money demand in the us a quantitative review.docx

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Carnegie-Roch ester Conf erence Series on Pub lic Polic y 29 (1988) 137-168

North-Holland

MONEY DEMAND IN THE UNITED STATES: A QUANTITATIVE REVIEW

ROBERT E. LUCAS, JR.1

The University of Chicago

I. INTRODUCTION

Allan Meltzer's research career has been so productive and so varied that it

would be an act of folly, not friendship, to attempt to review it in a single paper. Yet

I do want to talk about his research on this occasion, for research is what Allan's

career is mainly about, and I want to do so in detail, because details are the way

scholarship is carried out. Accordingly, I will focus my attention mainly on a singlepaper, one that has influenced my own thinking on monetary economics a great

deal, Meltzer's "The Demand for Money: The Evidence from Time Series," published

in the Journal of Political Economy in 1963.

Meltzer's "Demand for Money" was one in a series of his empirical studies in

monetary economics, much of which involved joint research with Karl Brunner. It

followed earlier work by Latane and others, especially Friedman, and helped to

stimulate closely related later contributions by Laidler and others.* The shared

objective of this research program was, in Friedman's (1956) terms, to demonstrate

that the demand for money is a "highly stable function" of a limited number of

variables, to discover the most useful, operational measures of money and these

other variables, and (again citing Friedman) to work "toward isolating the numerical

'constants' of monetary behavior." Meltzer's paper was the first to estimate an

income

1This paper was prepared for the Novembert 1987 Carnegie-Rochester Conference. I would like to

thank John Cochrane, Thomas Cooley, Milton Friedman, Lars Peter Hansen, Robert King, Leonardo

Leiderman, Bennett McCallum, Sherwin Rosen, Thomas Sargent and Lawrence Summers for helpful

discussions and/or comments on an earlier draft. I also benefitted from a stimulating discussion at the

Conference. P.S. Eswar-Prasad provided excellent research assistance.

'Two important sequels to this paper are Brunner and M^lfzer (1963) and Laidler (1966). Of course,

this and other work on money demand wss closely related to rther contemporary research, especially

the ear l ier contr ibut ions of Fr iedman (1956) a n d h i s s t u d e n t s , , a n d Friedman (1959). See Laidler

(1977)and,more recently, McCallum and Goocf r i end (1987) for some of the relevant background.

0 167 - 2231/88/S3.50 1988 Elsevier Science Publishers B.V. (North-Holland)


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(or wealth) elasticity and an interest elasticity simultaneously from time series data

from a single country (the U.S.). The objective of the present paper will be to review

and replicate these results, to reconsider how they might be interpreted

theoretically, and to see how well they stand up to the 25 years of new data that

have become available since Meltzer wrote.

An estimated money demand function provides answers to two important

questions of economic policy. The income elasticity, in a setting in which long run

real output growth is both fairly predictable and insensitive to changes in monetary

policy, provides the answer to the question: What rate of growth of money is

consistent with long run price stability? The interest elasticity is the key parameter

needed to answer the question: What are the welfare costs to society of deviations

from long run price stability? Purely qualitative answers to these questions, along

the lines of "Inflation rates are significantly related to money growth rates" or

"Inflation reduces welfare" are interesting and useful, perhaps, but surelypropositions such as "An HI growth rate of 3 percent per year will bring about price

stability" or "A ten percent annual inflation rate has a social cost equivalent to a 0.5

percent decline in real income" are more interesting and, if accurate, much more

useful.

Though the objective of an economics that provides quantitative answers to

important questions of economic policy is now very widely subscribed to, it is

remarkable how little attention is paid in many of our discussions to the substance

of parameter estimation, and how little honor is paid to those few economists who

do it well. All of us have sat through many discussions of econometric work in

which the theoretical underpinnings of the relationships estimated and tested and

the econometric methods used are subjected to intense scrutiny and yet no one

seems to care what the numerical results were! Even in Laidler's (1977) survey of

the evidence on money demand, or in McCallum and Gcodfriend's (1987) more

recent summary, it is difficult to find clear statements of what the money demand

function is. As quantitative economists we often seem to be, in Samuel son's (1947)

phrase, "like highly trained athletes who never run a race, and in consequence

grow stale."

Meltzer ran this particular race, in 1963, and turned in his two numbers. Much

has happened since to monetary theory and to the development of econometric

methods, and almost three decades of new data have since become available. In

Section II I will sunmarize the evidence on the income (or wealth) and interest

elasticities of money demand from 1900-58 data, essentially identical to those

Meltzer used. Section III introduces a


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utility-theoretic framework for thinking about money demand, from which I will

conclude that there is some reason to view these two parameters as structural.

Section IV reviews U.S. time series evidence from the 1958-85 period, a period

during which nominal interest rates reached levels about twice the highest levels

attained in the U.S. in the earlier years of the century. Remarkably, in view of the

stringent nature of the experiment, these new data precisely confirm the estimates

Meltzer obtained in 1963.

I I . R E V I E W O F T H E E V I D E N C E F R O M 1 9 0 0 - 1 9 5 8

The hypothetical household decision problem underlying the results reported

in Meltzer (1963) is that of allocating a given stock of wealth across different assets,

given a vector of asset returns. I will come back to this problem in more detail in

Section III, but I have said enough to rationalize a demand function for money of the

form

= f(r,w) .

Throughout his paper, Meltzer used the log-linear form:

in(mt) = a - b?n(rt) + can(wt) + ut, (1)

where m^. is the stock of real balances at t, w tis real wealth or real income, rtan

interest rate, utis an error term, and a, b and c are parameters. Meltzer used a long

term interest rate to measure rt, treated as a stand-in for the entire vector of returns

on alternative assets. He experimented with a very wide variety of income and

wealth variables as measures of real wealth, and with both Ml and M2 as measures

of the money stock. The sample period was 1900-1958, with results also reported

for the two subperiods 1900-1929 and 1930-1958.

lne experimental approach Meltzer used for measuring money and wealth is

obviously suitable: we do not have theories that single out particular measures asclearly superior to others. One could indeed criticize the paper for reporting too few

results, since the single interest rate he used to represent asset returns was

arbitrarily chosen. But much of this experimentation indicated that the choice of

wealth and money aggregates was not critically important. This finding has been

confirmed by much subsequent research, as described in Laidler (1977). I will

therefore report and


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replicate only a small subset of the results reported by Meltzer (1963).

