Moral Hazard: Hidden Action

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1 Moral Hazard: Hidden Action Part of these Notes were taken (almost literally) from Rasmusen, 2007 UIB Course (UIB) MH-Hidden Actions Course / 29

2 A Principal-agent Model. The Production Game Two individuals: Principal (P) and Agent (A) P + A generate a surplus that depends on the e ort of A, namely e, and possibly on the realization of a random variable θ (a state of the world) P proposes a contract c to A, which can be accepted or refused. Accepts. A chooses an e ort and a surplus q (e, θ) is generated. A receives a payment w (c). P obtains q (e, θ) w (c). Does not accept. No surplus is generated (We could assume that A makes the o er) e 2 E R θ 2 Ω R is chosen according to a probability density f (θ). q : E Ω! R is increasing in e. (UIB) MH-Hidden Actions Course / 29

3 The preferences Satisfy the von Neumann-Morgenstern assumptions (?) Principal: Agent: V (q w) denotes her elementary utility function, de ned over her income. V 0 > 0. Increasing with income. V Risk-neutral if V 00 = 0 and risk-averse when V 00 < 0. U (w, e) denotes her elementary utility function U w (w, e) > 0. Increasing with income U e (w, e) < 0. Decreasing in e. U ww (w, e) 0. Risk-neutral if U ww = 0 and risk-averse if U ww < 0. U ee (w, e) 0. Usually we will assume separability: U (w, e) = u(w) v(e). (UIB) MH-Hidden Actions Course / 29

4 The contract, utilities and the reservation utility Veri able e ort level: c = w (q, e), where w (q) denotes the wage received by the agent exerting e ort e when the outcome is q. Unveri able e ort level: c = w (q) If two or more Principals compete to employ an agent then their equilibrium pro t equals zero (Bertrand competition). If many agents compete to work for the Principal then the agent s equilibrium utility is at least equal to what they can obtain by not accepting the job, called the reservation utility, U. (UIB) MH-Hidden Actions Course / 29

5 Production game I: Full information The Principal makes the o er. Assume that e is observed by P! c = w (q (e, θ), e) = w (e) and q (e, θ) = q e, θ 0 The problem of the Principal: max V (q (e) w) e,w s.to : U (w, e) U As V 0 (q (e) e : V 0 (q (e) w)q 0 U (w, e) (e) + λ = 0 e w (e) : V 0 U (w, e) (q (e) w) + λ = 0 w λ : λ U (w, e) U = 0, λ 0 w) > 0 and U (w,e) w > 0 we obtain λ > 0 so that U (w, e ) U = 0. (UIB) MH-Hidden Actions Course / 29

6 Thus, EFFICIENCY! 0 = V 0 (q (e ) w ) " =) MRS P e,w = q 0 (e ) = U (w,e ) e U (w,e ) w U (w (e ),e ) e U (w (e ),e ) w q 0 (e) + # = MRS A e,w The Principal must design a contract such that w (e ) = w that induces agent to choose e. There are many ways: 1 Forcing contract: w (e) = w if e = e and zero otherwise. 2 Threshold contract: w (e) = w if e e and zero otherwise. 3 Continuous contract: w (e) satisfying U (w (e ), e ) w w (e ) e + U (w (e ), e ) e = 0 SOC < 0 (UIB) MH-Hidden Actions Course / 29

7 Production game II: Full information. Agent moves rst The agent makes the o er e is observed by P! c = w (q (e)) = w (e). The problem of the Agent: The FOC yield (EFFICIENCY!) max U (w, e) e,w s.to : V (q (e) w) 0 MRS P e,w = q 0 (e ) = V (q (e ) w ) = 0 U (w (e ),e ) e U (w (e ),e ) w = MRS A e,w I.e., the contract is e cient (as before), and the Principal obtains her "reservation pro ts" of zero. (UIB) MH-Hidden Actions Course / 29

