Global Gains from Reduction of Trade Costs
Edwin L.-C. LAIy
Haichao FANz
Han (Ste¤an) QIx
This draft: 5 September 2016 (v. 10f5)
Abstract
We develop a simple formula for computing the global welfare e¤ect of reduction of bilateral
trade costs, such as shipping costs or the costs of administrative barriers to trade. The formula is
applicable to a broad class of perfect-competition and monopolistic competition models and settings,
including perfect competition with single-stage and multi-stage production and Melitz’s (2003) model
with general …rm productivity distribution. We prove that the underlying mechanism is the envelope
theorem. We then extend our analysis to models with non-constant markup. Finally, we carry out
some empirical applications to show the user-friendliness of the formula.
Keywords: global welfare, trade cost, gains from trade
JEL Classi…cation codes: F10, F12, F13
An earlier version of this paper was circulated under the title “Global Gains from Trade Liberalization” (CESifo
working paper no. 3775, March 2012). We would like to thank Davin Chor, Gene Grossman, Stephen Yeaple, Tim
Kehoe, Jaime Ventura, Jonathan Vogel, Hamid Sabourian, Ralph Ossa, Arnaud Costinot and seminar and conference
participants in University of Melbourne, University of New South Wales, University of Hong Kong, City University of Hong
Kong, HKUST, Shanghai University of Finance and Economics, National University of Singapore, Singapore Management
University, Asia-Paci…c Trade Seminars, AEA Annual Meeting in Philadelphia in 2014, and World Congress of Econometric
Society in Montreal in 2015, for their helpful comments. Naturally, all errors remain ours. The work in this paper has
been supported by the Research Grants Council of Hong Kong, China (General Research Funds Project no. 642210 and
691813) and by the Shanghai Pujiang Program (15PJC041).
y
Lai: Corresponding author. CESifo Research Network Fellow. Department of Economics, Hong Kong University of
Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Phone: +852-2358-7611; Fax: +852-2358-2084; Email:
[email protected] or [email protected]
z
Fan: School of International Business Administration, Shanghai University of Finance and Economics, Shanghai,
China. Phone: +86-21-65907307. Email: [email protected]
x
Qi: Department of Economics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong. Phone: +8523411-5682; Email: ste¤[email protected]
0
1
Introduction
One major theme of international trade economics is the gains from trade. There is a presumption in
all trade models that the more integrated is the world, the larger are the gains from trade. Therefore,
the lower the trade barriers, the greater global welfare should be. But, how much do trade barriers
matter to the world quantitatively? For example, what is global bene…t in monetary terms of a one
percent reduction in all bilateral trade costs worldwide? How sensitive is the answer to this question to
the trade model being used? Answers to these questions would help us evaluate issues such as the global
bene…t of improvement of transportation technology or that of the reduction of administrative barriers
to global trade. Understanding the global welfare impact of such changes is important as a larger pie
means that there is more room to make every country better o¤ by proper side payments. In fact,
international organizations recognize the importance of evaluating the global welfare gains from the
reduction of administrative trade barriers and shipping time. OECD, World Bank and World Economic
Forum found that trade facilitation could yield large economic gains to the world.1 This paper derives
a simple equation for computing the global welfare e¤ect of simultaneous reduction of multiple bilateral
trade costs, such as shipping costs or the costs of administrative barriers to trade. We …nd that the
equation is applicable to a broad class of models and settings. We then carry out some empirical
applications. We …nd the estimates to be reasonable and consistent with others’in the literature.
We derive a measure of the percentage change in global welfare based on the concept of equivalent
variation and Kaldor-Hicks’concept of welfare change for a group. We rigorously demonstrate that the
P
bi (the expenditure-share-weighted average percentage change of country welfare) is
expression ni=1 YEwi U
a reasonable measure of the percentage change in global welfare, where Ei is the aggregate expenditure
bi is a small percentage change
of country i, Y w is the global GDP of the n countries in the world, and U
in welfare of country i. This expression for the percentage change in global welfare has been used in
the literature, such as in Hsieh and Ossa (2011), Atkeson and Burstein (2010) (hereinafter abbreviated
as AB) and Burstein and Cravino (2015) (hereinafter abbreviated as BC). However, as far as we are
aware, we are among the …rst to present a justi…cation for its use.2
Then, we derive a simple equation for computing the total global welfare e¤ect of simultaneous small
reduction in multiple bilateral trade costs. We …nd that the percentage change in global welfare is given
by
Pn Pn Xij
j=1
i=1 w bij ,
Y
(1)
where Xij is the total value of exports from country i to country j, and bij is a small percentage change
in the iceberg cost,
1
2
ij ,
of exporting from i to j. This turns out to be just the total saving in trade
See for example, World Economic Forum (2013) and OECD (2003).
Note also that though our formal analysis is based on in…nitesimal changes, the equation should be a su¢ ciently good
approximation as long as all percentage changes of Pi and Ui are less than, say, ten percent as a rule of thumb. Therefore,
the equation should applicable to many yearly changes.
1
costs divided by global GDP, keeping the values of all bilateral trade ‡ows unchanged. In other words,
only the direct e¤ect matters, as it is …rst order. The indirect e¤ects, such as changes in allocation of
resources to di¤erent goods and the resulting changes in input costs, do not matter as they are second
order. The key reason is that the allocation of resources to di¤erent goods were already optimally chosen
before the changes in trade costs take place.
Expression (1) re‡ects the fact that, in the absence of externalities and price distortions, the market
response to changes in bilateral trade costs is identical to the optimal response of a global planner who
maximizes global income. Therefore, one can invoke the envelope theorem when evaluating the e¤ect of
changes in bilateral trade costs on the change in global welfare. The envelope theorem turns out to be
a very powerful tool for evaluating the global welfare impact of changes in bilateral trade costs under
rather general condition.
The envelope theorem can be applied to many models and settings. Examples are 1. models of perfect competition (hereinafter PC): with variable extensive margins of trade, with incomplete/complete
specialization, and with multi-stage production, e.g. Dixit and Norman (1980, Chapters 3-5) with trade
costs, Heckscher-Ohlin model with trade costs, Dornbusch-Fisher-Samuelson (DFS, 1977 and 1980), the
Armington (1969) model, Eaton-Kortum (2002) (hereinafter EK2002), Melitz and Redding (2014), Yi
(2003), Yi (2010); 2. monopolistic competition (hereinafter MC) models with constant markup, including Krugman (1980) (hereinafter K1980), Melitz (2003) with general …rm productivity distribution
(hereinafter MwG).
The reason that the envelope theorem applies to the above models is that the market is e¢ cient as
there are no externalities and no price distortions. This is clear for the PC models. For MC models, we
extend the proof of Dhingra and Morrow (2014) that the Melitz model is e¢ cient in the global economy.
Intuitively, the reason that equation (1) applies to MwG is that as cuto¤ productivities change due to
changes in trade costs, the e¤ect on the average productivity of …rms serving each market (productivity
e¤ect) and the e¤ect on the mass of …rms serving each market (…rm mass e¤ect) completely o¤set each
other from the global welfare’s point of view, regardless of the distribution of …rm productivity. The
reason is that, for each exporting country, the labor constraint dictates that the change in average
productivity and change in …rm mass in each market go in opposite directions, and they o¤set each
other when summing up over all markets.
The intuition for equation (1) to hold for multi-stage production (i.e. fragmentation) is that whenever
there is an amount of terms of trade gain by an exporter of a good at any stage, there is an equal amount
of terms of trade loss by an importer of the same good. Thus, the global e¤ect of terms of trade changes
is nil. Therefore, only the direct e¤ect, which is the total saving in trade costs in all stages, matters
for global welfare change. Furthermore, any saving in trade cost at any stage is eventually passed on
to the …nal stage. As a result, the saving in trade cost in each stage shows up as the gain in global
2
income in the …nal stage. Given that fragmentation is pervasive in practice, the fact that our formula
is applicable to such a setting is important, as it demonstrates that our formula is relevant to the real
world.
Ossa (2015) found that assuming symmetric trade elasticity across sectors leads to a gross underestimation of the gains from trade. In the extensions section, we show that if the sectors with low
elasticities of substitution (i.e. low trade elasticities) tend to be associated with positive weighted sum
of the …rm mass e¤ect and productivity e¤ect, then there would be additional global gains from the
reduction of trade costs beyond the benchmark result given by (1). Apparently, the …nding of Ossa
(2012) shows that this is true empirically.
Our work is inspired by Arkolakis, Costinot & Rodriguez-Clare (2012) (hereinafter abbreviated
as ACR). They show that for a class of trade models which include Armington (1969), Krugman
(1980) (hereinafter K1980), Melitz (2003) with Pareto distribution of …rm productivity, and EK2002,
a country’s gains from trade are always given by the same simple formula that contains two su¢ cient
statistics: (i) the share of expenditure on domestic goods, and (ii) the trade elasticity. The focus of
our analysis is, however, di¤erent from that of ACR. In contrast to ACR, our goal is to calculate the
global welfare change caused by small changes in bilateral trade costs. The underlying mechanisms of
our results are also quite di¤erent. Whereas ACR’s result is based on the special properties of a class of
quantitative trade models, our result is based on the envelope theorem, which is valid as long as there is
constant markup and there are no externalities arising from the actions of economic agents. Nonetheless
our two papers are related in some way. The mapping between our equation (1) and the gains from
trade equation in ACR is that if one adopts the assumption that the import demand system is CES
(i.e. Assumption R3 in ACR), then our global gains formula (1) is precisely equal to an expenditureshare-weighted average percentage change of individual country’s gains from trade as given by the ACR
formula. Therefore, ACR’s equation implies our equation if their assumptions are adopted. Indeed,
there is a mapping between our main result and that of ACR, which is presented in Appendix E. Thus,
our work complements that of ACR in that while ACR’s equation can predict the change in welfare of
an individual country based on certain set of assumptions, our equation can predict the global welfare
change based on a set of less restrictive assumptions. This is because the indirect e¤ects cancel each
other in aggregation and we do not have to account for them when we calculate the global welfare
impact. Based on a less restrictive set of assumptions, our equation is applicable to a broader set of
models and settings, e.g. our equation can be applied to MwG and PC with multi-stage production.
Our work is also inspired by Atkeson and Burstein (2010) (AB), who prove that the details of …rms’
responses are of secondary importance to the estimation of the welfare impact of trade costs reduction
for an individual country. Under symmetry, they …nd that though changes in trade costs can have a
substantial impact on heterogeneous …rms’ exit, export, and process innovation decisions, the impact
of these changes on a country’s welfare largely o¤set each other. In the end, only the “direct e¤ect”
3
of trade cost reduction matters. Our paper follows a similar line of thinking as AB, but our focus is
very di¤erent. Whereas AB focus on proving that the individual and global welfare gains from changes
in trade costs depend only on the direct e¤ect based on a particular two-country model, we focus on
establishing a simple general formula for the global welfare gains induced by changes in bilateral trade
costs that can be applied to as general a condition as possible. Thus, our work complements that of
AB by further proving the power of their insight that for small changes only the direct e¤ect matters.
Our work has some overlaps with the independent work of BC, who …nd that, changes in world real
GDP in response to changes in variable trade costs coincide, up to a …rst order approximation, with
changes in world theoretical consumption. Like us, they have an equation that associates changes in
world real GDP with two set of su¢ cient statistics, namely, changes in bilateral variable trade costs
and export shares of continuing exporting producers. Thus, our work complements each other — we
corroborate each other’s …nding, though we start from di¤erent model environments. In contrast with
BC, we show that the formula applies to MwG, not just MwP (i.e. Melitz with Pareto productivity
distribution); moreover, we show that it applies to multi-stage production under perfect competition as
well. Distinct from both AB and BC, we rigorously prove that the underlying mechanism for the result
is that in the absence of externalities and price distortions, the envelope theorem can be applied to the
maximization problem of a global planner to obtain the market outcome.
One of the key assumptions for the main result given by (1) is constant markup in the MC model.
Later in the paper, we relax this assumption to allow for non-constant markup. We examine two cases.
The …rst is MC with multiple sectors and di¤erent elasticities of substitution across sectors. We …nd
that there is an extra term which depends on the combination of the …rm mass e¤ect and productivity
e¤ect. Unlike in the case of MC with constant markup, these two e¤ects do not cancel each other. The
intuition is that the sectors with lower elasticities of substitution would set higher markups, and if these
sectors tend to have negative (positive) combined productivity e¤ect and …rm mass e¤ect, then there
would be negative (positive) overall impact on global welfare gains from reduction of trade costs. This
makes sense as sectors with higher markups are associated with greater distortion. The e¤ect of sectors
with greater distortion would dominate the e¤ect of sectors with smaller markups (and hence smaller
distortion).
