imperfect commitment santiago truffa. agenda 1.“authority and communication in organizations”...

Imperfect commitment

Santiago Truffa

Agenda

1. “Authority and communication in Organizations” Wouter Dessein, RES 1999

2. “Contracting for information under imperfect commitment”, Krishna-Morgan (2008)

Authority and communication in Organizations

Introduction:

• Delegation as an alternative to communication

• Information is dispersed in the hierarchy of an organization

• Why an un-informed principal may grant formal decision making rights to an agent who is better informed but may have other objectives?

– To avoid noisy communication - loss of information


Introduction:

• But by delegating authority principal commits never to reverse the agents decision (agency problem)

• Difference in objectives implies strategic behaviour when communicating (cheap talk-since there is no commitment, no mechanism design can elicit the truth), whereas the bias is systematic and predictable (no uncertainty).

• Central trade-off:

Loss of control (under delegation) v/s Loss of information (under comunication


The Model:

• The principal must screen several projects • The agents is better informed, but is biased• Projects cannot be contracted uppon• The principal can only contract on the authority of the project• The the preincipal has 2 choices:

– Delegate (agency problem)

– Order after consulting the agent (strategic inf. transmition)

• Main result:

As long as the divergence in preferences (relative to the principal uncertainty) is not too large, is optimal to delegate control.


The Model:

• There is a continum of projects, only 1 can be selected

• Utility of the principal is max at y=m

–

• Utility of the agent is max at y=m+b

–

rvmRymyUU pp ,,,

0,,, bbmyUU AA

0)0(0);(,, llmylmmUmyU pp

0)0(0);)((,, llbmylmbmUmyU AA


Information structure:

• Only agent (A) observes m, F(m), f(m) supported in • All other parametes are common knowledge• Soft information (not verifiable) Incomplete contracting

approach• Resources Principal (P)

Timming:

1. (P) decides Delegate (D) / Not delegate (ND)

2. (A) observes m. Initiates project if (D), can be asked if (ND)

3. (A) implements project

LL,


Communication v/s Delegation: separate problems

Delegation:

• P and A cannot communicate • If (A) has control implements y=m+b• Then:

• (P) delegates if b<b´ given by:

)(),( blmmUU pp

L

Ly

mdFmylbl )()(min´)(


Communication:• (P) cannot commit to let (A) decide, and communication is feasible• Since the project is non contractible communication can only

change (P) beleives• Bayesian equilibrium:

– i) Family of signaling rules (cond. Prob.)

– ii) Decision rule • if is in the support of q(), then max the utility of the agent

given y() • y(n) max the expected utility of (P) a partition of

)( mnqRNny :)(

LLm , n n

n LL,


Communication:• Let denote a partition of with N steps• • Define

• Proposition 1: If b>0, Then

Exists at least one equilibria where:

(1)

),...,( 0 Naaa LL,

LaaaL N ...10

aaLLaa ,,

dmmfmyUaaya

a

p )(,maxarg),(

)(1)( bNNNstbN )(),( mnqny

iiii aamifmaayq ,1),( 11


Communication:• Proposition 1:

(A)

(2)

(3)

1,..,10),,(),,( 11 NiaaayUaaayU iiiAiiiA

),(),()( 11 iiii aanifaayny

LaLa N 0


Communication:• Equilibrium selection (multiple equilibria):

Pick the equilibria where (P) and (A) utilities are ex-ante maximize

N(b) as large as possible.

• Minimal average size of partition elements (loss of inf):

• Sufficient condition for delegation:

)(

2)(

bN

LbA

4)(

4

)()()(

b

bAbAmyEbmyE


Communication: • Lemma 1

Partition are increasingly large, and a function of b.

Comments: • exagerating is noisy, thus costly to the agent. • Higher the bias, higher the cost of exageration• Higher N(b) better communication… the worse communication

performs relative to delegation

ibaaaa ii 4011

bbNibbN

aabAbN

i

1)(

101 1)(24

)(

1)(


Communication: • Proposition 2:

If F() is Uniformly distributed over , (P) prefers delegation to communication whenever b is sucha that communication is feasible (N(b)=2)

• Corollary:

If F() is Uniformly distributed over , (P) delegates control rights to the agent iff , where b´ is such that the principal is indifferent between an un-informed decision and a biased decision:

LL,

LL,´bb

1

0

)(1

´)( dmmlL

bl


Communication:• Proposition 3:

