《货币经济学》课程文献阅读：Credit Ration with Imperfect Information（Stiglitz and Weiss,1981）.pdf_大学文库

Credit rationing in Markets with Imperfect Information By JOSEPH E STIGLITZ AND ANDREW WEISS* Why is credit rationed? Perhaps the most they receive on the loan, and the riskiness of basic tenet of economics is that market equi- the loan. However, the interest rate a bank if demand should exceed suang demand; that charges may itself affect the riskiness of the librium entails supply equal rise, decreasing demand and/or increasing borrowers(the adverse selection effect); or 2) supply until demand and supply are equated affecting the actions of borrowers( the incen at the new equilibrium price. So if prices do tive effect). Both effects derive directly from their job, rationing should not exist. How- the residual imperfect information which ver, credit rationing and unemployment do present in loan markets after banks have in fact exist. They seem to imply an excess evaluated loan applications. When the price demand for loanable funds or an excess (interest rate)affects the nature of the trans supply of workers. action, it may not also clear the market One method of "explaining "these condi- The adverse selection aspect of interest tions associates them with short-or long-term rates is a consequence of different borrowers isequilibrium. In the short term they are having different probabilities of repaying viewed as temporary disequilibrium phenom- their loan. The expected return to the bank a; that is, the economy has incurred an obviously depends on the probability of re- xogenous shock, and for reasons not fully payment, so the bank would like to be able explained, there is some stickiness in the to identify borrowers who are more likely to prices of labor or capital (wages and interest repay. It is difficult to identify"good bor rates)so that there is a transitional period rowers, "and to do so requires the bank to during which rationing of jobs or credit use a variety of screening devices. The inter- urs. On the other hand, long-term un- est rate which an individual is willing to pay employment( above some“ natural rate”) may act as one such screening device: the ose credit rationing is explained by governmen- who are willing to pay high interest rates tal constraints such as usury laws or mini- may, on average, be worse risks; they are mum wage legislation willing to borrow at high interest rates be The object of this paper is to show that cause they perceive their probability of re- in equilibrium a loan market may be char- paying the loan to be low. As the interest acterized by credit rationing. Banks making rate rises, the average"riskiness"of those loans are concerned about the interest rate who borrow increases, possibly lowering the bank's profits and Bell Laboratories, Inc, respectively. We Similarly, as the interest rate and other Bell Telephone Laboratories, Inc, and Princeton ms of the contract che the behavior of to thank Bruce Greenwald, Henry Landau, the borrower is likely to change. For in- Rob Porter, and Andy Postlewaite for fruitful cor stance, raising the interest rate decreases the Science Foundation is gratefully acknowledged. An return on projects which succeed. We will version of this paper was presented at the spring how that higher interest rates induce firms 977 meetings of the Mathematics in the Social Science to undertake projects with lower probabili ties of success but higher payoffs whe Indeed, even if markets were not competitive one cessful hopolistic bank to raise In a world with perfect and costless infor the interest rate it charges on loans to the point where mation, the bank would stipulate precisely excess demand for loans was eliminated all the actions which the borrower could

Credit Rationing in Markets with Imperfect Information By JOSEPH E. STIGLITZ AND ANDREW WEISS* Why is credit rationed? Perhaps the most basic tenet of economics is that market equilibrium entails supply equalling demand; that if demand should exceed supply, prices will rise, decreasing demand and/or increasing supply until demand and supply are equated at the new equilibrium price. So if prices do their job, rationing should not exist. However, credit rationing and unemployment do in fact exist. They seem to imply an excess demand for loanable funds or an excess supply of workers. One method of "explaining" these conditions associates them with short- or long-term disequilibrium. In the short term they are viewed as temporary disequilibrium phenomena; that is, the economy has incurred an exogenous shock, and for reasons not fully explained, there is some stickiness in the prices of labor or capital (wages and interest rates) so that there is a transitional period during which rationing of jobs or credit occurs. On the other hand, long-term unemployment (above some "natural rate") or credit rationing is explained by governmental constraints such as usury laws or minimum wage legislation.' The object of this paper is to show that in equilibrium a loan market may be characterized by credit rationing. Banks making loans are concerned about the interest rate they receive on the loan, and the riskiness of the loan. However, the interest rate a bank charges may itself affect the riskiness of the pool of loans by either: 1) sorting potential borrowers (the adverse selection effect); or 2) affecting the actions of borrowers (the incentive effect). Both effects derive directly from the residual imperfect information which is present in loan markets after banks have evaluated loan applications. When the price (interest rate) affects the nature of the transaction, it may not also clear the market. The adverse selection aspect of interest rates is a consequence of different borrowers having different probabilities of repaying their loan. The expected return to the bank obviously depends on the probability of repayment, so the bank would like to be able to identify borrowers who are more likely to repay. It is difficult to identify "good borrowers," and to do so requires the bank to use a variety of screening devices. The interest rate which an individual is willing to pay may act as one such screening device: those who are willing to pay high interest rates may, on average, be worse risks; they are willing to borrow at high interest rates because they perceive their probability of repaying the loan to be low. As the interest rate rises, the average "riskiness" of those who borrow increases, possibly lowering the bank's profits. Similarly, as the interest rate and other terms of the contract change, the behavior of the borrower is likely to change. For instance, raising the interest rate decreases the return on projects which succeed. We will show that higher interest rates induce firms to undertake projects with lower probabilities of success but higher payoffs when successful. In a world with perfect and costless information, the bank would stipulate precisely all the actions which the borrower could *Bell Telephone Laboratories, Inc. and Princeton University, and Bell Laboratories, Inc., respectively. We would like to thank Bruce Greenwald, Henry Landau, Rob Porter, and Andy Postlewaite for fruitful comments and suggestions. Financial support from the National Science Foundation is gratefully acknowledged. An earlier version of this paper was presented at the spring 1977 meetings of the Mathematics in the Social Sciences Board in Squam Lake, New Hampshire. 'Indeed, even if markets were not competitive one would not expect to find rationing; profit maximization would, for instance, lead a monopolistic bank to raise the interest rate it charges on loans to the point where excess demand for loans was eliminated. 393

