Modeling the Demand for Liquid Assets: An Application to Canada

Donal Donovan

doi:10.5089/9781451930443.024.A003

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Modeling the Demand for Liquid Assets: An Application to Canada

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Donal Donovan

Donal Donovan
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Publication Date:: 01 Jan 1978

eISBN:: 9781451930443

Language:: English

Keywords:: SP; elasticity estimate; TML term deposit; asset demand; normalized price vector; expenditure elasticity; Price elasticity; Private savings; Total expenditures; Food prices

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IN RECENT YEARS, monetary economists have shown considerable interest in attempting to measure both the substitutability/complementarity relationships among the liabilities of various financial institutions and their elasticities of demand with respect to changes in total wealth. The reasons for their interest are clear. Knowledge of these relationships is important for a wide range of interrelated policy questions, such as, for example, the appropriate definition of money and the stability of a narrowly defined demand for money function, the effects of interest rate competition among various financial intermediaries, and the desirability of subjecting the nonbank deposit-taking financial institutions to monetary controls similar to those applied to commercial banks. The answers to these questions involve empirical judgments, and often depend in part upon the particular numerical values assumed for the relevant set of price and wealth elasticities.

Abstract

Thus, by this time, a large body of empirical literature exists dealing with money/near-money relationships. In most of the research undertaken, however, investigators have adopted the approach of estimating a wide variety of essentially ad hoc, unrelated specifications of liquid asset demand functions. ¹ The resulting lack of an explicit unifying theoretical framework, in addition to being methodologically unsatisfactory, renders it difficult to interpret and compare alternative results obtained by different researchers in a meaningfully consistent manner.

Motivated in part by these considerations, a number of researchers have recently shown considerable interest in a technique first introduced by Chetty (1969). Rather than specifying arbitrary liquid asset demand functions for estimation purposes, Chetty suggested that one derive the demand functions from an explicit model of utility-maximizing behavior, where the services of liquid assets are assumed to enter as variables in some underlying preference function. This assumption is essentially the “direct utility of money” hypothesis advanced by Milton Friedman (1956), which, it can reasonably be argued, has in fact formed the implicit basis for most of the empirical studies in this area. ² However, Chetty was the first author to attempt to consider the important implications in empirical work of such an hypothesis. Adopting this method, Chetty estimated the elasticities of substitution between various liquid assets using U. S. time-series data. Subsequently, a number of more recent papers have appeared using essentially the same model; for the United States, further work has been undertaken by Bisignano (1974), Short and Villanueva (1975), and Moroney and Wilbratte (1976), while the article by Short and Villanueva (1977) contains a similar application to Canadian data.

The use of an explicit theoretical model represents a potentially important contribution to the literature on the issue of money/nearmoney substitutability. The basic motivation underlying the original Chetty model is to be welcomed; nevertheless, there are a number of fundamental problems in his formulation of the utility-maximization problem and his estimating procedures. These difficulties relate principally to the nature of the relevant price variables, the formulation of the corresponding budget constraint, the fact that Chetty does not in fact estimate demand functions as conventionally defined, and his failure to take account of the constraints implied by the use of an explicit demand theory model. Furthermore, these problems have not been recognized or dealt with by any of the recent papers just cited. In view of the continued usage of the Chetty approach in the literature, it is desirable to point out and clarify the nature of these problems. This forms the first objective of the present paper.

Second, as will become clear from the following discussion of the Chetty type of model, the problems present in his and others’ work stem essentially from a failure to take advantage of many of the concepts and techniques of modern consumer theory. Thus, the second objective of this paper is to rigorously derive a system of aggregate liquid asset demand equations from a household model of utility-maximizing behavior by applying the concept of the rental price of a durable good to the case of money and also by exploiting recent developments in the use of duality theory. Finally, the model is applied to annual household Canadian data for 1952–74. It may be pointed out at this stage that the empirical results, in addition to providing some interesting and useful information regarding different measures of liquid asset substitutability and wealth ³ elasticities (and their stability over time), also permit of an explicit test of the utility-maximizing hypothesis underlying the model.

The remainder of the paper is organized as follows. Section I explains and discusses the problems present in the model used by Chetty (1969) and other adherents of his approach. Section II presents an alternative approach using the same underlying assumption and derives aggregate liquid asset demand equations, as well as the corresponding elasticity measures, from a specific representation of individual household preferences. Section III is concerned with econometric and data considerations, while Section IV presents and discusses the estimation results. Finally, Section V contains a brief summary of the main empirical findings as well as some concluding comments.

I. The Chetty Model

The model first introduced by Chetty (1969) is described by the following equations for the representative household ⁴

Maximize

\begin{matrix} U = (β_{0} {X_{0}}^{- ρ_{0}} + β_{1} {X_{1}}^{- ρ_{1}} + … + β_{n} {X_{n}}^{- ρ_{n}}) \frac{- 1}{ρ_{0}} & (1) \end{matrix}

with respect to X₀, X₁, … , X_n

Subject to

\begin{matrix} W = \frac{X_{0}}{1 + r_{0}} + \frac{X_{1}}{1 + r_{1}} + … + \frac{X_{n}}{1 + r_{n}} & (2) \end{matrix}

where U is a generalized constant elasticity of substitution (CES) utility function, and X_i, r_i, (β_i, ρ_i) are the dollar values (at the beginning of the next period), interest rates, and parameters, respectively, associated with the ith liquid asset (i = 0, 1, … , n). Conventionally, X₀ is assumed to be noninterest-bearing money, and thus r₀ equals zero. W is the dollar value of this period’s financial (liquid asset) wealth.

Using the Lagrangean method to solve the maximization problem given by (1) and (2) yields (N + 1) first-order marginal utility conditions given by (3), together with the first-order “wealth constraint” condition (4)

\begin{matrix} \begin{matrix} \frac{\partial U}{\partial X_{t}} = \frac{- 1}{ρ_{0}} {(Σ_{i = 0}^{n} β_{i} {X_{i}}^{- ρ i})}^{\frac{- 1}{ρ_{0}} - 1} (- ρ_{i} β_{i} {X_{i}}^{- ρ_{i} - I}) = \frac{λ}{1 + r_{i}} \end{matrix} \\ i = 0, 1, …, n & (3) \end{matrix}

\begin{matrix} w = Σ_{i = 0}^{n} \frac{X_{i}}{1 + r_{i}} & (4) \end{matrix}

Chetty now solves the (N +1) equation system (3) by eliminating the Lagrangean multiplier λ to obtain the following N equation system (5)

\begin{matrix} \begin{matrix} \log X_{i} = \frac{- 1}{ρ_{i} + 1} \log \frac{β_{0} ρ_{0}}{β_{i} ρ_{i}} - \frac{1}{ρ_{i} + 1} \log \frac{1}{1 + r_{i}} + \frac{ρ_{0} + 1}{ρ_{1} + 1} \log X_{0} \end{matrix} \\ i = 1, …, n & (5) \end{matrix}

The system described by (5) forms the basis for Chetty’s empirical work.

There are a number of problems associated with Chetty’s procedure. The first point to notice is that the equation system (5) is not a system of asset demand equations in the sense of relationships expressing the endogenous quantity of each liquid asset demanded as a function only of exogenous prices (interest rates) and “income” (or wealth, in this case). The system derived as (5) does not take into account the last first-order condition, namely, the “budget constraint” (4) and thus wealth (W) does not appear as an exogenous variable. A far more satisfactory approach would consist of solving the entire set of (N + 2) first-order conditions (3) and (4) to obtain a set of (N + 1) estimating equations for X₀, X₁, …, X_n, each containing as arguments the exogenous variables, prices (interest rates), and wealth. ⁵ As is explained in the following section, the use of results derived from duality theory allows this to be achieved very easily for any generalized preference ordering.

The second difficulty concerns the nature of the so-called budget constraint (2). The arguments in the utility function fundamentally represent the flows of monetary services yielded by each asset. It is normally assumed, for example, that the monetary services yielded by each asset are proportional to the average stock held of that asset, and thus it is legitimate to speak equivalently of the demand for an average stock and the demand for the associated service flow. In formulating the budget constraint, however, the relevant prices must be the prices of the service flows yielded by the liquid assets, also referred to as the rental prices of the assets. The budget constraint should then express the condition that the sum of the expenditures on the service flows yielded by each asset (that is, the sum of rental expenditures) be less than or equal to total expenditure on monetary services. The concept of the rental price of a liquid asset is developed more precisely in the following section. It suffices here to note that Chetty’s so-called budget constraint (4) does not express this condition.

