Gains from Trade: Does Sectoral Heterogeneity Matter?
Rahul Giri∗
Kei-Mu Yi†
Hakan Yilmazkuday‡
December 8, 2016
[PRELIMINARY]
Abstract: This paper assesses the quantitative importance of sectoral heterogeneity in computing the gains
from trade. Our framework draws from Caliendo and Parro (2015) and Alvarez and Lucas (2007) and has
sectoral heterogeneity in several dimensions, including the elasticity of trade to trade costs, the value-added
share, and input-output structure. The key parameter we estimate is the sectoral trade elasticity, and we use
the Simonovska and Waugh (2015) simulated method of moments estimator using actual price data from 1990.
Our estimates range from 2.97 to 8.94, in contrast to a range of 4.26 to 35.55 with the Eaton and Kortum
(2002) price-based method. Our calibrated model has 21 OECD countries and 20 sectors. From our benchmark
model with sectoral heterogeneity, we conduct two sets of exercises. In the first, we start from our benchmark
model and then eliminate heterogeneity, one dimension at a time, leaving other parameters unchanged. We
then raise trade costs to induce autarky and compare the results to the benchmark model. In the second,
we calibrate a model with no sectoral heterogeneity and compare it to the benchmark model. In both sets
of exercises, we find that the presence of sectoral heterogeneity affects the gains from trade calculation, but
it does not always lead to an increase in the gains from trade – contrary to the findings of several recent
papers. It depends on whether, for example, the sectors with the low trade elasticities (hence, implying high
gains from trade), are also the sectors with low value added shares and with high trade shares to begin with.
JEL Codes: F10, F11, F14, F17
Keywords: gains from trade, trade elasticities, gravity, international price dispersion, multi-sector trade
∗
†
International Monetary Fund. E-mail: [email protected]
Department of Economics University of Houston; Federal Reserve Bank of Dallas and Minneapolis; and
NBER. Email: [email protected]
‡
Department of Economics, Florida
[email protected]
International
University,
Miami,
FL,
USA.
E-mail:
Introduction
Arkolakis, Costinot and Rodriguez-Clare (2012, hereafter ACR) showed that for a wide
class of international trade models – differing in the form of competition between firms, on
the importance of the external margin vs. the internal margin in trade, and on whether there
was entry and exit – the gains from trade could be captured by a simple formula involving
just two numbers, the elasticity of trade with respect to trade costs, and the share of total
spending on domestic goods.1 The publication of this paper sparked a series a papers devoted
to frameworks and counterfactuals where the ACR results would not hold. Several of these
papers, including Levchenko and Zhang (2014), Costinot and Rodriguez-Clare (2014), and
Ossa (2015) demonstrated that sectoral heterogeneity could lead to higher gains from trade
compared to the ACR formula.
The goal of this paper is to comprehensively and quantitatively assess whether more
sectoral heterogeneity indeed delivers these higher gains. We develop a rich multi-country,
multi-sectoral trade model that draws from Caliendo and Parro (2015) and Alvarez and
Lucas (2007). We estimate and calibrate the parameters of the model to match key features
of the price, output, and trade data in 1990.
In this class of models, as Caliendo and Parro (2015) discuss, the gains from trade
can be divided into two categories.2 The first is the direct gains from increased trade.
Within each sector, these gains are greater the larger the increase in trade, the lower the
trade elasticity, and the smaller the value-added share. To the extent these three forces
are correlated across sectors, the gains are larger, as well. The second is from intersectoral
linkages. These linkages show up as changes in relative prices of inputs, mediated by their
share in the production of the sectoral good. The more the relative price of inputs decline, the
larger the gains. Accordingly, based on the interaction of these forces, the welfare gains from
trade can be higher or lower when sectoral heterogeneity is considered. Careful estimation
and calibration of the parameters is needed.
In this context, a key contribution of our paper is that we estimate the elasticity of trade
1
The gains from trade exercise is one in which the current equilibrium based on actual data is compared
against a counterfactual equilibrium with sufficiently high trade costs that autarky occurs.
2
Ossa (2015) shows how these gains can be collapsed to a single formula, following Costinot and
Rodriguez-Clare (2014)
1
with respect to trade costs for each of 19 sectors using the simulated method of moments
(SMM) methodology of Simonovska and Waugh (2015). This methodology retains the virtue
of using micro-price level data, but corrects the bias introduced by a small sample of price
observations, in estimating the elasticity with the Eaton and Kortum (2002) methodology.3
To our knowledge, this is the first application of the Simonovska and Waugh SMM estimator
to estimate the trade elasticity at the sector level.4 We use the Eurostat surveys of retail
prices.5 The data cover 21 OECD countries and 19 three-digit ISIC traded good sectors for
1990. Caliendo and Parro (2015) also estimate structurally consistent estimates of sectoral
trade elasticities using a triple-difference approach. However, they utilize sector-level data –
sector-level tariff rates – to estimate their trade elasticities.
Our sectoral trade elasticity estimates range from 2.97 (ISIC 341 and 342, Paper and
products; printing and publishing) to 8.94 (ISIC 371, Iron and steel); and the median is
4.38 (ISIC 355, Rubber products). We also estimate the sectoral trade elasticities with
the original Eaton and Kortum (EK) method and the minimum, maximum, and median
elasticities are 4.26, 35.55, and 10.29. So, our SMM estimates are clearly lower, as SW
obtained in their paper with a one-sector framework. As SW found, the “bias” is larger the
smaller the sample size. For example, ISIC 352, Other chemicals, has a sample size of 4,
while ISIC 311, Food products, has a sample size of 343. Our SMM estimates are similar
across these two industries, 3.75 and 3.57, respectively, but the EK estimates are 11.93 and
4.28, respectively.
Our calibrated model has 21 countries and 20 sectors (19 traded sectors and one nontraded goods sector). We calibrate the other parameters to match their data counterparts
or to be consistent with sectoral outputs and trade flows. With our calibrated model, we
compute the gains from trade. We follow the recent literature by examining the gains
obtained in the equilibrium corresponding to the actual data relative to a counterfactual
autarky equilibrium. With that as our benchmark, we conduct two sets of exercises. In the
first set of exercises, we eliminate one source of sectoral heterogeneity; in its place, we employ
3
Eaton, Kortum and Kramarz (2011) also employ this type of estimator.
Simonovska and Waugh (2014b) go further by showing that, given the data on trade flows and microlevel prices, different models have different implied trade elasticities and welfare gains.
5
See Crucini, Telmer, and Zachariadis (2005), for example.
4
2
a homogeneous parameter that is common across all sectors. For example, we replace the
estimated sectoral trade elasticities with a single elasticity common to all sectors. We then
compute the gains from trade relative to the autarky equilibrium, and compare these gains
to the benchmark model gains. In the second set of exercises, we calibrate and estimate
a model with no sectoral heterogeneity in trade elasticities, value-added shares, and inputoutput structure. We then compute the gains from trade relative to autarky, and compare
those gains to the gains in the our benchmark calibrated model.
Our benchmark calibrated model delivers gains from trade ranging from 0.38 percent
in Japan to 9.59 percent in Ireland. The median gain in going from autarky to the calibrated
equilibrium is 4.25 percent (Finland). We decompose our “benchmark” gains by sector, and
we find that in most countries only a few sectors account for most of the welfare gains. In
addition, we find that the direct trade effect is positive and the intersectoral linkage effect
is negative in all countries. The presence of the non-traded goods sector is critical to this
result.
We then turn to the role of sectoral heterogeneity. In our benchmark model, we replace
our estimated sectoral trade elasticities with the median estimate, as well as our estimate
from a one-sector (the 19 traded sectors are aggregated into a single traded sector, and the
non-traded sector) version of the model (2.37). When we use the median elasticity, the gains
from trade are slightly larger than in the benchmark model. When we use the elasticity
estimate from the one-sector model, the gains from trade are considerably larger than in the
benchmark model. The reason is simple. A low trade elasticity leads to large gains from
trade. We also investigate a counterfactual in which we impose a common value-added share
across all sectors in the benchmark model. The gains are considerably larger in this case over
the benchmark model. The reason is that in the benchmark model, those sectors with low
value-added shares are the sectors with either high trade elasticities or small increases in trade
– limiting the gains from trade. However, when we impose a common input-output structure
across all sectors, we find the gains are slightly smaller than in the benchmark model. Our
conclusion from these exercises is that the gains depend on the source of heterogeneity. In
the cases of the trade elasticity and the value-added shares, homogeneity, not heterogeneity,
leads to larger gains.
