Misallocation Under Trade Liberalization
Yan BaiKeyu Jin

Dan Lu ∗University of Rochester

LSE

University of Rochester

NBER

CEPR

August 2, 2019
Abstract
This paper incorporates firm-level distortions into a Melitz model and characterizes welfare
under misallocation. We derive an analogue to the well-known ACR result in an economy
with distortions. We highlight a channel through which trade can reduce welfare by exacerbating misallocation. A key statistic to infer welfare is the gap between input and output
shares. Using Chinese manufacturing data for quantitative analysis, we show that tradeintegration can lead to a 18% welfare loss coming from a reduction in allocative efficiency.
The overall gains to trade is substantially smaller than implied by standard calculations.
Keywords: Capital and labor wedges, misallocation, trade liberalization, gains from trade,industrial policy
JEL classification: E23 F12 F14 F63 L25 O47∗ Bai: Department of Economics, University of Rochester; email: yanbai06@gmail.com. Jin: Department of

Economics, London School of Economics; email: k.jin@lse.ac.uk. Lu: Department of Economics, Universityof Rochester; email: danlv27@gmail.com. Acknowledgments: We thank George Alessandria, Mark Bils, Oleg
Itskhoki, Narayana Kocherlakota, Ezra Oberfield, Michael Song, Kjetil Storesletten, Heiwai Tang, Fabrizio
Zillibotti, Daniel Xu for their invaluable comments, as well as seminar participants in NBER Summer Institute,
NBER China meeting, Tsinghua growth and institute, AWSOME conference of McMaster U., Philadelphia Fed
Trade Workshop, St. Louis Fed Macro-Trade Conference, IMF, Fed Board, Chinese U. of Hong Kong, U. of
Hong Kong, HongKong U. of Science and Technology, SHUFE, Fudan U. and NSE of Peking Unveristy.

1

Introduction

The question of how much developing countries benefit from opening up to goods trade isa time-honoured subject, both of practical import and intellectual interest. Much has been
understood about the nature and type of gains to trade, thanks to the remarkable progress
made in the field of international trade in recent decades. Less clear, however, is why
certain developing countries have benefited from trade more than others, and why certain
countries have seemingly not benefitted much at all.1 New trade theories suggest thatdeveloping countries have the most to gain from trade: if trade liberalization can induce
reallocation of resources from less to more productive firms, aggregate productivity andwelfare will rise in turn.
But developing countries are different in another respect: they are also subject to prevalent policy and institutional distortions. Examples include taxes and subsidies to certain
firms, implicit guarantees and bailouts, preferential access to land and capital, and industrial policy and export promotion policies—common themes in developing countries. In thecase of China, for instance, this explains why inefficient but politically-connected private
firms and state-owned companies (SOE) have survived and even thrived. Implicit and explicit support for these firms combined with limited exit mechanisms for many SOEs have
weakened firm selection effects, the upshot of which is a drag on aggregate productivity.
Many believe that joining the WTO can potentially alleviate some of these problems byinviting more Foreign direct competition and helping the more productive firms expand.
How effective is this mechanism? Can trade necessarily improve allocations? Does
trade necessarily lead to welfare gains for developing countries? These issues are far fromobvious as alluded to by Rodríguez-Clare (2018), “ [a] complication that may matter forthe computation of the gains from trade is the presence of domestic distortions.” Thus, thegoal of the paper is to provide an answer to this question using the discipline of a generalequilibrium model of trade that incorporates idiosyncratic firm-level distortions. We thenuse Chinese manufacturing data to quantify the effect of trade liberalization as well as toconduct empirical investigations relevant for our mechanisms.
1 For example, Waugh (2010) shows, in large sample of countries, that poor countries do not systematicallygain more from trade.

2Our modelling framework incorporates firm-specific distortions into a two-country Melitz
model. There are two dimensions of heterogeneity at the firm-level: productivity and distortions. These distortions are assumed to be exogenous output wedges or factor wedges.
They drive differences in the marginal products across firms.
In this framework, we derive a general expression for welfare under trade that encompasses the case with and without distortions. We show that a special case in which there
is only heterogeneity in productivity, which follows a Pareto distribution, yields the wellknown result of ACR (after Arkolakis, Costinot, and Rodríguez-Clare (2012)): a special
case in which there is only heterogeneity in distortion with Pareto distribution yields ananalogue to ACR under misallocation. These two are special cases in which firm selection
is driven either solely by productivity or solely by distortions. This explicit expression for
welfare yields insight on exactly why there could be a loss associated with trade. A crucialsufficient statistic is the gap between the aggregate input and output share. If the required
inputs used for production is greater than the output share it yields, then the reduction inallocative efficiency arising from a reallocation of resources can bring about a welfare loss.
In the special case, welfare is always lower in an open compared to a closed economy.
The welfare formula also nests a more general case without misallocation, corresponding to Melitz and Redding (2015). In the general case with distortions, productivity anddistortions are competing in their impact on firm selection. The relative strength of thetwo depends on their joint distribution, and micro-level information still matters for welfare. The impact of allocative efficiency arising from a reallocation of resources can again
be captured by the input-output gap, along with additional structural microeconomic parameters. To our knowledge, the paper is the first to theoretically characterize welfare
gains to trade in a model with misallocation. The decomposition of welfare into a ‘puretechnology effect’ and a ‘resource reallocation effect’ in this instance resonates with thedecomposition in the recent work of Baqaee and Farhi (2017)–an analysis that takes place
in a closed-economy with distortions but without firm entry.
Contrary to the mechanism underpinning the Melitz (2003) model and its extensions,i.e. that trade can induce a reallocation of resources from low productivity to high productivity firms, the presence of distortions can bring about the opposite and exacerbate3

misallocation. The reason is simple: distortions (for instance, tax and subsidies) act as aveil to a firm’s true productivity. A firm may be producing in the market not because
it is inherently productive, but because it is sufficiently subsidized. A mass of highlysubsidized but not adequately productive firms will export and expand at the cost of othermore productive firms. The high productivity/ high tax firms which were marginally able
to survive in the domestic market would be driven out as the other firms gain market shareand drive up costs. In other words, the selection effect which brings about efficiency gains
in the Melitz-type model is no longer based solely on productivity; it is determined jointly
by firm productivity and distortions. Trade may thus lower the average productivity offirms. Depending on the prevalence and nature of distortions, different countries may havedifferent experiences with trade liberalization.
Second, we provide supportive empirical evidence on some of the key implications ofour model. Although the study can apply to any group of countries, we focus on Chinese
manufacturing data for the simple reason that this country is well known for its prevalentState interventions and policies; that a body of work has shown (see references below) that
idiosyncratic distortions explain the majority of the dispersion in marginal products; andthat trade liberalization has been an important recent phenomenon.
Employing Chinese manufacturing firm-level data, we first use a ‘measured wedge’ following the customary approach to examine some of its key features. These measured
wedges reveal that there is substantial misallocation at the firm level. Their patterns arealso consistent with common perception on what types of firms might enjoy special privileges, and hence have ’lower taxes’. We then evaluate whether key implications of themodel receive empirical support. Specifically, the model implies that selection will drive apositive relationship between productivity and taxes for the reason that firms facing higher
taxes must be more productive to survive. This is true in the data—there is a large dispersion of the measured wedges across firms, as well as a robust positive relationship between
wedges and productivity.2 Second, the model indicates that given productivity, firms that
2 Different from Hsieh and Klenow (2009), this correlation matters.

Typically, this correlation is taken
to be exogenous. In works that assume a perfect correlation between distortions and productivity (such as
Costa-Scottini (2018) and Ho (2010) ), there are always TFP and welfare gains to trade, in contrast to losses
that can arise from our model. We demonstrate why this correlation cannot be taken to be exogenous when
there is firm selection and fixed cost. Moreover, this assumption counters key empirical facts. We discuss this4

face lower taxes will tend to export. Data indicates that controlling for productivity, exporters indeed have lower wedges.
Third, fully recognizing the limitations of the empirical methods to infer wedges (though
it provides a robustness check and serves as supportive evidence), we employ a structuralmodel and combine it with micro data to conduct a quantitative analysis of the impactof trade on welfare and aggregate productivity. We run counterfactual experiments for
local changes in trade cost, as well as counterfactual experiments for domestic reforms.
Our main conclusion is that welfare gains are much smaller when taking into accountdistortions; that there is a TFP loss of 3% as opposed to a TFP gain of 13.3% in the casewithout distortions, and that allocative inefficiency can induce a welfare loss of 18%.
It is important to point out that in the quantitative analyses we do not use directly theempirically-measured wedges, observed correlations, or distributions in the data to assess
the impact of trade on welfare. The reason is that the observed statistics are not the underlying ones: existing firms have been subject to selection and thus their observed distributions are not the true ones. The same reasoning goes for the observed correlation between
productivity and wedges. Both selection and fixed costs can drive a positive relationshipbetween the two. For these reasons, the approach adopted in the quantitative exercises is toestimate the underlying joint distribution of wedges and productivity, costs of producing
and exporting so as to match the observed patterns of firms’ outputs, inputs, and exports.
On this basis, we evaluate how the presence of distortions change the impact of trade on
productivity and welfare, and how much trade has contributed to Chinese growth in a decomposition exercise. This contrasts with the reduced-form approach adopted in Berthou,
Chung, Manova, and Bragard (2018), which uses empirically measured revenue productivity to assess the impact of trade reforms on aggregate productivity under misallocation. 3
Our works are broadly complementary, as we focus on theoretical analyses and a structural
approach to inferring welfare gains for China, whilst they focus on an empirical assessmentof trade on aggregate productivities for 14 European countries.
at the end of Section 3.
3 Berthou et al. (2018) theoretically and empirically assess the impact of trade reforms for 14 European
countries and 20 industries over the period 1998-2011. They find that trade reforms have ambiguous effects
on measured revenue productivity in the theory, while they are positive in the data.

5

An important point is the fact that ‘misallocation of resources’ goes beyond the observed
misallocation among a set of operating firms. Because policy distortions also act as a barrier
to entry (and exit), there is also misallocation among potential entrants and incumbents–
firms that should have entered the market in an efficient economy that couldn’t, and firmsthat should have otherwise exited but have not. This reallocation along the entry/exit
margin can also be significant. The endogenous mechanism of entry/exit and the attendant firm selection effect, along with trade, makes this different from the important worksof Hsieh and Klenow (2009), Baily, Hulten, and Campbell (1992), Restuccia and Rogerson (2008), Bartelsman, Haltiwanger, and Scarpetta (2009), etc. Empirical works have also
demonstrated the importance of entry and exit for China’s growth.4
The source of misallocation of resources is not the focus of our work, though we takethe particular stance that firm-specific ’distortions’ due to policy and institutional features
is a major cause of misallocation. Distortions in China manifest themselves in the form of
substantial privileges of state owned enterprises over private firms, of connected private
firms, or of firms belonging to particular locations. Specific policies that can drive thesewedges include implicit subsidies such as soft budget constraints, favorable costs of capital,
preferential tax treatments and implicit guarantees. Firms with political connections having
access to special deals and receiving substantial benefits are also widely documented (seeGuo, Jiang, Kim, and Xu (2013) and Bai, Hsieh, and Song (2019)). Wu (2018) conducts
an empirical analysis and finds that policy distortions can be explained by investment
promoting programs that favor such firms.
There is also substantial evidence coming from a number of papers that idiosyncratic
firm-distortions due to policy and institutional features account for a large part of theobserved dispersion in marginal products across firms in China. In principle, misallocation can arise from a variety of factors; but different approaches to disentangle them havecome to similar conclusions that policy distortions are elemental.5 Furthermore, we con4 Brandt, Van Biesebroeck, and Zhang (2012) find that net entry accounts for roughly half of Chinese
manufacturing productivity growth. The creation and selection of new firms in China’s non-state sector has
been particularly important.
5 These could be technological frictions, such as adjustment costs, information frictions, financial frictions, or markups. Wu (2018) finds that policies account for the majority of the observed misallocation ofcapital, as opposed to financial frictions. Using a different approach and modeling framework, David andVenkateswaran (2017) find also that firm-specific distortions, rather than technological or information fric-6

duct empirical analyses to show how these wedges systematically relate to certain firm
characteristics in a way that echoes prior findings. To ensure that measured dispersion is
not due to measurement error, we use three alternative approaches utilizing panel data todemonstrate the minor role it plays.6
Many of these distortions are presumably idiosyncratic and unrelated to trade. Examples are in the legion: Chinese firms such as the car manufacturing company Chery, haveenjoyed easy access to land and capital from the Wuhu government. Foxconn, the world’s
largest electronics contracter manufacturer, has enjoyed substantial tax breaks from many
provinces including industrial land at significantly discounted prices. Tesla has recently
received free land and subsidies from the local government of Shanghai. A recent study byChen and Kung (2018) demonstrate the firms that are connected with political elites wereable to obtain land at 80 to 90 percent discount over the period 2004-2016.
For the reasons above, the baseline model in the paper is to focus on domestic policy
distortions. Still, one can ask whether some of the large dispersion of marginal products
reflects endogenous distortions—those that can potentially change with trade liberalization. As a robustness check we examine a model of endogenous distortions with variablemarkup, and ask whether trade can mitigate these distortions and the misallocation of resources. Section 5.1 takes up a variable markup model. We show that these models 1)
yield some obvious counterfactual predictions on the relationship between exporters andwedges; 2) that markup alone also explains little of the dispersion in wedges. To match theobserved correlation and dispersion one would still need to include exogenous distortions.
Moreover, the attendant pro-competitive effects in a model with endogenous markup may
be ‘elusive’ as pointed out by Arkolakis, Costinot, Donaldson, and Rodríguez-Clare (2018).7
In this framework, positive firm selection is the central driving force for gains to trade.
As such, it abstracts from other types of gains to trade, such as trade-induced technologitions, account for the majority of the observed dispersions in marginal products. Bai, Lu, and Tian (2018)
disciplines financial frictions with firms’ financing patterns, sales distribution and change of capital. Theyfind that financial frictions cannot explain the observed relation between firms’ measured distortions andsize.
6 However, Bils, Klenow, and Ruane (2017) cannot rule out multiplicative measurement errors.
7 This paper makes the point that when more productive firms expand at the expense of less productive
ones, thanks to trade, the aggregate markup tends to rise. Thus, overall, trade models with endogenous
markups do not necessarily generate higher gains from trade.

