A general equilibrium study of the monetary mechanism

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 74-13495
ISBN 0-8130-0407-1

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 sectors-the 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-
Schwartz-Cagan and Brunner-Meltzer approaches to the money supply
are the best known today.
The Friedman-Schwartz-Cagan4 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)

High-powered 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 Friedman-Schwartz-Cagan 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., 1867-1960 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 hypotheses-linear 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
right-hand 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
interest-bearing 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- c-h (1-c-h 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)
1-c-h

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 Brookings-SSRC 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 short-run constraints on asset holding.
Solving this system simultaneously, de Leeuw derives the following
expression for the money stock (p. 518):

RESNBC
M 1-RDD + 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 three-month 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 right-hand 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 sectors-the 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 inputs-capital and labor. A unit of labor is
indistinguishable from any other unit of labor. Capital is also perfectly
homogeneous.
2. There are only two outputs-capital and the consumption good.
The consumption good is perfectly homogeneous.
3. Firms fall into two categories-those 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 full-employment 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 full-employment 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 full-employment 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 2.14-16
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 1-6 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, 1-6 are
not sufficient to insure that the equilibrium output vector is also a
full-employment output vector. This is simply because nothing in these
conditions implies that the total stocks of capital and labor are being
used. That is, 1-6 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 full-employment equilibrium.
This condition is simply that the outputs of capital and the consumer
goods that satisfy 1-6 also satisfy

7. Xk X

where Xk and Xc are the outputs resulting from satisfying 1-6. 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 assets-cash, demand deposits,
time deposits, government securities, and deposits in intermediaries-in
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 so-called 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 nt-1 (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

Int-2)] + (Int- 1 Int- 2)

Et =3(Kt) (Kt- 1).

28 EQUILIBRIUM STUDY OF THE MONETARY MECHANISM

The firms' decision-making 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=t-n N i=t- n N

abbreviated ELf. Debt retirement equals

t (1 + rfc(i))Fc(i)
i=t-n 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 together-that 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. Reallocation-All 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 regulation-Fiscal 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 open-market 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 one-year 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 open-market 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 supplier-absorber 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 noninterest-bearing 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 forms-cash, 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 day-to-day
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 reserves-e.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 low-yielding 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 short-run, 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= b-S
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 Lb2-Db 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 Db-Sb 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 = rbpt-1 + ap(Lpt-1 Lpt- 1) +

bp(rbpt-1 rnpt-1)

Db -Sb
rbft = rbft-1 + af(Lt- 1 Lft-1 +

bf(rbft- rnft-l1)

(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
(Lt-1 L-t- 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=t-n

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 n-m
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+l-n 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 one-year 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+l-k k i=m-k

(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 one-half of L -
Ln. If all three rates are equal, the increase in demand for each asset
will equal one-third 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=t-n- 1

r Ln. t- 1
n i=Pt-n-)
n i= t -n-1

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=m-k

(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 self-explanatory.
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 return-earning 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 return-earning
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
downward-sloping 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, Ct-1, with the
desired change in this period. Note that Ct-1 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 bank-loan 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 bank-loan 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 bank-loan 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)

--- n--------I--------------

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 one-to-one 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 a-full-employment 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 + Int-1 + s[PctXct + PktXkt] + nt-1 (2.35)

ED= f3(Kt) O(Kt-1). (2.52)

2. Desired level of financing:

FD= nt nt-1. (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 (Int-2). (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 = r-G + 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(GD-1 GD-2).

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 b-S

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 = rbpt-1 + apLt-1 pt-b +

bp(rpt-1 rnpt-1) (2.108)

-Db -Sb
rbft = rbft- 1 + af(t-L1 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 n-Sn
t pLt a18(rnpt- rnft). (2.131)

3. Desired level of government securities:

S = g-N 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 supplier-absorber 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 3.6-3.11 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 = rnft-1 + al9(Lft-1 Lft ) + b9
(rnft-1 rbft-1) (2.138)

rhpt = rpt-1 + a20(Lnpt- Lnpt-1) + b20
(rnpt-1 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,
(rbpt-1 rnpt-1) (2.108)

Db -Sb
rbft = rbft- 1 + af(t 1 Lft- 1) + bf

(rbft-1 rnft-1) (2.109)

S D
rf = rft-1 + f(Bft-1 Bt-1) (3.18)

rg = rgt-1 g(Gt 1- GD-1). (3.19)

Differentiation of these relations with respect to the r's reveals that
the terms in the "interest-interaction" 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 fifty-six equations
in the fifty-six interest-interaction 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 arn-p

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 right-hand 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
self-financed 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 interest-interaction terms ari/arj enter the expressions via the last
three terms on the right-hand 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 one-dollar 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 (nfdn-the 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"
effect-the 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 5-Continued

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 6-Continued

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 n-commodity 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 right-hand 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.

	Front Cover
	Acknowledgement
	Table of Contents
	Review of the literature
	The model
	Solution with a passive govern...
	Solution with an active govern...
	Results of the study
	Appendix: Definitions
	Literature cited
	Back Cover

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