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, the to compete for depositors, then p will be the analysis for Theorem 5 is unaffected by the interest rate received by depositors. In the multiplicity of modes. There will be credit upper right quadrant we plot LS as a func- rationing at interest rate A. The rationed tion of F, through the impact of f on the borrowers will not be able to obtain credit return on each loan, and hence on the inter- by offering to pay a higher interest rate est rate p banks can offer to attract loanable On the other hand, if p>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 r<rm, the analysis for Theorem 5 is unaffected by the multiplicity of modes. There will be credit rationing at interest rate r. The rationed borrowers will not be able to obtain credit by offering to pay a higher interest rate. On the other hand, if r>rm, 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