Monetary Shocks and the Dynamics of Inflation

It is a standard property of neoclassical growth models involving money that in the long run, or steady state, the rate of growth of money will be equal to the rate of growth of real income and the growth of prices. 1 Expressed in terms of real money balances; this property ensures that the growth of real balances will be proportional only to the growth of real income, and that changes in the growth of nominal money will have no effect. In the short run, however, it has been observed that an increase in the rate of monetary emission results in a larger initial stock of real money balances, or, what amounts to the same thing, that velocity tends initially to move in the opposite direction of the change in monetary growth. This phenomenon has been detected empirically in the studies by Harberger (1963) on Chile, Diz (1970) on Argentina, and Pastore (1975) on Brazil. 2

Abstract

It is a standard property of neoclassical growth models involving money that in the long run, or steady state, the rate of growth of money will be equal to the rate of growth of real income and the growth of prices. 1 Expressed in terms of real money balances; this property ensures that the growth of real balances will be proportional only to the growth of real income, and that changes in the growth of nominal money will have no effect. In the short run, however, it has been observed that an increase in the rate of monetary emission results in a larger initial stock of real money balances, or, what amounts to the same thing, that velocity tends initially to move in the opposite direction of the change in monetary growth. This phenomenon has been detected empirically in the studies by Harberger (1963) on Chile, Diz (1970) on Argentina, and Pastore (1975) on Brazil. 2

It is a standard property of neoclassical growth models involving money that in the long run, or steady state, the rate of growth of money will be equal to the rate of growth of real income and the growth of prices. 1 Expressed in terms of real money balances; this property ensures that the growth of real balances will be proportional only to the growth of real income, and that changes in the growth of nominal money will have no effect. In the short run, however, it has been observed that an increase in the rate of monetary emission results in a larger initial stock of real money balances, or, what amounts to the same thing, that velocity tends initially to move in the opposite direction of the change in monetary growth. This phenomenon has been detected empirically in the studies by Harberger (1963) on Chile, Diz (1970) on Argentina, and Pastore (1975) on Brazil. 2

In general, this short-run response of real money balances to a monetary change can be explained by the lag between the change in the supply of money and the response of inflation (or nominal income). With the existence of such a lag, an expansion in money would raise the stock of nominal money balances and, since prices would not respond to the change immediately, real money balances would increase. Empirical evidence on this lag is fairly common, and two recent examples are the studies by Vogel (1974) on a group of Latin American countries and Von Furstenberg and White (1980) on industrial countries.

Despite the empirical evidence, and the recognition by Friedman (1970) that, in the transition phase, real money balances would be expected to respond positively to changes in the growth of nominal money, standard money demand models continue to be formulated on the basis of instantaneous and equiproportional adjustment of prices and, therefore, real money balances. Equilibrium (long-run) models necessarily have this feature and therefore cannot be faulted. The criticism is directed essentially at the “dynamic” or “disequilibrium” models that purport to describe the time path between two equilibria, namely, the models that involve some type of partial-adjustment mechanism or an adaptive expectations generating scheme. 3 Neither of these two methods of introducing dynamics into the system is able to provide a satisfactory explanation as to why real money balances rise initially after a monetary change. Their basic feature is the ability to trace the time path of real money balances after a change in exogenous variables other than the growth of the nominal money supply.

Clearly, theoretical considerations alone would provide sufficient motivation for the development of money demand models that would describe this phenomenon of the initial rise in real cash balances. In addition, the importance of demand for money models in the actual formulation of monetary policy makes an examination of this issue directly relevant from a policy point of view. 4 For example, if the aim of the authorities is to affect the rate of inflation, then the use of a conventionally formulated money demand function, while perhaps yielding the correct policy in the long run, may give misleading signals during the approach to this long-run position if real money balances happen to move initially in the same direction as the change in the nominal money supply. Short-run policy, thus, may have to be designed to compensate for this effect.

It is the awareness of this issue that has led to various attempts to develop alternative theoretical models, either through the adoption of more complicated expectations formation schemes (e.g., Frenkel (1975), Auernheimer (1979)), or the introduction of the concept of temporary disequilibrium in the holding of money balances (Sjaastad (1972), Auernheimer (1974), Pastore (1975)). The purpose of this paper is to consider an alternative approach that relies on the idea of a monetary “shock,” or “surprise.” A simple adjustment model is designed in which it is hypothesized that, while prices adjust instantaneously to expected, or anticipated, changes in the growth of money, unexpected increases initially show up in increases in the public’s money holdings. Eventually, these excess money holdings are worked off and prices begin to respond. The model is in the spirit of the rational expectations literature 5 and remains consistent with the long-run monetarist proposition contained in Friedman (1970). It enables one to trace out more accurately the path that real money balances will take in reaching the new equilibrium, and consequently the dynamic behavior of inflation, after there is a change in the money supply.

Instead of dealing with the issue simply on a theoretical level, the model is estimated for a group of 11 developing countries, 7 Latin American—Argentina, Brazil, Colombia, Costa Rica, the Dominican Republic, Mexico, and Uruguay—and 4 Asian—India, Malaysia, the Philippines, and Thailand. Since the monetary and inflationary experience of these countries is fairly diverse, the general applicability of the model is put to a fair test. While there is no particular reason for concentrating exclusively on developing countries, it will be seen that certain simplifying assumptions made in the course of the analysis are somewhat easier to justify in the context of these types of economies. 6

The outline of the paper is as follows. In Section I the standard money demand models are discussed within the framework of the basic hypothesis of this paper and then compared with the model proposed here. The results from estimating the model, and dynamic simulations demonstrating both the effects of monetary shocks on inflation, and, conversely, the appropriate monetary policy to achieve a target rate of inflation, are shown in Section II. The broader policy implications of the exercise, and a summary of the findings, are contained in the concluding section.

I. Theoretical Framework

Following the extensive number of theoretical and empirical studies on the subject, a general consensus has emerged that the demand for real money balances is essentially determined by a scale variable and the opportunity cost of holding an asset (money) that does not yield a positive nominal rate of return. 7 Since substitution can occur both between money and goods and between money and financial assets, the relevant opportunity costs would be the (expected) rate of inflation and the (expected) rate of interest on alternative financial assets, respectively. In addition, one could also introduce an “own” rate of return of money into the formulation. This would be particularly relevant if money was defined to include time and savings deposits paying an interest rate. In (semi) log-linear terms, the function can be written as

logmtd=a0+a1logyta2Πta3Rte+a4Rdte(1)

where md is the demand for real money balances, defined as the nominal demand, Md, deflated by the “expected” price level, Pe. The variable y is real income, 8 II the expected rate of inflation, and Re and Rde the expected rates of interest on financial assets and time and savings deposits, respectively. In this specification, all expectations of variables in period t are assumed to have been formed in the previous period. Written in the form of equation (1), the parameters are all expected to be positive.

While equation (1) is a suitable representation of a general model of money demand, it is not clear whether the inclusion of the two interest rates makes it fully relevant for developing countries. In such economies, the relatively thin markets for financial securities make the substitution between money and goods, or real assets, quantitatively more important. Added to this general absence of adequate financial assets is the problem created by the authorities who maintain controls, or ceilings, over the interest rates on the financial assets that are available. 9 Since changes are made fairly infrequently, the interest rate series display very little variation over time, and thus make it exceedingly difficult to empirically detect any systematic relationship between money and interest rates. For developing countries, it would seem that equation (1) could be specified without the interest rate variables, 10 that is, as

logmtd=a0+a1logyta2Πt(2)

In equilibrium, or the long run, the demand for real money balances will be equal to the actual stock, and the expected and actual inflation rates would be identical (and constant):

md = m = Md/Pe = M/P

Π = ΔlogP

where ΔlogP is the actual rate of inflation and Δ is a difference operator. Then in equilibrium equation (2) would be

logMtlogPt=a0+a1logyta2ΔlogPt(3)

In terms of rates of growth, and since Δ2log Pt = 0, this function can be expressed as

ΔlogMtΔlogPt=a1Δlogyt(4)

and the effect on inflation of the growth in money can be determined from the equation

ΔlogPt=a1ΔlogytΔlogMt(5)

which has the property that ∂(Δlog Pt)/∂(Δlog Mt) = 1.

It should be stressed that this is a steady-state property, when all the adjustments have been completed. 11 There is substantial evidence, however, that in the short run the relationship between the growth of money and inflation cannot be characterized as instantaneous. Typically, one observes that the time path of real money balances after a monetary increase appears as shown in Figure 1. At time t0, there is a stepwise increase in the rate of monetary growth, and initially the stock of real money balances rises, reaching a maximum at point t¯. As inflation catches up with the growth of money at t*, real money balances are once again at their previous level. However, the process continues beyond t* as inflation overshoots the rate of growth of money. This phenomenon of later overshooting has been noted by Harberger (1978, p. 519) who states:

Figure 1.
Figure 1.

Time Path of Real Money Balances in Response to Monetary Shock

Citation: IMF Staff Papers 1980, 002; 10.5089/9781451946864.024.A002

  • Somewhere in the process of an economy’s adjusting to a new and higher rate of monetary expansion there must be a period wherein the rate of price increase rises above the rate of monetary expansion. (Italics in original.)

Figure 1 clearly illustrates the existence of a lag between changes in money and the rate of inflation, and there is empirical evidence supporting the hypothetical time path that is drawn. For example, Vogel (1974, p. 112), in extending the earlier Harberger (1963) study on Chile to a group of 16 Latin American countries, concludes “that an increase in the rate of growth of the money supply causes a proportionate increase in the rate of inflation within two years.” Similar lags, averaging between one and two years, were found by Aghevli and others (1979) for a group of Asian countries; for industrial countries, Von Furstenberg and White (1980) report a lag of a little over one year.

Standard Models of Lagged Adjustment12

Within the framework of money demand models, lags are generally introduced through imposing a partial-adjustment, or error-learning, mechanism on the structure. In this approach, popularized by Chow (1966), and utilized by Mundell (1971) to deal with the particular problem considered in this paper, the actual stock of real money balances is assumed to adjust proportionally to the difference between the demand for real money balances in the current period and the actual stock in the previous period:

Δlogmt=λ[logmtdlogmt1](6)

where λ is the coefficient of adjustment, 0 ≤ λ ≤ 1.

While in equation (6) the rate of inflation can respond in a lagged fashion to changes in the arguments contained in md, it is fairly straightforward to show that adjustment to the growth in the nominal money supply is identical to the equilibrium case, since:

ΔlogPt=λ[logmtdlogmt1]+ΔlogMt(7)

and from equation (7) it follows that δ(Δlog Pt)/δ(Δlog Mt) = 1.

In an attempt to allow for lagged adjustment between inflation and the growth of money, Goldfeld (1973) and White (1978) have proposed that the partial-adjustment function be specified in nominal rather than real terms. When expressed in a comparable form to equation (6), that is, in real terms, this equation is written as follows: 13

Δlogmt=λ[logmtdlogmt1]+(1λ)ΔlogPt(8)

or, when inflation is the endogenous variable, as

ΔlogPt=[logmtdlogmt1]+1λΔlogMt(9)

Apart from implying that changes in the arguments in md have an instantaneous effect on inflation, since λ will generally be less than unity, this model further argues that inflation will initially overshoot the growth in money. 14 Real money balances would, thus, initially decline. This is at variance with Sjaastad’s (1972, p. 5) conclusion that

  • an increase in the rate of monetary emission usually results in an increase, rather than decrease, in the real quantity of money; empirical evidence for Chile and Argentina is quite strong in this connection.

Another popular way of introducing lags into the model is through arguing that expectations (of inflation) are formed adaptively along the lines of the Cagan (1956) model, where expectations of inflation are revised in proportion to the error between the actual rate of inflation in the previous period and the rate that was expected to prevail:

ΔΠ=β[ΔlogPt1Πt1](10)

Here, β, the coefficient that measures the degree to which the error affects the revision of expectations, lies within the interval (0, 1).

Combining equations (2), (6), and (10) yields the following reduced form:

Δlogmt=λa0+λa1Δlogyt+βλa1logyt1βλa2ΔlogPt1(λ+β1)Δlogmt1βλlogmt2(11)

Again, as in the partial-adjustment model, equation (6), the effect on inflation of the growth in money corresponds to the steady-state solution, 15 that is,

ΔlogPt=λa0λa1Δlogyt+βλa1logyt1+βλa2ΔlogPt1+(λ+β1)Δlogmt1+βλlogmt2+ΔlogMt(12)

Adjustment Model Incorporating Monetary Shocks

Clearly, it is the inability of standard dynamic models to reproduce the behavior of real money balances, as outlined in Figure 1, that has led to a number of studies that attempt to correct for this deficiency. Generally, the literature has followed two lines of direction. Frenkel (1975) and Auernheimer (1979) respecify the expectations function to include both regressive and extrapolative elements, 16 while others, such as Sjaastad (1972), Auernheimer (1974), Pastore (1975), and Brillembourg (1978), stress the role of temporary disequilibrium within the framework of the partial-adjustment model. A related line to this second approach has been the one taken by Barro (1978), Carr and Darby (1978), and Coats (1978), which considers the role of monetary shocks or surprises on the stock of real money balances. 17 The model proposed here also utilizes this distinction between the effects of anticipated, or expected, changes in the money supply on inflation and the effects of unexpected shocks, and combines it with the temporary disequilibrium concept. There is, thus, a great deal of similarity between the analysis here and the models of Pastore (1975) and Carr and Darby (1978).

The basic model takes as its starting point the standard partial-adjustment function written with inflation as the endogenous variable, that is, equation (7), and proceeds to argue that the growth of money can be decomposed into its expected and unexpected components. The expected growth in the money supply has an instantaneous and equiproportional effect on inflation, while changes that are unexpected act only with some delay. More formally, equation (7) is reformulated as

ΔlogPt=λ[Δlogmtdlogmt1]+ΔlogMte+γ[ΔlogMt+ΔlogMte](13)

where Δlog Me is the expected, and [Δlog M - Δlog Me] the unexpected, growth in the nominal money supply. It is the latter variable that is regarded as the shock variable, and the parameter γ measures its effect on inflation. If the effect is instantaneous, γ will be equal to one, and a value of zero would imply that there would be no impact. Generally, this parameter would be expected to lie between these two extreme values.

From equation (13) it is straightforward to derive the equation for the change in real money balances:

Δlogmt=λ[Δlogmtdlogmt1]+(1γ)[ΔlogMt+ΔlogMte](14)

Equation (14) thus differs from the standard model by allowing for the possibility of flow disequilibrium, in addition to stock disequilibrium, given by [Δlogmtdlogmt1]. If the parameter γ is zero, real money balances will rise by the full amount of the unexpected growth in money, and if it is unity the model reduces to the standard partial-adjustment model given in equation (6).

Before discussing the formal properties of this model further, it may be useful to consider briefly some of its more general features. The theory underlying the model is by no means new, having been discussed at length by Friedman (1969, Ch. I) and (1970) and, more recently, by the adherents of the rational expectations school. 18 As long as the growth of money is fully anticipated, the rate of inflation will be equal to that rate of growth. 19 The delayed response of inflation to unexpected changes, or shocks in the rate of monetary growth, however, results from prices being “sticky” in the short run and thus not reacting immediately. 20 Also in developing countries, the absence of financial markets puts the main burden of adjustment to unexpected changes in the supply of money on the goods market, since there is no meaningful interest rate that could change and thus clear the flow disequilibrium. As spending decisions take time, part or all of the unexpected increase in money is initially absorbed into cash balances, thereby raising the stock of real money balances. Only as this excess stock is worked off over a period of time do prices start to rise.

Certain assumptions made in the analysis should be mentioned. In the model it is assumed that the growth in the supply of money, whether expected or unexpected, is strictly exogenous with respect to inflation. In other words, the possibility of a feedback from inflation to monetary growth is ignored. There are, of course, two obvious channels through which inflation can affect the domestic money stock. First, a causal relationship running from inflation to money could emerge as a consequence of government deficit financing operations. Inflation, once triggered, could result in an increase in the fiscal deficit if the government attempted to maintain real expenditures while revenues lagged behind the movement in prices. If, then, these deficits were financed by the banking system, the domestic money stock would become effectively endogenous to domestic prices. 21 Second, changes in the money supply could occur through the balance of payments. It is well known that, in an open economy operating under a fixed exchange rate regime, any disequilibrium in the money market created by, say, a change in domestic prices, could result in international reserve flows that affect the overall stock of money.22 The assumption made here does not, of course, imply that the supply of money is necessarily under the effective control of the authorities. Indeed, it would be difficult to make such a claim in most developing countries, since they generally have open economies and are, therefore, vulnerable to changes in the money supply emanating from the foreign sector. What is simply assumed for the analysis here is that inflation does not affect monetary growth within the same quarter. A lagged relationship running from inflation to money could certainly exist without affecting the estimation of the relevant parameters of the model. In addition, it is also argued that real income is unaffected by changes, again expected or unexpected, in the money supply. These assumptions are necessary to stay within the framework of a single-equation model. Relaxing them would mean the relationships would have to be modeled explicitly in some way. 23

Turning now to an examination of the formal properties of the model, it must first be determined how money holders form their expectations. In the case of inflation, the simplest procedure would be to utilize the adaptive expections model shown in equation (10). Expectations regarding the growth of nominal money could be formed in the same way as inflation expectations, or they could be generated through some other, more general, distributed lag method. 24 However, in keeping with the concepts contained in the rational expectations literature, there must be consistency in the formation of expectations of the different variables that are in the model. In the context of the model under consideration here, the consistency principle would imply that expectations of inflation and monetary growth cannot be formed independently and thus have to be linked in some way. In a complete rational expectations framework, if expectations of inflation are generated by an adaptive expectations scheme such as in equation (10), then for these to be rational in the sense of Muth (1961) implies a particular process for money supply growth. 25 While it is possible, as shown by Sargent and Wallace (1973) and Sargent (1977), to derive this money supply process for the simple Cagan (1956) model, 26 any expansion of the model, as Fernandez (1977) demonstrates, makes the problem fairly complex.

The approach here is a much simpler one, which, though not strictly “rational,” nevertheless retains somewhat loosely the concept of consistency in the formation of expectations of both inflation and monetary growth. To maintain this consistency concept, it was postulated that the expected rate of monetary growth is identical with the growth in nominal money demand. In other words, money holders expect that the necessary money will be supplied to satisfy the growth in their nominal demand. 27 This nominal demand would, from equation (2), be

logMtd=logPte+a0+a1logyta2Πt(15)

and in rates of growth

ΔlogMtd=ΔlogPte+a1logyta2ΔΠt(16)

Since ΔlogPte=Πt, and assuming that Δlog yt and ΔΠt can be incorporated in a constant term, the following equation can be written:

ΔlogMte=ΔlogMtd=Πk(17)

where k is a constant equal to (a, Δlog yt - a2 ΔΠ). Substituting equations (2) and (17) into equation (14), and ignoring the constant k, 28 and also, for the time being, the other constant, a0:

Δlogmt=λ[a1logyta2Πtlogmt1]+(1γ)[ΔlogMtΠt](18)

Collecting terms and using lag-operator notation, that is, where Lixt=xti,

(1(1λ)L)logmt=λa1logyt(λa2+(1γ))Πt+(1γ)ΔlogMt(19)

Similarly, the adaptive expectations equation for inflation, equation (10), can be specified as follows: 29

Πt=βL(1(1β)L)ΔlogPt(20)

Replacing Πt in equation (19) with equation (20), and multiplying through by (1 -(1 - β)L):

(1(1β)L)(1(1λ)L)logmt=(1(1β)L)λa1logytβ(λa2+(1γ))LΔlogPt+(1(1β)L)(1γ)ΔlogMt(21)

The model is closed by the identity that the rate of inflation is equal to the growth in nominal money less the growth in real money balances:

ΔlogPt=ΔlogMt(1L)logmt(22)

The complete model can thus be represented by the two-equation system:

((1(1β)L)(1(1λ)L)1Lβ(λa2+(1γ))L1)(logmtΔlogPt)=((1(1β)L)λa10(1(1β)L)(1γ)1)(logytΔlogMt)(23)

The solution of this model in terms of the exogenous variables is given by

logmt=A1(L)Detlogyt+A2(L)DetΔlogMt(24)
ΔlogPt=A3(L)Detlogyt+A4(L)DetΔlogMt(25)

where

A1(L)=λa1(1β)λa1L
A2(L)=(1γ)[(1β)(1γ)+β(λa2+(1γ))]L
A3(L)=λa1+λa2(2β)Lλa1(1β)L2
A4(L)=γ[(1β)+(1λ)(1γ)(1γ)(1β)]L+(1β)[(1λ)(1γ)]L2

and

Det1[(1λ)+(1β)+β(λa2+(1γ))]L+[(1λ)(1β)+β(λa2+(1γ))]L2

In this model, the partial derivative of inflation with respect to the growth in nominal money will not necessarily be unity but rather ∂(Δlog Pt)/∂(Δlog Mt) = γ. Clearly, the impact effect will depend on the value of γ, and thus vary between zero and one.

The dynamic stability of the model will be based on the characteristic roots of Det, and, since β and λ lie between zero and unity, the necessary and sufficient condition for stability is

(β)(1λ)+β(λa2+(1γ))<1(26)

Provided this is satisfied, the system will converge to the long-run equilibrium given by

logmt=a1logyta2LΔlogMt(27)
ΔlogPt=a1ΔlogytΔlogMt(28)

This solution, it can be seen, is similar to the one yielded by the long-run equilibrium model discussed earlier (equations (3) and (5)).

It may be useful at this point to trace out verbally what happens in this model when there is an increase in Δlog M of, say, 10 per cent, and this is a once-and-for-all type of change. Initially, real money balances will rise by (1 - γ)10 per cent as individuals absorb this proportion into their money holdings. In other words, nominal money balances rise by the full 10 per cent but prices only by γ10 per cent. However, as prices begin to rise, the expected rate of inflation increases and starts to exert downward pressure on the demand for money, and real balances start to decline. At some point the rate of inflation will rise by an amount greater than 10 per cent, and thus reduce real money balances below their original level. Eventually, if the system is stable, inflation will settle down to be equal to the rate of growth of money. 30

Thus, the time path of real money balances shown in Figure 1, or the path of inflation discussed in Friedman (1969, Ch. I), can be reproduced through the theoretical model. It allows for the possibility of inflation initially undershooting, and then later overshooting, the growth of money. The size of these movements, and the length of time it takes for the overshooting to occur and for the system to eventually settle down, will obviously depend on the particular values assigned to the parameters involved in the system.

II. Empirical Estimates

With the introduction of the constant, 31 and a stochastic term, vt, the function to be estimated becomes 32

(1(1β)L)logmt=λa0+(1(1β)L)λa1logyt+β(λa2+(1γ))LΔlogPt+(1(1β)L)(1λ)logmt1+(1(1β)L)(1γ)ΔlogMt+υt(29)

If an error term, ut, had been included in the equilibrium money demand equation (2), then vt would have a moving-average structure of the form

υt=λ[1(1β)L]ut(30)

Since equation (29) is nonlinear in parameters, it has to be estimated by a restricted least-squares method that allows for identification of the individual parameters. One such method is the search procedure suggested by Dhrymes (1971). In this paper, a grid search was performed for the expectations coefficient, β, varying it over the interval (0, 1) in increments of 0.05, and then choosing the one that maximized the log-likelihood function. For a given value of β, the equation can be estimated by ordinary least squares, and all the parameters can be readily identified. It is well known that the resulting estimates of the parameters (including β have asymptotic maximum-likelihood properties. 33

The only additional adjustment made was to allow for the possibility of serial correlation in υt by specifying that it follow a first-order autoregressive process, that is,

υt=ρυt1+ϵt(31)

where ρ is the coefficient of autocorrelation, |ρ| < 1, and ϵ is an error term that has classical properties. Equation (29) was estimated subject to this restriction using the Cochrane-Orcutt method.

The countries in the sample were Argentina, Brazil, Colombia, Costa Rica, the Dominican Republic, Mexico, Uruguay, India, Malaysia, the Philippines, and Thailand. The choice of countries was dictated principally by data considerations, although an effort was made to cover as diverse a group of countries as possible, both in terms of structure of the financial system and in experience with inflation. Table 1 shows the average of the annual rates of inflation, and the average ratios of money to income, over the period 1962–76 for each of the countries. The latter variable has often been taken as a rough indicator of financial development. 34 The numbers in Table 1 indicate that the aim for diversity, particularly in the rate of inflation, is clearly met. The values for the inverse of velocity are difficult to interpret, but they do indicate that there are substantial differences across countries, 35 and hence it can be argued that if the model is equally applicable to these various countries it is not country-specific and is relevant to different types of developing countries.

Table 1.

Average Annual Rate of Inflation and Ratio of Money to Income, 1962–76

(In per cent)

article image

Percentage change in the consumer price index.

Money plus quasi-money deflated by gross domestic product.

The estimation was done over the period 1962–76 on a quarterly basis. If one is to try to identify lags with any degree of precision, it is advisable to work with as small a time unit of observation as possible. Using annual data, as for example has been done by Vogel (1974) and Von Furstenberg and White (1980), restricts the shortest lag to at least one year, and any additional lags are necessarily a multiple of this. The definitions of the variable used, and the sources of the data, are given in the Appendix. The only particular point to mention here is that the testing was conducted with money being broadly defined (M2), that is, including currency, demand deposits, and time and savings deposits. As there are some countries in the sample where the rate of interest paid on time and savings deposits is flexible, 36 one could wish to include this rate in the specification. This was not done here for two basic reasons: First, it is not clear what sign one should expect for the coefficient when the monetary variable is taken to be M2. An increase in the deposit rate may simply result in a shift in composition without affecting the total stock, since currency and demand deposits fall while time and savings deposits rise. Second, in most of the countries the period of flexible interest rates has generally been fairly brief, and this lack of an adequate number of observations makes it difficult to estimate the effect with any reasonable degree of confidence. 37 The choice of broad money was motivated by the fact that it is the monetary aggregate that is generally regarded as having a closer relationship with the ultimate targets of the monetary authorities, namely, prices in this case. Also, in many of the countries monetary policy is exercised through direct credit controls, that is, through the placing of ceilings on the domestic assets of the banking system. In the balance sheet identity, the counterpart to this is the total liabilities of the banking system, including all deposits and not simply demand deposits. 38

Results

The estimates for real money balances, conditional upon the value of the parameter β, are shown in Table 2. Seasonal dummies were included in the equation, but their respective coefficients are not reported in the table. The values of the parameters are presented both in the composite form in which they appeared in equation (29) and individually. As mentioned previously, with known β, the model is exactly identified and it is easy to calculate the individual values. Also shown in the table are various goodness-of-fit statistics, that is, the coefficient of determination (adjusted for degrees of freedom), R¯2, the Durbin-Watson test statistic, D-W, calculated after the adjustment for first-order autocorrelation given by equation (31) was made, and the standard error of the estimated equation, SEE.

Table 2.

Money Demand Results, First Quarter 1962-Fourth Quarter 1976 1

article image

The t-values are in parentheses.

As it is essentially the introduction of the growth of money into the specification that distinguishes the model from more standard formulations, clearly it is the value obtained for the parameter γ that is important. 39 This parameter turns out to be positive and significantly different from unity at the 5 per cent level in all cases, 40 thus implying that the effect of unanticipated monetary growth on inflation is not instantaneous. The particular values range from a low of 0.311 (for the Philippines) to a maximum of 0.615 (for India), with most of the estimates clustered around a value of 0.5. Based on the estimates, an increase of, say, 10 per cent in the growth of money will initially raise the rate of inflation by 5 per cent, and thus the stock of real money balances by the same amount.

In all the estimates the coefficient of the lagged dependent variable, 1 – λ, is positive and significantly different from zero at the 5 per cent level. However, the adjustment of the actual stock of real balances to changes in demand appears to be generally slow. In fact, in a number of cases the hypothesis that λ is significantly different from zero, 41 which would indicate that adjustment is never actually fully completed, cannot be rejected at a reasonable level of confidence. This rather puzzling result appears consistently in money demand studies, whether on developing or developed countries. 42 There appears to be no particular reason for this occurrence, but one possible explanation could be that the problem lies in misspecification of the error structure, the consequences of which are seen in an abnormally large value for the coefficient of the lagged dependent variable. The allowance for more general error schemes, rather than restricting the process to a first-order autoregressive one, could possibly reduce or eliminate this bias. 43

As expected, the short-run income elasticities are positive, but they all tend to be somewhat small. At the impact level, only in Argentina, Colombia, and India are they even significantly different from zero. On the other hand, the corresponding long-run, or equilibrium, elasticities given by the value of a1 are substantially larger. In 7 of the countries these are greater than unity, and in fact the value of a1 was greater than two in the estimates for Costa Rica and Malaysia. A finding of a long-run income elasticity of greater than unity is a standard result for developing countries. For financially developed economies, one would expect a proportional relationship between real income and real money balances, 44 but in developing countries, as monetization occurs, the demand for cash balances may well rise at a faster rate than income. 45

The effect of an increase in the expected rate of inflation is to reduce the stock of real money balances, 46 but the estimated values of the long-run coefficient are more widely dispersed across countries than any of the other parameters contained in the model. They range from a low of 0.427 for Mexico, to over 15 for Malaysia. A more meaningful parameter to consider is the expected inflation elasticity, which was calculated at the respective sample means of the inflation rates for each of the countries. 47 Excluding the results for India and Malaysia, which appear to be definite outliers, there is a pattern to the sizes of this elasticity in that they seem to be positively related, to some degree, to the level of inflation. In countries with higher rates of inflation, that is, Argentina, Brazil, and Uruguay, money holders appear to respond more to a given change in the expected rate of inflation. This could well be a reflection of a greater awareness of the costs that inflation imposes on money holdings in these countries. 48 There is no such pattern to be observed in the values of the coefficient of expectations, β, although the revision of expectations tends to be uniformly rapid. The mean time lag in the revision of expectations ranges between two and three quarters. The fit of all the equations, except perhaps for Argentina, is close, and there is no evidence of first-order serial correlation in the estimated residuals. 49

Using these estimates of the parameters in Table 2, the dynamic stability of each of the models using the condition in equation (26) can be evaluated:

(1 − β)(1 − λ) + βa2 + (1 − γ)).

The values obtained are shown in Table 3. In all cases it can be seen that the model satisfies the condition that the expression be less than unity, and thus the models can be considered stable. The approach to equilibrium is, however, generally oscillatory. 50

Table 3.

Stability Condition of Model

article image

(1β)(1λ)+β(λa2+(1γ))

Simulations

Having shown that the model formulated here apparently fits the data fairly well, and that it does converge to a stable equilibrium, it is now possible to examine whether the estimated versions generate the type of response in the stock of real money balances and inflation to a monetary shock, as postulated in Section I. Essentially, the focus is on tracing the path that inflation would take from the point that the monetary shock is assumed to occur. This is done by performing a simulation experiment in which, starting from a position of initial equilibrium, the system is subjected to a shock change in the nominal money supply, and then observing what happens to inflation both initially and in subsequent periods. 51

Since this simulation exercise was done only for illustrative purposes, a set of estimated parameters that related to a particular country was not utilized. Instead, an approximate average of the values of the parameters in Table 2 was used. 52 The specific values chosen for the simulation are shown in Table 4.

Table 4.

Parameter Values for Simulation Experiment

article image

These parameter values were then introduced into the reduced form equations (24) and (25). For simplicity, it was assumed that real income was constant. Allowing real income to grow at some hypothetical constant rate would not alter the conclusions in any significant way. 53 Also, only the results for inflation are reported, because it is the response of this variable that is the cornerstone of the analysis. 54 The actual simulation was conducted over 60 periods (each corresponding to a quarter) on the grounds that this would be a sufficient length of time to allow the system to approximate steady-state equilibrium.

Simulation I

Assume that, in equilibrium, prices and nominal money were growing at zero per cent up to time t0, and at this point the rate of growth of money was increased to 1 per cent per quarter and held there. The time path of inflation after this shock is shown in Figure 2. Following the monetary shock, prices start to rise as well, but initially at a rate of less than 1 per cent. Based on the parameter values used in this exercise, it takes about five quarters for the rate of inflation to catch up with the growth of money at t¯. 55 After t¯, inflation starts to rise above the growth in money—the phenomenon of overshooting. The inflation rate peaks at a rate of 1.5 per cent in about three years after the shock (t*), and then starts to fall toward the growth of money. Some undershooting occurs, but eventually the growth rates of prices and money become equal again (at 1 per cent). 56 The path of inflation shown in Figure 2 is an empirical reproduction of the hypothetical example shown in Friedman (1969, Fig. 4, p. 13). Friedman regards these oscillations as a “key element in monetary theories of cyclical fluctuations” (p. 13) and gives a variety of explanations as to why it may occur, such as lags in adjustment of money balances, inelastic price expectations, and effects of unexpected changes in money on output. It would seem that the model here also provides a further plausible explanation of these movements in inflation.

Figure 2.
Figure 2.

Time Path of Inflation

Citation: IMF Staff Papers 1980, 002; 10.5089/9781451946864.024.A002

It is fairly easy to determine from Figure 2 that the stock of real money balances will behave in the same manner as outlined in Figure 1. Initially these will rise, and then start to decline, reaching a constant (lower) level at the point where inflation and the growth of money become equal again.

It should be stressed that, while this simulation result is based on a particular set of parameters, it is independent of the values chosen for the growth of money. In other words, the time form of inflation would be the same if monetary growth was reduced from 1 per cent to zero per cent, or from 10 per cent to 5 per cent. 57

Simulation II

Given that there is empirical evidence confirming that inflation reacts slowly to monetary shocks, how, then, should monetary growth be set to achieve a desired rate of inflation? To determine the appropriate policy, the model was solved for the nominal growth in money, thereby making it the endogenous, and inflation the exogenous, variable. In a sense, this simulation—with the roles of inflation and monetary growth reversed—is approximately a transposition of the time paths of inflation and monetary growth. 58

Assume that prices and money have been growing at a rate of 1 per cent a quarter. If the authorities wish to reduce the rate of inflation to zero per cent in period t0, and hold it there, then the growth of money will have to be negative (-1 per cent) at that point (Figure 3). After that initial shock, the authorities will be able to expand from that position until the growth of nominal money equals a rate of zero per cent, at t¯. However, the model indicates that the authorities should not stop at this point but should proceed to expand the money supply further. This result may seem somewhat counterintuitive, but it is a perfectly consistent outcome stemming from the underlying theory of the model. This initial undershooting in the money supply is the counterpart of the overshooting in inflation observed in Figure 2, and the overshoot in monetary growth after t¯ has to compensate for the earlier undershoot. This overshooting occurs basically from the way expectations of inflation are formed, and how they influence the current rate of inflation. When inflation is reduced at t0, expectations of inflation are revised downward only gradually. Even when inflation has been stabilized at a constant rate, the expected rate of inflation still continues to decline and consequently exerts independent downward pressure on prices. To hold the inflation rate constant in the face of this pressure, the authorities would have to expand monetary growth. Eventually the expected rate of inflation would itself stabilize at the same constant rate as observed inflation, and monetary growth could then be set equal to both.

Figure 3.
Figure 3.

Time Path of Growth of Money: Sudden Reduction in Inflation

Citation: IMF Staff Papers 1980, 002; 10.5089/9781451946864.024.A002

Taking a somewhat more realistic case, within the framework of this type of simulation exercise, assume that the authorities wish to bring the rate of inflation down from 1 per cent to a rate of zero per cent gradually in the course of a year. 59 The required monetary policy is indicated in Figure 4. Again, a shock would be necessary at t0, followed by a series of further smaller shocks until inflation equaled the desired rate. After t¯, as before, monetary growth would have to be increased and, later, overshooting, though smaller in degree, would still be a necessary outcome. When Figures 3 and 4 are put together they indicate that the more gradually the authorities wish to achieve their inflation target the smaller the monetary shocks that would be required. In the extreme case, monetary growth and inflation could be identical, with no overshooting or undershooting being necessary. This follows from the fact that, as the time horizon of the authorities is extended far enough, the model approaches the characteristics of the equilibrium model.

Figure 4.
Figure 4.

Time Path of Growth of Money: Gradual Reduction in Inflation

Citation: IMF Staff Papers 1980, 002; 10.5089/9781451946864.024.A002

It may be useful to summarize some of the implications of the simulation exercises that were conducted. First, changes in the rate of monetary expansion will not have an immediate impact on inflation, and initial undershooting of the rate of inflation will occur. Second, the time horizon of the authorities is a crucial element in determining the type of monetary policy that must be followed in attaining the objective. If the period over which the inflation target is to be achieved is sufficiently long, then, following a gradualist policy indicated by equilibrium models—that is, setting the rate of monetary growth equal to the desired, or target, rate of inflation 60 at each point in time—would be appropriate. On the other hand, if the objective has to be reached within a particular time period, say one year, then the policy cannot be a gradual one and must involve a single monetary shock or a series of monetary shocks. Furthermore, and perhaps more important, the authorities would have to change the direction of their policy as soon as the objective was reached. While in the context of the model here, these gyrations in policy pose no particular problem, in a more general framework, they could have undesirable effects on other variables in the system. Particular reference is made to the effects on real output and the balance of payments, which are usually the other major objectives of the authorities, that such a policy could have.

III. Conclusion

The basic purpose of this paper was to formulate a model capable of reproducing the empirical observation that the stock of real money balances tends to rise initially, or that the income velocity of money falls, when there is an increase in the nominal supply of money. This phenomenon, which results from the presence of lags between the change in money and the effects on inflation or nominal income, is generally not captured by standard dynamic models of money demand. Even in the short run these models adhere to the longer-run monetarist proposition that the growth in prices and nominal income will be equal to the growth of money, thereby leaving velocity or the stock of real money balances unaffected by changes, anticipated or unanticipated, in the rate of change of the nominal money stock.

Relying on the distinction between the effects of expected and unexpected (“shock”) changes in the supply of money, a simple monetary adjustment model was specified. This model, while remaining consistent in the long run with the conclusions of standard monetary models, allowed for real money balances to react positively in the short run to unexpected changes in the money supply. Tests of the model, based on a sample of 11 developing countries, provided empirical support for the basic hypothesis. Generally, it was found that only about one half of the change in money was reflected in an increase in prices in the initial quarter, and that it would take about five quarters before the stock of real money balances, or, if real income was growing, the income velocity of money, returned to a constant value.

These empirical tests have certain general implications for the conduct of monetary policy, and an attempt was made to bring these out through simulation experiments. If policy is geared toward the objective of reducing the rate of inflation, the only target considered in this paper, then the analysis here stresses the importance of the time horizon chosen to achieve this policy. Attempting to reach this target within a reasonable period of time, say, one year, could require fairly sharp variations in the growth of money. Further, it should be recognized that once the target rate of inflation is reached, the direction in the growth of money may have to be reversed. The simulations performed highlighted these aspects of the model. While these simulation experiments were conditional on a particular set of parameters, because of the similarity of these parameters across countries, it does seem that the results have some general validity and are not entirely country-specific. Furthermore, even allowing for small changes in the parameters would probably not cause any major qualitative changes in the policy conclusions reached.

Given the importance of the money demand function in the formulation of monetary policy in both developed and developing countries, and the fact that such policy is framed within a period of one to two years at most, close attention should be paid to the dynamics of the system. If the monetary authorities are to set some type of target growth for the money supply over a medium-term period, then it is quite possible that there would be times when the target was breached simply as a natural outcome of the underlying dynamics. This feature should be recognized and allowance made for this possibility. Also, without necessarily attempting to raise the “gradualism versus shock” debate again, the results here point toward a more shock-type policy if reaching the inflation target rapidly is the basic goal. However, the authorities would have to take into account the effects that this type of policy might have on other variables in the economy. Consideration of these effects may call for a more gradualist policy.

Finally, some caveats to the analysis here ought to be made. First, in deriving the theoretical model, a fairly simple expectations formation scheme for money supply growth was assumed, namely, that money holders expect the authorities to keep monetary growth in line with the growth in nominal demand. Since this nominal demand is essentially determined by an expected inflation rate generated by an internal adaptive process, information on the future conduct of monetary policy is not directly utilized. To the extent that expectations are formed differently, or that the model generating these expectations undergoes some type of change, the results could clearly be different. Second, it was also assumed that the supply of money was unaffected by inflation, and, further, did not affect real income, within the same quarter. This was necessary to stay within the confines of a single-equation model, since the specification of a complete model was not the aim of this paper. Furthermore, the focus here has been exclusively on the relationship between inflation and the growth of the money supply, an approach that would be strictly valid only for a closed economy, or one with a freely floating exchange rate. However, the analysis could be cast in terms of inflation and domestic credit expansion, which for open economies operating a fixed exchange rate system is certainly the more relevant relationship. In the framework of this paper, this would mean that the authorities would have to follow a credit policy that would be aimed at achieving a monetary growth consistent with their inflation target, taking into account the balance of payments effects that would be bound to occur. Provided that in the short run the changes in domestic credit were not completely offset by international reserve movements, one would expect that qualitatively the time path of domestic credit growth would be similar to that of overall monetary growth, as outlined here. Third, the inflationary process has been assumed to be essentially monetary in character, and thus monetary policy was considered as the only instrument for controlling inflation. Nonmonetary factors, such as wage changes and import price and exchange rate changes, can be independently important in affecting prices, and inflation can often be controlled by suitable fiscal, exchange rate, and incomes policies. Excluding all these probably leaves the analysis incomplete, but it should be stressed that this does not invalidate the essential nature of the dynamic relationship between money and inflation that has been investigated in this paper.

APPENDIX: Data Sources and Definitions

All the data used in this paper were taken from International Monetary Fund, International Financial Statistics (IFS). The basic series, covering the period 1962 to 1976, are centered at the middle of the quarter. The rate of inflation (Δlog Pt) and the growth of money (Δlog Mt) are thus centered at the beginning of the quarter. The definitions of the variables are as follows:

M = money plus quasi-money; IFS, lines 34 and 35

P = consumer price index; IFS, line 64

y = real income

For Argentina, India, Mexico, the Philippines, and Uruguay, gross domestic product at 1975 prices (IFS, line 99b. p) was used; for the remaining countries, nominal gross domestic product (IFS, line 99b) deflated by the consumer price index had to be used. Since these series on income are not available on a quarterly basis, the annual data (in real terms) was interpolated by a linear interpolation method.

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*

Mr. Khan, Assistant Chief of the Financial Studies Division of the Research Department, is a graduate of Columbia University and the London School of Economics.

The author is grateful to Tomas Balitño, Willem Buiter, Rudiger Dornbusch, Jacob Frenkel, David Laidler, and John Williamson for helpful comments on this paper.

1

See, for example, Friedman (1969, Ch. I) and also (1970). This result requires that the income elasticity of money demand be unity. If not, trend changes in velocity would also have to be taken into account.

2

See also Sjaastad (1972) for a discussion of the evidence.

3

See Laidler (1977) for an extensive review and references to the use of such models.

4

For a discussion of the role of the demand for money function in the operation of monetary policy, see Organization for Economic Cooperation and Development (1979) and, in relation to Fund stabilization programs, Robichek (1967).

6

Carr and Darby (1978) and Coats (1978) have proposed similar models for the United States.

8

In the interests of theoretical consistency, the scale variable should also be expected, or “permanent,” real income. As in practice it does not appear to matter greatly which particular scale variable is used, for simplicity current real income is utilized.

10

To the extent that interest rates paid on time and savings deposits are indexed to the rate of inflation, the elimination of Rde from the specification could result in some bias in the coefficient of expected inflation, a2. In the particular sample here, only Brazil has consistently had a system of indexation of interest rates, with Colombia and Costa Rica having only short-lived experiments with such schemes. The cases of these countries, and also those where the rates are somewhat flexible, are discussed in Section II.

