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38 the production function, (3.1), into the expression for capital accumulation, (3.4), yields k(t) = sd   at)i "/(t)«A(f) " )!". (3.13) Dividing both sides by K{t) and defining = s{l  )"{1  aiV" gives us kit) K(t) = Ck A{t)Ut) L Kit) J (3.14) Thus whether Qk is rising, falling, or holding steady depends on the behavior of AL/K. The growth rate of this ratio is gA + n  Qk Thus is rising if 0A + tl  is positive, falling if this expression is negative, and constant if it is zero. This information is summarized in Figure 3.6. In ( , ) space, the locus of points where is constant has an intercept of n and a slope of one. Above the locus, is falling; below the locus, it is rising. Similarly, dividing both sides of equation (3.3) by A(t) yields an expression for the growth rate of A: 0A(f) = CAnwAit)" (3.15) FIGURE 3.6 The dynamics of the growth rate of capital in the general version of the model
Case 1: /8 + 0 < 1 If /3 + e is less than 1, (1  e) 3 is greater than 1. Thus the locus of points where = 0 is steeper than the locus where = 0. This case is shown in FIGURE 3.7 The dynamics of the growth rate of knowledge in the general version of the model where s . Aside from the presence of the Kl term, this is essentially the same as equation (3.8) in the simple version of the model. Equation (3.1 S) implies that the behavior of depends on + +(  1)  Qa is rising if this expression is positive, falling if it is negative, and constant if it is zero. This is shown in Figure 3.7. The set of points where is constant has an intercept of yn 3 and a slope of (1  e) 3 (the figure is drawn for the case of e < 1, so this slope is shown as positive). Above this locus, is rising; and below the locus, it is falling. The production function for output (equation [3.1J) exhibits constant returns to scale in the two produced factors of production, capital and knowledge. Thus whether there are on net increasing, decreasing, or constant returns to scale to the produced factors depends on their returns to scale in the production function for knowledge, equation (3.3). As that equation shows, the degree of returns to scale to and A in knowledge production is /3 + e: increasing both and by a factor of X increases by a factor of xjius the key determinant of the economys behavior is now not how e compares with 1, but how /3 + 6 compares with 1. As before, we discuss each of the three possibilities.
Figure 3.8. The initial values of Qa and Qk are determined by the parameters of the model and by the initial values of A, , and I. Their dynamics are then as shown in the figure. The figure shows that regardless of where qa and begin, they converge to Point E in the diagram. Both and are zero at this point. Thus the values of 0 and al Point E, which we denote gX and g, must satisfy 0 + "  0 = 0 (3.16) pg + yn + {  l)gX = 0. (3.17) Rewriting (3.16) as g = gX n and substituting into (3.17) yields + + r)n + (e  i)0A = 0, (3.18) * P + 7 (3.19) FIGURE 3.8 The dynamics of the growth rates of capital and knowledge when p + e < 1
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