Problem 1. Two gas stations, A-Ok Oil and X-on Service Station, compete to sell gas in a very small town. Every day, they choose whether to charge $3.00 or $2.50 for a gallon of gas. If they both choose the same price they split the market and each sells 500 gallons. If they choose different prices, the station with the lower price sells 800 gallons, while the station with the higher price sells only 200 gallons. The marginal cost of gasoline is $1 per gallon. The firms discount future profits by a factor of ? = .9 per day.
1. If this game were played for a single day, what would be the static Nash equilibrium profile?
2. What is the present discounted value of profits of A-Ok Oil if both firms play the static Nash equilibrium profile in every period? Is this equilibrium of the dynamic game?
3. What are the static Nash equilibrium strategies for this market? What are profits for a single period in this case?
4. Suppose the two firms agree to maximize joint profits rather than individual profits and share the proceeds equally. How many chips does each firm agree to make? What are firms profits for a single period?
5. Suppose the firms colluded to maximize joint profits in every period. What is the present discounted value of profits of a single firm in this case?
6. Suggest a strategy profile which is an equilibrium and is able to maximize joint profits in every period.
7. Show that the strategy profile you suggest is in fact equilibrium.
8. Suppose that the government decides to restrict the ability of gas stations to change their price. It regulates the industries so that stations are only allowed to change their price every 3 days. Does this change your analysis of the game? If so, how? If not, why not?
Problem 2. Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1chips and firm two produces q2chips and the firms face a demand curve of P = 50 ? 10Q, where Q = q1 + q2. Both firms have a constant marginal cost of $20 per chip, C(qi) = 20qi. Suppose the same industry exists for an infinite number of periods, the discount factor for profits is ? = .8. Firms are considering the following strategy:
• As long as no firm has cheated, play according to the agreement to optimize joint profits and split the proceeds.
• If either firm has cheated in any earlier period, play the static Nash equilibrium in every period from now on.
1. What are the static Nash equilibrium strategies for this market? What are profits for a single period in this case?
2. Suppose the two firms agree to maximize joint profits rather than individual profits and share the proceeds equally. How many chips does each firm agree to make? What are firms profits for a single period?
3. Suppose the firms have agreed to maximize joint profits, but while firm 2 produces according to the agreement, firm 1 decides to cheat and maximize individual profits instead. How many chips does firm 1 decide to produce? What are profits for each firm?
4. If both firms follow the proposed strategy, what is the present discounted value of each firm’s profits?
5. Based on your previous answers in this question, is the proposed strategy an equi- librium? Why or why not?
6. What is the lowest discount factor, ? that the firms could have for the proposed strategy to be an equilibrium?
Problem 3. Suppose there is a 1 mile beach where a mass of 1 beach goers uniformly distributed. There are two ice cream vendors. Each day ice cream vendors need to determine where they want to locate. After their locations become common knowledge, they then simultaneously determine how much to charge for their ice cream. Assume that the ice creams are identical so the beach goers only care for the distance that they need to travel and the price of the ice cream. Suppose that it is very hot and beach goers values an ice cream so high that they are going to purchase one ice cream. The transport costs increase with the square of distance:
1. Write down the utility of a consumer whose position x from purchasing from firm 1 located at l1
2. Write down both firms’ optimization problem in the price stage, assume firm 1 is the firm on the left, i.e., l1< l2 3. Find the price equilibrium given locations of both firms l1< l2 4. Write down firms profits given the equilibrium pricing strategy: 5. Solve for the locations that both firms choose in the equilibrium.