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

* Consider an economy where there is a continuum of entrepreneurs, normalized to be of size 1. Each entrepreneur is endowed with an investment project that requires a capital inflow of $1, but has no private resources, so that she must look to a lender to obtain this financing. Projects pay off an amount R with probability [theta], and 0 with probability 1 - [theta]. We assume that this outcome is perfectly observable and contractible by both parties, but that the parameter describing the probability of success, [theta], is unknown to either the borrower or the lender before entering into a credit relationship. [theta] is uniformly distributed between 0 and 1, with average success probability = 1/2. We assume that once a borrower obtains a loan from a lender, that lender learns the borrower's type [theta]. Neither the lender nor the entrepreneur can credibly communicate the type information to other lenders.

The market is composed of [lambda]. new borrowers and 1 - [lambda] old borrowers. Both of these groups have the same distribution over types given above. We assume that lenders are unable to distinguish between new borrowers and borrowers that are being rejected by a competing lender or who are simply switching lenders to take advantage of lower rates. Although this is a strong assumption, as borrowers often carry with them some kind of credit history which may be available to competitor lenders, it captures the idea that a borrower's prior lender possesses better information than what is available on a credit record. (2) This information may be gathered through either monitoring or having access to books or simply through being able to better observe the kind of projects in which a borrower invests (see Lummer and McConnell, 1989, for evidence that banks may gather private information about borrowers over the course of the relationship). In this sense, the borrower's current lender has an informational advantage over competing lenders, as other lenders are only able to less precisely determine an applicant borrower's type. (3)

There are two lenders in the market. Lender 1 (informed) has a pre-existing market share of 100%, and thus perfect information about all the old borrowers. (4) Lender 1 also has access to an unlimited supply of funds at a constant gross interest rate given by 1 + [[delta].sub.1]. Lender 2 (uninformed) has a pre-existing market share of 0, and thus no information about old borrowers, but has access to an equally unlimited supply of funds at a constant gross rate 1 + [[delta].sub.2], where [[delta].sub.1] [greater than or equal to] [[delta].sub.2] [greater than or equal to] 0. (5)

The timing of the model is as follows. Competition for borrowers occurs in two stages. First, both lenders simultaneously choose an interest rate for the pool of borrowers unknown to them, which for lender 1 consists simply of the [lambda] new borrowers, whereas for lender 2 it consists of the whole market. Lenders choose their gross interest rates from the set [0, R] [union] {D}, where D represents not offering a loan (denying credit). Then, after observing the realized rates for all lenders, lender 1 chooses an interest rate for its old customers. Essentially, we allow all borrowers to seek competitive offers, but assume that the informed lender may make a counteroffer to any old customers it wishes to retain. (6) These rates may be type contingent, as they apply to borrowers whose quality is known to their current lender. Borrowers act last by choosing the lowest interest rate offered to them.

3. Analysis of a cost increase

* We solve the game by backward induction. We first characterize the equilibrium of the subgame after lenders have submitted a bid to all unknown borrowers and lender 1 can now offer a counterbid to its old customers. Then we solve the first stage, where both lenders compete for new borrowers.

It is straightforward to show that, in equilibrium, lender 1 is able to retain all of its borrowers of sufficiently good quality, and denies financing to all borrowers for whom lending at prevailing interest rates would yield it losses. To be explicit, let [r.sub.i] be the interest rate charged by lender i to all unknown customers, and let [r.sub.1[theta]] be the interest rate charged by lender 1 to an old customer of type [theta] (rates are denoted as gross interest rates, that is, net interest plus principal). To retain a customer, lender 1 needs to offer that customer a rate no higher than that being offered by lender 2. Therefore, all old borrowers retained by lender 1 are charged a matching rate, [r.sub.1[theta]] = [r.sub.2]. At this rate, however, lender 1 will want to retain only borrowers of sufficiently high quality, for whom [r.sub.2[theta]] [greater than or equal to] 1 + [[delta].sub.1]. We can therefore define the threshold or cutoff quality level of old borrowers who obtain financing from lender 1 as [??] [equivalent to] 1 + [[delta].sub.1]/[r.sub.2], as long as lender 2 bids. If lender 2 does not make an offer, lender 1 can charge its old customers the maximum rate R without fear of losing them, so that the cutoff value [??] becomes 1 + [[delta].sub.1]/R. Note that, because all retained customers obtain the same matching rate, we can now drop the type dependency and use the notation [r.sub.OLD] to refer to the rate offered to all of lender 1's old customers.

