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The Genuine Meaning of Keynes’ Multiplier*
By Ezra Davar
Abstract
This paper shows that Keynes’ investment multiplier and Kahn’s employment
multiplier are different: (1) Kahn’s approach is compatible with the second phase of
the investment process, when fixed capital (investment) is determinant; while Keynes’
approach is compatible with the first phase when investment (saving) is determinate;
(2) The source of an additional increment of investment in Kahn’s approach is
exogenous, whilst in Keynes’ approach is endogenous.
Also, Keynes’ investment “multiplier” is a reciprocal of the marginal
propensity to invest (save); therefore, its genuine meaning is that of a requirement,
which indicates the quantity of national income required to realize one unit of
investment (saving).
Keywords: Multiplier, Requirement, Walras, Keynes, Kahn, General
Equilibrium Theory
JEL Classifications: B, D5, and E2
Author: Dr, Ezra Davar, Ben-Gurion University of the Negev
Yehuda Hanasi St. 15/17
Netanya 42444
Israel
Telephone: 972-9-8349790
E-mail: [email protected]
* The paper was written to the 2003 UK HET Conference, Leeds, 3-5 September
The Genuine Meaning of Keynes’ Multiplier
By Ezra Davar
“The traditional analysis has been aware that saving depends on income
but it has overlooked the fact that income depends on investment, in such
fashion that, when investment changes, income must necessarily change in
just that degree which is necessary to make the change in saving equal to
the change in investment.” Keynes, The General Theory, p.1841
1. Introduction
The multiplier is one of the main tools for establishing a relationship between
income, investment, consumption and employment developed by Keynes in his “The
General Theory”. Since its very first appearance, the attitude towards the multiplier
has been ambiguous. One group of economists (mainly Keynes’ followers) state that
the Keynesian multiplier is a new paradigm in economic theory and that it is a
milestone in the development of macroeconomic theory. Blaug writes (p.189): ‘The
principal novel prediction of Keynesian economics is that the value of the
instantaneous multiplier is greater than unity’ (see also Pasinetti, p.40). Yet, some
scholars stated that "This is, no doubt, in a general way, correct" (Pigou, 1936). And
"In order to avoid any possible misunderstanding, let me conclude this paper by
emphasizing that there can be no doubt about the importance of Kahn's 1931
multiplier article in the history of the development of macroeconomic theory. Nor can
there be any doubt that his multiplier formula is logically equivalent to the theory of
effective demand" (Patinkin, pp.324-5). But at the same time, there were economists
who indicated doubt regarding the multiplier "... it seems to me doubtful whether, for
the analysis of a fluctuating world, the "multiplier" constitutes much advance over
more crudely "monetary" weapons of thought".(Robertson, 1936); Another group
posed serious doubts concerning the multiplier’s correctness (Ahiakpor, 2001; Hazlitt,
1959). There are authors who considered multiplier as a dynamic phenomenon
(process) (Goodwin, 1947; Hansen, 1953) and V. Chick claims that Keynes’
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multiplier has two views: equilibrium and dynamic (Chick, 1983). Many arguments
were raised against it, such as: the multiplier is severely static (Schumpeter, p.1174)
unrealistic, characterized by instability, and so forth. Keynes’ followers have been
attempting to "improve" the theory (multiplier) of Keynes in order to silence such
critics. A central argument, often used by them, is that the multiplier has rests on a
solid mathematical foundation (Minsky, 1975). In addition, several papers show that
‘the balanced-budget expenditure has a multiplier of exactly one;’ (Samuelson, 1966;
see also Samuelson, 1974; Salant, 1974 (1942); Haavelmo, 1943; Turvey, 1953 and so
on). But this “discovery” does not solve the problem because according to Keynes’
theoretical definition and practical estimation of multiplier, the “indicator’s”
magnitude is strongly greater than one and, investment does not produce income itself
(vide infra). Recent debate between Ahiakpor (2000) and Dimand (2000) and Darity
and Young (2000) have been showing continuing controversial interpretation of the
multiplier and how much the solution of this problem is required. Yet, this unresolved
argument has serious implications, such as its continued presence in textbooks despite
its unresolved status. Nevertheless, "Like many other tools of economic theory, the
multiplier must be used with care, but there is no case for its demise. It remains a very
important relationship in post-Keynesian monetary macroeconomics" (Gilbert, 1982).
