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14 Iny* + ~1 5 ln(n + g + d). (1.36) I  a 1  a Mankiw, D. Romer, and Weil (1992) estimate equation (1.36) empirically usmg crosscountry data. Their basic specification is Iny, = a + b[lns, ~ ln(n, b 0 I g)] b f,, (1.37) here i indexes countries. Finding empirical counterparts for y, s, and n :s fairly straightforward. Mankiw, Romer, and Weil measure as real GDP per person of working age in 1985, 5 as the average share of real private and government investment in real GDP over the period 19601985, and n as the average growth rate of the population of working age over the same period. Finally, g  S is set to 0.05 for all countries. The results for the broadest set of countries considered by Mankiw, Romer, and Weil are: Iny, = 6.87 + 1.48[1 5,ln(ni bO.05)], (0.12) (0.12) (1.38) = 0.59, s.e.e. = 0.69. Sa\Tng and population growth enter in the directions predicted by the model and are highly statistically significant, and the regression accounts for a large portion of crosscountry differences in income. In this sense, the model is a success. There is one major difficulty, however: the estimated effect of saving and population growth is far larger than the model predicts. The estimate of i) = 1.48 implies a == 0.60 (with a standard error of 0.02).23 Thus the relationship between saving and population growth and real income is far stronger than the model predicts for reasonable values of the capital share, and the data are grossly inconsistent with the hypothesis that a is in the \icinity of onethird. Thus, Mankiw, Romer, and Weils results confirm the conclusion that the Solow model carmot account for important features of crosscountry income differences. The data are from the Summers and Heston (1988) crosscountry data set. See Summers and Heston (1991) for a more recent version. Finding estimates and standard errors for parameters that are nonlinear functions of regression coefficients is straightforward. In the case of (1.36)41.38), solving =  a) tor a yields a = fo/(l + b). The estimate of a = 0.60 is thus obtained by computing a = fo/(l + b) = 1.48/(1 + 1.48). In addition, a firstorder Taylorseries approximation of a = fo/(l + b) around b = b yields a [fo/(I + b)] + [1 /(1 + ]{ ~ b). Thus the difference between the true J and a is approximately 1/(1 + b), or 0.16, times the difference between the true b and b. The standard error of a is therefore approximately 0.16 times the standard error of b, or 0.16(0.12) = 0.02. (Because of the nonlinearity and the use of approximations, the formal econometric justification for these procedures rehes on asymptotic theory. See, for example, Greene, 1993, Section 10.3.3; or Judge et al.. 1985, Section 5.3.4.)
Problems 1.1. Consider a Solow economy that is on its balanced growth path. Assume for simplicity that there is no technological progress. Now suppose that the rate of population growth falls. {a) What happens to the balancedgrowthpath values of capital per worker, output per worker, and consumption per worker? Sketch the paths of these variables as the economy moves to its new balanced growth path. (b) Describe the effect of the fall in population growth on the path of output (that is, total output, not output per worker). 1.2. Suppose that the production function is CobbDouglas. (a) Find expressions for k*, y*, and c* as functions of the parameters of the model, s, n, S, g, and a. ib) What is the goldenrule value of k? (c) What saving rate is needed to yield the goldenrule capital stock? 1.3. Consider the constant elasticity of substitution (CES) production function, Y = [X<ri)i<T + ( )(<7l)/< ]<7/(l) here 0 < cr < 00 and ( 1.(( Is the elasticity of substitution between capital and effective labor. In the special case of o > 1, the CES function reduces to the CobbDouglas.) (a) Show that this production function exhibits constant returns to scale. (b) Find the intensive form of the production function. (c) Under what conditions does the intensive form satisfy f{) > 0, /"(•) < 0? (d) Under what conditions does the intensive form satisfy the Inada conditions? 1.4. Consider an economy with technological progress but without population growth that is on its balanced growth path. Now suppose there is a onetime jump in the number of workers. (a) At the time of the jump, does output per unit of effective labor rise, fall, or stay the same? Why? ib) After the initial change (if any) in output per unit of effective labor when the new workers appear, is there any further change in output per unit of effective labor? If so, does it rise or fall? Why? (c) Once the economy has again reached a balanced growth path, is output per unit of effective labor higher, lower, or the same as it was before the new workers appeared? Why? 1.5. Find the elasticity of output per unit of effective labor on the balanced growth path, y*, with respect to the rate of population growth, n. If { *) = \, g = 2%, and S = 3%, by about how much does a fall in n from 2% to 1% raise y*7 1.6. Suppose that, despite the political obstacles, the United States permanently reduces its budget deficit from 3% of GDP to zero. Suppose that initially s = 0.15 and that investment rises by the full amount of the fall in the deficit. Assume that capitals share Is
(a) By about how much does output eventually rise relative to what it would have been without the deficit reduction? (b) By about how much does consumption rise relative to what it would have been without the deficit reduction? ic) What Is the immediate effect of the deficit reduction on consumption? About how long does it take for consumption to return to what it would have been without the deficit reduction? 1.7. Factor payments in the Solow model. Assume that both labor and capital are paid their marginal products. Let w denote dF(K,AL)/dL and r denote dF(K,AL)ldK. (a) Show that the marginal product of labor, w, is A[f(k)  kf(k)]. (b) Show that if both capital and labor are paid their marginal products, constant retums to scale implies that the total amount paid to the factors of production equals total output. That is, show that under constant returns, wL+rK =F(K,AL). (c) Two additional stylized facts about growth listed by Kaldor (1961) are that the return to capital (r) is approximately constant and that the shares of output going to capital and labor are each roughly constant. Does a Solow economy on a balanced growth path exhibit these properties? What are the growth rates of w and r on a balanced growth path? (d) Suppose the economy begins with a level of less than k*. As moves toward k*,isw growing at a rate greater than, less than, or equal to its growth rate on the balanced growth path? What about r? 1.8. Suppose that, as in Problem 1.7, capital and labor are paid their marginal products. In addition, suppose that all capital income is saved and all labor income is consumed. Thus = [ { , 1)/ ] ~ SK. {a) Show that this economy converges to a balanced growth path. (b) Is on the balanced growth path greater than, less than, or equal to the goldenrule level of k? What is the intuitton for this result? 1.9. The HarrodDomar model. (See Harrod, 1939, and Domar, 1946.) Suppose the production function is Leontief, Y(t) = minlcKKit), ClCLit)], where , Cl, and g are all positive. As in the Solow model, L(t) = nl(f) and Kit) = sY{t)SKit). Finally, assume (0) = dUP). (a) Under what condition does ( ) = CiceLit) for all t? If , Cl, g, s, S, and n are determined by separate considerations. Is there any reason to expect that this condition holds? (b) If CieLit) is growing faster than CKK(t) (and if the excess labor is assumed to be unemployed), what happens to the unemployment rate over time? (c) If CKK(t) is growing faster than cieBL(t) (and if the excess capital is assumed to be unused), what happens to the fraction of the capital stock that is used over time? 1.10. Natural resources in the Solow model. At least since Malthus, some have argued that the fact that some factors of production (notably land and natural
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