Dynamic modelling of consumption patterns using LA-AIDS: a comparative study of developed versus developing countries

Abstract

The traditional applied demand system estimation assumes that when consumer income and commodity prices change, the consumers instantaneously fully adjust to a new consumption equilibrium level and use a static demand system such as Static Linear Almost Ideal Demand System (Static LA-AIDS) for estimation. However, in real-life situations, such an assumption does not hold as consumers take time to settle to a new consumption equilibrium level. Hence, dynamic demand system estimations generate more nuanced insights into the short-run dynamics of consumer demand. This paper estimates two forms of dynamic versions of AIDS, Dynamic LA-AIDS and the error-corrected LA-AIDS, and compares the results with the Static LA-AIDS. The paper also models the consumption patterns of consumers in developed and developing countries and provides a comparative analysis of implied elasticities using recent data. The results show that the dynamic models support demand theory hypotheses—demand homogeneity and Slutsky symmetry—more than the static model. The estimated mean own-price elasticities reveal that the demand in the short-run and long-run is price inelastic across all countries. Long-run and short-run income elasticities demonstrated notable variation across various commodity groups and country groups. For example, the restaurant meal is a luxury in the short-run for all countries. In the long-run, it is a necessity in developed countries and a luxury in developing countries. Food and housing are necessities; durables, transport and recreation are luxuries in developed and developing countries in the long-run and short-run.

1 Introduction

The interest in the analysis of consumer demand dates back to the last century and has continued to be of great interest (Barten 1964; Clements and Gao 2015; Deaton and Muellbauer 1980a; Engel 1895; Selvanathan and Selvanathan 1994; Theil 1965). With the rapidly changing consumer behaviour resulting from income growth, price changes followed by domestic and international shocks, global mass production and technological changes, the analysis of consumer demand and the decision making around production and government policy has become severely complicated. As such, obtaining reliable estimates of income and price elasticities that provide vital inputs for various policy decision making has become more crucial than ever before.

Much of the demand analysis research heavily depends on the standard static demand system models. Although widely utilised, the static demand system, such as the standard static linear Almost Ideal Demand System (Static LA-AIDS), has some drawbacks. In particular, Static LA-AIDS does not account for consumer adjustments to consumer income and commodity price changes that take place in the short-run. This is because static demand systems assume that when income and prices change, the consumers instantaneously fully adjust to a new consumption equilibrium level. It is well acknowledged that in real-life situations, such an assumption does not hold. The consumers take time to settle to a new equilibrium consumption level in each time period due to distinct short-run consumption behaviour driven by short-run adjustment costs, unexpected real price levels, false expectations and habit formation (Hassan et al. 1977; Rathnayaka et al. 2019). For example, when petrol price increases, although consumers may make some consumption adjustments to manage their budgets, in the short-run, most of the consumers will continue to pay higher prices and use their car. This is because there are various costs involved in fully adjusting the consumption for a price change. The lack of alternatives could also be another factor that consumer may not be able to completely switch their consumption in the short-run. However, when they realise that the situation is no-longer sustainable, in the long-run, consumers tend to adjust their consumption, such as switch to an electric car, or to a scooter or motorcycle or public transport. This is in particularly common for costly items, such as housing and durable goods as consumers make substantial financial commitments to acquire these goods. For example, while demand for housing is inelastic in the short- run, in the long-run it may become demand elastic when consumers move out of expensive housing in the city to cheaper housing in outer city area and decide to incur higher transport expenses. For this reason, obtaining the short-run elasticity estimates and the speed with which these estimates reach their long-run values is a critical consideration in most business and government policy decision making processes (Eakins and Gallagher 2003). Considering this requirement, while not denying the importance of estimating the long-run elasticity estimates, the literature on consumer demand analysis highlights the need for obtaining the counterpart short-run elasticities (see, for example, Anderson and Blundell 1983; Edgerton et al. 1996). For example, government policy related to Goods and Services Tax (GST) on a commodity requires long-run elasticity estimates, while decision making related to the marketing of a commodity requires short-run elasticity estimates.

Although production and consumption decision making require short-run elasticity estimates, Static LA-AIDS will not best serve such a purpose. The use of dynamic demand systems for estimations, however, provides a resolution for this concern. This is because the dynamic demand systems allow for intertemporal rationality of consumer behaviour in the demand system estimation by explicitly incorporating the mechanism underlying the short-run adjustment process in the model specification.

With the globalisation of economic activities, it has become important to know the income and price elasticity estimates of consumer goods at the single-country level so that importers/exporters, producers and manufacturers can make informed production and marketing decisions when and where these elasticities may differ across countries. Therefore, obtaining short-run and long-run elasticity estimates at the individual country level has also become valuable information for informing decisions related to various trade-related economic activities. There is a considerable body of the literature on dynamic demand systems and elasticity estimations of single-country settings (see, for example, Alessie and Kapteyn 1991; Blanciforti et al. 1986; Karagiannis and Velentzas 1993; Molina 1994; Rathnayaka et al. 2019; Singh et al. 2011). Nevertheless, so far, no attention has been paid to estimating dynamic demand systems for a group of countries and comparing the performance among the dynamic and static demand systems.

