chapter 7: economic growth i:chapter 7: economic growth i: capital …dxs093000/macro/chap7... ·...
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Chapter 7: Economic Growth I:Chapter 7: Economic Growth I:Capital Accumulation and Population GrowthGrowth
0CHAPTER 1 The Science of Macroeconomics
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Why growth matters
Data on infant mortality rates: 20% in the poorest 1/5 of all countries 20% in the poorest 1/5 of all countries 0.4% in the richest 1/5
I P ki t 85% f l li l th $2/d In Pakistan, 85% of people live on less than $2/day. One-fourth of the poorest countries have had
f i d i th t 3 d dfamines during the past 3 decades. Poverty is associated with oppression of women
and minoritiesand minorities.Economic growth raises living standards and reduces poverty
1CHAPTER 7 Economic Growth I
reduces poverty….
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Income and poverty in the world selected countries, 2000selected countries, 2000
90
100 Madagascar
India
70
80
tion
ay o
r les
s India
BangladeshNepal
40
50
60
of p
opul
atn
$2 p
er d
a
Botswana
Peru
China
Kenya
20
30
40
% o
livin
g on Mexico
Thailand
Peru
0
10
$0 $5 000 $10 000 $15 000 $20 000
ChileS. Korea
Brazil Russian Federation
$0 $5,000 $10,000 $15,000 $20,000
Income per capita in dollars
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links to prepared graphs @ Gapminder.org
notes: circle size is proportional to population size, notes: circle size is proportional to population size, color of circle indicates continent, press “play” on color of circle indicates continent, press “play” on bottom to see the cross section graph evolve over timebottom to see the cross section graph evolve over timebottom to see the cross section graph evolve over time, bottom to see the cross section graph evolve over time, click here for one-page instruction guide
Income per capita andIncome per capita andIncome per capita andIncome per capita and
Life expectancy
Infant mortality
Malaria deaths per 100,000
Adult literacy
Cell phone users per 100,000Cell phone users per 100,000
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Why growth matters
Anything that effects the long-run rate of economic y g ggrowth – even by a tiny amount – will have huge effects on living standards in the long run.
percentage increase in standard of living after
annual growth rate of
…100 years…50 years…25 years
standard of living after…gincome per
capita
1,081.4%243.7%85.4%
624.5%169.2%64.0%
2.5%
2.0%
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,
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Why growth matters
If the annual growth rate of U.S. real GDP per capita had been just one tenth of one percentcapita had been just one-tenth of one percent higher during the 1990s, the U.S. would have generated an additional $496 billion of incomegenerated an additional $496 billion of income during that decade.
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The lessons of growth theory…can make a positive difference in the lives of hundreds of millions of people.p p
These lessons help us understand why poorunderstand why poor
countries are poor design policies that
can help them grow learn how our own
th t i ff t dgrowth rate is affected by shocks and our government’s policies
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government s policies
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The Solow model
due to Robert Solow,won Nobel Prize for contributions towon Nobel Prize for contributions to the study of economic growth
j di 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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How Solow model is different from Ch t 3’ d lChapter 3’s model1. K is no longer fixed:
investment causes it to grow, depreciation causes it to shrink
2. L is no longer fixed:population growth causes it to grow
3. the consumption function is simpler
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How Solow model is different from Ch t 3’ d lChapter 3’s model4. no G or T
(only to simplify presentation; we can still do fiscal policy experiments)
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
Output per worker, yo e , y
f(k)
MPK = f(k +1) f(k)1
MPK = f(k +1) – f(k)
Note: this production function exhibits diminishing MPK. Note: this production function exhibits diminishing MPK.
Capital per
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p pworker, k
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The national income identity
Y = C + I (remember, no G )
In “per worker” terms: y = c + i
h C/L d i I /Lwhere 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 toits uppercase version divided by L
Consumption function: c = (1 s)y Consumption function: c = (1–s)y(per worker)
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Saving and investment
saving (per worker) = y – c= y – (1–s)y= sy
National income identity is y = c + iRearrange to get: i = y c = syRearrange to get: i = y – c = sy
(investment = saving, like in chap. 3!)
