Trade, Capital Redistribution and Firm Structure
Larry D. Qiu and Wen Zhou∗
The University of Hong Kong
November 6, 2009

Abstract
A model of heterogeneous firms with multiple products and two production factors (labor and capital)
is used to study how trade liberalization affects firms’ choices through both product and factor markets.
Trade liberalization is shown to always redistribute capital toward more efficient firms and always toimprove an industry’s total factor productivity. However, it may reduce capital prices and cause laborproductivity to drop. Low efficiency firms are affected mainly by changes in the factor market, while highefficiency firms are affected mainly by changes in the product market. In response to trade liberalization,
low efficiency firms always reduce their product scope, but high efficiency firms may expand their scope.
The model demonstrates the importance of the interplay between product and factor markets.
Keywords: firm heterogeneity, trade liberalization, multiproduct, multifactor, firm structure, scale,scope, mergers and acquisitions
JEL Code: F12, F13, F15, L11, L25∗
Correspondence: larryqiu@hku.hk and wzhou@business.hku.hk. This is a substantially revised version of an earlier paper
entitled “Globalization, Acquisitions and Endogenous Firm Structure”. We thank Jiahua Che, Jee-Hyeong Park, Alan Spearot,
Wing Suen, Shangjin Wei, Stephen Yeaple, and seminar participants at the Chinese University of Hong Kong, Fudan University,
Hong Kong University of Science and Technology, Peking University, Seoul National University, Shanghai University of Financeand Economics, and the University of Hong Kong for their helpful comments. We also benefited from presentations at the 2008
International Industrial Organization Conference, the 2009 Asia-Pacific Trade Seminars, the 9th Conference of the Society for
the Advancement of Economic Theory, the 2009 European Trade Study Group (ETSG) meeting, the 2009 Summer Workshop
in Industrial Organization and Management Strategy, and the 36th European Association for Research in Industrial Economics(EARIE) conference.

1

1 Introduction
Recent studies of international trade have focused on how trade liberalization improves overall efficiency
by redistributing resources across heterogenous firms.1 These studies have shown that international tradeexerts its impacts through intensified competition in the factor market or in the product market. In his
seminal paper, Melitz (2003) demonstrated how, by bidding up the wage rate, trade liberalization forces theleast efficient firms to quit. The intensified competition in the product market, however, does not play any
role, as a CES preference was used.2 This is changed in Melitz and Ottaviano (2008), who used a linear
demand function and showed that exit of the least efficient firms was caused by tougher competition in the
product market. But in that model, the factor market was dormant because the wage rate was assumed to be
fixed.
In this research, we studied the joint impacts of intensified competition in both the product and factor
markets. New results and new forces emerge when both markets are at work. In particular, we found that
intensified competition in the product market may reduce the factor price, and substitution between factors
may cause labor productivity to drop. Firms respond to changes in the product and factor markets in a nonuniform way: less efficient firms are affected mainly by changes in the factor market, while more efficientfirms are affected mainly by changes in the product market. In response to trade liberalization, low efficiency
firms always shrink the scope of their products, but high efficiency firms may expand it.
In our model, heterogeneous firms produce differentiated products using two inputs, labor and capital.
While labor is readily available at an exogenous wage rate, industry-specific capital is in fixed supply andthe capital price is endogenized. Demand is linear a la Melitz and Ottaviano (2008), so competition in
the product market also affects firms’ choices. Firms play a two-stage game. In the first stage, they trade
capital in a perfectly competitive acquisition market before allocating capital to the production of multiple
products. Each firm chooses the number of products (i.e., its scope) and the amount of capital applied toeach product (i.e., the scale). In the second stage, all products compete monopolistically in the productmarket. Each firm’s structure (i.e., scope and scale) and the distribution of inputs and outputs across firms
will be determined by the tradeoff between the product and acquisition markets.
We start with two identical countries originally in autarky due to, say, prohibitively high trade costs. A
shock then completely wipes out all the trade costs. This trade liberalization affects the acquisition market
both directly, and indirectly through the product market. The exporting opportunity induces firms to produce
more, pushing up the aggregate demand for capital. This is the direct effect. The increased competitionin each country’s product market reduces demand for each product, depressing the aggregate demand forcapital. This is the indirect effect. We have shown that the relative strength of the two effects depends
crucially on the substitutability between products. When the products are close substitutes, the reduction in
product demand can be so great that the indirect effect dominates, in which case the capital price will drop
as a result of trade liberalization.
1

See the survey by Helpman (2006).
“In the isoelastic case, the demand level has no effect on the cutoff because any shift in demand is offset by entry, in contrast
to the case of linear demand.” (Helpman, 2006, p.603)
22

For individual firms, trade liberalization brings exporting opportunities as well as the challenges of intensified product competition. Although all firms face the same opportunities and challenges, they responddifferently. Low efficiency firms can make only small profits from each product; having exporting opportunity will not add much. They are affected mainly by the intensified competition, so they sell capital andreduce their scope and scale. High efficiency firms, in contrast, are more capable of making use of the exporting opportunity; they buy capital and expand their scope and/or scale. Capital moves from low to highefficiency firms.
Along with capital, labor will also flow toward more efficient firms. The redistribution improves industry
total factor productivity as measured by the joint contribution of labor and capital. For labor productivity,
however, there is an extra effect due to the substitution between the two inputs. Since both inputs are
optimally chosen by each firm and all firms face the same wage rate and capital price, they all choose thesame labor/capital ratio. The ratio will be affected by trade liberalization, which changes the capital price
but not the wage rate. When capital becomes more expensive, firms will substitute labor for capital, morelabor will be employed, and labor productivity at the firm level will drop. Such a substitution effect (withineach firm) is distinct from the resource reallocation effect (across firms), so even when firm-level laborproductivity drops, industry-level labor productivity may still improve.
This analysis clarifies the fundamental tradeoff that shapes firms’ choices. In models where the impactof trade liberalization is through the factor market, the tradeoff is between product profits and various fixed
costs (Melitz, 2003). In models where the impact is through increased competition in the product market,
the tradeoff is between product profits and variable costs (Melitz and Ottaviano, 2008). In both cases,
trade liberalization has a uniform impact on firms (i.e., firms’ product market profits all move in the same
direction), and all adjustments take place at the margin: The marginal exporters expand, reducing the productdemand or pushing up the factor price, which in turn forces the marginal producers to quit.3 In our model,
both product and factor markets are at work, and the fundamental tradeoff faced by heterogeneous firms
is better viewed as the one between products and resources.4 Firms differ in their efficiency in turning
resources into products. While a resource is equally costly to all, it generates different revenues in the
product market depending on a firm’s efficiency. Trade liberalization brings shocks to both markets andchanges the tradeoff between products and resources. Further, the change depends on a firm’s efficiency,
leading to non-uniform responses to trade liberalization.
While the study of heterogeneous firms started with single product firms, attention has recently turned
to multiproduct firms. According to Bernard, Jensen and Schott (2005) and Bernard, Redding and Schott
3
Given the constant marginal cost of production assumed in all the models, domestic sales and exports are independent andadditive activities governed by different fixed or variable costs. Firms self select into different activities: those with the lowest
efficiencies exit; those with moderate efficiencies produce only for the domestic market; and those with the highest efficiencies
produce for both their domestic market and the export. When trade costs are lowered, exporting profits increase, so more firms
export. This will intensify product competition or bid up the factor price. Because the marginal producers do not export, they bear
the burden of intensified competition in the product or factor market without reaping the benefits of increased exporting profits, sothey are forced to quit.
4
Such a view is not in conflict with the two approaches taken in previous studies, as fixed costs and variable costs are all costsof resources needed in production. Between the two approaches, the one by Melitz and Ottaviano (2008) is closer to ours becauseof the linearity of demand and the absence of fixed costs.

3

(2008), only 41 percent of U.S. manufacturing firms produce more than a single product, but these firms
account for 91 percent of U.S. manufacturing output and 95 percent of U.S. exports. Theoretical modellingof multiproduct firms extended heterogeneity to products within a firm. The basic force remains the same:
trade liberalization eliminates the marginal products, so all firms reduce scope uniformly (Bernard, Redding
and Schott, 2009; Eckel and Neary, 2009; Mayer, Melitz and Ottaviano, 2009).5 In our model, products arehomogeneous within a firm, and the scope is proportional to each variety’s profits. We have shown that afirm’s choice of product scope in response to trade liberalization then depends on its efficiency.
Empirical studies of multiproduct firms have turned up mixed results about product scope. Although
Bernard, Redding and Schott’s (2009) theory predicts uniform reduction in scope, their own (2008) empiricalinvestigation of U.S. manufacturing firms between 1987 and 1997 had different findings: High efficiency
firms increased scope while low efficiency firms reduced scope, which is consistent with our theory. Baldwin
and Gu (2006) found that both exporters and non-exporters in Canada reduced product diversification from1973 to 1997, but the reduction does not seem to have been related to tariff cuts, even though Canada
underwent two rounds of trade liberalization during that period. Iacovone and Javorcik (2008) found that
Mexican firms invested in developing new products for export as a response to trade liberalization. Goldberg
et al (2009) found that Indian firms did not reduce their product scopes during the 1989-2003 period, which
included profound trade and other reforms.6 Our model shows the possibility of and the conditions for
expansions in scope after trade liberalization.
Capital trading was modeled explicitly in this study.7 Such trading enables firms to add or delete production facilities, and thus can be regarded as a partial merger or acquisition. Mergers and acquisitions (M&As)
constitute a major method of industrial restructuring (UNCTAD, 2000) and are the quickest and least costly
way to respond to external shocks such as trade liberalization. Waves of mergers have been documented
as a consequence of trade liberalization and other such shocks (Mitchell and Mulherin, 1996). Breinlich(2008) found that the Canada-United States Free Trade Agreement of 1989 increased domestic Canadian
M&A activity by over 70%. Using data on Swedish firms for the period 1980-1996, Greenaway, Gullstrand
and Kneller (2008) have shown that intensified international competition induced M&As. Maksimovic andPhillips (2001) contended that “industry shocks alter the value of the assets and create incentives for transfers to more productive uses”, and they showed that productive assets tend to move from less efficient tomore efficient firms when an industry undergoes a positive demand shock. Our analysis has now provided atheoretical explanation for these empirical findings.
We will first present the model and analyze the autarky equilibrium. The equilibrium after trade liberalization will then be found and compared with the autarky case. All the proofs are collected in Appendix
5
Baldwin and Gu (2006) reached the same reduced scope conclusion assuming homogeneous products within each firm. Withinfirm homogeneity has also been assumed by Nocke and Yeaple (2006), who further assumed an exogenous tradeoff between productscope and firm-level productivity. They predicted that trade liberalization would induce high efficiency firms to shrink their productlines and low efficiency firms to expand them.
6
Goldberg et al (2009) attributed the discrepancy between their findings and the predictions of prevailing theories to regulationsin India, which prevented the optimal allocation of resources.
7
Spearot (2008) also allowed firms to trade capital, but assumed single-product firms and capital transactions in single unit. His
conclusion that only moderately efficient firms acquire capital is thus very different from our predictions.

