Economic Growth Selective material from Macroeconomics chapters 7 and 8 by N. Gregory Mankiw

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Economic GrowthSelective material from
Macroeconomics chapters 7 and 8 by N. Gregory
Mankiw
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We will cover

• the closed economy Solow model
• how a countrys standard of living depends on its
  saving and population growth rates
• how to incorporate technological progress in the
  Solow model
• how to use the Golden Rule to find the optimal
  saving rate and capital stock

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Why growth matters

• Data on infant mortality rates
• 20 in the poorest 1/5 of all countries
• 0.4 in the richest 1/5
• In Pakistan, 85 of people live on less than
  2/day.
• One-fourth of the poorest countries have had
  famines during the past 3 decades.
• Poverty is associated with oppression of women
  and minorities.
• Economic growth raises living standards and
  reduces poverty.

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The Solow model

• due to Robert Solow,won Nobel Prize for
  contributions to the study of economic growth
• a major paradigm
• widely used in policy making
• benchmark against which most recent growth
  theories are compared
• looks at the determinants of economic growth and
  the standard of living in the long run

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The Solow model

• 1. K is capital it is not fixedinvestment
  causes it to grow, depreciation causes it to
  shrink
• 2. L is labor is not fixedpopulation growth
  causes it to grow
• 3. the consumption function depends on income
• 4. For simplification purposes it is assumed no G
  or T
• 5. cosmetic differences

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The production function

• In aggregate terms Y F (K, L)
• Define y Y/L output per worker
• k K/L capital per worker
• Assume constant returns to scale zY F (zK,
  zL ) for any z 0
• Pick z 1/L. Then
• Y/L F (K/L, 1)
• y F (k, 1)
• y f(k) where f(k) F(k, 1)

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The production function
Note this production function exhibits
diminishing MPK.
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The national income identity

• Y C I (remember, no G )
• In per worker terms y c i where
  c C/L and i I /L

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The consumption function

• s the saving rate, the fraction of
  income that is saved
• (s is an exogenous parameter)
• Note s is the only lowercase variable that
  is not equal to its uppercase version divided by
  L
• Consumption function c (1s)y (per worker)

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Saving and investment

• saving (per worker) y c
• y (1s)y
• sy
• National income identity is y c i
• Rearrange to get i y c sy
• Using the results above, i sy sf(k)

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Output, consumption, and investment
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Depreciation
? the rate of depreciation the fraction
of the capital stock that wears out each period
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Capital accumulation

• The basic idea Investment increases the capital
  stock, depreciation reduces it.

Change in capital stock investment
depreciation ?k i ?k Since
i sf(k) , this becomes
?k s f(k) ?k
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The equation of motion for k
?k s f(k) ?k

• The Solow models central equation
• Determines behavior of capital over time
• which, in turn, determines behavior of all of
  the other endogenous variables because they all
  depend on k. E.g.,
• income per person y f(k)
• consumption per person c (1s) f(k)

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The steady state
?k s f(k) ?k

• If investment is just enough to cover
  depreciation sf(k) ?k ,
• then capital per worker will remain constant
  ?k 0.
• This occurs at one value of k, denoted k,
  called the steady state capital stock.

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The steady state
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Moving toward the steady state
?k sf(k) ? ?k
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Moving toward the steady state
?k sf(k) ? ?k
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Moving toward the steady state
?k sf(k) ? ?k
k2
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Moving toward the steady state
?k sf(k) ? ?k
k2
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Moving toward the steady state
?k sf(k) ? ?k
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Moving toward the steady state
?k sf(k) ? ?k
k2
k3
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Moving toward the steady state
?k sf(k) ? ?k
SummaryAs long as k exceed depreciation, and k will continue to grow
toward k.
k3
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An increase in the saving rate
An increase in the saving rate raises investment
causing k to grow toward a new steady state
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Prediction

• Higher s ? higher k.
• And since y f(k) , higher k ? higher y .
• Thus, the Solow model predicts that countries
  with higher rates of saving and investment will
  have higher levels of capital and income per
  worker in the long run.

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The Golden Rule Introduction

• Different values of s lead to different steady
  states. How do we know which is the best
  steady state?
• The best steady state has the highest possible
  consumption per person c (1s) f(k).
• An increase in s
• leads to higher k and y, which raises c
• reduces consumptions share of income (1s),
  which lowers c.
• So, how do we find the s and k that maximize c?

