Monopolistic Competition: CES Redux?
Paolo Bertolettiy
Paolo Epifaniz
Pavia University and IEFE
Bocconi University and BAFFI
February 2014
Abstract
We study the competitive and reallocation e¤ects of trade opening in monopolistic
competition. To this purpose, we generalize the Melitz (2003) setup with heterogeneous …rms and …xed and variable trade costs beyond the CES to the case of additively
separable utility functions. We …nd that extensive margin (Melitz-type selection) effects are robust to relaxing the CES assumption. Intensive margin e¤ects (market
share reallocations across inframarginal …rms) and competitive (markup) e¤ects are
instead fragile. An important implication is that measured productivity gains from
trade opening are no longer ensured with non-CES preferences. We discuss our results in the light of alternative setups featuring non-additive preferences, strategic
interaction and consumers’preference for an ideal variety.
JEL Classi…cation: F1; Keywords: Monopolistic Competition; CES Preferences;
International Trade; Competitive and Reallocation E¤ects.
1
Introduction
In this paper, we explore the robustness of the competitive and reallocation e¤ects of trade
opening in monopolistic competition. Our study is motivated by the fact that the Dixit
and Stiglitz (1977, henceforth D-S) setup with constant elasticity of substitution (CES)
preferences, so far the workhorse of trade economists, has been put under …re by a recent
The paper has bene…ted enormously from the comments of Bob Staiger (the Editor) and two anonymous referees. We are also grateful to Kristian Behrens, Fabrice Defever, Swati Dhingra, Federico Etro,
Eileen Fumagalli, Gino Gancia, Dennis Novy, Marco Ottaviani, Marcello Pagnini, Clara Poletti, Luca
Salvatici, Piero Tedeschi and seminar participants at many venues for helpful comments and discussions.
All errors are our own.
y
Department of Economics and Management, Università di Pavia, Via San Felice 5, 27100 Pavia (Italy),
and IEFE, Bocconi University. E-mail: [email protected]
z
Department of Economics and BAFFI, Università Bocconi, Via Röntgen 1, 20136 Milan (Italy). Email: [email protected].
1
literature.1 The traditional critique that CES preferences make monopolistic competition
little interesting, due to the implied invariance of markups to trade opening, has been
recently revived by Neary (2004, 2009a, 2010), who proposes to study competitive e¤ects
in oligopoly. Dhingra and Morrow (2012, henceforth DM) and Zhelobodko, Kokovin,
Parenti and Thisse (2012, henceforth ZKPT) propose instead to relax the CES assumption
in the standard D-S setup. In particular, ZKPT shows that CES preferences are a knifeedge between cases yielding opposite competitive and selection e¤ects, thereby concluding
that a theory of monopolistic competition cannot be built on CES preferences, due to
the "peculiarity" of its implications.2 We argue instead that the critique of the CES
assumption in monopolistic competition has gone too far. First, because although CES
preferences are clearly peculiar, relaxing the CES assumption does not obviously lead to
less peculiar competitive and reallocation e¤ects. Second, because selection e¤ects à la
Melitz (2003) are indeed robust to relaxing the CES assumption.
In Section 2, we generalize the Melitz (2003) model to the case of additively separable
utility functions by exploiting his approach of relating the behavior of active …rms to
that of a cuto¤ …rm. In Section 3, we then use the model to study the competitive and
reallocation e¤ects of trade opening, both in the stylized scenario of pure globalization
(formally equivalent to an increase in market size) and in the more general case of costly
trade. Our approach allows us to easily obtain some of the results independently found
by ZKPT and DM, to identify a new source of reallocations and to deal with the case
of costly trade. In particular, we show that reallocation e¤ects consist of two distinct
mechanisms: an extensive margin (or Melitz-type selection) e¤ect, whereby trade changes
the set of active …rms through entry and exit, and an intensive margin e¤ect, whereby
trade a¤ects the market shares of inframarginal …rms. By competitive e¤ects we refer
instead to the impact of trade on …rms’ price-marginal cost markups. We show how, in
non-CES environments, the e¤ects of trade opening depend on properties of the additive
functional forms that we characterize in this paper, and how the latter a¤ect aggregate
productivity.
Our main results can be summarized as follows. When trade costs are large enough
to induce a partitioning of …rms into exporters and non-exporters (arguably, the empiri1
In this paper, we refer to monopolistic competition as a market structure characterized by product
di¤erentiation, a downward sloping average cost curve and free entry. By Dixit-Stiglitz assumption we
refer instead to the absence of strategic interaction among …rms.
2
Another recent paper by Arkolakis, Costinot and Rodriguez-Clare (2012) argues that, provided that
certain conditions that crucially exploit the properties of CES preferences are satis…ed, gains from trade
are invariant to the microeconomic details of the underlying model (conditional on observed trade ‡ows).
It also shows, however, that with CES preferences two statistics (the expenditure share of domestic goods
and the elasticity of substitution) are su¢ cient to compute the welfare gains from a reduction in trade
costs.
2
cally relevant case), extensive margin e¤ects are robust to relaxing the CES assumption.
Intensive margin and competitive e¤ects are instead fragile. An important implication is
that measured productivity gains from trade opening are not ensured with non-CES preferences. We argue that these results arise from the fact that, in a D-S setup with additive
preferences (with or without heterogeneous …rms), markup changes do not fully re‡ect the
action of competitive forces. In Section 4, we therefore consider alternative monopolistic
competition environments. We …rst consider the example of quasi-linear quadratic preferences, as in Melitz and Ottaviano (2008), thereby relaxing the assumption of additive
preferences. Next, we relax the D-S assumption that …rms do not interact strategically
by studying both Bertrand and Cournot competition with additive preferences. Finally,
we reconsider Lancaster’s (1979) ideal variety approach to monopolistic competition. Surprisingly, we …nd that none of these setups delivers a strong and robust trade-induced
pro-competitive mechanism. In Section 5, we brie‡y conclude that one possible interpretation of our results is that they indirectly support the venerable D-S monopolistic
competition setup with CES preferences.
Our paper is related to the vast theoretical literature on monopolistic competition
and international trade, initiated by Dixit and Stiglitz (1977), Krugman (1979, 1980),
Lancaster (1979), Helpman (1981), and whose early contributions are systematized in
Helpman and Krugman (1985). It is also closely related to Melitz (2003), whose methodology and results we generalize to the case of additively separable utility functions, and
to a recent literature which independently studies some implications of non-CES preferences in a D-S monopolistic competition setup with heterogeneous …rms. In particular,
our setup is similar but more general than that of ZKPT, which leads us to reach di¤erent
conclusions. Speci…cally, ZKPT does not study the intensive margin e¤ects of trade opening and their impact on aggregate productivity, nor the cases of costly trade and strategic
interactions. DM studies the welfare e¤ects of an increase in market size, and is therefore
complementary to our paper, as we do not focus on welfare issues. Arkolakis, Costinot,
Donaldson and Rodriguez-Clare (2012, henceforth ACDR) is also complementary to our
work, as it shows some counter-intuitive welfare implications of variable markups in monopolistic competition.3 Finally, Mrázová and Neary (2012) complements our analysis by
introducing a distinction between …rst-order and second-order selection e¤ects (the former
involve a binary choice between performing or not a certain activity, whereas the latter
concern how to perform a certain activity) and showing that the former are more robust
3
In particular, ACDR shows that, with a Pareto distribution of …rm productivity and preferences
featuring log concavity of demand and a choke-o¤ price, the welfare gains from a reduction in trade costs
are smaller than those predicted by models with constant markups.
3
than the latter.
2
Setup
In this Section, we illustrate the Dixit-Stiglitz monopolistic competition setup with additive preferences and heterogeneous …rms. In the next Section, we will use it to study the
competitive and reallocation e¤ects of trade opening.
Demand. Consider an economy populated by L consumers/workers, whose wage is w =
1. They share the same additively separable and symmetric preferences, de…ned over a
continuum of varieties and represented by the utility function
U=
Z
N
u(ci )di;
0
where ci is individual consumption of variety i, and N is the mass of potential varieties.
Only n < N varieties are available for consumption. The sub-utility function u( ) is strictly
increasing and concave, and is at least thrice continuously di¤erentiable. In particular, we
assume that u0 (c) > 0 > u00 (c) for c > 0, and u(0) = 0.