4I. Appendix II analyzes unilateral trade liberalization in which a country faces competition from imports ofdifferentiated products but is not allowed to export this industry’s products.

2 Autarky
2.1 Production and inputs
Consider a world with two identical countries, between which there initially is no trade due to prohibitively
high trade costs. In each country, consumption consists of a numeraire good and differentiated products
produced by a continuum of firms in an industry. From a uniform distribution on [0, 1], each firm draws its
efficiency, ϕ, which is the only parameter that distinguishes firms.
A firm can produce multiple products by incurring a management cost, mv2 , where m > 0 and v is thenumber of varieties that the firm produces. Production of the numeraire good requires two units of laborfor one unit of output. Labor supply to this industry is perfectly elastic, so the wage rate, w, is equal to 12 .
Producing differentiated products requires both labor and capital as inputs. The production function of asingle variety is1
q = (ϕxl) 2 ,in which q is the output of the variety, x is its capital input and l is its labor input. In the short run, a plant’s
1capital is fixed, and thus its variable cost is minl (wl) subject to q = (ϕxl) 2 , which gives rise to a labor costw 2
q , orof ϕxq2.
c(q|ϕ, x) =2ϕxEach firm is endowed with one unit of capital. With this endowment and knowing their efficiency levels,
firms play a two-stage game. In the first stage (the acquisition stage), firms buy and sell capital in a perfectlycompetitive acquisition market. If a firm sells all of its capital, it becomes inactive in the product market.
An inactive firm still keeps its ϕ and may choose to re-enter the product market later by buying capital in
the acquisition market. Entry or exit incurs no extra cost. After the capital trading, every active firm chooses
its number of varieties and the allocation of capital to each plant. A firm’s structure therefore consists of
its product scope (the number of varieties) and plant-level capital scale (the amount of capital used in eachplant). For expositional convenience, scale and scope will be treated as continuous variables. Capital is
assumed to be perfectly divisible, and there is no minimum capital requirement for running a plant. In thesecond stage (the production stage), production is carried out and all varieties compete in a monopolisticallycompetitive product market.8
8

We assume away any cannibalization producing one variety has on the profitability of other varieties. Schwartz and Thompson(1986) and also Baye, Crocker and Ju (1996) have shown that, in an effort to gain market share from competing companies orforestall entry, many companies instruct their divisions to act as independent firms despite cannibalization. If this assumption
is relaxed to accommodate cannibalization, our major conclusions still hold. Feenstra and Ma (2008) have demonstrated that5

In this model, capital represents industry-specific physical assets that are needed in production. Unlike
financial assets, after the initial trading round, capital must be acquired through acquisition or merger rather
than from financial institutions. This implies that the supply of capital is perfectly inelastic at the industry
level. This is justified if the total amount of physical assets cannot quickly be increased, but the results still
hold even if new capital can be generated in response to an increase in the capital price. Assuming a perfectlycompetitive acquisition market implies that capital is homogeneous and acquisitions can be partial. That is,
instead of acquiring a stand-alone target firm, the acquirer buys some productive assets from other firms
and uses them along with its own assets. Jovanovic and Rousseua (2002) have argued that transactions in
the used-capital market work just like those in a M&A market. Maksimovic and Phillips (2001) report thatabout half of U.S. M&A transactions are partial acquisitions or divestitures by multi-product conglomerates.
Consider first the product market, then the acquisition market (also referred to as the factor market), andfinally the industry equilibrium.

2.2 The product market
Assume L identical consumers in each country. Each consumer has a quasi-linear preference (à la Melitz
and Ottaviano, 2008) for the numeraire good and all varieties in the industry:Z

1
qi di − βU = Q0 + α
2
i∈Ω

µZ

¶2
Z
1
qi di − γ
q 2 di,2 i∈Ω ii∈Ωwhere α, β, γ > 0, Q0 is the consumption of the numeraire good, Ω is the set of all varieties, and qi is theconsumption of variety i. A consumer maximizes his utility subjectZ to a budget constraint. As a result, theβmarket demand for variety i from all L consumers is pi = α − Lj∈Ω

qj dj − Lγ qi . For a given γ, whenβ is larger, other varieties’ outputs reduce the demand for variety i by a larger amount, meaning that thesubstitution between varieties is stronger. Therefore, β measures substitutability between varieties: larger βmeans stronger substitution.
In monopolistic competition, the seller of variety i regards itself as a monopolist, and the competition
between products is captured only in the vertical intercept of the demand function. In equilibrium, thedemand function for variety i is:

pi = A − bqi , where A =γαγ + βPand b = .
βM + γL(1)

R
In this demand function, pi is the price of variety i, M is the measure of Ω, and P = i∈Ω pi di is theaggregate price of all varieties. The slope b is exogenous, but the intercept A is endogenous, depending
cannibalization would not change the major findings of Melitz (2003).

6

on both the degree of product substitution (captured in β) and the degree of product market competition(captured in the endogenous P and M ).
Each firm takes A as given when choosing its output. If a firm with efficiency ϕ has amount xi of capitalin its plant to produce variety i, it chooses output qi to maximize its profit from this variety:max π i ≡ (A − bqi )qi −
qi ≥0

qi2
.
2ϕxi

(2)

The resulting quantity, price and profit for this variety are, respectively,qi (xi ) =ϕxi A(ϕbxi + 1)A
ϕxi A2, pi (xi ) =, and π i (xi ) =.
2ϕbxi + 1
2ϕbxi + 12(2ϕbxi + 1)

(3)

Greater demand (i.e., a larger A) leads to more output, a higher price and more profit for each variety. Sincea variety’s profit, π i (xi ), is increasing and concave in xi , a firm will always allocate its total capital amongits varieties equally. Consequently, the subscript i can be dropped in (3).

2.3 The acquisition market
Let R be the market price of capital. If a firm chooses scope v and scale x, its capital cost is (vx − 1)R,
where vx − 1 is the firm’s net demand for capital, which can be negative (meaning that the firm is selling
capital). The firm’s optimization problem in the acquisition market ismax Π(v, x) ≡ vπ(x) − (vx − 1)R − mv2 = vπ̃(x) + R − mv 2 ,x≥0, v≥0where π̃(x) ≡ π(x) − xR is the profit of each single plant, taking into account the capital cost but not the
management cost. Given the expression for π(x), a plant’s profit isπ̃(x) =

ϕxA2
− xR.
2(2ϕbx + 1)

It is as if the firm first sells its endowment of unit capital and then chooses how much capital to buy for eachof its plants. Since there is no transaction cost, selling and buying capital is fully reversible.
Given the above decomposition of the profit function, a firm’s optimization problem can be solved in
two steps: The optimal scale is x∗ = arg maxx π̃(x), which is independent of the choice of v, and the
optimal scope is then v∗ = arg maxv Π(v, x∗ ).

7

2.3.1 Plant-level capital scale
Define

A
y≡√ .
2R

(4)
2

The variable y reflects the connection between the product and factor markets. π(x) is proportional to A2 ,A2reflects the value of capital (in the product market) relative to the cost of capital (in the factor
so y2 = 2R
market).
The first-order condition ∂∂xπ̃ = 0 leads to the optimal scale for each plant (the asterisk is dropped forsimplicity of notation):
⎧
⎪
⎪
⎨ 0
for ϕ ≤ ϕ0 ,
(5)
x=
√
⎪
y
ϕ−1
⎪
0
⎩
for ϕ > ϕ ,2ϕbwhere ϕ0 ≡ y12 . The second-order condition is always satisfied. It will later become clear that y > 1 inequilibrium and therefore ϕ0 < 1. Expression (5) says that very inefficient firms will not operate in the
product market. When ϕ is small, a plant’s profit from the product market will also be small, and the firm
will earn a better payoff by selling all its capital and discontinuing production.9 Note that the cutoff point
for exit, ϕ0 , depends solely on y. When the product market is more profitable relative to the capital price,even a low efficiency firm will choose to operate, so more firms will be active in the product market. For
an active plant, the scale increases with y: When the product market becomes more profitable relative tothe capital price, each plant will operate on a larger scale. This is because the marginal benefit of capital is
2
proportional to A2 , while the marginal cost of acquiring capital is R.
Interestingly, more efficient firms do not necessarily have larger scale:⎧⎪√ ⎪2−y ϕ⎨ >0∂x=⎪∂ϕ
4ϕ2 b ⎪⎩ <0forϕ ∈ (ϕ0 , 4ϕ0 ),for

ϕ > 4ϕ0 .