Table 1 transcribes results in Meltzer (1963). Line 1 is equation (3) on p. 225,

with R1reported instead of R and "standard errors" instead of "t-statistics."^ Lines

2,3,5,6,7 and 8 are from Table 2, p. 232. Line 4 is from Table 1, p. 229. Of course, al l

regressions reported in this and al 1 other tables in this paper were estimated with

constant terms. Since the units of the dependent variable I used are not

meaningful, I wi 11 not report these constants.

The central findings in lines 1-3 of Table 1 (these and all subsequent

references are to tables in this paper), confirmed by other results in the original

paper, are the wealth or income elasticities of about unity and the strong, negative

effect of interest rates on real balances demanded. Notice that neither finding

shows up very clearly when the period is divided in two, as reported in lines 4-8 of

Table 1. For the early period, the income and wealth elasticities diverge, in different

directions, from unity and the interest elasticities are much reduced. Meltzer doesnot report the results with wealth only for 1930-1958. From what is reported,

however, it appears that the results for the full period were mainly dictated by

events in the latter half.)

Table 2 contains my replications of the results in Table 1. I dropped

1

The residuals from my replications of Meltzer's equations show very severe autocorrelation, and it is

clear from the Durbin-Watson statistics reported in Meltzer (1964) t viat this is also true of his original

regressions. As a result, I do not know how to interpret the "standard errors" reported in these tables. I

experimented with a variety of methods for correcting for serial correlation, but obtained only wildly erratic

elasticity estimates.

^For money, I used Ml throughout the paper. For 1900-14, this series is taken from Historical

Statistics (1960), series X267. From 1914-47, it is from Friedman and Schwartz (1970), pp. 704-718, column

7. For 1948-85, it is the "IMF series 3" from the International Monetary Fund's "International Financial

Statistics" tape. (The primary source for these IMF data is the Federal Reserve Bulletin.)

For 1900-49, real wealth is from Goldsmith (1956), Table W-3, column 1 ("total national wealth at 1929

prices"). For 1950-57, this series is from Historical Statistics (1960), series F446.

For 1884-1975, real income is real net national product from Friedman and Schwartz (1982), Table 4.8.

For 1976-85, it is taken from various July issues of the Survey of Current Business. The price level (used to

deflate Ml) is the implicit NNP deflator from the same sources. Permanent income is the geometrically

weighted sum of current and past real NNP's used in Friedman (1957). The weight on current income is .33.

The long term interest rate (used only for 1900-57) is the "basic yield on 20 year corporate bonds" in

Historical Statistics (1960), series X346. The short term rate for 190075 .s the "6 month commercial paper"

rate from Friedman and Schwartz (1982), Table 4.8, column 6. For 1976-85 I used Table B-68 in the Economic

Report of Jhe President (1987).


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141

Meltzer (1963) Results Dependent variable: n(Mj/P)

Coeff icients on:(standard errors)

regressions track this well (of course, with the interest rate also included as a

regressor). Real balances did not decrease nearly as much as did NNP in the 1930s,

but they increased much more than income in the 1940s. I conclude (though this is

the sort of issue reasonable people can disagree on) that current income induces

"too much" cyclical responsiveness in predicted money demand, relative to wealth,

and that wealth or some other "smoothed" income measure is preferred as the

regressor. This is also the conclusion reached by Laidler (1977).In Table 3, I report the consequences of some variations on Meltzer's results.

The objective of this experimentation is to locate a version of Meltzer's model that

is reasonably faithful, conceptually and quantitatively, to the original and is at the

same time inexpensive to test on more recent data.1

Line 1 in Table 3 uses permanent income (defined by Friedman's distributed

lag on current and past real NNP's) in place of wealth. This change does an

excellent job of reproducing line 1 of either Table 1 or 2. From a comparison of

Figure 1 with Figure 2, one can see that permanent income behaves more like

wealth than like current NNP in the 1930s.

Lines 2 and 3 report two variations on line 1. In line 2, the long interest rate

used by Meltzer is replaced by a short rate. I will explain my strong preference for

the latter in Section III. The short rate (over this period) varies sympathetically with

the long, but with more amplitude: hence its smaller coefficient. Otherwise, this

variation doesn't matter much. In line 3, I use an unlogged short rate. The issue

1The variations reported in Table 3 are very close to results in Laidler (1966). Laidle r used U.S. annual

series from 1892-1960, and deflated real balances and permanent income population. In his counterpart to

line 1 of Table 3 (his Table 2, A, p.548) he obtained permanent income and interest elasticities respectively

of 1.51 and .25. His counterpart of my line 2 (also Table 2, A in his paper) are 1.39 and .16. He did not try

unlogged interest rates .

Line ___________ Years ___________ ftn(r ) ___________ &n(W/P) __________ &n(Y/P)_____________ __ R*1 1900-58 -.949 1.11 .984

(.044) (.026)

2 1900-58 -.79 1.05 .960

(.083) (.041)

3 1900-58 -.92 .97 .13 .980

(.053) (.103) (.093)

4 1900-29 -.32 1.84 .960

(.107) (.114)

5 1900-29 -.05 .70 .960

(.094) (.45)

5 1900-29 -.22 .48 .3) .960

(.122) (.240) (.194)

7 1930-58 -.69 .94 .902

(.160) (.094)

8 1930-58 -1.15 1.35 -.10 .980

(.097) (.155) (.125)


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between the different functional forms in lines 2 and 3 is mainly aesthetic: the semi-

elasticity at the sample mean value of r (3.26 for 1900-57) is, from the estimate of

the elasticity in line 2, (-18)/(3.26) = .055. From line 3, this same semi-elasticity is

estimated at .07. (In this, as in all other economic applications with which I am

familiar, the choice of functional form is of little substantive consequence.) Thus I

will take Table 3 as justifying my referring to the model reported in line 3 as

"Meltzer's theory".


