informationdarp.lse.ac.uk/pdf/ec202/ec202_classes/microeconomicssolutions11.pdfmicroeconomics q f t...

Chapter 11

Information

Exercise 11.1 A �rm sells a single good to a group of customers. Each cus-tomer either buys zero or exactly one unit of the good; the good cannot be dividedor resold. However, it can be delivered as either a high-quality or a low-qualitygood. The quality is characterised by a non-negative number q; the cost of pro-ducing one unit of good at quality q is C(q) where C is an increasing and strictlyconvex function. The taste of customer h is �h � the marginal willingness topay for quality. Utility for h is

Uh (q; x) = �hq + x

where �h is a positive taste parameter and x is the quantity consumed of allother goods.

1. If F is the fee required as payment for the good write down the budgetconstraint for the individual customer.

2. If there are two types of customer show that the single-crossing conditionis satis�ed and establish the conditions for a full-information solution.

3. Show that the second-best solution must satisfy the no-distortion-at-the-topprinciple.

4. Derive the second-best optimum.

Outline Answer

1. If the consumer has income y then the budget constraint is

x+ �F (q) � y

where � is a variable taking the values 0 or 1, representing the cases �notbuy�and �buy�.

2. Assume that each person�s type is common knowledge.

(a) If there are two taste types �a; � b with

�a > � b (11.1)

The preferences are as shown in Figure 11.1.

195

Microeconomics CHAPTER 11. INFORMATION

qualityq

x

τb

τa

Figure 11.1: Preferences: quality

q

F

Π2 = F2 − C(q)

quality

Π1 = F1 − C(q)

Π0 = F0 − C(q)

increasingprofit

q

F

Π2 = F2 − C(q)

quality

Π1 = F1 − C(q)

Π0 = F0 − C(q)

increasingprofit

Figure 11.2: Isopro�t curves: quality

c Frank Cowell 2006 196

Microeconomics

q

F

τbq

τaq

qualityq*aq*b

F*b

F*a

•

•

q

F

τbq

τaq

qualityq*aq*b

F*b

F*a

•

•

Figure 11.3: Full-information solution: quality

(b) Given that each person�s type can be observed the �rm can tailorthe fee schedule exactly to personal characteristics, charging F a toan a-type and F b to an b-type The �rm�s pro�ts from each of the twogroups of consumers are

F a � C (qa) (11.2)

F b � C�qb�

(11.3)

and the isopro�t curves are as shown in Figure 11.2.

(c) If a person chooses not to buy the good then his utility is just y. Sothe �rm chooses qa and F a to maximise (11.2) subject to the a-type�sparticipation constraint

�aqa � F a + y � y

or, equivalently�aqa � F a � 0; (11.4)

it likewise chooses qb and F b to maximise (11.3) subject to the b-type�s participation constraint

� bqb � F b � 0: (11.5)

The full information solution is found at the tangency of an iso-pro�t curve with a reservation indi¤erence for each of the two types,as shown in �gure 11.3 Clearly, at the optimum (q�a; F �a) and�q�b; F �b

�, each of the participation constraints (11.4) and (11.5) is

binding and each person just gets his reservation utility y.

3. Now it is no longer possible to condition the fee schedule directly on aperson�s type.



q

F

τbq

τaq

qualityq*aq*b

F*b

F*a

•

•preference

Figure 11.4: Type a prefers type b contract

(a) If the full-information contracts from part 2 were available an a-typeperson would want to take a b-type contract since his utility wouldthen be �aq�b�F �b+y which, in view of the fact that � bq�b�F �b = 0,becomes �

�a � � b�q�b + y

which is strictly greater than y. See Figure 11.4.

(b) We need to �nd the second-best contracts that take account of thisincentive-compatibility problem. Incentive compatibility requires that,for a:

�aqa � F a � �aqb � F b (11.6)

and, for b:� bqb � F b � � bqa � F a (11.7)

Suppose it is known that there is a proportion �, 1 � � of a-typesand b-types, respectively. Now the �rm�s problem is to choose qa, qb,F a and F b to maximise

� [F a � C (qa)] + [1� �]�F b � C

�qb��

(11.8)

subject to the participation constraints (11.4) and (11.5) and theincentive-compatibility constraints (11.6) and (11.7). As in the textwe can simplify the problem by determining which constraints areslack and which are binding. Note that:

� (11.6) and (11.1) imply

�aqa � F a � �aqb � F b � � bqb � F b (11.9)

This implies that if constraint (11.5) were slack, then constraint(11.4) would also be slack; this cannot be true at the optimum


Microeconomics

since it would then be possible for the �rm to increase both F a

and F b and increase pro�ts. Hence (11.5) must be binding.� Given that F b > 0 at the optimum (11.5) then implies thatqb > 0: However (11.1). (11.9) and qb > 0 imply

�aqa � F a � �aqb � F b > � bqb � F b = 0 (11.10)

which implies that�aqa � F a > 0 (11.11)

So constraint (11.4) is slack and can be ignored.� If (11.6) were slack then, by (11.11) it would be possible to in-crease F a without violating the constraint. So (11.6) must bebinding.

