Product Mix and Firm Productivity Responses to Trade
Competition
Thierry Mayer
Marc J. Melitz
Gianmarco I.P. Ottaviano
Sciences-Po
CEPII and CEPR
Harvard University
NBER and CEPR
London School of Economics
U Bologna, CEP and CEPR
December 2014
PRELIMINARY AND INCOMPLETE DRAFT
1
Introduction
In this paper, we document how demand shocks in export markets lead French multi-product
exporters to re-allocate the product mix sold in those destinations. We develop a theoretical model
of multi-product …rms that highlight the speci…c demand and cost conditions needed to generate all
those empirical predictions. We then show how the increased competition from the demand shocks
in export markets – and the associated product mix responses – lead to substantial productivity
improvements for multi-product exporters.
Recent studies using detailed micro-level datasets on …rms, plants, and the products they produce have documented vast di¤erences in all measurable performance metrics across those di¤erent
units. Those studies have also documented that these performance di¤erences are systematically
related to participation in international markets (see, e.g., Mayer and Ottaviano (2008) for Europe,
and Bernard, Jensen, Redding and Schott (2012) for the U.S.): Exporting …rms and plants are
bigger, more productive, more pro…table, and less likely to exit than non-exporters. And better
performing …rms and plants export a larger number of products to a larger number of destinations.
Exporters are larger in terms of employment, output, revenue and pro…t. Similar patterns also
emerge across the set of products sold by multi-product …rms. There is a stable performance ranking for …rms based on the products’performance in any given market, or in worldwide sales. Thus,
better performing products in one market are most likely to be the better performing products in
any other market (including the global export market). This also applies to the products’selection
into a destination, so better performing products are also sold in a larger set of destinations.
Given this heterogeneity, trade shocks induce many di¤erent reallocations across …rms and
products. Some of these reallocations are driven by ‘selection e¤ects’that determine which products
are sold where (across domestic and export markets), along with …rm entry/exit decisions (into/out
of any given export market, or overall entry/exit of the …rm). Other reallocations are driven by
‘competition e¤ects’ whereby – conditional on selection (a given set of products sold in a given
market) – trade a¤ects the relative market shares of those products. Both types of reallocations
generate (endogenous) productivity changes that are independent of ‘technology’(the production
function at the product-level). This creates an additional channel for the aggregate gains from
trade.
Unfortunately, measuring the direct impact of trade on those reallocations across …rms is a
very hard task. On one hand, shocks that a¤ect trade are also likely to a¤ect the distribution
1
of market shares across …rms. On the other hand, changes in market shares across …rms likely
re‡ect many technological factors (not related to reallocations). Looking at reallocations across
products within …rms obviates some of these problems. Recent theoretical models of multi-product
…rms highlight how trade induces a similar pattern of reallocations within …rms as it does across
…rms. And measuring reallocations within multi-product …rms has several advantages: Trade
shocks that are exogenous to individual …rms can be identi…ed much more easily than at a higher
level of aggregation; Controls for any technology changes at the …rm-level are also possible; and
reallocations can be measured for the same set of narrowly de…ned products sold by same …rm
across destinations or over time. Moreover, multi-product …rms dominate world production and
trade ‡ows. Hence, reallocations within multi-product …rms have the potential to generate large
changes in aggregate productivity. Empirically, we …nd very strong evidence for the e¤ects of trade
shocks on those reallocations, and ultimately on the productivity of multi-product …rms. The
overall impact on aggregate French manufacturing productivity is substantial.
2
Previous Evidence on Trade-Induced Reallocations
In a previous paper, Mayer, Melitz and Ottaviano (2014), we investigated, both theoretically and
empirically, the mechanics of these reallocations within multi-product …rms. We used a comprehensive …rm-level data on annual shipments by all French exporters to all countries in the world
(not including the French domestic market) for a set of more than 10,000 goods. Firm-level exports
are collected by French customs and include export sales for each 8-digit (combined nomenclature
or NC8) product by destination country. Our focus then was on the cross-section of …rm-product
exports across destinations (for a single year, 2003). We presented evidence that French multiproduct …rms indeed exhibit a stable ranking of products in terms of their shares of export sales
across export destinations with ‘core’products being sold in a larger number of destinations (and
commanding larger market shares across destinations). We used the term ‘skewness’ to refer to
the dispersion of these export market shares in any destination and showed that this skewness
consistently varied with destination characteristics such as GDP and geography: French …rms sold
relatively more of their best performing products in bigger, more centrally-located destinations
(where competition from other exporters and domestic producers is tougher).
Other research has also documented similar patterns of product reallocations (within multiproduct …rms) over time following trade liberalization. For the case of CUSFTA/NAFYA, Baldwin
and Gu (2009), Bernard, Redding and Schott (2011), and Iacovone and Javorcik (2008) all report
2
that Canadian, U.S., and Mexican multi-product …rms reduced the number of products they produce during these trade-liberalization episodes. Baldwin and Gu (2009) and Bernard, Redding and
Schott (2011) further report that CUSFTA induced a signi…cant increase in the skewness of production across products. Iacovone and Javorcik (2008) separately measure the skewness of Mexican
…rms’ export sales to the US. They report an increase in this skewness following NAFTA: They
show that Mexican …rms expanded their exports of their better performing products (higher market shares) signi…cantly more than those for their worse performing exported products during the
period of trade expansion from 1994
2003.
As prices are rarely observed, there is little direct evidence on how markups, prices and costs
are related across products supplied with di¤erent productivity, and how they respond to trade
liberalization.1 A notable exception is the recent paper by DeLoecker, Goldberg, Pavcnik and
Khandelwal (2012) who exploit unique information on the prices and quantities of Indian …rms’
products over India’s trade liberalization period from 1989
2003. They also document that better
performing …rms (higher sales and productivity) produce more products. They then focus on
markups. Across …rms, they document that those better performing …rms set higher markups.
They also document a similar patter across the products produced by a given multi-product …rm,
which sets relatively higher markups on their better performing products (lower marginal cost and
higher market shares). In addition, they show strong evidence for endogenous markup adjustments
via imperfect pass-through from products’marginal costs to their prices: Only a portion of marginal
cost decreases are passed on to consumers in the form of lower prices, while the remaining portion
goes to higher markups. This is consistent with recent …rm-level evidence on exchange rate passthrough. Berman, Martin and Mayer (2012) analyze the heterogeneous reaction of exporters to
real exchange rate changes using a rich French …rm-level data set with destination speci…c export
values and volumes on the period 1995 2005. They …nd that on average …rms react to depreciation
by increasing their markup. They also …nd that high-performance …rms increases their markup
signi…cantly more –implying that the pass-through rate is signi…cantly lower for better performing
…rms.
3
Reallocations Over Time
We now document how changes within a destination market over time induce a similar pattern
of reallocations as the ones we previously described. More speci…cally, we show that demand
1
Prices are typically backed-out as unit values based on reported quantity information, which is extremely noisy.
3
shocks in any given destination market induce …rms to skew their product level export sales to that
destination towards their best performing products. In terms of …rst moments, we show that these
demand shocks also lead to strong positive responses in both the intensive and extensive margins
of export sales to that destination.
3.1
Data
We use the same data as Mayer, Melitz and Ottaviano (2014), the only di¤erence being multiple
years 1995
2005 instead of a single year 2003. Besides what we already discussed, the reporting
criteria for all …rms operating in the French metropolitan territory are as follows. For within EU
exports, the …rm’s annual trade value exceeds 100,000 Euros;2 and for exports outside the EU, the
exported value to a destination exceeds 1,000 Euros or a weight of a ton. Despite these limitations,
the database is nearly comprehensive. For instance, in 2003, 100,033 …rms report exports across
229 destination countries (or territories) for 10,072 products. This represents data on over 2 million
shipments.
We restrict our analysis to export data in manufacturing industries, mostly eliminating …rms
in the service and wholesale/distribution sector to ensure that …rms take part in the production of
the goods they export.3 This leaves us with data on over a million shipments by …rms in the whole
range of manufacturing sectors.4
3.2
Measuring Trade Shocks
Consider a …rm i who exports a number of products s in industry I to destination d in year
t. We measure industries (I) at the 3-digit ISIC level (35 di¤erent classi…cations across French
manufacturing). We consider several measures of demand shocks that a¤ect this export ‡ow. At
the most aggregate level we use the variation in GDP in d, log GDPd;t . At the industry level
I . We can also use our detailed
I, we use total imports into d excluding French exports, log Md;t
product-level shipment data to construct a …rm i-speci…c demand shock:
shockIi;d;t
s
log Md;t
for all products s 2 I exported by …rm i to d in year t0 ,
2
(1)
If that threshold is not met, …rms can choose to report under a simpli…ed scheme without supplying export destinations. However, in practice, many …rms under that threshold report the detailed export destination information.
3
Some large distributors such as Carrefour account for a disproportionate number of annual shipments.
4
In a robustness check, we also drop observations for …rms that the French national statistical institute reports as
having an a¢ liate abroad. This avoids the issue that multinational …rms may substitute exports of some of their best
performing products with a¢ liate production in the destination country, thus reducing noise in the product export
skewness. Results are quantitatively very similar in all regressions.
4
s represents total imports into d (again, excluding French exports) for product s. For
where Md;t
world trade, the …nest level of product level of aggregation is the HS-6 level (from UN-COMTRADE
and CEPII-BACI), which is slightly more aggregated than our NC8 classi…cation for French exports
(roughly 5,300 HS products per year versus 10,000 NC8 products per year). The construction of
I , except that
this last trade shock is very similar to the one for the industry level imports log Md;t
we only use imports into d for the precise product categories that …rm i exports to d.5 In order
to ensure that this demand shock is exogenous to the …rm, we use the set of products exported
by the …rm in its …rst export year in our sample (1995, or later if the …rm starts exporting later
on in our sample), and then exclude this year from our subsequent analysis. Note that we use
an un-weighted average so that the shocks for all exported products s (within an industry I) are
represented proportionately.
I ,M s , we compute the …rst di¤erence as the
For all of these demand shocks Xt = GDPd;t ; Md;t
d;t
Davis-Haltiwanger growth rate: ~ Xt
(Xt
Xt
1 ) = (:5Xt
+ :5Xt
1 ).
This measure of the …rst
di¤erence preserves observations when the shock switches from 0 to a positive number, and has a
maximum growth rate of
2 or 2. This is mostly relevant for our measure of the …rm-speci…c trade
s can often switch between 0 and positive values.
shock, where the product-level imports into d, Md;t
Whenever Xt
rates (j
1 ; Xt
> 0, ~ Xt is monotonic in
log Xt and approximately linear for typical growth
log Xt j < 2).6 We thus obtain our three measures of trade shocks in …rst di¤erences:
~ GDPd;t ; ~ M I , ~ M s . For the …rm i-speci…c shock ~ M s , we take the un-weighted average of
d;t
d;t
d;t
the growth rates for all products exported by the …rm in t
3.3
1.
