The Suntory and Toyota International Centres for Economics and Related Disciplines
A Micro-Macroeconomic Analysis Based on a Representative Firm
Author(s): Yew-Kwang Ng
Source: Economica, New Series, Vol. 49, No. 194 (May, 1982), pp. 121-139
Published by: Wiley on behalf of The London School of Economics and Political Science and The
Suntory and Toyota International Centres for Economics and Related Disciplines
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Economica, 49, 121-139
A Micro-macroeconomic Analysis based on a
Representative Firm
By YEW-KWANGNG
Monash University, Australia
In this paper, I develop a method of analysis incorporating elements of
micro- and macroeconomics as well as general equilibrium, in order to examine
the effects of economy-wide changes in demand, costs, expectations, etc. It
focuses on the microeconomics of a representative firm1but takes account of
the effects of macroeconomic variables (aggregate demand, aggregate output
and the price level) on the demand and cost functions of the firm. It thus goes
beyond a partial microeconomic analysis but stops short of a fully general
equilibrium analysis of the Arrow-Debreu type. It deals with aggregates and
averages but with the microeconomic foundation built-into the analysis.
Our method is based on a number of simplifications (some essential, some
purely for simplicity) outlined below. If we view the method as using the
response of the representative firm to approximate the response of the whole
economy, the simplifications involved seem reasonable. In the spirit of positive
economics, whether the approximation is acceptable can be settled only by
empirical testing of the conclusions of the theory. By way of conclusions, our
method not only provides strong qualitative results but in some cases also
quantitative results summarized into the four propositions below. Since our
method can be used either for the whole economy or for an industry (on
which see Ng, 1981a), it has significance for both macro- and microeconomic
problems.
I. SIMPLIFICATIONS
A theory abstracts from complicating features of the real world and
concentrates on the relationships that are important for the problem on hand.
This is especially true for an aggregative analysis, which must necessarily
involve some simplification in the procedure of aggregation. Our analysis is
no exception. First, we take a firm to represent the whole economy. A
theoretically most straightforward way to do this is to assume a number of
identical firms. Then, apart from changes in the number of firms, each firm
is representative of the whole economy. Even if firms are not identical, one
may still use a representative firm to approximate the whole economy if we
define the representative firm appropriately.
Consider the marginal cost curve (MCC) illustrated in Figure 1. Starting
from an initial profit-maximizing equilibrium A, suppose the marginal revenue
curve (MRC) moves from MR to MR': output will expand (abstracting from
any possible movement in MCC to be discussed below) to q1, q2, q3 respectively
if the MCC is MC1, MC2, MC3. Thus, for the case of three firms of similar
size with MC1, MC2 and MC3, we may take MC2 as the MCC of the
representative firm. The precise method of constructing this representative
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122
[MAY
ECONOMICA
MCC will have to be discussed in the actual empirical application or testing
of the theory but need not detain us here. Suffice it to understand here that
the MCC of the average firm is some kind of weighted average of the MCCs
of all firms (or a random sample).
$
MR'
MR\
\
0
\
X
q0
q,
FIGURE
q2
q3
~~~~~~MC,
q
1
Studies on the representability of consumers' demand by a single consumer
(or a single utility function) show that the conditions required are very
restrictive. In general, the representation is not a very good approximation.
By analogy, it may be thought that the use of the representative firm is open
to the same objection. However, there is an important difference here. (I owe
this observation to Kevin Roberts.) In the former case, the non-representability is produced in the presence of first-order redistribution among consumers. In the present case of the representative firm, no such redistribution
is involved. Nevertheless, one has to be very careful in the use of the
representative firm construction. On the one hand, one has to avoid the fallacy
of composition. For example, each single firm may be able to expand output
without affecting its marginal cost: this does not imply that all of them can
do so simultaneously. On the other hand, one has to avoid the reverse fallacy,
which may be called the fallacy of attribution. If a representative firm (which
may not actually exist) knows that it is representative (in a model of N identical
firms, each may know precisely that), it knows that, if it charges a price
according to its own profit-maximizing calculation, it will turn out to equal
the average price. Nevertheless, it cannot then assume that, whatever price
it charges, the average price will be equal to it. This would be the case only
if there is complete implicit collusion. In the absence of collusion, each firm
has to maximize with respect only to the variable under its control. It is a
fallacy to attribute what all firms can do together to a single (even if representative) firm. (In terms of the mathematics below, we should take aX/8p = 0 in
deriving the first-order condition for the firm but take dir = dp in the total
differentiation of the first-order condition, where p is the price of the firm
and ir the average price of the whole economy.)
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1982]
MICRO-MACROECONOMIC
ANALYSIS
123
Second, while the whole vector of prices, in general, affects the demand
for the product of a firm, we will simplify by taking account of just the price
of the product, the average price of the whole economy, (nominal) aggregate
demand, and the number of firms. Though this is a simplification, it is an
advance over the traditional partial equilibrium microeconomic analysis, and
the degree of simplification is no more (probably less) than aggregative
macroeconomics. Moreover, the microeconomic foundation is built into our
analysis, and hence is superior to the traditional aggregative macroeconomics
in this respect.
Third, we assume that the representative firm is small enough to have no
appreciable effects on the average price, aggregate demand and aggregate
output. The complications of size and oligopolistic interdependence will be
pursued elsewhere (Ng, 1981c). Moreover, we shall be using mainly comparative static analysis, and questions such as joint products, non-price competition, etc., will be ignored. Consumers are not explicitly analysed; they exert
their influence through the demand function faced by the firm and through
the implicit input supply functions. In this paper the number of firms is also
taken as given. (In Ng, 198 lb, the analysis is extended to the long run, taking
the number of firms as a variable.) Since the cost function is fairly general,
the analysis can be interpreted as a medium-run one (long-run cost function
with no change in the number of firms) as well as a short-run one (cost function
in the short run as well). Lastly, while changes in aggregate demand must
affect the quantity demanded of the representative firm, the elasticity of
demand is taken as unaffected. This possible effect will be examined in the
long-run analysis where changes in the number of firms must affect demand
elasticity.
