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An optimal monetary policy in an aggregate neoclassical model of economic growth
JOHN 2. DWICKI University
of Arizona
AKIRA TAKAYAMA Kyoto Unioersity
An Optimal Monetary Policy in an Aggregate Neoclassical Model of Economic Growth* This paper shows that the usual Ramsey-Cass-Koopmans optimal-growth theory is applicable to decentralized monetary economies and illustrates, with a simple model, how optimal growth can be achieved via a simple monetary policy. Securities and the endogeneity aspect of the money supply are explicitly introduced. This paper shows that the steady state under optimal growth is a saddle point, that the dynamic behavior of the capital-labor ratio and real per capita consumption is identical to that found in the usual literature in which money is not introduced, and that the optimal monetary policy is “counter-cyclical.”
1. Introduction The purpose of this paper is to investigate optimal monetary policy in terms of the well-known framework of optimal growth developed by Ramsey (1928), Koopmans (1965), Cass (1965), and others. In particular, we show that this framework, which is typically discussed in terms of a centralized nonmonetary economy, is applicable with proper modifications to a decentralized monetary economy, and we illustrate with a simple model how optimal growth can be achieved via simple monetary policy. Hence the present study, in a sense, follows the works by Uzawa (1966), Foley-Shell-Sidrauski (1969), and Foley-Sidrauski (1971, chap. 15), etc. Although there are many varieties of monetary macro models under a decentralized framework in the literature, we have identified one model in which the Cass-Koopmans-type results on optimal growth for a nonmonetary, centralized economy can be carried over. This model is an extension of the familiar neoclassical monetary-growth model B la Tobin (1965), in which the idea of Fisher’s real return on money and the Wicksellian cumulative process is juxtaposed with the Solow-Swan type of neoclassical descriptive model. ’ This line of thought, developed in terms of de*We are indebted to W.A. Brock, M.C. Kemp, Keizo Nagatani, and two anonymous referees for useful comments. ‘For excellent surveys on this topic see, for example, Burmeister-Dobell (1970, Chap. 6) and Nagatani (1978, Chap. 12). Journal of Macroeconomics, Winter 1983, Vol. 5, No. 1, pp. 53-74 Copyright 0 1983 by Wayne State University Press.
53
John 2. Drabicki
and Akira
Takayama
scriptive models of a decentralized economy, is made applicable to the theory of optimal growth.’ This type of money-and-growth literature [and also Uzawa (1966)] typically assumes that there are only two types of assets in the economy, physical capital and (outside) money, in which the role of the banking sector is virtually ignored. In the present study we explicitly introduce securities (bonds) into the analysis, and incorporate the endogeneity aspect of the money supply, which is derived from the behavior of the commercial banks and of the central bank. As it is shown by Drabicki-Takayama (1980) in terms of a descriptive model, the introduction of securities as an alternative means of a store of value provides additional “flexibility” in the model: Drabicki-Takayama (D-T) have shown that the steady state, under similar assumptions regarding the money supply, can be stable under perfect myopic foresight (contrary to the traditional result that it is a saddle-point). Here we utilize this framework of analysis developed in D-T. Although our model would enable us to investigate optimal-growth policy in terms of other policy parameters, we confine ourselves here to a simple monetary policy in which the policy authorities control 8, the rate of change of the (augmented) high-powered money stock. Among other results, this paper shows: (1) The optimal paths for different initial values define a two dimensional surface. For given initial conditions, the unique optimal path converges to the steady state which is a saddle point. (2) The familiar Cass-Koopmans-type results on optimal growth for a centrally planned economy can be carried over in a decentralized monetary economy. For+ example, the dynamic behavior of the capital-labor ratio and real per capita consumption is identical in these two types of economies. (3) There are no such things as the optimal rate of inflation, the optimal quantity of money, and the optimal value of 8, which are constant along the dynamic path, whereas the steady-state rate of inflation is equal to the optimal steady-state value of the rate of monetary expansion (0) minus the rate of population growth. Point (3) may be useful in shedding some light upon the literature concerning the “optimum quantity of money” by Friedman
‘Optimal growth in the context cussed by Hahn (1969) and Stein optimal growth in a decentralized (1971, Chap. 15) which is, in part, 54
of the money and growth literature is also dis(1970), f or example. A more complete study of monetary economy is given by Foley-Sidrauski an elaboration of Foley-Shell-Sidrausk (1969).
Optimal
Monetary
Policy
(1964) and others3 However, the focus of our paper and the framework of our analysis is different from the one which is typically found in that literature. Throughout the paper, we assume perfect myopic foresight. We also assume that households follow a simple rule of thumb, i.e., proportional savings behavior. Consequently, the problem of time inconsistency which may be encountered in the works of Calvo (1978), Turnovsky-Brock (1980), and others does not arise here.4 Namely, in this paper we simply argue that if the public follows such a rule, then the familiar Cass-Koopmans theorem for optimal growth can be validated for a monetary economy.
2. Framework and Basic Notation We consider here an economy consisting of four sectors: the government, the central bank, the commercial banks, and the public (households and firms). There are five types of assets: physical goods, currency, (demand) deposits, bank reserves, and interestyielding securities (bonds). In Table 1, we summarize the symbols used and the balance sheet of each sector, where all entries are measured in monetary units [see Tobin (1969)]. A positive entry represents an asset for a particular sector whereas a negative entry represents a liability. The symbols Sp, S b and SC represent, reTABLE
1.
The Summary
of the Balance Sheets Sectors
Public
Commercial Banks
Central Banks
Exogenous SUPPlY
Currency Deposits Bank Reserves Securities Physical Capital
L” Lb 0 SP pV
0 -Mb Rb Sb 0
-MC 0 -R SC 0
0 0 0 D” PK
Net worth
A”
0
0
D” + pK
Assets
‘See also Brock (1975), Clower (1968), Hahn (1971); Nagatani (1978, Chap. 13), Stein (1970), Tobin (1968), and Tsiang (1969). For a more recent criticism on this point in terms of stochastic models, see Yoshikawa (1981). ‘For a clear exposition of the time inconsistency problem, see Kydland-Prescott (1977). 55
John 2. Drabicki
and Akira
Takayama
spectively, the net holding (stock excess demand) of securities by the public, the commercial banks, and the central bank. The sum of a column for a particular sector equals the net worth of that sector which gives its balance sheet identity. The entry in the last column is the net exogenous supply of a particular asset. For physical capital (or equities in physical capital), where p denotes its price, its supply is the economy’s inheritance from the past. (Here, for simplicity, we assume away the problem involved in “Tobin’s 9, ” and thus p also signifies the price of current output.) The securities consist of both private and government bonds, which are assumed to be indistinguishable to the public in all respects (such as in the expected rate of return, risk, etc.). The government is a net debtor whose debt consists only of government bonds, the amount of which is equal to D” (“dollars”). All international transactions are assumed away. The first important relation can be obtained from Table 1, as is done in Tobin (1969), by noting that the sum of the final column is equal to the sum of the last row, i.e., D”+pK=A”,
or
D + KS
A;
D 5 D”/p,
A =AA”/p.
(1)
In short, the public’s nominal net worth is equal to the sum of the value of the physical stock of capital, pK, and the total outstanding government debt.5 Next, note that consolidation of the balance sheets of the commercial banks and of the central bank yields Sb + SC = M;
M = Mb + MC.
(2)
In (2) we assume the stock equilibrium of bank reserves, i.e., = R. Given this, we obtain the following important relation D”=L+Sp;
L=Lb-tLc,
Rb
(3)
assuming stock equilibrium of securities, Sp + Sb + SC = D”, and of money, M = L. Condition (3) states that the public’s holding of financial assets is equal in amount to the total national debt out5To simplify the analysis, we assume that the public and the do not discount their holdings of government bonds, in spite increases in taxation by the government to finance its debt. On long as such a discount factor is a positive constant, our analysis in any essential way. 56
commercial banks of possible future the other hand, as will not be altered
Optimal Monetary Policy standing [see Modigliani (1963) and Tobin (1969)]. Assume, following much of the money-supply literature, that the currency-deposit ratio (CY)is an exogenously given constant. Let y be the legal reserve ratio. Also, let iF and i, respectively, signify the central bank’s discount rate and the nominal interest rate on bonds. Then we can obtain the following “supply function of money”6
M = +(i, iF, y, cx)Sc; d+/di > 0, (4) &$/SF < 0,
&$/ay < 0,
@/&I
< 0 .
3. The Short-Run Equilibrium We assume that the single output Y is produced with labor N and capital K under a constant-returns-to-scale, diminishing-returns technology,’ i.e., y = y(k);
y' > 0,
y" < 0 ,
(5)
where y = Y/N and k = K/N. Here we assume full employment. Following much of the literature, we assume that the nominal demand for money L is written as
L = L[i, y’(k) + I?, pY, A”],
(6)
where & < 0, $ < 0, La > 0, and t, > 0, with the subscript j (i = 1, 2, 3, 4) d eno t ing the partial derivative with respect to the jth argument. Here i and y’(k) + ITS, respectively, signify the nominal rate of return on securities and on equities, where # denotes the expected rate of inflation. Note that in (6) the public’s portfolio choice of the three assets (securities, equities, and money) is made explicit. Following the money and growth literature, we assume that the money-demand function is homogeneous of degree one in its ‘For the derivation of this formula, see Takayama-Drabicki (1976) and Drabicki-Takayama (1980). The derivation of such a formula is along the line of Brunner-Meltzer (1964) and Takahashi (1971). TO illustrate the importance of the endogeneity of the money supply, we may simply observe that a change in the interest rate (which is obviously an endogenous variable), for example, affects the commercial banks’ behavior regarding free reserves and loans, and that this affects the money supply. ‘In the present paper we, for simplicity, ignore productive activities of the government, which, in turn, follows much of the literature. 57
John 2. Drabicki
and Akira
last two arguments.
