In the Dixit-Stiglitz model of monopolistic competition, entry of firms is socially too small. Other authors have shown that excess entry is also a possibility with other preferences for diversity. We show that the cost structure and workers's rents can also explain excess entry.monopolistic competition; product diversity

Eric Toulemonde

Pierre M. Picard

English

Research Papers in Economics

On monopolistic competition and optimal
product diversity: a comment on cost
structure and workers’ rents∗
Pierre M. Picard†
University of Manchester and Université catholique de Louvain
Eric Toulemonde‡
University of Lausanne
February 15, 2005
Abstract
In the Dixit-Stiglitz model of monopolistic competition, entry of
ﬁrms is socially too small. Other authors have shown that excess entry
is also a possibility with other preferences for diversity. We show that
the cost structure and workers’s rents can also explain excess entry.
∗W et h a n kC .d ’ A s p r e m o n t ,C .B e h r e n sa n dY asusada Murata for helpful comments.
†Université catholique de Louvain, CORE, and University of Manchester, SoSS, Oxford
Road, Manchester MP13 9PL, United Kingdom. E-mail: Pierre.Picard@man.ac.uk.
‡DEEP, HEC, Université de Lausanne, BFSH1, CH-1015 Lausanne, Switzerland.
Email: eric.toulemonde@unil.ch.
11 Introduction
Do ﬁrms oﬀer too many varieties under monopolistic competition? Diﬀerent
strands of economic literature have answered this question. Contributions
in industrial organization literature focus on partial equilibrium frameworks.
For instance, Chamberlin (1950) considers the case of ﬁrms selling perfect
substitutes and concludes that ﬁrms set production to the left of the point of
their minimum average cost so that too many ﬁrms enter. Dixit and Stiglitz
(1977) analyze the case of imperfect substitutes where ﬁrms set production
levels larger than the (unconstrained) social optimum so that entry is be-
low its social optimum. Generalizing this work, Benassy (1996) and Vives
(1999, p.172) show that entry can be t o ol a r g eo rt o os m a l la c c o r d i n gt o
the balance between consumers’ preferences for variety and for individual
consumption of each single variety. Other contributions have reconsidered
Dixit and Stiglitz’s (1977) model by allowing ﬁrms to behave strategically in
a general equilibrium setting. In particular, d’Aspremont et al. (1989), Yang
and Heijdra (1993) and d’Aspremont et al. (1996) consider that ﬁrms have
non zero masses and use their ability to alter price indices and incomes in
order to increase their own proﬁts.1 In such models, ﬁrms’ strategic behavior
may yield excess entry, which reverses the Dixit and Stiglitz’ (1977) result.
In this paper we present a general equilibrium model with imperfect la-
bor markets in which the Dixit and Stiglitz’ (1977) prediction can also be
reversed. The model includes a sector with constant returns to scale and a
sector with increasing returns to scale, allowing workers of the latter to cap-
ture a rent (either because these workers are unionized - see McDonald and
Solow, 1981 - or because of eﬃciency wage considerations - see Solow, 1979
and Akerlof and Yellen, 1990). Workers’ positive rents raise ﬁrms’ costs but
also inﬂate the product demand and ﬁrms’ revenues. The impact of rents on
proﬁts and entry then depends on how workers’ rents translate to costs and
to the demand. If rents are associated to ﬁxed costs activities, they decrease
proﬁts on a one-to-one basis. The eﬀect of rents on demand is then domi-
nated by their eﬀect on costs: proﬁts and entry fall with higher rents. By
contrast, rents associated to variable costs activities have a smaller impact
on proﬁts because ﬁrms are able to adapt their production levels. In this
case, the eﬀect of rents on demand can dominate their eﬀect on costs: proﬁts