In this paper, we address these important gaps and contribute to the consumer demand analysis literature in three ways. Firstly, to provide insights into short-run implied elasticities for a broad range of commodities, using the most recent data available, we estimate two forms of dynamic versions of Almost Ideal Demand System (AIDS), namely Dynamic Linear AIDS (Dynamic LA-AIDS) (Blanciforti and Green 1983; Edgerton et al. 1996; Kesavan et al. 1993) and the error-corrected linear approximated AIDS (EC-LA-AIDS) (Karagiannis et al. 2000; Nzuma and Sarker 2010). Secondly, to provide further insights into the long-run and short-run elasticity differences, we compare the performance of the Static LA-AIDS demand system estimation results with those of the dynamic demand systems. Thirdly, we estimate and compare cross-country static and dynamic demand systems for 40 developing and developed countries. Such comparative analysis of consumption patterns of major consumer goods between developed and developing countries will also facilitate understanding the differences in consumption behaviour based on the stage of development and assist with production and policy decisions in a globalised economic setting.

The organisation of the paper is as follows. In Sect. 2, we present a review of the literature on cross-country demand analysis and dynamic demand modelling. In Sect. 3, we present a preliminary analysis of the data at individual country level, group levels (developed and developing countries) and global levels. Section 4 introduces the dynamic demand systems for estimations: Dynamic LA-AIDS and EC-LA-AIDS. In Sect. 5, we select the preferred model for each country and present the income and price elasticities implied by the preferred demand systems. We provide concluding comments in the last section of the paper.

2 A review of the literature

2.1 Cross-country demand analysis studies

There is a large body of consumption economics literature on modelling consumer demand. The literature on cross-country consumption goes back to the pioneering work of Houthakker (1957), which represents the first study to compare the expenditure elasticities for different countries—developed and developing. The study estimated expenditure elasticities for food, clothing, housing and miscellaneous items based on double-log Engel curves using cross-sectional data from 30 countries. The results revealed that elasticities are similar across commodities but not equal. Since the study utilised cross-sectional data that does not capture price variation, the study eschews the estimation of price elasticities. Houthakker (1965) performed a similar study as Houthakker (1957) using time-series data for the period 1948–1959 for 13 OECD countries. The results suggested that while price elasticities show no uniformity across the commodities (food, clothing, rent, durables and miscellaneous), the expenditure elasticities are consistent.

Goldberger and Gamaletsos (1970) analysed the consumer expenditure patterns of 13 OECD countries for food, clothing, rent and durables over the period 1950–1961, employing the Linear Expenditure System (LES) introduced by Stone (1954) and the Constant Elasticity Demand System (CEDS) used by Houthakker (1965). Parks and Barten (1973) hypothesised that some of the differences in demand behaviour between countries could be explained by differences in the age composition of the population. The study estimated LES using time-series data of five commodities (food, clothing, housing, durables, others) for the years 1950–1967 for 14 OECD countries. The results revealed that population composition has a significant effect on the parameters of the demand model after correcting for the difference associated with the level of real income across the countries. The results also found that income elasticities are positive and own-price elasticities are negative in all countries.

Lluch and Powell (1975) estimated the LES for eight commodity groups using data from 19 countries. The cross-country analysis revealed some discernible patterns in the variation of price and expenditure elasticities as a function of GNP per head. Moreover, the study noted that the own-price and expenditure elasticity of food appeared to decline in absolute value as real income increases. Overall, the own-price elasticities and cross-price elasticities of food appear to account for most of the total price responsiveness (about 80%). Lluch et al. (1977) employed an extended LES where total consumption expenditure was assumed to be endogenous across 17 developing countries between 1955 and 1969. The study considered eight commodities (food, clothing, housing, durables, personal care, transport, recreation and miscellaneous) and found food and housing to be necessities, clothing to be borderline and durables, personal care, transport, recreation and miscellaneous goods to be luxuries.

Another leading work on cross-country application was conducted by Theil (1987) using data compiled by Kravis et al. (1982). These data, which are part of the International Comparisons Programme (ICP) sponsored by the United Nations and the World Bank, covered 34 countries and provided comparable price and volume indices for more than 100 detailed categories of consumption. Selvanathan and Selvanathan (1993) analysed the consumption of ten commodities across 18 OECD countries from 1960 to 1981. The study revealed that OECD consumers spend about half of their income on food, housing and transport. The results also found that food, housing and medical care are necessities and clothing, durables, transport and recreation are luxuries in most OECD countries. The demand for all goods considered was found to be price inelastic.

Clements and Theil (1996) used cross-sectional data from Kravis et al. (1978) from 16 countries to estimate a common system of demand equations for all countries. Compared to the usual time-series application for a given country, countries here played the role of time periods. This idea was introduced by Theil and Suhm (1981), which suggested that tastes are to be the same across countries. Although a rather bold assumption, it is the one advocated by Stigler and Becker (1977), who hypothesised that tastes neither change capriciously nor differ importantly between people. In the international context, this hypothesis implies that consumers in different countries have similar tastes irrespective of differences in language, religion, culture and geography. Pollak and Wales (1987) tested this using the quadratic expenditure system with time series/cross-country data for Belgium, the UK and the USA. Based on likelihood ratios and nonparametric (revealed preference) tests, the study concluded that the data from these countries could not be pooled to estimate a common demand system. Hence, the study rejected the hypothesis of identical tastes. Selvanathan and Selvanathan (1993) also found that OECD consumption data do not support Stigler and Becker’s (1977) hypothesis using consumption data for 18 OECD countries.

