Improving the Demand-for-Money Function in Moderate Inflation

A major concern in setting economic targets is to estimate the proper amount of monetary expansion during the next year. Improvements have been made in estimating the appropriate elasticities of demand for money with respect to interest, income, and prices, and in projecting the monetary increases in response to those variables over time. In particular, it has come to be recognized that the various elasticities of demand for money are lower in the short run than in the medium and long run.

Abstract

A major concern in setting economic targets is to estimate the proper amount of monetary expansion during the next year. Improvements have been made in estimating the appropriate elasticities of demand for money with respect to interest, income, and prices, and in projecting the monetary increases in response to those variables over time. In particular, it has come to be recognized that the various elasticities of demand for money are lower in the short run than in the medium and long run.

A major concern in setting economic targets is to estimate the proper amount of monetary expansion during the next year. Improvements have been made in estimating the appropriate elasticities of demand for money with respect to interest, income, and prices, and in projecting the monetary increases in response to those variables over time. In particular, it has come to be recognized that the various elasticities of demand for money are lower in the short run than in the medium and long run.

There are two customary means of handling elasticities that rise over time. The first assumes that the gradual adjustment in holdings of money to observed changes in the variables determining the demand for money reflects an instantaneous, full response of demand to the expected or “permanent” values of the independent variables (population, real income, prices, interest rates, wealth, expected inflation rate) combined with a gradual response of the expected values themselves to the observable current values. The second assumes that—in addition to any lag in the formation of expectations—there must be some lag in adjusting the actual holdings of money to the desired levels associated with current or expected values of variables. The approach is that of the “stock-adjustment lag.” As generally accepted, stock adjustment causes a fixed fraction of the discrepancy between the desired holdings of money and the pre-existing stock of money to be made good in each time period by additions to or subtractions from the pre-existing actual money stock.

Testing of both the stock-adjustment and the expectations-formation lags is most likely to be feasible in countries where quarterly data are available. Such countries, however, have until recently included only those with relatively little inflation. Inflation is now probably the strongest influence creating discrepancies between desired and actual holdings of financial assets, since it can abruptly reduce the real value of existing assets while leaving the desired real values relatively unaffected. It will be shown that the conventional stock-adjustment models need revision for use in inflationary conditions.

The stock-adjustment lag approach has been rejected by Milton Friedman on two grounds: (i) wealth, as he defines it, is a constant proportion of the relevant, “permanent” income, and (ii) other assets in the wealth portfolio can be quickly converted to money. The first implies that changes in the relevant income variable—permanent income—cannot produce discrepancies between the actual stock of assets and the stock of assets desired in the long run. The second implies that changes in the desired share of money in total assets, owing to changes in interest rates, are achievable with little or no lag. It will be argued in this study that the segment of real permanent wealth held in monetary form varies out of proportion with real permanent income when prices change. The desired amounts of other financial assets that are convertible into money with relative ease will themselves be subject to change in the same direction as the desired stock of money and therefore will, in great part, not be available to serve more than temporarily as a buffer stock for desired money holdings. Moreover, little of the important human part of total “permanent” wealth could be usefully converted into money. For these reasons, the stock of actual financial assets will fall below desired levels, and the shortfalls can be made good only by gradually accumulating financial assets as rates of saving increase.

This rejection of the Friedman argument of instant stock adjustment is consistent with the well-known Mundell-Johnson view that the price rise associated with a devaluation reduces the real value of monetary holdings and leads to a period of increased saving to rebuild them. Mundell and Johnson concluded that, during the period of reduced consumption spending, the extra saving would permit the devaluation to succeed without complementary demand-restraining measures. (See Johnson (1976), pp. 267, 273-75.) For econometric verification of the lagged enlargement of savings placed in monetary form following price increases, see the results of Townend (1976, pp. 56, 59) for the United Kingdom and his citations of results for the United States. See the Bank of England’s “Economic Commetary” (1978, p. 7) for tentative official acceptance of the relationship in the United Kingdom. Similar findings for four developing countries are reported by Morgan (1978, pp. 20-22). A recent study thought to reverse the U.S. econometric findings and to refute the logical case for the lagged wealth effect (Wachtel (1977)) is discussed in footnote 27.

Friedman’s short lag in adjustment to changed levels of interest rates is a special case. These rapid stock adjustments occur only because the desired addition to narrow money associated with a drop in interest rates must be accompanied by a desired reduction of an equal amount in other assets.

The fact that there was some apparent empirical support for the Friedman views made it easier to overlook the general case against zero adjustment lags in the demand for money. But the empirical findings consistent with zero stock-adjustment lags are inconclusive because of misspecifications in the conventional stock-adjustment models, which cause a downward bias in their lag estimates. In addition to technical biases, other downward biases are associated with the use of a single speed of adjustment to represent an average of (a) the normal adjustment lags and (b) the very short lags in adjusting to changes in interest rates, in adjusting to changes in desired transactions cash, and in correcting relatively small discrepancies between desired and actual money that are made good out of normal savings. The relatively large discrepancies that inflation has been producing in recent years are likely to require relatively long adjustment lags. It follows that if one is to obtain proper estimates of the correct time path for monetary variables, a stock-adjustment model must be estimated to accommodate zero adjustment lags for the effects of certain variables and a common lag for the effects of other variables. The length of the common lag may be an increasing function of the size of the discrepancy between actual and desired money holdings and also be influenced by whether the discrepancy is positive or negative.

There have been recent claims that exogenous increases in the supply of money cause it to depart from the “equilibrium” values given by the demand functions in the stock-adjustment-lag approach. But even when prices are stable, correction of the misspecifications of these functions may eliminate most of the appearance of monetary disequilibrium. For example, an exogenous increase in the money supply raises income and causes a large temporary jump in saving—a jump that is ignored by many conventional models. The extra saving makes it possible to attain the desired long-run increase in monetary holdings that is called for by the rise in income more quickly than the conventional model permits.

I. Formal Lag Models: Adjustment of Actual Real Money and Nominal Money to the Desired Levels

The demand-for-money function has traditionally been formulated in terms of a long-run equilibrium demand function. Equilibrium is approached after a time lag. The lag has been estimated by means of a stock-adjustment function based on the pioneering work of L. M. Koyck. It is customary to assume that a fixed proportion of the gap between the actual amount of the money held in the preceding time period and the amount of money currently desired is eliminated in the current period through adjustments of actual holdings. (The discussion of this paper is in terms of discrete time periods; where the advantages of a continuous time formulation are sought, the reader can easily introduce the appropriate transformations.)

In the past decade, leading investigators of monetary questions have introduced a simplification that has permitted them to disregard the price level by specifying the stock-adjustment model in terms of real money stocks. Several authors have found that the price-level variable in the demand function for either real or nominal money could be made superfluous; demand for nominal money rose pari passu with the price level. It was concluded by these investigators that they could omit the price variable by specifying their demand-for-money functions exclusively in real-money and real-income terms, thereby simplifying the estimating process and increasing the degrees of freedom. Studies following this reasoning are those by G. Chow (1968), E. Feige (1967), J. Frenkel (1977), D. Laidler and M. Parkin (1970), G. Laumas and Y. Mehra (1976), D. Starleaf (1970),1 and F. de Leeuw and E. Gramlich (1968) (on the Federal Reserve-MIT model).2

It is easily shown, however, that treating a demand-for-money function that contains lags in real-money terms constitutes a serious misspecification of the behavioral relationship found, on balance, when prices are changing.3 This misspecification may do no more than reduce goodness of fit. But it can easily introduce a large upward or a large downward bias in the estimated value of the speed-of-adjustment coefficient, depending on the size and sign of the covariances among the independent variables. With many of the earlier models fitted to periods of relatively stable prices, this misspecification may have been consistent with valid estimates in some countries. For models in less developed countries, however, and for all countries in the recent inflationary years, this misspecification causes a bias in the estimated parameters.

The misspecification is simple. It consists of disregarding any discrepancy between desired and actual money holdings caused by a change in the price level between period t-1 and period t that altered the real value in t of the stock of money already held in t-1.

Given the customary formula for the long-run or equilibrium level of desired money, M*,

Mt*=a(Y/P)tbrtcPte(1)=aYtbrtcPt(eb)(2)

or; in logarithms,

lnMt*=lna+bln(YP)t+clnrt+elnPt(1)=lna+blnYt+clnrt+(eb)lnPt(2)

where Y denotes money income, P denotes prices, and r denotes the interest rate.4 Some of the writers cited have found the price level elasticity e to be not significantly different from unity (See Frenkel and Johnson (1976, p. 24), who found the evidence “overwhelming.”).5 They have concluded that it is permissible to express equation (1) in real terms. Both sides of equation (1) could therefore be divided by the price level, so that the price term would drop out.

GivenmtMtPt:mt*=a(YP)rbrc(1)orlnmt*=lna+bln(YP)t+clnrt(2)

The preceding discussion refers to long-run equilibria, but the observations to which models are fitted must reflect short-run disequilibrium conditions. The traditional stock-adjustment model makes the actual stock of money rise from its level in the preceding period by a fraction of the amount of the desired rise above the actual level of the previous period. Only a part of the desired increase will be realized because of the increasing cost of acquiring monetary assets as the speed of acquisition increases.

The conventional stock-adjustment model using nominal values is

MtMt1=(Mt*Mt1)γ

Converting to logarithms,

lnMtlnMt1=γ(lnMt*lnMt1)(3)

Since homogeneity with respect to the price level was thought to have been established, this formulation has been commonly used in terms of real money. Each period’s monetary figure was deflated by the price index of that same period to yield

lnmtlnmt1=γ(lnmt*lnmt1)(4)

where m denotes the real value of M.

The error in this formulation is seen most clearly under the assumption that mt*=mt1 but that Pt > Pt-1. Here it is obvious that desired and actual money are equal by definition—so that no stock-adjustment process would be set under way—but, in fact, the equality is irrelevant; in the preceding period, the real value of the given nominal stock of money was equal to the real stock desired in the current period, but the intervening rise in prices means that the real value today of last period’s nominal money stock has fallen below today’s desired level. Therefore a new flow of money, whether measured in nominal or in real terms, must be generated.

In short, a model that makes the left and the right sides of equation (4) equal zero must misrepresent the actual stock-adjustment process. For equation (4) to be valid, it would be necessary to assume that lagged stock adjustment applied to changes in real income and other variables (such as the interest rate and inflation rate), but that when the stock disequilibrium was caused by a change in the price level, the adjustment took place without lag. This is seen through conversion of equation (4) into nominal money form by addition of ln Ptln Pt-1 to both sides,

lnMtlnMt1=γ(lnmt*lnmt1)+ΔlnPt(5)

Since the logarithmic formulation refers to percentage changes, equation (5) shows that the percentage change in money holdings will match, without lag, the percentage change in prices.

misspecification of inflation in stock adjustment as an explanation of findings of zero adjustment lag

Feige (1967, p. 471) claimed to have confirmed the Friedman claim of essentially zero lags with his finding that the fractional stock-adjustment coefficient was a little above unity for narrow money and just equal to unity for narrow money plus time deposits. Feige’s confirmation of Friedman’s view has led others to take this zero adjustment lag as a point of departure, so that the test for changes in lags refers only to the variability in income and interest-rate expectations formation. But Feige’s test antedated the period of significant inflation, and very rapid stock adjustment is the correct finding for small amounts of stock disequilibrium. Also, Goldfeld (1973) concluded that the standard stock-adjustment lags and the alternative expectations-formation lags are equally good explanations. The empirical evidence favoring Friedman is thus quite limited, and his a priori case for unlagged adjustment of actual money to desired money will be challenged in the next section. Here it will be indicated only that the mishandling of the price deflator in the stock-adjustment function is a likely explanation of any empirical confirmation of the Friedman/Feige expectation of a zero adjustment lag.6

The correct stock-adjustment formulation, equation (3), is made comparable with the conventional formula, equation (4), by converting it into real-money terms. Equation (3) can be restated as

lnmt+lnPtlnmt1lnPt1=γ(mt*+lnPtlnmt1lnPt1)(3)

Collecting terms,

lnmtlnmt1=γ(lnmt*lnmt1)(1γ)ΔlnPt(3)

The adjustment coefficient implied is

γr=lnmtlnmt1+(1γr)ΔlnPtlnmt*lnmt1(5)

where the subscript r distinguishes the correct formulation of γ from the conventional one. Dividing equation (5′) by the solution of equation (4) for γ,

γr/γ=1+(1γr)ΔlnPtΔlnmt(6)

Equation (6) permits some inference about the direction of bias in the conventional estimate of the speed of adjustment γ relative to the correct speed γr. The direction of bias depends on the sign of ΔPtmt (except when γr = 1, for which there is no bias). The change in real money, Δmt, is a function of mt*mt1, which, in turn, reflects the still-unadjusted parts of past changes in m* and mt-1 and the current change in variables determining mt*. Assuming away for simplicity any autocorrelation of variables, we may consider only the effects of current period ΔPt on mt*mt1 and, therefore, on mt.7

The direct effect of ΔPt will show a clearly negative value for the fraction ΔPtmt, as the rise in prices reduces the real value of the stock of money in the previous period. Since the stock adjustment in the current period is only a partial one, some of the initial fall in real money will survive. This means that mt will fall below mt-1, because of the rise in P.