1Such models give evidence of a ‘Ford’ eﬀect, in reference to Henry Ford who ﬁrstly
exploited the positive causality between wages and product demand.
2rise with higher rents and more ﬁr m se n t e r .W et h u ss h o wt h a tt h er e s p o n s e
of entry to a change in workers’ rents depends on the structure of costs.2
Furthermore, the equilibrium number of ﬁrms can become larger than the
social optimum.
2T h e m o d e l
We consider a general equilibrium model under imperfect competition where
m individuals consume an homogenous good produced under constant re-
turns to scale and a bundle of diﬀerentiated varieties produced under in-
creasing returns to scale. Labor productivity is normalized to one in the
production of the former good, which is taken as the numéraire. Individuals
share the same preferences: U = c1−µ
o ∗ (
R n
0 c(i)1−1/σdi)
µ σ
σ−1 where co is the
consumption of the numéraire and where c(i) is the consumption of a diﬀer-
entiated variety i ∈ [0,n]. The share of revenue spent on the diﬀerentiated
varieties is µ and the share of revenue spent on the numéraire is 1 − µ;t h e
elasticity of substitution among diﬀerentiated varieties is constant and equal
to σ>1. Accordingly, consumers’ demand for the diﬀerentiated variety i is
given by
c(i)=
µ
p(i)
P
¶−σ µE
P
where P ≡
µZ n
0
p(i)
1−σdi
¶ 1
1−σ
(1)
where E denotes the consumers’ expenditure, p(i) is the price of the diﬀer-
entiated variety i,a n dP is the price index of the diﬀerentiated varieties.
Individuals work either in the constant returns to scale sector or in the
increasing returns to scale sector. In the former sector, we assume unit labor
productivity so that the wage is equal to one, i.e. the price of the numéraire.
In the latter sector, there is a rent that can be bargained between employers
and employees so that workers earn larger wages in that sector (w>1). In
the Appendix, we extend the simple bargaining framework that justiﬁes such
rigid wages higher than in the constant returns to scale sector (see also Picard
and Toulemonde 2005). Larger wages can also result from incentives to
eﬀort as in eﬃciency models or from workers’ investment in industry speciﬁc
education.
2To our knowledge, no authors have analyzed the cost side eﬀects on diversity. Gans
(1997) develops a model with endogenous ﬁxed costs to capture ‘big push’ eﬀects, but
oﬀers no conclusion about diversity.
3Firm i’s proﬁtc a nb ew r i t t e na sπ(i) ≡ [p(i) − v]x(i) −f where v and f
denote variable and ﬁxed costs respectively. Firms use two inputs: unionized
labor and numeraire. The parameters α and β ∈ [0,1] describe the shares
of unionized labor in variable and ﬁxed costs so that v ≡ αw +( 1− α) and
f ≡ βw+( 1− β). Total expenditures in the economy are made of earnings
of individuals working in the constant returns to scale sector, and wages of
unionized workers3: E =( m − l)+lw where l is the total number of workers
paid at the unionized wage w.
3 Monopolistic competition
Firm i chooses the price p(i) that maximizes its proﬁts. Under monopolistic
competition, each ﬁrm takes the index P and the expenditures E as given.
Because the product demand is iso-elastic, ﬁrm i s e t si t sp r i c ea sam a r k u p
over the variable costs: p ≡ p(i)=vσ/(σ − 1). Given the manufactured
price index (P = pn−1/(σ−1)) and the demand for each variety, output and
proﬁts are equal to
x =
(σ − 1)µE
σvn
and π =
µE
σn
− f (2)
Under free entry, ﬁrms enter until their proﬁts fall to zero. Wages may
aﬀect proﬁts through their eﬀects on ﬁxed costs, on variable costs, and on
expenditures. First, larger wages increase the ﬁxed costs f, which clearly
reduces the equilibrium number of ﬁrms. Second, larger wages increase the
variable costs v, but this does not aﬀect π, as seen in (2). Indeed, with iso-
