1. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 1 INTRODUCTION MORAL HAZARD MORAL HAZARD ; When one party to a transaction may undertake certain actions that ; a- affect the other party’s valuation of the transaction but that ; b- the second party can not monitor or enforce perfectly. fire insurance use of incentives A classical example is fire insurance, where the insure may or may not exhibit sufficient care while storing flammable materials. The solution to a problem of moral hazard is the use of incentives. partial insurance This is structuring the transaction so that the party who undertakes the action will, in his own best interests, take actions the second party would prefer. Ex; partial insurance leasing a car hiring a person to do a hard work Another examples are ;leasing a car for a specific period and return it after that,or hiring a person to do a hard work at the tasks that are set for him perfect monitoring impossible. In each of these cases it is possible to monitor, but perfect monitoring and enforcement may be impossible. transaction might be restructured But the transaction might be restructured in such a way that the party taking the action has relatively greater incentives to act in away the second party prefers.

2. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 2INTRODUCTION party who is taking the hidden action bears fully the consequence of his action From the incentive point of view the transaction should be restructured in such a way that the party who is taking the hidden action bears fully the consequence of his action. For example ; 1 1-The insurance company may only insure only up to 90 percent. Or the company may refuse to insure the company. 2 2-The party who lease the car could share a part of the amount of sale if the car is sold after leasing. Or the company may sell the car instead of leasing it. 3 3-The worker’s wage may be tied to some observable measures of hardworking. For example piece rate paying system. inefficiencies But there are always inefficiencies in these kind of restructuring contracts. For example ; 1- 1- Selling the car may bear more tax than leasing it. 2- 2- A company with few individual share holders may be less willing to bear the risk of having a fire. 3- 3- The piece rate paying system may not be feasible. Quality as well as quantity may be important. Piece rate system may subject the worker to risks that he may be less well equipped.

3. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 3 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE principle principle ; the first party who hires a second party ; agent agent the second party which is performing some specific task for the principle. reservation level of utility ; agent is not willing to work hard reservation level of utility ; the utility which the agent could get from the next best opportunity. All other things being equal the agent is not willing to work hard. Whether the agent works hard or not determines the value to the principle of having this agents work. Reservation wage If the agent does work hard, then the principle will get enough out of the transaction to make it worthwhile for both sides Reservation wage ; a wage high enough so that combined with not working hard the agent’s net utility exceeds the reservation level of utility. If the agent does work hard, then the principle will get enough out of the transaction to make it worthwhile for both sides. U(w,a) = W 1/2 – a U(w,a) = W 1/2 – a = the agent’s overall utility from work. W = the amount which the agent receive. a = 5, if the agent works hard. a = 0, if the agent does not work hard. a = 0, if the agent does not work hard.

4. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 4 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE Reservation level of utility = U * = 9 If the agent works hard, the accomplished task worth $270 to the principle and if does not work hard it worth $70 to the principle. 1- If the agent does not work hard, ( U ≥ 9, a=0 ), then ; W 1/2 ≥ 9, or W ≥ $81, so cost to the principle ≥ 81, worth of work to principle = 70, there will be no deal. 2- if the agent works hard, (U ≥ 9, a = 5), then ; W 1/2 - 5 ≥ 9, → W ≥ $196, cost is $ 196, and worth of work is equal to $ 270. The deal is worth to do for the principal W 1/2 - 5 ≥ 9, → W ≥ $196, cost is $ 196, and worth of work is equal to $ 270. The deal is worth to do for the principal. The principle should try to make the agent to work hard. HOW ? how could he monitor his hard working ? The principle could pay an amount greater than 196 and less than 270 to induce him to work hard. But how could he monitor his hard working ? agent’s pay depend on how much effort he puts in. Another possibility is to offer a contract that calls for the agent’s pay depend on how much effort he puts in.

5. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 5 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE sign a contract to pay the agent any amount greater than $196 and less than 270 if he works hard and any amount less than $81 if he does not work hard. For example the principle could sign a contract to pay the agent any amount greater than $196 and less than 270 if he works hard and any amount less than $81 if he does not work hard. is this contract enforceable high transaction cost and high cost of monitoring, But, is this contract enforceable? For many reasons which could deal with high transaction cost and high cost of monitoring, this kind of contract might not be feasible, either for principle or for agent. indirect measure of effort to which wages can be tied. Even if the principle cannot tie the worker’s wage directly to his level of effort, the principle might be able to find some indirect measure of effort to which wages can be tied. For example suppose that : Agent is a salesman, who will be representing the principal to a particular client Agent is a salesman, who will be representing the principal to a particular client. two cases ; the agent ( the salesman) can or can not work hard; There are two cases ; the agent ( the salesman) can or can not work hard;

6. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 6 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE 1if the agents work hard at making the sale, then 1- if the agents work hard at making the sale, then ; a $400 sale result with 0.6 probability a $100 sale result with 0.3 probability a $0 sale result with 0.1 probability. 2 - if the agents does not work hard then ; a $400 sale result with 0.1 probability a $100 sale result with 0.3 probability a $0 sale result with 0.6 probability.. The principle is always risk neutralThe worth of the agent’s work ( $270 when working hard and $70 when does not working hard ) is the expected gross profit Unless the principle can observe the effort level of the agent directly, the data receive do not tell conclusively what level of effort was put in because data could be the outcome of case 1 or 2. ( $400 sale could be the result of working hard or working not hard) The size of the sale is observable and the agent’s wages can be made contingent upon this variable. The principle is always risk neutral. The worth of the agent’s work ( $270 when working hard and $70 when does not working hard ) is the expected gross profit. Unless the principle can observe the effort level of the agent directly, the data receive do not tell conclusively what level of effort was put in because data could be the outcome of case 1 or 2. ( $400 sale could be the result of working hard or working not hard) The probability distribution is the same for principle and agent

7. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 7 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE Case 1 – A risk neutral agent U(w,a)= w – a Utility function = U(w,a)= w – a U * = $81 Reservation level of utility =U * = $81 High effort level → a = 25 and low effort level → a=0 high work effort a=25, w ≥ (U * + a) → w ≥ 106 If high work effort could be guaranteed, then principal should pay an amount to agent which leave him a utility greater than the reservation level → a=25, w ≥ (U * + a) → w ≥ 106 then ; Principal profit = 270 – 106 = 164 at most. hard work can not be guaranteed a=0, w ≥ (U * + a)=81 If the hard work can not be guaranteed and the principal has to pay the agent a wage which leaves him a utility at least greater than the reservation level → a=0, w ≥ (U * + a)=81 then ; Principal’s profit = 70 – 81 = - 11, is there any other solution ? Principal’s profit = 70 – 81 = - 11, is there any other solution ? The difficulty and cost of monitoring the high level of effort is present The principal could offer the agent the following contract ;

8. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 8 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE 1 - 1 - If the agent makes no sale he should pay $164 to principle 2 - 2 - If the agent makes small sale (worth $100) he should pay $ 64 (164 – 100) to principle. 3 - 3 - If the agent makes large sale (worth $400) the principle pay agent $236 = (400 – 164) $164 worth of profit (,maximum profit when agent works hard) should be guaranteed for the principal any way $164 worth of profit (,maximum profit when agent works hard) should be guaranteed for the principal any way. The agent could do one of the followings ; A- A- turn down the contract and get reservation level equal to $81. B- B- take the contract and put in low effort; his net expected utility is; (0.1)(236) + (0.3)(-64) + (0.6)(-164) -0 = - 94 C- C- take the contract and work hard ; his net expected utility is ; (0.6)(236) + (0.3)(-64) + (0.1)(-164) -25 = 81 The agent is indifferent between option (a) and (c), because the reservation level of utility is $81. If principal could make option (c) a little more attractive option (c) will be chosen. What the principal has to do is to get the agent to internalize the effort of his decision.

9. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 9 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE Case 2 – A risk averse agent Agent’s utility =U = W 1/2 – aReservation level = U * =9 Agent’s utility = U = W 1/2 – a. Reservation level = U * =9 Working hard → a=5, not working hard → a= 0 contract based upon a effort level If we could write a contract based upon a effort level, If the agent works hard, the accomplished task worth $270 to the principle and if does not work hard it worth $70 to the principle the best one is ; A- the agent gets $196 or a little bit more, if he works hard A- the agent gets $196 or a little bit more, if he works hard, since if the agent works hard, (U ≥ 9, a = 5), then ; W 1/2 - 5 ≥ 9, → W ≥ $196, cost is $ 196, and worth of is equal to $ 270 for principle then principle’s profit will be$270 – $196 = $74 W 1/2 - 5 ≥ 9, → W ≥ $196, cost is $ 196, and worth of is equal to $ 270 for principle then principle’s profit will be $270 – $196 = $74 B- the agents gets almost zerto if he does not work hard The profit is zero B- the agents gets almost zerto if he does not work hard.there will be no deal The profit is zero. expected profit in this contract ( taking both A and B into account ) is equal to $74 So the expected profit in this contract ( taking both A and B into account ) is equal to $74. The cost of monitoring is still present contractbased upon the size of the sale. But the principal can make a contract (pay agent’s wage) based upon the size of the sale. When the agent was risk neutral it was possible But in this case it is not possible. When the agent was risk neutral it was possible to make the principal as well off as if principal could write a contract contingent on the actual effort level. But in this case it is not possible.

10. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 10 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE most efficient arrangement is the one in which the agent’s wage is certain. Because ;when one party to a transaction is risk averse and the other is risk neutral, then it is efficient for the risk neutral party to bear all the risks in this case when the principal is risk neutral and the agent is risk averse. the most efficient arrangement is the one in which the agent’s wage is certain. Because ; when one party to a transaction is risk averse and the other is risk neutral, then it is efficient for the risk neutral party to bear all the risks.The intuitive reasoning is as follows ; If the principle pays the agent a random wage (different wages with different probabilities ), then the agent evaluates the wage according to his expected utility. Being risk averse,if the wage is at all risky the agent values it at less than its expected value ( because of the concave shape of the utility function ). But being risk neutral, the principal values the cost of the wages paid at their expected value (utility function is a straight line ). agent’s wage had expected value equal to W * ; If we imagine that the agent’s wage had expected value equal to W * ; The principal see this as an outflow from his pocket equivalent to W * The agent see this as an inflow to his pocket of something less than W *, as long as there is any risk at all The agent see this as an inflow to his pocket of something less than W *, as long as there is any risk at all. On the other hand if the wage was riskless the agent has no incentive to work hard, and in this case the principal does not want to enter the transaction. What should be done ?

11. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 11 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE To answer this suppose that the agent is paid as follows ; AX o 2 no sale A- the agent is paid X o 2 if the no sale is obtained (prob. = 0.1) BX 1 2 small sale B- the agent is paid X 1 2 if the small sale is made (prob. = 0.3) C X 2 2 large sale C- the agent is paid X 2 2 if the large sale is made (prob. = 0.6) Offered this contract the agent has three choices ; 1- refusereservationU * = 9 1- refuse the contract and get reservation level equal to U * = 9. 2-take the contracthigh effort 2- take the contract and put it in high effort and get an expected utility equal to ; E(U) =Σ Π i (U i ) = Σ Π i (w i 1/2 – 5 )= 0.1(x o -5) + 0.3(x 1 -5) + 0.6(x 2 -5) = 0.6x 2 + 0.3x 1 + 0.1x 0 -5 0.6x 2 + 0.3x 1 + 0.1x 0 -5 3take the contractlow effort 3- take the contract and put it in low effort ; then ; E(U)= 0.6x 0 + 0.3x 1 + 0.1x 2 The best possible contract from the principle point of view is the one in which the agent take the contract and put in a high level of effort and optimum wage level for the principal is the one in which the expected wage was minimum subject to providing a utility level for the agent more than the reservation level U * = 9 ;

12. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 12 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE Min EXP (W) = 0.1x o 2 + 0.3x 1 2 + 0.6x 2 2 (for high effort) Sub to 0.1x o + 0.3x 1 + 0.6x 2 - 5 ≥ 9 0.1x o + 0.3x 1 + 0.6x 2 - 5 ≥ 0.6x o + 0.3x 1 + 0.1x 2 0.1x o + 0.3x 1 + 0.6x 2 - 5 ≥ 0.6x o + 0.3x 1 + 0.1x 2 the solution for x o, x 1, x 2 should be such that the expected utility obtained be greater than reservation level (first constraint ), and expected utility for high level of effort (hard working ) be greater than the low level of effort (second constraint ). The solution is as follows ; x 0 2 =$29.46, if no sale is made. x 1 2 = 196, if $100 sale is made. x 1 2 = 196, if $100 sale is made. x 2 2 = 238.04, if $400 sale is made x 2 2 = 238.04, if $400 sale is made. The expected wage bill is 0.1(29.46) + 0.3(196) + 0.6(238.04) = 204.56 Principal expected profit = 270 – 204.56 = 65.44 x 2 2

13. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 13 EFFICIENT INCENTIVES ; EXAMPLE EFFICIENT INCENTIVES ; EXAMPLE comparing this with the contract that the principle gives the agent a wage of $196 and relies on trust (or the compulsion of monitoring scheme) to ensure that the agent puts in a high level of effort reveals that, To give the agent right incentive we had to have him bear some of the risk by rewarding him in case of the outcome that is more likely if he puts in greater effort. This cost the principal an expected amount of $8.56. (204.56 – 196 ).

14. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 14 Finitely many action and outcomes Finitely many action and outcomes agentaction α A = {α 1, α 2, …. α n ). Suppose an agent who chooses an action α from the set of A = {α 1, α 2, …. α n ). principal imperfect) signal s (sale or profit for example) S = {s 1, s 2, s 3 ….. S M } The action is not observed by principal ; instead the principal sees an(imperfect) signal s (sale or profit for example) from S = {s 1, s 2, s 3 ….. S M } Π nm = the probability that signal s m is produced, if the agent chooses action a n Π nm = the probability that signal s m is produced, if the agent chooses action a n. Where for each n. Assumption 1. The probability Π nm >0 for all n and m. Every outcome is possible. agent utility = U(w, a ) = u(w) – d(a ) gent is a risk averse agent agent utility = U(w, a ) = u(w) – d(a ). Where w is wage and a is the action taken by the agent. Agent is a risk averse agent. B(a) = gross benefit to the principle of hiring the agent if action a is chosen by the agent. Assumption 2 ; The function U is strictly increasing, continuously differentiable, and concave. Principle net benefit = B(a) – exp(wage paid).

15. Finitely many action and outcomes Finitely many action and outcomes Solving the basic problem ; Step 1 ; for each a n € A, what is the minimum expected wage that must be paid to induce the agent to take the job and choose the action a n Step 1 ; for each a n € A, what is the minimum expected wage that must be paid to induce the agent to take the job and choose the action a n. signals m w(s m ) define x m If the signal chosen by agent is s m, and if w(s m ) is the wage paid to the agent, we could define x m as following for m=1,,,,M. x m = u{w( s m )} = value of utility as a function of wage paid when signal s m is choosen x m = u{w( s m )} = value of utility as a function of wage paid when signal s m is choosen. x m = decision variable which is the level of wage utility ( utility of wage ) paid to agent as a function of signal s then w(s m ) = v(x m ).= wage paid as a function of utility obtained when signal s m is chosen. And if v be the inverse of u, then w(s m ) = v(x m ).= wage paid as a function of utility obtained when signal s m is chosen..Expected wage which the principle must pay in terms of utility if the agent takes action a n : D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 15

16. Finitely many action and outcomes Finitely many action and outcomes 16

17. Finitely many action and outcomes Finitely many action and outcomes C(a n )= value of the expected wage at the optimal solution = the minimum expected cost of inducing the agent to select action a n. Step 2 ; we should search for the action (a i ) which maximize net benefit of the principle,B(a) - C(a n ). So, first we find the minimum cost way to induce action a for each a and then we choose the optimal a by comparing the benefits and costs For each action (a i, i=1,..n ) taken by the agent we could find C(a n ) and the principle will have an amount of benefit equal to B(a n ). Step 2 ; we should search for the action (a i ) which maximize net benefit of the principle,B(a) - C(a n ). So, first we find the minimum cost way to induce action a for each a and then we choose the optimal a by comparing the benefits and costs BASIC RESULTS AND ANALYSIS The agent is risk averse and the principal is risk neutral. So the cheapest way of guaranteeing the agent a given level of utility is with a fixed payment. So the cheapest way (C 0 (a n )) to convince the agent to accept the action a n is to offer him : It is evident that for all n. Since this way is the cheapest way. 17

18. Finitely many action and outcomes Finitely many action and outcomes We should then maximize for all n. The effort level that solves this is called the first best level of effort. Preposition 1 If the agent is risk averse, then the unique efficient risk-sharing arrangement is for the principal to bear all the risk ; the agent gets a sure wage. But if the agent is given a wage that is independent of the signal ( C 0 (a n ) ), he will choose the least onerous ( hardest) action. Because as it is mentioned the principal will choose to give the agent cheapest wage which is equal to C 0 (a n ) the reservation level. Preposition 2 For a risk neutral agent, max a B(a) – C 0 (a) = max a [B(a) – C(a)]. In fact, if a * achieves the maximum in, max a B(a) – C 0 (a), one scheme that imements this action is for the principal to pay the agent s m -B(a * )+ C(a * ) if the gross profit s are s m, so the principle receives [s m - {s m - B(a * ) + C(a * ) }= B(a * ) - C(a * ) ] for sure and the agent bears all the risk ( since for each a there are m signals and it is not clear that which one will be observed). 18

