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.
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.
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Diewert, W. E. (1974 b), “Applications of Duality Theory,” Ch. 3 in Frontiers of Quantitative Economics: Papers Invited for Presentation at the Econometric Society Winter Meetings, Toronto, 1972, Vol. 2, ed. by Michael D. Intriligator and David A. Kendrick (Amsterdam and New York, 1974), pp. 106–71.
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)| false ( Diewert, W. E. 1974 b), “Applications of Duality Theory,”Ch. 3 in Frontiers of Quantitative Economics: Papers Invited for Presentation at the Econometric Society Winter Meetings, Toronto, 1972, Vol. 2, ed. by ( Michael D. Intriligatorand David A. Kendrick Amsterdam and New York, 1974), pp. 106– 71.
Diewert, W. E. (1976 a), “On Symmetry Conditions for Market Demand Functions: A Review and Some Extensions,” Department of Economics, University of British Columbia (unpublished, 1976).
M. Avriel, and I. Zang (1977), “Nine Kinds of Quasi-Concavity and Concavity,” Discussion Paper No. 77–31, Department of Economics, University of British Columbia (unpublished, 1977).
Donovan, Donal J., “Consumption, Leisure and the Demand for Money and Money Subititutes” (unpublished doctoral dissertation, University of British Columbia, 1977).
Feige, Edgar L., and Douglas K. Pearce, “The Substitutability of Money and Near- Monies: A Survey of the Time-Series Evidence,” Journal of Economic Literature, Vol. 15 (June 1977), pp. 439–69.
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Khan, Mohsin S., and Pentti J.K. Kouri, “Real Money Balances as a Factor of Production: A Comment,” Review of Economics and Statistics, Vol. 57 (May 1975), pp. 244–46.
Moroney, John R., and Barry J. Wilbratte, “Money and Money Substitutes: A Time Series Analysis of Household Portfolios,” Journal of Money, Credit and Banking, Vol. 8 (May 1976), pp. 181–98.
Prais, Zmira, “Real Money Balances As a Variable in the Production Function: A Comment,” Journal of Money, Credit and Banking, Vol. 7 (November 1975), pp. 535–43.
Short, Brock K., and Delano P. Villanueva (1975), “The Stability of Substitution Between Money and Near-Monies” (unpublished, International Monetary Fund, August 13, 1975).
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Sinai, Allen, and Houston H. Stokes, “Real Money Balances: An Omitted Variable from the Production Functions?” Review of Economics and Statistics, Vol. 54 (August 1972), pp. 290–96.
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 βi 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.)
Problem I is a version of the Chetty model, where U is the utility function, W is wealth, Xi is the dollar amount of each asset i, and ri 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 + ri) rather than 11(1 + ri) 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)
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 Xis as functions of the set of ri and W, while solution of equations (10) and (11) leads to a system of demand equations for optimal Xis 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
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.
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.
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.
Assuming that (i) each consumer has expenditure y between
This form has been obtained without having to impose the strong restriction of homotheticity (namely, αi = 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).)
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 (σij) 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.
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
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 ρi, and subtracting the result from (40), to obtain a “classical” disturbance term ui(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 pi. 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 Rt in equation (24) remains constant, so that changes in the rental price are equivalent to changes in rit.
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.
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.”