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
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)

3. 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

4. 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

5. 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

6. 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

7. 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 >

8. 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

9. 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

10. 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

11. 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

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

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

14. 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

15. 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:

16. 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%