Chen and Clements (1996) analysed consumption patterns in 13 emerging/developing economies to identify key empirical regularities using a system-wide approach. Chen (2001) extended the work of Selvanathan and Selvanathan (1993) by adding 13 less-developed countries (giving a total of 31) during the same time period. Clements and Ye (2003) estimated a double-log model for 18 wealthy (OECD) countries and 13 developing countries. The study concluded that differences in income and relative prices explained a significant share of the variation in international consumption patterns. Selvanathan and Selvanathan (2003b) investigated the consumption patterns of the five strongest Asian economies (Hong Kong, Japan, Korea, Singapore and Taiwan). The results based on the Rotterdam model revealed that the consumption data of these five countries supported a number of empirical regularities, including the law of demand and Engel’s law. Clements et al. (2006a, b) constructed a new database that built upon that of Selvanathan and Selvanathan (2003a) and estimated the CBS model to analyse the extent to which the consumption basket was diversified, how this changed with income and whether a simple utility maximising model was capable of explaining the diversity of consumption patterns internationally. In a recent study, Rathnayaka et al. (2022) examined the similarities in consumption patterns in ten Asian countries, using the Rotterdam, CBS and AIDS models. The study noted a less diversified consumption basket for consumers in developing countries than those in relatively wealthy countries in Asia. The demand elasticity estimates suggested that food and housing are necessities, and clothing, durables and transport are luxuries for Asian consumers. The demand for all goods was found to be price inelastic.

Many advances in this important area of study have been achieved due to data made available through the International Comparison Program (ICP) since the early 1970s. Pioneering studies that applied a cross-country demand system to ICP data include Clements et al. (1979), Theil et al. (1980), Finke et al. (1983), Seale et al. (2003) and Seale and Regmi (2009).

2.2 Dynamic demand analysis

As noted above, all cross-country consumption studies discussed in the previous section are based on the static demand systems estimations. A summary of selected static cross-country consumption studies is found in Table 1. However, static demand systems used in these empirical studies assume that consumers immediately and fully adjust to a new equilibrium when either incomes or prices change. However, consumers are unlikely to have adjusted to equilibrium in each time period; hence, the assumption of instantaneous adjustments by consumers is potentially incorrect. Therefore, a number of studies, predominantly single-country studies, have recognised the importance of including dynamic adjustments in demand systems and have adopted a number of approaches.

Table 1 Major cross-country consumption studies

For instance, several consumer demand studies have introduced dynamic nature into the well-known AIDS by including the lagged budget share w_i,t-1 on the right-hand side of the AIDS equation (see Blanciforti and Green 1983 and Blanciforti et al. 1986 for the USA; Karagiannis and Velentzas 1993 for Greece; and Molina 1994 for Spain). Furthermore, a few other studies have estimated a more general dynamic AIDS model by including its own past budget shares and also that of all other goods (for example, Alessie and Kapteyn 1991 for the Netherlands; Edgerton 1997 for Sweden; Kesavan et al. 1993 for the USA; and Klonaris and Hallam 2003 for Greece).

Relatedly, Anderson and Blundell (1983) for the UK, Balcombe and Davis (1996) for Bulgaria, Edgerton et al. (1996) for Sweden and Karagiannis and Velentzas (1997) for Greece have incorporated dynamic elements into AIDS by relying on the statistical properties of the data. This has led to the application of the error-corrected linear approximated AIDS (EC-LA-AIDS) in a number of recent studies on demand for non-durable goods, such as food sub-categories. For example, using EC-LA-AIDS, Karagiannis et al. (2000) analysed the demand for meat in Greece. Eakins and Gallagher (2003) analysed the dynamics of alcohol expenditure in Ireland, while Fanelli and Mozzocchi (2002) estimated the demand for meat in Italy, and Nzuma and Sarker (2010) estimated the demand for major cereals consumed in Kenya. Singh et al. (2011) employed EC-LA-AIDS to estimate demand for major crustaceans at a disaggregated level in the USA. More recently, Rathnayaka et al. (2019) modelled the dynamic behaviour of Sri Lankan consumers in consuming eight broad commodity groups employing EC-LA-AIDS. A few studies have also employed EC-LA-AIDS to model tourism demand (Durbarry and Sinclair 2003; Wu et al. 2012).

In addition to the above studies based on dynamic versions of AIDS, Bushehri (2003) introduced a generalised dynamic Rotterdam model. A few consumption studies have employed this approach (see, for example, Muhammad and Jones 2009; Muhammad et al. 2015, for the USA and Greear and Muhammad 2021, for Japan).

In summary, according to the major findings from previous studies reported in Table 1, in most countries, food and housing are necessities. Moreover, OECD consumers spend about half of their income on food, housing and transport, while consumers in low-income countries spend more than half of their budget on food. Consumers in developing countries have a less diversified consumption basket than those in relatively wealthy countries. Also, the demand for all goods considered was found to be price inelastic. However, the above cross-country consumption studies assume that consumers immediately and fully adjust to a new equilibrium when either income or prices change, and their elasticity estimations are based on static demand systems.

