Global Gains from Reduction of Trade Costs
Haichao FANy
Edwin L.-C. LAIz
Han (Ste¤an) QIx
This draft: 18 November 2015
Abstract
This paper derives a measure of change in global welfare, and then develops a simple equation
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 equation is applicable to a broad class of
perfect-competition and monopolistic competition models and settings, including perfect competition with/without multi-stage production, Melitz (2003) with general …rm productivity distribution
and its extension to multi-sector setting. We prove that the underlying mechanism is the envelope
theorem. We then carry out some empirical applications. We …nd our estimates to be reasonable
and consistent with others’in the literature.
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
Fan: School of International Business Administration, Shanghai University of Finance and Economics, Shanghai,
China. Phone: +86-21-65907307. Email: [email protected]
z
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]
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 are 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 the global bene…ts of improvement
of transportation technology or 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 important of reduction of administrative trade barriers and shipping time to global welfare. OECD,
World Bank and World Economic Forum found that trade facilitation could yield enormous economic
gains to the world.1 This paper derives a simple equation 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,
among multiple trading partners. 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 compensating
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.
Then, we derive a simple equation for computing the total global welfare e¤ect of small reduction
in bilateral trade costs for multiple pairs of trading partners. 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,
ij ,
of exporting from i to j. This turns out to be just the total saving in trade costs
divided by global GDP, keeping the values of all bilateral trades unchanged. In other words, the only
…rst order e¤ect is the direct e¤ect. The indirect e¤ects, of changes in allocation of resources to di¤erent
goods, and changes in input costs, are second order, because allocation of resources across outputs and
exports were already optimally chosen when the changes in trade costs take place.
The formula re‡ects the fact that, in the absence of externalities and price distortions, the market
1
See for example, World Economic Forum (2013) and OECD (2003).
1
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 changes in global welfare. As a result, only the direct e¤ect matters;
the indirect e¤ects are negligible when changes are small, as they are second order e¤ects. The envelope
theorem turns out to be a very powerful tool for evaluating the global welfare impact of changes in
bilateral trade costs (and other exogenous variables, such as changes in population and technology)
under rather general condition.
The envelope theorem can be applied to a broad set of models and settings. Examples are 1.
Models of perfect competition: with …xed or variable extensive margins of trade, with complete or
incomplete specialization, and with multi-stage production, e.g. Dixit and Norman (1980, Chapters
3-5); Dornbusch-Fisher-Samuelson (DFS, 1977 and 1980); the Armington (1969) model; Eaton-Kortum
(2002) (hereinafter EK2002); Melitz and Redding (2014); Yi (2003); 2. Monopolistic competition models
with constant markup, including Krugman (1980) (hereinafter K1980), Melitz (2003) with general …rm
productivity distribution (hereinafter MwG) and their extensions to the multi-sector setting.
The reason that the envelope theorem applies to the above models is that there are no externalities
and no price distortions. Under perfect competition and monopolistic competition with constant markup
(i.e. CES preferences), there are no price distortions, the market resource allocation is the same as that
of the global planner, and so the market response is the same as the response of the global-welfaremaximizing planner. The monopolistic competition case has been proved by Dhingra and Morrow
(2014) for the closed economy. We extend the analysis to the global economy, and show that the
envelope theorem can be naturally applied to MwG and its extension to the multi-sector setting as well.
The reason that equation (1) applies to MwG is that as cuto¤ productivities change due to changes
in trade costs, the e¤ects on average productivity of …rms serving each market (productivity e¤ect)
and the e¤ects 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
intuition is that market a¤ects labor allocations, which determines the cuto¤ productivities, which in
turn determines average productivities and …rm masses simultaneously through the resource constraints.
For each exporting country, the labor constraint dictates that the changes in average productivity and
…rm mass go in opposite directions, and hence they o¤set each other.
The result that equation (1) continues to hold for the extension to multi-stage production with
tradeable intermediate goods (i.e. fragmentation) is non-obvious, important and new to the literature,
and therefore should be highlighted. The intuition of this result is that whenever there is terms of trade
gain by an exporter of a good at any stage, there is an equal terms of trade loss by an importer of the
same good. Thus, the indirect e¤ect of terms of trade changes is nil. Therefore, only the direct e¤ect,
which is the total saving in trade cost in all stages, matters for global welfare change. Furthermore, any
saving in trade cost at any stage is passed on to the following stages until the …nal stage. Eventually,
2
the saving in trade costs shows up as the gains in global income received by all countries in the …nal
stage. Given that fragmentation is pervasive, this result points to the relevance of our equation in that
it is applicable to a setting that captures an important aspect of the real world.
Ossa (2012) found that assuming symmetric trade elasticity across sectors leads to a gross underestimation of the gains from trade. Since our formula (8) is applicable to asymmetric sectoral trade
elasticities, it does not su¤er from this problem.
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 are no cross-border externalities arising from agents’pro…t-maximizing and utility-maximization
decisions. 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 Assumption R3 in ACR, then our
global gains formula (1) is precisely equal to an expenditure-share-weighted average percentage change
of individual country’s gains from trade as given by the ACR formula.2 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 G. 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 aggregate 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 in this paper. As the assumptions we made are less restrictive than those
of ACR, our equation is applicable to a broader set of models and settings, e.g. our equation can be
applied to MwG and its extension to multi-sectors and PC with multi-stage production.
Our work is also inspired by Atkeson and Burstein (2010) (hereinafter abbreviated as 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” 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
2
Assumption R3 of ACR is that the import demand system is CES.
3
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 many models and settings as we can …nd. 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 Burstein and Cravino (2015) (hereinafter
abbreviated as 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 and its
extension to multi-sector, 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. In addition, we
carry out some empirical applications to demonstrate how our equation can be applied. 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 class of models to which equation (1) is not applicable is the imperfect competition models
with endogenously variable markups. Such models can capture the pro-competitive gains from trade
which models with constant markups cannot. However, there is mixed empirical evidence on the size
of the pro-competitive gains from trade. Indeed, pro-competitive e¤ects are sometimes negative so that
variable markups reduce the gains from trade (see, for example, Arkolakis, Costinot, Donaldson and
Rodriguez-Clare, 2015). In fact, Edmond, Midrigan and Xu (2015) conclude that the size of the gains
from trade depends a great deal on the underlying micro-details. Therefore, there is no presumption
that our equation will bias the estimate upward or downward when applied to this class of models.
The structure of this paper is as follow. In section 2, we derive a measure of the percentage change
in global welfare. In section 3, we demonstrate that equation (1) is applicable to perfect competition
without fragmentation. In section 4, we prove that equation (1) holds under perfect competition with
fragmentation. In section 5, we explain intuitively how the envelope theorem is the underlying mechanism of the results in sections 3 and 4. In section 6, we demonstrate that the same formula holds under
MwG with single sector and multiple sectors, as well as explaining the underlying mechanism of the
result. Section 7 presents some empirical applications of the model. The last section concludes.
4
2
De…nition of percentage change in global welfare
In this section, we present a justi…cation for the use of an equation that we shall use as a measure
of percentage change in global welfare. Suppose in the world economy there are n countries that are
capable of producing m goods, which are all tradable. 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)
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. Note that 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).3
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.
De…ne Ei , Pi and Ui as the expenditure, exact price index and welfare of country i, respectively.
The utility function (or welfare function) of country i, given by Ui (qi ) (where qi = qi1 ; qi2 ; :::; qim is the
vector of quantities of goods consumed) 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 total utility of all consumers in country i (i.e. welfare of country i) is
given by Ui = Ei =Pi for all i. 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 equiproportional increase in welfare of all countries following reduction of trade
P
Pn P
Pn
P
Pn
costs. Hereinafter, i
i=1 ,
j
j=1 and
k
k=1 to simplify notation. Then,
X
(Pi + dPi ) (Ui + dUi ) =
i
3
X
(Pi + dPi ) ( + 1) Ui
i
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.
5
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.4
Re-arranging the above equation and simplifying, we have
P
i (Pi + dPi ) (dUi )
= P
(Pi + dPi ) Ui
P i
P
i i (dUi )
= P
+O 2
P
U
i i i
1 X b
= w
Ei Ui
Y
(2)
i
where O
2
bi
changes; U
denotes second order terms, which can be dropped, as we are considering only in…nitesimal
P
dUi
w
5
b dx
k Pk Uk is the GDP of the world. Hereinafter, x
Ui , Ei = Pi Ui and Y
x , which
we call the percentage change of x.
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 a 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.6 This expression
for the percentage change in global welfare has been used in the literature, such as in Hsieh and Ossa
(2011), AB and BC. However, as far as we are aware, we are the …rst to present a justi…cation for its
use.7
4
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)
5
See the appendix for a more detailed proof.
6
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.
7
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.
6
3
Perfect competition (PC) without fragmentation
3.1
Incomplete Specialization and …xed extensive margins of trade (PC-FEM)
Consider a many-country (n), many-good (m), many-factor (K ) trade model with perfect competition
in all goods and factor markets, with incomplete specialization (meaning that a country can import the
same good from more than one country) in each sector in each country, and …xed extensive margins of
trade.8 Suppose all goods are tradable, and there is only one single stage of production. Thus, there
is no fragmentation or vertical specialization. The extensive margin of country i’s exports to country
j is the set of goods exported from i to j, i.e.
of
ij
ij ,
where
= f1; :::; mg. We assume that each
ij
for all i and j is unchanged in response to changes in
ij
for all i and j. Denote pj (!) as the
price faced by consumers of good ! in country j. De…ne pij (!) as the price of variety ! consumed in
country j that is imported from country i. Note that pj (!) = pjj (!) when country j exports good !,
and pj (!) = pij (!) when country j imports good ! from country i. Therefore,
pij (!) = pii (!)
where
ij
is an iceberg trade cost such that
ij
ij
for ! 2
ij
units are shipped from the source country i for one unit
to arrive at the destination country j (assume that
ii
= 1).9 We can call pii (!) the F.O.B. price and
pij (!) the C.I.F. price of good !.
