back start next
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [ 39 ] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183]
39 yn FIGURE 3.9 The dynamics of the growth rates of capital and knowledge when p A e> 1 From above, is simply g + n. Equation (3.1) then implies that when A and are growing at these rates, output is growing at rate g Output per worker is therefore growing at rate g. This case is similar to the case when is less than 1 in the version of the model without capital. Here, as in that case, the longrun growth rate of the economy is endogenous, and again longrun growth is an increasing function of population growth and is zero if population growth is zero. The fractions of the labor force and the capital stock engaged in R&D, ai and , do not affect longrun growth; nor does the saving rate, s. The reason that these parameters do not affect longrun growth is essentially the same as the reason that at does not affect longnm growth in the simple version of the model. Case 2: p +0>1 In this case, the loci where and are constant diverge, as shown in Figure 3.9. As the phase diagram shows, regardless of where the economy starts, it eventually enters the region between the two loci. Once this occurs, the growth rates of both A and K, and hence the growth rate of output, increase continually. One can show that increases in s and n cause output per worker to rise above its previous trajectory by an everincreasing amount. See Problem 3.5 for a more detailed analysis of the impact of a change in the savmg rate in this model.
Case 3: p + 0 = 1 The final possibility is that p + equals 1. In this case, (1  )/ equals 1, and thus the = 0 and = 0 loci have the same slope. If n is positive, the = 0 line lies above the 0a = 0 line, and the dynamics of the economy are similar to those when /3 + e > 1; this case is shown in Panel (a) of Figure 3.10. FIGURE 3.10 The dynamics ofthe growth rates ofcapital and knowledge when p+e = l The effects of changes in Ul and are more complicated, however, since they involve shifts of resources between the two sectors. Thus this case is analogous to the case when exceeds 1 in the simple model.
See Problem 3.6. "At the aggregate level, Romers model differs in two minor respects from this. First, Ol and s are built up from microeconomic relationships, and are thus endogenous and potentially timevarying; in equilibrium they are constant, however. Second, his model distinguishes between skilled and unskilled labor; unskilled labor is used only in goods production. The stocks of both types of labor are exogenous and constant, however. If 71 is zero, on the other hand, the two loci lie directly on top of one another, as shown in Panel (b) of the figure. The figure shows that, regardless of where the economy begins, it converges to a balanced growth path. As in the case of e = 1 and n = 0 in the model without capital, the phase diagram does not tell us what balanced growth path the economy converges to. One can show, however, that the economy has a unique balanced growth path, and that the economys growth rate on that path is a complicated function of the parameters. Increases in the saving rate and in the size of the population increase this longrun growth rate; the uituition is essentially the same as the intuition for why increases in ai and L increase longrun growth when there is no capital. And, as in Case 2, increases in ai and have ambiguous effects on longrun growth. Unfortunately, the derivation of the longrun growth rate is tedious and not particularly insightful. Thus we will not work through the details. A specific example of a model of knowledge accumulation and growth whose macroeconomic side fits into this framework is P. Romers model of "endogenous technological change" (Romer, 1990; the microeconomic side of Romers model, which may be of more importance, is discussed in Section 3.4). As here, population growth is zero, and there are constant returns to scale to the produced inputs in both sectors. In addition, R&D uses labor and the existing stock of knowledge, but not physical capital. Thus in our notation, the production function for new knowledge is A{t) = BaiLAit). (3.20) Since all physical capital is used to produce goods, goods production is y(t) = :(f)"[(l  ai)LA(t)]i". (3.21) Our usual assumption of a constant saving rate (so k{t) = sY(t)) completes the model. This is the case we have been considering with /3 = 0, e = 1, and 7 = 1. To see the implications of this version of the model, note that (3.20) implies that A grows steadily at rate BaiL. This means the model is identical to the Solow model with n = 6 = 0 and with the rate of technological progress equal to BaiL. Thus (since there is no population growth), the growth rates of output and capital on the balanced growth path are Bail. This model provides a simple example of a situation where longrun growth is endogenous (and depends on parameters other than population growth), but is not affected by the saving rate.
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [ 39 ] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183]

