Economic growth and economic development 368

Introduction to Modern Economic Growth Differentiating this equation with respect to time, we obtain q˙ (t) = φ00 (i (t)) i˙ (t) (7.61) Substituting this into the second necessary condition, we obtain the following law of motion for investment: (7.62) i˙ (t) = [(r + δ) (1 + φ0 (i (t))) − f (k (t))] φ (i (t)) 00 A number of interesting economic features emerge from this equation First, as φ00 (i) tends to zero, it can be verified that i˙ (t) diverges, meaning that investment jumps to a particular value In other words, it can be shown that this value is such that the capital stock immediately reaches its state-state value (see Exercise 7.22) This is intuitive As φ00 (i) tends to zero, φ0 (i) becomes linear In this case, adjustment costs simply increase the cost of investment linearly and not create any need for smoothing In contrast, when φ00 (i (t)) > 0, there will be a motive for smoothing, i˙ (t) will take a finite value, and investment will adjust slowly Therefore, as claimed above, adjustment costs lead to a smoother path of investment We can now analyze the behavior of investment and capital stock using the differential equations (7.59) and (7.62) First, it can be verified easily that there exists a unique steady-state solution with k > This solution involves a level of capital stock k∗ for the firm and investment just enough to replenish the depreciated capital, i∗ = δk∗ This steady-state level of capital satisfies the first-order condition (corresponding to the right hand side of (7.62) being equal to zero): f (k∗ ) = (r + δ) (1 + φ0 (δk∗ )) This first-order condition differs from the standard “modified golden rule” condition, which requires the marginal product of capital to be equal to the interest rate plus the depreciation rate, because an additional cost of having a higher capital stock is that there will have to be more investment to replenish depreciated capital This is captured by the term φ0 (δk∗ ) Since φ is strictly convex and f is strictly concave and satisfies the Inada conditions (from Assumption 2), there exists a unique value of k∗ that satisfies this condition 354