3
In our second set of counterfactuals, we compare the benchmark model gains with a
re-calibrated and estimated one-sector model. In that one-sector model, the estimated trade
elasticity is, as noted above, 2.37. We find the gains from trade are slightly larger in the
one-sector case. Once again, this is primarily because the estimated elasticity is so low.
If we replace that elasticity with the median elasticity (4.37) from the benchmark model,
then the gains are considerably smaller than in the benchmark model. Again, we conclude
that heterogeneity does not necessarily imply larger gains from trade. The intuition is clear.
The gains from trade is a non-linear function of the existing sectoral domestic expenditure
shares, the sectoral trade elasticity, and the value-added shares, on the one hand, and the
input-output structure and relative prices, on the other hand.
How do we relate our results to those of Levchenko and Zhang (2014), Costinot and
Rodriguez-Clare (2014), and Ossa (2015), all of whom find that greater heterogeneity leads
to greater gains from trade? Levchenko and Zhang (2014) do not consider heterogeneity
in the trade elasticity. In all their calculations, they use a single aggregate trade elasticity.
Among the other features of their model, they add heterogeneity, one feature at a time,
and while overall the gains from trade increase, it is not monotonic. In terms of Costinot
and Rodriguez-Clare (2014) and Ossa (2015), they find that the gains from trade increase
as more heterogeneity is added. However, in both cases, they employ sectoral elasticities
(the former from Caliendo and Parro (2015) and the latter estimated), several of which are
less than two. In Ossa’s case a number of the elasticities are less than one. All else equal,
an elasticity of 1 delivers 10 times the (percentage) gains from trade as an elasticity of 10.
Hence, the gains from a world of two sectors with these two elasticities will be considerably
larger than in a world with one sector and an elasticity that is the median of these two. As
Ossa notes, this is an application of Jensen’s inequality. In our work, our lowest elasticity is
around three. Hence, we do not have sectors with the low elasticities that deliver large gains
from trade. We conclude that the absence of very low estimated elasticities is the primary
reason why our results do not show that greater sectoral heterogeneity leads to greater gains
from trade.
The rest of the paper is organized as follows. Section 2 lays out our model, and the
following section provides our calibration and estimation methodology. Section 4 presents
4
our results, including our elasticity estimates, our benchmark welfare gains, and our counterfactuals. Section 5 concludes.
Model
Our model draws from Alvarez and Lucas (2007) and Caliendo and Parro (2015), both of
which extend Eaton and Kortum (2002). There are N countries, and there are S sectors
producing tradable goods, and one sector producing a homogenous non-traded good.
Production
In each tradable goods sector s of country i, there is continuum of goods xsi ∈ [0, 1], and
each good is produced by combining labor and intermediate inputs through a Cobb-Douglas
production technology.
−θs
qis (xsi ) = zi (xs )
βs
[li (xs )]
" S+1
Y
#1−β s
ξ s,m
s
Qm
i (x )
m=1
zi (xs ) is a random cost draw for good xsi from an exponential distribution - exp (λsi ). λsi
is sector - and country-specific productivity parameter. These zi (xs )’s are assumed to be
independent across goods, sectors and countries, and they are amplified in percentage terms
by the parameter θs . The amount of labor used to produce good x of sector s in country
i is li (xs ), while the amount of composite good of sector m used as an intermediate input
s
s,m
capture the input-output structure of production
is denoted by Qm
i (x ). The shares ξ
PS+1 s,m
across sectors, and they satisfy m=1 ξ
= 1 for every sector s.
The individual goods in each sector are traded across countries. Sector s composite
good - Cis - is produced, using a constant elasticity of substitution (CES) production function,
by procuring each individual good at the lowest price from across the set of countries.
Cis =
Z
q si (xs )
η s −1
ηs
f s (xs ) dxs
s
ηsη−1
where q si (xs ) is the amount of good x of sector s used to produce the composite good. The
elasticity of substitution between the individual tradable goods of sector s is η s . Given
that the cost draws - zi (xs )’s -, characterizing goods in the continuum, are assumed to be
5
independent across goods, sectors and countries, the joint density function is given by:
N
Y
f s (xs ) = f s (z s ) =
!
λsi
exp −
i=1
N
X
!
λsi zis
i=1
The non-traded sector’s composite good is produced as follows:
CiS+1 = CiN
" S+1
#1−γ
Y N,m φN,m
γ
,
= AN
liN
Qi
i
m=1
where liN is the labor used in the production of the non-traded composite good, QN,m
is the
i
amount of sector m composite used as intermediate input, and the shares of φN,m satisfy
PS+1 N,m
= 1. AN
i represents the TFP in non-traded sector.
m=1 φ
Prices
We make the standard iceberg assumption - in order for country i to receive one unit of a
good of sector s, country j must ship τijs > 1 units. Given that the individual goods can be
bought from domestic or foreign producers, the price of good x of sector s in country i is
n
θs o
psi (xs ) = min B s Vjs zjs xsj τijs
(2.1)
j
where Vis is an input bundle cost given by:
Vis = [wi ]
βs
" S+1
Y
#1−β s
(Pim )
ξ s,m
,
(2.2)
m=1
s
with wi representing wages, and Pim reprsenting the price of Qm
i , and B represents a sector-
specific constant defined as follows:
s −β s
B s = (β )
s −(1−β s )
(1 − β )
" S+1
Y
m=1
6
#(1−β s )
(ξ
s,m −ξ s,m
)
.
Properties of the exponential distribution imply that
(
psi (xs )
1
θs
∼ exp (B s )
−1
θs
N
X
)
ψijs
j=1
where
ψijs = Vjs τijs
−1
θs
λsj .
(2.3)
Since sector s composite good is produced using CES technology, the price of the sector s
composite is
Pis
Z
1
psi
=
s 1−η s
(x )
dx
s
1
1−η
s
,
0
1
which, given the distribution of psi (xs ) θs , can be written as (see the Appendix for details)
N
X
Pis = As B s
!−θs
ψijs
,
(2.4)
j=1
where As is a sector-specific constant.6
The price of the non-traded composite good is:
PiN = E
[wi ]γ
hQ
S+1
m=1
(Pim )φ
N,m
i1−γ
,
AN
i
which implies,

γ
 [wi ]
PiN = E
hQ
S
m=1
(Pim )φ
N,m
i1−γ  1−φN,N1 (1−γ)
where
−(1−γ)
E = γ −γ (1 − γ)
S+1
Y
−φN,m
φN,m
m=1
6
As =
∞
R
uθ
s
(1−η s )
,


AN
i
1
1−η
s
exp (−u) du
is a Gamma function.
0
7
!(1−γ)
.
(2.5)
Consumption
In each country there is a representative household that derives utility from a final consumption good (Yi ).
Ui = Yi .
The final consumption good is a Cobb-Douglas aggregator of the sectoral composite goods.
Yi =
S+1
Y
δf,m
Yif,m
,
(2.6)
m=1
where Yif,m is the final consumption of sector m composite, and
PS+1
m=1
δ f,m = 1 . The price
of the final good, therefore, is given by
Pi =
S+1
Y
δ f,m
−δf,m
(Pim )δ
f,m
.
(2.7)
m=1
Market Clearing
The market clearing conditions for labor and sectoral composite goods are given by
S Z
X
s=1
S Z
X
m=1 | 0
|0
1
lis (xs ) dxs +liN ≤ 1 , i = 1, . . . , N ,
{z
}
lis
1
Qm,s
(xs ) dxs +QN,s
+ Yif,s ≤ Cis , i = 1, . . . , N , s = i = 1, . . . , S + 1 .
i
i
{z
}
Qm,s
i
Trade Flows
We now derive an expression for trade shares that will be important for our quantitative
work. The probability that good x of sector s is exported by country j to country i is:
πijs
s
(x ) = Pr B
s
Vjs zjs
s θs
(x )
τijs
n
o
s s s
s θs s
≤ min B Vn zn (x ) τin
.
{n6=j}
8
Applying properties of the exponential function yields
πijs (xs ) = πijs =
ψijs +
ψs
Pij
n6=j
s
ψin
ψijs
= PN
s
n=1 ψin
.