De…ne pj
(pj (1) ; :::; pj (m)) as a vector prices; qj
qj (1) ; :::; qj (m) as a vector of quantities
consumed in country j. Ej denotes the expenditure in country j, while Ej (pj ; Uj ) stands for the
expenditure function; Uj (qj ) denotes the welfare function of country j, which is homogeneous of degree
one; the GDP function Rj (pj ; Lj ) (where Lj is the vector of K factor endowments, which are assumed to
be …xed) denotes the total sales revenue of …rms in country j, which is also equal to total output Yj , or the
GDP of country j. Trade is assumed to be balanced for all countries. Therefore, Rj (pj ; Lj ) = Ej (pj ; Uj )
for all j.10 Totally di¤erentiating the equation and re-arranging terms, we have
P @Rj
P @Ej
@Ej
dUj =
dpj (!)
dpj (!)
@Uj
!2 @pj (!)
!2 @pj (!)
X P
X P
@Rj
@Ej
=
dpjj (!)
dpij (!)
!2 ji @pjj (!)
!2 ij @pij (!)
i
(3)
i
Uj (qj ) being homogeneous of degree one implies that
8
9
@Ej
bj
dUj = Ej U
@Uj
(4)
We would like to thank Gene Grossman for suggesting a two-country two-good version of this model to us.
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.
10
The result continues to hold if there exists a non-zero but exogenous trade balance.
7
bk for i; j; k 2 f1; :::; ng? The answer to this
The question is: what are the e¤ects of ij on Ek U
P
bk resulting from changes in ij for i; j 2 f1; :::; ng.
question will guide us to calculate Y1w k Ek U
We prove our result step by step below.
1.
pc
c
ij (!) = p
ii (!) + bij for ! 2
(5)
ij
2. In the absence of externalities, the competitive market outcome is the same as the solution
of country j’s national social planner’s problem of choosing fyji (!) 8i and ! 2
subject to the endowment constraints and fpjj (!) for ! 2
jg
ji g
to maximize Rj
, where yji (!) denotes the quantity of
good ! produced in j that is exported to country i. Therefore, applying envelope theorem, we have
@Rj
@pjj (!)
= yji (!) for ! 2
ji .
3. The representative consumer in j chooses fqij (!) 8i and ! 2
ij g
to minimize expenditure Ej
taking fpij (!) 8ig and Uj as given, where qij (!) denotes the quantity of good ! consumed in j that
is imported from country i. Similar to Step 2 above, by envelope theorem, we have
!2
@Ej
@pij (!)
= qij (!) for
ij .
4. The iceberg trade cost links qij (!) with yij (!) :
ij qij
(!)
yij (!)
for ! 2
ij
which implies
pij (!) qij (!)
pii (!) yij (!)
xij (!)
for ! 2
(6)
ij
where xij (!) is the export of good ! from i to j. Aggregate exports from i to j is denoted by
P
xij (!).
Xij
!2
ij
5. Invoking (3) and (4), the e¤ect on country j’s welfare of changes in prices of exports and imports
of country j induced by changes in trade costs is given by
bj = P
Ej U
!2
=
ji
X
X P
|
i
!2
!2
i
ji
P
yji (!) dpjj (!)
xji (!) pbjj (!)
{z
ij
X P
i
!2
terms of trade e¤ect
X
ij
qij (!) dpij (!)
i
xij (!) pbii (!)
}
X
|
i
by (3) and steps 2 and 3
Xij bij
{z
direct e¤ect
by (5) and (6)
(7)
}
Explanation of the various e¤ects on the last line: The …rst term re‡ects the increase in
welfare of j resulting from increases in F.O.B. prices of goods that j exports, while the second term is
the decrease in welfare of j resulting from increases in F.O.B. prices of goods that j imports. The sum
of the …rst and the second term can be positive or negative. It captures the change in purchasing power
induced by the general equilibrium adjustments in relative factor prices, which in turn a¤ects the terms
of trade. The third term captures the saving in trade costs, which we refer to as “direct e¤ect”.
8
bj over j, we have
Summing Ej U
X
j
bj =
Ej U
XX P
i
=
j
!2
XX
j
i
ji
XX P
xji (!) pbjj (!)
i
Xij bij
j
!2
ij
xij (!) pbii (!)
X
i
Xij bij
The second line follows from the fact that the terms of trade gains of the exporters exactly o¤set the
terms of trade losses of the importers from the global welfare point of view.
Therefore, percentage change in global welfare is given by
X Ei
bi =
U
Yw
i
where Y w =
P
k
Ek =
P
k
X X Xij
i
j
Yw
Yk = GDP of the world.
bij
(8)
Note that the RHS of (8) is equal to the total saving in trade costs, controlling for the volumes of
exports, divided by global GDP.11 This is the only …rst order e¤ect. As we shall show in the appendix,
the key to obtaining the simple result in (8) is the envelope theorem.
We summarize the main result in the following proposition.
Proposition 1 Under perfect competition (with complete or incomplete specialization) and …xed extensive margins of trade, the change in global welfare associated with small changes in bilateral trade costs
is given by equation (8).
Examples of models to which this proposition applies include: neoclassical model based on comparative advantage (e.g. Dixit and Norman, 1980), Armington (1969).
3.2
Complete specialization and variable extensive margins of trade (PC-VEM)
To deal with variable extensive margins of trade, we note that under incomplete specialization, it is
possible that the global welfare impact of the adjustments of extensive margins of trade is …rst order,
not second order. In that case, we have to take into account the adjustments of extensive margins of
trade when calculating the global impact of the reduction of trade costs. In contrast, under complete
specialization the impact of the adjustments of extensive margins of trade on global welfare is of second
i
pii (!) qij (!) ( ij 1). Therefore,
hP
i
the total saving in exporting costs from i to j, controlling for trade ‡ows, is equal to
!2 ij pii (!) qij (!) d ij =
11
P
The total variable exporting costs for exporting from i to j is equal to
!2 ij
pij (!)qij (!)
ij
d
ij
= Xij bij .
9
hP
!2 ij
order.12 For equation (8) to hold, therefore, it is su¢ cient to assume complete specialization so that
the e¤ects of changes in extensive margins of trade is rendered second order, leaving the direct e¤ect as
the only …rst order one.
The following proof is based on the assumption of perfect competition and complete specialization.
Without loss of generality, we assume that there is balanced trade, though the results will be qualitatively
the same when non-zero trade balances are exogenous. The utility function only needs to be constant
returns to scale. Furthermore, without loss of generality, we assume that labor is the only factor input.
P
Balanced trade implies that Ej = Yj = i Xji = wj Lj 8j.
The percentage change in welfare of country j:
bj =E
bj
U
bj =Ej E
bj
Ej U
)
Pbj
Ej Pbj
Balanced trade implies that the terms of trade (TOT) e¤ect on j when it is an exporter is given by:
X
bj =
Ej E
Xji w
bj
i
Complete specialization implies that the goods consumed by j can be partitioned into disjoint sets
each of which is imported from a distinct source. Therefore, the change in price index is the summation
of changes resulting from changes in prices of imports from various sources:13
X Xij
pbij
Pbj =
Ej
i
X
X
=) Ej Pbj =
Xij pbij =
Xij (w
bi + bij )
i
i
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 is given by
X
X
bj =
Ej U
(Xji w
bj Xij w
bi )
Xij bij
) Global welfare e¤ect:
X
j
i
bj =
Ej U
XX
|
j
i
i
Xji w
bj
{z
}
TOT e¤ect on exporters
12
XX
|
j
i
Xij w
bi
{z
}
TOT e¤ect on importers
XX
|
j
i
Xij bij
{z
}
direct e¤ect of trade costs
A detailed proof can be found in ACR (2012) pp.119-122. Although restriction R3 is assumed, all the proof needs is
complete specialization, which is implied by R3. As explained on p.107 of their paper, “under perfect competition, the
welfare e¤ect of changes at the extensive margin is necessarily second-order since consumers must be initially indi¤erent
between the ‘cuto¤’goods produced by di¤erent countries.”
R
13
The consumer price index Pj can be write as follows: Pj = min 0 pj (!) qj (!) d! s.t.
Uj (f qj (!)j ! 2
g)
Ej
where pj (!) = pij (!) if variety ! is imported from country i; qj (!) = qij (!) if variety ! is imported from country i.
R
Since the utility function is homogeneous of degree one, envelope theorem implies that Pbj = 0 j (!) pbj (!) d! where
Pn
b
bij =
j (!) denotes the expenditure share on good ! in country j. Complete specialization implies that Pj =
i=1 ij p
Pn Xij
(
w
b
+
b
)
where
denotes
the
expenditure
share
of
country
j
on
goods
from
country
i.
i
ij
ij
i=1 Ej
10
Again, the TOT e¤ect on exporters and TOT e¤ect on importers exactly cancel out from the global
welfare point of view. Consequently, we have
1 X b
Ej Uj
Yw
j
|
{z
}
=
Percentage change in global welfare
Therefore, we have
|
1 XX
Xij bij
Yw
j
i
{z
}
direct e¤ect of trade costs
Proposition 2 Under perfect competition, variable extensive margins of trade and complete specialization, (8) holds.
Examples of models to which this proposition applies are DFS1977 and EK2002.
We next show that equation (8) continues to hold assuming multi-stage production.
4
Perfect competition with fragmentation
In this section we allow some goods to be used only as intermediate inputs into the production of other
goods in a multi-stage production setting and the intermediate goods of all stages are tradeable (i.e.
there is fragmentation or vertical specialization). The …nal good is produced in F sequential stages.14
Without loss of generality, we assume that there is complete specialization in all stages and each country
produces a single variety at each stage. The …nal result will be qualitatively the same when each country
produces more than one variety at each stage and we have …xed extensive margins of trade or variable
extensive margins of trade with complete specialization at each stage. Furthermore, we assume that
there is balanced trade (Ej = Yj ), though the results will be qualitatively the same when non-zero trade
P
PF
balances are exogenous.15 Labor is the only factor input (Yj = wj Lj ). Hereinafter, s
s=1 to
simplify notation. The utility function is constant returns to scale, given by:
Uj = Uj
F
qij
j8i
,
F is country j’s consumption of the …nal good imported from i. For s = f2; 3; :::; F g, the stage-s
where qij
production (which shall be called “stage-s good”) requires the input of previous stage’s output and
labor. 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 market structure for all goods is
14
Here, we assume all …nal goods are produced in F sequential stages. If we assume that the number of production
stages for di¤erent countries is di¤erent, our results continue to hold.