Takayama
Then noting
(l), we rewrite
(6) as,
L/W) = Ui, y’(k) + a’>y(k), k + 4,
(7)
where p is the price level and d = D”/(pN) represents the real per capita national debt. In an optimization model, it would be reasonable to assume # = 7~ (perfect myopic foresight), where 7~ denotes the actual rate of inflation. Using (4) and (7) with v = $, we obtain Ad = L[i, y’(k) + a, y(k), k + d] ,
(8)
where X = SC/D” and L = i/4 with Z.,r < 0, L2 < 0, L3 > 0, L4 b 0. Here CL, iF, and y are assumed to be constants. Namely, to simplify the analysis, we assume that monetary policy is carried out by open market operations, as represented by changing the value of A. Here we assume that SC consists only of government securities, and thus that X signifies the proportion of the national debt held by the central bank. Turning to the goods market, the equilibrium relation is written as
(9)
Y=C+Z+G,
where C, I, and G, respectively, denote consumption, investment, and government expenditures on goods (all in real terms). Consumption is assumed, for the sake of simplicity, to be a constant fraction of real disposable income Yd; i.e., c = (1 - s)Yd; Letting yd = Yd/N, we obtain
0 < s< 1.
o = Z/N, and g = G/N
y = (1 - s)yd + 2) + g *
(10)
and using (9) and (lo),
(11)
The public’s real disposable income Yd is output (Y) minus taxes net of transfer payments (T) plus interest payments and capital gains (or losses) on the outstanding national debt (which are all expressed in real terms)? ‘The details of the derivation of (12) are explained in Drabicki-Takayama (1980). This exercise is analogous to the derivation of the expression for disposable income 58
Optimal yd = Y - T + iD - ITD;
D E D”/p
Monetary
Policy
.
(1%
Note that capital gains or losses associated Gith price changes incorporated into Y d.’ By subsuming the interest payments on public debt into the variable T”, i.e., by letting T” = T - iD, may simplify (12) as Yd = Y - T’ - ITD. Letting t” 3 TX/N, then have yd = y - tX - rd. For the present fi.mction’O
study we postulate
I = Z[i - IT, y’(k), Yd];
I1 < 0,
are the we we
(13) the following
IQ > 0,
investment
13 > 0 .
(14)
That is, (real) investment depends on the real rate of return on bonds, on the real rate of return on physical capital, and on disposable income. The dependence on (i - IT) and y’(k) reflects the firm’s portfolio choice regarding the physical stock of capital, bonds, for the outside money case that appears in the usual money and growth literature [e.g., Burmeister-Dobell (1970), pp. 166-671. It is assumed that all bonds have a fixed money price (say, unity) realizable on demand with a variable interest rate [see Modigliani (1963), Foley-Sidrauski (1971), etc.] By assuming away the problem of Tobin’s 9, we may avoid the problem of capital gains or losses from holding physical assets due to a change in 9. ‘Foley-Shell-Sidrauski (F-S- S) (1969) and Foley-Sidrauski (F-S) (1971) both consider the two output (consumption and capital goods) economy, where pt and pm, respectively, denote the prices of the capital good and money in terms of the consumption good. Both of these works assume that pm is kept constant by governmental policy. F-S-S assume that the economic agents expect pk to be constant (static foresight), while pk actually does change along the optimal growth process. Hence their expectations are unwarranted and irrational. F-S also contains a similar problem where the planner’s utility function (which we denote by u) is nonlinear. For the case of a linear u, F-S obtain the result that p, is constant along the optimal path. In this case, static expectations are warranted, but the problems associated with inflation are avoided as pk and p, are both constant. In any case, both F-S-S and F-S avoid the problem of capital gains or losses associated with price change altogether. We may also note that it is not clear why society wishes to choose the governmental policy of keeping p, constant. “‘An alternative formulation would be that investment demand depends exclusively on Tobin’s 9. [See Tobin (1969), f or example.] Such a formulation is left to the interested reader. There seems to remain some doubt regarding the plausibility of such a specification (especially in aggregate growth models). See, for example, recent exchanges among von Furstenberg, Lovell, and Tobin in the Brookings Papers (1977), pp. 347-408. 59
John Z. Drabicki
and Akira
Takayama
and money. To simplify the analysis, ment function can be written as” u=v[i-~~,y’(k),y~];
q
we assume that the invest-
u2>0,
u3>0,
where v = Z/N. Using (13) and (15), we may then rewrite SYd = u[i - T, y’(k), yd] + g - tX - Td,
(15) (11) as 06)
where the expression for y* is given in (13). Equations (8) and (16) d enote the short-run equilibrium relations of our model which, respectively, correspond to the LMand the IS- relations: they determine the short-run equilibrium values of n and i for given values of k and d and the specified values of the parameters g, ta and A. The model is completed with the specification of the accumulation (or decumulation) equations of the two assets, K and D”, namely, K=
Z
and
on=
p(G - T’),
(17)
where depreciation of the physical asset is assumed away. Here the dot signifies the time derivative. Assuming that N grows at a constant rate n, we obtain the following two equations from (17) k = u[i - T, y’(k), yd] - nk ,
084
d=g-t”-nd-nd.
(18b) M
4. Optimal Growth The problem of optimal growth in the context of the present model is to choose the (piecewise continuous) time path of g, t* and h so as to maximize a certain target functional, subject to (18) and to the short-run equilibrium conditions. Jumps in the state variables [e.g., Arrow-Kurz (1970), pp. 51-571 are ignored. In this paper we confine ourselves to a simple monetary policy rule, which is specified by
“This “homogeneity” assumption, though it is restrictive, turns out to be a very useful specification in simplifying the analysis. For an example of such a homogeneity specification, we may recall Friedman (1971, p. 325).
60
Optimal
Monetary
Policy
where A is an exogenously given constant and 0 is considered the policy parameter. Note that the policy specified in (19) may be interpreted as one where the “(augmented) high powered money,” MC + R, is increased (or decreased) at a rate 8, where we may recall D” = SC/X = (MC + R)/X.‘2 From (19), we can easily obtain -t” = ed, so that (13), (16), (18a) and (18b) are, respectively, modified as yd = y(k) + (cl - lT)d ;
(13’)
SYd = u[i - 97, y’(k), yd] + (Cl - 7r)d ;
(16’)
h = sy - nk - (1 - s)(0 - IT)~ ; ci = (cl -
7r -
n)d
;
where, to obtain (20), we use (Isa), (13’), and (16’). Using (8) and (16’) together with (13’), we obtain run equilibrium values of IT and i a?
IT = n(k, d; Our problem of optimal as to maximizeI
0, A); i = i(k, d;
growth
8, A) .
the short-
(22)
is to choose the time path of 8 so
‘*This specification corresponds to the usual specification in the money and growth literature in which there is no government spending or taxation except for the transfer of (outside) money, which grows at a constant rate 8. If we assume 8 is constant, then we obtain a descriptive model whose properties are studied elsewhere [Drabicki-Takayama (1980)], while in the present study tl is a policy parameter. Needless to say, given the framework developed in the previous section, we can consider more complex policy rules which maximize a certain target functional. r3For the sake of simplicity, we assume the existence of a unique short-run equilibrium as well as continuous differentiability of the functions n and i. ‘*Equation (23) is the standard objective functional of the optimal growth literature, where p is assumed to be a positive constant signifying the discount rate. We assume u’(c) > 0, u”(c) < 0 for all c 2 0 and u’(0) = a. On the other hand, this type of utility function is different from the one that appears in the usual literature on the optimum quantity of money, where money enters into the utility function. For skeptical comments on that type of utility function, see Clower (1968) and Hahn (1971, pp. 63-64). Incidentally, in the present maximization problem, we shall not impose the constraint I t 0, i.e., physical capital is allowed to be “eaten up.” The reader should be able to easily incorporate this constraint if he or she so desires. 61
John 2. Drabicki
and Akira
Takayama m
u(c)eWPtdt ,
rJE I
subject to (20), (2I), where
(22) and the usual nonnegativity
c = (1 - s)yd = (1 - s){y(k) + [e - T(k, d; Define
(23)
0
the Hamiltonian H = u(c) + j&k,
conditions,
8, X)]d} .
(24)
by d;
0, A) + vd(k, d;
8, A),
(25)
where p and Y are the multipliers signifying the shadow prices of the accumulation (or decumulation) of the physical good and of the national debt, respectively. Among the necessary conditions for an interior solution we haveI [u’(c)(l
- s) - ~(1 - s) + u](l - mO)d = 0 ;
Ii=pl.l-aH/ak, lim peeptk = 0 , t-am
(26)
iJ=pv-aHlad;
(27)
lim ueeP’d = 0 , t-b-
(28)
where 7~~ = aT/ae. Assume d > 0. Since we can show 1 - nTTe < 0 (see Appendix), we obtain from (26) u’(c)(l
- s) - p(l - s) + V = 0.
Using this expression we can compute the second equation of (27): i = (p + n)v .
the following
(2% relation
from
(30)
Hence, in view of the second relation of (28) with d > 0, we can obtain v = 0 for all t. It then follows from (29) that we have “For a discussion on the transversality condition, such as in (28), see Benveniste-Scheinkman (1982). Postulating the conditions which guarantee lim k > 0 and t--rlim d > 0, (28) reduces to I-+lim Fe -P’ = 0, lim ye-@ = 0 @3’) t-r 1-I 62
Optimal
Monetary
u’(c) = p I
Policy
(31)
which is the familiar Ramsey-Cass-Koopmans rule (or the “KeynesRamsey rule”) which equates the loss of marginal utility in giving up one unit of consumption to the shadow price of the accumulation of the physical good. Using v = 0 and (31), we can compute the following relation from the first equation of (27) with k > 0: & = (p + n - y’)p .
(32)
From (32), we may at once conclude that in the steady state, in which b = 0, the capital-labor ratio assumes the familiar “modified golden-rule” value k*, which is uniquely determined by y’(k*)
= p + n .
(33)
Also, by setting 1 = 0 and d = 0, respectively, in (20) and (21), we obtain the following steady-state values of d and 8: d* = [sy(k*)
- nk*]/[(l
Cl* = n + n*,
- s)n] ,
T* = r(k*,
d*;
(34) 8*, A) .
(35)
Real per capita consumption in the steady state (c*) is obtained from (24) and (35) as c* = (1 - s)[y(k*) + nd*] .