The indirect effects of ΔPt, are mixed. Quarterly changes in real income are probably positively correlated with the simultaneous changes in prices, and the change in income tends to cause desired, and therefore actual, real money to change in the same direction (i.e., ΔPm > 0). It is probable that this indirect effect of prices will cancel out the direct effect described above, unless γr < 1/2 or unless the percentage change in real income is smaller than the associated change in prices. The latter is likely in conditions of near full employment, and then a negative overall correlation and hence an upward bias in γ could be expected. If the income demand for money depends on permanent or expected income, then a negative value for the overall correlation between ΔP and Δm* or Δm is also likely.

Increases in interest rates reduce the desire to hold money and therefore tend to make Δm negative. Increases in interest rates also tend to be associated with rising prices, because the expected inflation rate is strongly influenced by the current inflation rate, and thus itself influences interest rates, or because the authorities themselves pursue an interest rate policy that is related to current and expected inflation rates. This consideration strengthens the likelihood that the fraction in equation (6) will be negative.8 Similarly, insofar as desired real money is a negative function of the expected, and thereby of the actual, inflation rate, a negative value is to be expected for the fraction.

In sum, there appears, on balance, to be a tendency for the current inflation rate to be negatively correlated with the current change in real money stocks. This implies a likelihood that the conventional, misspecified estimates of the stock-adjustment coefficient have overestimated the true coefficient, so that the conventional evidence for a coefficient as high as unity is inconclusive.

Conflicting empirical evidence on the effects on the length of the adjustment lag of correcting the conventional model has been found by a major contributor to money-demand modeling. In a 1973 study, Goldfeld pointed out the need for the correction, tested it once, found no significant change in parameter estimates, and abandoned its use. The explanation for finding no change in parameter estimates was that the adjustment lag at issue was (predominantly) the expectations-formation lag, and for that lag the conventional estimating equation (e.g., equation (4)) is correct (See Goldfeld (1973), pp. 611-12.).

Three considerations invalidate this interpretation. First, Goldfeld lost sight of the fact that he had also tested the equivalent of equation (3ʺ) above (treating it as the conventional equation plus a term that reflected changes in current prices, which served as a proxy for the influence of the expected inflation rate on desired holdings) and found not only a very good fit but also our expected reduced speed of adjustment. (He overlooked this finding because of the reasonable conclusion that a public opinion survey provided a better proxy for the expected inflation rate than did the current inflation rate; he replaced the latter and found a weak role for expected inflation but an increased speed of adjustment.) Second, if the adjustment lag really is chiefly an expectations-formation lag, the imposition of a correction that was (assertedly) necessary only when a stock-adjustment lag was involved should have worsened the results; Goldfeld’s unchanged results must therefore imply a mixture of the two kinds of lag, with the stock-adjustment component still requiring correction of the conventional formula and, therefore, still justifying the presumption of a reduced speed of adjustment. Finally, in an updating of his work that permitted the faster inflation of the mid-1970s to be reflected in the estimated parameters, Goldfeld found it necessary to correct the conventional model. In addition to improving the fit, that correction markedly reduced the speed of adjustment (See Goldfeld (1976), p. 692, footnote and text.).

For periods when rising prices have become accepted by the public, there is a further source of upward bias in the speed of stock adjustment. Long-run desired money is made a function of either current money income or of “permanent” (expected) money income. But the former ignores the public’s recognition that prices will keep rising during the adjustment period and will immediately make inadequate any long-run desired money target based on today’s prices. And because it is compared with an imposed, understated target, the actual fractional stock adjustment is made to appear a larger fraction than it actually is. This bias is only partly eliminated under the Friedman approach of estimating “permanent” prices using the error-learning formulation of equation (7). That formulation can be shown to yield the expected prices of only one quarter ahead. But, when the price trend is rising, Friedman’s theoretical concept of permanent prices—the discounted present value of future prices—almost certainly implies prices that are well above the level expected to prevail a mere three months ahead.

exogenous money: a cause of “disequilibrium” money stock and a challenge to the corrected model

The stock-adjustment models have recently been criticized on the grounds that, when changes in nominal money supply are exogenous, the public is, at times, “forced” to hold disequilibrium amounts of money—holdings that are not on either the long-run or the short-run demand curve (See, for example, Starleaf (1970), pp. 748-49.). While some imprecision is logically possible, it seems reasonable that most of the estimated disequilibria are explainable by the misspecifications in the stock-adjustment models used, which are described in this study. In particular, an exogenous rise in money that raises real income will cause a disproportionately large initial jump in savings and, therefore, a voluntary enlargement of stock via either the consumption function’s “ratchet effect” or the parallel effect under the “permanent” income approach. And where the exogenous rise in money is achieved by open market operations and merely depresses interest rates, there seems no reason why wealth holders should depart significantly from their interest-demand functions in selling bonds in exchange for cash.

The corrected model has beeen challenged by Aghevli and Khan (1978, footnote 13, p. 388) for the case in which the nominal money supply is exogenous: “… specifications, such as the nominal money variant … imply that the public can collectively adjust the nominal stock of money ….” But addition of Δln P to both sides of the model in nominal terms that is shown in equation (3″) yields an equation in nominal money and in the change in prices that has exactly the same variables as the conventional model (See formulation in equation (5).) supported by Aghevli and Khan. (The only difference is that the unit coefficient of Δln P in equation (5) is replaced by γ.) Hence, this criticism is invalid.9

There is, however, one special case of the stock-adjustment process for which the conventional, real-terms model would be the preferred one: nominal money is fixed; an exogenous change in desired money occurs, which is caused by a shift in liquidity preference; the price change induced by the effects of the liquidity-preference shift on the flow of spending is not offset by an upward trend of prices owing to cost-push factors. When those conditions are met, the induced price change provides the desired stock adjustment and therefore does not enter into the stock-adjustment process with any sort of lag. Equation (5) then yields the correct result, with Δ ln Mt = 0, and γΔlnmt*=ΔlnPt10 But it can be shown that most of the factors generating stock-adjustment processes—a shift in the investment function, an exogenous change in the nominal money supply, exogenous cost-push increases in prices—cause the adjustment process to be dominated by observed price changes in the direction that aggravates (rather than eliminating) the stock disequilibrium. For any of these three stock-adjustment processes, it is a question of eliminating the extra stock disequilibrium created by the observed price change. Given that this involves a change in spending relative to income—a change that entails adjustment costs—the proper equation must be one that reflects the lagged stock-adjustment process implied by the introduction of γ into the price-change term in equation (5).

Ideally, a combination of the two models, with relative weights that change radically from quarter to quarter according to the nature of the current stock disequilibrium, should be used. But that seems impractical, especially in view of the further complications that this study will show to be necessary for a good model. Since the corrected model is superior, on average, its exclusive use (except at points where flights from money owing to high inflation can be identified) should be the preferred procedure.

II. Friedman’s Case Against Stock Adjustment

The preceding evidence in favor of a significant stock-adjustment lag is not conclusive, and other empirical evidence of varying quality can be introduced on both sides of the question. It seems worthwhile, therefore, to examine the Friedman argument that economic agents can logically ignore stock adjustment; they attain desired money holdings simply and quickly by reallocation of their wealth portfolios. As cited by Feige, Friedman believes that a buffer stock is available to make possible the desired holdings of narrow money or narrow money plus time deposits in the form of “other balance sheet items such as personal debt, consumer credit, and perhaps securities.”11 Friedman, of course, supports the existence of a lag in the adjustment of money to current income, but that is not a lag in adjusting to desired money holdings. Instead, it is the lag of changes in the variable to which desired money responded—“permanent” or expected income—after the changes in current income.

The nonmonetary portion of total assets would not provide a good source of desired increases in the money stock except that, by definition, total assets (wealth) are assumed to vary in direct proportion with permanent income. This means that there can be no shortage of the other assets for conversion into monetary form. Indeed, if the monetary assets have risen less than in proportion to permanent income, it must follow that the nonmonetary assets have risen more than proportionately, and a reallocation toward money will therefore also be desired to restore the balance. Wealth can move in step with expected income, at least when interest rates are constant, because the discounted present value of the expected stream of future earned income (“human” wealth) by definition moves in parallel with expected income; the interest rate establishes the same fixed ratio between the portions of expected income derived from real assets and from monetary assets and their discounted present value (“nonhuman,” “capitalistic” wealth).

While this construction permits wealth to move exactly in proportion to the expected value of income (at least when interest rates and the age distribution of the working population are unchanged), this by no means proves that the stock of money adjusts to expected income instantaneously. Friedman would argue that when inflation causes the value of monetary assets to fall substantially relative to expected money income, an untagged shift from the increased money value of “human” wealth (the increased level of future money wage rates) into monetary form would be made by borrowing against the stream of future earned income. But such unsecured loans tend to be expensive and also to be frowned on as irresponsible. Moreover, such borrowing would often fail to accomplish its purpose, for the desired ratio of monetary assets to income is commonly a desired ratio of net monetary assets to income, and cannot be attained by adding to debt. Finally, Friedman (1969, p. 137) himself has conceded that desired money might be more a function of conventionally-defined wealth than of “permanent” wealth. That concession requires acceptance of the assumption that inflation will depress that portion of wealth held in monetary form (and, hence, the entirety of conventional wealth) relative to income. The desired ratio of money to income can be fully restored to its desired position only through increased savings—the stock-adjustment process identified as the “Pigou effect,” or “wealth effect.”

One of Friedman’s zero stock-adjustment lags—the zero lag in adjusting to interest rate changes—is acceptable. A change in interest rates alters the desire to hold money in one direction and it also alters, by a roughly equal amount, the desire to hold interest-bearing assets in the opposite direction.12 But that is a unique case. The important causes of stock disequilibria are changes in income and, especially in today’s world, changes in the price level. In these circumstances, a gradual adjustment can be brought about only by a change in the rate of saving—and not by a mere reshuffling of existing assets.

The zero adjustment lag for the interest rate variable can be easily introduced into the standard estimating equation for Mt. In equation (3), for example, the modification would consist of removing c Δ ln rt, within the parentheses (where the lagged-effect variables belong) and adding the same term outside the parentheses (where the unlagged variables belong). In the unlikely case that this instant adjustment of money to the interest rate depends on the expected, rather than the actual, interest rate, an expectations-formation lag can be introduced for the interest variable. Such a modification would contribute to a form of the composite-lag estimating equation that is described in the next section.

III. Composite Expectations/Stock-Adjustment Lag Versus Separate Lags

The preceding discussion has presented two kinds of lag in the adjustment of money to observable economic variables: (i) the lag in adjusting actual money to the desired level; and (ii) the lag in the formation of expected value after a change in the actual value (insofar as expected or “permanent” values are among the causal factors determining desired money). If the “adaptive expectations” or “error learning” expectations-formation model (See equation (7) later in this section.) is considered acceptable for the expectations-formation model lag, the estimated lag distribution determining expected income, etc., will have the same general form (Koyck’s exponentially-declining weights for actual values becoming increasingly more remote in time) as the one implied by the standard stock-adjustment lag. Except for the serial correlation introduced into the error term, the stock-adjustment estimating equation found by substitution of equation (3) into equation (2’) will look the same as the expectations-formation equation that uses a single expectations-formation lag for its independent variables. This suggests that where both kinds of lag are or may be present, the same model can be used. Its partial adjustment coefficient y would then represent the combination of the expectations-formation and the stock-adjustment lags.13

An alternative procedure that yields separate estimates for the two kinds of lags, provided by Feige (1967), complicates the estimation procedure, but it may have some advantages, on balance. First, there may be an advantage from the correct specification of the autoregression in the error term when the roles of each kind of lag are distinguished. Second, if it is desired to retain the formulation of the model in real terms, the correction of the deflation of previous-period prices described earlier is needed only for that part of the total lag that is a stock-adjustment lag; the conventional real-terms model might remain acceptable for any expectations-formation lag. For example, as Goldfeld (1973, pp. 610-11) argued, if there is exclusively an expectations-formation lag, mt always equals mt* and there is no need to consider the discrepancy between mt* and mt-1 and, therefore, no need to be concerned with whether Mt-1 should be deflated by Pt-1 or by Pt.

