chapter04 (2)

1.

macro
Economic Growth
macroeconomics
N. Gregory Mankiw

2. Chapter 4 learning objectives

Learn the closed economy Solow model
See how a country’s standard of living
depends on its saving and population
growth rates
Learn how to use the “Golden Rule”
to find the optimal savings rate and capital
stock
Technological progress in the Solow model
Policies to promote growth

3. 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

4. How Solow model is different from Chapter 3’s model

1.
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.
4.
No G or T
(only to simplify presentation;
we can still do fiscal policy experiments

5. 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)

6. The production function

Output per
worker, y
f(k)
1
MPK =f(k +1) – f(k)
Note:
Note: this
this production
production function
function
exhibits
exhibits diminishing
diminishing MPK.
MPK.
Capital per
worker, k

7. 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

8. 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 = (1–s)y
(per worker)

9. Saving and investment

saving (per worker)
= y – c
= y – (1–s)y
=
sy
National income identity is y = c + i
Rearrange to get: i = y – c = sy
(investment = saving, like in chap. 3!)
Using the results above,
i = sy = sf(k)

10. Output, consumption, and investment

Output per
worker, y
f(k)
c1
sf(k)
y1
i1
k1
Capital per
worker, k

11. Depreciation

Depreciation
per worker, k
== the
the rate
rate of
of depreciation
depreciation
== the
the fraction
fraction of
of the
the capital
capital stock
stock
that
that wears
wears out
out each
each period
period
k
1
Capital per
worker, k

12. Capital accumulation

The
Thebasic
basicidea:
idea:
Investment
Investmentmakes
makes
the
thecapital
capitalstock
stockbigger,
bigger,
depreciation
depreciationmakes
makesititsmaller.
smaller.

13. Capital accumulation

Change in capital stock
k
=
i
= investment – depreciation
–
k
Since i = sf(k) , this becomes:
k = s f(k) – k

14. The equation of motion for k

k = s f(k) – k
the Solow model’s 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:
consump. per person:
y = f(k)
c = (1–s) f(k)

15. 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 constant value, denoted k*, is called the
steady state capital stock.

16. The steady state

Investment
and
depreciation
k
sf(k)
k*
Capital per
worker, k

17. Moving toward the steady state

Investment
and
depreciation
k = sf(k) k
k
sf(k)
k
investment
depreciation
k1
k*
Capital per
worker, k

18. Moving toward the steady state

Investment
and
depreciation
k = sf(k) k
k
sf(k)
k
k1 k2
k*
Capital per
worker, k

19. Moving toward the steady state

Investment
and
depreciation
k = sf(k) k
k
sf(k)
k
investment
depreciation
k2
k*
Capital per
worker, k

20. Moving toward the steady state

Investment
and
depreciation
k = sf(k) k
k
sf(k)
k
k2 k3 k*
Capital per
worker, k

21. Moving toward the steady state

Investment
and
depreciation
k = sf(k) k
k
sf(k)
Summary:
Summary:
*
As
As long
long as
as kk << kk*,,
investment
investment will
will exceed
exceed
depreciation,
depreciation,
and
and kk will
will continue
continue to
to
**.
grow
toward
k
grow toward k .
k3 k*
Capital per
worker, k

22. Moving toward the steady state

A numerical example
Production function (aggregate):
Y F (K , L ) K L K
L
1/2 1/2
To derive the per-worker production function,
divide through by L:
1/2
1/2 1/2
Y K L
K
L
L
L
Then substitute y = Y/L and k = K/L to get
y f (k ) k
1/2

23. Moving toward the steady state

A numerical example, cont.
Assume:
s = 0.3
= 0.1
initial value of k = 4.0

24. A numerical example

Approaching the Steady State:
A Numerical Example
y
Assumptions:
Year
Year
11
0.200
0.200
22
0.195
0.195
33
0.189
0.189
kk
yy
s 0.3;
k;
cc
ii
k
k
4.000
4.000
2.000
2.000
1.400
1.400
0.600
0.600
0.400
0.400
4.200
4.200
2.049
2.049
1.435
1.435
0.615
0.615
0.420
0.420
4.395
4.395
2.096
2.096
1.467
1.467
0.629
0.629
0.440
0.440
k
k

