Credit Rationing by Loan Size

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Credit Rationing by Loan Sizein Com mercial Loan Markets

Stacey L.. Schreft and Anne P. Vi~~arni~

I NTRODUCTION

Ample evidence exists suggesting that banksration credit w ith respect to loan size. Fo r exam -ple, Evans an d Jovanovic (1989) find evidence of loansize rationing in data from the National Longitudina lSurvey of You ng Me n.2 Further, the Federal R eserveBoards quarterly Survey of Terms of Ban k Lendingconsistently indicates tha t the average interest ratecharged on comm ercial loans (i.e., the rate per dollarlent) is inversely related to loan size. This e videncesuggests two questions. First, why might loan sizerationing occur? Second, why might loan size ration-ing have the particular interest rate and loan size pat-tern reported in the Survey of Terms of Ban kLending? Econom ists generally believe that higheraverage interest rates are charged on smalle r loansbecause sma ll borrowers are greater credit risks orbecause loan administration costs are being spreadover a smalle r base. This paper presents a counter-examp le to that belief. It show s that, even if creditrisk and loan administration costs are the samefor all borrowers, a lender with market power and

Stacey Schreft is an economist at the Federal Reserve Ban kof Richmon d, and Anne Villamil is an economics professor atthe University of Illinois at Cham paign-Urban a. The authorsgratefully ackno wledge the financial support of the Universityof Illinois and the National Science Foundation (SES 89-09242).They wish to thank D an Bechter, Kathryn Combs, Bill Cullison,Michael Dotsey, Donald Hodgman, Charles Holt, DavidHum phrey, Ayse Imroho roglu, Peter Ireland, Jeffrey Lacker,David Mengle, Loretta Mester, Neil Wallace, Roy Webb andseminar pa rticipants at the Federal Reserve Ban k of Richmo ndfor comments on an earlier version of this paper. Th e view sexpressed in this paper do not necessarily reflect the views ofthe Federal Reserve Svstem or the Federal Reserve Bank ofRichmond. Jaffee and Stiglitz (19 90) present alternative definitions ofcredit rationing. Broadly defined, credit rationing occurs whenthere exists an excess dem and for loans because quoted interestrates differ from those that would equate the demand andsupply of loans.* This evidence is contrary to most recent theoretical modelsof credit rationing. That literature derives loan quantig ration-ing, whereby some borrowers obtain loans while other observa-tionally identical borrowers do not, in the spirit of Stiglitz andWeiss (1981). While some quantity rationing does occur, theevidence suggests that size rationing is more comm on.

imperfect information about borrowers characteristicsstill will offer quantity-depen dent loan interest ratesof exactly the type reported in the Survey of Termsof Bank Lending.3

The quantity-depe ndent loan interest rates that wederive a re a form of second-degree price discrimina -tion. P rice disc rimination is said to occur in a marke twhen a seller offers different units of a good to buyersat different prices. This type of pricing is com-monly used by private firms, governm ents and publicutilities. For exam ple, many firms have bulk ratepricing schemes, whereby they offer lower ma rginalrates for large quantity purcha ses. The income taxrates in the U.S. federal income tax schedule dependon the level of reported income; higher m arginal taxrates are levied on higher-income taxpayers. Inaddition, the price per unit of electricity oftendepends on how much is used.

Both ma rket p ower-a firms ability to affect itsproducts price-and imperfect information regardingborrowers c haracteristics are essential for producingthe loan size-interest rate patterns observed in com-mercial loan markets.4 To see why, suppose that alender ha s ma rket pow er and perfect informationabout borrowers loan demand. In this case, we wouldobserve first-degree (or perfect) price discrimin a-tion: the lender wou ld charge each borrower the mosthe/she is willing to pay and w ould lend to all thatare willing to pay at least the m arginal cost of theloan. Suppose instead that a lender has imperfect in-formation and operates in a compe titive ma rket.Milde and Riley (1988, p . 120) have shown that sucha lender may not ration credit, even if borrowers cansend the lende r a signal about their c haracteristics.