�aqa � F a = �aqb � F b (11.12)

� If (11.7) were binding, then this and (11.12) would imply

qb��a � � b

�= qa

��a � � b

�but, given (11.1).this can only be true if

qb = qa:

But this implies a pooling outcome and we know that the �rmcan do better than a pooling outcome by forcing the high-valueconsumers to reveal themselves. This implies that constraint(11.7) is slack and can be ignored.

q

F

τbq

τaq

qualityq*aqb

Fb

Fa •

preference

Figure 11.5: Second-best solution: quality

(c) The Lagrangean is therefore

� [F a � C (qa)] + [1� �]�F b � C

�qb��

+�� bqb � F b

�+��aqa � F a � �aqb + F b

� (11.13)



where � and � are the Lagrange multipliers for the constraints (11.5,11.6) respectively. The �rst-order conditions are

� �Cq (qa) + ��a = 0 (11.14)

� [1� �]Cq�qb�+ �� b � ��a = 0 (11.15)

� � � = 0 (11.16)

1� � � �+ � = 0 (11.17)

From (11.16) and (11.17) and we have � = � and � = 1. Using thesevalues in (11.14) and (11.15) we have

Cq (qa) = �a (11.18)

Cq�qb�= � b � �

1� ��a � � b

�< � b (11.19)

Clearly the a-type�s consumption is at the point where marginal cost- marginal willingness to pay for quality �see Fig 11.5.


Microeconomics

Exercise 11.2 An employee�s type can take the value �1 or �2, where �2 > �1.The bene�t of the employee�s services to his employer is proportional to z, theamount of education that the employee has received. The cost of obtaining zyears of education for an employee of type � is given by

C (z; �) = ze�� :

The employee�s utility function is

U(y; z) = �e�y � C (z; �)

where y is the payment received from his employer. The risk-neutral employerdesigns contracts contingent on the observed gross bene�t, to maximise his ex-pected pro�ts.

1. If the employer knows the employee�s type, what contracts will be o¤ered?If he does not know the employee�s type, which type will self-select the�wrong�contract?

2. Show how to determine the second-best contracts. Which constraints bind?How will the solution to the second-best problem compare with that in part1?

Outline AnswerThis is a problem of hidden information: The type of the employee � is

exogenously given, but private information. The problem for the employer is todesign a contract that leads the employee to reveal his true type.The employer is interested in truthful revelation since it is less costly for em-

ployees of type �2 to attain a given level of education. Hence, the employer wouldhave to reward �2 types by less to get the same level of education. However,type �2 would like to pretend to be of type �1 to receive the higher payment.The reservation utility of the employee corresponds to the case where the

employee receives 0 years of education, for which he receives 0 from the employer:

� = �1:

This yields participation constraints for each type:

1� e�y � ze�� 0

For further reference, we can deduce the shape of the indi¤erence curves for agiven type. The �rst and second derivatives, respectively, are:

dy

dz= ey�� i > 0 (11.20)

d2y

dz2= ey�� i

dy

dz= e2[y�� i] > 0

See Figure 11.6 for an illustration of the reservation indi¤erence curves for thetwo types, which go through the origin. The shaded areas are the acceptancesets for the two types of individuals. Note that as dy

dz is independent of z, theindi¤erence curves for a given type must be horizontally parallel in (z; y) space.Also note that the single crossing property is satis�ed.



0

y

z

υ1_υ1_

υ2_υ2_

y1*y1*

*y2*y2

*z2*z2

*z1*z1

•

•

Figure 11.6: Full-information contracts

1. If the employer knows the employee�s type, then, as before, the employercan implement full-information contracts, maximising the return fromeach type separately. Let � be the type. Then the problem of the employeris to

maxz;y

� := z � y

subject to the participation constraint

1� e�y � ze�� 0 (11.21)

We can now set up the Lagrangean associated with this maximizationproblem

L (z; y; �) := z � y + ��1� e�y � ze��

�where � is the Lagrange multiplier for the constraint (11.21). The FOCsyield:

@L@z

= 1� �e�� = 0 (11.22)

@L@y

= �1 + �e�y = 0 (11.23)

@L@�

= 1� e�y � ze�� 0 (11.24)

Combining equations (11.22) and (11.23), we see that e� = ey and hencethat

y� = �

Those equations also imply that � > 0, and hence that the participationconstraint (11.24) must bind. Hence

z� =1� e�ye��

= e� � 1 > 0


Microeconomics

Thus the employer o¤ers two di¤erent contracts to the two types, wherethe level of education is z�i = e� i � 1, and the compensation is y�i = � i.Since the slope of the indi¤erence curve in Figure 11.6 was given byequation 11.20, we see that the contracts are located where the slope ofthe reservation indi¤erence curves are equal to 1.

2. Now assume that the employer cannot observe the type of the employee.

(a) Given the contracts found in part 1 an employee of type 2 will self-select the wrong contract. To see this, compare the utility of type 2when choosing the type-2 contract

�2 = �e��2 � [e�2 � 1] e��2

to that when choosing the type-1 contract:

�2 = �e��1 � [e�1 � 1] e��2

Since �1 < �2 it is clear that �2 > �2. Hence, type 2 would self-selectthe type 1 contract, and hence incentive compatibility is violated.

(b) If there is an incentive for type 2 to self-select the wrong contract,then any second-best contract has to insure that the incentive com-patibility constraint for type 2 is not violated. Type 1 will never havean incentive to select the type 2 contract, hence the incentive compat-ibility constraint for type 1 will not be binding. Furthermore, sinceany non-trivial contract for type 1 would enable type 2 to achieve autility level greater than his reservation utility level, the participa-tion constraint for type 2 cannot be binding. However, we can keeptype 1 on his reservation utility. Hence, we expect that the two con-straints which will be binding in the second-best contract will be theparticipation constraint for type 1

1� e�y1 � z1e��1 � 0 (11.25)

and the incentive-compatibility constraint for type 2:

� e�y2 � z2e��2 � �e�y1 � z1e��2 (11.26)

This is as in the standard adverse selection model:

