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 ν of borrowers (called type 1) have sure projects with return h for an investmen investmentt of 1. A proportion 1 ν of borrowers (called type 2) have (stochastically independent) projects with return h only with probability θ in (0, (0, 1) and return 0 with probability 1 θ, for an invest investmen mentt of 1. If he does not apply apply for a loan, the borrow borrower has an outside opportunity utility level of u.
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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., θh > r + u 1- Expl Explai ain n wh why y ther theree is no loss loss of gener generali ality ty in cons consid ider erin ingg the the menu menuss of contr contract actss (r1 , P 1 ), (r2 , P 2 ) where P i 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 , P 1 ); (r2 , P 2 ) to maximize its expected profit under the borrower’s participation and incentive constraints (for simplicity assume that if a borrower applies for a loan he loses his outside opportunity u).
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}
3- Show that the optimal contract entails a non-random allocation of loans (i.e., P i is either 0 or 1, i = 1, 2). Characterize the optimal contract. Discuss its properties. 3
4
EXERCISES
Bundling with Asymmetric Information
We consider a continuum of consumers who have the following independent willingnesses 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 θ ε, θ or θ + ε (with θ > ε).
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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 P B . Determine the optimal P B 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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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
v > 0, v < 0
V N = qN + nP v(qP ),
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 of yN . 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 + nP θv(qP ) where qR is his consumption and θ in θ, θ¯ a parameter describing how altruistic the rich in the South are. There are nR rich in the South, each one with an endowment of yR .
{ }
The preference parameter θ 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 θ 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. A Determine the optimal level of aid qP 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 U A (θ). Special case : v( ) = log( ).
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2- Suppose first that the North knows θ and brings to the South a level of aid nR a 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 aid when a can be made conditional on the consumption level of the poor qP . Special case : v( ) = log( )
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¯ Determine the 3- Assume now that the North does not know θ and let ν = P r(θ = θ). optimal menu of contracts (¯a, q¯P ); (a, q P ), 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 θi the outside utility level that worker i can obtain. θi can take one of two positive values θ, θ¯ with Δθ = θ¯ θ > 0. These outside opportunities are identically and independently distributed between workers on θ, θ¯ with ν = Pr(θi = θ) and 1 ν = ¯ Pr(θi = θ).
{ }
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{ }
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A new allocation of labor is characterized by the proportions p (resp. p) ¯ of workers of ¯ who remain in the public firm. Assuming that workers are risk-neutral, type θ (resp. θ) a downsizing mechanism can be viewed as a pair of contracts ( p,t);(¯ p, t¯) specifying for ¯ a probability to remain in the public firm and a payment each announcement θ or θ, (unconditional on remaining or not in the public firm).
{
}
Total production of the public firm is then q = νp + (1 ν ) p. ¯ Let us first assume that the values of outside opportunities are public knowledge. In other words the government is under complete information. The workers accept to play the downsizing mechanism if the government pays them, t for type θ, t¯ for type θ¯ and satisfies their participation constraints
−
t + (1 t¯ + (1
− p)θ ≥ − p)¯ θ¯ ≥
θ, or t pθ, ¯ or t¯ p¯θ. ¯ θ,
≥ ≥
If there is a cost 1 + λ of public funds, social welfare is equal to: ¯ Social value of the public good S (νp + (1 ν ) p) (1 + λ)(νt + (1 ν )t¯) Social cost of transfers +ν (t + (1 p)θ) Welfare of workers. ¯ +(1 ν )(t¯ + (1 p) ¯ θ)
−
−
− −
− −
If λ > 0, the participation constraints are binding and social welfare can be reduced to S (νp + (1
− ν ) p)¯ − (1 + λ)(νpθ + (1 − ν ) p¯θ)¯ + νθ + (1 − ν )θ.¯
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The social optimum is then characterized by: ¯ if S (ν ) > (1 + λ)θ, S (ν + (1
− ν ) p)¯ = (1 + λ)θ¯ and p = 1,
if (1 + λ)θ¯ > S (ν ) > (1 + λ)θ, p = 1 and p¯ = 0, if S (ν ) < (1 + λ)θ, S (νp) = (1 + λ)θ 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
{
}
− p)¯ θ¯ ≥ t + (1 − p)θ ≥ t¯ + (1
t + (1 t¯ + (1
− p)θ¯ ¯ − p)θ.
It satisfies the participation constraint if
− p)¯ θ¯ ≥ t + (1 − p)θ ≥ t¯ + (1
θ¯ θ.
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 the incentive 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 social welfare. Distinguish three cases according to the value of S (ν ). 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 θ¯ worker produces ρ units of public good while a type θ worker still produces one
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EXERCISES
unit. Characterize the allocation of labor which maximizes expected social welfare when ¯ ρθ > θ.
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Labor Contracts
A firm designs a labor contract for a worker. The utility function of the worker is, U = u(c) θl, where c is consumption and l is labor supply, θ in θ, θ¯ is the preference ¯ The parameter known to the worker and u( ) in an increasing concave function; θ < θ. proportion of workers with low disutility of effort, θ = θ, is ν . The agent’s optimal choice must satisfy the budget restriction c t, where t is the payment he receives from the firm. The firm has the following utility function: U P = f (l) t, where f (l) is a decreasing returns to scale production function. A
−
{ }
·
≤
−
1- Assume the firm knows θ. Characterize the solution to the firm’s 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 θ are observed by him. Show that the First-Best solution is not implementable. 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 θ¯ 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 of V . Characterize the first-best and second-best (asymmetric information) solutions in this case. Take into account that the solution will depend on the size of V . For Δθ = θ¯ θ, consider the cases in which ¯lSB Δθ V (SB = second-best), ¯lSB Δθ V t∗ Δθ ( = first-best), ¯l∗ Δθ V l∗ Δθ and V l∗ Δθ. Which are the binding constraints in each case and what type of distortion is needed?
·
≤ ≤
− ∗
≤ ≤
≥
≥
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EXERCISES
Control of a Self-Managed Firm
Consider a firm that has a production function y = θl 1/2 , where l is the number of workers (considered to be a continuous variable: l in R+ ) and θ > 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 U SM =
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 if w is too large? Why is the size of the self-managed firm not optimal in general? 3- Assume that the government knows θ. Consider the case in which w is small enough to justify the presence of a self-managed firm. Compute the per unit tax τ 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 the economy). 4- Suppose now that the government does not know θ which can take one of two values ˜ t(θ)) ˜ which associates θ or θ¯ with Δθ = θ¯ θ > 0. It uses a regulatory mechanism (l(θ), ˜ and a transfer t(θ) ˜ to the firm’s announcement θ. ˜ The firm’s objective a labor input l(θ) is now ˜ 1/2 + t(θ) ˜ pθ(l(θ)) A SM U = . ˜ l(θ)
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Characterize the set of regulatory mechanisms which induce truthful revelation of θ. Derive the implementability condition on . 5- Assume that ν = Pr(θ = θ). Suppose that the government wishes to maximize the expectation of U G = pθl 1/2 wl.
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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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EXERCISES
Information and Incentives
An agent (natural monopoly) produces a quantity q of a good at a variable cost θq with θ in θ, θ¯ , Δθ = θ¯ θ. The principal gets an utility S (q)(S > 0, S < 0) from this production and gives a transfer t to the agent.
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The principal’s utility function is V = S (q) t and the agent’s utility function is U = t θq. Furthermore, the agent’s status quo utility payoff is normalized at 0.
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1- Characterize the optimal contract of the principal under complete information about θ. 2- θ is now private information of the agent and ν = Pr(θ = θ). Characterize the optimal contract of the principal under incomplete information with interim participation constraints 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 σ in σ, σ¯ with
{ }
¯ μ = Pr(σ = σ θ = θ) = Pr(σ = σ¯ θ = θ)
|
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≥ 12 .
Determine the updated principal’s beliefs about the efficiency of the agent, that is ¯ ). ν = Pr(θ = θ σ = σ); ν¯ = Pr(θ = θ σ = σ
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Characterize the optimal contract for each σ. 4- Show that an increase of μ corresponds to an improvement of information in the sense of Blackwell. 5- Show that an increase of μ has two effects on the principal’s expected utility, the classical effect (that we will call the Blackwell effect) and an effect due to the direct impact of μ on the expected utility of the principal conditional on σ. 6- Show that nevertheless the expected utility of the principal increases with μ.
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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 administration, 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. 2 Let us call q the decrease of delay that the official can provide at a cost (q−2Q) for him where Q is a constant. We assume that there is a proportion ν (resp. 1 ν ) of type 1 (resp. type 2) citizens ¯ who derive a benefit from a decrease q of delay equal to θq(θq). Citizens are willing to bribe the official to decrease of delays.
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Characterize the optimal bribing contract that the official will offer to the citizens.
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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, θ), with C x < 0, C xx > 0 and θ is a parameter known only to the firm. θ can take two values θ, θ¯ and ν = Pr(θ = θ) is common knowledge.
≥
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1- The level of pollution x∗ (θ) corresponding to the complete information optimum is characterized by D (x) + C x (x, θ) = 0. Show that, if the regulator is not obliged to satisfy a participation constraint of the firm, he can implement x∗(θ) 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 + λ)t + t − C (x, θ)
(1)
where t is the transfer from the regulator to the firm and 1 + λ is the opportunity cost of social funds. When the regulator must satisfy the firm’s participation constraint
− C (x, θ) ≥ 0 for all θ, characterize the decision rule x ˆ(θ), θ in {θ, θ¯}, which maximizes W under complete infort
mation. Compare with question 1.
3- We assume in addition that C θ < 0 and C xθ < 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 θ is distributed according to the distribution F (θ) with ¯ with density f (θ) on the interval [θ, θ] d dθ
− 1
F (θ) f (θ)
< 0, C xxθ
≥0
and
C θθx
≤ 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, θ 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 θ can take two values θ or θ¯ with θ¯ knowledge proportion of type θ consumers.
− θ = 1 and let ν be the common
Consumers have large resources in good 1, x∗ , so that their behavior is always characterized 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∗ (θ), θ in θ, θ¯ which corresponds to an interior Pareto optimal allocation is the solution of:
{ }
max ν [θ log q(θ) + x(θ)] + (1
− ν )
subject to νx(θ) + (1
− ν )x(θ)¯ = x¯ − c
Determine q∗ (θ).
¯ + x(θ) ¯ , θ¯ log q(θ)
νq(θ) + (1
− ν )q(θ)¯ − K.
2- Let (q, t);(¯ q, t¯) the direct revelation mechanism which elicits the parameters θ from consumers. Characterize the direct revelation mechanisms which are truthful.
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}
3- Write the optimization program of the monopoly when he is constrained to provide a non negative utility level to each consumer θ log q(θ)
− t(θ) ≥ 0
for θ in θ, θ¯ .
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Characterize the truthful direct revelation mechanism which is optimal for the monopoly. 4- The government uses now a linear tax τ 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 monopoly’s profits (with a weight σ larger than 1) and of taxes (with a weight λ > 0 such that σ λ), show that the optimal tax is negative.
≥
5- Optional. Questions 2 and 3 when θ is distributed according to the distribution F (θ) ¯ with d 1−F (θ) < 0. Consider the special with positive density f (θ) on the interval [θ, θ] dθ f (θ) case of a uniform distribution on [2, 3] and c = 1 and obtain the associated nonlinear price.
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EXERCISES
Shared Information Goods, Majority Voting and Optimal 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. 2
Let θi q q2 t the utility function of consumer i, i = 1, 2, 3 where q is the quantity of the good and t the payment. The θi are independently drawn from the uniform distribution on [1, 2] and private information of the consumers.
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1- Assuming first that the monopolist can prevent the consumers to share the good, characterize 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 of three independent draws from the distribution F ( ) is: G(x) = [F (x)]2 [3 2F (x)]).
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3- Show that the expected profit of the monopolist is higher when consumers share the good.
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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 θ (θ in θ, θ¯ ) suffers a disutility of working equal to ψ(θy).
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In other words, a worker of type θ must work units of labor (with = θy) which create a disutility of labor ψ(), ψ ( ) > 0, ψ ( ) 0.
·
·≥
If he receives a compensation t from the employer his net utility is U = t The employer’s utility function is V = y t.
−
− ψ(θy).
1- Apply the Revelation Principle and characterize the truthful direct revelation mechanisms. 2- Assume now that ν (resp. (1 ν )) equals the probability that the worker is of type θ ¯ Characterize the optimal contract of an employer who maximizes his expected (resp. θ). utility under the incentive and interim participation constraint of the worker.
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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 probability 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 ψ 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 the agent.
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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. This probability of success depends upon the effort e in eH , eL of the entrepreneur: it is pH if effort is high (eH ) and pL if e = eL ; 1 > pH > p L = 0. A loan contract specifies a given payment P if the income is R and 0 if the income is 0.
{
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}
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 economy’s rate of interest is equal to 0. 1- Show that the project is financed if and only if pH R
≥ B + I
(1)
2- Suppose that (1) is not satisfied but pH R > 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¯ e Monitor pH P m, pH (R P )) ( pH R m, Not monitor ( pH P, pH (R P )) (0, B)
−
−
−
−
0)
Assume m < p H R < 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: pH R
≥ m + I.
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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 ¯ with probability a risky project. In the absence of information, the risky project yields S ¯ (1 ν )S = 0. ν and S with probability 1 ν . We will assume that ν S +
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By incurring an effort with cost ψ, the agent can learn a signal σ future realization of the risky project. ¯) = Pr(σ S ) = θ, with θ We will assume that Pr(¯ σ S precision of the signal.
|
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∈]
1 , 1] 2
∈ {σ, σ¯ } on the
being interpreted as the
1- As a benchmark, suppose that the principal uses the technology for information gathering himself. Show that the project is done only when ¯σ 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 ¯ (resp. S ) realizes. t0 is the transfer he receives agent if he chooses the risky project and S if he chooses the safe project. Write the incentive constraints needed to have the risky project being chosen if and only if σ¯ 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 ψ(e) (ψ > 0, ψ > 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 holds. The principal benefits from q agent.
| f e (q e) f (q e)
|
>0
− t(q) where t(q) is the transfers he makes to the
1- Characterize the first-best effort. 2- Write the agent’s incentive and participation constraints when e is non observable by the principal. Use the first-order approach. 3- Write the Lagrangian of the principal’s 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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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 θ˜ + e where θ˜ is a random variable with zero mean. ˜ Compute 1- Suppose that the agent chooses his effort before knowing the realization of θ. 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 ψ > 0 and ψ > 0 imply that the second best optimal effort RSB < eSB . Conclude.
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Raising Liability Rule
We conside considerr a lender lender-bor -borro rowe werr relati relations onship hip und under er moral moral hazard. hazard. The risk-n risk-neut eutral ral borrower wants to borrow I from a lender to finance a project with safe return V . V . The project may with probability probability 1 e harm a third-part third-party y. The amount of safety safety care e costs ( ψ > 0, ψ > 0, ψ > 0). The harm harm has value alue h. A financi financial al ψ(e) to the borrower with (ψ contract is a pair (t, ( t, t¯) where t (resp. t¯) is the borrower’s reimbursement to the bank if there is no (resp. one) environmental damage.
