TECHNICAL NOTE 2: MATHEMATICAL DESCRIPTION OF HEDONIC METHODS

In the index number context, the hedonic function pi,t = ht(zi) for a product with multiple varieties—where pi,t is the price of the ith variety in period t and zi is a vector of the ith variety’s characteristics or attributes—plays the same conceptual role as the (scalar) price plays for an undifferentiated good. In the present context, the hedonic function can be viewed as a menu from which individual consumers make choices.

A typical hedonic specification for econometric estimation uses the natural logarithm of an item’s price as the dependent variable and several characteristics as the explanatory variables. The model may contain discrete variables, indicating whether or not a model has a feature, such as a CD drive on a computer, as well as continuous variables, such as the thread count of a fabric. Control variables, such as purchase location or outlet type, may also be included. When, as is typically the case, the explanatory variables are included linearly (rather than, say, logarithmically), the coefficients can be interpreted as giving proportional changes in price associated with a one-unit change in the quality characteristic or from a switch in the dichotomous variable. If explanatory variables enter non-linearly, these proportional changes depend on the values of the explanatory variables.

There is a large theoretical literature on the properties of observed hedonic functions (see, e.g., Rosen, 1974; Muelbauer, 1974; Feenstra, 1995; Barry et al., 1995; Diewert, 2001). Much of this literature is concerned with the extent to which ht provides information on producers’ costs and consumers’ preferences under various assumptions about the nature of competition. This is not our concern here: in general, hedonic functions are reduced-form reflections of details of tastes, technologies, endowments, and strategic behavior in differentiated product markets. In particular, when competition is imperfect, it is generally not possible to infer marginal costs from the observed hedonic functions. We follow most of the theoretical literature and assume what Pollak (1983) calls “Houthakker’s ‘heterogeneous’ or ‘H-characteristics’” approach, which fits products for which consumers purchase one and only one variety. (The alternative, “Lancaster’s ‘linear and additive’ or ‘L-characteristics’” approach, applies when consumers purchase multiple varieties and care about the total amount of each characteristic supplied by all.)

The use of hedonics in the index number context rests on being able to interpret the ht functions as summarizing the menu of alternatives faced by consumers in period t. This raises the general problem that different consumers in fact face different prices and have different stocks of information about their alternatives. Moreover, when price is not linear in the values of characteristics about which consumers care (see Muelbauer, 1974, for some relevant theory), which most hedonic studies seem to find, it follows that, even if ht is a smooth



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