7. A risk-neutral principal hires an agent to work on a project at wage te. The agent's utility function is U(w,e₁)=√w-g(e) where g(e) is the disutility associated with the effort level e, exerted on the project. The agent can choose one of two possible effort levels, ej or ez, with associated disutility levels g(e) = 4, and g(ez)= 2.5. If the agent chooses effort level e, the project yields 100 with probability 1/2, and 0 with probability 1/2. If he chooses ez, the project yields 100 with probability 1/4 and 0 with probability 3/4. The reservation utility of the agent is 0. (a) If effort is observable, which effort level should the principal implement? Derive the best wage contract that implements this effort. Does this involve sharing risk with the agent? Explain your answer. (b) If effort is not observable, which effort level should the principal imple- ment? What is the best wage contract that implements this effort? (c) Suppose an insurance company observes the wage contract offered to the agent in part (b), and knows the probabilities associated with each wage level, and assumes these to be fixed. Suppose the insurance company offers the agent full insurance at a premium of 24 (the premium is paid by the agent to the insurance company whether the wage is high or low, and the insurance company covers the loss in income if the wage is low). Woul- the agent accept this, and would the insurance company make a profit from this?
7. A risk-neutral principal hires an agent to work on a project at wage te. The agent's utility function is U(w,e₁)=√w-g(e) where g(e) is the disutility associated with the effort level e, exerted on the project. The agent can choose one of two possible effort levels, ej or ez, with associated disutility levels g(e) = 4, and g(ez)= 2.5. If the agent chooses effort level e, the project yields 100 with probability 1/2, and 0 with probability 1/2. If he chooses ez, the project yields 100 with probability 1/4 and 0 with probability 3/4. The reservation utility of the agent is 0. (a) If effort is observable, which effort level should the principal implement? Derive the best wage contract that implements this effort. Does this involve sharing risk with the agent? Explain your answer. (b) If effort is not observable, which effort level should the principal imple- ment? What is the best wage contract that implements this effort? (c) Suppose an insurance company observes the wage contract offered to the agent in part (b), and knows the probabilities associated with each wage level, and assumes these to be fixed. Suppose the insurance company offers the agent full insurance at a premium of 24 (the premium is paid by the agent to the insurance company whether the wage is high or low, and the insurance company covers the loss in income if the wage is low). Woul- the agent accept this, and would the insurance company make a profit from this?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:7. A risk-neutral principal hires an agent to work on a project at wage to. The
agent's utility function is
U(w,e₁)=√w-g(₁)
where g(e) is the disutility associated with the effort level e, exerted on the
project.
The agent can choose one of two possible effort levels, ej or ez, with associated
disutility levels g(e) = 4, and g(₂) 2.5. If the agent chooses effort level e,
the project yields 100 with probability 1/2, and 0 with probability 1/2. If he
chooses e2, the project yields 100 with probability 1/4 and 0 with probability
3/4.
The reservation utility of the agent is 0.
(a) If effort is observable, which effort level should the principal implement?
Derive the best wage contract that implements this effort. Does this involve
sharing risk with the agent? Explain your answer.
(b) If effort is not observable, which effort level should the principal imple-
ment? What is the best wage contract that implements this effort?
(c) Suppose an insurance company observes the wage contract offered to the
agent in part (b), and knows the probabilities associated with each wage
level, and assumes these to be fixed. Suppose the insurance company offers
the agent full insurance at a premium of 24 (the premium is paid by the
agent to the insurance company whether the wage is high or low, and the
insurance company covers the loss in income if the wage is low). Would
the agent accept this, and would the insurance company make a profit from
this?
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