Plant scale increases with a firm’s efficiency when the efficiency is low, but if 4ϕ0 < 1 (or equivalently,y > 2), the scale decreases with efficiency when the efficiency is sufficiently high, generating an inverse
U-shape. The reason is that when ϕ is small, each variety is sold at a high price, where demand is highly
elastic. Increasing x will bring a large benefit. But when ϕ is large, each variety is already sold at a low
9In many formulations, least efficient firms exit due to fixed production costs (e.g., Melitz, 2003). Without fixed costs, exit
happens with a linear demand function (Melitz and Ottaviano, 2008), as the constant marginal costs that some firms draw turn out
to be larger than the intercept of the demand. In our model, demand is linear as in Melitz and Ottaviano (2008) and there is no fixed
production cost. A firm’s marginal cost can be arbitrary small (when its output is close to zero) even if it is very inefficient. It will
always generate some profit, however small, in the product market. A firm exits only because it can earn a better payoff by sellingits capital in the factor market. The tradeoff between product and factor markets is the driving force here.

8price, where the demand is barely elastic. Increasing x will not bring much benefit. Note that the inverse U
relationship does not exist with CES preferences.

2.3.2 Product scope√R(y ϕ−1)2

for ϕ >
Given the optimal scale, a firm’s profit from each variety (after paying for capital) is π̃ =2ϕb
∂Π0
ϕ , which is strictly increasing in ϕ. Given π̃, the first-order condition for optimal scope ∂v = π̃ −2mv = 0π̃orleads to v = 2m
⎧
⎪
⎪
⎨ 0
for ϕ ≤ ϕ0 ,
v=
(6)
√
⎪
2
⎪
⎩ R(y ϕ−1) for ϕ > ϕ0 .
4ϕbm
The second-order condition is always satisfied. Notice that∂v
> 0 for all ϕ > ϕ0 .
∂ϕThus, more-efficient firms maintain larger scopes. The marginal benefit from adding a variety is π̃, while
the marginal cost is 2vm. Since the marginal benefit increases with ϕ, the optimal scope should be larger
when ϕ is higher.
For ϕ > ϕ0 , the scope can be rewritten as

v=

³ √
√ ´2
A ϕ − 2R
8ϕbm

.

(7)

The numerator relates a firm’s product scope to the product and acquisition markets, or more precisely tothe value of capital in the product market vis-a-vis the cost of capital in the acquisition market. The valueof capital is positively related to ϕ, while the cost of capital is independent of ϕ. Therefore, if there is any
change to the two markets due to, say, trade liberalization, firms will respond differently depending on theirefficiencies even though they face the same changes in A and R. In particular, high efficiency firms areaffected mainly by the value of capital and thus by changes in the product market, while low efficiency firms
are affected mainly by the cost of capital and thus by changes in the acquisition market.

92.3.3 Firm-level capital
A firm’s capital is xv. For an active firm,
⎧
⎪
⎪
0
√
R(y ϕ − 1)2∂xv√ ⎨ > 0 for ϕ < 16ϕ ,=(4 − y ϕ)⎪∂ϕ
16ϕ3 b2 m⎪⎩< 0 for ϕ > 16ϕ0 .
Because scope increases with ϕ while scale is inverse U-shaped, firm capital is also inverse U-shaped withefficiency. Note that the turning point for firm capital (at 16ϕ0 ) is larger than that for plant scale (at 4ϕ0 ).
In any case, the important message is that more efficient firms do not necessarily require more capital. Also
note that the envelope theorem indicates that a firm’s profit always increases with ϕ even if its capital scaledoes not.

2.3.4 Outputs
With the optimal choice of x, a plant’s output isAq=2bwhile a firm’s total output is

µ
¶
1
1− √
,
y ϕ

µ
¶3
A3
1
qv =
1− √
.
16b2 my ϕBoth outputs increase with efficiency.

2.3.5 Productivity
Three measures of productivity will be calculated: labor productivity, total factor productivity (or TFP) of
labor and capital, and overall productivity that takes into consideration management costs.
q2
Labor productivity is the ratio between output and labor input. At the plant level, labor input is l = ϕx.
Given the optimal choices of x and q, the plant-level labor productivity is:
q
λ ≡ =
l
ϕ

r

ϕ
.
2R

Since all plants of a firm are identical, the firm-level labor productivity is the same. Capital price R mattersfor labor productivity because it affects the relative price between labor and capital and thus the optimal
10combination of the two inputs in production.
In this model, both labor and capital are used in production, so we need a measure that relates outputs to√the two inputs. Given the production function at the plant level, q = ϕxl, define total factor productivityasq√
ωϕ ≡ √ = ϕ,xlwhich is the same at both the plant and firm levels. Unlike labor productivity, TFP does not depend on
market conditions such as input prices, substitutability between products, or degree of competition. To theextent that firm-level productivity should reflect technologies only and should be independent of changes inmarket conditions, TFP is a more useful measure than labor productivity. Although fixed at the firm level,
TFP at the industry level will change as labor and capital is redistributed among firms, thus providing ameasure of the impact of trade liberalization.
The two productivity measures defined above ignore the management costs of maintaining multiple
products.10 To take that into consideration, define overall productivity as the reciprocal of average cost.
Plant-level cost consists of labor and capital costs, and the overall productivity isqμ ≡=wl + xR
p

r

ϕ
,
2R

which equals the plant-level labor productivity.11 At the firm level, management cost is included and theoverall productivity is
vq
4
´.
= ³
μϕ ≡
vwl + vxR + mv 2
A 1 + √3y ϕ

The overall productivity increases with ϕ at both the plant and firm levels, and the difference between thetwo reflects the impact of management cost.
To summarize:
Proposition 1. Given A and R, capital scale at both the plant and firm levels increases with a firm’s
efficiency ϕ when ϕ is small, but may decrease with ϕ when ϕ is large. Scope, output and productivity all
increase with ϕ.
Since a firm’s scope and each plant’s output are both increasing in ϕ, the two are positively correlated.
That is, intensive margins (i.e., each variety’s output) and extensive margins (i.e., the number of varieties)
are positively correlated, confirming what Bernard, Redding and Schott (2009) have found.
10

We view the management cost as a labor cost, but this labor is skilled labor, unlike the (unskilled) labor that is an input toproduction. More specifically, v 2 can be viewed as the physical units of skilled labor needed to manage v varieties, and m is theexogenous wage rate for skilled labor. Throughout this discussion, labor refers to unskilled labor.
11
This is not an coincidence. Given the Cobb-Douglas production function, the expenditure on the two inputs will be equalized
when labor and capital are chosen optimally. Then wl + xR = 2wl = l given that w = 12 .

11

2.3.6 Capital priceNow to determine the equilibrium capital price R for a given A. Market clearing in the acquisition market
requires total capital supply to equal capital demand at the industryZlevel. Since each firm is initially endowed
1

1dϕ = 1. The total capital demand iswith one unit of capital, the total capital supply in the industry is

0

K≡

Z 1

xvdϕ =

0Rρ, where ρ = 2y3 + 3y 2 (1 − 2 ln y) − 6y + 1.
8b2 m

Given A, it can be proved (see Appendix I) that there exists a unique equilibrium capital price R satisfying the equilibrium condition K(R) = 1, from which R can be derived as a function of y:

R(y) =

8b2 m
.
ρ

(8)

Note that this expression is not a reduced form solution for R, as y is defined on R.

2.4 Industry equilibrium
The measure of varieties is

M≡

Z 1

vdϕ =

ϕ0

R
ψ, where ψ = y2 − 4y + 3 + 2 ln y.
4bm

(9)

Given the choice of x, each single plant’s price and quantity are:
A
p=
2

µ
¶
µ
¶
1
A
1
1+ √
and q =
1− √
y ϕ
2b
y ϕ

for all ϕ > ϕ0 . Therefore, the aggregate price is

P ≡

Z 1

ϕ0

vpdϕ =AR φ, where φ = y3 − 2y 2 − 2 + 3y − 2y ln y.
8bm y

12

(10)

By the definition of A in (1): A = αγ+βP
βM+γ , and using (9) and (10) we have
A=

α
β η
1+ L
yρ

,

(11)

where η = y 3 − 6y 2 + 2 + 3y + 6y ln y. The three equilibrium values A, R and y are jointly determined byequations (4), (8) and (11). Use (8) and (11) in (4) to yield the following equation expressed in terms of y
only:
ρ3
16b2 mZ(y) = 0, where Z(y) ≡ ³
.
(12)
´2 −
α2
β
yρ + Lη

It can be shown that Z 0 (y) > 0, Z(1) < 0, and Z(y) > 0 when y is sufficiently large. Therefore, there
exists a unique y > 1 satisfying (12). Once y is determined, R is determined from (8) and A is determinedfrom (4) or (11). As a result,
Proposition 2. A unique equilibrium A and R exist in autarky.
The industry-level productivity can now be calculated. The industry’s total output is