The Impact of Demand Shocks on Trade Margins and Skewness
Before focusing on the e¤ects of the demand shocks on the skewness of export sales, we …rst show
how the demand shocks a¤ect …rm export sales at the intensive and extensive margins (the …rst
moments of the distribution of product export sales). Table 1 reports how our three demand shocks
(in …rst di¤erences) a¤ect changes in …rm exports to destination d in ISIC I (so each observation
represents a …rm-destination-ISIC combination). We decompose the …rm’s export response to each
shock into an intensive margin (average exports per product) and an extensive margin (number of
exported products). We clearly see how all three demand shocks induce very strong (and highly
5
There is a one-to-many matching between the NC8 and HS6 product classi…cations, so every NC8 product is
s
assigned a unique HS6 classi…cation. We use the same Md;t
data for any NC8 product s within the same HS6
classi…cation.
6
Switching to …rst di¤erence growth rates measured as log Xt (and dropping products with zero trade in the
trade shock average) does not materially a¤ect any of our results.
5
signi…cant) positive responses for both margins. This con…rms that our demand shocks capture
important changes in the local demand faced by French exporters.7
Table 1: Demand shocks and local exports
Dependent Variable
~ GDP Shock
log Exports per Product
0.486a
(0.046)
~ Trade Shock
log # Products Exported
0.147a
(0.016)
0.273a
(0.009)
~ Trade Shock - ISIC
Observations
396740
402522
0.075a
(0.004)
0.038a
(0.005)
402522
396740
402522
0.014a
(0.002)
402522
Standard errors in parentheses: c < 0:1; b < 0:05; a < 0:01. All regressions include year
dummies, and standard errors clustered at the level relevant for the variable of interest:
destination country for columns (1) and (4), …rm-destination for columns (2) and (5) and
ISIC-destination for columns (3) and (6).
We now investigate the consequences of those demand shocks for the skewness of export sales
(independent of the level of product sales). In Mayer, Melitz and Ottaviano (2014), we focused on
those e¤ects in the cross-section across destinations. Here, we examine the response of skewness
within a destination over time using our new demand shocks. We rely on the Theil index as our
measure of skewness due to its aggregation properties: We will later aggregate the export responses
at the destination-ISIC level up to the …rm-level – in order to generate predictions for …rm-level
productivity. Thus, our measure of skewness for the distribution of …rm i’s exports to destination
d in industry I, xsi;d;t ; is the Theil index:
I
Ti;d;t
X xsi;d;t
xI
s2I i;d;t
log
xIi;d;t
xsi;d;t
!
;
xIidt
X
xsi;d;t :
(2)
s2I
Table 2 reports regressions of this skewness measure on all three demand shocks (jointly) – at
the …rm-destination-ISIC level. In the …rst column, we use a speci…cation in (log) levels (FE), and
use …rm-destination-ISIC …xed e¤ects to isolate the variation over time. In the second column,
we return to our speci…cation in …rst di¤erences (FD). In the third column we add the …rmdestination-ISIC …xed e¤ects to this speci…cation in …rst di¤erences (FD-FE). This controls for
7
Speci…cations using the log levels of the shocks and …rm-destination-ISIC …xed-e¤ects yield similar results. Other
speci…cations including the three covariates show that those demand shocks are di¤erent enough to be estimated jointly
while each keeping its positive sign and statistical signi…cance.
6
any trend growth rate in our demand shocks over time. Across all three speci…cations, we see
that positive (negative) demand shocks induce a highly signi…cant increase in the skewness of …rm
export sales to a destination.8 Next, we construct a measure of the change in skewness restricted
to the subset of products s exported in both periods (for the …rst di¤erence speci…cations). This
alternate skewness measure
I;const
Ti;d;t
isolates changes that are driven only by the intensive margin
of exports. The two …rst-di¤erence speci…cations using this alternate skewness measure are reported
in the last two columns. The e¤ects of our …rm-level trade shock is still highly signi…cant (well
beyond the 1% level), whereas the e¤ects based on the ISIC-level trade shock are teetering at the
5% signi…cance level. However, those columns show that the GDP shocks tend to proportionately
change the export sales of ‘incumbent’ products. The e¤ect of the GDP shock on skewness thus
comes entirely from the extensive margin (new products exported in response to higher GDP levels
in a destination).9
Table 2: Demand shocks and local skewness
Dependent Variable
Speci…cation
GDP Shock
I
Ti;d;t
FE
0.076a
(0.016)
Trade Shock
0.047a
(0.005)
Trade Shock - ISIC
0.002a
(0.000)
I
Ti;d;t
FD
FD-FE
FD
I;const
Ti;d;t
FD-FE
~ GDP Shock
0.067a
(0.012)
0.068a
(0.016)
-0.005
(0.008)
-0.004
(0.009)
~ Trade Shock
0.036a
(0.005)
0.032a
(0.006)
0.012a
(0.003)
0.012a
(0.003)
~ Trade Shock - ISIC
0.006a
(0.002)
396740
0.004
(0.003)
396740
0.002
(0.001)
437626
0.004b
(0.002)
437626
Observations
474506
Standard errors in parentheses: c < 0:1; b < 0:05; a < 0:01. All regressions include year
dummies, and standard errors clustered at the level of the destination country.
8
The e¤ect of all three shocks are weakened a little bit due to some collinearity. Even in the FD-FE speci…cation,
the ISIC-level shock remains signi…cant at the 1% level when it is entered on its own.
9
These newly exported products have substantially smaller market shares than the incumbent products and
therefore contribute to an increase in the skewness of export sales.
7
4
Theory
In the previous section we documented the pattern of product reallocations in response to demand
shocks in export markets. We now develop a theoretical model of multi-product …rms that highlights
the speci…c demand conditions needed to generate this pattern. Our theory shows that these
demand conditions imply that the demand shocks lead to increased competition for exporters in
those markets. In addition, we show how this increased competition and the associated product
mix responses lead to higher …rm productivity. Finally, we also show that the demand conditions
needed to predict the observed pattern of reallocations to demand shocks are the same that are
needed to predict the relation among …rm/product performance measures discussed in Section 2.
In particular, we …rst present a ‡exible model with both single- and multi-product …rms characterizing the properties of the demand system that are needed to predict the observed relation
between …rm/product performance measures. We then show that these properties necessarily imply the documented the pattern of product reallocations in response to demand shocks in export
markets. This is our main point of departure relative to the recent literature focusing on more
‡exible demand systems in monopolistic competition models of trade.10
4.1
Closed Economy
To better highlight the role played by the properties of the demand system, we initially start with
a closed economy. We will then move to the open economy to discuss the trade shocks and their
e¤ects.
Multi-Product Production with Additive Separable Utility
Consider an economy populated by Lw identical households, each consisting of
workers. Thus,
each worker supplies 1= e¢ ciency units of labor inelastically so that labor supply equals Lw while
the total number of consumers equals Lc = Lw . Labor is the only productive factor and e¢ ciency
units of labor per worker are chosen as numeraire so that each consumer earns unit wage (and
household income is ). We think of a demand shock as a change in
for given Lw , as this changes
the number of consumers Lc for given number of workers.11
10
See, for instance, Arkolakis, Costinot, Donaldson and Rodriguez-Clare (2012), Zhelobodko, Kokovin, Parenti and
Thisse (2012), Fabinger and Weyl (2012 and 2014), Mrazova and Neary (2014), Parenti, Ushchev and Thisse (2014).
11
In the same vein, Parenti, Ushchev and Thisse (2014) consider a number of consumers Lc = L, each inelastically
supplying y e¢ ciency units so that labor supply equals Lw = yL = yLc . Clearly, imposing = 1=y delivers our
parametrization. However, they then study the comparative statics with respect to changes in L and y. Both changes
8
Utility is assumed to be additive separable over a continuum of imperfectly substitutable products indexed i 2 [0; M ] where M is the measure of products available. The typical consumer in any
household solves the following utility maximization problem:
max
xi 0
Z
M
u(xi )di s.t.
Z
M
pi xi di = 1;
0
0
where u(xi ) is the sub-utility associated with the consumption of xi units of product i and expenditure equals unit wage. We assume that this sub-utlity exhibits the following properties:
(A1) u(xi )
0 with equality for xi = 0; u0 (xi ) > 0 and u00 (xi ) < 0 for xi
0.
The …rst order conditions for the consumer’s problem determine the inverse demand function:
pi =
u0 (xi )
; with
=
Z
M
u0 (xi )xi di;
(3)
0
where
> 0 is the marginal utility of income. Larger
shifts inverse demand downwards, reducing
the price the consumer is willing to pay for any level of consumption. Concavity of u(xi ) ensures
that xi satisfying (3) also meets the second order condition for the consumer’s problem. Note that
is an increasing function of M and xi .
Products are supplied by …rms that may be single- or multi-product. Market structure is
monopolistically competitive as in Mayer, Melitz and Ottaviano (2014) in that each product is
supplied by only one …rm and each …rm supplies a countable number of the continuum of products.
Technology exhibits increasing returns to scale associated with a …xed production cost, along with
a constant marginal cost. The …xed cost f is the same for all products while the marginal cost v
di¤ers across them. For a given …rm, products are indexed in increasing order m of marginal cost
from a ‘core product’ indexed by m = 0. Moreover, entry incurs a sunk cost f e . Only after this
cost is incurred, entrants randomly draw their marginal cost levels for their core products from a
common continuous di¤erentiable distribution with density function (c) and cumulative density
function
(c) de…ned over the support [0; 1). We use M (c) to denote the number of products
supplied by a …rm with core marginal cost c and v(m; c) to denote the marginal cost of its product
m. We assume v(m; c) = cz(m) with z(0) = 1 and z 0 (m) > 0.12
thus a¤ect labor supply, which we want instead to hold constant.
12
The assumption z 0 (m) > 0 will generate the within-…rm ranking of products discussed in Section 2. In the limit
case when z 0 (m) is in…nite, all …rms are single-product.
