Economic growth and economic development 399

Introduction to Modern Economic Growth capital-labor ratio, consumption and output are constant Therefore, c˙ (t) = From (8.20), this implies that as long as f (k∗ ) > 0, irrespective of the exact utility function, we must have a capital-labor ratio k∗ such that (8.21) f (k ∗ ) = ρ + δ, which is the equivalent of the steady-state relationship in the discrete-time optimal growth model.2 This equation pins down the steady-state capital-labor ratio only as a function of the production function, the discount rate and the depreciation rate This corresponds to the modified golden rule, rather than the golden rule we saw in the Solow model (see Exercise 8.8) The modified golden rule involves a level of the capital stock that does not maximize steady-state consumption, because earlier consumption is preferred to later consumption This is because of discounting, which means that the objective is not to maximize steady-state consumption, but involves giving a higher weight to earlier consumption Given k∗ , the steady-state consumption level is straightforward to determine as: (8.22) c∗ = f (k∗ ) − (n + δ)k∗ , which is similar to the consumption level in the basic Solow model Moreover, given Assumption 4’, a steady state where the capital-labor ratio and thus output are constant necessarily satisfies the transversality condition This analysis therefore establishes: Proposition 8.2 In the neoclassical growth model described above, with Assumptions 1, 2, and 4’, the steady-state equilibrium capital-labor ratio, k∗ , is uniquely determined by (8.21) and is independent of the utility function The steadystate consumption per capita, c∗ , is given by (8.22) As with the basic Solow growth model, there are also a number of straightforward comparative static results that show how the steady-state values of capital-labor 2In addition, if f (0) = 0, there exists another, economically uninteresting steady state at k = As in Chapter 2, we ignore this steady state throughout Moreover, we will see below that starting with any k (0) > 0, the economy will always tend to the steady-state capital-labor ratio k ∗ given by (8.21) 385