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1- Suppose that e is observable. observable. Compute Compute the first-best first-best level of safety safety care and assume that the project is socially valuable when the interest rate is r. 2- Suppose now that e is not observable. observable. We suppose that the bank is competitive competitive and that the borrower borrower has sufficient liability liability.. Show that the first-best first-best is still implementab implementable le 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 borrower’s expected payoff subject to the bank’s zero profit constraint, the borrower’s incentive constraint and his limited liability constraint. 5- Show that raising the bank’s 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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EXERCISES
Risk-Averse Principal and Moral Hazard
Suppose that Suppose that a risk-a risk-ave verse rse principa principall delegat delegates es a task task to a risk-n risk-neut eutral ral agent. agent. With With probability e (resp. 1 e) the outcome is q¯ (resp. q < q¯). The risk-averse principal utility is v (q t) where t is the agent’s transfer and v ( ) is a CARA von Neumann-Morgenstern utility function. Effort costs ψ (e) to the agent (ψ ( ψ > 0, ψ > 0).
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−
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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 where he 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, [0, 1] of iden identi tica call agen agents ts.. Th Thee income of an healthy agent is w. Each agent becomes (independently) sick with probability π0 , in which case he has only a survival income of w. Let δ 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 utilit utility y function function of a sick sick agent who has not been treated is uM ( ) with uM > 0, uM < 0.
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1- Let p the insurance premium premium that only healthy healthy and treated treated sick agents can pay. pay. We have potentially three types of agents:
• a proportion 1 − π of healthy agents with utility level u(w − p), p), • a proportion δπ of treated sick agents with utility level u(w − p), p), • a proportion (1 − δ)π of non treated sick agents with utility level u 0
0
0
M (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 δ and p (assume an interior solution). Determine the comparative statics of the optimal solution with respect to m, w and π0 . Let W ( W (π0 , p∗ (π0 ), δ ∗ (π0 )) denote optimal social welfare. 2- Suppose now that with a non observable non monetary cost of health care ψ any agent can decrease his probability of becoming sick from π0 to π1 , with Δπ Δπ = π0 π1.
−
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 ψ in order to decrease their probability of being sick. 3- We consider now pairs of agents who observe each other’s effort of health care, and we assume that agents agents can perfectl p erfectly y coordinate their health care behavior. behavior. Assuming 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
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.
√
27
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 entrepreneur’s 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 ψ 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 entrepreneur’s share of output, his expected utility is ¯ − ψ if he exerts effort ¯e, • px • 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 ψ > r > pz.
−
1- Determine the optimal contract offer to an entrepreneur when there is a single entrepreneur. 2- There are now two entrepreneurs who do not observe each other’s 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 other’s effort level and coordinate their effort levels. Consider the program of the bank which implements effort by both entrepreneurs with group lending contracts: max p¯2 (2z
¯ − p)(2z ¯ − 2x) + p(1 − 2y) − 2r
s.t. 2¯ ¯ p2 x + 2 p(1
¯ − 2ψ ≥ − p)y ≥ ≥
2 p2 x + 2 p(1
− p)y 2 p¯ ¯ + p(1 ¯ − p)y − ψ px + p(1 − p)y 0,
(1) (2) (3)
28
EXERCISES
x
≥ 0,
y
≥ 0.
Find the optimal contract. Show that it is better than individual contracts for the bank. Explain why.
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 θ over which the principal has property rights. The principal’s 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 agent’s utility function is u(θ
− q) − t.
1- Under complete information about θ, determine the optimal contract offered to the agent by a principal who maximizes his utility under the participation constraint of the agent u(θ
− q) − t ≥ 0.
2- θ can take one of two values θ, θ¯ with respective probabilities (1 ν, ν ) , with Δθ = θ¯ θ > 0 and is now private information of the agent. Suppose that the principal offers a contract to the agent before the agent’s search, i.e., before the agent learns θ. Characterize the optimal contract that the principal offers to the agent when he maximizes his expected utility under the agent’s incentive constraints and the agent’s ex ante participation constraint.
−
{ }
−
3- We assume now that the contract is imperfectly enforced. When the agent discovers θ¯ (resp. θ), he can exit the relationship with a quantity Δθ (resp. 0) of the resource. Therefore, the principal is faced with the following ex post participation constraints: u(θ u(θ¯
− q) − t ≥ − q¯) − t¯ ≥
0
(1)
u(Δθ),
(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” θ, and the incentive constraint of the “efficient type” θ¯ 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 Δθ > θ. Characterize the optimal contract and denote W (ν ) the principal’s expected welfare.
30
EXERCISES
5- We assume now that by exerting an effort which costs him ψ the agent increases the ¯ probability of a θ-discovery from ν 0 to ν 1 > ν 0 with Δν = ν 1 ν 0 . Show that W (ν ) is decreasing in ν . Write the program of a principal who selects a contract which discourages ψ effort. Solve it when u(Δθ) < Δν . Discussion.
−
31
III- MIXED MODELS
Political Economy of Regulation
We consider a firm which realizes two projects, of gross value S 1 and S 2 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: C i = β
−e
i
where β is the efficiency parameter of the firm. The efficiency of the firm is the same for both projects. ¯ with ν = Pr(β = β ). Parameter β can take values in two values β, β
{ }
The cost reducing efforts create a disutility to the firm equal to 1 ψ(e1 , e2 ) = (e21 + e22 ) + γe 1 e2 , 2
γ > 0.
A regulator reimburses the observable costs C 1 and C 2 and pays a net transfer t to the firm which has utility U = t ψ(e1 , e2 ).
−
Social welfare is S 1 + S 2
− (1 + λ)(t + C + C ) + U. 1
2
1- Determine the optimal regulation under complete information. 2- Determine the optimal regulation under incomplete information when β 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 α the proportion of stakeholders. If α > 1/2 the majority belongs to stakeholders who choose regulation to maximize their objective function: α(S 1 + S 2
− (1 + λ)(t
1
+ C 1 + t2 + C 2 )) + U.
32
EXERCISES
If α < 1/2 the majority belongs to non-stakeholders who choose regulation to maximize instead: (1 α)(S 1 + S 2 (1 + λ)(t1 + C 1 + t2 + C 2 )).
−
−
Determine in each case the optimal regulation under incomplete information.
33
Regulation of Quality
We consider a natural monopoly which has the cost function C = (β + s e)q where q is the production level, e is the effort level of the manager, s is the quality of the product ¯ is a cost parameter. and β in β, β
−
{ }
We assume that the regulator observes only the cost C and the firm’s revenue, and, in addition, pays a transfer t net of cost and revenue. If ψ(e) (with ψ > 0, ψ > 0, ψ 0) is the disutility of effort for the manager, it implies that the net utility of the manager is U = t ψ(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 B 2 (ks hθ)2 S (q,s,θ) = (A + ks hθ)q q 2 2 where A, B, h, k are positive constants and θ in θ, θ¯ is a demand parameter known by the firm, but not by the regulator.
−
− − { }
−
Let p(q) the inverse demand function. The utility derived by consumers is then: V = S (q,s,θ)
− p(q)q − (1 + λ)(t + C − p(q)q)
where 1 + λ 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 θ and β can be reduced to a one dimensional adverse selection problem with the parameter γ = β + hk θ. 2- Assume that γ is distributed according to a uniform distribution. Write the maximization 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.
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 = (β
− e)q,
¯ with ν = where q is the production level, β is an adverse selection parameter in β, β Pr(β = β ) and e is a moral hazard variable which decreases cost, but creates to the manager a disutility ψ(e) with ψ > 0, ψ > 0, ψ 0.
{ }
≥
Consumers derive an utility S (q), S > 0, S < 0 from the consumption of the natural monopoly’s good. Let p( ) the inverse demand function and tˆ the transfer to the firm from the regulator. The firm’s net utility is:
·
U = tˆ + p(q)q
− (β − e)q − F − ψ(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 = tˆ + p(q)q (β e)q F . Accordingly, the participation constraint of the firm can be written:
− −
U = t
−
− ψ(e) ≥ 0.
To finance the transfer t, the government must raise taxes with a cost of public funds 1 + λ, λ > 0. Hence, consumers’ net utility is V = S (q)
− p(q)q − (1 + λ)tˆ.
Utilitarian social welfare writes then: W = U + V = S (q) + λp(q)q
− (1 + λ)((β − e)q + F + ψ(e)) − λU.
1- Under complete information, the regulator maximizes social welfare under the firm’s participation constraint. Characterize the optimal solution. 2- Suppose now that the regulator cannot observe the effort level e and does not know β . However, he can offer a contract to the firm before the latter discovers its type (see Figure 1 for the timing).
35
E Time The regulator offers the regulatory contract.
The firm accepts or not the contract.
The firm discovers its type.
Production and transfer take place.
Figure 1 Explain why the regulator can restrict his contract to a pair (t, c); (t¯, c¯) (where c = is average cost) which satisfy incentive constraints.
{
}
C q
Write the incentive constraints and the firm’s ex ante participation constraint. Solve for the optimal contracts. ¯ in question 3- We assume now that if the firm has a negative ex post utility (as firm β 3) it attempts to renegotiate its regulatory contract. However, with a probability π(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 π(0) = 0, π > 0, π < 0 with the Inada conditions π (0) = and limc↔+∞ π(c) = 1.
∞
With probability 1 π(c) the regulator is forced to accept a renegotiation. To model this 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 .
−
−
¯ which wants to renegotiate. Therefore, Assume that it is only the inefficient type β costly bargaining takes place under complete information. Its outcome solves: ¯) ¯ E + F )(δ W (¯ ˆ q, e¯, β max (U
¯E} {q¯,¯ e,U
− λU ¯
E
+ H )
where δ in (0, 1) models the cost of renegotiation and ¯) = S (q¯) + λp(¯ ˆ q, e¯, β W (¯ q )¯ q
− (1 + λ)((β ¯ − e¯)¯q + F + ψ(¯e)).
¯ E ). Compute the outcome of renegotiation (¯ qE , e¯E , U 4- Write the firm’s new participation constraint which takes into account that with probability 1 π(c) there will be renegotiation.
−
Substitute the outcome of renegotiation into the regulator’s objective function and solve for the optimal contract and the optimal level of enforcement expenses c. Discuss.
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 = β e, where β in ¯ is an efficiency parameter and e is a level of effort which creates a disutility ψ(e) β, β (ψ > 0, ψ > 0, ψ 0) for the firm’s 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 + λ. The manager of the firm is risk averse, and has the utility function u(t u > 0, u < 0).
− ψ(e)) with
e and β are not observable by the regulator, but it is common knowledge that ν = Pr(β = β ). ¯) = t¯, C (β ¯) = C ¯ ; t(β ) = t, C (β ) = C , write the 1- For the revelation mechanism t(β incentive and participation constraints of the firm.
{
}
2- Expected social welfare is defined as
− (1 + λ) ν (t + C ) + (1 − ν )(t¯ + C ¯ ) ¯ − C ¯ ). with π = t − ψ(β − C ) and π ¯ = t¯ − ψ(β W = S
+ u−1 (νu(π) + (1
− ν )u(¯π)) ,
Interpret this social welfare function and, assuming that it is concave in (π, π ¯ ), determine 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 firm’s manager is risk neutral. 4- Consider the special case
1 u(x) = (1 ρ
−ρx
−e
).
¯ is decreasing in ρ. Show that the effort level required from type β
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 C 1 = β 1 e1 where β 1 is an efficiency parameter which is common knowledge and e1 is the manager’s effort level which creates a disutility 14 e21 for him. Ex ante β 1 is unknown for everybody and drawn from 2, 3 with Pr(β 1 = 2) = Pr(β 1 = 3) = 1/2.
−
{ }
Firm 2 has the cost function C 2 = kβ 2 e2 , k 1, where β 2 is also drawn ex ante (independently of β 1 ) from 2, 3 with Pr(β 2 = 2) = Pr(β 2 = 3) = 1/2. However, the value of β 2 is only observed by the manager of firm 2; e2 is the manager’s effort level which creates a disutility 14 e22 for him.
{ }
−
≤
C 1 and C 2 are observed by the regulator. The outside opportunity utility levels are normalized to zero for both firms. The timing of events is summarized below: ex ante
interim β 1 observed by all, β 2 observed by firm 2
Regulator offers contracts
E t
Firms accept or reject
Contracts are executed
The regulator is utilitarian and can use transfers with a price of public funds 1 + λ, λ > 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 the values of λ and k such that firm 2 is chosen. 4- Assuming that the regulator selects the firm at the interim stage, characterize the values of λ, k and β 1 such that firm 2 is chosen. 2
−k) 5- Suppose that k can be chosen ex ante at the cost (1+λ)(3 by the regulator; what is 2 the optimal strategy of the regulator at the ex ante stage?
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 + αqc ) with α [0, 1].
∈
The firm has a utility function with constant absolute risk aversion so that his utility is −ρ[R(q0 +αqc )− p0 q0 −c˜qc ]
−e
.
1- Assuming that c˜ is a normal random variable with mean μ and variance σ2 , compute the demand functions q0 ( p0 ), qc ( p0 ) of a firm maximizing its expected utility. What is the M 2 optimal monopoly price pM 0 ? Study the dependence of p0 with respect to ρ, σ , μ , α , c. Suppose 1 R(q0 + αqc ) = a(q0 + αqc ) b(q0 + αqc )2 . 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 α,μ,σ 2 , c (Hint: use the envelop theorem). ˜ ˜ be social welfare under monopoly. Obtain ∂ W Let W . Let μ = c0 + f where c0 is the ∂μ reproducing cost and f an expected fine. Supposing that the cost of implementing the fine f is 12 δf 2 , determine the optimal fine. 3- What is the optimal two part tariff of the monopolist? 4- Assume that ρ is private information of the firm and can take two values ρ or ρ¯ with ρ¯ > ρ and ν = Pr(ρ = ρ). We will look for the optimal nonlinear price or the optimal direct revelation mechanism (t, q0 , t¯, q¯0 ). First show that the Spence-Mirrlees property holds for the surrogate utility function: 1 2 2 V (t0 , q0 , ρ) = max R(q0 + αqc ) t0 μqc ρσ qc . qc 2
− −
−
Write the firm’s incentive constraints for the function V . Solve for the optimal direct revelation mechanism. Discuss.
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 θq with θ in θ, θ¯ , θ¯ > θ. Let t be the transfer from the principal to the agent ; then, the agent’s utility is U = t θq.
{ }
−
The principal’s 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 θ at a cost ψ 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 θ 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 1 ) which induce the agent to choose e = 0 and those (class 2 ) which induce the agent to choose e = 1.
C
C
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 1 or in the class 2 of contracts.
C
C
2- Show that a lower bound for the principal is obtained by constraining contracts to ¯q 0. Derive from this result that the interesting contracts to consider t θq 0 ; t¯ θ¯ ¯q entail t¯ θ¯ 0. Show that the principal can always mimic a contract in the class 2 with a contract in the class 1 .
− ≥ − ≥ − ≤
C
C
3- Determine the optimal contract in the class 1 . (Distinguish three cases depending on whether the ex ante participation constraint is binding, the moral hazard constraint is binding or both constraints are binding according to the value of ψ).
