Variable effort, the marginal value of net worth, and the pooling of equity:-

In the fixed-investment model, the shadow price of entrepreneurial net worth is equal to 0 almost everywhere and is infinite at the threshold A = A. A more continuous response arises when the entrepreneur's effort is continuous rather than discrete. The object of this exercise is to show that the shadow price is positive and decreasing in A in the range in which the entrepreneur is able to finance her project but must borrow from investors. It then applies the analysis to the internal allocation of funds between two divisions.

An entrepreneur has cash A and wants to invest I>A into a fixed-size project. The project yields R with probability p and 0 with probability 1 - p. Reaching a probability of success p requires the entrepreneur to sink (unobservable) effort cost 1 2p2 (there is no private benefit in this version). The borrower is risk neutral and is protected by limited liability. Investors are risk neutral and the market rate of interest is 0. Assume that .

(i) Note that, had the borrower no need to borrow (A ≥ I), the borrower's net utility would be independently of A.

(ii) Find the threshold A under which the project is not funded. (Hint: write the pledgeable income as a function of the entrepreneur's reward Rb in the case of success. Argue that one can focus attention on the values of Rb that exceedR. Do not forget that the NPV must be nonnegative.) Letting V (A) denote the NPV in the region in which the entrepreneur's project is financed. Show that the shadow price of net worth, V (A), satisfies

(iii) Following Cestone and Fumagalli (2005), consider two entrepreneurs, each with net worth A. Theywill each have a project described as above, but with random investment cost. For simplicity, one of them will face investment cost IH and the other IL, where

but it is not known in advance who will face which investment cost (each is equally likely to be the lucky entrepreneur). Investment costs, however, will become publicly known before the investments are sunk. Assume that

so that the only binding constraint for financing in question (ii) is the investors' breakeven constraint; and that

and so both projects can be financed by pooling resources. Do the entrepreneurs, behind the veil of ignorance, want to pool their resources and commit to force the lucky firm to cross-subsidize the unlucky one?

(Hint: show that under pooling, and, if both invest, the net worth is split so that both entrepreneurs have the same stake in success.)