This paper presents two counter-examples to the Keynesian features attributed to imperfect competition in general equilibrium models. In particular, by considering indirect tax rates, a non positive and monotonically non-increasing relationship between the magnitude of both the balanced budget and welfare multipliers and market-power is obtained.

Ramón J. Torregrosa Montaner

English

Research Papers in Economics

BALANCED BUDGET MULTIPLIER 
WITH INDIRECT TAXES 
UNDER IMPERFECT COMPETITION 
Ramón J. Torregrosa Montaner* 
Universidad de Salamanca 
Resumen: Se presentan dos contraejemplos a las propiedades keynesianas atri-
buidas a la competencia imperfecta en modelos de equilibrio general. 
En particular, bajo los dos tipos habituales de impuestos indirectos, se 
obtiene una relación no positiva y no creciente entre las magnitudes del 
multiplicador con presupuesto equilibrado y del bienestar con respecto 
al poder de mercado. 
Abstract: This paper presents two counter-examples to the Keynesian features 
attributed to imperfect competition in general equilibrium models. In 
particular, by considering indirect tax rates, a non possitive and mono 
tonically non-increasing relationship between the magnitude of both 
the balanced budget and welfare multipliers and market-power is ob-
tained. 
JEL Classifications: E12, E62, L16 
Fecha de recepción: 01 IV 2002 Fecha de aceptación: 27 VIII 2002 
* I am grateful to C. Pita for her helpful comments and suggestions. I am also 
indebted to the Instituto Valenciano de Investigaciones Económicas (IVIE). The 
usual caveat applies.rtorregr@usal.es 
3 4 ESTUDIOS ECONÓMICOS 
1. Introduction 
General equilibrium models with imperfect competition have been 
used as an explanation of some Keynesian features with fully flexi-
ble prices. In this line, papers such as Hart (1982), Blanchard and 
Kiyotaki (1987), Dixon (1987) and, Mankiw (1988), among others, 
explore the effect of different market power settings on the macroeco-
nomic multipliers, reaching a positive and monotonically increasing 
relationship between the balanced budget multiplier and the degree 
of market power. A common setup of these models is that the govern-
ment can resort to profits or lump-sum taxation to balance its budget. 
In this framework imperfect competition works as the only source of 
inefficiency which generates a space for public intervention. This in-
sight is supported on the basis that fiscal policy does not distort rela-
tive prices in the margin. This statement calls into question whether 
these Keynesian features of the multiplier remain unchanged under 
distortionary tax schemes. Within this trend, Molana and Moutos 
(1992), and Heijdra, Ligthart and Ploeg (1998) find non-positive mul-
tipliers for labor income tax rates, whereas Torregrosa (1998), for the 
same tax rates, proves that this multiplier can be monotonically de-
creasing with respect to market power. 
Considering this point of departure, this paper deals with the re-
lationship between the balanced budget multiplier and market power, 
for indirect (ad-valorem and excise) tax rate schemes, providing an-
other counter-example to the Dixon-Mankiws monotonicity result. 
The paper is structured as follows. In section 2, the model is pre-
sented and both the output and the welfare multipliers are calculated 
in their general form. Sections 3 and 4 develop these multipliers for 
both the ad-valorem and excise tax rates respectively. Finally, section 
5, summarizes with the final comments. 
2. The Model and Multipliers 
Let us consider an economy formed by two commodities: leisure (con-
sidered as the numeraire) and a composed commodity produced from 
labor; n + 2 independent agents: the representative consumer, the 
government and n non-competitive firms. The former two agents 
constitute the demand-side of the economy and the latter the supply-
side, according to the following assumptions: 
(i) Household preferences are represented by a separable utility 
function. On the one hand, a Cobb-Douglas sub-utility function over BALANCED BUDGET MULTIPLIER 5 
consumption of the produced good (C) and leisure (L) and, on the 
other, a sub-utility function over the publicly-provided good (g) 
u(C,L,g) = C
aL
1-
a + P(g), (1) 
where a € (0,1); ¿0(0) = 0, Beta' > 0 and ¡3" < 0. Let us denote by T 
