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An empirical test of homogeneity and symmetry in a demand system with taste changes
Structural Change and Economic
PRACTITIONERS’
Dynamics, vol. 3, no. 1, 1992
CORNER:
AN EMPIRICAL TEST OF HOMOGENEITY AND SYMMETRY IN A DEMAND SYSTEM WITH TASTE CHANGES ATSUSHI
MAKIlY2
The objectives of this paper are twofold: to examine the law of demand and to examine the possibility of changing demand empirically in the era of high economic growth of the Japanese economy. The homogeneity and symmetry conditions of the Slutsky equation are important in testing the correspondence between demand functions and utility functions. These restrictions are tested in this paper. It is found that homogeneity, symmetry, and the joint restrictions of homogeneity and symmetry, are empirically supported by a consistent data set based on the National Accounts when taste changes are introduced. In Japan, the period of observation includes the era of high economic growth during the 1960s and is thus an especially appropriate period for observing a dynamic shift, specified as a time trend, in consumption patterns.
1. INTRODUCTION
The homogeneity and symmetry conditions of the Slutsky equation are known as the ‘Law of Demand’ and are important in testing the correspondence between demand functions and utility functions. The law of demand is the fundamental principle connecting consumer demand behaviour and the utility maximization principle. It is also the foundation of microeconomic theory and the basis for empirical analysis. Determining whether or not the estimated parameters of the demand functions statistically satisfy the law of demand tests the utility maximizing behaviour of consumers. Tests of the law of demand have been conducted by using models such as the Rotterdam model (Barten 1967, 1969; Deaton, 1974) the transcendental logarithmic utility function model (Christensen et al., 1975), and the almost ideal demand system (AIDS) (Deaton and Muellbauer, 1980b). In a series of analyses estimating the parameters of demand functions, the maintained hypothesis that the law of demand holds was almost always rejected ’ Faculty of Business and Commerce, Keio University, Japan. 2 I am grateful to Russel Cooper, Richard Cornes, Charles Horioka, Mike McAleer, David Ryan, Paul Sheard, seminar participants at the Australian National University and the University of Western Sydney, and two anonymous referees for helpful comments and discussions. An earlier version of this paper was written during my stay at Osaka University and the Australian National University. The research was funded by the Nomura Research Promotion Foundation. (Q Oxford University Press 1992
167
168 A.
MAKI
(1977) summarizes the results of 10 separate findings by the data. Barten conducted before 1977. In summary, the homogeneity, symmetry, and negativity conditions are not rejected (see Barten, 1967). In a subsequent paper, Barten (1969) found that both the homogeneity and symmetry conditions are rejected. In the two analyses based on the Rotterdam model, Barten used Dutch data from 1922 to 1961 excluding the years 1940-1948. The difference between Barten’s two analyses rests in the classification of total consumption expenditure: in the first paper it is classified into four categories, while in the second paper it is classified into sixteen. Deaton (1974) analysed UK data classified into nine categories from 1900 to 1970 using the Rotterdam model, and obtained results rejecting the homogeneity condition, but not rejecting the symmetry and negativity conditions. Christensen et al. (1975) examined the theoretical restrictions using US data classified into three categories from 1929 to 1972, specifying a transcendental logarithmic utility function, and found that the symmetry condition is rejected. After rejecting the symmetry condition, they tested further restrictions such as additivity and homogeneity upon imposing the law of demand. Following Barten’s survey (1977), Deaton and Muellbauer (1980b) conducted an analysis using British data from 1954 to 1974, classified into eight categories using the AIDS model. Their findings rejected the homogeneity, symmetry and negativity conditions. Suruga (1980) estimated the Rotterdam model using annual data of Japanese households classified into five categories from 1953 to 1971, and obtained results but not rejecting the conditions of homogerejecting the negativity condition, neity, joint homogeneity and symmetry, and symmetry under the assumption of homogeneity. His findings also rejected the structural changes of the consumption Hashimoto (1986) analysed the Japanese households data demand system. classified into five categories from 1963 to 1983 using the Rotterdam model, and obtained results rejecting the symmetry condition under the assumption of homogeneity. Why is the law of demand confounded by empirical analysis? First, there is an approximation error in the functional form of the estimated demand equations. In the Rotterdam model, a discrete approximation has been made in place of the differential equations of the true relations, whereas in the transcendental logarithmic utility function, the function has been approximated by a Taylor series which uses only the first and second derivatives of the true utility function. Second, there is an aggregation problem. In theory, the consumer unit is an individual. However, the data used for estimation are based on aggregate data from the Family Income and Expenditure Survey or the National Income Statistics. Thus, the estimates of individual consumption behaviour do not correspond strictly to theoretical assumptions. Third, static theory does not capture the dynamics of consumer behaviour. The utility function may shift over time and structural changes may alter consumption demand behaviour, for example, due to the effects of habit formation and However, when the estimated demand functions household characteristics. include only total consumption expenditure and prices as systematic variables, the
HOMOGENEITY
AND
SYMMETRY
IN
A DEMAND
SYSTEM
169
estimated parameters of demand functions will be biased, leading to negative findings concerning the law of demand. This paper focuses on the third factor. The objectives of this paper are twofold: to examine the law of demand and to examine the empirical effects of changing demand during the high economic growth era of the Japanese economy.3 Structural changes in consumption demand behaviour are considered empirically, in addition to changes in total consumption expenditure and prices using Japanese data. The Rotterdam model is estimated and statistical tests of various restrictions are presented.4 This paper reports that there is a data set and a model which can satisfy the law of demand. The plan of the paper is as follows. Section 2 explains the theoretical model, while Section 3 describes the data used in the estimation and reports the empirical results. Finally, Section 4 presents some concluding remarks.
