Introduction to Modern Economic Growth ¯ defined by into a level of capital-labor ratio greater than k, ¡ ¢ δ k¯ = f k¯ , since this is the capital-labor ratio that would sustain itself when consumption is set ¯ it can never exceed k ¯ If it starts equal to If the economy starts with k (0) < k, ¯ it can never exceed k (0) Therefore, without loss of any generality, with k (0) > k, h i we can restrict consumption and capital stock to lie in the compact set 0, k , where ¡ © ê â ê k f max k (0) , k¯ + (1 − δ) max k (0) , k¯ Consequently, Theorems 6.1-6.6 immediately apply to this problem and we can use these results to derive the following proposition to characterize the optimal growth path of the one-sector infinite-horizon economy Proposition 6.1 Given Assumptions 1, and 3’, the optimal growth model as specified in (6.33) and (6.34) has a solution characterized by the value function V (k) and consumption function c (k) The capital stock of the next period is given by s (k) = f (k)+(1 − δ) k−c (k) Moreover, V (k) is strictly increasing and concave and s (k) is nondecreasing in k Proof Optimality of the solution to the value function (6.35) for the problem (6.33) and (6.34) follows from Theorems 6.1 and 6.2 That V (k) exists follows from Theorem 6.3, and the fact that it is increasing and strictly concave, with the policy correspondence being a policy function follows from Theorem 6.4 and Corollary 6.1 Thus we only have to show that s (k) is nondecreasing This can be proved by contradiction Suppose, to arrive at a contradiction, that s (k) is decreasing, i.e., there exists k and k0 > k such that s (k) > s (k0 ) Since k0 > k, s (k) is feasible when the capital stock is k0 Moreover, since, by hypothesis, s (k) > s (k0 ), s (k0 ) is feasible at capital stock k By optimality and feasibility, we must have: V (k) = u (f (k) + (1 − δ) k − s (k)) + βV (s (k)) ≥ u (f (k) + (1 − δ) k − s (k0 )) + βV (s (k0 )) V (k ) = u (f (k0 ) + (1 − δ) k0 − s (k0 )) + βV (s (k0 )) ≥ u (f (k0 ) + (1 − δ) k0 − s (k)) + βV (s (k)) 293