Economic growth and economic development 105

Introduction to Modern Economic Growth stock for three distinct reasons The first is again the depreciation at the rate δ The second is population growth at the rate n, which reduces capital per worker The third is Harrod-neutral technological progress at the rate g Recall that we now need to keep the effective capital-labor ratio, k, given by (2.43) constant Even if K/L is constant, k will now decline because of the growth of A Thus the replenishment of the effective capital-labor ratio requires investments to be equal to (δ + g + n) k, which is the intuitive explanation for equation (2.46) The comparative static results are also similar to before, with the additional comparative static with respect to the initial level of the labor-augmenting technology, A (0) (since the level of technology at all points in time, A (t), is completely determined by A (0) given the assumption in (2.41)) We therefore have: Proposition 2.13 Suppose Assumptions and hold and let A (0) be the initial level of technology Denote the balanced growth path level of effective capital-labor ratio by k∗ (A (0) , s, δ, n) and the level of output per capita by y ∗ (A (0) , s, δ, n, t) (the latter is a function of time since it is growing over time) Then we have ∂k∗ (A (0) , s, δ, n) ∂k∗ (A (0) , s, δ, n) = 0, > 0, ∂A (0) ∂s ∂k∗ (A (0) , s, δ, n) ∂k∗ (A (0) , s, δ, n) < and < 0, ∂n ∂δ and also ∂y ∗ (A (0) , s, δ, n, t) ∂y ∗ (A (0) , s, δ, n, t) > 0, > 0, ∂A (0) ∂s ∂y ∗ (A (0) , s, δ, n, t) ∂y ∗ (A (0) , s, δ, n, t) < and < 0, ∂n ∂δ for each t Ô Proof See Exercise 2.14 Finally, we also have very similar transitional dynamics Proposition 2.14 Suppose that Assumptions and hold, then the Solow growth model with Harrod-neutral technological progress and population growth in continuous time is asymptotically stable, i.e., starting from any k (0) > 0, the effective capital-labor ratio converges to a steady-state value k∗ (k (t) → k∗ ) 91