- Email: [email protected]
Distribution effects and the business demand for money
GEORGE M. KATSIMBRIS University
of Bridgeport
STEPHEN M. MILLER University
of
Connecticut
Distribution Effects and the Business Demand for Money* This paper examines whether and how changes in an industry’s firm-size distribution affect the per-firm demand for money. The size distribution of an industry potentially affects the demand for money through several channels. We examine four of those channels: 1) economies of scale; 2) decentralization in cash management; 3) cost of credit; and 4) compensating balances. We conclude that increasing the size inequality increases the industry’s per-firm demand for money.
1. Introduction The works of Baumol (1952) and Tobin (1956), hereafter referred to as B-T, generated intense interest in “inventory-theoretic” models of the demand for money. At the theoretical level, the B-T model has been extended by 1) the introduction of decentralized cash management [see Sprenkle (1969)] ; 2) the introduction of the cost of credit [see Lewis (1974), Litzenberger (1971), Sastry, (1970) and (1971), and Wrightsman and Terniko (1971)J ; 3) the consideration of compensating balances [see Frost (1970), Hodgman (1961), and Sprenkle (1971)] ; 4) the inclusion of more than two assets [see Feige and Parkin (1971) ] ; 5) the inclusion of interest earnings in income [see Johnson (1970)] ; and 6) the introduction of stochastic elements [see Miller and Orr (1966) and Whalen (1966)]. On the empirical side, the presence of economies of scale in cash management and the interest-rate elasticity of transaction balances have received major attention. This paper focuses on the business demand for money. In Meltzer (1963), money demand was estimated for each of fourteen industries in each of nine years, using a sample stratified by asset-size class. Beginning with a money demand dependent on wealth, Meltzer derived that *This paper was developed from the first author’s dissertation. We would to thank William F. Lott, William A. McEachern, and Anthony A. Romeo for comments. The University of Connecticut Computer Center provided support the econometric work. Journal of Macroeconomics, @ Wayne State University
Fall Press,
1980, 1980.
Vol.
2, No.
4, pp.
287305
like their for
287
George M. Katsimbris
and Stephen M. Miller
Mij = Y*,sifP, where M, i is the money demand for the ith firm in the ith industry; S, i is sales; and yi, and l3 are parameters. Meltzer [(1963), p. 4201 concluded, “. . . the cross section demand for money by firms is a function of sales, to a first approximation linear in logarithms and unit elastic.” Whalen [(1965), p. 4341 noted, “Also required for the validity of the results of the preceding statistical investigation is the requirement that firms of differing size in the same industry are similar in other respects as well.” Meltzer assumed that the market rate of interest, the variable reflecting the demand for the firm’s product and the capital/labor ratio (K,,), and the internal rate of return for an industry or a class of firms (p,) are constant at a point in time within a given industry. However, Whalen noted that these terms may vary with firm size, in which case the estimated coefficients would be biased. Meltzer’s theoretical derivation implied that
where W,, is the non-human assets invested in the ith firm in the jth industry. Thus, the variable (S,,/ W,,) would pick up the effect of changes in (K,, p,) across asset classes within an industry. After introducing this variable into the regression equations, Whalen found that the sales elasticity was reduced in every industry. Whalen [(1965), p. 4371 concluded, “The results do indicate that even for firms in the same industry cross section analysis must allow for structural changes associated with differences in the size of business operations.” Vogel and Maddala (1967) investigated the question of heterogeneity across industries and asset-size classes in cross sections and across asset-size classes in time series. They found heterogeneity in all cases; however, their most dramatic results were obtained with asset-class dummy variables. In both the cross-section and time-series regressions, the introduction of asset-class dummy variables caused the sales elasticity to become significantly less than one while the intercept term increased with increasing asset-size class.’ Thus, Vogel and Maddala provided additional evidence that ‘Note that (1965) before 288
Vogel them,
and use
Maddala (1967), as well as Meltzer one variable, assets, to classify firms
(1963) and according
Whalen to size,
Distribution
Effects
and Money Demand
firm size affects business demand for money. Using a procedure suggested by Kuh (1963), Shapiro (1969) also investigated the firm’s demand for money, employing a sample drawn from twenty-one manufacturing industries over a fifteen-year period. The data were not stratified by asset size, and Shapiro found that the firm’s demand for money was unstable over time. He conjectured that the instability might be explained by interest rates. The empirical evidence indicates significant differences in the behavior of firms in the same industry across asset-size classes. In this paper, we examine the relationship between the firm-size distribution in an industry and its per-firm demand for money. Since the size distribution affects the firm’s demand for money indirectly, it serves as a surrogate for variables omitted in previous studies and thereby allows us to obtain a less biased estimate of the sales elasticity. Section 2 employs a technique suggested by Blinder (1975) to analyze the effect of the firm-size distribution of an industry on the industry’s demand for money. Section 3 presents empirical results. Section 4 concludes our analysis.
2. Theoretical Considerations This section examines size-distribution effects on business demand for money via four channels: 1) the economies-of-scale effect; 2) the decentralization-in-cash-management effect; 3) the cost-of-credit effect; and 4) the compensating-balances effect. Economies of Scale Assume an industry consists of n firms, and each firm’s demand for money depends only on its size.2 The demand for money of each firm is specified as:
but they use another variable, business receipts (which proxies for sales), to measure the scale of operation. If one is interested in measuring the business receipts elasticity, then one might prefer a decomposition of firms by business-receipts class. Until 1963, this was not possible. In our analysis, we use both business receipts and asset distributions. ‘In the economies-of-scale sections, size refers to transactions, which we proxy with business receipts. Some of the distribution effects discussed, however, might be more appropriately related to assets. Consequently, throughout Section 2, we discuss the size variable without specifying whether it is transactions or assets. In Section 3, we employ inequality measures based both on business receipts and asset distributions. 289
George M. Katsimbris
and Stephen M. Miller m = m(S) ; m’ (S) > 0;
and m”(S)
$ 0,
(3)
where S is the size of the firm; m is the firm’s demand for money; and primes refer to first and second derivatives. Further, we specify z (S, g) as the probability density function of firm sizes in the industry, where g represents the concept of a “mean preserving spread” [see Rothschild and Stiglitz (1970)]. Also, 2 (S, g) is the cumulative distribution of z (S, g). Finally, let t, 1, and h represent the average, lowest, and highest size in the industry. For convenience, we assume that 1 and h are unaffected by changes in g. For these specifications, we obtain h
t=h-
Z (S, g) dS .
(4)
Z, (s, g) dS = 0 >
(5)
I Given the definition
of g, h
atlag = I
where Z, is a continous function on the interval 2 s S s h. The expected value of the firm’s demand for money is h
A=
4s)
z (S, g) dS .
(6)
I Differentiating by parts yields
equation
(6) with respect to g and integrating
h
aA/ag= -
m’(S) Z, (S, g) dS . 1
To evaluate the sign of equation S * such that 290
(7)
(7), we note that there exists an
Distribution
Effects and Money Demand
Z,(S,g)~O
for
Z~S~Ss”,
Z,(S,g)SO
for
S*ZZSZZh.’
and
Given equation
(8)
(5) and that m’(S) is positive,
we have the result:
aA/ag 2 0 as m”(S)
2 0.
