1. Intermediate Macroeconomics
Chapter 3
Long-Run Economic Growth

2. Long-run Economic Growth
Growth accounting
Empirical results
Neoclassical growth model
Neoclassical growth model golden rule
Endogenous growth model
Government policy and growth
Intermediate Macroeconomics

3. 1. Growth Accounting Growth in six countries
United States
Canada
U.K.
Hong Kong
Japan
Brazil
Intermediate Macroeconomics

4. Growth Accounting Growth accounting equation
ΔY = ΔA + εK ΔK + εL ΔL
Y A K L
Output growth rate
= Productivity growth rate, ΔA/A
+ Capital growth rate, ΔK/K
x elasticity of output with respect to capital, εK
+ Labor growth rate, ΔL/L
x elasticity of output with respect to labor, εL
Intermediate Macroeconomics

5. Empirical Results Solow and Denison studies
Period covered
1909 – 1949
Output growth
2.9%
Capital growth
0.3%
0.6%
Labor growth
1.1%
1.3%
Productivity growth
1.5%
1.0%
Intermediate Macroeconomics

6. 3. Neoclassical Growth Model Growth in Actual and Potential U.S. GDP
Actual GDP cycles around the long-run GDP growth rate (also called potential or full-employment GDP)
Long run growth rate
Actual Real GDP
Source: Penn World tables (
Intermediate Macroeconomics

7. Neoclassical Growth Model Start with growth accounting equation
Assumption 1. No technological change:
ΔY = εK ΔK + εL ΔL
Y K L
where, ΔA = 0 A
The "neoclassical" theory of economic growth was originally developed in the 1950s by Robert M. Solow, for which he was awarded the Nobel Prize in The foundation had been laid in the 1940s by Sir Roy Harrod, an English economist, and Evsey Domar, who later taught at MIT
Intermediate Macroeconomics

8. 3. Neoclassical Growth Model Steady State
Assumption 2. Steady State - a condition of constant rates of growth in economic measures. With no technological change, a steady state is represented by identical constant growth rates in population, total output, and the level of capital.
ΔL = ΔK = n = population growth rate
L K
A steady state with no technological change implies that output per worker and the capital-labor ratio are constant. Capital must grow at the same rate as the population. If there is no technological change there should be no reason to increase or decrease the amount of capital every worker is given. New entrants to the labor force will be given a level of capital identical to all other workers.
Intermediate Macroeconomics

9. 3. Neoclassical Growth Model Constant returns to scale
Assumption 3. Constant returns to scale production function.
εK + εL = 1
We assume the production function has constant returns to scale. Constant returns to scale simply means that if population grows at 2% per year and capital grows at 2% per year (the capital-labor ratio in this steady state model is constant) then output also grows at 2% per year.
Malthus: Declining returns to sale in macroeconomic growth models implies that as population grows (and the level of capital grows with it) a country would get poorer in terms of real income per capita. Output would not increase as fast as the population. Thomas Malthus applied the concept of declining returns when he conjectured that the world population would eventually outgrow the capability to produce food.
Increasing returns to scale would imply that output grows faster than the population. The only thing a country would need to do to become wealthier is to increase its labor force while maintaining the capital-labor ratio.
Assume an economy is a single room with 10 sewing machines, 10 workers, with total daily output of 10 suits (1 suit/worker/day).
Build an identical second room with 10 sewing machines for 10 new workers.
Labor force grows 100%
Capital grows 100%
Output grows 100%. Declining returns typical of microeconomic models of a single industry.
Intermediate Macroeconomics

10. Neoclassical Growth Model Neoclassical growth equation result
ΔY = εK ΔK + εL ΔL
Y K L
= εK n + εL n
= (εK + εL ) n
= n
= population growth rate
The growth rate of output is independent of savings or the level of capital.
Intermediate Macroeconomics

11. 3.. Neoclassical Growth Model Simple model implications
Aggregate output grows at the same rate as population.*
Per worker output remains unchanged.*
The level of capital has no effect of aggregate or per worker output growth rates in steady state.**
* Unless there is technological change, i.e. an increase in productivity.
** An increase in the savings rate and capital-labor ratio provides a temporary boost to aggregate output and per worker output growth rates. Growth rates will return to original steady state levels but at a permanently higher output per worker level.
Intermediate Macroeconomics

12. 3. Neoclassical Growth Model Constant returns to scale
Production function:
Y = f(K, L)
If constant returns:
z Y = f(z K, z L)
If z = 1/L:
Y = f(K, 1)
L L
Or,
Y/L = f(K/L)
Intermediate Macroeconomics