C
40
EXERCISES
Better Information Structures and Incentives
We consider a natural monopoly which realizes a public project valued S at a cost ¯ is a parameter which is privately known by the manager with C = β e when β in β, β ¯ β > 0 and ν = Pr(β = β ) is the common knowledge probability that the firm is Δβ = β a low cost firm; e is the manager’s effort which has a disutility ψ(e) with ψ > 0, ψ > 0, 0. The cost C is observable by the regulator and reimbursed to the monopoly. ψ Accordingly the firm’s utility is U = t ψ(e)
− −
{ }
≥
−
where t is the net monetary transfer from the regulator to the firm and consumers’ welfare is V = S (1 + λ)(t + β e)
−
−
where λ > 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 firm’s incentive and participation constraints (it is assumed that the regulatory contract is offered 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 Σ = σ1 , σ2 , . . . , σI of signals and conditional probabilities Pr(σi β ) i = 1, . . . , I . Denote νˆ i the posterior belief that the firm has a low cost after signal σi , i.e., νˆi = Pr(β = β σi ), i = 1, . . . , I .
{
}
|
|
Characterize for each σi the optimal effort level e¯i requested from the high cost firm and denote e¯i = Z 1−νˆ iνˆ i the solution e¯i as a function of the ratio 1−νˆ iνˆ i .
Show that, if after each σi , 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 shut down the high cost firm.
41
Competitive Pressure and Incentives
We consider the case of a monopoly (producing good 1 in quantity q1 ) with a competitive fringe producing a differentiated good 2 in quantity q2 . The consumers’ utility function is S (q1 + q2 ) + θq1 q2 where θ is a measure of complementarity of the two goods. The monopoly’s cost function is C 1 = (β
− e)q , 1
¯ , β¯ > β , is an adverse selection parameter (with ν = Pr(β = β )) and e where β in β, β is an effort level which decreases cost with a disutility for the manager of ψ(e), ψ > 0, ψ > 0, ψ 0.
{ } ≥
The fringe’s 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 ) + θq1 q2 + λp1 (q1 , q2 )q1
− (1 + λ)(ψ(e) + (β − e)q ) − cq − λU, 1
2
where U = t ψ(e) is the utility of the monopoly where t is the net transfer (in addition to reimbursement of cost) from the regulator to the firm.
−
2- Characterize the optimal regulation under asymmetric information when the monopoly’s status quo utility level is zero. 3- We say that the two goods are strategic complements (substitutes) if S q1 + S + θ > 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 λ 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 (θ decreases). If the products are strong enough substitutes, the effort of the regulated firm increases with the degree of substitution.
42
EXERCISES
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 θ in (0, 1) and zero with probability 1 θ 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 (P 1 , r1 ); (P 2 , r2 ) which are incentive compatible :
{
P 1 (h
−r ) ≥ P θ(h − r ) ≥ 1
P 2 (h
2
2
1
}
−r ) P θ(h − r ).
(1)
2
(2)
1
2- The bank maximizes its expected profit under the incentive and participation constraints, i.e., solves max
{(P 1 ,r1 );(P 2 ,r2 )}
νP 1 (r1
− r) + (1 − ν )P (θr − r) 2
2
s.t. (1), (2) and P 1 (h
−r ) ≥ P θ(h − r ) ≥ 1
u
(3)
2
2
u
(4)
43
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 θ, (1) and (2) imply P 1 (h
− r ) = P (h − r ). 1
2
2
(5)
Since θ is in (0, 1), (4) is binding and not (3). From (4) we have r2 = h
− θP u
if P 2 = 0.
2
From (5) we have r1 = h
− θP u . 1
Substituting these expressions into the bank’s expected profit we get νP 1 (h
− r) + (1 − ν )P (θh − r) − νuθ − (1 − ν )u. 2
Since θh > r, maximizing with respect to P 1 and P 2 gives P 1 = P 2 = 1 and r1 = r2 = h uθ .
−
Therefore, we obtain a pooling contract with an expected profit for the bank of
−−
ν h
r
u + (1 θ
Type 1’s ex post information rent is h
− ν )(θh − r − u).
−r −u= 1
(1−θ) θ u.
Consider now case 2. The bank offers a loan intended only for type 1. It is only constrained by type 1’s 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 u
− with an expected profit for the bank ν (h − r − u).
Case 2 is better for the bank than case 1 if and only if ν (h
− r − u) > ν
or (1
−− h
r
u + (1 θ
− ν )(θh − r − u)
− ν )(θh − r − u) < ν (1 −θ θ) u,
i.e., the expected revenue made with type 2 borrowers is less than the expected rent which must be given up to type 1’s borrowers because of the presence of type 2 borrowers.
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,θ) = P θ(h
− r).
The Spence-Mirrlees condition writes ∂U ∂θ
∂U ∂P ∂U ∂r
∂U = ∂θ
(h
− r)θ
Pθ
=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).
46
SOLUTIONS
Bundling with Asymmetric Information
1- Since consumers want to consume only one unit and since only deterministic mechanisms are considered, there is no way to screen types with either the quantities they consume 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 θ ε, θ or θ + ε, for each good.
−
If p = θ + ε only 1/3 of the consumers buy and revenue is 13 (θ + ε). If p = θ, 23 of the consumers buy and revenue is 23 θ. If p = θ ε all consumers buy and revenue is θ ε.
−
−
Since ε < θ, the optimal single good price is (for each good): p∗ = θ =θ
if θ 3ε if θ > 3ε
≤
−ε
with a total revenue for both goods R∗ = 43 θ = 2(θ
− ε)
if θ 3ε if θ > 3ε.
≤
2- Suppose now that the monopolist can offer a price P B 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
1 9
{θ + ε, θ + ε} {θ + ε, θ} {θ, θ} {θ + ε, θ − ε} {θ, θ − ε} {θ − ε, θ − ε}
2θ + 2ε
2 9 1 9 2 9 2 9 1 9
2θ + ε 2θ 2θ 2θ
−ε 2θ − 2ε
Again, the relevant posted prices for the bundle correspond to the willingnesses to
47
pay. It is immediate to see that P B = 2θ = 2θ = 2θ
if θ if θ if θ
−ε − 2ε
≤ 2ε ∈ [2ε, 5ε] ≥ 5ε,
with respective revenues worth 4 θ 3 16 9 θ
8 9ε
− 2θ − 2ε Therefore, for θ in [2ε, 5ε], P B = 2θ good prices.
if θ if θ if θ
≤ 2ε ∈ [2ε, 5ε] ≥ 5ε.
− ε provides more revenue than the optimal single
3- For the optimal prices p∗ we can obtain the surplus made by each type of consumer and the associated revenue of the monopolist:
Population
Preferences
1 9
{θ + ε, θ + ε} {θ + ε, θ} {θ, θ} {θ + ε, θ − ε} {θ, θ − ε} {θ − ε, θ − ε}
2 9 1 9 2 9 2 9 1 9
p∗ = θ p∗ = θ ε Surplus Profit Surplus Profit
−
2ε
2θ
4ε
ε
2θ
3ε
0
2θ
2ε
ε
θ
2ε
0
θ
ε
0
0
0
2(θ
− ε) 2(θ − ε) 2(θ − ε) 2(θ − ε) 2(θ − ε) 2(θ − ε)
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 θ < 3ε and p∗ = θ. A bundle price of 2θ ε makes positive profit since θ > ε and attracts all consumers except the (θ ε, θ ε) types. It also makes more profit than the optimal single good prices when 8 4 (2θ ε) > θ 9 9 i.e. θ > 2ε.
−
−
−
−
Adams and Yellen (1976).
48
SOLUTIONS
Incentives and Aid
1- The rich in the South solve A maxA qR + nP θv(qP )
{qR ,qP }
subject to A = nR yR nR qR + nP qP
hence A ) = 1, nR θv (qP
(1)
with a status quo utility level of yR
− nn
P A qP R
A + nP θv(qP ) = U A (θ).
For v( ) = log( ),
·
·
A = nR θ, qP
U A (θ) = yR + nP θ(log nR θ
− 1).
2- The rich of the South solve A maxA qR + nP θv(qP )
{qR ,qP }
A = nR yR + nR a nR qR + nqP
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 ,qP a}
qN + nP v(qP )
subject to nR qR + nP qP = nR yR + nR a qR + nP θv(qP )
A
≥ U (θ)
49
nN qN = nN yN
−n
Ra
or (nN + nR θ)v (qP ) = 1.
(2)
Comparing (1) and (2) we see that the poor in the South consume more. For v( ) = log( ), qP = nN + nR θ instead of nR θ.
·
·
3- To the previous program we must add incentive constraints. For given a and qP , the rich’s utility is nP yR + a qP + nP θv(qP ). nR
−
The incentive constraints are
− nn n + ¯a − n
P
yR + a yR
R P
q P + nP θv(q P )
R
¯ qP ) q¯P + nP θv(¯
≥ ≥
− nn n +a− n
P
yR + ¯a yR
q¯P + nP θv(¯ qP )
R P
¯ ). qP + nP θv(q P
R
Since the status quo utility level U A (θ) depends on the type, there is a potential problem of countervailing incentives. However we will guess that it is, as usual, the ¯ which is binding, as well as the incentive constraint of the one who wants to lie, θ, participation constraint of type θ. The problem of the principal becomes max
¯,¯ qP ;a,qP } {a
ν yN
− nn n +a− n
P
yR + ¯a yR
− R P R
nR a¯ + nP v(¯ qP ) + (1 nN
− ν )
¯ qP ) = yR + a q¯P + nP θv(¯
yN
− nn
−
P R
nR a + nP v(qP ) nN
¯ ) q P + nP θv(q P
qP + nP θv(qP ) = U A (θ).
(3) (4)
Solving (3) and (4) for ¯a and a, inserting in the objective function and maximizing with respect to q¯P , qP yields
{
}
¯ (¯ (nN + nR θ)v qP ) = 1 (nN + nR θ)v (q P ) = 1 +
ν 1
−
ΔθnR v (q P ). ν
Asymmetric information leads to a decrease in the poor’s consumption when the rich of the South are of type θ, i.e., have a low altruistic behavior.
50
SOLUTIONS
With v( ) = log( ), we get:
·
·
q¯P = nN + nR θ¯ qP = nN + nR θ
− 1 −ν ν Δθn
R.
It remains to check that the participation constraint of type θ¯ is satisfied as well as type θ’s incentive constraint. Note that the above solutions are in fact only valid as long as a 0, a ¯ 0. In fact when ν is too large, a = 0 and the poor in the South get the same utility as under autarky. It happens when
≥
≥
nN
≤ 1 −ν ν Δθn
Azam and Laffont (2002).
R
or
ν
nN . N + ΔθnR
≥n
51
Downsizing a Public Firm
1- Let us rewrite the constraints as ¯ = t¯ p¯θ¯ U
≥ t − pθ¯ = U − pΔθ ¯ + pΔθ ¯ = U ¯ ≥ t¯ − pθ ≥ 0 ≥ 0.
− U = t − pθ ¯ = t¯ − p¯θ¯ U U = t − pθ
(1) (2) (3) (4)
Then, we are back to the familiar formulation and (2) + (3) imply (4). Now, the “good” type is the θ-type. 2- Adding (1) and (2) implies p p. ¯ For any implementable pair ( p, p), ¯ we have the following constraints ( p p)θ ¯ ¯ θ¯ t t¯ ( p p)
≥
−
≤ − ≤ −
and (3) implies ¯ t¯ p¯θ.
≥
The government minimizes the expected transfers νt + (1 t = t¯ + ( p
− ν )t¯ by choosing t¯ = p¯θ¯ and
− p)θ ¯ = pθ + Δθ p. ¯
3- Expected social welfare can be rewritten as: S (νp + (1
− ν ) p)¯ − (1 + λ)νpθ − (1 + λ)(1 − ν ) p¯θ¯ − λν Δθ p¯
up to a constant. If we have S (ν ) > (1 + λ)θ¯ + λ then, p¯ solves S (ν + (1
ν
1
− ν Δθ,
− ν ) p)¯ = (1 + λ)θ¯ + 1 λν − ν Δθ
and p = 1. If we have instead ν (1 + λ)θ¯ + λ Δθ > S (ν ) > (1 + λ)θ 1 ν
−
we get p = 1 and p¯ = 0.
(5)
52
SOLUTIONS
If, finally S (ν ) < (1 + λ)θ, then p solves S (νp) = (1 + λ)θ and p¯ = 0. As (5) shows, for low levels of downsizing, asymmetric information (Δθ) 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 ¯ + (1 t¯ = p¯w
− p)s = pθ + Δθ p¯ ¯ − p)¯ ¯ s = p¯θ.
(6) (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 = θ and w¯ = θ¯ ensures the first point. Then, from (6) and (7) s¯ = 0
and
s = Δθ
p¯ 1
− p > 0.
5- The incentive constraints remain unchanged so that the optimization program is now W = S (νp + (1
− ν )ρ¯ p) − (1 + λ)νpθ − (1 + λ)(1 − ν ) p¯θ¯ − λν Δθ p¯
∂W = ν [S (1 + λ)θ] ∂p ∂W = (1 ν ) ρS (1 + λ)θ¯ ∂ ¯ p
−
−
As long as ρθ > θ¯ + be replaced by
λ ν Δθ, 1+λ 1−ν
ρS (ν + (1
−
ν
− λ 1 − ν Δθ
.
the solution is as in question 3 except that (5) must Δθ − ν )ρ¯ p) = (1 + λ)θ¯ + λν . 1 − ν
λ ν When ρθ < θ¯ + 1+λ Δθ, the first-order conditions would call for p < p¯ which is 1−ν in conflict with the implementability condition p p. ¯ Then, we have bunching at the optimal contract which solves
≥
max S ((ν + (1 p
− ν )ρ) p) − (1 + λ)(νθ + (1 − ν )θ)¯ p − λν Δθp
53
i.e.,
S ((ν + (1
−
(1 + λ)(νθ + (1 ν )ρ) p) = ν + (1
Jeon and Laffont (1999).
− ν )θ)¯ + λν Δθ . − ν )ρ
54
SOLUTIONS
Labor Contracts
1- The firm’s problem is max f (l) {l,t}
− t subject to u(t) − θl ≥ 0.
