laffont marchimort excersises

8/6/2019 Laffont Marchimort Excersises

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INTRODUCTION TO INCENTIVE THEORY

Jean-Jacques Laffont & David Martimort

October 21, 2003


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EXERCISES

I- ADVERSE SELECTION

Lending with adverse selection

There is a continuum of risk neutral borrowers with no personal wealth and limited

liability. A proportion of borrowers (called type 1) have sure projects with return h for

an investment of 1. A proportion 1 of borrowers (called type 2) have (stochasticallyindependent) projects with return h only with probability in (0, 1) and return 0 with

probability 1 , for an investment of 1. If he does not apply for a loan, the borrowerhas an outside opportunity utility level of u.

There is a single risk neutral bank available for loans which has a financing cost of r.

The bank offers contracts to maximize its expected profit. For simplicity, we assume that

all projects are socially valuable, i.e.,

h > r + u

1- Explain why there is no loss of generality in considering the menus of contracts

(r1, P1), (r2, P2) where Pi is the probability of obtaining a loan and ri is the repayment to

the bank when the investment succeeds if the borrower announces that he is of type i.2- Write the maximization program of the bank which chooses the menu {(r1, P1); (r2, P2)}to maximize its expected profit under the borrowers participation and incentive con-

straints (for simplicity assume that if a borrower applies for a loan he loses his outside

opportunity u).

3- Show that the optimal contract entails a non-random allocation of loans (i.e., Pi is

either 0 or 1, i = 1, 2). Characterize the optimal contract. Discuss its properties.

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4 EXERCISES

Bundling with Asymmetric Information

We consider a continuum of consumers who have the following independent willing-

nesses to pay for goods 1 and 2 and desire only one unit of each good. For each good, a

consumer has an equal probability of having valuations , or + (with > ).

The two goods are sold by a monopolist who has a zero marginal cost for each of them.

1- Determine the optimal pricing policy for each good and the associated revenue. In this

exercise consider only deterministic pricing strategies.

2- Suppose that the monopolist can offer only a bundle of the two goods at a price PB.

Determine the optimal PB and show conditions under which it raises more revenue than

the optimal single good prices.

3- Show that there exist prices for the bundle of the goods which improve revenue even in

the presence of the optimal single good prices. (Hint: Draw the table of the surpluses that

the different types of consumers derive from the optimal single prices with the associated

profits of the monopoly. Exhibit a price for the bundle which attracts some consumers

and makes more revenue from these consumers than the optimal single good prices).


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5

Incentives and aid

We consider the problem of the North willing to aid the poor in the South.

The utility function of a representative agent in the North is

VN = qN + nPv(qP), v > 0, v

< 0

where qN is the consumption, nP is the number of poor in the South and qP is the per

capita consumption of the poor in the South. Each agent of the North has an endowment

ofyN. The consumption of the poor exerts a positive externality on the rich in the North

who are nN in number.

The representative rich of the South has the utility function

U = qR + nPv(qP)

where qR is his consumption and in {, } a parameter describing how altruistic the richin the South are. There are nR rich in the South, each one with an endowment of yR.

The preference parameter is private information of the rich in the South. Given

that the North must use the rich in the South as an intermediary (because they control

the government) to help the poor we want to study how the incomplete information on

affects the optimal level of aid chosen by the North.

1- Suppose first that the South lives in autarky and that the poor of the South are only

helped by the rich in the South. We assume that the poor have no endowment.

Determine the optimal level of aid qAP when it is determined by a representative rich

of the South. The corresponding level of utility obtained by the rich will be called the

status quo utility level UA(). Special case : v() = log().2- Suppose first that the North knows and brings to the South a level of aid nRa

to increase the incentives of the rich in the South to help the poor. Show that if a

is unconditional it does not affect the level of aid. Determine the optimal level of aidwhen a can be made conditional on the consumption level of the poor qP. Special case :

v() = log()3- Assume now that the North does not know and let = P r( = ). Determine the

optimal menu of contracts (a, qP); (a, qP), specifying aid conditional on the consumption

of the poor, which maximizes the expected utility of the North under the incentive and

participation constraints of the South. Discuss. Special case : v() = log()


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6 EXERCISES

Downsizing a Public Firm

We consider a public firm which is producing a public good with a continuum of

workers of mass 1. Each worker produces one unit of public good. A mass q in [0, 1] of

workers produces an output q which has social value

S(q), with S > 0 and S < 0.

New outside opportunities appear for workers, calling for a downsizing of the public

firm. Let i the outside utility level that worker i can obtain. i can take one of two

positive values {, } with = > 0. These outside opportunities are identicallyand independently distributed between workers on

{,

}with = Pr(i = ) and 1

=

Pr(i = ).

A new allocation of labor is characterized by the proportions p (resp. p) of workers of

type (resp. ) who remain in the public firm. Assuming that workers are risk-neutral,

a downsizing mechanism can be viewed as a pair of contracts {(p,t);(p, t)} specifying foreach announcement or , a probability to remain in the public firm and a payment

(unconditional on remaining or not in the public firm).

Total production of the public firm is then q = p + (1 )p. Let us first assume thatthe values of outside opportunities are public knowledge. In other words the government

is under complete information. The workers accept to play the downsizing mechanismif the government pays them, t for type , t for type and satisfies their participation

constraints

t + (1 p) , or t p,t + (1 p) , or t p.

If there is a cost 1 + of public funds, social welfare is equal to:

S(p + (1 )p) Social value of the public good(1 + )(t + (1 )t) Social cost of transfers+(t + (1 p)) Welfare of workers.+(1 )(t + (1 p))

If > 0, the participation constraints are binding and social welfare can be reduced

to

S(p + (1 )p) (1 + )(p + (1 )p) + + (1 ).


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The social optimum is then characterized by:

if S() > (1 + ),

S(+ (1 )p) = (1 + ) and p = 1,

if (1 + ) > S() > (1 + ),

p = 1 and p = 0,

if S() < (1 + ),

S(p) = (1 + ) and p = 0.

From now on, we suppose that the values of the outside opportunities are observed by

the workers, and not by the government.

A downsizing mechanism {(p,t);(p, t)} is incentive compatible if and only if

t + (1 p) t + (1 p)t + (1 p) t + (1 p).

It satisfies the participation constraint if

t + (1 p) t + (1 p) .

1- Discuss these constraints and show that one condition is redundant with the other

three conditions.

2- We will say that an allocation of labor (p, p) can be implemented if there exists a

pair of transfers (t, t) such that the downsizing mechanism {(p,t);(p, t)} satisfies all theincentive compatibility and participation constraints.

a) Show that (p, p) can be implemented if and only if p p.b) For any implementable (p, p), what is the pair of transfers (t, t) that minimizes the

social cost of transfers?

3- Determine the allocation of labor and the transfers that maximize expected socialwelfare. Distinguish three cases according to the value of S().

4- Is it possible to structure the payments in such a way that each type does not regret

to have participated in the mechanism? (Hint: differentiate the payments between those

who stay and those who leave)

5- Suppose that workers differ by the quality of their production in the public firm. A

type worker produces units of public good while a type worker still produces one


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8 EXERCISES

unit. Characterize the allocation of labor which maximizes expected social welfare when

> .


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Labor Contracts

A firm designs a labor contract for a worker. The utility function of the worker is,

UA = u(c) l, where c is consumption and l is labor supply, in {, } is the preferenceparameter known to the worker and u() in an increasing concave function; < . Theproportion of workers with low disutility of effort, = , is . The agents optimal choice

must satisfy the budget restriction c t, where t is the payment he receives from thefirm. The firm has the following utility function: UP = f(l) t, where f(l) is a decreasingreturns to scale production function.

1- Assume the firm knows . Characterize the solution to the firms problem which

maximizes its utility function under the participation constraint of the worker. Call this

solution the First-Best solution.

2- Now assume that labor supply is observed by the employer (and is verifiable), but

neither U nor are observed by him. Show that the First-Best solution is not imple-

mentable. The firm can now offer menus of contracts: (t, l) and (t, l), where (t, l) is the

contract chosen by the worker with disutility of effort and (t, l) is the contract for l.

Find the optimal contract. Compare this (Second-Best) solution to the First-Best.

3- Now suppose that the worker with low disutility of effort has an outside opportunity

that gives him a utility level ofV. Characterize the first-best and second-best (asymmetric

information) solutions in this case. Take into account that the solution will depend on thesize of V. For = , consider the cases in which lSB V (SB = second-best),lSB V t ( = first-best), l V l and V l. Which are thebinding constraints in each case and what type of distortion is needed?


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10 EXERCISES

Control of a Self-Managed Firm

Consider a firm that has a production function

y = l1/2,

where l is the number of workers (considered to be a continuous variable: l in R+) and

> 0 is a productivity parameter known by the firm. There are fixed costs of production

A. Let p be the price of the good produced competitively by the firm.

1- Assume that the firm is managed by the workers and that its objective function is

USM =py A

l.

Determine the optimal size (for workers) of the self-managed firm.

