Chapter 11Growth and Technological Progress- The Solow-.ppt
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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
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