Project size increase at an intermediate date:-
An entrepreneur has initial net worth A and starts at date 0 with a fixed-investment project costing I. The project succeeds (yields R) or fails (yields 0) with probability p ∈ {pL, pH}. The entrepreneur obtains private benefit B at date 0 when misbehaving (choosing p = pL) and 0 otherwise. Everyone is risk neutral, investors demand a 0 rate of return, and the entrepreneur is protected by limited liability.
The twist relative to this standard fixed-investment model is that, with probability λ, the size may be doubled at no additional cost to the investors (i.e., the project duplicated) at date 1. The new investment is identical with the initial one (same date-2 stochastic revenue; same description of moral hazard, except that it takes place at date 1) and is perfectly correlated with it.
That is, there are three states of nature: either both projects succeed independently of the entrepreneur's effort, or both fail independently of effort, or a project for which the entrepreneur behaved succeeds and the other for which she misbehaved fails. Denote by Rb the entrepreneur's compensation in the case of success when the reinvestment opportunity does not occur, and by Rb that when both the initial and the new projects are successful. (The entrepreneur optimally receives 0 if any activity fails.) Show that the project and its (contingent) duplication receive funding if and only if