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The Golden Rule capital stock

• the Golden Rule level of capital, the steady
  state value of k that maximizes consumption.

To find it, first express c in terms of k c
y ? i f (k) ? i f
(k) ? ?k
In the steady state i ?k because ?k 0.
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The Golden Rule capital stock
Then, graph f(k) and ?k, look for the point
where the gap between them is biggest.
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The Golden Rule capital stock

• c f(k) ? ?kis biggest where the slope of
  the production function equals the slope of
  the depreciation line

MPK ?
steady-state capital per worker, k
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The transition to the Golden Rule steady state

• The economy does NOT have a tendency to move
  toward the Golden Rule steady state.
• Achieving the Golden Rule requires that
  policymakers adjust s.
• This adjustment leads to a new steady state with
  higher consumption.

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Population growth

• Assume that the population (and labor force) grow
  at rate n. (n is exogenous.)
• EX Suppose L 1,000 in year 1 and the
  population is growing at 2 per year (n 0.02).
  
• Then ?L n L 0.02 ? 1,000 20,so L 1,020
  in year 2.

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Break-even investment

• (? n)k break-even investment, the amount of
  investment necessary to keep k constant.
• Break-even investment includes
• ? k to replace capital as it wears out
• n k to equip new workers with capital
• (Otherwise, k would fall as the existing capital
  stock would be spread more thinly over a larger
  population of workers.)

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The equation of motion for k

• With population growth, the equation of motion
  for k is

?k s f(k) ? (? n) k
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The Solow model diagram
?k s f(k) ? (? n)k
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The impact of population growth
Investment, break-even investment
(? n1) k
An increase in n causes an increase in break-even
investment,
leading to a lower steady-state level of k.
k1
Capital per worker, k
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Prediction

• Higher n ? lower k.
• And since y f(k) , lower k ? lower y.
• Thus, the Solow model predicts that countries
  with higher population growth rates will have
  lower levels of capital and income per worker in
  the long run.

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The Golden Rule with population growth
To find the Golden Rule capital stock, express
c in terms of k c y ? i f
(k ) ? (? n) k c is maximized when
MPK ? n or equivalently, MPK ? ?
n
In the Golden Rule steady state, the marginal
product of capital net of depreciation equals
the population growth rate.
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Incorporating Technology Change

• In the previous Solow model
• the production technology is held constant.
• income per capita is constant in the steady
  state.
• Neither point is true in the real world
• 1904-2004 U.S. real GDP per person grew by a
  factor of 7.6, or 2 per year.

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Examples of technological progress

• From 1950 to 2000, U.S. farm sector productivity
  nearly tripled.
• The real price of computer power has fallen an
  average of 30 per year over the past three
  decades.
• Percentage of U.S. households with 1 computers
  8 in 1984, 62 in 2003
• 1981 213 computers connected to the
  Internet2000 60 million computers connected to
  the Internet
• 2001 iPod capacity 5gb, 1000 songs. Not
  capable of playing episodes of Desperate
  Housewives.
• 2005 iPod capacity 60gb, 15,000 songs. Can
  play episodes of Desperate Housewives.

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Technological progress in the Solow model

• A new variable E labor efficiency
• Assume Technological progress is
  labor-augmenting it increases labor efficiency
  at the exogenous rate g

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Technological progress in the Solow model

• We now write the production function as

• where L ? E the number of effective workers.
  
• Increases in labor efficiency have the same
  effect on output as increases in the labor
  force.

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Technological progress in the Solow model

• Notation
• y Y/LE output per effective worker
• k K/LE capital per effective worker
• Production function per effective worker y
  f(k)
• Saving and investment per effective worker s y
  s f(k)

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Technological progress in the Solow model

• (? n g)k break-even investment the
  amount of investment necessary to keep k
  constant.
• Consists of
• ? k to replace depreciating capital
• n k to provide capital for new workers
• g k to provide capital for the new effective
  workers created by technological progress