Using the results above, i = sy = sf(k)
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Output, consumption, and investment
Output per worker y
f(k)worker, y
sf(k)c1
sf(k)y1
i
Capital perk
i1
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Capital per worker, k
k1
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Depreciation
Depreciation per worker k
= the rate of depreciation = the fraction of the capital stock
= the rate of depreciation = the fraction of the capital stockper worker, k = the fraction of the capital stock
that wears out each period= the fraction of the capital stock
that wears out each period
k
1
Capital per
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Capital per worker, k
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Capital accumulation
The basic idea: Investment increases the capital
Change in capital stock = investment – depreciation
stock, depreciation reduces it.
Change in capital stock = investment – depreciationk = i – k
Since i = sf(k) , this becomes:
k = sf(k) – k
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The equation of motion for k
k = sf(k) – k The Solow model’s central equation
D t i b h i f it l ti
( )
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 = (1–s) f(k)
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The steady state
k = sf(k) – kIf investment is just enough to cover depreciation [sf(k) = k ]
( )
[sf(k) = k ], then capital per worker will remain constant:
k = 0k = 0.
This occurs at one value of k, denoted k*, , ,called the steady state capital stock.
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The steady state
Investment and
depreciation sf(k)
k
sf(k)
Capital perk*
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Capital per worker, k
k
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Moving toward the steady state
Investment k = sf(k) kand
depreciation sf(k)
k
sf(k)
d i ti
kinvestment
Capital perk*
depreciation
k1
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Capital per worker, k
kk1
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Moving toward the steady state
Investment k = sf(k) kand
depreciation sf(k)
k
sf(k)
Capital perk*k1
k
k2
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Capital per worker, k
kk1 k2
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Moving toward the steady state
Investment k = sf(k) kand
depreciation sf(k)
k
sf(k)
kinvestment
depreciation
k
Capital perk*k2
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Capital per worker, k
kk2
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Moving toward the steady state
Investment k = sf(k) kand
depreciation sf(k)
k
sf(k)
kk
Capital perk*k2
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Capital per worker, k
kk2
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Moving toward the steady state
Investment k = sf(k) kand
depreciation sf(k)
k
sf(k)
Capital perk*k2
k
k3
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Capital per worker, k
kk2 k3
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Moving toward the steady state
Investment k = sf(k) kand
depreciation sf(k)
k
SS sf(k)Summary:As long as k < k*,
investment will exceed
Summary:As long as k < k*,
investment will exceedinvestment will exceed depreciation,
and k will continue to *
investment will exceed depreciation,
and k will continue to *
Capital perk*k3
grow toward k*.grow toward k*.
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Capital per worker, k
kk3
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NOW YOU TRY:
Approaching k* from aboveApproaching k* from above
Draw the Solow model diagramDraw the Solow model diagram, labeling the steady state k*.
O th h i t l i i k l t th k*On the horizontal axis, pick a value greater than k*
for the economy’s initial capital stock. Label it k1.
Show what happens to k over time. Does k move toward the steady state or away from it?
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A numerical example
Production function (aggregate):
1 /2 1 /2( , )Y F K L K L K L
T d i th k d ti f ti
1 /21 /2 1 /2Y K L K
To derive the per-worker production function, divide through by L:
/ /Y K L KL L L
1 /2( )y f k kThen substitute y = Y/L and k = K/L to get
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( )y f k k
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A numerical example, cont.
