Chapter 21

A bicycle manufacturer (the "buyer," abbreviated B) wishes to procure a new robotic system for the production of mountain-bike frames. The firm contracts with a supplier (S), who will design and construct the robot. The contractual relationship is modeled by the following game: The parties first negotiate a contract specifying an externally enforced price that the buyer must pay. The price is contingent on whether the buyer later accepts delivery of the robot (A) or rejects delivery (R), which is the only event that is verifiable to the court. Specifically, if the buyer accepts delivery, then he must pay \(p_{1}\); if he rejects delivery, then he pays \(p_{0}\). After the contract is made, the seller decides whether to invest at a high level (H) or at a low level (L). High investment indicates that the seller has worked diligently to create a high-quality robot-one that meets the buyer's specifications. High investment costs the seller 10. The buyer observes the seller's investment and then decides whether to accept delivery. If the seller selected \(\mathrm{H}\) and the buyer accepts delivery, then the robot is worth 20 units of revenue to the buyer. If the seller selected \(\mathrm{L}\) and the buyer accepts delivery, then the robot is only worth 5 to the buyer. If the buyer rejects delivery, then the robot gives him no value. (a) What is the efficient outcome of this game? (b) Suppose the parties wish to write a "specific-performance" contract, which mandates that the buyer accept delivery at price \(p_{1}\). How can \(p_{0}\) be set so that the buyer has the incentive to accept delivery regardless of the seller's investment? Would the seller choose H in this case? (c) Under what conditions of \(p_{0}\) and \(p_{1}\) would the buyer have the incentive to accept delivery if and only if the seller selects H? Show that the efficient outcome can be obtained through the use of such an "option contract." (d) Fully describe the negotiation equilibrium of the game, under the assumption that the parties have equal bargaining weights.

Estelle has an antique desk that she does not need, whereas Joel and his wife have a new house with no furniture. Estelle and Joel would like to arrange a trade, whereby Joel would get the desk at a price. In addition, the desk could use restoration work, which would enhance its value to Joel. Specifically, the desk is worth 0 to Estelle (its current owner), regardless of whether it is restored. An unrestored desk is worth \(\$ 100\) to Joel, whereas a restored desk is worth \(\$ 900\). Neither Joel nor Estelle has the skills to perform the restoration. Jerry, a professional actor and woodworker, can perform the restoration at a personal cost of \(\$ 500\). Jerry does not need a desk, so his value of owning the restored or unrestored desk is 0 . (a) Suppose Estelle, Jerry, and Joel can meet to negotiate a spot contract specifying transfer of the desk, restoration, and transfer of money. Model this as a three-player, joint-decision problem, and draw the appropriate extensive form. Calculate the outcome by using the standard bargaining solution, under the assumption that the players have equal bargaining weights \(\left(\pi_{\mathrm{E}}=\pi_{\text {Jerry }}=\pi_{\text {Joel }}=1 / 3\right)\). Does the desk get traded? Is the desk restored? Is this the efficient outcome? (b) Suppose spot contracting as in part (a) is not possible. Instead, the players interact in the following way. On Monday, Estelle and Jerry jointly decide whether to have Jerry restore the desk (and at what price to Estelle). If they choose to restore the desk, Jerry performs the work immediately. Then on Wednesday, regardless of what happened on Monday, Estelle and Joel jointly decide whether to trade the desk for money. Model this game by drawing the extensive form. (Hint: The extensive form only has joint-decision nodes.) Assume the parties have equal bargaining weights at all joint-decision nodes. Determine the negotiation equilibrium. Compare the outcome with that of part (a). (c) Now suppose the players interact in a different order. On Monday, Estelle and Joel jointly decide whether to trade the desk for money. Trade takes place immediately. On Wednesday, if Joel owns the desk, then he and Jerry jointly decide whether to have Jerry restore the desk (and at what price to Joel). If they choose to restore the desk, Jerry performs the work immediately. Model this game by drawing the extensive form. (Hint: Again, the extensive form only has joint-decision nodes.) Assume the parties have equal bargaining weights at all jointdecision nodes. Determine the negotiation equilibrium. Compare the outcome with that of parts (a) and (b). (d) Explain the nature of the hold-up problem in this example.

Recall that "human capital" refers to skills and expertise that workers develop. General human capital is that which makes a worker highly productive in potential jobs with many different employers. Specific human capital is that which makes a worker highly productive with only a single employer. What kind of investment in human capital should you make to increase your bargaining power with an employer, general or specific? Why? Do valuable outside options enhance or diminish your bargaining power?