(c) Assume that the probability of encountering a type 1 employee is� 2 (0; 1). We may now set up the maximization problem for themonopolist, which consists of choosing z1; y1; z2; y2 to maximise :

� := � [z1 � y1] + [1� �] [z2 � y2]

subject to (11.25) and (11.26). The Lagrangean is

L (z1; y1; z2; y2; �; �) := � [z1 � y1] + [1� �] [z2 � y2]+� [1� e�y1 � z1e��1 ]+� [e�y1 � e�y2 + z1e��2 � z2e��2 ]



where � and � are the Lagrange multipliers associated with the con-straints (11.25) and (11.26), respectively. Maximising this yields theFOCs

@L@z1

= � � �e��1 + �e��2 = 0 (11.27)

@L@y1

= �� + [�� ] e�y1 = 0 (11.28)

@L@z2

= [1� �]� �e��2 = 0 (11.29)

@L@y2

= � [1� �] + �e�y2 = 0 (11.30)

As � � 0 by Kuhn-Tucker conditions, and � > 0, equation 11.27implies that � > 0. Hence, constraint (11.25) will bind. Consider-ing equation 11.29, we see similarly that � > 0, so that constraint(11.26) will bind. Since both constraints will bind, and the �rst orderconditions are satis�ed, we have a system with six equations and sixvariables. Call the solutions to this system (z1; y1) and (z2; y2).

(d) Since the incentive-compatibility constraint (11.26) is binding, thesetwo contracts will lie on the same type 2 indi¤erence curve. But thetype-1 incentive compatibility constraint is slack, so type 1 strictlyprefers (z1; y1) to (z2; y2). Given the Single Crossing Property andthe fact that the type 2 indi¤erence curves are �atter that of type1, we must have z1 < z2 and y1 < y2. Consider equations 11.29and 11.30. We �nd that y2 = �2, the �no distortion at the top�result. Clearly, the marginal rate of substitution for type 2 at thispoint is

�e��2=e�y2 = 1;which is the same as the slope of the isopro�t contour. Now comparethe second-best contracts with the full-information solution. Type 2was on his reservation indi¤erence curve �2 under the full-informationcontract. But under the second-best, the participation constraint fortype 2 is slack, and hence he is now on an indi¤erence curve I2 above�2. But since

y2 = y�2 = �2;

this must mean that z2 < z�2 .

(e) Finally, how far has the solution moved from �2? There are twopossibilities: The new (z1; y1) could be below and to the left of thefull-information (z�1 ; y

�1), or above and to the right. We will show

that z1 < z�1 and y1 < y�1 (recall that (11.25) is still binding, so thenew solution will be on the reservation indi¤erence curve for type 1).The slope of the indi¤erence curve �1 at (z1; y1) is

e��1

e�y1:

From equations (11.27) and (11.28) we have

�e�y1 = � + �e�y1

�e��1 = � + �e��2


Microeconomics

hence the slope of the indi¤erence curve is

� + �e��2

� + �e�y1< 1 (11.31)

This follows because y1 < y2 = �2 which implies e��2 < e�y1 . Wefound that the slope of the type 1 indi¤erence curve at (z�1 ; y

�1) was 1,

so condition (11.31) implies that z1 < z�1 and y1 < y�1 . See �gure 11.7.

0z2z2

y

zz1z1

y1y1

υ1_υ1_

υ2_υ2_

y2y2

•

•

Figure 11.7: Second-best contracts



Exercise 11.3 A large risk-neutral �rm employs a number of lawyers. For alawyer of type � the required time to produce an amount x of legal services isgiven by

z =x

�

The lawyer may be a high-productivity a-type lawyer or a low-productivity b-type:�a > � b > 0. Let y be the payment to the lawyer. The lawyer�s utility functionis

y12 � z

and his reservation level of utility is 0. The lawyer knows his type and the �rmcannot observe his action z: The price of legal services is 1.

1. If the �rm knows the lawyer�s type what contract will it o¤er? Is it e¢ -cient?

2. Suppose the �rm believes that the probability that the lawyer has low pro-ductivity is �: Assume � b � [1� �] �a: In what way would the �rm thenmodify the set of contracts on o¤er if it does not know the lawyer�s typeand cannot observe his action?

Outline Answer.The problem is one of adverse selection with hidden information.

0

y

x

•

•

slope = 1

slope = 1

x*a = 2x*b = ½

y*b = ¼

y*a = 1

υb_

_υa

0

y

x

•

•

slope = 1

slope = 1

x*a = 2x*b = ½

y*b = ¼

y*a = 1

υb_υb_

_υa_υa

Figure 11.8: Full-information contracts

1. Full-information.

(a) The principal knows the type and so maximises x� y subject topy � x

��

where� = 0


Microeconomics

for each individual type. We know that the participation constraintbinds and that there is no distortion. So

� =py � x

�(11.32)

Di¤erentiate to �nd the slope of the indi¤erence curve:

dy

dx

��=�

=2py

�(11.33)

Since there is no distortion (11.33) must be equal to 1 which implies

y� =1

4�2: (11.34)

Using the fact that � = � and substituting (11.34) into (11.32) weget

x� =1

2�2: (11.35)

In this case the optimal contracts are (x�a; y�a) and�x�b; y�b

�as

shown in Figure 11.8.

(b) Since there is no distortion the solution is e¢ cient. There is nochange which could make one person better o¤ without making theother worse o¤.

