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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 es

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 infinite-dimensional problems, because it makes sure that there are no beneficial simul-taneous changes in an infinite number of choice variables In contrast, in finite-dimensional 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:

t→∞βt∇x(t)U (x∗(t) , x∗(t + 1))· x∗(t) = 0, where “·” denotes the inner product operator In the one-dimensional case, we have the simpler transversality condition:

t→∞βt∂U (x

∗(t) , x∗(t + 1))

∂x (t) · x∗(t) = 0

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 solu-tion 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(x∗

t), 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

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