EconS 424 - Signalling Games III
Transcript of EconS 424 - Signalling Games III
EconS 424 - Signalling Games III
FÈlix MuÒoz-GarcÌa
Washington State University
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 2 / 24
Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 3 / 24
Watson, Ch. 29 Exercise 9
Find the managerís optimal contract o§er.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 4 / 24
Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 5 / 24
Watson, Ch. 29 Exercise 9
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
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 6 / 24
Watson, Ch. 29 Exercise 9
How do e and x depend on parameters?
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 7 / 24
Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 8 / 24
Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 9 / 24
Watson, Ch. 29 Exercise 9
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
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 10 / 24
Watson, Ch. 29 Exercise 9
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?
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 11 / 24
Watson, Ch. 29 Exercise 9
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.
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Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 13 / 24
Watson, Ch. 29 Exercise 9
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?
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 14 / 24
Watson, Ch. 29 Exercise 9
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)
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 15 / 24
Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 16 / 24
Watson, Ch. 29 Exercise 9
Compute the managerís optimal menu (eL, xL) and (eH , xH ).
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 17 / 24
Watson, Ch. 29 Exercise 9
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.
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 18 / 24
Watson, Ch. 29 Exercise 9
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
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 19 / 24
Watson, Ch. 29 Exercise 9
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
(
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 20 / 24
Watson, Ch. 29 Exercise 9
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
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 21 / 24
Watson, Ch. 29 Exercise 9
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?
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 22 / 24
Watson, Ch. 29 Exercise 9
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
FÈlix MuÒoz-GarcÌa (WSU) EconS 424 - Recitation 12 April 28, 2014 23 / 24
Watson, Ch. 29 Exercise 9
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 ).
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