2. Saving, Capital Accumulation, and Output
The effects of the saving rate - the ratio of saving to GDP – on capital and output per capita are the topics of this chapter.
An increase in the saving rate would lead to higher growth for some time, and eventually to a higher standard of living in the United States.

3. 11-1 Interactions between Output and Capital
At the center of the determination of output in the long run are two relations between output and capital:
The amount of capital determines the amount of output being produced.
The amount of output determines the amount of saving and, in turn, the amount of capital accumulated over time.

4. 11-1 Interactions between Output and Capital
Capital, Output, and Saving/Investment
Figure

5. 11-1 Interactions between Output and Capital
The Effects of Capital on Output
Under constant returns to scale, we can write the relation between output and capital per worker as follows:

6. 11-1 Interactions between Output and Capital
The Effects of Capital on Output
Since the focus here is on the role of capital accumulation, we make the following assumptions:
The size of the population, the participation rate, and the unemployment rate are all constant.
There is no technological progress.

7. 11-1 Interactions between Output and Capital
The Effects of Capital on Output
With these two assumptions, our first relation between output and capital per worker, from the production side, can be written as
In words, higher capital per worker leads to higher output per worker.

8. 11-1 Interactions between Output and Capital
The Effects of Output on Capital Accumulation
To derive the second relation, between output and capital accumulation, we proceed in two steps:
We derive the relation between output and investment.
We derive the relation between investment and capital accumulation.

9. 11-1 Interactions between Output and Capital
The Effects of Output on Capital Accumulation
Output and Investment
We make three assumptions to derive the relation between output and investment:
We assume the economy is closed.
We assume public saving, T – G, is equal to zero.
We assume that private saving is proportional to income, so
Combining these two relations gives:

10. Output, consumption, and investment
Output per worker, Y/N
Capital per worker, K/N
f(K/N)
Y/N1
K/N1 c1
sf(K/N) i1
CHAPTER 7 Economic Growth I

11. Depreciation  = the rate of depreciation
Depreciation per worker, K/N
Capital per worker, k
 = the rate of depreciation
= the fraction of the capital stock that wears out each period
 K/N  1
CHAPTER 7 Economic Growth I

12. 11-1 Interactions between Output and Capital
The Effects of Output on Capital Accumulation
Investment and Capital Accumulation
The evolution of the capital stock is given by:
 denotes the rate of depreciation
Combining the relation from output to investment, and the relation from investment to capital accumulation, we obtain the second important relation we want to express, from output to capital accumulation:

13. 11-1 Interactions between Output and Capital
The Effects of Output on Capital Accumulation
Investment and Capital Accumulation
Output and Capital per Worker:
Rearranging terms in the equation above, we can articulate the change in capital per worker over time:
In words, the change in the capital stock per worker (left side) is equal to saving per worker minus depreciation (right side).

14. 11-2 The Implications of Alternative Saving Rates
Our two main relations are:
First relation: Capital determines output.
Second relation: Output determines capital accumulation
Combining the two relations, we can study the behavior of output and capital over time.

15. 11-2 The Implications of Alternative Saving Rates
Dynamics of Capital and Output
From our main relations above, we express output per worker (Yt/N) in terms of capital per worker to derive the equation below: =
Investment during year t _
Depreciation during year t
Change in capital from year t to year t + 1

16. 11-2 The Implications of AlternativeSaving Rates
Dynamics of Capital and Output =
Investment during year t _
Depreciation during year t
Change in capital from year t to year t + 1
This relation describes what happens to capital per worker. The change in capital per worker from this year to next year depends on the difference between two terms:
If investment per worker 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.

17. 11-2 The Implications of Alternative Saving Rates
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.

18. 11-2 The Implications of Alternative Saving Rates
Dynamics of Capital and Output
At K*/N, output per worker and capital per worker remain constant at their long-run equilibrium levels.
Investment per worker increases with capital per worker, but by less and less as capital per worker increases.
Depreciation per worker increases in proportion to capital per worker.

19. 11-2 The Implications of Alternative Saving Rates
Dynamics of Capital and Output
Capital and Output Dynamics
Figure
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.

