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Credit rationing with an individual short-side rule : estimation for business loans in Canada Crawford, Allan Charles 1985

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CREDIT RATIONING WITH AN INDIVIDUAL SHORT-SIDE RULE ESTIMATION FOR BUSINESS LOANS IN CANADA by ALLAN CHARLES CRAWFORD B.A., Queen's University, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1985 © A l l a n Charles Crawford, 1985 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Economics  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date February 27, 1986 A B S T R A C T It i s g e n e r a l l y accepted that rationing occurs i n loan markets with demand of some borrowers exceeding desired supply by lenders at p r e v a i l i n g i n t e r e s t r a t e s . P r e v i o u s e m p i r i c a l s t u d i e s of c r e d i t r a t i o n i n g use e s t a b l i s h e d d i s e q u i l i b r i u m econometric methods to estimate s t r u c t u r a l models of business l o a n markets. This study argues that e x i s t i n g d i s e q u i l i b r i u m techniques are not s u i t a b l e f o r analyzing c r e d i t rationing since they ignore features of loan markets emphasized i n the t h e o r e t i c a l l i t e r a t u r e . While recent theory d i s t i n g u i s h e s between e q u i l i b r i u m and d i s e q u i l i b r i u m c a t e g o r i e s of c r e d i t r a t i o n i n g , e x i s t i n g e m p i r i c a l work a l l o w s o n l y the l a t t e r to e x i s t . In a d d i t i o n the t r a d i t i o n a l e m p i r i c a l model does not d e r i v e l o a n equations from micro foundations d e s p i t e a t h e o r e t i c a l focus on loan determination at the i n d i v i d u a l borrower l e v e l . Instead, equations are c o n s t r u c t e d by assuming aggregate l o a n q u a n t i t y corresponds to the minimum of aggregate s u p p l y and demand. These inconsistencies with theory suggest that estimates of r a t i o n i n g from the t r a d i t i o n a l model are u n r e l i a b l e . Unlike t r a d i t i o n a l methods the empirical model developed i n t h i s study derives aggregate equations from a micro approach to l o a n d e t e r m i n a t i o n . I n d i v i d u a l loan sizes are determined by the minimum of borrower-specific supply and demand functions and e x p l i c i t aggregation across a l l borrowers gives estimating equations with desired properties. An a t t r a c t i v e feature of the new model i s that i t y i e l d s the f i r s t estimates of e q u i l i b r i u m c r e d i t rationing. This allowance for both equilibrium and d i s e q u i l i b r i u m r a t i o n i n g , together with the micro f o u n d a t i o n s , means that the proposed model p r o v i d e s g r e a t e r c o n s i s t e n c y between t h e o r e t i c a l and applied work than has been previously p o s s i b l e . The new model i s a p p l i e d to the market f o r b u s i n e s s loans from Canadian banks f o r the p e r i o d 1968 to 1979. R e s u l t s i n d i c a t e that r a t i o n i n g i s e m p i r i c a l l y s i g n i f i c a n t as t o t a l r a t i o n i n g averages approximately one-third of aggregate flow demand for loans. Equilibrium r a t i o n i n g a p p e a r s to be an i m p o r t a n t phenomenon s i n c e i t exceeds d i s e q u i l i b r i u m rationing each period. However, intertemporal f l u c t u a t i o n s i n t o t a l r a t i o n i n g are caused p r i m a r i l y by changes i n d i s e q u i l i b r i u m r a t i o n i n g . A comparison of the new and t r a d i t i o n a l models shows that r a t i o n i n g estimates are g r e a t e r i n the new approach w i t h much of the d i f f e r e n c e a t t r i b u t a b l e to the amount of e q u i l i b r i u m r a t i o n i n g i n that model. i i i TABLE OF CONTENTS Abstract i i L i s t of Tables v i L i s t of Figures v i i Acknowledgement v i i i CHAPTER 1 Introduction 1 1.1 Credit Rationing 1 1.2 A P r i o r i Evidence 4 1.3 Business Loans Market 6 1.4 Contents 10 2 Theories of Credit Rationing 12 2.1 A v a i l a b i l i t y Doctrine 12 2.2 Customer Relationship Theories 14 2.3 Equilibrium Credit Rationing 17 2.3.1 Jaffee (1971) 17 2.3.2 Jaffee-Russell (1976) 19 2.3.3 S t i g l i t z - W e i s s (1981) 20 2.3.4 Fried-Howitt (1980) 22 2.4 Disequilibrium Credit Rationing 23 3 Imperfect Interest Rate D i f f e r e n t i a t i o n 25 3.1 Loan Offer Curve 26 3.2 Discriminating Monopoly P r i c i n g 34 3.3 Rationing with Constrained D i f f e r e n t i a t i o n 35 3.4 Iso- P r o f i t Curves 42 3.5 Imperfect D i f f e r e n t i a t i o n and Demand Uncertainty 45 4 Econometric Studies of Disequilibrium Rationing 54 4.1 Proxy Method 54 4.2 Structural Estimation 57 4.3 Aggregate Rationing with an Individual Short-Side Rule 64 5 An Econometric Model of Aggregate Credit Rationing 72 5.1 Individual Loan Size Determination (Flows) 72 5.2 Derivation of the Aggregate Loan Equation 76 5.3 Interest Rate Equation S p e c i f i c a t i o n 81 5.4 Equilibrium and Disequilibrium Credit Rationing 84 5.5 Empirical Results 86 5.6 Model Comparisons 98 6 Conclusions 108 Bibliography 113 i v , V f Appendix 1: Properties of the Loan Offer Curve 115 Appendix 2: Properties of I s o - P r o f i t Curves 118 Appendix 3: Cumulative Densities and Expectations 123 Appendix 4: Modelling Predetermined Loans i n an Aggregate Loan Equation 124 v LIST OF TABLES Table 1: Business Loans Outstanding from Chartered Banks 8 2: Business Loans Outstanding from Fi n a n c i a l I n s t i t u t i o n s 9 3: Parameter Estimates from the Individual Short-Side Rule Model 90 4: Per Capita Rationing Estimates 92 5: Interest Rate Adjustment i n the Individual Short-Side Rule Model 97 6: Parameter Estimates from Aggregate Short-Side Rule Models 101 7: Interest Rate Adjustment i n AG-1 and AG-2 102 8: Per Capita Disequilibrium Rationing i n Aggregate and Individual Short-Side Rule Models 105 v i LIST OF FIGURES Figure 1: Loan Offer Curve to Borrower i 33 2: Default Risk and Rationing Status 40 3: Potential Equilibrium Positions with Individual Demand Uncertainty 47 4: Credit Rationing with L = min(L d,L s) 63 5: Individual Loan Determination 77 6: Aggregate Credit Rationing with L^ = min(LV.LT) 85 v i i A c k n o w l e d g e m e n t I acknowledge with much g r a t i t u d e the a s s i s t a n c e p r o v i d e d by Professor Keizo Nagatani as my d i s s e r t a t i o n supervisor. His guidance and encouragement were important on numerous o c c a s i o n s . I am a l s o t h a n k f u l to P r o f e s s o r T e r r y Wales f o r v a l u a b l e s u g g e s t i o n s made at c r i t i c a l stages of t h i s study. G r a t i t u d e i s a l s b extended to P r o f e s s o r s Robbie Jones and Ken White who se r v e d as members of the supervisory committee at various times. T h i s work i s d e d i c a t e d i n memory of my mother, E i l e e n Ruth Crawford, with c o n t i n u i n g a p p r e c i a t i o n f o r her kin d support and i n s p i r a t i o n . v i i i CHAPTER 1 INTRODUCTION 1.1 Credit Rationing Since the e a r l y 1950s there has been frequent controversy concerning the r e l a t i v e importance of p r i c e versus n o n - p r i c e a l l o c a t i o n i n l o a n markets. Numerous authors have i d e n t i f i e d c h a r a c t e r i s t i c s of loan markets that a l l e g e d l y prevent the i n t e r e s t r a t e from a d j u s t i n g s u f f i c i e n t l y to eliminate a l l excess demand for loans. This imperfect price f l e x i b i l i t y means that some i n d i v i d u a l s experience u n s a t i s f i e d demands at the rate of in t e r e s t quoted f o r t h e i r loans. More formally, "cred i t r ationing" occurs whenever an i n d i v i d u a l ' s desired loan demand i s greater than the lender's a c t u a l s u p p l y at the i n t e r e s t r a t e set by a l e n d e r . It i s nec e s s a r y f o r lenders to use non-price c r i t e r i a to determine whether a given customer i s rationed or receives the loan si z e requested. The e x i s t e n c e of c r e d i t r a t i o n i n g appears to v i o l a t e c o n v e n t i o n a l economic a n a l y s i s which emphasizes the a l l o c a t i v e function of the price mechanism. In a competitive market non-price rationing would not occur i n equilibrium since excess demand induces price increases u n t i l supply and demand are equalized. S i m i l a r l y , rationing does not develop i n standard a n a l y s i s of monopoly p r i c i n g since the quantity traded i s determined along the demand curve. Consequently, i t must be considered why transactions may occur i n l o a n markets at i n t e r e s t r a t e s c o n s i s t e n t with excess demand. Theoretical explanations of cre d i t rationing, to be discussed i n Chapter 2, must address the fundamental i s s u e of the r a t i o n a l i t y of l e n d e r s a l l o c a t i n g loans (at l e a s t i n part) by non-price means. If c r e d i t r a t i o n i n g does e x i s t i t has p o t e n t i a l r e p e r c u s s i o n s i n se v e r a l areas. The a l l o c a t i o n of loanable funds among d i f f e r e n t types of 1 borrowers may d i f f e r s y s t e m a t i c a l l y according to whether loan decisions are guided by a pure p r i c e system or some combination of p r i c e and n o n - p r i c e fac t o r s . Therefore, c r e d i t rationing might influence the r e l a t i v e growth of d i f f e r e n t sectors i n the economy by a f f e c t i n g the r e l a t i v e a v a i l a b i l i t y of l oans to these s e c t o r s . The p r e c i s e nature of these a v a i l a b i l i t y e f f e c t s would depend on the p a r t i c u l a r non-price c r i t e r i a used to decide which borrowers are rationed. The p o s s i b i l i t y of r a t i o n i n g a l s o poses i n t e r e s t i n g q u e s t i o n s concerning the impact of a quantity constraint i n one f i n a n c i a l market on a firm's t r a n s a c t i o n s i n other f i n a n c i a l and r e a l markets. P o t e n t i a l borrowers may have access to s e v e r a l a l t e r n a t i v e sources of f i n a n c e and adjust t h e i r demands among these sources i f i n i t i a l plans are not r e a l i z e d . As an example, a firm seeking to finance an investment project may t r y to o b t a i n a d d i t i o n a l funds from non-bank f i n a n c i a l i n t e r m e d i a r i e s i f i t i s rationed by a bank. The ultimate impact on r e a l spending of rationing i n the bank l o a n market would depend on the r a t i o n e d customers' success i n other f i n a n c i a l markets and on the cost of such a l t e r n a t i v e financing. The macroeconomic consequences of c r e d i t rationing have been noted i n discussions of monetary p o l i c y e f f e c t i v e n e s s . The p r i n c i p l e channel of monetary p o l i c y impact on r e a l v a r i a b l e s i s often I d e n t i f i e d with i n t e r e s t rate effects.^ Contractionary monetary p o l i c y which raises i n t e r e s t rates w i l l decrease output and employment provided some component of aggregate demand i s i n t e r e s t - s e n s i t i v e . However, i f c r e d i t r a t i o n i n g i s an e m p i r i c a l l y s i g n i f i c a n t phenomenon, i t r e p r e s e n t s a second route f o r monetary p o l i c y to influence the r e a l sector. If monetary contraction i s accompanied by lagged adjustment of loan i n t e r e s t rates, the quantity of l oans granted can be c o n s t r a i n e d by the l e n d e r s ' w i l l i n g n e s s to s u p p l y 1. A l t e r n a t i v e channels of monetary p o l i c y influence on the r e a l sector are discussed i n Park (1972). loans r a t h e r than borrowers' demands at the c u r r e n t r a t e of i n t e r e s t . T h i s n o n - p r i c e r a t i o n i n g may decrease the aggregate volume of r e a l expenditures by imposing financing constraints on a c t i v i t i e s of borrowers. The a d d i t i o n of c r e d i t r a t i o n i n g to the l i s t of monetary p o l i c y transmission mechanisms i s s i g n i f i c a n t for two reasons. Even i f investment spending i s not very i n t e r e s t - e l a s t i c , so that the i n t e r e s t rate channel i s weak or n o n - e x i s t e n t , monetary p o l i c y c o u l d s t i l l have r e a l e f f e c t s by changing the volume of c r e d i t rationing. In addition, the speed with which monetary p o l i c y operates on r e a l v a r i a b l e s should depend on the strength of the rationing channel. Tucker (1968) notes that lagged adjustment of loan i n t e r e s t rates a f t e r c o n t r a c t i o n a r y monetary p o l i c y has two c o n f l i c t i n g t e ndencies f o r investment demand. The immediate e f f e c t on investment through the i n t e r e s t r a t e channel i s reduced s i n c e movement to h i g h e r i n t e r e s t r a t e s hxas been d e l a y e d . However, as d e s c r i b e d p r e v i o u s l y , the p a r t i a l adjustment of the loan i n t e r e s t rate creates non-price rationing which has the opposite e f f e c t of strengthening the response of investment spending to monetary p o l i c y . The r e l a t i v e strength of these two opposing f o r c e s i n f l u e n c e s the speed of output adjustment a f t e r c o n t r a c t i o n a r y monetary p o l i c y . ' E m p i r i c a l s t u d i e s are r e q u i r e d to gain i n s i g h t s i n t o such i s s u e s as the q u a n t i t a t i v e s i g n i f i c a n c e of rationing, the degree to which rationing i n one l o a n market i s o f f s e t by i n c r e a s e d a c t i v i t y i n other f i n a n c i a l 2~. This point i s considered by Park (1972) and Scott (1957a). 3. Tucker (1968) analyses the r e l a t i o n s h i p between lagged i n t e r e s t rate adjustment and the speed of monetary p o l i c y e f f e c t s with a dynamic IS-LM model. He concludes (p.83) that " i f c r e d i t r a t i o n i n g has a s i g n i f i c a n t l y s t r o n g impact on investment demand, and i f there i s a s i g n i f i c a n t m a r k e t - c l e a r i n g l a g i n the product market, then the economy w i l l respond more r a p i d l y to monetary c o n t r a c t i o n when the i n t e r e s t r a t e i s s t i c k y t h an when i t b e h a v e s w i t h extreme f l e x i b i l i t y . " 4. An i n d i c a t i o n that monetary a u t h o r i t i e s do perceive the operation of both i n t e r e s t r a t e and r a t i o n i n g c h annels i s found i n Rasminsky (1969), p. 14. 3 markets, and the u l t i m a t e impact of c r e d i t r a t i o n i n g on r e a l expenditures. However, adequate i n v e s t i g a t i o n of these i s s u e s has been impeded by d i f f i c u l t i e s i n o b t a i n i n g a s a t i s f a c t o r y e m p i r i c a l measure of c r e d i t r a t i o n i n g . In response to t h i s d e f i c i e n c y the p r e s e n t s t u d y d e v e l o p s a methodology t h a t can p r o v i d e q u a n t i t a t i v e e s t i m a t e s of the volume of r a t i o n i n g . The model i s then a p p l i e d t o the b u s i n e s s l o a n s market of Canadian chartered banks. Before proceeding to these o b j e c t i v e s i t w i l l be b e n e f i c i a l t o o u t l i n e a p r i o r i e v i d e n c e f o r the e x i s t e n c e of c r e d i t r a t i o n i n g i n Canada and d e s c r i b e the b u s i n e s s l o a n s market of Canadian banks. 1.2 A Priori Evidence S e v e r a l sample surveys c o n t a i n i n f o r m a t i o n which suggests that c r e d i t r a t i o n i n g i s an e m p i r i c a l phenomenon i n Canada. The most comprehensive s u r v e y was u n d e r t a k e n as p a r t of a r e c e n t study on the r o l e of c h a r t e r e d banks i n s m a l l business f i n a n c i n g . R e s u l t s of t h i s survey, summarized i n Hatch, Wynant, and Grant (1982), were d e r i v e d from t h r e e s o u r c e s of i n f o r m a t i o n . F i r s t , 400 s m a l l b u s i n e s s e s responded to a q u e s t i o n n a i r e c o n c e r n i n g t h e i r d e a l i n g s w i t h Canadian c h a r t e r e d banks. S e c o n d l y , i n t e r v i e w s were c a r r i e d out w i t h 120 bank employees to d e t e r m i n e the lenders' p e r s p e c t i v e on the loan process. F i n a l l y , 2,300 a c t u a l loan f i l e s f o r both s m a l l and l a r g e businesses were reviewed. A n a l y s i s of the t h i r d i n f o r m a t i o n source i n d i c a t e d t h a t on average firms i n the sample had r e c e i v e d approximately 90 per cent of the amounts r e q u e s t e d on f o r m a l l o a n a p p l i c a t i o n s . However, the 10 per cent u n s a t i s f i e d demand on these formal a p p l i c a t i o n s probably understates the o v e r a l l magnitude of r a t i o n i n g . I n t e r v i e w s w i t h bank branch managers suggested that about 25 per cent of a l l loan i n q u i r i e s are r e j e c t e d before r e a c h i n g the f o r m a l a p p l i c a t i o n s t a g e of the l o a n p r o c e s s . I t i s a l s o 4 p o s s i b l e that some loan sizes a c t u a l l y requested on form a l a p p l i c a t i o n s were l e s s than the amounts o r i g i n a l l y sought by the borrowers. Hence, the 10 per cent f i g u r e d e r i v e d from a c t u a l a p p l i c a t i o n s may be regarded as a lower bound estimate of the prevalence of cr e d i t rationing. A f i n a l piece of evidence from the Hatch-Wynant-Grant study was a find i n g that one-third of a l l firms responding to the questionnaire had been denied a loan request at some time during the preceding three year period.-* A d d i t i o n a l a p r i o r i i n d i c a t i o n s of cr e d i t rationing i n Canada may be i n f e r r e d from comments of J.A. G a l b r a i t h who has been a c t i v e l y i n v o l v e d w i t h i n the f i n a n c i a l s e c t o r . G a l b r a i t h observed circumstances when in t e r e s t rates did not adjust completely to changes i n market conditions and non-price rationing was necessary. "When the monetary p o l i c y of the day r e s t r i c t s the lending resources of the banks, the banks must r e s t r i c t t h e i r l e n d i n g a c t i v i t i e s . In circumstances such as those faced by Canadian banks at the end of the 1960s, lending a c t i v i t y at the branches has to be c u r t a i l e d . When l e n d i n g r a t e s are not r a i s e d s u f f i c i e n t l y to disc o u r a g e the demand f o r loans or when hi g h e r r a t e s a p p a r e n t l y f a i l to r e s t r i c t demand to the a v a i l a b l e resources, i n s t r u c t i o n s have to be communicated to the branches to c u r t a i l l o a n s . A set of p r i o r i t i e s i s developed to be used as a guide by the branches." A statement by the Canadian Bankers' A s s o c i a t i o n l i s t s some of the non-price guidelines used by lenders to e s t a b l i s h loan p r i o r i t i e s during these peri o d s . "... during periods when c r e d i t i s l e s s e a s i l y a v a i l a b l e a change i n a t t i t u d e must take p l a c e . Lending must become more s e l e c t i v e . Several categories of loans come 5. Sears (1972) discusses some survey evidence on cr e d i t r a t ioning from the e a r l y 1960s i n Canada. J a f f e e (1971), pp. 159-161, presents survey information from the United States. 6. Galbraith (1970), p. 255. 5 under r e s t r i c t i o n , the s e v e r i t y of which depends on the tightness of the c r e d i t squeeze. For example, during a period of cr e d i t stringency a l l forms of lending r e l a t e d to s p e c u l a t i v e a c t i v i t y , such as t r a d i n g i n l a n d and s e c u r i t i e s , come under e a r l y r e s t r i c t i o n . Applications f o r new or i n c r e a s e d c r e d i t are a l s o g i v e n c l o s e s c r u t i n y . C e r t a i n types of programs, such as consumer i n s t a l m e n t l o a n s , are c u r t a i l e d , with p r o m o t i o n a l a c t i v i t y being discontinued. New lending programs under consideration are postponed. Even borrowers with long-e s t a b l i s h e d l i n e s of c r e d i t are asked to review t h e i r requirements under e x i s t i n g commitments i f the l i q u i d i t y p o s i t i o n of the i n d i v i d u a l bank requires such action." 1.3 B u s i n e s s L o a n s M a r k e t The evidence presented i n the previous section suggests that c r e d i t rationing does occur i n Canada and that further empirical examination i s warranted. As noted p r e v i o u s l y the p r i n c i p a l focus of t h i s study i s an a n a l y s i s of r a t i o n i n g i n the market f o r business l o a n s from Canadian c h a r t e r e d banks. Some b a s i c d e t a i l s w i l l e s t a b l i s h the i n s t i t u t i o n a l background and r e l a t i v e importance of th i s f i n a n c i a l market. A c t i v i t i e s of Canadian banks are governed by fede r a l l e g i s l a t i o n under the Bank Act. In the 1967 r e v i s i o n of th i s l e g i s l a t i o n there were s e v e r a l Q major changes a f f e c t i n g the l e n d i n g f u n c t i o n of banks. P r i o r to t h i s r e v i s i o n there was a c e i l i n g of 6% on the i n t e r e s t r a t e t h a t c o u l d be charged on a l o a n by c h a r t e r e d banks. Thi s maximum r a t e p r o v i s i o n was removed i n stages under the 1967 Bank Act u n t i l r e g u l a t i o n of i n t e r e s t r a t e s was e l i m i n a t e d c o m p l e t e l y by the beginning of 1968. A second r e g u l a t o r y change i n 1967 i n c r e a s e d the a b i l i t y of c h a r t e r e d banks to engage i n mortgage lending. These Bank Act re v i s i o n s represented a s i g n i f i c a n t s t r u c t u r a l change having important implications for any empirical study of c r e d i t rationing. 7. Quoted i n G a l b r a i t h (1970), pp. 253-254, from a p r e s e n t a t i o n of the Canadian Bankers' Association to the Committee on Finance, Trade and Economic A f f a i r s , September 1969. 8. These changes are discussed i n Shearer, Chant and Bond (1984), p.361. 6 Comments of the Governor of the Bank of Canada during that period indicated a b e l i e f t h a t the l e g i s l a t i v e c e i l i n g on l o a n r a t e s had i n c r e a s e d the incidence of non-price rationing of business loans. "A good case c o u l d be made that i n the past the l e g a l r e s t r i c t i o n s on chartered bank lending rates backfired on the groups of borrowers they were intended to protect - such as s m a l l businessmen - because the banks were d e p r i v e d of any p r o f i t i n c e n t i v e to make a s e r i o u s e f f o r t to i n c r e a s e t h e i r access to bank c r e d i t ... I t used to be the case th a t monetary r e s t r a i n t a f f e c t e d l e n d i n g p o l i c i e s j u s t about as much as i t a f f e c t e d i n t e r e s t rates. F l e x i b i l i t y i n the banks' lending rates and consequently i n t h e i r deposit rates was l i m i t e d by the Bank Act, and when the banks' l i q u i d i t y was under downward pr e s s u r e they were f o r c e d to adopt more s e l e c t i v e l e n d i n g p o l i c i e s , i n other words to r a t i o n loans more c l o s e l y . " ^ The l e g i s l a t i v e c o n s t r a i n t on l o a n i n t e r e s t r a t e s was a source of c r e d i t r a t i o n i n g unique to the pre-1968 p e r i o d . In order to i s o l a t e the magnitude of rationing a r i s i n g from n o n - l e g i s l a t i v e sources the time period selected for subsequent empirical work begins i n 1968. In nominal terms the v a l u e of business l o a n s o u t s t a n d i n g from c h a r t e r e d banks i n c r e a s e d from $6.9 b i l l i o n at the end of 1967 to $55.4 b i l l i o n i n 1980 (see Table 1). These loans are an important component of the banks' asset p o r t f o l i o s . Approximately 20% of t o t a l bank assets were held as business loans over the 1967-1980 period. Table 2 indicates that chartered banks are the primary source of a l l loans to businesses i n Canada. It i s estimated that i n 1979 approximately 80% of the t o t a l value of business loans from f i n a n c i a l i n s t i t u t i o n s were granted by c h a r t e r e d banks. Th i s s t a t i s t i c suggests that a study of r a t i o n i n g by c h a r t e r e d banks sho u l d g i v e a r e l i a b l e i n d i c a t i o n of the o v e r a l l degree of c r e d i t r a t ioning confronting Canadian businesses. 9. Rasminsky (1969), pp. 13-14. 7 TABLE 1 Business Loans Outstanding from Chartered Banks (m i l l i o n s of d o l l a r s , end of period values) Business Loans Total Bank Ratio of Business Outstanding Assets Loans to To t a l Assets 1967 6,929 31,669 .219 1968 7,589 36,746 .207 1969 8,654 42,632 .203 1970 8,900 47,307 .188 1971 11,068 54,428 .203 1972 13,461 63,222 .213 1973 17,135 79,754 .215 1974 20,568 97,015 .212 1975 23,228 108,378 .214 1976 28,218 126,403 .223 1977 31,323 150,477 .208 1978 34,441 189,100 .182 1979 44,866 229,440 .196 1980 55,385 281 ,244 .196 Source: Bank of Canada Review 8 TABLE 2 Business Loans Outstanding from F i n a n c i a l I n s t i t u t i o n s ( m i l l i o n s of d o l l a r s , end of 1979) Chartered Banks Trust Companies Mortgage Loan Companies Credit Unions Fi n a n c i a l Corporations F i n a n c i a l Leasing Corporations Business Financing Corporations Total Business Loans In s t i t u t i o n s Chartered Banks Trust Companies Mortgage Loan Companies Credit Unions F i n a n c i a l Corporations, F i n a n c i a l Leasing Corporations, Business F i n a n c i a l Corporations 44,866 276 173 581 5,718 333 4,286 56,233 De f i n i t i o n s of Business Loans Business loans outstanding Other c o l l a t e r a l business loans Other c o l l a t e r a l business loans Non-mortgage loans: commercial, i n d u s t r i a l , and co-operative enterprises Business loans: commercial; R e t a i l sales financing: i n d u s t r i a l and commercial; Wholesale financing Sources: Bank of Canada Review, A p r i l 1980, Table 10. F i n a n c i a l I n s t i t u t i o n s : F i n a n c i a l S t a t i s t i c s , A p r i l 1982, S t a t i s t i c s Canada, Tables 8, 16, 21, 23, 33, 42 and 46. 9 1 . 