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42 "The model in P. Romer (1986) that launched the new growth theory fits fairly well into this category. There are two main differences. First, the role played by physical capital here is played by knowledge in Romers model: privately controlled knowledge both contributes directly to production at a particular firm and adds to aggregate knowledge, which contributes to production at all firms. Second, knowledge accumulation occurs through a separate production function rather than through foregone output; there are increasing returns to knowledge in goods production and (asymptotically) constant returns in knowledge accumulation. As a result, the economy converges to a constant growth rate. Since kit) = sY(t), the dynamics of are given by k(t) = sBK(t)K(t) "L(r)i". (3.25) In our model of knowledge accumulation without capital in Section 3.2, the dynamics of A are given by A(f) = 1 .1( )]> ( )" (equation [3.7]). Comparing equation (3.25) of the learningbydoing model with this equation shows that the structures of the two models are similar. In the model of Section 3.2, there is a single productive input, knowledge. Here, we can think of there also being only one productive input, capital. As equations (3.7) and (3.25) show, the dynamics of the two models are essentially the same. Thus we can use the results of our analysis of the earlier model to analyze this one. There, the key determinant of the economys dynamics is how compares with 1. Here, by analogy, it is how a \ (1 a) compares with 1, which is equivalent to how compares with 1. If is less than 1, the longrun growth rate of the economy is a function of the rate of population growth, n. If is greater than 1, there is explosive growth. And if equals 1, there is explosive growth if n is positive and steady growth if n equals 0. Once again, a case that has received particular attention is = 1 and n = 0. In this case, the production function (equation [3.24]) becomes Y(t) = bK{t), fofii"!". (3.26) Capital accumulation is therefore given by (f) = sbK(t). (3.27) As in the similar cases we have already considered, the dynamics of this economy are straightforward. Equation (3.27) immediately implies that grows steadily at rate sb. And since output is proportional to it also grows at this rate. Thus we have another example of a model in which longrun growth is endogenous and depends on the saving rate. Here it occurs because the contribution of capital is larger than its conventional contribution: increased capital not only raises output through its direct contribution to production (the K" term in [3.24]), but also by indirectly contributing to the development of new ideas and thereby making all other capital more productive (the K term in [3.24]). Because the production function in these models is often written using the symbol "A" rather than the "b" used in (3.26), these models are often referred to as " = AK" models.i
eP~dt, p>0, a>0, (3.28) f=0 1  o where is the households consumption, p is its discount rate, and a is its coefficient of relative risk aversion. (Except for the use of a rather than and the fact that the size of the household is normalized to 1, this is identical to equations [2.1][2.2].) Capital and labor are paid their private marginal products. Households take their initial wealth and the paths of interest rates and wages as given, and choose the path of consumption to maximize U. When = 1, the aggregate production function, (3.24), is = "Ki"!! «. Recall that the K" term is capitals direct contribution to output, and that the K"" term is its indirect contribution through the accumulation of ideas. Thus the production function for a single firm, firm I, is Yi(t) = Bi«X,(t)«X(t)i"L,(f)i«, (3.29) Readers who have not read Chapter 2 may wish to skip this section. Making saving endogenous in the cases either of multiple produced inputs or nonconstant returns is considerably more complex. Mulligan and SalaiMartin (1993) analyze the case of two produced inputs and no population growth, with constant returns to the two inputs. Romer (1986) is an example of a model with a single produced input, nonconstant returns, and endogenous saving. 3.5 Endogenous Saving in Models of Knowledge Accumulation: An Example The analysis in the previous sections, following the spirit of the Solow model, takes the saving rate as given. But again we sometimes want to model saving behavior as axising from the choices of optimizing individuals or households, particularly if we are interested in welfare issues. Making saving endogenous in models like the ones we have been considering is often difficult. Here we consider only the simplest case: a single produced input, constant returns to that input, and no population growth. That is, we consider the case of e = 1 and n = 0 in the model with knowledge but without physical capital, or the case of = 1 and n = 0 in the learningbydoing model. For concreteness, the discussion is phrased in terms of the learningbydoing model. Assume that the division of output between consumption and saving is determined by the choices of infinitelylived households like those of the Ramsey model of Chapter 2. Since there is no population growth, we can assume that each household has exactly one member. Thus the utility function of the representative household is
where K, and L, are the amounts of capital and labor employed by the firm and is the aggregate capital stock, which the firm takes as given. Thus the private marginal product of capitalthe contribution of an additional unit of capital employed by firm / to the firms outputis Qfii "Lj". The firm hires capital up to the point where the private marginal product of capital equals the real interest rate. In equilibrium, the capitallabor ratio is equated across firms. Thus Kf /I, must equal the aggregate capitallabor ratio, which is /L. Substituting this fact into the expression for the private marginal product of capital gives us = ab (3.30) = r, where the second line uses the definition of b as B~"L~". Thus with constant returns to capital and no population growth, the real interest rate is constant. Similarly, the wage is given by the private marginal product of labor: w(f) = (1  a)£«/,(f)«K(f)i«L,(f)" = (la)B"K(t)L", (3.31) = (1  a)bK{t)/L, where the second line again uses the fact that, in equilibrium, each firms capitallabor ratio equals the aggregate ratio, K/L. Thus the real wage is proportional to the capital stock. From Chapter 2, we know that the consumption path of a household whose utility is given by (3.28) satisfies C(0 a (see equation [2.19]). Since r is constant and equal to r, consumption grows steadily at rate {r  )/( . Let 0 denote this growth rate, and assume that it is less than 7. The fact that consumption grows at rate 0 suggests that the capital stock and output also grow at this rate: if they did not, the saving rate would be continually rising or continually falling. To see if this is indeed the case, we need to check whether assuming a growth rate of the capital stock of 0 causes households to choose a level of consumption that actually causes capital to grow at this rate. That is, our procedure is to guess that and grow at the same rate as consumption, and then to verify that this is an equilibrium.
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