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

2. 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.

3. 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

4. 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

5. 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

6. 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

7. 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

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

9. 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

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

11. 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

12. 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

13. 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