EC487 Advanced Microeconomics, Part I: Lecture 9 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 9

Leonardo Felli

32L.LG.04

24 November 2017

Bargaining Games:

Recall

◮ Two players, i ∈ {A, B} are trying to share a surplus. ◮ The size of the surplus is normalized to 1. ◮ Payoffs to the players in case of disagreement are normalized

to 0.

◮ Denote (δA, δB) the parties’ discount factors:

0 ≤ δi ≤ 1, ∀i ∈ {A, B};

◮ Denote:

◮ x the share of the pie to party A; ◮ (1 − x) the share of the pie to party B. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 2 / 65

T periods alternating offers bargaining

◮ Assume for simplicity that δA = δB = δ. ◮ Assume now that the game lasts for T periods where T is

even.

◮ We compute the Subgame Perfect Equilibrium of the

bargaining game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 3 / 65

T periods alternating offers bargaining (cont’d)

Extensive form:

◮ Odd periods n ≤ T − 1:

Stage I: A makes an offer xA to B; Stage II: B observes the offer and can accept or reject it; If the offer is accepted then x = xA and the game terminates and the players payoffs are: ΠA(xA, y) = δn−1 xA ΠB(xA, y) = δn−1 (1−xA) If the offer is rejected the game moves to Stage I of the following period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 4 / 65

T periods alternating offers bargaining (cont’d)

◮ Even periods τ < T:

Stage I: B makes an offer xB to A; Stage II: A observes the offer and can accept or reject it; If the offer is accepted then x = xB and the game terminates and the players payoffs are: ΠA(xB, y) = δτ−1 xB ΠB(xB, y) = δτ−1 (1−xB) If the offer is rejected the game moves to Stage I of the following period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 5 / 65

T periods alternating offers bargaining (cont’d)

◮ Period T:

Stage I: B makes an offer xB to A; Stage II: A observes the offer and can accept or reject it; If the offer is accepted then x = xB and the game terminates and the players payoffs are: ΠA(xB, y) = δT−1xB ΠB(xB, y) = δT−1(1−xB) If the offer is rejected the game ends and the players’ payoffs are: ΠA(xB, n) = 0 ΠB(xB, n) = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 6 / 65

T periods alternating offers bargaining (cont’d)

Notice that:

◮ In the T-th period (even) B makes an offer and the size of

the pie is δT−1. Player B makes a take-it-or-leave-it offer: ΠA = ΠB = δT−1.

◮ In the (T − 1)-th period (odd) A makes an offer and the share

• f the pie that will not be available any more next period is

(δT−2 − δT−1). Player A makes a take-it-or-leave-it offer on this share: ΠA = + (δT−2 − δT−1) ΠB = δT−1 + 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 7 / 65

T periods alternating offers bargaining (cont’d)

◮ .

. .

◮ In the second period (even) B makes an offer and the share of

the pie that will not be available any more next period is (δ − δ2). Player B makes a take-it-or-leave-it offer on this share: ΠA = + (δT−2 − δT−1) + . . . + ΠB = δT−1 + + . . . + (δ − δ2).

◮ In the first period (odd) A makes an offer and the share of the

pie that will not be available any more next period is (1 − δ). Player A makes a take-it-or-leave-it offer on this share: ΠA = (δT−2 − δT−1) + . . . + + (1 − δ) ΠB = δT−1 + . . . + (δ − δ2) + 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 8 / 65

T periods alternating offers bargaining (cont’d)

In other words, the SPE payoffs are: ΠA = (1 − δ) + (δ2 − δ3) + . . . + (δT−2 − δT−1) ΠB = (δ − δ2) + (δ3 − δ4) + . . . + (δT−3 − δT−2) + δT−1. We can re-write both payoffs as: ΠA = (1 − δ) (1 + δ2 + δ4 + . . . + δT−2) ΠB = (1 − δ) δ (1 + δ2 + δ4 + . . . + δT−4) + δT−1.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 9 / 65

T periods alternating offers bargaining (cont’d)

Recall that: 1 + a + a2 + a3 + . . . + an = 1 − an+1 1 − a Let now a = δ2 then we have: (1 + δ2 + δ4 + . . . + δT−2) = 1 − (δ2)( T−2

2

+1)

1 − δ2 = 1 − δT 1 − δ2 (1 + δ2 + δ4 + . . . + δT−4) = 1 − (δ2)( T−4

2

+1)

1 − δ2 = 1 − δT−2 1 − δ2

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 10 / 65

T periods alternating offers bargaining (cont’d)