Strictly speaking, the conventional real-terms equation also misspecifies the expectations-formation lag model. In deflating Mt* by Pt, it is assumed that expected prices always equal actual prices, whereas the underlying adaptive-expectations model requires that the two prices generally differ. In the 1960s and early 1970s, the assumption that expected prices equal actual prices may have been an acceptable simplification, because prices were no longer expected to drop back to any pre-existing normal level. But now that rising prices have become the standard expectation, Pe is greater than Pt, and the simplification of assuming them to be equal is once more unacceptable. The adaptive-expectations approach should be reintroduced for prices, although, in the current inflationary climate, it should be applied to the expected rate of price rise rather than the expected price level.

If separate determination of the two components of the delay from change in observable variables to change in money proves unmanageable,14 then the direct measures of income and price expectations available for some countries can be tested; they would bypass the expectations-formation lag and leave stock adjustment as the only lag to be estimated. Where the direct series on expected values are not usable, the simple composite lag will have to be accepted. This shortcut is especially likely to prove necessary if the further complications proposed in this paper are to be tested.

Whether one composite lag or separate lags are used, the procedure may be improved through introduction of flexibility in the shape of the distributed lag pattern obtainable with explicit polynomial distributed lags in place of the Koyck lag implied by equations such as (3) and (4).15 Particularly for the expectations part of the total lag, it is reasonable that, instead of continuously declining weights for increasingly remote values of the observable independent variable, there will be increasing weights at first.

The nature of the several alternatives will be represented here only by derivation of separate lag estimates for the expectations and stock-adjustment processes under the Koyck lag approach.

illustration of estimating separate expectations and stock-adjustment koyck lags

The Koyck approach to the composite lag is illustrated below by the case of a single expectations-formation lag (with “ln” signs and error terms omitted for simplicity).

Given the “error-learning” version of the expectations-formation lag for the income variable:

Yt*=Yt1*+λ(Y1Yt1*)(7)

where Yt* is the value of Y that is expected, in period t, to prevail in period t + 1.

And given the stock-adjustment lag of actual to desired money,

MtMt1=γ(Mt*Mt1)(8)

and the long-run equilibrium demand for money,

Mt*=a+bYt*(9)

The unobservable Y* variable can be eliminated, through use of equations (7) and (9), by first subtracting from equation (9) its own value lagged one period and multiplied by (1-γ). That yields a form of Yt* and Yt1*, to which equation (7) can also be converted by transposition of terms. These two steps permit replacement of the Y* terms by the known variable Y. The resulting equation still retains unknown M*s but M* is found in terms of current and lagged values of M (See equation (8).), so that all the unobservable (*) variables are eliminated.

The final equation, combining the distributed lag and the stock-adjustment lag, is

Mt=λγ(a+bYt)+[2(λ+γ)]Mt1(1γ)(1λ)Mt2(10)

Inclusion of another variable whose expected value (formulated as is equation (7)) contributed to the determination of desired money would cause the addition of that variable’s expectations-formulation coefficient λ′ and of Yt-1 and Mt-3.

IV. Misspecification Through Neglect of the Zero Lag Adjustment for Desired Transactions Cash

While the logic of the stock-adjustment approach requires that changes in desired assets be achieved only gradually, transactions cash balances are a special case. These are funds that are tied—notionally, at least—to the movement of goods through the production-consumption process. In the simplest interpretation, money income cannot rise without this accompanying rise in transactions cash balances. The modification of placing temporarily idle transactions funds in interest-earning assets when interest rates are high enough, which was introduced by Baumol and Tobin, will be shown to leave the case for a zero lag in transactions-cash adjustment unaffected.

Zero adjustment lags for the transactions part of M1 could be a partial explanation of why some of the conventionally estimated lags are short enough to appear consistent with Friedman’s zero adjustment lag for all of M1. Therefore, a separate lag estimate should be made for the nontransactions part of desired money.

Assuming an approximately fixed ratio k0 of transactions cash to gross national product (GNP), the desired nontransactions cash in period t and the actual figure in period t-1 are, respectively, (Mt*k0Yt)and(Mt1k0Yt1).16 The difference between these two values yields the part of the excess of desired over actual money that has to be adjusted by the partial adjustment process, with the remainder of the discrepancy between desired and actual being adjusted fully in the given time period. In ordinary arithmetic form,

MtMt1=γ(Mt*Mt1k0ΔYt)+k0ΔYt

or

MtMt1=γ(Mt*Mt1)+(1γ)k0ΔYt(11)

This formula obviously differs to a significant extent from the conventional one, since the k0 term on the right is absent from the conventional model. Two considerations make it plausible that k0 could be large enough to deserve attention. First, M1 tends to be largely in the hands of the household sector. Second, among households, the upper-income group—which dominates household financial assets and presumably holds cash more for convenience than for current transactions—is a secondary factor in household M1.

The above segregation of the unlagged, transactions-cash part of M1 may seem to be challenged by the Baumol-Tobin reasoning, which justifies a significant interest elasticity of demand for even the “pure” transactions part of M1. An interest elasticity of - 1/2 is claimed even for the cash holdings created by employees’ weekly or monthly pay, which tend to be held solely in the form of cash until completely used up at the end of the pay period. Baumol’s justification is important because, when interest rates are not too low, these funds account for an important part of total narrow money. At a high enough interest rate, it is argued, the employee is willing to lend out at least that part of the paycheck that will be temporarily idle during the earlier part of the pay period, receiving repayment of the loan in time to finance the living expenses of the remainder of the week or month. But in practice, very high interest rates are necessary to attract such temporarily idle and very small-scale funds; the interest elasticity of employees’ transactions balances should be zero up to an implausibly high level of interest rates.17

If the interest elasticity were significant, then correct specification of the model would be attained, not by ignoring the unlagged demand for part of money but by specifying it as a function of the interest rate or by dealing in terms of broad money, an aggregate that should capture both forms of holding transactions funds and leave the total invariant with respect to the interest rate.

superiority of M2 and misspecification of M1 when M2 is buffer for M1

Two studies have found that separate demand functions for m1 and real time deposits other than certificates of deposit (CDs) at commercial banks in the United States yielded parameters that were startlingly inconsistent with those of an aggregate m2 function. In one case, the simulation forecasts of aggregate m2 were inferior to those of the sum of the two separate equations’ forecasts. The first of the studies is inappropriate for further consideration because it used expectations-formation lags for income and interest, but no stock-adjustment lag. However, the study cites an obvious explanation for these unsatisfactory results—the development of CDs at the beginning of the 1960s caused a change in the structure of the demand for the non-CD time deposits counted in M2; a sharp change in structure is, in fact, reported for 1961 by the second study.18

There is a further explanation of these inconsistencies that should be important for money-demand modeling because it implies that the time-deposit function and the M1 function were both misspecified, with only the M2 function having the proper treatment of stock adjustment. A part of desired narrow money is linked to transactions, although not so tightly linked as “pure” transactions cash. When income rises in association with “pure” transactions cash, this desired quasi-transactions cash will rise also, and it is plausible that it will have a prior claim on financial assets over the other, more “idle” holdings of broad money. To that extent, other holdings, such as time deposits, will play a buffer stock role, permitting the more urgent demands for quasi-transactions cash to be temporarily satisfied at the expense of the parallel desired stock adjustment of other forms of financial wealth. This seems to be exactly what Friedman is implying in his argument that the entirety of M1 adjusts with a zero lag because of the ease of shifting funds out of the rest of the asset portfolio. (See the quotation at the beginning of Section II.) Friedman’s position seems incorrect only in its neglect of the fact that the other assets are simultaneously experiencing the same direction of change in desired amount and will defer to the M1 change only with respect to that part of the desired change in M1 that is relatively urgent—the quasi-transactions part of M1. In short, at least apart of M1 (in addition to the “pure” transactions cash discussed in the preceding section) will show a negligible adjustment lag because it draws on the buffer stock holdings found in the time-deposit portion of M2.19

The mirror image of a near-zero adjustment lag for a segment of M1 is a negative initial-period stock adjustment for the time-deposit portion of M2. Such a relationship implies that the customary M1 and time-deposit models are misspecified, although the M2 models remain correct.

The customary stock-adjustment equations for M1 and time deposits should each be modified to allow for a shift of funds between the two categories equal to the fraction k1 of the change in transactions, as reflected by the change in income. We then require

TDtTDt1=γ(TDt*TDt1+k1ΔYt)k1ΔYt(12)

The shift of funds out of time deposits, equal to k1ΔYt, reduces the rise in TD between t-1 and t by an equal amount (See the final term in equation (12).). Since that loss of time deposits enlarges the excess of desired over actual holdings by the same amount, k1ΔYt is also added to the term within parentheses.

Collecting terms,

TDtTDt1=γ(TDt*TDt1)(1γ)k1ΔYt(13)

Similarly,

M1tM1t1=γ(M*1tM1t1k1ΔYt)+k1ΔYt(14)

The addition of the k1ΔYt term—the final term in equation (14)—to the observed change in M1 reflects the shift of funds out of time deposits and into M1. Since that much of the desired rise in stock has been attained, the still-unsatisfied excess of desired over actual—the expression within parentheses—is reduced by that same k1ΔYt.

Collecting terms,

M1tM1t1=γ(M*1tM1t1)+(1γ)k1ΔYt(14)

Summing equations (13) and (14′) to obtain an equation in M2 restores the conventional formulation,

M2tM2t1=γ(M2t*M2t1)(15)

assuming, as seems reasonable once the k1 term is allowed for, that the two gammas are equal.

Provided that k1 is significantly different from zero, as the Friedman analysis requires, the contradiction between the aggregated and the disaggregated broad-money equations should be resolved in favor of the aggregated equation. This equation has been correctly specified, but its component equations have been misspecified.

It will be noted that the k0 adjustment in equation (11) has been omitted for simplicity. The omission is repaired by simply adding (1–γ)k0ΔYt to the expressions for both M1 and M2 in equations (14′) and (15).

The advantage of simplicity in use of broad money (equation (15)) rather than narrow money (equation (14′)) is reinforced by the prospect that the interest-rate variable will be unimportant or unnecessary in the broad or very broad formulations. The major part of the effect of the interest rate on M1 will consist of shifts between M1 and the interest-bearing segments of the broader definitions of money; the interest elasticity of demand for broad money will move much closer to zero, and errors in estimating that elasticity will become much less significant. The advantage of eliminating or minimizing the interest elasticity of demand is particularly great because it reduces the number of different distributed lags to be estimated and avoids the alternative of arbitrary imposition of a zero interest lag that would misrepresent the actual lag.20

The broad-money alternative also has policymaking advantages. A reduction of demand deposits owing to a shift of “idle” money into time deposits when the time-deposit yield increases implies that credit expansion can be accelerated when policy is set in terms of M1. But no change in real economic conditions has been caused by this reclassification of a given total of bank deposits. The broader the definition of money that is used, the less serious is the distortion of policy caused by shifts of funds owing to changes in interest rates.21

The points made in this and the preceding section imply that the past findings of values for money-adjustment lags that were consistent with zero lags need not constitute support for the Friedman zero lag. They may, in reality, reflect a zero lag for the transactions and quasi-transactions part of M1 but some lag for the rest of M1 and longer lags for time deposits and especially buffer-stock time deposits. The following sections present two alternative corrections of the conventional model, which are aimed at two different kinds of error. It is plausible that they will provide a rough correction of all three kinds of lag misspecifications.

V. Varying the Adjustment Lag with Size and Sign of Desired Adjustment

It has been a convenient simplification, in the Koyck models of the demand for financial assets, to use the same delay γ in the adjustment of actual assets to their desired levels, regardless of the size or the direction of adjustment. This simplification has been provided with a theoretical justification: the balancing of the increasing cost of more rapid adjustment to desired positions against the loss of utility owing to the remaining deviation from the desired position. Given the customary quadratic cost function, this balancing requires that the same proportion of the existing discrepancy be eliminated, regardless of the size of that discrepancy.

But the reasoning of the previous sections has introduced a new consideration: the discrepancies between actual and desired holdings of financial assets can be made extremely large because inflation reduces the real value of existing financial holdings, relative to income, much more than has been recognized in most of the customary models. Even models that did permit recognition of the stock disequilibria caused by inflation—models with nominal money or continuous time—have failed to show the inadequacy of fixed lags because the time periods used have included little role for the years of rapid price change; for less recent periods, reliance on the average size of the actual adjustment lags may have been consistent with adequate fits.