A number of other more straightforward problems present in the Chetty (1969) model should also be noted. The generalized CES utility function, while not homothetic (thereby implying unitary “wealth” elasticities), is strongly separable. That is, the marginal rate of substitution between any two assets i and j is independent of the level of any other asset k. This condition may be unduly restrictive from a behavioral point of view and deserves to be relaxed. ⁶ Furthermore, in his empirical work, Chetty does not simultaneously estimate the system of equations given by (5). That is, from an econometric point of view, no account is taken of possible contemporaneous correlation among the residuals in the N equations. In addition, and this is an economic issue, Chetty does not impose the across-equation restrictions on β₀ and ρ₀, implied by (5). However, this defeats one of the main purposes of the explicit utility function approach, namely, to derive various elasticity estimates consistent with one underlying preference ordering.

The problems just discussed are also present in those papers subsequently adopting the Chetty model as a basis for empirical analysis (Bisignano (1974), Short and Villanueva (1975; 1977)). In a recent paper, Moroney and Wilbratte (1976) employ a similar type of liquid asset model that is based on an alternative theoretical structure but, they suggest, leads to the same estimating equations as those of Chetty. The alleged similarity between the two models, however, results solely from a failure to consider the last first-order condition in either case. The implied demand equations from the two models, when derived taking that condition into account, will be quite different. ⁷

To summarize, although the models that have been discussed are all based explicitly on a utility-maximizing hypothesis, their authors have not exploited many of the standard methods and techniques of modern applied consumer demand theory. There is need for a model in which liquid asset demand equations (in the conventional sense of the term) are derived carefully from a generalized representation of household preferences, and are estimated taking into account the economic and econometric considerations implied by the adoption of the utility-maximizing framework. Furthermore, it is desirable to investigate, by examining the properties of the resulting estimated preference ordering, whether the observed data are consistent with the basic utility-maximization hypothesis underlying the model. The development and application of such a model to Canadian data constitute the remaining sections of this paper.

II. The Theretical Model

the individual consumer and the rental price of money

The representative consumer is assumed to maximize the following utility function (12) subject to a budget constraint given by (13)

\begin{matrix} Maximize & U = f (x) & (12) \end{matrix}

with respect to x

\begin{matrix} Subject to & p^{t} x = y & (13) \end{matrix}

where U = f(x) is a well-behaved ⁸ preference ordering defined over the services yielded by an n x 1 vector of money and money substitutes x, p is an n x 1 vector of the prices of the service flows yielded by x, and y is a total expenditure on the services of money and money substitutes (t denotes vector transposition). The model summarized by (12) and (13) assumes that liquid assets are weakly separable from consumption services and leisure in the consumer’s utility function; that is, the marginal rate of substitution between any two liquid assets is independent of any “real” choice variable of the household. This assumption may be unduly restrictive, and the consequences of its relaxation have been analyzed both theoretically and empirically in other work by the author (Donovan (1977)). For the purposes of this paper, however, the more conventional approach of analyzing the liquid asset allocation decision independently of other household decisions is adopted.

The service flows yielded by money and money substitutes are assumed to be proportional to the real value ⁹ of the stocks of each form of asset held. As a result, the demand for the flow of services from x may be represented as a demand for the real stock of x held. The relevant prices for inclusion in the budget constraint (13), however, are the prices of the services yielded by x, or (what is equivalent, owing to the stock/service flow proportionality assumption) the one-period imputed holding cost or rental price of the liquid asset.

To further explain the concept of the rental price of money, it is useful to examine the analogous concept for a “real” consumer durable good. ¹⁰ Consider the nth consumer good, x_n, with which is associated a one-period depreciation rate, δ_n, 0 ≤ δ_n ≤ 1. If δ_n = 1, then x_n is an ordinary (nondurable) consumer good. It is assumed that a depreciated durable good is considered equivalent to the new good, except that it is smaller in quantity. Denote the current price for the services of x_n by p_nt. Let $p_{n t}^{*}$ be the current purchase price of x_n, and let $p_{n, t + 1}^{*^{e}}$ be the expected resale price of x_n in period t + 1. Denote the nominal rate of return on “bonds” by R_t. Then, the rental price, or imputed one-period holding cost, P_nt, is given by the cost of the purchase of x_n in this period minus the discounted expected resale ¹¹ value of the depreciated good in the next period, that is,

\begin{matrix} p_{n t} = p_{n t}^{*} - \frac{(1 - δ_{n}) p_{n, t + 1}^{*^{e}}}{1 + R_{t}} & (14) \end{matrix}

Consider now formula (14) applied to money and near money. The problem is examined under two alternative assumptions about household behavior: (i) households are assumed to hold static expectations as regards the general price level; (ii) households expect some positive (or negative) rate of inflation to occur.

Static expectations

Let us assume that there is only one real good, namely, “food,” consumed by households and, furthermore, that the factor of proportionality linking monetary services to the real value of the liquid asset is unity. Both of these assumptions are made purely for expositional convenience and do not affect the results. Consider first noninterest-bearing money. Denote the price of food at time t by p_ft, expressed in dollars per unit of food. By assumption, $p_{f, t + 1}^{e}$ , the expected price of food in period t + 1, equals P_ft.

Suppose that the consumer holds X nominal dollars of money form i in period t, denoted X_it, which implies that he holds x_it (≡ X_it/p_ft) units of monetary services. The first term in equation (14), the dollar purchase price, denoted $p_{x i t}^{*}$ , of a unit of monetary services is then equal to p_ft, the price of food. That is, in order to be able to buy one unit of food (or, equivalently, to hold one food unit of monetary services), the consumer must pay p_ft dollars.

Now examine the second term in (14). In the next period, since the price of food is expected to remain the same, the consumer expects to still hold x_it food units of monetary service (that is, he expects to be able to buy X_it/P_ft units of food). Thus, unlike ordinary durable goods, where it is typically assumed that the service flow is reduced in the next period owing to physical depreciation, for nominal money balances, with no change in the general price level assumed, there is no depreciation. Furthermore, the value of each unit in the next period is expected to be unchanged, equaling p_ft. Thus, from equation (14), for noninterest-bearing money, the rental price of monetary services, $p_{x_{i t}}$ , is then

\begin{matrix} p_{x_{i t}} = p_{f t} - \frac{p_{f t}}{(1 + R_{t})} & (15) \end{matrix}

Suppose now that the money in question, X_j, earns nominal interest at a rate r_jt per period. In this case, the consumer at the end of the period does not hold X_jt/p_ft(≡ x_jt) units of monetary services, but rather X_jt (1 + r_j)/p_ft units; X_jt therefore appreciates in terms of its ability to yield monetary services. Each unit of monetary services, however, is still valued at p_ft. For interest-bearing money X_j, the rental price expression (14) then becomes

\begin{matrix} p_{x_{j t}} = p_{f t} - \frac{(1 + r_{j t}) p_{f t}}{1 + R_{t}} & (16) \end{matrix}

Nonstatic expectations

Consider now the situation where the price of food, p_ft, is not expected to remain constant, and assume initially that the money X_i earns no interest. In the current period, the purchase price of a food unit of monetary services remains p_ft. In the next period, however, while the consumer will still hold X_it dollars, he does not expect to hold the same number of units of monetary services. That is, in the next period he expects to hold $X_{i t} / p_{f, t + 1}^{e}$ , which will be less than X_it/p_ft (his current holdings) if the expected inflation rate is positive. To derive the exact expression for the expected depreciation rate on X_it, δ_t, equation (17) may be solved for δ_t

\begin{matrix} \frac{X_{i t}}{p_{f, t + 1}^{e}} = (1 - δ_{t}) \frac{X_{i t}}{p_{f t}} & (17) \end{matrix}

to obtain

\begin{matrix} δ_{t} = \frac{p_{f, t + 1}^{e} - p_{f t}}{p_{f, t + 1}^{e}} & (18) \end{matrix}

Thus, δ_t is the expected change in the price of food, expressed as a proportion of the expected price of food.

The one-period depreciation rate on money, δ_t, is given by (18). Consider now the expected value of each unit of monetary services in the next period. Although the consumer holds fewer units of monetary services owing to the depreciation factor, δ_t, each unit of monetary service is now worth more to the consumer because of the expected rise in p_ft. Thus, the rental price becomes

\begin{matrix} {\begin{matrix} p \end{matrix}}_{x_{i t}} = p_{f t} - \frac{(1 - δ_{t}) p_{f, t + 1}^{e}}{(1 + R_{t})} & (19) \end{matrix}

Substituting for δ_t, from (18), this reduces to

\begin{matrix} {\begin{matrix} p \end{matrix}}_{x_{i t}} = p_{f t} - \frac{p_{f t}}{(1 + R_{t})} & (20) \end{matrix}

Equation (20) is identical to (15), the rental price expression for noninterest-bearing monetary services when static price expectations are assumed. By an argument similar to that outlined earlier, the rental price for interest-bearing monetary services, X_j, becomes

\begin{matrix} {\begin{matrix} p \end{matrix}}_{x_{i t}} = p_{f t} - \frac{(1 - δ_{t}) p_{f, t + 1}^{e}}{(1 + R_{t})} & (21) \end{matrix}

where δ_t is defined by (22)

\begin{matrix} \frac{X_{j t} (1 + r_{j t})}{p_{f, t + 1}^{e}} = \frac{(1 - δ_{t}) X_{j t}}{p_{f t}} & (22) \end{matrix}

Equation (22) reduces to

\begin{matrix} p_{x_{j t}} = p_{f t} - \frac{(1 + r_{j t}) p_{f t}}{1 + R_{t}} & (23) \end{matrix}

which is also identical to the static expectations formula (16).

