post-war consumption in canada: a first look at the aggregates

Post-War Consumption in Canada: A First Look at the AggregatesAuthor(s): Alan PowellSource: The Canadian Journal of Economics and Political Science / Revue canadienned'Economique et de Science politique, Vol. 31, No. 4 (Nov., 1965), pp. 559-565Published by: Wiley on behalf of Canadian Economics AssociationStable URL: http://www.jstor.org/stable/139831 .

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POST-WAR CONSUMPTION IN CANADA: A FIRST LOOK AT THE AGGREGATES*

ALAN POWELL Monash University, Australia

This paper is a preliminary report of a study of post-war patterns of aggregate consumption in Canada. The data consist of the eight gross categories of personal consumption recorded in the official statistics, and the corresponding price indicators.' A fifteen-year time-series spanning the years 1949 to 1963 has been used to fit a complete system of consumer demand equations for the eight categories using a method more fully reported elsewhere.2 Basically, this method makes the simplest plausible assumption about the form of the utility function, namely, that of "directly additive preferences," and utilizes the consequent dependence between substitution and income effects to sharpen the specification of the model.3 In the absence of such specifying restrictions, analysis of short time series is not likely to prove very fruitful.

It is not claimed that "directly additive preferences" provides the optimum set of restrictions (though it certainly provides the simplest). More sophisti- cated utility schemes, such as Mr. Barten's "almost-additive" preference con- cept,4 are likely to yield more reliable results. In this sense, the present report is preliminary. There are also other areas even within the framework of additive preferences in which further improvements could be sought; some of these are indicated later. Finally, the available arbitrary classification of consumption expenditure is in no sense ideal.

1 / The Model5

The set of expenditure equations fitted is linear in prices, income, and time. Thus the system has the form

8

(1)ist = aipj + bim + cit + ct (i = 1 * 8),

*The research reported here was completed while the author held the University of Chicago's Post-doctoral Fellowship in Political Economy. Computations were financed, in part, by the National Science Foundation. 'For consumption and population data, see: Dominion Bureau of Statistics, National Accounts -Income and Expenditure (Ottawa, 1963, and earlier annual editions), Table 47 and Appendix, Table 1. Seven components of the Consumer Price Index are tabulated by the Dominion Bureau of Statistics, in the official yearbook, Canada 1963 (Ottawa). A disaggregation of the indicator for the price of "Housing" into components "Shelter" and "Household Expenses" may be obtained from the Bank of Canada's Statistical Summary, Supplement 1963 (Ottawa), 133. 2Alan Powell, "A Complete System of Consumer Demand Equations for the Australian Econo- my Fitted by a Model of Additive Preferences," Econometrica, 34, no. 3 (July 1966) (in press). 3H. S. Houthakker, "Additive Preferences," Econometrica, 28, n. 2 (April 1960), 244-57. 4A. P. Barten, "Consumer Demand Functions under Conditions of Almost Additive Prefer- ences," Econometrica, 32, no. 1-2 (Jan.-April 1964), 1-38. 5For a fuller account, see Powell, "A Complete System."

XXXI, no. 4, November/novembre, 1965



560 ALAN POWELL

where Vit is per capita expenditure during period t on the ith commodity, pjg is the price of the jth commodity at t,

8

ml:-E Vit j=l is total money consumption outlay per capita during t, and the aj, bi, and cj are constants. The fit are disturbances. In these equations, the trend is centred on the mid-year of the data, so that 1956 corresponds to t = 0. Before (1) can be fitted, however, the system must be transformed into a form which does not demand too many parameter estimates (there are 64 aij's alone) from so small a set of data. To overcome this problem, Leser postulated that partial cross-elasticities of substitution were equal for all pairs of commodities.6 However, a somewhat more plausible (and, at all events, more easily interpreted) restriction is to assume directly additive preferences. Under this assumption, substitution effects are proportional to income derivatives7; that is, the income-compensated cross price derivatives are

