Additive Preferences and Cost of Living Indexes : An Empirical Study of the Australian Consumer’s Welfare*

I Movements in the consumer price index and the impact of such

movements on the Australian economy have been subjected to thorough analysis in recent years (see for example [4] and [Sl l ) . A funda- mental assumption implicit in all the studies so far carried out is that the ‘true’ cost of living index-an ordinal concept-can be approxi- mated by a fixed arithmetic mean weight Laspeyres index without reference to the ‘true’ level of maximum satisfaction enjoyed by the ‘representative’ consumer in the personal expenditure section of the economy. Recently, however, a cardinal utility function in the form of a direct or indirect separable income-price-quantity relationship for broadly defined commodities has been widely adopted in empirical studies of consumer demand [2], [ 51, [9], [ 101, [ 121 and has proved helpful in dealing with the problem of measuring the level of con- sumer satisfaction.

A cost of living index for Australia as a whole and also for each of the six States is presented in this paper, exploiting the concept of directly additive preferences [2], [5] for seven complete sets of consumer demand equations. Use is made of a Stone-Geary cardinal utility function to derive a cost of living index which, in contrast to conventional consumer price indexes, takes proper account of the substitution which accompanies commodity price changes. The corre- lation between our cost of living index and various accepted indicators of welfare is also briefly examined.

*The author is indebted to Professors Alan A. Powell and Arthur S. Gold- berger for stimulating his interest in this subject. Thanks are also due to Dr Bill Howard for helpful suggestions.

1Figures in square brackets relate to the references listed at the end of the article.

432

SEPT., 1969 AUSTRALIAN CONSUMER '6 WELFARE 433

I I (a) A Model of Additive Preferences

functions for a 'representative' consumer's n budget periods Consider a complete demand model of say k linear expenditure

L xi: = 2 %P,t+Bi~:+yi~t+ei t (i = 1-9 * * * 3 k)

j = 1 ( t = l , . . . , n) (1)

where, throughout time t , Xdt is the personal consumption expenditure per head of the population on the ith commodity; Prt the current price of the j th commodity ; Mt = Sk, X4t, the total expenditure outlay per head of the population; Tt a time function representing the consumer's changing tastes and habits ; a+j, pr, yc the behavioural parameters to be estimated; and qt the random errors. Further, it will for convenience

be assumed that T = ; L"t Tt = 0, and this property will underline the results reported below.2

Under directly additive preferences, Houthakker [ 51 has shown that the substitution effect between any pair of commodities is propor- tional to the income (total expenditure) derivatives of both commodi- ties and to what Houthakker has termed the income flexibility A. In view of this theorem of direct utility functions, Powell [9] has demon- strated that the system (l), in terms of sample mean prices {Fc} and overage quantum (consumption level) indexes a: can be conveniently: transformed into

1

xi: = Pt:~i+hzit+BtW:+y,Tt+clt (2) in which

and k

wt = M t - z PjtO, (i = 1, . . . , k) (4)

A computer programme is available to estimate the transform (2), which is non-linear in its behavioural parameters.

( b ) Economic Properties It is a feature of the transformed model (2) that, whenever the

marginal value share {/I*} is less than zero, the corresponding income elasticity will become negative (the case of inferior goods) and the own price elasticity invariably positive (the Giffen paradox). As a

j = l ( t = 1,. . . , n )

- 2 The assumption that T = 0 is necessary to maintain the adding-up property

3 The quantum index (0,) is calculated as <a') = { ~ , / ~ ~ } , where denotes of the model formulated below. See [lo] for a fuller account.

sample mean expenditure. H

434 THE ECONOMIC RECORD SEPT.

result of this constraint, inferior goods in this model will always appear to be complementary in consumption with other normal goods. Further, in the situation of general inflation where increases in total expenditure are due solely to uniform increases in prices in all the individual commodities, it has been shown that the marginal value shares {p%} will approach the average budget shares { X J M } . A direct consequence is that income elmticities in our model will be biased towards unity [ 101. However, in a recent study of consumer demand for Australia and the six States [12], the extent to which all com- modity prices changed proportionately was found to be negligible. It is therefore safe to assume, in the present exercise, that bias of elasticity estimation due to uniform price changes is generally absent.

I I I (a) TVelfare Indicators

The income elasticity of the marginal utility of money or the money flexibility &, introduced by Frisch [2] as a measure of welfare and now popular in view of its conceptually ordinal and empirically computable flavour, is defined as

where t is the marginal utility of money. In view of the definition of Houthakker ’s income flexibility A, we note immediately the expression (also evaluated at sample mean income B)

Since the marginal utility of money t is presumably very high a t low income levels and approaches zero a t high income levels, Frisch’s money flexibility will then move from large negative values a t low income levels to small negative values a t high income levels. In other words, the absolute value of may be used &s an inverse indicator of consumers’ welfare. This concept has been taken up by various authors for time series as well as cross country studies in demand [3] , [9].

