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The Theory of Economic Growth: The Solow Growth Model Reading: DeLong/Olney: Macroeconomics; McGraw-Hill; 2006; Chapter 4 4-1 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Questions • What are the causes of long-run economic growth? • What is the “efficiency of labor”? • What is an economy’s “capital intensity”? • What is an economy’s “balancedgrowth path”? • What can we say about convergence to the “balanced growth path”? 4-2 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Questions • How important is faster labor-growth as a drag on economic growth? • How important is a high saving rate as a cause of economic growth? • How important is technological and organizational progress for economic growth? 4-3 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Long-Run Economic Growth • We classify the factors that generate differences in productive potentials into two broad groups – differences in the efficiency of labor • how technology is deployed and organization is used – differences in capital intensity • how much current production has been set aside to produce useful machines, buildings, and infrastructure 4-6 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Efficiency of Labor • The efficiency of labor has risen for two reasons – advances in technology – advances in organization • Economists are good at analyzing the consequences of advances in technology but they have less to say about their sources 4-7 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Capital Intensity • There is a direct relationship between capital-intensity and productivity – a more capital-intensive economy will be a richer and more productive economy 4-8 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Standard Growth Model • Also called the Solow growth model • Consists of – variables – behavioral relationships – equilibrium conditions • The key variable is labor productivity – output per worker (Y/L) 4-9 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Solow Growth Model • Balanced-growth equilibrium – the capital intensity of the economy (K/Y) is stable, but (K/L) and (Y/L) grow – the economy’s capital stock and level of real GDP are growing at the same rate – the economy’s capital-output ratio is constant 4-10 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Solow Growth Model • The balanced-growth path: – if the economy is on its balanced-growth path, the present value and future values of output per worker will continue to follow the balanced-growth path – if the economy is not yet on its balancedgrowth path, it will head towards it 4-11 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Production Function • The production function tells us how the average worker’s productivity (Y/L) is related to the efficiency of labor (E) and the amount of capital at the average worker’s disposal (K/L) (Y/L) F[(K/L), E] • Cobb-Douglas production function (Y/L) (K/L) (E)1- 4-12 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Production Function (Y/L) (K/L) (E)1- • measures how fast diminishing marginal returns to investment set in – the smaller the value of , the faster diminishing returns are occurring 4-13 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.2 - The Cobb-Douglas Production Function for Different Values of 4-14 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Production Function (Y/L) (K/L) (E)1- • The value of the efficiency of labor (E) tells us about the placement of the production function – a higher level of E means that more output per worker is produced for each possible value of the capital stock per worker 4-15 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.3 - The Cobb-Douglas Production Function for Different Values of E 4-16 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Production Function: Example • E = $10,000 • = 0.3 • K/L = $125,000 Y K 1 0.3 0.7 E (125,000) (10,000) $21,334 L L 4-17 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Production Function: Example • If K/L rises to $250,000 Y K 1 0. 3 0 .7 E (250,000) (10,000) $26,265 L L – the first $125,000 of K/L increased Y/L from $0 to $21,334 – the second $125,000 of K/L increased Y/L from $21,334 to $26,265 4-18 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • The net flow of saving is equal to the amount of investment • Remember from National Accounts that real GDP (Y) can be divided into four parts – consumption (C) – investment (I) – government purchases (G) – net exports (NX = GX - IM) 4-19 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation C I G GX IM Y C I (G T ) GX IM Y T I ( Y T C) (G T ) (IM GX ) 4-20 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation I ( Y T C) (G T ) (IM GX ) • The right-hand side shows the three pieces of total saving – household saving (SH) – government saving (SG) – foreign saving (SF) I SH SG SF 4-21 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • Let’s assume that total saving is a constant fraction (s) of real GDP S S S s Y • Therefore, it must be true that H G F I sY 4-22 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • We will refer to s as the economy’s saving rate – we will assume that it will remain at its current value as we look far into the future – s measures the flow of saving and the share of total production that is invested and used to increase the capital stock 4-23 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • The capital stock is not constant • We will let – K0 will mean the capital stock at some initial year – K2003 will mean the capital stock in 2003 – Kt will mean the capital stock in the current year – Kt+1 will mean the capital stock next year – Kt-1 will mean the capital stock last year 4-24 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • Investment