Can cost increases increase competition? Asymmetric information and equilibrium prices.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

because [partial derivative][F.sub.2](r)/[partial derivative] [[delta].sub.1]) < 0 (Lemma 1). Q.E.D.

The effect on lender 1's share of old customers through increases in its own cost [[delta].sub.1] is direct: as its cost increases, the threshold quality level [??] increases for each interest rate offered by lender 2. Therefore, in equilibrium, the expected threshold increases, and lender 1's expected share of old customers, 1 - E[[??]], decreases. Conversely, as the cost of the uninformed lender, [[delta].sub.2], increases, any cost advantage lender 1 has gets reduced. This leads to an equilibrium where the uninformed lender bids less aggressively, allowing lender 1 to profitably retain a larger fraction of its old customers. In other words, the number of old borrowers financed by the informed lender, 1 - E[[??]], increases as lender 2's cost advantage decreases.

Given the asymmetric effect stemming from increases in [[delta].sub.1] versus [[delta].sub.2], a natural question is to ask what would happen if both lenders' costs increased at the same time. However, since by assumption [[delta].sub.1] > [[delta].sub.2], there is no unambiguous way of defining what constitutes an "equal increase" in both lenders' costs. One simple way to do this is to assume that cost increases are proportional by requiring lender 2's cost to increase in the same proportion as that of lender 1. Formally, define [beta] [equivalent to] [[delta].sub.2]/[[delta].sub.1] < 1, and assume that any change in [[delta].sub.1] must be matched by a proportional change in [[delta].sub.2]: [DELTA][[delta].sub.2] = [beta][DELTA][[delta].sub.1]. We can now take the derivative of E[[??]] with respect to [[delta].sub.1], as above. The only thing that is different here is the derivative of the lower bound [r.bar], which is now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Note that for [beta] [right arrow] 0, expression (14) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

At the other extreme, for [beta] [right arrow] 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

From this it is clear that there must be a value [[beta].bar] [member of] (0, 1) such that, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as before. (9) Our results therefore extend to the case where both lenders' costs increase proportionally, as long as lender 2's cost structure is sufficiently more favorable than lender 1's (i.e., [beta] is sufficiently small).

The intuition for this result is useful for establishing our main result.

Proposition 2. For [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), the expected rate lender l's old borrowers pay, [r.sub.OLD], is

(i) decreasing in [[delta].sub.1] the cost of funds for lender 1: [partial derivative]E[[r.sub.OLD]]/[partial derivative][[delta].sub.1] < O;

(ii) increasing in [[delta].sub.2], the cost of funds for lender 2: [partial derivative]E[[r.sub.OLD]]/[partial derivative][[delta].sub.2] > 0.

Proof. The expected rate paid by lender l's old customers is E[[r.sub.OLD]] = E[[r.sub.2]] whenever lender 2 bids, and R otherwise. Therefore,

E[[r.sub.OLD]] = [F.sub.2](R)E [[r.sub.2] | [r.sub.2] < R] + (1 - [F.sub.2](R))R = [??] rd [F.sub.2](r) + (1 - [F.sub.2](R))R. (17)

We first integrate this to get a simpler expression,

E[[r.sub.OLD]] = R - [??] [F.sub.2](r) dr. (18)

Parts 1 and 2 now follow from Lemma 1 and the fact that [F.sub.2]([r.bar]) = 0. Q.E.D.

The proposition illustrates our main result, that the equilibrium expected interest rate paid by old borrowers reacts asymmetrically to changes in the two lenders' costs. In particular, higher costs for lender 1 imply lower interest rates paid by old borrowers. This result is driven by the effect of the information asymmetry between lenders in competition in the loan market. As established in Proposition 1, lender l's information advantage provides it with a competitive advantage that limits the extent to which lender 2 is willing to bid in this market. However, if lender l's cost is higher, the number of marginally creditworthy borrowers lender 1 is able to profitably serve at prevailing interest rates will be lower (Corollary 1). Many of these borrowers not served by lender 1 may nevertheless be ones that yield a positive profit for lender 2. This improvement in the pool of borrowers denied credit by lender 1 reduces the adverse selection problem that lender 2 faces, and allows it to compete more aggressively. Because lender 2's profit must remain at zero, the net effect of this increased competition is to lower the rate lender 2 offers to all borrowers, thus lowering the rate that lender 1 is forced to match in order to retain its old customers. (10)

The intuition for this result is illustrated in Figure 1. An increase in the cost of funds for lender 1, [[delta].sub.1], leads to an increase in [??], the marginal old customer served by lender 1. However, these borrowers rejected by lender 1 will instead be financed by lender 2. But the addition of these borrowers improves the average quality of the pool of old borrowers financed by lender 2, because they are of higher-than-average quality for lender 2 (in fact, they are the best old borrowers financed by lender 2). The qualitative improvement in lender 2's pool of borrowers as a result of the informed lender's higher cost allows lender 2 to offer a lower interest rate to all borrowers.