THEAMERICAN ECONOMIC REVIEW JUNE 1981 no competitive forces leading supply to equal ￥Pz2 demand, and credit is rationed But the interest rate is not the only term of the contract which is important. The amount of the loan and the amount of collateral or quity the bank demands of loan applicants, will also affect both the behavior of bor rowers and the distribution of borrowers in Section Ill, we show that increasing the col- INTEREST RATE lateral requirements of lenders(beyond some point)may decrease the returns to the bank, JRE L THERE EXISTS AN INTEREST RATE WI by either decreasing the average degree of XIMIZES THE EXPECTED RETURN TO THE B/ risk aversion of the pool of borrowers; or in a multiperiod model inducing individual in- vestors to undertake riskier projects Consequently, it may not be profitable to raise the interest rate or collateral require undertake (which might affect the return to ments when a bank has an excess demand the loan). However, the bank is not able to for credit; instead, banks deny loans to bor directly control all the actions of the bor- rowers who are observationally indi rower; therefore, it will formulate the terms tinguishable from those who receive loans of the loan contract in a manner designed to It is not our argument that credit rationing induce the borrower to take actions which will always characterize capital markets, but re in the interest of the bank as well rather that it may occur under not implausi- attract low-risk borrowers ble assumptions concerning borrower and For both these reasons, the expected re- lender behavior turn by the bank may increase less rapidly This paper thus provides the first theoret than the interest rate; and, beyond a point, ical justification of true credit rationing. Pre- may actually decrease, as depicted in Figure vious studies have sought to explain why The interest rate at which the expected each individual faces an upward sloping in- return to the bank is maximized we refer to terest rate schedule. The explanations offered as the "bank-optimal"rate, F* are (a) the probability of default for any Both the demand for loans and the supply particular borrower increases as the amount of funds are functions of the interest rate borrowed increases(see Stiglitz 1970, 1972 (the latter being determined by the expected Marshall Freimer and Myron Gordon; return at P). Clearly, it is conceivable that at Dwight Jaffee; George Stigler), or(b) the P* the demand for funds exceeds the supply mix of borrowers changes adversely (see of funds. Traditional analysis would argue Jaffee and Thomas Russell). In these circum- that, in the presence of an excess demand for stances we would not expect loans of differ loans, unsatisfied borrowers would offer to ent size to pay the same interest rate, any pay a higher interest rate to the bank, bid more than we would expect two borrowers, ding up the interest rate until demand equals one of whom has a reputation for prudence supply. But although supply does not equal and the other a reputation as a bad credit demand at F, it is the equilibrium interest risk, to be able to borrow at the same interest rate! The bank would not lend to an individ- rate ual who offered to pay more than f*. In the We reserve the term credit rationing for bank's judgment, such a loan is likely to be a circumstances in which either (a) among loal worse risk than the average loan at interest applicants who appear to be identical some rate F*, and the expected return to a loan an interest rate above f is actually lower our attention was than the expected return to the loans the drawn to W. Keeton's book In chapter 3 he develops bank is presently making. Hence, there are incentive argument for credit rationing

