A risk neutral principal hires a risk averse agent to produce output. The agent can exert costly effort to increase output. The output is
y=μ+e+ε,
where ε∼N(0,σ²). μ and σ² represent the exogenous expected payoff and the riskiness of the given production method separately; while e is effort exerted by the agent. The effort cost is c(e).
Contract. The effort is not observable. The only observable and contractible variable is the output y. Assume that the principal uses a linear contract
w=w₀+αy,
where w₀ is the fixed salary and α is the power of incentive.
Given the contract, the agent choose e to maximize his certainty equivalent
CE=w₀+αμ+αe-((rα²σ²)/2)-c(e).
F.O.C with respect to e gives the following incentive compatible condition
IC₁:c'(e)=α.
The participation constraint requires
IR₁:CE>=0.
The participation constraint should be binding. Otherwise, the principal can be better off by reducing the fixed salary w₀. By adjusting the fixed salary, the principal extracts all the surplus. His problem is then to choose the power of incentive α in order to maximize the total surplus:
max_{α} μ+e-c(e)-((rα²σ²)/2)
s.t.IC₁.
Solving this problem, one can get the optimal power of incentive:
α₁^{∗}=(1/(1+rσ²c'')).
<proposition/>There are both a negative relationship between risk and performance pay and a negative relationship between agent's risk aversion attitude and performance pay.
For the principal, there is a trade off between inducing effort and providing insurance. The benefit of a higher power of incentive is that it induces a higher effort. The cost is that the principal needs to pay the agent a higher risk premium. As the risk or the agent's risk averse attitude increases, a higher power of incentive becomes more costly. Accordingly, the principal will reduce the power of incentive.
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