19. Finitely many action and outcomes Finitely many action and outcomes It is as the principle “sold the venture “ to the agent, who is now propitiator and is working for himself, and who now chooses the optimal action in his own sole and best interest. Now suppose that the actions are levels of effort that the agent might select ordered in terms of increasing utility according to the utility function U(w, a ) = u(w) – d(a) or ; U(w, a ) = u(w) – a d(a 1 ) < d(a 2 ) < ….. < d(a N ) or a 1 < a 2 < ….. < a N we also assume that higher level of efforts result in higher gross expected profit for the principle. To write this assumption we should have ; s 1 < s 2 < …. < s M. To assume that higher level of effort leads to higher level of profit we have to assume that is increasing in n. But some thing should Increase in effort results in higher probability of higher levels of profit be assumed that is Increase in effort results in higher probability of higher levels of profit. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 19

20. Finitely many action and outcomes Finitely many action and outcomes ‘ MORAL HAZARD AND INCENTIVES 20

21. Finitely many action and outcomes Finitely many action and outcomes Choosing effort lelvels a 1 : S 1 with, and S 2 with and S 3 with a 2 : S 1 “ “= 0.4 “ S 2 “ 0.1 S 3 “ 0.5 By assumption 3 higher effort level leads to higher probabilities better outcomes. Exp( gross profit for a 1 ) = 2001.5 Exp ( gross profit for a 2 ) = 5000.6,so he chooses a 2 by Ass. 3. The first order condition for the expected wage utility minimization (page 16) will be as follows ; v ’ (x 1 ) = λ + η(1- (0.5/0.4)) = λ – 0.25 η v ’ (x 2 ) = λ + η(1- (0.3/0.1)) = λ – 2η v ’ (x 3 ) = λ + η(1- (0.2/0.5)) = λ + 0.6 η D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 21

22. Finitely many action and outcomes Finitely many action and outcomes Where x 1 is the wage utility corresponding to s 1 = $1 “ x 2 “ s 2 = $2 ‘ x 3 “ s 3 = $10000 λ is the multiplier on the participation constraint η is the multiplier on the relative incentive constraint. if η>0 → v ’ (x 1 ) > v ’ (x 2 ), → x 1 > x 2 → w(x 1 )>w(x 2 ) which is not. In order for this not to happen the ratio of probabilities in brackets should change in right way as stated in Ass. 4 ; Assumption 4 : the monotonic-likelihood ratio property. For any two effort level and such that, and for any two gross profit level, the relative likelihood of the better outcome under the higher effort level to the lower is at least as large as this likelihood ratio for the lower outcome. Or ; D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 22

23. Finitely many action and outcomes Finitely many action and outcomes Is assumption 4 enough to get an affirmative answer to the question ? We will see that it is not. To see why, we should take into account the first order condition ; As it is seen as long as n ’ m and V’ (x m ) n we have while m ’ > m, then the above relation may not hold. What is needed in addition to assumption 4 is an assumption that the only binding relative incentive constraint for the optimal effort level a n are constraints corresponding to levels of effort lower than a n. Assumption 5 : if are effort levels such that for each m=1,2,3….M then : 23

24. Finitely many action and outcomes Finitely many action and outcomes. Let ; This is called the concavity of distribution function condition. Let ;. = probability of seeing a gross profit level at least as large as s m if effort level a n is taken. The assumption says that increase in the effort level ( measured by disutility) have decreasing marginal impact ( concave shape) on the probabilities of better outcome (a n” is the lowest level of effort) If, then we could see that, Increase in disutility in moving from the lowest level of effort to the intermediate level is the same as increase in moving from to. And according to the assumption it should be; 24

25. Finitely many action and outcomes Finitely many action and outcomes Preposition 3 : if assumptions 1,..5 all hold and U is strictly concave, then the optimal wage incentive scheme for the principal has wages that are non decreasing functions of the level of gross profit Preposition 3 : if assumptions 1,..5 all hold and U is strictly concave, then the optimal wage incentive scheme for the principal has wages that are non decreasing functions of the level of gross profit. Proof is not needed. THE CASE OF TWO OUTCOME An intuitive case might be made that the principal would always choose to implement a level of effort less than or equal to the first-best level, since risk-sharing between the agent and the principal entails shielding the risk averse agent from some of the consequences of his effort in terms of profits, while he bears fully its utility. Suppose gross profits can take only two forms ; s 1 and s 2 > s 1. Agent’s effort results in either failure or success of some venture. Checking for the assumptions ; Ass 1 ; both success and failure outcomes are possible. Ass 2 ; utility function should be strictly increasing and continuously differentiable and concave. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 25

