Asymetries of Information: Moral Hazard and Adverse Selectiona

8/16/2019 Asymetries of Information: Moral Hazard and Adverse Selectiona

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Asymmetries of Information:

Moral Hazard and Adverse Selection

Daniel Hojman

This version: July 24, 2013


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In many markets and economic relationships -perhaps most- the parties involved in a trans-

action or the signing of a contract do not have equal access to information directly relevantto the exchange. For example, when we buy a computer, it is hard to know in advance if theprocessor is fast and reliable. The computer producer or the seller are likely to know muchmore about the quality of their product. This happens with all ” experience” goods, suchas a car or a movie. Learning about the quality of the good happens after purchasing andexperiencing consumption. In these cases, the supplier has better information than potentialbuyers. On the hand, when an individual demands health insurance, the information she hasabout his or her own health history, family factors, and habits is much better than the one apotential insurer may have. In this case the demand side has private information relevant tothe supplier. Regardless of who holds the private information, in all of these examples, theasymmetry of information is associated with a characteristic or quality of the good. There is a

hidden characteristic .

Sometimes asymmetries of information are associated with behaviors difficult to observerather than unobserved characteristics. For example, unless an employer has access to a perfect-monitoring system -which could be ethically or legally questionable- it might be difficult todetermine if an employee is doing a good job. The sales of a shop are affected by a number of variables: the efforts of employees and other factor beyond their control such as the location of the shop, the weather or the state of the economy. If sales are low, it could be due to a pooreffort but also to external factors that lead to a low demand. There are many examples of this nature. It can be difficult to tell if a manager is carrying out a strategy that benefits theshareholders of a company or one that may involve a short-term private benefit to the manager

or benefit friends. In politics, it can be hard to assess if a public official allocates resourcesappropriately -in ways the public he or she is supposed to represent might support- as opposedto sustaining a patronage network or corruption. In all of these examples, the asymmetry of information is associated with a hidden action .

In general, information asymmetries may lead to efficiency losses, market and governmentfailures. In this chapter we consider two phenomena that can arise with asymmetric informa-tion: the moral hazard and adverse selection . Moral hazard concerns information asymmetriesrelated to behavior. If actions cannot credibly be contracted upon and their is conflict of in-terest, the risk of opportunistic behavior by one of parties after signing a contract may leadto inefficient outcomes. If characteristics of a good are unobserved before signing a contract,

this may lead to lower trade -rationing or no trade whatsoever. With hidden characteristics,self-selection is likely to determine the quality of the goods that end up being exchanged.

The tools developed here should be useful to analyze a wide range of economic and socialphenomena. Information asymmetries play a central role in organizing labor markets, financialmarkets, insurance, experience goods, health and education, within organizations, the govern-ment, or even the family.


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Chapter 1

Moral Hazard and the

Principal-Agent Model

The term ”moral hazard” was used in British insurance markets in the late nineteenth cen-tury. The idea was that someone who buys insurance may engage in opportunistic or recklessbehaviors precisely because he or she is insured.

To illustrate this point, suppose that if someone parks a car on the street there is a 10%chance of it being stolen. This could motivate someone to buy insurance. The moral hazard

problem is that once the individual has insurance, he might be less careful when closing thecar or choosing a safe place to park. Moral hazard occurs because the insured can affect theprobability of the event that triggers the insurance payment. The problem arises from theinability of an insurer to verify the behavior of the insured. If reckless behavior were observedand most importantly, if these actions were verifiable by a court of law, one could write contractsthat specify a coverage level contingent on the behavior: ”If you make an effort to reduce poorperformance, coverage insurance is $1,000, if reckless, their coverage is $0”.

How can insurance companies deal with the potential of moral hazard? A common responseis the use of deductibles and co-payments. These instruments aim to induce the insured to ac-tions that reduce the incidence of a bad outcome. By assuming part of the risk, the individual’sincentives are more more aligned with those of the insurance company. However, this happens

at a cost: lower insurance. Further, the individual may be totally alien to the occurrence of an accident or theft, he could be extremely careful but must settle for a lower coverage, as thisbehavior cannot be verified by the insurer. In general, moral hazard leads to higher insurancepremiums and lower coverage than would be efficient.

Beyond this classic example, there are many economic and social situations in which moralhazard can play a central role and lead to inefficiencies or contracts carefully designed to min-imize the ”incentive” problems associated with the potential of opportunistic behavior. Animportant family of cases are agency problems . Agency problems arise when a principal dele-gates a task to another person, the agent. For example, in a company the shareholders delegate

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Harvard Kennedy School - FEN, Universidad de Chile 3

the management-management-agents to make decisions affecting the destiny of the company.

The principal-agent problem arises if the goal of the agent differs from that of the principal,the agent can have a significant impact on the well-being of the principal and informationasymmetries make it hard to verify if the actions taken by the agent are those that the principalwould choose to maximize her utility. In sum, moral hazard can arise when there is a conflictof interest and information asymmetries hinder the possibility of verifying an opportunisticbehavior.

We analyze a model that is helpful in a wide range of moral hazard problems, the Principal-Agent model. We explore the impact that different contracts may have in aligning the objectivesof the agent to the interest of the principal. At the end of the chapter, we then discuss a number

of real-world examples, including applications to microfinance, social policy and politics.

1.1 The Principal-Agent Model

The core elements of the Principal-Agent problem can be illustrated with a basic model:

• There is a principal who delegates a task to an agent. The task is functional to theprincipal’s objectives (e.g. sales, private value, public value).

• The agent’s action affects the outcome of this task (e.g effort, choice of a project).

• The action is not observable by the principal. More importantly, even if it were observ-able, the action is not verifiable by a court of law. This means that the compensationcontract offered by the principal, cannot explicitly specify the action to be performed bythe agent. (Even if the contract specified what the agent should do, it would be ”deadlaw”.) The contract can only be contingent on results that are observable and verifiableex-post . For example, the effort of a seller may not be observable but sales are. If thereis a statistical relationship between effort and sales, one way to align incentives might beto make the compensation contract contingent on sales.

• The principal offers a contract that the agent can accept or reject. The principal cannotforce the agent to accept the contract. To induce the agent to accept a contract, the prin-cipal must offer a contract that gives the agent at least the same utility as her outsideoption (e.g. an alternative job or home production).

• If the agent accepts the contract, he chooses what action to take.

We formalize the above description using a model. To fix ideas, suppose that the principal,Paula, owns a car dealership. The agent, Andrew, is a potential seller. Sales are a randomvariable x̃ that depend on agent’s effort and other exogenous factors that are not controlled by

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either the principal or the agent such as the macroeconomy or the weather. For simplicity, we

assume that there are two levels of sales A and B , with A > B.

In this model, a contract is a compensation scheme contingent on observable sales x̃ ∈ {A, B}.That is, a contract specifies compensation for each level of sales. Given our assumption of twolevels of sales, a contract is a pair (wA, wB), where wA is the compensation if sales are highand wB is compensation if sales are low.

1

After observing the offer, the agent can accept it or reject it. If the offer is rejected, we as-sume that the agent prefers an outside option and gets a reservation utility u0. If the agentaccepts the contract, the agent chooses the level of effort e ∈ {eL, eH }, where eH is the highlevel of effort and eL. This effort affects the probability of successful sales. Let P r(x̃|e)be the probability of a sales level x̃ given the agent chooses a level of effort e. Note thatP r(x̃ = A|e) + P r(x̃ = B|e) = 1.

The following table describes the statistical relationship between sales -observed outcome-and effort -unobserved action.

Effort (e) P r(x̃ = A|e) P r(x̃ = B|e)eH pH 1 − pH eL pL 1 − pL

Naturally pH

> pL

. That is, high sales are more likely if effort is high than if it is low.

We complete the description of the model by specifying the utility functions of the principaland the agent. We assume that the principal is risk neutral and she only cares about her netprofit, so that her utility is Π̃(x̃, w̃) = x̃ − w̃. If the agent’s effort level is e, the expected utilityfor the principal is

Π(e, wA, wB) = P r(x̃ = A|e)(A − wA) + (1 − P r(x̃ = A|e))(B − wB).For example, if e = eH ,

Π(eH , wA, wB) = pH (A − wA) + (1 − pH )(B − wB).

