economic growth -...

(c) Copyright 1999 by Douglas H. Joines 1

Economic Growth


Module Objectives

• Know what determines the growth rates ofaggregate and per capita GDP

• Distinguish factors that affect theeconomy’s growth rate from those thatmerely shift the level of GDP

• Explain the concept of convergence


If South Korea and Taiwan continue togrow as fast as they have over the pastdecade, then they could overtake America’sincome-per-head within the next quarter-century or so.

- The Economist, October 16, 1993

Focus on growth rates of:

• aggregate output (the best indicator of the size of a market)

• per capita output

• per capita consumption

• real wages (the last three are indicators of the standard of living)


U.S. Real GDP

0

1000

2000

3000

4000

5000

00 10 20 30 40 50 60 70 80 905.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

00 10 20 30 40 50 60 70 80 90

U.S. Real GDP(billions of 1987 dollars)

Log of U.S. Real GDP

Year Year

Time-series data are most commonly displayed on a line graph, also known asa time-series plot.

What are the time-series characteristics of U.S. real GDP?


International GDP Comparison

0

1000

2000

3000

4000

5000

1 2 3 4 5 6 7 8 9

GDP, 1992USA

ChinaJapan

W.Ger.India

MexicoBrazil

Belg. Zimb.

Billions of 1985 U.S. dollars

GDP is the single factor that most influences the size of a market.

Why is there so much variation in GDP across countries?

If we can’t explain these large variations, we can’t hope to explain the muchsmaller variations due to business cycles.


Production Function

Y = f(K, L, F | Technology)

Production Process

K

L

F

Y

The box represents production technology.

K = capital, L = labor, F = fixed factors (e.g., land).

F can vary across countries.

K and L can vary across countries and over time within a country.


U.S. GDP and Capital Stock

2000

4000

60008000

100001200014000

00 10 20 30 40 50 60 70 80 90

Capital

GDP

Billions of 1987 dollars (log scale)Year


U.S. GDP and Labor Input

1000

2000

3000

40005000

0.05

0.10

0.15

0.20

0.25

00 10 20 30 40 50 60 70 80 90

Labor

GDP

GDP: Billions of 1987 dollars per year (log scale)Labor: Trillions of hours per year (log scale)

Year

GD

P

Labo

r

Making quantitative statements about the importance of capital and laborinputs requires a specific form for the production function.


Cobb-Douglas Production Function

Yt = At Ktα Lt

1-α

• A is called total factor productivity (TFP)

• How can we measure A ?

At = Yt /(Ktα Lt

1-α)

The time subscript indicates variables that can change over time.

Technology (TFP) can vary over time.


Growth Accounting

d Y d A d K d Lln ln ln ( ) ln= + + −α α1

GDP Capital Labor TFP1992 4919.9 12020.3 .24391 570.341896 264.4 851.7 .06093 185.96Ratio (1992/1896) 18.61 14.11 4.00 3.07d ln (1992 − 1896) 2.924 2.647 1.387 1.121Growth share 1.000 .299 .318 .383

How much of U.S. real GDP growth is attributable to K, L, and A?

d ln denotes change in natural log of a variable between 1896 and 1992

The figures in this slide are calculated using α = 0.33 and 1−α = 0.67rather than the values of 0.3 and 0.7 used in class.

The table indicates that ∆ln(GDP) = 2.924. From the formula, we have

∆ln(GDP) = ∆ln(TFP) + α∆ln(K) + (1−α)∆ln(L) , or

2.924 = 1.121 + 0.33*2.647 + 0.67*1.387

= 1.121 + 0.874 + 0.929

Dividing through by 2.924 gives the fraction of the total increase in outputattributable to each of the three causes

1.0 (i.e., 100%) = 0.383 + 0.299 + 0.318


Labor Productivity

Y/L = A(K/L)α

• What causes variations in labor productivity?

* Variations in A

* Variations in K/L

How does output per worker (or per worker hour) in the U.S. compare withthat in other countries?

What accounts for these differences?

What can lead to international differences in A?


U.S. Labor Productivity andTotal Factor Productivity

5000

10000

15000

20000

25000

200

300

400

500

600

00 10 20 30 40 50 60 70 80 90

A

Y/L

Year

Y/L A


Distinguish increase in slope of growth path from parallel upward shift ingrowth path.

Faster growing variable will eventually become larger, no matter what theinitial levels.

Implies that, within an economy, nothing can grow faster than GDP forever.

Can economies grow forever at different rates?

How long does it take for the level of a variable to double?

Rule of 72: no. of periods = 72/g

Example: 12% annual growth ⇒ 6 periods to double

2% annual growth ⇒ 36 periods to double

Level vs. Growth Rate

time time

ln Y ln Y

Increase in Level Increase in Growth Rate


Malthusian Model Add more L to fixed K and land

Y

f ( L )

L

Will L grow forever under these circumstances?

-- population growth stops at subsistence level of output


Adjustment to Equilibrium• population adjusts to equilibrium level

• no long-run growth

Y/L

y_

SubsistenceOutput

f(L)/L

L0 L

Birth and death rates depend on Y/L

Subsistence level of output is output at which birth rate and death rate areequal, giving stable population.

-- growth rate is zero in Malthusian model, i.e., growth path is flat

Subsistence output is affected by factors that affect birth and death rates.

What would be the effect of programs to improve public health, e.g.,immunization?

• would lower death rate at the old subsistence y

• population would increase

• implies lower subsistence (and thus actual) y

• parallel downward shift in growth path


Predictions of Malthusian Model

• The model’s predictions are generallycontradicted by evidence

• Time series: per capita output

• Cross section: per capita output vs.population density

• Time series: fertility rates

Malthusian model emphasizes fixed factors of production.