8 Production game III: A at wage under certainty Assume now that the P cannot condition the wage neither to the output nor the e ort. The outcome is obvious and (generally) ine cient. If w is such that U (w, 0) U then clearly the agent accepts the job and exerts zero e ort. So, P o ers w satisfying U (w, 0) = U if V (q (0), w ) 0 and/or ew satisfying U ( ew, 0) < U when V (q (0), w ) < 0 so that the contract is not accepted. This is a clear example of moral hazard: the agent chooses an action that goes against the interests of the Principal, as she cannot use any contract to punish him. As we will see next, a clever contract can overcome moral hazard by conditioning the wage on "something" that is observable and correlated with e ort (such as output). If there is nothing to condition the wage, however, the agency problem cannot be solved (UIB) MH-Hidden Actions Course / 29

9 Production game IV: An output based wage under certainty Assume that the principal cannot observe e but she can observe q and specify w (q). It is easy to check that the outcome is the same than that obtained in Production Game I. This is due to our assumption that q 0 (e) > 0 so that conditioning the wage to q is equivalent to condition it to the e ort level. Thus, the true agency problem occurs when q and e are not perfectly correlated. (UIB) MH-Hidden Actions Course / 29

10 Production game V: An output based wage under Uncertainty The Principal cannot observe e but she can observe q and speci es w (q). q (e, θ) 6= q e, θ 0 for some θ, θ 0 2 Ω and there is (e, θ), e 0, θ 0 such that q (e, θ) = q e 0, θ 0! Hidden action Remember that θ is chosen (by Nature) according to some f (θ). A given output can now be produced by several e orts. Thus, a forcing contract will not necessarily induce the desired e ort. (UIB) MH-Hidden Actions Course / 29

11 First/Second Best contracts In Production Game V Under Uncertainty and Asymmetric Information we observe a trade o : Due to uncertainty, the Principal wants to insurance the agent against risk as this will allows her to o er a lower acceptable expected wage. By doing so she reduces the incentives of the agent to put e ort, since (due to asymmetric information), e ort is not contractible. We will distinguish between two types of optimality: 1 "First-Best Contract" will be referred to the allocation that is optimal when both players have the same information sets and all variables are contractible. 2 "Second-Best Contract" will be referred to the contract that is optimal given information asymmetry and constraints in writing contracts. The di erence in welfare between the two contracts is the Cost of the Agency Problem. (UIB) MH-Hidden Actions Course / 29

12 The ICC and The PC In the PG V, the principal s objective is to maximize her utility knowing that: 1 The agent is free to reject the contract. 2 The agent is free to choose the e ort level. The constraints that guarantee that A accepts the contract and that the agent chooses the desired e ort e ort level are named the Participation constraint and the Incentive Compatibility constraint, respectively. The Principals problem is, then: max E θv (q (e, θ) e,w (q) w (q (e, θ))) ICC : e 2 arg max E θ U (w (q (e, θ)), e) e PC : E θ U (w (q (e, θ)), e) U (UIB) MH-Hidden Actions Course / 29

13 How to solve the problem Three-Step Procedure (when the Principal is risk-neutral) 1 For any e, determine the set of wage contracts that induce the agent to choose such an e ort level. 2 For any e, nd the contract which supports this e ort level at the lowest cost to the Principal. 3 Choose the e ort level that maximizes pro ts. In general, when P is not risk-neutral, we must use the First Order Approach, consisting on substituting the ICC by the FOC and the SOC of the Agent s maximization problem. When e is a discrete variable we apply a Two-Step Procedure: (2) i (3). (UIB) MH-Hidden Actions Course / 29

14 Optimal Contracts: the Broadway Game This game illustrates a peculiarity of optimal contracts: they are not necessarily monotone. There is a producer and investors. The game is as follows 1 Investors o er a wage contract w(q) as a function of the revenue q. 2 The producer either accepts or reject the contract. 3 The producer selects Embezzle or Do not Embezzle. 4 Nature chooses the state of the world to be Success or Failure with equal probability. Revenues are shown in the next table(s) (UIB) MH-Hidden Actions Course / 29

