1. Economic Growth I: Capital Accumulation and Population Growth
PART III Growth Theory: The Economy in the Very Long Run
Economic Growth I: Capital Accumulation and Population Growth
Chapter 8 of Macroeconomics, 8th edition, by N. Gregory Mankiw
ECO62 Udayan Roy

2. The Solow-Swan Model This is a theory of macroeconomic dynamics
Using this theory,
you can predict where an economy will be tomorrow, the day after, and so on, if you know where it is today
you can predict the dynamic effects of changes in
the saving rate
the rate of population growth
and other factors

3. Two productive resources and one produced good
There are two productive resources:
Capital, K
Labor, L
These two productive resources are used to produce one
final good, Y

4. The Production Function
The production function is an equation that tells us how much of the final good is produced with specified amounts of capital and labor
Y = F(K, L)
Example: Y = 5K0.3L0.7
Y = 5K0.3L0.7
labor 10 20 30
capital 1
25.06
40.71
54.07 2
30.85
50.12
66.57 3
34.84
56.60
75.18 4
37.98
61.70
81.95

5. Constant returns to scale
Y = 5K0.3L0.7
labor 10 20 30
capital 1
25.06
40.71
54.07 2
30.85
50.12
66.57 3
34.84
56.60
75.18 4
37.98
61.70
81.95
Y = F(K, L) = 5K0.3L0.7
Note:
if you double both K and L, Y will also double
if you triple both K and L, Y will also triple
… and so on
This feature of the Y = 5K0.3L0.7 production function is called constant returns to scale
The Solow-Swan model assumes that production functions obey constant returns to scale

6. Constant returns to scale
Definition: The production function F(K, L) obeys constant returns to scale if and only if
for any positive number z (that is, z > 0)
F(zK, zL) = zF(K, L)
Example: Suppose F(K, L) = 5K0.3L0.7.
Then, for any z > 0, F(zK, zL) = 5(zK)0.3(zL)0.7 = 5z0.3K0.3z0.7L0.7 = 5z K0.3L0.7 = z5K0.3L0.7 = zF(K, L)
At this point, you should be able to do problem 1 (a) on page 232 of the textbook.

7. Constant returns to scale
CRS requires F(zK, zL) = zF(K, L) for any z > 0
Let z be 1/L.
Then,
So, y = F(k, 1) .
From now on, F(k, 1) will be denoted f(k), the per worker production function.
k = K/L denotes per worker stock of capital
y = Y/L denotes per worker output
So, y = f(k) .

8. Per worker production function: exampley 1 5 2
6.156 3
6.952 4
7.579
8.103 6
8.559 7
8.964 8
9.330 9
9.666 10
9.976 11
10.266 12
10.537 13
10.793 14
11.036 15
11.267 16
11.487 17
11.698 18
11.900 19
12.095 20
12.282
At this point, you should be able to do problems 1 (b) and 3 (a) on pages 232 and 233 of the textbook.
This is what a typical per worker production function looks like: concave

9. The Cobb-Douglas Production Function
Y = F(K, L) = 5K0.3L0.7
This production function is itself an instance of a more general production function called the Cobb-Douglas Production Function
Y = AKαL1 − α ,
where A is any positive number (A > 0) and
α is any positive fraction (1 > α > 0)

10. Per worker Cobb-Douglas production function
Y = AKαL1 − α implies y = Akα = f(k)
In problem 1 on page 232 of the textbook, you get Y = K1/2L1/2, which is the Cobb-Douglas production with A = 1 and α = ½.
In problem 3 on page 233, you get Y = K0.3L0.7, which is the Cobb-Douglas production with A = 1 and α = 0.3.

11. Per worker production function: graph

12. Income, consumption, saving, investment
Output = income
The Solow-Swan model assumes that each individual saves a constant fraction, s, of his or her income
Therefore, saving per worker = sy = sf(k)
This saving becomes an addition to the existing capital stock
Consumption per worker is denoted c = y – sy = (1 – s)y

13. Income, consumption, saving, investment: graph

14. Depreciation But part of the existing capital stock wears out
This is called depreciation
The Solow-Swan model assumes that a constant fraction, δ, of the existing capital stock wears out in every period
That is, an individual who currently has k units of capital will lose δk units of capital though depreciation (or, wear and tear)

15. Depreciation
Although 0 < δ < 1 is the fraction of existing capital that wears out every period, in some cases—as in problem 1 (c) on page 219 of the textbook—depreciation is expressed as a percentage.
In such cases, care must be taken to convert the percentage value to a fraction
For example, if depreciation is given as 5 percent, you need to set δ = 5/100 = 0.05

16. Depreciation: graph

17. dynamics

18. Dynamics: what time is it?
We’ll attach a subscript to each variable to denote what date we’re talking about
For example, kt will denote the economy’s per worker stock of capital on date t and kt+1 will denote the per worker stock of capital on date t + 1

19. How does per worker capital change?
A worker has kt units of capital on date t
He or she adds syt units of capital through saving
and loses δkt units of capital through depreciation
So, each worker accumulates kt + syt − δkt units of capital on date t + 1
Does this mean kt+1 = kt + syt − δkt?
Not quite!

