1、1,Chapter 11 Growth and Technological Progress: The Solow-Swan Model, Pierre-Richard Agnor,The World Bank,2,Basic Structure and Assumptions The Dynamics of Capital and Output A Digression on Low-Income Traps Population, Savings, and Steady-State Output The Speed of Adjustment Model Predictions and E
2、mpirical Facts,3,Basic Structure and Assumptions,4,Solow-Swan model assumptions: closed economy, producing one good using both labor and capital; technological progress is given and the saving rate is exogenously determined; no government and fixed number of firms in the economy, each with the same
3、production technology; output price is constant and factor prices (including wages) adjust to ensure full utilization of all available inputs.,5,Four variables considered: flow of output, Y; stock of capital, K; number of workers, L; knowledge or the effectiveness of labor, A. Aggregate production f
4、unction given by,Y = F(K,AL).A and L enter multiplicatively, where AL is effective labor, and technological progress enters as labor augmenting or Harrod neutral.,6,Assumed characteristics of the model: Marginal product of each factor is positive (Fh 0, where h = K, AL) and there are diminishing ret
5、urns to each input (Fhh 0). Constant returns to scale (CRS) in capital and effective labor:F(mK,mAL) = mF(K,AL). m 0 (1)Inputs other than capital, labor, and knowledge are unimportant. Model neglects land and other natural resources.,7,Intensive-form production function: Output per unit of effective
6、 labor, y, and capital per unit of effective labor, k, are related by setting m = 1/AL in (1), F(K/AL, 1) = 1/AL F(K, AL) (2)Let k = K/AL, y = Y/AL, and f(k) = F(k,1). Equation (2) is then written as y = f(k), f(0) = 0 (3),8,Intensive-form production function: In (3), f (k) is the marginal product o
7、f capital, FK, marginal product of capital is positive. marginal product of capital decreases as capital (per unit of effective labor) rises, f “(k) 0.,k0,k,Intensive-form satisfies Inada conditions:lim f (k) = , lim f (k) = 0,9,Cobb-Douglas Function: A production function that satisfies Solow-Swan
8、model characteristics:Y = F(K, AL) = K(AL)1- , 0 0, f “(k) = - (1 - )k - 2 0.,10,Labor and Knowledge Labor and knowledge determined exogenously with constant growth rates, and :,Savings and Consumption Output divided between consumption, C, and investment, I: Y = C + I (9),11,Savings, S, defined as
9、Y - C, a constant fraction, s, of output: S Y - C= sY, 0 s 1 (10)Savings equals investment such that:S = I = sY (11),12,with denoting the rate of capital stock depreciation Consumption per unit of labor, c, is determined by:c= (1 - s)k , i = sk (13)with, c = C/AL and i = I/AL See Figure 11.1 for gra
10、phical distribution of output allocated among consumption, saving, and investment.,Savings and Consumption Capital stock, K, changes through time by:K = sY - K (12),.,13,14,The Dynamics of Capital and Output,15,Differentiating the expression k = K/AL with respect to time yields,Economic growth deter
11、mined by the behavior of the capital stock.,Substituting (4), (8), and (12) into (14) results ink = sk - ( + + )k, + + 0, (15)a non-linear, first-order differential equation.,.,16,Equation 15: the key to the Solow-Swan model; the rate of change of the capital stock per units of effective labor as th
12、e difference between two terms: sk: actual investment per unit of effective labor. Output per units of effective labor is k, and the fraction of that output that is invested is s. ( + + )k: required investment or the amount of investment that must be undertaken in order to keep k at its existing lev
13、el. Figure 11.2.,17,18,Reasons why investment is needed to prevent k from falling (see (15), capital stock is depreciating by k; effective labor is growing by (n + )k; therefore if sk is greater (lower) than (n + + )k, k rises (falls).,19,k/s 0, e.g. increase in savings rate increases k. (15) is glo
14、bally stable: capital stock always adjusts over time such that k converges to k. With k constant at k, capital stock grows at the rate,Equilibrium point, k, found by setting k = 0, in (15), with the solution:,.,gK = = A/A + L/L = + ,.,.,Figure 11.3.,20,21,At k, output grows at a rategY = A/A + L/L =
15、 + ,.,.,Since Y = ALy and y is constant at k , output grows at the rate of growth of effective labor. Growth of capital per worker and output per worker (labor productivity), are given by,gK/L = gK - L/L = , gY/L = gY - L/L = (20),.,.,Because K equals ALk, capitals growth rate will equal the growth