15
The analysis based on the more general assumptions is available from the corresponding author upon request.
11
assumed to be perfect competition. The production function for stage-s good is given by the following
Cobb-Douglas form:16
yjs
=
8"
< X
:
1
s
s s 1
i qij
s
i
#
s
s
1
9
=
s
Asj ljs
;
1
s
, for s = 2; 3; :::; F ;
(9)
yjs = Asj ljs , for s = 1;
s
where yjs is country j 0 s output of stage-s good; qij
good from country i for producing stage-s good;
1
is country j 0 s use of imported input of stage-s
s
i
1
parameterizes the quality or productivity of the
intermediate input from country i for production in stage s; Asj is a measure of country j 0 s total factor
productivity associated with the production of stage-s good and ljs is country j 0 s labor input in the
production of stage-s good. The unit cost function is given in the appendix.
s denote the value of exports of stage-s good from country i to country j, and E s =
Let Xij
j
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
s = ps q s , where ps is the import price in country j of
expenditure on …nal good). Note that 1. Xij
ij ij
ij
stage-s good from country i; 2. the total value of production of stage-s good in country j is equal to
P s
s s
s
i Xji = pjj yj for s = 1; 2; :::; F , where pjj is the unit cost of stage-s good produced by country j; 3.
psij = psii
origin; 5.
s
ij ;
yjs
s s = y s which is the quantity of stage-s good exported from i to j measured at the
4. qij
ij
ij
P s
= i yji .
In addition, the total value of production of stage-s good in country j can be decomposed into two
parts for s = 2; 3; :::; F : …rst, the value-added by country j labor in the production of stage-s good,
P s
given by Vjs = (1
s)
i Xji ; second, the total expenditure on intermediate inputs for the production
P 1
P s
as no intermediate
of stage-s good in country j, given by Ejs 1 = s i Xji
. Moreover, Vj1 = i Xji
input is required for the …rst stage. GDP, Yj , which is equal to Ej (due to trade balance), is equal to
the sum of value-added over all stages:
Ej =
X
Vjs for j = 1; 2; :::; n
s
As in the case of one-stage production with complete specialization,
bj = E
bj
U
Pbj
Intuitively, this says that an increase in wj tends to increase Uj through the increase in Ej resulting
from the TOT e¤ect when j exports a good. On the other hand, an increase in foreign wage wi tends
to reduce Uj through the increase in Pj resulting from the TOT e¤ect when j imports a good from i.
Thus, the change in global welfare is given by
16
The function draws from the one used by Melitz and Redding (2014) and Yi (2003, 2010). If
becomes the function given by Melitz and Redding (2014). If
given by Yi (2003).
12
s
s
= 1 for all stages, it
= 0 for all stages and F = 2, it becomes the function
Ej Pbj
bj
bj = Ej E
Ej U
The global welfare e¤ect resulting from changes in prices of exports:17
X
j
bj =
Ej E
XX
|
j
Ej Pbj =
XX
j
i
{z
(10)
}
TOT e¤ect on exporters over all stages
whereas18
X
s
j
Vjs w
bj
F c
pFii + bijF
Xij
due to complete specialization.
Complete specialization further implies that 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 the stages, we get the global cost
e¤ect on the F.O.B. prices of the …nal goods, which is the accumulation of the changes in trade costs
and TOT e¤ect on importers (i.e. changes in the F.O.B. prices of goods) over the stages:19
XX
j
i
F F
Xij
pbii =
F
X1 X X
s=1
j
i
s s
Xij
bij +
XX
s
i
Vis w
bi
(11)
Using the last two equations, we can express the global welfare e¤ect resulting from changes in prices
of imports as follows:20
X
j
Ej Pbj =
XXX
|
s
j
i
{z
s s
Xij
bij
XX
+
|
}
i
s
{z
Vis w
bi
(12)
}
TOT e¤ect on importers over all stages
direct e¤ect on trade in goods over all stages
Thus, 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 (10) and (12),
we conclude that
X
j
bj =
Ej U
|
XXX
s
j
{z
i
s s
Xij
bij
P
bj = P P Xji w
Compare with j Ej E
bj in the one-stage case.
P
Pj Pi
18
b
Compare with j Ej Pj = j i Xij (b
pii + bij ) in the one-stage case.
19
Refer to the appendix for a detailed proof.
XX
XX
P
20
b
Compare this with
Xij bij
+
Xij w
bi
j Ej Pj =
17
|
j
i
}
direct e¤ect on trade in goods over all stages
{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
13
in the one-stage case.
P
At each stage, the TOT e¤ect on exporters exactly o¤set the TOT e¤ect on importers, i.e.
sb
j
j Vj w
s
If c
ij for all s, we have:
ij = c
s
s Xij .
sb
i
i Vi w
=
at each s. Thus, at each stage, the only welfare impact from the global point of view is the
direct e¤ect.
P
P
P
Ej b
j Y w Uj
=
Thus, we have the following proposition:
P P P
s
i
s
Xij
s
c
j Y w ij
=
P P
i
Xij
ij ,
j Yw c
where Xij =
Proposition 3 Under multiple-stage production with PC-FEM at each stage or PC-VEM with complete
s = c for all s, (8) holds.
specialization at each stage, and c
ij
ij
Multi-stage production tends to increase the global gross trade ‡ows. However, global welfare gains
from trade costs reduction still solely arises from the direct e¤ect. The TOT e¤ect is related to total
value added, rather than gross production. But the TOT e¤ect on importers will o¤set the TOT e¤ect
on exporters over each stage, because for each amount of TOT gain for an exporter, there is an equal
TOT loss for the importer.
Note that a PC model with multi-stage production and one without multi-stage production but with
the same deep parameters will in general give rise to di¤erent Xij for the same set of bilateral trade
costs (with larger Xij under multi-stage production) and therefore the one with multi-stage production
gives rise to larger global welfare gains for the same percentage changes in bilateral trade costs.
Given that fragmentation is 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.
5
Application of Envelope Theorem to PC case
It is well-known that the market equilibrium is e¢ cient under PC as long as there are no externalities.
This is the case with PC models with and without fragmentation. Thus, 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. The detailed proof is put in the appendix. Below we give some
brief explanation of the intuition.
In Appendix C, we present the envelope theorem for the case without fragmentation. Equation (2)
and footnote 4 imply that we can also evaluate percentage change in global welfare by using pre-change
market prices and the concept of equivalent variation. The global planner calculates the GDP function
of the world by choosing factor allocation to production of each good to maximize the global GDP
given factor endowment of each country and all market prices. (See, for example, Feenstra 2004 for the
14
idea of GDP function used here.) Applying envelope theorem, we show that as the optimally chosen
factor allocations and market prices are kept constant, the increase in global GDP resulting from small
changes in trade costs is just equal to the saving in trade costs, i.e. only the direct e¤ect matters.
In Appendix D, we present the envelope theorem for the case with fragmentation. As before, the
global planner maximizes global income by choosing the allocation of labor to production of goods in
various stages, taking trade costs at various stages and the market prices of the …nal goods as given.
Applying the envelope theorem, we show that when the optimally chosen labor allocations at various
stages and market prices of …nal goods are kept constant, the change in global income as trade costs
change slightly is just the direct e¤ect, i.e. total saving in trade costs. The initial cost impact of a small
s at stage-s + 1 received by country j due to
change of trade cost of d s is equal to a gain of X s c
ij
ij ij
the change in the cost of its intermediate good. 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, the cost saving is passed on to
all countries. 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 .
6
Melitz model with general productivity distribution (MwG)
6.1
One-sector MwG
Extending Dhingra and Morrow’s (2014) analysis to the global economy, we show that the global market
equilibrium is e¢ cient under MwG, which assumes CES preferences. For this reason, the envelope
theorem is applicable to this model, and so (8) continues to hold under MwG. We present the proof
below.
We consider a world economy consisting of n countries indexed by i = 1; :::; n, with a single factor
input, labor, which is inelastically supplied and immobile across countries. There is a continuum of
goods indexed by ! 2
that can potentially be made available to consumers. We use Li and wi to
denote the total endowment of labor and the (endogenous) wage level in country i respectively. We
assume that there is balanced trade (Ej = Yj ), though the results will be the same when non-zero trade
balances are exogenous.
Preferences, Technology and Market Structure
Preferences: We assume CES preferences, which lead to constant markup, which in turn implies
that relative prices and relative quantities are not distorted (i.e. they are the same as under perfect
competition). In each country j, a consumption bundle f qij (!)j 8i; !g is chosen to maximize utility
15
subject to an expenditure constraint. The utility function is given by
Uj =
where
"
XR
!2
ij
1
qij (!)
i
> 1 is the elasticity of substitution, and
d!
#
1
(13)
is the set of goods exported from i to j.
ij
Therefore, the ideal price index is given by
Pj =
Technology: For every good ! 2
"
XR
!2
ij
pij (!)
i
d!
#
1
1
, there is a blueprint that has been acquired by a …rm through
R&D. If a …rm from country i produces y
function is given by
Ci (wi ; q; !) =
1
X
[
fyij (!)g units of good ! to be sold to country j, its cost
ij wi ai (!) yij
(!) +
ij wi
1 (yij (!) > 0)]
j
and 1 (yij (!) > 0) is an indicator function, wi is the wage in country i,
marginal cost inclusive of trade cost, and
ij
ij wi
ij wi ai (!)
denotes the constant
denotes the …xed cost of exporting from i to j, where
0 8 i; j is exogenous. Note that we treat a country-i …rm serving market i as exporting from i to
i. We assume that
ii
can be non-zero. The exogenous unit labor requirement parameter ai (!) re‡ects
the heterogeneity of productivities across blueprints.21
Market Structure: We consider monopolistic competition with free entry. There is a large number of
…rms in each country, and all markets for goods and factors clear. A …rm from country i needs to hire fe
units of labor to develop a blueprint, which confers it with monopoly power. The measure of number of
entrants Ni (successful or not) in country i is endogenously determined by the free entry condition. The
number of …rms in country i serving market j, Nij , is determined by the zero cuto¤ pro…t conditions. In
equilibrium, the entry cost, wi fe , is equal to the expected pro…t of each …rm. Because of the existence
of …xed cost of exporting and of serving the domestic market, some entrants do not survive, and not all
of those who survive export to all countries, depending on their productivity.
