Presentation Transcript

1. PPS231S.01Law, Economics, and Organization

   Spring 2012 III.1 risk sharing and Incentives
2. Why Contracts? Introduction Recall that we discussed how incentive contracts hold individuals partially responsible for the results of their actions, even though doing so exposes them to risk. This section of the lecture notes develops a detailed theory of efficient incentive contracts in the presence of moral hazard.
3. Why Contracts? Incentive Contracts as a Response to Moral Hazard There are generally more options open to us than choosing between insuring a risk or not. Real-life insurance contracts are also incentive contracts: they restrict and condition claim payments in ways that provide better incentives than full insurance without removing the essential part of insurance coverage.
4. Why Contracts? Ex.: Homeowner’s insurance has a deductible, so the insured bears the initial part of any loss while remaining insured against large financial losses. Ex.: Health insurance (often) requires a co-payment; the insurance pays a fraction of the costs, the rest is borne by the insured.
5. Why Contracts? Ex.: Automobile insurance is experience-rated, so that more accidents mean higher rates. Ex.: Credit card contracts are also experience-rated. Recent case of AmEx and Wal-Mart. All but the last are examples of features that aim at deterring moral hazard. What other examples are there?
6. Why Contracts? Within a firm, similar concerns arise. Compensation must strike the right balance between motivating the employees and protecting them from fluctuations in the value of the firm. Bottom line: (Second-best) efficient contracts balance the costs of risk bearing against the incentive gains that result.
7. Why Contracts? Sources of Randomness If the principal could always observe agent effort perfectly, tying pay to performance would not generate any risk-bearing costs. There would be no risk in the employee’s pay, simply because the outcome would be completely under the employee’s control – the incentives would be as “high-powered” (Williamson, 1985) as possible.
8. Why Contracts? In truth, tying pay to performance means exposing employees to risk, since perfect measures of performance/productivity/behavior are unavailable. To make things worse, outcomes are often affected by things outside the employee’s control.
9. Why Contracts? Then, the problem is that randomness in outcomes results in randomness in compensation, i.e., the agent bears (income) risk. A second source of randomness arises when performance itself is measured, but the evaluation measure has some subjectivity. This is also a source of “risk”.
10. Why Contracts? Finally, another source of randomness comes from outside events that affect the employee’s ability to perform as contracted (e.g., health problems; weather; market conditions; etc.) Again, performance-based compensation becomes random because of all these sources of randomness – this is what we typically call “risk”. To make things even more complex, “risk” and “uncertainty” are (generally) synonyms in economics.
11. Why Contracts? Striking the Right Balance Is it possible to protect employees against all risk by making compensation unrelated to performance outcomes? Yes, but then, there are no incentives to perform in more than the most perfunctory fashion. Most of us have held jobs that paid a flat wage.
12. Why Contracts? Efficient contracting strikes the right balance between incentives and risk sharing – neither too much of either, nor too little. Ex.: A lawyer who sues for damages on behalf of a client receives a contingency fee (% of settlement), which provides an incentive to work hard. Given that outcome of lawsuit is uncertain, both client and litigator incomes are uncertain.
13. Why Contracts? Performance assessment can be made (more) precise. Supervision and monitoring, however imperfect, can be very useful. Ex.: Seth Sanders’ call centers paper. This is the topic of our next set of lecture notes. For now, we assume, like Marshall, that enforcement is “prohibitively costly”.
14. Speaking of Alfred Marshall… Marshall (1920), in his Principles of Economics offered a diagrammatic proof of moral hazard. w, MPL w MPL aMPL Labor
15. Why Contracts? Decisions Under Uncertainty and the Evaluation of Financial Risk In this course, we’ll consider only the special case where the risk is financial. We describe (financial) risk using two concepts: mean and variance, and so we define preferences over the mean and variance of any contract.
16. Why Contracts? Formulas Suppose we have N outcomes xi, each with probability pi, with i in {1,…,N}. Then, Mean: (First Moment) Variance: (Second Moment)
17. Here, the mean is 2,000 and the variance is 5,000,000. Why Contracts?
18. Why Contracts? Certainty Equivalents and Risk Premia Hypothesis: Most people are risk-averse (empirical and experimental fact), some are risk-neutral, no one is generally risk-seeking. What this means is: They would prefer receiving a certain income Ī to receiving a random income I whose mean is Ī.
