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

(i) The uninformed lender (lender 2) makes zero expected profits. The informed lender (lender 1) makes strictly positive expected profits from any new customers, as well as from its old customers.

(ii) Lender 2 refrains from bidding with positive probability, so that 1 - [F.sub.2](R) > 0 (the probability that lender 2 plays the strategy D is positive).

(iii) Lender 1 bids [r.sub.1] = R with positive probability.

Proof As many of the arguments are fairly standard, we provide only a sketch of the proof. An undercutting argument proves that there cannot be an equilibrium with pure strategies, or one where both lenders make positive expected profits, as long as [[delta].sub.2] is not too low relative to [lambda], that is, for [lambda] < [??]. For X within the specified bounds, the proof that the distribution functions [F.sub.1] and [F.sub.2] are continuous and strictly increasing can be found in various sources (see, e.g., Fudenberg and Tirole, 1991), and can be constructed along the lines of Dell'Ariccia, Friedman, and Marquez (1999) or von Thadden (2004). Moreover, the same arguments can be used to show that both lenders mix over the same interval.

For this range of [lambda], we can show that the relative disadvantage of lender 2 leads it to make zero profits, whereas lender 1 makes positive profits from its private information. To see this, suppose to the contrary that equilibrium profits for lender 1, [[pi].sup.*.sub.1], are equal to zero. At the lowest price offered in equilibrium, [r.bar], lender 1 would have the lowest rate with probability 1, so that its profit would be [[pi].sub.1]([r.bar] | w) = [lambda]([bar.r][bar.[theta]] - (1 + [[delta].sub.1])) = 0 [??] [r.bar] = 1+[[delta].sub.1]/[bar.[theta]]. We can now substitute this lower bound into lender 2's profit expression to obtain

[[pi].sub.2]([r.bar] | w) = [lambda]([[delta].sub.1] - [[delta].sub.2]) + (1 - [lambda])1/4 ([[delta].sub.1] - 1 - 2[[delta].sub.2]). (5)

which is less than zero for [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), contradicting the assumption that lender 2 is willing to participate. Because one of the lenders must make zero profits, it therefore must be that lender 2 makes zero profits, as stated. We can now calculate the lower bound of the mixing distributions, [r.bar], directly from the zero-profit condition for lender 2, [[pi].sub.2]([r.bar] | w) = 0 [??] [lambda]([bar.r]1/2 - (1 + [[delta].sub.2])) + (1 - [lambda]) 1+[[delta].sub.1]/[r.bar]1/2([[delta].sub.1] - 1 - 2[[delta].sub.2] = 0, which when solved for [bar.r] yields the expression above.

The distribution functions [F.sub.1] and [F.sub.2] are obtained from the equilibrium conditions for the lenders' profits. Because each lender must be indifferent between all the rates that it offers in equilibrium, it must be that, for any given rate r [member of] [[bar.r], R), the lender's expected profit must be equal to its equilibrium profit. For lender 2, this profit is just 0. For lender 1, the equilibrium expected profit is given by the expression [[pi].sub.1]([bar.r] | w), defined above, because at the lowest possible rate, [bar.r], lender 1 can be sure of having the best rate and attracting all the new customers. Therefore, the expected profits in equilibrium for lenders 1 and 2 are

[[pi].sub.1](r) = (1 - [F.sub.2](r))[[pi].sub.1](r | w) = [[pi].sub.1] ([r.bar] | w) (6)

[[pi].sub.2](r) = (1 - [F.sub.1](r))[[pi].sub.2](r | w) = [F.sub.1] (r)[[pi].sub.2](r | w) = 0. (7)

These equations can be inverted to obtain [F.sub.1](r) = 1 + (1-[lambda]) (1+[[delta].sub.1])([[delta].sub.1] - 1 - 2[[delta].sub.2])/2[r][delta] (r[bar.[theta]-(1+[[delta].sub.2])) and [F.sub.2](r) = [bar.[theta](r-[r.bar])/r[bar.[theta]-(1+[[delta].sub.1]). Note that r = R is the highest rate that can be offered in equilibrium because R is the maximum return on the project, and that both [F.sub.1](R) and [F.sub.2](R) < 1. An undercutting argument demonstrates that both lenders cannot offer the same rate r = R with positive probability, and as lender 2 obtains zero profits in equilibrium, it can be shown that the difference 1 - [F.sub.2](R) represents a probability of not bidding.

As a final step, we calculate the bound [bar.[lambda]] from the participation constraint for lender 1, [[pi].sub.1]([bar.r] | w) [greater than or equal to] 0. Substituting in for [bar.r], we can solve for the value of [lambda] that satisfies this expression with equality to obtain [bar.[lambda]] [equivalent to] 2[[delta].sub.2]+1-[[delta].sub.1]/3[[delta].sub.1]-2[[delta].sub.2]+1. To find the lower bound [[lambda].bar]. we evaluate lender 2's profits at the highest possible price, R, assuming that no lender offers a lower price, [[pi].sub.2](R | w). In order for [[pi].sub.2](R | w) to at least be equal to zero, we need [lambda] [greater than or equal to] [bar.[lambda]] = (1+[[delta].sub.1])(2[[delta].sub.2]+1- [[delta].sub.1])/[R.sup.2]-R(1+[[delta].sub.2])+(1+[[delta].sub.1]) (2[[delta].sub.2]+1-[[delta].sub.1]), desired. Q.E.D.

