In this paper we present a fixprice model in which private and public consumption show some degree of substitution. We offer formulae for the Keynesian multiplier which depend on this degree of substitution. We also show that there is a Pigou effect and that, sometimes, this effect is larger than the Keynesian multiplier.Keynes, Multiplier, Unemployment, Pigou

Luis Carlos CorchÃ³n

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 1 
  
The Keynesian multiplier and the Pigou effect under substitution between 
private and public consumption 
  
 
 
Luis Carlos Corchón  
Universidad Carlos III 
Abstract 
In this paper we present a fixprice model in which private and public consumption show some degree of substitution. 
We offer formulae for the Keynesian multiplier which depend on this degree of substitution. We also show that there 
is a Pigou effect and that, sometimes, this effect is larger than the Keynesian multiplier.
The author acknowledges comments from Alberto Alonso and a referee and financial support from SEJ2005-06167/ECON 
Citation: Luis Carlos Corchón, (2010) ''The Keynesian multiplier and the Pigou effect under substitution between private and public 
consumption'', Economics Bulletin, Vol. 30 no.1 pp. 829-836. 
Submitted: Sep 14 2009.   Published: March 25, 2010. 
 
     1. INTRODUCTION
Current economic events have fueled a lively discussion about the best way
to ￿ght unemployment. This paper is an attempt to discuss this issue in a
simple static macroeconomic model in which:
1. Agents maximize utility subject to budget constraint and given prices.
2. Goods and money markets clear but, for exogenous reasons that are not
discussed in this paper, the labor market does not clear.1
Our model is very stylized: There is only one input, labor, which produces
a consumption good under constant returns to scale. The latter assumption
simpli￿es the supply side of the economy by making supply in￿nitely elastic
and together with our assumption of perfect competition implies that there are
no pro￿ts. Since neither the responsiveness of supply to demand nor the distri-
bution of income are at the stake in current discussions this is not a harmful as-
sumption. Money is the only asset and it is demanded because it is an argument
in the utility function. This choice of modelling is dictated by simplicity, see
Blanchard and Fischer (1989). The government obtains all revenue on a linear
income tax again an assumption brought about by simplicity. Finally, monetary
wages are given.2 Thus, important considerations for the understanding of the
crisis such as productivity growth, capital/investment, indirect taxation, mo-
nopolistic competition, stock market and increasing/decreasing returns to scale
are outside the scope of this paper.
Following an idea of Bailey (1971), see also Barro (1981), we consider the
possible substitution between private and public consumption. For instance
if publicly built schools are as good as privately built schools, an increase in
the former should have some e⁄ect on the private demand of the latter. Or if
the government runs a de￿cit, the citizens may feel that this de￿cit, sooner or
later, will fall on them.3 Thus in our model the utility of the representative
consumer is a function of two variables. On the one hand, a linear combination
of the privately and the publicly provided consumption good with weights 1
and a 2 [0;1]. On the other hand on money and the budget surplus of the
government with weights 1 and b 2 [0;1]. If a = 0, the consumer feels that
the publicly provided good, say bridges, is not related to the privately provided
good, say, beef. If a = 1, the consumer takes the publicly provided consumption
good, say public health care, as a perfect substitute for the privately provided
consumption good, say private health care. If b = 0, the consumer feels that the
debt (resp. surplus) incurred by the government will not be paid (resp. received)
by her, because it will be paid (resp. received) in a distant future or by a future
