Question $9$ options:

Consider the following principal$-$agent problem between a manager and a worker. The manager of a car dealership wants to sell cars. Each car is sold at the price P$=$$$1,000.$ The worker $($salesperson$)$ needs to put effort to try and sell the cars. If the worker chooses effort level e$,$ then with probability Prob$=50$e she is able to sell $10$ cars, and with probability Prob$=1-50$e she is only able to sell $5$ cars.

It is costly for the worker to put effort trying to sell cars. For the worker, the effort cost is given by the function

C$=62,500$e$^2$

The manager will pay the worker a bonus b for each car sold. The worker's expected utility equals her expected income minus her effort cost. The worker chooses the effort that maximizes her expected utility.

The car dealer's only cost is the bonus it pays the worker, and the revenue comes from the cars sold. The manager chooses the bonus that maximizes the firm's expected profit.

$1)$ Find the best response function of the worker $($note that your answer here should be a function, not a number$).$ The best response function of the worker is

$2)$ The manager will optimally offer a bonus to the worker that is b$=$