4 C o n t e n t s Recent t h e o r e t i c a l explanations of c r e d i t rationing have emphasized a d i s t i n c t i o n between e q u i l i b r i u m and d i s e q u i l i b r i u m categories of rationing. Whereas eq u i l i b r i u m models explain why excess demand for loans may p e r s i s t even at the e q u i l i b r i u m r a t e of i n t e r e s t , d i s e q u i l i b r i u m r a t i o n i n g i s a t r a n s i t o r y consequence of lagged adjustment of the i n t e r e s t r a t e to i t s e q u i l i b r i u m l e v e l . Current e m p i r i c a l s t u d i e s of r a t i o n i n g may be c r i t i c i z e d f o r neglecting to incorporate the t h e o r e t i c a l i n t e r e s t i n both e q u i l i b r i u m and d i s e q u i l i b r i u m c r e d i t r a t i o n i n g . E x i s t i n g econometric evidence has been obtained from e s t i m a t i o n of aggregate demand and s u p p l y equations f o r business loans. These two equations are derived from an "aggregate short-side r u l e " that assumes the observed aggregate loan volume i s the minimum of aggregate demand and supply. Furthermore, these studies postulate that the e q u i l i b r i u m i n t e r e s t r a t e occurs at the i n t e r s e c t i o n of the two aggregate functions. Taken together these two assumptions imply that the current method of estimating the volume of rationing focuses e x c l u s i v e l y on di s e q u i l i b r i u m rationing and ignores the equ i l i b r i u m category. The o b j e c t i v e of t h i s study i s to reduce the c u r r e n t d i s c r e p a n c y between theory and econometric p r a c t i c e by e s t i m a t i n g a model which combines e q u i l i b r i u m and d i s e q u i l i b r i u m c r e d i t r a t i o n i n g . In order to construct t h i s u n i f i e d model the aggregate short-side r u l e described above i s r e p l a c e d by an " i n d i v i d u a l s h o r t - s i d e r u l e " found i n the t h e o r e t i c a l l i t e r a t u r e . The l a t t e r r u l e s t a t e s that the l o a n s i z e r e c e i v e d by an i n d i v i d u a l i s the minimum of b o r r o w e r - s p e c i f i c l o a n demand and s u p p l y functions at the current i n t e r e s t rate. The contents of the study are as f o l l o w s . Chapters 2 and 3 summarize and c r i t i q u e t h e o r e t i c a l e x p l a n a t i o n s f o r the e x i s t e n c e of c r e d i t 10 r a t i o n i n g . E x i s t i n g econometric models of d i s e q u i l i b r i u m r a t i o n i n g are presented i n Chapter 4. Chapter 4 a l s o e v a l u a t e s these d i s e q u i l i b r i u m models u s i n g the t h e o r e t i c a l i n d i v i d u a l s h o r t - s i d e r u l e . Chapter 5 proposes a two-equation system that s a t i s f i e s the s t a t e d o b j e c t i v e of i n c o r p o r a t i n g both r a t i o n i n g c a t e g o r i e s i n a u n i f i e d model. It i s shown that parameter estimates from t h i s system may be used to i n f e r the separate magnitudes of e q u i l i b r i u m and d i s e q u i l i b r i u m r a t i o n i n g f o r each time p e r i o d . Estimates of these q u a n t i t i e s f o r chartered bank business loans are presented and d i s c u s s e d . F i n a l l y , Chapter 6 comments on p o t e n t i a l a p p l i c a t i o n s of the model to the study of other i s s u e s r e l a t e d to c r e d i t r a t i o n i n g . 11 CHAPTER 2 Theories of Credit Rationing C r e d i t r a t i o n i n g occurs when a l e n d e r s u p p l i e s a l o a n s i z e that i s l e s s than borrower demand at the i n t e r e s t r a t e quoted f o r the l o a n . Conceivably t h i s non-price a l l o c a t i o n could have been replaced by a p o l i c y of r a i s i n g i n t e r e s t rates u n t i l a l l excess loan demand i s eliminated. The t h e o r e t i c a l c r e d i t rationing l i t e r a t u r e explains the existence of rationing by i d e n t i f y i n g f a c t o r s that might prevent the degree of i n t e r e s t r a t e f l e x i b i l i t y necessary to remove a l l excess demand. A l t e r n a t i v e theories may be d i f f e r e n t i a t e d a c c o r d i n g to the manner i n which they r a t i o n a l i z e t h i s i m p e r f e c t f l e x i b i l i t y of l o a n r a t e s . The v a r i o u s e x p l a n a t i o n s can a l s o be d i s t i n g u i s h e d on the b a s i s of t h e i r p r e d i c t i o n s c o n c e r n i n g the c h a r a c t e r i s t i c s of p o t e n t i a l borrowers most l i k e l y to be rationed. These points are demonstrated i n the survey of the t h e o r e t i c a l l i t e r a t u r e which fol l o w s . This review w i l l a l s o c l a r i f y the concepts of equ i l i b r i u m and d i s e q u i l i b r i u m c r e d i t r a t i o n i n g which are important i n l a t e r e m p i r i c a l a n a l y s i s . * 2.1 Availability Doctrine Some of the e a r l i e s t discussions of c r e d i t rationing originated with the a v a i l a b i l i t y doctrine. This doctrine was developed during the 1950s as an argument that monetary p o l i c y c o u l d be e f f e c t i v e even i f borrower behavior i s i n s e n s i t i v e to va r i a t i o n s i n i n t e r e s t rates. According to this viewpoint, contractionary monetary p o l i c y can decrease the volume of loans granted even i f loan demand i s p e r f e c t l y i n e l a s t i c , and the primary channel of monetary p o l i c y e f f e c t i v e n e s s i s through changes i n the a v a i l a b i l i t y of 1. Baltensperger (1978) and Jaffee (1971) contain surveys of t h e o r e t i c a l studies. 12 c r e d i t from lenders rather than the cost of borrowing. A representative t h e o r e t i c a l exposition of the a v a i l a b i l i t y doctrine i s g i v e n by Scott (1957a). He a n a l y s e s the p o r t f o l i o d e c i s i o n of a f i n a n c i a l intermediary faced with the problem of a l l o c a t i n g funds between government s e c u r i t i e s and p r i v a t e l o a n s . Loans are assumed to be the higher r i s k asset since government s e c u r i t i e s are not subject to d e f a u l t r i s k on i n t e r e s t payments and repayment of p r i n c i p a l . It i s a l s o assumed that i n v e s t o r u t i l i t y depends on the mean and variance of p o r t f o l i o y i e l d . S c o t t shows that an open-market s a l e of government bonds by the c e n t r a l bank w i l l r a i s e expected return on the bonds and decrease the proportion of funds invested i n p r i v a t e loans. Since i t i s assumed that expected return on loans i s constant due to "the s t i c k i n e s s of customer l o a n r a t e s , " contractionary monetary p o l i c y reduces the quantity of loans supplied at the o r i g i n a l l o a n i n t e r e s t r a t e by i n c r e a s i n g the r e l a t i v e y i e l d on government bonds. The r e l a t i o n s h i p between the a v a i l a b i l i t y d o c t r i n e and c r e d i t rationing can be demonstrated by i n t e r p r e t i n g Scott's r e s u l t i n terms of a supply-demand diagram f o r l o a n s . An i n c r e a s e i n expected y i e l d on government bonds w i l l s h i f t the l o a n s u p p l y curve to the l e f t at each i n t e r e s t r a t e f o r l o a n s . If the i n i t i a l l o a n r a t e occurred at the i n t e r s e c t i o n of supply and demand, the leftward s h i f t of the supply curve must reduce the quantity of loans transacted and cause non-price rationing i f the l o a n r a t e remains unchanged as assumed by S c o t t . R a t i o n i n g would develop even with price f l e x i b i l i t y provided the loan rate did not move immediately to the new supply-demand i n t e r s e c t i o n . A major conclusion from Scott's a n a l y s i s i s that monetary p o l i c y can 2. Roosa (1951) and Scott (1957b) p r o v i d e g e n e r a l d i s c u s s i o n s of the a v a i l a b i l i t y doctrine. 3. Scott (1957a), p. 46. 13 be e f f e c t i v e even i f investment demand i s I n t e r e s t - i n e l a s t i c by reducing the quantity of loans supplied by lenders. Although c r e d i t rationing i s given a r o l e i n t h i s process, the a v a i l a b i l i t y doctrine i s not considered an adequate theory of rationing. It has been c r i t i c i z e d for not providing a t h e o r e t i c a l explanation for the lender's f a i l u r e to r a i s e the loan rate to eliminate excess demand. For example, Jaffee (1971) states that Scott's assumption of r i g i d i t i e s i n the l o a n r a t e " i s more a statement of a necessary condition for his conclusion than an explanation."^ 2 . 2 C u s t o m e r R e l a t i o n s h i p T h e o r i e s Hodgman (1961) and Kane and M a l k i e l (1965) d e v e l o p e d t h e o r i e s of rationing that consider the multi-dimensional nature of the r e l a t i o n s h i p between an i n d i v i d u a l borrower and lender. In these customer r e l a t i o n s h i p t h e o r i e s the i n c i d e n c e of r a t i o n i n g i s r e l a t e d to f a c t o r s such as the s t a b i l i t y of the borrower-lender association and the p r o f i t a b i l i t y to the lender of any non-loan transactions. Hodgman considers whether a loan applicant's demand deposit balances, by p r o v i d i n g non-loan income to the l e n d e r , w i l l a f f e c t that customer's success i n o b t a i n i n g c r e d i t . There i s an I n c e n t i v e f o r banks to compete f o r p r o f i t a b l e d e p o s i t accounts by o f f e r i n g l oans to owners of these deposits at i n t e r e s t rates below the l e v e l charged non-depositors.~* In a c o m p e t i t i v e market the s t r u c t u r e of l o a n r a t e s would a d j u s t u n t i l the lender i s i n d i f f e r e n t between a loan to a borrower-depositor at a reduced i n t e r e s t r a t e and a l o a n to a n o n - d e p o s i t o r at a h i g h e r r a t e . Thus, i f loan rates were competitively determined, an i n d i v i d u a l ' s deposits would Jaffee (1971), p. 20. 5. Hodgman assumes an i n s t i t u t i o n a l s e t t i n g where i n t e r e s t payments on demand d e p o s i t s are p r o h i b i t e d by law. In t h i s s i t u a t i o n a bank regards a g i v e n d e p o s i t o r ' s b a l a n c e s as p r o f i t a b l e i f revenue from a s s e t s supported by those b a l a n c e s exceeds s e r v i c i n g c o s t s on the account. >. 14 i n f l u e n c e the l e v e l of the loan i n t e r e s t rate but would not be a source of non-price c r e d i t r ationing. A connection between deposit balances and c r e d i t rationing arises when Hodgman introduces p r i c i n g constraints into the loan process. He asserts that banks i n an o l i g o p o l i s t i c market l i m i t i n t e r - b a n k c o m p e t i t i o n f o r deposits by i m p l i c i t l y agreeing not to grant any loans at rates below some minimum such as the prime r a t e . With t h i s p r i c i n g c o n s t r a i n t some customers are charged the prime rate but would have received a lower loan rate under competitive conditions due to t h e i r p r o f i t a b l e deposit balances. T h e r e f o r e , the bank's o v e r a l l r a t e of r e t u r n from a l l business of such i n d i v i d u a l s i s above the g e n e r a l market r e t u r n and these customers have p r e f e r r e d s t a t u s . During p e r i o d s of high demand l o a n requests from preferred borrowers are accommodated f i r s t and other i n d i v i d u a l s could be rationed. In p r i n c i p l e a non-preferred customer could avoid rationing by paying a g r e a t e r l o a n r a t e to match the bank's o v e r a l l r e t u r n from a p r e f e r r e d b o r r o w e r . However, Hodgman a r g u e s t h a t i n t e r t e m p o r a l considerations would induce lenders to use non-price rationing rather than r a i s e i n t e r e s t r a t e s immediately f o r n o n - p r e f e r r e d customers. In h i s opinion the former option would be l e s s l i k e l y to cause these customers to s h i f t t h e i r business to another bank. Kane and Malkiel's discussion p a r a l l e l s Hodgman's ana l y s i s and comes to s i m i l a r c o n c l u s i o n s . R e f u s a l of a l o a n request c o u l d a l i e n a t e the customer and l e a d to withdrawal of any d e p o s i t s h e l d with the l e n d e r . Since deposit v a r i a b i l i t y reduces expected p r o f i t and raises lender r i s k , ^ l o a n a p p l i c a n t s p o s s e s s i n g l a r g e and s t a b l e d e p o s i t b a l a n c e s r e c e i v e p r e f e r e n t i a l treatment from the bank. If each bank has im p e r f e c t 6~. Hodgman (1961), pp. 265-266. 7. Thi s c o n c l u s i o n i s obtained from a mo d i f i e d mean-variance model of p o r t f o l i o a l l o c a t i o n . 15 information on deposit c h a r a c t e r i s t i c s of customers c u r r e n t l y dealing with other banks, t h i s information asymmetry among banks impedes the competitive pressures toward lower loan rates for borrower-depositors. Consequently, holders of large and s t a b l e deposits are compensated through p r e f e r e n t i a l queuing during periods of excess loan demand rather than reduced i n t e r e s t r a t e s on l o a n s . I n d i v i d u a l s w i t h o u t t h e s e f a v o u r a b l e d e p o s i t c h a r a c t e r i s t i c s would be the f i r s t to be denied l o a n r e q u e s t s . Kane and M a l k i e l speculate that t h i s r a t ioning would be a short-run r e s u l t of lagged o l i g o p o l i s t i c price adjustment a f t e r a s h i f t i n loan demand. In the Hodgman and K a n e - M a l k i e l s t u d i e s l a r g e d e p o s i t s p r o v i d e p r o t e c t i o n from c r e d i t r a t i o n i n g because i n s t i t u t i o n a l or i n f o r m a t i o n problems prevent f u l l compensation to holders of deposits through e x p l i c i t loan or deposit i n t e r e s t rates. Blackwell and Santomero (1982) remove t h i s p r i c i n g r e s t r i c t i o n and reach o p p o s i t e c o n c l u s i o n s r e g a r d i n g the l i n k between deposits and the incidence of rationing. They analyse the case of a monopolistic lender that sets a profit-maximizing i n t e r e s t rate for each customer's loan. In t h i s s i t u a t i o n they argue that i n d i v i d u a l s with high d e p o s i t b a l a n c e s or s t r o n g i n t e r t e m p o r a l l o a n demand would be most s u s c e p t i b l e to c r e d i t r a t i o n i n g s i n c e these c h a r a c t e r i s t i c s reduce an i n d i v i d u a l ' s e q u i l i b r i u m loan rate. Evaluated at these i n d i v i d u a l rates, the lender's l o s s of p r o f i t from quantity rationing i s lowest for borrowers with high deposits and intertemporal loan demand. If rationing does take p l a c e these customers would be r a t i o n e d before other groups. However, B l a c k w e l l and Santomero o n l y c o n s i d e r the incidence of u n s a t i s f i e d loan demand under an assumption th a t i t does e x i s t . They do not e x p l a i n the e x i s t e n c e of r a t i o n i n g s i n c e the monopolist's d e c i s i o n to r a t i o n any customer i s unprofitable at the i n d i v i d u a l rates of i n t e r e s t derived from t h e i r model. 16 2 . 3 Equilibrium Credit Rationing Kane and M a l k i e l speculated that i n the long-run loan i n t e r e s t rates would r i s e to l e v e l s that eliminated excess demands and the need for non-price rationing. In the short-run they b e l i e v e d rationing would occur i f i n t e r e s t r a t e s a d j u s t e d with a l a g a f t e r an i n c r e a s e i n l o a n demand. Subsequent t h e o r e t i c a l studies have emphasized e q u i l i b r i u m c r e d i t rationing which r e f e r s to u n s a t i s f i e d l o a n demands at the e q u i l i b r i u m r a t e of i n t e r e s t . The o b j e c t i v e of these s t u d i e s i s to e x p l a i n why r a t i o n a l l e n d e r s might be w i l l i n g to set an e q u i l i b r i u m i n t e r e s t r a t e that i s c o n s i s t e n t w i t h excess demand f o r at l e a s t some borrowers. An important i m p l i c a t i o n of equ i l i b r i u m rationing i s that u n s a t i s f i e d demands may be a permanent feature of loan markets and not merely a short-run consequence of incomplete i n t e r e s t r a t e movements. Some c o n t r i b u t i o n s i n t h i s area i n c l u d e J a f f e e (1971), K o s k e l a (1976, 1979a, 1979b), J a f f e e and R u s s e l l (1976), Fried and Howitt (1980), and S t i g l i t z and Weiss (1981). 2 . 3 . 1 Jaffee (1971) Jaffee examines the loan decision under an assumption that the lender o can form e x p e c t a t i o n s of each i n d i v i d u a l borrower's d e f a u l t r i s k . In addition to t h i s risk-screening capacity i t i s assumed i m p l i c i t l y that the l e n d e r knows each customer's demand cu r v e . With these assumptions the e x i s t e n c e of c r e d i t r a t i o n i n g i s shown to depend on the nature of the p r i c e - s e t t i n g regime. Jaffee i n i t i a l l y c o n s i d e r s a m o n o p o l i s t i c l e n d e r that i s able to charge a d i f f e r e n t i n t e r e s t rate f o r each customer's loan to r e f l e c t i n d i v i d u a l demand and r i s k c h a r a c t e r i s t i c s . With t h i s p r i c i n g system the monopolist's p r o f i t - m a x i m i z i n g l o a n i s a l o n g the borrower's demand curve. Hence, eq u i l i b r i u m c r e d i t rationing would be non-existent 8. A f o r m a l p r e s e n t a t i o n of J a f f e e ' s model i s cont a i n e d i n Chapter 3. Koskela (1976, 1979a, 1979b) uses a s i m i l a r model. 17 since excess demand i s zero at the lender's optimal (equilibrium) rate of i n t e r e s t . A d i f f e r e n t c o n c l u s i o n i s reached i f there are c o n s t r a i n t s on a lender's a b i l i t y or w i l l i n g n e s s to set d i f f e r e n t i n t e r e s t r a t e s f o r d i f f e r e n t borrowers. In th i s regard Jaffee argues that i t may be r a t i o n a l for banks to group customers into a l i m i t e d number of borrower categories and charge each borrower w i t h i n a g i v e n category an i d e n t i c a l i n t e r e s t r a t e . One reason f o r t h i s p r i c i n g scheme i s that "the pr e s s u r e s of good w i l l and s o c i a l mores" tend to c o n s t r a i n banks from s e t t i n g "widely d i f f e r e n t " r a t e s f o r d i f f e r e n t customers. Furthermore, a c o m p a r a t i v e l y s i m p l e borrower c l a s s i f i c a t i o n system and i n t e r e s t r a t e s t r u c t u r e may promote c o o r d i n a t e d c o l l u s i v e p r i c i n g and minimize c o m p e t i t i v e p r i c e -cutting i n o l i g o p o l i s t i c loan markets.^ These factors could cause borrower clas s e s to be established with l i m i t e d d i f f e r e n t i a t i o n of i n t e r e s t rates among customers. Equ i l i b r i u m rationing i s shown to be p r o f i t a b l e i f the above factors induce a l e n d e r to charge an i d e n t i c a l i n t e r e s t r a t e to each i n d i v i d u a l i = l,...,n i n a g i v e n borrower category. At the group's e q u i l i b r i u m i n t e r e s t r a t e R, which maximizes the lender's o v e r a l l p r o f i t from a l l n customers, t h e r e w i l l be some borrowers with demand g r e a t e r than the lender's optimal supply at R.^u A l l borrowers i n th i s p o s i t i o n receive a supply-determined loan and experience equilibrium c r e d i t r a t ioning since i t would not be p r o f i t a b l e for the lender to increase loan sizes to s a t i s f y 9. Jaffee (1971), p. 48, elaborates on th i s point. "In order to prevent, or at l e a s t minimize, c o m p e t i t i v e u n d e r b i d d i n g of r a t e s they would need t a c i t agreement as to the a p p r o p r i a t e r a t e s t r u c t u r e f o r customers, and thus a c l a s s i f i c a t i o n scheme based on r e a d i l y v e r i f i -a b l e o b j e c t i v e c r i t e r i a would appear as an e f f i c i e n t and e f f e c t i v e d e v i c e . Furthermore, to make the whole arrangement manageable, the number of d i f f e r e n t rate c l a s s e s would have to be reasonably small." 10. The lender's o p t i m a l l o a n s u p p l y to an i n d i v i d u a l at a l t e r n a t i v e i n t e r e s t rates forms a loan o f f e r curve. It i s p o s i t i v e l y - s l o p e d i f the i n d i v i d u a l ' s p r o b a b i l i t y of d e f a u l t increases with loan size. 18 t h e i r demands at R. Default r i s k receives emphasis i n the Jaffee model as the c r i t e r i o n determining which borrowers are rationed. Since d e f a u l t r i s k reduces the lender's d e s i r e d s u p p l y at a g i v e n i n t e r e s t r a t e , the p o s s i b i l i t y t h a t a borrower's demand exceeds s u p p l y at R i s p o s i t i v e l y r e l a t e d to the i n d i v i d u a l ' s degree of d e f a u l t r i s k . Jaffee's conclusion that constrained d i f f e r e n t i a t i o n of i n t e r e s t rates causes c r e d i t r a t i o n i n g has been c r i t i c i z e d by A z z i and Cox (1976) f o r disregarding the r o l e of non-price terms of loan contracts. In response to excess loan demand lenders may adjust not only the e x p l i c i t i n t e r e s t rate but a l s o n o n - p r i c e terms such as e q u i t y , c o l l a t e r a l , or compensating b a l a n c e requirements. A z z i and Cox c oncluded th a t e q u i l i b r i u m c r e d i t rationing i s not optimal for a lender unless there are constraints on the lender's s e l e c t i o n of both i n t e r e s t r a t e and n o n - p r i c e terms f o r an i n d i v i d u a l ' s loan. While acknowledging that perfect d i s c r i m i n a t i o n of non-p r i c e terms among customers would e l i m i n a t e r a t i o n i n g , J a f f e e and M o d i g l l a n i (1976) and Koskela (1976) argued that this condition i s u n l i k e l y to be met for s e v e r a l reasons. In t h e i r opinions borrowers t y p i c a l l y are l i m i t e d i n the amounts of equity and c o l l a t e r a l they are able to provide. Furthermore, the same factors that l i m i t i n t e r e s t rate d i f f e r e n t i a t i o n a l s o cause impe r f e c t i n t e r - c u s t o m e r d i f f e r e n t i a t i o n of n o n - p r i c e terms. Consequently, the p o s s i b i l i t y of c r e d i t r a t i o n i n g i s not removed by considering non-price elements of loan contracts. 2.3.2 J a f f e e - R u s s e l l (1976) A second type of e q u i l i b r i u m r a t i o n i n g model was proposed by J a f f e e and R u s s e l l (1976) who consider a s i t u a t i o n where lenders have imperfect information about the d e f a u l t r i s k c h a r a c t e r i s t i c s of i n d i v i d u a l customers. By assumption, and i n c o n t r a s t w i t h the p r e v i o u s model of J a f f e e (1971), l e n d e r s cannot determine an i n d i v i d u a l ' s honesty ( d e f a u l t r i s k ) from a 19 p r i o r i information. A l l borrowers are i d e n t i c a l except for the degree of r i s k . "Honest" b o r r o w e r s i n t e n d to r e p a y t h e i r l o a n s under a l l circumstances but " d i s h o n e s t " borrowers d e f a u l t whenever t h i s b e h a v i o r r a i s e s t h e i r u t i l i t y . * * The lender's i n a b i l i t y to screen i n d i v i d u a l customer r i s k means that both types of borrowers face the same loan supply 1 9 schedule and are charged the same i n t e r e s t rate. The J a f f e e - R u s s e l 1 model suggests two p o t e n t i a l outcomes i n a c o m p e t i t i v e market s t r u c t u r e . C o m p e t i t i v e p r e s s u r e s may l e a d to an e q u i l i b r i u m with a l l customers r a t i o n e d at a l o a n s i z e a l o n g the s u p p l y c u r v e below the i n t e r s e c t i o n of s u p p l y and demand. Both honest and d i s h o n e s t i n d i v i d u a l s are r a t i o n e d s i n c e the l e n d e r cannot d i s t i n g u i s h between these groups i n advance of a c t u a l loan d e f a u l t s . There i s a l s o a p o s s i b i l i t y of i n s t a b i l i t y under competitive conditions with entry and e x i t of lenders i n response to short-run p r o f i t s and losses. These competitive predictions d i f f e r from the pure monopoly case where the lender's p r o f i t -maximizing s o l u t i o n i s on the l o a n demand curve and t h e r e f o r e p r e c l u d e s non-price r a t i o n i n g . 2.3.3 Stiglitz-Weiss (1981) A number of s t u d i e s have a n a l y s e d e q u i l i b r i u m i n markets where the q u a l i t y of the good being traded v a r i e s with the p r i c e due to adverse 11. More s p e c i f i c a l l y , a d i s h o n e s t borrower d e f a u l t s i f the l e v e l of contractual ( i n t e r e s t plus p r i n c i p a l ) payments exceeds that customer's cost of d e f a u l t . The model's conclusions are v a l i d f o r any s i t u a t i o n with uncertainty and imperfect information that causes the proportion of borrowers th a t d e f a u l t to i n c r e a s e with the s i z e of c o n t r a c t u a l payments (Jaffee-Russel 1 (1976), p. 657). 12. The loan supply schedule to an i n d i v i d u a l i s p o s i t i v e l y - s l o p e d (or p o s s i b l y backward-bending) i f the aggregate p r o p o r t i o n of a l l borrowers that d e f a u l t i n c r e a s e s with the s i z e of c o n t r a c t u a l payments. This explanation of supply d i f f e r s from the model of Jaffee (1971) where a p o s i t i v e l y - s l o p e d s u p p l y curve i s the r e s u l t of an i n c r e a s e i n the i n d i v i d u a l ' s p r o b a b i l i t y of d e f a u l t at higher l o a n s i z e s . 