We assume that the agent is risk averse and maximizes his expected utility. The Bernoulliutility of the agent is

U (w̃, e) = v(w̃) − C (e),where v(·) is an increasing and concave function and C (e) is the cost of effort. Without furtherloss of generality, we normalize C (eL) =0 and define C H = C (eH ) > 0. The expected utility of the agent as a function of the contract (wA, wB) is given by

1If there were three levels of sales, A, B and C , a contract would be a triplet (wA, wB, wC ).

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V (e, wA, wB) = E [U (w̃, e)|e] = P r(x̃ = A|e)v(wA) + (1 − P r(x̃ = A|e))v(wB) − C (e).

Note that

V (eH , wA, wB) = pH v(wA) + (1 − pH )v(wB) − C H and

V (eL, wA, wB) = pLv(wA) + (1 − pL)v(wB).

The extensive form of the sequential game that describes the relationship between the agent

and principal is illustrated. The subgame perfect equilibrium of this game can be found usingbackwards induction. Note that, conditional on optimal reactions of the agent, the principalcan determine the equilibrium in the subgame in which the agent moves.

[Insert figure game extensively]

This situation can be formulated as an optimization problem in which the principal chooses acontract and the effort level to be implemented constrained by the fact that the agent choosesoptimally whether to accept or reject the contract and, if he accepts, he decides the level of effort.

In formal terms, the equilibrium of the game corresponds to the solution of a constrained opti-mization problem: the principal maximizes her expected utility choosing a contract (wA, wB)and implementing a level of effort e subject to two constraints: a participation constraint (P )that indicates that the agent prefers to accept the contract and an incentive compatibility con-straint (IC ) that indicates that if the agent accepts the contract, the agent prefers the level of effort that the principal aims to implement. The constraints express the fact that the principalcan not force the agent to accept the contract and does not directly choose the effort level. Theproblem is then:

max(e,wA,wB)

Π(e, wA, wB)

subject toV (e, wA, wB) ≥ u0

V (e, wA, wB) ≥ V (e, wA, wB), e = e.

The optimal contract can be found in two steps:

1. First, for each level of effort e, find the contract (w∗A(e), w∗

B(e)) that minimizes the ex-pected value of the compensation subject to the participation (P) and the incentivecompatibility (IC) constraints. That is, for each e = eL, eH , solve

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min(wA,wB) P r(x̃ = A|e)wA + (1 − P r(x̃ = A|e))wB

subject to

V (e, wA, wB) ≥ u0V (e, wA, wB) ≥ V (e, wA, wB), e = e.

2. Second, using the solution of previous step, find the effort level e∗ that maximizes theutility of the principal. That is, compare ΠL = Π(eL, w

∗

A(eL), w∗

B(eL)) with ΠH =Π(eH , w

∗

A(eH ), w∗

B(eH )).

Before finding the general solution to this problem, we develop an example that illustratesthe economic intuition of the problem.

1.1.1 Example: The trade-off between incentives and insurance

In this example we assume that A = 1, 800 and B = 0. The probability of selling A if theeffort is low is pL = 0 and, if the effort is high, pH = 0.6. Andrew’s Bernoulli utility functionis v(w) =

√ w, C H = 9 and that the outside option is associated with a reservation utility of

u0 = 10.

We analyze the incentives associated to different compensation schemes. In principle, a contractmust achieve two objectives. On the one hand, provide incentives for the agent to do what isgood for the principal, ”achieve the task” or exert high effort. On the other, it must make thedeal attractive enough for the agent relative to her/his outside option. These objectives can beat odds as we aim to illustrate. The principal faces a trade-off between exposing the agent torisk to encourage a high effort level and insuring the agent to reduce the high premium requiredto compensate the agent for taking a risk. As a benchmark, we start with the ”first best” (FB)case in which there are no information asymmetries.

First Best: Observable effort

Assume that Paula observes the effort made by Andrew. Moreover, effort can be verified by acourt of law, which implies that a contract contingent on Andrew’s actions can be enforced. If Paula wanted to implement eH she can offer the following contract: ”If you shirk (e = eL) Iwill sue you, you will go to prison for a long time and I will also destroy your reputation”. ”If you try (e = eH ) gain $400 regardless of whether you sell or not.” The punishment is so strongthat we can be sure that Andrew would strive if he accepts the contract. Thus, if the effort isobservable, incentive compatibility is not an issue.

Would Andrew accept the contract? The participation constraint (P) is satisfied if and only if Andrew’s utility of accepting the contract, U (w = 400, eH ) is at least as large as u0 the utilityof the outside option. Noting that U (w = 400, eH ) = v(400)

−C H =

√ 400

−9 = 11 > 10, we

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see that the constraint is satisfied with slack. In short, Andrew would take the contract and

would choose eH .

Is this contract is optimal for the principal? As long as (P) is satisfied, the principal can reducethe compensation and increase her expected utility. Indeed, the optimal wage level correspondsto w0 such that v(w0) − C H = u0 or √ w0 = 19 w0 = 361. The expected utility of the principalis E [Π] = 0.6 · 1, 800 − 361 = $719.

We can therefore conclude that if effort is observable, the optimal contract is independentof sales. Hence, Andrew is completely insured.

Second Best: asymmetric information, non-observable effort

We return to the case in which effort is not observable and therefore it is not possible to enforcea contract contingent on it.

We consider three compensation schemes:

1. Flat salary :

Paula offers a salary of 400 independent sales, so that the agent faces no risk. We analyzethe two constraints, (IC) and (P). If Andrew accepts the contract will he choose eH ?Would Andrew accept the contract?

We start with the (IC). Andrew chooses eH if and only if V (w = 400, e = eH ) ≥ V (w =400, e = eL). This does not hold as v(400)−C H ≤ v(400), the (IC) is violated: if Andrewaccepted the contract, he would choose eL.

Observe that the participation constraint is satisfied. If Andrew takes the contract hispayoff would be V (w = 400, e = eL = 20) which exceeds u0 = 10.

In general, with asymmetric information and risk-free wage scheme, the agent will haveno incentive to strive.2

2. Sales contract

Suppose that Paula offers a contract fully contingent on sales: sales are shared equally.That is, w̃ = x̃/2, or equivalently, wA = 900 and wB = 0.

Under this scheme the wage is a random variable. The distribution depends on the effortlevel chosen. In particular, if e = eL, w̃|eL = wB = 0. If instead e = eH ,

w̃|e=eH =

wA = 900 with probability 0.6wB = 0 with probability 0.4

2In this section we consider intrinsic motivation, a central aspect of human motivations. The focus is onmaterial incentives.

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In contrast to the flat compensation scheme, if Andrew took the job, he would choose

eH . In this case, the expected utility associated to eH is V (eH , 900, 0) = 0.6 ·√ 900+0.4 ·√ 0 − 9 = 9. The utility of eL is V (eL, 900, 0) = 0 · √ 900 + 1 · √ 0 = 0. Thus, the (IC) is

satisfied as 9 ≥0.

However, the contract does not satisfy (P) as V (eH , 900, 0) = 9 ≥ u0 = 10. We concludethat under this scheme Andrew prefers the outside option.

The scheme incentivizes Andrew to exert effort for effort and is associated with an ex-pected compensation of $540 (= 0.6·$900), relatively high compared to $361, the compen-sation required to implement eH with complete information. Clearly, if Andrew receivedthe expected salary for sure he would accept the contract. The problem is that Andrew is

risk averse and this scheme is associated with a high volatility (the standard deviation of the compensation is σ = 441). In order for the agent to accept the job, the principal mustcompensate the agent with a high enough risk premium. In this case, it would require ahigher share of sales.

3. An intermediate scheme: Base + Bonus

Suppose now that Paula offers Andrew a contract with an ”intermediate” wage scheme,one that ensures him a certain amount but also exposes him to risk by offering a bonuscontingent on sales. The contract has a fixed component of $150 -the base- and a vari-able component -a bonus- equal to 1/3 of the profits from sales. That is, wA = $150 and

wB = $750. We see that V (eH , 750, 150) = 0.6 · √ 750+0.4 · √ 150 − 9 = 12.33. Moreover,V (eL, 750, 150) =

√ 150 = 12.24. Thus, under this scheme, the expected utility associated

to the high effort level s greater than the expected utility of low effort level, the (IC) issatisfied. (P) is also fulfilled as 12.33 ≥ 10. This contract induces Andrew to accept andchoose eH .