The preceding analysis has established that expected inflation does not alter the own rental price of monetary services (assuming that R_t is constant). Comparing equations (14) and (23), however, the rental price of real durable goods will fall relative to that of money if expected inflation is present $p_{n, t + 1}^{* e}$ will be greater than $p_{n t}^{*}$ ). This relative price change still occurs if it is assumed that R_t moves exactly in accordance with inflationary expectations, as in this case; the rental price of the durable remains the same, while that of monetary services rises. ¹²

Rearranging the notation in equation (23), in general, the rental price of money or money substitutes, P_it, is given by

\begin{matrix} p_{i t} = {\bar{p}}_{t} - \frac{(1 + r_{i t}) {\bar{p}}_{t}}{1 + R_{t}} \\ = \frac{{\bar{p}}_{t} (R_{t} - r_{i t})}{1 + R_{t}} & (24) \end{matrix}

where r_it is the interest rate earned on the ith asset (which may be zero), and ${\bar{p}}_{t}$ is some index of the consumer price level.

The rental price of each liquid asset, given by equation (24), should be used as the price variable in the budget constraint (13). This formulation of the budget constraint is considerably different from that adopted by the adherents of the Chetty model. As discussed already, however, it is not at all clear how their budget constraint (which is really a stock-accounting identity) relates to the underlying assumption of utility maximization over a set of arguments representing the service flows yielded by money and money substitutes.

solution of the maximization problem

Given an explicit functional form for the utility function in equation (12), the “primal” approach to solving a maximization problem of this nature consists of differentiating the associated Lagrangean function and solving the resulting first-order conditions to obtain optimal quantities of real assets demanded as functions of prices ¹³ and total expenditure, y. This approach becomes very unwieldy, however, for a generalized functional form that imposes, a priori, as few restrictions on preferences (such as, for example, homotheticity) as possible, as it is often very difficult, if not impossible, to explicitly solve the associated set of first-order conditions. ¹⁴ Instead, it is convenient to derive explicit demand equations using a “dual” approach, via the indirect utility function. ¹⁵

Define an n × 1 normalized price vector v = ply. The indirect utility function g(v) corresponding to the maximization problem (12)–(13) is defined as

\begin{matrix} g (v) \equiv M a x {f (x); v^{t} x < 1; x > 0_{n}} & (25) \end{matrix}

with respect to x

Assuming that the direct utility function satisfies certain regularity conditions, then g(v) will satisfy corresponding regularity conditions (and vice versa). Furthermore, it can be shown that there is a duality between f(x) and g(v)—given one function, the other is uniquely determined.

The usefulness of this relationship is apparent when we consider the theorem known as Roy’s identity (Roy (1947, p. 222)), which states that assuming the required regularity conditions are satisfied the solution to the maximization problem (12)–(13) is unique and is given by

\begin{matrix} x (v^{*}) = \frac{\nabla_{g} (v^{*})}{v^{* t} \nabla_{g} (v^{*})} & (\begin{matrix} 26 \end{matrix}) \end{matrix}

where the asterisk denotes any given normalized price vector v. Thus, starting from an indirect utility function, equation (26) provides us with the same set of demand functions that would have been obtained had the corresponding direct utility function been generated and the resulting maximization problem solved. The latter approach can be fairly intractable, while the application of Roy’s identity, on the other hand, is quite straightforward.

derivation of per capita market demand equations

The following functional form was chosen for the individual consumer’s indirect utility function in (25)

\begin{matrix} g (v) = Σ_{i = 1}^{n} Σ_{j = 1}^{n} β_{i j} v_{j}^{- 1 / 2} v_{j}^{- 1 / 2} + Σ_{i = 1}^{n} α_{i} \log (v_{i} / v_{i}^{*}) & (27) \end{matrix}

where

\begin{matrix} Σ_{i = 0}^{n} α_{i} = 0 and β_{i j} = β_{j i} & (28) \end{matrix}

The properties of this class of indirect utility functions are discussed extensively inDiewert (1976a). It may be shown that the indirect utility function defined by (27) and (28) is a second-order differential approximation ¹⁶ (at the point v^*) to any arbitrary function belonging to the nonhomothetic so-called Gorman polar form ¹⁷ class. Preferences will be homothetic, thereby implying that all expenditure elasticities are unity only if α _i = 0, all i. Furthermore, because of the “flexible” nature of the function, no restrictions are placed on the elasticities of substitution between assets. In turn, this implies that no separability restrictions are imposed on preferences. ¹⁸ Thus, the specification adopted here is more general than that used by Chetty (1969) and other writers discussed earlier.

Applying Roy’s identity (26) to (27), after some simplifications, the following demand equations are obtained for the individual consumer

\begin{matrix} x_{i} (p : y) = y \frac{Σ_{j = 1}^{n} β_{i j} {p_{i}}^{- 3 / 2} {p_{j}}^{- 1 / 2} - α_{i} p_{i}^{- 1}}{Σ_{k = 1}^{n} Σ_{m = 1}^{n} β_{k m} {p_{k}}^{- 1 / 2} {p_{m}}^{- 1 / 2}} i = 1, …, n & (29) \end{matrix}

Then the market demand X_i for the ith asset derived from (29) ¹⁹ can be written as

\begin{matrix} X_{i} (p : φ) = M \frac{y^{*} Σ_{j = 1}^{n} β_{i j} {p_{i}}^{- 3 / 2} {p_{j}}^{- 1 / 2} - α_{i} {p_{i}}^{- 1}}{Σ_{k = 1}^{n} Σ_{m = 1}^{n} β_{k m} {p_{k}}^{- 1 / 2} {p_{m}}^{- 1 / 2}} & (30) \end{matrix}

where $y^{*} = \int \frac{\overset{=}{y}}{\bar{y}} φ (y) d y$ is the average monetary service expenditure of the M consumers or, in per capita terms,

\begin{matrix} x_{i} (p : y^{*}) \equiv \frac{X_{i} (p : φ)}{M} = y^{*} \frac{Σ_{j = 1}^{n} β_{i j} {p_{i}}^{- 3 / 2} {p_{j}}^{- 1 / 2} - α_{i} ρ_{i}^{- 1}}{Σ_{k = 1}^{n} Σ_{m = 1}^{n} β_{k m} p_{k}^{- 1 / 2} p_{m}^{- 1 / 2}} & (31) \end{matrix}

where henceforth x_i denotes per capita demand.

The per capita demand functions x_i ≡ X_i (p:φ)/M are identical to the demand functions generated by a “representative” consumer facing prices p and average total expenditure y*, irrespective of the distribution of expenditure. This property is a direct consequence of using an indirect utility function belonging to the nonhomothetic Gorman polar form class. ²⁰

The per capita demand functions (31) may be converted to expenditure share form by multiplying by p_i/_y to obtain (32)

\begin{matrix} s_{i} \equiv \frac{p_{i} x_{i}}{y^{*}} = \frac{Σ_{j = 1}^{n} β_{i j} p_{i}^{- 1 / 2} p_{j}^{- 1 / 2} - α_{y i}^{*^{- 1}}}{Σ_{k = 1}^{n} Σ_{m = 1}^{n} β_{k m} p_{k}^{- 1 / 2} p_{m}^{- 1 / 2}} i = 1, …, n & (32) \end{matrix}

The equation system (32) forms the basis for the empirical results reported in Section IV.

elasticities

Price elasticities

Define the price elasticity of the ith demand $x_{t}^{*} (p^{*}, y^{*})$ with respect to a change in $p_{j}^{*}$ as

\begin{matrix} E_{i j}^{*} \equiv \frac{\partial x_{t}^{*} (p^{*}, y^{*})}{\partial p_{j}^{*}} . \frac{p_{j}^{*}}{x_{i}^{*}} & (33) \end{matrix}

where $x_{i}^{*}$ denotes the fitted per capita demand given by (31) corresponding to prices p^* and expenditure y^*.

