A Cash Expenditure Consumption Function

A Cash Expenditure Consumption FunctionAuthor(s): Alice BourneufSource: The Review of Economics and Statistics, Vol. 42, No. 1 (Feb., 1960), pp. 108-112Published by: The MIT PressStable URL: http://www.jstor.org/stable/1926105 .

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Io8 THE REVIEW OF ECONOMICS AND STATISTICS

sons perhaps as descriptive of changing industrial structure as the usual concentration ratios. The Prais analysis strongly suggests that there are a number of interesting ways in which statistical tech-

niques may be adapted to the analysis of changing market structures, but to label all of these techniques "measures of concentration"' is surely to confuse their meaning and limit their usefulness.

A CASH EXPENDITURE CONSUMPTION FUNCTION

Alice Bourneuf *

Several authorities have suggested that consumption and income as thought of by the man in the street may be more closely related to each other than consumption and income as measured by official statistics. The man in the street may think of his current income as his cash income, and current consumption as current cash expenditures on consumption. This paper examines the hypothesis that cash expenditures on consumption are more closely correlated with current income than the value of current purchases of consumer goods. Only one step is taken toward a full cash approach to consumption and income. Measures of consumption based on cash expenditures on consumer durables are compared with measures based on the value of consumers' durables purchases. No adjustment is made in the official income estimates.

The notion that consumers' durable goods purchases should receive special treatment in the analysis of consumption is widely accepted. Many econ- omists believe on a priori grounds that the flow of services from durables should be treated as consumption rather than current purchases of durables. The flow-of-services approach seems appropriate for long-period studies, implying that individuals aim at a flow of real satisfactions related to their real income and to other factors such as their real wealth. The cash expenditure approach may be more appropriate for short-period studies, implying that individuals have a substantial degree of "money illusion," and tend to adjust their dollar expenditures to their dollar incomes. If consumer cash expenditures are more closely correlated with income than consumer purchases, the practical question for income and employment theory is how to use this relationship between cash expenditures and income to analyze purchases of consumer goods.

II

To test the relative strength and stability of the short-period relationship to income of cash expenditure and purchase measures of consumption it is necessary to use some form of the consumption function. Recent studies of the consumption function based on prewar and postwar data have suggested that liquid assets and past consumption or income levels should be considered as independent variables, in addition to current income.' The depth of the depression of the I930'S, the abnormal accumulation of liquid assets during the war, and the strict war- time controls on durable purchases, undoubtedly have influenced the results obtained in these studies.

The cash expenditure hypothesis can be tested, however, on the basis of quarterly data for a relatively short postwar period. The period chosen for this study is I950-58. During this period there was no serious depression, no abnormal accumulation of liquid assets, and no release of pent-up consumer demand. It seems reasonable to assume, therefore, that current income may have been the single most important determinant of current consumption.

A simple short-period consumption function based on current income, even if it does not offer the most complete explanation of consumption, seems appropriate for testing the cash expenditure hypothesis. The simple regression equations probably offer the best comparison of the relative strength of the rela- tionships between income and the various measures of consumption. The effects of using different definitions of consumption are easily discernible.

Analysis of the period starting in I950 soon brings out the violent fluctuations in consumption that occurred in the third quarter of I950 and the first quarter of I95I as a result of changes in ex- pectations due to the Korean war. The analysis was therefore made also for the period beginning in I95I-II.

The data are United States quarterly data at current prices, seasonally adjusted and at annual rates. The period is too short to require adjustment for

* This study was financed by the Bureau of Business and Economic Research of the University of California. Several members of the Economics Department at the University of California were kind enough to read and criticize an earlier draft of this article.

' See Arnold Zellner, "The Short Run Consumption Function," Econometrica, xxv (October I957).



NOTES AND BOOK REVIEWS I09

population changes. The data are not adjusted for price changes because the hypothesis that individuals think in terms of cash expenditures on consumption implies some money illusion and suggests that undeflated data will lead to higher correlations. Adjustment for price changes would not substantially alter the results in a short period of relative price stability.

