EC516 Contracts and Organisations for Research Students: Lecture 1 - - PowerPoint PPT Presentation

EC516 Contracts and Organisations for Research Students: Lecture 1

Leonardo Felli LSE, D6 19 February 2009

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Course Outline

The Law and Economics of Contracts Lecture 1: Contracts and Enforcement: Complete

• Contracts. Coase Theorem and its failures.

Lecture 2: Transaction Costs, Why parties go to Court? Lecture 3: The role of Courts: insurance, filling the gaps, disclosure. Lecture 4: Legal Systems: efficiency and tradeoffs. Lecture 5: Enforcement, Power, Crime and Punishment.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Admin

My coordinates: S.478, x7525, lfelli@econ.lse.ac.uk PA: Gill Wedlake, S.379, x6889, g.m.wedlake@lse.ac.uk Office Hours:

Thursday 2:00-4:00 p.m.

• r by appointment (e-mail lfelli@econ.lse.ac.uk).

Course Material: available at: http://econ.lse.ac.uk/staff/lfelli/teaching

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

References: Contract Theory

Oliver Hart, Firms Contracts and Financial Structure, Oxford: Oxford University Press, 1995. Jean-Jacques Laffont and David Martimort, The Theory of Inncentives: The Principal-Agent Model, Princeton and Oxford: Princeton University Press, 2002. Patrick Bolton and Mathias Dewatripont, Contract Theory, Cambridge: M.I.T. Press, 2004.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

References: Contract Law

John D. Calamari and Joseph M. Perillo, Contracts, St. Paul: West Publishing Co., 1987. Stephen A. Smith and P.S. Atiyah, Atiyah’s Introduction to the Law of Contract, Oxford: Oxford University Press, Clarendon Law Series, 2006. P.S. Atiyah, The Rise and Fall of the Freedom of Contract, Oxford: Clarendon Press, 1979. Richard Craswell and Alan Schwartz, Foundations of Contract Law, New York: Foundation Press, 1994. Roy Kreitner, Calculating Promises: The Emergence of Modern American Contract Doctrine, Stanford: Stanford University Press, 2007.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The Contract

The first natural question that needs to be answered is: What is a contract? Definition A contract is the ruling of an economic transaction: the description of the performance that the contracting parties agree to complete at a (possibly future) date.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Example: a contract for the purchase of a specific item, say a meal. It specifies:

the restaurant’s performance (number of courses, quality

• f food, cooking details, etc. . . ),

the customer’s performance (payment in full upon completion).

Contracts involve not only the contracting parties, but also

• utsiders (enforcing authority: the court).

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

We distinguish between implicit and explicit contracts. A contract is implicit or self-enforcing whenever the environment in which the contracting parties operate corresponds to the extensive form of a game whose (unique) subgame perfect Nash equilibrium (PBE) exactly corresponds to the outcome the parties would like to implement. If you believe in SPE or PBE then there is no need for explicit communication. The two rational individuals will behave in the way required.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

If the outcome the parties would like to implement is not the subgame perfect Nash equilibrium of the environment in which they operate the parties might want to modify the environment. This is accomplished through and explicit contract. An explicit contract is a commitment device requiring:

an explicit agreement between the parties, the intervention of a third party: the court.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The role of the court is to force the parties to behave in a way that differs from the one that would arise in the absence of any agreement. An explicit contract therefore specifies a new extensive form corresponding to a new game for the parties. The usual way for the court to guarantee that the parties

• perate in this new environment is by modifying the

parties’ payoffs, when necessary. By agreeing to bring in a court in the game the parties commit to play a game that differs from the initial one they were in.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

To see how the presence of a court may work consider the following example: (Kreps, 1984) A buyer B and a seller S wish to trade an indivisible item at date 1. The buyer’s valuation: v, The seller’s delivery cost: c. Let v > c In other words, trade is socially efficient.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Let p be a reasonable price level (we abstract for the moment from bargaining) such that: v > p > c. B’s and S’s situation may be described by the following normal form: deliver not deliver pay p v − p, p − c −p, p not pay p v, −c 0, 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The unique Nash equilibrium (dominant solvable) is: (B does not pay, S does not deliver). This is clearly an inefficient outcome: no trade. The situation does not change if any of the following two extensive forms are played.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The unique SPE of the following game is: {B does not pay, S does not deliver at both nodes}.