2- Let w be the competitive wage rate in the rest of the economy, so that w is the

opportunity cost of labor in this economy. What is the optimal allocation of labor? What

happens ifw is too large? Why is the size of the self-managed firm not optimal in general?

3- Assume that the government knows . Consider the case in which w is small enough

to justify the presence of a self-managed firm. Compute the per unit tax on the good

produced by the firm that restores the optimal allocation of labor. Show that we could

also use a lump-sum tax T on the revenue of the self-managed firm to achieve the same

objective. (Assume that the firm is of negligible size with respect to the rest of theeconomy).

4- Suppose now that the government does not know which can take one of two values

or with = > 0. It uses a regulatory mechanism (l(), t()) which associatesa labor input l() and a transfer t() to the firms announcement . The firms objective

is now

USM =p(l())1/2 + t() A

l().

Characterize the set of regulatory mechanisms which induce truthful revelation of .Derive the implementability condition on .

5- Assume that = Pr( = ). Suppose that the government wishes to maximize the

expectation of

UG = pl1/2 wl.

Show that the solution of this problem does not satisfy the implementability condition;

so that the first-best is not implementable even if transfers are costless to the government.


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What is then the optimal regulatory mechanism, when the labor managed firm has a zero

outside opportunity level of utility?


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12 EXERCISES

Information and Incentives

An agent (natural monopoly) produces a quantity q of a good at a variable cost q

with in {, }, = . The principal gets an utility S(q)(S > 0, S < 0) from thisproduction and gives a transfer t to the agent.

The principals utility function is V = S(q) t and the agents utility function isU = t q. Furthermore, the agents status quo utility payoff is normalized at 0.1- Characterize the optimal contract of the principal under complete information about

.

2- is now private information of the agent and = Pr( = ). Characterize the

optimal contract of the principal under incomplete information with interim participationconstraints of the agent (suppose here and later that the value of the project is large

enough so that the principal always wants to obtain a positive level of production).

3- Suppose now that the principal has access to the information technology which allows

him to receive a signal in {, } with

= Pr( = | = ) = Pr( = | = ) 12

.

Determine the updated principals beliefs about the efficiency of the agent, that is

= Pr( = | = ); = Pr( = | = ).

Characterize the optimal contract for each .

4- Show that an increase of corresponds to an improvement of information in the sense

of Blackwell.

5- Show that an increase of has two effects on the principals expected utility, the

classical effect (that we will call the Blackwell effect) and an effect due to the direct

impact of on the expected utility of the principal conditional on .

6- Show that nevertheless the expected utility of the principal increases with .


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The Bribing Game

We consider an administration which is supposed to deliver with some fixed delay a

service to the citizens (passport, permits,...). With the normal functioning of the admin-

istration, citizens derive a benefit u0 which depends on their valuation of time.

With some additional effort the official can deliver the service with a shorter delay.

Let us call q the decrease of delay that the official can provide at a cost (qQ)2

2for him

where Q is a constant.

We assume that there is a proportion (resp. 1 ) of type 1 (resp. type 2) citizenswho derive a benefit from a decrease q of delay equal to q(q). Citizens are willing to

bribe the official to decrease of delays.

Characterize the optimal bribing contract that the official will offer to the citizens.


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14 EXERCISES

Regulation of Pollution

We consider a firm which has a revenue R, but creates a level of pollution x from

its activities. The damage created by the level of pollution x is D(x) with D(x) >

0, D(x) 0. The production cost of the firm is C(x, ), with Cx < 0, Cxx > 0 and isa parameter known only to the firm. can take two values {, } and = Pr( = ) iscommon knowledge.

1- The level of pollution x() corresponding to the complete information optimum is

characterized by

D(x) + Cx(x, ) = 0.

Show that, if the regulator is not obliged to satisfy a participation constraint of the firm,

he can implement x() by asking a transfer equal to the cost of damage up to a constant.

2- Suppose now that the firm can refuse to participate (but in this case has a zero utility

level) and assume that the regulator has (up to a constant) the following objective function

W = D(x) (1 + )t + t C(x, ) (1)

where t is the transfer from the regulator to the firm and 1 + is the opportunity cost of

social funds. When the regulator must satisfy the firms participation constraint

t C(x, ) 0 for all ,characterize the decision rule x(), in {, }, which maximizes W under complete infor-mation. Compare with question 1.

3- We assume in addition that C < 0 and Cx < 0. Determine the menu of contracts

(t, x), (t, x) which maximizes the expectation of (1) under participation and incentive

constraints of the firm.

4- Optional. Same problem when is distributed according to the distribution F() with

density f() on the interval [, ] with

d

d

1 F()

f()

< 0, Cxx 0 and Cx 0.


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Taxation of a Monopoly

We consider a monopoly facing a continuum [0, 1] of consumers. Each consumer is

characterized by his utility function, log q + x, where x is his consumption of good 1(chosen as the numeraire) and q is his consumption of good 2 produced by the monopoly.

The parameter can take two values or with = 1 and let be the commonknowledge proportion of type consumers.

Consumers have large resources in good 1, x, so that their behavior is always char-

acterized by the first-order conditions of their optimization programs.

The monopoly has a variable cost function C(q) = cq and must incur a fixed cost K.

1- Explain why the profile of consumption q

(), in {, } which corresponds to aninterior Pareto optimal allocation is the solution of:

max [ log q() + x()] + (1 ) log q() + x() ,subject to

x() + (1 )x() = x c q() + (1 )q() K.Determine q().

2- Let {(q, t);(q, t)} the direct revelation mechanism which elicits the parameters fromconsumers. Characterize the direct revelation mechanisms which are truthful.

3- Write the optimization program of the monopoly when he is constrained to provide a

non negative utility level to each consumer

log q() t() 0 for in {, }.

Characterize the truthful direct revelation mechanism which is optimal for the monopoly.

4- The government uses now a linear tax on the consumption of good 2 to control the

monopoly. Assuming that the government maximizes a weighted average of consumers

utility functions (with a weight 1), of the monopolys profits (with a weight larger than1) and of taxes (with a weight > 0 such that ), show that the optimal tax isnegative.

5- Optional. Questions 2 and 3 when is distributed according to the distribution F()

with positive density f() on the interval [, ] with dd

1F()f()

< 0. Consider the special

case of a uniform distribution on [2, 3] and c = 1 and obtain the associated nonlinear

price.


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16 EXERCISES

Shared Information Goods, Majority Voting and Op-timal Pricing

We consider a group of three consumers who share or not an information good sold

by a monopolist who has a zero marginal cost.

Let iq q22 t the utility function of consumer i, i = 1, 2, 3 where q is the quantity of thegood and t the payment. The i are independently drawn from the uniform distribution

on [1, 2] and private information of the consumers.

1- Assuming first that the monopolist can prevent the consumers to share the good, char-

acterize the optimal pricing policy of the monopolist (under the simplifying assumption

that almost all consumers must have a non-zero consumption level).

2- Suppose now that consumers share the good and the payment. Assuming that demand

is determined by the median consumer, characterize the optimal pricing policy of the

monopolist (hint 1: assume that, for the optimal pricing, consumers have single peaked

preferences and check it ex post), (hint 2: the distribution function G() of the median ofthree independent draws from the distribution F() is: G(x) = [F(x)]2[3 2F(x)]).3- Show that the expected profit of the monopolist is higher when consumers share the

good.


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17

Labor Contract with Adverse Selection

We consider a principal-agent relationship in which the principal is an employer and

the agent is a worker.

For a production level y the worker of type ( in {, }) suffers a disutility of workingequal to

(y).

In other words, a worker of type must work units of labor (with = y) which

create a disutility of labor (), () > 0, () 0.If he receives a compensation t from the employer his net utility is U = t (y).

The employers utility function is V = y t.1- Apply the Revelation Principle and characterize the truthful direct revelation mecha-

nisms.

2- Assume now that (resp. (1 )) equals the probability that the worker is of type (resp. ). Characterize the optimal contract of an employer who maximizes his expected

utility under the incentive and interim participation constraint of the worker.


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18 EXERCISES

II- MORAL HAZARD

Lending with moral hazard

We consider a cashless entrepreneur who wants to borrow and carry out the following

project. With an investment normalized to 1 unit he will get an output of z with prob-

ability P > 0 if he exerts an effort level of e and with probability P > 0 (P > P) if he

exerts no effort, and nothing otherwise. Let the cost of effort e for the entrepreneur.

Furthermore his status quo utility level is normalized to 0 and Pz < r.

A monopolistic bank with cost of fund r offers a loan of 1 unit for a reimbursement of

z x when the project is successful, where x is the share of production retained by theagent.

Determine the optimal loan contract of a bank which maximizes its expected profit

under the incentive and participation constraints of the entrepreneur.


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Moral Hazard and Monitoring

An entrepreneur who has no cash and no assets wants to finance a project which

costs I > 0. The project yields R with probability p and 0 with probability 1 p. Thisprobability of success depends upon the effort e in {eH, eL} of the entrepreneur: it is pHif effort is high (eH) and pL if e = eL; 1 > pH > pL = 0. A loan contract specifies a given

payment P if the income is R and 0 if the income is 0.