(b) Cost of Living Indexes Goldberger [3] has shown that the transformed model (2) can be

exactly derived by maximizing a Stone-Geary convex utility function of the form

where the minimum required consumption levels {N} are known a priori to satisfy a particular pattern

1969 AUSTRALIAN CONSUMER ’S WELFARE 435

However, equation (7 ) justifies a true cost of living index in the Klein- Rubin sense [ 61. That is to say, equation (7) will register exactly the rates of change in income so that the consumer’s optimal level of satisfaction (utility) may be maintained, in the face of commodity price changes, throughout the period of observation. Algebraically, the additive preferences cost of living index between, say, a com- parison period with prices {P(,} and a reference period with prices { P 4 0 } may be written as

which combines weighted arithmetic and weighted geometric means of the two sets of prices.

This is an exact cost of living (consumer price) index dependent on observable price changes and in strict accordance with our model of restricted directly additive preferences. It takes into account, as weights, the subsistence budget proportions {ht}, the marginal value shares {pi} and the consumer’s welfare indicator $. It has obvious advantages over the Paasche and Laspeyres indexes whose biases are well known. Since the estimates of the behavioural parameters {p4}, { y ( } , and X as formulated in (2) are consistent, the v a h e of the statisti- cal series as derived from (9) gives a consistent estimate of the ‘true ’ cost of living index on the assumption of directly additive preferences.

IV I n this section, we report an additive preferences estimate of the

cost of living for Australia as a whole and for the six States separately for the period 1948-49 to 196465. The sets of data required to com- pute the index as formulated in (9) were derived from recent esti- mates of demand elasticities as described in detail in [ 121. In general, a collection of 3even sets of national (aggregate) and State (regional) demand elastkities was computed from the additive preferences model (2) by applying an iterative version of Aitken’s generalized least- squares estimation methods. Each set consists of nine commodity groups representing all officially published items of national and reg- ional consumption expenditure. I n strict accordance with the concept of additive preferences, all nine commodities have been assumed to be independent in that the marginal rate of substitution of any pair of commodities is unrelated to the consumption levels of all other com- modi ties.

The estimated cost of living indexes for Australia and the six States, as derived from an additive preferences model of linear ex- penditure equations (1) and (9), are given in Table I. The correlation among the national and regional cost of living indexes may be estab- lished by using a zero-order correlation matrix. But the simple correla- tion among the cost of living indexes may be misleading in the sense that it registers only relative annual movements of the statistical series. This is apparent in the case of the series given in Table I where the

436 TEE ECONOMIC RECORD SEFT.

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33

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1969 AUSTRALIAN CONSUMER ’S WELFARE 437

derived simple correlation among the national and regional in- dexes appears, for all practical purposes, statistically perfect in each case. However, the average annual increase in the cost of living index for Australia and the six States for this period ranges from 6.0994 per cent (S.A.) to 7.0322 per cent (Tas.) as compared with 6.7971 per cent (S.A.) and 7.5062 per cent (Tas.) in the consumer price index.

While there is a lack of statistical evidence to support the conten- tion that the income elasticity of the marginal utility of money I: in- creases with a rise in real income,4 a monotonically decreasing relation- ship between 2: and the rate of incerase in the additive preferences cost of living index among the six States, as observed in Tables I1 and 111, is striking. This is tantamount t o saying that, in our exercise, Frisch’s money flexibility w records not only the consumer’s level of welfare but also links it with the movements of the ‘true’ cost of living index.

Table I1 reproduces, for comparison, various welfare indicators frequently referred to in Australia. These include the official consumer price index as published by the Commonwealth Statistician, personal disposable income per head of population, per capita personal con- sumption expenditure, the weekly basic wage rate set by the Common- wealth Conciliation and Arbitration Commission, average weekly earn- ings per employed unit, and finally our estimated cost of living index as derived from (9). Whilst we admit that the derived simple correla- tion between the cost of living index and the consumer price index fo r Australia and the six States appears almost perfect in all cases, the average annual increase in the cost of living index is consistently about 0.4913 per cent below the average annual increase in the con- sumer price index. This is not surprising since the consumer price index is a Laspeyres index using reference year weights and therefore having an upward bias. It suggests that increments in the weekly basic wage based solely on the movements of the consumer price index are generally overstated, if not misleading.

The movement of real income, compared with income and optimal satisfaction acquired in the reference period, is obtained in the con- ventional way by deflating nominal income in the comparison period by the ‘true’ cost of living index f o r the same period. The rate of change in real income is thus the percentage rate of change in nominal income less the percentage rate of change in consumer commodity prices. If the cost of living is overestimated, increases in real income will obviously be underestimated.