2. THEORETICAL
MODEL
A demand function is a relation between quantities demanded and total consumption expenditure, prices, and other factors. Define qi, pi and wi(=piq,/x) as the real expenditure, the price index and the average budget share of the ith category, respectively, and x (= cpiqi) denotes total consumption expenditure. The Rotterdam model appears to be one of the simplest of the demand functions and is also easy to estimate. The fundamental equation of the Rotterdam model is: wi d log qi = bj d logx*
where
+ 2~~~ d logp,,
i=1,2,...,n
(1)
d log x* in (1) is dlogx*
=dlogx-Cw,dlogpk=Xwkdlogqk.
When a discrete approximation is used for (l), to be constant, the empirical Rotterdam model
wl; A log qir = ai + biIZw:t A
log
qkt + XCik
(2)
and the bi’s and Q’S are assumed is given as follows: A
lOgpk,
i=l,2,...,n
(3)
where
A log qit = log qir - log qic-1 A log Pit
=
log Pit
Wz = (Wit +
-
log pit- 1
Wi[_l)/2*
(5) (6)
3 In this paper we are specifically interested in identifying a consistent data set which satisfies the law of demand and whether or not the structural changes in the consumer demand system is verified by the data. In Section 4 of the present paper, we consider possible causes of structural changes in consumption demand. 4 One of the objectives of this paper is to test the law of demand empirically; we estimated the Rotterdam model because of the simplicity of estimation. A comparison with alternative functional forms is beyond the scope of the present analysis.
170
A.
MAKI
Equation (3) includes the dynamic coefficients as indicated in the intercept term ai. If the ai are all zero, this indicates a static demand system. However, when one of the ai is not zero, the dynamics are introduced through the time trend. The a priori restrictions on the parameters are: Zak=O
(7)
Xbk
=1
CC,
=
(8) j = 1, 2, . . . , n.
0
The three restrictions in (7)-(9) indicate the adding-up condition functions as homogeneous functions of degree zero for consumer respect to total consumption expenditure and prices. The conditions neity, symmetry, and negativity are, respectively: XCj,
=
cij =
Cji
0
j = 1,2,
...,n
(9) and demand demand with for homoge-
(10) (11)
C = {cij} is negative
semi-definite
(12)
(see, e.g. Deaton and Muellbauer, 1980a). The likelihood ratio test is used to test the theoretical restrictions. When a parameter space is described by Q a subset of Q is described by S2,, and the maximized likelihood values for the spaces of Q and S2, are described by L(8) and L(Q), respectively, then the likelihood ratio is given by:
-2 log p, converges to the asymptotic chi-square The likelihood ratio statistic, distribution with r degrees of freedom, where r is the number of parametric restrictions.
3. THE
DATA
AND
EMPIRICAL
RESULTS
The data used for estimation are the seasonally adjusted quarterly series of the National Income Statistics from the second quarter of 1965 through to the first quarter of 1984. The total number of observations is 76. Total consumption expenditure is divided into five categories: Food, Clothing, Fuel and Electricity, Housing, and Miscellaneous, as compiled by the Economic Planning Agency.’ The first half of the period, 1965-1973, is called the era of high economic growth because real GNP growth averaged 9.8%. The latter half of the period, after the oil price shock of 1973, is called the era of the stable economy because economic growth averaged 3.8%, which is less than 50% of the earlier period. Japan experienced two different stages of economic growth before and after the oil price shock. 5 Though Suruga (1980) and Hashimoto (1986) used annual data, seasonally adjusted data is used in the present analysis in order to increase the number of observations. Applying the likelihood ratio test to hypothesis testings in a small sample set, Anderson’s (1958) small sample correction would be necessary.
HOMOGENEITY
AND
SYMMETRY
IN
A DEMAND
SYSTEM
171
The maximum likelihood method was used for the demand estimation. Since the covariance matrix of the error structure is singular (see, e.g. Barten, 1969), an arbitrary demand equation was omitted for purposes of estimation. The estimation results of the Rotterdam model with taste changes are reported in Table 1. Results are presented for: (i) no restrictions, (ii) subject to the homogeneity restriction, and (iii) subject to the homogeneity and symmetry restrictions jointly. Taste changes are reflected in the ai coefficients. TABLE 1. The Rotterdam Unrestricted
a1 a2
a3
a4
as bl b2
b3
b4
b5
Cl1
Cl2
Cl3
Cl4
Cl5
c21
c22
c23
c24
C2s
e31
iMode With Taste Changes Restricted
0i
Homogeneity (ii)
-0.00103 (1.18) -0.00113 (2.01) 0.000417 (2.23) 0.00242 (2.92) -0.000677 (0.654) 0.308 (10.1) 0.169 (8.69) 0.00478 (0.727) 0.186 (6.46) 0.332 (9.26) -0.0859 (2.30) 0.000998 (0.0332) 0.0136 (1.28) 0.0403 (0.698) 0.0547 (1.58) 0.0444 (1.86) -0.0682 (3.54) -0.00734 (1.08) -0.00011 (0.0029) 0.0203 (0.915) 0.0129 (1.62)
-0.000617 (1.15) -0.00132 (3.84) 0.000212 (1.83) 0.000792 (1.50) 0.000933 (1.44) 0.302 (10.5) 0.172 (9.33) 0.00771 (1.25) 0.209 (7.39) 0.309 (8.88) -0.0845 (2.26) 0.000493 (0.0164) 0.0126 (1.20) 0.0160 (0.386) 0.0.554 (1.59) 0.0438 (1.83) -0.0680 (3.53) -0.00690 (1.03) 0.0111 (0.418) 0.0200 (0.900) 0.0122 (1.52)
models Homogeneity and symmetry (iii) -0.000327 (0.626) -0.00118 (3.56) 0.000217 (1.97) 0.000865 (2.23) 0.000425 (0.703) 0.288 (10.2) 0.164 (9.28) 0.00790 (1.94) 0.202 (7.40) 0.338 (10.5) -0.105 (2.97) 0.0205 (1.09) 0.0127 (2.07) 0.0472 (l-64) 0.0246 (0.883) 0.0205 (1.09) -0.0661 (3.70) 0.000963 (0.216) 0.0229 (1.18) 0.0221 (1.17) 0.0127 (2.07)
172
A.
