Prerequisites Almost essential Risk MORAL HAZARD MICROECONOMICS Principles

Prerequisites Almost essential Risk MORAL HAZARD MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Moral Hazard 1

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 • for hidden information about personal characteristics • see Adverse Selection • see also Signalling § Similar issues arise in setting up the economic problem § Set-up based on model of trade under uncertainty April 2018 Frank Cowell: Moral Hazard 2

Overview Moral Hazard The basics Information: hidden-actions model A simplified model The general model April 2018 Frank Cowell: Moral Hazard 3

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 April 2018 Frank Cowell: Moral Hazard 4

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 April 2018 Frank Cowell: Moral Hazard 5

Principal-and-Agent: extensive-form game 0 p [RED] § “Nature” chooses a state of the world § Probabilities are common knowledge § Principal offers contract, not knowing state of world § Agent chooses whether to accept contract 1 -p Bill [NO] [BLUE] Bill [OFFER] [NO] Alf [low] April 2018 Alf [high] [low] [high] Frank Cowell: Moral Hazard 6

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 • 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 April 2018 Frank Cowell: Moral Hazard 7

Trade: p common knowledge b a x. RED § Certainty line for 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 p. RED – ____ p. BLUE x. BLUE p. RED – ____ p. BLUE • • Oa April 2018 • b x. BLUE a x. RED Frank Cowell: Moral Hazard 8

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 April 2018 Frank Cowell: Moral Hazard 9

Trade: Bill is risk neutral b a x. RED p. RED – ____ p. BLUE x. BLUE Ob • § Certainty line for Alf § Alf's indifference curves § Certainty line for Bill § Bill's indifference curves § Endowment point § CE prices + allocation § Alf's reservation utility § If Bill can exploit Alf • Oa April 2018 • b x. BLUE a x. RED Frank Cowell: Moral Hazard 10

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 hazard April 2018 Frank Cowell: Moral Hazard 11

Overview Moral Hazard The basics Lessons from the 2 x 2 case A simplified model The general model April 2018 Frank Cowell: Moral Hazard 12

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 April 2018 Frank Cowell: Moral Hazard 13

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 April 2018 Frank Cowell: Moral Hazard 14

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) April 2018 Frank Cowell: Moral Hazard 15

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 by p(z) • z is the effort supplied by the agent § The Agent chooses either • low effort z _ • high effort z April 2018 Frank Cowell: Moral Hazard 16

The 2 2 case: motivation § Agent's utility: • consumption of the single good xa ( ) • the effort put in, z ( ) • given v. NM preferences utility is Eua(xa, z) § Agent is risk averse • ua(·, ·) is strictly concave in its first argument § Principal consumes all output not consumed by Agent • xb = q – x a § (In the simple model) Principal is risk neutral • utility is Eq – xa § Can interpret this in the trading diagram April 2018 Frank Cowell: Moral Hazard 17

Low effort b x. RED a x. BLUE p. RED – ____ p. BLUE § Certainty line for Alf (Agent) b § Alf's curves Ob indifference O § 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 b x. BLUE Oa April 2018 a x. RED Switch to high effort Frank Cowell: Moral Hazard 18

* detail on slide can only be seen if you run the slideshow High effort b x. RED a x. BLUE p. RED – ____ p. BLUE § Alf: Cert. line and indiff curves § Bill: O Obb. Cert. Ob line and indiff curves § Endowment point § Alf's reservation utility §High effort tilts ICs, shifts equilibrium outcome §Contrast with low effort b x. BLUE Oa April 2018 a x. RED Frank Cowell: Moral Hazard Combine to get menu of contracts 19

Full information: max problem § Agent's consumption is determined by the wage § 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 April 2018 Frank Cowell: Moral Hazard 20

Full-information contracts – q b x. RED Ob a x. BLUE q – § Alf's low-effort ICs § Bills ICs § Alf's high-effort ICs § Bills ICs § Low-effort contract § High-effort contract – w(z) – b x. BLUE Oa April 2018 – w(z) – a x. RED Frank Cowell: Moral Hazard 21

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 April 2018 Frank Cowell: Moral Hazard 22

Second best: principles § Utility functions • as before § Wage schedule • effort is unobservable • cannot condition wage on effort or on the state-of-the-world • but resulting output is observable • 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 April 2018 Frank Cowell: Moral Hazard 23

Participation constraint § Principal can condition the wage on the observed output: _ _ • wage w if output is q § Agent will choose high or low effort • this determines the probability of getting high output • 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: _ _ _ a a • p(z) u (w, z) + [1 – p(z)] u (w, z) ua April 2018 Frank Cowell: Moral Hazard 24