elastic demands, an increase in variable costs is automatically matched by a
change in the production that leaves proﬁts unaﬀected. Third, larger wages
aﬀect expenditures:
E = m + n(αx + β)(w − 1) =
σ(m − n)(σw +1− α)
ασ (1 − µ)(w − 1) + σ − µ
(3)
Whereas larger wages reduce the output (and the employment level) (see
(2)), they also raise the earnings of unionized workers so that the net eﬀect
on expenditures E is ap r i o r iunknown. From the second equality in the
above expression, it is readily checked that an increase in wages raises the
earnings, which promotes the entry of new ﬁrms.
3We assume that there is free entry so that proﬁts fall to zero.
4Under free entry, the equilibrium number of ﬁr m si sg i v e nb y
ne =
µm
σf (1 − µ)+µ + µ(σ − 1)f/v
(4)
where
dne
dw
> 0 ⇐⇒
vµ(σ − 1) + v2σ(1 − µ)
µ(σ − 1) + v2σ(1 − µ)
>f
where we used the deﬁnitions of v and f to substitute for β and α.
T h en o v e l t yi nt h i sp a p e ri st h ea n a l y s i so ft h ee ﬀects that wages (costs)
have on entry via the expenditures in a general equilibrium framework. Since
under w>1,vis larger than one, it is readily veriﬁed that the left hand side
of the last inequality has the following properties: (i) it is larger than one,
(ii) it is larger than f if v>f(that is if α>β ), and (iii) it increases with
v - and thus with α - for reasonable values of union wage premium (i.e. for
w ∈ [1,1+
p
(σ − µ)/(σ(1 − µ))] which includes w ∈ [1,2]).
As a result of property (i), an increase in wages always promotes entry
if ﬁxed costs are paid in terms of the numéraire (f =1or β =0 ). As a
consequence of property (ii), the increase in wages also promotes entry if the
share of unionized labor is proportionally larger in variable costs than in ﬁxed
costs. Finally, because of property (iii) and because f increases with β,t h e
following proposition applies:
Proposition 1 An increase in unionized wages raises the number of vari-
eties if the share of unionized workers in the variable costs (α)i sh i g he n o u g h
or if their share in the ﬁxed costs (β) is low enough.
The natural next question is whether the economy operates with too much
or too little entry.
4 Entry: too much or too little?
We compare the competitive equilibrium with the (unconstrained) social op-
timum where a planner is able to choose the values of co, c(i) and n that
maximize utility under the resource constraint. In the constant returns
to scale sector, the economy uses co units of labor (for producing co)a n d
(1 − β)n+(1− α)
R n
0 c(i)di units of labor to produce the input for the other
sector. The increasing returns to scale sector also uses βn+α
R n
0 c(i)di units
5of labor. The resource constraint is therefore m = co +
R n
0 c(i)di + n. Sub-
stituting co from the constraint in the planner’s objective, maximizing with
respect to c(i) and n, and using symmetry (c(i)=c ∀i ) yields the ﬁrst best
levels of consumption and varieties: co = σ − 1 and no = µm/(σ − 1+µ).
Comparing ne to no gives
ne − no < 0 ⇐⇒
(σ − 1)v
σ(1 − µ)v + µ(σ − 1)
<f
If unionized workers do not manage to get larger wages than the workers
from the constant returns to scale sector (w =1 ), then v = f =1and the
above inequality holds. This is the standard result according to which there
is too little entry under perfect labor markets (Dixit and Stiglitz, 1977 and
Vives, 1999). Now when unionized workers get larger wages, the expenditures
E increase with wages, which may promote entry and may result in over
provision of varieties, particularly when wages have a strong eﬀect on the
number of ﬁrms ne. Indeed, it can be checked that the left hand side of the
condition increases in v, and thus in α, whereas the right hand side increases
with β. This gives the following proposition
Proposition 2 There is too much entry if the share of unionized workers in
