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10 /* ds fik*) in +g + S)sfik*) in+ g + S)k*f(k*) fik*)[in +g + S)~in+g + S)k* f ik*) / fik*)] k* fik*)/fik*) (1.21) l[k*fik*)/fik*)r k" fik*)/fik*) is the elasticity of output with respect to capital at = . Denoting this by axik*), we have s dy* axik*) 22) y* ds 1  { * If markets are competitive and there are no externalities, capital earns .*s marginal product. In this case, the total amount received by capital (per jut of effective labor) on the balanced growth path is k*fik*). Thus if :apital earns its marginal product, the share of total income that goes to capital on the balanced growth path is k* fik*)/fik*), or axik*). In most countries, the share of income paid to capital is about onethird. I: we use this as an estimate of *), it follows that the elasticity of output vith respect to the saving rate in the long run is about onehalf. Thus, for example, a 10 percent increase in the saving rate (from 20% of output to 22%, : instance) raises output per worker in the long run by about 5 percent dative to the path it would have followed. Even a 50 percent increase in s raises y* only by about 22 percent. Thus significant changes in saving have nlv moderate effects on the level of output on the balanced growth path. Intuitively, a small value of ax(k*) makes the impact of saving on output . for two reasons. First, it implies that the actual investment curve, sfik), rends fairly sharply; as a result, an upward shift of the curve moves its ntersection with the breakeven investment line relatively Uttle. Thus the jipact of a change in s on * is small. Second, a low value of axik*) means "at the impact of a change in * on y* is small. The Speed of Convergence In practice, we are interested not only in the eventual effects of some change such as a change in the saving rate), but also in how rapidly those effects occur. Again, we can use approximations around the longrun equilibrium *o address this issue. For simplicity, we focus on the behavior of rather tfian y. Our goal IS thus to determine how rapidly approaches *. We know that is determined by (see [1.13]); thus we can write = kik). When equals k*. s ay* s f{k*)f{k*)
(k  k*). (1.23) k=k*/ That is, is approximately equal to the product of the difference between and k* and the derivative of with respect to at = k*. Differentiating expression (1.13) for with respect to and evaluating the resulting expression at = k* yields 3k(k) dk k=k* = 5 ( *)( + 0 + 5) { + + ) * *) , (l) = [ { *)1]( + + 8), where the second line again uses the fact that sf{k*) = (n + g + S)k* to substitute for s, and where the last line uses the definition of  Substituting (1.24) into (1.23) yields k(f) [1  { * +g + S)[k(f)  k*]. (1.25) Equation (1.25) implies that, in the vicinity of the balanced growth path, capital per unit of effective labor converges toward k" at a speed proportional to its distance from k*. That is, defining x(f) = k(f)  k* and = (1  )( + g + S), (1.25) implies x(f) Ax(t): the growth rate of x is constant and equals A. The path of x is therefore given by x(t) = x(0)e "f, where x(0) is the initial value of x. In terms of k, this means k(f) k* (]«K)(»i+g+a)f((o)  k*). (1.26) One can show that approaches y* at the same rate that approaches k*; that is, y(f)  y* =  ( )  *]. We can calibrate (1.26) to see how quicldy actual economies are likely to approach their balanced growth paths, n + g + 8 is typically about 6% per year (this would arise, for example, with 1 to 2% population growth, 1 to 2% growth in output per worker, and 3 to 4% depreciation). If capitals share is roughly onethird, (1  )( + g + 8) is thus roughly 4%. and therefore move 4% of the remaining distance toward k* and y* each year, and take approximately eighteen years to gel halfway to their balancedgrowthpath values.Thus in our example of a 10% increase in the saving rate, output is "The time it takes for a variable (in this case, y*) with a constant negative growth rate to fall in half is approximately equal to 70 divided by its growth rate in percent (similarly, is zero. A firstorder Taylorseries approximation of k(k) around = k* therefore yields
the doubling time of a variable with positive growth is 70 divided by the growth rate). Thus in this case the halflife is roughly 70/(4%/year), or about eighteen years. More exactly, the halflife, t*. is the solution to e"* = 0.5, where A is the rate of decrease. Taking logs of both sides, t* = ln(0.5)/A = 0.69/A. These results are derived from a Taylorseries approximation around the balanced growth path. Thus, formally, we can rely on them only in an arbitrarily small neighborhood around the balanced growth path. The question of whether Taylorseries approximations provide good guides for finite changes does not have a general answer. For the Solow model with conventional production functions, and for moderate changes in parameter values (such as those we have been considering), the Taylorseries approximations are generally quite reliable. 0.04(5%) = 0.2% above its previous patfi after 1 year; is 0.5(5%) = 2.5% above after 18 years; and asymptotically approaches 5% above the previous path. Thus not only is the overall impact of a substantial change in the saving rate modest, but it does not occur very quickly. 1.6 The Solow Model and the Central Questions of Growth Theory The Solow model identifies two possible sources of variationeither over time or across parts of the worldin output per worker: differences in capital per worker (K/L) and differences in the effectiveness of labor (A). We have seen, however, that only growth in the effectiveness of labor can lead to permanent growth in output per worker, and that for reasonable cases the impact of changes in capital per worker on output per worker is modest. As a result, only differences in the effectiveness of labor have any reasonable hope of accounting for the vast differences in wealth across time and space. Specifically, the central conclusion of the Solow model is that if the returns that capital commands in the market are a rough guide to its contributions to output, then variations in the accumulation of physical capital do not account for a significant part of either worldwide economic growth or crosscountry income differences. There are two problems with trying to account for large differences in incomes on the basis of differences in capital. First, the required differences in capital are far too large. Consider, for example, a tenfold difference in output per worker. Output per worker in the United States today, for instance, is on the order of ten times larger than it was a hundred years ago, and than it is in India today. Recall that is the elasticity of output with respect to the capital stock. Thus accounting for a tenfold difference in output per worker on the basis of differences in capital requires a difference of a factor of 10"* in capital per worker. For ax = 5, this is a factor of a thousand. Even if capitals share is onehalf, which is well above what data on capital income suggest, one still needs a difference of a factor of a hundred. There is no evidence of such differences in capital stocks. One of the stylized facts about growth mentioned in Section 1.3 is that capitaloutput
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