9
An entrant supplying product i with marginal cost v solves the pro…t maximization problem:
max (qi ) = pi qi
qi 0
vqi
f;
subject to the market clearing condition for its output qi = xi Lc and inverse demand given by
(3).13 The optimal level of output qv = xv Lc satis…es the …rst order condition:
u0 (xv ) + u00 (xv )xv = v;
(4)
where r(xv ) = (xv )= is the marginal revenue associated with a given variety. Markup pricing is
revealed by rewriting (4) as
p(xv ) =
v
;
"p (xv )
(5)
u00 (xv )xv
u0 (xv )
(6)
1
where
"p (xv )
is the elasticity of inverse demand, such that "p (xv ) 2 (0; 1).14 This is a measure of the concavity
of u(xv ).15
The optimal level of output xv must also satisfy the second order condition for pro…t maximization:
0
(xv )
2u00 (xv ) + u000 (xv )xv < 0;
(7)
which can be restated is terms of the elasticity of marginal revenue "r (xv ) as:
"r (xv )
0 (x )x
v v
(xv )
> 0:
(8)
This requires the inverse demand to be not too convex and implies that, for any given , a unique
solution xv ( v) exists for (4).16 This is our second assumption:
(A2) "r (xv ) > 0.
13
This problem is faced by any entrant, no matter whether single- or multi-product, as our assumptions rule out
cannibalization within and between entrants’product ranges.
14
As in Zhelobodko, Kokovin, Parenti and Thisse (2012), we use "p (xv ) to denote the elasticity of p(xv ) with
respect to xv . This is the inverse of the price elasticity of demand that would be denoted "x (pv ).
15
This elasticity should not be confused with the elasticity of utility u0 (xv )xv =u(xv ), which is an inverse measure
of “love of variety”. As discussed by Neary and Mrazova (2014), this elasticity is important for welfare analysis. In
Zhelobodko et al (2012) "p (xv ) is called “relative love of variety”.
16
In Mrazova and Neary (2014), (8) is equivalently stated as (xv ) < 2 where (xv )
[u000 (xv )xv ] =u00 (xv ) =
2 "r (xv ) [1 "p (xv )] ="p (xv ) measures the convexity of inverse demand.
10
Hence, (A1) and (A2) are necessary and su¢ cient conditions for the consumers’ and …rms’ optimization problems. When satis…ed, they imply a unique output and price level for all varieties
xv > 0 and p(xv ) > 0, and for any given
> 0. As all admissible additive separable preferences
must satisfy (A1) and (A2), for all subsequent results we will assume that (A1) and (A2) always
hold without explicitly mentioning those assumptions each time.
Several implications of cost heterogeneity for product performance matching some key …ndings
discussed in Section 2 can be derived from those optimization conditions. In particular, as long as
(A1) and (A2) hold, lower cost …rms/products are associated with lower price, larger output, larger
revenue and larger pro…t.17
Free Entry Equilibrium
In equilibrium consumers maximize utility, …rms maximize pro…ts, and their optimal choices in the
product and labor markets are mutually compatible. To characterize this equilibrium outcome, it
is useful to make the dependence of maximized operating pro…t
v
and pro…t-maximizing output
xv on the endogenous marginal utility of income explicit. In particular, we de…ne
v
=
(v; )Lc
and xv = x ( v) with
(v; ) = max
u0 (x)
v x;
x
x ( v) = arg max
u0 (x)
v x;
x
so that, given (A1) and (A2), we obtain:
@x ( v)
= 00
@v
u (xv ) [2
@x ( v)
v
= 00
@
u (xv ) [2
(xv )]
(xv )]
< 0;
(9)
< 0:
By the envelope theorem, maximized pro…t is decreasing in both its arguments:
@
(v; )
=
@v
@ (v; )
=
@
x ( v) < 0;
u0 (x ( v))x ( v)
2
(10)
< 0:
The fact that maximized pro…t is decreasing in marginal cost implies that only products with
17
See the appendix for a proof.
11
marginal cost v below some cost cuto¤ vb can be pro…tably produced. At the same time, entrants
that do not …nd it pro…table to sell even their core products will decide not to produce at all. Thus,
the product cuto¤ level vb is also the …rm cuto¤ level b
c for core competency: entrants drawing a
core marginal cost c > b
c exit immediately without producing.
Given Lc = Lw and v(0; c) = c, the indi¤erence condition for the marginal producer is:
(b
c; ) Lw = f:
Since
(11)
(c; ) is decreasing in both c and , this cuto¤ condition has a unique solution b
c( ). For
a given measure of entrants Ne , b
c( ) determines the fraction of those entrants that eventually
produce:
(b
c( )).
All prospective entrants are identical ex-ante. Free entry then requires that expected pro…t
b
c earns pro…t:
equal the sunk entry cost. Post-entry, an entrant with a core cost draw c
M (c) 1
X
(c; )
(cz(m); ) Lc
[
f] ;
m=0
where M (c) is the number of products the entrant supplies with marginal cost cz(m)
upon entry, the expected pro…t of an entrant is
Z
0
b
c
(c; ) (c)dc =
=
8
Z bc <
X
0 :fmjcz(m)
"Z
1
b
c=z(m)
X
[
[
b
cg
(cz(m); ) Lc
(cz(m); ) Lc
f]
9
=
;
(c)dc
#
f ] (c)dc :
0
m=0
b
c. Hence,
The free entry condition can then be re-stated as
Z
0
b
c
(c; ) (c)dc =
"Z
1
X
m=0
0
b
c=z(m)
[
w
(cz(m); ) L
#
f ] (c)dc = f e
(12)
Equations (11) and (12) jointly determine the equilibrium cost cuto¤ b
c and the marginal utility
of income
. As both
(c; ) and b
c( ) decrease in , this solution (b
c ;
) exists and is unique.
The labor market clearing condition
Ne
(
e
f +
"Z
1
X
m=0
0
b
c=z(m)
w
#)
[cz(m)x (cz(m); ) L + f ] (c)dc
12
= Lw ;
(13)
evaluated for b
c=b
c , then determines the equilibrium number of entrants Ne and producers N =
(b
c ) Ne .
Reconciling Empirical Facts with Preferences
We have already argued that, as long as (A1) and (A2) hold, lower cost …rms/products are associated with lower price, larger output, larger revenue and larger pro…t. These implications match
some of the empirical relationships among …rm/product performance measures highlighted in Section 2, and are common across all preferences compatible with utility and pro…t maximization.
However, additional restrictions are needed in order to capture the additional empirical regularities
we described.
First, De Loecker, Goldberg, Pavcnik and Khandelwal (2012) show that lower costs are associated with larger markups so that cost advantages are not fully passed through to prices. As the
pass-through from cost to price can be expressed as
(xv ) =
"p (xv )
d ln p(xv )
=
;
d ln v
"r (xv )
(14)
a necessary and su¢ cient condition for (xv ) < 1 is that the elasticity of marginal revenue is
larger than the elasticity of inverse demand: "r (xv ) > "p (xv ).18 This condition can be equivalently
stated as requiring that the elasticity of inverse demand increases in consumption. This is our third
assumption:19
(B1) "0p (xv ) > 0:
Second, Berman, Martin and Mayer (2012) …nd that high-performance …rms react to a real
exchange depreciation by increasing signi…cantly more their markup. This happens when the
pass-through decreases with …rm performance:
0 (x
v)
< 0. Given (B1), a necessary condition for
decreasing pass-through to be predicted by our model is that marginal revenue is increasing in xv
18
See also proof in the appendix.
0
In the terminology of Neary and Mrazova (2014) "p (xv ) > 0 de…nes the “subconvex” case, with “subconvexity”
of an inverse demand function p(x) at an arbitrary point xc being equivalent to the function being less convex at that
point than a CES demand function with the same elasticity. In the terminology of Zhelobodko, Kokovin, Parenti and
Thisse (2012), in this case preferences are said to display an increasing “relative love of variety” (RLV) as consumers
care less about variety when their consumption level is lower. The RLV is, thus, increasing if and only if the demand
for a variety becomes more elastic when the price of this variety rises. Also the “Adjustable pass-through” (Apt)
class of demand functions proposed by Fabinger and Weyl (2012) satis…es (B1). This assumption is, instead, weaker
than the assumption by Arkolakis, Costinot, Donaldson and Rodriguez-Clare (2012) that the demand function of any
product is log-concave in log-prices. Log-concavity implies (B1) but not vice versa and, thus, log-concavity is not
necessary to associate lower cost with larger markups (see the appendix for proofs).
19
13
(hence decreasing in v).20 This is our fourth assumption:
(B2) "0r (xv ) > 0.
Third and last, empirically lower cost …rms/products are associated with larger employment.
This is the case if and only if the elasticity of marginal revenue is smaller than one, otherwise
products associated with negligible marginal cost would be produced with negligible employment.21
This is our …fth and …nal assumption:
(B3) "r (xv ) < 1
To summarize, for the model to be consistent with the existing evidence on heterogeneous …rm
performance, the following properties of demand have to hold: 0 < "p (xv ) < 1, 0 < "r (xv ) < 1,
0
0
"p (xv ) > 0 and "p (xv ) > 0, or equivalently 0 < "p (xv ) < "r (xv ) < 1 and "0r (xv ) > 0.22 Figure 1 provides a graphical representation of demand and marginal revenue curve satisfying these properties.
These additional properties of demand must be satis…ed in order to generate predictions that are
not counter-factual with respect to the empirical …ndings summarized in Section 2. Importantly,
those empirical …ndings rule out the case of CES and log-convex demand.
Demand Shock
We now discuss the implications of our model under these empirically relevant assumptions (B1)(B3). We focus on the e¤ects of a demand shock (a change in ) on the product range, the product
mix and the productivity of multi-product …rms. In so doing, we distinguish between ‘long run’
e¤ects when entry is free and ‘short run’e¤ects when entry is, instead, restricted.
Long Run The following proposition holds:23
20
See proof in appendix.
See appendix for proof.
22
Although our results have been derived for an additive separable utility function, they eventually depend on
the properties of the associated inverse demand. They thus may hold also for utility or expenditure functions that
are not additive separable but still share those properties. In this respect, our results suggest that the taxonomy of
demand systems proposed by Mrazova and Neary (2014) in terms of 1="p (xv ) and (xv ) –or equivalently in terms of
"p (xv ) and "r (xv ) –could be fruitfully enriched also in terms of 0 (xv ) –or equivalently in terms of "0r (xv ) –to cover
additional comparative statics implications that are crucial when products are associated with heterogeneous costs.
23
In the case of non-separable preferences, Parenti, Thisse an Ushchev (2014) characterize general conditions on
pro…ts such that larger market size leads to lower cuto¤, but point out that general conditions on demand are
unavailable due to dependence on the cost distribution.
21
14
log p, log mr
MR
D
log xv
Figure 1: Graphical Representation of Demand Assumptions
Proposition 1 (Closed Economy) Consider an additive separable utility function satisfying
(A1)-(A2) and (B1)-(B3). Then, a positive demand shock (i) increases the marginal utility of
income, (ii) reduces the …rm cost cuto¤ , (iii) reallocates output, revenue and employment from
higher to lower cost products, and (iv) increases (decreases) pro…t for low (high) cost products.
