CHAPTER
11
Saving, Capital
Accumulation,
and Output
Prepared by:
Fernando Quijano and Yvonn Quijano
And Modified by Gabriel Martinez
Saving and Growth
Country
1870
1913
1950
1979
2000
Annual %
change
1870-2000
Germany
1,205
2,320
5,005
18,014
23,247
2.3
3.1
963
1,825
2,216
16,899
24,772
2.5
4.8
2,843
6,745
11,921
22,480
32,629
1.9
2.0
Japan
United States
Annual %
change
1950-2000
Since 1950 the US saving rate (S/Y) has averaged
18%.
The German saving rate averaged 24% and the
Japanese, 34%.
Can this explain the growth differences? Probably not.
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Macroeconomics, 3/e
Olivier Blanchard
Preview





Suppose output rises.
A proportion of that, sY, is saved.
Assume S=I. Then I = sY.
Investment increases the capital stock.
With a greater capital stock, we can produce
more.
 Output rises.
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Olivier Blanchard
Preview
Output
Rises
S and
I
Rise
Capit
al
Stock
rises
Output
Rises
 Greater output leads to more saving, capital
accumulation, and therefore more output.
 Is this a perpetual cycle?
 No. There are diminishing returns to capital.
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Olivier Blanchard
Preview
Diminishing
Returns
Capital
Accumulation
A Fixed
Proportion
I = DK
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Olivier Blanchard
Preview
 There are diminishing returns to capital.
 More capital will increase output, but at a
decreasing rate.
 Suppose population growth is positive.
 Eventually, the increase in output won’t be
enough to increase output-per-capita.
 So there’s a “dynamic equilibrium”, a steady
state of capital accumulation in the long run.
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Olivier Blanchard
11-1
Interactions Between
Output and Capital
 Two important relations in the long run are:
– The amount of capital determines the amount of
output produced.
– The amount of output determines the amount of
saving and investment, and so the amount of
capital accumulated.
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The Effects of Capital on Output
 The aggregate production function is a
specification of the relation between
aggregate output and the inputs in
Y  F ( K, N )
production.
Y = aggregate output.
K = capital — the sum of all the machines, plants, and office
buildings in the economy.
N = labor — the number of workers in the economy.
The function F, tells us how much output is produced for given
quantities of capital and labor.
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The Effects of Capital on Output
 Constant returns to scale implies that we
can rewrite the aggregate production
function Y  F ( K , N ) as:
Y
 K N
K 
 F  ,   F  ,1
 N N
N 
N
 The amount of output per worker, Y/N
depends on the amount of capital per worker,
K/N.
 As capital per worker increases, so does
output per worker.
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The Effects of Capital on Output
 Under constant returns to scale, we can
write the relation between output and capital
per worker as follows: Y  F  K ,1
N
If we define
N 
 K
K 
f    F  ,1
 N
N 
Simplifying:
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Y
 K
 f 
 N
N
y  f (k )
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More capital per
worker produces
more output per
worker.
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Output per Worker and
Capital per Worker
Output and Capital
per Worker
Increases in capital
per worker lead to
smaller and smaller
increases in output
per worker.
An increase in capital
per worker, K/N,
causes a move along
the production
function.
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The Effects of Capital on Output
 Focus on the role of capital accumulation:
1. The size of the population, the participation
rate, and the unemployment rate are all
constant.
 Because this is the long run, it’s natural to assume
that ut=ut-1.
 This fit the concerns of the Classical economists:
What determines the Wealth of Nations in the long
run, when all prices are flexible?
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The Effects of Capital on Output

Focus on the role of capital accumulation:
2. There is no technological progress.

