1. Chapter 11: Saving, Capital Accumulation, and Output
More detailed analysis of the last chapter

2. Saving rates in USA

3. Saving rates in EU vs. USA

4. Important question at this point is:
How does saving and output correlate?
Does higher saving lead to higher output?
 Use a model to understand how they correlate

5. Effects of Capital on Output
𝑌 𝑁 =𝐹 𝐾 𝑁 , 𝑁 𝑁 = 𝐹 𝐾 𝑁 ,1
𝑓 𝐾 𝑁 = 𝐹 𝐾 𝑁 ,1
Assume that the following are constant to concentrate on the role of capital alone:
Population
Participation Rate
Unemployment Rate
Technology (for now)

6. Effects of Capital on Output
Adding time subscripts and imposing our assumptions:
𝑌 𝑡 𝑁 =𝑓( 𝐾 𝑡 𝑁 )
Output per worker is a function of capital per worker
Higher capital per worker leads to higher output per worker

7. Effects of output on capital accumulation
We know from chapter 3 that the following holds:
I = S + ( T – G )
What if there is no government sector?
T = G = O (zero)
S = I
S = s*Y where s = savings rate
 It = s*Yt
The higher the savings rate, the higher Investment

8. Investment and capital accumulation
What is the relation between investment and capital accumulation?
Kt+1 = (1-δ) Kt + It
Write it in per worker terms:
𝐾 𝑡+1 𝑁 = 1−𝛿 𝐾 𝑡 𝑁 + 𝑠 𝑌 𝑡 𝑁
Rewrite it a little to have a more convenient form:
𝐾 𝑡+1 𝑁 − 𝐾 𝑡 𝑁 =𝑠 𝑌 𝑡 𝑁 −𝛿 𝐾 𝑡 𝑁
In words: The change in capital stock per worker, is equal to savings per worker and depreciation of capital per worker.

9. Investment and capital accumulation
𝐾 𝑡+1 𝑁 − 𝐾 𝑡 𝑁 =𝑠 𝑌 𝑡 𝑁 −𝛿 𝐾 𝑡 𝑁
What can we say about the relationship between the savings rate, depreciation and growth of capital per worker?
As long as s*Yt > δKt , capital per worker grows
When s*Yt < δKt , capital decreases!
How is this important for output?
Since 𝑌 𝑡 𝑁 =𝑓( 𝐾 𝑡 𝑁 )
 A higher savings rate implies a higher level of output!

10. Investment and capital accumulation
Can you guess what the savings rate would be approximately for the graph above?

11. Investment and capital accumulation
Proportion of the French Capital Stock Destroyed by the End of World War II
Looking back at our graph, what happened to K/N and to Y/N?

12. Steady-state capital and output
Suppose that we look at the very long run (e.g years):
According to the convergence hypothesis, growth of variables will be equal to zero
Drop time subscripts
𝐾 ∗ 𝑁 − 𝐾 ∗ 𝑁 =𝑠 𝑌 𝑁 −𝛿 𝐾 ∗ 𝑁
𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁
Steady state value of capital per worker: savings rate is just sufficient to cover depreciation of capital per worker

13. What about different savings rates for the following equation?
𝐾 𝑡+1 𝑁 − 𝐾 𝑡 𝑁 =𝑠 𝑌 𝑡 𝑁 −𝛿 𝐾 𝑡 𝑁

14. Steady-state capital and output
What about different savings rates for this
𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁
Steady state value of capital per worker: savings rate is just sufficient to cover depreciation of capital per worker

15. 𝑌 ∗ 𝑁 = 𝑓 𝐾 ∗ 𝑁 Steady-state capital and output
We know from the beginning of the chapter that:
𝑌 ∗ 𝑁 = 𝑓 𝐾 ∗ 𝑁
Does anyone have a guess on why there is a gradual increase of output?

16. Steady-state capital and output
What if there is technological progress?
The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress

17. Steady-state capital and output
Question: What is the “optimal” savings rate?
Consider two extremes:
There are no (and never have been savings). So, according to this:
𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁
 Output = 0
2. All income generated is saved.
Consumption = 0
So, what number generates maximum consumption?
A “golden” number between 0 and 1!

