Graduate Microeconomics II Lecture 7: Moral Hazard. Patrick Legros

Transcription

1 Graduate Microeconomics II Lecture 7: Moral Hazard Patrick Legros 1 / 25

2 Outline Introduction 2 / 25

3 Outline Introduction A principal-agent model The value of information 3 / 25

4 Outline Introduction A principal-agent model The value of information Rent extraction Limited liability of the agent 4 / 25

5 Outline Introduction A principal-agent model The value of information Rent extraction Limited liability of the agent The exponential linear normal model 5 / 25

6 Outline Introduction A principal-agent model The value of information Rent extraction Limited liability of the agent The exponential linear normal model Moral hazard in teams 6 / 25

7 Introduction Prevent workers from shirking Insure that firms produce the right quality How much information about effort is embodied in the output that is observed? Role of monitoring? Value of additional information? Does joint production help or does it make things worse from an incentive point of view? 7 / 25

8 Introduction Prevent workers from shirking Insure that firms produce the right quality How much information about effort is embodied in the output that is observed? Role of monitoring? Value of additional information? Does joint production help or does it make things worse from an incentive point of view? As for adverse selection (screening), there are two basic constraints on the design of incentive schemes Incentive Compability agents must prefer to take the equilibrium action than another action. Voluntary Participation agents prefer to participate than refusing the contract. 8 / 25

9 A principal-agent model An agent (worker in a firm) thas an outside opportunity of u. If hired in a firm, she has to take an action a [0, ). The verifiable output is a random variable ỹ with distribution F (y, a). Higher actions are assumed to lead to higher expected output. More generaly, assume FOSD if a > a the F (, a) FOSD F (, a ). The agent is risk averse and her utility of a monetary payoff of x when she takes action a is u(x) c(a) where u is increasing concave and c is increasing convex. The principal (owner of the firm) is a residual claimant: if the output is y and the worker is paid x, he has utility v(y x) 9 / 25

10 A principal-agent model Full contractibility Suppose that the effort of the agent is verifiable. The contract between the principal and the agent can then specify an effort that the agent must exert and the output contingent wage w(y) that the agent will receive. max v(y w(y))f (y, a)dy a,{w(y)} s.t. u(w(y))f (y, a)dy c(a) u The constraint is the participation constraint of the agent. If λ is the coefficient of the constraint, the problem can be written [v(y w(y)) + λu(w(y))] f (y, a)dy λ[c(a) + u] max a,{w(y)},λ 0 10 / 25

11 Assuming that a > 0, pointwise maximization then yields y, v (y w(y)) + λu (w(y)) = 0 (1) [v(y w(y)) + λu(w(y))] f a (y, a)dy λc (a) = 0 (2) u(w(y))f (y, a)dy c(a) u = 0 (3) (1) implies that for each y, v (y w(y)) u (w(y)) = λ hence for each output level the ratio of marginal utility levels is constant. This is Borch rule. Note that by concavity of u and v, the optimal w(y) is increasing in y. Question: why is (3) binding? That is why is λ > 0? 11 / 25

12 A principal-agent model The risk-incentive tradeoff Assume now that a is not verifiable. The principal can no longer impose a value of a but can still design a contract {w(y)} in such a way that the agent will be induced to take the right value of a. The design must take into account the incentive problem. Faced with an output contingent compensation the agent will choose a to maximize u(w(y))f (y, a)dy c(a). Hence by choosing w( ), the principal effectively induces the agent to choose a that maximizes her expected utility. Hence the problem is now, max {w(y)},a v(y w(y))f (y, a)dy (4) s.t. u(w(y))f (y, a)dy c(a) u (5) a arg max u(w(y))f (y, â)dy c(â) (6) 12 / 25