Note that in this case the expected compensation is $510 (= 0.6 · 750 + 0.4·150) and itsstandard deviation is σ = 294. Comparing this scheme with the sales contract, we seethat in this case the expected cost of implementing high effort is lower, implying a greatersurplus for Paula (further, in the previous case (P) was not even satisfied). Reducing therisk faced by Andrew not only makes the contract more attractive relative to the out-

side option but it lowers cost of implementing high effort. The intuition is that the riskpremium paid by the principal get the risk averse agent to accept the contract is smallerbecause the risk is lower.

The example illustrates that with asymmetric information, exposing the agent to risk is nec-essary to encourage their effort. Unlike the case where effort is observable, the (IC) constraintto implement eH requires not to fully insure the agent. On the other, the higher the riskfaced by the agent (who is risk averse), the higher the risk premium the principal must pay tomake the contract attractive relative to the outside option. That is, increasing risk, implies ahigher expected compensation required to ensure that (P) is satisfied, increasing the cost for

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the principal. Consequently, the optimal contract balance the trade-off between exposing the

agent to risk to incentivize high effort and insuring the agent to reduce risk premium requiredto make the agent participate.

1.2 The optimal contract

We characterize the optimal contract for the general case. As before, we start with the case inwhich effort is observable.

1.2.1 First Best: Observable effort

If effort is observable and verifiable by a court then the (IC) constraint is irrelevant as thecontract can specify the level of effort. The optimal contract only considers the participationconstraint (P). In this case the principal can implement an effort level e that will give her thehighest profit by paying the lowest wage consistent with (P). This means that, at the optimalcontract, (P) binds (it holds with equality).

Then, to implement eL, the optimal wage is w0 = v−1(u0). In the above example, we have

that w0 =100. To implement eH optimal salary is wH = v−1(u0 + C H ). In the example, this

number is wH = $361.

Using these values, implementing eL yields the principal a payoff of

ΠL = pL · A + (1 − pL) · B − v−1(u0).

Implementing eH is associated with a utility for the principal of

ΠH = pH · A + (1 − pH ) · B − v−1(u0 + C H ).

In our example, ΠL =-100 and ΠH =719, from which it follows that it is optimal for theprincipal to implement eH .

1.2.2 Second Best: Asymmetric information, non-observable effort

We start by finding the contracts that minimize the cost of implementing each level of effort.Note that to implement the low effort level, eL, the principal faces no incentives/insurancetrade-off as she does not need to provide incentives to exert effort. Consequently it is optimalto offer a contract that fully insures the agent. As in the case of observable effort, the optimalcompensation is the smallest one satisfying (P). That is, the optimal contract to implement eLis w∗A(eL) = w

∗

B(eL) = w0, identical to the case of observable effort. The utility of the principalis also the same, ΠL = pL · A + (1 − pL) · B − v−1(u0).

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The optimal contract that implements eH solves

min(wA,wB)

pH · wA + (1 − pH ) · wB

subject to pH · v(wA) + (1 − pH ) · v(wH ) − C H ≥ u0

pH · v(wA) + (1 − pH ) · v(wH ) − C H ≥ pL · v(wA) + (1 − pL) · v(wH ).

This is an optimization problem with two variables and two constraints.

Proposition 1. In optimal problem of minimizing the cost of implementing eH , both con-

straints, participation (P) and incentive compatibility (IC), bind.

The proof is the Appendix. The figure illustrates the feasible region associated with bothconstraints, (P) and (IC). The equation pH · v(wA) + (1 − pH ) · v(wH ) − C H = u0 defines anindifference curve in the plane (wA, wB). All pairs above this curve are contracts that satisfy(P). Meanwhile, the corresponding equation (IC) defines another curve. To gain some intuition,we rewrite (IC):

∆ p · (v(wA) − v(wB)) ≥ C H ,

where ∆ p = pH −

pL. Hence, the (IC) says that a ”bonus” -a spread between the high and lowsales compensation levels- is required to induce high effort. The 45 degree line corresponds toequal wages or full insurance. The increasing curve above this line represents this minimumspread consistent with the (IC). Any contract above this curve satisfies the (IC). Finally, theobjective function of the problem is linear and the iso-cost curves are straight lines of the formwA =

−(1− pH ) pH

wB + Φ. Lower implementation costs correspond to iso-cost lines closer to theorigin. The optimal contract is the pair where corresponding to the intersection of curves whereeach of constraints bind.

Since both constraints bind at the optimum, the optimal contract can be solved using the twoequations defined by binding constraints. There are two equations and two unknowns. Solving

these equations, the optimal contract to implement eH is (w∗

A, w∗

B), such that

v(w∗B) = u0 + C H − pH ∆ p

C H

and

v(w∗A) = u0 + C H + 1 − pH

∆ p C H .

We can return to our sales example and use the above equations for the optimal contract thatimplements eH . We find that w

∗

A = $625 and w∗

B = $100. That is, the optimal contract is

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Figure 1.1: Optimal Contract

not flat and exposes the agent to the risk necessary to implement high effort. The agent’s

equilibrium utility is still u0. However, the expected compensation is $415 which is greaterthan $361, the compensation required to implement eH when effort is observable. The utilityof principal is then ΠH = $665, which is lower than the utility for the principal when effort isobservable (719). The welfare loss for the principal, the agency cost , is precisely the differencein the cost of compensation, namely, $54 (=719-665=415-361).

The optimal effort level

Having determined the optimal contracts to implement each effort level, the optimal effort level

is characterized. Using the previous results, implementing eL reports a profit ΠL = pL ·A+(1− pL)·B−v−1(u0) while implementing eH reports a profit ΠH = pH ·(A−w∗A)+(1− pH )(B−w∗B).The level of effort to implement is determined by comparing ΠLwith ΠH . Considering ∆x =A − B, we have that

ΠH − ΠL = ∆ p · ∆x − [ pH · w∗A + (1 − pH )w∗B − v−1(u0)]

If ΠH −ΠL ≥ 0, then it is optimal for the principal to implement eH , while if ΠH − ΠL ≥ 0,eL is optimal. In our example, ΠH = 665 > ΠL = −100, where eH is the optimal effort.

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1.3 Welfare Analysis

In this model, the utility of the agent in any optimal contract is always the same: the par-ticipation constraint (P) always binds which means that the agent obtains an expected utilityof u0 in any scenario. Either in the first best or with asymmetric information, regardless of whether it is optimal to implement eL or eH , the principal fully extracts the agent’s surplus

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Accordingly, welfare analysis can focus on the utility of the principal.

We saw that if it is optimal to implement to low level of effort, eL, the optimal compensationis the same in the first best (FB) and the second best (SB). In contrast, if the optimal effortis eH , (SB) requires a compensation whose expected value is greater than that of the first best(FB). Hence:

• If it is optimal to implement eL in the (FB) situation, it is also optimal in (SB). Theimplementation cost is the same in both cases.

• If it is optimal to implement eH in the (FB), it is not necessarily optimal in the (SB) asthe cost of implementation is greater in the second case.

• In the (SB), implementing eH requires exposing the agent to risk, whereas in the (FB)the agent is fully insured (although the expected utility, u0, is the same in both cases).

• The expected profit for the principal is higher in the (FB) as the expected value of

compensation is lower.• Relative to (FB), in the (SB) there is an efficiency loss on two margins (a) Distortion of

the effort choice: the principal may choose to implement eL rather than eH , even if eH is optimal when effort is observable; (b) Agency cost: if it is optimal to implement eH inthe (SB) there is an agency cost is reflected in a lower utility for the principal and, thus,a lower total surplus relative to (FB).

1.4 Applications

Agency problems arise in a wide variety of economic, political or social. Some examples are

discussed.

