ECON 444 ECON444 Midterm 2 Answers (Penn State University)
ECON 444 ECON444 Midterm 2 with Answers (Penn State University)
Problem 1. (35 points, each part is weighted equally.) Answer the following questions as best you can based on what you have learned from the course. Partial answers will receive partial credit.
Give an example of a game we have studied that exhibits strategic complementarily (that is, is a game of strategic complements). Explain your answer.
Explain why, in a two-firm homogeneous Bertrand pricing game where marginal cost is c for both firms, the strategy profile p1 = p2 = c is an equilibrium.
In the Hotelling model, is the cost of traveling an example of horizontal or vertical differentiation? Explain your answer.
You are playing a sequential game. Would you rather go first, second, or do you need more information to decide? Explain your answer.
Suppose you win 3 thousand dollars in the lottery. You can receive your winnings as 1 thousand dollars for three years, or an upfront payment of $2,500. Your discount factor is .8, which should you choose? Explain your answer.
In the Hotelling Model we studied in class, is consumer surplus higher or lower when the game is played sequentially rather than simultaneously.
What does it mean for a strategy profile to be a Subgame Perfect Nash Equilibrium (SPNE)?
Problem 2. (45 points) Suppose Delta and United are two companies operating flights from State College to Chicago. Delta’s flight departs at 1 PM while United’s departs at 5 PM. Assume there’re 200 consumers who would like to fly to Chicago and their preferred departure times are uniformly distributed between 1 PM and 5 PM. Each person values the flight at $400 and dislikes leaving at a time that’s not their preferred time by $50 per hour. The marginal cost of serving an additional passenger is zero for both airlines. The FAA regulations have changed such that Delta and United must simultaneously set a single price for all seats on the flight, these prices are pD and pU respectively. You may assume that all 200 consumers will purchase a ticket from one airline or the other.
What is the utility of flying United’s flight for a consumer whose preferred departure time is x PM?
Find the preferred time of the consumer who’s indifferent between flying Delta and
United given prices (pD , pU ). Call this marginal consumer’s preferred time xm (pD , pU ).
3. What’s the demand curve for Delta? What’s the demand curve for United?
4. Write down Delta and United’s profit functions, call them πD(pD,pU) and πU(pD,pU).
5. Given the profit functions, what’s their best response functions (in terms of the other airline’s price)?
6. What is the equilibrium prices? How much profit does each company make?
Now suppose Delta has purchased United. That is Delta is now a monopoly and operates both 1 PM and 5 PM flights. In order to approve the merger, the government requires that Delta set prices such that all 200 consumers still want to purchase. and Delta must charge the same price for both flights. Given that delta serves all consumers, write down Delta’s profits as a function of this, single, price?
Write down the highest price that Delta can legally set? (Hint: Consider the utility of the marginal consumer xm)
9. What is the optimal price for Delta? How much profit does Delta make?
Problem 3. (40 points) The demand for beer is P (Q) = 240 − Q at State College (Q is in terms of cases). Beer Belly’s and Nittany Beverage are duopolies in the market and com- peting in a squential Cournot game. Suppose Beer Belly’s has a better logistic management and sets its quantities qB first. Nittany then responses by choosing qN. Total quantity is then Q = qB + qN . Assume that the marginal cost of a case of beer is $20 for both stores.
Given that Beer Belly’s has chosen quantity qB, write down the profit function for Nittany as a function of qB and qN .
What is Nittany’s optimal quantity as a function of qB? Call it qN∗ (qB).
Beer Belly’s knows that Nittany’s strategy is given by the function you found above. Write down Beer Belly’s profit function in terms of its quantity qB only.
What is Beer Belly’s optimal quantity (qB∗ ) ?
Find Nittany’s equilibrium quantity (qN∗ ). What’s the equilibrium price?
How much profit does each store make?
Calculate the consumer surplus.
Is the outcome efficient? Why?
Problem 4. (35 Points) Suppose Cafe Buzz and Ishmael’s Cofee are playing a game by setting prices for iced coffee. Each of them has two choices: either a high price (H) or a low price (L). If they agree to “fix” prices (both play H), then each gets $5 (thousands) in profit. If one of the them plays L and the other plays H, the firm with lower price gets $10 (thousands) and the other gets $0. If they both play L, each receives profit $2 (thousands).
1. Draw the normal form of the game. Be sure to label the game completely.
What is the Nash Equilibrium if the game is played only once? Verify your answer.
Suppose the game is played twice, are the firms able to collude? What prices will they set in first and second period? Explain your answer
Now suppose the game is repeated with an infinite horizon. Firms value future profits with a discount factor of δ, where 0 ≤ δ < 1. State the Grim Trigger strategy.
What is the payoff for each firm if they both play the Grim Trigger strategy (as a function of δ)?
The discount rate is still δ. Assume the firms are attempting to collude using Grim Trigger strategy. What is the payoff for a firm that deviates from this strategy profile immediately (that is, deviates in the initial period)?
7. What is the smallest value for δ that facilitates collusion under Grim Trigger strategy?
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