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143

Let me conclude this section with a somewhat less formal summary of the

information on income and interest elasticities contained in this 190057 sample.

Over this period, real Ml balances grew at the annual rate of .03356 and real

permanent income at the rate .03126. Short term ir erest rates fluctuated between

.69 (during World War II) and 7.4 (in 1920) but with a negligible trend. Hence the

ratio of the money growth rate to the income growth rate, 1.07, is a good estimate

of the income elasticity. This is about the number obtained, under various

assumptions, in Table 3. Over long periods, it must always be the case that the

trend in the dependent variable must be "explained" by that subset of the

regressors that have trends. In this application, real income does and interest rates

do not.Now imposing an income elasticity of unity, the semi-elasticity of money

demand with respect to the interest rate is just the slope of a plot of ln(Ml/Pyp)

against r$. This plot is displayed in Figure 3. This "estimation method" - get the

income elasticity from money and income trends and then get the interest elasticity

from a two-variable regression

Rjplicat ions Dependent variable: ln(Mj/P)Line Years !n (r) Coeff icients on:

(standard errors)

tn(W/P)

in(Y/P) R

1 1900-57 -1 .v 1.32 .957

( . 0 1) (.056)

2 1900-57 6 7 1.04 .971

(.077) (.036)

3 1900-57 .90 .49 .68 .978

(.089) (.122) (.095)

4 1900 29 -.21 .86 .957

(.099) ( . 0 5 1 )

5 1900-29 -.07 .73 .932

(.119) (.057)

6 1900-29 -.20 .65 .19 .960

(.098) (.149) (.132)

7 1930-57 -1.72 1.53 .901

(.139) (.163)

8 1930 57 -.55 .93 .937

(.141) (.075)

9 1930-57 -.78 .34 .75 .939

.264 .332 .191


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(Q4>3*

>T30>

cCD

To3->

U


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145

Predicted Ml/P using current income (line 2, Table 1).

,Predicted Ml/P using wealth (line 1, Table 1).


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Actual Ml/P

Predicted Ml/P using current income (line 2, Table 1).

Predicted Ml/P using permanent income (line 1, Table 3).

Figure 2

Year


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TABLE 1

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ftn(r) n(rs)Line

rs

n(yp)

.9891.03

(.021

)

-.77

(.044)

96 61.07

(.039

)

2 -.18

(.025)

96 31.06

(.042

)

3 -.07

( . 0 1 1

)

Variations on Table 2 for 1900-1957

Dependent variable: ftn(M1/P)

Coefficients on: (standard errors)

- does not depend very critically on our ability to characterize the residuals

accurately, or even on the residuals having a common structure over the entire

period. Since we have much more reason, to which I will turn in the next section,

for believing these elasticities to be stable than we have reason to believe anything

in particular about the residuals, this seems to me a desirable feature.^

Of course, no estimation method is satisfactory under all assumptions about

the errors, and the critical assumption here is that the errors are trend-free. If there

were important technical changes, not occurring in response to interest rate

movements, permitting agents to economize on their use of Ml balances my

method (and Meltzer's too) has understated the income elasticity. I do not see how

one can learn snore about this possibility by examining the series at hand.

'These informal remarks are not intended as a substitute for econometric theory. One would

certainIy have a better understanding of the estimates reported here and below if one could write down a

be I ievable stochastic model and use it to derive the proper t i es of these estimates explicitly. But I have

not done this and so am obIiged to follow a second best route and explain why I proceeded as I did in alooser (and hence less informative) way.


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Short-terrr Interest Rate

Figure 3 : 1900-57

2 3 4 5 6


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I I I . A T H E O R E T I C A L F R A M E W OR K

As an aid in interpreting the results reported in the last section and the

additional results to be reported in Section IV, I will introduce a simple theoretical

framework based on the model analyzed in Lucas and Stokey (1987). The

framework has the advantage (relative to the framework Meltzer used) of being

explicit about the connection between the portfolio and transactions demands for

money, and the disadvantage of being unrealistically stylized about the way trading

occurs. It will take some care to exploit the explicitness of this model without being

led too far astray by its unrealistic features.

We consider an economy in which the representative agent has the ultimate

objective of maximizing the discounted expected utility from consumption of

goods,

3t E{z8U(c.)} .

t=0 z

This agent lives in a Markovian world, the state of which at t is summarized by a

vector s^.. The distribution ofSj.+j, given s^., is given by a fixed transition function

F(s,A) = Pr{sule A|s = s) .

In this setting, all equilibrium date-t prices and quantities will be fixed (no time

subscript) functions of the current state, s^.Agents are assumed to alternate between securities trading and goods

trading in lockstep fashion. At the beginning of each period, all agents trade in

securities, including money, in a single centralized market, all with full knowledge

of the current realization of st. When securities trading is concluded, all agents

disperse either to produce or to purchase consumption goods. Some of these

goods can only be purchased with money acquired during the course of securities

trading: This transactions requirement is the sole reason for including cash in a

portfolio, in preference to interest bearing claims to future cash.

Consider first the decision problem facing an agent who is engaged in

securities trading at a time in which the state of the economy is s and his personal

wealth in dollar terms is W. (In a centralized securities market all assets are priced,

so the single number W summarizes his asset position fully.) Let v(s,W) denote the

value of this agent's expected,


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discounted utility if he proceeds optimally from this point on.