To the best of our knowledge, no other published comprehensive econometric studies analyse the cross-country consumption patterns of broad commodity groups using dynamic demand models with more recent data. The current study, therefore, adds to the literature by bridging this gap by estimating dynamic AIDS and EC-LA-AIDS for 40 developed and developing countries using the data up to 2019 (for most of the countries) and comparing the performance of the two dynamic AIDS models with that of the static AIDS model.

3 Data description

3.1 Data sources

The annual consumption expenditures (in current and constant prices) and the population for 40 countries, consisting of 27 developed and 13 developing countries (selected based on data availability), are compiled from the National Accounts of OECD Countries (OECD: Paris, various issues) and various individual country Government Statistical Department websites and publications.¹

In this paper, we classify the consumer basket into nine commodity groups.² When compiling data, it was found that statistical agencies of some countries publish information about commodities in less than nine commodity groups. Relatedly, during the estimation, it was found that the sample size (time periods) was too small for the dynamic model estimation for nine commodity groups as lagged variables were involved. To overcome these issues, we combined some commodities into a single group when performing the estimation.

3.2 Data summary

Table 2 provides details about the database. Figure 1 plots the food share (column 9) against log of per capita GDP (column 6). As can be seen, in general, the higher a country’s per capita GDP, the lower the budget share on food, which gives support to the Engel’s (1987) law. In particular, a slope coefficient of − 0.13 is not inconsistent with the “strong version” of Engel’s law.

Table 2 Country characteristics, 45 countries

Fig. 1

Mean food share versus per capita GDP across 40 countries

Let there be n commodities and p_it be the price, and q_it be the per capita quantity consumed of commodity i. We define price and quantity log-changes, for commodity i, as

$$\begin{array}{*{20}c} {Dp_{it} = \, \log p_{it} - \, \log p_{it - 1}, } & {Dq_{it} = \, \log q_{it} - \, \log q_{it - 1} ,} & {t = 1, \, \ldots , \, T,i = 1, \, \ldots ,n} \\ \end{array}$$

respectively. When these price and quantity log-changes are multiplied by 100, they can be interpreted as percentage growth rates in price and quantity, respectively, from year t-1 to t. Here and elsewhere, log refers to the natural logarithm.

Let M_t be the total expenditure (income for short) on the n commodities during period t. Therefore, the total expenditure M_t = $\sum\nolimits_{i = 1}^{n} {p_{it} q_{it} .}$ The proportion of total expenditure devoted to commodity i, called the budget share of i, is ${w}_{it}=\frac{{p}_{it}{q}_{it}}{{M}_{t}}$ (t = 1,…,T, i = 1, …, n), where T is the sample size.

To measure the overall growth in consumption and prices, we use the Divisia volume and price indices defined below. The Divisia volume index, DQ_t, and Divisia price index, DP_t, for period t are defined as $D{Q}_{t}=\sum_{i=1}^{n}{\overline{w} }_{it}D{q}_{it,}$ and $D{P}_{t}=\sum_{i=1}^{n}{\overline{w} }_{it}D{p}_{it,}$ respectively (t = 1,…,T), where $\overline{w}_{it} = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {w_{it} + w_{it - 1} } \right)$ is the arithmetic average of the budget shares of commodity i in periods t and t − 1.

The Divisia volume index and Divisia price index averaged over the whole sample period for each country are calculated as

$$D\overline{Q} = \frac{1}{T}\sum\limits_{t = 1}^{T} { \, DQ_{t} } \;{\text{and}}\;D\overline{P} = \frac{1}{T}\sum\limits_{t = 1}^{T} { \, DP_{t} }$$

(1)

The Divisia quantity index (DQ_t) and the Divisia price index (DP_t) are first-order moments. The second-order moments and the Divisia quantity and price variances are defined as

$$\begin{array}{*{20}c} {{\rm K}_{t} = \sum\limits_{i = 1}^{n} { \, \overline{w}_{it} } \left[ {Dq_{it} - DQ_{t} } \right]^{2} ,} & {\Pi_{t} { = }\sum\limits_{i = 1}^{n} {\overline{w}_{it} } \left[ {Dp_{it} - DP_{t} } \right]^{2} ,} & {t = {1}, \, \ldots ,{\text{ T}}} \\ \end{array}$$

These two variances measure the degree to which the quantities and prices of the individual commodities change disproportionately. When all quantities and prices change proportionately, these two variances will vanish.

To measure the co-movement of prices and quantities, the Divisia price-quantity correlation is defined as ρ_t = $\frac{{\Gamma_{t} }}{{\sqrt {{\rm K}_{t} \Pi_{{\text{t}}} } }}$, where Γ_t = $\Sigma_{i = 1}^{n} \overline{w}_{it} (Dp_{it} - DP_{t} )(Dq_{it} - DQ_{t} )$ is the Divisia price-quantity covariance between prices and quantities. Since relative prices and relative consumptions are expected to move in the opposite direction, we would expect Divisia price-quantity correlation to be negative. The corresponding sample period averages for each country are calculated as

$$\begin{array}{*{20}c} {{\overline{\text{K}}} = \frac{1}{T}\sum\limits_{t = 1}^{T} {{\text{K}}_{t} ,} } & {\overline{\Pi } = \frac{1}{T}\sum\limits_{t = 1}^{T} {\prod_{t} } } & {{\text{and}}\quad \overline{\rho } = \frac{1}{T}\sum\limits_{t = 1}^{T} { \, \rho_{t} } .} \\ \end{array}$$