The entrepreneur enjoys a private benefit B > 0 if effort is low and 0 if effort is high.

There is a competitive loan market and the economys rate of interest is equal to 0.

1- Show that the project is financed if and only if

pHR B + I (1)

2- Suppose that (1) is not satisfied but pHR > I. Introduce a monitoring technology.

By spending m > 0, the lender can catch the entrepreneur if the effort is low and reverse

the decision to obtain a high level of effort; in this case the entrepreneur is punished and

receives 0. The lender and the entrepreneur choose simultaneously whether to monitor

and whether to select a high effort level. The expected payoff matrix for this game is

thus:

e eMonitor pHP m, pH(R P)) (pHR m, 0)Not monitor (pHP, pH(R P)) (0, B)

Assume m < pHR < B .

a. Show that the only equilibrium for P < R is in mixed strategies. Find the

equilibrium strategies.

b. Argue that the project is financed if and only if:

pHR m + I.


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20 EXERCISES

Inducing Information Learning

We consider a principal-agent problem in which the risk-neutral principal wants to

delegate to a cashless risk-neutral agent protected by limited liability, the acquisition of

soft information about the quality of a risky project as well as the decision to engage or

not in the risky project.

There is a safe project which yields 0 to the principal with probability 1. There is also

a risky project. In the absence of information, the risky project yields S with probability

and S with probability 1 . We will assume that S+ (1 )S = 0.By incurring an effort with cost , the agent can learn a signal {, } on the

future realization of the risky project.

We will assume that Pr(|S) = Pr(|S) = , with ] 12

, 1] being interpreted as the

precision of the signal.

1- As a benchmark, suppose that the principal uses the technology for information gath-

ering himself. Show that the project is done only when is observed. Write the condition

under which the learning of information is optimal.

2- Suppose now that the agent decides to adopt or not the risky project. The principal

uses a contract (t , t ,t0) to incentivize the agent. t (resp. t) is the transfer received by the

agent if he chooses the risky project and S (resp. S) realizes. t0 is the transfer he receives

if he chooses the safe project. Write the incentive constraints needed to have the risky

project being chosen if and only if is observed.

3- Write the incentive constraint needed to induce the agent to learn information.

4- Find the optimal contract offered to the agent and determine the t, t0 and t which

induce information learning.

5- Find the second-best rule followed by the principal.


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Optimal Contract and Limited Liability

We consider a risk-neutral principal who delegates a task to a risk-neutral agent

protected by limited liability. His effort e is a continuous variable which costs him

(e) ( > 0, > 0). The return to the principal q follows the distribution (F(|e)with density f(|e) on [0, q] such that the MLRP property

q

fe(q|e)f(q|e)

> 0

holds. The principal benefits from q t(q) where t(q) is the transfers he makes to theagent.

1- Characterize the first-best effort.2- Write the agents incentive and participation constraints when e is non observable by

the principal. Use the first-order approach.

3- Write the Lagrangian of the principals problem and optimize when the transfer t(q)

belongs to [0, q]. Show that the optimal contract involves a cut-off q such that t(q) = 0

for q < q and t(q) = q for q > q.


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22 EXERCISES

The Value of Information under Moral Hazard

We consider the simple model of contracting with limited liability of Section 5.1.2

except that the probability of success writes as + e where is a random variable with

zero mean.

1- Suppose that the agent chooses his effort before knowing the realization of. Compute

the second-best optimal effort eSB .

2- Suppose that the agent wants to guarantee a probability of success R, but can fine

tune the choice of his effort as a function of the realized state of nature that he observes.

Show that > 0 and > 0 imply that the second best optimal effort RSB < eSB .

Conclude.


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Raising Liability Rule

We consider a lender-borrower relationship under moral hazard. The risk-neutral

borrower wants to borrow I from a lender to finance a project with safe return V. The

project may with probability 1 e harm a third-party. The amount of safety care e costs(e) to the borrower with ( > 0, > 0, > 0). The harm has value h. A financial

contract is a pair (t, t) where t (resp. t) is the borrowers reimbursement to the bank if

there is no (resp. one) environmental damage.

1- Suppose that e is observable. Compute the first-best level of safety care and assume

that the project is socially valuable when the interest rate is r.

2- Suppose now that e is not observable. We suppose that the bank is competitive and

that the borrower has sufficient liability. Show that the first-best is still implementable if

the bank must reimburse h to the third-party in case of an accident.

3- Suppose that the bank must reimburse c < h to the third-party. We denote by w the

initial assets of the borrower. Show that as w diminishes, the first-best level of effort can

no longer be implemented.

4- Compute the second-best optimal level of effort maximizing the borrowers expected

payoff subject to the banks zero profit constraint, the borrowers incentive constraint and

his limited liability constraint.

5- Show that raising the banks liability c leads to a lower expected welfare.

6- Show that this result no longer holds when the bank is a monopoly.


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24 EXERCISES

Risk-Averse Principal and Moral Hazard

Suppose that a risk-averse principal delegates a task to a risk-neutral agent. With

probability e (resp. 1 e) the outcome is q (resp. q < q). The risk-averse principal utilityis v(q t) where t is the agents transfer and v() is a CARA von Neumann-Morgensternutility function. Effort costs (e) to the agent ( > 0, > 0).

1- Suppose that e is not observable, compute the optimal contract with a risk-neutral

agent.

2- Suppose that the agent is protected by limited liability. Compute the second-best level

of effort.

3- Analyze the two limiting cases where the principal is infinitely risk-averse and wherehe is risk-neutral. Explain your findings.


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Poverty, Health Care and Moral Hazard

We consider an economy composed of a continuum [0, 1] of identical agents. The

income of an healthy agent is w. Each agent becomes (independently) sick with probability

0, in which case he has only a survival income of w. Let the proportion of sick agents

who benefit from a medical treatment which costs m per capita: then, these agents recover

their normal income w. Finally, the utility function of an healthy agent or of a sick agent

who has received a treatment is u() with u > 0, u < 0. The utility function of a sickagent who has not been treated is uM() with uM > 0, uM < 0.1- Let p the insurance premium that only healthy and treated sick agents can pay. We

have potentially three types of agents:

a proportion 1 0 of healthy agents with utility level u(w p),

a proportion 0 of treated sick agents with utility level u(w p),

a proportion (1 )0 of non treated sick agents with utility level uM(w).

Suppose income redistribution and price discrimination of insurance are not possible.

Write thee optimization program of a utilitarian social welfare maximizer who must satisfy

budget balance of the health sector. Write and interpret the first order conditions of this

program with respect to and p (assume an interior solution). Determine the comparativestatics of the optimal solution with respect to m, w and 0. Let W(0, p(0), (0))

denote optimal social welfare.

2- Suppose now that with a non observable non monetary cost of health care any agent

can decrease his probability of becoming sick from 0 to 1, with = 0 1.Determine the various regimes obtained when the expected social welfare is maximized

under the budget constraint and the moral hazard constraint that agents find valuable to

spend in order to decrease their probability of being sick.

3- We consider now pairs of agents who observe each others effort of health care, and we

assume that agents can perfectly coordinate their health care behavior. Assuming now

that everybody is treated, let t11 the insurance premium of an healthy agent paired with

a healthy agent, t12 (resp. t21) the one paid by a healthy (resp. sick) agent paired with a

sick (resp. healthy) agent and t22 the one paid by a sick agent paired with a sick agent.

Maximize the expected social welfare of a pair of agents under the incentive constraints

that the pair prefers exerting two efforts of health care rather than zero or one and under


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26 EXERCISES

an expected budget balance equation (guess that the local incentive constraint is binding

and that the other incentive constraint is satisfied for u(x) =

2x). Show that the

solution obtained dominates the solution with individual contracts.


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Group Lending with Moral Hazard

We consider two entrepreneurs each of whom can carry out a project with the following

characteristics. Investing 1 generates a stochastic output which can take two values, z > 0

or 0. The probability of success (i.e., of getting z) depends in the entrepreneurs effort, e,

which can take also two values e > 0 or 0. The probability of success is p for a high level

of effort e and p for no effort with p > p > 0. The disutility of effort is for e and zero

for e = 0.

Each entrepreneur has no wealth and must borrow to invest. He can only repay his

loan if he succeeds. Denoting by x the entrepreneurs share of output, his expected utility

is

px if he exerts effort e,

px if he exerts no effort.

Funds are supplied by a profit-maximizing bank which has a cost of funds r. We

assume that:

pz > r > pz.

1- Determine the optimal contract offer to an entrepreneur when there is a single en-

trepreneur.

2- There are now two entrepreneurs who do not observe each others effort level. A

group lending contract calls for a payment x when the partner succeeds and y when it

fails. Consider a group lending contract which induces effort of both entrepreneur as a

Nash equilibrium. Show that a group lending contract does not perform better than the

individual contracts considered in question 1.