(2.8)
Thus, every good x of sector s has the same probability of being exported by country j to
country i. As a result, the expenditure of country i on sector s goods of country j is
Xijs = πijs Xis ,
where Xis is the total per capita expenditure of country i on sector s goods. Hence, the share
of country j in country i’s total expenditure on sector s goods is:
s
Dij
=
Xijs
ψijs
s
.
=
π
=
P
ij
N
s
Xis
n=1 ψin
Using (2.4), we can rewrite this expression as:
s
Dij
s
1
s − θs
= (A B )
Vjs τijs
Pis
−1
θs
λsj
(2.9)
which is decreasing in source country input bundle cost Vjs , trade costs τijs and increasing in
the productivity parameter λsj .
Expenditure on Sectoral Goods, Wages and Labor Allocations
How is the per capita expenditure on a sector’s goods determined? To see this, start with
the market clearing condition for composite good of sector s and multiply both sides by its
price. This yields
Xis
=
Pis Cis
=
S
X
Pis Qm,s
+ Pis QN,s
+ Pis Yif,s ,
i
i
(2.10)
m=1
where the first term on the right hand side is the expenditure of all traded good sectors
on sector s composite, the second term is the expenditure of the non-traded good sector
on sector s composite, while the last term is final consumption expenditure on sector s
composite. Due to Cobb-Douglas production technologies, these components of expenditure
9
of sector s composite (s = 1, . . . , S + 1) are given by
Pis Yif,s = δ f,s Pi Yi = δ f,s wi Li ,
Pis Qm,s
= (1 − β m ) ξ m,s
i
N
X
m
Lj Xjm Dji
, m = 1, . . . , S ,
j=1
Pis QN,s
= (1 − γ) φN,s Li XiN ,
i
where XiN = PiN CiN . Thus, the total expenditure on sector s = 1, . . . , S + 1 composite is
given by
Li Xis = δ f,s wi Li + (1 − γ) φN,s Li XiN +
S
X
(1 − β m ) ξ m,s
m=1
N
X
m
Lj Xjm Dji
.
(2.11)
j=1
Wages are determined by using the balanced trade condition.
S X
N
X
s
Lj Xjs Dji
=
s=1 j=1
S
X
Li Xis .
(2.12)
s=1
Once we have the per capita expenditure on a sector’s goods and the wage, labor
allocation is derived from
Li wi lis = β s
N
X
s
Lj Xjs Dji
,
(2.13)
j=1
Li wi liN = γLi XiN .
(2.14)
Equilibrium
Competitive equilibrium is the price indices of sectoral composites ({Pis }Ss=1 , PiN ), per capita
expenditures on each sector’s goods ({Xis }Ss=1 , XiN ), wages (wi ), and labor allocations ({lis }Ss=1 , liN )
that provide a solution to the system of equations - (2.4), (2.5), (2.9), (2.11), (2.12), (2.13),
and (2.14).
10
Gains from Trade
We first construct a baseline equilibrium to infer λsi - from (3.6) and AN
i from the data
on real GDP per capita. In this baseline equilibrium, (i) trade costs are estimated using
the gravity equations for each sector ((3.3) and (3.4)), (ii) prices of sectoral intermediate
composites (Pis ) are computed from (3.5) using the source country coefficients and trade costs
s
obtained from gravity equation, (iii) trade shares (Dij
) are taken from data, (iv) equilibrium
wages (wi ), equilibrium per capita expenditures (Xis ), and equilibrium labor allocations (lis
and liN ) are determined using (2.12), (2.11), (2.13) and (2.14), respectively.
In order to assess the welfare gains from trade, we use real GDP per capita as the
s
measure of welfare. We take the estimated λsi , AN
i , and τij , and solve for baseline equilibrium
wages, per capita expenditures, labor allocations, prices of sectoral intermediate composites,
and trade shares. Using the same λsi and AN
i we solve for an autarky equilibrium by setting
trade costs to 100 times their baseline level. Then, using real GDP per capita, Wi = wi /Pi ,
as the measure of welfare, we compute the change in welfare as ln Wia /Wib × 100, where
Wia denotes the autarky value and Wib denotes the baseline value. Pi is the price of the final
consumption good, given by (2.7).
Sources of Gains from Trade
What are the sources of gains from trade in our model with inter-sectoral linkages?
Using (2.7) for the price of the final consumption good yields the following expression for
real GDP per capita:
δf,s
S+1 1 Y wi
Wi =
,
H s=1 Pis
where
H=
S+1
Y
δ f,s
−δf,s
(2.15)
.
s=1
Thus, real income is a geometric average of the real wage expressed relative to the price
of sector s composite, with the weight being each sector’s weight in the final consumption
good. Combining (2.9) with the expression for unit cost of the input bundle (Vis ) yields the
11
following expression for wi /Pis for the tradable goods sectors:
wi
=
Pis
1
s
A Bs
β1s Diis
λsi
s
−ξs,m (1−β s )
−θ
S+1 m βs Y
βs
P
i
m=1
Pis
, s = 1, . . . , S
For the non-traded sector, the analogous expression is given by
"
φN,s (1−γ) # 1−φN,N1 (1−γ)
S Y
wi
w
AN
i
i
=
.
E s=1 Pis
PiN
Substituting the last two expressions in the expression for real wage gives us
Wi = δ f,N
δf,N
φN,s (1−γ)
 δf,N N,N

+δ f
s
−ξs,m (1−β s )
δ f,N
−θ
1
1−φ
(1−γ)
S
S+1
s
s
s
s
m
N
N,N
Y
Y
β
β
δf,s
Dii β
Pi
Ai 1−φ (1−γ)
1


δ f,s
s
s
s
s
E
A
B
λ
P
i
i
s=1
m=1
(2.16)
Then the logarithm of the change in real GDP per capita is given by





S+1
S
X
 δ f,N φN,s (1 − γ)
 s
(1 − β s ) X s,m
s
f,s   θ
b
c

ξ ln
ln Wi =
 1 − φN,N (1 − γ) +δ  − β s ln Dii −
βs

m=1
s=1
| {z } |
|
{z
}
{z
non-traded sector effect
trade effect
Pbim
Pbs
inter-sector linkage effect
i
!

 .


}
(2.17)
As discussed in Caliendo and Parro (2015), there are two main sources of gains from
trade.7 When there is reduction in trade barriers, trade increases, i.e., sectoral home expenb s < 0). This leads to an increase in welfare.
diture shares, Diis , will tend to decline (ln D
ii
The “trade effect”represents the standard gains driven by increased specialization and trade.
A given decline in the sectoral home expenditures share, Diis , leads to a larger increase in
welfare if θs /β s is high. A higher θs , which corresponds to a smaller trade elasticity for sector
s, implies larger welfare gains essentially because goods are less substitutable. In addition, a
lower β s , or a higher intermediate input share in sector s leads to greater gains from trade.
This owes to the “round-trip” effect discussed in Jones (2011).
The second source of gains stems from the change in relative prices of intermediates;
7
There is also a third source of gains from trade - change in sectoral productivities, λsi . However, as in
Caliendo and Parro (2015) , we assume that sectoral productivities do not change.
12
we call it the “inter-sectoral linkages effect”.8 As trade barriers decline, the costs of imports
decline, which shows up as lower prices of sectoral intermediate goods. To the extent these
prices decline relative to the price of the sector s output good that is being produced from
these intermediates, there are gains from trade. A given relative price decline has a larger
impact on welfare, the larger the share of the intermediate good in the output good’s production process, i.e., the larger is ξ s,m . In addition, the welfare gains are larger, the greater
the intermediate input share in the sector s good 1 − β s .
For both sources of gains, there is an additional effect coming from the weight of sector
s goods in final consumption, which is captured by δ f,s . Finally, the non-traded final good
has two partially offsetting effects. On the one hand, the presence of the non-traded good
dampens the gains from trade coming from the inter-sector linkages. This negative effect
of the non-traded sector on welfare is partially offset, because increased trade reduces the
cost of the traded intermediate goods that are used to produce the non-traded good. The
overall effect depends on the relative importance of intermediate composite of sector s in
production of non-traded good, and the weight on the non-traded good in final consumption
good. This effect is captured by the term
δ f,N φN,s (1−γ)
.