394 THE AMERICAN ECONOMIC REVIEW JUNE 1981 z m 1Z w I- / 0 a- 2 @ - w w w r INTEREST RATE FIGURE 1. THERE EXISTS AN INTEREST RATE WHICH MAXIMIZES THE EXPECTED RETURN TO THE BANK undertake (which might affect the return to the loan). However, the bank is not able to directly control all the actions of the borrower; therefore, it will formulate the terms of the loan contract in a manner designed to induce the borrower to take actions which are in the interest of the bank, as well as to attract low-risk borrowers. For both these reasons, the expected return by the bank may increase less rapidly than the interest rate; and, beyond a point, may actually decrease, as depicted in Figure 1. The interest rate at which the expected return to the bank is maximized, we refer to as the "bank-optimal" rate, Pr. Both the demand for loans and the supply of funds are functions of the interest rate (the latter being determined by the expected return at r*). Clearly, it is conceivable that at r the demand for funds exceeds the supply of funds. Traditional analysis would argue that, in the presence of an excess demand for loans, unsatisfied borrowers would offer to pay a higher interest rate to the bank, bidding up the interest rate until demand equals supply. But although supply does not equal demand at r*, it is the equilibrium interest rate! The bank would not lend to an individual who offered to pay more than r*. In the bank's judgment, such a loan is likely to be a worse risk than the average loan at interest rate P*, and the expected return to a loan at an interest rate above r* is actually lower than the expected return to the loans the bank is presently making. Hence, there are no competitive forces leading supply to equal demand, and credit is rationed. But the interest rate is not the only term of the contract which is important. The amount of the loan, and the amount of collateral or equity the bank demands of loan applicants, will also affect both the behavior of borrowers and the distribution of borrowers. In Section III, we show that increasing the collateral requirements of lenders (beyond some point) may decrease the returns to the bank, by either decreasing the average degree of risk aversion of the pool of borrowers; or in a multiperiod model inducing individual investors to undertake riskier projects. Consequently, it may not be profitable to raise the interest rate or collateral requirements when a bank has an excess demand for credit; instead, banks deny loans to borrowers who are observationally indistinguishable from those who receive loans.2 It is not our argument that credit rationing will always characterize capital markets, but rather that it may occur under not implausible assumptions concerning borrower and lender behavior. This paper thus provides the first theoretical justification of true credit rationing. Previous studies have sought to explain why each individual faces an upward sloping interest rate schedule. The explanations offered are (a) the probability of default for any particular borrower increases as the amount borrowed increases (see Stiglitz 1970, 1972; Marshall Freimer and Myron Gordon; Dwight Jaffee; George Stigler), or (b) the mix of borrowers changes adversely (see Jaffee and Thomas Russell). In these circumstances we would not expect loans of different size to pay the same interest rate, any more than we would expect two borrowers, one of whom has a reputation for prudence and the other a reputation as a bad credit risk, to be able to borrow at the same interest rate. We reserve the term credit rationing for circumstances in which either (a) among loan applicants who appear to be identical some 2After this paper was completed, our attention was drawn to W. Keeton's book. In chapter 3 he develops an incentive argument for credit rationing

VOL. 7I NO. 3 STIGLITZ AND WEISS: CREDIT RATIONING receive a loan and others do not, and the of projects; for each project 0 there is a rejected applicants would not receive a loan probability distribution of(gross)returns R even if they offered to pay a higher interest We assume for the moment that this distri- rate;or(b) there are identifiable groups of bution cannot be altered by the borrower individuals in the population who, with a Different firms have different probability given supply of credit, are unable to obtain distributions of returns. We initially assume loans at any interest rate, even though with a that the bank is able to distinguish projects larger supply of credit, they would. 3 with different mean returns so we will at In our construction of an equilibrium first confine ourselves to the decision prob- model with credit rationing, we describe a lem of a bank facing projects having the market equilibrium in which there are man same mean return however. the bank can- banks and many potential borrowers. Both not ascertain the riskiness of a project. For borrowers and banks seek to maximize prof- simplicity, we write the distribution of re- its, the former through their choice of a turns*as F(R, 0)and the density function as project, the latter through the interest rate f(R, 0), and we assume that greater 0 corre- they charge borrowers and the collateral they sponds to greater risk in the sense of mean require of borrowers(the interest rate re- preserving spreads(see Rothschild-Stiglitz) ceived by depositors is determined by the i.e., for 8,>82,if zero-profit condition). Obviously, we are not discussing a"price-taking"equilibrium. our (1)Rf(R, 1)dR equilibrium notion is competitive in tha R(R, 02)dR banks compete; one means by which they compete is by their choice of a price(interest then for y>0 rate)which maximizes their profits. The reader should notice that in the model pre (2)F(R, )dr> F(R,02)dR the demand for loanable funds equals the supply of loanable funds. However, these are If the individual borrows the amount B, and not, in general, equilibrium interest rates. If, the interst rate is f, then we say the individ at those interest rates, banks could increase ual defaults on his loan if the return R plus their profits by lowering the interest rate the collateral C is insufficient to pay back charged borrowers, they would do so. he promised amount, i. e, if Although these results are presented in the context of credit markets, we show in Section ( 3) C+R≤B(1+P) V that they are applicable to a wide class of principal-agent problems (including those describing the landlord-tenant or employer These are subjective probability distributions; the employee relationship) perceptions on the part of the bank may differ from I. Interest Rate as a Screening device Michael Rothschild and Stiglitz show that condi- tions (1)and(2)imply that project 2 has a greater arance than project 1, although the converse is not In this section we focus on the role of true. That is, the mean preserving spread criterion for interest rates as screening devices for dis- measuring risk is stronger than the increasing varianc tinguishing between good and bad risks. We criterion. They also show that(1)and(2)can be in assume that the bank has identified a group terpreted equally well as: given two projects with equal every risk averter pre I to pro definition. a firm There is another form of rationing which is might be said to be in default if R<B(I+A). Nothing subject of our 1980 paper: banks make the provision however, that if the firm defaults, the bank has first credit in later periods contingent on performance in claim on R+C. The analysis may easily be generalized earlier period; banks may then refuse to lend even when to include bankruptcy costs. However, to simplify the hese later period projects stochastically dominate earlier analysis, we usually shall these projects which are financed this section we assume that the project is the sole project