It is necessary to stress that Keynes himself in the letter to Beveridge claimed that
“The theory of the multiplier. You write here as though this was a matter on which I
had only dogmatized and not discussed. But not only have I given a long chapter
expressly to this subject, but about half the book is really about it.” (Keynes, 1973,
p.57)
This paper will present and discuss the arguments made by both sides, and also
present a third, new perspective on the Keynes’ multiplier. It will be shown that
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Keynes’ investment multiplier and Kahn’s employment multiplier are different not
only as regards their theoretical meaning, but also with respect their practical
magnitude because of the following reasons: (1) Kahn’s approach is compatible with
the second phase of the investment process, when fixed capital (investment) is
determinant; therefore, the calculation of Kahn’s multiplier is a dynamic process,
albeit though the time-lag is not taking into account; while in Keynes’ approach, when
investment (saving) is determinate, is compatible with the first phase of the
investment process, therefore, the calculation is static; (2) The source of an additional
increment of investment in Kahn’s approach is exogenous, whilst in Keynes’
approach is endogenous; (3) Kahn’s multiplier is a technological-economic
phenomenon, while Keynes’ multiplier is a psychological phenomenon.
Also, it will be shown that Keynes’ investment “multiplier” is a reciprocal of the
marginal propensity to invest (save); therefore, its genuine meaning is that of a
requirement, which indicates the quantity of national income required to realize one
unit of investment (saving).
The paper consists of five sections. Following the introduction, the second section
describes a commonly used general equilibrium theory approach and discusses two
possibilities to income increases: endogenous and exogenous. In the third section,
Kahn’s original definition of multiplier is presented. The forth section reviews
Keynes’ definition of the multiplier. In the final section the genuine meaning of
Keynes’ multiplier will be presented, followed by conclusions.
2. Produced Income and Used Income
For the reasons of convenience, general equilibrium theories state that in an
equilibrium position the total value of all employed services (employed quantities of
services multiplied by their according prices) equals the total value of demand
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(consumption) of all commodities (demanded quantities of commodities multiplied by
their according prices). The first is known as produced (created) national income or
national income in terms of factors’ (services’) prices Yp, while the latter is known as
used (consumed) national income or national income in terms of commodities’ market
prices Y. This statement is known as Walras’ law, which all schools of thought of
economics recognize and, in addition, and something which is very important – it is
compatible with empirical accounts of national income. At the same time, it is
necessary to point out that there is a difference between Walras’ own law and the
post-Walrasian version of this law. Walras’ law means that the law is only satisfied
for equilibrium prices established for each goods and services separately in the
condition that the total demand is equal to the total supply for services and the cost of
production (the supply price) is equal to the demand prices for goods. In addition,
prices are strongly positive even some services are underemployed in equilibrium
state. The Walras followers’ law is satisfied for any prices and if there is unemployed
service, then its price is equal zero. Therefore, the post-Walras’ law is a by-product of
his original law, but Walras’ own law is compatible to economical reality, whilst the
post-Walras’ law is incompatible with reality, whereas this law is fundamental to the
post-Walrasian approach (Davar, 1994). Hence, the Walras’ law can be presented as
n
m
1
1
2  1
 Oi pi   D j  j
Where
pi – is equilibrium price of service i;
Oi – is equilibrium employed (required) quantity of service i, which is either
equal to or less than its available (existence) quantity Qi, i.e., Oi  Qi;
j – is equilibrium price of commodity j;
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Dj – is equilibrium demanded quantities of commodity j.
For the following discussing there are two things must be pointed out: (1) the
employed (required) quantities of services are obtained on the basis of inverse
coefficients of factors (Walras, 1954, pp.240-1; Leontief, 1986, pp.394-5; Davar,
1994, pp.17-8), which means that they include the direct and indirect inputs of
services. So, these required (employed) quantities of services include the total (direct
plus indirect) quantities of services required for the production of demand goods; (2)
the list of demand commodities in general include all kinds of commodities, “public
work”. If there are some commodities that are not included in the list of demand
commodities, it has to be added and required quantities for its production would be
obtained according to the previous note.
The left side expression of (2-1) is national income in terms of factors’ (services’)
prices Yp (produced income) and if we take into account the results of the individuals
model’ solution it have divided into two components: commodities for consumption
(Cd) and saving (S). The right side expression is national income in terms of
commodities prices (used income) and it is also combined by two components:
commodities for consumption (Cd) and new capital goods (investment - I). So, in the
equilibrium situation there are:
Yp =  Oipi = Cd + S,
(2-2)
Y =  Dii = Cd + I,
(2-3)
Yp = Y and Cd + S = Cd + I, therefore S = I,
(2-4)
Equation (2-4) indicates that in an equilibrium situation saving equals investment.
But, individuals who save differ from individuals who invest. This means saving may
differ from investment. Therefore, in order to establish equilibrium, saving must be
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equal to investment by means of changing the interest rate (Chick, 1997, p.174;
Sawyer, p.52).
