Economic growth and economic development 397

Introduction to Modern Economic Growth and substituting for c (t) into this lifetime budget constraint in this iso-elastic case, we obtain (8.19) Z c (0) = ả ả µ ¸ Z ∞ (1 − θ) r¯ (t) ρ − + n t dt a (0) + exp − w (t) exp (− (¯ r (t) − n) t) dt θ θ 0 as the initial value of consumption ∞ 8.2.3 Equilibrium Prices Equilibrium prices are straightforward and are given by (8.5) and (8.6) This implies that the market rate of return for consumers, r (t), is given by (8.8), i.e., r (t) = f (k (t)) − δ Substituting this into the consumer’s problem, we have c˙ (t) = (f (k (t)) − δ − ρ) (8.20) c (t) εu (c (t)) as the equilibrium version of the consumption growth equation, (8.14) Equation (8.19) similarly generalizes for the case of iso-elastic utility function 8.3 Optimal Growth Before characterizing the equilibrium further, it is useful to look at the optimal growth problem, defined as the capital and consumption path chosen by a benevolent social planner trying to achieve a Pareto optimal outcome In particular, recall that in an economy that admits a representative household, the optimal growth problem simply involves the maximization of the utility of the representative household subject to technology and feasibility constraints That is, Z ∞ exp (− (ρ − n) t) u (c (t)) dt, max∞ [k(t),c(t)]t=0 subject to k˙ (t) = f (k (t)) − (n + δ)k (t) − c (t) , and k (0) > 0.1 As noted in Chapter 5, versions of the First and Second Welfare Theorems for economies with a continuum of commodities would imply that the solution to this problem should be the same as the equilibrium growth problem of 1In the case where the infinite-horizon problem represents dynastic utilities as discussed in Chapter 5, this specification presumes that the social planner gives the same weight to different generations as does the current dynastic decision-maker 383