(2)

Furthermore, if we regress the relative consumption $(Dq_{it} - DQ_{t} )$ against relative prices $(Dp_{it} - DP_{t} )$, under the assumption of unitary income elasticity, we can show that the weighted least squares estimate of the slope of the regression equation is the measure of income flexibility (for example see, Selvanathan and Selvanathan 2003b) given by

$$\hat{\phi }_{t} = \frac{{\Gamma_{t} }}{{\Pi_{t} }} = \rho_{t} \sqrt {\frac{{K_{t} }}{{\prod_{t} }}} .$$

(3)

Table 3 presents the collection of the above average values for each country, averaged over the respective time periods given in Eqs. (1)–(3).

Table 3 Mean Divisia price and quantity indices and variances, price-quantity correlation and income flexibility, 40 countries

As can be seen from column (2), the overall growth in consumption in the 40 countries vary between 0.3% (Italy) and 7.5% (Ireland) with an overall average of 2.5% per annum across all countries (see last row of the table). The average consumption growth across the developing countries (3.2%) is slightly higher than that of the developed countries (2.1%). The overall annual price growth in the 40 countries presented in column (3) of the table varies between − 3.6% (Ireland) and 9.4% (South Africa) with an overall cross-country average of 3.4% per annum. It is also observable that the overall cross-country average price growth in developing countries (5%) is twice that of the growth in the developed countries (2.6%). Columns (4) and (5) reveal that the Divisia quantity variances systematically exceed that of the corresponding Divisia price variances. To confirm this relationship, in Fig. 2, we plot $\sqrt {\overline{K}^{c} }$ against $\sqrt {\overline{\Pi }^{c} }$ for c = 1,…,40. As can be seen, most of the points lie above the 45° line indicating that, on average, the quantity variance systematically exceeds the price variance. This pattern agrees well with the results reported in Chen (2001), Clements et al. (2020), Meisner (1979), Rathnayaka et al. (2019), Selvanathan and Selvanathan (1993; 2003a; 2005) and Theil and Suhm (1981). This is mainly because prices are mostly sticky and take time to adjust, and it is the quantity consumed that adjusts to shocks mostly (for example, see Clements 2019).

Fig. 2

Quantity standard deviation versus price standard deviation, all countries (at sample means)

As can be seen, all Divisia price-quantity correlations presented in Table 3 are negative with a cross-country average of − 0.4. The negative correlation values reflect the tendency of the consumer to move away from those commodities that have above average price increases. All the income flexibility ϕ estimates in column (7) are negative as they should be, and the cross-country average is close to − 0.5, which is well in agreement with previous studies (Clements et al. 2020; Rathnayaka et al. 2019; Selvanathan and Selvanathan 1993, 2003a, 2005 and Theil and Suhm 1981).

4 Dynamic demand systems

Below we consider three versions of Almost Ideal Demand System (AIDS) developed by Deaton and Muellbauer (1980b) to analyse the consumption pattern of 40 countries under consideration in this study, to obtain the income and price elasticities for each country in the short-run and the long-run.

4.1 Static LA-AIDS

The linear version of the standard static Almost Ideal Demand System (Static LA-AIDS) developed by Deaton and Muellbauer (1980b) is given by

$$\begin{array}{*{20}c} {w_{i} = \alpha _{{\text{i}}} + \beta _{{\text{i}}} {\text{log}}\frac{M}{P} + \sum\limits_{{j = 1}}^{n}{\gamma }_{ij} {{\text{log}}\;p_{{\text{j}}} ,} } & {i = 1,2,{\mkern 1mu} \ldots ,n} \\ \end{array}$$

(4)

where w_i = (p_iq_i/M) is the budget share of commodity i in the total consumer expenditure $M=\sum_{i=1}^{n}{p}_{i}{q}_{i}$, p_i is the price of commodity i and $\mathrm{log}P=\sum_{k=1}^{n}{w}_{k}\mathrm{log}{p}_{k}$ is the Stone price index and the variables α_i, β_i and γ_ij are constants satisfying the adding up conditions; $\sum_{i=1}^{n}{\alpha }_{i} =1$; $\sum_{i=1}^{n}{\beta }_{i} =0$; and$\sum_{i=1}^{n}{\gamma }_{ij}=0.$³ The Static LA-AIDS also satisfy the demand homogeneity, $\sum_{j=1}^{n}{\gamma }_{ij}=0,$ and symmetry conditions, ${\gamma }_{ij}={\gamma }_{ji}.$

The income and uncompensated price elasticities implied by Static LA-AIDS are

$\begin{array}{*{20}c} {{{ \eta }}_{i} = {1} + \frac{{\beta_{i} }}{{w_{i} }},} & {{{ \eta }}_{ij} = - {\delta}_{ij} + \frac{1}{{w_{i} }}\left[ {\gamma_{ij} - \beta_{i} \left\{ {w_{j} } \right\}} \right],} & {i,j = {1},{2}, \ldots ,n.} \\ \end{array}$