16、rate of effective labor at k.,22,On the balanced growth path, the growth rate of output per worker is determined solely by growth of technological progress. Rate of return on capital, r, and wage rate, w, are given by:r = k - 1 - , andw = (1 - )Ak ,23,A Digression on Low-Income Traps,24,Endogenously
17、 determined labor supply can lead to a dynamically stable, low-steady-state level of y. Figure 11.4 displays a situation where n(k) is negative at low levels of k, positive at intermediate levels of k, and again negative at higher values of k, leading to multiple equilibria and the possibility of a
18、low level trap.,Setting n = n(k) in (15),k = sk - (n(k)+ + ),.,25,26,Rosenstein-Rodan (1961) and Murphy, Shleifer, and Vishny (1989) show that a big push, e.g. an exogenous increase in savings, can bring the economy up to a higher, more stable equilibrium path.,27,Population, Savings, and Steady-Sta
19、te Output,28,With exogenous population growth: A decline in population growth from nH to nL causes the following results in Solow-Swan See Figure 11.5:,k increases until reaching kL kH, output per worker, Ak rises, but in the long run, Y/L grows at .,If the saving rate, s, increases: sk shifts up an
20、d to the left, k rises from kL to kH , Ak rises, but in the long run, Y/L grows at . See Figure 11.6.,29,30,31,Savings and Consumption,Consumption per unit of effective labor, c, is given byc = (1 - sL)k , i = sk (21)On the balanced growth path, with sk = (n + + )k,c = k - (n + + )k and,c/s = k - 1
21、- (n + + ) k/s/0,32,c/s is positive (negative) when marginal product of capital, k - 1 , is greater (lower) than (n + + ).,Elasticity of y with respect to s, , given by: = / (1 - ) (23)Low implies that y/k is low, that the actual investment curve, sk, bends sharply and that FK is low.,33,Speed of Ad
22、justment,34,Important to assess the speed at which adjustment to a new equilibrium proceeds. Changes in k can be approximated by,35,Solution of (24) given byk - k e t(k0-k)and y - y e t(y0-y) (25),(25) indicates that * 100 percent of the initial gap between output per worker and its steady state lev
23、el is closed every year.,36,Adjustment ratio, , is the fraction of change from y0 to y completed after t years:,using (25) = 1 - e t Time taken to achieve a given fraction of the adjustment is given by t* = ln(1 - )/ - /,37,Model Predictions and Empirical Facts,38,Predictions of the Solow-Swan Model
24、: Capital-effective labor ratio, marginal product of capital, and output per units of effective labor, are constant on the balanced growth path. Steady-state growth rate of capital per worker K/L and output per worker Y/L are determined solely by the rate of technological progress. In particular, ne
25、ither variable depends on the saving rate or on the specific form of the production function. Output, capital stock, and effective labor all grow at the same rate, given by the sum of the growth rate of the labor force and the growth rate of technological progress.,39,Reduction in the population gro
26、wth rate raises the steady-state levels of the capital-effective labor ratio and output in efficiency units and lowers the rate of growth of output, the capital stock, and effective labor. Rise in the saving rate also increases the capital-effective labor ratio and output in efficiency units in the
27、long run, but has no effect on the steady-state growth rates of output, the capital stock, and effective labor.,40,The Empirical Evidence: Support Data from industrial countries suggests that growth rates of labor, capital, and output are each roughly constant (Romer, 1989). Growth of output and cap
28、ital are about equal and are larger than the labor growth rate. In industrial countries output per worker and capital per worker are rising over time. As shown in Figures 10.2 and 10.3, evidence on the relationship between the level of income and both population growth and savings are consistent wit
29、h the model.,41,The Empirical Evidence: Baggage Strong empirical relationship between growth of per capita income and both the savings rate and the share of investment in output. Solow-Swan predicts no association. Observed differences in y are far too large to be accounted for by differences in phy
30、sical capital per worker. Attributing differences in output to differences in capital without differences in the the effectiveness of labor implies large variations in the rate of return of capital (Lucas, 1990).,42,The only source of variation in income per capita growth rates across countries is labor effectiveness, . But the model is incomplete because this driving force of long-run growth is exogenous to Solow-Swan.,