1−φN,N (1−γ)
It is worth noting that it cannot be immediately inferred from (2.17) that more heterogeneity automatically implies greater gains from trade. It depends on the details of the
sectoral heterogeneity.
Calibration and Estimation Methodology
We now describe how we calibrate the model. The key parameters and variables to be calibrated are the θs , the trade costs, τijs , and the productivity parameters, λsi . Of these, the most
important is calibrating the sector-specific parameter of θs that represents trade elasticities
with respect to trade costs. We employ the simulated method of moments methodology for
estimating these elasticities introduced by Simonovska and Waugh (2014a), hereafter, SW,
θs at the sectoral level. The estimation procedure employs micro-level price data, categorized
into sectors, and sectoral trade flow data, with the latter captured by a ‘gravity’ equation
linking sectoral trade shares to source and destination fixed effects and to trade costs. We
8
Caliendo and Parro (2015) call this the “sectoral linkages effect”.
13
also describe our data sources, as well as the calibration of the other parameters. Finally, a
number of our counterfactuals involve a one-sector model, and we describe how we calibrate
and estimate that.
Sector-Level Trade Elasticity
To estimate the trade elasticity - 1/θ - for each sector, we use the Simulated Method of
Moments (SMM) estimation methodology developed by SW. A summary of the methodology
is provided here; a detailed description is provided in the Appendix.
To obtain the equation to be estimated, according to (2.9), we can write:
s
1
s − θs
V s τ s − θ1s
j ij
s s − θ1s
s
(A B )
λsj
Dij
Pj τij
Pis
=
=
1
s
−
1
Djj
Pis
V sτ s
θs
λsj
(As B s )− θs jP sjj
.
j
The log version of this expression can be estimated for each sector individually to obtain
Note that if we had only one sector, we would have
1
θs
1
.
θs
= θEK where θEK represents θ in EK.
The log version of this expression can be written as:
log
s
Dij
s
Djj
1
= − s log
θ
Pjs τijs
Pis
,
(3.1)
maxx {rij (xs )}
,
Hs
P
s
s
[rij (x )] /H
(3.2)
and similar to EK and SW, we use
log
Pjs τijs
Pis
=
j=1
where rij (xs ) = log psi (xs ) − log psj (xs ), maxx means the highest value across goods, and H s
is the number of goods in sector s of which prices are observed in the data.
Using (3.1), for each sector s, individually, we employ an SMM estimation as follows
(of which technical details are provided in the Appendix):
1. Estimate θs using trade and price data in (3.1) and (3.2) by the method of moments
s
(MM) estimator as in EK. Call this θEK
.
2. Estimate gravity equation given by (3.3) and (3.4). Since the data include zero-trade
14
observations, we use poisson pseudo maximum likelihood (PPML) estimation as advocated in Silva and Tenreyro (2006). The gravity equation estimates provide measures
of sector-source-destination trade costs subject to the determination of θs .
s
3. For a given θs , say, θG
, use source dummies in the gravity equation to estimate source
marginal costs.
4. Use the trade cost and marginal cost estimates to compute the set of all possible
destination prices and select the minimum price for each destination. These prices are
the simulated prices. We allow for 50, 000 goods in each sector; we randomly draw
goods prices from each of these pools.9
5. Using the simulated prices, calculate the model-implied trade shares, and call them as
the simulated trade shares.
6. Using the simulated trade shares and simulated prices, estimate θs with the MM estimator. Call the estimate θSs . Repeat this exercise 1, 000 times.
s
s
and the mean θSs . The
that minimizes the weighted distance between θEK
7. Find the θG
s
s
selected θG
is the SMM estimate of θs . Call this θSM
M.
Following Eaton, Kortum, and Kramarz (2011) and SW, we calculate standard errors
using a bootstrap technique, taking into account both sampling error and simulation error.
In particular, we proceed as follows:
1. Using the fitted values and residuals in the gravity equation of (3.3), resample residuals
with replacement and generate a new set of data using the fitted values. This is very
similar to Step 5 in SMM estimation, above.
2. For each resampling b, with the generated data set, estimate θs using MM estimator
(as in EK) together with trade and price data in (3.1). Call this θbs .
3. To account for simulation error, set a new seed to generate a new set of model-generated
s
moments; i.e., follow Steps 2-7 for SMM estimation above to estimate θb,SM
M for each
bootstrap b.
9
In SW, the total number of goods in the one-sector model is 100,000.
15
4. Repeat this exercise 25 times and compute the estimated standard error of the estimate
s
of θSM
M as follows:
# 12
25
X
1
0
s
s
s
s
s
θb,SM
θb,SM
S.E. (θSM
M − θSM M
M − θSM M
M) =
25 b=1
"
s
where θb,SM
M is a vector with the size of (25 × 1).
Sector-Level Bilateral Trade Costs: Gravity Equation Estimation
Given estimates of the trade elasticities, 1/θs , we can compute bilateral trade costs for every
sector by estimating (3.3) below. A gravity-type expression for trade at the sector-level can
be obtained using (2.9):
s
Dij
=
Diis
Vjs τijs
Vis
− θ1s
λsj
λsi
1
Let Ωsi = (Vis )− θs λsi and Tis = ln (Ωsi ). Then
ln
s
Dij
Diis
= Tjs − Tis −
1
s
ln
τ
ij
θs
(3.3)
We estimate (3.3), where, as in Waugh (2010), we specify trade costs as follows:
ln τijs = distI + brdr
+ tblkG +
|{z} + lang
| {z }
| {z }
|{z}
distance
border
language
trade block
srcs
|{z}i
+ εsij ,
(3.4)
source effect
where distI (I = 1, . . . , 6) is the effect of distance between i and j lying in the Ith interval,
brdr is the effect of i and j sharing a border, lang is the effect of i and j sharing a language,
tblkG (G = 1, 2) is the effect of i and j belonging to a free trade area G, and srci (i = 1, . . . , n)
is a source effect. The error term εsij captures trade barriers due to all other factors, and
is assumed to be orthogonal to the regressors. The errors are assumed to be normally
distributed with mean zero and variance, σε . The six distance intervals (in miles) are:
[0, 375); [375, 750); [750, 1500); [1500, 3000); [3000, 6000) and [6000, maximum]. The two
free trade areas are the European Union (EU) and the North-American Free Trade Agreement
(NAFTA). Tis is captured as the coefficient on source-country dummies for each sector s.
16
Sector-Level Technology Parameters
To infer the technology parameters of the traded goods sectors, λsi , we use the full information
from the gravity equation, as done in Waugh (2010). The technology parameter of the nontraded goods sector, AN
i , is chosen to match the GDP per capita in the data. The procedure
is outlined below:
Step 1 - Estimate trade costs, τijs , and country dummies, Tis , using the gravity equation (3.3) and (3.4).
1
Step 2 - Since Ωsi = (Vis )− θs λsi and Tis = ln (Ωsi ), compute price of sector s composite good
as
Pis = As B s
N
X
!−θs
Tjs
e τijs
− θ1s
.
(3.5)
j=1
s
Step 3 - Start with a guess of AN
i . Taking Li and Dij from the data, solve for price of
non-traded sector’s composite good, per capita expenditures, wages, and labor allocations
using (2.5), (2.11), (2.12), (2.13), and (2.14).
Step 4 - Then update AN
i using the expression for GDP per capita (data comes from Penn
World Tables (PWT))
δf,s
S+1 1 Y wi
Wi =
,
H s=1 Pis
where
H=
S+1
Y
δ f,s
−δf,s
.
s=1
If distance between starting guess and updated values of AN
i is smaller then a threshold then
move to the next step, else replace the guess in step 3 by the updated values and repeat
steps 3 and 4.
Step 5 - Lastly, the productivity parameter for sector s in country i, λsi , is obtained using
the relationship
1
s
Ωsi = (Vis )− θs λsi = eTi ,
where Vis is the factor cost, given by (2.2).