Increasing Y/L over time requires increasing K and/or T fast enough to offsetthe increase in L, if L increases at all

-- for now, hold L fixed to see how K and T affect living standards


Neoclassical (Solow) Model

• Emphasizes reproducible capital andtechnical progress

• Population grows at rate n

• TFP increases at a constant rate γ– (for now, assume n = γ = 0)

• Closed economy, no government


Production FunctionY

f ( K )

K

Question: Can capital accumulation alone sustain growth?


Depreciation Capital depreciates at a constant rate.

Y

K

δδK

Kt+1 = Kt + It - δKt


Definition of Equilibrium

• Long-run equilibrium is called a steady state

• Capital remains constant over time

• This requires that investment equaldepreciation

• C = Y − I

Kt+1 = Kt implies It - δKt


Output, Consumption, andInvestment

Y

Y = f ( K )

K

I = δδK

Implies maximum sustainable capital stock of Kmax

-- at Kmax, consumption is zero


Result 1

• Capital accumulation alone cannot sustaingrowth


Maximum Consumption(Golden Rule)

Y

f ( K )

K

δδK

K*

Implies MPK = δ, or MPK = δ

What determines actual steady-state capital?


Aggregate Saving

• Assume consumption is given by

C = βY

• Thus, saving is

S = (1 − β)Y

• In equilibrium,

S = I


Steady-State Equilibrium

Y

Y

0 K

(1-ββ )Y

δδK

K

In equilibrium, actual saving (investment) equals investment required toreplace worn-out capital.

Steady-state equilibrium is stable.

Growth rate is zero in this simple version of Solow model:

• growth path is flat

• aggregate quantities constant over time

• per capita quantities also constant

• real wage rate (MPL) constant

• net MPK constant

Question: What would happen if the saving rate increased?

• economy would move to higher steady-state capital stock

• parallel upward shift in growth path

• long-run growth rate remains zero


Result 2 - Convergence

• Countries with the same technology andsaving rate will converge to the same capitalstock and output level

What would happen to 2 countries with fixed A and L, where A is the same inthe 2 countries, but where Y/L differs across countries?

• why might Y/L initially differ across countries?

• because of differences in K/L

• countries converge to same K/L if they have the same saving rate

Convergence occurs because of adjustment of capital stock. This takes a longtime.

Evidence supports convergence for developed countries.

Barro evidence on speed of convergence, with applications to East Germany,China, Japan.

What might generate growth? Try population growth.


Population Growth

y

f ( k )

k

(1-ββ )y

(δδ+n)k

k 0

Suppose population grows at a constant rate n.

The same model as before can be used, with 2 changes:

• the relevant variables are now per capita quantities y=Y/L, k=K/L, etc.

• more investment is required to keep K/L constant

• must equip new workers as well as replace worn-out capital

In a steady state:

• all per capita quantities are constant over time (flat growth path)

• all aggregate quantities grow at rate n (upward-sloping growth path)

• the real wage is constant

• the net marginal product of capital is constant

Golden Rule now requires MPK - δ = n


Result 3

• Other things equal, higher populationgrowth:– raises the growth rate of aggregate output

– has no effect on the growth rate of per capitaoutput

– results in a lower level of per capita output

I.e., it changes the slope of the growth path of aggregate output, but causesonly a parallel shift of the growth path of per capita output.

Evidence seems to support this prediction. See the scatter plot of Y/L vs. n inMankiw’s text.

We still do not have growth of per capita output. What can cause suchgrowth? Technical progress is the only remaining explanation.


Technical Improvement Raises theSteady-State Capital Stock

Y

K

f ( K )

δδK(1-ββ)f (K)

(1-ββ)f (K)

f ( K )

0

0

0

1

1

K K 1

Examine a one-time improvement in technology.

• This raises the production function.

• This raises the steady-state levels of capital, income, etc.

• I.e., it causes a parallel upward shift in the economy’s growth path.

The one-time technical improvement illustrates an important point: Technicalprogress stimulates capital accumulation, which reinforces the beneficialeffects of the technical progress.


Result 4

• A higher rate of technical improvementresults in:– a higher rate of growth of the capital stock

– a higher rate of growth of aggregate output

– a higher rate of growth of per capita output

Now consider ongoing technical progress. Assume that total factorproductivity grows at the rate γ.

Because technical progress stimulates capital accumulation, per capita outputgrows at an even faster rate of g = γ/(1-α).

In the steady state:

• all per capita quantities grow at rate g

• all aggregate quantities grow at rate n + g

• real wage grows at rate g

• net MPK is constant over time

Golden Rule now requires MPK - δ = n + g


Productivity Slowdown

AVERAGE ANNUAL GROWTH RATES

GDP Capital Labor TFP1896-1923 3.59 3.14 2.09 1.15

1923-53 2.95 2.07 0.87 1.681953-73 3.31 3.67 1.23 1.291973-90 2.35 2.45 1.90 0.27

1896-1990 3.10 2.79 1.49 1.19


Tax on Income from Capital

• Forms of tax on income from capital– Personal income tax

• capital gains tax

– Corporate profits tax

– Property tax

• In general, taxation exerts substitutioneffects that reduce the amount of the taxedactivity


Taxation, Saving, and Investment

S

I

r

r

r(1−τk)

Revenue

S, II0I1

The before-tax rate of return is r. The after-tax rate of return is r(1−τk). Thehigher the tax rate, the wider the gap between these two rates of return and thesmaller the level of saving and investment.


Taxation and Growth

• A capital income tax reduces saving andinvestment

• A lower saving rate implies a smallersteady-state capital stock and lower income

• If taxation reduces the rate of technicalprogress, then the long-run growth rate isalso reduced

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