Normal savings out of current income may be adequate for providing the normally desired rises in real and financial assets associated with rising real income and a modest uptrend in prices. But, as will be shown, an additional price rise of 16 per cent per annum would require a doubling of the normal rate of savings if the observed quarterly γ were to be prevented from dropping below its pre-inflation estimated value, even if that value were as low as one third. A reduced speed of adjustment is more likely than such a drastic rise in the propensity to save.

Apart from the matter of varying the lag according to the size of adjustment, the corrected model should permit testing to determine whether there is a shorter lag for stock reductions than for stock increases.

theoretical basis for variable adjustment lag

The standard formulation of the optimum change of money holdings per time period balances the costs of disequilibrium (an excess of desired over actual) against the costs of adjustment (raising the actual above its previous-period value). The marginal cost of holding any actual amount of money is expressed in terms of the effects of a unit change in actual holdings on the adjustment cost and the disequilibrium cost. Raising the given period’s holdings has a negative marginal cost for disequilibrium and a positive marginal cost for adjustment. The optimum speed of adjustment is the speed that yields current-period money holdings such that the sum of the two marginal costs is equal to zero.

The standard formula is

TotalCost=a(Mt*Mt)2+b(MtMt1)2(16)
(dTC/dM)t=2a(MtMt*)+2b(MtMt1)(17)

Subtracting 2a(MtMt-1) from the first term of equation (17) and adding it to the second,

(dTC/dM)t=2a(Mt1Mt*)+(2b+2a)(MtMt1)(18)

and, setting dTC/dM equal to zero for the minimum total cost value of Mt,

MtMt1Mt*Mt1=aa+b=γ(19)

The right side, which is our stock-adjustment coefficient y, is a constant. That result is the consequence of the assumption that the two marginal costs have equal elasticities with respect to their two kinds of disequilibrium—that is, equal exponents for the two terms in parentheses on the right side of equation (16). But there is a presumption that the marginal cost of adjustment changes faster than that of disequilibrium, so that the second term in equation (16) should have a higher exponent than the first. If that presumption holds good, it is appropriate to assign an exponent of 4 to the second term in equation (16). That will yield an exponent of 3 for the second term in equation (17), with the speed-of-adjustment coefficient being

MtMt1Mt*Mt1=11+(2b/a)(MtMt1)2(19)

This formulation yields a γ that is a decreasing function of the absolute change in actual money holdings, with an upper limit of unity as the change in money holdings approaches zero. The formula is deficient, in that the actual, rather than the desired, change in money appears on the right side. However, because the right side is always positive, the two changes must always have the same sign; and, as just indicated, the actual change is always smaller than, and a decreasing ratio of, the desired one. That means that the formula is adequate for showing the diminishing speed of adjustment when the size of desired adjustment is increasing.

It follows that replacing (Mt – Mt-1)2 with an expression in terms of the needed (Mt*Mt1) would call for the latter to have a lower exponent than 2. Unity may seem a reasonable assumption for computational convenience. However, substitution for γ of equation (19′) as modified by that simplification would yield a quite complicated stock-adjustment model.

There is less complication if we make γ vary directly with (M* – M-1)1-η, where 2>η>1, thus yielding a more concise stock-adjustment equation

lnMtlnMt1=γ0|(lnMt*lnMt1)|2ηMt*Mt1|(Mt*Mt1)|(20)

The final term is required for restoring the direction of change in actual money holdings.22

reasons for higher elasticity of marginal adjustment cost

The assumption of a higher elasticity of marginal adjustment cost than of marginal disequilibrium cost requires explanation. Since we are considering those stock increases that are achieved by increases in saving, and therefore by reductions in consumption, we are dealing with an increasing cost of further compressing consumption to enlarge the amount of stock adjustment. There is some point in the reduction of consumption toward the starvation level where an individual will experience infinite costs of further accelerating the speed of stock adjustment. As that point is approached, an accelerating rise in marginal adjustment cost is obviously created. Since different individuals will approach their minimum levels of consumption at different points in time, the acceleration will occur gradually in the aggregate.

There seems to be no comparable accelerated rise in the marginal cost of disequilibrium. An irreducible need for transactions cash might seem to exist, but that is excluded from consideration by the finding earlier in this paper that there is no adjustment lag for the satisfaction of demand for transactions cash.

The relevant part of desired money should, therefore, be reducible (i.e., should have finite marginal disequilibrium cost) even as the amount held approaches zero. Thus, a given percentage shortfall in stocks would leave stocks further from the irreducible point than consumption would be when cut below normal by the same percentage. Moreover, the issue is stock adjustment during a period of one quarter, and the money supply is larger than one quarter’s consumption. Hence, the percentage reduction of consumption will be larger than the associated percentage shortfall of stocks. It follows that, even if the rises in marginal cost were equal for equal percentage changes, the marginal cost of adjustment would still be higher than that of stock disequilibrium.

The likelihood that the percentage reduction of consumption will exceed the accompanying percentage of stock shortfall is shown by data for the United States, where the relevant stocks of nominal assets of households are equivalent to four times one quarter’s disposable personal income.23 In these circumstances, a quarterly fractional adjustment coefficient of 1/3 would require the percentage reduction of the quarter’s consumption to be 1⅓ times the initial percentage shortfall of stock. Eliminating one third of a 4 per cent shrinkage in the real value of the stock in nominal terms would require a reduction in the quarter’s personal consumption of (1⅓ × 4 =)5⅓ per cent of disposable personal income. That would require close to a doubling of the savings/income ratio. So large a change is very rarely observed, and realizing it in one quarter would be an unheard-of feat.24

shorter lags for stock reductions

Faster stock adjustment is achieved through having more of the total change in consumption take place in the current quarter and/or reducing the number of quarters that are needed to complete the adjustment. Thus, faster stock adjustment implies more change in consumption relative to the norm in the current quarters and less change relative to the norm in the more remote quarters. The two adjustments exactly compensate in terms of quantity of consumption, but, because of the law of diminishing marginal utility, they do not compensate in terms of utility. While this means that faster stock adjustment in either direction reduces utility, it can be shown that the loss of utility is larger for a stock increase/consumption decrease. Moving upward and leftward along the marginal utility curve of current consumption implies that the curve is increasingly steep as it approaches the irreducible level of consumption. Since stock increases imply less consumption, and therefore positions higher up on the marginal utility curve, they also imply that a larger loss from oscillations around any given level will be found for steeper ranges of the curve.25

The difference in adjustment speed according to the sign of stock adjustment could be tested by use of a dummy variable reflecting the sign applied to the γ term in equation (20). For most countries in the last two decades, practically all stock adjustments may prove to be positive—at least once the role of rising prices in creating a need to add to stocks is accepted—or, at any rate, to have negative adjustments that are so close to zero as to promise insignificantly shorter lags than the lags for stock increases of equal size. In that situation and provided that the model’s estimated lag will not be used for cases of appreciable stock reduction, the directional complication can be bypassed.

zero-lag adjustment for high-yield financial assets

There is a special class of inflation-proof financial assets for which instant stock adjustment must be expected and for which the adjustment is achieved without any reduction of consumption relative to the pertinent measure of consumer income. The share of such assets in total financial assets is low for the United Kingdom and the United States, but in other countries where the proportion is higher, there will be a reduced chance of finding large desired stock adjustments that require reduced speeds of adjustment. It is necessary to examine, therefore, the nature of the inflation-proof financial assets.

Inflation ceases to shrink the real value of some financial assets once the inflation process has been prolonged enough to free sophisticated asset holders from money illusion. They then begin to treat as income only the real part of the high nominal interest rate received on marketable securities. Such wealth holders will regard the rest of the nominal interest income merely as compensation for the shrinkage of the real value of their financial claims rather than as spendable income. Therefore—in the misleading terms of national accounts statistics—they will automatically “save” most of such receipts of non-real interest income, placing them in financial assets without a lag.26

Lagged stock adjustment should still have a significant role, even in these conditions. The shrinkage in real spendable income owing to inflation will elicit less than equal reductions in consumption, so that some net decline in real assets must occur that will require correction with a lag via the wealth effect. The initial net shrinkage in the real value of assets should be larger than would be indicated by the shortfall of the aggregate marginal propensity to consume below unity. That is so because the high ratios of financial assets to income in developed countries imply very high ratios for the wealthier groups experiencing the greater part of the shrinkages and, therefore, quite large reductions of their spendable income; moreover, the resulting desire to cut consumption by only a small fraction of the cut in income can be easily gratified since these groups’ liquid assets are more than ample to finance long-term dissavings.

The role for lagged stock adjustment is further enlarged by the consideration that, until after the peak rate of inflation has been reached, adaptive expectations require that the expected inflation rate be lower than the one actually experienced. To that extent, the loss in real assets is recognized too late for instant correction and is likely to be treated as an unexpected, “windfall” loss—the case par excellence for lagged adjustment via the wealth effect.

Even where the actual inflation is correctly forecast, consumption will not be adjusted to the associated reduction of real spendable income, insofar as a lower inflation rate is foreseen for the long run. In such cases, the shrinkage of spendable income is seen as only temporary, and temporary reductions of income yield (under a variety of approaches to the consumption function) much smaller decreases in current consumption than permanent income reductions do.

For broad money, the asset that is of chief concern in this study, there is reason to expect that almost the entirety of the shrinkage of real value will not be accompanied by reductions of consumption. The fact that broad money and the other popular savings media, such as special government securities, offer low nominal interest rates even in inflationary conditions implies that they are held more for convenience and precautionary reasons than as sources of income. The decline in that income to a negative real figure is therefore likely to be ignored by the bulk of the asset holders when they decide on consumption, even though they recognize that the real value of the holdings has become inadequate. This behavior should play a particularly large role in aggregate broad money after inflation has become established; by that time, the asset holders who are sensitive to rates of return will already have shifted funds out of broad money into assets that provide a better return.27

A first-approximation assumption for purposes of estimation restricts lagged stock adjustment to holdings of narrow money (minus “transactions” cash) and time deposits other than CDs, to holdings of government “savings” bonds, and perhaps to the cash values of life insurance and similar policies owned. For the United States, this exclusion of all marketable, interest-bearing securities reduced the relevant total household stock of financial assets cited earlier by only one sixth.

VI. Reviving the Chow Two-Speed Lag Model

A neglected part of the widely cited study by Chow is a proposed combination of a zero lag for the change in money created by the part of normal savings that is normally placed in monetary form and a conventional lag adjustment for the excess of desired over actual money. This simple two-lag formulation contains a worthwhile insight into the rapidity of stock adjustment when the desired adjustment is small and can be handled out of current savings.28 Moreover, it might be a useful substitute for the variable lag proposed above if the latter proves to be unmanageable statistically. The Chow model also provides insights into the role of wealth in the demand for money. (See Chow (1966), pp. 112-14.)

Chow’s preferred model is

mtmt1=γ(mt*mt1)+d(realwealthtrealwealtht1)(21)

where d is the proportion of savings normally placed in money form. The change in real wealth is assumed to equal the savings in the given period. Those savings are made equal to the difference between current income and a fraction of permanent income,

mtmt1=γ(mt*mt1)+d(ytθytperm)29(22)

This formulation constitutes a misspecification of the deflation for price changes; the change in the real value of wealth cannot equal the current flow of savings when wealth is partly financial and prices are changing. This problem will not be examined now. Of interest here are the contributions that the Chow model might make.

In their criticism of the Chow model, Brunner and Meltzer (1968, pp. 1234-40) pointed out that it required that new savings be placed in monetary form (second term) even when desired money equaled money in the previous period (mt*mt1=0).30 Chow (1968, pp. 1240-43) replied that, since desired money increased along with the wealth provided by current savings, no such contradiction could exist. But that would require that m* already reflect a wealth variable that included the current additions to wealth. And in that case, the model cannot show the change in wealth separately with the coefficient d, which is defined as the share of current savings normally placed in monetary assets. The Chow equation “double-counts” the desired rise in money. The correct formulation for separate presentation of current savings first requires subtraction of d times current savings from m*. The corrected equations are then

mtmt1=γ[mt*d(ytθytperm)mt1]+d(ytθytperm)(23)
mtmt1=γ(mt*mt1)+(1γ)d(ytθytperm)(23)

The subtraction from m* of the extra money desired because of current-period saving is necessary, of course, because the appearance of that amount of extra money with a positive sign outside the gamma term implies that part of the stock adjustment, m* – mt-1 is instantly completed and therefore is no longer eligible for the partial adjustment process of the gamma term.

