DOMAR GROWTH MODEL
The Framework
The basic premises of the Domar model are as follows:
1,Any change in the rate of investment flow per year I(t) will
produce a dual effect,it will affect the aggregate demand as well
as the productive capacity of the economy.
2,The demand effect of a change in I(t) operates through the
multiplier process,assumed to work instantaneously,Thus an
increase in I(t) will raise the rate of income flow per year Y(t) by
a multiple of the increment in I(t),The multiplier is k=1/s,
where s stands for the given (constant) marginal propensity to
save,On the assumption that I(t) is the only (parametric)
expenditure flow that influences the rate of income flow,we can
then state that
sdt
dI
dt
dY 1? ( 1 )
3,The capacity effect of investment is to be measured by the
change in the rate of potential output the economy is capable
of producing,Assuming a constant capacity-capital ratio,we
can write
K ( = a constant )
where (the Greek letter kappa) stands for capacity or
potential output flow per year,and (the Greek letter rho)
denotes the given capacity-capital ratio,This implies,of
course,that with a capital stock K(t) the economy is
potentially capable of producing an annual product,or income,
amounting to dollars,Note that,from
K K
(the production function),it follows that,and
dKd
IdtdKdtd ( 2 )
In Domar's model,equilibrium is defined to be a situation in
which productive capacity is fully utilized,To have
equilibrium is,therefore,to require the aggregate demand to
be exactly equal to the potential output producible in a year;
that is,,If we start initially from an equilibrium
situation,however,the requirement will reduce to the
balancing of the respective changes in capacity and in
aggregate demand; that is
Y
dt
d
dt
dY ( 3 )
What kind of time path of investment I(t) can satisfy this
equilibrium condition at all times?
Finding the Solution
To answer this question,we first substitute (1) and (2) into the
equilibrium condition (3),The result is the following
differential equation:
IsdtdI1
or
sdtdII1
( 4 )
Since (4) specifies a definite pattern of change for I,we should
be able to find the equilibrium (or required) investment path
from it.
In this simple case,the solution is obtainable by directly
integrating both sides of the second equation in (4) with
respect to t,The fact that the two sides are identical in
equilibrium assures the equality of their integrals,Thus,
s d tdtdtdII?1
By the substitution rule and the log rule,the left side gives us
1||ln cIIdI )0(?I
Whereas the right side yields ( being a constant)s?
2csts d t
Equating the two sides and combining the two constants,we
have
cstI||ln ( 5 )
To obtain | I | from ln| I |,we perform an operation known as
“taking the antilog of ln| I |,” which utilizes the fact that
,Thus,letting each side of (5) become the exponent
of the constant,e,we obtain
xe x?ln
)(||ln cstI ee
or
stcst AeeeI||
where
ceA?
If we take investment to be positive,then | I | = I,so that the
above result becomes,where A is arbitrary.
To get rid of this arbitrary constant,we set t = 0 in the
equation,to get This
definitizes the constant A,and enables us to express the
solution--the required investment path--as
stAetI)(
stAetI)(,)0( 0 AAeI
steItI?)0()(?
where I(0) denotes the initial rate of investment.
( 6 )
This result has a disquieting economic meaning,In order to
maintain the balance between capacity and demand over time,
the rate of investment flow must grow precisely at the
exponential rate of,along a path such as illustrated in Fig 1.
Obviously,the larger will be the required rate of growth of
investment,the larger the capacity-capital ratio and the
marginal propensity to save happen to be,But at any rate,
once the values of and s are known,the required growth
path of investment becomes very rigidly set.
s?
The Razor's Edge
It now becomes relevant to ask what will happen if the actual
rate of growth of investment --call that rate r -- differs from
the required rate,s?
Domar's approach is to define a coefficient of utilization
)(
)(lim
t
tYu
t
[u = 1 means full utilization of capacity.]
and show that,/ sru so that
1u,sr
as
In other words,if there is a discrepancy between the actual
and required rates ( ),we will find in the end
either a shortage of capacity or a surplus of
capacity,depending on whether r is greater of less
than
sr )(t
)1(?u
)1(?u
.s?
We can show,however,that the conclusion about capacity
shortage and surplus really applies at any time t,not only
as,For a growth rate of r implies thatt
rteItI )0()(? and rt
erIdtdI )0(?
therefore,by (1) and (2),we have
rteIsrdtdIsdtdY )0(1
rteItIdtd )0()(
The ratio between these two derivatives,
s
r
dtd
dtdY

should tell us the relative magnitudes of
the demand-creating effect and the capacity-generating
effect of investment at any time t,under the actual growth
rate of r,If r (the actual rate) exceeds (the required rate),
then,and the demand effect outstrip the
capacity effect,causing a shortage of capacity,Conversely,
if,there will be a deficiency in aggregate demand and,
hence,a surplus of capacity.
s?
dtddtdY
sr
The curious thing about this conclusion is that if
investment actually grows at a faster rate than required
the end result will be a shortage rather than a surplus of
capacity,It is equally curious that if the actual growth of
investment lags behind the required rate,we will
encounter a capacity surplus rather than a shortage,Indeed,
because of such paradoxical results,if we now allow the
entrepreneurs to adjust the actual growth rate r (hitherto
),( sr
)( sr
taken to be constant) according to the prevailing capacity
situation,they will most certainly make the "wrong" kind of
adjustment,In the case of for instance,the emergent
capacity shortage will motivate an even faster rate of
investment,But this would mean a increase in r,instead of
the reduction called for under the circumstances.
Consequently,the discrepancy between the two rates of
growth would be intensified rather than reduced.
,sr
t
I(t)
O
I(0)
steItI?)0()(?
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