Below, we prove the following proposition:
Proposition 4 Under the monopolistic competition model of Melitz (2003) featuring general distribution of …rm productivity, (8) holds.
Calculating Global Gains
21
In fact, equation (8) will continue to be valid even if we adopt more general assumptions on the technology such as
1
P
1
Ci (wi ; q; !) = n
qij (!) + ij wi bij (!) mij (tj ) 1 (qij (!) > 0) , similar to that of ACR. Please refer
ij wi aij (!) tj
j=1
to ACR (2012) for detail. We make simpler assumptions here so as to highlight our intuition more clearly.
16
Labor productivity is a random variable denoted by '
1=ai (!). 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. According to Melitz (2003), the average productivity of a …rm in country
1
1
R1
1
1
(')
g
(')
d'
. Thus '
eij is a function of 'ij .
i serving market j is given by '
eij
i
'
1 Gi ('ij )
ij
h
i
The number of …rms in country i serving market j, Nij = Ni 1 Gi 'ij is a function of 'ij and Ni .
The value 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
1
price index in country j is given by
Pj =
(
X
Nij
1
i
)
1
wi ij
'
eij
wi ij
'
eij .
Therefore, the expected
1
1
Totally di¤erentiating the logarithm of the above equation, invoking complete specialization and
re-arranging yield:
Ej Pbj =
X
1
Xij w
bi + bij
i
1
c
'
e
ij
bij
N
(14)
The e¤ect of the change in cuto¤ productivity 'ij on the average productivity '
eij can be derived as
follows:
('
eij )
=)
1
1
=
1
2
Gi 'ij
6
c
'
e
ij = 41
'ij
('
eij )
Z
1
(')
1
gi (') d'
'ij
1
1
3
7
5
gi 'ij d'ij
1
1
On the other hand, the labor market clearing conditions Ni
h
i
Nij = Ni 1 Gi 'ij imply that22
bij =
N
P
j ij
P
j
1
1 ('
eij ) 1 = 'ij
h
i
1
G
'
('
eij )
ij
i
ij
“Productivity e¤ect”.
1
Gi 'ij
hP h
j
1
Gi 'ij
i
ij
+ fe
(15)
i
= Li and
gi 'ij d'ij
1
1
= 'ij
“Firm mass e¤ect”
(16)
(15) and (16) together give us the following important relationship:23
!
X
X
bij
Xij N
c
=
Xij '
e
Firm mass e¤ect plus productivity e¤ect = 0
ij
1
j
j
22
Refer to the appendix for detailed derivation.
23
Note also that Xij = Nij Rij ('
eij ) = Nij Rij 'ij ('
eij )
1
= 'ij
1
= Ni 1
Gi 'ij
wi
ij
('
eij )
based on (20) and zero cuto¤ pro…t condition. This, together with (15) and (16), give us this relationship.
17
1
= 'ij
1
The intuition is: A change in
ij
will have e¤ect on 'ij for all i; j 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) but a decrease in Nij (fewer …rms from
i serving the market in j — LHS), leading to counteracting (but not completely o¤setting) e¤ects on
Ej Pbj , according to (14). When summing over all j, the two e¤ects o¤set each other completely from a
global welfare point of view.24 Thus naturally, 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.
Summing up (14) for all possible destination j, global welfare change is given by
X
j
X
bj =
Ej U
j
XX
=
|
j
i
bj
Ej E
Xji w
bj
{z
Pbj
XX
j
i
Xij w
bi
}
TOT e¤ect on all importers and exporters = 0
=)
X Ej
bj =
U
Yw
j
X X Xij
i
j
Yw
bij
|
XX
i
j
{z
Xij bij +
direct e¤ect
}
XX
|
j
i
c
Xij '
e
ij
{z
bij
Xij N
1
!
}
global …rm mass e¤ect plus productivity e¤ect = 0
Only the direct e¤ect matters.
bij and 'c are equal to zero
Note that the same equation holds under K1980 simply because both N
ij
c
for any i and j. In MwG, if there is no change in the cuto¤ productivities, i.e. 'ij = 0 for all i, j, then
bij will also be equal to zero as in K1980, according to the labor market clearing condition. A change
N
c
b
in 'ij leads to counteracting e¤ects on '
e
ij (“productivity e¤ect”) and Nij (“…rm mass e¤ect”), which
o¤set each other from the global welfare point of view.
As Melitz and Redding (2015) have shown, a model from K1980 and a model from 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 ) and therefore the MwG model gives rise to larger global welfare gains
than the K1980 model for the same percentage changes in bilateral trade costs.
6.2
Application of Envelope Theorem to MwG
Under monopolistic competition with constant markup (i.e. CES preferences) and with the absence of
externalities, there are no price distortions, and so the market resource allocation is the same as that of
the global planner. We have also shown in the online appendix that the market equilibrium is e¢ cient
under MwG. So, the market response to changes in trade costs is the same as the response of the global
24
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. Let us 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
= X11 '
e11 + X12 '
e12 .
1
1
18
planner. Thus, the envelope theorem can be applied to MwG and its extension to the multi-sector setting
as well. The detailed proof is given in the appendix. Like before, the global planner maximizes global
income by choosing allocation of labor to each good, taking trade costs and the market prices as given.
As in the case of perfect competition, when the optimally chosen labor allocations and market prices
are kept constant, the change in global income as trade costs change slightly is just the direct e¤ect, i.e.
total reduction in trade costs. The allocation of labor implicitly determines the cuto¤ productivities
n o
'ij and the number of potential entrants fNi g through the resources constraints, which in turn lead
to the “productivity e¤ect”and “…rm mass e¤ect”. The two e¤ects cancel each other when the changes
n o
in 'ij are small, as the two e¤ects together re‡ect the extensive margin e¤ect, which is second order
given complete specialization.
6.3
MwG with multiple sectors
The virtue of extending to the multiple sectors setting is that we can allow for di¤erent trade costs,
di¤erent trade elasticities and di¤erent …xed costs in di¤erent sectors, which is an important empirical
fact. Under PC-FEM and PC-VEM with complete specialization, it is simple to extend the model to
the multiple sectors.25 Here we only present the extension of MwG to multiple sectors, which is less
straightforward.
Suppose that the set of goods ! 2
and
z
z
is separated into groups denoted by
where z = 1; :::; Z,
is referred to as sector z. Consumers in country j have their preferences represented by the
following utility function:
Uj =
hP
Z
z=1 Uj
(z)
1
i
1
where Uj (z) =
hP
n R
i=1 !2
z
ij
z
qij
(!)
1
z
z
d!
i
z
z
1
z (!) denotes the consumption of variety ! in sector z in country j of goods originating from
where qij
country i.
For the production side, the marginal cost inclusive of trade cost is
z
ij wi ai (!),
where
z
ij
is the
iceberg trade cost from country i to j in sector z. The …xed exporting cost from country i to j in sector
z is a labor cost of
ijz wi .
Each …rm from country i needs to pay a labor cost of fez wi to acquire a
blueprint in sector z. The exogenous unit labor requirement parameter ai (!) re‡ects the heterogeneity
of productivities across blueprints. Labor productivity is a random variable denoted by '
1=ai (!).
Giz (') and giz (') are the cdf and pdf respectively of ' in sector z at country i. We prove the following
proposition in the appendix.
Proposition 5 If there are multiple sectors and in each sector there is the monopolistic competition
model Melitz (2003) featuring general distribution of …rm productivity, the percentage change in global
25
The analysis for perfect competition is available upon request.
19
welfare is given by
Pn
Therefore, if we assume that
Ej b
j=1 w Uj =
Y
z
ij
=
ij
z
Xij
z
c
z=1 w ij .
Y
Pn Pn PZ
j=1
i=1
for any z = 1; :::; Z, the change in global welfare again
reduces to (8). The above result is in fact quite intuitive though the proof seems complicated. With the
assumed Cobb-Douglas aggregation of utility from each sector, the percentage change in global welfare
is a (constant) weighted average of the percentage change in global welfare of each sector. Regardless
of how trade costs, trade elasticity and …xed costs might di¤er across sectors, equation (8) holds at the
sectoral level. Summing up the sectoral global welfare gains for all sector z leads to the above equation.
7
Empirical Applications
The First Empirical Application. We consider a counterfactual experiment of a one-percent reduction in all bilateral trade costs in the year 2010. This reduction can be due to technological improvements
or other exogenous changes such as 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. The
data shows that global trade value was equal to 22.8% of world GDP in the year 2010. It follows from
(8) that the elasticity of global welfare with respect to trade cost is equal to 0.228. In other words, one
percent reduction in all bilateral trade costs would increase global welfare by 0.228% of world GDP in
2010, i.e. approximately USD 149 billion. In comparison, in 2013, the OECD estimated that reducing
global trade costs by 1% (not necessarily uniform, and not iceberg costs) would increase worldwide
income by more than USD 40 billion.
The Second Empirical Application. Equation (8) can be easily implemented empirically if
we are able to estimate the small changes in bilateral trade costs within a period. We do not use
gravity-model-based approach to estimate trade costs (such as that suggested by Anderson and Van
Wincoop, 2004) as some of the models to which our formula is applicable does not lead to bilateral
trade ‡ows expressible in the gravity form. Instead, we estimate the global welfare impact of changes in
one component of trade cost, the freight cost, in the last few decades, and compare it with the impact
of changes in other components of trade cost. Note that we do not pretend that freight cost captures all
or even a large percentage of trade cost. We just evaluate how a change in this particular component
of trade cost a¤ects global welfare.
We …rst estimate the annual change in bilateral freight cost between each country pair among the
thirty largest trading countries for the years 1960-2010. Then we apply these estimated changes to
equation (8) to obtain an estimate of the percentage change of global welfare in each year. Then we
add up the impacts over the years to obtain the cumulative impact over the 50 years.
The estimation procedure is simple. We …rst obtain the bilateral trade ‡ow data from the Direction
20
of Trade Statistics (DOTS) of IMF. The DOTS contains bilateral trade ‡ows based on f.o.b. prices
reported by the exporters and the bilateral trade ‡ows based on c.i.f. prices reported by the importers.
