Lecture 8: Incomplete Contractsccfour/EO8.pdf · Lecture 8: Incomplete Contracts Cheng Chen School...

Lecture 8: Incomplete Contracts

Cheng Chen

School of Economics and Finance

The University of Hong Kong

(Cheng Chen (HKU)) Econ 6006 1 / 23

Introduction

Motivation

Development of microeconomics theory:1 General equilibrium theory (Arrow, Debreu, Scarf, Mas-Colell...)2 Economics of uncertainly (Von Neumann, Morgenstein...)

3 Complete contract and incentive theory (Mirrlees, Akerlof, Stiglitz,Spence, Myerson, Maskin, Holmstrom, Milgrom)

4 Dynamic contract and renegotiation (Tirole, Tirole, La�ont...)5 Incomplete contract (Grossman, Hart, Moore, Bolton...)


Introduction

Motivation

Development of microeconomics theory:1 General equilibrium theory (Arrow, Debreu, Scarf, Mas-Colell...)2 Economics of uncertainly (Von Neumann, Morgenstein...)3 Complete contract and incentive theory (Mirrlees, Akerlof, Stiglitz,

Spence, Myerson, Maskin, Holmstrom, Milgrom)4 Dynamic contract and renegotiation (Tirole, Tirole, La�ont...)

5 Incomplete contract (Grossman, Hart, Moore, Bolton...)


Introduction

Motivation

Development of microeconomics theory:1 General equilibrium theory (Arrow, Debreu, Scarf, Mas-Colell...)2 Economics of uncertainly (Von Neumann, Morgenstein...)3 Complete contract and incentive theory (Mirrlees, Akerlof, Stiglitz,

Spence, Myerson, Maskin, Holmstrom, Milgrom)4 Dynamic contract and renegotiation (Tirole, Tirole, La�ont...)5 Incomplete contract (Grossman, Hart, Moore, Bolton...)


Introduction

Several Concepts

Transaction-cost economics and boundary of �rm: Coase (1937),

Williamson (1975, 1985).

Hold-up problem: Klein, Crawford and Alchian (1978).

Ex post haggling (Simon) and ex ante ine�ciency (property-rights

theory).

Property-rights theory (Grossman and Hart, 1986; Hart and Moore,1990):

I Asset owner is residual claimant of ownership, not pro�t.I Observability and Veri�ability (ex ante investment).I Unforseen contingencies and cost of writing a contract.I Cost and bene�t of integration and threat point.


Introduction

Several Concepts





theory).




Introduction

Several Concepts





theory).


I Asset owner is residual claimant of ownership, not pro�t.I Observability and Veri�ability (ex ante investment).

I Unforseen contingencies and cost of writing a contract.I Cost and bene�t of integration and threat point.


Introduction

Several Concepts





theory).




Ownership and Property-Rights theory of the Firm (Section11.2.1) Grossman-Hart-Moore Approach

A General Framework

A printer (agent 1) and a publisher (agent 2).

Two assets: {a1, a2}: Both are essential for production.

Investment to increase value of payo�: x .

Cost of investment: ψi (xi ).

Payo�s:I V (x1, x2) ≡ V ({1, 2}; {a1, a2}

∣∣x1, x2): total payo� if two agents worktogether and two assets are used for production.

I Φ1(x1, x2) ≡ V ({1}; {a1, a2}∣∣x1, x2): payo� to agent 1 if he owns

both assets.I Φ2(x1, x2) ≡ V ({2}; {a1, a2}

∣∣x1, x2): payo� to agent 2 if he ownsboth assets.

I V ({1}; {∅}∣∣x1, x2): payo� to agent 1 if he does not own any asset.



A General Framework





Payo�s:I V (x1, x2) ≡ V ({1, 2}; {a1, a2}








A General Framework





Payo�s:I V (x1, x2) ≡ V ({1, 2}; {a1, a2}








A General Framework





Payo�s:I V (x1, x2) ≡ V ({1, 2}; {a1, a2}



both assets.


both assets.I V ({1}; {∅}

∣∣x1, x2): payo� to agent 1 if he does not own any asset.