The second case is a one-sector MC model with variable markup. We use the simplest possible model
to illustrate the e¤ect of variable markup on the global gains formula (1). In this case, we …nd that there
is an extra term which depends on the change in markup of …rms. If …rms with large market shares
tend to reduce (raise) their markups following reduction of trade costs, the global gains would be larger
(smaller) than the benchmark case. This makes sense as markups are distortions, and lower markups
lead to high e¢ ciency and thus higher global welfare gains. This result is consistent with the empirical
…nding of, for example, Edmond, Midrigan and Xu (2015), who report that there are additional gains
from reduction of trade costs due to the existence of variable markup as …rms with larger market shares
4
(i.e. domestic …rms) tend to lower their markups.
Clearly, once we depart from constant markup, the global gains formula becomes a lot more complicated, and one needs to use a lot more information, including …rm-level data, to calculate the global
gains from reduction of trade costs.
The structure of this paper is as follows. In section 2, we state the general setting and provide a
justi…cation of the de…nition of the percentage change in global welfare. Then, we state and explain the
general result of the paper. We state three assumptions and two propositions and present the sketches
of the proofs. The two propositions together indicate that the underlying mechanism for formula (1) is
the envelope theorem. In section 3, we analyze how the market responds to exogenous changes in trade
costs under the three models that satisfy the three assumptions stated in section 2, viz. PC with singlestage and multi-stage production and MwG, and verify that expression (1) gives the percentage change
in global welfare in all models under the market. Section 4 presents the extensions to multi-sector MwG
and a one-sector MC model with variable markup, together with some empirical applications of the
formula. The last section concludes.
2
General Results
2.1
General Setting
Suppose in the world economy there are n countries (the set of countries is denoted by N ) that are
capable of producing …nal goods ! 2
F
where the superscript “F ” is assigned to variables pertaining
to “…nal good”whenever it is necessary to avoid confusion. The set of goods that country i is capable of
producing is
F.
i
Assume that there is complete or incomplete specialization (complete specialization
means that a country cannot import the same good from more than one country) in all sectors for all
countries, and variable extensive margins of trade. Assume that all goods are tradable. The extensive
margin of country i’s exports to country j is denoted by
F.
ij
Therefore,
F
ij
F
i
F.
De…ne Ei , Yi , Pi and Ui as the expenditure, income, exact price index and welfare of country i,
n
o
F (!) ! 2 F , i 2 N
respectively. De…ne qF
q
as a vector of quantities of …nal goods consumed
ij
ij
j
F (!) is the quantity of …nal good ! consumed in j that is imported from country
in country j (where qij
n
o
F (!) ! 2 F , i 2 N
i), pF
p
as a vector of the corresponding prices (where pFij (!) is the price
ij
ij
j
of …nal good ! consumed in country j that is imported from country i). We shall assume that ! is
continuous unless otherwise stated.3 The utility function (or welfare function) of country i, given by
F
Ui qF
i , is assumed to be homogeneous of degree one in qi . Consequently, we can de…ne an exact price
index Pi , which stands for the cost a consumer has to pay to obtain one unit of utility. Therefore, the
3
F
Calling qF
j and pj “vectors” is a slight abuse of language when ! is continuous. But since there is no ambiguity, we
shall use it for simplicity of exposition.
5
total utility of all consumers in country i (i.e. welfare of country i) is given by Ui = Ei =Pi for all i.
Without loss of generality, we assume that labor is the only factor input, and marginal cost of
production is assumed to be invariant with output. The variable Lj denotes country j’s …xed labor
supply while wj denotes its labor wage. There is an iceberg trade cost such that
ij
units are shipped
from the source country i for one unit to arrive at the destination country j (assume that
ii
= 1).4
Therefore,
pFij (!) = pFii (!)
F (!) = w
Since pc
ci for all !, we have
ii
ij
for ! 2
F (!) = w
pc
ci + bij for ! 2
ij
F
ij
F
ij
F (!) denotes the quantity of good ! produced in i that is exported to j. The iceberg
The variable yij
F (!) with y F (!) :
trade cost links qij
ij
F
ij qij
(!)
F
yij
(!)
for ! 2
F
ij
which implies that
F
pFij (!) qij
(!)
F
pFii (!) yij
(!)
xFij (!)
for ! 2
F
ij
where xFij (!) denotes the exports of …nal good ! from i to j. Aggregate exports of …nal goods from i
R
F
F
to j is denoted by Xij
!2 F xij (!) d!.
ij
De…nition of percentage change in global welfare
Next, we present a justi…cation for the use of an expression that we use to measure the percentage
change in global welfare.We want to de…ne a measure of percentage change in global welfare resulting
from small changes (in…nitesimal ones in the formal analysis) in bilateral trade costs. We would
like to have a concept of change of global welfare such that an increase in global welfare signi…es an
enlargement of the global pie so that potentially every country can be made better o¤ by some proper
income transfers between countries. Note that income is transferable but utility is not transferable.
Therefore, the sum of utility of all countries is not a good measure of global welfare based on this
concept.
As the world consists of a number of sovereign countries, each of which only cares about its own
welfare, we have to de…ne the improvement in global welfare based on the concept of Pareto improvement.
More speci…cally, the improvement to global welfare resulting from a shock (such as reduction of trade
costs) should be measured by how much potential Pareto improvement is a¤orded to the world after
the change. Thus, we de…ne the percentage increase in global welfare (following trade costs reduction)
4
The amount
ij
1 can be called the “wastage due to shipping” per unit arriving at the destination, but it should
also include the ad-valorem trade cost equivalent of any administrative delay or other non-tari¤ barriers.
6
as the maximum potential equiproportional increase in welfare of all countries after some proper lump
sum income transfers between countries. It measures the potential amount of Pareto improvement to
the countries of the world as a whole. In principle, this amount can be negative. The above concept
of the change in global welfare is consistent with that of Kaldor and Hicks (see, for example, Friedman
1998).5 Consistent with Kaldor-Hicks’concept of e¢ ciency, an outcome is more e¢ cient if those that are
made better o¤ could in principle compensate those who are made worse o¤, so that a Pareto improving
outcome can potentially result. This concept of Pareto improvement does not require compensation
actually be paid, but merely that the possibility for compensation exists.
Following the reduction of trade costs, the vector of price-cum-welfare of the countries changes
from (P1 ; :::; Pn ; U1 ; :::; Un ) to (P1 + dP1 ; :::; Pn + dPn ; U1 + dU1 ; :::; Un + dUn ). Let
be the potential
P
equiproportional increase in welfare of all countries following reduction of trade costs. Hereinafter, i
Pn
P
P
i=1 to simplify notation. Then,
i (Pi + dPi ) (Ui + dUi ) =
i (Pi + dPi ) ( + 1) Ui . The LHS is the
total global expenditure before lump-sum transfers while the RHS is the total global expenditure after
lump-sum transfers that leads to an equiproportional increase in welfare for all countries by a fraction .
Note that this scheme of lump-sum transfers is equivalent to summing up the compensating variations
of all countries and then distribute them equiproportionally across countries. However, because the
changes in Pi and Ui are in…nitesimal, this scheme is the same as summing up the equivalent variations
of all countries and then distribute them equiproportionally across countries.6 That is,
X
Pi (Ui + dUi ) =
i
X
Pi ( + 1) Ui
i
In the rest of the paper, we shall use the concept of equivalent variation to evaluate the percentage
change in global welfare. Re-arranging the above equation and simplifying, we have
P
Pi dUi
= Pi
i Pi Ui
X Pi Ui
dUi
P
=
P
U
Ui
k k k
i
1 X b
= w
Ei Ui
Y
(2)
i
bi
where U
dUi
Ui ,
Ei = Pi Ui and Y w
P
Pk Uk is the GDP of the world. Note that the sum of
P
dx
bi . Hereinafter, x
equivalent variations of all countries is equal to ni=1 Ei U
b
x , which we call the
k
percentage change of x.
5
Because we are considering small changes, the use of the Kaldor-Hicks compensation criterion does not carry with it
some of the shortcomings cited by its critics.
6
This is because (Pi + dPi ) (Ui + dUi ) (Pi + dPi ) Ui is the compensating variation of country i. Note also that the
equivalent variation of country i, Pi (Ui + dUi )
Pi Ui , is the same as the compensating variation in the current context,
because the changes are in…nitesimal. For the concepts of compensating variation and equivalent variation, see, for example,
Varian (1992, pp.160-163)
7
Thus,
P
Ei b
i Y w Ui
(or the expenditure-share-weighted average percentage change of welfare of all coun-
tries) is the percentage change in global welfare. This makes sense as the importance of a country as
indicated by its size should be re‡ected in the calculation of the change in global welfare. Note that we
do not have to de…ne what is the global welfare function. We only need to de…ne what is the percentage
change in global welfare. Note also that utility is cardinal, not ordinal, in this model. Intuitively, the
welfare impact of a percentage change in global welfare of
is equivalent to having all consumers in
the world potentially increasing their consumption of each good by a fraction of , if the same sets of
goods are produced, traded and consumed by each country before and after the shock.7
Note that
F
Ej = Pj Uj = pF
j qj .
F
F
Moreover, the ideal price index Pj is a function of pF
j . Thus, given pj and qj , we can calculate Pj
F
and Uj . In other words, there is a one-to-one mapping from pF
j ; qj
onto (Pj , Uj ), and therefore a
F
F
F
one-to-one mapping from pF
1 ; :::; pn ; q1 ; :::; qn onto (P1 ; :::; Pn ; U1 ; :::; Un ). The equivalent variation
F
of j, given by Pj dUj , is equal to pF
j dqj . So, the sum of equivalent variations of all countries, given by
P F
P
F
j pj dqj . Thus, the percentage change in global welfare can also be written
j Pj (dUj ), is equal to
as
1
1 0
0
X
X
FA
FA @
:
=
=@
pF
pF
j qj
j dqj
j
j
2.2
(3)
Speci…c Setting
We consider three types of setting below. The environment stated above in section 2.1 is satis…ed in all
settings. In addition, some more structure is imposed on each model.
P
In the rest of the paper, where it is useful to simplify notation, we shall use
a;b;c to denote
P P P
a
b
c where each summation is over all the possible values that the dummy variable can take, e.g.
P
Pn
i
i=1 if i is the dummy for a country and there are n countries in the world.
1. PC with single-stage production.
Preferences. The utility function, which is homogeneous of degree one in the …nal goods consumed,
n
o
F (!) 8 i, ! 2 F
F (!) is country j’s consumption of …nal good !
is given by Uj = Uj
qij
, where qij
ij
imported from i.
Technology. The production function is homogeneous of degree one in the factor input, i.e. yjF (!) =
'j (!) lj (!) where yjF (!) is output of …nal good ! from country j, lj (!) is labor input and 'j (!)
7
This is because we assume that the utility function of each country is homogeneous of degree one in quantities of all
goods consumed in that country. Therefore, the percentage change in global welfare is homogeneous of degree one in the
percentage change of quantities of all goods consumed in the world as a whole.
8
is productivity. More generally, there can be multiple factor inputs, but the main result would be
qualitatively the same.
Market Structure. The market structure for all goods is assumed to be perfect competition. When
F
is divided into sets each of which is interpreted as a sector, this is a model with multiple sectors.
2. PC with multi-stage production. The multi-stage production case subsumes single-stage production as a special case.
Preferences. The utility function, which is homogeneous of degree one in the …nal goods consumed,
n
o
F (!) 8 i, ! 2 F
F (!) is country j’s consumption of the …nal
is given by Uj = Uj
qij
, where qij
ij
good (which is also stage-F good) ! imported from i.
Technology. Goods produced in each stage other than the …nal stage are used as intermediate inputs
in the production of goods in the next stage, and all intermediate goods and …nal goods are tradeable.
The …nal good is assumed to be produced in F sequential stages.