26. Finitely many action and outcomes Finitely many action and outcomes Ass 3 ; higher levels of efforts leads to increased chances of success. Ass 4 ; is not needed, when Ass.3 holds. Ass 5 ; The probability of success should be a concave function of the effort level. In this special case with Ass. 3 Ass. 5 is not needed. With Ass. 3 alone we can formulate the problem as follows ; The agent has a base wage denoted by b 1 to which added a bonus of b 2 ≥ 0 in the event of success. If we write The agent wage with success outcome is should be positive if principal wants to implement any efforts level above minimum ( s 1 ). Preposition 4 ; if U is strictly concave, then at the optimal wage- incentive schedule. Otherwise the principle prefers the outcome to be failure which can not be a scheme. Proof is not needed. D.M. KREPS CH 16 MORAL HAZARD AND INCENTIVES 26

27. Finitely many action and outcomes Finitely many action and outcomes An example which shows how the first best solution may not work. Two outcome and two possible effort level ; a 1 = 0 with P F = 0.9 and P S = 0.1 and slightly different effort level, a 2 = 0.1 with P F = 0.85 and P S = 0.15 U( w,a ) = ln ( w) – a, with u 0 = 0, and Success worth $10 and Failure worth $0 to the principal. 1- To implement the effort level a 1, the principle expected profit ; Exp(profit) = (0.9)($0) + ( 0.1)($10) = $1, and wage cost is ; ln(w)=0, w = e 0 =1 and net profit = 1-1 = 0 2- To implement the effort level a 2 = 0.1, the principle should pay a wage equal to ln(w)= 0.1, w= e 0.1 = $1.105, regardless of action and his exp(profit) = 0.85($0) + (0.15)($10) = $1.5, when a 2 =0.1, and his gross expected profit = 1.5 – 1.105 = $0.395 hence the first best effort level is a 2. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 27

28. Finitely many action and outcomes Finitely many action and outcomes Working out the optimal wage incentive program ( effort level a 2 ) and solving the first order equations we will get w 1 = 0.8187 if outcome is failure and w 2 = 6.04965 if outcome is success. Now exp(wage )= (0.8187)(0.85) + (6.04965)(0.15)=$1.6034 effort level a 2 is chosen so expected profit = $1.5 net expected profit = $1.5 - $1.6034 = - $0.1034. So the principle has to choose the other option ( effort level a 1 ) which is not optimal, that is paying the fixed wage of $1. So as it is seen the principle choose an effort level less than the first-best one. CONTINOUS ACTION : First order approach the agent can choose any effort level drawn from some interval {a 0,a 1 ] and that the range of possible gross profit levels is R or some appropriate subinterval. as an example we use the model of two possible outcome ; success and failure and action choice that bears the interpretation of effort. D.M. KREPS CH 16MORAL HAZARD AND INCENTIVES 28

29. CONTINOUS ACTION Effort is chosen from an interval [ a 0, a 1 ] where effort is measured in terms of disutility. U(w,a) = u(w) – a. If effort level a is chosen, then the probability of a success outcome is π(a), where π(.) is a strictly increasing function. If we want to discover the minimal cost of implementing a *, we should solve Min (1- π(a * )) v(x f ) + π(a * )v(x s ) sub (1- π(a * )) x f + π(a * ) x s - a * ≥ u 0, (1- π(a * )) x f + π(a * ) x s - a * ≥ (1- π(a)) x f + π(a) x s - a for all a € [ a 0, a 1 ]. The last constraint contains of infinite number of constraints. In order to make it operational the last constraint could be written as follows ; U(a) = (1- π(a)) x f + π(a) x s - a, should be maximized at a = a *, or π’(a * ) [x s - x f ] =1. Then substitute this for the last constraint ; Min (1- π(a * )) v(x f ) + π(a * )v(x s ) sub (1- π(a * )) x f + π(a * ) x s - a * ≥ u 0 π’(a * ) [x s - x f ] =1, two equations in two unknowns could be solved if we assume that both constraints are binding MORAL HAZARD AND INCENTIVES 29