(36)
Utilizing (34), we may rewrite (36) as c* = y(k*) - nk* .
(36’)
This value of c* is precisely the same as that which is obtained in the usual optimal-growth literature. Note also that k*, d*, and c* in (33), (34), and (36’) are determined independently of X. Thus, an open-market operation (i.e., a change in A), in the course of optimal growth, has no enduring effect on k*, d*, or c*, though it would accompany a corresponding change in the optimal path of 8. We now summarize some of the results obtained in the above analysis. PROPOSITION 1: (i) The optimal-growth path for the present manetary economy follows
the familiar
Ramsey-Cass-Koopmans
rule
63
John 2. Drabicki
and Akira
Takayama
(31). (ii) The steady-state values of the capital-labor ratio and of real per capita consumption, under optimal growth in the present monetary economy, are identical to those obtained in the familiar Cuss-Koopmans-type model (in which money is not explicitly introduced).‘6 (iii} The steady-state rate of inflation (price change) is equal to the steady-state value of “monetary expansion” (p) minus the rate of population growth. (iv) Open-market operations have no effect on the (optimal) steady-state values of the capital-labor ratio, real per capita national debt, and real per capita consumption. We now wish to obtain an adequate characterization of the optimal-growth process. For this purpose we need to obtain the signs of the partial derivatives for the functions T and i, which are determined from the short-run equilibrium conditions. Postponing such a discussion to the Appendix, we will simply record the main results here:
ik > 0,
id < 0
if
6 - n 2 0,
iO < 0,
ih < 0 ,
W’b)
where nk = an/ak, rd = an/ad,ik = ailak, ih = Z/ah, etc. Here 7rg > 0, ?rh > 0, ia < 0, and i, < 0, for example, mean that monetary expansion via an increase in 8 or via an increase in A (an open-market purchase) increases the short-run rate of inflation and reduces the short-run interest rate. We may call TI, < 0 and ik > 0 the “capacity effect” and the “illiquidity effect,” respectively, as suggested to us by K. Nagatani.17 Substituting the expression for c found in (24) into (31), we may now solve for the optimal value of 8 as a function of k, d, p and h, i.e.,
8 = W, d, CL; A) ,
(38)
where we can easily compute
‘% is well known
that Ramsey (1928) confined
his analysis to the case of p =
0. “Note that (37) holds whether or not 0 is chosen at the optimal value. The relations in (37) are, in fact, obtained by us elsewhere [Drabicki-Takayama (1980)], where the economic interpretations of the signs of partial derivatives such as mk and 718 are also provided. 64
Optimal Monetary Policy
,_
(e -
pkd Tejd
>
fld =
-
IT) - rrdd (1 - n&d ’
ek =
-(“1
ep =
1 Oh= -EL 1 - TrTg ' u"(1 - s)(l - m,)d ’
(39) (40)
where ok = X)/ak, ed = se/ad, etc. Recalling I - 7~~ < 0, we can at once conclude from (37a), (39), and (40) that ok > 0,
ed < 0
if
0 - r 2 0,
0)~> 0,
and
oh < 0 .
(41)
Since ok > 0, ?rk < 0 and ik > 0 from (41) and (37), an increase in k implies a “counter-cyclical” monetary policy in the sense that a fall in the rate of inflation rr and an increase in i must, ceteris paribus, accompany an increase in the rate of monetary expansion 8. (An increase in 6 in turn tends to offset the effects on 7~ and i through pg > 0 and ie < 0.) Similarly, an increase in d which implies a fall in P and an increase in i should, ceteris paribus, accompany an increase in 6. The ceteris paribus nature of the above may have to be emphasized. Namely, since
the (optimal) time path of 0 obviously does not depend exclusively on the signs of ok and t&. On the other hand, the above consideration also indicates that there are no such things as the optimum quantity of money and an optimum fixed value of 6. This proposition might be of some interest in view of the popular argument that monetary authorities should follow a certain fixed “rule,” such as that of increasing the money supply in accordance with the average secular growth of aggregate economic output, instead of “indulging” in discretionary policies. In summary, we may conclude: PROPOSITION 2. The signs of the partial derivatives for the func-
tions n, i, and 8 are determined as in (37) and (41). In particular, the optimal monetary policy concerning the choice of 0 is “‘counter-cyclical” in the ceteris paribus sense described above. The optimal value of 8 would not, in general, be constant along the growth process. 65
John 2. Drabicki
and Akira
Takayama
5. Dynamic path Substitute and 2 as
(38) into (20) and (21) and define
the functions
k
r; = i(k, d, JJ,; A) = sy(k) - nk - (1 - s){e(k, d, p;
- dk, 4
x)
W , d, I.C A>,AlId,
(42)
and d = d(k, d, p;
x)
= {W, d, P; A) - T[k, d; Also, from (32), define
e(k, d, p;
the function
A), A] - n}d .
(43)
& as
$ = lik 14= [P + n - dW1~ .
(44)
Given (28), the optimal path of (k, d, CL) is completely described by the system of differential equations (42), (43), and (44), where A is assumed to be constant. To study the behavior of the dynamic path, we first compute from (42)
ai/ak = I./ -
n,
a/i/ad= 0, ai/ap =
-l/tP
(45)
where we utilize the expressions for Ok, fld, and 8, that are recorded in (39) and (40). Also, from (43) we may compute
ad/ak= -I/,
ad/ad= -71, ad/al*= i/(1 - s)u’,
where we again utilize (39) and (40). From (44), we further al;/&
= &‘,
a&/ad = 0,
a$/ak
= p + n - y’ .
(46)
compute
(47)
Observe from (45) and (47) that ailed = 0 and a&/ad = 0. This means that equations (42) and (44) are independent of d, and as such they form a subsystem of two differential equations in the 66
Optimal
Monetary
Policy
two variables k and CL.Thus the optimal time paths of k and i.~ can be determined independently of equation (43) which specifies the behavior of d. Using the rest of the relations found in (39) and (41), and recalling the transversality condition (28), we may illustrate the dynamic optimal path of (k, i.rJ in Figure 1 by the curves a and a’, which are the stable branches of the system of differential equations (42) and (44). It can be seen from Figure 1 that, regardless of the initial real per capita debt d, if the initial value of the capital-labor ratio is less than (greater than) the steady-state value k*, the multiplier p should be chosen in such a way as to enter the time path labelled “a” (“a’ “). In such a case, along the optimal path the capital-labor ratio will increase (decrease) monotonically over time and approach k* in the limit, and the corresponding value of the multiplier i.~will decrease (increase) monotonically, approaching i.~* in the limit. From (31), we may also conclude that real per capita consumption c will increase (decrease) in this process while monotonically approaching c *. If the initial value of k is equal to k*, it is optimal for the economy to remain at (k*, CL*, c*). These conclusions are identical to those obtained in the usual literature a
The Optimal
Figure 1 Behavior of k and p
k
67
John 2. Drabicki
and Akira
Takayama
la Cass and Koopmans where money is not introduced. This result, together with statements (i) and (ii) of Proposition 1, shows that the familiar Cass-Koopmans-type results on optimal growth for a centralized nonmonetary economy can, in essence, be validated for a decentralized monetary economy. We may call this result the “ditheorem chotomy theorem, ” in the sense that the optimal-growth for the real sector can be “dichotomized” from the repercussions from the monetary sector.” To study the behavior of the three-dimensional trajectory of (k, d, p), we will confine ourselves to its behavior in the neighborhood of the steady state. For this, we consider the following coefficient matrix of the differential equations (42)-(44)
al; ak
al;
ai -
ad
G
a$ ii
w ak
y’--n
=
w ali, ad av..
0
-Y’
-n
-by”
0
-l/d
l/(1
- S)d
(48)
p+n-y’
in which the explicit forms of the components are obtained in (45)(47). We then compute the three eigenvalues of the above matrix as follows: (p +- VP” + 4&‘/u”)/2
and
-n.
(49)
Namely, two of the eigenvalues are negative while one eigenvalue is positive, and thus we may conclude the saddle-like nature of the optimal path. More specifically, the steady state is a “saddle point of type 2,” i.e., the optimal paths for different initial values define a two-dimensional surface in the three-dimensional space (k, d, p.). Under the usual circumstances, the optimal path is obviously unique for given initial values of k and d. “It can be shown that the dichotomy theorem holds even for a much simpler economy, in which securities are assumed away. See Drabicki-Takayama (1978). Note that the present paper, as well as D-T (1978), prooes the dichotomy theorem, which may be sharply contrasted with F-S-S (1969) and F-S (1971), where they simply assume it (implicitly). 68
Optimal Monetary Policy Given the optimal trajectory of (k, d, IL), the optimal value of the monetary parameter 8 can be computed from condition (38). The optimum time path of the interest rate i and the rate of price change IT can then be extracted from (22) and (37). As a general rule, the nature of such an optimal time path of (t3, IT, i) will depend on the initial values of k and d. In addition, none of the time paths of 8, IT, or i need be monotone. Furthermore, at each instant of time, for given values of k, d, and p+ (38) specifies an optimal value of 8 and hence an “optimal” rate of inflation (price change) IT. As these change over time, so will the optimal rate of inflation. In short, (although it may be contrary to a popular belief), there is no such thing as the optimum rate of inflation or deflation (which is constant during the growth process). Similarly, there are no such things as an optimum quantity of money and an optimum value of 9, which are constant along the growth process. On the other hand, if we confine ourselves to the steady state, we have T* = 0* - n, as observed earlier (see Proposition 1). We may now summarize some of the above results.
3. (i) The steady state in the present monetary economy is a saddle point, and given the initial values of k and d, we may choose a unique time path of t? which achieves optimal growth (with which the time path of (k, d, p) is associated). (ii) The dynamic behavior of the optimal values of the capital-labor ratio (k), the shadow price of the accumulation of physical capital (p), and real per capita consumption (c) is identical to the familiar behavior found in the standard literature (for p > 0), such as Cuss (1965) and Koopmans (1965), in which money is not introduced. The optimal trajectory of (k, p, c) in the present monetary economy is independent of the time path of real per capita national debt (d), i.e., optimal monetary policy is “neutral.” (iii) There are no such things as the optimal rate of inflation, the optimum quantity of money, and the optimum value of 9, which are constant in the growth process, while in the steady state we have n* = 8* - n.”