8
(For a single-stage production
model, F = 1.) For s = f2; 3; :::; F g, the output of stage-s production (which shall be called “stage-s
good”) requires the input of labor and the previous stage’s output. The production of stage-1 good
requires only labor. The outputs at all stages are tradable, and all countries possess the technologies of
production for all stages. The production function for stage-s good is assumed to be constant returns
to scale, and is given by:
8
>
< 's (!) f
j
s
yj (!) =
>
:
where
yjs (!)
is country
s 1
yij
(! 0 )
s 1
ij
s 1
ij
i 2 N , !0 2
'sj (!) ljs (!)
for ! 2
for s = 1
output of stage-s good !;
s 1
yij
(! 0 ) = ijs 1
=
s 1
qij
(! 0 )
1 good ! 0 from country i for producing stage-s good !;
margin of exports of stage-(s-1) goods from i to j;
stage-s good !;
for s = 2; 3; :::; F
s
j
(4)
j0s
imported input of stage-s
'sj (!)
; ljs (!)
ljs (!)
is country
j0s
is country
s 1
ij
j0s
use of
is the extensive
labor input in the production of
productivity.9
is
Market Structure. The market structure for all goods is assumed to be perfect competition.
3. MC with heterogeneous …rm productivity. We assume that there is a large number of …rms
producing di¤erentiated goods so that any single …rm’s choice of price would not a¤ect the demand
8
Here, we assume all …nal goods are produced in F sequential stages. Even if we assume that the number of production
stages for di¤erent countries is di¤erent, our results continue to hold.
9
Constant
returns
to
scale
means
that
f
f
'sj
s 1
qij
(!)
8
>
*
>
<
>
>
:
0
0
(! ) i 2 N , ! 2
P R
i
1
!0 2 s
ij
s 1
ij
s 1
qij
0
(! )
;
ljs
(!)
i
d!
0
i
1
+
1
1
(! 0 ) i 2 N , ! 0 2
s 1
ij
;
ljs (!)
An example of such a production function is
9 1
>
>
1=
s
+ lj (!)
for s = 2; 3; :::; F .
>
>
;
> 0.
i
1
i
8
s
qij
9
=
yjs
(!) =
curve faced by other …rms.
Preferences. The utility function is homogeneous of degree one in the …nal goods consumed.
Technology. There is only one stage of production and all goods are …nal goods. For every …nal
good ! 2
i,
there is a blueprint that has been acquired by a …rm through R&D. If a …rm from country
n
o
F
F
i produces y
yij (!) units of good ! to be sold to country j, its cost function is given by
Ci wi ; yF ; ! =
X
F
ij wi ai (!) qij
(!) +
ij wi
F
1 yij
(!) > 0
j
F (!) > 0 is an indicator function, a (!) denotes the unit labor requirement in producing
where 1 yij
i
good ! in country i, and
from i to j, where
ij
ij
denotes the labor requirement that underlies the …xed cost of exporting
0 8 i; j.
Market Structure. Ni is the measure of number of entrants (successful or not) in country i, which
is endogenously determined by the free entry condition so that net pro…t for any …rm is equal to zero.
A …rm from country i needs to hire fe units of labor to develop a blueprint, which confers it with
monopoly power. In equilibrium, the entry cost, wi fe , is equal to the expected pro…t of each …rm. The
number of …rms in country i serving market j, Nij , is determined by the zero cuto¤ pro…t conditions.
Labor productivity is a random variable denoted by '
1=ai (!). From now on, ' and ! are used
interchangeably to index goods. Functions Gi (') and gi (') are the cdf and pdf respectively of '.
De…ne 'ij as the cuto¤ productivity for a …rm in country i to be able to pro…tably export to country j.
2.3
Results
Below we state a few assumptions and two propositions. Then, based on these assumptions and propositions, we prove that we can obtain expression (1) by invoking the envelope theorem.
Assumptions: 1. Trade balance in all countries are zero.10
2. Price is constant markup over marginal cost.11
3. There are no externalities.
Proposition 1 If Assumptions 1-3 are satis…ed by a trade model, then the market is e¢ cient.
Proof of Proposition 1
10
11
Trade balance can be non-zero if it is exogenous. This assumption implies that dEj = dYj 8j.
Perfect competition implies that the constant markup is equal to one.
10
PC with single-stage production
The result follows from the First Fundamental Theorem of Welfare Economics.
PC with multi-stage production
The result follows from the First Fundamental Theorem of Welfare Economics.
MC with heterogeneous …rm productivity
In order to satisfy all the assumptions stated above, we need price to be constant markup over
marginal cost. Therefore, more structure has to be imposed on preferences. Speci…cally, we have to
assume that the utility in country j is given by
Uj =
where
"
XR
!2
i
F
ij
1
F
qij
(!)
d!
#
1
(5)
> 1 is the elasticity of substitution. Pro…t maximization by …rms and utility maximization by
consumers subject to the budget constraint implies that the demand for each variety is given by
wi
1
F
qij
(') =
ij
'Pj
Ej
Pj
(6)
where Ej denotes total expenditure in country j, and Pj denotes the ideal price index in country j.
F (') is given by l (') =
Correspondingly, the labor used to produced qij
ij
F (')
qij
'
ij
.
In online appendix A, we prove that the global market is e¢ cient, in the sense that the market
allocation of resources flij (')g is identical to the allocation of a global planner who maximizes global
income subject to the labor constraint of each country. Conceptually, the proof is an extension of
Dhingra and Morrow’s (2014) analysis to the global economy. Intuitively, with constant markup, there
is no distortion in the relative prices. So, in the absence of externalities, the market is e¢ cient.
Next, we state the core proposition of this paper.
Proposition 2 If Assumptions 1-3 are satis…ed by a trade model, then the percentage change in global
P
bi =
welfare induced by changes in bilateral trade costs is given by expression (1). In other words, Y1w i Ei U
P Xij
j;i Y w bij , where Xij is the total value of exports from country i to country j, and bij is a small
percentage change in the iceberg cost,
ij ,
of exporting from i to j, and Y w is the global GDP of the n
countries in the world.
Note that with multi-stage production, Xij =
s goods from i to j; and we assume that
production, Xij =
F
Xij
and
ij
s
ij
=
P
s
s Xij
ij
s is the value of exports of stagewhere Xij
8s in the above proposition. With single-stage
applies to trade in …nal goods.
11
General Proof of Proposition 2
Proposition 1 implies that the market response to changes in bilateral trade costs is identical to
the optimal response of a global planner who maximizes global income. Therefore, one can invoke the
envelope theorem when evaluating the e¤ect of changes in bilateral trade costs on global welfare. In
fact, Proposition 2 can be proved by invoking the envelope theorem and analyzing the response of the
global planner to the reduction in trade costs. Below we give a general proof of Proposition 2. Speci…c
proofs of the three models are relegated to the appendix.
Based on the argument presented in section 1, we shall use the pre-change market prices and the
concept of equivalent variation to evaluate the global welfare change. We now consider a global planner
calculating the global GDP function given a set of shadow prices (which are also the market prices), the
labor supplies of countries and bilateral trade costs.12 As she chooses labor allocation to the production
of each good to maximize global income, she would automatically maximize the income of each country
subject to its resource constraint and the set of shadow prices. Thus, the national GDP resulting from
the global planner’s labor allocation is also the market-determined national GDP. Simultaneously, on
the demand side, the utility of each country is maximized by the combination of quantities of goods
available for consumption (which are determined by the allocation of labor to the production of various
goods) subject to the set of shadow prices and the GDP of the country.
For generality, the following proof can accommodate the cases of single-stage or multi-stage production under PC or single-stage production under MC (i.e. MwG), with stage indexed by s = 1; 2; :::; F .
The stage-F good stands for the …nal good. The multi-stage production model subsumes the singlestage production model. For a single-stage production model (either under PC or MC), F = 1. Let
s (!) be the variable labor input used to produce the stage-s good ! that is exported from i to j.
lij
F (!) is also denoted by l (!) for simplicity of notation. At the shadow
For a single-stage model, lij
ij
n
o
s
s
s
s
s
pij (!) i 2 N , ! 2 sij , the quantities demanded and
prices p fp1 ; :::; pn g for all s, where pj
quantities supplied of all goods are equalized.
Maximization of global income
The global planner maximizes global income W by choosing a vector of choice variables
is described below), taking trade costs
ij
for all i; j, and the shadow prices
are also market prices), as given. De…ne l
costs. The de…nitions of l, p and
12
(which
(!) for all i; j; ! (which
s (!) ; 8i; j; s; ! as a vector of all labor allocations;
lij
(psij (!)); 8i; j; s; ! as a vector of all shadow prices; and
p
pFij
s
ij
; 8i; j; s as a vector of all trade
di¤er for di¤erent models as explained in the speci…c proofs in the
This is analogous to a country’s social planner calculating the market-determined national GDP function based on the
market prices faced by the country. (See, for example, Feenstra 2004, pp. 6-8, for the idea of GDP function used here.)
12
appendix. Thus, she solves
max W =
f g
X
i;j
R
!2
F
ij
F (!) (
pFij (!) yij
ij ) d!
F
ij
subject to the labor constraint of each country and the production functions of all goods
F (!) (
yij
ij )
8i; j; !, where the vector of inputs
ij
and the vector of choice variables
for di¤erent models. For PC with single-stage production,
production,
ij
= fl; g ; for MwG,
ij
are di¤erent
= lij (!); for PC with multi-stage
= lij (!), which is also denoted by lij (') (as ' and ! are used
n n oo
s
= l;
for PC with single-stage production and PC with
ij
n
n o
o
= flij (') ; 8i; j; 'g ; 'ij ; fNi g for MwG. The maximized value of W
ij
goods).13
interchangeably to index
multi-stage production;14
before changes in trade costs is denoted by Y w .
The expressions for (i) the exports of …nal goods from i to j, given by
R
!2
F
ij
F (!) (
pFij (!) yij
F
yij
(ii) the labor constraint in each country, and (iii) the production function, given by
(!) (
ij ),
ij ) d!,
di¤er
for di¤erent models, as explained in the speci…c proofs in the appendix.
Note that we can also write global income in the following way:
X
W =
F
Xij
=
i;j
X
F
pF
j qj
j
We shall make use of these equalities later.
, denoted by e , is a function of and pF . Note that pF is a function of
a¤ects e directly as well as indirectly through pF . Formally,
The optimal value of
too. Thus, a change in
e =g
where g is some function.
In evaluating the total e¤ect of
indirect e¤ects of how
and
pF
account (1) the direct e¤ect of
indirect e¤ect of
s
ij
!
pF
!
s
ij
; pF ( )
on W , we have to evaluate the direct e¤ect of
are a¤ected by the change of
s
ij
s
ij .
s
ij
as well as the
In other words, we have to take into
! W ; (2) plus the indirect e¤ect of
! W ; (4) plus the indirect e¤ect of
s
ij
s
ij
!
!
! W ; (3) plus the
pF
! W . However,
as we explain below, only e¤ects (1) through (3) are relevant for calculating the sum of equivalent
13
F
F
For PC with multi-stage production, yij
(!) is a¤ected not only by labor input lij
(!) but by intermediate inputs
(which are functions of l and
F
yij
single-stage production,
14
s
is included in
ij
F
) and labor inputs (l) in all stages. Thus yij
(!) is a function of l and
F
lij
. For PC with
(!) is a function of
(!) only, and is independent of .
when extensive margins of trade are variable under PC; it is not included when extensive
margins are …xed.
13
variations of all countries. To see this, note that the total e¤ect of
dW
=
d ijs
@W
@W
+
s
@ ij
@
|{z}
|
E¤ect (1)
=
@
@ ijs
{z
E¤ect (2)
@W
@ ijs
|{z}
+
E¤ects (1)+(2)+(3)
+
=e
}
@
@pF
@pF @ ijs
{z
@W
@
|
@W @pF
@pF @ ijs
|
{z
E¤ect (4)
E¤ect (3)
since
=e
+
=e
@W
@
}
}
s
ij
on W can be written as
@W @pF
@pF @ ijs
|
{z
E¤ect (4)
= 0 as
=e
}
has been optimally chosen.
=e
(7)
Accordingly, e¤ects (2) and (3) are equal to zero, as they are second order e¤ects, and the exogenous
changes are in…nitesimal. This is precisely the principle underlying the envelope theorem. On the other
P
F
s
hand, since W = j pF
j qj , the total e¤ect of ij on W can also be written as
X
X dpF
dqF
dW
j
j
F
=
+
qF
p
j .
j
s
s
s
d ij
d ij
d ij
j
j
|
|
{z
}
{z
}
E¤ects (1)+(2)+(3)
E¤ect (4)
Recall from subsection 2.1 and equation (3) that the …rst term on the RHS of the above equation
corresponds to the sum of equivalent variations of all countries caused by a change in
s
ij ,
and it is the
only term we care about in order to calculate the percentage change in global welfare. The rationale
is that, as we are using pre-change market prices and concept of equivalent variation to evaluate the
global welfare change, the direct e¤ect of pF , i.e. E¤ect (4) above, can be ignored.
