1. Introduction to Economic Growth
Mr. Vaughan
Income and Employment Theory (402)
Last Updated: 2/9/2009

2. Lecture Outline Stylized Facts about Economic Growth
Explaining the Stylized Facts – Part 1
Derive Solow Growth Model (SGM)
“Shock” exogenous variables in SGM and observe results.
Assess empirical performance of SGM (preliminarily)
Last Updated: 2/9/2009
Total Slides: 42

3. Horizontal axis plots real GDP per capita (constant 2000 U. S
Horizontal axis plots real GDP per capita (constant 2000 U.S. dollars), and vertical axis shows number of countries with each real per capita GDP (151 total). Representative countries are indicated for ranges (histogram bars).
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Total Slides: 42

4. Rich/Poor Nations in 2000
U.S. with real per capita GDP of $34,800* second richest nation (Luxembourg with $45,900* the richest).
20 of 25 richest nations were Organization for Economic Cooperation and Development (OECD)
OECD includes Western Europe, U.S., Canada, Australia, and Japan
Other 5: Singapore (3rd), Hong Kong (6th), Macao (14th), Cyprus (23rd), and Taiwan (25th)
Poorest country is Congo (Kinshasa) in sub-Saharan Africa with real GDP per capita of $238.
Luxembourg has real GDP per capita 193 times larger; U.S. 146 times.
23 of 25 countries with lowest real per capita GDPs are in sub-Saraha.
*Constant 2000 U.S. dollars
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Total Slides: 42

5. Horizontal axis plots real GDP per capita (constant 2000 U. S
Horizontal axis plots real GDP per capita (constant 2000 U.S. dollars), and vertical axis shows number of countries with each real per capita GDP (113 total). Representative countries are indicated for ranges (histogram bars).
Last Updated: 2/9/2009
Total Slides: 42

6. Rich/Poor Nations in 1960
U.S. with real per capita GDP of $12,800* second richest nation (again); Switzerland with $15,600* the richest.
20 of 25 richest nations were OECD members (again)
OECD includes Western Europe, U.S., Canada, Australia, and Japan
Difference from 2000: No Asian countries in top 25.
Poorest country was Tanzania in sub-Saharan Africa with real GDP per capita of $400.
Switzerland had real GDP per capita 39 times larger; U.S. 32 times.
“Only” 19 of 25 countries with lowest real per capita GDP in sub-Saraha.
5 of poorest in Asia (Pakistan, China, Nepal, India, and Indonesia).
High growth in Asia and low growth in Sub-Saharan Africa from 1960 to 2000 were major events in trends in world living standards!
*Constant 2000 U.S. dollars
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7. Horizontal axis plots growth rate of real GDP per capita (constant 2000 U.S. dollars); vertical axis shows number of countries with each real per capita growth rate (112 total). Representative countries are indicated for ranges (histogram bars). Unweighted average growth rate was 1.8% per year.
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Total Slides: 42

8. High/Low Growth Nations 1960-2000
112 country average = 1.8% (growth of real per capita GDP)
Fastest = Taiwan (6.4%); Asia in general (8 of top 12 growth rates, top 20 included Singapore, South Korea, Hong Kong, Thailand, China, Japan, Malaysia, and Indonesia) .
Slowest = Congo (Kinshasa, -3.6%); Sub-Saharan Africa in general (18 of 20 worst growth performances).
Take-away: Low 2000 levels of real GDP per capita can be explained by
Countries started off with low standards of living.
Countries turned in poor growth performance.
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Total Slides: 42

9. Poverty and Inequality
Terms often used interchangeably but have different meanings.
Poverty: minimum acceptable standard of living
World Bank’s $1 per day (individual living in poverty if his income < $1 per day, or $570 in constant 1996 dollars)
Inequality: unequal distribution of income
Fraction of income received by persons in lowest (highest) quintile.
U.S. (1990) 6.5% of income received by lowest quintile (about world average); 39% for highest quintile (below world average)
Inequality can remain constant while poverty declines.
Suppose everyone’s real income doubled
Income inequality would not change
Percentage of people with less than $1 per day income would fall.
Implies poverty is more meaningful measure of welfare than inequality.
1970 to 2000, number of people below $1 per day poverty line fell from 700 million (20% of world population) to 398 million (7%)
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10. World Income Distributions in 1970
Horizontal axis plots growth rate of real income (constant 1985 U.S. dollars) on proportionate scale; vertical axis shows number of people in world/country with each income level. Vertical line at $1/day notes World Bank poverty measure. Fraction of world’s population in poverty equals area under red line to left of $1 line, divided by total area under red line. (Note: FSU is Former Soviet Union.)
Last Updated: 2/9/2009