Q≡

Z 1

vqdϕ =

ϕ0AR η,
8b2 m yand the industry’s total labor input is

Lb ≡

Z 1

ϕ0

A2 R ρ
q2
dϕ = 2.
ϕx
8b m y 2

λ≡

Q
yη
,
=
Lb
Aρ

v

So the industry’s labor productivity is

and industry TFP is
Q
yη
ω≡√
= 3,
KLb
ρ2
where K = 1 is the industry’s capital input. Overall productivity at the industry level is

μ≡

R+

Z 1

ϕ0

Q
(vwl)dϕ +Z 1=
mv 2 dϕ4y η, where θ = y 4 + 12y 2 (1 − ln y) − 16y + 3.
Aθ

ϕ0

13

3 Trade Liberalization: The Impacts of Imports and Exports
Now suppose that trade liberalization reduces all trade costs (fixed and variable) to zero.12 Such a tradeliberalization gives producers in each country both the opportunity to export and the challenge of intensifiedcompetition. We are interested in how firms respond to trade liberalization by adjusting their capital structure, and how productivity changes as a result of resource redistribution. Cross-border M&A is excluded.13

3.1 Equilibrium after trade liberalization
Since the two countries are symmetrical, there must be a symmetric equilibrium. Consider one of thecountries and defineAt
1and ϕ0t ≡ 2 ,yt ≡ √
yt
2Rt
where the subscript t stands for trade. Variables in autarchy are denoted by subscript a. In what follows, wewill use fi to denote f (yi ) for any given function f (y), where i = a, t.
Given At in each country, a firm chooses the quantity of each variety, qt , which is allocated equally tothe domestic and export markets to maximize the variety’s total profit from the two markets:³
q2
qt ´ qt− t .
max π t ≡ 2 At − b
qt ≥0
2 2
2ϕxt

(13)

The optimal output, price and profit are, respectively,

qt =ϕxt At(ϕbxt + 2)At
ϕxt A2t, pt =, and π t =.
ϕbxt + 1
2(ϕbxt + 1)
2(ϕbxt + 1)

A firm chooses its optimal scale for each plant to maximize the plant-level profit π̃ t = π t − xt Rt . The
solution is
⎧
⎪
⎪
⎨ 0,
if ϕ ≤ ϕ0t ,
(14)
xt =
√
⎪
⎪
⎩ yt ϕ−1 , if ϕ > ϕ0t .
ϕb

The firm chooses its scope, vt , to maximize its total profit vt π̃ t (xt ) − mvt2 + Rt , yielding the following
12

If trade costs are positive but small enough to allow for trade, the qualitative results will not change.
Because the two countries are symmetrical, cross-border trading of capital will not happen even if it is allowed. Cross-border
M&A will matter if the two countries are asymmetric or if the trading in the product market is one directional.
13

14

optimal scope
vt =

⎧
⎪
⎪
⎨ 0,

2
√
⎪
⎪
⎩ Rt (yt ϕ−1) , if ϕ > ϕ0t .
2ϕbm

The aggregate demand for capital is Kt ≡market, Kt = 1, leads toif ϕ ≤ ϕ0t ,

Z 1

ϕ0t

(15)

xt vt dϕ = 2bR2tm ρt . Market clearing in the acquisition

Rt (yt ) =

2b2 m
.
ρt

(16)

Z 1

t ψt
vt dϕ = R2bm
, while the aggregate variety
ϕ0t
Z 1t Rt φtconsumed in each country is 2Mt . The aggregate price is Pt ≡ 2
vt pt dϕ = A2bmy. Plug Pt , Mt andt

The aggregate variety produced in a country is Mt ≡ϕ0tαγ+βPtRt into At = 2βM
to yieldt +γ

At (yt ) =

α
β ηt
1+ L
yt ρ

.

(17)

t

Equations (16) and (17) and the definition of yt jointly determine the three unknowns At , Rt and yt as theequilibrium after trade liberalization.
In fact, instead of going through these steps, there is an alternative and much simpler way to obtain theequilibrium. Note that the plant’s maximization problem (13) can be rewritten as¶µbq2
max π t = At − qt qt − t .
qt
2
2ϕxt
This problem is identical to that in the autarky case, (2), except that the demand slope b is replaced by 2b . In
other words, if the equilibrium solutions are p(A, ϕ, x, b), q(A, ϕ, x, b) and π(A, ϕ, x, b) under autarky, thenafter trade liberalization the solutions will be pt = p(A, ϕ, x, 2b ), qt = q(A, ϕ, x, 2b ) and π t = π(A, ϕ, x, 2b ).
It is straightforward to reach the following conclusion: if ya = y(b), Aa = A(b) and Ra = R(b) are
¡ ¢
¡ ¢
¡ ¢
the equilibria in autarky, then the trade equilibria are yt = y 2b , At = A 2b , and Rt = R 2b . Thisconnection between the equilibria before and after trade liberalization proves very convenient in studying
the impact of trade liberalization.
These two approaches are not only mathematically equivalent, but also economically isomorphic. Without any trade cost, a firm views the two product markets (domestic and export) as identical. At the sametime, a consumer is assumed not to treat domestic varieties differently from imported varieties. Comparedto autarky, therefore, a firm’s demand under trade liberalization is doubled (population increases from L
to 2L), which is captured by the change of the slope from b to 2b (because b = Lγ ). From the firm’s pointof view, for given A and R, the change in the slope of the demand curve is the only difference between

15capital scale

product scopeϕϕ

outputϕFigure 1: Impacts of Trade Liberalization
autarky and liberalized trade, so its optimal scale and scope are doubled after trade. Of course, A, R andconsequently y will be different from those in autarky.

3.2 Equilibrium comparison
A proof Appendix I shows that yt < ya . The following proposition summarizes the effects of trade liberalization.
Proposition 3. After trade liberalization, the following changes happen:(1) Product market: the demand in each country drops: At ∈ ( A2a , Aa );(2) Capital redistribution: capital moves from low to high efficiency firms; the least efficient firms exit:
0ϕt > ϕ0a ;(3) Firm structure: low efficiency firms reduce scope and scale, while high efficiency firms expand scope
and/or scale;(4) Aggregate scope: the total number of varieties consumed in each country increases: 2Mt > Ma ;(5) Outputs: low efficiency firms reduce plant and firm outputs, while high efficiency firms increase these
outputs; industry output increases.
Proof. See Appendix I.
Some of the changes are illustrated in Figure 1, where the solid lines represent autarky and the dottedlines represent the situation after liberalization. After trade liberalization, there is more competition in the
product market. More products are sold in each country, so the demand for each variety, represented by A,
drops. Later it will be shown that the capital price R may rise or drop. In either case, due to the increased
16

competition, the product market in each country becomes less profitable relative to the capital price, i.e., ydrops.
Plant capital scale is affected by trade liberalization through two effects. First, because product markets
become less profitable relative to capital prices, each plant will reduce its scale (recall that scale increases
with y). Second, because each variety is sold in both countries, plant scale will be doubled. Low efficiency
firms have small scales; doubling the scale will not add much. So the first effect dominates for low efficiency
firms, and they reduce their scales after trade liberalization. Firms with very low efficiency will sell theircapital and exit. For high efficiency firms, the second effect dominates, so they expand their scale. Depending on the parameters, it is possible that all firms reduce scale, but if a firm expands its scale, all firms withhigher efficiency will also expand their scale.
Product scope is proportional to each plant’s profit, and it is affected by trade liberalization in the same
way as scale. Because the product market becomes more competitive, the profit will drop. Because eachvariety is sold in both countries, the profit will double. Again, the first effect dominates for low efficiency
firms, so they reduce scope; the second effect dominates for high efficiency firms, so they expand scope. It
is possible that all firms reduce scope, but if a firm expands its scope, all firms with higher efficiency will
also expand their scope.
Now consider the redistribution of capital. Low efficiency firms reduce both scale and scope, so theymust sell capital; the least efficient firms sell all of their capital and exit. High efficiency firms, by contrast,
expand scale and/or scope, and they buy capital. It is possible that all firms reduce scale, or all firms reduce
scope,14 but not both at the same time. Otherwise all firms would sell capital, which cannot happen at
equilibrium.
The effect of trade liberalization on the aggregate scope (i.e., the total number of varieties) producedin each country is unclear, as low efficiency firms reduce scope while high efficiency firms may or may
not expand it. Nevertheless, the aggregate scope consumed in each country, produced by both countries, ishigher after trade liberalization. The effect on outputs is not surprising: low efficiency firms reduce plantand firm outputs, while high efficiency firms increase these outputs even when they reduce scale or scope.
Total industry output increases.
The overall picture is clear: trade liberalization brings both exporting opportunities and intensified competition. Facing the same opportunities and challenges, different firms respond differently. High efficiency
firms are more capable of turning inputs into outputs, but in autarky they are constrained by the limited
demand in the product market. Trade liberalization enlarges the markets for their products, and they takethe opportunity to expand. Low efficiency firms, by contrast, have limited ability to take advantage of theexporting opportunity and are affected mainly by the increased competition. Different product/factor tradeoffs lead to different choices: low efficiency firms reduce scale, scope, outputs and even exit, while highefficiency firms do the opposite. Capital moves from low to high efficiency firms.
A prediction of many previous multiproduct models has been that all firms reduce product scope after2

βWhen Lis larger or Lγ2 αm2 is smaller, the cutoff point of ϕ for scale moves to the left (i.e., more firms increase scale) while thecutoff point of ϕ for scope moves to the right (i.e., fewer firms expand scope).
14

