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

1

Introduction to Economic Growth

Mr. VaughanIncome and Employment Theory (402)

Last Updated: 1/31/2009

Lecture Outline

• Stylized Facts about Economic Growth

• Explaining the Stylized Facts – Part 1– Deriving the Solow Growth Model

2Last Updated: 1/31/2009 Total Slides: 43

3Last Updated: 1/31/2009

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).

Total Slides: 43

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

Last Updated: 1/31/2009 4-43Total Slides: 43


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).

Total Slides: 43

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



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.

Total Slides: 43

High/Low Growth Nations1960-2000

• 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.


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%)



Horizontal axis plots growth rate of real income (constant 1985 U.S. dollars) on a proportionate scale; vertical axis shows number of people in the 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 given by area under red line to the left of $1 line, divided by total area under red line. (Note: FSU is Former Soviet Union.)

Trends in World Poverty• Biggest improvements occurred in Asia (particularly China and

India, which accounted for 40% or 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.


Long-Term Growth in U.S. and other Rich Countries

• Recall: Main reason U.S. and other OECD countries are rich is they already had high per capita real GDP levels in 1960.

• Source of Prosperity: High (but not explosive) long-term growth rates of per capita real GDP

– U.S., 1869-2005 = 2.0% (doubled every 35 years)

– OECD (countries with data), 1820-2000 = 1.8% (doubled every 39 years)

• Growth rates not constant over time, though.


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Long-Term Economic Growthin OECD Countries


Note: The decline in growth of real GDP per person from 3.1% per year for 1960–1980 to 1.8% per year for 1980–2000 is sometimes called productivity slowdown.

Total Slides: 43

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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 start with Solow Growth Model. First, we need a key building block – the production function.

Last Updated: 1/31/2009 Total Slides: 43

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Production Function(Building Block of Growth Models)

Y = A· F(K, L) (3.1)

– A Technology Level

– K Capitol Stock (machines & buildings used by firms).

– L Labor Force – number of workers



Marginal Product of Capital (MPK) = ΔY/ΔKAssumption: MPK > 0, but diminishing


Marginal Product of Labor (MPL) = ΔY/ΔLAssumption: MPL > 0, but diminishing

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Constant Returns to Scale (CRS, Type of Production Function)

• Definition: Multiple K and L “λ”, Y rises by “λ”Example: If 5 machines & 5 workers produce 100 units output, then 10 machines & 10 workers will produce 200.

• Note: CRS production functions commonly used as starting point in growth modeling (particularly in Solow Growth Model).

• Implication: Multiply K and L by 1/L, Y changes by 1/L

Y/L = A· F(K/ L, L/L)or

Y/L = A· F(K /L, 1)


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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)

• y output per worker• k capital per worker



Note: Prior assumptions about diminishing marginal products imply output per worker declines as capital per worker increases.

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Growth Accounting

• Production Function: relationship between levels of input and level of output.

• Growth Accounting: relationship between growth rates of inputs and growth rate of output.

∆Y/Y = ∆A/A + α·(∆K/K) + β·(∆L/L) (3.1)

In words…

Growth rate of real 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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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%)

• Point 2 implies: 0 < α < 1, 0 < β < 1• Assuming closed economy and negligible depreciation:

– real GDP = real national income– Capital income share ≈ α, and labor income share ≈ β– Payments to capital and labor exhaust total real income.

• If α + β = 1, β = 1 – α. Substituting into (3.1) yields a key equation:

∆Y/Y = ∆A/A + α·(∆K/K + (1 - α )·(∆L/L ) (3.4)


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Solow Growth Model (SGM)

• Story thus far: growth of real GDP depends on growth of technology and

weighted average of growth of capital and labor.

• Goal: 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:

– Growth of labor force (L) = population growth rate


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Solow Growth Model

Additional Assumptions: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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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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• Useful to shift focus to real GDP per worker.

• From per worker production function

∆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.

• Similarly

∆k/k = ∆K/K − ∆L/L (3.7)

In words…

Growth in capital per worker equals growth of capital minus growth of workers.


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Now, for a bit of algebraic manipulation…

1. Recall: ∆Y/Y= α·(∆K/K) + (1−α)·(∆L/L) (3.5)

2. ∆Y/Y = α·(∆K/K) − α·(∆L/L) + ∆L/L

3. ∆Y/Y − ∆L/L = α · (∆K/K − ∆L/L)

4. ∆y/y = α·(∆k/k) (3.8)

In words,

Growth of real output per worker depends only on growth rate of capital per worker.

Implication: To explain growth of real output per worker, just look at factors driving growth of capital stock and labor force


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• Goal: Show ∆K depends on total saving only.

• Assumption: Fixed saving rate (s)– Each household divides real income in fixed proportion between

saving (S) and consumption (C)

• Assumption: Capital depreciates at constant rate (δ)

– Implication: δK is total annual depreciation of capital


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Deriving Solow Growth Model(why real net investment = real saving)

• Logically:

– Real saving = s (Real national income or NDP) or

– Real saving = s (Y − δK), where “Y” is real GDP

• And real national income (NDP) can either 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…real goods/services either real consumer goods or capital goods


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• Note: In Y = C + I , “I” is real gross investment.

