Adverse selection and ratings:-
A borrower has assets A and must find financing for a fixed investment I>A. As usual, the project yields R (success) or 0 (failure). The borrower is protected by limited liability. The probability of success is pH or pL, depending on whether the borrower works or shirks, with ?p ≡ pH - pL > 0. There is no private benefit when working. The private benefit enjoyed by the borrower when shirking is either b (with probability α) or B (with probability 1 - α). At the date of contracting, the borrower knows her private benefit, but the market (which is risk neutral and charges a 0 average rate of interest) does not know it. Assume that pLR + B
(i) Interpret conditions (1) and (2) and show that there is no lending in equilibrium.
(ii) Suppose now that the borrower can at cost r (x) = r x (which is paid from the cash endowment A) purchase a signal with quality x ∈ [0, 1]. (This quality can be interpreted as the reputation or the number of rating agencies that the borrower contracts with.) With probability x, the signal reveals the borrower's type (b or B) perfectly; with probability 1 - x, the signal reveals nothing. The financial market observes both the quality x of the signal chosen by the borrower and the outcome of the signal (full or no information). The borrower then offers a contract that gives the borrower Rb and the lenders R - Rb in the case of success (so, a contract is the choice of an Rb ∈ [0, R]). The timing is summarized in Figure 6.5.
Look for a pure strategy, separating equilibrium, that is, an equilibrium in which the two types pick different signal qualities.79
- Argue that the bad borrower (borrower B) does not purchase a signal in a separating equilibrium.
- Argue that the good borrower (borrower b) borrows under the same conditions regardless of the signal's realization, in a separating equilibrium.
- Show that the good borrower chooses signal quality x ∈ (0, 1) given by
- Show that this separating equilibrium exists only if r is "not too large."