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TABLE l-continued Unrestricted
Restricted
models
0i
Homogeneity (ii)
Homogeneity and symmetry (iii)
d-w(l) d-w(2) d-w(3) d-w(4) d-w(5)
0.00593 (0.925) -0.0103 (4.55) -0.0319 (2.59) 0.011 (1.58) 0.0776 (2.19) 0.0137 (0.481) -0.0168 (1.68) -0.183 (3.35) 0.0160 (0.487) -0.0490 (1.11) 0.0476 (1.34) 0.0208 (1.67) 0.175 (2.57) -0.102 (2.50) 2.13 2.43 2.11 2.10 2.31
0.00618 (0.952) -0.0098 (4.34) -0.0200 (2.24) 0.0114 (1.51) 0.0724 (1.97) 0.0157 (0.529) -0.0131 (1.27) -0.0885 (2.17) 0.0135 (0.374) -0.0439 (0.973) 0.0456 (1.25) 0.0172 (1.35) 0.0814 (0.354) -0.1004 (2.38) 2.11 2.43 2.10 2.00 2.28
0.000963 (0.216) - 0.00890 (4.12) -0.0147 (2.23) 0.0098 (1.55) 0.0472 (1.64) 0.0229 (1.18) -0.0147 (2.23) - 0.0902 (2.25) 0.0350 (1.15) 0.0246 (0.883) 0.0221 (1.17) 0.0098 (1.55) 0.0350 (1.15) -0.0915 (2.28) 2.03 2.47 2.11 2.06 2.35
log L
1500.22
1495.60
C32
c33
C34
c35
C41
c42
C43
%I C4s
(31 G2
G.3
G4
C5s
1492.33
The figures in parentheses denote asymptotic t-ratios. denotes the maximized log-likelihood value. The d-w(i) Durbin-Watson ratio of the ith category.
log L is the
Let us examine the case including taste changes in Table 1. In the no restriction case (i),the ai coefficients of Clothing, Fuel and Electricity, and Housing are significant at the 5% level. The coefficients of income bi are all significant except for the Fuel and Electricity category. Turning to the Cii)s corresponding to the the diagonal elements Cii’S are all significant, satisfy the price coefficients, theoretical sign restrictions and, therefore, the negativity condition. There is also obvious evidence of no autocorrelation in the residuals. These findings are unchanged when the homogeneity (case (ii) in Table l), and the joint homogeneity and symmetry conditions (case (iii)) are imposed on the
HOMOGENEITY TABLE
AND
2. Test
of
Taste
SYMMETRY
IN
Change Variables Restrictions
Degrees of freedom
Model No restriction Homogeneity Homogeneity symmetry The critical
A DEMAND
Likelihood
4 4 4
and values
are x (4,0.05)
Under
SYSTEM
173
Different ratio test
18.88 22.40 19.96 = 9.49 and x(4,0.01)
= 13.28.
taste change Rotterdam models. The magnitudes of the parameter estimates in the three cases do not indicate drastic changes and are also stable. Therefore, in all of the three cases, the negativity condition is satisfied. The theoretical restrictions are then tested using the likelihood ratio test. Table 2 presents the results of testing the taste change variables in the demand functions for the same three cases. All three results indicate that the introduction of the taste change variable is necessary for the specification of the model? Next we examine whether or not the law of demand is supported in the taste change model, with the results presented in Table 3. Here the number of degrees of freedom is equivalent to the number of parametric restrictions. The results in Table 3 are different from those of previous studies. In Barten (1969) and Deaton (1974), as described in Section 1, the homogeneity restriction was rejected, but this restriction is not rejected in the present analysis. The restriction of symmetry under the assumption of homogeneity and the joint restriction of homogeneity and symmetry are not rejected at the 5%, indicating that the law of demand is not rejected in the present analysis. If the taste change variables are omitted from the analysis, the homogeneity restriction is rejected. Finally, we reconsider the cases including taste changes shown in Tables 1 and 3. The observation period is divided into two periods based on two distinctive growth rates: the periods before and after the oil price shock. The results of the two demand functions are estimated and presented in Table 4. The law of demand is satisfied in the era of high economic growth, but is rejected when economic growth declined. This finding may indicate one reason for differences between the present and previous studies. In Barten (1969) Deaton (1974) and Christensen et al. (1975), TABLE
3. Tests of the Theoretical
Restrictions
Degrees of freedom
Homogeneity Symmetry under homogeneity Homogeneity and symmetry The critical 6 All the results
in Table
values
2 indicate
Likelihood test
Restrictions ratio
Decision (5 % level >
4 6
9.24 6.54
not rejected not rejected
10
15.8
not rejected
are x(6,0.05)
= 12.59 and x(10,0.05)
that the tests of the hypotheses
= 18.31.
are significant
at the 1% level.
174
A.
MAKI TABLE
4.