3- We suppose now that entrepreneur observe each others effort level and coordinate

their effort levels. Consider the program of the bank which implements effort by both

entrepreneurs with group lending contracts:

max p2(2z 2x) + p(1 p)(2z 2y) 2r

s.t.

2p2x + 2p(1 p)y 2 2p2x + 2p(1 p)y (1) 2ppx + p(1 p)y + p(1 p)y (2) 0, (3)


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28 EXERCISES

x 0, y 0.

Find the optimal contract. Show that it is better than individual contracts for the

bank. Explain why.


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29

Incentives and Discovery

We consider a principal-agent relationship in which a principal delegates to an agent

the search for a resource of unknown magnitude over which the principal has property

rights.

The principals utility function is u(q) + t with u > 0, u < 0, u(0) = 0 and where q

is the quantity of the resource obtained by the principal and t is the monetary payment

made by the agent to the principal for being allowed to search.

The agents utility function is u( q) t.1- Under complete information about , determine the optimal contract offered to the

agent by a principal who maximizes his utility under the participation constraint of theagent

u( q) t 0.

2- can take one of two values {, } with respective probabilities (1 , ), with = > 0 and is now private information of the agent. Suppose that the prin-cipal offers a contract to the agent before the agents search, i.e., before the agent learns

. Characterize the optimal contract that the principal offers to the agent when he maxi-

mizes his expected utility under the agents incentive constraints and the agents ex ante

participation constraint.3- We assume now that the contract is imperfectly enforced. When the agent discovers

(resp. ), he can exit the relationship with a quantity (resp. 0) of the resource.

Therefore, the principal is faced with the following ex post participation constraints:

u( q) t 0 (1)u( q) t u(), (2)

if he wants to maintain both types of agent in the relationship.

Show that the regime with the participation constraint of the inefficient type , and

the incentive constraint of the efficient type cannot be optimal (hint: assume the

contrary).

4- Consider the regime where only the participation constraints (1) and (2) are binding

(i.e. the incentive constraints are slack). Show that this is the relevant regime when

> . Characterize the optimal contract and denote W() the principals expected

welfare.


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30 EXERCISES

5- We assume now that by exerting an effort which costs him the agent increases the

probability of a -discovery from 0 to 1 > 0 with = 1 0. Show that W() isdecreasing in . Write the program of a principal who selects a contract which discourages

effort. Solve it when u() <

. Discussion.


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31

III- MIXED MODELS

Political Economy of Regulation

We consider a firm which realizes two projects, of gross value S1 and S2 for the

consumers. The firm can provide an effort ei in order to reduce the cost associated with

project i, i = 1, 2. The cost function of the firm for project i is:

Ci = ei

where is the efficiency parameter of the firm. The efficiency of the firm is the same for

both projects.

Parameter can take values in two values {, } with = Pr( = ).The cost reducing efforts create a disutility to the firm equal to

(e1, e2) =1

2(e21 + e

22) + e1e2, > 0.

A regulator reimburses the observable costs C1 and C2 and pays a net transfer t to

the firm which has utility

U = t (e1, e2).

Social welfare is

S1 + S2 (1 + )(t + C1 + C2) + U.

1- Determine the optimal regulation under complete information.

2- Determine the optimal regulation under incomplete information when is private

information of the firm.

3- We assume now that the regulatory mechanism is determined by the political majority.

Agents are of two types. Either they are stakeholders in the regulated firm, i.e., they share

the rent of the regulated firm. Or they are non-stakeholders and do not share the rent.

Let the proportion of stakeholders. If > 1/2 the majority belongs to stakeholders

who choose regulation to maximize their objective function:

(S1 + S2 (1 + )(t1 + C1 + t2 + C2)) + U.


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32 EXERCISES

If < 1/2 the majority belongs to non-stakeholders who choose regulation to maximize

instead:

(1 )(S1 + S2 (1 + )(t1 + C1 + t2 + C2)).

Determine in each case the optimal regulation under incomplete information.


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33

Regulation of Quality

We consider a natural monopoly which has the cost function C = (+ s

e)q where q

is the production level, e is the effort level of the manager, s is the quality of the product

and in {, } is a cost parameter.We assume that the regulator observes only the cost C and the firms revenue, and, in

addition, pays a transfer t net of cost and revenue. If (e) (with > 0, > 0, 0)is the disutility of effort for the manager, it implies that the net utility of the manager is

U = t (e). His outside opportunity utility level is normalized at zero.We assume that the consumers get a gross surplus from the consumption of q units

equal to

S(q,s,) = (A + ks h)q B2

q2 (ks h)22

where A, B, h, k are positive constants and in {, } is a demand parameter known bythe firm, but not by the regulator.

Let p(q) the inverse demand function. The utility derived by consumers is then:

V = S(q,s,) p(q)q (1 + )(t + Cp(q)q)

where 1 + is the opportunity cost of public funds.

1- We consider a utilitarian regulator who wants to maximize expected social welfare(V + U). Show that the adverse selection problem with two parameters and can be

reduced to a one dimensional adverse selection problem with the parameter = + hk

.

2- Assume that is distributed according to a uniform distribution. Write the maxi-

mization problem of the regulator under the incentive and participation constraints of the

firm.

3- Show that the optimal regulation can be implemented with indirect mecanisms which

are functions of a variable which aggregates the cost and quality dimensions.

4- Study the dependence of the optimal regulatory mechanism with respect to the concern

for quality.


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34 EXERCISES

Enforcement and Regulation

We consider a natural monopoly which, in addition to a fixed cost F which is common

knowledge, has a variable cost function

C = ( e)q,

where q is the production level, is an adverse selection parameter in {, } with =Pr( = ) and e is a moral hazard variable which decreases cost, but creates to the

manager a disutility (e) with > 0, > 0, 0.Consumers derive an utility S(q), S > 0, S < 0 from the consumption of the natural

monopolys good. Let p(

) the inverse demand function and t the transfer to the firm

from the regulator. The firms net utility is:

U = t + p(q)q ( e)q F (e).

We assume that cost is ex post observable by the regulator as well as the price and the

quantity. So, we can make the accounting assumption that revenues and cost are incurred

by the regulator, who pays a net transfer t = t + p(q)q ( e)q F. Accordingly, theparticipation constraint of the firm can be written:

U = t

(e)

0.

To finance the transfer t, the government must raise taxes with a cost of public funds

1 + , > 0. Hence, consumers net utility is

V = S(q) p(q)q (1 + )t.

Utilitarian social welfare writes then:

W = U + V = S(q) + p(q)q (1 + )(( e)q + F + (e)) U.

1- Under complete information, the regulator maximizes social welfare under the firms

participation constraint. Characterize the optimal solution.

2- Suppose now that the regulator cannot observe the effort level e and does not know

. However, he can offer a contract to the firm before the latter discovers its type (see

Figure 1 for the timing).


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35

E TimeThe regulator

offers theregulatory contract.

The firmaccepts or not

the contract.

The firmdiscoversits type.

Productionand transfertake place.

Figure 1

Explain why the regulator can restrict his contract to a pair {(t, c); (t, c)} (where c = Cqis average cost) which satisfy incentive constraints.

Write the incentive constraints and the firms ex ante participation constraint. Solve

for the optimal contracts.

3- We assume now that if the firm has a negative ex post utility (as firm in question

3) it attempts to renegotiate its regulatory contract. However, with a probability (c),the regulator is able to impose the implementation of the agreed upon contract. This

probability depends on the expenses c incurred to set up an enforcement mechanism.

We assume that (0) = 0, > 0, < 0 with the Inada conditions (0) = andlimc+ (c) = 1.

With probability 1 (c) the regulator is forced to accept a renegotiation. To modelthis renegotiation we use the Nash bargaining solution but assume that renegotiation is

costly (because it takes time say). The status quo payoffs which obtain if the negotiation

fails are determined as follows: The firm loses its fixed cost F. The regulator is also

penalized by a loss of reputation and obtains the utility level H.Assume that it is only the inefficient type which wants to renegotiate. Therefore,

costly bargaining takes place under complete information. Its outcome solves:

max{q,e,UE}

(UE + F)(W(q, e, ) UE+ H)

where in (0, 1) models the cost of renegotiation and

W(q, e, ) = S(q) + p(q)q (1 + )(( e)q + F + (e)).

Compute the outcome of renegotiation (qE, eE, UE).

4- Write the firms new participation constraint which takes into account that with prob-

ability 1 (c) there will be renegotiation.Substitute the outcome of renegotiation into the regulators objective function and

solve for the optimal contract and the optimal level of enforcement expenses c. Discuss.


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36 EXERCISES

Regulation of a Risk Averse Firm

We consider a utilitarian regulator who wishes to realize a public project which has

social value S. A single firm can undertake the project for a cost, C = e, where in{, } is an efficiency parameter and e is a level of effort which creates a disutility (e)( > 0, > 0, 0) for the firms manager.

The cost C is observed by the regulator who can give a transfer t to the firm with a

cost of public funds 1 + .

The manager of the firm is risk averse, and has the utility function u(t (e)) withu > 0, u < 0).

e and are not observable by the regulator, but it is common knowledge that =Pr( = ).

