perfect bayesian equillibirum

8/6/2019 Perfect Bayesian Equillibirum

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ECON 407 Lecture NotesContract Theory:

Signalling.

Philip Gunby

Department of Economics

University of Canterbury

[email protected]

cPhilip P. Gunby 2005

DISCLAIMER: If you do not attend lectures, you are duly warned that these notes

do not contain all of the material presented in class. You do not attend lectures at

your own peril! Also note that not all material in these notes may be covered in class.

Finally, no matter how carefully I proof-read these lecture notes, there are still boundto be mistakes in them!

1. INTRODUCTION

We have seen that the presence of asymmetric information between the principal

and the agent results in a distortion of contracts, since the more informed party, the

agent, attempts to take advantage of the asymmetry. We have found the following

results:

Moral hazard: The agent is no worse off from the information asymmetry, be-

cause they still get their reservation utility level. The principal is certainly worse

off though as she has to pay some potential surplus to ensure that the agent

does not cheat on the contract.

Adverse selection when a principal has some local monopoly power and the good

agent has a cost advantage: The bad agent is no worse off from the information

asymmetry compared to symmetric information, and the good agent is in fact

better off. The principal is worse off under asymmetric information because she

has to distort the bad agents contract and pay a rent to the good agent to stop

the good agent passing himself off as a bad agent.

Adverse selection with competing principals and the good agent has a productiv-

ity advantage: The bad agent is no worse off from the information asymmetry

than under symmetric information. The good agent, however, is worse off be-

cause their contract is distorted to ensure that bad agents dont want to pass

themselves off as good agents. The principals make zero expected profits under

both symmetric and asymmetric information and are therefore indifferent.

In each case it is possible that one of the parties has an incentive to take a costly

(and possibly imperfect) action to reduce their loss over the symmetric information

outcome. Principals have the incentive to take the following actions:

Monitoring (in the first instance).

Example: video cameras, keyboard monitoring, supervisors, etc.

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The economic intuition behind the results are as follows:

(a) The relative costs of sending a signal are sufficiently different so it only

makes sense for the good agents to send the signal, it is economically attrac-

tive for the good agents to actually send the signal, and the only sensiblebeliefs that principals can have given this structure is that only good agents

will send a signal.

(b) The cost of sending a signal is still high for a bad agent, although the

cost of sending a signal for good agents, while still economically attractive, is

not as overwhelming as in (a). As a result, good agents will find it desirable

to send a signal if principals believe that signals are informative but also

dont find deviating from a pooling equilibrium attractive if principals are

not convinced about the informativeness of any signal sent. (c) The relative costs of sending a signal are sufficiently similar so both agents

find it equally attractive to send a signal and the only sensible beliefs that

principals can have given this structure is that the signal is uninformative.

4. Contracts As Signals

The emphasis so far has been on the role of pre-contractual actions to signal the

type of agent. It is possible that the contracts themselves can be used to signal

information to potential parties to a contract. Possible examples of this practice

are:

1. The quality of a good can be signalled by the inclusion of guarantees in a

contract or the price of the good.

2. The ratio of debt to equity in a world of asymmetric information may signal

the future actions of the firms managers to banks or shareholders.

4.1 Model

Assume the following economic environment:

There are two types of people considering entering into a contractual relationship:

a risk-neutral principal and a risk-averse or risk-neutral agent.

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The principal has private information about the type of job being offered to the

agent (which is pre-determined by nature):

1. The job is easy and productivity is low, in which case the profit and utility

functions are: B1(w, e) = (e) w

U1(w, e) = u(w) v(e).

2. The job is difficult and productivity is high, in which case the profit and

utility functions are:

Bk(w, e) = k(e) w

Uk(w, e) = u(w) kv(e).

with k > 1.

U = reservation wage of the agent.

(w, e) = a contract.

In this situation effort is observable by everyone, but the disutility is not. This

creates a problem in designing contracts as not only does the agent have preferences

regarding the effort they have to expend, but they also care about the difficulty of

the job. Note that this could also be interpreted as safety!

