Moral Hazard

1. Prerequisites Almost essential Risk Moral Hazard MICROECONOMICS Principles and Analysis Frank Cowell June 2004

2. The moral hazard problem • A key aspect of hidden information • Information relates to actions. • Hidden action by one party affects probability of favourable/unfavourable outcomes. • Hidden information about personal characteristics is dealt with... • ... under “adverse selection.” • ... under “signalling.” • However similar issues arise in setting up the economic problem. • Set-up based on model of trade under uncertainty. Jump to “Adverse selection” Jump to “Signalling”

3. Overview... Moral Hazard The basics Information: hidden-actions model A simplified model The general model

4. Key concepts • Contract: • An agreement to provide specified service… • …in exchange for specified payment • Type of contract will depend on information available. • Wage schedule: • Set-up involving a menu of contracts • The Principal draws up the menu • Allows selection by the Agent • Again the type of wage schedule will depend on information available • Events: • Assume that events consist of single states-of-the-world • Distribution of these is common knowledge • But distribution may be conditional on the Agent’s effort

5. Strategic foundation • A version of a Bayesian game. • Two main players • Alf is the Agent. • Bill is the Boss (the Principal) • An additional player • Nature is “player 0” • Chooses a state of the world • Bill does not observe what this is...

6. Principal-and-Agent: extensive-form game • "Nature" chooses a state of the world • Probabilities are common knowledge • Principal may offer a contract, not knowing the type 0 p 1-p • Agent chooses whether to accept contract [RED] [BLUE] Bill Bill [OFFER] [OFFER] [NO] [NO] Alf Alf [low] [high] [low] [high]

7. Extension of trading model • Start with trading model under uncertainty • There are two states-of-the world • So exactly two possible events • Probabilities of the two events are common knowledge • Assume: • A single physical good… • …so consumption in each state-of-the world is a distinct “contingent good”. • Two traders Alf, Bill • CE in Edgeworth box determined as usual: • Draw a common tangent through the endowment point. • Gives equilibrium prices and allocation • But what happens in noncompetitive world? • Suppose Bill can completely exploit Alf

8. pRED – ____ pBLUE pRED – ____ pBLUE • a b b a xRED xRED xBLUE xBLUE Trade: p common knowledge • Certainty line for Alf • Alf's indifference curves Ob • Certainty line for Bill • Bill's indifference curves • Endowment point • CE prices + allocation • Alf's reservation utility • If Bill can exploit Alf... • • Oa

9. Outcomes of trading model • CE solution as usual potentially yields gains to both parties • Exploitative solution puts Alf on reservation indifference curve • Under CE or full-exploitation there is risk sharing • Exact share depends on risk aversion of the two parties. • What would happen if Bill, say, were risk neutral? • Retain assumption that p is common knowledge • We just need to alter the b-indifference curves The special case

10. pRED – ____ pBLUE • a b b a xRED xRED xBLUE xBLUE Trade: Bill is risk neutral • Certainty line for Alf • Alf's indifference curves Ob • Certainty line for Bill • Bill's indifference curves • Endowment point • CE prices + allocation • Alf's reservation utility • If Bill can exploit Alf... • • Oa

11. Outcomes of trading model (2) • Minor modification yields clear-cut results • Risk-neutral Bill bears all the risk • So Alf is on his certainty line • Also if Bill has discriminatory monopoly power • Bill provides Alf with full insurance • But gets all the gains from trade for himself • This forms the basis for the elementary model of moral hazad.

12. Overview... Moral Hazard The basics Lessons from the 2x2 case A simplified model The general model

13. Outline of the problem • Bill employs Alf to do a job of work • The outcome to Bill (the product) depends on • A chance element • The effort put in by Alf • Alf's effort affects probability of chance element. • High effort – high probability of favourable outcome • Low effort – low probability of favourable outcome • The issues are: • Does Bill find it worth while to pay Alf for high effort? • Is it possible to monitor whether high effort is provided? • If not, how can Bill best construct the contract? • Deal with the problem in stages