15 The Broadway Game I Conditional probabilities: F (1/2) S(1/2) E DNE =) E 1/2 1/2 0 DNE 1/2 0 1/2 Payo s: The producer is risk-averse and the investors are risk-neutral. U (w (q) + 50) if E U p = U (w (q)) if DNE, U p = q w (q), U = U (100) Contracts: Investors only observe q so the possible contracts are c = (w ( 100), w (100), w (500)). (UIB) MH-Hidden Actions Course / 29

16 The Broadway Game I Expected utility of the producer (A): 1 U p = 2 U (w ( 100) + 50) U (w (+100) + 50) if E 1 2 U (w ( 100)) U (w (+500)) if DNE The ICC is: 1 2 U (w ( 100)) + 1 U (w (+500)) U (w ( 100) + 50) + 1 U (w (+100) + 50) 2 The PC is: 1 2 U (w ( 100)) + 1 U (w (+500)) U (100) 2 (UIB) MH-Hidden Actions Course / 29

17 The Broadway Game I Since investors want to impose as little risk as possible to A (allows to reduce the expected wage) they will ideally select w ( 100) = w (+500) satisfying the PC with equality, which provides full insurance. In general, this may go against incentives. However, as Pr (+100jDNE ) = 0 and Pr (+100jE ) > 0 this is not the case and the optimal solution of investors is easily achieved by selecting This yields so that it is pro table. w ( 100) = w (+500) = 100 w (+100) = Up e = 1 2 ( 100) + 1 (+500) 100 = 100 > 0 2 (UIB) MH-Hidden Actions Course / 29

18 The Broadway Game I This contract illustrates the idea that the principal rewards inputs, not output. If the contract was monotone with respect to output, this would mean that she rewards Nature, not the agent. Thus, the optimal wage structure would depend on the information about the choice of the agent that can be inferred by observing the outcome. Sometimes, as in the previous example, there is a su cient statistic to exactly infer the agent s choice (if +100 is observed, it means E). In general, when there is no su cient statistic for wrong actions, (i) the rst best cannot be achieved and (ii) such an extreme boiling-in-oil contract cannot be used. However, when the e orts level alters the distribution of the outcomes, the optimal contracts would also include such a dependence. Consequently, the contract is not necessarily monotone. (UIB) MH-Hidden Actions Course / 29

19 The Broadway Game II =) F (0.5) ms(0.3) MS(0.2) E DNE E DNE w ( 100) = w (+450) = w (+575) = 100 and w (+400) = (or smaller than...) (UIB) MH-Hidden Actions Course / 29

20 The Broadway Game III. How Public Information may hurt both players Assume that before the agent takes his action both players can tell whether the show will be a major success or not. I.e., they know that they will be either in scenario A or B given below: A : B : F (0.625) ms(0.375) E DNE MS(1) E +400 DNE +575 Expected pro ts if DNE: ( ) (0.625 ( 100) ). However, in case A the agent will choose E... (UIB) MH-Hidden Actions Course / 29

21 A nite set of outcome and two e ort levels X = fx 1,...x n g e 2 e H, e L, that induce p i e H = p H i and p i e L = p L i. v e H > v e L U (w, e) = u(w) v(e) Constraints of P when selecting c = (w 1,..., w n ) in order to induce e L 1 Participation: n i =1 p i (e L )u(w i ) v(e L ) U. 2 Incentives: n i =1 p i (e L )u(w i ) v(e L ) n i =1 p i (e H )u(w i ) v(e H ). To induce e L is simple. Any xed wage satisfying PC would. (MAYBE IT IS NOT OPTIMAL!!) Di culties appear when P wants to induce e H. (UIB) MH-Hidden Actions Course / 29

22 Inducing a high e ort L(w 1,..., w n, λ, µ) = n i=1 pi H B(x i w i ) +λ n i=1 pi H u(w i ) v(e H ) U +µ n i=1 p H i pi L u(wi ) v(e H ) + v(e L ), FOC: 1 pi H B 0 (x i w i ) + λpi H u 0 (w i ) + µ pi H pi L u 0 (w i ) = 0 for all i = 1,..., n 2 0 = λ n i=1 pi H u(w i ) v(e H ) U, λ = µ n i=1 p H i u(wi ) v(e H ) + v(e L ), µ 0. p L i (UIB) MH-Hidden Actions Course / 29