20 s e l e c t i o n e f f e c t s . i J It i s concluded that when such a l i n k e x i s t s between price and q u a l i t y the eq u i l i b r i u m price does not n e c e s s a r i l y coincide with the i n t e r s e c t i o n of supply and demand schedules. S t i g l i t z and Weiss (1981) i n v e s t i g a t e whether s i m i l a r b e h a v i o r can explain the presence of excess demand for loans at the eq u i l i b r i u m i n t e r e s t rate. In the S t i g l i t z - W e i s s model def a u l t r i s k e x i s t s because returns from investment p r o j e c t s of borrowers are u n c e r t a i n . By assumption l e n d e r s cannot screen customer r i s k d i r e c t l y since they cannot determine the degree of r i s k associated with a p a r t i c u l a r project. However, the i n t e r e s t rate may act as an i n d i r e c t screening device i f the r i s k i n e s s of loans granted i s known to i n c r e a s e with the l o a n r a t e . S t i g l i t z and Weiss demonstrate that t h i s r e l a t i o n s h i p can a r i s e from two sources. As the i n t e r e s t r a t e increases a loan i s no longer p r o f i t a b l e f o r lower r i s k firms and they w i l l not apply f o r l o a n s . ^ Consequently, the higher i n t e r e s t rate s h i f t s the o v e r a l l mix of loan applicants toward higher r i s k firms. In addition to t h i s adverse s e l e c t i o n e f f e c t , each borrower's s e l e c t i o n of investment projects may be affected unfavourably from the lender's perspective. If a firm i s i n i t i a l l y i n d i f f e r e n t between two p o t e n t i a l projects an increase i n the loan rate would induce the firm to s e l e c t the higher r i s k project. Thus, by changing the mix of applicants and t h e i r choice of projects, an increase i n i n t e r e s t rate can raise the general l e v e l of def a u l t r i s k i n the lender's l o a n p o r t f o l i o . As a r e s u l t the lender's expected r a t e of r e t u r n may reach a maximum at some i n t e r e s t r a t e and d e c l i n e at hi g h e r i n t e r e s t r a t e s . T h i s non-monotonic r e l a t i o n s h i p can cause e q u i l i b r i u m c r e d i t r a t i o n i n g since the eq u i l i b r i u m i n t e r e s t rate, set by lenders at the l e v e l that maximizes t h e i r expected return, may be below the i n t e r s e c t i o n 13. An example of t h i s type of study i s Wilson (1980). 14. S t i g l i t z and Weiss measure r i s k by the mean-preserving spread c r i t e r i o n . 2 1 of the aggregate supply and demand schedules. Even with excess demand at t h i s i n t e r e s t rate there would be no incentive f o r the lender to increase the loan rate since t h i s action would ra i s e the average r i s k on loans and lower expected l e n d e r r e t u r n . Although d i r e c t s c r e e n i n g has not been pos s i b l e , r i s k has been c o n t r o l l e d i n d i r e c t l y because borrower b e h a v i o r responds to the lender's choice of i n t e r e s t rate. The lender's i n a b i l i t y to evaluate i n d i v i d u a l r i s k i s the underlying cause of r a t i o n i n g i n the S t i g l i t z - W e i s s model. T h e r e f o r e , i f there i s aggregate excess demand at the e q u i l i b r i u m i n t e r e s t r a t e the l i s t of customers to be rationed cannot be determined on the basis of i n d i v i d u a l d e f a u l t r i s k as i n the J a f f e e (1971) case. The l e n d e r w i l l a r b i t r a r i l y r a t i o n some f i r m s even though they appear to be i d e n t i c a l to other applicants that do receive t h e i r loan requests. 2.3.4 Fried-Howitt (1980) The e q u i l i b r i u m models of Jaffee, J a f f e e - R u s s e l l , and S t i g l i t z - W e i s s emphasize the r e l a t i o n s h i p between d e f a u l t r i s k and c r e d i t rationing under various assumptions about the p r i c e - s e t t i n g regime and the lender's a b i l i t y to screen i n d i v i d u a l r i s k . F r i e d and Howitt (1980) take a d i f f e r e n t approach by c o n s i d e r i n g a form of u n c e r t a i n t y other than d e f a u l t r i s k . They use i m p l i c i t contract theory arguments to explain c r e d i t rationing as a possible r e s u l t of an equ i l i b r i u m risk-sharing agreement between a lender and i t s customers. In the absence of any agreements a borrower faces the r i s k of v a r i a t i o n s i n the loan rate over time i f the lender's cost of funds i s a random v a r i a b l e . Given t h i s i n t e r t e m p o r a l r i s k , borrowers might accept a h i g h e r average i n t e r e s t r a t e i f the l e n d e r agrees to lower customers' r i s k by l i m i t i n g f l u c t u a t i o n s i n the l o a n r a t e . E q u i l i b r i u m contracts may emerge with c r e d i t rationing e x i s t i n g f o r some i n d i v i d u a l s i f agents cannot change trading partners with zero cost. If switching costs were zero, so that borrowers could move c o s t l e s s l y among banks i n search of a loan at the lowest po s s i b l e "spot" rate, the lender would be u n w i l l i n g to s t a b i l i z e the in t e r e s t rate and non-price rationing would not occur. 2.4 Disequilibrium Credit Rationing E q u i l i b r i u m c r e d i t r a t i o n i n g has been d e f i n e d as any excess demand that e x i s t s at the e q u i l i b r i u m r a t e of i n t e r e s t . The source of t h i s r a t i o n i n g may o r i g i n a t e from i m p e r f e c t i n f o r m a t i o n on d e f a u l t r i s k by lenders (Jaffee-Russel 1, S t i g l i t z - W e i s s ) or constraints on inter-customer i n t e r e s t r a t e d i f f e r e n t i a t i o n ( J a f f e e ) . There i s a second source of r a t i o n i n g i f the l o a n r a t e a d j u s t s with a l a g to a s h i f t i n s u p p l y or demand c o n d i t i o n s . With non-instantaneous adjustment the q u a n t i t y of rationing a c t u a l l y observed at a d i s e q u i l i b r i u m i n t e r e s t rate d i f f e r s from the e q u i l i b r i u m r a t i o n i n g t h a t would have oc c u r r e d w i t h complete adjustment. Any u n s a t i s f i e d demand r e s u l t i n g from non-instantaneous movement to the lo n g - r u n e q u i l i b r i u m l o a n r a t e may be d e f i n e d as d i s e q u i l i b r i u m c r e d i t rationing. As the in t e r e s t rate adjusts over time d i s e q u i l i b r i u m rationing approaches zero and i n the l i m i t a l l excess demand i s a t t r i b u t e d to equ i l i b r i u m rationing. T h e o r e t i c a l a n a l y s i s of the causes of incomplete short-run movements i n i n t e r e s t r a t e s has been l i m i t e d . T u c k e r (1968) examined some implications f o r the speed of monetary p o l i c y e f f e c t s on the r e a l sector but gave l i t t l e c o n s i d e r a t i o n to the u n d e r l y i n g source of imper f e c t f l e x i b i l i t y . Lags i n m a r k e t - c l e a r i n g were a s s o c i a t e d w i t h l o a n market imperfections and slow-adjusting administered p r i c i n g by lenders. Jaffee's 15. Without s w i t c h i n g c o s t s a customer would borrow at an i m p l i c i t agreement's f i x e d l o a n r a t e o n l y when spot market r a t e s , and the lender's cost of funds, were high r e l a t i v e to the fixed rate. Lender p r o f i t would be n e g a t i v e f o r these t r a n s a c t i o n s so an agreement to s t a b i l i z e the loan rate would be unprofitable for the lender. (1971) a n a l y s i s of d i s e q u i l i b r i u m r a t i o n i n g was a l s o based on an argument that i n t e r e s t r a t e responses to d i s t u r b a n c e s tend to be lagged i n o l i g o p o l i s t i c loan markets. A more r i g o r o u s a n a l y s i s of the i n t e r e s t r a t e adjustment process i s attempted by K o s k e l a (1976, Chapter 6).*^ He examines op t i m a l l e n d e r behavior when aggregate loan demand i s stochastic and lump-sum costs are incurred by the lender each time the i n t e r e s t rate i s changed. With these assumptions the i n t e r e s t rate i s not a l t e r e d a f t e r a s h i f t i n demand unless the benefits from adjustment exceed the lump-sum adjustment cost. Demand can f l u c t u a t e w i t h i n some range (determined by the adjustment c o s t and other parameters of the model) without inducing a price change. Further d e t a i l on d i s e q u i l i b r i u m rationing i s presented i n Chapter 3 where eq u i l i b r i u m and d i s e q u i l i b r i u m rationing are combined within a s i n g l e model. 16. K o s k e l a d i r e c t l y examines the o p t i m a l change i n n o n - p r i c e l o a n c o n t r a c t s but the same model c o u l d be used to e x p l a i n rate v a r i a t i o n s . terms of i n t e r e s t CHAPTER 3 Imperfect Interest Rate D i f f e r e n t i a t i o n Models of equilibrium c r e d i t rationing were discussed i n Chapter 2 f o r the separate cases of perfect and imperfect screening of r i s k by lenders. J a f f e e - R u s s e l l and S t i g l i t z - W e i s s considered s i t u a t i o n s i n which the lender has imperfect information concerning the d e f a u l t r i s k a s s o c i a t e d with an i n d i v i d u a l loan. This information deficiency can cause non-price rationing since i t removes the lender's a b i l i t y to discriminate among customers i n i t s supply and i n t e r e s t rate behavior. In c o n t r a s t , J a f f e e assumed tha t d e f a u l t r i s k of each p o t e n t i a l borrower can be evaluated by the bank (perfect screening). His model was then based on the premise t h a t a p r i c e - s e t t i n g l e n d e r w i l l engage i n l i m i t e d d i f f e r e n t i a t i o n of i n t e r e s t r a t e s among i t s customers. This i m p e r f e c t d i f f e r e n t i a t i o n was r a t i o n a l i z e d by r e f e r e n c e s to moral or o l i g o p o l i s t i c p r i c i n g c o n s i d e r a t i o n s . Ba 1 1 e n s p e r g e r ( 1 978), a f t e r e x p r e s s i n g d i s s a t i s f a c t i o n with moral or i n s t i t u t i o n a l r a t i o n a l i z a t i o n s , c a l l e d f or a l t e r n a t i v e explanations of constrained d i f f e r e n t i a t i o n i n the perfect screening case. The present chapter i s devoted to this issue. In p a r t i c u l a r , i t i s shown that even i f moral or c o l l u s i v e factors were non-e x i s t e n t , a l e n d e r may decide to charge a l l customers ( w i t h i n a g i v e n borrower category) the same l o a n r a t e i f there i s i m p e r f e c t a p r i o r i i n f o r m a t i o n on borrowers' l o a n demand c u r v e s . A common i n t e r e s t r a t e system may benefit the lender by avoiding bargaining problems which would develop with p e r f e c t l y discriminating monopoly p r i c i n g . This explanation does not deny the p o t e n t i a l importance of the sources of constrained i n t e r -customer d i f f e r e n t i a t i o n suggested p r e v i o u s l y . However, i t does provide an a d d i t i o n a l j u s t i f i c a t i o n that does not r e l y on moral or c o l l u s i v e reasons. This a l t e r n a t i v e perspective on c r e d i t rationing i s discussed i n section 3.4 a f t e r a formal presentation of the Jaffee (1971) model. 3.1 Loan Offer Curve The occurrence of c r e d i t rationing depends on the l e v e l of loan demand r e l a t i v e to desired lender supply at the p r e v a i l i n g rate of i n t e r e s t . This s e c t i o n a n a l y s e s the lender's o p t i m a l s u p p l y b e h a v i o r by c o n s i d e r i n g a r i s k - n e u t r a l bank that faces n p o t e n t i a l borrowers. Each of these customers seeks to finance an investment project using some combination of loans and equity. The end-of-period gross return from firm i's project i s a random v a r i a b l e defined by (2): (1) A i = L± + E t i = l , . . . . , n (2) xj_ = p[A±]y±k± i = l , . . . . , n with A = t o t a l investment i n the project, L = loan s i z e , E = equity, X = random gross return, y = random rate of return v a r i a b l e , and P[A] = non-random scale f a c t o r . Gross return x i s the product of t o t a l investment A and a random rate of return on Investment denoted by Py. The random v a r i a b l e y i s m u l t i p l i e d by a non-random f u n c t i o n p[A] to a l l o w the r a t e of r e t u r n to v a r y s y s t e m a t i c a l l y with the siz e of investment. The function P[A] i s assumed to have the f o l l o w i n g properties. 1. The f o l l o w i n g summary of the model i s based on J a f f e e (1971) and Koskela (1976, 1979a, 1979b). 26 (3)(a) P'[A] = (b) q[A] = %[A]A = P[A] + p'[A]A > 0 (c) q'[A] = ^ L M = 2p»[A] + P"[A]A < 0 Assumption (3)(a) a s s e r t s t h a t there i s e i t h e r constant expected r e t u r n s to s c a l e ( P1 = 0) or d e c r e a s i n g r e t u r n s (P'< 0). The remaining conditions ensure that the marginal return on investment <5pyA/ 6 A = yq i s non-negative ( i m p l i e d by (3)(b) ) but n o n - i n c r e a s i n g i n A ( i m p l i e d by ( 3 ) ( c ) ). Due to the uncertainty of each borrower's t o t a l return the bank must consider the degree of d e f a u l t r i s k when c a l c u l a t i n g expected p r o f i t from a loan. It forms a subjective e v a l u a t i o n of the p r o b a b i l i t i e s of d i f f e r e n t outcomes f o r the project's random rate of return y. These expectations are d e s c r i b e d by a d e n s i t y f u n c t i o n g^(y). I t i s assumed that there e x i s t rates of return v^ and V^, 0 < v^ < < °° , such that (4) g ± ( y ) = 0 for y £ v± or y >^  V±, (5) G ±[y] =0 for y < v± G ±[y] =1 for y > V 1 and m with G^[m] gi<y)dy = cumulative density function evaluated at m. 2. The lender's a b i l i t y to perceive a density function g^(y) s p e c i f i c to each i n d i v i d u a l i = l,..,n i n d i c a t e s the p e r f e c t r i s k s c r e e n i n g assumption. 27 The v a l u e s v^ and may be i n t e r p r e t e d as the minimum and maximum rates of return (neglecting the s c a l e adjustment p [A]) from the borrower's proj ect. The lender i s interested i n i s o l a t i n g the range of outcomes over which loan payments are defaulted. Borrower i defa u l t s i f gross returns are l e s s than loan p r i n c i p a l plus i n t e r e s t payments, that i s d e f a u l t occurs i f J (6 ) x = pyA < with = (1 + r^) = i n t e r e s t r a t e f a c t o r f o r borrower i , and r^ = i n t e r e s t rate on borrower i's loan. From ( 6 ) a c r i t i c a l r a t e of r e t u r n (g) from the p r o j e c t can be c a l c u l a t e d which i s j u s t s u f f i c i e n t to a v o i d any d e f a u l t on p r i n c i p a l and i n t e r e s t payments. (7) 3 , R i L i PA p(L i+E i) At a l l rates of return y above the c r i t i c a l rate 3 there i s no default. If y i s l e s s than g there i s e i t h e r p a r t i a l or complete d e f a u l t by the borrower. The p r o b a b i l i t y of r e a l i z i n g some de f a u l t outcome i s G[&] = g(y)dy 3 . D e f a u l t c o n d i t i o n ( 6 ) assumes there are no c o l l a t e r a l terms or a d d i t i o n a l sources of revenue for loan payments. If c o l l a t e r a l C i s p r o v i d e d on the l o a n d e f a u l t i s no l o n g e r synonymous wit h the project's return not covering l o a n payments. The d e f a u l t c o n d i t i o n becomes p yA+C<RL and the no-default rate of return i s ^  c=(RiLi~C)/P A. The p r o b a b i l i t y of d e f a u l t i s reduced at g i v e n l o a n s i z e s but p r o p e r t i e s of o f f e r c u r v e s d e s c r i b e d below are maintained when m o d i f i c a t i o n s to ( 6 ) are c o n s i d e r e d . For convenience ( 6 ) i s used i n subsequent a n a l y s i s and borrower subscripts are included only f o r the in t e r e s t rate and loan s i z e . 28 Expected l e n d e r p r o f i t P ^ on a l o a n to f i r m i may be re p r e s e n t e d by ( 8 ) : V ( 8 ) P f = g(y)dy + P A yg(y)dy - IL± v with I = (1 + j ) , and j = lender's opportunity cost of the loan. The i n i t i a l two terms i n ( 8 ) w i t h i n t e g r a l s g i v e the bank's expected revenue. The lender a c t u a l l y receives a l l contractual payments R^L^ only i f no d e f a u l t occurs, i.e. when y > 3. Th e r e f o r e , the n o - d e f a u l t revenue R ^ L ^ i s weighted by the p r o b a b i l i t y of no d e f a u l t i n the f i r s t term. The second component of ( 8 ) i n v o l v e s lender revenue i n the event of de f a u l t . It i s assumed that the bank claims a l l gross returns PyA of the firm when these r e t u r n s are i n s u f f i c i e n t to meet l o a n payments, i.e. when y <3 . Each de f a u l t outcome i s weighted by the appropriate p r o b a b i l i t y to form the middle term i n ( 8 ) . The f i n a l term accounts for the lender's opportunity cost of supplying the loan. A co n v e n i e n t form of the expected p r o f i t f u n c t i o n i s obt a i n e d by 6 V adding and subtracting R-JLJ g(y)dy from ( 8 ) and noting that g(y)dy = 1 v v from (5). ( 9 ) P| = R ^ - qCRi.Li) with 2 9 (10) C i(R ±J. i) = ( R J L J - pyA)g(y)dy + IL± D + I L . This formulation i s o l a t e s a cost function ^(R^, L ^ ) which represents the lender's cost of granting a loan to borrower i . Cost at a given loan s i z e depends on the o p p o r t u n i t y c o s t I and expected d e f a u l t l o s s e s . D e f a u l t losses equal to contractual payments R ^ L ^ minus r e a l i z e d gross- returns pyA are i n c u r r e d by the l e n d e r whenever y < 3. T h e r e f o r e expected d e f a u l t losses are D = ( R ^ - pyA)g(y)dy and t o t a l cost i s C i(R i,L i) = D + IL±. Equations (9) and (10) may be used to determine the lender's o p t i m a l l o a n s u p p l y to f i r m i at a g i v e n i n t e r e s t r a t e . Expected p r o f i t i s maximized at a g i v e n R by d i f f e r e n t i a t i n g (9) w i t h r e s p e c t to L ^ and se t t i n g the r e s u l t equal to zero. ( I D 3Pf = R - 9C± = 0 3L, 9 L i with 9Ci 3L. (%. - qy)g(y)dy + i D' = 8 0 D + I , = marginal default l o s s , and 3L, q as defined by (3)(b) 30 The l o c u s of o p t i m a l l o a n s i z e s at d i f f e r e n t i n t e r e s t r a t e s , i m p l i c i t l y given by (11), i s defined as the bank's optimal loan o f f e r curve L i ( R i ) to borrower i . It i n d i c a t e s the lender's d e s i r e d l o a n at each in t e r e s t rate independent of any demand constraint. The c h a r a c t e r i s t i c s of an o f f e r curve are described below where 6g / 6R^ > 0, 6g/6L.^ > 0, and ( qg - R i) < 0. 4 (12)(a) L? = 0 for R± < I (b) 0 < Lf < ^ BA for R. = I and I-vp > 0 — 1 — I-vp x -(i-G[e]+(qe-R ) 6£_ g ( B ) ) (c) 9L? 1 6R ± (qe-R,)j5£_ g(g)+ q' 6L. yg(y)dy > 0 i f 1-G[B] + (qg-R,)!^ g(6) > 0 6R. l (d) 6L? • L 61 (qB-R ) 5Bg(B)+q' 6L. i v s yg(y)dy (e) , lim L![ = N where N i s f i n i t e R.-*» l f or R.>I 3R. . R i l <0 4. These p r o p e r t i e s are d i s c u s s e d i n Appendix 1. A n a l y s i s c o u l d be extended to consider the impact on o f f e r curves of r i s k aversion and p o r t f o l i o d i v e r s i f i c a t i o n p r i n c i p l e s . With r i s k aversion o f f e r curves are obtained by the l e n d e r maximizing u t i l i t y U = U(P s,a^) where P s and cr are the mean and v a r i a n c e of l e n d e r p r o f i t , U^=3U/9P S > 0, u*2 = 3u/3a ^ < o, and the lender's b a l a n c e sheet c o n s t r a i n t i s s u b s t i t u t e d i n t o the d e f i n i t i o n s of P s and a^. The o f f e r c u r v e f o r borrower i i s d e f i n e d by U, 13P s/3L i + U 2 3a 2/3L i = 0. Since 3P S/3L ± ( d e f i n e d by (11)) and 3a^3L^ depend on i n d i v i d u a l i ' s r i s k c h a r a c t e r i s t i c s borrower-specific o f f e r c u r v e s a l s o e x i s t with r i s k a v e r s i o n . F o l l o w i n g J a f f e e and K o s k e l a , the f o l l o w i n g d i s c u s s i o n considers the r i s k - n e u t r a l case. 31 The loan o f f e r curve to i n d i v i d u a l i depicted i n Figure 1 i s based on the p r o p e r t i e s summarized by (12). The h o r i z o n t a l segment at = I corresponds to a l l loan sizes for which there i s ho p o s s i b i l i t y of d e f a u l t losses for the bank. Along t h i s segment of L? the c r i t i c a l no-default rate of r e t u r n 3 d e f i n e d by (7) i s l e s s than the investment p r o j e c t ' s minimum r a t e of r e t u r n v. Consequently, from (11) the o f f e r c u r v e reduces to R^  = I over t h i s range. The p o s i t i v e l y - s l o p e d portion above I represents l o a n s i z e s with p o s i t i v e d e f a u l t risk.-* An i n c r e a s e i n the lender's o p p o r t u n i t y cost would s h i f t the o f f e r c u r v e to the l e f t a c c o r d i n g to (12)(d). F i n a l l y , o f f e r curves to d i f f e r e n t i n d i v i d u a l s w i l l vary due to d i f f e r e n c e s i n i n d i v i d u a l d e f a u l t r i s k . An i n c r e a s e i n d e f a u l t r i s k at g i v e n l o a n s i z e s r a i s e s 9 C . / 9 L . = D' + I and s h i f t s the o f f e r c u r v e i x upward. T h e r e f o r e , the o f f e r c u r v e to a h i g h - r i s k borrower i s above the o f f e r curve to a lower-risk customer. S e c t i o n s 3.2 and 3.3 use o f f e r c u r v e s to examine the p o s s i b i l i t y of c r e d i t r a t i o n i n g under various p r i c i n g systems. 5. From (12)(c) the o f f e r curve may be backward-bending over some range i f the i n d i v i d u a l ' s p r o b a b i l i t y of d e f a u l t G[ 3 ] and the term g (3 ) become s u f f i c i e n t l y large as loan s i z e increases. 32 FIGURE 1: Loan Offer Curve to Borrower i 33 3.2 Discriminating Monopoly Pricing A lender acts as a discriminating monopolist i f i t can set a d i f f e r e n t i n t e r e s t rate fpr each loan. Interest rates w i l l vary among customers to e x p l o i t i n d i v i d u a l demand c o n d i t i o n s and compensate f o r d i f f e r e n c e s i n d e f a u l t r i s k . Subject to a c o n s t r a i n t t h a t an i n d i v i d u a l ' s l o a n cannot exceed d e s i r e d demand at the i n t e r e s t r a t e charged, the d i s c r i m i n a t i n g monopolist chooses the i n t e r e s t rate and loan s i z e which maximize expected p r o f i t P?. The optimization problem i s (13) max Pf = - C i(R i,L i) subject to Lj_ - L ^ R ^ < 0 i = l , . . . , n with C^(R^,L^) = l e n d e r s cost f u n c t i o n f o r g r a n t i n g l o a n s to borrower i ( d e f i n e d by (10)), and L^(R^) = borrower i ' s loan demand function with SL^/5 < 0 The Lagrangian functions are Z i = R i L i " ^ ( R i . V " X i ( L i " L i ( R i ) ) i = l , . . . , n where i s a Lagrangian m u l t i p l i e r . Kuhn-Tucker conditions for a maximum are: SZ- SC. SL. (14)(a) _± = L i + A _ i = 0 i = 1 n SR. i 6R. l SR. U J-,.-.,n i l l , . sz. sc. l l (c) A i >_ 0 i = l , . . . , n 34 (d) L. - L d(R.) < 0 1 1 iJ — •,i = 1,.. . ,n (e) x.(L. - L^(R.)) =0 i = l , . . . , n If l o a n s i z e i s p o s i t i v e then i t f o l l o w s from (14)(a) th a t Xi>0 s i n c e L± - 6C i/6R 1 = 1^(1 - G[B]) > 0 and 6 L | / 6 R 1 < 0. From-condition (14)(e) i f > 0 then L^=L4(R^). Thus, a d i s c r i m i n a t i n g m o n o p o l i s t would not ration any borrowers since the optimal loan s i z e to any borrower i i s always determined along the demand curve L^(R^). This r e s u l t demonstrates Jaffee's conclusion that e q u i l i b r i u m rationing i s never p r o f i t a b l e f o r a lender that i s able to charge d i f f e r e n t i n t e r e s t rates to a l l customers. 3.3 Rationing with Constrained Differentiation As d e s c r i b e d i n Chapter 2 J a f f e e shows that e q u i l i b r i u m c r e d i t r a t i o n i n g i s p r o f i t a b l e f o r the l e n d e r i f i t i s c o n s t r a i n e d to charge heterogeneous customers the same r a t e of i n t e r e s t . He c o n s i d e r s a bank that maximizes expected p r o f i t by s e l e c t i n g o p t i m a l l o a n s i z e s f o r n customers and a common i n t e r e s t r a t e R f o r each of these n l o a n s . Realized loan s i z e can be no greater than the borrower's desired demand at a given i n t e r e s t rate. The bank's decision problem i s ( 1 5 ) ( R L 1 1 1^ L / = I P i = ^ f R L i " C i ( R ' V 5 i R ' L i ' " - ' n 1 i = l 1 i = l 1 1 1 subject to L ± - L^(R i) < 0 i = 1,... ,n 6. The r a t i o n a l i t y of th i s behavior was discussed i n Chapter 2 and i s re-examined i n 3.5. The f o l l o w i n g a n a l y s i s d i r e c t l y concerns a s i n g l e group of n po t e n t i a l borrowers. However, the same conclusions hold i f lenders e s t a b l i s h a la r g e r number of customer categories and charge each member within a given category the same i n t e r e s t rate (see Jaffee (1971), pp. 45-47). 35 Total expected p r o f i t of the lender P s i s the summation of expected p r o f i t s from i n d i v i d u a l l o a n s as d e f i n e d by (9). Borrower h e t e r o g e n e i t y i s introduced through inter-customer d i f f e r e n c e s i n l o a n demand and d e f a u l t r i s k . The Lagrangian function for (15) i s n n , Z = I (RL ± - C ±(R,L i) ) - I X i ( L ± - L-J(R) ) i=l i=l and the Kuhn-Tucker conditions for a maximum are: * 7 n 5C. 6L d ( 1 6 ) ( a ) | f = J ( L i - ^ + A ± ^ ) = 0 i=l fi <sc. (b) = R - r r - i - X. < 0 i = l , . . . , n oL. 6 L, l — l l (c) \ ± > 0 i = 1,... ,n •(d) L ± _ L ^ R ) < 0 i = l , . . . , n (e) X±CL± ~ L J ( R ) ) =0 i = l , . . . , n This set of Kuhn-Tucker conditions i s i d e n t i c a l to the preceding conditions for a discriminating monopolist with the exception of (16)(a) which defines the optimal value of the group's common in t e r e s t rate R. The p r o f i t a b i l i t y of equilibrium c r e d i t r a t ioning i s now shown for the common i n t e r e s t r a t e system. It i s p r o f i t a b l e f o r the l e n d e r to r a t i o n customer i at the equilibrium i n t e r e s t rate R only i f the marginal cost of the loan exceeds R when loan s i z e i s evaluated along the demand curve, that i s , ^ 6C,(L?