11. World Income Distribution in 2000
Horizontal axis plots growth of real income (constant 1985 U.S. dollars) on proportionate scale; vertical axis shows number of people in world/country with each level of income. Vertical line at $1/day corresponds to World Bank poverty measure. Fraction of world’s population in poverty equals area under red line to left of $1 line, divided by total area under red line. (Note: FSU is Former Soviet Union.)
Last Updated: 2/9/2009

12. Trends in World Poverty
Biggest improvements occurred in Asia (particularly China and India, which accounted for 40% of world population in 2000).
In 1970, Asia accounted for 80% persons living below $1 per day poverty.
In 2000, Asia accounted for only 19% of persons living in poverty.
Reflects strong Chinese growth in 1990s; strong Indian growth in 1980s.
Biggest declines in sub-Saharan Africa
In 1970, sub-Saharan Africa accounted for 13% persons living below $1 per day.
In 2000, sub-Saharan Africa accounted for 74% of persons below $1 per day.
Poverty has shifted from an Asian problem to an African problem!
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Total Slides: 42

13. Long-Term Growth in OECD Countries
Recall: Reason U.S./other OECD countries are rich is they already had high per capita real GDP levels in 1960.
Source of Prosperity: Impressive (but not explosive) growth rate of per capita real GDP over very long horizon.
U.S., = 2.0% (doubled every 35 years)*
OECD (countries with data), = 1.8% (doubled every 39 years)*
* Rule of 70: Years required for number to double = 70 / (growth rate x 100). So real GDP per capital growing at 2% doubles in 35 years [70 / (0.02) (100)]. Sometimes this formula is expressed as “Rule of 69” or “Rule of 72,” depending on compounding frequency. For 99% of practical applications, “Rule of 70” does best job of trading off computational ease and precision.
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Total Slides: 42

14. Long-Term Economic Growth in OECD Countries
Note:
Decline in growth of real GDP per capita from 3.1% per year for 1960–80 to 1.8% per year for 1980–2000 is sometimes called productivity slowdown. The “outlier,” however, is unusually large growth from
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15. Growth Questions
What caused some countries to grow fast, others to grow slowly over periods like 1960 to 2000?
In particular, why did Asia do better than Africa?
How did countries such OECD members sustain real GDP per capita growth of ≈ 2% per year for 100+ years?
Could policy boost growth rates of real GDP per capita?
Answers require a model. We’ll start with Solow Growth Model.
We will ultimately discover (i) growth in real GDP depends on growth in capital per worker and (ii) growth in capital per worker depends on level of technology, average productivity of capital, saving rate, depreciation rate and population-growth rate.
First, we need a key building block – the production function.
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Total Slides: 42

16. Production Function Building Block of Growth Models
Y = A· F(K, L) (3.1)
where:
Y  Real output (in this case, real GDP)
A  Level of technology
K  Capital stock (machines & buildings used by firms)
L  Labor force (number of workers)
F  Function mapping capital and labor into output
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17. Marginal Product of Capital (MPK) = ΔY/ΔK
Assumption: MPK > 0, but diminishing
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18. Marginal Product of Labor (MPL) = ΔY/ΔL
Assumption: MPL > 0, but diminishing
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19. Constant Returns to Scale Type of Production Function
DEFINITION: Multiple K and L “λ”, Y rises by “λ”
Example: If 5 machines and 5 workers produce 100 units output, then 10 machines and 10 workers will produce 200.
NOTE: CRS production functions common starting point in growth modeling (particularly Solow Growth Model).
IMPLICATION: Multiply K and L by 1/L, Y changes by 1/L as well.
Y/L = A· F(K/ L, L/L) or
Y/L = A· F(K /L, 1)
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Total Slides: 42

20. CRS Production Function (Building Block of Growth Models)
“1” can be ignored.
So production function can be re-written in “per worker” terms.
y = A · f(k) (3.2)
where:
A  level of technology
y  output per worker
k  capital per worker
f  function mapping capital/worker into output/worker
Nomenclature Alert:
Upper case means “total,” lower case means “per worker.” All magnitudes “real” (as opposed to nominal) unless otherwise noted.
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Total Slides: 42