1- For the revelation mechanism {t() = t, C() = C; t() = t, C() = C}, write theincentive and participation constraints of the firm.

2- Expected social welfare is defined as

W = S (1 + ) (t + C) + (1 )(t + C) + u1 (u() + (1 )u()) ,with = t ( C) and = t ( C).

Interpret this social welfare function and, assuming that it is concave in (, ), de-

termine the optimal regulation under incomplete information when the regulator offers a

contract at the interim stage.

3- Compare the result of question 2 with the case where the firms manager is risk neutral.

4- Consider the special case

u(x) =1

(1 ex).

Show that the effort level required from type is decreasing in .


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37

Technological versus Informational Advantage

We consider a project which has value S for consumers. This project can be realized

by two different firms run by two different managers.

Firm 1 has the cost function C1 = 1 e1 where 1 is an efficiency parameter whichis common knowledge and e1 is the managers effort level which creates a disutility

14 e

21

for him. Ex ante 1 is unknown for everybody and drawn from {2, 3} with Pr(1 = 2) =Pr(1 = 3) = 1/2.

Firm 2 has the cost function C2 = k2 e2, k 1, where 2 is also drawn ex ante(independently of 1) from {2, 3} with Pr(2 = 2) = Pr(2 = 3) = 1/2. However, thevalue of 2 is only observed by the manager of firm 2; e2 is the managers effort level

which creates a disutility 14 e22 for him.

C1 and C2 are observed by the regulator. The outside opportunity utility levels are

normalized to zero for both firms. The timing of events is summarized below:

E tex ante interim

1 observedby all, 2

observed by

firm 2

Regulator

offerscontracts

Firmsaccept

orreject

Contractsare

executed

The regulator is utilitarian and can use transfers with a price of public funds 1 + ,

> 0.

1- Assuming that the regulator offers a contract only to firm 1, at the interim stage, what

is the optimal regulation?

2- Assuming that the regulator offers only a contract to firm 2 at the interim stage, what

is the optimal regulation?

3- Assuming that the regulator selects the firm at the ex ante stage, characterize thevalues of and k such that firm 2 is chosen.

4- Assuming that the regulator selects the firm at the interim stage, characterize the

values of , k and 1 such that firm 2 is chosen.

5- Suppose that k can be chosen ex ante at the cost (1+)(3k)2

2 by the regulator; what is

the optimal strategy of the regulator at the ex ante stage?


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38 EXERCISES

Piracy and Optimal Pricing

We consider a firm which can buy from a monopoly with marginal cost c software in

quantity q0 at price p0 or pirate software in quantity qc at a random cost c which includes

the illegal reproducing cost itself and a random fine. The value for the firm of these

purchases is

R(q0 + qc) with [0, 1].

The firm has a utility function with constant absolute risk aversion so that his utility

is

e[R(q0+qc)p0q0cqc].

1- Assuming that c is a normal random variable with mean and variance 2, computethe demand functions q0(p0), qc(p0) of a firm maximizing its expected utility. What is the

optimal monopoly price pM0 ? Study the dependence of pM0 with respect to ,

2,,,c.

Suppose

R(q0 + qc) = a(q0 + qc) 12

b(q0 + qc)2.

2- For a social welfare function W which adds profit to the certainty equivalent of the

consumers utility level, characterize the (constrained) optimal q0 and qc. Study the

comparative statics of W with respect to ,,2, c (Hint: use the envelop theorem).

Let W be social welfare under monopoly. ObtainW . Let = c0 + f where c0 is the

reproducing cost and f an expected fine. Supposing that the cost of implementing the

fine f is 12

f2, determine the optimal fine.

3- What is the optimal two part tariff of the monopolist?

4- Assume that is private information of the firm and can take two values or with

> and = Pr( = ).

We will look for the optimal nonlinear price or the optimal direct revelation mechanism

(t, q0, t, q0). First show that the Spence-Mirrlees property holds for the surrogate utility

function:

V(t0, q0, ) = maxqc

R(q0 + qc) t0 qc 1

22q2c

.

Write the firms incentive constraints for the function V. Solve for the optimal direct

revelation mechanism. Discuss.


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39

Gathering information before signing a contract

We consider a principal agent problem in which the agent produces a quantity q of a

good at a cost q with in {, }, > . Let t be the transfer from the principal to theagent ; then, the agents utility is U = t q.

The principals utility is

V = S(q) t with S > 0, S < 0 and S(0) = 0.

At date 1 the principal offers a menu of contracts (t, q), (t, q).

At date 2 the agent decides to learn at a cost or not. Let e be this decision : e = 1

if he learns, e = 0 if not. e is a moral hazard variable not observed by the principal.

At date 3 he accepts or not the contract.

At date 4 the agent learns if he has decided not to learn it at date 2.

At date 5 the contract is executed.

We will consider two sets of contracts : those (class C1) which induce the agent tochoose e = 0 and those (class C2) which induce the agent to choose e = 1.1- Write the optimization program of a principal who maximizes his expected utility

under the incentive and participation constraints of the agent, either in the class C1 or inthe class C2 of contracts.2- Show that a lower bound for the principal is obtained by constraining contracts to

t q 0 ; t q 0. Derive from this result that the interesting contracts to considerentail t q 0. Show that the principal can always mimic a contract in the class C2with a contract in the class C1.3- Determine the optimal contract in the class C1. (Distinguish three cases depending onwhether the ex ante participation constraint is binding, the moral hazard constraint is

binding or both constraints are binding according to the value of ).


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40 EXERCISES

Better Information Structures and Incentives

We consider a natural monopoly which realizes a public project valued S at a cost

C = e when in {, } is a parameter which is privately known by the manager with = > 0 and = Pr( = ) is the common knowledge probability that the firm isa low cost firm; e is the managers effort which has a disutility (e) with > 0, > 0,

0. The cost C is observable by the regulator and reimbursed to the monopoly.Accordingly the firms utility is

U = t (e)where t is the net monetary transfer from the regulator to the firm and consumers welfare

is

V = S (1 + )(t + e)where > 0 is the social cost of public funds.

Social welfare is defined as U + V.

1- Characterize the regulation which maximizes expected social welfare under the firms

incentive and participation constraints (it is assumed that the regulatory contract is of-

fered to the firm at the interim stage and that its status quo utility level is zero).

2- The regulator benefits ex ante from an information structure J with a set =

{1, 2, . . . , I

}of signals and conditional probabilities Pr(i

|) i = 1, . . . , I . Denote

i the posterior belief that the firm has a low cost after signal i, i.e., i = Pr( = |i),i = 1, . . . , I .

Characterize for each i the optimal effort level ei requested from the high cost firm

and denote ei = Z

i1i

the solution ei as a function of the ratio

i1i

.

Show that, if after each i, the regulator wants to keep both types of firms, and if Z

is concave, the expected power of incentives decreases when the regulator has access to

the information structure J.

Discuss the more general case where after some signals the regulator may want to shutdown the high cost firm.


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41

Competitive Pressure and Incentives

We consider the case of a monopoly (producing good 1 in quantity q1) with a com-

petitive fringe producing a differentiated good 2 in quantity q2. The consumers utilityfunction is

S(q1 + q2) + q1q2

where is a measure of complementarity of the two goods.

The monopolys cost function is

C1 = ( e)q1,where in {, }, > , is an adverse selection parameter (with = Pr( = )) and e

is an effort level which decreases cost with a disutility for the manager of (e),

> 0, > 0, 0.

The fringes cost function is

C(q2) = cq2

where c is common knowledge.

Let p1(q1, q2) the inverse demand function of good 1.

1- Show that for a utilitarian social welfare maximizer the social welfare function can be

written as

S(q1 + q2) + q1q2 + p1(q1, q2)q1 (1 + )((e) + ( e)q1) cq2 U,where U = t (e) is the utility of the monopoly where t is the net transfer (in additionto reimbursement of cost) from the regulator to the firm.

2- Characterize the optimal regulation under asymmetric information when the monopolys

status quo utility level is zero.

3- We say that the two goods are strategic complements (substitutes) if Sq1 + S + >

0(< 0).

Show that if the two goods are strategic substitutes a reduction in marginal cost c

reduces effort for the regulated firm. If the products are strategic complements and is

large enough, then a reduction in c increases effort for the regulated firm.

4- Show that, if the two goods are strategic complements then effort of the regulated firm

decreases with an increase in the degree of substitution ( decreases). If the products are

strong enough substitutes, the effort of the regulated firm increases with the degree of

substitution.


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42 EXERCISES


43/114

SOLUTIONS

I- ADVERSE SELECTION

Lending with Adverse Selection

1- It is a principal-agent problem with adverse selection. The principal is the bank and

the agent is the borrower. The agent has two possible types: Type 1 obtains h for sure for

an investment of 1. Type 2 obtains h with probability in (0, 1) and zero with probability

1 for an investment of 1. The action space is the probability granting a loan and,when a loan is granted, the level of repayment if the project succeeds (if the project does

not succeed the borrower has no revenue and no wealth). So we have

A = {(P, r) : P [0, 1]; r IR+}

where P is the probability of receiving a loan and r is the repayment (in case of success).