(R) ) ( 1 7 ) R - - V <0 i 36 is a necessary condition for equilibrium rationing. The possibility of rationing is seen from (18) which is obtained by substituting (16)(b) into (16)(a), imposing L ± = L^, and noting that 6 = L^G[B ] . n <5C. 6L d ( 1 8 ) t v i - s r ' h j r ) 1 = 1 n S C . <SL4 = I (L d(l-G[ B ]) + ( R - ^ ) 5 ^ ) =0 1=1 i The expression after the summation sign in (18) is positive for some potential borrowers and negative for others since the total summation across a l l n customers is zero. Some individuals with positive values w i l l have combinations of demand and risk parameters such that R - 6 0^ /6 < 0 at the loan sizes given by their demand curves at R. Thus, necessary condition (17) for equilibrium rationing is satisfied for some borrowers when banks set an identical interest rate for a group of customers. The solution to (16)(a) - (16)(e) implies that quantity determination follows a switching rule. If the demand constraint is binding (X^ > 0) then = from (16)(e). If the demand constraint is not binding, so that A\ = 0 and < L.4 , then from (16)(b) R = 6 C±/<$ determines loan quantity. In this event the loan size is along the lender's offer curve f^I>i = L|) since R = 6C^/6L^ defines L? from (11). These two results form an individual short-side rule (19) which states that customer i's loan size is the minimum of borrower-specific demand and offer curves at the quoted interest rate. (19) L t = min(Lj,L|) A l l individuals with supply-determined loans at the current interest rate 37 are r a t i o n e d w i t h = L | < L ^ . E q u i l i b r i u m r a t i o n i n g occurs f o r those customers with L | = min ( L ^ , L ? ) at the equilibrium i n t e r e s t rate R implied by (16)(a). Insight into the incidence of equilibrium r a t i o n i n g i s obtained from the necessary condition (17) and the expression for marginal cost: 6C. 1 6 L . 1 (R - qy) g(y)dy + I = D' + I from (11) Any borrower i w i t h m a r g i n a l c o s t 6 ( ^ / 6 1 ^ above R at l o a n s i z e = L^(R) i s rationed i n equilibrium. Since <S C^/6 L - ^ i s p o s i t i v e l y r e l a t e d to the d e f a u l t r i s k measure D', i t i s e v i d e n t that h i g h e r - r i s k i n d i v i d u a l s have a greater l i k e l i h o o d of rationing at the common i n t e r e s t rate R. The r o l e of def a u l t r i s k as a determinant of rationing status i s i l l u s t r a t e d by F i g u r e 2. F i g u r e (2)(a) c o n t a i n s l o a n o f f e r c u r v e s f o r two i n d i v i d u a l s w i t h i d e n t i c a l l o a n demand but d i f f e r e n t d e f a u l t r i s k . By assumption, customer 1 has hi g h e r r i s k than customer 2 ( S C ^ / S L ^ > 6 C 2 / 6 L 2 a t a l l l o a n sizes) so o f f e r curve i s above L | . At R short-side r u l e (19) indicates that the high-ri s k customer 1 i s rationed with loan s i z e but the lower-r i s k borrower 2 i s unrationed at the demand-determined quantity I^. Figure (2)(b) presents the polar case of a customer with zero d e f a u l t r i s k at a l l loan sizes. The lender's marginal cost i s constant f o r t h i s case so the o f f e r c u r v e to customer 3 i s h o r i z o n t a l at R = I. With R > I t h i s r i s k -f r e e borrower i s not r a t i o n e d s i n c e the r e a l i z e d l o a n s i z e i s a l o n g the demand curve. In summary, at equ i l i b r i u m i n t e r e s t rate R L1 = L^(R) = min ( L^,L^) L 2 = I^fR) = min(L2,L2> 38 L 3 = L^(R) = min(L^,L§> and t o t a l e q uilibrium c r e d i t rationing (ER) i s (20) ER = L d(R) - L i • An important comparative s t a t i c s p r e d i c t i o n of Jaffee's model i s that there i s no systematic r e l a t i o n s h i p between the general l e v e l of i n t e r e s t rates and the t o t a l magnitude of equilibrium rationing. An increase i n the lender's opportunity cost I has two e f f e c t s with c o n f l i c t i n g repercussions f o r the amount of r a t i o n i n g . I n d i v i d u a l l o a n o f f e r c u r v e s s h i f t to the l e f t (from (12)(d)) and t o t a l u n s a t i s f i e d demand increases at the i n i t i a l e q u i l i b r i u m i n t e r e s t r a t e . However, the e q u i l i b r i u m i n t e r e s t rate w i l l r i s e a f t e r an increase i n opportunity cost, and t h i s adjustment i n the loan r a t e tends to o f f s e t the i n i t i a l p o s i t i v e impact on r a t i o n i n g from the s h i f t s i n o f f e r c u r v e s . As a r e s u l t the net change i n e q u i l i b r i u m rationing i s indeterminate. An increase i n aggregate loan demand a l s o has ambiguous e f f e c t s on e q u i l i b r i u m r a t i o n i n g s i n c e the h i g h e r demand i s accompanied by an increase i n R. The preceding comparative s t a t i c s r e s u l t s suggest that intertemporal v a r i a t i o n s i n r a t i o n i n g are not r e l a t e d s y s t e m a t i c a l l y to changes i n equilibrium rationing. Instead, the primary source of these v a r i a t i o n s i s d i s e q u i l i b r i u m r a t i o n i n g (DR) which i s the d i f f e r e n c e between t o t a l r a t i o n i n g (TR) at the c u r r e n t i n t e r e s t r a t e and t h a t which would have occurred at the equilibrium rate R, i.e. DR = TR - ER.^ This conclusion i s shown u s i n g the three borrowers c o n s i d e r e d i n F i g u r e 2. Suppose the current period's i n t e r e s t rate R t i s below the equilibrium rate R. Whereas only customer 1 would be rationed at R, both r i s k y borrowers i n Figure 2(a) 7. S e c t i o n 2.4 d i s c u s s e d reasons why the i n t e r e s t r a t e may not a d j u s t immediately to i t s equilibrium l e v e l . 40 now have excess demand at R t and t o t a l rationing during the current period i s (21) TR = (L^(R t) - L f ( R t ) ) + (L^R,.) - L f ( R t ) ) . Examination of F i g u r e 2 toge t h e r with (20) and (21) i n d i c a t e s t h a t t o t a l rationing exceeds equilibrium rationing when the i n t e r e s t rate i s below i t s e q u i l i b r i u m value. This r e s u l t s i g n i f i e s that d i s e q u i l i b r i u m rationing i s p o s i t i v e when R t < R since DR = TR - ER. However, as R r i s e s toward R over time d i s e q u i l i b r i u m rationing approaches zero and t o t a l r a t i o n i n g f a l l s toward the l e v e l of equ i l i b r i u m rationing given by (20). It i s a l s o apparent from F i g u r e 2 that when the c u r r e n t r a t e of i n t e r e s t i s above R a c t u a l r a t i o n i n g at R t i s l e s s than the q u a n t i t y of e q u i l i b r i u m r a t i o n i n g that would have o c c u r r e d at R. T h e r e f o r e , d i s e q u i l i b r i u m r a t i o n i n g i s n e g a t i v e with R t > R t, s i n c e at the higher c u r r e n t r a t e fewer borrowers would be r a t i o n e d and those borrowers that continue to be rationed would have smaller excess demands. The preceding conclusions are summarized below. Proposition 1: There i s no systematic r e l a t i o n s h i p between the absolute l e v e l s of i n t e r e s t rates or aggregate loan demand and the magnitude of eq u i l i b r i u m c r e d i t rationing. Proposition 2: (a) I n t e r t e m p o r a l v a r i a t i o n s i n t o t a l r a t i o n i n g are due p r i m a r i l y to v a r i a t i o n s i n d i s e q u i l i b r i u m rationing, (b) Disequilibrium rationing i s p o s i t i v e ( n e g a t i v e ) when the c u r r e n t i n t e r e s t r a t e R t i s l e s s than ( g r e a t e r than) the e q u i l i b r i u m r a t e R, and the a b s o l u t e v a l u e of DR i s p o s i t i v e l y r e l a t e d to the difference (R - R t). 41 F i n a l l y , the incidence of d i s e q u i l i b r i u m rationing depends on def a u l t r i s k . In the example of F i g u r e 2 the r i s k - f r e e borrower 3 i s s t i l l u n r a t i o n e d at Rfc but borrower 2 , who would not be r a t i o n e d at R, now has excess demand at the d i s e q u i l i b r i u m rate Rt. This r e s u l t i s generalized as P r o p o s i t i o n 3. Proposition 3: As the c u r r e n t i n t e r e s t r a t e f a l l s r e l a t i v e to the 3.4 Iso-Profit Curves C h a r a c t e r i s t i c s of i s o - p r o f i t curves f o r borrowers and lenders w i l l now be presented for l a t e r use i n section 3.5. An i s o - p r o f i t curve of the s lender, denoted by n , consists of a l l combinations of borrower i's loan si z e and i n t e r e s t rate y i e l d i n g a constant expected p r o f i t to the lender. The s l o p e of an i s o - p r o f i t c u r v e i s found by s e t t i n g expected p r o f i t (8) s equal to a constant and d i f f e r e n t i a t i n g . Along the i s o - p r o f i t curve II , e q u i l i b r i u m rate, the r e s u l t i n g increase i n rationing i s di s t r i b u t e d among customers on the basis of d e f a u l t r i s k ( c e t e r i s p a r i b u s ) , with low r i s k borrowers the l e a s t l i k e l y to experience u n s a t i s f i e d demand. 6R. l 6Pf/SL. i l (22) 6L. - ( R i ( l - G [ 3 ] ) v L . ( l - G [ 3 ] ) The f o l l o w i n g propositions describe the lender's i s o - p r o f i t map.' 8 8. These propositions are proven i n Appendix 2. Proposition 4: An i s o - p r o f i t curve of the lender i s p o s i t i v e l y - s l o p e d to the r i g h t of the o f f e r curve L| and n e g a t i v e l y - s l o p e d to the l e f t of l | . Proposition 5: Expected lender p r o f i t increases along an o f f e r curve L| as the i n t e r e s t rate increases. Therefore, i s o - p r o f i t curves i n t e r s e c t i n g L | at s u c c e s s i v e l y h i g h e r i n t e r e s t r a t e s represent s u c c e s s i v e l y greater l e v e l s of expected lender p r o f i t . I s o - p r o f i t c urves of the borrower are d e r i v e d i n a s i m i l a r manner. Borrower p r o f i t from the investment opportunity i s uncertain since end-of-period gross return defined by (2) i s unknown. (2) x = p[A]yA The firm's e x p e c t a t i o n s f o r the random r a t e of r e t u r n y from the p r o j e c t are described by a density function h(y).^ It i s assumed that there e x i s t r a t e s of r e t u r n k and K, 0 < k < K < oo, such that (23) h(y) = 0 f o r y < k (24) H[y] = 0 f o r y < k H[y] = 1 for m y > K or y > K, and with H[m] = h(y)dy = cumulative density function evaluated at m. 9. The borrower's expectations represented by h(y) may d i f f e r from those of the l e n d e r g(y). These e x p e c t a t i o n s e n t e r i n t o the firm's expected p r o f i t f u n c t i o n (25). If d e f a u l t on the loan does not occur, with y greater than or equal to the n o - d e f a u l t r a t e of r e t u r n B, borrower p r o f i t e q u a l s gross r e t u r n s pyA minus the contractual loan payments R^L^. Each no-default r e s u l t i s weighted by the a p p r o p r i a t e p r o b a b i l i t y g i v e n by h(y). If the borrower does de f a u l t with y<8 i t i s assumed that the lender receives a l l returns from the investment project. Therefore, borrower i's expected p r o f i t P^ i s K (25) Pf = (pyA - R iL i)h(y)dy The borrower maximizes expected p r o f i t at a g i v e n i n t e r e s t r a t e by choosing an optimal loan s i z e . From (25) the f i r s t - o r d e r condition i s K 6P d (26) _ J , 6L. l (qy - R i)h(y)dy = 0 w i t h q d e f i n e d by (3)(b). The s o l u t i o n to (26) at a l t e r n a t i v e i n t e r e s t rates provides borrower i's loan demand curve L^(R^). Along L d (27) 8Ld i_ 6RT I 1 - H[B] + (qB - R ) 3B_ h(B) 6R. l K q' yh(y)dy - (qB -RO 6B h(B) 6L. l An i s o - p r o f i t curve of the borrower n i s a l o c u s of i n t e r e s t r a t e s and loan sizes that give a constant l e v e l of expected p r o f i t to borrower i . The s l o p e of t h i s c u r v e i s found by e q u a t i n g (25) to a constant and d i f f e r e n t i a t i n g the r e s u l t i n g expression. 44 1 1 (28) n d q yh(y)dy - R.(1 - H[B] ) 3 4(1 - H[B] ) Propositions 6 and 7 describe the borrower's i s o - p r o f i t map. Proposition 6: An i s o - p r o f i t curve of the borrower i s p o s i t i v e l y - s l o p e d to the l e f t of the l o a n demand curve L d and n e g a t i v e l y -sloped to the ri g h t of L d. Proposition 7: Expected p r o f i t of borrower i i n c r e a s e s as the i n t e r e s t 3.5 Imperfect D i f f e r e n t i a t i o n and Demand un c e r t a i n t y S e c t i o n s 3.1 - 3.3 d e v e l o p e d p r e v i o u s arguments that i n d i c a t e non-price r a t i o n i n g i s p r o f i t a b l e f o r a lender i f there are moral or c o l l u s i v e c o n s t r a i n t s on the lender's a b i l i t y to p r i c e d i s c r i m i n a t e . An i m p l i c i t assumption of th i s l i t e r a t u r e i s that the lender has perfect information about each borrower's l o a n demand. The c u r r e n t s e c t i o n suggests that l i m i t e d i n t e r e s t rate d i f f e r e n t i a t i o n may be an endogenous outcome of the model once t h i s information assumption i s relaxed. I s o - p r o f i t curves of the lender and borrower i are combined i n Figure 3. Slope c h a r a c t e r i s t i c s are i n f e r r e d from P r o p o s i t i o n s 4 and 6 w h i l e 10. These propositions are proven i n Appendix 2. r a t e f a l l s a l o n g L^. T h e r e f o r e , borrower i s o - p r o f i t c u rves i n t e r s e c t i n g i/j at s u c c e s s i v e l y lower i n t e r e s t rates represent s u c c e s s i v e l y g r e a t e r l e v e l s of expected borrower p r o f i t . 45 Propositions 5 and 7 indicate that II R A > II R and nR* < nR. Point A i s the conventional e q u i l i b r i u m for discriminating monopoly p r i c i n g discussed i n s e c t i o n 3.2. Subject to the demand c o n s t r a i n t £ L d(R^) the l e n d e r reaches the highest a t t a i n a b l e i s o - p r o f i t curve at the loan contract given S H by the tangency of n R A and L^ . At the d i s c r i m i n a t i n g i n t e r e s t r a t e R* there i s no rationing since the lender's profit-maximizing loan si z e L^(R*) i s demand-determined. If instead the lender charges a group of borrowers a common i n t e r e s t r a t e R, the borrower of F i g u r e 3 i s r a t i o n e d at p o i n t B since L| = min (LD,L?) at R. F i g u r e 3 i l l u s t r a t e s t h a t a common i n t e r e s t r a t e p o l i c y can induce c r e d i t r a t i o n i n g but the r a t i o n a l i t y of t h i s p r i c i n g s t r a t e g y c o u l d be c h a l l e n g e d . Comparison of p o i n t s A and B shows that movement from i n d i v i d u a l discrimination to uniform p r i c i n g reduces the lender's expected s s p r o f i t from II ^ to IIR . This conclusion i s repeated for every borrower i = 1, ... , n s i n c e the l e n d e r reaches the h i g h e s t a t t a i n a b l e i s o - p r o f i t c u rves o n l y at the i n t e r e s t r a t e s R£ ( i = l , ... ,n). T h e r e f o r e , t o t a l expected p r o f i t P s has decreased r e l a t i v e to the i n d i v i d u a l discrimination case with n n P S(R*) = I P?(Rj) > P S(R) = I P?(R) i=l i=l Based on points A and B the common in t e r e s t rate p o l i c y appears to be i r r a t i o n a l i n the absence of o l i g o p o l i s t i c p r i c i n g arguments. However, i t i s not c e r t a i n that the actual outcome with discriminatory p r i c i n g would be A i n Figure 3. The Jaffee model i m p l i c i t l y assumes (a) the lender has information on each borrower's demand curve, and (b) the demand curve p e r c e i v e d by the l e n d e r i s the schedul e LD implied by (26), which gives the borrower's profit-maximizing loan sizes at given i n t e r e s t rates. F I G U R E 3: Potential Equilibrium Positions with Individual Demand Uncertainty 47 With r e s p e c t to (a), the i n f o r m a t i o n on l o a n demand c o u l d be obtained i n s e v e r a l ways. The l e n d e r might ask each borrower f o r d e s i r e d l o a n quantities at various i n t e r e s t rates. A l t e r n a t i v e l y , loan demand could be i n f e r r e d from observed customer behavior i n current and previous periods. In e i t h e r case i t s h o u l d be noted that the lender's p e r c e p t i o n of demand r e l i e s on i n f o r m a t i o n gathered from the borrower. An important i s s u e i s whether borrower b e h a v i o r , and t h e r e f o r e the p e r c e i v e d demand curve, i s l i k e l y to be independent of the p r i c i n g system established by the lender. In f a c t , there i s reason to b e l i e v e that a borrower would p r e f e r not to r e v e a l the conventional demand function L | to the lender under a system of i n d i v i d u a l price discrimination. If t h i s i s true then point A i n Figure 3 may not be the a c t u a l s o l u t i o n f o r discriminatory p r i c i n g . This argument and i t s i m p l i c a t i o n s f o r n o n - p r i c e r a t i o n i n g are seen from Figure 3. Suppose borrower i could s e l e c t any combination of i n t e r e s t rate and loan s i z e subject only to a r e s t r i c t i o n that loan si z e could not exceed the o f f e r curve L | . The highest a t t a i n a b l e i s o - p r o f i t curve would 'd be reached at C where i s o - p r o f i t c u r v e n~ i s tangent to L ? . Having i d e n t i f i e d C as the borrower's optimal loan contract, i t i s i n s t r u c t i v e to determine conditions under which t h i s outcome would be r e a l i z e d . As shown e a r l i e r C i s not the e q u i l i b r i u m when the lender maximizes expected p r o f i t s u b j e c t to L D ( R ^ ) . With i n d i v i d u a l p r i c e d i s c r i m i n a t i o n and m a r g i n a l revenue equals marginal cost p r i c i n g , C i s chosen by the lender only i f i t i s b e l i e v e d that demand i s an i n f i n i t e l y e l a s t i c schedule il4 at i n t e r e s t rate R ^ . Therefore, there i s an i n c e n t i v e under discriminatory p r i c i n g f or the borrower to behave according to the pseudo-demand function L D rather than the conventional r e l a t i o n s h i p since a higher i s o - p r o f i t curve i s d d - d achieved (n ^ > n ^* ) i f L° i s the perceived demand curve. To t h i s p o i n t i t has been assumed tha t the bank p a s s i v e l y accepts 48 whatever demand curve i s revealed by the customer and price discriminates subject to th i s function. However, as Figure 3 i l l u s t r a t e s , the borrower's decision to r e v e a l L d rather than has reduced lender p r o f i t by moving the l e n d e r from n|* at A to a lower i s o - p r o f i t curve H | (not shown i n Figure 3) through C. It i s u n l i k e l y that the bank would not perceive t h i s adverse p o s s i b i l i t y and ant i c i p a t e the borrower's e f f o r t s to conceal actual w i l l i n g n e s s to pay. One p o t e n t i a l counter-response of lenders would be to m a i n t a i n i n d i v i d u a l p r i c e d i s c r i m i n a t i o n but abandon the p a s s i v e o p t i m i z a t i o n s u b j e c t to the i n i t i a l demand expressed by the borrower. A b i l a t e r a l bargaining process could develop i n which the lender attempts to avoid unfavourable outcomes such as C by acquiring information on actual a b i l i t y to pay as d e s c r i b e d by L^. Ac c o r d i n g to t h i s approach, the lender's optimum at A i s unknown to the lender at the st a r t of negotiations and i s attained only i f perfect information on demand i s c o l l e c t e d during n e g o t i a t i o n s . The borrower, i n c o n t r a s t , t r i e s to e x p l o i t the i n i t i a l asymmetry i n i n f o r m a t i o n by b a r g a i n i n g f o r an outcome as c l o s e to i s o -p r o f i t curve n jj, as p o s s i b l e . With such a bargaining process i t appears that the eventual e q u i l i b r i u m would be indeterminate under i n d i v i d u a l price d i s c r i m i n a t i o n and demand uncertainty. The conventional s o l u t i o n at A and the borrower's optimum at C may be regarded as p o l a r cases that maximize expected p r o f i t of the lender and borrower r e s p e c t i v e l y . An a l t e r n a t i v e lender response to s t r a t e g i c borrower behavior would be to replace i n d i v i d u a l price discrimination with the system whereby a group of borrowers i s charged an i d e n t i c a l r a t e R. The consequences of t h i s p r a c t i c e can be examined by mo d i f y i n g a n a l y s i s i n s e c t i o n 3.3 to i n c o r p o r a t e demand u n c e r t a i n t y . For s i m p l i c i t y i t i s assumed tha t the lender has no a p r i o r i knowledge of i n d i v i d u a l demand curves but can form expectations of demand by a representative i n d i v i d u a l from the group. The 49 p r e v i o u s assumption that the l e n d e r can s c r e e n d e f a u l t r i s k of each customer i s maintained. In these circumstances a two-stage l e n d e r optimization might circumvent the problem of demand r e v e l a t i o n observed under i n d i v i d u a l p r i c e d i s c r i m i n a t i o n . In the f i r s t stage the common int e r e s t rate R i s set before borrowers express t h e i r demands. Once R i s announced borrowers r e v e a l l o a n demands and a c t u a l l o a n s i z e s are determined. At the p r e l i m i n a r y stage of o p t i m i z a t i o n the l e n d e r forms i t s expectations of demand by a representative i n d i v i d u a l and determines the optimal i n t e r e s t rate. The maximization problem i s (15)' max P S = I (RLf - C±(R, L f ) ) {R,Lf,...,L*} i=l su bject to L| - L d ' e ( R ) j< 0 i = l , . . . , n with L| = lender's ex ante expectations of borrower i's loan s i z e , and L d , e = lender's e x p e c t a t i o n s of demand by a r e p r e s e n t a t i v e i n d i v i d u a l from a given borrower category. The Lagrangian function i s Z= I (RLf - qCR.Lf)) - I -X (L? - L d ' e ( R ) ) i=l i=l and Kuhn-Tucker conditions for a maximum are s n <SC X T d ' e 1=1 A 7 &Ci (b) — = R - X. < 0 i = l , . . . , n 6L? 6L? X l l (c) A . - 0 i = 1, . . . ,n l (d) L| - L d> e(R) <_0 i = l , . . . , n 50 (e) ^ ( L f - L d ' e ( R ) ) = 0 i = l , . . . , n . Equation (29)(a) d e f i n e s the e q u i l i b r i u m i n t e r e s t r a t e R and (29)(b) d e f i n e s the l o a n s i z e s expected by the l e n d e r before a c t u a l demands are known. In contrast to i n d i v i d u a l price discrimination, i t i s evident from (29)(a) t h a t borrower i ' s i n t e r e s t r a t e i s independent of the demand e v e n t u a l l y expressed by t h i s i n d i v i d u a l . If the i n t e r e s t rate i s given to the i n d i v i d u a l then the profit-maximizing loan demand i s determined by the demand f u n c t i o n L d d e f i n e d by (26). Consequently, i f the l e n d e r e s t a b l i s h e s a common i n t e r e s t r a t e p o l i c y there i s no i n c e n t i v e f o r the borrower to conceal the conventional demand function L d i n an attempt to reach a higher i s o - p r o f i t curve. This conclusion demonstrates that a f t e r R i s announced borrower i w i l l submit a request to the l e n d e r f o r the l o a n si z e given by L d(R). At the second stage of optimization the lender has knowledge of actual demands and s e l e c t s loan, s i z e s t h a t maximize expected p r o f i t from each borrower s u b j e c t to R. = R and <^  L?(R) f o r a l l i = 1, ... , n. The remaining problems to be solved are (30) max Z. = RL. - C, (R,L,) - *,(L, - L d(R)) {L ±} 1 f o r a l l i = 1, ... , n. The s o l u t i o n s to (30) are c o n s i s t e n t with p r e v i o u s c o n c l u s i o n s i n 3.3 f o r a common i n t e r e s t r a t e system. Borrower i's l o a n s i z e i s determined by the short-side r u l e L^ = min (Ld,L|) and any customer with L|(R) = min ( L d ( R ) , L|(R) ) i s r a t i o n e d at the e q u i l i b r i u m i n t e r e s t r a t e R. The p o t e n t i a l advantage to the lender from the common i n t e r e s t rate p o l i c y i s seen from F i g u r e 3. At R the l e n d e r reaches i s o - p r o f i t curve II- which i s l e s s favourable than the l e v e l n R A attained i n standard R 51 a n a l y s i s of i n d i v i d u a l d i s c r i m i n a t i o n . However, with imperfect information on borrowers' demand curves the r e l a t i v e p r o f i t a b i l i t y of the two p r i c i n g regimes should not be assessed by comparing points A and B. The outcome at A assumes that demand schedu l e L d i s known by the l e n d e r . I f t h i s assumption i s removed, a borrower facing price discrimination attempts to convey pseudo-demand conditions and benefits from t h i s behavior as long as the eventual bargaining s o l u t i o n l i e s below i s o - p r o f i t curve Jl^. . From the lender's perspective the p r o f i t a b i l i t y of R = R depends on a comparison of the u n c e r t a i n b a r g a i n i n g s o l u t i o n under d i s c r i m i n a t i o n and the R so l u t i o n at B. A common i n t e r e s t rate p o l i c y improves lender p r o f i t from borrower i i f i t has a v o i d e d a d i s c r i m i n a t o r y l o a n c o n t r a c t w i t h i n area IQSB where n s < IP • However, due to the u n c e r t a i n t y of b a r g a i n i n g R outcomes with i n d i v i d u a l price discrimination, the r e l a t i v e p r o f i t a b i l i t y of the two p r i c i n g regimes i s ambiguous with ^ n - > ~ n -(31) P S(R) = I P|(R) < P S(R*) = I pf(RJ) i=l i=l where P S(R) = t o t a l expected lender p r o f i t with a common i n t e r e s t r a t e R, and P S(R*) = t o t a l expected lender p r o f i t with i n d i v i d u a l price d i s c r i m i n a t i o n and i m p e r f e c t i n f o r m a t i o n on i n d i v i d u a l loan demand schedules. 11. The p r o f i t f u n c t i o n s i n (31) ig n o r e any c o s t s to the l e n d e r of processing loan a p p l i c a t i o n s . It i s expected that these costs would be greatest with i n d i v i d u a l price discrimination since the lender must a l l o c a t e time and res o u r c e s to gain i n f o r m a t i o n about borrowers' underlying demand conditions. With R^  = R there i s no inc e n t i v e f o r borrowers to withhold t h i s information so processing costs should be s m a l l e r . I f p r o c e s s i n g c o s t s are added to (31) the r e l a t i v e p r o f i t a b i l i t y of a common i n t e r e s t rate p o l i c y should improve. 