21. Note: Prior assumptions about diminishing marginal products imply output per worker increases with capital per worker, but at a decreasing rate.
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22. ∆Y/Y = ∆A/A + α·(∆K/K) + β·(∆L/L) (3.1)
Growth Accounting
Production Function: Relationship between levels of technology/inputs and level of output.
Growth Accounting: relationship between growth rates of technology/inputs and growth rate of output.
∆Y/Y = ∆A/A + α·(∆K/K) + β·(∆L/L) (3.1)
In words…
Growth rate of GDP (∆Y/Y) = Growth rate of technology (∆A/A) + contribution from growth of capital [α·(∆K/K)] and labor [β ·(∆L/L)].
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Total Slides: 42

23. Contribution to GDP Growth
∆Y/Y = ∆A/A + α·(∆K/K) + β·(∆L/L) (3.1)
Note:
Y is proportional to A so coefficient on ∆A/A is 1.
α + β = 1 (Suppose ∆A/A = 0, and ∆K/K = ∆L/L = 1%, CRS implies ∆Y/Y = 1%)
Assuming closed economy and negligible depreciation:
GDP = Total national income
Capital income share ≈ α, and labor income share ≈ β
Payments to capital and labor exhaust national income.
If α + β = 1, then β = 1 – α. Substituting into (3.1) yields:
∆Y/Y = ∆A/A + α·(∆K/K) + (1 - α )·(∆L/L ) (3.4)
In words… GDP growth depends on growth of technology and weighted average of growth of capital and labor.
Last Updated: 2/9/2009
Total Slides: 42

24. Solow Growth Model (SGM)
Story thus far:
GDP growth depends on growth of technology and weighted average of growth of capital and labor.
What Now?
Explain growth of technology, capital, and labor.
SGM assumptions (in addition to CRS Production Function):
Labor input = labor force (i.e., constant zero unemployment rate)
Labor force participation rate is constant
Labor input (or Labor Force, L) can be written:
(labor force/population) · population, where (labor force/population) is labor-force participation rate
Implication of SGM assumptions about labor:
Growth rate of labor force (L) = growth rate of population
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Total Slides: 42

25. Solow Growth Model Additional Assumptions Realistic? No!
Ignore:
Government
Implies: no taxes, public expenditures, debt, or money
International Trade
Implies: No trade in goods or financial assets
Realistic? No!
But let’s see how far we get…
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Total Slides: 42

26. Deriving Solow Growth Model
Starting with:
∆Y/Y = ∆A/A + α·(∆K/K) + (1 - α )·(∆L/L) (3.4)
New Assumption: ∆A/A = 0 (ignore technology for now)
Implication:
∆Y/Y = α·(∆K/K) + (1−α)·(∆L/L) (3.5)
In words… Growth of real GDP is weighted average of growth capital and labor.
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Total Slides: 42

27. Deriving Solow Growth Model
Useful to shift focus to real GDP per worker.
From per worker production function, note:
∆y/y = ∆Y/Y − ∆L/L (3.6)
In words: Growth in real output per worker equals growth of real output minus growth of workers (i.e., population).
Similarly:
∆k/k = ∆K/K − ∆L/L (3.7)
In words: Growth in capital per worker equals growth of capital minus growth of workers.
Last Updated: 2/9/2009
Total Slides: 42

28. Deriving Solow Growth Model
Now, for a bit of algebraic manipulation…
Recall: ∆Y/Y = α·(∆K/K) + (1−α)·(∆L/L) (3.5)
∆Y/Y = α·(∆K/K) − α·(∆L/L) + ∆L/L
∆Y/Y − ∆L/L = α·(∆K/K − ∆L/L)
∆y/y = α·(∆k/k) (3.8)
In words… Growth of output per worker depends only on growth rate of capital per worker.
Implication
To explain growth of output per worker, look at factors driving growth of capital and labor (because ∆k/k = ∆K/K − ∆L/L). Let’s start with ∆K/K.
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Total Slides: 42

29. Deriving Solow Growth Model What determines growth rate of capital stock (∆K/K)?
Let’s take a slight detour…
In principles, you learned more investment (purchases of capital goods) led to more output, and more saving meant more investment. Let’s formalize this...
Goal: Show how saving affects growth (rate) of capital stock (∆K/K).
Assumptions:
Fixed saving rate (s)
Each household divides real income in fixed proportion between consumption (C) and saving (S)
Capital depreciates at constant rate (δ)
Implication: δK is total annual depreciation of capital stock
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Total Slides: 42