So, for θ = θ: u(t∗ ) = θ l ∗ θ f (l∗ ) = ∗ u (t ) ¯ for θ = θ: u(t¯∗ ) = θ¯ ¯l∗ θ¯ ¯∗ f (l ) = ¯∗ . u (t ) 2- The first-best allocation (t∗ , l∗); (t¯∗ , ¯l∗ ) is not implementable. Suppose this menu were offered under asymmetric information. Then, the θ worker will select (t¯∗, ¯l∗ ). Indeed, his utility is then u(t¯∗ ) θ¯l∗ = Δθ¯l∗ > 0
{
}
−
instead of zero for (t∗ , l∗ ). Under asymmetric information, the optimal contract is the solution of max ν (f (l)
l);(t,l)} {(t¯,¯
− t) + (1 − ν )(f (¯l) − t¯), subject to
u(t) u(t¯)
−θ l ≥ − θ¯ ¯l ≥
u(t¯) 0,
− θ ¯l
where we just consider the relevant incentive and participation constraints. Let ψ( ) be the inverse function of u( ), i.e., t = ψ(u). The program can be rewritten as:
·
·
max
{(¯ u,¯ l);(u,l)}
ν [f (l)
− ψ(u)] + (1 − ν )[f (¯l) − ψ(¯u)]
− θ l ≥ u¯ − θ ¯l u¯ − θ¯ ¯l ≥ 0. u
55
The two constraints are binding. Therefore u¯ = θ¯ ¯l and u = θ l + Δθ¯l. Substituting these values in the employer’s objective function and maximizing with respect to ( l, ¯l) yields θ ; u(tSB ) = θl SB + Δθ ¯lSB SB u (t ) Δθ θ¯ ν ¯SB ; u(t¯SB ) = θ¯ ¯lSB . f (l ) = ¯SB + SB u (t ) 1 ν u (t ) f (lSB ) =
−
Since u (tSB ) < u (ψ(θ l SB )) f (lSB )u (ψ(θ l SB )) > θ = f (l∗ )u (ψ(θ l ∗ )), which implies lSB < l ∗ since u < 0 and f < 0. Since θ¯ u (t¯) f (¯lSB )u (ψ(θ¯ ¯lSB )) > θ¯ = f (¯l∗ )u (ψ(θ¯ ¯l∗ )), f (¯lSB ) >
we finally get ¯lSB < ¯l∗ . 3- The firm’s problem is now: max f (l) {t,l}
− t s.t. u(t) − θl ≥ V
yields u(t∗ ) = θl ∗ + V f (l∗ ) u (t∗ ) = θ.
·
Under asymmetric information, the employer’s program is now: max
{(¯ u,¯ l);(u,l)}
ν (f (l)
u
− ψ(u)) + (1 − ν )(f (¯l) − ψ(¯u))
− θl ≥ ¯ − θ¯¯l ≥ u
max(V, u ¯ 0.
− θ¯l)
If Δθ¯lSB V , the solution of question 2 is unchanged, since the rent of asymmetric information is greater or equal to the outside opportunity level of utility.
≥
56
SOLUTIONS
¯l∗ Δθ, there is no reason to distort so much the production level If Δθ¯lSB < V of the inefficient type (the θ-incentive constraint and both participation constraints are binding). The firm’s problem becomes:
≤
max
u,¯ l);(u,l)} {(¯
ν (f (l)
− ψ(θl + V )) + (1 − ν )(f (¯l) − ψ(θ¯¯l)) V
− Δθ¯l ≥ 0
(λ).
It admits the solution: θ , u(t) = θl + Δθ¯l u (t) θ¯ λ Δθ, u(t¯) = θ¯¯l. f (¯l) = ¯ + u (t) 1 ν f (l) =
−
If Δθ¯l∗ < V < Δθl ∗ , 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) = f (¯l) =
θ u (t)
; t = θl + V
θ¯ ; t¯ = θ¯¯l. ¯ u (t)
¯ However, if V becomes greater than Δθl ∗ , the θ-incentive constraint is binding and the firm’s problem becomes: max
u,¯ l);(u,l)} {(¯
ν (f (l)
− ψ(u)) + (1 − ν )(f (¯l) − ψ(¯u))
− θ¯¯l ≥ u − θl¯ = V − Δθl u − θl = V ¯ − θ¯¯l ≥ 0. u ¯ u
¯ There exists two subcases. In the first one, the θ-incentive constraint and both participation constraints are binding. max ν (f (l) {l,¯ l}
− ψ(θl + V )) + (1 − ν )(f (¯l) − ψ(θ¯¯l + V )) −V + Δθl ≥ 0
(λ)
θ¯ ; u(t¯) = θ¯¯l ¯ u (t) θ λ Δθ ; u(t) = θl + V. f (l) = u (t) ν f (¯l) =
−
57
l is increased to reduce the information rent to zero. SB
Let ˆl
be defined by 1 ν Δθ θ SB ˆlSB + ΔθˆlSB ; ) = u(t θ SB ν u (t¯SB ) u (t ) θ¯ f (¯lSB ) = ¯SB ; u(t¯SB ) = θ¯ ¯lSB . u (t ) SB f (ˆl ) =
− −
SB ¯ If V > Δθˆl , then only the θ-incentive constraint and the θ-participation constraint are binding and the solution is characterized by:
1 ν Δθ θ ; u(t) = θl + V u (t) ν u (t¯) θ¯ f (¯l) = ¯ ; u(t¯) = θ¯¯l + V Δθl. u (t) f (l) =
− −
−
58
SOLUTIONS
Control of Self-Managed Firm
1- The optimal size of the self-managed firm is the solution to max
pθ1/2
i.e. LM
=
−A
2
2A pθ
.
2- The optimal allocation of labor is determined by equating the marginal product of labor to the wage w, 2 1 −1/2 pθ ∗ = w or = pθ . 2 2w
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
− τ )θ − A
hence =
2A ( p τ )θ
−
,
2
.
To achieve efficiency we need to equate this term to ∗ , i.e. τ = p
− 4Aw . pθ 2
If instead we use a lump sum tax, the LM firm solves max
pθ1/2
hence =
− A − T ,
2(A + T ) pθ
To achieve efficiency we need p2 θ2 T = 4w
− A.
2
.
59
4- Incentive constraints are pθ¯ ¯1/2 + t¯ ¯ pθ 1/2 + t
pθ¯ 1/2 + t pθ ¯1/2 + t¯ ¯
−A ≥ −A ≥
−A
(1)
− A.
(2)
Adding those two incentive constraints we get p(θ¯
− θ)(¯
−1/2
−1/2
−
)
≥0
¯
or
≤ .
5- The government’s program is max ν ( pθ 1/2
¯ t¯)} {(,t);(,
¯ − w) + (1 − ν )( pθ¯ ¯ − w), 1/2
subject to (1)-(2) and pθ¯¯1/2 + t¯ ¯ pθ1/2 + t
−A ≥ −A ≥
0
(3)
0.
(4)
The firm’s objective function is ∂ ∂θ
U (,t,θ) = pθ−1/2 A−1 1 ∂U/∂ = p−1/2 > 0, 2 ∂U/∂t
−
−1
− t
.
and we could expect the usual constraints to be binding. However, the benevolent government is only interested in efficiency and would like to implement 2 2 pθ pθ¯ ∗ ∗ ¯ ; = = 2w 2w i.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(νθ + (1 {}
or
¯ p − w, − ν )θ)¯
p(νθ + (1 = 2w
1/2
− ν )θ)¯
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
SOLUTIONS
Information and Incentives
1max S (q)
− t subject to t − θq ≥ 0
{(t,q)}
yields S (q) = θ ;
t = θq.
2max
{(t¯,¯ q );(t,q)}
ν (S (q)
− t) + (1 − ν )(S (q¯) − t¯)
subject to t t¯
− θq ¯q − θ¯ t − θq ¯q t¯ − θ¯
≥ t¯ − θ¯q ≥ t − θq¯ ≥ 0 ≥ 0,
yields S (q ∗ ) = θ ; t = θq ∗ + Δθ¯ q ν ¯q. Δθ ; t¯ = θ¯ S (¯ q ) = θ¯ + 1 ν
−
3νμ νμ + (1 ν )(1 μ) ν (1 μ) ¯) = ν¯ = Pr(θ = θ σ = σ . ν (1 μ) + (1 ν )μ ν = Pr(θ = θ σ = σ) =
|
− − − − −
|
For each value of σ (σ and σ¯ ), the optimal contract is characterized as in question 2, where ν is replaced by ν and ν¯ respectively. 4- For any μ, the information structure is characterized by the matrix F =
¯ Pr(σ = σ θ = θ) Pr(σ = σ θ = θ) ¯ Pr(σ = σ¯ θ = θ) Pr(σ = σ¯ θ = θ)
| |
or 1
F =
| |
μ 1
−μ
1
−μ μ
.
61
For μ
− Δμ
F 2 =
μ 1
− Δμ
1
− μ + Δμ
− μ + Δμ μ − Δμ
.
F 1 is an improvement in the sense of Blackwell if there exists a bistochastic matrix B such that F 2 = BF 1 . Take B=
1
Δμ 2μ−1 Δμ 2μ−1
−
1
Δμ 2μ−1 Δμ 2μ−1
−
.
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 (μ) denote the expected utility of the principal (with obvious notations):
| | | |
W (μ) = Pr(σ = σ) Pr(θ = θ σ = σ)(S (q∗ ) +Pr(θ = θ¯ σ = σ) S (q¯(σ, μ)
−
− θq
σ
∗
¯q (¯σ, μ)) − θ¯
This expression can be written symbolically
−
¯q (σ, μ)) θ¯
+ Pr(σ = σ ¯ ) Pr(θ = θ σ = σ ¯ )(S (q∗ ) +Pr(θ = θ¯ σ = σ¯ ) S (q¯(¯σ, μ)
∗
− θq − Δθ¯q(σ, μ) Δθ¯ q (¯σ, μ) .
|
F (q,θ,μ,σ)dG(θ σ) dH (σ).
θ |σ
The difference with a classical decision problem is that F ( ) depends here directly on the precision μ of the signal and the signal σ, because of the presence of the information rents Δθ¯ q(¯σ, μ) and Δθ¯ q (σ, μ) in the principal’s objective function.
·
Therefore
dW ∂W = dμ ∂μ
+
σ
μ fixed in F (q,θ,μ)
θ |σ
Blackwell’s effect
∂F dG(θ σ)dH (σ) . ∂μ
|
6- The expected utility of the principal can be rewritten W = νμ[S (q∗ )
∗
New effect
− θq − Δθ¯q(σ, μ)] ¯q(σ, μ)] +(1 − ν )(1 − μ)[S (q¯(σ, μ)) − θ¯ +ν (1 − μ)[S (q ) − θq − Δθ¯ q (¯σ, μ)] ¯q(¯σ, μ)]. +(1 − ν )μ[S (q¯(¯σ, μ)) − θ¯ ∗
∗
62
SOLUTIONS
By the Envelop Theorem we have: dW = dμ
¯q (σ, μ)] −ν Δθ¯q(σ, μ) − (1 − ν )[S (q¯(σ, μ)) − θ¯ ¯q (¯σ, μ)] +ν Δθ¯ q(¯σ, μ) + (1 − ν )[S (q¯(¯σ, μ)) − θ¯ ν (1 − ν ) Δθ dq > 0, S (q) − θ¯ + 1 − ν
q¯(¯ σ,μ)
=
q¯(σ,μ)
¯ since q¯(¯σ, μ) > q¯(σ) and S (¯ q (¯σ, μ)) > θ.
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 principal’s program is max
q );(t,q)} {(t¯,¯
−
ν t
(q
− Q)
2
2
+ (1
− ν )
− t¯
(¯ q
2
− Q) 2
subject to θq ¯q θ¯
−t − t¯ θq − t ¯q − t¯ θ¯ ¯q As usual t = θq + u0 ; t¯ = θ¯
≥ ≥ ≥ ≥
− Δθq + u
0
θ¯ q ¯ θq u0
− t¯ −t
u0 .
hence the solution
q¯ = Q + θ¯ q = Q+θ
− 1 −ν ν Δθ.
Agents who value more time are offered a higher decrease of delay. This exercise ??? from Saha (2001).
64
SOLUTIONS
Regulation of Pollution 1- Let x∗ (θ) the solution of D (x) + C x (x, θ) = 0.
(1)
If the firms must pay t(x) = D(x) + K it solves min D(x) + C (x, θ) + K x
{
}
yielding (1). 2max {x,t}
{−D(x) − (1 + λ)t + t − C (x, θ)} subject to t
Let U = t
− C (x, θ) ≥ 0.
− C (x, θ). The program can be rewritten max{−D(x) − (1 + λ)C (x, θ) − λU } subject to U
≥0
hence, the solution is D (x) + (1 + λ)C x (x, θ) = 0, t = C (x, θ). The participation constraint requires now the use of public money which has an opportunity cost of 1 + λ. So the marginal disutility of pollution is equal to the social marginal cost of depollution (1 + λ)C x (x, θ) 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
ν ( D(x)
−
− (1 + λ)C (x, θ) − λU ) + (1 − ν ) −D(¯x) − (1 + λ)C (¯x, θ)¯ − λU ¯ subject to
,
65
≥ U ¯ + C (¯x, θ)¯ − C (¯x, θ) ≥ U + C (x, θ) − C (x, θ)¯ U ≥ 0 ¯ ≥ 0, U ¯ ¯ = t¯ − C (¯x, θ). where we use the notation U = t − C (x, θ); U
(2)
U ¯ U
(3) (4) (5)
¯ = C (x, θ) As usual since C θ < 0, θ¯ is the efficient type and U = 0 and U
¯ − C (x, θ).
Hence, the solution ¯ =0 D (¯x) + (1 + λ)C x (¯x, θ) 1 ν ¯ = 0. D (x) + (1 + λ)C x (x, θ) + λ[C x (x, θ) C x (x, θ)] ν
−
−
Since D 0, C xx > 0 and C xθ < 0, x¯ is greater than the full information level. Depollution is more costly because of the information rent which must be given up to elicit the information.
≥
4- Let U (θ) = t(θ) C (x(θ), θ) the level of utility of the truthful firm when it is faced with the DRM (t(θ), x(θ)). The local incentive constraints are
−
˙ (θ) = U
−C (x(θ), θ) ˙ ≥ 0, x(θ)
(6)
θ
since the first-order condition of incentive compatibility is ˙ t(θ)
− C (x(θ), θ)x(θ) ˙ =0 x
and (6) follows from the Envelope Theorem. The second order condition is
˙ xθ (x(θ), θ)x(θ)
−C
≥ 0 or x˙ ≥ 0 since C
xθ
The regulator’s optimization program is θ¯
max
{x(·),U (·)}
{−
D(x(θ))
θ
− (1 + λ)C (x(θ), θ) − λU (θ)} dF (θ) subject to
˙ (θ) = U ˙ x(θ) U (θ)
≥ ≥
−C (x(θ), θ) (−μ(θ)) θ
0
0 for all θ.
Since C θ < 0, the participation constraint reduces to U (θ)
≥ 0.
< 0.
66
SOLUTIONS
The Hamiltonian is H = ( D(x)
−
− (1 + λ)C (x, θ) − λU )f (θ) − μC (x, θ). θ
¯ the transversality condition implies μ(θ) ¯ = 0. Since there is no constraint at θ = θ, From the Pontryagin condition ˙ = μ(θ)
= λf (θ) − ∂H ∂U
hence ¯ μ(θ)
− μ(θ) = λ(1 − F (θ))
or μ(θ) =
−λ(1 − F (θ)).
Maximizing H with respect to x we obtain finally D (x(θ)) + (1 + λ)C x (x(θ), θ)
− F (θ)) C − λ(1 f (θ)
θx (x(θ), θ)
¯ and since C θx < 0 and D There is no distortion at θ = θ, distortion of pollution for all the other types.
≥ 0, C
xx
= 0. > 0, an downward
It remains to check that the second order condition is satisfied, i.e., x(θ) ˙ sufficient condition is d dθ
− ≤ 1
F (θ) f (θ)
0 ;
C θθx < 0; C xxθ
≥ 0.
See Groves and Loeb (1975), Aspremont and G´erard-Varet (1979).
≥ 0.
