Nonparametric Models in Consumer Behaviour

In its most basic form, the classical theory of consumer behaviour describes how a consumer allocates a given budget to a set of goods and services, while taking as given the prices of these goods and services. Although the most important implications of this theory have been known at least since Hicks (1939)’ Value and Capital, it has taken another fifteen years before the theory was brought in its entirety to real world data. This was done by Richard Stone (1954), who applied the Linear Expenditure System (LES) to British aggregate demand data. As shown by Geary (1950), the underlying preferences of the LES are of the Stone-Geary type. In other words, if one maximizes a direct utility function that presents Stone-Geary preferences subject to the consumer’s budget constraint, then one obtains the LES as the relation between the quantities purchased by the consumer, and her budget and the prices she is faced with. As is well-known, this system of Marshallian demand functions will satisfy all the theoretical implications of the maximization of rational preferences subject to a linear budget constraint. Firstly, it will satisfy adding-up, which implies that the sum of all the expenses on the different goods and services purchased by the consumer will be equal to the consumer’s budget. Secondly, the demand will be homogeneous of degree zero in prices and budget, which implies that if one multiplies all prices and the budget by, say, two, that the quantities purchased will remain unaffected (this implies that the consumer does not suffer from money illusion). Finally, the Slutsky matrix, that contains all the Hicksian or compensated price effects, will be symmetric and negative semidefinite. The latter implies, among others, that the consumer’s Hicksian or compensated demand of a given good can never increase following a price increase of that good, ceteris paribus. Stone’s work thus implies that for the first time a demand system was estimated that in principle could satisfy all the theoretical implications of the classical theory of consumer behaviour. On its turn, this implies that, for the first time, an estimate of, in Stone’s case, an average consumer’s preferences was obtained based on real-world data. An important feature of the LES is that it makes specific assumptions on the relation between the quantities purchased by the consumer, her budget and the prices she is faced with. These, on their turn, imply a particular specification for the consumer's preferences (in this case of the Stone-Geary type). These specific assumptions are far from innocuous. They potentially rule out consumer behaviour that is theoretically possible. For example, income elasticities that are derived from the LES, will all be positive, which implies that the LES can only capture consumer behaviour for goods and services that are normal. Inferior goods cannot be modelled by means of the LES. The same applies to the substitutability pattern between the modelled goods and services: with Stone-Geary preferences, all goods are substitutes for each other; complementary goods are ruled out by construction.Throughout the years, more general systems of demand equations have been proposed in the literature that allow the econometrician to capture richer behavioural patterns than those that can be modelled by means of the LES. Examples of such, often widely used, demand systems are the Rotterdam model of Barten (1964) and Theil (1965), the translog model of Christensen, Jorgenson and Lau (1975), the Almost Ideal Demand System of Deaton and Muellbauer (1980), the Quadratic Almost Ideal Demand System (QUAIDS) of Banks, Blundell and Lewbel (1997) and the Exact Affine Stone Index (EASI) demand system of Lewbel and Pendakur (2009). Still, all these demand systems have in common that they impose additional structure on the form consumer behaviour can take, which goes beyond the pure theory of consumer behaviour. In other words, theory gives only little guidance on the specific functional form of demand or the consumer’s preferences. The approach just described can be coined a parametric approach: the functional form of the preferences or the demand system is assumed to be known from the outset by the econometrician, while the unknown parameters in this functional form are to be estimated by means of econometric techniques. The latter are to be applied to the data at hand that captures observed consumer behaviour. The strength of the parametric approach is that it not only allows econometricians to easily apply and test the theory of the consumer’s utility maximizing behaviour, but that it also opens up a toolbox that contains plenty of instruments that are directly useful to evaluate economic policy. Think about the estimation of price and income elasticities, or the calculation of Hicksian equivalent and compensation variations to evaluate the distributional effects of price changes coming from, for example, indirect tax reforms like an increase in the taxes on gasoline or the introduction of a sugar tax. The main disadvantage of the parametric approach, though, is that it is prone to misspecification. As mentioned before, the particular functional form for the demand system used by the econometrician goes beyond the pure theory of consumption behaviour. The parametric approach implies additional assumptions, on top of other, mainly statistical assumptions, to bring the theory to the data. And these assumptions might not fit well with the data at hand. A rejection of, say, Slutsky symmetry, may either be due to the theory of consumer behaviour that is not appropriate to explain observed demand behaviour, or it may be due to the use of a functional specification that is not suitable for the data at hand. The nonparametric approach is an alternative way to bring the theory of consumer behaviour to the data. In a nutshell, the nonparametric approach aims to analyse consumer behaviour by starting from the pure theory of consumer behaviour while imposing only minimal additional assumptions that are needed to bring the theory to the data. Most importantly, it aims to analyse consumer behaviour without making any assumptions on the specific system of demand equations applicable to the consumer or without assuming specific preferences of that consumer. The term “nonparametric approach” has multiple meanings though. In what follows, we will concentrate on two meanings that figured prominently in the applied demand literature. The first meaning refers to the theory of revealed preference, that was initially proposed by Samuelson (1938, 1948). The second meaning refers to applications of consumer behaviour, whereby the relation between demand, income and/or prices can be of a very general shape that does not refer to known parametric shapes of demand or preferences. This general shape is then typically estimated by means of nonparametric regression techniques. We will end this encyclopedia entry with a discussion of a final nonparametric approach that combines the revealed preference approach with nonparametric (or semi-parametric) regression.