Tests of the Theoretical Restrictions for Different Sample Periods
Restrictions
Degrees of freedom
19653~1973:3 Homogeneity symmetry under homogeneity homogeneity and symmetry 197314-198411 Homogeneity Symmetry under homogeneity Homogeneity and symmetry
Likelihood ratio test
Decision level)
(5%
4 6
4.34 4.49
not rejected not rejected
10
8.83
not rejected
4 6
10.4 9.71
rejected not rejected
10
20.1
rejected
Dutch, UK and US data, respectively, were used. The growth rates of these countries have been lower than that in Japan and the possible effects of a changing structure over time were not considered. For the early part of the high growth period, many new products and improved products, such as electrical products, automobiles and services, were constantly being offered to consumers through the advertising of new products and retail services from the suppliers. Steady changes in consumer demand for the five surveyed categories are described by the intercepts of the dynamic version of the Rotterdam model. However, during the era of stable economic growth, it is difficult to pick up the influences of the time trend factor. Therefore, a more complex specification and careful selection of taste change variables is needed to pick up the structural changes in consumption demand since there is a substantial difference in the law of demand depending on levels of economic growth. 4. CONCLUDING
REMARKS
In this paper we tested the law of demand using consumption data for Japan. Total consumption was divided into five categories with 76 observations from the second quarter of 1965 to the first quarter of 1981. We succeeded in identifying a consistent data set in which the introduction of the taste change variable is warranted, and homogeneity, symmetry, and the joint restriction of homogeneity and symmetry are satisfied when taste changes are introduced. The likelihood ratio test was used to test the law of demand. Three restrictions, namely homogeneity, symmetry, and the joint restriction of homogeneity and symmetry, are satisfied when taste changes are considered. This result strongly validates the dynamics of the consumption function specification. In Japan, the period of observation includes the era of high economic growth during the 196Os, which is an appropriate period to observe dynamic shifts,
HOMOGENEITY
AND
SYMMETRY
IN A DEMAND
SYSTEM
175
patterns In Eu .ropean countries and specified as the time trend, in consumption growth was slower t:han in Japan, and thus it is the US, the pace of economic patterns by the trend changes in consumption difficu It to capt ure the structural factor from . the ava ilable data. Similar findings supporting the law of demand during periods of high economic approach. growth were o btained by Varian (1982) using the non-parametric In addition to total cons umpWhat, then, is the nature of changing demand? tion expenditure and prices as major factors in determining consumption demand, economists have considered other factors in changing demand patterns. For example, Veblen (1924) proposed the hypothesis of conspicuous consumption to (1949) introduced intra- and explain US consumption behaviour . Duesenberry inter-society habit formation (i.e. the relative income hypothesis) as an autonomous shift of the utility function, and Houthakker and Taylor (1970) estimated demand functions based on the state adjustment hypothesis. Prais and Houthakker (1955) and Pollak and Wales (1978) stressed the family characteristics effect which directly affects the demand patterns of consumers.7 Our analysis suggests that it is probable that the law of demand is supported by empirical analysis if the effect of structural changes due to factors such as variations in family attributes and habit formation is incorporated. Here, there is need for further research and analysis.
REFERENCES ANDERSON, G. and BLUNDELL, R. (1983). ‘Consumer
non-durables in the U.K.: A dynamic demand Economic Journal (Supplement), 94, 35-44. ANDERSON, T. W. (1958). An Introduction to Multivariate Statistical Analysis. John Wiley, New York. BARTEN, A. (1967). ‘Evidence on the Slutsky conditions for demand equations’, Review of Economics and Statistics, 49, 77-84. (1969). ‘Maximum likelihood estimation of a complete system of demand equations’, European Economic Review, 1, 7-73. (1977). ‘The system of consumer demand functions approach: A review’, Econometrica, 45, 23-51. BLANCIFORTI,L. and GREEN, R. (1983). ‘An almost ideal demand system incorporating habits: An analysis of expenditures on food and aggregate commodity groups’, Review of Economics and Statistics, 65, 511-5. CHRISTENSEN, L., JORGENSON, D. W. and LAU, L. (1975). ‘Transcendental logarithmic utility functions’, American Economic Review, 65, 367-83. DEATON, A. (1974). ‘The Analysis of Consumer Demand in the United Kingdom’, 1900-1970’, Econometrica, 42, 341-67. and MUELLBAUER, J. (1980a). Economics and Consumer Behavior. Cambridge University Press, Cambridge, UK. and (1980b). ‘An almost ideal demand system’, American Economic Review, 70, 312-26. DUESENBERRY,J. (1949). Income, Saving and the Theory of Consumer Behavior. Harvard University Press, Cambridge, MA. HASHIMOTO, N. (1986). ‘Tests of demand theory by the Rotterdam model’ (in Japanese), Economic Studies Quarterly, 37, 271-5. HOUTHAKKER, H. S. and TAYLOR, L. D. (1970). Consumer Demand in the U.S.: Analyses and Projections. Harvard University Press, Cambridge, MA. system’,
7 Anderson and Blundell (1983) and Blanciforti and Green (1983) partly succeeded in the dynamic specification of the consumer demand system. However, both were unable to verify the law of demand.
A.
MAKI
POLLAK, R. A. and WALES, T. J. (1978). ‘Estimation of complete demand system from household budget data: The linear and quadratic expenditure system’, American Economic Review, 68, 348-59. PRAIS, S. J. and HOUTHAKKER, H. S. (1955). The Analysis of Family Budget. Cambridge University Press, Cambridge. SURUGA, T. (1980). ‘Testing the Rotterdam demand model on the Japanese expenditure pattern’, Economic Review, 31, 368-74. VARIAN, H. (1982). ‘The nonparametric approach to demand analysis’, Economefrica, 50, 945-73. VEBLEN, T. (1924). The Theory of the Leisure Class: An Economic Study of Institutions. George Allen, London.