25. A numerical example, cont.

Approaching the Steady State:
A Numerical Example
y
Assumptions:
Year
Year kk
11 4.000
4.000
22 4.200
4.200
33 4.395
4.395
44 4.584
4.584
……
10
10 5.602
5.602
……
25
25 7.351
7.351
……
100
1008.962
8.962
……
9.000
9.000
yy
cc
s 0.3;
k;
ii
k
k
k
k
2.000
2.000
2.049
2.049
1.400
1.400
1.435
1.435
0.600
0.600
0.615
0.615
0.400
0.400
0.420
0.420
0.200
0.200
0.195
0.195
2.367
2.367
1.657
1.657
0.710
0.710
0.560
0.560
0.150
0.150
2.706
2.706
1.894
1.894
0.812
0.812
0.732
0.732
0.080
0.080
2.994
2.994
2.096
2.096
0.898
0.898
0.896
0.896
0.002
0.002
3.000
3.000
2.100
2.100
0.900
0.900
0.900
0.900
0.000
0.000
2.096
2.096
2.141
2.141
1.467
1.467
1.499
1.499
0.629
0.629
0.642
0.642
0.440
0.440
0.458
0.458
0.189
0.189
0.184
0.184

26. Approaching the Steady State: A Numerical Example

Exercise: solve for the steady state
Continue to assume
s = 0.3, = 0.1, and y = k 1/2
Use the equation of motion
k = s f(k) k
to solve for the steady-state values of
k, y, and c.

27. Approaching the Steady State: A Numerical Example

Solution to exercise:
k 0
def. of steady state
s f (k *) k *
eq'n of motion with k 0
0.3 k * 0.1k *
using assumed values
k*
3
k *
k*
Solve to get: k * 9 and y * k * 3
Finally, c * (1 s )y * 0.7 3 2.1

28. Exercise: solve for the steady state

An increase in the saving rate
An increase in the saving rate raises investment…
…causing the capital stock to grow toward a new steady state:
Investment
and
depreciation
k
s2 f(k)
s1 f(k)
k
*
1
k
*
2
k

29. Solution to exercise:

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.

30. An increase in the saving rate

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) f(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* ?

31. Prediction:

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*
= f (k*) i*
= f (k*) k*
In general:
i = k + k
In the steady state:
i* = k*
because k = 0.

32. The Golden Rule: introduction

The Golden Rule Capital Stock
steady state
output and
depreciation
Then, graph
f(k*) and k*,
and look for the
point where the
gap between
them is biggest.
*
*
y gold
f (k gold
)
k*
f(k*)
*
c gold
*
*
i gold
k gold
*
k gold
steady-state
capital per
worker, k*

33. The Golden Rule Capital Stock

c* = f(k*) k*
is biggest where
the slope of the
production func.
equals
the slope of the
depreciation line:
k*
f(k*)
*
c gold
MPK =
*
k gold
steady-state
capital per
worker, k*

34. The Golden Rule Capital Stock

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?

35. The Golden Rule Capital Stock

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

36. The transition to the Golden Rule Steady State

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

37. Starting with too much capital

Population Growth
Assume that the population--and labor force-grow at rate n. (n is exogenous)
L
L
n
EX: Suppose L = 1000 in year 1 and the
population is growing at 2%/year (n = 0.02).
Then L = n L = 0.02 1000 = 20,
so L = 1020 in year 2.

38. Starting with too little capital

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)

39. Population Growth

The equation of motion for k
With population growth, the equation of
motion for k is
k = s f(k) ( + n) k
actual
investment
break-even
investment

40. Break-even investment

The Solow Model diagram
Investment,
break-even
investment
k = s f(k) ( +n)k
( + n ) k
sf(k)
k*
Capital per
worker, k

41. The equation of motion for k

The impact of population growth
Investment,
break-even
investment
( +n2) k
( +n1) k
An increase in n
causes an
increase in breakeven investment,
leading to a lower
steady-state level
of k.
sf(k)
k2*
k1* Capital per
worker, k

42. The Solow Model diagram

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.

43. The impact of population growth

The Golden Rule with Population Growth
To find the Golden Rule capital stock,
we again express c* in terms of k*:
c* =
y*
i*
= f (k* ) ( + n) k*
c* is maximized when
MPK = + n
or equivalently,
MPK = n
In
In the
the Golden
Golden
Rule
Rule Steady
Steady State,
State,
the
the marginal
marginal product
product of
of
capital
capital net
net of
of
depreciation
depreciation equals
equals the
the
population
population growth
growth rate.
rate.

