Economic growth and economic development 357

Introduction to Modern Economic Growth above, a natural conjecture might be that, as in the finite-horizon case, the transversality condition should be similar to that in Theorem 7.1, with t1 replaced with the limit of t → ∞, that is, limt→∞ λ (t) = The following example, which is very close to the original Ramsey model, illustrates that this is not the case; without further assumptions, the valid transversality condition is given by the weaker condition (7.39) Example 7.2 Consider the following problem: Z ∞ [log (c (t)) − log c∗ ] dt max subject to k˙ (t) = [k (t)]α − c (t) − δk (t) k (0) = and lim k (t) ≥ t→∞ where c∗ ≡ [k∗ ]α − δk∗ and k∗ ≡ (α/δ)1/(1−α) In other words, c∗ is the maximum level of consumption that can be achieved in the steady state of this model and k∗ is the corresponding steady-state level of capital This way of writing the objective function makes sure that the integral converges and takes a finite value (since c (t) cannot exceed c∗ forever) The Hamiltonian is straightforward to construct; it does not explicitly depend on time and takes the form H (k, c, λ) = [log c (t) − log c∗ ] + λ [k (t)α − c (t) − δk (t)] , and implies the following necessary conditions (dropping time dependence to simplify the notation): − λ (t) = c (t) ¡ ¢ Hk (k, c, λ) = λ (t) αk (t)α−1 − δ = −λ˙ (t) Hc (k, c, λ) = It can be verified that any optimal path must feature c (t) → c∗ as t → ∞ This, however, implies that lim λ (t) = t→∞ > and lim k (t) = k∗ t→∞ c∗ 343