In addition, it has been shown that the classical and the Walrasian approach -- as
well as Keynes approach -- assumes that in an equilibrium position, might be
unemployed services (including labor) (Davar, 1994), It is necessary to stress that
there is a difference between Walras’ and Keynes’ unemployment albeit that they are
interwoven. The former is observed as voluntary unemployment, while the latter as
involuntary underemployment (Davar, 2002). If the unemployed share of services is
considerable, it will be desirable to attempt to shift equilibrium position, if it is
possible, in order to decrease unemployment and, consequently to increase income
(investment).
2.1 Investment (Saving) as Determinate and Fixed Capital (Investment) as
Determinant
Keynes stated that ‘Thus the traditional analysis is faulty because it has failed to
isolate correctly the independent variables of the system. Saving and Investment
are the determinates of the system, not the determinants. They are the twin results
of the system’s determinants, namely, the propensity to consume, the schedule of
the marginal efficiency of capital and the rate of interest (Keynes, 1936, pp.1834).’ This is a very important statement, which is based on Walras’ theory, but
Keynes destroyed it in the multiplier theory, where investment becomes a
determinant (vide infra).
For the following discussion it is necessary to make clear whether new capital goods
(investment) produced in a certain period (year) transform services (fixed capital)
either in that year or the next year. It is, in general, assumed that new capital goods
produced during a certain year become services the next year. One of the central
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assumptions of Keynes’ analysis is ‘the existing quality and quantity of available
equipment, the existing technique,’ (Keynes, 1936, p.245). This means that new
capital goods do not create (produce) national income in the year when they are
produced. So in this case there is no direct connection between investment and the
creating of national income. There is a connection between them in such way so as to
produce new capital goods (invest) so that according national income must be created.
This is the first phase of the whole process of investment, i.e. when investment is
created, therefore, in this phase, investment (saving) is determinate. It must be
stressed that investment might be increased without any changes of created income,
and moreover it might be increased if created income is decreased, by decreasing of
the value of consumers’ commodities (vide infra). Even if assuming that new capital
goods (investment) became services in the same year when they are produced, then
created income might be increased according the magnitude of its employment and
according to the required employed quantities of other services (especially labour)
determined by the current technology.
In the second phase, investment is transformed into fixed capital and produces
income in combination with other services (labour, land, money) according to the
technology of the sector (industry) where fixed capital (investment) is planned.
So, in this phase fixed capital (investment) is determinant. The first round of
producing income continues over the number of years that the fixed capital might be
served (5 years in our case). The income produced, in turn, makes up investment
according to the marginal propensity to consume (invest) (vide infra) in each year.
The latter produces new income in the second round and continues on the serving
years of fixed capital (investment). This process continues infinitely (see Appendix 1,
where we confine ourselves to describing only three rounds) (see also Sawyer, p.52).
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There are two things to be noted: (1) starting from the second round (beginning from
second year) income has a cumulative character. The same is true for investment from
the third round (beginning from third year); (2) It is assumed, for all rounds (years),
that the new fixed capital will be fully employed and required quantities of other
factors (labour, land, money) will be satisfied accordingly.
2.2 Endogenous and Exogenous Increment of Investment
Let us to point out, firstly, that the problem of investment increment is compatible
with the first phase of investment process, when investment (saving) is determinate.
There are two sources for increasing investment: endogenous and exogenous. Let us
to start with the first - endogenous source. On the basis of (2.3) we can conclude that
there are three possibilities for investment increasing: 1) when value of consumption
is decreased and income does not change; 2) value of consumption is constant (no
change) and investment increase by increasing income; 3) both, consumption and
investment, are increased by increasing income, either they are increased with
variable marginal propensity to consume, as Keynes assumed in the General Theory
(Keynes, 1936, p.114, see also Hawtrey, p.179n) or they are increased by constant
marginal propensity to consume. Instead of that, Keynes considered the first version
of investment increasing with others ‘Assuming that the decisions to invest become
effective, they must in doing so either curtail consumption or expand income. Thus
the act of investment in itself cannot help causing the residual or margin, which we
call saving, to increase by a corresponding amount.’ (Keynes, 1936, p.64) let us
discuss only two latter cases, because not only the first happens very rarely in reality,
but also Keynes discussed only the following. Assume that the value of the
consumption and investment commodities are increased, so that Cd1 (>Cd) and I1 (>I).
