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Introduction to Modern Economic Growth of w) Naturally, the value of the individual at time t can in turn be written as V (y (t)) = max c(t)+b(t)≤y(t) {u (c (t)) + βV (b (t) + w (t + 1))} , which defines the current value of the individual starting with income y (t) and takes into account what the continuation value will be We will see in the next chapter that this is the canonical form of a dynamic programming representation of an infinite-horizon maximization problem In particular, under some mild technical assumptions, this dynamic programming representation is equivalent to maximizing ∞ X β s u (ct+s ) s=0 at time t Intuitively, while each individual lives for one period, he cares about the utility of his offspring, and realizes that in turn his offspring cares about the utility of his own offspring, etc This makes each individual internalize the utility of all future members of the “dynasty” Consequently, fully altruistic behavior within a dynasty (so-called “dynastic” preferences) will also lead to an economy in which decision makers act as if they have an infinite planning horizon 5.4 The Representative Firm The previous section discussed how the general equilibrium economy admits a representative household only under special circumstances The other assumption commonly used in growth models, and already introduced in Chapter 2, is the “representative firm” assumption In particular, recall from Chapter that the entire production side of the economy was represented by an aggregate production possibilities set, which can be thought of as the production facility set or the “production function” of a representative firm One may think that this representation also requires quite stringent assumptions on the production structure of the economy This is not the case, however While not all economies would admit a representative household, the standard assumptions we adopt in general equilibrium theory or a dynamic general equilibrium analysis (in particular no production externalities and competitive markets) are sufficient to ensure that the formulation with a representative firm is without loss of any generality This result is stated in the next theorem 229