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47 FIGURE 3.16 The effects of an increase in the saving rate 3.10 Implications Qualitative Implications When the economy reaches Point E, it is on a balanced growth path. On the balanced growth path, k,h, and are constant; total physical capital, human capital, and output (K, H, and Y) are growing at rate n+g; and physical capital per worker, human capital per worker, and output per worker (K /I, H/L, and Y/L) are growing at rate g. Thus, as in the Solow model, the longrun growth rate of output per worker is determined by the exogenous rate of technological progress. To see how changes in saving affect the economy, suppose that initially it is on a balanced growth path, and that sk increases. Equations (3.49) and (3.50) imply that this change affects the A: = 0 locus but not the h = 0 locus. The = 0 locus shifts up; this is shown in Figure 3.16. The old balanced growth path. Point E, is on the h = 0 locus but is below the new A: = 0 locus. Thus initially h is constant and is rising, and so the economy moves upward in {h, k) space. This moves the economy above the h = 0 locus, and so h also begins to rise. Thereafter and h both increase, and the economy moves up and to the right in (h, k) space until it reaches the new balanced growth path. Point £. We can write output per worker, Y/L, as A{Y/AL), or Ak"hl. During the transition between the two balanced growth paths, output per worker is
Quantitative Implications The model does not, however, share the Solow models implications concerning the magnitude of the effects of changes in saving rates and population growth. To see this, it is helpful to solve for the level of on the balanced growth path, y*Let k* and h* denote the values of and h on the balanced growth path. Since A; = h = 0 on the balanced growth path, (3.49) and (3.50) imply %k*«h*P(n 0) *, (3.51) SHk*"h*P ={n + g)h*. (3.52) Taking logs of these two equations: InsK + * + plnh* =ln(n 10)4 Ink*, (3.53) In Sh + alnk* + pin h* =ln(n +g) + lnh*. (3.54) We can solve these two linear equations for Ink* and Inh*; this yields Ink* = InsK + InsHjln(n+g), (3.55) lap lap 1ap Inh* = ~ InsK + InsH ln(n + g). (3.56) 1  a  p I  a  p 1  a  p Finally, the production function (3.43) implies Iny* = alnk*i;8 Inb*. Substituting (3.55) and (3.56) into this expression and combining terms yields: Iny* = , " „lnsK + InsH  " ln(n + g). (3.57) 1ap lap 1ap The analogous expression for the Solow model is the same as (3.57) with p An alternative approach, paralleling the analysis in Section 1.5, is to assume a general production function in place of (3.43) and then consider approximations around the balanced growth path. This yields essentially the same results. rising both for the usual reason that A is rising and because and h are rising. Thus output per worker is growing at a rate greater than g. When the economy reaches the new balanced growth path, and h are again constant, and so the growth rate of output per worker returns to g. Thus the permanent increase in the saving rate leads to a temporary increase in the economys growth rate. In short, the models qualitative implications are almost identical to the Solow models.
Inysoiow = 7 InsK  ln(n + g). (3.58) 1 (X 1 I To assess the models quamitatlve implications, we need a rough estimate of /8, human capitals share. There are various ways to obtain such a figure. For concreteness, consider the United States. Kendricks (1976) estimates of the value of the human capital stock are slightly larger than estimates of the value of the physical capital stock; this suggests that (3 is slightly more than . The wage earned by unskilled workers, as approximated by the minimum wage, is typically between and of the average wage. This suggests that between and of the total payments to labor represent returns to human capital, or that (1  a) < /3 < f (1  a). This implies avalue of /8 between and . In the era before comprehensive coverage of minimumwage laws, unskilled immigrants to the United States earned roughly I of the average wage. This suggests ;8  . To see the importance of human capital, suppose that /8 is 0.4 and a is 0.35. Equation (3.57) implies that with these parameter values, output has elasticities of 1.4 with respect to sk, 1.6 with respect to sh, and 3 with respect lo n + g.ln the model without himian capital, in contrast, a value for a of 0.35 implies that outputs elasticity with respect to sk is 0.54 and its elasticity with respect to n  g is 0.54. Because of the large elasticities of output with respect to its underlying determinants, the model has the potential to account for large crosscountry income differences. Consider, for example, two countries with the same production function and technology, and continue to assume a = 0.35 and jS = 0.4. Suppose that sk and sh are twice as large in the second country as in the first, and that n  g is 20 percent smaller; differences of these magnitudes do not appear uncommon in practice. Equation (3.57) implies that these differences lead to a difference in log output per worker on the balanced growth path of "There is a sense in which essentially all of the payments to labor must reflect return to human capital: the marginal product of a person with no childrearing or education would be virtually zero. There are two possible responses to this observation. One, suggested by Mankiw, Romer, and Weil (1992), is to argue that there is some mmimum level of human capitalthe ability to talk, to read and wTite, and so onthat most individuals obtain more or less automatically. This component of human capital accumulation would not be well described by equation (3.47), but instead would simply grow with population. This component could therefore be included in L. The second response, which is developed in Problem 3.1 5, is to accept the argument that raw labor is not directly useful m producing output, but to argue that it is useful in producing human capital: raw labor (that is, children and students) is an important input into childrearing and education. set to zero (see Problem 1.2):
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