Sep 19, 2017

(MORE ON POOLING IN CREDIT MARKETS). CONSIDER THE MODEL OF EXERCISE 6.1, IN WHICH THE BORROWER…

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(more on pooling in credit markets). Consider the model of Exercise 6.1, in which the borrower has private information about her bene- fit of misbehaving, except that the borrower’s type is drawn from a continuous distribution instead of a binary one. We will also assume that there is a monopoly lender, who makes a credit offer to the borrower. The borrower has no equity (A = 0). Only the borrower knows the private benefit B of misbehaving. The lender only knows that this private benefit is drawn from an ex ante cumulative distribution H(B) on an interval [0, B]¯ (so, H(0) = 0, H(B)¯ = 1). (Alternatively, one can imagine that lenders face a population of borrowers with characteristic B distributed according to distribution H, and are unable to tell different types of borrower apart in their credit analysis.) The lender knows all other parameters. For a loan agreement specifying share Rb for the borrower in the case of success, and 0 in the case of failure, show that the lender’s expected profit is

the proportion of “high-quality borrowers” (that is, of borrowers who behave) is endogenous and increases with Rb; 78

• adverse selection reduces the quality of lending (if lending occurs, which as we will see cannot be taken for granted);

• there is an externality among different types of borrower, in that the low-quality types (B large) force the lender to charge an interest rate that generates strictly positive profit on high-quality types (those with small B);

• the credit market may “break down,” that is, it may be the case that no credit is extended at all even though the borrower may be creditworthy (that is, have a low private benefit). To illustrate this, suppose that pL = 0 and H is uniform (H(B) = B/B¯). Show that if

(which is the case for B¯ large enough), no loan agreement can enable the lender to recoup on average his investment.

Exercise 6.1

(privately known private benefit and market breakdown). Section 6.2 illustrated the possibility of market breakdown without the possibility of signaling. This exercise supplies another illustration. Let us consider the fixed-investment model of Section 3.2 and assume that only the borrower knows the private benefit associated with misbehavior. When the borrower has private information about this parameter, lenders are concerned that this private benefit might be high and induce the borrower to misbehave. In the parlance of information economics, the “bad types” are the types of borrower with high private benefit. We study the case of two possible levels of private benefit (see Exercise 6.2 for an extension to a continuum of possible types). The borrower wants to finance a fixed-size project costing I, and, for simplicity, has no equity (A = 0). The project yields R (success) or 0 (failure). 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 B enjoyed by the borrower when shirking is either BL > 0 or BH > BL. The borrower will be labeled a “good borrower” when B = BL and a “bad borrower” when B = BH. At the date of contracting, the borrower knows the level of her private benefit, while the capital market puts (common knowledge) probabilities α that the borrower is a good borrower and 1−α that she is a bad borrower. All other parameters are common knowledge between the borrower and the lenders. To make things interesting, let us assume that under asymmetric information, the lenders are uncertain about whether the project should be funded

Assume that investors cannot break even if the borrower shirks:

(i) Note that the investor cannot finance only good borrowers. Assume that the entrepreneur receives no reward in the case of failure (this is indeed optimal); consider the effect of rewards Rb in the case of success that are (a) smaller than BL/∆p, (b) larger than BH/∆p, (c) between these two values.

(ii) Show that there exists α∗, 0 ∗

• no financing occurs if 

• financing is an equilibrium

(iii) Describe the “cross-subsidies” between types that occur when borrowing is feasible



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