(9)
In the B-T model, the size variable is transactions and the second derivative of money demand is negative, indicating economies of scale. Thus, an increase in the inequality of transactions, ceteris paribus, leads to a reduction in the industry’s per-firm demand for money. Any aggregation of firm data-be it for the whole industry or for classes within an industry-introduces a bias into the estimate unless an inequality measure is also introduced. Decentralization in Cash Management The B-T model implicitly assumes that the firm manages one centralized account. It does not allow for multiple accounts, some of which might be too small to control optimally. Sprenkle (1969) developed several models of decentralized cash management which purport to demonstrate the uselessness of transactionsdemand models. He assumed, as we do, that the larger the firm, the larger the number of accounts. We write a general demand for money function for the firm as m = f [S,
k(S)],
(10)
where k is the number of accounts and k’(S) is positive. Proceeding along the lines indicated in the previous we can obtain this relationship:
sSince an increase in g represents firms, the new cumulative distribution for low (high) values of S.
an increase in must lie above
section,
the inequality of S among (below) the old distribution
291
George M. Katsimbris
and Stephen M. Miller h
aA/ag = leading
to the conclusion
I
(dmlds) Z, (S,g) dS I
(11)
that
as (d2m/dS2) Differentiating d’m/dS”
2 0 I
(12)
(lo), = a2m/aS”
+ (a2m/ak2)(k’)’
+ 2k’ (a2m/aSak)
+ (am/ak)k”.
(13)
The first term in (13) corresponds to the pure economies-of-scale effect. That is, if the number of accounts is constant and if all accounts are optimally managed, then the money demand increases at a decreasing rate. But, as Sprenkle (1969) argued, if some fraction of the accounts is not optimally managed and if this fraction increases with firm size, then it is possible for the first term to be positive. The sign of the second term depends upon the sign of (a”m/ak”). Sprenkle demonstrated that as the number of accounts increases, holding S constant, the money demand increases at a decreasing rate when all accounts are optimally managed. But again, non-optimally managed accounts could cause the term to be positive. Finally, the signs of the third and fourth terms depend upon the signs of (a”m/aSak) and (k”), respectively. We have no a priori expectations about the sign of either term. Note that the sign of (k”) determines whether the number of accounts increases at an increasing or a decreasing rate. Thus, we cannot specify the sign of (aA /ag) a priori. Cost of Credit Sastry (1970) developed a model of the transactions demand for money which allowed firms to finance expenditures with credit. The cost of credit thereby became directly related to the business demand for money. Several criticisms and modifications of Sastry’s work [e.g., Lewis (1974), Litzenberger (1971), Sastry (1971), and 292
Distribution
Effects
and Money Demand
Wrightsman and Terniko (1971)] consider the impact of the cost of credit on the interest-rate elasticity of money demand. We return to this point in the discussion of our empirical results. Nevertheless, for the individual firm, we anticipate a positive relationship between the cost of credit and the demand for money. Several authors have discussed the relationship between firm size and the cost of credit. Selden [(1961), p. 5061 remarked, “For a variety of reasons, the cost of funds is much higher for small firms.” Whalen stated [(1965), p. 4331, “ . . . the widely accepted observation is that the cost of borrowing is lower for large firms than for small.” Gupta (1972) developed a theoretical model and utilized empirical observations to reach the same conclusion. Since the cost of borrowing affects the demand for money, and firm size affects the cost of borrowing, the size distribution within an industry becomes a determinant of the average cost of credit, and thus of the demand for money. The expected value of the firm’s cost of credit is h r,
=
h r(W’(S)z(S,
g)dS
I
D(S)@>g)dS I
1
= E [r(S)D(S)1lE [D(S)1> where E [ ] is of credit; r(S) the outstanding defined. The effect
>
(14)
the expected value operator; rA is the average cost is the cost of credit to a firm of size S; D(S) is debt of the firm; and other variables are as previously of a change in g on rA is given by
ar,lQ = {E [D(S)1@/@HE[r(W(VI) - E MW(S)I (a/WE [D(S)1)1/E P(W” .
(15)
Now, if the cost of credit is independent of size, then r(S) can be factored out of the expectations operator, and the partial derivative (ar,/ag) is equal to zero. However, we can not specify a priori the sign of (arA/ag).4 We can conclude only that 4Although Gupta’s (1972) work implied that r’(S) is negative and although we expect a lower bound on T causing s”(S) to be positive, we also know that D’(S) is positive and that D”(S) could be of either sign. Thus, the sign of (ar,/dg) is ambiguous.
George M. Katsimbris
and Stephen M. Miller
as
E [W)l @lW{E b(WNI
> 5 E [r(W(S)I @lW{E [&%I 1 . (16)
In words, an increase in g causes rA to rise if the expected value of debt times the change in the expected interest cost of the debt exceeds the expected interest cost of the debt times the change in the expected debt. Compensating Balances It is common practice for commerical banks to require business customers to hold a minimum cash balance in order to compensate for granting loans at favorable terms, for establishing a credit line, or for rendering other banking services. Thus, the total demand for compensating balances is a derived demand from the demand for loans, credit lines, and other banking services. For expository purposes, we divide the total demand for compensating balances into two components: 1) the demand resulting from loans and credit lines; and 2) the demand resulting from other banking services. In both. instances, we assume a direct relationship between firm size and the compensating-balance demands. Sprenkle [(1971), p. 1571 argued that the demand for compensating balances resulting from 1) increases at a decreasing rate while the demand resulting from 2) could increase at either an increasing or a decreasing rate. This latter result occurs because, according to Sprenkle, other banking services include “routine” banking services which increase at a decreasing rate, and more specialized services which increase at an increasing rate. The total demand for compensating balances by a firm is given by b = b(S) ; b’(S) > 0 ; and b”(S) $ 0 , where b” is negative unless the demand for more specialized dominates. 294
services
Distribution
Effects and Money Demand
The expected value of the firm’s balances is given by
demand
for compensating
1 h
B = Differentiating parts gives us
equation
b(S) z (S, g) dS .
(18) with
respect to g and integrating
(18) by
h
aB/ag= 1 Thus, we conclude
b'(S) Z, (s, g) dS .
(19)
that
as b”(S) $
0 .
(W
That is, an increase in g causes the demand for compensating balances to increase (decrease) if the firm’s demand increases at an increasing (decreasing) rate with the size of the firm. Finally, how does the incorporation of compensating balances into our analysis affect the industry’s per-firm demand for voluntary balances? Nothing can be said a priori. It depends upon the size of the voluntary cash balances as compared to required compensating balances. If voluntary balances are less than the required compensating balances, the industry’s per-firm demand for money will respond to the size-distribution’s effect on compensating balances. If voluntary balances are much larger than required compensating balances, they will absorb increased balance requirements, leaving the per-firm demand for money unchanged. If compensating-balance requirements are stated as a minimum requirement rather than an average requirement, the firm may be forced to increase its balances, even if voluntary cash holdings exceed compensating-balance requirements. The relationship between voluntary and compensatingbalance requirements remains in need of further study [see Orr (1974) and Sprenkle (1977)]. After analyzing the implications of firm-size distribution for the business demand for money via 1) the economies-of-scale effect;
George M. Katsimbris
and Stephen M. Miller
2) the decentralization-in-cash-management effect; 3) the cost-ofcredit effect; and 4) the compensating-balances effect, we are unable to determine the direction of the size-distribution effect. In the next empirical section, we introduce variables which account for the firm-size distribution across industries and, thus, reduce the bias inherent in previous estimates of the scale elasticity. To the extent that differences across industries are due to factors unrelated to firm size, our estimates will remain biased.