7

cal diffusion (Alvarez, Buera, and Lucas Jr (2013) and Buera and Oberfield (2016)), adoption (Perla, Tonetti, and Waugh (2015) and Sampson (2015)) and innovation (Atkeson andBurstein (2010)). While these mechanisms in principle work to increase the gains to trade,with its quantitative significance a subject to debate,8 it does not detract from the fact thatthe distortionary impact on allocation efficiency still induces large welfare losses, which
is what we are interested in. Of course, distortions can also interact with some of these
additional channels. For instance, in a model with firm innovation, one would need toconsider the fact that distortions not only affect production decisions, but potentially alsoinnovation decisions. These considerations go beyond the scope of this paper but deserve
further consideration. We also do not consider how trade can reduce domestic distortions,for example if concurrent domestic reforms are requisite for joining the WTO or if quotas
are removed (see Khandelwal, Schott, and Wei (2013)). As a robustness check, however, weallow for firms to face a different distribution of distortions when they start to export andexamine welfare and efficiency gains therein.
A key message of this paper is that in order for developing countries to reap the full gainsof trade, simultaneous or antecedent domestic reforms aimed at reducing policy distortions
may be crucial. Based on our structural model, trade contributed only 8% to growth overthe period of 1998 to 2005 in China. This echoes with the findings of Tombe and Zhu(2019) that find similar numbers despite employing a very different framework with migration flows. The policy implication drawn from this framework is consistent with works
indicating that policies aimed to neutralize domestic distortions may be complementary to
trade liberalization (Chang, Kaltani, and Loayza (2009) and Harrison and Rodríguez-Clare(2010)). It counters other claims that trade liberalization should take precedence owing topositive firm selection (Asturias, Hur, Kehoe, and Ruhl (2016)).9
8 Perla, Tonetti, and Waugh (2015) and Atkeson and Burstein (2010), for instance, find that trade gains are

not too different from ACR gains. In Perla, Tonetti, and Waugh (2015), there is trade-induced within-firm
productivity improvements. However, their aggregate growth effects come with costs—losses in variety andreallocation of resources away from goods production. Thus, the aggregate effect on welfare is similar to
ACR gains. Atkeson and Burstein (2010) show that general equilibrium effects limits the first-order effects on
aggregate productivity even when there is firm-level innovation.
9 They show that the best sequence of reforms is to first decrease trade costs, then to improve contract
enforcement, and, finally, to decrease the cost of firm creation.The reason is that an increase in competition
leads to an expansion of productive firms and crowding out of less efficient ones. By liberalizing international
trade first so as to impose firm selection early, inefficient firms are prevented from entering later whencontract enforcement and firm entry costs are reformed. In contrast, we show that the selection mechanism

8In sum, this paper shows that experiences of trade liberalization in developing countries
should not be considered to be independent of micro-level distortions to which they are
subject. Our paper demonstrates that the presence of policy distortions have a first-order
quantitative effect on the gains to trade. The organization of the paper is as follows: Section
2 derives a theoretical framework of trade gains under misallocation. Section 3 investigates
patterns of measured wedges among Chinese manufacturing firms. Section 4 provides aquantitative assessment on the impact of trade liberalization under misallocation, with various extensions of the benchmark framework. Section 5 discusses a model with endogenous
distortions. Section 6 concludes.

2

Theoretical Framework

2.1

Baseline Model

The world consists of two large open economies, Home and Foreign, with heterogenous
firms. The two economies can differ in the size of labor and distribution of firms. Labor isimmobile across countries and inelastic in supply.

Consumers. A representative consumer in the Home country chooses the amount of final
goods C in order to maximize utility u(C ), subject to the budget constraint
PC = wL + Π + T,
where L is labor, w is wage rate, Π is dividend income, and T is the amount of lump-sumtransfers received from the government.
Final Goods Producers. Final goods producers are perfectly competitive, and combine
intermediate goods using a CES production function

Q=

Z
ω ∈Ω

q(ω )

is substantially weakened in the presence of distortions.

9

σ −1
σ

 σ

σ −1

dω

,

where σ is the elasticity of substitution across intermediate goods, and Ω is the endogenous
set of goods. The corresponding final goods price index is thus

P=

Z
ω ∈Ω

p(ω )

1− σ

 1

1− σ

dω

,

where p(ω ) is the price of good ω in the market. The individual demand for this good isthus given byq(ω ) =p (ω )−σQ.
P−σIntermediate Goods Producers. There is a competitive fringe of potential entrants (in bothcountries) that can enter by paying a sunk entry cost of f e units of labor. Potential entrants
face uncertainty about their productivity in the industry. They also face a stochastic revenue
wedge τ, which can be seen as a tax (>1) or subsidy (<1) on every pq earned.10 Once thesunk entry cost is paid, a firm draws its productivity ϕ and τ from a fixed joint distribution,g( ϕ, τ ) over ϕ ∈ (0, ∞), τ ∈ (0, ∞).11
Firms are monopolistically competitive. Production of each intermediate good entails
fixed production cost of f units of labor and a constant variable cost that depends on firm
productivity. The total labor required to produce q( ϕ) units of a variety is therefore: 12

`= f+

q
.
ϕ

Productivity ϕ is idiosyncratic and independent across firms. The existence of a fixed
production cost means that only a subset of firms produces—those that draw a sufficiently
low productivity cannot generate enough variable profits to cover the fixed production cost.
If firms decide to export, they face a fixed exporting cost of f x units of labor and iceberg
10 It is equivalent to an input wedge on all the input a firm uses.
11 The model equilibrium is equivalent to a stationary equilibrium of a model allowing for constant exoge-

nous probability of death δ and entry cost f e /δ.
12 We can easily extend the production to include capital, i.e. k α `1−α . The unit cost for producing q
or fixed cost is α−α (1 − α)α−1 w1−α rkα where rk is the rental cost of capital. In our model, we introduce
one heterogeneous distortions at the firm level, and our τ is an output distortion, but it includes all input
distortions that increase the marginal products of capital and labor by the same proportion as an output
distortion. In the data, there are distortions that affect both capital and labor and distortions that change themarginal product of one of the factors relative to the other. In our quantitative exercises, we include both
capital and labor, and the distortions on both factors.

10

variable costs of trade τx , which is greater than 1. Firms with the same productivity anddistortion behave identically, and thus we can index firms by their (ϕ, τ) combination.
An intermediate goods firm thus solves the following problem
max
p,q

pq w
− q − wfτϕ

p−σ

subject to the demand function q = P−σ Q, henceforward suppressing ω for convenience.
From here it is clear that a revenue tax is equivalent to a tax on all input costs incurred by
the firm.
Firms are infinitesimally small, and thus take the aggregate price index as given. Equating the after-tax marginal revenue with marginal costs yields the standard result that equilibrium prices are a mark-up over marginal costs:
p=

σ wτ
.
σ−1 ϕ

(1)

Optimal profits are thenπ = σ−σ (σ − 1)σ−1 Pσ Qτ −σ w1−σ ϕσ−1 − w f .

(2)

It immediately follows that given the fixed cost of production, there is a zero-profit cutoff
productivity below which firms would choose not to produce, and exit the market. Thus,a firm would choose to produce only if ϕ ≥ ϕ∗ (τ ). This cutoff productivity level satisfies

 1
σ
σ
σ σ −1w f σ −1
σ −1 .
ϕ (τ ) =wτ
σ − 1 Pσ Q∗

(3)

The cutoff productivity is now a function of the firm-specific distortion, and differs acrossfirms facing different levels of distortions. Firms with a higher tax τ will have a higher
cutoff for productivity. This means that low productivity firms that would have been otherwise excluded from the market can now enter the market and survive if sufficiently
subsidized.
The government’s budget is balanced so that
11

T=



Z
ω ∈Σ

1
1−
τ


p(ω )q(ω )dω,

where Σ is the endogenous set of home products. With trade, firms now have the option ofexporting abroad. If a Home firm exports to the Foreign economy, it solves the following
problemmax

px q fw− τx q f − w f xτϕp−σ

subject to the FForeignoreign demand function q = Px−σ Q f , where Pf and Q f denote the agf

gregate price index and demand in Foreign. Given the same constant elasticity of demandin the domestic and export markets, equilibrium prices in the export market are a constantmultiple of those in the domestic market:
p x ( ϕ, τ ) =σ wτx τ,σ−1 ϕThe optimal profit from servicing the Foreign market,
π x = σ−σ (σ − 1)σ−1 Pfσ Q f τ −σ (wτx )1−σ ϕσ−1 − w f x ,(4)yields an optimal cutoff for exporting:"# 1
σσ −1 σ −1
σ −1
σ
σ
w
f
τ
x x
ϕ∗x (τ ) =wτ σ−1 .
σσ−1Pf Q f(5)

Consumer love of variety, a fixed production cost and additional fixed cost of exporting,mean that firms would never export without also selling in the domestic market. There
are, hence, two cutoff productivities relevant for the domestic economy: one for entering
the domestic market as given by (3) and one for entering the Foreign market, as givenby (5). To the extent that taxes τ are constant across firms, the ratio ϕ∗x (τ )/ϕ∗ (τ ) is aconstant and is greater than 1 so long asτxσ−1 f x Pσ Q
f
Pσf Q f > 1. Analogously, firms in the Foreigncountry, which draw their productivity from a distribution g f ( ϕ, τ ), are subject to two
cutoff productivities, one for servicing their domestic market, and one for exporting to the

12Home economyσσ σ −1ϕ∗f (τ ) =
σ−1

"

wf f
Pfσ Q f

# 1

σ −1
σ

w f τ σ −1 ,

(6)

"
# 1
σ
σ −1 σ −1
σ −1
w
f
τ
σ
σ
x
f
x
ϕ∗x f (τ ) =w f τ σ −1 .
σ
σ−1
P Q

(7)

where w f denotes the Foreign wages, and the fixed cost of producing and exporting areassumed to be identical in the two economies.
The equilibrium features a constant mass of firms entering Me and producing M, alongwith a ex-post distributions of productivity and distortion among operational firms µ( ϕ, τ ).
The ex-post distribution µ( ϕ, τ ) is a truncation of the ex-ante productivity-distortion distribution, g( ϕ, τ ), at the zero-profit cutoff productivity given by Eq.3:g( ϕ, τ )ϕ∗ (τ ) g ( ϕ, τ ) dϕdτµ( ϕ, τ ) = R R ∞(8)

if ϕ ≥ ϕ∗ (τ ); and µ( ϕ, τ ) = 0 otherwise. The denominator is the probability of successful
entry, denoted as
ωe =

Z Z ∞
ϕ∗ (τ )g ( ϕ, τ ) dϕdτ.

(9)In equilibrium, the measure of producing firms equals the product of measure of entrants
and the probability of entering, i.e.
ωe Me = M.
We define the probability of exporting conditional on entry as

ωx =

Z Z ∞
ϕ∗x (τ )R R∞

ϕ∗ (τ ) g ( ϕ, τ ) dϕdτµ ( ϕ, τ ) dϕdτ = R R ∞xϕ∗ (τ ) g ( ϕ, τ ) dϕdτ.

In an equilibrium with positive entry, the free entry condition requires thatZ Z
ϕ∗ (τ )π ( ϕ, τ ) g( ϕ, τ )dϕdτ +Z Z
ϕ∗x (τ )π x ( ϕ, τ ) g( ϕ, τ )dϕdτ = w f e .