After matching these two values for each country pair, we calculate the iceberg transport cost from
country i to j in year t as
tij;t =
cifij;t
f obij;t
The DOTS contains bilateral trade ‡ows reported by 208 countries, which almost include all countries
in the world. To prevent inconsistent reporting by the small countries, we focus on the 30 countries with
the largest trading values in these 50 years. To ensure the trade values in di¤erent years are comparable,
we de‡ated the trade values by the in‡ation rate of the United States gathered from the World Bank
WDI database, as the trade ‡ows are measured in USD. We then choose the following 30 economies
with the largest aggregate de‡ated trade values over the years 1960-2010.26
United States
Germany
China
Japan
France
United Kingdom
Italy
Netherlands
Canada
South Korea
Taiwan
Saudi Arabia
Mexico
Switzerland
Russia
Spain
Belgium
Malaysia
Sweden
Singapore
Australia
Brazil
India
Austria
Indonesia
Thailand
Norway
Ireland
UAE
Hong Kong
Following Limao and Venables (2001), we drop the observations with tij;t < 1 and those with tij;t =
1:10. The reason for the latter is that IMF will impute the ratio to be 1.1 when the cifij;t is not reported
by the importer j.27 As a result, we obtain a total of 20,984 observations, with an almost-balanced
panel of around 450 observations per year after 1978. Then we calculate the annual percentage change
t
t
in iceberg freight cost from i to j at t as b
tij;t = ij;ttij;tij;t1 1 . The average annual percentage change
(weighted by trade ‡ows) of tij;t in each decade are listed in the following table.
Year
b
tij;t
1960-1970
1.00%
1970-1980
0.55%
1980-1990
1.52%
1990-2000
0.83%
2000-2010
0.07%
We observe a clear increasing trend of average freight cost over these 50 years.
26
The total trade value among these 30 countries constitutes 80.36% of the cumulative de‡ated global trade value for
1960-2010. In terms of the nominal trade value, the total cumulative trade value among these 30 countries constitutes
81.22% of the cumulative global trade value for the given period.
27
We follow the approach in Limao and Venables (2001) to eliminate the observations with CIF/FOB between 1.0909
and 1.101. In fact, the result will not change much if we include these observations with tijt = 1:1 as the imputed value
does not change over time.
21
We then apply the estimated annual changes in freight costs b
tij;t for i, j in each year t to equation
(8) to obtain an estimate of the percentage change of global welfare in each year t:
bt =
U
X X Xijt
i
j
Ytw
b
tij;t
where the Xijt is the trade ‡ow from country i to j based on c.i.f. prices and Ytw is the global GDP
obtained from the World Bank WDI database. Summing up the annual percentage change in global
welfare over the 50 years, we calculate the cumulative global income gains from changes in freight costs
over the 50 years 1960-2010 to be
2:18%, i.e. approximately a loss of USD 1429 billion in the year
2010. The following table decomposes the global welfare change by decade.
Year
Global Welfare Gains
1961-1970
-0.34%
1971-1980
-0.32%
1981-1990
-0.96%
1991-2000
-0.51%
2001-2010
-0.06%
Total
-2.18%
Why has freight cost been increasing? One possible reason is the faster growth of air shipping than
ocean shipping, which “grew 2.6 times faster than ocean cargo”but “[its costs are] on average 6.5 times
higher than ocean freight charges” according to Hummels and Schauer (2013). However, freight cost
is only one component of shipping costs. Another notable component of shipping costs is shipping
time. 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. How does the global impact of reduction in shipping
time compare with that of increases in freight costs over the years 1960-2010? Any such comparison
requires an estimate of how much bilateral shipping times have been reduced each year over this period
of time, and it is not easy to …nd data. But we can do a rough estimate. A back-of-envelope calculation
following Hummels’(2001) approach shows that the cumulative global gains from savings in shipping
times for 1960-2010 is about [2:98%; 8:04%] of global income in 2010.28 Comparing it with the -2.18%
from increases in freight costs, we can conclude that advances in shipping technology in the last 50 years
28
Following 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. At that time, 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. Thus, the average shipping days dropped from 40 to 0:5
sea) + 0:5
20 (by
1 (by air) = 10.5 days during the period 1960-2010 (assuming that air freight takes just one day). As we
assume the cost reduction process to be gradual (i.e. the rate of reduction 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 trade in GDP from
22
have contributed to an increase of [0:80%; 5:86%] of global income in 2010, which is approximately USD
[522, 3824] billion, with the mid-point being USD 2.2 trillion. It seems that advances in transportation
technology that saved shipping time have more than o¤set the increases in freight costs and lowered the
overall shipping costs, thus delivering net gains to the world in the last …ve decades.
8
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 of reduction of
bilateral trade costs to world welfare. 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.
We have derived a simple equation for computing the global welfare gains from small reduction of
bilateral trade costs, such as shipping costs or the costs of administrative barriers to trade. Surprisingly,
the equation is very general and is applicable to a broad class of models and settings. We then carry
out a couple of empirical applications. We …nd the estimates to be reasonable and consistent with other
estimates in the literature.
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 compensating variation. Third, we are able to demonstrate the user-friendliness
of our formula by carrying out a couple of simple empirical applications.
One class of models to which this paper does not apply is the imperfect competition models with
endogenous and variable markups. However, the empirical evidence about whether the existence of such
pro-competitive e¤ect leads to upward or downward bias of the estimate of welfare impact is mixed so
far. Another e¤ect our model does not capture is the dynamic gains from trade costs reduction, such
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 [2:98%; 8:04%]. We obtain this estimate
by calculating the annual percentage increase in global welfare each year from 1960 through 2009, and then add them up
to obtain the cumulative global welfare gain, as each annual gain is very small.
23
as learning by doing or learning by exporting. This is an interesting and potentially important source
of gains from trade. It is left for future research.
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, 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.
24
Appendix
In the following appendixes, to simplify notation, we de…ne
P
a;b;c
P P P
a
tion is over all the possible values that the dummy variable can take, e.g.
for a country.
A
P
i
b
c
Pn
where each summa-
i=1
if i is the dummy
Proof related to derivation of equation (2)
P
P
(Pi + dPi ) (dUi )
i
i Pi (dUi )
P
P
Pi Ui
i (Pi + dPi ) Ui
P
Pi
P
P
[ i (Pi + dPi ) (dUi )] [ i Pi Ui ] [ i Pi (dUi )] [ i (Pi + dPi ) Ui ]
P
P
=
[ i (Pi + dPi ) Ui ] [ i Pi Ui ]
P
P
P
P
[ i Pi (dUi )] [ i Pi Ui ] [ i Pi (dUi )] [ i Pi Ui ]
P
P
=
+O 2
[ i (Pi + dPi ) Ui ] [ i Pi Ui ]
=0
where the last line stems from ignoring
o
hP P
i nP
P
(P
+
dP
)
U
]
P
U
, which is a residual that
O 2
dP
P
(dU
U
U
dU
)
=
[
i
i
i
i
j
i j
i
j
i
j j j
i
j
contains all terms of order higher than the …rst.
B
Derivation of Equation (11)
The unit cost function for producing stage-s good can be written as:
psjj
=
1
s
s
1
s)
(1
s
82
>
< X
4
>
: i
psjj =
where
"
P
i
s 1 s 1
ij pii
s
i
1
s
#
s 1 s 1
ij pii
s
i
!1
wj
, for s = 1;
Asj
s
3
1
1
s
5
9
>
=
>
;
s
wj
Asj
!1
s
, for s = 2; 3; :::; F ;
1
1
s
denotes the aggregate price of intermediate goods in country j for the
production of stage-s variety, and
s 1
ij
is associated iceberg trade costs for exporting of intermediate
goods from i to j for the production of stage-s good. Totally di¤erentiating the above equation, we
have:
pbsjj
=
s
"
X
i
s 1
Xij
P s
i Xij
1
bijs 1
+
pbsii 1
#
+ (1
pbsjj = w
bj , for s = 1
25
bj ,
s) w
for s = 2; 3; :::; F
as the aggregation of these intermediate goods is CES. This equation implies that
X
i
)
s
s 1 s
Xij
pbjj =
X
i
s s
Xji
pbjj
=
s
X
s
Dividing both sides by
|
j
i
1
i
X
i
XX
s
Xij
bijs
s 1 s 1
Xij
pbii
s,
s s
Xij
pbii
1
+
+ pbsii
s
1
X
i
+ (1
s)
X
i
s 1 s 1
Xij
bij
+ (1
s 1
Xij
w
bj
s) s
X
i
s
Xji
w
bj
as
X
s
Xij
1
=
i
s
X
s
Xji
i
summing over all j, and re-arranging terms, we have
XX
j
{z
i
s 1 s
Xij
pbii
increment of the cost e¤ect for
importers from one stage to the next
1
}
=
XX
|
j
i
s 1 s
Xij
bij
{z
1
}
increases in trade costs
+ (1
|
s)
XX
j
{z
i
s
Xij
w
bi
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
XX
|
j
i
Fc
Xij
pFii
{z
}
=
F
X1 X X
s=1
|
j
i
{z
s s
Xij
bij
+
}
price e¤ect on importers
direct e¤ect on trade in goods of
of …nal goods
all stages other than the …nal one
XX
|
j
i
1
Xij
w
ci +
F
X
s=2
(1
{z
s)
XX
j
i
TOT e¤ect on
importers over all stages
s
Xij
w
bi
}
which gives us (11).
C
Envelope theorem for PC without fragmentation
Without loss of generality, we focus on the PC-FEM model in section 3.1. All notations follow those
of sections 2 and 3.1 unless otherwise stated. As in section 3.1, we assume that there are n countries
and m goods and K factor inputs. Equation (2) and footnote 4 imply that we can evaluate percentage
change in global welfare, , by using pre-change market prices and the concept of equivalent variation.
Here is the reason: Referring to section 2, if we use the pre-change market prices to evaluate income,
P
global income before the change is given by j Pj Uj . The sum of equivalent variations of all countries
P
P
P
is equal to j Pj (dUj ). Thus, = j Pj (dUj ) = j Pj Uj . For ease of exposition, we shall use the
pre-change market prices and equivalent variation to evaluate the global welfare change, even though
our original motivation of the change in global welfare explained in section 2 is based on the concept of
compensating variation.