19. Why Contracts? 1. A person’s WTP to make this switch is called the risk premium with random income I. 2. The WTP depends on two things: (i) the riskiness of the income; and (ii) the degree of risk aversion of the person. The amount left after the risk premium is paid is called the certainty equivalent of the random income – the amount of income, payable for certain, that the person regards as equivalent in value to the original, random income.
20. Why Contracts? The certainty equivalent is estimated as follows: Ī – 1/2 · r(Ī) · Var(I), where Ī and Var(I) are the mean and variance of I, and r(Ī) is the individual’s Arrow-Pratt coefficient of absolute risk aversion (ARA) for gambles with mean Ī. Note: This makes 1/2 · r(Ī) · Var(I) the risk premium. Why?
21. Why Contracts? No risk or uncertainty means Var(I) = 0. What does the formula tell us then? Ī – 1/2 · r(Ī) · Var(I) = Ī, and so the certainty equivalent is the mean of the “gamble”. Note: Different values of coefficient of ARA. In other words, people have heterogeneous risk preferences.
22. Why Contracts? Risk Premia and Value Maximization In what we will do in this chapter, a contract will be efficientiff it maximizes the total certain equivalent wealth of all parties involved. Implicit assumptions: (1) No liquidity constraints; and (2) CARA.
23. Why Contracts? The CARA assumption means that r is constant in wealth. Is this realistic? Moreover, it means that Expected Income = Ī, Risk Premium = 1/2 · r · Var(I), and Certainty Equivalent = Ī – 1/2 · r · Var(I). We will use these formulas to calculate the benefits of insurance and the costs of the risk bearing required to provide incentives.
24. Why Contracts? Risk Sharing and Insurance When several people face statistically independent risk, then risk sharing can reduce the cost of risk bearing. Two risks are statistically independent if knowing the realized value of one risk gives you no information about the value that the other will achieve. In addition – and more formally – independence implies a correlation of zero. Does the converse hold true?
25. Why Contracts? Risk Sharing and Insurance Many institutions assist people in risk sharing. An important one is insurance companies: many policyholders face independent risks, which allows companies to reduce individual risk. The more policyholders and the greater the independence between idiosyncratic risks, the more effective the insurance scheme (e.g., auto insurance.)
26. Why Contracts? Ex.: Two individuals, A and B. Let IA and IB be their (random) incomes; ĪA and ĪB are the respective means; Var(IA) and Var(IB) are the respective variances; and rA and rB are their respective coefficients of risk aversion. Every efficient risk sharing contract maximizes the total certain equivalent income of all the parties, and every such contract is an efficient one.
27. Why Contracts? Without contracting, the total cost they incur (“social cost”) is the total risk premium: 1/2rAVar(IA) + 1/2rBVar(IB). They agree on a risk sharing contract in which A receives a fraction a (b) of income IA (IB), and B receives the remainder of the risk. Also, A receives a payment c from B to bear the risk (which could be negative… why?)
28. Why Contracts? Then, A receives aIA + bIB + c and B receives (1 – a)IA + (1 – b)IB – c The total income sums up to IA + IB.
29. Why Contracts? The total risk premium for both parties is then 1/2rAVar[aIA + bIB + c] + 1/2rBVar[(1 – a) IA + (1 – b)IB – c] Efficient arrangements are those that minimize this equation – or maximize social surplus. The total risk premium is minimized when a/(1 – a) = b/(1 – b) = rB/rA
30. Why Contracts? To help think through this, let 1/r be an individual’s degree of risk tolerance. In the preceding example, A’s share of each risk is equal to A’s share of the total risk tolerance. So if rA = 2 and rB = 4, then A’s share of each risk is equal to [0.5(0.5 + 0.4)] = 0.667. Thus, when risks are shared efficiently, the share of risk each party bears is equal to their share of the total risk tolerance of the group – a pretty intuitive result.
31. Why Contracts? Moreover, when risks are allocated efficiently, the total risk premium is 1/2Var[IA + IB]/[(1/rA) + (1/rB)] With efficient risk sharing, the group is less risk averse than the people comprising it, and so there is interest in pooling risks.
