The Role of Savings in Flow Demand for Money: Alternative Partial Adjustment Models

ARTURO BRILLEMBOURG *

Abstract

ARTURO BRILLEMBOURG *

ARTURO BRILLEMBOURG *

Economic theory is usually better suited to describing the long-run values of variables than their short-run or dynamic properties, although the latter may be of equal, if not greater, interest. In lieu of a strong theoretical foundation, economists have often resorted to ad hoc mechanisms that transform an otherwise static theory into a dynamic one. Nowhere has this been more prevalent than in studies of asset demand functions, particularly those concerned with the demand for money.

In studies that treat the flow demand for an asset, it is common practice to postulate a static or long-run stock demand function for the asset and then to transform this function into a dynamic function by invoking the partial stock adjustment model. This model states that the change in the demand for an asset (the flow demand) is proportional to the difference between the desired and the actual levels of the stock of that asset. While this model has been found to lack a strong theoretical foundation, its simplicity in application has given it a predominant role in empirical studies.1

In many cases, the dynamic properties of the flow demand functions have been augmented by postulating more complex formulations of the partial adjustment model and/or by assuming autocorrelation of the estimated residual errors. In general, this practice too has had no theoretical foundation and can be justified only on grounds of empirical validity.

The purpose of this paper is to reformulate the theoretical underpinnings of the partial adjustment model in such a way as to make this last step more theoretically defensible. In particular, this paper reformulates a simple but common version of the partial adjustment model under conditions of uncertainty. In doing so, it highlights the role of savings in the adjustment toward asset equilibrium. Indeed, perhaps the main contribution of this paper is that, by giving some theoretical foundation for a prevailing practice, it brings together two strands of the literature on the dynamic specification of asset demand functions: one that emphasizes the role of savings and the other that grows out of empirical pragmatism. It is hoped that, rather than providing some theoretical justification for a proliferation of variations on the partial adjustment model, this paper will induce a research effort to provide a more consistent and stronger theoretical underpinning for the alternative modifications of the partial adjustment model.

The paper is divided into five sections. Section I introduces the issues raised by the paper by discussing the role of savings in the adjustment to asset demand disequilibria. Section II derives, in the context of uncertainty, a general partial adjustment that takes into account the role of savings. Nowhere is the application of the partial adjustment mechanism more prevalent than in the empirical literature of the flow demand for money. It is, therefore, useful to illustrate, in Section III, the application of the general partial adjustment model to the demand for money by formulating various models of flow demand. Section IV estimates these various models using quarterly data for the United States during the period extending from the first quarter of 1960 to the first quarter of 1977. Finally, the conclusions of the paper are given in Section V.

I. Role of Savings in Adjustment to Asset Disequilibria

For expositional convenience, it is useful to consider a simple example in which an unforeseen capital loss causes a divergence between the desired demand for, and the existing stock of, an asset. This disequilibrium can be thought of as an opportunity cost that asset holders seek to minimize. Assuming that the existing stocks of, and the desired demands for, all other assets are in equilibrium, asset holders can minimize their opportunity costs by spreading the disequilibrium among many assets, that is, by selling part of the stock of the assets in excess supply and by replenishing (in part) the stock of the asset that suffered the capital loss and is in excess demand. Depending on the opportunity costs associated with the disequilibrium in each asset, asset holders can minimize their losses by spreading the disequilibrium among all the assets, but they will have to weigh the benefits thus derived against the transactions costs incurred in the selling and buying of assets. The partial adjustment model has been developed to model such behavior by the asset holder.

The common representation of the partial adjustment model, however, neglects a second alternative left to the asset holder, namely, the reallocation of savings. Typically, the asset holder receives these savings in the form of a menu of assets, although in many cases, it may only include liquid assets. As a reaction to the capital loss incurred in one asset, asset holders may be able to replenish this asset by acquiring the savings in the form of this asset or, alternatively, by selling the assets acquired in saving and buying the asset that needs replenishment. Consequently, reacting to asset disequilibrium by reallocating savings rather than the existing stock of assets may lead to lower transactions costs and faster adjustment.

Additionally, there is another, often neglected but possibly very important, alternative open to the asset holder. In reaction to the stock disequilibrium, the asset holder can reduce present consumption, that is, increase the actual flow of savings and invest these additional savings in the asset whose stock needs replenishment. In this case, asset holders will weigh the benefits of decreasing the asset disequilibrium against the costs of reducing present consumption as well as, perhaps, incurring some transactions costs. This third alternative may be a very important link between asset disequilibria and current consumption. It may be that this alternative provides a strong link between the effects of monetary and fiscal policy in the financial markets and the response of current consumption to these effects.

In summary, there are two possible adjustment reactions to asset demand disequilibria—reallocating the existing portfolio and directing the flow of savings to the assets that need replenishment. The latter reaction may take place by revising the menu of assets in which savings are accrued and/or revising the total amount of savings. The most common formulation of the partial adjustment model accounts for only the portfolio reallocation.

This deficiency has been noted in the literature. Brainard and Tobin (1968) were among the first to introduce savings explicitly into the partial adjustment model. By assuming, however, that the allocation of savings is predetermined, they remove its role in the adjustment process. Savings are given an active role in the adjustment process by Modigliani (1972) and Friedman (1977). In both cases, however, the authors restrict the analysis to the role of the reallocation of savings in the adjustment process. Modigliani takes a single-equation approach to the allocation of savings while Friedman takes a more general portfolio approach. Purvis (1978) endogenizes the savings/consumption decision in the context of a portfolio model but assumes that the allocation of savings is predetermined.

There does not seem to be a wholly satisfactory way of producing an empirically tractable model of the partial adjustment mechanism that includes the three possible reactions to asset demand disequilibrium.2 One can, nevertheless, reformulate the partial stock adjustment mechanism in a general form that separates the role of savings (and capital gains and losses) from that of portfolio reallocation in reaction to a disequilibrating shock. This reformulation is done in the next section.

II. A General Partial Adjustment Model Under Uncertainty

A model describing an asset holder’s behavior is most meaningfully developed in the context of uncertainty. In order to do so, however, one must model this context under a set of simplifying assumptions or stylized facts, so as to keep the model manageable. For this purpose, it is useful to assume that all asset holders have a common and fixed horizon period over which asset-holding decisions are made.3 In this context, asset holders form all expectations and enter into all contracts at the beginning of the period. At the end of the period, all flows are received and all contracts executed.

The institutional framework assumed in this paper can be outlined as follows. The essential assumption is that all assets are sold through auction markets in which participants make bids at the beginning of the period to buy or sell an asset at the end of the period. These bids or contracts may specify a range of quantities and/or prices over which the asset holder is willing to buy or sell. At the end of the period, the market executes all contracts and transactions for which there were matching buying and selling bids. It is evident that under this arrangement, not all bids need result in transactions. Moreover, depending on the institutional framework, there may be excess demands or supplies at the end of the period.4

The contracts for the allocation of savings are also entered into at the beginning of the period. In this case, the asset holder specifies the types of assets among which the flow of savings is to be allocated. For instance, in exchange for their labor services, asset holders may specify payment to be made in the form of liquid assets, pension fund contributions, mortgage or consumer credit repayments, etc. In payment for their capital services, they may receive liquid assets, capital gains, stock options, etc. There is a great potential variety of arrangements for the allocation of savings. For the purposes of this paper, however, the important characteristic of these arrangements is that either they do not entail any transaction costs, or that these costs are substantially less than those incurred in the reallocation of the total portfolio.

In this context, the task of the asset holder is to enter into contracts at the beginning of the period that will minimize the costs incurred at the end of the period. Following the traditional arguments for the partial adjustment models, the asset holder minimizes two types of costs: disequilibrium costs, which are incurred when there exist excess demands or supplies at the end of the period, and transactions costs, which are incurred in the execution of contracts that lead to transactions.