VOL. 71 NO. 3 STIGLITZ AND WEISS: CREDIT RATIONING 395 receive a loan and others do not, and the rejected applicants would not receive a loan even if they offered to pay a higher interest rate; or (b) there are identifiable groups of individuals in the population who, with a given supply of credit, are unable to obtain loans at any interest rate, even though with a larger supply of credit, they would.3 In our construction of an equilibrium model with credit rationing, we describe a market equilibrium in which there are many banks and many potential borrowers. Both borrowers and banks seek to maximize profits, the former through their choice of a project, the latter through the interest rate they charge borrowers and the collateral they require of borrowers (the interest rate received by depositors is determined by the zero-profit condition). Obviously, we are not discussing a "price-taking" equilibrium. Our equilibrium notion is competitive in that banks compete; one means by which they compete is by their choice of a price (interest rate) which maximizes their profits. The reader should notice that in the model presented below there are interest rates at which the demand for loanable funds equals the supply of loanable funds. However, these are not, in general, equilibrium interest rates. If, at those interest rates, banks could increase their profits by lowering the interest rate charged borrowers, they would do so. Although these results are presented in the context of credit markets, we show in Section V that they are applicable to a wide class of principal-agent problems (including those describing the landlord-tenant or employeremployee relationship). I. Interest Rate as a Screening Device In this section we focus on the role of interest rates as screening devices for distinguishing between good and bad risks. We assume that the bank has identified a group of projects; for each project 6 there is a probability distribution of (gross) returns R. We assume for the moment that this distribution cannot be altered by the borrower. Different firms have different probability distributions of returns. We initially assume that- the bank is able to distinguish projects with different mean returns, so we will at first confine ourselves to the decision problem of a bank facing projects having the same mean return. However, the bank cannot ascertain the riskiness of a project. For simplicity, we write the distribution of returns4 as F(R, 0) and the density function as f(R, 0), and we assume that greater 6 corresponds to greater risk in the sense of mean preserving spreads5 (see Rothschild-Stiglitz), i.e., for , >2,Jif 00 0 (1) fRf(R, 01) dR= Rf(R, 2) dR then for y O, (2) j F(R,01)dR> jF(R,02)dR If the individual borrows the amount B, and the interst rate is r, then we say the individual defaults on his loan if the return R plus the collateral C is insufficient to pay back the promised amount,6 i.e., if (3) C+R<B(I +P) 3There is another form of rationing which is the subject of our 1980 paper: banks make the provision of credit in later periods contingent on performance in earlier period; banks may then refuse to lend even when these later period projects stochastically dominate earlier projects which are financed. 4These are subjective probability distributions; the perceptions on the part of the bank may differ from those of the firm. 5Michael Rothschild and Stiglitz show that conditions (I) and (2) imply that project 2 has a greater variance than project 1, although the converse is not true. That is, the mean preserving spread criterion for measuring risk is stronger than the increasing variance criterion. They also show that (I) and (2) can be interpreted equally well as: given two projects with equal means, every risk averter prefers project I to project 2. 6This is not the only possible definition. A firm might be said to be in default if R < B(1 + P). Nothing critical depends on the precise definition. We assume, however, that if the firm defaults, the bank has first claim on R+ C. The analysis may easily be generalized to include bankruptcy costs. However, to simplify the analysis, we usually shall ignore these costs. Throughout this section we assume that the project is the sole project

THEAMERICAN ECONOMIC REVIEW JUNE /98I Thus the net return to the borrower (R, r) an be written as (4a)T(R, A)=max(R-(1+P)B;-c) The return to the bank can be written as (4b)P(R, )=min(R+C; B(1+r)) that is, the borrower must pay back either FIGURE 2a. FIRM PROFITS ARE A CONVEX FUNCTION OF THE RETI N THE PROJECT the promised amount or the maximum he can pay back(R+C) For simplicity, we shall assume that the borrower has a given amount of equity(which he cannot increase), that borrowers and lenders are risk neutral, that the supply of loanable funds available to a bank is unaf fected by the interest rate it charges bor owers, that the cost of the project is fixed, and unless the individual can borrow the difference between his equity and the cost of the project, the project will not be under FIGURE 2b. THE RETURN TO THE BANK IS A CONCAVE taken, that is, projects are not divisible. For FUNCTION OF THE RETURN ON THE PROJEC notational simplicity, we assume the amount borrowed for each project is identical,so that the distribution functions describing the The value of 6 for which expected profits number of loan applications are identical to are zero satisfies those describing the monetary value of loan (5)I(f,6)≡ would make the amount borrowed by each individual a function of the terms of the max[R-(P+1)B;-q]dF(R,6)=0 contract; the quality mix could change not only as a result of a change in the mix of applicants, but also because of a change in interest rates could cause the returns to the the relative size of applications of different bank to decrease with increasing interest rates roups.) hinged on the conjecture that as the interest We shall now prove that the interest rate rate increased e mix of applicants became acts as a screening device; more precisely we worse; or THEOREM 2: As the interest rate increases THEOREM 1: For a given interest rate the critical value of 8, below which individuals there is a critical value 6 such that a firm do not apply for loar ns. Increases borrows from the bank if and only if0>0 This follows immediately upon differenti This follows immediately upon observing ating(5) that profits are a convex function of R, as in Figure 2a. Hence expected profits increase with risk dF(R, 0) (6) 1+P)B-C aI/06 0 undertaken by the firm (individual) and that there is limited liability. The equilibrium extent of liability is derived in Section Ill For each 6, expected profits are decreased