From the Revelation Principle (Proposition 2.2), we know that we can restrict the

analysis to truthful direct revelation mechanisms, i.e., pairs {(P1, r1); (P2, r2)} which areincentive compatible :

P1(h r1) P2(h r2) (1)P2(h r2) P1(h r1). (2)

2- The bank maximizes its expected profit under the incentive and participation con-

straints, i.e., solves

max{(P1,r1);(P2,r2)}

P1(r1 r) + (1 )P2(r2 r)

s.t. (1), (2) and

P1(h r1) u (3)P2(h r2) u (4)

43


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44 SOLUTIONS

if it wishes to give a loan with positive probability to both types (case 1). Alternatively,

it might offer a loan only accepted by type 1 (case 2).

3- Let us first consider case 1 where the bank contracts with both types of borrowers.

Dividing (2) by , (1) and (2) imply

P1(h r1) = P2(h r2). (5)

Since is in (0, 1), (4) is binding and not (3). From (4) we have

r2 = h uP2

if P2 = 0.

From (5) we have

r1 = h uP1

.

Substituting these expressions into the banks expected profit we get

P1(h r) + (1 )P2(h r) u

(1 )u.

Since h > r, maximizing with respect to P1 and P2 gives P1 = P2 = 1 and r1 = r2 =

h u

.

Therefore, we obtain a pooling contract with an expected profit for the bank of

h r u

+ (1 )(h r u).

Type 1s ex post information rent is h r1 u = (1) u.Consider now case 2. The bank offers a loan intended only for type 1. It is only

constrained by type 1s participation constraint. The bank will obviously make this

constraint binding, leave no information rent to type 1, and provide a loan with probability

one:

r1 = h uwith an expected profit for the bank (h r u).

Case 2 is better for the bank than case 1 if and only if

(h r u) >

h r u

+ (1 )(h r u)

or

(1 )(h r u) < (1 )

u,

i.e., the expected revenue made with type 2 borrowers is less than the expected rent which

must be given up to type 1s borrowers because of the presence of type 2 borrowers.


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45

Why do we obtain a pooling contract if the principal wants both agents to participate?

The utility function of the agent can be written

U(P,r,) = P (h r).

The Spence-Mirrlees condition writes

U

UPUr

=

U

(h r)

P

= 0

and we obtain pooling because there is no way to screen apart both types. Indifferences

curves of the two types in the (P, r) plan do not cross.

De Meza and Webb (1987), Stiglitz and Weiss (1981), Laffont (2003).


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46 SOLUTIONS

Bundling with Asymmetric Information

1- Since consumers want to consume only one unit and since only deterministic mech-

anisms are considered, there is no way to screen types with either the quantities theyconsume or the probabilities of receiving the good. The seller can only post a price that

a subset of consumers accept to pay.

Given the willingnesses to pay, the three relevant prices that can be posted are , or + , for each good.

If p = + only 1/3 of the consumers buy and revenue is 13 ( + ). Ifp = ,23 of the

consumers buy and revenue is 23

. If p = all consumers buy and revenue is .Since < , the optimal single good price is (for each good):

p = if 3= if > 3

with a total revenue for both goods

R = 43 if 3= 2( ) if > 3.

2- Suppose now that the monopolist can offer a price PB for a bundle made of one unit

of each good. Since the willingnesses to pay for the two goods are independent, we have

six possible types of consumers with the following aggregate willingnesses to pay.

Population Preferences Aggregate willingness to pay

19

{ + , + } 2 + 229

{ + , } 2 + 19

{, } 229

{ + , } 229

{, } 2 19 { , } 2 2

Again, the relevant posted prices for the bundle correspond to the willingnesses to


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47

pay. It is immediate to see that

PB = 2 if 2= 2 if [2, 5]= 2 2 if 5,

with respective revenues worth

43

if 2169 89 if [2, 5]

2 2 if 5.

Therefore, for in [2, 5], PB = 2 provides more revenue than the optimal singlegood prices.

3- For the optimal prices p we can obtain the surplus made by each type of consumer

and the associated revenue of the monopolist:

p = p = Population Preferences Surplus Profit Surplus Profit

19

{ + , + } 2 2 4 2( )29

{ + , } 2 3 2( )19

{, } 0 2 2 2( )29

{ + ,

} 2 2(

)

29

{, } 0 2( )19

{ , } 0 0 0 2( )

A successful bundle price must attract some consumers and make more profit with

them than with the optimal single good prices.

Consider the case where < 3 and p = .

A bundle price of 2 makes positive profit since > and attracts all consumersexcept the ( , ) types. It also makes more profit than the optimal single goodprices when

8

9(2 ) > 4

9

i.e. > 2.

Adams and Yellen (1976).


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48 SOLUTIONS

Incentives and Aid

1- The rich in the South solve

max{qR,q

AP}

qR + nPv(qAP)

subject to

nRqR + nPqAP = nRyR

hence

nRv(qAP) = 1, (1)

with a status quo utility level of

yR nPnR

qAP + nPv(qAP) = U

A().

For v() = log(),

qAP = nR,

UA() = yR + nP(log nR 1).

2- The rich of the South solve

max{qR,q

AP}

qR + nPv(qAP)

nRqR + nqAP = nRyR + nRa

with the same result for the consumption of the poor. Unconditional aid is ineffective in

helping the poor.

The North offers now a contract (qP, a) which specifies the level of aid for a level of

consumption of the poor. It is accepted by the South as long as it gets as much as in the

autarky regime. Hence the program of the North:

max{qN,qR,qPa}

qN + nPv(qP)

subject to

nRqR + nPqP = nRyR + nRa

qR + nPv(qP) UA()


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49

nNqN = nNyN nRaor

(nN + nR)v(qP) = 1. (2)

Comparing (1) and (2) we see that the poor in the South consume more. For v() =log(),

qP = nN + nR instead of nR.

3- To the previous program we must add incentive constraints. For given a and qP, the

richs utility is

yR + a nPnR

qP + nPv(qP).

The incentive constraints are

yR + a nPnR

qP

+ nPv(qP) yR + a nPnR

qP + nPv(qP)

yR + a nPnR

qP + nPv(qP) yR + a nPnR

qP

+ nPv(qP).

Since the status quo utility level UA() depends on the type, there is a potential

problem of countervailing incentives. However we will guess that it is, as usual, the

incentive constraint of the one who wants to lie, , which is binding, as well as the

participation constraint of type .

The problem of the principal becomes

max{a,qP;a,qP}

yN nR

nNa + nPv(qP)

+ (1 )

yN nR

nNa + nPv(qP)

yR + a nPnR

qP + nPv(qP) = yR + a nPnR

qP

+ nPv(qP) (3)

yR + a nPnR

qP

+ nPv(qP) = UA(). (4)

Solving (3) and (4) for a and a, inserting in the objective function and maximizingwith respect to {qP, qP} yields

(nN + nR)v(qP) = 1

(nN + nR)v(q

P) = 1 +

1 nRv(q

P).

Asymmetric information leads to a decrease in the poors consumption when the rich

of the South are of type , i.e., have a low altruistic behavior.


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50 SOLUTIONS

With v() = log(), we get:

qP = nN + nR

qP

= nN + nR 1 nR.

It remains to check that the participation constraint of type is satisfied as well as

type s incentive constraint. Note that the above solutions are in fact only valid as long

as a 0, a 0. In fact when is too large, a = 0 and the poor in the South get thesame utility as under autarky. It happens when

nN 1 nR or

nNnN + nR

.

Azam and Laffont (2002).


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51

Downsizing a Public Firm

1- Let us rewrite the constraints as

U = t p t p = Up (1)U = t p t p = U + p (2)U = t p 0 (3)U = t p 0. (4)

Then, we are back to the familiar formulation and (2) + (3) imply (4). Now, the

good type is the -type.

2- Adding (1) and (2) implies p

p. For any implementable pair (p, p), we have the

following constraints

(p p) t t (p p)and (3) implies

t p.

The government minimizes the expected transfers t + (1)t by choosing t = p and

t = t + (p p) = p + p.

3- Expected social welfare can be rewritten as:

S(p + (1 )p) (1 + )p (1 + )(1 )p p

up to a constant.

If we have

S() > (1 + ) +

1 ,then, p solves

S(+ (1 )p) = (1 + ) + 1 (5)

and p = 1.

If we have instead

(1 + ) +

1 > S() > (1 + )

we get p = 1 and p = 0.


52/114

52 SOLUTIONS

If, finally

S() < (1 + ),

then p solves

S(p) = (1 + )

and p = 0.

As (5) shows, for low levels of downsizing, asymmetric information () increases the

level of downsizing and leaves it unchanged otherwise.

4- Let w the wage if the worker remains in the firm and s his severance payment if he

quits. By definition

t = pw + (1 p)s = p + p (6)t = pw + (1 p)s = p. (7)

A worker does not regret to have participated if his wage in the firm is larger than his

outside opportunity and his severance payment is non-negative.

