Economic growth and economic development 104

Introduction to Modern Economic Growth income per capita is constant, but for balanced growth paths, where income per capita grows at a constant rate, while some transformed variables such as yˆ (t) or k (t) in (2.44) remain constant Since these transformed variables remain constant, balanced growth paths can be thought of as steady states of a transformed model Motivated by this, in models with technological change throughout we will use the terms “steady state” and balanced growth path interchangeably Substituting for K˙ (t) from (2.42) into (2.44), we obtain: sF [K (t) , A (t) L (t)] k˙ (t) = − (δ + g + n) k (t) K (t) Now using (2.43), (2.45) k˙ (t) sf (k (t)) = − (δ + g + n) , k (t) k (t) which is very similar to the law of motion of the capital-labor ratio in the continuous time model, (2.32) The only difference is the presence of g, which reflects the fact that now k is no longer the capital-labor ratio but the effective capital-labor ratio Precisely because it is the effective capital-labor ratio, k will remain constant in the balanced growth path equilibrium of this economy An equilibrium in this model is defined similarly to before A steady state or a balanced growth path is, in turn, defined as an equilibrium in which k (t) is constant Consequently, we have (proof omitted): Proposition 2.12 Consider the basic Solow growth model in continuous time, with Harrod-neutral technological progress at the rate g and population growth at the rate n Suppose that Assumptions and hold, and define the effective capital-labor ratio as in (2.43) Then there exists a unique steady state (balanced growth path) equilibrium where the effective capital-labor ratio is equal to k ∗ ∈ (0, ∞) and is given by δ+g+n f (k∗ ) = (2.46) k∗ s Per capita output and consumption grow at the rate g Equation (2.46), which determines the steady-state level of effective capital-labor ratio, emphasizes that now total savings, sf (k), are used for replenishing the capital 90