The SPE payoffs are then: ΠA = (1 − δ) 1 − δT 1 − δ2 ΠB = (1 − δ) δ 1 − δT−2 1 − δ2 + δT−1. Recall also that (1 − δ2) = (1 − δ) (1 + δ).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 11 / 65

T periods alternating offers bargaining (cont’d)

When T is even agreement is reached in the first period and the SPE payoffs are: ΠA = 1 − δT 1 + δ ΠB = 1 − ΠA = δ + δT 1 + δ Strategies for t ≤ T remaining periods are: If t is odd

◮ A offers share xA = 1 − δt

1 + δ ;

◮ B accepts any share x′ ≤ (1 − δt)

1 + δ and rejects x′ > (1 − δt) 1 + δ If t is even:

◮ B offers share xA = δ − δt

1 + δ ;

◮ A accepts any share x′ ≥ (δ − δt)

1 + δ and rejects x′ < (δ − δt) 1 + δ

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 12 / 65

T periods alternating offers bargaining (cont’d)

When T is odd agreement is reached in the first period and the SPE payoffs are: ΠA = 1 + δT 1 + δ ΠB = δ − δT 1 + δ Strategies for t ≤ T remaining periods are: If t is odd:

◮ A offers share xA = 1 + δt

1 + δ ;

◮ B accepts any share x′ ≤ (1 + δt)

1 + δ and rejects x′ > (1 + δt) 1 + δ If t is even:

◮ B offers share xA = δ + δt

1 + δ ;

◮ A accepts any share x′ ≥ (δ + δt)

1 + δ and rejects x′ < (δ + δt) 1 + δ

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 13 / 65

Alternating offers bargaining ∞ horizon (Rubinstein 1982):

Consider the general case where δA and δB might differ. Extensive form:

◮ Odd periods:

Stage I: A makes an offer xA to B; Stage II: B observes the offer and can accept or reject it; If the offer is accepted then x = xA and the game terminates; If the offer is rejected the game moves to Stage I of the following period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 14 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ Even periods:

Stage I: B makes an offer xB to A; Stage II: A observes the offer and can accept or reject it; If the offer is accepted then x = xB and the game terminates; If the offer is rejected the game moves to Stage I of the following period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 15 / 65

Alternating offers bargaining ∞ horizon (cont’d)

Payoffs:

◮ If parties agree on x in period n:

ΠA(σA, σB) = δn−1

A

x, ΠB(σA, σB) = δn−1

B

(1 − x),

◮ or if they do not agree:

Πi(σA, σB) = 0 i ∈ {A, B}.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 16 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ Nash equilibria:

any share of the surplus x ∈ [0, 1].

◮ Strategies:

◮ A offers share xA = x ∈ [0, 1] in the first period; ◮ B accepts any share x′ ≤ x and rejects any share x′ > x in the

first period;

◮ player i offers share xi = x, for i ∈ {A, B} in any other period; ◮ player −i accepts any offer xi, i ∈ {A, B} such that xA ≥ x or

xB ≤ x and rejects all other offers in every other period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 17 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ Subgame Perfect Equilibrium Outcome:

Agreement is reached in the first period and the equilibrium payoffs are: ΠA = 1 − δB 1 − δAδB and ΠB = δB (1 − δA) 1 − δAδB

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 18 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ Strategies:

◮ A always offers share xA =

1 − δB 1 − δAδB in odd periods;

◮ B accepts any share x′ ≤ (1 − δB)

1 − δAδB and rejects any share x′ > (1 − δB) 1 − δAδB in odd periods;

◮ B offers share xB = δA (1 − δB)

1 − δAδB in even periods;

◮ A accepts any offer x′ ≥ δA (1 − δB)

1 − δAδB and rejects any share x′ < δA (1 − δB) 1 − δAδB in even periods.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 19 / 65

Alternating offers bargaining ∞ horizon (cont’d)

Proof: (Shaked and Sutton 1984)

◮ Notice that the game is stationary: the continuation game

starting at every odd period is the same, the continuation game starting in every even period also looks identical.

◮ Denote xH i

the highest equilibrium share player A can get in a subgame starting in a period in which player i ∈ {A, B} makes an offer.

