Is the price elasticity of money demand always unity?

Evans, Paul ; Wang, Xiaojun

I. INTRODUCTION

For much of history, money largely consisted of monetary metals. In particular, during late nineteenth and early twentieth centuries, many countries adopted gold standards and gold served as money--either directly as coins held by the public or as claims on bullion held by commercial and central banks. At the same time that large stocks of gold were being held for monetary uses, even larger stocks were being held for nonmonetary uses owing to gold's superior luster, reflectivity, malleability, conductivity, ductility, and resistance to corrosion. (1) In its monetary uses, it was valued only in terms of its ability to purchase consumption goods. As a result, asset holders demanded real gold balances, that is, the nominal stock deflated by an appropriate index of prices for goods. By contrast, in its nonmonetary uses, it was valued in terms of its physical units, that is, its undeflated nominal stock. These dual uses of gold imply that the demand for monetary gold should not be expected to be unit-elastic with respect to the price level, a result that we demonstrate formally in the next section. (2)

Empirical researchers have not noticed that money demand behaves differently under commodity standards from how it behaves under fiat standards. Since the available historical data are dominated by observations from fiat standards, the characteristics of money demand under commodity standards are largely concealed when the researchers use data that span both commodity and fiat standards. More specifically, the hypothesis of long-run price homogeneity is usually not rejected. For example, Meltzer (1963) used U.S. data from 1900 to 1958 to estimate money demand functions formulated in both nominal and real terms and found little evidence against price homogeneity. Laidler (1971) carried out a similar test using both U.K. and U.S. data over the period of 1900-1965 and obtained a similar result notwithstanding a different specification. Friedman and Schwartz (1982) also found support for price homogeneity for both United Kingdom and United States over the period of 1867-1975 using the method of phase averaging to extract the long-run correlation between nominal money demand and the price level. Hendry and Ericsson (1991) applied cointegration methods to the annual U.K. data from Friedman and Schwartz (1982) to confirm price homogeneity. Applying the same approach to annual U.S. data from 1874 to 1975, MacDonald and Taylor (1992) found that price homogeneity cannot be rejected at the .05 statistical significance level but can be rejected at the .10 level.

[FIGURE 1 OMITTED]

The United Kingdom and United States were on fiat standards for the bulk of the sample periods that these researchers employed. (3) For this reason, their findings do not serve as evidence for price homogeneity under commodity standards. In Section II, we present a simple theoretical model that employs the money-in-utility framework augmented by also giving utility to holdings of nonmonetary gold. We demonstrate that under fairly general conditions, the price elasticity of money demand should be less than 1 under commodity standards. Section III uses three data sets to provide empirical evidence supporting this theoretical prediction. Section IV draws some final conclusions.

II. THEORETICAL MODEL

Consider a representative household that chooses the paths for its consumption C, stock of monetary gold M, stock of nonmonetary gold N, and real stock of assets A taking as given the paths for non-asset income Y, the price level P, and the nominal and real interest rates i and r that it earns on A. In making its choices, it seeks to maximize the objective function:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to a sequence of budget and portfolio constraints of the form: (4)

(2) [C.sub.t] + [i.sub.t][([M.sub.t] + [N.sub.t])/[P.sub.t]] + [A.sub.t] [less than or equal to] [Y.sub.t] + [r.sub.t][A.sub.t], t [member of] E [0, [infinity],

where

(3) [A.sub.t] = [([M.sub.t] + [N.sub.t])/[P.sub.t],] + [K.sub.t]

and a sequence of borrowing constraints sufficiently tight to rule out Ponzi schemes but sufficiently loose to never bind and a fixed initial value for A. In Equations (1)-(3), t [member of] [0, [infinity]) is a continuous index of time, [rho] [member of] (0, 1) is a subjective discount rate, and K is the real stock of assets other than gold. We assume that the instantaneous utility function U is increasing, strictly concave, and twice continuously differentiable and satisfies Inada conditions in all its arguments.

We have chosen to induce a demand for monetary gold by inserting M/P into the instantaneous utility function. This modeling choice has a long pedigree in monetary economics largely because of its simplicity and ability to yield sensible demand functions similar to what other modeling choices yield. We could have employed a shopping time or a cash-in-advance model and obtained similar results.

