econs 424 - signalling games iii

EconS 424 - Signalling Games III

FÈlix MuÒoz-GarcÌa

Washington State University

[email protected]

April 28, 2014

FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 1 / 24

Watson, Ch. 29 Exercise 9

A manager and a worker interact as follows:

The manager would like the worker to exert some e§ort on a project.

Let e denote the workerís e§ort.Each unit of e§ort produces a unit of revenue for the Örm; that is,

revenue is e.The worker bears a cost of e§ort given by ae2, where a is a positiveconstant.

The manager can pay the worker some money, which enters their

payo§s in an additive way.

Thus, if the worker picks e§ort level e and the manager pays theworker x , then the managerís payo§ is e ! x and the workerís payo§ isx ! ae2.Assume that e§ort is veriÖable and externally enforceable, meaning

that the parties can commit to a payment and e§ort level.



Imagine that the parties interact as follows:

First, the manager makes a contract o§er to the worker.

The contract is a speciÖcation of e§ort e and a wage x .Then the worker accepts or rejects the o§er.

If she rejects, then the game ends and both parties obtain payo§s of 0.

If she accepts, then the contract is enforced (e§ort e is taken and x ispaid).

Because the contract is externally enforced, you do not have to

concern yourself with the workerís incentive to exert e§ort.



Find the managerís optimal contract o§er.



Setting up the maximization problem,

maxe ,x

e ! x (1)

s.t. x ! ae2 " 0 =) x = ae2 (2)

The constraint (2) on the maximization problem for the manager is

the minimum payment that the worker requires in order to participate

(Participation Constraint).

The Manager wants to minimize the payment that he makes to the

Worker (to maximize proÖts), so we can assume that the constraint is

binding and he selects x = ae2.



We can now substitute that value back into the maximization

problem in order to make it unconstrained, as follows:

maxe

e ! ae2

Taking F.O.C.s with respect to e, yields

1! 2ae = 0

Solving for e we obtain an optimal e§ort e = 12a .

Plugging this e§ort back into equation (2), x = ae2, yields x = 14a



How do e and x depend on parameters?



As a is in the denominator, an increase in a decreases both e and x .

This is also the e¢cient outcome from a Social Plannerís perspective:

The sum of both playersí utility functions results in the Managerís

unconstrained maximization problem (which we just solved).

Indeed, the sum of the Managerís and Workerís utilities yields:

(e ! x) + (x ! ae2) = e ! ae2

which coincides with the objective function in the above maximization

problem.)

Hence, the e¢cient outcome and the Managerís ideal outcome

coincide.



Let e and x denote the equilibrium contract in the case in which

a = 18, and

let e and x denote the equilibrium contract in the case in which

a = 38.

Calculate these four values.



High type, aH =38, equilibrium values:

e =1

2 $ aH=4

3and x =

1

4 $ aH=2

3

Low type, aL =18, equilibrium values:

e =1

2 $ aL= 4 and x =

1

4 $ aL= 2



Suppose that a is private information to the worker.

The manager knows only that:

a = 18 with probability

12 , and

a = 38 with probability

12 .

Suppose that the manager o§ers the worker a choice between

contracts (e, x) and (e, x)ñthat is, the manager o§ers a menu ofcontractsñin the hope that the high type will choose (e, x) and thelow type will choose (e, x).

Will each type pick the contract intended for him? If not, what will

happen and why?



High type: ! aH =38, When he chooses the contract meant for the

low-type worker, (e, x), his payo§ is:

x ! aHe2 = 2!3

8$ 42 = !4 < 0

If, in contrast, he chooses the contract meant for him, (e, x), thehigh-type worker obtains:

x ! aH e2 =2

3!3

8

!4

3

"2= 0

So the high type chooses the contract meant for him (e, x), since ityields a higher utility (0) than the contract meant for the low-type

(e, x), -4.



Low Type: ! aL =18When he chooses the contract meant for the

him, (e, x), his payo§ is:

x ! aLe2 = 2!1

8$ 42 = 0

When he chooses the contract meant for the high-type worker, (e, x),his payo§ is:

x ! aLe2 =2

3!1

8

!4

3

"2=4

9> 0

So the low type chooses the contract meant for the high type, (e, x),since it yields a higher utility (4

9) than the contract meant for himself

(e, x), 0.



Suppose that the manager o§ers a menu of two contracts (eL, xL)and (eH , xH ), where he hopes that the Örst contract will be acceptedby the low type and the second will be accepted by the high type.

Under what conditions will each type accept the contract intended for

him?



The Incentive Compatibility Conditions for the low and high types,respectively, are:

Low Type: !

Payo§ from beinga low typez }| {xL !

1

8e2L "

Payo§ from pretendingto be a high typez }| {xH !

1

8e2H (3)

High Type: ! xH !3

8e2H

| {z }Payo§ from beinga high type

" xH !3

8e2H

| {z }Payo§ from pretendingto be a low type

(4)



The Participation Constraints are:

Low Type: !

Low type workeríspayo§ from participatingz }| {

xL !1

8e2L "

Low type workerís payo§from not participatingz}|{

0 (5)

High Type: ! xH !3

8e2H

| {z }High type worker

payo§ from participating

" 0|{z}High type workerís payo§from not participating

(6)

When inequalities 3-6 hold, each type will accept the contract that is

intended for him.



Compute the managerís optimal menu (eL, xL) and (eH , xH ).



The Manager wants to maximize his expected payo§:

1

2[eH ! xH ] +

1

2[eL ! xL]

where ( 12) represents the probabilities of the Worker being either the

High or Low types...

something that the Manager cannot observe when o§ering the two

possible contracts to the Worker.



From part (d), equation (3) is binding (Recall that the low type would

rather pretend to be the high type) and yields:

xL = xH +1

8e2L !

1

8e2H

and equation (6) is binding (The high type has the higher reservation

utility) and yields:

xH =3

8e2H

Then substituting the latter equation into the former we obtain:

xL =3

8e2H +

1

8e2L !

1

8e2H

Combining terms:

xL =1

4e2H +

1

8e2L



Substituting for xL and xH into the Managerís expected payo§ yieldsthe following unconstrained maximization problem:

maxeL ,eH

1

2

'eH !

3

8e2H

(+1

2

'eL !

1

4e2H !

1

8e2L

(



F.O.C.s with respect to eL:

∂L∂eL

=1

2!1

8eL = 0 ! e$L = 4

F.O.C.s with respect to eH :

∂L∂eH

=1

2!3

8eH = 0 ! e$H =

4

5

which implies

x$H =3

8e2H =

3

8$!4

5

"2=6

25

x$L =1

4e2H +

1

8e2L =

1

4$!4

5

"2+1

8$ 42 =

54

24



Comment on the relation between the solution to the managerís

problem when there is complete versus incomplete information.

How does the optimal meny under asymmetric information distort away

from e¢ciency?



Complete Information Incomplete Information

(e, x)(4, 2)

(e, x))43, 23

* (e$L , x$L ))

4, 5424

* (e$H , x$H ))

45, 625

*

Low Type

aL =18

0 14

425

High Type

aH =38

0 ! 154

0



We can now use the above summarizing table in order to compare the

e§ort for each type of worker:

Low-type worker:

The low-type worker exerts the same e§ort under both information

contexts (eL = 4),However, he is paid more under incomplete info, 5425 , than under

complete info, 2, in order for him to have incentives to reveal his type

by voluntarily selecting the contract meant for him.

Low-type worker:

The high-type worker exerts less e§ort under incomplete information,

( 45 <43 ),

but is paid less, ( 625 <23 ).


econs 424 - signalling games iii

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