1. Saving, Capital Accumulation, and Output The Long Run
Observation:
The savings rate since 1950
U.S % Germany 24.6% Japan 33.7%
What do you think…
Would increasing the U.S. savings rate lead to sustained higher U.S. growth in the future?
…NO!

2. Interactions between Output and Capital
Two long-run relations between output and capital
The amount of capital determines the amount of output being produced
The amount of output determines the amount of savings and investment and, thus, the amount of capital

3. Interactions between Output and Capital
Capital, Output, and Saving/Investment
The amount of capital (K)  amount of output (Y)
The amount of output (Y)  the amount of savings (S) & investment (I = S when G-T=0)  amount of capital (K)

4. Interactions between Output and Capital
When
employment (N) is constant and there is no technological progress (f is constant)
I = S in closed economy where G – T = 0
S = sY: Private saving is proportional to income s = Savings rate (between 0 & 1)
Investment and Capital Accumulation:
Kt+1 = (1-d) Kt + It
Where d = depreciation rate

5. Interactions between Output and Capital
Investment and Capital Accumulation
Per worker output and capital accumulation
Capital/worker in t+1 = Capital/Worker in t, adjusted for depreciation and investment Investment/worker = Savings rate x Output/worker in t

6. Interactions between Output and Capital
The Effects of Output on Capital Accumulation
Investment and Capital Accumulation
The relation between output and capital accumulation
Reorganizing:
Savings/worker - depreciation
Change in

7. Implications of Alternative Saving Rates
Dynamics of Capital and Output: The change in Kapital/worker from t to t+1 depends on Investment/Worker – Depreciation/Worker -
Change in capital from year t to year t+1 =
Invest- ment during year t
depreciation during year t

8. Implications of Alternative Saving Rates
Dynamics of Capital and Output Graphically
Output per worker
f(Kt/N)
Depreciation per worker
Kt/N
Investment per worker
sf(Kt/N)
Y*/N A B
Output per worker, Y/N C
AB = Output/worker
AC = Investment/worker D
AD = Depreciation
AC > AD
(Ko/N)
K*/N
Capital per worker, K/N

9. Implications of Alternative Saving Rates
Steady-State Capital and Output
Steady-State Value of Capital/Worker:
Investment just offsets depreciation
Steady-State Value of Output/Worker

10. Implications of Alternative Saving Rates
The Saving Rate and Output
What are the effects of the saving rate on the rate of output per worker?
- The saving rate has NO effect on the long run growth rate of output/worker
Growth rate = ZERO in long run!
- An increase in the saving rate leads to a higher growth of output/worker for some time, but not forever.
- The saving rate determines the level of output/worker in the long run

11. Implications of Alternative Saving Rates
The Effects of Different Saving Rate
Depreciation per worker
Kt/N
Output per worker, Y/N
Capital per worker, K/N
Investment
s1f(Kt/N)
K1/N
Investment
s0f(Kt/N)
(K0/N)

12. Implications of Alternative Saving Rates
The Effects of Different Saving Rate
Investment
s0f(Kt/N)
s1f(Kt/N)
Output per worker, Y/N
Capital per worker, K/N
Depreciation per worker
Kt/N
Output per worker
f(Kt/N) D
Y1/N
K1/N B A
(K0/N)
Y0/N C
I >

13. Implications of Alternative Saving Rates
The Effects of Different Saving Rate
Output per worker, Y/N
Time
(No technological progress)
Associated with saving rate s1 > s0
Y1/N
Y0/N
Associated with saving rate s0 t

14. Implications of Alternative Saving Rates
The Effects of Different Saving Rate
(Technological progress)
Output per worker, Y/N
(log scale)
Time
Associated
with saving
rate s1 > s0
Associated with saving rate s0 t

15. Implications of Alternative Saving Rates
The Savings Rate and the Golden Rule
Does an increase in saving lead to an increase in consumption in the long run?
Two Scenarios:
Saving Rate = 0
Capital = 0
Output = 0
Consumption = 0
Saving Rate = 1
Consumption = 0
Output replaces depreciation

16. Implications of Alternative Saving Rates
Maximum steady state
Consumption per worker:
At Golden Rule Level of Capital
Consumption per worker, C/N
Saving rate, s sG 1

17. Getting a Sense of Magnitudes
Assume:
(Constant return to scale and decreasing returns to either capital or labor)

18. Getting a Sense of Magnitudes
Then
In steady-state is constant and the left side = 0 and:

19. Getting a Sense of Magnitudes
The Effects of the Saving Rate on Steady-State Output
Steady-State Output/Worker:
Higher saving rate and lower depreciation both lead to higher and in the long run.
Double s  Quadruple K/N and double Y/N

20. Getting a Sense of Magnitudes
The Dynamic Effects of an Increase in the Saving Rate

21. Getting a Sense of Magnitudes
The Dynamic Effects of an Increase in the Saving Rate

22. Getting a Sense of Magnitudes
The U.S. Saving Rate and the Golden Rule
What saving rate that would maximize steady-state consumption?
In Steady-State:

23. Getting a Sense of Magnitudes
The Saving Rate and the Steady-State Levels of Capital, Output, and Consumption per Worker
Capital Output Consumption Saving Rate, per Worker, per Worker, per Worker, s K/N Y/N C/N

24. Getting a Sense of Magnitudes
The U.S. Saving Rate and the Golden Rule
Observation
If s < .50: increasing s will increase long-run consumption
In the U.S., s < 20%

25. Physical Versus Human Capital
Extending the Production Function
Measuring the Impact of Human Capital:
Education and On-the-Job Training