ECO421: Moral hazard. Marcin P ski. March 26, 2018

Transcription

1 ECO421: Moral hazard Marcin P ski March 26, 2018

2 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

3 Introduction Moral hazard Moral hazard: two agents with misaligned interests, one of them chooses an action that matters for both of them, the other can oer a contract to change the incentives of the rst agent.

4 Introduction Moral hazard: Insurance Circumstance that increases the probability of occurrence of a loss, or a larger than normal loss, because of a change in an insurance policy applicant's behavior after the issuance of policy.

5 Introduction Moral hazard: Financial markets Moral hazard is a situation in which one party gets involved in a risky event knowing that it is protected against the risk and the other party will incur the cost. It arises when both the parties have incomplete information about each other. In a nancial market, there is a risk that the borrower might engage in activities that are undesirable from the lender's point of view because they make him less likely to pay back a loan. more specic applications of the general principle from the previous slide

6 Introduction Moral hazard: Job Two players: Ann: principal, boss Bob: agent, worker the principal wants the agent to take certain action the interests of the principal and the agent are misaligned so, the agent may not necessarily take the action that is optimal for the principal. To align incentives, the principal and the agent may agree on a binding contract. the contract has to provide the agent's incentives, and it must be preferrable to no contract.

7 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

8 How to pay a worker Example A principal hires a worker. The principal designs contract w (.) The worker decides whether to accept the contract. If not, the worker receives outside option U 0. The principal recives 0. If yes, the worker chooses eort level e, receives wage w (e), and pays the cost of eort c (e). The eort is observable. The principal's payo is equal to u (e) w (e), where u (e) are the prots from the worker with eort e, and w (e) is the paid out wage.

9 How to pay a worker Fully observable eort We assume that u > 0 : principal likes the eort, c < 0: agent does not. Outside option U 0 0, alternative employment, cost of time, etc. Any contract w (.) chosen by the principal induces the agent's best eort e, it is convenient to consider e as chosen by the principal as well and add IC ocnstraints to the problem.

10 How to pay a worker Fully observable eort Principal's problem: max w(.),e u (e ) w (e ) st. choose contract w (e) and eort level e so to maximize principal's payos, st.

11 How to pay a worker Fully observable eort Principal's problem: max u (e) w (e) st. w(.),e IC:w (e ) c (e ) w (e) c (e) for each e, choose contract w (e) and eort level e so to maximize principal's payos, st. the agent has incentives to choose e, and

12 How to pay a worker Fully observable eort Principal's problem: max u (e) w (e) st. w(.),e IC:w (e) c (e) w ( e ) c ( e ) for each e, IR:w (e) c (e) U 0 choose contract w (e) and eort level e so to maximize principal's payos, st. the agent has incentives to choose e, and the agent wants to accept the contract.

13 How to pay a worker Fully observable eort Two questions: What is the optimal eort e, i.e, the eort that maximizes the principal's prots? How to nd the cheapeast way to provide the agent incentives to do e?

14 How to pay a worker Fully observable eort Dene the social surplus as Π (e) = (u (e) w (e)) }{{} principal's payo = u (e) c (e) U 0 + (w (e) c (e) U 0 ) }{{} agent's payo agent's payo takes into account the opportunity costs U 0. It the contract {w (.), e } is IR, it must be that u (e ) w (e ) u (e ) w (e ) + (w (e ) c (e ) U 0 ). = u (e ) c (e ) U 0 =: Π (e). the expression in the [...] brackets is positive because of IR.

15 How to pay a worker Fully observable eort Let e be the socially optimal (rst-best) choice of eort: e max u (e) c (e) U 0. e Π (e ) is an upper bound on the principal's prots. The guess is if e maximizes the upper bound on prots, maybe it also maximizes the proftis. Indeed, we will show that it is the case and the upper bound prots can be achieved:

16 How to pay a worker Fully observable eort Take IR holds: w (e) = { U 0 + c (e ), if e = e, 0, otherwise. w (e ) c (e ) = U 0 + c (e ) c (e ) = U 0 U 0, IC holds: for each e e FB, w (e) c (e) = c (e) 0 U 0 = w (e ) c (e ). Finally, the principal's prots are equal to the upper bound: u (e ) w (e ) = u (e ) c (e ) U 0 = Π (e ) max Π (e). e We found an optimal contract! Optimal contract: general procedure: Step 1: Find the cheapest way of making sure that the agent chooses eort e. Step 2: Find (principal)-optimal level of eort.

