EC537 Microeconomic Theory for Research Students, Part II: Lecture 4 - - PowerPoint PPT Presentation

EC537 Microeconomic Theory for Research Students, Part II: Lecture 4

Leonardo Felli

CLM.G.4

29 November, 2011

Transaction Costs

We now return to the Coase Theorem and consider how the result is affected by the presence of transaction costs.

Theorem (Strong version of the Coase Theorem)

The Coase theorem guarantees efficiency:

1 regardless of the way in which property rights are assigned, and 2 whenever the mutual gains from trade exceed the necessary

transaction costs.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 2 / 66

We are going to show that this is not necessarily the case. The reason is the strategic role of transaction costs. Key factor: some transaction costs have to be paid ex-ante, before the negotiation starts. These ex-ante transaction costs generate an inefficiency known as a hold-up problem. The hold-up problem yield an outcome that is constrained inefficient.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 3 / 66

Numerical Example

Potential Surplus = 100, Ex-ante cost to each negotiating party = 20, Distribution of bargaining power = (10%, 90%), Ex-ante Payoff to party A = (10% 100 − 20) = −10, Ex-ante Payoff to party B = (90% 100 − 20) = 70, Social surplus = 60. Coasian negotiation opportunity is left unexploited.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 4 / 66

Natural question: whether it is possible to find a Coasian solution to this inefficiency. In other words we are asking whether the parties can agree ex-ante on a transfer contingent on each party entering a future negotiation. We are going to show that under plausible conditions a Coasian solution of this form may not be available. The reason is that any new negotiation may itself be associated with (possibly small) ex-ante transaction costs.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 5 / 66

In the Numerical Example above Party B makes a transfer to party A contingent on the cost of 20 being paid by A. Assume that B makes a take-it-or-leave-it offer to A, Ex-ante costs to each party associated with this ‘agreement contingent on future negotiation’ = 1, A accepts this transfer if: 10% 100 − 20 + x ≥ 0,

• r x ≥ 10.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 6 / 66

There always exists an equilibrium in which x = 10. Ex-ante payoff to party A πA = 10% 100 − 20 + 10 − 1 = −1 No negotiation (contingent or not) will take place. A Coasian solution is not available.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 7 / 66

What are the transaction costs?

Ex-ante transaction costs: time to arrange and/or participate in a meeting, time and effort to conceive and agree upon a suitable negotiation language, time and effort to collect information about the legal environment in which the agreement is enforced, time to collect and analyze background information for the negotiation, time and effort to think about the negotiation at hand.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 8 / 66

These costs can be monetized through the hiring of an outside party: typically a lawyer. The problem does not disappear if the lawyer needs to be paid independently of the success of the negotiation: no contingent fees. Indeed, monetizing the costs may increase the magnitude of the inefficiency: moral hazard.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 9 / 66

Simple Coasian Negotiation:

Consider the following simple coasian negotiation: two agents, i ∈ {A, B}; share a surplus, size of the surplus normalized to one, parties’ payoffs in case of disagreement to zero. Each party faces ex-ante costs: (cA, cB).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 10 / 66

Assume that (cA, cB) are: complements: each party i has to pay ci for the negotiation to be feasible; affordable: party i’s endowment covers ci; efficient: cA + cB < 1.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 11 / 66

Timing:

✲ ✲ s s s

t = 0 t = 1 Contract Negotiated enforced

• Simult. Decisions
• n (cA, cB)

Contract t = 2

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 12 / 66

We solve the model backward. We start with a simple bargaining rule:

Let λ be the bargaining power of A. The division of surplus at t = 1 is then (λ, 1 − λ).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 13 / 66

Result

For any given λ there exists a pair (cA, cB) of affordable and efficient ex-ante costs such that the unique SPE is (not pay cA, not pay cB)

Result

For any pair (cA, cB) of affordable and efficient ex-ante costs there exists a value of λ such that the unique SPE is (not pay cA, not pay cB)

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 14 / 66

Proof: The ‘reduced form’ of the two stage game is: pay cB not pay cB pay cA λ − cA, 1 − λ − cB −cA, 0 not pay cA 0, −cB 0, 0 A pays cA iff λ ≥ cA and B pays cB, A does not pay cA if B does not pay cB, B pays cB iff 1 − λ ≥ cB and A pays cA, B does not pay cB if A does not pay cA. Therefore the result holds when λ < cA or 1 − λ < cB.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 15 / 66

Assume now that (cA, cB) are: substitutes: either party has to pay ci; affordable: party i’s endowment covers ci; efficient: min{cA, cB} < 1.

Result

Both results above hold. In the second result only one type of inefficiency may occur.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 16 / 66

Proof: The reduced form game is now: pay cB not pay cB pay cA λ − cA, 1 − λ − cB λ − cA, 1 − λ not pay cA λ, 1 − λ − cB 0, 0 A pays cA iff λ ≥ cA and B does not pay cB, A does not pay cA if B pays cB, B pays cB iff 1 − λ ≥ cB and A does not pay cA, B does not pay cB if A pays cA.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 17 / 66

We then have two types of inefficiencies: an inefficiency that leads to a unique SPE with no agreement: λ < cA, and (1 − λ) < cB an inefficiency that leads to an agreement obtained paying too high a cost: if cA < cB, λ < cA, and (1 − λ) > cB Results 1 and 2 also generalize to the case in which (cA, cB) are substitutes and strategic complements.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 18 / 66

The Impossibility of a Coasian Solution

Is there a Coasian solution to this hold-up problem? Consider the following simple contingent agreement: A transfer σB ≥ 0 (σA ≥ 0) payable contingent on whether the

• ther party decides to pay cA, (cB).

Key assumption: this new negotiation is associated with a fresh set of ex-ante costs (c1

A, c1 B);

the two sets of ex-ante costs are assumed to be complements, affordable and efficient.

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Assume: λ < cA

s s s s s s ✲ ✲ ✲ ✇

−2 −1 1 B makes offer to A A Accepts/Rejects

• n (c1

A, c1 B)

• n (cA, cB)

Tranfers Contract A/B does

✻ s

Negotiation

• Simult. Decision

not pay

• Simult. Decision

enforced 2

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Result

There always exists a SPE of this game in which both agents pay neither the second tier, (c1

A, c1 B), nor the first tier, (cA, cB), of ex-ante costs.

Proof: At each stage the two agents decide simultaneously and independently whether to pay their ex-ante costs. An agreement is achieved only if both agents pay (c1

A, c1 B) and (cA, cB).

Either agent will never pay if he expects the other not to pay his ex-ante cost.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 21 / 66

Result

There always exists a SPE of this game in which both agents pay the second tier, (c1

A, c1 B), and the first tier, (cA, cB), of ex-ante costs and an

agreement is successfully negotiated . Proof: Assume that: both parties have paid the ex-ante costs (c1

A, c1 B) at t = −2 and

party A has accepted the transfer σB ≥ 0.

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The parties’ continuation game is then: pay cB not pay cB pay cA λ − cA + σB, 1 − λ − cB − σB −cA, 0 not pay cA 0, −cB 0, It follows: A pays cA if B pays cB and λ + σB > cA and B pays cB if A pays cA and 1 − λ − σB > cB.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 23 / 66

Therefore if 1 − λ − cB > σB > cA − λ the subgame has two Pareto-ranked equilibria:

• ne in which an agreement is successfully negotiated,

an other one in which an agreement does not arise.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 24 / 66

We can then construct a SPE of the model such that at t = 0 if λ + σB ≥ cA + c1

A

the (constrained) efficient equilibrium is played. if λ + σB < cA + c1

A

the no-agreement equilibrium is played.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 25 / 66

In this equilibrium necessarily σB = cA + c1

A − λ

Therefore the agreement is successfully negotiated. Notice that: All equilibria of the model are constrained inefficient: costs paid are inefficiently high. The equilibrium described in last result is not renegotiation-proof.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 26 / 66

Definition (Benoˆ ıt and Krishna, 1993)

A SPE of the model is renegotiation-proof (RP) if and only if the equilibria played in every proper subgame are not strictly Pareto-dominated by any

• ther equilibrium of the same subgame.

Result

The unique RP SPE of the game involves both agents paying neither the second (c1

A, c1 B) nor the first (cA, cB) tier of ex-ante costs.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 27 / 66

Proof: (intuition) In any RP equilibrium the costs (c1

A, c1 B) (if paid) are

‘strategically sunk’. Therefore in equilibrium σB = cA − λ. Hence, A does not pay c1

A.