52 The p r e c e d i n g argument suggests the p o s s i b i l i t y that i t may be p r o f i t a b l e f o r a bank to set a common loan rate f o r a group of borrowers i f there i s imperfect information on i n d i v i d u a l demand curves. As shown in 3.3 e q u i l i b r i u m c r e d i t r a t i o n i n g can occur when d i f f e r e n t customers are charged a uniform i n t e r e s t r a t e . Thus, the e x i s t e n c e of e q u i l i b r i u m r a t i o n i n g i n the p e r f e c t r i s k - s c r e e n i n g case may not depend on moral or c o l l u s i v e constraints on i n t e r e s t rate d i f f e r e n t i a t i o n . The above e x p l a n a t i o n f o r r a t i o n i n g shares a common theme with the S t i g l i t z - W e i s s (1981) a n a l y s i s of i m p e r f e c t r i s k - s c r e e n i n g . In each example the lender reacts to an information deficiency by e s t a b l i s h i n g loan terms that have d e s i r e d e f f e c t s on borrower b e h a v i o r . Despite the i n a b i l i t y to screen i n d i v i d u a l r i s k i n the S t i g l i t z - W e i s s model, a lender c o u l d induce f a v o u r a b l e borrower a c t i o n s and c o n t r o l r i s k through an appropriate choice of the l e v e l of i n t e r e s t rate. S i m i l a r l y , i n the above context of unknown i n d i v i d u a l demand schedules, the lender's s e l e c t i o n of a uniform p r i c i n g regime over price discrimination may be a l o g i c a l response to the i n f o r m a t i o n problem. By adopting uniform p r i c i n g the l e n d e r structures the loan process to remove the borrower's incentive to e x p l o i t a superior information p o s i t i o n . Chapters 2 and 3 have presented various explanations f o r the non-price r a t i o n i n g b e l i e v e d to e x i s t i n l o a n markets. T h i s theory i s used i n the f o l l o w i n g chapter to assess the t h e o r e t i c a l merit of e x i s t i n g empirical models of c r e d i t r a t i o n i n g . 53 Chapter 4 Econometric Studies of Disequilibrium Rationing It would be useful to have information on the empirical s i g n i f i c a n c e of c r e d i t r a t i o n i n g f o r a number of reasons as d i s c u s s e d i n Chapter 1. However, comprehensive d i r e c t measures are not a v a i l a b l e since the actual volume of u n s a t i s f i e d loan demand i s not recorded. The sample survey of Hatch-Wynant-Grant (1982) provides a p a r t i a l i n d i c a t i o n of the importance of c r e d i t r a t i o n i n g i n Canada but i t i s i n c a p a b l e of r e s o l v i n g many questions, such as: (a) the aggregate volume of r a t i o n i n g and the p a t t e r n of intertemporal v a r i a t i o n s i n t h i s t o t a l , and (b) t h e r e l a t i v e m a g n i t u d e s of e q u i l i b r i u m and d i s e q u i l i b r i u m rationing in each period. Given the l i m i t e d information obtainable from d i r e c t sources s e v e r a l i n d i r e c t methods have been developed to test for the existence of c r e d i t r a t i o n i n g . One approach attempts to examine the i s s u e w i t h the a i d of proxy v a r i a b l e s b e l i e v e d to be c o r r e l a t e d with the u n o b s e r v a b l e a c t u a l l e v e l of r a t i o n i n g . A second approach uses d i s e q u i l i b r i u m econometric techniques to estimate s t r u c t u r a l l o a n s u p p l y and demand equations and o b t a i n i n f o r m a t i o n on r a t i o n i n g . T h i s chapter d i s c u s s e s and e v a l u a t e s these a l t e r n a t i v e methods. 4.1 Proxy Method The proxy method, developed by Jaffee (1971) and subsequently followed by Rimbara and Santomero (1976), constructs a measure of rationing to serve as a dependent v a r i a b l e f o r t e s t s of the J a f f e e t h e o r e t i c a l model. The i d e a l measure CR would be the r a t i o of t o t a l rationing to demand of r i s k y firms.* 1. The d e f i n i t i o n of rationing i n (1) depends only on D^  and since the model predicts r i s k - f r e e customers are never rationed. ( 1 ) CR = D l - L l D l where = t o t a l demand of r i s k y firms, and Lj = actual volume of loans granted to r i s k y firms Since excess demand (D^ - L^) i s un o b s e r v a b l e some proxy r e l a t e d to the actual measure CR must be i d e n t i f i e d . According to Propositions 2 and 3 of Chapter 3 , the l e v e l of d i s e q u i l i b r i u m (and t o t a l ) r a t i o n i n g i s p o s i t i v e l y c o r r e l a t e d with the proportion of t o t a l loans granted to r i s k -free borrowers. Therefore, CRp i s proposed as a proxy f o r CR. ( 2 ) CRp = L 2 where I^ = volume of loans granted to r i s k - f r e e firms. After noting that loans to r i s k - f r e e firms are demand-determined (L^ = D 2 ) in the t h e o r e t i c a l model, the r e l a t i o n s h i p between the proxy CRp and CR can be d e r i v e d from ( 1 ) and ( 2 ) . ( 3 ) CRp = 1 1 + ( 1 - CR) D 2 From ( 3 ) i t i s seen that the proxy i s p o s i t i v e l y r e l a t e d to the a c t u a l 9 measure CR. In order to make the proxy operational the volume of loans to r i s k - f r e e firms i s represented by a v a i l a b l e data on loans made at the prime i n t e r e s t r a t e . Time series data f o r CRp can be used to test s e v e r a l predictions from the J a f f e e model. In p a r t i c u l a r , the model p r e d i c t s that v a r i a t i o n s i n r a t i o n i n g are due p r i m a r i l y to changes i n d i s e q u i l i b r i u m r a t i o n i n g , and 2 . One problem with the proxy i s that i t can vary independently of CR i f there i s a change i n the r e l a t i v e demand D^ / D 2 of the r i s k y and r i s k -f r e e borrowers. that d i s e q u i l i b r i u m r a t i o n i n g depends on the d i f f e r e n c e between the equilibrium and current i n t e r e s t rates. These concepts may be tested with regression equation (4) which uses rationing measure CRp as the dependent v a r i a b l e . (4) CR p = a Q + ajCR,.-- Rfc) + e where CRp = proportion of t o t a l loans granted at the prime in t e r e s t rate, R t = equilibrium i n t e r e s t rate (measured as an average across a l l borrowers), Rt = current (average) i n t e r e s t rate, and e = random error term. If the c u r r e n t i n t e r e s t r a t e i s below (above) the e q u i l i b r i u m there w i l l be p o s i t i v e ( n e g a t i v e ) d i s e q u i l i b r i u m r a t i o n i n g . T h e r e f o r e , a p o s i t i v e r e l a t i o n s h i p i s expected between the rationing proxy CRp and ( R f -R t). The regression equation i s completed by specifying determinants of the equilibrium loan rate Rt. P o r t f o l i o theory suggests that the e q u i l i b r i u m r e t u r n on loans w i l l e q u a l the y i e l d from any other asset h e l d by the l e n d e r a f t e r adjustment f o r other f a c t o r s such as l i q u i d i t y and r i s k . Thus, R't may be s p e c i f i e d as a function of the y i e l d on treasury b i l l s and other v a r i a b l e s which attempt to adjust for l i q u i d i t y differences between loans and treasury b i l l s . Jaffee's a p p l i c a t i o n of equation (4) to the commercial loans market of U.S. commercial banks gave r e s u l t s that were consistent with the underlying theory. With the estimated c o e f f i c i e n t for a^ having the expected p o s i t i v e sign and s t a t i s t i c a l l y s i g n i f i c a n t , r i s k - f r e e borrowers received a higher p r o p o r t i o n of t o t a l l o ans granted as (Rfc - Rfc) i n c r e a s e d . T h i s f i n d i n g suggests that d i s e q u i l i b r i u m rationing does ex i s t and that the incidence of rationing i s r e l a t e d to d e f a u l t r i s k . However, despite this evidence on existence, the proxy method does not provide measures of e i t h e r the t o t a l volume of c r e d i t rationing or the r e l a t i v e s i g n i f i c a n c e of the equilibrium and d i s e q u i l i b r i u m components. 4.2 S t r u c t u r a l Estimation Econometric techniques have been developed for estimating supply and demand s c h e d u l e s i n markets which may be i n d i s e q u i l i b r i u m . L a f f o n t -Garcia (1977), Sealey (1979), and Ito and Ueda (1981) use these methods to examine r a t i o n i n g i n the business loans markets of v a r i o u s c o u n t r i e s . Given t h e i r hypothesis that the loan rate may not adjust to equate supply and demand within the current period, a short-side r u l e must be s p e c i f i e d to determine the quantity transacted when the i n t e r e s t rate d i f f e r s from the m a r k e t - c l e a r i n g l e v e l . Each of the above s t u d i e s assumes that the observed market quantity i s the minimum of notional aggregate demand and aggregate supply at the current i n t e r e s t rate: (5) L t = min ( L d , Lp where L t = observed value of t o t a l loans at time t, L d = n o t i o n a l aggregate demand, and = notional aggregate supply d <3 These authors then s p e c i f y arguments i n L and L and combine these schedules with an i n t e r e s t rate equation to estimate s t r u c t u r a l supply and demand parameters and examine d i s e q u i l i b r i u m behavior. For example, Ito and Ueda consider the f o l l o w i n g system in conjunction with (5). (6) L d = c i ^ + a 2R t + u: l t 3. Quandt (1982) sur v e y s the l i t e r a t u r e on econometric d i s e q u i l i b r i u m models. 57 (7) L| - 3 i z t + 0 2 R t + u 2 t (8) R t = yR t_ 1 + ( i - y)R* o <. y < l where X,Z = vectors of predetermined v a r i a b l e s , R t = observed i n t e r e s t rate at time t, Rj? = eq u i l i b r i u m i n t e r e s t rate at time t, and u l t , u 2 t = error terms. Equation (8) i s a p a r t i a l p r i c e adjustment mechanism i n which the current i n t e r e s t rate i s a weighted average of the preceding period's value and the e q u i l i b r i u m r a t e f o r the c u r r e n t p e r i o d . The parameter P i s i n v e r s e l y r e l a t e d to the degree of i n t e r e s t rate f l e x i b i l i t y . If p = 0 the i n t e r e s t rate adjusts completely to i t s equilibrium R* within the current period such that the loan market i s in equilibrium each period. If 0 < y < 1 there i s p a r t i a l adjustment in the short-run and d i s e q u i l i b r i u m p r e v a i l s i n the l o a n market throughout the p e r i o d of t r a n s i t i o n . The e q u i l i b r i u m i n t e r e s t rate Rfc i s determined simply by the i n t e r s e c t i o n of the aggregate demand and supply schedules. From (6) and (7),^ * 1 (9) \ = (oijX - 3jZ + ul - u 2) . (3 2 : - ot2) Bowden (1978) proposed (8) as an a l t e r n a t i v e to the usual adjustment scheme (10) where the change i n price i s proportional to excess demand: (10) AR t - M L d - L*) 0 < x < 0 0 4. For convenience time subscripts w i l l be deleted from a l l v a r i a b l e s other than i n t e r e s t rates. 58 One c r i t i c i s m of (10) i s that the s i z e of the adjustment speed parameter X depends on the units of measurement for the i n t e r e s t rate and quantity v a r i a b l e s . ^ i n c o n t r a s t , the parameter u i n (8) i s independent of the u n i t s of measurement which makes i t an e a s i l y i n t e r p r e t e d measure of the degree of pr i c e f l e x i b i l i t y . Equations (6) and (7) cannot be estimated i n t h e i r present form since (5) indicates that one of the dependent v a r i a b l e s i s unobservable when the market i s i n d i s e q u i l i b r i u m . However, estimating equations for aggregate supply and demand schedules can be derived from (5)-(9). During a period of aggregate excess demand, i d e n t i f i e d by a r i s i n g i n t e r e s t r a t e , the s h o r t - s i d e p r i n c i p l e (5) s t a t e s t h a t the observed q u a n t i t y i s aggregate supply. Although demand i s not observed d i r e c t l y i t equals the r e a l i z e d quantity plus excess demand. Therefore, ( l l ) ( a ) L s = L (b) L d = L + ( L d - L s) i f R t > Rt_2 A f t e r r e a r r a n g i n g ( l l ) ( b ) and e x p r e s s i n g the excess demand component i n terms of (6) and (7), (12) L = L d - ((a2 - B 2)R t + c^ X - BjZ + U j - u 2) = L d + ( a 2 - B 2)(R* - R t) us i n g (9). I t can a l s o be shown from (8) that (13) (R* - Rt> = 7 ^ ( R t " Rt-1>-( l - y ) 5. L a f f o n t - G a r c i a and S e a l e y use the t r a d i t i o n a l adjustment mechanism (10). 59 F i n a l l y , the e s t i m a t i n g e q u a t i o n f o r demand d u r i n g excess demand periods i s obtained by sub s t i t u t i n g (6) and (13) into (12). (14) L = o^X + cx2Rt + _H _ ( a 2 - 3 2)(R t " Rt-0 + u l In (14) the observed change i n i n t e r e s t r a t e a c t s as an i n d i c a t o r of unobserved excess demand with rationing an increasing function of (R t-R t_^) for a given adjustment speed parameter y . For a p e r i o d w i t h aggregate excess s u p p l y , i d e n t i f i e d by a f a l l i n g i n t e r e s t rate, the short-side r u l e indicates that the quantity exchanged i s aggregate demand and aggregate s u p p l y equals the observed quantity plus excess supply. (15)(a) L d = L (b) L s = L + ( L s - L d ) • l f R t < R t - 1 By f o l l o w i n g s i m i l a r steps to those used i n the excess demand case, (16) i s d e r i v e d as the e s t i m a t i n g e q u a t i o n f o r s u p p l y i n the presence of market excess supply. (16) L = BjZ + 3 2R t + ^ ) ( 3 2 - a 2 ) ( R t - R ^ ) + u 2 Since the i n t e r e s t r a t e change (R t - R t _ i ) appears i n the demand (s u p p l y ) e s t i m a t i n g e q u a t i o n o n l y when there i s aggregate excess demand (supply), the general forms f o r these equations are:^ 6. The e s t i m a t i n g equations have a s t r u c t u r e s i m i l a r to (17) and (18) when the conventional p r i c e adjustment mechanism (10) i s used i n place of the Bowden process. These a l t e r n a t i v e equations can be derived as (17)' L = ajX + a 2R t - j AR+ + u j 1 (18)' L = BjZ + 3 2R t + ^ AR + u 2 60 (17) L = c^X +a2R. + ( a 2 " V A R t + U l (18) L = B l Z +3 2R t + T T ^ y (S 2 - a 2) AR^ + u 2 where ARj AR t Information concerning the s t r u c t u r a l demand and supply functions i s obtained from parameter estimates of » 3i> a n < l $2 ^ r o m t ^ i e system defined by (17) and (18). The hypothesis that the loan market i s always i n e q u i l i b r i u m i s t e s t e d under the n u l l hypotheses V = 0.^ For the case y = 0 the d i s e q u i l i b r i u m model (17) and (18) c o l l a p s e s to a standard e q u i l i b r i u m model wit h the observed l o a n q u a n t i t y a t t r i b u t e d to the i n t e r s e c t i o n of L d and L s . This s t r u c t u r a l approach meets one of the desired c h a r a c t e r i s t i c s of an e m p i r i c a l model by p e r m i t t i n g the volume of c r e d i t r a t i o n i n g to be estimated as the difference ( L d - L s) for those periods when the current i n t e r e s t rate i s below equilibrium. For example, i f the present loan rate i s R t the magnitude of excess demand would be (b-a) i n F i g u r e 4. D e s p i t e t h i s improvement over the proxy method, the model of Laffont-Garcia et a l 7. P r e v i o u s s t u d i e s have reached c o n f l i c t i n g conclusions regarding the p r e v a l e n c e of c r e d i t r a t i o n i n g . L a f f o n t - G a r c i a (1977), p. 1198 concluded that i n the Canadian business loans market "the main feature was a downward s l u g g i s h n e s s of p r i c e s l e a d i n g to an e s s e n t i a l l y demand-determined market". S e a l e y (1979), p. 697 found t h a t i n the U.S. " b u s i n e s s l o a n s a r e e s s e n t i a l l y s u p p l y d e t e r m i n e d w i t h i n t e r m i t t e n t p e r i o d s of demand determination". Ito-Ueda, u s i n g the fo r m a l t e s t i n v o l v i n g the parameter y , accepted the e q u i l i b r i u m h y p o t h e s i s f o r the U.S. but r e j e c t e d the e q u i l i b r i u m h y p o t h e s i s f o r Japan. R t " R t - 1 i f R t > R t_ x otherwise R t " R t - 1 i f R t < R t - 1 otherwise 61 i s inadequate since i t a r b i t r a r i l y excludes the p o s s i b i l i t y of e q u i l i b r i u m rationing. It i s r e c a l l e d that two major assumptions were used to derive the model. The aggregate short-side r u l e assumes that the observed market quantity i s the minimum of aggregate demand L d and supply L s at the current rate of i n t e r e s t . In addition, the e q u i l i b r i u m rate occurs at the i n t e r -s e c t i o n of the two aggregate s c h e d u l e s . These assumptions have two i m p l i c a t i o n s : (a) r a t i o n i n g can occur o n l y when the c u r r e n t i n t e r e s t r a t e i s below i t s equilibrium, and (b) there i s no rationing at the equilibrium i n t e r e s t rate since the e q u i l i b r i u m q u a n t i t y (c i n F i g u r e 4) must correspond to the i n t e r s e c t i o n of L and L . These conclusions demonstrate that p r e v i o u s e m p i r i c a l s t u d i e s focus e x c l u s i v e l y on d i s e q u i l i b r i u m rationing and make no allowance for the e q u i l i b r i u m rationing discussed i n the t h e o r e t i c a l l i t e r a t u r e . 62 6 3 4.3 Aggregate Rationing with an Individual Short-side Rule The o b j e c t i v e of t h i s s t u d y i s to r e d u c e the p r e s e n t inconsistency between t h e o r e t i c a l and applied studies by estimating a general model incorporating both equilibrium and d i s e q u i l i b r i u m c r e d i t rationing. It would be d e s i r a b l e to construct an aggregate empirical model from an e x p l i c i t micro e x p l a n a t i o n of l o a n markets. J a f f e e ' s (1971) theory, presented i n Chapters 2 and 3, was s e l e c t e d to represent these micro-foundations and serve as a basis for c r i t i q u i n g the t h e o r e t i c a l v a l i d i t y of the d i s e q u i l i b r i u m models of L a f f o n t -Q Garcia and others. A d i s t i n g u i s h i n g feature of the Jaffee model i s that an i n d i v i d u a l ' s l o a n s i z e i s the minimum of that borrower's demand and supply (offer) curves: (19) L ± = min(Lj, L?) The implications of short-side r u l e (19) were not taken into account by the s t r u c t u r a l models of d i s e q u i l i b r i u m rationing. A f t e r d e r i v i n g p o t e n t i a l e s t i m a t i n g equations f o r aggregate demand and s u p p l y functions from this i n d i v i d u a l short-side r u l e , i t w i l l be shown that the a g g r e g a t e s h o r t - s i d e p r i n c i p l e (5) u t i l i z e d by p r e v i o u s researchers i s v a l i d only under very r e s t r i c t i v e conditions. Consider the f o l l o w i n g s p e c i f i c a t i o n . (19) L± = min(Lf, L?) n n (20) L = I L ± = 1 min(L d, Lf) i=l i=l 8. I t s h o u l d be noted th a t J a f f e e , u n l i k e J a f f e e - R u s s e l 1 (1976) or S t i g l i t z - W e i s s (1981), assumes l e n d e r s can form e x p e c t a t i o n s of d e f a u l t r i s k for each customer i n d i v i d u a l l y . 64 n n (21) I 4 -I < a x . + a2.Rt + u ) i=l i=l n n (22) I I* -.J ( 3 ^ + 32.Rt +u2.) i=l i=l (23) uR t_ 1 + ( l - y ) R t - 0 < y < 1 with L = observed aggregate volume of loans, L d = notional aggregate demand, L s = notional aggregate supply, ^ i ' ^ i = v e c t o r s of predetermined v a r i a b l e s i n customer i's demand and supply functions, u l i ' u 2 i = e r r o r t e r m s > a°d Rr = equilibrium i n t e r e s t rate This system d i f f e r s from the c o n v e n t i o n a l aggregate model i n s e v e r a l important r e s p e c t s . F i r s t l y , n o t i o n a l aggregate demand and su p p l y s c h e d u l e s are expressed e x p l i c i t l y as summations across n customers. In many markets i t would not be r e a l i s t i c to d e f i n e a s e l l e r ' s supply function f o r each i n d i v i d u a l but, f o l l o w i n g J a f f e e (1971), i n d i v i d u a l l o a n s u p p l y c u r v e s are j u s t i f i e d g i v e n i n t e r -customer v a r i a t i o n s i n d e f a u l t r i s k and r i s k - s c r e e n i n g by l e n d e r s . Secondly, a c c o r d i n g to the i n d i v i d u a l s h o r t - s i d e r u l e (19) the aggregate quantity observed (20) i s the sum of n i n d i v i d u a l minimums, rather than the minimum of aggregate demand and supply as assumed by the econometric studies of d i s e q u i l i b r i u m r a t i o n i n g . Equation (23) restates the p a r t i a l price adjustment mechanism which represents the current i n t e r e s t rate as a weighted average of l a s t period's value and the current equilibrium rate. Equations (16)(a)-( 16)(e) i n Chapter 3 suggested that d u r i n g a 65 given period some borrowers receive loans on t h e i r demand functions d s L i while others are rationed along supply functions L^. P a r t i t i o n the former group to a set D and the l a t t e r to set S. N o t i o n a l aggregate demand may then be defined as L d = I Ld± + I (L| + (4 - Lf)) ieD ieS where ieD i f L{ = min(L4,L?) ieS i f Lf = min(L d,L?) Since Lj_ = min (L^.Ll), J. ^ n ^ i and J G L | are the o b s e r v a b l e l o a n quantities for the two sets of borrowers. Therefore, (24) L d = I j£ + I L | + I ( L d _ LS) ieD ieS ieS = L + I (Lf - Lf) ieS w i t h e g ( L d - L^) > 0 equal to t o t a l excess demand of r a t i o n e d borrowers. A f t e r s u b s t i t u t i n g f o r L d and L^ from (21) and (22), and s o l v i n g for the i n t e r e s t rate corresponding to the i n t e r s e c t i o n of i n d i v i d u a l i's demand and supply functions, (24) may be expressed as (25) L d = L + I ( B 2 ± - c t 2 1 ) C r l t - R F C) ieS where . r l t = ( _ } ( a X - fl z + u - u ), 2i 2 i ( r ± t - R ) > 0 for a l l ieS, and B 2. > 0, a 2 . < 0. 66 If the e q u i l i b r i u m i n t e r e s t r a t e R t i s added and s u b t r a c t e d from ( r i t - R t), the excess demand term i n (25) i s expanded to (26) J (3 2 i - a 2 . ) ( r . t - R £) + J (B,. - a ) d - R ) = J& ( B 2 i - a 2 . ) ( r . t - R t) + ^ (B,. - a , . ) ^ ^ - R ^ ) using (13). F i n a l l y , from (25) and (26) notio n a l aggregate demand may be w r i t t e n as (27) when q u a n t i t y d e t e r m i n a t i o n i s governed by the i n d i v i d u a l short-side r u l e . (27) L = L + I ( 3 2 i - a 2 i ) ( r i t - R t) ieS + I C02i " a 2 i > -f^r < Rt ' Rt-1> i£S C l - U ) T o t a l excess demand of r a t i o n e d borrowers, g i v e n by the two summations ov e r ieS , i n (27), c o n t a i n s both e q u i l i b r i u m and d i s e q u i l i b r i u m rationing. Unlike the model of section 4.2 used by Laffont-Garcia et a l , the r e a l i z e d loan quantity does not correspond to demand i f the l o a n market i s i n e q u i l i b r i u m . If y = 0 so that the i n t e r e s t r a t e adjusts completely to eq u i l i b r i u m within the current period, or i f there has been f u l l a d j u s t m e n t to p a s t d i s t u r b a n c e s w i t h R t = R t_^, di s e q u i l i b r i u m rationing does not ex i s t and the second summation i s zero. Nevertheless, i f i n d i v i d u a l i's demand and supply curves i n t e r s e c t at an in t e r e s t rate r ^ t greater than the equ i l i b r i u m rate Rfc , then ( B 2^ - a 2£ ) ( - R t ) i s the magnitude of excess demand t h i s borrower would experience at Rt. The f i r s t summation i n (27) would give the t o t a l amount of e q u i l i b r i u m c r e d i t rationing. 67 S i m i l a r l y , the notional aggregate supply schedule consistent with (19) can be derived as (28) L s = L + I ( c t 2 1 - 3 2 i ) ( r i t - Rfc) ieD + J < a2i ~ B2i> 7TI7o(Rt " Rt-1> ieD v M / using the steps outlined above. Desired lender supply to a borrower may be g r e a t e r than the i n d i v i d u a l ' s demand at the c u r r e n t i n t e r e s t r a t e . Since r e a l i z e d l o a n s i z e cannot exceed d e s i r e d demand, these r e l a t i o n s h i p s are characterized by excess s u p p l y . The summations i n (28) represent t o t a l excess supply to unrationed borrowers ieD. In order to f a c i l i t a t e comparison, the systems d e r i v e d from the a l t e r n a t i v e short-side p r i n c i p l e s L = min(L d, L s) and = min(L d, L|) are restated as A. and B. r e s p e c t i v e l y . A . A g g r e g a t e S h o r t - s i d e R u l e S y s t e m (5) L = min(L d, L s ) (17) L = cijX + a 2 R t + T J _ y - g 2 ) A R + + ^ (18) L = B XZ + e 2R t + v." • ( g 2 _ a ) AR" + u £ (1-y) where A B.% and AR^ are as defined above. 68 B. Individual Short-side Rule System a 2 . ) ( r . t - R t) (from (21), (27) ) 3 2 i ) ( r i t - R t ) (from (22), (28) ) A major d i s s i m i l a r i t y between the two systems concerns the treatment of e q u i l i b r i u m c r e d i t rationing. As d e t a i l e d p r e v i o u s l y , the t r a d i t i o n a l system A. a b s t r a c t s from the p o s i t i o n s of i n d i v i d u a l borrowers and precludes the existence of equ i l i b r i u m rationing. In contrast, system B. i s based on the u n d e r l y i n g s u p p l y and demand f u n c t i o n s of i n d i v i d u a l customers. If l e n d e r s face c o n s t r a i n t s on i n t e r - c u s t o m e r i n t e r e s t r a t e d i f f e r e n t i a t i o n , the equ i l i b r i u m rate g e n e r a l l y w i l l not coincide with the in t e r s e c t i o n of an i n d i v i d u a l ' s supply and demand functions. Consequently, some borrowers are rationed at the equ i l i b r i u m rate of in t e r e s t i n system B. The a l t e r n a t i v e systems d i f f e r s i g n i f i c a n t l y i n another important respect. According to the aggregate short-side r u l e system, the observed market q u a n t i t y always l i e s on at l e a s t one of the aggregate demand and supply schedules. However, examination of (29) and (30) indicates that L d s i s positioned on L or L only under extreme conditions; during the current n n d T s, (20) L = I L = I min(L?,L?)i = l i=l (29) L = J ( a u X . + a 2 . R T + u u ) ~ I ($2i " i=l ieS " I ( p 2 i " a 2 i } T R 0 - < R t " R t - 1 } ieS (30) L = j ( B U Z . + e 2 . R T + u 2.) - I <a2. -1=1 leD " I ( a 2 i " *2i> OPyT ( R t " R t - 1 } i£D period either a l 1 i n d i v i d u a l s belong to the unrationed set D or a l l must belong to the rationed set S. The theory associated with the i n d i v i d u a l s h o r t - s i d e r u l e suggests i n s t e a d that i n a l l p e r i o d s there w i l l be some customers i n each of these s e t s . Given t h i s p r e d i c t e d d i v e r s i t y i n rationing status, i t i s r e a d i l y shown that r e a l i z e d market quantity belongs to neither L D nor L S , but rather i s l e s s than both n o t i o n a l functions. n n d s With L = 1 L. = £min (L. ,L. ), i t follows immediately that i=l i=l 1 n L = ( I L* + I LS.) < L D = I L D ieD ieS - i=l s d since L. < L. for a l l ieS, and 1 l ' L = ( I L d • + I L?) < L S = J L? ieD ieS i=l d s since L. < L . for a l l ieD. I I Hence, (32) L < m i n ( L D , L S ) with L± = min(L4, L|) This l a s t r e s u l t indicates that the observed quantity understates both aggregate demand and s u p p l y i n a l l p e r i o d s . Consequently, e x i s t i n g econometric s t u d i e s which u t i l i z e s h o r t - s i d e r u l e (5) m i s s p e c i f y these aggregate r e l a t i o n s h i p s and give biased estimates f o r parameters ( i n c l u d i n g the speed of i n t e r e s t rate adjustment). E q u a t i o n s (29) and (30) from s y s t e m B. have been u s e f u l i n i l l u s t r a t i n g the d e f i c i e n c i e s of the current method of estimation. However, (29) - (30) cannot be estimated s i n c e these equations are not i d e n t i f i e d . If the d e f i n i t i o n of r i t i s substituted into (29) and (30), i t i s seen that 9. T h i s requirement r e s t r i c t s the c u r r e n t i n t e r e s t r a t e to be e i t h e r above the Intersection of i n d i v i d u a l supply and demand functions for every borrower or, a l t e r n a t i v e l y , below t h i s i n t e r s e c t i o n for every borrower. 