Taking w = and w = ensures the first point. Then, from (6) and (7)

s = 0 and s = p

1 p > 0.

5- The incentive constraints remain unchanged so that the optimization program is now

W = S(p + (1 )p) (1 + )p (1 + )(1 )p p

W

p= [S (1 + )]

W

p= (1 )

S (1 + )

1

.

As long as > + 1+

1

, the solution is as in question 3 except that (5) must

be replaced byS(+ (1 )p) = (1 + ) +

1 .

When < + 1+

1, the first-order conditions would call for p < p which is

in conflict with the implementability condition p p. Then, we have bunching at theoptimal contract which solves

maxp

S((+ (1 ))p) (1 + )( + (1 ))p p


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53

i.e.,

S((+ (1 ))p) = (1 + )( + (1 )) + + (1 ) .

Jeon and Laffont (1999).


54/114

54 SOLUTIONS

Labor Contracts

1- The firms problem is

max{l,t}

f(l) t subject to u(t) l 0.

So, for = :

u(t) = l

f(l) =

u(t)

for = :

u(t) = l

f(l) =

u(t).

2- The first-best allocation {(t, l); (t, l)} is not implementable. Suppose this menu wereoffered under asymmetric information. Then, the worker will select (t, l). Indeed, his

utility is then

u(t) l = l > 0instead of zero for (t, l).

Under asymmetric information, the optimal contract is the solution of

max{(t,l);(t,l)}

(f(l) t) + (1 )(f(l) t),

subject to

u(t) l u(t) lu(t) l 0,

where we just consider the relevant incentive and participation constraints.

Let () be the inverse function of u(), i.e., t = (u). The program can be rewrittenas:

max{(u,l);(u,l)}

[f(l) (u)] + (1 )[f(l) (u)]

u l u lu l 0.


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55

The two constraints are binding. Therefore u = l and u = l + l. Substituting

these values in the employers objective function and maximizing with respect to ( l, l)

yields

f(lSB) =

u(tSB); u(tSB) = lSB + lSB

f(lSB) =

u(tSB)+

1

u(tSB); u(tSB) = lSB .

Since u(tSB) < u(( lSB))

f(lSB)u(( lSB)) > = f(l)u(( l)),

which implies lSB < l since u < 0 and f < 0.

Since

f(lSB) >

u(t)

f(lSB)u(( lSB)) > = f(l)u(( l)),

we finally get

lSB < l.

3- The firms problem is now:

max{t,l} f(l) t s.t. u(t) l V

yields

u(t) = l + V

f(l) u(t) = .

Under asymmetric information, the employers program is now:

max{(u,l);(u,l)} (f(l) (u)) + (1 )(f(l) (u))

u l max(V, u l)u l 0.

If lSB V, the solution of question 2 is unchanged, since the rent of asymmetricinformation is greater or equal to the outside opportunity level of utility.


56/114

56 SOLUTIONS

If lSB < V l, there is no reason to distort so much the production levelof the inefficient type (the -incentive constraint and both participation constraints are

binding). The firms problem becomes:

max{(u,l);(u,l)}

(f(l) (l + V)) + (1 )(f(l) (l))

V l 0 ().

It admits the solution:

f(l) =

u(t), u(t) = l + l

f(l) =

u(t)+

1 , u(t) = l.

If l < V < l, there is no reason to distort the production level of the inefficient

type and the solution is characterized by (only the participation constraints are binding)

f(l) =

u(t); t = l + V

f(l) =

u(t); t = l.

However, if V becomes greater than l, the -incentive constraint is binding and

the firms problem becomes:

max{(u,l);(u,l)} (f(l) (u)) + (1 )(f(l) (u))

u l u l = V lu l = Vu l 0.

There exists two subcases. In the first one, the -incentive constraint and both par-

ticipation constraints are binding.

max{l,l}

(f(l) (l + V)) + (1 )(f(l) (l + V))

V + l 0 ()

f(l) =

u(t); u(t) = l

f(l) =

u(t)

; u(t) = l + V.


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57

l is increased to reduce the information rent to zero.

Let lSB

be defined by

f(lSB

) =

u(tSB) 1

u(tSB); u(tSB) = l

SB+ l

SB

f(lSB) = u(tSB)

; u(tSB) = lSB .

If V > lSB

, then only the -incentive constraint and the -participation constraint

are binding and the solution is characterized by:

f(l) =

u(t) 1

u(t); u(t) = l + V

f(l) =

u(t); u(t) = l + V l.


58/114

58 SOLUTIONS

Control of Self-Managed Firm

1- The optimal size of the self-managed firm is the solution to

max

p1/2 A

i.e.

LM =

2A

p

2.

2- The optimal allocation of labor is determined by equating the marginal product of

labor to the wage w,1

2p1/2 = w or =

p

2w

2.

If w is small, the self-managed firm, which maximizes added value per capita and not

profit, restricts the size of the firm with respect to the optimal size. If w is larger than

the per capita added value for LM the workers of the labor managed will quit to benefit

from the high wage elsewhere in the economy. From now on we assume that w is small

enough to justify the presence of the labor managed firm (LM firm).

3- The LM firm solves now

max

(p )1/2 A

,

hence

=

2A

(p )2

.

To achieve efficiency we need to equate this term to , i.e.

= p 4Awp2

.

If instead we use a lump sum tax, the LM firm solves

max p

1/2

A T ,hence

=

2(A + T)

p

2.

To achieve efficiency we need

T =p22

4w A.


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59

4- Incentive constraints are

p 1/2 + t A

p 1/2 + t A

(1)

p 1/2 + t A

p 1/2 + t A

. (2)

Adding those two incentive constraints we get

p( )(1/2 1/2) 0 or .

5- The governments program is

max{(,t);(,t)}

(p 1/2 w) + (1 )(p 1/2 w),

subject to (1)-(2) and

p1/2 + t

A

0 (3)p1/2 + t A

0. (4)

The firms objective function is

U(,t,) = p1/2 A1 t1.

U/

U/t

=

1

2p1/2 > 0,

and we could expect the usual constraints to be binding.

However, the benevolent government is only interested in efficiency and would like to

implement

=

p

2w

2; =

p

2w

2i.e. > . This first-best allocation is not implementable.

Therefore, the optimal solution entails bunching and does not exploit the information

of the agent. It is obtained from the program:

max{}

( + (1

))p1/2

w,

or

=

p( + (1 ))

2w

2.

Here, there is a total conflict between the implementability condition and the profile

of allocations that the principal is interested in.

Guesnerie and Laffont (1984).


60/114

60 SOLUTIONS

Information and Incentives

1-

max{(t,q)} S(q) t subject to t q 0yields

S(q) = ; t = q.

2-

max{(t,q);(t,q)}

(S(q) t) + (1 )(S(q) t)

subject to

t q t qt q t qt q 0t q 0,

yields

S(q) = ; t = q + q

S

(q) = +

1 ; t = q.

3-

= Pr( = | = ) = + (1 )(1 )

= Pr( = | = ) = (1 )(1 ) + (1 ).

For each value of ( and ), the optimal contract is characterized as in question 2,

where is replaced by and respectively.

4- For any , the information structure is characterized by the matrix

F =

Pr( = | = ) Pr( = | = )Pr( = | = ) Pr( = | = )

or

F1 =

1

1

.


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61

For F2 =

1 +

1 +

.

F1 is an improvement in the sense of Blackwell if there exists a bistochastic matrix B

such that F2 = BF1.

Take

B =

1

21

21

211 21

.

In a classical statistical decision problem, the utility of the decision-maker increases

for an improvement of his information in the Blackwell sense. Here it is not necessarily

the case.

5- Let W() denote the expected utility of the principal (with obvious notations):

W() = Pr( = )

Pr( = | = )(S(q) q q(, )+Pr( = | = ) S(q(, ) q(, ))

+ Pr( = )

Pr( = | = )(S(q) q q(, )+Pr( = | = ) S(q(, ) q(, )) .

This expression can be written symbolically

| F(q,,,)dG(|) dH().The difference with a classical decision problem is that F() depends here directly on

the precision of the signal and the signal , because of the presence of the information

rents q(, ) and q(, ) in the principals objective function.

Therefore

dW

d=

W

fixed in F(q,,) Blackwells effect

+

|

F

dG(|)dH()

New effect.

6- The expected utility of the principal can be rewritten

W = [S(q) q q(, )]+(1 )(1 )[S(q(, )) q(, )]+(1 )[S(q) q q(, )]+(1 )[S(q(, )) q(, )].


62/114

62 SOLUTIONS

By the Envelop Theorem we have:

dW

d= q(, ) (1 )[S(q(, )) q(, )]

+q(, ) + (1 )[S(q(, )) q(, )]

= (1 )q(,)q(,)

S(q) + 1 dq > 0,since q(, ) > q() and S(q(, )) > .


63/114

63

The Bribing Game

Let (t, q) and (t, q) the contracts offered where t (resp. t) is the bribe requested for a

decrease of delay q (resp. q).The principals program is

max{(t,q);(t,q)}

t (q Q)

2

2

+ (1 )

t (q Q)

2

2

subject to

q t q tq

t

q

t

q t u0q t u0.