A General Framework





Payo�s:I V (x1, x2) ≡ V ({1, 2}; {a1, a2}








A General Framework





Payo�s:I V (x1, x2) ≡ V ({1, 2}; {a1, a2}








Insights

Suppose investment x is made ex ante.

Ex post bargaining on realized payo� V ({1, 2}; {a1, a2}∣∣x1, x2) is

e�cient. I.e., negotiation does not break up, and max. payo� is

distributed to both agents.

Assume Nash bargaining rule for ex post bargaining.

Key: di�erence in threat points under di�erent ownership structures.

No ex post ine�ciency. However, ex ante ine�ciency is key.



Insights










Insights










Insights










Non-Integration

Agent 1 owns asset one; agent 2 owns asset two.

If ex post negotiation breaks up, payo� is zero to both agents.

Suppose bargaining power is 1/2 for each agent.

Agent 1's incentive to invest ex ante:

maxx1

1

2[V (x1, x2)− 0] + 0− ψ1(x1).


maxx2

1

2[V (x1, x2)− 0] + 0− ψ2(x2).



Non-Integration





maxx1

1

2[V (x1, x2)− 0] + 0− ψ1(x1).


maxx2

1

2[V (x1, x2)− 0] + 0− ψ2(x2).



Non-Integration





maxx1

1

2[V (x1, x2)− 0] + 0− ψ1(x1).


maxx2

1

2[V (x1, x2)− 0] + 0− ψ2(x2).



Printer-Integration

Assume ∂Φi (xi , xj )/∂xj = 0: the other agent's investment does not

a�ect my own outside option.

See Che and Hausch (1999) on this point.

If printer owns all assets (printer-integration)I Agent 1's incentive to invest ex ante:

maxx1

1

2[V (x1, x2)−Φ1(x1)] + Φ1(x1)− ψ1(x1).

I Agent 2's incentive to invest ex ante:

maxx2

1

2[V (x1, x2)−Φ1(x1)] + 0− ψ2(x2).



Printer-Integration





maxx1

1

2[V (x1, x2)−Φ1(x1)] + Φ1(x1)− ψ1(x1).


maxx2

1

2[V (x1, x2)−Φ1(x1)] + 0− ψ2(x2).



Printer-Integration





maxx1

1

2[V (x1, x2)−Φ1(x1)] + Φ1(x1)− ψ1(x1).


maxx2

1

2[V (x1, x2)−Φ1(x1)] + 0− ψ2(x2).



Publisher-Integration

If publisher owns all assets (publisher-integration)I Agent 1's incentive to invest ex ante:

maxx1

1

2[V (x1, x2)−Φ2(x2)] + 0− ψ1(x1).


maxx2

1

2[V (x1, x2)−Φ2(x2)] + Φ2(x2)− ψ2(x2).

We assume that ex ante is relationship-speci�c. I.e.

∂V (x1, x2)

∂x1> Φ

′1(x1) for all x2

and∂V (x1, x2)

∂x2> Φ

′2(x2) for all x1.





maxx1

1

2[V (x1, x2)−Φ2(x2)] + 0− ψ1(x1).


maxx2

1

2[V (x1, x2)−Φ2(x2)] + Φ2(x2)− ψ2(x2).


∂V (x1, x2)

∂x1> Φ

′1(x1) for all x2

and∂V (x1, x2)

∂x2> Φ






maxx1

1

2[V (x1, x2)−Φ2(x2)] + 0− ψ1(x1).


maxx2

1

2[V (x1, x2)−Φ2(x2)] + Φ2(x2)− ψ2(x2).