With Chow’s estimates for d now reinterpreted as (1 - γ)d in equation (23′), his statistical results can be seriously re-examined (leaving aside the question of price deflation) and may provide a useful alternative method for handling the different and varying adjustment lags proposed earlier. If useful results are hindered by the value close to zero for the coefficient of the savings term—(1 - γ)d instead of d—retention of equation (23) instead of equation (23′) may solve the difficulty.31

There remains one further correction for the Chow model, which is necessary because it is expressed in real terms. Since the model is in non-log form, the correction is made simply by multiplying mt-1 by Pt-1/Pt. Because the model is based on the “permanent” approach, consistency requires that expected or “permanent” prices be used in place of P.

VII. Refining the Test for Interest-Rate Role of Expected Inflation Rate

The increased inflation rates of recent years have drawn attention to the need to test models that make savings and investment a function of the real interest rate rather than merely a function of the nominal interest rate. The real interest rate has no role to play in the demand function for money, however, for desired money is presumably a decreasing function of the sum of the nominal interest rate and the expected inflation rate, rather than of their difference. Wealth holders will want to shift from money to bonds when the nominal yield on bonds rises and from money to hoardable goods when the expected inflation rate rises. These shifts will be made even if the real interest rate is unchanged.

The discussion so far has implicitly assumed that the role of the interest rate in the demand for money was filled by the nominal interest rate alone. That assumption is in accord with the traditional view that the nominal interest rate is the rate of exchange between bonds and money and that the inflation rate is irrelevant to that exchange rate because the real yields on these two alternatives are equally diminished by the expectation of rising prices. But insofar as holders of money are able to consider holding real assets (hoarded goods, equity claims on businesses, houses) as a further alternative to money, then a change in the expected inflation rate can change the desired amount of nominal assets associated with any given level of nominal interest rates. The nominal yield on real assets is, by definition, increased by the amount of the expected increase in the inflation rate, whereas the nominal yields on bonds and money are, by assumption, unchanged.

reasons against the inflation-rate elasticity

This deductive argument is insufficient to demonstrate the need to add an expected-inflation-rate variable to the money-demand function. There are four other factors, one or more of which could weaken or abolish the influence of the expected inflation rate:

(i) Where the nominal interest rate tends to reflect fully the expected rate of inflation, even high substitutability between money and real assets will not justify the addition of the inflation-rate variable. The rise in the opportunity cost of holding money in place of either real assets or bonds is fully reflected by the nominal interest rate alone. The nominal rate of return on real assets equals their real rate of return plus the inflation rate. That rate of return rises by the amount of rise in the expected inflation rate. And that rise in the nominal rate of return is exactly the same as the rise in the nominal rate of return on bonds. This tendency is especially strong in countries suffering from hyperinflation, so that movements in a free-market nominal interest rate—if available—would be a good proxy for movements in the expected rate of inflation.

(ii) The real-assets alternative need not be widely available to holders of money and bonds, and real assets may be considered poor substitutes for nominal assets because of their illiquidity and tendency to deteriorate or—in the case of common shares, which can be moderately liquid—because of the volatility of their market prices.32

(iii) The supply of real investment assets and equities may be sufficiently less elastic compared with the supply of output in general in the short to medium term to produce a situation in which the effort to shift from monetary to real assets will raise the prices of real assets relative to general prices; that price rise may reduce the yield on real assets by enough to choke off any desired asset shift before much shifting out of monetary assets can take place.

(iv) Where, as Friedman argues, desired money is a function of “permanent” income or expected real income and prices rather than current income and prices, it follows that, once the norm of rising prices has displaced the norm of stable prices, a higher expected inflation rate must imply a tendency toward higher current holdings of money. Thus, in addition to appearing with its theoretically negative exponent, under the prevalent practice of using the actual value of current prices, the expected inflation rate must also be introduced with a positive exponent to yield the price level expected in the future. Instead of the customary

M*=aypermbPrc(ΔPeP)d(24)

we require

M*=aypermbP(1+ΔPeP)rc(ΔPeP)d(25)

where ΔPe/P is the expected rise in prices from quarter t to quarter t + 4. The term (1 + ΔPe/P) has an exponent of unity and therefore a unitary elasticity with respect to the future price level. Collection of the ΔPe/P terms leads to the following inflation-rate elasticity of desired nominal (or real) money:

ηM*ΔPe/Pd+ΔPe/P(25)

where d < 0, ΔPe/P > 0.

The composite inflation-rate elasticity given by equation (25′) must be the elasticity estimated by the standard equation (24). Models like the one specified by equation (24) should therefore yield zero elasticities of demand for money with respect to the inflation rate when the true elasticity d is approximately equal to the expected inflation rate and has the opposite sign. With a higher expected inflation rate, the standard model should yield a positive inflation-rate elasticity of demand for money. Since either of these results would be considered by users of equation (24) to be contrary to common sense, it can be suspected that such results were found but rejected and were replaced by variants of the standard model that happened to yield the negative elasticities sought. But the cumulative effect of points (i)–(iv) makes it reasonable to consider models suspect if they do not yield elasticities with respect to the expected inflation rate that are close to zero or even slightly positive.33

See the discussion immediately preceding the final subsection of Section I for the reason why the expected price rise in equations (25) and (25′) was not the customary one expected for just one quarter ahead. The reader who prefers prices expected for two or three quarters ahead can apply the coefficient 0.5 or 0.75, respectively, to the ΔPe/P term in equation (25′).

A statistical problem in introducing the expected inflation rate is the intercorrelation between the interest-rate and inflation-rate variables in those countries in which expected inflation is thought to influence the level of interest rates. In addition to the difficulty of finding significant regression coefficients for collinear variables, there is the presumption that those observations that break the normal correlation and permit the estimation of partial regression coefficients may have represented abnormal economic events. For example, in the recent period of wide movements in both short-term interest rates and inflation rates, instances of rises in the latter that were not accompanied by much rise in interest rates may reflect temporary shock factors, such as a devaluation, that are expected to accelerate prices, but only for a short time. In this case, the elasticity of the demand for money with respect to the (near-future) expected inflation rate underestimates the desired concept. Similarly, when interest rates fall while expected inflation is undiminished, a recession is likely to be underway; and the estimated nominal interest elasticity will be the reflection of cyclical variables that are not explicitly represented in the model.

improving the test for the inflation rate

These logical problems with the expected inflation variables have been compounded by shortcomings in the testing of the variables. Improvements in testing include the following: improving the usefulness of the variables derived from public opinion surveys by selecting the responses from those sectors of the public that hold disproportionately large shares of total financial assets; where the surveys are not available, modifying the expectations series given by past price changes (as has recently been done for hyperinflation) by varying the rapidity of expectations formation in step with the current inflation rate, and adjusting expectations ad hoc when current economic shocks should be changing them relative to what past experience alone would imply; if these adjustments are insufficient to yield a usable expectations series (and the preferred survey figures are unavailable), then a modified rational-expectations approach may be worth testing, although derivation of the basic series needed will require some effort.

(i) The first step in estimating the role of the expected inflation rate in determining the demand for real money is correction of the procedure used for deflating prices. As was shown earlier in the derivation of equation (3″), the misspecification in the conventional model can be corrected by addition of the term - (1 - γ) Δln Pt. It follows that introducing into the customary version an expected price-change variable that is positively correlated with the actual price change must tend to yield a significant and negative coefficient for that variable, even if it plays no valid causal role. The inflation-rate variable is found to be significant only because it serves to correct the model’s error in deflation to real terms. This is confirmed by Goldfeld’s test, which adds expected inflation to both the correct and the incorrect models. The latter model assigned expected inflation a negative sign and a t-ratio of over 4, while the correct model gave inflation a nonsignificant value.34

(ii) In the past, most models have derived their expected price-change series via adaptive expectations models; these make the expected price level or price change a weighted average of current and past price changes in the same way (discussed earlier) that permanent or expected income is derived from observed values of income. The disadvantages of this method, which result from its assumption that expectations are always based in the same way on past experience, can now be avoided for many countries because series based on opinion surveys are available for deriving the expected rate of inflation. These series avoid the problems of inability to capture change in the expectations-formation process during the period of observation and of inability to allow for the influence of current economic shocks—devaluations and changes in government policy orientation, for example—on price expectations.

As was noted earlier, survey series for expectations—for income as well as prices—can contribute also to the solution of the problem of a mixture of stock-adjustment and expectations-formation lags within the total observed adjustment lag.35

The expectations surveyed usually concern periods of 6 months or less, although they sometimes cover periods as long as 12 months. Over these periods, consumers’ opinions about price prospects may be influenced by prospects of changes in food prices associated with climatic conditions, the known shock impact of the 1974 increases in oil prices, and the like. But these changes tend to be seen as transitory, and temporary changes in the inflation rate can fail to justify the transactions costs involved in shifting into and out of real assets. This erratic element in the consumer survey series for expected consumer prices can be corrected by use of a combination of the consumer survey figures with the smoother adaptive expectations series or (better—if they are available) with the accompanying surveys of industrialists on the expected change in prices of manufactured goods. The same consideration may make manufactured goods price expectations an acceptable substitute for consumer price expectations in countries where the latter are not available.

In the absence of opinion surveys that focus directly on the expected inflation rate, there is often a substitute survey series that also might prove superior to the moving average of past observations. An example of that series is found in the cyclical indicators section of the Organization for Economic Cooperation and Development’s Main Economic Indicators; expected price changes are constructed, where necessary, by regression of the actual rate of price change on the results of the popular qualitative opinion surveys, which ask, “Will prices rise faster than, as fast as, or more slowly than, they have been rising?”

A further refinement of the survey figures for the expected inflation rate is either excluding or giving little weight to socio-economic groups that control relatively little of the country’s stock of monetary assets. In France, the official surveys of frequency of expectations of acceleration or deceleration of inflation permit the exclusion of, or the assignment of low weights to, responses by groups such as the retired, farmers, and laborers. In the United States, the University of Michigan’s Survey Research Center’s quarterly series on households’ expected inflation rates permits separation of households with more than $15,000 of annual income from the total. That group controls a disproportionately large share of monetary assets and is more likely to manage its assets in accordance with the expected inflation rate it perceives than the lower-income group is.36

(iii) Where current and lagged actual price changes must be used as a proxy for expected changes, an innovation that was recently tested for conditions of hyperinflation is worth considering in the context of the moderate inflations that are at issue here. Khan (1977) has demonstrated the variable adjustment lag suggested by Cagan, varying the weight assigned to the current rate of price rise in the adaptive-expectations model (cf. equation (7)) according to the size of the current price rise.37 Money holders realized, after a period of continuous rise in inflation rates, that adjustments of the expected rate that were made to correct for past errors in expectations were increasingly inadequate. Therefore, they had to give increased weight to the highest of the observed inflation rates, that of the current period. A refinement of the reweighting rule would make the weight assigned to the current period’s inflation rate an increasing function of the current and recent-past positive serial correlation of forecast errors.

Insofar as observed jumps in the inflation rate are owing to shocks, such as the oil crisis or exchange devaluations or poor harvests, the current inflation rate may be assigned a reduced weight because relatively more of its influence is expected to be temporary. The serial correlation procedure, perhaps supplemented by dummy variables, may make this problem manageable, however. If explicit allowance for the economic and political shocks affecting price expectations is desired, some dummy variables may be added to the adaptive expectations model.

Some economic and political shocks cause long-lasting changes in expected inflation rates and should be assigned high weights. Although there is the risk of subjectivity, the researcher may consider the addition of a separate class of dummy variables for those long-lasting shocks that can be identified.38

(iv) Modifying the adaptive expectations model for known economic shocks that promise long-lasting effects suggests going further in the direction of the “rational expectations” models. These apply known values of causal economic variables to a macroeconomic model that yields logical results for future values of prices and incomes. Econometric testing of the expectations of inflation formed “rationally” in a succession of quarters will require a great deal of preparatory work, because the formation of rational economic expectations depends in great part on what economic consultants have told wealth holders about economic prospects. Since different econometric forecasting services often issue conflicting forecasts for any given quarter, and since they change their own models over time as they learn from their own errors and in response to the progress of economics, it is necessary to reconstruct a weighted average of the actual values of past forecasts. The weight assigned to each forecasting service might be in proportion to clients’ total payments for subscriptions or in proportion to the service’s ranking in terms of forecasting success in the recent past. Since economic agents know that the forecasting services disagree with one another and change their own models over time, subscriber skepticism may exist; this may make it necessary to use an average of the econometricians’ rational series and the adaptive expectations series.39

Some rational expectations models base current expectations about the inflation rate on policy variables, such as the rate of monetary expansion currently established by the authorities. It is assumed that this rate of money growth is expected to continue over the relevant time period. If it is an inflationary rate of money growth, the associated rate of inflation will be rationally foreseen. But this set of models suffers from the prospects that the government’s target rate of money growth will prove not to be inflationary (if only because a cyclical contraction happens to develop) or else will be abandoned by the government when it is discovered to be inflationary, or will be stopped when the government falls because it has caused inflation.40 At the extreme, these considerations would lead to the belief that future prices are randomly correlated with current monetary policy, so that expectations would be based exclusively on past experience reflected in the adaptive expectations approach.

VIII. Improving Deflation by Population

Just as the ability of a society to save depends on the amount of its per capita income rather than solely on its aggregate income, so the desire to hold cash for any given aggregate of income is likely to depend on the amount of income per capita or per family. Only in the special case in which real-income elasticity of demand for money approximates unity can the deflation by population be ignored.

Despite its strong theoretical grounding, the population variable has not yielded good results, in either the population version or the number-of-households version. See, for example, Goodhart and Crockett (1970, p. 195) and Goldfeld (1973, p. 625). A further issue calling for examination is the claim that no stock-adjustment lag exists where the desired rise in money is owing to a rise in population.

The standard model for long-run desired money makes allowance for the population variable in the following way:

m*=a(y/n)bnrc(26)
m*=aybn(1b)rc(27)

where y denotes aggregate real income and n denotes the number of households or the population.

The inadequacy of the model in the population definition of n can be seen by considering a normal case in which b, the real income elasticity of demand for money, is less than 1. Then a rise in population with income unchanged reduces desired real money per capita; but, because it raises the number of persons in the population, the composite effect, 1–b, is a net rise in desired money. This result misrepresents a frequently approximated situation: the population increase consists of a rise in the number of dependents per spending unit (household) with the number of households and aggregate income unchanged. Clearly, the standard of living of the head of the household falls on average, and his desired money holdings are reduced. Because the dependent members of the household tend to pool “their” transactions and other cash holdings with those of the head of the household, desired per-household money should be multiplied by the number of households. And—with number of households by assumption unchanged—the aggregate amount of desired money must be reduced by the rise in the dependent segment of the population.

The alternative procedure of using the number of households for n is also unsatisfactory, although better than using population; it implies no change in standards of living when the number of children or other dependents increases and, therefore, rules out the fall in desired money holdings just described.

The correct deflator (divisor) of total income to a per-unit figure starts with the number of households but combines that variable with an index of the weighted number of persons per household. Dependent members, who presumably have lower consumption needs, are assigned lower weights than active adult members of the household (e.g., 1/2 of the adult weight). The per-unit income needed for the term in parentheses in equation (26) is therefore

yH(popwH)b=ypopwwhenb=1.

where H denotes the number of households and popw denotes the population with dependents assigned lower weights than productive adult members. The exponent b′ ≤ 1 is introduced to allow for the extent to which desired money is transactions cash, which is rigidly tied to the level of money income and little affected by the size of household. In addition to this per-weighted-household income variable, the separate n term in equation (26) is obviously the number of these households H. The revised desired-money equation for b′ = 1 is thus

m*=a(y/popw)bHrc(28)

This formulation is exact when new households are established by the newly employed. But enlargement of the number of households through splitting off equal proportions of the employed and of the dependent members from existing households should leave desired money unaffected—contrary to what equation (28) and other “households” models imply. It is therefore necessary to redefine the households variable H to exclude those households formed by a member who is already employed. But that adjustment is likely to be impracticable. Therefore, in countries where new households are normally formed by persons who are already employed, the total male labor force could be used for H.

An annual weighted-population series is easily constructed, and quarterly interpolation should be manageable. If data on households do not exist, figures for the male labor force, the working age population, or at least the total adult population should be manageable as proxy variables; their availability implies the availability of a proxy for the weighted average population as well.41

The discussion so far has assumed that the population variable’s effect on desired money holdings would be realized via the normal stock-adjustment process. However, that assumption has been challenged by Goodhart and Crockett (1970, p. 195): “… there is little theoretical justification for expecting lagged adjustment in the case of the population variable…. Additions to the population will not affect the behaviour of existing money holders; nor are they likely to ‘adjust gradually to their own existence’.” But the case of no stock adjustment lag is that described previously in which a part of existing income (and money) is transferred out of the old household into the newly-formed one, with equal and opposite changes in the desired moneys of the two units and, hence, the kind of aggregate stock adjustment that can be instantaneous—a zero stock adjustment. It is when the new household is formed by a new income earner, with a disproportionately small claim on the money of any old household, that the desired aggregate stock changes. And then the stock adjustment must be a gradual one because it is financed out of the flow of savings.

This inference of error from imposing a zero stock-adjustment lag on the population variable is consistent with the statistical results: the fit found by Goodhart and Crockett (their equation (5)) was so weakened by the unlagged household variable that the investigators finally dropped population deflation completely. Their rationale was that the proxy they used for households, which was based on the population of working age or older, was invalidated by a post-World War II shift in age distribution. That explanation is reasonable. But an equally plausible explanation exists in the combination of incorrect deflation and the suppression of the deflator’s lagged stock adjustment.

REFERENCES

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SUMMARIES

Government Deficits and the Inflationary Process in Developing Countries—bijan b. aghevli and mohsin s. khan (pages 383-416)

The purpose of this paper is to examine the relationship between increases in the money supply and inflation in four developing countries (Brazil, Colombia, the Dominican Republic, and Thailand) over the period 1961-74. It is first shown that the growth in the money supply and inflation are linked in a two-way relationship in these countries, and then a dynamic model is designed that explicitly introduces the link in the form of reactions of the government fiscal deficit to inflation. The basic hypothesis is that an increase in the rate of inflation, whatever its cause, increases the real value of the fiscal deficit, owing to the fact that money expenditures keep pace with inflation while nominal revenues tend to lag behind. The financing of this deficit increases the supply of money, thus generating further inflation. In this framework, one would expect to observe increases in the supply of money both causing inflation and, at the same time, being positively affected by it.

The model is estimated for the four countries, and the empirical results tend to validate the hypothesis. In particular, it is found that fiscal deficits play an important role in the inflationary process, and that increases in these deficits are largely owing to the differences in the lags of government expenditures and revenues.

Two basic policy conclusions emerge from this study: first, the tendency of government budgetary positions to be automatically destabilizing in developing economies underscores the need for an actively anti-inflationary fiscal policy in these economies. Second, developing countries should attach priority to tax reforms designed to eliminate revenue lags.

Inflation, Real Tax Revenue, and the Case for Inflationary Finance: Theory with an Application to Argentina—vito tanzi (pages 417-51)

The literature on inflationary finance has dealt only with the case in which inflation leaves the real revenue from the tax system unaffected. However, in most cases, inflation brings about changes in real tax revenue. If the average lag in tax collection is short and the price elasticity of the tax system is greater than unity, inflation will be accompanied by increases in real tax revenue. If, on the other hand, the average lag in tax collection is long and the price elasticity of the tax system is equal to, or lower than, one, inflation will be accompanied by falls in real tax revenue. These effects on tax revenue must be taken into account in a truly general theory of inflationary finance. This paper develops such a theory in connection with the situation in which inflation leads to falls in real tax revenue, which is the situation prevailing in most developing countries.

A theoretical model, for a situation in which the price elasticity of the tax system is unity and the average collection lag is significant, is developed. It is shown that total government revenue—from taxes and inflationary finance—at given rates of inflation will depend on the values of the following: (a) the ratio of total tax revenue to national income at zero inflation, (b) the average collection lag for the tax system, (c) the ratio of money to income at zero inflation, and (d) the sensitivity of the demand for money with respect to the rate of inflation. A simulation exercise using realistic figures for the above variables shows that the net gain from inflationary finance is likely to be significantly less than would be expected from traditional theory. Finally, the model is applied to Argentina for the 1968-76 period, and simulated and actual results are compared. It is shown that the model performs quite well for that period.

Relative Price Distortions and Inflation: The Case of Argentina, 1963-76—ke-young chu and andrew feltenstein (pages 452-93)

This paper analyzes the interrelationship between distortions in a country’s relative price structure and rate of inflation. The model developed in the paper is applied to the recent period of high inflation in Argentina. A monetary model is constructed in which demand for real balances is determined by inflationary expectations, while the supply of money is a function of the balance of payments, the nominal deficit of the central government, and the level of credit to the private sector. This credit is, in turn, a function of distortions in the economy’s relative price structure, which, for Argentina, were brought about by the imposition of selective price controls. These distortions are measured as the difference between the observed price of a given commodity and a zero-profit price, which is derived from a dynamic input-output estimate of the economy. If an industry’s actual output price is held below its zero-profit price, then the industry is making a loss on each unit of output and will be unable to pay its factors of production unless it receives loans from the banking system. These loans, which, in Argentina, were given almost passively, lead to credit creation that, in turn, leads to an increase in the reserve base and hence to an increase in the money supply.

This model, which is nonlinear, is estimated for the period 1963-76 on a quarterly basis, and the aggregate losses owing to price distortions are found to be highly significant in explaining the rate of growth of the money supply. A number of policy simulations are run, and it is found that the rate of inflation remains unacceptably high, even if the budget is balanced, if price distortions are not removed. These distortions, which have a lagged impact, are also shown to have vanished almost immediately after the removal of price controls in mid-1976.

Wage Subsidies and Employment: An Analysis of the French Experience—george f. kopits (pages 494-527)

In recent years, faced simultaneously with high rates of unemployment and inflation, several countries—mainly in the industrialized world—adopted various types of temporary marginal wage subsidies, which are provided to employers to offset a portion of wage payments to workers who would otherwise be unemployed. After a review of wage subsidies implemented in the Federal Republic of Germany, France, the United Kingdom, and the United States, and a discussion of such subsidies in general, the paper focuses on France’s prime d’incitation à la création d’emploi (Incentive Bonus for Job Creation).

A labor input model is specified and estimated using quarterly data (disaggregated by major industrial branch) for France’s nonfinancial nonfarm sector. The resulting parameter estimates are used to forecast levels of employment for comparison with actual levels, beyond the estimation period, in order to assess the subsidy’s effectiveness. The ex post simulation for the third and fourth quarters of 1975 suggests that the Bonus led to a relatively modest rise in employment (about one seventh of the potential effect), concentrated in construction and public works. This outcome can be ascribed to administrative difficulties, stiff eligibility requirements for employees, and some degree of government intervention in the labor market.

A major conclusion of the paper is that a marginal wage subsidy, aimed principally at reducing cyclical unemployment, should be provided to firms for adding to their work forces (above a variable base level) rather than for retaining redundant employees. The specifics of the payment (amount, duration, etc.), and of employee eligibility should be calibrated to prevailing labor market conditions; also, special categorical features may be incorporated in the definition of the target population, in an attempt to direct the subsidy to the hardest-hit groups of unemployed, identified by age, region, or skill. With appropriate timing, such a subsidy can retard layoffs toward the end of a decline in economic activity and can accelerate hiring during a recovery, although it will have practically no impact during a deepening recession. A preferable instrument during a recession would be a general wage subsidy applied to the wages of the entire labor force.

Capital Requirements for Commercial Banks: A Survey of the Issues—brock k. short (pages 528-63)

This paper covers many issues that have arisen in the course of the debate over capital adequacy. One of the preliminary tasks is to define capital itself. The choices range from paid-up capital alone to the sum of it and other items such as retained earnings, undivided profits, general reserves, reserves for specific contingencies, and subordinated debt. Another topic is the relation between bank liquidity and bank solvency. Bank capital performs several functions. To bank owners and managers, capital is a means of attracting depositors and a source of funds, as well as protection for themselves against their bank’s failure. Capital is a concern of bank depositors and creditors since it provides protection for their funds, of bank borrowers since it provides protection of the banking system as a source of credit, and of the whole economy since it provides protection for the financial system.

Given this emphasis on the protective purpose of bank capital, the risks to which banks are exposed are considered. These range from the risk that deposit withdrawals will all come at once, through the risks of losses on loans and investments, to the risk of criminal acts. With this background established, the divergent interests in, and perspectives on, capital adequacy can be determined. From here, it is easy to see the possibility of continual divergence of opinion between bank shareholders and managers, on the one hand, and regulators, on the other.

Debate on capital adequacy has included the questions of whether a minimum amount of capital, a capital ratio, or no capital requirement at all should be imposed and what is an appropriate level for the requirement, if any. Within the past decade, bank earnings have been proposed as the first line of defense against losses. Finally, the paper surveys capital requirements in 42 countries as illustrations of the different proposals for regulating capital.

Improving the Demand-for-Money Function in Moderate Inflation—william h. white (pages 564-607)

Inflation reduces not only the real value of money holdings but also the real value of nonmonetary financial assets. Rebuilding stocks of both money and other financial assets must be a gradual process achieved through an increased rate of saving. It is argued that this dependence on saving implies that the customary constant speed of stock adjustment, regardless of the size and the direction of the necessary adjustment, should be abandoned.

Similarly, models specifying monetary variables in real terms may give erroneous results in periods of changing prices. If such models represent stock-adjustment lags, they understate the size of the desired adjustment. If they represent expectations-formation lags, they imply an instant “expectation” response to the price-level variable. That can yield the proper deflator for current nominal values only in the transition period between expectation of a normal level of prices, when inflation is not anticipated, and expectation of a normal level of the rate of price rise, when inflation has become the norm.

Other improvements suggested include an allowance for some zero adjustment lags—for the desired rise in financial assets provided by the normal rate of saving (itself an increasing function of the current rate of increase of real income); for transactions cash; for other parts of desired narrow money holdings that benefit from time deposits as a temporary buffer stock; and perhaps for the response to changing interest rates. Additional improvements may be obtained through rejecting models that yield elasticities of the expected inflation rate that are as high as the interest elasticity, and through deflating by a combination of population and number of households rather than by only one of those two variables. Some of the proposed corrections should account for most of the evidence provided by existing models of “disequilibrium” money holdings when the money supply is exogenous.

RESUMES

RESUMENES

In statistical matter (except in the résumés and resúmenes) throughout this issue,

Dots (…) indicate that data are not available;

A dash (—) indicates that the figure is zero or less than half the final digit shown, or that the item does not exist;

A single dot (.) indicates decimals;

A comma (,) separates thousands and millions;

“Billion” means a thousand million;

A short dash (-) is used between years or months (e.g., 1971-74 or January-October) to indicate a total of the years or months inclusive of the beginning and ending years or months;

A stroke (/) is used between years (e.g., 1973/74) to indicate a fiscal year or a crop year;

Components of tables may not add to totals shown because of rounding.

International Monetary Fund, Washington, D.C. 20431 U.S.A.

Telephone number: 202 477 7000

Cable address: Interfund

GOVERNMENT FINANCE STATISTICS YEARBOOK

Volume II

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Detailed data are provided for five years for most countries, extending at least through 1976 for three fourths of the countries and through 1977 for 25 countries. In addition, derivation tables for about half the countries indicate the national sources drawn upon and the detailed adjustments necessary to bring the data to the internationally comparable standards used in the Yearbook. The derivation tables are designed to facilitate future compilation of such statistics and to help to identify differences between data in the Yearbook and in other sources. The standards followed in the Yearbook are explained in a brief introduction, which is presented in French and Spanish as well as in English, as are the main statistical headings. Summary data on state and local government finances are provided for about 50 countries.

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*

Mr. White is a Senior Economist in the Financial Studies Division of the Research Department. He received undergraduate and doctoral degrees from Harvard University, and has been on the Senior Staff of the Brookings Institution. He has published articles in many economic journals.

1

Starleaf (1970) uses the misspecified real-terms model, rather than the equally serviceable nominal-terms model, as a point of departure for his attempt to reverse partial stock adjustment and to make elasticities higher in the short run.

2

See de Leeuw and Gramlich (1968). In their time-deposits equation on page 31, nominal values are used for demand deposits, but time deposits are deflated by money income and are made a function of time deposits lagged one quarter and deflated by money income lagged one quarter. This constitutes parallel errors of deflation for both prices and real income.

3

The first published recognition of this problem came in 1970, but it received little attention because of its terseness and its placement near the end of the final appendix of a long article. See Goodhart and Crockett (1970), p. 195. See also an almost encyclopedic study of demand-for-money functions in the United States, Goldfeld (1973).

4

If a wide range of interest rates is at issue and approximations to Keynes’ infinite demand for money at low rates are rejected, ln r in equation (2′) can be replaced by r. This implies that interest elasticity of demand diminishes as the rate declines, thus holding the demand for money to finite levels, even at negative interest rates.

5

A unitary long-run price elasticity is accepted for purposes of discussion in this study. However, its present empirical basis is not well founded because of the following: (i) the misspecifications of the models on which it is based, (ii) the confusion of type I and type II errors (reliance on the 5 per cent chance that a non-unitary estimated price elasticity was consistent with a true elasticity of unity and disregard of the mirror-image 5 per cent chance that it was also consistent with a true elasticity more different from unity than the estimated one), and (iii) occasional reliance on near-unitary values for the coefficient of P in equation (2′) rather than in equation (1′).

6

A simple cause of upward bias of the speed of adjustment in quarterly models is the expedient of using data from the end of the month to calculate the average money stock for the quarter. The correct average would be made up of the three figures from the middle of the month (or of daily averages, if they were available). But the easy procedure represents the quarterly average for a quarter starting one-half month after the calendar quarter. Hence, it would cause a true one-half month (discrete) adjustment lag to be recorded as a zero lag. The reduction in the estimated lag is sufficient to make a quite significant adjustment lag appear to be not significantly different from zero.

7

Insofar as mtmt-1, is positively correlated with mt1*mt2 but ΔPt is uncorrelated with mt*mt1, γ yields an unbiased estimate of γr.

8

The role of the short-term interest rate may seem a minor one, because interest elasticities are usually found to be quite small. Goldfeld (1973) and Khan (1974) have found that the income elasticities of demand for money were 3½ to 4 times the comparable interest elasticities.

However, the volatility of short-term interest rates is much higher than that of real GNP. To some extent, that greater volatility should hold also for the quarterly changes in short-term interest rates and in real income that are associated with quarterly changes in prices; this would make it more plausible that the sum of the direct price effect and the indirect effect via interest rates would more than offset the opposite indirect effect via real income.

9

Laidler (1977, pp. 143-44) has tentatively made a more developed form of this same kind of criticism, but he directs it against the conventional model: Individuals’ efforts to raise money holdings when nominal money is exogenous can be realized in the aggregate only through depression of the price level. But the conventional, stock-adjustment model in real terms did not seem suited for tracking the dynamic path of real stock adjustments achieved through price changes. As is shown in the next paragraph of the text, there is a case of exogenous rise in desired real money for which the conventional model is the correct one in other respects. And use of the observed, partly endogenous values of the variables remains consistent with correct specification of the dynamic adjustment path because mt* is a function of the unlagged, simultaneous values of the other, mutually-determined variables like yt and rt.

10

Where γΔlnmt* is the percentage of period t’s money receipts that is hoarded rather than returned to the flow of spending. It will be noted that γ Δln m* is smaller than the short-run percentage change in liquidity preference; it is an observed, partly endogenous value, reflecting the negative feedback from accompanying changes in interest rates and income on mt*.

11

See Feige (1967), p. 465. The passage cited actually refers to the buffer-stock role of non-M1 assets in maintaining the desired M1/Yperm ratio in the face of fluctuations of savings between positive and negative as current income fluctuates around permanent income. But that buffer role for non-M1 seems reasonably translatable into the role cited by Feige. (See Friedman (1959), p. 333.)

12

Evidence in support of an exclusive role for the expectations-formation lags that is offered by Dickson and Starleaf (1972, p. 1039) seems, in fact, to support a stock-adjustment lag even where the interest rate is concerned. A 4- to 5-quarter polynomial distributed lag in the effects of current income on money may be interpreted as solely the expectations-formation lag. But very little of the “remarkably similar” 4- to 5-quarter distributed lag in the effects of the short-term interest rate can be attributed to expectations-formation delays. For the most part, shifts between M1 and short-term, interest-bearing assets depend only on the instantly-known, current level of short-term rates. To some extent, the short-term rate may be serving as a proxy for the long-term rate, but in this case expectations-formation lags imply exceptionally rapid portfolio reallocations. For example, if the expected long-term interest rate lags behind a current rise in the long-term rate, the indicated shift to long-term bonds is accelerated, because a fall in long-term rates yielding a capital gain on bonds is expected.

13

Where a separate expectations-formation lag coefficient is used for each of the expected or “permanent” variables, the estimating equation will, of course, be different from the stock-adjustment form of the estimating equation; the latter will contain only a single partial adjustment coefficient. But the adjustment coefficients in the expectations equation could still represent a forced combination of an adjustment lag common to all of them and varying amounts of true expectations-formation lag.

14

Goldfeld mentioned trying Feige’s means of measuring the two kinds of lag separately, but his use of the procedure did not yield differences in results that were worth the effort; he had difficulty with the nonlinear constraints needed for estimating the separate lag. (See Goldfeld (1973), p. 601, footnotes 40, 43.)

15

A little more flexibility can be introduced into the traditional Koyck procedure by delaying the start of stock adjustment or of expectations formation for, say, one time period: MtMt1=γ(Mt1*Mt2).

16

The use of money income does not conflict with the view that the real income and price level elasticities should be permitted to differ from each other. As far as transactions cash is concerned, the funds needed for “turning over” money income are independent of the prices/real income composition of that money income. If the traditional optimum inventory formula were applied, the price level elasticity would be unity, but the real income elasticity would be constrained to be ½. (See footnotes 17 and 18.)

Because the second of these two expressions cannot show a stable relationship between its two terms, the following equations are not conveniently expressible in log form. Equations (11)(14’) in this paper are therefore valid in arithmetic form only.

17

Twenty per cent simple interest earned for one-half week on one half of a weekly salary would yield interest income equal to a mere 1/10 of 1 per cent of the salary (i.e., 10 cents on a $100 salary!). Such small amounts would not compensate for the transactions or inconvenience costs incurred. The ability to earn even those amounts is, of course, restricted by the exclusion of small savers from the organized short-term money markets and by the practice—often found among banks receiving small savings—of paying interest only on the minimum deposit balance during a month or a quarter or not paying interest on funds withdrawn before the end of the quarter.

18

See (1) Dickson and Starleaf (1972, p. 1042) and (2) Goldfeld (1973, pp. 594-95). A comparable change in structure, which has been ignored in econometric work, is the rapid elimination, between the end of the 1960s and about 1973, of the demand deposits of large U.S. enterprises that were “idle” for even one day. Banks automatically transfer title to part of their holdings of government securities for a brief “overnight” period. See Garvy and Blyn (1969), pp. 75 and 87; also see Rose (1977), p. 205. In his second paper, Goldfeld (1976, p. 721) cited this factor as a possible explanation for the downward shift in the mid-1970s in U. S. business’ demand function for M1; but, perhaps because he was unaware of its pervasiveness by the mid-1970s, Goldfeld did not test it econometrically. (By early 1978, nonfinancial corporations held these deposit substitutes in an amount equal to one third of their holdings of currency and demand deposits.) Combined with the demonstration in footnote 17 that holders of small amounts of money cannot reflect the Baumol-Tobin interest elasticity of temporarily idle transactions cash, this development may be the coup de grace for the “inventory-theoretic” analysis of demand for money in the United States.

19

The “idle” part of M1 could play a part of the buffer stock role, but such disaggregation of M1 is not feasible. The measurable amount of quasi-transactions cash will have to exclude the quasi-transactions cash that depends on an M1 buffer stock.

20

The use of broad money might seem to entail the complication of adding Eurobank deposits in the same currency to attain a stable money demand function. However, Day and Heller (1977, p. 707) have found that Eurodeposits are owned primarily by banks and that much of the rest of deposits are of a term that makes them eligible for inclusion only in very broad money. Eurodeposits would make an almost imperceptible addition to broad money.

At times of substantial exchange rate speculation, part of the money stocks of fixed-exchange or managed-floating-exchange countries may be shifted outside the country or may be inflated by foreigners’ shifts of their funds into the country. This may create a problem in obtaining good fits from the model used, especially where broad money is the dependent variable. The problem exists even if the capital shifted into a foreign country’s bank deposits is held in the form of domestic nonmoney assets, such as money market securities (unless the securities are sold to the central bank). To deal with this problem, the yields on foreign placements (interest rate plus expected rate of exchange appreciation) can be introduced into the model, or published estimates of amounts of money that are shifted from one nation to another during periods of exchange speculation can be used to construct a total of money owned by residents.

21

A partial offset to this advantage for broad money will exist where unforeseeable rises in the savings rate accelerate the growth of time deposits at the expense of M1. Maintaining the M2 target means failing to pass on the extra savings through extra credit expansion, thereby causing deflation. The M1 target will elicit the needed credit expansion. This disadvantage of using the M2 target may be outweighed by the previously-mentioned advantage of using it in developed countries where substantial rises in the propensity to save are concentrated on assets other than the time deposits included in M2.

22

It will be noted that the customary collection of the lagged endogenous terms Mt-1, is not possible in this formulation. However, a maximum-likelihood technique such as that developed by C. R. Wymer (1973) makes it possible to estimate the equation in this form.

