Introduction to Modern Economic Growth grows at the rate n, starting with L (0) = 1, so that total population is (8.2) L (t) = exp (nt) All members of the household supply their labor inelastically Our baseline assumption is that the household is fully altruistic towards all of its future members, and always makes the allocations of consumption (among household members) cooperatively This implies that the objective function of each household at time t = 0, U (0), can be written as Z ∞ exp (− (ρ − n) t) u (c (t)) dt, (8.3) U (0) ≡ where c (t) is consumption per capita at time t, ρ is the subjective discount rate, and the effective discount rate is ρ − n, since it is assumed that the household derives utility from the consumption per capita of its additional members in the future as well (see Exercise 8.1) It is useful to be a little more explicit about where the objective function (8.3) is coming from First, given the strict concavity of u (·) and the assumption that within-household allocation decisions are cooperative, each household member will have an equal consumption (Exercise 8.1) This implies that each member will consume C (t) L (t) at date t, where C (t) is total consumption and L (t) is the size of the representative c (t) ≡ household (equal to total population, since the measure of households is normalized to 1) This implies that the household will receive a utility of u (c (t)) per household member at time t, or a total utility of L (t) u (c (t)) = exp (nt) u (c (t)) Since utility at time t is discounted back to time with a discount rate of exp (−ρt), we obtain the expression in (8.3) We also assume throughout that Assumption 4’ ρ > n This assumption ensures that there is in fact discounting of future utility streams Otherwise, (8.3) would have infinite value, and standard optimization techniques would not be useful in characterizing optimal plans Assumption 4’ makes sure that 374