44. Prediction:

Tech. progress in the Solow model
So far,
the production technology is held constant
income per capita is constant in the steady
state.
Neither point is true in the real world.

45. The Golden Rule with Population Growth

Tech. progress in the Solow model
A new variable: E = labor efficiency
Assume:
Technological progress is laboraugmenting: it increases labor efficiency
at the exogenous rate g:
E
g
E

46. 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.

47. 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 y = s f(k)

48. 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

49. 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

50. Tech. progress in the Solow model

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

51. Tech. progress in the Solow model

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
In the
the Golden
Golden
Rule
Rule Steady
Steady State,
State,
the
the marginal
marginal
product
product of
of capital
capital
net
net of
of depreciation
depreciation
equals
equals the
the
pop.
pop. growth
growth rate
rate
plus
plus the
the rate
rate of
of
tech
tech progress.
progress.

52. Steady-State Growth Rates in the Solow Model with Tech. Progress

Policies to promote growth
Three policy questions:
1. Are we saving enough? Too much?
2. What policies might change the saving
rate?
3. What policies might encourage faster
technological progress?

53. The Golden Rule

1. Evaluating the Rate of Saving
Use the Golden Rule to determine whether
our saving rate and capital stock are too high,
too low, or about right.
To do this, we need to compare
(MPK ) to (n + g ).
If (MPK ) > (n + g ), then we are below the
Golden Rule steady state and should increase s.
If (MPK ) < (n + g ), then we are above the
Golden Rule steady state and should reduce s.

54. Policies to promote growth

1. Evaluating the Rate of Saving
To estimate (MPK ), we use
three facts about the U.S. economy:
1. k = 2.5 y
The capital stock is about 2.5 times one year’s
GDP.
2. k = 0.1 y
About 10% of GDP is used to replace
depreciating capital.
3. MPK k = 0.3 y
Capital income is about 30% of GDP

55. 1. Evaluating the Rate of Saving

1. k = 2.5 y
2. k = 0.1 y
3. MPK k = 0.3 y
To determine , divided 2 by 1:
k
0.1 y
k
2.5 y
0.1
0.04
2.5

56. 1. Evaluating the Rate of Saving

1. k = 2.5 y
2. k = 0.1 y
3. MPK k = 0.3 y
To determine MPK, divided 3 by 1:
MPK k
k
0.3 y
2.5 y
0.3
MPK
0.12
2.5
Hence, MPK = 0.12 0.04 = 0.08

57. 1. Evaluating the Rate of Saving

From the last slide: MPK = 0.08
U.S. real GDP grows an average of 3%/year,
so n + g = 0.03
Thus, in the U.S.,
MPK = 0.08 > 0.03 = n + g
Conclusion:
The
The U.S.
U.S. is
is below
below the
the Golden
Golden Rule
Rule steady
steady state:
state:
ifif we
we increase
increase our
our saving
saving rate,
rate, we
we will
will have
have faster
faster
growth
growth until
until we
we get
get to
to aa new
new steady
steady state
state with
with
higher
higher consumption
consumption per
per capita.
capita.

58. 1. Evaluating the Rate of Saving

2. Policies to increase the saving rate
Reduce the government budget deficit
(or increase the budget surplus)
Increase incentives for private saving:
reduce capital gains tax, corporate income
tax, estate tax as they discourage saving

59. 1. Evaluating the Rate of Saving

3. Encouraging technological progress
Patent laws:
encourage innovation by granting temporary
monopolies to inventors of new products
Tax incentives for R&D
Grants to fund basic research at universities
Industrial policy:
encourage specific industries that are key for
rapid tech. progress
(subject to the concerns on the preceding slide)

60. 2. Policies to increase the saving rate

Chapter Summary
1. The Solow growth model shows that, in the
long run, a country’s standard of living depends
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.

61. 3. Encouraging technological progress

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.

62. Chapter Summary

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
the U.S. has much less capital than the
Golden Rule steady state
5. Ways to increase the saving rate
increase public saving (reduce budget
deficit)
tax incentives for private saving