At this point, the agent is faced by a vector Q(s) of securities prices (in

dollars, so the price of money is unity). He must choose money holdings M and a

vector of securities holdings z, subject to a portfolio constraint:

N + Q(s) z < W . (2)

Let G(M,z,s) be the indirect utility function he uses to make this choice. (Clearly G

will depend on s, since the current state variable includes all the information he has

about the returns from these securities.) Then v(s,W) must satisfy:

v(s,W) = max G(M,z,s) subject to (2). (3)M,z

I call (3) the agent's portfolio problem.

Now where does this indirect utility function G come from? Having completed

securities trading, the agent is about to engage in purchasing a vector c of

consumption goods. He will also receive an endowment y(s) of goods, but this he

must sell for cash or future cash: He cannot consume his own endowment. The

rules of trading in this goods market are summarized by a vector of constants a,

where a^ e [0,11 is the fraction of purchases of good i that must be covered by

money. It will be an expositional simplification in what follows to postulate a

technology together with a choice of units for measuring goods such that all goods

sell for the same nominal price P(s). In this case, the agent's Clower- or cash-in-

advance constraint is:

P(s)a c < M . (4)

The outcome (M,z) of the portfolio decision plus the outcome (c,y(s)) of his

goods trades plus a given vector D(s') of nominal returns (dividends, interest,

principal) on securities will determine this agent's nominal wealth position W as of

tomorrow, conditional on tomorrow's state s'. He begins next period with his dollar

holdings as of today, M, plus the dividends and resale value of his securities,

(Q(s')+D(s'))"z, plus the dollar value of his endowment, P(s)S y (s), less the dollar

value of his goods purchases, P(s)S c . That is:


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W = H + [Q(s')+D(s')lz + p(s)fIyi(s)-C|I

These considerations determine what I call the transactions problem:

G(M,z,s) = max U(c) + 3/ v(s',W)F(s,ds') subject to (4) , (6)c

where W is defined in (5).

Eliminating the function G between (3) and (6) defines a functional equation

in the value function v. See Lucas and Stokey (1987) for an analysis of this

equation and its use in constructing an equilibrium for this economy. My purpose

here is not so much analysis as it is clarifying what we mean by a "demand

function for money," and hence in understanding what an empirical money

demand function might mean. Let me begin with what I think Meltzer (1963) and

certainly Hamburger (1977) meant by a "demand function for money."From the portfolio problem (3) one obtains the first order conditions:

G(yj(M,z,s) = v , (7)

G (M,z,s) = (hv , j = l,...,m , (8)J J

where \> is the multiplier associated with the wealth constraint (?) and where j

indexes the m available securities. These m+1 equations together with (2) can be

solved to obtain the demand functions for the assets (M,z) which have as

arguments the prices Q and wealth W. Singling out the demand function (in this

sense) for money:

M = f(Q,W,s) . (9)

Note that the entire vector Q of securities prices enters on the right of (9). In

practice, as in any empirical application of demand theory, one would focus on theprices of securities thought to have strong substitution or complementary

relationships with money. In this spirit, Meltzer used a long term bond yield in his

econometric work. In the same spirit, Hamburger (1977) experimented with equities

yields and other securities returns in his.Certainly (9) is a respectable bas i s for an empirical study,

(5)


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consistent with what we knew then about monetary theory and, I would say,consistent with what we know now. Yet it does not seem to me that one would haveany confidence that the demand function (9), based on portfolio considerations onlyas in my derivation, would remain stable over time. Included as suppressedarguments in this functions f are all variables s characterizing the current state ofthe system, including all the information used by agents in forecasting futurereturns on all securities. Moreover, if the stochastic environment in which agentsoperate (the "regime," as it is often called) should change from time to time, these

changes too will induce shifts in f. Surely shifts in the realizations of informationalvariables and/or in the processes assumed to generate these realizations must havebeen substantial over so long a period as 1900-1958.

To decide whether the fact that the functions f are not likely to be structural is

an important objection to the empirical application of (9), consider the fact that by

exactly the above argument on money demand, we could derive a demand function

of the same form as (9) for any portfolio item. Would one, for example, attempt to

estimate a demand function for Brazilian government securities, including as

arguments only their own current yield and another interest rate standing in for the

composite security consisting of all other portfolio items, and expect thisrelationship to be stable over a 60 year period? I think there is more to Meltzer's

money demand theory than portfolio considerations alone.

To see what this is, turn to the transactions problem (6), which also defines

the indirect utility function G. The first order conditions for the n consumption

goods in this problem are:

Mc)= B fvw(s',W')P(s)F(s,ds') + yP(s)ai, i=l,...,n, (10)

where y is the multiplier associated with the cash-in-advance constraint (4). One

can also calculate the derivatives of the function G from (6):

G|y|(M,z,s) = u + 0/ vw(s',W')F(s,ds') , (11)

Gz.(M,z,s) =B fvw(s',W')[Qj(s')+Oj(s')]F(s,ds'), j=l,...,m. (12)

That is, the value (in utiIs) of a dollar is its "liquidity" value y during

goods trading plus the marginal valueof nominal wealth one period hence. The value

of any other security is the v/alue of the increment it provides


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to future wealth. Equations (11) and (12) thus reduce the values of securities,

money included, to the values of their associated "fundamentals."

Now suppose that among the m available securities is a (nominal) risk free,

dollar denominated, one period bond. For this security,Qj(s') = 0

and Dj(s') = 1. Let its current price be i+r(s) , so r(s) is the one

period nominal interest rate. Then combining (7) and (8) from the portfolio problem

and (11) and (12) from the transactions problem (where both (8) and (12) are

specialized to this one period bond) and inserting into the first order

conditions (10) we obtain:

Uj(c) - P(s)ulai+ ^ry| , i =

That is to say, the relative "prices" of these consumption goods, as seen by

consumers (normalized so that the prices of each received by sellers are all equalto P(s)) depend on the cash holdings required to purchase them together with the

opportunity cost of holding cash, as measured by the nominal interest rate.