17 How to pay a worker Fully observable eort Step 1: There are many other contracts that achieve the same payo: forcing contract: { U 0 + c (e ), if e = e, w (e) = 0, otherwise, threshold contract: w (e) = { U 0 + c (e ), if e e, 0, otherwise. linear contract (assuming that c is convex, c > 0): w (e) = U 0 + c (e ) (e e ). Check FOC of the agent problem to verify the IC constraint. Piece rate

18 Education Separating equilibrium w linear contract U 0 w c(e) = U 0 threshold contract forcing contract e e

19 How to pay a worker Fully observable eort Step 2: as long as the cost function is strictly convex and utility is strictly concave, the optimal level of eort is unique

20 How to pay a worker Fully Unobservable eort Next, suppose the eort is completely unobservable. Contract w cannot depend on wage = at wage! The agent will choose e 0 arg max w c (e). If c > 0, e 0 = 0. IR is satised if w U 0 + c (0). There is no need to worry about IC. Why?

21 How to pay a worker Fully Unobservable eort If u (0) > U 0 + c (0), then the Principal oers wage w 0 = U 0 + c (0) if u (0) > w 0. Otherwise, no contract.

22 How to pay a worker Summary If the eort is observable, the rst-best eort (i.e., the eort that maximizes social surplus) is a solution to the principal's problem. If the eort is completely not observable, only at wage possible, and 0 eort. Next, partially observable eort.

23 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

24 How to pay an inventor Partially observable eort Example Two agents: principal and agent (inventor). Inventor may achieve breakthrough. The probability of the breakthrough q (e) depends on the inventor's eort. The eort is costly c (e), and not observable by the principal. The principal observes whether there is a breakthrough i = 0, 1, receives payo u i and pays out wages w i that depend on the existence of breakthrough i = 0, 1. We assume u 1 > u 0. Also, q > 0, q 0 and c, c > 0. Special case c (e) = 1 2 e 2, q (e) = e.

25 How to pay an inventor Partially observable eort Example Contract w 0, w 1. Interpretation: w 0 is the salary, w = w 1 w 0 is a breatkthough bonus. Question: how to pay the inventor? salary vs bonus?

26 How to pay an inventor Inventor's (agent's) problem Agent's problem: max q (e) w 1 + (1 q (e)) w 0 c (e). e FOC: q (e) (w 1 w 0 ) c (e) = 0. Let w = w 1 w 0 denote breakthrough bonus. Then, q w = c. let e A ( w) be the solution to the above equation, optimal eort depends only on the bonus.

27 How to pay an inventor Principal's problem Principal's problem: max w 0,w 1,e q (e) (u 1 w 1 ) + (1 q (e)) (u 0 w 0 ) st.

28 How to pay an inventor Principal's problem Principal's problem: max w 0,w 1,e q (e) (u 1 w 1 ) + (1 q (e)) (u 0 w 0 ) st. IR :q (e) w 1 + (1 q (e)) w 0 c (e) U 0,

29 How to pay an inventor Principal's problem Principal's problem:also, q > 0 and c, c > 0. max w 0,w 1,e q (e) (u 1 w 1 ) + (1 q (e)) (u 0 w 0 ) st. IR :q (e) w 1 + (1 q (e)) w 0 c (e) U 0, IC :e arg max e q ( e ) w 1 + ( 1 q ( e )) w 0 c ( e ).

30 How to pay an inventor Principal's problem Using the agent's solution, we can restate the principal's problem as max w 0, w q (e A ( w)) (u 1 w 1 ) + (1 q (e A ( w))) (u 0 w 0 ) st. IR :q (e A ( w)) w 1 + (1 q (e A ( w))) w 0 c (e A ( w)) U 0, IC condition disappears because we use the solution to the agent's problem, similarly, e disappears from the optimization.

