Start-up and venture capitalist exit strategy:-

There are three periods, t = 0, 1, 2. The rate of interest in the economy is equal to 0, and everyone is risk neutral. A start-up entrepreneur with initial cash A and protected by limited liability wants to invest in a fixed-size project. The cost of investment, incurred at date 0, is I > A . The project yields, at date 2, R > 0 with probability p and 0 with probability 1 - p.

The probability of success is p = p_H if the entrepreneur works and p = p_L = p_H - ?_p (?p > 0) if the entrepreneur shirks. The entrepreneur's effort decision is made at date 0. Left unmonitored, the entrepreneur obtains private benefit B if she shirks and 0 otherwise. If monitored (at date 0), the private benefit from shirking is reduced to b 0 when monitoring the start-up and 0 otherwise (the subscript "A" refers to "active monitoring"). The twist is that the venture capitalist wants his money back at date 1, before the final return, which is realized at date 2 (technically, the venture capitalist has preferences c0+c1, while the entrepreneur and the uninformed investors have preferences c₀ + c₁ + c₂, where ct is the date-t consumption). Assume that

(i) Assume first that the financial market learns (for free) at date 1 whether the project will be successful or fail at date 2. Note that we are then in the standard two-period model, in which the outcome can be verified at date 1 (one can, for example, organize an IPO at date 1, at which the shares in the venture are sold at a price equal to their date-2 dividend). Show that the entrepreneur cannot be financed without hiring a venture capitalist. Write the two incentive constraints in the presence of a venture capitalist and show that financing is feasible. Show that the entrepreneur's utility is

(ii) Assume now that at date 1 a speculator (yet unknown at date 0) will be able to learn the (date-2) realization of the venture's profit by incurring private cost cP, where the subscript "P" refers to "passive monitoring.

At date 0, the venture capitalist is given s shares. The date-0 contract with the venture capitalist specifies that these s shares will be put for sale at date 1 in a "nondiscriminatory auction" with reservation price P. That is, shares are sold to the highest bidder at a price equal to the highest of the unsuccessful bids, but no lower than P. If left unsold, the venture capitalist's shares are handed over for free to the date-0 uninformed investors (the limited partners) in the venture.

(a) Find conditions under which it is an equilibrium for the speculator (provided he has monitored and received good news) to bid R for shares, and for uninformed arbitrageurs to bid 0 (or less than P).

(b) Write the condition on (s, P ) under which the speculator is indifferent between monitoring and not monitoring. Writing the venture capitalist's incentive constraint, show that P satisfies

How should the venture capital contract be structured if these conditions are not satisfied?