- Email: [email protected]
The flow equilibrium and the neutrality of money in a neoclassical monetary growth model
AKRA YAKITA Mie University,
japan
The Flow Equilibrium and the Neutra/ity of Money in a Neoclassical Monetary Growth Model* We reconstruct the one-sector neoclassical monetary growth theory to make it internally consistent in the sense of Hayakawa’s (1982) argument. This requires that the theory be formulated in terms of a flow equilibrium model in which the rate of change in the inflation rate is determined as a function of real balances per capita, the capital-labor ratio, and the intlation rate. We show that the super-neutrality of money can be established even though household disposable income is defined to include real balance flows as in the traditional case.
1. Introduction Monetary growth models can be classified into two types: the neoclassical and the Keynes-Wicksellian. The former assumes that investment is always equal to savings and that asset markets are always in equilibrium; in this framework, the price movement is determined by the equilibrium conditions in asset markets. Contrastingly, in the latter, separate investment and saving functions are postulated and commodity prices are assumed to change if and only if the commodity market is in disequilibrium. This paper deals with the one-sector neoclassical model as represented in a descriptive manner. Money is reported to be nonneutral in this model in the sense that the steady-state capital-labor ratio is affected by a change in the growth rate of money supply. According to Hahn (1969), this non-neutrality is due to the fact that disposable income, defined to include real balance flows, exceeds income that is actually produced. The balanced growth path, however, is a saddle-point under perfect foresight, so that the capitallabor ratio and real balances per capita are adjusted monotonically
*The author is deeply indebted to an anonymous referee for his helpful and heartwarming comments on an earlier draft of this paper. Any errors are the sole responsibility of the author.
]ournul of Macroeconomics, Spring 1988, Vol. 10, No. 2, pp. 201-216 Copyright 0 1988 by Louisiana State University Press n.-, .3-n, I_” **. e..
201
Akira
Yakita
along the converging trajectories toward equilibrium (see, for example, Nagatani 1970). These results need to be reexamined in light of the on-going discussions on two alternative specifications of asset market equilibrium. Foley (1975) argues that the end-of-period and the beginningof-period equilibrium specifications are equivalent provided that expectations satisfy the perfect foresight condition and the desired stocks of assets are always attained instantaneously. However, in a recent paper, Hayakawa (1982) points out an inconsistency in the way income flow variables are treated in the traditional stock equilibrium formulations of demands for assets. To avoid this inconsistency, he suggests an alternative specification of a continuous time macroeconomic model by dwelling on Walras’ Law in flow form, which requires that the values of the instantaneous excess flow supply rates of goods and assets add up to zero. Then, it is shown that such a reformulation leads to different dynamic implications horn traditional analysis. We, therefore, attempt to construct a neoclassical monetary growth model that is internally consistent in the sense of Hayakawa’s (1982) argument and to analyze the comparative statics of such a model. The household disposable income is defined to include real balance flows as in the traditional case, and a constant fraction of this disposable income is assumed to be saved. The stock demand for real balances is specified as a function of flow income, the desired holdings of wealth, and the expected nominal rate of return on capital. The nominal money stock grows at a constant rate, and the government purchases a constant fraction of output produced. The most important feature of our model is that the money market equilibrium condition is specified in flow equilibrium terms. This specification leads to the dynamics of the rate of change in the inflation rate given as a function of the real balances per household, the capital-labor ratio, and the inflation rate. We show that these dynamics are responsible for establishing the super-neutrality of money in our flow equilibrium model. Under the assumption of perfect foresight, the steady-state path is characterized as a saddlepoint; this requires that the inflation rate be treated as a jump variable.
2. Model A one-sector economy is considered with no distinction between capital and consumption goods. There are three groups of 202
Flow Equilibrium
and Neutrality
of
Money
decision makers: the government, households, and firms. Households are identical and the number grows at a constant exponential rate, n(>O). Each household supplies one unit of effective labor at each time. There is only one &an&l asset, outside money, of which all is held by households alone. Perfect foresight is assumed to prevail in predicting commodity prices and the real rate of return on capital.’ Portfolio adjustment costs are nonexistent, and physical capital is not distinguished from equity claims on capital. The budget constraint of a representative household at time t is given by
y(t)
-
7(t) = c(t)
Mf@>t) + -Kf(t, t) .
+ PWW
N(t)
’
(1)
where y(t), T(t), and c(t) stand for rates of income flow, taxes (transfers if negative), and consumption, respectively; p(t) is the commodity price level at time t; N(t) is the number of households at time t; Mf(t, t)/p(t)N(t) and K!(t, t)/N(t) are the flow demands for real balances and capital desired by the household at time t, respectively. We define flow disposable income rate, y”(t), as y”(t) = y(t) - T(t) - +)m(t)
,
(2)
where n(t) is the rate of inflation of commodity prices and m(t) = M(t)/p(t)N(t) is the stock of real balances per household at time t. We can write the flow saving rate, s(t) = y”(t) - c(t), as s(t) = W;‘@, t)
_ M;‘(t>t) I Kf@,t) - +)m(t) , N(t) pW(t)
(3)
where W;‘(t, t) is the rate of change of the desired holdings of wealth ‘In this paper, we employ the notion of end-of-period equilibrium. Therefore, as is stated by Hayakawa (1982), we are imposing perfect foresight axiom on expectations, which is defined by Tumovsky and Burmeister (1977) as p*(t + h, t) = p(t + h) for ail h 2 0, where p*(t + h, t) is the commodity price level expected by households at time t to hold at time t + h; and p(t + h) is the actual price level at tie t + h. Perfect foresight implies perfect myopic foresight that Iim-[p*(t + h, t) - p(t)]/h = lim&p(t + h) - p(t)]/h = e(t) and also the weak consistency axiom that p*(t, t) = p(t). The similar holds for the real rate of return on capital.
203
Akira
Yakita
with respect to the first time index at time t. Then, assuming that a constant fraction, 1 - c, of disposable income is saved, we have s(t) = (1 - c)[y(t) - 7(t) - lT(t)m(t)] .
(4)
The technology is assumed to be invariant over time represented by the regular neoclassical production function:
y(t)= f [WI >
k(t) = W /W),
f’ ’ 0,
and is
f” < 0 ;
(5)
where K(t) is the stock of capital existing at time t. The rate of return on capital is assumed to equal the marginal productivity of capital, so that r(t) = f’[k(t)]. The budget constraint of the government is written as
Nt)
g(t) - 7(t) = PW@)
’
(6)
where g(t) is the planned government expenditure flow per household and k(t) = &4(t)/& is the flow supply rate of money. Budget deficits are financed entirely with money. We assume for simplicity that the government purchases and consumes a constant fraction of output; that is,
g(t) = rfPW1,
y = constant (20) .
We also assume that the government expenditure and money issue plans are realized. Transfers (taxes) to households are given by (6). Note that when y = 0, newly issued money is transferred to households as gifts as in the traditional case. (For the derivation of (1) through (6) from the discrete time version, see Appendix A.) From the budget constraints-(l), (S), and (6)-we obtain Walras’ Law as
- c(t) - g(t) - @$}
+
--&
&f(t) - M;‘(t,41= 0 .
