Lectures on the Theory of Contracts in Discrete and

Lectures on the Theory of Contracts in Discrete and Continuous Time Jaeyoung Sung Ajou University, Korea E-mail: [email protected]

This version: August, 2007

Preface This very preliminary and incomplete manuscript is based on lectures that I have given to graduate students at the University of Illinois at Chicago and at Pohang University of Science and Technology, Pohang, Korea. The objective is to introduce methodological aspects of contracting theories and their applications in corporate finance problems. The target audience for this manuscript are students and researchers who have basic background in mathematics including introductory stochastic calculus. Background in finance is desirable but not required. Chapter 1 provides a brief overview of various areas of finance, and describes how agency problems can arise in corporate management. Then we examine moral hazard and adverse selection problems using discrete-time and continuous-time models in chapters 2 to 5. In particular, the moral hazard problems are investigated with discrete-time formulations in chapter 2 and 3, and with a continuous-time formulation in chapter 4. In chapter 5, we study interactions between moral hazard and adverse selection problems in both discrete-time and continuous-time formulations. Then we discuss dynamics of multi-period contracting problems in chapter 6. In chapter 8 to 10, other important contracting issues are introduced, including career concern, signalling and incomplete contracting.

Contents 1 Introduction 1.1 Corporate Investment Decisions . . 1.2 Principal-Agent Problems . . . . . 1.2.1 Examples: . . . . . . . . . . 1.3 How to Control Moral Hazard . . . 1.3.1 Monitoring . . . . . . . . . 1.3.2 Bonding . . . . . . . . . . . 1.3.3 Explicit incentive contracts

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2 Moral Hazard and Incentive Contracts: Discrete-Time 2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Feasible contracts . . . . . . . . . . . . . . . . . . . . . . 2.3 The First-Best Contract . . . . . . . . . . . . . . . . . . 2.4 The Second Best Contract with a Risk Neutral Agent . . 2.5 The Second Best Contract with Limited Liability . . . . 2.6 The Second Best Contract with a Risk Averse Agent . .

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Approach I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 15 16 17 18

3 Moral Hazard and Incentive Contracts: Discrete-Time Approach II 3.1 The Basic Model with Continuous Outcome Distribution . . . . . . . 3.2 The First-Best Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Second-Best Solution: the case of moral hazard . . . . . . . . . . 3.4 The Shape of the Second-best Contract . . . . . . . . . . . . . . . . . 3.5 The Value of Informative Signal in Contracting . . . . . . . . . . . . 3.5.1 Application to Insurance Deductibles . . . . . . . . . . . . . . 3.6 The Validity of the First-order Approach . . . . . . . . . . . . . . . . 3.6.1 Normally Distributed Outcome . . . . . . . . . . . . . . . . .

21 21 22 23 26 28 29 29 32

4 Moral Hazard and Incentive Contracts: Continuous-Time Approach 4.1 The First-best Contract . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Second-best Contract . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Unobservable Project Selection . . . . . . . . . . . . . . . . .

35 36 38 45 47

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5 Adverse Selection and Moral Hazard 5.1 The Model: Pure Adverse Selection . . . . . . . . . . . . . . . . . . 5.2 The Two-Type Case . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The first best . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Asymmetric information: The second best . . . . . . . . . . 5.2.3 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Continuum of Types . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Managerial Contracting under Adverse Selection and Moral Hazard 5.4.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .

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51 52 53 53 54 56 57 60 65

6 Dynamics of Contracts under Moral Hazard 6.1 Full Commitment . . . . . . . . . . . . . . . . 6.1.1 The two-outcome model . . . . . . . . 6.2 Moral Hazard with Noncommitment . . . . . 6.2.1 Case 1 . . . . . . . . . . . . . . . . . . 6.2.2 Case 2 . . . . . . . . . . . . . . . . . .

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69 69 71 77 78 78

Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Dynamics of Contracts under 7.1 Full Commitment . . . . . . 7.2 Noncommitment . . . . . . 7.2.1 The two-type case . .

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8 Career Concern

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9 Signaling 9.1 The Market for Lemons . . . 9.2 Job Market Signaling . . . . . 9.2.1 Separating Equilibria . 9.2.2 Pooling Equilibria . . . 9.2.3 Job market equilibrium

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A A.1 Dynamic Programming Equation with Exponential Utility . A.2 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . A.2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Optimal Contracts . . . . . . . . . . . . . . . . . . . A.3 Backward Stochastic Differential Equation (BSDE) Method . A.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . A.3.2 Representation of Incentive-compatible Contracts . . A.3.3 The Principal’s Problem . . . . . . . . . . . . . . . . A.4 Time Additive Utility . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1 Introduction There are many areas in finance. Basic areas include corporate finance, investments and financial intermediation. Corporate finance is concerned with investors’ investing and financing behaviors in product markets. Investments deals with the same issues in capital markets. In product markets, real assets are traded. Examples of real assets are factories, machinery, lands, production technology, etc. Major issues in corporate finance include capital budgeting (choice of right projects), capital structure (choice of financing methods), dividend policy, and mergers/acquisitions. In capital markets, financial assets such as stocks, bonds, and derivatives are traded. Major issues in investments include asset pricing, consumption/investment decisions, and market efficiency. A popular subarea of investments is derivatives which is concerned with pricing derivative assets such as options, futures and swaps Financial intermediation is also an important area to facilitate investors’ various investing and finance activities in both product and capital markets. This area is concerned with the efficiency of security trading systems (market microstructure), and brokerage functions of various financial institutions such as investment banking and insurance. In typical MBA programs, the major three areas of finance are covered in many different finance courses. such as corporate finance, investments, options and futures (financial engineering), mergers and acquisitions, international finance, money and banking, and insurance. Sometimes, real estate is also included in the finance program. In this manuscript, we mainly focus on the agency relationships between the manager and investors in corporate management. The agency theories provide a major paradigm to study modern corporate finance. 3

1.1

Corporate Investment Decisions

The values of financial assets fundamentally depend on the value of underlying assets. For example, the fundamental determinant of corporate securities such as stocks, bonds and their derivatives is the economic value of the corporation. The firm creates its value by choosing right projects, financing with right kinds of capital market instruments, and managing for the best outcomes (profits). In order to examine how the value is created, let us consider a simple example of corporate investment decision-making problem. Suppose that the current riskfree interest rate is 5%, and that the firm has an opportunity to invest in a project with the following prospect of future cashflows: t=0 t=1 t=2 −1,000 500 800 Then the firm’s problems is to decide whether it should accept this project. Since the opportunity cost of taking on this project (the cost of capital) is the riskfree interest rate, the decision rule can be as simple as follows: If the project yields higher return than the riskfree interest rate, then accept the project. Since 800 500 + − 1000 = 201.81 > 0, 1.05 1.052 the firm should accept the project. In other words, since the present value (PV) of the future cashflows from the project is greater than the initial required investment, the project should be accepted. By accepting this project, the manager increases the value of the firm by $201.81. The above practice for the project selection decision is called “the positive-NPV rule.” The NPV (net present value) of the project is defined as follows: NPV := PV of future cash inflows − PV of future cash outflows. The NPV is the amount of value created by the project, and it is also sometimes called the “economic value added.” It is well-known that the optimal project selection rule is: take a project if its NPV is positive; and otherwise reject. The reason why the positive-NPV rule is called “optimal” is that the rule leads to the maximization of shareholders’ wealth. However, typically the decision maker is the manager or the CEO of the firm, not shareholders. The positive-NPV rule implicitly assumes that the manager always behaves in the best interest of shareholders. Then one may ask whether the assumption can be valid in reality. In particular, one may ask whether the manager may want to use the positive-NPV rule even when his project selection cannot be unobserved by shareholders at all. Given that the manager is an independent economic agent, there is no a priori reason why the manager should behave to maximize the shareholders’ wealth unless the manager has incentives to do so. Then the question is: What kind of incentive systems are needed to motivate the manager to work for shareholders’ best interests? 4

In order to obtain an insight into this question, we need to look at the relationship between the manager and shareholders.

1.2

Principal-Agent Problems

The manager and shareholders are related to each other via a managerial compensation contract. In the literature, the party like the manager is called the agent and the party like shareholders the principal. A problem arises because the agent acts in his own best interest, not in the best interest of shareholders. If the principal can costlessly monitor the agent’s actions, then the principal will specify in the contract all courses of actions the agent must follow. If the agent deviates from the specification, then the agent will be punished so severely that he has no incentives to go against it. If the principal cannot costlessly monitor the agent’s actions, the principal cannot enforce the detailed action plan to the agent. As a result, the agent may have discretion on some actions. The structure of the contract affects the agent’s behavior, particularly on actions that are unobservable to the principal. The agent carries out the unobservable actions to his own interest, not to the principal’s best interest. This kind of problem is called “moral hazard problem.” Note however that the term is a misnomer. Instead, it should be called “morale hazard problem.” In general, the principal-agent problems can arise because of the following reasons: • Hidden action: moral hazard Hidden information (information asymmetry after contracting) • Adverse selection (information asymmetry before contracting) In brief, the principal-agent problem arises because of information asymmetry between the two parties before or after the contract is signed. Consequently, both the principal and agent play different kinds of games against each other depending upon different kinds of information asymmetry problems. The structures of different games are illustrated in Figure 1.1 through 1.5.

1.2.1

Examples:

It is instructive to go over a few real life examples of agency problems. The examples in this subsection are adapted from Milgrom and Roberts [1992, p170]. The U.S. Savings and Loan Crisis Savings and loan Associations (S&Ls) are financial institutions that borrow money from the public in the form of deposits and invest it primarily in residential mortgage loans. The deposits of individual investors in an S&L are insured by the Federal 5

Savings and Loan Insurance Corporation (FSLIC), a U.S. federal government agency, until 1990. Each S&L purchases the deposit insurance. The size of the insurance premiums were not linked to the riskiness of the S&Ls’ portfolio of loans and other investments. In case of default by an S&L, the depositors will get paid by the FSLIC. In 1980s, the S&Ls turned to riskier investments, including loans on commercial real estate and “junk bonds” (high-yielding but risky corporate bonds). Unfortunately, a lethal combination of events occurred: (1) Commercial real estate market collapsed in several parts of the country; and (2) defaults by some corporations on their junk bonds. As a result, over 500 S&Ls slipped into bankruptcy. Because of this massive scale of the bankruptcy problem, the FSLIC’s reserves were inadequate to cover its promises to protect depositors and U.S. tax payers ended up with paying the bill for hundreds of billions of dollars. Who or what is to blame? Milgrom and Roberts [1992] argue that moral hazard is to blame. The deposit insurance program was designed so poorly that its low capital requirement gave S&Ls managers incentives to take risks aggressively at the expense of depositors’ and U.S. tax payers’ wealth. In 1980s, S&L capital requirements (the amount of the S&L owners’ own money at risk) was as low as 3% of its total investments. To see how this low capital requirement creates moral hazard problems, let us consider the following example. Suppose a S&L can choose between to possible investments, “safe” and “risky.” Both investments require an initial outlay of $100, which consists of $97 from deposits and $3 from the S&L’s own capital. Assume that the interest rate on the deposits is zero. The safe investment returns either $100 or $110 with equal probabilities. The risky investment returns either $125 or $65 with equal probabilities. In this case, although the safe investment is optimal for the society, the S&L has a very strong incentive to choose the risky investment. To make matters worse, competition among S&Ls intensified managers’ risk-taking behaviors. This S&L case illustrates a moral hazard problem which arose in financial institutions. Similar problems can also occur in corporations. Managerial Moral Hazard Consider a widely-held public firm which is characterized by disperse ownership across small investors. Thus no individual shareholder has any real incentives to monitor managers in order to ensure that they are running the firm in shareholders’ interests. In this case, there can be many potential sources of corporate moral hazard problems as follows. • Corporate executives may invest firm’s earnings in low-value projects to expand their empires. • They may pay themselves exorbitantly and lavish expensive perquisites upon themselves. 6

• They may run ongoing operations in a way to pursue their own personal goals other than maximizing the value of the firm. • They may resist attempts to force more profitable operations, especially by resisting takeovers that threaten their jobs. To see why manager’s takeover defense can be viewed as a moral hazard problem, let us look at the history. During the 1980s, there was a wave of hostile takeovers in the U.S. A hostile takeover is the acquisition of enough number of shares in a company to give a controlling ownership in the firm, where the offer to acquire the firm is opposed by the target company’s executives and directors. Successful hostile takeover attempts generally resulted in the replacement of target firms’ senior management and the naming of new board of directors. Many observers have interpreted the hostile takeovers as a corrective response to managerial moral hazard or managerial incompetence. The prices paid for the stock of firms in hostile takeovers in this period on average represented a 50% premium over the target’s original market value. This indicates that manager’s takeover defense can pose a great obstacle in creating extra wealth for shareholders. A major form of takeover defense is poison pills. An example of poison pills is “Shareholder Rights Plans.” For example, existing managers sell “Shareholder Rights Plans” to their shareholders, where the plans give rights to their holders to buy shares of the (target) firm at very low prices if the takeover occurs. By selling these plans, the managers in effect remove the ability of the owners of the firm to sell their shares to a corporate raider. Current systems allow boards of directors to adopt poison pills without shareholders’ approval. The empirical evidence is that adopting a poison pill typically reduces firm’s share price. It suggests that the adoption does not serve shareholder’s interests.

1.3

How to Control Moral Hazard

Two major conditions for a moral hazard problem to arise between the principal and agent are: (1) conflicts of interests, and (2) inability to write enforceable contracts covering all crucial elements of transactions. There are several ways to mitigate a moral hazard problem.

1.3.1

Monitoring

Shareholders may devote resources to monitoring and verification, and use the results of monitoring as the basis for rewards and penalties. They may use monitoring in preventing inappropriate behavior directly catching it before it occurs. E.g. U.S. firms are not allowed to publish financial statements until they have been verified by independent auditors. 7

1.3.2

Bonding

In some industries, it is common to require the posting of bonds to guarantee performance. The bond is a sum of money to be forfeited in the event that inappropriate behavior is detected. For example, the capital provided by the owners of S&L acts like a bond because in the event of losses the capital must be paid out to meet obligations.

1.3.3

Explicit incentive contracts

Sometimes monitoring is too costly. It may be impossible to monitor individual managerial actions and efforts in various managerial activities, but it may still possible to measure the outcome of managerial actions and efforts. The outcome may be the realized (or reported) annual profit of the firm. Unfortunately, perfect connections between unobservable actions and observed outcome are rare. Nevertheless, the principal may use the outcome to provide the agent incentives to work toward the principal’s interests by rewarding good outcomes. For example, in practice, explicit incentives contracts are constructed so that the bonus depends on various quantities including accounting earnings per share (EPS), return on equity (ROE), return on asset (ROA), economic value added (EVA, net operating profit after tax - cost of capital), stock options with fixed exercise prices, stock options with fixed exercise prices, stock options with adjustable exercise prices (exercise price increases at a rate equal to cost of capital minus dividend yield minus compensation-risk premium), and stock ownerships. The bonus plans based on above quantities however may not elicit desired managerial behaviors, because the plans can sometimes induce the manager to go against shareholders’ interests. For example, the managerial bonus plans in Xerox were based on the EPS. According to the W.S.J. article (June 1, 2001, C1) by Maremont and Bandler, “Concession by Xerox May Not Satisfy the SEC,” the SEC found that Xerox inflated its revenues by using a low discount rate, say 6%, in valuing lease contracts with companies in Latin America when the local interest rates were as high as 30% in Brazil. It was argued that Xerox managers used lower discount rates in an attempt to receive large bonuses for high revenues. However, this kind of misuses of accounting rules costs shareholders dearly in higher corporate taxes as well as larger executive bonuses. The bonus plans based on the other quantities can also sometimes produce undesirable managerial behaviors. Exercise 1.1. Stock options and ownerships are popular ways used in practice to reduce managerial moral hazard problems. Discuss why they can effectively or ineffectively serve the purpose. Then what will be the right quantity for the bonus plan to be based on? The bonus plan should be depend on the outcome of managerial effort and decisions. Note that typically the outcome is interpreted as the overall profit of the firm. However, the overall profit can be a sum of market-dependent profit and firm-specific 8

profit. Of the two components, the market-dependent profit has nothing to do with managerial effort and decisions. Suppose that the firm’s profit in a particular year has increased or decreased because of favorable or unfavorable market situations. Then the manager should not be rewarded or penalized because of increased or decreased market-dependent portion of the overall profit. What the manager can affect by his effort and decisions is the firm-specific portion of the overall profit. Thus the managerial bonus plan has to depend only on the the firm-specific portion of the overall profit, not on the overall profit. In this sense, a bonus plan tied with stock price is not efficient, because the manager can be rewarded or penalized for market movements. In this manuscript, we focus on the design of incentive contracts based on firmspecific portion of the overall profit. Finally, gaming structures of agency relationships are illustrated in the following figures.

