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Page i Page ii Acknowledgement Page iii Page iv Table of Contents Page v Page vi Review of the literature Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 The model Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Solution with a passive government Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Solution with an active government Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Page 133 Page 134 Page 135 Page 136 Page 137 Page 138 Page 139 Results of the study Page 140 Page 141 Page 142 Page 143 Page 144 Appendix: Definitions Page 145 Page 146 Page 147 Page 148 Page 149 Page 150 Literature cited Page 151 Page 152 Page 153 Page 154 Back Cover Page 155 Page 156 

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A General Equilibrium Study of the Monetary Mechanism David L. Schulze A UNIVERSITY OF FLORIDA BOOK THE UNIVERSITY PRESSES OF FLORIDA GAINESVILLE / 1974 __ __ C_ __ __ __ __ __ _ EDITORIAL COMMITTEE Social Sciences Monographs WILLIAM E. CARTER, Chairman Director, Center for Latin American Studies IRVING J. GOFFMAN Professor of Economics MANNING J. DAUER Professor of Political Science BENJAMIN L. GORMAN Professor of Sociology VYNCE A. HINES Professor of Education HARRY W. PAUL Associate Professor of History Library of Congress Cataloging in Publication Data Schulze, David. 1939 A general equilibrium study of the monetary mechanism. (University of Florida social sciences monograph no. 51) "A University of Florida book." Bibliography: p. 1. Money supply. 2. Equilibrium (Economics). 3. Money supply Mathematical models. I. Title. II. Series: Florida. University, Gainesville. University of Florida monographs. Social sciences, no. 51. HG221.S358 332.4 7413495 ISBN 0813004071 30cf. Z F , 3 u COPYRIGHT 1974 BY THE BOARD OF REGENTS OF THE STATE OF FLORIDA PRINTED BY BOYD BROTHERS, INCORPORATED PANAMA CITY, FLORIDA c. 1. Acknowledgments I SHOULD like to express my appreciation to the National Science Foundation for providing funds to support this work and to Iowa State University for actually allocating the funds under the NSF grant. I am also deeply indebted to Professor Dudley G. Luckett for his guidance throughout the course of the project. Special thanks go to my wife for her cheerful suffering of the effects of my frustrations. Thanks must go also to the Graduate School of the University of Florida for making possible the publication of this monograph. Contents 1. Review of the Literature 1 2. The Model 12 3. Solution with a Passive Government 68 4. Solution with an Active Government 96 5. Results of the Study 140 Appendix: Definitions 145 Literature Cited 151 1. Review of the Literature T HE CLASSICAL dichotomy between the real and monetary vari ables in the economy is, in one form or another, an extremely hardy beast. One of its milder reincarnations is the idea that an examination of the determinants of the stock of money is, at best, only an intellectual game, since the chain of causality runs from income and prices to the money stock. The demand for money is visualized as primarily a func tion of the level of national income, and any correlation between income and prices and money is due solely to the "pull" of income on the money stock.1 No important feedback from the money stock to income and prices is believed to exist. With the great deal of work done in the 1950s and early 1960s providing a convincing theoretical basis for the existence of a chain of causality running from the money stock to the real variables in the economy,2 not to mention Keynes' work (31, 32), economists began, in the early 1960s, to investigate more thoroughly the determination of the money stock. The forces affecting the money stock were important, since the money stock in turn affected the level of prices and income. The primary purpose of this work is to examine the processes through which the money stock is determined. In addition, further theoretical support will be provided for the position that changes in the stock of money affect the level of economic activity, and the effects and effec tiveness of the various tools of monetary policy will be examined. The framework in which this will be carried out is a general equilibrium model of the economy composed of five sectorsthe public, manufac turing firms, banks, nonbank financial firms, and the government. 1. See, for example, Goldsmith (26) and Klein and Goldberger (33). Full information on literature cited begins on p. 151. For definitions of symbols used in the model, see Appendix, p. 145. 2. Patinkin's Money, Interest, and Prices (47) served as both a milestone and a stimulus for further work in this area. 2 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM This work is primarily an extension of what is commonly referred to as "money supply theory." The basic idea of the approaches to be discussed is to generate expressions for the stock of money in terms of the variables of whatever economic model is postulated, and to derive statements about the effects of changes in these variables on the stock of money. These expressions for the money stock are called money supply equations.3 The study of the supply of money began with the early work of C. A. Phillips (48) and others (1, 35, 39, 50) in the 1920s and 1930s. Their work culminated in the standard textbook money multipliers, with which we are so familiar, like AM = (1/r) times the original change in the money stock (where r is the average reserve requirement). No real advance in this area occurred until the 1960s and the appearance of the works of Milton Friedman and Anna Schwartz (24), Phillip Cagan (10), and Karl Brunner and Alan Meltzer (4, 5, 6, 8, 41). The Friedman SchwartzCagan and BrunnerMeltzer approaches to the money supply are the best known today. The FriedmanSchwartzCagan4 approach is based on two simple defi nitions. The money stock, M, is equal to total currency holdings, C, and total demand deposits: M = C + D. (1.1) Highpowered money, H, defined as the total of all types of money that can be used as currency or reserves, is simply H=C+R (1.2) where R is simply reserves. The basic FriedmanSchwartzCagan result is obtained by simply divid 3. They are not supply equations in the normal sense of the term, since they all purport to give the actual stock of money when the values of their parameters are known. If they were true supply equations, the actual stock of money would be given, not by the "supply" equation alone, but by simultaneous solution of the aggregate demand for money equation and a "true" supply equation. For this reason, we choose to speak of the monetary mechanism implying a simultaneous determination of the money stock, rather than the supply of money alone. 4. Cagan's tautology for the money stock is slightly different from that pre sented in Appendix B to A Monetary History of the U.S., 18671960 by Friedman and Schwartz (24). Cagan's formulation is based on the tautology derived by Friedman and Schwartz described in the text. REVIEW OF THE LITERATURE ing Equation 1.1 by Equation 1.2 which yields, after a few simple algebraic manipulations,5 M = H D/R (1 + D/C)(1.3) D/R + D/C Equation 1.3 is a tautology, being derived from the definitions of M and H. In this approach the money stock is determined by the decisions of three sectors: the government, which determines H; the public, by determining its deposit to currency ratio, D/C; and the banks, by determining the deposit to reserve ratio, D/R. Friedman and Schwartz (24) call H, D/R, and D/C the "proximate determinants" of the money stock (p. 791). The factors underlying these proximate determinants are spelled out only vaguely. D/C is said to depend upon the "relative usefulness" of deposits and currency, the costs of holding these assets, and "perhaps income" (p. 787). D/R is a function of legal reserve requirements and precautionary reserves (p. 785). The determinants of H are not spelled out specifically, even though a large portion of the book is devoted to describing and analyzing various actions by the monetary authorities. Brunner and Meltzer actually present two hypotheseslinear and non linear. Their linear hypothesis is based on the reaction of the banking system to the presence of surplus reserves, defined as the difference between actual and desired reserves, the portfolio adjustments caused by these surplus reserves (4, 8), and the process by which surplus reserves are generated or absorbed. The total portfolio response of the banking system to the presence of surplus reserves is given by 1 dE = S (1.4) 5. The derivation of Equation 1.3 is: (1) M/H = C + D/C + R. Multiplying numerator and denominator by D yields (2) M/H = DC + D2/DC + RD. Then the righthand side of (2) is multiplied by RC/RC, yielding DC+D2 DC D2 D D2 D D + + (1 + M RC RC RC R RC R C H DC+RD DC RD D D D D RC RC RC R C R Multiplying both sides by H gives the desired result. 4 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM where E is the value of the banks' portfolio, S is the amount of surplus reserves, and X is the average loss coefficient (i.e., X measures the amount of surplus reserves lost per dollar of portfolio adjustment). X is less than 1 since the banking system will generate added deposits (and thus reserves) as it attempts to eliminate surplus reserves by buying interestbearing assets. p is equal to (1 n)p, where p is the average spillover of deposits from the expanding bank (the one trying to elimi nate surplus reserves) to other banks and n is a linear combination of average spillover into currency and time deposits. Thus p reduces the average loss coefficient and the term (X p)1 is Brunner and Meltzer's money multiplier for responses to surplus reserves. Surplus reserves are given by the relation S = Ao dB + dL A1 dC0 + A2 dto + A3 dE dVd; (1.5) B is the monetary base (the amount of money issued by the govern ment); L is the total of changes in required reserves resulting from changes in the average reserve requirement and from the redistribution of deposits among various classes of banks; E is a parameter measuring the structure of interbank deposits; dC0 represents changes in the public's demand for currency occurring independently of changes in the public's monetary wealth; dto represents changes in the public's demand for time deposits occurring independently of the public's wealth; and dV0 represents changes in the banks' demand for cash assets in excess of required reserves occurring independently of changes in the level of banks' deposits. The Ai are positive constants. Then the change in M2 (defined as currency plus demand deposits plus time deposits) is given by dM2 = m2s + q dB (1.6) where m2 is the surplus reserve (or money) multiplier and q is the proportion of a change in the money base that affects bank reserves and deposits simultaneously. The change in M1 (M2 T) is dM1 = m's + q dB dto (1.7) where m1 is the money multiplier for the definition of M excluding time deposits. REVIEW OF THE LITERATURE Replacing s in Equations 1.6 and 1.7 with Equation 1.5 and integrat ing yields the linear hypothesis' expressions for M1 and M2: M2 = mo + m2(B + L) m2A1Co + m2A2to m2V (i) (1.8) M1 = no + m(B + L) m'A1Co [1 m1A2 to  m Vd (i) (1.9) where B + L is the "extended monetary base," mo and no are positive constants, and the notation Vd(i) is used to express the dependence of the money stock on interest rates through the impact of interest rates on the banks' asset portfolio. m' and m2 are the money multipliers. Behind the terms Co, to, and Vo lie the public's demands for currency and time deposits, which depend upon the public's money wealth, nonmoney wealth, and all interest rates, as well as the banks' demand for "available cash assets," which depends on the relevant interest rates and the level of deposit liabilities. Again the money stock depends upon the decision of three sectors: the government in determining B + L, the public in determining Co and to, and the banking system in determining Vd (i). Implicit in this hy pothesis is the assumption, as Fand has pointed out (19), that the marginal propensities to hold time and demand deposits (with respect to changes in M) are constant. Brunner and Meltzer's nonlinear hypothesis centers on the credit market. The money stock and interest rates emerge from the interaction of the public's supply of assets to the banks and the banks' resulting portfolio readjustment. The banks' desired rate of portfolio readjustment, Es, is given by Es=h(R R) (1.10) where R is actual reserves and Rd is desired reserves. Rd = Rd (D, T, i, p) (1.11) where i is a vector of all interest rates and p is the discount rate. Excess reserves, Re, are given by Re = Re (i, p, D + T). (1.12) 6 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Re is assumed to be homogeneous of degree one in D + T so that we can write Re = e (i, p) (D + T). (1.13) The public's rate of supply of assets to the bank, E is given by d = f (i, W, E) (1.14) where W is the public's wealth and Ed the public's desired portfolio of liabilities to the banks. The public's desired rates of change in currency holdings and time deposits are given by CP = q1 (kD Cp) (1.15) S =q2 (tD T) (1.16) where k is the desired currency to demand deposit ratio, t is the desired time deposit to demand deposit ratio, and D is the level of demand deposits. The banks' desired rate of change of indebtedness to the Federal Reserve System is A a [b (D + T) A] (1.17) where b is the desired indebtedness ratio. Based on the above, Brunner and Meltzer write: B = A + Ba (1.18) B = R + C (1.19) R = (r + e) (D + T) (1.20) CP =kD (1.21) T = tD (1.22) A = b (D + T) (1.23) E = E (i, W) (1.24) REVIEW OF THE LITERATURE where BA in the adjusted or "relatively exogenous" base and all other symbols have been previously defined. This system of seven equations is then reduced to two through substitution and with the help of the assumption that Ba and W are given exogenously. These equations are: M2 = m2 Ba (1.25) (m2 1) Ba = E (i, W) (1.26) where m2, the money multiplier, is given by l+k+t m2'= +(1.27) (r + e b) (1 + t) + k From Equation 1.26, one of the rates of interest, say i1 (using their notation), can be determined in terms of the rest of the interest rates, p, W, Ba, r, and k. Then this solution for i1 can be substituted into 1.25,6 giving M2 as a function of interest rates is (s : 1), p, Ba, r, and W. In other words, the solution to their two equations yields M2 and one rate of interest, i1. Equations 1.25 and 1.26 are the expression of the nonlinear hypothesis. Of the many other works that might be mentioned briefly,7 we shall concentrate on the models of Ronald L. Teigen (51) and Frank de Leeuw (16). The Teigen model is based on the proposition that the total level of reserves in the Federal Reserve System, various rules (such as the reserve requirements), and regular behavioral relations (between currency levels and the total money stock, etc.) "determine a maximum attainable money stock at any given time, and that this quantity (M**) can be considered to be the sum of two parts: one part which is considered to be exogenous and is based on reserves supplied by the Federal Reserve System (RS),8 and the other based on reserves created by member bank borrowing (B),9 and therefore considered endogenous" (p. 478). Teigen's goal is to explain the ratio of the observed money stock (M) to the exogenous segment of the total money supply (M*). He asserts that 6. Since all the terms in m2 are functions of i and/or p. 7. See, for example, Grambley and Chase (27), Meigs (40), Modigliani (44), and Goldfeld (25). These and several other studies cited in the bibliography will not be discussed because of their highly specialized nature. 8. This is M*. 9. This is B*. 