Or, at least, we don’t know where it comes from.
3. Saving equals investment

Financial markets work perfectly.
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The Effects of Capital on Output
 Under these assumptions, the first
important relation we want to express is
between output and capital per worker:
Yt
 Kt 
 f 
 N
N
In words, higher capital per
worker leads to higher output
per worker.
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The Effects of Output on
Capital Accumulation
Output and Investment:
 The equations below describe the relation
between private saving and investment:
I  S  (T  G )
If T  G  T  G  0  I  S
S  sY
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0 s 1
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The Effects of Output on
Capital Accumulation
 The saving rate (s) is the proportion of
income that is saved (sY).
S  sY
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0 s 1
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The Effects of Output on
Capital Accumulation
Output and Investment:
 In the long run, private saving is equal to investment,
and proportional to income.
I t  sYt
 Therefore, investment is proportional to output:
 Higher output → higher saving → higher investment.
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The Effects of Output on
Capital Accumulation
Investment and Capital Accumulation:
 Suppose that capital is eternal (once installed, it’s
there forever)
 Then the evolution of the capital stock is given by:
Kt 1  Kt  I t
 Investment adds to capital.
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The Effects of Output on
Capital Accumulation
Investment and Capital Accumulation:
 More realistically, capital depreciates.
 The evolution of the capital stock is given by:
Kt + 1 = (1- δ )Kt + It
 Investment adds to capital.
  denotes the rate of depreciation.
 A proportion (1- ) of capital remains
from the previous period.
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The Effects of Output on
Capital Accumulation
Investment and Capital Accumulation:
 Combine the relation from output to investment,
I t  sYt ,
and the relation from investment to capital
accumulation, Kt 1  (1   ) Kt  I t
we get
Kt 1  (1   ) Kt  sYt
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The Effects of Output on
Capital Accumulation
Investment and Capital Accumulation:
Kt 1  (1   ) Kt  sYt
 If we divide this equation by N, we get
K t 1
Kt
Yt
 (1   )
s
N
N
N
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The Effects of Output on
Capital Accumulation
Output and Capital per Worker:
K t 1
Kt
Yt
 (1   )
s
N
N
N
 Can we use this equation to know what will
happen to capital per worker over time?
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The Effects of Output on
Capital Accumulation
Output and Capital per Worker:
K t 1
K
Y
 (1   ) t  s t
N
N
N
 We can articulate the change in capital per
worker over time by rearranging terms in the
equation above.
Kt + 1
N
-
Kt
N
= s
Yt
N
- δ
Kt
N
In words, the change in the capital stock per worker (left
side) is equal to saving per worker minus depreciation
(right side).
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The Effects of Output on
Capital Accumulation
Output and Capital per Worker:
Kt + 1
N
-
Kt
N
= s
Yt
N
- δ
Kt
N
The stock of capital per worker will increase if
The total amount of saving in the economy
exceeds the part of the capital stock that is worn
out.
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Olivier Blanchard
Implications of
Alternative Saving Rates
11-2
 Our two main relations are:
Yt
 Kt 
 f 
 N
N
First relation:
Capital determines
output.
Kt 1 Kt
Yt
Kt

 s 
N
N
N
N
Second relation:
Output determines capital
accumulation
 Combining the two relations, we can study the behavior
of output and capital over time.
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Dynamics of Capital and Output
Yt
K 
 f t
 N
N
Kt 1 Kt
Yt
Kt

 s 
N
N
N
N
 From our main relations above, we express
output per worker (Y/N) in terms of capital
per worker to derive the equation below:
Kt 1 Kt


N
N
change in capital from
year t to year t+1
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 Kt 
sf  
N
investment
during year t
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Kt
 
N
depreciation
during year t
Olivier Blanchard
Dynamics of Capital and Output
K t  1 Kt

N
N
change in capital from
year t to year t+1
=
K 
sf  t 
 N
investment
during year t