30. Deriving Solow Growth Model What determines ∆K/K?
Logically:
Saving = s (NDP*) or
Saving = s (Y − δK), where “Y” is GDP
* NDP is correct measure of national income (i.e., depreciation should be “netted out” because it represents purchases of capital necessary to replace worn-out capital)
And NDP must be consumed or saved, so:
Y − δK = C + S
Y − δK = C + s (Y − δK) (3.9)
Recall:
We assumed (a) closed economy and (b) no government
Implication:
Y = C + I
In words, goods/services produced are either consumer or capital goods.
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Total Slides: 42

31. Deriving Solow Growth Model What determines ∆K/K?
Note: In Y = C + I , “I” is gross investment.
Subtracting depreciation (δ K) from both sides yields:
( Y− δ K) = C+ (I − δ K) (3.10)
In words, NDP = consumption + net investment
Recall: ( Y− δ K) = C+ s· ( Y− δ K) (3.9)
Eq. 3.9 and 3.10 together imply:
(I − δ K) = s· ( Y− δ K) (3.11)
In words, net investment = saving
Last Updated: 2/9/2009
Total Slides: 42

32. Deriving Solow Growth Model What Determines ∆K/K?
Logically: ∆K = I − δ K
In words: Change in capital stock equals gross investment minus depreciation, or change in capital stock equals net investment.
Substituting ∆K into (3.11) yields:
∆K = s· (Y− δ K) (3.12)
In words: Change in capital stock = real saving
To get growth rate of capital stock, divide through by K:
∆K/K = s·(Y/K) − sδ (3.13)
Goal met!
growth rate of capital stock depends on three factors:
saving rate (s)
average productivity of capital (Y/K)
depreciation rate (δ)
We have one of two pieces needed to determine growth rate of capital per worker (∆k/k = ∆K/K − ∆L/L). We now turn to growth rate for labor (∆L/L)…
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Total Slides: 42

33. (Labor Force/Population) x (Population)
Deriving Solow Growth Model What determines growth rate of labor input (∆L/L)?
Recall prior assumptions about labor inputs:
Constant, zero unemployment rate
Constant labor force participation rate
Because labor input (L) is given by:
(Labor Force/Population) x (Population)
Growth rate of labor input (∆L/L) equals growth rate of population.
Now, suppose (new assumption) population growth rate is exogenous and constant. If “n” is “given” population growth rate (where n > 0), then:
n = ∆L/L (3.14)
In words… Growth rate of labor input equals growth rate of population.
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Total Slides: 42

34. Deriving Solow Growth Model Returning to growth rate of capital per worker (∆k/k)
Collecting prior results:
∆y/y = α·(∆k/k) (3.8)
∆k/k = ∆K/K − ∆L/L (3.7)
∆K/K = s·(Y/ K) − sδ (3.13)
n = ∆L/L (3.14)
Substituting (3.13) and (3.14) into (3.7) yields: ∆k/k = s·(Y/K) − sδ − n (3.15)
Now, rewriting average product of capital in terms of per capita output and capital per worker [Y/K = (Y/L)/(K/L) = y/k] and substituting into 3.15 yields Solow Growth Model:
∆k/k = s·(y/k) - sδ - n (3.16)
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Total Slides: 42

35. Solow Growth Model Intuition
Key Equation (Solow Growth Model):
∆k/k = s (y/k) − sδ − n (3.16)
Economic intuition?
Growth in per capita GDP (Δy/y) depends on growth in capital per worker (Δk/k), holding capital-income share, α, constant. And growth in per capita capital stock depends on three exogenous factors (or “givens”):
Saving rate (s)
Depreciation rate (δ)
Population growth rate (n)
And one variable:
Average product of capital (Y/K or y/k)
Note: Only reason ∆k/k varies over time is average product of capital (APK = Y/K or y/k) varies with capital per worker (k).
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Total Slides: 42

36. Solow Growth Model How Does “k” Affect “APK”?
Note:
MPK for given “k” is slope of line tangent to production function at “k”.
APK for given “k” is slope of ray from origin through production function at “k”.
APK falls as “k” rises.
Note: Diminishing MPK implies diminishing APK
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Total Slides: 42

