Engel curves and systems of demand functions

The functional forms of the Engel curves implied by several popular complete systems of demand functions are discussed. Among the systems are those generated by generalized quadratic indirect utility (the translog system, the generalized Leontief system, etc.), the linear and quadratic expenditure systems, and Deaton and Muellbauer's PIGL and AIDS. A new system, which can produce the Engel curves of all these systems as special cases, is suggested. The different Engel curves are tested against data from a Norwegian survey of consumer expenditure. This is done both in a complete system of Engel curves for eight commodities and commodity by commodity.     

Applications of robust estimation techniques in demand analysis

Estimates are presented of the demand for tobacco in Italy in the context of a demand system approach. The grouping of goods is based on Varian's non-parametric test for weak separability. The parametric demand model is Banks, Blundell and Lewbel's QUAIDS model, which combines the empirical flexibility of quadratic logarithmic Engel curves with integrability. Estimation is by iterative GMM procedure. The estimates are used in an analysis of welfare-improving tax reforms which incorporates distributional data. The results are discussed with respect to the special features of the Italian tobacco market.     

Explaining shapes of Engel curves: the impact of differential satiation dynamics on consumer behavior

This paper presents a formal model in which differential satiation dynamics of various consumer needs explain (not only describe) the shapes of Engel curves. In the model, individuals allocate their income to various consumption categories proportional to corresponding need deprivation states, a decision making process called matching. The model allows explaining some empirical regularities that other models have difficulties accounting for. It can, for example, reconstruct that income elasticities for food tend to decrease with rising income, and that goods that are luxuries at relatively low income levels can become necessities at higher income levels. Moreover, the paper compares the Engel curves obtained from the matching model with Engel curves obtained from a utility maximization model. While both types of Engel curves are relatively similar at high income levels, at lower income levels matching and maximization lead to very different allocations of income. The paper shows that a given amount of income redistribution leads to less additional welfare when individuals follow matching behavior than when they maximize their utility. Accordingly, to obtain a given amount of additional welfare more income redistribution is needed than a policy maker who (mistakenly) assumes that individuals rationally maximize their utility predicts.     

A conditional version of working's model

Working's (1943) model is shown to imply conditional Engel curves of the Working form when the income elasticity of demand for the group of goods as a whole is about unity.     

Is the Engel curve approach viable in the estimation of alternative PPPs?

Conventional estimates of purchasing power parities (PPP) rely on cross-country price data. Using Engel curves, Almås (Am Econ Rev 102:1093–1117, 2012) was, however, able to show that PPPs contain substantial bias. Since constructing conventional estimates is expensive and time consuming, Almås’ idea of employing Engel curves is welcome. This article examines the viability of the Engel curve approach to PPP and its sensitivity to differences in relative prices and preferences by estimating Engel curves not only between countries but also for regions within a given country. My empirical evidence from the United States and Norway suggests that the differences can be problematic, but not sufficiently to discredit the new methodology. A pragmatic approach to PPP estimation between countries that are different is to compute a PPP band, rather than a point estimate. I present a practical example of this using expenditure data from 2001, which yields a band for NOK and US dollar.     

Demographic versus expenditure flexibility in Engel curves

We consider the effects of demographic and expenditure variables on consumer demand in a system of Engel curves using a smooth coefficient semiparametric model where the expenditure effects on the budget shares vary nonparametrically with demographic variables such as the age of head and number of children in the household. Our findings, based on UK micro data, suggest that with a smooth coefficient semiparametric model there is no need for nonlinear logarithmic expenditure effects in the budget shares. Furthermore, we find evidence of a trade-off between demographic and expenditure effects in Engel curves and that a rank-2 system of Engel curves where the logarithmic expenditure effects are allowed to vary with demographic characteristics either nonparametrically or as a third degree polynomial function cannot be rejected against a rank-3 (quadratic logarithmic) model. The implications on household behavior and welfare are also examined. We would like to thank an anonymous referee and Baldev Raj, the editor, for useful comments and suggestions. We would also like to thank the University of Cyprus for financial support, Theofanis Mamuneas for stimulating discussions and the Office of National Statistics for making available the UK Family Expenditure Survey data through the ESRC Data Archive. The last author would also like to acknowledge the financial support from SSHRC of Canada.     

A non-separable generalisation of the linear expenditure system allowing non-linear Engel curves

This paper proposes and estimates a new generalisation of the LES that allows for both non-linear Engel curves and non-separable preferences. The empirical results decisively reject linear Engel curves on pooled cross section data and separable preferences on pure time series data.     

On the estimation of 'an implicitly additive demand system'

Motivated by a lack of Engel flexibility in commonly used demand systems, Rimmer and Powell developed a new demand system. This system, referred to AIDADS, is implicitly, directly additive, possesses marginal budget shares and thus Engel elasticities, that vary nonlinearly with expenditure such that predicted budget shares are restricted to the [0,1] interval. Due to these attractive Engel properties, AIDADS represents a significant contribution to the literature on demand analysis. This paper presents an alternative estimation procedure to the one used by Rimmer and Powell and examines its properties via a case study. The proposed approach avoids a linear approximation employed in Rimmer and Powell's estimation framework. Based on a small Monte Carlo study, it appears that the approach produces more accurate estimates of the parameters and Engel elasticities.     

Innovation,structural change and demand evolution: does demand saturate?

In this paper the exploration of the interaction mechanisms between the evolution of demand and the overall process of differentiation of the economic system is pursued in a twofold way: (i) the question is asked whether demand indeed saturates and whether such saturation is required for an ongoing economic development; (ii) the impact of the income distribution on the evolution of demand and thus on economic development is investigated. To study the existence and impact of demand saturation, Engel curves are calculated under different conditions. To study the impact of income distribution, TEVECON distinguishes within the population of the economic system two social classes, which differ in their level and quality of education, and thus for their human capital and wages.     

Imposing local curvature in the QUAIDS

In this paper, we build on Ryan and Wales (1998), Moschini (1999), and Serletis and Shahmoradi (2007) and impose curvature conditions locally on the quadratic Almost Ideal Demand System (AIDS) model of Banks et al. (1997), an extension of the simple AIDS model of Deaton and Muellbauer (1980) that can generate quadratic Engel curves [that is, rank-three demand systems, in the terminology of Lewbel (1991)]. In doing so, we exploit the Slutsky matrix of second order derivatives of the indirect utility function.     

Using inverted Engel curves to estimate material standards of living in a household

This article demonstrates how to estimate latent total consumption expenditure or material standard of living in households by inverting estimated Engel curves. While the conventional estimator, total purchase expenditure, is unbiased for latent total household consumption expenditure, it is not variance minimizing since it is an un-weighted sum. In two stages, this article derives a variance-minimizing, unbiased estimator by first estimating and inverting Engel curves; then combining the estimators from the inverted Engel curves. The employed latent variable method allows for utilization of non-expenditure relations. The suggested method may help improve the accuracy in studies of consumption inequality and tax evasion.      

The finite basis property and exact aggregation

An individual demand function is said to satisfy the finite basis property when its set of Engel curves has a finite spanning subset. In the present letter we discuss why this property underlies the theory of exact aggregation.     

Intertemporal propensity to consume

Cross-sectionally estimated Engel curves tend to exhibit shapes that imply ultimately decreasing marginal propensities to consume as income rises. This suggests at first sight a tendency to slow down in private consumption. This tendency has to be reconsidered taking into account the fact that Engel curves usually shift significantly over time. We introduce the notion of “intertemporal propensity to consume” to shed more light on the drivers and patterns of the evolution of Engel curves. Using this notion, we are able to identify and measure the direction of the expansion path of consumption as time goes by, while controlling for household characteristics and price levels. Using German household data, we show that, over time, the intertemporal propensity to consume systematically offsets the consumption paths implied by the marginal propensity to consume that is derived cross-sectionally at one point in time. We claim that this difference between marginal and intertemporal propensity to consume should be ascribed to variations in the socioeconomic structures that occur over time and that may be linked both to the supply side (innovation and technical change) and the demand side (learning dynamics of consumers). We also show that changes in relative prices play a minor role in the evolution of Engel curves.     

A globally flexible,quadratic almost ideal demand system: an application to demand for meats and fish in Taiwan

A new demand system, called a globally flexible, quadratic almost ideal demand system (GF–QUAIDS), integrates both the quadratic Engel curve and the Fourier series. This novel GF–QUAIDS model is applied to study the demand for meats and fish in Taiwan. Empirical results show that the GF–QUAIDS fits the data well and that, by using the likelihood ratio test, its nested models are rejected, including the locally flexible functional form and the linear Engel curve. Nevertheless, other empirical model-selection measures indicate that the GF–QUAIDS can be as good as its famous nested models such as the AIDS, QUAIDS and GF–AIDS.     

