Transcript Lecture 2: Savings, Capital Accumulation and Output

Lecture 2: Savings, Capital
Accumulation and Output
• Key Issues on Savings and Growth
1. Understanding the effects of the savings rate on capital
and output per capita
2. Key findings are that an increase in the saving rate leads
to:
• higher growth in output for some time,
• a higher standard of living
• no permanent increase in the rate of growth
3. Understanding the “Golden Rule” which indicates the
savings rate (and associated level of capital) that maximises
consumption in the steady state
Structure of the Lecture
1. Interactions between output and capital
2. Understanding the long-run equilibrium (or
steady state)
3. Implications of alternative savings rates
4. Implications for consumption and the
“golden rule”
5. Getting a sense of magnitudes
6. Physical Capital vs. Human Capital
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 (or a two-way causality running
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
Interactions between Output and Capital
Figure 11 - 1
Capital, Output, and
Saving/Investment
First Causality - Capital’s effect on output
• Higher capital per worker leads to higher
Yt
output per worker
 Kt 
 f 
 N
N
• Since the focus 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
Second Causality - Output’s effect on Capital
• To derive the second relation, between output
and capital accumulation, we proceed in two
steps:
– Step 1: We derive the relation between output
and investment (Inv fn of Y)
– Step 2: We derive the relation between
investment and capital accumulation (K fn of Inv)
Step 1: Output and Investment
• Three assumptions are made to derive the relation between output and
investment:
– We assume the economy is closed:
– Instead of I + G + X = S + T + M, we exclude the external sector (X and M),
therefore: I + G = S + T , or I = S + (T – G)
– We assume public saving is equal to zero i.e. T – G = 0, therefore: I = S
– We assume that private saving is proportional to income, so: S = sY,
where s is the savings rate between 0 and 1
– Combining the two relations gives: I = sY (that is investment is
proportional to output i.e. higher output implies higher investment (and
higher savings)
Step 2: Investment and Capital accumulation
• The evolution of the capital stock is given by (where
∂ denotes the rate of depreciation):
Kt 1  (1   ) Kt  I t
• Note: Investment (I) is a flow which contributes to
capital (K) a stock
• Combining the relation I=sY and the relation from
investment to capital accumulation, the expression
from output to capital accumulation is given (per
worker) as:
Kt 1
Kt
Yt
 (1   )
s
N
N
N
Investment and Capital accumulation
Kt 1
Kt
Yt
 (1   )
s
N
N
N
• Capital per worker at the beginning of the year t+1 is equal to
capital per worker at the beginning of year t, adjusted for
depreciation, plus investment per worker during year t (which
is equal to the savings rate times output per worker during
year t)
• Rearranging terms, we can express the equation as follows:
Kt 1 Kt
Yt
Kt

 s 
N
N
N
N
• That is, the change in the capital stock per worker (left side)
is equal to savings per worker minus depreciation (right side)
2. Understanding long-run equilibrium
or steady state
• Our two main relations are:
Yt
 Kt 
 f 
 N
N
 First relation:
Capital determines
output
(Y on left hand side)
Kt 1 Kt
Yt
Kt

 s 
N
N
N
N
 Second relation:
Output determines
capital accumulation
(K on left hand side)
• Combining the two relations (substituting the
first into the second), we can study the
behavior of output and capital over time (or
the dynamics of capital and output)
Capital formation, savings and depreciation
K t 1 K t

N
N
Change in capital
from year t to year t + 1
=
 Kt 
sf  
N 
=
Investment
during year t
_
_
Kt

N
Depreciation
during year t
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
 Key drivers of capital accumulation are:
 The rate of savings (s) which is positively associated with the capital
accumulation of capital stock
 The rate of depreciation (or the proportion of capital stock that breaks down or
becomes useless) which is negatively associated with the capital accumulation
of capital stock
Capital formation, savings and depreciation
In Fig 11.2 output per
worker is on the
vertical axis and capital
per worker on the
horizontal axis
Blue – f(K/N)
Green - sf(K/N)
Purple – ∂(K/N)
Change in capital = CD
(as investment >
depreciation)
For any particular
savings rate (s) capital
formation will tend
toward a steady state
where investment =
depreciation
Notes on Fig 11.2
 Output per worker f(K/N) increases with capital per worker,
but by less and less as capital per worker increases due to
decreasing returns to capital
 Investment per worker sf(K/N) has the same shape as the
output per worker relation but is lower by factor s ( between
0 and 1) (the savings rate)
 Depreciation per worker ∂K/N increases in proportion to
capital per worker (a straight line with slope ∂)
 At K*/N, output per worker and capital per worker remain
constant at their long-run equilibrium levels
 When capital and output are high (above K*/N), investment is less
than depreciation, and capital decreases
Notes on Fig 11.2
 AB gives output per worker at the initial level of capital per worker
(K0/N)
 AC gives the level of investment per worker at the initial level of
capital per worker (K0/N) (due to s)
 AD gives the level of depreciation per worker at the initial level of
capital per worker (K0/N) (due to ∂)
 As CD = AC –AD and CD>0 therefore investment per worker
exceeds depreciation per worker so capital per worker
increases
 Over time output per worker and capital per worker increases
due to the process of capital accumulation until a point of
long-run equilibrium (or steady state) is reached at K*/N
where output per worker and capital per worker remain
constant at Y*/N and K*/N (where savings and related
investment equals the rate of depreciation)
Steady state capital and output
Kt 1 Kt
K
K 

