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46 The Dynamics of the Economy The analysis of the dynamics of this economy parallels the analysis of the Solow model. The main difference is that, instead of just considering the dynamics of physical capital, we now consider the dynamics of both physical and human capital. Specifically, define = K/AL, h = H/AL, and = Y/AL. These definitions and (3.43) imply y{t) = k(t)h{t). (3.48) Consider first. The definition of and the equations of motion for K, L, and A imply It is more appealing to interpret (3.47) as saying not that some output is "saved" in the form of human capital, but that the technology for producing new human capital combines physical capital, human capital, and raw labor in the same way as the technology for producing goods. That is, suppose that H = ,° lAIfl"", where K, He, and Le denote the quantities of physical capital, human capital, and raw labor devoted to education; and suppose that Ke = sK, Hp = ShH, and Lt = SuL. Then it immediately follows that H = Sh[KHP{AL)p]. cal capital accumulation, and where we again assume no depreciation for simplicity. In addition, because our goal here is not to explain worldwide growth, the model follows the Solow model and assumes constant and exogenous technological progress: Ait) = gAit). (3.46) Finally, for simphcity, human capital accumulation is modeled in the same way as physical capital accumulation: H(t) = shy{t), (3.47) where sh is the fraction of resources devoted to human capital accumulation.26 This model can be generalized in several ways without affecting its central messages. The CobbDouglas function can be replaced by a general production function y = F(K, H, AL) that exhibits constant returns to scale and that, in intensive form, satisfies a twovariable analogue of the Inada conditions. The assumption of exogenous technological progress can be replaced by a model of endogenous growth of knowledge along the lines of the models in Part A of this chapter. And the assumption that the technology for producing new human capital is the same as the technology for producing output can be relaxed. None of these changes affect the models central messages about crosscountry differences in incomes.
FIGURE 3.13 The dynamics of physical capital per unit of effective labor K(t) A(f)I(f) [A(f)I(f)]2 Sk Y{t) Kit) [A(t)Ut) + L{t)A(t)] A{t)Ut) A{t)Ut) = SKy(t){n +g)k{t) Ut) Ait) (3.49) = SKkit)"hit)l in+g)kit). Thus is zero when SKkh = in + g)k. This condition is equivalent to ki« = [sK/in + g)]hi, or = sk/(h + g)]!""" hPi"\ The combinations of h and satisfying this condition are shown in Figure 3.13; since /8 < 1  a, the second derivative of with respect to h along this locus is negative. In addition, (3.49) imphes that is increasing in h. Thus to the right of the = 0 locus is positive, and to the left it is negative. Now consider h. Reasoning parallel to that used to derive (3.49) yields hit) = SHkit)"hit)P in+g)hit). (3.50) is zero when 5 = (n +g)h,ork = [(n +0)/ ]" hi)/", xhis set of points is shown in Figure 3.14; since 1  /8 > a, its second derivative is positive, h is positive above this locus and negative below. The initial values of K, H, A, and I determine the initial levels of and h, which then evolve according to (3.49) and (3.50). Figure 3.15 shows the dynamics of and h together. Point E is globally stable: whatever the economys initial position, it converges to Point E. Once it reaches E, it remains there.2 As in Cliapter 1, we ignore tiie possibility of beginning without capital. If the initial value of or ft is zero, the economy converges to = h = 0.
h = 0 ih>0) (h<0) FIGURE 3.14 The dynamics of human capital per unit of effective labor
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