37. Solow Growth Model Transition to Steady State
Note:
∆k/k = s(y/k) − sδ − n, so ∆k/k = s(y/k) − (sδ + n)
s, δ, n = constants, so (sδ + n) is horizontal line.
- (sδ + n) implies difference between curves is ∆k/k.
Let K(0), L(0) be initial capital stock and labor force, so k(0) is initial capital per worker. At k(0), ∆k/k (growth rate of capital per worker) is positive, so k rises (moves rightward). As k rises, ∆k/k falls. In words, growth of capital per worker slows down over time. At k*, ∆k/k = 0. This is the steady state.
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38. Solow Growth Model What does Steady State Look Like?
In steady state, ∆k/k = 0%, so:
s(y*/k*) − sδ − n = 0
Moving “n” to right side, factoring out s/k* on left side yields:
(s/k*)·(y* − δk*) = n*
Multiplying through by “k*” yields:
s·(y* − δk*) = nk* (3.17)
Economic Intuition?
Left Side (3.17): s·(y*−δk*) = s·[(Y/L)*− δ(K/L)*] = s·[(Y* - δK*)/L*] = [s·(Y*- δK*)]/L*
Steady-State Saving Per Worker
Right Side (3.17): Population (labor force) growing at “n” rate, k* is capital per worker, so nk* is additional capital needed for each new worker in steady state or steady-state capital provided for each new worker.
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Total Slides: 42

39. Solow Growth Model Putting It All Together
We set out to identify factors affecting a country’s level/growth rate standard of living (i.e., real per capita GDP). We have learned:
∆y/y = α·(∆k/k) (3.8)
∆k/k = s (y/k) − sδ − n (3.16)
For given “α”·(capital-income share < 1), growth in per capita GDP is driven by growth in capital per worker (3.8).
A country in steady-state equilibrium (k1*) experiences no growth in capital per worker or GDP per capita (Δk/k = Δy/y = 0%).
Suppose a country in equilibrium experiences a shock so that s(y/k) > (sδ + n) at current level of capital per worker (former steady-state level, k1*). At this level [call it k(0)], growth of capital per worker (Δk/k) and real GDP per capita (Δy/y) become positive. Put another way, levels of capital per worker (k) and real GDP per capita (y) start rising.
As capital per worker (k) rises, growth rates of capital per worker (Δk/k) and real GDP per capita (Δy/y) fall. But as long as growth rate of capital per worker (Δk/k) is positive, the level of real GDP per capita (y) keeps rising.
Eventually, growth rates of capital per worker (Δk/k) and real GDP per capita (Δy/y) fall to zero – a new steady state (k2*, y2*). But capital per worker and real GDP per capita are higher in the new steady state (k2*, y2*) than initial steady state (k1*, y1*).
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40. Solow Growth Model What’s Next?
Use SGM to analyze “shocks,” specifically impact of changes in:
Saving rate (s)
Depreciation rate (δ)
Population growth rate (n)
Level of labor force (L)
on steady-state capital per worker (k*) and per capita GDP (y*) , as well as transition path to that new steady state.
We can also analyze impact of changes in technology (A) which we previously assumed away. Recall:
∆k/k = s (y/k) - sδ - n (3.16)
y = A·f(k) (3.2)
So,
∆k/k = s [A·f(k)]/k - sδ - n (4.3; SGM + technology)
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41. Solow Growth Model In Action
Real Business Cycle (RBC) macroeconomists use Solow Residuals to identify technology shocks. Recall:
∆Y/Y = ∆A/A + α·(∆K/K) + (1 - α )·(∆L/L) (3.4)
Rearranging yields:
∆A/A = ∆Y/Y – [α·(∆K/K) + (1 - α )·(∆L/L)] (3.22)
Level of technology (A) is not observable, so technology shocks (∆A/A) cannot be directly measured. But terms on right side of (3.22) can be quantified with national income account data.
RBC economists use Solow Residuals (∆A/A) to “shock” general- equilibrium models, then compare behavior of artificial economy with actual economy.
Result: Behavior of artificial economy looks remarkably like actual post-WWII U.S. economy. (By construction, time path of output in artificial economy is Pareto Optimal, so RBC’s reason, time path of U.S. economy must be as well.
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42. Introduction to Economic Growth?
Questions over
Introduction to Economic Growth?
Mr. Vaughan
Income and Employment Theory (402)
Last Updated: 2/9/2009