∂V (x1, x2)

∂x1> Φ

′1(x1) for all x2

and∂V (x1, x2)

∂x2> Φ




Incentives to InvestFor non-integration:

1

2

∂V (xNI1 , xNI

2 )

∂x1= ψ

′1(x

NI1 );

1

2

∂V (xNI1 , xNI2 )

∂x2= ψ

′2(x

NI2 )

For printer-integration:

1

2

[∂V (xPI1 , xPI2 )

∂x1+ Φ

′1(x

PI1 )]= ψ

′1(x

PI1 );

1

2

∂V (xPI1 , xPI2 )

∂x2= ψ

′2(x

PI2 )

For publisher-integration:

1

2

∂V (xpI1 , xpI2 )

∂x1= ψ

′1(x

pI1 );

1

2

[∂V (xpI1 , xpI2 )

∂x2+ Φ

′2(x

pI2 )]= ψ

′2(x

pI2 )

All investment level is below FB level:

∂V (xFB1 , xFB2 )

∂x1= ψ

′1(x

FB1 );

∂V (xFB1 , xFB2 )

∂x2= ψ

′2(x

FB2 ).




1

2

∂V (xNI1 , xNI

2 )

∂x1= ψ

′1(x

NI1 );

1

2

∂V (xNI1 , xNI2 )

∂x2= ψ

′2(x

NI2 )


1

2

[∂V (xPI1 , xPI2 )

∂x1+ Φ

′1(x

PI1 )]= ψ

′1(x

PI1 );

1

2

∂V (xPI1 , xPI2 )

∂x2= ψ

′2(x

PI2 )


1

2

∂V (xpI1 , xpI2 )

∂x1= ψ

′1(x

pI1 );

1

2

[∂V (xpI1 , xpI2 )

∂x2+ Φ

′2(x

pI2 )]= ψ

′2(x

pI2 )


∂V (xFB1 , xFB2 )

∂x1= ψ

′1(x

FB1 );

∂V (xFB1 , xFB2 )

∂x2= ψ

′2(x

FB2 ).




1

2

∂V (xNI1 , xNI

2 )

∂x1= ψ

′1(x

NI1 );

1

2

∂V (xNI1 , xNI2 )

∂x2= ψ

′2(x

NI2 )


1

2

[∂V (xPI1 , xPI2 )

∂x1+ Φ

′1(x

PI1 )]= ψ

′1(x

PI1 );

1

2

∂V (xPI1 , xPI2 )

∂x2= ψ

′2(x

PI2 )


1

2

∂V (xpI1 , xpI2 )

∂x1= ψ

′1(x

pI1 );

1

2

[∂V (xpI1 , xpI2 )

∂x2+ Φ

′2(x

pI2 )]= ψ

′2(x

pI2 )


∂V (xFB1 , xFB2 )

∂x1= ψ

′1(x

FB1 );

∂V (xFB1 , xFB2 )

∂x2= ψ

′2(x

FB2 ).




1

2

∂V (xNI1 , xNI

2 )

∂x1= ψ

′1(x

NI1 );

1

2

∂V (xNI1 , xNI2 )

∂x2= ψ

′2(x

NI2 )


1

2

[∂V (xPI1 , xPI2 )

∂x1+ Φ

′1(x

PI1 )]= ψ

′1(x

PI1 );

1

2

∂V (xPI1 , xPI2 )

∂x2= ψ

′2(x

PI2 )


1

2

∂V (xpI1 , xpI2 )

∂x1= ψ

′1(x

pI1 );

1

2

[∂V (xpI1 , xpI2 )

∂x2+ Φ

′2(x

pI2 )]= ψ

′2(x

pI2 )


∂V (xFB1 , xFB2 )

∂x1= ψ

′1(x

FB1 );

∂V (xFB1 , xFB2 )

∂x2= ψ

′2(x

FB2 ).



Equilibrium Ownership Structure

Firm chooses ownership structure maximize ex post payo�:

V (x1, x2)− ψ1(x1)− ψ2(x2).

If Φ′1(x1) > 0, non-integration is never optimal.

If printer's investment matters more for �nal payo�, I.e.

∂V (x1, x2)

∂x1>>

∂V (x1, x2)

∂x2,

then printer-integration is optimal and vice versa.

It is possible that Φ′1(x1) < 0. Thus, non-integration might be

optimal.

Cost and bene�t of integration. Not just transaction costs (i.e., costs

associated with market transactions)!





V (x1, x2)− ψ1(x1)− ψ2(x2).



∂V (x1, x2)

∂x1>>

∂V (x1, x2)

∂x2,



optimal.