13 A principal-agent model Two difficulties 1. (6) may be difficult to deal with: in particular, there may be different optimal values of effort for a given incentive scheme w. If the objective function of the agent is concave in â, then it is possible to replace the incentive compatiblity constraint (6) by the corresponding first order condition. Conditions under which this is appropriate have been derived by Mirrlees, Holmström, Rogerson, Jewitt. One such (pretty strong) condition is that F (y, a) is concave in a. Grossman and Hart (1982) analyze the problem by assuming finitely many states and action levels and bypass the first order problem altogether. 2. If the agent and the principal have large levels of wealth that can be used in contracting, then it may be possible to get very close to the first best situation by imposing very large penalties for low levels of output (Mirrlees). 13 / 25

14 A principal-agent model The first order approach It is convenient to replace the FOC by a weak inequality (Jewitt). The problem of the principal is, max {w(y)},a v(y w(y))f (y, a)dy (7) s.t. u(w(y))f (y, a)dy c(a) u (8) u(w(y))f a (y, a)dy c (a) 0 (9) Let λ the coefficient for the IR constraint and let µ be the coefficient for the IC constraint (9). 14 / 25

15 Pointwise optimization with respect to w(y) leads to v (y w(y)) u (w(y)) = λ + µ f a(y, a) f (y, a) where f a (y, a)/f (y, a) is the likelihood ratio. Note that w(y) is increasing in y only if f a /f is increasing (MLRP condition). This condition holds for many usual distributions (uniform, normal in particular). 15 / 25

16 A principal-agent model The cost of IC: µ > 0 If µ = 0, we are back to Borch rule and the first best can be implemented. However this is impossible if F (y, a) is a nontrivial function of a and if the first best effort is positive. Going back to the first best problem, if the optimal a is positive, it solves [v(y w(y)) + λu(w(y))] f a (y, a)dy λc (a) = 0 Since the IC condition (9) holds we must have v(y w(y))f a (y, a)dy = 0, that is the principal does not benefit from an increase in the effort of the agent. 16 / 25

17 A principal-agent model Good news-bad news As long as MLRP holds, that is when f a (y, a)/f (y, a) is increasing in a, the second best compensation w(y) is increasing in y. Since f a (y, a)dy = 0, y there exists ŷ such that the likelihood ratio is equal to zero at ŷ. Hence we have w(y) > w(ŷ) if y > ŷ [Good news] w(y) < w(ŷ) if y < ŷ [Bad news] In other words, with respect to the first best compensation, the compensation involves an increasing bonus as the output is higher and an increasing malus as the output is lower than ŷ. 17 / 25

18 A principal-agent model The value of additional information Suppose that in addition to output y there is an additional signal z and that there is a joint distribution F (y, z, a). Starting from a world where only y is contractible (hence F (y, a) = z F (y, z, a)dz), the principal and agent have second best payoffs of V, U. A question raised by Holmström (1979) is the following: if z is contractible in addition to y is it the case that the resulting second best optimum will lead to higher payoffs for the principal and the agent? 18 / 25

19 Answer can be glanced via the FOCs with and without z With y only as a basis for compensation schemes: v (y w(y)) u (w(y)) = λ + µ f a(y, a) f (y, a) With y and z as a basis for compensation schemes: v (y w(y, z)) u (w(y, z)) = λ + µ f a(y, z, a) f (y,, z, a) Hence, w(y, z) is a non-trivial function of z only if the likelihood is a non-trivial function of z. ratio fa(y,z,a) f (y,,z,a) 19 / 25

20 Proposition: The additional signal z leads to a Pareto improvement if and only if z is informative for a. That is if and only if we cannot write f (y, z, a) = g(y, a)h(y, z) 20 / 25

21 Principal-agent models The rent extraction motive In the previous problem the principal distorts the compensation scheme with respect to the first best (Borch) in order to create incentives for the agent: since the agent is risk-averse, under MLRP, more variance in compensation induces the agent to exert high effort in order to avoid the low compensation corresponding to low output levels. Obviously this incentive creation has force only if the agent is risk averse. With there is in fact no incentive problem if the agent is risk neutral! Why? 21 / 25