396 THE A MERICAN ECONOMIC REVIEW JUNE 1981 Thus the net return to the borrower 7T(R, r) can be written as (4a) 7(R, r) =max(R-(1 +r)B; -C) The return to the bank can be written as (4b) p(R,fr)=min(R+C; B(1+r)) that is, the borrower must pay back either the promised amount or the maximum he can pay back (R+ C). For simplicity, we shall assume that the borrower has a given amount of equity (which he cannot increase), that borrowers and lenders are risk neutral, that the supply of loanable funds available to a bank is unaffected by the interest rate it charges borrowers, that the cost of the project is fixed, and unless the individual can borrow the difference between his equity and the cost of the project, the project will not be undertaken, that is, projects are not divisible. For notational simplicity, we assume the amount borrowed for each project is identical, so that the distribution functions describing the number of loan applications are identical to those describing the monetary value of loan applications. (In a more general model, we would make the amount borrowed by each individual a function of the terms of the contract; the quality mix could change not only as a result of a change in the mix of applicants, but also because of a change in the relative size of applications of different groups.) We shall now prove that the interest rate acts as a screening device; more precisely we establish THEOREM 1: For a given interest rate r, there is a critical value 0 such that a firm borrows from the bank if and only if 0>0. This follows immediately upon observing that profits are a convex function of R, as in Figure 2a. Hence expected profits increase with risk. (1+r)B-C / --~R -C FIGURE 2a. FIRM PROFITS ARE A CONVEX FIJNCTION OF THE RETURN ON THE PROJECT C R (1 + r) B -C FIGURE 2b. THE RETURN TO THE BANK IS A CONCAVE FUNCTION OF THE RETURN ON THE PROJECT The value of 0 for which expected profits are zero satisfies (5) r(IA) E f max[R-(r+ 1)B; -C] dF(R, ) 0 Our argument that the adverse selection of interest rates could cause the returns to the bank to decrease with increasing interest rates hinged on the conjecture that as the interest rate increased, the mix of applicants became worse; or THEOREM 2: As the interest rate increases, the critical value of 0, below which individuals do not apply for loans, increases. This follows immediately upon differentiating (5): BJ dF(R,O) (6) do I1 +rP)B- C >0 dr ari/ao For each 0, expected profits are decreased; undertaken by the firm (individual) and that there is limited liability. The equilibrium extent of liability is derived in Section III

VOL. 71 NO. 3 STIGLITZ AND WEISS: CREDIT RATIONING BOTH TYPES APPLY HIGH RISK APPLY FIGURE 4. DETERMINATION OF THE MARKET EQUILIBRIUM sive group drops out of the market, there is a FIGURE 3. OPTIMAL INTEREST RATE r discrete fall in p(where p() is the mean return to the bank from the set of applicants at hence using Theorem 1, the result is im- the interest rate F) We next show Other conditions for nonmonotonicity of () will be established later. Theorems 5 THEOREM 3: The expected return on a loan and 6 show why nonmonotonicity is so im- a bank decreasing function of the portant riskiness of the loan THEOREM 5: Whenever p() has an interior PROOF: mode, there exist supply functions of fun From(4b)we see that p(R, f)is a con- such that competitive equilibrium entails credit cave function of R. hence the result is im- rationin mediate. The concavity of p(R, F)is il- This will be the case whenever the "Wal- Theorems 2 and 3 imply that, in addition rasian equilibrium"interest rate- the one at to the usual direct effect of increases in the which demand for funds equals supply-is interest rate increasing a bank,'s return, there such that there exists a lower interest rate fc is an indirect, adverse-selection effect acting which p, the return to the bank, is higher in the opposite direction. We now show that In Figure 4 we illustrate a credit rationing this adverse-selection effect may outweigh equilibrium. because demand for funds de the direct effect pends on F, the interest rate charged by To see this most simply, assume there are banks, while the supply of funds depends on two groups; the"safe"group will borrow p, the mean return on loans, we cannot use a only at interest rates below r, the"risky" conventional demand /supply curve diagram group below r2, and r,<r2. When the inter- The demand for loans is a decreasing func est rate is raised slightly above r, the mix of tion of the interest rate charged borrowers applicants changes dramatically: all low risk this relation L is drawn in the upper right applicants withdraw. (See Figure 3. By the quadrant. The nonmonotonic relation be same argument we can establish tween the interest charged borrowers, and the expected return to the bank per dollar THEOREM 4: If there are a discrete number loaned p is drawn in the lower right quadrant of potential borrowers(or types of borrowers) In the lower left quadrant we depict the each with a different 0, p(r)will not be a relation between p and the supply of loana monotonic function of f, since as each succes- ble funds L.(We have drawn L' as if