20. 11-2 The Implications of Alternative Saving Rates
Steady-State Capital and Output
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:
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:

21. Inferences in steady state
如果储蓄率s、劳动增长率n都为常数，则人均实际收入（产出）在长期内无法增长，也就是，在这些条件下生活水平（人均实际收入）长期得不到提高。
从长期看，当经济处于稳态(y*=f(k*)),所有实际总量会按劳动力增长率增加。
K=k*N,由于k*=常数，N按n增长，K也按n增长
Y=y*N=f(k*)N, Y也按n增长。
同理，I=sY=sf(k*)N, C=(1-s)f(k*)N,也按n增长。
故长期内，如果储蓄率、劳动增长率和全要素（技术）增长率为常数，各个经济总量的增长率取决于劳动增长率。(Solow growth model is an exogenous model.)

22. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Output
Three observations about the effects of the saving rate on the growth rate of output per worker are:
The saving rate has no effect on the long run growth rate of output per worker, which is equal to zero.
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.
An increase in the saving rate will lead to higher growth of output per worker for some time, but not forever.

23. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Output
The Effects of Different Saving Rates
Figure
A country with a higher saving rate achieves a higher steady- state level of output per worker.

24. Prediction:
Higher s  higher k*. A new steady state. And since y = f(k) , higher k*  higher y* .
Thus, the Solow model predicts that
More higher rates of saving , higher levels of investment, capital, and income per worker in the long run.
The growth rate of total variables don’t change.
But it takes times to change from one steady state to another new steady state.
After showing this slide, you might also note that the converse is true, as well: a fall in s (caused, for example, by tax cuts or government spending increases) leads ultimately to a lower standard of living.
In the static model of Chapter 3, we learned that a fiscal expansion crowds out investment. The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment. If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labor productivity, and income per capita to fall toward a new, lower steady state. (If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)
This, of course, is relevant because actual U.S. public saving has fallen sharply since 2001.

25. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Output
The Effects of an Increase in the Saving Rate on Output per Worker
Figure
An increase in the saving rate leads to a period of higher growth until output reaches its new, higher steady-state level.

26. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Output
The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress
Figure
An increase in the saving rate leads to a period of higher growth until output reaches a new, higher path.

27. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Consumption
An increase in the saving rate always leads to an increase in the level of output per worker. But output is not the same as consumption.
Different values of s lead to different steady states. How do we know which is the “best” steady state?
The “best” steady state has the highest possible 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), which lowers c*.
So, how do we find the s and k* that maximize c*?

28. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Consumption
The level of capital associated with the value of the saving rate that yields the highest level of consumption in steady state is known as the golden-rule level of capital.

29. 11-2 The Implications of Alternative Saving Rates
The Saving Rate and Consumption
The Effects of the Saving Rate on Steady- State Consumption per Worker
Figure
An increase in the saving rate leads to an increase and then to a decrease in steady-state consumption per worker.

30. The Golden Rule capital stock
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 the steady state: i* = k* because k = 0.
CHAPTER 7 Economic Growth I

31. The Golden Rule capital stock
steady state output and depreciation
steady-state capital per worker, k*
 k*
Then, graph f(k*) and k*, look for the point where the gap between them is biggest.
f(k*)
Students sometimes confuse this graph with the other Solow model diagram, as the curves look similar.
Be sure to clarify the differences:
On this graph, the horizontal axis measures k*, not k. Thus, once we have found k* using the other graph, we plot that k* on this graph to see where the economy’s steady state is in relation to the golden rule capital stock.
On this graph, the curve measures f(k*), not sf(k).
On the other diagram, the intersection of the two curves determines k*. On this graph, the only thing determined by the intersection of the two curves is the level of capital where c*=0, and we certainly wouldn’t want to be there.
There are no dynamics in this graph, as we are in a steady state. In the other graph, the gap between the two curves determines the change in capital.
CHAPTER 7 Economic Growth I

32. The Golden Rule capital stock
c* = f(k*)  k* is biggest where the slope of the production function equals the slope of the depreciation line:
 k*
f(k*)
If your students have had a semester of calculus, you can show them that deriving the condition MPK =  is straight-forward:
The problem is to find the value of k* that maximizes c* = f(k*)  k*.
Just take the first derivative of that expression and set equal to zero:
f(k*)   = 0
where f(k*) = MPK = slope of production function and  = slope of steady-state investment line.
MPK = 
steady-state capital per worker, k*
稳态下的黄金律人均资本不会自动达到，需要一个特殊的
储蓄率（黄金律储蓄率）支持
CHAPTER 7 Economic Growth I

33. Population growth
Assume that the population (and labor force) grow at rate n. (n is exogenous.)
EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02).
Then L = n L = 0.02  1,000 = 20, so L = 1,020 in year 2.

