Economic growth and economic development 299

Introduction to Modern Economic Growth Example 6.4 Consider the following optimal growth, with log preferences, CobbDouglas technology and full depreciation of capital stock max {c(t),k(t+1)}∞ t=0 subject to ∞ X β t ln c (t) t=0 k (t + 1) = [k (t)]α − c (t) k (0) = k0 > 0, where, as usual, β ∈ (0, 1), k denotes the capital-labor ratio (capital stock), and the resource constraint follows from the production function K α L1−α , written in per capita terms This is one of the canonical examples which admits an explicit-form characteriza- tion To derive this, let us follow Example 6.1 and set up the maximization problem in its recursive form as V (x) = max {ln (xα − y) + βV (y)} , y≥0 with x corresponding to today’s capital stock and y to tomorrow’s capital stock Our main objective is to find the policy function y = π (x), which determines tomorrow’s capital stock as a function of today’s capital stock Once this is done, we can easily determine the level of consumption as a function of today’s capital stock from the resource constraint It can be verified that this problem satisfies Assumptions 6.1-6.5 The only non-obvious feature here is whether x and y indeed belong to a compact set The argument used in Section 6.6 for Proposition 6.1 can be used to verify that this is the case, and we will not repeat the argument here Consequently, Theorems 6.1-6.6 apply In particular, since V (·) is differentiable, the Euler equation for the one-dimensional case, (6.22), implies xα = βV (y) −y The envelope condition, (6.23), gives: αxα−1 xα − y 285 V (x) =