4.2 Dynamic version of LA-AIDS

Blanciforti and Green (1983) incorporated habit effects along the line of Pollak and Wales (1969) and developed the following dynamic version of (4):

$${w}_{it}={\phi }_{i}+\sum_{j=1}^{n}{\theta }_{ij}{w}_{jt-1}+{\beta }_{i}\mathrm{log}\frac{{M}_{t}}{{P}_{t}}+\sum_{j=1}^{n}{\gamma }_{ij}\mathrm{log}{p}_{jt}, \quad i=1,\dots , n; t=1,\dots , T$$

(5)

where $\sum_{i=1}^{n}{\phi }_{i} =1, \sum_{i=1}^{n}{\theta }_{ij}=0$; $\sum_{i=1}^{n}{\beta }_{i} =0$; and $\sum_{i=1}^{n}{\gamma }_{ij}=0.$

4.3 Error-corrected linear approximated AIDS (EC-LA-AIDS)

The error-corrected version of Eq. (4), EC-LA-AIDS, can be written as

$$\Delta {w}_{it}\!=\!{\alpha }_{i}+{\theta }_{i}\Delta {w}_{jt-1}+{\beta }_{i}\Delta\mathrm{log}\frac{{M}_{t}}{{P}_{t}}+\sum\nolimits_{j=1}^{n}{\gamma }_{ij}+{\lambda }_{i}{EC}_{it-1} \quad i\!=\!1, 2, \dots , n$$

(6)

where ∆ is the difference operator and EC_it-1 is the error correction term that measures the adjustment of the decision error made in the previous period, which is estimated using the residual term from the long-run Static LA-AIDS model given by Eq. (4). We have incorporated habit effects into Eq. (6) by including the lagged dependent variable on the right-hand side of (6).

5 Estimating the AIDS models

In this section, for each country, we estimate the three versions of AIDS model, Static LA-AIDS, Dynamic LA-AIDS and EC-LA-AIDS given by Eqs. (4), (5) and (6). Some of the previous research suggests that selecting a nonlinear form of AIDS models is crucial when there is a high-income inequality (Banks et al. 1997). Therefore, countries are listed in Table 2 with Gini coefficient higher than 0.4; namely, for Hong Kong, the USA, Sri Lanka and South Africa, we estimate the nonlinear AIDS model (QAIDS). We also test the demand theory hypotheses, demand homogeneity and Slutsky symmetry and select the preferred model based on the information inaccuracy measure. We then evaluate the implied income and price elasticities for each country based on its preferred model.

5.1 Testing for stationarity and co-integration

Before estimation, we first investigate whether the time-series variables w_it, ln(M_t/P_t) and ln(p_it), i = 1,2, …, n are stationary, using the Lee and Strazicich (2003) unit root tests with multiple breaks. The results revealed that about 98% of the series are stationary.⁴ Therefore, we consider that the estimation results are overall non-spurious.

5.2 Testing demand theory hypotheses

The traditional approach of testing demand theory hypotheses, demand homogeneity and Slutsky symmetry is based on the asymptotic tests such as Wald test, likelihood ratio test or Lagrange multiplier test. Using simulations, a number of previous research studies have showed that such asymptotic tests are bias towards rejecting the null hypothesis, especially when applied to large demand systems with fewer observations like ours (Balcombe and Davis 1996; Laitinen 1978; Meisner 1979; Selvanathan and Selvanathan 1993; Clements and Selvanathan 1995). Therefore, in this paper we use the sample-size corrected test developed by Court (1968) and Deaton (1974) to test demand homogeneity and Slutsky symmetry. This test statistic has been used in many studies in the literature over time (for example see, Chambers 1990; Rathnayaka et al. 2019; Wu et al. 2012) and is given by

$$TS=\frac{{tr\left({\Omega }^{R}\right)}^{-1}\left({\Omega }^{R}-{\Omega }^{U}\right)/q}{{tr\left({\Omega }^{R}\right)}^{-1}{\Omega }^{U}/\left(n-1\right)\left(T-k\right)}$$

(7)

where ${\Omega }^{R}$ and ${\Omega }^{U}$ are the unrestricted and restricted estimated residual covariance matrices, respectively; k is the number of parameters estimated in each equation; q is the number of restrictions imposed; n is the number of equations in the system; and T is the number of observations. The test statistic TS follows an F-distribution with q and (n − 1)(T − k) degrees of freedom.

Tables 4, 5, 6 present the hypotheses tests’ results for demand homogeneity and Slutsky symmetry for the three versions of AIDS model, the Static LA-AIDS, Dynamic LA-AIDS and EC-LA-AIDS.⁵ Table 7 gives the percentage acceptance of each hypothesis (which is calculated as 100 percentage rejections) calculated from Tables 4, 5, 6.⁶ As can be seen, under Static LA-AIDS, homogeneity is acceptable for 67% of all countries, symmetry is acceptable for 47% of all countries and homogeneity and symmetry combined is acceptable only for 32% of all countries. Under Dynamic LA-AIDS, homogeneity is acceptable for 95% of all countries, symmetry is acceptable for 87% of all countries and homogeneity and symmetry combined is acceptable for 67% of all countries. Under EC-LA-AIDS, homogeneity is acceptable for 98% of all countries, symmetry is acceptable for 72% of all countries and homogeneity and symmetry combined is acceptable for 72% of all countries. The overall conclusion is that in terms of demand theory hypotheses, in general, the Dynamic LA-AIDS model outperforms the Static LA-AIDS and EC-LA-AIDS models.

