Contracting Under Imperfect Commitment

PhD 279B December 3, 2007

Agenda – Selected Main Results1. Authority and Communication in

Organizations (Dessein, 2002)

2. Contracting for Information under Imperfect Commitment (Krishna and Morgan, February 2006 working paper)

Dessein’s Assumptions Starts with Crawford-Sobel (1982) model

m=state of nature or private information /type only known to sender – uniformly distributed

b = bias between preferences of Receiver and Sender y = project / output chosen by Receiver (function of m)

Looks at communication in a principal-agent setting Assumes bias systematic and predictable: b>0 Imperfect contracting approach: Projects cannot be contracted upon,

but the principal can commit to never overruling agent by delegating to

agent control over the critical resources (using contracts, job descriptions, company charters, sale of assets, etc.)

Principal precommits to delegate or communicate

Dessein, 2002

Principal’s Key Tradeoffs

Utility tradeoff between loss of control when principal delegates authority and loss of information when principal and agent communicate

Expected cost of loss of control is the bias, b Expected cost of loss of information under

communication is the difference between the state of the world and the principal’s belief about the state of the world after receiving the messageDessein, 2002

m

Delegate?AgentImplementsy

n

AgentImplements

From CS, a message is a partition Figure 1

Lemma 1: The size of a partition element is always 4b larger than the size of the preceding one a2-a1=a1-a0+4b

Proof y1=(a0+a1)/2 and y2=(a1+a2)/2

At dividing point m=a1 agent is indifferent between y1 and y2

Thus, a1= (y1 + y2 )/2 – b

(y1 + y2 )/2 = a1+ b

y1 + y2 = 2(a1+ b)

(a0+a1)/2 + (a1+a2)/2= 2(a1+ b)

a0+2a1 +a2= 4(a1+ b)

a2-a1=a1-a0+4b QED

Dessein, 2002

a2

Why does partition size matter? N(b)=number of partitions Good communication requires lots of partition elements. Bias gets

smaller faster as N(b) grows. Thus, the average size of the partition element will be very large relative to the bias b

For what N(b) is communication informative? When N(b)>3, A(b)>4b and When N(b)=2,

Principal chooses y*= a0+(a1 -a0 )/2 if mϵ(a0,a1)

Principal chooses y*= a1+(a1 -a0 )/2 +2b if mϵ(a0,a1)

and

Delegation is superior when communication is feasible (and b is small)

Intuition from prior slide: N(b)>2 implies communication feasible but also that information loss exceeds bias

Proposition 2: If F(m) is uniformly distributed over [-L,L], the principal prefers delegation to communication whenever b is such that informative communication is feasible

Dessein, 2002

Also, compare utility of communication vs. delegation Principal’s expected utility under communication

Find N that maximizes EUp

For N>2, EUp<-b2

Principal’s expected utility under delegation U(y,m)=-(y-m)2

If delegating, E(U)=∫-(m+b-m)2=-b2

Dessein, 2002

What happens when the bias is large? Informative communication is not

possible Specifically, when b> ¼, N(b)=1 The agent does not communicate The principal optimally takes an

uninformed decision

Dessein, 2002 and Krishna and Morgan, 2004

Empirical Tests?

See “The Flattening Firm: Evidence from Panel Data on the Changing Nature of Corporate Hierarchies” (Rajan and Wulf, 2006)300 companies10 years data

Critique of Dessein

Omits issues of imperfect commitment: principal is assumed to be able to commit not to intervene in the project chosen by the agent (Krishna and Morgan, 2004)

(Note: Dessein suggests intermediary as a commitment device)

Krishna and Morgan presentation overview

Main results we will cover under imperfect commitmentFull revelation contracts are always feasible

(Prop 3)Full revelation contracts are never optimal

(Prop 4) General compare and contrast to perfect

commitment

Why care about imperfect commitment?

Perfect commitment almost never exists in the real world Principals often retain the value to exercise their

discretion, regardless of the agent’s message e.g. CEOs and investment bankers

How Krishna and Morgan explore imperfect commitment

Perfect commitment has been a critical assumption in mechanism design because it allows us to use the standard revelation principle

However, under imperfect commitment, if agents reveal truthfully, principals are free to use the information to their own advantage. Knowing this, agents will generally be better off not revealing the whole truth

In imperfect commitment, Krishna and Morgan assume the set of types Θ is a continuum, and derive a partial version of the revelation principle to get their results

Why the Revelation Principle matters

Two reasons we use the revelation principle Restrict attention to direct mechanisms Allows us to only consider truth-telling equilibria

Two major weaknesses of the revelation principle Assumes players commit to their strategies Assumes Θ is static

Prop 3: Under imperfect commitment, full revelation contracts are always feasible

If truth-telling is the best response for the agent, then the following must be true

Take the first order condition:

Agent gives truthful message:

Agent’s utility Transfer

Prop 3 Proof (cont’d)

First order condition

We know the following:

So, all full revelation contracts are downward sloping The least-cost full revelation contract is

Prop 3 Proof (cont’d)

Intuition is that transfers can be used to get truth-telling

Main idea of the proof of Prop 3: There exists some amount you can pay the agent to get him to tell the truth

θ axisθ

Transferto agentto get truth

1Stylized illustration:

Prop 3 Proof (cont’d)