In this paper, we provide an explicit analysis ofthe information aspects of price discrimina tion in3 Of course, the theory we will present is not inconsistent withdifferential credit risk and loan administration costs, althoughthese factors are not necessary to obtain the observed interestrate-loan size pattern.4 See Jaffee and Modigliani (196 0) for an early distinction be-tween types of price discrimination and credit rationing.

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Electronic copy available at: http://ssrn.com/abstract=2125124

comm ercial loan markets. We interpret a lendersprice discrimination with respect to loan size as a formof credit rationing that limits borrowing by all butthe largest borrowers. Further, because we show thatsuch credit rationing arises from rational, profit-maxim izing lender behavior, our analysis hasnormative implications. We find that small borrowersare more credit constrained than large borrowers andthus bear a larger share of the distortion induced bythe market imperfections.5 In the next section, w edescribe a simple prototype econom y with a singlelender and many different types of borrowers aboutwhom the lender has limited information. We thenpresent the lenders profit-maximization problem.Section III follows, describing the loan size-totalrepaymen t schedule that solves the lenders problemand explaining why the solution involves creditrationing w ith respect to loan size. Section IVconcludes.

1I A SIMPLEMODELECONOMY

Consider an endowm ent econom y with a singlelender that ma y be thought of either as a localmono polist or as a price leade r in the industry.6Suppose also that there are n types of borrowers,where n is a positive and finite number. There areNi borrowers of each type i (i = 1 ...,n) who livefor only two periods. The borrowers may be thoughtof as privately owned firms that differ only withrespect to their fixed endowm ents of physical good.All firms have the same first-period endowm ent:wi = 0 for all i;* however, higher-ind.ex firms havelarger second-period endowm ent: wi+ > wi. Inaddition, each firms second-period endowm ent ispositive and known with certainty at the beginningof the first period.

5 Price discrimination in loan markets is facilitated by banks useof base rate pricing practices: banks quote a prime rate (thebase) and price other loans off that rate. W ith a base rate pric-ing scheme, banks p rice loans competitively for large borrowerswith direct access to credit m arkets, while they act as price-setterson loans to smaller borrowers. Goldberg (1982, 1984) findssubstantial evidence for such pricing practices.6 The changing of the prime rate has been interpreted by bank-ing industry insiders as an example of price leadership andcalled the biggest game of follow-the-leader in Americanbusiness [Leander (199O)l. This interpretation is consistent with Prescott and Boyd (1987),which models the firm as a coalition of two-period lived agentswith identical preferences and endowments; the coalition in ourmodel consists of only one agent.8 We assume w; = 0 for simplicity to guarantee that firms bor-row in the first period.

We assume that the w elfare of each type i firm (i.e.,borrow,er) is represented by a utility function,u(xf, xi), where xi is the am ount of period t goodconsum ed by the owner of the firm, for t = 1,2. Theutility function U(O) ndicates the satisfaction that theowner gets from various combinations of consum p-tion in the two time periods. We assume that theowners utility function is twice differentiable, strictlyincreasing and strictly concave. T hese mathem aticalproperties imply that the owner prefers mo re con-sumption to less and prefers relatively equal levelsof consum ption in the two time periods. We alsoassume that xi is a normal good, w hich mean s thatowner is dema nd for good x increases with his/herincome. Given these assumptions and the endow-ment pattern specified, all firms will borrow in thefirst period and higher-index firms will be largerborrowers.9

The econom ys single lender wishes to maxim izeprofit, which is the difference between revenues (i.e.,funds received from loan repayments) and costs(funds lent). Assum e that the lenders capital at time1, measured in units of physical good, is sufficientto support its lending policy, and suppose that thefollowing information restriction exists: the lender andall borrowers know the utility function, the endow-ment pa ttern, and the numb er of borrowers of eachtype, but cannot identify the type of any individualborrower. Thus, a borrowers type is private infor-mation. This information restriction prevents perfectprice discrimination by the lender but allows for thepossibility of imperfect discrimination via policies thatresult in borrowers correctly sorting themselves in-to groups by choosing the loan package designed fortheir type.lO Finally, we assume that borrowers areunable to share loans.

The lenders problem is to choose a total repay-ment (i.e., principal plus interest) schedule for period2, denoted by P(q), suc h that any firm that borrowsamoun t q in period 1 must repay am ount P in period2. Let Ri(q) denote the reservation outlay for loansof size q by a type i borrower; that is, Ri(q) indicates9 Because endowm ent patterns are deterministic, there is nodefault risk in this model if the lender induces each type of bor-rower to self-select the correct loan size-interest rate package.We will specify self-selection constraints to ensure that all agentsprefer the correct package. C onsequently, we obtain pricediscrimination in the form of quantity discounts despite theabsence of differences in default risk across borrowers.lo With complete information about borrowers endowments,the lender would use perfect price discrimination, offering eachborrower a loan at the highest interest rate the borrower wouldwillingly pay.

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Figure 7OPTIMAL QUAN TITY DEPENDENT INTEREST RATE SCHEDULE

Total Loan Outl ays P

consumersurpl usextractedf romthel owest-i ndexborrower

e

I I I II II I

I/ /I I I

1 II II IIai+t , III II I4 w Loan Si ze qqi qi+t

Note: Unlike a typical d emand function, the total outlay schedule in Figure 1 slopes upward. This occurs because the loan outlayschedule, P qi) = prqt, is the total amount that a borrower pays for a loan of size qt. In contrast, an ordinary demand functionrepresents the size of a loan requested as a function of price only pi). The total outlay schedule in Figure 1 is quantitydependentin the sense that any quantity increase implies a decrease in the average interest rate charged by the lender, CX~ P qr)/qi. Thus, inFigure 1 the average interest rate charged on a loan of size q. +, is lower than the average interest rate charged on a smaller)loan of size qr. Of course total outlays are higher for the larger loan qr+,) than the smaller loan qr). The average price will be theperfectly competitive price i.e., a constant, or uniform, per unit price) only when the outlay schedule is a straight l ine through theorigin.

i=l ,...,n - 1 borrowers exceeds the marginal costof the loan.

Equation (4) and M,+ r = 0 indicate that b orrowersof type n (those with the largest endowm ent) clear-ly obtain the same loan size that they would receivein a perfectly comp etitive market. However, thedegree of credit rationing experienced by borrowersfrom all other groups, i = 1 ...,n - 1, is regressive(i.e., inversely related to their index). To establishthat the pattern of distortion is regressive, we provein the appendix that our assumptions on preferencesand net worth im ply that Rfi+,(qi) > Rfi(qi), whichmean s that higher-index borrowers have a higherimplicit value for a loan of size qi than lower-indexborrowers. This result and the restrictions on thedistribution of borrower types (i.e., on the Nr) mean

that equation (4) implies that low-index (small) bor-rowers are relatively more constrained than high-index (large) borrowers. I1 This is confirmed bythe first term on the right-hand side of equation (4),which is relatively higher for low-index groups.2

The final result pertains to the welfare propertiesof the discriminatory price and quantity sch eme given I See Spence 1980, p. 824) for a discussion of constraints onthe distribution of consumer types.2 For example, suppose Ni = 10 for all borrower groups. Fur-ther, consider an economy with only two different bo rrowergroups, i = 1,2. Let MZ = 0.1 and Ms = 0.9. Then clearlyMa/(Nt + Ma) = O.l/lO.l, which exceeds Ms/(Na + Ms) =0.9/10.9, showing that the implicit marginal price of the loanto group 1 is higher than the implicit marginal price to group2; the marginal cost is one in both cases. This pricing patternis a general feature of the policy.

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by equations (3) and (4). For any single price dif-ferent from ma rginal co st, there is a discriminatoryoutlay schedule that benefits, or at least does notharm, all borrowers and the lender without sidepaymen ts.13 In other words, if the borrowers andlender were given a choice between (i) any singleinterest rate policy that differs from the compe titiveinterest rate and (ii) a quantity-dependent array ofinterest rates, with one rate appropriate for eachgroup, then they w ould all prefer or at least be indif-ferent to the latter policy without coercion. Thisresult indicates that there exists some quantity-dependen t interest rate policy that makes all indi-viduals at least as well off as any uniform interest ratepolicy, except for the single rate that prevails in acompetitive market.

Two other features of the solution warrant discus-sion. Because imperfect information prevents perfectprice discrimination, the lender must ensure that theloan size-interest rate package designed for eachgroup satisfies equation (2). The ordering of loan sizesso that qi 2 qi-i for all i, which is illustrated inFigure 1, is necessary for this constraint to besatisfied. This cond ition states that the lender m ustoffer loans to high-index (i.e., large-endowment) bor-rowers that are at least as large as those offered tolow-index borrowers. Further, P(q)/q is weaklydecreasing in q, which indicates that large borrowerspay lower average interest rates than small borrowers;the declining sequence of ai in Figure 1 illustratesthis. These features of the solution stem from thelenders need to ensure that each group selects thecorrect loan size-interest rate package. The lendermust make the selection of a small loan undesirablefor high-index borrowers. It does this by allowingthe average interest rate to fall with loan size, thusletting larger borrowers keep some of their gainsfrom trade. T he lender m ust also ensure that smallborrowers do not select loans designed for largeborrowers. Such loan sharing is ruled out by assump-tion here.

We interpret the preceding results on loan size andinterest rate distortions as credit rationing. All butthe largest borrowers are prohibited from obtainingloans as large as they w ould choose if the lender hadno market power and all agents h ad perfect infor-mation. Further, the lower a borrowers net worth,the more troublesome (i.e., distorting) the loan sizeI3 See Spence (1980, pp. 823-2 4) for a formal proof.

constraints imposed. These theoretical predictionsappear to be consistent with the empirical resultsnoted in the introduction. The intuition behind themis as follows. The mode l con sists of nume rous bor-rowers w ho differ along a single dimension, name ly,second-period endowm ent. The lender has marketpower and wishes to maxim ize profit. It knows thedistribution of borrower types in the econom y, butdoes not know the identity of any particular borrower.This information restriction prohibits policies suchas perfect price discrimination. However, the lendercan exploit the correlation of borrowers marketchoices with their endowment and does so by offer-ing a discriminatory interest rate schedule that ra-tions loan sizes to all but the largest group. The in-formation implicitly revealed by borrowers c hoicesallows the lender to partially offset its inability,because of imperfect information about borrowercharacteristics, to design borrower-specific interest-rate schedules. Thus, the quantity constraints, whichwe interpret as credit rationing, arise endogenouslyas an optimal response to the information restrictionin an imperfectly comp etitive market.

IV CONCLUSION

This paper has presented a theoretical mode l ofa com mercial loan market characterized by imperfectinformation and imperfect com petition. The modelshows that a profit-maximizing lender operating insuch a market will choose to price discriminate (orcredit ration) by setting a n inverse relationship be-tween the loan sizes offered and the interest ratescharged. This loan size-interest rate pattern is con-sistent with empirical evidence regarding com mer-cial lending. In addition, it is a good examp le of how,as Friedrich von Hayek a rgued, the price system caneconom ize on information in a way that brings aboutdesirable results. Hayek (194.5, pp. 526-27) notedthat the most significant fact about [the price] systemis the econom y of knowledge with which it operates,or how little the individual pa rticipants need to knowin order to be able to take the right ac tion. Theanalysis here shows that a lender with imperfect in-formation about borrower types can set an interestrate schedule that reveals borrowers characteristicsthrough their borrowing dec isions. Interestingly, allloan market participants-the lender and all bor-rowers-are at least as well off with this discriminatoryinterest rate schedule as they wou ld be if faced withany uniform interest rate other than the compe titiverate.

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APPENDIX

We adapt an argument in Villamil(1988) to showthat o ur model is a special case of the widely usedSpence nonuniform pricing mode l. Recall thatRi(q) = pq is the borrowers reservation outlay func-tion, where p denotes the reservation interest ratethat a borrower would be willing to pay for a loanof size q. We prove that the assum ptions of our modelimply reservation outlay functions that satisfySpences (1980, pp. 82 1-22) assumptions. We sup-press the qi and pi notation because it is unnecessary;indeed, we prove our result for every n onnegativeloan amount q. In equilibrium each q is associatedwith a particular p. Thus, the index i is implicit.