We have also chosen to induce a demand for nonmonetary gold by also inserting N into the instantaneous utility function. This modeling choice also has a long pedigree since nonmonetary gold is simply one type of consumer durable. Even though cars and washing machines as well as jewelry are not demanded for their own sake, their stocks do generate service flows that have many of the same characteristics as nondurable consumer goods and can be plausibly entered as arguments in momentary utility functions.

The budget constraint (2) imposes a relative price of 1 between monetary and nonmonetary gold because both are produced by the same production process. Furthermore, under the gold standard, nonmonetary gold could always be minted into monetary gold or simply stored as bullion in bank vaults and monetary gold could always be melted down and put to nonmonetary uses. This property distinguishes nonmonetary gold from other consumer durables, which have variable relative prices that are endogenous in general equilibrium.

The first-order conditions for the household's problem imply that

(4) [U.sub.m](C,M/P,N) = i[U.sub.c](C,M/P,N)

and

(5) [U.sub.n](C,M/P,N) = (i/P)[U.sub.c](C,M/P,N),

where we have suppressed the t subscript and appended subscripts c, m, and n to the function U in order to indicate derivatives with respect to C, M/P, and N. According to Equations (4) and (5), households equate the marginal utility of their real stock of monetary gold to the marginal utility of the nominal interest that they forgo from holding it, while they equate the marginal utility of the physical stock of nonmonetary gold to the marginal utility of the real nominal interest that they forgo from holding it.

Equations (4) and (5) and the implicit function theorem imply that M/P is related to C, i, and P by a function of the form: (5)

(6) M/P = [LAMBDA](C, i, P).

Totally differentiating Equations (4) and (5) and solving for [[LAMBDA].sub.p], the partial derivative of A with respect to log P, yield

(7) [[LAMBDA].sub.p] = (i/P)[U.sub.c]([U.sub.mn] - i[U.sub.cn])/D,

where

(8) D [equivelent to] ([U.sub.mm] - i[U.sub.cm])[[U.sub.nn] - (i/P)[U.sub.cn]] -- ([U.sub.mn] - i[U.sub.cn])[[U.sub.mn], - (i/P)[U.sub.cm]].

[LAMBDA].sub.p] cannot be signed in general without making further assumptions about the instantaneous utility function U. It is reasonable, however, to suppose that nonmonetary gold and monetary gold are Edgeworth substitutes since gold originally became money for many of the same reasons that it is attractive as a nonmonetary asset. On the assumption that nonmonetary gold is also a sufficiently weak Edgeworth substitute for C ([U.sub.cn] is either nonnegative or not too negative), we then have [LAMBDA]p < 0 since D > 0 is a necessary condition for a maximum in the household's problem. In other words, the monetary stock of gold should be less than unit-elastic with respect to the price level.

The intuition behind this phenomenon is straightforward. Eliminating i[U.sub.c] between Equations (4) and (5) gives us:

[U.sub.m](C,M/P,N) = P[U.sub.n](C,M/P,N).

An increase in the price level raises the relative price of real balances in terms of nonmonetary gold. It is therefore natural to expect gold to shift from monetary to nonmonetary uses. This tendency offsets some part of the usual proportional response of M to P. In terms of the above equation, M/ P tends to fall and N tends to rise in response to an increase in P.

III. EMPIRICAL ANALYSIS

We present three sets of empirical results for the world, the United Kingdom, and the United States. For each data set, we estimate the log-log demand function

(9) log [M.sub.t] = [[lambda].sub.0] + [[lambda].sub.p] log [P.sub.t] + [[lambda].sub.y] log [Y.sub.t] + [u.sub.t]

for two specifications of the error term u,. In our data, the nominal interest rate is stationary, while log M, log P, and log Y are cointegrated. (6,7) We therefore employ the levels specification (9), which excludes the nominal interest rate. (8) We are primarily interested in the elasticity [[lambda].sub.p], which the theory of the previous section claims is less than 1. For this reason, we shall be testing the null hypothesis that it is 1 against the one-tailed alternative hypothesis that it is less than 1. Rejection of the null will support the theory. We also expect the elasticity [[lambda].sub.y] to be appreciably larger than 0 and will regard our estimates as problematical if our estimates of [[lambda].sub.y] are not.