◮ Denote xL i the lowest equilibrium share player A can get in a

subgame starting in a period in which player i ∈ {A, B} makes the offer.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 20 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ Then:

xH

B ≤ δA xH A

1 − xL

A ≤ δB

• 1 − xL

B

• ◮ moreover:

xL

B ≥ δA xL A

1 − xH

A ≥ δB

• 1 − xH

B

• ◮ Substituting we get:

1 − xL

A ≤ δB

• 1 − δA xL

A

• 1 − xH

A ≥ δB

• 1 − δA xH

A

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part II 24 November 2017 21 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ and

xL

B ≥ δA

• 1 − δB
• 1 − xL

B

• xH

B ≤ δA

• 1 − δB
• 1 − xH

B

• ◮ From these four inequalities we get:

xL

A ≥

1 − δB 1 − δAδB ≥ xH

A

and xL

B ≥ δA[1 − δB]

1 − δAδB ≥ xH

B

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 22 / 65

Alternating offers bargaining ∞ horizon (cont’d)

◮ These inequalities are therefore satisfied with equality:

xA = 1 − δB 1 − δAδB and xB = δA[1 − δB] 1 − δAδB

◮ These are the offers in odd and even periods respectively. We

also get: xB = δA xA and 1 − xA = δB(1 − xB)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 23 / 65

Robustness of Rubinstein’s SPE

◮ We here identify the sense in which the unique SPE

equilibrium of the Rubinstein’s game is a robust and meaningful solution.

◮ Let us consider the Rubinstein’s equilibrium payoffs in the

case: δA = δB = δ: ΠA = 1 1 + δ, ΠB = δ 1 + δ. (1)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 24 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ Recall also the equilibrium payoffs in the case, seen above,

where the alternating offer bargaining game terminates in T (even) periods: ΠA = 1 − δT 1 + δ (2) ΠB = δ + δT 1 + δ = (1 − δT−2) 1 − δT δ 1 − δT 1 + δ + δT−1 (3)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 25 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ The payoffs of the two games coincide in the limit T → ∞. ◮ Indeed

lim

T→∞

1 − δT 1 + δ = 1 1 + δ lim

T→∞

δ + δT 1 + δ = δ 1 + δ

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 26 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ Recall

ΠA = 1 − δT 1 + δ ΠB =

• δ 1 − δT−2

1 − δT 1 − δT 1 + δ + δT−1

◮ The last term of ΠB, δT−1 can be interpreted as the last

mover advantage.

◮ Clearly the last mover advantage converges to zero as T

increases.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 27 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ The coefficient of ΠB

δ (1 − δT−2) 1 − δT can be interpreted as the first mover advantage, (second mover disadvantage).

◮ To understand this coefficient better we need to consider a

version of Rubinstein’s game that eliminates asymptotically such an advantage.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 28 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ Denote ∆t the length of the time period that lapses between

each two offers.

◮ The two discount factors are then:

δA = exp{−rA ∆t} δB = exp{−rB ∆t}

◮ Taylor expansion implies that in a neighborhood of ∆t = 0 we

have that: δA ≃ (1 − rA ∆t) δB ≃ (1 − rB ∆t)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 29 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ Substituting we get that in a neighborhood of ∆t = 0:

ΠA = rB ∆t 1 − (1 − rA ∆t)(1 − rB ∆t) = rB rA + rB − rA rB ∆t ΠB = rA ∆t (1 − rB ∆t) 1 − (1 − rA ∆t)(1 − rB ∆t) = rA − rA rB ∆t rA + rB − rA rB ∆t

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 30 / 65

Robustness of Rubinstein’s SPE (cont’d)

◮ If we now consider the limit of the above payoffs for ∆t → 0

we get: ΠA = rB rA + rB ΠB = rA rA + rB

◮ Notice that if rA = rB = r we get:

ΠA = 1 2, ΠB = 1 2

◮ This is clearly an intuitive result in the case there are no first

mover or last mover advantages.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 31 / 65

Bargaining Power

Notice that each party’s “bargaining power” is determined by:

◮ each party’s discount factor δi or interest rate ri, ◮ the extensive form of the bargaining game: first mover, last

mover advantage etc...

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 32 / 65

Coase Theorem

What could parties achieve in an economic environment in which they can costlessly negotiate a contractual agreement?

Theorem (Coase Theorem: Coase (1960))

In an economy where ownership rights are well defined and transacting is costless gains from trade will be exploited (a contract will be agreed upon) and efficiency achieved whatever the distribution of entitlements. That is rational agents negotiate agreements that are individually rational and Pareto efficient. An agreement is individually rational if each contracting party is not worse off by deciding to sign the contract rather then choosing not to sign it.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 33 / 65

Freedom of Contract

◮ This is the reflection of a basic principle of a well functioning

legal system known as: freedom of contract.