Comparing the last lines of the above two equations, we conclude that the sum of equivalent variations of all countries induced by an in…nitesimal change of
s
ij
is equal to the partial derivative @W=@
Thus, we have the following key lemma of this paper.
Lemma 1 The change in global welfare induced by a small change in
s
ij
is equal to @W=@
s
ij .
Thus, according to (3), the percentage change in global welfare is given by
F
X pF
j dqj
P F F
j pj qj
j
1 X @W s
d
= w
Y
@ ijs ij
=
s;i;j
=
s
X Xij
s;i;j
=
s
ij
Yw
X Xij c
ij
i;j
Y
w
1
d
s
ij
s
if c
ij for all s, and Xij =
ij = c
14
X
s
s
Xij
.
(1)
s
ij .
R
where W =
s
ij
!2
psij
P
F
s
i;j Xij and Xij
s
(!) yij
(!) (
ij ) d!
hR
!2
s (!) (
psij (!) yij
s
ij
ij ) d!
i
s
ij .
=
Although the expression for
is di¤erent for di¤erent models, expression (1) holds for all models. For
single-stage production under PC and MwG, the third line of the above calculation is obvious. For
multi-stage production under PC, the third line stems from the fact that the change in total global
value of outputs at stage F is equal to the change in total global value of inputs at an earlier stage s + 1
induced by a decrease in
s
ij
(where s < F ).15 This …nishes our general proof of Proposition 2.
In appendixes A and B, we provide speci…c proofs for PC with multi-stage production and MC with
heterogeneous …rms to demonstrate that this general principle holds.
3
Analysis of market response to reduction of trade costs
In this section, we analyze the market’s response under each speci…c model to changes in bilateral trade
costs, and show that the percentage change in global welfare is given by (1) under the decentralized
market. Thus, we verify that the change of the value of the objective function under the market is
identical to that under the global planner. By analyzing the market’s response instead of the global
planner’s response, we are able to identify e¤ects that are canceled out when we consider the impact on
global welfare instead of welfare of individual countries.
From now on, in order to simplify notation, we shall omit the superscript “F ” for most of the
variables in the context of discussion of single-stage production (PC or MC) whenever such omission
would not cause any confusion.
3.1
PC with single-stage production
The percentage change in welfare of country j as
changes is given by
ij
bj = E
bj
U
Pbj .
Intuitively, this says that, when j exports a good, an increase in wj tends to raise Uj through the
increase in Ej resulting from the increases in the prices of its exports, i.e. the terms of trade (TOT)
e¤ect on j as an exporter. On the other hand, when j imports a good from a foreign country i, the
prices of its imports are a¤ected in two ways. First, an increase in foreign wage wi tends to reduce Uj
through the increase in Pj resulting from the TOT e¤ect on j as an importer. Second, Pj is further
a¤ected by the change in trade cost
15
That is,
@W
s
@ ij
d
s
ij
=
@
s
@ ij
P P
i
j
F
Xij
d
ij .
s
ij
Thus, the expenditure-weighted change in welfare of country j
=
@
s
@ ij
P P
i
j
s
Xij
d
s
ij
=
s
@Xij
s
@ ij
d
s
ij
is the total global value of inputs at stage-(s + 1). See appendix A for more detail.
15
=
s
Xij
s
ij
1
d
s
ij ,
where
P P
i
j
s
Xij
is given by
bj =
Ej U
bj
Ej E
| {z }
Ej Pbj
| {z }
.
TOT e¤ect on j when
Welfare e¤ect on j when
it is an exporter
it is an importer
Balanced trade implies that dEj = dYj = Lj dwj = wj Lj w
cj = Yj w
cj =
e¤ect on j when it is an exporter is given by:
bj =
Ej E
X
i
P
bj .
i Xji w
Thus, the TOT
Xji w
bj
It can be seen that we need balanced trade so that the change in total expenditure is equal to the change
in income. The change in income is in turn tied to the change in the wage of the country weighted by
the total value of its exports, as shown above.
The change in the price index is the weighted sum of changes resulting from changes in prices of
imports from various sources:
Pbj =
X Xij
Ej
i
pbFij
which implies that the welfare e¤ect on j when it is an importer is given by
Ej Pbj =
X
i
Xij pbFij =
X
i
Xij (w
bi + bij )
This is the TOT e¤ect on j when it is an importer, plus the direct e¤ect of changes in trade costs.
Summing up the two e¤ects, the expenditure-weighted change in welfare of j is given by
bj =
Ej U
X
i
(Xji w
bj
X
Xij w
bi )
i
Xij bij .
Thus, to calculate the welfare change of an individual country, we need to calculate the net TOT gains
of the country in addition to the saving in trade costs. However, when we calculate the global welfare
e¤ect, we get the following:
Global welfare e¤ect:
X
j
bj =
Ej U
X
j;i
|
Xji w
bj
{z
}
TOT e¤ect on exporters
X
j;i
|
Xij w
bi
{z
}
TOT e¤ect on importers
X
j;i
|
Xij bij
{z
}
direct e¤ect of trade costs
The TOT e¤ect on exporters and TOT e¤ect on importers exactly cancel out from the global welfare
point of view. This is no surprise, as for each amount of TOT gain by an exporter, there is an equal
amount of TOT loss by the importer. Consequently, we have
1 X b
Ej Uj
Yw
j
|
{z
}
Percentage change in global welfare
=
|
1 X
Xij bij .
Yw
j;i
{z
}
Direct e¤ect
16
(1)
3.2
PC with multi-stage production
The utility function and the production at each stage are as given in section 2.2. Let Ejs
P
s
i Xij
denote the total expenditure on stage-s good in country j. (For the …nal stage, EjF Ej is country j’s
P s 1 P
s
total expenditure on …nal goods). De…ne sj
i Xij =
k Xjk the cost share of intermediate goods
R
s =
s
s
in the total cost of stage-s output produced by country j. Note that 1. Xij
!2 s pij (!) qij (!) d!,
ij
where psij (!) is the import price in country j of stage-s good ! from country i; 2. the total value of
R
P s
production of stage-s good in country j is equal to i Xji
= !2 s psjj (!) yjs (!) d! for s = 1; 2; :::; F ,
j
where psjj (!) is the unit cost of stage-s good ! produced by country j; 3. psij (!) = psii (!)
s (!)
qij
s
ij ;
4.
s
ij
s (!) which is the quantity of stage-s good ! exported from i to j measured at the origin;
= yij
P
s
5. yjs (!)
i yji (!).
P
In addition, the total value of production of stage-s good (for s = 2; 3; :::; F ) in country j, given by
s
i Xji ,
can be decomposed into two parts: …rst, the value-added by country j labor in the production
j P
s
of stage-s good, given by Vjs = 1
s
i Xji ; second, the total expenditure on intermediate inputs
P s
P 1
for the production of stage-s good in country j, given by Ejs 1 = sj i Xji
. Moreover, Vj1 = i Xji
as
no intermediate input is required for the …rst stage. GDP, Yj , which is equal to Ej (due to balanced
trade), is equal to the sum of value-added over all stages:
X
Ej =
Vjs for j = 1; 2; :::; n
s
Similar to the single-stage production case, we have
bj = E
bj
U
and
bj = Ej E
bj
Ej U
Pbj .
Ej Pbj .
In contrast to the single-stage production case, we make use of balanced trade to infer that dEj =
P
P
bj , as Yj = s Vjs . Thus, the TOT e¤ect on j when it is an
dYj = Lj dwj = wj Lj w
cj = Yj w
cj = s Vjs w
exporter is given by:
bj =
Ej E
X
s
Vjs w
bj
Thus, the global welfare e¤ect resulting from changes in prices of exports is given by:
X
XX
bj =
Ej E
Vjs w
bj
j
|
j
s
{z
(8)
}
TOT e¤ect on exporters over all stages
whereas the global welfare e¤ect resulting from changes in prices of imports is given by
X
X
X
s s
Ej Pbj =
Xij
bij
+
Vjs w
bj
j
s;j;i
|
{z
}
j;s
Direct e¤ect on trade in goods over all stages
17
|
{z
}
TOT e¤ect on importers over all stages
(9)
This is the TOT e¤ect on all the importers, plus the direct e¤ect of changes in trade costs borne by the
importers. The derivation of (9) is given in appendix C.16
The global welfare e¤ect is the di¤erence between the global e¤ect resulting from changes in prices of
exports and the global e¤ect resulting from changes in prices of imports. From (8) and (9), we conclude
that
X
j
bj =
Ej U
|
X
s;j;i
s s
Xij
bij
{z
}
Direct e¤ect on trade in goods over all stages
At each stage, the TOT e¤ect on exporters exactly o¤sets the TOT e¤ect on importers. This is no
surprise, as for each amount of TOT gain by an exporter, there is an equal amount of TOT loss by the
importer. Thus, at each stage, the only welfare impact from the global point of view is the direct e¤ect.
s
P s
P
P Xij
P Ej b
Xij
s =
s = c for all s, we have:
c
X ), which
If c
w Uj =
w
w c
ij , (where Xij =
ij
j Y
ij
s;i;j Y
ij
s
i;j Y
ij
is expression (1).
The multi-stage production model includes single-stage production models as a special case. Examples of the single-stage version of this model to which Proposition 2 applies include: neoclassical model
based on endowment-driven comparative advantage (e.g. Dixit and Norman (1980) with trade costs
or Heckscher-Ohlin model with trade costs), Armington (1969), DFS1977, DFS1980 and EK2002, and
indeed any Ricardian model.
Examples of models to which the multi-stage version of this model applies are Melitz and Redding
(2014) and Yi (2003, 2010). In fact, it applies to all PC models with multi-stage production.
Note that a PC model with more stages of production and one with fewer stages of production but
with the same production function at each overlapping stage will in general give rise to di¤erent Xij for
the same set of bilateral trade costs (with larger Xij under the setting with more stages of production).
Therefore the model with more stages of production gives rise to larger global welfare gains for the same
percentage change in bilateral trade costs. Thus, the global welfare impact of reduction in trade costs
is greater, the larger is the number of stages of production.
Given that multi-stage production and fragmentation are pervasive in global trade, the result we
prove in this section reassures us that the our formula (1) can be applied to estimate global welfare
gains even when there is a lot of trade in intermediate goods.
16
When there is single-stage production, (8) and (9) become
XX
XX
Xij bij
+
Xij w
bi
respectively.
|
j
i
{z
}
D ire c t e ¤e c t o f tra d e c o sts
|
j
i
{z
}
T O T e ¤e c t o n im p o rte rs
18
P
j
bj
Ej E
=
P P
j
i
Xji w
bj and
P
j
Ej Pbj
=
3.3
MC with heterogeneous …rm productivity (MwG)
According to Melitz (2003), the average productivity of a …rm in country i serving market j is given
1
1
R1
1
1
by '
eij
. Thus '
eij is a function of 'ij . The number of …rms in
(')
g
(')
d'
i
1 Gi ('ij ) 'ij
h
i
country i serving market j, Nij = Ni 1 Gi 'ij , is a function of 'ij and Ni . The values of '
eij and
Nij , in turn, directly a¤ects Pj , as shown below. Constant markup implies that the average price of a
good sold in j imported from i is given by
j is given by
Pj =
(
X
1
wi ij
'
eij .
Nij
1
i
Therefore, the expected price index in country
wi ij
'
eij
1
)
1
1
Totally di¤erentiating the logarithm of the above equation and re-arranging yield:
Ej Pbj =
X
i
Xij w
bi + bij
1
1
bij
N
c
'
e
ij
(10)
c
We call '
e
ij the “productivity e¤ect” — the e¤ect of the change in cuto¤ productivity 'ij on the
bij the “…rm mass e¤ect” — the e¤ect of the
average productivity '
eij . On the other hand, we call N
change in cuto¤ productivity 'ij on …rm mass Nij . It turns out that, given any exporting country i,
the two e¤ects are related in the following way when summing over all the importing countries:17
!
X
X
bij
Xij N
c
=
Xij '
e
i.e. …rm mass e¤ect plus productivity e¤ect = 0.
