ME II, Prof. Dr. T. Wollmershäuser Chapter 12 Saving, Capital Accumulation, and Output Version: 23.06.2010
Saving, Capital Accumulation, and Output The effects of the saving rate the ratio of saving to GDP on capital and output per capita are the topics of this chapter. An increase in the saving rate would lead to higher growth for some time, and eventually to a higher standard of living. We have to extend our production function by an additional factor of production, capital. ME II, Prof. Dr. T. Wollmershäuser, Slide 2
Interactions between Output and Capital The aggregate production function is a specification of the relation between aggregate output Y and the inputs in production: Yt = F( Kt, Nt, At) state of technology (a set of blue prints defining the range of products and the techniques available to produce them, A t ) labor (employment, the number of workers in the economy, N t ) capital (the sum of all the machines, plants, and office buildings in the economy, K t ) The function F, which tells us how much output is produced for given quantities of capital and labor, is the aggregate production function. Here we focus on the accumulation of capital: 1. The size of the population, the participation rate, and the unemployment rate are all constant: N t = N 2. There is no technological progress: A t = A = 1 Y = F( K, N) t t ME II, Prof. Dr. T. Wollmershäuser, Slide 3
Interactions between Output and Capital At the center of the determination of output in the long run are two relations between output and capital: 1. capital production The amount of capital determines the amount of output being produced. 2. production capital accumulation The amount of output determines the amount of saving and, in turn, the amount of capital accumulated over time. ME II, Prof. Dr. T. Wollmershäuser, Slide 4
Interactions between Output and Capital Y = F( K, N) K = +Δ + 1 K K I = S = sy Δ K = I δ K 1 ME II, Prof. Dr. T. Wollmershäuser, Slide 5
The Effects of Capital on Output Constant returns to scale is a property of the economy (of the production function) in which, if the scale of operation is doubled that is, if the quantities of capital and labor are doubled then output will also double: 2 Y = F(2 K,2 N) Or more generally, for any number x: xy = F( xk, xn) Constant returns to scale implies that we can rewrite the aggregate production function as: xy xk xn xk K Y = F, F,1 F,1 xn xn xn = = = xn N N ME II, Prof. Dr. T. Wollmershäuser, Slide 6
The Effects of Capital on Output Y t N Kt = F,1 N The amount of output per worker, Y t /N depends on the amount of capital per worker, K t /N. As capital per worker increases, so does output per worker. Simplifying: Y t N Kt = f N where: f Kt Kt F,1 N N ME II, Prof. Dr. T. Wollmershäuser, Slide 7
The Effects of Output on Capital Accumulation To derive the second relation, between output and capital accumulation, we proceed in two steps: 1. We derive the relation between output and investment. 2. We derive the relation between investment and capital accumulation. ME II, Prof. Dr. T. Wollmershäuser, Slide 8
The Effects of Output on Capital Accumulation Output and Investment: In a closed economy the following relationship between private saving and investment holds: I = S+ ( T G) If T = G T G = 0 I = S S = sy where 0 < s < 1 Investment is equal to private saving; private saving is proportional to income: I = sy t Thus, investment is proportional to income: The higher production, the higher income, private saving and investment. t ME II, Prof. Dr. T. Wollmershäuser, Slide 9
The Effects of Output on Capital Accumulation Investment and Capital Accumulation: The evolution of the capital stock is given by: K = 1 (1 + δ ) K + I t t t δ denotes the rate of depreciation. Combining the relation from output to investment, I t = sy t, and the relation from investment to capital accumulation, we obtain the second important relation we want to express, from output to capital accumulation: K = + + 1 (1 δ ) K sy t t t ME II, Prof. Dr. T. Wollmershäuser, Slide 10
The Effects of Output on Capital Accumulation Output and Capital per Worker: Kt+ 1 Kt Yt = (1 δ ) + s N N N Rearranging terms in the equation above, we can articulate the change in capital per worker over time: Kt+ 1 Kt Yt Kt = s δ N N N N In words, the change in the capital stock per worker (left side) is equal to saving per worker minus depreciation (right side). ME II, Prof. Dr. T. Wollmershäuser, Slide 11