4.2 Symmetric Information Contracts

If the principal is the easy job type then she solves the following problem:

Max

(w, e)(e) w s.t. u(w) v(e) U ,

which gives the familiar looking FOCs,

u(w1) v(e1) = U

(e1)

v(e1)=

1

u(w1).

There is a corresponding for the difficult job type of principal:

Max(w, e)

k(e) w s.t. u(w) kv(e) U,

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3. Signalling Can Look Inefficient Ex-Post.

The signals typically involve costly actions which dont seem to increase produc-

tivity or quality or whatever. So after the fact, these look silly socially inefficient

actions to take.Example: some forms of advertising.

This analysis ignores the fact that the actions exist in a second-best informa-

tion constrained world. Taking this fact into account means the actions are not

socially inefficient ex p ost.

4. There is Evidence That Supports The Basic Conclusions Of Signalling Theory.

Some of this evidence is just from looking around us. For example:

Education.

Guarantees and Warranties for durable products.

Initial prices below cost for new products for non-durable products.

Advertising.

Other evidence is from more formal studies:

We have seen evidence that for secondary school at least that additional

classes doesnt seem to matter much in terms of wages and by implication

productivity (and hence implicitly at this level schooling seems to count little

for learning).

We have seen that the lifetime earnings profiles fit the implications of the de-

sign of contracts that are consistent with a separating equilibrium operating

through signalling/screening.

There is also other evidence not gone into here giving similar types of results.

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References

Altonji, J. (1995). The Effects of High School Curriculum on Education and Labor Market Outcomes.

Journal of Human Resources. 30(3):409-438.

Chiappori, P. and B. Salanie. (2000). Testing Contract Theory: A Survey of Some Recent Work.Invited lecture, World Congress of the Econometric Society, Seattle, August, 2000.

Kreps, D. (1990). A Course in Microeconomic Theory. Princeton, New Jersey: Princeton University

Press.

Macho-Stadler, I. and D. Perez-Castrillo. (1995). An Introduction to the Economics of Information.Translated by Richard Watt. Oxford: Oxford University Press. [Chapter 5]

Milgrom, P. and J. Roberts. (1992). Economics, Organization, and Management. Englewood Cliffs,New Jersey: P rentice-Hall. [pp. 154-156, 342-343, 505-508]

Molho, I. (1997) The Economics of Information: Lying and Cheating in Markets and Organizations.Oxford: Blackwell. [Chapters 5-8]

Riley, J. (1979). Testing the Educational Screening Hypothesis. Journal of Political Economy.87(5):S227-S252.

Salanie, B. (1998). The Economics of Contracts: a Primer. Cambridge, Mass.: MIT Press. [Chapter4]

Spence, M. (1973). Market Signalling: Information Transfer in Hiring and Related Processes. Cam-

bridge, Mass.: Harvard University Press.

Weiss, A. (1995). Human Capital vs. Signalling Explanations of Wages. Journal of Economic

Perspectives. 9(4):133-154.

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SIGNALLING

1. Introduction and Example

Signalling: Is any action taken that demonstrates to others the agents intentions or

abilities or some other characteristics about which the agent has private, unverifiable

information. The agents desire to signal such information may or may not be

deliberate.

Model

Assume the following: Workers come in two types: high or low productivity.

The marginal product of high-productivity workers = $50/hour.

The marginal product of low-productivity workers = $20/hour.

The value of the outside options to each type of worker = $20.

No Signalling Mechanisms Available

Example: Say the proportion of high-productivity workers is 30% of the population

then the wage paid is:

(0.3 $50) + (0.7 $20) = $29.

Signalling Mechanisms Available

Say the workers can attain an education where,

CL = cost of a unit of education to a low-productivity type.

CH = cost of a unit of education to a high-productivity type.

EL = amount of education a low-productivity type obtains.

EH = amount of education a high-productivity type obtains.

WL = competitive industry wage paid to a low-productivity worker.

WH = competitive industry wage paid to a high-productivity worker.

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with the following self-selection constraints:

$50 CL EH $20 CL EL,

or,CL(EH EL) 30;

and,

$50 CH EH $20 CH EL;

or,

CH(EH EL) 30.