3. Empirical Resul?s The empirical work of the present section is based on a sample of eleven manufacturing industries. The data are drawn from the Internal Revenue Service, Statistics of Income for the years 19631971, excluding 1967. The distributions for 1963 and 1964 are collapsed to correspond to the classes in the remaining years. The final distributions possess information across twelve size classes for business receipts, and eleven for assets. Since the information in 1967 contains only eight size classes, it was omitted.’ The variables used are defined, where i refers to the industry and t refers to the year, as: M,, = cash, in millions of dollars; BR,, = business receipts, in millions N,, = number of firms;
of dollars;
‘We employ business receipts and asset distributions. These data are available by subcategory within manufacturing from 1963 through 1971. We required that, for each industry selected, the number of class sizes be as complete as possible. Industries were excluded if their inclusion unduly reduced the number of class sizes. The industries chosen include: 1) Food and Kindred Products; 2) Textile Mill Products, 3) Apparel and Other Fabricated Textile Products; 4) Lumber and Wood Products, except furniture; 5) Printing, Publishing, and Allied Industries; 6) Chemicals and Allied Products; 7) Stone, Clay, and Glass Products; 8) Fabricated Metal Products, except machinery and transportation equipment; 9) Machinery, except electrical; 10) Electrical Machinery, Equipment and Supplies; and 11) Scientific Equipment, Photographic Equipment, Watches, and Clocks. The data are subject to a variety of problems. First, since they are drawn from balance sheets and income statements of individual firms, they are subject to end-of-year “window dressing.” Second, though the data are annual, they refer to different intervals during the year. Third, the money variable is categorized as cash which is primarily currency and demand deposits. To the extent that the reported numbers include time deposits held by firms at non-member banks [see Meltzer (1963), p. 4091, we may have additional bias. Finally, annual data may obscure important information. 298
Distribution GBR,, GA,, VLBR,, VLA,, Rst R Lt
= = = = = =
Effects and Money Demand
Gini coefficient of business receipts;8 Gini coefficient of assets; variance of the logarithms of business receipts;7 variance of logarithms of assets; the market yield on 3-month Treasury bills; and the market yield on AAA corporate bonds.*
We are using business receipts as a proxy for transactions and four measures of size distribution: GBR; GA; VLBR; and VLA. The Gini coefficient and the variance of the logarithms usually move together. In fact, if the distribution is log-normal, then there is a one-to-one relationship between these two measures of size inequality. The use of the Gini coefficient or the variance of logarithms has the advantage of conserving on the degrees of freedom in the regression equations. Both also possess, however, the wellknown disadvantage of assuming that all possible redistributions which change the variables by the same amount have the same effect on the firm’s demand for money, ceteris paribus. Clearly, this assumption need not be true. Vogel and Maddala (1967) attempted to account for differences between firms by employing a pooled regression across industries with asset-class dummies. They concluded that the sales elasticity ‘The Gini suppressed):
coefficient
was
calculated
using
the
formula
(i and
t subscripts
are
where a, is the cumulative number of firms from the first to the ith class; 1 and a,,, are the means of the jth and (i + 1)st class of business receipts (or asset) distribution, respectively; N is the total number of firms; TV is the overall mean of business receipts (or assets); and J is the number of classes. ‘The variance of logarithms was calculated using the formula
VL = i
,=I
N, (In, -
i@/N ,
where In,& the natural logarithm of average business receipts (or assets) in the jth class; In is the geometric mean of business receipts (or assets); N, is the number of firms in the ith class; N is the total number of firms; and J is the number of classes. 8Both R,, and R,, were obtained from various issues of the Federal Reserue Bulletin. 297
George M. Katsimbris
and Stephen M. Miller
TABLE 1. Pooled Regressions with Znterest Rates” Constant
WR,,I
N,,)
WGBR,,)
WX,)
Al
-3.8426" (-14.3944)
0.7421" (20.7807)
A2
-3.9057+ (-14.8477)
0.7482* (20.9687)
Bl
-3.2263' (-14.9247)
(14.0075)
-3.3292" (-16.0268)
(14.4961)
Cl
-3.4141Q (-15.6751)
0.5658" (15.3018)
3.1829" (7.3228)
c2
-3.5709' (-17.5768)
0.5670' (16.1382)
(8.0462)
Dl
-3.2442' (-10.4707)
0.6005" (11.0596)
D2
-3.2912* (-10.2482)
0.6154" (11.1709)
El
-3.2572" (-7.1634)
0.5506" (9.6022)
E2
-3.3363" (-7.3740)
(9.7614)
B2
0.5298’
0.5349”
2.2887" (8.0227) 2.3124* (8.3272)
3.3494*
0.6595”
“Numbers in parentheses are t-statistics. Zn(BR,,/N,), In(&), and It@,,). Equations while the other equations have 84. ’ Significant at the one per cent level.
(Al)
All tests and (A2)
are two-tailed have 85 degrees
except for of freedom
estimatesof other studies were biased due to significant differences in the intercept terms across asset classes. Our analysis follows a different procedure for two reasons.First, we prefer to use the business-receiptsdistribution in searching for a size-distribution effect, and this distribution is not available for individual industries 298
Distribution TABLE
1.
ln(VLBR,,)
Effects
and Money Detnund
Continued ln(VLA,,)
WW
ln(R,,)
-0.2479” (-2.9444) -0.2864 * (-3.2300) -0.2204* (-3.4523) -0.2723. (-4.1236) -0.2630” (-3.9715) -0.3427. (-5.0856) 0.4574’ (3.3305)
-0.2512* (-3.1548)
0.4219’ (3.0661)
-0.2684. (-3.1654) 0.3336 (1.5833)
-0.2361+ (-2.8174)
0.3222 (1.5407)
-0.2728” (-3.0861)
R”
F
0.8327
217.5
0.8358
222.4
0.9042
274.6
0.9090
290.6
0.8966
252.6
0.9062
281.0
0.8504
165.9
0.8505
166.1
0.8356
148.4
0.8384
151.5
prior to 1963. Second, for those years that we do have a business-receipts distribution, cash is not reported for each size class. Thus, we are constrained to analyzing models using per-firm cash holdings at the industry level; consequently, the use of the Gini coefficient or the variance of logarithms is required. The regression equations are of the form ln(M,,/N,,)
= 0~~+ a, ln(BR,,INit)
+ a2 Wg,,)
+ a3 lnB, + c, (21) 299
George M. Katsimbris
and Stephen M. Miller
where g,, is either the Gini coefficient of business receipts or assets or the variance of the logarithms of business receipts or assets, and R, is either the short- or long-term interest rate.’ Regressions are calculated both with and without gdt and/or R,.“’ The form of equation (21) does not allow the size-distribution variable to go to zero. We also ran regressions with g,* replacing Zn(g,,) and the results are almost identical. Additionally, comparing the 8” values of the two functional forms, equation (21) performs marginally better. Table 1 presents the results of the pooled regression. In each case, the scale variable is the per-firm business receipts. Equations (Al) through (A2) omit size-distribution variables. Equations (Bl) through (E2) include the Gini of business receipts, the Gini of assets, the variance of the logarithms of business receipts, and the variance of the logarithms of assets, respectively. The corresponding numbers include (1) and (2), the short- and long-term interests rates, respectively. All of the scale elasticities are significantly less than one at the one per cent level, indicating economies of scale in cash management (Table 2). The inclusion of size-distribution variables reduces the estimate of the scale elasticity in each case. We performed t-tests to determine whether or not the scale elasticities with size-distribution variables are significantly less than without these variables. The scale elasticity when GBR and GA are introduced is significantly less than the scale elasticity in equations (Al) and (A2) at the one per cent level. The scale elasticity when VLBR is introduced is significantly less than its counterpart in (Al) and (A2) at the five per cent level. On the other hand, the scale elasticity is not significantly altered when VLA is introduced. The simple B-T model implies a scale elasticity of one-half. Without any size-distribution variable, the scale elasticity is significantly different from one-half at the one per cent level (Table 2). ‘Notice that the regressions are on a per-firm basis. We ran regressions of M,, variables. In each case, the coefficient of In N,, was on s,,, N,,, and the other not significantly different from one minus the coefficient of In S,. Thus, we report only the per-firm regressions. “We first ran cross-section regressions in each of the eight years. However, after running two pooled regressions with R,, and RLt, respectively, we were unable to reject the null hypothesis of homogeneity over time at the ten percent level. The cross section regressions are similar to our pooled results; consequently, we report only the latter. Cross section results are available from the authors upon request. All regressions are computed using ordinary least squares. 300
Distribution 2.