(2) Kij = X(OxO/Om)Q(xO/am) (i J j), where the x's are quantum indexes of consumption, and X is a single constant. As a consequence of this restriction, all price coefficients can be formulated as functions of a single price parameter X, the income coefficients {bi}, and observed variables. In fact, it is shown elsewhere8 that

(3) aij = Xbibjlpj - bj;j (i j), and

(4) aii = (bi - 1)[(Xbj/Pj) -i],

where the additive-preference restriction has been imposed at mean prices {lpi and expenditures {vh}, and the {:i} denote the "typical" consumption bundle obtained by forming j = -V/lp. Substitution from (3) and (4) into (1) yields

(5) Yit = biut + cit + Eit,

in which 8

(6) yt= -vit-pii - bi E bj(pj/tjl - Pit/Pi) j=l

and 8

(7) ut = MlE PjltXis j=l In the above equations, the variable Yit may loosely be interpreted as the

deviation from its mean of current dollar consumption of the ith commodity at time t after all substitution effects have been removed. The regressor u,

6C. E. V. Leser, "Commodity Group Expenditure Functions for the United Kingdom, 1948- 1957," Econometrica, 29, no. 1 (Jan. 1961), 24-32. 7Houthakker, "Additive Preferences." 8Powell, "A Complete System."



Post-War Consumption in Canada 561

is a measure of "supernumerary" money income after certain "typical" consumption needs have been met at current prices.9 Deflation by pi, would thus yield a measure of "real" supernumerary income in terms of purchasing power of the ith good in the case of ut, whilst deflating Yit would yield a quantity index for the ith commodity. The constant X has the interpretation

(8) X = d / P a 4)/am' where 4 is the marginal utility of consumption outlay, i.e., "the marginal utility of money." It follows that we can estimate Frisch's'0 "money flexibi- bility" co by (9) estimate X = -mf/(estimate X).

The system (5)-(7) is solved for the bi's, ci's, and X by an iterative procedure. Thus the total number of parameters in the transformed systems is 17. Additivity is preserved identically in this model, whilst homogeneity and symmetry are satisfied only approximately. The statistical features of the estimating procedure have not been investigated, but approximate significance tests are available from classical least-squares theory in the case of the marginal value shares bi and linear trend coefficients ci.

2 / Results

The main results of this analysis are summarized in Table I.

STATISTICAL RELIABILITY

Statistically, there is perhaps only one sense in which the over-all results could be described as "satisfactory," and that is in terms of goodness of fit. The R's found for the individual transformed equations range upward from .898, except for two equations which do not fit the model at all well. The regressands involved are Y3t (adjusted clothing expenditure) and Y8

(adjusted miscellaneous expenditures). The proportions of the variance explained in these cases are only 48 and 34 per cent respectively. Surprisingly, two of a total of six "significant" coefficients come from the clothing equation.

In terms of the original expenditure variables Vit, however, the situation is much better. The lowest R2 in terms of these variables occurs for the "miscellaneous" category, a value of .88; R2's otherwise are above 0.96.

The estimated errors, however, are very poorly behaved from the view- point of statistical efficiency. The only equation clearly displaying an absence of autocorrelation in the residuals is that for transportation. Two other cases are in doubt, whilst the remainder exhibit positive autocorrelation at the 5 per cent significance level. Eight of twenty-eight estimated contemporaneous covariances (eiej) are "significant" at the 5 per cent level under classical tests. 'This interpretation, drawn from a similar variable in a related model, is due to Richard Stone. See his "Linear Expenditure Systems and Demand Analysis: An Application to the British Pattern of Demand," Economic Journal, 64, no. 255 (Sept. 1954), 511-27. ?'Ragnar Frisch, "A Complete Scheme for Computing All Direct and Cross Demand Elasti- cities in a Model of Many Sectors," Econometrica, 27, no. 2 (April 1959), 177-96.