8 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM this ratio is a function of the profitability of bank lending. The impor tant conclusions of the Teigen model are derived from his definitions of the money stock and the public's demand for currency and demand deposits (which he assumes are a constant proportion of the actual money stock). kk 1 M = (R Re) + (B D) (1.28) =1 ch (1ch k g where k is the reciprocal of the weighted average reserve ratio, c is the fraction of M held as currency by the public, h is the fraction of M held by the public as demand deposits in nonmember banks, Re is excess reserves, and Dg is U.S. government deposits in member banks. M* = k (Rs) = k*Rs (1.29) 1ch and M X (rc, r) (1.30) where r is a measure of the return on bank loans and rc is a measure of the cost of bank loans. ax/ar > 0 indicating that as the return on loans increases, the en dogenous portion of M increases relative to M*. ax/arc < 0 indicating that as the cost of loans increases, M* becomes a larger proportion of M. Thus, Teigen breaks the money stock down into endogenous and exogenous portions and attempts to explain the relation between the actual money stock and the exogenous portion in terms of the returns and costs of bank loans. Changes in these factors presumably change the quantity of loans banks are willing to supply and thus result in portfolio readjustment by the banking system, fueling changes in the actual stock of money. The de Leeuw model is part of the BrookingsSSRC model. His portion of the overall model deals with the financial sector. There are seven markets: bank reserves, currency, demand deposits, time deposits, U.S. securities, "savings and insurance," and private securities. The sectors included are banks, nonbank financial, the Federal Reserve, the Treasury, and the public. This submodel (of the SSRC model) assumes that the value of real variables is known and does not consider the REVIEW OF THE LITERATURE effects of changes in the various rates of interest, or the money stock on the real variables in the model. (Their effects are measured elsewhere in the Brookings model.) The model itself is composed of nineteen simultaneous equations, four of which are identities (the reserve identity, etc.) and the rest of which express the desired changes in assets in terms of lagged asset holdings, rates of return, and various shortrun constraints on asset holding. Solving this system simultaneously, de Leeuw derives the following expression for the money stock (p. 518): RESNBC M 1RDD + 0.84 [RRRDD] [RDD + RDDG + 0.82 [RRRDT] [RDT] + [0.011 RMFRB  0.010 RMGBS3 0.007] [RDD + RDTI] (1.31) where SM is the money supply (private demand deposits [DD] and currency); RDD = DD/S; RDT = DT/SM; RDD DDGF/SM; DT is private time deposits; RESNBc is unborrowed reserves plus currency held by member banks; RRRDD is a weighted average of required reserve ratios against demand deposits; RRRDT is a weighted average of required reserve ratios against time deposits; DDGF is government de mand deposits; RMFRB is the discount rate; and RMGBS3 is the average market yield on threemonth Treasury bills. Substituting the definitions for RDD and RDT into 1.31 and using the ai to replace constants, we have S RESNBC M DD DD+DDGF +a S + a RRRDD ( +DD +a2 RRRDT S~SM 2SM DT)[ RM RMSB DD + DT (1.32) SM [a RMFB G 3 SM which clearly shows the dependence of the righthand side of Equation 1.31 on SM, supposedly given by Equation 1.31. Solving Equation 1.32 for SM yields 10 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM SM = RESNBC + DD 0.84 [RRRDD] [DD + DDGF]  0.82 RRRDT DT [0.011 RMFRB 0.010 RMFBS3 0.007] [DD + DT]. (1.33) Equation 1.33, derived from de Leeuw's expression for the money stock, says that the money supply, SM, is smaller in size than unbor rowed reserves plus currency held by the banks plus private demand deposits. This is a nonsense result and throws suspicion on the entire de Leeuw model. For the month of April 1969, the appropriate figures (taken from the July 1969 Federal Reserve Bulletin) are (in billions of dollars): Total reserves 27.079 Borrowings .996 Unborrowed reserves 26.083 Total demand deposits 152.8 Government demand deposits 5.1 Private demand deposits 147.7 Discount rate 5.5 per cent Yield on three month bills 6.11 per cent Time deposits 201.6 Plugging these figures into Equation 1.33 and performing the arithmetic, we find that de Leeuw's equation gives a money supply of $150.64 billion. The actual money supply for April 1969 was $196.7 billion. The difference between de Leeuw's prediction and the actual money stock is primarily the $43.9 billion of currency in circulation in April. As can be seen from Equation 1.33, this component of the money supply has been lost in de Leeuw's formulation. Of the models reviewed here, the de Leeuw model is most similar to the approach taken in this study. The other models tend to be deficient in two respects. First, they are too aggregative in the sense that the economy is broken down into only three sectorsthe government, banks, and the public. No distinctions among households, manufacturing firms, and nonbank financial firms are drawn. Second, they all hide the general equilibrium nature of the monetary mechanism. In the first two models reviewed, the behavior functions of the various sectors are not explicitly specified. The Teigen model, while specifying the public's REVIEW OF THE LITERATURE 11 demands for demand deposits and time deposits, does so in terms of the total money stock, and takes the total stock of money as the independ ent variable in these functions. While such a formulation will probably yield significant empirical results, from a theoretical point of view it seems awkward to visualize the public changing their holdings of de mand and time deposits in response to a change in M rather than because of changes in income, prices, and interest rates. Hiding the general equilibrium nature of the problem also precludes description of the effects of changes in the money stock on the real variables of the economy. (This can be done in the SSRC model, but not by the de Leeuw submodel itself.) The purpose of this work is, however, not to repudiate any of the existing work in this area, but rather to extend and amplify the analysis begun by these more narrow and specialized studies. 2. The Model THE MODEL is made up of five sectors: public, manufacturing (the firms), banking (the banks), nonbank financial (the intermediaries), and government. This chapter contains details of each sector and the relationships among sectors. The solution to the model will be con sidered in chapters 3 and 4. The behavioral relations for each sector are given in both implicit and explicit form. For simplicity two assumptions are made: most of the explicit forms are linear and reflect either utility or profit maximizing behavior, and the individual units in each sector are homogeneous so that, in most cases, aggregate levels can be obtained by summing the representative functions. PRODUCTION, INVESTMENT, AND GROWTH Technology is assumed to be characterized by increasing opportunity costs and is constant over time. For simplicity these assumptions are made: 1. There are only two inputscapital and labor. A unit of labor is indistinguishable from any other unit of labor. Capital is also perfectly homogeneous. 2. There are only two outputscapital and the consumption good. The consumption good is perfectly homogeneous. 3. Firms fall into two categoriesthose that produce only the capital good and those that produce only the consumer good. Each firm within each category is identical to every other firm in the group. There is a large enough number of firms in each category so that, coupled with freedom of entry and exit, each firm is a perfect competitor in the output market. 4. Individuals in the economy have identical endowments of capital and labor. No organization controls the supply of either capital or labor. 12 Thus, the capital good firms are also perfect competitors in the input market and the labor market is perfectly competitive. Production The aggregate production function for the capital good is given by X = Xk (Lk, Xkk) (2.1) where Lk is the amount of labor used in the production of capital and Xkk is the amount of capital used in the production of capital, axa a a2 Xa 2 Xa S >0, k >0, k <0, k <0, 8Lk )Xkk aLk2 aXkk2 and a2 Xa k > 0. 8Lk axkk If there are n firms producing capital, the production function for the ih individual firm is Xa L X Xki= =Xk ( ). (2.2) The aggregate production function for the consumer good firms is given by Xa = Xc(L,, Xk) (2.3) where LC is the amount of labor used in the production of the con sumption good and Xkc is the amount of capital used in the production of the consumption good. 2 Xa aXa a2 Xa a2Xa C >0, C >0, c <0, <0, a8Lc Xkc a,8L2 Xkc2 and 13 THE MODEL 14 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM a2 Xa Xkc aL < If there are m firms producing the consumer good, the production function for the jth firm is Xa L Xk Xcj c = Xc( c) (2.4) The transformation curve is shown in Figure 1. The transformation function is given by Xc = T(Xk) (2.5) where 1. T(0) = aci 2. T(aki) = 0 3Xk 3. < 0. axc aci is the maximum production of the consumption good in period i while OCki represents the maximum production of capital for the same period. Defining the transformation function implicitly we have T'(Xk, Xc) = 0. (2.6) Thus X = (Xk, Xc) is a fullemployment output vector if T'(Xk, Xc) = 0. (2.7) X' = (X'k, X'c) is less than full employment if T'(X'k, X'c) < 0. (2.8) If T' < 0 the unused productive potential of the economy is measured by the negative value of T. The economy is in fullemployment equilib rium if 2.6 is satisfied and if Pc/Pk = dXk/dXc. For purposes of the model it is assumed that the explicit form of 2.5 is X2 + Xi2 2 ci ki ki (2.9) and a = ki for all i. Thus the explicit transformation curve assumed is a quarter circle in the positive quadrant. Writing 2.9 in a form equiva XK aKi  Xc aC, Fig. 1. The transformation curve lent to 2.2 we have vector if that X = (Xi, Xki) is a fullemployment output S2 '2 0 Xki ki Xci, ki 0. The marginal rate of transformation is dX X MRT dXc = + Xk dXk X)1/2 (2.10) (2.11) When 2.10 is satisfied, output is at full employment and where, in addition, 2.11 is equal to the price ratio, output is also an equilibrium output. The explicit form of 2.7 is simply X2i + X2i i2 = 0. ci ki ki 15 THE MODEL (2.12) 16 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Growth and Investment The labor force is assumed to grow at the same rate as the population. This rate is assumed to be a function of the rate of change of the real output of the consumption good over time. dL dXc /L =X= () (2.13) where X is the rate of growth of the labor force, X(dXc/dt) is functional notation, and where 1> dh >0. dX d( CdX) dt The rate of growth of the capital stock, k, is not tied to the rate of growth of the labor force. Gross and net investment are determined by the interaction of the supply and demand for capital. In general, how ever, k is assumed to be a function of the price of capital, the price of the firms' output, the firms' profit expectations, the various rates of interest, the rate of depreciation, and the existing stock of capital. The demand for capital is composed of both a stock demand for capital, Dk, and a flow demand, dK.1 The stock demand is given by Dk = Dk(Pk, r, 0) (2.14) where r is a vector of interest rates (r = (rf, rg, rt, etc.)) and 0 is a profit expectation function; also aDk <0 Dk <0, and DK >0. aPk ar 30 The flow demand for capital, dK, is given by dK = nK (2.15) where n is the rate of depreciation, 0 < n < 1, and K is the existing capital stock. 1. See P. Davidson (13). The supply of capital is also composed of a stock supply and a flow supply. The stock supply, Sk, is simply equal to the existing capital stock, while the flow supply, Sk, is assumed to be dependent on the price of capital. Sk = Sk(Pk) (2.16) where dSk > 0. dPk In Figure 2, Dk + dK is the market (stock + flow) demand for capital and Sk + Sk is the market (stock + flow) supply of capital. Pk is the PK K K, I K, SK +SK DK+dK  K Fig. 2. Supply and demand for capital equilibrium price of capital. At this price gross investment is equal to K2 K and net investment equal to K, K. Note that it is the rates of interest relevant for financing and deter mining relevant discount rates that, along with profit expectations, determine the exact locations of Dk. As rates fall, Dk shifts outward, ceteris paribus. Although it would be more elegant to consider gross and net invest 17 THE MODEL 18 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM ment for each group of firms separately, we shall assume that 2.1416 are defined in such a manner that their solution as shown in Figure 2 represents the aggregate levels of gross and net investment for both groups of firms combined. The rate of growth of the capital stock, k, is, in terms of Figure 2, K+ Ki K K1 k = (2.17) K K We must now consider the effects of changes in the capital stock and the labor force on the transformation curve, given that technology is constant. The question is basically this: if Lt, Kt give kt = at, what will act + 1 and akt + equal if L grows by Xt per cent and K grows by kt per cent, Xt >/< kt, e.g., what relation will the transformation curve in t + 1 bear to the curve in t? Will relation 2.9 hold over time? If not, what other assumptions about the nature of production must we make to insure that it does? Writing the total differentials of the production functions, we have ax ax dX=. c dKc + dLc (2.18) aKc L c dXk = Xk dKk + XkdL. (2.19) aKk a Lk From the definitions of k and X, K dK k = /K (2.20) K dt L dL X /L, (2.21) L dt we have dK = kK (2.22) dt and THE MODEL dL = XL, dt from which it follows that dK = kKdt and dL = XLdt. Substituting 2.24 and 2.25 into 2.18 and 2.19 we have dX c= kKdt + x Ldt c K aL and ___ aXk dXk k kKdt + a XLdt k K aL (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) from which it follows that dXc Xc kK + a XL dt aX 3L and aXk Xk kK + k XL. at LK L (2.29) Equations 2.28 and 2.29 tell us how the maximum possible outputs of the consumer good and the capital good change over time if the entire increase in the stocks of labor and capital is used in one good or the other. In order for the transformation curve to shift in a parallel way as a result of the growth of capital and labor, it is necessary and sufficient that 2.28 equal 2.29. Since we are starting from a position where ac = ak, only the rates of change need be equal to insure the increase in ac is equal to the increase in k. Thus we have 19 20 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM kX aX L aXk aX c kK + aXL = kK + k L. (2.30) aK aL, 3Kk aLk Equation 2.30 can be rewritten in two ways, both of which express the condition necessary for a parallel shift of the transformation curve. (1) kK( c ak ) + XL( cax = 0 ( Kc aKk aKL aLk Regardless of the relative sizes of k and X and of their signs, if the marginal product of capital in production of the consumer good is equal to its marginal product in producing capital, and if the same is true of the marginal products of labor in its two uses, (1) will be satisfied. If these marginal products are not equal, (2) expresses the condition that must be satisfied for parallel shifts. Number (2) is also derived directly from 2.30. axk axc (2) = aKk 8Kc axc aXk aLc aLk Since (2) places no unrealistic constraints on the production processes, we assume that it is satisfied for all X and k between 1. The Labor Market The aggregate supply of labor is a function of the wage rate, PQ, and the size of the population. If it is assumed that the labor force is a constant percentage of the population, we may write S= S(Pk, L) (2.31) where aS aS S> 0and > 0. aP aL At any point in time, the supply of labor may be taken to be a function of only the price of labor. The aggregate demand for labor is composed of the demand of the capital good firms and the consumer good firms, as well as the demands of the banks, government, and intermediaries. No attempt will be made to specify explicitly the demand functions for these sectors. (This is in keeping with the practice of not specifying these sectors' demand for the capital good explicitly.) This demand is included by adding a constant, E, to the sum of 2.32 and 2.33. These demands are D = MPkPk (2.32) D = MPPc (2.33) since all firms are perfect competitors. The aggregate demand is simply D = MP kPk + MPcPXc + E. (2.34) Under our assumptions on production, the aggregate demand curve will be downward sloping. Since labor is homogeneous, it must be paid the same wage in each use. Thus we have Figure 3. Pg is the equilibrium PL LL SL I D +DK +E DL +DL DLL I L Fig. 3. Supply and demand for labor 21 THE MODEL 22 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM price of labor. The amounts employed by each group of firms can be read from the diagram: L is the total amount employed, LC the amount used by the consumer good firms, and L Lc the amount employed by the capital good firms. No restrictions are placed on dPc/dt, e.g., no assumption of wage inflexibility is made. Thus, in one sense, labor will be fully employed so long as the existing Pk is an equilibrium price. If Pc > P), involuntary unemployment exists. If Pk < PQ, labor is fully employed, even though there is a positive excess demand for labor. Production Equilibrium and Full Employment These conditions must be satisfied for the productive sector of the economy to be in equilibrium: 1. Pg is equal in both uses. 2. Pk is equal in both uses. 3. Pg = MPkPk = MPQcPc. 