In the environment I have been describing, in which no new information

reaches agents after they have switched from securities trading to goods trading,

agents will plan money holdings so that the cash-in-advance constraint (4) holds

with equality: In the theory, as in fact, cash is dominated by nominal bonds as a

store of value. In this case (13) and (4) (with equality) form a system of n+1

equations in the consumption vector c and the multiplier p. It is not quite a demand

system (since the "prices" in (13) are not the same as the "r.rices" in (4)) but it can

be

treated just as if it were and solved fp Vn consumption vector c as a

. ' 'function of iVP(s) and r(s), say: ^

C = g ( p . r ) .

Thus we obtain, from transactions considerations, ar exact relationshipbetween agents' desired consumption mix, their demand for real balances, and the

nominal interest rate. Noticc that no other securities prices or returns enter into

this relationship, nor does the state s

(13)

(14)


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(except through the two prices P(s) and r(s)).^ Changes in information or in the

information structure of the system will not shift these curves. They will be stable

over time provided only that preferences are and that the trading technology as

sunmarized in the coefficients a^,...,anis stable.

It seems to me a violation of common usage to call the relationship (14) a

"demand function for money." It is a relationship among complementary choice

variables that the demand functions must satisfy. Whatever one calls it, however, it

is a relationship that must obtain in equilibrium and it seems more likely to be an

empirically stable one than does the "true" demand function (9). Why not provide

an operational specification of these coefficients a^ and try to estimate it

econometrically? This is the approach taken in a recent paper by Mankiw and

Sumners (1986), with very interesting results that I wil 1 come back to in the next

section, first, however, it will be useful to go into more detail about the connections

between (9) and (14).Meltzer's estimated income and wealth elasticities are around unity,

suggesting (under the utility-theoretic framework I am using here) that the current

period utility function U takes the form of a constant relative risk aversion function

of a homogeneous of degree one function of consumption. Let us impose this on

the model above. Then equations (13) can be solved for the ratios c^/c of

consumption of each good to total consumption c = c^ = g^(r)c , say.

Substituting into the cash constraint gives:

= Eiaigi(r) c = h(r)c , (15)

where the second equality defines the function h. This is just a consolidated

special case of (14), stil1 not a demand function for money. Under these same

assumptions, the "true" demand function for total

^Th i s rationale for (14) is essen t i a I I y the same as that used for a simila! purpose by McCallum

and Goodfriend (1987). See Ando, Modigliani and Shell (1975) for the earliest derivation of (14) along these

lines that I have found. These writers draw the same conclusion I have in the text: that on Iy the short rate

ought to appear on the right side of a money demand function. Hamburger (1977) views (14) as a

"Keynesian" formulation, explicitly contrasting it to the "monetarist" emphasis on portfolio considerations.

If he is right, then my use of (14) to derive Mel^er's equat ion (18) is a very "un-monetar i si" argument. But

one of the purposes of this sect ion is exactly to argue that port folio and transactions considerations are

complementary in thinking about money demand.


19/

(16)

155

consumption c takes the form:

c - k(Q,s) .

Then combining (15) and (16), we have shown that, under this homotheticity

assumption, the true demand function for money (9) takes the form:

p = h(r)k(Q,s) p .

Now there is no theoretical reason to expect (17) to be more stable

empirically than (9): They are the same relationship! But empirically, total

consumption has been found to be a fairly stable function of permanent income,

suggesting that k(Q,s)/r is nearly constant over a wide range of circumstances. If

so, then:

F *(r>*p

where


20/

156

characterization of transactions demand that leads to a relationship between real

balances, short term interest rates and permanent income or wealth that one might

want to view as structural. This characterization was made tractable by the

assumption that everyone engages in securities trade at the same time, all with the

same fixed period. That this assumption is unrealistic is obvious. That it is

unrealistic in a way that is critical to the theory of money demand was shown by

Grossman and Weiss (1983) and Rotemberg (1984), who examined theoretical

settings in which only a subset of agents is engaged in securities trading at any

time. This modification alters the way the system responds to open market

operations, because when the central bank issues money for bonds, interest rates

must move so that the subset of private agents on the other side of this exchange

is willing to acquire a disproportionate share of the economy's new money supply.

This alteration introduces a Keynesian "liquidity preference" element into money

demand that is entirely absent from the formulation I have sketched. Cochrane(1988) appears to have identified these liquidity effects, for periods up to a year, in

post-1979 U.S. weekly series on Treasury bill rates and money growth rates. (I say

"appears" because the connections between theoretical models of the Grossman-

Weiss-Rotemberg type and the estimation methods used by Cochrane have not

been worked out in any detail.)

By using annual data, it seemed possible that Meltzer's results and mine

might avoid contamination from these "liquidity preference" effects. We will see in

the next section, however, that this hope is not confirmed, at least for post-1958

data. The trick will thus be to get as much as we can out of a money demand theory

that is not adequate to account for some short run events.

I V . H O N E Y D E M A N D S I N C E 1 9 5 8

Econometric research on money demand has undergone considerable

development since the early 1960s. In the main, this work (with the notable excption

of Friedman and Schwartz's (1963) and (1982) studies of long U.S. and U.K. timeseries) has focused on evidence from postwar U.S. quarterly series. Meltzer's work

is not cited in Judd and Scadding's (1982) review article (though they do make

repeated use of Laidler (1977), which was in turn heavily influenced by Meltzer's

work) and, in general, the research cited in this survey is not much concerned with

comparison of postwar evidence with evidence from the earlier years of the

century.


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157

The pioneering paper in this "modern" era of money demand studies is

Goldfeld (1973), which introduced distributed lag methods that seem to be needed

to obtain close fits to quarterly data. Subsequent work has, in large part, been

devoted to the refinement of Goldfeld's studies and to dealing with the fact

(stressed most forcefully by Goldfeld (1976)) that his equations deteriorated in fit

on data outside the original sample period.

