Credit rationing as a (temporary) suboptimal equilibrium with imperfect information

European Econcmic Review 23 ( 1983) 195-201. North-Holland

CREDIT

RATIONiNG AS A (TEMPORARY) SUBOPTPMAL IUM WITH 1: PERFECT INFORMATION

John Peter D. CHATEAU* McGill University, Montreal, Canada H3A IG5

Received October 1982, final version received May 1983 Jaffee and Russell (1976) characterized partial rationing as a contract rate and contract size below those of the no-rationing equilibrium. Their non-price rationing here obtains as a s&optimal equilibrium for a risk-averse rate-setting intermediary with marginal increasing, cost and a monopolistically competitive loan demand. The temporary (dis)equilibrium corresponds to a dominated strategy, and the wider class of models of which J-R’s solution is but a singular case is touched upon.

1. Intductim

Ac~rding ac Jaffee and Russell (J-R) (1976), ‘credit rationing occurs when lenders quote an interest rate on loans and then proceed to supply a smaller loan size than that demanded by the borrowers’. They then show that for (honest?) loaf1 grantees, partial rationing results in a contract rate and contract quantity below the no-rationing equilibrium, in the absence of institutional arrangements. They also reckon that the temporary situation is part of an oscillating long-run market solution that depends critically on dynamic interactions and assumptions about entry. It is shown here that their rationing solution obtains as a (temporary) suboptimal equilibrium for the risk-averse rate-setting intermediary within a monopolistic loan market. This is done in two steps. The intermediary first sets its loan rate because it faces its own sloping loan demand at disequilibrium rates [in the sense of Arrow’s (1959) temporary disequilibrium market). Next, the institution proceeds with partial rationing on the basis of asymmetric and incomplete information. The non-market clearing solution hinges on the fact that the riskiness induced by incomplete information is embodied in the loan size, a surrogate for various non-price characteristics and so on. The such as default, borrowers” trustworthiness and collateral, *I wish to thank Professors E. Losq and W. Sealey. Editor H. Glejser and an anonymous referee for helpful comments ar,d valuable suggestions that led to a marked improvement of an earlier version. The usual caveat applies. ‘Incompletely informed (rate-taking) borrowers have to incur costly search to discoler the distribution of rates charged elsewhere. This opens thr uoor to dispersed and!or disequilibrium loan rates.

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196

intermediary will eventually revert to rate-setting since rationing is a (relativelv) domina,ted strategy that depends critically on the nature of the cost function. In view of the foregoing, section 2 lays out the first stage of the model, rate-setting, and derives a lemma for the no-rationing equilibrium. Section 3 character&& the second stage, rationing, with its partial incidence captured in Proposition 2. Section 4 discusses the solution’s temporary character, the strategy the intermdiary is likely to revert to: as well as a wider class of rationing models. 2 Stage 11.: Tbe model and risky ratesetting

equiPbrim

The monopolistically competitive intermediary faces its own, slightly differentiated demand schedule (as in J-R’s single-contract, rationing/no rationing’ eq-uibbrium) z=L(r,, U) with

L; ~0

and

UU>O,

(1)

where z denotes the stochastic loan demand, rl the loan rate, u the asymmetrical information-based uncertainty whose unknown ex ante subjective probability is dF(u), and L1 and LL are the partial derivatives (denoted by primes) of us) with respect to its (subscript) arguments. Given (l), the intermediary’s officers set the loan rate that maximizes the expected utility, EU, of ex post short-run profit, rc, over the single-period horizon. Analytically, that is to

maxEfW&h

~)--Wh

41 -Bl},

(2)

‘I

where r&(.) stands for the loan proceeds function, CCL(.)] represents the variable cost’ of loans, and B refers to the intermediary’s branch network (fixed) cost Optimizing (interior solution) expression (2) with respect to rl yields firstand second-order conditions, E(U’&jL;