17

trade liberalization (Bernard, Redding and Schott, 2009; Eckel and Neary, 2009; Mayer, Melitz and Ottaviano, 2009). The conclusion of this analysis is different: high efficiency firms may expand their scope.
Previous models considered products as heterogeneous within a firm, and the impact of trade liberalization
worked through either the factor market or the product market, but not both. The export opportunity bids
up labor price or raises product market competition, forcing each firm to drop its marginal products, just as
trade liberalization forces marginal firms to exit in single product models. This study assumes products arehomogeneous within a firm and the optimal scope is proportional to each plant’s profit. Trade liberalization
changes a firm’s product/factor tradeoff differently depending on the firm’s efficiency, leading to different
responses in terms of scope. Efficient firms can make better use of exporting opportunities. Their profits
may increase, leading to scope expansion. This points to the possibility that high efficiency firms may expand their scope after trade liberalization. Furthermore, they are more likely to do so when products are lessvaluable (small α) or more differentiated (small β), the market size is small (small L), or managing varieties
is more costly (large m). It is therefore an empirical question as to whether or not high efficiency firms
expand their scope after trade liberalization and, if so, under what conditions.
A major purpose of studying heterogeneous firms is to investigate how trade liberalization changes
industry productivity. We have defined three measures of productivity.
Proposition 4. After trade liberalization,(1) Total factor productivity improves: ωt > ω a ;
q
λϕ
λϕ
λt
a
t
;
(2) Labor productivity has the following properties: λtϕ = R
Rt ; λa > λϕ
a
a
q
p
ϕ
μ
μ
μp
μp
μϕ
a
(3) Overall productivity has the following properties: μpt = R; μtϕ > μpt ; μμt > μpt . In addition, μtϕ
R
t
a
a
a
a
a
a
increases with ϕ.
Proof. See Appendix I.
√
TFP is fixed at ϕ at both the plant and firm level, and therefore does not change after trade liberalization. However, because capital, labor and output redistribute to more efficient firms, the industry’s TFP
improves.
Labor productivity at the plant and firm level is inversely related to the capital price—trade liberalization
improves labor productivity if and only if capital becomes cheaper. This inverse relationship is easy tounderstand. When capital is cheaper, firms will substitute capital for labor, so for the same output, the labor
input will decrease, improving labor productivity. Note that the proportional change in labor productivity is
the same for all firms. This is because, in this two-stage game, all firms choose the optimal combinationsof labor and capital. Because they face the same wage rate and capital price, the combination is the samefor all firms. On the industry level, because labor moves to more efficient firms, the improvement of laborproductivity is greater than at the individual firm level.
Overall productivity at the plant, firm and industry levels are all inversely related to the capital price.
That is simply because capital is part of the cost, so more expensive capital affects overall productivity
inversely. At the plant level, the proportional change in overall productivity depends only on the proportionalchange in the capital price and is therefore the same for all firms. The improvement in firm productivity is
18

μϕ

μp

greater than that at the plant level ( μtϕ > μpt ), indicating rationalization within firms. Because the twoaameasures of overall productivity differ only in the management cost, the rationalization indicates that tradeliberalization is conducive to the management of multiple products. Furthermore, the rationalization isμϕgreater for more efficient firms ( μtϕ increases with ϕ). If overall productivity improves for firm ϕ0 , then itaalso improves for all firms with ϕ > ϕ0 . The improvement in overall industry productivity is also greaterthan that at the plant level.
The three measurements, TFP best captures the redistribution of resources among firms, labor productivity reflects substitution between labor and capital, and overall productivity captures the joint effectiveness ofmanaging varieties and the production of each variety. In the latter two cases, although all the measures depend on the capital price, which may rise or drop, rationalization is still evident in comparisons between themeasures. Therefore, the general picture is that trade liberalization induces more efficient use of resources.
The two inputs, labor is supplied perfectly elastically, so the wage rate is fixed. Capital supply, however,is perfectly inelastic and the capital price is endogenized. So, we have seen, the effect of trade liberalization
depends crucially on changes in the capital price, which may increase or decrease after trade liberalization.
2ββ
is small, and decreases when L
is large while Lγ2 αm2 is small.
Proposition 5. Capital price increases when L

Proof. See Appendix I.
Whether or not the capital price increases depends crucially on product substitutability represented byγβ. Recall that the demand for each variety in autarky is pi = A − bqi , where A = αγ+βP
βM+γ and b = L . Tradeliberalization brings two changes to product demand: A drops, meaning that competition in the productmarket is intensified; and b is halved, meaning that the market size for each product is doubled. If β is verysmall, for example, imagine β = 0 (each variety is a monopoly), then A = α, and trade liberalization will
not reduce A. In that case, every firm produces for a larger market with no more competition. The extra
production raises demand for capital, which bids up the capital price. Conversely, when β is large, products
are close substitutes. Because competition between products is strong, trade liberalization will intensify
competition greatly, leading to a much smaller A. This may dominate the double-market-size effect andlead many firms to sell capital, in which case the aggregate demand for capital drops and the capital price2
declines. Note that for the capital price to drop, in addition to a large β, another condition is a small Lγ2 αm2 ,which implies that y must be small both before and after trade liberalization.15 That is, capital must be
scarce in the general economy.
In this model, the only use of capital is for production, so the capital price relative to the value of the
product produced can be a measure of capital scarcity. On that measure, the effect of trade liberalization isunambiguous: capital becomes more expensive relative to the demand in the product market as y ≡ √A2R
decreases.
15

Recall that ya is solved from equation (12) and yt is solved from the same equation with b being replaced with 2b . In both2

cases, y is increasing in Lγ2 αm2 .

19The finding that factor prices may drop in response to trade liberalization is a surprise. In all previous
studies in which factor price was not fixed (Melitz, 2003; Bernard, Redding and Schott, 2009), trade liberalization was invariably found to raise factor prices. There are two reasons for the difference. Melitz (2003)
derived demand from CES preference and thus had a constant and exogenous markup, and there was freeentry (i.e., repeated drawing of efficiencies) so that any disturbance to the market demand was exactly offset
by entry. In this analysis, by contrast, demand is linear, and there is no free entry to restore A, the aggregate
variable representing the degree of market competition. Note that even when the capital price drops, theleast efficient firms still exit, which was impossible in previous formulations.
Combining propositions 4 and 5, we have
Corollary:2β
is large while Lγ2 αm2 is small, labor productivity and overall productivity improve at all levels,(1) When L
plant, firm and industry.
β
is small, labor productivity drops at the plant and firm levels, but may rise at the industry(2) When Llevel. Overall productivity drops at the plant level, but may rise at the firm and industry levels.
As has been discussed, trade liberalization always improves industry TFP, as resources move from low
to higher efficiency firms. Other measurements depend on the capital price. The corollary says that when thecapital price declines, all efficiency measures improve, indicating a more efficient distribution of resources
both across and within firms. Even when the capital price increases, rationalization is evident in the fact thatthe improvement at the firm and industry levels is greater than the improvement at the plant level.

4 Concluding Remarks
This study looked at heterogeneous firms and the tradeoff between products and resources. Both productand factor markets are at work in shaping firms’ choices. Consistent with prior work in this area, we haveshown that trade liberalization improves industry productivity by redistributing resources toward more efficient firms. However, the joint consideration of product and factor markets produces new results. Factor
prices may drop as a result of trade liberalization, which is impossible if the two markets are considered separately. Although all firms face the same changes in the product and factor markets, they respond differently
depending on their efficiencies. While capital is homogenous and affects firms uniformly (i.e., the cost of
acquiring capital is the same for all firms), the product market affects more efficient firms more heavily than
it affects less efficient firms.
Most prior work has assumed a single production factor such as labor, and hence has focused on laborproductivity.16 Firms in our model use two inputs, and the analysis demonstrates that labor productivity
16
An exception is the work of Bernard, Redding and Schott (2007), who investigated the impact of trade liberalization in a
general equilibrium framework with different relative factor abundance across countries. But they used CES demand functions andtherefore trade liberalization affected firms’ choices only through the factor markets.

20

depends crucially on the substitution between labor and capital. This represents a new channel throughwhich productivity can be affected by trade liberalization. Furthermore, we have demonstrated that thesubstitution between factors depends on the relative factor prices, which in turn depends on the degree of
product substitution in consumers’ demand function.
The bilateral trade liberalization considered here brings exporting opportunities and intensified competition. To further demonstrate how the tradeoff between product and factor markets shapes firms’ choices,
we study in Appendix II a different scenario in which firms face intensified competition without enjoying
the benefit of exporting. This will be the case if trade liberalization is unilateral, i.e. a country imports theindustry’s products but cannot export them. Then the direct effect disappears; trade liberalization affects theacquisition market only indirectly through the product market. Because imports reduce the demand for eachdomestic product, the aggregate demand for capital drops unambiguously, reducing the capital price. In fact,
we show that the capital price must drop more than the demand does so that the product actually becomes
more profitable relative to the capital price. All plants expand scale, but the change in product scope is again
non-uniform. High efficiency firms are affected mainly by changes in the product market, so they reduce
scope and sell capital. Low efficiency firms are affected mainly by changes in the acquisition market, sothey increase scope and buy capital.
To allow the product market to affect firms’ choices, we have followed Melitz and Ottaviano (2008)
in assuming a linear demand function. An alternative is the CES demands assumed by Melitz (2003) andmost subsequent studies.17 A CES demand has the peculiar feature of exogenous and constant markups thatare independent of costs, demands or market sizes. This is not only unrealistic (Helpman, 2006), but also
leads to the special property that firm choices depend only on the ratio between demand and the factor price,leading to uniform impacts of trade liberalization even when product and factor markets are both considered.
With linear demand, firms’ choices will depend on both demand and the factor price rather than the ratio
between the two. Trade liberalization causes non-uniform responses from firms because only the productmarket impact depends on a firm’s efficiency. Admittedly, a linear demand function is highly specific, but a
CES function is equally so. Our model demonstrates that some of the fine details of the implications drawn
from previous models based on CES functions will not survive under alternative forms of demand.
We have assumed that production requires two inputs, labor and capital, which are treated asymmetrically: Labor is supplied perfectly elastically at a fixed wage rate, while capital is supplied perfectly inelastically at a price that is determined endogenously. Our qualitative results will not change if the wage rate isalso endogenized, or if the supply of capital is elastic, as long as capital price is endogenous.
Since two inputs are used, it seems that rationalization between the two inputs may provide anotherchannel for trade liberalization to improve productivity. However, this turns out not to be the case in thisformulation. The substitution between the two inputs in the production function is assumed to be the samefor all firms. Given the optimal choice of capital and labor in the two-stage game and the same wage rate andcapital price faced by all firms, the labor/capital ratio is uniform across firms. In fact, a composite input (atthe optimal labor/capital ratio) can be used in place of the two inputs to generate the same results. Although17

CES preference has also been used by Bernard, Redding and Schott (2009) and Feenstra and Ma (2008), while the linear
demand was used by Baldwin and Gu (2005), Eckel and Neary (2009), and Nocke and Yeaple (2006).

21

trade liberalization changes the capital price and thus the labor/capital ratio, the same ratio applies to all
firms, so there is no rationalization between labor and capital. Nevertheless, such a rationalization will bepossible if firms differ in the labor/capital substitutability of their production functions. This is a refinement
worth further investigation.
We have assumed away the possibility of cross-border capital movement. When the two countries are
symmetric, relaxing this assumption will not change any result for bilateral trade liberalization, as the capital
price will be the same in any case. If the countries are asymmetric, or if trade liberalization is unilateral,cross-border M&A will allow an extra channel through which resources are rationalized. Such a setting will
be particularly suitable for investigating trade liberalization between countries that are asymmetric in size,
preferences, or labor/capital endowments, and represents a second direction for future work.

Appendix I: Proofs
Proof of the existence and uniqueness of R for given A
ρ(y)
0
A2
A2
Since R = 2y
2 , we have K = 16b2 m ξ(y), where ξ(y) = y 2 and ξ (y) > 0. Hence,
∂K
A2 0
∂y
=
ξ (y)
< 0.
∂R
16b2 m∂R
2

When R = A2 , we have y = 1, ξ = 0 and so K = 0. When R is sufficiently close to zero, y is large, whichmeans ξ can be large enough that K > 1. Thus, given A, there exists a unique equilibrium capital price, R,
satisfying K(R) = 1.
Q.E.D.
Proof of Proposition 3
All variables will be expressed as functions of y. We first prove yt < ya . ya is solved from equation
ρ3t
ρ3a16b2 m4b2 m= 0. yt is solved from Zt (yt ) ≡= 0. Obviously,(12): Za (ya ) ≡2 −2
2 − α2
β
β
α
y
ρ
+
η
y
ρ
+
η
( a a L a)
( t t L t)
Zt (y) > Za (y). Then Zt (ya ) > Za (ya ) = 0 = Zt (yt ). Because Zt0 (·) > 0, we have ya > yt .
To compare a particular variable before and after trade liberalization, express the variable as a function
2of y independent of b. To do so, make use of the relations R = 8bρm , A = αβ η (which does not contain
1+ L yρ√√ yb) or A = y 2R = 4b m √ρ (which is linear in b). A comparison is possible because the expression of aparticular variable differs between the two cases only in the value of b. With an expression independent ofb, we need only determine whether the expression is increasing or decreasing in y, as we already know thatyt < ya .
(1) Product market:ηdecreases in y. Therefore, At = A(yt ) < A(ya ) =A(y) = αβ η . It is straightforward to verify that yρ
1+ L yρ√ yt)
Aa . On the other hand, A = A(b, y) = 4b m √ρ , so Aa = A(b, ya ) while At = A( 2b , yt ) = A(b,y. Then2y2At = A(b, yt ). Because √ρ decreases in y, we have 2At > Aa .
22

(2) Capital redistribution:
Since yt < ya , we have ϕ0t > ϕ0a . Therefore, for ϕ ∈ [ϕ0a , ϕ0t ], xa va > 0 while xt vt = 0, i.e., thesefirms sell capital.
³ ´1
√
y ϕ−1
Ra 3
. The left hand
A firm buys capital, i.e., xt (ϕ)vt (ϕ) > xa (ϕ)va (ϕ), if and only if yat √ϕ−1 > 4Rtside of this inequality is increasing in ϕ because yt < ya , while the right hand side is independent of ϕ.
Therefore, if the inequality holds for ϕ0 , it also holds for all ϕ > ϕ0 . Because some firms (at least those
who exit) sell capital, there must be some firms who buy capital. Thus, the threshold of ϕ must be strictly
smaller than 1.
(3) Firm structure:
q
√
y ϕ−1
Ra
. Thus, a firm expands scope if and only if its ϕScope: vt (ϕ) > va (ϕ) if and only if yat √ϕ−1 > 2Rtexceeds some threshold.
√Scale: xt (ϕ) > xa (ϕ) if and only if (2yt − ya ) ϕ > 1. If 2yt < ya , all firms reduce scale. If 2yt > ya ,a plant expands scale if and only if its ϕ exceeds some threshold.
Note that the thresholds for x, v and xv are in general different, and that the thresholds for x and v may
be 1, i.e., all plants reduce scale, or all firms reduce scope, but the two will not happen at the same time, as
all firms would reduce capital, which cannot occur in equilibrium.
(4) Aggregate scope:b
M = M (b, y) = 2bψρ , so Ma = M (b, ya ) and Mt = M ( 2 , yt ), which means 2Mt = M (b, yt ). Becaused( ψ)ρdy < 0 and yt < ya , we have 2Mt > Ma .

(5) Outputs:
³
´
´
´
³
√ ³
A
1 − y√1 ϕ = 2 m √yρ − √ρ1√ϕ . Then qt > qa if and only if √1ρ − √1ρ √1ϕ <Plant level: q = 2b
t

y
√

a

d ρ
√yt − √ya . Because dρ > 0,
ρt
ρa
dy
y

< 0, and yt < ya , we have √1ρ > √1ρ and √yρt > √yρa . Then qt > qatataif and only if ϕ exceeds some threshold. Because firms with ϕ ∈ [ϕ0a , ϕ0t ] exit and therefore reduce their
outputs, the threshold is greater than ϕ0t (i.e., the least efficient firms that survive must reduce plant-level
√
d y−1
√
ρ
√ . Because
output). For the most efficient firm, ϕ = 1 and therefore q = 2 m y−1< 0, we haveρyqt > qa for ϕ = 1, i.e., the most efficient firm must increase plant-level output.
³
´3 ³
´3
3
α
0
Firm level: qv = 16bA2 m 1 − y√1 ϕ = y − √1ϕβ . Then, if qt vt > qa va for ϕ , it must be trueyρ+ L ηfor all ϕ > ϕ0 . So firm-level output increases if and only if ϕ exceeds some threshold. The least efficient3. It canfirms that survive must reduce output. For the most efficient firm, ϕ = 1 and therefore qv = α(y−1)β

yρ+ L η(y−1)3βbe shown that
decreases with y for any positive value of L . Thus, firm-level output increases for the
β
yρ+ L
η

most efficient firm.
yρηαIndustry level output: Q = 8bAR
2 m y = yρβ . Because η increases with y, industry output increases.
η+L

Q.E.D.
Proof of Proposition 4(1) TFP: ω = yη3 , which decreases with y. Because yt < ya , we have ω t > ω a .
ρ2(2) Labor productivity:23Plant and firm level: λϕ =qqyηIndustry level: λ = Aρ
= ηρ
λϕ
Ra
t
=
.
Rt
λϕ
a

λϕ
ϕ
t
=
2R , so λϕ
a

q

q

1
2R . Because

Ra
R .
 t
d ηρ
dy

< 0, we have ηρt > ηρa . Then, λλat = ηρt
t

a

t

.

ηa
ρa

q

Ra
Rt >

(3) Overall productivity:
q
q
μp
ϕ
a
Plant level, μp = 2R
, so μpt = RRt .
aq ya + √3
q
μϕ
μpt
ϕ
Ra
Ra
Firm level, μϕ =  4 3  = √  4 3  . Then, μtϕ =>=. Because3
Rt yt + √Rt
μpaaA 1+ y√ϕ
2R y+ √ϕ
ϕ
q
√
μϕ
μϕ
a ya√ ϕ+3
t
t
= R
and
y
<
y
,
increases with ϕ.
t
a
R
y
ϕ+3
μϕ
μϕ
t t
a
a
q
d( ηθ )
μpt
ηt
ηa
μt
η
Ra
√4 η . Because
Industry level, μ = 4y
=
<
0,
we
have
>
.
Then,
>
=
.
A θ
dy
θt
θa
μa
Rt
μpa
2R θ
Q.E.D.
Proof of Proposition 5
A2
Capital price R(y) = 2y
2 =

2
βη
β
 α 2 . It can be easily verified that y +L ρ increases with y when L isβ η
2 y+ L ρ
2βsmall, but decreases with y when L
is sufficiently large and y is small, which happens when Lγ2 αm2 is small.
Here is a numerical example: When α = β = γ = L = m = 1, Ra = 0.0019 and Rt = 0.0079 > Ra .
When α = 20, β = 100, and other parameters remain the same, Ra = 0.034 and Rt = 0.032 < Ra .
Q.E.D.

