Frank Cowell Microeconomics January 2007 Exercise 10 12

Frank Cowell: Microeconomics January 2007 Exercise 10. 12 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 10. 12(1): Question Frank Cowell: Microeconomics n n purpose: Set out a one-sided bargaining game method: Use backwards induction methods where appropriate.

Ex 10. 12(1): setting Frank Cowell: Microeconomics n n Alf offers Bill a share g of his cake Bill may or may not accept the offer u u n Two main ways of continuing u u n if the offer is accepted game over if rejected game continues end the game after a finite number of periods allow the offer-and-response sequence to continue indefinitely To analyse this: u u use dynamic games find subgame-perfect equilibrium

Ex 10. 12(1): payoff structure Frank Cowell: Microeconomics n Begin by drawing extensive form tree for this bargaining game u u n Note that payoffs can accrue u u u n start with 3 periods but tree is easily extended either in period 1 (if Bill accepts immediately) or in period 2 (if Bill accepts the second offer) or in period 3 (Bill rejects both offers) Compute payoffs at each possible stage u discount all payoffs back to period 1 the extensive form

Ex 10. 12(1): extensive form Frank Cowell: Microeconomics Alf period 1 §Alf makes Bill an offer [offer g 1] §If Bill accepts, game ends §If Bill rejects, they go to period 2 Bill [accept] §Alf makes Bill another offer [reject] §If Bill accepts, game ends (1 g 1, g 1) §If Bill rejects, they go to period 3 Alf period 2 [offer g 2] §Game is over anyway in period 3 Bill [accept] [reject] (d[1 g 2], dg 2) period 3 §Values discounted to period 1 (d 2[1 g], d 2 g)

Ex 10. 12(1): Backward induction, t=2 Frank Cowell: Microeconomics n n n Assume game has reached t = 2 Bill decides whether to accept the offer g 2 made by Alf Best-response function for Bill is u u n u wants to maximise own payoff this offer would leave Alf with 1 − dg Should Alf offer less than dg today and get 1 − γ tomorrow? u u n if g 2 ≥ dg otherwise Alf will not offer more than dg u n [accept] [reject] tomorrow’s payoff is worth d[1 − g], discounted back to t = 2 but d < 1, so 1 − dg > d[1 − g] So Alf would offer exactly g 2 = dg to Bill u and Bill accepts the offer

Ex 10. 12(1): Backward induction, t=1 Frank Cowell: Microeconomics n n Now, consider an offer of g 1 made by Alf in period 1 The best-response function for Bill at t = 1 is u u n u receiving 1 − d 2 g in period 1 receiving 1 − dg in period 2 But we find 1 − d 2 g > d[1 − d g] u n (same argument as before) So Alf has choice between u n if g 1 ≥ d 2 g otherwise Alf will not offer more than d 2 g in period 1 u n [accept] [reject] again since d < 1 So Alf will offer g 1 = d 2 g to Bill today u and Bill accepts the offer

Ex 10. 12(2): Question Frank Cowell: Microeconomics method: n Extend the backward-induction reasoning

Ex 10. 12(2): 2 < T < ∞ Frank Cowell: Microeconomics n Consider a longer, but finite time horizon u u n Use the backwards induction method again u u n increase from T = 2 bargaining rounds… …to T = T' same structure of problem as before same type of solution as before Apply the same argument at each stage: u u u as the time horizon increases the offer made by Alf reduces to g 1 = δT'γ which is accepted by Bill

Ex 10. 12(3): Question Frank Cowell: Microeconomics method: n Reason on the “steady state” situation

Ex 10. 12(3): T = ∞ Frank Cowell: Microeconomics n Could we use previous part to suggest: as T→∞, g 1→ 0? u u n Instead, consider the continuation game after each period t u n the game played if Bill rejects the offer made by Alf This looks identical to the game just played u u n this reasoning is inappropriate there is no “last period” from which backwards induction outcome can be obtained there is in both games… …a potentially infinite number of future periods This insight enables us to find the equilibrium outcome of this game u use a kind of “steady-state” argument

Ex 10. 12(3): T = ∞ Frank Cowell: Microeconomics n Consider the continuation game that follows if Bill rejects at t u u n n Thus, given a solution (1 g, g), Alf would offer g 1 = dγ Now apply the “steady state” argument: u u n if γ is a solution to the continuation game, must also be a solution to the game at tl so g 1 = g It follows that u u n suppose it has a solution with allocation (1 γ, γ) so, in period t, Bill will accept an offer g 1 if g 1 ≥ δγ, as before g=dg this is only true if γ = 0 Alf will offer g = 0 to Bill, which is accepted

Ex 10. 12: Points to remember Frank Cowell: Microeconomics n n Use backwards induction in all finite-period cases Take are in “thinking about infinity” u u u if T→∞ there is no “last period” so we cannot use simple backwards induction method