Prof. Dr. Fritz Helmedag
Technische Universität Chemnitz
Fakultät für Wirtschaftswissenschaften
Lehrstuhl Volkswirtschaftslehre II (Mikroökonomie)
D-09107 Chemnitz
Phone:
Fax:
E-Mail:
+49/(0)371/531-4185
+49/(0)371/531-4352
Fritz.Helmedag@wirtschaft.tu-chemnitz.de
How national income and profits depend
on expenditures and total wages
Abstract
Macroeconomic analysis often employs a uniform saving rate. Yet, this
approach is only consistent with two special cases: either all households
spend the same fraction of earnings or the shares in national income are held
constant by assumption. Both premises appear misleading. It is shown that
fluctuations in investments (as a synonym for autonomous demand) generally
affect distribution. In addition, the impacts of changing wages on the social
product (‘purchasing power argument’) or rather profits (‘wage-profit-tradeoff’) are revealed.
Keywords:
Wages, Profits, Income, Saving rates, Profit sharing
JEL-classification: E12, E25
How national income and profits depend
on expenditures and total wages
1. Effective demand in short supply
Popular textbooks on macroeconomics create the impression that the study of
multiplier effects and the theory of income determination in the tradition of
John M. Keynes, Michal Kalecki, Nicholas Kaldor and others are nowadays
doomed to a shadow existence.1 However, considerations in this vein should
be acknowledged as indispensable since convincing explanations for the
volume of social product and its distribution must fulfil the flow-balance
conditions.
It is a salient feature of exchange economies that all expenditures turn into
income, whereas the converse does not hold: non-spending defined as saving
brings about the accumulation of wealth. Opinions differ whether or not the
formation of reserves jeopardizes the level of employment. Even in this issue
mainstream economists stress the strength of the laws of supply and demand,
but show disregard for the mutual dependencies and interactions between
macroeconomic aggregates.
1
Cf. e.g. Blanchard (2003), Burda / Wyplosz (1997), Heijdra / van der Ploeg
(2002), Minford / Peel (2002), Mankiw (2000), Romer (1996).
2
How national income and profits depend on expenditures and total wages
The aim of the present paper is to provide a clear-cut and general account
of how spending decisions and the wage bill influence the value of net output
and profits. Moreover, the ‘purchasing power argument’, quite often put forth
by trade unions’ representatives, is scrutinized. They claim that an increase in
total wages raises expenditures and thus the demand for labour. In contrast,
employers resort to a ‘wage-profit trade-off’. A higher pay and, therefore, an
allegedly lower profit entails a decline in investments and soaring prices. As
a consequence, inflation and dismissals would occur. Again, such widespread
doctrines deserve to be examined meticulously.
To keep the analysis as simple as possible, the economy under
investigation is assumed to be closed and the government does not perform
any economic activity. The society consists of workers and employers (or
capitalists) who carry out the entrepreneurial function. Saving habits are
class-specific and do not refer to the sources of income, i.e. workers follow a
uniform propensity to consume even if they earn a part of profits.
Finally, only two types of spending are distinguished: on the one hand the
households’ purchases as a fraction of income and on the other hand
autonomous demand alternatively called ‘investment’. These exogenous
expenditures can be financed either by cash, the liquidation of assets, or
outside means. Price and quantity reactions are not separated; in principle
both adjustments are possible.
How national income and profits depend on expenditures and total wages
3
Within this framework, three models are presented. In the first setting, the
society’s consumption hinges on a uniform saving rate. Next, specific
spending habits for workers and capitalists are postulated. In the third variant,
profits are divided between the two groups according to their respective
savings. For each of the three alternative set-ups, the equations for the
national income and the profits are established. Subsequently, the
consequences of changing investments and wages are ascertained.
2.
A single class society: model 1
Usually s denotes the saving rate. Yet, the standard literature fails to inform
the reader whether this symbol indicates the saving rate of identical
households or the weighted average of class-specific ones. In the latter case s
inevitably varies with the distribution of income. Since textbooks do not offer
such an interpretation, their world seems to be populated by households
which share a common propensity to save with 0 < s ≤ 1. Then, total
consumption C in money terms amounts to C1 = (1 − s )Y1 where Y1 stands for
the nominal national income.2 Because the government and the rest of the
world are not involved here, only investments (I ) representing autonomous
demand have to be considered. Equating the value of net output and total
2 The
index refers to the corresponding model.
