Moral Hazard: Part 2. April 16, 2018

Transcription

1 Moral Hazard: Part 2 April 16, 2018

2 The basic model: A is risk neutral We now turn to the problem of moral hazard (asymmetric information), where A is risk neutral. When A is risk neutral, u (t) is linear. Assume u (t) = t. Then, u 1 (ψ) = ψ. The P now has to take into account the ICC of A. He needs to make sure that A is willing to exert the high effort.

3 The basic model: A is risk neutral P s problem becomes: max π 1 ( S t ) + (1 π 1 ) (S t) t, t s.t. PC : π 1 t + (1 π 1 ) t ψ 0 ICC : π 1 t + (1 π 1 ) t ψ π 0 t + (1 π 0 ) t, where S = S ( q) and S = S (q). As before, we still need to make sure that P prefers A to exert the high effort level, i.e. V 1 V 0. Strategy of P: spend as little as possible to incentivize A; and ensure A with the outside option, E (U) = 0.

4 PC and ICC Both constraints can be made binding! First, rewrite the ICC as: π 1 t π 0 t + (1 π 1 ) t (1 π 0 ) t ψ (π 1 π 0 ) t + (1 π π 0 ) t ψ (π 1 π 0 ) t (π 1 π 0 ) t ψ t ψ π Assume PC is binding, i.e. E (u) = 0: π 1 t + (1 π 1 ) t ψ = 0 or t = 1 π 1 t ψ π 1 π Then, one can increase (decrease) t and correspondingly reduce (increase) t by 1 π1 π 1 until t = ψ π.

5 Optimal contract, with A risk neutral P optimally chooses t and t to satisfy PC and ICC. This is a system of 2 equations with 2 unknowns: π 1 t + (1 π 1 ) t ψ = 0 π 1 t + (1 π 1 ) t ψ π 0 t + (1 π 0 ) t. The solution is: t = 1 π 0 π 1 π 0 ψ (Reward); t = π 0 π 1 π 0 ψ (Punishment). P is able to first best incentivize A! P incurs no extra cost to induce A to exert the high effort: the expected transfer to P is π 1 t + (1 π 1 ) t = ψ, i.e. exactly covering the disutility of effort of A. A s expected gain from exerting effort is t π = ψ, i.e. exactly compensating for the disutility of effort.

6 A risk neutral: discussion Even with moral hazard, when A is risk neutral it is possible to induce the first best effort: if effort is optimal under complete information, P can incentivize effort at no extra cost; if effort is not optimal, P can just give no transfers. No moral hazard related issues emerge for P: optimal effort can be decentralized. Here, delegating optimal effort is costless. However, now t t : A is not perfectly insured against risk! Luckily, A is risk neutral...

7 P s final choice As before, we still need to discuss whether P wants to induce A to exert high effort or not. As in the complete information case, V 1 V 0 iff: π S u 1 (ψ) = ψ.

8 A risk neutral: a sell-off implementation Up to now, we assumed ICC is met with equality: A is indifferent between exerting high or low effort. Yet, one can implement first best effort also by making A strictly prefer exerting effort. Consider the transfers: t = S T and t = S T, where T is an upfront payment made by A before output realizes. These transfers strictly satisfy ICC since: π ( t t ) = π S > ψ, provided exerting effort is optimal. Note that T is P s ex-ante expected payoff: π 1 S + (1 π1 ) S ψ and, also, the ex-ante payment from A to P. In fact, P is selling off her property rights over production to A.

9 Extention: Limited liability The first best implementation requires a punishment (negative transfer t < 0) in the bad state of the world. The availability of punishment may not be taken for granted. Limited liability is the condition by which shareholders are legally responsible for the debts of a company only to the extent of the nominal value of their shares. In our model, limited liability takes the form of two additional inequality constraints: t l, t l, where l 0 captures the liability constraint. If l is large, nothing happens and the first best contract may still be feasible, i.e. if t = π0 π 1 π 0 ψ l. If l is sufficiently small, the first best contract may not be feasible, i.e. t < l.

10 Extention: limited liability Assume t < l, i.e. l < π0 π 1 π 0 ψ and our first best contract is not feasible. The constraints are now: PC : π 1 t + (1 π 1 ) t ψ 0 ICC : π 1 t + (1 π 1 ) t ψ π 0 t + (1 π 0 ) t LLC : t l LCC : t l. Assume optimal mechanism is of the reward/punishment type. Then: LCC implies LLC; and PC cannot be binding. In fact, LCC and ICC are now the only relevant and binding constraints: t SB = l; and t SB = l + ψ π.