A
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(θ),x(θ));(q(θ),x( θ))
ν [θ log q(θ) + x(θ)] + (1
¯ − ν )[θ¯ log q(θ)¯ + x(θ)]
subject to νx(θ) + (1
¯ − K − ν )x(θ)¯ = x¯ − c[νq(θ) + (1 − ν )q(θ)]
i.e., q(θ) =
θ c
and
¯ ¯ = θ. q(θ) c
2- A θ-consumer’s utility function writes θ log q + x∗ t where x∗ is a fixed parameter and t is the payment made to the monopoly. Incentive constraints are then
−
θ log q θ¯ log q¯
−t ≥ − t¯ ≥
θ log q¯ θ¯ log q
− t¯ − t.
(1) (2)
3- The monopoly’s maximization program is then max
{(t,q);(t¯,¯ q )}
ν (t
− cq) + (1 − ν )(t¯ − c¯q)
subject to (1)-(2) and θ log q θ¯ log q¯
−t ≥ − t¯ ≥
0
(3)
0.
(4)
We can expect (3) and (2) to be binding. Hence t = θ log q and t¯ = θ¯ log q¯ Δθ log q. Inserting in the objective function of the monopoly and maximizing with respect to q and q¯, we obtain θ¯ θ (1 ν ) Δθ ; q= q¯ = . c c ν c
−
− −
4- When τ is the tax, the monopoly’s problem becomes max
q )} {(t,q);(t¯,¯
ν (t
− cq) + (1 − ν )(t¯ − c¯q) subject to
θ log q
− t − τq ≥ 0
68
SOLUTIONS
θ¯ log q¯
− t¯ − τ ¯q ≥ θ¯ log q − t − τ q
where we just write the relevant constraints or
max ν θ log q {q, q¯}
hence
− (c + τ )q
+ (1
− ν )
θ¯ θ ; q= q¯ = c + τ c + τ
θ¯ log q¯
− (c + τ )¯q − Δθ log q
− (1 −ν ν ) cΔθ . + τ
(5)
Everything happens as if the marginal cost was ˜c = c + τ instead of c. The government’s problem is then
− ν )Δθ log q + σ ν (θ log q − (c + τ )q) + (1 − ν )(θ¯ log q¯ − (c + τ )¯q − Δθ log q) +λ(νq + (1 − ν )q¯)τ
max (1 τ
subject to (5).
The maximand can be rewritten Δθ(1
− σ)(1 − ν ) log q + σν θ log q + σ(1 − ν )θ¯ log q¯ + (λτ − σ(c + τ ))(νq + (1 − ν )q¯).
The derivative with respect to τ is
(1
− σ)(1 − ν )Δθ + σν θ + ν (λτ − σ(c + τ )) q
q
+(λ
dq σ(1 ν )θ¯ + + (1 dτ q¯
−
− ν )(λτ − σ(c + τ ))
− σ)(νq + (1 − ν )q¯).
Using (5) this expression becomes
dq (1 dτ
− ν )Δθ + νλτ q
+
d¯ q (1 dτ
− ν )λτ + (λ − σ)(νq + (1 − ν )q¯).
For λ > 0 and σ λ, (6) is negative for τ > 0. Also τ is bounded below by (5)). The interior solution is then:
≥
(λ
− σ)(νq + (1 − ν )q¯) + τ = − λ ν + (1 − ν )
dq dτ
dq 1−ν Δθ q dτ
d¯ q dτ
(6)
−c (see
< 0.
5- Let U (θ) = θ log q(θ) t(θ) be the utility of a θ-consumer. By the envelope theorem, the incentive constraints are
−
˙ (θ) = log q(θ) U
(7)
˙ q(θ)
(8)
≥
0
for all θ.
d¯ q dτ
69
The monopoly’s maximization program is θ¯
max
{t(·),q(·),U (·)}
(t(θ)
θ
θ¯
− cq(θ))dF (θ) =
(θ log q(θ)
θ
subject to (7)-(8) and U (θ)
− cq(θ) − U (θ))dF (θ)
≥ 0.
The Hamiltonian is H = (θ log q(θ)
− cq(θ) − U (θ))f (θ) + μ(θ)log q(θ).
¯ = 0. From the Since U is increasing, there is no constraint at θ = θ¯ so that μ(θ) Pontryagin principle ˙ = μ(θ)
= f (θ). − ∂H ∂U
(9)
¯ μ(θ) = F (θ) ¯ F (θ) or μ(θ) = Integrating (9) between θ¯ and θ we get μ(θ)
−
−
Maximizing with respect to q( ) we obtain finally
·
1 F (θ) 1 θ =c+ q(θ) f (θ) q(θ)
−
or q SB (θ) =
θ
−
1−F (θ) f (θ)
c
with no distortion at the top only. In the case of a uniform distribution on [2, 3], F (θ) = θ q(θ) = 2θ
−3
or
θ(q) =
− 2 and c = 1:
3+q 2
− − − θ
t(θ) = θ log q(θ)
log q(u)du
θ
T (q) = t(θ(q)) =
3 q + log q 2 2
1 3 1 dT = + + log q 2 2q 2 dq 3+a d2 T = . 2q2 dq2
1 log 2
3 q + 2 2
log q(u)du
2
3 q + 2 2
2
−
T ( ) is concave. There is a discount for buying more (see figure below).
·
−(1−F (θ)).
70
SOLUTIONS
T
3
•
q (θ) 2 q SB (θ) 1 ∗
•
•
1
2
3
E
θ
71
Shared Information Goods, Majority Voting and Optimal Pricing
1- The monopolist’s optimization program is:
2
max 3
{q(·);U (·)}
1
q(θ)2 θq(θ) 2 subject to
−
− U (θ)
dθ
˙ (θ) = q(θ) U ˙ q(θ) yielding q(θ) = 2(θ
≥
0,
− 1) and an expected profit for the monopolist 3 [2(θ − 1)] dθ = 2. π = 2
2
I
2
1
2- Let G(θ) the cumulative distribution function of the median of the types and g(θ) its density function. The monopolist’s program is:
2
max
{q(·);U (·)}
3 θq(θ)
1
−
q(θ)2 2
− U (θ)
dG(θ)
˙ (θ) = q(θ) U ˙ q(θ) yielding q(θ) = θ
−
1−G(θ) g(θ)
≥
0,
and an expected profit for the monopolist 3 π II = 2
Given that G(θ) = (5 2θ)(θ be its inverse function. As
−
− 2
1
θ
− G(θ) g(θ)
1
dG(θ).
2
− 1) , we can check that q(θ) is increasing in θ. Let θ(q) 2
U (θ) = θq(θ)
− q(θ) − t(θ) 2
−
θ(q) q2 T (q) = t(θ(q)) = θ(q)q q(u)du. 2 1 Then each agent’s utility function is single-peaked, and the majority rule yields the choice of the median agent. Indeed
−
V (θ, q) = θq
q2 2
− − T (q)
yields V qq (θ, q) = 3- G(θ) = (θ
2
− 1) (5 − 2θ). Then π
I
−θ (q) < 0. = 2 and π ≈ 9.25. II
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 agent’s incentive constraints are t t¯
− ψ(θy) ≥ t¯ − ψ(θ¯y) ¯y) ≥ t − ψ(θy). ¯ − ψ(θ¯
(1) (2)
2- The principal’s optimization program is: max
{(y,t),(¯ y,t¯)}
ν [y
− t] + (1 − ν )[¯y − t¯]
subject to (1)-(2) and t t¯
− ψ(θy) ≥ ¯y) ≥ − ψ(θ¯
0
(3)
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 ¯y ) and t = ψ(θy) + ψ(θ¯ ¯y) t¯ = ψ(θ¯
− ψ(θ¯y).
Substituting these solutions in the principal’s objective function and maximizing with respect to y, y¯ we obtain: 1 ψ () = ψ (θy) = ; θ ¯ () ¯ θψ () 1 ν θψ ¯ ¯ ψ () = ψ (θ¯ y) = ¯ . θ 1 ν θ¯
−
− −
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 (1). (2) becomes: ¯ + ψ(θy) + ψ(θ¯ ¯y) ψ(θ¯ 0 ψ(θy) y)
≥−
or
−
θ¯
0
≥ y¯
θ
ψ (θ¯ y)dθ
θ¯
− y
θ
ψ (θy)dθ
73
or θ¯
0
≥ (¯y − y)
ψ (θ¯ y )dθ
θ
θ¯
− y
y
θ
θ
ψ (θy)dydθ.
(5)
y¯
From the optimal solution > ¯ implies y > y¯ hence (5) holds. Indeed the Spence-Mirrlees condition writes here. If U = (t ∂ ∂θ
∂U ∂y ∂U ∂t
=
∂ ( θψ (θy)) = ∂θ
−
−ψ − θyψ
− ψ(θy)),
< 0,
and we know from Section 2.11 that the problem is well behaved and the constraints are binding as we guessed. Chari (1983), Green and Kahn (1983), Hart (1983a), Hart and Holmstr¨om (1987).
74
SOLUTIONS
II- MORAL HAZARD Lending with Moral Hazard
The optimization problem of the bank is: max p(z ¯ x≥0
− x) − r
subject to ¯ px
−ψ ≥ ¯ −ψ ≥ px
px
(1)
0.
(2)
ψ The incentive constraint (1) implies x = p¯− . From (2) the expected rent of the p entrepreneur is: pψ R= . p¯ p
−
The bank’s expected profit is ¯ pz
¯ − p¯ pψ − p − r.
Hence, the optimal individual contract entails a loan which induces effort if ¯ pz Otherwise, there is no lending.
≥ ψ + r + p¯ pψ − p .
75
Moral Hazard and Monitoring
1- Since pL = 0, it is worth financing the project only if one can design a contract which induces effort eH , satisfies the entrepreneur’s participation constraint and ensures non negative profit, i.e., pH (R
B
(1)
pH
0
(2)
I.
(3)
− P ) ≥ (R − P ) ≥ p P ≥ H
From (1), P = R
−
B pH
and from (3) pH R
≥ B + I .
2- pH R > I but pH R < B + I and m < pH R. It is immediate to check that there is no pure strategy equilibrium. Let us compute the mixed strategy equilibrium. Let μ1 = Pr(e = eH ), μ2 = Pr (Monitor). The entrepreneur must be indifferent between the two effort levels, i.e. μ2 pH (R
− P ) + (1 − μ ) p 2
H (R
− P ) = μ × 0 + (1 − μ ) · B 2
2
or pH (R
− P ) = (1 − μ )B. 2
The lender must be indifferent between monitoring or not, i.e. μ1 [ pH P
− m] + (1 − μ )( p R − m) = μ p 1
1 H P
H
or m = (1
− μ ) p 1
H R.
The participation constraint of the entrepreneur is pH (R
− P ) ≥ 0.
So the bank can now choose a best repayment of P = R. The mixed strategy equilibrium entails μ2 = 1 and μ1 = 1 pHmR . But then the effort is always high since the lender always monitors. Lending occurs if
−
pH R
− m ≥ I.
76
SOLUTIONS
Inducing Information Learning
1- From Bayes rule ¯) Pr(S ¯) Pr(¯ σ S ¯ Pr(S σ ¯) = = Pr(¯ σ) ¯ (1 θ)(1 θν S + ¯) = E (S σ θν + (1 θ)(1 ¯ θ(1 (1 θ)ν S + E (S σ) = (1 θ)ν + θ(1
|
| |
− −
|
− −
θν θν + (1 θ)(1 ν ) ¯ (2θ 1)ν S ν )S = >0 ν ) θν + (1 θ)(1 ν ) ¯ (1 2θ)ν S ν )S = < 0. (1 θ)ν + θ(1 ν ) ν )
−
− − − −
−
− − − −
− −
So, the principal chooses the risky project if σ = σ¯ and the safe project if σ = σ. His expected utility if he decides to learn information is (2θ
− 1)ν S ¯ − ψ,
and learning is optimal if this expression is positive. 2- The agent chooses the risky project if he learns σ ¯ when (1 θ)(1 ν )t θν t¯ + θν + (1 θ)(1 ν ) θν + (1 θ)(1 ν )
−
−
−
−
−
He chooses the safe project if he learns σ when (1 θ)ν t¯ θ(1 ν )t + (1 θ)ν + θ(1 ν ) (1 θ)ν + θ(1
−
−
−
−
−
− ≥t .
(1)
− ν ) ≤ t . 0
(2)
t0
(3)
0
3- Ex ante, the agent decides to learn information if θν t¯ + (1
− θ)(1 − ν )t + [(1 − θ)ν + θ(1 − ν )]t − ψ ≥ ≥ 0
νt + (1
− ν )t¯.
(4)
(3) (resp. (4)) says that he prefers to learn the information (and follow the optimal choices induced by (1), (2)) rather than not learning and always choosing the safe project (resp. not learning and always choosing the risky project). 4- But, note that (3) implies (1) and (4) implies (2). If the principal induces the agent to learn information then the agent’s decision rule is optimal. Since the principal wants to minimize payments he chooses t = 0 t0 = t¯ =
(2θ
−
(2θ
−
ψ 1)(1 ν ) ψ . 1)ν (1 ν )
−
−
77
The principal’s expected utility is then (2θ
− 1)ν S ¯ − θν t¯ − [(1 − θ)ν + θ(1 − ν )]t ¯ − ψ · θ + (1 − θ)ν + θ(1 − ν ) = (2θ − 1)ν S (2θ − 1)(1 − ν ) 0
.
(5)
5- It is worth inducing information revelation only if (5) is positive. This happens less often than if the principal was using the information technology himself since (5) can be rewritten ψ ¯ ψ (2θ 1)ν S . (2θ 1)(1 ν )
−
− −
Otherwise, he should not use the agent.
−
Inspired from Gromb and Martimort (2002).
−
78
SOLUTIONS
Optimal Contract and Limited Liability
1- The optimal contract solves the following problem
q¯
max e
qf ( qf (q e)dq
|
0
− ψ(e)
Assuming the f ( f ( ) satisfies CDFC, we have a strictly concave objective function and the optimal first-best effort solves
·
qf e (q e∗ )dq = ψ (e∗ ).
t(q)f e (q e)dq = ψ (e),
q¯
|
0
2- The incentive constraint is q¯
(1)
|
0
provided the agent’s objective function is strictly concave. It is when t (q ) > 0, and f ( f ( ) satisfies CDFC.
·
The participation constraint is
q¯
t(q)f ( f (q e)
| − ψ(e) ≥ 0.
0
(2)
3- The principal’s problem becomes:
q¯
max
0≤t(q )≤q
(q
0
− t(q))f ))f ((q |e)dq
subject to (1) and (2). Writing the Lagrangian, we get: L(q,t,e) q,t,e) = (q
)], − t)f ( f (q|e) + λ[tf (q |e) − ψ (e)] + μ[tf ( tf (q |e) = ψ (e)], for t ∈ [0, [0, q ] where λ and μ are respectively the multipliers of (1) and (2) (μ ( μ ≥ 0 because e
(2) is an inequality.
The objective is linear in t and the solution has a bang-bang feature if if if
q
(μ (μ (μ
1)f ((q|e) + λf (q|e) − 1)f − 1)f 1)f ((q|e) + λf (q|e) 1)f ((q|e) + λf (q|e) − 1)f e e e
>0 <0 =0
then t = q then t = 0 then t [0, [0, q ].