31 How to pay an inventor Principal's problem Let's take u = u 1 u 0. Using the fact that qa + (1 q) B = q (A B) + B, we can restate the principal's problem further as max w 0, w q (e A ( w)) ( u w) + (u 0 w 0 ) st. IR :q (e A ( w)) w + w 0 c (e A ( w)) U 0.

32 How to pay an inventor Principal's problem max w q (e A ( w)) ( u w) + (u 0 w 0 ) st. IR :q (e A ( w)) w + w 0 c (e A ( w)) U 0. Because lowering w 0 increases principal's payo (and it does not aect the optimal eort), we can assume that IR binds, or: q (e A ( w)) w + w 0 c (e A ( w)) = U 0, or U 0 + c (e A ( w)) q (e A ( w)) w = w 0

33 How to pay an inventor Principal's problem When we substtitue w 0 into the principal's objective, we obtain max w q (e A ( w)) ( u w) + u 0 + q (e A ( w)) w c (e A ( w = max w q (e A ( w)) u + u 0 c (e A ( w)) U 0. and principal's problem has no more constraints.

34 How to pay an inventor Optimal contract max w q (e A ( w)) u + u 0 c (e A ( w)) U 0 FOC: q e A u c e A = 0. We can divide by the derivative e A, and recall that 1 = (q w) /c, to get which implies that Optimal contract! u w = 1, w = u, and e = e A ( u).

35 How to pay an inventor Optimal contract Summary: How did we solve the principal's problem. Step 1: Solve the agent's problem e A (.) substite the solution into the principal's problem and IR constraints, takes care of the IC. Step 2: show that the IR constraint is binding substitute w 0 into the principal's problem, takes care of the IR constraint. Solve unconstrained problem.

36 How to pay an inventor Optimal contract Let's look at the optimal contract more carefully: e = e A ( u). w 0 = U 0 + c (e ) q (e ) u, w 1 = w 0 + u. The decision is optimal and it maximizes the social welfare: W (e) =q (e) u 1 + (1 q (e)) u 0 c (e) =q (e) u + u 0 c (e). Indeed, e A ( u) = arg max W (e).

37 How to pay an inventor Optimal contract Worker's utility is U 0. Principal's utility: if there is no breakthrough:u 0 w 0,if there is a breakthrough u 1 w 1 = u 1 w 0 (u 1 u 0 ) = u 0 w 0. The same! Also, notice that u 0 w 0 = u 1 w 1 = q (e ) u + u 0 c (e ) U 0 = W (e ) U 0. Hence, principal gets the value of the social welfare minus the inventor's outside option.

38 How to pay an inventor Optimal contract The contract is equivalent to selling the rm: principal sells the rm to the inventor for W (e ) U 0. the worker receives payo u i depending on the breakthrough, chooses socially optimal action, Why selling the rm solves the moral hazard problem! agent (inventor) works for himself, no incentive problem, Magical solution.

39 How to pay an inventor Optimal contract Can we always sell the rm to solve the moral hazard problem? Unfortunately no. Agent is not always able to realize the same benet from the breakthrough as the principal (not able to market the invention, etc) maybe both agents and principal are necessary to generate the payos which one of the should be the principal? Agent does not always have enough money to buy the rm cash- or credit-constraints, in our problem, i may mean that w 0 0, or that the worker cannot have negative wages, added constraint.

40 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

41 How to pay CEO Alternative to selling the rm is to allign the agent incentives with the principal by transferring ownership of the share of the rm. stocks, high-powered incentives For instance, the ownership of x% of the rm. Is x% enough? It depends on the CEO incentives (eort, empire building, fame) vs the size of the rm If the value of the rm is really large to swamp anything else that the CEO cares about (power, cost of eort, etc), yes.