11 Extention: limited liability We next show that the other constraints hold: LLC: t SB = l + ψ l. π PC: when l π 0 ψ, π π 1 t SB + (1 π 1) t SB ψ = l + π 1 ψ π 1 ψ π 0 ψ > 0 π π π A now gets a positive liability rent. Mechanism: P is here limited in punishment in the bad state of the world; without enough punishment capacity, the only way to induce high effort of A is to increase the reward in the good state liability rent!

12 Extension: limited liability l = 0 At the extreme scenario for no punishment possibilities, i.e. l = 0, the indirect utilities of P are as follows: V 1 = π 1 S + (1 π1 ) S π 1 π ψ; V 0 = π 0 S + (1 π0 ) S. Thus, the optimal effort inducing choice depends on V 1 V 0, which requires: π S }{{} expected gain of effort π 1 π ψ = ψ }{{}}{{} FB cost of effort expected cost of effort + π 0 π ψ }{{} liability rent

13 Example: agriculture in developing countries Consider an agriculture-based economy A Landlord owns the land, which is farmered by risk-neutral peasants By exerting effort, the farmer increases the probability of high output: q {q, q}; e = {0, 1}; Pr ( q = q e = 1 ) = π1 > π 0 = Pr ( q = q e = 0 ). Moreover, assume that high effort is socially optimal with complete information, i.e. π q ψ. Consider different contract arrangements: which one is optimal? when? why? SB contract (non-linear contract) rental contract: the landlord rents the land to the farmer Sharecropping: the landlord and the farmer share the output of the land (common in some European regions since middle-age, in Africa since colonial age, in US South after abolition of slavery)

14 Agriculture: non-linear contract The problem of P is: max π 1 ( q t) + (1 π 1 ) (q t) t, t s.t. PC : π 1 t + (1 π 1 ) t ψ 0 ICC : π 1 t + (1 π 1 ) t ψ π 0 t + (1 π 0 ) t Solving for the constraints gives: t = 1 π 0 π ψ; and t = π 0 π ψ With LLC, i.e. the farmer is extremely poor (hand-to-mouth), t, t 0. Then, t LL = ψ π ; and tll = 0

15 Agriculture: rental contract The farmer gives a rent R to the landlord and keeps all the output. P solves: max R R s.t. PC : π 1 q + (1 π 1 ) q ψ R 0 ICC : π 1 q + (1 π 1 ) q ψ R π 0 q + (1 π 0 ) q R ICC always holds since π q ψ. Also, R cancels out of ICC. R SB emerges from PC being binding: R SB = π 1 q + (1 π 1 ) q ψ. Assume limited liability, i.e. R q and the farmer cannot pay more than the product of land. Then, LLC is binding, while PC is not. PC not binding comes from: R SB = π 1 q + (1 π 1 ) q ψ π 0 q + (1 π 0 ) q > q. R LL = q.

16 Agriculture: sharecropping contract Here, the farmer keeps a share α (0, 1) of output, while the landlord gets the rest. The problem of P is now: max α (1 α) [π 1 q + (1 π 1 ) q] s.t. PC : απ 1 q + α (1 π 1 ) q ψ 0 ICC : απ 1 q + α (1 π 1 ) q ψ απ 0 q + α (1 π 0 ) q ICC implies PC: the right hand side of ICC is positive! Then, ICC must be binding: α SB = ψ π q What with limited liability? Never an issue: the farmer always gets a reward.

17 Agriculture: optimal contract without LL Compute the indirect utility of the landlord corresponding to each contract: EV non linear = π 1 q + (1 π 1) q ψ; EV Rental = π 1 q + (1 π 1) q ψ; and EV Sharecropping = π 1 q + (1 π 1) q ψ (π1 q + (1 π1) q). π q All contracts incentivize high effort Then: EV non linear = EV Rental > EV Sharecropping

18 Agriculture: optimal contract with LL Compute the indirect utility of the landlord corresponding to each contract: EV non linear,ll = π 1 q + (1 π 1) q π 1 π ψ; EV Rental,LL = q; and EV Sharecropping,LL = π 1 q + (1 π 1) q ψ (π1 q + (1 π1) q). π q All contracts incentivize high effort. Then: EV non linear,ll > EV Rental,LL EV non linear > EV Sharecropping,LL