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]
7 The Dynamics of Because the economy may be growing over time, it turns out to be convenient to focus on the capital stock per unit of effective labor, k, rather than the unadjusted capital stock, K. Since = K/AL, we can use the chain rule to find* Kit) Ut) Kit) A(t) (1.11) Ait)L(t) A(t)Ut)Ut) AmWAit) K/AL is simply . From (1.8) and (1.9), 1/L and A/Aaren and g.K is given by (1.10). Substituting these facts into (1.11) yields (1.12) Yit) =  ~ "(  9 . That is, since is a function of K, L, and A, each of which are functions of f, then dk dk dk k =  + I + A sL SA insights about particular features of the world. If a simplifying assumption causes a model to give incorrect answers to the questions it is being used to address, then that lack of realism may be a defect. (Even then, the simplificationby showing clearly the consequences of those features of the world in an idealized settingmay be a useful reference point.) If the simplification does not cause the model to provide incorrect answers to the questions it is being used to address, however, then the lack of realism is a virtue: by isolating the effect of interest more clearly, the simplification makes it easier to understand. 1.3 The Dynamics of the Model We want to determine the behavior of the economy we have just described. The evolution of two of the three inputs into production, labor and knowledge, is exogenous. Thus to characterize the behavior of the economy we must analyze the behavior of the third input, capital.
1.3 The Dynamics of the Model 13 Breakeven investment {n + g + s)k k* FIGURE 1.2 Actual and breakeven investment Finally, using the fact that Y/AL is given by f(k), we have kt) = sfmt)) ~{n+g + S)k(t). (1.13) Equation (1.13) is the key equation of the Solow model. It states that the rate of change of the capital stock per unit of effective labor is the difference between two terms. The first, sf(k), is actual investment per unit of effective labor: output per unit of effective labor is f{k), and the fraction of that output that is invested is s. The second term, (n+g + 8)k, is breakeven investment, the amount of investment that must be done just to keep at its existing level. There are two reasons that some investment is needed to prevent from falling. First, existing capital is depreciating; this capital must be replaced to keep the capital stock from falling. This is the 8k term m (1.13). Second, the quantity of effective labor is growing. Thus doing enough investment to keep the capital stock (K) constant is not enough to keep the capital stock per unit of effective labor (k) constant. Instead, since the quantity of effective labor is growing at rate n+g, the capital stock must grow at rate n\g\.o hold steady. This is the (n f g)k term in (1.13). When actual investment per unit of effective labor exceeds the investment needed to break even, is rising. When actual investment falls short of breakeven investment, is falling. And when the two are equal, is constant. Figure 1.2 plots the two terms of the expression for as functions of k. Breakeven investment, {n + g + 8)k, is proportional to k. Actual investment, sf(k), is a constant times output per unit of effective labor. Since f{0) = 0, actual investment and breakeven investment are equal at = 0. The Inada conditions imply that at = 0, f(k) is large, and thus that The growth rate of a variable, x, refers its proportional rate of change, x/x.lt is easy to verify that the growth rate of the product of two variables, , is the sum of their growth rates, xj /i + x/x. Similarly, the growth rate of the ratio of two variables, xi / 2, IS the difference of their growth rates, xi /xi  xz / 2. Thus, the growth rate oik = /al is ik  (A/a + L/l). It follows that keeping constant requires / = n t g.
FIGURE 1.3 The phase diagram for in the Solow model the sf(k) line is steeper than the (n + g + 8)k Une. Thus, for small values of k, actual investment is larger than breakeven investment. The Inada conditions also imply that fik) falls toward zero as becomes large. At some point, the slope of the actual investment line falls below the slope of the breakeven investment line. With the sf(k) line flatter than the (n + g + S)k line, the two must eventually cross. Finally, the fact that f"(k) < 0 implies that the two lines intersect only once for > 0. We let * denote the value of where actual investment and breakeven investment are equal. Figure 1.3 summarizes this information in the form of a phase diagram, which shows as a function of k. If is initially less than k*, actual investment exceeds breakeven investment, and so is positivethat is, is rising. If exceeds k*,kis negative. Finally, if equals k*,kis zero. Thus, regardless of where starts, it converges to k*} The Balanced Growth Path Since converges to *, it is natural to ask how the variables of the model behave when equals k*. By assumption, labor and knowledge are growing at rates n and g, respectively. The capital stock, K, equals ALk; since is constant atk*,K is growing at rate n + g (that is, K/K equals n + g). With both capital and effective labor growing at rate n + g, the assumption of constant returns implies that output, Y, is also growing at that rate. Finally, capital per worker, K/L, and output per worker, Y/L, are growing at rate g. Thus the Solow model impUes that, regardless of its starting point, the economy converges to a balanced growth patha situation where each "If is initially zero, it remains there. We ignore this possibility in what follows.
[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]