V (x1, x2)− ψ1(x1)− ψ2(x2).



∂V (x1, x2)

∂x1>>

∂V (x1, x2)

∂x2,



optimal.




The Holdup Problem (Section 12.3.1)

Setup

References: Goldberg (1976), Klein, Crawford, Alchian (1978), and


A buyer and a seller.

Quantity of trading: q ∈ [0, 1].

Value: v ∈ {vL, vH}, vL < vH and Prob(vH) = j .

Cost: c ∈ {cL, cH}, cL < cH and Prob(cL) = i .

Ex post payo�s:

vq − P − ψ(j)

and

P − cq − φ(i).

cH > vH > cL > vL

Ex post e�cient level of trade is q = 1 if θ = (vH , cL) and 0 otherwise.



Setup







Ex post payo�s:

vq − P − ψ(j)

and

P − cq − φ(i).

cH > vH > cL > vL




Setup







Ex post payo�s:

vq − P − ψ(j)

and

P − cq − φ(i).

cH > vH > cL > vL




Setup







Ex post payo�s:

vq − P − ψ(j)

and

P − cq − φ(i).

cH > vH > cL > vL




FB and the Holdup ProblemFB is

maxi ,j{ij(vH − cL)− ψ(j)− φ(i)}

Solution:

i∗(vH − cL) = ψ′(j∗)

and

j∗(vH − cL) = φ′(i∗).

However, assume investment happens ex ante, and both agents

bargain over generated payo� through using a Nash bargaining rule.

Assume they have equal bargaining power. Investment level is

1

2iSB(vH − cL) = ψ

′(jSB)

and1

2jSB(vH − cL) = φ

′(iSB).

Follow-up research: Di�erent assumptions on the extent to which level

of trade is contractable. E�cient ex post renegotiation.





Solution:

i∗(vH − cL) = ψ′(j∗)

and

j∗(vH − cL) = φ′(i∗).




1


′(jSB)

and1


′(iSB).







Solution:

i∗(vH − cL) = ψ′(j∗)

and

j∗(vH − cL) = φ′(i∗).




1


′(jSB)

and1


′(iSB).




Real and Formal Authority (Section 12.4.2)

Setup

Reference: Aghion and Tirole (1997).

Formal authority 6= real authority.

P: Principal. A: Agent.

N potential projects k ∈ {1, 2, ...,N}.P has one most preferred project with payo� H and βh to P and A

respectively.

A has one most preferred project with payo� αH and h to P and A

respectively.

Congruence parameters (con�ict of interests): α, β.



Setup




N potential projects k ∈ {1, 2, ...,N}.

P has one most preferred project with payo� H and βh to P and A

respectively.


respectively.




Setup





respectively.


respectively.




Setup





respectively.


respectively.




Setup





respectively.


respectively.




Setup (cont.)

P knows which project she prefers most with Prob E , if she exerts

e�ort at cost ψP(E ).

A knows which project she prefers most with Prob e, if she exerts

e�ort at cost ψA(e).

∃ One bad project generating extremely negative payo� to both P and

A → Don't choose any project, if both P and A don't know state.



P-Control

Assume P has formal authority.

With Prob. E : P has both formal and real authority.

With Prob. (1− E )e: P has formal authority, while A has real

authority.

Payo�s:

UP = EH + (1− E )eαH − ψP(E )

and

UA = Eβh+ (1− E )eh− ψA(e).



P-Control

Assume P has formal authority.

With Prob. E : P has both formal and real authority.

With Prob. (1− E )e: P has formal authority, while A has real

authority.

Payo�s:

UP = EH + (1− E )eαH − ψP(E )

and

UA = Eβh+ (1− E )eh− ψA(e).



P-Control (cont.)

FOC:

(1− αe)H = ψ′P(E )

and

(1− E )h = ψ′A(e)

Key parameters: α and β.

Key economic force: crowding-out e�ect.

Substitutability between e and E .

E�ect of α and β on A's e�ort choice.



P-Control (cont.)

FOC:

(1− αe)H = ψ′P(E )

and

(1− E )h = ψ′A(e)







E-Control

Assume A has formal authority.

