A firm has applied for a loan with the goal to fund a risky project. The bank is considering whether to grant a loan to the firm. The bank knows that the firm may choose between two projects: A or B$.$ The return of these two projects is described below.

A: Invest $40$ and obtain $100$ with probability $0\mathrm{.}6$ and $50$ with probability $0\mathrm{.}4.$

B: Invest $40$ and obtain $110$ with probability $0\mathrm{.}5$ and $0$ with probability $0\mathrm{.}5.$

The choice, made by the firm, of which project to invest in is not verifiable by the bank.However, the outcome of the project $($i$.$e$.,$ whether the firm obtains $100,110,50,$ or $0)$ is verifiable. The bank is a monopolist in the lending market $($that is$,$ there is no other bank that can grant a loan to the firm$)$ and the bank can guarantee to obtain $45$ for lending $40$ of its funds elsewhere. The firm does not have funds of its own to carry out the project, so that the firm gets a value of $0$ if not funded by the bank. The firm operates with limited liability. That is$,$ a loan contract specifies that the firm will have to pay $$R in case the project

realized return is larger than $$R$.$

a$)$ If R$=60,$ calculate the firm$$s profit for choosing project A and project B$.$ Which

project would the firm choose?

b$)$ If R$=30,$ calculate the firm$$s profit for choosing project A and project B$.$ Which

project would the firm choose?

c$)$ What amount R should the bank charge in order for the firm to be indifferent

between choosing project A or project B$?$

d$)$ Which is the profit$-$maximizing amount R that the bank should charge the firm?

Which project will be implemented? What is the firm$$s expected profit under this

contract?