Rn
1, where pi is the
Utility maximization subject to the budget constraint 0 pi ci di
Rn 0
0
price of variety i, yields u (ci ) = pi , where = 0 u (ci )ci di is the marginal utility of
income. Under the D-S assumption that …rms treat
demand for a variety is thus
p(c) =
u0 (c)
as a constant, the inverse individual
;
(1)
where we have dropped the variety index, as all …rms face the same demand function. The
price elasticity of demand, "(c), is given by
"(c) =
p(c)
=
p0 (c)c
u0 (c)
;
u00 (c)c
(2)
and can be shown to equal the elasticity of substitution between any two varieties, (c),
when they are consumed in the same amount c.4 Krugman (1979) assumed that
0, i.e., a decreasing elasticity of substitution (henceforth, DES preferences);
0 (c)
0 (c)
<
> 0
represents instead the case of an increasing elasticity of substitution (henceforth, IES
preferences) and
0 (c)
= 0 is the well-known CES case.5
4
Note that " is independent of , and that it equals the reciprocal of the elasticity of marginal utility with
respect to individual consumption. See also Blackorby and Russell (1989) on the elasticity of substitution.
5
As far as we know, the case of monopolistic competition with IES preferences was …rst studied by
Bertoletti, Poletti and Fumagalli (2008). Bertoletti (2006) and Behrens and Murata (2012) discuss instead
a speci…c functional form for DES preferences.
4
Production. Producing a di¤erentiated variety involves a …xed cost
and a marginal
cost . As in Melitz (2003) we assume that, upon paying a …xed entry cost
their marginal cost
2
with density g( ) and
e,
…rms draw
; 1 from a common, continuous cumulative distribution G( )
> 0. All costs are in terms of labor.
Using (1), …rm revenue equals p(c)cL = r(c)(L= ), where r(c) = u0 (c)c. Marginal
revenue and the derivative of marginal revenue therefore equal, respectively, r0 (c)= and
r00 (c)=( L), where
r0 (c) = u0 (c) + u00 (c)c;
r00 (c) = 2u00 (c) + u000 (c)c:
(3)
The …rst-order condition for pro…t maximization yields:
r0 (c) =
;
(4)
where we assume that r0 (c) > 0 > r00 (c) for all c > 0 to obtain a well-behaved solution.6
Note that r0 > 0 implies
> 1.
Equations (1) and (4) imply that the pro…t-maximizing price rule can be written as
p = m(c) ;
m(c) =
(c)
u0 (c)
=
;
0
r (c)
(c) 1
(5)
where m(c) is the price-marginal cost markup. Note that m0 ? 0 ,
0
(c) ? 0 ,
r0 (c)u00 (c)
(c)
=
? 1;
00
0
r (c)u (c)
(c)
0
7 0, and that
r0 (c)
> 0;
r00 (c)c
(c) =
where (c) is the reciprocal of the elasticity of marginal revenue in absolute value. Thus,
markups are increasing in individual consumption c with DES preferences (
), decreasing in c with IES preferences (
0
>0,
0
<0,
<
> ) and constant in the CES case
( = ). Note also, for later reference, that the elasticity of m(c) can be written as:
1
d ln m(c)
=
d ln c
(c)
Finally, using (5), the expression for variable pro…ts
v(
; c; L) = [m(c)
1] Lc =
6
1
:
(c)
v
(6)
can be written as:
u0 (c)
r0 (c)
1
Lc:
(7)
We also assume that an interior solution to the problem of pro…t maximization exists. Su¢ cient (but
not necessary) conditions are: limc!1 r0 (c) = 0 and limc!0 r0 (c) = 1.
5
Di¤erentiating (7) yields that
v
is increasing in c, with
@ ln v
(c)
=
> 0:
@ ln c
(c)
Individual consumption. Denote by
of
(8)
the marginal cost cuto¤, namely, the value
satisfying the zero cuto¤ pro…t condition (
consumption of a cuto¤ …rm’s good, and
; c ; L; ) = 0, where c is individual
( ; c; L; ) =
v(
; c; L)
is total pro…t.
Thus, using (7),
v(
which implicitly de…nes c (
; c ; L) = [m(c )
1]
c L= ;
(9)
; L; ).7 Following Krugman (1979) and Melitz (2003), we
can now relate the behavior of an active
the marginal utility of income
-…rm to that of a cuto¤ …rm by eliminating
from the relevant …rst-order conditions for pro…t max-
imization. Formally, r0 (c ) =
by (4). Solving for
and substituting back into (4)
yields:
r0 (c) = r0 (c )
:
(10)
Eq. (10) is key to the characterization of the equilibrium, as it implicitly de…nes the
individual consumption of a -…rm’s good as c( ;
show how c( ;
;c (
; L; )) varies with
; c ). Using c (
, L and
; L; ), we can then
. In the Appendix, we prove the
following
Lemma 1 Individual consumption of a -…rm’s good, c( ;
; L; ), is increasing in
and
and decreasing in L, with
(c)
@ ln c
@ ln c
=
> 0;
=
@ ln
m(c )
@ ln L
@ ln c
=
@ ln
Free entry. Free entry implies that expected pro…ts,
E
where c = c( ;
=
Z
@ ln
@ ln
E
=
Z
equal the …xed entry cost,
( ; c; L; )dG( ) =
; L; ). Di¤erentiating
gives:
E,
(c)
< 0:
(c )
E
with respect to
@ ln v @ ln c
dG( ) =
@ ln c @ ln
Z
e;
e:
(11)
, using (8) and Lemma 1,
(c)
dG( ) > 0:
m(c )
(12)
7
Note, from (9), that this and the following results actually depend on L= . For expositional purposes,
however, in this Section we treat L and as separate arguments.
6
Thus, as in Melitz (2003), expected pro…ts are increasing in
(11) uniquely pins down
value of c( ; L; ;
as a function of L,
and
e,
and the free-entry condition
thereby de…ning the equilibrium
e ).
Firm size, pro…ts and markups. From now on, we index …rms by their marginal
cost
and denote by c( ) the equilibrium individual consumption of a -…rm’s good. Let
m( ) = m(c( )), p( ) = m( ) and
v(
) = [m( )
1] c( )L be the optimal markup,
price and variable pro…ts of a -…rm.8 As shown in the Appendix:
d ln c( )
d ln
d ln v ( )
d ln
=
= 1
d ln p( )c( )
=1
(c( )) < 0;
d ln
d ln m( )
(c( ))
(c( )) < 0;
=
1?0,
d ln
(c( ))
(c( )) < 0;
(13)
0
(c) ? 0; (14)
where
(c) = 1 + (c) 1
1
> 1;
(c)
(c) ? (c) ,
0
(c) ? 0:
(15)
Note that, as in Melitz (2003), high-productivity (low- ) …rms are larger, both in terms
of output and revenue, and more pro…table. Unlike the Melitz model, however, where
preferences are CES and markups are constant, in this setup high-productivity …rms charge
higher markups with DES preferences and lower markups with IES preferences. Moreover,
=
=
for
0
= 0. With non-CES preferences, instead,
6=
6=
and all these
variables are functions of c, thereby implying that output, revenue and pro…t pro…les are
governed, respectively, by ,
3
and .
Competitive and Reallocation Effects of Trade Opening
Having laid down the setup of the model, we can now study the extensive margin, intensive
margin and competitive e¤ects of trade opening. We begin by analyzing the impact of an
increase in market size L, which can be interpreted as pure globalization. This will allow
us to clarify the basic forces at work in our setup and to better compare our results to
previous works. Then we will study the more realistic case of costly trade, in which trade
opening leads to a reduction in …xed or variable trade costs.
8
The measure of active …rms n is determined by the budget condition, requiring average expenditure
to equal 1=n. Thus,
"Z
# 1
dG( )
p( )c( )
:
n=
G( )
7
3.1
Pure Globalization
Extensive margin e¤ects. The extensive margin (selection) e¤ects of pure globalization are captured by the impact of an increase in L on the marginal cost cuto¤
shown in the Appendix, d
=dL ? 0 ,
0
. As
? 0. Thus, pure globalization leads to a stan-
dard selection e¤ect with DES preferences. With IES preferences, instead, it leads to an
anti-selection e¤ect, whereby less productive …rms can survive in a larger market.9 The
mechanics behind these results can be summarized as follows. First, using Lemma 1, the
total impact of an increase in L on individual consumption equals
@ ln c
@ ln c d ln
d ln c( )
=
+
=
d ln L
@ ln L @ ln
d ln L
where
;L
= 1= (c )
(1=m(c )) (d ln
(c( ))
;L ;
(16)
=d ln L) is a term common to all …rms. Next,
using (8) and (16), the total impact of an increase in market size on variable pro…ts equals
@ ln v d ln c
d ln v ( )
=1+
=1
d ln L
@ ln c d ln L
Finally, totally di¤erentiating
Z
which implies that
E,
;L :
(17)
) = 0;
(18)
and using (17), yields:
d ln v
dG( ) =
d ln L
;L
(c( ))
Z
[1
(c( ))
;L ] dG(
> 0, and hence that c( ) is decreasing in L. The intuition is
that, ceteris paribus, an increase in market size raises variable pro…ts proportionally by
increasing all market demands; individual demands must therefore fall to restore the zero
expected pro…t condition. Note, also, that
;L
= 1=
and d ln
CES case. With non-CES preferences, instead, the sign of d ln
: when
0
v(
v(
)=d ln L = 0 in the
)=d ln L depends on
< 0, a rise of L leads to an increase in the relative pro…ts of more productive
…rms. In this case, the pro…ts of a
-…rm become negative and the marginal cost cuto¤
must fall to restore the zero cuto¤ pro…t and free entry conditions. Eq. (18) also implies
that the pro…ts of more productive …rms increase. Conversely, when
0
> 0, an increase
in L raises the relative pro…ts of less productive …rms, and thus leads to a rise of
and
a reduction in the pro…ts of more productive …rms.