17
(3.6)
Data for Estimating Sectoral Trade Elasticities, Trade Costs, and Productivities
The data on prices of goods, needed for the estimation of sectoral θ’s, come from Eurostat
surveys of retail prices in the capital cities of EU countries for the year 1990. The data
set has been compiled by Crucini, Telmer, and Zachariadis (2005) and used by Giri (2010),
Inanc and Zachariadis (Forthcoming), and Yorukoglu (2000).10 We use price data for 12
countries included in the surveys - Austria, Belgium, Denmark, France, Germany, Greece,
Ireland, Italy, Netherlands, Portugal, Spain and United Kingdom. The goods maintain a
high degree of comparability across locations; typical examples of item descriptions are 500
grams of long-grained rice in carton, or racing bicycle selected brand. Hence, the level of
detail goes down to the level of the same brand sampled across locations. This enables exact
comparisons across space at a given point in time. The retail price of a good in a given
country is the average of surveyed prices across different sales points within the capital city
of that country. Furthermore, the effect of different value added tax (VAT) rates across
countries has been removed from the retail prices. Although the price data cover 1896 goods
for the year 1990, we use 1410 of these good prices that are mapped into our trade data
covering 19 ISIC sectors. The sample size of prices in each sector (H s ) is given in Table 4.
Our sample of countries includes 21 OECD countries - Australia, Austria, Belgium,
Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Mexico, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, United Kingdom, and United States.
Data on output and bilateral trade by sectors for 1990 come from the Trade, Production and Protection database of the World Bank. They provide a broad set of data covering
measures of trade, production and protection for 28 manufacturing sectors corresponding
to the 3-digit level International Standard Industrial Classification (ISIC), Revision 2. Out
of the 28 manufacturing sectors, we use data for 21 sectors, because, for the other sectors,
there are many missing observations on trade volumes. For the same reason, we also combine
sectors 313 (Beverages) and 314 (Tobacco) into one single sector and sectors 341 (Paper and
paper products) and 342 (Printing and Publishing) into another single sector. The description of the 19 sectors is provided in the appendix in Table 6.11 The data on trade barriers 10
11
The data can be downloaded from http://www.aeaweb.org/articles.php?doi=10.1257/0002828054201332.
Sectors dropped due to missing data on trade volumes are Industrial chemicals, Petroleum refineries,
18
distance, border and language - come from Centre D’Etudes Prospectives Et D’Informations
Internationales (http://www.cpeii.fr).
To compute the trade shares for a sector s - share of country j in country i’s total
expenditure on sector s goods - total exports of a country are subtracted from its gross
output. This gives each country’s home purchases for a sector (Xiis ). Adding home purchases
and total imports of a country gives the country’s total expenditure on sector s goods (Xis ).
Normalizing home purchases and imports of an importing country from its trading partners
s
by the importer’s total expenditure creates the expenditure shares - Dij
- that are used in
the gravity equation estimation.
Calibration of Other Parameters
In this section, we explain how the other parameters of the model are calibrated. β s is
computed as the, average across countries, share of value added in gross output of a sector.
The data on value added and gross output by sector come from the World Bank Trade,
Production and Protection database. γ is computed as the, average across countries, share
of value added in the gross output of non-traded goods’ sector. The non-traded sector
includes everything except manufacturing. To compute γ we use data from the OECD
STAN Structural Analysis database (STAN Industry, ISIC Rev. 2 Vol 1998 release 01). We
find γ to be 0.61.
ξ s,m , which represents the share of sector m = 1, . . . , S + 1 in the expenditure of traded
goods sectors s = 1, . . . , S on intermediate inputs is computed using the input-output table
of the United States for the year 1990. The same data and computations are also used to
get φN,m - the share of sector m = 1, . . . , S + 1 in the expenditure of non-traded goods sector
on intermediate inputs. Lastly, we use the same data to compute δ f,s as the share of sector
s = 1, . . . , S + 1 in final domestic expenditure.
η s is the elasticity of substitution between goods of a sector. Anderson and Wincoop
(2004) find that elasticity of demand for imports at sector level is between 5 and 10. However,
as in EK, this parameter does not have any implications for the results of the model. We
choose η s = 2 for all sectors. This is the highest value that ensures that the Gamma function
Miscellaneous petroleum and coal products, Non-ferrous metals, Machinery, except electrical, Professional
and scientific equipment, and Other manufactured products.
19
determining As is well defined for all sectors.
Table 1 summarizes the calibration of all the parameters of the model and Table 2
provides more detail on the sector level value added shares in production, and shares in final
domestic expenditure.
Table 1: Summary of Parameters
Parameter
s
τij
1/θs
λsi
AN
i
γ
βs
ξs, m
φN, m
δ f,s
ηs
Description
trade cost
trade elasticity of sector
Frechet parameter for traded goods sectors
productivity in non-traded goods sector
Value
Table
Table
Table
Table
value added share in gross output for non-traded good sector
value added share in gross output for traded goods sectors
expenditure on intermediates by traded goods sectors
expenditure on intermediates by non-traded goods sector
share of sector in final domestic expenditure
elasticity of substitution within traded goods sectors
0.61
Table
Table
Table
Table
2
Table 2: β s and δ f,s
ISIC Code
βs
δ f,s
311
313,314
321
322
323
324
331
332
341,342
352
355
356
361
362
369
371
381
383
384
400
0.2603
0.4911
0.3878
0.4269
0.3296
0.3925
0.3565
0.4150
0.4285
0.4421
0.4380
0.4085
0.5829
0.5003
0.4512
0.3391
0.4180
0.4176
0.3469
0.6090
0.0232
0.0232
0.0049
0.0049
0.0049
0.0049
0.0036
0.0036
0.0139
0.0135
0.0014
0.0014
0.0003
0.0003
0.0003
0.0000
0.0003
0.0171
0.0493
0.8288
20
??, Table ??
4
??
??
2
7
7
2
Source
estimated
estimated
computed
GDP per
(section 3.2)
(section 3.1)
(section 3.3)
capita (PWT) (sec-
tion 3.3)
OECD STAN data
World Bank data
US IO table 1990
US IO table 1990
US IO table 1990
EK
Calibration of One-sector Model
One set of counterfactual exercises that we perform involve an aggregated version of our
model that collapses the 19 traded goods sectors of our benchmark model into a single
traded good sector. There continues to be a non-traded good sector.12 We first estimate the
single 1/θ for this model, and then we estimate the trade costs and technology parameters
following the same procedure as in our benchmark model, except we now use aggregated
trade data we view all the micro-prices as coming from a single traded sector. With this
model, we examine the gains from trade, and compare it to the benchmark model. Table 3
shows the key parameters in this version of the model.
Table 3: Parameters for One Sector Model
1/θ
β
δ
γ
2.37
0.37
0.17
0.61
PS f,s
Note: δ =
= 1 − δ f,N . β is computed as average across countries of the ratio of sum of value added of all traded goods sectors to the sum of gross output o
1 δ
Results
We first present our estimated trade elasticities and then turn to the welfare gains from our
benchmark model, as well as from our two sets of exercises.
Estimated Trade Elasticities
Table 4 presents the results of the estimation of trade elasticites - 1/θs . The column labeled
”SMM-PPML” shows the estimates coming from our application of the SW SMM methodology with PPML estimation of the gravity equation.13 Standard errors for each estimate
are in parentheses. The column labeled “EK” provides the estimates resulting from the EK
estimator. The last column shows the number of goods in each sector (after mapping the
individual goods into the 3 digit ISIC sector categories.
For each sector, our SMM estimate of the trade elasticity is smaller than the estimate
obtained by the EK methodology. This extends the results found by SW to the sector level.
In some sense, the small sample issue identified by SW is a bigger problem at the sector level.
12
We continue to have an input-output structure, but now it is just a 2 x 2 matrix.
We also estimated the elasticity by OLS. The estimates are very similar to the PPML estimates. This
suggests that the issue of zeros in trade-flow data is not significant with our data.
13
21
The range of elasticity estimates is still large with the highest estimated elasticity, 8.94, about
three times larger than the lowest estimated elasticity, 2.97. It is worth pointing out that
our lowest estimated elasticity is considerably larger than the lowest estimated elasticity in
Caliendo and Parro (2015), 0.39, and in Ossa (2015), 0.54. This will be important for our
interpretation of our welfare results vis-a-vis these other papers later.
For our benchmark model we use the estimates of 1/θs from the SMM-PPML column,
and the corresponding estimates of trade costs and productivity parameters.
