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37 In FIGURE 3.4 The impact of a rise in the fraction of the labor force engaged in R&D when e > 1 growth path. Intuitively, here knowledge is so useful in the production of new knowledge that each marginal increase in its level results in so much more new knowledge that the growth rate of knowledge rises rather than falls. Thus once the accumulation of knowledge beginswhich it necessarily does in the modelthe economy embarks on a path of everincreasing growth. The impact of an increase in the fraction of the labor force engaged in R&D is now dramatic. From Equation (3.8), an increase in ai causes an immediate increase in Qa, as before. But qa is an increasing function of Qa, thus Qa rises as well. And the more rapidly Qa rises, the morerapidly its growth rate rises. Thus the increase in ai leads to an everwidening gap between the new path of A and the path it otherwise would have followed. This is depicted in Figure 3.4. Case 3: 0 = 1 When is exactly equal to 1, the expressions for gA and gA simplify to 0A(f) = BalUtr, (3.11) Mt) = yngA(t). (3.12)
FIGURE 3.5 n > 0 The dynamics of the growth rate of knowledge when = 1 and If population growth is positive, is growing over time; in this case the dynamics of the model are similar to those when e > 1. Figure 3.5 shows the phase diagram for this case. If population growth is zero (or if is zero), is constant regardless of Its initial situation. In this case, knowledge is just useful enough in producmg new knowledge that the level of A has no impact on its growth rate. Thus there is no adjustment toward a balanced growth path: no matter where It begins, the economy immediately exhibits steady growth. As equations (3.6) and (3.11) show, the growth rates of knowledge, output, and output per worker are all equal to BalU in this case. Thus in this case ai affects the longrun growth rate of the economy. Since the output good in this economy has no use other than in consumption, it is natural to think of it as being entirely consumed. Thus 1  at is the fraction of societys resources devoted to producing goods for current consumption, and at is the fraction devoted to producing a good (namely knowledge) that is useful for producing output in the future. Thus one can think of ai as a measure of saving in this economy. "One slightly awkward feature of using the generalized CobbDouglas production function is that, in the cases of 6 > 1 and of e = 1 and n > 0, it imphes not merely that growth IS increasing, but that it rises so fast that output reaches inhnity in a ftnite amount of time. Consider, for example, the case of e > iwithn = 0. One can check that ( ) = Ci/(C2t)" with Cl = 1 /[(e  DBfli IT]" * and Cz chosen so that .4(0) equals the initial value of A, satisfies (3.7). Thus A explodes at time cj. Since output cannot reach infinity in a finite time, rhis imphes that the generalized CobbDouglas production function must break down at ome point. But it does not mean that the function cannot provide a good description over !ie relevant range. Indeed, Section 3.7 presents evidence that a model similar to this one provides a good approximation to historical data over thousands of years.
The Dynamics of Knowledge and Capital As mentioned above, when the model includes capital, there are two endogenous stock variables, A and K. Parallehng our analysis of the simple model, here we focus on the dynamics of the growth rates of A and K. Substituting With this interpretation, this ease of the model provides a simple example of a model where the saving rate affects longrun growth. Models of this form are known as linear growth models; for reasons that wUl become clear in Section 3.4, they are also known as = AK models. Because of their simplicity, linear growth models have received a great deal of attention in work on endogenous growth. The Importance of Returns to Scale to Produced Factors The reason that these three cases have such different implications is that whether is less than, greater than, or equal to 1 determines whether there are decreasing, increasing, or constant returns to scale to produced factors of production. The growth of labor is exogenous, and we have eliminated capital from the model; thus knowledge is the only produced factor. There are constant returns to knowledge in goods production. Thus whether there are on the whole increasing, decreasing, or constant returns to knowledge in this economy is determined by the returns to scale to knowledge in knowledge productionthat is, by . To see why the returns to the produced input are critical to the behavior of the economy, suppose that the economy is on some path, and suppose there is an exogenous increase in A of 1 percent. If is exactly equal to 1, A grows by 1 percent as well: knowledge is just productive enough in the production of new knowledge that the increase in A is selfsustaining. Thus the jump in A has no effect on its growth rate. If exceeds I, the 1 percent increase in A causes more than a 1 percent increase in A. Thus in this case the increase in A raises the growth rate of A Finally, if is less than 1, the 1 percent increase in A results in an increase of less than 1 percent in A, and so the growth rate of knowledge falls. 3.3 The General Case We now want to reintroduce capital into the model and determine how this modifies the earlier analysis. Thus the model is now described by equations (3.1) and (3.3)(3.5) rather than by (3.5)(3.7).
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