Differentiating (26), after some manipulation, yields

\begin{matrix} E_{i j} = \frac{g_{i j v_{j}}}{x_{i} d} - \frac{v_{j} d_{j}}{d} - x_{j} v_{j} & (34) \end{matrix}

where

\begin{matrix} d \equiv v^{t} \nabla_{g (v)} = Σ_{i = 1}^{n} v_{i} g_{i} & (35) \end{matrix}

\begin{matrix} {\begin{matrix} d \end{matrix}}_{j} \equiv Σ_{m = 1}^{n} v_{m} g_{m j} j = 1, …, n & (36) \end{matrix}

and g_i and g_ij correspond to the first-order and second-order partial derivatives of the indirect utility function (27). The asterisks have been omitted for notational convenience. Equation (34) is used to calculate the n × n matrix of own and cross-price elasticities.

Expenditure elasticities

Define the elasticity of the ith fitted demand with respect to total expenditure, y, as

\begin{matrix} E_{i j} \equiv \frac{\partial_{x_{i}} (p, y)}{\partial y} . \frac{y}{x_{i}} i = 1, …, n & (37) \end{matrix}

A well-known result is that

\begin{matrix} {\begin{matrix} E \end{matrix}}_{i v} = - Σ_{j = 1}^{n} E_{i j} & (\begin{matrix} 38 \end{matrix}) \end{matrix}

where E_ij is defined by (34). If the function g(v) is homothetic, then E_iv defined by (38) will be unity, all i.

Elasticities of substitution

The Allen partial elasticity of substitution between two goods, i and j, σ_ij, can be derived as ²¹

\begin{matrix} σ_{i j} = \frac{E_{i j}}{s_{j}} - Σ_{j = 1}^{n} E_{i j} & (39) \end{matrix}

where s_j(≡ p_jx_j/y) is the jth fitted share of total expenditure y. The n × n matrix of elasticities of substitution (σ_ij) is symmetric and has rank (n–1). These restrictions serve as useful computational checks.

tests of utility-maximizing behavior

The following restrictions on the form of the indirect utility function are implied by the theory of utility-maximizing behavior and may be tested using the estimates obtained of the preference ordering (27).

The positivity and monotonicity restrictions are checked by direct computation of the values of the fitted demands and the gradient vector of the estimated indirect utility function, respectively. The curvature conditions are tested by examining the computed (σ_ij) matrix to establish whether the determinant conditions required for negative semi-definiteness hold. ²²

III. Econometric Estimation, Hypothesis Testing, and Data

estimation

The market expenditure share equations corresponding to the n asset demand equations are given by equation (32). The actual shares are assumed to deviate from the “true” shares by an additive disturbance term, e_i(t). That is,

\begin{matrix} s_{i} (t) = \frac{Σ_{j = 1}^{n} β_{i j} p_{i}^{- 1 / 2} p_{j}^{- 1 / 2} - α_{i} y^{* - 1}}{Σ_{k = 1}^{n} Σ_{m = 1}^{n} β_{k m} p_{k}^{- 1 / 2} p_{m}^{- 1 / 2}} + e_{t} (t) & \underset{t = 1, …, T}{\begin{matrix} i = 1, …, n \end{matrix}} & (40) \end{matrix}

where T is the total number of observations (years) in the sample; e _i(t) is assumed to be due to errors in the utility-maximizing process or in aggregation over assets and consumers. Define an n x 1 column vector of additive disturbances at time t as

\begin{matrix} e (t) = [e_{1} (t), e_{2} (t), …, e_{n} (t)] t = 1, …, T & (\begin{matrix} 41 \end{matrix}) \end{matrix}

e(t) is assumed to be distributed normally, independent of the variables on the right-hand side of (40). Furthermore, it is assumed that

\begin{matrix} (i) & E [e_{t} (t)] = 0 & i = 1, …, n \\ t = 1, …, T \end{matrix}

\begin{matrix} (i i) E [e (s) e {(t)}^{t}] = {\begin{matrix} Ω f o r s = t \\ {\begin{matrix} 0 f o r s \neq t \end{matrix}}^{, a l l s, t} \end{matrix} & (42) \end{matrix}

where Ω is an n × n covariance matrix. Assumption (42) permits contemporaneous correlation among the disturbances at time t, but rules out any intertemporal correlation. This is the so-called classical stochastic specification.

Since the shares s_i(t) on the left-hand side of (40), by definition, add to unity at each observation, it follows that $Σ_{i = 1}^{n} e_{i} (t) = 0$ at each observation. Furthermore, the disturbance covariance matrix and also the estimated covariance matrix will be singular at each observation. To solve this problem, one share equation is arbitrarily dropped, and a truncated (n–1) × (n–1) disturbance covariance matrix Ω* is defined as

\begin{matrix} E [e^{*} (s) {(t)}^{t}] = \begin{matrix} {\begin{matrix} Ω \end{matrix}}^{*} f o r s = t \\ 0 f o r s \neq t' \end{matrix}, a l l s, t & (43) \end{matrix}

where e*(s) and e*(t) are (n –1) × 1 vectors of additive disturbances at times s and t. Define the ijth element of Ω* by w_ij. A maximum likelihood estimate of w_ij, ${\hat{w}}_{i j}$ is given by

\begin{matrix} {\hat{w}}_{u} = \frac{e_{i}^{t} e_{j}}{T} & (44) \end{matrix}

where e_i and e_j are T × 1 vectors of computed residuals from the ith and jth equations, respectively; ${\hat{Ω}}^{*}$ is the resulting estimate of Ω*.

Since e*(t) is normally distributed, the log-likelihood function may be written as

\begin{matrix} \log L = \frac{n - 1}{T} \log (2 π + 1) - \frac{T}{2} \log | {\hat{Ω}}^{*} | & (45) \end{matrix}

Maximizing the likelihood function is equivalent to minimizing $| {\hat{Ω}}^{*} |$ . Furthermore, it may be shown (Barten (1969)) that the likelihood function, in a singular system, is invariant to the equation deleted. Thus, the parameter estimates of the various elasticities, and also the test statistics, are invariant to the equation deleted.

The computational algorithm that was used to minimize $| {\hat{Ω}}^{*} |$ is a generalized least-squares adaptation of the Gauss-Newton method. ²³ The properties of the algorithm have been discussed by Malinaud (1972) andBerndt, Hall, Hall, and Hausman (1974). It can be shown that if convergence is achieved the resulting estimator converges numerically to (i) Zellner’s minimum distance estimator, and (ii) the maximum likelihood estimator, provided that the disturbances are normally distributed.

hypothesis testing

Apart from examining the results to establish whether the data are consistent with the assumption of utility-maximizing behavior, the hypothesis of homotheticity (α_i=0, all i) was tested for statistically. The likelihood-ratio (LR) test statistic, equal to twice the difference between the log of the likelihood function under the null hypothesis (H₀) and under the alternative hypothesis (H₁), was calculated by estimating both constrained (i.e., setting α_i=0, all i) and unconstrained versions. The LR test statistic is distributed asymptotically as X² with degrees of freedom equal to the difference in the number of free parameters under H₀ and H₁.

data

The model requires the construction of rental price and per capita quantity series for the services of money and money substitutes held by households during the period 1952–74. Four assets were considered: chartered bank personal savings deposits; trust and mortgage loan company (TML) savings deposits; TML term deposits of more than one year; and Canada Savings Bonds. Excluded from consideration in the empirical analysis for a variety of reasons were personal checking accounts; demand deposits; nonpersonal term and notice deposits; and swapped deposits (all of chartered banks); currency held by households; and TML time deposits of less than one year. ²⁴

Personal checking accounts were introduced by the chartered banks only in 1958; they were of a relatively insignificant order of magnitude throughout most of the data period. ²⁵ Demand deposits and swapped deposits generally are not held by households, ²⁶ while nonpersonal term and notice deposits, by definition, are not held by households. Unfortunately, no data exist regarding currency holdings by households. The ownership issue is more difficult to resolve for TML liabilities. Judging from the discussion in Clinton (1974), it appears that noncheckable TML savings deposits are held predominantly by households, while, on the other hand, TML term deposits of less than one year are regarded as close substitutes for chartered bank nonpersonal term and notice deposits, and thus by inference typically do not belong in the household sector. However, while households and small savers appear to form a high proportion of those holding checkable TML savings deposits and TML term deposits of over one year, the assumption that these are held exclusively by households involves a certain error. Nevertheless, in the absence of any more precise information, this assumption can be viewed as a reasonable working approximation. The final asset considered, Canada Savings Bonds, poses few ownership problems, as firms are not permitted to hold this asset; they can be held to only a very limited extent (recently) by institutions. In general, the ownership assumptions employed in this study constitute, it is felt, a significant improvement over the work of both Chetty (1969) and Short and Villanueva (1977), where the question of ownership was not considered. ²⁷

The procedure adopted was, first, to calculate the average per capita stocks of the four assets held and, second, to construct the corresponding rental price series based on equation (24). The annual quantity series were derived by taking an arithmetic mean of either quarter-end or month-end figures (depending on availability of data), while the corresponding annual interest rate series were calculated as weighted averages of monthly or quarterly rates. A series on the population of Canada aged 15 years and over was used to derive a per capita figure. The basic data were obtained from a variety of statistical sources, and, particularly for earlier years, a number of adjustments were ncesssary to achieve as great a degree of consistency as possible. The sources and methodology used for each asset are explained in detail in Donovan (1977).

