Economic growth and economic development 290

Introduction to Modern Economic Growth Let xt+1 = (x∗ (t + 1) , x (t + 2) , ) ∈ Φ (x∗ (t + 1)) be any feasible plan starting with x∗ (t + 1) By definition, xt = (x∗ (t) , xt+1 ) ∈ Φ (x∗ (t)) Since V ∗ (x∗ (t)) is the supremum starting with x∗ (t), we have ¯ t) V ∗ (x∗ (t)) ≥ U(x ¯ t+1 ) = U(x∗ (t) , x∗ (t + 1)) + β U(x Combining this inequality with (6.14), we obtain ¯ ∗t+1 ) ≥ U(x ¯ t+1 ) V ∗ (x∗ (t + 1)) = U(x for all xt+1 ∈ Φ (x∗ (t + 1)) This establishes that x∗t+1 attains the supremum start- ing from x∗ (t + 1) and completes the induction step, proving that equation (6.13) holds for all t ≥ Equation (6.13) then implies that ¯ ∗t ) V ∗ (x∗ (t)) = U(x ¯ ∗t+1 ) = U(x∗ (t) , x∗ (t + 1)) + β U(x = U(x∗ (t) , x∗ (t + 1)) + βV ∗ (x∗ (t + 1)) , establishing (6.3) and thus completing the proof of the first part of the theorem Now suppose that (6.3) holds for x∗ ∈ Φ (x (0)) Then substituting repeatedly for x∗ , we obtain V ∗ (x (0)) = n X β t U (x∗ (t) , x∗ (t + 1)) + β n+1 V ∗ (x (n + 1)) t=0 In view of the fact that V ∗ (·) is bounded, we have that ¯ ∗) = U(x lim n→∞ ∗ n X β t U (x∗ (t) , x∗ (t + 1)) t=0 = V (x (0)) , thus x∗ attains the optimal value in Problem A1, completing the proof of the second part of the theorem Ô We have therefore established that under Assumptions 6.1 and 6.2, we can freely interchange Problems A1 and A2 Our next task is to prove that a policy achieving the optimal path exists for both problems We will provide two alternative proofs 276