Table 4: Estimates of Trade Elasticities by Sector (1/θs )
ISIC Code
311
313,314
321
322
323
324
331
332
341,342
352
355
356
361
362
369
371
381
383
384
Sector Description
Food products
Beverages and Tobacco
Textiles
Wearing apparel, except footwear
Leather products
Footwear, except rubber or plast
Wood products, except furniture
Furniture, except metal
Paper and products and printing and publishing
Other chemicals
Rubber products
Plastic products
Pottery, china, earthenware
Glass and products
Other non-metallic mineral products
Iron and steel
Fabricated metal products
Machinery, electric
Transport equipment
Minimum
Maximum
Average
Median
SMM-PPML
EK
3.57 (0.31)
3.57 (0.22)
3.27 (0.2)
4.41 (0.33)
5.28 (0.3)
4.77 (0.49)
4.17 (1.26)
4.47 (0.31)
2.97 (0.18)
3.75 (0.22)
4.38 (0.54)
3.87 (1.41)
5.94 (0.59)
5.61 (0.67)
3.87 (0.47)
8.94 (1.55)
5.07 (1.42)
3.27 (0.21)
4.47 (0.8)
4.28
5.36
5.21
5.17
8.14
10.29
15.45
15.37
6.57
11.93
8.02
16.00
19.79
19.08
14.10
35.55
18.50
4.26
6.50
Sample Size
of Prices
343
93
36
143
20
20
8
5
14
4
14
8
14
6
7
16
11
416
232
2.97
8.94
4.51
4.38
4.26
35.55
12.08
10.29
4.00
416.00
74.21
14.00
Welfare Gains of Benchmark Multi-Sector Model
We now study the gains from trade in our benchmark model. We compute the gains
from trade in the same way as done recently in ACR, Levchenko and Zhang (2014), Costinot
and Rodriguez-Clare (2014), Ossa (2015), and other papers – we compute the welfare loss
in going from the 1990 level of openness to autarky. In practice, we implement autarky by
raising trade costs to 100 times their level in the benchmark model. Our measure of welfare
22
is the real-wage (or per capita GDP) implied by the model, which is the appropriate measure
for our framework. Specifically, it is ln Wia /Wib × 100, where Wia denotes the autarky value
and Wib denotes the baseline value. Pi is the price of the final consumption good, given by
(2.7).
The changes in welfare in moving from the baseline sector-level home trade shares to
autarky are given in column (1) in Table 5 and in Figure 1 below for each country. Hereafter,
we will refer to the welfare change as the gain in going from autarky to the baseline home
trade share. The welfare gain ranges from 0.38% for Japan to and 9.59% for Netherlands with
a median (mean) gain of 4.25% (4.60%). The gains for small countries are about an order
of magnitude larger than for the largest countries. These gains are of the same magnitude,
roughly, as those found in EK, for example.
Figure 1: Welfare Gains from Trade (relative to Autarky)
Figure 2 illustrates the contribution of each sector to the welfare gain of each country.
In most countries, only a few sectors account for most of the welfare gains from trade. For
example, in Japan, sectors 383 (machinery, electric) and 384 (transport equipment) account
for almost 90% of the gains. These two sectors, along with sectors 341 (paper products and
publishing) and 311 (food products) account for the majority of the gains from trade in most
23
countries. That said, the figure shows there is considerable heterogeneity across countries in
the importance of particular sectors to the gains from trade. Thus, comparative advantage
at the sector-level is an important factor determining the welfare gains from trade.
Figure 2: Contribution of Sectors to Welfare Gains - Benchmark Case (sector-specific θ’s)
We also decompose our results into a the direct trade owing to changes in home expenditure shares Dii and an intersectoral linkage effect owing to changes in intersectoral prices,
as discussed above. From equation (2.17), the direct trade effect is captured by the first term
in brackets, and the intersectoral linkage effect is captured by the second term. Figure 3
shows the results. For every country, the direct trade effect in going from the actual equilibrium to autarky is negative, while the intersectoral price effect is positive, and the absolute
magnitude of the direct effect is larger; hence, the overall effect is negative. Why does the
24
intersectoral price effect work in the opposite direction from the direct trade effect? Consider
the movement towards autarky; here, the direct trade effect is negative as domestic expenditures shares become 1. The intersectoral linkage effect is driven by changes in the prices
of the sectoral inputs relative to the output price of a sectoral good. Owing to the raising of
trade costs to autarky levels, the traded sectoral prices rise, some more than others. However, the non-traded sectoral good is different for two reasons. First, the importance of the
wage (the value-added share γ) in the non-traded good is considerably higher than in trade
goods, and second, the non-traded good is the dominant input in traded goods (as captured
by a large ξ s,N coefficient). Because the real wage decreases in going from our baseline scenario to autarky, nominal wages must rise by less than the appropriate geometric-weighted
average of prices. Hence, the non-traded good price will rise by less than the traded goods
prices, which implies that, on average, input prices relative to the sectoral output price fall
when the economy moves to autarky. Thus, the inter-sectoral linkage effect will be welfare
improving.
Figure 3: Role of Intersectoral Linkages
20%
Intersectoral Linkage Effect
Trade Effect
Change in Real GDP Per Capita
15%
10%
5%
0%
‐5%
‐10%
25
Role of Sectoral Heterogeneity
Our framework has five types of sectoral heterogeneity – trade elasticities, input-output
structure, value-added shares, trade costs, and productivities – and they all influence the
gains from trade, as captured by (2.17). Two assess the importance of this heterogeneity
in the gains from trade calculations, we conduct two sets of counterfactuals. The first set
of counterfactuals addresses the importance of heterogeneity in the trade elasticities, valueadded shares, and input-output structure. Here, we start from the benchmark model, and
then alter particular parameters. For example, to assess the importance of heterogeneity
in the sectoral trade elasticities 1/θs , we replace the sector level 1/θs with a single aggregate θ. We do not change any other parameters. We then raise trade costs to autarky
levels and compute the gains from trade.14 The second set of counterfactuals compares our
benchmark model to the calibrated one-sector model. This provides a comprehensive look
at the importance of sectoral heterogeneity and is similar to the exercise in Ossa (2015).
However, this approach does not reveal the intricacies of the importance of particular forms
of heterogeneity.
Replacing Sector-specific Parameters with Single Parameter in Benchmark Model
We start by examining the role of heterogeneity in the elasticity of trade 1/θs , because
this parameter has received the most attention in the existing research, and according to
frameworks like those in ACR, it is the single most important parameter. Figure 4 compares
the welfare gains country-by-country in the benchmark model (see column (1) in Table 5)
against those implied by the same model but with the sector-specific 1/θ’s replaced by a single
theta. The figure gives three cases, the median of our sectoral theta estimates, 1/θ = 4.37,
the estimate from our one-sector model, 1/θ = 2.37, and one of the main estimates from EK,
1/θ = 8.28. The countries in the figure are ordered left to right in descending order of gains
in the benchmark model.
The figure shows that when the sector-specific 1/θ’s are replaced by the median of
14
Note that this exercise will alter both the baseline level of welfare, as well as the autarky level of welfare.
Because of this, we focus on the difference between the baseline and autarky levels of welfare. By focusing
on the difference, we are, in some sense, differencing out the bias resulting from replacing some parameters
in a model, but not re-estimating and re-calibrating the entire model.
26
the sector-specific 1/θ’s, the welfare gains in the median case are slightly larger. On the
other hand, if we replace the sector-specific 1/θ’s with the actual estimate of 1/θ from our
one-sector model (2.37), then, we can see that the gains from trade with the lower 1/θ are
several times larger than in our benchmark model. The importance of the value of 1/θ that
is used in this counterfactual can also be seen when we compute the welfare gains from using
the estimated 1/θ from EK: 8.28. In this case, the gains from trade are only about one-half
to two-thirds as large as in the benchmark model. Hence, it is clear that the value of 1/θ
that is used in the comparison matters.
We now show the effects of heterogeneity in sector-level value-added shares of gross
output in tradable sectors β’s. The benchmark welfare gains illustrated in Figure 1 and
shown in column (1) in Table 5 involves sector-specific β’s. Column (5) in Table 5 depicts the
welfare gains for a common value of β = 0.37 across sectors. Comparing these two columns
– also illustrated in Figure 5 – shows clearly that welfare gains in the common β case are
roughly 1.5 times larger. Therefore, the sectoral heterogeneity through sector-specific β’s
results in lower welfare gains, all else equal. The reason is that in the benchmark model,
those sectors with low value-added shares are the sectors with either high trade elasticities
or small increases in trade.
