Mini Case $4\mathrm{.}2$ The company Golden Goose invests into a project that will either have a free cash flow of EUR $65,000$ or EUR $37,000.$ The probability of the good realisation depends on the effort of the project manager. If she exerts a lot of effort, the project will generate a cash flow of EUR $65,000$ with a probability of $80\%.$ If she does not put in high effort, the probability of success is only $30\%.$

The manager decides whether to put effort in a project based on her expected utility $($hint: the manager chooses to exert the effort if this maximizes her expected utility$).$ The manager$$s utility function is:

U$($compensation$)=$ log$($compensation$)$ High effort reduces the manager$$s utility by $0\mathrm{.}05.$

Equity investors are risk$-$neutral and pay the manager out of the cash flow from the project. The manager receives a fixed wage of EUR $4,000($independent of the outcome of the project$).$ Assume that the cost of equity capital is $0\%.$

a$.$ What is the expected value of equity, if the manager exerts low effort?

b$.$ What is the expected value of equity, if the manager exerts high effort?

c$.$ Will the manager choose to exert high or low effort?

Equity holders are thinking about a compensation schedule for the manager that depends on the success of the project. The manger gets $25\%$ of the project$$s value. There is no fixed wage.

d$.$ In this case, will the manager choose to exert high or low effort?

e$.$ What is the expected value of equity, if the manager exerts high effort?

f$.$ Which type of compensation will the equity investors offer the manager?

Instead of paying the manager based on the project$$s value, the equity holders can incentivize the manager by giving him a fixed amount of bonus in addition to the fixed salary if the project succeeds:

g$.$ What is the minimum bonus that makes the manager indifferent between exerting high or low effort?