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Frank Cowell: Microeconomics Exercise 10.12 MICROECONOMICS Principles and Analysis Frank Cowell January 2007

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Frank Cowell: Microeconomics Ex 10.12(1): Question purpose: Set out a one-sided bargaining game purpose: Set out a one-sided bargaining game method: Use backwards induction methods where appropriate. method: Use backwards induction methods where appropriate.

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Frank Cowell: Microeconomics Ex 10.12(1): setting Alf offers Bill a share of his cake Alf offers Bill a share of his cake Bill may or may not accept the offer Bill may or may not accept the offer if the offer is accepted game over if rejected game continues Two main ways of continuing Two main ways of continuing end the game after a finite number of periods allow the offer-and-response sequence to continue indefinitely To analyse this: To analyse this: use dynamic games find subgame-perfect equilibrium

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Frank Cowell: Microeconomics Ex 10.12(1): payoff structure Begin by drawing extensive form tree for this bargaining game Begin by drawing extensive form tree for this bargaining game start with 3 periods but tree is easily extended Note that payoffs can accrue Note that payoffs can accrue 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 Compute payoffs at each possible stage discount all payoffs back to period 1 the extensive form

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Frank Cowell: Microeconomics Alf [accept][reject] Bill (1 1, 1 ) [offer 1 ] Bill Alf [offer 2 ] [accept][reject] ( [1 2 ], 2 ) ( 2 [1 ], 2 ) period 1 period 2 period 3 Ex 10.12(1): extensive form Alf makes Bill an offer If Bill accepts, game ends Alf makes Bill another offer If Bill accepts, game ends If Bill rejects, they go to period 2 If Bill rejects, they go to period 3 Values discounted to period 1 Game is over anyway in period 3

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Frank Cowell: Microeconomics Ex 10.12(1): Backward induction, t=2 Assume game has reached t = 2 Assume game has reached t = 2 Bill decides whether to accept the offer 2 made by Alf Bill decides whether to accept the offer 2 made by Alf Best-response function for Bill is Best-response function for Bill is [accept]if 2 ≥ [reject]otherwise Alf will not offer more than Alf will not offer more than wants to maximise own payoff this offer would leave Alf with 1 − Should Alf offer less than today and get 1 − γ tomorrow? Should Alf offer less than today and get 1 − γ tomorrow? tomorrow’s payoff is worth [1 − ], discounted back to t = 2 but [1 − ] So Alf would offer exactly 2 = to Bill So Alf would offer exactly 2 = to Bill and Bill accepts the offer

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Frank Cowell: Microeconomics Ex 10.12(1): Backward induction, t=1 Now, consider an offer of 1 made by Alf in period 1 Now, consider an offer of 1 made by Alf in period 1 The best-response function for Bill at t = 1 is The best-response function for Bill at t = 1 is [accept]if 1 ≥ 2 [reject]otherwise Alf will not offer more than 2 in period 1 Alf will not offer more than 2 in period 1 (same argument as before) So Alf has choice between So Alf has choice between receiving 1 − 2 in period 1 receiving 1 − in period 2 But we find 1 − 2 > [1 − ] But we find 1 − 2 > [1 − ] again since < 1 So Alf will offer 1 = 2 to Bill today So Alf will offer 1 = 2 to Bill today and Bill accepts the offer

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Frank Cowell: Microeconomics Ex 10.12(2): Question method: Extend the backward-induction reasoning Extend the backward-induction reasoning

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Frank Cowell: Microeconomics Ex 10.12(2): 2 < T < ∞ Consider a longer, but finite time horizon Consider a longer, but finite time horizon increase from T = 2 bargaining rounds… …to T = T' Use the backwards induction method again Use the backwards induction method again same structure of problem as before same type of solution as before Apply the same argument at each stage: Apply the same argument at each stage: as the time horizon increases the offer made by Alf reduces to 1 = δ T' γ which is accepted by Bill

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Frank Cowell: Microeconomics Ex 10.12(3): Question method: Reason on the “steady state” situation Reason on the “steady state” situation

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Frank Cowell: Microeconomics Ex 10.12(3): T = ∞ Could we use previous part to suggest: as T→∞, 1 →0? Could we use previous part to suggest: as T→∞, 1 →0? this reasoning is inappropriate there is no “last period” from which backwards induction outcome can be obtained Instead, consider the continuation game after each period t Instead, consider the continuation game after each period t the game played if Bill rejects the offer made by Alf This looks identical to the game just played This looks identical to the game just played there is in both games… …a potentially infinite number of future periods This insight enables us to find the equilibrium outcome of this game This insight enables us to find the equilibrium outcome of this game use a kind of “steady-state” argument

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Frank Cowell: Microeconomics Ex 10.12(3): T = ∞ Consider the continuation game that follows if Bill rejects at t Consider the continuation game that follows if Bill rejects at t suppose it has a solution with allocation (1 γ, γ) so, in period t, Bill will accept an offer 1 if 1 ≥ δγ, as before Thus, given a solution (1 , ), Alf would offer 1 = γ Thus, given a solution (1 , ), Alf would offer 1 = γ Now apply the “steady state” argument: Now apply the “steady state” argument: if γ is a solution to the continuation game, must also be a solution to the game at tl so 1 = It follows that It follows that = this is only true if γ = 0 Alf will offer = 0 to Bill, which is accepted Alf will offer = 0 to Bill, which is accepted

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Frank Cowell: Microeconomics Ex 10.12: Points to remember Use backwards induction in all finite-period cases Use backwards induction in all finite-period cases Take are in “thinking about infinity” Take are in “thinking about infinity” if T→∞ there is no “last period” so we cannot use simple backwards induction method

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