= sf  t    t
 N
N
N
N
•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:
K*
 K *
sf 
 
N
 N 
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 (where the * is
indicative of a steady state):
Y*

N
 K *
f

N


3. The Implications of Alternative Saving Rates
• Three observations about the effects of the saving rate on the growth rate
of output per worker are:
1. The saving rate has no effect on the long run growth rate of output per
worker, which is equal to zero (in the absence of technological progress).
1.
2.
As the economy converges on a steady state of output per worker, the long run growth rate of output is equal to
zero no matter what the savings rate.
Furthermore, the economy would have to save a greater and greater portion of its output to due to decreasing
returns to capital. (Krugman’s “Stalinist growth” – increasing state-controlled savings rate)
2.
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 (and a higher standard of living) in the long
run (Fig 11.3 Y1/N > Y0/N).
3.
An increase in the saving rate will lead to higher growth of output per
worker for some time, but not forever (Fig 11.4 (without technological
progress) and 11.5 (with technological progress)
The Implications of Alternative Saving Rates
Figure 11 - 3
The Effects of Different
Saving Rates
A country with a higher
saving rate achieves a
higher steady state level of
output per worker.
The Implications of Alternative Saving Rates
Figure 11 - 4
The Effects of an Increase in
the Saving Rate on Output
per Worker
An increase in the saving rate
leads to a period of higher
growth until output reaches its
new, higher steady-state level
(and growth returns to 0,
indicated by a 0 slope).
The Implications of Alternative Saving Rates
Figure 11 - 5
The Effects of an Increase in
the Saving Rate on Output
per Worker in an Economy
with Technological Progress
Where there is technological
progress the economy will have
positive growth of output per
worker even in the long run.
Therefore, an increase in the
saving rate leads to a period of
higher growth until output
reaches a new path where the
rate of growth is equal to the
initial growth rate as AA has the
same slope as BB)
4. Implications for consumption and the “golden rule”
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 as output is either consumed or
saved i.e. Y = C + S.
Consider what happens for two extreme values of the saving rate:

An economy in which the saving rate is (and has always been) 0 is an economy in
which capital is equal to zero. In this case, output is also equal to zero, and so is
consumption. A saving rate equal to zero implies zero consumption in the long
run, as both capital and labour are needed for production via the production
function).

Now consider an economy in which the saving rate is equal to one (100%): People
save all their income. The level of capital, and thus output, in this economy will be
very high. But because people save all their income, consumption is equal to zero.
A saving rate equal to one also implies zero consumption in the long run.

The level of the saving rate (not 0 or 1) that yields the highest level of
consumption in steady state is known as the “golden-rule” level of capital.
Implications for consumption (Fig 11.6)
• The golden rule savings rate is given as SG where consumption per worker is
maximised (in Fig 11.6)
•Increases in the savings rate (when below the “golden rule” level) results in
lower consumption initially and increased consumption in the long run (or
increased steady state consumption)
• Increases in the savings rate (when above the “golden rule” level) decreases
consumption not only initially but also in the long run (steady state)
• Above the golden rule level the increase in capital due to the increased
savings rate leads to only a small increase in output (due to negative return
to factors) which is too small to cover the increased depreciation. The
economy carries too much capital.
Implications for consumption
Figure 11 - 6
The Effects of the Saving
Rate on Steady- State
Consumption per Worker
An increase in the saving rate
leads to an increase and then
to a decrease in steady-state
consumption per worker.
Policy issues
• In practice, most economies operate below the golden rule SG
• The public policy challenge is to attempt to try and get savings up
and closer to SG
• The trade-off: an increase in the savings rate toward SG will result in
lower consumption now but higher consumption later
• This raises inter-generational issues i.e. current generations will
reduce consumption and future generations will gain (due to the
larger capital accumulation)
• Politically, future generations do not vote!
• In SA the debate has been about the appropriateness of budget
surplus which contributes to savings
• In the US about the funding of the Social Security systems and how
the impacts on US savings
US Case: Social Security and Savings
 One way to run a social security system is the fully-funded system,
where workers are taxed, their contributions invested in financial
assets, and when workers retire, they receive the principal plus the
interest payments on their investments. This contributes to savings –
even though govt savings grow in composition, but there is no
negative effect on capital accumulation e.g. government pension
funds invest in capital projects based on returns for future retirees).
 Another way to run a social security system is the pay-as-you-go
system, where the taxes that workers pay are the benefits that current
retirees receive. This results in reduced saving and reduced capital
accumulation.
• In anticipation of demographic changes, the Social Security tax rate
has seen increases, and contributions are now larger than benefits,
leading to the accumulation of a Social Security trust fund. Bt this
is not enough as the trust fund is far smaller than the value of the
benefits promised to future retirees. As each year taxes are used to
pay these benefits the US system is basically a pay-as-you-go
system contributing to loser savings rates.
5. Getting a Sense of Magnitudes
2
Y*
K*
s
s