TABLE I PRINCIPAL RESULTS: ADDITIVE PREFERENCE MODEL OF CANADIAN CONSUMPTION, 1949-1963

Proportion of total sum of squares

Proportion of Annual Annual shifts Elasticity of demand "explained" in case of additional shift Value of Student's t as percentage with respect tot Residual

outlay spent in of mean sum of final original Durbin- Commodity group on group demand for for expenditure own squares regressand variable Watson

s 100 cl /vi income price EQ t2 yit vit statistic

1. Food 0.1431 0.0514 2.017 0.351 0.181 0.581 -0.464 1.847 .948 .990 0.6639t 2. Tobacco and alcohol 0.0555 0.0087 1.540 0.117 0.108 0.793 -0.538 0.478 .899 .969 0.6831t 3. Clothing 0. 0804* -0. 1439* 2.752 -2.387 -1.148 0.740 -0.519 0.314 .485 .965 1.0415? 4. Shelter 0.0721 0.4648* 1.465 4.575 2.865 0.512 -0.379 0.891 .992 .997 1.2009? 5. Household expenses 0. 2703* -0.3658* 4.497 -2.950 -2.506 2.134 -1.275 1.327 .898 .980 0. 6926t 6. Transportation 0. 2685* -0.1269 6.524 -1.494 -0.953 2.325 -1.365 0.622 .988 .994 2.0127 7. Personal, medical -.0052 0.2464 -0.085 1.940 2.904 -0.071 0.051 1.391 .918 .983 0.3882t 8. Miscellaneous 0.1153 -0.1347 0.881 -0.499 -0.986 0.973 -0.671 6.285 .337 .882 0.5614t

Total 1 0

*Significantly different from zero at 5 per cent level under classical assumptions (approximate test only).

tSignificant positive autocorrelation at 5 per cent level.

tElasticities evaluated at mean prices and expenditures. ?Test inconclusive.



TABLE II PRICE COEFFICIENTS FOR LINEAR PREDICTING EQUATIONS

Coefficient with respect to the price* of

Tobacco and Household Trans- Personal, Miscel- Equation Food alcohol Clothing Shelter expenses portation medical laneous

1. Food 13.0183 -0.5102 -0.8584 -1.1933 0.6713 0.7587 -0.9538 -0.5718 2. Tobacco and

alcohol -0.8438 3.3682 -0.3333 -0.4633 0.2606 0.2946 -0.3703 -0.2220 3. Clothing -1.2216 -0.2868 5.5180 -0.6707 0.3773 0.4264 -0.5361 -0.3214 4. Shelter -1.0957 -0.2572 -0.4328 7.7396 0.3384 0.3825 -0.4809 -0.2883 5. Household

expenses -4.1065 -0.9640 -1.6220 -2.2547 -3.4239 1.4335 -1.8023 -1.0804 6. Transportation -4.0788 -0.9575 -1.6111 -2.2395 1.2598 -3.8794 -1.7902 -1.0731 7. Personal, medical 0.0794 0.0186 0.0314 0.0436 -0.0245 -0.0277 6.7024 0.0209 8. Miscellaneous -1.7514 -0.4111 -0.6918 -0.9616 0.5409 0.6114 -0.7687 3.5361

Sum 0 0 0 0 0 0 0 0

*Price indicators shown are components of the Consumer Price Index.



564 ALAN POWELL

The upshot of these considerations is that the estimators used here are far from efficient. Ideally, we might seek to employ generalizations of least-squares theory in order to harness the correlations exhibited in time" and between commodities.'2 In fact, only the former type of correlation can be removed from this model using standard techniques. The structure of the transformed system (5) is one of the special cases for which the Aitken least-squares estimators are identically the simple least squares estimates already computed. In this special case, these are also the maximum likelihood estimators. This occurs because the set of regressors gut, t} is the same for each of the eight transformed equations.'3 However, it is possible that removal of the autocorrelation would also eliminate some or all of the inter-equation correlations. But in either event, provided the specification which enabled the system (5)-(7) to be constructed were not at fault, the bi's and ci's obtained would remain maximum likelihood estimators in view of the structural feature noted above.14

However, the details involved in removing the serial correlations remain to be worked out. In particular, care would be needed to ensure that the techniques employed did not result in estimates which violated the desirable economic properties of the model. For example, it is most desirable that the estimated bi's should add to unity and that the sum of the c,'s should vanish. Work is proceeding along these lines. For the present time it must be remarked, however, that since the evidence against the classical least-squares assumptions is so ample, the "significance" of six coefficients reported in Table I is somewhat conjectural.