(11)
ij
1
j
j
The intuition is: A change in
ij
for any i; j will have e¤ect on 'ij through its e¤ect on wages. For any
given i, from country j’s welfare point of view, an increase in 'ij leads to an increase in '
eij (average
productivity of each …rm from i serving j is higher — RHS of the above equation without the summation
over j) but a decrease in Nij (fewer …rms from i serving the market in j — LHS of the above equation
without the summation over j), leading to counteracting (but not completely o¤setting) e¤ects on Ej Pbj ,
according to (10). When summing over all j, the two e¤ects o¤set each other completely from a global
welfare point of view.18 Thus, the productivity e¤ect and …rm mass e¤ect completely o¤set each other
from the global welfare point of view, when summing over all i and j.
17
Refer to appendix D for detailed derivation.
18
For example, in the symmetric two-country case (with countries 1 and 2) in Melitz (2003), a reduction of
=
12
=
21
raises the wage in each country. Suppose we focus on i = 1. This raises the domestic productivity cuto¤ '11 (thus
raising '
e11 and lowering N11 ) but lowers exporting productivity cuto¤ '12 (thus lowering '
e12 and raising N12 ) . Moreover,
b11
b12
X11 N
X12 N
d
d
e11 + X12 '
e12 .
= X11 '
1
1
19
Summing up (10) for all possible destination j, global welfare change is given by
X
X
bj Pbj
bj =
Ej E
Ej U
j
j
cj
Ej E
z }| {
X
Xji w
bj
=
j;i
|
{z
X
j;i
Xij w
bi
}
|
X
i;j
Xij bij +
{z
TOT e¤ect on all importers and exporters = 0 Direct e¤ect
=)
X Ej
bj =
U
Yw
j
X Xij
i;j
Yw
bij
}
X
j;i
|
!
bij
Xij N
c
+ Xij '
e
ij
1
{z
}
Global …rm mass e¤ect plus productivity e¤ect = 0
bij and
which is expression (1). Note that the same equation holds under K1980 simply because both N
c
'
e
ij are equal to zero for any i and j.
As Melitz and Redding (2015) have shown, a model based on K1980 and a model based on MwG
with the same deep parameters will in general give rise to di¤erent Xij for the same set of bilateral trade
costs (with MwG yielding larger Xij ). Therefore, the MwG model gives rise to larger global welfare
gains than the K1980 model does for the same percentage changes in bilateral trade costs.
4
Extensions and Empirical Applications
In this section, we analyze two special cases for which Assumption 2 is violated so that prices are not
constant markup over marginal cost. These are not general cases of variable markup, but they are
interesting extensions to the general results reported in section 2. Then, we carry out a couple of simple
empirical applications.
4.1
MwG with multiple sectors
Suppose that the set of goods ! 2
is separated into sectors denoted by
z
where sectors are indexed
by z = 1; :::; Z. Consumers in country j have their preferences represented by the following utility
function:
Uj = U (f uj (z)j z = 1; 2; ::; Zg) where uj (z) =
hP R
i !2
z
ij
z
qij
(!)
1
z
z
d!
i
z
z
1
z (!) denotes the consumption
where the function Uj is homogeneous of degree one in uj (z),19 and qij
of variety ! of sector z in country j of goods originating from country i. In general, the elasticity of
substitution
z
can be di¤erent across sectors.
The …xed exporting cost from country i to j in sector z is equal to
exporting cost from country i to country j in sector z is
19
An example is Uj =
hP
z uj
(z)
1
i
1
.
20
z
ij .
ijz
in units of labor; the iceberg
Each …rm needs to pay a …xed entry cost
of fez in units of labor to acquire a blueprint to produce in sector z. Then the …rm gets a productivity
draw '. Unit labor requirement of good ! is denoted by ai (!)
1='. Thus, ! and ' can be used
interchangeably to index a good. The functions Giz (') and giz (') are the cdf and pdf respectively of
' in sector z in country i.
We prove the following proposition in online appendix B.
Proposition 3 Suppose there are multiple sectors and the market structure of each sector is monopolistic competition as per Melitz (2003) with general …rm productivity distribution, the percentage change
in global welfare is given by
Moreover,
h
z (
X
j;i;z ij
P
P Ej b
j w Uj =
Y
z
X Xij
j;i;z
Yw
i
z = 0.
d
b
1) '
e
+
N
ijz
ij
z
Besides the direct e¤ect,
P
z
Xij
z
j;i;z Y w bij ,
z +
c
ij
z
X Xij
j;i;z
bz
N
ij
d
'
e
ijz +
Yw
there is one more term,
z
1
!
P
z
Xij
j;i;z Y w
.
d
'
e
ijz +
bz
N
ij
z
1
, which is
the combination of …rm mass e¤ect and productivity e¤ect. Moreover, it can be shown that the labor
i
h
P
z =
z (
d
b
1)
'
e
+
N
market clearing condition, together with the free entry condition, lead to j;i;z Xij
z
ijz
ij
0, i.e. the weighted sum of the combination of …rm mass e¤ect and productivity e¤ect over all sectors
(with the weight being
If
z
1) is equal to zero from the global point of view.
z
is the same across sectors,
P
z
Xij
j;i;z Y w
d
'
e
ijz +
bz
N
ij
z
1
is equal to zero as the …rm mass e¤ect
and productivity e¤ect completely o¤set each other (as shown in equation (11) in section 3.3). In that
z
P
P E b
Xij
z
case, the change in global welfare is given by j Y wj U
j =
j;i;z Y w bij , which reduces to expression
(1) when
z
ij
=
ij
for all z. If
z
are di¤erent across sectors, price markups are di¤erent across
sectors. As a result, relative prices are distorted and market resource allocation is not e¢ cient. If
h
i
P
z (
d
b z tends to be positive (negative) in the sector with small z (i.e. higher
X
1)
'
e
+
N
z
ijz
ij
ij
j;i
bz
P
N
ij
z '
d
markup), then j;i;z Xij
e
would be positive (negative), and the combination of the …rm
ijz + z 1
mass e¤ect and productivity e¤ect on global welfare would be positive (negative). In other words, the
e¤ect of the sectors with higher markups dominate. This makes sense as sectors with higher markups are
associated with greater distortion. The e¤ect of sectors with greater distortion would dominate over the
e¤ect of sectors with smaller markup (and hence smaller distortion). Ossa (2015) …nds that assuming
symmetric trade elasticity across sectors leads to a gross underestimation of the gains from trade. Thus,
h
i
P
d
b z tends to be
we can infer that Ossa’s (2012) empirical …nding implies that
X z ( z 1) '
e
ijz + N
j;i
positive (negative) when
z
ij
ij
is small (large).
In principle, we should be able to calculate the value of the extra term, but then we need a lot of
…rm level data in order to do it.
21
4.2
MC with single sector and variable markup
To use the simplest possible model to illustrate how the existence of variable markup a¤ect our benchmark result, consider the case with heterogeneous …rms and there is a discrete number of …rms that
serve each market in equilibrium. As there is a discrete number of …rms, and the changes in trade costs
are in…nitesimal, the number of …rms that serve each market by each country is unchanged after the
reduction of trade costs. Thus, there is neither …rm mass e¤ect nor productivity e¤ect. We continue
to assume that each …rm only produces one variety. As we shall see, the result would be sharper if we
consider a case when a signi…cant market share is concentrated in a small number of …rms from a small
number of countries.
We assume that the utility in country j is given by:
2
6X X F
qij (!)
Uj = 4
i
where
!2
1
F
ij
3
1
7
5
(12)
> 1 is the elasticity of substitution. Then, consumer optimization yields the demand function
for good !:
F
qij
(!) =
where Pj =
P P
i
!2
F
ij
h
pFij
i1
(!)
pFij (!)
Pj1
Ej
(13)
1
1
is the consumer price index in country j. Let ' be the labor
productivity associated with variety !.
Given (13), pro…t maximization yields
ij
where
ij
(') =
pF
ij (')
cij (')
(') = 1 +
1
1) [1 sij (')]
(
is the markup by a …rm and cij (') =
ij wi
'
is the marginal cost, and sij (')
xij (') =Ej is the market share of a product by a …rm with productivity ' from country i in country j,
and xij (') is the exports of the …rm with productivity ' from country i to country j.
Clearly,
ij
(') increases with sij ('), which in turn increases with '. Obviously, Pbj would be
a¤ected not just by w
bi + bij (which a¤ects cij (') ), but also by bij (').
We prove the following proposition in online appendix C.
Proposition 4 Suppose there is a single sector and single stage of production and the market structure
is monopolistic competition with variable markup, the percentage change in global welfare is given by
2
3
X Xij
X Ej 6 X
X Ej U
bj
7
=
bij
sij (') bij (')5
4
w
w
Y
Y
Yw
F
j
i;j
i;j
22
!2
ij
Compared with the benchmark result, there is an extra term which depends on the changes in
markups of …rms. If …rms with large market shares tend to reduce (raise) their markups following
reduction of trade costs, the global gains would be larger (smaller) than the benchmark case. This
makes sense as markups are distortions, and lower markups lead to high e¢ ciency and thus higher
global welfare gains. This is consistent with the …nding of, for example, Edmond, Midrigan and Xu
(2015), who report that reduction of trade costs tend to raise the markups of foreign …rms but lower the
markups of domestic …rms, but since the market shares of domestic …rms are larger, there are additional
gains from reduction of trade costs due to the existence of variable markup. Note that in the case where
countries are symmetric (with …xed exporting costs) where sij (') for i 6= j is the same for any given ',
the second term is trivial given that there is a large number of countries n. However, if we consider a
highly asymmetric case when a signi…cant market share is concentrated in a small number of …rms from
a small number of countries, then the second term can be non-trivial compared with the …rst term.
Again, in principle, we should be able to calculate the value of the extra term, but then we need a
lot of …rm level data in order to do it.
4.3
Empirical Applications
To illustrate the user-friendliness of our formula (1), we o¤er two empirical applications.
Empirical Application One. First, we demonstrate that we can easily calculate the elasticity of
global welfare with respect to a uniform reduction of all bilateral iceberg trade costs. This reduction
can be due to technological improvements or other exogenous changes such as implementation of trade
facilitation measures. Based on our theory, we only need to know the share of total trade value in world
GDP in order to calculate the impact on global welfare. For example, in 2003, the data indicate that
total value of world merchandise trade was approximately equal to 20.0% of world GDP. It follows from
(1) that the elasticity of global welfare with respect to a uniform percentage reduction in all bilateral
iceberg trade costs is equal to 0.200. In other words, 1% reduction in all bilateral iceberg trade costs
would increase global welfare by 0.200% of world GDP in 2003.
How is the global welfare change estimated from our formula (1) compared with other estimates in
the profession? The OECD (2003, p.4) estimates that, in 2003, assuming that trade facilitation leads
to a reduction in border procedure-related trade transaction costs (TTCs) by 1 per cent of the value of
world trade, global welfare gains would be about USD 40 billion. Let us compare the two estimates.
By de…nition, the bilateral TTCs (denoted by Tij ) between i and j as a fraction of the value of
exports from i to j can be written as Tij =
ij
1
tij , where tij denotes other sources of bilateral trade
costs (such as transport costs, tari¤s, etc) as a fraction of the value of bilateral exports. Therefore, a
reduction of bilateral TTCs from i to j by 1% of the value of exports from i to j means that dTij =
which implies that d
ij
=
0:01,
0:01 when tij stays unchanged. Anderson and van Wincoop (2004) estimate
23
that the tax equivalent total international cost is about 74%, which implies that ij = 1:74. With
P
d ij = 0:01, we have d ij = ij = 0:00575. The value of world trade in 2003, i;j Xij , is approximately
USD 7.7 trillion. Thus, in 2003, a uniform reduction of Tij by 1% of the value of exports from i to j for
P
all i, j leads to an increase in global income of i;j Xij c
ij , which is equal to USD 44.3 billion, according
to our theory. Therefore, our estimate is not that di¤erent from that of OECD.
Empirical Application Two.
