Phillips’ stabilization model for a closed economy, an economic application of second-order differential equations.
The level of national output (income) is determined, as we know from elementary
macroeconomics, by the level of aggregate demand. The latter is made up of a
part originating from private economic agents and of a part originating from the
government. The government manoeuvres its expenditure in order that aggregate
demand is such that national income attains a given desired level. Thus we can
talk of stabilization of aggregate demand. Phillips analyses this economic policy
problem from a dynamic point of view.
Let us assume that national income is initially ar the desired level and that an
exogenous decrease in aggregate demand occurs. The variables are measured as
deviations from their desired levels, so that a negative value simply means that the
actual is smaller than the desired value. Before introducing the stabilization policy
we must know, to have a standard of comparison, the ‘spontaneous’ behaviour of
the economic system, i.e. its behaviour when there is no government expenditure.
The basic dynamic mechanism is the multiplier.
The fundamental assumption is that producers react to excess demand by making adjustments in output: if aggregated demand exceeds (falls short of) current
output, the latter will be increased (decreased). Obviously this mechanism operates
independently of the origin of excess demand, and so it is the same both without
and with government expenditure. In formal terms,
Y 0 = α(D − Y ) ,
α > 0,
(5.1)
where Y is national output, D is aggregate demand and α is a reaction coefficient,
representing the velocity of adjustment to a discrepancy between aggregate demand
and current output.
Aggregate private demand is a function of national income:
D = (1 − l)Y ,
(5.2)
where (1 − l) is the marginal propensity to spend (i.e. the marginal propensity
to consume plus the marginal propensity to invest). This propensity is assumed
smaller than unity, whence 0 < l < 1. Introducing the exogeneous disturbance u
we have
D = (1 − l)Y − u .
(5.20 )
Substituting (5.20 ) into (5.1) and giving to u a unit value, we have
Y 0 + αlY = α .
(5.3)
This is a first-order equation, whose solution is
Y (t) = Ae−αlt −
1
.
l
(5.4)
We assumed that in the initial period national income was at the desired level, so
that Y (0) = 0 and so A = 1/l, whence
1
Y (t) = − (1 − e−αlt ) .
l
(5.40 )
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Since αl > 0, Y (t) tends monotonically to −1/l, i.e. to the value obtained applying
the multiplier 1/l to the exogeneous decrease in expenditure (−1).
We may now go on to examine the effects of a stabilization policy. Phillips
enumerates three types of stabilization policy. They are the following:
Proportional stabilization policy: government expenditure is proportional and of
opposite sign to the deviation between the actual and the desired value of output,
i.e. G∗ = −fp Y , where fp > 0 is the coefficient of proportionality.
Derivative stabilization policy: government expenditure is proportional and of
opposite sign to the variation in (that is, to the derivative of) current output, i.e.
G∗ = −fd Y 0 , where fd > 0 is the coefficient of proportionality.
Integral stabilization policy: government expenditure is proportional and of opposite sign to the sum (in continuous terms, to the integral) of all the differences
that have occurred, from time zero to the current moment, between tha actual and
the desired values of output, i.e.
∗
G = −fi
Z
t
Y (s) ds ,
0
where fi > 0 is the coefficient of proportionality.
The reader will have noted that we have marked with an asterisk the government
demand. The reason is that the various values of such demand indicated in the
enumeration of the various policies are the theoretical or potential values, i.e. the
values that define in theory the different policies. Now, in Phillips’ words (Phillips,
1954, p. 294), “the actual policy demand will usually be different from the potential
demand, owing to the time required for observing changes in the error, adjusting the
correcting action accordingly and for changes in the correcting action to produce
their full effects. . . . whenever such a difference exists the actual policy demand
will be changing in a direction which tends to eliminate the difference and at a
rate proportional to the difference”. Thus, using the symbol G to indicate actual
government demand, we have
G0 = β(G∗ − G) ,
β > 0,
(5.5)
where β is a reaction coefficient, indicating the speed of response to a discrepancy
between potential and actual public expenditure.
Now, when government demand is present, equation (5.20 ) becomes
D = (1 − l)Y + G − u .
(5.200 )
The stabilization model is made of equations (5.1), (5.200 ), (5.5) and of one or
more of the relations defining G∗ . We now manipulate the model to reduce it to a
single equation. From (5.5) we have
G0 + βG = βG∗ .
(5.50 )
D0 = (1 − l)Y 0 + G0 .
(5.2000 )
Differentiating (5.200 ) we obtain
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Multiplying both members of (5.200 ) by β and adding the result to eq. (5.2000 ) we
have
D0 + βD(1 − l)Y 0 + β(1 − l)Y + G0 + βG − βu ,
(5.6)
whence, using (5.50 )
D0 + βD = (1 − l)Y 0 + β(1 − l)Y + βG∗ − βu .
(5.60 )
From (5.1) we obtain
Y 0 + αY
D=
,
α
whence, differentiating both members,
D0 =
Y 00 + αY 0
.
α
(5.10 )
(5.100 )
Now multiply both members of (5.10 ) by β and add the result to (5.100 ), obtaining
D0 + βD =
Y 00 + (α + β)Y 0 + αβY
.
α
(5.7)
Equating the right-hand members of (5.7) and of (5.60 ) we obtain
Y 00 + (α + β)Y 0 + αβY
= (1 − l)Y 0 + β(1 − l)Y + βG∗ − βu ,
α
(5.8)
so that, multiplying through by α and rearranging terms, we have, considering a
unit decrease in aggregate demand,
Y 00 + (αl + β)Y 0 + αβlY − αβG∗ = −αβ .
(5.9)
Eq. (5.9) is the basic differential equation of the model. Inserting the various
relations defining G∗ we can determine the time path of output and so study the
effects of the single stabilization policies or of combinations of them when two or
more are used simultaneously. As examples we shall study the ‘pure proportional’
case and the ‘mixed proportional-derivative’ case. The introduction of the integral
stabilization policy, alone or in conjunction with others, will have to wait until Part
II, ch.7, since it gives rise to a third-order equation.