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Neo-Classical Growth Model

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1 Neo-Classical Growth Model

2 Large Variations in Labor per Person (

3 Variation in Labor Force Participaton

4 Main Differences in Countries are Due to Variation in Labor Productivity

5 Plan Come up with separate theories governing both labor productivity and hours worked. First, labor productivity.

6 Rule of Thumb Growth Rate Rule of Thumb
Productivity growth is output growth minus labor growth.

7 Agricultural Era Prior to about 1775 or so, GDP per capita remained stagnant in virtually every country in the world. There were many technological advances during this period. Greater, productivity per unit of land tended to go into increasing the population level.

8 GDP per capita through history
Year Population GDP per Capita Macroeconomics by J. Bradford DeLong, Chap. 5

9 Pre-Industrial Revolution Source: Angus Madisson, Measuring the Chinese Economy

10 Industrial Revolution Spreads to NA, W, Europe and East Asia

11 Capital Productivity

12 Productivity Catch Up: Europe Source: Groningen Growth & Development Center
1950 % of USA 2003 Growth Rate U.S.A 12.00 100.0% 33.97 2.00% France 5.63 46.9% 37.75 111.1% 3.46% Germany 4.36 36.3% 30.01 88.3% 3.95% UK 7.49 62.4% 28.01 82.5% 2.91% Spain 2.60 21.7% 22.21 65.4% 4.94% 1990 US$, Average Output per Hour (Y/L)

13 Productivity Catch Up: Latin America Source: Groningen Growth & Development Center
1950 % of USA 2003 Growth Rate U.S.A 12.00 100.0% 33.97 2.00% Argentina 6.16 51.4% 10.57 31.1% 1.04% Brazil 2.48 20.7% 7.81 23.0% 2.21% Chili 4.66 38.9% 14.07 41.4% 2.12% Mexico 3.56 29.7% 10.24 30.1% 2.03%

14 Productivity Catch Up: East Asia Source: Groningen Growth & Development Center
1950 % of USA 2003 Growth Rate U.S.A 12.00 100.0% 33.97 2.00% Japan 2.30 19.2% 24.78 73.0% 4.57% 1973 Hong Kong 7.49 35.0% 22.28 65.6% 4.74% Korea 3.64 17.0% 14.25 42.0% 5.93% Singapore 6.80 31.8% 19.63 57.8% 4.61% Taiwan 4.37 20.4% 18.77 55.2% 6.33%

15 Growth by Region The World Economy, A Millienial Perspective by Angus Madisson

16 USA in Industrial Era Average Productivity of Capital, shows no trend upward or downward. The shares of income devoted to capital and labor show no trend. The average growth rate of output per person has been positive and relatively constant over time.

17 USA Factor Productivity 1980-2003

18 Productivity Growth in Korea and the USA
Compare the post-war growth of labor productivity of Korea and the US. The US started out with much higher labor productivity than the Korea. Both countries have seen positive productivity growth, Korea’s has been much faster.

19 Korean Labor Productivity goes from less than 10% of USA to more than 40%.

20 Capital Productivity: How do you measure capital
Option: Count it. Take surveys of industries, firms and households to find out the value of the capital that they own. Problem: Expensive Use the perpetual inventory method, to calculate the capital stock.

21 Capital Accumulation Capital is accumulated through investment and is lost through depreciation. Depreciation is not measured either. We might assume a constant rate of depreciation, δ.

22 Perpetual Inventory Method
Steps Estimate capital depreciation rate (usually δ≈.08 for annual). Guess initial capital stock (e.g. K1950 = I1950/ δ). Solve recursively forward.

23 Relatively Stable Capital Productivity in USA, decline in South Korea

24 Data Labor productivity is from All series derived from this database need to be referred to as: “Groningen Growth and Development Centre and The Conference Board, Total Economy Database, August 2004, Capital productivity data can be downloaded at Center for International Comparisons at University of Pennsylvania. URL Address:

25 Labor Force Growth Typically, we expect to see growth in the labor force due to population growth. This has been true in the US and Korea. Note that the labor force growth rate has been roughly constant over time in each country (though faster in Korea).