9

Figure 1.1: Hidden action: moral hazard Outcome:Y˜ = e + ε˜ Agent’s cost = C(e) Information: Principal observes Y˜ High Effort

Accept P

- N

ContractA

Low Reject

j

*$

j$

End

Example: After managerial contracting, the principal cannot observe managerial effort.

Figure 1.2: Hidden information Outcome:Y˜ = e + I˜ Agent’s cost = C(e) Information: Principal observes Y˜

Good message & Effort

Good

-$

: Accept P

N *

Bad

ContractA Reject

j

A z -$ Bad message & Effort

End

Example: After contracting, the manager knows more about the firm than the principal.

Figure 1.3: Adverse Selection Outcome:Y˜ = e + a ˜ Agent’s cost = C(a, e) Information: Principal observes Y˜

Choose High : N

A

- P

Effort

- $

Contract menu A

Low z Reject j End Example: Before contracting, the manager knows better about his own ability.

10

Figure 1.4: Moral Hazard (hidden action) and Adverse Selection Outcome:Y˜ = e + a ˜ + ε˜ Agent’s cost = C(a, e) Information: Principal observes Y˜

Choose

High : N1

A Low

Contract menu - P A

EffortN2

High* $ Low j $

z j Reject End

Example: Before contracting, the manager knows his own ability. After contracting, the managerial effort is unobservable.

Figure 1.5: Hidden action + Hidden information Outcome:Y˜ = e + I˜ + ε˜ Agent’s cost = C(e) Information: Principal observes Y˜ Good message & Effort - N2

Good : Accept *

N1

A Bad

ContractP A

z

- N2 Bad message & Effort

High $ * -$ Low High$ Lowj

Reject j

End

Example: After contracting, the manager knows more about the firm than the principal, and exerts unobservable effort.

11

$

12

Chapter 2 Moral Hazard and Incentive Contracts: Discrete-Time Approach I Although the economic significance of contractual relationships between economic agents have long been recognized since Adam Smith in the 18th century, it was not until only recently that economists started formal rigorous economic analyses. Pioneers in modern contracting theories include Ross [1973], Mirrlees [1974, 1976] and Holmstrom [1979]. These researchers tried to formulate contracting problems by using discrete-time models. In this manuscript, we also start with discrete-time models in two chapters, and then we will discuss continuous-time models. For discrete-time models, we discuss two groups of models: one with discrete outcome distributions in this chapter, and the other with continuous outcome distributions in the next chapter.

2.1

The Model

There are two discrete time periods, 0 and 1. The contract between the principal and agent can only be written on verifiable information. The principal is risk neutral and the agent is risk averse. The principal precommits to contract S, i.e. she makes a take-it-or-leave-it offer to the agent whose reservation utility is exogenously given as θ ∈ R. Recall that the agent’s reservation utility is the level of his utility that he can achieve by rejecting the offer and taking a job elsewhere. We normalize θ to zero. Contract S can be enforced costlessly and accurately (by court). There is no financial markets for the contract, and thus the manager cannot trade his contract in the market in an attempt to undo the incentives implied by the contract. If the agent accepts the offer at time 0, the principal delegates the agent the production of a good until time 1. Then the agent exerts effort (a hidden action) 13

a ∈ {0, 1} which affects the outcome x ∈ {xl , xh }, with xh − xl = ∆x > 0. The influence of the agent’s effort on the outcome is described by probabilities as follows: P r(x = xh | a = 0) = π0 , P r(x = xh | a = 1) = π1 , and ∆π = π1 − π0 > 0. The agent’s effort is unobservable to the principal. The outcome x will be realized at time 1. The contract S can be contingent on the outcome x, but not on a. One may view x as the corporate economic profit realized at the end of year. The agent’s utility for wealth is V (w) with u increasing and concave (V 0 > 0, V 00 < 0). Let H be the inverse function of V , i.e. H = V −1 . Then H is increasing and convex (H 0 > 0, H 00 > 0). On the other hand, the agent’s effort incurs a disutility D(a), which we normalize by setting D(0) = 0 and D(1) = D. We assume that his total utility is separable between wealth and effort, V (S) − D(a). The principal’s problem is to design S(x) in order to maximize her expected utility, or max E[U (x − S(x))], while making S(x) acceptable by the agent. In designing the contracts, the following issues arise: (1) the optimal sharing rule; (2) the effect of moral hazard to the principal’s welfare (agency cost); and (3) the agent’s optimal responses (effort/decisions) given the optimal sharing rule.

2.2

Feasible contracts

Note that the risk-neutral principal’s expected utility is E[U ] = π1 (xh − S(xh )) + (1 − π1 )(xl − S(xl )) if the agent’s effort is 1; and E[U ] = π0 (xh − S(xh )) + (1 − π0 )(xl − S(xl )) if the agent’s effort is 0. From now on we write S(xh ) = Sh and S(xl ) = Sl . The agent’s effort decision depends on the compensation scheme (Sl , Sh ). When it induces the agent to exert high effort, the contract satisfies π1 V (Sh ) + (1 − π1 )V (Sl ) − D ≥ π0 V (Sh ) + (1 − π0 )V (Sl ).

(2.1)

The above condition is called the incentive compatibility condition for the high effort. On the other hand the contract should guarantee the agent his reservation utility. π1 V (Sh ) + (1 − π1 )V (Sl ) − D ≥ 0.

(2.2)

The above constraint is called the participation constraint to induce the agent to accept the contract. This constraint ensures that if the agent exerts the high effort, he will achieve at least his reservation utility. 14

2.3

The First-Best Contract

We first look at a benchmark by assuming that the agent’s effort is observable. Then the principal can ignore the incentive compatibility condition. If she wants to induce effort, her problem becomes max Sl ,Sh

s.t.

π1 (xh − Sh ) + (1 − π1 )(xl − Sl ) π1 V (Sh ) + (1 − π1 )V (Sl ) − D ≥ 0.

The Lagrangian is L = π1 (xh − Sh ) + (1 − π1 )(xl − Sl ) + λ{π1 V (Sh ) + (1 − π1 )V (Sl ) − D}. The first order conditions (FOCs) are −π1 + λπ1 V 0 (Sh∗ ) = 0 −(1 − π1 ) + λ(1 − π1 )V 0 (Sl∗ ) = 0. Therefore, we have λ = 1/V 0 (Sh∗ ) = 1/V 0 (Sl∗ ) > 0, and thus Sh∗ = Sl∗ . That is, the principal provides the agent a full insurance; the agent receives the same compensation regardless of the realized outcome. In particular, the participation constraint implies Sh∗ = Sl∗ = H(D). Consequently, when she tries to induce the high effort, the principal’s optimal expected utility is π1 xh + (1 − π1 )xl − H(D). H(D) can be interpreted as the expected cost to the principal to implement the positive effort level. If she induces zero effort, the principal utility is π0 xh + (1 − π0 )xl . Therefore, she induces the agent exert the high effort, if π1 xh + (1 − π1 )xl − H(D) ≥ π0 xh + (1 − π0 )xl , or ∆π∆x ≥ H(D). The above simply implies that if the expected gain from positive effort is greater than the expected cost of inducing the effort, then the principal is better off with positive effort. 15

2.4

The Second Best Contract with a Risk Neutral Agent

Now we examine the compensation contract when the agent’s effort is unobservable and thus not contractible. To simplify the problem, we first look at the case when the agent is risk neutral. Then in the next section we will investigate the effect of the agent’s risk aversion on the optimal contract. Suppose the risk-neutral principal wants to induce positive effort from the agent. Then the principal’s problem is as follows: max Sl ,Sh

s.t.

π1 (xh − Sh ) + (1 − π1 )(xl − Sl ) π1 Sh + (1 − π1 )Sl − D ≥ π0 Sh + (1 − π0 )Sl π1 Sh + (1 − π1 )Sl − D ≥ 0.

Note that both constraints are binding. Solving the constraints with equalities, we have π0 Sh∗ + (1 − π0 )Sl∗ = 0, π1 Sh∗ + (1 − π1 )Sl∗ − D = 0. Or

1 − π0 π0 D, Sl∗ = − D. (2.3) ∆π ∆π That is, the agent is rewarded if production is high, and punished if production is low. As is the case with the first best, the expected compensation to the agent is D, and that the principal utility under the above optimal contract is Sh∗ =

π1 xh + (1 − π1 )xl − D. Thus, both the principal and agent are as well off as in the first best case. That is, when both the principal and agent are risk neutral, moral hazard problem does not affect their welfare. Suppose that the principal wants to induce zero effort from the agent. Then her problem is max Sl ,Sh

s.t.

π0 (xh − Sh ) + (1 − π0 )(xl − Sl ) π0 Sh + (1 − π0 )Sl ≥ π1 Sh + (1 − π1 )Sl − D π0 Sh + (1 − π0 )Sl ≥ 0.

There are infinite number of contracts that satisfy both constraints including a constant wage contract and the above (Sl∗ , Sh∗ ) as in (2.3).1 However the principal expected utility with zero effort is π0 xh +(1−π0 )xl . Therefore, we have the exactly same condition to motivate the principal to induce positive effort. That is, ∆π∆x ≥ D. 1 In many cases, economists assume that if the agent is indifferent between two choices, then he will choose one that will maximize the principal’s utility.

16

2.5

The Second Best Contract with Limited Liability

Recall that the optimal contract in (2.3) requires the principal to punish the agent for a low outcome. However, in reality, the agent is protected by the limited liability. Thus, there is a limit with which the principal can punish the agent. Now we assume that the principal has extra constraints, Sl ≥ −l,

Sh ≥ −l,

where l > 0. The principal is limited in punishing the agent for a bad outcome. If she wants to induce positive effort, the principal’s problem is max Sl ,Sh

s.t.

π1 (xh − Sh ) + (1 − π1 )(xl − Sl ) π1 Sh + (1 − π1 )Sl − D ≥ π0 Sh + (1 − π0 )Sl π1 Sh + (1 − π1 )Sl − D ≥ 0 Sl ≥ −l, Sh ≥ −l.

One can immediately see the following results. (i) If l >

π0 D, ∆π

then the limited liability constraints are not binding.

π0 (ii) If 0 ≤ l ≤ ∆π D, then the incentive constraint is binding but the participation constraint is not binding. (One can show that the solution to the principal’s problem under the IC constraint and Sl ≥ −l satisfies the participation constraint and Sh ≥ −l. Thus the participation constraint is not binding.) The optimal compensation scheme is given by

Sl∗ = −l,

Sh∗ = −l +

D . ∆π

Moreover the agent’s expected utility with limited liability is π1 Sh∗ + (1 − π1 )Sl∗ − D = −l +

π0 D ≥ 0. ∆π

π0 When 0 ≤ l ≤ ∆π D and the principal wants to induce positive effort, her optimal expected utility is   π1 D π1 xh + (1 − π1 )xl − −l ∆π

If the principal induces zero effort, her expected utility is simply π0 xh + (1 − π0 )xl . Thus she want to induce positive effort if   π1 D π0 D ∆π∆x ≥ −l =D+ −l . ∆π ∆π 17

2.6

The Second Best Contract with a Risk Averse Agent

If she wants to induce positive effort, the principal problem is max Sl ,Sh

s.t.

π1 (xh − Sh ) + (1 − π1 )(xl − Sl ) π1 V (Sh ) + (1 − π1 )V (Sl ) − D ≥ π0 V (Sh ) + (1 − π0 )V (Sl ) π1 V (Sh ) + (1 − π1 )V (Sl ) − D ≥ 0.

The Lagrangian is L = π1 (xh − Sh ) + (1 − π1 )(xl − Sl ) + λ{π1 V (Sh ) + (1 − π1 )V (Sl ) − D} +q{∆πV (Sh ) − ∆πV (Sl ) − D} The FOCs are −π1 + λπ1 V 0 (Sh∗ ) + q∆πV 0 (Sh∗ ) = 0 −(1 − π1 ) + λ(1 − π1 )V 0 (Sl∗ ) − q∆πV 0 (Sl∗ ) = 0. Therefore, we have 1

∆π V π1 ∆π 1 =λ−q . ∗ 0 V (Sl ) 1 − π1 0 (S ∗ ) h

And we have λ=

V

=λ+q

π1 0 (S ∗ ) h

(1 − π1 )π1 q= ∆π



+

1 − π1 > 0, V 0 (Sl∗ )

1 1 − 0 ∗ ∗ 0 V (Sh ) V (Sl )

 .

Note that the incentive constraint implies ∆π (V (Sh∗ ) − V (Sl∗ )) ≥ D > 0, Thus V (Sh∗ ) − V (Sl∗ ) > 0, which in turn, together with V 00 < 0, implies that q > 0. The fact that V (Sh∗ ) − V (Sl∗ ) > 0 also implies that the optimal contract cannot be a constant wage scheme. By the participation constraint, π1 (V (Sh∗ ) − V (Sl∗ )) + V (Sl∗ ) = D. 18

By the incentive constraint, V (Sh∗ ) − V (Sl∗ ) =

D . ∆π

By substitution, π1

D + V (Sl∗ ) = D, ∆π

Or Sl∗ Thus

or V (Sl∗ ) = − 

=H

π0 D. ∆π

 −π0 D . ∆π

1 − π0 D = D, ∆π ∆π   1 − π0 ∗ Sh = H D . ∆π

(2.4)

V (Sh∗ ) = V (Sl∗ ) + or

(2.5)

Recall that in the first best ShF B = SlF B = H(D). In the second best, the agent is paid higher (lower) with high (low) outcome than he is in the first best. Since the payment is risky, the agent must be paid a risk premium in order to induce his participation. Indeed, in the second best, the participation constraint implies that D = π1 V (ShSB ) + (1 − π1 )V (SlSB ) < V (π1 ShSB + (1 − π1 )SlSB ). The last inequality is from Jensen’s inequality. But D = π1 V (ShF B ) + (1 − π1 )V (SlF B ) = V (H(D)). That is, H(D) < π1 ShSB + (1 − π1 )SlSB . Since the (expected) compensation in the first best is H(D), one can say that the expected compensation in the second best is higher than in the first best. The difference between the first and second best is due to the risk premium. Finally, if she induces positive effort, the principal’s optimal expected utility is π1 (xh − ShSB ) + (1 − π1 )(xl − SlSB ). On the other hand, an optimal compensation scheme to induce zero effort is a constant wage contract such that Sh = Sl = 0, and her expected utility is simply π0 xh + (1 − π0 )xl . Thus she induces positive effort if     −π0 1 − π0 SB D + (1 − π1 )H D . ∆π∆x ≥ E[S ] = π1 H ∆π ∆π Since H is convex, the above condition implies that E[S SB ] ≥ H(D). Therefore, in the second best, the principal requires a higher amount of benefit, or a higher value, in ∆π∆x, for her to induce positive effort. Hence, positive effort is more likely in the first best than in the second best. 19

Exercise 2.1. A risk-averse entrepreneur wants to start a project requiring an initial investment of I. He does not have cash on his own, and must raise money from a risk-neutral bank. The return on the project will be either xh with probability π(a) or xl with probability 1 − π(a), where a ∈ {0, 1} is the entrepreneur’s effort. Assume zero discount rate. The cost of effort is C > 0 with a = 1 and 0 with a = 0. The financial contract consists of repayments are zl and zh when the project returns are xl and xh , respectively. The realized project return x is publicly observable. 1. Assume that the entrepreneur’s effort is observable. (a) Write the risk-neutral bank’s problem if it wants to induce the entrepreneur to exert high effort. (b) Solve for the optimal repayment schedule. (c) Under what condition(s), the bank may want to induce the entrepreneur to exert high effort? 2. Assume that the entrepreneur’s effort is unobservable. (a) Write the risk-neutral bank’s problem if it wants to induce the entrepreneur to exert high effort. (b) Solve for the optimal repayment schedule. (c) Under what condition(s), the bank may want to induce the entrepreneur to exert high effort? (d) Assume that the cashless entrepreneur is protected by the limited liability law. i. Suppose that the bank wants to induce the entrepreneur to exert high effort. Solve for the optimal repayment schedule. ii. Under what condition(s), the bank may want to induce the entrepreneur to exert high effort?