To conclude, as also argued by ZKPT and DM, selection e¤ects seem to crucially
depend on the assumptions about preferences. In the next Section we show, however,
9
It is also possible to show that, independent of consumer preferences, an increase in the …xed production
cost leads to a selection e¤ect, whereas an increase in the …xed entry cost e leads to an anti-selection
e¤ect.
8
that the fragility of Melitz-type selection e¤ects is more apparent than real, as the above
results hold only in the absence of trade frictions.
Intensive margin e¤ects. ZKPT and DM seem to argue that selection e¤ects are the
only source of reallocations induced by an increase in L. Note, however, that with nonCES preferences an increase in market size also reallocates market shares (both in terms of
sales and revenue) across inframarginal …rms. This follows immediately from (16), which
implies that a rise of L leads to an increase in the relative output of high-productivity
…rms when
0
0
< 0, and to an increase in the relative output of low-productivity …rms when
> 0. Similarly, using (6) and (16), the total impact of L on p( )c( ) equals:
d ln m
+1
d ln c
d ln fp( )c( )g
=
d ln L
d ln c
=
d ln L
(c( ))
;L
< 0:
Evidently, an increase in market size leads to an increase in the relative revenue of
0
high-productivity …rms when
0
productivity …rms when
< 0, and to an increase in the relative revenue of low-
> 0.10 We summarize our main results in the following
Proposition 1 The extensive margin e¤ ects of pure globalization are governed by the sign
of
0.
Instead, the intensive margin e¤ ects in terms of sales and revenue are governed,
respectively, by the signs of
0
and
0.
Note also that:
0
(c) =
r00 (c)2 c
[r000 (c)c + r00 (c)] r0 (c)
)
(r00 (c)c)2
0
(c) ? 0 ,
r000 (c)c
1 + (c)
?
;
00
r (c)
(c)
where r000 = u0000 c + 3u000 . Moreover, di¤erentiating (c) yields:
0
(c) =
(c)
(c)
0 (c)
(c)
( (c)
1) +
0 (c)
(c)
:
Consequently, the intensive margin e¤ects seem to depend on properties of preferences
that are hard to pin down and are not determined by the sign of
0.
As discussed in the
10
The marginal utility of income, , provides an alternative way of illustrating the model mechanics. To
see how, note once again that, ceteris paribus, an increase in market size raises variable pro…ts proportionally by increasing all market demands. By (1), must therefore increase to reduce individual demands
and pro…ts. Note also, from (4), that a rise of L can a¤ect the pro…t-maximizing level of c( ) only through
, and that and play a symmetric role in determining c( ). Therefore, the results in (13) and (14)
immediately tell us that, with non-CES preferences, the e¤ects of an increase in market size L on relative
…rm output, revenue and pro…ts are determined by the signs of 0 , 0 and 0 . Actually, it is easily shown
that:
d ln c( )
d ln c d ln
d ln
=
=
(c( ))
;
d ln L
d ln d ln L
d ln L
thereby implying that ;L is the elasticity of with respect to L.
9
Appendix, an important implication is that pure globalization has an ambiguous impact
on a measure of aggregate productivity when the implied extensive and intensive margin
e¤ects point in opposite directions. For instance, the selection e¤ect implied by
may be associated with a reduction in measured productivity if
0
0
<0
> 0.11
We record this result in the following
Remark 1 If
0 and
0
do not agree in sign, a selection e¤ ect may be associated with
a reduction in measured productivity when preferences are DES; an anti-selection e¤ ect
may instead be associated with an increase in measured productivity when preferences are
IES.
Competitive e¤ects. The competitive e¤ects of pure globalization depend on how it
a¤ects the behavior of inframarginal …rms and the selection of …rms through entry or
exit. The impact on inframarginal …rms entirely depends on how an increase in L a¤ects
individual consumption. Since c is decreasing in L, an increase in market size leads
inframarginal …rms to charge lower markups when
0
< 0 and higher markups when
0
>
0.12 Moreover, the anti-competitive e¤ect associated with IES preferences is strengthened
by the concomitant anti-selection e¤ect, as the latter implies entry of low-productivity,
high-markup …rms. In contrast, with DES preferences there is a countervailing increase in
average markups due to exit of low-productivity, low -markup …rms. Finally, the impact
on the average markup weighted by market shares also depends on the intensive margin
e¤ects discussed above.
3.2
Costly Trade
We now show how the e¤ects of a reduction in trade cost are di¤erent from those of
pure globalization. To this purpose, we consider two identical countries separated by
symmetric trade costs. Variables related to the foreign market will be indexed by an x,
whereas variables related to the domestic market will not be indexed. Selling in the foreign
market involves a …xed cost of exporting
x
0 and a variable iceberg trade cost
1.
As in Melitz (2003), we focus on the more interesting case in which trade costs are large
enough to induce a partitioning of …rms into exporters and non-exporters. This requires
11
Although measured productivity changes do not directly measure welfare changes, they provide the
standard way of quantifying empirically the gains from trade due to reallocations across heterogeneous
…rms: see, e.g., Melitz and Tre‡er (2012, p. 109) on this point.
12
Similarly, it is possible to show that an increase in or e are pro-competitive with IES preferences
and anti-competitive with DES preferences.
10
>
x,
where
x
is the marginal cost cuto¤ for exporting …rms.13
A -…rm’s pro…ts in the domestic and foreign markets are given, respectively, by ( ) =
v(
)
and
x(
)=
of each market, and
v
v(
1(
)
x,
x)
v
where
1(
v(
) = [m(c( ))
1] c( )L, L is the size
) to satisfy the condition
14
x.
Thus, the
marginal cost cuto¤ for exporters is implicitly given by:
x( x)
= [m(c(
x ))
1]
x c(
x )L
x
= 0:
(19)
Finally, the free-entry condition is now given by:
E
We use (19) to de…ne
=
x(
Z
; ;
Extensive margin e¤ects.
( )dG( ) +
x ),
Z
x
x(
)dG( ) =
e:
and (9), (10) and (20) to pin down
(20)
.15
As shown in the Appendix, applying the implicit function
theorem to (20) yields:
m(c )
@ ln
@ ln
=Z
Evidently, a reduction in
x
x
c(
p( )c( )LdG( ) +
Z
leads to a reduction in
in the Appendix we show that @
exporting
Z
=@
x
)LdG( )
> 0:
(21)
x
p(
)c(
)LdG( )
, a standard selection e¤ect. Moreover,
> 0, i.e., that a reduction in the …xed cost of
also leads to a selection e¤ect.
The intuition for why extensive margin e¤ects are robust to the sign of
0
is that, as
in the Melitz model, a reduction in trade costs increases expected pro…ts by increasing
exporting …rms’pro…ts at the expense of non-exporting …rms, which face instead a tougher
competition in the domestic market. Consequently, a cuto¤ …rm’s pro…ts become negative
13
Given that we consider two identical countries, balanced trade requires the wage to be equalized, and
we can use it as the numeraire. Moreover, our strategy of eliminating the (common) marginal utility of
income proves as useful as in the case of pure globalization.
14
Note that a domestic …rm producing only for the foreign market would incur an overall …xed cost equal
to + x , which implies that this case cannot arise in equilibrium. For analytical convenience, we can
therefore apportion the …xed cost to domestic pro…ts, in this following the heterogeneous-…rm literature.
15
The measure of active …rms is given by:
# 1
"Z
Z x
dG( )
dG( )
+
:
n=
p( )c( )
p( )c( )
G( )
G( x )
11
and the marginal cost cuto¤ has to fall to restore the equilibrium. We record these results
in the following
Proposition 2 When …xed (
x)
and variable ( ) trade costs induce a partitioning of …rms
into exporters and non-exporters, a reduction in either
x
or
leads to a reduction in
.
The extensive margin e¤ ects of a reduction in …xed and variable trade costs are therefore
robust to the sign of
0.