The most common way to model the disequilibrium costs is to postulate that they are proportional to the square of the difference between the actual (At) and the desired (At*) stock of the asset at the end of the period. Similarly, the transactions costs are assumed to be proportional to the square of the total value of transactions in an asset that take place at the end of the period.5 In the above context, the amount of these transactions must be the difference between the actual stock held at the end of the period and the stock held at the beginning of the period, adjusted for both capital gains (losses) and savings (dissavings) incurred in the asset in question. For notational ease, this adjusted stock at the beginning of the period, At0 is referred to as the passive stock of an asset at the end of the period. Adding the two costs together, the overall cost at the end of the period, Ct, is

Ct=a(AtAt*)2+b(AtAt0)2(1)

While equation (1) is similar to the common representation of the costs that lead to the partial adjustment model, it differs from the latter in two important ways. First, the passive stock is usually made identical to the stock at the beginning of the period. This assumption excludes both savings and capital gains from performing their roles in adjustment to asset equilibrium. For example, if asset holders expect wealth to grow at 10 per cent, they could avoid all transactions costs by acquiring assets that yielded an expected capital gain of 10 per cent rather than those that returned 10 per cent in liquid assets at the end of the period.6 Since both capital gains and savings reduce the amount of rebalancing of the portfolio needed at the end of the period, the neglect of these factors leads to an overestimation of the transactions costs involved.

The second way this model differs from the common representation is that the costs at the end of the period are unknown to asset holders at the beginning of the period. At the beginning of the period, the asset holders are uncertain about the actual value of transactions incurred, the actual stock of the asset, and even the desired stock of the asset at the end of the period. For example, if asset holders desire to hold a fixed proportion of their wealth in the form of a given asset, the desired stock of this asset is unknown to them at the beginning of the period because the value of wealth at the end of the period is uncertain. Thus, in view of this uncertainty at the beginning of the period, asset holders seek to minimize the costs expected at the end of the period.

In this paper, it is assumed that expectations are formed rationally. This assumption implies that the actual stock at the end of the period, At, is equal to the expected stock EAt, plus an error term, which represents the unexpected shocks to the stock of the asset at the end of the period. This error term has an expected mean of zero and is serially uncorrelated,

At=EAt+et(2)

At the beginning of the period, the expected costs to asset holders are derived by taking expectations of equation (1), using the definition of variance,

ECt=a(EAtEAt*)2+b(EAtEAt0)2+aV(AtAt*)+bV(AtAt0)(3)

where E is the expectations operator and V is the expected variance operator.

Asset holders will minimize the expected loss by setting equal to zero the partial derivative of the expected loss with respect to the expected stock.7

ECtEAt=2a(EAtEAt*)+2b(EAtEAt0)=0(4)

By rearranging equation (4) and substituting into equation (3), the general partial adjustment model under uncertainty is derived,

At=θEAt*+(1θ)EAt0+et(5)

where θ = a/(a+b).

In the general partial stock adjustment model, the actual stock at the end of the period is a weighted average of the desired and passive stocks expected at the end of the period. Consequently, the use of this model requires the specification of both the desired and the passive stocks. This paper is most concerned with the specification of the latter, taking the specification of the desired stock to be outside its scope. The next section presents a number of alternative approaches to the specification of the present stock.

III. Alternative Partial Stock Adjustment Models: The Flow Demand for Money

As mentioned in Section I, a theoretically satisfactory description of the saving process that is also empirically manageable has not yet emerged in the literature. All empirically oriented formulations involve some simplifying assumptions. The task of this section is to present a variety of partial adjustment models derived from the general formulation in Section II under different specifications of the saving process. The models presented here are most appropriate in the specifications of single-equation, rather than portfolio, models of the flow demand for an asset. Since the single-equation approach is most prevalent in the empirical literature on the demand for money, it seems useful to derive alternate models of the flow demand for money using various approaches for the hoarding (defined as savings in the form of money) process.

For the flow demand for money, the general partial adjustment model presented in the previous section can be rewritten as

mt=θEmt*+(1θ)Emt0+et(6)

where the actual stock mt is a weighted average of the expected desired stock Emt* and the expected passive stock Emt0, and where all stock variables are in logarithms.