the endowment of time, p the price of the produced commodity, and 
7r the total profits of the firms. The household budget constraint is 
given by 
pC = T-L + n. (2) 
Consumer's choice is related only to C and L. Thus, the solution 
for the maximization of (1) subject to (2) is 
C = ——•—- (3) 
V 
L= (l-a)(r + 7r) (4) 
(ii) The government's role is modeled in the usual Keynesian 
fashion: a (indirect) tax revenue R is used to finance the amount g 
of government purchases. Thus, given the price p, the government 
budget's constraint is 
R = pg = G (5) 
Adding equations (3) and (5), the total expenditure in the econ-
omy is given by 
Y = a{T + TT) + G, (6) 
which represents the demand side of the economy. 
(in) The industry is formed by n non-competitive firms produc-
ing an amount qj (j = 1,2,...n) of output from labor. Without loss 
of generality, let us assume the simple constant returns technology 
qj = Nj (N = f^Nj). It is also assumed that the labor market is 
competitive and firms' choices are independent, although households 
are the firms' owners. Then the goal of the representative firm is to 
maximize 
PQj ~ Ii ~ Rj (
7) 
where Rj represents the amount of taxes levied on the jth firm and 6 ESTUDIOS ECONÓMICOS 
n 
refers to total tax revenue. Section 3 is devoted to the ad-valorem 
tax rate case, where Rj = tvqj with 0 < t < 1, while section 4 is 
concerned with the excise tax rate case Rj = tqj with 0 < t. 
The first order condition for equation (7) can be written as 
p(l_At)=1+i^ii (9) 
dqj 
where n € (0, 1) is interpreted as an ad-valorem measure of market 
power: when p. tends to one, industry behaves as monopolist (perfect 
collusion); when y. tends to zero, firms behave as Bertrand oligopolists 
(perfect competition); when fi equals 1/en, where e is elasticity of 
demand, firms behave "a la Cournot". Finally, given the better firm's 
choice q* which fulfills equation (9), the supply-side of the economy 
is represented by total output 
l 
and aggregated profits in the economy equal 
n 
* = I>j. (ii) 
i 
and here TT, = P(Q)q* - q* - R, is the representative firm's profit in 
equilibrium. Finally,' general equilibrium requires the usual market 
clearing condition 
Y = pQ, (12) 
which implies, according to equation (6), that 
Q = g+
 Q(
T + 7r) (13) 
P 
Both 7T and p depend on g due to equations (5), (7) and (8). 
Then, differentiating equation (13) with respect to g, taking into ac-
count equation (3), the output balanced budget multiplier is BALANCED BUDGET MULTIPLIER 7 
dQ a dn 
— - 1 + 
dg p dg 
C dp 
p dg 
(14) 
The increase in output due to a raise in government purchases 
is affected, first, by an income effect through the change in profits 
and, second, by a price effect that results from an increase in the tax 
rate needed to finance the higher government purchases. Notice that 
equation (14) captures the main difference with the lump-sum taxa-
tion models quoted in section 1. In fact the first two terms of equation 
(14) are just the Dixon-Mankiw's multiplier. The last term adds the 
distortion due to indirect tax rates, whose effect is the opposite. 
Finally, it is interesting to study the effect on welfare of this 
boost to the economy. Substituting equations (3) and (4) in equation 
(1) the indirect utility function is obtained 
where 7 = Q
q(1 -a)
1"". Differentiating with respect to g and taking 
into account equation (3) we obtain 
which represents the impact of the balanced budget expansionary 
policy on welfare. As can be observed, the positive effect on welfare 
due to larger government purchases is diminished by the change in 
consumption. This change is motivated by a price increase and a 
decrease in profits both generated by the change in the tax rate. It 
is necessary to remark that this effect on welfare is the opposite to 
that predicted by Keynes. This is because, according to Keynes, 
the balanced budget expansionary policy should not cause changes in 
welfare. 
The next sections are devoted to computing these effects for both 
the ad-valorem and the excise tax rates. 
3. Balanced Budget Expansionary Policy Under Ad-valorem 
Tax Rates 
In this case Rj = tp(Q)qj with 0 < t < 1. Thus, according to equation 
V(p,*,g) = l{T + n)p~
a+p(g), 
(15) 
(9), the equilibrium price is 
V = 
(1-0(1-/*)' 
(16) 8 ESTUDIOS ECONÓMICOS 