26 Labor Force Growth World Bank Global Development Database.

27 Objective Construct an economic theory that is consistent with these growth facts: In mature economies (like the USA) Labor productivity grows at a roughly constant rate. Capital productivity stays roughly constant. In developing economies (like Korea) Labor productivity grows faster than mature economies. Capital productivity shrinks over time.

28 Neo-classical Productivity Function (see Branson, p.576)
We assume constant returns to scale production function Because of constant returns we can write this as a productivity function Define and then,

29 Example: Cobb-Douglas
Divide both sides by L The labor productivity function is in capital per labor unit and technology.

30 Marginal Product It is true (because of CRTS) that the marginal effect of capital on output is equal to marginal product of capital per labor unit on labor productivity. Example: Cobb-Douglas

31 Labor Productivity Function (Constant A)
f(k,A) Slope = MPK= k

32 Diminishing Returns The production function has diminishing returns to capital and so does the productivity function have diminishing returns to capital per labor unit.

33 Cobb-Douglas Capital Productivity Function
We can also write the capital productivity function as a function of the capital labor ratio. In the Cobb-Douglas case, average productivity of capital

34 Capital Productivity Capital productivity is a decreasing function of the capital/labor ratio. Intuition: If you give more and more capital to the same amount of workers, the output that each machine will produce will go down.

35 Capital Productivity Function (Constant A)

36 Reading Supplement The best intermediate macroeconomics text on growth theory is Delong, Chapter 4. The main difference between these notes and Delong is that what we call technology, At, Delong calls Et. Delong emphasizes the importance of the capital output ratio. We emphasize capital productivity which is the inverse of the capital-output ratio.

37 Capital Accumulation The increase in the capital stock that occurs in every period is gross investment, It, net of depreciation, δKt.

38 Growth Rates vs. Continuous Growth Rates
The growth rate over a period of time is written as the difference in the variable over the period over its initial start value. When the change in the variable is not too large or the length of the time period is not too long, the growth rate is close to the continuous growth rate.

39 Investment Rate (see Branson, p. 579)
Define the Investment rate, The growth rate of capital is a function of the investment rate, capital productivity and the depreciation rate.

40 Growth Rate of Capital Per Labor Unit
The growth rate of capital per labor unit is equal to capital growth rate minus labor growth rate

41 Investment per Labor Unit
We can define investment per labor unit as Assume a constant investment rate

42 Investment per Worker Function (Constant A)
y sF(k,A) k

43 Replacement Investment per Worker
Assume you had a constant growth rate of labor If you invest just enough per worker to keep the capital-labor ratio constant, you need to replace depreciated capital and equip new workers. The greater is k, the more replacement investment you need to do.

44 Replacement Investment per Worker Function (Constant s,n)
y,rep sF(k,A) (δ+n)k k

45 Capital per Worker as an Engine of Labor Productivity Growth.
As long as investment per worker is greater than replacement investment per worker, the capital stock per worker will be growing. Holding A constant, this implies output per worker will be growing.

46 Steady State Capital Stock
Investment per worker is an increasing function the capital per worker since it is proportional to output per worker. However, both output per worker and investment per worker are diminishing functions of k. Investment per worker increases with k at a non-diminishing rate. Implication: There is a steady state capital stock where investment per worker is exactly equal to replacement investment per worker.

47 Replacement Investment per Worker Function (Constant s,n)
y,rep sF(k,A) (δ+n)k k k*

48 Steady State Capital per Worker
Define steady-state k* will solve the function Cobb-Douglas Example

49 Determinants of Long-term Capital Productivity
Investment Rates: When economies invest a high percentage of their output, they can “support” a high level of capital per worker. To maintain a steady level of capital per worker, investment must be done in every period to replace depreciated equipment and equip new workers. If investment levels are high, a high level of depreciated capital can be replaced. A high ratio of capital to labor implies a low level capital productivity and a relatively high level of labor productivity.

