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 bank398 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 func￾tion of r, through the impact of r on the return on each loan, and hence on the inter￾est 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 in￾creases, the excess demand for funds de￾creases, but the interest rate charged remains unchanged, so long as there is any credit ra￾tioning. 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 equi￾librium for such cases is described by Theo￾rem 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 inter￾est 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 re￾jected 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 distribu￾tion 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 inter￾est 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 eventu￾ally 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