Intensive margin e¤ects. Consider …rst a reduction in the …xed cost of exporting
Note that
in
x
x
a¤ects individual consumption only through
, and that
x.
is increasing
according to Proposition 2. Moreover, individual consumption is increasing in
according to Lemma 1. It follows that a reduction in
cx = c(
x
reduces both c = c( ) and
), with:
(c( )) @ ln
d ln c( )
=
d ln x
m(c ) @ ln
In both expressions,
is the only
> 0;
x
d ln c(
d ln
)
=
x
(c( )) @ ln
m(c ) @ ln
> 0:
(22)
x
-…rm speci…c term. Consequently, as in the case of
pure globalization, the intensive margin e¤ects of a reduction in
x
depend on the sign of
0 .16
Consider now a reduction in the variable trade cost . By Proposition 2 and Lemma
1, this leads to a reduction in
and thus in c:
(c( )) @ ln
d ln c( )
=
d ln
m(c ) @ ln
As for cx , a reduction in
through
> 0:
(23)
has both a positive direct e¤ect and a negative indirect e¤ect
. Using Lemma 1 and (13) we can write:
d ln c(
d ln
)
=
@ ln c
@ ln c @ ln
+
@ ln
@ ln
@ ln
=
(c(
1 @ ln
m(c ) @ ln
)) 1
< 0;
(24)
where the inequality follows directly from (21). Thus, (23) and (24) imply that a reduction in
reduces individual consumption of domestic varieties c and increases individual
consumption of foreign varieties cx . Moreover, in both expressions
speci…c term. Consequently, independent of the sign of
0,
is the only
a reduction in
-…rm
a¤ects the
relative output of inframarginal …rms in opposite directions in the domestic and foreign
market. For instance, if
0
< 0, a reduction in
leads to an increase in the relative sales
of high-productivity …rms in the domestic market, and to a reduction in the relative sales
16
For brevity, in this Section we focus on intensive margin e¤ects in terms of sales.
12
of high-productivity inframarginal exporters in the foreign market.17 Just the opposite
results hold for
0
> 0. We record these results in the following
Proposition 3 The intensive margin e¤ ects (in terms of sales) of a reduction in …xed
and variable trade costs are governed by the sign of
0.
A change in the variable trade cost
leads to opposite intensive margin e¤ ects in the domestic and foreign market.
Competitive e¤ects. According to (22), a reduction in
x
reduces both c and cx , thus
leading inframarginal …rms to charge lower markups with DES preferences and higher
markups with IES preferences. Conversely, by (23) and (24), a reduction in
reduces
c and increases cx , leading inframarginal …rms to charge lower markups in the domestic
market and higher markups in the export market when preferences are DES. The opposite
result holds instead when preferences are IES.18
Finally, note that
d ln c( ) d ln c(
+
d ln
d ln
)
=
(c( ))
=1
@ ln
@ ln
2
m(c )
where the inequality follows from (21), which implies that @ ln
evaluated at
= 1. Thus, by continuity, a small increase in
1
< 0;
(25)
=1
=@ ln
around
< m(c )=2 when
= 1 leads all infra-
marginal exporting …rms to reduce average sales, and therefore to charge a lower average
markup across markets with DES preferences (a higher markup with IES preferences).19
Equation (25) also implies that, with non-CES preferences, average markups may be nonmonotonically related to trade costs. This follows from the fact that a move from costless
trade to autarky is equivalent to a reduction in market size and therefore leads …rms to
charge higher markups with DES preferences (and lower markups with IES preferences),
as discussed in the previous Section. We record these results in the following
Proposition 4 With DES preferences, a reduction in the …xed cost of exporting
17
x
leads
A reduction in trade costs can also be shown to increase x , thereby increasing the set of exporters.
As discussed in the previous Section, also the extensive margin e¤ects impact on average markups.
In particular, the distribution of markups is a¤ected by variation in the cuto¤s induced by changes in
x and , possibly leading to countervailing e¤ects on average markups (see also ACDR on this point).
For instance, as in the case of pure globalization, a reduction in with DES preferences forces the exit
of some low-productivity …rms in the domestic market. Given that these …rms charge lower markups,
their exit tends to increase the average markup charged by domestic …rms. Conversely, the entry of new
low-productivity, low-markup exporters may countervail the increase in the average markup charged by
inframarginal exporters.
19
If in addition x =
(namely, if all …rms export), the above results apply to the average markup
charged by all inframarginal …rms. Finally, given that the size of marginal …rms entering or exiting the
domestic and foreign markets after the increase in trade costs is smaller than the average, the overall
impact on the average of markups weighted by market shares is likely to be negative with DES preferences
(and positive with IES preferences).
18
13
inframarginal …rms to charge lower markups in the domestic and foreign market. Instead,
a reduction in the variable trade cost
leads inframarginal …rms to charge lower markups
in the domestic market and higher markups in the foreign market. The opposite results
hold with IES preferences. In both cases, the average markup of inframarginal …rms is
non-monotonically related to .
3.3
Discussion
Our results suggest that, although the extensive margin e¤ects of trade opening are robust
to relaxing the CES assumption, the intensive margin and competitive e¤ects are instead
fragile. An important implication is that gains from trade opening in terms of measured
productivity are no longer warranted in our setup.
The reason why allowing for variable markups in the Melitz model opens up a Pandora’s
box of contrasting and possibly counter-intuitive reallocation e¤ects is closely related to
the nature of competitive e¤ects in a Dixit-Stiglitz setup (with or without heterogeneous
…rms), in which markups are entirely driven by the level of individual consumption c. This
turns out to be restrictive for many reasons. The …rst is that the relationship between
individual consumption and markups is not robust to the assumptions about preferences:
markups are increasing in individual consumption with DES preferences and decreasing
with IES preferences. This explains why pure globalization is largely pro-competitive with
DES preferences and anti-competitive with IES preferences.20
Second, the equilibrium level of individual consumption is naturally decreasing in the
marginal cost, which leads to a pass-through e¤ect whereby markups vary with marginal
costs. For instance, markups are decreasing in marginal costs with DES preferences (incomplete pass-through), which explains why a reduction in the marginal cost of exporting
leads to higher markups in the foreign market. This also explains why a genuinely procompetitive shock, such as a reduction in variable trade costs, may have a non-monotonic
e¤ect on average markups. Finally, it implies that, whatever the assumptions about preferences, a reduction in variable trade costs leads to opposite intensive margin e¤ects in
20
These peculiar implications are perhaps surprising also because the IES assumption is prima facie no
less reasonable than the DES assumption. Indeed, a possible rationalization of IES preferences is that, by
their very nature, di¤erentiated varieties of some product can be used to perform either generic or more
speci…c tasks. For instance, a blue pencil can be used either to write down a laundry list (for which a red
pencil would be equally appropriate) or, jointly with a red pencil, to mark di¤erent types of comments on
an exam paper. Hence, a fall in the symmetric endowment of varieties, by reducing the opportunity to use
varieties to perform speci…c tasks, may also reduce their substitutability. Moreover, consider a situation
in which you are endowed with two red pencils and two blue pencils, and compare it with a situation in
which you are endowed instead with ten red and ten blue pencils: if you perceive a red and a blue pencil as
more substitutable in the latter situation, then your preferences are IES. See however Mrázová and Neary
(2012) for an argument in support of DES preferences which appeals to the so-called Marshall’s Second
Law of Demand.
14
the domestic and foreign market.
The direct dependence of markups only on individual consumption implies that the
e¤ects of trade opening are partly unrelated to pro-competitive forces. To deepen these issues, in the next Section we will explore the robustness of competitive e¤ects in alternative
monopolistic competition environments.
4
Alternative Environments
One possible explanation for the lack of robust competitive e¤ects in a D-S monopolistic
competition setup with additive preferences is that the latter implies that markups do
not directly depend on either the behavior of competitors or on their number n. This
is because the Dixit-Stiglitz assumption that …rms do not interact strategically implies
that the demand elasticity that they perceive equals the elasticity of substitution, and the
latter is independent of the number of varieties when preferences are additive.21 Moreover,
in this setup the number of product characteristics equals the number of varieties/…rms
and thus an increase in n does not "crowd" the variety space.22 It follows that a procompetitive e¤ect can arise only from variation in c, which does not fully capture, as
argued above, the operation of competitive forces.
In this Section, we therefore consider monopolistic competition environments in which
symmetric …rms produce a …nite number of varieties n which directly a¤ects the equilibrium elasticity of demand.23 We …rst analyze the case of quasi-linear quadratic preferences,
as in Melitz and Ottaviano (2008), then study the implications of strategic interaction à
la Bertrand and Cournot, and …nally discuss Lancaster (1979)’s ideal variety approach
to monopolistic competition. Perhaps surprisingly, we …nd that none of the above setups
seems to yield a robust pro-competitive mechanism in monopolistic competition.
