THEAMERICAN ECONOMIC REVIEW JUNE 198/ (a)if limR-kg(R)#0, oo then a sufficient the bank-optimal interest rate. High interest condition is X>K-D, or equivalently, rates may make projects with low mean re- limRkP(r)+P(r)X<O turns-the projects undertaken by risk averse if g(k)=0, g (K)#0, oo then a suffi- individuals-infeasible, but leave relatively lently, limR-xP(R)+2p(R)X<o quiva- unaffected the isky projects. The mean re turn to the bank, however is lower on the (c)if g(k)=0,g(K)=0, g(K)#0, then riskier projects than on the safe projects. In a sufficient condition is 3X>K-K-D ing example, it is systematic dif equivalently, limR_xP(R)+3P(r)X<O ferences in risk aversion which results in Condition(a)implies that if, as 1+pK, there being an optimal interest rate the probability of an increase in the interest Assume a fraction A of the population is rate being repaid is outweighed by the infinitely risk averse; each such individual dead weight loss of riskier loans, the bank undertakes the best perfectly safe project will maximize its return per dollar loaned at which is available to him. within that group an interest rate below the maximum rate at the distribution of returns is G(R)where which it can loan funds(K-1). The condi- G(K)=1. The other group is risk neutral tions for an interior bank optimal interest For simplicity we shall assume that they all rate are significantly less stringent when face the same risky project with probability g(K)=0 of success p and a return, if successful, of R">K; if not their return is zero. Letting 3. Differences in Attitudes Towards Risk R=(1+r)B the (expected) return to the loan applicants are clearly more risk bank is averse than others These differences will be flected in (R)+(1-A)p G(R)+(1-入) P) P(J)=[J-D+XIIT-D] +RB g(r)dR G(R)+(1-入) Differentiating and collecting terms Hence for R<K, the upper bound on re turns from the safe project T-d a +[J-D+XJ dIn g(r)dR (2)32am(1+)=1- g(J) Kg(r) (1-入)(1-p)入g(R)R J-D/g( R)dR+g() (1-入G(R)((1-G(R)+p(1-入) g(r)dR A sufficient condition for the existence of an interior bank optimal interest rate is again Using IHopital's rule and the assumption that g(k)+ that lim dp/aF<0 or fr limRkg(R)R>P/1-P. The greater is th K-D riskiness of the risky project(the lower is p), the more like n interior bank optimal im t- d ae =sign( K-D-x) interest rate. Similarly, the higher is the rela tive prop rtion of the risk averse individuals affected by increases in the interest rate to Conditions 2 and 3 follow in a similar manner risk neutral borrowers, the more important is400 THE A MERICAN ECONOMIC REVIEW JUNE 1981 (a) if fimR-Kg(R)=O0, so then a sufficient condition is X> K- D, or equivalently, limR-Kp(R)+p'(R)X<0 (b) if g(K) = 0, g'(K) 7 0, so then a suffi￾cient condition is 2X>K-D, or equiva￾lently, IimR,Kp(R)+2p (R)X<O (c) if g(K)=O, g'(K)=0, g"(K)57z0, then a sufficient condition is 3X>K-K-D, or equivalently, limR,Kp(R) + 3p'(R)X< 0 Condition (a) implies that if, as 1 + rP- K, the probability of an increase in the interest rate being repaid is outweighed by the deadweight loss of riskier loans, the bank will maximize its return per dollar loaned at an interest rate below the maximum rate at which it can loan funds (K- 1). The condi￾tions for an interior bank optimal interest rate are significantly less stringent when g(K) = 0. 3. Differences in Attitudes Towards Risk Some loan applicants are clearly more risk averse than others. These differences will be reflected in project choices, and thus affect the bank-optimal interest rate. High interest rates may make projects with low mean re￾turns- the projects undertaken by risk averse individuals-infeasible, but leave relatively unaffected the risky projects. The mean re￾turn to the bank, however, is lower on the riskier projects than on the safe projects. In the following example, it is systematic dif￾ferences in risk aversion which results in there being an optimal interest rate. Assume a fraction X of the population is infinitely risk averse; each such individual undertakes the best perfectly safe project which is available to him. Within that group, the distribution of returns is G(R) where G(K)=1. The other group is risk neutral. For simplicity we shall assume that they all face the same risky project with probability of success p and a return, if successful, of R* > K; if not their return is zero. Letting R =(1 + r)B the (expected) return to the bank is (11) p(r) -{ X(l -G(R^))+ (I -X)p } (I +r) X(l1-G(RA))+(1- X)(?) r[1 _ (1 -p)(l-X) 1 R 1X (1-G(R))+(1-A)d B Hence for R<K, the upper bound on re￾turns from the safe project (12) d lnj -1- dIn(1 +rP) (1-X)(1 -p)Ag(R)R (1 -XG(R))(X(I - G(R)) +p(l -X)) A sufficient condition for the existence of an interior bank optimal interest rate is again that limRK K / ar<O, or from (12), X/1 -X fimR,Kg(R)A>p/l-p. The greater is the riskiness of the risky project (the lower is p), the more likely is an interior bank optimal interest rate. Similarly, the higher is the rela￾tive proportion of the risk averse individuals affected by increases in the interest rate to risk neutral borrowers, the more important is expected profit per dollar loaned may be rewritten as JK g(R dR J,R-D d p(J)=[J-D+X][T-D] K +D-X j g(R)dR Differentiating, and collecting terms JK g( dR T-D aJ rK R ) +[J-D+X] + T XD aJ f Kg(R) dR [ -() jfg(R)dR+g(J) ) dR] Using l'Hopital's rule and the assumption that g(K) - 0,oo I a d1 K-D?X J-K T-D aJ / ( K-D 2(K-D) ) or sign( lim ap ) sign (K-D-X) Conditions 2 and 3 follow in a similar manner