1 / 13

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 of his cake

nara
Télécharger la présentation

Exercise 10.12

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 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 1 - 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 (g1, 1 - g1) Alf period 2 • Values discounted to period 1 • Game is over anyway in period 3 [offer 1 - g2] Bill [accept] [reject] (dg2, d[1 - g2]) (d2g, d2[1 -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 1  g2 made by Alf • Best-response function for Bill is • [accept] if [1 g2] ≥ d [1  g] • [reject] otherwise • Alf will not offer more than [1  g]d • wants to maximise own payoff • this offer would leave Alf with 1 − d + dg • Should Alf offer less than d[1 − g] today and get γ tomorrow? • tomorrow’s payoff is worth dg, discounted back to t = 2 • but d < 1, so 1 − d + dg > dg • So Alf would offer exactly 1 − g2 = d[1 − g] 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 1 −g1 ≥ d2 [1 − g] • [reject] otherwise • Alf will not offer more than d2g in period 1 • (same argument as before) • So Alf has choice between • receiving 1 − d2[1 − g] in period 1 • receiving 1 − d[1 − g] in period 2 • But we find 1 − d2[1 − g] > 1 − d[1 − g] • again since d < 1 • So Alf will offer 1 −g1 = d2[1 − g] 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 1 −g1 = δT' [1 − 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 = ∞ • Can we use previous part to suggest: as T→∞, 1 −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 (γ, 1 γ) • so, in period t, Bill will accept an offer 1g1 if g1 ≥ δγ, as before • Thus, given a solution (g, 1  g), Alf would offer • 1  g1 = d[1 γ] • 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 offer1  g = 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

More Related