Contracts and Incentives in Organizations

Contracts and Incentives in Organizations

Ram Singh

Lecture 9

Contexts

Two players: Principal and Agent Example

Principal as a Government Department and Agent as an Employee Principal as a firm and Agent as a worker

Principal as the owner(s) and Agent as the Manager Principal as a landlord and Agent as a Tenant

Principal as a client and Agent as a Professional service provide

SB: Linear Contracts I

Assumptions:

q(e, ) =e+, where∼N(0, σ²).

Principal is risk-neutral.V(q,w) =q−w

Agent is risk-averse.u(w,e) =−e^{−r(w−ψ(e))},r >0, whereψ(e)is the (money) cost of efforte.

r =−^u_u⁰⁰0 >0, i.e., CARA ψ(e) =¹₂ce²,c>0.

Contract: w(q) =t+sq, wheres>0.

w¯ =Certainty equivalent of the reservation (outside) wage

SB: Linear Contracts II

Noteu(w,e)is increasing inw and decreasing ine.

The First Best: The first best is solution to

maxe,t,s E(q−w)

s.t.−e^{−r(w−ψ(e))}=−e^−rw¯, i.e.,w−ψ(e) = ¯w, i.e.,w = ¯w+ψ(e).

Therefore, the first best is solution to

maxe E(e+−w¯ −ψ(e)),i.e., maxe {e−1

2ce²},

sinceE() =0. Therefore, the first best effort level is given by the following foc

SB: Linear Contracts III

ce^∗ =1,i.e.,e^∗= 1

c. (1)

Whenecontractible, the following contract can achieve the first best:

w = ¯w+_2c¹ ife=¹_c; w=−∞otherwise.

Second Best: eis not contractible butqis. The principal solves maxe,t,s E(q−w)

s.t.

E(u(w,e)) =E(−e^{−r(w−ψ(e))}) ≥ −e^{−r( ¯w)}=u( ¯w) (IR) e = arg max

ˆe E(−e^{−r(w−ψ(ˆe))}) (IC)

SB: Linear Contracts IV

Note that−r(w−ψ(e)) =−r(t+sq−ψ(e)) =−r(t+s(e+)−ψ(e)), i.e.,

−r(w−ψ(e)) =−r(t+se−ψ(e))−rs. Therefore,

E(−e^{−r(w−ψ(e))}) =−E(e−r(t+se−ψ(e))−rs),i.e.

E(−e^{−r(w−ψ(e))}) =−E(e−r(t+se−ψ(e)).e^−rs),i.e.

E(−e^{−r(w−ψ(e))}) =−e−r(t+se−ψ(e))E(e^−rs).

Since for a random variablex is such thatx ∼N(0, σ_x²), so E(e^γx) =e^γ2

σ2 x 2 .

Therefore, we have

E(−e^{−r(w−ψ(e))}) =−e^−r(t+se−ψ(e))

.e^r2s2σ

2 2 ,i.e.,

SB: Linear Contracts V

E(−e^{−r(w−ψ(e))}) =−e−r(t+se−ψ(e))+r²s^2σ₂². (2) Remark

Let’s define

−e^−rw(e)ˆ =E(−e^{−r(w−ψ(e))}) (3) From(2)and(3)

−rw(e) =ˆ −r(t+se−ψ(e)) +r²s²σ² 2 ,i.e.,

wˆ(e)

| {z }

certainty−equivalent wage

= t+se

| {z }

expected wage

−1

2ce²− rs²σ² 2

| {z }

risk−premium

SB: Linear Contracts VI

Therefore, the agent will chooseeto solve max

ˆe {ˆr w(e) =r(t+se−ψ(e))−r²s²σ² 2 }.

the foc for which iss−ec=0, i.e.,

e^SB= s

c (4)

Therefore, the Principal’s problem can be written as maxe,t,s E(q−w), i.e., max

e,t,sE(e+−(t+sq)), i.e., maxe,t,s E(e+−t−s(e+)), i.e.,

maxe,t,s(e−t−se) s.t.