(10)

The first term is the expected profits from domestic sales conditional on entry, multiplied by
13

the probability of entry. The second term is the expect profits from export sales conditionalon exporting, multiplied by the probability of exporting. The free entry condition requiresthat their sum be equal to the entry costs (in terms of labor).
The free entry condition (10), combined with optimal profit functions (2) and (4) gives
an expression for the price index P:
"

σ
P=
M
σ−1

Z Z ∞



ϕ∗ (τ )wτϕ

 1− σµ ( ϕ, τ ) dϕdτ + M f

Z Z ∞ϕ∗x f (τ )w f τx τϕ# 1 1− σ1− σµ f ( ϕ, τ ) dϕdτ(11)where M and M f denote the measure of operating firms in Home and Foreign. The Foreign
price index Pf takes a similar form.
Goods market clearing. The assumption of a balanced trade results in
Pfσ Q f M

Z Z ∞ϕ∗x (τ )



wτx τ
ϕ

 1− σσµ ( ϕ, τ ) dϕdτ = P QM f

Z Z ∞ϕ∗x f (τ )w f τx τϕ 1− σµ f ( ϕ, τ ) dϕdτ.
(12)Labor market clearing. In an equilibrium in which the mass of firms M are constant inboth economies, the labor market condition yields
L

M=


σ

fe
ωe + f + ω x f x

.

(13)

Normalizing the Home country wage rate to 1, there are eleven equations, the zero cutoff
productivities for domestic production and exporting (3), (5), and its Foreign counterparts,
the free entry conditions (10) along with its Foreign counterpart, the definition of the Homeand Foreign price indices (11), and a goods market clearing/balanced trade equation (12),along with the measure of firms (13) and its Foreign counterpart. These equations yield
the equilibrium consisting of eleven unknowns { ϕ∗ (τ ), ϕ∗x (τ ), ϕ∗f (τ ), ϕ∗f x (τ ), P, Pf , Q, Q f ,w f , M, M f }. A detailed derivation of the model is provided in Appendix A.
Proposition 1. The allocations, entrants, and cutoff functions { Q, Q f , M, M f , ϕ∗ (τ ), ϕ∗f (τ ),ϕ∗x (τ ), ϕ∗x f (τ )} are independent of mean wedge τ̄. Prices { P, Pf , w f } change proportionally withτ̄, τ̄ f , i.e. P(τ̄1 )/P(τ̄2 ) = τ̄1 /τ̄2 , and similarly for Pf and w f .
14,

The proof of the proposition is straightforward.

2.2

Theoretical Comparative Static

We proceed to analyze welfare and efficiency with distortions. The welfare, or consumption, is given by13"
# σ−1 1 σ −1 σ −1Z Z
Pfσ Q f Z Z
σ−1
ϕ MRPLMRPLϕW=
Meg( ϕ, τ )dϕdτ + Me σg( ϕ, τ )dϕdτ
σ
MRPLτ
P Q
τx MRPLτϕ∗ (τ )ϕ∗x (τ )(14)

where MRPLτ is the firm specific marginal revenue product of labor MRPLτ = wτ, and
MRPL is the economy wide marginal revenue product of labor, MRPL = PQ/wL. Hencethe welfare relates some weighted productivity of firms with relative distortions as theweight. In an efficient case without distortions, all firms have the same marginal revenue
product, MRPLτ = MRPL for any τ. Hence, Equation (14) shows that the source of welfare
loss in the presence of firm-level distortions can arise from a misallocation of resources,
captured by dispersions in MRPL/MRPLτ , and a misallocation caused by selection andentry mechanisms as captured by Me , ϕ∗ , ϕ∗x being different from their respective efficientlevels. Change in trade cost affects the economy through the selection and entry and themisallocations.
Next, we derive a general expression for changes in welfare associated with changes
in trade costs. Arkolakis, Costinot, and Rodríguez-Clare (2012) demonstrates that in theabsence of distortions, welfare changes across a wide class of models can be inferred using
two variables: (i) changes in the share of expenditure on domestic goods; and (ii) the elasticity of bilateral imports with respect to variable trade costs (the trade elasticity). Different
trade models can have different micro-level predictions, sources of welfare gains, and different structural interpretations of the trade elasticity. But conditional on observed tradeflows and an estimated trade elasticity, the welfare predictions are the same. The generalityof this formulation, however, relies on a certain set of macro-level restrictions. Melitz andRedding (2015) (henceforth MR) show that under more general distribution functions for
productivity, the trade elasticity is no longer invariant to trade costs and across markets,
13 Home’s welfare depends on foreign exports, which are rewritten by using the balanced trade condition.

15

and therefore no longer a sufficient statistic for welfare. Micro-level information is still
important for welfare.
In the analysis below, we first consider a fall in trade costs in an open economy equilibrium. The following proposition provides a general representation of welfare:
Proposition 2. The change in welfare associated with an iceberg cost shock ish1
d ln W =− d ln λγs + σ − 1( ACR)(15)+ d ln Me( MR)
σ+(γs − γd )d ln Me
σ
−
1




i
σγd
σγs
d ln λ + σ − 1 +d ln S− σ−1+σ−1
σ−1( Reallocation)

where λ is the share of domestic output; S is the share of variable labor used in producing domestic
goods; γd is the elasticity of the cumulative domestic market share for firms above the domestic cutoff,with respect to the cutoff; and γs is the same elasticity for S. 14
1. Without domestic distortions, S = λ and γs = γd . If productivity follows a Pareto distribution
with parameter θ, γd = θ − σ + 1 and dlnMe = 0. Hence, dlnW = 1θ [−dlnλ] as in ACR.
2. Under a general distribution function and without domestic distortions, S = λ, γs = γd ,1
but they are not constant, and dlnMe 6= 0. Hence, dlnW = γ ( ϕ∗ )+[−dlnλ + dlnMe ]. Microσ −1dstructure matters for γd ( ϕ∗ ) and welfare as in MR.
3. With homogenous productivity and Pareto-distributed domestic distortion 1/τ with parameter1σ −1σθ, γd = σ−σ ( θ − σ + 1) and γs = σ ( θ − σ ). Hence, dlnW = σ−1 [ dlnS − dlnλ ].
14 The elasticities are also known as the hazard functions. The elasticity related to outputs or the hazard

function of the distribution of log firm size within the domestic market is given byR  ϕ ∗ ( τ )  σ −1g( ϕ∗ , τ ) ϕ∗ dτ
τ
R
R
γd =
.
ϕ σ −1g( ϕ, τ )dϕdτϕ∗ (τ ) ( τ )The elasticity related to inputs or the hazard function for the distribution of log after-tax firm size, which isalso proportional to the distribution of firm variable input, within the domestic market is given byR  ϕ ∗ ( τ )  σ −1
γs = R R

τ/τg( ϕ∗ , τ ) ϕ∗ dτ

.
ϕ σ −1/τg( ϕ, τ )dϕdτϕ∗ (τ ) ( τ )16

4. In general, for a local change in trade cost, information on the change of domestic (sales andvariable labor) shares and the measure of entrants, the joint distribution of firms sales and variable
inputs, and the marginal firms (hence we know γd and γs ) are sufficient for computing the associatedwelfare change.
PROOF: Appendix B.1.
The above proposition encapsulates welfare results for four different cases. The special
case in which there is only heterogeneity in productivity, Pareto-distributed, recovers the
ACR formula (point 1). The case without distortion and under a more general productivity
distribution gives rise to MR (point 2). With misallocation, the special case in which there
is only heterogeneity in distortions which is Pareto distributed gives an analogue formulato ACR: the difference in the change in the domestic labor and output share provides a
sufficient statistics for welfare (point 3). In the general welfare representation, the first termis referred to as ACR, the third and fourth terms are brought about by distortions, and isreferred to as an entry and reallocation effect.15
Note that the proposition applies to both symmetric and asymmetric countries and takes
into consideration the impact of the Foreign distribution of firms on the Home country. It
also shows the effect of domestic distortions on a Foreign country. In the case that theForeign economy is devoid of distortions, the third and fourth terms in Foreign’s welfare
formula go to zero. Thus, Home’s domestic distortions affect Foreign only through Foreign’s λ, Me , the cutoffs, and hence γ.
In the two special cases, firm selection is either driven solely by productivity, or solely
by distortions. The former implies that there is always a welfare improvement when theeconomy opens up to trade, whereas the latter implies that there is an unambiguous loss
(see Lemma below). In the more general case, both productivity and distortions jointly
determine firm selection. The resource reallocation is both one amongst existing firms (lastterm), and along the entry dimension (third term). Without distortions, a firm’s shareof input is equal to its share of output, so that in aggregate, S = λ. In the presence of
15 Note that we can write the welfare change using different general equilibrium variables, e.g. d ln W =γd / ( σ − 1 )1γd +σ −1 [− d ln λ + d ln Me ] + ( γd +σ −1 + 1) d ln PQ.

Trade changes the allocation and misallocation of theresources, which in turn affects the aggregate expenditure (and thus selection), and hence welfare in theeconomy. Replacing d ln PQ by equilibrium conditions, one arrives at Proposition 2, which provides moretransparent indications of how exporting/selection reallocates resources and affects misallocation.

17

distortions, the two are no longer equal. The gap between input and output shares isinformative about changes in allocative efficiency. If the change in required inputs exceeds
the change in output it produces, i.e. dlnS < dlnλ, resources reallocation has induced anefficiency loss.16 This reallocation comes from the flows that take place among a given setof producers/exporters on the intensive margin, captured by (σ − 1)(dlnS − d ln λ); it alsocomes from a reallocation on the extensive margin, induced by the fact that cutoffs havechanged. This effect is summarized by the terms with the hazard rates γs and γd .
Lemma 1. Under homogenous productivity and Pareto-distributed domestic distortion 1/τ withparameter θ,
1. Moving from a closed economy to an open economy entails a welfare loss.
2. In the open-economy equilibrium, the reallocation term is always negative.
PROOF: Appendix B.2.
This Lemma presents two important features under the special case. First, comparedto the close economy, an open economy with any level of finite iceberg trade cost always
has a lower welfare as long as there is selection to export. Second, among the equilibriaof the open economy, for a marginal reduction of iceberg cost, as long as there are larger
fraction of exporters, the reallocation term is always negative, implying a worsening of
misallocation after a reduction in trade costs.
The intuition for why the open economy has a lower welfare than in the closed economy
is made transparent by this special case: under homogenous productivity, the efficient
allocation for the closed economy dictates that firms have identical market shares— either
all firms export or none of them export when the economy opens up to trade. Hence,the ex-post efficient allocation should also be equal shares for all firms. However, with
distortions, the relatively subsidized firms produce more than in the efficient case, with thedispersion of sales (employment) reflecting the distortions. When opening up to trade, it is
the relatively subsidized firms that export, in turn making the firm’s distribution even moreskewed—misallocation is exacerbated. The share of labor required in producing domestic
goods ends up being less than the domestic output share. When firm selection is purely
16 Note that d ln S ≤ 0 and d ln λ ≤ 0 in response to a trade cost reduction.

18

driven by distortions, allocative efficiency deteriorates when moving from autarky to an
open economy.
Point 2 in the above Lemma focuses on a local change in trade costs in the open economyσequilibrium, where dlnW = σ−
1 [ dlnS − dlnλ ]. The difference in the change in the share of

expenditure on domestic goods and the share of variable labor used in producing domestic
goods constitutes a sufficient statistics for the welfare change. This change of welfare
reduces to two terms: the standard ACR term, where −d ln λ > 0, and the reallocation
term, which is negative.17 Overall, the change in welfare in the open economy equilibrium
displays a U-shape pattern: for high levels of trade cost, there is a welfare loss; and for low
levels of trade cost, there is a welfare gain associate with a marginal change in trade cost.
The reason is that firms selection driven by distortions are less significant when trade costs
are small (at zero trade cost all firms export), and the welfare gains dominate the losses
associated with reallocation.
These implications carry a similar flavor to findings in Baqaee and Farhi (2017), to which
we draw an analogy in order to further fix ideas in our open economy. Their work studies
the macroeconomic impact of microeconomic shocks to productivity and wedges, and showthat the effect on output can be decomposed into a “pure technology effect” and a “resource
allocation effect”. The former is the change in output holding fixed the share of resources
going to each user; the latter is the change in output resulting from the reallocation ofshares of resources across users. To invoke a closer comparison to theirs, one can do asimilar analysis in our closed-economy setting—-that is, decomposing the aggregate effectof a firm-specific productivity shock to a pure technology effect and a resource reallocation
effect occasioned by distortions.18 This decomposition demonstrates why an increase in a
firm’s productivity may not raise aggregate productivity: if a firm is relatively subsidized,then firm i’s labor share is larger than its sales share, and its expansion means that the
17 In

the welfare
expression, thesecond and third terms cancel out, and the fourth term

σγd 
s
− σ − 1 + σσγ
d
ln
λ
+
σ
−
1
+
d
ln
S = (1 − θ )d ln λ + θd ln S < 0.
−1
σ −1
18 The aggregate impact of an exogenous firm-specific productivity shock is given by dlnQ
dlnϕi

qj

qj

dj

+

qσ−1 τ1q σ −1(σ − 1)[ R iσ−1

q σ −1= R iσ−1

dj

i qi
i qi
− R iσ−1 1i ] = pPQ+ (σ − 1)[ pPQ− `Li ], which shows that it depends on firm i’s sales share

qj

τj dj

and the gap between its sales share and labor share. A firm is relatively subsidized if 1/τi is larger than the
average level of distortion. In a closed-economy, with no free entry and exogenous markup, the gap is thechange of labor income share.

