Economic growth and economic development 298

Introduction to Modern Economic Growth x ∈Φ (x (0)), define ∆x ≡ lim T →∞ T X t=0 β t [U(x∗ (t) , x∗ (t + 1)) − U (x (t) , x (t + 1))] as the difference of the objective function between the feasible sequences x∗ and x From Assumptions 6.2 and 6.5, U is continuous, concave, and differentiable By definition of a concave function, we have ∆x ≥ lim T →∞ T X t=0 β t [∇Ux (x∗ (t) , x∗ (t + 1)) · (x∗ (t) − x (t)) +∇Uy (x∗ (t) , x∗ (t + 1)) · (x∗ (t + 1) − x (t + 1))] for any x ∈Φ (x (0)) Using the fact that x∗ (0) = x (0) and rearranging terms, we obtain ∆x ≥ ( T X β t [∇Uy (x∗ (t) , x∗ (t + 1)) + β∇Ux (x∗ (t + 1) , x∗ (t + 2))] · (x∗ (t + 1) − x (t + 1)) lim T →∞ t=0 ª + β ∇Uy (x∗ (T ) , x∗ (T + 1)) · (x∗ (T + 1) − x (T + 1)) T Since x∗ satisfies (6.21), the terms in first line are all equal to zero Therefore, substituting from (6.21), we obtain ∆x ≥ − lim β T ∇Ux (x∗ (T ) , x∗ (T + 1)) · (x∗ (T ) − x (T )) T →∞ ≥ − lim β T ∇Ux (x∗ (T ) , x∗ (T + 1)) · x∗ (T ) ≥ T →∞ where the second inequality uses the fact that from Assumption 6.4, U is increasing in x, i.e., ∇x U ≥ and x ≥ 0, and the last inequality follows from (6.25) This implies that ∆x ≥ for any x ∈Φ (x (0)) Consequently, x∗ yields higher value than any feasible x ∈Φ (x (0)), and is therefore optimal Ô We now illustrate how the tools that will so far can be used in the context of the problem of optimal growth, which will be further discussed in Section 6.6 284