Kt
N
depreciation
during year t
 If investment per worker (sY) exceeds
depreciation per worker, the change in capital per
worker is positive: Capital per worker increases.
 If investment per worker is less than depreciation
per worker, the change in capital per worker is
negative: Capital per worker decreases.
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Dynamics of Capital and Output
Capital and Output
Dynamics
 Output per worker
increases with capital
per worker, but by less
and less as capital per
worker increases.
 Investment per worker
increases by a
proportion of output per
worker.
More capital per worker leads to more investment, but at a diminishing rate.
Suppose a country made a great effort to produce K/N and to save a bunch
of output to invest it. The “extra bang for the extra buck” diminishes.
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Dynamics of Capital and Output
Capital and Output
Dynamics
When capital and output
are low, investment
exceeds depreciation,
and capital increases.
When capital and output
are high, investment is
less than depreciation
and capital decreases.
 At low levels of K/N, the “extra bang for the extra buck” invested is
large, larger than what is being taken away by depreciation.
 If the country already has a very large K/N, the “extra bang for the
extra buck” invested is small and is overwhelmed by depreciation.
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Dynamics of Capital and Output
 At K0/N, capital
per worker is
low, investment
exceeds
depreciation,
thus, capital per
worker and
output per
worker tend to
increase over
time.
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Dynamics of Capital and Output
 At K*/N, output
per worker and
capital per worker
remain constant
at their long-run
equilibrium levels.
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Dynamics of Capital and Output
 At K1/N, capital
per worker is
too high. Here
investment is
overwhelmed
by
depreciation.
Therefore K/N
and Y/N will
fall
over
time.
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B
C
D
A
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Dynamics of Capital and Output
 This model suggests convergence:
– Take a poor country (one with low K/N) and a
rich country (that has a high K/N).
– The poor country will probably be farther away
from K*/N than the rich country.
– Then the poor country should grow faster than
the rich country and catch up.
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Dynamics of Capital and Output
 This model suggests convergence:
– Given the same level of technology and human
capital, same institutions, etc., …
– This model says that all countries should
converge to the same level.
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Dynamics of Capital and Output
 This model suggests convergence:
– High capital accumulation is not enough to
sustain growth forever: if K/N is too high,
diminishing returns will make investment to be
lower than depreciation.
 Compare Singapore with Hong Kong (case study).
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Steady-State Capital and
Output
Steady-State Capital and Output
Kt + 1
N
-
Kt
N
= s
Yt
N
- δ
Kt
N
 The state in which output per worker and capital
per worker are no longer changing is called the
steady state of the economy. In steady state, the
left side of the equation above equals zero, then:
*
 K t* 
Kt
  
sf 
N
 N 
 The important point is to notice that there is
one and only one value of K*/N that satisfies
this equation.
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Steady-State Capital and Output
 There is only one value of K*/N that satisfies this
equation:
 K* 
K*
sf  t    t
N
 N 
 Given the steady state of capital per worker
(K*/N), the steady-state value of output per
worker (Y*/N), is given by the production
function:
 Y *
 K *

  f

 N
 N 
 There is only one steady-state value of Y*/N.
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Steady-State Capital and Output
 There is only one value of K*/N that satisfies this
equation:  K * 
*
Y *
K *
K


