Introduction to Modern Economic Growth However, the term in square brackets has a limit of the form 0/0 Let us next write this as ln (1 + ∆t · r) r/ (1 + ∆t · r) = lim = rT ∆t→0 ∆t→0 ∆t/T 1/T where the first equality follows from l’Hopital’s rule Therefore, lim v (T ) = exp (rT ) Conversely, $1 in T periods from now, is worth exp (−rT ) today The same reasoning applies to discounting utility, so the utility of consuming c (t) in period t evaluated at time t = is exp (−ρt) u (c (t)), where ρ denotes the (subjective) discount rate 5.6 Welfare Theorems We are ultimately interested in equilibrium growth But in general competitive economies such as those analyzed so far, we know that there should be a close connection between Pareto optima and competitive equilibria So far we did not exploit these connections, since without explicitly specifying preferences we could not compare locations We now introduce these theorems and develop the relevant connections between the theory of economic growth and dynamic general equilibrium models Let us start with models that have a finite number of consumers, so that in terms of the notation above, the set H is finite However, we allow an infinite num- ber of commodities, since in dynamic growth models, we are ultimately interested in economies that have an infinite number of time periods, thus an infinite number of commodities The results stated in this section have analogues for economies with a continuum of commodities (corresponding to dynamic economies in continuous time), but for the sake of brevity and to reduce technical details, we focus on economies with a countable number of commodities â ê Therefore, let the commodities be indexed by j ∈ N and xi xij j=0 be the â ê consumption bundle of household i, and ω i ≡ ω ij j=0 be its endowment bundle In addition, let us assume that feasible xi ’s must belong to some consumption set X i ⊂ R∞ We introduce the consumption set in order to allow for situations in which an individual may not have negative consumption of certain commodities The consumption set is a subset of R∞ since consumption bundles are represented by infinite 233