50 Steady State Capital in Two Countries, B and D , sB > sD (Constant A,n)
sB F(k,A) sD F(k,A) (δ+n)k k kD* kB*

51 Determinants of Long-term Capital Productivity
Labor Growth Rates: When economies have high population growth, a large share of investment must be used to equip new workers with capital. This means that less investment can be used to build up the capital allocated to each worker. When the capital-labor ratio is low, capital productivity is high and labor productivity is low.

52 Steady State Capital in Two Countries, D and B , nD > nB (Constant A,s)
(δ+nB)k (δ+nD)k sF(k,A) k kD* kB*

53 Growth If k < k*, investment per worker is greater than replacement investment implying that capital per worker and output per worker are growing. If k > k*, investment per worker is less than replacement investment needs and capital per worker and output per worker must fall.

54 The End of Growth Capital cannot be the engine of labor productivity growth because it has diminishing returns. An economy eventually develops a capital stock so large that the amount of investment needed in every period to replace depreciated capital is greater than the extra investment capacity generated by extra productivity. But, in the early industrializers, the UK USA, labor productivity growth has continued for 200 years. So, the theory is incomplete.

55 Long-Term Productivity Growth

56 Phenomena we can understand.
We can understand why Korea has had a higher growth rate than the USA. Korea had a higher product of capital and thus Korean investment had a bigger impact on output in that country. We can also explain why Korean capital productivity fell over time. As Korea accumulated capital, output per unit of capital fell. Eventually, the high Korean investment rate leads to lower capital productivity.

57 Phenomena we can’t understand
We can’t explain why the US has maintained constant labor productivity growth while still maintaining a relatively constant level of output per capital. The missing element is technology growth. Here we assume that A is fixed. But we observe A growing over time (as in the case of Ireland).

58 Exogenous Growth Most closely follows Delong “Macroeconomics,” Section 4.3.
We assume that the technology level grows over time as a natural and costless by-product of economic activity. Assume a constant growth rate of technology:

59 Productivity and Rules of Thumb
Growth Rules of Thumb If 2. If Therefore

60 Growth Rate of Labor Productivity
The growth rate of productivity in the Cobb-Douglas case is a weighted average of capital per worker growth and technology growth.

61 Labor Productivity Growth Rate
This implies that the growth rate of output and the growth rate of capital is a function of the average productivity of capital.

62 Implications If the growth rate of capital per worker is faster than the growth rate of technology, the growth rate of capital per worker will be higher than the growth rate of labor productivity. This, in turn, will imply that capital productivity (the ratio of output to capital) will fall. This will in turn imply that labor productivity growth & capital per worker growth will slow down.

63 Capital Productivity & Growth
gk gy gA

64 Two Phases of Growth Transition Path – Emerging economy with high capital productivity experiences capital-investment led growth in which the growth rate of labor productivity is increasing faster than the world frontier of technology. Along the transition path, capital is growing faster than output and capital productivity is falling. During much of the post-war period, Korea was on its transition path.

65 Two Phases of Growth pt. 2 Balanced Growth Path – On the balanced growth path, labor productivity and capital per worker are each growing at the same rate as the world technology frontier. During the post-war period, the US was on its balanced growth path. All balanced growth paths should increase at the same rate (the growth rate of world technology frontier, gA). However the positions of labor productivity on the growth path may be different.

66 Transition to Balanced Growth Path

67 Dynamics of Productivity: High Growth in Labor Productivity, but slowing to technology growth rate
yBG Balanced Growth Path Transition Path time

68 Initially high capital productivity, but shrinking to steady state level.
Balanced Growth Path Transition Path time

69 Balanced growth path capital productivity
We can solve for capital productivity and labor productivity along the balanced growth path.

70 Capital Productivity and Labor Productivity
Holding technology constant, there is a negative relationship between capital productivity and labor productivity. If you have a high ratio of machines to workers, capital productivity will be low and worker productivity high and vice versa.

71 Capital productivity Different countries are likely to have different investment growth rates and labor force growth rates. Thus, they will have different capital productivity levels in steady state. Note that capital productivity along the balanced growth path does not depend on the level of technology.