While we have made a number of simplifications, we have also achieved
some generalizations. Apart from the basic feature of combining macro- and
microeconomics and taking account of secondary repercussions, we allow the
representative firm to be a perfect or a non-perfect competitor (monopolistic
competitor or even a monopolist). It is hoped that these generalizations and
the substantive results obtained more than justify the simplifications involved.
II.
THE
MODEL
The representative firm has a demand function in which the quantity
demanded (q) is a (twice-differentiable) function of its price (p), the average
price (ir) of all other firms in the industry or economy, and (nominal) aggregate
demand (a).
(1)
q =F(p, r, a).
Strictly speaking, the actual average price X should be replaced by the
expected average price ir. Each firm cannot simultaneously observe the prices
of others before fixing its own price. Nevertheless, in an equilibrium, the
expected equals the actual average price. Since we will be concerned only
with equilibria rather than with the path of adjustment, we will use ir in (1),
thus ensuring the realization of expectations (and hence also making our
analysis consistent with rational expectations). However, when we come to
analyse price expectations, Ir will be replaced by 'r.
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124
ECONOMICA
[MAY
Since we use the firm to represent the whole economy, it is convenient to
have (by suitable definition of units if required), at the initial equilibrium,
(2)
p= r.
Moreover, as p changes in response to economy-wide changes, ir also
changes by the same extent. Hence, (2) must also hold in a new equilibrium.
Thus one may use dir = dp in the analysis. (But each firm takes X as beyond
its control.) If our hypothesis that the response of the economy can be
approximated by that of the representative firm is incorrect, we may not
actually have d7rr= dp. But since our whole analysis is based on this hypothesis,
it is consistent, and imposes no further restriction to have d7r= dp. This is
not tautological, since we show how the change in p is determined by the
maximizing behaviour of the firm with Xras only one of the influencing factors.
If nominal aggregate demand a and all (nominal) prices change by the
same proportion, quantity demanded should remain unchanged. Hence (1)
may be taken as homogeneous of degree zero in (p, xT,a). We thus have
(3)
q=F
p
a
c
which may be written (after putting the effect of the constant r/r into the
functional form) as
(4)
q f(P,')
In other words, the quantity demanded depends on the relative price and real
aggregate demand. Alternatively, since we ignore changes in the relative
prices of all other goods, we may lump them together into a single composite
good and use it as a numeraire. We still have (4) from (1).
Consider now a change in real aggregate demand a/l while the relative
price p/lr is being held constant. As real aggregate demand increases by x
per cent, the demand for the product of a firm may increase by more or by
less than x per cent or even decrease. But for the representative firm, it must
increase by x per cent at p = r (i.e. if it is charging the representative price).
Otherwise it is not representative. To illustrate this point graphically, it is
simplest to consider the case of an x per cent increase in a with Xr remaining
unchanged. In Figure 2, dd is the initial demand curve of the firm. As this
demand curve plots q as a function of p, it must in general shift as X and/or
a changes. Here, as a increases by x per cent with ir held constant, if p is
also held constant, q must also increase by x per cent from A to B for the
firm to be representative. But for points p $ x, q may not increase by exactly
x per cent unless the new demand curve (d'd') remains isoelastic at each
price in comparison with dd.2 Should the demand curve become more (d'd')
or less (d3d3) elastic, this is no longer so. There are some considerations
suggesting that the demand curve may become more elastic and some suggesting the reverse as real aggregate demand increases. Moreover, if the number
of firms (a variable held constant in this paper) changes with a/7r, demand
elasticity will also be affected. These considerations are pursued in Ng (198 lb),
where the number of firms is taken as a variable. Here, I will adopt a simplifying
assumption that the demand curve will not become more elastic or less elastic.
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1982]
MICRO-MACROECONOMIC
125
ANALYSIS
$
2
d
\
q
0
FIGURE 2
It is believed that this is a reasonable approximation except for cases where
there are reasons suggesting important changes in demand elasticity in one
direction. Thus, we take the demand for the product of the representative
firm as of unitary (real) aggregate demand elasticity. This means that q is
homogeneous of degree one in ax/I, given p/IT, or
(5)
3-h()
q =aft(P1)
which says that q is a function of the relative price and a proportionate
function of real aggregate demand. It can be seen that (5) is homogeneous
of degree one in
of degree zero in (a, p, I) and homogeneous
as desired.
a,,
In terms of the demand curve, when only a increases by x per cent, the
demand curve moves horizontally to the right by x per cent since q increases
by x per cent at given p. When both a and ir increase by x per cent, the
demand curve moves vertically upward by x per cent, since q remains
unchanged if p also increases by x per cent. Both cases may be seen from (5).
We also have, in equilibrium,
(6)
= pqN
TQ
= a
where Q = aggregate output, N = the given number of firms.
The firm is assumed to maximize its profits (for revenue maximization,
see Ng, 1981a), which may be written as
(7)
aih( P
C(qq
T2Q, )
where C is a twice-differentiable total cost function and E is some exogenous
(set of) factor(s). The firm is taken as small enough to ignore its own influence
on ir and Q. The average price r may affect costs directly through the prices
of material inputs (if we allow for intermediate goods) and indirectly through
its effect on (money) wage-rates. Aggregate output Q may affect costs by
raising wage-rates (through a higher demand for labour) and through external
economies/diseconomies. Other factors that may affect costs, such as
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126
[MAY
ECONOMICA
exogenous changes in wage-rates (owing, say, to increased union militancy
instead of changes in ir and Q), external (to the industry/economy) prices
and technological changes are captured by the variable e.3 The firm takes
Q, ir, a, E as given and maximizes (7) with respect to p or q. Differentiating
(7) in the respect to p, we obtain the first-order condition (the second-order
condition and the non-shutting down condition p > A VC are assumed
satisfied)
(8)
+
h()
h2Pht()
2h(P-)
.
c(q, iT,Q,
E)=0
where c aC/aq is the marginal cost.
