Question: Your company is employing both single and family men. They are equally productive, producing output \(Y=e+X+1\), where effort \(e\) has cost \(C(e)=\frac{1}{2} e^{2}\), and the random factor \(X\) is equally likely to be \(-1\) or 1 . Revenue is $p Y$ with \(p=3\). Family men are more risk averse with risk aversion \(r_{f}=2\) while singles have \(r_{s}=1 / 2\).

1. What is expectation and variance of $X ?$ As in class, denote the variance by \(V\). Given \(e\), what is the expectation and variance of output \(Y\) ?
2. How much effort will single and family men exert under a linear contract \(W=\alpha+\beta Y ?\)
3. You can offer different contracts to single men \(\left(\alpha_{s}, \beta_{s}\right)\) and family men \(\left(\alpha_{f}, \beta_{f}\right)\). You want to choose each contract to maximize profit subject to the worker accepting. Assume that the outside option gives each type of worker utility \(\bar{u}=4\). What are the optimal contracts? \(^{1}\) Explain why these differ.
4. Calculate the expected compensation and expected profit for both types of employee. Explain why these differ.
5. Suppose a family man could pretend to be single, or a single man could pretend to be married. Would they want to?

(b) The agent needs to solve the following maximization problem, assuming that a linear contract is given:

Hence, we need to maximize

which leads directly to the optimal effort \(e*=\beta \).

(c) Now, given the response of the agent, the principal can decide on the optimal linear contracts \(\left( {{\alpha }_{F}},{{\beta }_{F}} \right)\) and \(\left( {{\alpha }_{S}},{{\beta }_{S}} \right)\) for family and single workers, considering the different risk aversion exhibited by the two different types of workers.

For family men

The principal will choose a binding IR condition, where \(U\left( agent \right)=\bar{u}\), and \(e=\beta \), so then, the optimal linear contract coefficients are:

which turns into:

For single men

The process is absolutely the same, and we get:

which turns into:

(d) For family men, the optimal effort e = 1/3. Hence, the expected profit is

For single men, the optimal effort e = 2/3. Hence, the expected profit is

(e) No, they would not, because their contract they would be offer would reduce their utility, with respect to the case of getting the right contract for their actual type.

Q2.

You employ two salespeople: One in Anaheim and one in Burbank. The Anaheim agent's sales, \(Y_{A}=e_{A}+X_{A}+X_{C}\), depend on his effort \(e_{A}\), the state of the Anaheim economy \(X_{A}\) and the state of the California economy \(X_{C}\) (both of which are random and out of his control). The Burbank agent's sales, \(Y_{B}=e_{B}+X_{B}+X_{C}\), depends on her effort \(e_{B}\), the state of the Burbank economy \(X_{B}\), and \(X_{C}\). All random variables \(X_{A}, X_{B}, X_{C}\) are independent. On top of the Anaheim manager's base wage \(\alpha\), you want to reward him for his absolute sales \(Y_{A}\) and/or his relative sales compared to Burbank \(Y_{A}-Y_{B}\).

1. What is the variance of \(Y_{A}\) and \(Y_{A}-Y_{B} ?^{2}\) If you can only choose one option, does paying on the basis of absolute sales or relative sales expose the Anaheim manager to less risk and therefore require the lower risk premium? How does this depend on the variance of \(X_{A}, X_{B}\) and \(X_{C} ?\)
2. Now you consider combinations of absolute sales and relative sales, i.e. \(\delta Y_{A}+(1-\) \(\delta\) ) \(\left(Y_{A}-Y_{B}\right)\). What level of \(\delta\) minimizes the risk to the Anaheim manager? What happens if \(X_{B}\) is much more uncertain than \(X_{C} ?\) Interpret your result.

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