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Assume also that there is a possibility to achieve the next state of equilibrium
according for this new data. Hence, for this case we also have
Cd1 + S1 = Cd1 + I1,
(2-5)
Comparing (2-2) and (2-5) we obtain
∆Cd + ∆S = ∆Cd +∆I,
(2-6)
From this condition to hold we must conclude that an increment in demand for
commodities, either consumption or investment or both has occurred, i.e., the
increment of used income stipulates an increment in produced income of the same
magnitude, and vice versa. In other words the increment in investment or
consumption must be financed by equivalent increment in produced income
(increment in value by the additional employed services). In addition, the increment in
investment must be equal the increment of saving. The relationship between
consumption and investment (marginal propensity to consume) does not influence this
conclusion. The latter influences only on the magnitude of the increase in income
(vide infra). So, we can conclude that in all cases where the increment in investment
(income) is a result of endogenous sources, the magnitude of the increment of
consumed (used) income equals the magnitude of produced income and vice versa
(Negishi, 1979, p.182; Samuelson, 1948 and 1974; Salant, 1974 (1942 and 1964);
Haavelmo, 1943; Turvey, 1953).
The second source for increasing investment - exogenous - means that the increment
is obtained by means of borrowing from either domestic sources (banks and other
similar institutions) or foreign sources (governments and banks), or both of them. In
the case of exogenous increment of investment, the calculation of the national income
created and the additional (produced) investment, generally, is divided into to stages:
(1) this is the same as was described for the second phase of the investment process
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i.e., in the next year investment is transformed into fixed capital, and starts the infinite
process of income and investment creation (see 2.1, p.7); (2) because of this, the
source of increment of investment is exogenous in the year of the investment, also
create national income in the branches where capital products are produced; and the
quantity of income created is equal to the value of investment minus the intermediate
input of goods. But this additional produced income is not used to finance additional
consumption and investment in the year of the investment as in the previous case, just
because its source is exogenous. So, the additional produced income is “free” and
might be use for the new additional purchase of commodities for consumption and
investment. Thus, this created income divided between consumption and investment
in according to propensity to consume (save) and following the calculation of
additional income and investment, is the same as the previous cases.
In addition to the two notes (see p.7) there is one more note, namely, the question of
the efficiency of borrowing and the problem of loan repayment are not discussed here.
Despite these obstacles, we can conclude that in the case of an exogenous increment
in used investment, the increase of desirable produced income is to be greater than the
increment in used income. In other words, here, the coefficient of produced income
increment might be more than one, whereas in the case of an endogenous increment
of investment, the coefficient always equals one.
3. Kahn’s definition of Multiplier
In his well-knowing paper “The Relation of Home Investment to Unemployment”,
published in the June 1931 edition of The Economic Journal, Kahn introduced the
concept of the employment multiplier into the economic literature. At that time,
almost all European countries were characterized by very high levels of
unemployment. Therefore, economists were not the only ones trying to find ways to
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overcome unemployment and modify its results. Kahn suggested increasing the rate of
home investment by “public work” and described its effects as decreasing
unemployment for throughout the economy:
The increased employment that is required in connection actually with the
increased investment will be described as the “primary” employment. It
includes the “direct” employment, and also, of course, the “indirect”
employment that is set up in the production and transport of the raw materials
required for making the new investment. To meet the increased expenditure of
wages and profits that is associated with the primary employment, the
production of consumption-goods is increased. Here again wages and profits
are increased, and the effect will be passed on, though with diminished
intensity. And so on ad infinitum. The total employment that is set up in this
way in the production of consumption-goods will be termed the “secondary”
employment. The ratio of secondary to primary employment is a measure of
these “beneficial repercussions” that are so often referred to (Kahn, 1931,
p.173).
A careful analysis of this statement helps us understand Kahn’s genuine meaning
and show that this is compatible with the second phase of the investment process,
because of the description of production of output (employment). In Kahn’s approach,
investment is exogenous ‘throughout this article it will be supposed that the necessary
funds are raised by means of borrowing. … For it is always within the power of the
banking system to advance to the Government the cost of the roads without in any
way affecting the flow of investment along the normal channels’ (Kahn, 1931, p.174).
So, Kahn assumes that one sources of net (additional) investment is borrowing from
banks (ibid. p.179). As to the second source of net investment, Kahn implicitly
11
determined that foreign sources may also be used in order to increase net investment
either by a reduction in foreign lending or by an increment in foreign borrowing (ibid.
p.189). At the same time, it is necessary to stress that Kahn categorically refused to
finance the additional investment by increasing the quantity of money, as some
modern authors have suggested. Kahn wrote: ‘There was no reason why additional
expenditure on public works needed to be financed by the creation of additional
money as against borrowing from the public. … The increase of employment was not
the result of an increasing quantity of money.’ (Kahn, 1984, p.104)
It appears then, that Kahn considered the problem of open economy and assumed
that the source of the increase of investment is exogenous. Therefore, the multiplier,
expressing the relation between primary and secondary employment stipulated by the
exogenous additional investment, might be more than one. In order to calculate the
exact magnitude of such a multiplier, much more complicated tools (models) are
needed than those available to Kahn. But, for our purposes his method is sufficient. It
is necessary to point out that Kahn indicated that ‘…the increase of efficiency should
certainly be taken into account if one wants to consider how a policy of public works
at the present time would affect the budgetary problems of the future’ (ibid. p.193).