Notice that if the parties pay the ex-ante costs sequentially, rather than simultaneously the SPE of the model is unique and satisfies the last result (coordination problem solved).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 28 / 66

Continuous Costs

Consider now the case with continuous ex-ante costs (Holmstr¨

• m

1982): the more detailed the agreement is, the higher the surplus. The surplus is a monotonic and concave function of costs: x(cA, cB) the costs are complements: ∂2x(cA, cB) ∂cA∂cB > 0.

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Payoff to party A is: λ x(cA, cB) − cA while the payoff to B is: (1 − λ)x(cA, cB) − cB.

Result

Given λ, every equilibrium is such that c∗

A < cE A

c∗

B < cE B .

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 30 / 66

Proof: The equilibrium costs (c∗

A, c∗ B) are such that:

max

cA

λ x(cA, cB) − cA max

cB

(1 − λ) x(cA, cB) − cB. The first order conditions of both these problems are: ∂x(c∗

A, c∗ B)

∂cA = 1 λ and ∂x(c∗

A, c∗ B)

∂cB = 1 1 − λ

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 31 / 66

The (constrained) efficient level of costs (cE

A , cE B ) are such that:

max

cA,cB x(cA, cB) − cA − cB

The first order conditions of this problem are: ∂x(cE

A , cE B )

∂cA = 1 and ∂x(cE

A , cE B )

∂cB = 1 Concavity of x(·, ·) and the fact that: ∂2x(cA, cB) ∂cA∂cB > 0.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 32 / 66

imply c∗

A < cE A

c∗

B < cE B .

One key different between this result and the one we found for discrete costs is that this result holds for every λ ∈ (0, 1). In other words, when costs are continuous the inefficiency is more pervasive.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 33 / 66

Alternating Offer Bargaining

We now consider an infinite horizon, alternating offers bargaining game with discounting: two agents, i ∈ {A, B}; share a surplus, size of the surplus normalized to one, payoffs in case of disagreement are zero.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 34 / 66

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, (cA, cB) the costs that need to be paid by the parties to negotiate in every period.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 35 / 66

Extensive form:

Odd periods: Stage I: both parties decide, simultaneously and independently, whether to pay (cA, cB); if either or both parties do not pay the game moves to Stage I of the following period; Stage II: if both parties pay, A makes an offer xA to B, Stage III: B observes the offer and can accept or reject; 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) EC537 Advanced Microeconomics 29 November, 2011 36 / 66

Even periods: Stage I: both parties decide, simultaneously and independently, whether to pay (cA, cB); if either or both parties do not pay the game moves to Stage I of the following period; Stage II: if both parties pay, B makes an offer xB to A, Stage III: A observes the offer and can accept or reject; 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) EC537 Advanced Microeconomics 29 November, 2011 37 / 66

Payoffs: If parties agree on x in period n + 1: ΠA(σA, σB) = δn x − CA(σA, σB), ΠB(σA, σB) = δn (1 − x) − CB(σA, σB), if they do not agree: Πi(σA, σB) = −Ci(σA, σB).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 38 / 66

Result

Whatever the values of δi and ci for i ∈ {A, B}, there exists an SPE of the game in which neither player pays his participation cost in any period, and therefore an agreement is never reached. Proof: By construction: in Stage I parties do not pay their costs; in Stage II party i demands the entire surplus (xA = 1, xB = 0 or xA = 0, xB = 1); in Stage III party i accepts any offer x ∈ [0, 1].

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 39 / 66

Result

The game has an SPE in which an agreement is reached in finite time if and only if δi and ci for i ∈ {A, B} satisfy δA(1 − cA − cB) ≥ cA and δB(1 − cA − cB) ≥ cB

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 40 / 66

First type of inefficiency:

cB cA

✻ ✲

1 δB 1 + δB δA 1 + δA 1

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳

SPE

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 41 / 66

Proof: (only if:) Notice that ∀i ∈ {A, B}: cA ≤ xi ≤ 1 − cB moreover: xH

B ≤ δA

• xH

A − cA

• 1 − xL

A ≤ δB

• 1 − xL

B − cB

• by substitution we get:

δA(1 − cA − cB) ≥ cA δB(1 − cA − cB) ≥ cB

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 42 / 66

Proof: (if:) We construct a SPE where xA = 1 − cB, xB = cA in every period players pay their costs;

• dd periods: the offer is 1 − cB and B accepts any x ≤ 1 − cB;

even periods: the offer is cA and A accepts any x ≥ cA; if either player does not pay the cost then we switch to (0, 0);

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 43 / 66

if either player rejects an offer he is supposed to accept we switch to (0, 0). Notice that A cannot gain by accepting an offer x < cA since, by waiting until the next period, he gets δA(1 − cB − cA) ≥ cA The same is true for B from (1).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 44 / 66

Assume that: δA(1 − cA − cB) ≥ cA δB(1 − cA − cB) ≥ cB

Result

There exists an SPE of the A subgames in which xA is agreed immediately, if an only if xA ∈ [1 − δB(1 − cA − cB), 1 − cB] (1) There also exists an SPE of the B subgames in which xB is agreed immediately, if and only if xB ∈ [cA, δA(1 − cA − cB)] (2)

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 45 / 66

However there also exists a huge set of inefficient SPE of the game.