The objective is to compare the closeness of the relation to current income of two essentially different types of consumption measures, cash expenditure measures and purchase measures. In all, however, five measures are used. The cash expenditure measures are extremely rough.2

i. Purchase Measures. (Official United States Department of Commerce

data.) P-i. Total consumer purchases. P-2. Consumer purchases excluding durable goods.

2. Cash Expenditure Measures. C-i. Total consumer purchases adjusted to a cash

expenditure basis for automobiles only. The aim here is to approximate a figure for cash expenditures on all durables. The adjustment is made for automobiles only because better data are available on automobile purchases than on other durable purchases.3 The adjustment deducts from the total value of total consumer purchases the value of car purchases, and adds, instead, an estimate of the total cash payments being made on both current and past car purchases.4 Cash payments may be for current cash purchases, for down payments on current credit purchases, or for instalment payments due on past credit purchases.

C-2. Total consumer purchases adjusted to a cash expenditure basis for all durable goods. The adjustment aims at deducting from total consumer purchases the total value of all current purchases of durables and adding, instead, an estimate of total cash payments on durables, whether on current or past purchases. (See footnote 4.) Two separate methods of adjustment were used, C-2-a and C-2-b, because the data are rough and inadequate. Method

C-2-a assumes that durables other than cars, furniture, and household appliances tend to be purchased for cash. Method C-2-b assumes they tend to be purchased on the same terms as furniture and household appliances.

III

The results of a correlation analysis of total consumption and total income, for all five definitions of consumption, are presented in Table i. Cash ex-

penditure measures of consumption are more closely correlated with current income than consumer purchase measures for the period from I950-I to I958-I. The coefficients are all very high and are very close to each other. Two of the cash expenditure measures are more closely correlated with income than the consumer purchase estimates for I95I-II to I958-I, but the differences between the coefficients are smaller. The other cash expenditure measure (C-2-b) has the lowest correlation coefficient, which may indicate that it is a less accurate measure.

The regression equations on the whole look reasonable. There is positive autocorrelation of the residuals from the I95I-II to I958-I regression equations for all five measures of consumption according to the Durbin-Watson test applied to graphically-deter- mined residuals. For the period, I950-I to I958-I the same test does not indicate positive correlation of residuals because of the violent fluctuations of the Korean period. The standard errors of the re-

TABLE I. - CORRELATION COEFFICIENTS AND 95 PER

CENT CONFIDENCE INTERVALS

Totals of Consumption and Income for Alternative Measures of Consumption

1950-I to 1958-I, 1951-II to 1958-I, 33 Observations 28 Observations

95 Per Cent 95 Per Cent Consumption Confidence Confidence

Measures R interval a R interval a

Purchases P-i Total .9946 .9890-.9967 .9977 .995I-.9990 P-2 excl.

durables .996I .9903-.998I .9970 .9935-9986

Cash Ex- penditures

C-i Adjusted for autos only .9966 .993I-.9983 .998I .9959-..999I

C-2 Adjusted for all durables

Method a .9972 .9942-.9986 .9983 .9963-9992 Method b .99 73 .9944-.9987 .9964 .9922-.9983

a The confidence intervals were calculated by using the z transfor- mation. The values of z between 3 and 5 were interpolated graphically from values shown in R. A. Fisher and Frank Yates, Statistical Tables (London, 1948).

2 A full description of the estimates in mimeographed form may be obtained from the author.

'Since fluctuations in automobile purchases account for a large part of the fluctuations in durable goods purchases, the adjustment to a cash basis for automobiles only might approximate the results of adjustment for all durables.

Instalment payments data are Federal Reserve Board monthly gross instalment repayment data. Estimates of cash sales and cash down payments are based on rough in- dications from scattered data in the various volumes of the Federal Reserve Board study of Consumer Instalment Credit, I957.



IIO THE REVIEW OF ECONOMICS AND STATISTICS

gression coefficients are very small and the standard errors of estimate as a percentage of total variance run from about 7 per cent to i i per cent. The regression coefficients for the total purchase measures are .94 and .99, and for the cash expenditure measures a little above I.O in each case, reflecting the inclusion of payments on instalment debt.

Although the correlation coefficients for cash expenditures measures of consumption are higher, at the 95 per cent confidence level we cannot reject the hypothesis that almost all of the coefficients for either of the two periods analyzed are sample estimates of the same parameter. For the period beginning in I950-I the differences in the coefficients are greater, and, for the two different types of measures, appear to be significant. There is a relatively small overlap between the 95 per cent confidence interval for the total consumer purchase coefficient and the 95 per cent confidence interval for the two cash expenditure measures adjusted to a cash basis for all durables. For the period beginning in I95I-II the correlation coefficients for the various measures of consumption are closer together, the confidence intervals overlap more, and the intervals by them- selves suggest that the differences are not significant. But the fact that the rank order of the various correlation coefficients is very close to that for the longer period, combined with the fact that the differences seem to be significant in the longer period, suggests that the differences in the shorter period may also be significant. Furthermore, with a larger number of observations, the confidence intervals might be smaller and overlap less.

IV

The short-period relation between changes in consumption and changes in income is much less stable than the relation between total consumption and total income. A better test of the closeness of the relation between various measures of consumption and income, therefore, lies in the correlation of quarter to quarter changes, or first differences.5 The correlations of first differences bring out sharply the short period instability of the consumption-income relation, but they also reveal the relatively much greater stability of the cash expenditure measure.

The correlation coefficients for first differences of consumption and income are low; the coefficients for the period beginning in I95o-I range from .26

for total consumer purchases to .35 for one of the cash expenditure measures (see Table 2). All three cash expenditure measures lead to higher coefficients

than the two purchase measures. But the 95 per cent confidence intervals of the coefficients are wide and overlap almost completely. The coefficients, viewed apart from other evidence, are not significantly different from zero or from each other. But the rank order of the various coefficients is so close to the order found in the correlations of totals that one can hardly hold it to be accidental.

The coefficients for the shorter, and more normal, period beginning in I95i-II are much higher, rang- ing from .65 to .79. Again, cash expenditures are more highly correlated with income than purchases. The rank order of the coefficients is the same as for the longer period. The coefficients of determination show that changes in income "explain" from 57 to 62 per cent of the changes in cash expenditures in the same quarter. The percentages "explained" are only 43 per cent for purchases including durables and 47 per cent for purchases excluding durables. These differences are large enough to suggest that the closer relation of cash expenditures to income in a relatively normal period is of considerable practical importance. But the coefficients are close together; ithe 95 per cent confidence intervals in fact overlap to a large degree (see Table 3). The differences in the coefficients can not be taken, by them- selves, as evidence of significant differences in the correlations for the various measures of consumption. But again the rank order of the coefficients is virtually the same as that found in analyzing the totals, and also the same as that found for the first differences in the longer period. This consistent pattern of rank orders suggest that the differences are significant. Furthermore the confidence intervals might be much smaller if more observations were available.

The correlation coefficients for the shorter and more normal period beginning in I95I-II must be considered in the light of the regression equations

TABLE 2. - CORRELATION COEFFICIENTS AND COEFFI- CIENTS OF DETERMINATION, FIRST DIFFERENCES OF CONSUMPTION AND INCOME, ALTERNATIVE MEASURES OF CONSUMPTION

1950-I to 1958-I 1951-II to 1958-I 32 Observations 2 7 Observations

Measures of Consumption R R2 R R2

Purchases P-i Total .2637 .o695 .6525 .4258 P-2 Excluding durables .2660 .o708 .6877 .4729

Cash Expenditures C-i Adjusted for

autos only .3io6 .o964 .7700 .5929 C-2 Adjusted for

all durables Method a .3470 .1204 .7859 .6176 Method b .32i6 .io34 .7558 .57I2

I See G. H. Orcutt and S. F. James, "Testing the Signifi- cance of Correlations Between Time Series," Biometrika, xxxv (December I948), 397-4I3.



