A Consider The Two Person Rubinstein Stahl Model Of Section 4 4 The Two Players Bar 2198967

(a) Consider the two-person Rubinstein-Stahl model of section 4.4. The two players bargain to divide a pie of size 1 and take turns making offers. The discount factor is 6. Introduce “outside options” in the following way: At each period, the player whose turn it is to make the offer makes the offer; the other player then has the choice among (1) accepting the offer, (2) exercising his outside option instead, and (3) continuing bargaining (making an offer the next period). Let x0 denote the value of the outside option. Show that, if x0 ≤δ/(1 + δ), the outside option has no effect on the equilibrium path. Comment. What happens if x0 > δ/(1 + δ)?

(b) Consider an alternative way of formalizing outside options in bargaining. Suppose that there is an “exogenous risk of breakdown” of re-negotiation (Binmore et al. 1986). At each period t, assuming that bargaining has gone on up to date t, there is probability (1 — x) that bargaining breaks down at the end of period t if the period-t offer is turned down. The players then get xo each. Show that the “outside opportunity” x0 matters even if it is small, and compute the subgame-perfect equilibrium.