Education will be able to signal the productivity type of a worker if,

CH(EH EL) 30 (= WH WL) CL(EH EL).

Education is more likely to signal the productivity type of a worker if:

CL is large.

CH is small.

EH EL is not too big nor too small.

WH WL is not too big nor too small.

2. Equilibrium Definition

Perfect Bayesian Equilibrium

Perfect Bayesian Equilibrium (from Kreps (1990, p. 435))

Consider the following class of extensive form games called signalling games.

There are two players: player 1 and player 2.

Nature moves first, selecting one of a number T of possible initial nodes

according to a strictly positive probability distribution p.

Player 1 knows natures choice and can send a signal about this to player 2

who does not know natures choice.

S = the set of signals available to player 1.

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Player 2 takes an action a A in response to the signal not knowing which

t T was chosen by nature.

Both players utility functions depend on the choice of nature, player 1s

signal, and player 2s action: u(t ,s,a) = player 1s utility function.

v(t,s,a) = player 2s utility function.

An equilibrium for general signalling games consists of strategies for players 1

and 2 where:

1(s; t) = the probability that player 1 sends message s if nature chooses t

and, sS

1(s; t) = 1 for each t T

2(a; s) = the probability that player 2 chooses action a upon observing

signal s and,

aA

2(a; s) = 1 for each s S

Define a signal s to be along the path of play if,tT

(t)1(s; t) > 0

or that there is some circumstance under which s is sent by player 1. For any

such signal s player 2 uses Bayes rule to compute her beliefs about t given that

player 2 observes s:

(t | s) =(t)1(s; t)

t

T

(t)1(s; t)

recalling that from Bayes Theorem with two events A and B,

P(A | B) =P(A B)

P(B)=

P(A)P(B | A)

P(B)if P(B) = 0.

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In such a setting a pair of strategies (1, 2) is a perfect Bayesian equilibrium if,

(a) For each t, if 1(s; t) > 0, then s maximises,

aA

2(a; s)u(t, s, a) s S;

(b) For each s that is along the path of play, 2(a; s) > 0 implies that a

maximises, tT

(t | s)v(t ,s,a) a A

where is computed as above from 1 by Bayes rule.

Intuitive Criterion

Consider the following generic signalling game structure:

There are two players: the leader and the follower, and the leader possesses

private information.

k K, where K = {1, . . . , n}, denotes the leaders type out the n possible types

and H K.

m M is a particular message sent by the leader.

q(m) = the followers belief about the leaders type if they send message m.

q (q1(m), . . . , qn(m)) and q1(m) + . . . + qn(m) = 1.

BR(H, m) = set of pure-strategy best responses of the follower when m has been

sent and beliefs are defined on H, or that,

kHq

k

(m) = 1.

When the utility of an agent doesnt depend explicitly on m then BR(H, m)

also doesnt depend on m, but if the utility of agent does depend on m then

so does BR(H, m).

BR(H, m) includes all the strategies that could be optimal when the follower

knows the leader is some type in H but doesnt know exactly which type.

BR(H, m) excludes all the strategies that are not optimal for any beliefs q

in which qk(m) = 0 when k H.

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U(k,m,r) = the utility of a type k leader when m is sent and when the

followers action is r.

Consider an equilibrium in which messages sent are m(k), reactions are

r

(m), and the equilibrium utility of a type k leader is,

U(k) = U(k, m(k), r(m(k))) .

For message m that is never sent in equilibrium define S(m) as the set of

leader types k such that,

U(k) > max{U(k,m,r) | r BR(m(k))}.

Intuitive Criterion: An equilibrium does not survive the application of theintuitive criterion if k K such that,

U(k) < min{U(k, m , r) | r BR(K S(m), m)}.

3. GENERAL NATURE OF THE PROBLEM

Model

The basic structure is as follows:

There are two types of agents: good and bad.

pG = probability of success for the good agents.

pB = probability of success for the bad agents.

pG > pB.

q = proportion of good agents.

S = value to principals of a success.

F = value to principals of a failure.

The agents before entering a contract can undertake an activity which is not

inherently beneficial to either party:

t = amount of activity undertaken by an agent and for simplicity t {0, t}

with t > 0.

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vG(t) = cost of good agent of undertaking t units of the activity with vG(0) UG

UB

and vG < UG

UG.

3. The pooling equilibrium fails the intuitive criterion test if,

vB > UG

UB

and vG < UG

UG.

4. The pooling equilibrium meets the intuitive criterion if either,

vB < UG

UB

or vG > UG

UG.

5. Referring back to the separating equilibrium note that since UB

< UG that,

UG

UG < UG

UB

,

so that

vG < UG

UG vG < UG

UB

.

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Summary

Result: In the proposed market using the concept of p erfect Bayesian

equilibrium and the intuitive criterion:

(a) If vB UG

UB

and vG < UG

UG then only a separating

equilibrium exists.

(b) If vB UG

UB

and vG [UG

UG, UG

UB

] then both sepa-

rating and pooling equilibria exist.

(a) IfvB < UG

UB

and vG UG

UG then only a pooling equilibrium

exists.

4. Contracts As Signals

Model

Assume the following economic environment:

There are two types of people: a risk-neutral principal and a risk-averse or risk-

neutral agent.

The principal has private information about the type of job:

1. The job is easy and productivity is low:

B1(w, e) = (e) w

U1(w, e) = u(w) v(e).

2. The job is difficult and productivity is high:

Bk(w, e) = k(e) w

Uk(w, e) = u(w) kv(e).

with k > 1.

U = reservation wage of the agent.

(w, e) = a contract.

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Symmetric Information Contracts

If the principal is the easy job type then she solves the following problem:

Max

(w, e) (e) w s.t. u(w) v(e) U ,

which gives FOCs,u(w1) v(e1) = U

(e1)

v(e1)=

1

u(w1).

There is a corresponding problem for the difficult job type of principal:

Max

(w, e)k(e) w s.t. u(w) kv(e) U,

giving the FOCs,u(wk) kv(ek) = U

(ek)

v(ek)=

1

u(wk).

The optimal contracts, C1

and Ck

, require that,

ek

< e1

s.t. wk

> w1

.

Asymmetric Information Contracts

Separating Equilibria

Define the following notation:

q = the agents prior probability that a principal is type 1.

q(w, e) = the agents prior posterior that a principal is type 1.

Result: In any perfect Bayesian separating equilibrium, with {C1, Ck}

{(w1, e1), (wk, ek)}, it must always be true that Ck = Ck

, that is to say,

(wk, ek) = (wk , ek).

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We know that any type 1 contract must be accepted by an agent, or

u(w1

) v(e1

) U, (1)

and that it cant appeal to a type k principal, or

k(ek

) wk

k(e1) w1, (2)

and that the type 1 principal must prefer it over Ck

, or

(e1) w1 (ek

) wk

. (3)

Now define,

B =max

(w, e)(e) w s.t. u(w) kv(e) U.

Now consider the class of contracts (w1, e1) that guarantee a type 1 principal profits

of at least B:

(e1) w1 B. (4)

Note that since (ek

, wk

) satisfies u(wk

) kv(ek

) U, we know that B

(ek

) wk

and so (4) implies (3).

Result: Consider a contract C1 (w1, e1) that satisfies equations (1), (2),

and (4). Then the following strategies and beliefs are a perfect Bayesian

separating equilibrium:

(i) The principal chooses C1 (Ck

) if she is type 1 (k).

(ii) q(C1) = 1, q(Ck

) = 0, and q(w, e) = 0 if (w, e) = C1 and (w, e) = Ck

.

(iii) The agent accepts both contracts C1 and Ck

. He also accepts any

other (w, e) such that u(w) kv(e) U. He rejects all other contracts.

There are many such contracts in this set, but they all have the same common

characteristics:

w1 < wk

and e1 < ek

.

and require that,

wk w1 is small enough to cause type k principals to prefer ek to e1.

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wk

w1 is large enough to cause type 1 principals to prefer e1 to ek

.

Result: Let C1

(w1, e1) be the contract defined by the following two

equations:u(w1) v(e1) = U ,

and,

k(ek

) wk

= k(e1) w1.