(7)
That is, the values of the excess flow supply rates of goods and money sum to zero. Hence, one of the flow equilibrium conditions can be deleted in specifying the equilibrium state of the economy. 204
Flow Equilibrium
and Neutrality
of Money
3. Equilibrium in Money Market Deleting the flow equilibrium condition in the goods market, we have as the equilibrium condition of the system:
G(t) -=PWW
M;‘(t,t) P(mv) ’
where the left- and right-hand sides are the flow supply and the flow demand rates of money in real terms, respectively. With the rate of increase in the stock of money denoted by 8, the money supply rate can be written as
G(t)- MO)
-=&)
-
Pww)
W)
= em(t) .
(9)
Pw(t)
The flow demand rate, on the other hand, can be specified as follows. Consider the demand function for real balances in the discrete time context in which the market interval is denoted by (t, t + h) and the planning horizon of each decision maker is only h time units long into the future. After Foley (1975) and Hayakawa (1982), we write the quantity of money, which households plan at time t to hold at time t + h, in per household terms as Md(t + h, t)
= L[Y(t + h, t), W”(t + h, t),
p*(t + h, W(t) r’(t
+ h, t) + r*(t
+ h, t)] ;
(10)
where Y(t + h, t) is income flow per household over period (t, t + h), Wd(t + h, t) is the desired holdings of wealth per household at time t for time t + h, p*(t + h, t) is the expected commodity price for time t + h formed at time t, r”(t + h, t) is the expected rate of return on capital for time t + h formed at time t, and m*(t + h, t) = [p*(t + 2h, t) - p*(t + h, t)]/hp*(t + h, t) is the expected inflation rate for time t + h formed at time t. As can be seen in (3), the rate of change of the desired wealth holdings is measured as the planning horizon increases at a given planning date. Then, if the planning horizon shrinks to zero, we have, under the assumption of perfect foresight, 205
Akira
Yakita
t) = uo, Wdkt),f-0)+ 4t)l , -MdO. pW@)
(11)
where n(t) = T*(t, t) = [p*(t + h, t) - p(t)]/hp(t). This is not the market demand under the notion of end-of-period equilibrium because the desired wealth at any point in time for that time must coincide identically with the actual wealth at that time; that is, Wd(t, t) = W(t), or Md(t, t) = M(t) and Kd(t, t) = K(t).’ Note that while the planning horizon is distinguished conceptually from the market interval in the literature (including Foley 1975 and Tumovsky 1977), they are not distinguished effectively in the analysis (except Buiter 1980); that is, it has been argued that if there are no transactions costs, the planning horizon need not be longer than the market interval. Following this argument, we assume that the planning horizon is of the same length as the market interval. Under this assumption, we can obtain the flow demand rate of real balances by subtracting both sides of (11) from those of (lo), dividing the digerences by h, and letting h * 0:
Mb t)
WW
= L,y(t) + L,Wf@, t)
+
Ls[?yt)
L, > 0,
+
e(t)]
O
where Mf(t, t) = lim &Md(t limheOIWd(t + h, t) - Wd(t,
f
7r(t)m(t)
)
L,
(12)
+ h, t) - M(t)]/h;W;l(t, t) = t)]/h; y(t) = limh+Y(t + h, t)/h;
‘As is shown by May (1970), when one moves to the limit, the single budget constraint gives rise to two constraints, the stock constraint and the flow constraint. In our model, the stock constraint or the balance sheet identity is given as
M(t) + K(t) = Md(t, t)
w(t) = PW@)
w
Pw(~)
Kd(t, t) + = wd(t, t) N(t)
W-1)
Hayakawa (1984) shows that under the notion of end-of-period equilibrium, when the household budget constraint is written in terms of notional demands, Md(t + h, t) and Kd(t + h, t), Wd(t, t) = W(t) if and only if M ’(t, t) = M(t) and K’(t, t) = K(t). The balance sheet identity, however, should be distinguished from the beginning-of-period or stock equilibrium condition in which the inflation rate is determined by a function of real balances per capita and capital-labor ratio.
206
Flow Equilibrium
and Neutrality
of Money
ir(t) = Tf(t, t) = lim h-Jn*(t + h, t) - n*(t, t)]/h; and the dot over any variable denotes its time derivative.” Then, using (9) and (12), we may rewrite (8) as
[e - 4dlmW = by@)+ L2Wt, where Wf(t,
t)
+ L&W + e(t)] ,
(13)
t) is given by (3) and (4) as
WV, t) = (1 - c)W - u)f[Wl + 10- ?rWlm(N and f(t) = f”i(t) from
can be determined,
k(t) = (1 - C)(l - y)f[k(t)]
in view of Walras’ maw, (7),
- c[O - 7r(t)]m(t) - nk(t) .
(14)
The expression (13) is similar with Foley’s asset market clearing condition (1975, Eq. 5a*).4 Given the expectations about the second time derivatives of prices and the rate of return on capital, the dynamics of the commodity price level are determined by (13); in particular, with perfect foresight, the rate of change in the inflation rate, k(t), is given as a function of r(t), k(t), and m(t).’ As in Hayakawa (1982), we assume that Li’s are constant (i = 1, 2, 3).6 Because the end-of-period equilibrium condition for money in the discrete time context is given by Md(t + h, t) = M(t + h) and because p*(t + h, t) = p(t + h) under perfect foresight, (15) holds at time t + h: Md(t + h, t) p*O + h, WW
N(t) =
M(t + h)
N(t + h)
PO + WV) [=m(t
+ h)N(t + h)] .
(15)
Se author is grateful to an anonymous referee for suggesting an introduction of the expected inflation rate in deriving (12) from (lo) and (11). ‘Unlike Foley (1975), income flow variable is inserted; that is, the transactions money is taken into account. The treatment of income flow variable is the one that is suggested by Hayakawa (1982). ‘Chand (1981) derives a neoclassical formula for actual rate of inflation that clears flow market of real balances. Our model is similar with his in spirit, though he postulates sluggish adaptive expectations. %near homogeneity of demand function for real balances with respect to real income and real weaith, which is often assumed in the literature on monetary growth models, does not necessarily hold. However, Tobin’s adding-up criterion is not violated. See Drabicki and Takayama (1982) and Hayakawa (1983).
207
Akira
Yakita
Subtracting M(t)/p(t + h)N(t) f rom both sides and using the conditions Md(t, t) = M(t) and Mf(t, t) = k(t), we have (see Appendix B)
Tit(t)= [e - ?r(t)- n]m(t).
(16)
4. The Long-Run Equilibrium The dynamic paths of the system are described simultaneously by three equations-(13), (14), and (16)-+&h two variables (m, k), the initial values of which are historically predetermined, and a variable T, the value of which is not predetermined. The long-run equilibrium is defined by the steady-state path where ir = 0, 1 = 0, and ti = 0. On the steady-state path, taking account of savings rate, (3), we have s(t)[-W;“(t,
t)] = n[m(t) -I- k(t)]
and
dt) = h/(t) + -UnCN f W I1 . The first equation implies that in steady state the savings rate is just enough to keep the per household holdings of wealth, real balances, and capital constant. The second equation implies that in the money market the flow demand is met by the amount of flow supply that keeps the stock of real balances per household constant.7 There exists a unique saddle path converging toward the longrun equilibrium provided that there are two negative roots corresponding to the two predetermined variables and one positive root corresponding to the non-predetermined variable. Approximating the system linearly, we state the saddle-point condition as
‘At each point in capital are achieved neutrality of money tributed to this fact.
208
time the desired stocks of wealth holdings, real balances, and through savings, s(t) = W:(t, t) = n[m(t) + k(t)]. The supersimilar with that in normative neoclassical models may be atFor normative models, see, for example, Sidrauski (1967).
Flow Equilibrium detJ = {-cf’[Ll - [(l -
and Neutrality
of Money
+ LZ(l - c)(l - r)] - r)f’
c)(l
-
nl[Gdl - 4 - llHmn/L) > 0,
(necessary) ;
(174
and trJ = [l + L3f”c
- L(l
+ [(l - c)(l - r)f’ (sufficient,
- c)l(-m/L,) - nl 5 0,
once (17a) is satisfied) ;
(17b)
where J is the Jacobian of the linearized system and the elements of J are evaluated at the equilibrium (‘rr*, k*, m*). The conditions for (17) are satisfied when (1 - c)(l - y)f’ - n < 0 and ((1 + cLJ”)m - L,[(l - c)(l - r)f’ - n]}/(l - c)m 5 L2. The former condition correspond? to the stability condition in the nonmonetary growth model of Solow (1956), and the latter condition means that the wealth effects on demand for real balances, L,, is sufficiently large.* We assume that these conditions are satisfied. Then, we consider the effects of changes in the growth rate of money supply, 0, and the government purchase ratio, y. The steady-state effects are dT/de
= 1,
(184
dk/de
= 0,
Wb)
dm/df3 = 0,
W4
drr/dr
(194
= 0,
dk/d-y = ((1 - c)fm[l
- L2 + (1 - n)
(fNL3 + L,)c]}/L,D
< 0 ,
and
W)
‘Since money demand depends on W “(t + h, t) not on W(t), L3 stands for the substitution effect between capital and money.
209
Akira
Yakita d??z/dy = [(l - c)fm(L$
+
L,n)IlLD < 0 ;
where D = det ](>O) and n < 1 (we assume). That is, money is neutral in the long run in that the capital-labor ratio of the balanced growth path is not affected by an increase in the growth rate of money supply. On the other hand, fiscal policy in the form of a reduction in the government purchase ratio increases the capitallabor ratio and, hence, output per household without raising the long-run inflation rate. The results may be interpreted as follows. Since the capitallabor ratio and the stock of real balances per household at any given point in time are predetermined historically, IT must jump upward by de in order for the money market to restore its flow equilibrium instantaneously at the time the money supply rate is changed. On the other hand, a reduction in the government purchase ratio with the money supply rate remaining unchanged implies an increase in disposable income, thereby leading to an increase in the capitallabor ratio and a reduction in the real rate of return on capital. The changes in disposable income and in the rate of return on capital give rise tofchanges in commodity prices, but the inflation rate is not aEected in the long run.g Turning to the short-run effects of policy changes, we now analyze a transition path from one steady state to another.” Since monetary policy is not effective in the long run, we consider an expansionary fiscal policy of raising the government purchase ratio without issuing money. Let the system be on a steady-state path initially and let the government purchase ratio, y, be raised. The inilation rates of the initial and new steady states are denoted by ~8 and IT?, respectively. From (19a), IT? - ~0” = 0. Then, denoting the negative roots by p and u, 7~ - ~8 = (c+e”’ + c,ve”f)dy for cI + c2 < 0. For the system to adjust along a convergent path toward the new steady state, it is necessary that the inflation rate, IT, jumps so as to satisfy n - n&o = (clp + c,v)dy. We assume that this condition is satisfied. There are two cases to be considered (see Appendix C). ‘I ‘Here we assume that the system is on the stable saddle path toward the new steady-state equilibrium. “The methodology of the paper owes much to Shah (1984), who examines the adjustment paths of Tobin’s “4” and capital accumulation. “When either c, or cz is positive and sufficiently large, it seems possible that Suppose that 71 7F - 7rlClllo = C,(L + c,u < 0; that is, m initially falls downward. falls downward at the time y is raised. A rise in y implies a tax increase and, hence,
210
Flow Equilibrium
and Neutrality
of Money
Case (I). Initially, n jumps upward discontinuously and then converges monotonically toward its original value. Since at the outset k/m = 0, the movement of n implies that lit/m < 0 on the transition path toward the new steady state. As a result, the stock of real balances on the new steady state is smaller than that on the initial steady state; that is, rnf < m$. This may be interpreted as follows. Since an increase in the government purchase ratio implies a tax increase for a given money supply rate, the saving rate (that is, the rate of change in the desired stock of wealth holdings) is reduced. Therefore, the flow demand rate for real balances is lowered through wealth effects. Since the flow equilibrium condition of the money market on a steady-state path can be written as 0m = L, y + &(l
- c)[(l - r)f
+ (8 - T)ml + 7rrn ,
n must jump upward to restore this equilibrium at the time y is raised. Since the flow demand for real balances at each point in time depends upon the rate of return on capital, 7~ + r, and since disposable income includes real balance flows, (0 - rr)m, households start decreasing savings (that is, the desired flow demand for wealth) and substituting capital for real balances; they continue to do so until n is restored to its long-run equilibrium value, where k/m = 0. Case (2). Initially T jumps upward and then converges nonmonotonically toward its original value. In this case, the rate of change in the stock of real balances per household, k/m, is initially negative, but it can be positive at a later time. The stock of real balances per household on the new steady state is smaller than that on the initial steady-state path.
5. Conclusion We have constructed a descriptive neoclassical monetary growth model under the notion of flow equilibrium or end-of-period equilibrium, and have shown, in contrast to the conventional result,
a reduction in the household disposable income on the one hand, and a decline in n implies a rise in real balance flows on the other. Since the equilibrium condition on a steady-state path is (8 - n)m = L,y + L(1 - c)[(l - y)f + (tl - n)m], the money market cannot restore its flow equilibrium instantaneously as long as both wealth effects on money demand, 4, and saving rate, 1 - c, are less than unity. Therefore, II must jump upward initially.
211
Akira
Yakita (ii>
(i>
Figure
1.
that money is neutral in the sense that the steady-state capital-labor ratio is not affected by a change in the growth rate of money supply. Also, we have found that the stock of real balances per household may be adjusted nonmonotonically along an adjustment trajectory toward steady state. Received: June 1986 Final version: November 1987
References Buiter, W.H. “Walras Law and All That: Budget Constraints and Balance Sheet Constraints in Period Models and Continuous Time Models.” International Economic Review 21 (February 1980): l16. Chand, S.K. “Stocks, Flows, and Market Equilibrium in Neoclassical Monetary Growth Models.” Journal of Monetary Economics 8 (February 1981): 117-29. Drabicki, J.Z., and A. Takayama. “The Symmetry of Real Purchasing Power and the Neoclassical Monetary Growth Model.” Journal of Macroeconomics 4 (Summer 1982): 357-62. Foley, D.K. “On Two Specifications of Asset Equilibrium in Macroeconomic Models.” Journul of Political Economy 83 (April 1975): 305-24. Hahn, F. “On Money and Growth.” Journal of Money, Credit, and Banking 1 (May 1969): 172-87. Hayakawa, H. “Conservation Principles and an Alternative Formulation of a Continuous Time Macro Model.” Economic Studies Quarterly 33 (August 1982): 111-25. 212
Flow Equilibrium
and Neutrality
of Money
-.
“Rationality of Liquidity Preferences and the Neoclassical Monetary Growth IModel.” Journal of Macroeconomics 5 (Fall 1983): 495-501. -. “Balance Sheet Identity and Walras’ Law.” Journul of Economic Theory 34 (August 1984): 187-202. May, J. “Period Analysis and Continuous Analysis in Patinkin’s Macroeconomic Model.” Journul of Economic Theory 2 (March 1970): l-9. Nagatani, K. “A Note on Professor Tobin’s ‘Money and Economic Growth’. ” Econometrica 38 (January 1970): 171-75. Shah, A. “Crowding Out, Capital Accumulation, the Stock Market, and Money-Financed Fiscal Policy.” Journal of Money, Credit, and Banking 16 (November 1984): 461-73. Sidrauski, M. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Reuiew 57 (May 1967): 53444. to the Theory of Economic Growth.” Solow, R. M. “A Contribution Quarterly Journal of Economics 79 (February 1956): 64-94. Tumovsky, S.J. “On the Formulation of Continuous Time Macroeconomic Models With Asset AccurHulation.” International Economic Reuiew 18 (February 1977): l-28. Tumovsky, S. J., and E. Burmeister. “Perfect Foresight, Expectational Consistency, and Macroeconomic Equilibrium.” Journal of Political Economy 85 (June 1977): 379-93.
Appendix A. Equations (I), (9, (3), (5), and (6) are derived from the discrete time version of the end-of-period equilibrium model. The budget constraint of a representative household over the discrete time market period (t, t + h) is given by
M(t) +Ko+ Pww
w
1 p*@ + k t)
Md(t + h, t) = p*(t + h, @V(t)
1
--
p(t)
+ Kd(t + h, t)
1
MM + Y(t i h, t) W
+ C(t + h, t) + T(t + h, t) ;
(A-l)
w
where Kd(t + h, t) = quantity of capital time t + h,
planned
at time
t to hold
at
213
Akira
Yakita
C(t + h, t) = planned consumption flow over period (t, t + h), and T(t + h, t) = flow of tax payments (net of transfer payments) over period (t + h, t). We define the amount of wealth the household to hold at time t + h, Wd(t + h, t), as W”(t + h, t) =
Md(t + h, t)
of wealth
at time t
+ Kd(t + h, t)
p*(t + h, W(t) and the holdings
planned
N(t)
at time t, W(t), as
W (t)= - M(t) w Pww + % * Then utilizing the definitional relationship-S(t + h, t) = Y”(t + h, t) - C(t + h, t) where Y”(t + h, t) and S(t + h, t) are flows of household disposable income and savings over period (t, t + h), respectively-it follows that S(t + h, t) = W”(t + h, t) - W(t)
(A-2)
if and only if we define
Y”(t + h, t) = Y(t + h, t) +
1
--
p*(t + h, t) - T(t + h, t) .
1 p(t)
1
-M(t) N(t)
(A-3)
That is, planned savings is equal to the desired accumulation of wealth if and only if household disposable income is defined to include the expected capital gains on money. The output of the economy over period (t, t + h) is Y(t + h, t)N(t) = hF[K(t), N(t)] , where F(K, N) is the supply function for flow of goods. By the assumption of constant returns to scale, we can rewrite it as 214
Flow Equilibrium
and Neutrality
of Money
Y(t + h, t) = hF [~>l]=W[~]. The government + h, t)N(t), for period T(t + h, t)N(t), or by plans of money issue constraint is
(A-4)
is assumed to have an expenditure plan, G(t (t, t + h), which is financed through taxes, printing money. Assuming that government are always realized, the government budget
M(t + h) - M(t)
= G(t -I- h, t) - 7’(t + h, t) .
(A-5)
p*(t + h, t)N(t)
We now proceed to take limits and let h + 0 (the planning horizon is defmed by h). Taking h 4 0 in (A-l), we obtain the balance sheet identity (N-l) in Footnote 2. Then dividing (A-l) by h, taking h -+ 0, and making use of (N-l) in this process, we obtain (1); where y(t) = limh,OY(t + h, t)/h, c(t) = limb-&(t + h, t)/h, and T(t) = lim,+J’(t + h, t)/h. Dividing (A-3) and (A-2) by h and taking h --, 0, we obtain (2) and (3), rqpectively, where s(t) = limh,,&t + h, t)/h. Similarly, dividing (A-4) and (A-5) by h and taking h -+ 0, we also have (5) and (6), respectively.
Appendix C. The two negative roots are denoted acteristic vector (xl, x2, xJ associated with
by t.~ and v. The charp can be obtained by 1
[1+ Lx&C
L
+ L,(l
Lw - 4
1 (1 -CM -a-’ -n-p - J%l - c)lm - c)U- r)lf’ + L3Kl - c)(l -P - r)f’ - nlf’
- Lsf”C - l]n
[1[I Xl
x2
-cn
r
-m
0
x3
=
0 0 0
-lJ -I
We obtain from the last equation x1/x3 = -p/m, and from the second equation xl/x, = -~[(l - c)(l - r)f’ - n - ~]/cm(p + n). The characteristic vector associated with p (up to a factor of proportionality) is 215
Akira
Yakita
-cdJ, + 4 [ lJ3
(1 - c)(l - r)f’
- n - /.L’ -m I
Similarly, we obtain the characteristic to a factor of proportionality) as -cm(v
vector
associated with
v (up
+ n)
[ ” (1 - c)(l - v)f’ - n - v’
em I .
Along the stable path IT - Tr* = c,pe
w + c2vevf ;
WdP + 4 ept (1 - c)(l - r)f’ - n - p
k-k*=
cpzm(v + n)
and
- (1 - c)(l - -y)f’ - fl - v eYt ; m - m* = -c,me”
- cgn.eYf .
Using the results of comparative statics and from (m - rnr)jtqo = M-J - rnr and (k - k:)j,=, = Ic$ - kr, where the subscripts 0 and 1 denote the initial steady-state variables and the new steady-state values, respectively, we have c
+ 1
c
= 2
(1 -
-
4&f’
+
L2n)fdy
~
A
<
o
D
and cm(v + n) ~74~ + 4 =dk
japan
The Flow Equilibrium and the Neutra/ity of Money in a Neoclassical Monetary Growth Model* We reconstruct the one-sector neoclassical monetary growth theory to make it internally consistent in the sense of Hayakawa’s (1982) argument. This requires that the theory be formulated in terms of a flow equilibrium model in which the rate of change in the inflation rate is determined as a function of real balances per capita, the capital-labor ratio, and the intlation rate. We show that the super-neutrality of money can be established even though household disposable income is defined to include real balance flows as in the traditional case.
1. Introduction Monetary growth models can be classified into two types: the neoclassical and the Keynes-Wicksellian. The former assumes that investment is always equal to savings and that asset markets are always in equilibrium; in this framework, the price movement is determined by the equilibrium conditions in asset markets. Contrastingly, in the latter, separate investment and saving functions are postulated and commodity prices are assumed to change if and only if the commodity market is in disequilibrium. This paper deals with the one-sector neoclassical model as represented in a descriptive manner. Money is reported to be nonneutral in this model in the sense that the steady-state capital-labor ratio is affected by a change in the growth rate of money supply. According to Hahn (1969), this non-neutrality is due to the fact that disposable income, defined to include real balance flows, exceeds income that is actually produced. The balanced growth path, however, is a saddle-point under perfect foresight, so that the capitallabor ratio and real balances per capita are adjusted monotonically
*The author is deeply indebted to an anonymous referee for his helpful and heartwarming comments on an earlier draft of this paper. Any errors are the sole responsibility of the author.
]ournul of Macroeconomics, Spring 1988, Vol. 10, No. 2, pp. 201-216 Copyright 0 1988 by Louisiana State University Press n.-, .3-n, I_” **. e..
201
Akira
Yakita
along the converging trajectories toward equilibrium (see, for example, Nagatani 1970). These results need to be reexamined in light of the on-going discussions on two alternative specifications of asset market equilibrium. Foley (1975) argues that the end-of-period and the beginningof-period equilibrium specifications are equivalent provided that expectations satisfy the perfect foresight condition and the desired stocks of assets are always attained instantaneously. However, in a recent paper, Hayakawa (1982) points out an inconsistency in the way income flow variables are treated in the traditional stock equilibrium formulations of demands for assets. To avoid this inconsistency, he suggests an alternative specification of a continuous time macroeconomic model by dwelling on Walras’ Law in flow form, which requires that the values of the instantaneous excess flow supply rates of goods and assets add up to zero. Then, it is shown that such a reformulation leads to different dynamic implications horn traditional analysis. We, therefore, attempt to construct a neoclassical monetary growth model that is internally consistent in the sense of Hayakawa’s (1982) argument and to analyze the comparative statics of such a model. The household disposable income is defined to include real balance flows as in the traditional case, and a constant fraction of this disposable income is assumed to be saved. The stock demand for real balances is specified as a function of flow income, the desired holdings of wealth, and the expected nominal rate of return on capital. The nominal money stock grows at a constant rate, and the government purchases a constant fraction of output produced. The most important feature of our model is that the money market equilibrium condition is specified in flow equilibrium terms. This specification leads to the dynamics of the rate of change in the inflation rate given as a function of the real balances per household, the capital-labor ratio, and the inflation rate. We show that these dynamics are responsible for establishing the super-neutrality of money in our flow equilibrium model. Under the assumption of perfect foresight, the steady-state path is characterized as a saddlepoint; this requires that the inflation rate be treated as a jump variable.
2. Model A one-sector economy is considered with no distinction between capital and consumption goods. There are three groups of 202
Flow Equilibrium
and Neutrality
of
Money
decision makers: the government, households, and firms. Households are identical and the number grows at a constant exponential rate, n(>O). Each household supplies one unit of effective labor at each time. There is only one &an&l asset, outside money, of which all is held by households alone. Perfect foresight is assumed to prevail in predicting commodity prices and the real rate of return on capital.’ Portfolio adjustment costs are nonexistent, and physical capital is not distinguished from equity claims on capital. The budget constraint of a representative household at time t is given by
y(t)
-
7(t) = c(t)
Mf@>t) + -Kf(t, t) .
+ PWW
N(t)
’
(1)
where y(t), T(t), and c(t) stand for rates of income flow, taxes (transfers if negative), and consumption, respectively; p(t) is the commodity price level at time t; N(t) is the number of households at time t; Mf(t, t)/p(t)N(t) and K!(t, t)/N(t) are the flow demands for real balances and capital desired by the household at time t, respectively. We define flow disposable income rate, y”(t), as y”(t) = y(t) - T(t) - +)m(t)
,
(2)
where n(t) is the rate of inflation of commodity prices and m(t) = M(t)/p(t)N(t) is the stock of real balances per household at time t. We can write the flow saving rate, s(t) = y”(t) - c(t), as s(t) = W;‘@, t)
_ M;‘(t>t) I Kf@,t) - +)m(t) , N(t) pW(t)
(3)
where W;‘(t, t) is the rate of change of the desired holdings of wealth ‘In this paper, we employ the notion of end-of-period equilibrium. Therefore, as is stated by Hayakawa (1982), we are imposing perfect foresight axiom on expectations, which is defined by Tumovsky and Burmeister (1977) as p*(t + h, t) = p(t + h) for ail h 2 0, where p*(t + h, t) is the commodity price level expected by households at time t to hold at time t + h; and p(t + h) is the actual price level at tie t + h. Perfect foresight implies perfect myopic foresight that Iim-[p*(t + h, t) - p(t)]/h = lim&p(t + h) - p(t)]/h = e(t) and also the weak consistency axiom that p*(t, t) = p(t). The similar holds for the real rate of return on capital.
203
Akira
Yakita
with respect to the first time index at time t. Then, assuming that a constant fraction, 1 - c, of disposable income is saved, we have s(t) = (1 - c)[y(t) - 7(t) - lT(t)m(t)] .
(4)
The technology is assumed to be invariant over time represented by the regular neoclassical production function:
y(t)= f [WI >
k(t) = W /W),
f’ ’ 0,
and is
f” < 0 ;
(5)
where K(t) is the stock of capital existing at time t. The rate of return on capital is assumed to equal the marginal productivity of capital, so that r(t) = f’[k(t)]. The budget constraint of the government is written as
Nt)
g(t) - 7(t) = PW@)
’
(6)
where g(t) is the planned government expenditure flow per household and k(t) = &4(t)/& is the flow supply rate of money. Budget deficits are financed entirely with money. We assume for simplicity that the government purchases and consumes a constant fraction of output; that is,
g(t) = rfPW1,
y = constant (20) .
We also assume that the government expenditure and money issue plans are realized. Transfers (taxes) to households are given by (6). Note that when y = 0, newly issued money is transferred to households as gifts as in the traditional case. (For the derivation of (1) through (6) from the discrete time version, see Appendix A.) From the budget constraints-(l), (S), and (6)-we obtain Walras’ Law as
- c(t) - g(t) - @$}
+
--&
&f(t) - M;‘(t,41= 0 .
(7)
That is, the values of the excess flow supply rates of goods and money sum to zero. Hence, one of the flow equilibrium conditions can be deleted in specifying the equilibrium state of the economy. 204
Flow Equilibrium
and Neutrality
of Money
3. Equilibrium in Money Market Deleting the flow equilibrium condition in the goods market, we have as the equilibrium condition of the system:
G(t) -=PWW
M;‘(t,t) P(mv) ’
where the left- and right-hand sides are the flow supply and the flow demand rates of money in real terms, respectively. With the rate of increase in the stock of money denoted by 8, the money supply rate can be written as
G(t)- MO)
-=&)
-
Pww)
W)
= em(t) .
(9)
Pw(t)
The flow demand rate, on the other hand, can be specified as follows. Consider the demand function for real balances in the discrete time context in which the market interval is denoted by (t, t + h) and the planning horizon of each decision maker is only h time units long into the future. After Foley (1975) and Hayakawa (1982), we write the quantity of money, which households plan at time t to hold at time t + h, in per household terms as Md(t + h, t)
= L[Y(t + h, t), W”(t + h, t),
p*(t + h, W(t) r’(t
+ h, t) + r*(t
+ h, t)] ;
(10)
where Y(t + h, t) is income flow per household over period (t, t + h), Wd(t + h, t) is the desired holdings of wealth per household at time t for time t + h, p*(t + h, t) is the expected commodity price for time t + h formed at time t, r”(t + h, t) is the expected rate of return on capital for time t + h formed at time t, and m*(t + h, t) = [p*(t + 2h, t) - p*(t + h, t)]/hp*(t + h, t) is the expected inflation rate for time t + h formed at time t. As can be seen in (3), the rate of change of the desired wealth holdings is measured as the planning horizon increases at a given planning date. Then, if the planning horizon shrinks to zero, we have, under the assumption of perfect foresight, 205
Akira
Yakita
t) = uo, Wdkt),f-0)+ 4t)l , -MdO. pW@)
(11)
where n(t) = T*(t, t) = [p*(t + h, t) - p(t)]/hp(t). This is not the market demand under the notion of end-of-period equilibrium because the desired wealth at any point in time for that time must coincide identically with the actual wealth at that time; that is, Wd(t, t) = W(t), or Md(t, t) = M(t) and Kd(t, t) = K(t).’ Note that while the planning horizon is distinguished conceptually from the market interval in the literature (including Foley 1975 and Tumovsky 1977), they are not distinguished effectively in the analysis (except Buiter 1980); that is, it has been argued that if there are no transactions costs, the planning horizon need not be longer than the market interval. Following this argument, we assume that the planning horizon is of the same length as the market interval. Under this assumption, we can obtain the flow demand rate of real balances by subtracting both sides of (11) from those of (lo), dividing the digerences by h, and letting h * 0:
Mb t)
WW
= L,y(t) + L,Wf@, t)
+
Ls[?yt)
L, > 0,
+
e(t)]
O
where Mf(t, t) = lim &Md(t limheOIWd(t + h, t) - Wd(t,
f
7r(t)m(t)
)
L,
(12)
+ h, t) - M(t)]/h;W;l(t, t) = t)]/h; y(t) = limh+Y(t + h, t)/h;
‘As is shown by May (1970), when one moves to the limit, the single budget constraint gives rise to two constraints, the stock constraint and the flow constraint. In our model, the stock constraint or the balance sheet identity is given as
M(t) + K(t) = Md(t, t)
w(t) = PW@)
w
Pw(~)
Kd(t, t) + = wd(t, t) N(t)
W-1)
Hayakawa (1984) shows that under the notion of end-of-period equilibrium, when the household budget constraint is written in terms of notional demands, Md(t + h, t) and Kd(t + h, t), Wd(t, t) = W(t) if and only if M ’(t, t) = M(t) and K’(t, t) = K(t). The balance sheet identity, however, should be distinguished from the beginning-of-period or stock equilibrium condition in which the inflation rate is determined by a function of real balances per capita and capital-labor ratio.
206
Flow Equilibrium
and Neutrality
of Money
ir(t) = Tf(t, t) = lim h-Jn*(t + h, t) - n*(t, t)]/h; and the dot over any variable denotes its time derivative.” Then, using (9) and (12), we may rewrite (8) as
[e - 4dlmW = by@)+ L2Wt, where Wf(t,
t)
+ L&W + e(t)] ,
(13)
t) is given by (3) and (4) as
WV, t) = (1 - c)W - u)f[Wl + 10- ?rWlm(N and f(t) = f”i(t) from
can be determined,
k(t) = (1 - C)(l - y)f[k(t)]
in view of Walras’ maw, (7),
- c[O - 7r(t)]m(t) - nk(t) .
(14)
The expression (13) is similar with Foley’s asset market clearing condition (1975, Eq. 5a*).4 Given the expectations about the second time derivatives of prices and the rate of return on capital, the dynamics of the commodity price level are determined by (13); in particular, with perfect foresight, the rate of change in the inflation rate, k(t), is given as a function of r(t), k(t), and m(t).’ As in Hayakawa (1982), we assume that Li’s are constant (i = 1, 2, 3).6 Because the end-of-period equilibrium condition for money in the discrete time context is given by Md(t + h, t) = M(t + h) and because p*(t + h, t) = p(t + h) under perfect foresight, (15) holds at time t + h: Md(t + h, t) p*O + h, WW
N(t) =
M(t + h)
N(t + h)
PO + WV) [=m(t
+ h)N(t + h)] .
(15)
Se author is grateful to an anonymous referee for suggesting an introduction of the expected inflation rate in deriving (12) from (lo) and (11). ‘Unlike Foley (1975), income flow variable is inserted; that is, the transactions money is taken into account. The treatment of income flow variable is the one that is suggested by Hayakawa (1982). ‘Chand (1981) derives a neoclassical formula for actual rate of inflation that clears flow market of real balances. Our model is similar with his in spirit, though he postulates sluggish adaptive expectations. %near homogeneity of demand function for real balances with respect to real income and real weaith, which is often assumed in the literature on monetary growth models, does not necessarily hold. However, Tobin’s adding-up criterion is not violated. See Drabicki and Takayama (1982) and Hayakawa (1983).
207
Akira
Yakita
Subtracting M(t)/p(t + h)N(t) f rom both sides and using the conditions Md(t, t) = M(t) and Mf(t, t) = k(t), we have (see Appendix B)
Tit(t)= [e - ?r(t)- n]m(t).
(16)
4. The Long-Run Equilibrium The dynamic paths of the system are described simultaneously by three equations-(13), (14), and (16)-+&h two variables (m, k), the initial values of which are historically predetermined, and a variable T, the value of which is not predetermined. The long-run equilibrium is defined by the steady-state path where ir = 0, 1 = 0, and ti = 0. On the steady-state path, taking account of savings rate, (3), we have s(t)[-W;“(t,
t)] = n[m(t) -I- k(t)]
and
dt) = h/(t) + -UnCN f W I1 . The first equation implies that in steady state the savings rate is just enough to keep the per household holdings of wealth, real balances, and capital constant. The second equation implies that in the money market the flow demand is met by the amount of flow supply that keeps the stock of real balances per household constant.7 There exists a unique saddle path converging toward the longrun equilibrium provided that there are two negative roots corresponding to the two predetermined variables and one positive root corresponding to the non-predetermined variable. Approximating the system linearly, we state the saddle-point condition as
‘At each point in capital are achieved neutrality of money tributed to this fact.
208
time the desired stocks of wealth holdings, real balances, and through savings, s(t) = W:(t, t) = n[m(t) + k(t)]. The supersimilar with that in normative neoclassical models may be atFor normative models, see, for example, Sidrauski (1967).
Flow Equilibrium detJ = {-cf’[Ll - [(l -
and Neutrality
of Money
+ LZ(l - c)(l - r)] - r)f’
c)(l
-
nl[Gdl - 4 - llHmn/L) > 0,
(necessary) ;
(174
and trJ = [l + L3f”c
- L(l
+ [(l - c)(l - r)f’ (sufficient,
- c)l(-m/L,) - nl 5 0,
once (17a) is satisfied) ;
(17b)
where J is the Jacobian of the linearized system and the elements of J are evaluated at the equilibrium (‘rr*, k*, m*). The conditions for (17) are satisfied when (1 - c)(l - y)f’ - n < 0 and ((1 + cLJ”)m - L,[(l - c)(l - r)f’ - n]}/(l - c)m 5 L2. The former condition correspond? to the stability condition in the nonmonetary growth model of Solow (1956), and the latter condition means that the wealth effects on demand for real balances, L,, is sufficiently large.* We assume that these conditions are satisfied. Then, we consider the effects of changes in the growth rate of money supply, 0, and the government purchase ratio, y. The steady-state effects are dT/de
= 1,
(184
dk/de
= 0,
Wb)
dm/df3 = 0,
W4
drr/dr
(194
= 0,
dk/d-y = ((1 - c)fm[l
- L2 + (1 - n)
(fNL3 + L,)c]}/L,D
< 0 ,
and
W)
‘Since money demand depends on W “(t + h, t) not on W(t), L3 stands for the substitution effect between capital and money.
209
Akira
Yakita d??z/dy = [(l - c)fm(L$
+
L,n)IlLD < 0 ;
where D = det ](>O) and n < 1 (we assume). That is, money is neutral in the long run in that the capital-labor ratio of the balanced growth path is not affected by an increase in the growth rate of money supply. On the other hand, fiscal policy in the form of a reduction in the government purchase ratio increases the capitallabor ratio and, hence, output per household without raising the long-run inflation rate. The results may be interpreted as follows. Since the capitallabor ratio and the stock of real balances per household at any given point in time are predetermined historically, IT must jump upward by de in order for the money market to restore its flow equilibrium instantaneously at the time the money supply rate is changed. On the other hand, a reduction in the government purchase ratio with the money supply rate remaining unchanged implies an increase in disposable income, thereby leading to an increase in the capitallabor ratio and a reduction in the real rate of return on capital. The changes in disposable income and in the rate of return on capital give rise tofchanges in commodity prices, but the inflation rate is not aEected in the long run.g Turning to the short-run effects of policy changes, we now analyze a transition path from one steady state to another.” Since monetary policy is not effective in the long run, we consider an expansionary fiscal policy of raising the government purchase ratio without issuing money. Let the system be on a steady-state path initially and let the government purchase ratio, y, be raised. The inilation rates of the initial and new steady states are denoted by ~8 and IT?, respectively. From (19a), IT? - ~0” = 0. Then, denoting the negative roots by p and u, 7~ - ~8 = (c+e”’ + c,ve”f)dy for cI + c2 < 0. For the system to adjust along a convergent path toward the new steady state, it is necessary that the inflation rate, IT, jumps so as to satisfy n - n&o = (clp + c,v)dy. We assume that this condition is satisfied. There are two cases to be considered (see Appendix C). ‘I ‘Here we assume that the system is on the stable saddle path toward the new steady-state equilibrium. “The methodology of the paper owes much to Shah (1984), who examines the adjustment paths of Tobin’s “4” and capital accumulation. “When either c, or cz is positive and sufficiently large, it seems possible that Suppose that 71 7F - 7rlClllo = C,(L + c,u < 0; that is, m initially falls downward. falls downward at the time y is raised. A rise in y implies a tax increase and, hence,
210
Flow Equilibrium
and Neutrality
of Money
Case (I). Initially, n jumps upward discontinuously and then converges monotonically toward its original value. Since at the outset k/m = 0, the movement of n implies that lit/m < 0 on the transition path toward the new steady state. As a result, the stock of real balances on the new steady state is smaller than that on the initial steady state; that is, rnf < m$. This may be interpreted as follows. Since an increase in the government purchase ratio implies a tax increase for a given money supply rate, the saving rate (that is, the rate of change in the desired stock of wealth holdings) is reduced. Therefore, the flow demand rate for real balances is lowered through wealth effects. Since the flow equilibrium condition of the money market on a steady-state path can be written as 0m = L, y + &(l
- c)[(l - r)f
+ (8 - T)ml + 7rrn ,
n must jump upward to restore this equilibrium at the time y is raised. Since the flow demand for real balances at each point in time depends upon the rate of return on capital, 7~ + r, and since disposable income includes real balance flows, (0 - rr)m, households start decreasing savings (that is, the desired flow demand for wealth) and substituting capital for real balances; they continue to do so until n is restored to its long-run equilibrium value, where k/m = 0. Case (2). Initially T jumps upward and then converges nonmonotonically toward its original value. In this case, the rate of change in the stock of real balances per household, k/m, is initially negative, but it can be positive at a later time. The stock of real balances per household on the new steady state is smaller than that on the initial steady-state path.
5. Conclusion We have constructed a descriptive neoclassical monetary growth model under the notion of flow equilibrium or end-of-period equilibrium, and have shown, in contrast to the conventional result,
a reduction in the household disposable income on the one hand, and a decline in n implies a rise in real balance flows on the other. Since the equilibrium condition on a steady-state path is (8 - n)m = L,y + L(1 - c)[(l - y)f + (tl - n)m], the money market cannot restore its flow equilibrium instantaneously as long as both wealth effects on money demand, 4, and saving rate, 1 - c, are less than unity. Therefore, II must jump upward initially.
211
Akira
Yakita (ii>
(i>
Figure
1.
that money is neutral in the sense that the steady-state capital-labor ratio is not affected by a change in the growth rate of money supply. Also, we have found that the stock of real balances per household may be adjusted nonmonotonically along an adjustment trajectory toward steady state. Received: June 1986 Final version: November 1987
References Buiter, W.H. “Walras Law and All That: Budget Constraints and Balance Sheet Constraints in Period Models and Continuous Time Models.” International Economic Review 21 (February 1980): l16. Chand, S.K. “Stocks, Flows, and Market Equilibrium in Neoclassical Monetary Growth Models.” Journal of Monetary Economics 8 (February 1981): 117-29. Drabicki, J.Z., and A. Takayama. “The Symmetry of Real Purchasing Power and the Neoclassical Monetary Growth Model.” Journal of Macroeconomics 4 (Summer 1982): 357-62. Foley, D.K. “On Two Specifications of Asset Equilibrium in Macroeconomic Models.” Journul of Political Economy 83 (April 1975): 305-24. Hahn, F. “On Money and Growth.” Journal of Money, Credit, and Banking 1 (May 1969): 172-87. Hayakawa, H. “Conservation Principles and an Alternative Formulation of a Continuous Time Macro Model.” Economic Studies Quarterly 33 (August 1982): 111-25. 212
Flow Equilibrium
and Neutrality
of Money
-.
“Rationality of Liquidity Preferences and the Neoclassical Monetary Growth IModel.” Journal of Macroeconomics 5 (Fall 1983): 495-501. -. “Balance Sheet Identity and Walras’ Law.” Journul of Economic Theory 34 (August 1984): 187-202. May, J. “Period Analysis and Continuous Analysis in Patinkin’s Macroeconomic Model.” Journul of Economic Theory 2 (March 1970): l-9. Nagatani, K. “A Note on Professor Tobin’s ‘Money and Economic Growth’. ” Econometrica 38 (January 1970): 171-75. Shah, A. “Crowding Out, Capital Accumulation, the Stock Market, and Money-Financed Fiscal Policy.” Journal of Money, Credit, and Banking 16 (November 1984): 461-73. Sidrauski, M. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Reuiew 57 (May 1967): 53444. to the Theory of Economic Growth.” Solow, R. M. “A Contribution Quarterly Journal of Economics 79 (February 1956): 64-94. Tumovsky, S.J. “On the Formulation of Continuous Time Macroeconomic Models With Asset AccurHulation.” International Economic Reuiew 18 (February 1977): l-28. Tumovsky, S. J., and E. Burmeister. “Perfect Foresight, Expectational Consistency, and Macroeconomic Equilibrium.” Journal of Political Economy 85 (June 1977): 379-93.
Appendix A. Equations (I), (9, (3), (5), and (6) are derived from the discrete time version of the end-of-period equilibrium model. The budget constraint of a representative household over the discrete time market period (t, t + h) is given by
M(t) +Ko+ Pww
w
1 p*@ + k t)
Md(t + h, t) = p*(t + h, @V(t)
1
--
p(t)
+ Kd(t + h, t)
1
MM + Y(t i h, t) W
+ C(t + h, t) + T(t + h, t) ;
(A-l)
w
where Kd(t + h, t) = quantity of capital time t + h,
planned
at time
t to hold
at
213
Akira
Yakita
C(t + h, t) = planned consumption flow over period (t, t + h), and T(t + h, t) = flow of tax payments (net of transfer payments) over period (t + h, t). We define the amount of wealth the household to hold at time t + h, Wd(t + h, t), as W”(t + h, t) =
Md(t + h, t)
of wealth
at time t
+ Kd(t + h, t)
p*(t + h, W(t) and the holdings
planned
N(t)
at time t, W(t), as
W (t)= - M(t) w Pww + % * Then utilizing the definitional relationship-S(t + h, t) = Y”(t + h, t) - C(t + h, t) where Y”(t + h, t) and S(t + h, t) are flows of household disposable income and savings over period (t, t + h), respectively-it follows that S(t + h, t) = W”(t + h, t) - W(t)
(A-2)
if and only if we define
Y”(t + h, t) = Y(t + h, t) +
1
--
p*(t + h, t) - T(t + h, t) .
1 p(t)
1
-M(t) N(t)
(A-3)
That is, planned savings is equal to the desired accumulation of wealth if and only if household disposable income is defined to include the expected capital gains on money. The output of the economy over period (t, t + h) is Y(t + h, t)N(t) = hF[K(t), N(t)] , where F(K, N) is the supply function for flow of goods. By the assumption of constant returns to scale, we can rewrite it as 214
Flow Equilibrium
and Neutrality
of Money
Y(t + h, t) = hF [~>l]=W[~]. The government + h, t)N(t), for period T(t + h, t)N(t), or by plans of money issue constraint is
(A-4)
is assumed to have an expenditure plan, G(t (t, t + h), which is financed through taxes, printing money. Assuming that government are always realized, the government budget
M(t + h) - M(t)
= G(t -I- h, t) - 7’(t + h, t) .
(A-5)
p*(t + h, t)N(t)
We now proceed to take limits and let h + 0 (the planning horizon is defmed by h). Taking h 4 0 in (A-l), we obtain the balance sheet identity (N-l) in Footnote 2. Then dividing (A-l) by h, taking h -+ 0, and making use of (N-l) in this process, we obtain (1); where y(t) = limh,OY(t + h, t)/h, c(t) = limb-&(t + h, t)/h, and T(t) = lim,+J’(t + h, t)/h. Dividing (A-3) and (A-2) by h and taking h --, 0, we obtain (2) and (3), rqpectively, where s(t) = limh,,&t + h, t)/h. Similarly, dividing (A-4) and (A-5) by h and taking h -+ 0, we also have (5) and (6), respectively.
Appendix C. The two negative roots are denoted acteristic vector (xl, x2, xJ associated with
by t.~ and v. The charp can be obtained by 1
[1+ Lx&C
L
+ L,(l
Lw - 4
1 (1 -CM -a-’ -n-p - J%l - c)lm - c)U- r)lf’ + L3Kl - c)(l -P - r)f’ - nlf’
- Lsf”C - l]n
[1[I Xl
x2
-cn
r
-m
0
x3
=
0 0 0
-lJ -I
We obtain from the last equation x1/x3 = -p/m, and from the second equation xl/x, = -~[(l - c)(l - r)f’ - n - ~]/cm(p + n). The characteristic vector associated with p (up to a factor of proportionality) is 215
Akira
Yakita
-cdJ, + 4 [ lJ3
(1 - c)(l - r)f’
- n - /.L’ -m I
Similarly, we obtain the characteristic to a factor of proportionality) as -cm(v
vector
associated with
v (up
+ n)
[ ” (1 - c)(l - v)f’ - n - v’
em I .
Along the stable path IT - Tr* = c,pe
w + c2vevf ;
WdP + 4 ept (1 - c)(l - r)f’ - n - p
k-k*=
cpzm(v + n)
and
- (1 - c)(l - -y)f’ - fl - v eYt ; m - m* = -c,me”
- cgn.eYf .
Using the results of comparative statics and from (m - rnr)jtqo = M-J - rnr and (k - k:)j,=, = Ic$ - kr, where the subscripts 0 and 1 denote the initial steady-state variables and the new steady-state values, respectively, we have c
+ 1
c
= 2
(1 -
-
4&f’
+
L2n)fdy
~
A
<
o
D
and cm(v + n) ~74~ + 4 =dk