Result

Consider any xA as in (1) and choose any odd number n. Then there exists an SPE of the A subgames with (continuation) payoffs ΠA = δn

A(xA − cA)

ΠB = δn

B(1 − xA − cB)

(3)

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 46 / 66

Result

Consider any xB as in (2) and choose any even number n. Then there exists an SPE of the A subgames with (continuation) payoffs ΠA = δn

A(xB − cA)

ΠB = δn

B(1 − xB − cB)

(4) The symmetric result holds for B subgames. Second inefficiency: When δA(1 − cA − cB) ≥ cA and δB(1 − cA − cB) ≥ cB there exist both efficient and inefficient equilibria with arbitrarily long delays.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 47 / 66

Robustness

Finite horizon version of the game ΓN:

Result

Let any finite N ≥ 1 be given. Then the unique SPE outcome of ΓN is neither player pays his participation cost in any period and hence agreement is never reached.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 48 / 66

Sequential payment of costs: Define ΓS the bargaining game with the following: Stage I: player i first decides whether to pay ci, the other player j observes in’s decision and decides whether to pay cj;

Result

The game ΓS always has an SPE in which neither player ever pays his participation cost, and hence agreement is never reached.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 49 / 66

Robustness of the first inefficiency to a random turn in making offers. Assume that Stage I of the game is modified so that: in odd periods A makes offer xOA with probability p while B makes

• ffer xOB with probability (1 − p);

in even periods B makes offer xEB with probability p while A makes

• ffer xEA with probability (1 − p).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 50 / 66

Result

This new game has an SPE in which an agreement is reached in finite time if and only if δi and ci for i ∈ {A, B} are such that: cA ≤ p and cB ≤ p and p δB(1 − cA − cB) ≥ cB − (1 − p) and q δA(1 − cA − cB) ≥ cA − (1 − q);

• r

cA ≤ p and cB ≤ (1 − p)

• r

cB ≤ p and cA ≤ (1 − p).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 51 / 66

If p > 1

2 then:

cA cB 1 1 p p

• .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• ✲

(1 − p) (1 − p)

• .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• SPE
• ❛

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ✲ ✻

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 52 / 66

If p = 1

2 then:

cA cB 1 1 p = 1

2

p = 1

2

• SPE
• ✻

✲

• ❍❍❍❍❍❍❍❍❍❍❍❍

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 53 / 66

Selecting Efficient Equilibria

Natural question: since the players bargain with complete information will they find a way to ‘agree’ to play the efficient equilibrium? In other words will the players renegotiate out of inefficient equilibria? First approach to renegotiation: in a Coasian fashion we attempt simply to select the efficient equilibria. Minimal consistency requirement: it should be done in every proper subgame.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 54 / 66

Definition

An SPE (σA, σB) is Consistently Pareto Efficient (CPE SPE) if and only if it yields a Pareto-efficient outcome in every possible subgame.

Result

The set of CPE SPE for this game is empty. Proof: Assume there exists an CPE SPE. By the definition above, agreement must be immediate in every period (subgame).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 55 / 66

Adapting Shaked and Sutton (1984), we obtain: xA = xH

A = xL A = 1 − δB + δAcA

1 − δAδB and xB = xH

B = xL B = δA[1 − δB(1 − cB) − cA]

1 − δAδB The two equalities above imply that: xB = δA(xA − cA) (5) and 1 − xA = δB(1 − xB − cB) (6)

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 56 / 66

Both A and B need to be willing to pay their costs in Stage I of every

• period. This implies:

xB − cA ≥ δA(xA − cA) (7) and 1 − xA − cB ≥ δB(1 − xB − cB) (8) a contradiction of (5) and (6).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 57 / 66