We fit Equation (9) using Johansen's method of estimating vector error correction models as well as dynamic ordinary least squares (DOLS) as described by Hamilton (1994, 602-608). In the latter method, the error term [u.sub.t] is modeled as taking the form:

(10) [u.sub.t] = [q.summation over (j = -q)] [[pi].sub.j] [DELTA]log [P.sub.t+j] + [q.summation over (j = -q)] [[eta].sub.j][LAMBDA]log [Y.sub.t+j] + [v.sub.t]

with

(11) [v.sub.t] = [n.summation over (j=1)] [[phi].sub.j][v.sub.t-j] + [e.sub.t],

where v, is an error term; the [[pi].sub.s], [[eta].sub.s], and [[phi].sub.s] are parameters; and [e.sub.t] is an independently and identically distributed error term with a zero mean and finite variance. Estimation then proceeds in three steps. First, OLS is applied to the regression

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

an order to obtain estimates of {[v.sub.t]}. Second, Equation (11) is fitted to these residuals in order to obtain {[[??].sub.j]}, the estimates of {[[phi].sub.j]). Third, OLS is applied to the regression

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

to obtain consistent estimates of the [[lambda].sub.s] and their standard errors. In Equation (13),

log [[??].sub.t] [equivalent to] log [M.sub.t] - [n.summation over (j=1)] [[??].sub.j] log [M.sub.t-j]

log [[??].sub.t] [equivalent to] log [P.sub.t] - [n.summation over (j=1)] [[??].sub.j] log [P.sub.t-j]

log [[??].sub.t] [equivalent to] log [Y.sub.t] - [n.summation over (j=1)] [[??].sub.j] log [Y.sub.t-j]

In our first data set, M is the world stock of monetary gold, P is the world price level, and Y is the world output. The world monetary gold stock comes from Warren and Pearson (1935). Our measure of world output is the sum of Maddison's (1995) estimates of real gross domestic product (GDP) for France, Germany, Italy, United Kingdom, and United States. (9) Our measure of the world price level is the real GDP-weighted average of these countries' price levels. (10,11) The series are annual and span the period 1880-1913, the heyday of the international gold standard.

The first three rows of Table 1 report statistics for testing whether log M, log P, and log Y are cointegrated for the data described in the previous paragraph. These statistics indicate the existence of exactly one cointegrating vector for the world as a whole.

The first two rows of Table 2 report two sets of estimates for [[lambda].sub.p] and [[lambda].sub.y], one based on Johansen's method and the other on DOLS. The former estimate of [[lambda].sub.p] is less than 1 though not statistically significantly so. The latter estimate, however, is significantly less than 1 at the .01 level. Both estimates of [[lambda].sub.y] are sensible and precisely estimated, significantly positive, and close to 1.

Under the gold standard, deposits and banknotes were nearly perfect substitutes for monetary gold. As a result, the demand for the stock of money should have the properties that the theory in the previous section identified for monetary gold. We should therefore find that Equation (9) with [[lambda].sub.p] < 1 characterizes the demand for money in countries that were on the gold standard.

M, P, and Y in our second and third data sets come from Tables 4.8 and 4.9 of Friedman and Schwartz (1982) and are the M2 money supplies, the implicit deflators, and real net national products for the United Kingdom and the United States. The last six rows of Table 1 report statistics for testing whether log M, log P, and log Y are cointegrated. The statistics indicate the existence of one cointegrating vector for both countries.

We used both Johansen's method and DOLS to estimate [[lambda].sub.p] and [[lambda].sub.y] for both the United Kingdom and the United States, and the last four rows of Table 2 report our estimates. In both cases, the price elasticities are significantly less than 1 and the income elasticities take on plausible and highly significant values.

IV. CONCLUSION

We have established a theoretical presumption that the price elasticity of money demand is less than unity under commodity monetary standards. Using data from the heyday of the international gold standard, we have also provided evidence supporting the theory.

A price elasticity differing from 1 has implications for the general equilibrium properties of economies on the gold standard. In particular, the quantity theory of money may not accurately describe the evolution of the price level. In other words, although a doubling of the stock of gold would no doubt have been associated with an increase in the price level, it would be unlikely to have been associated with an exact doubling.