From (8) we have
irh(p/lr)+{p-c(q,
r,
Q, E)}h'(p/ir)=0
whence
(8')
P+
h (p irr)= c (q, r,Q,
E).
Totally differentiate this equation to obtain, on rearranging, and substituting h/h' = (c -p)/7r from (8'),
(9)
2
) P (rCf
2
2
d7r=Cqdq+codQ+de
where a subscript denotes a partial derivative, and dce cEde. This equation
shows that the way in which the representative firm will change its price (dp)
depends also on its expectation of the average price change (dir). If dp ? dir,
the expectation will be frustrated and further adjustment will ensue. To ensure
a new equilibrium, let us impose dp = dI (i.e. the total differentiation of
equation (2)). Thus our analysis below is consistent with the realization of
expectations (and hence also with rational expectations), and the results will
not be reversed owing to the frustration of expectations. Substituting p =
dQ =Ndq = (Q/q) dq (from Q = Nq), we have, on rearranging
ir, dp = d7rr,
and dividing through by c to cast in elasticity form,4
(10)
(1 _ qc,)
dplp
_ (,cq
+, Ncl)
dqlq =dic/
cq cqq/c and ?cQ = CQQ/Care, respectively, the elaswhere 7rc-ir/c,
ticities of marginal cost with respect to the average price xr, output q and
aggregate output Q. While Cqand r,cq are the slope and the elasticity of the
marginal cost curve, CQ and qcQ refer to the shift in MCC. It may also be
noted that (10) may also be derived by using the inverse demand function
p = irg(irq/a) obtained from (5) with g as the inverse function of h.
We need an extra equation specifying the determination of aggregate
demand a to close the system. We adopt a very general function
(11)
a = a(r, Q, X)
where X is some exogenous set of (nominal) factors probably including the
money supply, fiscal policy variables and other autonomous (independent of
ir and Q) factors affecting spending (consumption and investment). The only
a, ir/ a > -, 1> 7aQ -acQQ/a > -1,
restrictions placed on (11) are 1 >
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1982]
MICRO-MACROECONOMIC
127
ANALYSIS
to avoid an explosive (unstable) system. (Note that ai" < 1 is not unreasonable,
since X, including the money supply, is being held constant. Also, q'Q < 1 is
the common restriction that the marginal propensity to spend is smaller than
one.) Equation (11) is quite general, including various forms of monetarist,
Keynesian and other theories of aggregate demand determination as special
cases. For example, for a simple monetarist theory that a equals a constant
multipleof moneysupply,we have q =i aQ _~0 aX = 1 whereX = money
supply.
Totally differentiate (6) and (11), and divide through by a = rQ = pqN,
(12)
da
a
(13)
da
a
dp dq dirr dQ
q
p
Q
ir
ar)j d
ir
da
QdQ
Q
a
where ds =axdX is the exogenous change in aggregate demand.5 Substitute
d r/Tr = dp/p, dQ/Q = dq/q and the first equation in (12) into (13),
(1- _7aQ)
(1)
dq+(1 -q
dp dai
av)
a
p
q
Substituting dq/q and dp/p from (14) in turn into (10), we obtain the following
basic equations:
(15)
{(1-
C7T)(1 - 71aQ) + (cq
=(~+
(16)
Q)
d/a
-_,aQ)
{(1 -_Cc)(1
+
+
(cq
+
cQ)(1
+ ,cQ)(1
dc/la - (1 -qa)
= -(_`Z)
ir)}
dp/p
dc/c
(1- _aQ)
_
a)I
dq/q
dic/
For the purpose of comparative static analysis, the bracketed term on the
left-hand side in (15) and (16) may be taken as non-negative; otherwise the
system is explosive. This does not mean that the term cannot be negative at
a particular point. But the explosion must end (negativity reversed) before a
new equilibrium is reached. Hence, for the purpose of determining the
direction of change from the old equilibrium to the new, the term may be
taken as positive.
since
It may be noted that q may be replaced by q
(17
Q
a/v-
a__
Qaa/1
(17)
aa(. )/r
Q
Q
a/i7r
aa
Q
aQ
aQ
If we are prepared to restrict the aggregate demand function (11) to one
that is specified in real terms (no money illusion), we have
a = ir4(Q, X/ir)
where
(11 )
X
is some function for real aggregate demand. We may then derive
aa/aX
= 02, aa/air
= k -Xk2/Ir,
qa
= (7
-X02)/a
Or,
(18)
1 -,air
aX
a
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= 1_
aX
128
ECONOMICA
III.
EXOGENOUS
[MAY
COST CHANGES
With (15) and (16), we are now prepared to examine the comparative
static effects. First, consider a change in the exogenous variable E or j which
may be due to an exogenous change in wage-rates (unrelated to changes in
vT and Q), exogenous changes in the prices of material inputs (such as oil)
purchased from abroad/other industries, or changes in production technology.
To isolate the effect of an exogenous cost change, we take da = 0. This is not
a partial analysis since endogenous changes in demand and costs are allowed
'
cq
Q
C1T
,CQ.
through qai,
and
From (15) and (16) we may derive, by
multiplying both sides by c/di and putting da = 0,
-
p'c_
(19)
(1
(20)
0'qJ
f
C1
(1 - C71)(1
) + (cq
--
- 7Q)
+ (cq
+
+7)
cQ)(1
-
-a1
air)
)
where o- c-(dp/di)
(c/p), a = (dq/di) (c/q) are respectively the elasticities
of price and output with respect to the exogenous change in marginal cost.