2. Types unknown.

(a) If it is impossible to monitor the lawyer�s type it is no longer viableto o¤er the e¢ cient contracts (x�a; y�a) and

�x�b; y�b

�from part 1. If

a type-a lawyer accepts the e¢ cient contract meant for him he getsutility

py�a � x�a

�a=1

2�a �

12 [�

a]2

�a= 0

But if a type-a lawyer were to get a type-b contract he would getutility

py�b � x�b

�a=1

2� b �

12

�� b�2

�a=1

2� b�1� � b

�a

�> 0:

So a type a would prefer to take a type-b contract.

(b) Given this problem the best that the �rm can do is to maximiseexpected pro�ts subject to an incentive-compatibility constraint forthe a types:

pya � xa

�a�pyb � xb

�a

Let � be the the probability that the lawyer is of type a. Thenexpected pro�ts are

� [xa � ya] + [1� �]�xb � yb

�c Frank Cowell 2006 207


In the problem of maximising expected pro�ts under these conditionswe know from previous exercises (such as Exercise 11.2) that theparticipation constraint (11.32) for type b will be binding

pyb � xb

� b= �b (11.36)

as is the incentive-compatibility constraint for type:

pya � xa

�a=pyb � xb

�a: (11.37)

So the relevant Lagrangean is

L�xa; ya; xb; yb; �; �

�:= � [xa � ya] + [1� �]

�xb � yb

�+�hp

yb � xb

�b

i+�hp

ya � xa

�a �pyb + xb

�a

i9>>=>>;(11.38)

where � and � are the Lagrange multipliers for the constraints (11.36)and (11.37) respectively.

(c) Let the solution values for maximising (11.38) be denoted xa, ya, xb,yb. Di¤erentiating (11.38) the FOC are:

� � + �

2pya

= 0; (11.39)

� [1� �] + �

2pyb� �

2pyb

= 0; (11.40)

� � �

�a= 0; (11.41)

1� � � �

� b+

�

�a= 0: (11.42)

From (11.41) and (11.42) we have

� = ��a

� = � b;

and substituting these values in (11.39) and (11.40) we get

�� + ��a

2pya

= 0

� [1� �] + � b

2pyb� ��a

2pyb

= 0

which implies

ya =1

4[�a]

2

yb =

�� b � �a�2 [1� �]

�2c Frank Cowell 2006 208

Microeconomics

And so, using the participation constraint (11.32) we have

xb = � bpyb

=� b

2 [1� �]�� b � �a�

�which is non-negative by assumption. Finally, from the incentive-compatibility constraint we get

xa =[�a]

2

2� �a � � b

� bxb:

0xa

y

xxb

yb

υb_

_υa

ya

•

•

0xaxa

y

xxbxb

ybyb

υb_υb_

_υa_υa

yaya

•

•

Figure 11.9: The second-best solution in the adverse selection problem

(d) Note that

xa < x�a

xb < x�b

ya = y�a

yb < y�b

�see Figure 11.9. This is the no-distortion-at-the-top result. Notethat indi¤erence curves for a given type are horizontally parallel be-cause utility is linear in x. So ya = y�a is immediate.



Exercise 11.4 The analysis of insurance in the text (section 11.2.6) was basedon the assumption that the insurance market is competitive. Show how theprinciples established in section 11.2.4 for a monopolist can be applied to theinsurance market:

1. In the case where full information about individuals�risk types is available.

2. Where individuals�risk types are unknown to the monopolist.

Outline Answer

1. See Figure 11.10

xBLUE

xRED

0

profits, πb

κa

ya_ya_

ya_ya_

•

profits

, πb

y

y − L •

Figure 11.10: Insurance: iso-pro�t

(a) The endowment point for both types of individual is at (y; y � L).Given that the probability of an accident for high-risk type a is �a

an insurance �rm would break even if it sold insurance against theloss L for a premium �a where

�a = �aL: (11.43)

De�neya := y � �aL = �a: (11.44)

The line with slope 1��a�a passing through the endowment point and

the point (ya; ya) is an isopro�t contour for the �rm when dealingwith the high-risk types. Pro�ts must increase to the �South-West�(consider the impact on pro�ts if the �rm were able to charge a higherpremium �, all other things being equal).


Microeconomics

(b) For the low-risk types the isopro�t contours are a family of lines withslope 1��b

�b.

(c) Clearly the reservation indi¤erence curve for each type of person isgiven as the contour passing through the endowment point. It hasslope 1��h

�h; h = a; b where it intersects the 45� line � see Figure

11.11.

xBLUE

xRED

(y, y − L)•

•

0

•

(ya, ya)_ _

(yb, yb)_ _

xBLUE

xRED

(y, y − L)•

•

0

•

(ya, ya)_ _

(ya, ya)_ _

(yb, yb)_ _

(yb, yb)_ _

Figure 11.11: Insurance: monopoly with full information

(d) So the full-information outcome is where the high-risk types are lo-cated at (ya; ya) and the low-risk types at

�yb; yb

�: the monopoly

insurance �rm rationally o¤ers better insurance terms to the low-risk.

2. Take the case where the individual�s risk type is unknown to the insurer.

(a) Given that there is imperfect information it is clear that a high-risktype would like to masquerade as a low-risk type and so take advan-tage of the more favourable terms. In Figure 11.12 see the a-type in-di¤erence curve passing through the point

�yb; yb

�.

The monopolist must take account of this possibility in setting upthe second-best optimisation problem.

(b) The solution to this problem will be of the following form: restrict thelow-risk b-types in the amount of insurance that they can purchase sothat they choose the prospect ~P which lies on the b-type reservationindi¤erence curve in Figure 11.12; o¤er full insurance at point (~ya; ~ya)



xBLUE

xRED

(y, y − L)

•

0

••

•

•

(yb, yb)_ _

(ya, ya)~ ~

Pb~

xBLUE

xRED

(y, y − L)

•

0

••

•

•

(yb, yb)_ _

(yb, yb)_ _

(ya, ya)~ ~(ya, ya)~ ~

Pb~

Pb~

Figure 11.12: Insurance: monopoly second-best solution

�only the a-types will wish to take up this o¤er. The formal analysisclosely follows that of section 11.2.4.