+ L) - c’(L)L;]) = 0,

LI=a{E[U’(lr)a~/i7r,])Jar, ~0,

(3) satisfied for all r,,

(4)

‘In keeping with the limited loan differentiation of the rationing literature, expression (1) can also be refixred to as the intermediary’s qgregte (or at least, group) !orn demand. See, for ~~~~~~ ,““ed and Hewitt (1980). mxrtainty of iiabi!ity-financed lc;ms is assumed to be independent of deposit-supply ~~~~~ty, for the sake of expositional ckr-ity. Following Chateau (1981, 1982), tota! variable cmd watts in output form comprises both input and resource costs.

J.P.D. Chateau, Credit

rationing

197

where U’ Es au(n)jax > 0,

C(L) = dC[L( .)]/tYL,

and the arguments of the utility and loan functions are suppressed for notational convenience. To focus on demand global uncertainty, (3) is recast in terms of the financial market equilibrium theory, that is EIMS(rl, u)] = - cov (U’, MS( .))/E[ U’],

(5)

where the original intermediation spread MS( .) is [(r,L; =tI,)-C’(L)L;]. The effect of nonlinear risk regimes on (optimal) risky rate-setting obtains when the sign of ihe covariance term in (5) is determined. It can easily be shown4 that cov(U’,MS(.))>(

=)-CO,

(6)

according to whether U is convex, linear or concave, respectively. Applying (6) to (5) yields E[(r,L; + L) - C’(Z&;-J> ( =) < 0,

(7)

or equivalently

according to whether U” <( =)>O, respectively. As for the state-dependent inequality, the superscripts u, it and p refer to aversion to, neutrality to arid preference for risk, respectively, and the asterisk to an equilibrium solutiori. The above result may accordingly be stated as follows: Lemma. The equilibrium rate set by the risk-averse (preferring) intermediary will unambig,lously be below (above) that aptimally defined for the risk-neutral state qf nature when (i) loan demand is additively separable in r6 and u, and when (ii) loar! margina cost is non-decreasing, C”(L)zO.

The lemma enables us to move to credit rationing, a mixed strategy in which the rate becomes sticky because quantity rationing entails a nonmarket clearing equilibrium. ‘The derivation of eq. (6) shows that aU’/& has the sign of U” and that 8[c3n(r,.u)/dr,]lhr is positive when L(r, ,u) is additive in rl and II and when C”(L) 2 0. In general, however, the sign of J[i?n/ar,J/& is ambiguous and depends critically on how changes in uncertainty - for instance, in the Pratt-.4rrow index of absolute risk aversion r’(x)>( =) ~0 - affect the profitability of rate changes.

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3. Stage Ik Credit rationing

This temporary disequilibrium is said to prevail when the rate-setting intermediary also decides to ration loan quantity (supply) prior to the revelation of the true demand conditions. If actual demand turns out to be below rationing, the intermediary is left with surplus loanable funds to be diverted most probably to short-term liquid assets, However, if actual demand exceeds the chosen supply, some borrowers are left (partially or completely) unsatisfied. Starting with the demand E=L(r!, u) corresponding to the rate-setting mode, the intermediary may also find a U, for any r, such that L, = L(rr, u,), when demand is increasing with 2?.It thes follows that when the optimal loan rate and credit allocation are chosen, L(rl, u _) CL, c L.(rl,u’), where u _ and u+ refer to lower and upper bounds of u, respectively. Given that u-
n = rtL(r,, 4 - CWr,, us)]- B, n,

E IX =

rrL(rr,

u,) - C[L(r,, u,)] -B,

It is necessary to introduce Hr E[U(n)] so that HI = d[EU(@]/8rI,

US%

(8)

u > us.

some new notation at this point. Let

H,, = 82[EU(7c)]/ar, au,,

and so on. Also, let r be the Pratt-Arrow index of risk aversion and let the risk-averse utility fYnction be approximated by the quadratic function U(n) = kx-nr2. We are interested in the changes in loan rate and credit allocation that accompany changes in risk aversion, starting from the initial and local position of risk neutrality, namely from fl, u,”and r = 0, k remaining constant. Totally differentiating the first-order conditions for rf and us, gives