We may divide the effects on price and output of an exogenous change
in marginal cost as specified in (19) and (20) into three parts: (a) the primary
effects, (b) the secondary cost effects through further endogenous shifts in the
marginal cost curve, owing to the terms qC` and qcQ, and (c) the secondary
demand effects through q"ai and q`Q. To examine the primary effects first,
c
air and 'Q (all taken as zero). We
let us ignore for the moment
71c",Q,
thus have
(19')
(20')
ope (primary) = 1/(1
qe
+ ,cq)
(primary) = -1/ (1 + 71cq)
cq is the elasticity of MC (marginal cost) with respect to q and is
positive/zero/negative if MCC (the marginal cost curve of the representative
firm) is upward/horizontal/downward sloping. The value of q cq may conceivably range from minus infinity to plus infinity. Nevertheless, it is very unlikely
that ?,cq< 1, which requires MCC to be sufficiently downward sloping as to
become inelastic. In any case, this would produce an explosive system which
we may ignore for a comparative static analysis.
From (19') and (20'), a-P (primary) and _0qc (primary) are smaller
than/equal to/larger than unity as r cq is positive/zero/negative. In other
words, an exogenous increase in marginal cost increases price and reduces
output by less than/exactly/more than proportionately (i.e., by the same
percentage as that of the cost increase) if MCC is upward/horizontal/downward-sloping, as far as the primary effects are concerned. The borderline case
of a horizontal MCC is illustrated in Figure 3, where the initial demand curve
dd is drawn as linear (not required for our results) for ease of drawing. A 10
per cent increase in MC to MC' increases the price by less than 5 per cent
(from A to B) before the effect of a higher ir on the demand curve is taken
into account. However, as the exogenous cost increase is not confined only
to this one firm but is economy-wide, X increases as p increases, shifting the
demand curve upward. (Note that this is included in the primary effect and
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1982]
MICRO-MACROECONOMIC
129
ANALYSIS
d I
d'
=
-MCI
q
FIGURE 3
is not the secondary demand effect discussed below.) This leads to a further
increase in p and hence ir, and so on (but successively by smaller and smaller
amounts). A final equilibrium E is reached when both p and ir have increased
by 10 per cent, and output q has fallen by 10 per cent to q. (If firms foresee
this, the full adjustment may be instantaneous.) It is not difficult to see that,
if MCC is upward/downward sloping, the changes in p and q will be smaller/larger. If rqcq< 1 (extremely unlikely), the effects become cumulative,
failing to reach an equilibrium until the condition no longer applies.
The secondary cost effects refer to the endogenous shifts in MCC as ir
and Q change. A higher ir increases c through higher input prices, including
wage-rates. In the absence of money illusion, lags, etc., c may respond fully
(proportionately) to ir and we have qc' = 1. It is also likely that 'qCQ is positive
as an increase in output tends to push up input prices unless there are
substantial unemployed resources and/or substantial external economies when
7 CQmay be zero or even negative. With full employment, 71co, for an expansion
in Q, is likely to be very large. With a perfectly inelastic labour supply, it
may even be infinite; if labour is the only variable input or a limitational input
(in the sense that no substitution of other factors is possible), we have the
extreme case of NqCQ = cc, making (20) equal zero. Then an exogenous decrease
in cost cannot increase output. Neither can it decrease prices. What happens
is that the exogenous reduction in MCC first tends to reduce prices and
increase output; but the simultaneous attempt by firms to expand, in the
absence of spare resources, pushes up wage-rates and prices, and hence MCC.
If labour is not a limitational factor, some increase in output is possible as
firms increase employment of the lower priced exogenous inputs.
In the presence of unemployed resources, q may be very small, even
zero or negative (in the presence of external economies). In any case, it is
then possible that the secondary effects may reinforce the primary effects as
(1- qar)p7cQ -(1 -q aQp)1cT may be negative; i.e., c is more responsive to ir
than to Q, after appropriate weighting. This weighting of both the secondary
cost effects and the primary effects is required since a may respond
endogenously as ir and Q change and hence cause further adjustments of
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130
ECONOMICA
[MAY
prices and/or output. These are the secondary demand effects, which are
similar to the Keynesian income multiplier effect except that the latter is
confined to real income (= output Q in equilibrium). Given rqa" and q'o,
the smaller q7cq+ qcQ and the larger qC1, the larger the total effects of changes
in exogenous costs on prices and outputs. The speed and extent with which
unions keep pace with price increases (a high r(" or a low or zero 1 -qC),
the reluctance of unions to accept a lower wage-rate even with significant
unemployment (a low 0cQ) and the evidence of the prevalence of non-upwardsloping MCC (low qcq) suggest that the denominator may be fairly small. This
explains the enormous effects of the oil crisis of 1973-1974. A doubling in
the price of oil involves an exogenous increase in the MCC of a representative
firm by only a small fraction, since oil composes only a fraction of the costs
of most firms. But if the denominater of (19) and (20) is small, a small increase
in c could lead to a large increase in p and a large fall in q, even though the
exogenous demand factors (including money supply) have been held
unchanged (dcx= 0) to isolate the effects of the exogenous cost change.
It is true that, if the cost-induced unemployment persists for a sufficiently
long time and the workers accept a lower wage-rate at the given employment
level, output and employment may increase towards the original level. In our
model, this can be interpreted either as a very large qcQ in the long run or
as an indication that the eventual shift in the (short-run) labour supply curve
offsets the exogenous increase in costs. The preceding discussion may be
summarized into a proposition.