42 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

43 How to give grades Grading is dicult. In the rst-best world, the teacher could trust that students would learn for the love of learning, for the usefulness of the knowledge, and it would choose the eort level rationally. We are not in the rst best world. The teacher uses grading policy to design incentives for students to put eort. The problem is that grades mix eort and ability. Simple model

44 How to give grades Example A student chooses eort level a {a l, a h }. The high eort costs c > 0. The eort is unobservable. The eort a leads to an exam score q (θ) = aθ, where θ is the student's ability. Grades are observable. { Teacher designs grading policy c (q q 0, if q < q, ) = 1, if q q. Student receives utilityu > c if she passes the course. Teacher wants to maximize the number of students who choose high eort. We consider a various combinations: both student and teacher observe θ, neither teacher nor student observes θ, only the student observes θ (mix of adverse selection and moral hazard).

45 How to give grades Example Version I: Ability θ is observed by both student and teacher.

46 How to give grades Ability is jointly observed If θ is jointly observed, the grading policy may depend on ability q (θ). By taking q (θ) = a H θ, we ensure that every student will choose high eort. recall that u > c.

47 How to give grades Example Version II. Ability is not observed by anybody.

48 How to give grades Ability is not observed by anybody Suppose that neither teacher nor student observe θ. θ F (.), where F (.)is the c.d.f. The probability that the student passes with grading policy q is equal to ( ) q 1 F. a

49 How to give grades Ability is not observed by anybody Student will choose high eort if ( ( )) ( ( )) q q 1 F u c > 1 F u, a h a l or F ) ) (q 1al F (q 1ah > c u. (1) notice that c u (0, 1) because c < u, maybe everybody, maybe nobody chooses high eort. Any q such that (1) holds is optimal, it depends on the distribution F whether such q exists.

50 How to give grades Example Version III. Ability is observed by student, but not the teacher.

51 How to give grades Ability is observed by student, but not by the teacher The student will choose high eort if it will allow him to pass, and low eort will lead to the failure. a l a h a l q a h q a l θ no way to pass pass only pass with with high eort low eort

52 How to give grades Ability is observed by student, but not by the teacher Grading policy q implies that fraction ( ) ( ) 1 1 F q F q a l a h of students puts an eort. For any grading policy, there are abilities who can never pass, even with high eort, and who can pass even with low eort, Any such abilities will choose low eort. In general, there is no way to ensure that all students put an eort.

53 How to give grades Ability is observed by student, but not by the teacher Special case: uniform distribution onθ [0, 1]. In such a case, ( ) ( ) 1 1 F q F q a l a h 0, q < 0 ( ) 1 a = l 1 a h q, q [0, a L ] 1 1 a h q q [a l, a h ], 0, q > a h.

54 How to give grades Ability is observed by student, but not by the teacher Optimal choice q opt = a L. All the students who pass the test choose the high eort more precisely, all but θ = 1 who is the only ability type who can pass with low eort. This property depend son the distribution. If the ability density is decreasing, it might be optimal to choose low threshold that motivates lower ability student, but

55 How to give grades Ability is observed by student, but not by the teacher Special case: consider density f (x) = c c 2 2 x for x [ 0, 2 ]. c Let γ = a l a h < 1. Find the optimal grading policy as a function of γ and c.

56 How to give grades Ability is observed by student, but not by the teacher Optimal choice q opt = a L. All the students who pass the test choose the high eort more precisely, all but θ = 1 who is the only ability type who can pass with low eort. This property depend son the distribution.

57 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

58 How to pay a VP Next, a small variation on the above problem. Example A VP in a large company responsible for a part of the market chooses eort level a {a l, a h }. The high eort costs c > 0. The eort is unobservable by the CEO. The eort a leads to the output equal to q (θ) = aθ, where θ is the market condition. θ is observed by the VP, but not by the CEO. The CEO has a budget u that can be used to motivate the VP.

59 How to pay a VP The previous discussion suggests that there is no way to design the wages as functions of q to ensure that the department chief always chooses high eort. For very bad or very good market conditions, the low eort is optimal. But, sometimes we can do better.

60 How to pay a VP Example There are two departments. Both department are headed by a dierent VP and the CEO has a budget u to motivate each VP. The outputs in the two departments are aected by the same market condition θ.

61 How to pay a VP Relative performance Instead of working with two VPs separately, we combine the problems together. Relative performance. Why would it help? We are learning something from the behavior of one agent, that may help us to provide better incentives for the other one.