TABLE
Additional
0.2579
0.5 - ci;
-0.2421
1 - o;
0.4702 -0.0298
0.5 - a; 1 - II;
0.4342
0.5 - a;
-0.0658
1 - a;
0.3995 -0.1005
0.5 - a: 1 - CY; 0.5 - a; A B a1 ‘4 a1 A 011
-
a1 c
-
a1 D
-a1
A ‘yl
E -
a1
and Money Demand
Tests on Scale Coefficients” Pooled w/R,
Pooled w/R, 1 - a;
Effects
0.3494 -0.1506
(7.222)’
0.2518 -0.2482
(-6.780)’
0.4651
(12.430)” (-0.7886)
-0.0349 0.4330
(11.742)” (-1.780)**”
-0.0670 0.3846
(7.358)* (-1.851)**” (5.156)
-0.1154 *
(-2.223)‘”
0.2123
(3.894)’
0.1763
(3.898)
0.1416 0.0915
0.3405 -0.1595
(7.058)” (- 6.955) (12.605)* (-0.9457) (12.324)” (-1.907)““* (6.982)
*
(-2.094)“” (5.04O)O (-2.361)**
0.2133
(3.958)’
0.1812
(3.485)*
(2.244)“”
0.1328
(2.002)“a
(1.191)
0.0887
(1.158)
*
*
“Numbers in parentheses are t-statistics. For (1 - ai) and (of - Q:) which are expected to be positive, the t-tests are one one-tailed; other tests are two-tailed. a, is the coefficient of In (B&/N,,) and superscripts refer to the appropriate equation in Table 1. ‘Significant at the one per cent level. * Significant at the five per cent level. ’ Significant at the ten per cent level. l l
l
When GBR is introduced, however, the scale elasticity is not significantly different from one-half. For GA, VLBR, and VLA, it is significantly different from one-half at either the five or ten per cent levels. The coefficients of the size-distribution variables are uniformly positive and, except for VLA, they are significantly different from zero at the one per cent level. The positive sign on the coefficients of the size-distribution variable is counter to the presumption implied by the economies-of-scale effect. Thus, the pooled regression estimates indicate that, taken together, the effects of the degree 301
Distribution
Effects and Money Demand
of decentralization in cash management, the cost of borrowing, and compensating balances are positive and more than offset the opposing, negative effect of economies of scale. Finally, both the short- and long-term interest-rate elasticities are negative and significant at the one percent level. However, they are also significantly different from minus one-half as implied by the simple B-T model (Table 3). In a model developed by Lewis (1974) which allows the cost of credit to vary with the market interest rate, the interest-rate elasticity of the demand for money becomes less negative under certain plausible conditions. Based on fi2 and F values, the equations with the long-term rate are marginally superior to the equations with the short-term rate. 4. Conclusion The primary objective of this paper has been to develop and test a model of the firm’s demand for money which explicitly considered the industry’s firm-size distribution. The question was whether and how changes in an industry’s size distribution affects its per-firm demand for money. In Section 2, we analyzed the potential aggregation bias introduced by ignoring the size distribution of industries. The impact on the per-firm demand for money from this source was related to: 1) economies of scale; 2) decentralization in cash management; 3) cost of credit; and 4) compensating balances. Because of counteracting influences, we were unable to specify, a priori, the direction of the size-distribution effect. For our sample, we found economies of scale in cash management before introducing our size-distribution variables. But after introducing the size-distribution variables, the scale elasticities were usually lowered significantly. In fact for GBR, the new elasticity was not significantly different from one-half. The results also indicate that increasing size inequality increases the per-firm demand for money of an industry. This finding is consistent with those of Vogel and Maddala (1967). Received:
February,
1980
References Baumol, W.J. “The Transactions Demand for Cash: An Inventory Theoretic Approach.” Quarterly Journal of Economics 66 (November 1952): 545-56. 303
George M. Katsimbris
and Stephen M. Miller
Blinder, A.S. “Distribution Effects and the Aggregate Consumption Function.” JournaZ of Political Economy 83 (June 1975): 447-75. Board of Governors of the Federal Reserve System. FederaE Reserve Bulletin. Number 12 (1963-71): 49-57. Feige, E.L. and M. Parkin. “The Optimal Quantity of Money, Bonds, Commodity Inventories, and Capital.” American Economic Review 61 (June 1971): 335-49. Frost, P.A. “Banking Services, Minimum Cash Blanaces, and the Firm’s Demand for Money.” Journal of Finance 25 (December 1970) : 1029-39. Gupta, M.C. “Differential Effects of Tight Money: An Economic Rationale.” Journal of Finance 27 (September 1972): 825-38. Hodgman, D. “The Deposit Relationship and Commercial Bank Investment Behavior.” Review of Economics and Statistics 43 (August 1961): 257-68. U.S. Treasury Department, Internal Revenue Service. Statistics of Income: Corporation Zncome Tax Returns. Washington, D.C.: U.S. Government Printing Office, 1963-1971, excluding 1967. Johnson, H.G. “A Note on the Theory of Transactions Demand for Cash.” Journal of Money, Credit and Banking 2 (August 1970): 383-84. Kuh, E. Capital Stock Growth: A Microeconomic Approach. Amsterdam: North-Holland 1963. Lewis, K.A. “A Note on the Interest Elasticity of the Transactions Demand for Cash.” Journal of Finance 29 (September 1974): 1144-52. Litzenberger, R.H. “The Effect of Credit on the Interest Elasticity of the Transactions Demand for Cash: Comment.” Journal of Finance 26 (December 1971): 1161-62. Meltzer, A.H. “The Demand for Money: A Cross-Section Study of Business Firms.” Quarterly Journal of Economics 77 (August 1963) : 405-22. Miller, M.H. and D. Orr. “A Model of the Demand for Money by Firms.” Quarterly Journal of Economics 80 (August 1966): 413-35. Orr, D. “A Note on the Uselessness of Transaction Demand Models.” Journal of Finance 29 (December 1974): 1565-72. Rothschild, M. and J.E. Stiglitz. “Increasing Risk: I. A Definition.” Journal of Economic Theory 2 (September 1970): 225-43. Sastry, A.S.R. “The Effect of Credit on Transactions Demand for Cash.” Zournal of Finance 25 (September 1970): 777-81. -. “The Effect of Credit on the Transactions Demand for 304
Distribution
Effects
and Money Demand
Cash: Reply.” journal of Finance 26 (December 1971): 1163-65. Selden, R.T. “The Postwar Rise in the Velocity of Money: A Sectoral Analysis.” Journal of Finance 16 (December 1961): 483-545. Shapiro, A.A. “Interindustry Differentials in the Demand for Money and Government Securities.” Southern Economic Journal 36 (October 1969): 165-70. Sprenkle, C.M. “The Uselessness of Transactions Demand Models.” Journal of Finance 24 (December 1969): 835-47. -. Effects of Large Firm and Bank Behauior on the Demand for Money of Large Firms. New York: American Bankers Association, 1971. -* “The Uselessness of Transaction Demand Models: Comment.” Journal of Finance 32 (March 1977): 227-30. Tobin, J. “The Interest Elasticity of Transactions Demand for Cash.” Review of Economics and Statistics 38 (August 1956): 241-47. Vogel, R.C. And G.S. Maddala. “Cross-Section Estimates of Liquid Asset Demand by Manufacturing Corporations.” Journal of Finance 22 (December 1967): 557-75. Whalen, E.L. “A Cross-Section Study of the Business Demand for Cash.” Journal of Finance 20 (September 1965): 423-43. -. “A Rationalization of the Precautionary Demand for Cash.” Quarterly Journal of Economics 80 (May 1966): 314-24. Wrightsman, D. and J. Terniko. “On the Measurement of Opportunity Cost in Transactions Demand Models.” Journal of Finance 26 (September 1971): 947-50.