4
How national income and profits depend on expenditures and total wages
expenditure gives Y1 = C1 + I . Inserting in this expression the households’
consumption yields:
Y1 =
I
s
(1)
Independent of wages, a rise in investments will entail a higher equilibrium
social product. The ‘simple’ multiplier relation reads:
dY1 1
=
dI s
(2)
As long as price and quantity adaptations are not isolated, it remains unclear
to what extent employment is correlated with changes in autonomous
demand. If the supply of goods and services is inelastic, prices will more or
less rise.
It is possible to establish a link between the money values of the wage bill
(W) and profits (P). By definition, the latter consists of the difference
between national income and the remuneration for the labour power:
P1 = Y1 − W
(3)
From an economic perspective, however, the two income components are not
on equal footing. Under capitalistic conditions, aggregate wages generally do
not suffice to purchase the social product. Expenditures stemming from other
sources must appear so that profits arise: “However great the margin of profit
on a unit of output, the capitalists cannot make more in total profits than they
How national income and profits depend on expenditures and total wages
5
consume and invest (inclusive of accumulation of unsold goods)” (Kalecki,
1942, p. 260).
Combining equations (1) and (3) leads to:
P1 =
I − sW
s
(4)
Obviously, profits are positive if and only if:
I > I1 : = sW
(5)
As a consequence, for profits to exist at all autonomous demand has to
exceed the savings out of wages.
Whenever the wage bill increases, profits decrease by the same amount in
the present scenario. Equation (3) verifies this statement. The derivative of
profits with respect to total wages expresses the widespread idea of an
existing one-to-one trade-off between the income categories:
dP1
= −1
dW
(6)
Since the social product is determined by s and I alone, wages do not affect
the value of net output. Hence, an augmented pay causes an equal reduction
in profits. Thus, the fight for income shares tends to be extremely fierce.
Equation (4) provides information on how an investment variation alters
profits:
6
How national income and profits depend on expenditures and total wages
dP1
=
dI
dW
dI
s
1− s
(7)
Dividing profits (4) by social product (1) gives the share of income accruing
to the capitalists:
P1
sW
= 1−
Y1
I
(8)
To keep the share of the profits in income constant, a very special condition
P 
d  1  − s  dW I − W 
Y
dI
 = 0 immediately follows:
is required. From  1  = 
2
dI
I
dW W
=
dI
I
(9)
National income and its components change with the same percentage if and
only if total wages and autonomous demand have identical growth rates. This
situation arises exclusively by pure chance. In a capitalist society, the righthand side of (9) is necessarily positive. Rising investments, however, have no
predetermined effect on the wage bill. Frequently, an increase in autonomous
demand raises the remuneration of the working class. In contrast, it is also
possible that labour saving investments reduce their income, i.e.
dW
< 0.
dI
Indeed, cost minimization drives the choice of technique. Finally a variation
of investments may even be without impact on total wages. This is
conceivable in cases of excess capacity, when more or less output is produced
How national income and profits depend on expenditures and total wages
7
with the same staff. In such situations, fluctuations in autonomous demand
are completely reflected in profits which change social product by the same
amount. At any rate, condition (9) either cannot be fulfilled or only by some
amazing fluke. Most likely, the share of profits in social product changes
with investments, though it is uncertain in which direction.
The preceding analysis rests on the simplifying assumption of a uniform
saving rate. In this regard a more realistic approach is necessary.
3.
Class-specific saving behaviour: model 2
According to the ‘fundamental psychological law’ (Keynes, 1936, p. 96) the
propensity to consume falls with income. If capitalists receive a relatively
high per-capita income, their saving rate (sP) will be larger than the workers’
one (sW):
0 ≤ sW < sP ≤ 1
(10)
Aggregate consumption is determined by C2 = (1 − sW )W + (1 − sP ) P2 , and
the social product becomes Y2 = (1 − sW )W + (1 − sP )(Y2 − W ) + I . Solving this
equation for Y2 yields:
Y2 =
I + ( sP − sW )W
sP
(11)
8
How national income and profits depend on expenditures and total wages
Contrary to model 1, the national income now depends on the wage bill. An
increment in pay implies a higher social product:
(10) sP − sW (11) dY2
≤1
=
0 <
sP
(12)
dW
Under such circumstances, the purchasing power argument applies in
principle. Clearly, total income forms an upper limit on the wage bill.
Profits amount to P2 = Y2 − W =
I + ( sP − sW )W
− W ≥ 0 . Rewriting
sP
creates more clarity:
P2 =
I − sW W
sP
(13)
Equation (13) states that profits can be increased by a higher autonomous
demand and less savings. Kaldor put it in a nutshell: “[…] Keynes regards
entrepreneurial incomes as being the resultant of their expenditure decisions,
rather than the other way round – which is perhaps the most important
difference between ‘Keynesian’ and ‘pre-Keynesian’ habits of thought”
(Kaldor 1955/56, p. 94, note).
In this scenario profits are positive whenever investments overcompensate
workers’ savings:
I > I 2 : = sW W
(14)
How national income and profits depend on expenditures and total wages
9
The profit function (13) describes the relation between wages and profits.
Forming the derivative of profits with respect to the wage bill yields
dP2
s
= − W . Due to condition (10), the range of the ‘reaction coefficient’ is:
dW
sP
−1 <
s
dP2
=− W ≤0
dW
sP
(15)
This economy seems to be a more harmonic place than the one of model 1.
Apparently, increasing wages do not require equally decreasing profits as
before. For sP > 2 sW , higher wages cause the social product to rise by an
amount larger than the corresponding decline in profits. Therefore, the
struggle over distribution of income may be moderated.
Next on the agenda is multiplier analysis. Differentiating equation (11)
with respect to autonomous demand ensues in:
dY2
=
dI
1 + ( sP − sW )
dW
dI
sP
(16)
Surprisingly, social product does not always grow with investments. In case
labour is substituted by machinery, Y2 falls as long as ( sP − sW )
3
dW
< −1. 3
dI
Strictly speaking, dW represents the present value of the original and all
(discounted) subsequent reductions in wages.
10
How national income and profits depend on expenditures and total wages
Equation (13) makes information about the profit reaction available:
dP2
=
dI
1 − sW
dW
dI
sP
(17)
Unexpectedly again, an additional autonomous demand compresses profits in
situations where sW
dW
> 1.
dI
It is interesting to find out how the independence between the average rate
of savings and investments is established. From s2 = sW
W
P
+ sP 2 , we get
Y2
Y2
after some obvious manipulations:
s2 =
sP I
I + ( sP − sW )W
(18)
Regularly, the level of investments affects the average saving rate. The
exception requires:
dW


( sP − sW ) 
sP  W ( sP − sW ) − I
ds2
dI
 =0
= 
2
dI
( I + W ( sP − sW ) )
(19)
Investments exert no influence on the average saving rate provided that
dW W
= . Amazingly enough, condition (9) must hold strictly even in this
dI
I
model to ensure that income shares and thereupon the average saving rate do
How national income and profits depend on expenditures and total wages
11
not vary with investments.4 Apart from such a pure coincidence, it is unclear
at the outset whether the share of profits will move in the same or in the
opposite direction as autonomous demand. Yet, owing to labour costs curbing
process innovations, i.e. for
dW
< 0 in equations (16) and (17), profits rise
dI
particularly strong, whereas social product may even shrink.
4.
Profit sharing: model 3
It is plausible that employees who save have an unearned income as well.
Pasinetti (1962) postulated a division of profits between workers (PW) and
capitalists (PP) equal to the ratio of their respective savings.5 Once this
premise is accepted, from
4
PP
sP PP
=
follows:
PW sW (W + PW )
In his often cited survey, Kaldor assumes full employment. Therefore “[…]
a rise in investment […] will raise prices and profit margins […]” (1955/56,
p. 95). At the same time, however, income is held constant. Thus, according
to Kaldor, an increase in investment seems to raise the share of profits in
social product.
5
Pasinetti’s analysis has been intensively debated; the discussion is reviewed
by Ahmad (1991, ch. 13).