62 How to pay a VP Relative performance Consider the following scheme. If one VP has a higher output, she gets bonus 2u, and the other VP gets nothing. If the two VPs have the same output, both of them get bonus u. In this scheme, the value of the bonus depends on the performance of the other VP game!

63 How to pay a VP Relative performance We are going to show that high eort is strictly dominant for each θ! Indeed, if the other VP chooses low eort, then ai = a h leads to payo of 2u c, vs ai = a l leads to payo of u. because u > c, choosing a h is better. If the other VP chooses high eort, then ai = a h leads to payo of u c, vs ai = a l leads to payo of 0. because u > c, choosing a h is, again, better.

64 How to pay a VP Relative performance High eort is chosen for any θ! Thus, a big improvement on any scheme that treated the VP problems separately. Thus, using the information about the other VP is benecial. We assumed that the VPs can observe θ. but the argument actually does not depend on it. High eort is dominant even if the market condition cannot be observed by the VPs.

65 How to pay to a VP Tournaments The above scheme is a special case of a tournament. one way to pay VP is to reward the better one with a job of CEO. let the best one win. Tournaments are the most common, oldest relative performance incentive schemes. Tournaments are great when there is a common shock, the relative performance allows to isolate the eect of the common shock from the incentives. The might be observed by the participants (adverse selection), but it does not have to be. Tournaments are great way to provide incentives for scientic research. patent races!

66 How to pay to a VP Tournaments Examples of famous tournaments: Longitude Act, 1714 series of rewards: 10,000 (worth over 1.33 million in 2016[4]) for anyone who could nd a practical way of determining longitude at sea to an accuracy of not greater than one degree of longitude (equates to 60 nautical miles (110 km; 69 mi) at the equator). The reward was to be increased to 15,000 if the accuracy was not greater than 40 minutes, and further enhanced to 20,000 if the accuracy was not greater than half a degree.[5] John Harrison, inventor of chronometer Scottish book, 1930ies, Lvov For problem 153, closely related to Stefan Banach's "basis problem", Stanisªaw Mazur oered the prize of a live goose. This problem was solved only in 1972 by Per Eno. The winner got the goose. Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000, Google Lunar X prize.

67 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

68 How to pay a teacher Huge increase in use of standardized tests to evaluate teaching performance for the last 20 years or so. precise measures, can isolate impact of individual teachers, seems like a perfect tool to measure teaching quality. However, such tests do not measure all that is important to teaching: arithmetic ability, but not general problem solving skills, creativity, etc. Using the test outcomes to incentivize the teachers may cause them to teach for the test, but not focus on skills that may be much more important in the long-run.

69 How to pay a teacher Example (Multitasking) A teacher picks eort allocation into two tasks e 1, e 2 0. Only the rst eort e 1 is observed. The teacher (i.e., agent) and the school district (principal) have dierent preferences over the tasks: π teacher (e 1, e 2 ; w) = w + a (e 1 + e 2 ) c (e 1, e 2 ), π district (e 1, e 2, w) = w + e 1 + βe 2, where β > 0 are, r the teacher's and the district's value for task i, and c (e 1, e 2 ) = 1 2 (e 1 + e 2 ) 2 is the cost of the two activities. We assume that β > 1, or that unobserved activity is more important.

70 How to pay a teacher Linear contracts Suppose that the school district oers a linear contract: w (e 1 ) = w 0 + αe 1. w0 is a (xed) salary, α 0 is a performance bonus. Simple type of wage contract, well-approximates real life, It turns out, in this example, without loss of generality. we can replicate any behavior under general contracts by linear contracts.

71 How to pay a teacher District's problem The district's problem: max w w 0,α,e1 0 αe,e 1 + e 1 + βe 2 2 st.(ic) : (e1, e2) arg max π teacher (e 1, e 2 ; w 0, α), e 1,e 2 Solution procedure: (IR) : π (e 1, e 2; w 0, α) U 0. we separately consider α = 0 and α > 0, we solve the teacher's problem - takes care of IC, we use the IR to gure out w 0, we solve the unconstrained disctrict problem, compare payos between α = 0 and α > 0.

