1 Bargaining Markets u Two populations: Buyers and Sellers u A sellers has 1 indivisible unit to...

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Bargaining & Markets

Two populations: Buyers and Sellers A sellers has 1 indivisible unit to sell A Buyer has 1 unit of divisible money If they agree on a price at period t, their

utilities are: δtp, δt(1-p). Random matching of pairs (buyer &

seller) Bargaining breaks down after 1 period,

if there was no agreement

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The Populations: Continuous or finite sets B, S

Bargaining: Nash Bargaining Solution, with the expected continuation values δVS , δVB as the disagreement point.

A pair that reached agreement leaves the market

Matching: If B > S then all sellers are matched and a buyer is matched with probability S/B.

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Model 1: The populations sizes B, S do not change with time. i.e. new individuals enter the market to replace those that left.

( Assume B > S )

We seek an equilibrium in stationary and anonymous strategies

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Bargaining & MarketsModel 1:Model 1: B, S constantB, S constant

B S B S

S B

S1 - δV

if δV ++ δV 1 - δV

δV < 1 then :- δVp* = = + δV

2 2

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B

B

B

S B

S BB

S

S S(1 - p*)+ (1 - )δVB B

VδV

if δV + δV < 1

=if δV

1 - δV

+ δ

+

V 1

p*

>

δV=2

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S B

S

S

B

B

S

S

if δV + δV < 1=

if δV +

p*V

δV

1 - δV

δV

+p

>

δV* =

1

2

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B

B

B

S

S

S

B

S B

S B

B

S B

S

if δV + δV < 1

=if δV + δV > 1

if δV + δV < 1=

if δ

S S(1 - p*)+ (1 - )δVB B

VδV

p*V

δV

1 - δV

V + δ

+

V > 1

δVp* =2

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B

B

B

S

S

S

B

S B

S B

B

S B

S

if δV + δV < 1

=if δV + δV > 1

if δV + δV < 1=

if δ

S S(1 - p*)+ (1 - )δVB B

VδV

p*V

δV

1 - δV

V + δ

+

V > 1

δVp* =

2

0,0similarly

an

if then

d

B B

B

S B

S

S B

δV + δV > 1

δV + δV

V = δVV

V< 1.

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S B

B B

S

B S

S SV = (1 - p*)+ (1 - )δVB B

V = p*1 - δV + δVp* =

2δV + δV < 1

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S B

B B

S

B S

S SV = (1 - p*)+ (1 - )δVB B

V = p*1 - δV + δV

p* =2

δV + δV < 1

1p* = S2 - δ + δB

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1p* =S2 - δ + δB

121

Bas δ , p*B + S

21B > S , p* >

12 .as S , p*

12 .as δ 0, p*

Find p when S >B !!!!

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q

p

S

1

B

Supply

Demand

Microeconomics I ??

p = 1p = 11p* =

S2 - δ + δB

121

Bas δ , p*B + S

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Model 2: No new individuals enter the market. The populations shrink over time.

Let the initial populations be of sizes B0 , S0. Since equal numbers of buyers & sellers leave the market, the populations B,S at any period satisfy: B -S = B0 - S0.

( Assume B0 > S0 )

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Bargaining & MarketsModel 2: No new entryModel 2: No new entry

If there is agreement then all pairs leave and only B0 - S0 buyers remain.

If there is agreement and one pair deviates then there will remain B0 - S0 +1 buyers and 1 seller in the market

The disagreement point for all pairs is:

B 0 0 S 0 0δV 1,B - S + 1 , δV 1,B - S + 1

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B S1 - δV 1,B - S + 1 + δV 1,B - S + 1p* (S,B) =

2

SV S,B p* (S,B)

BSV S,B = 1 - p* (S,B)B

set S = 1

B S1 - δV 1,B + δV 1,Bp* (1,B) =

2

SV 1,B p* (1,B)

B1V 1,B = 1 - p* (1,B)B

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SV 1,B p* (1,B)

B1V 1,B = 1 - p* (1,B)B

B S1 - δV 1,B + δV 1,Bp* (1,B) =

2

11 - δBp* (1,B) = 12 - - δ

B

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B S1 - δV 1,B - S + 1 + δV 1,B - S + 1p* (S,B) =

211 - δBp* (1,B) = 12 - δ - δ

B

δB-S+1

δB-S+1

1 -p* (1,B - S + 1) =

2 - δ - S= V (1,B - S + 1)

1B B-S+1V (1,B - S + 1) = (1 - p* (1,B - S + 1))

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δB-S+1

δB-S+1

1 -p* (S,B) =

2 - δ -

S < B, δ 1, p* (S,B) 1

12S = B p* (S,B) =

12S < B p* (S,B) >

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δB-S+1

δB-S+1

1 -p* (S,B) =

2 - δ -

lim

1δ 12

1 if S < Bp* (S,B) =

if S = B

Find p* when S >B !!!!

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Justifying the assumptions of Models 1 & 2

The MarketModel 1 or 2

Individuals borneach period

B > Sε - cost of entry

B* > S *

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Model 1 (constant populations)

Equal numbers of buyers & sellers should enter each period

The MarketModel 1

Individuals borneach period

B > Sε - cost of entry

B* > S *

Bif V (S*,B*) > ε then all B will enter,

and there is not enough S to keep nos. equal

B V (S*,B*) = ε

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B > S ε - cost of entry B* > S *

B V (S*,B*) = ε

Justifying Model 1Justifying Model 1

Could S* > B* ??

1

B 2B*S*

1ε = V (S*,B*) = >2 - +

12if ε < then S* < B*

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B V (S*,B*) = ε


S* < B*

S*B*

B S*B*

V (S*,B*) = = ε2 - +

S S*

B*

1 1 -p* = V (S*,B*) = = > ε2 - + 2 -

2 - δ εS * =B* 1 - δε

S* = S

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B V (S*,B*) = ε


S* < B* 2 - δ εS * =

B* 1 - δεS* = S

1 - δεB* = S2 - δ ε

is B* < B ??

S ,S enter each period

S*,B* 0,B* S * S*,S *

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B V (S*,B*) = ε


S* < B* 2 - δ εS * =

B* 1 - δε

1 -p* =2 -

S ,S S*, S *

0, S * 1as 0 : p*B* 2 - δ

as 0 , δ 0 : p* 1

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δB-S+1

δB-S+1

1 -p* (S,B) =

2 - δ -

assume δ 1

lim

1δ 12

1 if S < Bp* (S,B) =

if S = B

Entry has a large effect on prices

Assume that an entrant anticipates his

effect on prices

?

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if S* < B* then p* ~ 1 and B's will not enter

lim

1δ 12

1 if S < Bp* (S,B) =

if S = BAssume that an

entrant anticipates his effect on prices

?

12if S* > B* then p* <

and B's will enter until S* = B*

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Let B* = S* = E12p* =

if another B enters price will be2 - δ p* = ~ 1

4 - 3δ δB-S+1

δB-S+1

1 -p* (S,B) =

2 - δ -

B* = S *

and his gain will be less than ε i.e. he w notill enter

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