1. Company 1: The shareholders of a company delegate management to managers. In thiscase the shareholders (the principal) do not know if the projects being decided by amanager (the agent) maximize the value of the company or if, instead, they are projectsassociated with a greater private benefits to the manager. Private benefits can either

3In this model the agent’s surplus is always extracted by the principal. This is associated with the structureof game. In particular, the principal offers a ”take it or leave” contract to the agent, there is no possibility for theagent to make a counter-offer. This implies that the principal has all the bargaining power. A more balancedbargaining game would leave some surplus to the agent, but the qualitative conclusions regarding efficiency-overall surplus loss and the potential distortion of the level of effort- would remain.

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take the form of low-effort (poor project choice), inefficient projects that benefit friends

or are perks associated with in kind to the manager (fame, cars, large headquarters, ...).

2. Company 2: Managers delegate tasks and function to eployees. This example is verysimilar to that developed in the chapter. In this case the principal is a manager andlower-ranked employees are agents.

3. Real estate: Homeowner delegate home sales to real estate agents. A real estate agentspends on average 10% less time selling your house than he/she spends selling her own,resulting in 3% lower prices (for example, $300,000 rather than $310,000).4

4. Democracy: Citizens delegate social decisions to government officials and parliamentari-

ans. In this example, citizens are the principals and politicians act as agents or represen-tatives. Typically, citizens do not have the same ideological preferences and therefore itmight be difficult to identify the principal’s objective in this example. Nonetheless, it isclear that most citizens would agree in condemning corrpution. The fact that corruptionexists and that it can be significant in many countries, provides clear evidence of agencyproblems in government.

It is worth noting that the alignment of politicians’ incentives with those of citizens (or atleast a significant group), may depend on institutions such as transparency laws, politicalfinancing and the electoral system. Some political and regulatory systems induce moreaccountability than others.

5. Parents and children. Education and nutritional decisions are made by parents andimpact greatly the wellbeing of children. In spite of parental authority, in this example,the principal are the children and the agent or agents, are parents. While most parentscare for their children, raising and taking care of children requires considerable effort andresources. Child abuse is an extremely sad example that illustrates the limitations of parents altruism and how parental motivations may not necessarily along with the bestinterest of the children.

Some of the social programs with the greatest impact in the last decade, conditional cashtransfers, explicitly take into account potential moral hazard problems. In Latin Amer-ica, programs such as Chile Solidario (Chile), Bolsa Familia (Brazil), Progresa (Mexico),among others, make subsidies are conditional on observable actions such as school at-

tendance or yearly medical checkups. These observables at least ensure some minimumcaring standards for the children.

6. Moral Hazard and sub-prime loans. The” Great Recession” that has affected financialmarkets and the global economy since 2007, began with the subprime loans’ crises. Partof the problem was due to moral hazard in the screening of mortgages, the subprimeloans. In practice, many of these loans were aimed at a population of individuals thathad a high probability of not repaying loans.

4Dubner and Levitt (2005) Freakonomics.

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What explains that banks gave out loans with a high likelihood of default? From the early

nineties banks started a process of securitization: banks created securities by tranchingthe original mortgages into slices and then repackaging tranches of thousands of loansinto an asset; once sold to other financial intermediaries they gave holders claims to thetranches of the loans originally generated by the bank. This meant that banks adopteda strategy of generating mortgage loans and distributing securities based on those loansto the financial market. This allowed banks to gain liquidity and, at the same time, todiversify the idiosyncratic risks associated to the population they served. The downsidewas that banks were ”overinsured”: since the risk of the original loans were distributedin the market, the banks faced little risk associated the loans they generated, reducingthe incentive to screen loans. Banks had an incentive o generate loans regardless thesolvency of the customer. The financial reform passed during the Obama administration

now requires banks to retain a significant fraction of the loans originated and face thatrisk, solving, in part, this moral hazard problem.

7. Moral hazard in the credit market. The agent is an entrepreneur who needs a loan tofinance a profitable investment project. The principal is the bank, interested in the re-payment of the credit and associated interest. A project can succeed or fail dependingon the effort and luck of the entrepreneur. The analysis in this chapter suggests that thebank should provide incentives to reward effort in case of success and ”punish” the agentif the project fails. In practice, the ”punishment” bad results is normally associated withthe requirement of a collateral. The risk of losing the collateral can be a strong incentivefor the entrepreneur to maximize the probability of success of the project.

However, contracts based on the existence of a collateral do not work for small businessesor poor individuals who simply are wealth and liquidity constrained. In this context,good projects that are in the hands of liquidity-constrained agents would receive no fund-ing. This is inefficient and it has direct effects on the income distribution and poverty.In other words, a market failure in the credit market may have important social andproductive consequences if the poorest individuals are rationed out due to asymmetriesof information.

The “microfinance movement” has emerged, in part, as an attempt to correct this marketfailure. It uses information and collaterals that arise naturally in a community. Indeed,many microfinance programs rely on group credits with joint liability: If two people getloans and one of them fails, the other is partially responsible for the repayment. Thisformula helps reduce to moral hazard because although the bank cannot monitor theefforts made by its customers, the community has better information on the behavior of its members. Further, if one of the involved parties failed to pay the credit it can receiveimportant social sanctions. For example, an individual can be ostracized and lose accessto social favors and social insurance (e.g. food, child care). This is a form of “socialcollateral”, a collateral that is available only in the community (not for the bank). Thismechanism illustrates how social monitoring and sanctions may serve to overcome moralhazard problems.

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1.5 Concluding remarks

We conclude this chapter by pointing out that there is a vast literature that seeks to extendthe agency model to incorporate important aspects of reality that were left out of our analysis.

For example, many jobs and enterprises are associated not only with one action but several.In principle, a teacher’s efforts are directed not only to teach skills in a particular area butmay potentially involve many dimensions -teach math, teamwork, tolerance, moral reasoning,communication, critical skills, tolerance, frustration management, etc... When an agent mustdevote her attention and efforts to multiple tasks and efforts -multitasking- it is not obvioushow to encourage a task without discouraging others. For example, rewarding teachers whosestudents achieve good scores in a standardized test focus the complex task of teaching to onedimension crowding out attention from other dimensions. Even worse, it could induce somedegree of corruption such as preventing worst-performing students to attend school when thereis a standardized test or a black market for test questions. A different example involves compa-nies that make decisions both about quality and cost innovations. If it is not easy to measurethe quality of a service, a provider may have incentives to reduce costs at the expense of quality.

Other aspects that we have ignored are related to intrinsic motivation. Many times peoplestrive at their jobs not because they have bonuses or ”high-powered” incentive contracts. Theydo so because it is the right thing to do or because they are passionate about what they do. It isnot obvious how important monetary incentives are to motivated agents. Recent advances seek

to establish precisely when monetary are incentives important and when not, and to identifycircumstances in which monetary incentives can crowd out intrinsic motivation.

1.6 Appendix

Proof of Proposition.To prove the statement, we use a change of variables. Finding wB and wA is equivalent todetermining vB = v(wB) and ∆v = vA − vB = v(wA) − v(wB). Let φ(.) = v−1(.) which isstrictly increasing and convex (for vis strictly increasing and concave). Using this notation werewrite the problem as

min(vB,∆v)

Φ(vB, ∆v) = pH · φ(vB + ∆v) + (1 − pH )φ(vB)

subject to∆ p∆v ≥ C H

pH ∆v + vB − C H ≥ u0

Once again, we see that (IC) is related to the establishment of a ”incentive bonus” ∆v. Givena bonus level, (P) determines the ”base compensation” vB.

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To see that (P) is a binding constraint at the optimum, assume, towards a contradiction, that

it is not. We find a profitable deviation for the principal, contradicting optimality. If (P) doesnot bind, pH ∆v + vB − C H > u0. Then, it is possible reduce vB and i vA by a small amount > 0. For small enough, (P) is not violated (as we assumed it is not active). In addition, thischange does not alter the (IC) as ∆v = vA − vB + − = vA − vB. Thus, if the participationconstraint is not binding, it is possible to reduce the fixed component of the compensationwithout affecting the effort incentives. Clearly it is beneficial to reduce the compensation forthe principal since it reduces the expected cost of implementation. This profitable deviation,yields the desired contradiction. We conclude that (P) must bind at the optimum.