There is no doubt that recent work is based on a much more sophisticated

awareness of econometric issues specific to time series analysis than was the

research of the 1950s and 60s. At the same time, the substantive results have been

disappointing. Judd and Scadding refer to "the observed instability in the demand

for money after 1973," and endorse the conclusion reached earlier by Cooley and

LeRoy (1981) "that the negative interest elasticity of money demand reported in the

literature represents prior beliefs much more than sample information." The unit

income (or wealth) elasticity is no longer regarded as wel1-established, and mostrecent work has focused on find i "scale variables" that sharpen short-term

forecast errors rather than on estimates of the income elasticity that stand up well

over different data sets. In short, one gains the impression that subsequent

research has generally failed to support Meltzer's findings, that the income and

interest elasticities he estimated are inconsistent with more recent evidence and

were even, perhaps, as much the product of his "prior" as they were inferences

drawn from the time series he studied.

I think al 1 of these conclusions, or impressions, are incorrect. In this section

I will argue that Meltzer's 1963 results are not only qualitatively but quantitatively

consistent with observations since 1958: that even if one takes the income and

interest elasticities estimated, by his methods, from pre-1958 data alone one

obtains a more useful account of money demand in the 25 year period sinee than is

obtained from more recent distributed lag formulations. Moreover, I will exhibit the

information on the interest elasticity of money demand contained in 1900-1985 data

in such a way as to concentrate even Cooley and LeRoy4s posterior distribution on

Meltzer's 1963 co .usion.

At the sair.. time, this application of Meltzer's equation to more recent data

will also reveal repeated, systematic patterns in the residuals. These are patterns

that are not consistent with the theoretical model reviewed in Section II (and hence

not consistent with Meltzer's theory as I have interpreted it). I think it will be easy to

see why these patterns motivated Goldfeld and others to resort to distributed lag

methods. But I

4 1900-85 -.07 .97 .967

(.044) (.019)


22/

158

will argue that these methods have served to obscure rather than reveal both the

sense in which this theory helps to understand recent events and the sense in

which it falls short.

Table 4 provides results for the entire 1900-85 period and for the recent

subperiod 1958-85. Line 1 is exactly the same regression as line 3, Table 3 for the

full period. Line 3 of Table 4 is the same regression for the period 1958-85 only. One

can see that simply adding the later years to the full sample results in virtually no

change in the estimated elasticities. However, the results for the later years taken

by themselves show a drastic deterioration in fit and large changes in estimated

coefficients as compared to the 1900-57 period. In lines 2 and 4 of Table 4, the

income elasticity is constrained to be unity (so no "standard error" is reported).

Line2is, not surprisingly, the same as line 1, but so too is line 4.

Examining trends over the later period (as I did in Section II for the earlier

years) helps in interpreting Table 4. In the27 year period 195885, real money

balances grew at an annual rate of .004 while real income grew at a rate of .03.

Short term interest rates increased (though not at all smoothly) from around 3

percent to around 9 percent, or at a rate of about.22percentage points per year. To

fit these trends, the interest semi-elasticity nrand the income elasticity have to

lie on the

line: nr= -.02 + (.14)ny With an income elasticity of unity, this implies an interest

semi-elasticity of .12. This pair of estimates is roughly

TABLE 4

Results from 1900-85 Dependent variable: n(Ml/P)

Coeff icients on:(standard errors)

2

Line Years r^ R consistentwith the estimates 1.06 and .07 reported in line 3 of Table

3. It is also consistent with the estimates .97 and .07 in line 1 of Table4, and with the constrained estimates in lines 2 and 4 of Table 4. Similarly, the

unconstrained estimates .21 and -.01 on line 3 of Table 4 lie roughly on this line.

One can account for the divergent trends in income and real balances over the

1953-85 period either with the 1900-57 estimated income and interest elasticities or

with much lower income and interesto

elasticities.

Figure 4 illustrates, in part, why I prefer tu?. constrained estimates reported

on lines 2 and 4 of Table 4 to the unconstrained estimates on line 3. This figure


23/

159

plots the log of P/Pyp against the short term interest rate for the entire 1900-85

period, with the post 1957 observations indicated by different symbols from the

1900-57 observations. One can see that if one constrains the income elasticity for

the entire period to be unity, one gets in return a single interest semi-elasticity for

the entire period. The most recent points lie exactly or the line defined by the

earlier ones and, since interest rates behaved so differently in the recent period,

the estimate is greatly sharpened by the new observations.

Let me try to summarize the sense in which Figure 4 confirms both Meltzer1s

hypothesis that real money demand is a stable function of permanent income (or

wealth) and interest rates and the numerical estimates h? obtained. Meltzer

estimated these two parameters by least squares. As Figure 2 shows, the estimated

income elasticity is mainly dictated by the common trend of real balances and

income. At this estimated value of unity, Figure 3 shows that the interest elasticity

is determined by a reasonably tight scatter of an(Ml/Pyp) against rs. If one imposesthe same income elasticity of unity on the 1953-85 period, this same scatter.,

reproduced as Figure 4, confirms the original interest elasticity estimates and since

interest rates were so much higher in the later period, the new experiment is a very

good one. Notice that there is nothing arbitrary or experimental about Figure 4: It is

precisely the scatter one would want to look at in view of the estimates Meltzer

obtained using pre-1958 data only.

However, as 1ine 3 of Table 4 shows, these two elasticity estimates cannot be

recovered from the 1958-85 data using least squares (as Meltzer


24/

Figure 4 :1900-65

160

6

J * 5| " .

5 - denotes observations from 1900 to 1957. 0 - denotes

observations from 1958 to 1985.