Finally, we show the effects of heterogeneity in the input-output structure across sectors. We impose a common input-output structure (common φ’s and ξ’s) for all sectors, and
then compute the welfare gains relative to autarky. The results can be seen by comparing the
benchmark case in column (1) with column (6) in Table 5, and also in Figure 5. The table
and figure show that the welfare gains from a common input-output structure are slightly
lower than in the benchmark case. Hence, heterogeneity in sectoral input-output structures
does deliver slightly larger gains from trade.
Our main conclusion from the three exercises here is that the effects of sectoral heterogeneity, operating through sector-specific parameters, can raise or lower welfare, depending
on the particular form of sectoral heterogeneity.
27
Figure 4: Role of Sectoral Heterogeneity in Trade Elasticity (1/θ) in Benchmark Model
IRL NLD DNK GRC BLX NOR SWE PRT AUT NZL FIN CAN MEX GBR AUS ITA FRA ESP DEU USA JPN
30%
Change in Real GDP Per Capita
25%
20%
15%
10%
5%
0%
Benchmark (Sector‐Specific 1/θ's)
1/θ = 8.28
1/θ = 4.37
1/θ = 2.37
Figure 5: Role of Value-Added, and Input-Output Structure, Heterogeneity in Benchmark Model
IRL NLD DNK GRC BLX NOR SWE PRT AUT NZL FIN CAN MEX GBR AUS ITA FRA ESP DEU USA JPN
16%
Change in Real GDP Per Capita
14%
Benchmark (Multi‐Sector Model)
Multi‐Sector Model, Same β Multi‐Sector Model, Common IO across Sectors
12%
10%
8%
6%
4%
2%
0%
28
Comparing Benchmark Model to One-Sector Model
We now turn to our second set of counterfactuals. We calibrate and estimate a one-sector
version of the model. Hence, all five sources of heterogeneity mentioned at the beginning of
this section are eliminated. The main result from this calibration and estimation is that the
estimated elasticity of trade is 2.37. This is a low value – considerably lower than the median
elasticity from the benchmark model. Consequently, as Figure 6 shows, the gains from trade
in the one-sector model are typically slightly larger than in the benchmark model. This is
perhaps our most surprising, and definitive, result.
Figure 6: Role of Sectoral Heterogeneity
IRL NLD DNK GRC BLX NOR SWE PRT AUT NZL FIN CAN MEX GBR AUS ITA FRA ESP DEU USA JPN
12%
Change in Real GDP Per Capita
10%
8%
6%
4%
2%
0%
Benchmark
One Sector Model, 1/θ=2.37
One‐Sector Model, 1/θ=4.37
Discussion
How do we reconcile our results from those of Levchenko and Zhang (2014), Costinot and
Rodriguez-Clare (2014), and Ossa (2015), all of which find that more sectoral heterogeneity is
associated with higher gains from trade? Levchenko and Zhang (2014) focus on heterogeneity
arising from differences in comparative advantage across sectors. They do not study the
effects of heterogeneity in within sector comparative advantage, as captured by sector-specific
29
1/θ’s. In addition, their main exercise is somewhat different from ours. They calibrate a
model with a great deal of sectoral heterogeneity (other than the sector-specific 1θ’s), and
they compute the welfare gains from their calibrated model, and then they evaluate several
one-sector and multi-sector formulas for the gains from trade that originate from simpler
models. They find that the multi-sector formulas that allow for the most heterogeneity, and
adjust for the importance of the non-traded sector, come the closest to matching the welfare
gains from the calibrated model.
Ossa (2015) and Costinot and Rodriguez-Clare (2014) conduct exercises in which, like
our study, there is more focus on heterogeneity in the sectoral trade elasticities. The key
difference between their sectoral elasticities and ours is that their lowest sectoral trade elasticity is considerably lower than ours, as mentioned above. Moreover, in our one-sector
model, as we highlighted above, we use an estimated elasticity that happens to be low. By
contrast, Ossa (2015) and Costinot and Rodriguez-Clare (2014) use a median-type elasticity
from the sector-specific estimates as their one-sector elasticity. The bottom line is that the
sector-specific elasticities in their calculations have some very low elasticities, while their
one-sector elasticity is quite large, compared to our sector-specific elasticities and one sector elasticity, respectively. We conjecture that these two reasons explain at least partially
the reason why their exercises implies greater welfare gains from the model with sectoral
heterogeneity relative to a model without it.
Finally, there is at least one element of consistency between Ossa (2015) and us. As
discussed above, Ossa’s exercise involves comparing his benchmark model to a one-sector
model in which the one-sector elasticity is the trade-weighted average of the benchmark
sector-specific elasticities. We do a simiilar comparison, as well. We compare our benchmark
model with our one-sector model, but instead of using our estimated elasticity (2.37), we
use the median elasticity from the benchmark model (4.37). Figure 6 shows the results.
The gains from our benchmark model are roughly two times larger than from this version
of the one-sector model, which is similar to Ossa’s result of a three times larger gain. Our
conclusion, then, is that part of the answer to the question on the importance of sectoral
heterogeneity for gains from trade calculations is that it depends on exactly what is being
compared.
30
Table 5: Welfare Gains from Trade
Multi-Sector Model
(S = 19 Traded Goods Sectors, 1 Non-traded Goods Sector)
Parameterization
θ
β
γ
δ
I-O
One Sector Model
(1 Traded Goods Sector (T ), 1 Non-traded Goods Sector (N ))
(1)
Sectoral
Sectoral
0.61
Sectoral
20 × 20
-
(2)
8.28
Sectoral
0.61
Sectoral
20 × 20
-
(3)
4.17
Sectoral
0.61
Sectoral
20 × 20
-
(4)
2.37
Sectoral
0.61
Sectoral
20 × 20
-
(5)
Sectoral
0.37
0.61
Sectoral
20 × 20
-
(6)
Sectoral
Sectoral
0.61
Sectoral
20 × 20
rows same
(7)
2.37
0.37
0.61
T, N
2×2
-
(8)
8.28
0.37
0.61
T, N
2×2
-
(9)
4.17
0.37
0.61
T, N
2×2
-
(10)
4.17
0.37
0.61
T, N
2×2
diagonal
(11)
8.28
0.37
0.61
T, N
2×2
diagonal
(12)
2.37
0.61
T, N
2×2
diagonal
Countries
AUS
AUT
BLX
CAN
DEU
DNK
ESP
FIN
FRA
GBR
GRC
IRL
ITA
JPN
MEX
NLD
NOR
NZL
PRT
SWE
USA
2.73%
5.52%
7.05%
3.09%
1.82%
8.40%
2.01%
4.25%
2.44%
2.85%
7.23%
9.59%
2.44%
0.38%
3.02%
8.87%
7.01%
5.33%
5.71%
5.81%
1.00%
2.37%
2.77%
4.56%
1.67%
1.11%
4.62%
1.03%
2.32%
1.31%
1.55%
4.99%
5.17%
1.19%
0.40%
2.21%
4.57%
4.52%
3.58%
3.18%
3.12%
0.69%
