Moral Hazard: Characterization of SB

Transcription

1 Moral Hazard: Characterization of SB Ram Singh Department of Economics March 2, 2015 Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

2 Characterization of Second Best Contracts I General Model: Suppose = (e, θ), where Θ is the set of states of nature and captures randomness. e E R and θ Θ (e,θ) e 0. Payoff functions: Principal is risk neutral or risk-neutral. So, let V (, w) = w, V > 0, V 0 Agent is (weakly) risk-averse. So, let u(w, e) = u(w) ψ(e), u > 0, u 0, where ψ(e) is the dis-utility of effort e, ψ > 0 and ψ 0. Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

3 Characterization of Second Best Contracts II Contract: The set of contracts is A = {(, w) : R +, w() R}. w = Certainty euivalent of the reservation (outside) wage ū = u( w), the reservation utility When u = 0 holds, i.e., when the agent is risk-neutral, the FB can be achieved by selling the output to the agent. So assume that the agent is risk-averse, i.e., u < 0. Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

4 Characterization of Second Best Contracts III The P will solve: s.t. IR max E{V ( w())} w() and However, E{u(w()) ψ(e)} ū e = arg max{e{u(w() ψ(ê)}} ê (IR) (IC) For given level of e, output can be treated as a random variable. Assume [, ]. Let, F ( e) is a conditional cumulative distribution of f ( e) is the associated conditional density function Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

5 Characterization of Second Best Contracts IV Note: F( e) is a distribution induced by the distribution of θ on Θ. F ( e) is induced through the production technology function = (e, θ) Note: (e,θ) e 0 F e ( e) 0 and (e,θ) e > 0 F e ( e) < 0 We will assume that F( e) satisfies First Order Stochastic Dominance. Definition Distribution F ( e) satisfies First Order Stochastic Dominance w.r.t effort if ( )[F e ( e) 0] & ( )[F e ( e) < 0] > 0 F e ( e) < 0 is sufficient for F( e) to satisfy the First Order Stochastic Dominance. Clearly (e,θ) e Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

6 Characterization of Second Best Contracts V Therefore, in the above setup, the P will solve: s.t. IR and max{ w() V ( w())f ( e)d} u(w())f ( e)d ψ(e) ū. (1) e = arg maxê{ u(w())f ( ê)d ψ(ê)} (IC) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

7 Characterization of Second Best Contracts VI For the time being assume that the agent s payoff function is concave. So, replacing IC with the foc and the relevant soc, we get Form the Lagrangian using (1) and (2) L() = u(w())f e ( e)d ψ (e) = 0 (2) u(w())f ee ( e)d ψ (e) < 0 (3) + λ[ + µ[ V ( w())f ( e)d u(w())f ( e)d ψ(e) ū] u(w())f e ( e)d ψ (e)] (4) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

8 Characterization of Second Best Contracts VII the foc w.r.t. w() is ( )[ V ( w()) u (w()) Moreover, when the P is risk-neutral, the foc is Note that risk-sharing is FB only if = λ + µ f e( e) f ( e) ] (5) 1 ( )[ u (w()) = λ + µf e( e) f ( e) ] (6) ( )[ V ( w()) u (w()) (7) follows from the Borch Rule. = constant], i.e., 1 ( )[ u = constant] (7) (w()) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

9 Characterization of Second Best Contracts VIII Some Conclusions: (7) reuires that both w() and w() are (weakly) increasing functions of ; Assuming P is risk-neutral, the FB risk sharing is independent of the distribution function F( e) for the random variable ; From (7), it can be (by differentiating) shown that 0 w () < 1; (8) Risk-sharing will be as reuired by (7) only if µ = 0, or if for some real number k. ( )[ f e( e) f ( e) = k] (9) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

10 Characterization of Second Best Contracts IX Since, for all e, f ( e)d = 1 holds, therefore, f ( e)kd = k and f e( e)d = 0. That is, (9) implies (can hold only if) 0 = that is, k = 0. That is, (9) holds only if f e ( e)d = ( )[f e ( e) = 0] f ( e)kd = k, But, we are not interested in such a scenario. Moreover, as we prove below, µ > 0. Therefore, risk sharing in not FB. Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