20. Population Growth
The Solow-Swan model assumes that each individual has n kids in each period
The kids become adult workers in the period immediately after they are born
and, like every other worker, have n kids of their own
and so on

21. Population Growth
Let the growth rate of any variable x be denoted xg. It is calculated as follows:
Therefore, the growth rate of the number of workers, Lg, is:

22. How does per worker capital change?

23. Dynamics: algebra

24. Dynamics: algebra
Now we are ready for dynamics!

25. Dynamics: algebra A = 10 α = 0.3 δ = 0.1 n = 0.2 k0 = 12 s = t kt
yt = Aktα
syt
δkt 12
1.2 1 2 3 4 5 6 7 8 9 10

26. Dynamics: algebra A = 10 α = 0.3 δ = 0.1 n = 0.2 k0 = 12 s = t kt
yt = Aktα
syt
δkt 12
1.2 1 2 3 4 5 6 7 8 9 10
At this point, you should be able to do problems 1 (d) and 3 (c) on pages 232 and 233 of the textbook. Please give them a try.

27. Dynamics: algebra to graphs
Although this is the basic Solow-Swan dynamic equation, a simple modification will help us analyze the theory graphically.
Now we are ready for graphical analysis.

28. Dynamics: algebra to graphs
This version of the Solow-Swan equation will help us understand the model graphically.

29. Dynamics: algebra to graphs
This version of the Solow-Swan equation will help us understand the model graphically.
2. The economy shrinks if and only if per worker saving and investment [sf(kt)] is less than (δ + n)kt.
3. The economy is at a steady state if and only if per worker saving and investment [sf(kt)] is equal to (δ + n)kt.
1. The economy grows if and only if per worker saving and investment [sf(kt)] exceeds (δ + n)kt.
4. (δ + n)kt is called break-even investment

30. Dynamics: graph

31. Investment and break-even investment
Capital per worker, k
(δ + n)kt
sf(kt) k* k1
Steady state

32. Moving toward the steady state
Investment and depreciation
Capital per worker, k
(δ + n)kt
sf(kt) k*
investment k1
k1
Break-even investment
Steady state

33. Moving toward the steady state
Investment and depreciation
Capital per worker, k
(δ + n)kt
sf(kt) k* k1
k1 k2
Steady state

34. Moving toward the steady state
Investment and depreciation
Capital per worker, k
(δ + n)kt
sf(kt) k*
investment
k2 k1
Break-even investment k2
Steady state

35. Moving toward the steady state
Investment and depreciation
Capital per worker, k
(δ + n)kt
sf(kt) k* k1
k2 k2 k3
Steady state

36. Moving toward the steady state
Investment and depreciation
Capital per worker, k
(δ + n)kt
sf(kt) k*
As long as k < k*, investment will exceed break-even investment, and k will continue to grow toward k* k1
The dynamics are reversed when k1 > k*. k2 k3
Steady state

37. Steady state

38. The steady state: algebra
The economy eventually reaches the steady state
This happens when per worker saving and investment [sf(kt)] is equal to break-even investment [(δ + n)kt].
For the Cobb-Douglas case, this condition is:
At this point, you should be able to do problems 1 (c) and 3 (b) on pages 232 and 233 of the textbook. Please give them a try.

39. Steady State and Transitional Dynamics: algebra10
α =
0.3
δ =
0.1
n =
0.2
k0 = 12
s =
k* = t kt
yt = Aktα
syt
δkt 12
1.2 1 2 3 4 5 6 7 8 9 10

40. Solow-swan predictions for the steady state

41. There is no growth! The Solow-Swan model predicts that
Every economy will end up at the steady-state; in the long run, the growth rate is zero!
That is, k = k* and y = f(k*) = y*
Growth is possible—temporarily!—only if the economy’s per worker stock of capital is less than the steady state per worker stock of capital (k < k*)
If k < k*, the smaller the value of k, the faster the growth of k and y

42. There is no growth!

43. From per worker to total
We have seen that per worker capital, k, is constant in the steady state
Now recall that k = K/L
and that L increases at the rate of n
Therefore, in the steady state, the total stock of capital, K, increases at the rate of n

44. From per worker to total
We have seen that per worker income y = f(k) is constant in the steady state
Now recall that y = Y/L
and that L increases at the rate of n
Therefore, in the steady state, the total income, Y, must also increase at the rate of n
Similarly, although per worker saving and investment, sy, is constant in the steady state, total saving and investment, sY, increases at the rate n

45. From per worker to per capita
Recall that L = amount of labor employed
Suppose P = amount of labor available
This is the labor force
But in the Solow-Swan model everybody is capable of work, even new-born children
So P can also be considered the population
Suppose u = fraction of population that is not engaged in production of the final good.
u is assumed constant
Then L = (1 – u)P or L/P = 1 – u

46. Per worker to per capita

47. Steady state: summary Variable Symbol Steady state behavior
Capital per worker k
Constant
Income per worker
y = f(k)
Saving and investment per worker sy
Consumption per worker
c = (1 – s)y
Labor
L = (1 – u)P
Grows at rate n
Capital K
Income
Y = F(K, L)
Saving and investment sY
Population P
Capital per capita
(1 – u)k
Income per capita
(1 – u)y
Saving and investment per capita
(1 – u)sy
Consumption per capita
(1 – u)c

48. Solow-swan predictions for changes to the steady state

49. A sudden fall in capital per worker
A sudden decrease in k could be caused by:
Earthquake or war that destroys capital but not people
Immigration
A decrease in u
What does the Solow-Swan model say will be the result of this?