With Prob. e: A has both formal and real authority.

With Prob. (1− e)E : A has formal authority, while P has real

authority.

Payo�s:

UP = eαH + (1− e)EH − ψP(E )

and

UA = eh+ (1− e)Eβh− ψA(e).



E-Control

Assume A has formal authority.

With Prob. e: A has both formal and real authority.

With Prob. (1− e)E : A has formal authority, while P has real

authority.

Payo�s:

UP = eαH + (1− e)EH − ψP(E )

and

UA = eh+ (1− e)Eβh− ψA(e).



E-Control (cont.)

FOC:

(1− e)H = ψ′P(E )

and

(1− βE )h = ψ′A(e)





When α, β→ 1: A-control is better?



E-Control (cont.)

FOC:

(1− e)H = ψ′P(E )

and

(1− βE )h = ψ′A(e)








E-Control (cont.)

FOC:

(1− e)H = ψ′P(E )

and

(1− βE )h = ψ′A(e)







Incomplete Contract and Entry Barriers (Section 13.2)

Setup

Reference: Aghion and Bolton (1987).

Key insight: In a dynamic model, incumbent �rm and consumer can

sign a long-term contract to prevent entry of new �rm (tradeo�

between rent and allocative e�ciency)

Two periods with discounting (t = 0, 1).

Incumbent sells one good to consumer in both periods.

Valuation of consumer: v = 1.

Cost of incumbent: cI ≤ 12 (deterministic)

Entrant may enter when t = 1, and its cost realization cE ∼ U [0, 1].



Setup












Setup












Setup












Spot Contract

When only spot contract is available.

Price p0 = 1 and p1 = 1 when realized cost cE < cI . (entry decision

is made before pricing decision)

Consumer's payo�: (1− cI )cI .

Incumbent's payo�: 1− cI + (1− cI )2.



Spot Contract

When only spot contract is available.

Price p0 = 1 and p1 = 1 when realized cost cE < cI . (entry decision

is made before pricing decision)

Consumer's payo�: (1− cI )cI .

Incumbent's payo�: 1− cI + (1− cI )2.



Long-Term Contract

Suppose incumbent and consumer can make a long-term contract (p0,p1). Punishment d for breaking contract.

Now we assume that entry decision is made after p1 is announced.

Contract is broken, if

1− pE ≥ 1− p1 + d .

Entry happens with Prob. p1 − d .



Long-Term Contract

Suppose incumbent and consumer can make a long-term contract (p0,p1). Punishment d for breaking contract.

Now we assume that entry decision is made after p1 is announced.

Contract is broken, if

1− pE ≥ 1− p1 + d .

Entry happens with Prob. p1 − d .



Long-Term Contract (Cont.)

Incumbent's ex ante payo� and objective function:

maxp0,p1,d

p0 − cI + (p1 − cI )(1− p1 + d) + d(p1 − d).

s.t. PC for consumer:

(1− p0) + (1− p1) ≥ (1− cI )cI (PC ).

We can set p0 = 1 (maybe sub-optimal). Express p1 in terms of cIusing (PC ).

Solutions:

d∗ =1+ (1− cI )(1− 2cI )

2

and

Prob(entry) = p1 − d∗ =cI2.



Long-Term Contract (Cont.)

Incumbent's ex ante payo� and objective function:

maxp0,p1,d

p0 − cI + (p1 − cI )(1− p1 + d) + d(p1 − d).

s.t. PC for consumer:

(1− p0) + (1− p1) ≥ (1− cI )cI (PC ).

We can set p0 = 1 (maybe sub-optimal). Express p1 in terms of cIusing (PC ).

Solutions:

d∗ =1+ (1− cI )(1− 2cI )

2

and

Prob(entry) = p1 − d∗ =cI2.



Discussion

Long-term contract always dominates spot contract (binding PC +(d = 0) + (P0 = 1) + same timing assumption).

Entry is deterred, since Prob(entry) = cI2 .

Obviously, not socially e�cient.

No way to improve, since contract is fully enforceable.

If entrant could promise something when t = 0, what would happen?



Discussion








Discussion







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