22 Principal-agent models The rent extraction motive In the previous problem the principal distorts the compensation scheme with respect to the first best (Borch) in order to create incentives for the agent: since the agent is risk-averse, under MLRP, more variance in compensation induces the agent to exert high effort in order to avoid the low compensation corresponding to low output levels. Obviously this incentive creation has force only if the agent is risk averse. With there is in fact no incentive problem if the agent is risk neutral! Why? Because the agent is risk neutral, Borch rule requires that the principal gets full insurance (that is the agent bears all the risk). That is the principal gets a fixed payoff P and the agent gets the residual y P. Letting a be the first best effort, it is clear that with the first best compensation the agent will choose a, that is the first best is implemented in the second best! 22 / 25

23 However this construction supposes that the agent is able to pay P to the principal for all possible output realizations. This may not be possible for low output levels if the agent does not have additional wealth or cannot easily borrow). If a fixed compensation P is not feasible, then the principal will not sell the firm to the agent and the resulting second best solution will not be first best. The source of inefficiency here is akin to a rent extraction motive. 23 / 25

24 Rent extraction motive A simple example Suppose two output levels (R and 0), risk neutral principal and agent, probability of high output is π(a) where a {0, 1} is the action of the agent. The cost of action is a. Let s R, s 0 be the contract. Assuming that the agent has limited liability, we need s y 0, y. For a = 0, the easiest way is to pay the agent a fixed wage s R = s 0 = w. For participation, we need w = u and the principal gets V a=0 = π(0)r u. For a = 1, the IC constraint π(1)s R + (1 π(1))s 0 1 π(0)s R + (1 π(0))s 0 (π(1) π(0))(s R s 0 ) 1 24 / 25

25 Incentives depend on the gap s R s 0. Since the expected utility of the agent is U = π(1)(s R s 0 ) + s 0 1 and since s 0 0, the lowest second best payoff to the agent conditional on her taking action a = 1 is U m = π(1) π(1) π(0) 1 If U m < u, the principal needs to increase the compensation to the agent (keeping the gap large enough), e.g., by increasing s 0. If U m u, the principal must give a rent (equal to U m u if he wants action 1 to be taken 25 / 25

26 The payoff to the principal when a = 1 is then V a=1 = π(1)r max{u m, u} 1 This has to be contrasted with V a=0 = π(0)r u. Clearly, the principal prefers to implement a = 1 if and only if (π(1) π(0))r max{u m u, 0} + 1 Efficiency would require that a = 1 when (π(1) π(0))r 1. However it is clear that if U m u is large enough that the principal will choose to implement a = 0 because providing rents to the agent becomes too costly from the point of view of the principal. 26 / 25

27 Principal-agent The exponential linear normal model Output is y = a + ɛ where ɛ is normally distributed, ɛ N(0, σ 2 ) and where a is effort. The principal is risk neutral and the agent has utility exp r(x c(a)) where r is the degree of absolute risk aversion of the agent and c is the cost of effort. Limit attention to linear sharing rules w(y) = s + by. The optimal sharing rate is then b = decreasing in r, in σ 2 and c rσ 2 c 27 / 25

28 Moral hazard in teams Two agents and joint production. Actions are a i, i = 1, 2 and cost for agent i is c(a i ). There is joint production and only the final output y(a 1 + a 2 ) is contractible. A sharing rule defines shares for each agent s 1 (y), s 2 (y). Budget balancing requires s 2 (y) = y s 1 (y). The first best would require to choose a to maximize y(a) c(a), hence that y (a ) = c (a i ), i = 1, 2 28 / 25

29 For any differentiable sharing rule, a is a Nash equilibrium if each agent i maximises s i (y(a i, a j ) c(a i), hence y (a )s 1(y(a )) = c (a ) (10) y (a )s 2(y(a )) = c (a ) (11) By budget balancing, s 1 (y) + s 2 (y) = 1 and therefore (11) is and adding to (10) we have y (a ) y (a )s 1(y(a )) = c (a ), y (a ) = 2c (a ) which contradicts the definition of a. 29 / 25