Proof. See appendix.
Hence, we generate a clear cut result for the e¤ects of a demand shock along the intensive
…rm/product margin. Note, however, that, since these e¤ects are accompanied by a changing
number of entrants, the impact of the demand shock on the number of producers and products
supplied is ambiguous without further assumptions on the distribution of marginal cost (c).
Proposition 1 implies that a positive demand shock induces multi-product incumbents to (weakly)
shed some more costly non-core products (and some single-product incumbents to stop producing
altogether). It also induces multi-product incumbents to shift output, revenue and employment
towards better-performing products (with lower marginal cost). Since those products already had
larger output, revenue and employment before the shock, this leads to an increase in the ‘skewness’ of output, revenue and employment. Given our demand assumptions, higher skewness, in
turn, leads to higher …rm-productivity (the average productivity across the constant range of products produced both before and after the demand shock). Speci…cally, let M be the number of
these products and index them by m = 0; :::; M
15
1 in increasing order of marginal cost so that
v0 < v1 < ::: < vM
1.
=
The average productivity computed for this set of products is then
PM
1
m=0 xm
PM 1
m=0 vm xm
=
PM
1 1
m=0 vm vm xm
PM 1
m=0 vm xm
=
PM
1 1
m=0 vm `m
PM 1
m=0 `m
=
M
X1
sm
m=0
1
;
vm
where `m = vm xm is employment in product m, xm is its output, and sm = `m =
product’s employment share. Hence,
PM
1
m=0 `m
is the
is the employment weighted average of products’produc-
tivity levels 1=vm ’s. When the marginal utility of income increases due to the positive demand
shock, …rm productivity
increases.24
Short Run So far we have consider a long-run scenario in which …rms can freely enter the market.
We now consider an alternative short-run scenario in which the number of incumbents is …xed at
N . In this scenario (12) no longer holds, and we assume that the sunk entry costs have already
been incurred (thus no employment associated with the entry cost f e ). The short-run equilibrium
is then characterized by two conditions. The …rst is the zero cuto¤ pro…t condition (11):
(b
c; ) Lw = f:
The second is the labor market clearing condition obtained from (13) after imposing Ne = N and
f e = 0:
N
"Z
1
X
m=0
0
b
c=z(m)
#
[cz(m)x (cz(m); ) L + f ] (c)dc = Lw :
These conditions pin down b
c and
w
(15)
for …xed N . They imply that the all results in the previous
proposition also hold in the short run, except (ii). This is because a positive demand shock has
an ambiguous impact on the …rm cost cuto¤ in the short-run (this also depends on distributional
assumptions for the core cost draw).
5
Open Economy
With our empirical application in mind, we consider a simpli…ed three-country economy consisting
of a Home country (H: France) and a Foreign country (F : RoW) both exporting to a Destination
country (D). For l; h 2 fH; F; Dg trade from country l to country h is subject to both a variable
iceberg cost
24
lh
> 1 and a …xed export cost fhx .
See appendix for proofs.
16
Country D is the focus of the analysis and is assumed to be ‘small’from the point of view of
both H and F so that changes in destination D-speci…c variables do not a¤ect equilibrium variables
for either H or F (apart from the export cuto¤ to D and thus the margins of trade to D). Since
wages in H and F are …xed, we choose labor as the numeraire. The short-run equilibrium is then
characterized by D’s zero cuto¤ pro…t condition with a …xed numbers of incumbents for D, H and
F , and …xed domestic cuto¤s for H and F . The long-run equilibrium is characterized by D’s zero
cuto¤ pro…t and free conditions with …xed domestic cuto¤s and …xed numbers of incumbents for
H and F .
To emphasize competition in the destination market D, we focus on a situation in which D does
not export (and trade is thus unbalanced). The same qualitative results hold when one allows also
for exports from D.25 Under the assumption that …rms located in D do not export, the long run
equilibrium is characterized by the following four conditions: the zero cuto¤ pro…t for domestic
sales in D
cDD ;
DD (b
D)
Lw
D = f;
(16)
the zero cuto¤ pro…t for export sales from l 2 fH; F g to D
clD ;
lD ( lD b
the free entry in D
"Z
1
X
m=0
0
b
cDD =z(m)
[
DD
D)
x
Lw
D = fD ;
(cz(m); ) Lw
D
(17)
#
f ] (c)dc = f e ;
(18)
and the labor market clearing in D
e
ND
(
e
f +
"Z
1
X
m=0
0
b
cDD =z(m)
[cz(m)xDD (cz(m); )
Lw
D
#)
+ f ] (c)dc
= Lw
D:
(19)
Given
DD (c;
D)
conditions (16) and (18) pin down b
cDD and
xDD (c;
25
D)
= max
x
D.
u0 (x)
c x;
(20)
D
Then, given
= arg max
x
See our appendix for this extension.
17
u0 (x)
D
c x;
(21)
e and producers N p =
condition (19) pins down the number of entrants ND
D
e.
(b
cDD ) ND
The measure of products sold in D includes also those exported from l 2 fH; F g. This is a
fraction
i
(b
clD ) of the …xed measure of incumbent products M l in l 2 fH; F g:
i
x
NlD
=
To determine b
clD ; we write
lD (
lD c;
D)
(b
clD ) M l :
= max
x
u0 (x)
lD c
x
(22)
D
and
xlD (
with
lD
> 1 . Then, given
D,
lD
D c) = arg max
x
u0 (x)
lD c
x;
(23)
D
X > f (17) also implies
condition (17) pins down b
clD . With fD
cHD
lD b
<b
cDD :
Consider now the e¤ects of raising . By comparing the corresponding expressions in the closed
and the open economies, we see that all closed economy results fully apply to D variables (b
cDD ,
e , N p ) in the open economy. In particular, by (16) and (18), larger Lc increases
ND
D
D
D
D
,
and decreases
b
cDD when the elasticity of inverse demand is increasing in consumption. Hence, accounting also
for the response of the export cuto¤ b
clD , the following result holds:
Proposition 2 (Open Economy) Consider an additive separable utility function satisfying (A1)(A2) and (B1-B3). Then a positive demand shock in an export market (i) increases the local
marginal utility of income, (ii) reduces the local …rm cost cuto¤ , (iii) reallocates output, revenue
and employment from higher to lower cost exported products, and (iv) increases (decreases) pro…t
for low (high) exported cost products. Moreover, as long as the export …xed cost is large enough, the
positive demand shock also decreases the export cost cuto¤ and increases the number of exported
products, and aggregate exports.
Proof. See appendix.
When the …xed export cost is large enough, the demand shock in D has opposite e¤ects on the
domestic cuto¤ (which increases) and the export cuto¤ (which decreases). This is due to the fact
that the demand shock, in equilibrium, induces a rotation of the residual demand curve faced by
18
each product. Thus, this demand shifts in for worse-performing products while it shifts out for the
better performing ones. When the …xed export cost is high enough, the cuto¤ exported product
has a relatively larger market share and thus bene…ts from the outward shift of a portion of the
demand curve. Thus, even though competition and selection is tougher in D following the demand
shock, an exporting …rm will then respond by increasing the set of products exported to D.
i
In the short run the number of incumbent …rms in D is …xed at N D , (18) does not hold,
e . Then the short-run equilibrium is
and there is no employment associated with the entry cost fD
characterized by the following three conditions: the zero cuto¤ pro…t condition (16) for domestic
sales in D; the zero cuto¤ pro…t (17) for export sales to D; and the labor market clearing condition
i
ND
(
"Z
1
X
0
m=0
b
cDD =z(m)
#)
[cz(m)xDD (cz(m); ) Lw
D + f ] (c)dc
Conditions (16) and (24) pin down b
cDD and
= Lw
D:
(24)
i
D
for …xed N D . These are the same as in the
short-run closed economy, so all corresponding short-run comparative statics results for larger
apply here too. Given
D,
export cuto¤s b
cHD and b
cF D are the determined by (17).
To conclude, we have shown that, under our demand assumptions, a positive demand shock in a
destination market D will generate an increase in competition that induces exporters to reallocate
their export sales towards their better performing products. This reallocation, in turn, will generate
an increase in …rm productivity. We now empirically test this prediction linking trade shocks and
…rm productivity.
6
Trade Competition and Productivity
Our theoretical model highlights how our measured demand shocks induce increases in competition for exporters to those destinations; and how the increased competition generates increases in
productivity by shifting market shares and employment towards better performing products. We
now directly test this link between increased trade competition (induced by the demand shocks
in export markets) and productivity. We focus on the portion of the product reallocations and
induced productivity increases within the …rm. We can then use our constructed demand shocks
(exogenous to the …rm) and trace out their impact on …rm productivity and product reallocations
(at the …rm level).
We obtain our measure of …rm productivity by merging our …rm-level trade data with …rm-level
19
production data. This latter dataset contains various measures of …rm outputs and inputs. As
we are interested in picking up productivity ‡uctuations at a yearly frequency, we focus on labor
productivity measured as de‡ated value added per worker (using sector-speci…c price de‡ators).
We then separately control for the impact of changes in factor intensities and returns to scale (or
variable utilization of labor) on labor productivity. Note that this …rm-level productivity measure
aggregates (using labor shares) to the overall de‡ated value-added per worker for manufacturing.
So long as our sector-speci…c price indices are accurately measured, this aggregate productivity
measure accurately tracks a welfare-relevant quantity index –even though we do not have access to
…rm-level prices. In other words, the e¤ect of pure markup changes at the sector level are netted-out
of our productivity measure. We can thus report a welfare-relevant aggregate productivity change
by aggregating our …rm-level productivity changes using the observed changes in labor shares.26
6.1
Firm-Level Trade Shock
We cannot separately measure productivity at the destination or product level – only at the …rm
level. We thus need to aggregate our destination-industry measures of demand shocks to the
…rm-level. We use the …rm’s market shares in those destinations and industries to perform this
aggregation.27 In a …rst step, we perform this aggregation over all export destinations and obtain
our …rm-level demand shock in (log) levels and …rst di¤erence:
shocki;t
X xIi;d;t
d;I
where xi;t
P
d;I
0
xi;t0
~ shocki;t =
shockIi;d;t ;
X xIi;d;t
d;I
xi;t
1
1
~ shockI
i;d;t ;
xIi;d;t represents …rm i’s total exports in year t. As was the case for the construc-
tion of our …rm-level destination shock (see 1), we only use the …rm-level information on exported
products and market shares in prior years (the year of …rst export sales t0 for the demand shock in
levels and lagged year t
1 for the …rst di¤erence between t and t
1). This ensures the exogeneity
of our constructed …rm-level demand shocks (exogenous to …rm-level actions in year t > t0 for
levels, and exogenous to …rm-level changes
t
for …rst di¤erences). In particular, changes in the
26
At the …rm-level, an increase in markups across all products will be picked-up in our …rm productivity measure
–even though this does not re‡ect a welfare-relevant increase in output. But if this is the case, then this …rm’s labor
share will decrease, and its productivity will carry a smaller weight in the aggregate index. In addition, increases in
…rm productivity are very strongly correlated with increases in labor shares –so this cannot be a pervasive component
of our measured productivity changes.
27
Later, when we focus on the impact of trade competition on skewness, we will show that those market shares are
the relevant weights to aggregate the destination-speci…c skewness measures.
20
set of exported products or exported market shares are not re‡ected in the demand shock.28
Ideally, this aggregation would also incorporate a …rm’s exposure to demand shocks in its
domestic (French) market. This is not possible for two reasons: most importantly, we do not
observe the product-level breakdown of the …rms’sales in the French market (we only observe total
domestic sales across products); in addition, world exports into France would not be exogenous to
…rm-level technology changes in France. Therefore, we need to adjust our export-speci…c demand
shock using the …rm’s export intensity to obtain an overall …rm-level demand shock:
shock_intensi;t =
xi;t0
shocki;t ;
xi;t0 + xi;F;t0
~ shock_intens
i;t
=
xi;t
xi;t 1
1 + xi;F;t
~ shocki;t ;
1
where xi;F;t denotes …rm i’s total (across products) sales to the French domestic market in year
t (and the ratio thus measures …rm i’s export intensity). Once again, we only use prior year’s
information on …rm-level sales to construct this overall demand shock. Note that this adjustment
using export intensity is equivalent to assuming a demand shock of zero in the French market and
including that market in our aggregation by market share relative to total …rm sales xi;t + xi;F;t .
6.2
Impact of the Trade Shock on Firm Productivity
In this section, we investigate the direct link between this …rm-level demand shock and …rm productivity. Here, we focus exclusively on our …rm-speci…c demand shock from (1), as this is our only
shock that exhibits …rm-level variation within destinations. Our measure of productivity is the log
of de‡ated value-added per worker. In order to control for changes in capital intensity, we use the
log of capital per worker (Kit =Lit ). We also control for unobserved changes in labor utilization and
returns to scale by using the log of raw materials (including energy use), Rit . Then, increases in
worker e¤ort or higher returns to scale will be re‡ected in the impact of raw materials use on labor
productivity. As there is no issue with zeros for all these …rm-level variables, we directly measure
the growth rate of those variable using simple …rst di¤erences of the log levels.
We begin with a graphical representation of the strong positive relationship between …rmlevel productivity and our constructed demand shock. Figure 2 illustrates the correlation between
those variables in …rst di¤erences for the largest French exporters (representing 50% of French
exports in 1996). Panel (a) is the unconditional scatter plot for those variables, while panel (b)
shows the added-variable plot for the …rst-di¤erence regression of productivity on the trade shock,
28
Lileeva and Tre‡er (2010) and Hummels et al (2014) use a similar strategy to construct …rm-level trade-related
trade shocks.
21
with additional controls for capital intensity, raw materials (both in log …rst-di¤erences) and time
dummies. Those …gures clearly highlight the very strong positive response of the large exporters’
productivity to changes in trade competition in export markets (captured by the demand shock).
-1.5
-1.5
change in ln labour productivity
-1
-.5
0
.5
1
cond. change in ln labour productivity
-1
-.5
0
.5
1
Figure 2: Exporters Representing 50% of French Trade in 1996: First Di¤erence 1996-2005
-.2
-.1
0
.1
change in ln trade shock
-.2
.2
-.1
0
.1
cond. change in ln trade shock
.2
regression line: coef = 1.094, se = .263, N = 1040
sample: 167 firms representing 50% of French exports in 1996
standard errors clustered by firm
regression line: coef = .783, se = .198, N = 1063
sample: 167 firms representing 50% of French exports in 1996
(a) Unconditional
(b) Conditional
Table 3 shows how this result generalizes to our full sample of …rms and our three di¤erent
speci…cations (FE, FD, FD-FE). Our theoretical model emphasizes how a multi-product …rm’s
productivity responds to the demand shock via its e¤ect on competition and product reallocations
in the …rm’s export markets. Thus, we assumed that the …rm’s technology at the product level
(the marginal cost v(m; c) for each product m) was exogenous (in particular, in respect to demand
‡uctuations in export markets). However, there is a substantial literature examining how this
technology responds to export market conditions via various forms of innovation or investment
choices made by the …rm. We feel that the timing dimension of our …rst di¤erence speci…cations –
especially our FD-FE speci…cations which nets out any …rm-level growth trends – eliminates this
technology response channel: It is highly unlikely that a …rm’s innovations or investment responses
to the trade shock in a given year (especially the innovation in the trade shock relative to trend)
would be re‡ected contemporaneously in the …rm’s productivity. However, we will also show some
additional robustness checks that address this potential technology response.
The …rst three columns of Table 3 show that, across our three timing speci…cations, there is
22
a stable and very strong response of …rm productivity to the trade shock. Since our measure
of productivity as value added per worker incorporates neither the impact of changes in input
intensities nor the e¤ects of non-constant returns to scale, we directly control for these e¤ects
in the next set of regressions. In the last 3 columns of Table 3, we add controls for capital per
worker and raw material use (including energy). Both of these controls are highly signi…cant: not
surprisingly, increases in capital intensity are re‡ected in labor productivity; and we …nd that
increases in raw materials use are also associated with higher labor productivity. This would be
the case if there are increasing returns to scale in the value-added production function, or if labor
utilization/e¤ort increases with scale (in the short-run). However, the very strong e¤ect of the
trade shock on …rm productivity remains when these controls are added –and they remain highly
signi…cant, well beyond the 1% signi…cance level. (From here on out, we will keep those controls in
all of our …rm productivity regressions.)
Table 3: Baseline Resultsl: Impact of Trade Shock on Firm Productivity
Dependent Variable
Speci…cation
log (trade shock export intens.)
~ (trade shock
log prod.
FE
0.094a
(0.019)
log prod.
FD
FD-FE
0.134a
(0.024)
log prod.
FE
0.073a
(0.018)
0.108a
(0.024)
0.096a
(0.028)
log capital stock per worker
0.327a
(0.008)
0.358a
(0.009)
log raw materials
0.100a
(0.004)
0.093a
(0.004)
174931
174931
export intens.)
0.116a
(0.028)
log prod.
FD
FD-FE
log capital stock per worker
0.228a
(0.007)
log raw materials
0.091a
(0.004)
Observations
213877
188328
Standard errors (clustered at the …rm level) in parentheses:
c
< 0:1;
188328
b
< 0:05;
201627
a
< 0:01
We now describe several robustness checks that further single-out our theoretical mechanism
operating through the demand-side product reallocations for multi-product …rms. In the next table
we regress our capital intensity measure on our trade shock; the results in Table 4 show that there
23
is no response of investment to the trade shock. This represents another way to show that the
short-run timing for the demand shocks precludes a contemporaneous technology response: if this
were the case, we would expect to see some of this response re‡ected in higher investment (along
with other responses along the technology dimension).
Table 4: K/L Does Not respond to Trade Shocks
Dependent Variable
Speci…cation
log (trade shock export intens.)
~ (trade shock
ln K=L
FE
-0.018
(0.018)
export intens.)
Observations
Standard errors in parentheses:
212745
c
< 0:1;
b
< 0:05;
a
ln K=L
FD
ln K=L
FD-FE
-0.003
(0.017)
-0.005
(0.020)
186171
186171
< 0:01
Next we use a di¤erent strategy to control for the e¤ects of non-constant returns to scale
or variable labor utilization: in Table 5, we split our sample between year intervals where …rms
increase/decrease employment. If the e¤ects of the trade shock on productivity were driven by scale
e¤ects or higher labor utilization/e¤ort, then we would expect to see the productivity responses
concentrated in the split of the sample where …rms are expanding employment (and also expanding
more generally). Yet, Table 5 shows that this is not the case: the e¤ect of the trade shock on
productivity is just as strong (even a bit stronger) in the sub-sample of years where …rms are
decreasing employment; and in both cases, the coe¢ cients have a similar magnitude to our baseline
results in Table 3.29
In order to further single-out our theoretical mechanism operating through the demand-side
product reallocations for multi-product …rms, we now report two di¤erent types of falsi…cation
tests. Our …rst test highlights that the link between productivity and the trade shocks is only
operative for multi-product …rms. Table 6 reports the same regression (with controls) as our
baseline results from Table 3, but only for single-product exporters. This new table clearly shows
that this there is no evidence of this link among this subset of …rms. Next, we show that this
productivity-trade link is only operative for …rms with a substantial exposure to export markets
(measured by export intensity). Similarly to single-product …rms, we would not expect to …nd a
29
Since we are splitting our sample across …rms, we no longer rely on the two speci…cations with …rm …xed-e¤ects
and only show results for the FD speci…cation.
24
Table 5: Robustness to Scale E¤ects
Sample
Dependent Variable
Speci…cation
~ (trade shock export intens.)
Employment Increase
log productivity
FD
0.135a
(0.035)
Employment Decrease
log productivity
FD
0.156a
(0.045)
log capital stock per worker
0.288a
(0.012)
0.332a
(0.013)
log raw materials
0.091a
(0.005)
69642
0.097a
(0.005)
65268
Observations
Standard errors in parentheses:
c
< 0:1;
b
< 0:05;
a
< 0:01
signi…cant productivity-trade link among …rms with very low export intensity. This is indeed the
case. In Table 7, we re-run our baseline speci…cation using the trade shock before it is interacted
with export intensity. The …rst three columns report the results for the quartile of …rms with the
lowest export intensity, and highlight that there is no evidence of the productivity-trade link for
those …rms. On the other hand, we clearly see from the last three columns that this e¤ect is very
strong and powerful for the quartile of …rms with the highest export intensity.30
30
Since the trade shock as not been interacted with export intensity, the coe¢ cients for this top quartile represent
signi…cantly higher magnitudes than the average coe¢ cients across the whole sample reported in Table 3 (since export
intensity is always below 1). This is also con…rmed by a speci…cation with the interacted trade shock restricted to
this same top quartile of …rms.
25
Table 6: Reduced Form: Single Product Firms
Dependent Variable
log prod.
log prod.
Speci…cation
FE
FD
FD-FE
log (trade shock export intens.)
0.005
(0.050)
log capital stock per worker
0.269a
(0.016)
log raw materials
0.101a
(0.010)
~ (trade shock
-0.021
(0.062)
-0.138c
(0.079)
log capital stock per worker
0.368a
(0.020)
0.415a
(0.028)
log raw materials
0.114a
(0.010)
0.090a
(0.013)
25330
25330
export intens.)