NOTES AND BOOK REVIEWS III

TABLE 3. - CORRELATION COEFFICIENTS AND 95 PER

CENT CONFIDENCE INTERVALS, FIRST DIFFERENCES

OF CONSUMPTION AND INCOME, ALTERNATIVE MEAS-

URES OF CONSUMPTION, 27 OBSERVATIONS I95I-II TO

I958-I

95 Per Cent Measures of Consumption R Confidence Intervals

Purchases P-I Total .6525 .3627-.8275 P-2 Excluding durables .6877 .4I36-.8455

Cash Expenditures C-I Adjusted for Autos Only .7700 .55II-.8896 C-2 Adjusted for all Durables

Method a .7859 .5784-.8977 Method b .7558 .5299-.8832

and standard errors (see Table 4). The standard errors of the regression coefficients are within reasonable limits.6 The standard errors of estimate are high, but they are somewhat smaller for cash expenditure measures than for the total purchase measure of consumption. The Durbin-Watson test indicates no positive autocorrelation of residuals from these regression equations.

On the whole it seems clear that changes in cash expenditures are much more closely correlated than changes in consumer purchases with changes in per- sonal disposable income. The instability of consumption-income relations from quarter to quarter is much less if consumption is measured by cash expenditures on consumption.

V

Even for the shorter period, and for the cash expenditure measure of consumption adjusted for all durables, the first difference analysis sugges;ts that current income changes do not offer a full explanation of current consumption changes. The period is one in which income changes might be expected to play a predominant role. Yet the correlations show current income changes "explaining" at most 62 per cent of current changes in cash expenditures on consumption.

The very high correlation for total cash expenditures on consumption and total current income, and the low correlation for first differences, could result from a simple lag in the adjustment of cash expendi-

tures on consumption to income changes.7 Time series charts for totals and for first differences, however, do not indicate a simple lag of consumption behind income for any of the consumption measures. The time series chart for first differences indicates that changes in consumption, however measured, are more closely correlated with changes in income of the same quarter than with changes in income of the preceding, or any other quarter. Peaks and troughs coincide more often than not, and there are instances of both leads and lags.

The time series chart for first differences of consumption and income suggests that a more complex lag does exist, however. Smoothing the jagged peaks and troughs in the first differences of income would raise the correlation with consumption, however measured, because consumption changes less violent- ly from quarter to quarter than income. The time series for first differences of consumption looks somewhat like moving averages of income changes. In a series of quarters in which there is a succession of substantial increases in income, cash expenditures tend to lag progressively behind income. In a series of quarters in which income is relatively stable or decreasing, cash expenditures increase more than income or decrease less. This behavior in periods of relatively stable or falling income is clearly consist-

TABLE 4. -REGRESSION EQUATIONS FOR CORRELATIONS

OF FIRST DIFFERENCES OF CONSUMPTION AND INCOME,

ALTERNATIVE MEASURES OF CONSUMPTION, 27 OBSER-

VATIONS FROM I95I-II TO I958-Ia

Measures of Consumption a b Sva Sc

Purchases P-I Total I.29 .585 I.8O .77I

(.35) P-2 Excluding

durables 1.71 .334 .94 .74I

(.070) Cash Expenditures

C-I Adjusted for autos only I.56 .558 I.23 .65I

(.093) C-2 Adjusted for

all durables Method a I.69 .563 I.I8 .632

(.o89) Regression equations are of the form

C=a+bY where C=CT1-CTO and Y = YT1 - YTO

a Amounts indicated under a and SCV are in billions of dollars. Scv is the standard error of estimate of C on the basis of the regression equation; Sc is the total variance of C.

6 The b terms in the regression equations for the first differences are not to be confused with any measure of the marginal propensity to consume. The b term measures the amount by which the change in consumption increases as the absolute size of the change in income increases. Since the a terms are positive and large compared to the average change in consumption, and the b's are not high, it is clear that larger increases in income are accompanied by propor- tionately smaller increases in consumption.

7In this case, however, positive autocorrelation of the residuals from the regression equations for first differences would probably be clearly apparent.