❜ ❅ ❅ ❅ ❅ ❅ ❅ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ q

• ☞

☞ ☞ ☞ ☞ ☞ ☞ ☞ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ q q q q q

B S S (v − p, p − c) (−p, p) (v, −c) (0, 0) pay p not pay p deliver not deliver not deliver deliver

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The unique SPE of the following game is: {S does not deliver, B does not pay at both nodes, }.

❜ ❅ ❅ ❅ ❅ ❅ ❅ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ q

• ☞

☞ ☞ ☞ ☞ ☞ ☞ ☞ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ q q q q q

S B B (p − c, v − p) (−c, v) (p, −p) (0, 0) deliver not deliver pay p not pay p not pay p pay p

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Solution: to this inefficiency is an explicit contract enforced by a court. It specifies:

the payment p that B is supposed to make contingent on S delivering the item, the punishment FB > p (implicit in the legal system) imposed by the court on B in the event that S delivers and B does not pay, the punishment FS > c (implicit in the legal system) imposed by the court on S in the event that B pays but S does not deliver.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

In this case the normal form describing the contracting parties problem once the contract is in place is: deliver not deliver pay p v − p, p − c FS − p, p − FS not pay p v − FB, FB − c 0, 0 The unique Nash equilibrium is now: (B pays p, S delivers). Notice that the particular contract considered is budget balanced off-the-equilibrium-path: it is renegotiation

• proof. The latter property does not always hold.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Consider now an environment in which when the plaintiff goes to court detection is costly (κ) and is successful only with probability π. The payoffs associated with (not pay p, deliver) are: v − π (FB + κ), π FB − (1 − π)κ − c The payoffs associated with (pay p, not deliver) are: π FS − (1 − π)κ − p, p − π (FS + κ)

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Notice that as deterrence goes: the detection probability (policing, monitoring) π and the size of the punishment, FB and FS, are substitutes (Becker 1968). The game is solved assuming that court’s costs κ are paid by the loosing party (British system). If court’s costs κ are too high the game has multiple Nash equilibria: (pay p, deliver) and (not pay p, not deliver).

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

This example clearly shows the need for an enforcement mechanism. This mechanism may be due to:

the parties being involved in a repeated relationship relationship/implicit contracting, (multiplicity might be a problem). the presence of a legal system that through a court enforces the parties agreement (explicit contracting).

Notice that according to this interpretation the court is essentially a commitment device available to the parties that can be used when the parties agree to call it in.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

An alternative interpretation is that the court itself is one

• f the players of the game.

It should therefore be endowed with a payoff function and an action space and should be explicitly considered in the analysis of the contractual situation (we will come back to this). It should be mentioned that using this line of argument

• ne could obtain a rather extreme interpretation of a

contract (a law) (Mailath, Morris and Postlewaite 2000).

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The view is that enforcement/punishment is the only relevant activity. A contract (a law) can at best be interpreted as cheap talk that allows the parties to coordinate on a particular equilibrium of the game. No new equilibrium is introduced by the parties agreeing

• n a contract or by the parliament passing a law.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

From now on we will assume that the two (or more) parties involved in the contractual relationship operate in a market economy with a well functioning legal system. Whatever contract the parties agree to it will be enforced by the court. The penalties for breaching the contract will be assumed to be sufficiently severe that no contracting party will ever consider the possibility of not honoring the contract. We will abstract from explicitly specifying these penalties.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Coase Theorem:

Once we have established what a contract is and how it works the next natural question is: What could parties achieve in an economic environment in which they can costlessly negotiate a contractual agreement? The answer to this question is the celebrated Coase Theorem.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

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 write contracts that are individually rational and Pareto efficient. A contract is individually rational if each contracting party is not worse off by deciding to sign the contract rather then choosing not to sign it.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

This is the reflection of an other 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 exists an

• ther feasible contract that makes at least one of the

contracting party strictly better off without making any

• ther contracting party worse off.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

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

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

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 (x > 0) imposed by A on B. B’s payoff function is then: ΠB(eB) − γ eA = RB(eB) − c eB − γ eA

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Assume first that the parties choose the amounts of input eA and eB simultaneously and independently without any prior agreement. 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

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Consider now the social efficient amounts of input e∗

A and

e∗

B.