A number of specific points regarding the construction of the data should be made. First, R_t the discount rate required in equation (24), was chosen to be the average annual yield on ten industrial bonds. Second, for estimating purposes, the price and quantity series for each asset were converted to index number form: that is, the rental price series was adjusted so that the value of the index was unity in 1961, and the quantity series was adjusted accordingly, so as to leave the value of total expenditure in each period unchanged. Third, for chartered bank personal savings deposits from 1968 onward, data are available on the three different subcomponents of this category, namely, fixed-term, checkable, and noncheckable savings deposits. Accordingly, three different price and quantity indices were constructed corresponding to each component, and then an aggregate price and quantity series for personal savings deposits was obtained using a Divisia aggregation method. ²⁸ An identical procedure was followed for TML checkable and noncheckable savings deposits. In this regard, it is worth remarking that, despite the fact that most of the literature on the empirical demand for money is based implicitly upon the nature of money-yielding utility, the implication of utility analysis for aggregating different forms of asset has generally been ignored. Finally, ${\bar{P}}_{t}$ , the general price index used both to deflate the nominal dollar asset values and to construct the rental price series based on equation (24), was derived as a Divisia index from disaggregated consumption price series constructed by the author for use in a model containing only consumption goods. ²⁹

IV. Empirical Results

The parameter estimates of the four-asset model are given in Table 1. The model was estimated for the period 1952–74. Since the Durbin-Watson statistics were low, ³⁰ an attempt was made to re-estimate the model allowing for a specific form of autocorrelation. ³¹ However, despite repeated attempts, after a large number of iterations, a maximum value of the likelihood function could not be attained, and thus the estimates reported here are based solely on a “classical” stochastic specification. ³²

Table 1.

Canadian Liquid Asset (Four-Asset) Model: Parameter Estimates, 1952–74¹

^¹t-statistics are in parentheses.

The first issue to be discussed is the extent to which the parameter estimates are consistent with the utility-maximizing hypothesis underlying the model. As was noted earlier, the estimated indirect utility function is required to satisfy at each observation the properties of (i) positivity (i.e., the fitted asset demands must be positive), (ii) monotonicity, and (iii) quasi-convexity. All three properties were rejected for 2 years, 1952 and 1973. ³³ Apart from these 2 years, however, positivity and monotonicity were always satisfied, and the own elasticities of substitution were generally negative (a necessary, but not sufficient, condition for quasi-convexity). The full set of restrictions, (i), (ii), and (iii), were satisfied in the years 1953, 1956–62, 1964–70, and 1974—that is, in 16 of the 23 years. In assessing the apparent rejection of the theory during the other 7 years, in a significant number of cases the required sign conditions on the matrix of elasticities of substitution were only barely violated. Unfortunately, no attempt has been made in this study (or in any other demand study, to this author’s knowledge) to construct confidence intervals for the determinants of the estimated elasticities of substitution matrix. Although this approach would be a more appropriate procedure for statistically testing the utility-maximizing hypothesis, computational and statistical complexities prevent its application at present. It should be borne in mind, however, that the use of point estimates could tend to overstate the degree of rejection of the theory during many of the “problem” years.

Notwithstanding the above-mentioned interpretation difficulty, the results are of considerable interest. To this author’s knowledge, this is the first explicit test in the literature of the hypothesis that households act as if maximizing a well-behaved preference ordering defined over monetary services. As discussed in Section I, this hypothesis not only forms the explicit basis for the work of those authors who have followed the Chetty approach but also has been adopted implicitly by many authors in the literature on the empirical demand for money. The assumptions underlying this approach deserve to be spelled out and tested in a manner similar to that employed in this paper. The results, as have been indicated, provide a reasonable amount of support for this type of model.

Apart from throwing some important light on the preceding methodological question, the principal empirical usefulness of the model consists of providing a comprehensive set of price elasticities, expenditure elasticities, and elasticities of substitution. These elasticities are presented in Table 2 for the years 1956, 1965, and 1974 (years in which all the utility-maximizing conditions were fulfilled).

Table 2.

Canadian Liquid Asset (Four-Asset) Model: Elasticity Estimates, 1956, 1965, and 1974

price elasticities

The own price elasticities (E_ii) are negative in all years. Personal savings deposits (E₁₁ and Canada Savings Bonds (E₄₄) are clearly price inelastic, with the former category somewhat more inelastic. The own price elasticity of Canada Savings Bonds appears to have increased between 1965 and 1974, which could be interpreted as a result (or cause?) of the increasing advertising effort undertaken each year, which stresses the interest advantages attached to holding this asset. On the other hand, both savings deposits and term deposits of trust and mortgage loan companies appear to be considerably more sensitive to changes in their own interest rate, ³⁴ generally exhibiting elasticities greater than unity.

Turning to possible (gross) substitutability/complementarity relationships, E₁₂ and E₂₁ have different signs. A fall in the rental price (brought about, for example, by a rise in the interest rate) of TML savings deposits leads to a decline in personal savings deposits (E₁₂ > 0). This evidence supports the Gurley and Shaw (1960) hypothesis of money/near-money ³⁵ substitutability; however, the elasticity estimate is really quite small (less than 0.1) in all years. On the other hand, E₂₁ is less than zero, thereby implying that an increase in the interest rate offered on personal savings deposits is associated with a rise in the demand for TML savings deposits. The asymmetrical responses between these two asset categories are explained by the presence of the “income effect.” (See elasticities of substitution.)

Other gross substitutability relationships are indicated by positive signs for E₂₃ in all years and E₄₃ in a majority of years. While the sizes of these elasticities are generally also less than 0.1, there is some evidence of increasing substitutability between 1965 and 1974. All the other elasticity estimates indicate complementarity, with the exceptions of E₁₃ in 1956 and E₃₂ in 1974; here again, the latter are less than 0.1. Thus, over all, it appears that there is not a great deal of evidence to support the hypothesis of a high degree of (gross) substitutability between liquid assets in Canada throughout most of the period under consideration. Any substitutability that has been discovered is of a low order of magnitude, even toward the end of the period, when it is present to a slightly greater extent.

expenditure elasticities

The estimated expenditure elasticities conform to a fairly clear-cut pattern. Personal savings deposits are a “normal good,” with an expenditure elasticity less than one but which tends to rise slightly over time. The “near-money” expenditure elasticities are all high, with TML term deposits being the highest. Clearly, “near-money” is a “luxury good.” This is perhaps intuitively quite plausible, since near-monies may be regarded as more “sophisticated” financial instruments, compared with the competing liabilities of chartered banks, and thus the demand for them might be expected to respond relatively strongly to growth in total portfolio size accompanied by increased awareness on the part of households. All the near-money expenditure elasticities appear to decline over time, especially those of TML term deposits. Coupled with the general lack of evidence of substitutability cited in the previous paragraph, it seems clear that the relatively faster overall growth of near-bank liquid asset liabilities held by households in Canada during the postwar period cannot be explained by relative price changes in favor of financial institutions other than chartered banks. The explanation lies rather in higher expenditure elasticities.

The formal test for homotheticity (implying unitary expenditure elasticities) indicated decisive rejection, as the LR test statistic was 95.764, compared with a X² critical value of 11.34. This finding is consistent with the significance of the α_i coefficients on the basis of individual t-tests (Table 1).

elasticities of substitution

The estimated elasticities of substitution (σ_ij) reported in Table 2 are generally positive, thereby indicating (compensated) substitutability among assets. The apparent conflict between these estimates and the generally complementary evidence cited earlier for the gross price elasticities (E_ij) is resolved easily when one considers the very high expenditure elasticities for all the near-monies. It is useful to rearrange the expression for σ_ij stated in Section II (equation (39))

\begin{matrix} E_{i j} = s_{j} [σ_{i j} - E_{i j}] & (47) \end{matrix}

Only if σ_ij (corresponding to the “pure substitution effect”) > E_iy, (corresponding to the “income effect”) > 0 will E_ij (the gross price elasticity) have the same sign as σ_ij. Given our evidence that E_iy is quite high for near-monies, analysis of money/near-money relationships in terms of the elasticities of substitution alone (the procedure of Chettey (1969), Bisignano (1974), Moroney and Wilbratte (1976), and Short and Villanueva (1975; 1977) could be quite misleading.