PRACTITIONERS’
Dynamics, vol. 3, no. 1, 1992
CORNER:
AN EMPIRICAL TEST OF HOMOGENEITY AND SYMMETRY IN A DEMAND SYSTEM WITH TASTE CHANGES ATSUSHI
MAKIlY2
The objectives of this paper are twofold: to examine the law of demand and to examine the possibility of changing demand empirically in the era of high economic growth of the Japanese economy. The homogeneity and symmetry conditions of the Slutsky equation are important in testing the correspondence between demand functions and utility functions. These restrictions are tested in this paper. It is found that homogeneity, symmetry, and the joint restrictions of homogeneity and symmetry, are empirically supported by a consistent data set based on the National Accounts when taste changes are introduced. In Japan, the period of observation includes the era of high economic growth during the 1960s and is thus an especially appropriate period for observing a dynamic shift, specified as a time trend, in consumption patterns.
1. INTRODUCTION
The homogeneity and symmetry conditions of the Slutsky equation are known as the ‘Law of Demand’ and are important in testing the correspondence between demand functions and utility functions. The law of demand is the fundamental principle connecting consumer demand behaviour and the utility maximization principle. It is also the foundation of microeconomic theory and the basis for empirical analysis. Determining whether or not the estimated parameters of the demand functions statistically satisfy the law of demand tests the utility maximizing behaviour of consumers. Tests of the law of demand have been conducted by using models such as the Rotterdam model (Barten 1967, 1969; Deaton, 1974) the transcendental logarithmic utility function model (Christensen et al., 1975), and the almost ideal demand system (AIDS) (Deaton and Muellbauer, 1980b). In a series of analyses estimating the parameters of demand functions, the maintained hypothesis that the law of demand holds was almost always rejected ’ Faculty of Business and Commerce, Keio University, Japan. 2 I am grateful to Russel Cooper, Richard Cornes, Charles Horioka, Mike McAleer, David Ryan, Paul Sheard, seminar participants at the Australian National University and the University of Western Sydney, and two anonymous referees for helpful comments and discussions. An earlier version of this paper was written during my stay at Osaka University and the Australian National University. The research was funded by the Nomura Research Promotion Foundation. (Q Oxford University Press 1992
167
168 A.
MAKI
(1977) summarizes the results of 10 separate findings by the data. Barten conducted before 1977. In summary, the homogeneity, symmetry, and negativity conditions are not rejected (see Barten, 1967). In a subsequent paper, Barten (1969) found that both the homogeneity and symmetry conditions are rejected. In the two analyses based on the Rotterdam model, Barten used Dutch data from 1922 to 1961 excluding the years 1940-1948. The difference between Barten’s two analyses rests in the classification of total consumption expenditure: in the first paper it is classified into four categories, while in the second paper it is classified into sixteen. Deaton (1974) analysed UK data classified into nine categories from 1900 to 1970 using the Rotterdam model, and obtained results rejecting the homogeneity condition, but not rejecting the symmetry and negativity conditions. Christensen et al. (1975) examined the theoretical restrictions using US data classified into three categories from 1929 to 1972, specifying a transcendental logarithmic utility function, and found that the symmetry condition is rejected. After rejecting the symmetry condition, they tested further restrictions such as additivity and homogeneity upon imposing the law of demand. Following Barten’s survey (1977), Deaton and Muellbauer (1980b) conducted an analysis using British data from 1954 to 1974, classified into eight categories using the AIDS model. Their findings rejected the homogeneity, symmetry and negativity conditions. Suruga (1980) estimated the Rotterdam model using annual data of Japanese households classified into five categories from 1953 to 1971, and obtained results but not rejecting the conditions of homogerejecting the negativity condition, neity, joint homogeneity and symmetry, and symmetry under the assumption of homogeneity. His findings also rejected the structural changes of the consumption Hashimoto (1986) analysed the Japanese households data demand system. classified into five categories from 1963 to 1983 using the Rotterdam model, and obtained results rejecting the symmetry condition under the assumption of homogeneity. Why is the law of demand confounded by empirical analysis? First, there is an approximation error in the functional form of the estimated demand equations. In the Rotterdam model, a discrete approximation has been made in place of the differential equations of the true relations, whereas in the transcendental logarithmic utility function, the function has been approximated by a Taylor series which uses only the first and second derivatives of the true utility function. Second, there is an aggregation problem. In theory, the consumer unit is an individual. However, the data used for estimation are based on aggregate data from the Family Income and Expenditure Survey or the National Income Statistics. Thus, the estimates of individual consumption behaviour do not correspond strictly to theoretical assumptions. Third, static theory does not capture the dynamics of consumer behaviour. The utility function may shift over time and structural changes may alter consumption demand behaviour, for example, due to the effects of habit formation and However, when the estimated demand functions household characteristics. include only total consumption expenditure and prices as systematic variables, the
HOMOGENEITY
AND
SYMMETRY
IN
A DEMAND
SYSTEM
169
estimated parameters of demand functions will be biased, leading to negative findings concerning the law of demand. This paper focuses on the third factor. The objectives of this paper are twofold: to examine the law of demand and to examine the empirical effects of changing demand during the high economic growth era of the Japanese economy.3 Structural changes in consumption demand behaviour are considered empirically, in addition to changes in total consumption expenditure and prices using Japanese data. The Rotterdam model is estimated and statistical tests of various restrictions are presented.4 This paper reports that there is a data set and a model which can satisfy the law of demand. The plan of the paper is as follows. Section 2 explains the theoretical model, while Section 3 describes the data used in the estimation and reports the empirical results. Finally, Section 4 presents some concluding remarks.