Yet, ‘I am here considering the position in the final position of equilibrium when
everything has settled down. But some time will, of course, elapse between the point
when the primary employment begins and the point when the secondary employment
reaches its full dimensions became wages and profits are not spent quite as soon as
they are earned. I do not enter the question of this time-lag.’ (ibid. p.183n) But Kahn
could not take into account these problems when he calculated the multiplier’s
magnitude because, to repeat, the limited tools used by him, prevented such a
possibility. Nevertheless, the method describing the multiplier’s calculation by Kahn
12
shows that his employment multiplier is a technological-economic phenomenon
(Kahn, 1931, pp.183-6).
Let us now summarize the main characteristics of Kahn’s employment multiplier:
(1) this is compatible with the second phase of the investment process, when
investment (fixed capital) is determinant; (2) this is a special case, because the source
of investment increment is external to the system; and (3) the additional new
investment is directed into “public work” such as roads; (4) the calculation of the
multiplier is dynamic process, although a time-lag is not take into account; and
finally, (5) the employment multiplier is a technological-economic phenomenon.
4. Keynes’ Definition of Multiplier
4.1 Keynes’ Equilibrium Theory
Before we define Keynes’ version of the multiplier, let us be clear as to what kind of
equilibrium theory is incorporated in Keynes’ approach. This is a very crucial point,
because a number of authors have stated that Keynes is not among Walras’ followers
(Blaug, p.188; Clower, 1985, p.110; Kaldor, 1983, p.9; Leijonhufvud, 1998, pp.175-6;
Meltzer, 1983, p.69; Negishi, 1979, p.1). Keynes himself remarked that ‘Walras’s
theory and all others along those lines are little better than nonsense’ (Skidelsky,
1996, p.615). In addition, as was shown, Kahn’s conception of the multiplier is based
on an equilibrium theory similar to Walras.
Proposition (4) of the summation of Keynes’ theory in General Theory states:
(4) Since D1 + D2 = D = (N), where  is the aggregate supply function, and
since, as we have seen in (2) above, D1 is function of N, which we may write
(N), depending on the propensity to consume, it follows that (N) - (N) =
D2 (Keynes, 1936, p.29)
where D1 – is the amount which the community is expected to spend on consumption;
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D2 – is the amount, which it is, expected to devote to new investment;
D – is what we have called above the effective demand. (Ib.29)
The first equation of this proposition (D1 + D2 = D = (N)) is a simplified version of
Walras’ law (vide supra), based on the fact that the right side of this equation
expresses total value of demanded commodities (used income) and the left side – total
value of employed labour (produced income). It is necessary to stress that there are
some controversial points associated with this model, but they do not impinge on the
subject of this paper; therefore, we will not discuss them here. This means that
Keynes’ theory is, generally, an equilibrium theory just like that classical theory and
Walras’ theory. Hence, all the properties of classical and Walrasian equilibrium
theory are compatible with Keynes’ theory, too.
The second equation ((N) - (N) = D2), which is obtained from the first equation, is
correct from the point of mathematics but incorrect from the point of view of
economics. As we have shown previously though saving equals investment in
equilibrium, each of them is determined independently of one another and generally
differs from each other. When they equal one another then equilibrium is established
and the first equation is obtained. This means that, in general, it is incorrect to
determine the amount of investment by deducting the amount of consumption
from the amount of income. Despite this statement, for our purposes, let us assume
that the second equation is also correct for equilibrium state, as Keynes assumed.
Adopting this assumption as authentic allows us to discuss Keynes’ definition of the
multiplier.
4.2 Keynes’ Definition of the Multiplier
The main argument of Keynes’ theory is that ‘when the genuine income of the
community increases or decreases, its consumption will increase or decrease but not
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so fast" (Keynes, 1936, p.114). From this we can conclude that income is equivalent
to the cause and consumption, as well as the saving (investment), is effect. Keynes, in
the following, stated that the previous statement "... can, therefore, be translated ...
into the propositions that ∆Cw and ∆Yw have the same sign, but ∆Yw>∆Cw, ... Let us
define, then, dCw/dYw as the marginal propensity to consume" (ibid. pp.114-5).
Keynes continues "This quantity is of considerable importance, because it tells us
how the next increment of output will have to be divided between consumption and
investment" (ibid. p.115). Here Keynes confirms once more our statement (vide
supra) that consumption and investment are effects and that income (output) is a
cause.