Finally, the model was estimated for two subperiods, 1952–62 and 1963–74. The resulting parameter estimates and calculated elasticities are not reported here for lack of space. ³⁶ A formal test of the hypothesis of parametric stability between the two periods indicated rejection. ³⁷ Comparing the elasticity estimates for the same years from the 1952–74 model and the relevant submodel, however, the differences turned out to be not very large. In particular, the substitution/complementarity patterns are quite similar; taking the price elasticities and the elasticities of substitution together, of the 78 relevant points of comparison (the 26 elasticities in each of three years) only 7 alter in sign comparing the 1952–74 model and the submodels. A corresponding conclusion holds as regards whether an asset is a “normal” or a “luxury” good. Thus, over all, the qualitative conclusions discussed earlier in large measure do not depend upon the period used in estimation.

V. Summary and Concluding Remarks

The objective of this paper has been to develop and apply the direct utility model of the demand for liquid assets, utilizing recent developments in applied demand theory, in particular, the theory of duality and the use of indirect utility functions. It has been argued that while the underlying approach is not new in the monetary literature, those authors adopting this approach, either implicitly or explicitly, have not exploited many of these propositions, which are well known to economists working in a different field. The first part of the paper contained therefore a model of aggregate liquid asset demand carefully derived from a microeconomic model of utility-maximizing behavior for the individual household. In constructing the model, special attention was devoted to deriving the rental price of monetary services, which is the conceptually correct price variable for inclusion in the household budget constraint.

As regards the empirical application of the model to Canadian data, the principal conclusions can be summarized as follows:

(i) The model fits reasonably well, although some autocorrelation problems of an undetermined nature may still be present.

(ii) For almost two thirds of the estimation period, the results were consistent with the maximization of an underlying well-behaved preference ordering defined over the services of money and money substitutes.

(iii) There is some evidence of gross substitutability among certain liquid assets. Most assets are gross complements, however, and when substitutability exists, the price elasticity estimate does not exceed 0.1.

(iv) Money (chartered bank personal savings deposits) is a normal good, while near-monies are luxury goods. The finding of compensated substitutability relationships among most assets is consistent with the afore-mentioned gross complementarity relationships when these “income effects” are taken into account.

(v) Although the hypothesis of parametric stability between the two subperiods 1952–62 and 1963–74 is rejected, the preceding qualitative conclusions still hold after examining the pattern of the estimated elasticities from the two subperiod models.

Thus, over all, it is felt that this type of model can provide some useful and important information; however, some limitations should be noted. As a fundamental point, the model does not provide any insights as to what constitutes the utility of the services derived from money or money substitutes, no more than the traditional model of consumer theory attempts to explain why people like food or clothing. For liquid assets, the answer lies in both the various disutilities attached to being without money or money substitutes (i.e., the “time and trouble” associated with going to the brokers to cash “bonds”) ³⁸ and the different transactions costs and services attached to using each type of liquid asset. ³⁹ Unfortunately, there is little quantitative information available regarding these costs (most of which are nonpecuniary in nature), and thus models attempting to deal explicitly with transactions costs have not progressed very far empirically. Second, the model implicitly assumes that the interest rates on the assets are known with certainty, and thus risk considerations are ignored. While this assumption represents a reasonable working approximation for the liquid assets considered in this paper, it effectively precludes the inclusion of such assets as government bonds and equities. In the case of the latter assets, elements of an uncertainty model of the Markowitz-Tobin variety are required. Finally, the question of lags and adjustment costs could be explored in more detail. Such an extension is in principle quite possible in the direct utility model, but requires the development of more efficient computational techniques—especially if the fairly complex nonlinear functional forms for the demand functions are to be retained.

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Mr. Donovan, economist in the African Department when this paper was prepared, is currently economist in the Trade and Payments Division of the Exchange and Trade Relations Department. He is a graduate of Trinity College, Dublin, and received his doctorate from the University of British Columbia.

This research was initially undertaken as part of the author’s doctoral dissertation (Donovan (1977)) and was supported in part by a Canada Council Fellowship. Apart from his Fund colleagues, the author wishes to acknowledge the many helpful comments of C. F. Boonekamp, W. E. Diewert, D. E. Rose, Gopal Yadav, and J. F. Helliwell, He is naturally solely responsible for any remaining errors or omissions.

For a review of the time-series evidence on the substitutability between money and near-money for the United States, see Feige and Pearce (1977).

That is to say, as Feige and Pearce (1977, p. 441) state, “Most of the empirical studies … rely upon the demand theory [direct utility] approach for selection of the relevant variables to be included in the demand function for money.”

The term “wealth elasticity,” used loosely here, takes on a special meaning in the context of the direct utility model. (See Section II.)

In fact, Chetty does not make clear whether the model refers only to households. See the discussion of this point in Section III. The notation used by Chetty has been altered slightly for expositional convenience.

This will also avoid the obvious simultaneity problem caused by the presence of X₀ (an endogenous choice variable) on the right-hand side of equation (5). Furthermore, it is not possible to estimate all the parameters in (5)—β₀ and β_i are not identifiable. It appears that Chetty (1969) must have used some normalizations; however, their nature or justification is not made clear.

For example, in Canada, the rate of substitution between chartered bank personal savings deposits and trust and mortgage loan company savings deposits may be affected by the level of trust and mortgage loan company term deposits or Canada Savings Bonds held.

Moroney and Wilbratte (1976) consider the following two problems. (The verbal argument advanced by them has been translated here into more explicit mathematical form.)

$\begin{matrix} \Pr oblemI: Maximize U = f (X_{0}, X_{1}, …, X_{n}) \\ withrespectto X_{0}, X_{1}, …, X_{n} & } (6) \\ Subject to W = Σ_{i = 0}^{n} r_{i} X_{t}, r_{0} = 1 \end{matrix}$

$\begin{matrix} \Pr o b l e m II: M a x i m i z e W = Σ_{i = 0}^{n} r_{i} X_{i} \\ withrespectto X_{0}, X_{1}, …, X_{n} & } (7) \\ S u b j e c t to T = f (X_{0}, X_{1}, …, X_{n}) \end{matrix}$

Problem I is a version of the Chetty model, where U is the utility function, W is wealth, X_i is the dollar amount of each asset i, and r_i is now (1 +) the interest rate on the ith asset (i = 0,1, … , n); X₀ is money; and thus r₀ = 1. Note that (6) is not identical to (2), since (1 + r_i) rather than 11(1 + r_i) appears in the “budget constraint.” Moroney and Wilbratte do not discuss any issues of timing. Problem II states that the household maximizes wealth (W) subject to a “transactions constraint,” where T is defined as “the anticipated volume of transactions that can be accomplished during a given period” (Moroney and Wilbratte (1976, p. 185)).

The first-order conditions for problems I and II are given by equations (8)–(11)

$\begin{matrix} \frac{\partial f}{{\partial X}_{i}} - {λ r}_{i} = 0 & i = 0, 1, …, n & } I & (8) \\ W - Σ_{i = n}^{n} r_{i} X_{i} = 0 & (9) \end{matrix}$

$\begin{matrix} r_{i} - λ^{'} \frac{\partial f}{{\partial X}_{i}} = 0 & i = 0, 1, …, n & } I I & (10) \\ T - f (X_{0}, X_{t}, … X_{n}) = 0 & (11) \end{matrix}$

Moroney and Wilbratte state that provided the f( ) functions in (6) and (7) are of the same parametic form, “the alternative behavioral models yield identical derived asset demand equations” (1976, p. 183). The basis for the authors’ claim lies in the fact that the ratio of the marginal utility conditions (8) and (10) are identical in both problems. However, the asset demand equations derived from each model will be entirely different. That is, equations (8) and (9), when solved, yield a system of demand equations for optimal X_is as functions of the set of r_i and W, while solution of equations (10) and (11) leads to a system of demand equations for optimal X_is as functions of the set of r, and T. While the functional form for these demand equations will be identical, the parameter values obtained in estimation will be quite different, depending upon the data series used for W and T. Furthermore, there is no “duality” between problems I and II, as the authors claim (1976, p. 183). Such a duality would exist only if, in problem II, the problem was to minimize $W = Σ_{i - 0}^{n} r_{i} X_{i}$ . Another way of expressing the same point is to note that the second-order (curvature) conditions for the f( ) functions in problems I and II will be entirely different. Moroney and Wilbratte (1976) do not estimate a complete demand system, but rather (following Chetty’s procedure) a set of equations similar to equation (5), obtained from solving only the marginal utility conditions in problem II. They also neglect to impose any across-equation restrictions on the system described by (10).

The precise regularity conditions required for f(x) are stated in the subsection, tests of utility-maximizing behavior.