2. THEORETICAL
MODEL
A demand function is a relation between quantities demanded and total consumption expenditure, prices, and other factors. Define qi, pi and wi(=piq,/x) as the real expenditure, the price index and the average budget share of the ith category, respectively, and x (= cpiqi) denotes total consumption expenditure. The Rotterdam model appears to be one of the simplest of the demand functions and is also easy to estimate. The fundamental equation of the Rotterdam model is: wi d log qi = bj d logx*
where
+ 2~~~ d logp,,
i=1,2,...,n
(1)
d log x* in (1) is dlogx*
=dlogx-Cw,dlogpk=Xwkdlogqk.
When a discrete approximation is used for (l), to be constant, the empirical Rotterdam model
wl; A log qir = ai + biIZw:t A
log
qkt + XCik
(2)
and the bi’s and Q’S are assumed is given as follows: A
lOgpk,
i=l,2,...,n
(3)
where
A log qit = log qir - log qic-1 A log Pit
=
log Pit
Wz = (Wit +
-
log pit- 1
Wi[_l)/2*
(5) (6)
3 In this paper we are specifically interested in identifying a consistent data set which satisfies the law of demand and whether or not the structural changes in the consumer demand system is verified by the data. In Section 4 of the present paper, we consider possible causes of structural changes in consumption demand. 4 One of the objectives of this paper is to test the law of demand empirically; we estimated the Rotterdam model because of the simplicity of estimation. A comparison with alternative functional forms is beyond the scope of the present analysis.
170
A.
MAKI
Equation (3) includes the dynamic coefficients as indicated in the intercept term ai. If the ai are all zero, this indicates a static demand system. However, when one of the ai is not zero, the dynamics are introduced through the time trend. The a priori restrictions on the parameters are: Zak=O
(7)
Xbk
=1
CC,
=
(8) j = 1, 2, . . . , n.
0
The three restrictions in (7)-(9) indicate the adding-up condition functions as homogeneous functions of degree zero for consumer respect to total consumption expenditure and prices. The conditions neity, symmetry, and negativity are, respectively: XCj,
=
cij =
Cji
0
j = 1,2,
...,n
(9) and demand demand with for homoge-
(10) (11)
C = {cij} is negative
semi-definite
(12)
(see, e.g. Deaton and Muellbauer, 1980a). The likelihood ratio test is used to test the theoretical restrictions. When a parameter space is described by Q a subset of Q is described by S2,, and the maximized likelihood values for the spaces of Q and S2, are described by L(8) and L(Q), respectively, then the likelihood ratio is given by:
-2 log p, converges to the asymptotic chi-square The likelihood ratio statistic, distribution with r degrees of freedom, where r is the number of parametric restrictions.
3. THE
DATA
AND
EMPIRICAL
RESULTS
The data used for estimation are the seasonally adjusted quarterly series of the National Income Statistics from the second quarter of 1965 through to the first quarter of 1984. The total number of observations is 76. Total consumption expenditure is divided into five categories: Food, Clothing, Fuel and Electricity, Housing, and Miscellaneous, as compiled by the Economic Planning Agency.’ The first half of the period, 1965-1973, is called the era of high economic growth because real GNP growth averaged 9.8%. The latter half of the period, after the oil price shock of 1973, is called the era of the stable economy because economic growth averaged 3.8%, which is less than 50% of the earlier period. Japan experienced two different stages of economic growth before and after the oil price shock. 5 Though Suruga (1980) and Hashimoto (1986) used annual data, seasonally adjusted data is used in the present analysis in order to increase the number of observations. Applying the likelihood ratio test to hypothesis testings in a small sample set, Anderson’s (1958) small sample correction would be necessary.
HOMOGENEITY
AND
SYMMETRY
IN
A DEMAND
SYSTEM
171
The maximum likelihood method was used for the demand estimation. Since the covariance matrix of the error structure is singular (see, e.g. Barten, 1969), an arbitrary demand equation was omitted for purposes of estimation. The estimation results of the Rotterdam model with taste changes are reported in Table 1. Results are presented for: (i) no restrictions, (ii) subject to the homogeneity restriction, and (iii) subject to the homogeneity and symmetry restrictions jointly. Taste changes are reflected in the ai coefficients. TABLE 1. The Rotterdam Unrestricted
a1 a2
a3
a4
as bl b2
b3
b4
b5
Cl1
Cl2
Cl3
Cl4
Cl5
c21
c22
c23
c24
C2s
e31
iMode With Taste Changes Restricted
0i
Homogeneity (ii)
-0.00103 (1.18) -0.00113 (2.01) 0.000417 (2.23) 0.00242 (2.92) -0.000677 (0.654) 0.308 (10.1) 0.169 (8.69) 0.00478 (0.727) 0.186 (6.46) 0.332 (9.26) -0.0859 (2.30) 0.000998 (0.0332) 0.0136 (1.28) 0.0403 (0.698) 0.0547 (1.58) 0.0444 (1.86) -0.0682 (3.54) -0.00734 (1.08) -0.00011 (0.0029) 0.0203 (0.915) 0.0129 (1.62)
-0.000617 (1.15) -0.00132 (3.84) 0.000212 (1.83) 0.000792 (1.50) 0.000933 (1.44) 0.302 (10.5) 0.172 (9.33) 0.00771 (1.25) 0.209 (7.39) 0.309 (8.88) -0.0845 (2.26) 0.000493 (0.0164) 0.0126 (1.20) 0.0160 (0.386) 0.0.554 (1.59) 0.0438 (1.83) -0.0680 (3.53) -0.00690 (1.03) 0.0111 (0.418) 0.0200 (0.900) 0.0122 (1.52)
models Homogeneity and symmetry (iii) -0.000327 (0.626) -0.00118 (3.56) 0.000217 (1.97) 0.000865 (2.23) 0.000425 (0.703) 0.288 (10.2) 0.164 (9.28) 0.00790 (1.94) 0.202 (7.40) 0.338 (10.5) -0.105 (2.97) 0.0205 (1.09) 0.0127 (2.07) 0.0472 (l-64) 0.0246 (0.883) 0.0205 (1.09) -0.0661 (3.70) 0.000963 (0.216) 0.0229 (1.18) 0.0221 (1.17) 0.0127 (2.07)
172
A.