13. 3. Neoclassical Growth Model Per worker production function
Output
= f(K/L)
We assume that technological change, A, is zero and constant returns to scale, which allows us to convert the aggregate production function in equation (7) to a per capita production function by diving though by L
Intermediate Macroeconomics

14. 3. Neoclassical Growth Model Savings per worker
Output
= f(K/L)
Savings
= s • f(K/L)
Second, we assume the national savings rate is some fixed fraction of total output
Intermediate Macroeconomics

15. 3. Neoclassical Growth Model Investment per worker
Output
= f(K/L)
Steady State Equilibrium
Investment
= (n+d) • (K/L)
Slope = n + d
Savings
= s • f(K/L)
Third, investment is by definition equal to the change in the capital stock, ΔK/K, plus depreciation. We must not only equip new workers with capital but must also replace existing equipment that wears out or becomes obsolete. Let d represent the capital depreciation rate, or the fraction of capital that wears out each year. Given the result in our previous section that the rate of growth in the capital stock is equal to the population growth rate, n, we have:
 I = n • K + d • K
where,
I = total investment per period d = the depreciation rate on existing capital stock.
Investment on a per capita basis is presented in equation.
I = (n + d) • K L                 L
Intermediate Macroeconomics

16. Neoclassical Growth Model Effect of an decrease in population growth rate
Investment Slope = n + d
gets smaller
Output
= f(K/L)
Investment 1
Investment 2 A
Savings B
Third, investment is by definition equal to the change in the capital stock, ΔK/K, plus depreciation. We must not only equip new workers with capital but must also replace existing equipment that wears out or becomes obsolete. Let d represent the capital depreciation rate, or the fraction of capital that wears out each year. Given the result in our previous section that the rate of growth in the capital stock is equal to the population growth rate, n, we have:
 I = n • K + d • K
where,
I = total investment per period d = the depreciation rate on existing capital stock.
Investment on a per capita basis is presented in equation.
I = (n + d) • K L                 L
Intermediate Macroeconomics

17. 3. Neoclassical Growth Model Effect of an decrease in population growth rate
U.S.
Source: Penn World Tables (
International Monetary Fund, World Economic Outlook databases (
Intermediate Macroeconomics

18. 3. Neoclassical Growth Model Effect of an increase in the savings rate
Savings curve shifts upward
Output B
Investment
Savings 2
Savings 1
The economy is initially in steady-state equilibrium at point A, where saving matches the investment required to replace depreciated capital and equip new workers.
Suppose the government introduces policies that strengthen the incentives for saving . This causes an upward shift of the saving schedule, Savings 1, to the new dashed schedule, Savings 2.
Saving has now risen relative to the investment requirement, and as a consequence, more is saved than is required to maintain capital per worker constant. The capital stock per worker will keep rising until we reach point B. At B, the higher amount of saving is just enough to maintain the higher stock of capital.
The new higher steady state capital-labor ratio corresponds to the intersection of the new saving curve and the investment line (point B). In the new steady state, output per worker and consumption per worker will be higher than in the original steady state. However, at point B, the economy has returned to its steady-state total output growth rate equal to the population growth rate.
Increase in savings rate
-> increase in investment
-> increase in capital-labor ratio
-> increase in output per capita
Note an increase in productivity, which would represent an upward shift of the production function (output curve) has the same effect as an increase in savings because the savings curve also shifts upward by the fraction, s, times the increase in productivity. A
Intermediate Macroeconomics

19. 3. Neoclassical Growth Model Effect of an increase in the savings rate
Steady State 2
Steady State 1
An increase in the savings rate leads to a temporary increase in the growth rate of output until a new steady state is achieved at a higher capital-labor ratio. Growth rate of output returns to its original rate.
Steady State 1
Steady State 2
Intermediate Macroeconomics

20. 3. Neoclassical Growth Model Declining marginal productivity of capital
Output
= f(K/L)
Larger increase in output per worker with one unit increase in labor
We assume that technological change, A, is zero and constant returns to scale, which allows us to convert the aggregate production function in equation (7) to a per capita production function by diving though by L
Intermediate Macroeconomics

21. Neoclassical Growth Model Rich and Poor Convergence
Poor countries:
Low capital-labor ratio
Each unit of new capital yields larger increase in output per worker
Best place to invest is in labor markets with greatest marginal increase in output for each new unit of capital.
Investment and wealth should grow faster in poor than rich countries.
Intermediate Macroeconomics

22. Neoclassical Growth Model Golden Rule Summary
Capital/Labor ratio that maximizes consumption in steady state.
Implies an optimal rate of savings – a country can have too high a rate of savings.
Intermediate Macroeconomics