Microeconomics

Exercise 11.5 Good second-hand cars are worth �a1 to the buyer and �a0 to

the seller where �a1 > �a0. Bad cars are worth �b1 to the buyer and �

b0 to the

seller where �b1 > �b0. It is common knowledge that the proportion of bad cars is�. There is a �xed stock of cars and e¤ectively an in�nite number of potentialbuyers

1. If there were perfect information about quality, why would cars be tradedin equilibrium? What would be pa and pb, the equilibrium prices of goodcars and of bad cars respectively?

2. If neither buyers nor sellers have any information about the quality of anindividual car what is p, the equilibrium price of cars?

3. If the seller is perfectly informed about quality and the buyer is uninformedshow that good cars are only sold in the market if the equilibrium price isabove �a0.

4. Show that in the asymmetric-information situation in part 3 there are onlytwo possible equilibria

� The case where pb < �a0: equilibrium price is pb.

� The case where p � �a0: equilibrium price is p.

Outline Answer.

1. If there is perfect information about quality then:

(a) Cars of whatever quality will always be traded if the value of thebuyer is greater than that to the seller, as in the question.

(b) Equilibrium prices are

pa = �a1

pb = �b1

2. Given that neither party can verify the quality of a car ex ante, but bothknow that the probability of a bad car is �:

(a) The expected value of a car to the buyer is

[1� �] �a1 + ��b1 (11.45)

and the expected value of a car to the seller is

[1� �] �a0 + ��b0 (11.46)

(b) Given that (11.45) is greater than (11.46) the equilibrium price is

p = [1� �] �a1 + ��b1 (11.47)

3. Sellers will only be willing to supply good cars to the market if the priceis at least as great as their private valuation �a0 .



4. Consider two cases.

� If the price is less than �a0 then, from part 3, the sellers will only sup-ply bad cars and the buyers will be well aware of this. The fact that,after purchase, the quality is revealed as bad con�rms the buyers�beliefs. Hence we have an equilibrium with price set at the buyers�valuation of bad cars:

p = �b1

� If the price is not less than �a0 then both types of car may be suppliedto the market. Once again the buyers will be aware of this and, inthe absence of further information will estimate the value as givenby (11.45). We have an equilibrium with price

p = p

where p is given by (11.47) if this price is at least as high as thesellers�valuation of good cars

[1� �] �a1 + ��b1 � �a0 : (11.48)


Microeconomics

Exercise 11.6 In an economy there are two types of worker: type-a workershave productivity 2 and type-b workers have productivity 1. Workers produc-tivities are unobservable by �rms but workers can spend their own resources toacquire educational certi�cates in order to signal their productivity. It is com-mon knowledge that the cost of acquiring an education level z equals z for type-bworkers and 1

2z for type-a workers.

1. Find the least-cost separating equilibrium.

2. Suppose the proportion of type-b workers is �. For what values of � willthe no-signalling outcome dominate any separating equilibrium?

3. Suppose � = 14 . What values of z are consistent with a pooling equilibrium?

Outline Answer

1. There is some z� such that �rms believe a worker to be of type a if z � z�

and type b otherwise. Beliefs are self-con�rming if type-a workers choosez = z� and type-b workers choose z = 0. This is satis�ed if

2� 12z� � 1

1 � 2� z�

in other words1 � z� � 2:

Any z� in [1; 2] does the job so that there is a continuum of separatingequilibria. z� = 1 is the least-cost separating equilibrium. The net payo¤to an a-type in the least-cost separating equilibrium is 2� 1

2z� = 1 12 .

2. Given the de�nition of � the expected productivity of a randomly chosenworker is

� + 2 [1� �] = 2� �:This would give a better payo¤ to a-types in the separating equilibrium if

2� � > 112

so that � < 12 .

3. If �z is education in the pooling equilibrium then the net payo¤ to a-typeworkers is

2� � � 12�z = 1

3

4� 12�z

and for b-type workers is

2� � � �z = 134� �z

The b-type workers are better o¤ in the pooling equilibrium if

13

4� �z > 1

�z <3

4



The a-type workers are better o¤ in the pooling equilibrium if

13

4� 12�z > 1

1

2

�z <1

2


Microeconomics

Exercise 11.7 A worker�s productivity is given by an ability parameter � >0. Firms pay workers on the basis of how much education, z, they have: thewage o¤ered to a person with education z is w (z) and the cost to the worker ofacquiring an amount of education z is ze�� .

1. Find the �rst-order condition for a type � person and show that it mustsatisfy

� = � log�dw (z�)

dz

�(11.49)

2. If people come to the labour market having the productivity that the em-ployers expect on the basis of their education show that the optimal wageschedule must satisfy

w (z) = log (z + k) (11.50)

where k is a constant.

3. Compare incomes net of educational cost with incomes that would prevailif it were possible to observe � directly.

Outline Answer

1. Individual income, net of educational costs is

w (z)� ze�� :

Maximising this with respect to z gives the FOC

wz (z�)� e�� = 0

from which (11.49) immediately follows.

