Testing an alternative habit persistence model.

• Author: Basmann, Robert L.

1. Introduction

   Marshall long ago noted the phenomenon of past consumption being highly correlated with current consumption. If "habit" was indeed a major determinant of current consumption behavior, then the neoclassical theory of utility maximization should take it into account. A rigorous incorporation of habit formation into consumer theory began with Gorman |12~. Pollak |21; 22; 23~ laid out the theoretical conditions that needed to be satisfied in order for "habit persistence" to be consistent with the utility maximizing model of consumer behavior.

   Competent empirical work in this area was done initially by Houthakker and Taylor |14~ with subsequent studies by Manser |16~, Phlips |18; 19~, Pollak and Wales |24~, Anderson and Blundell |2~ and Blanciforti and Green |8~. These papers have focused on examining the impact of measured past consumption on current consumption. Blundell |9~ provides a comprehensive review of this literature.

   The purpose of this paper is to analyze aggregate consumption behavior by constructing an alternative model of habit persistence. The usual approach in the microeconomics literature is that of Pollak which assumes the argument of the utility function is a transformation of quantities consumed. We choose an alternative tact in that the parameters of the utility function depend upon past consumption, an approach suggested by Gorman |12~ and Peston |17~. Specifically, we develop and test a model which allows us to examine the impact of past consumption behavior on the current structure of preferences. Our model differs from earlier work in that we not only allow "habit" to influence current consumption as in the aforementioned studies, but we also allow "habit" to alter the consumer's preference structure. Our model allows past ratios of expenditures on various commodity groups and lagged quantities of commodities to impact the elasticity of the marginal rate at which consumers substitute one commodity group for another. The model is presented in section two. Section three presents the empirical results of our study. The study is summarized in section four.

2. The Model

   In the standard neoclassical approach, quantities of commodities appear only as arguments of the utility function and do not affect parameters. We hypothesize that past consumption impacts consumer's preference structure. That is, habit formation can be analyzed by examining changes in consumer's preferences which subsequently lead to changes in consumer demand. In order to test this hypothesis we develop a model in which past consumption behavior can have some effect on current preferences.

   Basmann, Molina and Slottje |5~ presented a framework for empirically testing for preferences which depend on prices and expenditures as well as quantities. This methodology is extended here to investigate the existence of habit persistence impacts on consumer preferences as well as consumer demand. Following Phlips and Spinnewyn |20~, we assume perfect information by consumers or deterministic expectations by not explicitly modelling for uncertainty. The theory of stochastic preference changers we discuss in the section on estimation allows us to econometrically incorporate uncertainty. Furthermore, it is assumed that individuals take aggregate expenditures as given and that these expenditures affect their preferences when aggregated. Abel |1~ and Constantinides |10~ have discussed these issues in arguing for habit persistence models vis-a-vis time-separable utility models. We proceed by summarizing the methodology and developing our extension.

   Let U (X;|Alpha~) be a direct utility function with continuous second partial derivatives with respect to X, where |Alpha~ designates the vector of all its parameters. Let |Mathematical Expression Omitted~ designate the marginal rate of substitution of |X.sub.n~ for |X.sub.i~ at the point X. Let ||Alpha~.sub.k~, k = 1,..., m be an observable magnitude different from X and its components. Assume that the direct utility function and all its first and second partial derivatives, |U.sub.i~ and |U.sub.ij~, are differentiable at least once at all points (X) of the budget domain with respect to each of the ||Alpha~.sub.k~. Then each of the marginal rates of substitution |Mathematical Expression Omitted~ is differentiable at every point (X) of the domain with respect to each preference-changing variable ||Alpha~.sub.k~, k = 1, 2,..., m. Following Ichimura |15~ and Tintner |25~, we interpret ||Alpha~.sub.k~ as a preference-changing variable for U (X;|Alpha~) at X, and

   |Mathematical Expression Omitted~

   for at least one i at (X).

   We can express the effect of a change of one economic magnitude on another in terms of mathematical elasticities. Let the elasticity of the marginal rate of substitution (M.R.S.) of |X.sub.n~ for |X.sub.i~ with respect to a change in ||Alpha~.sub.k~ be defined as

   |Mathematical Expression Omitted~

   Recalling that

   |Mathematical Expression Omitted~

   where |U.sub.i~ and |U.sub.n~ are marginal utilities of |X.sub.i~ and |X.sub.n~ respectively, we can define ||Sigma~.sub.h,||Alpha~.sub.k~~ as the elasticity of the marginal utilities with respect to a preference-changing variable ||Alpha~.sub.k~:

   ||Sigma~.sub.h,||Alpha~.sub.k~~ = (||Alpha~.sub.k~/|U.sub.h~)(|Delta~|U.sub.h~/||Delta~||Alpha~.sub.k~) h = 1, 2,..., n and k = 1, 2,..., m. (4)

   The elasticities of the M.R.S. (2.2), with respect to a change in ||Alpha~.sub.j~ are in general:

   |Mathematical Expression Omitted~.

   To test for the existence of habit persistence effects on consumer preferences we use a direct utility function in which past consumption can have an explicit impact on the parameters, |Alpha~, of the utility function. We specify a functional form for U (X;|Alpha~) for which the preference changing variables are a function of past period expenditures, quantities and interest rates. One such form is the Generalized Fechner-Thurstone (GFT) direct utility function. The GFT direct utility function is designed for intertemporal comparisons of consumer equilibria. Time periods are sufficiently long to allow budget constraints, i.e., prices and total expenditure to change significantly. The fundamental assumption is that periods sufficiently long for such changes to occur are also sufficiently long for significant changes of tastes to occur as well. As elsewhere in science, this kind of correlation does not entail that budget constraint changes "cause" or produce the accompanying taste changes. Consider

   |Mathematical Expression Omitted~

   |summation of~||Theta~.sub.i~ = |Theta~. (7)

   The exponents ||Theta~.sub.i~, i = 1, 2,... n, vary from period to period. Their numerical values may be determined directly from the price and expenditure data, Basmann and Slottje |6~ describe this calculation. The ratios ||Theta~.sub.i~/||Theta~.sub.j~ are defined to be equal to |M.sub.i~/|M.sub.j~ where |M.sub.i~ represents expenditure share for good i. This condition is deductively implied by the first order conditions for the utility function given in (6) and (7).

   Once the numerical values of the ||Theta~.sub.i~ have been computed for each time period, t, their variations may be modelled. To do this we specify that

   ||Theta~.sub.i~ = ||Theta~*.sub.i~(P, M, |Psi~)|e.sup.|u.sub.i~~,

   where M is nominal income, |u.sub.i~ is a random variable and ||Psi~.sub.i~ is a systematic function of specified economic and demographic variables. We do not use this result here. We only mention it to help clarify that the GFT form is not a Cobb-Douglas form as we...