Introduction to Modern Economic Growth In this case, we have a finite-dimensional optimization problem and we can simply look at first-order conditions Moreover, let us again assume that the optimal solution lies in the interior of the constraint set, i.e., x∗ (t) > 0, so that we not have to worry about boundary conditions and complementary-slackness type conditions Given these, the first-order conditions of this finite-dimensional problem are exactly the same as the above Euler equation In particular, we have ∂U (x∗ (t + 1) , x∗ (t + 2)) ∂U(x∗ (t) , x∗ (t + 1)) +β = 0, for any ≤ t ≤ T − 1, ∂x (t + 1) ∂x (t + 1) which are identical to the Euler equations for the infinite-horizon case In addition, for x (T + 1), we have the following boundary condition (6.32) x∗ (T + 1) ≥ 0, and β T ∂U (x∗ (T ) , x∗ (T + 1)) ∗ x (T + 1) = ∂x (T + 1) Intuitively, this boundary condition requires that x∗ (T + 1) should be positive only if an interior value of it maximizes the salvage value at the end To provide more intuition for this expression, let us return to the formulation of the optimal growth problem in Example 6.1 Example 6.6 Recall that in terms of the optimal growth problem, we have U (x (t) , x (t + 1)) = u (f (x (t)) + (1 − δ) x (t) − x (t + 1)) , with x (t) = k (t) and x (t + 1) = k (t + 1) Suppose we have a finite-horizon optimal growth problem like the one discussed above where the world comes to an end at date T Then at the last date T , we have ∂U (x∗ (T ) , x∗ (T + 1)) = −u0 (c∗ (T + 1)) < ∂x (T + 1) From (6.32) and the fact that U is increasing in its first argument (Assumption 6.4), an optimal path must have k∗ (T + 1) = x∗ (T + 1) = Intuitively, there should be no capital left at the end of the world If any resources were left after the end of the world, utility could be improved by consuming them either at the last date or at some earlier date 290