Proposition 1. The primary effects of an exogenous increase/decrease in
marginal costs are to increase/decrease prices and reduce/expand output by
less than/exactly/more than proportionately if MCC (the marginal cost curve
of the representative firm) is upward/horizontal/downward-sloping. The
secondary effects through endogenous shifts in MCC are reinforcing/offsetting
to the primary effects if (1 - 71")71cQ - (1 - rQ )p1Tc is negative/positive. The
total effects are larger the larger/smaller are the (proportionate) responses
of marginal cost to prices/outputs and the less/more upward/downward
sloping is MCC.
IV.
EXOGENOUS
CHANGES
IN AGGREGATE
DEMAND
Exogenous changes in (nominal) aggregate demand a may be caused by
changes in monetary/fiscal policies and other exogenous factors. To isolate
the effects of an exogenous change in aggregate demand, we take dc = 0. This
is not a partial analysis, since endogenous changes in c are not excluded and
are reflected in 71cq, rc andT co. Substituting dc= O into (15) and (16), we
have, after rearrangements,
(21) P&
cq+ cQ
=
(21) ~
(cq
)
(1 `_ C)(1-q
(22)
qaQ&
=
(1 -_
)+
+ , 1o)
Co
1
Cc)(1 _ TiaQ) + (7,cq +
cQ)(1
_-
alm)
These say that whether an exogenous increase in aggregate demand
increases the price level and/or output depends on the value of (1 - Tic) and
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1982]
(Cq
+
MICRO-MACROECONOMIC
q)
131
ANALYSIS
(The magnitudesof the effects depend also on the endogenous
changes in a through q1 and Q) In particular, the following four cases
may be identified: (a) 1 =0, 71cq + coQ> 0 when an increase in aggregate
demand increases only the price level without affecting output; (b) 1- cA >
0o,7cq +,cQ = 0 when output increases with the price level unchanged;(c)
1 _ 7cf > 0, qcq + ,cQ > 0 when both price and output increase; (d) 1- ,c1T =
0, 7cq + 71cQ = 0 when the outcome is indeterminate. In this last case, further
analysis shows that the outcome depends entirely on price expectations. If
firms expect prices to go up by y per cent, they will find it profit-maximizing
to increase their prices by y per cent, irrespective of the value of y. (Compare
the analysis in the next section.)
S~~~~~
Mc
d
MRMR'
q
0
FIGURE 4
Ignoring first the endogenous changes in a, case (a) is illustrated in Figure
4, where dd is the initial demand curve of the representative firm. As aggregate
demand a increases exogenously by x per cent, both the demand curve and
MRC (marginal revenue curve) move rightward by x per cent if the firm does
not foresee an increase in prices, i.e. if vr is not expected to go up with a.
The new profit-maximizing price increases from A to B if MCC is upwardsloping (, cq >0) as depicted and/or shifts upward owing to a positive qcQ
(not depicted). But as p increases, so does ir, which leads to an upward shift
in the demand curve. This leads to a further increase in p and 7r, and so on.
If 1c" = 1, MCC also moves up by the same percentage as vr.A final equilibrium
is reached at E, when both the demand curve, MRC and MCC have all moved
upward by x per cent to d"d",MR" and MC' respectively. This involves an x
per cent increase in p and no change in q. Had the firm foreseen the inevitable
increase in prices as a increases under the conditions of 1- 71c" = o, cq + qcQ >
0, its demand curve may jump directly to d"d" and it may adjust its price
from A to E in one go.
The inclusion of the endogenous changes in a does not affect the qualitative
results, since they have the same effects as the exogenous change and hence
affect only the magnitude of the total effects. For example, in the present
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132
[MAY
ECONOMICA
case (a), the endogenous change in a reinforces/partially offsets the increase
in prices discussed above if q" is positive/negative. Reinforcement may be
taken to be the rule, since q " (and also q 'o for other cases) may be reasonably
taken as positive or at least non-negative. This reinforcement through qTi
may be called the price-multiplier effect and may be illustrated similarly to
the income-multiplier effect. A change in X (say money supply) increases a
by x per cent (this is the initial exogenous increase) at existing income and
price levels. In the present case (a), this leads to a x per cent increase in
prices with no effect on output. If Ta, > 0, aggregate demand increases further
even if X does not change further. If Ti 7 =4,the final equilibrium will involve
an increase in prices by 2x per cent, the price-multiplier being 1/(1 -, air).
This price-multiplier effect is different from the price-cost-price effect through
the effect of ir on the demand and cost curves illustrated in Figure 4. This
price-cost-price spiral operates even if a,T = 0.
It may be mistakenly believed that a price-multiplier larger than one is
inconsistent with the (simple) monetarist theory of aggregate demand determination. However, with no money illusion, we have shown in (18) above that
1
aX
aX
d
as
a. . If
= 2.1 Thus, to get an exogenous increase in a by
7
71= 77=
]2, 7
=1 per cent, X (money supply here) has to be increased by 2 per cent to begin
with. In the presentcase of Ti = 1, ceq + -ac > , we have, from (21),
p&
a
da p
(da/aX) dX p
dp X
ada X
a
dp
_dp
PX
1
aX
dX p/ aXa 1aa
1
x
/T
x 1 irrespective of the value of the price multiplier.
Hence we actually have -pX
This is quite consistent with monetarism with X - money supply. Our more
general formulation is useful in allowing for the possibility of factors (including
but not exclusively money supply) affecting a initially by x per cent and finally
multiplied into kx per cent.
Now consider the contrasting case (b), illustrated in Figure 5. The value
of eq + TicQ equals zero if either MCC is horizontal (cq = 0) and does not
S
P=P,
N
O
d
X
q
q/
q
FIGURE 5
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1982]
MICRO-MACROECONOMIC
ANALYSIS
133
shift (7 cQ = 0), or if the slope of MCC is just balanced by its reverse movement.