The average annual increases in real personal disposable income from 1948-49 to 196465, with respect to the official consumer price index and to the additive preferences cost of living aa formulated in (9), are given in Table 111. The general picture which emerges is that the use of the consumer price index as an approximation to the ‘true’

4 See for example [ 121.

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1969 AUSTRALIAN CONSUMER 'S WELFARE 439

cost of living leads consistently to the underestimation of the average annual increases in real personal disposable income. The discrepancy between the average annual increases in real personal disposable in- come using, as deflators, the consumer price index and the cost of living varies respectively between 0.0395 per cent (W.A.) and 0.1792 per cent (S.A.) for the six States. For the whole economy, the differ- ence is about 0.1214 per cent a year. This means that, at an annual groffth rate of real personal disposable income of say 1.7120 per cent, the downward bias of the annual increase in real income resulting from deflating by the consumer price index is about 7.0911 per cent.:

From the average annual rates of change of the various indicators given in Table IT, it would appear that the aggregate weekly basic wage has been rising some 2.2670 per cent a year faster than the consbimer price index or about 2.7728 per cent a year faster than the cost of living index. However, the real annual increase in the weekly basic wage with respect to the consumer price index and the cost of living ranges respectively from 1.2610 per cent (Qld.) to 1.4013 per cent (S.A.) and from 1.3284 per cent (Vic.) to 1.5618 per cent (S.A.). Thus using the consumer mice isdex and the cost of living as deflators, the average annual rate of increase in real national personal disposable income is approximately 0.2735 per cent and 0.2943 per cent greater than the average annual rate of increase in the national weekly basic wage set by the Commonwealth Conciliation and Arbitration Com- mission. Significantly, the discrepancy between the rates of increase in averagG weekly earnings and in personal consumption expenditure due to employing the, consumer price index as deflator is 0.1628 per cent a year; yet the discrepancy is negligible when the cost of living is used as the deflator.

v I n this paper, we have attempted to exploit the concept of additive

preferences within the framework of a Stone-Geary utility function to $7 For the post-Korean war period (1952-53 to 1964-65), we obtained the

following average annual percentage rates of change in the consumer price index, the cost of living, and the personal disposable income per head of population:

Indicator Aust. N.S.W. Vie. Q'ld. S A. W.A. Tas.

Consumer price index 2.5333 2.4oOo 2.6833 2.8250 2.3833 2.3000 2.8000

Personal disposable Cost of living 2.5971 2.6088 2.7229 2.7410 2.2570 2,4026 2.7102

income 5.8475 6,3050 5.6191 5.7783 4.8666 4.6883 5.4325

Soitrcc : See Table I and notes to Table 11.

Thus the discrepancy between the average annual rates of increase in real Ixrsonal disposable income with respect to the consumer price index and the cost of living is 0,0567 per cent for the whole economy, and ranges from -0.1143 per cent (S.A.) to 0.2102 per cent (N.S.W.) for the six States. The data should he interpreted with caution, since here we have assumed that the cstinzuafcd consumer demand structure i e identical for both sample sizes : 1945-49 to 1964-65 and 1953-53 t o 1964-65.

440 THE ECONOMIC RECORD SEPT., 1969 derive a statistical measure of real income or the ‘true’ cost of living index f o r Australia. We have also examined the relationship between the ‘true’ cost of living index and various indicators frequently used to gauge the level of community welfare. Unlike fixed weight index numbers, our approach takes account simultaneously of such things as average budget proportions, subsistence budget shares, marginal value shares, changing tastes and habits, and the substitutability among all individual commodities with respect to commodity price changes. This approach would seem to be preferable to other summary methods where only simple averages or aggregates of the relative commodity price changes are used.

M o w h University TRAN V m HOA

REFERENCES Allen, R. G. D., Mathematical Analysis for Economists (Macmillan, London, 1938). Frisch, R., ‘A Complete Scheme for Computing All Direct and Cross Demand Elasticities in a Model with Many Sectors’, Econometrica, Vol. 27, April 1959, pp. 177-96. Goldberger, A. S., ‘Functional Form and Utility: A Review of Consumer Demand Theory’, Social Systems Research Institute Workshop Paper 6703 (University of Wisconsin, mimeographed, 1%7). Hancock, K. J., ‘The Basic Wage and the Cost of Living’, Australian Econ- omic Papers, Vol. 1, S5ptember 1962, pp. 42-56. Houthakker, H. S., Additive Preference’, Econometrica, Vol. 28, April

Klein, L. R. and Rubin, H., ‘A Constant-Utility Index of the Cost of Living’, Review of Economic Studies, Vol. XV, No. 2, 1948-49, pp. 84-7. Leser, C. E. V., ‘Demand Functions for Nine Commodity Groups in Aus- tralia’, Australian journal of Statistics, Vol. 2, November 1960, pp. 102-13. Pitchford, J. D., An Analysis of Price Movements in Australia 1947-68’ (mimeographed, 1968). Powell, A. A., ‘A Complete System of Consumer Demand Equations for the Australian Economy Fitted by a Model of Additive Preferences’, Econo- nzetrica, Vol. 34, July 1966, pp. 661-75.

, Tran Van Hoa and Wilson, R. H., ‘A Multi-Sectoral Analysis of Consumer Demand in the Post-War Period’, Southern Economic Journal, Vol. XXXV, October 1968, pp. 109-20. Samuelson, P. A., Foundations of Economic Analysis (Harvard University Press, Cambridge, Mass., 1947), especially Chapters V and VIE. Tran Van Hoa, ‘Inter-regional Elasticities and Aggregation Bias : A Study of Consumer Demand in Australia’, Australian Economic Papers, Vol. 7, December 1968, pp. 206-26.

1960, pp. 244-57.