An entrepreneur has a project that requires an investment outlay of I = 55 and g
ID: 3048041 • Letter: A
Question
An entrepreneur has a project that requires an investment outlay of I = 55 and generates profits X that can take values {XL, XH}, with XL = 30 and XH = 120. The entrepreneur can exert high effort eH or low effort eL. Let p(e) be the probability that profits are XH when the entrepreneur puts effort e. Assume that p(e H) = 0.8 and p(e L) = 0.2. The cost of high effort is c(e H) = 27 and the cost of low effort is c(e L) = 0. Assume throughout that the entrepreneur is risk neutral.
(i) Suppose the entrepreneur has initial wealth A I (i.e., she has enough money to undertake the project without borrowing). If she invests her own money to undertake the project, what level of effort will she choose (i.e., what effort level maximizes her expected utility net of effort cost)? Will she find it profitable to invest her own money to undertake the project?
(ii) Suppose now that the entrepreneur has no resources of her own (i.e., A = 0) and considers financing the project by selling equity. She offers a fraction [0, 1] of the project’s final profits to an outside investor who is risk neutral, while retaining 1 for herself. Show that, for 1/2 the entrepreneur finds it optimal to put high effort, and for > 1/2 the entrepreneur finds it optimal to put low effort.
(iii) Using your answer to part (ii), show that for any [0, 1], the outside investor will get negative profits. Conclude that the investor is not willing to finance the project with equity.
(iv) Suppose now that the entrepreneur finances the project by offering to repay the investor XL = 30 if profits are XL and to repay Z (XL, XH) if profits are XH. Hence, in this case the entrepreneur earns 0 if profits are XL and earns XH Z if profits are XH. Show that, for any Z < 75, the entrepreneur finds it optimal to put high effort.
(v) Suppose that Z = 70. Show that, under this repayment scheme, the investor makes positive expected profits, and is thus willing to finance the project.
Explanation / Answer
a) When the entrepreneur has initial wealth more than the investmnet required then he is not worried and so puts low effort because it doesnt cost him any thing and gets a return of 30.
b) Now without initial wealth he needs to generate 55 for investment. And suppose he highers outside investor then
if he offers the outside investor less than or equal half then he needs to generate more to fulfill and so he has to put more efforts and that also cost him and so he goes to put more effort. If he offers outside investor more than half then his need of investment can be satisfied with low efforts also and so he puts low efforts at no cost.