Economic growth and economic development 297

Introduction to Modern Economic Growth However, this Euler equation is not sufficient for optimality In addition we need the transversality condition The transversality condition is essential in infinitedimensional problems, because it makes sure that there are no beneficial simultaneous changes in an infinite number of choice variables In contrast, in finitedimensional problems, there is no need for such a condition, since the first-order conditions are sufficient to rule out possible gains when we change many or all of the control variables at the same time The role that the transversality condition plays in infinite-dimensional optimization problems will become more apparent after we see Theorem 6.10 and after the discussion in the next subsection In the general case, the transversality condition takes the form: (6.25) lim β t ∇x(t) U (x∗ (t) , x∗ (t + 1)) · x∗ (t) = 0, t→∞ where “·” denotes the inner product operator In the one-dimensional case, we have the simpler transversality condition: (6.26) lim β t t→∞ ∂U(x∗ (t) , x∗ (t + 1)) ∗ · x (t) = ∂x (t) In words, this condition requires that the product of the marginal return from the state variable x times the value of this state variable does not increase asymptotically at a rate faster than 1/β The next theorem shows that the transversality condition together with the transformed Euler equations in (6.21) are sufficient to characterize an optimal solution to Problem A1 and therefore to Problem A2 Theorem 6.10 (Euler Equations and the Transversality Condition) Let X ⊂ RK + , and suppose that Assumptions 6.1-6.5 hold Then the sequence ∗ ∗ {x∗ (t + 1)}∞ t=0 , with x (t + 1) ∈IntG(xt ), t = 0, 1, , is optimal for Problem A1 given x (0), if it satisfies (6.21) and (6.25) Proof Consider an arbitrary x (0) and x∗ ≡ (x (0) , x∗ (1) , ) ∈ Φ (x (0)) be a feasible (nonnegative) sequence satisfying (6.21) and (6.25) We first show that x∗ yields higher value than any other x ≡ (x (0) , x (1) , ) ∈ Φ (x (0)) For any 283