1.8

1.6


25/

161

1612 144 6 8 10

Short-Term Interest Rate

m_ e * t 4 .

f o f l o * n r> * C

0.8

0.6

1.4

1.2


26/

162

recovered them from the earlier data). There is a reason why these estimates came

out as they did, as Figure 5 shows. Interest rates were not only increasing

dramatically over the 1958-85 period but were also highly erratic. The relatively high

interest semi-elasticity on line 3 reconciles the trends with a high income elasticity,

but the cost of this reconciliation is that the "predicted" path of real balances from

the constrained estimates is much too interest-sensitive to fit observed, year-to-year

movements. Actual real balances move in the predicted direction in response to

interest rate changes, but by much less than is predicted. These lead to large

residuals, which are also strongly correlated with interest rates. This is why the

order revealed in Figure 4 cannot be discovered using unconstrained least squares.

Mankiw and Summers (1986) recover exactly an income elasticity of unity and

an interest semi-elasticity of .05 from least squares applied to 1960-84 U.S. quarterly

series. They do so using consumption in place of permanent income (justified in part

by the kind of argument I used in Section III) and by using Almon lags to average the

independent variables over time. One can conjecture from Figure 5 that averaging

interest rates will "work," and Mankiw and Summers1s results confirm this. (I

suspect that long interest rates worked as well as they did in Meltzer's study for

much the same reason: Long rates are a kind of average of short rates.)

V. CONCLUSIONS

This paper has had three main objectives. As reported in Section II, I first

replicated some of the results in Meltzer (1963), using his 1900-1957 sample period,

and showed that two variations of interest to me are empirically indistinguishab1e

from the model he used. Second, in Section III, I reviewed a theoretical model of

money demand in which the two parameters Meltzer estimated could be expected to

be "structural." Thirc?, in Section IV, I compared the predictions of Meltzer's model,

with his original parameter estimates, to post-1958 data, and concluded that this

comparison yields additional confirmation of the theory and of these two estimates.

Meltzer (1963) was criticized (for example, by Courchene and Shapiro (1964))

for, among other things, his failure to correct his estimates for severely serially

correlated residuals and his failure, despite great emphasis on the "stability" of the

money demand function, to apply standard statistical tests for the stability of

parameter estimates across different


27/

163

Actual Ml/P.

Predicted Ml/P from line 4, Table 4.

Figure 5

Year


28/3

164

sample periods. These two criticisms can certainly be applied as well to the present

paper, for I share Meltzer's emphasis on the "stability" of the money demand

function.

But I agree with Meltzer (1964) that these econometric criticisms are very badly

off the economic point. We begin with a simple economic model that suggests a two-

parameter description of money demand. When we hypothesize that this relationship

is "stable," we mean that we expect these two parameters to reflect relatively stable

features of consumer preferences and the way in which business is carried out, and

we expect them not to shift around as monetary or other policies are altered over

time. This theory does not suggest that the residuals can be characterized in a

simple, elegant fashion over a given time period, or even that the stochastic structure

of the residuals should be stable over time. Accordingly, there is little point in testing

the theory by maintaining an extreme hypothesis on the residuals that is not implied

by any theoretical considerations and then performing a Chi-square test for the

equality of coefficients over subperiods. One needs a maintained hypothesis in

which one has more, not less, confidence than one has in the hypothesis being

tested.

Thus Meltzer argued, and I agree, that we can only test the theory by comparing

its numerical predictions to as wide a variety of data as we can find. In carrying out

such tests, it is of no interest whatever to let the two crucial elasticities isolated by

the theory change arbitrarily from one data set to the next. The theory is of no

interest or use unless these two parameters are stable under a wide range of

circumstances.

Over the time period Meltzer studied, in which income has a strong trend and

interest rates had none, the method of least squares isolates an income elasticity of

unity, just as does a comparison of income and real balance trends. With this income

elasticity, one can see from Figure 3 that there is enough interest variability to trace

out a fairly clear demand curve. Over the more recent period, interest rates have a

very strong upward trend, as does income, so that there are many combinations of

elasticities that are consistent with trends in the holding of real balances. Least

squares picks out a combination of elasticities that is very different from the pair thatis consistent with earlier evidence. Yet imposing the same elasticities in the later

period is also consistent with long term trends and, as Figure 4 shows, traces out a

demand function that is consistent with theearlierdata, and much clearer than was

possible


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165

with those data alone.^ This picture did not arise by chance!

The evidence from the post-1960 years also reveals strong patterns in the

residuals from this estimated demand function that did not appear in the earlier

years of the century. It is clear that, as investigators sinee Goldfeld have

concluded, the portfolio adjustment process is subject to lags in a way that neither

the theory Meltzer had in mind nor the cash-in- advance model I sketched in

Section III helps to understand. This fact is hardly surprising: One is, if anything,

surprised that this simple model captures as much as it does.

In these circumstances, it seems to me that it is the econometrician1s job to

display as clearly as he can the respects in which the model he has is a good

approximation to reality and the sense in which it is not. This is what Meltzer did in

his 1963 paper, and it is what I have tried to do in this one. I hope Figure 4

convinces anyone who sees it that the interest semi-elasticity of money demand

has remained stable at something between .05 and .10 for nearly a century in the

U.S. I hope Figure 5 helps to stimulate someone, perhaps along the lines suggested

by Grossman, Weiss and Rotemberg, to discover the short run dynamics that can

reconcile this fact with year-to-year or even quarter-to-quarter movements in

observed money holdings.

^An income elasticity of unity is a I so consi stent wi th the cross-section evidence reported in Meltzer

(1963b). The interest semi-eIast i c i t i es est imated from U.S. t i me series are a I so consistent wi th the

range ot estimates Cagan (1956) found in his study of hyper i nfI at ions.


30/

166

REFERENCES

Ando, A., Modigliani, F., and Shell, K.

1975 Some Reflections on Describing Structures of Financial Sectors, in Gary

Froran and Lawrence R. Klein, (eds.)Brookings Model Perspective and

Recent Developments .Amsterdam: North- Holland, 524-563.

Stunner, K. and Meltzer, A.H.