2. If the employer�s expectations are ful�lled then the revealed marginal prod-uct � equals the wage paid w (z). Using this result in (11.49) we obtain

� log�dw (z)

dz

�= w (z) (11.51)

which is a �rst-order di¤erential equation in z. Rearranging we have

ew(z)dw (z)

dz= 1 (11.52)

Integrating over z we getew(z) = z + k (11.53)

where k is a constant of integration. From this (11.50) follows.

3. In this model the costs of education ze� are a net loss to the workers whowould have been paid according to their type � anyway, if only � couldhave been observed.



Exercise 11.8 The manager of a �rm can exert a high e¤ort level z = 2 ora low e¤ort level z = 1. The gross pro�t of the �rm is either �1 = 16 or�2 = 2. The manager�s choice a¤ects the probability of a particular pro�toutcome occurring. If he chooses z, then �1 occurs with probability � = 3

4 , but ifhe chooses z then that probability is only � = 1

4 . The risk neutral owner designscontracts which specify a payment yi to the manager contingent on gross pro�t�i. The utility function of the manager is u(y; z) = y1=2�z, and his reservationutility � = 0.

1. Solve for the full-information contract.

2. Con�rm that the owner would like to induce the manager to take actionz.

3. Solve for the second-best contracts in the event that the owner cannotobserve the manager�s action.

4. Comment on the implications for risk sharing.

Outline Answer.

1. Under the full-information contract, both manager and owner can observethe action of the manager. So we can solve the optimization problem forthe owner for both high and low e¤ort levels separately, and then let theowner choose the one that yields higher expected pro�ts.

(a) Formally, denoting the expected pro�t to the owner under high andlow e¤ort levels by the manager as � and � respectively, we have thefollowing pair of optimisation problems for the owner:

maxy1;y2

� = � [�1 � y1] + [1� �] [�2 � y2] (11.54)

subject to�u (y1; z) + [1� �]u (y2; z) � 0 (11.55)

in the high-e¤ort case and

maxy1;y2

� = �i [�1 � y1] + [1� �] [�2 � y2] (11.56)

subject to�u (y1; z) + [1� �]u (y2; z) � 0 (11.57)

in the low-e¤ort case, where (11.55) and (11.57) are the participationconstraints that will be binding at the optimum.

(b) Since the owner can observe the manager�s action, and since themanager is risk-averse, the owner will set y1 = y2 = y in the solutionto (11.54, 11.55) and we can solve for y simply by setting

u (y; z) = 0

which impliesy1=2 � z = 0

Likewise in the solution to (11.56,11.57) we can solve for y from theequation

y1=2 � z = 0


Microeconomics

(c) Using the numerical values given in the question, we obtain

y = 4

y = 1

2. From part 1 the optimal payments in the case of high and low e¤ort yieldexpected pro�ts to the owner of

� = 8:5

� = 4:5

Thus, the owner would indeed like to induce the manager to take actionz. By construction, the manager is indi¤erent between taking z or z, sincehe receives his reservation utility in either case.

3. However, if the manager could convince the owner that he was using z,and get the owner to pay him y, while in fact only using z, his payo¤would be

u(y; z) = y1=2 � z = 1 > 0

and hence the incentive-compatibility constraint would be violated. Thus,we now consider the second-best contract, where the owner cannot observethe action of the manager, but can induce him to take the �right�e¤ortlevel.

(a) Under the second-best contract, the owner has the choice of inducingthe manager to choose either high or low e¤ort levels. Since thereis no incentive compatibility problem with the low e¤ort level, thefull-information solution continues to hold. The interesting case isinducing the manager to take the high e¤ort level with a second-bestcontract.

(b) Under the second-best contract, the participation constraint contin-ues to hold. In addition, however, we have an incentive compatibilityconstraint that guarantees that the manager would choose z over z,given the contract. Thus, the owner has to solve the problem

maxy1;y2

� = � [�1 � y1] + [1� �] [�2 � y2]

subject to the participation constraint

�y121 + [1� �] y

122 � z � 0

which becomes3y

121 + y

122 � 8 (11.58)

and the incentive-compatibility constraint

�y121 + [1� �] y

122 � z � �y

121 + [1� �] y

122 � z

which becomesy121 � y

122 � 2 (11.59)



(c) We may set up the appropriate Lagrangean as:

L(y1; y2; �; �) =3

4[16� y1]+

1

4[2� y2]+�

h3y

121 + y

122 � 8

i+�hy121 � y

122 � 2

iwhere � and � are the Lagrange multipliers on constraints (11.58) and(11.59) respectively. Using the Kuhn-Tucker conditions the FOC fora maximum we have

@L@y1

= �34+ �

3

2y� 12

1 +1

2�y

� 12

1 = 0 if y1 > 0 (11.60)

@L@y2

= �14+ �

1

2y� 12

2 � 12�y

� 12

2 = 0 if y2 > 0 (11.61)

�@L@�

= �h3y

121 + y

122 � 8

i= 0 (11.62)

�@L@�

= �hy121 � y

122 � 2

i(11.63)

From (11.60) and (11.61) we have

[6�+ 2�] y� 12

1 = 3 (11.64)

2 [�� ] y�12

2 = 1 (11.65)

We need to determine whether the constraints will be binding. Con-sider � = 0. By equations (11.64) and (11.65), this implies that

y� 12

1 = y� 12

2 =1

2�

and hence that y1 = y2. But this would violates the incentive-compatibility constraint, as before. Thus, � > 0, and, by (11.62)the incentive-compatibility constraint (11.59) must bind. But equa-tion (11.65) implies that � > �, and hence (11.58) must bind aswell. We can now solve for y�1 and y

�2 using the participation and

incentive-compatibility constraints, and obtain that

y�1 =25

4

y�2 =1

4

It can be shown that the expected pro�ts associated with this con-tract are higher than those under a �rst best contract with low e¤ortlevel, but lower than those under �rst best contract with high e¤ortlevel.

4. We observe that y�1 > y > y�2 . The manager receives more than under�rst best in the good outcome, but less in the bad outcome. While it wasrational for the owner to bear all the risk under �rst best, given that hewas risk-neutral while the manager was risk-averse, under the second-bestcontract the risk is shared between the owner and manager. This inducesthe manager to take the action which yields a higher probability of a goodoutcome.


Microeconomics

Exercise 11.9 The manager of a �rm can exert an e¤ort level z = 43 or z = 1

and gross pro�ts are either �1 = 3z2 or �2 = 3z. The outcome �1 occurs withprobability � = 2

3 if action z is taken, and with probability � =13 otherwise. The

manager�s utility function is u(y; z) = log y � z, and his reservation utility is� = 0. The risk neutral owner designs contracts which specify a payment yi tothe manager, contingent on obtaining gross pro�ts �i.

1. Solve for the full-information contracts. Which action does the owner wishthe manager to take?

2. Solve for the second-best contracts. What is the agency cost of the asym-metric information?

3. In part 1, the manager�s action can be observed. Are the full-informationcontracts equivalent to contracts which specify payments contingent on ef-fort?

Outline Answer

1. The structure of the problem is identical to that of Exercise 11.8 so wemay follow the same logic.

(a) Since the owner can observe the manager�s action, and since themanager is risk-averse, the owner will set y1 = y2 = y in the solutionto the counterpart to (11.54, 11.55) and we can solve for y by setting

u (y; z) = 0:

In the present case, this implies

log y � z = 0

Likewise we can solve for y from the equation

log y � z = 0

Substituting in the speci�c values given in the questions, we imme-diately obtain that the full-information contract puts

y = e4=3 (11.66)

andy = e: (11.67)

(b) To ascertain which action the owner would like the manager to take,compare expected pro�ts under the two actions:

� =

�2

3�1 +

1

3�2 � e4=3

�(11.68)

and

� =

�1

3�1 +

2

3�2 � e

�: (11.69)



Clearly

�� = 1

3[�1 ��2]�

he4=3 � e

iwhich is positive. Hence, the owner would like the manager to takez.

2. To solve for the second-best contract, we again use the structure developedin the Exercise 11.8.

(a) The goal is to �nd a contract that induces the manager to take thehigh e¤ort level. From the Exercise 11.8, we know that both theparticipation and incentive-compatibility constraints will be binding.Hence, denoting v(y) = log(y), we have the participation constraint:

� log (y1) + [1� �] log (y2)� z = 0 (11.70)

and the incentive-compatibility constraint:

� log (y1) + [1� �] log (y2)� z = � log (y1) + [1� �] log (y2)� z(11.71)

Rearranging equations (11.70) and (11.71), we �nd

log (y1)� log (y2) =z � z� � � (11.72)

Substituting in the values given in the question, the right-hand sideof (11.72) is 1 and so we have

log (y1) = 1 + log (y2) (11.73)

Hence, using (11.70) we have

� [1 + log (y2)] + [1� �] log (y2)� z = 0

Substituting in the values given in the question we have

y�2 = e23 (11.74)

Using this value in (11.73) we get

y�1 = e53 (11.75)

� [�1 � y�1 ] + [1� �] [�2 � y�2)]2

3

h3z2 � e 53

i+1

3

h3z � e 23

i(b) The agency cost of asymmetric information is given by the di¤erence

of pro�ts under a full information contract, given by (11.68) andthose under a second-best contract. So the agency cost is

� [�1 � y] + [1� �] [�2 � y]| {z }[under full information]

� � [�1 � y�1 ] + [1� �] [�2 � y�2 ]| {z }[under second-best]

> 0


Microeconomics

which becomes� [y�1 � y] + [1� �] [y�2 � y]

substituting in from (11.66),(11.67), (11.74) and (11.75) we get

2

3

he53 � e 43

i+1

3

he23 � e 43

i=2

3e53 +

1

3e23 � e 43

which is positive. Hence, the agency cost of asymmetric informationis greater than zero, as expected. The owner incurs an agency cost,which arises from delegating the decision making to the manager andnot observing the manager�s e¤ort level.

3. Full-information contracts are equivalent to contracts which specify pay-ments contingent on e¤ort, since in both kinds of contracts, the owner caninduce the manager to implement the high e¤ort level, and expected grosspro�ts are the same. However, the high-e¤ort manager extracts a positiverent, so we do not have equivalence in payo¤ structure.



Exercise 11.10 A risk-neutral �rm can undertake one of two investment projectseach requiring an investment of z. The outcome of project i is xi with probability�i and 0 otherwise, where

�1x1 > �2x2 > z

x2 > x1 > 0

�1 > �2 > 0:

The project requires credit from a monopolistic, risk-neutral bank. There islimited liability, so that the bank gets nothing if the project fails.

1. If the bank stipulates repayment y from any successful project what is theexpected payo¤ to the �rm and to the bank if the �rm selects project i?

2. What would be the outcome if there were perfect information?

3. Now assume that the bank cannot monitor which project the �rm chooses.Show that the �rm will choose project 1 if y � y where

y :=�1x1 � �2x2�1 � �2

4. Plot the graph of the bank�s expected pro�ts against y. Show that the bankwill set y = y if �1y > �2x2 and y = x2 otherwise.

5. Suppose there are N such �rms and that the bank has a �xed amount Mavailable to fund credit to the �rms where

z < M < Nz

Show that if �1y > �2x2 there will be credit rationing but no credit ra-tioning otherwise.