◮ This is equivalent to assume that the action space of the

contracting parties always contains the option not to sign the contract.

◮ A contract is Pareto efficient if there does not exist an other

feasible contract that makes at least one of the contracting party strictly better off without making any other contracting party worse off.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 34 / 65

A Model of Production Externality

◮ Consider the following simple model of a production

externality.

◮ Consider two parties, labelled A and B. ◮ Party A generates revenue RA(eA) (strictly concave) by

choosing the input eA at a linear cost c eA (c > 0).

◮ A’s payoff function is then:

ΠA(eA) = RA(eA) − c eA

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 35 / 65

A Model of Production Externality (cont’d)

◮ Party B generates revenue RB(eB) (strictly concave) by

choosing the input eB at the linear cost c eB (c > 0).

◮ Party B also suffers from an externality γ eA (γ > 0) imposed

by A on B.

◮ B’s payoff function is then:

ΠB(eB) − γ eA = RB(eB) − c eB − γ eA

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 36 / 65

Social Efficient Outcome

◮ Consider first the social efficient amounts of input e∗ A and e∗ B. ◮ These solve the Central Planner’s problem:

max

eA,eB ΠA(eA) + ΠB(eB) − γ eA ◮ In other words (e∗ A, e∗ B) are such that:

R′

A(e∗ A) = c + γ

R′

B(e∗ B) = c

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 37 / 65

No Agreement Outcome

◮ Assume now that parties choose the amounts of input eA and

eB simultaneously and independently.

◮ Party A’s problem:

max

eA

ΠA(eA)

◮ Party B’s problem:

max

eB

ΠB(eB) − γ eA

◮ In equilibrium the inputs chosen (ˆ

eA, ˆ eB) are: R′

A(ˆ

eA) = c, R′

B(ˆ

eB) = c

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 38 / 65

Gains form Trade

◮ Comparing (ˆ

eA, ˆ eB) and (e∗

A, e∗ B) we obtain using concavity of

RA(·): e∗

B = ˆ

eB, e∗

A < ˆ

eA

◮ In other words:

[ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A] − [ΠA(ˆ

eA) + ΠB(ˆ eB) − γ ˆ eA] = = [ΠA(e∗

A) − ΠA(ˆ

eA)] + γ (ˆ eA − e∗

A) > 0 ◮ The joint surplus is reduced by the inefficiency generated by

the externality.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 39 / 65

Gains form Trade (cont’d)

◮ Assume now that the two contracting parties get together and

agree on a contract before the amounts of input are chosen.

◮ A reduction of input eA from ˆ

eA to e∗

A generates:

◮ a decrease in the net revenues from A’s technology:

ΠA(e∗

A) < ΠA(ˆ

eA)

◮ reduction in the negative externality

γ e∗

A < γ ˆ

eA

and the former effect is more than compensated by the latter

• ne

γ (ˆ eA − e∗

A) > [ΠA(ˆ

eA) − ΠA(e∗

A)]

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 40 / 65

Negotiation and Ownership Rights

◮ This may create room for negotiation. ◮ For simplicity normalize to 1 the total size of the surplus that

is available to share between the two contracting parties (parties negotiate on which percentage of the surplus accrues to each one).

◮ To establish a well defined negotiation ownership rights need

to be specified.

◮ Entitlements/ownership rights define the outside option of

each party to the contract.

◮ In other words they define the payoff each party is entitled to

without need for the other party to agree.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 41 / 65

Bargaining

◮ Denote wA and wB the entitlements of party A, respectively B

where: wA + wB < 1.

◮ In general, the Coase Theorem is stated without a specific

reference to the extensive form of the costless negotiation between the two parties.

◮ In what follows we will show the result for two examples of a

bargaining game with outside options.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 42 / 65

Bargaining (cont’d)

Once again denote:

◮ δ the parties’ common discount factor, ◮ x the share of the pie to party A, ◮ (1 − x) the share of the pie to party B. ◮ wA and wB each party’s outside option.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 43 / 65

Take-it-or-leave-it offer by Party A

Extensive form:

◮ A makes an offer x ∈ [0, 1] to B; ◮ B observes the offer x and decides whether to accept or reject

it.