PROPOSITION
Lastly,
an interesting
comparative-dynamics
question
in this
]@A simple characterization of dynamic behaviors of d and 6 may be obtained if we assume. that k and TVwere set at k* and TV*. In this case, d = d(k*, d, p*; A). Noting &i/ad = --n, we may conclude that d converges monotonically to d*. Since 8d < 0 in a neighborhood of the steady state, we may also conclude that f~ must decrease (i.e., a contractionary monetary policy) along the path in which d increases (which naturally accompanies monetary expansion) in a “sufficient” neighborhood (see Appendix) of the steady state, where k and d are set at k* and d*. 69
John 2. Drabicki
and Akira
Takayama
context may be the effect of a change in A (open-market operation) on the optimal rate of monetary expansion 8. To investigate this question, observe
al;/ax = ad/ax = ah/ax = 0, which follows from (42), (43) and (44) and the second expression of (40). Thus, for each fixed value of (k, d), a change in A induces no corresponding changes in the k = 0, the d = 0 and the 6 = 0 loci in the phase space, which in turn implies no change in the time path of IL; namely, the indirect effect of h on 0 via p vanishes. Thus, for each given (k, d), the effect of A on 9 is simply given by - no) [see (JO)], w h’ic h is negative. Thus we may con4 = nJ(l clude: PROPOSITION 4. An expansionary monetary policy uia an open-market purchase must be offset by a corresponding “contractionary” monetary policy associated with r-educing 8 in order for the economy to be on the optimal path. Similarly, a contractionary manetary policy via an open-market sale must be accompanied by an expansionary monetary policy as represented by increasing 0. Received:January, 1981 Final version received: June, 1982
References Arrow, K.J. “Applications of Control Theory to Economic Growth.” In Mathematics of the Decision Sciences. G.B. Dantzig and A.F. Veinott, eds. Part 2. Providence, R.I.: American Mathematical Society, 1968, 85-119. Arrow, K.J. and M. Kurz. Public Investment, The Rate of Return, and Optimal Fiscal Policy. Baltimore: Johns Hopkins Press, 1970. Benveniste, L.M. and J.A. Scheinkman. “Duality Theory for Dynamic Optimization Models of Economics: The Continuous Time Case.” Journal of Economic Theory 27 (June 1982): 1-19. Blinder, A.S. and R.M. Solow. “Does Fiscal Policy Matter?’ Journal of Public Economics 2 (November 1973): 319-37. Brock, W.A. “A Simple Perfect Foresight Monetary Model.” Journal of Monetary Economics 1 (April 1975): 133-50. Brunner, K. and A.H. Meltzer. “Some Further Investigations of Demand and Supply Functions for Money. ” Journal of Finance 19 (May 1964): 240-83. 70
Optimal
Monetary
Policy
Burmeister, E. and R.A. Dobell. LMathematical Theories of Economic Growth. New York: Macmillan, 1970. Calvo, G.A. “On the Time Consistency of Optimal Policy in a Monetary Economy. ” Econometrica 46 (November 1978): 1411-28. Cass, D. “Optimum Growth in an Aggregate Model of Capital Accumulation.” Review of Economic Studies 32 (July 1965): 23340. Clower, R.W. “Comment on Papers by Tobin and Marty.” Journal of Political Economy 86 (July/August 1968, Part II): 876-80. Drabicki, J.Z. and A. Takayama. “Endogenous Supply of Money, Myopic Perfect Foresight, and Economic Growth.” Unpublished. November, 1980. and -. “Money, Inflation and Optimal Economic Growth. ” Economics Letters 1 (1978): 55-58. Foley, D.K., K. Shell, and M. Sidrauski. “Optimal Fiscal and Monetary Policy and Economic Growth. ” Journal of Political Economy 77 (July/August 1969, Part II): 698-719. Foley, D. K., and M. Sidrauski. Monetary and Fiscal Policy in a Growing Economy. New York: Macmillan, 1971. Friedman, M. The Optimum Quantity of Money and Other Essays. Chicago: Aldine, 1969. “A Monetary Theory of Nominal Income. ” Journal of Political Economy 79 (March/April 1971): 323-37. Hahn, F.H. “On Money and Growth. ” Journal of Money, Credit, and Banking 1 (May 1969): 127-87. “Professor Friedman’s Views on Money. ” Economica 38 (February 1971): 61-80. Koopmans, T.C. “On the Concept of Optimal Economic Growth.” In The Econometric Approach to Development Planning. Pontificiae Academiae Scietiarum Scripum Varia. Amsterdam: North Holland, 1965. Kydland, F.E. and E.C. Prescott. “Rules Rather than Discretion: The Inconsistency of Optimal Plans. ” Journal of Political Economy 85 (June 1977): 473-91. Modigliani, F. “Monetary Mechanism and Its Interaction with Real Phenomena. ” Review of Economics and Statistics 65 (February 1963): 79-107. Nagatani, K. “A Note on Professor Tobin’s ‘Money and Economic Growth. ’” Econometrica 38 (January 1970): 171-75. -. Monetary Theory. Amsterdam: North-Holland, 1978. Ramsey, F.P. “A Mathematical Theory of Saving. ” Economic Journal 38 (December 1928): 543-59. 71
]ohn 2. Drabicki
and Akira Takayama
Sidrauski, M. “Inflation and Economic Growth. ” Journal of Political Economy 75 (December 1967): 796-810. Stein, J.L. “The Optimun Quantity of Money.” Journal of Money, Credit, and Banking 2 (November 1970): 397-419. Takahashi, F. “Money Supply and Economic Growth.” Econometrica 39 (March 1971): 285-303. Takayama, A. and J.Z. Drabicki. “On the Endogenous Supply of Money. ” Keizai Kenkyu (Economic Review) 27 (October 1976): 336-48. Tobin, J. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84. -. “A General Equilibrium Approach to Monetary Theory. ” Journal of Money, Credit, and Banking 1 (February 1969): 1529. -. “Notes on Optimal Monetary Growth.” Journal of Political Economy 76 (July/August 1968, Part II): 833-59. Turnovsky, S.J. and W.A. Brock. “Time Consistency and Optimal Government Policies in Perfect Foresight Equilibrium. ” Journal of Public Economics 13 (April 1980): 183-212. Tsiang, S.C. “A Critical Note on the Optimum Supply of Money. ” Journal of Money, Credit, and Banking 1 (May 1969): 266-80. Uzawa, H. “An Optimal Fiscal Policy in an Aggregative Model of Economic Growth.” In The Theory and Design of Economic Deuelopment. I. Adelman and E. Thorbecke, eds. Baltimore: The Johns Hopkins Press, 1966. Yoshikawa, H. “Alternative Monetary Policies and Stability in a St07 chastic Keynesian Model. ” Znternational Economic Review 22 (October 1981): 541-65.
Appendix. The Derivation
of (37)
The short-run equilibrium of a and i which are defined by (22) are obtained from (8) and (16’) with (13’). A straightforward differentiation of these equations yields:
aT/ak = -{[u2y’ + (u3 ai/ak =
{(Us
+
-
s)Y’]L,
-
U&~ZJ”
+
L,y’
+
W/J~
ad)(Lzy” + Ly’ + L) + [uzy” + (us - S)Y‘WI
aT/ad = 72
(A.14
-[II&
-
I&)
+
(e -
T+JL~]/J,
3 (A. lb) (A. lc)
Optimal Monetary Policy ailad
= -[(q
+ od)(h - L4) - (0 - 7r)aLJJ
an/a0 = -L,ud/]
(A. Id)
,
, ai/ae = L,ud/J ,
(A. le)
an/ax = --old/] , ailax = -(q + ad)d/J , ] = -[(q
(A. lf)
+ ad)& + ulLz] ,
64.2)
where u = (1 - s) + u3 signifies the marginal propensity to spend out of current disposal income, which we assume to be positive. In order to determine the signs of these partial derivatives, it becomes necessary to appeal to the correspondence principle. To this end, we hypothesize the adjustment process that an excess demand for goods increases IT, while an excess demand for money increases, i, i.e.,
dn/dT = a{u[i - IT, y’(k), yd] + (tl - n)d - syd} = @(IT, i), di/dT = b{L[i, y’(k) + IT, y(k), k + d] - Ad1 E ‘P(~F,i), where yd = y(k) + (0 - v)d, and stants. Here k and d, as well as 8 time. The necessary and sufficient ear approximation system of (A.3)
(A.3) (A.4)
where a and b are positive conand A, are fixed, while T signifies conditions for stability and (A.4) are
of the lin-
where Q,.,= a@/av,V!, = aq/ai, and are evaluated at a particular short-run equilibrium point. Although (A.5) provides only a sufficient condition for stability of the original system we, following the usual practice of the correspondence principle, assume (A.5). The partial derivatives a,,, Wi, etc., are readily computed as Q,, = -a(u,
+ ud),
@i = au1 < 0, 64.6)
‘PT= bL,
qi=
bL,
(A.5) as
@‘, + qi = -a(q
+ ad) + bL, < 0,
(A. 7) 73
John Z. Drabicki and Akira Takayama a),** - 4+*,, = -ab[(ul + ad)& + v,Lz] > 0
(A.@
(i.e., J > 0.) Since ur < 0, L1 < 0 and L2 < 0, the “trace condition” (A.7) requires (ur + ad) > 0 in which case the “Jacobian condition” (A.8) is automatically satisfied. Also J > 0 requires d > 0 or, more specifically, J > 0 if and only if d > -q(L1 + L&/(aL1)(>O), which we assume throughout the paper. From this discussion, we can specify the sign of the partial derivatives in (A.1) as they are recorded in (37) in the text, where we assume X > L4 to obtain the signs of ?rd and id. The assumption of A > L4 means that an increase in d increases the demand for money to a lesser degree than it increases the supply of money. That 1 - 1~~ < 0 is obtained easily from (A.le). As mentioned earlier, the results here and their economic interpretations are obtained elsewhere by Drabicki-Takayama (1980). The discussion in this Appendix is only included to make the present paper sufficiently self-contained.