VOL. 71 NO. 3 STIGLITZ AND WEISS: CREDITRATIONING 397 TYPES APPLY ONL / /HIG~H RISK / / ~APPLY rl ? FIGURE 3. OPTIMAL INTEREST RATE r1 hence using Theorem 1, the result is immediate. We next show: THEOREM 3: The expected return on a loan to a bank is a decreasing function of the riskiness of the loan. PROOF: From (4b) we see that p(R, r) is a concave function of R, hence the result is immediate. The concavity of p(R, r) is illustrated in Figure 2b. Theorems 2 and 3 imply that, in addition to the usual direct effect of increases in the interest rate increasing a bank's return, there is an indirect, adverse-selection effect acting in the opposite direction. We now show that this adverse-selection effect may outweigh the direct effect. To see this most simply, assume there are two groups; the "safe" group will borrow only at interest rates below r,, the "risky" group below r2, and r, <r2. When the interest rate is raised slightly above r,, the mix of applicants changes dramatically: all low risk applicants withdraw. (See Figure 3.) By the same argument we can establish THEOREM 4: If there are a discrete number of potential borrowers (or types of borrowers) each with a different 0, p(r) will not be a monotonic function of r, since as each succesL~~~~~~~~~L L X LD 0 ~ ~ irm ' '~ ~ --------- FIGURE 4. DETERMINATION OF THE MARKET EQUILIBRIUM sive group drops out of the market, there is a discrete fall in - (where p(r) is the mean return to the bank from the set of applicants at the interest rate r). Other conditions for nonmonotonicity of p(r) will be established later. Theorems 5 and 6 show why nonmonotonicity is so important: THEOREM 5: Whenever p(r) has an interior mode, there exist supply functions of funds such that competitive equilibrium entails credit rationing. This will be the case whenever the "Walrasian equilibrium" interest rate- the one at which demand for funds equals supply-is such that there exists a lower interest rate for which p, the return to the bank, is higher. In Figure 4 we illustrate a credit rationing equilibrium. Because demand for funds depends on r, the interest rate charged by banks, while the supply of funds depends on p, the mean return on loans, we cannot use a conventional demand/supply curve diagram. The demand for loans is a decreasing function of the interest rate charged borrowers; this relation LD is drawn in the upper right quadrant. The nonmonotonic relation between the interest charged borrowers, and the expected return to the bank per dollar loaned - is drawn in the lower right quadrant. In the lower left quadrant we depict the relation between - and the supply of loanable funds LS. (We have drawn LS as if it

THE AMERICAN ECONOMIC REVIEW JUNE 198 Figure 5 illustrates a p()function multiple modes. The nature of the librium for such cases is described by THEOREM 6: If the p(r)function has several modes, market equilibrium either be characterized by a single rate at or below the market-clearing level, or by two nterest rates, with an excess demand for credit at the lower one FIGURE 5. A TwO- INTEREST RATE EQUILIBRIUM PROOF Denote the lowest Walrasian equilibrium were an increasing function of p. This is not interest rate by m and denote by f the inter- necessary for our analysis. If banks are free est rate which maximizes p(r). If /rm, then loans funds may be made at two interest rates, denoted A credit rationing equilibrium exists given by r, and r2. r, is the interest rate which the relations drawn in Figure 4; the demand maximizes p(r)conditional on rsrm: r, is for loanable funds at f* exceeds the supply the lowest interest rate greater than m such of loanable funds at /* and any individual that p(r2)=p(r). From the definition of r bank increasing its interest rate beyond A* and the downward slope of the loan demand would lower its return per dollar loaned. The function, there will be an excess demand for excess demand for funds is measured by Z. loanable funds at r,(unless n,=m, in which Notice that there is an interest rate m at case there is no credit rationing). Some re- which the demand for loanable funds equals jected borrowers(with reservation interest the supply of loanable funds; however, 'm is rates greater than or equal to r2) will apply not an equilibrium interest rate. a bank could for loans at the higher interest rate. Since increase its profits by charging F* rather than there would be an excess supply of loanable r: at the lower interest rate it would attract funds at r, if no loans were made at r, and at least all the borrowers it attracted at rm an aggregate excess demand for funds if no and would make larger profits from each loans were made at r2, there exists a distribu- loan(or dollar loaned) tion of loanable funds available to borrowers Figure 4 can also be used to illustrate an at r, and r2 such that all applicants who are important comparative statics property of rejected at interest rate r, and who apply for our market equilibrium: loans at r2 will get credit at the higher inter est rate. Similarly, all the funds available at COROLLARY 1. As the supply of funds in- p(r,) will be loaned at either r, or r2.(There creases,the excess demand for funds de- is, of course, an excess demand for loanable creases, but the interest rate charged remains funds at r, since every borrower who eventu unchanged, so long as there is any credit ra- ally borrows at r2 will have first applied for tioning credit at rr There is clearly no incentive for small deviations from r, which is a local Eventually, of course, Z will be reduced to maximum of p(r).a bank lending at an rO; further increases in the supply of funds interest rate rs such that p(r3)<p(r) would hen reduce the market rate of interest not be able to obtain credit Thus, no bank

