EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 - - PowerPoint PPT Presentation

EC537 Microeconomic Theory for Research Students, Part II: Lecture 1

Leonardo Felli

CLM.G.4

8 November 2011

Course Outline

Topics in Contract Theory Lecture 1: Contracts what are they? The Coase Theorem, Hidden Information, Bilateral trading. Lecture 2: Hidden Action, Multi-Tasking, Informed Principal, Intertemporal Incentives. Lecture 3: Implementation and the Foundations of Contracting with Unverifiable Information. Lecture 4: Transaction Costs. Lecture 5: Hold-Up Problem, Specific Investments and Competition.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 2 / 79

Admin

My coordinates: S.478, x7525, lfelli@econ.lse.ac.uk PA: Elizabeth Mirhady, S.686, e.mirhady@lse.ac.uk. Office Hours:

Wednesday 3:30-4:30 p.m.

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

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 3 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 4 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 5 / 79

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

the restaurant’s performance (number of courses, quality of food, cooking details, etc. . . ), the customer’s performance (payment in full upon completion).

Contracts involve not only the contracting parties, but also outsiders (enforcing authority: Court or Enforcer).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 6 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 7 / 79

If the outcome the parties would like to implement is not the subgame perfect Nash equilibrium of the environment in which they

• perate the parties might want to modify the environment.

This is accomplished through an explicit contract. An explicit contract is a commitment device requiring:

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 8 / 79

The role of the enforcer 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 enforcer to guarantee that the parties operate in this new environment is by modifying the parties’ payoffs, when necessary. By agreeing to bring in an enforcer in the game the parties commit to play a game that differs from the initial one they were in.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 9 / 79

To see how the presence of an enforcer 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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 10 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 11 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 12 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 13 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 14 / 79

Solution: to this inefficiency is an explicit contract enforced by a third party (enforcer). 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 enforcer 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 enforcer on S in the event that B pays but S does not deliver.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 15 / 79

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). This particular contract is budget balanced off-the-equilibrium-path (renegotiation proof). The latter property does not always hold.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 16 / 79

Consider now an environment in which when a party goes to the enforcer (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)

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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, below, assumes that the enforcer’s costs κ are paid by the loosing party (British system): deliver not deliver pay p v − p, p − c π FS − (1 − π)κ − p, p − π (FS + κ) not pay p v − π (FB + κ), π FB − (1 − π)κ − c 0, 0 If court’s costs κ are too high the game has multiple Nash equilibria: (pay p, deliver) and (not pay p, not deliver).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 18 / 79

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 enforces the parties agreement (explicit contracting).

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

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An alternative interpretation is that the enforcer itself is one of 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 one could

• btain a rather extreme interpretation of a contract (a law) (Mailath,

Morris and Postlewaite 2000).

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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 on a contract or by the parliament passing a law.

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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.

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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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 23 / 79

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.

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

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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

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Party B generates revenue RB(eB) (strictly concave) by choosing the input eB at the linear cost c eB (c > 0). Party B also suffers from an externality γ eA (γ > 0) imposed by A on B. B’s payoff function is then: ΠB(eB) − γ eA = RB(eB) − c eB − γ eA

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 27 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 28 / 79

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

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Comparing (ˆ eA, ˆ eB) and (e∗

A, e∗ B) we obtain using concavity of RA(·):

e∗

B = ˆ

eB, e∗

A < ˆ

eA In other words: ΠA(e∗

A) + ΠB(e∗ B) − γ e∗ A − [ΠA(ˆ

eA) + ΠB(ˆ eB) − γ ˆ eA] = = [ΠA(e∗

A) − ΠA(ˆ

eA)] + γ (ˆ eA − e∗

A) > 0

The joint surplus is reduced by the inefficiency generated by the externality.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 30 / 79

Assume now that the two contracting parties have the opportunity 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

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The former effect is more than compensated by the latter one. 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.

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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.

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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.

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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 option wB and the game terminates; he can reject the offer and do not take his outside

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

period.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 35 / 79

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 option wA and the game terminates; he can reject the offer and do not take his outside

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

period.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 36 / 79

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 option in period n + 1: πA(σA, σB) = δnwA, πB(σA, σB) = δn wB.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 37 / 79

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 options are never exercised.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 38 / 79

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

• }

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 39 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 40 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 41 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 42 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 43 / 79

In either case the result above implies that we would get the efficient

• utcome: (e∗

A, e∗ B).

However, the share that accrues 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.

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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

• f asymmetric information between the parties.

This is a situation in which each party has private information on his

• wn 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).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 45 / 79

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

• f externalities therefore if inefficiency arises in the absence of

externalities we can conclude that the Theorem fails.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 46 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 47 / 79

Consider the following simple model of bilateral trade (double auction) (Chatterjee and Samuelson, 1983). Two players, a buyer and a seller: N = {b, s}. The seller names an asking price: ps. The buyer names an offer price: pb.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 48 / 79

The action spaces: As = {ps ≥ 0}, Ab = {pb ≥ 0}. The seller owns and attaches value vs to an indivisible unit of 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 opponent v−i takes values in the unit interval.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 49 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 50 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 51 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 52 / 79

• 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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 53 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 54 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 55 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 56 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 57 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 58 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 59 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 60 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 61 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 62 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 63 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 64 / 79

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 of the revelation mechanisms (Dewatripont, 1989).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 65 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 66 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 67 / 79

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)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 68 / 79

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]

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 69 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 70 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 71 / 79

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)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 72 / 79

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)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 73 / 79

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)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 74 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 75 / 79

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 76 / 79

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)

• Leonardo Felli (LSE)

EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 77 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 78 / 79

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 1 8 November 2011 79 / 79