Portfolio Adjustment and Monetary Growth

The Review of Economic Studies, Ltd.

Portfolio Adjustment and Monetary GrowthAuthor(s): Lewis JohnsonSource: The Review of Economic Studies, Vol. 43, No. 3 (Oct., 1976), pp. 475-481Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297226 .

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Portfolio Adjustment and

Monetary Growth LEWIS JOHNSON

Federal Reserve System

A common feature of disequilibrium monetary growth models (those in which markets need not be cleared at any given moment in time) is the assumption of a simple portfolio adjustment process which generates a flow demand for money from a discrepancy between desired and actual holdings of real balances. However, some care must be taken in incorporating this process into the specification of asset market equilibrium. An implication of the portfolio adjustment process that has been uniformly overlooked is that anticipated real balance flows (e.g. due to inflation or government transfers) during the adjustment period will alter the extent to which wealth holders will need to enter the market to achieve a given adjustment. This is significant because the anticipated accommodation of desired reductions in real balances by inflation during the period of adjustment is potentially an important stabilizing force. As will be shown, this stabilizing force may be sufficiently potent to permit stability of an equilibrium growth model even if inflationary expectations are formed with perfect foresight (a possibility that is excluded in the existing literature). Thus the dynamics of monetary growth have been misunderstood. The insight that there must be sufficient friction in the system if the economy is to be stable is correct, but it has been overlooked that the portfolio adjustment process itself may be capable of providing that friction.

It will be sufficient to examine a simple model with three markets, for money, bonds and goods (on the assumption that the wage rate adjusts to continually clear the labour market). It will be apparent that the results generalize to more elaborate economies. The model utilized will be sufficiently general to include several important classes of monetary growth models (in terms of adjustment processes) as special cases, and it will be shown that proper specification of asset market equilibrium will significantly relax the stability conditions in all of these cases.

1. CREDIT MARKET EQUILIBRIUM

Financial markets will be assumed to be adequately characterized by a loanable funds theory of interest rate determination, as exposited by Rose [8]. The main point of this study could, however, also be made within a liquidity preference framework, or in the Robertson- Tsiang variant of the loanable funds model (although some injustice must be done to the Robertsonian model to represent it in continuous time).

Accordingly it is assumed that " the " interest rate responds to the flow excess demand for credit. By Walras Law the excess demand for credit, valued in units of goods, equals the sum of the real excess demands for goods (XDG) and money (XDM). The loanable funds theory may be expressed formally as

dr/dt-r = b(XDG + XDM), b'( )>O, ... .(1)

where r is the market (nominal) rate of interest. The excess demand for goods is the excess of planned investments over planned savings.

It will be convenient to express these magnitudes as fractions of the capital stock. It is 475



476 REVIEW OF ECONOMIC STUDIES

assumed that capital accumulation planned per unit of capital is a non-decreasing function of the ratio of labour to capital (x -LK) and a decreasing function of the real rate of interest (p), and that savings planned per unit of capital is an increasing function of the ratio of labour to capital:

I/K =A(x, p) A1 _ O, A2<0,

S/K=g(x) gx>O.

(A more general savings relation including the rate of interest and real balances would complicate the exposition without altering the basic results.)

It will be necessary to pay particular attention to the concept of the flow excess demand for money, because failure to specify this term correctly has generated invalid conclusions about the dynamics of monetary growth. For simplicity, assume that the rate at which the community wishes to acquire real balances per unit of capital (m _ M/pK) is proportional to the discrepancy between the desired portfolio composition and the actual composition. Then the desired flow of real balances (per unit of capital) to the community will be:

md' = A(L(x, r) -m) L 1> 0, L2 <0, ... (2)

where r is the nominal rate of interest. The function L(x, r) is the ratio of real balances to capital that the community would choose to hold if portfolio adjustment were costless, and has the usual arguments and partial derivatives.

A stock adjustment model of portfolio behaviour of this type is common in the monetary growth literature (e.g. Hadjimichalakis [2], [3], Stein [10], and Stein and Nagatani [11]). However, in this literature expression (2) is uniformly identified as the excess demand for money. This is implausible, as Mackay [6] points out, since (2) incorporates no flow supply term. Mackay specified M/pK as the flow supply of money, giving an excess demand of (md - M/pK). Intuition suggests that this cannot be correct either, since it implies that even in long-run equilibrium in which anticipations are fully realized there will exist a perpetual discrepancy between desired and actual portfolio compositions for all non-zero rates of inflation.

