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

* The analysis so far has assumed that [lambda], the degree of borrower turnover, falls in some well-defined intermediate range ([[lambda].bar], [bar.[lambda]]). As such, the result that interest rates may decrease when the cost of one of the firms increases is predicated on the assumption that both lenders find it optimal to compete in this market. Because lender 1 is assumed to have all the knowledge concerning existing borrowers, one interpretation of this lender is as an incumbent bank. Lender 2 can be thought of as either an entrant bank, or an anonymous public debt market with no private information about the market.

It is worth illustrating what happens if there is no effective competition in this market, such as if lender 2 were to choose not to enter, or if lender 1, because of a large cost disadvantage, were to find it too costly to lend. For [lambda] > [bar.[lambda]], lender 2 is essentially a limit-pricing monopolist, charging a rate determined by the minimum rate lender 1 would be willing to offer that would still allow it to break even, [r.sub.2] = 1+[[delta].sub.1]/[bar.[theta]]. Contrary to our earlier result, this rate is increasing in [[delta].sub.1] because an increase in lender l's cost further increases lender 2's market power, allowing it to charge a higher price. For [lambda] < [[lambda].bar], lender 2 does not compete in this market at all, in which case lender 1 offers the monopolist's rate of [r.sub.1] = R, which does not depend on its cost St. However, which old borrowers get credit does depend on lender l's cost of funds, because it will choose to deny credit to any old borrowers identified as having a repayment probability lower than 1+[[delta].sub.1]/R.

The results from Section 3 can be also extended to consider the exit of lender 1 if we interpret lender 2 as a competitive fringe of entrant lenders, none of which have any information about the market beyond the publicly available information concerning the distribution of borrowers. In this case, we can interpret a sufficiently large cost increase for lender 1 as a forced exit from the market, so that only the uninformed competitive lending market remains. We state the following result as a corollary to the previous results.

Corollary 2. Suppose that lender 1 competes with a fringe of uninformed lenders. Assume that lender 1 exits the market and makes no loans to either old or new borrowers. Expected interest rates will then be lower than in the case where lender 1 does not exit.

The corollary is immediate from the previous discussion and results. The exit by lender 1 eliminates the adverse selection problem faced by lender 2. If this "lender 2" represents a competitive credit market lending on the basis of purely public information, competition under symmetric information will drive interest rates down to their competitive, break-even rate, r = 1+[[delta].sub.2]/[bar.[delta]], which is lower than any rate charged in the presence of lender 1. (11)

There is an additional implication resulting from the exit of lender 1 on the quantity of loans granted. Proposition I establishes that, for [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), lender 2 sometimes refrains from bidding. This implies that, in equilibrium, there is an expected mass of borrowers (of size 1 - [F.sub.2](R)) with repayment probability [theta] < 1+[[delta].sub.1]/R that does not obtain financing. When lender 1 's cost is higher, however, lender 2's (or the competitive fringe's) probability of bidding increases, thus increasing the overall fraction of borrowers that obtain financing. In the limit, if lender 1 exits the market, lender 2 will finance all borrowers in the market, thus increasing the number of loans granted. We summarize this in the following corollary.

Corollary 3. A cost increase for lender 1 increases the (expected) quantity of loans granted by either lender. If lender 1 exits the market, all borrowers are financed by lender 2.

This result demonstrates that there is a quantity as well as a price effect in the model, in that the increase in competition as a result of a cost increase for lender 1 (an increase in [[delta].sub.1]) leads not only to lower prices but also to a greater supply by competing lenders. However, whether this increase in supply is efficient or not is unclear. Much of the expansion in lending stems from financing borrowers rejected by lender 1, which by their nature are worse-than-average borrowers. Nevertheless, many of these borrowers are creditworthy from the standpoint of lender 2, because [[delta].sub.2] < [[delta].sub.1]. There is therefore a tradeoff in that, although information may restrict the supply of credit, it can be useful for increasing the efficient allocation of this credit.

