Asymmetric information 3: post contract

Even under a symmetric information, there can be private information about either:
What the agents do
The outcome of agents’ actions
The relevant circumstances facing the agents

In contracts, there are three trade-offs
1. incentives vs risk sharing( we will focus on this)
2. incentives vs rent extraction
3. incentive power vs incentive balance

And three solutions:
1. monitoring
2. incentive contracts
3. regulation (task design)

Model: principle(an employer) and agent(an worker)

Player: a risk neutral employer, a risk averse worker.
Rules:
Worker’s action is unobservable to the employer, only the outcome can be observed.
w wage = α＋βz
e effort
z observed outcome (profit) = e+x
x random variable (mean zero)
r measure of (absolute) risk aversion (worker not only care about wage but also care about risk)
C(e) workers’ disutility of effort
u(worker’s ulitity) is wbar-1/2*rVar〔ｗ〕

Outcomes:
The agent’s utility is
UA=α＋β(e+x)- C(e) -1/2*rVar〔α＋β(e+x)〕=α＋βe- C(e) -1/2*r2Var〔x〕
(because mean x is zero, andαand e is constant)
The principle utility is
UP=e-(α＋βe)
Method to optimum:
We should get β becauseα can allocate surplus in any desired way.
We want to maximize UA+UP
Two steps:
(i) Find the agent’s optimal choice of e for anyβ
β-c”(e)=0
(ii) Find principal’s optimal choice ofβ
Considering about (i), UA+UP = e-C(e)-1/2*r(C”(e))2Var〔x〕
(iii) find the condition which maximize UA+UP by differentiation.
e-C(e)-1/2*r(C”(e))2Var〔x〕is 1-C”(e)-rC”(e)C””(e)Var〔ｘ〕＝0
UsingC“(e)= β, this condition can be rewritten as
β=1/(1+rC““(e)Var〔ｘ〕)
Emprics:
(i)CEO’s saraly positively relate to his firm’s stock return.
(ii)For low return variability, the number is 20-25, and for high variability, it is around 2-5.
In general, CEO’s salary should connect to not only firm’s performance but also depend on negatively industries performance. but in reality, CEO’s salary is positively related industries performance but if there is big share holder, the relation becomes weaker.

Other extensions:
Multiple tasks: Incentive power vs incentive balance
( if focus on one measure, it breaks the balance of incentive)
Monitoring
Multiple agents
-yardstick competition(ideal competition among different markets)
-credibility and tournaments

Appendix: Informativeness Principle and Filtering

Suppose pay can be made own output and competitor’s output.
z is own output = e+x
z2 is competitor’s = e2＋y
w is wage = α＋βｚ＋γｚ２

the worker’s expected utility is then
UA= α＋β(e+x)＋γ（ebar2+ybar）-C(e)-1/2rVar〔α＋β(e+x)＋γ（ebar2+y）〕
= α＋βe＋γebar2-C(e)-1/2r(β2Var〔x〕＋γ2Var〔y〕＋2βγCov〔x、y〕)

The problem is to chooseγso as to minimize
γ2Var〔ｙ〕＋２βrCov〔x,y〕

First order condition is

2γVar〔y〕+2βrCov〔x,y〕=0

Is γ=－βCov〔x、y〕/Var〔y〕