Assume:
s = 0.3
0 1 = 0.1
initial value of k = 4.0
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Approaching the steady state: A numerical exampleA numerical exampleAssumptions: ; 0.3; 0.1; initial 4.0y k s k
Year k y c i k Dk1 4.000 2.000 1.400 0.600 0.400 0.2002 4 200 2 049 1 435 0 615 0 420 0 1952 4.200 2.049 1.435 0.615 0.420 0.1953 4.395 2.096 1.467 0.629 0.440 0.189
4 4 584 2 141 1 499 0 642 0 458 0 1844 4.584 2.141 1.499 0.642 0.458 0.184…10 5.602 2.367 1.657 0.710 0.560 0.150…25 7.351 2.706 1.894 0.812 0.732 0.080…
100 8 962 2 994 2 096 0 898 0 896 0 002
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100 8.962 2.994 2.096 0.898 0.896 0.002… 9.000 3.000 2.100 0.900 0.900 0.000
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NOW YOU TRY:
Solve for the Steady StateSolve for the Steady State
Continue to assumeContinue to assume s = 0.3, = 0.1, and y = k 1/2
Use the equation of motion k = s f(k) ks ( )
to solve for the steady-state values of k, y, and c.
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ANSWERS:
Solve for the Steady StateSolve for the Steady State
def. of steady statek 0 def. of steady statek 0
eq'n of motion with s f k k k ( *) * 0
using assumed valuesk k0.3 * 0.1 *
*k *3 *
*k kk
and y k * * 3Solve to get: k * 9
Finally c s y * (1 ) * 0 7 3 2 1Finally, c s y * (1 ) * 0.7 3 2.1
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An increase in the saving rateAn increase in the saving rate raises investment……causing k to grow toward a new steady state:
Investment and dk
g g y
and depreciation
s1 f(k)
s2 f(k)
s1 f(k)
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*k 2
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Prediction:
Higher s higher k*. g g
And since y = f(k) , hi h k* hi h *higher k* higher y* .
Thus the Solow model predicts that countriesThus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income perwill have higher levels of capital and income per worker in the long run.
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International evidence on investment rates and income per personand income per person
100,000Income per person in
10,000
2003 (log scale)
,
1,000
1000 5 10 15 20 25 30 350 5 10 15 20 25 30 35
Investment as percentage of output (average 1960-2003)
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The Golden Rule: Introduction
Different values of s lead to different steady states. H d k hi h i th “b t” t d t t ?How do we know which is the “best” steady state?
The “best” steady state has the highest possible ti * (1 ) f(k*)consumption per person: c* = (1–s) f(k*).
An increase in s leads to higher k* and y*, which raises c* reduces consumption’s share of income (1–s),
hi h l *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, *goldk p ,
the steady state value of kthat maximizes consumption.
gold
To find it, first express c* in terms of k*:
c* = y* i*c y i
= f (k*) i*
f (k*) k*In the steady state:
* *= f (k*) k* i* = k*
because k = 0.
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The Golden Rule capital stocksteady state output and
depreciation k*
Then, graph f(k*) and k*Then, graph f(k*) and k*
depreciation
f(k*)
k*
f(k ) and k , look for the point where
f(k ) and k , look for the point where
( )
pthe gap between them is biggest.
pthe gap between them is biggest.
*goldc
* *i k
steady-state capital per
*goldk
gold goldi k* *( )gold goldy f k
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capital per worker, k*
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The Golden Rule capital stock
c* = f(k*) k*c* = f(k*) k* k*c = f(k ) kis biggest where the slope of the
c = f(k ) kis biggest where the slope of the f(k*)
k*
production function equals
the slope of the
production function equals
the slope of the
( )
the slope of the depreciation line: the slope of the depreciation line:
*goldc
MPK
steady-state capital per
*goldk
MPK =
39CHAPTER 7 Economic Growth I
capital per worker, k*
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The transition to the Golde R le te d t teGolden Rule steady state
The economy does NOT have a tendency to y ymove toward the Golden Rule steady state.
Achieving the Golden Rule requires thatAchieving the Golden Rule requires that policymakers adjust s.
This adjustment leads to a new steady state with This adjustment leads to a new steady state with higher consumption.
But what happens to consumption during the transition to the Golden Rule?
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Starting with too much capital
If goldk k* *
then increasing c*
requires a fall in s. then increasing c*
requires a fall in s.