19

resource allocation component is negative.
The same reasoning applies for the trade cost shock. Associated with a lowering of trade
costs is a positive "pure technology" effect (the first two terms of welfare in Proposition 2),
as well as a distortion-induced entry and reallocation effect (the last two terms ), where γdand γs and the change in S summarize the reallocation of resources towards or away from
more distorted firms. When a relatively subsidized firm expands due to trade, allocative
efficiency deteriorates, and it is welfare-reducing.
Having established sufficiency conditions in the special case, we can also now examine
the general case. The necessary condition for the resource allocation term to be negative
is: either γs ≤ γd or d ln S ≤ d ln λ. Intuitively, misallocation happens when the input
elasticity is smaller than the output elasticity, i.e. more resources are used to produced thesame unit of output. The following Lemma presents a sufficient condition for γs ≤ γd fora more general distribution for productivity and distortions.
Lemma 2. Suppose (τ, ϕ) are jointly log-normally distributed with standard deviations of στ and1
σϕ and correlation ρ. When στ ≥ σ−σ ρσϕ ,

1. the cumulative labor share distribution stochastically dominates the cumulative sales share
distribution according to the likelihood ratio order.
2. the hazard functions γs ≤ γd and shares S ≤ λ at any cutoff, hence moving from a closed
economy to an open economy, the reallocation term is always negative.
PROOF: Appendix B.3.
1
Under the condition στ ≥ σ−σ ρσϕ , the cumulative labor share distribution stochasticallydominates the cumulative sales share distribution according to the likelihood ratio order.
We can prove the input elasticity is always smaller than the output elasticity at any cutoff,and the share of variable labor used in producing domestic goods is always smaller than
the share of expenditure on domestic goods, which implies from close to open d ln S is
more negative than d ln λ. The reallocation term is always negative.

20

2.3

Numerical Example

To unpack the theoretical results and to provide more intuition for the mechanisms that
underpin these results, we next turn to a numerical example of the benchmark model withsymmetric countries. The joint distribution between productivity and distortions is taken
to be joint log-normal with standard deviations of στ = σϕ = 0.5 and correlation of ϕ and τof ρ = 0.8. The elasticity of substitution σ = 2, and the fixed costs are f = 0.03, f x = 0.035,
f e = 0.01.
First, the example illustrates that welfare (Eq. 14) can fall when the economy opens up
to trade. Figure 1.1 (a) plots the level of welfare against import shares under the alternative
scenarios: the efficient case without distortions, the case with distortions, and when theeconomy is closed or open. Three observations immediately follow: 1) that there is a
welfare loss in the case with distortions compared to the case without; 2) opening up
to trade leads to welfare gains in the efficient case; however, 3) opening up engenders awelfare loss in the presence of distortions. Taking the differences between the open andclose economy in either case, with or without distortion, we plot the welfare change after
trade in Figure 1.1 (b). It is clear that there is welfare loss with distortions in our benchmark.
Figure 1.1: Welfare and the Change from Trade
0.2

Gain from Trade

0.15
0.1
efficient

0.05
0
-0.05
-0.1

bench
0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Import Share(a) Welfare(b) Gain from Trade

Second, we show the welfare decompositions according to our welfare formula (15) in
Proposition 2. Figure 1.2 (a) displays the three components in our benchmark model— theACR term, entry Me , and reallocation. Through the impact of technology, welfare increases
with lower trade cost, i.e the ACR term increases with the import share. Trade, however,21exacerbates the misallocation and reduces allocative efficiency. Hence, the reallocation term
becomes more and more negative as the import share increases. Overall, the resource misallocation effect dominates, and the benchmark shows welfare losses with trade. In thecurrent example, the entry effect plays very small role. Figure 1.2 (b) shows the decomposition without distortions. In this case, there is no misallocation and the resource allocation
term remains zero. The entry effect becomes more negative with trade, which mitigates theincrease in ACR. Overall, there is a welfare gain after trade.
The numerical example also shows using import share to infer welfare changes when
distortions are present gives rise to markedly different results. Figure 1.2 (a) shows that
ACR invariably predicts gains to trade, rather than losses. Figure 1.2 (b) shows that in
the absence of distortions, ACR is a good approximation for welfare in the efficient case.
But using ACR under distortions leads to a large departure: our benchmark results in thisexample predicts welfare losses rather than gains. Using aggregate observables to infer
welfare gains as in ACR can thus be very misleading in the presence of distortions.
Figure 1.2: Welfare Decomposition
0.3

0.3
0.2

ACR

0.1

Gain from Trade

Gain from Trade

0.2

Me

0
-0.1

Bench

-0.2
Reallocation

-0.3
-0.4

0

0.1

0.2

ACR

0.1

Efficient

Reallocation

0

Me

-0.1
-0.2
-0.3

0.3

-0.4

0.4

0

0.1

0.2

0.3

Import Share

Import Share(a) With Distortion(b) Without Distortion

0.4

Third, we examine the precise mechanisms that explain how trade can reduce aggregate efficiency. In the same numerical example, Figure 1.3 illustrates how distortions affectfirm selection. The density of firms is shown by a heat map of firms that lie along a
positively sloped distortion-productivity line. It is clear that the productivity cutoff for
production and exports is no longer determined solely by productivity, but also by domestic distortion. Only firms below the cutoff line can operate. In this figure, a large mass ofhighly-productive firms are excluded from servicing the market altogether. As the econ22

omy opens up, the cutoff line is shifted further downward. Even if firms have the samelevel of productivity, some with higher taxes may be displaced while those with lower ones
will survive. This downward shift of the cutoffs allows for some low productivity and highsubsidy firms to survive and gain market share.

3

Figure 1.3: Selection Effects
.025

log t

.015

1

.01

(mean) den

2

.02

.005

0

0
-1

-.5

0
.5
log Productivity

1

1.5

Autarky production cutoff
Open production cutoff
Export cutoff
correlation=0.8

Another way to show the impact on selection is to examine firms’ market share. The twopanels in Figure 1.4 plot the market share of firms, both in the closed and open economy.
The left panel is the case without distortions. Firms with the same productivity level
have the same marginal cost; their market share, above a cutoff productivity, rises with
their productivity. Comparing the blue and red lines show that above the export cutoff,
more productive firms have higher market shares in the open economy than in the closed
economy, demonstrating that these firms expand under trade liberalization. This happensat the cost of displacing other less productive firms’ market share, or driving them out of
the market entirely. Here, the example clearly demonstrates that resources move from lessproductive to more productive firms as an economy opens up to trade.
The right panel shows the firm’s market share in the case with distortions. Firms may
share the same marginal cost and face the same potential revenues. However, their after-tax
profits may differ, and thus their market share can also differ. Consider the point at which
23

Figure 1.4: Selection Effects
Market Share (no distortion)

0.08

Market Share (with distortion)

0.06

0.07

0.05

0.06

0.04

0.04

Percent

Percent

0.05
Open

0.03

Open

0.03

low , low

0.02

0.02

0.01

0.01
0
-2

Medium

Close
-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0
-3

High , high

Close
-2.5

-2

-1.5

-1

-0.5

0

log(1/mc)=log( / )log(1/mc)=log( )(a) Without Distortion(b) With Distortionthe (log) effective productivity level (ϕ/τ) is at −1. At this point, a firm with high, medium
and low level of productivity face the same marginal costs. However, the high productivity
firm is also subject to high taxes and thus low after-tax profit, and does not make the cut
for production. The medium-tax-medium-productivity firm has positive market share but
loses out to the low-tax-low-productivity firm when the economy opens up. Resources
are reallocated from the more productive to the less productive firms. Also, there is nolonger a neat line up of market shares according to productivity: there is a wide range of
productivities for which production is excluded.19
Distribution of Distortions. The distribution of distortions is an important determinant
to the gains to trade. There are two key parameters: ρ, the correlation of τ and ϕ, and στ ,the dispersion of τ. The correlation of distortion and productivity is important insofar asa higher correlation means that more productive firms are more likely to be excluded fromthe market. But reductions in welfare is possible even when the correlation is negative. The
reason is that for any given productivity, it is always the more subsidized firms can export
and highly taxed firms exit, leading to a possible welfare loss. Figure 1.5 (a) illustrates this.
It compares the gain from trade for ρ = 0.8, under our benchmark numerical example, andfor ρ = −0.8, where productivity and distortion are highly negatively correlated and otherparameters are the same as in the benchmark example. Under ρ = −0.8, the welfare gain
19 This is also true if the distortions are input wedge on all the labor a firm uses. Firms face higher input

wedge would have a lower profit in a market.

24

(loss) from trade is always larger (smaller) than that in the case of ρ = 0.8. But when theimport share is below 20%, there are still losses from trade.
Figure 1.5: Gains/Loss from Trade
0.25

0

0.2

-0.02

Gain from Trade

Gain from trade

0.15
0.1
=-0.8

0.05
0
-0.05

-0.06
-0.08

t

=0.5

=0.8

-0.1

-0.1
-0.15

-0.04

-0.12
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t

0.05

=0.6

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Import Share

Import Share(a) Correlation Matters(b) Dispersion Matters

Figure 1.5 (b) compares the gain from trade under different στ and other parameters are
the same as in the benchmark example. The welfare gain (loss) from trade is always larger(smaller) when στ is smaller.
In summary, the size of welfare loss after opening up depends on the correlation of ϕ andτ and the dispersion of τ. The firm level data helps us identify these parameters. Specifically, in the quantitative section, we will use the firm-level output and use its dispersion
and its correlation with firm inputs to estimate ρ and στ .

3

Empirical Results

We proceed to investigate whether key empirical implications of our model is supportedby the data. Inferring firm-level distortion directly from the data has its challenges. Still,
key properties and relationships of these measured wedges and productivity support core
implications of our model. Though our focus is to estimate a structural model to infer
welfare and productivity gains to trade, without directly using these empirical measures,the exercise that follows is of independent value, as well as providing a robustness check.

25

Data. Our data for Chinese firms are from an annual survey of manufacturing enterprises
collected by the Chinese National Bureau of Statistics. The dataset includes non-state firms
with sales over 5 million RMB (about 600,000 US dollars) and all of the state firms for the
1998-2007 period. We have information from the balance sheet, profit and loss statements,
and cash flow statements, which incorporate more than 100 financial variables. The raw
data consist of over 125,858 firms in 1998 and 306,298 firms by 2007.

Customary Measured Distortions. To back out factor and output distortions we adopt a
Cobb-Douglas production function for a firm i in industry j, y ji = ϕ ji kαji `1ji−α , where y ji ,` ji , k ji represent the output, labor, and capital stock of firm i in sector j. The marginal
revenue product of labor and capital is ∂( p ji y ji )/∂(` ji ) and ∂( p ji y ji )/∂(k ji ), and with firm
profit maximization, yields
MRPL ji ≡p ji y ji
σ−1(1 − α j )= τji` w jσ` ji

MRPK ji ≡σ − 1 p ji y ji
αj= τjik r j ,σk ji

where w j and r j denote industry-level wages and interest rates. These marginal products
are proportional to the average products, assuming common markups and capital elasticities within the industry, and no fixed cost as in Hsieh and Klenow (2009) (HK). Firms
equalize the after-tax marginal revenue products of factors. In the absence of distortions,revenue per person should be equalized across firms. In the presence of distortions, afirm that faces higher taxes will end up with a higher marginal revenue product and lesscapital/labor than an otherwise identical firm facing a subsidy. Equilibrium allocationsyield"
p ji y ji ∝# σ −1ϕ ji(τjik )α j (τji` )1−α j,from which firm-level productivity can be inferred asϕ ji = Pjσ−1 X j 1 ( p y ) σ−σ 1
ji ji
1− σ

26

α j 1− α jk ji ` ji,(16)

which is also referred as TFPQ as HK.
It is the relative marginal revenue and relative productivities—deviations from the industry mean—that matters. Thus, the measured relative marginal revenue product, or
the relative average product (ARPK ji ), is calculated as log p ji yij /k ij − log ARPK j where
ARPK j is the industry mean of the average product. The same holds for the measured
marginal revenue of labor. The elasticity of output with respect to capital in each industry
is taken to be 1 minus the labor share in the corresponding industry in the U.S, following
HK. The reason that labor shares are not computed from Chinese data is that the prevalenceof distortions would affect these elasticities, and industry-level elasticities and distortions
cannot be separately identified. The U.S. is taken to be the benchmark as the relatively
undistorted economy. These labor share comes from the U.S. NBER productivity database,which is based on the Census and the Annual Survey of Manufactures (ASM). We takethe benchmark elasticity of substitution parameter σ to be 3, but experiment with othervalues within the conventional range. Different from HK, we take a firm’s employment tomeasure ` ji rather than the firm’s wage bill. This addresses the problem that Chinese wage
data implies too low of a labor share as measured by input-output tables and the national
accounts. We define the capital stock as the book value of fixed capital net of depreciation.