◮ If the offer is accepted the game ends and the players payoffs

are: PA = x [ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A],

PB = (1 − x)[ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A] ◮ If the offer is rejected the game ends and the players’ payoffs

are: PA = wA [ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A],

PB = wB [ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A]

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 44 / 65

Take-it-or-leave-it offer by Party A (cont’d)

◮ Subgame Perfect Equilibria Outcome:

Shares: x = 1 − wB (1 − x) = wB

◮ SPE Strategies:

◮ A offers share 1 − x = wB; ◮ B accepts any share 1 − x′ ≥ wB.

Proof: backward induction

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 45 / 65

Take-it-or-leave-it offer by Party A (cont’d)

◮ The Payoffs associated with this equilibrium agreement are

then: PA = (1 − wB) [ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A],

PB = wB [ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A] ◮ Clearly, efficiency applies:

PA + PB = [ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A] ◮ In other words input choices are efficient (e∗ A, e∗ B). ◮ The ownership rights/entitlements of player B determine the

shares of the two parties.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 46 / 65

Two Periods Alternating Offers

Period 1: Stage I: A makes an offer xA to B, Stage II: B observes the offer and has three alternatives:

◮ he can accept the offer, then x = xA and the

game terminates;

◮ he can reject the offer and take his outside

• ption wB and the game terminates;

◮ he can reject the offer and not take his outside

• ption, then the game moves to Stage I of the

following period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 47 / 65

Two Periods Alternating Offers (cont’d)

Period 2: Stage I: B makes an offer xB to A, Stage II: A observes the offer and has two alternative choices:

◮ he can accept the offer, then x = xB and the

game terminates;

◮ he can reject the offer and take his outside

• ption wA and the game terminates.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 48 / 65

Two Periods Alternating Offers, Equilibrium

◮ Subgame Perfect Equilibrium Outcome:

Agreement is reached in the first period with payoffs: (1 − max{wB, δ(1 − wA)}, max{wB, δ(1 − wA)})

◮ SPE Strategies:

◮ A offers share 1 − xA = max{wB, δ(1 − wA)} in period 1; ◮ B accepts any share 1 − x′ ≥ max{wB, δ(1 − wA)} in period 1; ◮ B rejects any share 1 − x′ < max{wB, δ(1 − wA)} in period 1; ◮ B offers share xB = wA in the period 2; ◮ A accepts any share x′ ≥ wA in the period 2. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 49 / 65

Efficiency and Ownership Rights

◮ Notice that an efficient agreement is reached in all cases

independently of the size of the entitlements (wA, wB).

◮ Clearly in all cases the result above implies that we would get

the efficient outcome: (e∗

A, e∗ B). ◮ However, the share that accrues to each party depends on the

entitlements wA and wB.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 50 / 65

Comments

◮ If each party is entitled to the choice of his input, then:

wA = ΠA(ˆ eA) ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A

wB = ΠB(ˆ eB) − γ ˆ eA ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A ◮ It is important to recall that the parties need to agree to the

contract before choosing the investments (eA, eB).

◮ The Coasian contract specifies: their choice of investments

(e∗

A, e∗ B) and the transfers that the parties have to make to

each other.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 51 / 65

Costly Bargaining

◮ Recall that the Coase Theorem explicitly stated that

bargaining was costless.

◮ Let us now instead consider bargaining when there exists an

• pportunity cost of time: spending the period bargaining has

a positive cost.

◮ Assume that this costs are (cA, cB). ◮ If either party decides not to pay the cost than negotiation

breaks down since either party does not spend the time at the negotiation table.

◮ Assume wA = wB = 0 and discount factors are δA and δB.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 52 / 65

Costly Bargaining (cont’d)

Consider first the take-it-or-leave-it offer: Stage 0 both parties decide, simultaneously and independently, whether to pay (cA, cB); if either or both parties do not pay the game ends; Stage I if both parties pay, A makes an offer xA to B, Stage II B observes the offer and can accept or reject it; if the offer is accepted then x = xA and the game terminates; if the offer is rejected the game ends.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 53 / 65

Costly Bargaining (cont’d)

◮ Payoffs: if parties agree on x:

ΠA = x − cA ΠB = (1 − x − cB),

◮ if they do not agree and both pay the costs :

ΠA = −cA, ΠB = −cB,

◮ if only one party, i ∈ {A, B} pays the cost:

Πi = −ci, Π−i = 0.