Outline Answer.

1. The expected payo¤ to the �rm if project i is selected is given by

�i [xi � y] (11.76)

The expected pro�t for the bank is

�iy � z

2. If there were perfect information then the bank can observe which projectis carried out and whether or not it succeeds. Only project 1 (with thehigher probability of success) will be funded and carried out and the bankwill require a repayment y = x1. The �rm is e¤ectively forced on to itsreservation indi¤erence curve so that no �rm gets less in the absence ofcredit than with credit; there is no credit rationing.


Microeconomics

0

y_y_• y

Π∗(y)

Π

x2

π2π1

Figure 11.13: Credit rationing

3. The �rm will choose the project that gives the greatest pro�ts so that,given z, project 1 is chosen if and only if:

�1 [x1 � y] � �2 [x2 � y]

which is equivalent to the condition y � y where

y :=�1x1 � �2x2�1 � �2

4. The bank�s pro�ts are given by

�� (y) :=

8<: �1y � z if y � y

�2y � z if y < y � x2

�see Figure 11.13. There are clearly two local maxima for �� and thebank would set y = y if

�1y > �2x2 (11.77)

and set y = x2 if�1 < �2x2: (11.78)

5. Distinguish between the situations at the two local maxima

� Take the case (11.78). From (11.76) the �rm�s expected payo¤ fromthe loan is

�2 [x2 � x2] = 0:

So �rms are indi¤erent between taking and not taking the loan.There is no credit rationing.



� Take the case (11.77). From (11.76) the �rm�s expected payo¤ fromthe loan is

�1 [x1 � y] > 0:

All �rms would wish to apply for the loan, so that the total demandis Nz. However, the amount available is M so that there is creditrationing.


Microeconomics

Exercise 11.11 The tax authority employs an inspector to audit tax returns.The dollar amount of tax evasion revealed by the audit is x 2 fx1; x2g. Itdepends on the inspector�s e¤ort level z and the random complexity of the taxreturn. The probability that x = xi conditional on e¤ort z is �i(z) > 0 i = 1; 2.The tax authority o¤ers the inspector a wage rate wi = w(x), contingent onthe result achieved and obtains the bene�t B (x� w). The inspector�s utilityfunction is

U(w; z) = u(w)� v(z)

and his reservation level of utility is �. Assume

B0(�) > 0; B00(�) � 0; u0(�) > 0; u00(�) � 0; v0(�) > 0; v00(�) � 0:

where primes denote derivatives. Information is symmetric unless otherwisespeci�ed.

1. For each possible e¤ort level �nd the �rst-order conditions characterisingthe optimal contract wi i = 1; :::; n.

2. What is the form of the optimal contract when the tax-authority is risk-neutral and the inspector is risk-averse? Comment on your solution andillustrate it in a box diagram.

3. How does this optimal contract change if the inspector is risk-neutral andthe tax-authority is risk-averse? Characterise the e¤ort level that the taxauthority will induce. State clearly any additional assumptions you wishto make.

4. As in part 2 assume that the tax authority is risk-neutral and the taxinspector is risk-averse. E¤ort can only take two possible values z or zwith z > z. The e¤ort level is no longer veri�able. Because the agencycost of enforcing z is too high the tax authority is content to induce z.What is the optimal contract?

Outline Answer.

1. The problem is

max2Xi=1

�i(z)B(xi � wi)

over z and wi such thatX�i(z)u(wi)� v(z) � �:

First �x z, write down the Lagrangian, and di¤erentiate with respect towi to obtain

��i(z)B0(xi � wi) + ��i(z)u0(wi) = 0

so

8i : � = B0(xi � wi)u0(wi)

> 0

so that the participation constraint binds.



w1 x1

x2

Inspector

w2

x1 w1

x2 w2

Authority

45o

υ

Figure 11.14: Inspector-authority equilibrium

2. Risk neutrality implies that B0(�) is constant and so u0(wi) is constant.As u00(�) 6= 0 this implies w(x1) = w(x2) = w, say. The tax authoritybears all the risk �see Figure 11.14. ThereforeX

�i(z)u(w)� v(z) = �

w = u�1 (� + v(z))

Of course the wage rate does depend on the e¤ort demanded.

3. In this case we haveu0(�) = constant

so B0(xi � wi) is constant 8i. Therefore

xi � wi = constant > 0

which impliesw(xi) = xi � k

This is a franchise contract where the inspector keeps the result xi butpays a �xed amount k for the privilege independent of the results. Theparticipation constraint implies

k(z) =X

�i(z)xi � � � v(z)

Thus X�i(z)B(xi � wi) = B(k(z))

X�i(z)

= B(k(z))


Microeconomics

x1

x2

Inspector w1

w2

x1 w1

x2 w2

Authority

45o υ

Figure 11.15: Inspector is risk-neutral

As B0(�) > 0 the tax authority seeks to induce an e¤ort level which max-imises k. This implies that it chooses z to maximiseX

i

�i(z)xi � v(z)

so Xi

�0

i(z)xi � v0(z) = 0

which has the interpretation �expected marginal payo¤ = marginal cost�.The second-order condition isX

i

�00

i (z)xi � v00(z) � 0

and so a su¢ cient condition isX�00

i (z)xi � 0

Note that this is trivially satis�ed if z is discrete. See 11.15.

4. O¤er wL = u�1(�+v(zL)) irrespective of xi:This guarantees the inspectorhis reservation level of utility. As

u(wL)� v(zL) � u(wL)� v(zH)

the incentive compatibility constraint is satis�ed. There is no moralhazard problem here.


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