To show that (IC) binds, assume that it is not. Then ∆ p∆v > C H . It is possible to reducethe bonus ∆v by a small amount > 0 and raise vB by pH

·. For small enough, the strict

inequality assumed for (IC) is reduced but without violating this constraint. Furthermore, thischange -by construction- does not affect (P). To conclude that (IC) binds it must be that thischange in compensation generates less cost to the principal, so that it is profitable. Using afirst-order Tayor approximation for small, we obtain Φ(vB + pH · , ∆v − ) ≈ Φ(vB, ∆v) +∂ Φ∂vB

· pH − ∂ Φ∂ ∆v . It follows that the two middle terms are

Φ(vB + pH · , ∆v − ) − Φ(vB, ∆v) = ∂ Φ∂vB

· pH − ∂ Φ∂ ∆v

= (1 − pH ) pH · [φ(vB) − φ(vB + ∆v)].

The convexity of φ implies φ(vB + ∆v) > φ(vB), which combined with the above yieldsthe conclusion that Φ(vB + pH · , ∆v − ) < Φ(vB, ∆v). Therefore, the change in the contractdesigned reduces the value of the expected compensation and is beneficial for the principal.This allows to conclude that (IC) binds. The proof is complete.

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Chapter 2

Adverse Selection, Signaling and

Screening

In this chapter we analyze asymmetries of information associated with a hidden characteristicor quality. Frequently, the characteristic that defines a good or service is better known by oneof the sides of a market, that is, one party has private information on the quality of that goodor service. In the used cars market, the supplier has better information about the quality of the product. In the labor market, a worker may know better her own skills than a potentialemployer. In a medical insurance market, a buyer has more information regarding his healthrisks than an insurer.

In all of these markets, the asymmetry of information impedes the existence of separatemarkets for each of the different ”types”. In the used car market, cars in good condition arepotentially confounded with cars sin poor condition. Similarly, prior to hiring someone, itmight be hard to distinguish between high and low productivity workers for a particular jobwithout incurring in a cost -a selection process, for example. An insurer may not (and, perhaps,should not) distinguish between individuals with different risk profiles. As we will see, privateinformation on hidden characteristics can lead to adverse selection, i.e., it can reduce markettransactions to a subset of types, lead to rationing or even no transactions at all. We discusssome government remedies to adverse selection. Market solutions such as signaling or screeningthat allow for private information to be disclosed in a market equilibrium are also analyzed.As we discuss later, this information revelation will typically be associated with potentiallysignificant transaction costs.

Unlike moral hazard, in which the information asymmetry is linked to an endogenous de-cision, in the case of hidden characteristic the private information is linked to an exogenouscharacteristic. Another difference is that hidden characteristic lead affect to selection and ra-tioning prior to the exchange or signing of a contract while the moral hazard problem associatedwith a hidden action takes place after the signing of the contract.

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We begin with simplified model of ”the market for lemons” (Akerlof, 1970), a seminal

model. Two additional examples follow, the insurance and labor markets. The rest of thechapter analyzes policy and market solutions to the adverse selection problem.

2.1 The Market for Lemons

Consider a market for used cars. There are an equal number of buyers and sellers, one hundred.It is public knowledge one half of the cars are low quality, lemons , and the other half are highquality, peaches . It is assumed that buyers and sellers are risk neutral.1

If p is the market price of a used car, the utility of a seller is U T = p if he trades the car.Otherwise, if he does not trade, the owner gets a reservation value u0 associated to keepingthe car. Hence, a seller is willing to sell if and only if p ≥ u0. The reservation value is higherfor peaches than it is for lemons. For illustration, it is assumed that u0 = $1, 000 if the car islemon and u0 = $2, 000 if it is a peach.

When the buyer is certain about quality of the car, buying the car gives him a utility of U = θ − p, where θ is the consumer’s valuation of the car. The valuation of the car for a poten-tial buyer varies according to the quality of the product. Hence, we can identify the unobservedquality of the car with the valuation of the consumer. We assume that θ = 2, 400 if the car is

peach and θ = 1, 200 for a lemon. The valuations of buyers and sellers are summarized in thefollowing table.

Valuations Lemons Peaches

Sellers reservation value (u0) 1, 000 2, 000

Consumer valuation (θ) 1, 200 2, 400

In principle, the potential buyer does not observe the quality of the car. Hence, in decidingwhether or not to purchase a car he considers the expected utility EU = µe − p, where µeis the expected valuation of the cars that are traded. If Θ is the set of cars that are traded,

then µe = E [θ|Θ]. For example, if all the cars are traded, then Θ = { peaches, lemons} andµe = E [θ|Θ] = 12 · 1, 200 + 122, 400 = 1, 800. If instead, only lemons are traded, Θ = {lemons}and µe = E [θ|Θ] = 1, 200. It is assumed that if the potential buyer does not participate thebusiness, will have a utility equal to zero. Therefore, a consumer will buy a car if and only if EU ≥ 0 or equivalently, µe ≥ p.

An important assumption of this model is that buyers have rational expectations . This meansthat after observing the market price p, a consumer takes into account the optimal behavior of

1Aside from simplifying the exposition, this shows that, in contrast to the principal-agent model, risk aversionis not central to the adverse selection problem.

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sellers to infer which type of cars would be offered at that price. In particular, for each price p,

the potential buyers infer the set Θ( p) of types that would trade at p and, from this inference,they determine their maximum willingness to pay µe( p) = E [θ|Θ( p)].2

A market equilibrium price p∗ is such that:

1. For those sellers who trade, the price weakly exceeds their reservation value: for eachθ ∈ Θ( p∗), u0(θ) ≤ p∗.

2. Consumers willingness to pay weakly exceeds the price: µe( p∗) ≥ p∗

We can define the ”willingness to accept” function, W A( p), as the maximum reservationvalue of the types that participate in the market at a given price p. Formally, W A( p) =max{u0(θ)|θ ∈ Θ( p)}. Any price p∗ that satisfies

µe( p∗) ≥ p∗ ≥ W A( p∗),

is an equilibrium price. Thus, in equilibrium, the willingness to pay of consumers is at leastthe market price and the willingness to accept of sellers is at most that same price.

We start by analyzing the market equilibrium for the benchmark case in which there is noprivate information, i.e., car quality is observed by the buyer (First Best). We next studythe case in which there is private information and only the seller knows the quality of the car(Second Best).

2.1.1 First Best: Complete information, observable quality

In rigor, peaches and lemons are different goods. If the quality of the car is public knowledge,lemons and peaches can be distinguished as different goods. Hence, without asymmetries of information, there will be two markets, one for each type of car.

1. Lemons market

The supply is determined by the seller’s reservation valuation, which can also be inter-preted as an opportunity cost. The demand curve can be identified with the consumerswillingness to pay for a lemon, namely 1,200. Any price p∗ ∈ [1, 000, 1, 200] clears themarket. Different equilibrium prices determine how the total surplus from exchange isdistributed between sellers and buyers but it does not affect total surplus from exchange

2Note that, in models without asymmetries of information, such as classical consumer theory, the willingnessto pay reflects only consumer preferences (e.g. the marginal or incremental utility of a good). In contrast, inthis case the willingness to pay depends µe( p) on the market price because the price delivers information aboutthe unobserved quality of the goods traded.

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peaches are traded. The information asymmetry produces adverse selection, the market

for used cars is a market for lemons, peaches are crowded out of the market.

3. Suppose p ∈ (1, 200, 2, 000)a) Sellers: Only lemons owners would be willing to sell, since 1000 < p


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used car market is a market for lemons and there is a welfare loss associated to the fact that

peaches are not traded.

2.1.3 Analysis

By comparing the results of the First Best to the Second Best we conclude that when thequality of the traded good is not publicly known, the market may fail to allocate resourcesefficiently. Only lemons are traded while the willingness to pay for peaches is greater thantheir opportunity cost. The loss of efficiency is measurable, it is the difference between socialsurplus obtained with asymmetric information and the social surplus in the first best. Itcorresponds exactly to the social surplus associated with the exchange of peaches:

Inefficiency = 30, 000 − 10.000 = $20, 000.

This market failure is usually referred as adverse selection because only low quality cars traded.