72 How to pay a teacher α = 0, Teacher's problem First, α = 0. The teacher's problem max w 0 + a (e 1 + e 2 ) 1 2 (e 1 + e 2 ) 2. Optimal choice: e 1 + e 2 = a.

73 How to pay a teacher α = 0, IRs constraint, IR constraint: w 0 + a (e 1 + e 2) 1 2 (e 1 + e 2) 2 U 0, or, using the solution to the teacher's problem, w 0 + a a2 U 0, Because the IR constraint is going to bind, we have w 0 = U a2.

74 How to pay a teacher α = 0, principal's problem School's payos w 0 + e 1 + βe 2, where e 1, e 2 0 and e 1 + e 2 = a. Hence, using the formula for wages, U a2 + (a e 2 ) + βe 2 = U a2 + a + (β 1) e 2, subject to e 2 (0, a). If β > 1, the payos are maximized by e 2 = a and equal to U a2 + a + (β 1) a.

75 How to pay a teacher α > 0, teacher's problem Case α > 0. The teacher's problem max w 0 + αe 1 + a (e 1 + e 2 ) 1 2 (e 1 + e 2 ) 2. Optimal choice: e 2 = 0: no eort put into the unobservable activity, to see why, notice that if e 2 > 0, then the teacher can increase his payof by taking e 1 = e 1 + e 2 and e 2 = 0, if e 2 = 0, e 1 can be derived from the FOCs: e 1 = α + a.

76 How to pay a teacher α > 0, IR constraint IR constraint: w 0 + αe 1 + a (e 1 + e 2) 1 2 (e 1 + e 2) 2 U 0, or using the teacher's solution w 0 + α (α + a) + a (α + a) 1 2 (α + a)2 U 0. Because the IR constraint is going to bind, we have w 0 = U (α + a)2.

77 How to pay a teacher α > 0, principal's problem Case α > 0. Then, e 1 = a + α, the school's payos are π school = w 0 αe 1 + e 1 + βe 2 = U (a + α)2 α (α + a) + a + α. FOC (wrt. α): a + α α (a + α) + 1 = 0, or α = 1. α = 1 aligns the teacher and the school's incentives. this is a familiar result (spelling rm).

78 How to pay a teacher α > 0, principal's problem School's payo: π school = U (a + 1)2 (a + 1) + a + 1 = U (a + 1)2 2 = U a2 + a

79 How to pay a teacher We can compare α = 0 : U a2 + a + (β 1) a. α > 0 : U a2 + a Thus, if βa > a + 1 2, the school's payos are higher if α = 0.

80 How to pay a teacher Lemma Suppose that (β 1) a > 1. Then, the optimal contract does not 2 involve any incentives α = 0. If β is high, the payo from the unobserved activity is high relative to the benet of incentivizing the teacher. If a is high, the teacher has lots of internal motivation. In both cases, the optimal choice of the remuneration is a at wage contract, without any incentive components. βa > a + 1 2, the school's payos are higher if α = 0.

81 How to pay a teacher Optimality of at wage sheds some light on the puzzle: moral hazard is prevalent, but most of the wage contracts have no explicit incentive component. Trade-o between incentivizing observable activity, and making sure that unobservable tasks are also executed. monitoring is dicult, in general, there are tasks that are very dicult to monitor, but they are valuable to principal, incentivizing only observable task may result in shifting eort away from unobservable tasks, which is bad for the principal.

82 How to pay a teacher Examples: sales: maximizing the number of sales vs. customer satisfaction and good will, services: time in which service is accomplished vs attention to detail and care, production: speed vs quality, production: the output vs taking care of the machines and/or safety.

83 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

84 Eciency wages Even if a wages do not explicitly depend on performance, there are other ways of providing incentives: keeping vs ring the worker, promotion. If the red worker can immediately nd an equivalent job, he is not scared of being red, so, no incentives. Moral hazard model lead to a simple theory of unemployment.