12
How national income and profits depend on expenditures and total wages
PW =
sW W
sP − sW
(20)
A higher workers’ saving rate raises their profits relative to the remuneration
for labour services as long as sW < sP (see Bortis, 1993, p. 109). In total,
they get:

sW 
sPW
W + PW = W  1 +
=
 sP − sW  sP − sW
(21)
Given the income of workers and capitalists, national consumption reads
 s W 

s PW 
C3 = (1 − sW )  P
 + (1 − s P )  Y3 −
 = s PW + (1 − s P )Y3 = Y3 − s P P3 .
s P − sW 
 s P − sW 

Adding investments yields the entire expenditure: Y3 = C3 + I = Y3 − sP P3 + I .
Solving this expression for aggregate profits results in:
P3 =
I
sP
(22)
Apparently, profits arise when investments are positive independent of the
workers’ saving rate. Although the employees obtain unearned income, the
sum of profits in this economy hinges exclusively on the capitalists’
decisions. Accordingly, no wage-profit-trade-off exists in this setting. Profits
remain unchanged by wage bill variations:
dP3
=0
dW
National income is calculated with recourse to the profit function (22):
(23)
How national income and profits depend on expenditures and total wages
Y3 = P3 + W =
I + sPW
sP
13
(24)
It can directly be seen that:
dY3
=1
dW
(25)
If both classes of the society receive profits, the purchasing power argument
applies in the strictest sense. Profits accruing to capitalists come to:
PP = P3 − PW =
s W
I
− W
sP sP − sW
(26)
This equation portrays the two-stage process in which the profits are
generated and allocated. In the first step, the entrepreneurs alone determine
the level of entire profits via I and sP. Subsequently, the workers participate
by performing a specific saving behaviour. For 0 < sW < sP , positive
capitalists’ profits require a higher autonomous demand than before:
I3 >
sP sW W (14) sP
I2 > I2
=
sP − sW
sP − sW
(27)
From social product (24) the effect of an investment variation on the net
output is derived:
dY3
=
dI
1 + sP
sP
dW
dI
(28)
14
How national income and profits depend on expenditures and total wages
As experienced in the previous model, an expanding autonomous demand
will not necessarily raise social product: whenever
dW
1
<−
the nominator
dI
sP
of (28) becomes negative. Yet, according to equation (22) aggregate profits
always increase:
dP3 1
=
dI
sP
(29)
Again the question arises, how to isolate the average rate of savings from
fluctuations in investments. Substituting equations (21), (24) and (26) in
s3 = sW
W + PW
P
+ sP P leads to:
Y3
Y3
s3 =
sP I
I + sPW
(30)
Remarkably, the saving behaviour of the workers does not contribute to the
average saving rate here. The condition for its constancy requires:
dW 

s p 2 W − I

ds3
dI 

=
=0
dI
( I + sPW )2
Once more, the proviso
(31)
dW W
=
must hold true, which hardly ever happens.
dI
I
Otherwise, income shares and the average saving rate are connected to
investments and cannot be considered as given. Moreover, in this model a
possible ‘contraction setting’ should be noted. Inspecting (26) reveals:
How national income and profits depend on expenditures and total wages
sW dW
dPP
1
=
−
dI
sP sP − sW dI
15
(32)
The second term on the right-hand side of equation (32) reflects the alteration
of the workers’ profit (20) due to a changing investment. Thus, if
dW sP − sW
, the repercussion on the capitalists’ profit is negative.
>
dI
sP sW
Paradoxically, augmenting autonomous demand diminishes their income. The
efforts to offset such losses intensify the entrepreneurs’ urge to control costs.
If, by means of labour saving technical progress, the capitalists succeed to
reduce the wage bill, their profits soar whereas social product possibly
declines. This, in return, aggravates the battle over distribution.
5.
A concluding assessment
The table below appears useful to summarize and annotate the findings of
this enquiry. The first four rows display the equations for national income
and profits together with the respective multiplier formulae. For differing
consumption patterns, the average saving rates presented in row five depend
both on the wage bill and on investments. A uniform saving rate excludes
these interrelations.
The sixth row demonstrates that the purchasing power argument gains
ground when considering model 1 through 3. In the third scenario, a rising
wage bill increases the national product by 100 %. At the same time, there
16
How national income and profits depend on expenditures and total wages
exists no trade-off between the income categories: varying wages do not
affect the level of profits. In model 3, only the capitalists decide on this part
of the social product. Thus, a profit sharing system seems to be more
receptive to an increment in wages.