These solve the 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

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral 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.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Assume now that the two contracting parties have the

• pportunity to get together and agree on a contract before

the amounts of input are chosen. There exists strictly positive gains from trade. A reduction in the amount of input eA from ˆ eA to e∗

A will

generate:

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

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The former effect is more than compensated by the latter

• ne. This may create room for negotiation.

Normalize for simplicity the total size of the surplus that is available to share between the two contracting parties to have size 1 (simple normalization). 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.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Denote wA and wB the entitlements of party A, respectively B where: wA + wB < 1. We assume the following extensive form for the costless negotiation between the two parties: Infinite horizon, alternating offers bargaining game with discounting and outside options.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

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.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Extensive Form:

Odd periods: 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 do not take his

• utside option, then the game moves to

Stage I of the following period.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Even periods: Stage I: B makes an offer xB to A, Stage II: A observes the offer and ha three 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;

he can reject the offer and do not take his

• utside option, then the game moves to

Stage I of the following period.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Payoffs:

If parties agree on x in period n + 1: πA(σA, σB) = δnx, πB(σA, σB) = δn (1 − x), If they do not agree and either party takes his outside

• ption in period n + 1:

πA(σA, σB) = δnwA, πB(σA, σB) = δn wB.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Theorem (Deal Me Out) For any discount factor δ < 1, and any pair (wA, wB), wA + wB < 1, the bargaining game has a unique subgame perfect equilibrium. Agreement between the parties is immediate and the outside

• ptions are never exercised.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Proof: (sketch) Denote xH

i , respectively xL i , i ∈ {A, B}, the highest,

respectively the lowest, possible share that A can receive in a subgame that starts with i making the offer. We then have that: xH

B ≤ max{wA, δ xH A },

1 − xL

A ≤ max{wB, δ

• 1 − xL

B

• }

Moreover: xL

B ≥ max{wA, δ xL A},

1 − xH

A ≤ max{wB, δ

• 1 − xH

B

• }

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Solving these inequalities we obtain: xH

A = xL A = xA,

xH

B = xL B = xB

We also obtain that:

If wA ≤ δ 1 + δ , wB ≤ δ 1 + δ then xA = 1 1 + δ , xB = δ 1 + δ

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

If wA ≥ δ 1 + δ, wB ≤ δ(1 − wA) then xA = 1 − δ(1 − wA), xB = wA If wA ≤ δ(1 − wB), wB ≥ δ 1 + δ then xA = 1 − wB, xB = δ(1 − wB) If wA ≥ δ(1 − wB), wB ≥ δ(1 − wA) then xA = 1 − wB, xB = wA

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

These offers characterize a pair of strategies (σA, σB). It is easy to show that these strategies constitute the unique subgame perfect equilibrium of the bargaining game. Notice that an efficient agreement is reached independently of the size of the entitlements.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

In particular if each party is entitle 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

If instead party B is entitled to preclude party A from operating his technology, then: wA = 0, wB = ΠB(ˆ eB) ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

In either case the result above implies that we would get the efficient outcome: (e∗

A, e∗ B).

However, the share that accrue to each party depends on the entitlements wA and wB. The equilibrium contract specifies a transfer between the two parties and A’s choice of input e∗

A.

Also the transfer depend on the entitlements wA and wB.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

From now on we are going to focus on models in which the Coase Theorem fails. The classic cause for the failure of the Coase Theorem is the presence of asymmetric information between the parties. This is a situation in which each party has private information on his own preferences (hidden information model). Recall that a game of incomplete information (a player does not know the preferences of one opponent) can always be recast as a game of imperfect information (a player does not know the history of the game) (Harsanyi 1967).

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Asymmetric Information:

We are going to consider first a very simple model of bargaining under bilateral asymmetric information (a specific extensive form) with no externalities. We will show that in this situation efficiency cannot be achieved. Recall that the Coase Theorem implies efficiency even in the presence of externalities therefore if inefficiency arises in the absence of externalities we can conclude that the Theorem fails.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Notice however that this does not imply that we cannot find an extensive form that will achieve efficiency. Fortunately an other fundamental principle of contract theory will help in this case: Revelation Principle. Using the revelation principle we will be able to conclude that efficiency cannot be achieved whatever extensive form governs the bargaining between the two parties under bilateral asymmetric information.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Consider the following simple model of bilateral trade (double auction). Two players, a buyer and a seller: N = {b, s}. The seller names an asking price: ps. The buyer names an offer price: pb.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The action spaces: As = {ps ≥ 0}, Ab = {pb ≥ 0}. The seller owns and attaches value vs to an indivisible unit

• f a good.