Incentive-compatibility constraint § Assume that Agent will actually participate _ _ • wage w if output is q § Agent will choose high or low effort § To ensure that high effort is chosen, set wages so that: _ _ _ a a • p(z) u (w, z) + [1 – p(z)] u (w, z) _ _ a p(z) u (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 April 2018 Frank Cowell: Moral Hazard 25

Second-best contracts b § Alf's low-effort ICs b O § Bills ICs § Alf's high-effort ICs § Bills ICs § Full-information contracts § Participation constraint § Incentive-compatibility constr. § Bill’s second-best feasible set § Second-best contract x. RED a x. BLUE ua –w §Contract maximises Bill’s utility over second-best feasible set b x. BLUE Oa April 2018 – w a x. RED Frank Cowell: Moral Hazard 26

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 April 2018 Frank Cowell: Moral Hazard 27

Overview Moral Hazard The basics Extending the “first-order” approach A simplified model The general model April 2018 Frank Cowell: Moral Hazard 28

General model: introduction § Retain assumption that it is a two-person contest • same essential roles for Principal and Agent • 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 April 2018 Frank Cowell: Moral Hazard 29

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) April 2018 Frank Cowell: Moral Hazard 30

Effect of effort §Support of the distribution §Output distribution: low effort §Output distribution: high effort f(q, z) §Higher effort biases frequency distribution to the right q – April 2018 q– q Frank Cowell: Moral Hazard 31

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 § Principal consumes output after wage is paid • but we allow for non-neutral risk preference • Eub(xb) = Eub(q – w) • ub(·) is concave April 2018 Frank Cowell: Moral Hazard 32

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 E ua(w(w), z) ua § Lagrangian is • Eub(f(z, w) – w(w)) + l[E ua(w(w), z) – ua] April 2018 Frank Cowell: Moral Hazard 33

Optimisation: outcomes § The Lagrangian is • E ub(xb) + l[E ua(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) + l. Euza(w(w), z) = 0 § Combining we get • uxb(xb) / uxa(xa) = l xa = w(w) uza(xa, z) b b • E ux (x )fz(z, w) + E ux (x ) = 0 uxa(xa, z) xb = f (z, w) – w(w) April 2018 Frank Cowell: Moral Hazard 34

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 • • • April 2018 uza(xa, z) b b E ux (x )fz(z, w) + E ux (x ) = 0 uxa(xa, z) in each state Bill’s (the Principal’s) MU is used as a weight special case where Bill is risk-neutral: this weight is the same in all states. Then we have: uza(xa, z) E fz(z, w) = – E uxa(xa, z) Expected MRT = Expected MRS for the Agent Frank Cowell: Moral Hazard 35

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 April 2018 Frank Cowell: Moral Hazard 36

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 April 2018 Frank Cowell: Moral Hazard 37

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 Lagrangian technique • assumes problem is “well-behaved” • this may not always be appropriate April 2018 Frank Cowell: Moral Hazard 38

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” : • E ua(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 April 2018 Frank Cowell: Moral Hazard 39

Second-best: optimisation § Problem is then • choose w(·) to max Eub(q – w(q)) • subject to E ua(w(q), z) ua • and E (ua(w(q), z)bz) + Euza(w(q), z) = 0 § Lagrangian is • E ub(q – w(q)) + l [E ua(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 Lagrangian 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 April 2018 Frank Cowell: Moral Hazard 40

Second-best: FOCs § Use a simplifying assumption: • uxza(·, ·) = 0 § Lagrangian 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): • FOC 1: – uxb(xb) + luxa(xa, z) + muxa(xa, z)bz = 0 § Differentiating with respect to z: • FOC 2: E ub(xb)bz+ m[ 2 E (ua(xa, z)) / z 2 ] = 0 April 2018 Frank Cowell: Moral Hazard 41

Second-best: results bz is +ve where xb is large § From FOC 2: 2 nd derivative – Eub(xb)bz • m = ——————— is negative 2 E(ua(xa, z))/ z 2 • m>0 • so the incentive-compatibility constraint is binding § From FOC 1: • uxb(xb) / uxa(xa, z) = l + m bz • 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 FOC 1: • because uxb(xb) / uxa(xa, z) = l + m bz • 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) April 2018 Frank Cowell: Moral Hazard 42

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 April 2018 Frank Cowell: Moral Hazard 43