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36 FIGURE 3.1 The dynamics of the growth rate of knowledge when < 1 Since and at are constant, whether qa is rising, falling, or constant defends on the behavior of VA". In particular, (3.8) implies that the growth Ate of is times the growth rate of I plus e  1 times the growth rate of A Thus, Mt) = [yn + (e  1)0 ( )]0 ( ). (3.9) The initial values of I and A and the parameters of the model determine the imtial value of (by [3.8]). Equation (3.9) then determines the subsequent behavior of  The production function for knowledge, (3.7), imphes that is always positive. Thus is rising if +(  1)0 is positive, falling if this quantity is negative, and constant if it is zero, is therefore constant when Qa = 1  (3.10) To describe further how the growth rate of A behaves (and thus to characterize the behavior of output per worker), we must distinguish among the cases e < 1, e > 1, and e = 1. We discuss each in turn. Case 1: 0 < 1 Equation (3.9) implies that, when is less than 1, 0a is falhng if it exceeds gX and is rising if it is less than gX Thus, regardless of the initial conditions, 4 converges to gX The phase diagram is shown in Figure 3.1. Once reaches 04, both A and YIL grow steadily at this rate; thus the economy is on a balanced growth path. •To denve (3.9) formally, differentiate (3.8) with respect to time to hnd g, and then use the dehmtion of 0 .
See Problem 3.1 for an analysis of how the change in fl; affects the path of output. This model is our first example of a model of endogenous growth. In this model, in contrast to the Solow, Ramsey, and Diamond models, the longrun growth rate of output per worker is determined within the model rather than by an exogenous rate of technological progress. The model imphes that the longrtm growth rate of output per worker, gX, is an increasing function of the rate of population growth, n. Indeed, positive population growth is necessary for sustained growth of output per worker. This may seem troubling; for example, the growth rate of output per worker is not on average higher in countries with faster population growth. If we think of the model as one of worldwide economic growth, however, this result is reasonable. A natural interpretation of the model is that A represents knowledge that can be used anywhere in the world. With this interpretation, the model does not imply that cotmtries with greater population growth enjoy greater income growth, but only that higher worldwide population growth raises worldwide income growth. And it is plausible that, at least up to the point where resource limitations (which are omitted froln the model) become important, higher population is beneficial to the growth of worldwide knowledge: the larger the population is, the more people there are to make new discoveries. What the result about the necessity of positive population growth to sustained growth of output per worker is telling us is that, if adding to the stock of knowledge becomes more difficult as the stock of knowledge rises (that is, if e < 1), growth would taper off in the absence of population growth. We discuss this issue further (and even consider some relevant empirical evidence) in Section 3.7. Equation (3.10) also implies that although the rate of population growth affects longrun growth, the fraction of the labor force engaged in R&D (oi) does not. This too may seem surprising: since growth is driven by technological progress and technological progress is endogenous, it is natural to expect an increase in the fraction of the economys resources devoted to technological progress to increase longrun growth. The reason that this does not occur is that, because is less than 1, the increase in ul has a level effect but not a growth effect on the path of A. Equation (3.8) implies that the increase in at causes an immediate increase in  But as the phase diagram shows, because of the limited contribution of the additional knowledge to the production of new knowledge, this increase in the growth rate of knowledge is not sustained. Thus, paralleling the impact of a rise in the saving rate on the path of output in the Solow model, the increase in ai results in a rise in g followed by a gradual return to its initial level; the level of A therefore moves gradually to a parallel path higher than its initial one. This is shown in Figure 3.2.
FIGURE 3.2 The impact of a rise in the fraction of the labor force engaged in R&D when e < 1 Case 2: >1 The second case to consider is greater than 1. In this case, (3.9) implies that is increasing in ; and since is necessarily positive, it also implies that must be positive. The phase diagram is shown in Figure 3.3. The implications of this case for longrun growth are very different from those of the previous case. As the phase diagram shows, the economy now exhibits everincreasing growth rather than converging to a balanced FIGURE 3.3 The dynamics of the growth rate of knowledge when > 1
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