4. Pk = MPkkPk = MPkP Condition 4 clearly implies that the marginal product of capital in the production of capital must be 1 in equilibrium. This is not startling, since the capital good firms would obviously increase their own use of capital if MPkk > 1 and reduce it if MPkk < 1. 5. The marginal rate of technical substitution of labor for capital equals the input price ratio for both groups of firms. a. MRTSk = MPkk k MPkk Pk c MPkc Pk b. MRTSk/ = M M P MP c Pk 6. MRTS/k = MRTS/Q = When conditions 16 are satisfied, the productive sector is in equilib rium in the sense that, given the total amounts of the inputs being used, it is impossible to increase output of the earlier commodity by redistrib uting the capital and labor being used between the two groups of firms THE MODEL without reducing the output of the other commodity. However, 16 are not sufficient to insure that the equilibrium output vector is also a fullemployment output vector. This is simply because nothing in these conditions implies that the total stocks of capital and labor are being used. That is, 16 may be satisfied under conditions of unemployed labor and/or capital. In this case, even though redistribution of the inputs actually being used cannot increase the output of one commodity without reducing the output of the other, it is entirely possible that increasing the total use of capital and/or labor can lead to an increase in the production of both goods. Thus, another condition must be added to insure that the equilibrium is also a fullemployment equilibrium. This condition is simply that the outputs of capital and the consumer goods that satisfy 16 also satisfy 7. Xk X where Xk and Xc are the outputs resulting from satisfying 16. Note that the aggregate level of consumption demand, not yet considered, has an impact on these conditions through its influence on Pc and Xc and, of course, may prevent condition 7 from being satisfied. THE MANUFACTURING SECTOR (FIRMS) The firms are divided into two groups: one produces only the capital good while the other produces only the consumer good. Each group is assumed to be perfectly competitive. The only interfirm purchases are those of capital goods. Each group will be treated in the aggregate rather than on an individual firm basis. Production and sales for each firm in a group are identical (see the section on production in this chapter). Each firm has a desired level of retained earnings such that the aggregate desired level is given by ED = aKt + t 1 + s[P,, X, + Pkt Xkt]. (2.35) In 2.1, aKt represents depreciation; Pt Xt + Pkt Xkt is, of course, aggregate sales in t; and s is a constant, 0 < s < 1. This term is included to reflect the demand for retained earnings arising from the desire of the firm to insure itself from the unexpected. Such risks are simplistically assumed to grow in proportion to total sales. Int 1 is a factor 23 24 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM reflecting the influence of past net investment on the level of retained earnings, a portion of which, it is assumed, are kept to meet the demand for financing net investment in future periods. For simplicity only one previous net investment figure is used in 2.35, although greater realism could be obtained by perhaps using averages of several past periods. It is assumed that 0 is greater than zero and less than one. Equation 2.35 also gives the desired level of financial assetscash, demand deposits, time deposits, government securities, and deposits in intermediariesin the aggregate for period t, since it is in these forms that retained earnings are held. At the end of each period the actual and desired stocks of retained earnings are equalized by adjustment of the profit payments to the owners of the firms. Only when profit payments are zero would it be possible for the actual stock of retained earnings to be less than the desired level. In no case will the actual stock exceed the desired level. Before discussing the desired distribution of the stock of retained earnings, another factor influencing the actual stock must be discussed. This is the relationship between desired financing and actual financing. The replacement demand for capital, Ig In, is assumed to be paid for completely out of retained earnings. Only a portion of net investment, equal to 3Int 1 is paid for from retained earnings. The remainder, Int  3Int 1 creates the socalled demand for financing. This demand is the basis for the firms' demand for bank loans and loans from inter mediaries and for their desire to issue more debt (the supply of firms' nonownership securities). We have F D = In nt1 (2.36) where FD is the desired level of financing in period t. L = f2(r, Ft) (2.37) explicitly, ft = fFo + A3?, (2.38) where Lft is firms' total demand for loans in t, A3 is a vector of constants, rf is a vector of all interest rates, and Ft the total amount of financing desired. F f3 (rf, F ) (2.39) explicitly, s D Ft = bf Ft + A4if (2.40) S where Ft is the supply of firms' securities in t and all other symbols are as defined above. The firms' demand for loans, Lt, is broken down into Dbft a demand for bank loans, Lft and a demand for loans from inter Dn mediaries, Lft, Db Db f f D Lf, = Lf (rbt, rnt, Lft) (2.41) Dn Dn f D Lt = Lf(rbft, rnft, Lt). (2.42) In explicit form, Db f f D Lft = a(rbt rt) + b, Lft (2.43) Lf = a2(rbt rnt)+ b2 Lft (2.44) where a1 = a2, b1 + b2 = 1, a a < 0, a2 > 0. These restrictions on the constants in 2.43 and 2.44 insure that the sum of 2.43 and 2.44 equals 2.37. The coefficients of LfD are assumed to be constant (and not necessarily equal) to allow for the possibility that the firms may want to borrow different amounts from the two lending sectors even though rb f Srn. This mix of desired borrowing is assumed to be constant over time. If Lft is the net increase in borrowing and Ft the net increase in securities outstanding, then Int Int 1 (Lft + Bft) > 0. (2.45) Lft and Ft are determined by the interaction of the demand for loans (the supply of securities) and the supply of loans to firms (the total demand for firms' securities). If 2.41 equals zero, then the financing demand is satisfied completely by increasing the firms' debt. If, how ever, 2.41 is positive, the difference is made up by a temporary reduc tion in the stock of retained earnings below their desired level. (Note that the capital good firms do not themselves extend credit to their 25 THE MODEL 26 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM purchasers.) This discrepancy between desired and actual retained earn ings is made up by a reduction in profit payments as discussed before. Let Int Int 1 (Lft + Bft) = AI(Aft) > 0 (2.46) where AI(Aft) represents the unintended change in the firms' financial asset position caused by insufficient financing to meet investment de mand. AI(Aft) represents a redistribution of income away from the owners of the consumer good firms to the owners of capital good firms. As we shall see, aggregate public income is not reduced. In summary, Et EA+ = E_ +AE (2.47) where Et, is the actual stock of retained earnings (or, equivalently, the actual stock of financial assets, Aft+) at the end of period t; Ea_ is the actual stock of retained earnings at the beginning of period t, which is equal to the actual and desired stock of retained earnings at the end of period t 1; and AEt is the desired change in the stock of retained earnings during t. AE can be expressed as follows: AED = ont 1 + s[Pct Xct + Pkt Xkt Pt 1 Xct 1 Pkt 1 Xkt 1] + nt 1 Int 2). (2.48) Furthermore, Pct Xct + Pkt Xkt can be written as a function of Kt: Pct Xct + Pkt Xkt = f3(Kt). (2.49) The function f3 is really a reduced form of the production function. The level of output is determinate, given the amount of capital used under the assumption that, for a particular input price ratio, the least cost combination of labor and capital is used. Thus, 2.44 becomes AEt = aI 1 + s[f3(Kt) f3(Kt 1)] + (2.50) O(Int Int 2) THE MODEL or = aInt 1 + s[f3(Kt 1 + nt 1) 3(Kt + (2.51) or = lnt 1 + s[f3(Kt 1 + nt 1) f (Kt 2+ nt 2)] +(nt 1 nt 2) (2.52) = aInt 1 + s[f3(Kt Kt 2+Int 1 Int 2)] + a(Int 1 Int 2) = alnt 1 + s[f3(lnt 1)1 +(lnt 1 nt 1). (2.53) (2.54) Letting (a + 0) (Int 1 ) + sf3 (nt I) equal f4 (nt 1) we have AED =f4(nt 1) nt 2) (2.55) which expresses the dependency of desired changes in retained earnings on net investment. Summing over t in 2.55 (or integrating when time is assumed to be continuous) leads to the dependency of the stock of retained earnings on the capital stock as expressed in 2.35. Rewriting 2.35 in an analogous manner leads to Et= aKt + sf3(Kt) + (Kt Kt 1) or =(a + f) Kt + sf3(Kt) (Kt 1) or, letting (a + I) Kt + sf3(Kt) = f3(K), (2.56) (2.57) or or 27 Int2)] + (Int 1 Int 2) Et =3(Kt) (Kt 1). 28 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The firms' decisionmaking process in regard to the size and distribution of retained earnings is visualized in this manner. First a decision is made regarding the desired stock of retained earnings and the necessary adjust ments of profit payments made to realize this goal. Second, after desired size has been achieved, the firm decides on the desired distribution of retained earnings (financial assets) as described below. Letting DED represent the desired distribution of retained earnings in t, we have: DEtD D DAD = f4(r, PX) (2.58) where DAft is the desired distribution of financial assets, r is a vector of all interest rates, and PX = Pc Xc + Pk Xk. DAft is itself a vector, the elements of which are cash balances, demand deposits, time deposits, government securities, and deposits in intermediaries. Firms are assumed to hold no debt instruments issued by other firms. Thus, D D D DG D D( DA = (CD, Df, TD, Gft, Nft). (2.59) Obviously, 5 D D(2.60) 2 (DAft)i Aft. (2.60) i=1 Equation 2.53 can be broken down into an interdependent system of equations each giving the desired level of one asset. The desired level of D cash balances, Cft, is assumed to depend only on the level of sales. Thus, CD [Pct +=] (2.61) ft = as PtXct + PktXkt (2.61) where as is a constant, 0 < as < 1. Changes in rates of interest are assumed not to affect desired currency balances, although they are assumed to influence the desired level of demand deposits. Equation 2.61 is designed to reflect the assumption that holding currency balances is a nuisance to the firms and such balances are held to an absolute minimum. The desired level of demand deposits is assumed to be a function of the level of sales, the rate of interest on time deposits, and the rate of interest on government securities. Thus, DDt = df(PX) + Af . Dft 6rf (2.62) The desired levels of time deposits and government securities are also functions of the same variables. Thus, Tft = tf(PX) + A7rf,; (2.63) G = gf(PX) + A8rf; (2.64) N = nf(PX) + A9Tf. (2.65) We now turn to an examination of the sources and uses of income for the firms in the consumer good group. There are four sources of income for these firms: sales, interest on time deposits, interest on government securities, and interest on deposits in intermediaries. Let Rft be the receipts (income) of these firms in period t. Then Rfct = PXt + t Gtt + r Tct + t Nct. (2.66) ct rgt Gt t Tct rnt Nfct The uses of income include these six: payments to labor, profit pay ments ("dividends"), changes in financial asset holdings, loan repay ments, debt (security) retirement, and investment expenditure. The first two are treated strictly as residuals and are represented by Yfe (income received by the public from consumer good firms). Changes in financial D asset holdings are equal to AECt. Loan repayments equal t (1 + rb f(i))L (i) t (1 + rf(i))L(i) [+ 1+ ] i=tn N i=t n N abbreviated ELf. Debt retirement equals t (1 + rfc(i))Fc(i) i=tn N abbreviated ZFc. Gross investment equals Igc. Let Ufc represent the sum of one through five. Then Uct =Yfct + AAfct + + Lfc + IF,. (2.67) However, Rfct < Ufct since part of net investment must be financed. Let Sf be total spending power of the consumer good firms in t. Then C THE MODEL 29 30 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Sfct = Rfct + Lfct + Fct (2.68) and Sft = Uct. (2.69) Inclusion of the capital good firms allows the elimination of the c subscript in 2.65 to 2.68. It is assumed a typical capital good firm, even though it satisfies its demand for capital goods from its own output, has a financing demand identical to the consumer good firm, and behaves otherwise in the manner described above. Thus, aggregation over all firms yields the detailed statement of 2.69: PXt + rt Gft + rtt Tft + rnt Nft + Lft + Bft A(Aft) + Lft + lBft + PktXkft + Yft (2.70) Yft = PXt + rgt Gft + rtt Tft + rt Nft + Lft + Bft A(Aft) LLft Fft PktXkft. (2.71) THE GOVERNMENT SECTOR All levels of government are treated togetherthat is, as if there were only one government. No attempt is made to reflect the actual institu tional constraints under which "government" operates. There are two functions our government performs. 1. ReallocationAll physical production is assumed to take place in the manufacturing sector. The government buys capital and the con sumption good from the firms. A portion of these goods is consumed by the government and a portion is distributed to the public free of charge. 2. Economic regulationFiscal policy is not consciously used to regu late the level of economic activity. Government spending is limited to the acquisition of the amount of goods necessary for the operation of the government and for making the (exogenously determined) transfer payments. Tax receipts are assumed to be equal to these expenditures plus the interest payments on government securities. Conscious eco nomic regulation is attempted through monetary policy exclusively. The standard monetary tools are available openmarket operations, changes in the discount rate, and changes in reserve requirements, each of which will be considered in detail in chapter 4. Taxes and Government Spending All taxes are assumed to be paid by the individuals in the economy. No taxes are explicitly levied on the banks, firms, or intermediaries. All profits over and above the requirements for retained earnings to meet future investment are paid to the individual owners of these enterprises. This income is taxed at the same rate as income received from other sources (wage and interest payments). We have then T = tY (2.72) where T is total tax receipts, t the tax rate, and Y aggregate public income before taxes. It is assumed that t is constant to reflect an earlier assumption of no conscious fiscal policy. Furthermore, government spending is given by T = rG + PkXkg + PcXcg (2.73) where rg and G represent the coupon rate and aggregate face value of government securities outstanding, respectively (see the next section). The amounts of capital and the consumer good purchased are deter mined residually: PkXkg= 0 (T rgG) (2.74) PcXcg = (1 0) (T gG) (2.75) where 0 is a positive constant less than 1. These relations insure that the budget is balanced. Government Securities and Monetary Affairs The government issues only one type of security with oneyear maturity and fixed face value of $1. The coupon rate is fixed at Tg. The actual rate in any period, rg, may, of course, differ from the coupon rate depending on whether or not the bond is sold at its face value. If Pg is 31 THE MODEL 32 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM the price of one security, then rg = Pgrg (2.76) or rg. (2.77) g Pg At the end of each year the holder of a bond receives $(1 + Tg) payment of interest and principal. The government is not required to buy back unmatured bonds, although they may be freely traded among individuals and corporate entities. The amounts and timing of govern ment sales and purchases of government securities may be determined either by purely passive reaction to the net demand of the nongovern ment sectors or may be determined by conscious monetary policy goals. This area will be examined in detail in