De…ne the pre-change vector of market prices of goods sold in country j as pj
and pre-change vector of market quantity consumed by country j as qj
(pj (1) ; :::; pj (m))
(qj (1) ; :::; qj (m)).
We now consider a global planner calculating the global GDP function based on the market prices,
26
the factor endowments and trade costs faced by countries in the world.29 It is a well-known result that,
in the absence of externalities, the global competitive equilibrium yields the same allocation of resources
to the production of di¤erent goods as the solution of a global planner’s global-income maximization
problem, given the market prices. 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. Thus, the GDP resulted from global planner’s labor allocation is also the marketdetermined GDP.
Maximization of income of each country subject to its resource constraint
k (!) be the kth element
Let lij (!) be the vector of K factor inputs for producing output yij (!), lij
of lij (!), Li be the vector of K factor endowments in country i, f ! (lij (!)) be the production function
for producing yij (!) units of good ! by country i for sales in country j.30 Note that
ij qij
(!)
yij (!)
for all i; j; !. In each country j,
Ej = Yj = f GDP (f pjk (!)j 8k; !g ; f
jk
(!)j 8k; !g ; Lj )
which is the GDP function of j, the outcome of the maximization of income of the country by the global
planner subject to its resource constraint.31 The dual of this problem is that the global planner solves
max Uj (qj ) s.t. pj qj = Ej
qj
where pj and expenditure Ej is the value of Yj obtained from the above GDP function. Therefore,
Pj (pj ), the price index in country j, is also exogenous. Note that
Ej = Pj Uj = pj qj .
Thus, there is a one-to-one mapping from (pj ; qj ) onto (Pj , Uj ), and therefore a one-to-one mapping
from (p1 ; :::; pn ; q1 ; :::; qn ) onto (P1 ; :::; Pn ; U1 ; :::; Un ). For given pj , an increase in Uj by a factor of
induced by an increase in Ej is equivalent to equi-proportional increases in all elements of qj by a
P
P
P
P
factor of . Thus, we can show that = j Pj (dUj ) = j Pj Uj = j pj dqj = j pj qj .
Maximization of global income
As the global planner maximizes global income W , she solves the following problem, by choosing
k
lij
(!) ; 8i; j; k; !, taking trade costs
29
ij
for all i; j, and the market prices pij (!) for all i; j; !, as given.
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.)
30
Here we assume that the production function of the same good is the same in all countries. We could assume that
production functions of the same good are di¤erent across countries, but the conclusion will remain the same.
P P pjk (!)f ! (ljk (!))
P P
3 1 GDP
f
(:) =
max
s.t. Lj = k !2 jk ljk (!) and f ! (ljk (!)) = yjk (!) 8k; !.
k
!
jk (!)
f ljk (!)j8k;!g
27
Thus, she solves
max
s.t.
Li =
De…ne l
k (!);8i;j;!
flij
g
P P
j
!2
ij
W =
X X X pij (!) f ! (lij (!))
i
j
!2
ij
ij
lij (!). The value of W evaluated based on pre-change prices is equal to Y w .
(lij (!)) ; 8i; j; ! as a vector of all factor input allocations; p
of all market prices; and
(
ij ) ; 8i; j
as a vector of all trade costs.
(pij (!)); 8i; j; ! as a vector
The optimal value of l, denoted by el, is a function of
prices p is a function of
and market prices p. Note that the market
a¤ects el directly as well as indirectly through p.
too. Thus, a change in
Formally,
el = g [ ; p ( )]
where g is some function. However, since we are using pre-change market prices and equivalent variation
to evaluate the global welfare change, the direct e¤ects of market prices p will be ignored as we evaluate
the change in W following changes in .
In evaluating the e¤ect of
ij
on W , we have to evaluate how l is a¤ected, then calculate its e¤ects
on W . In other words, we have to take into account 1. the direct e¤ect of
indirect e¤ect of
ij
! l ! W ; 3. plus the indirect e¤ect of
Therefore, the total e¤ect of
ij
ij
ij
! W ; 2. plus the
! p ! l ! W.
on W , ignoring the direct e¤ects of all market prices p, can be
calculated as
@l @p
@W
@W @l
@W
dW
=
+
+
d ij
@ ij
@l @ ij l=el
@l
@p @ ij l=el
@W
@W
= 0:
=
by envelope theorem, since
@ ij
@l l=el
P
Thus, the sum of equivalent variations of all countries, j Pj (dUj ), is equal to
X X @W
XX X
d ij =
pij (!) f ! (lij (!)) ij 2 d ij
@ ij
i
j
i
=
j
!2
XX
i
j
ij
Xij c
ij
as
X pij (!) f ! (lij (!))
!2
ij
Global income evaluated based on pre-change market prices,
=
X X Xij c
ij
i
D
j
= Xij
ij
P
j
Pj Uj , is equal to Y w . Therefore,
Yw
Envelope theorem for PC with fragmentation
Here, we focus on the model in section 4. All notations follow those of sections 2 and 4 unless otherwise
stated. Although we have multiple stages of production here, under perfect competition, in the absence
28
of externalities, the global competitive equilibrium yields the same allocation of resources to the production of di¤erent goods as the solution of a global planner’s global-income maximization problem,
given the market prices.
The proof follows similar logic as that in Appendix C. De…ne the pre-change vector of market
(pF1j ; :::; pFnj ) and pre-change vector of market quantity
prices of …nal goods sold in country j as pj
F ; :::; q F ). As before, as the global planner maximizes
(q1j
nj
of …nal goods consumed by country j as qj
global income subject to the resource constraints and the production functions at all stages, she would
automatically maximize the income of each country subject to its resource constraint.
Maximization of income of each country subject to its resource constraint
In each country j,
Ej = Yj = f GDP
F
jk
pFjk 8k ;
8k ; Lj
which is the GDP function of j, the outcome of the maximization of income of the country subject to
its resource constraint by the planner.32 The dual of this problem is that the planner solves
max Uj (qj ) s.t. pj qj = Ej
qj
where Ej is the value of Yj obtained from the above GDP function. Note that Ej = Pj Uj . Following
similar logic as in Appendix C, we can show that
P
P
P
P
P
P
F=
F F
= j Pj (dUj ) = j Pj Uj = j pj dqj = j pj qj = i;j pFij dqij
i;j pij qij .
Maximization of global income
s ; 8j; k; s, taking trade costs
The global planner maximizes global income W by choosing ljk
i; j,s and the market prices of …nal goods
pFij
for all i; j as given:
max
s.t. (i) Li =
s ;8j;k;s
fljk
g
P
s
yjk
s
j;s lij
W =
F
X pFij yij
F
ij
i;j
for all i , (ii) the production function of stage-s good exported from j to k:
82
>
< X
= 4
>
: i
s 1
vis yij;k
s 1
ij
!
1
s
s
3
s
s
5
1
9
>
=
s
>
;
s
Asj ljk
1
s
for s=2, 3, ..., F and all j, k
and (iii) the production of stage-1 good exported from j to k:
1
1
yjk
= A1j ljk
32
f GDP (:) = n max
ls
8k;s
jk
s
ij
o
P
k
F
pF
jk yjk
F
jk
s.t. Lj =
P P
k
s
s ljk
for j, k = 1, ..., n
and the production function at each stage.
29
for all
s is the quantity of labor used in combination with the quantities of inputs exported from i to
where ljk
Pn s
PF s
s 1
s . Thus,
s
j, yij;k
, to produce output to be exported to k, yjk
k=1 ljk = lj and
s=1 li = Li for all i; s.
In equilibrium, it can be easily seen that
s
yjk
= yjs
De…ne l
s
ljk
ljs
82
>
< X
= 4
>
: i
s 1
vis yij
s 1
ij
1
s
s
3
s
s
5
1
9
>
=
s
>
;
1
Asj ljs
s
s
ljk
ljs
s ; 8i; j; s as a vector of all labor input allocation; p
lij
s
ij
prices of …nal goods; and
(pFij ); 8i; j as a vector of all market
and market prices p. Note that the market
too. More speci…cally, if we denote ps
ps 1
for s=2, 3„..., F and all j, k (17)
; 8i; j; s as a vector of all trade costs at all stages.
The optimal value of l, denoted by el, is a function of
prices p is a function of
ps
!
s.
pF
(psij ); 8i; j and
1 ; :::;
is a function of
and
Thus, p =
is a function of
a¤ects el directly as well as indirectly through p. Formally,
F.
s
s
ij
; 8i; j, then
As a result, a change in
el = g [ ; p ( )]
where g is some function. However, since we are using pre-change market prices and equivalent variation
to evaluate the global welfare change, the direct e¤ects of market prices p will be ignored as we evaluate
the change in W following changes in .
s
ij
In evaluating the e¤ect of
on W , we have to evaluate how l is a¤ected, then calculate its e¤ects
on W . In other words, we have to take into account 1. the direct e¤ect of
indirect e¤ect of
s
ij
! l ! W ; 3. plus the indirect e¤ect of
Therefore, the total e¤ect of
s
ij
s
ij
s
ij
! W ; 2. plus the
! p ! l ! W.
on W , ignoring the direct e¤ects of all market prices p, can be
calculated as
@W
@W
dW
=
+
s
s
d ij
@ ij
@l
@l
@ ijs
@W
+
@l
e
l=l
@l @p
@p @ ijs
@W
@W
=
by envelope theorem, since
s
@ ij
@l
Now, we try to evaluate @W=@
s
ij .
!
l=el
= 0:
l=el
s for all i; j; s are optimally
We start from a state where all the lij
chosen given the exogenous variables, and evaluate the e¤ects of changes in
s
ij
on global income W .
s and ps as the c.i.f. price of output y s . Therefore, ps
De…ne psii as the f.o.b. price of yij
ij
ii
ij
Thus global income is given by
W =
X
F
pFii yij
=
i;j
n
X
i=1
30
pFii yiF :
s
ij
= psij .
Cost minimization leads to
s
vis yij
s 1
y
psij 1 ijs
s 1
ij
where
s 1
ij
P
=
s 1
s 1 yij
s
i pij
1
P
s
vis yij
1
s
1
s
s 1
ij
i
ij
s
s 1
ij
1
ij
1
s
1
is country j’s cost share on stage-s-1 intermediate goods imported from country i in the
total costs of stage-s-1 intermediate goods imported by j.