1963 Predicting Velocity: Implications for Theory and Policy. Journal of

Finance, 18: 319-354.

Cagan, P.

1956 The Monetary Dynamics of Hyperinflation, in Milton Friedman, (ed.),

Studies in the Quantity Theory of Money. Chicago: University of Chicago

Press.

Cochrane, J.H.

1988 The Return of the Liquidity Effect: A Study of the Short Run Relation

Between Money Growth and Interest Rates.Journal of Business and

Economic Statist ics. Forthcoming.

Cooley, T.F. and LeRoy, S.F.

1981 Identification and Estimation of Money Demand. Amer ican Economic

Review, 71: 825-844.

Courchene, T.J. and Shapiro, H.T.

1964 The Demand for Money: A Notes from the Time Series.Journal of

Pol i t ical Econom y, 72: 498-503.

Economic Repor t of the President

1987 Washington D.C.

Friedman, M.

1956 The Quantity Theory of Money - A Restatement," in Milton Friedman,

(ed.),Studies in th e Quantity Theor)> of Money. Chicago: University of

Chicago Press.


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A Theory of the Consumpt ionrunct ion.Princeton: Princeton University

Press, for the National Bureau of Economic Research.

1959 The Demand for Money - Some Theoretical and Empirical Results.Journal r f

Pol i t ica l Econom y,67: 327-351.

________ and Schwartz, A.J.

1963A Monetary History of the United States, 1867-1960.Princeton:

Princeton University Press, for the National Bureau of Economic

Research.

________ and _________

(1970)Monetary Stat ist ics of th e United States.New York: Columbia University Press

for the National Bureau of Economic Research.

________ and _________

1982Monetary Trends in the United States and the United Kingdom.

Chicago: University of Chicago Press, for the National Bureau of

Economic Research.

Goldfeld, S.M.

1973 The Demand for Money Revisited.Brookings Papers on Economic A ct iv i ty

, 577-638.

1976 The Case of the Missing Money."Brookings Papers on Economic Act iv i ty, 683-

730.

Goldsmith, R.W., et al.

1956A Study of Saving in th e United States,Vol. III. Princeton: Princeton University

Press, for the National Bureau of Economic Research.

Friedman, H.

1957


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Grossman, S. and Weiss, L.

1983 A Transactions-Based Model of the Monetary Transmission Mechanism."

Amer ican Economic Review, 73: 871-880.

Hamburger, M.J.

1977 Behavior of the Money Stock: Is There a Puzzle?Journal of MonetaryEconomics, 3: 265-288.

Judd, J.P. and Scadding, J.L.

1982 The Search for a Stable Money Demand Function.Journal of Economic

Literature, 20: 993-1023.

Laidler, D.E.W.

1966 The Rate of Interest and the Demand for Money - Some Empirical

Evidence.Journal of Pol i t ica l Economy,74: 545-555.

1977The Demand for Mo ney: Theoret ical and Empir ical Evidenc e. Second Edition. New

York: Dun-Donneley.

Lucas, R.E., Jr. and Stokey, N.L.

1987 Money and Interest in a Cash-in-Advance Economy. Econometr ica, 55: 491-

514.

Mankiw, N.G. and Summers, L.H.

1986 "Money Demand and the Effects of Fiscal Policies.Journal of Money,

Credit , and Bankin g, 18: 415-429.

McCallum, B.T. and Goodfriend, M.S.

1987 Money: Theoretical Analysis of the Demand for Money." Paper

prepared for The New Palgrave: A Dict ionary of Economic Theory and

Doctr ine.

Meltzer, A.H.

1963 The Demand for Money: The Evidence from the Time Series. Journal of

Pol i t ical Econom y, 71: 219-246.


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Meltzer, A.H.

1963b The Demand for Money: A Cross-Section Study of Business Firms.

Quarter ly Jo urnal of Econom ics,77: 405-422.

1954 A Little More Evidence from the Time Series.Journal of Pol i t ica l Economy,72:

504-508.

Poole, W.

1970 Whither Money Demand?Brookings Papers on Economic A ct iv i ty , 485-501.

Rotemberg, J.J.

1984 A Monetary Equilibrium Model with Transactions Costs.Journal of Polit ical

Economy,92: 40-58.

Samuel son, P.A.

(1947)Foundat ions of Econom ic Analysis.Cambridge: Harvard University Press.

U.S. Bureau of the Census.

1960Histor ical Stat ist ics of the United States, Colonial Times to 1957. Washington D.C.

1958 from the sample because I could not find w for that year. Otherwise, I attempted

to follow the sources and procedures described in Meltzer (1963). One can see that

lines 1 and 2 from Tables 1 and 2 are very close, though closer for the income

regression than the wealth regression. When both variables are included (line 3) I

obtained very different results from his, for reasons I cannot explain. Notice,

however, that Meltzer*s and my estimates of the sum of these coefficients are very

close: I suspect this is all either of us is estimating with much precision. The other

striking difference is in line 4 of Tables 1 and 2: my wealth elasticity for this

subperiod is well below one; Meltzer's is 1.8.

I wanted to use a graphical device to help me see how different a theory one

obtains with different wealth or income measures. I know this question is not very

well posed, but Figure 1 seems to me helpful. It exhibits three series, all for the fullperiod 1900-1957. They are: actual Ml/P; the "predicted" Ml/P from line 1 of Table 2;

and the predicted Ml/P from line 2 of Table 2. One can see that real balances followed

a different trend from 1930 on than in the earlier years. Both the income and wealth


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170

2 190C-85 -.09 1.0 ~

(.001) -------------------

3 1958-85 -.01 .21 .328

(.005) (.059)

4 1958-85 -.07 1.0 ---

(.008)


35/

171

Q

Poole (1570) argued much earlier that cnc- needs to constrain the income elasticity in ot der to obtain an

interest elasticity from post-World War II data that is consistent vith pre-war evidence.

money demand in the us a quantitative review.docx

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