As usual t = q + u0; t = q q + u0 hence the solution

q = Q +

q = Q + 1

.

Agents who value more time are offered a higher decrease of delay.

This exercise ??? from Saha (2001).


64/114

64 SOLUTIONS

Regulation of Pollution

1- Let x() the solution of

D(x) + Cx(x, ) = 0. (1)

If the firms must pay t(x) = D(x) + K it solves

minx

{D(x) + C(x, ) + K}

yielding (1).

2-

max{x,t}{D(x) (1 + )t + t C(x, )}subject to

t C(x, ) 0.

Let U = t C(x, ). The program can be rewritten

max{D(x) (1 + )C(x, ) U}

subject to

U 0hence, the solution is

D(x) + (1 + )Cx(x, ) = 0,

t = C(x, ).

The participation constraint requires now the use of public money which has an oppor-

tunity cost of 1 + . So the marginal disutility of pollution is equal to the social marginal

cost of depollution (1 + )Cx(x, ) which includes the financial cost.3- Under incomplete information the regulator maximizes expected social welfare under

the incentive and participation constraint of the firm

max{(x,U);(x,U)}

(D(x) (1 + )C(x, ) U) + (1 ) D(x) (1 + )C(x, ) U ,subject to


65/114

65

U U + C(x, ) C(x, ) (2)U U + C(x, ) C(x, ) (3)U 0 (4)U 0, (5)

where we use the notation U = t C(x, ); U = t C(x, ).As usual since C < 0, is the efficient type and U = 0 and U = C(x, ) C(x, ).Hence, the solution

D(x) + (1 + )Cx(x, ) = 0

D(x) + (1 + )Cx(x, ) +1

[Cx(x, ) Cx(x, )] = 0.

Since D 0, Cxx > 0 and Cx < 0, x is greater than the full information level.

Depollution is more costly because of the information rent which must be given up toelicit the information.

4- Let U() = t() C(x(), ) the level of utility of the truthful firm when it is facedwith the DRM (t(), x()). The local incentive constraints are

U() = C(x(), ) (6)

x() 0,since the first-order condition of incentive compatibility is

t() Cx(x(), )x() = 0

and (6) follows from the Envelope Theorem.

The second order condition is Cx(x(), )x() 0 or x 0 since Cx < 0.The regulators optimization program is

max{x(),U()}

{D(x()) (1 + )C(x(), ) U()} dF()

subject to

U() = C(x(), ) (())x() 0

U() 0 for all .

Since C < 0, the participation constraint reduces to U() 0.


66/114

66 SOLUTIONS

The Hamiltonian is

H = (D(x) (1 + )C(x, ) U)f() C(x, ).

Since there is no constraint at = , the transversality condition implies () = 0.

From the Pontryagin condition

() = HU

= f()

hence

() () = (1 F())or

() = (1 F()).

Maximizing H with respect to x we obtain finally

D(x()) + (1 + )Cx(x(), ) (1 F())f()

Cx(x(), ) = 0.

There is no distortion at = , and since Cx < 0 and D 0, Cxx > 0, an downwarddistortion of pollution for all the other types.

It remains to check that the second order condition is satisfied, i.e., x() 0. Asufficient condition is

dd1 F()

f() 0 ; Cx < 0; Cxx 0.

See Groves and Loeb (1975), Aspremont and Gerard-Varet (1979).


67/114

67

Taxation of a Monopoly

1- Because utility functions are quasi-linear all interior Pareto optima are obtained by

maximizing the utilitarian criterion under the resource constraint of the economy, i.e.,

max{(q(),x());(q(),x())}

[ log q() + x()] + (1 )[ log q() + x()]

subject to

x() + (1 )x() = x c[q() + (1 )q()] Ki.e.,

q() =

cand q() =

c.

2- A -consumers utility function writes log q + x t where x is a fixed parameterand t is the payment made to the monopoly. Incentive constraints are then

log q t log q t (1) log q t log q t. (2)

3- The monopolys maximization program is then

max{(t,q);(t,q)}

(t cq) + (1 )(t cq)


log q t 0 (3) log q t 0. (4)

We can expect (3) and (2) to be binding. Hence t = log q and t = log q log q.Inserting in the objective function of the monopoly and maximizing with respect to q and

q, we obtain

q =

c ; q =

c (1

)

c .

4- When is the tax, the monopolys problem becomes

max{(t,q);(t,q)}

(t cq) + (1 )(t cq)

subject to

log q t q 0


68/114

68 SOLUTIONS

log q t q log q t qwhere we just write the relevant constraints or

max{q,q}

log q (c + )q + (1 ) log q (c + )q log qhence

q =

c + ; q =

c + (1 )

c + . (5)

Everything happens as if the marginal cost was c = c + instead of c.

The governments problem is then

max

(1 ) log q +

( log q (c + )q) + (1 )( log q (c + )q log q)

+(q + (1 )q)

subject to (5).

The maximand can be rewritten

(1 )(1 )log q + log q + (1 ) log q + ( (c + ))(q + (1 )q).

The derivative with respect to is(1 )(1 )

q+

q+ ( (c + ))

dq

d+

(1 )

q+ (1 )( (c + ))

dq

d

+( )(q + (1 )q).

Using (5) this expression becomes

dq

d

(1 )

q+

+

dq

d(1 ) + ( )(q + (1 )q). (6)

For > 0 and , (6) is negative for > 0. Also is bounded below by c (see(5)). The interior solution is then:

= ( )(q + (1 )q) +1

q

dq

d

dq

d+ (1 ) dq

d

< 0.5- Let U() = log q() t() be the utility of a -consumer. By the envelope theorem,the incentive constraints are

U() = log q() (7)

q() 0 for all . (8)


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69

The monopolys maximization program is

max{t(),q(),U()}

(t() cq())dF() =

( log q() cq() U())dF()

subject to (7)-(8) and U()

0.

The Hamiltonian is

H = ( log q() cq() U())f() + ()log q().

Since U is increasing, there is no constraint at = so that () = 0. From the

Pontryagin principle

() = HU

= f(). (9)

Integrating (9) between and we get ()() = F()F() or () = (1F()).Maximizing with respect to q() we obtain finally

q()= c +

1 F()f()

1

q()

or

qSB() = 1F()f()

c

with no distortion at the top only.

In the case of a uniform distribution on [2, 3], F() = 2 and c = 1:

q() = 2 3 or (q) = 3 + q2

t() = log q()

log q(u)du

T(q) = t((q)) = 3

2

+q

2 log q 32

+ q2

2

log q(u)du

dT

dq=

1

2+

3

2q+

1

2log q 1

2log

3

2+

q

2

2d2T

dq2= 3 + a

2q2.

T() is concave. There is a discount for buying more (see figure below).


70/114

70 SOLUTIONS

E

T

1

2

3

1 2 3

q()

qSB()


71/114

71

Shared Information Goods, Majority Voting and Op-timal Pricing

1- The monopolists optimization program is:

max{q();U()}

3

21

q() q()

2

2 U()

d

subject to

U() = q()

q() 0,yielding q() = 2( 1) and an expected profit for the monopolist

I =3

2 2

1

[2(

1)]2d = 2.

2- Let G() the cumulative distribution function of the median of the types and g() its

density function. The monopolists program is:

max{q();U()}

21

3

q() q()

2

2 U()

dG()

U() = q()

q() 0,yielding q() =

1G()

g()

and an expected profit for the monopolist

II =3

2

21

1 G()

g()

dG().

Given that G() = (5 2)( 1)2, we can check that q() is increasing in . Let (q)be its inverse function. As

U() = q() q()2

2 t()

T(q) = t((q)) = (q)q q2

2(q)

1

q(u)du.

Then each agents utility function is single-peaked, and the majority rule yields the choiceof the median agent. Indeed

V(, q) = q q2

2 T(q)

yields

Vqq(, q) = (q) < 0.

3- G() = ( 1)2(5 2). Then I = 2 and II 9.25.


72/114

72 SOLUTIONS

Labor Contract with Adverse Selection

1- Since the observables of the principal are y and t

A = {y,t, y IR+ and t IR} .

From the Revelation Principle, we can restrict the analysis to the pair of contracts

(y, t)(y, t). The agents incentive constraints are

t (y) t (y) (1)t (y) t (y). (2)

2- The principals optimization program is:

max{(y,t),(y,t)}

[y t] + (1 )[y t]


t (y) 0 (3)t (y) 0. (4)

We can expect the participation constraint of the inefficient type (4) and the incentive

constraint of the efficient type (1) to be binding. Hence

t = (y) and t = (y) + (y) (y).

Substituting these solutions in the principals objective function and maximizing with

respect to y, y we obtain:

() = (y) =1

;

() = (y) =1

1

() ()

.

The marginal disutility of labor is equated to its marginal productivity for the efficient

type. It is distorted downwards for the inefficient type. (3) is implied by (4) and (

laffont marchimort excersises

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