We use this threshold [??] to define the pool of loan applicants to lender 2 as all new borrowers plus all borrowers rejected by lender 1, that is, all old borrowers with repayment probability [??] [less than or equal to] [??]. Consider then the competition for these borrowers. Lender 1's profit on unknown borrowers, conditional on having the strictly lowest ("winning") rate, is

[[pi].sub.1]([r.sub.1] | w) = [lambda] [[integral].sup.1.sub.0] ([r.sub.1][theta] - (1 + [[delta].sub.1]))d[theta] = [lambda]([r.sub.1][??] - (1 + [[delta].sub.1])), (1)

because it only competes for new borrowers.

For lender 2, its profit conditional on having the strictly lowest interest rate offer comprises two terms, one for the new borrowers and the second for lender 1's rejected borrowers,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

= [lambda]([r.sub.2][??] - (1 + [[delta].sub.2])) + (1 - [lambda]) (1 + [[delta].sub.1]/[r.sub.2]) 1/2([[delta].sub.1] - 1 - 2[[delta].sub.2]). (3)

The second term in equation (3) is negative for [[delta].sub.2] > [[delta].sub.1] - 1/2, because lender 1 only casts out those borrowers for which [theta][r.sub.2] < 1 + [[delta].sub.1]. (7)

Conditional on having the higher rate ("losing"), lender 1 extends no new loans and therefore makes zero profits on new borrowers. Lender 2, however, makes loans to lender 1's rejected borrowers, so that its payoff is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Note that whereas [[pi].sub.2](r | l) (equation 4) is negative, [[pi].sub.2](r | w) (equation 3) can be positive for sufficiently large values of [r.sub.2] or low values of [[delta].sub.2]. This is simply because no adverse selection effects operate with respect to the new borrowers (the first term in equation (3)), so that their expected repayment probability is the mean of the full distribution of borrowers.

In order to focus on situations where there is actual competition between the two lenders, we assume that [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), where 0 < [[lambda].bar] < [bar.[lambda]]. < 1. For [lambda] > [bar.lambda]], lender 1's information advantage is very small, so that any cost advantage lender 2 has will enable it to bid a low rate and completely squeeze lender 1 out of the market. This is equivalent to the case where lender 2 could offer a rate [r.sub.2] [less than or equal to] 1 + [[delta].sub.1]/[bar.[theta]] and still at least break even, which lender 1 clearly could not match. Conversely, for [lambda] < [[lambda].bar], lender 1's information advantage is so large that lender 2 is shut out of the market completely as it would face an insurmountable adverse selection problem were it to enter. This case is obtained from the participation constraint for lender 2, that [[pi].sub.2](R | w) [greater than or equal to] 0. These bounds are given explicitly by [[lambda].bar] [equivalent to] (1+[[delta].sub.1])(2[[delta].sub.2] + 1 - [[delta].sub.1])/[R.sup.2] - 2 (1+[[delta].sub.2]) + (1+[[delta].sub.1])(2[[delta].sub.2]+1-[[delta].sub.1]) and [bar.[lambda]], and are obtained in the proof of Proposition 1 below. We discuss in Section 4 how cost increases affect prices if [lambda] is outside this region.

For [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), we can now state the following proposition regarding the equilibrium of the full game. A well-known result of models of competition under asymmetric information is that the equilibrium often involves competitors playing mixed strategies. (8) This is also true in our model.

Proposition 1. For [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), a unique equilibrium to the two-stage game exists and is characterized by a distribution function over strategies (interest rates and credit denial probability) for each lender, [F.sub.i](r), i = 1, 2, where [F.sub.i](r) = prob([r.sub.i] [less than or equal to] r). These distribution functions are continuous and strictly increasing on [[bar.r], R), where [bar.r] = (1 + [[delta].sub.2]) + 1/lambda][square root of [[lambda].sup.2][(1 + [[delta].sub.2]).sup.2] - [lambda](1 - [lambda])(1 + [[delta].sub.1])([[delta].sub.1] - 1 - 2[[delta].sub.2]). The equilibrium has the following additional properties: