Government Deficits and the Inflationary Process in Developing Countries
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

While monetarist explanations of inflation have tended to follow the classic study of hyperinflation by Cagan (1956) in stressing that it is changes in money that in some sense cause changes in prices, there have recently been a number of papers questioning the unidirectional nature of the Cagan model. For example, after re-examining the hyperinflation cases studied by Cagan (1956), Sargent and Wallace (1973) concluded that it was more appropriate to view the causation as running both ways.1 Similar conclusions about two-way causality between money and prices during periods of hyperinflation were also reached by Frenkel (1977) and Jacobs (1977).2

Abstract

While monetarist explanations of inflation have tended to follow the classic study of hyperinflation by Cagan (1956) in stressing that it is changes in money that in some sense cause changes in prices, there have recently been a number of papers questioning the unidirectional nature of the Cagan model. For example, after re-examining the hyperinflation cases studied by Cagan (1956), Sargent and Wallace (1973) concluded that it was more appropriate to view the causation as running both ways.1 Similar conclusions about two-way causality between money and prices during periods of hyperinflation were also reached by Frenkel (1977) and Jacobs (1977).2

While monetarist explanations of inflation have tended to follow the classic study of hyperinflation by Cagan (1956) in stressing that it is changes in money that in some sense cause changes in prices, there have recently been a number of papers questioning the unidirectional nature of the Cagan model. For example, after re-examining the hyperinflation cases studied by Cagan (1956), Sargent and Wallace (1973) concluded that it was more appropriate to view the causation as running both ways.1 Similar conclusions about two-way causality between money and prices during periods of hyperinflation were also reached by Frenkel (1977) and Jacobs (1977).2

Each of the later studies contains a common explanation for increases in the money supply in response to inflation—namely, the role of the government’s fiscal operations. Attempts by the government to extract real resources at a faster rate than was sustainable at a given rate of inflation would result in increases in the money supply and further inflation.3 In a recent paper by Jacobs (1977), this hypothesis also appears to be empirically verified for a number of countries that have had hyperinflationary episodes.

Until very recently, studies concerned with the monetary aspects of the inflationary process in developing countries generally took the Cagan model as their basic foundation, that is, they treated the money supply as strictly exogenous (or policy determined). Inflation is, therefore, essentially caused by the expansion in the money supply and there is no feedback.4 Lately, however, some papers, both theoretical and empirical, have recognized that the expansion in the money supply may not be independent of inflation. They have explicitly introduced the idea that inflation results in a widening of fiscal deficits financed through the banking system (in particular, by the central bank), leading to further increases in the money supply and further increases in prices. This self-perpetuating process was first formalized by Olivera (1967) and was later applied empirically by Dutton (1971) and Aghevli and Khan (1977) in order to explain the episodes of high inflation experienced by Argentina and Indonesia, respectively.

The purpose of this paper is to examine the relationship between increases in the money supply and inflation in four developing countries. We first show that these two variables are linked in a two-way relationship, and we then design and apply a model that explicitly introduces the link in the form of the reactions of the government fiscal deficit to inflation. The lags in government expenditures and revenues, on which the link basically hinges, are not imposed a priori but are estimated within the model. Furthermore, while previous applications of this type of model have been restricted to countries experiencing rapid inflation,5 we consider a group of countries that have had considerable variety in their experience with inflation. We postulate that while the basic phenomenon may well be accentuated in countries with high inflation rates, the model would be equally applicable to developing countries with moderate rates of inflation.

The countries we examine are Brazil, Colombia, the Dominican Republic, and Thailand. The countries were chosen primarily with a view to the range of inflation rates they experienced,6 although the availability of data was obviously also an important consideration. The four countries can be considered reasonably representative of developing countries in general, since they all have limited domestic capital markets and tax bases. The period of study is 1961-74, except for Brazil, where we were limited, by the availability of data on government revenues and expenditures, to the period 1964-74.

The outline of the paper is as follows: in Section I, we examine the relationship between inflation and the growth of the money supply in our sample of countries, and then we describe a dynamic model that incorporates this relationship. Section II contains the results obtained from estimating this model for each of the four countries. Section III contains a simulation experiment conducted to determine whether the model is able to capture the historical behavior of prices in these countries. Questions regarding the dynamic stability of the estimated models are considered in Section IV. Our overall results, and their implications for policy, are summarized in the concluding section. Appendices I and II discuss more formally the causality tests used and the theoretical stability conditions of the model, respectively. Appendix III contains the sources and the definitions of the data used.

I. A Model of the Inflationary Process

Before formulating the model, we examine the nature and extent of the relationship between the rate of growth of the money supply and changes in the price level in the countries we have selected. A casual comparison of the averages of the annual rates of inflation and growth in the money supply for each of the countries (presented in Table 1) would seem to support the contention that there is a positive relationship between the two variables. Clearly, this cursory examination of average growth rates is merely suggestive and cannot be interpreted as constituting the presence of a significant relationship (let alone a causal relationship) between the two variables across the four countries. In order to examine the relationship between the two variables more rigorously, we subjected them to the formal tests of causality that have been developed by Pierce (1977) and Pierce and Haugh (1977). Causality in the framework of these tests is interpreted in the sense of leads and lags between the two series. A series, say prices, is viewed as “causing” the other series, money, if it leads money over time. If, on the other hand, prices consistently lag behind money, then money “causes” prices. Two-way causality, or feedback, is the reflection of one of the series leading, as well as lagging behind, the other series.

Table 1.

Averages of Annual Inflation Rates and Growth of Money Supply

(In per cent)

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The basic procedure of these tests is fairly simple and can be outlined briefly. We first put each of the two series through a filter to remove any existing autocorrelation. This prefiltering removes the effect on a series of its own past values and converts it into one with white-noise properties.7 Cross-correlating the two series after the pre-whitening tests whether they are independent or related. Since the cross-correlation is performed with both the lagged and the future values of the transformed series, this test can also give some indication of the direction of causality. For example, a significant value for the correlation coefficient between the current price level and the current money supply would lead one to reject the null hypothesis that the two series are independent. If only the lagged values of the rate of growth of the money supply were significantly correlated with the current values of the price level, one could argue that increases in money cause prices. On the other hand, if only the future values of the rate of growth of the money supply were significantly correlated with the current value of inflation, one could interpret that to imply that inflation causes increases in the money supply. Based on this reasoning, a significant correlation between current inflation and both the past and future values of the growth of the money supply would imply two-way causality.8 While the complete results are presented in Appendix I, here we present in Table 2 the cross-correlation coefficients between the current rate of inflation and the rate of growth of the money supply in the previous quarter, the current quarter, and the first subsequent quarter.9

While in no instance is the contemporaneous relationship statistically significant,10 it is fairly clear that current inflation is correlated with the past value of the rate of growth of money, as well as with the future value, in three of the four countries. For the Dominican Republic, both the lags and leads were longer, but there was evidence pointing to the existence of a two-way relationship between the two series.11

Table 2.

Cross-correlation Coefficients Between Inflation and Rate of Growth of Money Supply1

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The asterisk (*) indicates that the estimated coefficient is twice the value of its standard error. See Appendix I.

Having established some evidence on the nature of the relationship, we now proceed to design a model that explicitly introduces the link that is postulated to explain this two-way causality. The basic model involves four stochastic equations explaining, respectively, the price level, government expenditures, government revenues, and the supply of money, and one definitional equation dealing with the formation of expectations regarding inflation. Each of these equations will be discussed in turn.

1. determination of prices

In order to determine prices, we first formulate the demand for real money balances (defined as broad money balances) as a function of the level of real income and the opportunity cost of holding assets in the form of money. In most developing countries, where a broad range of financial assets does not exist as an alternative to money, the substitution between money and physical assets becomes more important. Thus, for money holders in developing countries, the relevant opportunity cost is the rate of return on physical assets or goods—namely, the expected rate of inflation—and the rate of return on financial assets can be ignored.

Therefore the demand for real money balances can be specified in log-linear terms as:

log(M/P)tD=a0+a1logYta2πta1,a2>0(1)

where

M = stock of nominal money balances

P = price level

Y = level of real income

π = expected rate of inflation

The superscript “D” signifies demand.

The function has been specified in logarithmic form essentially for reasons of convenience, that is, because elasticities can be obtained directly. Furthermore, there is empirical evidence that this is also a preferred functional form.12

The actual stock of real money balances is assumed to adjust proportionally to the difference between the demand for real money balances and the actual stock in the previous period,13

Δlog(M/P)t=λ[log(M/P)tDlog(M/P)t1](2)

where λ denotes the coefficient of adjustment, 1 > λ > 0.

The expected rate of inflation is assumed to be generated by the adaptive-expectations or error-learning mechanism proposed by Cagan (1956),

Δπt=β[ΔlogPtπt1]1>β>0(3)

where β denotes the coefficient of expectations and Δlog Pt denotes the current rate of inflation.14

Substituting equation (1) into (2) and solving for the level of real money balances, we obtain

log(M/P)t=λa0+λa1logYtλa2πt+(1λ)log(M/P)t1(4)

Since we are interested in the price level, we can solve to obtain

logPt=λa0λa1logYt+λa2πt(1λ)log(M/P)t1+logMt(4)

2. government sector

Assuming that the “desired” real expenditures of the government are related to the level of real income,15 we can specify the function in logarithms as

log(G/P)tD=g0+gtlogYtgt>0(5)

where G denotes nominal government expenditures, and the other variables are defined as before. It may be reasonable to assume that, in the long run, the government wishes to increase its expenditure proportionately with the growth of real income and, therefore, we would expect gi the real income elasticity of government expenditure, to be equal to unity.16

Actual real expenditures are specified as adjusting to the difference between the desired real expenditures and actual real expenditures in the previous period,

Δlog(G/P)t=γ[log(G/P)tDlog(G/P)t1](6)

where γ is the coefficient of adjustment, 1 > γ > 0.

The particular assumption contained in equation (6) is that the government attempts to keep its real expenditures constant in the face of an increase in the price level.

This adjustment function could also be specified in terms of nominal government expenditures as

ΔlogGt=γ[logGtDGt1](6)

which states that the government adjusts its nominal expenditures rather than its real expenditures, as given by equation (6). If we convert equation (6) into nominal terms, that is,

ΔlogGt=γ[logGtDlogGt1]+(1γ)ΔlogPt(6)

then as γ → 1, the two adjustment functions become identical. In this case, specifying the adjustment of expenditures in real or nominal terms makes no difference.

Substituting equation (5) into (6), we obtain a solution for the level of real expenditures:17

log(G/P)t=γg0+γg1logYt+(1γ)log(G/P)t1(7)

From this equation, one can obtain the mean or average time lag in the adjustment of real government expenditures. This lag is defined simply as (1 - γ)/γ.

Equation (7) can be written in terms of nominal expenditures,

logGt=γg0+γg1logYt+(1γ)log(G/P)t1+logPt(8)

Desired nominal revenues of the government are assumed to be functionally related to the level of nominal income,

logRtD=t0+t1(logYt+logPt)t1>0(9)

where R denotes the level of nominal revenue and other variables are defined as before. We expect that the elasticity of revenues, t1, will be positive.18

Actual revenues adjust to the difference between desired revenue and the actual revenue obtained in the previous period,

ΔlogRt=τ[logRtDlogRt1](10)

where τ is the coefficient of adjustment, 1 > τ > 0.

Substituting equation (9) into (10), we obtain an equation for nominal revenues,19

logRt=τt0+τt1(logYt+logPt)+(1τ)logRt1(11)

If, in the long run, government revenues grow at the same rate as nominal income, we would expect t1 to be equal to unity.20 It should be noted that, in this framework, even if we start from a balanced budget, as nominal income rises, we will observe an increasing divergence between expenditure and revenue if the former adjusts faster. That is, the nominal deficit will be a function of the increase in the price level, provided that τ < γ, even though t1 = g1.

There are plausible reasons for expecting government expenditures in developing countries to adjust faster than revenues to nominal income increases arising from inflation. Even if governments fully recognize the need to restrain expenditures during periods of inflation, they find it difficult to reduce their commitments in real terms. On the other hand, in contrast to the situation in most developed countries, where nominal revenues often more than keep pace with price increases, in developing countries they lag substantially behind.21 The contrast arises both because of low nominal income elasticities of tax systems and long lags in tax collection in developing countries. For income taxes, the income to which tax payments in a given period relate is typically less current than in developed countries. In any event, tax systems in developing countries depend rather heavily on indirect taxes and, in particular, on foreign trade taxes. The markedly lesser progressivity of indirect taxes compared with direct taxes is well known. Further, indirect taxes in developing countries are often specific, and even when they are ad valorem, the adjustment of base values for some of these taxes is not frequent enough to keep pace with inflation. For example, the adjustment of the exchange rate to domestic inflation is sluggish, which has adverse effects on revenue collection from trade taxes, and the prices of some basic products, such as fuel and tobacco, may be controlled. Finally, long delays also occur in the periodic reassessment of values needed for property taxes.

3. supply of money

The supply of money, M, can be multiplicatively related through the money multiplier, m, to the stock of high-powered money, H,

Mt=mtHt(12)

Changes in high-powered money can occur through changes in international reserves, changes in the central bank’s claims on the government (ΔCG), and changes in the central bank’s claims on commercial banks and the private sector. If we consolidate changes in international reserves and changes in claims on the private sector into one composite variable (ΔOA), we can write

ΔHt=ΔCGt+ΔOAt(13)

or

Ht=ΔCGt+ΔOAt+Ht1(13)

If changes in central bank claims on the government are simply a reflection of the fiscal deficit, equation (13′) can be specified as

Ht=GtRt+Et(14)

where Et = ΔOAt + Ht–1. An increase in the fiscal deficit is thus assumed to result in an equal change in the stock of high-powered money. This would be true to the extent that government deficits were financed by borrowing from the central bank or using cash balances held with the central bank, by borrowing abroad, or by borrowing from commercial banks with the banks immediately replenishing reserves by recourse to the central bank. On the other hand, when deficits are financed by borrowing from commercial banks without the latter discounting at the central bank and by borrowing from the nonbank private sector, this assumption would not be valid. Despite recent progress in many developing countries, the scope for the latter types of borrowing remains limited in these countries, so that our previous assumptions can be regarded as being generally satisfied. The money supply equation (12) can then be written as:

Mt=mt(GtRt+Et)(15)

The use of identity (15) with our other equations would make the model nonlinear in variables; and since it is more convenient to work with linear models from an estimation point of view,22 as well as when dealing with questions of dynamic stability both theoretically and empirically, we approximated equation (15) by a relationship linear in logarithms, which was obtained by linearizing about sample means.23 The resulting linear equation is

logMt=logmt+k0+k1logGtk2logRt+k3logEt(16)

where the parameters k0, k1, k2, and k3 are all functions of the sample means of log G, log R, and log E.24 The values of the parameters in equation (16) can be obtained by calculating the sample means of the logarithms of the variables G, R, and E, and then deriving them, or by estimating them within the model along with the other parameters.

4. complete model

The complete model can be outlined as follows:

logPt=λa0λa1logYt+λa2πt(1λ)log(M/P)t1+logMt(17)
logGt=γg0+γg1logYt+(1γ)log(G/P)t1+logPt(18)
logRt=τt0+τt1(logYt+logPt)+(1τ)logRt1(19)
logMt=logmt+k0+k1logGtk2logRt+k3logEt(20)
πt=βΔlogPt+(1β)πt1(21)

The variables are defined again for convenience:

Endogenous

P = domestic price level

G = nominal government expenditures

R = nominal government revenues

M = stock of money

π = expected rate of inflation

Exogenous

Y = level of real income

m = money multiplier

E = residual item containing the change in international reserves, changes in central bank claims on the private sector, the stock of high-powered money in the previous period, and the error involved in ΔCG being different from G–R.

The inflationary process inherent in this system of equations can be briefly outlined. Assume initially that there is an increase in the money supply through, say, an increase in E in equation (2). This would raise the price level from equation (17),25 and result in increases in both government expenditures (equation (18)) and revenues (equation (19)). If the increase in expenditures is greater than that in revenues, the fiscal deficit will increase. This will cause the money supply to increase further, and the process will be repeated.26 One of the interesting questions that emerges is whether this process is stable, in the sense that the self-perpetuating nature of price increases tends to produce a stable rate of inflation or an accelerating rate. This issue will be explored in Section IV.

II. Estimates of Structural Model

Since the basic model given by equations (17)(21) is linear, we estimated it by three-stage least squares (3SLS). This method allows one to take into account all a priori restrictions inherent in the specification. The estimation results are shown in Table 3, where we report the values of the estimated coefficients (in composite form) and the ratios of the coefficients to their respective standard errors in parentheses.27 For each equation, we also present the value of the coefficient of determination and the standard error of the estimated equation, while recognizing that the formal meaning of these goodness-of-fit statistics is unclear when using 3SLS.28

Table 3.

Structural Equation Estimates1

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Figures in parentheses are f-ratios.

Table 4.

Individual Parameter Estimates

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In Table 4, we show the values of the individual parameters. These include the values of the adjustment coefficients and the equilibrium, or long-run, values of the relevant elasticities. The coefficients in the money supply equation are, of course, approximately equal to the coefficients one would obtain in the linearization process.29 Moreover, the value of coefficient β was fixed based on the criterion of maximizing the likelihood function of equation (17) when it was estimated singly for each country using the ordinary-least-squares (OLS) method of estimation.

Considering the price equation, we find that increases in real income would lower the price level while expectations of increased inflation would obviously increase it in all cases. Also, the higher the previous quarter’s stock of real money balances, the lower would be the current price level. This results, of course, from the original money demand function, which implies that the higher the stock of real balances in the previous quarter relative to a given demand for these balances, the smaller will be the required adjustment and, thus, the smaller will be the price reduction. All estimated coefficients are significant at the 5 per cent confidence level except the coefficients of expected inflation for the Dominican Republic and Thailand, which, nevertheless, have the correct sign and are by no means negligible. The long-run income elasticities30 of the demand for money are, in general, higher than unity (See Table 4.) except for Brazil.31 The higher value of the income elasticity for the Dominican Republic probably reflects the rapid monetization that is taking place in that economy. In general, the specification of the demand for money seems reasonable, as the fit of the price equation is very good for all the countries in the sample.

The effect of income on government expenditures and revenues in the short run is positive and significant at the 1 per cent confidence level in all the countries under study. The adjustment coefficient of government expenditures, γ, is not significantly different from unity (The calculated values are all above 0.9.) in all of the four countries, which confirms the hypothesis that nominal government expenditures are adjusted upward almost automatically to keep pace with inflation.32 As for τ, since our revenue data have not been adjusted to exclude the revenue effects of discretionary tax charges, it is possible that the calculated value of τ is biased upward. It is interesting to note that the value of τ is smaller than the corresponding value of γ and is also inversely related to the average value of inflation in the four countries.33 That is, those countries that had a more sluggish system of tax collection experienced a higher rate of inflation, as postulated in our theoretical formulation.34 The long-run income elasticities of government expenditure and revenue are also not significantly different from unity, as expected, implying that, in the long run, both government expenditures and revenues would move proportionately with inflation.

Expectations of inflation appear to be revised fairly rapidly in all cases, as the estimated coefficient is close to unity. For Brazil, the estimated value is substantially larger than the value obtained by Khan (1977 b); this result is presumably a reflection of further experience of the public with continuing inflation.35 It can be assumed that expectations are revised faster, the longer the inflationary episode.36

As for the mean time lags, the results of the price equation indicate that real money balances in the Dominican Republic take over seven quarters to adjust to the difference between the demand for money and the actual money stock in the previous quarter37 (See Table 5.). While the average lags in adjustment of real money balances are about two quarters in both Brazil and Thailand, the lag in Colombia is about one quarter. The average lags in the adjustment of real government expenditures to the difference between desired and actual real expenditures in the previous period turn out to be quite close to zero in all four countries. This implies that, for all practical purposes, government expenditures were adjusted promptly in order to keep pace with increases in income and prices. On the other hand, the lags in the adjustment of nominal revenues are generally longer.38 The average rate of inflation prevailing over the period of our study varies directly with these lags. This result is a confirmation of one of our original hypotheses, namely, that the longer is the lag in revenue, the higher will be the inflation. It is interesting to note that this does not depend at all on the particular rate of inflation in the country, but appears to be perfectly general. Furthermore, since the average lags in the adjustment of revenues are longer than the corresponding government expenditure lags, this constitutes empirical verification of our view that the main reason for the existence of two-way causality between money and prices hinges crucially on a shorter time lag in the adjustment of nominal (and real) government expenditures than in the adjustment of revenues.

Table 5.

Average Time Lags

(In quarters)

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The average lags in the revision of expectations of inflation are very short. These lags are also very similar across countries that have had very dissimilar experiences with inflation.

III. Simulation Results

In order to simulate the behavior of the endogenous variables over the same period, we constructed a dynamic simulation in which the values of the lagged endogenous variables were themselves generated by the simulation process. The structural model can be written in matrix notation as follows:

AXt+BXt1+CZt=Utt=1,,T(22)

where Xt, is a vector of endogenous variables, Xt–1 is a vector of lagged endogenous variables, and Zt is a vector of exogenous variables, and Ut, is a vector of error terms. The A and B are matrices of parameters of endogenous and lagged endogenous variables, and C is a matrix of parameters of exogenous variables. The reduced form of equation (22) is

Xt=ΩXt1+ΓZt+Vt(23)

where Ω = – A-1B, Γ = – A-1C, and V = – A-1U.

In the dynamic simulation, we use equation (23) in the following way:

X^t=Ω^X^t1+Γ^Zt(23)

where the “^” indicates the estimated, or predicted, values of the parameters and variables.

This particular simulation experiment is used primarily to test the goodness of fit of the model as a whole. The coefficients of determination we presented earlier in Table 3 (Section II), are applicable only to individual equations. The comparison between the actual and simulated values of the endogenous variables is thus, in a sense, comparable to an R2 calculated for the complete model. However, since the simulation experiment is dynamic, it is a more stringent goodness-of-fit test than a simulation in which the vector of lagged endogenous variables is assumed to be predetermined. This is because prediction errors in the dynamic simulation case can quite easily be compounded, namely because a large error in any period between the actual and simulated values of the endogenous variables, also affects the size of the error in the following period. By allowing the lagged endogenous variables to be determined within the simulation, the dynamic simulation can also be used as a test of stability of the estimated model.

While we worked with all of the five endogenous variables in the simulation, we have chosen to present the results for only the most relevant variable, that is, the price level. The results of the simulations for the four countries, as well as the actual values of the price indices, are shown in Charts 1 to 4.

In these charts, we find that the simulated path of the price index tends to track the general movement of the actual price index over the sample period fairly well in all cases. Large errors, however, appear on several occasions, and we do find that the actual price index over the sample period in each of the countries tends to be much smoother than does the simulated price index. The correlation coefficients between the actual and simulated values are shown in Table 6. For Brazil and Colombia, these correlation coefficients indicate that the relationship between the actual and simulated price indices is a close one. While the correlation between these two series for the Dominican Republic and Thailand is also high, it is not quite as high as for the other two countries.

On average, however, the models do well in tracking the historical behavior of prices. For example, the average rates of inflation obtained by the simulated price indices for the four countries are very close to the actual average rates of inflation.39

Chart 1.

Brazil: Consumer Price Index, Third Quarter, 1964-Fourth Quarter, 1974

(1970 = 100)

A01ct01
Chart 2.

Colombia: Consumer Price Index, Third Quarter, 1961-Fourth Quarter, 1974

(1970 = 100)

A01ct02
Chart 3.

Dominican Republic: Consumer Price Index, Third Quarter, 1961-Fourth Quarter, 1974

(1970 = 100)

A01ct03
Chart 4.

Thailand: Consumer Price Index, Third Quarter, 1961—Fourth Quarter, 1974

(1970 = 100)

A01ct04
Table 6.

Correlation Between Actual and Simulated Prices

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IV. Dynamic Stability of the Model

In order to analyze the dynamic stability of the model, we first rewrite the system in terms of the deviations of the endogenous variables from their steady-state values. Let (*) denote the deviation of the endogenous variable from its corresponding steady-state value. The homogeneous part of the system of linear difference equations (17) to (21) can then be rewritten as follows:40

log(M/P)t*=λa2πt*+(1λ)log(M/P)t1*(17)
log(G/P)t*=(1γ)log(G/P)t1*(18)
log(R/P)t*=(1τ)logRt1*(1τ)ΔlogPt*(19)
log(M/P)t*=k1log(G/P)t*k2log(R/P)t*(1k1k2)ΔlogPt*+(1k1k2)log(M/P)t1*(20)
πt*=βΔlogPt*+(1β)πt1*(21)

Since the lagged deviations of the inflation rate from its steady-state value (i.e., ΔlogPt1* do not appear in the above system, we can eliminate the logPt* term by substituting from equation (21′) into equations (19′) and (20′). The resulting first-order system of difference equations can be written in matrix notation:

Xt(4×1)=D(4×4)Xt1(4×1)(24)
whereXt=[log(M/P)t*log(G/P)t*log(R/P)t*πt*](25)

The elements of the matrix D are defined in Appendix II. It can be shown that a necessary condition for the model to be stable is

1βa2λ+τt1k2k1>0(26)

This stability condition differs from the one derived from the Cagan (1956) model. In our notation, the necessary and sufficient condition for stability in the Cagan model is

1βa2λ>0(27)

A comparison of expressions (26) and (27) reveals that, in general, incorporating the feedback of inflation into the money supply process will result in more instability. For lower values of τ (i.e., slower adjustment of government revenue), it is quite conceivable that the expression (26) is negative while expression (27) is positive.

Clearly, on the basis of the assumed signs of the parameters, we cannot unambiguously determine the sign of expression (26). We have, therefore, evaluated expression (26) by substituting the numerical values of the estimated parameters for all four countries (See Table 7.). In all four cases, the necessary condition is satisfied. We now proceed to examine the necessary and sufficient conditions for the stability of the dynamic system by considering the eigensystem of the set of first-order difference equation (24). Let θj be the jth eigenvalue of the matrix D; then the necessary and sufficient condition for stability is that the absolute values of the moduli of all eigenvalues | θj | are less than unity, that is,41

|θj|<1forj=1,,r(28)

The model has two real eigenvalues and two complex ones, which are complex conjugates of each other. In Table 8, we present these eigenvalues for each country, as well as their respective moduli. In all four of the countries, the moduli are less than unity, thus confirming our conclusion drawn on the basis of an examination of only the necessary condition shown in Table 7.

The use of only eigenvalues to determine the stability of the dynamic system has one particular drawback, in that one cannot isolate the cause of instability, should this be indicated. In other words, we are not able to ascertain the sensitivity of the model to changes in a particular parameter, or a set of parameters. A numerical technique, however, enables one to gauge the relative importance of parameters in determining the stability or instability of the system by taking the (numerical) derivative of the calculated eigenvalues with respect to the parameters of the system.42 For example, if ηi is the ith parameter in the system and τj is the jth eigenvalue, we calculate the partial derivative

|θ^j|η^i
Table 7.

Estimated Values of Determinant, |D|

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Table 8.

Eigenvalues of Estimated Model

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By examining the value of this partial derivative, we can detect the influence on the stability of the estimated model of marginal changes in any of the estimated parameters.

In our particular case, we have seen from Table 8 that there are 3 moduli;43 and in our system, we have 11 parameters (excluding the constants). We therefore calculated the partial derivatives of these moduli with respect to each of the parameters in the system. In Table 9, we report only those partial derivatives of the three moduli with respect to the parameters that have at least one non-zero value. For example, the partial derivatives of each of the moduli with respect to the parameters a1, g1, t1, and k3 are zero because these parameters are the coefficients of exogenous variables.

Table 9.

Sensitivity Matrix of Moduli with Respect to Estimated Parameters

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In examining the results shown in Table 9, we find that the stability of the model depends essentially on the values of the parameters λ and β in the cases of Brazil and the Dominican Republic, and the parameters τ and β in the cases of Colombia and Thailand, rather than on the small changes of the other parameters.44

Thus, having identified to some extent the key parameters relevant for establishing the question of dynamic stability, we can now examine the degree of correspondence between the value of the determinant (the necessary condition) and the calculated values of the characteristic roots (representing both the necessary and sufficient conditions). This is done by conducting a simple sensitivity experiment where the values of the parameters are allowed to vary with changes in the value of the determinant of the structural model, and then the eigenvalues corresponding to this new value of the determinant are calculated. In essence, our procedure is the following: we take the necessary conditions for stability contained in equation (26) and vary those parameters that have a crucial influence on the stability of the model as determined in Table 9. In other words, we vary the composite coefficient βa2λ and the parameter τ. For purposes of this exercise, we fixed t1 and g1 to be equal to unity and k1 = 0.36 and k2 = 0.32. These values were chosen so that they would correspond approximately to the average of the values estimated for the four countries.

The value of the product of βa2λ was varied from 0.0 to 1.0 in intervals of 0.1.45 For each value of this product, the revenue coefficient of adjustment τ was allowed to vary from 0.3 to 1.0 in steps of 0.1.46 For each of these combinations, we calculated the eigenvalues of the model to establish whether the sufficient conditions for stability were also satisfied.

The correspondence between the necessary condition and the values of the moduli is shown in Table 10. In this table, the entry “s” indicates that the moduli are less than unity (i.e., the model is stable), while the entry “u” indicates that the model is unstable. The asterisk implies that the necessary condition is greater than zero, indicating that the necessary condition for stability is satisfied, while the calculated eigenvalues imply that the model is unstable. The asterisk, therefore, indicates a conflict between stability determined only by observing the necessary condition and that determined by calculating both the necessary and sufficient conditions. Apart from the borderline cases shown in Table 10, it is clear that, when one is evaluating the dynamic stability of this model, looking only at the necessary condition would not result in too serious an error.

It is also clear from Table 10 that an increase in τ will stabilize the system by reducing the collection lag of revenue, leading to smaller government deficits for any given inflation rate. On the other hand, any increase in the parameter βa2λ will reduce the demand for real balances for a given rate of inflation, leading to further inflationary pressures and thus destabilizing the system.47 This implies that any values of βa2λ larger than unity will always result in instability regardless of the value of τ, the revenue collection lag parameter. It should be pointed out, however, that in this paper we have concentrated on the destabilizing nature of the process of inflationary finance. In a more general framework, we must take into account two other important factors that tend to stabilize the system. First, it is likely that the authorities would make a conscious effort to reduce credit in the face of rapid inflation. Second, the openness of the economy would provide another leakage, which would tend to stabilize prices at the level of international prices through the movements of capital as well as direct goods arbitrage.48

V. Conclusion

The purpose of this paper has been to extend to a sample of developing countries with a variety of inflationary experiences some of the recent studies examining the two-way causality between money and prices in high-inflation countries.49 The basic hypothesis was that, while government expenditures rise concomitantly with inflation, government revenues would tend to fall behind in real terms owing to collection lags. The financing of this inflation-induced deficit would then increase the money supply and generate further inflation. Thus, the increase in the supply of money would both cause inflation and would be the result thereof, a phenomenon that was confirmed by our formal tests of causality between the two variables.

Table 10.

Correspondence Between Values of Determinant and Modulus1

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The s indicates that the moduli are less than unity (i.e., the model is stable).

The u indicates that the moduli are greater than unity (i.e., the model is unstable).

An asterisk indicates that 1 - βa2λ + τt1k2 - k1, > 0.

A simple dynamic model was constructed containing the main elements of the above process,50 and it was demonstrated that it had sensible properties, in the short run as well as the long run. The model was then estimated for a heterogeneous group of developing countries: Brazil, Colombia, the Dominican Republic, and Thailand. In general, the results supported our theoretical expectations regarding the values of the parameters, the behavior of the price level over the period of study, and the dynamic stability properties of the estimated model.

In particular, we found that mean lags in the adjustment of expenditures were negligible in all cases, while the corresponding revenue lags were substantially longer, leading to higher deficits in more inflationary periods. Moreover, the countries with longer revenue lags were also the ones that had higher inflation rates over the sample period. Thus, the basic theory is consistent with our observation of differing rates of inflation in the four countries.

The conclusion of the paper, thus, is that, in developing countries, fiscal policy tends to be automatically destabilizing, the principal built-in destabilizer being the various revenue lags indicated earlier. A passive fiscal policy in times of inflation is, therefore, hazardous. The control of inflation requires deliberate action by budgetary authorities to eliminate budgetary deficits or even to achieve surpluses, if the burden on monetary policy is not to be excessive. In any event, these countries should also give priority to reshaping the revenue system so as to mitigate the various lags. It is beyond the scope of this paper to review the efficacy of different methods of achieving this, since these methods lie generally in the area of tax administration. It would seem, however, that the possibility should be explored of indexing the nominal value of certain taxes, particularly personal and corporate income taxes and property taxes, which tend to have the longest lags.51

APPENDICES

I. Causality Tests

Following the methodology of Sims (1972) and Pierce (1977), the two variables, money and the price level, for each of the four countries, were pre-whitened prior to testing in order to remove the influence of their respective past values on their current values. For purposes of the tests, we utilized the filter proposed by Sims (1972), which resulted in the following transformed series:

mt = (1-0.75L)2 log Mt

pt = (1-0.75L)2 log Pt

This filter, according to Sims (1972), flattens the spectral density of most economic time series so that m and p can be considered as having white-noise properties.

Using the pre-whitened series, we cross-correlated pt with up to 8 quarters of the lagged and future values of mt. We also correlated the current values of these two variables. This is the test of causality outlined by Pierce (1977) and Pierce and Haugh (1977). The sample cross-correlation coefficients are shown in Table 11.

The results in this table cover the period 1964-74 (quarterly) for Brazil and 1961-74 (quarterly) for the other three countries. The asterisks beside the coefficients indicate that their value is greater than twice their respective standard errors. The standard error of the correlation coefficient is defined as 1/N, where N is the number of observations. Thus, for Brazil, the standard error is 0.151; for the other countries, it is 0.134.

There does appear to be evidence of significant correlation between current prices and both past and future values of the supply of money. We interpret this result to imply two-way causality between these two variables.

II. Dynamic Stability of the Model

The dynamic model was specified in matrix form according to equation (24). The matrix D in equation (24) can be written, in terms of the parameters of the system, as follows:

D=1|Det|=[d11d21d12d22d31d41d32d42d13d23d14d24d33d43d34d44](29)

where

d11 = (1 – λ)(1 – k1 + k2τ)/βλa2(1 – k1 + k2)

d12 = 0

d13 = (1 – τ)(k1k2λ)/β

d14 = –(k1k2λ)

d21 = –a2λk1(1 – γ)

d22 = |Det|(1 – γ)

d23 = –k1(1 – τ)(1 – γ)

d24 = k1(1 – γ)

d31 = a2λk2(1 – γ)

d32 = 0

d33 = (1 – τ)(1 – k1k2βa2λ)/β

d34 = –k2(1 – τ)

d41 = –(1 – β)a2λ(1 – k1 + k2τ)/β

d42 = 0

d43 = –(1 – τ)(1 – β)a2λ/β

d44 = (1 – β)(1 – k1 + k2τ)/β

and where the determinant |Det| = (1 – βa2λ + k2 τ - k1)/β

A necessary and sufficient condition for the stability of this system of difference equations is that the moduli of all the eigenvalues of matrix D be less than unity. An inspection of the second row of the matrix reveals that one of the eigenvalues of the matrix D is equal to (1 - γ), which is between zero and unity.52 Thus, it is apparent that the parameter γ will not affect the stability of the system. The stability condition is, thus, reduced to the examination of the eigenvalues of the remaining three by three matrix, H, which is formed by eliminating the second column and row of matrix D.

Let hij denote the ijth element of matrix H and define the following expressions:

H1=Trace(h)=h11+h22+h33H2=[h11h12h21h22]+[h11h13h31h33]+[h22h23h32h33]H3=Det|H|

It can be shown that if

|Det| = (1 – βa2λk1 + k2 τ)/β > 0

then necessary and sufficient conditions for stability are reduced to53

(i) b0 = 1 + H1 + H2 + H3 > 0

(ii) b1 = 3 + H1 - H2 - 3H3 > 0

(iii) b2 = 3 - H1 - H2 + 3H3 > 0

(iv) b3 = 1 - H1 + H2 + H3 > 0

Clearly, the values of the above expressions depend on the various parameter values, and we have therefore computed the eigenvalues of the system numerically.

Next, we establish the existence of a steady-state solution for the dynamic system. Clearly, in the steady state, all nominal variables must grow at the same rate as long as there are no exogenous factors present. Thus, let μ denote the steady-state rate of inflation that is also equal to the steady-state rates of expected inflation, and the steady-state rates of growth of nominal money supply, nominal government expenditure, and nominal government revenue. Let a bar above a variable (x¯) denote the initial value of the corresponding variable, (x). We can write the following steady-state solutions for the endogenous variables of the system of equations (17)(21):

M=M¯(1+μ)tR=R¯(1+μ)tG=G¯(1+μ)tP=P¯(1+μ)tπ=ΔlogP=ΔlogM=ΔlogR=ΔlogG=μ(30)

It can easily be shown that the above expressions constitute a solution to the dynamic system. Substituting the above relationships into equations (17)-(21) and solving for μ gives the following expression for the steady-state rate of inflation that is also the steady-state rate of growth of all the nominal variables:

μ=[k0+k1g1k2t0(k1k2)a0]/[1k2τ(k1k2)(1+a2)](31)

The above expression is positive for the range of estimated parameter values. It should be noted, however, that a necessary condition for the steady-state inflation rate to remain positive is that 1 – k2/τ > 0. Thus, for k2 = 0.32, which was the value used for the sensitivity analysis of the stability of the system in the test, τ was chosen to be larger than 0.3.54

III. Data Definitions and Sources

All data used in this study are taken from the International Monetary Fund’s monthly publication, International Financial Statistics. Line numbers refer to data on the relevant country pages of that publication. For Brazil, the data cover the period 1964-74, while for the other countries, the period covered was 1961-74. Quarterly data on government expenditures and revenues are not available prior to 1964 for Brazil. Seasonality in the data was taken into account by taking deviations from seasonal means.

P = consumer price index (1970 = 100) (Line 64)

G = government expenditures (Line 82)

R = government revenues (Line 81)

M = money plus quasi-money (Lines 34 and 35)

Y = level of real income (Line 99a divided by line 64). The series was interpolated on a quarterly basis using the quadratic interpolation described in Wymer (1976 a)

m = money multiplier (M divided by line 14)

E = residual item obtained from the definition

E = HG + R

where H is the stock of high-powered money (Line 14)

Table 11.

Cross-Correlations Between Prices and Money, with Prices as the Dependent Variable1

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An asterisk indicates that the value is greater than twice its standard error.

Period: 1964-74.

Period: 1961-74.

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*

Mr. Aghevli, Senior Economist in the Asian Department, is a graduate of Brown University.

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 authors would like to thank Phillip Cagan and Carlos Rodriguez for comments, and Sharon Foley for valuable research assistance.

2

The paper by Frenkel (1977) is different from the others, in that it utilizes the formal tests of causality proposed by Sims (1972) and Pierce (1977).

5

For example, see Dutton (1971) on Argentina and Aghevli and Khan (1977) on Indonesia.

6

The average annual rate of inflation for all developing countries during 1961-74 was about 14 per cent. Brazil’s rate of inflation was considerably higher than the average, while Colombia’s was slightly below the average. The other two countries experienced much lower rates of inflation than the average.

7

That is, consecutive values of the series are uncorrelated.

8

This very heuristic explanation is formalized in Pierce (1977) and Pierce and Haugh (1977).

9

In the actual exercises presented in Appendix I, we cross-correlated the current rate of inflation (after pre-whitening) with up to 8-quarter past and future values of the rate of growth of the money supply (also suitably pre-whitened).

10

This result may be owing to the existence of a lag between changes in wholesale prices and consumer prices. Even if the relationship between money growth and changes in wholesale prices is contemporaneous, the relationship with changes in consumer prices (the series we use) would involve a lag.

11

The correlation coefficient was statistically significant between current prices and the value of the money supply lagged two quarters and leading four quarters. See Appendix I.

12

See Zarembka (1968). Note, however, that following Cagan (1956), our equation (1) is really semi-logarithmic with respect to the expected rate of inflation. Frenkel (1977) has examined both the semi-logarithmic and double logarithmic forms of the money demand equations for the German hyperinflation of the early 1920s and concluded that there was no clear preference for one or the other. We have used the semi-logarithmic formulation, since the inflation rate in some quarters was negative.

13

This adjustment function assumes (with the money supply exogenous) that prices adjust to the excess demand for money. Alternative specifications, such as the nominal money variant proposed by White (1978), imply that the public can collectively adjust the nominal stock of money. Except in certain peculiar circumstances, this latter for mulation is fairly implausible. Furthermore, on a more technical level, the nominal adjustment model implies that inflation (the proper endogenous variable in both types of models) adjusts instantaneously to changes in the demand for money balances—real or nominal. Clearly, on both these counts, the nominal specification for the adjustment function must be viewed as inferior to the alternative real-terms specification that we have employed.

14

This particular formulation does not correspond exactly to the continuous-time mechanism proposed by Cagan (1956). However, it can be viewed as a possible difference equation approximation to the differential equation form. For a discussion of some of the problems contained in the standard adaptive-expectations mechanism, see Khan (1977 a), (1977 b).

15

While, for present purposes, we treat real income as exogenous, it could also be argued that there is a reverse causation running from government expenditure to income. For an analysis of cost and benefits associated with government investment financed by means of an inflation tax, see Aghevli (1977).

16

We do not, however, impose this as a constraint.

17

It should be noted that equation (7) could be derived differently by assuming that the government authorities maintained their real expenditures in some constant proportion to the level of real income that they “expected” to prevail in the future. In other words, we could respecify equation (5) formally as

log(G/P)t=g0+g1logYt*(5)

where Y* refers to real income expected in period t to prevail in the future, and expected real income adjusts adaptively to the error between the actual level of real income and the level of real income that was expected in the previous period, that is,

ΔlogYt*=γ[logYtlogYt1*](5)

where γ is now the coefficient of expectations, 0 < γ < 1.

Substituting equation (5″) into (5′), we would obtain a function identical in appearance to equation (7). Moreover, since both the partial-adjustment and the adaptive-expectations models are only specific formulations of a general distributed-lag function (with geometrically declining weights) one could specify

log(G/P)t=g0+g1(1γ)Σi=0γilogYti(5)

Using a Koyck transformation on equation (5‴), we would be left with an equation that would look exactly like the one derived from the partial-adjustment model, equation (6), or the one obtained from using an adaptive-expectations mechanism, as outlined in equation (5″). This general formulation clearly shows that, in non-stochastic form, the reduced-form equation (7) is consistent with two equally plausible structural models.