If goldk ky
q
In the transition to the Golden Rule,
q
In the transition to the Golden Rule,
c,
consumption is higher at all points
,consumption is higher at all points
i
in time.in time.timet0
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Starting with too little capital
th i i *th i i *If goldk k* *
then increasing c*
requires an increase in s.
then increasing c*
requires an increase in s.
yincrease in s. Future generations enjoy higher
increase in s. Future generations enjoy higher
c
j y gconsumption, but the current
i
j y gconsumption, but the current
ii
one experiences an initial drop in consumption.
one experiences an initial drop in consumption. timet0
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pp
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Population growth
Assume the population and labor force grow at rate n (exogenous):at rate n (exogenous):
L n
EX: Suppose L = 1,000 in year 1 and the l ti i i t 2% ( 0 02)
L
population is growing at 2% per year (n = 0.02).
Then L = nL = 0.021,000 = 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 outk to replace capital as it wears out nk to equip new workers with capital
(Otherwise k would fall as the existing capital stock(Otherwise, k would fall as the existing capital stock is 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)kk s f(k) ( +n)k
b kactual break-even investmentinvestment
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The Solow model diagram
Investment, break even
k = s f(k) ( +n)kbreak-even investment
(+ n )k
sf(k)
Capital perk*
46CHAPTER 7 Economic Growth I
Capital per worker, k
k
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The impact of population growth
Investment, break even (+n )kbreak-even investment
(+n1)k
(+n2)k
An increase in nAn increase in nsf(k)
An increase in ncauses an increase in break-
i t t
An increase in ncauses an increase in break-
i t teven investment,even investment,leading to a lower steady-state level
Capital perk *k *
yof k.
47CHAPTER 7 Economic Growth I
Capital per worker, k
k1k2
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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 havewith higher population growth rates will have lower levels of capital and income per worker in the long runthe long run.
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International evidence on population growth and income per personand income per person
Income per person in
100,000
2003 (log scale)
10,000
1,000
1000 1 2 3 4 5
Population growth (percent per year, average 1960-2003)
0 1 2 3 4 5
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The Golden Rule with population growthgrowth
To find the Golden Rule capital stock, express c* in terms of k*:
c* = y* i*y
= f (k* ) ( + n) k*
* i i i d hIn the Golden R l t d t tc* is maximized when
MPK = + nRule steady state, the marginal product of capital net of
or equivalently, MPK = n
of capital net of depreciation equals the population
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p pgrowth rate.
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Alternative perspectives on population growthgrowthThe Malthusian Model (1798) Predicts population growth will outstrip the
Earth’s ability to produce food, leading to the impoverishment of humanityimpoverishment of humanity. Since Malthus, world population has increased
sixfold yet living standards are higher than eversixfold, yet living standards are higher than ever. Malthus neglected the effects of technological
progress. p g
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Alternative perspectives on population growthgrowth
The Kremerian Model (1993) Posits that population growth contributes to
economic growth. More people = more geniuses, scientists &
engineers, so faster technological progress.E id f l hi t i l i d Evidence, from very long historical periods: As world pop. growth rate increased, so did rate
of growth in living standardsof growth in living standards Historically, regions with larger populations have
enjoyed faster growth
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enjoyed faster growth.
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Chapter SummaryChapter SummaryChapter SummaryChapter Summary1. The Solow growth model shows that,
i th l t ’ t d d f li iin the long run, a country’s standard of living depends: iti l it i t positively on its saving rate negatively on its population growth rate
2. An increase in the saving rate leads to: higher output in the long run faster growth temporarily but not faster steady state growth
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Chapter SummaryChapter SummaryChapter SummaryChapter Summary3. If the economy has more capital than the
G ld R l l l th d i i illGolden Rule level, then reducing saving will increase consumption at all points in time, making all generations better offmaking all generations better off.
If the economy has less capital than the G ld R l l l th i i i illGolden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the presentbut reduce consumption for the present generation.