t h ev a r i a b l ec o s t s( α)i sh i g he n o u g ho ri ft h e i rs h a r ei nt h eﬁxed costs (β)i s
low enough. Otherwise, there is too little entry.
This proposition readily applies to conﬁgurations where variable and ﬁxed
c o s t sa r em a d eo fas i n g l et y p eo fw o r k e r s ,i . e . w h e nα and β are equal to
0 or 1. Then, one can check that excess entry is supported only when the
variable cost is paid in terms of unionized labor whereas the ﬁxed cost is
paid in terms of the numéraire (α =1 , β =0 ). Two additional conditions
are then required: the share of revenues spent on the diﬀerentiated varieties
must be large enough (σµ > 1) and unionized wages must be large enough
(w>(µσ − µ)/(µσ − 1)). Because the right hand side of this last condition
increases in σ and decreases in µ, we conclude that excess entry is more
likely when varieties are lower substitutes (low σ) or when they are more
intensely consumed (high µ). In both cases, ﬁrms beneﬁt from both larger
market power and larger sales, which increases proﬁts and attracts new ﬁrms
that may enter in excess. Furthermore, excess entry is not inconsistent with
reasonable values of economic parameters.4
4These two conditions are fulﬁlled for instance if σ =4 , µ =0 .75 and w ≥ 1.125,
which are relevant values of parameters (see e.g. Hanson, 1998, Head and Mayer, 2004 for
65 References
Akerlof G.A. and Yellen J.L. (1990), The Fair Wage-Eﬀort Hypothesis and
Unemployment, Quarterly Journal of Economics, 105, 255-283.
Benassy J.P. (1996), Taste for Variety and Optimum Production Patterns in
Monopolistic Competition, Economic Letters, 52, 41-47.
Chamberlin E. (1950), Product Heterogeneity and Public Policy, American
Economic Review Proceedings, 40, 85-92.
d’Aspremont C., Dos Santos Ferreira R., Gerard-Varet L.A., (1989), Un-
employment in a Cournot Oligopoly Model with Ford Eﬀects, Recherches
Economiques de Louvain. 1989, 55, 33-60.
d’Aspremont C., Dos Santos Ferreira R., Gerard-Varet L.A. (1996), On the
Dixit-Stiglitz Model of Monopolistic Competition, American Economic Re-
view, 86, 623-629.
Dixit A.K. and Stiglitz J.E. (1977), Monopolistic Competition and Optimum
Product Diversity, American Economic Review, 67, 297-308.
Gans J.S. (1997), Fixed Cost Assumptions in Industrialization Theories, Eco-
nomic Letters, 56, 111-119.
G.H. Hanson (1998), Market Potential, Increasing Returns, and Geographic
Concentration, NBER Working Paper 6429.
K. Head and T. Mayer (2004), The Empirics of Agglomeration and Trade, in
Handbook of Urban and Regional Economics, Ed. J.V. Henderson and J-F
Thisse.
H. G. Lewis (1986), “Union Relative Wage Eﬀects: A Survey”, The Univer-
sity of Chicago Press, Chicago.
McDonald I.M. and Solow R.M. (1981), Wage Bargaining and Employment,
American Economic Review, 71, 896-908.
P. Picard and Toulemonde E. (2005), Unions and Firms Agglomeration,
Forthcoming in European Economic Review.
Solow, R.M. (1979), Another Possible Source of Wage Stickiness, Journal of
Macroeconomics,1 ,7 9 - 8 2 .
estimations of σ and µ, see Lewis, 1986 for estimations of w).
7Vives X. (1999), Oligopoly Pricing: Old Ideas And New Tools, MIT Press.
Yang X. and Heijdra B. (1993), Monopolistic Competition and Optimum
Product Diversity: Comment, American Economic Review, 83, 295-301.
86 Appendix: Endogenous wages
In the paper we derived conditions for excess entry under exogenous wages.
Still, one needs to be convinced that excess entry exists when wages are en-
dogenous. We already know that if wages in the increasing returns to scale
sector are determined on a perfectly competitive labor market (w =1 ), then
there is no excess entry. If labor markets are non competitive then labor
market conditions link the wages to the number of ﬁrms by a relationship