70 each equation contains an i d e n t i c a l l i s t of explanatory v a r i a b l e s . Thus, i t appears that a general model with equilibrium and d i s e q u i l i b r i u m c r e d i t rationing cannot be estimated d i r e c t l y from separate aggregate demand and s u p p l y e q u a t i o n s , which was the methodology used by the d i s e q u i l i b r i u m studies of Laffont-Garcia et a l . Chapter 5 maintains the r e l i a n c e on an i n d i v i d u a l short-side r u l e to propose an a l t e r n a t i v e method of estimation. 71 CHAPTER 5 AN ECONOMETRIC MODEL OF AGGREGATE CREDIT RATIONING The previous chapter demonstrated the problems i n v o l v e d i n attempting to make exi s t i n g empirical models of c r e d i t rationing consistent with the i n d i v i d u a l s h o r t - s i d e r u l e . An a l t e r n a t i v e approach i s r e q u i r e d which incorporates both equilibrium and d i s e q u i l i b r i u m rationing and i s based on a micro e x p l a n a t i o n of l o a n d e t e r m i n a t i o n . A system s a t i s f y i n g these o b j e c t i v e s i s d e r i v e d i n t h i s chapter by m o d e l l i n g i n d i v i d u a l l o a n determination i n 5.1 and then using t h i s micro a n a l y s i s i n 5.2 to obtain an aggregate loan equation. This loan r e l a t i o n s h i p , i n combination with an i n t e r e s t r a t e equation, forms a two-equation e m p i r i c a l model which i s applied to the Canadian business loans market. The chapter concludes with a comparison of r e s u l t s from aggregate and i n d i v i d u a l s h o r t - s i d e r u l e models. 5.1 Individual Loan Size Determination (Flows) The i n d i v i d u a l s h o r t - s i d e r u l e s t a t e s that the observed l o a n s i z e granted to borrower i i s the minimum of the demand and supply functions of borrower i at the p r e v a i l i n g i n t e r e s t rate. ( 1 ) L± = min(L.f,. L?) i - l , , n hf > 0, L? > 0 Due to the absence of micro data for the n loan customers i n d i v i d u a l demand and supply functions must be s p e c i f i e d with aggregate v a r i a b l e s as arguments. Th i s approach can be implemented by e x p r e s s i n g borrower i's demand and s u p p l y c h a r a c t e r i s t i c s i n terms of d e v i a t i o n s from the mean p o s i t i o n of the t o t a l p o p u l a t i o n of borrowers. For example, suppose 72 current period flow demand and supply for borrower i are (2) L4 = a Q + a Q l + ctj ( X + X ± ) + a 2 ( R + R ± ) (3) L ? = B Q + B o i + B 2 (2 + Z ± ) + g 2 ( R + R ± ) with X = mean value across a l l borrowers of exogenous v a r i a b l e X , Z = mean value across a l l borrowers of exogenous v a r i a b l e Z, R = mean i n t e r e s t rate on loans, and (ct 0j_, B Q i , X ^ , Z^, R^) measured as b o r r o w e r - s p e c i f i c (zero mean) deviations from corresponding elements i n the vector of population means ( a Q , B 0 , X," Z , R ) . 1 Equations (2) and (3) a l l o w i n t e r - c u s t o m e r h e t e r o g e n e i t y to occur through d i f f e r e n t i n t e r c e p t terms and d i f f e r e n t v a l u e s f o r e x p l a n a t o r y v a r i a b l e s . From ( 2 ) , borrower i's demand c o n s i s t s of one term v a l u e d at population means of the intercept and explanatory v a r i a b l e s and a second term e-^ which depends on deviations from means. ( A ) L J = ( a Q + ttlX + a 2 R ) + e 1 ± with e1± = a Q i + + a 2 R I It w i l l be assumed that e^ i s normally d i s t r i b u t e d over the population of 2 borrowers with mean zero and variance o ^ • e 2 " N(0, a f) 1. It i s conceivable that a borrower would r e c e i v e a loan from one bank a f t e r being r a t i o n e d by another l e n d e r . In the J a f f e e t h e o r e t i c a l model this event could occur i f lenders have d i f f e r e n t assessments of the i n d i v i d u a l ' s d e f a u l t r i s k . In such cases t h e l o a n s u p p l y c u r v e (3) c o u l d be i n t e r p r e t e d as t o t a l l o a n s u p p l y by a l l l e n d e r s to i n d i v i d u a l i i n a given period. 73 S i m i l a r l y , e q u a t i o n ( 3 ) can be rearranged to show that the lender's s u p p l y f u n c t i o n to borrower i c o n t a i n s one term composed of p o p u l a t i o n means and a b o r r o w e r - s p e c i f i c d e v i a t i o n e 2 i . Assume that the d e v i a t i o n component e 2 i s n o r m a l l y d i s t r i b u t e d a cross the p o p u l a t i o n of borrowers with mean zero and variance a 2. ( 5 ) L| = ( 6 Q + 3 X Z + B2R) + e 2. with e 2i o i + hZi + e 2 R i e 2 ~ N(0, a\) E x p l i c i t recognition of the non-negativity constraint on loan sizes g i v e s ( 6 ) and ( 7 ) as g e n e r a l r e p r e s e n t a t i o n s of i n d i v i d u a l demand and supply. From ( 4 ) and ( 5 ) , ( 6 ) L4 = a Q + a^X + a 2R + e-^ i f a Q + 04X + a 2R + e±± > 0 i . e . el± > -(an + c^X + a 9R) Q T U^ A -r a 2 r 0 otherwise i . e . e1± < - ( a Q + c^X + a 2R) ( 7 ) Lf = ! Q + B l Z + B2R + e 2. i f e 2 i > - ( B 0 + 8jZ + 3 2R) 0 otherwise It i s now p o s s i b l e to state the conditions which e s t a b l i s h whether the r e a l i z e d loan si z e of a p a r t i c u l a r borrower i s supply or demand-determined. 7 4 Define as the i n t e r e s t r a t e at which borrower i's s u p p l y and demand f u n c t i o n s i n t e r s e c t . If r ^ i s l e s s than the i n t e r e s t r a t e charged on borrower i' s l o a n , then a c c o r d i n g to s h o r t - s i d e r u l e (1) the l o a n i s demand-determined: L ± = L4 = min(L4, Lf) for r ± < R± with v± = - a-a o + a o i + a l < X + X i > " O l i -e^z + z±) and Rj. = R + R ± = i n t e r e s t rate charged on borrower i ' s loan. Unfortunately information on the values of r ^ and R^ i s u n a v a i l a b l e for each p o t e n t i a l borrower. However the requirement r^ < R^  f o r a demand-determined loan i s equivalent to e ^ <^  n f o r e 3 i = e l i " e 2 i and TI = ( & 2 - « 2 ) R " ao " a l X + Bo + h7" This a l t e r n a t i v e condition for a demand-determined loan i s convenient since a l l unobservable borrower-specific random components are i s o l a t e d on the l e f t s i d e of the i n e q u a l i t y as e-j^ whereas the r i g h t s i d e depends on more r e a d i l y obtainable mean values of explanatory v a r i a b l e s . Thus, from (1), (8) L ± = L i i f e 3 i £ n L? i f e 3 i > n . e 3 . N(0,a 3) ( r ± < R ±) ( r ± > R ±) 7 5 E q u a t i o n (9) combines (6) -with a non-negativity constraint. (8) to summarize loan s i z e determination L d = cx0 + oqX + oi2R + e n i f ( i ) e 3 1 _< n and ( i i ) e-^ > -( a Q + a^X + a 2 R ) = Y . L ! = eG + M + + e 2 i (9) 4 = 1 i f ( i ) e 3 i > n and ( i i ) e 2 i > - ( g Q + g lZ + g 2R) =. 9-0 i f ( i ) e 1 ± < - ( a 0 + a 2X + a 2R) or ( i i ) e 2 i < - ( B 0 + BjZ + 6 2 R ) Figure 5 i l l u s t r a t e s (9). The d i r e c t i o n of i n e q u a l i t y between e-j^ and n determines whether borrower i's loan i s demand-determined (as i n regions A and B) or supply-determined (C and D). N o n - n e g a t i v i t y c o n s t r a i n t s on l o a n demand and s u p p l y become b i n d i n g at s u f f i c i e n t l y low v a l u e s of borrower-specific random terms e ^ and e 2^ r e s p e c t i v e l y . 5.2 Deri v a t i o n of the Aggregate Loan Equation An aggregate loan flow equation consistent with the i n d i v i d u a l short-s i d e r u l e can be d e r i v e d u s i n g the above i n f o r m a t i o n on i n d i v i d u a l l o a n determination. Order the t o t a l set of p o t e n t i a l borrowers such that the f i r s t n Q ( i = l,...,n Q) r e c e i v e a new l o a n i n the c u r r e n t p e r i o d and the l a s t (n - n Q) do not o b t a i n a l o a n . Expected new l o a n s i z e of a representative borrower from the set i = l , . . . . , n i s (using (9)) 76 FIGURE 5: Individual Loan Determination 7 7 (10) E a j l ^ > 0) = I K E ( L d | L i > 0) Prob ( L ± = Lf > 0) + E(L®|L i = L? > 0) Prob ( L ± = L? > 0) I ( E C L ^ e ^ >Y, e3.< n) Prob ( e 1 ± >Y , e 3 ± <n)' + E ( L ? | e 2 i > 9, e 3 1 > n) Prob ( e 2 i > 6, e 3 i > n) with K = Prob(L i > 0) = proportion of potential borrowers that do receive a loan. A f t e r s u b s t i t u t i n g the d e f i n i t i o n s of L d and L | i n t o (10) and u s i n g ( i ) -( i v ) i n Appendix 3, the expectation i s stated as (11) where §i(ei» e3^ a n d g 2 ( e 2 , e 3) are b i v a r i a t e normal density functions. (11) E(L. L. > 0) = £ l 1 l K (aQ + oijX + a2R + gl (el, e^)de^e.^ + (B 0 + B XZ + B2R + e 2 ) g 2 ( e 2 , e 3 ) d e 2 d e 3 n e 78 For a p a r t i c u l a r borrower i within i = l,...,n 0, r e a l i z e d loan flow v a r i e s from the expected v a l u e (11) c a l c u l a t e d over a l l n Q borrowers by an amount v^. (12) L ± = E a j L . > 0) + v t i = l , . . . , n Q Summation of (12) over i = l,.,.,n 0 gives an equation for new loan flows i n period t. (13) L f = n D E a j L . > 0) + £° v. i = l 1 with = aggregate loan flow, and n Q = number of new loans. In ad d i t i o n to the new loan flows modelled by (13), the business loans data used i n t h i s study includes "predetermined" loans granted i n previous p e r i o d s and s t i l l o u t s t a n d i n g . A l though the p r e c i s e breakdown between predetermined and new loans i s unknown, Appendix 4 d e t a i l s how a v a i l a b l e information can be used to model predetermined loans by (14): (14) PL t = <)>(1 + h ) T L t _ 1 ( > l ) with PL t = value of predetermined loans at quarter t, <f) = unknown parameter to be estimated, h = ^ ' t - l ^ ) B L t - 1 ( > l ) BL t_j(<l) = t o t a l stock of outstanding loans less than $1 m i l l i o n at the end of t-1, BL t_j(>l) = t o t a l stock of outstanding loans of s i z e $1 m i l l i o n and greater, and 79 T L t _ j ( > l ) = t o t a l stock of term loans of s i z e $1 m i l l i o n and greater at the end of t-1. B r i e f l y , a high proportion of term loans ( o r i g i n a l maturity at l e a s t twelve months) outstanding at the end of quarter t-1 i s predetermined for quarter t since many of these long-term loans w i l l not mature during period t. Therefore, the lagged stock of term loans has informational value i n a s s e s s i n g the o v e r a l l amount of c u r r e n t predetermined l o a n s , and the parameter <J> i n (14) may be i n t e r p r e t e d as a f a c t o r of p r o p o r t i o n a l i t y between t o t a l term loans l a s t period T L t _ j and PL^. The a v a i l a b l e series on term loans i s r e s t r i c t e d to loans with a minimum s i z e of $1 m i l l i o n so T L t _ j ( > 1) i s augmented by the f a c t o r (1 + h) to o b t a i n a f i g u r e 2 for t o t a l term loans of a l l sizes. The aggregate l o a n e q u a t i o n used i n e s t i m a t i o n combines the f l o w r e l a t i o n s h i p (13) and the predetermined loan equation (14). (15) BL t = PL t + L f t = <j, (1 + h)TL J._ 1(> 1) + n oE(L i|L 1>0) + e L with BL = t o t a l outstanding stock of business loans, PL = predetermined loans, L^ = new loan flows, n Q = number of new loans, e-^  = random error term, and E ( L i | L i > 0) i s defined by ( l l ) . 3 2. The adjustment i s undertaken to remove a p o t e n t i a l source of s y s t e m a t i c i n t e r t e m p o r a l parameter v a r i a t i o n . Appendix 4 has additional discussion. 3. Equation (15) can be estimated without knowledge of the number of new l o a n s n Q. From (11), n QE(L^ | L^>0) i n v o l v e s the term n Q/K which i s equal to n (the t o t a l number of p o t e n t i a l borrowers). The data series for n i s described i n s e c t i o n 5.5 below. 80 It s h o u l d be noted that the proposed model c o n t a i n s a s i n g l e l o a n e q u a t i o n f o r t h e a c t u a l q u a n t i t y of l o a n s g r a n t e d . U n l i k e t h e d i s e q u i l i b r i u m models of Chapter 4 separate aggregate demand and aggregate s u p p l y equations are not estimated d i r e c t l y . More i m p o r t a n t l y , the new model i s preferred over the e a r l i e r approaches since i t does not constrain i n d i v i d u a l l o a n s to be e i t h e r e n t i r e l y demand-determined or e n t i r e l y supply-determined within a given period. 5.3 Interest Rate Equation S p e c i f i c a t i o n A second e q u a t i o n must be added to (15) to make the i n t e r e s t r a t e endogenous. One p o s s i b i l i t y i s to consider that the current i n t e r e s t rate i s a weighted average of the p r e c e d i n g period's r a t e and the c u r r e n t equilibrium rate.^ R t = yR t_! + (1 - y)R t 0 < y < 1 This approach requires an a p r i o r i judgment regarding the precise form of the equilibrium i n t e r e s t rate. For example, f o l l o w i n g the empirical models derived from the aggregate short-side r u l e , i t could be asserted that R t occurs at the i n t e r s e c t i o n of the aggregate schedules L and L . However, the t h e o r e t i c a l c r e d i t rationing l i t e r a t u r e has presented se v e r a l reasons why the e q u i l i b r i u m i n t e r e s t r a t e may not equate n o t i o n a l s u p p l y and d ema nd. An a l t e r n a t i v e s p e c i f i c a t i o n was chosen which does not c o n s t r a i n equilibrium to occur at the i n t e r s e c t i o n of aggregate supply and demand, although t h i s r e s u l t i s contained as a s p e c i a l case when 6 = 0. 4. This equation was discussed p r e v i o u s l y i n Chapter 4's a n a l y s i s of the Ito-Ueda model of d i s e q u i l i b r i u m r a t i o n i n g . 81 (16) ARt = 6 Q + Sjaj! - L p + ej n t AR t = R t R t - 1 > 6o < 0 ' 61 > 0 Equation (16) suggests that the current period change i n in t e r e s t rate depends p o s i t i v e l y on the dif f e r e n c e between aggregate demand and supply with the t o t a l number of p o t e n t i a l borrowers n t used to s c a l e t h i s d i f f e r e n c e r e l a t i v e to market s i z e . The s i g n of the constant term 6 determines whether equilibrium exists above or below the i n t e r s e c t i o n of L d and L s. The equilibrium i n t e r e s t rate for period t can be found from (16) by s o l v i n g for the in t e r e s t rate at which AR t = 0. Expressions for notional aggregate demand and supply are required to complete s p e c i f i c a t i o n of the i n t e r e s t r a t e equation. Aggregate f l o w demand at a given i n t e r e s t rate i s the summation of i n d i v i d u a l flow demands of a l l borrowers with p o s i t i v e demand at that rate. This quantity can be determined by m u l t i p l y i n g the number of po t e n t i a l borrowers with p o s i t i v e demand by the mean value of demand for this group: L d(R) = n Prob(L d(R) > 0) E(L d|L d(R) > 0) where n = t o t a l number of potential borrowers, Prob(L d(R) > 0) = proportion of po t e n t i a l borrowers with p o s i t i v e demand at in t e r e s t rate R, n Prob(L d(R) > 0) = number of borrowers with p o s i t i v e demand at R, and E(L d|L d(R) > 0) = average demand for borrowers with p o s i t i v e demand at R. L d may be written i n terms of known quantities using information from (6). 82 (17) L D = n P r o b ( e u > y) E ( a Q + oijX + o^R + e 1 1 | e l i >y ) = n (1 - F (y/q)) ( a Q + c^X + a 2&) + n a 1f(y/a 1) with Y = - ( a Q + ajX + a 2 R ) , Prob ( L ^ > 0) = Prob(e 1 : i >y ) = 1 - F(y/ai) , ai f(y/oi) E < e l i l e l i > Y ) = 1-F ( v / d ) ' f ( ) = standard normal density function, and F( ) = standard normal cumulative d i s t r i b u t i o n function. S i m i l a r l y , aggregate flow supply at a given i n t e r e s t rate i s obtained by summing i n d i v i d u a l supplies for a l l borrowers with p o s i t i v e supply at that rate. L S ( R ) = n P r o b ( L | ( R ) > 0) E ( L ^ | L ? ( R ) > 0) where n Prob ( L ? ( R ) > 0) = number of borrowers with p o s i t i v e supply at R , and E ( L ? | L | ( R ) > 0) = average supply to borrowers with p o s i t i v e supply at R . The aggregate supply function may be expressed as (using (7)) (18) L S = n(Prob(e 2 i > 6) E ( 6 Q + BjZ + B 2 R + e 2 i | e 2 i > 6) = n ( 1 - F (e/o 2)) (B Q + BjZ + B 2 R ) + n a 2 f ( 9 / a 2 ) 83 with 0 - -(B Q + BjZ + B 2 R ) > Prob(Lf > 0) = P r o b ( e 2 i > 6) = 1 - F(e/a 2) , and E ( e 9 . | e ? i > 0) = °2 f(e/a 2) 1-F (0/a 2) The i n t e r e s t r a t e e q u a t i o n i s s p e c i f i e d completely with d e f i n i t i o n s (17) and (18) substituted into (16). 5.4 Equilibrium and Disequilibrium Credit Rationing Equations (15) and (16) form a system which i s s u i t a b l e f o r the e s t i m a t i o n of both e q u i l i b r i u m and d i s e q u i l i b r i u m c r e d i t rationing. The estimate of the aggregate loan equation (15) gives a " r e a l i z e d loan flow function" L^(R) which traces the volume of new loans a c t u a l l y granted at a l t e r n a t i v e rates of in t e r e s t . Since l/(R) i s derived from the i n d i v i d u a l short-side r u l e , i t has the property L f(R) < min(L d(R), L S(R)), as shown i n 5 d <? F i g u r e 6. The estimated n o t i o n a l aggregate f l o w s L and L can be constructed using d e f i n i t i o n s (17) and (18) and parameter estimates. Credit r a tioning during period t, defined as t o t a l u n s a t i s f i e d loan demand at the c u r r e n t i n t e r e s t r a t e , i s equal to the d i f f e r e n c e between aggregate demand and the quantity of new loans granted during period t as determined by L^(R): (19) Total rationing = L d ( R t ) - L f ( R t ) . This quantity can be c a l c u l a t e d for each sample period by evaluating the estimated aggregate demand f u n c t i o n (17) and the l o a n f l o w r e l a t i o n s h i p (13) at current period data values. 5. In Chapter 4 i t i s shown that the observed aggregate loan quantity i s l e s s than both aggregate demand and aggregate supply when i n d i v i d u a l loan sizes are determined by the r u l e = min(L i,L|). 84 FIGURE 6: Aggregate Credit Rationing With 1^  = min(L^,L|) Interest Rate b c d Aggregate Loan Flow 85 The t o t a l rationing f i g u r e from (19) can be decomposed into separate e q u i l i b r i u m and d i s e q u i l i b r i u m components. The volume of e q u i l i b r i u m r a t i o n i n g i s g i v e n by the d i f f e r e n c e between aggregate demand and the r e a l i z e d l o a n f l o w f u n c t i o n , where these f u n c t i o n s are e v a l u a t e d at the equilibrium i n t e r e s t rate implied by i n t e r e s t rate equation (16) at ARt=0. (20) Equilibrium Rationing = L d ( R t ) - L f ( R t ) with R t = equilibrium i n t e r e s t rate i n period t . F i g u r e 6 i l l u s t r a t e s a s i t u a t i o n where the c u r r e n t i n t e r e s t r a t e i s below e q u i l i b r i u m and t o t a l r a t i o n i n g (d-a) i s g r e a t e r than e q u i l i b r i u m r a t i o n i n g (c-b). The d i f f e r e n c e between these q u a n t i t i e s i s due to p o s i t i v e d i s e q u i l i b r i u m r a t i o n i n g (b-a) + (d-c) at the d i s e q u i l i b r i u m i n t e r e s t rate Rt < R£. 5.5 Empirical Results The pr e c e d i n g model was a p p l i e d to the b u s i n e s s l o a n s market of Canadian chartered banks using q u a r t e r l y data for the period 1968 - 1979. This c h o i c e of study p e r i o d between the 1967 and 1980 Bank Act r e v i s i o n s ensures that lending behavior i s examined under a given set of regulatory c o n s t r a i n t s . E x p l a n a t o r y v a r i a b l e s i n demand and supply functions were s i m i l a r to those included i n previous empirical studies of d i s e q u i l i b r i u m rationing by Laffont-Garcia and other authors. The s p e c i f i c a t i o n of l o a n demand r e f l e c t s the f a c t t h a t f i r m s have a l t e r n a t i v e means of f i n a n c i n g desired' investment expenditures. In a d d i t i o n to s e c u r i n g a bank l o a n , f i r m s might use r e t a i n e d p r o f i t s or obtain financing from non-bank sources. The demand for bank loans should be i n v e r s e l y r e l a t e d to the own i n t e r e s t rate and p o s i t i v e l y r e l a t e d to the rate on non-bank loans. These s u b s t i t u t i o n e f f e c t s are summarized by the 86 d i f f e r e n t i a l between the chartered banks' prime i n t e r e s t rate on business loans and the 90-day rate for prime corporate paper. An increase i n this i n t e r e s t d i f f e r e n t i a l should reduce l o a n demand by r a i s i n g the r e l a t i v e cost of bank l o a n s . The l e v e l of r e t a i n e d earnings from the pr e c e d i n g period, which measures the a b i l i t y of firms to substitute i n t e r n a l sources of financing for loans, i s a l s o included as an explanatory v a r i a b l e with an expected negative sign. A firm's d e s i r e d investment spending, and t h e r e f o r e d e s i r e d l o a n demand, w i l l depend on current economic conditions and anticipated future conditions. The lagged index of r e a l domestic product i s used as a proxy for these determinants of loan demand and i s anticipated to have a p o s i t i v e e f f e c t on demand. The l e v e l of predetermined l o a n s i s used as another s c a l e factor i n flow demand. It i s expected that current period demand for new loans would d e c l i n e i f the stock of outstanding loans c a r r i e d into the period increases. The banks' desired supply of business loans should increase with the p r o f i t a b i l i t y of these loans. This p r o f i t a b i l i t y , determined by the loan rate r e l a t i v e to the banks' cost of funds, i s measured by the d i f f e r e n t i a l between the prime rate on business loans and the in t e r e s t rate on chartered bank savings deposits. The s c a l e of desired loan supply i s a l s o r e l a t e d to the l e v e l s of bank deposits and predetermined loans. As deposits increase some p r o p o r t i o n of t h i s expansion i n bank funds i s a l l o c a t e d towards business loans. Conversely, an increase i n outstanding predetermined loans should have a negative effect on current period flow supply by chartered banks. An important feature of the model i s that the estimating equations are 6. The corporate paper rate i s taken as a representative a l t e r n a t i v e rate for money market sources of financing. C h a r a c t e r i s t i c s of corporate paper are described i n Binhammer (1982), p. 147 and Shearer, Chant and Bond (1984), p. 57. 87 derived from e x p l i c i t i n d i v i d u a l borrower demand and supply functions given by (2) and (3). These i n d i v i d u a l functions express an explanatory v a r i a b l e as a combination of a " p o p u l a t i o n mean" v a l u e and a b o r r o w e r - s p e c i f i c d e v i a t i o n from t h i s mean. The population mean of an explanatory v a r i a b l e such as retained p r o f i t s i s defined as t o t a l retained p r o f i t s of a l l firms d i v i d e d by the t o t a l number of f i r m s n.^ V a r i a b l e s i n the v e c t o r s of population means X and Z i n demand and supply functions are de f l a t e d by n wherever necessary to achieve the appropriate s c a l i n g . The e m p i r i c a l model suggests that the average i n t e r e s t r a t e on a l l l o a n s R i s the i d e a l i n t e r e s t r a t e v a r i a b l e f o r demand and s u p p l y equations. Data a v a i l a b i l i t y r e q u i r e s that the prime r a t e of i n t e r e s t , r e p r e s e n t i n g the minimum r a t e charged on loans to low r i s k borrowers, be used i n s t e a d of R. De s p i t e t h i s data c o n s t r a i n t the prime i n t e r e s t r a t e v a r i a b l e s h o u l d be s a t i s f a c t o r y s i n c e changes i n the g e n e r a l l e v e l s of Q rates are related c l o s e l y to fl u c t u a t i o n s i n the prime rate. Other f a c t o r s minimize p o t e n t i a l problems with the prime r a t e . The su p p l y p r i c e v a r i a b l e used to measure net p r o f i t a b i l i t y of lo a n s i s the d i f f e r e n t i a l (R - RD ) between the prime r a t e R and d e p o s i t r a t e RD. Although use of the prime r a t e i n p l a c e of R tends to u n d e r s t a t e the lender's revenue, t h i s deficiency causing (R - RD) to understate net lender p r o f i t a b i l i t y i s at l e a s t p a r t i a l l y a l l e v i a t e d since administrative loan 7. The t o t a l number of f i r m s r e p r e s e n t s the t o t a l number of p o t e n t i a l borrowers i n a business loans model. Data for the number of firms are c o n s t r u c t e d from t a x a t i o n s t a t i s t i c s f o r i n c o r p o r a t e d and unincorporated businesses. 8. Data are a v a i l a b l e for the average i n t e r e s t rate on new demand loans f o r part of the sample p e r i o d (1968 IV to 1979 IV). The f o l l o w i n g r e g r e s s i o n r e s u l t s i mply that movements i n the prime r a t e (R) are associated with v i r t u a l l y i d e n t i c a l changes i n the average rate on new demand loans (AD). Numbers i n parentheses are t - s t a t i s t i c s . AD = .467 + .985 R R 2 = .964 (1.77) (33.9) 88 costs are excluded from the measure (R - RD). T a b l e 3 pr e s e n t s r e s u l t s from maximum l i k e l i h o o d e s t i m a t i o n of equations (15) and (16) under the assumption that the error terms e ^  and e R have zero means and are s e r i a l l y u n c o r r e l a t e d . ^ Most c o e f f i c i e n t s are s t a t i s t i c a l l y s i g n i f i c a n t at the 1% l e v e l and o n l y the constant term i n su p p l y i s not s i g n i f i c a n t at the 5% l e v e l . 