These are shown as cases 1, 2, and 3 in Figure 5. An increase in aggregate
demand by x per cent moves the demand curve and hence MRC rightward
by x per cent if X is unchanged. The new MRC (i.e. MR') intersects the new
MCC (MC'1, MC'2 or MC'3 as the case may be) at an output level x per cent
higher, with the profit-maximizing price remaining unchanged, confirming the
original expectation of no price changes. If 1 - 7C' > 0, it can be shown that
this is the only expectation that will be realized in the present case of
cq+q co= 0. The endogenous change in aggregate demand now works
through q'Q and reinforces the increase in output just discussed as q'o can
be confidently taken as positive. This reinforcement effect through q 'Q (mar; see (17)) is in fact the Keynesian
ginal propensity to spend, q71Q =?
income-multiplier effect.
In case (d), if X is expected to increase by x per cent as a increases by x
per cent, the demand curve moves upward by x per cent as in Figure 4, with
the resulting x per cent increase in price, confirming the expectation. If X is
expected to remain unchanged as a increases by x per cent, the demand curve
moves right by x per cent as in Figure 5, with the resulting x per cent increase
in q and unchanged p again confirming the expectation. In this special case,
any expectation will be self-fulfilling. It then becomes rational to expect
whatever that is expected to be expected!
Anotherinterestingcase ariseswhen 1 - cCIT= o,7 cq + , cQ < 0. Here, from
the consideration of the responses of cost to output, it seems that a price
reduction is justified as a increases. But (21) and (22) dictate a price increase
with no response in output. What happens is that, if firms reduce prices in
view of the lower MC as output expands with a, this reduction in Xr will
further lower and flatten the demand curve, leading to further output
expansion and price reduction, failing to reach an equilibrium if 1- ,CI =0
and <cq+ 0cQ<0 persist. Somewhat surprisingly, the expectation that will be
realized under this condition is for 7rto increase by the same proportion as
a, leading to the outcome depicted in Figure 4. Nevertheless, if 77cq+ 77cQ < o
holds, it is more likely that cumulative price reduction will result until the
condition no longer holds. It is very unlikely that firms will revert to expecting
price increases even if that expectation will be realized if held (and if 1 - .7cI =
O).
6
The discussion above may be summarized into the following proposition.
Proposition 2. With an exogenous increase/decrease in aggregate demand
(excluding the price and income-multiplier effects, which reinforce the following responses), (a) the Quantity theory case; the average price increases/decreases by the same proportion with no change in output if MC (marginal
cost) is positively responsive to output (, cq +,, cQ>0) and proportionately
responsive to the average price (q c= 1); (b) the Keynesian case; output
increases/decreases by at least the same proportion with no increase/decrease
in the average price if MC is not positively responsive to output and less than
proportionately responsive to prices: (c) the intermediate case: both output
and the average price increase/decrease for cases in between (MC positively
responsive to output and less than proportionately responsive to prices);
(d) the Expectation Wonderland: if MC is not responsive to output but
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134
[MAY
ECONOMICA
proportionately responsive to prices, the outcome depends entirely on
expectation that will then be self-fulfilling.
IN PRICE EXPECTATIONS
V. CHANGES
It may seem a little odd to analyse the effects of changes in price expectations themselves. If demand and cost conditions do not change, should prices
not remain unchanged? Usually, changes in price expectations are due to
changes in the objective factors. But changes that are due mainly to subjective
factors or to wrong subjective estimates of objective factors are not inconceivable. We wish to concentrate here on the pure expectations effect as distinct
from the effects of changes in the objective factors (analysed above) that may
also trigger changes in price expectations. It may be thought that a purely
expectational change, without any objective base, cannot prove to be correct.
Any short-term effect is thus likely to be nullified subsequently. While this
is true in many cases, we shall see that there are some special sets of conditions
where a purely price-expectational change will be self-fulfilling, even if not
validated by changes in (nominal) aggregate demand. Moreover, this is true
even if the expectation is confined to the business sector (firms) and not shared
by consumers/input-suppliers.
To analyse the effect of price expectations by the business sector, replace
the actual average price Xr by the expected average price X' in deriving the
equilibrium condition for the representative firm. Then we have to replace
dT in (9) by dii, and we do not necessarily have d4T-= dp though we still have
= p (as we are starting from an equilibrium). Then, instead of (10), we
X =
have from the revised (9), after substituting in
i
hh"
q- aq- p- A
aA aA aq = ag2hI2ap
aq ap ap A ap q p
dQ =-dq
-n qnP/p,
q
as before, and dividing through by c =,
gq qP dp
(23)
(1_ nc
+
q
_
(cq
qqAP)
X
p
+
_
,cQ)
dq+ dJ
C
q
where
Au=marginal revenue (MR) =p + ih-(p/ )
niq
a
aq
h'(plr
91
A
is the elasticity of MRC, and nqp (aq/ap)(p/q) is the demand elasticity of
the representative firm.7 Substitute dq/q from (14) into (23) (taking dc =0
to isolate the effects of a purely expectational change), obtaining
2
(24)
i
o'"1
(71cq
+
+ 7 cQ)(oT1 + 71,
gq
1p(171)
-1) + (aQc+ (
c)(1 cq+
1)(1-71
-1)
where
p4
dp i_
dfr p
do X
7r
CY
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)
1982]
MICRO-MACROECONOMIC
135
ANALYSIS
For the change in price expectations to be fulfilled, we need a" = 1. The
term after one in the right-hand side of (24) must thus equal zero. The
numerator of this term equals zero if either (a) q'qf= 1 and ao* + ,q" = 1, or
(b) < "m= 1 and (71cq + cQ ) = 0. This term will also go to zero if the denominator
goes to plus-or-minus infinity but the numerator remains finite. This will be
the case if either (c) MRC is vertical, i.e. q q equals (minus) infinity; (d)
= 1 and either MCC is vertical (all firms producing at absolute
uT +q
1
capacity), or the input supply is perfectly inelastic, i.e. qcq and/or qcQ equals
infinity. If only ,cq + ,cQ = o but o-" + q' $ 1, the numerator also goes to
infinity as the denominator does, failing to make the term zero.