This proposition points out that, for the specified range for the borrower turnover parameter, the informational advantage of lender 1 also endows it with a competitive advantage. The information advantage allows the informed lender to reap profits not just from its old customers but also from borrowers to which it has not previously lent, which for lender 1 are just the new borrowers. This occurs because the informational advantage of lender 1 leads lender 2 to compete less aggressively, yielding higher profits to lender 1 on all its borrowers.

Having characterized the equilibrium, we proceed with the analysis of the effects of changes in each lender's cost of funds. As argued above, lender 1 grants credit to all old borrowers of type [theta] [greater than or equal to] [??] = 1+[[delta].sub.1]/[r.sub.2] as long as lender 2 bids, and 1+[[delta].sub.1]/R otherwise. Therefore, the expected quality of the marginal old borrower that obtains credit from the informed lender is E[[??]] = Pr(lender 2 bids)E[1+[[delta].sub.1]/[r.sub.2] | [r.sub.2] [less than or equal to] R] + (1 - Pr(lender 2 bids))1+[[delta].sub.1]/R. The following set of preliminary results will prove useful in establishing the main results that follow.

Lemma 1. The distribution functions over interests for lenders 1 and 2, [F.sub.1] and [F.sub.2], are increasing in lender 1's cost of funds, [[delta].sub.1], and decreasing in lender 2's cost of funds, [[delta].sub.2].

Proof From the proof of Proposition 1, the distribution function for lender 1 is [F.sub.1](r) = 1 + (1-[lambda])(1+[[delta].sub.1]) ([[delta].sub.1]-1-2[[delta].sub.2])/2r[lambda](r[bar.[theta]- (1+[[delta].sub.2])). Therefore, [partial derivative][F.sub.1](r)/ [partial derivative][[delta].sub.1] = (1- [lambda])(1+[[delta].sub.1])([[delta].sub.1]-[[delta].sub.2] [greater than or equal to] 0 for [[delta].sub.1] [greater than or equal to] [[delta].sub.2], with the inequality strict whenever [[delta].sub.1] > [[delta].sub.2]. Similarly, [partial derivative] [F.sub.1](r)/[partial derivative][[delta].sub.2] = (1-[lambda]) (1+[[delta].sub.1])([[delta].sub.1]+1-2r[bar.[theta]])/2r[lambda] [(r[bar.[theta]]-(1+[[delta].sub.2])).sup.2] < 0 because r [greater than or equal to] 1 + [[delta].sub.2] > 1 + [[delta].sub.2].

We also obtain [F.sub.2] from Proposition 1 as [F.sub.2](r) = [bar.[theta]](r-[r.bar])/[(r[bar.[theta]]- (1+[[delta].sub.1])).sup.2], which implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because the second term, ([bar.[theta]](r-[r.bar])/ [(r[bar.[theta]]-(1+[[delta].sub.1])).sup.2], is positive, the result will be established if [partial derivative][r.bar]/[partial derivative] [[delta].sub.1] [less than or equal to] 0. Recalling that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as [[delta].sub.1] [greater than or equal to] [[delta].sub.2]. Therefore, [partial derivative][F.sub.2](r)/[partial derivative][[delta].sub.1] > 0. Similarly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and it is immediate from the definition of [bar.r] that [partial derivative][bar.r]/[partial derivative][[delta].sub.2] > 0, implying that [partial derivative][F.sub.2](r)/[partial derivative][[delta].sub.2] < 0. Q.E.D.

We can now state and prove the following comparative statics result.

Corollary 1. The expected marginal old borrower (E[[??]]) obtaining credit from lender 1 is

(i) Increasing in [[delta].sub.1]: [partial derivative]E[bar.[theta]]/ [partial derivative][[delta].sub.1] > 0.

(ii) Decreasing in [[delta].sub.2]: [partial derivative]E[bar.[theta]]/ [partial derivative][[delta].sub.2] < 0.

Proof Note that E[[??]] = [F.sub.2](R)E[[1+[[delta].sub.1]/[r.sub.2] [less than or equal to] R] + (1 - [F.sub.2](R))1+[[delta].sub.1]/R, because [F.sub.2](R) is the probability of a bid by lender 2. We can write this expression as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

We first integrate this to get a simpler expression,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

The derivative of this expression is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

= 1/R + [[integral].sup.R.sub.[r.bar]] (1/[r.sup.2][F.sub.2] + 1 + [[delta].sub.1]/[r.sup.2] [partial derivative][F.sub.2]/[partial derivative][[delta].sub.1]) dr, (12)

as [F.sub.2]([r.bar]) is zero. From this, we see that because [partial derivative][F.sub.2](r)/[partial derivative][[delta].sub.1]) > 0 (Lemma 1), we have that [partial derivative]E[[bar.[theta]]/[partial derivative][[delta].sub.1]) > 0.

For the second part of the corollary,