Table 4 Hypothesis testing results: Static LA-AIDS model

Table 5 Hypothesis testing results: Dynamic LA-AIDS model

Table 6 Hypothesis testing results: EC-LA-AIDS model

Table 7 Percentage of acceptance (100 – percentage rejections)

Based on the hypothesis testing results presented in Tables 4, 5, 6, we can select the most appropriate restrictions we can impose for each country when estimating the respective demand systems. For example, for Australia, for Static LA-AIDS (Table 4) we estimate the unrestricted model because all three hypotheses were rejected by the data. For Dynamic LA-AIDS (Table 5), for Australia, all three hypotheses were accepted, and therefore, we use the model with homogeneity and symmetry restriction-imposed. However, for EC-LA-AIDS (Table 6), for Australia, we use only homogeneity restriction-imposed model as symmetry and homogeneity and symmetry restrictions were rejected by the data. Table 8 details which restrictions are acceptable for each country with respect to the three models. The results are presented in columns (3), (5) and (7) of Table 8, for each model, respectively.

Table 8 Information inaccuracies

5.3 Selecting the preferred model using information inaccuracy

To select the preferred model among the three versions of AIDS, we use the budget shares predicted by the three models. To measure the quality of the predictions, we use the concept from information theory, called the information inaccuracy. This measure has been extensively used in the literature (see, for example, Jayasinghe et al. 2019; Selvanathan and Selvanathan 2005; Theil 1996) and is well suited to analyse the fit of allocation models such as ours. In general, if we have n goods with budget share predictions $\hat{w}_{1t} ,$…,$\hat{w}_{nt} ,$ the information inaccuracy of these predictions is defined as

$$I_{t} = \sum\limits_{i = 1}^{n} {w_{it} \log \frac{{w_{it} }}{{\hat{w}_{it} }}}$$

(8)

where ${\widehat{w}}_{it}={w}_{it}-{e}_{it}$.

When predictions are perfect (i.e. $\hat{w}_{it}$ = w_it, i = 1,…,n), I_t = 0; otherwise, I_t > 0. Therefore, the smaller the information inaccuracy, the better the predictions. Consequently, the preferred demand system would be the one with the lowest value of information inaccuracy. If we define e_it = log (w_it /$\hat{w}_{it}$) as the relative error, then I_t = $\sum\nolimits_{i = 1}^{n} {w_{it} e_{it} }$ can be interpreted as the Divisia mean (i.e. the budget-share-weighted mean) of the relative errors. Thus, I_t × 100 is approximately equal to the weighted average percentage error.

Table 8 presents the information inaccuracies for each model, calculated based on Eq. (8). The column (8) of Table 8 gives the preferred model among the three models with appropriate restrictions for each country. As can be seen, based on information inaccuracies, for 80% of the countries, the preferred model among the three versions of AIDS is the Dynamic LA-AIDS model and EC-LA-AIDS for the remaining 20% of the countries. Static LA-AIDS is not preferred by any country.

5.4 Implied income and price elasticities

Tables 9 and 10 present the short-run and long-run income elasticities, and Tables 11 and 12 present the short-run and long-run own-price elasticities for the individual countries based on the preferred models reported in the last column of Table 8.

Table 9 Short-run income elasticities

Table 10 Long–run income elasticities

Table 11 Short-run price elasticities

Table 12 Long-run price elasticities

Table 13 presents a summary for the short-run and long-run income elasticities, and Table 14 presents a summary for the short-run and long-run price elasticities. As can be seen, in the short-run as well as in the long-run, across all countries, food and housing are necessities (less than 1); durables, transport and recreation are luxuries (greater than 1). In the short-run, clothing is a luxury for developed countries and a necessity for developing countries, while in the long-run it is a necessity for all countries. Medical is a necessity in the short-run and a luxury in the long-run. Restaurant is a luxury in the short-run for all countries, and in the long-run, it is a luxury for developing countries but a necessity for the developed countries. In the short-run, there is no difference in the characteristics of the commodities (food, housing, medical are necessities, and durables, transport, recreation and restaurant meals are luxuries) in developing and developed countries, except clothing, which is a luxury for developed countries and a necessity for developing countries. In the long-run, for both groups of countries, food, clothing and housing are necessities and durables, medical, transport and recreation are luxuries; restaurant meals is a necessity for the developed countries and is a luxury for the developing countries. Irrespective of short-run or long-run and developed or developing countries, the demand for all goods is price inelastic (less than one in absolute value). Overall, our results are in line with those of many previous studies (see, for example, Bustamante and Shimoga 2018; Clements et al. 2006a, b; Clements, et al. 2020).