Extensive Form Renegotiation

Celarly, a bargaining game is a model of how the negotiation proceeds between the two players. Therefore there should be no intrinsic difference between negotiation and renegotiation. We account explicitly for renegotiation possibilities in the extensive form. In this case the extensive form has to give players the chance to break

• ut of an inefficient outcome paths without making that an obligation;

However, the extensive form must allow for the outcome path to be inefficient, on- and off-the-equilibrium-path.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 58 / 66

Game of Imperfect Recall

We transform the model in a game of imperfect recall. In other words, we introduce a new stage: Stage O: Nature draws with probability p whether at the end of period n the players will forget the history of play; Dn ∈ {R, F} (i.i.d. across periods). The players are informed of the realization of Dn — denoted dn — only at the end of period n. If Dn = F all nodes leading to Stage I of period n + 1 are in the same information set. If Dn = R the game is the same as the one described above.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 59 / 66

No Absent Mindedness

Players know the date and the date at which they forgot the history last time. The game does not exhibit absent mindedness (Piccione and Rubinstein, 1997), Player i’s strategy for the game is σi = {s1

i , s2 i , s3 i . . .}

each element sn

i is contingent on Dn = F.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 60 / 66

Extensive Form

d2 d1

✚ ✚ ✚ ✚

• ❅

❅ ❅

• ❅

❅ ❅

• ❅

❅ ❅

• ❅

❅ ❅

d2

❩ ❩ ❩ ❩

F s3

i

❩ ❩ ❩ ❩

R F R

❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧

F R F F F R R

✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱

s3

i

✚ ✚ ✚ ✚

d3 s2

i

s2

i

d3 d3 d3 F R s1

i

s1

i

s3

i

s3

i

R s1

i

s1

i

s2

i

s2

i

s1

i

s1

i

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 61 / 66

Strategy and Payoffs

Nature’s strategy is σN with probability measure µ over the sequence of i.i.d. draws. New Payoffs: ΠE

i (σA, σB) =

• σN∈ΣN

µ(σN)Πi(σA, σB, σN) where if agreement x is reached in period m (m ≤ +∞): ΠA(σA, σB, σN) = δm

A x − CA(σA, σB, σN)

and ΠB(σA, σB, σN) = δm

B (1 − x) − CB(σA, σB, σN)

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 62 / 66

Definition (Perfect Bayesian Equilibrium (PBE))

A PBE is a pair of strategies and a set of beliefs such that, at every information set, the strategies are optimal given beliefs and beliefs are

• btained from equilibrium strategies and observed actions using Bayes’

rule. In equilibrium, players’ beliefs about Nature’s moves are correct.

Result

For any given pair of costs (cA, cB) there exists a ¯ p < 1 — independent of (δA, δB) — such that whenever p > ¯ p, the unique PBE of the game is such that, along any possible realization of Nature’s moves, both players never pay their participation costs in any period, and therefore an agreement is never reached.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 63 / 66

Proof: Assume by way of contradiction that for every p ∈ (0, 1) there exist a PBE in which agreement xB is reached for at least one σN in an even (odd symmetric argument) period n. In period n both players pay the costs in Stage I, B offers xB in Stage II and A accepts xB in Stage III. For xB to be an equilibrium we need A to reject any offer x ≤ xB x ≤ ΠE

A(x)

∀x < xB which implies xB ≤ pΠE

A(F) + (1 − p)δA (1 − cA)

(9)

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 64 / 66

We also need A to pay the cost cA in Stage I of period n: xB − cA ≥ pΠE

A(F)

(10) Inequalities (9) and (10) imply pΠE

A(F) + cA ≤ xB ≤ pΠE A(F) + (1 − p)δA (1 − cA)

which, in turn, gives us cA ≤ (1 − p)(1 − cA) set ¯ p to be: ¯ p = max 1 − 2cA 1 − cA , 1 − 2cB 1 − cB

• For every p > ¯

p we get a contradiction of either conditions (9) or (10).

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 65 / 66

Back to the Coase Theorem

When negotiation takes place in the presence of transaction costs the Coase theorem does not necessarily hold. In an alternating offers bargaining game under perfect information several types of inefficiencies may arise. Inefficiencies are pervasive for two reasons: It is impossible to select consistently the efficient equilibria of the model. If the parties are given sufficient opportunities to renegotiate out of inefficient outcomes, the only outcome which survives is the most inefficient one.

Leonardo Felli (LSE) EC537 Advanced Microeconomics 29 November, 2011 66 / 66