Due to equations (5) and (8), equilibrium government purchases 
are given by 
9=tQ, (17) 
and aggregate profits by 
7T=-^—Q. (18) 
1 /Li 
In order to compute the balanced budget multiplier defined in 
equation (14), the variations on profits and price of the balanced 
budget expansionary policy must be calculated. First, from equation 
(18), 
dn_ p. dQ 
~dg~ ~~ 1 - n~dg' ^ ' 
Second, to obtain the effect on price, let us start computing the 
effect of such an expansion on the tax rate consistent with the gov-
ernment's budget constraint given by equation (17), which is 
dt _ 1 / dQ\ 
dg~ Q \
1~
t~dg~)' ^ ' 
Thus differentiating equation (16) with respect to g, and taking 
into account equation (20), 
— = 7-^-r-(l-t—V (21) 
dg (1 -1) Q V dg J
 K ' 
Substituting equations (19) and (21) in equation (14), the output 
balanced budget multiplier equals zero, i.e., 
This means that a balanced budget expansionary policy has no effects 
on output (employment). The explanation of this total crowding out 
effect, under ad-valorem tax rates, is that this boost to the economy 
increases both government's demand and prices, in such an amount 
that the decrease in consumption equals the increase in the govern-
ment's demand. Hence, firms do not change either their output level 
or profits (substituting the result in equation (19), f< = 0). The BALANCED BUDGET MULTIPLIER 9 
effect on price is calculated differentiating equation (21) with respect 
to g, which is 
dg (l-t)Q 
This allows us to compute the effects of the balanced budget expan-
sionary policy on the welfare. Then, substituting the multipliers in 
equation (15), taking into account equation (13) and (17) and oper-
ating, the following equality holds 
^- = P'- 7P
1-" (22) 
dg 
As can be seen, a balanced budget expansionary policy under 
ad-valorem tax rate affects welfare in two ways: a positive effect de-
rived from the increase in government purchases, and a negative effect 
arising from the increase in price due to the increase in the tax rate. 
Finally, the direction in which the effect of the balanced budget ex-
pansionary policy on the welfare changes with respect to the degree 
of market power can be computed differentiating equation (22) with 
respect to n 
d^Jil = _7(i -a)p-°<-L. 
d\i dji 
Since, according to equation (16), 
dp 
dy.  1 - n 
> 0, 
the effect of the balanced budget expansionary policy on welfare is 
monotonically decreasing with respect to the degree of market power. 
4. Balanced Budget Expansionary Policy Under Excise Tax 
Rates 
In this case it,- = £?7- with 0 < t < 1.
1 Thus, according to equation 
(9), the equilibrium price is 
1 Despite the fact that t can be greater than orte, it is assumed that t < 1 This 
is because the model is expressed in units of leisure. Thus, t > 1 would mean an 
excise tax rate higher than the current wage.This condition is also compatible with 
the fact that household expenditure is higher than public expenditure. Indeed, 10 ESTUDIOS ECONÓMICOS 
Due to equations (5) and (8), equilibrium government purchases 
(24) 
V 
and aggregate profits are 
7T = pnQ. (25) 
In order to calculate the balanced budget multiplier defined in equa-
tion (14), the variations on profits and price of the balanced budget 
expansionary policy must be calculated. First, from equation (25), 
^=H
Q^
+vJ-
 (26
) 
Second, to obtain the effect on price, let us start computing the 
effect of such an expansion on the tax rate consistent with the gov-
ernment's budget constraint given by equation (24), that is 
ÉÎ = L(g
dl+p-.t
dQ\ (27) 
dg Q \ dg dg J 
Thus differentiating equation (23) with respect to g, and taking 
into account equation (27), 
dl=L(p-t*l\ (28) 
dg Q\ dg)'
 K ' 
Substituting equations (26), (27) and (28) in equation (14), tak-
ing into account equation (24) and operating, the output balanced 
budget multiplier under excise tax rates can be written as 
the ratio between public expenditure and total expenditure is given, according to 
equations (12) and (17) by, y = ^ and the ratio between household expenditure 
and total expenditure is given, according to equations (2) and (6), by 'y- = L — —. 
then 1 * ^ U ' t' C24) > i=i but  fi [ , j p > , t en p !> p. sing equation ( ), /i > 2t
 1 u 
since ¡1 > 0j then t < 1. BALANCED BUDGET MULTIPLIER 11 
dQ _ (l-a)y(l + t)
2 
dg (i_p)[i_(Q_(i_a)i2)/i]
 <u- W 
