Economic growth and economic development 371

Introduction to Modern Economic Growth Armed with these theorems, we can now investigate the transitional dynamics in the q-theory of investment To see that the equilibrium will tend to this steady-state level of capital stock it suffices to plot (7.59) and (7.62) in the k-i space This is done in Figure ?? The curve corresponding to k˙ = 0, (7.59), is upward sloping, since a greater level of capital stock requires more investment to replenish the depreciated capital When we are above this curve, there is more investment than necessary for replenishment, so that k˙ > When we are below this curve, then k˙ < On the other hand, the curve corresponding to i˙ = 0, (7.62), can be nonmonotonic Nevertheless, it is straightforward to verify that in the neighborhood of the steadystate it is downward sloping (see Exercise 7.22) When we are to the right of this curve, f (k) is lower, thus i˙ > When we are to its left, i˙ < The resulting phase diagram, together with the one-dimensional stable manifold, is shown in Figure ?? (see again Exercise 7.22 for a different proof) Starting with an arbitrary level of capital stock, k (0) > 0, the unique optimal solution involves an initial level of investment i (0) > 0, followed by a smooth and monotonic approach to the steady-state investment level of δk∗ In particular, it can be shown easily that when k (0) < k∗ , i (0) > i∗ and it monotonically decreases towards i∗ (see Exercise 7.22) This is intuitive Adjustment costs discourage large values of investment, thus the firm cannot adjust its capital stock to its steady-state level immediately However, because of diminishing returns, the benefit of increasing the capital stock is greater when the level of capital stock is low Therefore, at the beginning the firm is willing to incur greater adjustment costs in order to increase its capital stock and i (0) is high As capital accumulates and k (t) > k (0), the benefit of boosting the capital stock declines and the firm also reduces investment towards the steady-state investment level It is also informative to see why a level of initial investment other than i (0) would violate either the transversality condition or the first-order necessary conditions Consider, for example, i0 (0) > i (0) as the initial level The phase diagram in Figure ?? makes it clear that starting from such a level of investment, i (t) and k (t) would tend to infinity It can be verified that in this case q (t) k (t) would tend to infinity at a rate faster than r, thus violating the transversality condition, limt→∞ exp (−rt) q (t) k (t) = To see this more explicitly, note that since along a 357