Question 1: In Puebla there are two bicycle repair shops, (A and B) with unlimited capacity to repair bicycle tires, and the quality of their service is identical. There are 300 bicycle users in Puebla who have a flat tire. Each user has a RP of $5 for the repair of their tire. Since they only have one flat tire, no user is interested in getting more than one tire repaired.

100 of the bicycle users are loyal A customers. They are ignorant of the price that B charges and no matter what the price (as long as it is no more than $5) they always buy from A. 100 buyers are loyal B customers who are ignorant of the price that A charges. No matter what the price (as long as it is no more than $5) they always buy from B. The remaining 100 are price sensitive. They are aware of the prices that each shop charges and patronize the one that gives them the largest surplus. In case of ties, the price sensitive buyers flip a fair coin to decide which firm to patronize. In the questions below assume a 'one shot game'.

1. Explain why both shops advertising a price of $5 per bicycle repair and committing to charge the advertised price to everyone is not an equilibrium?

2. Suppose both shops announce an advertised price matching guarantee. Specifically, each shop guarantees to match the advertised price of the other. What price will each shop advertise in equilibrium? Only price sensitive buyers can take advantage of this guarantee. Assume each shop is committed to charge the advertised price to anyone who does not invoke the guarantee.

3. A continues to use an advertised price matching guarantee. B is contemplating a shift to a price beating guarantee. Specifically, B advertises a price and guarantees to refund 150% of the difference plus a penny to a customer if A has an equal or lower advertised price. For example, if A prices at $2 and B prices at $3, B must charge to every buyer who invokes the guarantee a price of $3 and refund them $1.51. Should B make the shift and if so what should their advertised price be? Once again, only price sensitive buyers can invoke the guarantee. Explain.

4. Would your answer to part (3) above change if A's price matching guarantee applied to the actual sale price instead? Explain.

Question 2: Two firms are competing in a market. Firm 1 and Firm 2 simultaneously announce quantities, q₁ and q₂. The price charged in the market is given by p = 1 - q₁ - 2q₂. Both Firm 1 and Firm 2 have 0 marginal cost of production.

1. What are equilibrium quantities?

2. Consider a different situation where there are two shareholders. Mr.A owns 80% of firm 1's stock and 20% of firm 2's stock. Thus, Mr.A is entitled to 80% of firm 1's profit and 20% of firm 2's profit. Additionally, Mr.A has full control over firm 1's quantity choice, q1. Mr.B owns 20% of firm 1's stock and 80% of firm 2's stock. Thus, Mr.B is entitled to 20% of firm 1's profit and 80% of firm 2's profit. Additionally, Mr.B has full control over firm 2's quantity choice, q₂.

Two shareholders simultaneously announce quantities.

What are equilibrium quantities?

Question 3: Bruno and Pascal work together in a partnership, sharing profits equally. Let b denote Bruno's effort level and p denotes Pascal's effort level. The profits from the firm that they jointly own are given by:

π = 4(b + p + γbp)

Assume 0 < γ < 1/4. The cost to Bruno of exerting b units of effort is b². The cost to Pascal of exerting p units of effort is p².

The profit of the firm are split evenly between Bruno and Pascal. Assume that Bruno and Pascal simultaneously choose effort, and that both b and p are between 0 and 4.

1. If Bruno chooses b units of effort and Pascal chooses p units of effort, write down Bruno's payoff.

2. If Bruno chooses b units of effort and Pascal chooses p units of effort, write down Pascal's payoff.

3. Derive Bruno's and Pascal's reaction functions.

4. What is the effort level each would choose in equilibrium?

5. What effort levels would maximize total payoff?

6. Why aren't Bruno and Pascal choosing the effort levels that would maximize their payoff?

7. Suppose that you want to increase Bruno's and Pascal's payoff in equilibrium. How would you change γ?

Question 4: Cocos Inc. is the monopoly supplier of cocos. The demand for cocos is

D(p) = 100 - 2p

and Cocos Inc's marginal cost of production of cocos is $10.

Mr. Smith wants to sell cocos, but he has a marginal cost of production of $20. If Mr. Smith sells cocos, him and Cocos Inc. will simultaneously choose quantities to sell (each choosing a quantity to sell, and will face the demand function shown above). Everybody knows all this information.

Cocos Inc. wishes to disuade Mr. Smith from entering the market. What deal can it offer Mr. Smith to deter him from entering the market?

Assume that if Mr. Smith enters the market firms will choose a quantity to sell, but the demand does not change, ie the demand above is the total demand in the market, and both firms will post the same price. This means that inverse demand is p(qtot) = 100-q_tot/2, where q_tot is the total quantity supplied by Mr. Smith and Cocos Inc.

Assume also that if Mr. Smith is indifferent between accepting the deal and rejecting, he will reject.