SB: Linear Contracts VII

w(e) =ˆ t+se−ψ(e)−rs²σ²

2 ≥ w¯ (IR)

e = s

c (IC)

That is,

maxt,s {s

c −t−ss c} s.t.

t+ss c −c

2 s²

c² −rs²σ² 2 = ¯w That is,

maxs {s c +s²

c − s²

2c −rs²σ² 2 −s²

c }

SB: Linear Contracts VIII

The foc w.r.t. sis

s= 1

1+rcσ² (5)

Remark

r >0⇒s<1, and s<1⇒e^SB<e^∗. r =0⇒s=1,i.e., e^SB =e^∗.

s∝ ¹_r, s∝ ¹_c and s∝ _σ¹.

Remark

Linear Contracts are not most efficient contracts

Non-linear contracts can achieve the better outcome for the Principal However, a Second Best contract will satisfy other above properties

Sub-optimality of Linear Contracts I

Suppose:

q=q(e, ) =e+

The error term∈[−k,k], where 0<k <∞

For instance, assumehas uniform distribution over[−k,k] Principal is risk-neutral.V(q,w) =q−w

Agent is risk-averse.u(w,e) =u(w)−ψ(e), whereu⁰>0,u⁰⁰<0 and ψ(e), is the dis-utility of efforte;ψ⁰(e)>0,ψ⁰⁰(e)>0

Lete^FB=e^∗

Letw^∗solveu(w^∗) =ψ(e^∗).

Sub-optimality of Linear Contracts II

Note sinceq=q(e, ) =e+,

q∈[e^∗−k,e^∗+k]ife=e^∗. q<e^∗−k only ife<e^∗.

So, when the output has bounded support which depends on the effort,qcan serve as a perfectly informative aboute.

Recallw^∗solvesu(w^∗) =ψ(e^∗).

Now consider the following contract:

w(q) =

w^∗, ifq∈[e^∗−k,e^∗+k];

−∞, ifq6∈[e^∗−k,e^∗+k].

This contract ensures the FB outcome; it implementse^∗as well, and provides full insurance to the risk-averse agent.

Linear Contracts: Sharecropping I

Model:

q=output;q=q(e, );q∈ {q_L,q_H},q_L<q_H. Monetary worth ofq=q(assume price is 1) =a random variable, a noise term;

e=effort level opted by the agent;e∈ {0,1}.

ψ(0) =0 andψ(1) =ψ.

p_H =Pr(q=q_H|e=1)is the probability of the realized output beingq_H; andp_L=Pr(q =q_H|e=0).

w=wage paid by the principal to the agent;w(.) =w(q).

Let the wage contractw(q) =sqbe linear; say, 0≤s≤1.

Linear Contracts: Sharecropping II

Assume that both parties are risk-neutral. So Payoff functions are:

Principal:V(x) =x,V⁰ >0,V⁰⁰=0;

Agent:u(w,e) =u(w)−ψ(e), whereu⁰ >0,u⁰⁰=0.

Optimum Linear Contract:

Suppose the P wants to inducee=1. Then, risk-neutral P will solve maxs {(1−s)[p_Hq_H+ (1−p_H)q_L]}

s.t.

s[pHqH+ (1−pH)qL]−ψ ≥ 0 (6) s[p_Hq_H+ (1−p_H)q_L]−ψ ≥ s[p_Lq_H+ (1−p_L)q_L] (7)

Linear Contracts: Sharecropping III

Notes>0 and(7)implies(6).

Let∆p=p_H−p_Land∆q=q₁−q₀. Exercise:

Ignoring IR, show that IC binds the foc w.r.t. sis

s^SB= ψ

∆p∆q Find out whether IR finds

Sub-optimality of Linear Contracts

Second Best:

Suppose the P wants to inducee=1. Then, risk-neutral P will solve

wmaxL,wH

{p_H[qH−wH] + (1−pH)[qL−wL]}

s.t.

pHwH+ (1−pH)wL−ψ ≥ 0 (8)

p_Hw_H+ (1−p_H)w_L−ψ ≥ p_Lw_H+ (1−p_L)w_L (9) Exercise:

The SB contract is superior to the sharecropping; that is linear contract is NOT Second Best

Compared to the SB, the agent is better-off under sharecropping contract

Find out whether IR finds