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# INTERTEMPORAL CHOICE AND THE CROSS-SECTIONAL VARIANCE OF MARGINAL UTILITY: Measuring marginal utility and its cross sectional variance

The problem that the parameter vector в is unknown can be handled in several ways. The simplest strategy is to drop the z variables, and to regress the variance of log consumption on the lagged variance. A more satisfactory measure of marginal utility is to assume that г includes only family size and to scale the consumption data using adult equivalent weights. This is equivalent to assuming that the utility function is defined on consumption per adult equivalent, rather than on consumption,

where Рц is the number of adult equivalents, Nul the number of household’s members and crthe elasticity of intertemporal substitution.

A third possibility is to posit more complex and flexible preference specifications that explicitly allow the utility function to depend on labor supply and demographic variables. Since the preference parameters are unknown, they have to be estimated from the data. We propose the following approach. In a first stage, we use the same cohort data and exploit the orthogonality conditions implied by the Euler equation to estimate the parameters of the utility function. In a second stage, we use the estimated parameters to compute the cross sectional variance in equation (9) and to test the hypothesis that /£= 1. In Section 3, rather than performing the first stage, we rely on previous estimates. In particular, we use preference parameters estimated by Euler equations that have been fitted to the same data sets used in this paper. A final econometric problem is that we estimate equations (9) and (10) for all cohorts simultaneously. Since the population groups (cohorts) might be characterized by different variances in ф and v, we always check if the results are sensitive to the introduction of dummies for cohort-specific intercepts.
Comparison with Deaton and Paxson methodology

Average cohort data lend themselves very well to the problem at hand. By definition, birth cohorts are groups with fixed membership. The most intuitive way of testing the implications of equation (5) is to track the cross-sectional variance of the marginal utility as the cohort ages. Indeed, in their seminal contribution, Deaton and Paxson constructed average cohort data for the US, the UK and Taiwan, considered the graphical representation of the cross-sectional variance and tested the hypothesis that the variance of (log) consumption increases with age.