On the Functional Form of Engel Curves: The Australian Household Expenditure Survey 1975–76*

This paper considers the properties of a class of models which is based on the Addilog demand system. These models not only fulfil the adding-up criterion but also ensure that the predicted expenditures are non-negative and a subset of these models permits the existence of saturation levels for certain commodities. A goodness of fit comparison between this class and other systems of Engel curves favours the Addilog model. The estimated expenditure elasticities compare favourably with previous estimates of Australian household expenditure patterns and it is observed that the elasticities for health and alcohol and tobacco expenditures have decreased substantially from their levels in the late 1960s.     

Empirical demand systems incorporating intertemporal consumption dynamics

This article presents and estimates demand systems by explicitly incorporating intertemporal consumption behavior as summarized by the Euler equation. Demand systems are characterized by two indirect utility functions which are effectively globally regular and can better approximate nonlinear Engel curves. Furthermore, an exact and nonlinear Euler equation is derived without a log approximation. This equation is estimated jointly with the demand functions by a careful implementation of the orthogonality conditions using generalized method of moments. We illustrate the techniques by estimating the demand system and Euler equation for Australian aggregate data. Results generally indicate that the proposed methods are promising. The estimated rate of time preference is fairly small while the restrictions producing the moment equations are not rejected. Estimated Frisch price elasticities, which are relevant in an intertemporal setting, appear reasonable, and the intertemporal elasticity of substitution for consumption is found to be small, which are consistent with the findings in earlier studies.     

Effective global regularity and empirical modelling of direct, inverse, and mixed demand systems

Abstract .  This paper utilizes the notion of 'effective global regularity' and the intuition stemming from Cooper and McLaren (1996)'s General Exponential Form to develop a family of 'composite' (product and ratio) direct, inverse and mixed demand systems. Apart from having larger regularity regions, the resulting specifications are also of potentially arbitrary rank, which can better approximate non-linear Engel curves. We also make extensive use of duality theory and a numerical inversion estimation method to rectify the endogeneity problem encountered in the estimation of the mixed demand systems. We illustrate the techniques by estimating different types of demand systems for Japanese quarterly meat and fish consumption.     

Recent approaches to estimating Engel curves

Classical approaches of estimating cross-section Engel curves are based on parametric models. However, misspecification of a parametric model implies that information of structural nature might be masked. An alternative avoiding problems related to predetermined functional relations is the nonparametric approach. This paper surveys recent advances of nonparametric statistics in their relevance to estimating cross-section Engel curves.     

Habit formation and long-run utility functions

This paper extends the work on habit formation of Pollak [Habit formation and dynamic demand functions. J. Polit. Econ. (1970)] and provides a critical counterexample to a conjecture of von Weizsäcker [Notes on endogenous changes of tastes, J. Econ. Theory (1971)] concerning the existence of a “long-run utility function.“ A linear specification of habit formation is applied to a general system of demand functions with linear Engel curves. It is shown that there exists a utility function which rationalizes the long-run demand functions if and only if they are the steady-state solution to a system of short-run demand functions generated by an additive utility function.     

Determinants of food consumption in rural Sierra Leone: Application of the quadratic expenditure system to the consumption-leisure component of a household-firm model

This paper reports the application of a quadratic expenditure system with demographic variables to the household consumption-leisure choice component of a household-firm model. A system of seven commodities is estimated: including five foods, non-food and leisure. Appropriate for such a level of disaggregation a demand system is used, the Quadratic Expenditure System, which allows for a flexible relationship between full income and commodity expenditures without sacrificing parsimony in parameters. Demographic data on households are explicitly incorporated into the model allowing for a richer specification than can be achieved by using per capita variables. The data are from a cross-section survey of households in rural Sierra Leone. Price variation exists by region, permitting estimation of a complete demand system. Engel curves are found to be significantly non-linear, with marginal expenditure on rice, the major staple, declining with higher income. Most foods are found to be reasonably price responsive with sizeable own price substitution effects, declining with higher income. Aggregate labor supply is found to be price inelastic.