Observations
Standard errors in parentheses:
32870
c
< 0:1;
26
b
< 0:05;
a
< 0:01
Table 7: Reduced Form: Low/High export intensity
Sample
Dependent Variable
Speci…cation
log trade shock
exp. intens. quartile # 1
log prod.
log prod.
FE
FD
FD-FE
0.009
(0.006)
exp. intens. quartile # 4
log prod.
log prod.
FE
FD
FD-FE
0.068a
(0.014)
log capital stock per worker
0.278a
(0.022)
0.217a
(0.015)
log raw materials
0.070a
(0.006)
0.128a
(0.010)
~ trade shock
0.000
(0.007)
-0.002
(0.009)
0.096a
(0.017)
0.100a
(0.021)
log capital stock per worker
0.323a
(0.016)
0.367a
(0.020)
0.325a
(0.014)
0.368a
(0.016)
log raw materials
0.070a
(0.006)
0.057a
(0.006)
0.129a
(0.008)
0.123a
(0.010)
38894
38894
46347
46347
Observations
Standard errors in parentheses:
49227
c
< 0:1;
b
< 0:05;
a
< 0:01
27
53125
7
Trade Competition and Product Reallocations at the Firm-Level
In order to further highlight the demand-side product reallocations channel for the productivitytrade link, we now show that our trade shock aggregated to the …rm level strongly predicts product
reallocations towards better performing products (higher market shares) at the …rm level; that
is the …rm’s global product mix (the distribution of product sales across all destinations). Our
theoretical model highlighted how (under our demand assumptions) those product reallocations
would then lead to higher …rm productivity.
Our previous results highlighted how demand shocks lead to reallocations towards better performing products at the destination-industry level. Intuitively, since there is a stable ranking of
products at the …rm level (better performing products in one market are most likely to be the
better performing products in other markets – as we previously discussed), then reallocations towards better performing products within destinations should also be re‡ected in the reallocations of
global sales/production towards better performing products; and the strength of this link between
the skewness of sales at the destination and global levels should depend on the importance of the
destination in the …rm’s global sales. Our chosen measure of skewness, the Theil index, makes this
intuition precise. It is the only measure of skewness that exhibits a stable decomposition from the
skewness of global sales into the skewness of destination-level sales (see Jost 2007).31 Speci…cally,
P s
let Ti;t be …rm i’s Theil index for the skewness of its global exports by product xsi;t
d xi;d;t .
(the sum of exports for that product across all destinations).32 Then this global Theil can be
decomposed into a market-share weighted average of the within-destination Theils Ti;d;t and a
“between-destination” Theil index that measures di¤erences in the distribution of product-level
market shares across destinations:33
Ti;t =
X xi;d;t
d
xi;t
Ti;d;t
X xi;d;t
d
31
xi;t
B
Ti;d;t
;
(25)
This decomposition property is similar –but not identical –to the within/between decomposition of Theil indices
across populations. In the latter, the sample is split into subsamples. In our case, the same observation (in this case,
product sales) is split into “destinations” and the global measure re‡ects the sum across “destinations”.
32
The Theil index is de…ned in the same way as the destination level Theil in (2).
33
For simplicity, we omit the industry referencing I for the destination Theils. The decomposition across industries
follows a similar pattern.
28
B is de…ned as
where the between-destination Theil Ti;d;t
B
Ti;d;t
=
X xsi;d;t
s
xi;d;t
log
xsi;d;t =xi;d;t
xsi;t =xi;t
!
:
Note that the weights used in this decomposition for both the within- and between-destination
Theils are the …rm’s export shares xi;d;t =xi;t across destinations d. The between-destination Theil
B measures the deviation in a product’s market share in a destination d, xs =x
Ti;d;t
i;d;t , from that
i;d;t
product’s global market share xsi;t =xi;t and then averages these deviations across destinations. It is
positive and converges to zero as the distributions of product market shares in di¤erent destination
become increasingly similar.
To better understand the logic behind (25), note that it implies that the average of the withindestination Theil indices can be decomposed into the sum of two positive elements: the global
Theil index, and the between-destination Theil index. This decomposition can be interpreted as
a decomposition of variance/dispersion. The dispersion observed in the destination level product
exports must be explained either by dispersion in global product exports (global Theil index), or
by the fact that the distribution of product sales varies across destinations (between-destination
Theil index).
A simple example helps to clarify this point. Take a …rm with 2 products and 2 destinations.
In each destination, exports of one product are x, and exports of the other product are 2x. This
leads to the same value for the within-destination Theil indices of (1=3) ln (1=3) + (2=3) ln(2=3),
and hence the same value for the average within-destination Theil index. Hence, if the same
product is the better performing product in each market (with 2x exports), then the distributions
will be synchronized across destinations and the between-destination Theil will be zero: all of the
dispersion is explained by the global Theil index, whose value is equal to the common value of the
two within-destination Theil indices. On the other hand, if the opposite products perform better
in each market, global sales are 3x for each product. There is thus no variation in global product
sales, and the global Theil index is zero. Accordingly, all of the variation in the within-destination
Theil indices is explained by the between-destination Theil.
The theoretical model of Bernard, Redding and Schott (2011) with CES demand predicts that
the between-destination Theil index would be exactly zero when measured on a common set of
exported products across destinations. With linear demand, Mayer, Melitz and Ottaviano (2014)
show (theoretically and empirically) that this between-destination index would deviate from zero
29
because skewness varies across destinations. We have shown earlier that this result holds for
a larger class of demand systems such that the elasticity and the convexity of inverse demand
increase with consumption. Yet, even in these cases, the between-destination Theil is predicted to
be small because the ranking of the product sales is very stable across destinations. This leads to
a prediction that the market-share weighted average of the destination Theils should be strongly
correlated with the …rm’s global Theil. Empirically, this prediction is strongly con…rmed as shown
in Figure 3.
Figure 3: Correlation Between Global Skewness and Average Local Skewness
This high correlation between destination and global skewness of product sales enables us to
move from our previous predictions for the e¤ects of the trade shocks on skewness at the destinationlevel to a new prediction at the …rm-level. By aggregating the trade shocks across destinations
using the …rm’s market share in each destination, we have constructed a …rm-level trade shock that
predicts changes in the weighted average of destination skewness Ti;d;t – and hence will predict
changes in the …rm’s global skewness Ti;t (given the high correlation between the two indices).
This result is con…rmed by our regression of the …rms’global Theil on our trade shock measures,
reported in the …rst three columns of Table 8. Our …rm-level trade shock has a strong and highly
signi…cant (again, well beyond the 1% signi…cance level) impact on the skewness of global exports.
In this regression, we have also added back our two more aggregated measures of demand shocks:
the industry (ISIC-3) level trade shock and the GDP shock (aggregated to the …rm-level using
the same market share weights across destinations). The industry level trade shock – which was
already substantially weaker than the …rm-level trade shock in the destination-level regressions
30
– is no longer signi…cant at the …rm level. The GDP shock is not signi…cant in the (log) levels
regressions, but is very strong and signi…cant in the two …rst-di¤erence speci…cations.
Table 8: The Impact of Demand Shocks on the Global Product Mix (Firm Level)
Dependent Variable
Speci…cation
log GDP shock
Ti;t
FE
-0.001
(0.004)
log trade shock
0.045a
(0.009)
0.014a
(0.003)
log trade shock - ISIC
-0.001
(0.001)
0.000
(0.000)
FD
Ti;t
FD-FE
Exp. Intensi;t
FE
0.003a
(0.001)
Exp. Intensi;t
FD
FD-FE
~ GDP shock
0.118a
(0.031)
0.107a
(0.038)
0.032a
(0.010)
0.035a
(0.012)
~ trade shock
0.057a
(0.011)
0.050a
(0.013)
0.019a
(0.003)
0.016a
(0.004)
~ trade shock - ISIC
-0.003
(0.005)
-0.010
(0.007)
0.002
(0.002)
0.000
(0.002)
(0.004)
117851
(0.004)
117851
(0.001)
107283
(0.001)
107283
Observations
(0.110)
117851
Standard errors in parentheses:
c
< 0:1;
b
< 0:05;
a
(0.030)
110565
< 0:01
Our global Theil measure Ti;t measures the skewness of export sales across all destinations,
but it does not entirely re‡ect the skewness of production levels across the …rm’s product range.
That is because we cannot measure the breakdown of product-level sales on the French domestic
market. Ultimately, it is the distribution of labor allocation across products (and the induced
distribution of production levels) that determines a …rm’s labor productivity – conditional on its
technology (the production functions for each individual product). As highlighted by our theoretical
model, the export market demand shocks generate two di¤erent types of reallocations that both
contribute to an increased skewness of production levels for the …rm: reallocations within the set of
exported products, which generate the increased skewness of global exports that we just discussed;
but also reallocations from non-exported products towards the better performing exported products
(including the extensive margin of newly exported products that we documented at the destinationlevel). Although we cannot measure the domestic product-level sales, we can measure a single
31
statistic that re‡ects this reallocation from non-exported to exported goods: the …rm’s export
intensity. We can thus test whether the …rm-level trade shocks also induce an increase in the …rm’s
export intensity. Those regressions are reported in the last three columns of Table 8, and con…rm
that our …rm-level trade shock has a very strong and highly signi…cant positive impact on a …rm’s
export intensity.34 The impact of the GDP coe¢ cient is also strong and signi…cant, whereas the
industry-level trade shock remains insigni…cant. Thus, our …rm-level trade shock and GDP shock
both predict the two types of reallocations towards better performing products that we highlighted
in our theoretical model (as a response to increased competition in export markets). Holding the
…rm’s product-level technology …xed, these reallocations both generate the subtantial increases in
…rm-level productivity that we previously documented.
8
Can the Measured Product Reallocations Explain the Entire Impact of
Trade Competition on Productivity?
To be completed...
9
Conclusion
To be completed...
10
References
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Trade Liberalization.” Quarterly Journal of Economics 126 (3): 1271-1318.
34
Since the export intensity is a ratio, we do not apply a log-transformation to that variable. However, speci…cations
using the log of export intensity yield very similar results.
32
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Empirics of Firm Heterogeneity and International Trade.” Annual Review of Economics 4
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33
Mayer, Thierry and Gianmarco I.P. Ottaviano. 2008. “The Happy Few: The Internationalisation
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34
Appendix
A
Higher cost is associated with higher price, lower output, lower revenue
and lower pro…t.