w = F(n). The equilibrium wage and number of ﬁrms is obtained by solving
this relationship with (4). In general, there is no closed form solution and
numerical simulations are needed. Yet we are able to present a simple ex-
ample of wage bargaining that yields excess entry. To simplify the analysis,
we focus on the simplest case in which there can be excess entry: we assume
that unionized workers contribute only to variable costs: α =1and β =0 .
We assume a decentralized wage setting with one independent, utilitarian
union per ﬁrm. First, the union and the ﬁrm bargain over wages. Then the
ﬁrm chooses employment given wages (the ﬁrm has the right to manage). We
consider the Nash solution, according to which wages maximize the following
product
N ≡
£
V (i) − V
¤φ
[π(i) − ¯ π]
1−φ
where φ is the union bargaining power, V (i) is the union utility, π(i) is the
ﬁrm’s proﬁts, V and ¯ π are the fall-back utilities and proﬁts.
Consider the ﬁrm i and its associated union i that bargain over the wage
w(i). On the one hand, union i maximizes the sum of utility that its workers
derive from the wages oﬀered by the ﬁrm. With the Cobb-Douglas-CES utility
speciﬁcation, worker’s utility is w(i)/PG where PG ≡ Pµµ−µ (1 − µ)
µ−1.I n
case of persistent disagreement with the ﬁrm, workers go in the constant
returns to scale sector where the wage is 1; they have a utility equal to 1/PG.
Hence, union utility levels under agreement and disagreement are equal to
V (i)=w(i)l(i)/PG and V (i)=l(i)/PG where l(i)=x(i) denotes the level
of employment in ﬁrm i.
On the other hand, ﬁrm i maximizes its proﬁt π(i)=( p(i) − w(i))x(i)−1.
Under right-to-manage, the ﬁrm chooses its price p(i) and consequently its
output level x(i). Maximization w.r.t. p(i) yields the price and the output
pe(i)=
σ
σ − 1
w(i) and xe(i)=µ
µ
σ
σ − 1
w(i)
P
¶−σ E
P
9In case of persistent disagreement with the union, we assume that the ﬁrm
still incurs the ﬁxed cost (¯ π(i)=−1). This is also what is implicitly assumed
in many models of wage bargaining in which the ﬁxed cost is set equal to
zero (see e.g. the seminal paper by McDonald and Solow, 1981). The ﬁrm’s
contribution to the Nash product is therefore
π(i) − ¯ π =
µ
σ
σ − 1
w(i)
P
¶−(σ−1)
E/σ.
In a Nash bargaining, w(i) maximizes the Nash product. Under monop-
olistic competition, there is a large number of ﬁrms and unions that consider
the price index P, and the earnings E as constants. The maximization gives
w(i)=w =1+
φ
σ − 1
for all ﬁrms i.T h u st h ew a g ei saﬁxed mark-up over the wage in the constant
returns to scale sector and is independent of the number of ﬁrm. The analysis
in Section 3 holds.
Finally we check whether excess entry can be supported by this simple
model. Let us take the previous example with σ =4and µ =0 .75. Then,
we have that w =1+φ/3 and excess entry occurs for φ ∈ [.375,1].
10

Another Possible Source of Wage Stickiness,

estimations of σ and µ, see Lewis,

Fixed Cost Assumptions in Industrialization Theories,

Market Potential, Increasing Returns, and Geographic Concentration,

Oligopoly Pricing: Old Ideas And New Tools,

Product Heterogeneity and Public Policy,

Taste for Variety and Optimum Production Patterns in Monopolistic Competition,

The Empirics of Agglomeration and Trade,

The Fair Wage-Eﬀort Hypothesis and Unemployment,

Unemployment in a Cournot Oligopoly Model with Ford Eﬀects, Recherches Economiques de Louvain.

Union Relative Wage Eﬀects: A Survey”,

Unions and Firms Agglomeration, Forthcoming in European Economic Review.

On Monopolistic Competition and Optimal Product Diversity: a Comment on Cost Structure and Workers' Rents

http://www.hec.unil.ch/deep/textes/05.02.pdf

On Monopolistic Competition and Optimal Product Diversity: a Comment on Cost Structure and Workers' Rents

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics