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 1(t) is the only(parametric) expenditure flow that influences the rate of income flow, we can then state that
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
dr d/1 dt dt s 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 K p (=a constant where k (the Greek letter kappa) stands for capacity or potential output flow per year, and p(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 K= pk dollars. Note that, from K=pK
dt s 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 dk=pdK, and dK dK dt (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,Y=k. 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 dy dK t What kind of time path of investment I(t) can satisfy this equilibrium condition at all times
(the production function), it follows that , and d = dK I dt dK dt d = = ( 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 dt s pl or i dt ps (4) Since(4)specifies a definite pattern of change for 1, 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
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: I dt s dI = 1 or s dt dI I = 1 ( 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 By the substitution rule and the log rule, the left side gives us hn|I|+c1(≠0 Whereas the right side yields(ps being a constant) psdt=pst+c2 Equating the two sides and combining the two constants, we ave ost c (5)
dt = sdt dt dI I 1 By the substitution rule and the log rule, the left side gives us = + 1 ln | I | c I dI (I 0) Whereas the right side yields ( being a constant) s = + 2 sdt st c Equating the two sides and combining the two constants, we have ln | I |= st + c ( 5 )
To obtain/ from In, we perform an operation known as taking the antilog of InI>which utilizes the fact that e=x. Thus, letting each side of (5)become the exponent of the constant e. we obtain ost+c 已 or I|=ee°=Ae where 三已 If we take investment to be positive, then 1=1, so that the above result becomes I(t)=Aepst, where A is arbitrary To get rid of this arbitrary constant, we set t=0 in the equation /(t)=Ae pst to get (0)=Ae= A. This definitizes the constant A. and enables us to express the solution--the required investment path--as (t)=(0)e where 1(0)denotes the initial rate of investment
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 e x x = ln ln|I| ( st c) e e + = or st c st I e e Ae | |= = where c A e 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 st I t Ae ( ) = st I t Ae ( ) = (0) . 0 I = Ae = A st I t I e ( ) = (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 ofps, along a path such as illustrated in 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 p and s are known, the required growth path of investment becomes very rigidly set 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 he requirea rate ps
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 u= 1 means full utilization of capacity. 1→0K(t) and show that u=r/ ps, so that u as r-ps ∞) either a shortage of capacity (u>1)or a surplus of capacity(u< 1), depending on whether r is greater of less than ps
Domar's approach is to define a coefficient of utilization ( ) ( ) lim t Y t u t→ = [u = 1 means full utilization of capacity.] and show that u = r / s, so that 1 u r s. 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 r s (t → ) (u 1) (u 1) s
We can show, however that the conclusion about capacity shortage and surplus really applies at any time t, not only ast→>∞. For a growth rate of r implies that ()=I(0)e =r(0)e therefore, by (1)and(2), we have dt s dt s (0)e dt pl(t)=pl(ver dK The ratio between these two derivatives dr/ dt r should tell us the relative magnitudes of dx dt ps
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 t → r implies that rt I(t) = I(0)e and rt rI e dt dI = (0) therefore, by (1) and (2), we have rt I e s r dt dI dt s dY (0) 1 = = rt I t I e dt d ( ) (0) = = The ratio between these two derivatives, s r d dt dY dt = 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 ps (the required rate) then dy/dt>dx/dt, and the demand effect outstrip the capacity effect, causing a shortage of capacity. Conversely ifr ps) 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 (r< ps), we will encounter a capacity surplus rather than a shortage. Inde because of such paradoxical results, if we now allow the entrepreneurs to adjust the actual growth rate r(hitherto
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 dY dt d dt r s 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(r s), (r s)