First, implicit di¤erentiation of (4) yields
dxv
=
dv
0 (x
v)
<0
(A.1)
where the sign is dictated by (A2). This shows that higher marginal cost leads to smaller output
qv = xv Lv . Second, by (3), under (A1) smaller output implies higher price as
u00 (xv )
dpv
=
<0
dxv
so higher marginal cost leads to higher price
dpv
dpv dxv
u00 (xv )
=
= 0
> 0:
dv
dxv dv
(xv )
Third, given "p (xv ) 2 (0; 1), higher marginal cost is associated with lower revenue rv Lv = pv xv Lv
as
dpv xv
drv
=
= [1
dv
dv
"p (xv )]
dxv
<0
dv
which shows that the negative e¤ect of cost on output dominates its positive e¤ect on price. Fourth,
by the envelope theorem, also pro…t is a decreasing function of marginal cost as
d v
=
dv
where
v
= (pv
xv < 0
(A.2)
v)xv Lc is operating pro…t. Hence, (A1) and (A2) are necessary and su¢ cient for
higher cost products to associated with higher price, lower output, lower revenue and lower pro…t.
B
Lower cost is associated with higher markup
From (5), the (multiplicative) markup equals
mv =
pv
=
v
1
35
1
"p (xv )
(B.1)
Given that in (a) we have already shown that xv is a decreasing function of v, markup mv will
0
be decreasing or increasing in v depending on whether the elasticity of inverse demand "p (xv ) is
0
positive or negative. Moreover, that "p (xv ) R 0 is equivalent to "r (xv ) R "p (xv ) can be seen by
noting that we can write
"0p (xv )xv
1 "p (xv )
=
["r (xv )
"p (xv )
"p (xv )
"p (xv )]
0
so that, given "p (xv ) 2 (0; 1), "p (xv ) R 0 if and only if "r (xv ) R "p (xv ).
Arkolakis, Costinot, Donaldson and Rodriguez-Clare (2012) assume that demand functions are
log-concave in log-prices for all di¤erentiated goods (this is their assumption "A3"). To compare
(B1) with this assumption, invert our inverse demand
pi =
u0 (xi )
; with
=
Z
M
u0 (xi )xi di
0
to obtain (direct) demand
xi = u0
1
eln pi +ln
.
(B.2)
Then, de…ne the function
d(y) = u0
1
(ey )
and rewrite (B.2) as
xi = d(ln pi + ln )
This is the way Arkolakis, Costinot, Donaldson and Rodriguez-Clare (2012) state their demand
system in the separable case. Clearly, under (A1) d(y) is strictly decreasing so that we have
d0 (y) < 0. Arkolakis, Costinot, Donaldson and Rodriguez-Clare (2012) de…ne “log-concavity” as
d00 (y) < 0
i.e. d00 (y) is concave. As the inverse of a strictly decreasing strictly concave real function is strictly
concave, strict concavity of d(y) implies strict concavity of its inverse
yi = ln u0 (xi ).
36
The concavity of this function depends on the sign of its second derivative
u000 (xi )u0 (xi ) u00 (xi )u00 (xi )
d2 ln u0 (xi )
=
;
2
dxi
[u0 (xi )]2
which, under (A1), is negative if and only if
u000 (xi )xi
u00 (xi )
u00 (xi )xi
>0
u0 (xi )
or equivalently
2
"r (xv )
>
"p (xv )
1
"p (xv )
"p (xv )
As the term on the right hand side is larger than one, the assumption of “log-concavity” in the
sense of Arkolakis, Costinot, Donaldson and Rodriguez-Clare (2012) is more restrictive than our
assumption (B1).
C
Pass-through is incomplete and lower cost is associated smaller
pass-through
De…ne the pass-through as
(xv )
d ln p(xv )
d ln p(xv ) d ln xv
=
d ln v
d ln xv d ln v
By de…nition we have
d ln p(xv )
=
d ln xv
"p (xv )
while implicit di¤erentiation of (3) gives
dxv
=
dv
0 (x
v)
so that
d ln xv
=
d ln v
(xv )
=
v )xv
0 (x
1
"r (xv )
where the …rst equality is granted by (3) and the second equality is granted by de…nition. Accordingly, the pass-through can be expressed as
(xv ) =
"p (xv )
"r (xv )
37
(C.1)
with (xv ) < 1 when (B1) holds. Di¤erentiating (C.1) show that
"0p (xv )xv
"0 (xv )xv
< r
"p (xv )
"r (xv )
is necessary and su¢ cient for
0 (x
v)
0 (x
v)
< 0. Then, given (B1) "0p (xv ) > 0, "0r (xv ) > 0 is necessary for
< 0.
In Berman, Martin and Mayer (2102) an exporter to destination market D maximizes
max
qi 0
D (qi )
=
pi
qi
subject to
pi =
vqi
f
D
u0 (xi )
; with
=
Z
M
u0 (xi )xi di
0
where pi is the price denominated in the destination’s currency, v is the marginal cost denominated
in the currency of the …rm’s country, and
D
is the exchange rate. The FOC for pro…t maximization
is
u0 (xi ) + u00 (xi )xi =
c
which implies that the pass-through from marginal cost to price is homomorphic to the pass-through
from exchange rate to price.
D
Lower cost is associated with larger employment
Employment in the supply of a product with marginal cost v equals vxv Lc . Di¤erentiation yields
dvxv
=
dv
1+
@xv v
@v xv
xv =
1
1
"r (xv )
xv
where the second equality is granted by (3) together with the de…nitions of "p (xv ) and "r (xv ).
Hence, employment decreases with marginal cost if and only if "r (xv ) < 1.
E
Long-Run Response to a Demand Shock in Closed Economy (Proposition 1)
The e¤ects of a demand shock on b
c and
can be characterized extending the analysis by Zhelo-
bodko et al (2012) to the case of multiproduct …rms. The reason is that, as these e¤ects are fully
determined by (11) and (12), changing
is paramount to changing Lc , which is the comparative
38
statics experiment studied by Zhelobodko et al (2012).
Too see this, rewrite expressions (10) in terms of elasticities as
=
=v (
( v) =
v) =
@
@
(v; )
@
(v; )
@v
(v; )
1
=
v
=
(v; )
1
"p (x ( v))
=
c
1
u0 (x ( v))
v
p(x ( v))
v
1
=
"p (x ( v))
"p (x ( v))
(E.1)
(E.2)
=dLc > 0 divide (12) by Lc to yield
To obtain d
1 Z
X
m=0 0
b
c=z(m)
(cz(m); ) (c)dc =
1
f X
fe
+
Lc Lc
b
c
z(m)
m=0
(E.3)
where regularity conditions are assumed on z(m) to make the series on the right hand side converge.
Then di¤erentiate with respect to Lc to get
1 Z
@ X bc=z(m) @
@Lc
0
1
(cz(m); )
@b
c X
(c)dc +
@
@Lc
m=0
=
fe
(Lc )2
f
(Lc )2
1 Z
@ X bc=z(m) @
@Lc
0
m=0
=
fe
(Lc )2
(b
c; )
m=0
1
X
m=0
b
c
z(m)
1
@b
c f X
+
@Lc Lc
m=0
1
(cz(m); )
@b
c X
(c)dc +
@
@Lc
(b
c=z(m))
z(m)
(b
c; )
m=0
1
f X
(Lc )2 m=0
(b
c=z(m))
z(m)
f
Lc
(b
c=z(m))
z(m)
b
c
z(m)
By (11), the term between square brackets equals zero while again (12) implies
fe
(Lc )2
1
f X
(Lc )2 m=0
b
c
z(m)
R bc=z(m)
m=0 0
P1
=
(cz(m); ) (c)dc
Lc
Exploiting these results, (E.4) can be restated in terms of elasticities as
1 Z
@ X bc=z(m) @
@Lc
0
m=0
(cz(m); )
(c)dc =
@
39
R bc=z(m)
m=0 0
P1
(cz(m); ) (c)dc
Lc
(E.4)
1 Z
@ Lc X bc=z(m) @ (cz(m); )
@Lc
@
(cz(m); )
m=0 0
Z
1
b
c=z(m)
X
=
(cz(m); ) (c)dc
(cz(m); ) (c)dc
m=0 0
=Lc
=Lc (
( b
c)
b
c) =
1 Z
X
m=0 0
b
c=z(m)
(cz(m); ) (c)dc =
=
1 Z
X
m=0 0
R bc=z(m)
m=0 0
P1 R bc=z(m)
m=0 0
P1
(cz(m); ) (c)dc
(cz(m); )
=
=
(c)dc
b
c=z(m)
(cz(m); ) (c)dc
R bc=z(m)
(cz(m); )
m=0 0
P1 R bc=z(m)
(cz(m); )
m=0 0
"p (x ( cz(m)))
P1
where the last equality is granted by (E.1). The restriction "p (x ( v)) 2 (0; 1) implies
(c)dc
(E.5)
(c)dc
=Lc
2 (0; 1).
This proves point (i) in the proposition.
To determine the sign of
@
b
c=Lc (
b
c), di¤erentiate (11) with respect to Lc and get
c
@ (b
c; ) @
(b
c; ) @b
+
=
c
@b
c
@L
@
@Lc
f
=
(Lc )2
(b
c; )
Lc
where the last equality is also granted by (11), or in terms of elasticities
=b
c b
c=Lc
+
=
=Lc
=
1
After substituting for (E.1), (E.2) and (E.5), this equation can be solved for
2
c
"p (x ( b
c))
"p (x ( b
c)) 4
=L
c) =
=
1
b
c=Lc ( b
1 "p (x ( b
c))
1 "p (x ( b
c))
P1
R bc=z(m)
b
c=Lc
to yield
3
(cz(m); )
m=0 0
"p (x ( b
c)) (c)dc
5
P1 R bc=z(m)
(cz(m); )
(c)dc
m=0 0
"p (x ( cz(m)))
The sign of this expression is dictated by the sign of the term in square brackets given "p (x ( v)) 2
(0; 1). As x ( v) > x ( b
c) for all v < b
c, then "p (x ( v)) > "p (x ( b
c)) for all v < b
c when
"0p (x ( b
c)) > 0. In this case,
b
c=Lc (
b
c) < 0 follows. Vice versa,
b
c=Lc (
b
c) > 0 follows from
"0p (x ( b
c)) < 0. This proves that larger Lc = Lw reduces (increases) the cost cuto¤ when the
elasticity of inverse demand is increasing in consumption, i.e. when (B1) holds. This proves point
(ii) in the proposition.
The e¤ects of a demand shock on intensive margin reallocations can be assessed as follows.
Start with point (iii) in the proposition. By result (e), larger Lc = Lw increases the marginal
utility of income . Hence, the impact of larger Lw on output, revenue and employment ratios
has the same sign as the impact of
on them.