This section proposes five different specifications of the expected passive stock and, therefore, five different models of the flow demand for money. In the first and simplest model (Model A), hoarding proceeds at a constant rate. This assumption leads to a model very close to a common specification of the flow demand for money. Models B and C are modifications of Model A in which hoarding reacts to past short-run disequilibria. These models are shown to be equivalent to Model A with the assumption of moving-average (Model B) or autoregressive (Model C) error terms. Model D assumes that hoarding depends on both short-run and medium-run past disequilibria. In assuming an adaptive expectations function, the last model (Model E) shows itself to be a special case of the more general class of models in which the passive money stock depends on a distributed lag of past actual money stocks.

MODEL A

Model A is the simplest of all models presented here, and, indeed, the rest of the models can be considered to be modifications of it. In this model, hoarding is assumed to grow at a constant rate g. Consequently, the actual passive money stock mt0 is the sum of the actual money stock at the beginning of the period, mt-1, less the rate of capital loss (P^t, the rate of inflation) plus the rate of growth of hoarding g,

mt0=mtkP^t+g(7)

The expected passive stock is derived by taking expectations of equation (7),

Emt0=mt1EP^t+g(8)

For notational convenience, the constant growth term will be dropped below, since it only affects the constant in the equation. For the same reason, a new variable is introduced—the adjusted lagged money stock m’t-1. It is, simply, the actual money stock at the beginning of the period less the expected rate of inflation,

Emt0=mt1=mt1EP^t(9)

Substituting equation (9) into equation (6), the flow demand for Model A is given by

mt=θEmt*+(1θ)mt1+et(10)

The reader may recognize that Model A has the same form as a common specification of the flow demand for money. However, it is most akin to the common partial adjustment model when it is specified in nominal, rather than real, terms.8 To see this, notice that the adjusted lagged money stock is the lagged money stock deflated by the expected price level. Rewriting equation (10),

MtPt=θEmt*+(1θ)(Mt1EP)+et(11)

where Pt is the logarithm of the price index. Rewriting equation (11),

MtMt1=θ(EMt*Mt1)+ut(12)

where Mt* is the desired nominal stock of money and ut = et + Pt—EPt.

MODELS B AND C

For many purposes, it has been found that Model A is not an acceptable representation of the flow demand for money. Recall that the general partial stock adjustment mechanism presented in the previous section does not assume short-run equilibrium. Consequently, it is reasonable to postulate that hoarding reacts to past disequilibria.

In Model B, hoarding reacts only to the unexpected shock experienced in the previous period, et-1. The sign of the reaction, however, cannot be determined on a priori grounds. If the allocation of savings among different assets is easily changed, one would expect that an excess supply of money in the previous period would lead to a decrease in expected hoarding during the current period. On the other hand, if the allocation of savings cannot be easily changed, it may be reasonable to expect that the shock that led to an excess of hoarding in the previous period may be repeated in the current period. Thus, in Model B, the expected passive money stock is equal to the adjusted lagged money stock less a proportion γ of the previous period’s shock et-1.

Emt0=mt1γet1γ0(13)

Substituting equation (13) into equation (6)

mt=θEmt*+(1θ)mt1+eλet1(14)

where λ = (l-θ)γ

Note that Model B is equivalent to Model A under the additional assumption that the residual error follows a first-order moving average. In this case, equation (14) is most easily estimated when it is transformed into the following form:

(L)mt=θ(L)Emt*+(1θ)(L)mt1+et(15)

where (L)=Σi=0λiLi;; the lag operator L is defined as LiXt = Xt-i; and – < γ < 1.

Model C can be considered to be a generalization of Model B. In Model C, hoarding reacts to all past short-run disequilibria. For simplicity, it is assumed that the reaction is an exponentially declining weighted average of all past shocks.

Emt0=mt1+α1γLet1=mt1+αΣi=0γiLiet1(16)

Substituting equation (16) into equation (6),

mt=θEmt*+(1θ)mt1+et+(1θ)α1γLet1(17)

Transforming equation (17) to make it amenable to estimation,9

(1γL)mt=θ(1γL)Emt*+(1γL)mt1+et(18)

It is interesting to note that Model C is equivalent to Model A after the Cochrane-Orcutt transformation has been applied to it. This transformation is performed when a model is suspected of yielding first-order autocorrelated residuals, an assumption usually made in the empirical literature on the flow demand for money.