t
t
  
sf 

  f

N
N
 N
 N 


 The steady-state values of K*/N and of Y*/N
are uniquely determined by the saving rate.
 A higher s will lead to a higher K*/N and a
higher Y*/N.
 But a higher s will leave the growth rate of
Y*/N unaffected.
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The Saving Rate and Output
The Saving Rate and Output
 The effects of the saving rate on the growth
rate of output per worker:
1. The saving rate has no effect on the long run
growth rate of output per worker, which is
equal to zero.
1. Output per worker and capital per worker are
constant in the steady state.
2. If an economy wanted to increase the steady
state K*/N every year it would have to increase
savings/output every year.
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The Saving Rate and Output
2. Nonetheless, the saving rate determines the
level of output per worker in the long run.
Other things equal, countries with a higher
saving rate will achieve higher output per
worker in the long run.
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The Saving Rate and Output
The Effects of
Different Saving
Rates
A country that raises its
saving rate achieves a
higher level of output
per worker in steady
state.
But, in the steady state,
output/worker does not
grow.
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The Saving Rate and Output
3. Therefore, an increase in the saving rate will
lead to higher growth of output per worker for
some time, but not forever.
 The saving rate does not affect the long-run
growth rate of output per worker.
 After an increase in the saving rate, growth
will end once the economy reaches its new
steady state.
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The Saving Rate and Output
The Effects of an
Increase in the Saving
Rate on Output per
Worker
An increase in the
saving rate leads to a
period of higher growth
until output reaches its
new higher steady-state
level.
The economy takes
some time to reach the
new steady state as it
accumulates capital.
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The Saving Rate and Output
The Effects of an Increase
in the Saving Rate on
Output per Worker in an
Economy with
Technological Progress
If there’s technological
progress, the growth
rate of Y/N is positive in
the steady state.
An increase in the
saving rate leads to a
period of higher growth
until output reaches a
new, higher path.
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(This is a logarithmic scale, so the slope of
the steady-state path is equal to the growth
rate of Y/N.)
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The Optimal Saving Rate and
Optimal Consumption
The Saving Rate and Consumption
 The previous section suggests that a
country could choose its steady state level
of capital per worker and output per worker.
 What steady state should a country choose?
 I’d think that the steady state that maximizes
consumption.
– (Output is nice, but I’d focus on eating).
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The Saving Rate and Consumption
Steady state
depreciation / worker, K*/N
Steady state
output / worker, Y*/N
Steady state output / worker,
f(K*/N)
?
Steady state investment /
worker, sf(K*/N)
Steady state capital / worker, K*/N
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The Saving Rate and Consumption
 Because people care about consumption
(and not about investment or output),
society should maximize steady-state
consumption.
Goldenrule saving
rate
Max
consumption
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Golden-rule
income
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Golden-rule
level of
capital/
worker
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The Saving Rate and Consumption
 The level of capital that causes the value of
the saving rate that yields the highest level
of consumption for all generations in steady
state is known as the golden-rule level of
capital.
Goldenrule saving
rate
Max
consumption
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Golden-rule
saving
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Golden-rule
level of
capital/
worker
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The Saving Rate and Consumption
 In the steady state, investment has to be
equal to depreciation.
*
*
t
K
I

N
N
– (If investment and depreciation were not equal,
capital would accumulate, which violates the
definition of the steady state.)
 So adding capital means more depreciation
(more capital has to be replaced).
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The Saving Rate and Consumption
 But adding capital also means more output
 K t* 
Y*

 f 
N
 N 
 Consumption, in a closed economy with no
government, is
C=Y–I
 From the above we know that
 K t* 
K t*
C*
  