4.1
Quasi-Linear Quadratic Preferences
Consider a D-S monopolistic competition setup with symmetric …rms and a utility function
U = c0 + u(c), where c0 is consumption of an outside good, c is the consumption vector
21
With non-additive preferences, the elasticity of substitution between any two varieties also depends
on the consumption of other varieties: at a symmetric equilibrium, this translates into a direct impact of
n on the elasticity of substitution for a given level of consumption c (see below).
22
See Pettengill (1979) on this point. He claims (p. 960) that "one’s normal idea of monopolistic competition is that as the number of products in the industry increases, they become closer and closer substitutes".
He therefore argues that Lancaster’s (1975) ideal variety approach to monopolistic competition is more
realistic in this respect.
23
Note that strategic interation cannot arise under the assumption, on which we relied in the previous
Section, of a continuum of varieties.
15
of n varieties of some product, and
u(c) = a
n
X
n
bX 2
cj
2
cj
j=1
j=1
with a; b;
@
n
X
j=1
12
cj A ;
(26)
> 0.24 The implied demand function for variety i is
ci =
where p =
2
0
1
n
Pn
j=1 pj
a
b+n
pi
n
1
+
p;
b
b+n b
(27)
is the average price of a variety. Under the D-S assumption that
…rms take n and p as given, …rm i’s perceived demand is linear in pi . Thus, at a symmetric
equilibrium the perceived demand elasticity can be written as
"=
p
1 ha
=
bc
b c
b
i
n :
(28)
Note that, perhaps surprisingly, for given c the number of …rms n has a negative direct
impact on the elasticity of demand. This example shows that, even if non-additive preferences introduce a direct channel whereby n a¤ects ", the resulting e¤ect is not obviously
positive.
Eq. (28) implies:
m(c) = 1 +
bc
Using (29), the free-entry condition (
v
; p = m (c)
= (m
= bc + :
(29)
p
b =L = ,
1) Lc = ) yields m = 1 +
thereby implying that equilibrium markups are decreasing in market size (or, more precisely, in L= ). However, given the negative direct impact of n on " for given c, the
pro-competitive e¤ect of pure globalization is entirely driven by the fact that, just as in
the case of DES preferences, the elasticity of substitution is decreasing in c.
Note also that m is decreasing in
. In the Appendix we show that, as with DES
preferences, this pass-through e¤ect implies that, under costly trade, a reduction in the
variable trade cost
leads …rms to charge lower markups in the domestic market and
higher markups in the foreign market.25 Moreover, a small increase in
from the free
trade leads to a reduction in average markups. We record these results in the following
24
We assume that c0 > 0, i.e., that an internal solution arises in equilibrium.
Similarly, when preferences are (homothetic) Translog, as in Feenstra (2003), it can be shown that a
reduction in trade costs leads …rms to charge higher markups in the foreign market. As shown by ACDR,
this may lead to smaller welfare gains from trade opening than in the case of constant markups. However,
Feenstra’s preferences imply that at a symmetric free trade equilibrium the elasticity of substitution is
linearly increasing in n and the markup is independent of the level of marginal cost .
25
16
Proposition 5 With quasi-linear quadratic preferences, as in the case of additive preferences, trade liberalization leads to opposite competitive e¤ ects in the domestic and foreign
market, and average markups are non-monotonically related to trade costs.
4.2
Strategic Interaction
The Dixit-Stiglitz assumption that each …rm treats the marginal utility of income, , as a
constant removes a direct channel whereby an increase in the number of …rms may raise
the perceived demand elasticity ". Instead, in a model with strategic interaction …rms
properly treat
as a function of prices, and the elasticity of demand no longer coincides
with the elasticity of substitution.
Consider …rst a setting with additive preferences and Bertrand competition, in which
…rms set prices. In a symmetric Bertrand-Nash equilibrium, …rms perceive the actual
demand elasticity, which can be shown to equal
(c) 1
:
n
"b (c; n) = (c)
(30)
Note that @"b =@n > 0, which captures the direct pro-competitive channel induced by
strategic interaction. Thus, the pricing rule is
p = mb (c; n) =
(n 1) (c) + 1
m(c) ;
(n 1) (c)
where @mb =@n < 0, and @mb =@c ? 0 , m0 ? 0 ,
0
(31)
7 0.
Next consider the case of Cournot competition, in which …rms set production levels.
As shown in the Appendix, in a symmetric Cournot-Nash equilibrium the pricing rule is
p = mc (c; n) =
n
n
1
m(c) ;
where, once again, @mc =@n < 0 and @mc =@c ? 0 , m0 ? 0 ,
(32)
0
7 0. Accordingly,
Cournotian …rms perceive a demand elasticity equal to26
"c (c; n) =
(c)n
:
(c) + n 1
We can use the pricing rules, (31) or (32), the free-entry condition (p =
(33)
+ =cL) and
the budget constraint (npc = 1) to characterize the symmetric Bertrand and Cournot
26
Note that "c (c; n) < "b (c; n) < "(c), i.e., with additive preferences Cournotian …rms perceive a lower
demand elasticity relative to Bertrand …rms, which in turn perceive a lower elasticity than under the
Dixit-Stiglitz assumption.
17
equilibria, summarized by the two-equation system
ms (cs ; ns )cs =
1
= + cs ;
ns
L
(34)
where s = b; c. As shown in the Appendix, di¤erentiating (34) and applying Cramer’s rule
yields:
@cs
@ ( =L)
0;
@cs
@ns
< 0;
< 0 and sign
@
@ ( =L)
@ns
@
= sign
0
;
(35)
where @cs =@( =L) = 0 only for s = c and n = 2.27 This allows us to study the competitive
e¤ects of pure globalization (a reduction in =L) and of a reduction in the marginal cost
. Note …rst that
@ms
@ms @cs
@ms @ns
=
+
;
@ ( =L)
@c @ ( =L)
@n @ ( =L)
(36)
where @ms =@n < 0 and (35) imply that the second term on the RHS is always positive.
It follows that @ms =@ ( =L) > 0 for
0
0, as in this case @ms =@c
0. Consequently,
an increase in market size is pro-competitive when preferences are DES or CES. Instead,
as shown in the Appendix, @ms =@ ( =L) < 0 for
0
> 0, unless n is small. Thus, pure
globalization is generally anti-competitive with IES preferences.
Finally, note that
@ms @cs @ms @ns
@ms
=
+
;
@
@c @
@n @
(37)
implying that the change in individual consumption and in the number of …rms induced
by a change in the marginal cost
a¤ect markups in opposite directions. In the Appendix
n so
= sign f 0 g, and hence that a reduction
we show, however, that in all cases sign @m
@
in the marginal cost is anti-competitive with DES preferences, and pro-competitive with
IES preferences. We summarize our main results in the following
Proposition 6 When …rms interact strategically, pure globalization is pro-competitive
with DES or CES preferences; with IES preferences, instead, a pro-competitive e¤ ect may
arise only when the number of …rms is small. Moreover, a reduction in the marginal cost is
anti-competitive with DES preferences, neutral with CES preferences, and pro-competitive
with IES preferences.
4.2.1
Discussion
With or without strategic interaction, additive non-CES preferences seem to yield the
same competitive e¤ects. In particular, pure globalization is generally anti-competitive
27
Although we treat n as a real number (thereby ignoring the integer problem), consistent with the
assumption of strategic interaction we focus on values of n equal or greater than 2.
18
when preferences are IES. Moreover, the competitive e¤ects of a reduction in the marginal
cost are generally the opposite than those of an increase in market size. This implies,
e.g., that with DES preferences markups increase when marginal costs fall, and thus the
pro-competitive e¤ect of a reduction in trade costs would still be contaminated by the
increase in markups on foreign sales due to an incomplete pass-through.
Finally note, from (30) and (33), that the competitive e¤ects arising from strategic
interaction can be relevant only when the number of …rms is small, or else the impact on
markups of an increase in n becomes negligible. However, as forcefully argued by Dixit
and Stiglitz (1993), assuming that n is small is problematic in a monopolistic competition
setting. In particular, if the number of …rms is small enough to induce them to interact
strategically, it is unclear why they do not also engage in collusion and entry deterrence,
thereby preventing the free entry of …rms. By the same token, it is actually unclear why we
should expect signi…cant competitive e¤ects in a market situation in which the number of
…rms is large enough to make sense of the free-entry condition postulated in monopolistic
competition models.
4.3
Ideal Variety Approach to Monopolistic Competition
Finally, one may argue that a pro-competitive e¤ect may naturally arise in a framework
in which an increase in the number of available varieties reduces their distance in a …xed
space of characteristics, thereby increasing their substitutability. In closing this Section
we show that, surprisingly, this need not be the case.