14. Simple version – the approach • Start with simple case • Two unknown events • Two levels of effort • Build on the trading model • Principal and Agent are the two traders • But Principal (Bill) has all the power • Agent (Alf) has the option of accepting/rejecting the contract offered. • Then move on to general model • Continuum of unknown events. • Agent has general choice of effort level

15. Power: Principal and Agent • Because Bill has power: • Can set the terms of the contract • ...constrained by the Alf’s option to refuse • Can drive Alf down to reservation utility • If the effort supplied is observable: • Contract can be conditioned on effort: w(z) • Get all the insights from the trading model • Otherwise: • Have to condition on output: w(q)

16. The 22 case: basics • A single good • Amount of output q is a random variable • Two possible outcomes • Failure q – _ • Success q • Probability of success is common knowledge: • given byp(z) • z is the effort supplied by the agent • The Agent chooses either • Low effort z _ • High effort z

17. The 22 case: motivation • The Agent's utility derives from • consumption of the single good xa () • the effort put in, z () • Given vNM preferences utility is Eua(xa, z) . • The Agent is risk averse • ua(•, •) is strictly concave in its first argument • The Principal consumes all output not consumed by Agent • xb = q – xa • (In the simple model) Principal is risk neutral • Utility is Eq – xa • Can interpret this in the trading diagram

18. pRED – ____ pBLUE b xBLUE a xRED a b xBLUE xRED Low effort • Certainty line for Alf (Agent) • Alf's indifference curves Ob Ob • Certainty line for Bill • Bill's indifference curves • Endowment point • Alf's reservation utility • If Bill exploits Alf then outcome is on reservation IC, ua • If Bill is risk-neutral and Alf risk averse then outcome is on Alf's certainty line. ua Switch to high effort Oa

19. pRED – ____ pBLUE b xBLUE a xRED a b xBLUE xRED High effort • Certainty line and indifference curves for Alf Ob Ob Ob • Certainty line and indifference curves for Bill • Endowment point • Alf's reservation utility • High effort tilts the ICs, shifts the equilibrium outcome. • Contrast with low effort Combine to get menu of contracts Oa

20. Full information: max problem • The Agent's consumption is determined by the wage paid. • The Principal chooses a wage schedule... • w = w(z) • ...subject to the participation constraint: • Eua(w,z) ua. • So, problem is choose w(•) to maximise • Eq – w + l[Eua(w,z) –ua] • Equivalently _ • Find w(z) that maximise p(z) q + [1 – p(z)] q– w(z)... _ • ... for the two cases z = z and z = z. • Choose the one that gives higher expected payoff to Principal

21. – q q – – w(z) w(z) – b xBLUE – a xRED w(z) w(z) – b a xRED xBLUE Full-information contracts • Alf's low-effort ICs Ob • Bills ICs • Alf's high-effort ICs • Bills ICs • Low-effort contract • High-effort contract Oa

22. Full-information contracts: summary • Schedule of contracts for high and low effort • Effort is verifiable • Contract specifies payment in each state-of-the-world • State-of-the-world is costlessly and accurately observable • Equivalent to effort being costlessly and accurately observable • Alf (agent) is forced on to reservation utility level • Efficient risk allocation • Bill is risk neutral • Alf is risk averse • Bill bears all the risk

23. Second best: principles • Utility functions • As before • Wage schedule • Because effort is unobservable… • ...cannot condition wage on effort or on the state-of-the-world. • But resulting output is observable... • ... so you can condition wage on output • Participation constraint • Essentially as before • (but we'll have another look) • New incentive-compatibility constraint • Cannot observe effort • Agent must get the utility level attainable under low effort Maths formulation

24. Participation constraint • The Principal can condition the wage on the observed output: _ _ • Pay wage w if output is q • Pay wage w if output is q • Agent will choose high or low effort. • This determines the probability of getting high output • ...and so the probability of getting a high wage. • Let's assume he would choose high effort • (check this out in next slide) • To ensure that Agent doesn't reject the contract... • ...must get the utility available elsewhere: _ _ _ _ _ • p(z) ua(w, z) + [1 – p(z)] ua(w, z) ua

