to invest in technology order to lower their variable costs. Without investment, each firm's total variable cost is: CN, j = Qj2 / 2,
where N stands for No Investment and j stands for either B (Bilge) or C (Chem).

With investment, each firm's variable cost is: CI, j = Qj2 / 6, where I stands for Investment and j stands for either B (Bilge) or C (Chem).

The fixed cost of investment equals \$20. If the firm chooses not to invest, fixed costs are zero.
1. Suppose both firms decide to invest. Write the profit functions in terms of QB and QC for the two firms. Use these to find the Nash equilibrium to the quantity-setting game. What are the equilibrium quantities and profits for both firms? What is the market price?
2. Now suppose both firms decide not to invest. What are the equilibrium quantities and profits for both firms? What is the market price?
3. Now suppose that Bilge decides to invest and Chem decides not to invest. What are the equilibrium quantities and profits for both firms? What is the market price?
4. Write out the 2 x 2 game table of the investment game between the two firms. Each firm has two strategies: Investment and No Investment. The payoffs are simply the profits found in parts a, b, and c. (Note the symmetry of the game.)
5. What is the subgame-perfect equilibrium to the overall two-stage game?
13. (Extra credit,15 points) DS, question 5.10.
14. (Extra credit, 15 points) Consider a game similar to the political-advertising game of Chapter 5. We have an election where the Democratic party chooses to spend x dollars on advertising, and the Republican party chooses to spend y dollars on advertising. The advertising shares determine the vote shares earned by each party. Suppose that the payoff to each party equals the percentage vote share minus the dollars spent on advertising, so that the Democratic payoff is 100x/(x+y)-x, and the Republican payoff is 100y/(x+y)-y.
1. Find the Nash equilibrium to this game.
2. Find the set of rationalizable strategies for each player, and show that the Nash equilibrium strategy is part of the set of rationalizable strategies for each player.