A notable component of shipping costs is shipping time. We
want to …nd the cumulative global welfare impact of the reduction of international shipping time in
the …fty-year period 1960-2010. Hummels and Schauer (2013) estimate that each additional day in
transit is equivalent to an increase in [0.6%, 2.1%] of ad valorem trade cost. We need to estimate how
much bilateral shipping time has been reduced each year over this period, and it is not easy to …nd
data. But we can do a rough estimate. Based on Hummels’(2001) estimate that “the introduction of
containerization in the late 1960s and 1970s results in a doubling of the average ocean ‡eet speed”, we
assume that average shipping time was 40 days for international trade in 1960, compared to the 20 days
in 2010 cited by him. In 1960, almost all international shipments were through ocean freight. We also
assume that there was gradual reduction of ocean shipping time and a gradual substitution towards air
freight since 1960. The average international ocean shipping time in 2010 was 20 days and air-shipped
trade rose from (approximately) 0% to 50% from 1960 to 2010.20 Thus, the average shipping days
dropped from 40 to 0.5*20 (by sea) + 0.5*1 (by air) = 10.5 days during the period 1960-2010 (assuming
that air freight takes just one day). If we assume the cost reduction process to be gradual (i.e. the
annual reduction of shipping days is constant throughout the …fty years), the number of shipping days
dropped by (40-10.5)/50 = 0.59 per year during the period 1960-2010, which is equivalent to a reduction
of ad-valorem trade cost by [0.354%, 1.239%] per year. Based on the data of the share of merchandise
trade in GDP from World Development Indicator (WDI) published by the World Bank, we calculate
from equation (1) the cumulative global welfare gains in the …ve decades 1960-2010 from the saving in
shipping time to be [2.71%, 9.81%], which is equivalent to an increase in global income in the range
USD [1709, 6183] billion in 2010, with the mid-point being USD 3946 billion.21 ,
20
22
These numbers are based on U.S. trade statistics. Given that the shipping industry has been very competitive, we
think it is reasonable to assume that these numbers also apply to all other countries of the world. A sharp speeding up of
ocean transport followed from the introduction of containerization in the late 1960s and 1970s. To simplify the calculation,
we assume that the annual rate of reduction of trade cost is constant during this period.
21
These estimates are roughly in line with the …nding of, say, Ossa (2014), who …nds that the average welfare gains
of moving from autarky to free trade for Brazil (10%), China (13.1%), European Union (12.6%), India (11.2%), Japan
(15.4%) and US (14.2%) is equal to about 11.0%.
22
percentage global welfare gain in year t. The cumulative percentage increase in global welfare is equal
t be the #
"Let
2009
Y
to
(1 + t )
1.
t=1960
24
5
Conclusion and Caveats
Our paper is motivated by the following question: How much does trade facilitation matter to the world?
Guided by this question, we derive a simple equation to evaluate the quantitative impact on global
welfare of small reduction of bilateral trade costs, such as shipping costs or the costs of administrative
barriers to trade, based on a set of simple assumptions. Although our equation cannot evaluate the
distribution of gains for di¤erent countries, it informs us of the increase in the size of the global pie.
Given that there are many channels for transfers between countries, a larger pie can very well make
every country better o¤. If global gains resulting from bilateral trade costs reduction are found to be
large, then it would provide stronger support to the advocates of global trade facilitation such as WTO,
OECD, and the World Bank.
Surprisingly, the equation is very general and is applicable to a broad class of models and settings.
We then carry out some extensions by relaxing the assumption of constant markup. We illustrate the
user-friendliness of the formula by carrying out a couple of simple empirical applications. We …nd the
estimates obtained from the applications to be reasonable and consistent with other estimates in the
profession.
Our paper distinguishes from other works in the literature in a few key aspects. First, not only have
we proved that only the direct e¤ect matters, but we have also proved that the underlying mechanism
driving the result is the envelope theorem. Thus, the formula is applicable to a broad variety of models
and settings as long as there are no externalities or price distortions. This intuition has not been
explained clearly in the literature. Second, we rigorously justify the use of the expenditure-shareweighted average percentage change of country welfare as a measure of the change of global welfare,
based on the concept of equivalent variation. Third, we investigate the implications of non-constant
markup on our benchmark result by carrying out a couple of extensions. The extensions further deepen
our understanding of how to evaluate quantitatively the global gains from reduction of trade costs.
Our simple equation provides a good benchmark for future research. The framework can be extended
to analyze more sophisticated settings and to answer more questions. Here are a few directions for future
research. First, our framework can also be applied to analyze the worldwide e¤ects of other shocks, such
as endowment changes or innovation originating from a set of countries. Second, while the global size of
gains from reduction of trade costs is independent of the indirect e¤ects based on our set of assumptions,
the global distribution of gains is not. It would be interesting to modify the present analysis to calculate
the distribution of global gains from trade cost reduction using di¤erent trade models, and compare
the results. Third, our present analysis can also be extended to calculate the welfare impacts of tari¤
reductions when the e¤ects of tari¤ revenues are properly accounted for. Finally, one e¤ect our model
does not capture is the dynamic gains from trade costs reduction, such as learning by doing or learning
by exporting. This is an interesting and potentially important source of gains from trade. It is left for
25
future research.
26
Appendix
A
Speci…c Proof of Proposition 2: PC with multi-stage production
The setting, preferences, technology and market structure are as described in sections 2.1 and 2.2. This
speci…c proof is a speci…c case (PC with multi-stage production) of the general proof of Proposition 2
presented in section 2.3. The multi-stage production model subsumes the single-stage production model.
So, we do not provide a separate proof for the single-stage production case. Recall that the pre-change
n
o
F (!) i 2 N , ! 2 F
vector of market prices of …nal goods sold in country j is pF
p
ij
ij and pre-change
j
n
o
F (!) i 2 N , ! 2 F . The
vector of market quantity of …nal goods consumed by country j is qF
q
ij
ij
j
P
P R
F =
F (!) dq F (!) d!.
sum of equivalent variations of all countries is equal to j pF
dq
p
ij
i;j ! F ij
j
j
ij
Maximization of global income
s ; 8j; k; s, and
The global planner maximizes global income W by choosing ljk
costs
s
ij
s
jk
8j; k; s taking trade
for all i; j,s and the shadow prices of …nal goods pFij for all i; j (which are also the market prices)
as given:
XZ
F (!)
pFij (!) yij
max
d!
W =
F
s ;8j;k;s
F
fljk
gf sjk ;8j;k;sg
ij
i;j ! ij
P R
s (!) d!
s.t. (i) labor constraint Lj = k;s !2 s ljk
for all j , (ii) the production function of stage-s
jk
good ! exported from j to k, given by
8
s 1
yij;k
(! 0 )
>
< 's (!) f
i 2 N , !0 2
s
1
j
s
ij
yjk
(!) =
>
:
's (!) ls (!)
j
s 1
ij
s (!)
; ljk
for s = 2; 3; :::; F
for s = 1
jk
for ! 2
s
jk
(14)
s (!) is the quantity of labor used in j in combination with the quantities
for all j; k = 1; :::; n, where ljk
s 1
of inputs imported from i to j, yij;k
(! 0 ) =
s 1
qij;k
(! 0 ), to produce stage-s output ! to be exported
P R
s (!), and 's is the productivity, which is constant. Thus,
s
s
from j to k, yjk
j
k !2 sjk ljk (!) d! = lj for
P
all j and s, and s lis = Li for all i.
Suppose there is a change in
s
ki
s 1
ij
s
such that c
ki < 0. We want to evaluate the e¤ect of this change on
s (!) for all i; j; s; ! are optimally chosen given the exogenous
W . We start from a state where all the lij
variables. According to (7), we only need to evaluate @W=@
keeping l and p unchanged, recalling that l
that psij (!) = psii (!)
s
ij .
s
ki ,
i.e. evaluate the e¤ect of
s (!) ; 8i; j; s; ! and p
lij
s
ki
on W
(psij (!)); 8i; j; s; !. Recall
Since the unit labor requirement for any p1ij (!) at stage 1 is constant, wi is
unchanged as all p1ij (!) are kept unchanged. We proof our result step by step below.
27
1. The total global value of inputs at stage s + 1 is equal to
!
X X
X
X
s+1
s
s
Xki + wi li
=
Xki
+
wi lis+1
i
where
s
Xki
=
k
R
!2
i
i;k
s
ki
s (!) d!
pski (!) yki
s
ki
for all s = 1; 2; :::; F .
is the value of stage-s output exported from k to i which is used as stage-(s + 1) input.
As
s
ki
s (!) unchanged for all s; k; i !, y s (!) is also unchanged.
varies while keeping pski (!) and lki
ki
Therefore, the e¤ect of
s
ki
s is given by
on Xki
s
@Xki
d s
s
@ ki ki
| {z }
=
Change in the value of stage-s
s c
s
Xki
ki
(15)
input exported from k to i
which is the direct saving in trade costs.
2. The value of stage-(s+1) output exported from i to j is given by
Z
s+1
s+1
Xij =
ps+1
ii (!) yij (!) d!.
!2
s+1
ij
At stage s + 1, the total global value of outputs is equal to the total global value of inputs:
X
i;j
|
s+1
Xij
{z
s
ki
s
Xij
X
+
i;j
}
| {z }
global value of outputs at stage s+1
As
X
=
wi lis+1 .
| i {z
(16)
}
global value-added at stage s+1
global value of inputs at stage s+1
is reduced, it increases the global value of inputs at stage s + 1,
P
i;j
s , through the direct
Xij
saving in trade costs, according to (14) for stage s+1 production. This in turn increases the global value
n
o
P
s+1
s+1
of outputs at stage s + 1, i;j Xij
, through the increases in yij
(!) according to (14). At stage
s+2, we have
X
s+2
Xij
=
i;j
X
s+1
Xij
+
i;j
X
wi lis+2 .
i
n
o
n
o
s+2
s+1
That is, increases in yij
(!) in turn lead to increases in yij
(!) according to (14) for stage s+2
P
s+2
production, which raises the global value of outputs at stage s+2, i;j Xij
. This process goes on until
n
o
P
F
F (!) . Thus, we have
the …nal stage F, leading to an increase in i;j Xij through increases in yij
X
F
Xij
i;j
| {z }
global income
=
X
s
Xij
+
i;j
| {z }
X
|
i
wi
F
X
m=s+1
{z
lim
}
cumulative global value-added from stages s+1 to F
global value of inputs at stage s+1
28
.
P
PF
s varies,
m
Since wi 8i and lim 8i, m are kept unchanged as ki
i wi
m=s+1 li is unchanged as well.
P
F , we have
As W = i;j Xij
0
0
1
1
s
X
X
@Xki
@ @
@W s
@ @
s
s c
FA
s
sA
s
s
d
=
X
d
=
X
d
=
ij
ij
ki
ki
ki
s
s
s
s d ki = Xki ki .
@ ki
@ ki
@ ki
@ ki
i;j
d
s
ij
i;j
The economic intuition is as follows. The initial welfare impact of a small reduction of trade cost of
sc
s
at stage s is equal to a gain of Xij
ij at stage s + 1 received by country j due to the reduction in
s denote the value of exports of stage-s good from country i
the cost of its intermediate good, where Xij
to country j. But since country j’s stage-s + 1 good is subsequently used by all other countries for their
production of goods at stage-s + 2, s + 3 and so on, the cost saving is passed on fully to all countries
in each later stage. This process will go on until the …nal stage. Eventually, the saving in trade cost
shows up as the gains in global income received by all countries in the …nal stage F . Therefore, the
sc
s
global welfare e¤ect of a change in trade cost ijs at stage s is equal to Xij
ij . Thus, the percentage
change in global welfare is equal to
|
i;j;s
Y
w
=
{z
X Xij
i;j
Yw
s
If c
ij for all s, and Xij =
ij = c
which is expression (1).
B
sc
s
X Xij
ij
P
c
ij ,
}
s
s Xij
Speci…c Proof of Proposition 2: MwG
The setting, preferences, technology and market structure are as described in sections 2.1 and 2.2. This
speci…c proof is a speci…c case of the general proof of Proposition 2 presented in section 2.3. Recall
that the labor productivity '
1=ai (!). So, each variety is indexed by ' or !, with a unique mapping
between the two. The two indexes are used interchangeably. Analogous to the general proof in section
n
o
n
o
F (') ; 8i; ' and qF = q F (') ; 8i; ' . The sum of equivalent variations of all
2.3, we de…ne pF
=
p
ij
ij
j
i
hP j R
P F
1 F
F (') g (') d' .