As explained above, as we evaluate the e¤ect of a small change in
s
ij
s and
on W , we can keep ljk
s gives us the
ljs for all s, j, k, as well as prices psij 8i; j; s constant. Thus, from (17), log-linearizing yjk
s caused by independent changes in y s 1 and
e¤ect on yjk
mj
s
ybjs = yc
jk =
s
X
s 1
mj
m
s 1
mj :
s 1
yd
mj
d
s 1
(18)
mj
Moreover, we know that
s 1
s ij
P
s
i pij
=
s 1
1 yij
psij
s 1
ij
P
psjj yjs
s 1
1 yij
s
i pij
s 1
ij
s 1
1 yij
s 1
ij
=
s
psii 1 yij
1
psjj yjs
which is the cost share of stage-s-1 intermediate good imported from country i in the total cost of a
stage-s good produced by country j.
A change in
s
ij
s (though q s = y s =
has no impact on yij
ij
ij
s
ij
s+1
for any k will
is a¤ected). However, yjk
be a¤ected (and so will yjs+1 ), which in turn a¤ect the values of outputs of all countries for later stages.
s
ij
Formally, setting m = i in (18), we obtain the e¤ect of a change in
s
ij
yjs+1
@yjs+1
@
s
ij
s
s+1 ij
=
=
on yjs+1 :
s
psii yij
s+1
ps+1
jj yj
Note that at stage s + 1, country j is the only country whose value of output is a¤ected by a change
in
s
ij .
Thus, the e¤ect of a change in
X
k
s
ij
on the value of global output in stage s + 1 is given by
s+1
ps+1
kk
@yj
@yks+1
= ps+1
d
jj
s
@ ij
@ ijs
=
s
psii yij
s
ij
d
s
ij
as
@yks+1
= 0 for k 6= j
@ ijs
s
ij ,
which is the total saving in trade costs resulting from a change in
31
(19)
s
ij .
Applying equation (18) to stage t > s + 1, we have
X
t 1d
t 1
ybkt =
t mk ymk as d
t 1
ij
=0
m
ptkk
)
X
)
k
ptkk
t 1
X
@ykt
t 1 @ymk
=
pmm
as
@ ijs
@ ijs
m
t 1
t mk
t 1
ptmm1 ymk
=
ptkk ykt
t 1
t 1
t 1
XX
X
X
@ykt
t 1 @yk
t 1 @ymk
t 1 @ym
=
p
=
p
=
p
mm
mm
kk @ s
@ ijs
@ ijs
@ ijs
ij
m
m
k
k
In other words, the e¤ects on the values of global output in all stages s + 1 and higher are the same.
That is, the e¤ect of a change in
s
ij
on the value of global output in each stage will pass on to the next
stage, until the …nal stage F . For the more economic intuition, refer to section 5 of the main text of
the paper. Thus,
dW =
P F @ykF
pkk s d
@ ij
k
s
ij
=
P F 1 @ykF 1
pkk
d
@ ijs
k
s
ij
= ::: =
X
s+1
ps+1
kk
k
@yj
@yks+1
= ps+1
d
jj
s
@ ij
@ ijs
s
ij
=
which is just the total saving in trade costs at stage s + 1 in country j. Thus, @W=@
Therefore, the sum of equivalent variations of all countries,
X @W
i;j;s
@
s
ij
d
s
ij
=
X
i;j;s
P
sc
s
Xij
ij
Global income evaluated based on pre-change market prices,
sc
s
X Xij
ij
=
i;j;s
E
j
s
psii yij
s
ij
s
ij
Pj (dUj ), is equal to
P
j
=
d
s
ij
=
sc
s
Xij
ij .
sc
s
Xij
ij
Pj Uj , is equal to Y w . Therefore,
Yw
Derivation of Equation (16)
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
('
eij )
'ij
1
(20)
1
The Free Entry Condition implies that expected pro…ts must be equal to the cost of entry:
X
Rij ('
eij )
1 Gi 'ij
= fe wi
ij wi
j
which, together with
P
eij ) = Ni
j Nij Rij ('
P h
j
1
Gi 'ij
i
Rij ('
eij ) = wi Li (total revenue is equal
to total income), implies the following labor market clearing condition:
8
9
<X
=
Ni
1 Gi 'ij
+
f
= Li .
ij
e
:
;
j
32
h
This equation, together with the facts that Nij = Ni 1
income, (20) and Rij 'ij =
ij wi
P
bij = n
N
P h
j
1
=P h
1
j
which leads to (16).
F
i
, total revenue is equal to total
(zero cuto¤ pro…t condition), give us (16), as shown below:
gi 'ij
1 Gi 'ij
P
j gi 'ij
=P h
j
j
Gi 'ij
ij d'ij
i
i
ij
+ fe
ij d'ij
o
g 'ij d'ij
1
Gi 'ij
g 'ij d'ij
1 Gi 'ij
Gi 'ij Rij ('
eij ) =
P
g 'ij d'ij
ij d'ij
j gi 'ij
,
i
1
1
1
G
'
i
Gi 'ij
(
'
e
)
=
'
ij
ij
ij
ij
Envelope theorem for MwG
Here, we focus on the one-sector MwG model in section 6.1. All notations follow those of sections
2 and 6.1 unless otherwise stated. We have shown in the online appendix that under monopolistic
competition with heterogeneous …rms and constant markup, the competitive market outcome is the
same as the global planner’s solution of her maximization problem.
The proof follows similar logic as that in Appendix C. As before, the global planner maximizes
global income subject to the resource constraints and the production functions. 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.
Without loss of generality, we assume that there is only one factor input labor. Let lij (!) denote
the labor input for producing output yij (!). 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.
Maximization of income of each country subject to its resource constraint
As in the previous appendixes, in each country j, expenditure Ej = Yj , where Yj is obtained from
the GDP function of the country, which is the outcome of the maximization of national income subject
to the national resource constraints and the production functions constraints. The dual of this is that
the global planner maximizes the country’s welfare by solving
Z 1
n
X
max Uj (fqij (')g) s.t.
Ni
pij (') qij (') gi (') d' = Ej
fqij (')g
'ij
i=1
33
where pij (') are exogenously given. Following similar logic as in the previous appendixes, there
is a one-to-one mapping from (fpij (')g ; fqij (')g ; 8i; ') onto (Pj , Uj ), and therefore a one-to-one
mapping from (fpij (')g ; fqij (')g ; 8i; j; ') onto (P1 ; :::; Pn ; U1 ; :::; Un ). For given fpij (') ; 8i; 'g, an
increase in Uj by a factor of
induced by an increase in Ej is equivalent to equi-proportional inP
P
creases in all fqij (') ; 8i; 'g by a factor of . Thus, we can show that = j Pj (dUj ) = j Pj Uj =
hP
i hP
i
R1
R1
i;j Ni ' pij (') dqij (') gi (') d' =
i;j Ni ' pij (') qij (') gi (') d' .
ij
ij
Maximization of global income
The global planner maximizes global income W by choosing lij (') ; 8i; j; ', taking trade costs
ij
for all i; j and the market prices pij (') for all i; j; ' as given:
max
flij (');8i;j;'g
8
<
X
s.t. Ni fe +
:
ij
W =
n X
n
X
Ni
Gi 'ij
j
1
pij (') f ! (lij (')) gi (')
+
d'
ij
'ij
i=1 j=1
1
Z
XZ
1
lij (') gi (') d'
'ij
j
9
=
;
= Li
for all i
where 'ij = min f' jlij (') > 0 g is the cuto¤ productivity in country i for a …rm serving destination
j, fe denotes the quantity of labor used in entry;
ij
denotes the labor units used as …xed cost for
exporting from i to j; Li denotes the labor endowment in country i and f ! (lij (')) is the production
function for producing yij (') =
ij qij
(') units of good by a …rm with productivity ' in country i for
sales to country j.33 Note that 'ij and Ni are implicitly determined by the choices flij (') ; 8i; j; 'g.
De…ne l
(lij (')) ; 8i; j; ' as a vector of all labor allocation; p
market prices; and
(
ij ) ; 8i; j
as a vector of all trade costs.34
(pij (')); 8i; j; ' as a vector of all
The optimal value of l, denoted by el, is a function of
prices p is a function of
too. Thus, a change in
Formally,
and market prices p. Note that the market
a¤ects el directly as well as indirectly through p.
el = h [ ; p ( )]
where h is some function. However, since we are using pre-change market prices and equivalent variation
to evaluate the global welfare change, the direct e¤ects of market prices p will be ignored as we evaluate
the change in W following changes in .
In evaluating the e¤ect of
ij
on W , we have to evaluate how l is a¤ected, then calculate its e¤ects
on W . In other words, we have to take into account 1. the direct e¤ect of
indirect e¤ect of
! W ; 2. plus the
! p ! l ! W . Note that in
n o
the second channel, l implicitly determines both the cuto¤ productivities 'ij and the …rm masses
33
ij
! l ! W ; 3. plus the indirect e¤ect of
ij
ij
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.
34
We abuse the notation a little here as the vector l and p includes in…nite elements (the upper bound for ' is 1).
34
fNij g.35 Thus, the indirect e¤ect of
ij
! l ! W re‡ects both the “productivity e¤ ect” and “…rm
mass e¤ ect” in Section 6.1. This explains why they cancel each other and do not appear in the …nal
e¤ect.
Therefore, the total e¤ect of
ij
on W , ignoring the direct e¤ects of all market prices p, can be
calculated as
@W
@W @l
@W
@l @p
dW
=
+
+
d ij
@ ij
@l @ ij l=el
@l
@p @ ij l=el
@W
@W
by envelope theorem, since
=
= 0:
@ ij
@l l=el
Thus, the sum of equivalent variations of all countries,
n X
n
X
dW
i=1 j=1
d
d
ij
=
ij
=
Z
n X
n
X
Ni
i=1 j=1
Xij bij as Ni
i=1 j=1
n X
n
X
1
P
j
Pj (dUj ), is equal to
pij (') f ! (lij (')) gi (')
2
ij
'ij
Z
1
G
d' = Xij
ij
'ij
X Xij c
ij
i;j
ij
pij (') f ! (lij (')) gi (')
Global income evaluated based on pre-change market prices,
=
d'd
P
j
Pj Uj , is equal to Y w . Therefore,
Yw
Mapping between our equation (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
ij
= "bij .