The intuition for the result relating to the cost of funds for lender 2 is more straightforward. This is just the usual result that when one competitor's cost is higher, some of that greater cost is likely to be passed on to consumers. Therefore, when lender 2's cost of funding is higher, it offers a higher interest rate on average, which lender 1 then matches for its retained customers.

We have focused so far on the effect of cost changes on the rate offered by lender 2 not only because this is the rate that in equilibrium applies to all the old customers but also because it is precisely lender 2 that provides the competition in this market and prevents lender 1 from appropriating all the rents. For completeness, however, we offer the following result that demonstrates that new customers also benefit from being in a market where lender 1 has a higher cost, which along with Proposition 2 implies that in fact all borrowers in this market are better off.

[FIGURE 1 OMITTED]

Proposition 3. An increase in the cost of funds for lender 1, [[delta].sub.1], leads to

(i) a decrease in the expected rate offered by lender 1, E[[r.sub.1]]; (ii) a decrease in the expected rate obtained by all new customers, E[min{[r.sub.1], [r.sub.2]}].

Proof. Denote the expected rate for lender 1 by E[[r.sub.1]]. Because E[[r.sub.1]] = R - [??] [F.sub.1](r) dr, then [partial derivative]E[[r.sub.1]]/[partial derivative][[delta].sub.1] [less than or equal to] 0 as [partial derivative][F.sub.1](r)/[partial derivative][[delta].sub.1] [greater than or equal to] 0 for [[delta].sub.1] [greater than or equal to] [[delta].sub.2] (Lemma 1), with the inequality strict if [[delta].sub.1] > [[delta].sub.2].

For the second part of the proposition, denote the expected rate for new borrowers by [??]. If lender 2 does not bid, which occurs with probability 1 - [F.sub.2](R), new borrowers receive a loan at the rate [r.sub.1]. If lender 1 were not to bid, new borrowers would pay [r.sub.2], assuming lender 2 bids. However, this is a zero probability event and can be ignored, as lender 1 always bids. Finally, both lenders might bid, which occurs with probability [F.sub.2](R), in which case new borrowers pay the rate [??] = min{[r.sub.1], [r.sub.2]}. Therefore, we have that

E[[??]] = (1 - [F.sub.2](R))E[[r.sub.1]] + [F.sub.2](R)E[min{[r.sub.1], [r.sub.2]} | [[r.sub.2] [less than or equal to] R]. (19)

Let us first focus on the last term. Because this must be calculated conditional on lender 2 bidding, the conditional distribution for a bid by lender 2 is [[??].sub.2](r) = [F.sub.2](r)/[F.sub.2](R). The distribution of the minimum rate is then F(r) = [F.sub.l](r) + [[??].sub.2](r) - [F.sub.l] (r)[[??].sub.2](r). We now substitute into the expression above to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Because [partial derivative][F.sub.1](r)/[partial derivative][[delta].sub.1] [greater than or equal to] 0 and [partial derivative][F.sub.2](r)/[partial derivative][[delta].sub.1] > 0, but [F.sub.2](r), [F.sub.1](r) < 1 for r < R, this implies that the expression inside the integral is negative, proving that [partial derivative]E[[??]]/[partial derivative][[delta].sub.1] < 0, as desired. Q.E.D.

The intuition for this result is the following. As for a monopolist facing a competitive fringe, the interest rate that lender 1 offers to the market is not driven by its own cost of funding, but rather by the interest rate offered by lender 2. It follows that when lender 1 has a higher cost of funding, adverse selection decreases, and hence lender 2 can bid more aggressively. Lender 1 is then forced to compete more fiercely for new borrowers even if it will have to accept a lower margin on each loan.

4. Entry and exit