18. Steady-state capital and output
Figure The Effects of the Saving Rate on Steady-State Consumption per Worker

19. Small detour: Social Security
Social security: Pension to retired workers.
- Cost: As of 2013, about 8% of GDP ≈ $1.2 trillion
- Keeps 20% of all Americans, age 65 or older, above the Federally defined poverty level
Two ways to fund:
Fully funded system: Workers pay today for future benefits, these funds will be invested and paid back as “pensions”.
Pay-as-you-go system: Workers pay today for current retirees’ pensions and next generation will pay for today’s workers’ pensions.
 Current system

20. Small detour: Social Security
Fully funded system: Workers pay today for future benefits, these funds will be invested and paid back as “pensions”.
Private savings↓ Why?
Ask yourself: if you knew that the state will pay you a pension once you retire, will you save less or more today?
(On average, people save less, hence, private savings go down)
What about a switch to this system?
Current workers would have to for themselves and for the current retirees.
 Double cost on current workers which is (especially now) not easy to implement

21. Small detour: Social Security
2. Pay-as-you-go system: Workers pay today for current retirees’ pensions and next generation will pay for today’s workers’ pensions.
Private savings↓ Why? Same argument as last slide.
But more problematic:
Population ages quickly but population does not increase to catch up
Either future retirees have to take less pensions (ask yourself, would you like to have that?)
Or current workers have to pay more today (ask yourself again, would you like to pay more today?)
Problem of social security……..

22. Figure 11-7B 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 (cont.)
Assignment:
Replicate these two figures using Excel.
Due date is next Friday
 Gain five points on your highest midterm!

23. Steady-state consumption and savings
But can we get this relationship between savings and consumption in a formal way?
Consider this:
We learned that in the steady-state the following holds:
𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁
There is no consumption! Any idea on how to get consumption?

24. Steady-state consumption and savings
Remember that (without a government):
Y = C + S (all income is either consumed or saved)
And 𝑌 𝑁 – 𝐶 𝑁 = 𝑆 𝑁 = 𝑠𝑓 𝐾 ∗ 𝑁
Hence, 𝑌 𝑁 – 𝐶 𝑁 = 𝛿 𝐾 ∗ 𝑁
So, 𝐶 𝑁 = 𝑌 𝑁 − 𝛿 𝐾 ∗ 𝑁

25. Steady-state consumption and savings
Remember that (given production function Y=(K*N)^0.5:
Y N = K N = s 2∗0.5 δ 2∗0.5 = s δ
C N = Y N − δ K ∗ N
C N = s δ − δ s 2 δ 2 = s(1−s) δ
Another way of writing consumption:
C N = Y N - s 𝑌 N = (1-s)* Y N = (1-s)* s δ = s(1−s) δ
Question: Which savings rate maximizes consumption per worker?
same

26. Steady-state consumption and savings
Max consumption at s = 0.5; after that, consumption goes down!

27. Understanding the bigger picture
You might be lost amid all the equations in regards to the “big picture”. What we should get out of this chapter is:
We now (hopefully) understand that:
Savings lead to capital accumulation
Capital accumulation leads to higher output
Higher output leads (ceteris paribus) to higher standard of living

28. Understanding the bigger picture
Sources of Mechanical Drive in American Manufacturing
Thousand Horsepower
Year
Steam Engines
Steam Turbines
Internal Combustion Engines
Water Wheels and Turbines
Electric Motors
Total
 Horsepower per Manufacturing Worker
1869
1216
1130
2346
 1.04
1879
2186
1225
3411
 1.10
1889
4581 9
1242 16
5848
 1.25
1899
8022
120
1236
475
9853
 1.57
1909
12026 90
592
1273
4582
18563
 2.25
1919
11491
465
856
970
15612
29394
 2.68
1929
6857
1112
722
623
33844
43158
 3.90
1939
4216
1736
866
394
44827
52039
 4.35
1948
86095
 6.60
1953
105007
 7.18
Source: Brad DeLong “Slouching Towards Utopia”