40
Consider …rst the impact on the output ratio xv =xv :
d
xv
xv
d
dxv
d xv
=
v
xv dx
d
(xv )2
The sign of the right hand side is dictated by the numerator. Implicit di¤erentiation of (4) can be
used to substitute for dxc =d and dxv =d , rewriting the numerator …rst as
c
(xv )
0 (x ) xc
v (xv )
v
(xv )
xv
0
(xv ) (xv )
and then, using (4) and the de…nition of the elasticity of marginal revenue in (8), as
"r (xv )
"r (xv ).
Repeating the same argument for any pair of values xv and xv such that xv > xv implies that
d xv =xv =d > 0 as long as "0r (xv ) > 0, which is the case under (B2).
Consider now impact on the revenue ratio
d rv =rv
@rv @xv
=
d
rv =rv
@xv @ rv
@rv @xv
= "r (xv )
@xv @ rv
rv
"r (xv )
rv
By result (a), we have xv > xv and rv > rv . Then, "0r (xv ) > 0 implies d rv =rv =d > 0.
Last, consider the impact on the employment ratio. Larger
implies a reallocation from the
higher to lower cost products when vxv =vxv > 1 and d vxv =vxv =d
> 0. By result (d), the
employment ratio is larger than one under (B3) "r (xv < 1. Moreover, as v and v are given, the
employment ratio increases with
when the output ratio increases in , that is, when "0r (xv ) > 0,
which is again the case under (B2).
Finally, consider point (iv) in the proposition. To see how changes in
di¤erentiate to obtain
@
(v; )Lc
=
@Lc
=
=
(v; ) @
Lc
@
@Lc
@ (v; )
@ Lc
(v; ) 1 +
@
(v; ) @Lc
(v; ) +
@
(v; ) 1 +
41
=
( v)
=Lc (
b
c)
a¤ect
(v; )Lc ,
which can be rewritten as
@
(v; )Lc
@Lc
Lc
=1+
(v; )Lc
=
( v)
=Lc (
and thus as
Lc =Lc (v;
where
=Lc (
;b
c) = 1 +
b
c) is common across products while
( v)
=
=
=Lc (
b
c)
b
c)
(E.6)
( c) is product-speci…c and is given by (E.1)
both in the short and in the long run. By (E.1), expression (E.6) can be restated as
Lc =Lc (v;
;b
c) = 1
b
c)
"p (x ( v))
=Lc (
with "p (x ( v)) 2 (0; 1). In the long run, by (E.5), we also have
=Lc (
increases (decreases) with Lc for low v and decreases (increases) with
0
(v; )Lc
b
c) 2 (0; 1) so that
for high v as long as
"p (xv ) > (<)0. In the short run, the same results hold when the change in Lc is driven by
@ =@ > 0 implies
F
=Lc (
(as
b
c) > 0).
A positive demand shock increases the productivity of multi-product …rms
By result (d), larger
increases the employment ratio `m =`m0 of a lower cost product m to a higher
cost one m0 . As the employment ratio equals the employment share ratio
`m
=
`m0
by increasing `m =`m0 , larger
`
PM m1
m=0
`m0
PM
`m
1
m=0 `m
=
sm
;
sm0
also increases sm relative to sm0 for any m < m0 . Given that pro-
ductivity levels 1=vm ’s are …xed, a higher share ratio in turn implies that larger
increases average
productivity as this is an average in which the weighting is being shifted from lower productivity
(higher cost) products like m0 towards higher productivity (lower cost) products like m.35
35
Everything would hold weakly if we relaxed the ranking condition v0 < v1 < ::: < vM
42
1
to v0
v1
:::
vM
1.
G
Short-Run Response to a Demand Shock in Closed Economy
To see what happens when
changes in the short run, di¤erentiate (11) to yield
c
(b
c; ) @
@ (b
c; ) @b
+
=
@
@
@b
c
@
@
f
2L
w
Then rewrite (15) as
1 Z
X
m=0 0
b
c=z(m)
1
f X
(b
c=z(m))
Lw
1
N
cz(m)x (cz(m); ) (c)dc =
m=0
and di¤erentiate to obtain
"
1 Z
X
m=0 0
Lw
#
1
X
@x (cz(m); )
@
cz(m)
(c)dc
+
@
@
m=0
P1
f N m=0 (b
c=z(m))
<0
2 Lw N
b
c=z(m)
=
where the negative sign is due to the fact that Lw
fN
f
Lw
b
cx (b
c; ) +
P1
(b
c=z(m)) @b
c
z(m)
@
(b
c=z(m)) > 0 is labor absorbed by
m=0
variable production costs. In matrix notation we can rewrite
2
4
@
(b
c; )
@
R bc @x ( c)
(c)dc
o c @
@
P1
m=0
h
(b
c; )
@b
c
b
cx (b
c; ) +
(b
c=z(m))
z(m)
f
Lw
32
@
@
i 54
@b
c
@
3
2
5=4
Lw f N
f
(Lw )2
P1
m=0
2 Lw N
(b
c=z(m))
3
5
which can be solved by Cramer rule to obtain
@
@
=
f
(Lw )2
@
hP
1
(b
c; )
@
and
m=0
hP
1
b
cx (b
c; ) +
m=0
Lw f N
@b
c
=
@
@
(b
c; )
@
hP
1
m=0
b
cx (b
c; ) +
P1
m=0
2 Lw N
(b
c=z(m))
z(m)
f
Lw
f
Lw
(b
c=z(m)) @
b
cx (b
c; ) +
(b
c) +
i
(b
c=z(m))
z(m)
(b
c; )
@
f
Lw
i
+
@
f
(Lw )2
(b
c=z(m))
z(m)
Lw f N
i
(b
c; )
@b
c
hR
@
P1
m=0
2L N
w
R bc
c @x@(
o
b
c @x ( c)
o c @
(b
c; )
@b
c
R bc
o
(b
c=z(m)) @
c)
i
(c)dc
c @x@(
c)
(b
c;g)
@b
c
>0
(c)dc
(c)dc
T0
where the sign of @ =@ is dictated by (9) and (10), while the sign of @b
c=@ depends on distributional
assumptions. The fact that the sign of @ =@ is positive while the sign of @b
c=@ is ambiguous implies
that result (f) as well as all results in Proposition 1 except (ii) also hold in the short run.
43
H
Response to a Demand Shock in Open Economy
To see what happens to the export cuto¤ when LcD changes due to changing , rewrite (17) as
lD (
clD ;
lD b
x
fD
LcD
D) =
with l 2 fH; F g. Di¤erentiation with respect to LcD gives
@
clD ;
lD ( lD b
D)
@b
clD
which can be restated as
@
@b
clD
=
@LcD
Given @
lD =@ D
< 0, @
clD
lD =@b
clD ;
lD ( lD b
@
@b
clD
+
@LcD
@
D
clD ; D )
lD ( lD b
@ D
@ D
@LcD
x
fD
@ D
=
@LcD
x
fD
LcD
2
2
(LcD )
clD ; D )
lD ( lD b
@b
clD
@
< 0 (as
D)
(v; ) is a decreasing function of v) and @
c
D =@LD
>0
x is large enough. Hence, as
when (A1)-(A2) hold, the export cuto¤ increases with LcD as long as fD
0
x . This leads to
rises, smaller b
cDD for "p (xc ) > 0 is accompanied by larger b
clD for large enough fD
x =
a larger number of exported products MlD
i
(b
clD ) M l .
Aggregate exports from country l 2 fH; F g are
EXPlD =
i
Nl
"Z
1
X
0
m=0
#
b
clD =z(m)
[xlD (cz(m); ) LcD + f ] (c)dc
i
i
where N l is the …xed measure of incumbent …rms in l. Dividing by N l and di¤entiating gives
1 @EXPlD
=
i
@LcD
N
l
+
+
(
"Z
1
X
b
clD =z(m)
0
m=0
"
Z
1
b
clD =z(m)
X
0
m=0
( 1
X
[xlD (b
clD ;
#)
@xlD (cz(m); ) c
@ D
LD (c)dc
@ D
@LcD
#
xlD (cz(m); ) (c)dc
) LcD
m=0
Given @
lD =@ D
< 0 and @
c
D =@LD
(b
clD =z(m))
+ f]
z(m)
)
@b
clD
@LcD
> 0 when (A1)-(A2) hold, the …rst term on the right hand
x is large enough to make @b
side is negative. Hence, we get @EXPlD =@LcD > 0 as long as fD
clD =@LcD
positive and large enough.
44
I
Allowing for the Destination Country to Export
In the main text we have analyzed a situation in which D does not export. This assumption is not
crucial, either in the long or the short run. Consider the long run. With exports from D the free
entry condition would become
"Z
1
X
fe =
0
m=0
"Z
1
b
cDH =z(m)
X
+
+
b
cDD =z(m)
0
m=0
"Z
1
b
cDF =z(m)
X
m=0
0
[
[
[
DD
DH
DF
(cz(m); ) Lw
D
(cz(m); ) Lw
H
Lw
F
(cz(m); )
#
f ] (c)dc
#
x
fH
] (c)dc
#
fFx ]
(c)dc
where the second and third terms on the right hand side are constant for our comparative statics
analysis with respect to . In particular, comparison with (18) reveals that allowing for exports
e to
from D is homomorphic to reducing the entry cost fD
e
fD
1
X
m=0
(Z
0
b
cDH =z(m)
[
DH
(cz(m); )
Lw
H
x
fH
]
(c)dc +
Z
0
b
cDF =z(m)
[
Lw
F
(cz(m); )
DF
fFx ]
(c)dc :
The assumption of D not exporting is also not crucial in the short run. With exports from D
the labor market clearing condition would become
Lw
D =
+
+
i
ND
(
fe +
"Z
1
X
"Z
1
X
m=0
b
cDH =z(m)
0
m=0
"Z
1
b
cDF =z(m)
X
m=0
0
0
b
cDD =z(m)
#
[cz(m)xDD (cz(m); ) Lw
D + f ] (c)dc
#
x
[cz(m)xDH (cz(m); ) Lw
H + fH ] (c)dc
[cz(m)xDF (cz(m); )
Lw
F
+
fFx ]
#)
(c)dc
where the second and third terms on the right hand side are again constant for our comparative
statics analysis with respect to . In particular, as comparison with (19) and (24) reveals, by
introducing another …xed source of employment for D’s workers, allowing for exports from D is
45
)
homomorphic to introducing some sort of …ctitious ‘entry cost’
"Z
1
X
e
fD?
m=0
+
Z
0
0
b
cDH =z(m)
b
cDF =z(m)
x
[cz(m)xDH (cz(m); ) Lw
H + fH ] (c)dc
[cz(m)xDF (cz(m); )
also in the short run.
46
Lw
F
+
fFx ]
#
(c)dc