First, examine H,,. Taking the total differential of the first-order condition H, =O, we have H,, du, + H,, dr = 0 (for rI fixed). Therefore, duJdrP’,=-=’ = - HJH,,,

(10)

Under reasonable conditions of absolute risk aversion, it can be shown [for instance, Sealey ( l%O)] that the level of loan credit decreases as risk aversion

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199

increases, that is du,/drlrf.con81ant is negative. Since H,, CO by second-order condition, it follows that H,, 0 holds unambiguously when Jn,‘&, is monotonic in u, and conditions (i) a:nd(ii) of the Lemma hold.

The only remaining sign required to carry out the comparative static experiment (9) is that of HIS, which may be obtained by taking the partial derivative of (3) with respect to u,. The resulting expression,

is positive’ for risk aversion and thle restrictive assumptions en-bodied in Proposition 1. Thus HI, ~0 results an.d as the Hessian matrix of coetlicients in (9) is negative definite, its diagonal elements are negative. Then the determinant of the soefficient matrix is positive and its inverse has all negative elements. The latter result will be a sufftcient condition to establish: Proposition 2. When faced with a loan demand additively separable in r, and u and with non-decreasing loan cost, the representative intermediary will simultaneously lower its loan rate and restrict its credit allocation, for a small increase in its risk aversion. 4. Discussion

The rationing solution corresponds to a suboptimal equilibrium to the extent that this (q-L,) mode is dominated by strategies such as creditsetting (L, mode) or rate-setting (rI behavior). It is in the sense that the intermediary may temporarily accept the (rI -L,) mode, before reverting to L, or rz strategies, that -we speak of a temporary disequilibrium. Sometimes intermediaries might find it advantageous to adhere to a mode (rl -L,) leaving no instrument of control ex post. For instance, rate-setting will not be preferred to credit rationing when the intermediary’s marginal cost of !oans is sufficiently increasing6 - as per our Lemma, namely when the expected cost of random intermediation (rI mode) exceeds that of a fixed intermediation (rI -L,) strategy. Beyond the transitional viability of c:,iedit rationing, there remains the ‘The I-..,t term is unambiguously positive since U’(a)>0 and it has been established that 8(&r/&,)/i%, is also positive according to Proposition 1. Given that U”(a) t0 for risk aversion, the second term will be positive when &c/r%-, and an/&, are of opposite signs, which [following Leland (1972)] can be shown to be the case for an imperfectly competitive loan market. “Cf. Lim (1980) for further detaili. The importance of cost of funds in rationing is also stressed by Fried and Howitt (1980) in a multiperiod framework.

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question of which strategy (r: or L,) to revert to. While we expect the intermediary to revert to the initial pricing mode, there are circumstances under which this variant is dominated by credit-setting - esl?e:ially for increasing marginal-cost institutions. This is the case when the L, fixedintermediation strategy allows for a specific market arrangement: the existence of some marketing - here discounting - agency [cf. Lim (1980, 1981)-J.To illustrate, banks could pass loans of honest as well as (and most probably) dishonest borrowers off on the lender of last resort, thus effectively shifting the risk of default and of credit extension to the latter. This classical theory of credit in which the @rural Bank was rediscounting intermediaries’ loans is (un)fortunately out of favor; instead the control of credit by openmarket operations forces the banks to consider risk sharing contracts in a rate-setting dynamic framework, The above institutional remark shaws that our characterization of J-R’s non-price rationing might be but a peculiar case of a wider class of (firm’s and/or market) disequilibrium explanations. As a prelude to further research on sequential signalling experiment, I briefly sketch here the intermediary’s strategy that could result in a disequilibrium explanation with credit rationing. Assume that the lender is to set his loan rate cx ante [e.g., at the same rate as in the (rI -L,) mode] and then supply a quantity of loans ex post which is less than or equal to the quantity demanded - an optimal dynamic programming procedure is also technically fashionable. Then, the corresponding objective function will be r

max E ‘I

max

J+‘tr,.4

U[r, L, - C(L,)] ,

(12)

rather than the just proposed alternative my EU[r, min (L,, LQI, u)) - C(L,)]. I’ L