II2 THE REVIEW OF ECONOMICS AND STATISTICS

TABLE 5. - CORRELATIONS OF CASH EXPENDITURE MEAS-

URES OF CONSUMPTION WITH WEIGHTED AVERAGE

OF CURRENT AND PAST INCOME, FIRST DIFFERENCES

AND TOTALS, 25 QUARTERLY OBSERVATIONS FROM

I95I-II TO I958-I

Scv

a b Scv Sc R R2

First Differences C-i Adjusted for

autos only .775 .762 I.09 .576 .8263 .6828 (.Io8)

C-2-a Adjusted for all durables .9I3 .764 I.04 .534 .837I .7007

(.I04) Totals

C-2-a Adjusted for all durables -26.I4 I.054 I.42 .054 .9984 .9968

(.004) Regression equation of the form C = a + b Y where C and Y

equal first differences of consumption and weighted first differences of income, or in the third case, totals of consumption and weighted totals for income.

ent with a Duesenberry-Modigliani hypothesis, and the progressively increasing lag in periods of rapid consecutive increases may also be.8

Any attempt to take into account the level of income, or changes in income, in previous quarters runs into the problem of serial correlation. Not only are time series for total consumption and total income positively serially correlated, but the first- difference-of-income series is also positively serially correlated. The inclusion of past as well as current income, or income changes, as independent variables will therefore lead to some bias in the regression coefficients. Notwithstanding the bias, a rough ex- periment was made with arbitrarily assigned weights. First differences of income "explained" a little over 6o per cent of cash expenditures on consumpition; this suggested that 6o per cent of the weight might be given to the current quarter. The first preceding quarter was given 30 per cent and the second preceding quarter iO per cent, on the ground that the influence varies inversely with time.

The use of the weighted-change-in-income term as the independent variable substantially raised the correlation with changes in consumption. The weighted-change-in-income term "explains" about 70 per cent of the changes in both cash expenditure measures of consumption, while the current quarter's

change in income "explained" about 6o per cent (see Table 5). The Durbin-Watson test indicates no positive autocorrelation of residuals from the weighted-change-in-income regression equations.

The weighted-total-income term raises the coefficient of determination for total cash expenditures on consumption from .9944 to .9968. The standard error of estimate is much lower in absolute terms, and also as a percentage of total variance, than the standard error for the same measure of consumption and the current quarter's total income. This equation for total consumption and total income might be useful even though open to formal criticism.9 It suggests that a more careful attempt to develop this type of equation might be worth while.

Summary and Conclusion

i. The statistical analysis lends considerable sup- port to the hypothesis that cash expenditures on consumption are more closely correlated with per- sonal disposable income than are total purchases of consumer goods.

2. Total cash expenditures on consumption are somewhat more highly correlated with current income than total consumer purchases, or total purchases excluding durable goods. But all the correlations are very high and the differences are slight.

3. Quarterly changes in cash expenditures on consumption are much more highly correlated with quarterly changes in income than are quarterly changes in total consumer purchases or in total purchases excluding durables.

4. The fact that cash expenditures on consumption are more closely correlated with income than consumer purchases, whether total or excluding durables, suggests that a cut in cash expenditures on durables, other things being equal, is likely to ac- company an increase in cash expenditures on non- durables, and vice-versa. It may be a mistake, therefore, to make independent estimates of durable and nondurable purchases.

5. A study of the first differences of consumption, according to the various measures, and the first differences of income, suggests that no simple lag of quarterly consumption changes behind income changes would raise the correlations. A weighted average of past income changes does raise the correlation for all measures of consumption.

' The different measures behave somewhat differently over time. First differences of consumer purchases including durables rise more than income in the early quarters of a period of successive increases, and more slowly, if at all, in the later periods.

9This correlation was made after the second quarter of I958 data for consumption and income became available, and includes data for this quarter. The Durbin-Watson test indicates positive autocorrelation of the residuals from the regression equation based on the weighted-total-income term. This may indicate that the weights are off, or that not even a weighted income term can explain all the movements of consumption.



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A Cash Expenditure Consumption Function