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Money, inflation, and optimal economic growth
Economics Letters 1 (1978) 0 North-Holland Publishing
55
55-58 Company
MONEY,INFLATION,ANDOPTIMALECONOMICGROWTH John Z. DRABICKI University of Arizona, Tucson, AZ 85 721, USA
Akira TAKAYAMA Purdue University, West Lafayette, IN 4 790 7, USA Received
June 1978
Using a two-asset model of a growing competitive economy, it is shown that the optimal growth path can be achieved via a policy operating exclusively on the rate of growth of the money stock.
The purpose of this Letter is to demonstrate that the well-known neoclassical framework of optimal growth a la Ramsey, Koopmans, Cass, and others, which is typically discussed in terms of a centralized economy, is applicable with proper modifications to decentralized economies. We illustrate, with a simple model, how optimal growth may be achieved via a simple monetary policy. To simplify our discussion we utilize the usual two-asset (outside money and physical capital) model which was developed in the ‘money and growth’ literature a la Tobin (1965) and others [cf. Burmeister-F-Dobell (1970)]. Assume that the single output Y is produced with labor N and capital K under a constant returns to scale and diminishing returns technology; i.e., Y =.Y(k) 7
y”
Y’>O,
(1)
where y = Y/N
and
k-K/N,
where we assume full employment. Let p be the price of the output and denote the real per capita supply of money by m f M/pN, where M is the nominal money stock. Then letting rre be the expected rate of price change, we can write the money market equilibrium relation as [cf. B-D (1970, p.l58)] m=Lb’(k)+#,y(k),
ktm],
(2)
where L 1 < 0, L2 > 0, and 0
J.Z. Drabicki and A. Takayama / Optimal economic growth
56
an optimization model, it is reasonable to assume rre = rr (perfect fores&ht) where ‘IIE c/p and the dot signifies the time derivative. Then we can rewrite (2) as ‘II= rr(k, m) ,
(3)
where nk>O
and
,rk = anlak,
n,
riz = [0 - n(k, m) - n]m = Gz(k, m; 0))
(4)
where 0 =2/M, n = h/N, and n is assumed to be a positive constant. Letting C and I, respectively, denote real consumption and real investment, may write the equilibrium condition of the goods market as
we
Y=C+I,
(5)
where government spending is assumed away. We adopt the usual consumption specification C = (1 - s) Yd, where Yd denotes real disposable income and 0
(6)
for simplicity,
we have I? = I. Then utilizing (5) and
i = sy(k) - nk - (1 - s)[e - n(k, m)] m E k(k, m; 0) . In this decentralized framework we assume that the policy authorities the time path of 0 (the rate of monetary growth) so as to maximize lJ=
s 0
O”u(c) eFPrdt
,
(7) choose
(8)
[where c is given by (6)], subject to (4), (7) and the usual non-negativity constraints. Here p > 0 is a discount factor, and it is assumed that u’(c) > 0, U”(C) < 0 for all c & 0, and u’(0) = 00. For simplicity we assume that K can be ‘eaten up’. The reader should be able to easily modify the analysis for the case in which I& 0. Define the (current-value) Hamiltonian H by HE u(c) + & (k, m; 0) + cGz(k, m; e), where p and v are multipliers. Then among the necessary conditions for an interior solution we have
[u’(c)( 1 - s) - (1 - s) P + v] m = 0 , 1; =
r;=pp-dH/ak,
lim pe +‘k t-+-
=0 ,
VP -
(9) ,
(10)
=0 .
(11)
aHlam
lim ve+‘m t+-
J.Z. Drabicki and A. Takayama / Optimal economic growth
Following the usual procedure,
we obtain from (9)-(11)
- (ja t n)] u’/u” )
c = -b’(k)
57
that v = 0, /A= u’(c), and
ti = [P + n - Y’(k)1 /J = I.@, II> ,
e=e(k,m,I-o, ok = -0’
(13) 0,
- nkm)/m ,
where ok = M/ak, can compute
ak ak arit Z ajj Z
aic iii -aliz am ap am
= -(e
~ 77- x,)/m
+
,
8, = i/[(i
- S) WZU”], (14)
etc. To obtain (13) we utilize u’(c) = P and (6). Using (14) we
ai ap ah -ap ag ap_
,y’-n 1 = I ,
i
-l/u”
0
-Y’
-n
l/(1 -S)U”
-/.ly”
0
ptn-y’
for which the three eigenvalues are obtained (p 5 dp*
(12)
4/.~y’“/u”)/2
and
(15)
as
(16)
-n.
Namely, two of the eigenvalues are negative while one eigenvalue is positive. From this we may conclude the saddle-like nature of the optimal path. Namely, given the initial values of k and m, we may choose a unique time path of 0 which achieves optimal growth under the present decentralized scheme. Note that we may rewrite (7) as
k =y(k)
- nk - c ,
(17)
by virtue of (6). From this and (12) it easily follows that the dynamic behavior of (k, c, r_l)is identical to the familiar behavior a la Cass (196.5) and Koopmans (1965) for p > 0. In particular, the optimal trajectory of (k, c, p) is independent of the time path of m; i.e., optimal monetary policy is ‘neutral’. Let k*, c*, and.m*, respectively, signify the steady state values of k, c and m. Then we can easily obtain y’(k*)=p+n,
c*=y(k*)-nk*,
m*=[sy(k*)-nk*]/[(l
-s)n].
(18)
Note that k” is equal to the well-known ‘modified golden-rule value’. In order for m* to be positive, it is required that sy(k*) > nk*. From our analysis, it also follows that there are no such things as the optimal rate of inflation, the optimum quantity of money, and the optimum value of 0, which are constant in the growth process, while in the steady state we have 0 * = n(k*, m*> + n. Utilizing (15), we can show that the optimal trajectories of (k, m, p) define an ‘optimal surface’ in three-dimensional space. The projection of the optimal surface onto the k ~ p plane yields the aa’ loci in fig. 1, while projections of optimal trajectories onto the k - m plane yields configurations such as those of fig. 2.
58
J.Z. Drabicki and A. Takayama / Optimal economic growth
k*
k
Fig. 1.
k*
k
Fig. 2.
In the above analysis, we have assumed away interest bearing securities and the endogenous nature of the money supply. It can be shown, however, that these complications do not alter the basic conclusions obtained above [cf. DrabickiTakayama (1978)].
References Burmeister, E. and R.A. Dobell, 1970, Mathematical
theories of economic growth, ch. 7 (Macmillan, New York). Cass, D., Optimum growth in an aggregate model of capital accumulation, Review of Economic Studies XxX11, July. Drabicki, J.Z. and A. Takayama, 1978, An optimal monetary policy in an aggregate neoclassical model of growth, Unpublished manuscript, April. Koopmans, T.C., 1965, On the concept of optimal economic growth, in: The econometric approach to development planning, Pontiflciae Academiae Scientiarvm Scriptvm Varia (North-Holland, Amsterdam). Tobin, J., 1965, Money and economic growth, Econometrica 33, Oct.
55
55-58 Company
MONEY,INFLATION,ANDOPTIMALECONOMICGROWTH John Z. DRABICKI University of Arizona, Tucson, AZ 85 721, USA
Akira TAKAYAMA Purdue University, West Lafayette, IN 4 790 7, USA Received
June 1978
Using a two-asset model of a growing competitive economy, it is shown that the optimal growth path can be achieved via a policy operating exclusively on the rate of growth of the money stock.
The purpose of this Letter is to demonstrate that the well-known neoclassical framework of optimal growth a la Ramsey, Koopmans, Cass, and others, which is typically discussed in terms of a centralized economy, is applicable with proper modifications to decentralized economies. We illustrate, with a simple model, how optimal growth may be achieved via a simple monetary policy. To simplify our discussion we utilize the usual two-asset (outside money and physical capital) model which was developed in the ‘money and growth’ literature a la Tobin (1965) and others [cf. Burmeister-F-Dobell (1970)]. Assume that the single output Y is produced with labor N and capital K under a constant returns to scale and diminishing returns technology; i.e., Y =.Y(k) 7
y”
Y’>O,
(1)
where y = Y/N
and
k-K/N,
where we assume full employment. Let p be the price of the output and denote the real per capita supply of money by m f M/pN, where M is the nominal money stock. Then letting rre be the expected rate of price change, we can write the money market equilibrium relation as [cf. B-D (1970, p.l58)] m=Lb’(k)+#,y(k),
ktm],
(2)
where L 1 < 0, L2 > 0, and 0
J.Z. Drabicki and A. Takayama / Optimal economic growth
56
an optimization model, it is reasonable to assume rre = rr (perfect fores&ht) where ‘IIE c/p and the dot signifies the time derivative. Then we can rewrite (2) as ‘II= rr(k, m) ,
(3)
where nk>O
and
,rk = anlak,
n,
riz = [0 - n(k, m) - n]m = Gz(k, m; 0))
(4)
where 0 =2/M, n = h/N, and n is assumed to be a positive constant. Letting C and I, respectively, denote real consumption and real investment, may write the equilibrium condition of the goods market as
we
Y=C+I,
(5)
where government spending is assumed away. We adopt the usual consumption specification C = (1 - s) Yd, where Yd denotes real disposable income and 0
(6)
for simplicity,
we have I? = I. Then utilizing (5) and
i = sy(k) - nk - (1 - s)[e - n(k, m)] m E k(k, m; 0) . In this decentralized framework we assume that the policy authorities the time path of 0 (the rate of monetary growth) so as to maximize lJ=
s 0
O”u(c) eFPrdt
,
(7) choose
(8)
[where c is given by (6)], subject to (4), (7) and the usual non-negativity constraints. Here p > 0 is a discount factor, and it is assumed that u’(c) > 0, U”(C) < 0 for all c & 0, and u’(0) = 00. For simplicity we assume that K can be ‘eaten up’. The reader should be able to easily modify the analysis for the case in which I& 0. Define the (current-value) Hamiltonian H by HE u(c) + & (k, m; 0) + cGz(k, m; e), where p and v are multipliers. Then among the necessary conditions for an interior solution we have
[u’(c)( 1 - s) - (1 - s) P + v] m = 0 , 1; =
r;=pp-dH/ak,
lim pe +‘k t-+-
=0 ,
VP -
(9) ,
(10)
=0 .
(11)
aHlam
lim ve+‘m t+-
J.Z. Drabicki and A. Takayama / Optimal economic growth
Following the usual procedure,
we obtain from (9)-(11)
- (ja t n)] u’/u” )
c = -b’(k)
57
that v = 0, /A= u’(c), and
ti = [P + n - Y’(k)1 /J = I.@, II> ,
e=e(k,m,I-o, ok = -0’
(13) 0,
- nkm)/m ,
where ok = M/ak, can compute
ak ak arit Z ajj Z
aic iii -aliz am ap am
= -(e
~ 77- x,)/m
+
,
8, = i/[(i
- S) WZU”], (14)
etc. To obtain (13) we utilize u’(c) = P and (6). Using (14) we
ai ap ah -ap ag ap_
,y’-n 1 = I ,
i
-l/u”
0
-Y’
-n
l/(1 -S)U”
-/.ly”
0
ptn-y’
for which the three eigenvalues are obtained (p 5 dp*
(12)
4/.~y’“/u”)/2
and
(15)
as
(16)
-n.
Namely, two of the eigenvalues are negative while one eigenvalue is positive. From this we may conclude the saddle-like nature of the optimal path. Namely, given the initial values of k and m, we may choose a unique time path of 0 which achieves optimal growth under the present decentralized scheme. Note that we may rewrite (7) as
k =y(k)
- nk - c ,
(17)
by virtue of (6). From this and (12) it easily follows that the dynamic behavior of (k, c, r_l)is identical to the familiar behavior a la Cass (196.5) and Koopmans (1965) for p > 0. In particular, the optimal trajectory of (k, c, p) is independent of the time path of m; i.e., optimal monetary policy is ‘neutral’. Let k*, c*, and.m*, respectively, signify the steady state values of k, c and m. Then we can easily obtain y’(k*)=p+n,
c*=y(k*)-nk*,
m*=[sy(k*)-nk*]/[(l
-s)n].
(18)
Note that k” is equal to the well-known ‘modified golden-rule value’. In order for m* to be positive, it is required that sy(k*) > nk*. From our analysis, it also follows that there are no such things as the optimal rate of inflation, the optimum quantity of money, and the optimum value of 0, which are constant in the growth process, while in the steady state we have 0 * = n(k*, m*> + n. Utilizing (15), we can show that the optimal trajectories of (k, m, p) define an ‘optimal surface’ in three-dimensional space. The projection of the optimal surface onto the k ~ p plane yields the aa’ loci in fig. 1, while projections of optimal trajectories onto the k - m plane yields configurations such as those of fig. 2.
58
J.Z. Drabicki and A. Takayama / Optimal economic growth
k*
k
Fig. 1.
k*
k
Fig. 2.
In the above analysis, we have assumed away interest bearing securities and the endogenous nature of the money supply. It can be shown, however, that these complications do not alter the basic conclusions obtained above [cf. DrabickiTakayama (1978)].
References Burmeister, E. and R.A. Dobell, 1970, Mathematical
theories of economic growth, ch. 7 (Macmillan, New York). Cass, D., Optimum growth in an aggregate model of capital accumulation, Review of Economic Studies XxX11, July. Drabicki, J.Z. and A. Takayama, 1978, An optimal monetary policy in an aggregate neoclassical model of growth, Unpublished manuscript, April. Koopmans, T.C., 1965, On the concept of optimal economic growth, in: The econometric approach to development planning, Pontiflciae Academiae Scientiarvm Scriptvm Varia (North-Holland, Amsterdam). Tobin, J., 1965, Money and economic growth, Econometrica 33, Oct.