305
of Bridgeport
STEPHEN M. MILLER University
of
Connecticut
Distribution Effects and the Business Demand for Money* This paper examines whether and how changes in an industry’s firm-size distribution affect the per-firm demand for money. The size distribution of an industry potentially affects the demand for money through several channels. We examine four of those channels: 1) economies of scale; 2) decentralization in cash management; 3) cost of credit; and 4) compensating balances. We conclude that increasing the size inequality increases the industry’s per-firm demand for money.
1. Introduction The works of Baumol (1952) and Tobin (1956), hereafter referred to as B-T, generated intense interest in “inventory-theoretic” models of the demand for money. At the theoretical level, the B-T model has been extended by 1) the introduction of decentralized cash management [see Sprenkle (1969)] ; 2) the introduction of the cost of credit [see Lewis (1974), Litzenberger (1971), Sastry, (1970) and (1971), and Wrightsman and Terniko (1971)J ; 3) the consideration of compensating balances [see Frost (1970), Hodgman (1961), and Sprenkle (1971)] ; 4) the inclusion of more than two assets [see Feige and Parkin (1971) ] ; 5) the inclusion of interest earnings in income [see Johnson (1970)] ; and 6) the introduction of stochastic elements [see Miller and Orr (1966) and Whalen (1966)]. On the empirical side, the presence of economies of scale in cash management and the interest-rate elasticity of transaction balances have received major attention. This paper focuses on the business demand for money. In Meltzer (1963), money demand was estimated for each of fourteen industries in each of nine years, using a sample stratified by asset-size class. Beginning with a money demand dependent on wealth, Meltzer derived that *This paper was developed from the first author’s dissertation. We would to thank William F. Lott, William A. McEachern, and Anthony A. Romeo for comments. The University of Connecticut Computer Center provided support the econometric work. Journal of Macroeconomics, @ Wayne State University
Fall Press,
1980, 1980.
Vol.
2, No.
4, pp.
287305
like their for
287
George M. Katsimbris
and Stephen M. Miller
Mij = Y*,sifP, where M, i is the money demand for the ith firm in the ith industry; S, i is sales; and yi, and l3 are parameters. Meltzer [(1963), p. 4201 concluded, “. . . the cross section demand for money by firms is a function of sales, to a first approximation linear in logarithms and unit elastic.” Whalen [(1965), p. 4341 noted, “Also required for the validity of the results of the preceding statistical investigation is the requirement that firms of differing size in the same industry are similar in other respects as well.” Meltzer assumed that the market rate of interest, the variable reflecting the demand for the firm’s product and the capital/labor ratio (K,,), and the internal rate of return for an industry or a class of firms (p,) are constant at a point in time within a given industry. However, Whalen noted that these terms may vary with firm size, in which case the estimated coefficients would be biased. Meltzer’s theoretical derivation implied that
where W,, is the non-human assets invested in the ith firm in the jth industry. Thus, the variable (S,,/ W,,) would pick up the effect of changes in (K,, p,) across asset classes within an industry. After introducing this variable into the regression equations, Whalen found that the sales elasticity was reduced in every industry. Whalen [(1965), p. 4371 concluded, “The results do indicate that even for firms in the same industry cross section analysis must allow for structural changes associated with differences in the size of business operations.” Vogel and Maddala (1967) investigated the question of heterogeneity across industries and asset-size classes in cross sections and across asset-size classes in time series. They found heterogeneity in all cases; however, their most dramatic results were obtained with asset-class dummy variables. In both the cross-section and time-series regressions, the introduction of asset-class dummy variables caused the sales elasticity to become significantly less than one while the intercept term increased with increasing asset-size class.’ Thus, Vogel and Maddala provided additional evidence that ‘Note that (1965) before 288
Vogel them,
and use
Maddala (1967), as well as Meltzer one variable, assets, to classify firms
(1963) and according
Whalen to size,
Distribution
Effects
and Money Demand
firm size affects business demand for money. Using a procedure suggested by Kuh (1963), Shapiro (1969) also investigated the firm’s demand for money, employing a sample drawn from twenty-one manufacturing industries over a fifteen-year period. The data were not stratified by asset size, and Shapiro found that the firm’s demand for money was unstable over time. He conjectured that the instability might be explained by interest rates. The empirical evidence indicates significant differences in the behavior of firms in the same industry across asset-size classes. In this paper, we examine the relationship between the firm-size distribution in an industry and its per-firm demand for money. Since the size distribution affects the firm’s demand for money indirectly, it serves as a surrogate for variables omitted in previous studies and thereby allows us to obtain a less biased estimate of the sales elasticity. Section 2 employs a technique suggested by Blinder (1975) to analyze the effect of the firm-size distribution of an industry on the industry’s demand for money. Section 3 presents empirical results. Section 4 concludes our analysis.