What behavioural considerations can be employed to determine an appropriate specification of the excess flow demand for money? Consider an individual wealth holder. Equation (2) gives the rate at which this individual desires to alter the ratio of his real balances to his capital stock. But he need not enter the market to achieve all of this desired adjustment if he expects to receive an inflow of real balances per unit of capital. For example, he may expect a share of government transfer payments (this will be assumed to be the sole source of money creation). If the individual correctly anticipates these transfers, this term is accounted for by Mackay with the term M/pK. But surely the individual wealth holder recognizes that deflation (inflation) will also automatically increase (reduce) his holdings of real balances over time. This flow of " m " is acquired costlessly and reduces (increases) the degree to which the wealth holder must enter the asset markets. Finally, capital accumulation must be countered by acquiring real balances to achieve the desired flow of m. Thus the expected rate of change in the ratio of real balances to capital (me) is in a sense a flow supply to balance the flow demand. All agents in aggregate will plan to enter the market to obtain real balances to the extent that desired flow exceeds expected flow, so the market excess demand for money will be:

XDM = A(L(x, r) - m) - the. ... (3)

It will be assumed that policy makers adjust the rate of monetary expansion in response to the expected rate of inflation, q, that the policy rule (KMM = 0(q)) is known to wealth holders, and that capital is expected to grow at the (exogenous) rate of population growth, n. Consequently,

.he (O(q - q - nas..4



JOHNSON MONETARY GROWTH 477

The policy rule will be assumed to satisfy the condition that an increase in the expected rate of inflation will lead to a reduction in the expected rate of growth of real balances per unit of capital (i.e. d,iieIdq<O, or 0'(q) < 1).

On the assumption that financial markets are sufficiently well organized to be in momentary equilibrium at every instant of time (b'-> oo), the nominal rate of interest is determined as a function r(x, q, m) by the condition for credit market equilibrium: that the (flow) excess demands for goods and money sum to zero. By the preceding behavioural assumptions, and the observation that p = r-q, this condition is

A(x, r-q)-g(x) + A(L(x, r)-m)-(O(q)-q-n)m = 0. ...(5) Note that D(XDG + XDM)/Dr is negative, making rapid adjustment viable.

The comparative statics of the credit market equilibrium are readily derived. For the purpose of examining stability of the dynamic model to be developed in the next section, partial derivatives are evaluated at the steady state values of x, q and m, at which point me = 0. Solve (5) implicitly for the rate of interest to get the partial derivatives of the function r(x, q, m) at steady state:

rx=-Al-gx+AL, ;t0, ... (6) r < ... (6)

AZ2+AL2 2

A2 + m)LL21 r 2+m(O-) >0. ...(8)

q A2 + LL

The impact of a change in x (which may be thought of as a measure of income) on the interest rate is indeterminate, because it increases both the demand for credit (through increasing investment plans and desired acquisition of real balances) and its supply (through increasing savings).

As may be expected, the response of the real interest rate (p) to an increase in the expected rate of inflation will be of crucial importance to the stability of the economy. If the real rate rises when inflationary expectations rise (Pq > 0), it will discourage investment plans, reducing the pressure on prices, thus providing a powerful stabilizing force. In fact, it will be proved below that Pq > 0 will be sufficient to guarantee stability of the model. When will this happen?

Pqq A2 + AL2

Since the denominator is negative, this expression will be positive if and only if: m(O'-1)-;AL2 < . ..(10)

Inequality (10) may be interpreted as defining a class of policies which will provide the economy with the stabilizing effect of Pq > 0. It will be noted that for AL2 sufficiently small a " neutral " policy of expanding the money supply at a constant rate (0' = 0) will be consistent with pq>0 Simple manipulation reveals that this will be the case if:

- rL2/m< r/I.

That is, if the interest elasticity of the demand for money is less than the ratio of the nominal rate of interest to the speed of portfolio adjustment, then the real rate will rise with a rise in inflationary expectations.

Having determined the rate of interest as a function of x, q and m, it will be convenient to define the investment function in terms of only these same variables. Therefore, let

k(x, q, m) = A(x, r(x, q, m)-q); ...(11)

kx = Al+A2rx, kq = A2(rq-1), km = A2rm.