20

Chapter 3 Moral Hazard and Incentive Contracts: Discrete-Time Approach II 3.1

The Basic Model with Continuous Outcome Distribution

There are two discrete time periods, 0 and 1. The contract between the principal and agent can only be written on verifiable information. Assume that the principal and the agent are endowed with utilities of wealth given by U (w) and V (w), respectively where U, V : R → R are continuously twice differentiable, strictly increasing, and concave. Both the principal and agent are expected utility maximizers. The principal precommits to contract S, i.e. he makes a take-it-or-leave-it offer to the agent whose reservation utility is exogenously given as θ ∈ R. θ is the level of the agent’s utility that he can achieve by rejecting the offer and taking a job elsewhere. Contract S can be enforced costlessly and accurately (by court). There is no financial markets for the contract, and thus the manager cannot trade his contract in the market in an attempt to undo the incentives implied by the contract. If the agent accepts the offer at time 0, the principal gives him an asset to manage until time 1. Then the agent exerts effort (a hidden action) a ∈ A which affects the outcome x ∈ X. (Typically, A, X ⊂ R.) In particular, let X = [xl , xh ], where xl and xh are independent of a. The outcome x will be realized at time 1. One may view x as the corporate economic profit realized at the end of year. Note that economic profit 6= accounting profit. For effort a, the agent incurs the disutility of effort D(a), where D0 (a) ≥ 0 and 00 D (a) > 0. We further assume that the agent’s net utility is separable in wealth and effort and given by V (w) − D(a). Assume that x = h(a, ε), where ε is a real random variable. Let the pdf of x given by f (x, a). f (x, a) > 0, ∀(x, a). fa and faa exist and 21

are continuous. Assume that a is unobservable to the principal, and that x is observable and verifiable. Then, the contract S can only be based on observables such as x, but not on unobservable a. We call S(x) the salary scheme or the sharing rule. At time 1, the agent will be compensated for his effort according to the salary scheme S(x).

3.2

The First-Best Solution

Before tackling the above problem, let us look at the benchmark case in which both a and x are observable and verifiable. The principal’s problem can be stated as follows: Problem 3.1. Choose S and a to Z xh U (x − S)f (x, a)dx max xl Z xh V (S)f (x, a)dx − D(a) ≥ θ s.t. xl

The constraint is called the agent’s participation or individual rationality (IR) constraint. The Lagrangian is Z xh L= {U (x − S)) + λV (S)} f (x, a)dx − λD(a) − λθ xl

Assuming an interior solution, the first order conditions are U 0 (x − S) =λ ∀x ∈ X, V 0 (S(x)) Z xh {U (x − S)) + λV (S)} fa (x, a)dx − λD0 (a) = 0 Zxlxh V (S)f (x, a)dx − D(a) = θ xl

The first condition implies that the salary scheme S has to be designed so that the ratios of marginal utilities of wealth of the principal and agent are equalized across states. This rule is sometimes called the Borch rule, or the optimal coinsurance rule. Remark 1: Suppose that U 0 = 1, i.e. the principal is risk-neutral. Then the first FOC implies that V 0 (S(x)) should be constant for all states x. That is, S(x) has to be constant across all x’s, i.e. the principal pays a fixed salary to the agent for his level of effort a, and then receives x − S. The intuition is straightforward: Since the principal is risk-neutral and the agent is risk-averse, the principal can take risks more easily than the agent. Therefore, the principal bears all risks and the agent receives 22

a riskfree payoff. That is, the agent is fully insured. For optimal level of effort choice, let us look at the second FOC. Z xh 0 {U (x − S(x))) + λV (S(x))} fa (x, a)dx λD (a) = xl

= {U (x − S(x))) + λV (S(x))} Fa (x, a)|xxhl Z xh {U 0 (x − S(x)))(1 − S 0 (x))) + λV 0 (S(x))S 0 (x)} Fa (x, a)dx, − xl

where F is the distribution function. Note that since ∀a ∈ A, F (xl , a) = 0 and F (xh , a) = 1, we have Fa (xl , a) = Fa (xh , a) = 0. Therefore, substituting the first FOC, we have Z xh

λD0 (a) = −

U 0 (x − S(x)))Fa (x, a)dx,

xl

or 0

Z

xh

D (a) = −

V 0 (S(x)))Fa (x, a)dx.

xl

When the principal is risk-neutral and the agent is risk-averse, S is constant. Thus Z xh 0 0 Fa (x, a)dx D (a) = −V (S) xl Z xh 0 xfa (x, a)dx = V (S) xl

∂ = V 0 (S) E[x|a] ∂a The last quantity is the agent’s marginal (monetary) value of incremental output for an additional unit of effort, and D0 (a) the agent’s marginal (monetary) cost of effort. Thus the optimal level of effort is determined such that the agent’s marginal value of an additional output is equal to the agent’s marginal disutility of effort. Moreover, when the principal is risk-neutral and the agent is risk-averse, the last FOC implies V (S) − D(a) = θ.

3.3

The Second-Best Solution: the case of moral hazard

In this section, the agent’s effort a is unobservable to the principal. The principal’s problem is stated as follows: 23

Problem 3.2. Choose S and a to Z xh U (x − S(x))f (x, a)dx max xl Z xh V (S(x))f (x, a)dx − D(a) ≥ θ s.t. xl Z xh V (S(x))f (x, a ˆ)dx − D(ˆ a). a ∈ arg max a ˆ∈A

xl

The last condition is called “the incentive compatibility” (IC) constraint for effort. Although its statement appears to be simple, the solution to Problem 3.2 is not easy to obtain. In some cases, Problem 3.2 can be solved under highly restrictive assumptions on U , V , D, and f . Even then, solutions are algebraically too complex to interprete. To transform the problem into a manageable form, consider the IC constraint. Again assuming an interior solution, the IC implies that Z xh V (S(x))fa (x, a)dx − D0 (a) = 0, (3.1) xl Z xh V (S(x))faa (x, a)dx − D00 (a) ≤ 0. (3.2) xl

The above two are the local first- and second-order conditions. To solve Problem 3.2, one may try to replace the IC constraint with the agent’s first order condition in (3.1). The problem with the IC replaced with the FOC is called the principal’s relaxed problem, and such a practice is called “the first order approach” to incentive contracting problems. Since it is possible that the FOC in (3.1) is not even a necessary condition, it is not guaranteed that the relaxed problem yields an optimal solution. That is, the validity of the first order approach is not guaranteed. We will come back to this point shortly. For the time being, we ignore this validity issue. With the first order approach, the principal’s problem yields the following Lagrangian: Z xh  Z xh L = U (x − S(x))f (x, a)dx + λ V (S)f (x, a)dx − D(a) − θ xl xl Z xh  0 +µ V (S(x))fa (x, a)dx − D (a) xl

With the pointwise maximization, the first order condition with respect to S is U 0 (x − S(x)) fa (x, a) =λ+µ , 0 V (S(x)) f (x, a) 24

∀x ∈ X.

(3.3)

Since U 0 , V 0 > 0, we have λ > 0. To see this, multiply both sides of the above FOC by f (x, a) and integrate over [xl , xh ], then we have Z xh 0 U (x − S(x)) λ= f (x, a)dx > 0. V 0 (S(x)) xl As compared with the first-best case, it can be seen that if µ 6= 0, then the secondbest solution is different, and the risk sharing between the principal and the agent becomes less efficient. Theorem 3.1. (Holmstrom, BJE 1979) Assume that V 00 < 0, Fa (x, a) < 0 for all x ∈ (xl , xh ), and the first order approach is valid with an interior solution. Then µ > 0, i.e. the second-best solution 6= the first-best solution. Proof: Suppose not, i.e µ ≤ 0. From the FOC with respect to a,   Z xh Z xh 00 V (S(x))faa (x, a)dx − D (a) = 0. U (x − S(x)))fa (x, a)dx + µ xl

xl

By using the agent’s second order condition in (3.2), the above equation implies that Z xh U (x − S(x))fa (x, a)dx ≤ 0. (3.4) xl

However, the principal’s FOC implies otherwise as can be seen below. Contradiction. Therefore µ > 0. To see this, define ϕ(x, S) as follows: ϕ(x, S) :=

U 0 (x − S) . V 0 (S)

Then since U 0 , V 0 > 0, V 00 < 0, and U 0 ≤ 0, ϕ is strictly increasing in S. Let us also define Sλ (x) through the following equation: for each x, ϕ(x, Sλ (x)) = λ. Then it can be shown that Sλ0 (x) ∈ [0, 1). Now recall that the principal’s FOC with respect to S is U 0 (x − S(x)) fa (x, a) =λ+µ , 0 V (S(x)) f (x, a) Thus Z

S(x)

ϕS (x, S)dS = µ Sλ (x)

25

fa (x, a) , f (x, a)

∀x ∈ X.

∀x ∈ X.

Since ϕS (x, S) > 0, sign(S(x) − Sλ (x)) = sign(µfa (x, a)). We have two cases: µ < 0 and µ = 0. If µ < 0, then S(x) ≤ Sλ (x) if and only if fa (x, a) ≥ 0. Thus, (check yourself) U (x − S(x))fa (x, a) ≥ U (x − Sλ (x))fa (x, a), That is

Z

xh

Z

∀x ∈ X.

xh

U (x − Sλ (x))fa (x, a) dx

U (x − S(x))fa (x, a) dx ≥ xl

xl

However Z

xh

U (x − Sλ (x))fa (x, a) dx xl

= U (x − Sλ (x))Fa (x, a)|xxhl Z xh − U 0 (x − Sλ (x))(1 − Sλ0 (x))Fa (x, a) dx > 0. xl

The last inequality is from the fact that Fa (xl , a) = Fa (xh , a) = 0, Sλ0 (x) ∈ [0, 1), and Fa (x, a) < 0 for all x ∈ (xl , xh ). Therefore, it contradicts (3.4). When µ = 0, S(x) ≡ Sλ (x) for all x. Thus we reach the same contradiction. 

Remark 1: Theorem 3.1 tells us that under a set of very reasonable conditions, moral hazard problems can be serious enough to result in losses in social welfare. Remark 2: The assumption that the supports xl and xh are independent of a is important. If they shift as a changes, some outcomes can be definitely informative about effort and the principal can enforce the first best. Remark 3: Note that fa /f is the derivative of the log-likelihood function of a, ln f (x, a), given a realization of x.

3.4

The Shape of the Second-best Contract

The next question is: What should be the shape of the 2nd-best optimal sharing rule in general? Definition 3.1. The pdf f satisfies the monotone likelihood ratio property (MLRP) if and only if fa /f is increasing in x. 26

Theorem 3.2. Assume that µ > 0, V 00 < 0, and the FOC (3.3) holds. If f satisfies the MLRP, then S 0 (x) > 0, i.e. the higher the outcome, the higher the compensation. Proof: By redifferentiating the FOC with respect to x, U 00 V 0 · (1 − S 0 (x)) − U 0 V 00 · S 0 (x) ∂ =µ 02 V ∂x



fa f



Or ∂ (−U V − U V )S (x) = µ ∂x 00

0

0

00

0



fa f



V 02 − U 00 V 0 > 0.

Therefore S 0 (x) > 0.



Remark 1: Under the MLRP, a high value of x implies that a high value of a is more likely. Remark 2: (See Stole [1997].) If the MLRP does not hold, sometimes the optimal contract may punish the agent for some high outcomes. Suppose that there are two possible levels of effort al and ah , and three possible outcomes xl , xm , and xh , and that the pdf’s f (x, a) are given as follows:

ah al fa /f

xl 0.4 0.5 –0.25

xm xh 0.1 0.5 0.4 0.1 –3 0.8

(Note that the discrete-time version of fa /f = (f (x, ah ) − f (x, al ))/f (x, ah ).) In this case, if the principal wishes to induce the high effort ah , it may be necessary to punish the agent when xm is realized because xm is most indicative of the low effort al . Suppose that the principal is risk neutral and the agent is risk averse. Then the first best risk sharing rule would be as follows. The principal takes all the risk and the agent takes a riskfree share. That is, the first-best optimal contract is a complete insurance contract for the agent. However, with such a contract, the agent has no incentives to work. Therefore, to make the agent to work, the principal has to impose some risks to the agent. But then, the first best cannot be achieved. Therefore, the second-best contract has to be based on a tradeoff between risk sharing and incentives. Exercise 3.1. Suppose both the principal and agent are risk neutral. Explain why the second-best contract should be designed such that the agent takes all the risks and the principal is given a fixed income. 27

3.5

The Value of Informative Signal in Contracting

The value of informative signal in contracting was studied by Holmstrom [1979] and Shavell [1979]. As we have seen above, the shape of the contract depends on the information content of the outcome about the effort level. With the MLRP, a high outcome means a high effort level with a high probability. What if there is another signal s about effort in addition to x? Will it be useful in contracting? First, recall the concept of sufficiency encountered in mathematical statistics. Definition 3.2. T (x, s) is sufficient for a ∈ A in the family of distributions given by {f (x, s, a), a ∈ A}, if the conditional distribution of (x, s) given T (x, s) is independent of a. Lemma 3.1. (Fisher-Neyman Factorization Theorem) T (x, s) is sufficient for a ∈ A in the family of distributions given by {f (x, s, a), a ∈ A}, if and only if the joint pdf of (x, s) is factorized as below: f (x, s, a) = g(T (x, s), a)h(x, s). Definition 3.3. An additional signal s is informative about effort if and only if x is not sufficient. In the absence of additional signals other than the performance metric x, the optimal salary scheme should satisfy fa (x, a) U 0 (x − S) = λ + µ , V 0 (S) f (x, a)

∀x ∈ X.

When there is an additional signal s about the agent’s effort a, the optimal salary scheme solves U 0 (x − S) fa (x, s, a) =λ+µ , ∀x ∈ X, 0 V (S) f (x, s, a) If fa /f is independent of s, then the signal s is useless. If not, the signal in general is useful in improving the agency problem. Consider two signals s and s0 . Signal s0 is uninformative if f (x, s, s0 , a) = g(T (x, s), a)h(x, s, s0 ). In this case, fa /f is independent of s0 , and T (x, s) is a sufficient statistic. When there is a sufficient statistic T (x, s), the optimal salary scheme depends only on T (x, s) and nothing else. 28

3.5.1

Application to Insurance Deductibles

Suppose that the probability of an insured accident depends on the effort of the insured, but that conditional on an accident occuring, the size of the damage from the occured accident is independent of the effort of the insured. Then what should be a right form of insurance contract? Let x be the size of an accident. Assume that f (0, a) = 1−p(a), f (x, a) = p(a)g(x) for x < 0, and that p0 (a) < 0. In this setup, the probability of an accident depends on effort, but the loss from it is completely independent of effort. Note that for x ≤ 0,  f (x, a) = (1 − p(a))χ{x=0} + p(a)(1 − χ{x=0} )  · χ{x=0} + g(x)(1 − χ{x=0} ) , where χ{x=0} is an indicator function for event {x = 0}. Thus χ{x=0} is sufficient for a. Therefore, the optimal insurance contract in this case should only depend on χ{x=0} . To see this, note that fa /f = p0 (a)/p(a) < 0, for x < 0, independent of x, and fa /f = −p0 (a)/(1 − p(a)) > 0 for x = 0, again independent of x. To find an optimal insurance contract, let us assume that the principal (the insurer) is risk neutral. The FOC is ( p0 (a) if x = 0 1 (1−p(a)) = λ + µ × 0 (a) p V 0 (S) if x < 0 p(a) Thus, the optimal contract will be of the following form: S = αχ{x=0} + β(1 − χ{x=0} ),

α, β ∈ R.

By rewriting, S = α + (β − α)(1 − χ{x=0} ). α can be viewed as an insurance premium and β −α as a deductible amount conditional on an accident occuring. In other words, the insured (agent) pays to the insurance company (principal) −α as an insurance premium, and when an accident occurs, the agent pays to the insurance company an amount of −(β − α). (Note that α and (β − α) should be negative in reasonable settings. In fact, to make sense out of this insurance model, for x < 0, x < β − α.)

3.6

The Validity of the First-order Approach

The first-order approach is valid if the agent’s problem given a contract is concave in effort. Definition 3.4. A distribution function is said to have a convexity distribution function condition (CDFC) if and only if Faa (x, a) ≥ 0, for all (x, a). Remark 1: An interesting special case of the CDFC is the linear distribution function condition (LDFC): for a ∈ (0, 1), f (x, a) = ag1 (x) + (1 − a)g2 (x) 29

where g1 (x) first-order stochastically dominates g2 (x). That is, the pdf f is a convex combination of two distributions, one of which dominates the other in the sense of the first-order stochastic dominance. Theorem 3.3. (Rogerson [1985]) Assume that λ > 0, V 00 < 0, and Fa (x, a) < 0 for all x ∈ (xl , xh ). The first-order approach is valid if the distribution function F (x, a) satisfies the MLRP and CDFC. Heuristic Proof: Given S, the agent’s expected utility is Z xh V (S(x))f (x, a)dx − D(a) xl Z xh xh V 0 (S(x))S 0 (x)F (x, a)dx − D(a) = V (S(x))F (x, a)|xl − xl Z xh = V (S(xh )) − V 0 (S(x))S 0 (x)F (x, a)dx − D(a). xl

We need to show that the agent’s SOC is satisfied. But Z xh − V (S(x))S 0 (x)Faa (x, a)dx − D00 (a) ≤ 0, xl

because Faa (x, a) ≥ 0 and S 0 (x) > 0 under MLRP and µ > 0.