398 THE A MERICA N ECONOMIC RE VIEW JUNE 1981 I I I I I I r,, rm r2 r FIGURE 5. A TWO-INTEREST RATE EQUILIBRIUM were an increasing function of p. This is not necessary for our analysis.) If banks are free to compete for depositors, then - will be the interest rate received by depositors. In the upper right quadrant we plot LS as a function of r, through the impact of r on the return on each loan, and hence on the interest rate - banks can offer to attract loanable funds. A credit rationing equilibrium exists given the relations drawn in Figure 4; the demand for loanable funds at r* exceeds the supply of loanable funds at r* and any individual bank increasing its interest rate beyond r* would lower its return per dollar loaned. The excess demand for funds is measured by Z. Notice that there is an interest rate rm at which the demand for loanable funds equals the supply of loanable funds; however, rm is not an equilibrium interest rate. A bank could increase its profits by charging r* rather than rm: at the lower interest rate it would attract at least all the borrowers it attracted at rm and would make larger profits from each loan (or dollar loaned). Figure 4 can also be used to illustrate an important comparative statics property of our market equilibrium: COROLLARY 1. As the supply of funds increases, the excess demand for funds decreases, but the interest rate charged remains unchanged, so long as there is any credit rationing. Eventually, of course, Z will be reduced to zero; further increases in the supply of funds then reduce the market rate of interest. Figure 5 illustrates a p(r) function with multiple modes. The nature of the equilibrium for such cases is described by Theorem 6. THEOREM 6: If the -p(r) function has several modes, market equilibrium could either be characterized by a single interest rate at or below the market-clearing level, or by two interest rates, with an excess demand for credit at the lower one. PROOF: Denote the lowest Walrasian equilibrium interest rate by rm and denote by r the interest rate which maximizes p(r). If rrm, then loans may be made at two interest rates, denoted by r, and r2. r, is the interest rate which maximizes p(r) conditional on r<rm; r2 is the lowest interest rate greater than rm such that p(r2)=p(r,). From the definition of rm, and the downward slope of the loan demand function, there will be an excess demand for loanable funds at r, (unless r, =rm, in which case there is no credit rationing). Some rejected borrowers (with reservation interest rates greater than or equal to r2) will apply for loans at the higher interest rate. Since there would be an excess supply of loanable funds at r2 if no loans were made at r,, and an aggregate excess demand for funds if no loans were made at r2, there exists a distribution of loanable funds available to borrowers at r, and r2 such that all applicants who are rejected at interest rate r, and who apply for loans at r2 will get credit at the higher interest rate. Similarly, all the funds available at p(r,) will be loaned at either r, or r2. (There is, of course, an excess demand for loanable funds at r, since every borrower who eventually borrows at r2 will have first applied for credit at r,.) There is clearly no incentive for small deviations from r1, which is a local maximum of p(r). A bank lending at an interest rate r3 such that p(r3)<p(r,) would not be able to obtain credit. Thus, no bank

VOL. 71 NO. 3 STIGLITZ AND WEISS: CREDIT RATIONING would switch to a loan offer between n, and large if (g()/1-G(O)D(de/ dr)is large r2,A bank offering an interest rate r4 such that is, a small change in the nominal inter- that p(r4)>p(r,) would not be able to at- est rate induces a large change in the appli tract any borrowers since by definition r4> cant pool r. and there is no excess demand at interest rate r2 2. Two Outcome Projects Here we consider the simplest kinds of A. Alternative Sufficient Conditions fo projects(from an analytical point of view Credit Rationing those which either succeed and yield a return R, or fail and yield a return D. We normalize Theorem 4 provided a sufficient condition to let B=l. All the projects have the same for adverse selection to lead to a nonmono- unsuccessful value(which could be the value tonic p()function. In the remainder of this of the plant and equipment)while R ranges section,we investigate other circumstances between S and K(where K>S). We also under which for some levels of supply of assume that projects have been screened so funds there will be credit rationing that all projects within a loan category have he same expected yield, T, and there is no I. Continuum of projects collateral required, that is, C=0, and if P(R) riskiness 0, and p(0, r) be the expected re- a successful return of R succeeds, then with Let G(a) be the distribution of projects by represents the probability that a project turn to the bank of a loan of risk g and interest rate r. The mean return to the bank (9) P(R)R+[1-P(R)JD=T p(0,P)dG(6) In addition the bank suffers a cost of X per dollar loaned upon loans that default, (7)p(P) which could be interpreted as the difference 1-G(6 between the value of plant and equipment to the firm and the value of the plant and From Theorem 5 we know that dp()/drJ): "u-0+)-c)o(0)(0 1-G(6) P(J) g(r)dR From Theorems I and 3. the first term is negative(representing the change in the mix of applicants), while the second term (the 1-P(R)IID-Xlg(r)dR pool fixed, from raising the interest charges) Using l'Hopital's rule and(1), we can estab is positive. The first term is large, in absolute lish sufficient conditions for lim -x(ap(J) value, if there is a large difference between aJ)<0(and hence for the nonmonotonicity the mean return on loans made at interest of p) rate f and the return to the bank from the project making zero returns to the firm at interest rate F(itssafest"loan). It is also The proofs of these propositions are slightly com- plicated. Consider 1. Since P(R)=T-D/R-D, the