We find large dispersions in measured distortions in China, similar to the levels in HK
for the year 1998 and 2007. Measured distortions have come down over time, between 1998
and 2007, as evident in Table 1. There is also greater dispersion in the average product of
capital than there is in the average product of labor.
We next turn to investigating further what factors are systematically related to measured
distortions. Table 2 reports the regression results of the relative average product of capitalof a firm on a set of variables. The coefficient on firm-productivity is large and significant;
1 percent increase in relative productivity (TFPQ) is associated with a 0.7 percent increase
in relative distortion. Moreover, more than half of the variation in distortions is explainedby productivity alone. The positive relationship is consistent with the predictions of ourmodel, as it predicts that firms that face higher tax have to be more productive to survive. The same is true for the results on exporters: given productivity, firms must have27

lower taxes on average in order to export. What these wedges are and where they mightcome from is suggested by its systematic relationship with firm characteristics: state-owned
enterprises and Foreign-owned firms are subject to lower taxes on average, given productivity. If we perform these regressions using the average product of labor instead, in Table
3, results are similar.
Table 1: Dispersion of Distortions
1998 2001 2004 2007
std(ARPK) 1.348 1.306 1.241 1.185
std(ARPL) 1.184 1.039 0.940 0.923

Table 2: ARPK Regressions
VARIABLESln( TFPQ)(1)
ln( ARPK )(2)
ln( ARPK )(3)
ln( ARPK )(4)ln( ARPK )(5)ln( ARPK )0.652***(147.7)0.697***(153.0)0.706***(154.8)0.705***(153.9)-0.00178***(-8.772)-3.617***(-134.6)-3.280***(-60.38)-3.204***(-54.16)-3.173***(-53.37)0.707***
0.711***(160.3)(168.1)-0.00191*** -0.00174***(-9.477)(-9.386)-0.116***-0.109***(-3.388)(-3.313)-0.460***-0.379***(-19.74)(-20.60)-0.233***(-13.82)-3.049***-3.042***(-44.45)(-44.88)age
1.soe
1.foreignown
exporters
Constant(6)ln( ARPK )Observations 1,616,507
1,616,507
1,506,572
1,505,657
1,505,657
1,505,657
R-squared
0.566
0.628
0.640
0.640
0.655
0.659
Time FE
Yes
Yes
Yes
Yes
Yes
Yes
Industry FE
No
Yes
Yes
Yes
Yes
Yes
Location FE
No
No
Yes
Yes
Yes
Yes
Robust t-statistics clustered at the four-digit industry level in parentheses*** p<0.01, ** p<0.05, * p<0.1

Relationship between productivity and distortion. There is no apriori reason to believethat more productive firms are associated with higher wedges, as is shown in the regression
28

Table 3: ARPL Regressions
VARIABLESln( TFPQ)(1)
ln( ARPL)(2)ln( ARPL)(3)ln( ARPL)(4)ln( ARPL)(5)ln( ARPL)(6)ln( ARPL)

0.530***(110.7)0.570***(228.5)0.569***(222.5)0.568***
0.565***(224.2)(228.4)-0.00161*** -0.00140***(-9.072)(-8.783)-0.0840***(-7.136)
0.0615***(3.925)-3.593***(-123.2)-3.274***(-109.1)-3.229***(-103.2)-3.201***(-100.5)

0.567***(229.4)-0.00128***(-8.440)-0.0787***(-7.057)
0.123***(8.317)-0.175***(-27.08)-3.167***(-97.30)

age
1.soe
1.foreignown
exporters
Constant-3.172***(-95.80)

Observations 1,616,507
1,616,507
1,506,572
1,505,657
1,505,657
1,505,657
R-squared
0.619
0.691
0.699
0.700
0.701
0.705
Time FE
Yes
Yes
Yes
Yes
Yes
Yes
Industry FE
No
Yes
Yes
Yes
Yes
Yes
Location FE
No
No
Yes
Yes
Yes
Yes
Robust t-statistics clustered at the four-digit industry level in parentheses*** p<0.01, ** p<0.05, * p<0.1

29

results. To show this graphically, we next plot the relationship between TFPQ and ARPK in
Figure 1.6; a similar relationship holds for TFPQ and ARPL. Though this relationship was
not important in the special case of HK assuming a joint log normal distribution between
the two variables—it does matter for more general cases. Moreover, it is vital for ourquantitatively analysis.

Figure 1.6: Correlation Between Measured Distortion and Measured Productivity

Note: ln(1 + tauK ) denotes the logged measured distortion, ARPK, and ln z is
the logged measured productivity, TFPQ.

The observed correlation between TFPQ and ARPK (or ARPL) cannot be treated to be
the same as the underlying correlation between ϕ and τ for three reasons. First, the observed relationship may be an outcome of a selection process as our model demonstrates.
Even if the underlying correlation is negative, the selection mechanism can induce the observed correlation to become positive, for the simple reason that high-taxed firms must be
more productive in order to stay in the market. The selection mechanism will strengthen
any underlying correlation between the two variables. In order to compute the impact of
distortions on welfare and productivity gains, one would need to know the underlying
correlation ρ, and therefore one would need micro data and a structural model to uncoverit. What is also interesting about the relationship in the data is the line that seems to cut
firms from above—reminiscent of the cutoff line endogenously generated in the model.
Costa-Scottini (2018) and Ho (2010) assume a perfect correlation of (log) productivityand (log) wedges in their analysis. We have shown why the assumption of an exogenous

30

correlation may be misplaced. Moreover, this assumption counters findings in the data inthat 1) measured (log) productivity and (log) wedges are far from perfectly correlated, and2) exporters have lower wedges. As long as these two variables are not perfectly correlated,
selection will affect their measured correlation and one would need to estimate it from themodel.
Second, the observed dispersion is not the underlying dispersion because it is among
the operating firms. Lastly, with fixed cost we can only measure average revenue product.
This measured revenue productivity does not map onto actual productivity or wedge as it
affected by productivity, fixed costs and distortions altogether.
For these reasons, we do not use directly the empirically-measured wedges, observed
correlations, or distributions in the data to assess the impact of trade on welfare. Instead,
we estimate the underlying joint distribution of distortions and productivity, costs of producing and exporting so as to match the observed patterns of firms’ outputs, inputs, andexports. Then we evaluate how the presence of distortions change the impact of trade on
productivity and welfare, and how much trade has contributed to Chinese growth in a
decomposition exercise. We turn to this analysis in the subsequent section.

4

Quantitative Results

This section presents estimates of the quantitative effects of trade liberalization when taking
into account the existence of domestic distortions. The two countries, Home and Foreign,
are calibrated to data corresponding to China and the U.S.. The quantitative exercise isimportant for gauging welfare gains to trade because measured distortions following theconventional approach, though with independent value, has its limitations. Table 4 reports
the calibrated and chosen parameters. The Home labor L is normalized to 1 and Foreignlabor L f to 0.2 to match the relative labor force of US to China. Productivity levels are
set to match the relative GDP of US to China. Given that Foreign affects Home only
though aggregate variables, we can assume that Foreign is absent of distortions, while
taking f e , f , f x , τx , σϕ to be the same as those in Home. We set the elasticity of substitution
between varieties σ to be 3, the one taken in HK. This value is consistent with the estimates31

from plant-level US manufacturing data in Bernard, Eaton, Jensen, and Kortum (2003).
Table 4: Model Parameters
Parameter
Elasticity of substitution σHome labor L
Foreign Labor L f

Value
3
1
0.2

Identification
Normalization
Relative labor size of US to China

Internal Estimation
Entry cost f e
0.2
Fraction of firms producing(one year survive rate in the data)
Fixed cost of producing f
0.015 Mean-lowest 5% ln(kα `1−α )Fixed cost of export f x
0.12
Fraction of firm exporting
Iceberg trade cost τx
1.5
Export intensity
Std. productivity σϕ
1.2
Std of existing firms ln VA
Std. distortion στ
0.9
Std of existing firms ln(kα `1−α )Corr(distortion, productivity) ρ
0.86
Corr(ln VA, ln(VA/kα `1−α ))Mean Foreign prod µ f ϕ
5.5
Relative GDP of U.S. to China
Note: VA denotes value added, k capital, ` employment, Std standard deviation, Corr correlations.

Table 5: Data and Model Moments
Target Moments
Fraction of firms producing
Mean − lowest 5% for ln(kα `1−α )Fraction of firm exporting
Export intensity
Std of existing firms ln(VA)
Std of existing firms ln(VA/kα `1−α )Corr(ln VA, ln(VA/kα `1−α ))Relative real GDP of US to China

Data(2005)

Model

0.85
1.82
0.30
0.41
1.20
0.93
0.41
1.79

0.85
1.53
0.28
0.42
1.26
0.84
0.35
1.77

Note: VA denotes value added, k capital, ` employment, Std standard deviation,
Corr correlations.

The remaining 8 parameters are estimated jointly, to match the model moments with
their data counterparts. Table 4 and 5 reports the estimated parameters and the momentsin the data and model. The moments we choose are the ones that are most relevant andsensitive to variations in model parameters. Clearly, every parameter matters for the general equilibrium and affects other moments. However, there is by and large a clear correspondance between certain parameters and moments. The parameter most relevant for
matching the fraction of surviving firms is the entry cost f e , as ωe E[π (φ, τ )] = w f e . Lower
32

entry costs induces more entrants to pay the costs, and the result is a lower fraction ofsurvivors. Next, to identify the fixed cost f , one needs only to turn to the smallest firms,
which have their profit just about cover fixed cost. That is, the after-tax profit π = w f andw`min = (σ − 1)w f , and the mean of firms’ labor w`mean = (σ − 1)w( f e /ωe + f ). Hence,the difference between mean and lowest 5% of input helps identify f .
We calibrate f x and τx to match the export participation and intensity in Chinese manufacturing. The resulting parameter τx = 1.5 is inline with the estimate of 1.7 in Anderson
and Van Wincoop (2004), and the 1.83 in Melitz and Redding (2015). The dispersions in
productivity and distortions, and their correlation are important for matching the observed
joint distribution between value-added and inputs in the data. Table 5 shows that thediscrepancy between our model and data moments is reasonably small, though we underestimate the dispersion in distortions and slightly overestimate the dispersion in size.
An important variable is the correlation between value added and distortions, Corr (ln VA,
ln(VA/kα `1−α ). This variable is more positive the higher is ρ στϕ , where ρ is the underlyσ

ing correlation between distortions and productivity. A higher underlying correlation anda lower dispersion in distortions raise the observed correlation between value added andinputs.

4.1

Implied Gains from Trade and Loss in TFP

Table 6 reports the gains from trade and efficiency losses for both Home and Foreign.
The upper panel compares welfare and TFP in the open economy to those in the closed
economy. In the benchmark estimation, the gains from trade for Home is 4.4%. Without
distortions, the gains from trade is more than doubled (9.8%). Foreign’s gain from tradeis about 8.2% when Home has domestic distortions. Eliminating Home distortions allows
Foreign to benefit more—a 19% of welfare gain.
Note that Proposition 2 holds for asymmetric countries, so that we can decompose thewelfare change according to its main equation, (Eq.15). In our benchmark, the importshare is 30.8%, which implies that the change in domestic output share is −d ln λ = 0.368,
and the ACR term in the equation is 17.9%. But there is a large and negative reallocation
term showing up in China, amounting to −18.2%. Taking only the ACR component would
33

Table 6: Welfare and TFP
WelfareHome (%)
BenchmarkNo-distortion
Foreign(%)
BenchmarkNo-distortion

BenchmarkHome Closed-Economy

4.4
9.8

Open relative to close
TFP
Import Share
-2.9
13.3

30.8
20.8

8.2
12.9
17.9
18.9
13.3
35
TFP loss: Distortion relative to no-distortion
Overall loss
Misallocation
Entry-selection
140.4
119.2
21.2
124.2
118.7
5.4

Decomposition
ACR
Reallocation
17.9
11.4

-18.2
0

9.0
19.1

0
0

overestimate the gains to trade by 407%, according to our model. Moreover, if one were toignore the presence of distortions and followed the usual approach to ACR using a tradeelasticity– for instance of 4, estimated in Simonovska and Waugh (2014), then the resulting
welfare gains would be 9.2% – more than doubling the gains that our model predicts.
As Melitz and Redding (2015) show, trade elasticities vary with trade costs, so whether apartial, a full or an average theoretical trade elasticity is used will matter for the size of the
ACR gain. But regardless, the ACR term is positive no matter which elasticity is utilized.
What we show here is that our channel leads to a sizeable welfare loss.
We assume here that Foreign does not face any distortions, and thus the reallocation
term for Foreign is zero according to our Proposition 2. Since Foreign has a smaller importshare than Home, one would arrive at the conclusion that more gains would accrue toHome than to Foreign, according to ACR. But our benchmark model reveals the opposite:
Foreign has gains to trade that is almost double that of Home, whereas using conventional
ACR, Foreign’s gain is only half of that of Home. Without distortions, the gains to trade
would double for both economies, suggesting that countries would gain more from tradeby undertaking domestic reforms.
We also compare TFP before and after trade liberalization. To compute TFP, we construct
the real GDP using producer price index following Burstein and Cravino (2015). Our
benchmark results show that opening up leads to a 3% loss. In contrast, without distortions,
TFP increases by 13.3%. Hence, contrary to the standard predictions, trade liberalization34

can exacerbate rather than improve resource allocation, causing a decline rather than a risein TFP. As Foreign has no distortions, the TFP levels are basically the same between thetwo models.
The lower panel of Table 6 reports Home’s TFP losses due to distortions, both for aclosed and open-economy case scenario. TFP loss is defined as the difference between
efficient TFP when there are no distortions and that when there are distortions. Different
from Hsieh and Klenow (2009), our efficient TFP considers endogenous entry. We therefore
can decompose the deviation of TFP from its efficient level into a misallocation effect amonga fixed set of operating firms, and a misallocation effect generated by entry and selection
into producing and export:
log TFPe f f − log TFP = log TFPFX − log TFP + log TFPe f f − log TFPFX ,
|
{z
} |
{z
}
misallocation loss

entry and selection loss

where TFPe f f pertains to the case without distortions, TFPFX corresponds to the level inthe case with distortions but where M, ϕ∗ , and ϕ∗x are fixed, in a closed economy.20
Not surprisingly, there are large TFP losses for Home under domestic distortions. In theclosed-economy case, eliminating these distortions would increase China’s TFP by 124%.
The TFP losses are larger, 140%, in the open economy benchmark model where China
has an import share of more than 30%. Looking at its decomposition, the majority of thelosses appears to come from misallocation among existing firms. One reason is that theshare of surviving firms is high according to available data, but the survival rate could beoverestimated as the dataset includes firms only of a certain scale. If smaller firms wereobserved, then it is possible that a lower survival rate would make the losses coming fromthe entry and selection margin larger.
20 Here are the definitions of TFP
e f f and TFPFX

TFPe f f =


 1
 Z ∞
 1
Z ∞
σ −1
σ −1
σ−1
σ−1
Me f fϕσ−1 µe f f ( ϕ) dϕ, TFPFX =M
ϕσ−1 µ ( ϕ) dϕ.
σ
σ
ϕe f f ∗
ϕ∗

35

4.2

Decomposing China’s Growth from 1998-2005

The rapid growth in China over the last four decades has been one of the most remarkablephenomena the world has witnessed in recent history. In between 1998 and 2005, its real
GDP increased by 57%. Accompanying this development was a combination of domestic
reforms and opening up programs—policies that fostered trade and FDI inflows. As a result, both trade and technological progress increased over time, while measured domestic
distortions concurrently fell. A natural question is how much of the growth is attributed
to trade over this period. Other competing factors include technological improvement,
factor accumulation, and domestic reforms—that is, the allocative gains associated with areduction in distortions. In what follows, we perform a quantitative analysis to answer thisquestion. Specifically, we recalibrate the model parameters for the year 1998 and compare
the implied GDP and TFP levels to those in the benchmark year, 2005. Overall, our results
attribute the majority of China’s GDP and TFP growth to technological improvement, capital accumulation, and a mitigation of distortions. Trade alone contributes to only about 8%
of GDP growth.
Table 7 reports the moments for both 1998 and 2005. The starting year is taken to be
1998, as it is the first year in which firm-level data is available, and three years before
China joined the WTO. Compared to the year 2005, trade intensity was significantly lower
in 1998, both in terms of the fraction of firms that export, and also the export intensityof these firms. Distortions were large in the earlier years, as seen by the fact that thedispersion of measured distortion was about 20% higher in 1998 compared to 2005. Thisimplies a higher trade cost τx and dispersion of distortion στ in 1998— at about 43% and
20% higher than the level in 2005. The mean TFP in 2005 is about 45% higher than that in1998, reflecting technological improvements and factor accumulation over time.
These estimates are then used to run counterfactual experiments, in order to decompose
China’s growth in between 1998 and 2005. The factors considered include technologicalprogress, input accumulation, and the reduction of trade costs and domestic distortions. In
each experiment, the parameters for the year 1998 remain fixed, while each of the following
parameters– mean TFP µ ϕ , trade cost τx , or dispersion of distortion στ —are allowed tovary to its 2005 level. Table 8 shows that the increase of technology and inputs alone36

Table 7: Data, 1998 and 2005
Target Moments
Fraction of firms producing ωeMean − lowest 5% for ln(K α L1−α )Fraction of firm exporting
Export intensity
std of existing firms ln(VA)
std of existing firms ln(VA/K α L1−α )Corr(lnVA, ln(VA/K α L1−α ))Relative real GDP of US to China
Change of China’s real GDP

Data (1998)

Data (2005)

0.77
2.04
0.25
0.30
1.33
1.12
0.47
2.50

0.85
1.82
0.30
0.41
1.20
0.93
0.41
1.79
57%

lead to a 44% increase in GDP and a 46% increase in TFP. Reduction in trade costs would
independently boost GDP by 8% and TFP by only 3%. In contrast, lowering the dispersionof distortions increases GDP by 66% and TFP by 69%.21
A notable point of comparison is with Tombe and Zhu (2019), which, despite adopting
an altogether different approach, finds also small gains to trade. In their model that features
migration across regions and sectors in China, international trade contributes to only 7%of productivity growth in between 2000 and 2005. In other words, international trade has
led to very little allocative benefits of labor across regions and sectors—as compared todirect reforms that lower migration costs or reductions in internal trade costs. Their model
does not feature distortions at the firm level that can render trade’s allocative benefits evensmaller. This leads us to find an even smaller effect of trade on productivity in China overroughly the same period.
Of course, a caveat is that trade may also help reduce domestic distortions. If, say, theWTO requires certain kind of domestic reforms as a pre-condition for entry, then some of
the technological improvement and reductions in the level of distortions could be partiallyinduced by opening up policies. We do not consider this here—but examine this issue
in the following subsection. Also, this quantitative exercise of course also ignores otherpotential channels of gains to trade, such as pro-competition effect of trade, or potentiallytransfers of technology (Ramondo and Rodríguez-Clare (2013))–though these effects may
21 Note that the contributions to GDP or TFP increase don’t add up to 100% because the productivity

distribution and fixed costs have also changed from 1998 to 2005. Furthermore, there are interacting effects
on mean TFP, trade cost, and distortion dispersions.

37

still be quantitatively small. The point we make here is that in our benchmark framework,
the contribution of trade pales in comparison to the contribution of domestic policies andtechnological progress in accounting for China’s growth experience. Part of this reason
may be attributed to the lingering presence of distortions.
Table 8: Decomposition of China’s Growth between 1998-2005
BenchmarkCounterfactual Change from 1998-2005:
Technology and inputs alone (Increase mean ϕ)Trade alone (Decrease τx )Distortion alone (Decrease στ )4.3

Change of Real GDP

Change of TFP

57%

56%

44%
8%
66%

46%
3%
69%

Selection through Export: an Out of Sample Test and Extension

To examine whether Chinese firm characteristics change when they become exporters, we
examine the relationship between measured distortions and firm export status in both thecross-section and the time series dimension. According to the model, given productivity—
exporters face a lower distortion due to selection. This implication, as we have shown
previously, is broadly consistent with the data. We then use the time series data to check
whether ‘measured’ distortions change when firms enter the export market. Model implications are qualitatively consistent with the data. Finally, we consider model extensions
with export rebate and allow for different distortions when exporting.
The first two columns of Table 9 reports the data and the model regressions of measured
distortion on measured productivity TFPQ and a dummy of exporters. Both the modeland the data exhibit a pattern whereby exporters face a lower marginal product. Note thatthe differences between exporters and non-exporters were not targeted. There is a stronger
selection effect in the model than in the data—exporters’ marginal product is about 64%
lower than non-exporters in the model, compared to 26% in the data.
We further consider whether the measured marginal product or distortion vary with entering or exiting the Foreign market. This will help us to understand whether differences38between exporters and non-exporters stem from selection or additional/different distortions when exporting. Throughout the sample period, we sort exporters into three types:‘always exporters’ are those who are exporting throughout the sample years 1998 to 2007,‘starters’ are those who started to export after 1998, and ‘stoppers’, who stop exporting
sometime in the interim years. Entry effect measures the percentage difference of measured distortions for starters, between the post- and pre-exporting entry periods. Exit effectmeasures percentage difference of measured distortion for stoppers, between the post- andpre-exporting exit periods.
As shown in Column 3 of Table 9, in the data, ‘always exporters’ have a lower marginal
product. Firms’ measured marginal products decrease when they start exporting and increase when they exit. Note that although the firm-level distortions are exogenous in themodel, the measured distortions do change after firms start exporting— due to the fixed
costs of exporting.
In our benchmark model, if trade costs decreases from their 1998 level to their 2005
level, some firms would start to export, and their measured distortion would decrease—as shown in the entry-effect of Column 4. This is true in the data, as ‘always exporters’and ‘starters’ have lower measured marginal product. Our benchmark model shows arelatively large selection effect since the model does not have as large heterogeneity as in
the data, and a relative small entry effect. In reality, the change in a firm’s marginal productafter exporting could be driven by multiple factors. For example, exporters face different
distortions from non-exporters, or there are endogenous distortions that change with tradeliberalization. We proceed to explore these possibilities.
In Column 5, we introduce a 10% tax rebate after exporting. This generates an evenlarger selection effect and the measured wedge further decrease upon exporting. In thiscase, export subsidies from tax rebate benefit the Foreign country, and Home welfare gains
from trade decreases to 1.6% and TFP loss increases to 4.5%.
Column 6 considers an extension that firms face different distribution of distortion whenexporting. We use the standard deviation of export intensity to discipline the dispersionof the additional wedges when exporting. To be more precise, we estimate the mean andstandard deviation of the extra export distortions to match the standard deviation of export
39

intensity and the regression coefficient on always exporters. When the trade cost is reduced
as in the benchmark, the coefficient on starter and the entry effect are similar to the data.
Not surprisingly, the loss from trade is much larger in this extension than in the benchmark.
The reason is because there are additional direct distortions on the selection of firms to
Foreign market when open up to trade.
Table 9: Measured Marginal Products, data vs model
VARIABLES

(1)
Data

(2)
Benchmarkentry effectexit effectstarter
stopperalways exporters
log(TFPQ)
exporters
Constant

0.636***(250.5)-0.264***(-24.14)-3.258***(-106.2)

0.652

(3)
Data

(4)
Benchmark(5)
Export rebate(6)
Different τ in export-0.104***(-12.69)
0.0315***(4.574)-0.101***(-21.74)-0.0891***(-20.98)-0.301***(-23.47)
0.638***(254.9)-0.050-0.103-0.09-0.429-0.400-0.08-0.768-0.791-0.324

0.653

0.654

0.613

-3.255***(-107.0)

0.414

0.425

0.714

-0.6370.401

Observations
1,587,629
1,584,242
R-squared
0.823
0.826
Time FE
Yes
Yes
Industry FE
Yes
Yes
Note: Robust t-statistics in parentheses, *** p<0.01, ** p<0.05, * p<0.1
The dependent variable is measured average product, which is log( TFPR) in the data
and log(VA/(` + f )) in the model.

405Discussion

In this section, we explore a model with endogenous distortion arising from endogenous
markup, which has been extensively studied in the standard trade literature. We show thatthe endogenous markup model runs counter with the data in that exporters in the model
face a higher markup and distortion. We then address the issue that misallocation could
be driven by mismeasurement of inputs or outputs. We show that even taking out thestandard measurement errors, there are still large distortions remaining among Chinese
firms.

5.1

Endogenous distortions

Here we build a model with endogenous markup as in Edmond, Midrigan, and Xu (2018).
The consumer’s problem is the same as before.

Final goods producer Final goods producers are competitive and produce with intermediate goods with a Kimball aggregator


Z
ω ∈Ω

γ

q
Q


dω = 1,

where γ(·) follows Klenow and Willis (2016) specification as

γ

q
Q



" 
 

σ
1
σ 1
−
1
= 1 + (σ − 1) exp
εε
Γ
,
−Γ
ε
ε ε

ε

σ (q/Q) σ
,
ε
ε

!#
,

σ > 1, ε ≥ 0 and Γ(s, x ) denotes the upper incomplete Gamma function Γ(s, x ) =(17)

R ∞ s −1 − t
e dt.
x t

The demand function for each intermediate good producer is therefore given by

where D is a demand index, D =

p(ω ) = γ

0

hR



0
ω ∈Ω γ

q(ω )
Q

q(ω )
Q

41




PD,

i −1
q(ω )
dω
.
Q

(18)

Intermediate good producer

The problem of an intermediate good producer is similar as

before except it faces a demand function as in equation (18). The firm will choose the priceas a markup over the marginal cost,p=wτ
.
σ − (q/Q) ϕσεσ

Note that the markup is endogenous and depends on the size of the firm, the higher thequantity a firm sells, the higher the markup it charges. The firm’s optimal production andprofit increase with ϕ and decrease with τ. Hence there exists a cutoff ϕ∗ (τ ), firms produce
when ϕ ≥ ϕ∗ (τ ).
Figure 1.7: Measured MPL and Productivity in an Endogenous Markup Model
0.5closedmeasured log(MPL)

0.4exporter

0.3

non-exporter

0.2

0.1

0
-3

-2

-1

0

1

2

3

4

measured log(productivity)

To compare with the benchmark model, we choose ε as 0.08 to match the aggregate
marginal product of labor of 1.45 as in Edmond, Midrigan, and Xu (2018), while keeping
other parameters the same as in the benchmark. Figure 1.7 plots the relationship between
the measured log( MPL) and the measured log productivity in this model. First, higher
productivity firms produce more and end up with a higher endogenous markup. The
measured MPL is therefore higher. Hence, we observe an upward sloping line for theclosed economy. Second, this upward sloping patterns also show up in the open economy.
Moreover, exporters are more productive and face a higher wedge. Non-exporters facea more competitive market after opening up and charge a lower markup, the measured
wedge is smaller. Around the exporting cutoff, exporters face a lower MPL due to the42fixed cost of exporting. Overall, exporters face higher measured MPL.
In summary, if the observed wedges are purely driven by markups and they endogenously change with trade, we should see that: 1) exporters on average have higher markups,
hence higher—rather than lower—wedges; and given productivity, they should have thesame wedge; 2) measured log (MPL) and log (VA) will be almost perfectly correlated.
These implications are at odds with the regression results shown in section 4.3, where
exporters face lower wedges. Thus, even in this extended model, similar exogenous distortions are needed to match the observed dispersion and correlation. This is consistent
with Song and Wu (2015) and David and Venkateswaran (2017) that the heterogeneity inmarkup explains very limited MPK dispersion in China. Moreover, Arkolakis et al. (2018)
show that the gains from trade in a model with endogenous markup is similar to ACR. In
the current model, the excess welfare gain from trade when removing domestic distortions
is 3.2%, while it is 5.4% (9.8%-4.4%) in the benchmark model.

5.2

Detecting measurement error

Differences in measured average products need not imply differences in true marginal
products. The presence of fixed costs in producing and exporting in our model, for instance, means that the average revenue products differ from the marginal revenue products. For this reason, a model estimation is used to back out the true dispersion in marginal
products. Nevertheless, other types of mismeasurements in output and input may also generate a dispersion in the average revenue products, and thereby affect the measured TFPR—
as shown in Bils, Klenow, and Ruane (2017) and Song and Wu (2015). In this section, we
address the issues surrounding measurement error following the practices adopted in theliterature, and provide more robustness checks for the empirical results.
The main approach involves using panel data to improve the estimates on the true
marginal product dispersion, rather than simply employing cross-sectional data. With thismethod, we find that the measurement errors are small in China, accounting for only 18%of the variation in the average product.22 This 18% includes the mismeasurement of pro22 Bils, Klenow, and Ruane (2017) finds measurement errors can explain about half of variation of average

products in Indian, and about 80% of that in the U.S, but little for China.

43

duction inputs in the presence of fixed cost, which is accounted for in our benchmark.
We exploit three alternative methods to detect measurement error: average annual observations within firms, first differences over years within firms, and covariance between
first differences and average products. All three approaches point to the same conclusion:that 1) there is a large dispersion in marginal products in China; 2) measurement error
only accounts for a small fraction of the dispersion in the measured marginal products (i.e.
average products).
First, if measurement error were idiosyncratic across firms and over time, one can takethe time average of annual observations within firms to wash out these errors, drasticallyreducing the dispersion of average products. The upper panel of Table 10 reports thestatistics when we take the average within firms. The average standard deviation is 1.19
for the average marginal product of capital and 0.96 for the average marginal product oflabor. The standard deviations of value added and the average product of inputs are 1.19
and 0.94, where the correlation between the two variables is 0.4. These results mimic ourbenchmark moments. In particular, the dispersions of average products of inputs are stillhigh. This implies that measurement errors of the iid type cannot explain the observed
dispersions in the average products.
Second, as pointed out by Bils, Klenow, and Ruane (2017), the dispersion of first differences reflect the true distortion if marginal products are constant over time. Calculating
the first differences of value added 4VA, capital 4K, and labor 4 L, and then taking theratio 4VA/4K and 4VA/4 L gives us an alternative measure of marginal products. The
1% tails of both ratios are trimmed, and the results are displayed in the middle panel ofTable 10 for the year of 2001, 2004, and 2007. The dispersions are even higher than those in
Table 1 for the standard measured marginal product of inputs using this measure.
Moreover, the alternative measured marginal products are highly correlated with our
average products. Figure 1.8 plots the ln(4VA/4 I ) against the benchmark average product of input ln(VA/I ) where I is the composite of inputs, I = K α L1−α , where each dot
corresponds to one of 100 percentiles of ln(VA/I ). The regression coefficient at the firmlevel is 0.72, see Table A-1. Note that without measurement errors, the two measures are
perfectly correlated. For the case with only measurement error, the two measures have no44

Table 10: Detecting Measurement Errors
Average annual observation within firmstd(ln(APL)) std(lnVA) std(ln(VA/I ))corr (lnVA, ln(VA/I ))
0.96
1.19
0.94
0.4
First level differences
2001
2004
2007
std(ln(4VA/4K ))
1.82
1.78
1.76
std(ln(4VA/4 L))
1.68
1.60
1.61
RegressionΨΨ (1 − λ )0.53∗∗∗−0.0997∗∗∗(34.58)(−20.65)

std(ln(APK))
1.19

Note: This table reports three ways to detect measurement errors.
The upper panel reports the average annual levels within firms.
The middle panel reports the ratio of first differences as another measure of marginal product,
where 4VA denotes the first difference of value added.
The lower panel reports regression coefficient as in equation (19).
Robust t-statistics in parentheses.

correlation. Hence, the high correlation between the alternative measure and the average
products suggest small measurement errors and a large distortion-induced misallocation.

-2log(ΔVA/ΔI)
-1
0

1

2

Figure 1.8: Measured Marginal Product using First Differences vs TFPR

-3

data
All measurement error
No measurement error
-3

-2

-1

0
log(VA/I)

1

2

Lastly, we follow Bils, Klenow, and Ruane (2017) and run the following regression to further quantify the extent to which measured average products reflect true marginal products:
c i = Φ · log( TFPRi ) + Ψ · 4 Îi − Ψ(1 − λ) · log( TFPRi ) · 4 Îi + Ds + ξ i
4VA
45

(19)

c i and 4 Îi are the growth rate of measured value added and inputs respectively,
where 4VA
and log( TFPRi ) is the measured average products. The underlying assumption here is thatthe measurement errors are additive. The variable of interest in the regression is λ, theσ2variance of distortions relative to that of TFPR: λ = σ2 ln τ . The regression coefficientln( TFPR)

for Ψ is 0.53 and for the interaction of log( TFPRi ) and 4 Îi is -0.0997. Both are significant,and the robust t-statistics are reported in Table 10. The implied λ is therefore 0.81. Hence,
81% of variation in TFPR or average products is accounted for by distortion τ and 19% isdue to measurement errors. The results are robust if we weight the observations with theirshare of aggregate value added or if we control for higher orders of ln( TFPR) to allow forstationary shocks to firms productivity and distortions. See the Appendix D for details.23
In summary, the three alternative ways of sifting out measurement errors using panel
data all point to the result that the dispersion in the average product of inputs are mainly
driven by distortions rather than measurement error typically conceived.

6

Conclusion

This paper evaluates the impact of trade liberalization when the economy is subject tofirm-level distortions. Given its prevalence and importance in developing countries, it isreasonable to ask how trade might affect welfare when these distortions are taken into
account. This paper shows theoretically and quantitatively that opening an economy may
in fact reduce allocative efficiency and exacerbate the misallocation of resources, by helping
firms that are more subsidized (rather than those who are more productive) to expand. The
findings in these papers, nevertheless, in no way disclaims the potential wide variety ofsources and the magnitude of gains to trade beyond what is conventionally modelled andtaken up in the current framework. But it does highlight that these losses could be sizeable
23 Bils, Klenow, and Ruane (2017) also considers the following extension to allow for stationary shocks tofirms productivity and distortions:c i = Φ · log( TFPRi ) + Ψ · 4 Îi − Ψ(1 − λ) · log( TFPRi )4VA+ Γ · [log( TFPRi )]2 + Λ(1 − λ) · [log( TFPRi )]2 4 Îi+ Υ · [log( TFPRi )]3 + Λ(1 − λ) · [log( TFPRi )]3 4 Îi .

46

and comparable to major sources of welfare gains. We use Chinese manufacturing data
in a period of the economy’s rapid integration to demonstrate quantitatively that standard
calculations for welfare may grossly overestimate the gains.
The paper serves as a first attempt to understand the interactions between trade andidiosyncratic firm level distortions on a theoretical level. The assumption of exogenous
distortions is both reasonable empirically but also helpful in laying bare the key theoreticalmechanisms on their interaction with trade. Of course, the next step would be to examine
more thoroughly how trade might partially change the distribution and nature of these
distortions. Extensions of the work are inherently numerous and promisingly fruitful. One
can examine how distortions interact with other channels of gains to trade, such as innovation. One can also examine a dynamic model and the sequence of trade and domestic
reforms. Our work joins the growing body of work and interest on why developing countries’ experience with trade liberalization might have been so curiously diverse and uneven.
Our work hopefully lends itself as one explanation to such a question.

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50

Online Appendix to “Misallocation Under Trade
Liberalization"
by Yan Bai, Keyu Jin, and Dan Lu

A

Model Derivation

Closed Economy Equilibrium. In a closed economy, the free entry condition requires thatthe present value of producing equals the entry cost. The probability of entry ωe is given
by
ωe =

Z Z ∞
ϕ∗ (t)g ( ϕ, t) dϕdt,(A.1)

and the distribution of operating firms µ( ϕ, t)g( ϕ, t)g( ϕ, t)=ωeϕ∗ (t) g ( ϕ, t ) dϕdtµ( ϕ, t) = R R ∞if ϕ ≥ ϕ∗ (t); and 0 otherwise. Let the per-period expected profit conditional on producingbeE[π ( ϕ, t)] =Z Z ∞ϕ∗ (t)π ( ϕ, t)µ ( ϕ, t)dϕdt.

The free entry condition is given by
ωe E[π ( ϕ, t)] = w f e .

(A.2)

The optimal profit function (2) combined with the above expression in the free entry condition yields an equation for P,
PQ
σ



σ−1
P
σ

 σ −1
w

1− σ

Z Z

h
ϕ∗ (t)

ϕσ −1 − σtig( ϕ, t)dϕdt − w fZ Zϕ∗ (t)g( ϕ, t)dϕdt = w f e .
(A.3)

Let Me be the measure of new entrant. The labor market clearing condition requiresq+ f + Me f e ,L = ME
ϕ51

where the average labor demanded by firms E

h

q
ϕ + f

i

is given by Z Z ∞ 

q
q
E
+f =
+ f µ ( ϕ, t) dϕdt.
ϕϕ∗ (t) ϕ

In a stationary equilibrium, the number of entrants equals number of exits, such that
ωe Me = M and
L

M=


σ

fe
ωe + f.

(A.4)

Open Economy Equilibrium. Optimal prices and cutoff functions are straightforward
analogues of the closed economy case. The free entry condition for Home and Foreign
impliesZ Z ∞ hZ Z ∞iσ − 1 σ −1 1− σw
ϕσ−1 t−σ g ( ϕ, t) dϕdt − w fg ( ϕ, t) dϕdt
σϕ∗ (t)ϕ∗ (t)"#
Z Z ∞ hZ Z ∞i
Pf Q fσ − 1 σ −1+
Pf(τx w)1−σ
ϕσ−1 t−σ g ( ϕ, t) dϕdt − w f x
g ( ϕ, t) dϕdtσσϕ∗x (t)ϕ∗x (t)

PQ
σ



P

= w f e . (A.5)

Z Z ∞ hZ Z ∞iσ − 1 σ −1 1− σPf
wf
ϕσ−1 t−σ g f ( ϕ, t) dϕdt − w f f
g f ( ϕ, t) dϕdt
σϕ∗f (t)ϕ∗f (t)
#


Z Z ∞
Z Z ∞
PQ
σ − 1 σ −1+P(τx w f )1−σ
ϕσ−1 t−σ g f ( ϕ, t) dϕdt − w f f x
g f ( ϕ, t) dϕdtσσϕ∗x f (t)ϕ∗x f (t)Pf Q fσ"= w f f e . (A.6)

It follows that the aggregate prices of Home and Foreign are
P 1− σ =

P1f −σ =



σ
σ−1

σ
σ−1

 1− σ 1− σR R∞ Mw1−σϕ∗ (t)R R∞

g ϕ, t) dϕdt
ϕ∗ (t) (R R∞

 M f w 1− σf

( ϕt )σ−1 g ( ϕ, t) dϕdt

ϕ∗ (t)

R∞+ M f (τx w f )1−σ

( ϕt )σ−1 g f ( ϕ, t) dϕdtf

R R∞

ϕ∗f (t)g f ( ϕ, t) dϕdt

52( ϕ )σ−1 g f ( ϕ, t) dϕdt
ϕ∗x f (t) tR R∞g ϕ, t) dϕdt
ϕ∗f (t) f (R∞+ Mw1−σ τx1−σ


ϕ σ −1
)
g
ϕ,
t
dϕdt
(
(
)
t
.
R Rx ∞
g
ϕ,
t
dϕdt
(
)
∗
ϕ (t)

ϕ∗ (t)

Next we derive the measure of operating firms at Home and Foreign M and M f . Let Me
be the measure of new entrant, ωe be the entry probability given by (A.1), and ωx be theexport probability conditional on entry

ωx =

R R∞

ϕ∗ (t) g ( ϕ, t ) dϕdt

Z Z ∞µ ( ϕ, t) dϕdt = R R ∞xϕ∗ (t) g ( ϕ, t ) dϕdt

ϕ∗x (t).