∈
Note that MLRP ensures that there exists a unique q ∗ such that q ∗.
≥
Hence t = q iff q
∗
≥ q , t = 0 otherwise.
f e (q |e) f ( f (q |e)
>
1−μ λ
for
79
The Value of Information under Moral Hazard
1- The agent solves max E (θ˜ + e)t¯ e
θ˜
− ψ(e)
and we get ψ (e) = t. t¯. The principal solves ¯ + (1 max eS + S e
− e)S − ψ (e)e
which yields ΔS = ψ (eSB ) + eSB ψ (eSB ). 2- Now the agent adjusts his effort e to have always a probability of success of R, i.e., e+θ = R
or
e=R
− θ.
He must be paid E (ψ (e)) = E (ψ (R θ˜
θ˜
− θ˜))
to guarantee an effort R. The principal solves ¯ + (1 RS + S
− R)S − E (E ψ (R − θ˜))
θ˜
θ˜
and the first-order condition is ΔS = E (ψ (R θ˜
− θ˜) + Rψ (R − θ˜)). )).
From ψ > 0 and Jensen’s inequality, we have E (ψ (R θ˜))
θ˜
SB
≥ ψ (R) hence R
SB
.
− θ˜)) ≥ ψ (R) and E (ψ (R −
θ˜
80
SOLUTIONS
Raising Liability Rule
1- The optimal level of effort is determined by minimizing h(1 ψ (e∗) = h. Social welfare is then V
∗
− e) + ψ(e),
∗
− (1 + r)I − (1 − e )h − ψ(e ) > 0.
i. i.e. by
(1)
2- The proper incentive for the agent is obtained by choosing t¯ and t such that ψ (e∗ ) = t¯
− t.
The agent’s participation constraint can be made binding by choosing t high enough V
− (1 + r)I − e t¯ − (1 − e )t − ψ(e ) = 0. ∗
∗
∗
But then the payment to the bank e∗ t + (1
− e )t¯ = V − (1 + r)I − ψ(e ) > (1 − e )h ∗
∗
∗
is from (1) enough to pay h to the third party, when damage happens. 3- If the limited liability constraint of the borrower is binding, we have t¯ = w t = w
∗
− ψ (e ),
and the expected revenue of the bank is w
∗
∗
− (1 − e )ψ (e )
which must be larger than (1 + r)I + I + (1
∗
− e )c.
Clearly, as w decreases this may become impossible. 4- This contract solves max V
{e,t,t t¯,t}
− et − (1 − e)t¯ − ψ(e) subject to
t¯
−t
t¯
et¯ + (1
− e)t
= ψ (e)
≤
w
= (1 + r )I + (1
− e)c,
81
where the latter constraint is the bank’s zero profit condition. This yields max V e
− w + (1 − e)ψ (e) − ψ(e) subject to
w
− eψ (e) = (1 + r)I + (1 − e)c.
Therefore e is determined by the zero profit constraint (1
− e)c + eψ (e) = w − (1 + r)I.
5-
de = dc
for c small.
−e <0 − ψ +1 eψ −c
6- The bank’s program is then max et¯ + (1
{e,t¯,t}
− e)t − (1 − e)c
t¯ t = ψ (e) t¯ = w
−
which reduces to max w e
− eψ (e) − (1 − e)c
or eψ (e) + ψ (e) = c with
1 de = > 0. dc eψ + 2ψ
82
SOLUTIONS
Risk-Averse Principal and Moral Hazard
1- If the agent is risk neutral he will insure completely the principal so that: t=q
− k and t¯ = q¯ − k
which implies t¯
∗
− t = Δq and ψ (e ) = Δq.
Finally, k is chosen to extract all the agent’s surplus e∗ q¯ + (1
∗
∗
− e )q − ψ(e ) = k.
2- If the agent is protected by limited liability we get instead the standard results, t¯ = ψ (e).
t=0
The participation constraint of the agent is satisfied from the convexity of ψ(e) since ψ convex and ψ(0) 0 implies
≥
eψ (e)
− ψ(e) ≥ 0.
The principal solves max e e
− 1
exp( ρ(¯ q ρ
− − ψ (e)))
+ (1
− e)
− 1
exp( ρq) . ρ
−
This yields the following first-order condition Δq = ψ (eSB ) + If ρ
SB
SB
→ 0, ψ (e ) + e If ρ → ∞, e → e . SB
1 ln(ρ + eSB ψ (eSB )). ρ
ψ (eSB ) = Δq as in Section 5.1.2.
∗
As the principal is more risk averse, he finds it more interesting to get more insurance by having a constant payoff across states. In the limit of an infinite risk aversion, the agent’s rent is no longer viewed as costly and the first-best effort guarantees full insurance.
83
Poverty, Health Care and Moral Hazard
1- The problem of the utilitarian social welfare maximizer is max(1 ( p,δ)
+ δπ0 )u(w
−π
0
− p) + (1 − δ)π u
0 M (w)
subject to δπ 0 m = (1
−π
0
+ δπ 0 ) p.
Hence, the first order conditions hold: u(w
∗
− p ) − u
M (w)
= (m p∗ )u (w
−
∗
− p )
(1)
(1 π0 ) p∗ δ = , π0 (m p∗ )
−
∗
(2)
−
(1) determines p∗ and (2) determines δ∗ . Let g1 ( p) = u(w
M (w)
− p) − u
and g2 ( p) = (m p)u (w
−
− p).
Then g1 ( ) cuts only once g2 ( ) from above. Then, it is immediate to see that
·
·
and from (2)
dp∗ < 0, dm
dp∗ > 0, dw
dp∗ = 0, dπ0
dδ∗ < 0, dm
dδ∗ > 0, dw
dδ∗ < 0. dπ0
2- The problem becomes: max(1 ( p,δ)
−π
1
+ δπ 1 )u(w
− p) + (1 − δ)π u
1 M (q)
−ψ
subject to δπ 1 m = (1 (1
−π
1
+ δπ 1 ) p
(3)
− π + δπ )u(w − p) + (1 − δ)π u (w) − ψ ≥ (1 − π + δπ )u(w − p) + (1 − δ)π u (w). 1
1
1 M
0
0
0 M
(4)
Suppose (4) is satisfied. Then, we obtain a solution like in 1 with π1 instead of π0 . It is indeed the solution if (4) holds for the optimal solution p∗ (π1 ), δ ∗ (π1 ), i.e., if ψ
∗
∗
≤ Δπ(1 − δ (π ))(u(w − p (π )) − u 1
1
M (w)) =
ψ∗ .
84
SOLUTIONS
Then, the fact that poverty prevents treating every one is enough to create the incentives for health care. For ψ higher than ψ∗ , the incentive constraint becomes binding. Substituting (3) in (4) the optimal second best premium is such that
−
pSB (π1 )(1 π1 ) π1 (m pSB (π1 ))
Δπ 1
−
−
SB
u(w
− p
(π1 ))
M (w) = ψ,
−u
pSB (π1 ) (and therefore δ(π1 )) decreases with ψ. For ψ even higher, health care is given up. 3- Let u11 = u(w ϕ( ) u−1 ( ).
·≡
·
−t
11 ),
u12 = u(w
12 ),
−t
u21 = u(w
21 ),
−t
u22 = u(w
22 )
−t
and
Maximizing expected social welfare we have: max
u11 ,u12 ,u21 ,u22
(1
2
−π ) u 1
11
+ π1 (1
− π )(u 1
12
+ u21 ) + π12 u22
−ψ
subject to (1 (2
− π )u 1
11
+
2π1 1 (u12 + u21 ) 2
−
−π u ≥ 1 22
− π − π )u + (π + π − 1)(u + u ) − (π + π )u ≥ (1 − π ) ϕ(u ) + π (1 − π )(ϕ(u ) + ϕ(u )) + π ϕ(u ) ≤ 1
1
0
2
11
11
1
0
1
12
1
21
12
1
21
0
2 1
22
22
ψ , 2Δπ ψ Δπ w π1 m.
−
(5) (6) (7)
Suppose that the constraint (6) is slack, and let λ1 (resp. μ1 ) the multiplier of (5) (resp. (7)). Then, the first order conditions are 1 λ1 + μ1 μ(1 π1 ) 1 λ1 (2π1 1) + ϕ (u12 ) = ϕ (u21 ) = μ1 2π1 (1 π1 )μ1 1 λ1 ϕ (u22 ) = , μ1 μ1 π1 ϕ (u11 ) =
−
−
−
−
from which u12 = u22 , u11 > u 22 , u11 > u 12 and u12 > u22 . Note that (6) can be rewritten (1
− π )u 1
11
+
(2π1 1) (u12 + u21 ) 2
−
Hence for ϕ(u) = 12 u2 , u11 satisfied.
−u
12
Δπ ψ + (u − u − π u ≥ 2Δπ 2
+ u22
1 22
−u
21
11
12
+ u22
21 ).
−u
(8)
= 0 and the global incentive constraint is
85
In an individual contract we must solve: max(1
− π )u 1
1
+ π1 u2
ψ − u ≥ Δπ (1 − π )ψ(u ) + π ψ(u ) ≤ w − π m u1
1
1
1
2
2
1
(λ) (μ),
hence 1 λ + μ μ(1 π1 ) 1 λ ψ (u2 ) = , μ μπ1 ψ (u1 ) =
−
−
which is like having the constraints u11 = u12 and u21 = u22 in the previous problems. This can only happen if λ1 = 0 which is impossible when there is moral hazard. Laffont, J.J. (2003a).
86
SOLUTIONS
Group Lending with Moral Hazard
1- Since r > pz, the only potentially valuable contract is a contract which induces effort. The bank solves the problem: max p(z ¯ x) r
− −
x≥0
¯ px
−ψ ≥ ¯ −ψ ≥ px
px
(1)
0,
(2)
where (1) is the incentive constraint and (2) the participation constraint. Clearly (1) implies (2) and is binding so that ψ x= p¯ p
−
and the bank’s expected profit is ¯ pz
¯ − p¯ pψ − p − r.
(3)
If (3) is negative, then there is no lending. So, because the bank maximizes profit, it may not lend, when it is socially valuable to lend. We may indeed have ¯ pz
¯ − p¯ pψ ¯ − r. − p − r < 0 < pz
2- Effort provision is a Nash Equilibrium if ¯ px + (1 p(¯
− p)y) ¯ − ψ ≥ p(¯ ¯ px + (1 − p)y).
(4)
The bank’s expected profit per entrepreneur is ¯ p(z
¯ − (¯ px + (1 − p)y)) − r.
From (4) ¯ + (1 px hence ¯ p(z
ψ − p)y ¯ ≥ ; p¯ − p
ψ ¯ ¯ − r − p¯ − (¯ px + (1 − p)y)) − r ≤ pz . p¯ − p
3- Constraint (3) in the text is implied by (1). Suppose that the local incentive constraint is not binding. It can be rewritten: ¯ p¯2 x + p(1
2
¯ − ψ ≥ p x + p(1 − p)y. − p)y
87
The bank’s expected profit is 2¯ p(z
− (¯ px + (1 − p)y)) − 2r. ¯
ψ The solution of the maximization’s bank under constraint (1) is x = p¯2 − , y = 0, i.e., p2 the entrepreneurs are rewarded only if they both succeed. It remains to check that (2) is satisfied by this solution, i.e.:
2(¯ p2
− p¯ p) · p¯ −ψ p ≥ p¯ 2 ≥ p¯ + p 2
2
ψ 1 or p¯
≥ p
which holds. The bank’s expected profit per entrepreneur is ¯ pz
p¯2 ψ = pz ¯ p¯2 p2
−r− −
¯ − r − p¯ p+¯ p · p¯ pψ − p
greater than the expected projet in the optimal individual contract ¯ pz
¯ − r − p¯ pψ − p .
Group lending dominates. Under group lending, agents coordinate their choices of effort. When considering a deviation to zero effort, an agent takes into account the negative externality he exerts on the other since such a deviation makes it less likely that the other agent receive a reward. Incentives provision is easier with group lending. Laffont, J.J. and P. Rey (2003b).
88
SOLUTIONS
Incentives and Discovery
1- The principal solves max u(q) + t subject to
− q) − t ≥ 0,
u(θ hence the solution q =
θ 2
and t = u
2- The principal solves max
{(¯ q,t¯);(q,t)}
θ 2
.
ν (u(¯ q ) + t¯) + (1
− ν )(u(q) + t),
subject to ¯ = u(θ¯ U
− q¯) − t¯ ≥ U = u(θ − q) − t ≥ ν (u(θ¯ − q¯) − t¯) + (1 − ν )(u(θ − q) − t) ≥
u(θ¯
− q) − t u(θ − q¯) − t¯ 0.
From Chapter 2, we know that, under ex ante contracting with a risk-neutral agent, the principal achieves the first best θ¯ q¯ = , 2
θ q= ; 2
¯ ) which implement this and that there are many pairs of transfers (t, t¯) or rents (U , U allocation. ¯ 3- Suppose on the contrary that the “usual” constraints, i.e., the θ-incentive constraint and θ-participation constraint are binding. Then: U = u(θ ¯ = u(θ¯ U
− q) − t − q¯) − t¯
= 0 = u(θ¯ = =
− q) − t u(θ − q) − t + u(θ¯ − q) − u(θ − q) u(Δθ + θ − q) − u(θ − q).
When those constraints are binding, the optimization of the principal yields immediately: ν u (q) = u (θ q) + u (θ q) u (θ¯ q) . 1 ν
−
−
− −
−
89
Since u < 0, u (θ
− q) > u (θ¯ − q), hence u (q) > u (θ − q) hence q < θ − q or q <
θ . 2
Therefore q < θ and ¯ = u(Δθ + θ U
− q) − u(θ − q) < u(Δθ) − u(0) = u(Δθ),
so that the θ¯ ex post participation constraint (2) is not satisfied, a contradiction with the assumption that the optimal contract entails the postulated binding constraints. 4- If only the participation constraints are binding t = u(θ
t¯ = u(θ¯
− q);
The optimal solution entails θ q= 2
− q¯) − u(Δθ).
θ¯ q¯ = . 2
and
These are the relevant constraints when Δθ > θ. Indeed, let us check that the incentive constraints are then slack
−
θ¯ u 2
−
θ t¯ = u(Δθ) > u Δθ + 2
θ , 2
u
from the concavity of u( ):
·
−
θ u 2 since
t = 0 > u( Δθ)
θ¯ 2
u
−
−
θ¯ u , 2
> u(0) > u( Δθ).
−
The principal’s expected welfare is
θ¯ + (1 W (ν ) = 2 νu 2
− ν )u
− θ 2
νu(Δθ).
5- Note that for Δθ = θ or θ¯ = 2θ
− − −
dW (ν ) = 2 u(θ) dν = u(θ)
2u
θ 2
u
θ 2
u(θ)
< 0.
Note that
− − −
d dW dθ¯ dν
d θ¯ = ¯ 2 u 2 dθ
θ u 2
u(Δθ)
=u
θ¯ 2
u (Δθ) < 0,
90
SOLUTIONS
¯ iff θ2 < Δθ or θ¯ > 2θ or Δθ > θ.
So
dW (ν ) dν
< 0 for all θ¯ such that Δθ > θ.
