1. (25 points) (Simultaneous Move Games) Suppose a company needs to be able to sell to 15 percent of the Fortune 500 if it is to be able to maintain the revenue and market presence to be profitable in the market for tax preparation software for large companies. If a company does not expect to get 15 percent, the only way it can avoid losses is to exit the industry immediately.
Currently, Megasoft has 30 percent of the market, Brydox has 20 percent, Apex has 20 percent, and two other companies have 15 percent each. Megasoft has just had an idea. It will approach each of the Fortune 500 companies, including its own customers, and ask them to sign an exclusive contract, that requires the company not to buy from Brydox or other firms for the next 15 years. Megasoft will charge 4 million dollars per year for the software, which is the monopoly price and well above the current price of 1 million dollars. It will, however, give the customer a 500,000 dollar one-time bonus if the customer signs up within the day. Megasoft has planned to secretly sabotage the nation's phone lines for that day to prevent the customers from phoning each other to ask what is going on. (The Megasoft sales reps will go to the customers in person.) Megasoft has paid large campaign contributions to both Congress and President to suppress any legal trouble. But Megasoft is still not sure the idea will work, and if it doesn't, the customers will be angry enough that Megasoft will lose market share.
(a) What are two pure-strategy Nash equilibria for this game? Assume that all players are rational, and they all know that everyone understands the situation and knows all the possible payoffs.
ANSWER. Equilibrium 1. Megasoft offers the exclusive contracts. All 500 customers accept.
In Equilibrium 1, Megasoft has no incentive to deviate (except to a lower signing bonus). No customer does either. If any customer refuses to sign the exclusive contract, then since Megasoft will have locked up 499 other customers, Apex, Brydox, and the two other firms will go out of business anyway, and the non-signing customer will end up having to pay Megasoft's price anyway, but without a signing bonus.
Equilibrium 2. Megasoft does not offer the exclusive contracts. If it did, no customer would accept.
In Equilibrium 2, Megasoft has no incentive to deviate, because if it offered the exclusive dealing contract, no customer would sign it, and Megasoft would simply lose goodwill. No customer would accept, because each customer knows that no other customer will sign, and so he can go to Apex and get similar software for a much cheaper price.
(b) What argument should the Megasoft sales representatives make to get the customers to sign on?
ANSWER. The Megasoft sales rep should say that every other Fortune 500 company is going to buy this product, and so competitors like Brydox don't stand a chance and will disappear within the year. This is therefore the only software that is going to be available, and the customer is going to have to pay the monopoly price, with or without the signing bonus. So the customer might as well sign up now and get the bonus.
2. (25 points) (Nash Equilibrium) Apex Inc. must decide whether to build a new widget factory or not. If it does, then Apex and Brydox (a competitor) will decide simultaneously whether to produce metric widgets or American ones (the machinery can't be set to do both). The net revenues from this are shown below. In addition, Apex must pay 3 million if it enters. What pure strategy equilibria exist for this game?
Brydox
American Metric
Apex American 1,1 4,1
Metric 2,4 0,0
ANSWER. There are two pure-strategy equilibria. Equilibrium 1. Apex builds the new factory. Then Apex chooses American and Brydox chooses Metric.
This is an equilibrium because the payoffs are 1 for Apex and 1 for Brydox, and neither can do better by deviating. If Apex doesn't enter, it gets 0. If it switches to Metric, it gets -3. If Brydox switches to American, it still gets 1.
Equilibrium 2. Apex does not build the new factory. If Apex did, it would choose Metric and Brydox would choose American.
This is an equilibrium because the payoffs are 0 for Apex and 0 for Brydox, and neither can do better by deviating. If Apex does enter, it gets -1. If it enters and switches to American, it gets -2. If Brydox switches to Metric following entry, it still gets 1 compared to the 4 it would get if entry occurred and it played American.
Note that even tho metric is weakly dominated for Brydox, Brydox might still play it in a Nash equilibrium. Sometimes, the only equilibrium is a weak Nash equilibrium in which one or more players plays a weakly dominated strategy. The fact that it is weakly dominated does not matter, tho, because if the players really think the Nash equilibrium will be played out, they ignore the higher payoffs they could get from other strategies.
This question could be expanded to make it more difficult, by asking about mixed strategies and about forward induction.
3. (25 points) (Reaction Curves) Apex and Brydox are involved in still another titanic battle, this time in the market for beer. Each is advertising in the national market for its flagship brand, Apex for its Epoch and Brydox for its Doxbreider beer. Draw reaction curves for the firms' choice of advertising, and explain why you drew them the particular shape you did.
ANSWER. The reaction curves should be for advertising, not for prices or quantity. One way to draw them is as strategic substitutes, downward sloping. This means that if Apex increases its advertising, Brydox will advertise less. That would happen if Apex's advertising serves to weaken the benefits to Brydox of its own advertising-- for example, if consumers get sick of beer ads and stop watching them.
A second way to draw them is as strategic complements, upward sloping. This means that if Apex increases its advertising, Brydox will advertise more. That would happen if Apex's advertising serves to strengthen the benefits to Brydox of its own advertising-- for example, if consumers start drinking more beer, and are open to which brand they drink more of.
4. (25 points) (Moral Hazard) A company that sells mainframe computers relies heavily on the efforts of its salesmen. Every salesman can sell at least one computer per year. To sell a second computer, however, requires either luck, high effort, or a combination of the two. A salesman who works hard has a 40 percent probability of selling a second computer, while a salesman who slacks off has a 20 percent probability.
Salesmen shift companies frequently, so their contracts must be designed to get results on a year-by-year basis. Salesmen do not mind risk; they choose employers based only on expected wage and the disutility of effort, which is equivalent to $10,000 per year if they work hard and 0 if they slack off. Even if he slacks off, a salesman requires an average salary of at least $40,000 or he will go work for someone else.
a. Design an incentive contract for a salesman that will induce him to work hard to sell two computers.
ANSWER: Suppose the salesman is paid A if he sells one computer and B if he sells two computers.
The participation constraint is that 40000 = .6A + .4B. -10000, so .4B = 50000 -.6A, and B = 125000 - 1.5A.
The incentive constraint is that .6A + .4B. -10000 = .8A + .2B. This solves to .2B - .2A = 10,000, so B = 50000 +A.
Thus, 125000 -1.5A = 50000 +A, so 75000 = 2.5A, and A = 30000. Then, B= 80000.
If he works hard, his expected wage is .4(80) +.6(30) = 50000.
b. Another company faces a similar problem. This company, however, exists in an alternative universe in which the American South won the Civil War and slavery still exists. All of its salesmen are slaves, and unable to shift employers. There is, however, a law forbidding masters from whipping or otherwise physically punishing their slaves. Thus, in effect the employer cannot have negative wages as part of the contract. (a) Design an incentive contract for a salesman that will induce him to work hard to sell two computers. (b) What is the salesman's expected wage under this contract?
ANSWER: Suppose the salesman is paid A if he sells one computer and B if he sells two computers.
There is no participation constraint now, tho the company is constrained to make both A and B non-negative.
The incentive constraint is the same as before, that .6A + .4B. -10000 = .8A + .2B. This solves to .2B - .2A = 10,000, so B = 50000 +A.
The company should set A to be as low as possible, so set A=0. Then, B=50,000.
If the slave salesman works hard he has an expected wage of .6(0) +.4(50) =20,000.