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Stabilization effect of endogenous money supply in a descriptive neoclassical growth model
KAZUO MINO Hiroshima
Uniuefsity lawn
Stabilization Effect of Endogenous Money Supply in a Descriptive Neoclassical Growth Model* This paper develops a neoclassical monetary growth model with myopic perfect foresight in which monetary expansion rate changes with government budget deficit or surplus. We show the saddlepoint instability of the neoclassical monetary growth model assuming that the constant money growth policy can be avoided under the endogenous money supply rule. We also compare the purely money-financed regime where no government bonds exist with the mixed-financed strategy under which government deficit is partly financed by issuing interest-yielding securities. Our result suggests that the destabilizing effect of bond financing, which has been frequently observed in Keynesian macrodynamic models, does not necessarily hold in the neoclassical system.
1. Introduction The purpose of this paper is to demonstrate that endogenous money supply policy may promote stability of the neoclassical monetary growth model. It is well known that Tobin’s monetary growth model, which assumes the constant money growth rule, exhibits the saddlepoint instability under myopic perfect foresight. This paper shows that if money supply changes with government deficit or surplus, Tobin’s model with myopic perfect foresight could be stable. We first consider the case of a purely money-financed economy where no interest-bearing securities exist, and reveal that the economy can be stabilized if the real government deficit per capita is higher than a certain level at the steady state. We also consider a mixed-financed economy in which the government deficit is partly financed by issuing interest-bearing bonds. It will be seen that, unlike the standard Keynesian macroeconomic models, the neoclassical monetary growth model is not necessarily destabilized by introducing bond-financing policies. *I am very much indebted to Jerome L. Stein for his insightful comments on an earlier version of this paper. I also wish to thank Juichi Kan, Takayoshi Kitaoka, and anonymous referees for their valuable comments.
Journal of Macroeconomics, Winter 1988, Vol. Copyright 0 1988 by Louisiana State University 0164-0704/88/$1.50
10, No. Press
1, pp.
125-137
125
Kazuo Mino The stability issue of the Tobin model is addressed by many authors. In particular, ever since Nagatani (1970) revealed the saddlepoint property of the Tobin model with myopic perfect foresight, a number of authors have proposed sufficient conditions under which the model may be stable. Hadjimichalakis (1971a, 197Ib) generalizes the Tobin model to show that either expectation formation or adjustment process in the money market should be sluggish to hold stability. This line of research is pursued by Hadjimichalakis and Okuguchi (1979), Benhabib and Miyao (1981), Hadjimichalakis (1981), and Hayakawa (1984). Recently, Drabicki and Takayama (1984a, 1984b) found another source of stability. They introduce interestyielding securities into the Tobin model and assume that these securities are imperfect substitutes for money and capital. This assumption enables them to introduce an independent investment function into the neoclassical monetary growth model, which becomes a stabilizing factor. Inspection of these contributions indicates that some friction or imperfection should be assumed to establish the stability of the Tobin model if the constant money growth policy is assumed. ’ The result of this paper suggests that even if everything is perfect in the neoclassical sense, the economy will be stable under a very simple money supply rule. The paper proceeds as follows. Section 2 sets up the model. Section 3 examines the dynamic behavior of the purely money-financed system. Section 4 discusses the mixed-financed regime where the model includes government bonds. Concluding remarks are given in Section 5.
2. The Model The analytical framework ing equations: y = f(k)>
of this paper is given by the follow-
f’>O,f”
Y = (1 - s)[(l - T)(Y + b) - IT(~ + b/p)] + L + nk + g ;
(1) (2)
‘It should be emphasized that Hayakawa (1979) does not belong to this type of literature. He introduces real purchasing power into the liquidity preference f&ction as well as the consumption demand function, and shows that this generalization can resolve the instability problem of the Tobin model with myopic perfect foresight without assuming any sluggishness in market-adjustment mechanism or in expectation formation.
126
Stabilization m = ~[p, f’(k)
+ TTT, y, k + m +
b/PI ,
L1 < 0, Lz < 0, L3 > 0, 0 < L4 < 1 ; p =f’(k) ti + i/p
(3)
+ 7r ;
= g + b - ~(y + b) - (n + r)(m + b/p) ; b=Om.