11

It is generally necessary, as pointed out earlier, in the steady state to assume that the income elasticity of demand for money, al, is unity.

12

The focus here is only on discrete time adjustment models, including those that are approximated from continuous time models for estimation purposes. In theoretical continuous time models, one has to be more careful in distinguishing between the effects of a stepwise increase in the growth of money, and an increase in the stock of money, on the rate of inflation. The adjustment to the former is instantaneous, but may be delayed with respect to the latter.

13

See White (1978, p. 570). In order to derive this, it is assumed that Md is deflated by the actual, rather than the expected, price level.

14

This is because, from equation (9), (ΔlogPt)/(ΔlogMt)=1/λ>1.

15

The nominal-adjustment model, equation (8), continues to indicate an initial decline in real money balances even when the adaptive expectations formulation for inflation is introduced into the model.

16

In a rational expectations framework, Mussa (1975) has shown that a money supply process that contains regressive and extrapolative elements will also yield the path for real money balances shown in Figure 1.

17

This last approach has been discussed at length by Laidler (1979).

19

Assuming that real income is constant. If real income is increasing, the two will differ by the growth in real income times the income elasticity of the demand for money.

20

See Mussa (1976), Brillembourg (1979), and Buiter (1980) for a discussion of this issue.

21

For a discussion of such a model, see Aghevli and Khan (1978).

23

If there was two-way causality between inflation and the growth of money, a model along the lines proposed by Aghevli and Khan (1978) could be employed. Models such as the ones developed by Barro (1977) and (1978), Fernandez (1977), and Frenkel and Rodriguez (1977) could be considered if one was interested in the response of real income to monetary changes.

24

In the absence of direct observations on expectations, it is necessary to generate these on the basis of the past behavior of the series in question.

26

In the context of the Cagan model of hyperinflation, the process for the growth of the money supply has the form ΔlogMte=(β/(Δ+βL))ΔlogPt+ωt, where ω is a serially uncorrelated error term.

27

This implicitly assumes that people do not learn to change their way of forming expectations even in the face of unexpected monetary changes. They continue to believe that the authorities will conduct monetary policy in a manner that would validate their expectations.

28

Ignoring the effects of Δlog y and ΔΠ may certainly involve some mis-specification, but it does turn out that for the countries in the sample these variables generally displayed very little variation over time.

29

In a completely rational model, the expected rate of inflation would be equal to the actual rate, lit = Δlog Pt. For a discussion of some of the properties of the adaptive expectations scheme for inflation, see Khan (1977 a) and (1977 b).

30

Adjusted for the growth in real income times the income elasticity of money demand.

31

The constant a0 can be assumed to include the arguments that were contained in the variable k, defined earlier.

32

The equation is now written without imposing a negative sign on the coefficient of Δlog Pt, since this is to be freely estimated.

34

Actually, it is likely that this variable would have an inverted U-shape. As monetization occurs, this ratio would rise, and, as the economy became more sophisticated financially and alternative assets were available, the ratio of money to income would tend to fall.

35

Recalling the previous comments on the difficulty in simply comparing this ratio across countries. Two countries with exactly the same ratio of money to income may well be substantially different in terms of financial development.

37

Interest rates were freed in Brazil only in 1975, Argentina in mid-1977, Uruguay in 1977, and Malaysia in 1978. Rates in Brazil have been indexed to the rate of inflation since 1964, but even so real rates have been consistently negative. Interest rates were partially indexed in Colombia during 1972–74, and in Costa Rica in 1974.

38

Assuming that foreign assets are zero. If not, an adjustment would have to be made, but broad money would still be the relevant variable. The equation was also estimated with money narrowly defined, and the results of this exercise are available upon request from the author, whose address is Research Department, International Monetary Fund, Washington, D.C. 20431.

39

Were this parameter to turn out to be equal to unity, the model would reduce to the standard partial-adjustment equation (6).

40

Since 1 – γ is significantly different from zero at the 5 per cent level. For Malaysia, the use of the current value of the rate of growth of money in the equation yielded an estimate of the coefficient of adjustment, λ, that was negative. Therefore, the results reported for this country in Table 2 were obtained by using the one-quarter lagged value of the growth in money.

41

Even though the point estimates are greater than zero.

43

Examining the time-series properties of the errors in the model through the methods of Box and Jenkins (1970), and then imposing a similar time-series structure on the model, may be the appropriate strategy to follow.

44

In fact, if there were significant economies of scale in the holding of money because of the existence of financial alternatives, the elasticity could easily be less than unity. See Laidler (1977).

46

Since |(λa2+(1γ))|>|(1γ)|, it can be seen that ya2 is negative.

47

It should be noted that this elasticity is not independent of the unit of observation. The elasticity shown in Table 2 is evaluated at the sample mean of the quarterly rates of inflation.

49

Higher-order autocorrelation may still, of course, be present.

50

Whether the approach is oscillatory or not depends on

{(1β)+(1λ)+β(λa2+(1γ))}4{(1β)(1λ)+β(λa2+(1γ))}

Except in the Dominican Republic and Thailand, the movement to equilibrium displayed (damped) oscillations. The parameters in these two countries yielded two real roots, rather than a pair of complex conjugates.

51

Needless to say, since the interest here is in the dynamic path of inflation, the lagged values of the endogenous variables were themselves generated by the system.

52

The average for the parameter a2 was calculated excluding the abnormally large estimates obtained for India and Malaysia.

53

The stock of real money balances would grow at a constant rate, while the rate of inflation would be lowered by a constant factor.

54

Given the growth rates of nominal money and prices, the behavior of real money balances is readily calculated.

55

This lag is very close to the one reported by Von Furstenberg and White (1980), and somewhat shorter than the lag of two years obtained by Vogel (1974).

56

The oscillations observed were to be expected from the formal stability analysis.

57

A reduction in the growth of money supply is probably the more realistic type of policy.

58

It is not exact because the coefficients of the lagged endogenous variables in the reduced form differ between the two cases. Only in the initial period are the simulations an exact transposition. The reduced form for nominal money growth has the form Δlog Mt = −(A3(L)/ A4(L)) log yt + (Det / A4(L)) Δlog Pt.

59

It is assumed that the authorities wish to reduce the rate of inflation by 0.25 per cent in each period.

60

And, if necessary, the trend change in velocity.

IMF Staff papers: Volume 27 No. 2
Author: International Monetary Fund. Research Dept.