Explaining changes in Italian consumption of meat

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sation of a utility function selected from a family of acceptable alternatives. * Pier Luigi Rizzi ... (Translog) system (Christensen et al., 1975) and the Almost Ideal Demand ... preferences for meat and the weak separability of beef, pork and chicken .... tems, for example, ordinarily assume that groupings of goods are weakly.
Explaining changes in Italian consumption of meat: Parametric and non-parametric analysis* GABRIELE DONO Universita delta Tuscia

(received June 1991, final version received October 1993)

Summary Lewbeis composite model for comparing the AIDS and Translog demand systems is estimated using Italian meat-consumption data. Preliminary nonparametric diagnoses suggest that exogenous shifters of price and expenditure need not be introduced into a parametric model. By contrast, the parametric analysis demonstrates that demographic shifters can account for substantial changes in patterns of meat consumption. Although a parametric model without demographic variables performs adequately, likelihood ratio tests substantiate that an AIDS model with demographic variables performs even better. Keywords: Food-demand analysis; nested model; aggregation; demographic; non-parametric analysis.

1. Introduction An important issue in empirical research is the choice of the functional form to be used in representing consumer behaviour. One approach to choosing functional forms consists of directly specifying an empirical demand system which is restricted to ensure compatibility with the representative consumer's maximisation of a general utility function. An alternative approach is to obtain solutions to a representative consumer's income-constrained maximisation of a utility function selected from a family of acceptable alternatives *

Pier Luigi Rizzi and two anonymous reviewers provided helpful comments on earlier drafts of this paper. The authors take responsibility for any remaining errors and omissions.

European Review of Agricultural Economics 21 (1994) 175-198

0165-1587/94/0021-0175 © Walter de Gruyter, Berlin

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GARY THOMPSON University of Arizona

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Gabriele Dono and Gary Thompson

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(Dahlgran, 1987). In this latter case, two of the most commonly estimated demand systems are the exactly aggregable Transcendental Logarithmic (Translog) system (Christensen et al., 1975) and the Almost Ideal Demand System (AIDS) (Deaton and Muellbauer, 1980). Choosing between the Translog and the AIDS models based on the consistency with economic theory can be difficult because both systems have similar properties (Lewbel, 1989). They are second-order flexible in prices and total expenditures. Both have budget-share Engel curves that are linear in the logarithm of total expenditure which implies that they have similar exact aggregation properties. Both have indirect utility functions that are constructed with polynomials in logarithm of prices (Lewbel, 1989). In addition to their consistency with economic theory, alternative functional forms can be judged against one another according to statistical criteria such as accuracy of estimation and parsimony in parameters (Judge et al., 1985: 870-881). Alternative functional forms can also be compared by means of non-nested tests which ordinarily required estimation of a composite model (e.g., J Test) or auxiliary regressions (e.g., Cox Test). Implementation of non-nested tests and interpretation of results is not always straightforward. Composite models, for example, usually require additional parameters which can lead to problems of collinearity and lack of degrees of freedom. Running auxiliary regressions may be time consuming, particularly in the case of nonlinear models where convergence is not easily achieved and models must be run from various starting values to check for local solutions. Furthermore, with pair-wise comparisons of alternative models, a transitive ordering of preferred models does not always result. In addition, relatively fewer non-nested tests have been developed for system of equations (Pesaran and Deaton, 1978; Davidson and MacKinnon, 1983). While small sample corrections have been developed for the Cox statistic applied to single equation models (Godfrey and Pesaran, 1982), the small sample properties of many test statistics applied to systems of equations are not well known. For all these reasons, non-nested tests for choosing among alternative functional forms may not be completely satisfactory. As a way to circumvent some of the problems with non-nested tests, Lewbel has constructed a parsimonious composite model in which various AIDS and Translog models can be derived via linear restrictions on subsets of parameters (Lewbel, 1989). This composite model provides a straightforward way to compare the AIDS demand system against the Translog without the excessive number of parameters associated with most composite models. Although separate nonlinear regressions must be performed to test nested alternatives of the composite model, once stable solutions for the composite model are obtained, solutions to the nested models may be found easily using the starting values from the composite model. The objective of this paper is to apply Lewbel's approach for choosing among alternative models of food demand analysis. In particular, the com-

Italian meat consumption

177

posite demand model is exactly aggregated and then estimated using Italian meat consumption data. Meat consumption in Italy is of special interest because it now accounts for 33% of total food expenditure. Patterns of individual meat consumption have also changed substantially: in per capita terms, consumption of beef has remained stable while pork increased by

16,00 14,00 12,00 10,00 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 Source: 1STAT, Annuario Statistico Italiano; various years.

