Economics Growth II
slide 1
Recap

The Solow Model endogenizes K and L to
describe long run output per worker.

We found a unique and stable equilibrium
for exogenous parameters (n, ,s).

The model helps us to understand
differences in output per worker in the
world.
Today


The Golden Rule:
When is consumption maximized in the
long run?
Technological growth:
The key determinant of long run
growth

Government’s role for savings
economic
Review key concepts
Notation:
Savings are a constant fraction of output:
A
A
A
The Solow Model diagram
Investment,
break-even
investment
( + n ) k
sAf(k)
k*
Capital per
worker, k
The Golden Rule: introduction

Different values of s lead to different steady states.
How do we know which is the “best” steady state?

Economic well-being depends on consumption,
so the “best” steady state has the highest possible
value of consumption per person: c* = (1–s) Af(k*)

An increase in s
• leads to higher k* and y*, which may raise c*
• reduces consumption’s share of income (1–s),
which may lower c*

So, how do we find the s and k* that maximize c* ?
The Golden Rule Capital Stock
*
k gold
= 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*
= Af (k*) − i*
= Af (k*) − k*
In general:
i = k + k
In the steady state:
i* = k*
because k = 0.
The Golden Rule Capital Stock
steady state
output and
depreciation
Then, graph
Af(k*) and k*,
and look for the
point where the
gap between
them is biggest.
*
*
y gold
= f (k gold
)
k*
Af(k*)
*
c gold
*
*
i gold
=  k gold
*
k gold
steady-state
capital per
worker, k*
The Golden Rule Capital Stock
c* = Af(k*) − k*
is biggest where
the slope of the
production func.
equals
the slope of the
depreciation line:
k*
Af(k*)
*
c gold
MPK = 
*
k gold
steady-state
capital per
worker, k*
Finding the Golden Rule Savings Rate
MPK = 
MPK is a function of the capital stock per
worker:
MPK = g(k)
Replace k by its steady state value
k = h(s)
Solve for s.
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.
But what happens to consumption
during the transition to the Golden Rule?
Starting with too much capital
*
If k *  k gold
then increasing
c* requires a
fall in s.
y
In the transition
to the
Golden Rule,
consumption is
higher at all
points in time.
c
i
t0
time
Starting with too little capital
*
If k *  k gold
then increasing c*
requires an
increase in s.
Future generations
enjoy higher
consumption,
but the current one
experiences
an initial drop
in consumption.
y
c
i
t0
time
Technological progress
In the Solow model of Chapter 7,
▪
▪
the production technology is held constant
income per capita is constant in the steady state.
Neither point is true in the real world:
▪
▪
1929-2001: U.S. real GDP per person grew by a factor of
4.8, or 2.2% per year.
examples of technological progress abound
(see next slide)
slide 14
Technological progress
slide 15
Examples of technological progress

1970: 50,000 computers in the world
2000: 51% of U.S. households have 1 or more computers

The real price of computer power has fallen an average of
30% per year over the past three decades.

The average car built in 1996 contained more computer
processing power than the first lunar landing craft in 1969.

Modems are 22 times faster today than two decades ago.

Since 1980, semiconductor usage per unit of GDP has
increased by a factor of 3500.

1981: 213 computers connected to the Internet
2000: 60 million computers connected to the Internet
slide 16
Tech. 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:
g =
E
E
slide 17
Tech. progress in the Solow model

We now write the production function as:
Y = F (K , L  E )
▪ where L  E = the number of effective
workers.
– Hence, increases in labor efficiency have
the same effect on output as increases in
the labor force.
slide 18
Steady state

19
02/11/2021
Tech. 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 f(k)
sy

slide 20
Law of motion for capital
Write everything in efficiency units:
slide 21
Steady state
In steady state:
slide 22
Tech. 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
slide 23
Tech. progress in the Solow model
Investment,
break-even
investment
k = s f(k) − ( +n +g)k
( +n +g ) k
sf(k)
k*
Capital per
worker, k
slide 24
Steady-State Growth Rates in the
Solow Model with Tech. Progress
Variable
Symbol
Steady-state
growth rate
Capital per
effective worker
k = K/ (L E )
0
Output per
effective worker
y = Y/ (L E )
0
Output per worker
(Y/ L ) = y E
g
Total output
Y = y E L
n+g
slide 25
Sources of economic growth
Take the total derivative:

26
Output increases because
 Capital increases.
 The number of workers increases.
 The technology level increases.
02/11/2021
Sources of economic growth

The rate of output growth depends on



27
The rate of capital growth.
The rate of population growth.
The rate of technological progress.
02/11/2021
Growth accounting in practice

In the data, we can calculate




The capital growth rate.
The labor force growth rate.
The income shares.
We calculate the growth rate of technology as residual,
hence, we call it¨Solow residual¨:
28
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Sources of US output growth
250,0
200,0
150,0
Suma crecimiento PIB
Suma N cantidad
100,0
Suma N calidad
Suma K
Suma TFP
50,0
2015
2012
2009
2006
2003
2000
1997
1994
1991
1988
1985
1982
1979
1976
1973
1970
1967
1964
1961
1958
1955
1952
0,0
-50,0
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Sources of Taiwan output growth
500,0
450,0
400,0
350,0
300,0
Suma crecimiento PIB
Suma N cantidad
250,0
Suma N calidad
Suma K
200,0
Suma TFP
150,0
100,0
50,0
30
2015
2012
2009
2006
2003
2000
1997
1994
1991
1988
1985
1982
1979
1976
1973
1970
1967
1964
1961
1958
1955
1952
0,0
02/11/2021
Sources of Zimbabwe output growth
250,0
200,0
150,0
Suma crecimiento PIB
100,0
Suma N cantidad
Suma N calidad
Suma K
50,0
Suma TFP
2015
2012
2009
2006
2003
2000
1997
1994
1991
1988
1985
1982
1979
1976
1973
1970
1967
1964
1961
1958
1955
1952
0,0
-50,0
-100,0
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Macroeconomía de 02/11/2021
The Golden Rule With E
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.
slide 32
Questions

Are we saving enough? Too much?

Which policies can change the savings
rate?

Savings are exogenous, but let us think
the government may influence it.
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Savings in Spain
K/Y
3,50
3,00
2,50
2,00
1,50
K/Y
1,00
0,50
0,00
Evaluating the Golden Rule

Let us use the Golden rule to judge whether the capital
stock is too high, or too low in Spain.

If MPK > (n + g + ),
Spain is below the Golden Rule and needs to increase
savings s.

If MPK < (n + g + ),
Spain is above the Golden Rule and needs to decrease
savings s.
Evaluating the Golden Rule II
To estimate (MPK −  ), let us use the following stylized
facts:
1. k = 2,75 y
Capital is approximatly 2,75 times GDP.
2.
 k = 0,1 y
Capital depreciation is about 10% of GDP.
3. MPK  k = 0,3 y
The capital share of income is about 30%.
Evaluating the Golden Rule III
1. k = 2,5 y
2.  k = 0,1 y
3. MPK  k = 0,3 y
To determine  , divide 2 by 1:
δk
0.1y
0.1
=
=
= 0.036  δ = 0.036
k
2.75y 2.75
Evaluating the Golden Rule IV
1. k = 2,5 y
2.  k = 0,1 y
3. MPK  k = 0,3 y
To determine MPK, divide 3 by 1:
MPK k
0.3y
0.3
=
 MPK =
= 0.11
k
2.75y
2.75
Hence, MPK −  = 0,11 − 0,036 = 0,074
Evaluating the Golden Rule V
▪ We have: PMK −  = 0,074
▪ Annual GDP growth is about 3% per year:
n + g = 0,03
▪ Hence,
PMK −  = 0,074 > 0,03 = n + g
Spain is below the golden Rule. Increasing savings would
increase long run consumption per worker.
How to increase savings?

Reduce the government deficit (government
consumption).

Increase private savings incentives:
 Decrease capital taxation (Firm profits, housing taxes).
 Increase consumption taxation.
 Increase incentives for private retirement savings.
Chapter Summary
3.
If the economy has more capital than the Golden
Rule level, then reducing saving will increase
consumption at all points in time, making all
generations better off.
If the economy has less capital than the Golden
Rule level, then increasing saving will increase
consumption for future generations, but reduce
consumption for the present generation.
Chapter summary
4. Key results from Solow model with tech
progress
▪ steady state growth rate of income per
person depends solely on the exogenous
rate of tech progress
slide 42