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

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

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 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

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

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 April 2018 Frank Cowell: Moral Hazard 4

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 April 2018 Frank Cowell: Moral Hazard 5

6 Principal-and-Agent: extensive-form game [RED] π 0 1 π [BLUE] 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 Bill Bill [NO] [OFFER] [NO] [OFFER] Alf Alf [low] [high] [low] [high] April 2018 Frank Cowell: Moral Hazard 6

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 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

8 Trade: π common knowledge a x BLUE b x RED π RED π BLUE O b 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 π RED π BLUE O a a x RED b x BLUE April 2018 Frank Cowell: Moral Hazard 8

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 π is common knowledge we just need to alter the b-indifference curves The special case April 2018 Frank Cowell: Moral Hazard 9

10 Trade: Bill is risk neutral a x BLUE b x RED π RED π BLUE O b 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 O a a x RED b x BLUE April 2018 Frank Cowell: Moral Hazard 10

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 hazard April 2018 Frank Cowell: Moral Hazard 11

12 Overview Moral Hazard The basics Lessons from the 2x2 case A simplified model The general model April 2018 Frank Cowell: Moral Hazard 12

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 April 2018 Frank Cowell: Moral Hazard 13

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 April 2018 Frank Cowell: Moral Hazard 14

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) April 2018 Frank Cowell: Moral Hazard 15

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 by π(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

17 The 2 2 case: motivation Agent's utility: consumption of the single good x a ( ) the effort put in, z ( ) given vnm preferences utility is Eu a (x a, z) Agent is risk averse u a (, ) is strictly concave in its first argument Principal consumes all output not consumed by Agent x b = q x a (In the simple model) Principal is risk neutral utility is Eq x a Can interpret this in the trading diagram April 2018 Frank Cowell: Moral Hazard 17

18 Low effort a x BLUE b x RED π RED π BLUE Certainty O b line for Alf (Agent) Alf's indifference O b curves Certainty line for Bill Bill's indifference curves Endowment point Alf's reservation utility If Bill exploits Alf then outcome is on reservation IC, υ a O a υ a a x RED If Bill is risk-neutral and Alf risk averse then outcome is on Alf's certainty line b x BLUE Switch to high effort April 2018 Frank Cowell: Moral Hazard 18

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

20 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: Eu a (w,z) υ a So, problem is choose w( ) to maximise Eq w + λ[eu a (w, z) υ a ] Equivalently _ find w(z) that maximise π(z) q + _ [1 π(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

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

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 April 2018 Frank Cowell: Moral Hazard 22

23 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

24 Participation constraint Principal can condition the wage on the observed output: _ wage w if output is q 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: _ π(z) u a (w, z) + [1 π(z)] u a (w, z) υ a _ April 2018 Frank Cowell: Moral Hazard 24

25 Incentive-compatibility constraint Assume that Agent will actually participate _ wage w if output is q wage w if output is q Agent will choose high or low effort _ To ensure that high effort is chosen, set wages so that: _ π(z) u a (w, z) + [1 π(z)] u a (w, z) π(z) u a (w, z) + [1 π(z)] u a (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

26 Second-best contracts a x BLUE b x RED υ a O b Alf's low-effort ICs 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 w Contract maximises Bill s utility over second-best feasible set O a w a x RED b x BLUE April 2018 Frank Cowell: Moral Hazard 26

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 April 2018 Frank Cowell: Moral Hazard 27

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

29 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

30 Model components: output and effort Production depends on effort z and state of the world ω: q = φ(z,ω) ω Ω 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 β z := f z (q, z)/f(q, z) April 2018 Frank Cowell: Moral Hazard 30

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

32 Model components: preferences Again the Agent's utility derives from the wage paid, w ( ) the effort put in, z ( ) Eu a (w, z) u a (, ) is strictly concave in its first argument Principal consumes output after wage is paid but we allow for non-neutral risk preference Eu b (x b ) = Eu b (q w) u b ( ) is concave April 2018 Frank Cowell: Moral Hazard 32

33 Full information: optimisation Alf s participation constraint: Eu a (w,z) υ a Bill sets the wage schedule can be conditioned on the realisation of ω w = w(ω) To set up the maximand, also use Bill s utility function u b production function φ Problem is then choose w( ) to max Eu b (φ(z,ω)) subject to E u a (w(ω),z) υ a Lagrangian is Eu b (φ(z,ω) w(ω)) + λ[e u a (w(ω), z) υ a ] April 2018 Frank Cowell: Moral Hazard 33

