Introduction to Modern Economic Growth β ∈ (0, 1) This can be written as (6.33) max {c(t),k(t)}∞ t=0 subject to (6.34) ∞ X β t u (c (t)) t=0 k (t + 1) = f (k (t)) + (1 − δ) k (t) − c (t) and k (t) ≥ 0, with the initial capital stock given by k (0) We continue to make the standard assumptions on the production function as in Assumptions and In addition, we assume that: Assumption 3’ u : [c, ∞)→ R is continuously differentiable and strictly con- cave for c ∈ [0, ∞) This is considerably stronger than what we need In fact, concavity or even continuity is enough for most of the results But this assumption helps us avoid inessential technical details The lower bound on consumption is imposed to have a compact set of consumption possibilities We refer to this as Assumption 3’ to distinguish it from the very closely related Assumption that will be introduced and used in Chapter and thereafter The first step is to write the optimal growth problem as a (stationary) dynamic programming problem This can be done along the lines of the above formulations In particular, let the choice variable be next date’s capital stock, denoted by s Then the resource constraint (6.34) implies that current consumption is given by c = f (k) + (1 − δ) k − s, and thus we can write the open growth problem in the following recursive form: (6.35) V (k) = max {u (f (k) + (1 − δ) k − s) + βV [s]} s∈G(k) where G (k) is the constraint correspondence, given by the interval [0, f (k)+(1 − δ) k− c], which imposes that consumption cannot fall below c and that the capital stock cannot be negative It can be verified that under Assumptions 1, and 3’, the optimal growth problem satisfies Assumptions 6.1-6.5 of the dynamic programming problems The only non-obvious feature is that the level of consumption and capital stock belong to a compact set To verify that this is the case, note that the economy can never settle 292