 f 
N
N
 N 
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The Saving Rate and Consumption
Steady state output / worker, Y*/N
Steady state
depreciation / worker,
K*/N
Steady state
output / worker,
f(K*/N)
C*/N
Steady state capital / worker, K*/N
Consumption is given by the vertical distance between the production
function and the depreciation line. Since steady state K*/N is given by
the intersection of the depreciation line and the investment curve,
steady state consumption is the difference between Y*/N and K*/N.
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The Saving Rate and
Consumption
The Effects of the Saving Rate on
Consumption per Worker in Steady State
 At very low levels of steady-state saving, steadystate capital and output per worker are very low
and consumption is very low.
 At very high levels of steady-state saving, steadystate capital per worker is very high. But
because of diminishing returns, a very large
proportion of output has to be devoted to
replacing depreciated capital, and consumption is
low.
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The Saving Rate and Consumption
Steady state
depreciation / worker, K*/N
Steady state
output / worker, Y*/N
Steady state output / worker,
f(K*/N)
Steady state investment /
worker, sf(K*/N)
Steady state capital / worker, K*/N
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The Saving Rate and
Consumption
 Evidently, if s=0, there’s no
investment. Capital
depreciates to zero, and
output disappears.
 For s smaller than sG,
increases in the saving
rate (which increase I*/N),
lead to higher capital and
output per worker. Y*/N
increases so much that
C*/N = Y*/N – I*/N
increases.
 How do we find the Golden-Rule level of saving?
 Golden Rule: set saving so that this generation’s consumption equals that of all
future generations.
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The Saving Rate and
Consumption
 For s larger than sG,
increases in the saving
rate still lead to higher
capital and output per
worker, but lower
consumption per worker
because Y*/N increases
by less than I*/N.
 For s=1, capital and output
per worker are high, but all
of the output is used to
replace depreciation, so
C*=0.
 Golden Rule: set saving so that this generation’s consumption equals that of all
future generations.
 How do we find the Golden-Rule level of saving?
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Macroeconomics, 3/e
Olivier Blanchard
The Saving Rate and Consumption
Steady state output / worker, Y*/N
Steady state
depreciation / worker,
K*/N
Steady state
output / worker,
f(K*/N)
C*/N
Steady state capital / worker, K*/N
 How do we find the Golden-Rule level of capital/worker?
 Set K*/N such that the slope of f(K*/N) is equal to .
 Set marginal product of capital/worker = depreciation
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Social Security, Social Security Reform, and
Capital Accumulation in the United States
 One way to run a social security system is the
pay-as-you-go system, where the taxes that
workers pay are the benefits that current retirees
receive.
– There is little or no saving or investment.
 Another is the fully-funded system, where
workers are taxed, their contributions invested in
financial assets, and when workers retire, they
receive the principal plus the interest payments
on their investments.
– All contributions are saved and invested.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Social Security, Social Security Reform, and
Capital Accumulation in the United States
 The US has a pay-as-you-go system, so that retirees’
benefits depend on current workers’ contributions.
 But the ratio of workers to retirees has been falling.
 In anticipation of demographic changes, the Social
Security tax rate has been increased, and contributions
are now larger than benefits, leading to the accumulation
of a Social Security trust fund.
 But the trust fund is expected to be depleted in midcentury, so there’s need for more reform.
–
Raise the tax rate further, increase the retirement age, reduce
benefits.
–
Shift to a fully-funded system (more pain, but more saving).
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
11-3
Getting a Sense
of the Magnitudes
 To get an idea of how to connect the model
with reality, we need to do a numerical
simulation of the model.
 That is, we need to define the functions
precisely, and we need to give numbers for
all the parameters.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Getting a Sense
of the Magnitudes
 Assume the production function is:
Y
 Output per worker is:
K N
Y

N
K N

N
K

N
 That is, the first relation of the model
(capital/worker determines output) is
K
N
 Kt 
f  
 N
 And the second relation of the model
(output determines capital accumulation) is
Kt + 1
N
-
Kt
N
Kt
N
= s
Yt
N
- δ
Kt
N
 Then,
Kt + 1
N
© 2003 Prentice Hall Business Publishing
-
Kt
N
= s
Kt
N
-δ
Kt
N
Macroeconomics, 3/e
Olivier Blanchard
The Effects of the Saving Rate
on Steady-State Output
Kt + 1
N
-
Kt
N
= s
Kt
N
-δ
Kt
N
 In steady state, the left side equals zero:
*
*
2 K
2 K 

  
 Squaring both sides, s
N
 N 
s
K
K
= δ
N
N
2
2
s
K*

 Dividing by (K/N) and rearranging,   
N
 
The steady state capital per worker is equal to the square of the
ratio of the saving rate to the depreciation rate.
 Steady-State Output per
worker is given by:
© 2003 Prentice Hall Business Publishing
Y

N
2
K
s
s
   
N

 
Macroeconomics, 3/e
*
Olivier Blanchard
The Effects of the Saving Rate
on Steady-State Output
Y

N
2
K
s
s
   
N

 
*
 Steady-state output per worker is equal
to the ratio of the saving rate to the
depreciation rate.
 A higher saving rate and a lower
depreciation rate both lead to higher
steady-state capital per worker and
higher steady-state output per worker.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
The Dynamic Effects of an
Increase in the Saving Rate
Dynamic Effects of an
Increase in the Saving Rate
from 10 to 20% on the
Level and the Growth Rate
of Output per Worker
Y s
K
s
 and
 
N 
N  
*
2
Suppose =0.10 and s=0.10.
Now suppose s rises to 0.20.
K/N quadruples and Y/N doubles.
It takes a long time for output to
adjust to its new higher level after
an increase in the saving rate.
Put another way, an increase in
the saving rate leads to a long
period of higher growth.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
The U.S. Saving Rate
and the Golden Rule
 In steady state, consumption per worker is equal
to output per worker minus depreciation per
C Y
K
worker.
=
- δ
N
 Knowing that:
then:
K*  s 
 