MODEL D

Model D builds upon Model C by assuming that hoarding reacts to both short-run and medium-run past disequilibria, where the latter are measured by divergences between the desired and adjusted lagged money stocks. The justification for this extension is that the allocation of savings may be more responsive to medium-run than to short-run disequilibria.

EMt0=mt1+ψ(mt1mt1*)+α1γLet1(19)

Substituting into equation (6),

mt=θ(1λL)Emt*+[1(1+λ)θ]mt1+et+(1θ)α1γLet1(20)

where λ = (1–θ) ψ/θ.

Transforming into the estimation equation,

(1γL)mt=θ(1γL)(1λL)Emt*+[1(1+λ)θ](1γL)mt1+et(21)

MODEL E

In the previous models, hoarding has been assumed to react to short-run and/or medium-run disequilibria in the flow demand for money. Model E takes a somewhat different approach, in that it reacts to divergences between the actual stock of money at the beginning of the period and the passive money stock expected at the beginning of the period. This approach is very reasonable in view of the theory that leads to the general partial adjustment model. Recall from Section I that adjustments through changes in the hoarding process and/or through capital gains or losses are less costly than adjustments through portfolio reallocation. If this hypothesis is correct, then the asset holder seeks to minimize portfolio reallocation at the end of the period by changing the amount of hoarding expected to accrue during the period. Therefore, one can postulate that hoarding will increase in response to a discrepancy in the previous period between the adjusted money stock and the expected passive money stock.10

Emt0=mt1+γ(mt1Emt10)(22)

Transforming and substituting equation (22) into equation (6),

mt=θEmt*+(1θ)(1+γ)Φ(L)mt1+et(23)
where
Φ(L)=Σi=0(γ)iLi
.

Equation (23) brings out the fact that Model E is a special case of the more general case in which the distributed lag function ϕ(L) is more general than the one assumed by this model.

IV. Estimation of Various Models of Flow Demand for Money

The previous section developed various models of the flow demand for money by specifying various models of the expected passive stock of money. In this section, these models are estimated using a simple functional form for the desired demand for money. The expected desired stock of money Emt* is a function of the logarithm of real income yt and the nominal interest rate rt,11

Emt*=a0+a1y1+a2rt(24)

where a1 > 0 and a2 < 0.

Table 1 presents the estimation results of the various models of the flow demand for money, using the estimating equation derived in the previous section and substituting equation (24) for Emt*. In general, all the models give similar results. The estimates of both the short-run income elasticity and the short-run interest rate semi-elasticity show little variation from one model to another. There are, nevertheless, criteria that enable one to distinguish among these models.

The first such criterion is consistency with the underlying theoretical models. Except for model E, all models estimate a negative sign for the speed-of-adjustment coefficient θ. For Models A, C, and D, the estimate is significantly different from zero at a 95 per cent confidence level, as well as being negative. Unfortunately, a negative speed-of-adjustment coefficient violates the assumptions under which the partial adjustment model is derived. Only Model E estimates a speed-of-adjustment coefficient with a positive sign. However, even in this case, it is not significantly different from zero and has a point estimate (0.1) that is much too low to be reasonable, as it implies a mean adjustment lag of 25 years. On the other hand, the long-run income elasticity does seem to be reasonable with a point estimate of 1.6—which is not significantly different from one. Consequently, although it does not yield entirely satisfactory results, Model E is the only model that yields results consistent with the theory of the general partial adjustment model.

Table 1.

Alternative Flow Demand for Money Models: Quarterly U.S. Data, First Quarter of 1960 to First Quarter of 19771

article image

Figures in parentheses are standard errors of coefficients.

R-squared adjusted for degrees of freedom, calculated using the logarithmic change in real money balances as the dependent variable.

Sum of squared residuals.

Nonlinear estimation performed using a search procedure. ϕ(L) approximated by truncating after four lags.

Nonlinear estimation performed using the Cochrane-Orcutt technique.

Nonlinear estimation performed using a search procedure in combination with the Cochrane-Orcutt technique.