2. Theoretical Considerations This section examines size-distribution effects on business demand for money via four channels: 1) the economies-of-scale effect; 2) the decentralization-in-cash-management effect; 3) the cost-of-credit effect; and 4) the compensating-balances effect. Economies of Scale Assume an industry consists of n firms, and each firm’s demand for money depends only on its size.2 The demand for money of each firm is specified as:
but they use another variable, business receipts (which proxies for sales), to measure the scale of operation. If one is interested in measuring the business receipts elasticity, then one might prefer a decomposition of firms by business-receipts class. Until 1963, this was not possible. In our analysis, we use both business receipts and asset distributions. ‘In the economies-of-scale sections, size refers to transactions, which we proxy with business receipts. Some of the distribution effects discussed, however, might be more appropriately related to assets. Consequently, throughout Section 2, we discuss the size variable without specifying whether it is transactions or assets. In Section 3, we employ inequality measures based both on business receipts and asset distributions. 289
George M. Katsimbris
and Stephen M. Miller m = m(S) ; m’ (S) > 0;
and m”(S)
$ 0,
(3)
where S is the size of the firm; m is the firm’s demand for money; and primes refer to first and second derivatives. Further, we specify z (S, g) as the probability density function of firm sizes in the industry, where g represents the concept of a “mean preserving spread” [see Rothschild and Stiglitz (1970)]. Also, 2 (S, g) is the cumulative distribution of z (S, g). Finally, let t, 1, and h represent the average, lowest, and highest size in the industry. For convenience, we assume that 1 and h are unaffected by changes in g. For these specifications, we obtain h
t=h-
Z (S, g) dS .
(4)
Z, (s, g) dS = 0 >
(5)
I Given the definition
of g, h
atlag = I
where Z, is a continous function on the interval 2 s S s h. The expected value of the firm’s demand for money is h
A=
4s)
z (S, g) dS .
(6)
I Differentiating by parts yields
equation
(6) with respect to g and integrating
h
aA/ag= -
m’(S) Z, (S, g) dS . 1
To evaluate the sign of equation S * such that 290
(7)
(7), we note that there exists an
Distribution
Effects and Money Demand
Z,(S,g)~O
for
Z~S~Ss”,
Z,(S,g)SO
for
S*ZZSZZh.’
and
Given equation
(8)
(5) and that m’(S) is positive,
we have the result:
aA/ag 2 0 as m”(S)
2 0.
(9)
In the B-T model, the size variable is transactions and the second derivative of money demand is negative, indicating economies of scale. Thus, an increase in the inequality of transactions, ceteris paribus, leads to a reduction in the industry’s per-firm demand for money. Any aggregation of firm data-be it for the whole industry or for classes within an industry-introduces a bias into the estimate unless an inequality measure is also introduced. Decentralization in Cash Management The B-T model implicitly assumes that the firm manages one centralized account. It does not allow for multiple accounts, some of which might be too small to control optimally. Sprenkle (1969) developed several models of decentralized cash management which purport to demonstrate the uselessness of transactionsdemand models. He assumed, as we do, that the larger the firm, the larger the number of accounts. We write a general demand for money function for the firm as m = f [S,
k(S)],
(10)
where k is the number of accounts and k’(S) is positive. Proceeding along the lines indicated in the previous we can obtain this relationship:
sSince an increase in g represents firms, the new cumulative distribution for low (high) values of S.
an increase in must lie above
section,
the inequality of S among (below) the old distribution
291
George M. Katsimbris
and Stephen M. Miller h
aA/ag = leading
to the conclusion
I
(dmlds) Z, (S,g) dS I
(11)
that
as (d2m/dS2) Differentiating d’m/dS”
2 0 I
(12)
(lo), = a2m/aS”
+ (a2m/ak2)(k’)’
+ 2k’ (a2m/aSak)
+ (am/ak)k”.
(13)
The first term in (13) corresponds to the pure economies-of-scale effect. That is, if the number of accounts is constant and if all accounts are optimally managed, then the money demand increases at a decreasing rate. But, as Sprenkle (1969) argued, if some fraction of the accounts is not optimally managed and if this fraction increases with firm size, then it is possible for the first term to be positive. The sign of the second term depends upon the sign of (a”m/ak”). Sprenkle demonstrated that as the number of accounts increases, holding S constant, the money demand increases at a decreasing rate when all accounts are optimally managed. But again, non-optimally managed accounts could cause the term to be positive. Finally, the signs of the third and fourth terms depend upon the signs of (a”m/aSak) and (k”), respectively. We have no a priori expectations about the sign of either term. Note that the sign of (k”) determines whether the number of accounts increases at an increasing or a decreasing rate. Thus, we cannot specify the sign of (aA /ag) a priori. Cost of Credit Sastry (1970) developed a model of the transactions demand for money which allowed firms to finance expenditures with credit. The cost of credit thereby became directly related to the business demand for money. Several criticisms and modifications of Sastry’s work [e.g., Lewis (1974), Litzenberger (1971), Sastry (1971), and 292
Distribution
Effects
and Money Demand
Wrightsman and Terniko (1971)] consider the impact of the cost of credit on the interest-rate elasticity of money demand. We return to this point in the discussion of our empirical results. Nevertheless, for the individual firm, we anticipate a positive relationship between the cost of credit and the demand for money. Several authors have discussed the relationship between firm size and the cost of credit. Selden [(1961), p. 5061 remarked, “For a variety of reasons, the cost of funds is much higher for small firms.” Whalen stated [(1965), p. 4331, “ . . . the widely accepted observation is that the cost of borrowing is lower for large firms than for small.” Gupta (1972) developed a theoretical model and utilized empirical observations to reach the same conclusion. Since the cost of borrowing affects the demand for money, and firm size affects the cost of borrowing, the size distribution within an industry becomes a determinant of the average cost of credit, and thus of the demand for money. The expected value of the firm’s cost of credit is h r,
=
h r(W’(S)z(S,
g)dS
I
D(S)@>g)dS I
1
= E [r(S)D(S)1lE [D(S)1> where E [ ] is of credit; r(S) the outstanding defined. The effect
>
(14)
the expected value operator; rA is the average cost is the cost of credit to a firm of size S; D(S) is debt of the firm; and other variables are as previously of a change in g on rA is given by
ar,lQ = {E [D(S)1@/@HE[r(W(VI) - E MW(S)I (a/WE [D(S)1)1/E P(W” .
(15)
Now, if the cost of credit is independent of size, then r(S) can be factored out of the expectations operator, and the partial derivative (ar,/ag) is equal to zero. However, we can not specify a priori the sign of (arA/ag).4 We can conclude only that 4Although Gupta’s (1972) work implied that r’(S) is negative and although we expect a lower bound on T causing s”(S) to be positive, we also know that D’(S) is positive and that D”(S) could be of either sign. Thus, the sign of (ar,/dg) is ambiguous.
George M. Katsimbris
and Stephen M. Miller
as
E [W)l @lW{E b(WNI
> 5 E [r(W(S)I @lW{E [&%I 1 . (16)
In words, an increase in g causes rA to rise if the expected value of debt times the change in the expected interest cost of the debt exceeds the expected interest cost of the debt times the change in the expected debt. Compensating Balances It is common practice for commerical banks to require business customers to hold a minimum cash balance in order to compensate for granting loans at favorable terms, for establishing a credit line, or for rendering other banking services. Thus, the total demand for compensating balances is a derived demand from the demand for loans, credit lines, and other banking services. For expository purposes, we divide the total demand for compensating balances into two components: 1) the demand resulting from loans and credit lines; and 2) the demand resulting from other banking services. In both. instances, we assume a direct relationship between firm size and the compensating-balance demands. Sprenkle [(1971), p. 1571 argued that the demand for compensating balances resulting from 1) increases at a decreasing rate while the demand resulting from 2) could increase at either an increasing or a decreasing rate. This latter result occurs because, according to Sprenkle, other banking services include “routine” banking services which increase at a decreasing rate, and more specialized services which increase at an increasing rate. The total demand for compensating balances by a firm is given by b = b(S) ; b’(S) > 0 ; and b”(S) $ 0 , where b” is negative unless the demand for more specialized dominates. 294
services
Distribution
Effects and Money Demand
The expected value of the firm’s balances is given by
demand
for compensating
1 h
B = Differentiating parts gives us
equation
b(S) z (S, g) dS .
(18) with
respect to g and integrating
(18) by
h
aB/ag= 1 Thus, we conclude
b'(S) Z, (s, g) dS .
(19)
that
as b”(S) $
0 .
(W
That is, an increase in g causes the demand for compensating balances to increase (decrease) if the firm’s demand increases at an increasing (decreasing) rate with the size of the firm. Finally, how does the incorporation of compensating balances into our analysis affect the industry’s per-firm demand for voluntary balances? Nothing can be said a priori. It depends upon the size of the voluntary cash balances as compared to required compensating balances. If voluntary balances are less than the required compensating balances, the industry’s per-firm demand for money will respond to the size-distribution’s effect on compensating balances. If voluntary balances are much larger than required compensating balances, they will absorb increased balance requirements, leaving the per-firm demand for money unchanged. If compensating-balance requirements are stated as a minimum requirement rather than an average requirement, the firm may be forced to increase its balances, even if voluntary cash holdings exceed compensating-balance requirements. The relationship between voluntary and compensatingbalance requirements remains in need of further study [see Orr (1974) and Sprenkle (1977)]. After analyzing the implications of firm-size distribution for the business demand for money via 1) the economies-of-scale effect;
George M. Katsimbris
and Stephen M. Miller
2) the decentralization-in-cash-management effect; 3) the cost-ofcredit effect; and 4) the compensating-balances effect, we are unable to determine the direction of the size-distribution effect. In the next empirical section, we introduce variables which account for the firm-size distribution across industries and, thus, reduce the bias inherent in previous estimates of the scale elasticity. To the extent that differences across industries are due to factors unrelated to firm size, our estimates will remain biased.
3. Empirical Resul?s The empirical work of the present section is based on a sample of eleven manufacturing industries. The data are drawn from the Internal Revenue Service, Statistics of Income for the years 19631971, excluding 1967. The distributions for 1963 and 1964 are collapsed to correspond to the classes in the remaining years. The final distributions possess information across twelve size classes for business receipts, and eleven for assets. Since the information in 1967 contains only eight size classes, it was omitted.’ The variables used are defined, where i refers to the industry and t refers to the year, as: M,, = cash, in millions of dollars; BR,, = business receipts, in millions N,, = number of firms;
of dollars;
‘We employ business receipts and asset distributions. These data are available by subcategory within manufacturing from 1963 through 1971. We required that, for each industry selected, the number of class sizes be as complete as possible. Industries were excluded if their inclusion unduly reduced the number of class sizes. The industries chosen include: 1) Food and Kindred Products; 2) Textile Mill Products, 3) Apparel and Other Fabricated Textile Products; 4) Lumber and Wood Products, except furniture; 5) Printing, Publishing, and Allied Industries; 6) Chemicals and Allied Products; 7) Stone, Clay, and Glass Products; 8) Fabricated Metal Products, except machinery and transportation equipment; 9) Machinery, except electrical; 10) Electrical Machinery, Equipment and Supplies; and 11) Scientific Equipment, Photographic Equipment, Watches, and Clocks. The data are subject to a variety of problems. First, since they are drawn from balance sheets and income statements of individual firms, they are subject to end-of-year “window dressing.” Second, though the data are annual, they refer to different intervals during the year. Third, the money variable is categorized as cash which is primarily currency and demand deposits. To the extent that the reported numbers include time deposits held by firms at non-member banks [see Meltzer (1963), p. 4091, we may have additional bias. Finally, annual data may obscure important information. 298
Distribution GBR,, GA,, VLBR,, VLA,, Rst R Lt
= = = = = =
Effects and Money Demand
Gini coefficient of business receipts;8 Gini coefficient of assets; variance of the logarithms of business receipts;7 variance of logarithms of assets; the market yield on 3-month Treasury bills; and the market yield on AAA corporate bonds.*
We are using business receipts as a proxy for transactions and four measures of size distribution: GBR; GA; VLBR; and VLA. The Gini coefficient and the variance of the logarithms usually move together. In fact, if the distribution is log-normal, then there is a one-to-one relationship between these two measures of size inequality. The use of the Gini coefficient or the variance of logarithms has the advantage of conserving on the degrees of freedom in the regression equations. Both also possess, however, the wellknown disadvantage of assuming that all possible redistributions which change the variables by the same amount have the same effect on the firm’s demand for money, ceteris paribus. Clearly, this assumption need not be true. Vogel and Maddala (1967) attempted to account for differences between firms by employing a pooled regression across industries with asset-class dummies. They concluded that the sales elasticity ‘The Gini suppressed):
coefficient
was
calculated
using
the
formula
(i and
t subscripts
are
where a, is the cumulative number of firms from the first to the ith class; 1 and a,,, are the means of the jth and (i + 1)st class of business receipts (or asset) distribution, respectively; N is the total number of firms; TV is the overall mean of business receipts (or assets); and J is the number of classes. ‘The variance of logarithms was calculated using the formula
VL = i
,=I
N, (In, -
i@/N ,
where In,& the natural logarithm of average business receipts (or assets) in the jth class; In is the geometric mean of business receipts (or assets); N, is the number of firms in the ith class; N is the total number of firms; and J is the number of classes. 8Both R,, and R,, were obtained from various issues of the Federal Reserue Bulletin. 297
George M. Katsimbris
and Stephen M. Miller
TABLE 1. Pooled Regressions with Znterest Rates” Constant
WR,,I
N,,)
WGBR,,)
WX,)
Al
-3.8426" (-14.3944)
0.7421" (20.7807)
A2
-3.9057+ (-14.8477)
0.7482* (20.9687)
Bl
-3.2263' (-14.9247)
(14.0075)
-3.3292" (-16.0268)
(14.4961)
Cl
-3.4141Q (-15.6751)
0.5658" (15.3018)
3.1829" (7.3228)
c2
-3.5709' (-17.5768)
0.5670' (16.1382)
(8.0462)
Dl
-3.2442' (-10.4707)
0.6005" (11.0596)
D2
-3.2912* (-10.2482)
0.6154" (11.1709)
El
-3.2572" (-7.1634)
0.5506" (9.6022)
E2
-3.3363" (-7.3740)
(9.7614)
B2
0.5298’
0.5349”
2.2887" (8.0227) 2.3124* (8.3272)
3.3494*
0.6595”
“Numbers in parentheses are t-statistics. Zn(BR,,/N,), In(&), and It@,,). Equations while the other equations have 84. ’ Significant at the one per cent level.