1 1 The expected s i g n s are obtained except f o r the r e t a i n e d earnings and index of r e a l domestic product v a r i a b l e s i n f l o w demand. The p o s i t i v e c o e f f i c i e n t on r e t a i n e d earnings, a l t h o u g h unexpected, i s a l s o present i n p r e v i o u s r e s u l t s of Laffont-Garcia for Canadian business loans. One major r e s u l t , which w i l l be pertinent to l a t e r discussions of rationing, i s that flow supply i s much more s e n s i t i v e than flow demand to changes i n the rel e v a n t i n t e r e s t rate v a r i a b l e . An i n d i c a t i o n of the r e s p o n s i v e n e s s of the l o a n i n t e r e s t r a t e to demand pressures can be c a l c u l a t e d from the in t e r e s t rate equation. If the f i n a l year of the sample p e r i o d i s c o n s i d e r e d , an i n c r e a s e i n q u a r t e r l y notional excess demand of approximately 625 m i l l i o n d o l l a r s (nominal) would induce a one percentage p o i n t i n c r e a s e i n the l o a n r a t e . T h i s f i g u r e of 625 m i l l i o n would represent about 8.4% of average estimated r e a l i z e d flows for 1979. The nature and s i g n i f i c a n c e of business loans rationing i n Canada can be examined using the estimates of Table 3. 9. On the demand side, r e s u l t s reported below i n d i c a t e low s e n s i t i v i t y to the i n t e r e s t r a t e v a r i a b l e . T h e r e f o r e , e v a l u a t i n g demand at prime instead of an average rate should hot a f f e c t conclusions. 10. Non-price v a r i a b l e s are measured i n hundreds of d o l l a r s and expressed i n r e a l terms by d e f l a t i n g with the GNE p r i c e d e f l a t o r (1971 = 100). Annual r a t e s of change i n the Consumer P r i c e Index are used to construct real i n t e r e s t rates for the in t e r e s t rate equation. 11. The c o e f f i c i e n t on n o t i o n a l excess demand i n the i n t e r e s t r a t e equation i s s i g n i f i c a n t at the 5% l e v e l using a on e - t a i l t e s t . 89 Table 3: Parameter Estimates From The Individual Short-Side Rule Model (numbers i n parentheses are asymptotic t - s t a t i s t i c s ) A: Loan Equation ( i ) Flow Demand Constant R E t - l / n t P 1 V n t . I P t - l / ' n t R t-CPR t a l 336.36 .088 -.882 -1.244 -9.612 256.64 (6.51) (3.19) (4.23) (3.31) (3.01) (3.70) ( i i ) Flow Supply Constant D t/n t PL t/n t Rt~RDt -175.27 .326 -1.474 93.376 142.64 (1.97) (2.96) (4.55) (3.93) (3.35) ( i i i ) Predetermined Loans S p e c i f i c a t i o n TL 2.549 (26.69) B: Interest Rate Equation Constant ( L d - L p / n t -5.193 .255 (2.07) (1.96) RE = retained earnings (real) PL = predetermined loans ( r e a l ) , defined by (14) IP t_2 = lagged index of real domestic product R = prime rate CPR = commercial paper rate D = chartered bank deposits ( r e a l ) RD = i n t e r e s t rate on chartered bank non-chequable savings deposits TL = term loans ( r e a l ) , defined i n (14) L d = aggregate flow demand, defined by (17) L s = aggregate flow supply, defined by (18) n = number of businessess 90 Issue 1: Significance of Rationing Relative to Market Size Column 1 of T a b l e 4 l i s t s the estimated q u a n t i t y of r e a l per c a p i t a r a t i o n i n g PCR f o r each q u a r t e r . These q u a n t i t i e s suggest that the general l e v e l of rationing i s e m p i r i c a l l y s i g n i f i c a n t r e l a t i v e to various measures of market size. For example, t o t a l rationing would have amounted to 9.3% of the outstanding stock of business loans i f the average value of PCR (25.1) had prevailed i n the f i n a l quarter. Another indica t o r of the empirical magnitude of rationing i s the r a t i o of r a t i o n i n g to aggregate f l o w demand. This f i g u r e , which measures the p r o p o r t i o n of l o a n demand that i s not s a t i s f i e d by the banks, averages a p p r o x i m a t e l y .34 f o r the 48 q u a r t e r s i n the e s t i m a t i o n p e r i o d . These econometric r e s u l t s on u n s a t i s f i e d demand are r o u g h l y comparable to the Canadian survey r e s u l t s r e p o r t e d by Hatch, Wynant and Grant (1982) when t h e i r f i n d i n g of high i n f o r m a l r e j e c t i o n s of business l o a n requests i s considered. Issue 2: Equilibrium vs Disequilibrium Rationing S e c t i o n 5.4 d e s c r i b e d how t o t a l r a t i o n i n g i n any q u a r t e r can be separated i n t o i n d i v i d u a l estimates of e q u i l i b r i u m and d i s e q u i l i b r i u m c r e d i t rationing. One o b j e c t i v e of t h i s separation i s to provide evidence concerning the r e l a t i v e magnitudes of these two categories of rationing. S e c o n d l y , t h i s p r o c e d u r e may i s o l a t e t h e dominant c a u s e of t h e intertemporal v a r i a t i o n s i n t o t a l r a t ioning shown i n Table 4. Eq u i l i b r i u m per capita r a t i o n i n g ER, given by column 2 of Table 4, i s r e l a t i v e l y s t a b l e over time with v a l u e s ranging from 21.0 to 31.2. In contrast, d i s e q u i l i b r i u m r a t i o n i n g DR (column 3) shows greater v a r i a b i l i t y 12. Per capita rationing i s t o t a l r a t ioning (equation (19)) d i v i d e d by the number of businesses (scaled as hundreds of real d o l l a r s ) . 13. The Hatch, Wynant and Grant (1982) study i s discussed i n Chapter 1. 91 Table 4: Per Capita Rationing Estimates (hundreds of r e a l d o l l a r s ) (1) (2) (3) PCR ER DR 1968 I 31.6 30.9 .7 II 24.6 31.2 -6.6 III 33.2 29.8 3.4 IV 32.9 30.2 2.7 1969 I 17.9 28.8 -10.9 II 31.6 28.6 3.0 III 30.8 28.1 2.7 IV 36.2 28.4 7.8 1970 I 31.0 27.1 3.9 II 32.0 27.1 4.9 III 28.4 26.8 1.6 IV 23.8 27.2 -3.4 1971 I 19.1 25.3 -6.2 II 21.7 26.1 -4.4 III 23.5 24.6 -1.1 IV 22.7 23.6 -.9 1972 I 27.5 23.0 4.5 II 36.1 22.2 13.9 III 33.5 21.3 12.2 IV 33.4 21.2 12.2 1973 I 33.2 21.0 12.2 II 22.6 21.6 1.0 III 31.2 21.3 9.9 IV 17.4 22.0 -4.6 1974 I 25.9 21.6 4.3 II 33.1 22.2 10.9 III 25.8 22.0 3.8 IV 20.3 22.0 -1.7 1975 I 11.5 21.5 -10.0 II 15.7 22.1 -6.4 III 19.6 22.3 -2.7 IV 19.3 22.1 -2.8 1976 I 27.2 21.7 5.5 II 28.4 21.7 6.7 III 27.5 21.5 5.8 IV 41.0 21.5 19.5 92 1977 I 18.9 II 22.9 III 21.5 IV 20.2 21.4 -2.5 21.2 1.7 21.4 .1 21.5 -1.3 1978 I 10.3 II 23.0 III 11.7 IV 23.5 21.9 -11.6 22.5 .5 22.5 -10.8 22.5 1.0 1979 I 24.1 22.2 1.9 II 27.5 22.5 5.0 III 19.0 22.2 -3.2 IV 13.1 22.5 -9.4 NOTE: A l l quantities are expressed i n "per capita" terms by d i v i d i n g the relevant t o t a l by the number of businesses i n the period. PCR = t o t a l per capita r a t i o n i n g (from (19)) ER = per capita equilibrium rationing (from (20)) DR = PCR - ER = per capita d i s e q u i l i b r i u m rationing 93 with v a l u e s between 19.5 and -11.6. These two f i n d i n g s i n d i c a t e that v a r i a t i o n s i n t o t a l rationing can be a t t r i b u t e d p r i m a r i l y to d i s e q u i l i b r i u m r a t i o n i n g caused by i m p e r f e c t adjustment of the c u r r e n t i n t e r e s t r a t e toward i t s e q u i l i b r i u m l e v e l . A l t h o u g h e q u i l i b r i u m r a t i o n i n g i s an important feature of the business loan market ( i n every period ER exceeds the a b s o l u t e v a l u e of DR), i t i s not the major source of i n t e r t e m p o r a l fl u c t u a t i o n s i n c r e d i t rationing.*"* Further understanding of c y c l i c a l v a r i a t i o n s i s p o s s i b l e by examining the determinants of d i s e q u i l i b r i u m rationing as the loan rate diverges from i t s e q u i l i b r i u m Rj.. I t i s r e c a l l e d that with the i n d i v i d u a l short-side r u l e the q u a n t i t y of new l o a n s i s determined by a r e a l i z e d l o a n f l o w f u n c t i o n L^(R t) which i s d e r i v e d from the mix of s u p p l y and demand-determined loans. As Figure 6 i l l u s t r a t e s , the amount of d i s e q u i l i b r i u m r a t i o n i n g depends on the change i n ( L d - L^) as R t v a r i e s from R t. Interest rate changes and r a t i o n i n g are i n v e r s e l y r e l a t e d through the L d component of (L - L ) but the s e p a r a t e e f f e c t through L i s u n c e r t a i n since movements i n R t have opposite repercussions on demand and supply. If the p o s i t i v e supply effect dominates i n the neighbourhood of equilibrium 14. As described i n Chapter 3 d i s e q u i l i b r i u m rationing i s p o s i t i v e when R c i s l e s s than R t. In t h i s s i t u a t i o n the lower i n t e r e s t r a t e c r e a t e s a d d i t i o n a l excess demand r e l a t i v e to the l e v e l that would have been observed at the equilibrium i n t e r e s t rate. Hence, for these quarters a c t u a l r a t i o n i n g PCR exceeds e q u i l i b r i u m r a t i o n i n g . P e r i o d s with negative values for d i s e q u i l i b r i u m rationing have R t greater than Rt. Actual rationing i n these periods i s below the equilibrium l e v e l since some borrowers would have been r a t i o n e d at R t but are u n r a t i o n e d at the higher current i n t e r e s t rate. 15. This conclusion i s consistent with Jaffee's (1971) t h e o r e t i c a l model. "In terms of the comparative s t a t i c properties of the model, i t was shown that parameter changes that would i n i t i a l l y lead to more rationing were o f f s e t by the adjustment of the l o a n r a t e to i t s new e q u i l i b r i u m l e v e l ... e q u i l i b r i u m r a t i o n i n g w i l l g e n e r a l l y not be the source of c y c l i c a l v a r i a t i o n s i n the t o t a l amount of observed r a t i o n i n g . " Jaffee (1971), pp. 53-54. 94 the actual quantity of loans granted i s p o s i t i v e l y r e l a t e d to R t along L • In the Canadian business l o a n market the s u p p l y e f f e c t o p e r a t i n g on r e a l i z e d l o a n f l o w s i s s u f f i c i e n t l y great that most d i s e q u i l i b r i u m r a t i o n i n g i s r e l a t e d to s u p p l y - s i d e responses to c u r r e n t i n t e r e s t r a t e s r a t h e r than changes i n d e s i r e d l o a n demands. Thi s c o n c l u s i o n i s demonstrated by r e f e r r i n g to the f o l l o w i n g figures evaluated at R t and R t (with d i f f e r e n c e s i n parentheses) from the f i n a l q u a r t e r i n the study period. L d/n L s / n L f / n Rationing 68.12 66.18 55.07 13.05 69.46 49.13 46.94 22.52 (-1.34) (17.05) (8.13) (-9.47) During this quarter the i n t e r e s t rate was above equilibrium so actual r a t i o n i n g at R t was l e s s that the e q u i l i b r i u m q u a n t i t y . At the h i g h e r current rate there was some decrease i n loan demand r e l a t i v e to R t but the i n c r e a s e i n s u p p l y induced by R t > R t was c o n s i d e r a b l y g r e a t e r . Thus, through the s u p p l y e f f e c t on a c t u a l l o a n f l o w s , most of the decrease i n r a t i o n i n g at R t > R t was r e l a t e d to h i g h e r r e a l i z e d l o a n f l o w s (8.13) rather than decreases i n loan demand (-1.34). In a d d i t i o n to the p r e c e d i n g s t r u c t u r a l f a c t o r s the magnitude and d u r a t i o n of d i s e q u i l i b r i u m r a t i o n i n g depends on the c h a r a c t e r i s t i c s of i n t e r e s t rate adjustment. Table 5 provides some evidence concerning the effectiveness of i n t e r e s t rate f l e x i b i l i t y as an e q u i l i b r a t i n g mechanism. In 39 p e r i o d s the r e a l i n t e r e s t r a t e was r e q u i r e d to change by at l e a s t .25% to a c h i e v e e q u i l i b r i u m i n the c u r r e n t p e r i o d (column 1). However, a f t e r i n t e r e s t r a t e adjustments had taken p l a c e , t h e r e were o n l y 12 quarters for which the actual i n t e r e s t rates were at l e a s t .25% d i f f e r e n t 95 from the equilibrium rates implied by the model (column 2). Issue 3: Monetary P o l i c y and Intertemporal Fluctuations i n Rationing R e s u l t s from the model may be used to i d e n t i f y p e r i o d s when c r e d i t r a t i o n i n g was p a r t i c u l a r l y important or unimportant. T a b l e 4 i n d i c a t e s s u b s t a n t i a l intertemporal v a r i a t i o n i n PCR and groupings of periods with s i m i l a r l e v e l s of r a t i o n i n g . One q u e s t i o n suggested by these trends i s whether t h e r e e x i s t s a s t a b l e r e l a t i o n s h i p between f l u c t u a t i o n s i n r a t i o n i n g and the c u r r e n t d i r e c t i o n of monetary p o l i c y . For example, c e t e r i s paribus, i t might be argued that periods i d e n t i f i e d with low (high) r a t i o n i n g w o u l d be a s s o c i a t e d w i t h e x p a n s i o n a r y ( t i g h t ) monetary c o n d i t i o n s . P r o p o s i t i o n 1 of C h a p t e r 3 e s t a b l i s h e s t h a t such a r e l a t i o n s h i p i s not expected for equilibrium rationing since changes i n the equilibrium i n t e r e s t rate tend to o f f s e t the immediate effect on r a t i o n i n g from policy-induced s h i f t s i n loan supply schedules. Nevertheless, by i n f l u e n c i n g the quantity of d i s e q u i l i b r i u m r a t i o n i n g , monetary p o l i c y i s a p o t e n t i a l source of c y c l i c a l f l u c t u a t i o n s i n rationing. Consider a contractionary monetary p o l i c y which increases the e q u i l i b r i u m l o a n r a t e . I f the c u r r e n t period's i n t e r e s t r a t e r i s e s by a smaller amount than R t there w i l l be p o s i t i v e d i s e q u i l i b r i u m r a t i o n i n g . Therefore, g i v e n a tendency for under-adjustment of i n t e r e s t rates, there would be an inverse r e l a t i o n s h i p between t o t a l r a tioning and indicators of monetary expansion. However, i f i n s t e a d the c u r r e n t l o a n r a t e r e a c t s to the same d i s t u r b a n c e by o v e r - a d j u s t i n g above R t, the c o n t r a c t i o n a r y monetary p o l i c y would reduce c u r r e n t r a t i o n i n g by c r e a t i n g n e g a t i v e d i s e q u i l i b r i u m r a t i o n i n g . This t h e o r e t i c a l a n a l y s i s reveals that monetary p o l i c y would not have 1 6 . An a d d i t i o n a l 3 quarters had (absolute) values of (Rfc - R t) within the range .20 - .24. 96 Table 5: I n t e r e s t Rate Adjustment i n the I n d i v i d u a l Short-Side Rule Model (1) (2) I n i t i a l End-of-Period Interest Rate Disequilibrium"'" Disequilibrium^ D i f f e r e n t i a l (R t - Rt_!) (R t - R t) number of periods 0 - .24 9 36 .25 - .49 13 11 .50 - .74 . 6 1 .75 - .99 8 0 > 1.00 12 0 tInterest rate d i f f e r e n t i a l s are taken as absolute values. 97 a u n i - d i r e c t i o n a l impact on r a t i o n i n g i n a market where both under-adjustment and over-adjustment can occur i n the short-run. The Canadian business l o a n market i s such a market. Although 27 q u a r t e r s were c h a r a c t e r i z e d by under-adjustment of the c u r r e n t l o a n r a t e toward e q u i l i b r i u m , the remaining 21 q u a r t e r s experienced an o v e r - s h o o t i n g phenomenon with the i n t e r e s t rate moving past i t s equilibrium l e v e l . Since t h e r e i s e m p i r i c a l evidence of both types of adjustment b e h a v i o r , the c y c l i c a l r a t i o n i n g p a t t e r n s i n T a b l e 4 w i l l not e x h i b i t a uniform r e l a t i o n s h i p with monetary t r e n d s . I t i s noteworthy, however, that the model's p r e d i c t i o n of above-average rationing throughout much of the 1968-1970 p e r i o d agrees with J.A. G a l b r a i t h ' s p e r c e p t i o n s of t i g h t monetary conditions and excess loan demand during that era.*^ 5.6 Model Comparisons I t i s i n s t r u c t i v e to compare the above r e s u l t s from an i n d i v i d u a l s h o r t - s i d e r u l e model with those obtained from the standard aggregate short-side r u l e model represented by equations (17) and (18) of Chapter 4. Two s p e c i f i c a t i o n s of the l a t t e r model were c o n s i d e r e d s i n c e the data series for business loans contains both new and predetermined loans. In model AG-1 the dependent v a r i a b l e i s the t o t a l stock of business loans and the two-equation system of Chapter 4 i s modified to be consistent with BL t = PL t + = PL t + min ( L d , Lp with BL = t o t a l outstanding stock of business loans, PL = predetermined loans, 17 . J.A. Galbraith's d e s c r i p t i o n of t h i s period i s quoted i n Chapter 1. 98 new loan flows, aggregate flow demand, and aggregate flow supply. The two estimating equations of AG-1 have the general forms BL t = PL t + ctjX + a 2R t + ( l - y ) (a - 0 2 ) AR* + u 1 BL t = PL t + BjZ + B 2 R t + ( l - y ) a 2) AR + u 2 with PL t (defined by (14)) a function of term loans TL. One deficiency of the AG-1 s p e c i f i c a t i o n i s that parameters for PL i n flow supply and demand could not be i d e n t i f i e d so PL cannot be used as an explanatory v a r i a b l e i n the X and Z vectors. Given t h i s l i m i t a t i o n , rationing estimates from AG-1 may not be comparable with estimates i n Table 4 for the i n d i v i d u a l short-s i d e r u l e model. In response to t h i s concern the l e v e l s of PL estimated from the i n d i v i d u a l short-side r u l e model were used to construct a series f o r new l o a n f l o w s which was then us ed as the depend ent v a r i a b l e i n AG-2. Thus AG-2 i s a l s o d e r i v e d from the aggregate s h o r t - s i d e r u l e L^ = mi n ( L d , L s ) but with the l o a n f l o w s e r i e s as the dependent v a r i a b l e the c o e f f i c i e n t s on PL i n the X and Z vectors can be i d e n t i f i e d . Explanatory v a r i a b l e s are the same as those used i n the p r e v i o u s s e c t i o n . However, fo l l o w i n g the l i t e r a t u r e of the conventional models, v a r i a b l e s are never deflated by the number of p o t e n t i a l borrowers to obtain per capita s e r i e s . The t h r e e stage l e a s t squares estimates r e p o r t e d i n T a b l e 6 are s i m i l a r f or AG-1 and AG-2. A l l parameters possess the expected signs but approximately h a l f of the c o e f f i c i e n t s are not s t a t i s t i c a l l y s i g n i f i c a n t . Although c o e f f i c i e n t s on i n t e r e s t rate v a r i a b l e s are not s i g n i f i c a n t t h e i r magnitudes are consistent with the conclusion from the i n d i v i d u a l short-99 side r u l e model that supply i s more in t e r e s t s e n s i t i v e than demand. As d i s c u s s e d i n Chapter 4 the parameter u has been proposed f o r te s t i n g the hypothesis that the loan market i s i n equilibrium. Estimates of .016 and .046 from AG-1 and AG-2 imply that the c u r r e n t i n t e r e s t r a t e adjusts to eliminate 98.4% and 95.4% r e s p e c t i v e l y of the d i f f e r e n c e between l a s t period's r a t e and the c u r r e n t e q u i l i b r i u m r a t e R*. Sinc e these c o e f f i c i e n t s are not s t a t i s t i c a l l y s i g n i f i c a n t from zero the h y p o t h e s i s test suggests that i n t e r e s t rate adjustment i s s u f f i c i e n t l y rapid that the business l o a n market i n Canada reaches e q u i l i b r i u m each p e r i o d . I t i s in t e r e s t i n g to note that this conclusion i n favour of equilibrium exists despite end-of-period d i s e q u i l i b r i u m (R* - R t) being at l e a s t .25% for 29 quarters i n AG-1 and 30 quarters i n AG-2 (Table 7). The e v i d e n c e summarized i n T a b l e 7 suggests that a v a l u e of y not s i g n i f i c a n t l y d i f f e r e n t from zero may not be a s a t i s f a c t o r y t e s t of equilibrium. The cause of th i s d eficiency i s the prevalence of both under-adjustment and over-adjustment of i n t e r e s t r a t e s throughout the study p e r i o d . A comparison of a c t u a l i n t e r e s t r a t e changes with model predictions of changes that would g i v e equilibrium r e v e a l s 21(18) quarters with over-adjustment i n AG-1 (AG-2). T h i s d i v e r s e b e h a v i o r f o r i n t e r e s t rate movements might i n d i c a t e that the adjustment speed i s v a r i a b l e rather than constant and that the t r u e v a l u e of y i n a g i v e n p e r i o d can be p o s i t i v e or n e g a t i v e . I f und er-ad j us tment (y > 0) and over-adjustment ( y < 0) are both common i n i n d i v i d u a l periods, the value of y estimated for the e n t i r e sample period could tend towards zero (and thus the hypothesis test could be biased toward acceptance of equilibrium) even i f the market 18. Rearrangement of the i n t e r e s t r a t e e q u a t i o n R t = ^ R t - 1 + ( ^  ~ ^  ) R * gives AR = R t - R t_ 1 = ( l - y ) ( R * - R ^ j ) . From t h i s r e l a t i o n s h i p i t i s seen that for a period of over-adjustment, with the actual change , A R t g r e a t e r t h a n t h e change (R^ - R t _ i ) t h a t w o u l d r e s t o r e equilibrium, the parameter y i s negative. 100 Table 6: Parameter Estimates From Aggregate Short-Side Rule Models (numbers i n parentheses are t - s t a t i s t i c s ) Demand Model Constant R E t - l I P t - l Rt-CPRt PL, R 2 AG-1 -5169.46 (3.76) -.160 (.41) 137.77 (7.28) -111.55 (.71) .97 AG-2 -8859.76 (2.78) -.183 (.47) 190.00 (4.24) -171.39 (1.04) -.822 (5.06) .56 Supply Model Constant D t Rt-RDt P L T R 2 AG-1 -1547.71 (1.39) .219 (6.27) 1170.62 (1.60) .98 AG-2 -1272.15 (1.12) .207 (5.69) 1144.37 (1.57) -.601 (9.18) .73 Miscellaneous Model y TL AG-1 .016 (.16) .946 (6.01) AG-2 .046 (.46) 101 Table 7: Interest Rate Adjustment In AG-1 and AG-2 Interest Rate I n i t i a l End-of-period D i f f e r e n t i a l Disequilibrium^ Disequilibrium ( R * " Rt-1> ( R* " V number of periods AG-1 AG-2 AG-1 AG-2 0 - .24 10 4 19 18 .25 - .49 10 11 16 11 .50 - .74 9 10 8 11 .75 - .99 5 8 2 6 > 1.00 14 15 3 2 Interest rate d i f f e r e n t i a l s are taken as absolute values. 102 was f r e q u e n t l y i n d i s e q u i l i b r i u m . T h e r e f o r e , the proposed t e s t f o r equilibrium based on y may be inappropriate i n cases such as the business l o a n market where t h e r e i s e v i d e n c e of p e r i o d i c over-adjustment of the int e r e s t rate i n the current period. In these s i t u a t i o n s the estimate of >y. might be interpreted as only a measure of the average speed of adjustment. As described p r e v i o u s l y aggregate short-side r u l e models such as AG-1 and AG-2 do not a l l o w f o r e q u i l i b r i u m r a t i o n i n g but d i s e q u i l i b r i u m rationing may be calculated as DR L d - L s 0 i f L a - L s > 0 otherwise. C h a r a c t e r i s t i c s of d i s e q u i l i b r i u m rationing i n the AG models may be compared with r e s u l t s i n T a b l e 4 to show the consequences of u s i n g an i n d i v i d u a l s h o r t - s i d e r u l e to model the l o a n market. The fou r p o s s i b l e combinations of outcomes from d i f f e r e n t models and t h e i r implications f o r r e l a t i v e i n t e r e s t rates are shown below: (21)(a) DR A g > 0 D RInd> 0 Rt < Rt* R t < R t (b) DR A g = 0 D RInd< 0 Rt > V R t > R t (c) DR A g - 0 D RInd> 0 R* < R t < R t (d) DR A g > 0 D RInd< 0 Rt < R t < Rt* 103 with DR Ag' DR Ind d i s e q u i l i b r i u m rationing i n an aggregate short-side r u l e model and t h e i n d i v i d u a l s h o r t - s i d e r u l e model resp e c t i v e l y , and e q u i l i b r i u m i n t e r e s t r a t e s i n t h e a g g r e g a t e and i n d i v i d u a l short-side r u l e models r e s p e c t i v e l y . Disequilibrium r e s u l t s from the two types of models are q u a l i t a t i v e l y i n agreement i n (21)(a) and (b) where p o s i t i v e (zero) d i s e q u i l i b r i u m r a t i o n i n g i n AG corresponds w i t h p o s i t i v e ( n e g a t i v e ) v a l u e s from the i n d i v i d u a l s h o r t - s i d e model IND. In these s i t u a t i o n s the nature of d i s e q u i l i b r i u m i s i d e n t i c a l i n each model with the current rate of i n t e r e s t below (above) both e q u i l i b r i u m i n t e r e s t r a t e s R t* and Rt> P e r i o d s d e s c r i b e d i n (21)(c) or (d) demonstrate disagreement between mod e l s with r e s p e c t to the d i r e c t i o n of i n t e r e s t r a t e adjustment necessary to reach equilibrium. Table 8's comparison of d i s e q u i l i b r i u m r a t i o n i n g estimates shows much greater s i m i l a r i t y between AG-1 and AG-2 than between the i n d i v i d u a l short-side r u l e model and the aggregate models. In only three periods did AG-1 and AG-2 produce d i f f e r e n t p r e d i c t i o n s on whether or not excess demand e x i s t e d . However, when-IND i s compared with AG-2 the i n c o n s i s t e n c i e s d e s c r i b e d by (21)(c)-(d) occur i n 21 of 48 q u a r t e r s , with p o s i t i v e d i s e q u i l i b r i u m rationing occurring more frequently i n the i n d i v i d u a l short-side r u l e model (29 periods) than i n the aggregate model (22 periods). D i f f e r e n c e s p e r s i s t even i f a n a l y s i s i s r e s t r i c t e d to p e r i o d s when both models have p o s i t i v e rationing. During these quarters t o t a l rationing averages 29.6 i n IND (of which 6.3 i s d i s e q u i l i b r i u m r a t i o n i n g ) but o n l y 11.1 and 12.0 i n AG-1 and AG-2 r e s p e c t i v e l y . T h e r e f o r e , r e l a t i v e to the 104 Per Capita Disequilibrium Rationing i n Aggregate and Individual Short-Side Rule Models (hundreds of real dollars) AG-1 AG-2 IND 1968 I II III IV 0 0 0 0 0 0 0 0 .7 -6.6 3.5 2.7 1969 I II III IV 0 0 2.8 1.4 0 0 5.5 2.9 -10.9 3.0 2.7 7.8 1970 I II III IV 12.6 0 4.9 0 15.8 0 6.4 0 3.9 4.9 1.6 -3.4 1971 I II III IV 0 0 2.4 0 0 0 3.6 0 -6.2 -4.4 -1.1 .9 1972 I II III IV 3.8 0 6.4 0 2.3 0 0 0 4.5 13.9 12.2 12.2 1973 I II III IV 10. 0 16. 0 3. 0 12. 