1. Consider two firms, one of which is the sole producer of a factor input that is used solely by the second. The producer of this good is character­ ized by increasing marginal costs of production for the good in question, and the user obtains decreasing marginal benefit (measured in terms of impact on profits) as it increases the scale of its use of this good. Both firms seek to maximize their profits. (You may assume that the marginal cost to produce the first unit of this good is less than the maripnal value to the user of the first unit, and for some large enough amount the marginal cost exceeds the marginal value. You may also assume that marginal cost and marginal benefit functions are continuous.)

We asserted at the start of this chapter that this is a special case of an Edgeworth box. How is it special? (Hint: Your first task is to identify the second good.)

2. Consider the bargaining protocol described- one round of simultaneous demands, and then (if a deal is not struck in the first round) simultaneous declarations that players stand firm or accede.

(a) Build a subgame perfect, pure strategy equilibrium for this bargaining protocol in which agreement is not reached on the first round, and the
equilibrium split is n to player 1and 100 - n to player 2. Be careful in describing the strategies; this isn't hard, but it isn't trivial either.

(b) Show that for any pure strategy equilibrium that results in agreement, no money is left on the table. Show this as well for variations (a) and (b). Describe the pure strategy equilibria that leave money on the table (if any exist). Are the equilibria you describe subgame perfect?

(c) Build a subgame perfect equilibrium for this bargaining protocol in which money is left on the table and there is agreement with positive probability. (This will involve mixed strategies, according to part [b].)