Strategic leverage:-
(i) A borrower has assets A and must find financing for an investment I(τ) > A. As usual, the project yields R (success) or 0 (failure). The borrower is protected by limited liability. The probability of success is pH+τ or pL+τ, depending on whether the borrower works or shirks, with ?p = pH - pL > 0. There is no private benefit when working and private benefit B when shirking. The financial market is competitive and the expected rate of return demanded by investors is equal to 0. It is never optimal to give incentives to shirk.
The investment cost I is an increasing and convex function of τ (it will be further assumed that pHR > I(0), that in the relevant range pH + τ ∗, A∗, and τ∗∗ be defined by
Can the borrower raise funds? If so, what is the equilibrium level τ of "quality of investment"?
(ii) Suppose now that there are two firms (that is, two borrowers) competing on this product market. If only firm i succeeds in its project, its income is (as in question (i)), equal to R (and firm j's income is 0). If the two firms succeed (both get hold of "the technology"), they compete à la Bertrand in the product market and get 0 each. For simplicity, assume that the lenders observe only whether the borrower's income is R or 0, rather than whether the borrower has succeeded in developing the technology (showoffs: you can discuss what would happen if the lenders observed "success/failure"!). So, if qi ≡ pi +τi denotes the probability that firm i develops the technology (with pi = pH or pL), the probability that firm i makes R is qi(1 - qj ). (This assumes implicitly that projects are independent.) Consider the following timing.
(1) Each borrower simultaneously and secretly arranges financing (if feasible). A borrower's leverage (or quality of investment) is not observed by the other borrower.
(2) Borrowers choose whether to work or shirk.
(3) Projects succeed or fail.
Interpret τˆ.
- Suppose that the two borrowers have the same initial net worth A. Find the lower bound Aˆ on A such that (τ, ˆ τ) ˆ is the (symmetric) Nash outcome.
- Derive a sufficient condition on A under which it is an equilibrium for a single firm to raise funds. (iii) Consider the set up of question (ii), except that borrower 1 moves first and publicly chooses τ1.
Borrower 2 may then try to raise funds (one will assume either that τ2 is secret or that borrower 1 is rewarded on the basis of her success/failure performance; this is in order to avoid strategic choices by borrower 2 that would try to induce borrower 1 to shirk). Suppose that each has net worth A˜ given by
