Frank Cowell: Microeconomics
January 2007
Exercise 10.12
MICROECONOMICS
Principles and Analysis
Frank Cowell
Ex 10.12(1): Question
Frank Cowell: Microeconomics
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purpose: Set out a one-sided bargaining game
method: Use backwards induction methods where appropriate.
Ex 10.12(1): setting
Frank Cowell: Microeconomics
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Alf offers Bill a share g of his cake
Bill may or may not accept the offer
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Two main ways of continuing
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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:
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use dynamic games
find subgame-perfect equilibrium
Ex 10.12(1): payoff structure
Frank Cowell: Microeconomics
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Begin by drawing extensive form tree for this bargaining
game
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Note that payoffs can accrue
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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
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discount all payoffs back to period 1
the extensive
form
Ex 10.12(1): extensive form
Frank Cowell: Microeconomics
Alf
Alf makes Bill an offer
[offer g1]
period 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  g1, g1)
If Bill rejects, they go to period 3
Alf
period 2
[offer g2]
Game is over anyway in period 3
Bill
[accept]
[reject]
(d[1  g2], dg2)
period 3
Values discounted to period 1
(d2[1 g], d2 g)
Ex 10.12(1): Backward induction, t=2
Frank Cowell: Microeconomics
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Assume game has reached t = 2
Bill decides whether to accept the offer g2 made by Alf
Best-response function for Bill is
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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?
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if g2 ≥ dg
otherwise
Alf will not offer more than dg
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[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 g2 = dg to Bill
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and Bill accepts the offer
Ex 10.12(1): Backward induction, t=1
Frank Cowell: Microeconomics
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Now, consider an offer of g1 made by Alf in period 1
The best-response function for Bill at t = 1 is
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receiving 1 − d2g in period 1
receiving 1 − dg in period 2
But we find 1 − d2g > d[1 − d g]
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(same argument as before)
So Alf has choice between
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if g1 ≥ d2g
otherwise
Alf will not offer more than d2g in period 1
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[accept]
[reject]
again since d < 1
So Alf will offer g1 = d2g to Bill today
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and Bill accepts the offer
Ex 10.12(2): Question
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method:
 Extend the backward-induction reasoning
Ex 10.12(2): 2 < T < ∞
Frank Cowell: Microeconomics
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Consider a longer, but finite time horizon
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Use the backwards induction method again
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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:
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as the time horizon increases
the offer made by Alf reduces to g1 = δT'γ
which is accepted by Bill
Ex 10.12(3): Question
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method:
 Reason on the “steady state” situation
Ex 10.12(3): T = ∞
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Could we use previous part to suggest: as T→∞, g1→0?
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Instead, consider the continuation game after each period t
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the game played if Bill rejects the offer made by Alf
This looks identical to the game just played
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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
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use a kind of “steady-state” argument
Ex 10.12(3): T = ∞
Frank Cowell: Microeconomics
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Consider the continuation game that follows if Bill rejects
at t
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Thus, given a solution (1  g, g), Alf would offer g1 = dγ
Now apply the “steady state” argument:
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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
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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
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
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Use backwards induction in all finite-period cases
Take are in “thinking about infinity”
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if T→∞
there is no “last period”
so we cannot use simple backwards induction method