It may appear that the denominator will also go to infinity if the firm's
elasticity of demand _q , is infinitely large (the case of perfect competition).
However, as ? qp = _%, q = 0. And q qp(= qP) is undefinedin the case
of perfect competition. Nevertheless, if we replace the expression for total
revenue in (7) by iTq for the case of perfect competition, we can derive an
equation identical to (24) except that the term q qqqp(1 - q`) is absent. Then,
if qcr = 1, we have ao* = .ra/(1 - q`)
This equals one if and only if co" +
71q = 1. Hence this is the same as case (a) above. It may be thought that
partial validation is sufficient since o*" = 1 -7 t < 1 if 71X >0, as may be
reasonably assumed. However, o-" < 1 does not imply that the exogenous
demand factor (e.g. money supply) does not have to increase by the same
proportion as r. Write
s
da r (aa/aX)dX7r
a
a
d- r dar
aa
aX
X dX
a dir X71
"
xs
With no money illusion, q1tX= 1 - 7qax (equation (18)). Hence, o* + 1 = 1
implies `x0orx4+ 1 - I1ax = 1, which implies ox- = 1 or full validation. We may
safely conclude that, with perfect competiton and no money illusion, a
price-expectational change can be realized if and only if it is fully validated.
It may also be noted that the case of a vertical MRC (,qq = -_) involves
a kink in the demand curve and hence, strictly speaking, violates our assumption of differentiability. Nevertheless, a closer examination reveals that a
kinked demand curve (details to be discussed in Ng 1981c) is quite consistent
without analysis in this section.
Proposition3 (a) A price expectational change will be self-fulfilling if at least
one of the following holds: (i) (a trivial validated case): MC (marginal cost)
responds proportionately to prices (7 cT = 1) and aggregate demand also
responds proportionately (all in the same direction); (ii) MC responds proportionately to prices but not to output (, cq + , cQ = 0); (iii) the marginal revenue
curves of firms are vertical; (iv) the marginal cost curves and/or input supply
curves are vertical (71cq + , cQ = cx) and aggregate demand responds proportionately to prices (another validated case). (b) With perfect competition and
no money illusion, a price-expectational change can be realized if and only
if it is fully validated (oX4 = 1).
Case (b) involves the same conditions as the case of Expectation Wonderland discussed above with respect to changes in aggregate demand. Here we
see that any price expectation will be realized even in the absence of exogenous
(and endogenous if desired) changes in aggregate demand. Consider Figure 6,
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136
[MAY
ECONOMICA
dd~
~~
d'
$~~~~~~~~~~~~
MR
q 4q
O
q
FIGURE 6
where the representative firm is at the original equilibrium point A on the
demand curve dd.9 If both aggregate demand and the average price are expected
to increase by 10 per cent, the demand curve moves vertically upward by 10
per cent to d'd". But if only X+ increases by 10 per cent and a remains
unchanged, the demand curve moves up only to d'd', which, in comparison
with d'd", is 10 per cent less horizontally (as a is 10 per cent less and * is
the same in both cases). With 'q" = 1, MCC moves up by 10 per cent to MC',
intersecting MR' at an output (q') 10 per cent (roughly-slightly over 9 per
centto be strict)lowerthanthe originaloutputq.If cq + cQ = 0, the secondary
shift in MCC (due to nqC), if any, is balanced by the slope in MCC. (Figure
6 depictsthe case of qcq = IcQ = 0.) Hence the equilibriumstays at E, which
involves a price 10 per cent higher than at A and equal to that at F, the
equlibrium that would result had aggregate demand increased by 10 per cent
as well. The two validated cases i and iv in Proposition 3 are illustrated in
Figure 7. As both Xr and a increase by x per cent, the demand curve of the
representative firm moves vertically upward by x per cent. Then, if either
MCC is vertical or if MCC (any shape) moves upward by x per cent (qC' = 1),
d
MC
B
q
O
FIGURE 7
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19821
MICRO-MACROECONOMIC
ANALYSIS
137
the new equilibrium (E) will involve a price x per cent higher (and no change
in output), confirming the expected increase.
VI.
AN
APPLICATION
TO SALES TAXES
According to the traditional partial equilibrium supply-and-demand analysis, an increase in a sales tax on a commodity increases the price, but by less
than the amount of the increase in the tax as long as the industrial demand
curve is downward-sloping and the supply curve is upward-sloping. However,
most businessmen and consumers believe that a sales tax is usually fully
(sometimes more than fully) passed on to consumers even in the short run
before considering entry and exit. This puzzle can be explained by using our
analysis. (A sales tax on a specific industry has similar primary effects as a
general sales tax; see Ng, 1981a.)
Writing t as the rate of sales tax, the total revenue of the representative
firm has to be multiplied by (1- t) to arrive at the net-of-tax revenue. With
this modification we may derive, in a similar way as the derivation of (9),
(25)
(1-
CI) dp
p _(Cq+7co)
dq
q
dt
1-t
di
c
It can be seen that dt/(1 - t) enters the equation in exactly the same way
as dc/c. A change in sales tax by x per cent (of net-of-tax price) can thus be
analysed in the same way as an exogenous change in marginal costs. We thus
have:
Proposition 4. Proposition 1 above, on the effects of an exogenous change in
marginal costs, also holds for a change in sales taxes.