Appendix II: Unilateral Opening-up
Trade liberalization brings both opportunities and challenges for firms. This analysis has shown thathigh efficiency firms can make better use of exporting opportunities while low efficiency firms are affected
mainly by the challenges of increased competition from imports. To better understand how firm responddifferently depending on their efficiencies, consider unilateral trade liberalization: a country opens up toimports but cannot export to the other country. The one-way trading involves only differentiated products.
To balance the trade, the numeraire good is assumed to be traded both ways. We focus on the importing
country, which is also referred to as the home country, while the exporting country is referred to as theforeign country.
Firm structure
Let the subscripts h, f and e stand for home, foreign and export, respectively, and define the following
notation:Af
AhAh, yf ≡ p, ye ≡ p,(18)
yh ≡ √
2Rh
2Rf
2Rf
with ϕ0i ≡ y12 for i = h, f, e.
iHome firms face the same optimization problem as in autarky for any given Ah and Rh . Hence, theexpressions of xh and vh can be obtained by replacing A, R and y with Ah , Rh and yh in (5) and (6).

24

In the foreign country, each firm can sell its products in both the domestic and export markets. Supposethat a foreign firm sells qf of a particular variety in its domestic market and qe of the same variety in theexport market. These quantities solve the following optimization problem:max

qf ≥0, qe ≥0

π f ≡ (Af − bqf )qf + (Ah − bqe )qe −(qf + qe )2.
2ϕxf

The interior solution (i.e., when the non-negative constraints are not binding) is

qf =(2ϕbxf + 1)Af − Ah
(2ϕbxf + 1)Ah − Afand qe =.
4b(ϕbxf + 1)4b(ϕbxf + 1)

(19)

Af and Ah are endogenous, and it can be shown³ that ´Af > Ah at equilibrium. Hence qf is always positive,A
1
while qe is positive if and only if ϕxf > 2b Ahf − 1 . That is, a firm exports if and only if its efficiency is
sufficiently high. Thus, even without any (fixed or variable) trade costs, firms with low efficiency sell only
to their domestic market without exporting.18³´
Af
1
So a foreign firm with efficiency ϕ sells in both markets if xf > 2ϕb−
1, and the sales in the twoAhmarkets are given by (19). The corresponding prices and a variety’s total profit in the two markets are

pe =
πf

=

Ah (2ϕbxf + 3) + Af
Af (2ϕbxf + 3) + Ah,
pf =,4(ϕbxf + 1)
4(ϕbxf + 1)
2ϕbxf (A2h + A2f ) + (Ah − Af )2.
8b(ϕbxf + 1)

³
´
Af
1
If xf ≤ 2ϕb−
1, then qe = 0. The firm makes its choices as if there were no exporting opportunity,Ahand the results are the same as in a closed economy:
A2f ϕxf
Af ϕxf
Af (ϕbxf + 1)
, pf =, and π f =.
qf =2ϕbxf + 1
2ϕbxf + 12(2ϕbxf + 1)
Each plant’s profit net of capital cost is π̃ f = π f −xf Rf , where π f is given above depending on whether
18
The intuition is the following. For a firm in the foreign country, the domestic demand (in the foreign country) is higher thanthe export demand (in the home country): Af > Ah . The marginal benefit of selling to the domestic market is therefore higher
than that of exporting if the sales in the two countries are the same. When ϕ (and consequently ϕxf ) is low, output is small. The
marginal benefit of selling the last unit in the domestic market is still higher than the marginal benefit of selling the first unit in
the export market, so the product is not exported. When ϕ is larger, output is larger. After selling some quantity in the domestic
market, the marginal benefit of selling a further unit falls below the marginal benefit of selling the first unit in the export market,
and the firm starts to export.

25

1
or not xf > 2ϕb³´. The firm chooses xf to maximize π̃ f , which yields

Af
Ah − 1

xf =

⎧
⎪
⎪
⎪
0,
⎪
⎪
⎪
⎨ √
y

if ϕ ≤ ϕ0f ,

ϕ−1

(20)

f
if ϕ0f < ϕ ≤ ϕ0e ,
2ϕb ,
⎪
⎪
⎪
⎪
√
√
⎪
⎪
⎩ yf ϕ−1 + ye ϕ−1 , if ϕ > ϕ0 .
e
2ϕb
2ϕb

Note that ϕ0f < ϕ0e because Af > Ah .
Given this optimal scale, a firm chooses its scope, vf , to maximize its profit vf π̃ f − mvf2 + Rf . The
optimal choice is
⎧
⎪
⎪
⎪
0,
if ϕ ≤ ϕ0f ,
⎪
⎪
⎪
⎨
√
2
Rf (yf ϕ−1)
vf =,if ϕ0f < ϕ ≤ ϕ0e ,
4ϕbm
⎪
⎪
k
l
⎪
√
2
√
2
⎪
⎪
⎪
⎩ Rf (yf ϕ−1) +(ye ϕ−1) , if ϕ > ϕ0 .
e
4ϕbm

³
´
³
´
A
A
At the optimal scale, a foreign firm’s domestic sales are qf = 2bf 1 − yf 1√ϕ at pf = 2f 1 + yf 1√ϕ
³
´
³
´
A
A
for ϕ > ϕ0f . Its export sales are qe = 2bf yyfe − yf 1√ϕ at price pe = 2f yyfe + yf 1√ϕ for ϕ > ϕ0e . Theqϕfirm’s labor productivity is 2R
for ϕ > ϕ0f . Thus, exporters are more efficient in production, producefmore and earn more than firms that produce only for the domestic market, confirming well-establishedempirical findings.
Industry equilibrium
In the home country, the acquisition market equilibrium is derived for given Ah in exactly the same way
as in autarky. Hence,
8b2 m
.
(21)
Rh =
ρh
Then Mh ≡

Z 1

ϕ0h

vh dϕ is the measure of all home varieties and Me ≡

Z 1

ϕ0e

vf dϕ is the measure of allh= yyfe ,imported varieties. Substituting a firm’s optimal scale in its equilibrium price and making use of A
A
f
³
´
³
´
1
0
we obtain ph = A2h 1 + yh1√ϕ for ϕ ≥ ϕ0h in the home country and pe = A2h 1 + ye √ϕ for ϕ ≥ ϕe in
Z 1
Z 1
vh ph dϕ +vf pe dϕ.
the foreign country. The aggregate price in the home market is Ph ≡

ϕ0h

26

ϕ0e

In the foreign country, the total capital demanded is Kf ≡

Rf =

Z 1

xf vf dϕ, and from Kf = 1 we obtain

0

8b2 m
,
ρf + ρe + 2ρ̂(22)

where ρ̂ = 1+2ye yf −yf2 +2ye2 +(ye +yf )(yf ye −3)−2 ln(ye )(ye2 +4ye yf +yf2 ). The total product varieties
Z 1yehvf dϕ. Using xf from (20) and the fact that A
are Mf ≡Af = yf , the individual product price sold in
ϕ0f

A
the foreign market is pf = 2f

Z 1
³
´
1√
0
1 + yf ϕ for ϕ ∈ (ϕf , 1]. The aggregate price is Pf ≡vf pf dϕ.
ϕ0f

Finally, by definition, Ah and Af are jointly determined by

Ah =
Af

=

αγ + βPh,β(Mh + Me ) + γ
αγ + βPf.
βMf + γ(23)
(24)

Having expressed Rh and Rf as functions of yf , ye and yh (from (21) and (22)), we have five unknowns,
Af , Ah , yf , ye and yh , and five equations, (23), (24) and the definitions of yf , ye and yh (from (18)), which
jointly determine the equilibrium.
Equilibrium comparison
α22hFrom (23), Ah < αγ+βP
βMh +γ = 1+ β ηh . The equilibrium yh is solved from Ah − 2yh Rh , which means
L yh ρh¶2µα− 2yh2 Rh > 0. Since Z(ya ) = 0 and Z 0 (·) > 0, we have yh > ya .
Z(yh ) =β ηh1+ L y ρ2h hR(y) = 8bρm . Since ρ increases in y, we have Rh = R(yh ) < R(ya ) = Ra .
√√A(y) = y 2R = 4b m √yρ . Because √yρ decreases in y, we have Ah < Aa .
The comparison of all the other variables is easy, as the only difference between autarky and unilateral
opening-up is that home firms in the latter case face competition from imports, reflected in different expresα
sions for A: A =β η in autarky but not after opening-up. All other expressions as functions of y are
1+ L yρthe same. Many proofs can thus be borrowed from the trade liberalization case except that y increases in
unilateral opening-up but decreases in trade liberalization, leading to opposite conclusions. Care is needed
only if a proof makes use of the expression A = αβ η , which is not applicable in unilateral opening-up.
1+ L yρ