From an empirical perspective, the results in this section shed light on the debate concerning the recent wave of bank mergers. There has been concern among some of the regulatory agencies that as banks increase their size via merger, small business borrowers may be hurt as a result of shifts in banks' lending policies away from these borrowers and toward larger corporate customers. There is mixed evidence on this front, however, because, in many markets, the exit of merged banks from small business lending has triggered an increase in lending by the remaining smaller banks, sometimes more than compensating for the reduction in lending by the merged banks (see Berger et al., 1998). This effect is consistent with our model, as the exit of the larger bank allows the smaller banks to compete more aggressively for the exiting bank's market share.

5. Robustness considerations

* In this section, we consider two important generalizations of the model. First, whereas we have assumed so far that only lender 1 has private information about some subset of borrowers and can therefore make counteroffers to them, we extend the analysis to the case where there is private information on the side of lender 2 as well, thus assuming greater symmetry between the lenders. Second, we allow for product differentiation, so that loan offers from each lender are not perfect substitutes for each other, as well as simultaneous bidding for all borrowers.

* Private information for lender 2. Assume that, of the 1 - [lambda]. known borrowers in the market, lender 1 only knows the type of a fraction [[alpha].sub.1] of them, and that lender 2 knows the type of the remaining fraction [[alpha].sub.2] = 1 - [[alpha].sub.1], with [[alpha].sub.1] > [[alpha].sub.2]. We also adjust the extensive form of the game slightly by allowing each lender to make a counteroffer to the borrowers it knows in stage 2.

It is now straightforward to show that all our previous results concerning changes in the equilibrium interest rate as a function of increases in costs carry through to this more general setting as long as the information asymmetry across lenders, [[alpha].sub.1] - [[alpha].sub.2], is sufficiently large that lender 1's information advantage still translates into a competitive advantage. We summarize this assertion in the following proposition, whose development and proof are relegated to the Appendix.

Proposition 4. [partial derivative][F.sub.1]/[partial derivative] [[delta].sub.1] > 0, and there exists an [[??].sub.2] > 0 such that [partial derivative][F.sub.2]/partial derivative [[delta].sub.2] > 0 for [[alpha].sub.2] < [[??].sub.2].

Given this result, we can conclude that, for [[alpha].sub.2] < [[??].sub.2], [partial derivative]E[[r.sub.1]]/[partial derivative][[delta].sub.1], [partial derivative]E[[r.sub.2]/[partial derivative][[delta].sub.1] < 0, as long as the lower bound for the bidding distributions, [r.bar], is nonincreasing in lender 1 's cost of funds: [partial derivative][r.bar]/[partial derivative][[delta].sub.1] [less than or equal to] O. In the Appendix, we show that the lower bound in this setting is given by

[r.bar] = (1 + [[delta].sub.2] + [square root of ([lambda][([[delta].sub.2 + 1).sup.2] + ([lambda] - 1)[[alpha].sub.1](1 + [[delta].sub.1]([[delta].sub.1] - 1 - 2[[delta].sub.2])/[lambda])], (23)

from which we determine that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

for [[delta].sub.1] [greater than or equal to] [[delta].sub.2], thus establishing the result. Our model is therefore robust to introducing greater symmetry in the lenders' strategy space. The main requirement is that we continue to have one lender (lender 1) with an information advantage, which is limited by the other lender's (lender 2) cost advantage.

* Product differentiation and simultaneous bidding. The main result of the model, that an increase in cost for an informed competitor can lead to a decrease in equilibrium prices, is also robust to the introduction of product differentiation, as well as to simultaneous bidding by both lenders for all borrowers (i.e., a single stage of bidding, with no counteroffers). To illustrate this, we sketch here a model with both of these features. Note that, in order to do so, the extensive form of the game must be changed somewhat, and we describe precisely how in what follows.