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35 Part A Research and Development Models 3.1 Framework and Assumptions Overview The view of growth that is most in keeping with the models we have seen is that the effectiveness of labor represents knowledge or technology. Certainly it is plausible that technological progress is the reason that more output can be produced today from a given quantity of capital and labor than could be produced a century or two ago. The natural extension of Chapters 1 and 2 is thus to model the growth of A rather than to take it as given. To do this, we need to introduce an explicit research and development (or R&D) sector, and then model the production of new technologies. We also need to model the allocation of resources between conventional goods production and R&D. In our formal modeling, we will take a fairly mechanical view of the production of new technologies. Specifically, we will assume a largely conventional production function in which labor, capital, and technology are combined to produce improvements in technology in a deterministic way. Of course, this is not a complete description of technological progress. But it is reasonable to think that, all else equal, devoting more resources to research yields more discoveries; this is what the production function captures. Since we are interested in growth over extended periods, modeling the randomness in technological progress would give little additional insight. And if we want to analyze the consequences of changes in other determinants of the success of R&D, we can introduce a shift parameter in the knowledge production function and examine the effects of changes in that parameter. The model provides no insight, however, concerning what those other determinants of the success of research activity are. We make two other major simplifications. First, both the R&D and goods production ftmctions are asstmied to be generahzed CobbDouglas functions. Second, in the spirit of the Solow model, the model takes the fraction of output saved and the fractions of the labor force and the capital stock used in the R&D sector as exogenous and constant. These assumptions do not change the models main implications. Specifics The specific model we consider is a simplified version of the models of R&D and growth developed by P. Romer (1990), Grossman and Helpman
See also Uzawa (1965); Shell (1966, 1967); and Phelps (1966b). The fact that the function does not necessarily have constant returns is the reason for referring to it as a generalized CobbDouglas function. 1991a), and Aghion and Howitt (1992). The model, like the others we have studied, involves four variables: labor (i), capital (K), technology (A), and jutput ( ). The model is set in continuous time. There are two sectors, a goodsproducing sector where output is produced and an R&D sector where additions to the stock of knowledge are made. Fraction ai of the labor force .s used in the R&D sector and fraction 1  in the goodsproducing sector; similarly, fraction of the capital stock is used in R&D and the rest in joods production. Both sectors use the full stock of knowledge. A: because •he use of an idea or a piece of knowledge in one place does not prevent it from being used elsewhere, we do not have to consider the division of the stock of knowledge between the two sectors. The quantity of output produced at time f is thus nt) = [(1  aK)K(t)riA{t)a  at)I(f)]~ 0 < a < 1. (3.1) Aside from the I  and 1  at terms and the restriction to the CobbDouglas functional form, this production function is identical to those of jur earlier models. Note that equation (3.1) implies constant returns to capital and labor: with a given technology, doubling the inputs doubles the amoimt that can be produced. The production of new ideas depends on the quantities of capital and labor engaged in research and on the level of technology: A(f) = G{aKK{t),aLUt),A{t)). (3.2) Under the assumption of generalized CobbDouglas production, this becomes A(t) = BlaKK{t)f[aLUt)VA(t)>, > 0, /3 > 0, > 0, (3.3) uhere is a shift parameter. Notice that the production function for knowledge is not assumed to have constant returns to scale to capital and labor. The standard argument that there must be at least constant returns is a replication one: if the mputs double, the new inputs can do exactly what the old ones were doing, thereby doubUng the amount produced. But in the case of knowledge production, exactly replicating what the existing inputs were doing would cause the same set of discoveries to be made twice, thereby leaving A unchanged. Thus it is possible that there are diminishing returns in R&D. At the same tune, interactions among researchers, fixed setup costs, and so on may be important enough in R&D that doubling capital and labor more than doubles output. We therefore also allow for the possibility of increasing returns.
= BalUt)A(t) .ei The model contains the Solow model with CobbDouglas production as a special case: if /3, y. , and Ot are all equal to zero and e is 1, the production function for knowledge becomes A = BA (which imphes that A grows at a constant rate), and the other equations of the model simplify to the corresponding equations of the Solow model. In addition, there does not appear to be any strong basis for restricting how increases in the stock of knowledge affect the production of new knowledge; thus no restriction is placed on in (3.3). If e = 1, is proportional to A; the effect is stronger if e > 1 and is weaker if e < 1. As in the Solow model, the saving rate is exogenous and constant. In addition, depreciation is set to zero for simplicity. Thus, = sY(t). (3.4) Finally, we continue to treat population growth as exogenous: Ut) = nUt), fl > 0. (3.5) This completes the description of the model. Because the model has two stock variables whose behavior is endogenous, and A, it is more complicated to analyze than the Solow model. We therefore begin by considering the model without capital; that is, we set a and /3 to zero. This case shows most of the models central messages. We then turn to the general case. 3.2 The Model without Capital The Dynamics of Knowledge Accumulation When there is no capital in the model, the production function for output (equation [3.1]) becomes y(f) = A(f)(l  ). (3.6) Similarly, the production function for new knowledge (equation [3.3]) is now A(t) = BlaLUt)rA(t). (3.7) Population growth continues to be described by equation (3.5). Equation (3.6) implies that output per worker is proportional to A, and thus that the growth rate of output per worker equals the growth rate of A. We therefore focus on the dynamics of A, which are given by (3.7). The growth rate of A, denoted Qa, is A(f) AiF) (3.8)
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