32. Why Contracts? When individual risks are independent, this implies that risk sharing can be a very effective way of reducing the cost of risk bearing. As the number of “players” n gets larger, even substantial financial losses (e.g., the loss of a house to a fire) can be reduced to economic insignificance by sharing it efficiently with a group.
33. Why Contracts? Intermezzo: Is Risk-Sharing Monotonic in the Wealth of Parties? Commonly made assumptions: IARA: dr/dz > 0 CARA: dr/dz = 0 DARA: dr/dz < 0 Where r is absolute risk aversion and z is wealth.
34. Why Contracts? In addition, as the principal gets more (absolutely) risk averse, she bears more of the risk, and as the agent gets more (absolutely) risk averse, he bears more of the risk as well.
35. Why Contracts? So logically, it makes sense that one would expect risk sharing to vary in obvious directions as the parties get wealthier. This is the intuition behind many empirical tests (e.g., Laffont and Matoussi, 1995; Dubois, 2002; Ackerberg and Botticini, 2002; Fukunaga and Huffman, 2009; etc.)
36. Why Contracts? The problem is that this intuition is completely false! (See Bellemare, M.F., and Z.S. Brown (2010), “On the (Mis)Use of Wealth as a Proxy for Risk Aversion,” American Journal of Agricultural Economics 92(1): 273-282.)
37. Why Contracts? Bellemare and Brown: Tests relying on wealth as a proxy for risk aversion are completely unidentified, even under the unrealistic assumptions of CARA and risk-neutrality of the principal. The moral of the story, then, is “Beware of sketchy deductive chains.”
38. Why Contracts? Optimal Risk Sharing Ignoring Incentives With risk neutrality, r = 0, and the risk premium is also zero. So logically, one would expect all the risk to be borne by the risk-neutral party. This misses the mark, however. Why?
39. Why Contracts? Incentive Pay This is known as the principal-agent problem. Here, the principal is the employer, and the agent is the employee. Performance-based pay entails a loss from inefficiency. The money value of this loss is equal to the risk premium associated with compensation minus the risk premium from efficient risk sharing. Firms that use incentive pay (hope to) recoup this loss through increased productivity.
40. Why Contracts? To analyze principal-agent relationships, let us introduce the cost of effort to the agent, denoted by C(e). This represents the disutility of effort to the agent. The firm derives a profit P(e) from agent effort. This relationship can be random or subjective.
41. Why Contracts? Pay can only vary according to something that the principal can observe. Typically, effort is unobservable, which leads to moral hazard.
42. Why Contracts? A Simple Model of Incentive Compensation Indicator of effort: z = e + x, where E(x) = 0. Linear contract: w = a + b(e + x +cy) Note: y is not affected by e but may be correlated with x; and E(y) = 0. Here, a is base pay, and b captures the incentive power of the contract; c captures how much relative weight is given to the information variable y. For any c > 0, z + cy provides an estimate of e.
43. Why Contracts? So an important part of incentive pay is how to determine what c is equal to. We have assumed a linear contract above. This is not always sensible (theoretical requirements are very restrictive), but it’s empirically defensible. How so?
44. Why Contracts? Total Wealth Under a Linear Contract Incentive problems aside, it would be optimal for firms to completely insure their employees. Obviously, this would entail paying only a base pay, which would provide the most scope for moral hazard. As a result, optimal contracts are balancing acts.
45. Why Contracts? We define a contract by (e, a, b, c), which specify the effort level the employer expects and the compensation the agent receives based on performance. The employee’s CE from this contract is his (expected) compensation minus his cost of effort minus any risk premium: a + b(e + x + y) – C(e) – 1/2rVar[a + b(e + x + y)]
46. Why Contracts? Above, x = E(x) and y = E(y). To simplify, we assume here that both expectations are zero. Using variance formulas in the appendix, we get CEA = a + be – C(e) – 1/2rb2Var(x + cy) CEB = P(e) – (a + be) What’s the implicit assumption on the principal’s preferences here? And why is it made?