Spences assumptions areS. 1: Borrower types can be ordered so that for all

9, R,+, q) > R, q) and R:+, q) > R: q).S.2: Firms need not borrow, and if they do not,

P(0) = 0 and Ri(0) = 0.Property S. 1 implies that borrowers reservationoutlay schedu les can be ordered so that a schedulerepresenting Ri+i(q) as a function of q lies above aschedule representing Ri(q) and has a steeper slope.From-S.2, it follows that the consum er surplus ofa borrower of type i from a loan of size q 1 0,

Ri(q) - P(q), is at least as great as the reservationprice for purchasing nothing, which is zero. Thefollowing proposition shows that our mode l satisfiesthese assumptions.Pmposit;on:The assum ptions on preferences and en-dowm ents made in Section II imply reservation outlayfunctions for consum ption in excess o f endowm entin the first period that satisfy S.l and S.2.Pm08 Let p denote the per unit price of date t + 1good in terms of date t good. Let q denote theamou nt borrowed, i.e., the amou nt of first-periodconsum ption in excess of wi, and let hi(p) denotethe excess demand for first-period consum ption bya type i borrower. From the assump tions that u(e)is concave and that consum ption is a normal good,hi(p) is single-valued and decreasing in p wherehi(p) > 0. Thus, for all q 2 0, hi(p) has an inversethat we shall denote by R:(q). From the assumptionson preferences and net worth, h i+i(p) > hi(p),and consequently, Ri+ i (q) > R:(q) for all q 1 0.Further, letting R,(q) = jgORfi(z)dz, we have thatRi+l(q) > Ri(q) for all q 2 0. Clearly, S.l issatisfied. Property S.2 is also satisfied because anyborrower can refuse to apply for a loan, in which casehis/her repayment obligation and reservation outlayare zero [i.e., P(0) = Ri(0) = 01.

REFERENCES

Evans, D. and B. Jovanovic. Entrepreneurial Choice andLiquidity Constraints, JournaLof Po/iccaLEconomy, vol. 97(1989) pp. 808-27.

Goldberg, M. The Pricing of the Prime Rate, Journal ofBanking and Finance, vol. 6 (1982), pp. 277-96.

. The Sensitivity of the Prime Rate to MoneyMarket Conditions, Journal of Financial Research, vol. 7(1984), pp. 269-80.

Hayek, F. A. von. The Use of Knowledge in Society, Americanonomic Review, vol. 35 (1945), pp. 519-30.

Jaffce, D. and F. Modigliani. A Theo ry and Test of CreditRationing, American Economic Rewiew, vol., 59 (1960),pp. 850-82.

Jaffee, D. and J. Stiglitz. Credit Rationing, in Benjamin M.Friedman and Frank H. Hahn, eds., Handbook ofMonetaryEconomics, vol. 2. New York: North-Holland, 1990, pp.838-88.

Leander, T. Fine Tim e to Fall in Line with the Prime,American Banker, January 16, 1990.

Milde, H. and J. Riley. Signalling in Credit Markets,Quarterly Journalof Economics, February 1988, pp. 101-29.

Prescott, E. C. and J. H. Boyd. Dynamic Coalitions, Growth,and the Firm, in Edward C. Prescott and Neil Wallace,eds . , Contractual Arrangements for Intertemporal Trade, vol. 1,Minnesota Studies in Macroeconomics. Minneapolis:University of Minnesota Press, 1987, pp. 146-60.

Spence, M. Multi-Product Quantity-Dependent Prices andProfitability Constraints, Review ofEconomic Studies, vol.47 (1980), pp. 821-41.

Stiglitz, J. and A. Weiss. Credit Rationing in Markets withImperfect Information, American Economic Rewiew, vol. 7 1(1981), pp. 393-410.

Villamil, A. Price Discriminating Monetary Policy: A Non-Uniform Pricing Approach, Journal of Public Economics,vol. 35 (1988), pp. 385-92.

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Credit Rationing by Loan Size

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