18

This type of function has been suggested by Chelliah (1971),

19

As in the case of government expenditures, this semi-reduced form could have been derived from other basic structural models. The most reasonable model would be the general one in which government revenues are simply a distributed-lag function of nominal income,

logRt=t0+t1(1τ)Σi=0τi(logYti+logPti)(10)

Solving equation (10′) by using a Koyck transformation, we would obtain an equation exactly equivalent to equation (11).

20

This means that, in the long run, both government expenditures and revenues would increase by the same percentage.

22

Since one can then use a system method of estimation.

24

If logG¯, logR¯, and logE¯ are the respective sample means of log G, log R, and log E, then

k0=log(elogG¯elogR¯+elogE¯)1elogG¯elogR¯+elogE¯[elogG¯logG¯elogR¯logR¯+elogE¯logE¯]k1=elogG¯elogG¯elogR¯+elogE¯k2=elogR¯elogG¯elogR¯+elogE¯k3=elogE¯elogG¯elogR¯+elogE¯
25

Further increases would result owing to increased expectations of inflation. See equation (21).

26

We have assumed that real income is not affected by increases in government expenditures. Clearly, if real income increased, then the process might be dampened. For a discussion of this issue, see Aghevli (1977).

27

This ratio has an asymptotic normal distribution, so that tests of significance should be based on this distribution rather than the more commonly used t-distribution.

28

It will be noticed that we have reported no tests of autocorrelation—for example, the Durbin-Watson statistic or the h-test also designed by Durbin (1970)—precisely because they have no significance in simultaneous models. As a precaution, however, we did test for autocorrelation when we were checking the specification of the individual equations using ordinary least squares. In no case did we find that autocorrelation in the residuals was important enough to warrant respecification and/or adjustment of the respective equation.

29

See footnote 24.

30

The long-run income elasticity is derived by dividing the coefficient of income in the demand for money by the coefficient of adjustment, γ, when the latter coefficient is calculated from the coefficient of lagged real balances, which is (1–γ).

31

It should be noted that the estimate of the income elasticity of demand for money for Brazil is somewhat higher than the previous estimates obtained by Silveira (1973), Pastore (1975), and Khan (1977 b). The inflation (semi) elasticity a2 is, however, lower than the previous estimates.

32

This shows that the specification of the adjustment function for government expenditure in nominal terms rather than real terms would have yielded the same results, as far as the model is concerned.

33

Brazil, which has the highest average inflation rate, has the smallest value of τ, and Thailand, which has the lowest average inflation rate, has the largest value.

34

Another interesting result is the difference between the estimated values of γ and τ. Since it has been hypothesized that this difference aggravates inflationary pressures, it is comforting to observe a positive relationship between it and the average rate of inflation in the four countries. However, the above unidirectional causation may not be accurate—that is, in those countries that experience high inflation rates, there is more incentive to defer tax payments, which leads to a lower value of τ.

35

The sample period of the study by Khan (1977 b) ended in 1970.

36

This argument was also proposed by Cagan (1956). For a discussion of the effect of continuing inflation on expectations, see Khan (1977 a), (1977 b).

37

The average lag is calculated as (1 - λ)/λ. The resulting value gives the period in which 63 per cent of the adjustment is completed.

38

Adjustment for discretionary changes in taxes would possibly lengthen the lag with respect to inflation.

39

This is, of course, a weaker method of comparison than either the inspection of the charts or the calculation of correlation coefficients, since positive and negative errors between the actual and simulated paths of the price index over the sample period could cancel each other out and produce a mean similar to the actual average inflation. Keeping this reservation in mind, the simulated annual average rate of inflation was 24.5 per cent for Brazil, 11.5 per cent for Colombia, 5.6 per cent for the Dominican Republic, and 4.5 per cent for Thailand. Compare these values with the actual averages shown in Table 1.

40

The homogeneous part of the system is composed of only the endogenous variables of the system. Clearly, the autonomous part of the dynamic system does not have any relevance for the stability issue, and it can therefore be eliminated for the present purposes.

41

If θj = a + ib, then the modulus of θj is defined by the expression a2+b2. The eigenvalues of the estimated model were derived using a computer program authored by C. R. Wymer.

42

For a discussion of this sensitivity analysis, see C. R. Wymer (1976 b).

43

Since two of the eigenvalues are complex conjugates, they will have the same modulus.

44

We must stress that these partial derivatives measure only the change in eigenvalues with respect to marginal changes in the parameters. Large changes in the parameters could, of course, change our conclusions.

45

In the necessary condition, the parameters β, a2, and λ only enter in product form.

46

Values of τ less than 0.3, corresponding to mean lags of longer than seven months, were not considered, since such long lags of revenue collection will not result in a meaningful equilibrium of the system (See Appendix II.).

47

It is evident from equations (1) and (2) that higher values of β and a2 correspond to higher desired demand for real balances (for a given rate of inflation). Higher values of λ, however, increase inflationary pressures by causing the adjustment of prices to any excess supply of money to take place more rapidly.

48

In this study, we have assumed that the balance of payments is independent of the growth of the money supply. In a regime of fixed exchange rates, the two would be related. For alternative models concentrating on the relationship between money, prices, and the balance of payments, see Aghevli (1975) and Aghevli and Khan (1976).

49

See Sargent and Wallace (1973), Frenkel (1977), Dutton (1971), and Aghevli and Khan (1977).

50

The model is essentially an extension of the one proposed and tested by us for Indonesia. See Aghevli and Khan (1977).

51

For a discussion of the introduction of such an indexation in Brazil, see Kafka (1967). Tanzi (1977) describes the adoption of such policies in Argentina and Chile.

52

Since 1 > γ > 0.

53

See Kenkel (1974), p. 174.

54

Values of τ less than 0.3 would correspond to a negative rate of inflation (in other words, to deflation) that would imply that the government initially had a surplus. In such a circumstance, longer delays in the collection of taxes would, in fact, increase government revenue in real terms, since taxes are based on lagged nominal incomes. While this point is an interesting theoretical quirk, it does not have much relevance for practical situations facing the developing countries.