MAKI
TABLE l-continued Unrestricted
Restricted
models
0i
Homogeneity (ii)
Homogeneity and symmetry (iii)
d-w(l) d-w(2) d-w(3) d-w(4) d-w(5)
0.00593 (0.925) -0.0103 (4.55) -0.0319 (2.59) 0.011 (1.58) 0.0776 (2.19) 0.0137 (0.481) -0.0168 (1.68) -0.183 (3.35) 0.0160 (0.487) -0.0490 (1.11) 0.0476 (1.34) 0.0208 (1.67) 0.175 (2.57) -0.102 (2.50) 2.13 2.43 2.11 2.10 2.31
0.00618 (0.952) -0.0098 (4.34) -0.0200 (2.24) 0.0114 (1.51) 0.0724 (1.97) 0.0157 (0.529) -0.0131 (1.27) -0.0885 (2.17) 0.0135 (0.374) -0.0439 (0.973) 0.0456 (1.25) 0.0172 (1.35) 0.0814 (0.354) -0.1004 (2.38) 2.11 2.43 2.10 2.00 2.28
0.000963 (0.216) - 0.00890 (4.12) -0.0147 (2.23) 0.0098 (1.55) 0.0472 (1.64) 0.0229 (1.18) -0.0147 (2.23) - 0.0902 (2.25) 0.0350 (1.15) 0.0246 (0.883) 0.0221 (1.17) 0.0098 (1.55) 0.0350 (1.15) -0.0915 (2.28) 2.03 2.47 2.11 2.06 2.35
log L
1500.22
1495.60
C32
c33
C34
c35
C41
c42
C43
%I C4s
(31 G2
G.3
G4
C5s
1492.33
The figures in parentheses denote asymptotic t-ratios. denotes the maximized log-likelihood value. The d-w(i) Durbin-Watson ratio of the ith category.
log L is the
Let us examine the case including taste changes in Table 1. In the no restriction case (i),the ai coefficients of Clothing, Fuel and Electricity, and Housing are significant at the 5% level. The coefficients of income bi are all significant except for the Fuel and Electricity category. Turning to the Cii)s corresponding to the the diagonal elements Cii’S are all significant, satisfy the price coefficients, theoretical sign restrictions and, therefore, the negativity condition. There is also obvious evidence of no autocorrelation in the residuals. These findings are unchanged when the homogeneity (case (ii) in Table l), and the joint homogeneity and symmetry conditions (case (iii)) are imposed on the
HOMOGENEITY TABLE
AND
2. Test
of
Taste
SYMMETRY
IN
Change Variables Restrictions
Degrees of freedom
Model No restriction Homogeneity Homogeneity symmetry The critical
A DEMAND
Likelihood
4 4 4
and values
are x (4,0.05)
Under
SYSTEM
173
Different ratio test
18.88 22.40 19.96 = 9.49 and x(4,0.01)
= 13.28.
taste change Rotterdam models. The magnitudes of the parameter estimates in the three cases do not indicate drastic changes and are also stable. Therefore, in all of the three cases, the negativity condition is satisfied. The theoretical restrictions are then tested using the likelihood ratio test. Table 2 presents the results of testing the taste change variables in the demand functions for the same three cases. All three results indicate that the introduction of the taste change variable is necessary for the specification of the model? Next we examine whether or not the law of demand is supported in the taste change model, with the results presented in Table 3. Here the number of degrees of freedom is equivalent to the number of parametric restrictions. The results in Table 3 are different from those of previous studies. In Barten (1969) and Deaton (1974), as described in Section 1, the homogeneity restriction was rejected, but this restriction is not rejected in the present analysis. The restriction of symmetry under the assumption of homogeneity and the joint restriction of homogeneity and symmetry are not rejected at the 5%, indicating that the law of demand is not rejected in the present analysis. If the taste change variables are omitted from the analysis, the homogeneity restriction is rejected. Finally, we reconsider the cases including taste changes shown in Tables 1 and 3. The observation period is divided into two periods based on two distinctive growth rates: the periods before and after the oil price shock. The results of the two demand functions are estimated and presented in Table 4. The law of demand is satisfied in the era of high economic growth, but is rejected when economic growth declined. This finding may indicate one reason for differences between the present and previous studies. In Barten (1969) Deaton (1974) and Christensen et al. (1975), TABLE
3. Tests of the Theoretical
Restrictions
Degrees of freedom
Homogeneity Symmetry under homogeneity Homogeneity and symmetry The critical 6 All the results
in Table
values
2 indicate
Likelihood test
Restrictions ratio
Decision (5 % level >
4 6
9.24 6.54
not rejected not rejected
10
15.8
not rejected
are x(6,0.05)
= 12.59 and x(10,0.05)
that the tests of the hypotheses
= 18.31.
are significant
at the 1% level.
174
A.
MAKI TABLE
4.