Keynes continues " For ∆Yw = ∆Cw + ∆Iw, where ∆Cw and ∆Iw are the increments of
consumption and investment; so that we can write ∆Yw = k∆Iw, where 1 - 1/k is equal
to the marginal propensity to consume.
" Let us call k the investment multiplier. It tells us that, when there is an increment
of aggregate investment, income will increase by an amount which is k times the
increment of investment" (ibid. p.115).
There are several things here that require explanation. First, income and investment
have been reversed (replaced); investment now becomes the cause and income the
effect2. Moreover, here investment is determinant, which is opposite to Keynes’ above
cited statement that “Saving and Investment are determinates’. But, the theory of
causality teaches that such a reverse (replacement) is generally incorrect. However,
the relationship between cause and effect does allow us to stipulate the magnitude of
the cause required achieving a predicted magnitude of the effect. This means that
either Keynes' definition of the multiplier is incorrect, or the assumed causal
relationship between income, investment and consumption) is incorrect (vide infra).
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Second, a contradiction exists between the determinant of the marginal propensity
to consume and its use. Since, marginal utility depends on the magnitude of the
increase in income. This means that in the case where a discrete event is discussed,
the multiplier depends not only on the point (Yw), (as a continuous case), but also on
the magnitude of ∆Yw. Stated differently, this means that Keynes confused discrete
and continuous cases in his model. Yet, the marginal propensity to consume means
that each additional unit of income is divided between consumption and saving
(investment) in a way different from the previous instance of an additional unit of
income. Keynes, however, interprets the multiplier, as a tool to calculate income by
means of investment, assuming that the propensity to consume is constant.
Third, Keynes offers two different definitions for k: 1) k is the reciprocal of the
marginal propensity to invest (vide infra); and 2) k is the investment multiplier. But
this is impossible because these definitions contradict one another (vide infra).
Fourth, Keynes’ “multiplier” is, by definition, a psychological phenomenon. This
means that the magnitude of multiplier depends on the psychological behaviour of the
community on the condition that technological-economic conditions are given, and do
not change over the time-period under discussing. In addition, Keynes approach is
compatible with the first phase of the investment process and, therefore it is static.
Hence, we can conclude that k cannot be the multiplier. The question then becomes:
“what is the genuine meaning of Keynes’ multiplier? Before answering this question,
let us to discuss the relationship between Kahn’s and Keynes’ multipliers.
4.3 The Relationship between Kahn’s and Keynes’ Multipliers
Keynes indicated that there are two types of multipliers: his own and that of Kahn.
Keynes tried to show that these two multipliers are identical. In the opposite case, that
16
is, if can be shown that they are not identical, and then the theory of the investment
multiplier is incorrect. Why?
Understanding this fact Keynes tried to prove this identity (Keynes, 1936, p.115).
From this we can conclude that for a specific level of investment there is a single
multiplier k', other things being equal. This means that the magnitude of the multiplier
for the specific level of investment might vary, depending on the other conditions. But
when these conditions are given, the magnitude of the multiplier for a specific level of
investment has a single-value (vide infra).
Keynes tried to discuss the conditions under which these two different multipliers
might be equal (ibid. p.116). What is the condition that applies to the simplified case
of k = k'? In note 1, page 116, Keynes described these conditions.
Together with this strong, unrealistic assumption, another flaw can be found. These
multipliers are simply different.3 First, Kahn’s model differs from Keynes’ because
the first is open and the second is closed (see also Chick, 1997, p.165). Second, in
Keynes’ approach, the source of the additional investment is not clear. Because that
the multiplier connecting income and investment is calculated under internal
conditions by the way of consumption. So, the source of additional investment must
be endogenous. In other words, an exogenous source is excluded. Despite that,
Keynes as well as Kahn, assumed that: ‘It follows, therefore, if the consumption
psychology of the community is such that they will choose to consume, e.g., ninetenth of an increment of income, then multiplier k is 10; and the total employment
caused by (e.g.) increased public works will be ten times the primary employment
provided by the public works themselves, assuming no reduction of investment in
other directions’ (ibid. pp.116-117). The last statement indicates that Keynes also
assumed that the source of investment might be exogenous source, which contradicts
17
his approach’s basic principles. Finally, Keynes’ multiplier is a psychological
phenomenon and the unrealistic magnitude of the multiplier (10-times increase of
income) in Keynes’ approach is the main indicator of its unrealistic character (vide
infra) (see also Davidson, 1994, pp. 37-43).
So, Keynes' “multiplier” differs from that of Kahn. The next section will discuss
whether Keynes' “multiplier” is indeed a “multiplier”.