Thus
Pn Pn Xij
j=1
i=1 w bij =
Y
as
Pn
i=1 ij
=1)
Pn
i=1 d ij
Pn Pn Xij 1 b
bjj
ij
j=1
i=1 w
Y "
Pn Ej Pn Xij 1 b
bjj
=
ij
j=1 w
i=1
Y
Ej "
Pn Ej Pn
ij b
bjj
=
ij
j=1 w
i=1
Y
"
P
Ej 1
= nj=1 w bjj
Y "
= 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.
35
is because lij (') > 0 i¤ ' > 'ij , and o
'ij in turn determines Nij through the resource constraints
nThis P
P R1
Ni fe + j ij 1 Gi 'ij + j ' lij (') gi (') d' = Li and Nij = Ni 1 Gi 'ij .
ij
35
References
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37
Online Appendix
A
Proof that the global market outcome is e¢ cient under MwG
The global planner solves two problems simultaneously. First, she chooses labor allocation lij (') to
maximize the global GDP Y w . Second, she chooses qij (') to maximize the utility of each individual
country j, Uj . In other words, he makes the production decision as well as the consumption decision.
The choices of pij (') serve to equalize the quantities supplied and demanded of goods qij (') for 8i; j; '.
Maximization of utility of each country
She chooses qij (') to maximize Uj given Ni , 'ij , Ej and pij ('). Thus, she solves
Max Uj =
s:t:
X
i
Let
j
"
X
Ni
Ni
1
qij (')
1
gi (') d'
'ij
i
Z
Z
1
#
1
(21)
pij (')qij (')gi (') d' = Ej
'ij
be the Lagrange multiplier for Uj . Then, the Lagrangean is given by
"
#
X Z 1
Lj = Uj + j Ej
Ni
pij (')qij (')gi (') d'
'ij
i
Substituting the F.O.C. w.r.t. qij (') into equation (21) leads to
j
where Pj =
hP
i Ni
R1
'ij
(pij ('))1
gi (') d'
i
=
1
Pj
1
1
is the aggregate price index in country j. Substituting
the above relationship back into the F.O.C. w.r.t. qij (') we have
qij (') =
pij (')
Pj
Uj =
pij (')
Pj
Ej
Pj
This expression will be used in the other maximization problem.
Maximization of global income
She chooses labor allocation lij (') to maximize the global GDP Y w , given
38
ij ,
Li and pij ('). The
variables Ni , 'ij , qij (') and Ej are implicitly determined from the choices of lij ('). Thus, she solves
max Y w =
flij (')g
8
<
X
s.t. Ni fe +
:
i
1
ij
XX
Ni
+
j
yij (')
Li =
+
XX
i
ij
Ni
i
Z
1
1
'ij
Ej
Pj
pij (')
Pj
. Let
pij (')qij (')gi (') d'
(22)
9
=
ij
qij (')gi (') d' = Li for all i
;
'
*
Li
(23)
from utility maximization
be the Lagrange multiplier. Thus, the Lagrangean is given by
i
1
1
(Ej ) (Pj qij ('))
gi (') d'
'ij
j
X
i
'lij (')
=
ij
XZ
j
and qij (') =
where qij (') =
1
'ij
j
Gi 'ij
Z
8
<
X
Ni fe +
:
ij
1
Gi 'ij
+
j
XZ
1
'ij
j
ij
'
qij (')gi (') d'
From the F.O.C. w.r.t qij ('), we have
ij
i
qij (') = Ej
1
Pj
'
9+
=
;
1
Substituting this equation into the budget constraint (23) leads to
i
h
P
Li
f
1
G
'
e
i
ij
j ij
Ni
=
i
1
P R1
ij
Ej gi (') d'
j '
1
'Pj
ij
Combining this expression with equation (22), we have:
Yw =
X
i
Ni
8
<L
i
: Ni
fe
X
ij
1
Gi 'ij
j
The F.O.C. w.r.t Ni of above expression leads to
Ni =
9
=
1
;
Li
DP nh
1
j
Gi 'ij
2
XZ
4
'ij
j
i
1
ij
o
+ fe
ij
'Pj
1
31
Ej gi (') d'5
E ,
(24)
(25)
which is the same as the labor market clearing condition under the decentralized market. SubP R1
1 Li
1
stituting equation (25) into (23) leads to j ' 'ij qij (')gi (') d' =
of
Ni , which means that
ij
the labor 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 leads to
X Z 1 1
1
wi Li =
Ni
Ej [Pj qij (')]
gi (') d'
j
=
'ij
i Li
39
Thus
i
= wi , which implies that pij (') =
wi ij
' ,
1
which is the same expression as the market
outcome. Thus qij (') and the corresponding
lij (') =
qij (')
'
ij
wi
1
= Ej
1
ij
'Pj
all have the same expressions as the market outcome.
According to equation (24), maximizing Y w w.r.t. 'ij is equivalent to maximizing
1
ln
8
<L
i
fe
: Ni
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
pij ('ij )
Pj
= Ej
to the cuto¤ productivity will spend
1
9
=
;
2
Z
1 4X
+ ln
j
1
ij
Gi 'ij
1
'ij
1
1
Ej gi 'ij
'ij Pj
P R1
ij
j 'ij
Ej gi (') d'5 .
'Pj
ij
i=
3
1
1
Ej gi (') d'
'Pj
= pij 'ij qij 'ij . The marginal …rm corresponding
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
MwG with multiple sectors
Without loss of generalization, we also assume that trade is balanced. We proceed with the proof in
two steps. The consumer price index is given by
Pj =
hP
Z
z=1 Pj
(z)1
i
1
1
, where Pj (z) =
(
X
Nijz
i
where Pj (z) denotes the aggregate price index for sector z, '
eijz
z
z
1
1
wi ijz
'
eijz
1 Giz ('ijz )
R1
1
'ijz
z
(')
)
z
1
1
z
(26)
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
40
consumer price index (26), we get
P
Pj (z)1
Pbj = Z
Pbj (z)
P
z=1
Z
1
P
(z)
z=1 j
Pn
z
PZ
i=1 Xij
Pb (z)
= z=1 PZ Pn
z j
X
i=1 ij
z=1
Pn
z
Xz
PZ Pn
1
i=1 Xij
bz
Pn ij z w
= z=1 i=1 PZ Pn
bi + bijz
N
ij
z
X
1
z
i=1 ij
z=1
i=1 Xij
z
Xij
P Pn
1
d
bijz '
w
bi + bijz
= Z
N
e
P
P
ijz
z=1
i=1
n
Z
z
1
z
X
i=1 ij
z=1
d
'
e
ijz
Step 1: The relationship between …rm mass e¤ ect and productivity e¤ ect is given by
!
zN
bz
Xji
PZ Pn
P Pn
ij
z d
= Z
eijz
z=1
j=1
z=1
j=1 Xji '
1
z
(27)
Proof: The e¤ect of the change in cuto¤ productivity 'ijz on the average productivity '
eijz can be
derived as follows
1
d
'
e
ijz =
'ijz
z
1
1
= ('
eijz )
z
1
giz 'ijz
Giz 'ijz
d'ijz
1
z
“The productivity e¤ect”.
(28)
z (') as the revenue of a …rm exporting from i to j in sector z, with productivity '. Note
De…ne Rij
that CES preferences implies that
z ('
Rij
eijz )
z '
Rij
ijz
=
('
eijz )
'ijz
z
z
1
(29)
1
The Free Entry Condition (expected pro…ts must be equal to the cost of entry) together with
i
PZ P z h
z ('
Rij
eijz ) = wi Li (total revenue is equal to total income), implies the
N
1
G
'
iz
ijz
z=1
j i
following labor market clearing condition
h
PZ
z Pn
z=1 Ni
j=1 1
Giz 'ijz
ijz
+ fez
i
z
Totally di¤erentiating the previous equation implies
i
PZ hPn
PZ Pn
z bz
1
G
'
+
f
iz
ijz
ez
z Ni Ni =
ijz
z=1
j=1
z=1
j=1
h
PZ Pn
P
P
n
z bz
, z=1 j=1 Xij
Nij = Z
('
eijz ) z 1 = 'ijz
z=1
j=1 z ijz wi 1
= Li
z
z ijz Ni giz
1
z
i
(30)
'ijz d'ijz
Niz giz 'ijz d'ijz
(31)
d'ijz
1
z
(32)
The equation (28), together with equation (29), implies
PZ Pn
z=1
z d
eijz
j=1 Xij '
=
=
PZ Pn
z=1
z
j=1 Ni
PZ Pn
z=1
j=1 z
z
d
Giz 'ijz Rij
('
eijz ) '
e
ijz
h
('
eijz ) z 1 = 'ijz
ijz wi 1
1
41
z
1
i
Niz giz 'ijz
According to equation (31) and (32), we have
PZ Pn
z=1
j=1
zN
bz
Xji
ij
1
z
!
=
PZ Pn
z=1
z d
eijz
j=1 Xji '
Step 2: The change in global welfare is given by
Pn
z
Xij
z
z=1 w bij .
Y
Pn Pn PZ
Ej b
j=1 w Uj =
Y
j=1
i=1
Proof: The global welfare change is given by
Pn
bj
b = Pn Ej E
j=1
j=1 Ej Uj
=
Pn Pn PZ
j=1
=
i=1
z
bj
z=1 Xji w
PPP z
Xji w
bj
i z
j
|
{z
Pbj
Pn Pn PZ
j=1
PPP z
Xij w
bi
j
i z
}
TOT e¤ect on all importers and exporters = 0
=)
X Ej
bj =
U
Yw
j
i=1
|
w
bi + bijz
PPP z z
Xij bij +
j
i z
{z
direct e¤ect
z
Xij
z
z=1 w bij
Y
Pn Pn PZ
j=1
z
z=1 Xij
i=1
42
}
1
z
1
bijz
N
PPP
j
|
i z
(33)
d
'
e
ijz
z d
Xij
'
eijz
{z
zN
bz
Xij
ij
1
z
!
}
global …rm mass e¤ect plus productivity e¤ect = 0
Only the direct e¤ect matters.