Trade and the Internal Labor Markets ofMultiproduct Firms
Carsten Eckel and Stephen Yeaple
University of Munich and Penn State University

Abstract
A large empirical literature has shown that multiproduct …rms tend to be larger,
more productive, more likely to export, and to pay higher wages than single product…rms. The exact source of the advantages of these …rms has not been identi…ed,
however. In this paper, we attribute the cost advantage of multiproduct …rms totheir ability to use internal labor markets in a setting of information frictions tobetter compensate more able workers. Our model can account for many of thestylized facts and can provide an interpretation of the growing wage inequalitythat has developed with increased international trade.

PRELIMINARY: Cite with Care1

Introduction

Economic activity is highly concentrated in large conglomerates. Larger …rms tend to
produce a wide range of products, tend to display high levels of total factor productivity,
pay their workers higher wages, and are more likely to be engaged in international tradeand investment (Bernard, Redding, and Schott, 2011). These well documented empiricalregularities raise several critical questions. First, what is the source of the competitive
advantages of large multiproduct …rms? Second, how do these advantages interact todetermine the size and scope of these …rms? Third, how do changes in product markets
caused by lower trade barriers between countries a¤ect the value of these competitive
advantages? Finally, why are resources frequently allocated to competing ends within
the internal factor markets of large multi-product …rms rather than on external factor
markets between …rms?
In this paper, we develop a model in which multiproduct …rms’competitive advantages vis-a-vis single product …rms arises exclusively from their ability to overcome labormarket frictions. We consider an environment in which workers are heterogeneous in
their ability. Prior to employment by a …rm, a worker’s ability is private information.
Subsequent to employment by a …rm, a worker’s ability is observed by the employer but
cannot be veri…ed in a court of law so that an initial employment contract cannot specify
a wage as a function of a worker’s skill. The advantage of multiproduct …rms relative
to single product …rms in our model lies in the fact that the multiproduct …rm has aninternal labor market in which di¤erent product divisions compete for workers.1 The existence of this internal labor market allows multiproduct …rms to attract the most able
workers because it has in place a mechanism for compensating highly skilled workers,
while single product …rms can only commit to pay a worker a wage that re‡ects theaverage skill of its employees.
In our setting, multiproduct …rms face an interesting tradeo¤ in the choice of the1

Here we assume that the …rm can commit to allow the divisions to compete internally for theworkers’services.

1

number of divisions. On the one hand, adding a division has a positive impact on
the Multiproduct …rm’s aggregate pro…t as each division generates a positive level ofpro…t. On the other hand, adding a division makes the multiproduct …rm’s internal
labor market more competitive, driving up the …rm’s marginal cost and lowering pro…ts
across all divisions. The …rm’s optimal scope obtains when the direct e¤ect of adding
another division is exactly o¤set by the inframarginal impact of higher marginal costs.
Our model is able produce many of the stylized facts concerning the features ofmultiproduct …rms relative to those of single product …rms. First, multiproduct …rms’
ability to identify and keep talented workers means that they pay higher wages andexhibit higher labor productivity than single product …rms. Second, although multiproduct …rms pay higher wages than single product …rms, this wage does not entirelyre‡ect worker’s ability, allowing these …rms to enjoy a rent in the form of lower marginal
costs of production relative to single product …rms.2 As such, the sales of the divisionsof multiproduct …rms will be larger than those of single division …rms. Third, althoughmultiproduct …rms do not enjoy any inherent productivity advantage over single product…rms, their marginal cost advantage allows them to pro…tability serve foreign markets
while single product …rms con…ne their sales to the local market.
We then consider the impact of a multi-lateral trade liberalization. Lower variable
costs to international trade lead to a reallocation of resources within multi-product …rms
away from domestic sales to foreign sales and across …rms from single product …rms tomultiproduct …rms. This reallocation has the implication that the real wages of the mostskilled workers rise while the real wages of the least talented workers fall. Interestingly,
aggregate welfare may rise or fall depending on parameter values. This ambiguous e¤ectof trade on aggregate welfare is due to the inherent labor market imperfections analyzed
in the model.
This paper contributes to several strands of the literature. First, the paper contributes to the growing literature that analyzes within …rm reallocations in a multiprod2

Silva (2013) wages and labor productivity increasing in the number of divisions managed by multiproduct …rm.

2

uct …rm setting. As in Eckel and Neary (2010) we analyze a model in which multiproduct…rms are those that have technologies that allow them to engage in “‡exible manufacturing,”…rms may introduce additional varieties at progressively higher marginal costs.3
As in the case of a number of papers in this literature, multiproduct …rms in our modelinternalize the impact of adding product lines. In Feenstra and Ma (2008), and Dhingra(2013) the key internalization issues revolve around cannibalization e¤ects: new products
steal demand from existing products, while in Nocke and Yeaple (2014) internalization
involves the allocation of a …xed endowment of organizational capital within the internal organizational capital market within a …rm. In this paper, …rms internalize theendogeneous e¤ect of added product lines on the tightness of an internal labor market.
Multiproduct …rms are less aggressive adding product lines because doing so bids up thewage that must be paid to their workers.
Second, our paper contributes to the literature on internal versus external labormarkets. As in Greenwald (1986), we explore the role of adverse selection in labormarkets. As in Greenwald, workers that sort into a “common pool” where their talentcannot be veri…ed are paid a wage that re‡ects the average talent within the labor force.
Unlike Greenwald, we explicitly model the internal labor market as a mechanism forovercoming the problems associated with adverse selection. In contrasting external versus
internal labor markets our work is related to a recent literature on the bene…t of …rm
size when labor is allocated within the …rm (e.g. Papageorgiou, 2012). In Papageorgiou(2012), larger …rms are more diverse in the nature of tasks that they need done. Because
workers have idiosyncratic (and initially unknown) task abilities, a large …rm o¤ers has
a productivity advantage over a small …rm and this advantage is partially passed onto
its workers in the form of higher wages. In our model, the relevant dimension of …rm
size is the (endogeneous) number of divisions that it manages as this number determinesthe “tightness”of its internal labor market and thus its reward to talent.
Our paper also contributes to the literature on the e¤ect of trade on wages in the
3

Other papers in this literature include Bernard, Redding and Schott (2011); and Mayer, Melitz, andOttaviano (2012).