• - - CHICKEN Figure la.

Meat per capita consumption (kg/year)

140% and chicken consumption doubled (see Figure la). By 1988, the quantities consumed per capita of these three meats were nearly equal. Increasing relative prices of beef to pork and beef to chicken may explain much of the observed increase in pork and chicken consumption (see Figure lb). Despite measurable changes in relative prices, some observers suggest that increases in per capita consumption of chickv.. are caused by consumers' concerns about health and dietary fat. Before applying Lewbel's parametric model to a demand system for Italian meat consumption, a series of non-parametric tests, Varian's tests of Revealed Preferences, are utilised as preliminary diagnostic tests for the stability of preferences for meat and the weak separability of beef, pork and chicken from other food and non-food items (Varian, 1982, 1983). The principal advantage of the non-parametric tests for stability of preferences and weak separability is that they are not conditional upon a maintained parametric model. The non-parametric tests can indicate admissible specifications of

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Source: ISTAT, Conti Economici Nazionali, anni 1970-1988 BE/PO - - • - - BE/CK • • - — PO/CK Figure lb.

Meat-prices indices ratios (1970 = 100)

parametric models. For example, non-parametric tests for weak separability indicate how groupings of commodities may be considered for use in a demand system, independent of the functional form of the parametric model. Similarly, non-parametric tests for stability of preferences may pinpoint observations in the sample when preferences have shifted. Some caveats regarding the application of non-parametric tests as a diagnostic tool should be made. First, non-parametric test cannot be interpreted in the same fashion as classical parametric tests because the nonparametric tests are not based upon probability distributions as are their classical counterparts. The results of the non-parametric tests are essentially binary: either the data set does not generate violations of the relationship being tested, or it does; no significance level is associated with outcome of the test. Secondly, Varian's non-parametric tests were developed for testing whether aggregate or micro-level data are consistent with utility theory. Thus, the issue of aggregation across consumers and its possible impact on empirical analysis was not addressed in the theoretical exposition (Varian, 1982, 1983). The remainder of the paper is structured as follows. In section 2 the composite demand system which nests the exactly aggregable Translog system and the Almost Ideal Demand System is developed. Section 3 analyses the use of Varian's non-parametric tests as a preliminary diagnostic tool for

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135,00 T 130,00 125,00 120,00 115,00 110,00 105,00 100,00 95,00 90,00 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

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detecting changes in consumer preferences and weak separability. Section 4 explains the approach used to treat the aggregation issue. Sections 5 and 6 present data employed and the econometric results. In section 7 the implications of the parametric and non-parametric results for meat consumption are discussed.

2.

A composite demand system model

ln[t/(p,-,x)]= £ bjpj + ln\d+ £ ajPj +0.5 £ £ cyPiP; 7=1

L /

"

N

N

J=l

i=l7=l

N

\

,j.

"I

£ aJ+ I I CUPJ.)X\

\j=l

i=lJ=l

/

J

where p, = ln(P,) and x = \n(X) (Lewbel, 1989). With the parameter restrictions for adding up and homogeneity, £,-^ = 1, £;ft, = 0, and E.EjC.^O satisfied, a logarithmic version of Roy's identity gives the following budget share: N

N

N

+ X ajPj + 0.5 £ £ cijPiPj Icj + b, 1 + 1 j=l

\

Ec,.p, ) \

1 = 1 7=1

JT-

/J

(2)

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One approach to specifying empirical demand systems is to derive a set of equations from the solution to a representative consumer's income-constrained maximisation of utility. Two of the most commonly estimated demand systems derived from such an approach are the Translog and the AIDS. An interesting issue in empirical demand analysis is the relative adequacy of these two functional forms to the set of data. Both models have similar theoretical properties so that choice of a single model based on the consistency with economic theory can be difficult. The problem of choosing the correct or most plausible of these two functional forms may be handled statistically by specifying a composite model which nests AIDS and Translog. The theoretical model for deriving a complete demand system of share equations is based on specifying a well-behaved indirect utility function which can be suitably constrained to produce the AIDS or Translog specifications. To develop the composite model, denote the quantities of each of N goods as Qi with corresponding prices given by Pt. Total expenditure on all N commodities is given by X = £,PjQ, and the budget share for good i is represented by wt. The indirect utility function U(ph x) is specified as

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3.