Dynamics of Capital and Output Our two main relations are: Yt N K f N t = First relation: Capital determines output via production function K K Y K = s δ N N N N t+ 1 t t t Second relation: Output determines capital accumulation via private saving Combining the two relations, we can study the behavior of output and capital over time. ME II, Prof. Dr. T. Wollmershäuser, Slide 12
Dynamics of Capital and Output Y t N Kt = f N Kt + 1 Kt Yt Kt = s δ N N N N From our main relations above, we express output per worker (Y t /N) in terms of capital per worker (f(k t /N)) to derive the equation below: Kt+ 1 Kt Kt Kt = sf δ N N N N Change in capital from year t to year t+1 Investment during year t Depreciation during year t ME II, Prof. Dr. T. Wollmershäuser, Slide 13
Dynamics of Capital and Output Kt+ 1 Kt Kt Kt = sf δ N N N N Change in capital from year t to year t+1 Investment during year t Depreciation during year t This relation describes what happens to capital per worker. The change in capital per worker from this year to next year depends on the difference between two terms: If investment per worker exceeds depreciation per worker, the change in capital per worker is positive: Capital per worker increases. If investment per worker is less than depreciation per worker, the change in capital per worker is negative: Capital per worker decreases. ME II, Prof. Dr. T. Wollmershäuser, Slide 14
Dynamics of Capital and Output At K 0/N, capital per worker is low, investment exceeds depreciation (AC > AD), thus, capital per worker and output per worker tend to increase over time. Decreasing returns to capital refers to the property that increases in capital lead to smaller and smaller increases in output as the level of capital increases. Depreciation per worker increases in proportion to capital per worker. Private saving (a share of production) equals investment. AB = Output per worker AC = Investment per worker AD = Depreciation per worker ME II, Prof. Dr. T. Wollmershäuser, Slide 15
Dynamics of Capital and Output At K*/N, output per worker and capital per worker remain constant at their long-run equilibrium levels. Investment per worker increases with capital per worker, but by less and less as capital per worker increases (decreasing returns to capital). Depreciation per worker increases in proportion to capital per worker. ME II, Prof. Dr. T. Wollmershäuser, Slide 16
Dynamics of Capital and Output When capital and output are low, investment exceeds depreciation, and capital increases. When capital and output are high, investment is less than depreciation, and capital decreases. ME II, Prof. Dr. T. Wollmershäuser, Slide 17
Steady-State Capital and Output Kt+ 1 Kt Kt Kt = sf N N N δ N The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy. In steady state, the left side of the equation above equals zero, then: * * * * Kt+ 1 Kt * * * K K = 0 Kt+ 1 = Kt = K s f = δ N N N N Given the steady state of capital per worker (K*/N), the steady-state value of output per worker (Y*/N) is given by the production function: Y * K * = f N N ME II, Prof. Dr. T. Wollmershäuser, Slide 18
The Saving Rate and Output The Effects of Different Saving Rates A country with a higher saving rate achieves a higher steady- state level of output per worker. ME II, Prof. Dr. T. Wollmershäuser, Slide 19
The Saving Rate and Output Three observations about the effects of the saving rate on the growth rate of output per worker are: 1. The saving rate has no effect on the long run growth rate of output per worker, which is equal to zero. In the steady state output per worker is constant. ME II, Prof. Dr. T. Wollmershäuser, Slide 20
The Saving Rate and Output Three observations about the effects of the saving rate on the growth rate of output per worker are: 2. Nonetheless, the saving rate determines the level of output per worker in the long run. Other things equal, countries with a higher saving rate will achieve higher output per worker in the long run. ME II, Prof. Dr. T. Wollmershäuser, Slide 21
The Saving Rate and Output Three observations about the effects of the saving rate on the growth rate of output per worker are: Note: the saving rate cannot be increased infinitely! It has to below one, since households cannot save more than they earn (s 1). 3. An increase in the saving rate will lead to higher growth of output per worker for some time, but not forever. The steady-state output per worker increases with a higher saving rate. The saving rate has no effect on the long run growth rate of output per worker, which is equal to zero. An increase in the saving rate leads to a period of higher growth until output reaches its new, higher steady-state level. ME II, Prof. Dr. T. Wollmershäuser, Slide 22