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Technological progress in the Solow model
?k s f(k) ? (? n g)k
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Steady-state growth rates in the Solow model with
tech. progress
0
k K/(L?E )
Capital per effective worker
0
y Y/(L?E )
Output per effective worker
g
(Y/ L) y?E
Output per worker
n g
Y y?E?L
Total output
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The Golden Rule
To find the Golden Rule capital stock, express
c in terms of k c y ? i f
(k ) ? (? n g) k c is maximized
when MPK ? n g or equivalently, MPK
? ? n g
In the Golden Rule steady state, the marginal
product of capital net of depreciation equals the
pop. growth rate plus the rate of tech progress.
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Growth empirics Balanced growth

• Solow models steady state exhibits balanced
  growth - many variables grow at the same rate.
• Solow model predicts Y/L and K/L grow at the same
  rate (g), so K/Y should be constant.
• This is true in the real world.
• Solow model predicts real wage grows at same rate
  as Y/L, while real rental price is constant.
• This is also true in the real world.

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Growth empirics Convergence

• Solow model predicts that, other things equal,
  poor countries (with lower Y/L and K/L) should
  grow faster than rich ones.
• If true, then the income gap between rich poor
  countries would shrink over time, causing living
  standards to converge.
• In real world, many poor countries do NOT grow
  faster than rich ones. Does this mean the Solow
  model fails?

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Growth Empirics Convergence

• Solow model predicts that, other things equal,
  poor countries (with lower Y/L and K/L) should
  grow faster than rich ones.
• No, because other things arent equal.
• In samples of countries with similar savings
  pop. growth rates, income gaps shrink about 2
  per year.
• In larger samples, after controlling for
  differences in saving, pop. growth, and human
  capital, incomes converge by about 2 per year.

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Growth empirics Convergence

• What the Solow model really predicts is
  conditional convergence - countries converge to
  their own steady states, which are determined by
  saving, population growth, and education.
• This prediction comes true in the real world.

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Growth empirics Factor accumulation vs.
production efficiency

• Differences in income per capita among countries
  can be due to differences in
• 1. capital physical or human per worker
• 2. the efficiency of production (the height of
  the production function)
• Studies
• both factors are important.
• the two factors are correlated countries with
  higher physical or human capital per worker also
  tend to have higher production efficiency.

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Growth empirics Factor accumulation vs.
production efficiency

• Possible explanations for the correlation between
  capital per worker and production efficiency
• Production efficiency encourages capital
  accumulation.
• Capital accumulation has externalities that raise
  efficiency.
• A third, unknown variable causes capital
  accumulation and efficiency to be higher in some
  countries than others.

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Growth empirics Production efficiency and free
trade

• Since Adam Smith, economists have argued that
  free trade can increase production efficiency and
  living standards.
• Research by Sachs Warner

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Growth empirics Production efficiency and free
trade

• To determine causation, Frankel and Romer exploit
  geographic differences among countries
• Some nations trade less because they are farther
  from other nations, or landlocked.
• Such geographical differences are correlated with
  trade but not with other determinants of income.
• Hence, they can be used to isolate the impact of
  trade on income.
• Findings increasing trade/GDP by 2 causes GDP
  per capita to rise 1, other things equal.

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CASE STUDY The productivity slowdown
1972-95
1948-72
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Possible explanations for the productivity
slowdown

• Measurement problemsProductivity increases not
  fully measured.
• But Why would measurement problems be worse
  after 1972 than before?
• Oil pricesOil shocks occurred about when
  productivity slowdown began.
• But Then why didnt productivity speed up when
  oil prices fell in the mid-1980s?

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Possible explanations for the productivity
slowdown

• Worker quality1970s - large influx of new
  entrants into labor force (baby boomers,
  women).New workers tend to be less productive
  than experienced workers.
• The depletion of ideasPerhaps the slow growth
  of 1972-1995 is normal, and the rapid growth
  during 1948-1972 is the anomaly.

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Which of these suspects is the culprit?

• All of them are plausible, but its difficult to
  prove that any one of them is guilty.

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CASE STUDY I.T. and the New Economy
1995-2004
1972-95
1948-72
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CASE STUDY I.T. and the New Economy

• Apparently, the computer revolution did not
  affect aggregate productivity until the
  mid-1990s.
• Two reasons
• 1. Computer industrys share of GDP much bigger
  in late 1990s than earlier.
• 2. Takes time for firms to determine how to