47. Why Contracts? Note: - Employee’s CE consists of a plus a function of the other contract terms (b, c, e). - Employer’s CE consists of –a plus another function of these variables. - Contract transfers money from one party to the other, so we can raise one party’s CE and reduce the other by an equal amount (“no wealth effects” condition), so we can apply the value maximization principle.
48. Why Contracts? So any efficient contract must be such that it maximizes Total CE = P(e) – C(e) – 1/2rb2Var(x + cy) But we’re still not done: this merely specifies the objective function, i.e., what needs to be maximized.
49. Why Contracts? Principal-agent problems are constrained in two important ways: they have to acceptable to the agent (IR) and they have to provide good motivation to the agent (IC). Incentive compatibility (IC) constraint: b – C’(e) = 0 In other words, the agent must be a rational maximizer. An efficient contract must be an acceptable contract that maximizes total CE among all incentive compatible contracts.
50. Why Contracts? To ensure that the contract is incentive compatible, we usually proceed in two steps and solve the by backward induction: (i) Solve for e, which gives us the agent’s best-response function; and (ii) Maximize total CE conditional on the agent’s best-response function. What kind of equilibrium is this?
51. Why Contracts? Total CE does not depend on a. Putting aside feasibility, then, efficiency does not depend on a. As for c, the total CE is maximized when c is s.t. Var(x + cy) is as small as possible.
52. Why Contracts? Informativeness Principle: In designing contracts, total value is always increased by factoring into the determinant of pay any performance measure that, with appropriate weight, allows reducing the error with which the agent’s choices are estimated and by excluding performance measures that increase the error with which effort is estimated (e.g., purely random factors).
53. Why Contracts? In other words: anything that increases the signal-to-noise ratio is good. So a measure with low error variance is a better basis for performance than a measure with higher variance. So y should be used iff there is some value of c that makes Var(x + cy) smaller than Var(x). This is determined by minimizing the former w.r.t. c.
54. Why Contracts? To choose c, we solve mincVar(x + cy) ↔ mincVar(x) + c2Var(y) + 2cCov(x,y) FOC: 2cVar(y) + 2Cov(x,y) = 0 ↔ c* = – Cov(x,y)/Var(y) (If you have taken econometrics, does this last line remind you of anything?) So, in essence, if the covariance is zero, the correlation is zero, and so c* = 0.
55. Why Contracts? Relatedly, given c* = – Cov(x,y)/Var(y), if x and y are positively correlated, then c* should be negative, and if x and y are negatively correlated, then c* should be positive. Note that as the noise (denominator) increases, the weight put on the signal decreases.
56. Why Contracts? The next step is to determine how intense the incentives should be. So, we fix c at some generic level and let V = Var(x + cy). Incentive Intensity Principle: The optimal intensity of incentives depends on the incremental profit created by extra effort, precision of assessment, agent’s risk tolerance, and his responsiveness to incentives.
57. Why Contracts? The formula for optimal intensity (or the slope of the contract, or the incentive power, or the share of income, etc.) is: b* = P’(e)/[1 + rVC”(e)]
58. Why Contracts? Moral Hazard with Risk-Neutral Agents This section assumes that r = 0, i.e., the agent is risk-neutral. Intuitively, what should change? In other words, what should b equal to if the agent is risk-neutral?
59. Why Contracts? In that case, no risk premium is incurred, no matter how the risk is shared. So the agent can be perfectly motivated by setting b = 1, i.e., making him residual claimant on profit/output/income/etc. So this means the agent running the firm, i.e., a franchising contract.
60. Why Contracts? But it doesn’t always work that way. Several things can make the moral hazard problem more difficult to solve. 1. The agent can be liquidity-constrained (e.g., cannot pay the upfront costs to the principal.) 2. The risk is non-financial, and thus more difficult to transfer (e.g., one cannot eliminate the damage done by HIV-infected blood transfusions.)
61. Why Contracts? 3. Adverse Selection and Moral Hazard Ex.: Employee in a department store chain who invents a new product. Negotiates with firm to market the product. What should be the arrangement made to market the product? (Here, employee = principal, firm = agent.)
62. Why Contracts? So the firm can bear all of the risk, since it is risk-neutral. The problem, though, is that the firm may have better information about marketability of product.