The buyer attaches value vb to the unit of the good and is willing to pay up to vb for it. The valuations for the unit of the good of the seller and the buyer are their private information of each player. Player i ∈ {b, s} believes that the valuation of the

• pponent v−i takes values in the unit interval.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The type spaces: Ts = {0 ≤ vs ≤ 1}, Tb = {0 ≤ vb ≤ 1} Player i ∈ {b, s} also believes that the valuation of the

• pponent is uniformly distributed on [0, 1]:

µs = 1, µb = 1. The extensive form of the game is such that:

If pb ≥ ps then they trade at the average price: p = (ps + pb) 2 . If pb < ps then no trade occurs.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The payoffs to both the seller and the buyer are then: us(ps, pb; vs, vb) =    (ps + pb) 2 if pb ≥ ps vs if pb < ps and ub(ps, pb; vs, vb) =    vb − (ps + pb) 2 if pb ≥ ps if pb < ps Players’ strategies: ps(vs) and pb(vb). We consider strictly monotonic and differentiable strategies.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Consider now the seller’s best reply. This is defined by the following maximization problem: max

ps

Evb {us(ps, pb; vs, vb) | vs, pb(vb)} Consider now the seller’s payoff, substituting pb(vb) we have: us =    (ps + pb(vb)) 2 if pb(vb) ≥ ps vs if pb(vb) < ps

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

• r

us =    (ps + pb(vb)) 2 if vb ≥ p−1

b (ps)

vs if vb < p−1

b (ps)

The seller’s maximization problem is then: max

ps

p−1

b

(ps) vb=0

vs dvb + 1

vb=p−1

b

(ps)

(ps + pb(vb) 2 dvb

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Recall that by Leibniz’s rule: ∂ ∂y β(y)

α(y)

G(x, y)dx

• =

= G(β(y), y) β′(y) − G(α(y), y)α′(y) + + β(y)

α(y)

∂G(x, y) ∂y dx

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Therefore the first order conditions are: vs dp−1

b (ps)

dps − 1 2

• ps + pb(p−1

b (ps))

dp−1

b (ps)

dps + + 1

p−1

b

(ps)

1 2 dvb = 0

• r from ps = pb(p−1

b (ps)):

(vs − ps) dp−1

b (ps)

dps + 1 2

• vb

1

p−1

b

(ps) = 0

which gives: (vs − ps)dp−1

b (ps)

dps + 1 2

• 1 − p−1

b (ps)

• = 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

The buyer’s best reply is instead defined by: max

pb

Evs {ub(ps, pb; vs, vb) | vb, ps(vs)} Consider now the buyer’s payoff obtained substituting ps(vs): ub =    vb − (ps(vs) + pb) 2 if vs ≤ p−1

s (pb)

if vs > p−1

s (pb)

we then get max

pb

p−1

s

(pb) vs=0

• vb − (ps(vs) + pb)

2

• dvs

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Therefore the first order conditions are:

• vb − (ps(p−1

s (pb)) + pb)

2 dp−1

s (pb)

dpb + − 1 2 p−1

s

(pb) vs=0

dvs = 0

• r

[vb − pb] dp−1

s (pb)

dpb − 1 2

• vs

p−1

s

(pb)

= 0 which gives: (vb − pb)dp−1

s (pb)

dpb − 1 2 p−1

s (pb) = 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

To simplify notation we re-write p−1

b (·) = qb(·) and

p−1

s (·) = qs(·).

The two differential equations that define the best reply of the seller and the buyer are then: [qs(ps) − ps] q′

b(ps) − 1

2 [1 − qb(ps)] = 0 [qb(pb) − pb] q′

s(pb) − 1

2 qs(pb) = 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Solving the second equation for qb(pb) and differentiating yields: q′

b(pb) = 1

2

• 3 − qs(pb)q′′

s (pb)

[q′

s(pb)]2

• Substituting this expression into the first differential

equation we get: [qs(ps) − ps]

• 3 − qs(ps)q′′

s (ps)

[q′

s(ps)]2

• −
• 1 − ps − qs(ps)

q′

s(ps)