◮ if neither party pays the cost:

ΠA = 0, ΠB = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 54 / 65

Costly Bargaining (cont’d)

◮ We solve for the SPE using backward induction. ◮ Consider Stages I and II:

◮ The SPE of these stages is A offers xA = 1 and B accepts. ◮ Payoffs associated with the SPE of this subgame: (1, 0). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 55 / 65

Costly Bargaining (cont’d)

◮ Consider now Stage 0. The transformed game is described by

the following normal form: pay cB do not pay cB pay cA 1 − cA, −cB −cA, 0 do not pay cA 0, −cB 0, 0

◮ The unique NE of this game is:

(do not pay cA, do not pay cB).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 56 / 65

Costly Bargaining (cont’d)

◮ Therefore the unique SPE of the entire game is: ◮ Strategies:

◮ A does not pay cA; ◮ A offers xA = 1; ◮ B does not pay cB; ◮ B accepts any offer xA ≤ 1.

◮ The outcome is then a payoff of (0, 0) . ◮ Notice that this result applies for every (even very small) costs

cA ≥ 0 and cB > 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 57 / 65

Costly Bargaining (cont’d)

◮ Notice that we can choose cA + cB < 1 which means that is

efficient to reach an agreement, but agreement is not an equilibrium of the game.

◮ In other words, this is a failure of the Coase Theorem. ◮ The question is then whether it is the take-it-or-leave-it nature

• f the game that generates this very inefficient outcome.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 58 / 65

Two-Periods bargaining with Transaction Costs

First period: Stage 0 both parties decide, simultaneously and independently, whether to pay (cA, cB); if either or both parties do not pay the game ends; Stage I if both parties pay, A makes an offer xA to B, Stage II B observes the offer and can accept or reject it; if the offer is accepted then x = xA and the game terminates; if the offer is rejected the game moves to Stage I of the following period.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 59 / 65

Two-Periods bargaining with Transaction Costs (cont’d)

Second period: Stage I if both parties pay, B makes an offer xB to A, Stage II A observes the offer and can accept or reject it; if the offer is accepted then x = xB and the game terminates; if the offer is rejected the game ends.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 60 / 65

Two-Periods bargaining with Transaction Costs (cont’d)

◮ Payoffs: if parties agree on x in period n:

ΠA = δn−1

A

x − cA ΠB = δn−1

B

(1 − x) − cB,

◮ if they do not agree and both pay the costs:

ΠA = −cA, ΠB = −cB,

◮ if only one party, i ∈ {A, B} pays the cost:

Πi = −ci, Π−i = 0.

◮ If neither party pays the cost:

ΠA = 0, ΠB = 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 61 / 65

Two-Periods bargaining with Transaction Costs (cont’d)

◮ We solve for the SPE using backward induction. ◮ Consider the second period and Stages I and II of the first

period:

◮ This is a familiar subgame and the SPE of these stages is:

◮ A offers xA = 1 − δB in period 1; ◮ B accepts any offer x′ ≤ 1 − δB and rejects any offer

x′ > 1 − δB in period 1;

◮ B offers xB = 0 in period 2; ◮ A accepts any offer x′ ≥ 0 in period 2. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 62 / 65

Two-Periods bargaining with Transaction Costs (cont’d)

◮ Payoffs associated with the SPE of this subgame are:

(1 − δB, δB).

◮ Consider now Stage 0 of period 1. The transformed game is

described by the following normal form: pay cB do not pay cB pay cA 1 − δB − cA, δB − cB −cA, 0 do not pay cA 0, −cB 0, 0

◮ The Nash equilibrium of this game will depend on whether

(1 − δB) ≥ cA and δB ≥ cB.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 63 / 65

Two-Periods bargaining with Transaction Costs (cont’d)

◮ In particular the unique SPE if δB > 1 − cA or δB < cB is

such that:

◮ No player i ∈ {A, B} pays his cost ci; ◮ A offers xA = 1 − δB in period 1; ◮ B accepts any offer x′ ≤ 1 − δB and rejects any offer

x′ > 1 − δBin period 1;

◮ B offers xB = 0; ◮ A accepts any offer x′ ≥ 0.

◮ The outcome is a payoff for both players of (0, 0).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 64 / 65

Two-Periods bargaining with Transaction Costs (cont’d)

◮ Notice that the equilibrium we just described may be

inefficient.

◮ Indeed, we just envisage a situation where the strong version

• f the Coase Theorem fails.

◮ This corresponds to the case:

cA + cB < 1 δB > 1 − cA

• r

δB < cB

◮ The inefficiency does not depend on the take-it-or-leave-it

nature of the game.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 65 / 65