Introduction to Modern Economic Growth (2.46) in the previous chapter, sf (k∗ ) /k∗ = δ +g +n), the definition of the elasticity of the f function, εf (k (t)), and again the fact that sf (k∗ ) /k∗ = δ + g + n Now substituting this into (3.7), we obtain y˙ (t) ' g − εf (k∗ ) (1 − εf (k∗ )) (δ + g + n) (log k (t) − log k∗ ) y (t) Let us define y ∗ (t) ≡ A (t) f (k ∗ ) as the level of per capita output that would apply if the effective capital-labor ratio were at its steady-state value and technology were at its time t level We therefore refer to y ∗ (t) as the “steady-state level of output per capita” even though it is not constant Now taking first-order Taylor expansions of log y (t) would respects to log k (t) around log k ∗ (t) gives log y (t) − log y ∗ (t) ' εf (k∗ ) (log k (t) − log k∗ ) Combining this with the previous equation, we obtain the following “convergence equation”: (3.8) y˙ (t) ' g − (1 − εf (k∗ )) (δ + g + n) (log y (t) − log y ∗ (t)) y (t) Equation (3.8) makes it clear that, in the Solow model, there are two sources of growth in output per capita: the first is g, the rate of technological progress, and the second is “convergence” This latter source of growth results from the negative impact of the gap between the current level of output per capita and the steadystate level of output per capita on the rate of capital accumulation (recall that < εf (k∗ ) < 1) Intuitively, the further below is a country from its steady state capital-labor ratio, the more capital it will accumulate and the faster it will grow This pattern is in fact visible in Figure 2.7 from the previous chapter The reason is also clear from the analysis in the previous chapter The lower is y (t) relative to y ∗ (t), and thus the lower is k (t) relative to k∗ , the greater is the average product of capital f (k∗ ) /k∗ , and this leads to faster growth in the effective capital-labor ratio Another noteworthy feature is that the speed of convergence in equation (3.8), measured by the term (1 − εf (k∗ )) (δ + g + n) multiplying the gap between log y (t) and log y ∗ (t), depends on δ + g + n and the elasticity of the production function εf (k∗ ) Both of these capture intuitive effects As discussed in the previous chapter, the term δ + g + n determines the rate at which effective capital-labor ratio needs 109