Saving, Capital Accumulation,
and Output
Interactions Between Output and Capital

Two important relations in the long
run are:


The amount of capital determines the
amount of output being produced.
The amount of output determines the
amount of saving (=investment), and
so the amount of capital being
accumulated.
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:
 K
K 
f    F  ,1
 N
N 
The Effects of Capital on Output

For now, we make the following assumptions:
1.
2.

The size of the population, the participation rate, and the
unemployment rate are all constant.
There is no technological progress.
Under these assumptions, the first important relation
we want to express is between output and capital per
Yt
 Kt 
worker:
 f 
N
 N
Which means that, at each period, higher capital per worker
leads to higher output per worker.
Effects of Output on Capital Accumulation
Output and Investment:

The equations below describe the relation between
private saving and investment:
S  sY
0 s 1
 Private saving is equal to investment, and proportional to
income.
I t  sYt
Effects of Output on Capital Accumulation
Investment and Capital Accumulation:

The evolution of the capital stock is given by:
Kt + 1 = ( 1- δ )Kt + It
Effects of Output on Capital Accumulation

Rearranging terms in the equations above,
we can articulate the change in capital per
worker over time:
Kt + 1
N
-
Kt
N
= s
Yt
N
- δ
Kt
N
In words, the change in the capital stock per worker (left side) is
equal to saving per worker minus depreciation (right side).
Implications of Alternative Saving Rates

Our two main relations are:
Yt  Kt 
 f 
N  N
First relation:
Capital determines
output.
Kt 1 Kt
Yt
Kt

 s 
N
N
N
N
Second relation:
Output determines capital
accumulation
Dynamics of Capital and Output
Kt 1 Kt

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

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.
Dynamics of Capital and Output
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.
Dynamics of Capital and Output
Dynamics of Capital and Output
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.
 Y *
 K *
   f 
 N
 N
Three observations about the effects of
the saving rate:
1.
The saving rate has no effect on the long run
growth rate of output per worker
which is equal to zero.
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.
3.
An increase in the saving rate will lead to higher
growth of output per worker for some time.
(Growth will end once the economy reaches its
new – higher - steady state.)
The Saving Rate and Output
A country with
a higher
saving rate
achieves a
higher level of
output per
worker in
steady state.
The Saving Rate and Consumption
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.
The Saving Rate and Consumption
An increase in
the saving
rate leads to
an increase,
then to a
decrease, in
consumption
per worker in
steady state.
The Dynamic Effects of an Increase in
the Saving Rate
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.
Physical Versus Human Capital



The set of skills of workers 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.
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:
Y
 K H
 f , 
 N N
N
( ,  )
Social Capital

The set of institutions, governing laws,
norms and cultural habits that might affect
the growth potential of an economy.
Private property
 Underground economy
 Cultural norms

Human Capital, Physical Capital, and Output


An increase in how much society “invests” in
human capital—through education and on-the-jobtraining—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/invests but also
on how much it spends on education.
Steady State Growth



In the model we studied until now (with or
without human capital), there is no growth in the
steady state.
The same model with technological progress
will yield growth in the steady state (lecture 16).
Models without exogenous technological
progress that have steady state growth are called
endogenous growth models (lecture 17).