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Exercise 10.12

Exercise 10.12. MICROECONOMICS Principles and Analysis Frank Cowell. January 2007. Ex 10.12(1): Question. purpose : Set out a one-sided bargaining game method : Use backwards induction methods where appropriate. Ex 10.12(1): setting. Alf offers Bill a share g of his cake

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Exercise 10.12

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  1. Exercise 10.12 MICROECONOMICS Principles and Analysis Frank Cowell January 2007

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

  3. Ex 10.12(1): setting • Alf offers Bill a share g of his cake • Bill may or may not accept the offer • if the offer is accepted  game over • if rejected  game continues • 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: • use dynamic games • find subgame-perfect equilibrium

  4. Ex 10.12(1): payoff structure • Begin by drawing extensive form tree for this bargaining game • start with 3 periods • but tree is easily extended • 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 • discount all payoffs back to period 1 the extensive form

  5. Ex 10.12(1): extensive form Alf • Alf makes Bill an offer period 1 [offer g1] • If Bill accepts, game ends • If Bill rejects, they go to period 2 Bill • Alf makes Bill another offer [accept] [reject] • If Bill accepts, game ends • If Bill rejects, they go to period 3 (1 - g1, g1) Alf period 2 • Values discounted to period 1 • Game is over anyway in period 3 [offer g2] Bill [accept] [reject] (d[1 - g2], dg2) (d2[1 -g], d2 g) period 3

  6. Ex 10.12(1): Backward induction, t=2 • Assume game has reached t = 2 • Bill decides whether to accept the offer g2 made by Alf • Best-response function for Bill is • [accept] if g2 ≥ dg • [reject] otherwise • Alf will not offer more than dg • 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? • 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 g2 = dg to Bill • and Bill accepts the offer

  7. Ex 10.12(1): Backward induction, t=1 • Now, consider an offer of g1 made by Alf in period 1 • The best-response function for Bill at t = 1 is • [accept] if g1 ≥ d2g • [reject] otherwise • Alf will not offer more than d2g in period 1 • (same argument as before) • So Alf has choice between • receiving 1 − d2g in period 1 • receiving 1 − dg in period 2 • But we find 1 − d2g > d[1 − dg] • again since d < 1 • So Alf will offer g1 = d2g to Bill today • and Bill accepts the offer

  8. Ex 10.12(2): Question method: • Extend the backward-induction reasoning

  9. Ex 10.12(2): 2 < T < ∞ • Consider a longer, but finite time horizon • increase from T = 2 bargaining rounds… • …to T = T' • 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: • as the time horizon increases • the offer made by Alf reduces to g1 = δT'γ • which is accepted by Bill

  10. Ex 10.12(3): Question method: • Reason on the “steady state” situation

  11. Ex 10.12(3): T = ∞ • Could we use previous part to suggest: as T→∞, g1→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 • the game played if Bill rejects the offer made by Alf • 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 • use a kind of “steady-state” argument

  12. Ex 10.12(3): 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 g1 if g1 ≥ δγ, as before • Thus, given a solution (1  g, g), Alf would offer g1 = dγ • 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 g1 = g • It follows that • g = dg • this is only true if γ = 0 • Alf will offerg = 0 to Bill, which is accepted

  13. Ex 10.12: Points to remember • Use backwards induction in all finite-period cases • 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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