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

* Derivation of results for section on "product differentiation and simultaneous bidding". Take any borrower with type [theta] [member of] [0, 1 ]. Assuming that lender I makes an offer, the borrower can choose between two prices: [r.sub.1]([theta]) and [r.sub.F], where [r.sub.F] is the rate offered by the fringe. It will choose [r.sub.F] if and only if

[theta](R - [r.sub.2]) - x > [theta](R - [r.sub.1]([theta])) [left and right arrow] x < [theta]([r.sub.1]([theta]) - [r.sub.2]), (A15)

where x represents the switching cost it incurs if it decides to borrow from the fringe. This implies that for a given [r.sub.1]([theta]) and [r.sub.F] and x drawn uniformly from [s, S], the probability of a switch for a borrower is [theta]([r.sub.1]- [r.sub.F])-s/S-s. If lender 1 makes no offer, then the fringe gets the borrower if

[theta](R - [r.sub.f]) - x > 0 [left and right arrow] x < [theta](R - [r.sub.F]), (A16)

so that the probability the fringe grants the loan is [theta](R-[r.sub.F])-s/S-s.

Now consider lender 1, who only wants to bid if [theta] - [[delta].sub.1] > 0 [left and right arrow] [theta] > [[delta].sub.1]/R. This provides a lower bound on [theta], which we define as t = [[delta].sub.1]/R. For any given [theta], lender 1 gets this borrower as long as [theta] (R - [r.sub.F]) - x < [theta](R - [r.sub.1]) [left and right arrow] x > [theta] ([r.sub.1] - [r.sub.F]). Lender 1 's demand will therefore be [D.sub.1] ([r.sub.l], [r.sub.F]; [theta]) = S-[theta]([r.sub.1] - [r.sub.F])/S-s. Its profit can now be written as

[[PI].sub.1](r.sub.1], [r.sub.F]; [theta]) = [D.sub.1]([theta][r.sub.1] - [[delta].sub.1]).

Maximizing this expression with respect to [r.sub.l], we get the first-order condition

[partial derivative][D.sub.1]/[partial derivative][r.sub.1]([theta][r.sub.1] - [[delta].sub.1]) + [D.sub.1][theta] = 0.

We can use this to solve for the reaction function for lender 1 as [r.sub.1]([theta]) = min{R, [r.sub.F] + [[delta].sub.1] + S/2[theta]}, because the interest rate offer is bounded above by R. A useful way of stating this is that we can define a cutoff value of [theta] as an upper bound for the interval where lender 1 offers the maximal rate R as T = [r.sub.F] + [[delta].sub.1] + S/2R. Therefore, for any price offered by the fringe, it expects to face a total demand of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A19)

We rewrite this slightly as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A21)

where [r.sub.1] = [r.sub.F] + [[delta].sub.1] + S/2[theta].

We can now substitute this into the equation for the fringe's profit, which is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A22)

We now impose a zero profit constraint for the fringe. Defining

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A23)

we can now write the fringe's profit as [[PI].sub.F] = 1/S-s Q, which must equal zero in equilibrium. Differentiating this expression with respect to [[delta].sub.1] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A24)

Note, however, that [r.sub.1](T) = R, so that the last two terms cancel each other out, and we are left with just [partial derivative]Q/[partial derivative][[delta].sub.1] = [integral].sup.1.sub.T] [theta] [partial derivative][r.sub.1]/[partial derivative][[delta].sub.1] ([r.sub.F][theta] - [[delta].sub.2])d[theta] > 0 because [partial derivative][r.sub.1]/[partial derivative][[delta].sub.1] = 1/2[theta] > 0 and [r.sub.F][theta] - [[delta].sub.F] > 0 for [theta] [member of] (T, l] in order for the fringe to at least break even in equilibrium. Therefore, ceteris paribus, an increase in [[delta].sub.1] increases the fringe's profits, which must be compensated by a decrease in the interest rate it offers in order to satisfy the zero profit constraint: d[r.sub.F/d[[delta].sub.1] < 0, as desired.

We would like to thank Elizabeth Bailey, Tito Cordelia, Hans DeGryse, Adolfo de Motta, Vincent Hogan, Jonathan Levin, Christopher Snyder, and seminar participants at the University of Maryland and the U.S. Department of Justice for useful discussions and suggestions. The views expressed in this article are those of the authors and do not necessarily represent those of the IMF.

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(1) This is sometimes referred to as the "exclusion principle," which can also be observed in the context of auctions with reservation prices above the seller's valuation for the good.

(2) An alternative interpretation of [lambda]. is as the probability that a credit screen performed by a particular bank provides useful private information. "Old" borrowers represent those borrowers for whom the credit screen was informative, and "new" borrowers those for whom the screen generated no additional nonpublic information. (We thank Chris Snyder for suggesting this interpretation.)

(3) There are many ways of modelling competition under asymmetric information. This specification provides us with a tractable analysis and a very simple measure of the degree of information asymmetry, given by the probability that lender 1 has private information about a borrower.

(4) The case where each lender i has a positive market share [[alpha].sub.i], with [[alpha].sub.1] > [[alpha].sub.2] > 0, is studied in Section 5.