Preliminary diagnostic tests

Empirical studies using demand systems seldom subject the hypothesis of weak separability of goods or the existence of a single well-behaved utility function to statistical scrutiny. Researchers employing AIDS demand systems, for example, ordinarily assume that groupings of goods are weakly separable (e.g., Blanciforti and Green, 1983; Rossi, 1988; Fulponi, 1989). Even when weak separability is tested statistically (Eales and Unnevehr, 1988), the test results are conditional upon the specification of the parametric

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The AIDS model is derived from this form by imposing the restrictions EJcIJ = 0, V i. The linear approximation to the AIDS model (LA/AIDS) can be obtained by substituting the Stone index, In P* = Siw1p1, for the (d + I'jOjPj + 0.51,iI,jCijpipj) terms (Blanciforti and Green, 1983). Although the AIDS model is nested within the specification of the share equations in equation (2) the LA/AIDS and the AIDS model are nonnested models (Green and Alston, 1990). The exactly aggregable Translog model is obtained by imposing the restrictions bt = 0, V i. Also the Fractional Translog model, in which the terms p, are divided by x, can be obtained by relaxing the constraint Z, S,-cy = 0 (Lewbel, 1987). For ease in exposition, the unrestricted system of share equations in (2) is referred to hereafter as TRAIDS because it is capable of nesting both the Translog and AIDS systems. The consistency with theory implies a demand system that adds up to total expenditure, is homogeneous of degree zero in prices and total expenditure taken together, and satisfies Slutsky symmetry. By imposing a set of linear constraints E, a, = 1, E; b{ = 0, E, Sjc y = 0 and ctJ = cjt on the TRAIDS model adding up, Slutsky symmetry and zero degree homogeneity in prices and total expenditure are satisfied by construction. By further restricting equation (2) with the constraint bt = 0, V i, the exactly aggregable Translog and the fractional Translog are obtained. Finally, imposing both symmetry and homogeneity restrictions yields the further simplification that the exactly aggregable Translog and the Fractional Translog models yield the same share system. Besides being able to nest the AIDS and Translog models, the TRAIDS model has the attractive feature that it has only few more free parameters than does one of its component models, the AIDS model. In fact, the symmetric-homogeneous TRAIDS has ((N2 + 5N)/2) - 2, while in the AIDS the number of free parameters is ((N2 + 3JV)/2) — 1, and the Translog model contains ((N2 +3N)/2)-2. Parsimony in the parameters of the TRAIDS composite model comes at a cost, however, because the TRAIDS embodies more nonlinear relationships among parameters than does the AIDS. Despite the potential difficulties in estimating nonlinear systems, the TRAIDS model offers a relatively simple way to compare two popular, non-nested models.

Italian meat consumption 181 model. Put differently, hypothesis test results may be biased if the maintained parametric model is misspecified. Non-parametric tests for shifts in preferences and weak separability are not conditioned by the maintained hypothesis of a particular parametric model. In the following sections we employ non-parametric tests as a diagnostic tool prior to specifying parametric models such as the TRAIDS, AIDS, or Translog models. The information generated by the non-parametric tests is subsequently used in the specification of the parametric model.1 A test of revealed preference

The non-parametric test for changes in consumer preferences is synonymous with a test for the existence of a well-behaved utility function. The test for stability of consumer preferences determines whether a set of price and quantity observations is consistent with Varian's notion of the Generalized Axiom of Revealed Preference (GARP) (Varian, 1982). Violations of a transitive ordering of expenditures indicate that a single, well-behaved utility function consistent with the data does not exist. Note that failure to find violations of revealed preference orderings does not preclude a shift in preferences; some violations could have gone undetected.2 The power of the test to detect violations may be limited if real expenditure for a bundle of goods increases more than does the variation in relative prices. In this case consumption bundles will probably grow so much that each bundle is revealed preferred to all previous ones3 (e.g., Landsburg, 1981; Chalfant and Alston, 1988; Varian, 1985). In the present study, the data on food consumption in Italy indicate that the growth in real expenditure is extremely limited all over the period with many small increases in many successive years. By contrast, relative food prices vary substantially in some years. Moreover, violations of GARP have been identified for some groups of commodities such as fat products consumed during the same period in Italy (Dono, 1991). This result suggests that consistency with GARP for different groupings of meat commodities is not solely attributable to the lack of power of the tests applied to those data. The approach taken in the present study is that no violations of revealed preference orderings provide useful information for subsequently specifying a parametric model. Although the researcher never knows the 'true' datagenerating process, the existence of a well-behaved utility function - perhaps not the 'true' function - is a reassuring result based on sample information. For the sample period under consideration, consistency with GARP implies three important characteristics: (i) the data are consistent with utility maximisation; (ii) a well-behaved, albeit unknown, utility function exists which can be approximated with parametric estimation; and (iii) symmetry and homogeneity restrictions could justifiably be imposed on a demand system if the correct functional form were known (Chalfant and Alston, 1988: 398).