This contract is the only one for which the beliefs, q(C1) = 1, q(Ck

) = 0,

and q(w, e) = 0 if (w, e) = C1 and (w, e) = Ck

, satisfy the intuitive criterion.

5. PRICES THAT SIGNAL QUALITIES

Model

Assume the following economic structure:

A single monopoly firm produces a good, the quality of which is known to the

firm but not to the consumers.

The quality of the good is either high or low:

q = the consumers belief that probability that a good is high quality.

X = the surplus to consumers from a high quality good.

0 = the surplus to consumers from a low quality good.

cH = the per unit cost of producing a high quality good.

cL = the per unit cost of producing a low quality good.

cH > cL.

There are two periods (t = 1, 2) and consumers can buy 1 unit of the good in

each period.

If the consumer buys the good in t = 1 he knows the goods quality in t = 2.

Consumers who make no purchase in t = 1 do not purchase in t = 2.

(0, 1) = discount rate over profits/surplus in t = 2.

pt = the per unit price paid by the consumers for the good in period t.

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The Second Period

In t = 2 the consumers know the quality of the good and will:

1. Buy the good if it is high quality and are willing to pay p2 X.

2. Not buy the good if it is low quality.

p2 = X.

The First Period

Assume first that qX < cL.

Given this we know three characteristics of any separating equilibrium:

1. Type L firms do not produce any goods.

2. The high quality type firm will set a price p1 = cL.

3. It must be the case that (cL cH) (X cH) because we know:

In a separating equilibrium,

H

= (p1 c

H

) + (X c

H

).

and p1 = cL.

If (cL cH) > (X cH) then H < 0.

If (cL cH) (X cH) then H 0.

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6. DEBT LEVEL AS SIGNAL OF THE VALUE OFA FIRM

Model

Assume the following economic structure:

A manager manages a firm on behalf of investors over two periods (t = 0 and

t = 1).

The firms activities produce uncertain earnings, x, realised in t = 1.

The firms earnings are affected by how well the firm is placed with regard to its

environment.

x for a type-k firm is a random variable uniformly distributed on the interval

[0, k].

f(x) = 1/k and F(x) = x/k for x [0, k].

The manager knows the value ofk, the earnings distribution of the firm, but the

shareholders do not.

D = the level of debt used to finance the firm.

L = the fine paid by the manager if the firm goes bankrupt (where this means

the final value of the firm is less than D).

V0(D) = the value of the shares at time t = 0 if the debt level is D.

The manager chooses a debt level D that maximises the weighted sum of the

market value of the firm in the periods t = 0 and t = 1. That is, the manager

chooses D to maximise:

U(D) = (1 )V0(D) +

kD

x1

kdx +

D0

(x L)1

kdx

,

where is the relative weight of the firm in the second period. Solving the

integrals gives,

U(D) = (1 )V0(D) +

k

2

LD

k

.

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Symmetric Information

If information were symmetric then the manager would choose D = 0 and the

expected value of the firm in t = 0 is V0(D) = k/2.

Asymmetric Information

If investors do not have information about k, and they observe D = 0, then they do

not know whether or not the firm is a good or a bad type.

To see why this might be the case consider now two types of firms: good value (kG)

and bad value (kB) with kG > kB.

We know from our previous results that DB = 0 in a separating equilibrium and

the utility of the manager of a bad firm is thus:

U(D = 0) = (1 )kB

2+

kB

2=

kB

2.

For any separating equilibrium with DG = DB we know that the ICCs must hold:

(1 ) k

G

2 +

k

B

2 LD

G

kB

k

B

2 .

and,

(1 )kG

2+

kG

2

LDG

kG

(1 )

kB

2+

kG

2.

These conditions can be written as:

DG (1 )

(kG kB)

2LkB,

(1 )

(kG kB)

2LkG .

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5. Conclusion

1. Signalling Is Likely A Very Common Practice.

2. Credible Signals Mean Incurring Costs.

3. Signalling Can Look Inefficient Ex-Post.

4. There is Evidence That Supports The Basic Conclusions Of Signalling Theory.

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