For example, the primary effects of an increase/decrease in a sales tax by
x per cent (of net price) are to increase/decrease the average price by less
than/the same/more than x per cent if MCC is upward/horizontal/downwardsloping.8 It must however be noted that, for an economy-wide change in tax
rates, the government revenue may be affected, leading to feedbacks on
aggregate demand a which may thus respond differently to the case of an
exogenous change in marginal costs. In other words, oc and (a may be of
different values for the case of an economy.
VII.
CONCLUDING
REMARKS
The results we have obtained (summarized into the four propositions
above) are quite remarkable. Perhaps some readers may suspect that we are
able to obtain these results only by defining a representative firm in a way
that makes the results tautological. A careful examination should convince
them that such is not the case: otherwise the Expectation Wonderland would
always prevail, rather than applying when a rather specific condition is satisfied.
By construction, it must be tautologically true that the output and price
of the representative firm must be representative of the whole economy in
the initial equilibrium. With some disturbances (cost, demand changes, etc.),
a new equilibrium may be attained. Our analysis is based on the hypothesis
that the responses of the whole economy may be approximated by that of
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138
ECONOMICA
[MAY
the representative firm. This is certainly not a tautology, and the propositions
based on this hypothesis may be, at least in principle, verified or falsified. By
the way the representative firm is constructed and the fact that secondary
repercussions are taken account of, one may reasonably believe that the
approximation will be acceptable or at least superior to a purely partial analysis
or a purely aggregative analysis.
Our analysis indicates that the elasticities of the marginal cost curves and
the elasticities of marginal cost with respect to prices and with respect to
output are very important in determining the effects on output and prices of
almost all changes we analysed. (It is expected that average costs will also
play a role when the entry/exit of firms are included). The empirical findings
of these elasticities are thus very important. Research attention should perhaps
be directed towards these empirical findings. Our propositions may also be
tested empirically and our analysis extended and applied to many economic
problems. But these can hardly be done in one paper; the writer hopes to
extend the analysis to the case of an industry and revenue- maximizing firm
(Ng, 1981a), to introduce entry/exit of firms (1981b), and to analyse the
complication of oligopoly (1981c).
ACKNOWLEDGMENTS
I am gratefulto a referee and the followingpersonsfor helpfulcomments:David
Friedman,Joseph Greenberg, MurrayKemp, Simon Domberger, Kevin Roberts,
James Mirrlees,Peter Warr. The paper was drafted when I was visiting VPI and
ManchesterUniversity,and I wish to thank their financeof a visitingprofessorship
and a Simonseniorresearchfellowshiprespectively.
NOTES
The concept of a representative firm was first used by Marshall. However, he used it to
determine the normal supply price of a perfectly competitive industry. Here, the response of the
representative firm is used to approximate the response of a (typically non-perfect competitive)
industry or the whole economy. More importantly, the effects of macroeconomic variables and
secondary disturbances are included in the analysis here. Our non-perfect competition aspect
resembles the analysis of imperfect competition in some aspects. But the theory of imperfect
competition greatly needs to be cast in general equilibrium terms, as emphasized by Triffin
(1940), who himself has not gone much further than delineating specific cases (pure monopoly,
circular and atomistic homeopoly and heteropoly, etc.). On the other hand, modern studies of
monopolistic general equilibrium have to be based on some highly simplistic assumptions (not to
mention the loss of comparative static results). See Kuenne (1967, p. 219n.) on some restrictive
aspects of Negishi's (1961) pioneering analysis. Nikaido (1975) works with objective demand
functions (this is what he means by "effective demand", which is not aggregate demand in the
macroeconomic sense as in our analysis) and hence represents an improvement over Negishi's
analysis based on perceived demand functions. However, Nikaido has to adopt the very restrictive
Leontief system.
2 While p # 7rcannot be an equilibrium solution, we must consider these points on the firm's
demand curve as they determine its marginal revenue curve. Readers uncertain about this should
re-read the discussion on the fallacy of attribution above.
3 More traditionally, we may write the cost function as C(q, 7r,w, E) where w is the wage-rate.
But since w may be written as a function of 7r,E and L (labour), where L is itself a function of
7rand Q, our cost function avoids the explicit introduction of w. It can of course be introduced
explicitly at the cost of some notational complications without any changes in our results.
4 Alternatively we may note from (8') that the arguments of h and h' are constant at p =
7r,
whence dp + (h/h') dIr = cq dq - c0dQ + c, d7r+ dc from which (10) follows immediately on using
h/h' = (c - p)/7r.
5 The first equation in (12) may also be derived by the differentiation of (5) and substituting
in dp - d,,r. This also confirms the consistency of the micro-specification (5) with the aggregate
requirements (6).
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MICRO-MACROECONOMIC
ANALYSIS
139
6This proposition is confirmed in Ng (1980), where the microeconomics of non-perfect
competitive firms is incorporated into an explicitly macroeconomic model. The proposition is
retained here for completeness of the present more comprehensive analysis.
7To abstract from the complication of uncertainty (apart from the expectation about 7r),we
have reasonably assumed that the firm knows its cost function with certainty, making q7c*= c.
Hence we need not replace
8
C1Tc by -C1T
in (23).
The result for the special case of constant marginal costs is also obtained by Williamson
(1967) for the whole economy using Kalecki's model of price determination (by current costs
and previous prices), taking account of prices charged by other firms (a feature that forms a
central part of our analysis) but assuming linear demand curves (immaterial to the result) and
relegating endogenous cost changes to the wage equation (central to the result). I am indebted
to the anonymous referee for drawing my attention to Williamson's contribution.
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