Introduction to Modern Economic Growth Now, heuristically we can derive the transversality condition as an extension of condition (6.32) to T → ∞ Take this limit, which implies lim β T T →∞ ∂U(x∗ (T ) , x∗ (T + 1)) ∗ x (T + 1) = ∂x (T + 1) Moreover, as T → ∞, we have the Euler equation ∂U(x∗ (T ) , x∗ (T + 1)) ∂U (x∗ (T + 1) , x∗ (T + 2)) +β = ∂x (T + 1) ∂x (T + 1) Substituting this relationship into the previous equation, we obtain − lim β T +1 T →∞ ∂U(x∗ (T + 1) , x∗ (T + 2)) ∗ x (T + 1) = ∂x (T + 1) Canceling the negative sign, and without loss of any generality, changing the timing: lim β T T →∞ ∂U(x∗ (T ) , x∗ (T + 1)) ∗ x (T ) = 0, ∂x (T ) which is exactly the transversality condition in (6.26) This derivation also highlights that alternatively we could have had the transversality condition as lim β T T →∞ ∂U(x∗ (T ) , x∗ (T + 1)) ∗ x (T + 1) = 0, ∂x (T + 1) which emphasizes that there is no unique transversality condition, but we generally need a boundary condition at infinity to rule out variations that change an infinite number of control variables at the same time A number of different boundary conditions at infinity can play this role We will return to this issue when we look at optimal control in continuous time 6.6 Optimal Growth in Discrete Time We are now in a position to apply the methods developed so far to characterize the solution to the standard discrete time optimal growth problem introduced in the previous chapter Example 6.4 already showed how this can be done in the special case with logarithmic utility, Cobb-Douglas production function and full depreciation In this section, we will see that the results apply more generally to the canonical optimal growth model introduced in Chapter Recall the optimal growth problem for a one-sector economy admitting a representative household with instantaneous utility function u and discount factor 291