   
N
N
  
•Based on the workings at section 11.3, the 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.
• Numerical examples:
–If the savings rate and depreciation rate are both equal to 10% per year. Then both Y*/N and
K*/N =1
–If the savings rate doubles to 20% (and there is no change to the depreciation rate) then
Y*/N doubles to 2 and K*/N rises from 1 to 4.
• Conclusion: an increase in the savings rate leads to an increase in the
steady state level of output (But how long does it take for output to reach its
new steady state level? – See Fig 11.7)
Dynamic effects
Figure 11 - 7
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
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.
Dynamic effects
• In Fig. 11.7 (a) after an increase in saving from 10% to
20% (with no change in depreciation), the level of output
per worker (Y/N) increases from 1 to 2 over a number of
years, each year by a smaller amount (computed year
after year using equations 11.8 and 11.7)
• In Fig. 11.7(b) the growth rate of output per worker (Y/N)
is plotted, with a high initial growth rate returning to zero
over time (hence the conclusion that a change in the
saving rate will not lead to a permanent change to the
growth rate of output – even though it does effect the
growth rate for quite some time)
Magnitudes and the “golden rule”
•In steady state, consumption per worker is equal to what is left after
enough is put aside to maintain constant level of capital, i.e. output per
worker minus depreciation per worker.
C Y
K


N N
N
Knowing that:
K*  s
 
N  
2
2
and
Y*
K*
s
s

   
N
N
  
C s
s1  s
 s
    
 
N 

2
then:
These equations are used to derive Table 11-1 in the next slide, comparing the
effect of different rates of saving on consumption given a constant rate of
depreciation equal to 10%.
Magnitudes and the “golden rule”
Table 11-1
Saving Rate, s
The Saving Rate and the Steady-state Levels of
Capital, Output, and Consumption per Worker
Capital per
Worker, (K/N)
Output per
Worker, (Y/N)
Consumption per
Worker, (C/N)
0.0
0.0
0.0
0.0
0.1
1.0
1.0
0.9
0.2
4.0
2.0
1.6
0.3
9.0
3.0
2.1
0.4
16.0
4.0
2.4
0.5
25.0
5.0
2.5
0.6
36.0
6.0
2.4
–
–
–
100.0
10.0
0.0
–
1.0
Magnitudes and the “golden rule”
• In Table 11.1 steady state consumption per worker is
largest when s = 0,5 i.e. the “golden rule” is
associated with saving rate of 50%
• Below 50% savings - increases in savings lead to
increases in long run steady state consumption
• Above 50% savings – increases in savings leads to
lower levels of long run steady state consumption
• In reality, US Savings are around 17% of GDP, SA
around 16% of GDP, China around 30% of GDP
6. 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.
•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:
Y
 K H
 f , 
 N N
N
( ,  )
•An increase in capital per worker or the average skill of workers leads to an
increase in output per worker.
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:
H 150
H  [(50  1)  (50  2)]  150 

 15
.
N 100
•The Major implication is that in the long run, output per worker
depends not only on how much society saves but also how much it
spends on education.
Endogenous Growth
• The emphasis on human capital has motivated a new generation of growth
models – called endogenous growth models (such as those of Robert Lucas and
Paul Romer)
• Such endogenous growth models allow for steady growth even without
technological progress, where growth depends on variables such as the saving
rate and the rate of spending on education.
• In these models steady state growth – in the form of increases in output per
worker - can be generated by increases both of physical capital per worker and
human capital per worker. Therefore, both the savings rate and the fraction of
output spent on education and training become key policy variables.
•Next week – we will focus on growth models that allow for technological
progress (and there is always the possibility that improved human capital itself is a
key driver of technological progress)