Table 13 Summary of short-run and long-run income elasticities

Table 14 Summary of short-run and long-run price elasticities

We carried out a meta-analysis of the income and own-price elasticities for 40 countries presented in Tables 9, 10, 11 and 12 to investigate the cross-country heterogeneity in the estimates.⁷ The results indicate substantial heterogeneity in income and own-price elasticities in food, housing and medical between countries both in developed and developing country groups. However, the test of group differences indicates that the group-specific overall effect sizes are not statistically significantly different. The results also reveal substantial variability in the elasticity estimates by development status and level of inequality of countries. The boxplots of long-run income and price elasticity also reveal substantial variability in the elasticity estimates by development status and level of inequality of countries under consideration of this study.⁸

The relationship between price and income elasticities was first considered by Pigou (1910) and is associated with preference independence. Under preference independence, the consumers tastes can be described by a utility function which is the sum of n sub-utility functions, one for each commodity. Deaton (1974) showed that under preference independence, the own-price elasticities are approximately proportional to the income elasticities,

$${{\eta}}_{ii} = \phi {{\eta}}_{i}$$

where ϕ is the income flexibility we introduced in Sect. 3. Several studies such as Clements et al. (1984), Theil (1980) and Selvanathan (1993) found that support for the above relationship with ϕ, generally, lies between − 0.5 and − 0.6.

Scatter plots⁹ of the own-price elasticities against income elasticities for each of the 9 commodities individually and pooled across commodities demonstrate that there is a negative linear relationship between income and price elasticities. The slopes of the scatter diagrams, which are also the income flexibility ϕ, for the 9 commodities and pooled data are presented in Table 15. As can be seen, our estimates are also giving an overall average of − 0.5 and closer to the value reported for ϕ in the above studies.

Table 15 Income flexibility estimates

6 Concluding comments

It is well known that different commodities exhibit different income and price elasticities and that implied elasticities provide valuable information on consumption patterns which in turn influences policy, pricing and production decisions. To this end, this paper modelled the dynamic patterns of consumption behaviour using the most recent available time-series data of 40 developed and developing countries. The empirical analysis of this study involved an investigation of the empirical validity of homogeneity and symmetry constraints and model selection process based on the demand theory hypothesis testing and information inaccuracy estimations to select the preferred demand model out of a static and two dynamic versions of AIDS, namely Static LA-AIDS, Dynamic LA-AIDS and EC-LA-AIDS. The nonlinearities in consumption in countries with high-income inequality using QAIDS were also incorporated into the model estimations in this study. By doing so, this study contributed to the limited body of the literature on cross-country comparison of consumer demand patterns based on a dynamic demand system modelling approach. In particular, the dynamic demand modelling techniques facilitate distinguishing long- and short-run effects of income and prices on demand for various commodity groups. This study represents one of the few studies that provide a comparison of both long-run and short-run income and price elasticities for developed and developing countries.

Overall, the Dynamic LA-AIDS model appeared to be the preferred model for most countries, followed by the EC-LA-AIDS model. The implied elasticities reveal that, in general, there are some differences between the long-run/short-run elasticities and developed/developing countries. Such differences in long-run and short-run elasticities across countries must be acknowledged in the design of long-run policy instruments. This is because long-run policy instruments that rely on short-run implied elasticities lead to market distortions.

Further reinforcing the findings of previous studies, the results of the current study revealed some similarities in consumption patterns of consumers in developed and developing countries, such as food, clothing and housing being determined as necessities and durables, medical, transport and recreation being determined as luxuries, and the demand for all goods is price inelastic. On the other hand, some differences based on country classification were also observed; in the long-run, restaurant meals are a necessity for developed countries and a luxury for developing countries; and in the short-run, clothing is a luxury for developed countries and a necessity for developing countries.

Through a rigorous analysis to identify the appropriate dynamic structure to represent data from 40 different countries, the current study provides new insight into the consumption patterns of consumers in different countries. The findings of this paper can be used as inputs by policy analysts and researchers since the knowledge of price and income responses is undoubtedly an important element in the formulation of fiscal policy or any other type of economic control. This is primarily because consumption absorbs more than 60% of GDP in most economies and is the largest macroeconomic aggregate; it thus has great significance for the state of an economy as a whole.

The study also reveals more advanced and reliable information for industries and businesses on the possible impact of changes in income and prices on consumer demand for various goods. For instance, as income growth is usually larger for poorer countries than for rich countries, and it is expected that the demand for durables, medical, transport and recreation in the future will grow faster for the developing countries than for the developed ones on a per capita basis. This boom in consumption thus will create exciting opportunities for business and investors in developing countries.

Moreover, household consumption patterns are of greater interest to policymakers as these provide important signals for planning government budgetary allocations and tax policy design. For such planning, having up-to-date income and price elasticity estimates for consumer goods at the individual country level is important as they are the key inputs for a number of policy applications, such as public finance policies and economy-wide models (for example, CGE modelling) that are used in designing consumer welfare policies. An up-to-date understanding of household consumption behaviour is also important for manufacturing industries, import and export businesses and investors to understand consumer preferences, identify emerging market opportunities and invest in research and development. The scale and patterns of household consumption are important determinants of environmental impacts because households represent about two-thirds of the demand for raw materials and land as well as the waste flows mobilised by production activities, and their attendant environmental loads globally. The income elasticities estimated in the current study along with the environmental extended multiregional input–output (MRIO) models can be used to examine the impact of income changes on consumption, and how this will translate to changes in carbon footprints.

Though income induces changes in household spending patterns and thereby a country’s consumption patterns, this is also likely to be influenced by the economy’s level of openness, urbanisation and the demographic features of the population. Therefore, future research can be designed to explore the impact of the aforementioned factors on similarities and dissimilarities in consumption patterns of individual countries as well as of a regional group of countries around the globe.