72 High Investment Rates, Low Capital Productivity

73 Investment, Long-term Capital Productivity & Growth
gk gA


75 sA > sB, , AA = AB yA yB time

76 How do real wages and real capital rental rates behave over the long-term?
Factor prices (under perfect competition and Cobb-Douglas) are proportional to average productivity. Real wages are proportional to labor productivity. Along the transition path, real wages will grow faster than technology but slower than real capital per worker. Along the balanced growth path, real wages will grow at the same rate as technology. Capital rental rates will fall along the transition path due to diminishing returns to capital. Along the balanced growth path capital rental rates will remain constant as capital productivity is constant.

77 Forecasting the Long-term Growth Path
Choose parameters. Typically n is the average population or labor force growth. Then α ≈ ⅓-½, δ ≈ We might set the growth rate of efficiency as gA ≈ , Calculate the current technology level. Forecast the constant path of labor efficiency. Calculate the steady-state capital ratio and multiply by the future efficiency level.

78 Gap From Balanced Growth Path
Along the balanced growth path, output reaches a constant relative to technology Define the GAPt as the % difference between a countries productivity relative to technology and the long run balanced growth path.

79 Convergence We can show that holds in approximation.
Solving this differential equation from some initial starting point ζ0 The larger is the gap, the faster the country will be growing. The parameter λ is the rate at which the transition path closes called the convergence rate.

80 Convergence Rate For the Solow model, we can solve for the convergence rate as λ = (1-α) (n+gA+δ) A country converges because the diminishing returns to capital fail to keep up with replacement investment needs when a country is quickly accumulating capital. When returns to capital diminish quickly (when α is small) or when replacement investment costs increase quickly (when (n+gA+δ) is large) an economy will converge quickly.

81 Half Life is the number of periods for the gap to be cut in half.
Convergence Rate Numerical example: (α = ⅓, n = .01, gA = .01, and δ = .1) implies (λ = .04) Half Life is the number of periods for the gap to be cut in half. When λ = .04, the half life is about 16.5 years. After 48 years, only 1/8 of the initial gap should remain.

82 Will poor countries catch up with rich countries?
If all countries share the same technology, At, countries will converge to a balanced growth path along which they will grow at the same rate. The position of that balanced growth path is determined by capital productivity which is determined in steady-state by investment and population growth rates. If a poor country has the same s and n, it will catch up with the rich countries!

83 Education Levels So far, we have thought of technology as knowledge which grows exogenously. One source of knowledge is human capital (education, experience, etc.) Workers in different countries may have large differences in this area.

84 Variations in Education per Worker

85 Capital Productivity

86 Labor Productivity

87 Human Capital Rewrite production function as
Define H as the human capital level of the work force Productivity function is Along balanced growth path,

88 yearsA > yearsB, , AA = AB
yA yB time

89 Parameterization of human capital
In cross-sectional studies of individual workers, it is typically found (in many countries) that an additional year of schooling means 8-10% higher pay. Parameterize φ =

90 Numerical Exercise Assume that Brazil, Korea, and US have access to the same technology and have roughly the same capital productivity level. Calculate relative labor productivity as predicted by the Solow model.

91 World Technology Frontier?
Two problems with modeling At as some common level of technology available in the world which drives the long-run growth path. As a matter of theory, we haven’t really explained long-term growth. Long-term growth occurs through the costless (and exogenous magic of) technology. Some countries on the balanced growth path seem to have grown at different rates (for example, US labor productivity growth has been faster over the 20th century than British. There have been long periods of relatively slow technology growth in developed countries, such as the productivity slowdown of

92 Effects of Saving In a closed economy or in the long run, domestic investment is financed with domestic savings. Ct = (1-s)*Yt Along the BGP, consumption per person is written as Savings has counter-veiling effects on consumption. First, it directly reduces the share of output devoted toward consumption. Second, it increases output along the balance growth path.

93 Golden Rule Saving Rate
Which saving rate maximizes consumption per capita.

94 Productivity Slowdown

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