chapters 3 and 4. In any event, the effect of net changes in the amount of government securities outstanding will be to change the stock of money in the hands of the private sectors. Suppose PgtGt dollars in bonds were issued at the beginning of period t. At the end of the period (1 + Tg)Gt dollars are paid out in interest and principal. In t + 1, Pgt + Gt + 1 dollars worth of bonds are issued and interest and principal payments are (1 + rg)Gt + 1 at the end of the period. The initial impact on the money stock in period t + 1 is given by (1 + Tg)Gt Pgt + Gt + 1. If this is positive, the refunding increases the money stock by (1 + rg)Gt  Pgt + 1Gt +1 times the appropriate multiplier; if negative, the money stock is decreased. The effect of refunding in t + 2 will be given by (1 + rg)Gt + 1 Pgt + 2Gt + 2 times the appropriate multiplier, etc. The government's ability to control G, the number of bonds issued, and Pg (or rg) is the key element in openmarket operations. The size of the multiplier and the strength of the relations between the stock of money and real variables determine the effectiveness of this sort of monetary policy (see chapter 4). When solving the model in the absence of discretionary monetary policy, we shall assume that the government's supply of bonds is infinitely elastic at the current rate of interest (price of bonds). In chapter 4, this assumption will be removed. Thus, Gs GDa GC =G(.8 (2.78) where GS is the supply of bonds and GDa the aggregate demand for new bonds. This implies that the price of bonds (actual rate of interest on government securities) is constant over time. The rediscount mechanism is assumed to operate in the following manner. All loans made by the banks are discounts and are assumed to be eligible paper. The rediscount rate, rd, is a percentage of the face value of the notes held by the bank. The bank receives (1 rd)X when it rediscounts a note whose face is X dollars. If the total value of the bank's loan portfolio in any period is Y dollars, the maximum amount of rediscounting is (1 rd)Y. The government is assumed to rediscount as much paper as the banks offer at the current rediscount rate. The rate itself is set by the monetary authority (see chapter 4). Thus, for any rediscount rate, d d(rd) (2.79) where d is the actual amount of rediscounting and d the quantity of rediscounting demanded at rate rd. The effect of rediscounting is, as will be shown, to increase the quantity of bank loans supplied. The government (monetary authority) also establishes, and is free to charge, the reserve requirement, r. It is assumed that both time and demand deposits are subject to the same reserve requirements. Thus, the total amount of required reserves, R, is given by R = r (D + T) (2.80) where D is the aggregate level of demand deposits and T the aggregate level of time deposits. All banks in the economy are assumed to be subject to the regulation of the monetary authority. Effects of change in the reserve requirement on the stock of money are discussed in chapter 4. The government is strictly a passive supplierabsorber of currency. Thus, at any point in time, the stock of currency, Ct, is identical to the aggregate demand for currency, CDa. Thus, Ct CDa (2.81) Until monetary policy is considered explicitly, the government is essentially passive in the model. 33 THE MODEL 34 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM THE BANKING SECTOR Banks perform two major functions: they accept time and demand deposits from the public, the firms, and the intermediaries, and they make loans to the public and the firms. As adjuncts to these services they also hold currency and government securities as secondary reserves, engage in rediscounting, and hold primary reserves. Time and Demand Deposits Time deposits earn a yearly rate of interest, rt. This rate is paid on all time deposits regardless of their source (public, firm, or intermediary). The banks view all time deposits as homogeneous, regardless of their source. Even though some time deposits (or demand deposits) may be held as compensatory balances, no attempt is made to distinguish this portion of deposits from "ordinary" time or demand deposits. The banks' demand for time deposits is perfectly elastic at the current rate of interest on time deposits. Banks "buy" the total of time deposits willing to be "sold" by the other sectors at the prevailing rate on time deposits. Thus, letting TD represent the banks' demand for time de posits, we have D S S S T T + T + Tn (2.82) S S S where Tp, Tf, and T represent the quantity of time deposits by the three sectors. Tt is the total level of time deposits in period t; thus, T T +Tt +T (2.83) t=t D t ft Tnt The rate banks pay on time deposits is assumed to depend on the profitability of loans and the cost of obtaining reserves from alternate sources. The rates of loans to the public (rbp) and to the firms (rbf) minus the rate on time deposits are surrogates for profitability. The only other source of reserves under the control of the banks is the rediscount mechanism (see below). The rediscount rate (rd) measures the cost of reserves obtained in this manner. Thus, t = rt(rbp r, rbf r, d). (2.84) t p b ) Simplifying, t = rt(rbp, rbf rd). (2.85) The explicit form of 2.85 is rt = rt 1 + a(Lb Lp Lf) (2.86) where 1 > a > 0. Increases in rbf or rbp make loans more profitable and thus induce the banks to attempt to attract more time deposits by raising rt and vice versa. Increases in the rediscount rate tend to reduce rediscounting and thus induce the bank to look elsewhere for reserves to make up for the drop in rediscounting. Demand deposits do not earn a monetary return. Service charges are ignored. The banks accept all demand deposits offered them. Thus, the banks' demand for demand deposits is perfectly elastic. Letting DD represent the banks' demand for demand deposits, we have DD =Dss+D+D,. (2.87) DD = Dp + D + Dn. (2.87) The total amount of demand deposits in t, Dt, is given by Dt DD =Dpt + DS +Dt. (2.88) Demand and time deposits are the only liabilities of the bank that will be given explicit treatment.2 The only explicit recognition of capital account items is the assumption that all profits are paid out to the banks' owners. Loans, Reserves, and Rediscounting The legal reserve requirement, r, applies to both demand and time deposits. The total level of required reserves in period t, Rt, is given by Rt = rt(Dt + Tt). (2.89) Required services are all held in the form of noninterestbearing deposits at the monetary authority. For simplicity, vault cash, or cash held by the banks, is not assumed to be part of required, or primary, reserves. Secondary reserves are held in three formscash, securities issued by 2. Banks are assumed not to hold demand deposits in other banks. This in effect eliminates the correspondent banking system from consideration in the model. 35 THE MODEL 36 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM firms, and government securities. Desired cash balances, Cb, are given by Cb = y(D + T) (2.90) where 1 > 7 > 0. Interest rates are omitted from 2.90, reflecting the assumption that banks hold vault cash strictly to meet daytoday withdrawal requirements, and that any cash over the minimum needed for these requirements will be used to buy government securities as long as rg is greater than zero. This is equivalent to assuming that the banks do not have a speculative demand for money, in this case, cash. The desired level of government securities, Gb, is given by Gb = p(D + T) + A1orb, (2.91) where p is a positive constant less than 1. Its magnitude is determined by the "institutional" requirements for secondary reserves, such as seasonal fluctuations in deposits, etc. alo is a positive constant reflecting two assumptions: that as the yield on government securities increases, these securities become a more attractive form in which to hold secondary reservese.g., an increase in rg causes a larger portion of secondary reserves to be held in government securities, ceteris paribus; and that an increase in rg also causes the total amount of desired secondary reserves to increase, ceteris paribus. blo is a negative constant reflecting the fact that secondary reserves will be switched from govern ment's to firms' securities as the yield on these securities rises, ceteris paribus. flo, d1o, and elo are also negative constants reflecting two assumptions: that as rt rises, profit margins are squeezed, inducing the banks to shift funds from lowyielding secondary reserves to higher yielding loans; and that as the rates banks charge for loans increase (rbp and rbf), the increased profitability of loans also tends to reduce the level of desired secondary reserves. The converses of the above assump tions are also assumed to hold. The desired level of firms' securities, Fb, is given by Fb = p(D + T) + A11Tb. (2.92) /p is a positive constant less than 1. al 1 is a negative constant while b, is positive. f11, d11, and e 1 are also negative constants. The arguments here are the same as those given for the signs of the constant terms in 2.91 above. THE MODEL The total level of desired secondary reserves is given by the sum of 2.90, 2.91, and 2.92. At no time will the actual level of secondary reserves be less than the desired level. If, however, the banks are unable to make the total amount of loans they wish, actual secondary reserves may be greater than the desired level. Letting Ra be actual secondary reserves in t, Ra = Cbt + Gbt + Fbt + (Lt Lt) (2.93) where Ls is the total quantity of loans banks wish to make in t and Lt the amount actually lent by the banks in t. The term in the parentheses in 2.93 will be referred to as surplus reserves. It is assumed that all surplus reserves are held in the form of cash so that the banks' actual cash holdings in t are given by Ct = Cbt + L L (2.94) when Lt Lt is positive. Cb = Ct (2.95) when Ls = Lt. Surplus reserves are assumed to be held in cash rather than securities to reflect their transitory nature. That is, banks feel that such a situation is only temporary and do not wish to switch in and out of securities on a shortrun, unpredictable basis.3 Bank loans to both the public and firms are made for a period of n years. Loans made in period t carry a rate of interest of rbpt and rbft, respectively. The proceeds to the bank of a loan of X dollars are (1 + rb)X, repaid in n installments of (1 + rb)X /n dollars. Loans to both the public and the firms are assumed to be riskless. (Alternatively, one could think of rbpt and rbft as representing the net return per dollar lent after default and added collection expenses.) The aggregate amount banks wish to loan from unborrowed reserves in t is given by s = r(D + Tt) + Al2rb. (2.96) 3. In an attempt to keep the model as simple as possible, we have omitted the Federal Funds market, although it is recognized that this is the sort of situation that created this particular financial market. 37 38 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM nT is a positive constant less than 1. a12 and b12 are negative constants since increased yields on securities will, ceteris paribus, tend to reduce loans and increase secondary reserves. d12 is a positive constant since increased costs create pressure for shifting from secondary reserves to loans, f 2 and e12 are positive, reflecting the fact that the increased profitability of loans as interest rates rise will result in increased willing ness to lend. Surplus reserves in period t 1 (Ct Cbt ) clearly increase the banks' willingness to lend. The amounts banks wish to loan Sb Sb S to the public and to firms, Lpt and Lt depend on Lt, the difference between rbpt and rbft and institutional factors: Sb= bS Lpt = pLs + a13(rbpt rbft) (2.97) Ltb =t S a13(rbpt rbft) (2.98) b b where P + b = 1 and a13 is a positive constant. Institutional factors, such as the desire to meet the demands of existing customers, unwilling ness to loan more or less than some percentages of the total loan portfolio to any one type of borrower, etc., determine the relative sizes of pb and kb as well as the absolute size of a13. The smaller a13 is, the more bp and rbf must differ to induce a wide difference between ILt and tb. Sb Sb Lpt and Lft represent the initial quantity of loans the banks are willing to supply the public and the firms from unborrowed reserves. These figures do not necessarily represent the actual amount of loans made. Let LDb a LDb made. Let LDb and Lft represent the quantity of bank loans demanded by the public and the firms in period t. The actual amount of bank b b loans made to each sector in period t, Lpt and Lft, depends on the quantities demanded and the quantities supplied from unborrowed re serves as well as the banks' willingness to engage in, and ability to obtain, rediscounting. Sb Sb Db Db When L + LSb > L + Lt, no rediscounting occurs. In this case Pt ft Pt ft the final amounts lent are given by Lpt = Lpt (2.99) b Db Lft = Lft (2.100) This situation is illustrated in Table 1. Db Sb In this case, even though Lpt > Lpt, the entire public loan demand was satisfied by shifting a portion of the initial (and unlent) allocation for loans to firms over to public loans. Firms were also able to borrow the amount they wished. Sb Sb > Db Db TABLE 1. Lpt + L Lb >Dt + Lft Sb Lb2Db b Lp = $100LDb = $125 L = $125 j7Sb Db b Lt = $200 Lft = $160 Lft= $160 Sb Sb Db Db b b Lpt + Lft = $300 Lpt + Lft $285 Lpt + L = $285 Sb Sb Db Db When Lt + Lft < Lpt + Lft the final amounts lent depend on the amount of rediscounting. The banks' total demand for rediscounting is given by d Db Db Sb Sb 1 dt = Lt + Lft (Lpt + Lft ) do ( rbpt rdt di (). (2.101) bft rdt It is convenient to decompose 2.101 into the banks' demand for d rediscounting to make additional loans to the public, dd, and to make additional loans to firms, dd. d = Db Sb Sb Db) d (2.102) dot = pt pt t ( ,f _ (2.102) rbpt rdt d Db Sb Sb Db d1 dt = t Lt (Lt Lpt ) rb (2.103) rbft Tdt where d d d dt + ddt = dt. (2.104) Sb DbSb Db The terms (L tb L t) in 2.102 and (L, Lp) in 2.103 enter these equations only when they are negative, that is, only when the 39 THE MODEL 40 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM quantity of loans demanded by one sector is smaller than the quantity the banks are willing to lend to that sector. This situation allows the banks to "shift" more funds in amounts equal to (tL Lft ) or Sb Db (Lp L Db) to the other sector, thus reducing the banks' demand for rediscounting. The parameters do and di are positive and assumed to be greater than 1. Their sizes determine how responsive the demand for discounting is to differences between the rates) of interest on loans and the redis count rate. The closer rd comes to rbp(rbf), the greater is do/rbpt  rdt(dl/rbft rdt) and the smaller is the amount of discounting the banks are willing to engage in. Note that the constructions of 2.102 and 2.103 imply that the amount of rediscounting will not be sufficient to meet the entire excess demand for loans. The addition of a constant term to these equations would make this possible for certain combina tions of the loan rates and the rediscount rate. These terms have been omitted to reflect more strongly the banks' assumed reluctance to engage in rediscounting. Based on the previous assumption, the actual amount of rediscounting is always equal to the amount of discounting demanded: d, = ddt + dI. (2.105) d d We shall use the terms dot and dt to stand for both the quantities of rediscounting demand as well as the amounts of rediscounting made because of the excess demand for loans from either the public (ddt) or d Sb Sb the firms (dd) in period t. Thus, in the case where L + Lb < Lb + Db b tbby Lft the actual amounts lent to each sector, Lpt and 1t, are given by b Db Sb Sb Db +d1 Lpt =min LDbp, L + (Lt Lf ) + d t (2.106) and b m (Db 7Sb Sb Db d 4t min L It + (Lt Lt ) + ddt (2.107) Sb Db Note that the terms LE Lt do not enter 2.106 and 2.107 if they are negative. In the case being considered, only one of these terms may be positive, although both may be negative. In that event, the corres ponding rediscount term will be zero. Table 2 illustrates the situation in which one of these terms is positive while the other is negative. Equa tions 2.106 and 2.107 can also be used to express the final amounts lent Sb Sb Db Db when L +I, > L + Lf since neither type of borrower can be induced to borrow more than the quantity of loans he initially demands (LD or