• Subtracting depreciation (δ K) from both sides yields:

( Y− δ K) = C+ (I − δ K) (3.10)

In words, real NDP = real consumption + real 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, real net investment = real saving


Deriving Solow Growth Model(why real net investment = real saving)

Total Slides: 43

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Deriving Solow Growth ModelGrowth of Capital Stock (∆K/K )

Logically: ∆K = I − δ KIn words…

Change in real capital stock equals gross real investment minus depreciation, or change in real capital stock equals real net investment

• Substituting ∆K into (3.11) yields: ∆K = s· (Y− δ K) (3.12)

In words, change in real capital stock = real saving

• Dividing through by K yields:∆K/K = s·Y/K − sδ (3.13)

Note: This one of two pieces needed to determine growth rate of capital per worker (∆k/k = ∆K/K − ∆L/L).

Now, we turn to the growth rate of labor….


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Solow Growth Model(Growth Rate of Labor Inputs)

Recall previous assumptions:

– Constant, zero unemployment rate

– Constant labor force participation rate

Implication: Growth rate of population = growth rate of labor inputs

New Assumption:

– Population growth rate is exogenous and constant (n, where n > 0).

– Given prior assumptions, this implies n = ∆L/L (3.14)


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Dervining Solow Growth Model• Recall prior results:

∆y/y = α·(∆k/k) (3.8)

∆k/k = ∆K/K − ∆L/L (3.7)

∆k/k = s· (Y/ K) − sδ − n (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 yields: Y/K =(Y/L) / (K/L) = y/k (*)

• Substituting (*) in 3.15 yields:

∆k/k = s (y/k) − sδ − n (3.16)

[Key Equation (Solow Growth Model)]


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Solow Growth ModelIntuition

∆k/k = s (y/k) − sδ − n (3.16)

[Key Equation (Solow Growth Model)]

• What does this mean?Growth in per capita real output depends on growth in per capita capital stock (given capital-income share, α). And growth in per capita capital stock depends on three “givens”:– Saving rate (s)– Depreciation rate (δ)– Population growth rate (n)And one variable:– Average product of capital (Y/K or y/k)

• Note: The only reason ∆k/k varies over time is average product of capital (APK = Y/K or y/k) varies.


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Solow Growth Model


Note: Diminishing MPK implies diminishing APK

Total Slides: 43

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Solow Growth ModelImpact of Changes in ∆k/k


Note:1.∆k/k = s(y/k) − sδ − n, so ∆k/k = s(y/k) − (sδ + n)

2.s, δ, n = constants, so(sδ + n) is horizontal line.

3.Increase in “k” produces decline in ∆k/k (declining APK)

Let K(0) and 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 increases (moves rightward). As k increases, however, ∆k/k declines. In words, growth of capital per worker slows down over time. At k*, ∆k/k = 0. This is the steady state.

Total Slides: 43

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Solow Growth Model


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Solow Growth Model

• In steady state, ∆ k/k equals zero, so:

s(y*/k*) − sδ − n = 0• Moving “n” to right, and factoring “s/k*” out on

left yields:

(s/k)·(y* − δk*) = n*• Multiplying through by “k*” yields:

s(y* − δk*) = nk*In words,

Steady-state saving per worker = steady-state capital provided for each new worker


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Solow Growth Model

Recall, our goal was gaining insight into how per capita real GDP varies over time. Collecting what we’ve learned:

∆y/y = α·(∆k/k) (3.8)

For a given α·(capital income share < 1), growth in real per capita output varies with growth in capital per worker.

Starting with initial capital per worker, k(0), real per capita output grows with growth in capital per worker (but at a slower rate, b/c α < 1)

As capital per worker grows, growth rates of capital per worker and real GDP per capita decline (but level of real GDP per capita rises).

Eventually, capital per worker reaches steady state (k*) and stops growing. In steady state, real GDP per capita does not grow either (y*).


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Solow Growth ModelWhat’s Next?

• Now, we can use SGM to analyze changes in exogenous variables:

– Saving rate (s)– Depreciation rate (δ)– Population growth rate (n)

on steady-state per capita GDP (y*) and capital per worker (k*), as well as the transition.

• We can also analyze impact of changes in technology, A, (which we assumed away until now). 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


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Solow Growth ModelIn 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)

• The level of technology (A) is not directly observable, so one cannot directly measure technology shocks (∆A/A). But everything on the right-hand side of (3.22) can be obtained from national income accounts data.

• RBC economists use Solow Residuals (∆A/A) to “shock” general equilibrium models. Then, observe behavior of artificial economy with actual economy.


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Introduction to Economic Growth?

Mr. VaughanIncome and Employment Theory (402)


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