To make the point, we reconsider Lancaster’s (1975, 1979) ideal variety approach to
monopolistic competition. In this setting, each consumer has a most preferred ("ideal")
variety, and the market demand for each variety arises from the diversity of tastes.28
Formally, each variety is represented by a point ! on the unit length circumference
of a
circle, and preferences for the ideal product within a mass L of consumers are uniformly
distributed over
be:
. The utility function of a consumer with ideal variety !
e is assumed to
U=
X
!2
c(!)
;
h( (!; !
e ))
(38)
where (!; !
e ) is the shortest arc distance between ! and !
e , and h( ) is the so-called
Lancaster’s compensation function, assumed to be positive, non decreasing and generally
28
As described in Helpman and Krugman (1985, pp. 120-21), ideal variety means that "when the
individual is o¤ered a well-de…ned quantity of the good but is free to choose any potentially possible
variety, he will choose the ideal variety independently of the quantity o¤ered and independently of the
consumption level of other goods. Moreover, when comparing a given quantity of two di¤erent varieties,
the individual prefers the variety that is closest to his ideal product".
19
normalized so that h(0) = 1 (see Lancaster, 1975). Moreover, it is generally assumed
(see Helpman and Krugman, 1985) that h( ) is strictly increasing and convex, and that
h0 (0) = 0.
The above assumptions imply that the marginal rate of substitution between !
e and
! is given by M RS(!; !
e ) = h( (!; !
e )), and is thus an increasing convex function of .
Following Helpman (1981) and Helpman and Krugman (1985), it is possible to show that,
at a symmetric equilibrium, the price elasticity of market demand equals
"=1+
where
h
=
h0 (1=2n)
2nh(1=2n)
1
;
2 h (1=2n)
is the elasticity of the compensation function. Accordingly, an
increase in market size L, by increasing the number of …rms n, also increases " if
0
h
> 0,
thus yielding a pro-competitive e¤ect.29 Instead, an increase in market size reduces "
and is therefore anti-competitive if
0
h
< 0. Finally, if
0
h
= 0 (i.e., if h( ) is isoelastic),
" is independent of n, just as in the D-S "love for variety" approach when preferences
are CES. Thus, a pro-competitive e¤ect is unwarranted even in a framework in which an
increase in the number of …rms crowds the variety space. There are two main reasons for
this result. First, given (38), varieties are of the "perfect substitute" type, their elasticity
of substitution is constant, and a crowding of the variety space cannot a¤ect it. In this
respect, the model fails to capture a potentially genuine pro-competitive e¤ect of the
crowding of the variety space.
Second, the ideal variety approach does not impose su¢ cient restrictions on h( ) to pin
down the properties of
is the assumption
0
h
h,
and therefore the relationship between " and n.30 Speci…cally,
> 0 plausible? Note that M RS(e
! ; !) = h( (!; !
e )) implies that,
in order for Lancaster’s model to deliver a pro-competitive e¤ect, consumer preferences
must feature an ever increasing distance elasticity of the marginal rate of substitution
between !
e and !. It is hard to provide a rationale for this assumption, which seems no
more plausible than the opposite assumption of a decreasing distance elasticity, which
would however deliver an anti-competitive e¤ect. We summarize in the following
Proposition 7 When preferences are of the ideal variety type, as in Lancaster (1979),
pure globalization does not a¤ ect markups if the compensation function h( ) is isoelastic;
29
For instance, in a generalized model of ideal variety with income e¤ects, Hummels and Lugovskyy
(2012) assume, without discussion, a functional form for the compensation function that implies 0h > 0
and thus delivers pro-competitive e¤ects.
30
Only in the limit case in which n goes to in…nity, and due to the (rather ad hoc) assumptions h(0) = 1
and h0 (0) = 0, the elasticity of market demand is increasing in n (and goes to in…nite). This requires a
situation, unfeasible under a positive …xed cost, in which the circumference of the circle is full and the
demand for each …rm is in…nitesimal.
20
it is instead pro-competitive when h( ) features an increasing distance elasticity, and anticompetitive when the distance elasticity is decreasing.
5
Conclusion
We have studied the competitive and reallocation e¤ects of trade opening in monopolistic competition. We have shown that, in a Dixit-Stiglitz setup with additively separable
preferences, heterogeneous …rms and …xed and variable costs of exporting, the extensive
margin (i.e., Melitz-type selection) e¤ects of trade opening are robust to relaxing the CES
assumption, whereas the intensive margin e¤ects (i.e., market share reallocations across
inframarginal …rms) are fragile. Moreover, measured productivity gains from trade opening are not ensured in this setup. We have also argued that these results arise from the
fragility of competitive (i.e., markup) e¤ects, and that allowing for non-additive preferences, for strategic interaction and for the crowding of the variety space does not obviously
lead to stronger and more robust competitive e¤ects in monopolistic competition.
Our results suggest that the e¤ects of trade opening delivered by non-CES preferences
in monopolistic competition are more complex and articulated than generally believed.
Consistently, one should carefully look for empirical evidence in support, e.g., of the implied anti-competitive e¤ects. Alternatively, one can invoke a sort of second-best argument
in support of CES preferences, according to which relaxing one out of a number of restrictive assumptions does not necessarily lead to a better model. We think that the latter
option is not equivalent to assuming that competitive e¤ects are little relevant, but rather
that they should be studied in di¤erent trade models. In this respect, oligopoly settings
seem a prominent alternative, as suggested by Neary (2003, 2009b, 2010), who also warns
us about the di¢ culties of dealing with them in general equilibrium. We conclude that
more creative theoretical work is still needed to better our understanding of the competitive and reallocation e¤ects of globalization.
6
6.1
Appendix
Proof of Results in Section 2
Lemma 1. We begin by proving Lemma 1. Note …rst that, di¤erentiating (9) with
respect to L and using (8), yields:
1+
@ ln
(c ) @ ln c
; c ; L) @ ln c
=1+
= 0:
@ ln c
@ ln L
(c ) @ ln L
v(
21
Thus,
@ ln c
=
@ ln L
Next, di¤erentiating (10) with respect to
r00 (c)
@ ln c
< 0:
@ ln
(c )
@ ln c
=
=
(c )
@ ln
@c
=
@
2
(39)
gives:
r00 (c )
@c
@
r0 (c ) :
Rearranging terms and using (39), we obtain:
@ ln c
(c)
=
> 0:
@ ln
m(c )
Similarly, di¤erentiating (10) with respect to L and using (39) yields:
@ ln c
=
@ ln L
(c)
=
(c )
@ ln c
< 0;
@ ln
which completes our proof of Lemma 1.
We now prove the results in equations (13)-(15).
Firm size, pro…ts and markups.
Di¤erentiating (4) gives:
r00 (c( ))
dc( )
=
d
=
r0 (c)
:
Rearranging terms yields:
r0 (c( ))
d ln c( )
= 00
=
d ln
r (c( ))c( )
(c( )) < 0:
(40)
As for the elasticity of markups with respect to , using (6) and (40), we obtain:
d ln m( )
d ln m d ln c( )
=
=
d ln
d ln c d ln
1
(c( ))
1
(c( ))
(c( )) =
(c( ))
(c( ))
1:
Next, recalling that p( )c( ) = m(c( )) c( ), and using the above results, we obtain:
d ln fp( )c( )g
d ln
Note also that
?
d ln m( )
d ln c( )
(c( ))
+1+
=
d ln
d ln
(c( ))
1
= 1
1 + (c( )) 1
=1
(c( ))
=
is equivalent to
(c) ? (c), it follows that, (c) ?
,
+
0 (c)
? 0.
22
2
(c( ))
(c( )) < 0:
? 0. Recalling that
0 (c)
?0,
Finally, using (8) and (40), the elasticity of variable pro…ts can be written as:
d ln v ( )
@ ln v d ln c( )
=1+
=1
d ln
@ ln c d ln
(c( )) < 0:
This completes our proof of the results in (13)-(15).
6.2
Proof of Results in Section 3
6.2.1
Pure Globalization
Extensive margin e¤ects.
cost cuto¤
The impact of an increase in market size L on the marginal
can be computed by applying the implicit function theorem to (11):
d
=
dL
Note that @
E =@
@
@
E =@L
E =@
:
> 0 by (12). As for the term on the numerator, di¤erentiating the
free-entry condition (11) with respect to L, using (8) and Lemma 1, yields:
@ ln E
=
@ ln L
Z
Note that, for c < c,
@ ln v @ ln c
1+
@ ln c @ ln L
(c ) >
0
1
(c) with DES preferences, and
E =@
preferences, thereby implying that @ ln
0,
dG( ) =
Z
ln L 7 0 ,
0
(c( ))
(c )
dG( ):
(c ) <
? 0. Consequently, @
=@L ?