since t < 1 and y < 1. Equation (29) shows that the balanced budget 
expansionary policy has negative effects on output (employment). Let 
us show that this effect worsens when market power increases. Dif-
ferentiating (29) with respect to y and operating, 
(l-Q)(l-M)
2(l-(a-(l-a)tV) 
(l-Ai)
2[l-(Q-(l-a)i
2)M]
2  (30) 
This result makes up a counter-example to the Dixon-Mankiw's 
monotonicity result, where, as competition in the goods market be-
comes less perfect, the balanced budget multiplier increases. 
Finally let us calculate the effect of the balanced budget expan-
sionary policy on welfare. Substituting equations (26), (27) and (28) 
in equation (14) and operating, 
dV 
dg 
IP  (ß + t) 
dg 
l)+ß',  (31) 
which shows two opposite effects, as was remarked upon in section 2. 
Differentiating equation (31) with respect to market power 
dy 
7 (l-a)p" 
a9 1  -
a^r\{y + t)^ - 1] + P
1-*^ + (y + t,—~ 
dy dg dg dy 
< 0 
since 
dp 
dy 
> 0, 
dQ_ 
dg 
d(f ) 
dy 
< 0 and  < 0, 
according to equations (23), (29) and (30). Thus, as in the ad-valorem 
tax rate case, the effect of the balanced budget expansionary policy 
on welfare is monotonically decreasing with respect to the degree of 
market power. 12 ESTUDIOS ECONÓMICOS 
Conclusions 
In this paper, both the monotonicity of the output multiplier and the 
effects on welfare of a balanced budget expansionary policy have been 
analyzed under the two main indirect taxes. The main contributions 
of this paper are: First, for the ad-valorem tax rate scheme, the 
output multiplier equals zero, which means that changes in public 
purchases have no effect on output, reaching a total crowding out 
effect independent of the degree of market power. The explanation of 
this crowding out effect is that the government's expansionary policy 
increases the price (through taxes) in such way that consumption falls 
in the same proportion as the increase in government purchases. With 
respect to the effect on welfare, this is monotonically decreasing with 
market power, which is opposite to that predicted by Keynes. 
Secondly, in the case of an excise tax rate, the results are related 
to the market power in the opposite way to the conclusion reached 
by Dixon (1987) and Mankiw (1988) for non-distortionary taxation. 
The output balanced budget multiplier is negative and monotonically 
decreasing with respect to market power. The reason for this is that 
an increase in the excise tax rate distorts relative prices reducing out-
put in greater proportion than the increase in public expenditure. 
This negative effect also increases as the degree of market power in-
creases. Regarding welfare the effect of a government expansionary 
policy is monotonically decreasing with respect to market power as 
in the former case. 
In conclusion, Keynesian features attributed to the general equi-
librium models with fully flexible prices and imperfect competition 
depend almost entirely on the tax scheme considered. It is true that 
market power causes inefficiency but it is also true that some taxes 
might do so, too. In this sense, the government could be using an 
inefficient tool to amend a market failure. 
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Dixon H. (1987). A Simple Model of Imperfect Competition with Walrasian 
Features, Oxford Economic Papers, 39, pp. 143-160. 
Hart, O. (1982). "A Model of Imperfect Competition with Keynesian Features", 
The Quarterly Journal of Economics, 97, pp. 109-138. BALANCED BUDGET MULTIPLIER  13 
Heijdra, B. J., J. E. Ligthart.and F. van der Ploeg (1998). Fiscal Policy, Dis-
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A Model of Imperfect Competition with Keynesian Features&quot;, The Quarterly

A Note on Taxation, Imperfect Competition and the Balanced Budget Multiplier,

A Simple Model of Imperfect Competition with Walrasian Features,

Imperfect Competition and the Keynesian Cross&quot;,

Monopolistic Competition and the Effects of Aggregated Demand&quot;,

On the Monotonicity of Balanced Budget Multiplier Under Imperfect Competition&quot;,

Balanced budget multiplier with indirect taxes under imperfect competition

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Balanced budget multiplier with indirect taxes under imperfect competition

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