F
countries is equal to j pj dqj =
N
p
(')
dq
i
ij
ij
j;i i '
ij
Maximization of global income
n o
The global planner maximizes global income W by choosing flij (') ; 8i; j; 'g ; 'ij andfNi g, taking
trade costs
for all i; j and the shadow (market) prices pFij (') for all i; j; ' as given. Thus, she solves
X Z 1 pFij (') f ' (lij (')) gi (')
max
W =
Ni
d'
ij
flij (');8i;j;'g;f'ij g;fNi g
'ij
i;j
8
9
<
=
X
XZ 1
s.t. Ni fe +
Gi 'ij +
lij (') gi (') d' = Li for all i
ij 1
:
;
'ij
ij
j
j
29
F (') = f ' (l (')) is the production function for producing y F (') =
where yij
ij
ij
by a …rm with productivity ' in country i for sales to country
Analogous to the general proof in section 2.3, we de…ne l
allocation; p
trade
(pFij
costs.24
@W
=
@ ij
as Ni
R1
'ij
Ni
Z
1
(lij (')) ; 8i; j; ' as a vector of all labor
pFij (') f ' (lij (')) gi (')
2
ij
'ij
(') units of good
j.23
(')); 8i; j; ' as a vector of all shadow prices; and
Then,
F
ij qij
(
ij ) ; 8i; j
as a vector of all
Xij
d' =
ij
n o
'
pF
ij (')f (lij ('))gi (')
d'
=
X
.
As
fl
(')
;
8i;
j;
'g
;
'ij andfNi g have been optimally chosen,
ij
ij
ij
their e¤ects on W are second order. Thus, we can analogously invoke (7) to compute the percentage
change in global welfare as
1 X @W
d
Yw
@ ij
ij
=
i;j
i;j
which is expression (1).
C
X Xij c
ij
Yw
Derivation of Equation (9)
Note that pbsjj (!) = pbsjj 8j; s; !. From the production function (4), we get
pbsjj
j
s
=
"
X
i
s 1
Xij
P s
i Xij
bijs 1
1
i
)
j
s
s 1 s
Xij
pbjj =
X
i
s s
Xji
pbjj
=
j
s
X
X
Dividing both sides by
j;i
|
1
i
j
s
i
X
s
Xij
bijs
s 1 s 1
Xij
pbii
j
s,
s s
Xij
pbii
1
+
+ pbsii
j
s
1
X
i
j
s
+ 1
X
i
s 1 s 1
Xij
bij
+ 1
w
bj , for s = 2; 3; :::; F
s 1
Xij
w
bj
j
s
j
s
X
i
s
Xji
w
bj
as
X
s
Xij
i
1
=
j
s
X
s
Xji
i
summing over all j, and re-arranging terms, we have
X
j;i
{z
s 1 s
Xij
pbii
1
}
Increment of the cost e¤ect for
importers from one stage to the next
23
j
s
+ 1
pbsjj = w
bj , for s = 1
This equation implies that
X
+
#
pbsii 1
=
X
j;i
|
s 1 s
Xij
bij
{z
1
}
increases in trade costs
+
X
j;i
|
1
j
s
{z
s
Xji
w
bj
}
TOT e¤ect on importers
Here we assume that the production function is the same in all countries. We could assume that the production
function is di¤erent across countries, but the conclusion will remain the same.
24
We abuse the notation a little here as the vector l and p includes in…nite elements (the upper bound for ' is 1).
30
In other words, the increment of the cost e¤ect for importers from one stage to the next is the sum
of the increases in trade costs and increases in the factor costs of the importers (i.e. TOT e¤ect on
importers).
Summing up the e¤ect over all stages from stage 2 to stage F and noting that pb1ii = w
ci , we have
X
j;i
|
Fc
Xij
pFii
{z
=
F
X1 X
s=1 j;i
}
Cost e¤ect on importers
X
s s
Xij
bij +
j;i
1
Xij
w
ci +
F X
X
j
s
1
s=2 j;i
s
Xji
w
bj
of …nal goods
F
X1 X
=
s s
Xij
bij
s=1 j;i
{z
|
XX
+
}
|
s
j
Vjs w
bj
{z
(17)
}
Direct e¤ect over all stages
TOT e¤ect on importers
other than the …nal one
over all stages
The global welfare e¤ect resulting from changes in prices of imports is given by25
X
X
F c
Ej Pbj =
Xij
pF + bijF .
ii
j
j;i
Substituting from (17) into the above equation, we obtain (9).
D
Derivation of Equation (11)
De…ne Rij (') as the revenue of a …rm exporting from i to j, with productivity '. Note that CES
preferences implies that
Rij ('
eij )
=
Rij 'ij
1
('
eij )
(18)
1
'ij
Total revenue is equal to total income:
wi Li =
P
j Xij
=
which, together with equation (18) and Rij 'ij =
wi Li =
P
j Nij
P
j Nij Rij
ij wi
('
eij )
('
eij )
(zero cuto¤ pro…t condition), imply:
1
'ij
ij wi
1
Di¤erentiating the natural logarithm of this equation leads to
25
Compare with
P
j
Ej Pbj =
w
ci =
b +(
j Xij Nij
P P
j
P
i
1)
P
j Xij
c
'
e
ij
Xij (w
bi + bij ) in the one-stage case.
31
'c
ci
ij + w
P
which implies equation (11), since
j
1) 'c
ij = 0, which is a consequence of the free entry
Xij (
condition (FEC). To see this, …rst note that the FEC implies that the total expected …xed costs is equal
to the expected net revenue:
fe +
P
j
1
G 'ij
ij
h
X 1
=
Gi 'ij
wi
j
i
Rij ('
eij )
X
=
1
Gi 'ij
ij
('
eij )
'ij
j
1
1
Totally di¤erentiating the LHS and RHS of the above equation leads to
X
j
c
ij g 'ij 'ij 'ij =
X
ij g 'ij
1
G 'ij
X
('
eij )
ij
,
X
1 X
Nij
Ni
ij
1
'ij
G 'ij
ij
1
'ij
j
('
eij )
1
'ij
j
1
('
eij )
1
(
X
,
1) 'c
ij = 0
1
c
1
(
1 'ij 'ij
'ij
j
j
,
1
('
eij )
h
1) 'c
ij = 0
(
as Nij = Ni 1
c
1) '
e
ij
(
1) 'c
ij
i
G 'ij
1) 'c
ij = 0
Xij (
j
where the second equality stems from the e¤ect of the change in cuto¤ productivity 'ij on the average
productivity '
eij , which is given by:
1
c
'
e
ij =
E
1
'ij
1
= ('
eij )
1
gi 'ij 'ij
'c
ij
Gi 'ij
“Productivity e¤ect”.
1
(19)
Mapping between our expression (1) and ACR’s equation (1)
Assumption R3 in ACR is equivalent to bij bjj = d ln
ij
d ln
jj
= d ln Xij d ln Xjj = "d ln
Thus
X Xij
j;i
Y
b
w ij
X Xij 1
bij
Yw "
=
j;i
bjj
X Ej X Xij 1
bij
Yw
Ej "
=
j
i
X Ej X ij
bij
Yw
"
=
j
i
X Ej 1
bjj
=
Yw "
j
32
bjj
bjj
ij
= "bij .
as
P
i
ij
=1)
P
i d ij
= 0. In short, if one adopts Assumption R3 in ACR, then the global gains
formula is precisely equal to an expenditure-share-weighted average percentage change of individual
country’s gains from trade (as given by the ACR formula) where the weights are the respective country’s
shares in world expenditure.
33
References
[1] Anderson, James E. and Eric Van Wincoop. 2004. “Trade Costs.” Journal of Economic Literature
42(3), 691-751.
[2] Arkolakis, Costas, Arnaud Costinot and Andres Rodriguez-Clare. 2012. “New Trade Models, Same
Old Gains?” American Economic Review 102(1), 94-130.
[3] Armington, Paul. 1969. “A Theory of Demand for Products Distinguished by Place of Production.”
International Monetary Fund Sta¤ Papers, XVI (1969), 159-178.
[4] Atkeson, Andrew and Ariel Burstein. 2010. “Innovation, Firm Dynamics, and International Trade.”
Journal of Political Economy 118(3), 433-484.
[5] Burstein, Ariel and Javier Cravino. 2015. “Measured Aggregate Gains from International Trade,”
American Economic Journal: Macroeconomics, forthcoming.
[6] Dhingra, Swati and John Morrow. 2014. “Monopolistic Competition and Optimum Product Diversity Under Firm Heterogeneity.” Journal of Political Economy, forthcoming.
[7] Dixit, Avinash and Victor Norman. 1980. "Theory of International Trade." Cambridge University
Press.
[8] Dornbusch, Rudiger, Stanley Fisher and Paul A. Samuelson. 1977. “Comparative Advantage, Trade,
and Payments in Ricardian model with a Continuum of Goods.”American Economic Review 67(5),
823–839.
[9] Dornbusch, Rudiger, Stanley Fisher and Paul A. Samuelson. 1980. “Heckscher-Ohlin Trade Theory
with a Continuum of Goods.” The Quarterly Journal of Economics 95(2), 203-224.
[10] Eaton, Jonathan and Samuel Kortum. 2002. “Technology, Geography and Trade.” Econometrica
70(5), 1741–1779.
[11] Edmond, Chris, Virgiliu Midrigan and Daniel Yi Xu. 2015. “Competition, Markups, and the Gains
from International Trade.” American Economic Review 105(10), 3183-3221
[12] Fan, Haichao, Edwin L.-C. Lai and Han Ste¤an Qi. 2012. “Global Gains from Trade Liberalization.”
CESifo Working Paper No. 3775 (March 2012).
[13] Feenstra, Robert C. 2004. Advanced International Trade. Princeton University Press: Princeton,
New Jersey.
[14] Friedman, Allan M. 1998. "Kaldor-Hicks Compensation." in The New Palgrave Dictionary of Economics and the Law, Peter Newman (ed.), Palgrave Macmillan.
34
[15] Hsieh, Chang-Tai and Ralph Ossa. 2011. “A Global View of Productivity Growth in China.”NBER
Working Paper No. 16778.
[16] Hummel, David L. 2001. “Time as a Trade Barrier.”Unpublished working paper, Purdue University.
[17] Hummel, David L. and Schaur Georg. 2013. “Time as a Trade Barrier.”American Economic Review
103(7), 2935-2959.
[18] Krugman, Paul. 1980. “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.”American Economic Review 70(5), 950-959.
[19] Melitz, Marc J. 2003. “The Impact of Trade on Intraindustry Reallocations and Aggregate Industry
Productivity.” Econometrica 71(6), 1695-1725.
[20] Melitz, Marc J. and Stephen J. Redding. 2014. “Missing Gains from Trade?”, American Economic
Review Papers and Proceedings 104(5), 317-321.
[21] Melitz, Marc J. and Stephen J. Redding. 2015. “New Trade Models, New Welfare Implications.”
American Economic Review, 105(3), 1105-1146.
[22] OECD, Working Party of the Trade Committee, “Quantitative Assessment of the Bene…ts of Trade
Facilitation,” TD/TC/WP(2003)31 (19 September 2003).
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(2), 4104–4146.
[24] Ossa, Ralph, 2015. “Why Trade Matters After All.”Journal of International Economics 97 (2015)
266–277.
[25] Varian, Hal. 1992. Microeconomic Analysis. Third Edition, W.W. Norton: New York.
[26] World
Economic
Forum,
“Enabling
Trade,
Valuing
Opportunities,”
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[27] Yi, Kei-Mu. 2003. “Can Vertical Specialization Explain the Growth of World Trade?” Journal of
Political Economy 111(1), 52-102.
[28] Yi, Kei-Mu. 2010. “Can Multistage Production Explain the Home Bias in Trade?”American Economic Review 100(1), 364-393.
35
Online Appendix (not to be published)
A
Proof that the global market outcome is e¢ cient under MwG
F (') through the choice of
The global planner solves two problems simultaneously. First, she chooses qij
F (') to maximize the
labor allocation lij (') to maximize the global GDP Y w . Second, she chooses qij
utility of each individual country j, Uj . In other words, she makes the production decision as well as the
consumption decision. The shadow price pFij (') equalizes the quantity supplied and quantity demanded
F ('), for 8i; j; '.
of good qij
Maximization of utility of each country
F (') to maximize U given N , ' , E and pF ('). Thus, she solves
She chooses qij
j
i
j
ij
ij
"
Z
X
1
#
1
1
F
Max Uj =
gi (') d'
Ni
qij
(')
F (')
q
'ij
f ij g
i
X Z 1
F
s:t:
Ni
pFij (')qij
(')gi (') d' = Ej
j
(21)
'ij
i
Let
(20)
be the Lagrange multiplier. Then, the Lagrangian function is given by
"
#
X Z 1
F
F
Lj = Uj + j Ej
Ni
pij (')qij (')gi (') d'
'ij
i
F (') into budget constraint (21) yields
Substituting the F.O.C. w.r.t. qij
j
where Pj =
P
R1 h
i Ni 'ij
pFij
i1
(')
=
1
Pj
1
1
gi (') d'
is the ideal price index in country j. Note that
F (') we have
Uj = Ej =Pj . Substituting the above relationship back into the F.O.C. w.r.t. qij
F
qij
"
pFij (')
(') =
Pj
#
"
pFij (')
Uj =
Pj
#
Ej
Pj
This is a demand function, and it will be used in the global income maximization problem.
Maximization of global income
36
(22)
She chooses Ni , 'ij and labor allocation lij (') to maximize the global GDP Y w , given
pFij ('),
for 8i; j; '. The variables
F
qij
and lij ('). Since
(') =
'lij (')
ij
F
qij
f
F (') when
, the choice of lij (') is the same as the choice of qij
max
8
<
X
s.t. Ni fe +
:
Ni ; 'ij g
1
ij
Li and
(') and Ej are implicitly determined from the choices of Ni , 'ij
' are given. Thus, her problem is stated as:
F (');
qij
ij ,
Yw =
X
Z
Ni
'ij
i;j
Gi 'ij
+
j
1
XZ
1
'ij
j
F
pFij (')qij
(')gi (') d'
ij F
qij (')gi (') d'
'
9
=
;
ij
and
(23)
= Li for all i
(24)
!
F (')
p
Ej
ij
F
and qij
(') =
from utility maximization
Pj
Pj
X Z 1
F
pFij (')qij
(')gi (') d',
and Ej =
Ni
'ij
i
F (') =
where qij
F (')
yij
=
ij
'lij (')
ij
. Let
be the Lagrange multiplier. Substituting for pFij (') from (22)
i
into (23), we get the Lagrangian function
X Z 1
1
F
(')
L=
Ni
(Ej ) Pj qij
'ij
i;j
+
X
i
i
*
Li
8
<
X
Ni fe +
:
1
gi (') d'
ij
1
Gi 'ij
+
j
XZ
1
'ij
j
ij F
qij (')gi (') d'
'
F ('), we get
From the F.O.C. w.r.t qij
ij
i
F
qij
(') = Ej
1
Pj
'
9+
=
;
1
Substituting this equation into the labor constraint (24) yields
i
h
P
Li
f
1
G
'
e
i
ij
j ij
Ni
=
i
1
R
P 1
ij
Ej gi (') d'
j '
1
'Pj
ij
F (') in terms of E , P and other exogenous
Combining the last two equations, we get an expression for qij
j
j
F (') from the resulting expression into equation (23) and substituting for
variables. Substituting for qij
pFij (') from (22) into (23), we have:
Yw =
X
i
Ni
8
<L
i
: Ni
fe
X
ij
1
Gi 'ij
j
9
=
1
;
2
XZ
4
1
'ij
j
ij
'Pj
1
31
Ej gi (') d'5
(25)
Now we can maximize Y w by choosing the optimal combination of Ni and 'ij based on the above
equation. The F.O.C. w.r.t Ni of above expression leads to
Ni =
DP nh
1
j
Li
Gi 'ij
37
i
ij
o
+ fe
E ,
(26)
which can be easily obtained from the free entry condition under the decentralized market. Substituting
P R1
1 Li
1
F (')g (') d' =
equation (26) into (24) yields j ' 'ij qij
of the labor
i
Ni , which means that
ij
will be allocated to cover the …xed market entry cost, while the rest will be allocated to production.
Substituting this expression into the de…nition of each country’s GDP yields
X Z 1 1
1
F
Ej Pj qij
gi (') d'
wi Li = Ei =
Ni
(')
'ij
j
=
Thus
i
i Li
= wi , which implies that pFij (') =
wi ij
' ,
1
which is the same expression as the market
F (') and the corresponding
outcome. Thus qij
F (')
qij
lij (') =
'
ij
wi
1
= Ej
1
ij
'Pj
both have the same expressions as under market equilibrium (refer to (6) in the main text.)
According to equation (25), maximizing Y w w.r.t. 'ij is equivalent to maximizing
1
ln
8
<L
i
: Ni
fe
X
1
ij
Gi 'ij
j
At the optimal 'ij , we have
ij gi
1
Li
Ni
which is equivalent to
P
fe
ij wi
j
'ij
h
ij 1
= Ej
9
=
;
2
Z
1 4X
+ ln
j
1
ij
1
ij
Gi 'ij
pF
ij ('ij )
Pj
i=
1
j 'ij
F '
= pFij 'ij qij
ij
…rm corresponding to the cuto¤ productivity will spend
1
Ej gi 'ij
'ij Pj
P R1
1
Ej gi (') d'5 .
'Pj
'ij
ij
3
1
1
'Pj
Ej gi (') d'
by invoking (26). The marginal
of its revenue to pay for the market entry
cost, which is the same as the market equilibrium outcome.
As the global planner’s choices of Ni , 'ij and lij (') are exactly the same as the market equilibrium,
we can conclude that the global planner’s solution shown above is exactly the same as the
equilibrium outcome.
B
Proof of Proposition 3
Step 1: Derivation of the change in the ideal price index. The ideal price index Pj of Uj can be written
as:
Pj = min
P
z pj
(z) uj (z) s.t. Uj (f uj (z)j z = 1; 2; ::; Zg)
38
1
where pj (z) is the ideal price index for subutility uj (z). Since the utility function is homogeneous
of
P z
P
X
ij
degree one, totally di¤erentiating Pj implies that Pbj = z j (z) pbj (z) where j (z) = P i z denotes
z;i Xij
R
z
z
z
the expenditure share on sector-z goods in country j, where Xij
!2 z pij (!)qij (!)d! is the exports
ij
of sector-z goods from country i to country j, and
pj (z) =
(
X
wi ijz
'
eijz
z
Nijz
1
z
i
1
z
1
is the aggregate price index for sector z, and '
eijz
1 Giz ('ijz )
)
1
1
R1
z
(27)
'ijz
(')
z
1
giz (') d'
1
z 1
is the
average productivity of a …rm in country i serving market j in sector z, 'ijz is the cuto¤ productivity
for a …rm in country i to pro…tably export to country j in sector z. Totally di¤erentiating the consumer
price index (27), we get
1
pbj (z) = w
bi + bijz
Thus,
1
z
z
d
N
ij
P z
P
i Xij
b
Pj = z P
bj (z)
zp
z;i Xij
Ej Pbj =
)
P
z
z;i Xij
w
bi + bijz
d
'
e
ijz .
1
z
1
z
d
N
ij
Step 2: The change in global welfare is given by
X
j
=
bj =
Ej U
X
j;i;z
=
X
z
Xji
w
bj
j
X
j;i;z
|
bj
Ej E
X
j;i;z
Pbj
1
z
Xij
w
bi + bijz
z
Xji
w
bj
{z
X
j;i;z
1
z
z
Xij
w
bi
}
TOT e¤ect on all importers and exporters = 0
=
X
j;i;z
z z
Xij
bij +
X
j;i;z
z d
Xij
'
eijz +
|
zN
z
d
Xij
ij
z
1
z
d
N
ij
X
j;i;z
d
'
e
ijz
z z
Xij
bij +
{z
direct e¤ect
!
}
X
j;i;z
|
d
'
e
ijz
z d
Xij
'
eijz +
{z
zN
z
d
Xij
ij
z
1
!
}
global …rm mass e¤ect plus productivity e¤ect 6= 0
which leads to the main equation in Proposition 3.
Step 3: Proof of the following relationship stated in Proposition 3:
P
zd
z
z;j Xij Nij
=
P
z;j
39
(
z
z d
1) Xij
'
eijz
(28)
by:
Proof: The e¤ect of the change in cuto¤ productivity 'ijz on the average productivity '
eijz is given
1
z
'ijz
d
'
e
ijz =
1
1
= ('
eijz )
1
z
giz 'ijz 'ijz
'd
ijz
z
Giz 'ijz
“The productivity e¤ect”.
1
(29)
z (') as the revenue of a …rm with productivity ' exporting from i to j in sector z. CES
De…ne Rij
preferences implies that:
z ('
Rij
eijz )
('
eijz )
=
z '
Rij
ijz
z
1
'ijz
(30)
1
z
Total income is equal to total revenue:
wi Li =
P
z
z;j Xij
=
P
z z
z;j Nij Rij
z '
which, together with equation (30) and Rij
ijz =
wi Li =
P
z
z;j Nij
z ijz wi
('
eijz )
'ijz
z
z
('
eijz )
(zero cuto¤ pro…t condition), imply:
1
1 z ijz wi
Di¤erentiating the natural logarithm of this equation leads to
w
bi =
P
zd
z
z;j Xij Nij
P
which implies equation (28), since
z;j
condition (FEC), as shown below.
Proof that
P
z;j
(
z
+
(
P
z;j
z
(
z
z d
1) Xij
'
eijz
'd
bi
ijz + w
(31)
z'
d = 0, which is a consequence of the free entry
1) Xij
ijz
1) X zij 'd
ijz = 0. First, note that the FEC implies that the total expected
…xed costs is equal to the expected net revenue:
fez +
X
1
Giz 'ijz
ijz
=
j
X
1
z ('
Rij
eijz ) X
=
1
z wi
Giz 'ijz
('
eijz )
Giz 'ijz
'ijz
j
j
z
z
1
1 ijz .
Second, totally di¤erentiating the LHS and RHS of the above equation leads to
X
j
ijz giz
'ijz
'ijz 'd
ijz =
X
ijz giz
'ijz
'ijz
j
X
('
eijz )
1
Giz 'ijz
ijz
z
z
1
('
eijz )
'ijz
j
40
d
1 'ijz 'ijz
z
z
1
1
h
(
z
d
1) '
e
ijz
(
z
1) 'd
ijz
i
which, together with equation (29), imply:
X
1
Giz 'ijz
ijz
'ijz
j
1 X z
Nij
Niz
,
('
eijz )
ijz
z
z
'ijz
j
1
1
(
1
z
z
(
1
z
Xij
(
1) 'd
ijz = 0
z
1) 'd
ijz = 0
z
X
,
C
('
eijz )
as Nijz = Niz 1
Giz 'ijz
1) 'd
ijz = 0.
z
j
Proof of Proposition 4
F (!) and P
cj have been given in subsection 4.2. Given (13), the pro…t maxiThe expressions for Uj , qij
mization problem of a product ! is given by:
ij wi
pFij (!)
max
pF
ij (!)
h
'
i
pFij (!)
Ej .
Pj1
Since the number of varieties is …nite, a change in pFij (!) will a¤ect the price index Pj . The …rst-order
condition is therefore given by:
1
where pFij
ij wi
pFij
'
pFij
!
+(
1)
ij wi
pFij
'
pFij
!
pFij (!). Reorganizing (32), we have:
pFij (') = cij (') 1 +
(
Note also that
pFij
F (')
(') qij
P P
F (!)
pF (!) qij
i
!2 F
ij ij
sij (')
Hence,
pFij
P P
i
!2
F
ij
1
1) [1 sij (')]
1
h
.
h
i1
pFij (')
=P P
h
i1
F (!)
p
F
i
!2
ij
ij
8
>
X< X
9
>
=
b
Pj =
sij (') [bij (') + w
bi + bij ]
>
>
;
i :!2 F
ij
2
3
X6 X
7 X PXij
sij (') bij (')5 +
=
(w
bi + bij )
4
i Xij
F
i
where
P
!2
F
ij
sij (') =
PXij
i Xij
!2
i
ij
is the import share from country i in country j.
41
=0
i1
pFij (!)
.
(32)
Therefore, global welfare change is given by:
X
j
=
bj =
Ej U
X
j;i
=
X
j
Xji w
bj
X
i;j
Xij bij
bj
Ej E
Pbj
2
3
X6 X
7
Ej sij (') bij (')5
4
i;j
!2
2
F
ij
3
X6 X
7
xij (') bij (')5
4
i;j
!2
X
i;j
Xij (w
bi + bij )
F
ij
where xij (') = Ej sij ('). Thus, we have the equation stated in Proposition 4.
42