Unlike in the trade liberalization case, we do not need to worry about b.
³ ´1√
yh ϕ−1
Ra 3√. The left hand side of the inequality
A firm buys capital (xh vh > xa va ) if and only if ya ϕ−1 > Rh0
is decreasing in ϕ because yh > ya , so if the inequality holds for ϕ , it also holds for all ϕ < ϕ0 . That is,
low efficiency firms√buy capital while high efficiency firms sell capital.
y ϕ−1
Because x = 2ϕb and yh > ya , all firms expand their scale. A firm expands scope if and only if27q√yh ϕ−1Ra√>ya ϕ−1
Rh , i.e., if and only if its ϕ is below some threshold. It is possible that all firms reduce scope.
d( ψ
)
ρ
and
Aggregate scope: M = 2bψρdy < 0, so Mh < Ma .
³
³
q ´
q ´3
1
1
2R
Plant output is q = 2b
A − 2Randfirm
outputisqv=
A−. Both increase if andϕ
ϕ
16b2 m√3ηmρη − 2 , which decreases with y,
only if ϕ is below some threshold. Industry output is Q = 8bAR
2 m y = 4b
so industry output decreases.
Total factor productivity: ω = yη3 , which decreases with y, so ω h < ω a .
ρ2qϕ, which increases after opening up as R drops.
Labor productivity at the plant and firm level: λϕ = 2R

yη
1 √η
= 4b√
, which increases with y, so industry labor productivity
Industry labor productivity is λ = Aρm ρ
improves.
q
ϕ
4t
ϕ
 =
 4
at the firm
Overall productivity: μp =2R at the plant level, and μ =32R
A 1+ y√ϕ
A+3 ϕ
√
η ρ
η
4
1
level. Both increase. Overall industry productivity is μ = √2R θ = b√m θ , which increases with y, so μ

increases.
These comparisons can be summarized as follows:
Proposition 6. After unilateral opening-up, the importing country undergoes the following changes:(1) Markets: the product market becomes more competitive (Ah < Aa ); the capital price drops (Rh <Ra ).
(2) Capital redistribution: capital moves from higher to lower efficiency firms.
(3) Firm structure: all plants expand scale; less efficient firms expand their scope while higher efficiency
firms may reduce scope.
(4) Aggregate scope: the total number of varieties produced by home firms decreases (Mh < Ma ).
(5) Outputs: plant and firm outputs increase for low efficiency firms and decrease for high efficiency
firms. Industry output decreases.
(6) Productivity: total factor productivity decreases; labor and overall productivity increase on theplant, firm and industry levels.
Some of the changes are illustrated in Figure 2, where the solid lines represent autarky and the dottedlines represent the situation after opening-up.
Discussion
After unilateral opening up, foreign products enter the home market, so the demand for each variety
drops (A is lower). Since home capital can only be used for home production, there will be less demand forcapital, so capital becomes cheaper (R drops). In fact, the drop in R must be greater than the drop in A sothat product market becomes more profitable relative to the capital price (y must rise), otherwise all firms
would reduce both scale and scope, which cannot happen at equilibrium.
Because the product market becomes more profitable relative to the capital price, every plant expands
scale. For product scope, recall that high efficiency firms are mainly affected by changes in the productmarket while low efficiency firms are affected mainly by changes in the acquisition market (see expression

28

Capital scale

Product scopeϕϕ

outputϕFigure 2: Impacts of Unilateral Trade Liberalization(7)). After opening up, the product market is more competitive, which tends to reduce scope, while capital
becomes cheaper, which tends to expand scope. As a result, high efficiency firms reduce scope while low
efficiency firms expand it. In the end, capital redistributes toward less efficient firms. So does output. Firms
are distributed more evenly in the industry in terms of capitalization and output.
Because capital becomes cheaper, all firms substitute capital for labor, raising labor productivity at
the plant, firm and industry levels. Overall productivity, which is inversely related to average cost, also
increases. Because inputs and outputs move from high to low efficiency firms, the total factor productivityof the industry drops. This last conclusion depends on the no free entry assumption, and no cross-border
trading of capital even though capital is cheaper in the importing country than in the exporting country.
For the exporting country, analytical comparisons are intractable,19 but numerical calculations confirm
the intuition that almost every change is the opposite of what happens in the home country. Capital becomes
more expensive. The least efficient firms exit. High efficiency firms buy capital and increase their scope
while low efficiency firms sell capital and reduce scope, leading to a more skewed distribution of firm size
measured by capital or output. Every firm reduces scale.
In both trade liberalization and unilateral opening-up, we have demonstrated that high and low efficiency
firms respond differently in their choices of firms structure, even though they all face the same change inthe product and factor markets. In the unilateral opening-up case, the mechanism is simple, because allchanges in the importing country are induced by a negative demand shock in the product market.20 And
19
In the home country, although there are no closed-form solutions, analytical comparisons are possible because the expressions
for scale and scope are exactly the same as in the closed economy, and we can prove that yh > ya because (23) and (11) differ
only by an extra term. In the foreign country, by contrast, the expressions for scale and scope are different from those in the closed
economy, and the expressions contain yf and ye that cannot be easily compared with ya .
20
Given A and R, all expressions are the same as in autarky, and the aggregate demand for capital will also be the same, leading29

the major conclusion is that high efficiency firms are affected mainly by changes in the product market,
while low efficiency firms are affected mainly by changes in the acquisition market. With bilateral tradeliberalization, there is an extra effect, because firms not only face increased competition but also enjoy
increased exporting opportunities.21 The major conclusion is that high efficiency firms are better able to
make use of the exporting opportunities, while low efficiency firms are less able to do so.
Unilateral opening-up is a rather hypothetical setting. Real life tends to have general equilibrium effects
with free entry, while this analysis is of partial equilibrium without entry. In the model, the product market
in the importing country undergoes a negative demand shock, which induces a negative demand shock in
the acquisition market. We can imagine the opposite, such as a positive taste shock that raises A, which inturn will induce a positive shock in the acquisition market. The foregoing discussion predicts that all of theconclusions will be reversed.

References
Baldwin, J. and W. Gu, 2006, “The impact of trade on plant scale, production-run length and diversification”,
Economic Analysis Research Paper Series, No. 038, Statistics Canada.
Baye, M.R., K.J. Crocker and J. Ju, 1996, “Divisionalization, franchising, and divestiture incentives inoligopoly”, American Economic Review, 86(1), 223-236.
Bernard, A.B., J.B. Jensen and F.K. Schott, 2005, “Importers, exporters, and multinationals”, NBER Working
Papers no. 11404.
Bernard, A.B., S.J. Redding and F.K. Schott, 2007, “Comparative advantage and heterogeneous firms”,
Review of Economic Studies, 74, 31-66.
Bernard, A.B., S.J. Redding and F.K. Schott, 2008, “Multi-product firms and product switching”, NBER
Working Papers no. 12293.
Bernard, A.B., S.J. Redding and F.K. Schott, 2009, “Multi-product firms and trade liberalization”, American
Economic Review (forthcoming).
Breinlich, H., 2008, “Trade liberalization and industrial restructuring through mergers and acquisitions”,
Journal of International Economics, forthcoming.
Eckel, C. and J.P. Neary, 2009, “Multi-product firms and flexible manufacturing in the global economy”,
Review of Economic Studies, forthcoming.
Feenstra, R. and H. Ma, 2008, “Optimal choice of product scope for multiproduct firms”, forthcoming in
E. Helpman, D. Marin and T. Verdier, eds., The Organization of Firms in a Global Economy, Harvard
University Press. (NBER working paper no. 13703)
Goldberg, P.K., A. Khandelwal, N. Pavcnik, and P. Topalova, 2009, “Multi-product firms and productturnover in the developing world: Evidence from India”, Review of Economics and Statistics, forthcoming.
to the same expression for the capital price. Therefore, the equilibrium is disturbed only through the reduction in A.
21
For given A and R, a firm’s demand for capital will be different from that in autarky due to the exporting opportunity.

30

Greenaway, D., J. Gullstrand and R. Kneller, 2008, “Surviving globalization”, Journal of International
Economics, 74, 264-277.
Helpman, E., 2006, “Trade, FDI, and the organization of firms”, Journal of Economic Literature, XLIV(3),589-630.
Iacovone, L. and B.S. Javorcik, 2008, “Multi-product exporters: Diversification and micro-level dynamics”,
working paper.
Jovanovic, B. and P.L. Rousseau, 2002, “The Q-theory of mergers”, American Economic Review Papers andProceedings, 92(2), 198-204.
Maksimovic, V. and G. Phillips, 2001, “The market for corporate assets: Who engages in mergers and asset
sales and are there efficiency gains?”, Journal of Finance, 55(6), 2119-2065.
Mayer, T, M. Melitz and G. Ottaviano, 2009, “Market size, competition and the product mix of exporters”,
working paper.
Melitz, M., 2003, “The impact of trade on intra-industry reallocations and aggregate industry productivity”,
Econometrica, 71(6), 1695-1725.
Melitz, M. and G. Ottaviano, 2008, “Market size, trade and productivity”, Review of Economic Studies 75,
295-316.
Mitchell, M. and J. Mulherin, 1996, “The impact of industry shocks on takeover and restructuring”, Journalof Financial Economics, 41, 193-209.
Nocke, V and S. Yeaple, 2006, “Globalization and endogenous firm scope”, NBER Working Papers no.
12322.
Schwartz, M. and E.A. Thompson, 1986, “Divisionalization and entry deterrence”, The Quarterly Journalof Economics, 101(2), 307-322.
Spearot, A.C., 2007, “Acquisition, productivity, and trade liberalization”, working paper.
UNCTAD, 2000, World Investment Report 2000: Cross-border Mergers and Acquisitions and Development,
United Nations, NY and Geneva.

31