? 0. This proves that the selection/anti-selection e¤ects of pure globalization are
determined by the sign of
0.
Consider the "average marginal cost" e implicitly
E¤ects on measured productivity.
de…ned by the following property of total cost invariance:
n
"Z
It is easily seen that
where
(c) with IES
#
e c( )L dG( ) +
G( )
e=
( )=
Z
n
Z
"Z
dG( )
c( )L
+
G( )
d ( );
Z
c(z)dG(z)
23
c(s)dG(s)
#
:
is the distribution of
weighted by the corresponding production levels. A measure of
aggregate productivity is therefore provided by 1= e . Notice that e is determined by ( ),
which has support
;
and depends on G( ) and c( ).
A simple "stochastic dominance" argument implies that, when
0
and
we can predict the impact of market size on aggregate productivity. Let
density function associated with
; denote by h = d log =d
= =
0
agree in sign,
=
0
be the
the corresponding
so-called reverse hazard rate, and recall that, if two distributions have the same reverse
hazard rate, they are identical. We obtain:
h( ) = Z
c( )g( )
=Z
c(z)dG(z)
g( )
;
c(z)
c( ) dG(z)
which implies that the reverse hazard rate corresponding to a -…rm depends on its output
relative to that of all other …rms with a lower marginal cost. Thus, by governing the relative
output of inframarginal …rms, a¤ects how L impacts on e . However, pure globalization
a¤ects measured productivity also through the extensive margin e¤ects.
b and suppose that
In particular, consider an increase in market size from L to L,
0;
0
< 0. Denote by L and Lb the corresponding distributions, with supports ; L
i
h
; Lb respectively, where Lb < L due to the selection e¤ect implied by DES
and
h
i
preferences. For all 2 ; Lb (i.e., for all inframarginal …rms), 0 < 0 implies
Z
Z
cL (z)
dG(z)
cL ( )
and then hLb ( )
for all
2
cLb (z)
dG(z);
cLb ( )
hL ( ) (an intensive margin e¤ect). Moreover, since hLb ( ) = 0
i
;
b( )
L (due to the extensive margin e¤ect), it follows that hL
b
L
;
. But then
hL ( )
hL ( )
is "reverse-hazard rate dominated" by L (see, e.g.,
Shaked and Shanthikumar, 1994), and thus e Lb < e L , implying an increase in measured
for all
2
L
b
L
productivity. By a similar reasoning, if
0;
0
> 0, a rise in market size leads to an
unambiguous reduction in measured productivity.
On the contrary, if
0
and
0
do not agree in sign, the selection e¤ect (governed by )
and the intensive margin e¤ect in terms of …rm output (governed by ) point in opposite
directions. In this case, the impact of pure globalization on measured productivity becomes
ambiguous, as it will in general depend on the properties of the distribution G.
24
6.2.2
Costly Trade
Extensive margin e¤ects. Consider …rst a reduction in the variable trade cost
.
Applying the implicit function theorem to (20) yields:
@ ln
@ ln
Noting that d ( )=d =
we obtain:
@
@
E
=
@
@
E
=
=
E =@
:
E =@
(41)
c ( ) L by the Envelope Theorem, and recalling that
Z
Similarly, noting that @
recalling that (
@
@
=
x
@
v(
)
@
v =@c
=
dG( ) =
Z
L
x( x)
= 0,
x
c(
) dG( ) < 0:
Lm(c)= (c) by (7)and (8), using Lemma 1, and also
) = 0, gives:
Z
Z x
@ v @c
@ v @c
dG( ) +
dG( )
@c @
@c @
"Z
Z x
L
p( )c( )dG( ) +
p( )c(
m(c )
Using the above results in (41) gives (21), thereby proving that
Consider now a reduction in the …xed cost of exporting
#
)dG( ) > 0:
is increasing in .
x.
Applying the implicit
function theorem to (20) yields:
@
@
where @
E =@
6.3
x
E =@
x
E =@
;
> 0 and
@
@
implying that @
@
@
=
=@
x
E
=
x
Z
x
@
x(
@
)
dG( ) =
x
> 0. This proves that
G(
x)
< 0;
is increasing in
x.
Proof of Results in Section 4
6.3.1
Quasi-Linear Quadratic Preferences
We now prove the results in Proposition 5. To this purpose, consider two identical countries
separated by a symmetric iceberg trade cost
25
1. Indexing variables related to the
foreign market by an x and using (29) gives:
mx (cx ) = 1 +
b cx
; px = mx (cx )
To compute the free-entry condition (
v
x
v
+
= bcx +
:
(42)
= ), note …rst that, using (29) and (42),
variable pro…ts in the domestic and foreign market can be written as:
v
1] Lc = bLc2 ;
(c) = [m (c)
x
v (cx )
= [mx (cx )
(43)
Lcx = bLc2x =
1]
v
(cx ) :
Next, note that (27) and the expressions for p and px in (29) and (42) imply that
c
cx =
px
p
b
(
=
1)
2b
;
(44)
which de…nes cx as a function of c and , i.e., cx (c; ) = c
(
1) =2b. Using (43) and
(44) in the free-entry condition, which implicitly de…nes c, yields:
v
(c) +
v
(cx (c; )) = bLc2 + bL c
(
2
1)
2b
= :
(45)
Di¤erentiating (44) and (45) yields:
@c
cx
dcx
=
=
> 0;
@
2b c + cx
d
2b
cx
c + cx
1
< 0:
Thus, using (29) and (42):
dm
d
dmx
d
=
=
b @c
1 cx
=
> 0;
@
2 c + cx
b
dcx
1
cx =
2
d
2
cx
c + cx
bcx
1
2
< 0:
Consequently, after a reduction in trade costs, …rms charge lower markups in the domestic
market and higher markups in the foreign market.
Finally, computing the above derivatives in
@c
@
dmx
d
=
=1
=
=1
= 1 yields:
dcx
;
4b
d
=1
1 bc
1
dm
+
> =
4
4
d
=
:
=1
Thus, a small increase in trade costs around the free trade causes markups on foreign sales
26
to fall by more (in absolute value) than the increase in markups on domestic sales, thus
leading to a reduction in average markups.
6.3.2
Comparative Statics of the Bertrand and Cournot Equilibria.
We now prove the results in Proposition 6. Note …rst that, di¤erentiating (34), yields:
2
4
@ms s
@c c
@ms s
@n c
+ ms
+
1
(ns )2
1
(ns )2
32
@cs
@( =L)
@ns
@( =L)
54
@cs
@
@ns
@
3
5=
"
ms cs
0
1
cs
#
:
(46)
Thus, applying Cramer’s rule,
2
4
3
2
1
5=
4
Ds
@cs
@
@ns
@
@cs
@( =L)
@ns
@( =L)
ms cs
(ns )2
@ms s
+ (n1s )2
@n c
@ms s
s
@c c + m
+ cs
@ms s
+ (n1s )2
@n c
@ms
s 2
@c (c )
3
5;
(47)
where Ds is the determinant of the …rst matrix on the LHS of (46).
Bertrand competition. Di¤erentiating (31) yields:
@mb
@c
@mb
@n
n
=
n
1
m0 ;
(48)
1
=
1)2 (
(n
m
(n 1)2
=
1)
=
mb
:
1) [(n 1) + 1]
(n
Next, using (48) in (47) yields:
2
Db 4
2
@cb
@( =L)
@nb
@( =L)
z
1
nb
6
6
6
6
= 6
6
6 z
4
@cb
@
@nb
@
>0
}|
1
nb
3
5
(49)
m
(nb
2
1) mb
!{
<0
nb
1
h
n
b
}|
0 b
m c +m
1
z
<0
c
2
(nb )
1
mb
{
i
}|
(nb
nb
1) [(nb 1) + 1]
2
nb (cb )
nb 1
m0
{
3
7
7
7
7
7;
7
7
5
where
Db =
nb (nb
m0 cb + m
1)
and the inequalities follow from
0
v
1+
m
(nb
1) mb
> 0;
(c) > 0 (which implies m0 c + m
de…nition of mb . This proves that the inequalities in (35) hold for s = b.
27
1 > 0) and the
Next, using (49) in (36) yields:
@mb
@( =L)
=
=
=
nb m0 @cb
nb 1 @ L
(nb
(
"
m0
1
1
b
b
D
n
1 nb
m+
cb (nb
mb
@nc
1) [(nb 1) + 1] @ L
#
mb
m
+
2
(nb 1) mb
(nb
1
nb
m0 cb g(nb )
2
1) [ (nb
1) + 1] Db
nb m0 cb + m
2
1) [(nb
1
1) + 1]
)
;
where g(n) = [(n 1) =n] [(n 1) + 1] > 0 is a monotonically increasing function, with
p
g(1 + 1= ) = 1. Note that @mb =@ ( =L) > 0 if and only if m + m0 cb g nb > 1=nb , a
condition that is always satis…ed for m0
0 (i.e., with DES or CES preferences). Instead,
for m0 < 0 (IES preferences), @mb =@ ( =L) < 0 unless nc is small, i.e., su¢ ciently close to
p
1 + 1= .