(13)

Within the cotimes of (12), credit is rationed for certain realizations; c~pected (utility of) profit will probably be at least as large as in the mixed strategy (rI --t,) and it is an open question as to one also has a lower interest rate than Iunder other variants. Further, the new development m’ay elicit a disequilibrium explanation of credit rationing which depends only on uncertainty and not, for instance, on adverse selection as in Jaffee and Russell (19X), the implicit contract story of Fried and Howitt (1980), or the adverse selection/&eening model of Stiglitz and Weiss (1981). The ultimate success of this approach would reqrdire a sensible picture of the fundamental sources of untirtainty and the pr 3Gsion of a dynamic setting in which to bed the static d&equilibrium model. Speaking of dynamics, the discussion can be closed perhaps fittingly by

J.P.D. Chrctczau,Credit rationing

201

examining how to integrate the temporary equilibrium Gthin the dynamics of credit. Stiglitz and Weiss (198 I) notice the intermediary may want, in the face of an upsurge in credit, to complement its single-period pricing mode with limits on loan size. Thus this incentive (and preventive?) device ushers non-price rationing into a multiperiodic setting. The argument runs as follows. Although the bank controls L, (initial funding), it does not control directly the total (expected value) of its loans per customer. So the intermediary may lend more in the future, namely extend (revolving or not) credit lines to borrowers. For instance, low-default borrowers or eariierperiod performers will be offered credit lines - forward contracts - at lower rate - prime only -- than that for other classes/groups of borrowers. The latter (and often multiperiod) customer relationship stresses the fact that interest rates charged directly affect the quality and/or riskiness of loans in a manner which matters to the intermediary. Yet, as a stop-gap measure in the face of a transitional shock, setting a sticky rate with a non-market clearing equilibrium is not a ‘phantasm’. References Arrow, K.J., 1959, Towards a theory of price adjustment, in: Abramovitz et al., eds., The allocation of resources (Stanford University Press, Stanford, CA) 41-51. Chateau, J.P.D., 1981, On the theory of financia! intermediaries: Deposit rate-setting under supply uncertainty, in: H. Giippl and R. Henn, eds., Proceedings of the Karlsruhe symposium on money, banking and insurance, Vol. II (Athenaurn Verlag, Konigstein) 606-616. Chateau, J.P.D., 1982, On DFIs’ liability management: Deposit capacity, multideposit, and riskefficient rate-setting, Journal of Banking and Finance 6, no. 4, 533-549. Fried, J. and P. Howitt, 1980, Credit rationing and implicit contract theory, Journal of Money, Credit and Banking 12, no. 3,471-487. Jatfee, D.M. and Th. Russell, 1976, Imperfect information, uncertainty. and credit rationing, Quarterly Journal of Economics 90,651-666. Leland, H.E., 1972, Theory of the firm facing uncertain demand, American Economic Review 62, no. 3,278-291. Lim, C., 1980, The ranking of behavioral modes of the 3rm facing uncertain demand, American Economic Review 70, no. 1,217-224. Lim, C., 1981, Risk pooling and intermediate trading agents, Canadian Journal of Economics 14, no. 2,261-275. Sealey, C.W., Jr., 1980, Deposit rate-setting, risk aversion and the theory of depository linancisl intermediaries, Journal of Finance 35, no. 5, 1139-l 154. Stiglitz, J.E. and A. Weiss, 1981, Credit rationing in markets with imperfect information, American Economic Review 71, no. 3, 35’3410.