Importantly, rational expectations play a key role in this model. When a seller decides whetheror not to participate in the market, it changes the perception of the buyer with respect tothe quality of the cars that are exchanged. When a lemon car owner decides to participate inthe market, the willingness to pay falls, generating an ”informational externality”. This hurtspeaches car owners to the point of crowding them out the market.

Is there always adverse selection when the quality of the good is private information? No. For

example, assume that the proportion of lemons and peaches is 0.25 and 0.75, respectively. Inthis case E [θ] = 0.25 · 1, 200 + 0.75 · 2, 400 = 2, 100. Consequently, any price p∗ ∈ [2000, 2100]will be a market equilibrium and all cars are traded. In fact, these prices exceed the reservationvalue of the sellers of peaches and at the same time, Θ( p) = { peaches, lemons}, so that thewillingness to pay is µe( p) = 2, 100 ≥ p. In general, if the proportion of lemons is less than 13there are two equilibria: in addition to the equilibrium adverse selection, there is an equilibriumin which all cars are traded .

If there are multiple equilibria, the social surplus is higher in the equilibrium with more trade.

2.2 Adverse selection in insurance markets

We start with a numerical example of the market for car insurance. A more general model ispresented in the sequel. In this case the hidden characteristic is a driver’s ability. We assumethat in the market there are 2000 drivers, 50% are good drivers and 50% are bad. A gooddriver is characterized by a lower probability of an accident than the bad driver. Thus, for thesame coverage, a bad driver -who has a a higher likelihood of an accident- is willing to paymore for insurance. The probability of an accident and the valuation of insurance for each typeof driver is described by the table below.

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(a) α = 0.5 (b) α = 0.75

Figure 2.1: Equilibrium Analysis

Bad driver Good driver

Willingness to pay for insurance 11, 600 2, 975

Probability of an accident 0.20 0.05

An insurer obtains a (per unit) expected profit Π = M

−C , where M is the premium charged

for insurance and C is the expected cost of reimbursing a customer for an accident. We as-sume that insurance companies are risk neutral. In case of an accident, the insurance coversa fixed amount equivalent to a loss of $50,000. That is, the coverage is $50,000 if there isan accident and $0 otherwise. Finally, we assume that the insurance market is competitive.This means that, in equilibrium, firms will have a zero expected profit (Π = 0 ⇔ M = C ).It is also assumed that the seller has rational expectations. This implies that, if the insureris unable to distinguish between different types of drivers, he will calculate the expected costof coverage C taking into consideration the participation of good and bad drivers in the market.

As in the previous example, we first analyze the case of complete information.

2.2.1 First Best: Complete information

Suppose that insurance companies know with certainty the type of driver it faces. Since theseller can distinguish whether the client is a good or a bad driver, there are two separate in-surance markets: one for good drivers and one for bad drivers. In each market firms mustmake zero expected profits in equilibrium. Hence, the market premium is equal to expectedcost of coverage. For good drivers, the expected cost is C B = 0.05 · 50, 000 = $2, 500 and forbad drivers this cost is C A = 0.20 · 50, 000 = $10, 000. It follows that insurance premium ineach market are given by M B = C B = $2, 500 and M A = C A = $10, 000, respectively. In both

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markets, the prices are lower than the willingness to pay of potential customers (2 , 975 > 2, 500

and 11, 600 > 10, 000), therefore, all drivers are insured.

If the driver’s ability is public information, the total surplus per driver associated with bothtypes of drivers is 1000 · (11, 600 − 10, 000) + 1000 · (2, 975 − 2, 500) = $2.075 millions.

2.2.2 Second Best: Asymmetric information

In this case insurance sellers cannot distinguish the riskiness of drivers. In principle, good andbad drivers would buy their insurance in the same market at the same price. If all drivers

participate, the average expected value of repayment C depend on the distribution of good andbad drivers. Since there are as many good drivers as bad ones,

C = 0.5 · C B + 0.5 · C A = 0.5 · 0.20 · 50, 000 + 0.5 · 0.05 · 50, 000 = $6, 250.

In a competitive market this value would also be the insurance premium M .

However, good drivers will be unwilling to pay this premium, M = $6, 250, as it exceeds theirwillingness to pay, $2,975. They would not participate in the market. The seller, anticipatingthat a good driver would not buy insurance, would not offer this premium as it does not cover

the expected cost of covering the bad driver is C B = $10, 000. In fact, knowing that gooddrivers do not participate in the market, insurers would offer a premium M = C = $10, 000.Hence, only bad drivers are insured and there is rationing or adverse selection.

Just as in the market for lemons example, adverse selection involves an efficiency loss. In thiscase, the total surplus transactions is 1000 · (11, 600 − 10, 000) = $1.6 millions. The differencewith respect to the first best is $0.475 millions. This amounts to the social surplus associatedto good drivers, who are now crowded out of the market.

As before, for a different distribution of types there need not be adverse selection. For example,if the share of high-risk drivers is 5%, the average expected value of repayment is C = 0.05

·0.20 · 50, 000+0.95 · 0.05 · 50, 000 = $2, 850. In this case, low-risk individuals are willing to payfor insurance and a pooling equilibrium in which all individuals buy insurance and are chargedthe same premium exists.

2.3 Insurance market, a model

We consider a formal model of the insurance market. Assume that there are two types of individuals who want insurance, high risk and low risk individuals. The probability of anaccident of a high risk individual is θA and for a low risk this number is θB. It holds that

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0 < θB < θA < 1. The initial wealth of the individual is W and if an accident occurs he would

suffer a loss of L. If the individual is not insured and an accident occurs, his final wealth isW − L. (In the example of previous section L = 50, 000.)

In general, the wealth levels with or without an accident will depend on the insurance. In fact,an insurance lowers the difference in wealth between the good and bad states. Let W 1 and W 2be the wealth levels without and with an accident, respectively. If v(·) is the Bernoulli utilityfunction of a typical consumer, v(·) increasing and concave, his expected utility is given by

U (W 1, W 2) = (1 − θ) · v(W 1) + θ · v(W 2),

In particular, the utility without insurance is

U N (θ) = (1 − θ) · v(W ) + θ · v(W − L) = v(W ) − θ · [v(W ) − v(W − L)],

which is decreasing in the level of risk θ .

On the supply side, in principle, a seller can design a contract described by the pair ( M, D),where M is the premium and D is the deductible. The deductible is the amount of the loss isnot covered by insurance, so that the reimbursement in the event of an accident is L − D. Foreach insurance contract the seller gets a profit π = M −C = M −θ · (L − D). Note that if theapplicant purchases the insurance, then his wealth will be W 1 = W −M and W 2 = W −M −D.Thus, setting a contract (M, D) amounts to setting the individual wealth levels (W 1, W 2).

Full or perfect insurance is a contract such that the individual faces no risk. That is, W 1 = W 2or equivalently, a zero deductible, D = 0. In this case, the expected coverage is C = θL andthe seller’s expected profit is Π = M − θL. Hereafter, for simplicity, we consider that the onlyinsurance available in the market is perfect insurance. In this case, individual wealth levels aresuch that W 1 = W 2 = W − M irrespective of whether or not an accident occurs.

We note that the maximum willingness to pay for full insurance for an individual of type θcorresponds to the value M ∗(θ) that makes him indifferent between buying the insurance withthet premium and not buying the insurance:

v(W − M ∗(θ)) = U N (θ).

For illustration, if v(z) = √

z then M ∗(θ) = θ2(1 − θ)(W − √ W √ W − L) + θ2L. Moreover,suppose that W = 90, 000 and L = 50, 000. Then, for θ = 0.2 we obtain M ∗(θ) = 11, 600 andfor θ = 0.05, M ∗(θ) = 2, 975. These are precisely the values used in our numerical example.

It can be shown, by differentiating the above equation with respect to θ, that the willing-ness to pay for insurance grows with the probability of an accident θ. Intuitively, given thesame level of insurance, the individual more likely to have an accident values the insurance more.

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2.3.1 First Best: Perfect Information

In this case, the insurer can perfectly identify customers according to their risk level. Thisallows insurers to offer different contracts for each type of customer. If the market is perfectlycompetitive (Π = 0), the premium set for a customer of type θ is M (θ) = θL. If the consumertakes this contract, then W 1 = W 2 = W − θL , i.e., completely eliminate the risk. In fact,this contract allows the buyer to enjoy a wealth level W − θL with certainty. This is theexpected wealth level of the individual if he does not buy insurance. The difference is that,without insurance, this wealth level is not certain. By the definition risk aversion, a risk averseindividual will always take the insurance at that price. In mathematical terms,

v(E [ W̃ ]) = v(W

−θL) > E [v( W̃ )] = (1

−θ)

·v(W ) + θ

·v(W

−L)

This is always true for any function v(.) strictly concave.3

2.3.2 Second Best: Imperfect Information

In this case, the insurer cannot identify customers according to their level of risk, and there issingle insurance market for low and high risk individuals. Since high-risk agents have a higherwillingness to pay for the same insurance (only perfect insurance contracts are offered), we havethat if low-risk individuals are then secured so do the more risky. It follows that there can be

two types of equilibria.

1. Equilibrium with Rationing/Adverse selection In this case, only high-risk individuals (θ = θA) are insured. In the lemons market andthe numerical insurance examples seen earlier, this case corresponds to the equilibriumin which only sold lemons or just bad drivers are insured. The market premium isM AS = θAL. This adverse selection equilibrium requires that low-risk individuals preferremaining without insurance, i.e.,

U N (θB) > v(w

−θAL),

where U N (θB) = (1 − θB) · v(W ) + θB · (v(W − L)). This condition is equivalent awillingness to pay of low risk individuals smaller than the expected loss of a high riskindividual. That is, M ∗(θB) < θAL. In our numerical example, M

∗(θB) = 2, 975 andθAL = 10, 000, so this condition holds.

2. Pooling equiibrium In this case, all potential market buyers participate. Since the seller cannot distinguishthe two types of consumers, it offers a single contract. By assumption, we are only

3Since the right hand side is just the expected utility without insurance U N (θ), and the willingness to payM ∗(θ) satisfies v(W −M ∗(θ)) = U N (θ), the unequality is equivalent to v(W −θL) > v(W −M ∗(θ)). Equivalently,M ∗(θ) > θL, that is, a risk averse individual is always willing to pay more than the expectted loss.

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considering perfect insurance and, since the market is competitive, the premium of this

insurance must be M pooling

= E [θ] · L. Note that if the fraction of individuals of type θAis α, then E [θ] = αθA + (1 − α)θB, a number between θB and θA. Since, M AS = θAL,we have that M pooling < M AS .

For this to be an equilibrium, the low-risk individual should have incentives to take theinsurance, i.e.,

V (w − E [θ]L) ≥ U N (θB)This condition is equivalent to require the maximum willingness to pay of low-risk indi-viduals to exceed the average expected loss: M ∗(θB) ≥ E [θ]L. In our numerical example,M ∗(θB) = 2, 975 and, for α = 0.5, E [θ]L = 6, 250, so this condition is not satisfied.

However, if α = 0.05 then E [θ]L = 2, 850, and the condition holds.

In general, for sufficiently low values of α there will be multiple equilibria, i.e., both theadverse selection and the pooling equilibrium exist.

For some values of the parameters both equilibria are possible. If there are multiple equilib-ria, the equilibrium in which the two types of agents receive insurance Pareto dominates theequilibrium with rationing. To see why, we start by noting that in any equilibrium insurersget zero utility (competitive markets). Moreover, low-risk individuals get a utility U N (θB) inthe equilibrium with rationing and, by revealed preference, in the equilibrium in which all areinsured they get a higher utility. Finally, individuals of type θA get the same insurance in both

equilibria but in equilibrium with rationing they pay a higher premium as M AS

> M pooling

.

2.4 Signaling and Screening in the Labor Market

We want to illustrate two market responses to the problems associated with hidden character-istics: signaling and screening. The case of signaling involves a costly action -the acquisitionof a signal- from the part of the market holding the private information. The value of a signalto the informed party is that it allows this party to distinguish from other types. For example,a quality certification can separate one firm from another offering a similar but lower quality

product. A diploma or a college degree may signal skills or competencies that are not directlyobservable at the time of hiring.

Screening is associated with actions by the uninformed party in the market to learn theprivate information.

Both signaling and screening allow private information to be revealed in equilibrium. However,as we shall see, the fact that the information ends up being revealed ex-post does not mean thatthere are no efficiency losses associated with ex-ante information asymmetries. Both signalingand screening will typically be associated with transaction costs.

We illustrate these phenomena with a model of the labor market. We assume that when a firmhires a worker it does not know his or her innate ability or productivity. Worker productivity θ

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is private information and can be high and equal to θA or low and equal to θB, where θA > θB.

To fix ideas, we assume that θA = 300 and θB = 200. The fraction of high productivity workersis α and the fraction of low productivity workers is 1 − α.Firms use only labor to produce and have constant returns to scale. This means that the profitsof hiring a worker of productivity θ are given by θ − w, where w is the wage that the firm pays.We consider a competitive market in which firms compete to hire workers: in equilibrium, firmsmake zero profits.

A worker is risk neutral. If his or her type is θ, the Bernoulli utility is given by

u(w,a,θ) = w − C (a, θ)

where w is the wage and C (a, θ) is the cost of an action a. In the case of signaling, we assumethat this action corresponds to the choice of a level of education, the signal. In the case of screening, we assume that it corresponds to a task level chosen by the hiring firms.

2.4.1 Education as signaling

The classic model of Spence (1974) suggests that education can serve as an informative signal of the unobserved productivity of a worker. For simplicity, to highlight this role, we assume thateducation does not increase productivity at all, it just has informative value. The qualitativeconclusions would not change if education has also a productive role.

We consider a game with the following sequence of actions:

1. Nature chooses the type θ of the worker, only the worker observes his type.

2. After observing θ, the worker decides a level of education that can be 0 or ê > 0. Weinterpret this level as the years of higher education.

3. Firms observe the level of education and offer a wage w(e) contingent on e.

4. The worker decides whether or not to accept the offer.

The key assumptions are the following:

• It is costly to produce the signal: C (0, θ) = 0 y C (ê, θ) > 0• The cost of education is lower for high productivity workers than for low productivity

workers. In particular, assume that

C (e, θ) = c(θ)ê

where c(θ) is the unit cost per year of education. If cA = c(θA) is the cost for someonewith high productivity and cB = c(θB) of someone with low productivity, this translatesinto cA < cB.

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As a reference, we start by noting that if productivity were observable (no asymmetries of in-

formation), firms would pay a wage equal to productivity. That is, there would be two markets,one for each type of worker and the equilibrium wages in each market would be wA = θA = 300and wB = θB = 200.

We consider two possible types of equilibrium, pooling and signaling (or separating equilib-rium).

Pooling equilibrium

In a pooling equilibrium, both types of workers receive the same salary w pooling. The zero profit

condition for firms implies that the wage is equal to the expected productivity, i.e.,

w pooling = E [θ] = α × 300 + (1 − α) × 200.

The utility of each worker is precisely equal to u = E [θ], independent of the type. Relative to thecase where productivity is observable, low-productivity workers are better and high productivityare worse because their salaries are dragged down by the fact that their productivity is pooledwith the one of lower productivity workers.

Signaling equilibrium

In what follows we characterize the conditions for the existence of a separating equilibriumin which low-productivity workers chose not educate -no investment in signal, high productiv-ity workers educate -signal- and firms offer different wages depending on the signal they observe.

Before observing the signal, the a priori probability that a firm assigns to a high-productivityworker is α. In a signaling equilibrium, after observing the education level of a worker, the firmupdates its belief on the (unobserved) productivity of the worker. If µ(e) is the probabilitythat a firm assigns the worker to be highly productive after observing a level of education e, ina separating equilibrium, µ(e = ê) = 1 and µ(e = 0) = 0 . In a competitive market, firms offerwages equal to expected productivity. Consequently, if w(e) is the wage offered after observing

a level of education e, w(e = 0) = θB = 200 and w(e = ê) = θA = 300.

In separating equilibrium, no education (e = 0) must be a best response for a low productivityworker ((θB) and education (e = ê) must be a best response for a high productivity worker(θA).