Finally, using (49) in (37) yields:
@mb
@
@cb
mb
@nb
(n 1) [(n 1) + 1] @
nb 1 @
(
b
1
nb
0c
b
m
1
m
(nb 1) Db
nb
(nb 1) + 1
= m0
=
=
nb
mb m0 nb
+ b
(n
1) [(n
2
cb
1) + 1]
)
m0 cb 1 mb
;
nb (nb 1) Db
which implies that sign
n
@mc
@
o
=
Cournot competition. Using
sign fm0 g = sign f 0 g.
X
=
u0 (cj )cj in the …rst-order conditions for utility
j
maximization (u0 (ci ) = pi ) yields the following inverse demand system:
pi (c) = X
u0 (ci )
j
u0 (cj )cj
; i = 1; :::n:
(50)
Firm i’s revenue is pi (c)ci L; marginal revenue is therefore given by
X
0 (c )
r
u0 (cj )cj
i
@ (pi (c)ci )
j6=i
=
;
X
2
@ci
u0 (cj )cj
(51)
j
and is decreasing in ci under our assumptions that r00 < 0 and r0 > 0. In a Cournot-Nash
equilibrium, each …rm chooses its quantity to satisfy the …rst-order condition @ (pi (c)ci ) =@ci =
under a correct conjecture about the quantities produced by its competitors. Then (51)
28
implies that, in any symmetric Cournot-Nash equilibrium,
c=
(n 1) r0 (c)
n 1
= 2
:
2
0
n u (c)
n m(c)
(52)
Using (50) and (52) yields equation (32) in the main text.
Di¤erentiating (32) yields:
n
@mc
@mc
=
m0 ;
=
@c
n 1
@n
m
1)2
(n
mc
:
n (n 1)
=
(53)
Next, using (53) in (47) yields:
2
Dc 4
2
@cc
@( =L)
@nc
@( =L)
@cc
@
@nc
@
0
}|
z
3
5
(54)
6
nc 2
6
6
(nc )2 (nc 1)
= 6
6
<0
6 z
}|
{
4
c
n
0 c
mc +m
nc 1
where
Dc =
(nc )2
<0
z
{
cc
nc (nc
1)
}|
1
m
2
nc
nc (cc )2 0
nc 1 m
nc (m0 cc + m)
nc 1
1+
1
nc
1
{
3
7
7
7
7;
7
7
5
> 0:
(55)
Note that @cc =@( =L) = 0 only for nc = 2, and that the inequalities in (54) and (55) are
straightforward once recalling that m0 c + m
1 > 0. Thus, the inequalities in (35) hold
also for s = c.
Next, using (54) in (36) yields:
@mc
nc m0 @cc
= c
@( =L)
n
1 @( =L)
m
(nc
@nc
m + m0 cc (nc 1)
=
:
1)2 @( =L)
nc (nc 1)2 Dc cc
Note that @mc =@ ( =L) > 0 if and only if m + m0 cc (nc
always satis…ed for m0
1) > 0, a condition that is
0 (i.e., with DES or CES preferences). Instead, for m0 < 0 (IES
preferences), @mc =@ ( =L) < 0 unless nc is small, i.e., su¢ ciently close to 2.
Finally, using (54) in (37) yields:
@mc
nc m0 @cc
= c
@
n
1@
implying that sign
sition 6.
n
@mc
@
o
=
cc m0 1 m n1c
mc
@nc
=
;
nc (nc 1) @
(nc 1)2 Dc
sign fm0 g = sign f 0 g. This completes our proof of Propo-
29
References
[1] Arkolakis, C., A. Costinot and A. Rodríguez-Clare (2012). "New Trade Models, Same
Old Gains?," American Economic Review 102(1), pp. 94-130.
[2] Arkolakis, C., A. Costinot, D. Donaldson and A. Rodríguez-Clare (2012). "The Elusive Pro-Competitive E¤ects of Trade," mimeo, MIT.
[3] Behrens, K. and Y. Murata (2012). "Trade, Competition, and E¢ ciency," Journal of
International Economics 87(1), pp. 1-17.
[4] Bertoletti, P. (2006). "Logarithmic Quasi-Homothetic Preferences," Economics Letters 90, pp. 433-439.
[5] Bertoletti, P., E. Fumagalli and C. Poletti (2008). "Price-Increasing Monopolistic
Competition? The Case of IES Preferences," IEFE Working Paper No. 15, Bocconi
University.
[6] Blackorby, C. and R. R. Russell (1989). "Will the Real Elasticity of Substitution
Please Stand Up? (A comparison of the Allen/Uzawa and Morishima Elasticities),"
American Economic Review 79, pp. 882-888.
[7] Dhingra, S. and J. Morrow (2012). "Monopolistic Competition and Optimum Product
Diversity under Firm Heterogeneity," mimeo, London School of Economics.
[8] Dixit, A. and J. Stiglitz (1977). "Monopolistic Competition and Optimum Product
Diversity," American Economic Review 67(3), pp. 297-308.
[9] Dixit, A. and J. Stiglitz (1993). "Monopolistic Competition and Optimum Product
Diversity: Reply," American Economic Review 83(1), pp. 302-304.
[10] Feenstra, R. (2003). "A Homothetic Utility Function for Monopolistic Competition
Models, without Constant Price Elasticity," Economics Letters 78, pp. 79-86.
[11] Helpman, E. (1981). "International Trade in the Presence of Product Di¤erentiation, Economies of Scale and Monopolistic Competition," Journal of International
Economics 11, pp. 305-340.
[12] Helpman, E. and P. Krugman (1985). Market Structure and Foreign Trade, MIT
Press, Cambridge MA.
[13] Hummels, D. and V. Lugovskyy (2009). "International Pricing in a Generalized Model
of Ideal Variety," Journal of Money, Credit and Banking 41, pp. 3-33.
30
[14] Krugman, P. (1979). "Increasing Returns, Monopolistic Competition and International Trade," Journal of International Economics 9, pp. 469-479.
[15] Krugman P. (1980). "Scale Economies, Product Di¤erentiation and the Pattern of
Trade," American Economic Review 70(5), pp. 950-959.
[16] Lancaster, K. (1975). "Socially Optimal Product Di¤erentiation," American Economic Review 65(4), pp. 567-585.
[17] Lancaster, K. (1979). Variety, Equity and E¢ ciency, Columbia University Press, New
York.
[18] Melitz, M. J. (2003). "The Impact of Trade on Intra-Industry Reallocations and
Aggregate Industry Productivity," Econometrica 71(6), pp. 1695-1725.
[19] Melitz, M. J. and G. Ottaviano (2008). "Market Size, Trade, and Productivity,"
Review of Economic Studies 75(1), pp. 295-316.
[20] Melitz, M. J. and D. Tre‡er (2012). "Gains from Trade when Firms Matter," Journal
of Economic Perspectives 26(2), pp. 91-118.
[21] Mrázová, M. and J. P. Neary (2012). "Selection E¤ects with Heterogeneous Firms,"
Working Paper, University of Oxford.
[22] Neary, J. P. (2003). "Globalisation and Market Structure," Journal of the European
Economic Association 1, pp. 245-271.
[23] Neary, J. P. (2004). "Monopolistic Competition and International Trade Theory," in
S. Brakman and B. J. Heijdra (eds.): The Monopolistic Competition Revolution in
Retrospect, Cambridge University Press, Cambridge, pp. 159-184.
[24] Neary, J. P. (2009a). "Putting the ‘New’ into New Trade Theory: Paul Krugman’s
Nobel Memorial Prize in Economics," Scandinavian Journal of Economics 111(2),
pp. 217-250.
[25] Neary, J. P. (2009b). "International Trade in General Oligopolistic Equilibrium,"
Working Paper, University of Oxford.
[26] Neary, J. P. (2010). "Two and a Half Theories of International Trade," The World
Economy 33(1), pp. 1-19.
[27] Pettengill, J. S. (1979). "Monopolistic Competition and Optimum Product Diversity:
Comment," American Economic Review 69(5), pp. 957-960.
31
[28] Shaked, M. and J. G. Shanthikumar (1994). Stochastic Orders and Their Applications,
Academic Press, San Diego.
[29] Zhelobodko, E., S. Kokovin, M. Parenti and J.-F. Thisse (2012). "Monopolistic Competition: Beyond the Constant Elasticity of Substitution," Econometrica 80(6), pp.
2765-2784.
32