As θ goes to 1, the transfers needed to get truth-telling get smaller

As θ goes to 0, the transfers needed to get truth-telling get larger

θ axisθ

Transferto agentto get truth

x*(θ)

1

Proposition 4

Under imperfect commitment full revelation contracts are never optimal.A graphic example of the uniform quadraticProof

An Example: Uniform Quadratic

b=.2 E[x]=-.2

x(θ)

θ

An Example: Uniform Quadratic

b=.2 E[x]=-.2 E[U]=-.2

x(θ) &Utils

θ

An Example: Uniform Quadratic

b=.2 E[x]=-.2 E[U]=-.2 z=.8 xz =.03

E[xz]=-.158

x(θ) &Utils

θ

An Example: Uniform Quadratic

b=.2 E[x]=-.2 E[U]=-.2 z=.8 xz =.03

E[xz]=-.158

E[Uz]~-.152

x(θ) &Utils

θ

An Example: Uniform Quadratic

x(θ) &Utils

θ

b=.2 E[x]=-.2 E[U]=-.2 z=.8 xz =.03

E[xz]=-.158

E[Uz]~-.152

Proof: Part 1 - Feasible Contract

Agent’s utility of project z

Transfer from message z

Agent’s Utility of pooling project.

x(θ) that reveals θ when θ is in [0,z] z<1 and pools for θ in [z,1]. Let xz be the payment when θ=z For indifference at θ=z we need:

Where:

Given θ in [z,1] is the project selected from the pooling message By taking z close to 1,

so xz>0 Like the full revelation contract, Incentive compatibility over θ in [0,z]

requires:

which is positive, so this contract is feasible.

So the proposed contract is feasible, but is it better than full revelation?

The principal’s utility from this contract is:

When z=1, this becomes the full revelation contract, so we’ll check if some z<1 is better than z=1 by taking the derivative of V and evaluating at 1.

If the slope is negative the EU(z=1-ε)>EU(z=1)

Proof: Part 2 – Principal’sUtility

From revealed state less transfer From pooling information and no transfer.

Proof: Part 3 - Lots of Math

Evaluating at z=1

Sincethe derivative is negative.

So, at z slightly less than 1, Utility is higher than at z=1.

Substituting in x(θ) and evaluating the derivative

Since f(1)=0 and ∫1

0 _f(θ)dθ=1

Substituting in xz

and evaluating the derivative at z=1

Additional results for the uniform quadratic The optimal contract is full revelation

below some α in [0,1), and pooling (like C.S.) to the right of α.

In the optimal contract, the principal never pays a transfer for pooling messages (imprecise information).

Perfect commitment

Perfect commitment allows the principal to commit to both transfers x and projects y

Similarities with imperfect commitment Full revelation is also feasible but NOT optimal In optimal contracts, principal does not pay for imprecise

information Differences with imperfect commitment

In some states, a compromise project is selected (between θ and θ+b)

Summary Full revelation contracts are always feasible (Prop 3) Full revelation contracts are never optimal (Prop 4) In an optimal contract under imperfect information,

principal never pays for imprecise information (Prop 6) Perfect commitment results

Full revelation is also feasible and not optimal, principal does not pay for imprecise information

In some states, a compromise project is selected

Appendix

Informative communication (cont’d) Informative communication is feasible

when b<b’, given by

Dessein, 2002

Dessein: See paper for additional results

Topic Dessein’s Insight

Amount of private information

Delegation is more likely when the amount of private information of the agent is large

Risk aversion The more concave her utility function, the more attractive is the constant bias under delegation

Generalization to other distributions

For small or moderate biases, result generalizes to any distribution; for large bias, delegation inferior

Value of an intermediary

For moderate biases, the principal optimally delegates decision rights to an intermediary

Delegation with veto power

Cites political science paper by Gilligan and Krehbiel, 1987 - House of Representatives can’t amend committee proposal under closed rule). Keeping a veto-right typically reduces the expected utility of the principal unless the incentive conflict is extreme

Crawford-Sobel Model (Cheap Talk) Assumptions:

ϴ = state of nature / world (the information) or private information /type only known to sender – uniformly distributed

b = bias between preferences of Receiver and Sender (b assumed > 0) y = project / output chosen by Receiver (function of ϴ)

“Strategic Information Transmission” (Crawford and Sobel, 1982) and summary of Crawford and Sobel (Krishna and Morgan, 2004)

ϴ

messagemessage

Crawford-Sobel Utility (cont’d) Payoffs: quadratic loss functions

Receiver’s payoff (b=0) Sender’s payoff

b

ϴ ϴ+b

y

Utility

“Strategic Information Transmission” (Crawford and Sobel, 1982) and summary of Crawford and Sobel (Krishna and Morgan, 2004)

Revenue Equivalence Theorem

Klemperer, 2003

Crawford-Sobel Result

Preference divergence (b>0) leads to withholding of information by the sender (loss of information disappears when preferences are congruent b=0)

Positive integer N(b) is the upper bound on the “size” (# of subintervals) of an equilibrium partition; there exists at least one equilibrium of each size from 1 through N(b)

Of multiple equilibria, pareto-superior equilibrium (better communication) is that which has the largest number, N(b), of partition elements

Strategic Information Transmission (Crawford and Sobel, 1982) and summary of Crawford and Sobel (Dessein, 2002)

Revelation Principle

Wikipedia.org