34 Optimisation: outcomes The Lagrangian is E u b (x b ) + λ[e u a (x a,z) υ a ] where x a = w(ω) ; x b = φ(z,ω) w(ω) Each w(ω) and z can be treated as control variables Bill chooses w(ω) Alf chooses z, knowing the wage schedule set by Bill First-order conditions are u xb (φ(z,ω) w(ω)) + λu xa (w(ω),z) = 0 Eu xb (φ(z,ω) w(ω))φ z (z,ω) + λeu za (w(ω), z) = 0 Combining we get u xb (x b ) / u xa (x a ) = λ u za (x a, z) E u xb (x b )φ z (z,ω) + E u xb (x b ) = 0 u xa (x a, z) x b = φ (z, ω) w(ω) x a = w(ω) April 2018 Frank Cowell: Moral Hazard 34

35 Full information: results Result 1 u xb (x b ) / u xa (x a ) = λ because u xa and u xb are positive λ must be positive so participation constraint is binding ratio of MUs is the same (λ) in all states of nature Result 2 u za (x a, z) E u xb (x b )φ z (z,ω) + E u xb (x b ) = 0 u xa (x a, 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: u za (x a, z) E φ z (z,ω) = E u xa (x a, z) Expected MRT = Expected MRS for the Agent April 2018 Frank Cowell: Moral Hazard 35

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 ω and ω MRS a = MRS b Efficient allocation of effort in the case where Principal is risk neutral Expected MRTS zx = Expected MRS zx April 2018 Frank Cowell: Moral Hazard 36

37 Second-best: introduction Now consider the case where effort z is unobserved This is equivalent to assuming state-of-the-world ω unobserved Can work with the distribution of output q: transformation of variables from ω to q just use the production function q= φ(z,ω) 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

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

39 Second-best: problem Bill sets the wage schedule cannot be conditioned on the realisation of ω but can be conditioned on observable output w = w(q) Bill knows that Alf must get at least reservation utility : E u a (w(q),z) υ a participation constraint Also knows that Alf will choose z to maximise own utility so Bill assumes (correctly) that the following FOC holds: E(u a (w, z)β z ) + Eu za (w, z) = 0 this is the incentive-compatibility constraint April 2018 Frank Cowell: Moral Hazard 39

40 Second-best: optimisation Problem is then choose w( ) to max Eu b (q w(q)) subject to E u a (w(q),z) υ a and E(u a (w(q), z)β z ) + Eu za (w(q), z) = 0 Lagrangian is E u b (q w(q)) + λ [E u a (w(q), z) υ a ] + µ [E(u a (w(q), z)β z ) + Eu za (w(q), z) ] λ is the price on the participation constraint µ 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

41 Second-best: FOCs Use a simplifying assumption: u xza (, ) = 0 Lagrangian is Eu b (x b ) + λ[eu a (x a, z) υ a ] + µ[ E(u a (x a, z)) / z ] where x a = w(q) x b = q w(q) Differentiating with respect to w(q): FOC1: u xb (x b ) + λu xa (x a, z) + µu xa (x a, z)β z = 0 Differentiating with respect to z: FOC2: E u b (x b )β z + µ[ 2 E (u a (x a, z)) / z 2 ] = 0 April 2018 Frank Cowell: Moral Hazard 41

42 Second-best: results From FOC2: Eu b (x b )β z β z is +ve where x b is large 2 nd derivative is negative µ = 2 E(u a (x a, z))/ z 2 µ > 0 so the incentive-compatibility constraint is binding From FOC1: u xb (x b ) / u xa (x a, z) = λ + µ β z we know that β z < 0 for low q so if λ = 0, this would imply LHS negative for low q (impossible) hence λ > 0: the participation constraint is binding From FOC1: because u xb (x b ) / u xa (x a, z) = λ + µ β z ratio of MUs > λ if β z > 0; ratio of MUs < λ if β z < 0 so a-consumption is high if q is high (where β z > 0) April 2018 Frank Cowell: Moral Hazard 42

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 April 2018 Frank Cowell: Moral Hazard 43