Lf ). Equations 2.99 and 2.100 express this in much simpler form than the more general relationships 2.106 and 2.107. TABLE 2. L L <0, Lp LD >0 p S D S D L 100 L 150 L L = 50 LS = 100 Lp = 90 L L = 10 L + Ls = 200 < L + LD = 240 d _ d = 90 100 (100 150)  rbp rd =10 0 o < 0 =0 rbp rdt di dd = 150 100 (100 90) dl rbf rd = 50 10 = (by assumption on dl, rbf, rd) = 35 rbf rd Lt = min 190, 100 + (100 150) Lb = min 150 100 + (100 90) +0 + 35 = min 90, 100 + 0 = 90 = D = 100 + 10 + 35 = 145 Lft The rates of interest on bank loans in t are given by Db Sb rbpt = rbpt1 + ap(Lpt1 Lpt 1) + bp(rbpt1 rnpt1) Db Sb rbft = rbft1 + af(Lt 1 Lft1 + bf(rbft rnftl1) (2.108) (2.109) where ap and af are positive constants. These equations embody this conceptualization: At the beginning of period t, banks adjust their rates 41 THE MODEL 42 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM either up or down, depending on whether there was an excess demand b Sb Db Sb (Lt1 Lt 1 > 0) or an excess supply of loans (L 1 L_ 1 < 0) in the previous period. The amount of the adjustment depends not only on the size of the previous excess supply or demand, but also on the sizes of ap and af. Other rates of interest are not explicitly included in 2.108 and 2.109, as their impact on rbp and rbf is contained in the terms LDb and L b Concluding this section we have the simple statement that the banks' assets and liabilities must be equal. t b Tt + Dt = Cbt +Gbt+ Fbt + Rt + Z L  i=t n t Z di. (2.110) i=tn The banks' income statement in period t is more complex as it must take into account the effects of rediscounting on loan profitability and the possibility of capital gains or losses incurred on government and firms' securities. Let 7r1pl represent the total stream of profits from loans made in period 1 at maturity. Then, without rediscounting, under the assumption of no default, n7 = (1 + rbpl) Lpl Lpl = rbplLpl. (2.111) The profit from loans made in period 1 in any one period, again without discounting, is simply (1 + rbl)Lpl l = rbplLPl (2.112) n n n The total stream of profits at maturity from loans made in period 1, given that a portion of them are rediscounted in periods after period 1, is given by m nm r (1 + rbl)L + dm + l, + [( )(1 + rbl) [L ndm + l, ] L (2.113) [ (n m)(l rdm +)(1 +rb 1 m where n (1 + rbl)LI gives the repayment of interest and principal received by the bank prior to rediscounting a portion of L1; dm + 1, is the proceeds of rediscounting a portion of L, in period m + 1; THE MODEL n m ( )(1 + rbl) [LI ndm + 1, /(n m)(1 rdm + i)(1 + rbl)] L is the repayment of interest and principal of the portion of L1 not rediscounted in period m + 1; and L1 is simply the face value of the loans made in period 1. Equation 2.113 reduces to 1 l =rblLI + dm +, [1 ] rm (2.114) 1 rdm + 1 The last term in 2.114 will be negative since 0 < 1 rdm + I < 1 and measures the loss of profit on L loans caused by rediscounting a portion of them. Profit on LI loans for the period in which they are rediscounted is given by m + 1 ndm +1, i 1( b+ r I [ (n m)(l + rbl)(l rdm +1) n nd 1 S[L1 ndm + 1, I ( ) + dm + (n m)(1 + rbl)(1 rdm +l) n 1 [1 ] (2.115) 1 rdm + which reduces to m+1_ rblL 1 S7r1 n +dm+ 1, (n m)(1 + rbl)( rdm + ) 1 1 1 (2.116) (n m)(1 rdm + 1) 1 + rdm + 1 When no rediscounting occurs (dm + 1, = 0), 2.116 is obviously equiva lent to 2.111. Equation 2.116 serves as the basis for expressing the profit from all loans in any one period. Letting dji represent the .th unrepaid principal and interest of a j period loan rediscounted in period i, we have m m+i m +1 rbji > j(L dji) In I [ i= m n j=m+ln n 1 +dm+ 1,j n(1 + m+ d 1,+ +(n m)(1 + rbj)(1 rdm + 1) 43 44 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 1 1 )l. (2.117) (n m)(1 rdm +1) 1 rdm +2 7 Profits on loans to the public, p+ is given by using rbpj, dpji, and dpm + 1. j in 2.117, while lrf is obtained by using the corresponding rate and rediscounting measures for loans to the firms. Thus, +1 +p 1+ 7 (2.118) Profit in period m +1 from government securities, rrm ,+ is simply m +1 gm Gb + 1 (2.119) g =gm + 1 bm + V No capital gains or losses are made on government securities as a result of the assumptions of oneyear maturity and no intraperiod trading of the securities by the banks. Profit in m +1 from firms' securities, Tr + includes both interest and possible capital gains (losses). Thus, m + m+ 1 rfiBbfi m+1 = + I i=m+lk k i=mk (Pbfm + 1 Pbfi) (Fbfm + 1 Fbfi) (2.120) where k is the maturity of the firms' bonds and Fbfm + 1 Fbfi does not enter the equation unless it is negative, e.g., unless the banks actually sell period i bonds in m + 1 to actually realize accrued capital gains or losses. The banks' overall gross profit (before payments to owners, purchase of factors, and payment of interest on time deposits) in m + 1, 7rm + 1, is given by m+l m+l nm+l m+1 m+l 7b1'b 7rp irf +irrg +rf (2.121) The only portion of 1nm + that is not received directly by the public in the form of income in m + 1 is the portion used to purchase capital, Pkm + 1 Xkbm +1* Thus, the banks' contribution to the public's income in m+ 1, Y + 1, is Ym +l = m+l Yb =b km + 1 Xkbm + 1( (2.122) THE NONBANK FINANCIAL SECTOR (INTERMEDIARIES) The nonbank financial sector is assumed to have two major functions. It accepts deposits from the public and the firms and lends to each of these sectors. The insurance function of this sector will not be explicitly recognized. Rather, deposits will be taken to include not only the typical savings deposit at, say, a savings and loan institution but also insurance premiums. Payments on insurance claims will be included in any withdrawals of principal plus interest from the "savings" accounts. For simplicity, it is further assumed that the rates of interest paid on these deposits are the same for the public and the firms. The deposits in the nonbank financial sector are not assumed to be part of the stock of money. With the submersion of the insurance function, the major impact of the nonbank financial sector will be on the banking sector with which it competes both for deposits and for loans. The following relations describe the aggregate activities of the non bank financial sector. N=Ns ND. (2.123) The actual level of deposits, N, is identically equal to the demand by the firms and the public for them, i.e., the supply of deposits is perfectly elastic. N = L4 + L + Gn + Dn + Bn + Cn + Tn. (2.124) This is simply the balance sheet equation for the nonbank financial sector. Note that 2.124 reflects the assumption that the nonbank sector has no direct connection with the banks except to hold demand deposits and time deposits. S L N+A (2.125) L = VnN + A14rn (2.125) D = dnN + A6rn (2.127) FD = bnN+A17n (2.128) C cn n 1(272n D = Cn =c nN. 45 THE MODEL (2.129) 46 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Equations 2.125 through 2.129 are the basic decision functions of the nonbank sector. Equation 2.125 gives the maximum aggregate amount of loans the sector is willing to make while 2.126 to 2.129 represent the demand for nonloan assets, given that the nonbank sector is able to loan all it desires. Insufficient demand for loans will result in additional holdings of government securities, firms' securities, cash, and demand deposits as specified below. n, gn, dn, bn, and cn are positive constants, while rnp is the rate of interest on loans to the public, rnf the rate on loans to the firms, and rf the rate paid on deposits. All other symbols are as defined previously. The aggregate amount willing to be lent, LS, is broken down between the firms and public in a manner analogous to that of the banks. LPt = ps + a8 (rnpt rnft) (2.130) Sn tnSn ( rn) (2.131) stn QfLs al 8 (rnpt ( ft) (2.131) Lpt +Ln t Ln. (2.132) Equations 2.130 and 2.131 are analogous to 2.97 and 2.98 for the banking sector. n and nf represent institutional factors influencing the desired distribution of loans between the public and firms. + = 1. al g is a positive constant whose size determines the importance of interest rate differences in loan distribution. Adjustments in the initial breakdown occur if either the public's or the firms' demand for loans from the nonbank sector is less than the initial amount the nonbank sector is willing to lend while the demand from D S the other sector is greater than the initial amount. Thus, if Ln < Lnp D S and Lnf > Lnf, then Lnp, the actual amount lent to the public, is equal to L and Ls = min (L + L LDp L LD ). Similarly, if Lp np nf nf np Lnp' np n SLp and Lp, < Lnf then Lp = min (np + L L,, Ln LD ) In either case the amounts actually lent in each sector are given by Lnp=min Ls + (n LLD"), LD (2.133) and Lnf= min S +(Sn + LDn.),Ls. (2.134) The total amount lent, Ln, is simply the sum of 2.133 and 2.134. In no case can Ln be larger than Ln. It may, however, be smaller. In this case THE MODEL the holdings of interest earning assets are increased above the levels given by 2.75 through 2.127 as given below. S D L~ Ln = AG ifrg > r,,rf (2.135) S D L Ln = AB if r>r, rg (2.136) S D L Ln= AT if r > rg, rf. (2.137) If two of the rates (rg, rt, rf) are equal and the third smaller, the increase in demand for those assets will be equal to onehalf of L  Ln. If all three rates are equal, the increase in demand for each asset will equal onethird of Ln Ln. Since the supplies of government securities and time deposits are assumed to be perfectly elastic, no complications arise if either 2.135 or 2.137 hold. If 2.136 holds, it is possible that, since the supply of firms' securities is not perfectly elastic, the nonbank sector may not be able to acquire all the firms' securities it desires. In this case, either government securities or time deposits will be increased in an amount equal to the unsatisfied demand for firms' securities, depending on the relative sizes of rg and rt. To describe the determination of the various rates of interest associ ated with the nonbank sector, we again introduce the subscript t to represent time. Note that this subscript has been omitted from the first fifteen equations merely for convenience. We have Sn D rnft +1 = rnft + al9(Lft Lf) + b9l(rnft rbft) (2.138) where a19 > 0 and b19 <0. pt + 1 = rpt + (Lpt npt) + b2o(rnpt rbpt) (2.139) where a2o > 0 and b2o < 0. S D D nt + 1= rnt + a21(L Lp Ln,) (2.140) where a21, b21 < 0. For the nonbank sector to be in equilibrium these conditions must be satisfied: nt + i = rnt + i 1 47 (2.141) 48 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM rnft + i = rnft + i 1 rnpt + i = pt + i I Lnt = Lnt F =FD (2.142) (2.143) (2.144) (2.145) Note that desired holdings of time and demand deposits, government securities, and cash will be satisfied under all conditions. The intermediaries' "balance sheet" was given in Equation 2.124. Their "income statement" provides Yn, the contribution of the intermediaries to the income of the public in period t. For simplicity, it is assumed that all loans made by the intermediaries have a maturity of n years. Since we do not provide for any governmental sources of reserves for the intermediaries (no rediscounting of their loans), the profit on loans t in period t, n7r, is simply t t 1 i=tn 1 r Ln. t 1 n i=Ptn) n i= t n1 rnfif i n (2.146) Profit on government securities in t, gn, is given by (2.147) Profit on firms' securities is t m + 1 rfi Bnfi i = m + 1 k k + afn  i=m+k k m + 1 i=mk (Pbfm + 1 Pbfi)(Bnfm + 1 Bnfi) (2.148) t which is equivalent to 2.120. The intermediaries' gross profit in t, rn, is tn fin + + t t 7Tn = 7rjn + 7Tgn + ITffn" agn = gtGnt (2.149) The only portion of rtn not received directly by the public in the form of interest payments, labor payments, or dividends is the portion used to purchase the capital good, PktXknt. Thus, Y=t t PktXkt. (2.150) THE PUBLIC SECTOR (HOUSEHOLDS) The public sector is composed of all individuals in the economy acting in their roles as consumers and suppliers of factors. Only the aggregate behavior of this sector is considered; no attempt is made to distinguish among different individuals or groups of individuals. The basic relationships for the public sector are: Dpt = klYt + A22rp (2.151) Cpt = k2Yt + A23rp (2.152) Tpt = k3Yt + A24p (2.153) Cnt = cYt + A2 Tp (2.154) Gpt = gYt + A26rp (2.155) Db =kb + (Lbtl Db Lptb = pY, + A2rp + (L L 1) (2.156) Dn = n + + n( Dn L = nYt + A28,r p t l Lpt ) (2.157) Fp = fpYt + Arp (2.158) Npt = npYt + A3rp (2.159) Equations 2.151 through 2.159 express the public's demand for demand deposits, currency, time deposits, the consumption good, government securities, loans, and firms' securities in dollar terms. Pc is not only the price of the consumption good but, since there is by assumption only one composite consumer good, it also serves as the consumers' price index. Equations 2.151 through 2.153 require no further comment. Equation 2.154 expresses the dollar value of the public's purchases of the con 49 THE MODEL 50 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM summer good, e.g., Cnt = Pctcpt. It does not include the value of the consumer good transferred to the public sector by the government. Total consumption by the public sector of Xc in period t is given by Cnt + PctXcgt (see the section on the government sector). Equations 2.156 and 2.157 express the public's demand for loans; their sum is the aggregate demand. It is assumed, for simplicity, that an unsatisfied demand for loans from one sector will not increase the quantity of loans the public demands from the other sector in the same time period. Unsatisfied loan demand in t causes the quantity of loans demanded in t + 1 to increase by an amount equal to the excess demand. Equations 2.155 and 2.158 are selfexplanatory. Other relations for the public sector include: (1) the definition of the public's gross income, Yt: Yt = Ygt bt + Ynt + Yft + rgtGpt + rttTpt + rntNpt + rftFpt; (2.160) (2) the definition of disposable income, Yt: Yt = (1 t)Yt; (2.161) (see the section on the government sector). THE MARKETS In this section I attempt to connect the previous sections by examining the various markets in the model in greater detail and in isolation. Currency Market The price of currency is the opportunity cost of holding it, the income sacrificed by not holding a returnearning asset. For simplicity it is assumed that this cost can be represented by the largest of rf, rg, rn, and rt.4 Thus, the price of currency for the public is equal to I + max 4. Perhaps a more theoretically aesthetic way of viewing Pc is this: The price of currency is the opportunity cost of holding currency rather than a returnearning asset, e.g., time deposits, deposits in intermediaries, government securities, and firm securities. Let O0 be this cost. Then Oc = Oc(rf, rg, rn, rt) (deposit in intermedi aries) where the function describes the return that could be earned on an extra I rf, rg, rn, rt r while for the firms it is 1 + max Irg rn, r rt since firms do not, by assumption, hold debt instruments of other firms. PC for the intermediaries is 1 + max I rf, rg, rt, while for the banks, Pc is 1 + max rf, rg since banks hold neither time deposits nor deposits in intermediaries. If Pc is the price of currency, we have the typical downwardsloping demand curve. Equations 2.66, 2.90, 2.129, and 2.154 give the demand for currency explicitly. All variables in these equations except max j rf, rg, rn, rt f must be considered fixed when included in Figure 4. Increases in rates other than the maximum rate shift the demand curves downward, since narrowing the difference be tween interest rates makes the security with the higher rate relatively less attractive. Increases in Y, D + T, or PcXc + PkXk shift the curve outward. Note that the demand curve gives the demand for changes in currency holdings. At prices lower than Pco the public (bank, firm, or intermediary) wants to increase its currency holdings, while at prices above Pco, it wants to reduce them. The stock demand for currency can be readily found by combining last period's stock, Ct1, with the desired change in this period. Note that Ct1 establishes the lower limit for changes in currency holdings during period t, since the stock of currency cannot be negative. dollar distributed in the same percentage as the present distribution between time deposits, government securities, and firms' securities. For each sector we have: T G 1. = rt + r + cp Tp + Gp + Bfp + Np Tp + Gp + Bfp + Np Bfp rnNp rf + Tp + Gp +Bfp +Np Tp +Gp +Bfp + Np 2. Ob =rt Tb + r Gb + rf Bfb T + Gp + Bfp Tp + Gp + Bfp Tp + Gp + Bfp Tf Gf rgNf 3. Ocf= + r + 3 cf Tf + Gf g Tf+Gf Tf + Gf+Nf 4 = rtTn + rgGn + rfBfn Tn +Gn + Bfn Tn +Gn +Bfn Tn + Gn + Bfn For simplicity we have chosen not to use this definition of Oc. THE MODEL 51 52 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The supply of currency in this model is not independent of the demand for it, since by assumption we have the government passively issuing or absorbing currency in the aggregate amount demanded by the public, firms, and banks. Thus, we have D(dC) S(dC) (2.162) for all Pc. This aggregate demand for currency is obtained by summing the four sectors' demands for currency. Equation 2.162 holds both for each sector and for the aggregate. Thus, the currency market is in perpetual equilibrium. PC I PC2 I PCO PC D(dC) Ct_, Ct2 dCt $(dC) CC Fig. 4. Demand for currency The Market for Demand Deposits The price of demand deposits, PD, is defined in the same way as the price of currency, i.e., PD = 1 + max rf, rg, rn, rt [ for the public, equal to 1 + max rg, rn, rt t for the firms, and equal to 1 + max {rf, rg, r t for the intermediaries. The demand for demand deposits can be viewed in the same way as the demands for currency, with the obvious exception that in this case there is, again by assumption, no bank demand for demand deposits. Figure 5 is analogous to Figure 4. The banks are willing to accept any amount of new demand deposits and cannot prevent their withdrawal. Thus, again the supply of demand deposits is not independent of their demand, and we have THE MODEL D(dD) E S(dD) (2.163) for all PD. The aggregate demand is the sum of the public's, firms', and intermediaries' demands and 2.163 holds for the aggregate market so that the market for demand deposits is also in perpetual equilibrium. The Market for Time Deposits The price of time deposits, Pt, is also an opportunity cost. Since time deposits earn a rate of return, the price of one dollar in time deposits, Pt, is given by Pt = 1 + max rf, rg rt for the public, Pt = 1 + rg rt for the firms, and Pt = 1 + max rf, rg rt for the intermediaries. (2.164) (2.165) (2.166) P, PDO _I __ ED(dD) Dt_, dD, $(dD) I D, I Fig. 5. Demand for demand deposits The demand for time deposits is also analogous to the demand for currency and is shown in Figure 6. Summing over the public's, firms', and intermediaries' demands again yields the aggregate demand for time deposits. The banks again are assumed to be willing to supply an amount of time deposits and cannot prevent their withdrawal long enough to affect the analysis. Once again, then, 53 54 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM D(dT) S(dT) (2.167) for Pt and the aggregate market for time deposits is always in equi librium. The Market for Government Securities This market is in all respects similar to the ones already described. We have Pg = 1 + max rf, rt rg for the public; Pg = 1 + rf rg for the banks; Pg = 1 + max { rn, rt rg for the firms; P = 1 + max rf, r rg for the intermediaries. (2.168) (2.169) (2.170) (2.171) Pt Pt0 D(dT) Fig. 6. Demand for time deposits Then the demands can be shown in Figure 7. Aggregate demand is the sum of the four sectors' demands. The government is a passive supplier absorber of government securities so that in each market and in the aggregate D(dG) =S(dG) (2.172) for all Pg. The market for government securities is always in equilibrium. The Market for Bank Loans This is the first market to boast a supply function that is independent of demand. The price of bank loans, Pk, is 1 + rbp for the public and 1 + rbf for the firms. The demand for loans is again expressed in terms of desired changes in indebtedness to the banks. Thus, we have in general the situation shown in Figure 8. L 1 is the total indebtedness to the bank (unpaid principal plus interest on loans) at the beginning of period t, and dL represents the change in indebtedness during t. Note that if Pk = P0o, the desired change in indebtedness is zero, but that this does not mean that desired new loans in t are also zero. When Pk = Po0, desired PG PGO PG1 .D(dG) Gt dG $(dG) 1 GI Fig. 7. Demand for government securities loans in t are equal to ft [1 + rb(r)] L(r)dr/n, the amount of loan repayments in t. Thus, desired new loans are zero when PQ = P2 and the demand for loans is given by D(dR) in Figure 8. A graph like Figure 8 exists both for firms and for the public. At any point in time the aggregate quantity of loans the banks are willing to supply is given by the solution to Equation 2.96. The supply of loans to each sector is based on the aggregate figure. The discussion in the section on the banking sector can be shown graphically as in Figure 9. Here Qspi and Qsfi represent the initial amounts desired to be loaned to the public and the firms as given by Equations 2.97 and 2.98. In the situation drawn in Figure 9, excess supply exists in both loan markets and the actual amount of loans made will be QDp + QDf. In the next period both rates THE MODEL 55 PI, P21  L LD(d/) I i D(dL) dL, fL I L, Fig. 8. Demand for bank loans I QSp, = rbp) QSA \bp+bf/ I I I QSA QS,, I I I II I 1+rbF   1+ rbp D(d/L) $ Fig. 9. Disequilibrium in bankloan market will be reduced. An equilibrium situation is shown in Figure 10. Not only do rates fall, but the aggregate amount of desired loans by the banks is also reduced. Figure 11 is a graphical presentation of the adjustment occurring when there is excess demand in one market and excess supply in the other. Here QSff represents the final quantity of loans made to the firms and the two black arrows, ESf, the final total excess supply of loans. Equilibrium in the loan market clearly requires QSp = QDr, QSf = QDf, and QDp + QDs = QSa. Equilibrium is achieved through adjustments of the rates of interest with its impact both on quantity demanded and quantity supplied. QD,,= QS, IQD,=QS, QS, I+I I 1+ rb d/ I I= I Id/) D(d/ Fig. 10. Equilibrium in bankloan market QS,, Qs,., Qs, QSFF ES, ES I I I I 1+rbF QD D(d/), ') D(d/),, QDQ, QDF, $ QC QD$ Fig. 11. Excess demand and excess supply in bankloan market 57 THE MODEL 58 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM The Market for Intermediary Loans The analysis of the market for intermediary loans is identical to that given above for bank loans and will not be repeated. The Market for the Capital Good See the section on production, investment, and growth, and in particular Figure 2, for a discussion of the market for the capital good. The Market for Firms' Securities Debt instruments issued by the firms are held by the public, by inter mediaries, and by banks. The price is again defined in an opportunity cost sense. Thus, Pf, the price of firms' securities, is taken to be: Pf = 1 + max {rf, rn, rt rf for the public; (2.173) Pf = 1 + rg rf for the bank; (2.174) Pf = 1 + max rg, rt rf for the intermediaries. (2.175) The demands can be shown by Figure 12, which is completely analogous to Figure 7 (the demand for changes in holdings of government securi ties). In each period, the supply of securities by the firms is given by the solution of Equation 2.40. The desired change in securities outstand PF D(dF)  nI Fig. 12. Demand for firms' securities THE MODEL ing can be represented by a vertical line. Thus, combining these flow demands and supplies we have, in the aggregate, Figure 13. A Note on Aggregate Demands Prices in the financial markets described have all been framed in terms of opportunity costs. This results in different prices for the same item in different sectors. For example, the price of government securities for the public is assumed to be 1 + max rt rg for the public, but 1 + max rt, rnI rg for the firms. If rf > rt, different prices result. When aggregating sector demands, the price is assumed to be 1/1 + r, where r represents the rate on the item in question. This function does not contradict the sector prices since there is a onetoone relationship between, for example, 1/1 + rg and 1 + max rf, rt, rg and 1 + max rt, rn rg. Fig. 13. Aggregate demand for firms' securities RELATIONS OF THE MODEL We reproduce convenience.5 simply the key relationships for each sector here for Production, Investment, and Growth 1. Aggregate production functions: k = Xk(Lk,Xkk) (2.1) 5. See Literature Cited, items 3, 20, 22, 25, 29, 37, 42, 43, 47, 53, 55, and 57 for a more complete discussion of some of the demand and supply functions presented here. 59 60 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM XC = Xc(Lc, Xk). (2.3) 2. Transformation functions: Xc = T(Xk) (2.5) 2 2 2 (2.9) Xci + Xki = ki. (2.9) 3. Full employment: X = (Xi', Xki) is afullemployment output vector if X 2 2 x' >0. (2.10) Xki ki Xc, Xki (2.10) 4. Rate of growth of the labor force: dX X =X( C). (2.13) dt 5. Stock demand for capital: Dk = Dk(Pk, r, 0). (2.14) 6. Flow demand for capital: dK = nK. (2.15) 7. Stock supply of capital: Skt = Kt 8. Flow supply of capital: Sk = Sk(Pk). (2.16) 9. Balanced growth: Xc kK + XL k kK + k XL. (2.30) 3Kc 3LL aKk aL THE MODEL 10. Supply of labor: S= S (P, L). (2.31) 11. Demand for labor: D = MPkPk + MPkcPXc + E. (2.34) The Firms 1. Desired level of retained earnings: Et = aKt + Int1 + s[PctXct + PktXkt] + nt1 (2.35) ED= f3(Kt) O(Kt1). (2.52) 2. Desired level of financing: FD= nt nt1. (2.36) 3. Demand for loans: Db f f D Lb = al(r1t rnt)+ b1L, (2.43) Dn f f D S= a2(t rnt) + b2 L (2.44) Lf = fFt + A3rf. (2.38) 4. Supply of securities: FS = bfFD + A4 f. (2.40) 5. Desired change in retained earnings: AED = f4(I 1 (Int2). (2.55) 6. Desired distribution of retained earnings: D D D D D D DAft = (Cft, Dft, Tft Gft, Nft). (2.59) 61 62 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 7. Desired level of cash (currency) balances: C= a5 [PtXct + PktXkt] (2.61) = a5 PX. 8. Desired level of demand deposits: Dt = d,(PX) + Af,. (2.62) 9. Desired level of time deposits: TD = tf(PX) + A,7r. (2.63) 10. Desired level of government securities: Gf = gf(PX) + A8f. (2.64) 11. Desired level of deposits in intermediaries: N = nf(PX) + A9Ff. (2.65) 12. Contribution of firms to public's income: Yft = PXt + rgtGft + rttTft + rntNft + ft + Bft A(Aft) Ift Fft PktXkft. (2.71) The Government 1. Tax receipts: T = tY. (2.72) 2. Government spending: T = rG + PkXkg + PcXcg. (2.73) 3. Supply and demand for government securities: Gs GDa. (2.78) THE MODEL 4. Rediscounting: d ddrd). (2.79) 5. Stock of currency: Ct CDa. (2.81) 6. Rate on government securities: rgt = rgt g(GD1 GD2). The Banking Sector 1. Demand for time deposits: T = D + TS + TS. (2.83) T = TD T ftnt 2. The rate of interest on time deposits: r = rtt + a(Lb Lb L,). (2.86) 3. Demand for demand deposits: I) S S S Dt = DD t + Dft + D. (2.88) 4. Level of legal reserves: Rt = rt(Dt + Tt). (2.89) 5. Desired currency balances: Cb = y(D + T). (2.90) 6. Desired level of government securities: Gb = p(D + T) + AorFb. (2.91) 7. Desired level of firms' securities: Fb = p(D + T) + A11Tib. (2.92) 63 64 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM 8. Loan supply: S _^= r(Dt + Tt) + A12rb (2.96) Sb bS Lt = f Lt a3(rbpt rbft). (2.98) 9. Demand for rediscounting: d Db Db Sb ++Sb do dt = Lpt +ft (L +t ft)  rbpt rdt dl (2.101) rbft rdt d Db Sb (TSb Db d t = t Lt t L) do (2.102) bpt dt dd =Db Sb Sb Db dl d= 4 Lft (Lp Lt rb (2.103) bft dt2.103) 10. Actual amounts lent each sector: yb D Sb Sb D b) d Lb = min LDb LSbp + (LS L) + dot (2.106) bt = min Ltb, Lb + (L LDb) + ddt (2.107) 11. Rates of interest on bank loans: rbpt = rbpt1 + apLt1 ptb + bp(rpt1 rnpt1) (2.108) Db Sb rbft = rbft 1 + af(tL1 t 1) + bf(rbft rnft 1). (2.109) 12. Contribution to public's income: m+l =m+1 (2.122) b b km+l kbm+1 The Intermediaries 1. The supply of deposits: N=Ns ND (2.123) 2. Supply of loans: Ls = N + A14 (2.125) Sn n Sn Ls nfLsnt + ai8(rnpt rnft) (2.130) Sn nSn t pLt a18(rnpt rnft). (2.131) 3. Desired level of government securities: S = gN N+A r (2.126) 4. Desired level of demand deposits: Dn = dnN + A16. (2.127) 5. Desired level of firms' securities: D F = bN + A17rn. (2.128) 6. Desired currency balances: D Cn = cnN. (2.129) 7. Actual amounts lent: S Sn Dn D Lnp=min Lnp +(LS ),CL (2.133) (S n Dn S Ln = min Lnf + (Ln LD n), Ls (2.134) 8. Rates of interest: Sn Dn rnft + 1 = rnft + al 9(Lnft L ) + b19(rnft rbft) (2.138) THE MODEL 65 66 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM pt + 1 = npt + a20(Ln Lp)+ b2onp rbp) (2.139) S D D nt + 1 rnt + a 1(L L L). (2.140) 9. Intermediaries' contribution to public's income: t = t P X (2.150) n = PktXknt (2.150) The Public Sector 1. Desired level of demand deposits: Dpt = klYt + A22rp. (2.151) 2. Desired currency balances: Cpt = k2Yt + A23rp. (2.152) 3. Desired level of time deposits: Tpt = k3Yt + A24rp. (2.153) 4. Demand for the consumption good: Cnt = cYt + A25rp. (2.154) 5. Desired level of government securities: Gpt = gYt + A26rp. (2.155) 6. Demand for bank loans: Db b Lt = pYt + A27r. (2.156) 7. Demand for intermediary loans: Lt= nYt + A28rp. (2.157) 8. Demand for firms' securities: THE MODEL 67 Fpt = fpY, + A29rp. (2.158) 9. Demand for deposits in intermediaries: Np = npYt + A3orp. (2.159) 10. Gross income: Yt = Ygt + Ybt + Ynt + Yft + rgtGpt + rntNpt + rttTpt + rftFpt" (2.160) 11. Disposable income: Yt = (1 t) Yt. (2.161) 3. Solution with a Passive Government T HE EFFECTS of changes in the variables of the model on the stock of money, and vice versa, will be considered here, under the assumption that the government is essentially passive, that is, the govern ment does not engage in active monetary or fiscal policy. The reserve requirement, r, is fixed at r*; the discount rate is fixed at r*; and the government is a passive supplierabsorber of government securities. The solution to the model concentrates on two areas: the effects of changes in the variables in the model on the stock of money, and the effects of changes in the money stock on the variables of the model. In the first case the solution is designed to yield the following expressions: aM/ar for all r, 3M/aY, aM/aP for all P, aM/aX for all X, a total of thirteen expressions (M the dependent variable). In the second case, we consider aX/aM for all X, 3Y/IM, aP/aM for all P, and ar/aM for all r, thirteen more expressions (M the independent variable). Due to the complexity of the model to be solved it is not in general true that, for example, aM/aPXc = 1/aP/8aM. Thus, different methods of solution will be used in each case in order to avoid making such (possibly) erroneous assumptions. AN EXPRESSION FOR THE MONEY STOCK Time deposits and deposits in the intermediaries are not considered part of the stock of money. (These could be easily included in the analysis by simply adding T and N to Equation 3.1.) The stock of money in existence in period t, Mt, is thus simply the sum of all currency holdings and all demand deposits: Mt = Ct + Cbt + Cnt + Cft + Dpt + Dnt + Dft. (3.1) Substituting the appropriate expressions from chapter 2 for each of the 68 SOLUTION WITH A PASSIVE GOVERNMENT expressions in 3.1 and simplifying, we obtain M = (a5 + df)(PX) + (dn + cn)N + (k, + k2)Y + y(D + T) + F,(A22 + A23) + fA6 + nAl 6 (3.2) (The t subscript has been dropped in 3.2.) Substituting the expressions for N and D + T from chapter 2 yields an expression for M in terms of PX (the value of goods produced), Y (disposable income), the various rates of interest, and the parameters of the model. M = PX(as + df + nfdn + nfcn + ydf + ttf + ydnnf) + Y(dnp + cnnp + ki + k2 + ydnnp + 7ki + yk3) + rp[(dn + cn + ydn)A30 + (1 + y)A22 + A23 + yA24] + rf[(dn + cn + ydn)A9 + (1 + y)A6 + 7A7] + rn [( + y)A16]. (3.3) Using C1, ..., Cs for the parametric terms in 3.4, we have M = PX Ci + Y C2 + pC3 + TfC4 + nC. (3.4) This expression for the money stock plays a key role in the solution of the model. SOLUTION WITH M AS THE DEPENDENT VARIABLE The solution in this case begins with differentiation of Equation 3.4. This yields the following equations: aM Pax aPX aXk C, [ Xc + Pc + Xk + Pk] + arf arf fc ar, arf C2 +3 + C4 + (3.5)1 ar, ar f r, ar, 1. Equations 3.63.11 have not been reproduced because they follow in sequence from Equation 3.5. 