• = 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

This is a second-order differential equation in qs(·) that has a two parameter family of solutions. The simplest family of solution takes the form: qs(ps) = α ps + β Then the values α = 3/2 and β = −3/8 solve the second-order differential equation. The definition of qs(·) and qb(·) imply that: ps = 2 3 vs + 1 4, pb = 2 3 vb + 1 12

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

This is the (unique) Bayesian Nash equilibrium of this game. Notice now that it is efficient to trade whenever: vb ≥ vs However in this double auction game trade occurs whenever: pb ≥ ps

• r

2 3 vb + 1 12 ≥ 2 3 vs + 1 4

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

In other words, in equilibrium trade occurs whenever: vb ≥ vs + 1 4 ................................................... .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

✻ ✲

• ♣

1 vb vs 1 (0, 0) vs = vb

• Trade

✻

vb = vs + 1

4

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Revelation Principle:

The obvious question is now: how can we make sure that there does not exists an alternative way for the parties to achieve efficiency? The tool that allows us to give an answer to this question is: Revelation Principle The Revelation Principle (Green and Laffont 1977, Myerson 1979, Harris and Townsend 1981, Dasgupta, Hammond and Maskin 1979) greatly simplify the set of feasible mechanisms for the parties.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

This Revelation principle says that there is no loss in generality in restricting attention to direct revelation mechanisms that satisfy truth-telling constraints. Recall:

the indirect mechanism is the one in which parties agree to a trade set prices etc. . . the direct mechanism is the one in which parties report their private information to a mechanism designer who according to the reports enforces the mechanism.

Looking for the truth-telling equilibrium of the direct mechanism that maximizes the principal’s utility is the way to identify the best possible indirect mechanism from the principal’s view point.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Since every BNE of every indirect mechanism has an associated truth-telling BNE of a direct mechanism if we find the truth-telling BNE of the direct mechanism that maximizes the principal’s utility there cannot exist any BNE of the indirect mechanism that is better for the principal. Notice that this way to proceed does not require us to specify the space of all possible indirect mechanisms. It is critical that the principal can commit to the mechanism in advance: renegotiation may lead to a failure

• f the revelation mechanisms (Dewatripont, 1989).

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Bilateral Trade (Myerson and Satterthwaite 1983)

In our setting there is no principal, but the two parties at an ex-ante stage — before they learn their private information — will commit to a mechanism via the contract. They will choose their contract in a way that maximizes their ex-ante welfare. Assume further that this is a pure bilateral contract transfers cannot involve a third party. In jargon the contract has to be budget balancing.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

A seller and a buyer trade a single unit of a good. The seller’s cost of delivering is c and it is the seller’s private information: c ∼ PS(c), c ∈ [c, c] The buyer’s valuation is v and it is the buyer’s private information: v ∼ PB(v), v ∈ [v, v] A contract in this environment is a pair (φ, t) where

φ is the probability of trade, t is the transfer from the buyer to the seller.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

By revelation principle restrict attention to truth-telling direct mechanisms. The seller’s indirect utility is then: US(ˆ c, v|c) = t(ˆ c, v) − φ(ˆ c, v) c The buyer’s indirect utility is instead: UB(c, ˆ v|v) = φ(c, ˆ v) v − t(c, ˆ v) Denote: US(ˆ c) = Ev [t(ˆ c, v) − φ(ˆ c, v) c] = t(ˆ c) − φ(ˆ c) c UB(ˆ v) = Ec [φ(c, ˆ v) v − t(c, ˆ v)] = φ(ˆ v) v − t(ˆ v)

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Therefore every truth-telling BNE direct mechanism has to satisfy the following set of incentive compatibility constraints (IC): US(c) ≥ t(ˆ c) − φ(ˆ c) c ∀c, ˆ c ∈ [c, c] UB(v) ≥ φ(ˆ v) v − t(ˆ v) ∀v, ˆ v ∈ [v, v] Since once again we insist on freedom of contract we also require the following individual rationality constraints (IR) to be satisfied: US(c) ≥ 0, ∀c, ˆ c ∈ [c, c] UB(v) ≥ 0, ∀v, ˆ v ∈ [v, v]

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Notice now that both parties’ preferences satisfy the Spence-Mirrlees single crossing conditions: ∂ ∂v

• − ∂UB/∂t

∂UB/∂φ

• > 0

The following result helps us to write (IC) and (IR) constraints in a manageable form.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Theorem (Myerson and Satterthwaite 1983) For any probability φ(c, v) there exists a transfer function t(c, v) that satisfies (IR) and (IC) if and only if: Ec,v [φ(c, v) (JB(v) − JS(c))] ≥ 0 where JB(v) =