25. Incentive-compatibility constraint • Assume that the Agent will actually participate _ _ • Pay wage w if output is q • Pay wage w if output is q • Agent will choose high or low effort. • To ensure that high effort is chosen, set wages so the following holds: _ _ _ _ _ • p(z) ua(w, z) + [1 – p(z)] ua(w, z)  _ _ p(z) ua(w, z) + [1 – p(z)] ua(w, z) • This condition determines a set of w-pairs • a set of contingent consumptions for Alf • must not reward Alf too highly if failure is observed

26. w – b xBLUE – a xRED w a b xBLUE xRED Second-best contracts • Alf's low-effort ICs • Bills ICs Ob • Alf's high-effort ICs • Bills ICs • Full-information contracts ua • Participation constraint • Incentive-compatibility constraint • Bill’s second-best feasible set • Second-best contract • Contract maximises Bill’s utility over second-best feasible set Oa

27. Simplified model: summary • Participation constraint • Set of contingent consumptions giving Alf his reservation utility. • If effort is observable get one such constraint for each effort level • Incentive compatibility constraint • Relevant for second-best policy. • Set of contingent consumptions such that Alf prefers to provide high effort. • Implemented by making wage payment contingent on output • Intersection of these two sets gives feasible set for Bill • Outcome depends on information regime • Observable effort: Bill bears all the risk • Moral hazard: Alf bears some risk

28. Overview... Moral Hazard The basics Extending the “first-order” approach A simplified model The general model

29. General model: introduction • Retain assumption that it is a two-person contest. • Same roles for Principal and Agent. But… • Allow for greater range of choice for Agent • Allow for different preferences for Principal • Again deal with full-information case first. • Draw on lessons from 2×2 case • Same principles apply • Then introduce the possibility of unobserved effort. • Needs some modification from 2×2 case • But similar principles emerge

30. Model components: output and effort • Production depends on effort z and state of the world w: • q = f(z,w) • w  W • Effort can be anything from “zero” to “full” • z [0,1] • Output has a known frequency distribution • f(q, z) • Support is the interval [q, q] • Increasing effort biases distribution rightward • Define proportional effect of effort bz := fz(q, z)/f(q, z)

31. – q q – Effect of effort • Support of the distribution • Output distribution: low effort f(q, z) • Output distribution: high effort • Higher effort biases frequency distribution to the right q

32. Model components: preferences • Again the Agent's utility derives from • the wage paid, w () • the effort put in, z () • Eua(w, z) . • ua(•, •) is strictly concave in its first argument • The Principal consumes output after wage is paid • But we allow for non-neutral risk preference • Eub(xb) = Eub(q – w) • ub(•) is concave

33. Full information: optimisation • Alf’s participation constraint: • Eua(w,z) ua. • Bill sets the wage schedule. • Can be conditioned on the realisation of w • w = w(w) • To set up the maximand, also use • Bill’s utility function ub • production function f • Problem is then • choose w(•) • to max Eub(f(z,w)) • subject to Eua(w(w),z)ua. • Lagrangean is • Eub(f(z,w)– w(w)) + l[Eua(w(w), z)–ua]

34. Optimisation: outcomes • The Lagrangean is • Eub(xb) + l[Eua(xa,z) –ua] • where xa = w(w) ; xb = f(z,w)– w(w) • Each w(w) and z can be treated as control variables • Bill chooses w(w) . • Alf chooses z, knowing the wage schedule set by Bill. • First-order conditions are • – uxb(f(z,w)– w(w)) + luxa(w(w),z) = 0 • Euxb(f(z,w)– w(w))fz(z,w) + lEuza(w(w), z) = 0 • Combining we get • uxb(xb) / uxa(xa) = l  uza(xa, z)  • Euxb(xb )fz(z,w) + E uxb(xb) = 0  uxa(xa, z)  xa = w(w) xb = f(z,w)– w(w)

35. Full information: results • Result 1 • uxb(xb) / uxa(xa) = l • Because uxa and uxb are positive l must be positive. • So participation constraint is binding • Ratio of MUs is the same (l) in all states of nature • Result 2  uza(xa, z)  • Euxb(xb )fz(z,w) + E uxb(xb) = 0  uxa(xa, z)  • In each state Bill’s (the Principal’s) MU is used as a weight. • In the special case where Bill is risk-neutral... • ...this weight is the same in all states. Then we have:  uza(xa, z) • Efz(z,w) = –E   uxa(xa, z) • Expected MRT = Expected MRS for the Agent