69 70 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM aM Pac SC [ X3 + arnp anp aY arf C2 + C arnp arnp ax, arn c aPk axk + a Xk + ar arnp k np + 4 r arnp arn 5np arnp where the rf,p,n/ar's are vectors of partial derivatives; for example, aT ar ar ar arn ar arf art arnf ar 0,0,arf arf) aM aP X aX aPk aXk am C [ X + c + Xk + k P aY a aY aY aY C2 + C3 + C4 + Cs ay ay ay aM aPc axc C, [Xc + CP aPk PC + Xk c Pc k aY a+rf a 2 p+ C3 + C4 ap aPc aPc aM aPk axk + P] + 8Pc aTn n + C 5 Pc aM aP aPk a C [Pc + xc + a axe 1 axc axe aY a+ f ax axc + C4 ax, X Xk Xk + a Pk] + axC aT + C n Pk] + (3.12) (3.13) (3.14) (3.15) (3.16) SOLUTION WITH A PASSIVE GOVERNMENT 71 aM a .... (3.17) aXk This system of thirteen equations contains the following unknowns: ar 1. V i, j (=1 when i = j) (56 unknowns); arj aY 2. Vr ( 8 unknowns); ar ap 3. V P, r (4 unknowns); 8r ax 4. V X,r ( 4 unknowns); ar aP 5. VP ( 2 unknowns); aY ax 6. V X ( 2 unknowns); aY ar 7. ( 8 unknowns); aY aX 8. VX, P ( 4 unknowns); ap 9. i i j ( 2 unknowns); aP aY 10. VP ( 2 unknowns); aP ar 11. V r, P (16 unknowns); 8P ax. 12. 1 i j ( 2 unknowns); axj 72 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM aP 13. V P, X ( 4 unknowns); aX aY 14. V X ( 2 unknowns); ax ar 15. Vr, X (16 unknowns). ax Expressing the unknowns in (1) above in matrix form we have 1 rgf rnf rtf rbff rbpf rnff rnpf rfg 1 g rtg rbfg rpg rnfg rpg nf rgn 1 tn rbfn rbpn rnfn rnpn rft rgt rnt 1 rbft rbpt rnft rnpt rfbf rgbf rnbf rtbf 1 rbpbf rnfbf rnpbf rfbp rgbp rnbp rtbp rbfbp 1 rnfbp npbp rfnf rgnf rnnf rtnf rbfnf rbpnf 1 rnpnf fnp rgnp nnp tnp rbfnp rbpnp rnfnp where rij = ari/arj. For example, rbfnp = arbf/arnp. The series of l's down the principal diagonal are the rii. Below are the relations describing how the various rates of interest are assumed to change over time: Sn Dn nft = rnft1 + al9(Lft1 Lft ) + b9 (rnft1 rbft1) (2.138) rhpt = rpt1 + a20(Lnpt Lnpt1) + b20 (rnpt1 rbpt 1) (2.139) t= +a( L Ln) (2.140) t =rtt + a( L) (2.86) rt= rt1 +a(Lb 4 tbf (2.86) SOLUTION WITH A PASSIVE GOVERNMENT Db Sb rbpt = bpt 1 + ap(Lpt 1 Lpt 1) + b, (rbpt1 rnpt1) (2.108) Db Sb rbft = rbft 1 + af(t 1 Lft 1) + bf (rbft1 rnft1) (2.109) S D rf = rft1 + f(Bft1 Bt1) (3.18) rg = rgt1 g(Gt 1 GD1). (3.19) Differentiation of these relations with respect to the r's reveals that the terms in the "interestinteraction" matrix depend on the effects of changes in the r's on the quantity demanded and quantity supplied of loans and of firms' securities; the quantity demanded of government securities; on institutional linkages between various rates (such as be tween the rates charged by different sectors on loans to the public and the rates charged by the banks on loans to the various sectors); and on the sensitivity of rates on deposits to either an excess supply or demand for loans in the previous period (the sizes of a19, a2o, a a, ap, etc.). Differentiation of this system would yield a system of fiftysix equations in the fiftysix interestinteraction terms and aY/ar V r, aX/3r V r, X, and aP/3r V r, P. Differentiation with respect to Y will yield expres sions for the ar/aY. Multiplying this result by 3Y/3P V P will yield expressions for ar/3P V P, while multiplying the original result by aY/3X V X gives 3r/aX V r, X. This will be discussed later. The expressions for 3Y/ar, Y/3P, and WY/aX V r can be obtained by differentiating the expressions for Y, Y = PX + rb + 7r + rgG + rtTp + rNp + rfBfp  rbfLbf nfLnf with respect to each of the r's, P's, and X's. The expressions for aP/3r, aX/3r, aX/3Y, aP/3Y, and aPi/aPj can be obtained from the implicit supply and demand functions for Xc and Xk. These are 73 74 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM Sk = k(Pk, Pc ) (3.20) Dk = dk(Pk, Pc, ) (3.21) Sc= (Pk Pc, P Y) (3.22) De = dc(PkPC, 7, Y). (3.23) The general technique is to differentiate both the supply and demand equations for one good with respect to the r's (or Y) and then impose the equilibrium condition that Sx = Dx. For example, differentiating 3.20 and 3.21 with respect to rf yields ask k aPsk k + sk 8Pc + sk ar brf aPk arf aPc arf ar arf aDk adk aPk /adk Pc adk+ a arf aPk arf aPc a rf ar arf as, aD, At equilibrium k drf Dk drf, so that arf arf (ask Pk a + ak as P + ask ar )dr aPk arf aPc rf +a arf adk aPk adk aP + adk ar ( ( + + ) dr,. (3.24) aPk arf aPc arf aT /arf Canceling drf from both sides, and simplifying, aP adk ak s) 7 a dk ask aPk arf aP, aPc a) f ar ar (. (3.25) arf ask dk aPk aPk Differentiating 3.22 and 3.23 with respect to rf and following the same procedure, aPc arf SOLUTION WITH A PASSIVE GOVERNMENT aPk adc asc af ad ak ( P ) + ( ) arf aP, apk arf ar as, ad, aP aPc 75 (3.26) Equations 3.25 and 3.26 form a system of two equations that can be solved simultaneously for the unknowns aPk/arf and aPc/arf. Repeating this procedure will yield aP/ar V P, r. S2 S, I 2 l IX] =s d D, =d, =X, Fig. 14. Changes in P and X The same system (3.20 to 3.23) is used to solve for aX/ar. We start with an equilibrium situation where s, and dI (the K and C subscripts) have been omitted since the technique is the same for both. See Figure 14. We then imagine a change in one of the elements of r that results in a shift in both the demand and supply curves to D2 and S2. This results in changes in both X and P. The expression for aP/aX was developed in the last paragraph. The equilibrium change in X is obtained in this manner: X, = dk(P1 Pc, = Sk(P,Pc, ) aP aP aT X2 = dk(P + dr, Pc + dr, + dr) ar ar ar P, D, 76 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM aP aPe a8 Sk(Pl + dr, Pc + dr,r + dr). The change in X, given the change in r, is simply X2 X1 or aP aPe aF dk(P + dr, P + dr, + dr) dk(P,P,,) ar ar ar ar dk(Pl C) AX aP aP ar aP aP, a8 = dk (dr, dr, = Sk( dr, a dr, dr). Ar ar d r 8r 8r r ar (3.27) In the limit as Ar + 0, AX/Ar ~ aX/8r which is still given by either expression in 3.27. Repetition of this process yields aX/ar for all X and r. The expressions for 8P/8Y and ax/8Y are obtained in an analogous manner which will not be repeated here. The same technique is also used to obtain expressions for aPi/aPj. The expressions for aXk/aXc and aXc/aXk can be obtained directly from the transformation function. These relations obtained by differenti ating the transformation function hold only in a situation of full employment. At less than full employment, these rates of change may approach + oo if the economy begins to utilize previously unused capital and/or labor. The expressions for aXi/aPi are obtained in a manner analogous to the above by differentiation of the supply and demand functions and the imposition of the equilibrium condition that quantity supplied equals quantity demanded. Before commenting further on the solution with M as the dependent variable, we will consider the solution with M as the independent variable because of the close similarity of the technique in this case and that used above. SOLUTION WITH M AS THE INDEPENDENT VARIABLE The rates of change we wish to develop here measure the effects of changes in the stock of money on the key variables in the model. Thus, we are interested in obtaining expressions for aXk/aM, a8X/aM, aY/aM, aPc/aM, aPk/aM, and ar/aM V r. As indicated earlier, it is not sufficient to assume that these rates of change are simply the inverses of those obtained earlier due to the complexity of the model. They may be, but in general it cannot be expected that they will be. SOLUTION WITH A PASSIVE GOVERNMENT To obtain expressions for aP/aM and aX/3M, the implicit supply and demand functions 3.20 to 3.23 are again used. Differentiation of 3.20 and 3.21 with respect to M yields as aSk asPk ask aP _Sk r k +kk + ak (3.28) 3M aPk 3M aPc aM 3T aM aDk adk aPk adk aPc adk ar + + (3.29) aM aPk mM Pc a3M Ba aM Again, for equilibrium, aSk/aM dM = 3Dk/aM dM so that, by equating 3.28 and 3.29 and simplifying, we obtain aPc adk ak a ( adk ask aPk aM cP, aP, 3M a a (3.30) aM ask adk aPk aPk Proceeding in the same manner we obtain the expression for 3aP/aM. This system can then be solved for aPc/aM and aPk/aM in terms of the parameters of the supply and demand functions and 37/aM. The expressions for aX/aM we obtained in the same manner in which those for aX/ar were obtained in the section on solution with M as the dependent variable. Thus, aXk aP k aP aY d dM, dM dM) aM aM aM aM (3.31) aPk aPe aT =sk ( dM, dM, dM) aM 'M '3M and similarly for aXc/aM. The expressions for ar/aM are obtained from the relations in the section on solution with M as the dependent variable, describing the determination of various rates of interest. Each of these relations is differentiated with respect to M. In general, the expressions for ar/aM depend upon the effects of changes in the money stock on the demand and supply of loans, firms' securities, and government securities. These effects are, in turn, primarily dependent 77 78 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM upon the influences of changes in the money stock on the various rates of interest. This procedure yields a system of eight relations in ar/aM which can then be solved simultaneously, yielding solutions in terms of aY/aM, aX/aM, and aP/aM. The expression for aY/aM is obtained by differentiating the expression for Y, Y = PX + irb + r + rgG + rtTp + rnNp + rfBf  rbfLbf rnfLnf, with respect to M, yielding aY_ aX aPc aXk aPk air aM aM c + M c + k + Xk + aM aM aM +M aM aM arn+ aGp, arg art arp r + arN aMt g aM am aM am M aM N P Sr, + B + f r (r Lb + rb + aM a M f aM M f aM f arnf aL nf (3.32) aM aM rnf where aY ab a a n an aY aLn a3 aM MaM' aM mM' aM aM a ' Gp = f3(Y af aT aY ay aN aY a aM aM' aMa' aM aM' aM' aM aM aM aB mY m 7 am am amLs aBfp f ( a ) at aLbf f( P D ai aM aM' M' aM aM' aM aM Lnf = f8 X aLs a aM aM aM aM' SOLUTION WITH A PASSIVE GOVERNMENT 79 THE "SOLUTION" The preceding two sections together yield a system of simultaneous equations which could be solved yielding expressions for a /aM and aM/a (where represents the variables of interest) in terms of the parameters (the elements of Ai, etc.) alone. No attempt has been made to push the solution to this level. The size and complexity of the resulting expressions would obscure rather than illuminate their eco nomic meaning and significance. Consequently, we will continue to express these relations in terms of partial derivatives, indicating when necessary what variables they depend on. This procedure increases the ease with which the results can be interpreted. The key results from the previous sections are reproduced here. aM aP aX aPk aXk S C[ Xc + c P + pk x + k P + arf ar + arf arf arf aY a a C arn (3.5) C, +CC4 P+ C, (3.5) a rf ar, arf arf aM aP, ax, aPk axk ar [ arX+ a Pc+ ar Xk+ rgPk]+ C aY a+C3 C p + c n (3.6) arg arg arg arg aM aP, ax, aPk axk a =C, [ XC + + k X k + xk Pk] + arn arn arn c arn arn Yf a3 C (3.7) C2 +n + C4'D3 +5+C n (3.7) arn arn arn arn aM aP ax aPk axk C1 [ Xc + a P + Xk + Pk] + art art ar c art art C2 + C f + C + C (3.8) art ar, ar, art 80 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM aM aP, 8X C1 [ P Xc c arbf arbf arbf aY aFf Ca a+ f+ arbf arbf aM aP a = C c Xc arbp arbp aX + b arbp S aPk P c+ abf C r P 4 arbf aXk Xk + Pk] + rbf aTn + C arbf (3.9) aPk aXk PC + ar Xk + xk Pk c bp 3bp aY caf C2 3 + arbp drbp aM aP aN = C [ cX +  arnf arnf c ar, c aY ar S  + 3 nf arnf arnf aM rn p aPc aXe  a [c X + xc ar p arnp aY C, + arnp aPk arnf axk Xk + axk pk] + knf ac + arn Srnf arnf aPk P + Xk ac rnp axk arnp aTn +a n arnp C3 3 + C4 np r np larnp aM aP ax, aPk axk aY C x + Y Pc+ aY X + aY Pk] ay ay ay ay ay C + C a 2 3 aM a C1 [ x + aP c aY a8f C a + C '3 +PC 3 a aY axc aPk aPc + aP + C4  c4 a c~ aTn aY aXk Xk + Pk] + aPc c a arbp +c5, arbp (3.10) (3.11) (3.12) (3.13) (3.14) SOLUTION WITH A PASSIVE GOVERNMENT aM CaP aXc  = C,[ c xc + c aPk ap, a aY C2 pk aPk aM axc aPc = C, [ ax, a3 aPk + C4 aPk x aPk c ax axc axk Pc + Xk + a kPk + aPk an + C k 5Pk aXk Xk + xk k] + axc axc 3 a X aM aP, ax Ca C[ 1 Xc + axk axk axk aY aTf C2 + Ca +a 2Xk 3Xk aPk _ aM aP aM aM aP adk aM aP ask aPc aPk PC + k Xk+ Pk] + aXk Sax 4 aXk c ar + C , axk aF adk + a (a aM 8a? ask a ) 87 ask dk aPk aPk aPk adc asc aF adc as, ( > ) +  ( a7 ) aM aPk aP, aM aP a asc adc aPc aPc aX aPk aP axk dk( dM, dM, aM aM aM a? dM) aM aPk aPe a? Sk( dM, dM, dM) aM a 'aM (3.15) + C4 ax axc ac + s axc (3.16) (3.17) (3.30) (3.30a) (3.31) 81 82 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM a d(aPk aP a a aM aM aM aM aM =(aP aP, a aY) (3.31a) aM aM aM aM aY ax, a, aXk aPk aM aM c +M P Xc M k +" k+ b n + r + kG + tT + aMM aM M M aM am am aG ar arm aTr arn aN arf M rt + N+ N rn + rB + aM aM aM aBfp arbf aLf arnf rM ( aM Lf + bf + a L nf + a nf) (3.32) aM It has been indicated previously how the terms on the righthand side of these relations can be obtained. We now attempt to examine these relations in more detail and to breathe some economic meaning into them. The expressions aM/ar depend upon the effects of changes in interest rates on prices, real output, income, and all other interest rates (as well as institutional factors which are represented by the values of the constants Ci). Consider an increase in one rate of interest, r*. The following statements consider the effects on P from the supply side only. In general, aP/ar* > 0 when r* is a cost to the firm (rbf, rnf, and rf fall into this category). On the other hand, when r* represents a return to the firm (rg, rt, rn), the effects of an increase in r* on prices will be less. Conceivably, in some cases, aP/ar* for some P and r* could even be negative. When r* represents a rate not directly related to the firms (rbp or rnp), aP/ar* will be very near zero. From the demand side, an increase in r* represents a potential increase in Y when r* is rg, rn, rt, or rf. In these cases, increased Y will also increase the demand for goods and thus tend to increase P. When r* is either rbp or rnp, the net direct effect on Y of an increase in r* will be zero since increased SOLUTION WITH A PASSIVE GOVERNMENT interest payments will result in increased profit distributions to the owners of the financial sectors. (Indirect effects on Y may be positive or negative depending on the responsiveness of actual amounts lent to changes in r* and the multiplier effects of changes in loans on income.) In general, therefore, we would expect aP/ar > 0. We would expect aXk/ar* < 0 when r* was a cost of investment (rf, rbf, or rnf), in keeping with standard investment theory. Likewise aXc/ar* < 0 would be expected in this case since r* represents a cost to the firm. See Figure 15. ACo P=MR X Fig. 15. ax/ar <0 Here AC' (> AC0) is the average cost curve after an increase in r*. Notice that when the demand side is considered as well, it is necessary to point out that increased r* causes an increase in Y which would tend to increase demand and thus X. This effect in general would not be large enough to offset the reduction in X noted earlier, since that reduction itself causes Y to fall, ceteris paribus. When r* is not directly related to the firm (rbp or rnp), we assume aX/ar* = 0. When r* represents a source of income to the firm (rg, rt, rn), we expect that aX/lr* > 0 (although these effects are probably small). One straightforward way to think about these effects is to note that increases in these rates may reduce the firms' dependence on 83 84 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM financing provided by banks and intermediaries and thus permit greater selffinanced expansion. We have already referred to the effects of changes in r* on Y as a secondary effect in discussing aP/ar* and aX/ar*. It also enters the expressions for aM/ar* directly. The previous discussion will not be repeated here. The interestinteraction terms ari/arj enter the expressions via the last three terms on the righthand side of 3.5 through 3.13. We hypothesize that 3ri/3rj > 0 for all i,j. This is equivalent to saying that all interest rates tend to move in the same direction. Clearly, the size of the expression will vary, depending on the closeness of the relation between the two rates. For some pairs we would expect this relation to be quite strong (such as arbp/arbf), while for others it may be quite weak (such as art/arnf) We now turn to a discussion of the constant terms C1 through Cg. C1 = (as + df + nfdn + nfCn + ydf + ytf + ydnnf). Table 3 contains the definitions of these terms and their signs. Clearly, C1 > 0. The value of C, tells us by how much the money stock in creases, given a onedollar increase in PX, as a result of the firms' increase in demand for money (as and df) and their deposits in banks and intermediaries, which in turn cause these sectors to increase their demands for money (nfdnthe increase in the intermediaries' demands for demand deposits as a result of firms' increasing their deposits in the intermediaries, etc.). The last term, ydnnf, is a "third generation" effectthe increase in banks' demand for