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]


29

OLD k(

FIGURE 2.12 The effects of a fall in the discount rate

The Speed of Convergence

To analyze formally how the economy converges to its balanced growth path, once again we linearize the equation of motion around k*. Equation (2.61), kf+i = Dkf", implies

k,=k

(2.62)

Equation (2.61) also implies that k*, the value of kt such that kt+i = kt, is given by

(2.63)

Substituting (2.63) into (2.62) shows that dkt+i/dkt evaluated at h = k* is simply a. Replacing (2.61) by its first-order Taylor approximation around = k* therefore gives us

kt+i = k* + aikt - k*). (2.64)

Since we can write this as kf+i -k* a(kt - k*), (2.64) implies

kt-k* a{ko-k*). (2.65)

kf therefore converges smoothly to k*. If a is one-third, for example,

The properties of a system of form x,+i -x* = \{x, - x*) are determined by A. If A is between 0 and 1 (which is what happens in this model), the system converges smoothly. If A is between -1 and 0, there are damped oscillations toward x*: x alternates between



2.12 The Dynamics of the Economy 79

moves two-tfiirds of the way toward k* each period. (Note, however, that each period in the model corresponds to half of a persons lifetime.)

Expression (2.65) differs from the corresponding expression in the Solow model (and in a discrete-time version of the Solow model-see Problem 2.14). The reason is that although the saving of the young is a constant fraction of their income and their income is a constant fraction of total income, the dissaving of the old is not a constant fraction of total income. The dissaving of the old as a fraction of output is Kt/F{Kt,AtLt), or ktffikt). The fact that there are diminishing returns to capital implies that this ratio is increasing in k. Since this term enters negatively into saving, it follows that total saving as a fraction of output is a decreasing function of k. Thus total saving as a fraction of output is above its balanced-growth-path value when < k* and is less when > *. Asa result, convergence is more rapid than m the Solow model.

The General Case

Let us now consider what occurs when the assumptions of logarithmic utility and Cobb-Douglas production are relaxed. It turns out that, despite the simplicity of the model, a wide range of behaviors of the economy are possible. Rather than attempting a comprehensive analysis, we simply discuss some of the more interesting cases.2

To understand the possibilities intuitively, it is helpful to rewrite the equation of motion, (2.60), as

I. 1 .Af(kt)-ktf(kt)]

= a + n)a + gf -m--

Equation (2.66) expresses capital per unit of effective labor in period t + 1 as the product of four terms. From right to left, those four terms are the following: output per unit of effective labor at f, the fraction of that output that is paid to labor, the fraction of that labor income that is saved, and the ratio of the amount of effective labor in period t to the amount in period f + 1.

Figure 2.13 shows some possible forms for the relation between kt+i and kt other than the well-behaved case shown in Figure 2.11. Panel (a) shows a case with multiple k*s. In the case shown, k* and k are stable: if starts sUghtly away from one of these points, it converges to that level, kj* is unstable (as isk = 0). If starts slightly below 2*. fi is less than kt each period, and converges to k*. If begins slightly above k}, it converges to 3*.

To understand the possibility of multiple k*s, note that since output per unit of capital is lower when is higher (capital has a diminishing marginal

being greater than and less than x*, but each period it gets closer. If A is greater than 1, the system explodes. Finally, if is less than ~1, there are explosive oscillations. Galor and Ryder (1989) analyze some of these issues in more detail.



k2 k,

ka kf,

(c) (d)

FIGURE 2.13 Various possibilities for the relationship between kt and kt+i

product), for there to be two k*s the saving of the young as a fraction of total output must be higher at the higher k*. When the fraction of output going to labor and the fraction of labor income saved are constant, the saving of the young IS a constant fraction of total output, and so multiple k*s are not possible This is what occurs with Cobb-Douglas production and logarithmic utility. But if labors share is greater at higher levels of (which occurs if /() IS more sharply curved than m the Cobb-Douglas case) or if workers save a greater fraction of their income when the rate of return is lower (which occurs if 6* > 1), or both, there may be more than one level of at which saving reproduces the existing capital stock.

Panel (b) shows a case in which kf+i is always less than kf, and in which therefore converges to zero regardless of its initial value. What is needed for this to occur IS for either labors share or the fraction of labor income saved (or both) to approach zero as approaches zero.

Panel (c) shows a case in which converges to zero if its imtial value is sufficiently low but to a strictly positive level if its imtial value is sufficiently high. Specifically, if < kj*. approaches zero; if > kf, converges



[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]