The expected per-period profit includes the profit from both domestic production andinternational production,
Z Z
ϕ∗ (t)π ( ϕ, t)µ( ϕ, t)dϕdt +Z Z

π x ( ϕ, t)µ( ϕ, t)dϕdt

ϕ∗x (t)≡ Eπ + ωx Eπ x
where µ x ( ϕ, t) = µ( ϕ, t)/ωx . Using the average profits, free entry, stationary equilibrium
condition, we have
L

M=


σ


R R∞
fe
ϕ∗
(t) g ( ϕ,t ) dϕdt
x
R R∞
+ f + R R ∞ g( ϕ,t)dϕdt f x
ϕ∗ (t) g ( ϕ,t ) dϕdt
ϕ∗ (t)

(A.7)

and
Lf

Mf =
σ

!.
R R∞
ϕ∗ (t) g f ( ϕ,t ) dϕdt
fe
xf
R R∞
+ f + R R ∞ g ( ϕ,t)dϕdt f x
ϕ∗ (t) g f ( ϕ,t ) dϕdt
ϕ∗ (t) ff

B

ProofsB.1

Proof for Proposition 2f

To derive Proposition 2, we use the following equilibrium equations,

53

(A.8)

(a) Free entry condition:
1
σ



σ−1
σ

 σ −1
w

1− σ


Pσ Q

Z Z
ϕ∗ (τ )

ϕσ−1 τ −σ g( ϕ, τ )dϕdτ + Pfσ Q f τx1−σZ Z
ϕ∗x (τ )ϕσ−1 τ −σ g( ϕ, τ )dϕd

= w f e + w f H + w f x Hx(b) The labor market clearing condition:
Me =

L
.
σ ( f e + f + Hx f x )(c) Price index:"
P1−σ = con p × MeZ Z
ϕ∗ (τ )

ϕσ−1 τ 1−σ g( ϕ, τ )dϕdτ +Pfσ Q fPσ Q

Me τx1−σ#Z
ϕ∗x (τ )ϕσ−1 τ 1−σ g( ϕ, τ )dϕdτ

where con p is a constant that depends on the model parameters.
(d) Cutoff of producing:1ϕ∗ = conv × P−1 ( PQ) 1−σ τ σ−1σwhere conv is a constant that depends on the model parameters.
(e) Definition of domestic share λ:
1−λ
=
λ

hR R
i
Pσf Q f
1
−
σ
σ
−
1
1
−
σ
τ
g( ϕ, τ )dϕdτPσ Q τx
ϕ∗x ϕ
hR R

ϕ∗

ϕσ−1 τ 1−σ g( ϕ, τ )dϕdτi(f) Definition of the share of variable labor used in producing domestic goods S:

RR
Pσf Q f1− σ
σ−1 t−σ g ( ϕ, t ) dϕdt
τ
σ
1−S
P Q x
ϕ∗x (t) ϕRR=σ−1 t−σ g ( ϕ, t ) dϕdt
Sϕ∗ (t) ϕ54

Differentiating the above system of equations and using the relationship d ln W = d ln( PQ) −d ln P, we get Proposition 2 with the following definitions:R( ϕ∗ (τ )) σ−1 τ 1−σ g( ϕ∗ (τ ), τ ) ϕ∗ (τ )dτRRσ−1 τ 1−σ g ( ϕ, τ ) dϕdτϕ∗ (τ ) ϕR

( ϕ∗ (τ ))σ−1 τ −σ g( ϕ∗ (τ ), τ ) ϕ∗ (τ )dτRR.
σ−1 τ −σ g ( ϕ, τ ) dϕdτϕ∗
ϕ (τ )γd =γs =With homogenous productivity and if domestic distortion 1/τ follows a Pareto disσ −11tribution with parameter θ, γd = σ−σ ( θ − σ + 1) and γs = σ ( θ − σ ), hence dlnW =σσ −1 [ dlnS − dlnλ ]. Changes in the share of expenditure on domestic goods and the share ofvariable labor used in producing domestic goods are sufficient statistics for welfare change.

B.2

Proof for Lemma 11
σ −1
Under the special case, γd = σ−σ ( θ − σ + 1) and γs = σ ( θ − σ ), and the change of welfareσbecomes dlnW = σ−
1 [ dlnS − dlnλ ].

Welfare change from close to open:
Pfσ Q f

# −1

"

# −1

"
λ=

S=

σ −1 f P σ Q σ − γ −1x1−σ τxτ() σ +1xσ
P Q
f
Pfσ Q fPfσ Q fσ −1 f P σ Q σ − γ
x1−σ τxτ() σ +1Pσ Q xf
Pfσ Q fas long as there is selection to export, λ > S, and d ln S is more negative than d ln λ fromclosed to open. The open economy has an unambiguously lower welfare than in the closed
economy. Because with reallocation purely driven by distortions, in the open economy, theinput share used to produce for exports exceeds the export share.
The reallocation term is always negative: Substitute in γd and γs , the last term, the55

reallocation becomes (1 − θ )d ln λ + θd ln S. Furthermore,σ−1(1 − λ)(1 − σ − γx )d ln λ = −(1 − λ) 1 − σ −(θ − σ + 1)) d ln τx +d ln( Pfσ Q f /Pσ Q)
σσ−1= (1 − λ )θ+1τ σ −1 P σ Q
d ln x σσ
Pf Q f
σ−1(1 − S)(1 − σ − γsx )d ln S = −(1 − S) 1 − σ −(θ − σ) d ln τx +d ln( Pfσ Q f /Pσ Q)
σσ−1τ σ −1 P σ Qθ= (1 − S) d ln x σσ
Pf Q fSubstitute in d ln λ and d ln S, the reallocation termθ 2 (1 − S) − (θ 2 − 1)(1 − λ)τxσ−1 Pσ Q
(1 − θ )d ln λ + θd ln S =d lnσ
Pfσ Q f
θ 2 (1 − S) − (θ 2 − 1)(1 − λ) > 0, hence as long as the trade cost reduction induces larger
fraction of exporters, the reallocation term is always negative. Q.E.D.

B.3

Proof for Lemma 2
σ

σ
σ −1

Recall the producing cutoff is given by φ∗ (c, τ ) = cτ σ−1 where c = σσ−1

h

wf
Pσ Q

i 1

σ −1

w. Define

two functions F (c) and H (c) where F relates to the cumulative labor share, and H relates
to the cumulative output share,
F (c) =

H (c) =Z Z

φσ τ −σ g(φ, τ )dφdτφ∗ (c,τ )Z Z
φ∗ (c,τ )φσ τ 1−σ g(φ, τ )dφdτ.

Let f (c) = F 0 (c) and h(c) = H 0 (c). The hazard functions of γd and γs are
γd =

h(c)
,
1 − H (c)

56

γs =

f (c)
.
1 − F (c)When f /h increases with c, i.e. f is likelihood ratio dominates h, then γd ≥ γs . Letσx = log ϕ, y = log τ, then x = c + σ−
1 y. Under joint-normal distribution of ( x, y ),Rexp(σx ∗ − σy) g( ϕ∗ (c, τ ), y)dyf (c)=RV (c) ≡h(c)exp(σx ∗ + (1 − σ )y) g( ϕ∗ (c, τ ), y)dy
where

"

1
g( x, y) = exp −2(1 − ρ2 )x2
y2 2ρxy
+
−
σϕ στ
σϕ2
στ2

!#
.

σ
0
When ρ στϕ ≤ σ−
1 , V ( c ) ≥ 0. Q.E.D.
σC

Equilibrium under Endogenous Markup

A closed-economy equilibrium consists of aggregate ( P, Q, M) that satisfy:M=

R σσ/ε R τ̂ (q̂)



Q 0

Q

Z σσ/ε Z τ̂ (q̂)

"

0

M
ωe

ω L
e
dϕ(τ,q̂)σ q̂
ε ϕ g ( ϕ ( τ, q̂ ), τ )dq dτd q̂

σ−q̂ σ#q̂
dϕ(τ, q̂)dτdq̂ = ωe f + f eg( ϕ(τ, q̂), τ )εdq
σ − q̂ σ ϕ
εq̂ σ

Z σσ/ε Z τ̂ (q̂)
0

γ (q̂) g( ϕ(τ, q̂), τ )dϕ(τ, q̂)dτdq̂ = 1,
dq

where
ωe =

Z σσ/ε Z τ̂ (q̂)
0g( ϕ(τ, q̂), τ )dτdq̂

and
γ0



q
Q



σ−1
=
exp
σ

ε

1 − (q/Q) σε!
.

The open equilibrium consists of unknowns ( P, Q, M, Pf , Q f , M f , w f ) that satisfy:wτx τ= γ0 (q̂ x ) Pf D fϕ
σ − q̂ x
σ

ε
σ

57

"
πx =

ε

#
τx qˆxQ f − f x w,εϕσσ − q̂
q̂ xσxwhere we get the zero profit cutoff. The free entry condition becomes:"#q̂Q − f g( ϕ, τ )dτdϕεϕ∗ (τ ) σ − q̂ σ ϕ
#
"
ε
Z Z
q̂ xσ τx q̂ x+
Q f − f x g( ϕ x , τ )dτdϕ = f eεϕ∗x (τ ) σ − q̂ σ ϕ

Z Z

ε

q̂ σ

(A.9)

x

The labor market clearing condition is:
M=

RR

"

M
ωe


ϕ∗ (τ )

Z Z

ωe L

RR
σ q̂
g( ϕ, τ )dτdϕ +ε ϕQϕ∗ (τ )


σ−q̂ σxσ

τx q̂ xQfε
σ ϕ(A.10)



σ−q̂ xg( ϕ, τ )dτdϕ#

Mf Z Zγ (q̂) g( ϕ, τ )dτdq̂ +γ q̂ x f g f ( ϕ, τ )dτdϕ = 1
ωe fϕ∗ (τ )
ϕ∗x f (τ )(A.11)For Foreign,


ε


q̂ f

Q f − f  g f ( ϕ, τ )dτdϕεϕ∗f (τ ) σ − q̂ σ ϕ
f


ε
Z Z
q̂ xσ f τx q̂ x f

+Q − f x  g f ( ϕ, τ )dτdϕ = f e f
εϕϕ∗x f (τ ) σ − q̂ σ

Z Z

q̂ fσ

(A.12)xf

ωe L fMf =!

RR
ϕ f (τ )

"

σ q̂
ε ϕQf
σ−q̂ fσg f ( ϕ, τ )dτdϕ +RRϕ∗x f (q̂)σ τx q̂ x fε
ϕ Q
σ−q̂ xσf(A.13)!
g f ( ϕ, τ )dτdq̂

#Z Z
Mf Z ZM
γ qˆf g f ( ϕ, τ )dτdq̂ +γ (q̂ x ) g( ϕ, τ )dτdϕ = 1ωe f
ωe
ϕ∗f (τ )ϕ∗x (τ )58(A.14)

Finally, the goods market clearing condition is:
M
ωe

"

Z Z
ϕ∗x (τ )wq̂
Qf
σ − q̂ ϕ
σ

ε
σ

#



Mf Z Z
w
q̂
f xf
 σ ε
g( ϕ, τ )dτdϕ =Q g f ( ϕ, τ )dτdϕ∗
ωe f
ϕϕ x f (τ ) σ − q̂ σ
xf

(A.15)

D

Regressions for measurement errors
Table A-1: Measured Marginal Products using First Differences vs TFPR
VARIABLES
log( TFPR)
Constant

Observations
R-squared
Time FE
Industry FE(1)log( 44VA
I )(2)log( 44VA
I )(3)log( 44VA
I )0.718***(135.3)1.410***(78.31)0.715***(158.6)0.331***(17.49)0.718***(135.3)1.410***(78.31)

624,659
0.173
Yes
Yes

624,699624,659
0.269
0.173
Yes
Yes
Yes
Yes
Robust t-statistics in parentheses*** p<0.01, ** p<0.05, * p<0.1
Specification (2) weights all the observations with the absolute value of composite input growth.
Specification (3) weights all the observations with the share of aggregate value added.

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Table A-2: Estimate Measurement Error
(1)
c
4VA

(2)
c
4VA

(3)
c
4VA

0.0376***(22.62)0.0144***(9.170)0.0616***(16.07)-0.0128***(-6.110)0.00152***(4.008)[4input0.530***(34.58)0.523***(33.03)0.524***(31.13)[log( TFPR) × 4input-0.0997***(-20.65)-0.0954***(-19.16)-0.0893***(-6.420)

VARIABLES
log( TFPR)[log( TFPR)]2[log( TFPR)]3[[log( TFPR)]2 × 4input-0.00611(-0.919)[[log( TFPR)]3 × 4input

0.00108(1.040)-0.0241***(-3.592)

Constant-0.0207***(-3.125)

0.0551***(8.231)

Observations
R-squared
Time FE
Industry FE

1,106,982 1,106,914
1,106,982
0.044
0.042
0.044
Yes
Yes
Yes
Yes
Yes
Yes
Robust t-statistics in parentheses*** p<0.01, ** p<0.05, * p<0.1
Specification (2) weights all the observations with the share of aggregate value added.

60