74
of Arizona
AKIRA TAKAYAMA Kyoto Unioersity
An Optimal Monetary Policy in an Aggregate Neoclassical Model of Economic Growth* This paper shows that the usual Ramsey-Cass-Koopmans optimal-growth theory is applicable to decentralized monetary economies and illustrates, with a simple model, how optimal growth can be achieved via a simple monetary policy. Securities and the endogeneity aspect of the money supply are explicitly introduced. This paper shows that the steady state under optimal growth is a saddle point, that the dynamic behavior of the capital-labor ratio and real per capita consumption is identical to that found in the usual literature in which money is not introduced, and that the optimal monetary policy is “counter-cyclical.”
1. Introduction The purpose of this paper is to investigate optimal monetary policy in terms of the well-known framework of optimal growth developed by Ramsey (1928), Koopmans (1965), Cass (1965), and others. In particular, we show that this framework, which is typically discussed in terms of a centralized nonmonetary economy, is applicable with proper modifications to a decentralized monetary economy, and we illustrate with a simple model how optimal growth can be achieved via simple monetary policy. Hence the present study, in a sense, follows the works by Uzawa (1966), Foley-Shell-Sidrauski (1969), and Foley-Sidrauski (1971, chap. 15), etc. Although there are many varieties of monetary macro models under a decentralized framework in the literature, we have identified one model in which the Cass-Koopmans-type results on optimal growth for a nonmonetary, centralized economy can be carried over. This model is an extension of the familiar neoclassical monetary-growth model B la Tobin (1965), in which the idea of Fisher’s real return on money and the Wicksellian cumulative process is juxtaposed with the Solow-Swan type of neoclassical descriptive model. ’ This line of thought, developed in terms of de*We are indebted to W.A. Brock, M.C. Kemp, Keizo Nagatani, and two anonymous referees for useful comments. ‘For excellent surveys on this topic see, for example, Burmeister-Dobell (1970, Chap. 6) and Nagatani (1978, Chap. 12). Journal of Macroeconomics, Winter 1983, Vol. 5, No. 1, pp. 53-74 Copyright 0 1983 by Wayne State University Press.
53
John 2. Drabicki
and Akira
Takayama
scriptive models of a decentralized economy, is made applicable to the theory of optimal growth.’ This type of money-and-growth literature [and also Uzawa (1966)] typically assumes that there are only two types of assets in the economy, physical capital and (outside) money, in which the role of the banking sector is virtually ignored. In the present study we explicitly introduce securities (bonds) into the analysis, and incorporate the endogeneity aspect of the money supply, which is derived from the behavior of the commercial banks and of the central bank. As it is shown by Drabicki-Takayama (1980) in terms of a descriptive model, the introduction of securities as an alternative means of a store of value provides additional “flexibility” in the model: Drabicki-Takayama (D-T) have shown that the steady state, under similar assumptions regarding the money supply, can be stable under perfect myopic foresight (contrary to the traditional result that it is a saddle-point). Here we utilize this framework of analysis developed in D-T. Although our model would enable us to investigate optimal-growth policy in terms of other policy parameters, we confine ourselves here to a simple monetary policy in which the policy authorities control 8, the rate of change of the (augmented) high-powered money stock. Among other results, this paper shows: (1) The optimal paths for different initial values define a two dimensional surface. For given initial conditions, the unique optimal path converges to the steady state which is a saddle point. (2) The familiar Cass-Koopmans-type results on optimal growth for a centrally planned economy can be carried over in a decentralized monetary economy. For+ example, the dynamic behavior of the capital-labor ratio and real per capita consumption is identical in these two types of economies. (3) There are no such things as the optimal rate of inflation, the optimal quantity of money, and the optimal value of 8, which are constant along the dynamic path, whereas the steady-state rate of inflation is equal to the optimal steady-state value of the rate of monetary expansion (0) minus the rate of population growth. Point (3) may be useful in shedding some light upon the literature concerning the “optimum quantity of money” by Friedman
‘Optimal growth in the context cussed by Hahn (1969) and Stein optimal growth in a decentralized (1971, Chap. 15) which is, in part, 54
of the money and growth literature is also dis(1970), f or example. A more complete study of monetary economy is given by Foley-Sidrauski an elaboration of Foley-Shell-Sidrausk (1969).
Optimal
Monetary
Policy
(1964) and others3 However, the focus of our paper and the framework of our analysis is different from the one which is typically found in that literature. Throughout the paper, we assume perfect myopic foresight. We also assume that households follow a simple rule of thumb, i.e., proportional savings behavior. Consequently, the problem of time inconsistency which may be encountered in the works of Calvo (1978), Turnovsky-Brock (1980), and others does not arise here.4 Namely, in this paper we simply argue that if the public follows such a rule, then the familiar Cass-Koopmans theorem for optimal growth can be validated for a monetary economy.
2. Framework and Basic Notation We consider here an economy consisting of four sectors: the government, the central bank, the commercial banks, and the public (households and firms). There are five types of assets: physical goods, currency, (demand) deposits, bank reserves, and interestyielding securities (bonds). In Table 1, we summarize the symbols used and the balance sheet of each sector, where all entries are measured in monetary units [see Tobin (1969)]. A positive entry represents an asset for a particular sector whereas a negative entry represents a liability. The symbols Sp, S b and SC represent, reTABLE
1.
The Summary
of the Balance Sheets Sectors
Public
Commercial Banks
Central Banks
Exogenous SUPPlY
Currency Deposits Bank Reserves Securities Physical Capital
L” Lb 0 SP pV
0 -Mb Rb Sb 0
-MC 0 -R SC 0
0 0 0 D” PK
Net worth
A”
0
0
D” + pK
Assets
‘See also Brock (1975), Clower (1968), Hahn (1971); Nagatani (1978, Chap. 13), Stein (1970), Tobin (1968), and Tsiang (1969). For a more recent criticism on this point in terms of stochastic models, see Yoshikawa (1981). ‘For a clear exposition of the time inconsistency problem, see Kydland-Prescott (1977). 55
John 2. Drabicki
and Akira
Takayama
spectively, the net holding (stock excess demand) of securities by the public, the commercial banks, and the central bank. The sum of a column for a particular sector equals the net worth of that sector which gives its balance sheet identity. The entry in the last column is the net exogenous supply of a particular asset. For physical capital (or equities in physical capital), where p denotes its price, its supply is the economy’s inheritance from the past. (Here, for simplicity, we assume away the problem involved in “Tobin’s 9, ” and thus p also signifies the price of current output.) The securities consist of both private and government bonds, which are assumed to be indistinguishable to the public in all respects (such as in the expected rate of return, risk, etc.). The government is a net debtor whose debt consists only of government bonds, the amount of which is equal to D” (“dollars”). All international transactions are assumed away. The first important relation can be obtained from Table 1, as is done in Tobin (1969), by noting that the sum of the final column is equal to the sum of the last row, i.e., D”+pK=A”,
or
D + KS
A;
D 5 D”/p,
A =AA”/p.
(1)
In short, the public’s nominal net worth is equal to the sum of the value of the physical stock of capital, pK, and the total outstanding government debt.5 Next, note that consolidation of the balance sheets of the commercial banks and of the central bank yields Sb + SC = M;
M = Mb + MC.
(2)
In (2) we assume the stock equilibrium of bank reserves, i.e., = R. Given this, we obtain the following important relation D”=L+Sp;
L=Lb-tLc,
Rb
(3)
assuming stock equilibrium of securities, Sp + Sb + SC = D”, and of money, M = L. Condition (3) states that the public’s holding of financial assets is equal in amount to the total national debt out5To simplify the analysis, we assume that the public and the do not discount their holdings of government bonds, in spite increases in taxation by the government to finance its debt. On long as such a discount factor is a positive constant, our analysis in any essential way. 56
commercial banks of possible future the other hand, as will not be altered
Optimal Monetary Policy standing [see Modigliani (1963) and Tobin (1969)]. Assume, following much of the money-supply literature, that the currency-deposit ratio (CY)is an exogenously given constant. Let y be the legal reserve ratio. Also, let iF and i, respectively, signify the central bank’s discount rate and the nominal interest rate on bonds. Then we can obtain the following “supply function of money”6
M = +(i, iF, y, cx)Sc; d+/di > 0, (4) &$/SF < 0,
&$/ay < 0,
@/&I
< 0 .
3. The Short-Run Equilibrium We assume that the single output Y is produced with labor N and capital K under a constant-returns-to-scale, diminishing-returns technology,’ i.e., y = y(k);
y' > 0,
y" < 0 ,
(5)
where y = Y/N and k = K/N. Here we assume full employment. Following much of the literature, we assume that the nominal demand for money L is written as
L = L[i, y’(k) + I?, pY, A”],
(6)
where & < 0, $ < 0, La > 0, and t, > 0, with the subscript j (i = 1, 2, 3, 4) d eno t ing the partial derivative with respect to the jth argument. Here i and y’(k) + ITS, respectively, signify the nominal rate of return on securities and on equities, where # denotes the expected rate of inflation. Note that in (6) the public’s portfolio choice of the three assets (securities, equities, and money) is made explicit. Following the money and growth literature, we assume that the money-demand function is homogeneous of degree one in its ‘For the derivation of this formula, see Takayama-Drabicki (1976) and Drabicki-Takayama (1980). The derivation of such a formula is along the line of Brunner-Meltzer (1964) and Takahashi (1971). TO illustrate the importance of the endogeneity of the money supply, we may simply observe that a change in the interest rate (which is obviously an endogenous variable), for example, affects the commercial banks’ behavior regarding free reserves and loans, and that this affects the money supply. ‘In the present paper we, for simplicity, ignore productive activities of the government, which, in turn, follows much of the literature. 57
John 2. Drabicki
and Akira
last two arguments.