1For a possible explanation of why wages do not adjust supply and demand of labor see
Bewley (1999).
2Our model falls into the ￿xprice literature pioneered by Barro and Grossman (1976),
Benassy (1975), DrŁze (1975), Malinvaud (1977) and YounŁs (1975). See Silvestre (1982) for
a general view on this literature.
3This is the issue behind the "Ricardian equivalence".
1generation. If b = 1, the consumer takes the government debt (resp. surplus)
as an immediate decrease (resp. increase) in her wealth. b can be interpreted
as the percentage of the current de￿cit paid by the current generation. Similar
preferences have been considered by Heijdra, Ligthart and van der Ploeg (1998)
in the framework of Monopolistic Competition and by Linnemann and Schabert
(2003) in a New Keynesian model. Our main results are:
1. When the increase in public expenditure is ￿nanced entirely by taxes or
b = 1, i.e. the de￿cit is discounted as an immediate reduction in wealth,
the Keynesian multiplier is 1 ￿ a. When publicly and privately provided
consumption goods are not related (a = 0) this is the well-known result
that the multiplier of the balanced budget is one.
2. When the increase in public expenditure is ￿nanced by a de￿cit the mul-
tiplier is decreasing in both a and b. When the goods o⁄ered by the
government and the private sector are not related (a = b = 0) we obtain
a formula that is identical to the textbook Keynesian multiplier.
3. When written in terms of elasticities, instead of the usual formulation of a
ratio of increments, the Keynesian multiplier is always smaller than one.
This means that an increase in public expenditure of x% increases total
output in less than x%.
4. The Pigou e⁄ect, namely the e⁄ect on real wealth of a decrease in prices,
in terms of elasticities is, also, smaller than one. When the increase in
public expenditure is ￿nanced entirely by taxes or a = b = 1 the elasticity
associated with the Pigou e⁄ect is larger than the elasticity of the Key-
nesian multiplier. If a = b = 0 the Keynesian multiplier is larger than its
Pigouvian counterpart for "reasonable" values of the parameters.
2. THE MODEL
There are two goods consumed with prices denoted by p and 1. The second
good will be called money from now on. There is an input, labor, whose price
is denoted by !. Good 1 is produced under constant returns to scale and units
are chosen such that the marginal cost of this good is !. Assuming that the
economy is perfectly competitive,
p = !: (1)
The government raises funds by a tax on income (I) with a constant tax rate t,
and buys goods 1 and 2 in quantities G and S. While G is positive, S can be
positive (surplus) or negative (de￿cit). The government budget constraint is
pG + S = tI: (2)
2There is a representative consumer with preferences representable by a Cobb-
Douglas utility function
U = A(x + aG)￿(M + bS)1￿￿
where x is the consumption of good 1, M is the consumption of money, A > 0,
￿ 2 (0;1) and a;b 2 [0;1]. When a = b = 1 the consumer considers that G and
S are perfect substitutes of private consumption and wealth. When a = b = 0
the consumer does not take into consideration G and S. This may be because
she considers public expenditure as a total waste (like certain public works) and
that the debt arising from the de￿cit will paid in a distant future, perhaps by a
di⁄erent generation. Compare with Heijdra, Ligthart and van der Ploeg (1998)
equation (1) and Linnemann and Schabert (2003) equation (2).
The representative consumer has endowments of labor and money of ￿ L and
￿ M respectively. Her budget constraint is
px + M = I(1 ￿ t): (3)
Adding (2) and (3), we get that pG+S+px+M = I. Since pro￿ts are zero and
wages are obtained producing G or x, income is de￿ned as I = !G + !x + ￿ M.
Taking into account (1); the last two equalities imply that S + M = ￿ M. Thus,
when the public sector runs a surplus (S > 0), M < ￿ M. A public sector de￿cit
(S < 0) implies M > ￿ M:
From utility maximization at given prices and income, demand functions are