Effect
(4) and
(5) 63)
Here, notation is y = output per capita, k = capital intensisty, p = nominal interest rate, T = rate of inflation, m = real balances per capita, b = real interest payments per capita, 7 = tax rate (0 < 7 < l), g = real government purchases per capita, 8 = interest payment-money ratio, s = propensity to save, and ?z = rate of population growth. Equation (1) is the productivity function that satisfies the standard neoclassical properties. Equation (2) states that output per capita is equal to real consumption per capita, (1 - ;)[(l - ~)(y + b) - n(m + b/p)], plus real investment per capita, k + nk, plus real government expenditures per capita, g. Following Tobin (1965), we assume a simple consumption function. Consumption demand is proportional to the real disposable income, which is equal to real income after tax per capita less inflation on real financial assets per capita. It is also assumed that the stock of government bonds consists of perpetuities each paying $1.00 per annum in interest. Therefore, b also denotes the number of bonds per capita and the price of bond is l/p. Money market equilibrium condition is given by Equation (3). Demand for the real balances per capita is negatively related to the nominal interest rate, p, and the nominal rate of return, f’(k) + T, and positively related to the output per capita, y, and the real wealth per capita, k + m + b/p. Note that the anticipated rate of inflation included in the nominal rate of return on capital is equal to the actual rate of inflation due to the assumption of myopic perfect foresight. Capital market is assumed to be perfect in the sense that the physical capital is a perfect substitute for the government bonds, so that the arbitrage condition, (4), is held at each moment of time. Equation (5) denotes the government budget constraint. The real government deficit is real per capita government consumption, 127
Kazuo Mino g, plus real per capita interest payments, b, less taxes, r(y + b). The deficit should be financed by new money creation and/or issuing interest-bearing bonds. Thus, we have k/pN
+ (~/PNWP)
= g + b - T(Y + b)
>
where M, B, N, and p denote the nominal stock of money, the nominal interest payments, the population, and the price level, respectively. From the definitions m = M/pN and b = B/pN, the above can be expressed as (5). Finally, we assume that the interest-payment-money ratio, 8, is kept constant at a certain level. The model contains six endogenous variables-y, m, k, b, p, and IT. Variables g, 7, and 8 are control variables. Substituting (l), (4), and (6) into (3), and solving it with respect to IT, we can express the rate of inflation such that
7~ = n(k, m: 13). Partial derivatives
of this function
are
L$+L, IQ = (L,8m/p2)
(7)
- L1 - L,
-f’>o,
and
?r, = ~(1 + fm - 1 (L,0m/p2)
- L1 - L, ’
(9)
If no government bonds exist (0 = 0), then 7~,,,has a negative sign (so that the Wicksell effect holds). This case is the same as Tobin’s original model. The Wicksell effect may not hold if there are government bonds (0 > 0). Since 8 is assumed to be constant, an increase in the real balances per capita raises government bonds at the same time. Thus, if 8 is sufficiently large, the increase in per capita demand for real balances through the wealth effect could be larger than the increase in m, so that the rate of inflation will increase to equilibrate the money market.2 *Another approach to reverse the sign of mr, is proposed by Hayakawa (1979). As mentioned in Footnote 1, in specifying the liquidity preference function, he considers income effect arising from real balance flows to generate the possibility that P, has a positive sign. Also see Drabicki and Takayama (1981), and Hayakawa (1984) for further discussion of this approach.
128
Stabilization
Effect
However, it should be noted that if rational households fully anticipate the future tax liabilities on the current debt, the govemment securities fail to constitute a part of net wealth. In this case, even if 0 > 0, TTT,is strictly negative.3 Precise analysis of this problem requires an optimizing framework rather than the ad hoc model employed here. Since, in our model, the results obtained in the case of neutral bonds may be similar to those in the purely moneyfinanced economy (f3 = 0), we will assume that the net wealth includes government debt. By (l), (2), and (6), the capital intensity changes according to i = [$ + T(1 - s)]f(k) - m(1 - s)[(l - T)e - ~(k, m)z(k, m)] - nk - g ,
(10)
where z(k, m) = 1 + 0/p = 1 + e/v(k) + T(k, m)]. Using (l), 0% and (7), th e g ovemment budget restraint, (5), provides the dynamic behavior of the real balances per capita: ni = [m/z(k,
g - vW m
m)]
+ (1 -
7)e
- (n + ?r(k, m)) z(k, m) Thus the complete dynamic equations, (10) and (11).
1 .
(11)
system is given by a set of differential
3. A Purely Money-Financed
Economy
If there were no government bonds (0 = 0), then the moneyfinancing policy means that the money supply changes with the
3This may A(0 S A 5 1) liabilities are mulation, the
be discussed by defining the net wealth as k + m + X(b/p), where represents the degree of rationality of asset holders. If the future tax completely anticipated by the public, then A = 0. Given this forequilibrium rate of inflation becomes ~(k, m: 9, A) and we obtain
wr, =
Therefore, a negative
qTT, may sign.
be positive
L,(l
+ AO/p)
(L,fJAm/p2)
if A0 > p(l/L,
-
1
- L, - & -
1). When
A = 0, vr, obviously
has
129
Kazuo Mino government deficit. The growth and stability implication of this policy regime in a Keynes-Wicksell growth model is discussed by Infante and Stein (1980).4 In our neoclassical framework, the dynamic behaviors of the capital intensity and the real balances per capita in a purely money-financed economy are described by li = [s + $1 - s)]f(k) + m(1 - s)n(k, m) - nk - g ,
(159
and ti = g - Tf(k) - m[n + T(k, m)] . At the steady state, we have the following
(13) conditions:
[S + ~(1 - s)]f(k) + m(1 - s)n(k, m) = nk + g ;
(14)
g - Tf(k) = m[n + n(k, m)] .
(15)
and
Linearizing (12) and (13) about the steady-state equilibrium defined by (14) and (15), we find that the necessary and sufficient conditions for local stability are the following: (sf’ - n) + (1 - S)(Tfl + m7rJ -
g-7f m
+ mn,
< 0;
(16)
+ n(1 - s)(rfl -t m7rTTk) C 0 .
(17)
and (sy - n) b (
- Tf
+ rnT,
)
In what follows, we assume that Solow’s stability condition for a barter economy, $(k) - n < 0, holds on the steady-growth path. Under this assumption, it is easy to see that the purely moneyfinanced economy is stable, if and only if
g - $(k) >--mT,+A; m
4See also
130
chapter
5 of Stein
(1982).
(18)
Stabilization
Effect
where A=max
z
(7-f’ + m7rJ )
(1 - a)($’ + 7727~~) - (n - sf’)
.
Note that if the golden rule of capital accumulation is established at the steady state, then A = of’ + rnrk.’ Condition (18) demonstrates that the Tobin model with myopic perfect foresight will be (locally) stable if the steady-state rate of monetary expansion, which is equal to the high employment deficit divided by the real bal+ A(> 0). Hence, under the purely ances, is larger than -mT, money-financing policy, if the control variables, g and T, are set to satisfy (18) on the steady-growth path, then no active stabilization policies are needed to realize the steady growth. There is a clear implication for the reason why the endogenous money supply rule can stabilize the neoclassical system which assumes myopic perfect foresight. In Tobin’s original model, where the monetary expansion rate is assumed to be constant, the dynamic equation describing the behavior of the real balances per capita is ti = m [p = 7~(k, m) - n] in which p denotes a given growth rate of money supply. This means that at the steady state we always have ani/am = -mn,,, > 0. Thus, under Solow’s condition, which ensures the stability of the capital intensity, the above instability element in the real balances per capita generates the saddlepoint property of the Tobin model. In contrast, in the purely moneyfinanced economy discussed here, the rate of monetary expansion moves counter-cyclically if the government deficit is kept positive; that is, g - of > 0. For example, if the real balances per capita start to increase, the growth rate of money supply automatically diminishes as long as g - of > 0, and therefore, we will have ati/am < 0 at the steady state when [g - $(k)]/m > -mT,. This self-stabilizing behavior of the real balances per capita contributes to an increase in the economic stability of our system and the economy actually converges to the steady state if the steady-state monetary growth rate is higher than -mnTT, + A.6 If condition (18) is not satisfied, then the steady state is either totally unstable or a
mn,)
‘If f’ # n, then n(1 - s)(~f) - (n - a-f’) = (1 - s)[n(l
‘Friedman (1948) forcefully phasizing the self-stabilizing sidered a verification of his
-
+ mrk)/(n 7) + rnPJ.
sf’)
= m
+ nmA
recommends the purely money-financed effect of this policy regime. Our finding assertion in a neoclassical framework.
>
(1 -
s)(~f’
+
rule by emmay be con-
131
Kazuo Mino saddlepoint. When (17) holds but (16) does not, the economy shows the complete instability. On the other hand, if (17) is violated, the steady-state equilibrium exhibits the saddlepoint instability. It should be noted that if the golden rule of capital accumulation is held approximately at the steady state and thus A = of’ + mnk, then condition (16) always holds under (17). Thus, (18) is equivalent to (17). Since (17) implies that the eigenvalues of the linearized system of (12) and (13) have the same signs, the economy satisfying the goldenrule condition always shows the saddlepoint instability if (17) is not fulfilled.
4. A Mixed-Financed
Economy
When interest-yielding librium is characterized by
securities
exist, the steady-state
equi-
[s + 70 - s)lf(~) + m(1 - s)n(k, m)z(k, m) = (1 - s)(l - r)mtl + nk + g ;
(1%
and
g - d-(k) + m
(1 - $0 = [n + n(k, m)]z(k, m) .
(20)
Linearizing the dynamic system consisting of (10) and (11) around the steady state, it is seen that the local stability conditions are (21) and (22): [s + T(l - s)]f’ - n + m(1 - s)(nkz + m?&) - (l/z) + m(nz, + m,
[
g - Tf y
+ mm 2)
< 0,
(21)
+ m(nzk + nzk + nkz)] < 0 ;
(22)
I and g - T. [sf’ - n - nm(1 - s)zk] y + m(nzrn + m% + %&)
+ n(1 - S)(z + mz,,$Tf’ 132
I
Stabilization
Effect
where Zk = -qf’
+ lTJJ/p2 < 0 ;
2, = - edp2
;
nzk + rzk -t nkz = flk[l + fl(f’ - n)/p” - ef’(T + n)/#] no, + ~2, + P,Z = IT&
; and
+ elf’ - n)/p2] .
Since the above conditions are rather complicated, in order to obtain clear implication of the stability conditions, we assume that the golden rule is satisfied approximately at the steady state; that is, f’(k) = n. However, it should be noted that the central message stated below would not change even if f’(k) f n.’ Given the goldenrule condition, we obtain
nz, + 7~2, + 7F,z = 71, ;
sf’ - n - mn(1 - s)zk = -fl(l
and
- s)[l - me(f”
+ rk)/p2] .
First, suppose that the interest-payments-money ratio, 8, is small enough to satisfy 1 - &n(j” + q)/p2 > 0. Then, conditions (21) and (22), coupled with the golden-rule assumption, may be summarized as
g- - 7f > m
-mn,
+ max(&,
B,) ;
(23)
where
B1 = [n(l - r) + m(nk(l - n) - ef”/p)] 2 + mz, B2 = -1 + mzk [m(m - e(n + n)f”/p”)
the
‘Even though similar forms
f’(k) # n, we can as (23) and (24).
arrange
the
stability
,
and
+ ml .
conditions
(21) and
(22) in
133
Kazuo Mino Remembering that the steady-state monetary expansion rate under the mixed-financed regime is (g - ~f)/m + (1 - ~)e, (23) implies that the monetary growth rate should be larger than -m7~, + max (B,, B2) + (1 - 40 in the steady state to keep stability. Hence, the stability condition for this policy regime is essentially the same as the one for the purely money-financed economy, if 8 is not so large. When 8 is sufhciently large enough to hold 1 - 0mcf’ + ~k)/p’ < 0, conditions (21) and (22) can be rewritten as
-mq,,+B1<-
g - Tf-< m
-mq,,
+ B, .
That is, if e is large, the steady-state monetary growth rate has an upper bound to hold the stability. Of course, when B1 > Bz, the above condition is never met. The condition (g - +)/m < -mn, + B2 corresponds to (22), which means that the mixed-financed policy regime yields the saddlepoint instability if the monetary expansion rate is higher than -mn, + B2 + (1 - T)tl. As shown in the previous section, a high level of monetary growth rate (so a high rate of inflation) is a stabilizing factor for the real balances in the sense that we could have arit/am < 0 at the steady state if the nominal money supply rapidly grows. However, unlike the purely money-financed system, in the mixed-financed economy, self-stabilizing property of the capital intensity (that is, ak/dk < 0) may not hold even under the Solow stability condition when 8 is considerably large. Therefore, if this destabilizing factor exists in the motion of capital intensity, the monetary growth rate should not be high - (ati/ to hold condition (22), which ensures (ar;t/am)(ak/ak)
ak)(ak/am)
> 0.
We should also note that, as emphasized in Section 2, 7~,,,could have a positive sign when 8 is large. If 7~, > 0 and 1 - ecf’ + q)/p2 > 0, then from (23) a high rate of inflation is not needed to realize the steady growth, which indicates that a large 8 will contribute to promote the stability. On the other hand, if ~~ > 0 and 1 - 0m(f’ + n&/p2 > 0, we cannot exclude the possibility that (24) is violated because B2 could be smaller than B1 if’ 7~, > 0. (Note that if ~~ < 0, B, always dominates B,). In the Keynesian macrodynamic models, it is common to find that by introducing interest-yielding government bonds increases instability elements in the economy. However, judging from our stability analysis in this section, it cannot be concluded that the 134
Stabilization
Effect
existence of government bonds tends to make the steady state unstable in the neoclassical economy with myopic perfect foresight. Whether or not bond-financing policy destabilizes a neoclassical system crucially depends upon magnitudes of parameters and elasticities of various functions contained in the model.
5. Concluding
Remarks
In the paper, we have shown that if money supply endogenously changes with government deficit or surplus, Tobin’s monetary growth model could be stable even if we assume myopic perfect foresight. Our result suggests that the saddlepoint instability of the Tobin model is closely connected with the assumption that the monetary expansion rate is kept constant. The basic idea of our analysis is simple. Remember that the crucial factor for the saddlepoint instability of the Tobin model is the sign of ti/dm. If ati./ dm is positive around the steady state, then the steady growth path exhibits saddlepoint property. Under the constant money growth policy, d(ti/m)/dm = -mm, so that the stability conditions are violated unless ITS > 0. In contrast, when the growth rate of money supply changes according to the government budget deficits or surplus, dynamic behavior of real money balances may be written as ti = m[p(k, m: 8) - ~(k, m: 0) - n]. (The monetary expansion rate is denoted by p[*]). We thus obtain a(ni/m)/am = t.~, - nTT,. This means that the stability could be realized even though 7~,,, has a negative sign. We should note that we have shown the stabilizing effect of endogenous money supply by using a neoclassical growth model in which consumption and money demand functions of agents are formulated in an ad hoc manner. Recent literature on monetary dynamics emphasizes the optimizing framework in which every behavioral equation is derived by solving agents’ intertemporal plannings. In this framework, the steady-state capital intensity is generally insensitive to monetary policies, and thus stability condition may be different from those obtained,in our analysis.’ Second, if we assume long-run perfect foresight (or rational expectations) instead of myopic perfect foresight, stability implication will drastically change. It is usually assumed that under long‘It is well known that the super neutrality is generally established monetary growth models, if the planning horizon of the representative is infinite and the time discount rate is exogenously given.
in optimizing household
135
Kazuo Mino run perfect foresight the price level can instantaneously jump to keep the economy on the stable manifold of a saddlepoint system, so the number of the stable eigenvalues of the dynamical system must be equal to the number of the predetermined variables to yield a unique stable path.’ In our case m (= M/piV) is a jumpable variable, while k is a backward-looking variable under long-run perfect foresight. Thus, a saddlepoint system obtained under myopic perfect foresight is stable under long-run perfect foresight.” Re-examining our conclusion in these alternative frameworks deserves further investigation. Received : ]une 1985 Final version: September
1987
References Black, F. “Uniqueness of Price Level in Monetary Growth Models with Rational Expectations.” JournaE of Economic Theory 7 (January 1974): 53-65. Benhabib, J., and T. Miyao. “Some New Results on the Dynamics of the Generalized Tobin Model.” international Economic Review 22 (October 1981): 589-96. Drabicki, J.D., and A. Takayama. “The Symmetry of Real Purchasing Power and the Neoclassical Monetary Growth Model.” Journal of Macroeconomics 4 (Summer 1981): 357-62. -. “Money, National Debt, and Economic Growth.” Journal of Economic Theory 33 (December 1984a): 356-67. -. “The Stability of a Neoclassical Monetary Growth Model.” Economic Studies Quarterly 35 (December 1984b): 262-68. Friedman, M. “A Monetary and Fiscal Framework for Economic Stability.” American Economic Review 38 (May 1948): 356-67. Hadjimichalakis, M. G. “Equilibrium and Disequilibrium Growth With Money: The Tobin Models.” Review of Economic Studies (October 1971a): 457-79. -. “Money, Expectations and Dynamics: An Alternative View.” International Economic Review 12 (October 197Ib): 381-402. -. “Expectations of the ‘Myopic Perfect Foresight’ Variety in
‘Perfect foresight equilibria under endogenous money supply is first investigated by Black (1974). “Mine (1985) investigates this problem in an optimizing monetary model. Further discussion of alternative financing behavior of the government in a growing economy can be found in Mino and Stein (1984).
136
Stabilization
Effect
Monetary Dynamics.” Journal of Economic Dynamics and Control 3 (May 1981): 157-76. Hadjimichalakis, M.G., and K. Okuguchi. “The Stability of the Generalized Tobin Model.” Review of Economic Studies 46 (January 1979): 157-78. Hayakawa, H. “Real Purchasing Power in the Neoclassical Monetary Growth Model.” Journal of Macroeconomics 1 (Winter 1979): 19-31. -* “A Dynamic Generalization of the Tobin Model.” Journal of Economic Dynamics and Control 7 (September 1984): 20932. Infante, E.F., and J.L. Stein. “Money Financed Fiscal Policy in a Growing Economy.” Journal of Political Economy 88 (April 1989): 259-87. Mino, K. “Multiple Convergent Equilibria Under Perfect Foresight: The Case of Endogenous Money Growth.” Hiroshima University, Japan. Mimeo. Mino, K., and J. L. Stein. “Monetary Policy Regimes: Stability and Capital Formation.” Discussion Paper, Department of Economics, Brown University, 1984. Nagatani, K. “A Note on Professor Tobin’s Money and Economic Growth.” Econometrica 38 (January 1970): 171-75. Stein, J.L. Monetarist, Keynesian, and New Classical Economics. Oxford: Basil Blackwell, 1982. Tobin, J. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84.
137
Uniuefsity lawn
Stabilization Effect of Endogenous Money Supply in a Descriptive Neoclassical Growth Model* This paper develops a neoclassical monetary growth model with myopic perfect foresight in which monetary expansion rate changes with government budget deficit or surplus. We show the saddlepoint instability of the neoclassical monetary growth model assuming that the constant money growth policy can be avoided under the endogenous money supply rule. We also compare the purely money-financed regime where no government bonds exist with the mixed-financed strategy under which government deficit is partly financed by issuing interest-yielding securities. Our result suggests that the destabilizing effect of bond financing, which has been frequently observed in Keynesian macrodynamic models, does not necessarily hold in the neoclassical system.
1. Introduction The purpose of this paper is to demonstrate that endogenous money supply policy may promote stability of the neoclassical monetary growth model. It is well known that Tobin’s monetary growth model, which assumes the constant money growth rule, exhibits the saddlepoint instability under myopic perfect foresight. This paper shows that if money supply changes with government deficit or surplus, Tobin’s model with myopic perfect foresight could be stable. We first consider the case of a purely money-financed economy where no interest-bearing securities exist, and reveal that the economy can be stabilized if the real government deficit per capita is higher than a certain level at the steady state. We also consider a mixed-financed economy in which the government deficit is partly financed by issuing interest-bearing bonds. It will be seen that, unlike the standard Keynesian macroeconomic models, the neoclassical monetary growth model is not necessarily destabilized by introducing bond-financing policies. *I am very much indebted to Jerome L. Stein for his insightful comments on an earlier version of this paper. I also wish to thank Juichi Kan, Takayoshi Kitaoka, and anonymous referees for their valuable comments.
Journal of Macroeconomics, Winter 1988, Vol. Copyright 0 1988 by Louisiana State University 0164-0704/88/$1.50
10, No. Press
1, pp.
125-137
125
Kazuo Mino The stability issue of the Tobin model is addressed by many authors. In particular, ever since Nagatani (1970) revealed the saddlepoint property of the Tobin model with myopic perfect foresight, a number of authors have proposed sufficient conditions under which the model may be stable. Hadjimichalakis (1971a, 197Ib) generalizes the Tobin model to show that either expectation formation or adjustment process in the money market should be sluggish to hold stability. This line of research is pursued by Hadjimichalakis and Okuguchi (1979), Benhabib and Miyao (1981), Hadjimichalakis (1981), and Hayakawa (1984). Recently, Drabicki and Takayama (1984a, 1984b) found another source of stability. They introduce interestyielding securities into the Tobin model and assume that these securities are imperfect substitutes for money and capital. This assumption enables them to introduce an independent investment function into the neoclassical monetary growth model, which becomes a stabilizing factor. Inspection of these contributions indicates that some friction or imperfection should be assumed to establish the stability of the Tobin model if the constant money growth policy is assumed. ’ The result of this paper suggests that even if everything is perfect in the neoclassical sense, the economy will be stable under a very simple money supply rule. The paper proceeds as follows. Section 2 sets up the model. Section 3 examines the dynamic behavior of the purely money-financed system. Section 4 discusses the mixed-financed regime where the model includes government bonds. Concluding remarks are given in Section 5.
2. The Model The analytical framework ing equations: y = f(k)>
of this paper is given by the follow-
f’>O,f”
Y = (1 - s)[(l - T)(Y + b) - IT(~ + b/p)] + L + nk + g ;
(1) (2)
‘It should be emphasized that Hayakawa (1979) does not belong to this type of literature. He introduces real purchasing power into the liquidity preference f&ction as well as the consumption demand function, and shows that this generalization can resolve the instability problem of the Tobin model with myopic perfect foresight without assuming any sluggishness in market-adjustment mechanism or in expectation formation.
126
Stabilization m = ~[p, f’(k)
+ TTT, y, k + m +
b/PI ,
L1 < 0, Lz < 0, L3 > 0, 0 < L4 < 1 ; p =f’(k) ti + i/p
(3)
+ 7r ;
= g + b - ~(y + b) - (n + r)(m + b/p) ; b=Om.
Effect
(4) and
(5) 63)
Here, notation is y = output per capita, k = capital intensisty, p = nominal interest rate, T = rate of inflation, m = real balances per capita, b = real interest payments per capita, 7 = tax rate (0 < 7 < l), g = real government purchases per capita, 8 = interest payment-money ratio, s = propensity to save, and ?z = rate of population growth. Equation (1) is the productivity function that satisfies the standard neoclassical properties. Equation (2) states that output per capita is equal to real consumption per capita, (1 - ;)[(l - ~)(y + b) - n(m + b/p)], plus real investment per capita, k + nk, plus real government expenditures per capita, g. Following Tobin (1965), we assume a simple consumption function. Consumption demand is proportional to the real disposable income, which is equal to real income after tax per capita less inflation on real financial assets per capita. It is also assumed that the stock of government bonds consists of perpetuities each paying $1.00 per annum in interest. Therefore, b also denotes the number of bonds per capita and the price of bond is l/p. Money market equilibrium condition is given by Equation (3). Demand for the real balances per capita is negatively related to the nominal interest rate, p, and the nominal rate of return, f’(k) + T, and positively related to the output per capita, y, and the real wealth per capita, k + m + b/p. Note that the anticipated rate of inflation included in the nominal rate of return on capital is equal to the actual rate of inflation due to the assumption of myopic perfect foresight. Capital market is assumed to be perfect in the sense that the physical capital is a perfect substitute for the government bonds, so that the arbitrage condition, (4), is held at each moment of time. Equation (5) denotes the government budget constraint. The real government deficit is real per capita government consumption, 127
Kazuo Mino g, plus real per capita interest payments, b, less taxes, r(y + b). The deficit should be financed by new money creation and/or issuing interest-bearing bonds. Thus, we have k/pN
+ (~/PNWP)
= g + b - T(Y + b)
>
where M, B, N, and p denote the nominal stock of money, the nominal interest payments, the population, and the price level, respectively. From the definitions m = M/pN and b = B/pN, the above can be expressed as (5). Finally, we assume that the interest-payment-money ratio, 8, is kept constant at a certain level. The model contains six endogenous variables-y, m, k, b, p, and IT. Variables g, 7, and 8 are control variables. Substituting (l), (4), and (6) into (3), and solving it with respect to IT, we can express the rate of inflation such that
7~ = n(k, m: 13). Partial derivatives
of this function
are
L$+L, IQ = (L,8m/p2)
(7)
- L1 - L,
-f’>o,
and
?r, = ~(1 + fm - 1 (L,0m/p2)
- L1 - L, ’
(9)
If no government bonds exist (0 = 0), then 7~,,,has a negative sign (so that the Wicksell effect holds). This case is the same as Tobin’s original model. The Wicksell effect may not hold if there are government bonds (0 > 0). Since 8 is assumed to be constant, an increase in the real balances per capita raises government bonds at the same time. Thus, if 8 is sufficiently large, the increase in per capita demand for real balances through the wealth effect could be larger than the increase in m, so that the rate of inflation will increase to equilibrate the money market.2 *Another approach to reverse the sign of mr, is proposed by Hayakawa (1979). As mentioned in Footnote 1, in specifying the liquidity preference function, he considers income effect arising from real balance flows to generate the possibility that P, has a positive sign. Also see Drabicki and Takayama (1981), and Hayakawa (1984) for further discussion of this approach.
128
Stabilization
Effect
However, it should be noted that if rational households fully anticipate the future tax liabilities on the current debt, the govemment securities fail to constitute a part of net wealth. In this case, even if 0 > 0, TTT,is strictly negative.3 Precise analysis of this problem requires an optimizing framework rather than the ad hoc model employed here. Since, in our model, the results obtained in the case of neutral bonds may be similar to those in the purely moneyfinanced economy (f3 = 0), we will assume that the net wealth includes government debt. By (l), (2), and (6), the capital intensity changes according to i = [$ + T(1 - s)]f(k) - m(1 - s)[(l - T)e - ~(k, m)z(k, m)] - nk - g ,
(10)
where z(k, m) = 1 + 0/p = 1 + e/v(k) + T(k, m)]. Using (l), 0% and (7), th e g ovemment budget restraint, (5), provides the dynamic behavior of the real balances per capita: ni = [m/z(k,
g - vW m
m)]
+ (1 -
7)e
- (n + ?r(k, m)) z(k, m) Thus the complete dynamic equations, (10) and (11).
1 .
(11)
system is given by a set of differential
3. A Purely Money-Financed
Economy
If there were no government bonds (0 = 0), then the moneyfinancing policy means that the money supply changes with the
3This may A(0 S A 5 1) liabilities are mulation, the
be discussed by defining the net wealth as k + m + X(b/p), where represents the degree of rationality of asset holders. If the future tax completely anticipated by the public, then A = 0. Given this forequilibrium rate of inflation becomes ~(k, m: 9, A) and we obtain
wr, =
Therefore, a negative
qTT, may sign.
be positive
L,(l
+ AO/p)
(L,fJAm/p2)
if A0 > p(l/L,
-
1
- L, - & -
1). When
A = 0, vr, obviously
has
129
Kazuo Mino government deficit. The growth and stability implication of this policy regime in a Keynes-Wicksell growth model is discussed by Infante and Stein (1980).4 In our neoclassical framework, the dynamic behaviors of the capital intensity and the real balances per capita in a purely money-financed economy are described by li = [s + $1 - s)]f(k) + m(1 - s)n(k, m) - nk - g ,
(159
and ti = g - Tf(k) - m[n + T(k, m)] . At the steady state, we have the following
(13) conditions:
[S + ~(1 - s)]f(k) + m(1 - s)n(k, m) = nk + g ;
(14)
g - Tf(k) = m[n + n(k, m)] .
(15)
and
Linearizing (12) and (13) about the steady-state equilibrium defined by (14) and (15), we find that the necessary and sufficient conditions for local stability are the following: (sf’ - n) + (1 - S)(Tfl + m7rJ -
g-7f m
+ mn,
< 0;
(16)
+ n(1 - s)(rfl -t m7rTTk) C 0 .
(17)
and (sy - n) b (
- Tf
+ rnT,
)
In what follows, we assume that Solow’s stability condition for a barter economy, $(k) - n < 0, holds on the steady-growth path. Under this assumption, it is easy to see that the purely moneyfinanced economy is stable, if and only if
g - $(k) >--mT,+A; m
4See also
130
chapter
5 of Stein
(1982).
(18)
Stabilization
Effect
where A=max
z
(7-f’ + m7rJ )
(1 - a)($’ + 7727~~) - (n - sf’)
.
Note that if the golden rule of capital accumulation is established at the steady state, then A = of’ + rnrk.’ Condition (18) demonstrates that the Tobin model with myopic perfect foresight will be (locally) stable if the steady-state rate of monetary expansion, which is equal to the high employment deficit divided by the real bal+ A(> 0). Hence, under the purely ances, is larger than -mT, money-financing policy, if the control variables, g and T, are set to satisfy (18) on the steady-growth path, then no active stabilization policies are needed to realize the steady growth. There is a clear implication for the reason why the endogenous money supply rule can stabilize the neoclassical system which assumes myopic perfect foresight. In Tobin’s original model, where the monetary expansion rate is assumed to be constant, the dynamic equation describing the behavior of the real balances per capita is ti = m [p = 7~(k, m) - n] in which p denotes a given growth rate of money supply. This means that at the steady state we always have ani/am = -mn,,, > 0. Thus, under Solow’s condition, which ensures the stability of the capital intensity, the above instability element in the real balances per capita generates the saddlepoint property of the Tobin model. In contrast, in the purely moneyfinanced economy discussed here, the rate of monetary expansion moves counter-cyclically if the government deficit is kept positive; that is, g - of > 0. For example, if the real balances per capita start to increase, the growth rate of money supply automatically diminishes as long as g - of > 0, and therefore, we will have ati/am < 0 at the steady state when [g - $(k)]/m > -mT,. This self-stabilizing behavior of the real balances per capita contributes to an increase in the economic stability of our system and the economy actually converges to the steady state if the steady-state monetary growth rate is higher than -mnTT, + A.6 If condition (18) is not satisfied, then the steady state is either totally unstable or a
mn,)
‘If f’ # n, then n(1 - s)(~f) - (n - a-f’) = (1 - s)[n(l
‘Friedman (1948) forcefully phasizing the self-stabilizing sidered a verification of his
-
+ mrk)/(n 7) + rnPJ.
sf’)
= m
+ nmA
recommends the purely money-financed effect of this policy regime. Our finding assertion in a neoclassical framework.
>
(1 -
s)(~f’
+
rule by emmay be con-
131
Kazuo Mino saddlepoint. When (17) holds but (16) does not, the economy shows the complete instability. On the other hand, if (17) is violated, the steady-state equilibrium exhibits the saddlepoint instability. It should be noted that if the golden rule of capital accumulation is held approximately at the steady state and thus A = of’ + mnk, then condition (16) always holds under (17). Thus, (18) is equivalent to (17). Since (17) implies that the eigenvalues of the linearized system of (12) and (13) have the same signs, the economy satisfying the goldenrule condition always shows the saddlepoint instability if (17) is not fulfilled.
4. A Mixed-Financed
Economy
When interest-yielding librium is characterized by
securities
exist, the steady-state
equi-
[s + 70 - s)lf(~) + m(1 - s)n(k, m)z(k, m) = (1 - s)(l - r)mtl + nk + g ;
(1%
and
g - d-(k) + m
(1 - $0 = [n + n(k, m)]z(k, m) .
(20)
Linearizing the dynamic system consisting of (10) and (11) around the steady state, it is seen that the local stability conditions are (21) and (22): [s + T(l - s)]f’ - n + m(1 - s)(nkz + m?&) - (l/z) + m(nz, + m,
[
g - Tf y
+ mm 2)
< 0,
(21)
+ m(nzk + nzk + nkz)] < 0 ;
(22)
I and g - T. [sf’ - n - nm(1 - s)zk] y + m(nzrn + m% + %&)
+ n(1 - S)(z + mz,,$Tf’ 132
I
Stabilization
Effect
where Zk = -qf’
+ lTJJ/p2 < 0 ;
2, = - edp2
;
nzk + rzk -t nkz = flk[l + fl(f’ - n)/p” - ef’(T + n)/#] no, + ~2, + P,Z = IT&
; and
+ elf’ - n)/p2] .
Since the above conditions are rather complicated, in order to obtain clear implication of the stability conditions, we assume that the golden rule is satisfied approximately at the steady state; that is, f’(k) = n. However, it should be noted that the central message stated below would not change even if f’(k) f n.’ Given the goldenrule condition, we obtain
nz, + 7~2, + 7F,z = 71, ;
sf’ - n - mn(1 - s)zk = -fl(l
and
- s)[l - me(f”
+ rk)/p2] .
First, suppose that the interest-payments-money ratio, 8, is small enough to satisfy 1 - &n(j” + q)/p2 > 0. Then, conditions (21) and (22), coupled with the golden-rule assumption, may be summarized as
g- - 7f > m
-mn,
+ max(&,
B,) ;
(23)
where
B1 = [n(l - r) + m(nk(l - n) - ef”/p)] 2 + mz, B2 = -1 + mzk [m(m - e(n + n)f”/p”)
the
‘Even though similar forms
f’(k) # n, we can as (23) and (24).
arrange
the
stability
,
and
+ ml .
conditions
(21) and
(22) in
133
Kazuo Mino Remembering that the steady-state monetary expansion rate under the mixed-financed regime is (g - ~f)/m + (1 - ~)e, (23) implies that the monetary growth rate should be larger than -m7~, + max (B,, B2) + (1 - 40 in the steady state to keep stability. Hence, the stability condition for this policy regime is essentially the same as the one for the purely money-financed economy, if 8 is not so large. When 8 is sufhciently large enough to hold 1 - 0mcf’ + ~k)/p’ < 0, conditions (21) and (22) can be rewritten as
-mq,,+B1<-
g - Tf-< m
-mq,,
+ B, .
That is, if e is large, the steady-state monetary growth rate has an upper bound to hold the stability. Of course, when B1 > Bz, the above condition is never met. The condition (g - +)/m < -mn, + B2 corresponds to (22), which means that the mixed-financed policy regime yields the saddlepoint instability if the monetary expansion rate is higher than -mn, + B2 + (1 - T)tl. As shown in the previous section, a high level of monetary growth rate (so a high rate of inflation) is a stabilizing factor for the real balances in the sense that we could have arit/am < 0 at the steady state if the nominal money supply rapidly grows. However, unlike the purely money-financed system, in the mixed-financed economy, self-stabilizing property of the capital intensity (that is, ak/dk < 0) may not hold even under the Solow stability condition when 8 is considerably large. Therefore, if this destabilizing factor exists in the motion of capital intensity, the monetary growth rate should not be high - (ati/ to hold condition (22), which ensures (ar;t/am)(ak/ak)
ak)(ak/am)
> 0.
We should also note that, as emphasized in Section 2, 7~,,,could have a positive sign when 8 is large. If 7~, > 0 and 1 - ecf’ + q)/p2 > 0, then from (23) a high rate of inflation is not needed to realize the steady growth, which indicates that a large 8 will contribute to promote the stability. On the other hand, if ~~ > 0 and 1 - 0m(f’ + n&/p2 > 0, we cannot exclude the possibility that (24) is violated because B2 could be smaller than B1 if’ 7~, > 0. (Note that if ~~ < 0, B, always dominates B,). In the Keynesian macrodynamic models, it is common to find that by introducing interest-yielding government bonds increases instability elements in the economy. However, judging from our stability analysis in this section, it cannot be concluded that the 134
Stabilization
Effect
existence of government bonds tends to make the steady state unstable in the neoclassical economy with myopic perfect foresight. Whether or not bond-financing policy destabilizes a neoclassical system crucially depends upon magnitudes of parameters and elasticities of various functions contained in the model.
5. Concluding
Remarks
In the paper, we have shown that if money supply endogenously changes with government deficit or surplus, Tobin’s monetary growth model could be stable even if we assume myopic perfect foresight. Our result suggests that the saddlepoint instability of the Tobin model is closely connected with the assumption that the monetary expansion rate is kept constant. The basic idea of our analysis is simple. Remember that the crucial factor for the saddlepoint instability of the Tobin model is the sign of ti/dm. If ati./ dm is positive around the steady state, then the steady growth path exhibits saddlepoint property. Under the constant money growth policy, d(ti/m)/dm = -mm, so that the stability conditions are violated unless ITS > 0. In contrast, when the growth rate of money supply changes according to the government budget deficits or surplus, dynamic behavior of real money balances may be written as ti = m[p(k, m: 8) - ~(k, m: 0) - n]. (The monetary expansion rate is denoted by p[*]). We thus obtain a(ni/m)/am = t.~, - nTT,. This means that the stability could be realized even though 7~,,, has a negative sign. We should note that we have shown the stabilizing effect of endogenous money supply by using a neoclassical growth model in which consumption and money demand functions of agents are formulated in an ad hoc manner. Recent literature on monetary dynamics emphasizes the optimizing framework in which every behavioral equation is derived by solving agents’ intertemporal plannings. In this framework, the steady-state capital intensity is generally insensitive to monetary policies, and thus stability condition may be different from those obtained,in our analysis.’ Second, if we assume long-run perfect foresight (or rational expectations) instead of myopic perfect foresight, stability implication will drastically change. It is usually assumed that under long‘It is well known that the super neutrality is generally established monetary growth models, if the planning horizon of the representative is infinite and the time discount rate is exogenously given.
in optimizing household
135
Kazuo Mino run perfect foresight the price level can instantaneously jump to keep the economy on the stable manifold of a saddlepoint system, so the number of the stable eigenvalues of the dynamical system must be equal to the number of the predetermined variables to yield a unique stable path.’ In our case m (= M/piV) is a jumpable variable, while k is a backward-looking variable under long-run perfect foresight. Thus, a saddlepoint system obtained under myopic perfect foresight is stable under long-run perfect foresight.” Re-examining our conclusion in these alternative frameworks deserves further investigation. Received : ]une 1985 Final version: September
1987
References Black, F. “Uniqueness of Price Level in Monetary Growth Models with Rational Expectations.” JournaE of Economic Theory 7 (January 1974): 53-65. Benhabib, J., and T. Miyao. “Some New Results on the Dynamics of the Generalized Tobin Model.” international Economic Review 22 (October 1981): 589-96. Drabicki, J.D., and A. Takayama. “The Symmetry of Real Purchasing Power and the Neoclassical Monetary Growth Model.” Journal of Macroeconomics 4 (Summer 1981): 357-62. -. “Money, National Debt, and Economic Growth.” Journal of Economic Theory 33 (December 1984a): 356-67. -. “The Stability of a Neoclassical Monetary Growth Model.” Economic Studies Quarterly 35 (December 1984b): 262-68. Friedman, M. “A Monetary and Fiscal Framework for Economic Stability.” American Economic Review 38 (May 1948): 356-67. Hadjimichalakis, M. G. “Equilibrium and Disequilibrium Growth With Money: The Tobin Models.” Review of Economic Studies (October 1971a): 457-79. -. “Money, Expectations and Dynamics: An Alternative View.” International Economic Review 12 (October 197Ib): 381-402. -. “Expectations of the ‘Myopic Perfect Foresight’ Variety in
‘Perfect foresight equilibria under endogenous money supply is first investigated by Black (1974). “Mine (1985) investigates this problem in an optimizing monetary model. Further discussion of alternative financing behavior of the government in a growing economy can be found in Mino and Stein (1984).
136
Stabilization
Effect
Monetary Dynamics.” Journal of Economic Dynamics and Control 3 (May 1981): 157-76. Hadjimichalakis, M.G., and K. Okuguchi. “The Stability of the Generalized Tobin Model.” Review of Economic Studies 46 (January 1979): 157-78. Hayakawa, H. “Real Purchasing Power in the Neoclassical Monetary Growth Model.” Journal of Macroeconomics 1 (Winter 1979): 19-31. -* “A Dynamic Generalization of the Tobin Model.” Journal of Economic Dynamics and Control 7 (September 1984): 20932. Infante, E.F., and J.L. Stein. “Money Financed Fiscal Policy in a Growing Economy.” Journal of Political Economy 88 (April 1989): 259-87. Mino, K. “Multiple Convergent Equilibria Under Perfect Foresight: The Case of Endogenous Money Growth.” Hiroshima University, Japan. Mimeo. Mino, K., and J. L. Stein. “Monetary Policy Regimes: Stability and Capital Formation.” Discussion Paper, Department of Economics, Brown University, 1984. Nagatani, K. “A Note on Professor Tobin’s Money and Economic Growth.” Econometrica 38 (January 1970): 171-75. Stein, J.L. Monetarist, Keynesian, and New Classical Economics. Oxford: Basil Blackwell, 1982. Tobin, J. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84.
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