3

context of heterogeneous labor. As in Yeaple (2005) ex ante identical …rms choose amongdi¤erent technologies and this choice a¤ects the nature of their work force. We deviate
from this setting in two major ways. First, we assume that there are inherent frictionsin the labor market. Second, high …xed cost …rms have low marginal costs not because
they are using technologies for which skilled workers have a comparative advantage but
because these …rms can commit to compensate workers for at least part of their skill. In
fact, from a purely e¢ ciency point of view, …rms with endogenously low marginal costs
are in fact low productivity producers. While trade leads to a widening income gap inboth settings, the aggregate welfare e¤ects are startingly di¤erent.4
The remainder of this paper is organized into four sections. We begin by specifying
the closed economy version of our model in section two and analyze the equilibrium in
section three. We focus on how multiproduct …rms di¤er along a number of performancemeasures in this discussion. In section four, we allow for trade between two identicalcountries. We extend the model’s implications for cross-sectional performance measures,and conclude the section with a comparative static analysis of trade liberalization. In
section …ve, we conclude.

2

Closed Economy Model

The representative consumer’s preferences are de…ned over a composite di¤erentiatedgood X where the constant elasticity of substitution over varieties of the di¤erentiatedgood is> 1. There are two factors, capital and labor. Each factor is inelasticallysupplied by households. The economy’s endowment of capital is given by K. We normalize the price of a unit of capital to unity. Labor is supplied by workers who areheterogeneous in terms of their skill z. The distribution of skill in the population ofworkers, G, is de…ned on support [z; 1) where z

0. To develop clean comparative

statics, we impose a regularity condition on the distribution of G. Speci…cally, letting
4

Another related paper that focuses on labor market frictions (screening technology and search) ina trade setting is Helpman, Itskhoki, and Redding (2010). Unlike that paper, our paper abstracts fromcomplementarity between …rm’s inherent types and workers skills to focus on the role of …rm scope oninternal labor markets.

4R(a) =Ra
0

G(z)dz, we require2

R00 (a)R0 (a)

R000 (a)R00 (a)R0 (a)> 0, for all a.
R(a)

(1)

This assumption requires the density function associated with G, and given by g(z), to be
decreasing su¢ ciently fast relative to G(z). As shown in the appendix, these restrictions
are satis…ed in the case of the Pareto distribution. We assume that a worker’s ability z
is initially observable only to the worker.
Varieties of the di¤erentiated good can be produced by two di¤erent types of …rms.
First, there are single product …rms (SPF). These …rms pay a …xed capital cost f to
introduce a single variety of the di¤erentiated good. SPFs require one unit of e¤ective
labor to produce one unit of output. Importantly, single product …rms cannot direct
observe the ability of individuals in its workforce. Second, there are multiproduct …rms
(MPF) which are free to produce as many products as they want. Like SPF, MPF
must pay a …xed cost f for every variety introduced. Unlike SPF, MPF must pay an
additional …xed capital cost f m ; which we think of as confering upon the …rm greaterorganizational capability. This greater organizational capability has two features: (1)it allows the …rm to manage multiple products and (2) it allows the …rm to screen its
workers so as to perfectly observe each worker’s skill.5 As in Eckel and Neary (2010)
we assume that each …rm has a ‘core’variety and that each new variety introduced is“further” away from the core variety than the previous variety. We use ! to index the
distance of a variety from the …rm’s ‘core’variety (so the core variety has ! = 0). We
capture the relative ine¢ ciency of production of non-core varieties as an e¤ective unit
labor requirement given by (!), where (0) = 1,0(!) > 0.

Markets in our model display two imperfections. First, as is usual in the trade
literature, the di¤erentiated good is produced by monopolistically competitive …rms that
5

The …rst part of this assumption, that MPF can manage multiple products, is made for expositional
convenience and can be relaxed. The second part, that paying f m allows the …rm to observe worker
skill, is critical in what follows.

5charge a mark-up over marginal costs. We assume that all …rms, including MPFs, are too
small to have an impact on the aggregate price index. Second and unique to our setting,
SPF cannot condition wages paid to individual workers on the basis of their skill. MPFs
also cannot write contracts based on a worker’s skill ex ante but are able to reward hightalent workers because they have competitive internal labor markets. We assume thatonce a worker has been employed by an MPF, the worker’s skill is observed and that theindividual divisions (associated with a given variety) can commit to compete with one
another for labor within a perfectly competitive internal labor market.6
The timing of actions is as follows. First, we assume that …rms choose to enter either
as SPF or MPF, buying capital in the process. Second, independently of their skill types,
workers search in undirected fashion for MPF. Each worker meets a single MPF. Upon
meeting their MPF they decide whether to accept employment at the MPF. At thattime, no wage is promised. Instead, accepting a job at an MPF is accepting a place in
the internal labor market of the MPF. Those that do not accept a job at an MPF willthen enter a common labor market for SPF …rms. Third, MPF observe the e¤ective unitsof labor available in their internal labor market and decide how many divisions(varieties)
to establish.7 As the marginal cost of production is rising in the distance from the corevariety, this choice is akin to choosing the maximum distance variety ! d from the corevariety to produce. At this time additional …xed costs are also incurred by MPF andthe capital market must clear. Fourth, internal labor markets clear at MPF and thecommon SPF labor market clear. Finally, …rms sell their output.
6

In future versions of this paper, we will buttress this feature of the model by introducing a timeelement to the model. This will allow us to justify the ability to commit to allow competition between
divisions by a desire to maintain the …rm’s reputation.
7
This timing assumption makes the model much easier to analyze. It can be justi…ed by appealing
to the possibility in the real world that a …rm may close a division after individual workers have made…rm-speci…c investments.