x =
￿(I(1 ￿ t) + bS)
p
￿ (1 ￿ ￿)aG, (4)
M = (1 ￿ ￿)(I(1 ￿ t) + apG) ￿ ￿bS (5)
2. 1. Market Clearing Equilibrium
In a market-clearing equilibrium all markets clear. Denoting the variable,
say, z in a market-clearing equilibrium as zC (C for clearing) we have that
xC + G = ￿ L: (6)
IC = !C ￿ L + ￿ M. (7)
Notice that the real GDP of our economy is xC + G which, given our choice of
units, is just ￿ L. From (4), (6) and (7) prices and wages in a market-clearing
equilibrium are
pC = !C =
￿( ￿ M ￿ S(1 ￿ b))
(1 ￿ ￿)(￿ L ￿ G(1 ￿ a))
: (8)
Thus prices and wages increase with the money supply, the de￿cit (unless b = 1)
and the public expenditure (unless a = 1) and decrease with the labor supply.
3From (6) consumption of good 1 in a market-clearing equilibrium is
xC = ￿ L ￿ G (9)
Thus, the consumption of good 1 increases with resources and decreases with
public expenditure in a one-by-one basis independently of the parameter a.
2.2. Non Clearing Equilibrium
In a non clearing labor market equilibrium, non clearing for short, the supply
of labor exceeds demand of labor denoted by LN. All other markets clear. Let
us denote the variables in such an equilibrium by the superscript N (N for non
clearing). Thus,
LN = xN + G. (10)
pN = !N: (11)
Now the income of the consumer is determined by the actual production, namely
IN = !N(xN + G) + ￿ M: (12)
Taking into account (11) and (4) we have that
xN =
￿(IN(1 ￿ t) + bS)
!N ￿ (1 ￿ ￿)aG: (13)
Plugging (12) and (2) in (13) and using (10) we have that
xN =
￿( ￿ M ￿ S(1 ￿ b))
!N(1 ￿ ￿)
￿ aG (14)
LN =
￿( ￿ M ￿ S(1 ￿ b))
!N(1 ￿ ￿)
+ G(1 ￿ a) (15)
From (15) we see that ￿ L > LN i⁄
!N >
￿( ￿ M ￿ S(1 ￿ b))
(1 ￿ ￿)(￿ L ￿ G(1 ￿ a))
= !C (16)
Thus it can be said that our model embodies a Pigou e⁄ect where a decrease in
prices may increase GDP.
Let us make a simple calibration exercise. Despite the fact that, as noticed
in the introduction, our economy has very stylized features, this exercise might
throw some light on our model. In order to obtain possible values for the
parameters, divide both sides of (15) by the GDP, LN. We interpret L
N!
N
￿ M as
the velocity of money and we will denote this magnitude by v. De￿ning g ￿ G
LN
and s ￿ S
!NLN = t ￿ g as, respectively, the fractions of public expenditure and
de￿cit in GDP, (15) yields
(1 ￿ ￿)(1 ￿ g(1 ￿ a)) = ￿(
1
v
￿ s(1 ￿ b)): (17)
4From (13) we interpret ￿ as the marginal propensity to consume. Letting
￿ = :8; g 2 [:3;:5] and s 2 [￿:03;￿:05] in (17) we see that depending on the
values of a and b, v ranges between 4 and 13:333, not very di⁄erent from what
is shown in real data. Notice that (17) implies that if g and s are constant the
velocity of money is constant too.
3. THE KEYNESIAN MULTIPLIER
Recall that the government can not choose S;G and t simultaneously. Equa-
tion (15) involves the de￿cit and public expenditure, so it is the appropriate
equation to work with when the strategies of the government are these two vari-
ables. Thus in this case, the tax rate is adjusted to satisfy (2). In this case we
see that the Keynesian multiplier is
@LN
@G
= 1 ￿ a (18)
which in the best case scenario (i.e. when public consumption does not a⁄ect
the private consumption or a = 0) is one and in the extreme case in which
public and private consumption are perfect substitutes it is zero. The former
corresponds to the well-known result that the multiplier with a balanced budget
is one. Writing the multiplier in terms of elasticities we have that
@LN
@G
G
LN = (1 ￿ a)g < 1:
Suppose now that the strategies of the government are t and G. In this
case the de￿cit (or surplus) takes the toll of the adjustment. After lengthy
calculations we obtain that
LN =
￿
￿ M
!N (1 ￿ (1 ￿ b)t) + G(a￿ ￿ a ￿ b￿ + 1)
(1 ￿ ￿) + ￿(1 ￿ b)t
(19)
Now the Keynesian multiplier is
@LN
@G
=
￿a(1 ￿ ￿) + 1 ￿ b￿
(1 ￿ ￿) + ￿(1 ￿ b)t
: (20)
Notice that, as intuition suggests, the larger a the smaller the value of the
multiplier. In the extreme case in which a = b = 0 the multiplier is 1
1￿￿(1￿t)