Since W is decreasing in ν , the principal does not wish to encourage effort but in the contrary to discourage effort. He solves the program: max ν 0 (u(¯q )
(q,¯ q,t,t¯)
− t¯) + (1 − ν )(u(¯q) − t¯), 0
subject to
u(θ u(θ¯
− q) − t ≥ − q¯) − t¯ ≥ ν (u(θ¯ − q¯) − t¯) + (1 − ν )(u(θ − q) − t) ≥ 0
0
0
(1)
u(Δθ) ν 1 (u(θ¯
(2)
− q¯) − t¯) + (1 − ν )(u(θ − q) − t) −(3) ψ. 1
With (1) is binding, (2) and (3) become u(θ¯ or, since u(Δθ) <
− q¯) − t¯ ≥
ψ max u(Δθ), , Δν
ψ , Δν
t¯ = u(θ¯
ψ − q¯) + Δν
and
t = u(θ
− q). ¯
Therefore, effort is discouraged with the efficient sharing of resources q = 2θ , q¯ = 2θ . Laffont, J.J. (2003c).
91
III- MIXED MODELS Political Economy of Regulation
1max
{t,C 1 ,C 2 ,e1 ,e2 }
W = S 1 + S 2
− (1 + λ)(t + C + C ) + U 1
2
subject to t
−
1 2 (e1 + e22 ) + γe 1 e2 2
C i = β
0
i = 1, 2,
−e
i
hence e1 = e2 = 2- Let U = t
≥
1 . 1 + γ
− ψ(e , e ) = t − ψ(β − C , β − C ). 1
2
1
2
The incentive constraints are
− ψ(β ¯ − C ¯ , β ¯ − C ¯ ) ≥ t − ψ(β ¯ − C , β ¯ − C ) ¯ , β − C ¯ ), t − ψ(β − C , β − C ) ≥ t¯ − ψ(β − C
¯ = t¯ U
1
2
1
2
U =
1
2
1
2
or ¯ U U ¯ with Φ(C 1 , C 2 ) = ψ(β
≥ ≥
U Φ(C 1 , C 2 ) ¯ + Φ(C ¯1 , C ¯2 ), U
(1)
−
(2)
− C , β ¯ − C ) − ψ(β − C , β − C ). 1
2
1
2
The participation constraints are ¯ U U
≥ ≥
0
(3)
0.
(4)
Expected welfare can be written
−
− (1 + λ) (1 ν ) ψ(β ¯ − C ¯ , β ¯ − C ¯ ) + C ¯ + C ¯ ¯ ). +ν (ψ(β − C , β − C ) + C + C ) − λ(νU + (1 − ν )U
EW = S 1 + S 2
1
2
1
1
2
2
1
2
92
SOLUTIONS
Assuming that (2) and (3) are binding and optimizing we find 1 1 + γ 1 = e¯2 = 1 + γ
e1 = e2 = e¯1
− 1 +λ λ 1 −ν ν Δβ.
3- For a majority of non-shareholders we find 1 1 + γ 1 = e¯2 = 1 + γ
e1 = e2 = e¯1
− Δβ.
For a majority of shareholders we find 1 1 + γ 1 = e¯2 = 1 + γ
e1 = e2 = e¯1
−
(1 + λ)α 1 (1 + λ)α
−
ν
1
− ν Δβ.
So a majority of non-stakeholders (stakeholders) distorts downward too much (too little) effort, because they undervalue (overvalue) rents with respect to the maximization of social welfare. This democratic fluctuation of policies is detrimental and opens the possibility that decentralization of the regulation of each project in each region could dominate this centralized regulation. Laffont and Pouyet (2003).
93
Regulation of Quality
For the solution see Laffont and Tirole (1993), Chapter 4.
94
SOLUTIONS
Enforcement and Regulation
1- The regulator solves max S (q) + λp(q)q
(q,e,U )
− (1 + λ)((β − e)q + F + ψ(e)) − λU subject to U = t
− ψ(e) ≥ 0.
The solution is S (q∗ ) + λ( p (q∗ )q∗ + p(q ∗)) = (1 + λ)(β
∗
−e )
ψ (e∗ ) = q ∗ U = 0. 2- From the Revelation Principle we can restrict the analysis to the pair (t, c); (t¯, c¯) . It is incentive compatible if
{
U = t ¯ = t¯ U
}
− ψ(β − c) ≥ t¯ − ψ(β − c¯) − ψ(β ¯ − c¯) ≥ t − ψ(β ¯ − c).
The firm’s ex ante participation constraint is νU + (1
− ν )U ¯ ≥ 0.
From Chapter 2 we know that efficiency is achieved for adverse selection problems and ex ante participation constraints when the agent is risk neutral. There are many transfers (or rents) which implement the efficient allocation. 3- The outcome of renegotiation entails efficient levels of production q¯∗ and effort e¯∗ since ¯, and the firm’s rent renegotiation takes place under complete information when β = β level ˆ q ∗ , e¯∗ , β ¯) + H F δW (¯ E ¯ = U . 2λ 2
−
4- The firm’s new participation constraint is νU + (1
− ν )π(c)U ¯ + (1 − ν )(1 − π(c))U ¯ ≥ 0. E
95
The regulator solves now: ¯) − λU ˆ q, e¯, β ¯] − λU ] + (1 − ν )π(c)[W (¯ ¯) − λU ¯ ] − (1 + λ)c, +(1 − ν )(1 − π(c))[δW (¯ q , e¯ , β
ˆ max ν [W (q,e,β )
(q,e,¯ q,¯ e,c)
∗
∗
E
under the participation and incentive constraints. We obtain immediately
(1
Laffont, J.J. (2002).
qE = q∗
eE = e∗
q¯E = q¯∗
e¯E = e¯∗
E
− ν )π (c
)=
(1
−
1+λ . ¯) ˆ q ∗, e¯∗ , β δ)W (¯
96
SOLUTIONS
Regulation of a Risk Averse Firm
1- The participation constraints are
− ψ(β ¯ − C ¯ )) ≥ u(t − ψ(β − C )) ≥ u(t¯
0 0.
The incentive constraints are
− ψ(β ¯ − C ¯ )) ≥ u(t − ψ(β − C )) ≥ u(t¯
u(t u(t¯
− ψ(β ¯ − C )) − ψ(β − C ¯ )).
Since u > 0 and u(0) = 0 they reduce to
− ψ(β ¯ − C ¯ ) ≥ π = t − ψ(β − C ) ≥ ¯ = t¯ π
0
(1)
0
(2)
− ψ(β ¯ − C ¯ ) ≥ t − ψ(β ¯ − C ) ¯ ). t − ψ(β − C ) ≥ t¯ − ψ(β − C t¯
(3) (4)
2- Social welfare adds the utility of consumers when they pay for the cost (t + C ) from distortionary taxes with a social cost of public funds 1 + λ, and the certainty equivalent of the manager’s welfare. As usual (1) and (4) are the relevant constraints and they can be rewritten π¯ π
≥ ≥
0
(5)
¯ ¯ + ψ(β π
− C ¯ ) − ψ(β − C ¯ )
¯) H (C
The regulator’s optimization program reduces to max S
¯ π} {C, C,π,¯
− (1 + λ)
ν (π + ψ(β
(6)
− C ) + C ) + (1 − ν )(¯π + ψ(β ¯ − C ¯ ) + C ¯
+u−1 (νu(π) + (1
− ν )u(¯π))
subject to (5)-(6) or max S
¯} {C, C
− (1 + λ)
¯ ) + ψ(β ν (H (C
− C ) + C ) + (1 − ν )(ψ(β ¯ − C ¯ ) + C ¯ )
97
¯ ))), +u−1 (νu(H (C hence ψ (e) = 1 ν
ψ (¯e) = 1 +
1
−
¯) H (C ν
− (1 −
¯ ))H (C ¯) νu (H (C ¯ ))) ν )(1 + λ)u (u−1 (νu(H (C
3- When the firm is risk neutral, u(x) = x. Then u (c)˙ = 1, ψ (e) = 1 ν
ψ (¯e) = 1 +
1
−
¯) H (C ν
¯) H (C (1 ν )(1 + λ)
− −
¯ )) > ν u(H (C ¯ )) implies H (C ¯ ) > u−1 (νu(H (C ¯ ))) or u (H (C ¯ )) < u (u−1(νu(H (C ¯ )))). u(H (C Thus, ψ ( ) > 0 implies e¯ is smaller when the firm is risk-neutral.
·
41 (1 e−ρx ) ρ u (H ) = e−ρH u(x) =
ψ (¯e) = 1 +
ν 1
− ν
− 1
−
e−ρH H . ¯) ρH ( C − (1 + λ)(1 ν (1 e ))
−
−
If ρ = 0 ψ (¯e) = 1 + = 1
−
λ ν ¯) H (C 1 + λ 1 ν λ ν Φ (¯e). 1 + λ 1 ν
− −
−ρH
As ρ increases, e¯ decreases if (1−ν (1e−e−ρH (C¯) )) decreases with ρ (since H < 0). But this expression is equal to (1
−
1 ν )eρH (C ¯ ) + ν
which is a decreasing function of ρ. So effort decreases as risk aversion increases. Indeed, the regulator perceives more negatively the randomness of the firm’s rents and decreases effort to decrease the information rent.
98
SOLUTIONS
Technological versus Informational Advantage
1- Optimal regulation for firm 1: max S
(t1 ,C 1 )
− (1 + λ)(t
1
−
+ C 1 ) + t1
e21 4
subject to (P C )
t1
2 1
− e4 ≥ 0.
t1 is costly, therefore, P C will bind. Substituting P C into the objective function, the problem becomes: e21 max S (1 + λ) + β 1 e1 e1 4
−
FOC with respect to e1 : (1 + λ)
− (1 + λ)
e∗1 2
−
=0
e∗1 = 2.
(1)
If β 1 = 2, then C 1 = 2 e∗1 = 0 e∗12 = 1. t1 = 4
−
If β 1 = 3, then C 1 = 3 e∗1 = 1 e∗12 = 1. t1 = 4
−
(2)
Ex ante welfare (expected) is: 1 W 1 = [S 2
− λ − 1] + 12 (S − λ − (1 + λ) − 1)
or W 1 = S
− 32 (1 + λ).
(3)
2- Optimal contract for firm 2. Regulator solves:
− −
− −
1 max S (1 + λ)(t¯2 + 3k e¯2 ) + t¯2 {e¯2 ,e2 } 2 1 + S (1 + λ)(t2 + 3k e2 ) + t2 2
−
−
e¯22 4 e22 4
.
101
If k < R 1 , then choose firm 2 at the interim stage when β 1 = 2 for firm 1. Case 2: β 1 = 3. Can be done in a similar manner as Case 1. 5- For any k, W 2 is now given by
1 λ(1 + 2λ) 2 W 2 = k 2 4(1 + λ)
− (5 + 6λ)k
+ [S + (1 + λ)]
− k) − (1 + λ)(3 2
2
from (7). So the optimal k will now be given by: max W 2 k
−
(1 + λ)(3 2
− k)
2
gives k∗ =
2(1 + λ) . 7λ + 2λ2 + 4
(12)
The optimal strategy depends upon comparing W 2 with W 1 : W 2
−
1 λ(1 + 2λ) 2 W 1 = k 2 4(1 + λ)
or W 2
− W = 1
−
−
(1 + λ)(3 (5 + 6λ)k +[S +(1+λ)] 2
2λ2 7λ 4 2 1 k + k 8(1 + λ) 2
− −
−
− 2(1 + λ) < 0
− k) − S − 3 (1 + λ) 2
for any k in (0, 1).
Therefore, the optimal strategy is to always choose firm 1. Note that W 1 does not depend on k, so any k in (0, 1) is optimal.
.
102
SOLUTIONS
Piracy and Optimal Pricing
1max E
(q0 ,qc )
is equivalent to max
−
e−ρ(R(q0 +αqc )− p0q0 −c˜qc
−ρ[R(q0 +αqc )− p0 q0 −μqc − 12 ρqc2 σ 2 ]
(q0 ,qc )
−e
or R (q0 + αqc ) = p0 αR (q0 + αqc ) = μ + ρqc σ2 αp0 μ qc ( p0 ) = . ρσ2
−
1 2
Since R(q0 + αqc ) = a(q0 + αqc )
(1)
2
− b(q + αq ) a − b(q + αq ) = p 0
2
0
a
q0 ( p0 ) =
c
− p − α(αp − μ) b ρσ 0
0
(2)
2
aρσ2 + αμb = bρσ2 The monopoly maximizes ( p0
0
p − bρσ 0
2
(ρσ2 + α2 ).
− c)q ( p ), i.e. 0
( p0
0
dq − c) dp
0 0
+ q0 ( p0 ) = 0
1 aρσ2 + αμb M +c . p0 = 2 ρσ2 + α2 b ∂p M 0 ∂μ ∂p M 0 ∂c ∂p M 0 ∂α ∂p M 0 ∂σ 2
= = = =
(3)
1 αb >0 2 ρσ2 + α2 b 1 >0 2 1 ρσ2 b(μ 2aα) 2α2 μb2 2 (ρσ2 + α2 b)2 1 (ρσ2 + α2 b)aρ (aρσ2 + αμb)ρ 2 (ρσ2 + α2 b)2
−
−
−
103
∂p M ραb(aα μ) 0 = (ρσ2 + α2 b)2 ∂σ 2
−
∂p M αbσ2 (aα μ) 0 = (ρσ2 + α2 b)2 ∂ρ
−
2- W = ( p0
− c)q
0
+ R(q0 + αqc )
>0
if
aα > μ
<0
if
aα < μ.
<0
if
aα > μ
<0
if
aα < μ.
− p q − μq − 0 0
c
1 ρqc2 σ2 . 2
The optimal levels of q0 and qc are obtained by maximizing W with respect to q0 and qc . By the Envelope Theorem ∂W ∂α ∂W ∂μ ∂W ∂σ 2 ∂W ∂c
= qc R (q0 + αqc ) > 0 =
−q < 0 − 12 ρq −q < 0. c
2 c
= =
0
Welfare under monopoly is: ˜ = ( pM W 0
− c)q
M 0 +
M M 0 q0
− p
a(q0M + αqcM )
− 12 b(q
M 0 +
αqcM )2
− μq − 12 ρσ − (q M c
2
M 2 c ) .
where q0M , qcM and pM 0 are given by equations (2), (1) and (3) respectively. From the envelop theorem ˜ ∂ W = ( pM 0 ∂μ
−
∂q 0M + c) ∂μ
− − q0M
∂p 0 ∂μ
qcM .
(4)
From (2) and (3): ∂q 0M α = ∂μ ρσ2 1 ∂p M αb 0 = . 2 ρσ2 + α2 b ∂μ
Substituting in (4):
˜ 4ρσ2 + 3α2 b μ ∂ W = 4(ρσ2 + α2 b) ρσ2 ∂μ
−
[3(ρσ2 + α2 b)c + aρσ2 ]α . 4(ρσ2 )(ρσ2 + α2 b)
(5)
104
SOLUTIONS
Given that μ = c0 + f ,
˜ ˜ ∂ W ∂ W = . ∂μ ∂f
The optimal fine is given by:
˜ max W f
hence:
˜ ∂ W ∂f
or
−
1 2 δf , 2
− δf = 0
4ρσ2 + 3α2 b f + Ac0 4ρσ2 (ρσ2 + α2 b)
where
||
− Bα − δf = 0
−
A
3(ρσ2 + α2 b)c + aρσ2 B= 4ρσ2 (ρσ2 + α2 b)
or
Ac0 + Bα ,0 A δ
f = max
−
(6)
3- Two part tariff ( p0 , F ) max( p0 p0 ,A
+ F
− c)q
0
subject to R(q0 + αqc )
− p q − μq − 0 0
c
ρσ2 qc2 2
− F ≥ 0
(P C ).
