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49 fraction of the population of working age that is in secondary school over the years 19601985. This is clearly an imperfect measure of the fraction of a countrys resources devoted to human capital accumulation. Because Sh enters (3.57) logarithmically, if the true Sh is proportional to this measure, only the constant term of the regression will be affected. Nonetheless, measurement error in Sh is a concern. Rewriting (3.57) slightly, the equation that Mankiw, Romer, and Weil estimate is Iny, = a + bllnsKi  ln(n, + 0.05)] + cllnsni  ln(n, + 0.05)] + f,, (3.63) where i indexes cotmtries. The results for the broadest sample of countries are by, = 7.86 + 0.73 finln(n,b 0.05)] (0.14) (0.12) + 0.67 [InsM ln(n, +0.05)], (3.64) (0.07) N = 98, = 0.78, s.e.e. = 0.51, where the numbers in parentheses are standard errors. The values of a and /3 imphed by these estimates of b and (again with standard errors in parentheses) are a = 0.31 (0.04) and /3 = 0.28(0.03). In addition, when bi(n, + 0.05) is entered separately, its coefficient is approximately equal to minus the sum of the coefficients on 1 %, and Insnt, as the model predicts, and this restriction is not rejected statistically. Thus the model fits the data remarkably well: the implied shares of physical and human capital are reasonable, and the regression accounts for almost 80 percent of crosscountry variation in output per worker. A natural concem is that the saving rates, particularly sh , are endogenous: it may be that in countries that are wealthy for reasons not captured by the model, a larger fraction of the population is in school. But, as Mankiw, Romer, and Weil observe, this would cause upward bias in the estimate of /3; the fact that the estimated /3 is if anything somewhat below direct estimates of human capitals share is therefore inconsistent with this possible explanation of the results. Mankiw, Romer and Weil then turn to the issue of convergence, which we discussed in Chapter 1. They begin by noting that the model implies that countries with different levels of sk, Sh, and n have different levels of output per worker on their balanced growth paths; thus there is a component of crosscountry income differences that persists over time. But differences that arise because countries are initially at different points in relation to their balanced growth paths gradually disappear as the coimtries converge to those balanced growth paths. The model therefore predicts convergence
Barro and SalaiMartin (1991,1992) also investigate conditional convergence empirically. See Problem 3.19. controlling for the determinants of income on the balanced growth path, or conditional convergence? Specifically, one can show that the model implies that, in the vicinity of the balanced growth path, converges to y* at rate (1  a  I3){n +g) = : A[\nyWlny4 (3.65) dt Equation (3.65) implies that Iny approaches Iny* exponentially: Iny(f)  Iny* ef[Iny(O)  1 *], (3.66) uhere y(0)"is the value of at some initial date. (By differentiating [3.66], it IS straightforward to check that it implies that y(f) obeys [3.65].) If a and p are each and + 0 is 6 percent, is 2 percent; this implies that a country moves halfway to its balanced growth path in 35 years. Adding Iny* lny(O) to both sides of (3.66) yields an expression for the growth of income: Iny(f)  Iny(O) === (1  ef)[lny(0)  Iny*]. (3.67) Note that (3.67) implies conditional convergence: countries with initial incomes that are low relative to their balanced growth paths have higher growth. Finally, using equation (3.57) to substitute for Iny* yields: Iny(f)  Iny(O) (1  ef)lny*  (1  eOlny(O) = (1  e) " AlnsK  ln(n + 0)] (3.68) + (1  e%[InsH  ln(n + 0)] 1ap  a ~ " (0). Mankiw, Romer, and Weil estimate this equation, using the same data as before. The results are Iny,(t)Iny,(0) = 2.46 + 0.500 [Insa,ln(n, 40)] (0.48) (0.082) + 0.238 [InSh/ln(n,+0)]  0.299 lny,(0), (3.69) (0.060) (0.061) N = 98, = 0.46, s.e.e. = 0.33.
The implied values of the parameters are a = 0.48(0.07), p = 0.23(0.05), and = 0.0142 (0.0019). The estimates are broadly in line with the predictions of the model: countries converge toward their balanced growth paths at about the rate that the model predicts, and the estimated capital shares are broadly similar to what direct evidence suggests. Overall, the evidence suggests that a model that maintains the assumption of diminishing returns to capital but that adopts a broader definition of capital than traditional physical capital, and therefore implies a total capital share closer to 1 than to 5, provides a good first approximation to the crosscoimtry data. Problems 3.1. Consider the model of Section 3.2 with e < 1. (a) On the balanced growth path, A = gAd), where g is the balancedgrowthpath value of . Use this fact and equation (3.7) to derive an expression for A{t) on the balanced growth path in terms of B, Ul, y, , and ). (i>) Use your answer to part (a) and the production function, (3.6), to obtain an expression for ( ) on the balanced growth path. Find the value of ai that maximizes output on the balanced growth path. 3.2. Consider two economies (indexed by i = 1, 2) described by , (f) = K, ( )" and k,{t) = s, y,(t), where > 1. Suppose that the two economies have the same initial value of K, but that Si > Si. Show that ] / 2 is continually rising. 3.3. Lags in a model of growth with an explicit knowledgeproduction sector. Assume that finalgoods production is given by y(f) = A(r)(l  aOL, where fl/ is the fraction of the population engaged in knowledge production and L is population, at and L are exogenous and constant. Suppose that knowledge becomes useful in generating new knowledge only with a lag, so that A(r) = BaiLJ(t), where J is the stock of knowledge useful in generating new knowledge, /(f) is given by 1=(1  e~y)A(t  ) , where v > 0. A simple way to check if there is a balanced growth path is to guess that it is possible for A to follow A(f) = Ce, and look for candidate values of g. (a) Show that the equation for J(r) and the guess that ) = Ce*" imply J(r) = [v/(v + g)]A(t). {b) Is there any positive value of g suchthat,withj(t) given by the expression in part (a) and A(t) given by ( ), A(t) in fact follows A(t) = 7 How does that value of g depend on v? What is its value as v approaches infinity? Explain intuitively how it is possible for a temporary delay in the availability of knowledge to permanently reduce the growth rate of the economy. 3.4. Consider the economy analyzed in Section 3.3. Assume that + p < 1 and n > 0, and that the economy is on its balanced growth path Describe how
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