which, interpreting again ￿ as the marginal propensity to consume, is just the
textbook Keynesian multiplier. But when b = 1, i.e. when the e⁄ect of debt
is fully anticipated, the multiplier is, again, 1 ￿ a, never larger than one. Our
expressions (18) and (20) generalize those obtained by Bailey (1971), Chapter
9, Table 1 for the limiting cases where a and b are either 0 or 1.
Again writing the multiplier in terms of elasticity from (20) and noticing
that ￿ ￿ M(1 ￿ (1 ￿ b)t) > 0 we have that
@LN
@G
G
LN =
(￿a(1 ￿ ￿) + 1 ￿ b￿)pNG
￿ ￿ M(1 ￿ (1 ￿ b)t) + pNG(￿a(1 ￿ ￿) + 1 ￿ b￿)
< 1: (21)
54. THE PIGOU EFFECT
The ￿xprice model is an interesting scenario to study the so called "Pigou
e⁄ect" in which a decrease in prices/wages increases employment via increases
in real wealth.4 Let us consider ￿rst the case in which the de￿cit is constant.
In this case, from (15) the multiplier associated with the Pigou e⁄ect is:
￿
@LN
@pN =
￿( ￿ M ￿ (1 ￿ b)S)
(1 ￿ ￿)(pN)2 : (22)
Multiplying both sides of (22) by pN=LN, we obtain the Pigou multiplier written
in terms of elasticities,
￿
@LN
@pN
pN
LN =
￿( ￿ M ￿ (1 ￿ b)S)
￿( ￿ M ￿ (1 ￿ b)S) + (1 ￿ ￿)pNG(1 ￿ a)
= 1 ￿ g(1 ￿ a): (23)
Clearly, the elasticity associated with the Keynesian multiplier is larger than
the elasticity associated with the Pigouvian multiplier i⁄
2g(1 ￿ a) > 1: (24)
Even in the most favorable case for the Keynesian multiplier, i.e. a = 0, the
inequality (24) requires that the ratio public expenditure/GDP be larger than a
half. So if the de￿cit is constant, we should expect better results on employment
from a pro-competitive policy which facilitates price decreases rather than from
public expenditure.
Suppose now that the strategies of the government are t and G. In this case
the Pigouvian multiplier is
￿
@LN
@pN =
￿ ￿ M(1 ￿ (1 ￿ b)t)
((1 ￿ ￿) + ￿(1 ￿ b)t)(!N)2. (25)
Writing the Pigou multiplier in terms of elasticities, we have that
￿
@LN
@pN
pN
LN =
￿ ￿ M(1 ￿ (1 ￿ b)t)
￿ ￿ M(1 ￿ (1 ￿ b)t) + pNG(a￿ ￿ a ￿ b￿ + 1)
< 1: (26)
From (21) and (26) we obtain that the elasticity associated with the Keynesian
multiplier is larger than the elasticity associated with the Pigouvian multiplier
i⁄ (￿a(1 ￿ ￿) + 1 ￿ b￿)pNG > ￿ ￿ M(1 ￿ (1 ￿ b)t), or
(￿a(1 ￿ ￿) + 1 ￿ b￿)gv > ￿(1 ￿ (1 ￿ b)t) (27)
In this case the comparison is ambiguous. If a = b = 1 then the Keynesian
multiplier is zero whereas the Pigouvian one is positive and thus the Pigouvian
multiplier is larger than its Keynesian counterpart. If a = b = 0 the Keynesian
multiplier is larger than its Pigouvian counterpart if and only if gv > ￿(1 ￿ t).
This inequality holds for the kind of values of g;v and ￿ considered in the simple
calibration exercise performed at the end of Section 4.
4The current crisis has caused a moderate de￿ation in many countries. Thus, prices de-
creased in Great Britain .4% in March 2009 and .1% in Spain in the same period.
65. CONCLUSIONS
In this paper we have presented a simple ￿xprice model in which public and
private consumption might be substitutes. We have shown that the value of the
multiplier depends on this degree of substitution. In particular when de￿cits
are fully incorporated into wealth we get a "Ricardian Equivalence" result, see
Benassy (2007) for a similar result in a di⁄erent framework. We also have
shown that the Pigou e⁄ect exists in our ￿xprice model and that, sometimes, it
is stronger than the Keynesian multiplier. This is, of course, not an argument
against public expenditure as a means of increasing employment. But it raises
the point that, sometimes, a policy of de￿ ation may have even better results in
terms of increasing employment. The latter policy was considered dangerous by
Keynes (1936, Chapter 19) because of its e⁄ects on price expectations. We plan
to study this question more carefully in a subsequent paper.
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The Keynesian multiplier and the Pigou effect under substitution between private and public consumption

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