# INTERTEMPORAL CHOICE AND THE CROSS-SECTIONAL VARIANCE OF MARGINAL UTILITY: Cohort definition and descriptive analysis 3

The last set of figures summarizes the age effect in the cross-sectional variance according to the different measures of marginal utility available in each of the three countries. The age effect is obtained by regressing the cross-sectional variance against a fifth-order age polynomial, a full set of cohort dummies and a set of time dummies constrained to sum to zero and to be orthogonal to a time trend. The estimated age-profiles show that consumption inequality increases with age only for the variance of log consumption. Even in this case, however, the age effects show little spreading of consumption. In the UK, the variance increases only from 0.2 to 0.4 (see Figure 7), in the US from 0.35 to 0.5 (Figure 8) and in Italy from 0.15 to 0.30 (Figure 9). If instead consumption is defined in terms of adult equivalent, or if we directly impute marginal utility, we find very flat, or even declining, age profiles in each of the three countries.

The pattern of the estimated age effects contrast with the findings of Deaton and Paxson, particularly in the case of the US. Using the CEX, they find that the age effects in the US increase from about 0.2 to 0.5 for both the variance of the logarithm of consumption and the variance computed defining consumption per adult equivalent. Part of the difference may derive from differences in the treatment of the data, but not by the different sample periods. They also use quarterly cells, but limit the analysis to the 1980-90 period. However, computing the age effects dropping the 1991-92 years does not change the profiles in Figure 10.

Only under special assumptions does the theory delivers a closed form solution for consumption and allows to test the hypothesis that consumption inequality increases with age on average and over long periods of time. Furthermore, such a prediction of the theory is testable only by arbitrarily normalizing the time effects. Using Deaton and Paxson’s methodology, this prediction of the theory is rejected in each of the three data sets when we take into account the fact that part of the inequality in consumption over the life-cycle is explained by variation in household size and composition. We now turn to our alternative – and in some respect more general – test of the implication of intertemporal choices for the evolution of the cross sectional variance of the marginal utility of consumption.