2. THE DYNAMIC MODEL

In the preceding section the state of an economy has been characterized given values of x, m and q. On specifying the time paths of these variables, we will have completed the description of the process by which the economy evolves. Assuming that savings plans are realized, and full employment maintained, the rate of change in the labour-capital ratio is derived by differentiating x -LK with respect to time and substituting g(x) for the rate of capital accumulation:

x= [n-g(x)]x. ...(12)

Similarly, the rate of growth of real balances per unit of capital is:

m= [0(q)-pi/p-g(x)]m, ...(13)

since 0(q) is the current rate of expansion of the nominal money supply (M/M). Finally, the model is completed by assuming that prices adjust according to a modified Walrasian tatonnement process and that expectations are formed adaptively:

p/pp= =(k(x, q, m)-g(x))+iiq, ... (14)

4 = y(lp-q). ... (15)

Equation (14) with il = 0 is the conventional tatonnement process, with 1= 1 it is the alternative price adjustment process proposed by Hadjimichalakis [3] and also analysed by Stein in [10]. Equilibrium growth models are special cases of this model, with g-* oo. Equation (15) is Cagan's [1] adaptive expectations process, which includes (myopic) perfect foresight as a special case (y-* oo).

Substituting the expression (14) for price movements into (13) and (15), the model may be summarized by the following equations of motion:

S= x[n-g(x)], ...(12)

m m[0(q)-s(k(x, q, m)-g(x))- 1q-g(x)], ... (16)

4 y 4[e(k(x, q, m)-g(x))-(1- n)q] ...(17)

By definition, steady state is characterized by the condition that (12), (16) and (17) are simultaneously equal to zero. This implies the usual steady state properties: (a) the rate of capital accumulation must equal the rate of growth of the labour force, (b) the expected rate of inflation must equal the actual rate of inflation, and (c) the expected (equals actual) rate of inflation must equal the rate of growth of per capita cash balances. Note that in the case with i1 = 1 condition (b) requires that excess demand for goods must be zero in steady state. In all other cases (with e finite) of non-zero inflation, a steady state will imply perpetual excess demand (or supply) of goods. Thus il = 1 is the only case consistent with long run equilibrium.

Taking the Taylor's expansion of (12), (16) and (17) about the steady-state values, the matrix (J) of coefficients of the linearized system may be shown to be:

-xgx 0 0

J -[ekx + (1-e)gx]m -[skm]m -[ekq- (0'- )]m I

y -(kx-gx) ys(km) y[s(kq) -(1 - ] J

with partials all evaluated at the steady state. A necessary and sufficient condition for stability of the system is that the characteristic

roots of J have negative real parts. One of the roots is - xgx which is negative; the other two are the roots of the 2 x 2 matrix J' obtained by deleting the first row and column of J.




The roots of J' will be negative if and only if the trace of J' is negative and its determinant is positive. Computing the latter yields:

det J' = ysmkm(1-0'), ...(18)

which is positive. The trace of J' is:

tr it'=- -mkm + q[s(kid -( ) * . . .(19) Substituting from (11) for the partials of k(x, q, m), then substituting from (6), (7) and (9), tr J' can be derived as

tr j'A= +L2 - A m(O'- 1)+L2+ ( A2+2L2 ...(20)

This will be negative if and only if the second term in brackets is positive. Rearranging terms and multiplying by r, it is seen that for [1+ (1- i)/A2e] positive (as it must be for

= 1, or ;-? oo), the necessary and sufficient condition for stability is

_ JL2 <_ JI [(- )(mE] 1+(1 - n)/A28)-] ..(21)

For [1 + (1- ti)/A2e] negative the system is necessarily stable. The nature and implications of condition (21) will be revealed by considering its meaning for several specific classes of growth models that have been widely examined in the literature.

3. ANALYSIS OF STABILITY FOR SPECIAL CASES

(a) Equilibrium Growth with Perfect Foresight (y, e-? oo) This class of models includes the Tobin model (Tobin [12], analysed also in Hadjimichalakis [2], [3], Nagatani [7] and Stein [10]). A basic conclusion of this literature is that there must be sufficient friction in the system if it is to be stable. Equilibrium growth with perfect foresight is necessarily unstable in these models. What has been overlooked, of course, is that the very frictions which provide the rationale for holding cash balances may be sufficient to stabilize the economy. To see this point, examine the limit of condition (21) as Y, - oo. The necessary and sufficient condition for stability is then:

- rL2/m < (rl)(l - 0')- . . .(22)

This is just the condition (10) under which pq>O i.e. under which an increase in inflationary expectations raises the real rate of interest. It should be noted that this is a condition that is sufficient for stability regardless of the magnitude of e and y. Even with a neutral policy, the model will be stable if the interest elasticity of demand for money is less than the ratio of the nominal interest rate to the speed of portfolio adjustment. One can, therefore, postulate a stable equilibrium growth with perfect foresight model so long as there exist high enough costs of transacting in asset markets to make portfolio adjustment a sufficiently gradual process.

(b) Equilibrium growth (e-+ so) Sidrauski [9] introduced the adaptive expectations process into the basic Tobin equilibrium growth model and proved that the resulting model would retain the comparative statics of the Tobin model and yet could be stable if the Cagan [1] stability criterion was satisfied, that is, if:

-rL2/m< r/y. ...(23)

In a model with gradual portfolio adjustment, condition (23) is unnecessarily restrictive (as is already evident, since we have proved that this model can be stable even with y-? oo).




The necessary and sufficient condition for stability of the equilibrium growth model is derived by taking the limit of (21) as s-* oo. The model will be stable if and only if:

rL2 <r { + 1 ...(24) m y A

It is evident from (24) that the lag in portfolio adjustment plays precisely the same role as the lag in expectations formation in stabilizing the system.

(c) Perfect Foresight (y-- oo) Stein and Nagatani, in [11], attempted to analyse the role and impact of monetary stabilization policies in a growing economy. Their model was essentially the perfect foresight type derivable as a special case (y- >oo) of the general model constructed in Section 2. They concluded that monetary policy (of the type 0 = 0(q)) could not alter the stability of the system. Mackay [6] demonstrated that this conclusion was unwarranted: it results from misspecification of asset market equilibrium by identifying the flow demand for money as the flow excess demand. Mackay, however, failed to account for the effect of anticipated inflation on desired portfolio adjustments, resulting in an overly vigorous policy rule in his analysis. To see this, examine condition (21) under the assumption that y- oo. In the limit, the necessary and sufficient condition for stability will be:

rL2 < + (1- O][( + (1-)/A2e)] ... (25)

This is to be compared to the analogous rule which would result under Mackay's specification of flow excess demand:

m mg - ) ][(I + (1 - q)/A2e)] . . .(26) It is clear from the preceding analysis that gradual portfolio adjustment tends to

stabilize a macroeconomic system in essentially the same way as slow revision of expectations and sluggish prices, a point which has previously been obscured by faulty analysis. The significance of this effect (and consequently the magnitude of the error in overlooking it) depends, of course, on the speed of portfolio adjustment relative to the speeds of adjustment in prices and expectations. Although the response of prices to excess demands may be thought to be slow, it should be noted that this is stabilizing only to the extent that increased expectations are not passed through to price movements: i.e. to the extent that i1 falls short of one (a point that is implicit in Hadjimichalakis [3]).

First version received February 1974; final version accepted November 1975 (Eds.). I wish to acknowledge my indebtedness to Professor M. G. Hadjimichalakis to whom I owe my under-

standing of the structure and behaviour of macroeconomic systems. His influence is readily apparent here. I also thank P. J. Hammond, managing editor of this Review, for making several suggestions which have significantly improved the exposition in this article. Of course, they share no responsibility for errors. The views expressed herein are solely my own, and do not necessarily represent the views of the Board of Governors of the Federal Reserve System.

REFERENCES [1] Cagan, Phillip. " The Monetary Dynamics of Hyperinflation ", in Friedman, M., ed., Studies in

the Quantity Theory of Money (University of Chicago Press, 1956). [2] Hadjimichalakis, Michael. "Equilibrium and Disequilibrium Growth with Money-the Tobin

Models ", Review of Economic Studies (October 1971). [3] Hadjimichalakis, Michael. " Money, Expectations and Dynamics-An Alternative View ", Inter-

national Economic Review (October 1971). [4] Hadjimichalakis, Michael. " On the Effectiveness of Monetary Policy as a Stabilization Device ",

Review of Economic Studies (October 1973). [5] Johnson, Lewis. " Inflationary Expectations and Momentary Equilibrium ", American Economic

Review (June 1976).




[6] Mackay, R. J. " Stein and Nagatani on Stabilization Policies in a Growing Economy ", Review of Economic Studies (October 1973).

[7] Nagatani, Keizo. " A Note on Professor Tobin's ' Money and Economic Growth' ", Econometrica (January 1970).

[8] Rose, Hugh. " Liquidity Preference and Loanable Funds ", Review of Economic Studies (February 1957).

[9] Sidrauski, Miguel. " Inflation and Economic Growth ", Journal of Political Economy (December 1967).

[10] Stein, Jerome L. Money and Capacity Growth (Columbia University Press, 1971). [11] Stein, Jerome L. and Nagatani, K. " Stabilization Policies in a Growing Economy ", Review of

Economic Studies (April 1969). [12] Tobin, James. " Money and Economic Growth ", Econometrica (October 1965).



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