Considering the criterion of goodness of fit, Model E again turns out to be the preferred model. Calculated with respect to the change in the logarithm of real money balances, R-squared, adjusted for degrees of freedom, is about 0.7 for all models, ranging from 0.67 for Model A to 0.75 for Model E. The results for Model C are of particular interest, since this model is found so often in the empirical literature on the demand for money.12 This model assumes that the residual errors follow a first-order autoregressive process. The results here suggest that it is more reasonable to assume that they follow a first-order, moving-average process as in Model B. On goodness-of-fit grounds, however, Model E is preferred even over Model B, giving a significantly improved adjusted R-squared compared with the latter.

In summary, the estimation results indicate that Model E is the preferred model on grounds of both theoretical consistency and goodness of fit. On the other hand, Model C, the model most often used, yields theoretically inconsistent results and is a poor third best according to goodness-of-fit criteria. Even Model E, however, does not yield entirely satisfactory results, suggesting that further research is needed on the specification of the flow demand for money.

V. Summary and Conclusion

This paper attempts to provide an empirically manageable partial stock adjustment model that accounts for the role of savings in the adjustment process. It is shown that the management of the allocation among different assets, as well as the total flow of savings, can play an important role in minimizing the costs of adjustment to asset dis-equilibria. Unfortunately, great difficulties are encountered in deriving a theoretically valid but empirically manageable model that accounts for the role of savings.

This paper attempts to develop such a model by deriving a partial adjustment model that distinguishes the role, in the stock adjustment process, of portfolio reallocation from that of savings. Derived under conditions of uncertainty, a general partial adjustment model is presented. Application of this model, however, depends on the modelling of the savings process. To show various ways of modelling this process, the formulation of the flow demand for money is attempted. Various models are presented and tested using U.S. data for the period extending from the first quarter of 1960 to the first quarter of 1977.

The usefulness of the approach taken here is that it provides some theoretical foundation for the prevalent practice of assuming autocor-related residual errors in the flow demand for money. It is shown, however, that the model of the hoarding process that is compatible with this assumption is but one of various reasonable models. In fact, the estimation results show that using other models not only provides theoretically more defensible results but also gives better fits of the data.

REFERENCES

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*

Mr. Brillembourg, economist in the Special Studies Division of the Research Department, is a graduate of Harvard University and of the University of Chicago.

1

For an attack of this model, see Griliches (1967).

2

See Mukherjee and Zabel (1974) for a theoretical model of dynamic asset allocation under uncertainty.

4

If, for example, there are more offers to buy than to sell at a given price, all the assets will be sold, but there will still be an excess demand for that asset. The type of arrangement to distribute assets among demanders will depend on the market.

5

See Griliches (1967). For simplicity, the reallocation of savings is assumed to be costless.

6

In other words, in a world of transaction costs, there may be a premium paid for equity rather than debt control.

7

It is assumed here that the variances of the disequilibrium and transactions costs are constant. In a less restrictive context, they could be responsive to the horizon period. For example, in periods of turbulence, the costs may be decreased by making rebalancing decisions more often than in tranquil periods, since the variance of these costs depends positively on the number of time units ahead that the forecast of the costs is made. This generalization is left for future research.

9

The careful reader will note that the term [(l–θ)α–γ]et-1 has been dropped from the equation in order to simplify the estimating equation.

10

It may seem more natural to postulate a reaction to the divergence between the desired and the expected passive money stocks. Unfortunately, this leads to a complex nonlinear estimating equation.

11

All variables are centered in the middle of the period. The interest rate is a period average. Both the actual current and the adjusted lagged money stocks are averages of data at the end of the period. Data source is the Fund’s International Financial Statistics. It is noted that the interest rate is not in logarithms. A test of Model C using interest rates in logarithms showed slightly worse fits. The theoretical advantage of this formulation is that it treats the inflation rate and the interest rate in a parallel manner. In order to simplify the estimation procedures, equation (7) uses actual variables, rather than the expected variables required by the general partial adjustment model. Similarly, the lagged money stock is adjusted by the actual, rather than the expected, inflation rate.

12

For surveys of this literature, see Goldfeld (1973 and 1976).