N  
N
N
and
Y*

N
2
2
K*
s
s
   
N

 
C* s
s(1  s)
s
    
N 

 
© 2003 Prentice Hall Business Publishing
2
Macroeconomics, 3/e
Olivier Blanchard
The U.S. Saving Rate
and the Golden Rule
 We saw above that the golden
Y
K* s


N
N 
rule level of capital is found by
setting K*/N such that the slope of f(K*/N) is
equal to .
d (Y / N ) 1 N 1 
d (K / N )
1

2 s
1
s
2
=0.10

2 K
Y
s 0.5
 
5
N  0.1
2
K *  0.5 

  25
N  0.1 
© 2003 Prentice Hall Business Publishing
*

2s
C s (1  s ) 0.5(1  0.5)


 2.5
N

0.1
Macroeconomics, 3/e
Olivier Blanchard
The U.S. Saving Rate
and the Golden Rule
Table 11-2
Saving Rate, s
The Saving Rate and the Steady-state Levels of Capital,
Output, and Consumption per Worker-=10%
Capital per worker,
K/N
K*  s 
 
N  
2
Output per
worker, Y/N
Y
s
=
N δ
Consumption per
worker, C/N
C s (1- s )
N
=
δ
0.0
0.0
0.0
0.0
0.1
1.0
1.0
0.9
0.2
4.0
2.0
1.6
0.3
9.0
3.0
2.1
0.4
16.0
4.0
2.4
0.5
25.0
5.0
2.5
0.6
36.0
6.0
2.4
:
:
:
100.0
10.0
0.0
:
1.0
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
11-4
Physical Versus
Human Capital
 The set of skills of the workers in the
economy is called human capital.
 An economy with many highly skilled
workers is likely to be much more productive
than an economy in which most workers
cannot read or write.
 The conclusions drawn about physical
capital accumulation remain valid after the
introduction of human capital in the analysis.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Extending the Production
Function
 When the level of output per workers
depends on both the level of physical
capital per worker, K/N, and the level of
human capital per worker, H/N, the
production function may be written as:
Y
 K H
 f , 
 N N
N
( ,  )
 An increase in capital per worker or the
average skill of workers leads to an increase in
output per worker.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Human Capital, Physical
Capital, and Output
 An increase in how much society “saves” in the
form of human capital—through education and onthe-job-training—increases steady-state human
capital per worker, which leads to an increase in
output per worker.
 In the long run, output per worker depends not
only on how much society saves but also how
much it spends on education.
– How do we generate the institutions to improve the
quality of education?
– How do we reduce spending on current consumption to
increase spending on education?
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Human Capital, Physical
Capital, and Output
 In the United States, spending on education
comprises about 6% of GDP, compared to 16%
investment in physical capital. This comparison:
– Accounts for the fact that education is partly
consumption.
– Does not account for the opportunity cost of education.
– Does not account for the opportunity cost of on-the-jobtraining.
– Considers gross, not net investment. Depreciation of
human capital is slower than that of physical capital.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Endogenous Growth
 A recent study has concluded that output per
worker depends roughly equally on the amount of
physical capital and the amount of human capital
in the economy.
 Still, models with human and physical capital
conclude that higher levels of accumulation of H/N
or K/N (that is, more education or saving) lead to
higher levels of output/worker but not higher
growth rates.
– Saving/Income can’t be higher than 100%.
– The same holds for expenditure on education / income.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard
Endogenous Growth
 Models that generate steady growth even
without technological progress are called
models of endogenous growth, where
growth depends on variables such as the
saving rate and the rate of spending on
education (Lucas and Romer).
– These models note that higher levels of
education lead to better technology.
Continuous technological improvement can lead
to continuously positive growth rates.
© 2003 Prentice Hall Business Publishing
Macroeconomics, 3/e
Olivier Blanchard