In this chapter we examine dynamic models of consumer choice under uncertainty. We continue, as in the Ramsey model, to take the decision of the household with regard to labor supply as given, assuming that each household provides a unit of labor per period. We also assume that the household can borrow and lend freely in competitive capital markets.

In these inter-temporal models consumption does not depend on current household income but on total household wealth, which consists of its current portfolio of assets, plus the present value of expected future labor income. In this sense, consumption smooths temporary changes in income, as it depends on the “permanent income” (Friedman 1957) or the “life cycle income” of the household (Modigliani Brumberg 1954).

Analyzing the consumption function of a representative household under certainty in Chapter 2, we concluded that the household consumes a proportion of its total wealth that depends on the evolution of interest rates, the pure rate of time preference, the elasticity of inter-temporal substitution in consumption and the population growth rate. In this model, the impact of interest rates on the propensity to consume out of total wealth depends on the elasticity of inter-temporal substitution. An increase in interest rates has two kinds of effects on the propensity to consume out of total wealth. First, it induces the household to substitute future for current consumption, as it increases the cost of current consumption relative to future consumption. This is the result of inter-temporal substitution in consumption. Secondly, an increase in interest rates increases household income from capital, inducing it to increase both current and future consumption. This is the income effect. If the inter-temporal elasticity of substitution is greater than one, the propensity to consume out of total wealth decreases when interest rates rise, because the substitution effect prevails on the income effect. If the inter-temporal elasticity of substitution is less than unity, the propensity to consume out of total wealth increases when interest rates rise, because the income effect prevails upon the substitution effect. Finally, in the case in which the inter-temporal elasticity of substitution of consumption is equal to one, which corresponds to logarithmic preferences, the two results cancel each other out, and the propensity to consume out of total wealth is independent of the path of real interest rates, as it is equal to difference between the pure rate of time preference and the population growth rate. In addition, an increase in real interest rates leads to a decrease in the present value of future labor income, reducing the overall wealth of the household and leading to lower consumption, even if the case where the elasticity of inter-temporal substitution is equal to one. Essentially, the wealth effect of real interest rates on the present value of income from employment reinforces the substitution effect on current consumption.

The choice of consumption under conditions of uncertainty is linked to the portfolio allocation decisions of the household (Samuelson 1969, Merton 1969). Under uncertainty, consumption generally depends on the same factors as under certainty, only in the case of quadratic preferences, which guarantee certainty equivalence. In all other cases, we cannot go beyond the first order conditions under uncertainty, and solve explicitly for consumption, unless we make further restrictive assumptions about the preferences of households or the variability of labor income.

With regard to portfolio choice, this model results in the consumption capital asset pricing model. This suggests that, under quadratic preferences, the expected return premium of a risky asset is proportional to the covariance of its return with consumption. This factor of proportionality is sometimes referred to as a consumption beta, and can be used to explain the valuation of risky assets.

However, under uncertainty, both the permanent income hypothesis and the consumption capital asset pricing model rely on very restrictive assumptions. In addition, empirical studies suggest two puzzles that throw doubt on the validity of both. One is the “excess sensitivity” of consumption to changes in current income, and the second is the “equity premium” puzzle.

Thus, although the inter-temporal model of consumption under uncertainty is a useful framework for modeling consumption and portfolio allocation decisions, one needs to go beyond the permanent income hypothesis and the consumption capital asset pricing model and consider less restrictive set ups, with precautionary savings, borrowing constraints and capital market imperfections.

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