The Saving Rate and Output The Effects of an Increase in the Saving Rate on Output per Worker An increase in the saving rate leads to a period of higher growth until output reaches its new, higher steady-state level. ME II, Prof. Dr. T. Wollmershäuser, Slide 23
The Saving Rate and Consumption An increase in the saving rate always leads to an increase in the level of output per worker. But output is not the same as consumption. To see why, consider what happens for two extreme values of the saving rate: An economy in which the saving rate is (and has always been) zero is an economy in which capital is equal to zero. In this case, output is also equal to zero, and so is consumption. A saving rate equal to zero implies zero consumption in the long run. Now consider an economy in which the saving rate is equal to one: People save all their income. The level of capital, and thus output, in this economy will be very high. But because people save all their income, consumption is equal to zero. A saving rate equal to one also implies zero consumption in the long run. ME II, Prof. Dr. T. Wollmershäuser, Slide 24
The Saving Rate and Consumption What is the optimal saving rate? The saving rate is optimal, if consumption per worker (not output per worker) is maximized. The level of capital associated with the value of the saving rate that yields the highest level of consumption per worker in steady state is known as the golden-rule level of capital. ME II, Prof. Dr. T. Wollmershäuser, Slide 25
The Saving Rate and Consumption The Effects of the Saving Rate on Steady- State Consumption per Worker Changes in the saving rate lead to changes in the steadystate consumption per worker. Produktion je Beschäftigten Output Y/N Y 1 /N Y 0 /N B A S = 1: C/N = 0 C/N at s 1 Depreciation δ K t /N Output f(k t /N) Investment at s 1 s 1 f(k t /N) Investment at s 0 s 0 f(k t /N) Consumption C/N at s 0 : AB S = 0: C/N = 0 (K 0 /N) K 1 /N Capital K/N ME II, Prof. Dr. T. Wollmershäuser, Slide 26
The Saving Rate and Consumption The Effects of the Saving Rate on Steady- State Consumption per Worker An increase in the saving rate up to s G leads to an increase in steady-state consumption per worker. A further increase then leads to a decrease in steady-state consumption per worker. ME II, Prof. Dr. T. Wollmershäuser, Slide 27
Getting a Sense of Magnitudes Assume the production function is of the Cobb-Douglastype: Output per worker is: Y = K N Y Kt N K t t Kt = = = N N N N Output per worker, as it relates to capital per worker is: t Yt Kt Kt = f = N N N Capital per worker is given by: Kt+ 1 Kt Kt Kt = sf δ N N N N Using the production function: + 1 = s δ t K K K K t t t t N N N N ME II, Prof. Dr. T. Wollmershäuser, Slide 28
Getting a Sense of Magnitudes K K K K = s δ N N N N t+ 1 t t t In the steady state the lhs is equal to zero: K K s = δ N N Square both sides of the equation: Divide by (K/N) and rearrange terms: In the steady state capital per worker is equal to the square of the ratio of the saving rate to the depreciation rate. Output per worker is given by: Y K s = = s = N N δ δ 2 2 K δ 2 K s = N N 2 K = s N δ 2 ME II, Prof. Dr. T. Wollmershäuser, Slide 29
Getting a Sense of Magnitudes Y K s = = s = N N δ δ Steady-state output per worker is equal to the ratio of the saving rate to the depreciation rate. A higher saving rate and a lower depreciation rate both lead to higher steady-state capital per worker and higher steadystate output per worker. Example: Depreciation rate = 10%, doubling of the saving rate from 10% to 20% Result: In the long run output per worker doubles. 2 ME II, Prof. Dr. T. Wollmershäuser, Slide 30
The Dynamic Effects of an Increase in the Saving Rate K0 01, Y0 K0 = = 1 = = 1= 1 N 01, N N K1 K new K K = s δ N N N N K1 K0 new K0 K0 = + s δ = 1+ 0, 2 1 0, 1 1= 11, N N N N Y1 K1 = = 11, = 105, N N K2 K1 new K1 K1 = + s δ = N N N N Y N 2 0 0 0 K = = N 2 2 2 K 02, Y K = = 4 = = 4 = 2 N 01, N N ME II, Prof. Dr. T. Wollmershäuser, Slide 31
The Dynamic Effects of an Increase in the Saving Rate The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker It takes a long time for output to adjust to its new, higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth. ME II, Prof. Dr. T. Wollmershäuser, Slide 32
The Saving Rate and the Golden Rule In steady state, consumption per worker is equal to output per worker minus depreciation per worker: C Y K = δ = + N N N ( Y C I) K N s = δ Knowing that: and 2 Y K s = = s = N N δ δ 2 then: C s 1 δ s s( s) = = N δ δ δ 2 a parabola that opens downward These equations are used to derive the Table in the next slide. ME II, Prof. Dr. T. Wollmershäuser, Slide 33