will make the capital stock tend to grow • Depreciation makes the capital stock tend to shrink – the depreciation rate is assumed to be constant and equal to 4-25 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • Next year’s capital will be Kt 1 Kt investment depreciation K t 1 K t sYt K t • The capital stock is constant when sYt K t Kt s Yt 4-26 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • Suppose that the economy has no labor force growth and no growth in the efficiency of labor – if K/Y < s/, depreciation is less than investment so K and K/Y will grow until K/Y = s/ – if K/Y > s/, depreciation is greater than investment so K and K/Y will fall until K/Y = s/ 4-27 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • Thus, if the economy has no labor force growth and no growth in the efficiency of labor, the equilibrium condition of this growth model is Kt s Yt 4-28 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Saving, Investment, and Capital Accumulation • Remember that, in this particular case, we are assuming that – the economy’s labor force is constant – the economy’s capital stock is constant – there are no changes in the efficiency of labor • Thus, equilibrium output per worker is constant • Now we will complicate the model 4-30 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Adding in Labor Force and Labor Efficiency Growth • Growth in labor force (L) – assume that L is growing at a constant rate (n) L t 1 (1 n) L t – if this year’s labor force is equal to 10 million and the growth rate is 2% per year, next year’s labor force will be L t 1 (1.02) 10,000,000 10,200,000 4-31 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.6 - Constant Labor-Force Growth (at n = 2% per Year) 4-32 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Adding in Labor Force and Labor Efficiency Growth • Assume that the efficiency of labor (E) is growing at a constant proportional rate (g) Et 1 (1 g) Et – if this year’s efficiency of labor is $10,000 and the growth rate is 1.5% per year, next year’s efficiency of labor will be Et 1 (1.015) $10,000 $10,150 4-33 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.7 - Efficiency-of-Labor Growth at g = 1.5% per Year 4-34 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Balanced-Growth CapitalOutput Ratio • When we assumed that the labor force and efficiency were both constant (so that n and g are equal to 0), the equilibrium condition was K s Y • Since s and are constant, K/Y is constant 4-35 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Balanced-Growth CapitalOutput Ratio • If we assume that the labor force and efficiency grow at n and g, the equilibrium condition still requires that K/Y is constant – the economy is in balanced growth – output per worker is growing at the same rate as the capital stock per worker • both growing at the same rate as the efficiency of labor 4-36 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Balanced-Growth CapitalOutput Ratio • The economy will be in balanced growth equilibrium when K s Y (n g ) • This is the balanced-growth equilibrium capital-output ratio 4-37 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Balanced-Growth Output per Worker • Suppose that the economy is on its balanced-growth path – K/Y is equal to its balanced-growth equilibrium value • Let’s calculate the level of output per worker (Y/L) 4-40 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Balanced-Growth Output per Worker • Begin with the capital-output ratio version of the production function Yt K t L t Yt 4-41 1 Et Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Balanced-Growth Output per Worker • Since the economy is on its balancedgrowth path Yt s Lt n g 1 Et • Since s, n, g, , and are all constants, [s/(n+g+)]/1- is a constant 4-42 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Balanced-Growth Output per Worker • Along the balanced-growth path, output per worker is simply a constant multiple of the efficiency of labor • Over time, the efficiency of labor grows at a constant rate g – Y/L must be growing at the same proportional rate 4-43 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.8 – Balanced Growth: Output per Worker and the Efficiency of Labor 4-44 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Balanced-Growth Output per Worker • Capital intensity (K/Y) determines what multiple Y/L is of the current labor efficiency – things that increase K/Y make the balanced-growth equilibrium Y/L a higher multiple of the efficiency of labor – things that reduce K/Y make the balancedgrowth equilibrium Y/L a lower multiple of the efficiency of labor 4-45 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Balanced-Growth Output per Worker • Changes in the capital intensity shift the balanced-growth path up or down • But the growth rate of Y/L along the balanced-growth path is simply the rate of growth of the efficiency of labor – the material standard of living grows at the same rate of labor efficiency – changes in K/L alone will not accomplish this 4-46 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.9 – Calculating Balanced-Growth Output per Worker Output per Worker [K/Y = s/(n+g+)] Balanced-Growth Y/K current output per worker along the balancedgrowth path Y/L (function of Et) Capital Stock per Worker 4-47 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.9 – Calculating Balanced-Growth Output per Worker Output per Worker Balanced-Growth Y/K Y/L will rise Y/K’ Y/L Anything that increases the balancedgrowth capital-output ratio will rotate the capital-output line clockwise Capital Stock per Worker 4-48 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.9 – Calculating Balanced-Growth Output per Worker Output per Worker Y/K’ Balanced-Growth Y/K Y/L will fall Y/L