VOL. 71 NO. 3 STIGLITZ AND WEISS: CREDIT RATIONING the self-selection effect, and the more like is an interior bank optimal interest rate I. Interest Rate as an Incentive Mechanism (14) B(1-F(1+P)B-C) A. Sufficient Conditions Thus, if at some A. / the increase in F lowers the expected return to the borrower The second way in which the interest rate from the project with the higher probability affects the bank's expected return from a of paying back the loan by more than it loan is by changing the behavior of the bor- lowers the expected return from the project ower. The interests of the lender and the with the lower probability of the loan being borrower do not coincide. The borrower is repaid only concerned with returns on the invest- On the other hand, if the firm is indiffer ment when the firm does not go bankrupt; ent between two projects with the same mean, the lender is concerned with the actions of we know from Theorem 2 that the bank the firm only to the extent that they affect prefers to lend to the safer project. Hence the probability of bankruptcy, and the re- raising the interest rate above p could so turns in those states of nature in which the increase the riskiness of loans as to lower the firm does go bankrupt. Because of this, and expected return to the bank ecause the behavior of a borrower cannot be perfectly and costlessly monitored by the THEOREM 8: The expected return to the lender, banks will take into account the ef- bank is lowered by an increase in the interest fect of the interest rate on the behavior of rate at F if, at A, the firm is indifferent between two projects and k with distributions F(R) his section, we show that increasing the he and FK(R),j having a higher probability of rate of interest increases the relative at- bankruptcy than k, and there exists a distribu tractiveness of riskier projects, for which the tion F/(R) such that return to the bank be lower. Hence (a)F(r)represents a mean preserving raising the rate of interest may lead bor- spread of the distribution F/(R), and owers to take actions which are contrary to K(R) satisfies a first-order dominance the interests of the lender, providing another relation with F(R); i.e., F(R)>F(R) for all r than raise the interest rate when there is an excess demand for loanable funds PROOF We return to the general model presented Since has a higher probability of bank- jects, denoted by superscripts j and k. We and k, an increase in the interest rate r leads first establish and Theorem 3, the return to the bank on a THEOREM 7: If, at a given nominal interest project whose return is distributed as Fr(R) rater, a risk-neutral firm is indifferent be- is higher than on project j, and because of tween two projects, an increase in the interest ( b) the return to the bank on project k is rate results in the firm preferring the project higher than the return on a project distrib- with the higher probability of bankruptcy. uted as F(R) PROOF: B. An Example The expected return to the ith project is given by To illustrate the implications of Theorem 8. assume all firms are identical and have a (13)丌= R +P)B -O hoice of two projects, yielding, if successful returns R and R, respectively(and nothing

VOL. 71 NO. 3 STIGLITZ AND WEISS: CREDIT RATIONING 401 the self-selection effect, and the more likely is an interior bank optimal interest rate. II. Interest Rate as an Incentive Mechanism A. Sufficient Conditions The second way in which the interest rate affects the bank's expected return from a loan is by changing the behavior of the borrower. The interests of the lender and the borrower do not coincide. The borrower is only concerned with returns on the investment when the firm does not go bankrupt; the lender is concerned with the actions of the firm only to the extent that they affect the probability of bankruptcy, and the returns in those states of nature in which the firm does go bankrupt. Because of this, and because the behavior of a borrower cannot be perfectly and costlessly monitored by the lender, banks will take into account the effect of the interest rate on the behavior of borrowers. In this section, we show that increasing the rate of interest increases the relative attractiveness of riskier projects, for which the return to the bank may be lower. Hence, raising the rate of interest may lead borrowers to take actions which are contrary to the interests of the lender, providing another incentive for banks to ration credit rather than raise the interest rate when there is an excess demand for loanable funds. We return to the general model presented above, but now we assume that each firm has a choice of projects. Consider any two projects, denoted by superscripts jand k. We first establish: THEOREM 7: If, at a given nominal interest rate r, a risk-neutral firm is indifferent between two projects, an increase in the interest rate results in the firm preferring the project with the higher probability of bankruptcy. PROOF: The expected return to the ith project is given by (13 w-E axR'(I+-),_ so (14) d =-B(1-Fi((l+r')B-C)) Thus, if at some r, X} =7 k, the increase in r lowers the expected return to the borrower from the project with the higher probability of paying back the loan by more than it lowers the expected return from the project with the lower probability of the loan being repaid. On the other hand, if the firm is indifferent between two projects with the same mean, we know from Theorem 2 that the bank prefers to lend to the safer project. Hence raising the interest rate above r could so increase the riskiness of loans as to lower the expected return to the bank. THEOREM 8: The expected return to the bank is lowered by an increase in the interest rate at r if, at r, the firm is indifferent between two projects j and k with distributions Fj(R) and Fk(R), j having a higher probability of bankruptcy than k, and there exists a distribution F,(R) such that (a) Fj(R) represents a mean preserving spread of the distribution F,(R), and (b) Fk(R) satisfies a first-order dominance relation with F,(R); i.e., FI(R)>Fk(R) for all R. PROOF: Since j has a higher probability of bankruptcy than does k, from Theorem 7 and the initial indifference of borrowers between j and k, an increase in the interest rate r leads firms to prefer project j to k. Because of (a) and Theorem 3, the return to the bank on a project whose return is distributed as F,(R) is higher than on project j, and because of (b) the return to the bank on project k is higher than the return on a project distributed as F,(R). B. An Example To illustrate the implications of Theorem 8, assume all firms are identical, and have a choice of two projects, yielding, if successful, returns Ra and Rb, respectively (and nothing