Introduction to Modern Economic Growth to be replenished The higher is this rate of replenishment, the larger is the amount of investment in the economy (recall Figure 2.7 in the previous chapter) and thus there is room for faster adjustment On the other hand, when εf (k∗ ) is high, we are close to a linear–AK–production function, and as demonstrated in the previous chapter, in this case convergence should be slow In the extreme case where εf (k∗ ) is equal to 1, we will be in the AK economy and there will be no convergence Example 3.1 (Cobb-Douglas production function and convergence) Consider briefly the Cobb-Douglas production function from Example 2.1 in the previous chapter, where Y (t) = A (t) K (t)α L (t)1−α This implies that y (t) = A (t) k (t)α Consequently, as noted above, εf (k (t)) = α Therefore, (3.8) becomes y˙ (t) ' g − (1 − α) (δ + g + n) (log y (t) − log y ∗ (t)) y (t) This equation also enables us to “calibrate” the speed of convergence in practice– meaning to obtain a back-of-the-envelope estimate of the speed of convergence by using plausible values of parameters Let us focus on advanced economies In that case, plausible values for these parameters might be g ' 0.02 for approximately 2% per year output per capita growth, n ' 0.01 for approximately 1% population growth and δ ' 0.05 for about 5% per year depreciation Recall also from the previous chapter that the share of capital in national income is about 1/3, so with the Cobb-Douglas production function we should have α ' 1/3 Consequently, we may expect the convergence coefficient in front of log y (t) − log y ∗ (t) to be around 0.054 (' 0.67 × 0.08) This is a very rapid rate of convergence and would imply that income gaps between two similar countries that have the same technology, the same depreciation rate and the same rate of population growth should narrow rather quickly For example, it can be computed that with these numbers, the gap of income between two similar countries should be halved in little more than 10 years (see Exercise 3.4) This is clearly at odds with the patterns we saw in Chapter Using equation (3.8), we can obtain a growth regression similar to those estimated by Barro (1991) In particular, using discrete time approximations, equation 110