Yet, the third arrangement bears conflict because profits have to be split
between the parties. Differentiating equations (20) and (26) with respect to
wages results in
dPW
sW
dP
=
= − P . Obviously, rising wages divert
dW sP − sW
dW
capitalists’ income to the workers’ profits. So the labour force enjoys higher
earnings from both sources. Furthermore, provided that 0 < sW < sP , the
comparison with equation (15) shows that the capitalists will suffer a more
severe cutback in profits as opposed to a situation without profit sharing.
Generally, fluctuations in autonomous demand alter the distribution.
Then, the average saving rate varies too. This correspondence is invalidated
only under two special premises. Firstly, either a uniform propensity to
consume is simply supposed for all levels of income or, secondly, equation
(9) is taken for granted, i.e.
dW W
=
ensures constant income shares.
dI
I
However, theories concerning the determination of social product and its
development are wide of the mark as long as they are based on such counterfactual prerequisites.
17
How national income and profits depend on expenditures and total wages
Table: A survey of results
profit sharing
no
yes
class-specific saving rates
0 ≤ sW < s P < 1
uniform
saving rate s
model 1
national income
Y
I
s
income
multiplier
dY
dI
1
s
profits P
(1)
(2)
model 2
I + ( sP − sW )W
sP
1 + ( sP − sW )
dW
dI
sP
I − sW
(4)
s
I − sW W
sP
model 3
(11)
(16)
(13)
I + s PW
(24)
sP
1 + sP
sP
dW
dI (28)
I
sP
(22)
dW
profit multiplier
dW
1
1 − sW
1− s
(29)
dI
dP
dI (7)
(17)
sP
sP
s
dI
sP I
sP I
average
(18) s3 =
(30)
s2 =
s
I + ( sP − sW )W
I + sPW
saving rate
purchasing
power
s −s
argument
0 < P W ≤ 1 (12)
0
(1)
1
(25)
sP
dY
dW
wage-profittrade-off
dP
dW
−1
(6)
−1< −
sW
≤0
sP
(15)
0
(23)
18
How national income and profits depend on expenditures and total wages
Arguably, in most instances the assumptions of model 2 come closest to
reality. In case workers spend what they get, a higher pay raises national
product by the same amount, while profits remain the same. When employees
save, an augmented remuneration is transmitted into an increasing social
product. The rest has to be covered by profits. Of course, entrepreneurs
steadily try to improve their income position by realizing labour saving
process innovations. If these efforts cause total wages to fall, social product
may even decrease although profits rise. Therefore, economic policy should
attach great significance to those components of autonomous demand which
are most likely to promote the growth of national income.
References
Ahmad, S. (1991) Capital in Economic Theory, Neo-classical, Cambridge and
Chaos. Edward Elgar.
Blanchard, O. (2003) Macroeconomics. 3rd edition, Prentice Hall.
Burda, M. and Wyplosz, Ch. (1997) Macroeconomics. 2nd edition, Oxford
University Press.
Bortis, H. (1993) ‘Notes on the Cambridge equation’, Journal of Post Keynesian
Economics 16, pp. 105-126.
Heijdra, B. J. and van der Ploeg, F. (2002) Foundations of Modern
Macroeconomics. Oxford University Press.
Kaldor, N. (1955/56) ‘Alternative Theories of Distribution’, Review of Economic
Studies 23, pp. 83-100.
Kalecki, M. (1942) ‘A Theory of Profits’, Economic Journal 52, pp. 258-266.
19
How national income and profits depend on expenditures and total wages
Keynes, J. M. (1936) The General Theory of Employment, Interest and Money. In:
The
Collected
Writings
of
John
Maynard
Keynes,
vol.
VII,
Macmillan / Cambridge University Press 1978
Mankiw, N. G. (2000) Makroökonomik. 4. Auflage, Schäffer-Poeschel.
Minford, P. and Peel, D. (2002) Advanced Macroeconomics. Edward Elgar.
Romer, D. (1996) Advanced Macroeconomics. Mc-Graw-Hill.
Pasinetti, L. (1962) ‘Rate of profit and income distribution in relation to the rate of
economic growth’, Review of Economic Studies 29, pp. 267-279.