6

3

Equilibrium in the Closed Economy

We solve the model backwards beginning with product market competition. We thenconclude the section with an analysis of the model’s predictions over the di¤erences
between MPF and SPF and a discussion of the features of technology that govern thesize of the role of MPF in equilibrium.
Product Market Competition Letting w(z) be the equilibrium wage received
by a worker of skill z, aggregate expenditure in the economy is given by
E=L

Z 1

w(z)dG(z) + K:

(2)0

Expenditure on an individual variety of the di¤erentiated good is then given by
x(i) = EP

1

Z

1

where
P =

p(i)

;

1

p(i)

di

i2

is the price index andis the set of varieties available.

As each …rm is by assumption too small to e¤ect the aggregate price index and eachfaces iso-elastic demand for each variety, each …rm charges a constant mark-up overmarginal cost. If c(i) is the marginal cost of producing variety i, then the optimal pricecharged by the producer of variety i is p(i) = c(i)=(1) and the reduced form demandfor variety i will be given by
x(!; c) = (
where
A=

EP

1)Ac(i)1

1

1

is the (endogenous) mark-up adjusted level of demand.

7(3)

MPF’s internal labor market At this stage, workers have committed to either
work for the multiproduct …rm (MPF) that they have met or have entered the common
e unitslabor market pool and will work for a single product …rm (SPF). Suppose that Lof e¤ective labor has chosen to work for a representative MPF …rm and that this labor
force is composed of workers that vary in terms of their skill. Once they have become

part of the internal labor market of a MPF, a worker’s skill will be observed by all of the…rm’s divisions and there will be a common valuation of the worker’s skill given by w(z).
As workers productivity is linear in their type, no-arbitrage has the implication that theequilibrium wage by the …rm to a worker of skill z can be written: w(z) = cz, where c isan endogeneous constant that is determined by internal labor market clearing. In fact,c is the price of a unit of e¤ective labor that clears the internal labor market.
Internal labor market clearing for a …rm that has chosen to produce the products one of e¤ective labor embedded in its work force requires
the interval (0; ! d ) and which has L
Z !d
0

e
x(!; c) (!)d! = L:

where x(!; c) is the total output of a variety that is distance ! from its core competency,! d is the produced variety that is furthest from the …rm’s core competence, and(!)

is the additional units of e¤ective labor required to produce an unit of a variety that is
distance ! from the …rm’s core competence. Hence, the marginal cost of producing avariety that is distance ! from the …rm’s core product is c (!). Substituting for x(!; c)using (3) allows us to rewrite the internal labor market clearing condition as
(

1)Ac

Z !d
0

(!)1

e
d! = L:(4)e
Everything else equal, the price of labor in an MPF is decreasing in the supply of labor L,

increasing in product market demand A, and increasing in the range of goods producedby the …rm ! d .

8

MPF’s choice of product scope We now consider the …rm’s choice of the numberof varieties to produce. Given …rms’optimal pricing strategies, the reduced form pro…ts
that a MPF could earn from variety that is distance ! from the core product is given by(!) = Ac1(!)1f

Aggregating over all varieties produced by the MPF and accounting for the initial entry
cost, the total pro…ts earned by a MPF are given by(1= max Ac
!d

Z !d

(!)

1

d!

d

! f

f

m

0

)

(5)

;

where c is endogenous and determined by the …rm’s internal labor market clearing condition, given by (4). The …rst-order condition for the optimal choice of product range,! d , isd 1

Ac(! )

d 1

(! )

f

(

d

1)Ac(! )

Z !d

(!)1

0

d!

@c
=0
@! d

where the term in brackets is the direct e¤ect of having an additional product line andthe second term is the inframarginal of the higher labor costs on incumbent product lines.
Expanding along one dimension necessarily results in a “cannibalization-like” e¤ect on
incumbent products due to the tighter internal labor market conditions. Critically, thesecosts are internalized by the …rm. By totally di¤erentiating (4) with respect to c and! d , substituting the resulting expression into the …rst-order condition, and simplifying,
we obtain
Ac(! d )1(! d )1(6)= f:

The inframarginal e¤ect can be seen at work in this expression as the pro…ts earned
by the marginal product line ! d are strictly positive (i.e. Ac(! d )1(! d )1

f > 0).

Given their captive labor force, …rms exploit their monospony power by restricting thenumber of divisions that they manage. The size of this e¤ect is mediated by ; which9governs the extent to which higher labor costs can be passed onto consumers.
External Labor Markets Having characterized the equilibrium in the MPF and
SPF labor markets given an arbitrary assignment of workers into those markets, we now
step back to the …rst part of the period and derive the sorting behavior of individual
workers on the basis of their type. To begin, recall that workers are randomly matchedwith MPFs at the start of the labor market. The workers anticipate the number of
products to be managed by the MPF and use this information to infer the wage thatthey would receive in the internal labor market of the …rm, which as noted earlier must
be wm (z) = cz for a worker of ability z. The alternative to becoming employed by the
MPF is to reject the job o¤er and instead to enter the common labor pool for SPF. As
SPFs cannot condition their wage on a worker’s skill, perfect competition means thatthey must be paid a wage that re‡ects their average productivity. Hence, workers in thecommon labor pool for SPF receiving a wage of ws = cs z, where z is the average abilityof workers that have chosen to enter the SPF labor market pool and cs is the endogenous
cost of a unit of e¤ective labor to a SPF. In equilibrium a worker’s wage as function ofher skill is given by
w(z) = maxfcz; cs zg:(7)

As Figure 0 makes clear, the wage of a worker is strictly increasing in her skill in MPF
and is constant in SPF. Hence, there must exist z~ that satis…es wm (~z ) = ws such that
all workers with z > ze choose to accept a job at the MPF that they met while all otherschoose to enter the common labor pool. Substituting for the wage schedules, we obtaina key equilibrium condition of the model:cs z(ez ) = zec:

where
z (~
z)

1
G (~
z)

Z z~

10

z

zdG (z) :(8)

No frictionw(z)With MPF

No MPF

Skill, zNote that the assumption given in (1) implies that z (~z ) is strictly concave in z~ so that
there is must be a unique relationship between z~ and c=cs .
Given the sorting behavior implied by equation (8), we can fully characterize labormarket clearing. If there are n SPFs that have entered, then labor market clearing in
the common labor pool can be writtenn(1)A (cs )

=L

Z ze

zdG(z):(9)z

Turning to labor market clearing among MPF, undirected seach has the implication that
L=m workers are matched with each …rm.8 These workers will have the same distributionof skill across all …rms as they are randomly matched with MPF. As only workers withz > ze choose to accept employment anticipating that they will earn a wage in the internal

labor market of the …rm that exceeds the wage they can get in the common pool labormarket for SPF. Hence, the total supply of e¤ective labor available to a MPF will be

8

e= L
L
m

Z 1
ze

zdG(z):

In fact, as workers cannot observe each others’skill they cannot coordinate on which …rm to apply
for a job. Hence, the composition of the work force must be the same for each MPF even if there is no
search process.

11and aggregating over the individual labor markets for MPF, we have
m(

Z !d

1)Ac

(!)

1

d! = LZ 1

zdG(z):

(10)

ze

0

Free Entry MPF and SPF can coexist in equilibrium because they have distinct
advantages over employing di¤erent types of workers. For MPF, the high …xed cost ofentry is compensated by earning a rent on the high skill workers. While SPF have higher
marginal costs, this disadvantage is o¤set by their low …xed costs of being in operation.
Firms enter until MPF and SPF …rms each earn zero pro…ts. For MPF, this implies thatthe pro…ts given in equation (5) must equal zero so that
1

Ac

Z !d

(!)1

d! = ! d f + f m :

(11)

= f:

(12)

0

Similarly, zero pro…ts for SPF requireA(cs )1

As is the case in the standard Krugman model with CES preferences, the size ofa SPF is pinned down by the free entry condition. This is not the case, however, for
MPF as they optimally choose the number of varieties to produce. Instead, their size ispinned down by the …rst-order condition for the choice of the number of products. The
collective mass of varieties depends on the choices of MPF …rms and entry of both typeof …rms and capital market clearing:nf + m ! d f + f m = K:

(13)

De…nition 1 An equilibrium in the closed economy are (1) a number of entrants, m andn, and a …rm scope ! d , (2) marginal costs c, cs and a cuto¤ skill level ze, (3) a mark-up
adjusted demand level A that satisfy the free entry conditions (12) and (22), the …rst
order conditions for …rm scope (6), labor market clearing conditions (9), (10), and (8),12and the capital market clearing condition (13).
MPF versus SPF This section serves two purposes. First, we compare the differences in observable characteristics between MPF and SPF that arise in equilibrium.
Second, we the features of technology that determine the allocation of workers between
these types of …rms.
As noted in the introduction, MPF tend to pay higher wages and exhibit higher
productivity than SPF. As we now show, our model is qualitatively consistent withthese observations. We begin by characterizing the di¤erence between …rms in terms of
the wages paid. To pin down the wage gradiant given in (7) we …rst must solve for c andcs . Combining equations (12) through (13) we obtain the following relationship between
cs , c, and ze:

cs

Z ze

zdG(z) + c0Z 1

zdG(z) =

ze

1K
.
L

(14)

This expression, which we refer to as the combined factor market clearing condition or
FMC, re‡ects the fact that in this model, labor income accounts for a constant fractionof aggregate income due to the Cobb-Douglas expenditure shares and the monopolisticcompetition result that …xed costs account for a constant fraction of …rm costs.
Expression (14) combined with the Sorting condition, given by (8), determine cs andc as functions of z~. Given a cuto¤ z~; the FMC is a linear downward sloping curve in accs space. Changes in z~ lead to a rotation of the curve around the point of intersection

with the 45 line. The Sorting condition c~z = cs z (~z ) is an upward sloping ray from theorigin. Since z~ > z (~z ), the slope of this curve is less than one, and the intersection with
the FMC must be below the 45 line.
We can now state the following proposition:
Proposition 1 Multiproduct …rms pay higher wages and yet face a lower e¤ective costof labor relative to single product …rms.
By being able to commit to reward skill, multiproduct …rms earn a rent on their workforce. That is, the wage premium o¤ered by MPF over SPF for any given level of worker
13skill does not fully re‡ect the worker’s true ability. This rent confers a marginal costadvantage to MPF that compensates these …rms for their higher overhead costs f m .
MPF have lower e¤ective labor costs relative to SPF so that they must have lower
marginal costs in their core product relative to SPF. Unlike SPF, MPF also produce
a range of goods in which they are technically less pro…cient than SPF. The following
proposition shows that despite this technical disadvantage, MPF ultimately have a lower
marginal cost of production in all of their products relative to a SPF.
Proposition 2 Multiproduct …rms sell more units per product than single product …rms.
Proof. The proposition follows immediately from the optimal scope equation (6) and
SPF free entry condition (12):Afdc (! ) =

1
1

<

A
f

1
1

= cs :

Hence, the marginal cost of all products sold by a MPF are lower than the product soldby a SPF an MPF sells more of every good it produces than a SPF.
The fact that MPF tend to appear more productive in all of the products that they
produce relative to a SPF re‡ects the fact that MPF exploit monopsony power within
their internal labor market. Internalizing the “supply side cannibalization”, they restrict
their product o¤ering su¢ ciently that even their lowest productivity product has a lower
marginal cost (but in fact lower inherent TFP) than SPF. The proposition means that
MPF are larger than SPF not only because they can expand along the extensive margin
by adding products, but they also sell more per product than a SPF. This is consistent
with existing empirical evidence (see Bernard, Redding, and Schott, 2010).
We now turn to solving for the allocation of workers by skill to MPF and to SPF.
We begin working the free entry conditions for MPF (11) and SPF (12). Dividing thefree entry condition for MPF by the free entry condition for SPF, we obtain
c
cs

1

Z !d

(!)1

0

14

d! = ! d +fmfusing (6) and simplifying, this condition becomes
1

c
cs

=

(! d )

1(! d )z(e
z ):

1

(15)

:

Now, substituting using (8) we obtainze =Note assumption (1) combined with the

1

1
1

(16)

(! d ) > 1 and limze !z z(ez ) = z imply an

unique ze that satis…es this expression as a function of ! d . Knowing what determines the

optimal scope of a multiproduct …rm is su¢ cient for knowing what determines the shareof the labor force employed at MPF. This relationship is summarized in the following
proposition.
Proposition 3 The share of the labor force employed in MPF, summarized by 1

G(e
z ),

is independent of a country’s size and its capital stock. It is strictly decreasing in the…xed cost of managing a multiproduct …rm, f m , and strictly increasing in the …xed costof adding an additional product, f .
Proof. From equation (16), it follows that ze is strictly increasing in the optimal scopeof a MPF, ! d . The optimal scope of a MPF can be solved for directly by dividing thethe free entry condition (11) by …rst order condition (6) to obtain
d

(! )

1

Z !d

(!)1

d!

!d

fm
=0
f

(!)1
(! d )1d!

00Di¤erentiatingwith respect to ! d , we obtain@ (! d )=(@! d1) 1 +R !d0(! d )(! d )!

>0Note that lim!d !0 (! d ) < 0 so to establish that there is an optimal equilibrium ! d we15need to establish that lim!d !1 (! d ) > 0. We havelim

! d !1(! d ) = lim! d !1"!d= lim ! d
! d !1(! d )

1

R !d
0

(!)1

d!

!d

lim

(! d )

1

R !d
0

(!)1

!#

d!

!d

! d !1

fm
f

1

!

fm
f

1

Applying l’Hopital’s rule to the terms in brackets on the last line, we obtainlim! d !1(! d )

1

R !d
0

!d

(!)1

d!

=

+ (

1) lim! d !1

R !d
0

(!)1
(! d )1d!

(! d )>(! d )

0Hence, lim!d !1 (! d ) > 0. As ! d is independent of the size of the labor force, so too is
the share of labor employed by MPF.
In the closed economy, the share of MPF relative to SPF depends only on technological parameters such as the share of ( ) and the relative size of …xed costs f m =f , theelasticity of substitution , and the distribution of skill in the population as summarizedby G. This last observation has the implication that countries with a greater share ofunobserved skill in the population will have a larger share of production done by MPF
than countries with a smaller share of unobserved skill.
It is worth noting at this point that the equilibrium to our model is suboptimal froma planner’s perspective because MPFs are not e¢ cient producers relative to SPF. Theyuse more capital per variety produced than an SPF and their unit labor requirements
are higher as (!) > 1 for ! > 0. A social planner tasked with the job of maximizing
aggregate welfare would not allow the formation of multi-product …rms. In equilibrium,
however, as high skill workers would not be compensated for their skill by an SPF, theychoose to work for MPF. The MPF is willing to incur high …xed costs because its will
earn some of the rent on the talent in its internal labor market. The following proposition
demonstrates that aggregate welfare would be improved were MPF banned.
Proposition 4 Aggregate output is strictly higher in equilibrium in an economy where
16

MPF are banned relative to the same economy in which MPF are allowed.
Proof. We …rst solve for aggregate output in an equilibrium in which MPF are banned
and then show that this output is higher than in the equilibrium solved for above.
Aggregate output in a no MPF equilibrium is given by
XnoMPF = n

1

Xs :

In an equilibrium with no MPF, we have only n homogeneous single product …rms
producing Xs units of output. From factor market clearing we haven = K=f; andZ
L 1
zdG(z):Xs =n 0
Hence, aggregate output must beKf

XnoMPF =

1

Z 1

Lf
K

zdG(z):

0In an MPF equilibrium, aggregate output must be
XMPF =

m

Z !d

1

X(!)

1

1

d! + nXs;0where X(!) = (

1)Ac

(!)

d! is the output of a variety that is distance ! from

the MPF core competency. Rearranging the expression for aggregate output and substituting, we have

XMPF = Xs

m= Xs m

R !d
0

c1

1

X(!)
Xs
R !d
0

c1s

17

d!

1

+n

(!)1

d!

!

1

1

+n

Free entry for MPF and SPF implies
c1

R !d
0

(!)1

d!

cs1

=

!df + f m
f

and so we can write
XMPF = Xsm(! d f + f m ) + nff1:

Factor market clearing in an equilibrium with MPF requiresm(! d f + f m ) + nf = K
Z
Lf ze
Xs =
zdG(z)
nf 0
and so we have
XMPF =

K
f

Lf
nf

1

Z ze

zdG(z):

0

Suppose that the claim in the proposition is wrong and an MPF equilibrium generates
at least weakly higher aggregate output, or XMPF
K
f

1

Z
Lf ze
zdG(z)
nf 0
Z ze
1
zdG(z)
nf 0

XnoMPF . This requires

Z 1
1 Lf
K
zdG(z)f
K 0
Z
1 1
zdG(z):
K 0

Rearranging this inequality yieldsKm(! d f + f m ) + nf
=
nf
nf
R1
zdG(z)m(! d f + f m )Rzeze
nfzdG(z)

R1
R0ze

zdG(z)

zdG(z)
0

R1= 1 + Rzeze

zdG(z)

zdG(z)
0:0

where we have used factor market clearing on the L.H.S. of this inequality. Finally, using

18

free entry and labor market clearing for both SPF and MPF, we haveR1m(! d f + f m )c ze zdG(z)=:Rnfcs ze zdG(z)0Plugging this into the inequality we obtainR1c ze zdG(z)Rcs ze zdG(z)
0

c

R1
Rzeze
0

zdG(z)
zdG(z)cs :But, in an MPF equilibrium, we have shown that it has to be the case that c < cs .
Hence, we contradicted the claim that an MPF equilibrium generates at least as muchaggregate output as a no-MPF equilibrium.

4

The Open Economy

We now consider a simple open economy version of our model in which two identicalcountries trade varieties of the di¤erentiated good. Given the symmetry of the problem,the extension is relatively straightforward. The remainder of this section is organized asfollows. We …rst introduce the additional assumptions governing international trade. We
then explore the optimal exporting decisions of SPF and MPF in an open economy equilibrium and update the critical equilibrium conditions accordingly. Finally, we consider
the e¤ect of a reduction in variable trade costs between countries on resource allocation
both within and across …rms and on the distribution of income.

4.1

Assumptions

We assume that at the time that a …rm chooses how many products to manage it alsochooses how many of its goods to export. Further, we assume that exporting requires
the …rm to incur …xed cost fx (again in terms of the numeraire, capital) and icebergtype shipping costs per unit exported> 1. Following Melitz, we make the following

19restriction on parameter values:
fx

1

(17)

>f

As in Melitz, this assumption is necessary to generate varieties that are produced but
not traded. We begin our adjustments of the model to the trading environment by …rst
considering the optimal export choices of SPF and MPF.

4.2

Open Economy Equilibrium

We begin this section by investigating the exporting decisions of SPF and MPF. Then,after having characterized the optimal behavior of …rms as a function of industry levelvariables, we update the equilibrium conditions that determine these variables.
Lemma 1 Single product …rms do not export.
Proof. Given the …xed (fx ) and variable trade costs ( ), the potential additional pro…ts
that a SPF could earn are given by(cs ) = A ( cs )1

fx :Suppose that SPF …rms exported. Then, they must break even in equilibrium so that
A(1 +

1

)c1s

= f + fx :But, assumption (17) implies that …rms that do not export make positive pro…ts whichcannot happen with free entry. To see this, suppose they did not make positive pro…ts.
Then, Ac1s

f: From the zero pro…t condition, we have fx

f 1, which violates

the assumption in (17).
We now consider the pro…ts that MPF could earn by exporting a variety that is
distance ! from its core product. These pro…ts are(!; c) = A [ c (!)]1
20

fx :

The pro…ts of a MPF that operates varieties on the interval (0; ! d ) and exports on thevarieties on the interval (0; ! f ) are then

Z !d

= Ac1

(!)1

d! +

1

Z !f

(!)1

d!

0

0

!

!df

! f fxf m:(18)

The internal labor market clearing condition of …rm that operates varieties on theinterval (0; ! d ) and exports on the varieties on the interval (0; ! f ) now becomesZ !dx (!; c) (!) d! +

Z !f
0

0

e
x (!; c) (!) d! = L:

Substituting for x (!; c) and for x (!; c) using (3) and simplifying, the internal labormarket clearing in the open economy becomes

(

1) Ac

Z !d

(!)1

d! +

1

0

Z !f
0

(!)1

d!

!

e
= L:

(19)

As in the case of the closed economy, the choice of the optimal product range to
produce and to export depends on how the marginal cost of production is a¤ected throughthe internal labor market. The …rst order condition for the range of domestic production
continues to be given by equation (6), while the …rst order condition for the optimal
range of exports (assuming provisionally for an internal solution) is implicitly given by
Ac1

1

!f

1

= fx :(20)

We are now in a position to characterize the conditions under which MPF export a setof goods.
m such that all MPF
Lemma 2 There exists a level of the …xed entry cost for MPF fcm.
export if f m > fc

21

Proof. We establish that for su¢ ciently large entry cost that an MPF would optimallydeviate from an equilibrium in which MPF did not export. Suppose that MPF did notexport and were breaking even. In a no-export equilibrium, the free entry condition
would be the same as in the closed economy, and given by equation (11). Further, theoptimal product range condition would also continue to be given by equation (6). For a
MPF to not want to export it would have to be true that an increase in the pro…t due toexporting the core variety (! = 0) would be less than the additional strain on the …rm’s
internal labor market, orA ( c)1< fx :

Combining the optimal product range condition with this expression, we …nd that forexporting to be unpro…table, we must have(! d )

1

<

fx
f

1

:

From the proof of proposition 3 we know that for non-exporting MPF ! d is strictlyincreasing and unbounded in f m . As the function (!) is strictly increasing, it follows
is strictly increasing and unbounded in f m . Hence, there must exist am such that for f m > f
m MPF would want to deviate from a no-exportccritical level fc

that(! d )

1equilibrium.

Henceforth, we restrict attention to equilibria in which f m is su¢ ciently large to guarantee that MPF export. We now characterize the range of products that are produced
for the local market and that are exported. Using equations (6) and (20), we have
!f

1

=

f 1
fx

!d

1

:

(21)

The parameter restriction given by (17) guarantees that the range of goods exported isstrictly smaller than the range of goods sold domestically: ! f < ! d .
Finally, as was the case in the closed economy, the …rst order conditions for the cuto¤
product combined with the free entry condition for MPF pins down the range of products
22

produced and exported. In the open economy case, the free entry condition for MPF
becomes
Ac1

Z !d

(!)1

d! +

Z !f

1

0

(!)1

d!

0

!

= ! d f + ! f fx + f m :(22)

Using equations (6) and (20) we can rewrite this expression as
f

Z !d
0

1

(!)
(! d )

d! + fx

Z !f(!)(! f )

0

1

d! = ! d f + ! f fx + f m(23)

The optimal range of products produced and exported can be derived from equations(21) and (23). Note that equation (21) implies a positive relationship between ! d and! f while equation (23) implies a negative relationship.
As the sorting condition continues to be given by equation (8), the labor market
clearing condition for MPF must only be amended to include exported goods, or

m(

1) Ac

Z !d
0

1

(!)

d! +

1

Z !f

1

(!)

d!

0

!

=L

Z 1

zdG (z) :

(24)

z~

Finally, as MPF exporters require capital to export their products, we must adjust
the capital market clearing condition to account for this additional source of demand.
The open economy capital market clearing condition is now given by
K = m f m + ! d f + ! f fx + nf

4.3

(25)

The Impact of Falling Trade Costs

We now consider the impact of globalization on labor markets and on the structure of
production across …rms. We begin our analysis with the e¤ect of lower trade costs on theportfolio of products produced by MPF. We then consider the reallocation of resources
across …rms, showing that trade liberalization tends to suppress SPF, lowering the share23

of labor that works for SPF and lowering the number of varieties produced by SPF. We
conclude the section with a discussion of the impact of trade liberalization on the real
income of workers of di¤ering levels of skill.
As in many of the models of multiproduct …rms in the literature, trade induces …rms
to expand the range of goods that they export and to consolidate the number of varieties
produced by the …rm. This result is shown in the following proposition.
Proposition 5 A reduction in trade costs induces …rms to consolidate the range of products produced (! d falls) and increases the range of products that the …rm exports (! fincreases).
Proof. The proposition is proven by somewhat tedious computation. Totally di¤erentiating equations (21) and (23) and rearranging yields
d ln ! d
=
d ln

!f

1

Z !f

(!)1

d!"

!f + !f0

d ln ! f=d ln
where " (!)

1

1

!d

1

Z !d

(!)1

!

fx > 0;!

d!"! d + ! d f < 0;

d!"

!

fx

!d + !df0! 0 (!)= (!) > 0 and
="

!d

!f

1

Z !f

(!)1

!f + !f

0

+"

!f

!d

1

Z !d

(!)1

0

>0

d!"

!

If trade costs fall, MPFs expand the range of products exported and reduce theproduct range at home. In our Figure 2, a reduction inleads to an upward rotation of

the Optimal Scope locus, leading to an increase in ! f and a reduction in ! d .

24

Dividing equation for the optimal domestic production range, given by (6), by thecondition for SPF free entry, given by (12), we see that the relative cost di¤erence
between MPF and SPF, c=cs , continues to be given by equation (15). As a decrease in
decreases ! d , it follows immediately that c=cs must fall. Further, using (16), which also
continues to hold in the open economy, it follows immediately that a fall in trade costs
is associated with a decrease in ze as higher relative wages that will be paid by MPFs
attract workers out of SPFs. Mathematically we havez)d ln z~ z (~
=
d ln
z~

1

z~g (~z ) [~z z (~z )]G (~z ) z (~
z)

1

"

!d

d ln ! d
> 0,
d ln

(26)

g(~
z ) [~z z(~z )]where z~G(~< 1 by assumption. We summarize these results in the followingz ) z(~
z)

proposition.
Proposition 6 A reduction in trade costs, , induces a shift of labor out of single product…rms into multi-product …rms.
The proposition tells us that the labor available to SPF must shrink as a result of
falling trade barriers. Not surprisingly, it can be shown that the number of SPF falls
as trade costs fall. Without further assumptions it is not possible to know whether
the number of MPF rises, however. Whether the number of MPF expands or contracts
depend on the shape of the skill distribution G and on the shape of the competency
function . Looking at the capital market clearing condition (25), it is clear that eitherm, ! f , or both must rise as both n and ! d have fallen.
The rising demand for labor by MPF intuitively is associated with an increase in the
reward of skilled workers (workers that were previously employed by MPF) relative tothe reward of unskilled workers (workers that remain employed by SPF). In fact, it is not
only relative wages that move against unskilled workers. To see this, recall that workers
for MPF are paid a wage of w(z) = cz so that their wage (measured in terms of capital)25

Afterw(z)/P

BeforeSkill, zdepend only on the level of c. From (8) and (14) we haved ln c=d ln

z~G (~z)d ln z~R1< 0.
z~G (~z ) + z~ zdG (z) d ln

(27)

As a reduction in transport costs lowers the cuto¤, the wage of workers who previously
worked for MPF must rise. Now consider the wage of workers that had previously
worked for a SPF. These workers recieve a wage of ws = c~z . Totally di¤erentiating thisexpression, and substituting (27), we obtain
R1
zdG (z)d ln ws
d ln z~
z~
R1
=
> 0.
d ln
z~G (~z ) + z~ zdG (z) d ln

The impact of a reduction in tari¤s on the wage distribution is illustrated in the following…gure.
The solid lines correspond to the post trade liberalization wage distribution while
the broken lines correspond to the pre-liberalization wage distribution. Comparing thewages of SPF and MPF workers before and after the trade liberalization, it is clear thatthe gap widens. As in the standard Melitz (2003) framework, a reduction in trade costs
raises demand for labor for export production but lowers demand for local production.
As skilled labor is used by the exporting MPF while unskilled labor is used by the SPF
that serve only the domestic market, trade liberalization widens the wage gap.
26

We conclude with an analysis of the e¤ect of trade liberalization on real income forindividual workers and for the economy as a whole. The following proposition shows that
trade liberalization induces an increase in the real income of incumbent MPF employees
and a decrease in the real income of incumbent SPF employees:
Proposition 7 A reduction in trade costs lowers the real wage of a SPF worker andincreases the real wage of an MPF worker.
Proof. Starting with the free entry condition
A(cs )1=f

From the de…nition of A and the result that E is a constant, we have
cs
=
P
where

(

1)

1

E=f

1=(

1)

is a constant. This expression says that the freeentry condition pins down the real marginal cost facing a SPF. Note that the wage of
SPF worker is just ws = cs z(ez ), so we can write the real wage of an SPF worker asws=P
A reduction in

z(e
z ).

lowers z(ez ), hence SPF workers are made worse o¤. Now consider the

implications for a worker at an MPF. Using the maximization condition for domestic
product range, we have
c
=
P

1
:
(! d )

The wage of an MPF worker before and after trade liberalization is w(z) = cz so thereal wage is then
w(z)
=
P

z
:
(! d )A reduction in trade cost lowers ! d and so lowers (! d ) thereby making the real wage27

of MPF workers rise.
Trade creates winners and losers, but whether the winners gain more than losers losedepends critically on the initial conditions. A reduction in trade costs shift resources tothe relatively less e¢ cient MPF, which tends to lower aggregate output, but it induces
the incumbent MPF to trim their least e¢ cient products, which tends to raise aggregate
output. Which e¤ect dominates depends on how prevalent MPF were in the initial
equilibrium as summarized in the following proposition:
Proposition 8 If trade costs fall, welfare increases when the share of multi-product …rms
is large and decreases when this share is small.
Proof. Aggregate welfare can be written U = X =w
L: As noted earlier aggregate
1P

expenditure is …xed in terms of capital (the numeraire) and so we need only sign thee¤ect on the price index P , but d ln U =d ln P =d ln cs , so we only need to sign theimpact on cs . Because
cs

Z z~

zdG (z) + c

Z 1

zdG (z) = (

1)

Z 1

[~
z

z~

z

K
;
L

it follows that from di¤erentiation thatd ln cs
d ln z~Z z~z

zdG (z) =d ln c z (~z)
d ln z~ z~

zdG (z)z (~z )] z~g (~
z) ;

z~

which is positive for small values of ze and negative for large values of ze. The proposition
then follows directly from the observation that dez d > 0.

Intuitively, when MPF dominate the economy initially, most of the action in labor
reallocation is due to the internal trimming of products of these …rms. When MPF are
not a large part of the economy initially then most of the reallocation of labor is from
SPF to MPF.

28

5

Conclusion

Much of the reallocation of resources due to economic shocks occurs within the boundaries of large multiproduct …rms rather than on independent factor markets. This paper
explicitly models exactly such a phenomenon in the context of industrial conglomerates.
Consistent with the stylized facts, multiproduct …rms in our paper are larger, appear
more productive, are more likely to export, and pay higher wages than single product…rms. This occurs in equilibrium despite the fact that multiproduct …rms are inherently
less e¢ cient than single product …rms. Multiproduct …rms appear to be successful because their ability to reward skill allows them to earn a rent on their skilled employees.
The existence of multiproduct …rms is clearly in the interest of skilled workers because
it allows them to earn higher wages than their less skilled peers, but it is does not meanthat internal labor markets are good for economic e¢ ciency.

6

Appendix: Desirable properties of pareto

The point of this note is to demonstrate that the Pareto distribution with parametersz )=ez is monotonically decreasing. By de…nition, we(1; ) will imply that the function z(e
have
z(e
z)
1
=
ze
zeG(e
z)

Z ze

zdG(z):

1Di¤erentiating with respect to ze, the condition that must be satis…ed is
g(e
z)
G(e
z)

z(e
z)
ze

1
g(e
z)
+
<0
ze G(e
z)
zeg(e
z)
z(e
z)
<
G(e
z ) ze z(ez)

Using the Pareto assumption with G(z) = 1
z(e
z) =

z

1
11

29

and g(z) = z

ze1
ze

:

1

, we haveNow, substituting this into the condition, we have
ze
1 ze

<

1 ze1
1 1 ze1 ze1
1 1 zeze

After substantial manipulation, this simpli…es to
ze
ze

(e
z)

1
1

Using l’Hopital’s rule, it can easily be established thatlim (ez ) = lim ze
ze#1

ze#1

lim (ez ) = lim ze

ze#1

1

= ;

1

= 1:

ze#1

Hence, to establish that z(ez )=ez is strictly increasing, we only need to show thatmonotonic. Di¤erentiating(ez ) is(ez ), we obtain
0

(e
z) =

By L’hopital’s rule, we have limze#1

1+(

0

1)e
z
(e
z 1)2

ze

1

:

(e
z ) > 0. Further, we havez)
we have z(estrictly decreasing.
ze

00

(e
z ) > 0 for ze > 1. So,

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