11 Characterization of Second Best Contracts X Let w λ () solve V ( w()) u (w()) where λ is the same as in (5). Recall, w() solves (5), i.e., ( )[ V ( w()) u (w()) = λ, (10) = λ + µ f e( e) f ( e) ] Therefore, µ > 0 implies that the SB contract is such that: { w() wλ (), on Q + = { f e ( e) 0}; w() < w λ (), on Q = { f e ( e) < 0}. (11) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

12 Characterization of Second Best Contracts XI Remark Note that f e( e) f ( e) ln f ( e) e = f e( e) f ( e), i.e., is the derivative of the likelihood function ln f ( e); ln f ( e) = ln Prob{e }. Definition Continuous Output Case: Monotone Likelihood Ratio Property (MLRP): Distribution F ( e) satisfies MLRP if d d [f e( e) d 0], i.e., f ( e) ln f ( e) [ 0]. d e Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

13 Characterization of Second Best Contracts XII Definition Discrete Output Case: Monotone Likelihood Ratio Property (MLRP): Assuming two output levels; 1 and 2. Distribution F( e) satisfies MLRP if ( e > ē), π( i ē) π( i e) = f ( i ē) f ( i e) is (weakly) decreasing in i., i.e., if ( e > ē), f ( i e) f ( i ē) f ( i e) in i. Proposition is (weakly) increasing The contract satisfies monotonicity iff F( e) satisfies Monotone Likelihood Ration Property, i.e., dw d 0 d d [f e( e) f ( e) 0]. Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

14 Characterization of Second Best Contracts XIII Special Case: Let { L, H } and e {e L, e H }. Now, assuming that the P is risk-neutral and wants to induce e H, the foc can be written as 1 u (w( L )) 1 u (w( H )) = λ + µ[1 f ( L e L ) f ( L e H ) ] = λ + µ[1 f ( H e L ) f ( H e H ) ] Now, if H is more likely when e = e H, and the L is more likely when e = e L, we get w H > w L, i.e., [ f ( H e H ) f ( H e L ) > 1 and f ( L e L ) f ( L e H ) > 1] w H > w L. That is, the contract is monotonic in output. Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

15 Characterization of Second Best Contracts XIV Proposition Now when F( e) satisfies First Order Stochastic Dominance w.r.t effort, µ > 0, i.e., IC will bind. Proof: Suppose µ 0 holds. Differentiating (4), w.r.t. e gives λ[ µ[ V ( w())f e ( e)d + u(w())f e ( e)d ψ(e) ū] + u(w())f ee ( e)d ψ (e)] = 0 (12) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

16 Characterization of Second Best Contracts XV In view of (2) and (3), µ 0 implies: Let w λ () solve (10), i.e., Recall, w() solves (5), i.e., V ( w())f e ( e)d 0. (13) V ( w()) u (w()) ( )[ V ( w()) u (w()) = λ = λ + µ f e( e) f ( e) ] Therefore, µ 0 implies: { w() wλ (), on Q + = { f e ( e) 0}; w() > w λ (), on Q = { f e ( e) < 0}. (14) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

17 Characterization of Second Best Contracts XVI Therefore, we get V ( w())f e ( e)d V ( w λ ())f e ( e)d. (15) In view of F e (, e) = F e (, e) = 0, integration by parts gives us V ( w λ ())f e ( e)d = V ( w λ ())(1 w λ())f e ( e)d. (16) Hold RHS to be fixed (assume µ = 0) and differentiate (5) w.r.t. to get w λ() = V ( w λ ()) λu (w λ ()) + V ( w λ ()) Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

18 Characterization of Second Best Contracts XVII In view of λ > 0, this gives 1 > w λ () 0. Also, V > 0 and F ( e) satisfies FOSD. Therefore, V ( w λ ())(1 w λ())f e ( e)d > 0. That is, we get V ( w λ ())f e ( e)d > 0. (17) But (13) and (17) imply a contradictions. Therefore, µ > 0. Q.E.D. Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19

19 Non-monotonic Contracts I Example Consider the following probability density function: f ( L e) f ( M e) f ( H e) e L e H where H > M > L. Note here MLRP is violated. Exercise Show that the SB contact is such that w H > w L > w M, i.e., the contract is non-monotonic. Limitation of Non-monotonic Contracts? Ram Singh (Delhi School of Economics) Moral Hazard March 2, / 19