The above heuristic proof is circular, because it makes use of µ > 0, which holds under the assumption that the first-order approach is valid. To avoid this kind of circular argument, Rogerson proposes the “doubly relaxed principal’s problem by replacing the first-order equality condition with an (artificial) inequality condition as follows: Problem 3.3. (Doubly Relaxed Principal’s Problem) Choose S and a to Z xh max U (x − S(x))f (x, a)dx xl Z xh s.t. V (S(x))f (x, a)dx − D(a) ≥ θ xl Z xh ∂ V (S(x))f (x, a)dx − D0 (a) ≥ 0. ∂a xl The last constraint is weaker than the first-order condition with equality. If the problem yields a solution satisfying the last condition with inequality, then it does not solve the original principal’s problem. If the solution satisfies the constraint with equality and the agent’s SOC is satisfied, then one can say the solution solves the 30

original problem. In both cases, Rogerson shows that the SOCs are satisfied. Proof: The Lagrangian for the above doubly relaxed problem is Z xh  Z xh U (x − S(x))f (x, a)dx + λ V (S)f (x, a)dx − D(a) − θ L = xl xl  Z xh  ∂ 0 +µ V (S(x))f (x, a)dx − D (a) , ∂a xl and we now know that µ ≥ 0. If µ > 0, then the constraint is binding and thus the agent’s FOC has to be satisfied with equality. Suppose that µ = 0. Then λ > 0. The principal’s FOC with respect to S is U 0 (x − S(x)) = λ, V 0 (S(x))

∀x ∈ X.

Thus we have S 0 (x) ≥ 0. On the hand, let us rewrite the Lagrangian  Z xh Z xh V (S)f (x, a)dx − D(a) − θ L = U (x − S(x))f (x, a)dx + λ xl xl Z xh = U (xh − S(xh )) − U 0 (x − S(x))S 0 (x)F (x, a)dx xl   Z xh V (S)f (x, a)dx − D(a) − θ . +λ xl

Since λ > 0, the principal’s FOC is Z xh ∂L = − U 0 (x − S(x))S 0 (x)Fa (x, a)dx ∂a xl  Z xh ∂ +λ V (S)f (x, a)dx − D(a) − θ = 0. ∂a xl Since S 0 (x) > 0 and Fa (x, a) ≤ 0, the FOC implies that we must have  Z xh  ∂ V (S)f (x, a)dx − D(a) − θ ≤ 0. ∂a xl This relationship can be consistent with the last constraint of the doubly relaxed problem only if  Z xh  ∂ V (S)f (x, a)dx − D(a) − θ = 0. ∂a xl Thus whether the last constraint is binding or not, the agent’s FOC has to be satisfied with equality. 31

Now, looking at the local SOC of the agent’s problem in (3.2), Z

xh

V (S(x))faa (x, a)dx − D00 (a) xl Z xh xh V 0 (S(x))S 0 (x)Faa (x, a)dx − D00 (a) = V (S(x))Faa (x, a)|xl − xl Z xh V 0 (S(x))S 0 (x)Faa (x, a)dx − D00 (a) ≤ 0. =− xl

The last inequality is satisfied by the MLRP and CDFC. (Recall S 0 (x) > 0 by the MLRP.)  Remark 1: Note that the MLRP and CDFC conditions are so restrictive that popular distributions like normal distribution cannot satisfy these conditions. Remark 2: The CDFC is particularly restrictive. Jewitt [1988] shows that under CARA utility assumption, one can avoid CDFC. Examples are 1. a Gamma distribution with a mean of αa such that f (x, a) = a−α xα−1 e−x/α Γ(α)−1 . 2. a Poisson distribution with a mean of a such that f (x, a) = ax e−a /Γ(1+x). 3. a Chi-square distribution with a degrees of freedom such that f (x, a) = Γ(2a)−1 2−2a x2a−1 2−x/2 .

3.6.1

Normally Distributed Outcome

What can happen to the optimal contract if the first-order approach cannot be justified? Mirrlees [1974] constructs a very interesting case. Theorem 3.4. (Mirrlees [1974]) Suppose f (x, a) is a pdf from a normal distribution with a mean of a and a variance of σ 2 . Also suppose that the principal is risk neutral, and that limw→wl V (w) = −∞. Then in the second best world, the first-best solution can be approximated with an arbitrary precision. Proof: Note that (x−a)2 1 f (x, a) = √ e− 2σ2 , 2πσ

fa (x, a) x−a = , f (x, a) σ2

and also that faa (x, a) = {(x − a)2 − σ 2 }

1 f (x, a) > 0, σ4 32

for x < a − σ.

Let S ∗ and a∗ be the first-best salary and effort, respectively. Since S ∗ is constant, V (S ∗ ) − D(a∗ ) = θ. Consider a compensation scheme S in a following form of a step function: for some number η < a∗ − σ,  ∗ S if x ≥ η S= (η) if x < η. Let (η)(∈ R) be determined to satisfy the agent’s incentive constraint with effort level a∗ such that given any η < a∗ − σ, Z ∞ Z η ∗ ∗ V (S )fa (x, a )dx + V ((η))fa (x, a∗ )dx = D0 (a∗ ). −∞

η

That is ∗

Z

η

fa (x, a∗ )dx = D0 (a∗ ).

{V ((η)) − V (S )} −∞ ∗

Since fa (x, a ) < 0 for all x < η, we have for all x < η, V (S ∗ ) − V ((η)) > 0,

S ∗ > (η).

Thus one can interpret  as punishment for a bad outcome. On the other hand, the agent’s FOC is always satisfied, because faa > 0 for all x < a∗ − σ, and Z η ∗ faa (x, a∗ )dx − D00 (a∗ ) < 0. {V ((η)) − V (S )} −∞

Therefore, S implements the first best effort a∗ . Next we claim that this step-function contract can also satisfy the participation constraint arbitrarily closely as η approaches −∞. Given this contract, the agent’s utility is Z η Z ∞ ∗ ∗ V (S )f (x, a )dx + V ((η))f (x, a∗ )dx − D(a∗ ) −∞ η Z η =θ− {V (S ∗ ) − V ((η))}f (x, a∗ ) dx. −∞

It suffices to show that Let

Rη −∞

{V (S ∗ )−V ((η))}f (x, a∗ ) dx approaches zero as η → −∞. M (η) :=

fa (η, a∗ ) . f (η, a∗ )

Since η < a∗ , M (η) < 0; and since M 0 (η) > 0, for all x < η, fa (x, a∗ ) < M (η) < 0, f (x, a∗ )

or 33

1 fa (x, a∗ ) > f (x, a∗ ). M (η)

Thus, Z

η

{V (S ∗ ) − V ((η))}f (x, a∗ ) dx −∞ Z η 1 < {V (S ∗ ) − V ((η))}fa (x, a∗ ) dx M (η) −∞

The LHS of the above inequality is always Rgreater than zero. However, by the incentive ∞ constraint and using the fact that V (S ∗ ) −∞ fa (x, a∗ )dx = 0, we have Z

η

{V (S ∗ ) − V ((η))}fa (x, a∗ )dx = D0 (a∗ ).

−∞

Hence Z

η

0
0 for all θ ∈ Θ. Upon receiving goods from the agent, the principal reimburses the cost C and pays S to the agent. Let q be a constant, representing the value of the agent’s work to the principal. The principal’s utility is q − C − S. The agent’s utility is S(θ) − D(e), where D : A → R+ is the disutility of effort. (The agent exerts costly effort to reduce the cost.) We assume D0 > 0 and D00 > 0. The agent’s reservation utility is zero, i.e. in equilibrium U (θ) = S(θ) − D(e(θ)) ≥ 0.

5.2

The Two-Type Case

¯ The probability of θ is ν. • θ ∈ {θ, θ}. ¯ ¯

5.2.1

The first best

• The principal observes θ. For each θ, the principal’s problem is to max

q − (C(θ) + S(C(θ)))

s.t.

S(θ) − D(e(θ)) ≥ 0.

S,C,e

Since the IR constraint is binding, S = D(e). Thus the principal’s problem reduces to max q − (θ − e(θ) + D(e(θ))) e

¯ = e∗ (θ) = e∗ ∈ arg maxe e − D(e). Or That is, e∗ (θ) ¯ D0 (e∗ ) = 1. That is, the first-best levels of effort for both agents are the same and are determined by equating the marginal disutility of effort and the marginal cost reduction through effort. Consequently, C ∗ = θ − e∗ , ¯ ¯

C¯ ∗ = θ¯ − e∗ , 53

S ∗ = S¯∗ = S ∗ = D(e∗ ). ¯

The agent’s realized utility is S(C(θ)) − D(e∗ ) = 0 ¯ for θ = θ, and ¯

¯ − D(e∗ ) = 0 S(C(θ))

¯ for θ = θ. Therefore, before contracting, the principal’s ex ante first-best expected utility is
¯ + S ∗ (C ∗ (θ)) ¯ q − ν (C ∗ (θ) + S(C ∗ (θ))) − (1 − ν) C ∗ (θ) ¯ ¯ = q − νθ − (1 − ν)θ¯ − e∗ − D(e∗ ), ¯ where e∗ ∈ arg maxe e − D(e).

5.2.2

Asymmetric information: The second best

• The principal’s utility is
¯ + S(C(θ)) ¯ q − ν (C(θ) + S(C(θ))) − (1 − ν) C(θ)
¯ ¯ ¯ = q − ν (θ − e + S(C(θ))) − (1 − ν) θ¯ − e¯ + S(C(θ)) ¯ ¯ ¯ The incentive compatibility conditions are ¯ − D(θ − C(θ)) ¯ U = S(C(θ)) − D(θ − C(θ)) ≥ S(C(θ)) ¯ ¯ ¯ ¯ ¯ ¯ − D(θ¯ − C(θ)) ¯ ≥ S(C(θ)) − D(θ¯ − C(θ)). U¯ = S(C(θ)) ¯ ¯

(5.1) (5.2)

The incentive compatibility for θ also implies ¯ ¯ − D(θ − C) ¯ S(C ) − D(θ − C ) ≥ U¯ + D(θ¯ − C) ¯ ¯ ¯ ¯ = U¯ + Φ(¯ e), where Φ(¯ e) = D(¯ e) − D(¯ e − ∆θ). Note that Φ(¯ e) ≥ 0, and that since D00 > 0, Φ0 > 0. The quantity Φ(¯ e) is called “the information rent” to be paid, in order to induce agent θ to reveal his type ¯ correctly. On the other hand, the participation (individual rationality) constraints are U ≥ 0, ¯ U¯ ≥ 0. 54

(5.3) (5.4)

Since Φ(¯ e) ≥ 0, (5.1) and (5.4) imply (5.3), and thus (5.3) is not a binding constraint. However at this moment it is not clear whether (5.2) is binding or not. Nevertheless, we ignore (5.2) momentarily. Then, the principal’s problem is
¯ max q − ν (θ − e + S(C )) − (1 − ν) θ¯ − e¯ + S(C) ¯ ¯ ¯ s.t. S(C ) − D(e) ≥ U¯ + Φ(¯ e) ¯ ¯ ¯ − D(¯ S(C) e) ≥ U¯ . Since rents U are costly to the principal, the above two constraints are binding, and the principal’s problem can be rewritten as
e) max q − ν (θ − e + D(e) + Φ(¯ e)) − (1 − ν) θ¯ − e¯ + D(¯ e,¯ e ¯ ¯ ¯ Note that if D000 ≥ 0, then the principal’s objective function is concave. The FOCs are D0 (e) = 1, or e = e∗ , ¯ ν Φ0 (¯ e) D0 (¯ e) = 1 − 1−ν Since Φ0 > 0, the above FOCs imply that e > e¯, and therefore C¯ > C . Note ¯ ¯ that constraint (5.2) is satisfied by the solution to the above FOCs. To see this, U¯ ≥ U + D(θ − C ) − D(θ¯ − C ) ¯ ¯ ¯ ¯ = U¯ + Φ(¯ e) + D(θ − C ) − D(θ¯ − C )  ¯ ¯ ¯ = U¯ + Φ(¯ e) − D(θ¯ − C ) − D(θ¯ − C − ∆θ) ¯ ¯ ¯ − Φ(θ¯ − C ) = U¯ + Φ(θ¯ − C) ¯ Since Φ0 > 0 and C¯ > C , the above inequality is always satisfied with strict ¯ inequality. That is, constraint (5.2) is not binding. Thus the above FOCs correctly describe the optimal solution and the principal’s optimal utility is
q − ν (θ − e∗ + D(e∗ ) + Φ(¯ e)) − (1 − ν) θ¯ − e¯ + D(¯ e) , ¯ where e¯ is given by D0 (¯ e) = 1 −

ν Φ0 (¯ e). 1−ν

Example: Suppose that D(e) = (1/2)e2 . Then θ = 1. Moreover, since Φ0 (¯ e) = ¯ ∆θ, ν e¯ = 1 − ∆θ. 1−ν 55

S∗ 6

¯ =0 U

U =0 ¯

S0 ¯

C

S = S ∗ + Φ(¯ e) ¯

D B

¯∗ = S ∗ S∗ = S ¯

A E

¯ S

U = S0 − S∗ ¯ ¯ U = Φ(¯ e) ¯

C∗ ¯

¯∗ C

¯ C

-

C

Figure 5.1: The menu of contracts and agent’s indifference curves The equilibrium information rent is 1 2 Φ(¯ e) = {¯ e − (¯ e − ∆θ)2 } = 2

  1+ν 1− ∆θ ∆θ. 2(1 − ν)

Exercise 5.1. Solving the principal’s problem, we have assumed that IC constraint (5.1) is binding. How will the solution change if we assume constraint (5.2) is bonding or if both constraints (5.1) and (5.2) are bonding? At the optimum of the principal’s utility, which constraint should be binding (5.1) or (5.2), or both? Why?

5.2.3

Intuition

In order to understand, how the agent chooses a contract from the menu, let us look at the agent indifference curves in the (C, S)-space. Given (C, S), let the agent θ’s utility be k. Then U = S − D(θ − C) = k Then

dS d2 S = −D0 (θ − C) < 0, = D00 (θ − C) > 0. dC dC 2 Thus, indifference curves for type θ¯ are steeper than those for type θ. ¯ In order to intuitively understand the optimal second-best menu, let us look at ¯ respectively. figure 5.1. Note that contract A and B are first-best contracts for θ and θ, ¯ But given the two contracts, A and B, agent θ has strong incentives to report his ¯ type to be θ¯ by choosing B, which violates the truth-telling condition. 56

In order to implement the first-best effort levels from both agents, the principal has to offer B and C. But then, the information rent (S 0 − S ∗ ) can be too high for ¯ the principal. Thus the principal may decide to sacrifice the effort efficiency of the ¯ C) ¯ in order to reduce less efficient agent by assigning him a less efficient contract (S, the information rent to be paid to θ. Consequently, the optimal rent is determined so ¯ that the expected marginal information rent νΦ0 (¯ e) is equal to the marginal efficiency loss of the less efficiency agent’s effort (1 − ν)(1 − D0 (¯ e)). This optimal tradeoff can result in a menu consisting of D and E.

5.3

Continuum of Types

Now, θ ∈ [θL , θH ]. The principal’s expected utility is Z (q − C(θ) − S(θ)) f (θ)dθ. Θ

The principal’s problem is to design a menu of contracts for the agent specifying the compensation S(θ) and C(θ). Problem 5.1. Choose S(θ) and C(θ) to Z max (q − C(θ) − S(θ)) f (θ)dθ Θ

s.t.

