# INTERTEMPORAL CHOICE AND THE CROSS-SECTIONAL VARIANCE OF MARGINAL UTILITY: The cross-sectional variance of marginal utility

We take off from Deaton and Paxson and their exploration of the implications of the permanent income hypothesis for the evolution of consumption and income inequality. It is well known that if the utility function is quadratic and if the real interest rate equals the discount rate, optimal consumption for individual i follows a martingale:

where variances and covariances are computed over a cross-section of households whose composition is constant over time. If aggregate variables are part of the information set of each agent, Deaton and Paxson show that the time average of cov(c. ; ,£iV+I) is zero. Equation (2) then implies that in a group with fixed membership the variance of consumption of individuals aged t is stochastically dominated by the variance of consumption of the same individuals aged r+1 .In other terms, in a stationary population, consumption inequality increases with age on average and over long periods of time. Since £iit+j represents the annuity value of the innovations in permanent income, consumption inequality increases until individuals face income shocks. Such dispersion should at least slow down for older households if earnings shocks dominate interest rate shocks. Electronic Payday Loans Online

**Relaxing certainty equivalence**

Equations (1) and (2) can be generalized to more flexible preference specifications. Suppose that utility is intertemporally separable, and that the instantaneous utility function, defined over non-durable consumption, depends on a set of conditioning variables z (demographic and labor supply, let us say) and on an unobservable component v that captures unobserved heterogeneity. In this case it can be immediately shown that the Euler equation that applies to individual i is:

where //(.) is the marginal utility of consumption, rf+} the rate of return on a generic asset к, set of parameters of the utility function and p the discount factor. Note that both в and p are assumed to be time-invariant and common to all individuals. Inspection of equation (3) reveals that if one abandons certainty equivalence, it is not necessarily the case that the cross-sectional variance of marginal utility increases with age.