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3.1.

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3.2.

Weak separability test

Non-parametric tests for weak separability can provide useful aggregation across commodities in parametric models. The separability ensures that a well-behaved utility function of form is consistent with the data for an arbitrary partition of goods into two groups, qlt and q2> U{quq2)=U*lquV(q2y\

guidelines for test for weak the following the bundle of (3) Downloaded from http://erae.oxfordjournals.org/ at University of Arizona on October 6, 2014

where U*[qlt V~\ is denoted the macro utility function and V(q2) is the subutility function. The necessary conditions for weak separability of qx and q2 are that the price and quantity data for the partitioned bundles satisfy GARP. The sufficient conditions for weak separability require satisfaction of the Afriat inequalities for concavity of the macro and subutility functions (Varian, 1983: 105). Because a necessary condition for weak separability is that the partitioned commodities satisfy GARP, each choice of commodities to be included in the partition is subject to a test for the stability of preferences. The existence of weakly separable commodity groups provides useful information for specifying parametric models. Degrees of freedom can be enhanced because the focus of analysis can be narrowed to the second stage of a two-stage budgeting process. A second stage in the budgeting process may also satisfy strong decentralisability (Blackorby et al., 1978: 177-178) which means that the consumer can optimally allocate expenditures within a commodity category knowing only commodity prices within that category. The necessary and sufficient conditions for any 'second stage' which is strongly decentralisable are weak separability (Blackorby et al., 1988: 188). Thus existence of weak separability provides information on how to specify parametric demand systems of a two-stage budgeting problem. The contribution of non-parametric tests in specifying the parametric models is thus twofold: the researcher can decide (i) whether changes in consumer preferences are appropriate and (ii) which groupings of goods are acceptable for parametric analysis. While the non-parametric tests may provide useful information for specifying parametric models, one caveat should be made. As mentioned earlier, the theoretical issue of aggregation across consumers is not addressed in the non-parametric literature. Hence, parametric models which explicitly account for aggregation across consumers by including demographic variables, for example, may be capable of explaining more about consumer behaviour than can be gleaned by non-parametric tests based solely on observations of prices and quantities of goods. If the researcher suspected that consumer preferences had changed, the inclusion of demographic variables in a parametric model might be a useful way to capture such changes. Suppose that a particular data set embodied a shift in consumer preferences.

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Non-parametric tests would probably indicate violations of revealed preference orderings, yet a parametric model with demographic variables might attribute such shifts in preferences to changing demographic patterns. With this caveat regarding the issue of aggregation across consumers in nonparametric tests, we turn in the next section to the aggregation issue in the TRAIDS model. 4.

Aggregation across households

(Theil, 1967). Based on the Muellbauer aggregation criterion:

hence: In x° = In X - Z

(4)

It follows that with a perfectly equitable distribution:

fx\ Z = lnN

fx\

and x°=l — \, otherwise x° > ( —I

Demographic characteristics varying across consumers may be correlated with income and expenditure levels; in the empirical application, however, the distribution of demographic attributes is assumed to be uncorrelated with the distribution of expenditures. Although this simplifying assumption may be questioned, the two distributions are often assumed to be uncorrelated because data for characterising these distributions are lacking (Deaton and Muellbauer, 1980). Demographic attributes are introduced into the TRAIDS system of equation (2) by specifying a variable Dk which represents the proportion of consumers possessing the attribute k (Lewbel, 1989). These attributes can be suitably normalised so that the first attribute represents

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Most parametric modelling deals with aggregation by appealing to either an average or a representative consumer. However, aggregation across households with different expenditure levels and demographic characteristics is incorporated explicitly in specifying the TRAIDS model (Lewbel, 1989). For aggregation over expenditure levels in a PIGLOG model, Muellbauer shows (1975: 537) that representative total expenditure, x°, is given by deflating average expenditure (X/N) by a measure of income dispersion, Z/N,A where X.= Z,x, (i = 1,..., N), N is the number of consumers and Z is Theil's entropy measure

184 Gabriele Dono and Gary Thompson an intercept and all remaining variables Dk represent the proportions of the K demographic attributes of consumers (fc = l, ...,K). This procedure requires only time series data whereas some other approaches require at least one cross section of data for modelling demographic attributes (Rossi, 1988). A single demographic attribute - the percentage of employed women in total population (ISTATc) - was included in the empirical specification due to the large number of parameters to be estimated in the nonlinear-inparameters model.