23 Risk neutral Principal: B 00 () = 0 We have that h pi H + λpi H u 0 (w i ) + µ pi H p L i i u 0 (w i ) = 0 for all i = 1,..., n or, (we know that u 0 (w i ) > 0), pi H h u 0 (w i ) = λph i + µ pi H Adding up all these n FOC we obtain p L i i for all i = 1,..., n which implies That is, PC is binding. λ = n i=1 p H i u 0 (w i ) > 0 n pi H u(w i ) v(e H ) U = 0 i=1 (UIB) MH-Hidden Actions Course / 29

24 From we also obtain pi H h u 0 (w i ) = λph i + µ pi H 1 u 0 (w i ) = λ + µ 1 pi L pi H p L i i for all i = 1,..., n for all i = 1,..., n Can we have µ = 0? When u 00 = 0 it is possible. If u 00 < 0, this would imply that w i is constant =) e = e L. Thus, µ = 0 and ICC (for e H ) cannot be simultaneously satis ed, which implies that µ > 0. In these cases, the wages w i do not depend directly on x i but on p L i /ph i, referred as the Likelihood ratio. higher p L i /ph i =) higher u 0 (w i ) =) lower w i. (UIB) MH-Hidden Actions Course / 29

25 The likelihood ratio Pr x i j e L Pr (x i j e H ) = pl i p H i = Pr eh Pr (e L ) Pr el, x i Pr (e H, x i ) = Φ Pr el, x i Pr (e H, x i ) It means that Pr e L, x i / Pr e H, x i increases with the likelihood ratio. Thus, for any x i, as high is the likelihood ratio as low is the relative probability that e ort is e H and therefore as low is the wage. (UIB) MH-Hidden Actions Course / 29

26 Example so that crop A e L e H pi L/pH i x 1 2/3 1/3 2 x 2 1/3 2/3 0.5 crop B e L e H pi L/pH i x 1 4/5 1/5 4 x 2 1/5 4/ p1 L(B) p1 H (B) > pl 1 (A) p1 H (A) > pl 2 (A) p2 H (A) > pl 2 (B) p2 H (B) w B 1 < w A 1 < w A 2 < w B 2. A good crop receives more reward in case B than in case A. in case B, if the agent puts high e ort he obtains a good crop with probability 4/5 = 80% In case A this probability is only 2/3 = 67% =) obtaining a good crop in B reveals that it is highly probable that the e ort has been high. (UIB) MH-Hidden Actions Course / 29

27 Example. Non-monotone contracts e L e H pi L/pH i x 1 1/6 1/3 1/2 x 2 2/3 1/3 2 x 3 1/6 1/3 1/2 =) w 1 = w 3 > w 2 Stochastic dominance does not imply monotone contracts, too. Consider, for instance l (e 1 ) = (x 1, x 2, x 3 ; 1/4, 1/4, 1/2) l (e 2 ) = (x 1, x 2, x 3 ; 1/8, 3/8, 1/2) so that l (e 1 ) 1 l (e 2 ). The likelihood ratios are (2, 2/3, 1) and therefore w 2 > w 3 > w 1. (UIB) MH-Hidden Actions Course / 29

28 Optimal contracts: 4 generic cases 1 B 00 = 0, u 00 < 0! c? 2 B 00 = 0, u 00 = 0! c? 3 B 00 < 0, u 00 < 0! c? 4 B 00 < 0, u 00 = 0! c? Problem Find an example of optimal contracts for any of the previous situations when v(e) = e, U = 1 and the conditional probabilities are given next x 1 = 10 x 2 = 100 e L = e H = (UIB) MH-Hidden Actions Course / 29

29 Bibliography E. Rasmusen (2007), "Games and Information: An Introduction to Game Theory", chapter 7. Blackwell Publishing. D. Cardona (2012), "Apunts d Economia de la Informació (20619)". Arxius/Assignatures/EI/Apunts-EI-2013.pdf (UIB) MH-Hidden Actions Course / 29

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