Takayama
Then noting
(l), we rewrite
(6) as,
L/W) = Ui, y’(k) + a’>y(k), k + 4,
(7)
where p is the price level and d = D”/(pN) represents the real per capita national debt. In an optimization model, it would be reasonable to assume # = 7~ (perfect myopic foresight), where 7~ denotes the actual rate of inflation. Using (4) and (7) with v = $, we obtain Ad = L[i, y’(k) + a, y(k), k + d] ,
(8)
where X = SC/D” and L = i/4 with Z.,r < 0, L2 < 0, L3 > 0, L4 b 0. Here CL, iF, and y are assumed to be constants. Namely, to simplify the analysis, we assume that monetary policy is carried out by open market operations, as represented by changing the value of A. Here we assume that SC consists only of government securities, and thus that X signifies the proportion of the national debt held by the central bank. Turning to the goods market, the equilibrium relation is written as
(9)
Y=C+Z+G,
where C, I, and G, respectively, denote consumption, investment, and government expenditures on goods (all in real terms). Consumption is assumed, for the sake of simplicity, to be a constant fraction of real disposable income Yd; i.e., c = (1 - s)Yd; Letting yd = Yd/N, we obtain
0 < s< 1.
o = Z/N, and g = G/N
y = (1 - s)yd + 2) + g *
(10)
and using (9) and (lo),
(11)
The public’s real disposable income Yd is output (Y) minus taxes net of transfer payments (T) plus interest payments and capital gains (or losses) on the outstanding national debt (which are all expressed in real terms)? ‘The details of the derivation of (12) are explained in Drabicki-Takayama (1980). This exercise is analogous to the derivation of the expression for disposable income 58
Optimal yd = Y - T + iD - ITD;
D E D”/p
Monetary
Policy
.
(1%
Note that capital gains or losses associated Gith price changes incorporated into Y d.’ By subsuming the interest payments on public debt into the variable T”, i.e., by letting T” = T - iD, may simplify (12) as Yd = Y - T’ - ITD. Letting t” 3 TX/N, then have yd = y - tX - rd. For the present fi.mction’O
study we postulate
I = Z[i - IT, y’(k), Yd];
I1 < 0,
are the we we
(13) the following
IQ > 0,
investment
13 > 0 .
(14)
That is, (real) investment depends on the real rate of return on bonds, on the real rate of return on physical capital, and on disposable income. The dependence on (i - IT) and y’(k) reflects the firm’s portfolio choice regarding the physical stock of capital, bonds, for the outside money case that appears in the usual money and growth literature [e.g., Burmeister-Dobell (1970), pp. 166-671. It is assumed that all bonds have a fixed money price (say, unity) realizable on demand with a variable interest rate [see Modigliani (1963), Foley-Sidrauski (1971), etc.] By assuming away the problem of Tobin’s 9, we may avoid the problem of capital gains or losses from holding physical assets due to a change in 9. ‘Foley-Shell-Sidrauski (F-S- S) (1969) and Foley-Sidrauski (F-S) (1971) both consider the two output (consumption and capital goods) economy, where pt and pm, respectively, denote the prices of the capital good and money in terms of the consumption good. Both of these works assume that pm is kept constant by governmental policy. F-S-S assume that the economic agents expect pk to be constant (static foresight), while pk actually does change along the optimal growth process. Hence their expectations are unwarranted and irrational. F-S also contains a similar problem where the planner’s utility function (which we denote by u) is nonlinear. For the case of a linear u, F-S obtain the result that p, is constant along the optimal path. In this case, static expectations are warranted, but the problems associated with inflation are avoided as pk and p, are both constant. In any case, both F-S-S and F-S avoid the problem of capital gains or losses associated with price change altogether. We may also note that it is not clear why society wishes to choose the governmental policy of keeping p, constant. “‘An alternative formulation would be that investment demand depends exclusively on Tobin’s 9. [See Tobin (1969), f or example.] Such a formulation is left to the interested reader. There seems to remain some doubt regarding the plausibility of such a specification (especially in aggregate growth models). See, for example, recent exchanges among von Furstenberg, Lovell, and Tobin in the Brookings Papers (1977), pp. 347-408. 59
John Z. Drabicki
and Akira
Takayama
and money. To simplify the analysis, ment function can be written as” u=v[i-~~,y’(k),y~];
q
we assume that the invest-
u2>0,
u3>0,
where v = Z/N. Using (13) and (15), we may then rewrite SYd = u[i - T, y’(k), yd] + g - tX - Td,
(15) (11) as 06)
where the expression for y* is given in (13). Equations (8) and (16) d enote the short-run equilibrium relations of our model which, respectively, correspond to the LMand the IS- relations: they determine the short-run equilibrium values of n and i for given values of k and d and the specified values of the parameters g, ta and A. The model is completed with the specification of the accumulation (or decumulation) equations of the two assets, K and D”, namely, K=
Z
and
on=
p(G - T’),
(17)
where depreciation of the physical asset is assumed away. Here the dot signifies the time derivative. Assuming that N grows at a constant rate n, we obtain the following two equations from (17) k = u[i - T, y’(k), yd] - nk ,
084
d=g-t”-nd-nd.
(18b) M
4. Optimal Growth The problem of optimal growth in the context of the present model is to choose the (piecewise continuous) time path of g, t* and h so as to maximize a certain target functional, subject to (18) and to the short-run equilibrium conditions. Jumps in the state variables [e.g., Arrow-Kurz (1970), pp. 51-571 are ignored. In this paper we confine ourselves to a simple monetary policy rule, which is specified by
“This “homogeneity” assumption, though it is restrictive, turns out to be a very useful specification in simplifying the analysis. For an example of such a homogeneity specification, we may recall Friedman (1971, p. 325).
60
Optimal
Monetary
Policy
where A is an exogenously given constant and 0 is considered the policy parameter. Note that the policy specified in (19) may be interpreted as one where the “(augmented) high powered money,” MC + R, is increased (or decreased) at a rate 8, where we may recall D” = SC/X = (MC + R)/X.‘2 From (19), we can easily obtain -t” = ed, so that (13), (16), (18a) and (18b) are, respectively, modified as yd = y(k) + (cl - lT)d ;
(13’)
SYd = u[i - 97, y’(k), yd] + (Cl - 7r)d ;
(16’)
h = sy - nk - (1 - s)(0 - IT)~ ; ci = (cl -
7r -
n)d
;
where, to obtain (20), we use (Isa), (13’), and (16’). Using (8) and (16’) together with (13’), we obtain run equilibrium values of IT and i a?
IT = n(k, d; Our problem of optimal as to maximizeI
0, A); i = i(k, d;
growth
8, A) .
the short-
(22)
is to choose the time path of 8 so
‘*This specification corresponds to the usual specification in the money and growth literature in which there is no government spending or taxation except for the transfer of (outside) money, which grows at a constant rate 8. If we assume 8 is constant, then we obtain a descriptive model whose properties are studied elsewhere [Drabicki-Takayama (1980)], while in the present study tl is a policy parameter. Needless to say, given the framework developed in the previous section, we can consider more complex policy rules which maximize a certain target functional. r3For the sake of simplicity, we assume the existence of a unique short-run equilibrium as well as continuous differentiability of the functions n and i. ‘*Equation (23) is the standard objective functional of the optimal growth literature, where p is assumed to be a positive constant signifying the discount rate. We assume u’(c) > 0, u”(c) < 0 for all c 2 0 and u’(0) = a. On the other hand, this type of utility function is different from the one that appears in the usual literature on the optimum quantity of money, where money enters into the utility function. For skeptical comments on that type of utility function, see Clower (1968) and Hahn (1971, pp. 63-64). Incidentally, in the present maximization problem, we shall not impose the constraint I t 0, i.e., physical capital is allowed to be “eaten up.” The reader should be able to easily incorporate this constraint if he or she so desires. 61
John 2. Drabicki
and Akira
Takayama m
u(c)eWPtdt ,
rJE I
subject to (20), (2I), where
(22) and the usual nonnegativity
c = (1 - s)yd = (1 - s){y(k) + [e - T(k, d; Define
(23)
0
the Hamiltonian H = u(c) + j&k,
conditions,
8, X)]d} .
(24)
by d;
0, A) + vd(k, d;
8, A),
(25)
where p and Y are the multipliers signifying the shadow prices of the accumulation (or decumulation) of the physical good and of the national debt, respectively. Among the necessary conditions for an interior solution we haveI [u’(c)(l
- s) - ~(1 - s) + u](l - mO)d = 0 ;
Ii=pl.l-aH/ak, lim peeptk = 0 , t-am
(26)
iJ=pv-aHlad;
(27)
lim ueeP’d = 0 , t-b-
(28)
where 7~~ = aT/ae. Assume d > 0. Since we can show 1 - nTTe < 0 (see Appendix), we obtain from (26) u’(c)(l
- s) - p(l - s) + V = 0.
Using this expression we can compute the second equation of (27): i = (p + n)v .
the following
(2% relation
from
(30)
Hence, in view of the second relation of (28) with d > 0, we can obtain v = 0 for all t. It then follows from (29) that we have “For a discussion on the transversality condition, such as in (28), see Benveniste-Scheinkman (1982). Postulating the conditions which guarantee lim k > 0 and t--rlim d > 0, (28) reduces to I-+lim Fe -P’ = 0, lim ye-@ = 0 @3’) t-r 1-I 62
Optimal
Monetary
u’(c) = p I
Policy
(31)
which is the familiar Ramsey-Cass-Koopmans rule (or the “KeynesRamsey rule”) which equates the loss of marginal utility in giving up one unit of consumption to the shadow price of the accumulation of the physical good. Using v = 0 and (31), we can compute the following relation from the first equation of (27) with k > 0: & = (p + n - y’)p .
(32)
From (32), we may at once conclude that in the steady state, in which b = 0, the capital-labor ratio assumes the familiar “modified golden-rule” value k*, which is uniquely determined by y’(k*)
= p + n .
(33)
Also, by setting 1 = 0 and d = 0, respectively, in (20) and (21), we obtain the following steady-state values of d and 8: d* = [sy(k*)
- nk*]/[(l
Cl* = n + n*,
- s)n] ,
T* = r(k*,
d*;
(34) 8*, A) .
(35)
Real per capita consumption in the steady state (c*) is obtained from (24) and (35) as c* = (1 - s)[y(k*) + nd*] .
(36)
Utilizing (34), we may rewrite (36) as c* = y(k*) - nk* .
(36’)
This value of c* is precisely the same as that which is obtained in the usual optimal-growth literature. Note also that k*, d*, and c* in (33), (34), and (36’) are determined independently of X. Thus, an open-market operation (i.e., a change in A), in the course of optimal growth, has no enduring effect on k*, d*, or c*, though it would accompany a corresponding change in the optimal path of 8. We now summarize some of the results obtained in the above analysis. PROPOSITION 1: (i) The optimal-growth path for the present manetary economy follows
the familiar
Ramsey-Cass-Koopmans
rule
63
John 2. Drabicki
and Akira
Takayama
(31). (ii) The steady-state values of the capital-labor ratio and of real per capita consumption, under optimal growth in the present monetary economy, are identical to those obtained in the familiar Cuss-Koopmans-type model (in which money is not explicitly introduced).‘6 (iii} The steady-state rate of inflation (price change) is equal to the steady-state value of “monetary expansion” (p) minus the rate of population growth. (iv) Open-market operations have no effect on the (optimal) steady-state values of the capital-labor ratio, real per capita national debt, and real per capita consumption. We now wish to obtain an adequate characterization of the optimal-growth process. For this purpose we need to obtain the signs of the partial derivatives for the functions T and i, which are determined from the short-run equilibrium conditions. Postponing such a discussion to the Appendix, we will simply record the main results here:
ik > 0,
id < 0
if
6 - n 2 0,
iO < 0,
ih < 0 ,
W’b)
where nk = an/ak, rd = an/ad,ik = ailak, ih = Z/ah, etc. Here 7rg > 0, ?rh > 0, ia < 0, and i, < 0, for example, mean that monetary expansion via an increase in 8 or via an increase in A (an open-market purchase) increases the short-run rate of inflation and reduces the short-run interest rate. We may call TI, < 0 and ik > 0 the “capacity effect” and the “illiquidity effect,” respectively, as suggested to us by K. Nagatani.17 Substituting the expression for c found in (24) into (31), we may now solve for the optimal value of 8 as a function of k, d, p and h, i.e.,
8 = W, d, CL; A) ,
(38)
where we can easily compute
‘% is well known
that Ramsey (1928) confined
his analysis to the case of p =
0. “Note that (37) holds whether or not 0 is chosen at the optimal value. The relations in (37) are, in fact, obtained by us elsewhere [Drabicki-Takayama (1980)], where the economic interpretations of the signs of partial derivatives such as mk and 718 are also provided. 64
Optimal Monetary Policy
,_
(e -
pkd Tejd
>
fld =
-
IT) - rrdd (1 - n&d ’
ek =
-(“1
ep =
1 Oh= -EL 1 - TrTg ' u"(1 - s)(l - m,)d ’
(39) (40)
where ok = X)/ak, ed = se/ad, etc. Recalling I - 7~~ < 0, we can at once conclude from (37a), (39), and (40) that ok > 0,
ed < 0
if
0 - r 2 0,
0)~> 0,
and
oh < 0 .
(41)
Since ok > 0, ?rk < 0 and ik > 0 from (41) and (37), an increase in k implies a “counter-cyclical” monetary policy in the sense that a fall in the rate of inflation rr and an increase in i must, ceteris paribus, accompany an increase in the rate of monetary expansion 8. (An increase in 6 in turn tends to offset the effects on 7~ and i through pg > 0 and ie < 0.) Similarly, an increase in d which implies a fall in P and an increase in i should, ceteris paribus, accompany an increase in 6. The ceteris paribus nature of the above may have to be emphasized. Namely, since
the (optimal) time path of 0 obviously does not depend exclusively on the signs of ok and t&. On the other hand, the above consideration also indicates that there are no such things as the optimum quantity of money and an optimum fixed value of 6. This proposition might be of some interest in view of the popular argument that monetary authorities should follow a certain fixed “rule,” such as that of increasing the money supply in accordance with the average secular growth of aggregate economic output, instead of “indulging” in discretionary policies. In summary, we may conclude: PROPOSITION 2. The signs of the partial derivatives for the func-
tions n, i, and 8 are determined as in (37) and (41). In particular, the optimal monetary policy concerning the choice of 0 is “‘counter-cyclical” in the ceteris paribus sense described above. The optimal value of 8 would not, in general, be constant along the growth process. 65
John 2. Drabicki
and Akira
Takayama
5. Dynamic path Substitute and 2 as
(38) into (20) and (21) and define
the functions
k
r; = i(k, d, JJ,; A) = sy(k) - nk - (1 - s){e(k, d, p;
- dk, 4
x)
W , d, I.C A>,AlId,
(42)
and d = d(k, d, p;
x)
= {W, d, P; A) - T[k, d; Also, from (32), define
e(k, d, p;
the function
A), A] - n}d .
(43)
& as
$ = lik 14= [P + n - dW1~ .
(44)
Given (28), the optimal path of (k, d, CL) is completely described by the system of differential equations (42), (43), and (44), where A is assumed to be constant. To study the behavior of the dynamic path, we first compute from (42)
ai/ak = I./ -
n,
a/i/ad= 0, ai/ap =
-l/tP
(45)
where we utilize the expressions for Ok, fld, and 8, that are recorded in (39) and (40). Also, from (43) we may compute
ad/ak= -I/,
ad/ad= -71, ad/al*= i/(1 - s)u’,
where we again utilize (39) and (40). From (44), we further al;/&
= &‘,
a&/ad = 0,
a$/ak
= p + n - y’ .
(46)
compute
(47)
Observe from (45) and (47) that ailed = 0 and a&/ad = 0. This means that equations (42) and (44) are independent of d, and as such they form a subsystem of two differential equations in the 66
Optimal
Monetary
Policy
two variables k and CL.Thus the optimal time paths of k and i.~ can be determined independently of equation (43) which specifies the behavior of d. Using the rest of the relations found in (39) and (41), and recalling the transversality condition (28), we may illustrate the dynamic optimal path of (k, i.rJ in Figure 1 by the curves a and a’, which are the stable branches of the system of differential equations (42) and (44). It can be seen from Figure 1 that, regardless of the initial real per capita debt d, if the initial value of the capital-labor ratio is less than (greater than) the steady-state value k*, the multiplier p should be chosen in such a way as to enter the time path labelled “a” (“a’ “). In such a case, along the optimal path the capital-labor ratio will increase (decrease) monotonically over time and approach k* in the limit, and the corresponding value of the multiplier i.~will decrease (increase) monotonically, approaching i.~* in the limit. From (31), we may also conclude that real per capita consumption c will increase (decrease) in this process while monotonically approaching c *. If the initial value of k is equal to k*, it is optimal for the economy to remain at (k*, CL*, c*). These conclusions are identical to those obtained in the usual literature a
The Optimal
Figure 1 Behavior of k and p
k
67
John 2. Drabicki
and Akira
Takayama
la Cass and Koopmans where money is not introduced. This result, together with statements (i) and (ii) of Proposition 1, shows that the familiar Cass-Koopmans-type results on optimal growth for a centralized nonmonetary economy can, in essence, be validated for a decentralized monetary economy. We may call this result the “ditheorem chotomy theorem, ” in the sense that the optimal-growth for the real sector can be “dichotomized” from the repercussions from the monetary sector.” To study the behavior of the three-dimensional trajectory of (k, d, p), we will confine ourselves to its behavior in the neighborhood of the steady state. For this, we consider the following coefficient matrix of the differential equations (42)-(44)
al; ak
al;
ai -
ad
G
a$ ii
w ak
y’--n
=
w ali, ad av..
0
-Y’
-n
-by”
0
-l/d
l/(1
- S)d
(48)
p+n-y’
in which the explicit forms of the components are obtained in (45)(47). We then compute the three eigenvalues of the above matrix as follows: (p +- VP” + 4&‘/u”)/2
and
-n.
(49)
Namely, two of the eigenvalues are negative while one eigenvalue is positive, and thus we may conclude the saddle-like nature of the optimal path. More specifically, the steady state is a “saddle point of type 2,” i.e., the optimal paths for different initial values define a two-dimensional surface in the three-dimensional space (k, d, p.). Under the usual circumstances, the optimal path is obviously unique for given initial values of k and d. “It can be shown that the dichotomy theorem holds even for a much simpler economy, in which securities are assumed away. See Drabicki-Takayama (1978). Note that the present paper, as well as D-T (1978), prooes the dichotomy theorem, which may be sharply contrasted with F-S-S (1969) and F-S (1971), where they simply assume it (implicitly). 68
Optimal Monetary Policy Given the optimal trajectory of (k, d, IL), the optimal value of the monetary parameter 8 can be computed from condition (38). The optimum time path of the interest rate i and the rate of price change IT can then be extracted from (22) and (37). As a general rule, the nature of such an optimal time path of (t3, IT, i) will depend on the initial values of k and d. In addition, none of the time paths of 8, IT, or i need be monotone. Furthermore, at each instant of time, for given values of k, d, and p+ (38) specifies an optimal value of 8 and hence an “optimal” rate of inflation (price change) IT. As these change over time, so will the optimal rate of inflation. In short, (although it may be contrary to a popular belief), there is no such thing as the optimum rate of inflation or deflation (which is constant during the growth process). Similarly, there are no such things as an optimum quantity of money and an optimum value of 9, which are constant along the growth process. On the other hand, if we confine ourselves to the steady state, we have T* = 0* - n, as observed earlier (see Proposition 1). We may now summarize some of the above results.
3. (i) The steady state in the present monetary economy is a saddle point, and given the initial values of k and d, we may choose a unique time path of t? which achieves optimal growth (with which the time path of (k, d, p) is associated). (ii) The dynamic behavior of the optimal values of the capital-labor ratio (k), the shadow price of the accumulation of physical capital (p), and real per capita consumption (c) is identical to the familiar behavior found in the standard literature (for p > 0), such as Cuss (1965) and Koopmans (1965), in which money is not introduced. The optimal trajectory of (k, p, c) in the present monetary economy is independent of the time path of real per capita national debt (d), i.e., optimal monetary policy is “neutral.” (iii) There are no such things as the optimal rate of inflation, the optimum quantity of money, and the optimum value of 9, which are constant in the growth process, while in the steady state we have n* = 8* - n.”
PROPOSITION
Lastly,
an interesting
comparative-dynamics
question
in this
]@A simple characterization of dynamic behaviors of d and 6 may be obtained if we assume. that k and TVwere set at k* and TV*. In this case, d = d(k*, d, p*; A). Noting &i/ad = --n, we may conclude that d converges monotonically to d*. Since 8d < 0 in a neighborhood of the steady state, we may also conclude that f~ must decrease (i.e., a contractionary monetary policy) along the path in which d increases (which naturally accompanies monetary expansion) in a “sufficient” neighborhood (see Appendix) of the steady state, where k and d are set at k* and d*. 69
John 2. Drabicki
and Akira
Takayama
context may be the effect of a change in A (open-market operation) on the optimal rate of monetary expansion 8. To investigate this question, observe
al;/ax = ad/ax = ah/ax = 0, which follows from (42), (43) and (44) and the second expression of (40). Thus, for each fixed value of (k, d), a change in A induces no corresponding changes in the k = 0, the d = 0 and the 6 = 0 loci in the phase space, which in turn implies no change in the time path of IL; namely, the indirect effect of h on 0 via p vanishes. Thus, for each given (k, d), the effect of A on 9 is simply given by - no) [see (JO)], w h’ic h is negative. Thus we may con4 = nJ(l clude: PROPOSITION 4. An expansionary monetary policy uia an open-market purchase must be offset by a corresponding “contractionary” monetary policy associated with r-educing 8 in order for the economy to be on the optimal path. Similarly, a contractionary manetary policy via an open-market sale must be accompanied by an expansionary monetary policy as represented by increasing 0. Received:January, 1981 Final version received: June, 1982
References Arrow, K.J. “Applications of Control Theory to Economic Growth.” In Mathematics of the Decision Sciences. G.B. Dantzig and A.F. Veinott, eds. Part 2. Providence, R.I.: American Mathematical Society, 1968, 85-119. Arrow, K.J. and M. Kurz. Public Investment, The Rate of Return, and Optimal Fiscal Policy. Baltimore: Johns Hopkins Press, 1970. Benveniste, L.M. and J.A. Scheinkman. “Duality Theory for Dynamic Optimization Models of Economics: The Continuous Time Case.” Journal of Economic Theory 27 (June 1982): 1-19. Blinder, A.S. and R.M. Solow. “Does Fiscal Policy Matter?’ Journal of Public Economics 2 (November 1973): 319-37. Brock, W.A. “A Simple Perfect Foresight Monetary Model.” Journal of Monetary Economics 1 (April 1975): 133-50. Brunner, K. and A.H. Meltzer. “Some Further Investigations of Demand and Supply Functions for Money. ” Journal of Finance 19 (May 1964): 240-83. 70
Optimal
Monetary
Policy
Burmeister, E. and R.A. Dobell. LMathematical Theories of Economic Growth. New York: Macmillan, 1970. Calvo, G.A. “On the Time Consistency of Optimal Policy in a Monetary Economy. ” Econometrica 46 (November 1978): 1411-28. Cass, D. “Optimum Growth in an Aggregate Model of Capital Accumulation.” Review of Economic Studies 32 (July 1965): 23340. Clower, R.W. “Comment on Papers by Tobin and Marty.” Journal of Political Economy 86 (July/August 1968, Part II): 876-80. Drabicki, J.Z. and A. Takayama. “Endogenous Supply of Money, Myopic Perfect Foresight, and Economic Growth.” Unpublished. November, 1980. and -. “Money, Inflation and Optimal Economic Growth. ” Economics Letters 1 (1978): 55-58. Foley, D.K., K. Shell, and M. Sidrauski. “Optimal Fiscal and Monetary Policy and Economic Growth. ” Journal of Political Economy 77 (July/August 1969, Part II): 698-719. Foley, D. K., and M. Sidrauski. Monetary and Fiscal Policy in a Growing Economy. New York: Macmillan, 1971. Friedman, M. The Optimum Quantity of Money and Other Essays. Chicago: Aldine, 1969. “A Monetary Theory of Nominal Income. ” Journal of Political Economy 79 (March/April 1971): 323-37. Hahn, F.H. “On Money and Growth. ” Journal of Money, Credit, and Banking 1 (May 1969): 127-87. “Professor Friedman’s Views on Money. ” Economica 38 (February 1971): 61-80. Koopmans, T.C. “On the Concept of Optimal Economic Growth.” In The Econometric Approach to Development Planning. Pontificiae Academiae Scietiarum Scripum Varia. Amsterdam: North Holland, 1965. Kydland, F.E. and E.C. Prescott. “Rules Rather than Discretion: The Inconsistency of Optimal Plans. ” Journal of Political Economy 85 (June 1977): 473-91. Modigliani, F. “Monetary Mechanism and Its Interaction with Real Phenomena. ” Review of Economics and Statistics 65 (February 1963): 79-107. Nagatani, K. “A Note on Professor Tobin’s ‘Money and Economic Growth. ’” Econometrica 38 (January 1970): 171-75. -. Monetary Theory. Amsterdam: North-Holland, 1978. Ramsey, F.P. “A Mathematical Theory of Saving. ” Economic Journal 38 (December 1928): 543-59. 71
]ohn 2. Drabicki
and Akira Takayama
Sidrauski, M. “Inflation and Economic Growth. ” Journal of Political Economy 75 (December 1967): 796-810. Stein, J.L. “The Optimun Quantity of Money.” Journal of Money, Credit, and Banking 2 (November 1970): 397-419. Takahashi, F. “Money Supply and Economic Growth.” Econometrica 39 (March 1971): 285-303. Takayama, A. and J.Z. Drabicki. “On the Endogenous Supply of Money. ” Keizai Kenkyu (Economic Review) 27 (October 1976): 336-48. Tobin, J. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84. -. “A General Equilibrium Approach to Monetary Theory. ” Journal of Money, Credit, and Banking 1 (February 1969): 1529. -. “Notes on Optimal Monetary Growth.” Journal of Political Economy 76 (July/August 1968, Part II): 833-59. Turnovsky, S.J. and W.A. Brock. “Time Consistency and Optimal Government Policies in Perfect Foresight Equilibrium. ” Journal of Public Economics 13 (April 1980): 183-212. Tsiang, S.C. “A Critical Note on the Optimum Supply of Money. ” Journal of Money, Credit, and Banking 1 (May 1969): 266-80. Uzawa, H. “An Optimal Fiscal Policy in an Aggregative Model of Economic Growth.” In The Theory and Design of Economic Deuelopment. I. Adelman and E. Thorbecke, eds. Baltimore: The Johns Hopkins Press, 1966. Yoshikawa, H. “Alternative Monetary Policies and Stability in a St07 chastic Keynesian Model. ” Znternational Economic Review 22 (October 1981): 541-65.
Appendix. The Derivation
of (37)
The short-run equilibrium of a and i which are defined by (22) are obtained from (8) and (16’) with (13’). A straightforward differentiation of these equations yields:
aT/ak = -{[u2y’ + (u3 ai/ak =
{(Us
+
-
s)Y’]L,
-
U&~ZJ”
+
L,y’
+
W/J~
ad)(Lzy” + Ly’ + L) + [uzy” + (us - S)Y‘WI
aT/ad = 72
(A.14
-[II&
-
I&)
+
(e -
T+JL~]/J,
3 (A. lb) (A. lc)
Optimal Monetary Policy ailad
= -[(q
+ od)(h - L4) - (0 - 7r)aLJJ
an/a0 = -L,ud/]
(A. Id)
,
, ai/ae = L,ud/J ,
(A. le)
an/ax = --old/] , ailax = -(q + ad)d/J , ] = -[(q
(A. lf)
+ ad)& + ulLz] ,
64.2)
where u = (1 - s) + u3 signifies the marginal propensity to spend out of current disposal income, which we assume to be positive. In order to determine the signs of these partial derivatives, it becomes necessary to appeal to the correspondence principle. To this end, we hypothesize the adjustment process that an excess demand for goods increases IT, while an excess demand for money increases, i, i.e.,
dn/dT = a{u[i - IT, y’(k), yd] + (tl - n)d - syd} = @(IT, i), di/dT = b{L[i, y’(k) + IT, y(k), k + d] - Ad1 E ‘P(~F,i), where yd = y(k) + (0 - v)d, and stants. Here k and d, as well as 8 time. The necessary and sufficient ear approximation system of (A.3)
(A.3) (A.4)
where a and b are positive conand A, are fixed, while T signifies conditions for stability and (A.4) are
of the lin-
where Q,.,= a@/av,V!, = aq/ai, and are evaluated at a particular short-run equilibrium point. Although (A.5) provides only a sufficient condition for stability of the original system we, following the usual practice of the correspondence principle, assume (A.5). The partial derivatives a,,, Wi, etc., are readily computed as Q,, = -a(u,
+ ud),
@i = au1 < 0, 64.6)
‘PT= bL,
qi=
bL,
(A.5) as
@‘, + qi = -a(q
+ ad) + bL, < 0,
(A. 7) 73
John Z. Drabicki and Akira Takayama a),** - 4+*,, = -ab[(ul + ad)& + v,Lz] > 0
(A.@
(i.e., J > 0.) Since ur < 0, L1 < 0 and L2 < 0, the “trace condition” (A.7) requires (ur + ad) > 0 in which case the “Jacobian condition” (A.8) is automatically satisfied. Also J > 0 requires d > 0 or, more specifically, J > 0 if and only if d > -q(L1 + L&/(aL1)(>O), which we assume throughout the paper. From this discussion, we can specify the sign of the partial derivatives in (A.1) as they are recorded in (37) in the text, where we assume X > L4 to obtain the signs of ?rd and id. The assumption of A > L4 means that an increase in d increases the demand for money to a lesser degree than it increases the supply of money. That 1 - 1~~ < 0 is obtained easily from (A.le). As mentioned earlier, the results here and their economic interpretations are obtained elsewhere by Drabicki-Takayama (1980). The discussion in this Appendix is only included to make the present paper sufficiently self-contained.
74