Annual data on Italian consumption from 1971 to 1988 were used to perform the non-parametric tests. Observations on the per capita consumption in kilograms (kg) for beef, pork and chicken were extracted from Food Balances of Istituto per Studi e Ricerche sul Mercato Agricolo (ISMEA). The price series were obtained by multiplying the 1970 prices (beef - £1,786 per kg; pork - £1,628 per kg; chicken - £897 per kg) with the price indexes at consumption level of the three meats (ISTATa). The diagnostic non-parametric results are summarised in Figure 2 which displays a utility tree for two decentralised stages. Consumption behaviour at the national level in Italy does not display detectable changes in consumer preferences. No violations of GARP were detected for those groupings at any of the stages reported in Figure 2. Hence, at the national level, a wellbehaved utility function exists which may be approximated using a demand system based upon a flexible functional form - AIDS, Translog, or other. The separability test was performed using both the individual beef, pork and chicken data found in the Food Balances publication (ISMEA) and the meat consumption data from the National Account series (ISTATb). This last series includes 'other meats' consumption and does not consider out of home consumption; so the meat-consumption figures reported do not coincide exactly with the quantities of the three meats reported in Food Balances.

Msa | | H Daily Products H-M Fruit-V

Figure 2. Groups of goods satisfying GARP and weak separability* * Results from Nonpar software

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5. Data, non-parametric tests results and models specification

Italian meat consumption 185

ln(w)] (5) where x° is Muellbauer's representative expenditure; / is an entropy measure of dispersion of a household average for consuming meat out of food expenditure; t = E(), with the household's average propensity to consume meat out of food expenditure divided by average propensity to consume meat out of food in the economy; v is the representative food expenditure constructed from the time series observations of distribution data, where each decile is considered equivalent to a single consumer. This decomposition of representative meat expenditure explicitly treats the link between the distributions of total food and of meat expenditure6 (Lewbel, 1989: 351). In the empirical estimation the parameter / defined in equation (5), has been omitted because it cannot be identified. So the representative total meat expenditure was computed as: ln(x°) = ln(AT) +1 ln(»)

(6)

A single demographic attribute - the percentage of employed women in total population (ISTATd) - was included in the empirical specification due to the large number of parameters to be estimated in the nonlinear-in-parameters model. Although the percentage of women employed in the population appears low when compared to that of other western contries - 24.4% in

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However, the quantity consumed of other meat products in Italy is negligible and the food expenditure out of home, although increasing, is still marginal. Thus, the results from using the two data series should compare favourably. Parenthetically, using the aggregated commodity groupings in Stage I the conditions for homothetic separability were not satisfied. Hence, this stage can not be conceived as the first stage of a two stage budgeting model (Blackorby et al., 1978: 206). The commodity groupings for each of the stages do pass the non-parametric test for weak separability. These results hold using both the National Account aggregate data as well as the Food Balances series for individual meats. The fact that the three meat products constitute a weakly separable group suggests that the meat group is appropriately aggregated even though conceptually other meat products might be included in the group. With weak separability satisfied in the second stage, meat-product consumption can admissibly be modelled as 'second' stage of a two stage budgeting process.5 In order to exactly aggregate the models a meat-expenditure dispersion measure and a demographic characteristic were constructed. Time series data on the distribution of food expenditures are not available for Italy consumers although comparable data are available for a sample of households (ISTATc). The survey data on total food expenditure by total expenditure deciles were used to construct Theil's entropy measure for the 1971-1988 period. Lewbel's approach was adopted to derive the relationships between meat expenditure and total food expenditure:

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Italy versus 66.2% in the United States in 1988 - the temporal increase in this percentage may represent changes in the household structure and consumer behaviour of Italians. Most other plausible demographic variables, such as the number of single-headed households, are highly correlated with the percentage of women employed in the population, and all these variables are correlated with a time trend. Lewbel (Lewbel, 1989) suggests that a time trend may be included as a proxy for demographic changes. The composite budget share model in (2), may be specified with demographic variables as N

/

*=i

w

i

i=i

\

K

N

N

N

Z Z dikPjDk +0.5 Z Z *=i;=i

'

c

uPiPj

i=ij=i

=

+ e,