23

The gross total of households’ assets in nominal terms at the end of 1975 was 1⅔ times the annual, seasonally-adjusted rate of disposable income in the fourth quarter of 1975. (See Board of Governors of the Federal Reserve System (1976), p. 66.) Three kinds of deduction from gross assets in monetary terms were made to reach the relevant total used above: one half of households’ debts, all of their marketable interest-bearing assets, and all of their (roughly estimated) stock of transactions cash.

Only half of the households’ debt was deducted because owner-occupiers tend to regard their predetermined monthly payments for interest and amortization as equivalent to rent payments under a long-term lease, instead of regarding the principal amount as a deduction from their financial assets. In the same way, the predetermined amounts of delayed payments for purchased consumer durables tend to be treated simply as part of the price of those durables; moreover, most of such high-cost consumer debt is owed by those who have minor holdings of financial assets and who therefore are especially unlikely to become aware of an inconsistency between their evaluations of financial assets and liabilities.

In the United Kingdom in 1973, liquid assets (very broad money) were at their long-run normal level of nearly 90 per cent of disposable personal income (73 per cent if all of the nonhousing borrowings from banks and hire-purchase borrowings are deducted). The ratio has fallen markedly since, but it was expected that a high savings ratio would be continued into the period of slowed inflation, so that the depleted stock of assets could be rebuilt. See “Economic Commentary” (1978), p. 154; “Sector Financing: 1977” (1978), p. 208; and “The Personal Sector 1966-1975” (1977), pp. 31-32.

The French ratio of liquid assets to disposable income is closer to the U. S. figure. The officially published ratio of broad money to quarterly disposable income for 1976 is 3.2—or 3.0 if short-term consumer finance and bank loans received are deducted. See Institut National de la Statistique et des Etudes Economiques (1978).

24

The shrinkage of households’ direct and indirect monetary claims on private businesses does not yield a reduction in their real wealth, insofar as they also own shares in the debtor businesses. That is the reasoning behind the customary use of only “outside” financial wealth in the wealth variable. But, even if total real wealth were unchanged, its composition would have shifted away from monetary assets; and the desired shift back from shares to monetary assets would be lagged, both because of transactions costs and because of the recognition that the desired shift would temporarily depress share prices. Friedman’s concession that some delay might exist in shifting from long-term bonds to money implies even more delay where the shift is from shares to money. Moreover, households will tend not to feel any benefit from the large part of outstanding shares held by pension funds, life insurance companies, etc. Finally, as discussed in White (1976), pp. 6-8, the shrinkage in the real value of an enterprise’s debt owing to inflation benefits the enterprise only insofar as banks and other lenders are willing to match the shrinkage with additional loans; but, up to now, they have been willing to only a limited extent.

Even if all these considerations were assumed away, part of “inside” financial assets would still have to be counted as creating wealth effects and, hence, stock-adjustment lags. This follows from the dispersion in wealth portfolios’ ratios of equity to net debt in monetary terms. To simplify, holders of portfolios consisting entirely of claims in monetary terms on domestic enterprises would be unable to buy the extra debt the enterprises wanted to issue to restore the pre-existing real value of their debt. The “wealth effect” specifies that only a fraction of the current period’s shrinkage in real wealth owing to rising prices can be made good through extra saving out of current income in any period. And it is unlikely that the debtors would be willing to issue the full amount of new debt if they had to raise the yield offered by enough to induce specialists in equity to shift their portfolios into debt by the necessary amount.

25

The tendency for there to be an accelerating rise in marginal utility of consumption as consumption falls below normal toward its irreducible level might be counterbalanced by a tendency for the marginal utility to fall faster as consumption rises above the normal level toward satiation. The discussion above is based on the assumption that the normal consumption level is relatively far from the level of satiation. The view that additional consumption will always have some utility, and, therefore, that marginal utility approaches zero asymptotically, also weighs against an accelerated decline in utility above the normal level of consumption.

26

Further discussion of this and related points is found in White (1976), p. 16. Households’ treatment of only the real part of the market interest rate as yielding spendable income, combined with business treatment of much of the purely nominal part of the same interest as an additional cost of production, is found to be a cause of “stagflation.”

This divergence between lender and borrower in valuation of the same interest payment may imply that the relevant total GNP perceived by economic agents is a little smaller than the published GNP figure in inflationary periods, although the two GNP figures are identical when prices are stable. Correction of the published GNP variable may sometimes be worth attempting. Correction of the published figures for personal income is obviously worth attempting where a separate demand function is being estimated for households.

27

The U. S. evidence that inflation stimulates saving in the form of nominal assets has recently been challenged, on both logical and empirical grounds, by Wachtel (1977). Logically, the existence of the lagged wealth effect depends on expected inflation being lower than actual inflation—a condition that, in the long run at least, was not satisfied (p. 562). But since the equality between expected and actual inflation is established only after inflation rates have reached a peak and leveled off, that consideration is far from conclusive when quarterly econometric models are used. Moreover, Wachtel overlooked the three additional grounds for the wealth effect presented on this and the preceding page.

Econometrically, Wachtel’s model should have verified increased financial saving owing to inflation. Because he used the official definition of personal income (p. 572), his espousal of instant “saving” of the nonreal part of interest income (p. 562) makes the higher financial saving a necessity. The failure to confirm that effect compels rejection of the econometric results. The inadequacies of the allocation of the available data on quarterly changes in financial assets between households and other sectors may have precluded meaningful results.

Wachtel’s econometric case for substitution of the “uncertainty effect” of higher inflation on the savings rate for the extra stock adjustments made necessary by faster inflation also raises difficulties. This uncertainty effect of inflation was verified by a finding that the dispersion of the expected inflation rates reported in consumer surveys (a proxy for the degree of uncertainty about the mean inflation rate reported) yielded a better fit than the mean expected inflation rate. But, since the reasoning is that faster inflation causes more uncertainty about the inflation rate, the two variables “are highly correlated” (p. 573). Hence a better fit with the “uncertainty” alternative (or an inadequate /-ratio when the inflation-rate variable is added) should have been found consistent with important roles for both the wealth effect and the uncertainty effect on saving.

The effect of uncertainty/inflation on net financial saving was limited to one on borrowing to finance consumer durables. But that limitation does not constitute support for the uncertainty effect over the wealth effect. Wachtel states (p. 558) that his uncertainty effect on net saving cannot be trusted until it is confirmed by a parallel effect on saving in the form of financial assets. Not only is reduced borrowing insufficient to vindicate the uncertainty effect but it also seems a dubious result of uncertainty. The faster inflation that causes uncertainty also reduces the real burden of debt, making it reasonable to suppose that the many persons enjoying a linkage or near-linkage of their money incomes to the cost of living index would increase their debt as inflation accelerated.

28

After the preparation of the present paper, a study making allowance for current savings in a variety of ways was published by Brillembourg (See Brillembourg (1978).).

29

See Chow (1966), equations (2) and (3), pp. 113-14. The stock-adjustment equation is combined with the long-run or desired money demand (equation (1)) in the standard way. Since the variables are said to represent values in real terms provided that money illusion is absent, it follows that the model is misspecified with respect to inflation. The exclusive and explicit use of the conventional model in real terms in Chow’s “Reply” (See Chow (1968), pp. 1241-42.) confirms this interpretation.

30

Brunner and Meltzer were unenthusiastic about their proposal for correction of the model—making the coefficient of the savings term a variable that is a function of interest rates and adjustment costs. The varying coefficient was disregarded in their own re-estimation, which used simply the nominal-money variant of the Chow model.

31

This model might continue to yield unsatisfactory results because of its reliance on a real-wealth variable. Thus Goldfeld (1973, p. 615) reported that his test of adding the change in real net worth of the private sector to the conventional equation improved the fit a little but weakened the goodness of dynamic simulation forecasts. He attributed the difficulty to the poor quality of his data; and if that problem exists in the United States, it can be assumed to be prevalent elsewhere.

The change in the real value of wealth may be more reliably measured by the sum of current savings and the change in the real value of pre-existing nominal assets than by the difference between the deflated money values of assets in successive quarters.

A further source of difficulty with the representation of wealth by deflated net worth is that the net worth concept contains only “outside” financial assets and therefore excludes the shrinkage in “inside” nominal assets’ real value caused by inflation. As described earlier, the shrinkage in real value of their mutual loan contract owing to inflation affects the borrowing enterprise less than it affects the lending household. A net effect on the relevant value for real wealth is therefore plausible.

32

Although earlier tests ruled out the dividend yield on equities, that yield plus the long-term bond yield were found to rescue the U. S. models of the demand for money, fitted through the early 1970s, that had failed to fit the actual money supplies of the mid-1970s. (See Hamburger (1977).)

33

Frenkel (1977, p. 658) has recently found that regressing desired money on expected, rather than current, prices in a zero-lag hyperinflation model “does not yield estimates that are significantly different from each other.” But that finding is inconclusive. Frenkel does not state whether the two ds were significantly different economically—an important matter, since (See footnote 37.) the expected inflation proxy seems an inadequate one; and omitting the (misspecified) stock-adjustment lag was itself subsequently found to be a probable misspecification. (See Frenkel (1977), pp. 660, 663.)

34

See Goldfeld (1973, pp. 609, 611). He indicates briefly an algebraic explanation for such results (See p. 611, footnote 56.) that is equivalent to the explanation above, although, as explained earlier, he ignored that explanation when considering his own findings.

35

A pioneering use of this approach, which involves the addition of survey findings on households’ expected future financial condition to the current quarter’s income, can be found in David (1970). This procedure also makes possible an allowance for those expected declines in real income that reflect fear of unemployment (and, therefore, desire for increased holdings of liquid assets).

36

The two kinds of divergence of the low-income group from the high-income group call for different kinds of adjustment. The lower share that the low-income group has in total financial assets than it has in total population calls for a lower weighting of the low group’s index of expected inflation. The presumption that the low-income group is less sophisticated in applying its expected inflation rate to its decisions on asset holdings (or that it is less able to do so because its members lack access to the market for real assets) calls for a reduction of the amplitude of its expected inflation series before the series is weighted for combination with the high-income group’s expectations. The two procedures are similar, but do not have identical effects. The lowering of weighting for the low-income group has no effect on the weighted average of the expected inflation rate when both groups expect the same inflation rate, but the reduction of the amplitude of change always has an effect.

37

In rapid inflation, where nominal interest rates are distorted by credit rationing, the forward discount may serve as a proxy for what the excess of a free-market interest rate over foreign interest rates would have been if they had not been distorted. Frenkel (1977) successfully applied this innovative proxy for the expected inflation rate to the demand for money in the German hyperinflation of the early 1920s. However, the difficulties encountered (the extremely large standard error of the forward discount as forecaster of the one-month rate of exchange depreciation, foreign speculators’ expectations of less inflation than residents expected for two thirds of the period, rise of foreign currency to dominance in Germans’ holdings of currency, tightened exchange controls) suggest that other procedures for improving on the original Cagan results (cf. Khan (1977)) are superior.

38

Carlson and Parkin (1975, pp. 132-33) found at least borderline statistical significance for dummy variables representing a number of political and economic shocks in the United Kingdom when they were regressing an unsatisfactory expected inflation survey on the inflation rate given by adaptive expectations plus dummies. More suitable opinion surveys might have yielded more significant values for the dummies.

39

In the United States, the historical record of one leading service’s forecasts may come close to the appropriate average of several surveys’ forecasts and of adaptive expectations. The Wharton Econometric Forecasting Unit strikes a compromise between its model’s forecast and the views of the leading business enterprises that subscribe to the service. The latter views reflect a weighted average of the (slightly earlier) views of the other forecasting services subscribed to plus the enterprises’ independent impressions of prospects arrived at via procedures like adaptive expectations.

40

Forming rational expectations on the basis of the official target for monetary expansion is almost impossible in cases such as that of the United States, in which the target is a range of rates that embraces both inflation and deflation, leaving the authorities relatively free to do what seems appropriate at any moment. The record of continual violations of the upper limit of the target range is a further obstacle to the formation of expectations on the basis of official targets.

41

Aggregation problems could bar satisfactory results from even the corrected population deflator. Changes in dependent population may be concentrated in the lower-income classes, whose members have both relatively low money holdings per household and an income elasticity of demand for money that is peculiar to their group. Cash holdings may be almost exclusively transactions balances, which are not much affected by the number of dependents per household. For this lower-income group, desired money may be best represented by income undeflated by any “population” proxy.

IMF Staff papers: Volume 25 No. 3
Author: International Monetary Fund. Research Dept.