34. Break-even investment
( + n)k = break-even investment盈亏平衡投资, 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.)

35. 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
Of course, “actual investment” and “break-even investment” here are in “per worker” magnitudes.

36. The Solow model diagram
k = s f(k)  ( +n)k
Investment, break-even investment
Capital per worker, k
( + n ) k
sf(k) k*

37. The impact of population growth
Investment, break-even investment
( +n2) k
( +n1) k
An increase in n causes an increase in break-even investment,
sf(k)
k2*
leading to a lower steady-state level of k.
k1*
Capital per worker, k
CHAPTER 7 Economic Growth I

38. 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.
This and the preceding slide establish an implication of the model.
The following slide confronts this implication with data.

39. International evidence on population growth and income per person
100,000
per Person
in 2000
(log scale)
10,000
1,000
Figure 7-13, p Number of countries = 96. Source: Penn World Table version 6.1.
The model predicts that faster population growth should be associated with a lower long-run income per capital The data is consistent with this prediction.
So far, we’ve now learned two things a poor country can do to raise its standard of living: increase national saving (perhaps by reducing its budget deficit) and reduce population growth.
100 1 2 3 4 5
Population Growth
(percent per year; average )

40. The Golden Rule with population growth
To find the Golden Rule capital stock, express c* in terms of k*:
c* = y*  i*
= f (k* )  ( + n) k*
c* is maximized when MPK =  + n
or equivalently, MPK   = n
In the Golden Rule steady state, the marginal product of capital net of depreciation equals the population growth rate.
CHAPTER 7 Economic Growth I

41. 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 omitted the effects of technological progress.
This and the next slide cover new material in the 6th edition. They can be omitted without loss of continuity.

42. 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):

43. 11-3 Getting a Sense of Magnitudes
Assume the production function is:
Output per worker is:
Output per worker, as it relates to capital per worker is:
Given our second relation,
Then,

44. 11-3 Getting a Sense of Magnitudes
The Effects of the Saving Rate on Steady-State Output
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.

45. 11-3 Getting a Sense of Magnitudes
The Dynamic Effects of an Increase in the Saving Rate
The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker
Figure
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.

46. 11-3 Getting a Sense of Magnitudes
The U.S. Saving Rate and the Golden Rule
In steady state, consumption per worker is equal to output per worker minus depreciation per worker.
Knowing that:
and
then:
These equations are used to derive Table 11-1 in the next slide.

47. 11-3 Getting a Sense of Magnitudes
The U.S. Saving Rate and the Golden Rule
Table 11-1
The Saving Rate and the Steady-state Levels of Capital, Output, and Consumption per Worker
Saving Rate, s
Capital per Worker, (K/N)
Output per Worker, (Y/N)
Consumption per Worker, (C/N)
0.0
0.1
1.0
0.9
0.2
4.0
2.0
1.6
0.3
9.0
3.0
2.1
0.4
16.0
2.4
0.5
25.0
5.0
2.5
0.6
36.0
6.0 –
100.0
10.0

48. 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.

49. 11-4 Physical versus Human Capital
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:
An increase in capital per worker or the average skill of workers leads to an increase in output per worker.

50. 11-4 Physical versus Human Capital
Extending the Production Function
A measure of human capital may be constructed as follows: Suppose an economy has 100 workers, half of them unskilled and half of them skilled. The relative wage of skilled workers is twice that of unskilled workers.
Then:

51. 11-4 Physical versus Human Capital
Human Capital, Physical Capital, and Output
An increase in how much society “saves” in the form of human capital—through education and on-the-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.

52. 11-4 Physical versus Human Capital
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.
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.
Output per worker depends on the level of both physical capital per worker and human capital per worker.
Is technological progress unrelated to the level of human capital in the economy? Can’t a better-educated labor force lead to a higher rate of technological progress? These questions take us to the topic of the next chapter: the sources and the effects of technological progress.

53. Key Terms saving rate steady state golden-rule level of capital
fully funded system
pay-as-you-go system
trust fund
human capital
models of endogenous growth