85 Eciency wages Economy 1 competitive rm with production function f (L), L number of hired workers, f (L) = w is the wage workers and rms separate for random reasons with prob. b, N workers in the economy, u = N L N unemployment rate. Probability of nding a job is decreasing in u. #job openings p (u) = #separations + #unemployed bl = bl + N L = bl/n bl/n + (N L) /N b (1 u) = b (1 u) + u = 1 u b + u (1 b).

86 Eciency wages Moral hazard Each worker can either shirk or work hard c > 0 cost of working hard If the worker shirks, it does not generate any output is caught with prob. q and red, Firms will hire the worker only if they expect her to work hard.

87 Eciency wages Moral hazard Let Ve - value of being employed, Vu - value of being unemployed, β - discount factor Worker won't shirk if β [(1 b) (qv u + (1 q) V e ) + bv u ] c + β ((1 b) V e + bv u ), or := V e V u c (1 b) βq. value of being employed must be higher than the value of being unemployed to compensate for the eort, is the dierence between the two values.

88 Eciency wages Equilibrium Value of being employed (and working hard) V e = w c + β (1 b) V e + βbv u = w c + βv e βb or (1 β) V e = w c βb.

89 Eciency wages Equilibrium Value of being unemployed: the convention is that if a worker nds job, she immediately benets from being employed, otherwise, she waits for a whole preiod, and becomes unemployed again: V u = p (u) V e + β (1 p (u)) V u = βv u + βp (u) + (1 β) p (u) V e = βv u + βp (u) + p (u) (w c βb ) = βv u + p (u) β (1 b) + p (u) (w c), where we used the above expression for V e.

90 Eciency wages Equilibrium After subtracting the two equations and some algebra, we get and (1 β + βb pβ (1 b)) = (1 p (u)) (w c), = 1 p (u) (w c). 1 β (1 p (u)) (1 b) Putting the above into the incentive inequality, ( ) c 1 β (1 p (u)) (1 b) w c +. (1 b) βq 1 p (u) = c + c ( ) 1 βq (1 b) (1 p (u)) β.

91 Eciency wages Equilibrium Incentive inequality w c + c ( βq 1 (1 b) (1 p (u)) β ). wages have to be suciently high to stop the worker from shirking and risking being red if it is easy to nd job, p (u) 1, and the incentive wages go to hence, unemployment in equilibrium must exist.

92 Eciency wages Equilibrium Equilibrium unemployment is at intersection of labor demand: labor supply w c + c βq Full employment is impossible. f ((1 u) N) = w, ( ) 1 β (1 b). 1 p (u) There are workers who would be happy to work for wages below the market level, but they cannot nd jobs.

93 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

94 Career incentives Risk of being red and unemployment is not the only tool for the rm to provide incentives for the worker. One argument is that rms have other tools to provide incentives for the workers. For example, promotion, or progress-through-the-ranks. Increasing wage proles are common.

95 Career incentives A worker who is exerting full eort, is kept and receives a higher wage in the next period a worker may be paid below their productivity initially to pay up for future raise, because she is paid more than her productivity later, on average, the worker is paid its productivity. A worker who is caught shirking is red and needs to start again. this provides incentives to put an eort, as long as the cost of eort is smaller than the future raise. This works without unemployment. One of the criticisms of the eciency wage theory of unemployment.

96 Plan Introduction Worker Inventor CEO Grades VP Teacher (multi-tasking) Eciency wages

97 Conclusions What did we learn - concepts Principal agent problem contract as a function of observables, Agent's problem: IR constraint and incentive conditions, Principal's problem maximize objective subject to the agent's problem.

98 Conclusions What did we learn - Types of principal's agent problem Fully observable eort, rst-best allocation Observable output (but not eort) selling the rm contract, minimum wage-constraints, Moral hazard with adverse selection

99 Conclusions What did we learn - Types of principal's agent problem Relative performance tournaments, multitasking, optimality of at wages Unemployment as an incentive eciency wages, Multi-period moral hazard.

100 Conclusions What did we learn - skills Write down principal agent problem, Show optimality of simple contracts in given situations threshold contract (fully observable eort) selling the rm (observable output) at wage (multitasking) tournament (common shocks with two agents) Explain how unemployment and career prospects can be used to provide incentives.