0 12.2 1.0 9.9 -4.6 1974 I II III IV 17.5 12.1 27.9 1.9 18.9 11.2 33.7 4.2 4.3 10.9 3.8 -1.7 1975 I II III IV 1,7 0 11.4 0 4.5 0 14.5 0 •10.0 -6.4 -2.7 -2.8 1976 I II 10.0 0 11.6 0 5.5 6.7 105 I l l 19.5 23.9 5.8 IV 14.0 14.5 19.5 1977 I 5.4 7.5 -2.5 II 0 0 1.7 III 9.8 11.7 .1 IV 0 0 -1.3 1978 1 0 0 -11.6 II 0 0 .5 III 0 3.7 -10.8 IV 0 0 1.0 1979 I 2.6 6.6 1.9 II 0 0 5.0 III 0 .8 -3.9 IV 0 0 -9.4 106 i n d i v i d u a l s h o r t - s i d e r u l e model, aggregate models u n d e r s t a t e t o t a l r a t i o n i n g and o v e r s t a t e d i s e q u i l i b r i u m r a t i o n i n g . The g r e a t e r d i s e q u i l i b r i u m rationing i n aggregate models i s explained by end-of-period 19 i n t e r e s t rates tending to be farther from equilibrium i n AG than i n IND. The new model presented and estimated i n t h i s Chapter has proven to be u s e f u l i n s t u d y i n g business loans r a t i o n i n g by Canadian chartered banks. Two of the major r e s u l t s are that e q u i l i b r i u m r a t i o n i n g i s s i g n i f i c a n t r e l a t i v e to v a r i o u s measures of market s i z e but that d i s e q u i l i b r i u m rationing i s the primary cause of c y c l i c a l f l u c t u a t i o n s i n t o t a l rationing of business loans. A p p l i c a t i o n of the i n d i v i d u a l short-side r u l e model to other f i n a n c i a l markets and other countries would demonstrate whether these conclusions are r e p l i c a t e d i n a l t e r n a t i v e contexts. 19''. The average a b s o l u t e v a l u e of (R* - R t) i s .39 i n AG-1 and .44 i n AG-2. The average a b s o l u t e v a l u e of (R t - R t) i s .18 i n IND. G reater e f f i c i e n c y of i n t e r e s t rate adjustment i n IND i s a l s o evident from a comparison of Tables 5 and 7. 107 Chapter 6 C O N C L U S I O N S It i s apparent from survey evidence and casual empiricism that loan i n t e r e s t r a t e s may not a d j u s t to l e v e l s at which l e n d e r s are w i l l i n g to supply a l l quantities demanded by p o t e n t i a l borrowers. With t h i s r i g i d i t y i n i n t e r e s t rate movements some non-price c r i t e r i a are used to determine which borrowers are rationed with u n s a t i s f i e d demands at current i n t e r e s t rates. Recent t h e o r e t i c a l explanations of c r e d i t r a t i o n i n g , f o c u s i n g on l o a n s i z e d e t e r m i n a t i o n from a m i c r o e c o n o m i c p e r s p e c t i v e , have demonstrated that r a t i o n i n g i s o p t i m a l b e h a v i o r under a v a r i e t y of conditions regarding the lender's a b i l i t i e s to screen i n d i v i d u a l d e f a u l t r i s k s and d i f f e r e n t i a t e i n t e r e s t rates charged to d i f f e r e n t borrowers. In a d d i t i o n to an emphasis on micro a n a l y s i s the t h e o r e t i c a l l i t e r a t u r e i s noted for i t s recognition that r a t i o n i n g may develop from two sources. E q u i l i b r i u m c r e d i t r a t i o n i n g , defined as u n s a t i s f i e d demand at the e q u i l i b r i u m rate of i n t e r e s t , i s encountered by some borrowers i f there are c o n s t r a i n t s on the degree of i n t e r e s t r a t e d i f f e r e n t i a t i o n among borrowers. In models where lenders can evaluate i n d i v i d u a l d e f a u l t r i s k s constrained d i f f e r e n t i a t i o n may r e s u l t from o l i g o p o l i s t i c p r i c i n g behavior or attempts by lenders to avoid unsatisfactory bargaining outcomes given i m p e r f e c t i n f o r m a t i o n on i n d i v i d u a l demand c o n d i t i o n s . H i g h e r - r i s k borrowers are most l i k e l y to be rationed i n these models. A second source of r a t i o n i n g e x i s t s i f there i s non-instantaneous movement of i n t e r e s t rates to new equ i l i b r i u m values a f t e r disturbances. Excess demands from t h i s source are referred to as d i s e q u i l i b r i u m r a t i o n i n g s i n c e they are s h o r t - r u n consequences of inc o m p l e t e i n t e r e s t r a t e adjustments. D i s e q u i l i b r i u m r a t i o n i n g i s p o s i t i v e and t o t a l r a t i o n i n g 108 1 exceeds the equ i l i b r i u m quantity when the p r e v a i l i n g i n t e r e s t rate R t i s below equ i l i b r i u m R t since lenders are u n w i l l i n g to s a t i s f y a l l increases i n l o a n demand a r i s i n g from R t being below i t s e q u i l i b r i u m l e v e l . C o n v e r s e l y , d i s e q u i l i b r i u m r a t i o n i n g i s n e g a t i v e w i t h R t > Rfc s i n c e the high current i n t e r e s t rate raises desired supply and decreases demand, with the r e s u l t t h a t some borrowers are not r a t i o n e d at R t but would have experienced excess demand at R^ . E m p i r i c a l study of c r e d i t r a t i o n i n g i s made d i f f i c u l t by the complications i t imposes on econometric modelling. T y p i c a l l y a market i s modelled by assuming i t i s i n market-clearing e q u i l i b r i u m with the observed quantity always corresponding to the i n t e r s e c t i o n of aggregate supply and demand. Such an assumption obviously does not a l l o w r a t i o n i n g to exist. Previous empirical studies of c r e d i t r a t i o n i n g respond to t h i s problem by u t i l i z i n g an aggregate short-side r u l e which assumes quantity observed i s the minimum of aggregate s u p p l y and demand. Although t h i s methodology introduces the p o s s i b i l i t y of ra t i o n i n g the r e s u l t i n g model i s subject to serious c r i t i c i s m for ignoring c h a r a c t e r i s t i c s of loan markets prominent i n the t h e o r e t i c a l l i t e r a t u r e . Loan equations constructed from the aggregate s h o r t - s i d e r u l e are not d e r i v e d from micro foundations whereas theory emphasizes loan determination at the i n d i v i d u a l borrower l e v e l . Another de p a r t u r e from t h e o r e t i c a l a n a l y s i s i s a t o t a l n e g l e c t of e q u i l i b r i u m rationing and a r e s t r i c t i o n that a l l excess demand must be d i s e q u i l i b r i u m rationing. B r i e f l y stated, e x i s t i n g empirical models f a i l to incorporate factors b e l i e v e d to d i s t i n g u i s h the loan market from many other markets, and the i n c o n s i s t e n c i e s w i t h theory suggest that e s t imates of r a t i o n i n g from these models may be u n r e l i a b l e . Unlike t r a d i t i o n a l methods the empirical model developed i n t h i s study d e r i v e s aggregate equations from an e x p l i c i t micro e x p l a n a t i o n of l o a n 109 determination. Loan sizes are determined by an i n d i v i d u a l short-side r u l e under which an i n d i v i d u a l ' s loan i s the minimum of borrower-specific loan supply and demand functions. Since the i n d i v i d u a l s h o r t - s i d e p r i n c i p l e allows a mix of supply and demand-determined loans during any given period the aggregate loan quantity i s l e s s than both aggregate supply and demand. In contrast to the standard aggregate short-side r u l e model points along the market's n o t i o n a l demand and s u p p l y f u n c t i o n s are never observed. Nevertheless, aggregation over a l l borrowers using the i n d i v i d u a l short-s i d e r u l e does y i e l d equations s u i t a b l e f o r e s t i m a t i o n purposes. An a t t r a c t i v e f e a t u r e of the new model i s that i t can be used to o b t a i n the f i r s t estimates of e q u i l i b r i u m c r e d i t rationing. This allowance for both equ i l i b r i u m and d i s e q u i l i b r i u m rationing, together with the micro emphasis, means that the proposed e m p i r i c a l model p r o v i d e s g r e a t e r c o n s i s t e n c y between t h e o r e t i c a l and applied work than has been possible previously. The new empirical model was applied to the market for business loans from Canadian banks using q u a r t e r l y data from 1968 to 1979. S e l e c t i o n of t h i s study p e r i o d corresponds with a time h o r i z o n between Bank Act r e v i s i o n s and thus ensures uniform r e g u l a t o r y c o n s t r a i n t s on l e n d i n g b e h a v i o r . R e s u l t s i n d i c a t e t h a t b u siness l o a n s r a t i o n i n g occurs i n s i g n i f i c a n t q u a n t i t i e s i n Canada as the average r a t i o of q u a r t e r l y r a t i o n i n g to aggregate f l o w demand i s a p p r o x i m a t e l y o n e - t h i r d . The estimates of e q u i l i b r i u m r a t i o n i n g r e v e a l that t h i s category, ignored i n conventional empirical models, i s important s i n c e e q u i l i b r i u m r a t i o n i n g exceeds the (absolute) value of d i s e q u i l i b r i u m r a t i o n i n g i n each quarter. However, the evidence indicates that intertemporal f l u c t u a t i o n s i n t o t a l r a t i o n i n g are caused p r i m a r i l y by v a r i a t i o n s i n d i s e q u i l i b r i u m rationing. With low i n t e r e s t s e n s i t i v i t y of loan demand but high i n t e r e s t s e n s i t i v i t y of s u p p l y these v a r i a t i o n s i n r a t i o n i n g are r e l a t e d l a r g e l y to s u p p l y 110 responses of lenders as current i n t e r e s t rates diverge from equilibrium. A comparison of r e s u l t s from the new approach wi t h those from the t r a d i t i o n a l model shows that c o n c l u s i o n s on the importance of c r e d i t r a t i o n i n g do depend on the s h o r t - s i d e r u l e used to r e p r e s e n t q u a n t i t y determination. One d i f f e r e n c e i s that r a t i o n i n g e x i s t s every period with the i n d i v i d u a l short-side r u l e . For some periods there i s no r a t i o n i n g i n the aggregate model which i s c o n s i s t e n t with micro t h e o r i e s of l o a n determination only i f each borrower has loan demand s a t i s f i e d completely. When p e r i o d s w i t h p o s i t i v e r a t i o n i n g i n both approaches are c o n s i d e r e d rationing estimates i n the aggregate model average about 40% of l e v e l s i n the i n d i v i d u a l short-side r u l e model. Much of the d i f f e r e n c e corresponds to the s i z e of e q u i l i b r i u m r a t i o n i n g i n the new model. D i s e q u i l i b r i u m r a t i o n i n g i s a c t u a l l y l a r g e r i n the t r a d i t i o n a l method where the gap between current and equilibrium i n t e r e s t rates tends to be greater. Future r e s e a r c h may be extended i n s e v e r a l d i r e c t i o n s u s i n g the framework presented i n t h i s study. One d i r e c t i o n i s to examine the g e n e r a l i t y of r e s u l t s obtained f o r b u s i n e s s loans r a t i o n i n g by Canadian banks. Applications of the i n d i v i d u a l short-side model to the other loan markets and other countries would demonstrate whether conclusions d i f f e r i n other s e t t i n g s . The model a l s o c o u l d be used to a n a l y z e the e f f e c t s on a g i v e n l o a n market of changes i n banking r e g u l a t i o n s . In Canada the 1980 Bank Act r e v i s i o n r e l a x e d p r e v i o u s l i m i t a t i o n s on domestic l e n d i n g a c t i v i t y of foreign-owned banks. Although c o n s i d e r a b l e r e s t r i c t i o n s remain, n o t a b l y that t o t a l Canadian assets of a l l foreign banks cannot exceed 8% of t o t a l domestic a s s e t s of a l l banks, the 1980 r e v i s i o n g i v e s some i n c r e a s e i n c o m p e t i t i o n i n c e r t a i n areas i n c l u d i n g business l o a n s . I n f o r m a t i o n on l e v e l s of r a t i o n i n g under 1980 Bank Act r e g u l a t i o n s would i n d i c a t e 1 11 p o t e n t i a l e f f e c t s of further easing of r e s t r i c t i o n s on f o r e i g n - c o n t r o l l e d banks. Another area f o r r e s e a r c h would be to use r a t i o n i n g e s t i mates from Chapter 5 to examine the impact of r a t i o n i n g on r e a l e x penditures by businesses. Firms that do not receive desired loan sizes from chartered banks may face financing constraints that prevent some intended investment spending from being undertaken. Whether r a t i o n i n g by banks imposes these l i q u i d i t y c o n s t r a i n t s depends on the a b i l i t y of r a t i o n e d f i r m s to substitute non-bank sources of financing for bank loan financing. Previous e m p i r i c a l c o n s i d e r a t i o n of r e a l e f f e c t s has been impeded by problems i n obtaining s a t i s f a c t o r y measures of rationing. Given the advantages of the i n d i v i d u a l short-side model described e a r l i e r i t s estimates of rat i o n i n g should benefit a n a l y s i s of i n t e r - r e l a t i o n s h i p s between f i n a n c i a l and r e a l sectors. The p o t e n t i a l extensions l i s t e d above show that the i n d i v i d u a l short-s i d e r u l e model can make u s e f u l c o n t r i b u t i o n s to the study of numerous issues d e a l i n g with loan markets. 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Rimbara, Y. and A. Santomero, "A Study of C r e d i t R a t i o n i n g i n Japan", International Economic Review, 17, October 1976, pp. 567-580. Roosa, R., " I n t e r e s t Rates and the C e n t r a l Bank", i n Money, Trade and  Economic Growth, pp. 270-295, New York: MacMillan, 1951. Scott, I., "The A v a i l a b i l i t y Doctrine: Theoretical Underpinnings", Review  of Economic Studies, 25, October 1957a, pp. 41-48. Sc o t t , I., "The A v a i l a b i l i t y D o c t r i n e : Development and I m p l i c a t i o n s " , Canadian J o u r n a l of Economics and P o l i t i c a l S cience, 23, November 1957b, pp. 532-539. Sealey, C, "Credit Rationing i n the Commercial Loan Market: Estimates of a S t r u c t u r a l Model Under C o n d i t i o n s of D i s e q u i l i b r i u m " , J o u r n a l of  Finance, 34, June 1979, pp. 689-702. Sears, J., I n s t i t u t i o n a l F i n a n c i n g of Smal1 Business i n Nova S c o t i a , Toronto: University of Toronto Press, 1972. Shearer, R., J. Chant and D. Bond, The Economics of the Canadian F i n a n c i a l  System, 2nd e d i t i o n , Scarborough: Pr e n t i c e - H a l l , 1984. S t i g l i t z , J. and A. Weiss, " C r e d i t R a t i o n i n g i n Markets w i t h Imperfect Information", American Economic Review, 71, June 1981, pp. 393-410. Tucker, D., " C r e d i t R a t i o n i n g , I n t e r e s t Rate Lags, and Monetary P o l i c y Speed", Quarterly Journal of Economics, 82, February 1968, pp. 54-84. Wilson, W., "The Nature of Equilibrium i n Markets with Adverse Selection", B e l l Journal of Economics, 11, Spring 1980, pp. 108-130. 1 14 APPENDIX 1: Properties of the Loan Offer Curve The concept of a loan o f f e r curve was used i n Chapter 3's analysis of c r e d i t r a t i o n i n g . The p r o p e r t i e s of t h i s c urve, summarized by (12)(a) -(12)(e) i n section 3.1, are now presented i n greater d e t a i l . The o f f e r curve i s defined by (11) SL, = R i " 6Ci 6L, = R£(1-G[B]) + q yg(y)dy - 1 = 0 . v After i n t e g r a t i o n parts (11) may be denoted as (11)' Sps «L, R ± (1-G[g]) + q (BG[B] - G[y]dy) - I = R± - I + (qB -R i)G[g] - q G[y]dy = 0 qL with (q B- R ±) = R ± ( p ( L . + E ) - 1) <' 0, and q = = p+p'A >_ 0. Inspection of (11)' indicates that <5 P|/6 L^ < 0 at a l l loan s i z e s for ^  < I. Therefore, (12)(a) L| = 0 for R± < I. 1 1 5 For = I, (qB -R i)G[3 ] = q G[y]dy a l o n g the o f f e r curve a c c o r d i n g v to (11)'. Since G[y] >^  0 and (qB -R i) < 0, (11)' i s s a t i s f i e d o n l y i f GLB] = q G[y]dy = 0 which from (5) requires B< v along the o f f e r curve at v R = I. Manipulation of g= R . L . l l p(L.+E) -< v implies vpE (12)(b) 0 < Lf <'JZ^ f o r R. = I and I-v p > 0. The requirement that the firm's c r i t i c a l no-default rate of return B can not exceed the minimum rate of return v s i g n i f i e s that d e f a u l t r i s k i s zero along the h o r i z o n t a l section of a loan o f f e r curve. The curvature of the o f f e r curve at R^  > I i s found by d i f f e r e n t i a t i n g (11). <s L; 6R. :6 2P?/6L i6R i 6 2 p f / S L i 2 6B (12)(c) = - (1-G[B] + (qB " V ^ R T g(B) ) with SB 6B_ 6L, (qB - g(B) + q' i yg(y)dy 6Ri pA R i ( p A - qL i) ( P A ) 2 > 0 , and q' 1 0. 116 The denominator of (12)(c) i s negative so the sign of SLf/iSR^ i s i d e n t i c a l to the s i g n of ( 1 - G [ B ] + ( q B - R , ) — g ( B ) ). Since 0 < G[ B ] < 1 the o f f e r 6R^ c u r v e has a p o s i t i v e s l o p e u n l e s s G [ B ] and g ( B ) have s i g n i f i c a n t l y l a r g e values. The e f f e c t on l o a n s u p p l y of a change i n the o p p o r t u n i t y cost I i s determined by d i f f e r e n t i a t i n g (11). 6 L i 6 2P?/6L i6l 61 6 2P|/6L 2 < 0 ( q B - V l i r s ( e ) + q ' y g ( y ) d y Property (12)(e) of an o f f e r curve i s proven i n Jaffee (1971), pp. 60-62. 1 1 7 APPENDIX 2: Properties of Iso-profit Curves Propositions concerning borrower and lender i s o - p r o f i t curves used i n Chapter 3 are proven In t h i s appendix. From s e c t i o n 3 . 4 an i s o - p r o f i t curve of the lender II s has slope ( 2 2 ) - 6 R i 6L. SPf/fil^ •us 6 p f / 6 R i = - (R ±(1-G[B]) + q yg(y)dy - I) L ± (l-G[g]) The numerator of ( 2 2 ) i s zero at the l o a n o f f e r c u r v e L | d e f i n e d by 6P|/6L^ = 0 . Thus, the s l o p e of an i s o - p r o f i t curve of the l e n d e r i s zero as i t i n t e r s e c t s ' L|. The s i g n of & R^/5L^ at p o i n t s not a l o n g L | i s determined from. A ( l ) . A ( l ) : For a given i n t e r e s t rate factor R^ , the lender's expected p r o f i t decreases monotonically as loan si z e v aries from the o f f e r curve i n either d i r e c t i o n . Proof: The lender's o f f e r curve i s determined by (11) 6P| &C± = R ± - = 0 . 6 L ± 6 L± D i f f e r e n t i a t e (11) with respect to to obtain 118 — 4 = (qB " R . ) 6 B <5L. 1 1' SL. 1 g(B) + q' yg(y)dy. v Since (q B - R ±) < 0, SB/SI^ > 0, and q' <_ 0, S 2P|/6L? < 0 which proves A ( l ) . A ( l) i n d i c a t e s that a change i n l o a n s i z e away from the l o a n o f f e r curve must be accompanied by an i n c r e a s e i n i n t e r e s t r a t e to maintain a constant l e v e l of expected lender p r o f i t . Thus, Proposition 4: An i s o - p r o f i t curve of the lender i s p o s i t i v e l y - s l o p e d to the rig h t of the o f f e r curve Lf and negatively-sloped to the l e f t of L f . Proposition 5 w i l l now be proven. Proposition 5: Expected lender p r o f i t increases along an o f f e r curve Lf as the i n t e r e s t rate increases. Therefore, i s o - p r o f i t curves i n t e r s e c t i n g L | at s u c c e s s i v e l y h i g h e r i n t e r e s t r a t e s represent s u c c e s s i v e l y g r e a t e r l e v e l s of expected l e n d e r p r o f i t . Proof: Expected lender p r o f i t along the loan o f f e r curve to borrower i , denoted by P f [ L f ] , i s obtained by s u b s t i t u t i n g (11) into (9) and e v a l u a t i n g L ± as L f ( R ± ) . Pf[Lf] = (pA - qLf) yg(y)dy 119 The change i n expected p r o f i t as the in t e r e s t rate varies along the of f e r curve i s 6P?[L?] 1 1 .6R. 1 6L S = - L? z - r - i + ( p A - qL?)B ||- g(B) J i 6R. l V * 6R. with z = (qB - yg(y)dy Since z^Lf/SRi » - (1 - G[ B ] + (q B - R.) ^ - g((3 ) ) oR^ from (12)(c) and B = RL/PA from (6), i t can be shown that 6p|[L|]/6R i L | ( l - G[B ] ) > 0 which proves Proposition 5. An i s o - p r o f i t curve of the borrower H d has the slope 6R. l ( 2 8 ) 6L7 6P d/6L. L 1 <5Pd/SR. K yh(y)dy - R^l-HijB]) L i(l-H[B]> The numerator of (28) i s zero at the lo a n demand curve L d d e f i n e d by < 5 P d / = 0. This indicates that an i s o - p r o f i t curve of the borrower has a slope equal to zero as i t inte r s e c t s L d. The sign of 6R^/6L^ at points not along L d i s determined from A(2). A(2): For a g i v e n i n t e r e s t r a t e f a c t o r R^, the borrower's expected p r o f i t decreases m o n o t o n i c a l l y as lo a n s i z e v a r i e s from the demand curve i n eit h e r d i r e c t i o n . 120 Proof: The borrower's demand curve i s defined by K (26) SP. ] SL. (qy - R.)h(y)dy = 0 1 D i f f e r e n t i a t e (26) with respect to L- to obtain K yh(y)dy s 2P d K i = - (qg - R.) | f - h(g) + q' S L 2 1 6 L i < 0 from second order conditions which proves A(2). A(2) implies that a movement in loan s i z e away from the demand curve must be accompanied by a decrease i n i n t e r e s t r a t e to keep expected borrower p r o f i t constant. Therefore, Proposition 6: An i s o - p r o f i t curve of the borrower i s p o s i t i v e l y - s l o p e d to the l e f t of the loan demand curve L d and negatively-sloped to the r i g h t of L d . F i n a l l y , the d i r e c t i o n of increasing expected p r o f i t on the borrower's i s o - p r o f i t map i s known from Proposition 7. Proposition 7: Expected p r o f i t of borrower i increases as the i n t e r e s t rate f a l l s a l o n g L d . T h e r e f o r e , borrower i s o - p r o f i t c u rves i n t e r s e c t i n g L d at s u c c e s s i v e l y lower i n t e r e s t r a t e s represent s u c c e s s i v e l y greater l e v e l s of expected borrower p r o f i t . Proof: Expected borrower p r o f i t along the loan demand curve, denoted by 121 P d [ L d ] , i s c a l c u l a t e d by su b s t i t u t i n g (26) into (25) and eva l u a t i n g as L f ( R i ) . K p f [ L d ] = (pA - qL.) yh(y)dy The change i n expected p r o f i t as the i n t e r e s t rate v a r i e s along the demand curve i s 6P d[L d] , 6L d , 1 1 i K with w = - (qB - R i) jf- h(B) + q' yh(y)dy i Since w6L d/6R i = 1 - H[B] + (qB - R ^ S B / ^ h(B) from (27), S p f t L ? ] / ^ = - L d ( l - H[ B]) < 0 which proves Proposition 7, 122 APPENDIX 3: Cumulative Densities and Expectations ( i ) ProbCLj = > 0) = P r o b ( e u >y , e 3 ± < n) g 1 ( e 1 , e 3 ) d e 1 d e 3 ( i i ) Prob(L i = L? > 0) = P r o b ( e 2 i > 6 , e 3 i > n ) g 2 ( e 2 , e 3 ) d e 2 d e 3 where g^Ce^. e 3) and g2(e2» e 3 ) are b i v a r i a t e normal density functions, ( i i i ) E ^ w j e i i > y , e 3 ± < n ) e 1 g 1 ( e 1 , e 3 ) d e 1 d e 3 g 1 ( e 1 , e 3 ) d e 1 d e 3 ( i v ) E ( e 2 i e 2 i > 9 > r\ ) 3i OO OO e 2 g 2 ( e 2 , e 3 ) d e 2 d e 3 OO oo g 2 ( e 2 , e 3 ) d e 2 d e 3 n 8 123 APPENDIX 4: Modelling Predetermined Loans in an Aggregate Loan Equation The business loans data ser i e s contains both new loans granted during the current period and "predetermined" loans granted i n p r e v i o u s p e r i o d s and s t i l l outstanding. A v a i l a b l e information can be used to construct a loan equation which accounts for both loan categories. Predetermined l o a n s at the end of q u a r t e r t ( P L t ) can be w r i t t e n as the sum of three f i g u r e s : ( i ) t o t a l term loans ( o r i g i n a l term to maturity of twelve months or g r e a t e r ) o u t s t a n d i n g at the end of q u a r t e r t - l ( T L t _ ^ ) , minus ( i i ) l o ans i n T L t _ j which matured b e f o r e the end of q u a r t e r t ( T L m a t ) , plus ( i i i ) non-term loans ( o r i g i n a l maturity l e s s than twelve months) which were outstanding at the end of t-1 and did not mature before the end of quarter t(OL). (1) PL t = T L t _ 1 - T L m a t + OL Some information i s a v a i l a b l e f or the f i r s t term loans component of (1). However, p r i o r to 1973 o n l y term loans of $1 m i l l i o n and over were i n c l u d e d i n t h i s data s e r i e s . T h e r e f o r e , the d e f i n i t i o n f o r PL i s rewritten by expressing TL t_^ as two elements. (2) PL t = TL t_! ( > 1) + TLt_l ( < 1) - T L m a t + OL with TL t_^ ( > 1) = t o t a l value of term loans of size $1 m i l l i o n and over, 124 _ TL t_^ ( < 1) = t o t a l v a l u e of term l o a n s l e s s than $1 m i l l i o n , and (3) TLt_l = TL(._1 ( > 1) + T L ^ j ( < 1) The o n l y known part of (2) i s T L t _ j ( > 1). I t c o u l d be s p e c i f i e d that the unknown value of predetermined loans i s proportional to the known variable T L t _ 1 ( > 1). (4) PL t = p t T L t _ j ( > 1) The f a c t o r of p r o p o r t i o n a l i t y p i s determined by s e t t i n g (4) eq u a l to d e f i n i t i o n (2). (5) p t = T L t _ 1 - T L m a t + OL (using (3)) TL t _ 1 ( > 1) It i s expected that the factor p w i l l not be constant over time since o n l y term l o a n s of $1 m i l l i o n and over are i n c l u d e d i n the denominator. Over time a higher proportion of loans should enter the $1 m i l l i o n and over category (evaluated i n nominal d o l l a r s ) due to the impact of i n f l a t i o n on nominal loan s i z e s . Consequently, a downward time-trend i s expected for the v a l u e of p so that (4) s h o u l d not be used i n e s t i m a t i o n of the lo a n equation. Equation (4) can be m o d i f i e d to a v o i d t h i s s y s t e m a t i c parameter v a r i a b i l i t y . F i r s t , (5) i s a l t e r e d to g i v e an e q u i v a l e n t d e f i n i t i o n of 1„ From (3), T L t - 1 = T L t - 1 ( > 1) (1 + k f c , ) . Rearrange and s u b s t i t u t e into the denominator of (5) to obtain (b). 125 (6) p t = ( T L t _ 1 - T L m a c + OL) (1 + k ^ ) T L ~ i with (7) kt_l = T L t _ 1 (< 1) T L t - l <> 1) Thus, using (4) and (6), predetermined loans are represented by (8) PL t = <J> (1 + k ^ ) TL,..! (> 1) where ct = ( T L t _ ^ -. T L m a t + OL) i s an unknown parameter to T L t - l be estimated. The parameter <f> i s a factor of p r o p o r t i o n a l i t y between predetermined loans of p e r i o d t and t o t a l term loans o u t s t a n d i n g at the end of t-1. U n l i k e the o r i g i n a l p r o p o r t i o n a l i t y f a c t o r d e f i n e d by (5), c|> does not c o n t a i n any components dependent on the $1 m i l l i o n c r i t i c a l l o a n s i z e . T h e r e f o r e , the adjustment i n v o l v i n g (1 + k t _ j ) removes the s y s t e m a t i c parameter v a r i a t i o n d i s c u s s e d above. However, a proxy i s necessary f o r k f c - 1 = T L t _ 1 (< l ) / T L t - 1 (> 1) s i n c e complete data on TL f c_ 1 (< 1) i s u n a v a i l a b l e . The corresponding r a t i o for t o t a l (term plus non-term) loans i s c a l c u l a t e d for each period and used as a proxy. The aggregate l o a n e q u a t i o n of S e c t i o n 5.2 uses (8) to represent predetermined loans: BL t = PL t + L^ with BL f = t o t a l outstanding stock of business loans, 126 predetermined loans <(, (1 + h) T L t _ 1 ( > 1), BL ( < 1) t - 1 , and BL ( > 1) new loan flows of period 127 

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