36. Full information: lessons • Principal fully exploits Agent • Because Principal drives Agent down to reservation utility • Follows from assumption that Principal has all the power • (No bargaining) • Efficient risk allocation • Take MRS between consumption in state-of-the-worlds w and w • MRSa = MRSb • Efficient allocation of effort • In the case where Principal is risk neutral... • Expected MRTSzx = Expected MRSzx

37. Second-best: introduction • Now consider the case where effort z is unobserved • This is equivalent to assuming state-of-the-world w unobserved • Can work with the distribution of output q: • Transformation of variables from w to q • Just use the production function q= f(z,w) • Clearly effort shifts the distribution of output • Use the expectation operator E over the distribution of output. • All model components can be expressed in terms of this distribution

38. Second-best: components • Objective function of Principal and of Agent are as before. • Distribution of output f depends on effort z. • Probability density at output q is f(q, z) • Participation constraint for Agent still the same • Modify it to allow for redefined distribution • Require also the incentive-compatibility constraint • Builds on the (hidden) optimisation of effort by the Agent • Again use Lagrangean technique • Assumes problem is “well-behaved” • This may not always be appropriate

39. Second-best: problem • Bill sets the wage schedule. • Cannot be conditioned on the realisation of w • But can be conditioned on observable output • w = w(q) • Bill knows that Alf must get at least “reservation utility” : • Eua(w(q),z) ua. • participation constraint • Also knows that Alf will choose z to maximise own utility • So Bill assumes (correctly) that the following FOC holds: • E(ua(w, z)bz) + Euza(w, z) = 0 • This is the incentive-compatibility constraint.

40. Second-best: optimisation • Problem is then • choose w(•) to max Eub(q– w(q)) • subject to Eua(w(q),z)ua. • and E(ua(w(q), z)bz) + Euza(w(q), z) = 0 • Lagrangean is • Eub(q– w(q)) + l [Eua(w(q), z)–ua ] + m [E(ua(w(q), z)bz) + Euza(w(q), z) ] • l is the “price” on the participation constraint • m is the “price” on the incentive-compatibility constraint • Differentiate Lagrangean with respect to w(q) … • each output level has its own specific wage level. • ... and with respect to z. • Bill can effectively manipulate Alf’s choice of z ... • ... subject to the incentive-compatibility constraint.

41. Second-best: FOCs • Use a simplifying assumption: • uxza(•,•)= 0 • Lagrangean is • Eub(xb) + l[Eua(xa, z) –ua ] + m[ E(ua(xa, z)) / z ] • where • xa = w(q) • xb = q– w(q) • Differentiating with respect to w(q): • FOC1:– uxb(xb) + luxa(xa, z) + muxa(xa, z)bz= 0 • Differentiating with respect to z: • FOC2: Eub(xb)bz+ m[ 2E(ua(xa, z)) / z2 ] = 0

42. Second-best: results bz is +ve where xb is large 2nd derivative is negative • From FOC2: – Eub(xb)bz • m = ——————— 2E(ua(xa, z))/z2 • m > 0 • So the incentive-compatibility constraint is binding • From FOC1: • uxb(xb) / uxa(xa, z) = l + mbz • We know that bz < 0 for low q... • So if l = 0, this would imply LHS negative for low q (impossible) • Hence l > 0: the participation constraint is binding. • From FOC1: • Because uxb(xb) / uxa(xa, z) = l + mbz • Ratio of MUs > l if bz > 0; ratio of MUs < l if bz < 0 • So a-consumption is high if q is high (where bz > 0).

43. Principal-and-Agent: Summary • In full-information case: • participation constraint is binding • risk-neutral Principal would fully insure risk-averse Agent. • Fully efficient outcome • In second-best case: • (where the moral hazard problem arises) • participation constraint is binding • incentive-compatibility constraint is also binding • Principal pays Agent more if output is high • Principal no longer insures Agent fully.