5. The Genuine Meaning of Keynes’ Multiplier
Let us assume that national income is only divided between consumption and saving
(investment), as in Keynes' hypothetical case, summarized above. Than, by Keynes'
definition of marginal propensity to consume
c+i=1
or c = 1 - i
i=1–c
or
(5-1)
where
c – is the marginal propensity to consume and
i - is the marginal propensity to save (invest)
Substituting the latter into Keynes’ formula we obtain
∆Yw = ∆Cw +∆Iw = c∆Yw + ∆Iw
and
∆Yw = 1/(1-c) ∆Iw = 1/i ∆Iw = k ∆Iw,
(5-2)
From this we can conclude that Keynes’ "multiplier" is the reciprocal (inverse) of
the marginal propensity to invest (save) (Ahiakpor, 2001, p.746; Hawtrey, p.179;
Patinkin, 323). But the reciprocal of the marginal propensity to invest (saving)
indicates the required quantities of income for a unit increment of investment when
marginal propensity of both does not change. Therefore, the genuine meaning of
Keynes’ multiplier’s is a requirement, and not a multiplication. Hence, the
requirement indicates the quantity of national income needed to realize one unit of
investment (saving) when the marginal propensity to consume is constant. Keynes
18
himself wrote that “The traditional analysis has been aware that saving depends on
income but it has overlooked the fact that income depends on investment, in such
fashion that, when investment changes, income must necessarily change in just that
degree which is necessary to make the change in saving equal to the change in
investment.” (ibid. p.184, see also p.117) By this definition, again taking Keynes’
example, the equation must be expressed, as, in order to increase the investment by
one, the income is required to increase by 10, where 9 units is set to increase
consumption. In other words, since the relationship between consumption and
investment is given, the increase of investment stipulates the increase in consumption
according to a given proportion. Therefore, to obtain an increment at specific levels of
investment, the corresponding quantities of income are required. That income is
composed of the value of relevant additional employed services (not only available
fixed capital, which generally differs from investment, but also labours and land). At
the same time, it must be stressed that the corresponding increase of income might not
be possible at all, because of limitations of labor, fixed capital, scarce raw materials,
and so on (Hicks, 1974). This reflects the fact that in this case, the connection
(relationship) between national income and investment is reciprocal, whereas in the
previous case, the relationship is direct. Therefore, the difference between them is that
the marginal propensity to save indicates a realistic possibility, while the requirement
may not. This means that Kahn’s multiplier and Keynes’ requirement are
fundamentally different, and can never be identical. It becomes clear, in our opinion,
why Keynes did not use the marginal propensity to invest in general case for
determining the multiplier, and preferred to use the partial case (vide infra).
19
Now, let us to consider the quantitative relationship between the requirement and the
“multiplier”. In general, assuming that the one unit can be divided into k parts, then
we obtain
c = (k-n)/k and i = n/k, where n = 1,2,...,k-1;
(5-3)
This means that Keynes "multiplier" is k/n; therefore, the case considered by Keynes
is the partial case of the general approach, when n = 1 (vide supra).
Whereas, the magnitude of the employment multiplier is near to two according to
Kahn's computation (Kahn, 1931, p.186)4. The inverse of marginal propensity to
invest (Keynes’ requirement), however, might reach 20, 30, or even more (Hazlitt,
p.139, p.151). Therefore, Keynes' requirement (the inverse of the marginal propensity
to invest) differs from the Kahn's multiplier not only in substance, but also in
magnitude. We should point out that these magnitudes might coincide by chance.
At the same time, some circumstances intensify the difference between Keynes’
multiplier and Keynes’ requirement. First, the investment differs from the fixed
capital, one of services, which composed produced income. Second, different sources
for investment (credits from domestic and foreign banks) and the "mutation" problem
of investments can influence the magnitude. Finally, new fixed capital (investment)
might also be unemployed.
Conclusions
In this paper the genuine meaning of Keynes’ multiplier is discussed. It was shown
that Keynes’ investment “multiplier” is essentially a requirement, not a multiplier,
and:
1. The whole process of investment generally consists of two phases: first, when
investment (saving) is produced (created), as a part of used (produced) income,
i.e., when investment (saving) is determinate; and therefore, the causal relation
here is from income to saving (investment), this is, the increment in income
20
causes an increment in saving (investment), while the reverse relation means a
requirement, this is, in order to increase investment there is a required income
increment accordingly; second, when investment is transformed into fixed capital
and creates (produces) income together with other services (labour, land, money),
i.e., when investment (fixed capital) is determinant; and therefore, the causal
relation here is from fixed capital (investment) to income, this is, fixed capital
(investment) increase causes an increment income, while the reverse relation is a
requirement, this is, in order to increase income there is a required fixed capital
(investment) increment accordingly; also, this is the infinite and cumulative
process of creation, not only for income but also for investment.
