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 cost 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.