4\. This exercise asks you to combine the investment and hold-up issue from this chapter with the "demand" bargaining game explained in Exercise 4 of Chapter 19. Consider an investment and trade game whereby player 1 first must choose an investment level \(x \geq 0\) at a cost of \(x^{2}\). After player l's investment choice, which player 2 observes, the two players negotiate over how to divide the surplus \(x\). Negotiation is modeled by a demand game, in which the players simultaneously and independently make demands \(m_{1}\) and \(m_{2}\). These numbers are required to be between 0 and \(x\). If \(m_{1}+m_{2} \leq x\) (compatible demands, given that the surplus to be divided equals \(x\) ), then player 1 obtains the payoff \(m_{1}-x^{2}\) and player 2 obtains \(m_{2}\). In contrast, if \(m_{1}+m_{2}>x\) (incompatible demands), then player 1 gets \(-x^{2}\) and player 2 gets 0 . Note that player 1 must pay his investment cost even if the surplus is wasted owing to disagreement. (a) Compute the efficient level of investment \(x^{*}\). (b) Show that there is an equilibrium in which player 1 chooses the efficient level of investment. Completely describe the equilibrium strategies. (c) Discuss the nature of the hold-up problem in this example. Offer an interpretation of the equilibrium of part (b) in terms of the parties' bargaining weights.

Suppose that an entrepreneur is deciding whether or not to build a new highspeed railroad on the West Coast. Building the railroad will require an initial sunk \(\operatorname{cost} F\). If operated, the new railroad will generate revenue \(R\). Operating the railroad will cost \(M\) in fuel and \(n w\) in wages, where \(n\) is the number of full-time jobs needed to operate the new railroad and \(w\) is the career wage per worker. If a rail worker does not work on the new railroad, then he can get a wage of \(\bar{w}\) at some other job. Assume that \(R>M+F+n \bar{w}\), so it would be profitable to build and operate the new railroad even if rail workers had to be paid somewhat more than the going rate \(\bar{w}\). The entrepreneur, however, must decide whether to invest the initial sunk cost \(F\) before knowing the wages she must pay. (a) Suppose that if the railroad is built, after \(F\) is invested, the local rail workers' union can make a "take it or leave it" wage demand \(w\) to the entrepreneur. That is, the entrepreneur can only choose to accept and pay the wage demand \(w\) or to shut down. If the railroad shuts down, each worker receives \(\bar{w}\). Will the railroad be built? Why? (b) Next suppose that the wage is jointly selected by the union and the entrepreneur, where the union has bargaining weight \(\pi_{\mathrm{U}}\) and the entrepreneur has bargaining weight \(\pi_{\mathrm{E}}=1-\pi_{\mathrm{U}}\). Use the concept of negotiation equilibrium to state the conditions under which the railroad will be built. (c) Explain the nature of the hold-up problem in this example. Discuss why the hold-up problem disappears when the entrepreneur has all of the bargaining power. Finally, describe ways in which people try to avoid the hold-up problem in practice.

Describe a real-world setting in which option contracts are used.

Suppose that you work for a large corporation and that your job entails many hours of working with a computer. If you treat the computer with care, it will not break down. But if you abuse the computer (a convenience for you), then the computer will frequently need costly service. Describe conditions under which it is best that you, rather than your employer, own the computer. Discuss verifiability and incentives in your answer.

8\. Here is a description of interaction between two players who are considering a possible business partnership. First the players simultaneously choose whether to make an investment. Investment entails a personal \(\cos t\) of 3 ; not investing costs nothing. The investment choices become common knowledge. Then the players jointly decide whether to form a partnership firm and, if so, how to divide the profit from the firm. If both players invested, then the firm's profit is 16 . If exactly one player invested or if neither invested, then the firm's profit is 12 . If the players decide not to form the firm, then each player \(i\) gets a default payoff of \(x-3\) if player \(i\) invested and zero if player \(i\) did not invest. The default payoff of \(x-3\) includes the cost of investment plus some value \(x\) that represents what player \(i\) can obtain by using his investment in other endeavors. Assume that the players divide surplus according to the standard bargaining solution with equal bargaining weights. (a) What outcome maximizes the joint value? That is, what are the efficient investment choices? (b) Describe conditions on \(x\) such that there is a negotiation equilibrium in which both players invest. Show that this is an equilibrium. (c) In light of your answers to parts (a) and (b), briefly provide some intuition for your answers in relation to the "hold-up" problem.

Suppose that prior to negotiation with a firm, a worker chooses whether to invest (I) or to not invest (N). Investing entails a personal cost of 10 to the worker. Not investing entails a cost of zero. The manager of the firm observes the worker's investment choice and then negotiates with the worker on whether to hire him and, if so, at what salary. Assume that the outcome of their negotiation is given by the standard bargaining solution with equal bargaining weights. The benefit to the firm of hiring the worker depends on whether the worker has made the investment. Investment yields a benefit of 30 to the firm. Noninvestment yields a benefit of 16 to the firm. The firm's payoff is the benefit it receives less the salary it pays the worker. The worker's payoff is the salary received less the investment cost. If the worker is not hired, then the firm's payoff is zero and the worker's payoff is zero less the cost of investment. (a) In a negotiation equilibrium, what is the worker's investment decision and what is the outcome of the negotiation? Explain. (b) Is the outcome you found in part (a) efficient? Explain why or why not.