That is, the dollar value divided by some general price level.

The discussion in this paragraph is based on Diewert (1974 a).

Possibly selling to himself.

Presumably, this is the reason why researchers on the demand for money have considered the expected rate of inflation as a variable in their estimating equations. However, the precise role of the expectations variable within the framework of the direct utility approach generally has not been explained explicitly.

Hereinafter, by “prices” are meant the rental prices given by equation (24).

In fact, this was the source of the difficulties present in the model used by Chetty (1969) and others. (See the discussion in Section I.)

There is by now a large and rapidly growing literature dealing with duality theory, and only a summarized treatment is given here. An excellent exposition of the theoretical results and empirical applications in the area is presented in Diewert (1974 b).

“A function of n variables f differentially approximates f^* to the second order at a point x^* if f(x^*) = f^*(x^*) and the first and second order partial derivatives of the two functions also coincide at x^*; i.e., ∇f(x^*) = ∇f^*(x^*) and ∇²f (x^*) = ∇²f^*(x^*),” Diewert (1976 a, p. 14, footnote 21).

See Gorman (1953). The essential property of the latter class of preferences is that the Engel curves, although linear, do not pass through the origin; put another way, preferences are quasi-homothetic, that is, they are homothetic only beyond a certain committed quantity consumed.

See Berndt and Christensen (1973).

Assuming that (i) each consumer has expenditure y between $\bar{y}$ and $\overset{=}{y}$ , (ii) each consumer has the same set of preferences representable by the demand functions in equation (29), (iii) each consumer faces the same vector p of money and money substitute rental prices, and (iv) expenditure on monetary services is distributed according to the density function φ(y); that is, the number of consumers having expenditure between y₀ and Y₁ is $M \int_{v 0}^{v 1} φ (y) d y$ , where M is the number of consumers in the group.

This form has been obtained without having to impose the strong restriction of homotheticity (namely, α_i = 0, all i) on the indirect utility function (27). However, it can be shown that it is necessary to assume that each consumer has sufficient expenditure to achieve the minimum “committed quantity” implied by the use of a nonhomothetic Gorman polar form representation of preferences. (See Diewert (1976a).)

See Allen (1938, p. 512).

The curvature conditions may be tested in a number of equivalent ways. It is well known that quasi-convexity of the indirect utility function (and, hence, quasi-concavity of the direct utility function) implies that the Slutsky matrix of substitution effects will be negative semidefinite. In turn, it can be shown (Diewert, Avriel, and Zang (1977)) that, provided that the conditions of monotonicity hold, this implies that the matrix of elasticities of substitutions (σ_ij) will be negative semidefinite.

The iterative process commences by assuming that Ω^*=I (the identity matrix). Using the Gauss-Newton method with varying step sizes, the parameters are estimated by nonlinear least squares to obtain a new estimate of Ω^*. Using this estimate of Ω^*, the parameters are re-estimated using a generalized inverse procedure and a further estimate of Ω^* is obtained. This iterative process continues until “convergence” is attained. Convergence is declared when both the parameter estimates and the elements of the estimated covariance matrix change by less than 1 per cent from one iteration to the next. This computational algorithm has been programmed into time-series processor (TSP). The actual computations were carried out on an IBM 370-168 model at the University of British Columbia Computing Centre.

Also excluded were deposits in credit unions and caisses populaires, as no information is available on the interest rates earned on these liabilities.

For example, in 1967, personal checking accounts, on the basis of annual average data, were only 2.6 per cent of chartered bank personal savings deposits. An additional reason for not including this category relates to a desire to reduce the number of variables in the model. Compared with a four-asset model, a five-asset model would have increased the number of parameters (in the nonhomothetic version) from 11 to 18. The latter would have been extremely difficult to estimate successfully, owing to the complex nonlinearities present in the estimating system.

See Binhammer (1968, p. 116), Galbraith (1970, p. 83), and Bond and Shearer (1972, p. 173) for descriptions of the typical holders of these assets.

Short and Villanueva (1977) include an asset clearly not held by households, namely, nonpersonal term and notice deposits, in what they describe as a “community utility function.” The concept of a “community utility function” containing liquid assets as arguments, however, makes little theoretical sense. The appropriate analogue to the household “utility function model” in the case of firms is the “money in the production function” model, which has been investigated recently by (among others) Sinai and Stokes (1972), Kahn and Kouri (1975), and Prais (1975).

Specifically, a Divisia index, using arithmetic weights, where the quantity index is derived residually using the weak-factor reversal test. This index, also known as the Tornquist index, has been termed a “superlative” index number in that it is exact for a second-order (translog) approximation to any underlying utility function. (See Diewert (1976 b).)

In fact, to avoid a technical problem that arose in models where consumption and asset choices are considered simultaneously, the price index was calculated as a forecast index using autoregressive integrated moving average (ARIMA) methods. (See Donovan (1977, pp. 63–66).)

A rigorous statistical interpretation of the individual Durbin-Watson statistics is not possible in a multiequation system such as that estimated here, since, ideally, one would prefer one overall statistic for the entire equation system.

That is, a first-order autocorrelation structure of the following type was assumed

\begin{matrix} e_{t} (t) = ρ_{i} e_{t} (t - 1) + u_{t} (t) \begin{matrix} i = 1, …, n \\ t = 1, …., T \end{matrix} & (46) \end{matrix}

where

E [u_{t} (t)] = 0 i = 1, …, n

E [u_{t} (s) u {(t)}^{t}] = {\begin{matrix} φ for s = t \\ 0 for s \neq t \end{matrix}

The model described by equations (40) and (46) may be estimated in principle by the well-known method of lagging (40) by one period, multiplying by ρ_i, and subtracting the result from (40), to obtain a “classical” disturbance term u_i(t) on the right-hand side. Ordinary maximum likelihood methods may then be used to derive parameter estimates of the transformed equation system along with estimates of p_i. However, there are two problems with this approach. First, the transformed equation system is even more nonlinear than the original system (40), and thus may turn out to be extremely difficult to estimate successfully. Second, in a singular equation system such as (40) an important constraint is imposed on the first-order autocorrelation structure. Berndt and Savin (1975) have shown that, in this case, ρ₁ must take on the same value (= ρ) in all equations. This restriction may be quite inappropriate in our model, where different “degrees” of first- order correlation may be present. While it is possible in principle to relax the restriction of the “diagonal” model (40), by allowing for more complex autocorrelation schemes, at present this procedure is computationally infeasible with this equation system, owing to the extreme nonlinearities present.

In any case, the value for ρ (footnote 31) turned out to be statistically insignificant on the basis of an individual t-test at the final iteration attempted.

The “failure” of the model in 1952 could be explained perhaps by the poor quality of the data. A possible explanation for the rejection in 1973 lies in the relatively high inflation experienced during that year. This in turn could invalidate the separability hypothesis underlying the model; that is, the consumer’s allocation among liquid assets may be influenced by his holdings of durable goods.

Assuming, that is, that R_t in equation (24) remains constant, so that changes in the rental price are equivalent to changes in r_it.

Often the term “money” is reserved for noninterest-bearing checkable demand deposits of commercial banks (as in the United States). In Canada, however, for households, personal savings deposits (which were all checkable until 1967) represent the equivalent concept, as noninterest-bearing personal checking accounts are of a small order of magnitude. (See Section III.) Thus, the term “money” as used in this section refers to personal savings deposits, while “near-money” denotes the three other liabilities under consideration.

These are available upon request from the author, whose address is Exchange and Trade Relations Department, International Monetary Fund, Washington, D. C. 20431.

To undertake the appropriate LR test, the unconstrained value of the likelihood function is calculated as the sum of the two likelihood values obtained from the two separate submodels. The constrained value is derived by estimating the model for the entire period, while the number of degrees of freedom equals the number of free parameters in the model.

See Patinkin (1965) for a formal development of this argument.

As one example, in Canada there are considerably fewer branch offices of TML deposit-taking institutions in existence, relative to those of chartered banks, and thus use of TML deposit facilities may require relatively more “time and trouble.”