currency, caused by an in crease in intermediaries' demand for deposits, which was in turn a result of an increase in the firms' demand for deposits in the intermediaries. C2 = (dnp + cnnp + kI + k2 + ydnnp + yk1 + yk3). Table 4 gives the definitions and signs of the terms in C2 not in C1. Thus, C2 is clearly greater than zero. Its interpretation is analogous to that of C1, except that it measures the effects of an increase in Y on the public's demand for money and the effects of changes in the public's demand on the banks' and intermediaries' demands for money. C3 = [(dn + cn + 7dn)A30 + (1 + ^)A22 + A23 + 7A24] SOLUTION WITH A PASSIVE GOVERNMENT TABLE 3. Terms of C1 Term Sign Definition a5 > 0 Coefficient of PX in firms' demand for currency df > 0 Coefficient of PX in firms' demand for demand deposits nf > 0 Coefficient of PX in firms' demand for deposits in intermediaries tf > 0 Coefficient of PX in firms' demand for time deposits 7 > 0 Coefficient of D + T in banks' demand for currency cn > 0 Coefficient of N in intermediaries' demand for currency dn > 0 Coefficient of N in intermediaries' demand for demand deposits Table 5 gives the definitions and signs of the new terms in C3. Thus, with the exception of c30 and d24, all terms in A30, A22, A23, and A24 are negative since, with the exception of these two terms, they represent the coefficients of rates of interest on competing assets for the public. The interpretation of the actual terms in C3 is straightforward. For example, (dn + cn + ydn)A30 gives the impact of changes in the public's demand for deposits in intermediaries (resulting from a change in some element of Tp) on the intermediaries' [(dn + cn)A3o] and the banks' (dnA3o) demands for money. TABLE 4. Terms of C2 Term Sign Definition np > 0 Coefficient of Y in public's demand for deposits in intermediaries k1 > 0 Coefficient of Y in public's demand for currency k2 > 0 Coefficient of Y in public's demand for demand deposits k3 > 0 Coefficient of Y in public's demand for time deposits 85 86 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 5. Terms of C3 Term Sign Definition A30 Coefficient of rp in public's demand for deposits in intermediaries a30 < 0 Coefficient of rf in public's demand for deposits in intermediaries b30 < 0 Coefficient of rg in public's demand for deposits in intermediaries c30 > 0 Coefficient of rn in public's demand for deposits in intermediaries d30 < 0 Coefficient of rt in public's demand for deposits in intermediaries f30 < 0 Coefficient of rbp in public's demand for deposits in intermediaries h30 < 0 Coefficient of rnp in public's demand for deposits in intermediaries A22 Coefficient of rp in public's demand for demand deposits a22 < 0 Coefficient of rf in public's demand for demand deposits b22 < 0 Coefficient of rg in public's demand for demand deposits c22 < 0 Coefficient of rn in public's demand for demand deposits d22 < 0 Coefficient of rt in public's demand for demand deposits f22 < 0 Coefficient of rbp in public's demand for demand deposits h22 < 0 Coefficient of rnp in public's demand for demand deposits A23 Coefficient of rp in public's demand for currency a23 < 0 Coefficient of rf in public's demand for currency b23 < 0 Coefficient of rg in public's demand for currency c2 3 < 0 Coefficient of rn in public's demand for currency d23 < 0 Coefficient of rt in public's demand for currency f23 < 0 Coefficient of rbp in public's demand for currency h23 < 0 Coefficient of rnp in public's demand for currency SOLUTION WITH A PASSIVE GOVERNMENT TABLE 5Continued A24 Coefficient of Tp in public's demand for time deposits a24 < 0 Coefficient of rf in public's demand for time deposits b24 < 0 Coefficient of rg in public's demand for time deposits c24 < 0 Coefficient of rn in public's demand for time deposits d24 > 0 Coefficient of rt in public's demand for time deposits f24 < 0 Coefficient of rbp in public's demand for time deposits h24 < 0 Coefficient of rnp in public's demand for time deposits C4 = [(dn + Cn + Ydn)A9 + (1 + 7)A6 + yA7]. The new terms in C4 are given in Table 6. Once again, all terms but c9 and d7 are negative since they represent rates on competing assets for the firms. The interpretation of the actual terms in C4 is analogous to those of the previous C's. [(dn + cn + ydn)A9], for example, is the effect of the firms' changed demand for deposits in intermediaries on the intermediaries' and banks' demands for money. C5 = (1 + y)A16. A1 6 is the coefficient of Tn in the intermediaries' demand for demand deposits. Table 7 gives the signs and definitions of the elements of A 6. Thus, Cs indicates the effects of changes in an element of Tn on the intermediaries' demand for demand deposits as well as the secondary effect on the banks' demand for currency. The preceding discussion and descriptions provide the necessary mate rial to interpret any of the expressions for aM/ar* for any r*. The expression for 8M/3Y contains the same five constants, C1 through Cg, described above. Both the derivatives of prices and physical outputs with respect to income will be positive for obvious reasons. The signs of 3r*/3Y are given in Table 8. The only sign in Table 8 that can be specified exactly without making further assumptions is that of arg/aY, which will be negative as in 87 88 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM TABLE 6. Terms of C4 Term Sign Definition A9 Coefficient of rf in firms' demand for deposits in intermediaries a9 < 0 Coefficient of rf in firms' demand for deposits in intermediaries b9 < 0 Coefficient of rg in firms' demand for deposits in intermediaries c9 > 0 Coefficient of rn in firms' demand for deposits in intermediaries d9 < 0 Coefficient of rt in firms' demand for deposits in intermediaries eg < 0 Coefficient of rbf in firms' demand for deposits in intermediaries g9 < 0 Coefficient of rnf in firms' demand for deposits in intermediaries A6 Coefficient of rf in firms' demand for demand deposits a6 < 0 Coefficient of rf in firms' demand for demand deposits b6 < 0 Coefficient of rg in firms' demand for demand deposits c6 < 0 Coefficient of rn in firms' demand for demand deposits d6 < 0 Coefficient of rt in firms' demand for demand deposits e6 < 0 Coefficient of rbf in firms' demand for demand deposits g6 < 0 Coefficient of rnf in firms' demand for demand deposits A7 Coefficient of f in firms' demand for time deposits a7 < 0 Coefficient of rf in firms' demand for time deposits b7 < 0 Coefficient of rg in firms' demand for time deposits c7 < 0 Coefficient of rn in firms' demand for time deposits d7 > 0 Coefficient of rt in firms' demand for time deposits SOLUTION WITH A PASSIVE GOVERNMENT TABLE 6Continued 89 e7 < 0 Coefficient of rbf in firms' demand for time deposits g7 < 0 Coefficient of rnf in firms' demand for time deposits TABLE 7. Elements of A16(C5) Term Sign Definition a16 < 0 Coefficient of rf in intermediaries' demand for demand deposits b16 < 0 Coefficient of rg in intermediaries' demand for demand deposits c16 < 0 Coefficient of rn in intermediaries' demand for demand deposits dl6 < 0 Coefficient of rt in intermediaries' demand for demand deposits g16 < 0 Coefficient of rnf in intermediaries' demand for demand deposits h16 < 0 Coefficient of rnp in intermediaries' demand for demand deposits ar* TABLE 8. o3r Term 3rf aY arg aY 3rn aY art aY Sign <0 < > < < >0 Term arbf aY arbp aY arnf rnp aY Sign 0 < < <0 < 90 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM creased demand for government securities will bid their price up and the rate down. The signs of the other terms depend on the relation between the impact of changes in income on the demand for loans and firms' securities and the supply of loans and securities. In a strictly partial equilibrium sense, we can say that if the impact on these demands is greater than on the corresponding supplies, the correspond ing partial derivative will be positive. If the impact on the supplies is larger than on the demand, the derivative will be negative. The question of the relative sizes of these effects is not one that can be answered without specifying the actual values of the parameters in the appropriate demand and supply functions. Thus, the answer must be provided either MC AC S P=MR, Po = MRo X XoXi Fig. 16. 3Xc/aPc and aXk/aPk by assumption on the parameters (which would be only a tentative answer subject to empirical verification) which we have, and will avoid, or by empirical estimation. It could (and will be) argued, however, that, since it is to be expected that increases in income will tend to increase the money stock, if some of the ar/aY are in fact negative, they cannot be so negative as to cause the entire expression for aM/WY to be negative. The expressions for 8M/aP also contain C1 through Cs as well as the partial derivatives of Xc, Xk, Pc, Pk, Y, and the r's with respect to the P's. The terms aXc/aPc and XXk/aPk are positive under the assumptions of perfect competition. See Figure 16. 3Y/3Pk and aY/aPc are both positive because of the direct relation between PX and Y, and since the terms 3Xc/8Pc and aXk/aPk are (as argued earlier) positive. The terms SOLUTION WITH A PASSIVE GOVERNMENT aPc/aPk and aPk/aPc are assumed to be positive in deference to the widely observed phenomenon that prices tend to move together. Table 9 lists the remaining terms in the /M//P and their signs. The six terms whose signs are greater than zero simply reflect the fact that as prices increase, so does output, thus increasing the firms' de mands for both internal and external financing and thus, ceteris paribus, aM TABLE 9. Terms of Term Sign Term Sign Term Sign Term Sign arf aPc arg apc arn apc art /Pc arbf aPc arbn apc arnf aPc arnp 8Pc > o >0 >o arf aPk arg aPk art aPk arbf aPk arbn aPk 3rnf 3Pk arnp aPk >0 > o >o the rates of interest paid on the various types of financing. The notation "6 0" is used for the other terms to indicate that they are "nearly" zero, but must be positive since, by our assumption on the interest interaction terms, all interest rates move together. The causality may, however, not run directly from a change in Pc or Pk to a change in the particular interest rate being considered. Thus, the terms aM/aPc and aM/aPk are positive once the assumption 91 92 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM (not a very startling one) is granted that aXk /aPk > I aXk/P I Pk and aXcp/aPc > I aXc/aPk I P. pk aM/aXc and aM/aXk are the last of the key relations in which the five constants C1 through Cg enter. With the exception of the terms aXc/aXk and aXk/aXc all terms in these two expressions will also be positive for reasons analogous to those given in the previous argument. At less than full employment these two terms can also be positive (as noted), even though when operating on the transformation curve they must both be negative. Once again, there is no ambiguity about the signs of aM/aXc and aM/aXk as both will be positive, even with negative axi/axj. We turn now to an examination of the expressions in which M appears as the independent variable. The first two of these, aPk/aM and aPc/aM, we have been assured by many, many economists, must be positive. Examination of 3.30 and 3.30a should, we hope, reaffirm the quantity theory. Clearly, the denominators of both of these expressions are positive, since both adk/aPk and adc/aPc are negative if the demand curves for Xk and Xc are downward sloping. (These derivatives are simply the change in the quantities demanded given a change in price.) What about the numerators? The two terms ask/aT and asc/a7 are negative, since increases in the elements of F represent an increase in costs and shift the firms' supply curve (MC) to the left. adk/aF and adc/aF will both be positive as quantity demanded will increase, given the increase in income caused by increases in the elements of F (This effect will be somewhat dampened if increases in F reduce loans signifi cantly and thus, indirectly, reduce the amounts of Xk and Xc de manded.) adk/aP, and adc/aPk can both be expected to be positive, since increases in the P's will increase income and thus the quantities demanded. So far, all the elements of 3.30 and 3.30a have the proper sign. The only potential source of trouble is in the signs of the elements of aT/aM. In general, it is expected that these terms will be negative increases in the money stock, should it tend to reduce rates of interest. Thus, 3.30 and 3.30a will have the "proper" sign (positive) only so long as aPk/aM (adc/aPk aS/aPk) > I F/aM (adc/aF asc/a) I and likewise in the corresponding expression for aPk/aM. There seems to be little reason to think this inequality will not be satisfied, since price effects should be more important than interest rate effects on quantities supplied and demanded. The derivations of the expressions for aXk/aM and aXc/aM require no further comment since all they amount to is plugging in the equilibrium SOLUTION WITH A PASSIVE GOVERNMENT price change and subtracting from that expression the expression for the original quantity demanded or supplied. The signs of these terms depend on the direction of the effect of changes in M on the demand and supply curves, as well as the location and shape of the initial and final curves. I hypothesize that increases in M cause both demand curves to shift to the right, because of the impact of M on Y. In the event that P MCoD = SUPPLYoLin MC,,, = SUPPLY,,, ACCNEW I ACo,I X, X2 Fig. 17. Effect of dM on MC the supply curve shifts downward, both 3Xk/aM and aXc/aM will be positive. An upward shift in the supply curve is a necessary but not sufficient condition for aXk/aM or aXc/3M to be negative. For nega tivity the reduction in supply must be appropriately large. Whether the supply curves shift upward or downward depends on whether, on bal ance, an increase in M increases or reduces average cost (and thus marginal cost). Interest expenses will tend to fall while the costs of labor and capital tend to increase. On the whole it must be concluded that increases in M tend to increase AC and thus to shift the supply curves to the left for at least a portion of the curve. If changes in M shift not only the position of the AC curve but also affect its shape significantly, the new supply curve (MC curve) may lie above the old curve for other ranges. The argument may be made that increases in M cause such significant increases in demand that firms are induced to build larger plants (perhaps through 0, the profits expectations variable, as well as a result of increases in prices) once a situation like Figure 17 consequently results. In the case illustrated in Figure 17, aX/aM is clearly positive. In general, we will assume that aXc/aM and aXk/aM will be positive, although it is clearly not true that in an ncommodity world aX/3M V X need be positive. Indeed, at full employment in even a two commodity world, increases in M cannot result in changes in the 93 94 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM amounts of both commodities produced. This will be dealt with in more detail. The terms in 8Y/aM are all positive. Clearly aY/aM is itself positive. Since Y is defined as money rather than real income, this conclusion should be obvious. The relations described and discussed above contain the essential information provided by the model on the determination of the stock of money and of the effects of changes in the stock of money on the variables (prices, physical output, income, and interest rates) of the model. These expressions are, of course, rates of change and do not, by themselves, provide us with the actual amount of change in any particu lar circumstance. This information is derived in the following manner. We know that the money stock is, by definition, the sum of currency outstanding (C) and total demand deposits (D). Thus M = C + D. (3.33) Taking the total differential of this expression yields aC aC aC aC aC dM = dc + dY + dPk + dP + dX + 8ar aY aPk caP c a k aC aD aD aD aD dXc + dr + dY + dPk + dPc + axc ar aY aPk aPc aD aD dXk + dXc. (3.34) aXk aXc Equation 3.34 can be simplified (since D + C = M) to aM aM aM aM dM dr + dY + dXk + dX + aY axk Xc aM aM a dPk + dP (3.35) aPk aPc The partial derivatives on the righthand side of 3.35 are the terms developed previously. Equation 3.35 provides us with the vehicle to calculate the change in the money stock resulting from a change in one or any combination of the variables T, Y, Xk, Xc, Pc, or Pk once the size of the changes) is (are) known. 