ABBREVIATIONS

DOLS: Dynamic Ordinary Least Squares

GDP: Gross Domestic Product

OLS: Ordinary Least Squares

doi: 10.1111/j.1465-7295.2007.00113.x Online Early publication January 14, 2008 [c] 2008 Western Economic Association International

REFERENCES

Blanchard, O. J., and S. Fischer. Lectures on Macroeconomics. Cambridge, MA: The MIT Press, 1989.

Friedman, M., and A. J. Schwartz. Monetary Trends in the United States and the United Kingdom. Chicago, IL: University of Chicago Press, 1982.

Gordon, R. J. The American Business Cycle. Chicago, IL: University of Chicago Press, 1986.

Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

Hendry, D. F., and N. R. Ericsson. "An Econometric Analysis of UK Money Demand in Monetary Trends in the United States and the United Kingdom by Milton Friedman and Anna J. Schwartz." American Economic Review, 81, 1991, 8-38.

Homer, S., and R. Sylla. A History of Interest Rates. 3rd ed. New Brunswick and London: Rutgers University Press, 1991.

Kitchin, J. "Memorandum on Gold Production," in The International Gold Problem. London: Oxford University Press, 1931, 47-55.

Laidler, D. E. W. "The Influence of Money on Economic Activity: A Survey of Some Current Problems," in Monetary Theory and Policy in the 1970s, edited by G. Clayton, J. C. Gilbert, and R. Sedgwick. London: Oxford University Press, 1971, 75-135.

MacDonald, R., and M. P. Taylor. "A Stable US Money Demand Function, 1874-1975." Economics Letters, 39, 1992, 191-98.

Maddison, A. Monitoring the World Economy 1820-1992. Paris: OECD, 1995.

McCallum, B., and M. Goodfriend. "Demand for Money: Theoretical Studies." in The New Palgrave." A Dictionary of Economics, edited by J. Eatwell, M. Milgate, and P. Newman. New York: Stockton Press, 1987.

Meltzer, A. H. "The Demand for Money: The Evidence from the Time Series." Journal of Political Economy, 71, 1963, 219-46.

Mitchell, B. R. European Historical Statistics 1750-1975. 2nd ed. New York: Macmillan Press, 1980.

Niehans, J. Theory of Money. Baltimore, MD: Johns Hopkins Press, 1978.

Warren, G. F., and F. A. Pearson. Gold and Prices. London: John Wiley & Sons and Chapman & Hall, 1935.

(1.) According to Kitchin (1931), total world gold production from 1834 to 1889 amounted to 1,037 million pounds sterling, among which 49.6% were added to the monetary gold stock. Annual data become available beginning only in 1890 and are presented in Figure 1. The average fraction of annual gold production that was added to the monetary gold stock rises to 57.7% during the 1890-1913 period.

(2.) Niehans (1978, 143) asserted this result but provided no formal demonstration of it.

(3.) Specifically, after the onset of World War I in 1914 for both the United States and the United Kingdom and before 1879 for the United States. See Homer and Sylla (1991). We interpret gold exchange standards as a kind of fiat standard rather than a full-fledged commodity standard.

4. An alternative form in which Equation (2) can be written is

[C.sub.t] + ([M.sub.t], + [N.sub.t])l[P.sub.t] + [K.sub.t] [less than or equal to] [Y.sub.t] + [r.sub.t][K.sub.t].

We prefer the form in the text because it highlights that the opportunity cost in terms of consumption of holding each unit of gold per period is i/P and that all assets held in the portfolio yield the same real rate of return r after accounting for both pecuniary and nonpecuniary returns. See Blanchard and Fischer (1989, 189).

5. McCallum and Goodfriend (1987) distinguish between two forms of the money demand function. In the first, the stock of real balances is a function of wealth and current and all future relative prices, which are the variables that each household takes as given. In the second, it is a function of current relative prices and current consumption, which is a choice that each household is making jointly with its current holdings of assets. In this paper, we employ the second form, which they term the portfolio-balance equation.

(6.) Using monthly data from the National Bureau of Economic Research macrohistory Web site (www.nber. org/databases/macrohistory) on the U.S. commercial paper rate for the period from January 1880 to December 1913, we obtained an augmented Dickey-Fuller test statistic of -7.28, which is statistically significant at well under the .00001 significance level. We take this as evidence that nominal interest rates in the entire gold standard world were stationary over this sample period. The Dickey-Fuller regression contained an intercept but no time trend.

