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 transver-sality condition should be similar to that in Theorem 7.1, w

Introduction to Modern Economic Growth above, a natural conjecture might be that, as in the finite-horizon case, the transver-sality condition should be similar to that in Theorem 7.1, with t1 replaced with the limit of t → ∞, that is, limt→∞λ (t) = 0 The following example, which is very close to the original Ramsey model, illustrates that this is not the case; without fur-ther assumptions, the valid transversality condition is given by the weaker condition (7.39)

Example 7.2 Consider the following problem:

max

Z ∞

0

[log (c (t))− log c∗] dt subject to

˙k (t) = [k (t)]α

− c (t) − δk (t)

k (0) = 1 and

lim

t→∞k (t)≥ 0 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 sim-plify the notation):

Hc(k, c, λ) = 1

c (t) − λ (t) = 0

Hk(k, c, λ) = λ (t)¡

αk (t)α−1− δ¢

=− ˙λ (t)

It can be verified that any optimal path must feature c (t) → c∗ as t → ∞ This, however, implies that

lim

t→∞λ (t) = 1

c∗ > 0 and lim

t→∞k (t) = k∗ 343