S(θ) − D(θ − C(θ)) ≥ 0 θ ∈ arg max U (θ0 |θ) := S(θ0 ) − D(θ − C(θ0 )). θ0 ∈Θ

The second is the incentive constraint via the revelation principle. The IC implies for all θ0 ∈ Θ, U (θ) ≡ U (θ|θ) ≥ U (θ0 |θ) Theorem 5.2. S is incentive compatible if and only if ∀θ, θ0 ∈ Θ, 0

Z

U (θ ) − U (θ) = −

θ0

D0 (θ − C(θ))dθ,

θ

˙ and C(θ) ≥ 0 a.e. Proof: Necessity. By the IC, U (θ) ≥ U (θ0 |θ) = U (θ0 ) + {U (θ0 |θ) − U (θ0 |θ0 )}. 57

But, U (θ0 |θ) − U (θ0 |θ0 ) = D(θ0 − C(θ0 )) − D(θ − C(θ0 )) Thus we have U (θ) − U (θ0 ) ≥ D(θ0 − C(θ0 )) − D(θ − C(θ0 )) Also by symmetry, U (θ0 ) − U (θ) ≥ D(θ − C(θ)) − D(θ0 − C(θ)) Therefore, D(θ − C(θ0 )) − D(θ0 − C(θ0 )) ≥ U (θ0 ) − U (θ) ≥ D(θ − C(θ)) − D(θ0 − C(θ)) (5.5) By letting θ0 → θ, we have dU = −D0 (θ − C(θ)). dθ Eq. (5.5) also implies that θ0

Z

Z

C(θ0 )

0≤ θ

D00 (θˆ − C)dCdθˆ

C(θ)

Thus, if θ0 > θ, then C(θ0 ) ≥ C(θ), which implies C˙ exists a.e. and C˙ ≥ 0 a.e. Sufficiency. Suppose not. That is, suppose that when U 0 (θ) = −D0 (θ − C(θ)) and C˙ ≥ 0, it is sometimes possible that there exist θ and θ0 such that U (θ0 |θ) > U (θ|θ). Without loss of generality, assume that θ0 > θ. Then, U (θ0 |θ) − U (θ0 |θ0 ) > U (θ) − U (θ0 ). Thus 0

0

Z

0

θ

D(θ − C(θ )) − D(θ − C(θ )) > −

ˆ θ, ˆ D0 (θˆ − C(θ))d

θ0

or Z

θ0

D (θˆ − C(θ0 ))dθˆ − 0

θ

Z

θ0

ˆ θˆ > 0. D0 (θˆ − C(θ))d

θ

But the above inequality cannot be true because Z θ0 Z C(θ0 ) − D00 (θˆ − C))dC dθˆ < 0 θ

ˆ C(θ)

Contradiction.



By Theorem 5.2, we can rewrite Problem 5.1 as follows. 58

Problem 5.2. Choose U and e to Z max (q − θ + e − U (θ) − D(e)) f (θ)dθ Θ

s.t.

U˙ (θ) = −D0 (e) U (θ) ≥ 0 1 − e(θ) ˙ ≥0

Since U (θ) is decreasing, the second constraint can be replaced with U (θh ) ≥ 0. Since the higher the U , the lower the principal’s expected profit, we set U (θh ) = 0. We first ignore the last constraint, and later on we will see that this constraint is not binding. By treating U as the state variable and e as the decision variable, The Hamiltonian is H = (q − θ + e − U − D(e)) f (θ) − µD0 (e). By the Pontryagin maximum principle, the necessary conditions are µ˙ ∗ (θ) = −

∂H = f (θ) ∂U

µ∗ (θl ) = 0 e∗ (θ) ∈ arg max H(e0 , θ, U, µ) e0

Thus, assuming an interior solution µ∗ (θ) = F (θ) D0 (e∗ ) = 1 −

F (θ) 00 ∗ D (e ). f (θ)

Note that F (θl ) = 0, and therefore for the most efficient type, i.e., θl , the optimal effort level is the same as the first best level. For other types, the optimal effort levels are lower than the first-best. Differentiating the last equation, we have     d F (θ) F (θ) 000 ∗ 00 ∗ 00 D e˙ = −D (e ) . D + f (θ) dθ f (θ) Assume that

d dθ



F (θ) f (θ)

 ≥ 0.

(5.6)

This assumption is satisfied by popular distributions such as uniform, normal, exponential, and chi-squared. Also assume that D000 ≥ 0. 59

(5.7)

Under these two reasonable assumptions in (5.6) and (5.7), we have e˙∗ (θ) ≤ 0, i.e., effort decreases with θ. Thus the original constraint e˙∗ (θ) ≤ 1 is not binding. Since C ∗ (θ) = θ − e∗ (θ), the two assumptions also imply C˙ ∗ (θ) ≥ 0, i.e. the cost increases with θ. The necessary conditions also shed some light on the relationship between the compensation and the realized cost C. Recall that Z θh ∗ ˆ θˆ D0 (e∗ (θ))d U (θ) = θ ∗

S (θ) = D(e∗ (θ)) + U ∗ (θ) Z θh ∗ ∗ ˆ θ, ˆ D0 (θˆ − C ∗ (θ))d S (θ) = D(θ − C (θ)) + θ

or  dS ∗ dS ∗ dC ∗  0 = / = D (1 − C˙ ∗ ) − D0 /C˙ ∗ = −D0 ≤ 0 dC dθ dθ   ∗ 2 ∗ 1 e˙ dS 00 00 = −D − 1 = −D ≥0 dC 2 C˙ ∗ C˙ ∗ The compensation S ∗ is decreasing and convex in cost.

5.4

Managerial Contracting under Adverse Selection and Moral Hazard

This section is based on Sung [2000]. There are two time periods, 0 and 1. The principal is risk neutral and the agent is risk averse with constant absolute risk aversion r. The outcome at time 1 is given by Y = µ + σB, where B is a standard normal random variable, and µ, σ ∈ R+ . The agent exerts costly effort to control µ and σ. His cost is given by c(µ, σ, θ) with cµ , cµµ > 0. The parameter θ stands for the agent’s ability, which is known to the agent himself but not to the principal. The principal knows that θ is a static random variable, independent of B and distributed on a strictly positive compact interval Θ := [θL , θH ], with a probability density function given by h(θ), where h(θ) > 0 for all θ ∈ Θ. The principal can observe Y and σ. But µ and θ are unobservable. The principal’s problem is as follows: 60

S∗ 6

-

C

Figure 5.2: Compensation vs. Realized Cost Problem 5.3. Choose {µ(θ); θ ∈ Θ} and a menu of contracts {(S(Y, θ), σ(θ)), θ ∈ Θ} to maximize E [Y (θ) − S(Y, θ)] subject to the following constraints.

(i) ∀θ Y (θ) = f (µ(θ, θ), σ(θ)) + σ(θ)B ˆ θ), (ii) ∀(θ, h n n ooi ( 0 ˆ ˆ E − exp −r S(Y, θ) − c(µ , σ( θ), θ) θ ˆ θ) ∈ arg max µ(θ, 0 ˆ + σ(θ)B ˆ µ s.t. Y (θ) = f (µ0 , σ(θ)) h n n ooi ˆ ˆ ˆ (iii) ∀θ, θ ∈ arg max Eθ − exp −r S(Y, θ) − c(µ(θ, θ), σ(θ), θ) θˆ

(iv) ∀θ, Eθ [− exp {−r {S(Y, θ) − c(µ(θ, θ), σ(θ), θ)}}] ≥ − exp {−rW0 } .

Let us consider menus that consist of linear contracts, i.e., S(Y, θ) = α(θ)+β(θ)Y , where α(θ), β(θ) ∈ R for all θ. Under the linearity assumption, the manager’s expected utility is   Ψ(θ) = E −e−r(α+βY −c(µ,σ,θ)) 1 2 2 = −e−r(α−c(µ,σ,θ)+βf (µ,σ)− 2 rσ β ) 61

Let Ψ∗ (θ) = maxµ Ψ(θ), and define Q(θ) by Ψ∗ (θ) = −e−rQ(θ) . Then the salary function S that satisfy the incentive constraint (ii) can be rewritten as follows: r c2µ (µ, σ, θ) 2 cµ (µ, σ, θ) σ + (Y − µ). S(Y, θ) = Q(θ) + c(µ, σ, θ) + 2 fµ2 (µ, σ) fµ (µ, σ) Note that (µ, σ, θ) is short for (µ(θ), σ(θ), θ). If the agent chooses µ ∈ U at optimum, any linear salary function should be given in the form of (5.8). Furthermore, one can also show that given any salary function in the form of (5.8) with (µ, σ, θ), agent θ will choose µ at the optimum, if given (σ, θ), c is concave in µ. Therefore, if c is concave in µ for given σ, θ and if the agent chooses µ ∈ U at optimum, then the managerial incentive and participation constraints can be equivalently replaced with Eq. (5.8). ˆ θ) and σ(θ) are differnetiable in θˆ and θ. Then Proposition 5.1. Assume that µ(θ, given constraints (ii) and (iv), conditions (iii) in the principal’s problem implies that Z

θH

Q(θ) = W0 +

cθ (µ, σ, θ0 )dθ0

(5.8)

θ

Proof: Let us write S = S(µ, σ, θ) to emphasize its dependence on those variables. Then at (interior) optimum, the relevant class of managerial salary functions becomes S(µ, σ, θ) where (µ, σ, θ) ∈ U × Σ × Θ. Suppose that the manager of type θ chooses ˆ Then the manager’s maximum expected utility becomes S(ˆ µ, σ ˆ , θ). h n  r 2 2 ˆ θ) = max E − exp −r Q(θ) ˆ + cˆ − cˆµ µ ˆ cˆµ + cˆµ Y1 u(θ; ˆ+ σ µ 2 oi − c(µ, σ ˆ , θ) . n  o ˆ + cˆ − cˆµ µ = max − exp −r Q(θ) ˆ + cˆµ µ − c(µ, σ ˆ , θ) µ

That is, the agent solves ˆ θ) := cˆ − cˆµ µ max Φ(µ, θ, ˆ + cˆµ µ − c(µ, σ ˆ , θ) µ

ˆ θ) solves the above maximization problem. Then the incentive and truthLet µ∗ (θ, ˆ ≡ µ∗ (θ, ˆ θ), ˆ or µ(θ) ≡ µ∗ (θ, θ) for all θ. Then we telling conditions imply that µ(θ) have ∂ ∗ ˆ Φ(µ , θ, θ) = cθ (µ(θ), σ(θ), θ), (5.9) ˆ ∂ θˆ θ=θ where the RHS of the above equation is the partial derivative of c with respect to its third argument. To see this, let us write explicitly the maximum value of Φ when the 62

ˆ type-θ agent chooses S(Y, θ). ˆ θ), θ, ˆ θ) Φ(µ∗ (θ, ˆ σ(θ), ˆ θ) ˆ − cµ (µ(θ), ˆ σ(θ), ˆ θ)µ( ˆ θ) ˆ = c(µ(θ), ˆ σ(θ), ˆ θ)µ ˆ ∗ (θ, ˆ θ) − c(µ∗ (θ, ˆ θ), σ(θ), ˆ θ) +cµ (µ(θ), Thus, by the Envelop theorem, we have ∂Φ ∗ ˆ ˆ θ) (µ (θ, θ), θ, ∂ θˆ dˆ µ dσ dˆ µ dσ dˆ µ = cθ + cµ + cσ − cµµ µ ˆ − cµσ µ ˆ − cµθ µ ˆ − cµ dθˆ dθˆ dθˆ dθˆ dθˆ dˆ µ dσ dσ +cµµ µ∗ + cµσ µ∗ + cµθ µ∗ − cσ dθˆ dθˆ dθˆ Since µ∗ (θ, θ) = µ(θ), this proves Eq. (5.9). On the other hand, constraint (iii) implies that given θ, we have for all θˆ n  o ˆ + Φ(µ∗ , θ, ˆ θ) −e−rQ(θ) ≥ − exp −r Q(θ) (5.10) ˆ to S(Y, θ).) In particular, since (Otherwise the agent of type θ will prefer S(Y, θ) ∗ ∗ Φ (µ , θ, θ) = 0, n  o −rQ(θ) ∗ ˆ ˆ −e = max − exp −r Q(θ) + Φ(µ , θ, θ) θˆ

The FOC at θˆ = θ is d ˆ + ∂ Φ(µ∗ (θ, ˆ θ), θ, ˆ θ) Q(θ) ˆ ˆ dθ ∂θ d ˆ + cθ (ˆ = Q(θ) µ, σ ˆ , θ) dθˆ

0 =

Finally, we show that Q(θ) is differentiable a.e. By constraint (iii), ∗ ˆ ˆ ˆ −e−rQ(θ) ≥ −e−r{Q(θ)+Φ(µ (θ,θ),θ,θ)} ∗ ˆ ∗ ˆ ˆ ˆ ˆˆ = −e−r{Q(θ)+Φ(µ (θ,θ),θ,θ)−Φ(µ (θ,θ),θ,θ)} .

But since cθ > 0, for θˆ > θ, ˆ θ), θ, ˆ θ) − Φ(µ∗ (θ, ˆ θ), θ, ˆ θ) ˆ = c(ˆ ˆ − c(ˆ Φ(µ∗ (θ, µ, σ ˆ , θ) µ, σ ˆ , θ) > 0. ˆ Thus, −e−rQ(θ) > −e−rQ(θ) if θˆ > θ. That is, Q(θ) is decreasing, and therefore Q(θ) is differentiable, a.e. 

63

Now we can use Eq.(5.8) to simplify the principal’s problem under the adverse selection. By Eq.(5.8), the risk-neutral principal’s problem is relaxed and restated as Z θH   r 2 2 max µ(θ) − Q(θ) − c(µ, σ, θ) − σ cµ (µ, σ, θ) h(θ)dθ µ,σ 2 θL Z θH  r = max µ(θ) − c(µ, σ, θ) − σ 2 c2µ (µ, σ, θ) µ,σ 2 θL  Z θH ¯ ¯ −W0 − cθ (µ, σ, θ)dθ h(θ)dθ (5.11) θ

By changing variables, the problem in (5.11) can be restated as Z θH   r 2 2 max µ − c(µ, σ, θ) − σ cµ (µ, σ, θ) − x h(θ)dθ µ,σ 2 θL dx s.t. = −cθ (µ, σ, θ) dθ x(θH ) = 0 Then, the Hamiltonian is:   r 2 2 H = µ − c(µ, σ, θ) − σ cµ (µ, σ, θ) − x h(θ) − λcθ (µ, σ, θ) 2 The necessary conditions are dλ ∂H =− = h(θ) dθ ∂x λ(θL ) = 0 (µ(θ), σ(θ)) ∈ arg max H(µ0 , σ 0 , θ, x(θ), λ(θ))

(5.12)

(5.13)

(5.14) (5.15) (5.16)

µ0 ,σ 0

Rθ Eq.’s (5.14) and (5.15) imply λ(θ) = θL h(θ0 )dθ0 . Note that the effect of the adverse selection is captured by the last term of the Hamiltonian. Let us call (µ(θ), σ(θ)) in (5.16) the third-best solution to the principal’s problem.1 Since λ(θ) is zero when θ = θL , we can conclude that the choices of (µ, σ) by the manager with the most efficient technology are not affected by the presence of the adverse selection problem. That is, when θ = θL , the third best coincides with the second best. It is well known that linear contracts may not be optimal in the above discretetime setup. See Mirrlees [1974] and also Baron and Besanko [1987]. Nevertheless. the results in this section can be justified as optimal in a continuous-time setting. See Sung [2000]. 1

Recall that under the pure moral hazard situation as in Sung [1995], the second-best solution to the principal’s problem can be found by maximizing over µ and σ r µ − c(µ, σ, θ) − σ 2 c2µ (µ, σ, θ). 2

64

5.4.1

Applications

In corporate management, the manager (the agent) can improve the outcome with his effort and/or by choosing a project from a given project opportunity set. Let us write the outcome process as follows: Y = (µ + g(σ)) + (σ + b)B,

(5.17)

where the function g(σ) represents the efficient frontier of (existing) project opportunities. The manager chooses a value for µ with a cost of c(µt , θ) and a project (g(σ), σ) without incurring a cost. The project choice decision is costless but observable to investors (the principal).2 The effectiveness parameter of his effort, θ, may not be known to investors. (Such a case may more significantly occur in a venture-capitalist contracting.) The costly managerial effort may be directed toward either enhancing existing project opportunities or improving the profitability of ongoing operations. When project (g(σ), σ) is chosen, the expected profit or the net present value (NPV) of the firm improves by g(σ) and the volatility by σ. Note that our NPV is computed as an amount before the managerial compensation is paid out. Thus it is the NPV in the first-best sense. After the project choice, the total instantaneous volatility is (σ + b) where b is an uncontrollable positive number and σ > −b. Furthermore, for convenience, we set g(0) = 0 to represent the status quo without the project choice. The parameter b can be viewed as the instantaneous volatility generated from the ongoing operations of the firm. When the profit from a chosen project is highly negatively correlated with the profit from ongoing operations, σ can take on a negative number. Thus σ is not the volatility of the income generated by a chosen project, but the contribution of a chosen project to the overall profit of the firm. The efficient frontier of project opportunities may alternatively be interpreted as a capital budgeting opportunity set, delineating a risk-return tradeoff schedule for different levels of initial investments. Let I be the level of investment at time t that increases both the instantaneous NPV and volatility of the overall profit by q(I) and s(I), respectively. Assume that both q and s are strictly increasing, q is concave and s is convex in I.3 Let σ ≡ s(I) and g(σ) ≡ q(s−1 (σ)), where s−1 is the inverse of s. Then q is strictly increasing and concave in σ, and the outcome with I is given by Y = (µ + q(I)) + (s(I) + b)B, which is equivalent to Y = (µ + g(σ)) + (σ + b)B. Thus a decision to invest an amount of I = s−1 (σ) bears exactly the same interpretations as the choice of a project (g(σ), σ). 2 The assumption of this costlessness is to highlight the decision making aspect as a part of managerial activities. See Sung [1995] for more explanation. 3 Suppose that the firm needs to decide the level of investment I in a particular project, the risk of which is independent of the risk of the ongoing operation. Then s is strictly increasing and convex in I.

65

In order to understand why the capital budgeting opportunity set q(s−1 (σ)) may be increasing and concave in σ, consider the following hypothetical case. The manager has found a positive NPV investment opportunity with a required capital investment I. Initially, to increase the NPV on the investment, he may simply double the amount to double the NPV. Of course, doubling I will increase the risk as well. However, as the size of investment becomes larger and the corresponding risk keeps increasing, the incremental NPV may only approach zero because of capital market competitions and transactions costs including price impacts. Thus an investment opportunity requiring I can generate a set of investment opportunities that can be described by an increasing and concave schedule like q(s−1 (σ)). Whichever interpretation is chosen to be, the problem can be stated as follows: Problem 5.4. Choose {{µ(θ)}; θ ∈ Θ} and a menu of contracts {(S(Y, θ), {σ(θ)}); θ ∈ Θ} to maximize E [Y (θ) − S(Y, θ)] subject to the following constraints: (i) ∀θ, Y (θ) = (µ∗ (θ, θ) + g(σ(θ))) + (σ(θ) + b)B ˆ θ), (ii) ∀(θ, h n n ooi ( B ˆ − c(µ, θ) − exp −r S(Y, θ) E ˆ θ) ∈ arg max µ∗ (θ, ˆ + b)B µ s.t. Y (θ) = (µ + g(σ(θ))) + (σ(θ) (iii) ∀θ, E B [− exp {−r {S(Y, θ) − c(µ∗ (θ, θ), θ)}}] ≥ − exp {−rW0 } h n n ooi ˆ − c(µ∗ (θ, ˆ θ), θ) (iv) ∀θ, θ ∈ arg max E B − exp −r S(Y, θ) . θˆ

Problem 5.4 is a slightly modified case of Problem 5.3. Thus similarly to (5.13), the risk-neutral investors’ problem is to Rθ h(θ0 ) dθ0 r 2 θL 2 cθ (µ, σ, θ). max µ + g(σ) − c(µ, σ, θ) − cµ (µ, σ, θ)(σ + b) − m,σ 2 h(θ) The FOC necessary conditions are Rθ 1 − cµ − r(σ + b)2 cµ cµµ −

θL

h(θ0 )dθ0 h(θ)

cθ µ = 0

gσ − r(σ + b)c2µ = 0 The second condition tells us about the decision rule for project selection under adverse selection. The rule is identical to that of the pure moral hazard model. That is, the project is selected so that marginal NPV from the project before managerial compensation is equal to the marginal managerial compensation-risk premium. Note however that the chosen project under moral hazard and adverse selection can be 66

different from that of pure moral hazard, because the sensitivities of contracts cµ are in general different between the two cases. Figure 5.3 conceptually illustrates the optimal project selections for three different cases: A is the first-best (or the NPV-maximizing) project; B is the second-best; and C is the average third-best (with adverse selection). The horizontal axis represents projects that yield zero NPV in the absence of moral hazard, and the vertical axis NPV of the projects in terms of the increment of the mean of the overall outcome. The origin of the graph for g(σ) represents status quo or ongoing operations plus some new riskfree NPV-zero projects. The slopes of tangents in the figure are r(σ + b)c2µ , the marginal compensation-risk premium. As seen in the figure, the optimal σ increases as the sensitivity of the contract cµ decreases. The manager with the highest ability chooses the same project as he would without the adverse selection problem. (Since the first order conditions for the manager with the highest ability are the same as in the pure moral hazard, optimal cµ is the same and thus he chooses the same project.) When cµµµ ≥ 0, managers with lower ability however are more willing to take risk to increase the marginal NPVs or to do less costly hedging activities than they would in the absence of the adverse selection problem. On average, the more severe the adverse selection problem is, the lower the sensitivity of the managerial contract to the outcome.

67

NPV 6

A C B −b

K

g(σ), the efficient frontier of feasible projects - σ

I

status quo

A = the 1st-best project B = the 2nd-best project C = the average 3rd-best project

NPV 6

g 0 (σ)

rc2µ (µ2nd )(σ + b) rc2µ (µ3rd )(σ + b) -

−b

∗ ∗ σ2nd σ3rd

σ

Figure 5.3: Project selection rules under moral hazard and adverse selection

68

Chapter 6 Dynamics of Contracts under Moral Hazard 6.1

Full Commitment

See Rogerson [1985, Econometrica, “Repeated Moral Hazard,” Vol53, 69-76] • Two periods, T = 2. • A finite set of outcomes, {x1 , x2 , ..., xN }, with a pdf of f (xi ; a) > 0 for all (xi , a). • A long term contract given by S = {S1 (xi ), S2 (xi , xj )}. • At time 1, the agent either accept or reject the long term contract. Once accepted, no quitting is allowed at time 2, i.e both the principal and agent are committed for all two periods. • After accepting the long term contract, the agent’s actions over time are a = {a1 , a2 }. • The agent utility is given by U=

N X

( f (xi , a1 ) (u(S1 ) − D(a1 )) + δ

i=1

N X

) f (xj , a2 ; xi , a1 )(u(S2 ) − D(a2 )) .

j=1

• The (risk-neutral) principal’s utility is V =

N X

( f (xi , a1 ) (xi − S1 (xi )) + δ

i=1

N X j=1

69

) f (xj , a2 ; xi , a1 )(xj − S2 (xi , xj ) .

Theorem 6.1. Let (a∗ , S ∗ ) be optimal. Then the contract S ∗ over time satisfies the following martingale property: For all i = 1, 2, ..., N ,   1 1 xi . =E ∗ ∗ 0 0 u (S1 (xi )) u (S2 (xi , xj )) Proof: The proof is by a variation argument. Let us define a contract S(xi , xj ) = (S1 (xi ), S2 (xi , xj )) by changing from S ∗ (xi , xj ) = (S1∗ (xi ), S2∗ (xi , xj )) along xk as follows: For i 6= k, S1 (xi ) = S1∗ (xi ), and S2 (xi , xj ) = S2∗ (xi , xj ); for i = k, u(S1 (xk )) = u(S1∗ (xk )) − ∆, and for j = 1, 2, ...., N , ∆ . δ Note that contract S is identical to S ∗ as long as xk is not realized at time 1. Also, note that given S, the agent’s optimal effort still remains to be a∗ = (a∗1 , a∗2 ). To see this, suppose xk is realized. Then the agent expected utility is E[u(S2 ) − D(a2 )|xk ] = E[u(S2∗ ) − D(a2 )|xk ] + ∆/δ. Thus given S2 , the agent’s optimal effort decision has to be a∗2 . At time 1, the agent’s expected utility with S is the same as that with S ∗ . That is, ( ) N N X X ∗ ∗ U (S, a) = f (xi , a1 ) (u(S1 ) − D(a1 )) + δ f (xj , a2 ; xi , a1 )(u(S2 ) − D(a2 )) . u(S2 (xk , xj )) = u(S2∗ (xk , xj )) +

i=1

j=1

Since both S and S ∗ induce the agent to exert identical effort levels over time, and provide identical expected utility levels to the agent, the principal can choose ∆ to maximize her expected utility. That is ) ( N N X X max f (xi , a∗1 ) (xi − S1∗ (xi )) + δ f (xj , a∗2 ; xi , a∗1 )(xj − S2∗ (xi , xj )) ∆

j=1

i=1,i6=k

( +f (xk , a1 ) (xk − S1 (xk )) + δ

N X

) f (xj , a∗2 ; xk , a∗1 )(xk − S2 (xk , xj ))

j=1

Or min ∆

S1 (xk ) + δ

N X

f (xj , a∗2 ; xk , a∗1 )S2 (xk , xj )

j=1

=u

−1

(u(S1∗ (xk )

− ∆) + δ

N X

f (xj , a∗2 ; xk , a∗1 )u−1 (u(S2∗ (xk , xj ) +

j=1

70

∆ ) δ

By construction, we must have ∆ = 0 at optimum. The first order condition leads to the assertion.  The above theorem tells us that the marginal rates of substitution between the principal and agent are equalized in expectation across time. The above martingale property of intertemporal contracts can be illustrated using a simpler model with two periods, two outcomes and two effort levels. (See Laffont and Martimort [2002].)

6.1.1

The two-outcome model

• Two periods, t = 1, 2. • Two outcomes, xt ∈ {xl , xh }. x1 and x2 are independent. • Two effort levels, at ∈ {0, 1}. • P r(xt = xh | a = 0) = π(0) and P r(xt = xh | a = 1) = π(1). ∆π = π(1) − π(0). • The time line is as follows: – At time 1, the principal offers the agent (S1 , S2 ), an intertemporal compensation scheme, or a long-term contract. The agent accepts or reject the contract. – If the agent accepts the contract, he exerts effort a1 . – At the end of time 1, x1 is realized, and the agent is paid S1 (x1 ). – At time 2, the agent exerts effort a2 . – At the end of time 2, x2 is realized, and the agent is paid S2 (x2 , x1 ). • The risk-neutral principal’s utility is x1 − S1 (x1 ) + δ(x2 − S2 (x2 , x1 )). • The agent utility is V (S1 ) − D(a1 ) + δ(V (S2 ) − D(a2 )), where D is disutility of effort. As in an earlier chapter, we set D(1) = D and D(0) = 0. • Notation: St (xh ) = S¯t , St (xl ) = S t . ¯ V (S1 (xh )) = V¯t , V (S1 (xl )) = V t . ¯ V (S2 (xh , x1 )) = V¯2 (x1 ), V (S1 (xl , x1 )) = V 2 (x1 ). ¯ Analysis We assume that the principal wants to induce positive effort in both periods. 71

• The agent’s incentive compatibility condition in the second period is, for all x1 ∈ {xh , xl }, π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) − D ≥ π(0)V (S2 (xh , x1 )) + (1 − π(0))V (S2 (xl , x1 )). Or for all x1 ∈ {xh , xl }, V (S2 (xh , x1 )) − V (S2 (xl , x1 )) ≥

D . ∆π

(6.1)

• Thus, the agent’s first period incentive compatibility condition can be stated as follows: V (S1 (xh )) + δ{π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))} − {V (S1 (xl )) + δ{π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}} D ≥ . (6.2) ∆π To see this – Thus, the agent’s first period incentive compatibility condition is π(1)V (S1 (xh )) + (1 − π(1))V (S1 (xl )) − D +δE[π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) − D | a1 = 1] ≥ π(0)V (S1 (xh )) + (1 − π(0))V (S1 (xl )) +δE[π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) | a1 = 0]. – However, E[π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) − D | a1 = 1] = π(1){π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))} +(1 − π(1)){π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}, and E[π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) | a1 = 0] = π(0){π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))} +(1 − π(0)){π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}. – Thus, E[π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) | a1 = 1] −E[π(1)V (S2 (xh , x1 )) + (1 − π(1))V (S2 (xl , x1 )) | a1 = 0] = ∆π{π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))} −∆π{π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}. 72

Therefore, the agent’s first period incentive compatibility condition reduces to the form as claimed above. • The agent’s participation constraint is π(1) {V (S1 (xh )) + δE[V (S2 (x2 , xh )) | a1 = 1] − D} +(1 − π(1)) {V (S1 (xl )) + δE[V (S2 (x2 , x1 )) | a1 = 1] − D} ≥ 0. Or π(1) [V (S1 (xh )) + δ {π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))}] +(1 − π(1)) [V (S1 (xl )) + δ (π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}] −(1 + δ)D ≥ 0. (6.3) • The principal’s problem is to choose S1 (xh ), S1 (xl ), S2 (xh , xh ), S2 (xl , xh ), S2 (xh , xl ), and S2 (xl , xl ), in order to max

s.t.

π(1)(xh − S1 (xh )) + (1 − π(1))(xh − S1 (xh )) +δπ(1) {π(1)(xh − S2 (xh , xh )) + (1 − π(1))(xh − S2 (xl , xh ))} +δ(1 − π(1)) {π(1)(xh − S2 (xh , xl )) + (1 − π(1))(xh − S2 (xl , xl ))} D (1) V (S2 (xh , xh )) − V (S2 (xl , xh )) ≥ ∆π D (2) V (S2 (xh , xl )) − V (S2 (xl , xl )) ≥ ∆π (3) V (S1 (xh )) + δ{π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))} − [V (S1 (xl )) + δ{π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}] D ≥ ∆π (4) π(1) [V (S1 (xh )) + δ {π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh ))}] +(1 − π(1)) [V (S1 (xl )) + δ {π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl ))}] −(1 + δ)D ≥ 0.

• The FOCs for S1 (xh ) and S1 (xl ) are −π(1) + V 0 (S1 (xh )) {λ3 + π(1)λ4 } = 0 −(1 − π(1)) − V 0 (S1 (xl )) {λ3 − (1 − π(1))λ4 } = 0. By adding the above two, we have λ4 =

π(1) V

0 (S (x )) 1 h

By substitution back into the FOC,  λ3 = π(1)(1 − π(1))

+

(1 − π(1)) > 0. V 0 (S1 (xl ))

1 1 − 0 0 V (S1 (xh )) V (S1 (xl ))

73

 .

• The FOCs for S2 ’s are  −δπ(1)π(1) + V 0 (S2 (xh , xh )) λ1 + λ3 δπ(1) + λ4 π 2 (1)δ = 0 −δπ(1)(1 − π(1)) − V 0 (S2 (xl , xh ) {λ1 − λ3 δ(1 − π(1)) − λ4 δπ(1)(1 − π(1))} = 0 −δ(1 − π(1))π(1) + V 0 (S2 (xh , xl )) {λ2 − λ3 δπ(1) + λ4 δ(1 − π(1))π(1)} = 0  −δ(1 − π(1))2 − V 0 (S2 (xl , xl )) λ2 + λ3 δ(1 − π(1)) − λ4 δ(1 − π(1))2 = 0. – By adding the first two FOCs,  δπ(1)π(1) 2 + λ + λ δπ(1) + λ π (1)δ =0 1 3 4 V 0 (S2 (xh , xh )) δπ(1)(1 − π(1)) − 0 − {λ1 − λ3 δ(1 − π(1)) − λ4 δπ(1)(1 − π(1))} = 0 V (S2 (xl , xh )   (1 − π(1)) π(1) + > 0. (6.4) λ3 + λ4 π(1) = π(1) V 0 (S2 (xh , xh )) V 0 (S2 (xl , xh ) −

– By adding the last two FOCs, δ(1 − π(1))π(1) + {λ2 − λ3 δπ(1) + λ4 δ(1 − π(1))π(1)} = 0 V 0 (S2 (xh , xl ))  δ(1 − π(1))2 − 0 − λ2 + λ3 δ(1 − π(1)) − λ4 δ(1 − π(1))2 = 0. V (S2 (xl , xl )) −

 −λ3 + λ4 (1 − π(1)) = (1 − π(1))

(1 − π(1)) π(1) + 0 0 V (S2 (xh , xl )) V (S2 (xl , xl ))

 > 0. (6.5)

By adding (6.4) and (6.5), we have   (1 − π(1)) π(1) λ4 = π(1) + V 0 (S2 (xh , xh )) V 0 (S2 (xl , xh )   π(1) (1 − π(1)) +(1 − π(1)) + . V 0 (S2 (xh , xl )) V 0 (S2 (xl , xl )) By substituting back, 

(1 − π(1)) V V 0 (S2 (xl , xh )  π(1) (1 − π(1)) − 0 − . V (S2 (xh , xl )) V 0 (S2 (xl , xl ))

λ3 = π(1)(1 − π(1))

π(1)

0 (S (x , x )) 2 h h

+

– By plugging (6.4) into the first of the four FOCs,   1 1 λ1 = δπ(1)π(1)(1 − π(1)) − V 0 (S2 (xh , xh )) V 0 (S2 (xl , xh ) 74

– By plugging (6.5) into the third of the four FOCs,   1 1 2 λ2 = δπ(1)(1 − π(1)) − V 0 (S2 (xh , xl )) V 0 (S2 (xl , xl )) – Recall that

π(1) V

0 (S (x )) 1 h

= λ3 + π(1)λ4

Using (6.4), π(1) (1 − π(1)) 1 = + . V 0 (S1 (xh )) V 0 (S2 (xh , xh )) V 0 (S2 (xl , xh )) Also recall that

(1 − π(1)) = −λ3 + λ4 (1 − π(1)). V 0 (S1 (xl ))

Using (6.5), V

1 π(1) (1 − π(1)) = 0 + 0 . V (S2 (xh , xl )) V (S2 (xl , xl )) 1 (xl ))

0 (S

Therefore, the martingale property is confirmed. √ • Example: Suppose V (w) = (2/b) a + bw where a, b > 0. Then 1/V 0 (w) = √ a + bw = (b/2)V (w). The martingale property implies that V (S1 (xh )) = π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh )). V (S1 (xl )) = π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl )). Thus constraint (3) and (4) with equalities reduce to D (1 + δ)∆π (4) π(1)V (S1 (xh )) + (1 − π(1))V (S1 (xl )) − D = 0. (3) V (S1 (xh )) − V (S1 (xl )) =

The above two yield D(1 − π(1)) (1 + δ)∆π Dπ(1) V (S1 (xl )) = D − (1 + δ)∆π V (S1 (xh )) = D +

(6.6) (6.7)

Since δ > 0, the first-period incentive scheme in the two-period case is lowerpowered than that in the one-period case. The utility spread between high D D and low outcome is (1+δ)∆π in the two-period case, whereas it is ∆π in the 75

one-period case. In the two period problem, the agent considers the effect of his effort on the second period contract. Knowing that a high outcome is advantageous in the second period contracting, the agent has some incentives to exert high effort even without explicit 1st-period incentives. Thus from the principal’s perspective, in the two-period world, she does not have to give as strong incentives as she does in the one-period world, in order to induce a given level of effort. Consequently, the power of the first-period incentives in the two-period world is lower than that in the one-period world. On the other hand, constraints (1) and (2) are D ∆π D V (S2 (xh , xl )) − V (S2 (xl , xl )) = ∆π

V (S2 (xh , xh )) − V (S2 (xl , xh )) =

By the martingale property and Eq.’s (6.6) and (6.7), D(1 − π(1)) (1 + δ)∆π Dπ(1) π(1)V (S2 (xh , xl )) + (1 − π(1))V (S2 (xl , xl )) = D − . (1 + δ)∆π π(1)V (S2 (xh , xh )) + (1 − π(1))V (S2 (xl , xh )) = D +

Using constraints (1) and (2), the above equations reduce to D(1 − π(1)) (1 − π(1))D + (1 + δ)∆π ∆π D(1 − π(1)) π(1)D − V (S2 (xl , xh )) = D + (1 + δ)∆π ∆π Dπ(1) (1 − π(1))D V (S2 (xh , xl )) = D − + (1 + δ)∆π ∆π Dπ(1) π(1)D V (S2 (xl , xl )) = D − − . (1 + δ)∆π ∆π

V (S2 (xh , xh )) = D +

(6.8) (6.9) (6.10) (6.11)

– The 2nd-period utility spreads between high and low outcomes are D/∆π regardless of the 1st-period outcome. comparing with the 1st-period spread D/(1 + δ)∆π, one can see that the power of the 2nd-period incentives followed by high 1st-period outcome is higher than that of the 1st-period incentives. – As future matters more, or as δ becomes larger, the power of the 1st-period incentive becomes lower. – An early success (failure) results in a higher (lower) future compensation. 76

Extensions • Intertemporal limited liability constraints. The limited liability results in The power of the first-period incentives is higher with limited liability than without it. The 2nd-period compensation with limited liability followed by low 1stperiod outcome is higher than that without limited liability. • Saving. Implicit in the above two-period model example is that the agent consume all compensation amounts as soon as he receives them. If the agent is allowed to save, and if the agent’s utility exhibits “wealth effect,” then the agent’s saving amount in the first period can affect the agent’s second-period utility and second-period incentive compatibility condition. One can show that without loss of generality, the principal can offer a savingproof long-term contract, with the agent has no incentive to save. Unfortunately, results become complex. • Infinitely repeated moral hazard. It is known that when δ is close to 1, the first best allocations can be achieved. With infinitely repeated moral hazard problems, the principal can spread the agent’s rewards and punishments over time and let the agent bear only a small fraction of the risk associated with current effort in any given period. Alternatively, infinitely repeated moral hazard problems, the agent can be almost completely diversified, and almost behaves like a risk-neutral agent. Hence the first-best allocations over time.

6.2

Moral Hazard with Noncommitment

When she cannot commit herself to a long-term contract, the principal can renegotiate the contract with the agent at time 2. One may think of a renegotiation problem after observing the first-period outcome. But then, this case is basically a repetition of the first period problem. Nevertheless, the problem can be interesting if the second period problem is based on a different set of information. We consider the following two cases: Case 1: Renegotiation after the agent’s first-period effort but before the realization of the outcome. Case 2: Renegotiation after receiving additional a signal regarding the agent’s firstperiod effort, but before the realization of the outcome. Here, we assume the signal is observable but not verifiable. 77

6.2.1

Case 1

The principal is risk neutral and the agent is risk averse. Theorem 6.2. Assume that the contract cannot be randomized. Then the agent exerts the lowest level of effort at time 1. Proof: At the renegotiation stage, the optimal contract has to be a constant contract because no effort is involved. Knowing that the agent will have a constant contract, he has no incentive to exert an effort level other than the lowest. . Thus, the principal may want to consider a randomized contract. See Fudenberg and Tirole [1990].

6.2.2

Case 2

Hermalin and Katz [1991]. They consider an agent-led renegotiation problem. Theorem 6.3. Suppose that the signal s perfectly reveals a, and that the agent makes a take-it-or-leave-it renegotiation offer at the renegotiation stage. Then the first-best action is optimal and all rent goes to the principal.

78

Chapter 7 Dynamics of Contracts under Adverse Selection • There are two periods, 1 and 2. • At each period, the firm produces a good at cost Ct = θ − e,

t = 1, 2.

• Ct is observable. But θ and e are not separately observable. • The principal reimburses the cost C and pays S to the agent. • The principal’s realized utility is q − C1 − S1 − δ(q − C2 − S2 ), where δ is a discount factor. • The agent’s realized utility is S1 − D(e1 ) + δ(S2 − D(e2 )). ¯ is distributed with prior pdf f1 (θ) at time 1 and with posterior pdf • θ ∈ [θ, θ] ¯ f2 (θ) at time 2. • The game between the principal and the agent is played as follows: Posterior f2 (θ) Prior f1 (θ) N

High: z Low

A

Choose a contract & effort O1 *

? - P S1 (.)- A

Choose a contract & effort * O2

? - P S2 (.)- A

Rejectj End Rejectj End

79

7.1

Full Commitment

• Two periods, 1 and 2. ¯ θ does not change over time. Alternatively, θ1 and θ2 are perfectly • θ ∈ {θ, θ}. ¯ positively correlated, where θt is agent’s type at time t. • the cost of production at time t is Ct = et − θ. • The risk-neutral principal’s utility is q1 − S1 + δ(q2 − S2 ), where qt is the output at time t, St compensation at time t, and δ discount factor at time t. Here, S1 = S1 (C1 ), and S2 = S2 (C1 , C2 ). • The risk-neutral agent’s utility is U = S1 − D1 + δ(S2 − D2 ). • Incentive compatibility conditions. S 1 − D1 + δ(S 2 − D2 ) ≥ S¯1 − D(θ − C¯1 ) + δ{S¯2 − D(θ − C¯2 )} (7.1) ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 + δ(S¯2 − D ¯ 2 ) ≥ S 1 − D(θ¯ − C 1 ) + δ{S 2 − D(θ¯ − C 2 )} (7.2) S¯1 − D ¯ ¯ ¯ ¯ 0 The above conditions imply that if agent θ chooses a contract for θ at time 1, he has to choose again a contract for θ0 at time 2 as well. No second-period deviation is allowed. • The participation constraints of both types for both periods are U ≥0 ¯ U¯ ≥ 0

(7.3) (7.4)

• The principal’s problem is  
max ν q 1 − S 1 + δ(q 2 − S 2 ) + (1 − ν) q¯1 − S¯1 + δ(¯ q2 − S¯2 ) S,C ¯ ¯ ¯ ¯ s.t. S 1 − D1 + δ(S 2 − D2 ) ≥ S¯1 − D(θ − C¯1 ) + δ{S¯2 − D(θ − C¯2 )} ¯ ¯ ¯ ¯ ¯ ¯ U¯ ≥ 0. Note that using the participation constraint, the RHS of the incentive constraint can be rewritten as S¯1 − D(θ − C¯1 ) + δ{S¯2 − D(θ − C¯2 )} = Φ(¯ e1 ) + δΦ(¯ e2 ). ¯ ¯ Proposition 7.1. In the above two-period adverse selection model with perfectly correlated types, the optimal long term contract with full commitment for two periods is the repetition of the optimal static (one-period) contract over time. In particular, the results can be summarized as follows: 80

• Constraints (7.1) and (7.4) are the only binding ones. • The efficient agent exerts the first best level of effort in both periods, i.e., D0 (e1 ) = D0 (e2 ) = 1 = D0 (e∗ ). ¯ ¯ • Inefficient agent exerts less than the first-best effort in both periods. In fact, the agent exerts the same levels of effort in both periods as he does in the one-period case. Remark: When θ1 and θ2 are correlated and there are limited liability constraints, one can show that the repeatedly inefficient agent produces less than the static optimum in the second period. That is, the static optimum may not be repeated in the second period.

7.2

Noncommitment

Consider two types θ0 and θ00 . For the time-1 strategies of type θ0 , we can think of two cases: (1) choose a contract for θ0 and reveal his type, and (2) choose a contract for θ00 and conceal his type. Suppose that θ0 < θ00 . In the first case, θ0 will enjoy information rent at time 1, but no information rent at time 2. In the second case, θ0 will enjoy extra utility gains at both periods, 1 and 2, because of cost differentials. In particular, the utility gain at time 2 is δU2 (θ00 |θ0 ) > 0. Thus, in the second case, the utility of θ0 is θ0 q(θ00 ) − t(θ00 ) + δU2 (θ00 |θ0 ). From the perspective of θ00 , if he chooses a contract for θ0 and conceals his type, he may have an extra utility gain at time 1. (Why an extra gain? Why not a negative gain?) But since he declared his type as θ0 , at time 2, he will be assigned with a contract for θ0 , which will give θ0 the reservation utility, but θ00 less than the reservation utility. Therefore, at time 2, θ00 has to reject the contract. This set of strategies is known as a “take-the-money-and-run” strategy. On the other hand, if he chooses a contract for θ00 , then θ00 enjoys no extra rent. ¯ We first look at the continuum case. Suppose θ ∈ [θ, θ]. ¯ Theorem 7.1. For any first-period incentive scheme {S1 (θ)}, there exits no nonde¯ over which separation occurs. generate subinterval of [θ, θ] ¯ Proof: Suppose there is a subinterval [b, u] in which a full separation occurs. Choose θ and θ0 from the subinterval such that θ < θ0 . At time 1, type θ produces at cost C(θ) and receives S(θ), and type θ0 produces at cost C(θ0 ) and receives S1 (θ0 ). When the full separation for [b, u] is optimal, neither θ nor θ0 should have incentives to deviate from the full separation. Nevertheless, given the full separation contracts, 81

consider a deviation strategy as follows: If type θ deviates and produce at cost c(θ0 ), then it will receive a strictly positive rent in the second period (because type θ0 obtains no rent under the full separation.) Let the second period rent denoted by U (θ0 |θ) > 0. Under the full separation equilibrium, the above deviation strategy has to be suboptimal. Thus, we have S1 (θ) − D(θ − C1 (θ)) ≥ S1 (θ0 ) − D(θ − C1 (θ0 )) + δU (θ0 |θ).

(7.5)

Next, consider a deviation strategy for θ0 . If θ0 deviates and behaves like θ, then it does not gain a rent in the second period. (Because under the full separation contracts at time 2, both θ and θ0 receive zero rents by choosing right contracts. If θ0 chooses S2 (θ), the rent will be negative. Thus if θ0 deviates and behaves like θ in the first period, then it will reject the second period contract in order to receive a zero rent instead of a negative rent. This strategy for the less efficient type θ0 is called “the take-the-money-and-run” strategy.) Since the above deviation strategy for θ0 , or the take-the-money-and-run strategy, has to be suboptimal (when the menu of full separation contracts is optimal), S1 (θ0 ) − D(θ0 − C1 (θ0 )) ≥ S1 (θ) − D(θ0 − C1 (θ)).

(7.6)

By adding (7.5) and (7.6), we have D(θ − C1 (θ0 )) − D(θ − C1 (θ)) ≥ D(θ0 − C1 (θ0 )) − D(θ0 − C1 (θ)) + δU (θ0 |θ). Using the fact that U (θ0 |θ) > 0, the above implies that Z C1 (θ) Z θ ˜ C˜ > 0. ˜ θd D00 (θ˜ − C)d C1 (θ0 )

θ0

Since D00 > 0, θ < θ0 , we must have C1 (θ) < C1 (θ0 ) and thus C is differentiable a.e. for θ ∈ [b, u]. On the other hand, since C1 (θ) < C1 (θ0 ) and since D is decreasing, (7.5) implies that S1 (θ) has to be decreasing. Now, by (7.6) S1 (θ0 ) − S1 (θ) − (D(θ0 − C1 (θ0 )) − D(θ0 − C1 (θ))) ≥ 0. That is, Z θ

θ0

˜ dS1 (θ) dθ˜ + dθ

Z θ

θ0

˜ dC1 (θ) ˜ dD(θ0 − C1 (θ)) dθ˜ ≥ 0. de dθ

0

By dividing the above by θ − θ and letting θ0 approach θ, we have dS1 (θ) dD(θ − C1 (θ)) dC1 (θ) + ≥ 0. dθ de dθ

(7.7)

On the other hand, note that in (7.5), U (θ0 |θ) is the second period rent enjoyed by θ when it mimics θ0 . Let the second period contract for θ0 be (S2 (θ0 ), C(θ0 )). 82

By mimicking θ0 , θ exerts effort so that e = θ − C(θ0 ) and receives (S2 (θ0 ). But by the reservation utility for θ0 , S2 (θ0 ) − D(θ0 − C(θ0 )) = 0. Therefore, U (θ0 |θ) = S2 (θ0 ) − D(θ − C(θ0 )) = S2 (θ0 ) − D(θ0 − C(θ0 )) + D(θ0 − C(θ0 )) − D(θ − C(θ0 )) = D(θ0 − C(θ0 )) − D(θ − C(θ0 )) Z θ0 D0 (θ˜ − C(θ0 ))dθ˜ = θ

Thus, by (7.5), we have 0 ≥ S1 (θ0 ) − S1 (θ) + D(θ − C1 (θ)) − D(θ − C1 (θ0 )) + δU (θ0 |θ). Z θ0 Z θ0 Z θ0 0 ˜ ˜ 0 0 ˜ ˜ ˜ ˜ = S1 (θ)dθ + D (θ − C1 (θ))C1 (θ)dθ + δ D0 (θ˜ − C(θ0 ))dθ. θ

θ

θ

Again, by dividing the above by θ0 − θ and letting θ0 approach θ, we have ˜ + D0 (θ − C1 (θ))C 0 (θ) + δD0 (θ − C(θ)). 0 ≥ S10 (θ) 1

(7.8)

However, since D0 (θ − C(θ)) = 1 under separating contracts, (7.7) and (7.8) cannot hold simultaneously. Contradiction.  Exercise 7.1. Theorem 7.1 suggests that when there is a continuum of types of agents, ¯ in the first period, a pooling contract is optimal for almost all [θ, θ], ¯ and the [θ, θ], ¯ ¯ ¯ i.e., all efficient agents pretend to be the least efficient. Comment. pooling occurs at θ,

7.2.1

The two-type case

As seen in Theorem 7.1, with a continuum of types, full separation over any subinterval is not implementable. In the two-type case, we will see that separation may be implementable, but typically it is not optimal. See Laffont and Tirole [1993].

83

84

Chapter 8 Career Concern • Holmstrom [1982, WP], published in RES, 1999 • Holmstrom and Ricarti I Costa [1986, QJE] • Dewatripont, Jewitt, and Tirole [1999, RES]. • Incentives: – Explicit incentives, e.g., salaries, bonuses, and stock options. – Implicit incentives, e.g., career concerns, and promotions. • Example: – Three dates: t = 0, 1, 2. – Performance at time t is yt , for t = 1, 2, yt = θ + et + t , ¯ σ 2 ), et ∈ [0, A] is the agent’s effort at where θ is the agent’s ability ∼ N (θ, θ time t, and t is the noise at time t ∼ N (0, σ2 ). Random variables, θ, 1 and 2 are independent of each other. – The agent’s wages are S1 and S2 at t = 1 and 2, respectively. S1 is constant, and S2 is determined by labor market competition and is equal to the market’s expectation of the agent’s marginal productivity conditional on her performance at time 1. – The distribution of θ and y1 given e1 is bivariate normal with a pdf given by f (θ, y1 | e1 ) ∝ e 85

−

¯2 (θ−θ) 2σ 2 θ

−

e

(y1 −θ−e1 )2 2σ2

– The agent’s utility is S1 − D(e1 ) + δS2 . Note that since S2 does not depend on y2 , the agent’s optimal effort at time 2 is e∗2 = 0. Suppose that the market anticipates e01 . Then, S2 = E M [y2 | y1 ] = E M [θ + e2 + 2 | y1 ] = E M [θ | y1 ] = E M [θ | θ + e01 + 1 ], where E M is the market expectation operator. Thus the agent’s problem is max δE M [θ | θ + e01 + 1 ] − D(e1 ), e1

The FOC is Z

d δ de1

Z

! ! f (θ, y1 | e01 ) dθ fˆ(y1 | e∗1 )dy1 = D0 (e∗1 ). θ 0 ˆ f (y1 | e ) 1

Since e01 = e∗1 in labor market equilibrium, we have Z Z δ

∗ ˆ ∗ fe1 (y1 | e1 ) θf (θ, y1 | e1 ) dθdy1 fˆ(y1 | e∗1 )

= D0 (e∗1 ).

Since E[fˆe1 /fˆ] = 0, the FOC can be rewritten as ! fˆe1 = De1 (e∗1 ). δcov θ, ˆ f Therefore, in an equilibrium of the career concerns model, the marginal disutility (cost) of the agent’s effort is equal to the covariance of ability and the likelihood ratio. – Note that fˆ(y1 | e1 ) ∝ e and

2 ¯ (y1 −θ−e 1) 2(σ 2 +σ2 ) θ

,

¯ + 1 fˆe1 (y1 | e1 ) (θ − θ) = σθ2 + σ2 fˆ(y1 | e1 )

Therefore, D0 (e∗1 ) = δ

σθ2 . σθ2 + σ2

Since D is strictly is convex, the equilibrium is unique, and the effort e1 increases when the performance become more informative in the sense that the variance of the measurement error  decreases.

86

Chapter 9 Signaling Akerlof [1970, QJE] Spence [1973, QJE] Crawford and Sobel [1982, Econometrica] • We consider information asymmetry problems in an economy where there are informed and uniformed economic agents. • Adverse selection models: The informed moves first by offering the uninformed a menu of contracts. • Signaling: The informed moves first by sending the uninformed a signal about the private information the informed has.

9.1

The Market for Lemons

• There are good cars and bad cars in a used car market. The proportion of good cars is q and that of bad cars is 1 − q. • The value of a good car is g to the seller and G to the buyer. Assume g < G. • The value of a bad car is b to the seller and B to the buyer. Assume b < B. • Assume G > B and g > b. Also assume that the supply of used cars is finite, but the demand is infinite. • The perfect information case: When both sellers and buyers can observe the quality, the price of a good car is G and that of a bad car is B. • The zero information case: When neither sellers nor buyers know the quality of a given car, its equilibrium price is qG + (1 − q)B. 87

• An asymmetric information case: The seller knows the quality of his used car, but the buyer cannot observe the quality. – The seller of a good used car wouldn’t sell at a price below g. Thus, if the price is less than g, the car has to be a lemon. Hence the buyer wouldn’t pay more than B. On the other hand, if the price is greater than g, then no information about the quality can be inferred. In this case, the car worth qG + (1 − q)B to the buyer. – Therefore, there are two possible equilibria. ∗ If p = B < g, then only lemons are traded. ∗ If p = qG + (1 − q)B ≥ g, then both types are traded.

9.2

Job Market Signaling

• There are many job applicants and employers. Each applicant has productivity θ ∈ {θ1 , θ2 }, where θ1 < θ2 . The applicant can observe his own productivity, but employers cannot. • If applicant θ studies for e years and is hired at wage w, then his utility will be u(w) − C(e, θ). • θ is independent of e. But the longer the number of years e, the higher the cost C, i.e., ∂C/∂e > 0. The marginal cost increases with e, i.e., ∂ 2 C/∂e2 > 0. The higher the productivity, the lower the marginal cost, i.e., ∂ 2 C/∂e∂θ > 0. Moreover, we set C(0, θ) = 0. • The number of years e in school is publicly observed. • Each applicant with e will be offered a wage w(e) = µ(e)θ1 + (1 − µ(e))θ2 , if given e, employers think that the candidate is θ1 with probability µ(e). Note that each applicant is paid the market expectation of his productivity. • We look at a perfect Bayesian equilibrium in pure strategies consisting of (e∗1 , e∗2 , w∗ ), and a system of beliefs µ∗ such that – The optimal number of years in school for applicant θi , i = 1, 2, is e∗i ∈ arg max u(w(e)) − C(e, θi ) e

88

– Each applicant with e will be offered an equilibrium wage w∗ (e) = µ∗ (e)θ1 + (1 − µ∗ (e))θ2 – The beliefs µ∗ (e) are consistent with the strategies e∗ as follows: ∗ When e∗1 6= e∗2 , · if e = e∗1 , then µ∗ (e) = 1; and · if e = e∗2 , then µ∗ (e) = 0. ∗ When e∗1 = e∗2 , · if e = e∗1 = e∗2 , then µ∗ (e) = µ0 . Next we explore implications of the above two equilibria.

9.2.1

Separating Equilibria

In a separating equilibria, e∗1 6= e∗2 , and thus employers can infer the agent’s productivity by looking at e. Applicant θ1 gets a wage equal to θ1 , and applicant θ2 gets a wage equal to θ2 . That is, applicant θ1 finds that his costly education is useless to him and therefore chooses not to study at all, i.e., e∗1 = 0. Furthermore, for a separating equilibrium with e∗1 = 0, u(θ1 ) − C(0, θ1 ) ≥ u(θ2 ) − C(e∗2 , θ1 ), u(θ2 ) − C(e∗2 , θ2 ) ≥ u(θ1 ) − C(0, θ2 ).

(9.1) (9.2)

Inequalities (9.1) and (9.2) imply C(e∗2 , θ1 ) − C(0, θ1 ) ≥ u(θ2 ) − u(θ1 ) ≥ C(e∗2 , θ2 ) − C(0, θ2 ). Thus e∗2 ∈ [e, e¯], for some e and e¯. ¯ ¯

9.2.2

Pooling Equilibria

• In a pooling equilibrium, both θ1 and θ2 choose the same number of years in school, e∗ . Thus employers cannot update their beliefs and each applicant is offered a wage µ0 θ1 + (1 − µ0 )θ2 . • Define eˆ as follows. u(µ0 θ1 + (1 − µ0 )θ2 ) − C(ˆ e, θ1 ) = u(θ1 ) − C(0, θ1 ) From the low-productivity applicant’s perspective, he will decide to spend time in school only if he anticipates his utility as a result of schooling to be at least equal to the utility without schooling. 89

Thus if the low-productivity applicant ever chooses e1 > 0 in an attempt to confuse employers about his productivity level, then u(µ0 θ1 + (1 − µ0 )θ2 ) − C(e1 , θ1 ) ≥ u(θ1 ) − C(0, θ1 ) That is, in a pooling equilibrium with e∗ , we must have e∗ ≤ eˆ. • Remark: Note that if e∗ > 0, all workers are better off if education is banned. Then they get the same wage µ0 θ1 + (1 − µ0 )θ2 and save the cost of schooling.

9.2.3

Job market equilibrium

From the previous subsections, we know that in a separating equilibrium, e∗1 = 0 and e∗2 ∈ [e, e¯]; and that in a pooling equilibrium, e∗ ∈ [0, eˆ]. ¯ Thus there can be infinite number of separating and pooling equilibria. Then which equilibrium can be more reasonable or more stable? Given an equilibrium, can an agent be better off if the other agent deviates from the equilibrium strategy? The above infinite number of equilibria can be refined as follows: • Consider a pooling equilibrium with e∗ . Suppose that the wage schedule stipulates that the agent can receive wage θ2 with education e. Then by definition, u(µ0 θ1 + (1 − µ0 )θ2 ) − C(e∗ , θ1 ) ≥ u(θ2 ) − C(e, θ1 ). – Consider e0 such that u(µ0 θ1 + (1 − µ0 )θ2 ) − C(e∗ , θ1 ) > u(θ2 ) − C(e0 , θ1 ). Then e0 is clearly a dominated strategy for agent θ1 . (Also note that e∗ < e0 .) Thus intuitively, it is reasonable to imagine that under the pooling equilibrium, employers assign µ∗ (e0 ) = 0 and w∗ (e0 ) = θ2 . (This is suggested by Cho-Kreps [1987, QJE] “intuitive criterion.”) – If µ∗ (e0 ) = 0 and w∗ (e0 ) = θ2 , then a question arise: Will the other agent θ2 still happily remain at e∗ ? The answer is clearly no. He will find e0 with which he can easily increase his utility. Thus no pooling equilibria are stable. • Next we consider a separating equilibrium. – We know that in a separating equilibrium, e∗1 = 0 and e∗2 ∈ [e, e¯]; and ¯ w(e∗1 ) = θ1 and w(e∗2 ) = θ2 . – We know that u(θ2 ) − C(e, θ1 ) = u(θ1 ) − C(0, θ1 ). ¯ Thus, Cho-Kreps [1987, QJE] “intuitive criterion” suggests that employers assign µ∗ (e) = 0 and w∗ (e) = θ2 for e > e. ¯ 90

– Suppose e∗2 > e. Then agent θ2 will be better off by deviating from e∗2 to ¯ e. Thus e∗2 cannot be a stable equilibrium unless e∗2 = e. ¯ ¯ – Therfore, the only intuitive equilibrium is the separating equilibrium with e∗1 = 0 and e∗2 = e; and w(e∗1 ) = θ1 and w(e∗2 ) = θ2 . This equilibrium is ¯ called the least cost separating equilibrium.

91

92

Appendix A

A.1

Dynamic Programming Equation with Exponential Utility

Let Bt be n-dimensional standard Wiener processes, and also let F α f, β σ

: : : :

Rn −→ R [0, 1] × Rn × U −→ R [0, 1] × Rn × U −→ Rn [0, 1] × Rn × U −→ Rn×n +

Consider the following stochastic control problem: h i R1 R1 max E −e−r{F (Y1 )− 0 α(t,Yt ,ut )dt− 0 β(t,Yt ,ut )dBt } u

s.t.

dYt = f (t, Yt , ut )dt + σ(t, Yt , ut )dBt .

Assume {Yt } has a unique strong solution. ( Note that a unique strong solution is ensured if the drift and the diffusion rates of {Yt } satisfies the linear growth condition and the global Lipschitz condition, respectively.) The stochastic control problem of the form (A.1) is a general form that will cover our optimization problems in the text as special cases. Define h i R1 R1 > J(u; t, Ft ) = E −e−r{F (Y1 )− t α(s,Ys ,us )ds− t β (s,Ys ,us )dBs } | Ft Assumption A.1. A function V is continuously differentiable in t and twice continuously differentiable in Yt , and satisfies for some p > 2 the following conditions: Z

1 p

V (t, Yt ) dt, 0

Z

1

k

and 0

∂V (t, Yt )kp dt, ∂y

93

are in L1 (Ω, F, P ).

Lemma A.1. Assume that J(u; t, Ft ) = J(u; t, Yt ).1 Define the value function V as follows: V (t, Yt ) := sup J(u; t, Yt ). u

Also assume that α and β are bounded and that there exists an optimal control policy u∗ such that V (t, Yt ) = J(u∗ ; t, Yt ), and that V satisfies the Assumption A.1. Then the value function V necessarily satisfies the following dynamic programming equation for any time t ∈ [0, 1] a.e.:  >  ∂V tr ∂ 2 V ∂V r 2 > + max (f + rσβ) + 0≡ σσ + rV (α + kβk ) . u ∂t ∂y 2 ∂y 2 2 The terminal condition is V (1, Y1 ) = −e−rF (Y1 ) . Lemma A.2. Assume that α and β are bounded. Consider a function V such that V : [0, 1]×Rm+n → R, satisfying the Assumption A.1 and the following partial differential equation   > ∂V tr ∂ 2 V r ∂V > 2 0≡ + sup (f + rσβ) + σσ + rV (α + kβk ) ∂t ∂y 2 ∂y 2 2 u with the terminal condition being V (1, y) = −e−rF (y) . Assume that there exists u∗ such that   > tr ∂ 2 V r ∂V > 2 ∗ (f + rσβ) + σσ + rV (α + kβk ) . u ∈ arg max u ∂y 2 ∂y 2 2 Then u∗ is an optimal control and J(u∗ ; t, Ft ) = V (t, Yt ). Proof of Lemma A.1: For 0 ≤ t ≤ τ ≤ 1, let  uˆs if t ≤ s ≤ τ u= u∗s if τ < s ≤ 1 where u∗ is an optimal control, and uˆ is an arbitrary admissible control. Let us define φτ ≡ er(Xτ −Xt ) , where dXt = α(t, Yt , u)dt + β > (t, Yt , u)dBt . Then we have by using the Itˆo’s formula, Z τ Z 1 τ 2 φτ − 1 = rφs dXs + r φs kβk2 ds. 2 t t 1

This definition is possible if the controls of interest are Markovian.

94

Define Pτ := φτ · V (τ, Yτ ). Then E [Pτ | Ft ] = E [φτ · V (τ, Yτ ) | Ft ] . This implies V (t, Yt ) ≥ E [Pτ | Ft ] Since V ∈ C 1,2 by assumption, an application of the Itˆo’s formula yields Z τ Z τ ∂V ∂V > V (τ, Yτ ) = V (t, Yt ) + (s, Ys )ds + (s, Ys )dYsc ∂s ∂y  t Z τ t 2 tr ∂ V + (s, Ys )σσ > ds 2 2 ∂y t

(A.1)

Therefore we have τ

τ

τ

∂V > φs dVs + V (s, Ys )dφs + rφs (s, Ys )σβds ∂y t t t Z τ  ∂V ∂V > (s, Ys ) + (s, Ys )(f + rσβ) = V (t, Yt ) + φs ∂s ∂y t    tr ∂ 2 V 1 > 2 (s, Ys )σσ + + rV (s, Ys )(α + rkβk ) ds 2 ∂y 2 2 Z τ > ∂V + φs { (s, Ys )σ + rV (s, Ys )β > }dBt . (A.2) ∂y t Z

Z

Z

Pτ = V (t, Yt ) +

Now we want to show that the stochastic integrals are martingales. For this purpose it suffices to show that Z τ ∂V > E[ φ2s k (s, Ys )σ + rV (s, Ys )β > k2 ds] < ∞. (A.3) ∂y t But φs can be rewritten as follows: φs = er(Xs −Xt )  Z s  Z s > = exp r αdτ + β dBτ t t Z s   Z s  Z r 1 s 2 2 2 > = exp rβ dBτ − r kβk dτ · exp r (α + kβk )dτ . 2 t 2 t t Note that the first exponential is a Girsanov density G(s). Since β is bounded, the Lp R1 1 norm of G(s), i.e. {E[ t Gp (s)ds]} p , is finite for all p > 0. By H¨older’s inequalities, this implies that (A.3) holds. Therefore the stochastic integral is a martingale and its expectation is zero. Recall that 0 ≥ E[Pτ |Ft ] − V (t, Yt ) for all τ ∈ [t, 1]. But if u ≡ u∗ for all s ∈ [t, 1], then 0 ≡ E[Pτ |Ft ] − V (t, Yt ) for all τ ∈ [t, 1]. Hence by dividing E[Pτ |Ft ] − V (t, Yt ) 95

by τ − t and letting τ → t+, we obtain the statement of Lemma A.1.



Proof of Lemma A.2: Consider a function V that satisfies the conditions stated in the Lemma A.2. Define Pτ := φτ · V (τ, Yτ ), where φ is as defined in the proof of Lemma A.1. Then by the same calculation, we obtain the equation (A.2). But by the definition of V in the statement, the equation (A.2) implies that E[Pτ | Ft ] ≤ V (t, Yt ) for all τ (≥ t). In particular, V (t, Yt ) ≥ E[P1 | Ft ] = J(u; t, Yt ). for all nonanticipative arbitrary control law u. Furthermore notice that if u∗ as defined in the statement of Lemma A.2 is used for the time period between [t, τ ], then for all τ ∈ [t, 1], V (t, Yt ) = E[Pτ | Ft ]. But then V (t, Yt ) = E[P1 | Ft ] = J(u∗ ; t, Yt ). Therefore, u∗ is an optimal control, and V is the value function. 

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A.2

Weak Formulation

See Sch¨attler and Sung [1993], and Davis and Varaiya [1973]. Let {Wt } be a standard Wiener process on a filtered probability space (Ω, F, {Ft }, P ), where Ft is the augmentation of the filtration generated by Wt .

A.2.1

Preliminaries Rt

1

Rt

r

2

Theorem A.1. (Novikov) Let Zt = e 0 φs dWs − 2 0 φs ds . If for each t ≥ 0, E[e 2 ∞, then E[Zt ] = 1, and Zt is a positive martingale.

Rt 0

φ2s ds