The Solow Growth Model

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Oct 28, 2013 (3 years and 9 months ago)

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The Solow Growth Model


Production function
:


The Solow model defined production at time
t
as a function of capital stock
K

at
time
t
, technology level A at time
t
, and labor force
L

at time
t
.
A
and
L

enter the
model multiplicatively as effective labor
A
L

(this is known as labor augmenting
of knowledge, which indicts the belief that possession of knowledge directly
affects the effectiveness of labor force in the process of production.)
Mathematically, the model is expressed as the following:




In this model, constant returns to scale in both capital and effective labor is
assumed.

(*see Romer 10 for explanations)

This means effectively the following:




This implies


,

if w
e set


We know
. Now define the following:




We now can write output per unit of effective labor as a function of capit
al per
unit of effective labor, or the intensive form

of production.



or


This function is assumed to satisfy


Definition of Inputs
:


Define the following




*note: the phrase
the growth rate of X

refers to the quantity
, where
is a
shorthand for
. We may also write
.
Thus we may
note that
, and
, i.e. we assume
tha
t
L

and
A

grow exponentially.

--
end of note


We
also
assume output of production is divided between saving and
consumption. What is saved is invested in capital stock. Capital stock
depreciates at the rate of
. Thus:




The Dynamics of K
:


Combine the above expression and production function we defined previously,
we have:




Reduce it to the intensive form




We also know that




We have already defined the following previously,




Thus




This equation states effectively that the rate of change of the capital stock per unit
of effective labor is the difference betwee
n the actual investment per unit of
effective labor and the
breakeven investment
, or the amount of investment that
must be done just to keep
k

at its existing level.


L
et
k*

denote the level of capital stock at which actual investment equals
breakeven inve
stment. At this point,
dk

is equal to 0.


The Balanced Growth Path
:


The capital stock,
K
, equals
ALk
; since
k

is constant at
k
*, K is growing at the rate
of
n+g
,

(
i.e.
). Under constant return of scale (CRS), output
Y

is also

growing at
n+g
. Finally,
K/L

and
Y/L

are growing at
g,

i.e. on the balanced
growth path, the growth rate of output per worker is determined solely by the
rate of technological progress.


The Effect of Saving
:


Consider a Solow economy on a balanced growt
h path; suppose that there is a
permanent increase in
s
, the saving rate.


short term effect


Long run effect


A change in the saving rate has a level effect but not a growth effect: it changes
the ec
onomy’s balanced growth path, and thus the level of output per worker at
any point in time, but it does not affect the growth rate of output per worker on
the balanced growth path. In the Solow model only changes in the rate of
technological progress have

growth effects, all other changes have only level
effects.


Impact on Consumption (Household behaviour

and the Golden Rule
)


s

dk (dy)

k

k*
1

k*
2

s
1
f(k)

s
2
f(k)

Let
c*

denote consumption per unit of effective la
bor on the balanced growth
path so that:


,

and on t
he balanced growth path
, since




k*

is determined by
s

and the other parameters of the model,
n
,
g
, and
, thus




Whether the increase raises or lowers consumption in the long run depends on
whether
--

the marginal product of capital


is more or less than
n+g+δ
. If
is less than
, then the additional output from the increased capital
is not enough to maintain the capital stock at its higher level. In this
consumption must fall to maintain the higher capital stock. If
exceeds
, on the other hand, there is more than enough additional output to
maintain
k

at its higher level, and so consumption rises.


We can also look from another perspective




i.e. the real growth of the economy (
, the growth minus depreciation) must
equal the population growth and technological development.


We can present the previous ideas graphically:





The Quantitative Approach


Know that

is the level of output per unit of effective labor on the
balanced growth path
. The long run effect of a rise in saving on output is given
by:




1)

Find


On the balanced growth path
, thus




and take derivatives of the two sides with respect to
s




and therefore



consumption

k

c


k



Multiply both sides with



We also know that




The portion

is known as the elasticity of output with respect to capital
at
. If markets are competitive and ther
e are no externalities, capital earns
its marginal product. In this case, on the balanced growth path capital earns
, and the share of total income that paid to capital is
.


We denote the elasticity a
s
. Thus the previous expression can be
rewritten as




In most countries, the share of income paid to capital is about 1/3, which implies
that the elasticity of output with respect to the saving rate

in the long run is
about 1/2. (
).





Speed of Convergence

We want to determine how rapidly
k

approaches
k*
. Since
,
we know that
is determined by
k
. Thus
. When
,
. A first
order Taylor approximation of
around

is known as




Let
denote

. The above expression is rewritten as


.


Since
, when
k

is slightly below
k*

and
when it is slightly above.
is negat
ive and
is positive.



implies that in the vicinity of the balanced growth path,
k

moves toward
k*

at a speed approximately proportional to its distance from
k*
,
i.e. the growth rate of
is approximately constant and equal to
. Thus



The specific value of
is subject to the specifics of the model




We may also present of

the issue of convergence graphically:


We have derived previously that




so that
, where
is the average productivity of
received capital and
is th
e necessary investment to hold the capital
stock constant.


Absolute Convergence


Conditional Convergence



The Central Conclusion of the Solow Model


The central conclusion of the Solow model is that

if the returns that capital
commands in the market are a rough guide to its contributions to output, then
variations in the accumulation of physical capital do not account for a significant
part of wither worldwide economic growth or cross
-
country income
difference.

k



Growth




k

The Solow Model with natural resources and land


We start by defining two extra variables


T =

Land;


R =

Natural resources;


Now the production is a function of




Convert it into natural logarithm formation
:




Differentiate with respect to time
t
, (
, by the chaining
rule), we have the following:




We also know that
, and on the balanced path of growth the
proportion

is constant, thus
, i.e. the growth rate of capital is equal
to the growth of production and they both equal to a constant. Thus:




and


thus:




There are two interesting terms in the above expression:

against
.


Essentially it illustrates the technol
ogical development weighs against the
depletion of natural resource and land by population growth.