(Al)
All tests and (A2)
are two-tailed have 85 degrees
except for of freedom
estimatesof other studies were biased due to significant differences in the intercept terms across asset classes. Our analysis follows a different procedure for two reasons.First, we prefer to use the business-receiptsdistribution in searching for a size-distribution effect, and this distribution is not available for individual industries 298
Distribution TABLE
1.
ln(VLBR,,)
Effects
and Money Detnund
Continued ln(VLA,,)
WW
ln(R,,)
-0.2479” (-2.9444) -0.2864 * (-3.2300) -0.2204* (-3.4523) -0.2723. (-4.1236) -0.2630” (-3.9715) -0.3427. (-5.0856) 0.4574’ (3.3305)
-0.2512* (-3.1548)
0.4219’ (3.0661)
-0.2684. (-3.1654) 0.3336 (1.5833)
-0.2361+ (-2.8174)
0.3222 (1.5407)
-0.2728” (-3.0861)
R”
F
0.8327
217.5
0.8358
222.4
0.9042
274.6
0.9090
290.6
0.8966
252.6
0.9062
281.0
0.8504
165.9
0.8505
166.1
0.8356
148.4
0.8384
151.5
prior to 1963. Second, for those years that we do have a business-receipts distribution, cash is not reported for each size class. Thus, we are constrained to analyzing models using per-firm cash holdings at the industry level; consequently, the use of the Gini coefficient or the variance of logarithms is required. The regression equations are of the form ln(M,,/N,,)
= 0~~+ a, ln(BR,,INit)
+ a2 Wg,,)
+ a3 lnB, + c, (21) 299
George M. Katsimbris
and Stephen M. Miller
where g,, is either the Gini coefficient of business receipts or assets or the variance of the logarithms of business receipts or assets, and R, is either the short- or long-term interest rate.’ Regressions are calculated both with and without gdt and/or R,.“’ The form of equation (21) does not allow the size-distribution variable to go to zero. We also ran regressions with g,* replacing Zn(g,,) and the results are almost identical. Additionally, comparing the 8” values of the two functional forms, equation (21) performs marginally better. Table 1 presents the results of the pooled regression. In each case, the scale variable is the per-firm business receipts. Equations (Al) through (A2) omit size-distribution variables. Equations (Bl) through (E2) include the Gini of business receipts, the Gini of assets, the variance of the logarithms of business receipts, and the variance of the logarithms of assets, respectively. The corresponding numbers include (1) and (2), the short- and long-term interests rates, respectively. All of the scale elasticities are significantly less than one at the one per cent level, indicating economies of scale in cash management (Table 2). The inclusion of size-distribution variables reduces the estimate of the scale elasticity in each case. We performed t-tests to determine whether or not the scale elasticities with size-distribution variables are significantly less than without these variables. The scale elasticity when GBR and GA are introduced is significantly less than the scale elasticity in equations (Al) and (A2) at the one per cent level. The scale elasticity when VLBR is introduced is significantly less than its counterpart in (Al) and (A2) at the five per cent level. On the other hand, the scale elasticity is not significantly altered when VLA is introduced. The simple B-T model implies a scale elasticity of one-half. Without any size-distribution variable, the scale elasticity is significantly different from one-half at the one per cent level (Table 2). ‘Notice that the regressions are on a per-firm basis. We ran regressions of M,, variables. In each case, the coefficient of In N,, was on s,,, N,,, and the other not significantly different from one minus the coefficient of In S,. Thus, we report only the per-firm regressions. “We first ran cross-section regressions in each of the eight years. However, after running two pooled regressions with R,, and RLt, respectively, we were unable to reject the null hypothesis of homogeneity over time at the ten percent level. The cross section regressions are similar to our pooled results; consequently, we report only the latter. Cross section results are available from the authors upon request. All regressions are computed using ordinary least squares. 300
Distribution 2.
TABLE
Additional
0.2579
0.5 - ci;
-0.2421
1 - o;
0.4702 -0.0298
0.5 - a; 1 - II;
0.4342
0.5 - a;
-0.0658
1 - a;
0.3995 -0.1005
0.5 - a: 1 - CY; 0.5 - a; A B a1 ‘4 a1 A 011
-
a1 c
-
a1 D
-a1
A ‘yl
E -
a1
and Money Demand
Tests on Scale Coefficients” Pooled w/R,
Pooled w/R, 1 - a;
Effects
0.3494 -0.1506
(7.222)’
0.2518 -0.2482
(-6.780)’
0.4651
(12.430)” (-0.7886)
-0.0349 0.4330
(11.742)” (-1.780)**”
-0.0670 0.3846
(7.358)* (-1.851)**” (5.156)
-0.1154 *
(-2.223)‘”
0.2123
(3.894)’
0.1763
(3.898)
0.1416 0.0915
0.3405 -0.1595
(7.058)” (- 6.955) (12.605)* (-0.9457) (12.324)” (-1.907)““* (6.982)
*
(-2.094)“” (5.04O)O (-2.361)**
0.2133
(3.958)’
0.1812
(3.485)*
(2.244)“”
0.1328
(2.002)“a
(1.191)
0.0887
(1.158)
*
*
“Numbers in parentheses are t-statistics. For (1 - ai) and (of - Q:) which are expected to be positive, the t-tests are one one-tailed; other tests are two-tailed. a, is the coefficient of In (B&/N,,) and superscripts refer to the appropriate equation in Table 1. ‘Significant at the one per cent level. * Significant at the five per cent level. ’ Significant at the ten per cent level. l l
l
When GBR is introduced, however, the scale elasticity is not significantly different from one-half. For GA, VLBR, and VLA, it is significantly different from one-half at either the five or ten per cent levels. The coefficients of the size-distribution variables are uniformly positive and, except for VLA, they are significantly different from zero at the one per cent level. The positive sign on the coefficients of the size-distribution variable is counter to the presumption implied by the economies-of-scale effect. Thus, the pooled regression estimates indicate that, taken together, the effects of the degree 301
Distribution
Effects and Money Demand
of decentralization in cash management, the cost of borrowing, and compensating balances are positive and more than offset the opposing, negative effect of economies of scale. Finally, both the short- and long-term interest-rate elasticities are negative and significant at the one percent level. However, they are also significantly different from minus one-half as implied by the simple B-T model (Table 3). In a model developed by Lewis (1974) which allows the cost of credit to vary with the market interest rate, the interest-rate elasticity of the demand for money becomes less negative under certain plausible conditions. Based on fi2 and F values, the equations with the long-term rate are marginally superior to the equations with the short-term rate. 4. Conclusion The primary objective of this paper has been to develop and test a model of the firm’s demand for money which explicitly considered the industry’s firm-size distribution. The question was whether and how changes in an industry’s size distribution affects its per-firm demand for money. In Section 2, we analyzed the potential aggregation bias introduced by ignoring the size distribution of industries. The impact on the per-firm demand for money from this source was related to: 1) economies of scale; 2) decentralization in cash management; 3) cost of credit; and 4) compensating balances. Because of counteracting influences, we were unable to specify, a priori, the direction of the size-distribution effect. For our sample, we found economies of scale in cash management before introducing our size-distribution variables. But after introducing the size-distribution variables, the scale elasticities were usually lowered significantly. In fact for GBR, the new elasticity was not significantly different from one-half. The results also indicate that increasing size inequality increases the per-firm demand for money of an industry. This finding is consistent with those of Vogel and Maddala (1967). Received:
February,
1980
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