Introduction to Modern Economic Growth The expression in (5.1) ignores uncertainty in the sense that it assumes the sequence of consumption levels for individual i, {ci (t)}∞ t=0 is known with certainty If instead this sequence were uncertain, we would need to look at expected util- itymaximization Most growth models not necessitate an analysis of growth under uncertainty, but a stochastic version of the neoclassical growth model is the workhorse of much of the rest of modern macroeconomics and will be presented in Chapter 17 For now, it suffices to say that in the presence of uncertainty, we interpret ui (·) as a Bernoulli utility function, so that the preferences of household i at time t = can be represented by the following von Neumann-Morgenstern expected utility function: Ei0 ∞ X β ti ui (ci (t)) , t=0 where Ei0 is the expectation operator with respect to the information set available to household i at time t = The formulation so far indexes individual utility function, ui (·), and the discount factor, β i , by “i” to emphasize that these preference parameters are potentially different across households Households could also differ according to their income processes For example, each household could have effective labor endowments of ∞ {ei (t)}∞ t=0 , thus a sequence of labor income of {ei (t) w (t)}t=0 , where w (t) is the equilibrium wage rate per unit of effective labor Unfortunately, at this level of generality, this problem is not tractable Even though we can establish some existence of equilibrium results, it would be impossible to go beyond that Proving the existence of equilibrium in this class of models is of some interest, but our focus is on developing workable models of economic growth that generate insights about the process of growth over time and cross-country income differences We will therefore follow the standard approach in macroeconomics and assume the existence of a representative household 5.2 The Representative Household When we say that an economy admits a representative household, this means that the preference (demand) side of the economy can be represented as if there were a single household making the aggregate consumption and saving decisions 218