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IMF Staff papers: Volume 25 No. 4

Author:

International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1978: Issue 004

Publisher:: International Monetary Fund

ISBN:: 9781451930443

ISSN:: 1020-7635

Pages:: 234

DOI:: https://doi.org/10.5089/9781451930443.024

Keywords
I. The Chetty Model
II. The Theretical Model
III. Econometric Estimation, Hypothesis Testing, and Data
IV. Empirical Results
V. Summary and Concluding Remarks
REFERENCES

Subscripts:	(1) Chartered bank personal savings deposits (total)
	(2) Trust and mortgage loan company savings deposits (total)
	(3) Trust and mortgage loan company term deposits for more than one year
	(4) Canada Savings Bonds
Parameter	α ₁	α ₂	α ₃	α ₄	β ₁₁	β ₁₂	β ₁₃
	–0.006	0.002	0.001	0.003	–0.416	0.159	0.129
	(–15.906)	(6.996)	(4.492)	(7.676)	(–8.420)	(6.231)	(7.888)
Parameter	β ₁₄	β ₂₂	β ₂₃	β ₂₄	β ₃₄	β ₄₄
	0.569	–0.120	0.007	0.089	–0.806	0.035	–0.356
	(17.800)	(–4.505)	(0.755)	(7.806)	(–9.273)	(4.127)	(–15.946)
	$\begin{matrix} R_{1}^{2} \\ 0.939 \end{matrix}$	$\begin{matrix} R_{2}^{2} \\ 0.834 \end{matrix}$	$\begin{matrix} R_{3}^{2} \\ 0.733 \end{matrix}$
	$\begin{matrix} {D - W}_{1} \\ 0.888 \end{matrix}$	$\begin{matrix} {D - W}_{2} \\ 0.792 \end{matrix}$	$\begin{matrix} {D - W}_{3} \\ 0.893 \end{matrix}$
Log of likelihood function = 128.525

Subscripts:	(1) Chartered bank personal savings deposits (total)
	(2) Trust and mortgage loan company savings deposits (total)
	(3) Trust and mortgage loan company term deposits for more than one year
	(4) Canada Savings Bonds
Parameter	α ₁	α ₂	α ₃	α ₄	β ₁₁	β ₁₂	β ₁₃
	–0.006	0.002	0.001	0.003	–0.416	0.159	0.129
	(–15.906)	(6.996)	(4.492)	(7.676)	(–8.420)	(6.231)	(7.888)
Parameter	β ₁₄	β ₂₂	β ₂₃	β ₂₄	β ₃₄	β ₄₄
	0.569	–0.120	0.007	0.089	–0.806	0.035	–0.356
	(17.800)	(–4.505)	(0.755)	(7.806)	(–9.273)	(4.127)	(–15.946)
	$\begin{matrix} R_{1}^{2} \\ 0.939 \end{matrix}$	$\begin{matrix} R_{2}^{2} \\ 0.834 \end{matrix}$	$\begin{matrix} R_{3}^{2} \\ 0.733 \end{matrix}$
	$\begin{matrix} {D - W}_{1} \\ 0.888 \end{matrix}$	$\begin{matrix} {D - W}_{2} \\ 0.792 \end{matrix}$	$\begin{matrix} {D - W}_{3} \\ 0.893 \end{matrix}$
Log of likelihood function = 128.525

Subscripts:		(1) Chartered bank personal savings deposits (total)
		(2) Trust and mortgage loan company savings deposits (total)
		(3) Trust and mortgage loan company term deposits for more than one year
		(4) Canada Savings Bonds
		1956	1965	1974
Price Elasticities
	E ₁₁	–0.561	–0.595	–0.527
	E ₁₂	0.021	0.033	0.006
	E ₁₃	0.086	–0.002	–0.017
	E ₁₄	0.003	–0.059	–0.031
	E ₂₁	–1.562	–0.546	–0.698
	E ₂₂	–0.879	–1.022	–1.047
	E ₂₃	0.096	0.049	0.082
	E ₂₄	–0.611	–0.333	–0.089
	E ₃₁	–3.489	–5.770	–2.240
	E ₃₂	–0.066	–0.199	0.019
	E ₃₃	–5.981	–0.717	–1.281
	E ₃₄	–0.582	–1.578	–0.176
	E ₄₁	–1.193	–1.004	–0.633
	E ₄₂	–0.118	–0.112	–0.004
	E ₄₃	0.080	–0.030	0.058
	E ₄₄	–0.799	–0.511	–0.871
Expenditure Elasticities
	E_1y	0.451	0.623	0.568
	E_2y	2.957	1.852	1.752
	E_3y	10.118	8.264	3.676
	E_4y	2.031	1.657	1.450
Elasticities of Substitution
	S₁₁	–0.283	–0.183	–0.266
	S₁₂	0.915	1.112	0.647
	S₁₃	5.558	0.447	0.132
	S₁₄	0.471	0.297	0.448
	S₂₂	–16.551	–13.266	–12.020
	S₂₃	8.659	5.317	3.925
	S₂₄	–0.579	0.003	1.400
	S₃₃	–344.748	–42.559	–30.181
	S₃₄	6.755	–0.489	2.987
	S₄₄	2.590	–1.178	–1.975

Subscripts:		(1) Chartered bank personal savings deposits (total)
		(2) Trust and mortgage loan company savings deposits (total)
		(3) Trust and mortgage loan company term deposits for more than one year
		(4) Canada Savings Bonds
		1956	1965	1974
Price Elasticities
	E ₁₁	–0.561	–0.595	–0.527
	E ₁₂	0.021	0.033	0.006
	E ₁₃	0.086	–0.002	–0.017
	E ₁₄	0.003	–0.059	–0.031
	E ₂₁	–1.562	–0.546	–0.698
	E ₂₂	–0.879	–1.022	–1.047
	E ₂₃	0.096	0.049	0.082
	E ₂₄	–0.611	–0.333	–0.089
	E ₃₁	–3.489	–5.770	–2.240
	E ₃₂	–0.066	–0.199	0.019
	E ₃₃	–5.981	–0.717	–1.281
	E ₃₄	–0.582	–1.578	–0.176
	E ₄₁	–1.193	–1.004	–0.633
	E ₄₂	–0.118	–0.112	–0.004
	E ₄₃	0.080	–0.030	0.058
	E ₄₄	–0.799	–0.511	–0.871
Expenditure Elasticities
	E_1y	0.451	0.623	0.568
	E_2y	2.957	1.852	1.752
	E_3y	10.118	8.264	3.676
	E_4y	2.031	1.657	1.450
Elasticities of Substitution
	S₁₁	–0.283	–0.183	–0.266
	S₁₂	0.915	1.112	0.647
	S₁₃	5.558	0.447	0.132
	S₁₄	0.471	0.297	0.448
	S₂₂	–16.551	–13.266	–12.020
	S₂₃	8.659	5.317	3.925
	S₂₄	–0.579	0.003	1.400
	S₃₃	–344.748	–42.559	–30.181
	S₃₄	6.755	–0.489	2.987
	S₄₄	2.590	–1.178	–1.975

Subscripts:		(1) Chartered bank personal savings deposits (total)
		(2) Trust and mortgage loan company savings deposits (total)
		(3) Trust and mortgage loan company term deposits for more than one year
		(4) Canada Savings Bonds
		1956	1965	1974
Price Elasticities
	E ₁₁	–0.561	–0.595	–0.527
	E ₁₂	0.021	0.033	0.006
	E ₁₃	0.086	–0.002	–0.017
	E ₁₄	0.003	–0.059	–0.031
	E ₂₁	–1.562	–0.546	–0.698
	E ₂₂	–0.879	–1.022	–1.047
	E ₂₃	0.096	0.049	0.082
	E ₂₄	–0.611	–0.333	–0.089
	E ₃₁	–3.489	–5.770	–2.240
	E ₃₂	–0.066	–0.199	0.019
	E ₃₃	–5.981	–0.717	–1.281
	E ₃₄	–0.582	–1.578	–0.176
	E ₄₁	–1.193	–1.004	–0.633
	E ₄₂	–0.118	–0.112	–0.004
	E ₄₃	0.080	–0.030	0.058
	E ₄₄	–0.799	–0.511	–0.871
Expenditure Elasticities
	E_1y	0.451	0.623	0.568
	E_2y	2.957	1.852	1.752
	E_3y	10.118	8.264	3.676
	E_4y	2.031	1.657	1.450
Elasticities of Substitution
	S₁₁	–0.283	–0.183	–0.266
	S₁₂	0.915	1.112	0.647
	S₁₃	5.558	0.447	0.132
	S₁₄	0.471	0.297	0.448
	S₂₂	–16.551	–13.266	–12.020
	S₂₃	8.659	5.317	3.925
	S₂₄	–0.579	0.003	1.400
	S₃₃	–344.748	–42.559	–30.181
	S₃₄	6.755	–0.489	2.987
	S₄₄	2.590	–1.178	–1.975

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Abstract

I. The Chetty Model

II. The Theretical Model

the individual consumer and the rental price of money

Static expectations

Nonstatic expectations

solution of the maximization problem

derivation of per capita market demand equations

elasticities

Price elasticities

Expenditure elasticities

Elasticities of substitution

tests of utility-maximizing behavior

III. Econometric Estimation, Hypothesis Testing, and Data

estimation

hypothesis testing

data

IV. Empirical Results

price elasticities

expenditure elasticities

elasticities of substitution

V. Summary and Concluding Remarks

REFERENCES

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