23. Neoclassical Growth Model Golden Rule Consumption
Consumption is the difference between output and investment in steady state.
C = Y - I
Intermediate Macroeconomics

24. 4. Neoclassical Growth Model Golden Rule Consumption = Output - Investment
= f(K/L)
Consumption
Investment
= (n+d) • (K/L)
Third, investment is by definition equal to the change in the capital stock, ΔK/K, plus depreciation. We must not only equip new workers with capital but must also replace existing equipment that wears out or becomes obsolete. Let d represent the capital depreciation rate, or the fraction of capital that wears out each year. Given the result in our previous section that the rate of growth in the capital stock is equal to the population growth rate, n, we have:
 I = n • K + d • K
where,
I = total investment per period d = the depreciation rate on existing capital stock.
Investment on a per capita basis is presented in equation.
I = (n + d) • K L                 L
Intermediate Macroeconomics

25. 4. Neoclassical Growth Model Golden Rule Consumption per worker
Maximum
Consumption
Third, investment is by definition equal to the change in the capital stock, ΔK/K, plus depreciation. We must not only equip new workers with capital but must also replace existing equipment that wears out or becomes obsolete. Let d represent the capital depreciation rate, or the fraction of capital that wears out each year. Given the result in our previous section that the rate of growth in the capital stock is equal to the population growth rate, n, we have:
 I = n • K + d • K
where,
I = total investment per period d = the depreciation rate on existing capital stock.
Investment on a per capita basis is presented in equation.
I = (n + d) • K L                 L
Intermediate Macroeconomics

26. 5. Endogenous Growth Model
Productivity growth:
Neoclassical model: productivity growth is exogenous; i.e., is given to us.
Endogenous growth model attempts to explain productivity growth within the model.
Marginal productivity of capital:
Neoclassical model: decreasing
Endogenous growth model: constant or even increasing
The key feature of endogenous growth models is that the marginal productivity of capital is no longer assumed to be decreasing. In the neoclassical model, as physical capital increases the skills of the labor force do not increase with and we have declining marginal productivity of capital. In endogenous growth models, the growing skills of the labor force may complement increases in capital. The education, training, and skills of the labor force are referred to as human capital. As economies accumulate physical capital and become wealthier they devote more resources to education, training, and research and development. This investment in human capital increases productivity. If human capital increases at the same rate as physical capital then we could have constant or even increasing marginal productivity of physical capital.
Intermediate Macroeconomics

27. 5. Endogenous Growth Model Growth rate of output
Output is proportional to the level of capital, which implies non-decreasing marginal productivity of capital:
Y = α ٠ K
The change in output is proportional to the change in capital:
ΔY = α ٠ ΔK
The growth rate of output equals the growth rate of capital:
ΔY = ΔK
Y K
Intermediate Macroeconomics

28. 5. Endogenous Growth Model Savings = Investment
Savings is proportional to output:
S = s ٠ Y
= s ٠ α ٠ K
Investment equals additions to the level of capital, ΔK, plus depreciation, d:
I = ΔK + d ٠ K
Thus, s ٠ α ٠ K = ΔK + d ٠ K
Intermediate Macroeconomics

29. 5. Endogenous Growth Model Savings versus Investment
Annual averages,
Venezuela
Israel
Nicaragua
Source: Penn World Tables (
Intermediate Macroeconomics

30. 5. Endogenous Growth Model Growth of output and rate of savings
Rearrange to solve for the change in the capital stock:
ΔK = s ٠ α ٠ K - d ٠ K
Divide both side by K:
ΔK = s ٠ α – d K
Substitute into the equation for the growth rate of output:
ΔY = s ٠ α – d Y
The growth rate of output is now a
function of the savings rate.
Intermediate Macroeconomics

31. 5. Endogenous Growth Model Effect of investment rate on GDP growth
Annual averages,
Where savings rate = investment rate in steady state here we will look at relationship between investment and GDP growth rate. Graph in class text present savings rate versus GDP growth rate.
Regression Line: Each 10% increase in investment rate increase growth rate of real GDP per worker by 0.77%.
Source: Penn World Tables (
Intermediate Macroeconomics

32. Government Policies and Economic Growth Summary
Policies that promote savings and investment
Temporary boost to aggregate growth rates in simple model. Sustained boost in endogenous growth model.
Permanently higher output per worker as long as higher savings rate is sustained.
Current consumption sacrificed.
Policies that raise productivity
Temporary increase to aggregate growth rates unless productivity growth rates sustained.
Sustained increase in output per worker.
No sacrifice of current consumption.
Intermediate Macroeconomics