2. There
are two resources of investment increment: (1) endogenous to the system;
and (2) exogenous to the system: domestic (banking and other financial
institution) and foreign (governments and banking). It is necessary to stress that
the investment increment emanating from the first, endogenous, source is
characterized by both phases of the investment process, whilst the investment
increment generated by the second, exogenous, source is characterized only by the
second phase of the investment process. But, at the same time, in this case there is
a creation of additional income in the branches where capital products are
produced and the quantity of income created is equal to the value of investment
minus the intermediate input of goods; and this additional income allows us to the
additional stage for the new infinite and cumulative process of creation of income
and investment to take place.
3. Kahn’s definition of the multiplier is compatible with the second phase of
investment process, when investment (fixed capital) is determinant and the
increment in investment comes from an exogenous source (banking, domestic and
foreign, other governments), and it is a technological-economic phenomenon;
therefore, its magnitude might be more than one. The calculation of Kahn’s
multiplier is a dynamic process and in order to calculate the magnitude of the
multiplier, Kahn introduced simplifying assumptions.
4. It was shown that Keynes’ “multiplier” is compatible with the first phase of
investment process, when investment (saving) is determinate and is a reciprocal of
21
the marginal propensity to invest (save) and therefore, it is a psychological
phenomenon and static. Therefore, its genuine meaning is that of a requirement,
which indicates the quantity of national income required to realize one unit of
investment. Hence, Kahn’s employment multiplier and Keynes’ requirement are
different.
Notes
1
“…the multiplier tells us what level of income will be necessary to make savings equal to that value
of investment” (Hicks, 1937, p.153).
2
“Then, by some wild leap, this “functional” and purely formal or terminological relationship is
confused with a causal relationship. Next, the causal relationship is stood on its head and the amazing
conclusion emerges that the greater the proportions of income spent, and the smaller the fraction that
represents investment, the more this investment must ‘multiply” itself to create the total income!”
(Hazlitt, p.139)
3
As was already shown by Patinkin, though by another arguments. In his paper "Keynes and the
Multiplier" (1978) Patinkin discuss two questions: "... first, Keynes's empirical estimates of the
multiplier, second, the relation between Kahn's multiplier analysis and the General Theory" (p.322).
His conclusions in both cases are negatives. Patinkin's own words: "... the estimates (of multiplierK.D.) become 2.9, 1.8, and 2.1, respectively. And it is hard to say that these results justify Keynes's
conclusion that "the multiplier seems to have been less than 3 and probably fairly stable in the
neighborhood of 2.5" (GT, p.128)." and "I said this is the way we teach the multiplier today. ... But this
is not the way Richard Kahn first presented the multiplier ..." (pp.323-4).
4
It is necessary to point out that Keynes also considered the value of his multiplier closer to this
magnitude: ‘If single years are taken in isolation, the results look rather wild. But if they are grouped in
pairs, the multiplier seems to have been lass than 3 and probably fairly stable in the neighborhood of
2.5’. (p.128)
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Appendix 1 The process of calculation of income and investment in consequence of
incremented investment ∆I in year 0* assuming that Keynes’ approach is used.
The first round
1.1 Y11 = k1 K11; ∆I11 = i1Y11;
where K11 = ∆I;
1.2 Y12 = k1 K11; ∆I12 = i1Y12;
1.3 Y13 = k1 K11; ∆I13 = i1Y13;
1.4 Y14 = k1 K11; ∆I14 = i1Y14;
1.5 Y15 = k1 K11; ∆I15 = i1Y15;
The second round
2.2 Y22=k2K22(=∆I11); ∆I22 = i2 (Y12+Y22);
2.3 Y23=k2K23(=∆I11); ∆I23 = i2 (Y13+Y23);
2.4 Y24=k2K24(=∆I11); ∆I24 = i2 (Y14+Y24);
2.5 Y25=k2K25(=∆I11); ∆I25 = i2 (Y15 +Y25);
2.6 Y26=k2K26(=∆I11);
∆I26 = i2 Y26;
The third round
3.3 Y33=k3K33(=∆I12+∆I22); ∆I33 = i3 (Y13+Y23+ Y33);
3.4 Y34=k3K34(=∆I13+∆I23); ∆I34 = i3 (Y14+Y24+ Y34);
3.5 Y35=k3K35(=∆I14+∆I24);
∆I35 = i3 (Y15+Y25+ Y35);
3.6 Y36=k3K36(=∆I15+∆I25); ∆I36 = i3 (Y26+ Y36);
3.7 Y37=k3K37(=∆I26);
∆I37 = i3Y37;
* For simplification we assume that fixed capital is served 5 years for all rounds and that the discount
factor is missing. Where k – is the capital-output coefficient and indicates the amount of income
produced by one unit of fixed capital (investment); and i – is the marginal propensity to invest (save).
25