ECONOMIC INTERPRETATION

The most evident anomaly among the estimated parameters of Table I is the negative income elasticity found for personal and medical expenses (-0.071). Obviously, we cannot take the "discovery" that medical expenses are an inferior good very seriously; this coefficient, at all events, does not differ significantly from zero.'5

Very high (and significant) marginal value shares of 27 per cent are found for both household expenses (which must consist largely of durables) and transportation (which includes private automobiles). In elasticity terms these income effects exceed a value of 2, which seems inordinately high. The clothing elasticity, also significant, seems in view of results in other countries to be undeiestimated at .74.16

"See, e.g., J. Johnston, Econometric Methods (New York, 1963), 192 ff. 12Arnold Zellner, "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal of the American Statistical Association, 57, no. 298 (June 1962), 348-68. 131bid., 351. '4The above treatment glosses over the problem that in equation (1), as it is written, at least one of the ejt cannot be a random variable since the sum over commodities of these disturbances must vanish identically at any given date. 15As a result of the structural interdependence in this model, the price elasticity estimated for personal and medical expenses also has the "wrong" sign. '6For example, see Powell, "A Complete System," Barten, "Consumer Demand Functions," and Leser, "Commodity Group Expenditure Functions."



Post-War Consumption in Canada 565

In view of the foregoing remarks about statistical reliability, it may not be wise to dwell too long upon the economic implications of the estimates. But worth noting among the so-called significant coefficients is a strong negative trend against clothing expenditures (1 per cent per annum). Evidence of a negative trend against clothing in the post-war era has also occurred, and then under less doubtful circumstances, in at least three other western econo- mies.17 Again the strong (and significant) trends against household expenses and in favour of shelter constitute a puzzle. Changes in tastes would seem to provide less than an adequate explanation of such strong trends.

Finally, the estimate obtained for Co, the income elasticity of the marginal utility of income, is -1.55. Unfortunately, not even approximate confidence intervals are available for this parameter. However, the evidence from other studies suggests that this figure may be too low, a value of -2.0 being more in line with overseas estimates.18

3 / Conclusion

Under the usual (but nevertheless heroic) assumption of accurately measured data, the estimates obtained here remain unbiased, even if of less than optimal efficiency. The extent of autocorrelation in the residuals was a factor which could not be foreseen ex ante, but corrective action will be needed to procure efficient estimators.

But this tale of woe would surely be incomplete without a brief mention of the major issues which this paper could not, in any event, touch. These are the questions of savings versus consumption, the identification of the system (1) as a set of demand (rather than supply) relations, the aggregation problem, and the treatment of durables.

In spite of their manifold shortcomings, the estimates computed here may none the less be useful for short-term projective work-at least until some- thing better becomes available. For such purposes the system (1) is more suitable than a set of elasticity estimates. Accordingly, the estimates of the linear price coefficients as>, obtained by appropriately transforming the estimated parameters, are set out in Table II.

17See Powell, "A Complete System," Leser, "Commodity Group Expenditure Functions," and James R. Donald, Frank Lowenstein, and Martin S. Simon, The Demand for Textile Fibres in the United States, USDA Technical Bulletin no. 1301 (Washington, 1963), 99. 18See Ragnar Frisch, "Dynamic Utility," Econometrica, 32, no. 3 (July 1964), 418-24 (esp. 422); and Powell, "A Complete System."



post-war consumption in canada: a first look at the aggregates

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