The Saving Rate and the Golden Rule Table Saving Rate, s The Saving Rate and the Steady-state Levels of Capital, Output, and Consumption per Worker Capital per Worker, (K/N) K s = N δ 2 2 Output per Worker, (Y/N) Y s = N δ Consumption per Worker, (C/N) C N ( 1 ) = s s δ 0.0 0.0 0.0 0.0 0.1 0.2 1.0 4.0 1.0 2.0 0.9 1.6 0.3 9.0 3.0 2.1 0.4 16.0 4.0 2.4 0.5 25.0 5.0 2.5 0.6 36.0 6.0 2.4 1.0 100.0 10.0 0.0 ME II, Prof. Dr. T. Wollmershäuser, Slide 34
The Saving Rate and the Golden Rule 14 12 Y = K N t Depreciation rate 10% 2% t Consumption per Worker C/N 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Saving rate s ME II, Prof. Dr. T. Wollmershäuser, Slide 35
The Saving Rate and the Golden Rule Y = K N = K N Y 05, 05, t t t in more general terms: t * K N * Y N = K N α t s = δ s = δ 1 α 1 α α 1 α α 1 1 α 1 α * C s s = δ N δ δ Example for α= 1 3 1 ME II, Prof. Dr. T. Wollmershäuser, Slide 36
The Saving Rate and the Golden Rule 3 2.5 Y = K N t 13 23 t Depreciation rate 10% 2% Consumption per Worker C/N 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Saving rate s ME II, Prof. Dr. T. Wollmershäuser, Slide 37
The Saving Rate and the Golden Rule is the optimal saving rate always equal to the output elasticity capital α? α 1 * C s 1 α s 1 α N = δ δ δ * ( ) α 1 1 1 1 α 1 α C N α s 1 1 s 1 = δ = 0 s 1 α δ δ 1 α δ δ α 1 1 1 1 α 1 α 1 α s 1 s = δ1 α δ 1 α δ α 1 1 1 1 α 1 α α s s = δ δ δ 1 α 1 α 1 α 1 1 1 1 1 + 1 1 α 1 α 1 α 1 α 1 α 1 α α s s s = = δ δ δ δ s = α s s = = δ δ ME II, Prof. Dr. T. Wollmershäuser, Slide 38
The Saving Rate and the Golden Rule 45 40 35 30 25 20 15 10 5 0 Gross national saving in % of disposable income (1960-2008) 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Germany USA Japan Quelle: AMECO ME II, Prof. Dr. T. Wollmershäuser, Slide 39
Physical versus Human Capital The set of skills of the workers in the economy is called human capital. An economy with many highly skilled workers is likely to be much more productive than an economy in which most workers cannot read or write. The conclusions drawn about physical capital accumulation remain valid after the introduction of human capital in the analysis. ME II, Prof. Dr. T. Wollmershäuser, Slide 40
Extending the Production Function When the level of output per workers depends on both the level of physical capital per worker, K/N, and the level of human capital per worker, H/N, the production function may be written as: Y N ( + ) ( + ) K H = f(, ) N N An increase in capital per worker or the average skill of workers leads to an increase in output per worker. We continue to assume that returns to capital are decreasing. ME II, Prof. Dr. T. Wollmershäuser, Slide 41
Extending the Production Function A measure of human capital H may be constructed as follows: Suppose an economy has N=100 workers, half of them unskilled and half of them skilled. The relative wage of skilled workers is twice that of unskilled workers. Then: H 150 H = [( 50 1) + ( 50 2)] = 150 = = 1. 5 N 100 ME II, Prof. Dr. T. Wollmershäuser, Slide 42
Human Capital, Physical Capital, and Output An increase in how much society saves in the form of human capital through education and on-the-jobtraining increases steady-state human capital per worker, which leads to an increase in output per worker. In the long run, output per worker depends not only on how much society saves but also how much it spends on education. ME II, Prof. Dr. T. Wollmershäuser, Slide 43
Human Capital, Physical Capital, and Output In Germany spending on education comprises about 5.6% of GDP, compared to 18.4% gross investment in physical capital. This comparison is only an approximation due to measurement problems of education: Education also enters the national accounts as consumption. It does not account for the informal on-thejob-training. Considers gross, not net investment. Depreciation of human capital is slower than that of physical capital. ME II, Prof. Dr. T. Wollmershäuser, Slide 44
Conclusion This Chapter has shown that capital accumulation cannot by itself sustain growth, but that it does affect the level of output per worker. We assumed that the state of technology is given and constant over time. Is technological progress unrelated to the level of human capital in the economy? Can t a better-educated labor force lead to a higher rate of technological progress? These questions take us to the topic of the next chapter: the sources and the effects of technological progress. ME II, Prof. Dr. T. Wollmershäuser, Slide 45