50. A sudden fall in capital per worker
Investment and depreciation
Capital per worker, k
(δ + n)kt
sf(kt) k*
At this point, you should be able to do problem 2 on pages 232 and 233 of the textbook. Please give it a try. k1
2. A gradual return to the steady state
1. A sudden decline

51. An increase in the saving rate
Investment and break-even investment k
(δ + n)kt
s2f(k)
s1f(k)
An increase in the saving rate causes a temporary spurt in growth. The economy returns to a steady state. But at the new steady state, per worker capital, output, and saving are all higher. Per worker consumption is a bit trickier.
k1*
k2*

52. An increase in the saving rate
1. An increase in the saving rate raises investment …
2. … causing k to grow (toward a new steady state)
Investment and break-even investment k
(δ + n)kt
s2f(k)
s1f(k)
3. This raises steady-state per worker output y* = f(k*) and saving sy*.
4. The growth rate begins at zero, becomes positive for a while, and eventually returns to zero.
k1*
k2*

53. An increase in the saving rate
1. Recall that the saving rate is a fraction between 0 and 1.
2. What can we say about the steady state levels of k, y, and c when s = 0?
Investment and break-even investment
3. And when s = 1?
(δ + n)kt
4. So, how is consumption per worker, c, affected by changes in s?
f(k)
sf(k) c*
f(k*)
sf(k*) k k*

54. Being the grasshopper is not good!
1. Recall that the saving rate is a fraction between 0 and 1.
2. What can we say about the steady state levels of k, y, and c when s = 0?
Investment and break-even investment
3. They are all zero!
(δ + n)kt
f(k)
sf(k) when s = 0 k
k*= 0

55. Being a miserly any is not good either!
1. Recall that the saving rate is a fraction between 0 and 1.
2. What can we say about the steady state levels of k, y, and c when s = 1?
Investment and break-even investment
(δ + n)kt
f(k)
= sf(k)
c*= 0
sf(k*)
f(k*) k k*

56. Effect of saving on steady state consumption
For the Cobb-Douglas case, it can be shown that the Golden Rule saving rate is equal to capital’s share of all income, which is approximately 30% or 0.30.
Steady state consumption per worker, c* = (1 – s)f(k*)
The US saving rate is well below So, according to the Solow-Swan model, if we save more we will, in the long run, consume more too!
Golden Rule consumption per worker
Golden Rule Saving rate 1
Saving rate, s

57. A decrease in the saving rate
This graph says that c does not fall below initial level when s decreases. But the next graph says that c rises above the initial level when s increases. Neither result is general. For the Cobb-Douglas case, it all depends on where the initial level of s is in relation to α, the income share of capital. Steady state c is maximized at s = α. So, an increase in s would increase steady state c if s < α but decrease it if s > α.

58. An increase in the saving rate

59. What do we get for thrift?
In the long run, a higher rate of saving and investment gives us
A higher per worker income
But not a faster rate of growth
And consumption per worker, c* = (1 – s)y*, may increase or decrease or stay unchanged when s increases.
At this point, you should be able to do problem 4 on page 233 of the textbook. Please try it.

60. Whole lecture in one slide!
Solow-Swan Predictions Grid
kt+1, yt+1, Δkt
k*, y*, i* = sy*
c* = (1 – s)y* kt ? s + A n – δ

61. Effect of saving: evidence

62. Faster population growth
3. This reduces steady-state per worker output y* = f(k*), saving sy*, and consumption (1 – s)y*.
4. The growth rate begins at zero, becomes negative for a while, and eventually returns to zero.
5. The effect is the same if depreciation increases

63. What do we get for having fewer kids?
In the long run, a lower rate of population growth gives us
A higher per worker income
But not a faster rate of growth
At this point, you should be able to do problem 6 on page 233 of the textbook. Please try it.

64. Faster population growth: evidence

65. Alternative perspectives on population growth
The Malthusian Model (1798)
Predicts population growth will outstrip the Earth’s ability to produce food, leading to the impoverishment of humanity.
Since Malthus, world population has increased sixfold, yet living standards are higher than ever.
Malthus neglected the effects of technological progress.

66. Alternative perspectives on population growth
The Kremerian Model (1993)
Posits that population growth contributes to economic growth.
More people = more geniuses, scientists & engineers, so faster technological progress.
Evidence, from very long historical periods:
As world pop. growth rate increased, so did rate of growth in living standards
Historically, regions with larger populations have enjoyed faster growth.
Michael Kremer, “Population Growth and Technological Change: One Million B.S. to 1990,” Quarterly Journal of Economics 108 (August 1993):

67. Alternative perspectives on population growth: Kremer

68. Robert Solow and Trevor Swan