# INTERTEMPORAL CHOICE AND THE CROSS-SECTIONAL VARIANCE OF MARGINAL UTILITY: The cross-sectional variance of marginal utility 2

If the instantaneous utility function is isoelastic, function of the logarithm of consumption:

where a is the elasticity of intertemporal substitution, ei t+x an expectational error and ф( f+1 the higher conditional moments of the distribution of consumption in equation (3).2 The second equality in equation (4) follows by defining £M+i = ф(1+1 – ф( + t+], where ф. indicates the time-series average of the conditional moments. The cross-sectional variance of the two sides of equation (4) is:

Equation (5), as such, cannot be measured empirically. For one thing, the parameters в are unknown; second, the variable v is by definition unobservable; and finally, ф includes unknown parameters and moments of a distribution which has not yet been specified. But it does highlight a sharp prediction of the theory, i.e. that the coefficient of the lagged cross-sectional variance of marginal utility equals l.3 It is precisely because of this property that in the simple case of quadratic utility the cross sectional variance of consumption increases, on average, with age. In the remaining of this section we illustrate how we propose to test this important implication of intertemporal optimization.

To make equation (5) operational, we need to make a number of identifying assumptions. We deal with the first problem (that the 0parameters are unknown) below. For the present we assume that the parameter vector# is known. To address the fact that v and ф are unobservable, one can write equation (4) as:

Assuming that the real interest rate is common across households, the cross-sectional variance of the two sides of equation (6) is given by the following expression:

Only the first term on the right hand side can be measured empirically using cross sectional data. However, the second and third terms can be decomposed into a part that is constant over time (such as the variance of ф and its covariance with the other components) and a time-variant component. Equation (7) can then be re-written as: Electronic Payday Loans Online

where E(co{+\) denotes the time series average of 0)t+\, (Do includes the time-invariant terms of the right-hand-side of equation (7):

The terms and GJl+J are implicitly defined by the second equality in equation (8). Note that most components of 67,+/ do not vanish. An exception is cov(0y,£rr+I), which is zero under the permanent income hypothesis.