The incentive compatibility condition for a worker of type θB is:

u(w(0), e = 0, θB) ≥ u(w(ê), e = ê, θB) ⇐⇒ 200 ≥ 300 − cB ê ⇐⇒ cB ê ≥ 100. (IC B)

The condition simply says that, for θB, the cost of the signal is not compensated by the benefitof the signal corresponding to a wage increase measured by the difference between the two

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productivities.

The incentive compatibility condition for a worker of type θA is:

u(w(ê), e = ê, θA) ≥ u(w(0), e = 0, θA) ⇐⇒ 300 − cAê ≥ 200 ⇐⇒ cAê ≤ 100. (IC A)

For θA, the benefit of the signal exceeds its cost.

To fix ideas, suppose that cA = 20 and cB = 25. The (IC B) condition translates into ê ≥ 4while the (IC A) condition requires ê

≤ 5. That is, there is a range for the level of education

and to work as a signal: it must be greater than 4 years -sufficiently high to discourage θB-and less than five years -not too excessive for θA to be interested to incur in that cost.

Figure 2.2: Signaling

Welfare Analysis

How does the economic welfare of the agents in the signaling equilibrium compare with thecase in which productivity is directly observable?

• In this example, education has no productive value, its only function is informative.Given this, the cost of the signal is pure social loss. With the numbers above, ê = 4

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is the minimum level required for a separating equilibrium to work. For this case, the

equilibrium utility for each type of worker usA = 300 − 20 ∗ 4 = 220 and u

sB = 200.

• Accordingly, the type θA is strictly worse than if productivity were observable. There isa transaction cost equal to 80, the cost of the signal.

• Signaling allows for private information to be disclosed and, in practice, it recovers theexistence two labor markets. But this is not free, there is a cost associated with theoriginal information asymmetry.

How does the economic welfare of the agents in the signaling equilibrium compare to a poolingequilibrium?

• Low productivity workers are worse off. As before, the do not educate (i.e., do not incurthe cost of the signal) but are paid less.

• While motivation for signaling is that high productivity workers can distinguish them-selves from lower productivity ones, it is not clear that they are better in the signalingequilibrium vis a vis the pooling equilibrium. In fact, they are better only if:

u pooling = E [θ] = α × 300 + (1 − α) × 200 ≤ usA = 220 ⇐⇒ α ≤ 0, 2.

That is, if the fraction of high productivity is high (α > 0, 2), so that the average pro-ductivity is relatively high, a high productivity worker would prefer not to signal. In

this case, the pooling equilibrium Pareto-dominates the signaling equilibrium. Paradox-ically, individuals are trapped in an equilibrium that reveals the information, but it isunequivocally worse than one in which there is no disclosure.

2.4.2 Screening competitive in the labor market

The central idea of screening is that the uninformed party -firms in this example- can inducethe informed party -the worker- to disclose information by offering a menu of options. Thismenu leads different types to choose different options in the menu. Separation and informationdisclosure occurs by self-selection into different options. The screening mechanism is context-dependent. In the labor market, one way to screen is to offer contracts that include, in addition

to a compensation w, a task t that allows to discriminate (a screening test, for example, or aspecific task to reveal competencies).

We consider a game, similar to previous one, with the following sequence of actions:

1. Nature chooses the type θ of the worker, only the worker observes his type.

2. The firms offer a menu of contracts simultaneously. In particular, each firm offers twocontracts: C A = (wA, tA) and C B = (wB, tB) where w j is the compensation and t j is thelevel of the task. In principle, firms can offer different menus with different contracts,but in equilibrium all will offer the same menu. Each contract C j is designed to attracta worker of type θ j, j = A, B.

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3. The worker decides which contract to accept.

The utility of an agent of type θ if he accepts a contract (w, t) is u(w ,t,θ) = w − C (t, θ).The key assumptions are:

• A task level t has a cost C (t, θ) where C (0, θ) = 0 and C (t, θ) > 0 for t > 0.• The cost of the task is less for high productivity workers. In particular, assume that

C (t, θ) = c(θ)t.

If cA = c(θA) is the unit cost for someone with high productivity and cB = c(θB) of

someone with low productivity, this translates into cA < cB.

For simplicity, we assume that the task does not affect the profit of the firm. This meansthat the expected profits of the firm are α(θA − wA) + (1 − α)(θB − wB). That is, the firm isinterested in minimizing the expected compensation αwA + (1 − α)wB, which only depends onw j. This does not mean that the tasks t j are irrelevant. In fact, differentiating tasks allowscontracts to induce self-selection. For this to happen, contracts must satisfy the following in-centive compatibility constraints.

The incentive compatibility condition for a worker of type θB is:

u(wB, tB, θB) ≥ u(wA, tA, θB) ⇐⇒ wB − cBtB ≥ wA − cBtA (IC B)

The incentive compatibility condition for a worker of type θA is:

u(wA, tA, θA) ≥ u(wB, tB, θA) ⇐⇒ wA − cAtA ≥ wB − cAtB (IC A)

In principle, it could be that there are two types of equilibria: a pooling equilibrium in whichboth types are offered the same contract; and a separating equilibrium in which each type isoffered a different contract, inducing self-selection. The following proposition summarizes theresults of the screening model.

Proposition 2. In the model of screening in the labor market

1. There is no pooling equilibrium. If an equilibrium exists, it is a separating equilibrium.

2. In a separating equilibrium:

• The low productivity worker accepts the contract C B = (θB, 0).• The high productivity worker accepts the contract C A = (θA, tA) where tA = (θA −

θB)/cB.

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To see that there is no pooling equilibrium, suppose that there is one. We will see that there

are incentives for firms to deviate, which means it cannot be an equilibrium. Indeed, if there isa pooling equilibrium, as firms must make zero profit (competitive market), the wage offeredmust be equal to the expected productivity: w = E [θ] = αθA + (1 − αB)θB. This means thatfirms are making a strict gain on high productivity workers (θA − E [θ] for each) and a strictloss on low productivity (θB − E [θ] for each). The gain and the loss net out. Consequently,a firm can cream skim the market and attract only high-productivity workers by offering aslightly higher wage w + and a task t + η also slightly higher. As the cost of the tasks dif-fers across workers of different type, η can be chosen high enough so as to attract only highproductivity workers. Thus, the firm will make a profit θA − E [θ] − > 0 for small. There-fore, in a situation of pooling, cream skimming is a deviation that is strictly beneficial for a firm.

Figure 2.3: Skimming

In a separating equilibrium in which firms earn zero profits, firms must pay each type of workera wage equal to their productivity. That is, wA = θA and wB = θB. To understand why thetask of the type of low productivity is tB = 0, observe that if in equilibrium firms offered acontract with tB > 0, a firm could deviate by offering a contract with a lower task level tB = 0and improve its profit. For example, by offering a wage somewhat smaller than θB, say θB − the firm would continue to attract workers as they are more than compensated by a lower task.The firm would increase profits as it pays lower salaries. Therefore, for this deviation not tooccur, it is necessary that tB = 0. Intuitively, in the background firms compete a la Bertrand,pushing the task level down to a minimum.

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Figure 2.4: Competition

Finally, the task tA in the contract accepted by a high productivity worker is the minimumthat can satisfy incentive compatibility for type θB, (IC B). Indeed, the firm could offer lowerwages to a high productivity worker if it also decreases the level of the task to compensate him.This increases the firms profit. However, the firm cannot go to far as it needs to prevent lowproductivity workers to choose the high productivity contact.

Figure 2.5: Equilibrium Contracts

As in the case of signaling, we conclude that screening allows the disclosure of private informa-tion. In this case, a menu of contracts that satisfy incentive compatibility constraints induces

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self-selection, but it does so at a has a cost. Indeed, the high productivity agent pays cost of

doing a task without productive value. The transaction cost is cAtA = cAcB (θA − θB). If we use

the same numbers as in the case of signaling, cA = 20 and cB = 25, the transaction cost is0, 8(300 − 200) = 80. This amounts to the same as signaling cost that allows for a separatingequilibrium in our signaling example.

Documents

Asymetries of Information: Moral Hazard and Adverse Selectiona