Consider the most informative equilibrium given b. For any F() in the limit as b tends to zero, a principal who keeps control and communicates is on average an infinite times farther away from m tahn a principal who has delegated control:

Idea: since the partition depends on b (independently of the distribution), the result is robust to changes in probability distributions.

b

myE

b

)(lim

0

Contracting for information under imperfect commitment

Introduction:

• Optimal contracting under imperfect commitment

• Un-informed principal (with authority) and an informed but biased agent (agency problem)

• Principal can commit to pay for advice, but retains authority


Introduction:

• Delegation principle: Informed agent should make the decisions– Agency problem involved---Incentives

• There might be commitment problems associated with delegation of authority

• No commitment power, Cheap talk– What if we allow for the possibility of contractual monetary transfers?

– How does the structure of optimal contracts is affected by the degree to wich the principal can commit?

1. Perfect commitment

2. Imperfect commitment (principal retains power)

3. Full delegation


The model:• Project • State on nature , distributed f• Agent observes • , where is the bias parameter• U is a 2 times continously differentiable:

–

• Bias is commonly known. Principal bias is normalized to 0.• • • If U is a quadratic loss function • Agents gives “costless” advice m, after learning • After earing m, principal chooses y

We suppose quasilinear preferences, and allow for monetary transfers

Ry 1,0

ibyUU ,, ib

0,0,0 131211 UUU

),(maxarg)( yUy

ntdisagreemeybyU )(),(013 )(;),( ybby


The model:

• • I. Contracts with perfect commitment (benchmark) • The principal can specify the project and the transfer as a function

of the message

• Then the revelation principle applies, we can consider direct contracts that satisfy incentive compatibility restrictions

• A direct contract

liabilityitedtwheretyUU p lim,0;),( tbyUU A ),,(

1,0)(),( ty


I. Contracts with perfect commitment (benchmark)

• A direct contract is incentive compatible if is best for the agent to report truthfully

•

• Necessary and sufficient conditions for (IC) contract:

1. y(.) must be non decreasing

2.

)(,),(maxarg

tbyU

)(,),()( ,1

, ybyUt


I. Contracts with perfect commitment (benchmark)• Optimal contract: Is the solution to an optimal control problem:

s.t

Law of motion:

constraints: (FOC agent)

Limmited liability t>0

1

0

)(,max dftyUy

ubyUt ,,1, uy ,


• Proposition 1: Under perfect commitment, an optimal contract (y,t) has the following features

1. Projects y() are non-decreasing in , and st y() is constant over

2. Transfers t() are non-decreasing in , and t() is zero over

3. and if,

1z 1,z

1,z

),()( byy )()(0)( yyt


II. Compensation contracts: Imperfect commitment

• Now the principal can contract on transfers, but not on y()

– M set of messages– T() is the transfer scheme, where T(m)>0

• Perfect Bayesian Equilibrium:

1. Strategy for the agent:

2. Strategy for the principal:

3. A belief function (using bayes rule):

GYu ,,

Mu :RMY :

MG :


Modified revelation principle…

“Given an equilibrium of any (indirect) mechanism, There exists and equilibrium of a direct mechanism that is aoutcome equivalent”

• Proposition 2: In the contracting for information model, consider an indirect contract (M,T) with imperfect commitment and any equilibrium under this contract. Then there exists a pure strategy equilibrium under a direct contract which is outcome equivalent

Then with imperfect commitment, is it possible for the principal to design a contract that induces full revelation???


Truth telling condition :

• Since t is downward sloping, the cheapest contract that induces full revelation is:

);(,),()(,),( tbyUtbyU

)(,),()( 1

ybyUtFOC

)()()( yyu

dybyUt )(,),()(1

1


• Proposition 3: Full revelation contracts are always feasible.

• Proposition 4: Full revelation compensation contracts are never optimal

Idea: assume a contract with pooling on (z,1). For high states the indirect cost of obtaining information (raise the cost for all states), more than offset the small gains (very aligned)


• Optimal compensation contracts:

Full separation is cost effective only in low states

• Proposition 5: An optimal compensation contract involves separation in low states, and pooling in high states.

No Payment for imprecise information

• Proposition 6: In an optimal compensation contract the principal never pays for imprecise information

Imperfect commitment

Santiago Truffa

imperfect commitment santiago truffa. agenda 1.“authority and communication in organizations”...

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