(7.) The Dickey-Fuller statistics for our measures of log M, log P, and log Y are, respectively, -3.06, +1.43, and -2.7l for the world, 2.66, -1.87, and 2.89 for the United Kingdom, and -2.06, -1.80, and -3.01 for the United States. None of these are statistically significant at conventional levels. We selected the augmentation lags for each Dickey-Fuller regression in order to minimize the Schwarz informational criterion. Each regression contained both an intercept and a time trend. Table 1 provides the evidence for cointegration.

(8.) Cointegrating relationships cannot include stationary variables.

(9.) These series are directly summable because they are expressed in common base-year units, that is, 1990 Geary-Khamis dollars.

(10.) The price levels are the GDP or net national product deflator for Germany, Italy, and United Kingdom, the gross national product deflator for the United States, and a cost of living index for France. The data come from Mitchell (1980) except for those for the United States, which come from Gordon (1986).

(11.) Weighting with real GDP is equivalent to calculating an implicit deflator. For example, suppose that country j's output and price level in period t are [Y.sub.jt] and [P.sub.it], respectively. Then,

[P.sub.t] = [summation over (j)] ([Y.sub.jt]/[summation over (k)] [Y.sub.kt])[P.sub.jt] = [summation over (j)] [P.sub.jt][Y.sub.jt]/[summation over (k)] [Y.sub.kt],

which is just the ratio of the sum of nominal GDPs to the sum of the real GDPs.

PAUL EVANS and XIAOJUN WANG *

* We wish to thank the editor and two anonymous referees for helpful suggestions.

Evans: Professor, Department of Economics, Ohio State University, Columbus, OH 43210. Phone 1-614-292-0072, Fax 1-614-292-3906, E-mail evans.21@osu.edu

Wang. Assistant Professor, Department of Economics, University of Hawaii, Honolulu, HI 96822. Phone 1-808-956-7721, Fax 1-808-956-4347, E-mail xiaojun@hawaii.edu

TABLE 1 Johansen Tests for Cointegration of log M, log P, and log Y,
1880-1913

 Hypothesized Number Maximum
 of Cointegrating Trace Eigenvalue
Economy Vectors Eigenvalues Statistic Statistic

World 0 0.565 42.5 (a) 28.4 (a)
 [less than or equal 0.322 14.2 13.2
 to] 1
 [less than or equal 0.028 0.9 3.8
 to] 2
United 0 0.505 30.4 (b) 22.5 (b)
 Kingdom
 [less than or equal 0.218 8.0 14.1
 to] 1
 [less than or equal 0.002 0.1 3.8
 to] 2
United 0 0.701 51.4 (a) 38.6 (a)
 States
 [less than or equal 0.318 12.8 14.1
 to] 1
 [less than or equal 0.015 0.5 3.8
 to] 2

Notes: The specification for the world includes two lags, and those
for the United Kingdom and the United States include one lag. Each
specification was estimated using EViews, assuming trend in the series
but not in the cointegrating relationships.

(a) and (b) nindicate statistical significance at the .0l and .05
levels, respectively.

TABLE 2
Money Demand under the Gold Standard,
1880-1913

Location and Method [[lambda].sub.p] [[lambda].sub.y]

World, Johansen 0.877 (0.149) 0.969 (0.049)
World, DOLS 0.728 (0.091) 1.072 (0.028)
United Kingdom, 0.642 (0.127) 0.997 (0.026)
 Johansen
United Kingdom, DOLS 0.537 (b) (0.229) 0.992 (0.059)
United States, Johansen 0.309 (a) (0.137) 1.560 (0.044)
United States, DOLS 0.603 (a) (0.189) 1.539 (0.066)

Notes. The figures in parentheses are standard errors. The vector error
correction model had two lags for the world and one for the United
Kingdom and the United States. In the DOLS regressions for the world,
q = n = 2 and [[PHI].sub.1] and [[PHI].sub.2] were estimated to be
0.979 (0.180) and -0.550 (0.178), respectively. In the DOLS regression
for the United Kingdom, q = n = 1 and [[PHI].sub.1] was estimated to be
0.609 (0.155). In the DOLS regressions for the United States, q = n = 1
and [[PHI].sub.1] was estimated to be 0.653 (0.123). The lag lengths
were chosen on the basis of pretests.

(a) and (b) indicate statistical significance at the .01 and .05
levels, respectively.