Introduction to Modern Economic Growth in this model, the equilibrium of an economy is described by the following equations: (3.5) y (t) = A (t) f (k (t)) , and (3.6) k˙ (t) sf (k (t)) = − δ − g − n, k (t) k (t) where A (t) is the labor-augmenting (Harrod-neutral) technology term, k (t) ≡ K (t) / (A (t) L (t)) is the effective capital labor ratio and f (·) is the per capita pro- duction function Equation (3.6) follows from the constant technological progress and constant population growth assumptions, i.e., A˙ (t) /A (t) = g and L˙ (t) /L (t) = n Now differentiating (3.5) with respect to time and dividing both sides by y (t), we obtain (3.7) k˙ (t) y˙ (t) = g + εf (k (t)) , y (t) k (t) where f (k (t)) k (t) ∈ (0, 1) f (k (t)) is the elasticity of the f (·) function The fact that it is between and follows εf (k (t)) ≡ from Assumption For example, with the Cobb-Douglas technology from Example 2.1 in the previous chapter, we would have εf (k (t)) = α, that is, it is a constant independent of k (t) (see Example 3.1 below) However, generally, this elasticity is a function of k (t) Now let us consider a first-order Taylor expansion of (3.6) with respect to log k (t) around the steady-state value k∗ (and recall that ∂y/∂ log x = (∂y/∂x) · x) This expansion implies that for k (t) in the neighborhood of k ∗ , we have ả ả k (t) f (k ) k sf (k∗ ) f (k∗ ) − δ − g − n + (log k (t) − log k∗ ) ' − s k (t) k∗ f (k∗ ) k∗ ' (εf (k∗ ) − 1) (δ + g + n) (log k (t) − log k∗ ) The use of the symbol “'” here is to emphasize that this is an approximation, ignoring second-order terms In particular, the first line follows simply by differentiating k˙ (t) /k (t) with respect to log k (t) and evaluating the derivatives at k ∗ (and ignoring second-order terms) The second line uses the fact that the first term in the first line is equal to zero by definition of the steady-state value k∗ (recall that from equation 108