Anything that decreases the balancedgrowth capital-output ratio will rotate the capital-output line counterclockwise Capital Stock per Worker 4-49 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Off the Balanced-Growth Path • When K/Y > s/(n+g+) – K/Y is falling because investment is insufficient to keep K growing as fast as Y • When K/Y < s/(n+g+) – K/Y is rising because the growth in K outruns the growth in Y • K/Y and Y/L will converge to their balanced-growth paths 4-50 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.10 - Convergence to a BalancedGrowth Capital-Output Ratio of 4 4-51 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. How Fast the Economy Heads for Its Balanced-Growth Path • A fraction (1-)(n+g+) of the gap between the economy’s current position and its balanced-growth path will be closed each year – assume that this fraction is 4% – according to the rule of 72,the economy will move halfway to equilibrium in 72/4 or 18 years – the convergence does not happen quickly 4-52 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.11 – The Return of the West German Economy to Its Balanced Growth Path 4-53 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Labor Force Growth Rate • The faster the growth of the labor force, the lower will be the economy’s balanced-growth K/Y ratio – the larger the share of current investment that must go to equip new workers with the capital they need • Thus, a sudden, permanent increase in labor force growth will also lower Y/L on the balanced-growth path 4-54 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.12 – The Labor Force Growth Rate Matters 4-55 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. The Saving Rate and the Price of Capital Goods • The higher the share of real GDP devoted to saving and gross investment, the higher will be the economy’s balanced-growth K/L ratio – more investment increases the amount of new capital • A higher saving rate also increases Y/L along the balanced-growth path 4-56 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 4.13 - Investment Shares of Output and Relative Prosperity 4-57 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Growth Rate of the Efficiency of Labor • An increase in g will reduce the economy’s balanced-growth K/L ratio – past investment will be small relative to current output • Changes in g change the growth rate of Y/L along the balanced-growth path – these effects are overwhelmed by the direct effect of g on Y/L 4-58 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Growth Rate of the Efficiency of Labor • The growth rate of the standard of living can change if and only if g changes – other factors can shift Y/L up but do not permanently change the growth rate of Y/L 4-59 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Table 4.2 – Effects of Increases in Parameters on the Solow Growth Model When there is an increase in the parameter… 4-60 The Effect on… Equilibrium K/Y Level of Y Permanent Permanent Level Growth Growth of Y/L Rate of Y Rate of Y/L s Increases Up Up n Decreases Up Down Increases Decreases Down Down No change No change g Decreases Up Increases Up No change Increases No change No change Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • One principal force driving long-run growth in output per worker is the set of improvements in the efficiency of labor springing from technological progress 4-61 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • A second principal force driving longrun growth in output per worker is the increases in capital intensity – the ratio of the capital stock to output 4-62 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • The balanced-growth equilibrium in the Solow growth model occurs when the capital output ratio K/Y is constant – when K/Y is constant, the capital stock and real output are growing at the same rate 4-63 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • The Cobb-Douglas production function we use is Y/L = [K/L]E(1-) – this is equivalent to Y/L = [K/Y]/1-(E) – an increase in makes the production function steeper – an increase in E makes the production function shift up 4-64 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • In equilibrium, investment equals saving: I = S = SH+SG+SF – we assume S/Y = s = saving rate is constant 4-65 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • The balanced-growth equilibrium value of the capital output ratio K/Y is a constant equal to the saving rate s divided by the sum of the labor force growth rate n, the labor efficiency growth rate g, and the depreciation rate – in balanced growth: K/Y = s/(n+g+) 4-66 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • If the economy’s actual value of K/Y is initially greater than s/(n+g+), then K/Y will fall until it reaches its equilibrium value • If the economy’s actual value of K/Y is initially less than s/(n+g+), then K/Y will rise until it reaches its equilibrium value 4-67 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • It can take decades or generations for K/Y to reach its balanced-growth equilibrium value 4-68 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • An increase in the saving rate s, a decrease in the labor force growth rate n, or a decrease in the depreciation rate increases Y/L – the growth rate of Y/L will accelerate as the economy moves to its new higher balanced-growth path – once there, Y/L will grow at the same rate as it did initially 4-69 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Summary • In balanced-growth equilibrium, the growth rate of output per worker equals the growth rate of labor efficiency g – only increases in g can produce a lasting increase in the growth rate of output per worker 4-70 Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.