EC516 Contracts and Organisations for Research Students: Lecture 2 - - PowerPoint PPT Presentation

EC516 Contracts and Organisations for Research Students: Lecture 2

Leonardo Felli LSE, D6 26 February 2009

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

Transaction Costs

Consider now a different cause for the failure of the Coase Theorem: the presence of transaction costs. Of course for this to be an interesting argument we need to abstract from other sources of failure of the Coase Theorem such as asymmetric information. 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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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 usually known as a hold-up problem. The hold-up problem yield an outcome that is constrained inefficient.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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 the agreement if B accepts it and: 10% 100 − 20 + x ≥ 0,

• r x ≥ 10.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

What are the transaction costs?

Ex-ante transaction costs: time to arrange 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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

These costs can be monetized through the hiring of an

• utside 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

• f the inefficiency: moral hazard.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

Timing:

✲ ✲ s s s

t = 0 t = 1 Contract Negotiated enforced

• Simult. Decisions
• n (cA, cB)

Contract t = 2

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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)

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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 other 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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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),

• f 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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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 other 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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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 .

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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); in Stage III party i accepts any offer x ∈ [0, 1].

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

First type of inefficiency:

cB cA

✻ ✲

1 δB 1 + δB δA 1 + δA 1

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

SPE

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

(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);

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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)

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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)

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

Robustness

Finite horizon version of the game ΓN: Remark Let any finite N ≥ 1 be given. Then the unique SPE outcome

• f ΓN is neither player pays his participation cost in any period

and hence agreement is never reached.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

Sequential payment of costs: Define ΓS the bargaining game with the following: Stage I: player i first decides whether to pay ci, the

• ther player j observes in’s decision and

decides whether to pay cj; Remark The game ΓS always has an SPE in which neither player ever pays his participation cost, and hence agreement is never reached.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

Robustness of the first inefficiency to a random turn in making

• ffers.

Assume that Stage I of the game is modified so that: in odd periods A makes offer xOA with probability p while B makes offer xOB with probability (1 − p); in even periods B makes offer xEB with probability p while A makes offer xEA with probability (1 − p).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

If p > 1

2 then:

cA cB 1 1 p p

• .

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

• ✲

(1 − p) (1 − p)

• .

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

• SPE
• ❛

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

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

If p = 1

2 then:

cA cB 1 1 p = 1

2

p = 1

2

• SPE
• ✻

✲

• ❍❍❍❍❍❍❍❍❍❍❍❍

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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.

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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)

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

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

• f 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).

EC516 Contracts and Organisations for Research Students: Lecture 2 Leonardo Felli Transaction Costs

Outline Numerical Example What are transaction costs? SImple negotiation Coasian solution Continuous Costs Alternating Offers Bargaining Robustness

There is no way to select consistently the efficient outcomes in the game. The only robust equilibrium of this bargaining game with transaction costs is the one in which an extreme inefficiency is reached: no agreement. This is obviously a serious failure of the Coase Theorem.