Hence q0 = qc =
a
− p − α(αp − μ) b ρσ αp − μ . 0
0
2
0
ρσ2
The participation constraint will bind (because raising F is desirable for the monopolist). Therefore: F = a(q0 + αqc )
−
b (q0 + αqc )2 2
− p q − μq − 0 0
c
ρσ2 qc2 . 2
(7)
Substituting F , q0 and qc in the objective function, the problem becomes
− − − −
a p0 max cq0 + a p0 b
−
1 a p0 b 2 b
2
αp0 μ μ ρσ2
−
− ρσ2 2
αp0 μ ρσ2
−
2
105
p0 = c.
(8)
4- Substituting p0 = c in the expression for q0 , qc and (7) gives us the lumpsum payment F . Let us now show that the Spence-Mirrlees property holds:
V (t0 , q0 , ρ) = max R(q0 + αqc ) qc
Hence the F.O.C.: α(a
− b(q
0
2
+ αqc )) qc
− t − μq − 0
c
1 2 2 ρσ qc . 2
− μ − ρσ q = 0 α(a − bq ) − μ = c
0
(9)
bα2 + ρσ2
or
−
− −
α2 (a bq0 ) αμ V (t0 , q0 , ρ) = a q0 + bα2 + ρσ2
∂V = 1 ∂t 0 ∂V = a ∂q 0
−
μα(a bq0 ) μ2 bα2 + ρσ2
−
− b(q
0
−
∂ ∂V/∂q0 = ∂ρ ∂V/∂t0 V (t0 , q0 ; ρ) = aq0
−
−σ
−
1 2 α(a bq0 ) μ ρσ 2 bα2 + ρσ2
+ αqc ) = a
b α2 (a bq0 ) αμ q0 + 2 bα2 + ρσ2
−
− bq
0
2 b(α
−
−
2
−
max
−
−
−
−
0
0
subject to
V (t0 , q 0 , ρ) V (t¯0 , q¯0 , ρ¯) V (t0 , q 0 , ρ)
≥ ≥
t0
−
− cq ] + (1 − ν )[t¯ − c¯q ] ≥ ≥
2
(a bq0 ) αμ) < 0. (bα2 + ρσ2 )2
V (t¯0 , q¯0 , ρ¯)
2
−
1 2 1 (α(a bq0 ) μ)2 bq + 2 0 2 bα2 + ρσ2
ν [t0
−
α2 (a bq0 ) αμ b . bα2 + ρσ2
The monopolist’s problem is: {(q 0 ,¯ q0 ),(t0 ,¯ t0 )}
−
0
(P C )
0
(P C )
V (t0 , q 0 , ρ¯) V (t¯0 , q¯0 , ρ)
(IC ) (IC ).
0
−
t0 .
(10)
106
SOLUTIONS
(P C ) and (IC ) will bind. Using (10) and this fact we get: t¯0 = a¯ q0
− 12 b¯q
2 0
+
1 (α(a b¯ q0 ) μ)2 , 2 bα2 + ρ¯σ2
−
−
(11)
and t0 = aq 0
−
−
2 1 2 1 (α(a bq0 ) μ) bq + 2 0 2 bα2 + ρσ2
−
−
1 [α(a b¯ q0 ) μ]2 Δρσ2 . 2 (bα2 + ρ¯σ2 )(bα2 + ρσ2 )
−
−
(12)
Substituting (11) and (12) in the objective function and maximizing w.r.t. q 0 and q¯0 we get: ( αb) [α(a bq0 ) μ] c = 0 q0 : a bq0 + 2 bα + ρσ2
−
−
−
− −
(E )
q¯0 :
−
α(a b¯ q0 ) μ (αb)Δρσ2 2 2 2 2 (bα + ρ¯σ )(bα + ρσ ) αb +(1 ν ) a b¯ [α(a b¯ q0 q0 ) bα2 + ρ¯σ2
+ν
−
−
−
−
−
(D)
− − μ]
c=0
q0 is the same as under complete information case and ¯q0 is greater than that under the complete information case, i.e., there is an upward distortion in the quantity sold to the low type. (If the problem is concave, D will be a decreasing function of q¯0 and since E is positive we need q¯0 to be greater than that under complete information case for the above FOC to hold). Crampes, C. and J.J. Laffont (2002)
107
Gathering Information before Signing a Contract
1- If the agent learns θ he accepts the contract only if t θq is non-negative. His expected utility is then ¯q) ψ. ν max(0, t θq) + (1 ν ) max(0, t¯ θ¯
−
−
−
−
−
Accordingly the moral hazard constraint inducing no effort in gathering information is ν (t
¯q) ≥ ν max(0, t − θq) + (1 − ν ) max(0, t¯ − θ¯ ¯q) − ψ. − θq) + (1 − ν )(t¯ − θ¯
(1)
The optimization problems of the principal are formalized as shown below in the solution of question 2. ¯q < 0 the agent accepts only the contract 2- Consider a contract in the class 2 . Since t¯ θ¯ if θ = θ. The principal can mimic this contract by choosing t¯ = q¯ = 0 and the agent saves the information cost.
C
−
So the principal can always mimic a contract in the class class 1 .
C
C
2
with a contract in the
¯q If t θq 0 and t¯ θ¯ classical problem
− ≥
− ≥ 0, (1) is strictly satisfied. The principal must then solve the max
q,t¯);(q,t)} {(¯
ν (S (q)
− t) + (1 − ν )(S (q¯) − t¯)
subject to t t¯
− θq ¯q − θ¯ t − θq ¯q t¯ − θ¯
≥ t¯ − θ¯q ≥ t − θq¯ ≥ 0 ≥ 0,
(2) (3) (4) (5)
which yields the second-best solution S (q SB ) = θ;
S (¯q SB ) = θ¯ +
t = θq SB + Δθ¯ qSB ;
ν
1
− ν Δθ
¯qSB . t¯ = θ¯
The principal’s utility level is V SB , which can be improved by lowering the e and e¯ in the same amount ε. It is easy to check the new contract still belong to 1 .
C
108
SOLUTIONS
This solution is the one obtained when the agent is informed costlessly. So it cannot be dominated by a solution in which the agent is informed at the cost ψ. ¯q ≥ 0. So, the only other interesting − ≥ 0 if t¯− θ¯
From the incentive constraint (2), t θq ¯q < 0. case to consider is t θq > 0; t¯ θ¯
−
−
The moral hazard constraint becomes (1
¯q) ≥ −ψ. − ν )(t¯ − θ¯
The general optimization problem in the class max
{(q,t);(¯ q,t¯)}
C is ν (S (q) − t) + (1 − ν )(S (q¯) − t¯) 1
subject to
ν (t ν (t
t
− θq ≥ t¯ − θ¯q
incentive constraint
(6)
t¯
¯q ≥ t − θq ¯ − θ¯
incentive constraint
(7)
¯q ) ≥ 0 − θq) + (1 − ν )(t¯ − θ¯
participation constraint
(8)
¯q ) ≥ ν max(0, t − θq) + (1 − ν ) max(0, t¯ − θ¯ ¯q ) − ψ − θq) + (1 − ν )(t¯ − θ¯
(9)
moral hazard constraint. For the maximization in the class
C , (8) and (9) are replaced by ¯q) − ψ ≥ 0 ν max(0, t − θq) + (1 − ν ) max(0, t¯ − θ¯ ¯q) ≥ 0. ν max(0, t − θq) + (1 − ν ) max(0, t¯ − θ¯
The maximimand in the class
C
2
2
is ν (S (q)
(10) (11)
− t).
3- Let us come back to the maximization in the class 1 , with (9) replaced by (1 ν )(t¯ ¯q) θ¯ ψ. If (8) is binding and not (9), we have a classical problem with an ex ante participation constraint. We know that the principal can reach the first best S (q F B ) = θ; S (¯ qF B ) = θ¯ with a zero expected rent
C
≥−
ν (t
− θq
FB
) + (1
¯q − ν )(t¯ − θ¯
FB
−
−
) = 0.
This case is valid if (9) is not binding, i.e., if (1
¯q ) ≥ −ψ. − ν )(t¯ − θ¯ FB
(12)
109
¯q is obtained when the incentive constraint (6) is binding − θ¯ = t¯ − θ¯ hence from (8) t − θq q ¯q = −ν Δθ¯ t¯ − θ¯ q .
The highest value of t¯
FB
FB
FB
Then (12) becomes ψ
FB
FB
≥ (1 − ν )ν Δθ¯q
.
If (9) is binding and not (8) ¯q t¯ = θ¯ and from the incentive constraint
− 1 −ψ ν
t = θq + Δθ¯ q
− 1 −ψ ν .
The optimal quantities are the second-best ones and the welfare of the principal is V SB +
ψ
1
− ν .
This solution is valid if (4) is satisfied
ν Δθ¯ q or
ψ
− 1 − ν ψ
+ (1
− ν )
− ψ
1
SB
≤ ν (1 − ν )Δθ¯q
− ν
>0
.
Finally, when (8) (9) are binding as well as (6) we obtain S (q) = θ and q¯ =
ψ . ν (1 ν )Δθ
−
Summarizing we have: T 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
qF B qSB
E
ν (1
SB
− ν )Δθ¯q
ν (1
ψ
FB
− ν )Δθ¯q
The principal suffers from the ability of the agent to become informed about its type before contracting. If ψ = 0 it is as if contracting was at the interim stage. When ψ
110
SOLUTIONS
is large enough the principal can achieve the first-best and as ψ increase the principal’s utility increases between V SB and V F B . Cr´ emer and Khalil (1992), Demski and Sappington (1986), Sappington (1984).
111
Better Information Structures and Incentives
1- Social welfare is W = S
− (1 + λ)(t + β − e) + U = S − (1 + λ)(ψ(e) + β − e) − λU.
The regulator’s program is max
¯ } e,U ) {(e,U );(¯
−
ν S
(1 + λ)(β
−
e + ψ(e))
λU + (1
−
−
ν ) S
subject to U ¯ U
≥ ≥ ≥ ≥
U ¯ U with Φ(e) = ψ(e)
−
¯ (1 + λ)(β
−
e¯ + ψ(¯e))
−
¯ λU
¯ + Φ(¯ U e) U 0
− Φ(e + Δβ )
0,
− ψ(e − Δβ ).
Hence, the solution is: ψ (e) = 1 ; ψ (¯e(ν )) = 1 ¯ = 0. U = Φ(¯ e) ; U
− 1 +λ λ 1 −ν ν Φ (¯e(ν ))
Furthermore there is shutdown of the inefficient firm (with U = 0) if ν as the solution of (1
− ν ) S − (1 + λ) β ¯ − e¯(ν ) + ψ(¯e(ν )) ∗
∗
∗
∗
≥ ν
defined
= λν ∗Φ(¯e(ν ∗ )).
2- The same analysis as in 1 can be made for each signal σi with the posterior probabilities νˆi instead of ν , hence, for each i ψ (ei ) = 1 ψ (¯ei ) = 1 Let e¯i (ˆν i ) = Z
νˆi 1−νˆ i
− 1 +λ λ · 1 −νˆ νˆ · Φ (¯e ). i
i
i
the solution of (1).
When the regulator keeps both types of firms E σi νˆi = E σi E (1I {β } σi ) = E 1I {β ¯} = ν
|
(1)
112
SOLUTIONS
¯, by the low of iterated expectations. where 1I {β } = 1 if β = β , 1I {β } = 0 if β = β From Jensen’s inequality E σi e¯i (ˆνi )
≤ e¯(ν )
ˆ (νˆ i ) if Z
≡ Z 1 −νˆ νˆ i
,
i
is concave.
≡
νˆ i Since ψ > 0, ψ > 0, Z ( ) is a decreasing function. h(ˆνi ) is an increasing 1−νˆ i ˆ = Z h 2 + Z h . Therefore Z ˆ is concave if Z 0. Note that convex function. Hence Z ˆ is concave if ψ is quadratic or λ small enough. Z
·
≤
So, the expected power of incentives decreases if the regulator has access to the information structure and there is never shutdown after σi , i = 1, . . . , I .
I
Say that a signal is favorable (defavorable) if νˆ i > ν (νˆ i < ν ). Then incentives decrease (increase) for a favorable (defavorable) signal. Now, if after each favorable signal ν i∗ > ν ∗ , i.e., the regulator shuts down the inefficient firm, the incentives increase (since ψ (ei ) = 1) when the firm is active. Since, on the other hand incentives increase when the signal is unfavorable, incentives increase with the availability of the information structure. Boyer and Laffont (2003).
113
Competitive Pressure and Incentives
1- Consumers’ utility is V = S (q1 + q2 ) + θq1 q2
− p (q , q )q − p (q , q )q − (1 + λ)tˆ 1
1
2
1
2
1
2
2
where tˆ is the gross transfer to the firm. The monopoly’s utility is U = tˆ + p1 (q1 , q2 )q1
− (β − e)q − ψ(e) ≡ t − ψ(e). 1
The fringe’s utility is U F = p2 (q1 , q2 )q2
− cq . 2
Social welfare is V + U + U F = S (q1 + q2 ) + θq1 q2 + λp1 (q1 , q2 )q1
− (1 + λ)((β − e)q
1
+ ψ(e))
− cq − λU. 2
2- Optimal regulation is characterized by the solution of
max ν S (q 1 + q2 ) + θq 1 q2 + λp1 (q 1 , q 2 )q 1 +(1
− ν )
− (1 + λ)((β − e)q + ψ(e)) − cq − λU ¯ − e¯)¯ ¯ S (q¯ + q¯ ) + θ¯ q q¯ + λp (¯ q , q¯ )¯ q − (1 + λ)((β q + ψ(¯e)) − c¯ q − λU 1
2
1 2
1
1
2
1
2
2
2
2
subject to
U U U ¯ U with Φ(¯e) = ψ(¯e)
≥ ≥ ≥ ≥
¯ + Φ(¯ U e) ¯ Φ(e + Δβ ) U 0
−
0,
− ψ(¯e − Δβ ). Hence the first-order conditions ψ (e) = q1 ψ (¯e) = q¯1
S + θq 2 + λ
− 1 +λ λ 1 −ν ν Φ (¯e)
∂p 1 (q , q )q + p1 (q1 , q2 ) ∂q 1 1 2 1 S + θq 1 + λ
= (1 + λ)(β
− e)
∂p 1 (q , q )q = c ∂q 2 1 2 1
∂p 1 ¯ ¯ + θ¯ (¯ = (1 + λ)(β S q2 + λ q1 , q¯2 )¯ q1 + p1 (¯q1 , q¯2 ) ∂ ¯ q1 ∂p 1 ¯ + θ¯ (¯ S q1 + λ q1 , q¯2 )¯ q1 = c. ∂q 2
− e¯)