• v − 1 − PB(v)

pB(v)

• ,

JS(c) =

• c + PS(c)

pS(c)

• and

dφ(c) dc ≤ 0, dφ(v) dv ≥ 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Proof: From the (IC) constraints since for every ˆ v > v we must have: UB(v) ≥ φ(ˆ v) v − t(ˆ v) = UB(ˆ v) − (ˆ v − v) φ(ˆ v), UB(ˆ v) ≥ φ(v) ˆ v − t(v) = UB(v) − (v − ˆ v) φ(v), Summing the two inequalities we get: (ˆ v − v) φ(ˆ v) ≥ (ˆ v − v) φ(v) Dividing by (ˆ v − v) and letting ˆ v tend to v we obtain: U′

B(v) = φ(v)

Since φ(v, c) ∈ [0, 1] we obtain for every v ∈ [v, v]: UB(v) ≥ UB(v)

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

We therefore conclude that the only relevant (IR) constraint is: UB(v) ≥ 0 Notice that since for every ˆ v > v we have: (ˆ v − v) φ(ˆ v) ≥ (ˆ v − v) φ(v) we also obtain that: dφ(v) dv ≥ 0 Symmetrically for the seller we can prove that (IC) constraint implies: dφ(c) dc ≤ 0 Consider now the differential equation obtained above: U′

B(v) = φ(v)

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Integrating it we obtain: UB(v) = UB(v) + v

v

φ(ν)dν and symmetrically for the seller we obtain: US(c) = US(c) + c

c

φ(γ)dγ By budget balancing we now get: 0 = Ec [t(c)] − Ev [t(v)] = = c

c

• φ(c) c +

c

c

φ(γ)dγ

• pS(c) dc + US(c) +

+ v

v

v

v

φ(ν)dν − v φ(v)

• pB(v)dv + UB(v)

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Integrating by parts we get: US(c) + UB(v) = = − c

c

• c + PS(c)

pS(c)

• φ(c) pS(c) dc +

+ v

v

• v − 1 − PB(v)

pB(v)

• φ(v) pB(v)dv
• r

US(c) + UB(v) = Ec,v [φ(c, v) (JB(v) − JS(c))] Since (IR) is such that US(c) ≥ 0 and UB(v) ≥ 0 then: Ec,v [φ(c, v) (JB(v) − JS(c))] ≥ 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Sufficiency is a bit more complex to prove it requires us to solve the partial differential equation that is represented by the FOC of the (IC) constraints. The parties’ ex-ante problem is now: max

φi

Ec,v [φ(c, v) (v − c)] s.t. Ec,v [φ(c, v) (JB(v) − JS(c))] ≥ 0 dφ(c) dc ≤ 0, dφ(v) dv ≥ 0

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Ignoring monotonicity conditions and denoting µ the lagrange multiplier of the remaining constraint we get a lagrangian function that is linear in φi: Ec,v

• φ(c, v)
• (v − c) −

µ 1 − µ 1 − PB(v) pB(v) − PS(c) pS(c)

• The solution is to set φ = 1 if and only if the term in

brackets is strictly positive. In other words trade occurs if and only if, for µ ≥ 0: v − µ 1 − µ 1 − PB(v) pB(v)

• ≥ c +

µ 1 − µ PS(c) pS(c)

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

This φ(c, v) is weakly monotonic in: v − µ 1 − µ 1 − PB(v) pB(v)

• and

c + µ 1 − µ PS(c) pS(c)

• Then MHRP implies that both monotonicity conditions

are satisfied and hence local and global (IC) holds. Clearly if µ > 0 there will be inefficiencies in trade.

EC516 Contracts and Organisations for Research Students: Lecture 1 Leonardo Felli Introduction

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The Contract. The Contract. Coase Theorem Asymmetric info Revelation Principle Bilateral Trade

Theorem (Myerson and Satterthwaite 1983) If c > v and v > c then necessarily µ > 0. Proof: Immediate by substituting the efficient probabilities φi into the constraint of the parties problem and showing that is violated. Clearly with bilateral asymmetric information the Coase Theorem fails in a very relevant sense: efficiency is no longer guaranteed by the use of contracts.