Tests of the Theoretical Restrictions for Different Sample Periods
Restrictions
Degrees of freedom
19653~1973:3 Homogeneity symmetry under homogeneity homogeneity and symmetry 197314-198411 Homogeneity Symmetry under homogeneity Homogeneity and symmetry
Likelihood ratio test
Decision level)
(5%
4 6
4.34 4.49
not rejected not rejected
10
8.83
not rejected
4 6
10.4 9.71
rejected not rejected
10
20.1
rejected
Dutch, UK and US data, respectively, were used. The growth rates of these countries have been lower than that in Japan and the possible effects of a changing structure over time were not considered. For the early part of the high growth period, many new products and improved products, such as electrical products, automobiles and services, were constantly being offered to consumers through the advertising of new products and retail services from the suppliers. Steady changes in consumer demand for the five surveyed categories are described by the intercepts of the dynamic version of the Rotterdam model. However, during the era of stable economic growth, it is difficult to pick up the influences of the time trend factor. Therefore, a more complex specification and careful selection of taste change variables is needed to pick up the structural changes in consumption demand since there is a substantial difference in the law of demand depending on levels of economic growth. 4. CONCLUDING
REMARKS
In this paper we tested the law of demand using consumption data for Japan. Total consumption was divided into five categories with 76 observations from the second quarter of 1965 to the first quarter of 1981. We succeeded in identifying a consistent data set in which the introduction of the taste change variable is warranted, and homogeneity, symmetry, and the joint restriction of homogeneity and symmetry are satisfied when taste changes are introduced. The likelihood ratio test was used to test the law of demand. Three restrictions, namely homogeneity, symmetry, and the joint restriction of homogeneity and symmetry, are satisfied when taste changes are considered. This result strongly validates the dynamics of the consumption function specification. In Japan, the period of observation includes the era of high economic growth during the 196Os, which is an appropriate period to observe dynamic shifts,
HOMOGENEITY
AND
SYMMETRY
IN A DEMAND
SYSTEM
175
patterns In Eu .ropean countries and specified as the time trend, in consumption growth was slower t:han in Japan, and thus it is the US, the pace of economic patterns by the trend changes in consumption difficu It to capt ure the structural factor from . the ava ilable data. Similar findings supporting the law of demand during periods of high economic approach. growth were o btained by Varian (1982) using the non-parametric In addition to total cons umpWhat, then, is the nature of changing demand? tion expenditure and prices as major factors in determining consumption demand, economists have considered other factors in changing demand patterns. For example, Veblen (1924) proposed the hypothesis of conspicuous consumption to (1949) introduced intra- and explain US consumption behaviour . Duesenberry inter-society habit formation (i.e. the relative income hypothesis) as an autonomous shift of the utility function, and Houthakker and Taylor (1970) estimated demand functions based on the state adjustment hypothesis. Prais and Houthakker (1955) and Pollak and Wales (1978) stressed the family characteristics effect which directly affects the demand patterns of consumers.7 Our analysis suggests that it is probable that the law of demand is supported by empirical analysis if the effect of structural changes due to factors such as variations in family attributes and habit formation is incorporated. Here, there is need for further research and analysis.
REFERENCES ANDERSON, G. and BLUNDELL, R. (1983). ‘Consumer
non-durables in the U.K.: A dynamic demand Economic Journal (Supplement), 94, 35-44. ANDERSON, T. W. (1958). An Introduction to Multivariate Statistical Analysis. John Wiley, New York. BARTEN, A. (1967). ‘Evidence on the Slutsky conditions for demand equations’, Review of Economics and Statistics, 49, 77-84. (1969). ‘Maximum likelihood estimation of a complete system of demand equations’, European Economic Review, 1, 7-73. (1977). ‘The system of consumer demand functions approach: A review’, Econometrica, 45, 23-51. BLANCIFORTI,L. and GREEN, R. (1983). ‘An almost ideal demand system incorporating habits: An analysis of expenditures on food and aggregate commodity groups’, Review of Economics and Statistics, 65, 511-5. CHRISTENSEN, L., JORGENSON, D. W. and LAU, L. (1975). ‘Transcendental logarithmic utility functions’, American Economic Review, 65, 367-83. DEATON, A. (1974). ‘The Analysis of Consumer Demand in the United Kingdom’, 1900-1970’, Econometrica, 42, 341-67. and MUELLBAUER, J. (1980a). Economics and Consumer Behavior. Cambridge University Press, Cambridge, UK. and (1980b). ‘An almost ideal demand system’, American Economic Review, 70, 312-26. DUESENBERRY,J. (1949). Income, Saving and the Theory of Consumer Behavior. Harvard University Press, Cambridge, MA. HASHIMOTO, N. (1986). ‘Tests of demand theory by the Rotterdam model’ (in Japanese), Economic Studies Quarterly, 37, 271-5. HOUTHAKKER, H. S. and TAYLOR, L. D. (1970). Consumer Demand in the U.S.: Analyses and Projections. Harvard University Press, Cambridge, MA. system’,
7 Anderson and Blundell (1983) and Blanciforti and Green (1983) partly succeeded in the dynamic specification of the consumer demand system. However, both were unable to verify the law of demand.
A.
MAKI
POLLAK, R. A. and WALES, T. J. (1978). ‘Estimation of complete demand system from household budget data: The linear and quadratic expenditure system’, American Economic Review, 68, 348-59. PRAIS, S. J. and HOUTHAKKER, H. S. (1955). The Analysis of Family Budget. Cambridge University Press, Cambridge. SURUGA, T. (1980). ‘Testing the Rotterdam demand model on the Japanese expenditure pattern’, Economic Review, 31, 368-74. VARIAN, H. (1982). ‘The nonparametric approach to demand analysis’, Economefrica, 50, 945-73. VEBLEN, T. (1924). The Theory of the Leisure Class: An Economic Study of Institutions. George Allen, London.