5.59%
5.68%
7.51%
3.23%
1.86%
9.27%
2.15%
4.09%
2.38%
3.22%
11.29%
9.84%
2.32%
0.53%
5.13%
8.99%
9.85%
7.52%
6.33%
5.62%
1.58%
14.39%
11.86%
11.74%
6.32%
2.97%
18.88%
4.66%
7.10%
4.28%
6.94%
27.65%
18.58%
4.63%
0.61%
12.93%
17.90%
22.31%
16.57%
13.34%
9.94%
3.85%
4.37%
8.09%
10.09%
4.77%
2.72%
13.54%
3.04%
6.78%
3.64%
4.46%
11.09%
14.44%
3.64%
0.54%
5.37%
13.27%
10.98%
8.60%
8.28%
9.04%
1.43%
2.17%
4.68%
6.67%
2.97%
1.85%
7.20%
1.79%
3.65%
2.32%
2.56%
6.15%
8.30%
2.15%
0.36%
2.48%
7.70%
5.70%
4.45%
4.71%
5.15%
0.89%
3.07%
6.71%
2.07%
4.97%
2.81%
9.35%
3.15%
4.66%
3.81%
4.57%
6.97%
9.32%
3.75%
0.47%
3.49%
11.40%
7.81%
4.54%
6.35%
6.89%
1.58%
0.60%
1.71%
1.21%
1.51%
0.94%
2.52%
0.73%
1.49%
1.06%
1.08%
0.96%
2.83%
0.93%
0.21%
0.60%
2.82%
1.70%
1.23%
1.42%
2.14%
0.38%
1.31%
3.38%
1.72%
2.78%
1.65%
4.86%
1.50%
2.68%
2.00%
2.19%
2.42%
5.21%
1.86%
0.34%
1.39%
5.62%
3.57%
2.36%
2.95%
3.90%
0.77%
1.96%
4.19%
1.11%
2.80%
1.55%
5.71%
1.88%
2.67%
2.25%
2.65%
4.47%
5.60%
2.32%
0.26%
1.86%
7.36%
5.13%
2.70%
3.67%
4.16%
1.04%
0.79%
2.03%
0.93%
1.52%
0.89%
2.82%
0.86%
1.49%
1.15%
1.22%
1.48%
3.00%
1.10%
0.18%
0.71%
3.45%
2.25%
1.31%
1.61%
2.24%
0.48%
5.67%
9.11%
1.02%
5.15%
2.82%
11.97%
4.49%
4.74%
4.80%
6.31%
15.90%
10.62%
5.26%
0.34%
5.73%
16.97%
12.64%
6.07%
9.29%
7.84%
2.47%
Average
Median
Max
Min
3.46%
2.95%
0.52%
7.28%
2.25%
2.04%
0.36%
4.42%
4.35%
4.00%
0.60%
7.95%
7.76%
7.55%
0.89%
15.72%
5.33%
4.75%
0.75%
10.82%
3.04%
2.54%
0.45%
6.54%
4.12%
3.64%
0.10%
13.02%
1.32%
0.87%
0.02%
4.82%
2.44%
1.87%
0.05%
8.45%
2.57%
2.00%
0.05%
9.19%
1.36%
0.99%
0.02%
5.29%
4.77%
5.32%
0.12%
14.61%
s
Benchmark model trade costs are increased by a factor of 100 to achieve numerical approximation of autarky (Dii
< 1e − 6 ∀ i, s). For sectoral values of β and δ refer to
Table 2 . For sectoral values of θ refer to the SMM-PPML column of Table 4. δ N , of the one sector model, is taken to be equal to δ f,S+1 of the multi-sector model. δ T , of
the one sector model is simply equal to (1 − δ N ).
Conclusion
To be added
References
Alvarez, F., and R. E. Lucas (2007): “General Equilibrium Analysis of the Eaton-Kortum Model of
International Trade,” Journal of Monetary Economics, 54(6), 1726–1768.
Anderson, J. E., and E. V. Wincoop (2004): “Trade Costs,” Journal of Economic Literature, 42, 691–
751.
Caliendo, L., and F. Parro (2015): “Estimates of the Trade and Welfare Effects of NAFTA,” Review of
Economic Studies, 82(1), 1–44.
Crucini, M. J., C. I. Telmer, and M. Zachariadis (2005): “Understanding European Real Exchange
Rates,” American Economic Review, 95(3), 724–738.
Eaton, J., S. S. Kortum, and F. Kramarz (2011): “An Anatomy of International Trade: Evidence from
French Firms,” Econometrica, 79(5), 1453–1498.
Giri, R. (2010): “Local Costs of Distribution, International Trade Costs and Micro Evidence on the Law of
One Price,” Discussion Series Papers, Centro De Investigacion Economica, Instituto Tecnologico Autonomo
de Mexico.
Inanc, O., and M. Zachariadis (Forthcoming): “The Importance of Trade Costs in Deviations from the
Law of One Price: Estimates based on the Direction of Trade,” Economic Inquiry.
Jones, C. I. (2011): “Intermediate Goods and Weak Links in the Theory of Economic Development,”
American Economic Journal: Macroeconomics, 3(2), 1–28.
Levchenko, A. A., and J. Zhang (2014): “Ricardian Productivity Differences and the Gains from Trade,”
European Economic Review, 65(1), 45–65.
Silva, S. J. M. C., and S. Tenreyro (2006): “The Log of Gravity,” The Review of Economics and
Statistics, 88(4).
Simonovska, I., and M. E. Waugh (2014a): “The Elasticity of Trade: Estimates and Evidence,” Journal
of International Economics, 92, 34–50.
(2014b): “Trade Models, Trade Elasticities, and the Gains from Trade,” Working Paper.
Waugh, M. E. (2010): “International Trade and Income Differences,” American Economic Review, 100(5),
2093–2124.
Yorukoglu, M. (2000): “Product vs. Process Innovations and Economic Fluctuations,” Carnegie-Rochester
Conference Series on Public Policy, 52, 137–163.
Appendix
Appendix A: Price of Sector s Intermediate Composite
The price of sector s composite is given by
Pis
1
Z
psi
=
s 1−η s
(x )
dx
s
1
1−η
s
.
0
Let
(
R = psi (xs )
1
θs
∼ exp (B s )
−1
θs
N
X
)
ψijs
j=1
and
M = psi (xs )1−η
s
.
Then, R can be written as a function of M :
1
R = R (M ) = M θs (1−ηs ) .
Using
f (M ) = f (R (M )) × |R0 (M )|
we can write
(
s
f (M ) = exp − (B )
−1
θs
N
X
ψijs M
1
θ s (1−η s )
j=1
Define
u = (B s )
N
X
−1
θs
!
ψijs
)
1
M θs (1−ηs ) −1
.
θs (1 − η s )
1
M θs (1−ηs ) ,
j=1
and hence
du = (B s )
−1
θs
N
X
!
ψijs
j=1
1
M θs (1−ηs ) −1
dM .
θs (1 − η s )
Since
s
(Pis )1−η
Z
∞
=
0
Z
∞
M f (M ) dM
(
s
M exp − (B )
=
0
−1
θs
N
X
!
ψijs
M
1
θ s (1−η s )
)
j=1
Z
=
∞
θs (1−η s )
u
0
(
N
−1 X
s θs
exp (−u) du
(B )
ψijs
j=1
1
M θs (1−ηs ) −1
dM
θs (1 − η s )
)−θs (1−ηs )−1
,
it gives us
Pis = As B s
N
X
!−θs
ψijs
= As B s
j=1
N
X
−1
s θs
Vjs τij
!−θs
λsj
,
(6.1)
j=1
1
1−η
s
s
s s
e
where A = A (θ , η )
and
s
es (θs , η s ) =
A
∞
Z

0


s
s
uθ (1−η ) exp (−u) du

is a Gamma function.
Appendix B: Methodology for Estimating Sector-Level θ’s and τij ’s
Recall that
1
s − θs
s
Dij
= (As B )
Vjs τijs
Pis
−1
θs
λsj
which implies that
1
s − θs
s
V s τ s −1
θs
j ij
s s −1
s
(A B )
λsj
Dij
Pj τij θs
Pis
=
=
.
V s τ s −1
s
Djj
Pis
θs
− θ1s
j jj
s
s
s
(A B )
λj
Ps
j
This corresponds to equation (12) in EK. The log version of this expression can be estimated for each sector
individually to obtain
1
’s
θs
that correspond to θ in EK (i.e., if we had only one sector, we would have
1
θs
= θEK
where θEK represents θ in EK). The log version can be written as
log
s
Dij
s
Djj
1
= − s log
θ
Pjs τijs
Pis
,
(6.2)
and similar to EK and SW, we use
log
Pjs τijs
Pis
max {rij (xs )}
,
= Hs
P
s
s
[rij (x )] /H
j=1
where rij (xs ) = log psi (xs ) − log psj (xs ), maxx means the highest value across goods, and H s is the number of
goods in sector s of which prices are observed in the data. This corresponds to equation (13) in EK.
Using (3.1), we employ two methods to estimate sector-level θ’s: (i) method-of-moments (MM) estimator
used by EK; (ii) simulated-method-of-moments (SMM) estimator used by SW. While the former is the mean
of the left-hand-side variable over the mean of the right-hand-side variable in (3.1), the latter is much more
detailed. The SMM estimator can be obtained as follows for each sector s:
s
1. Estimate θs using MM estimator (as in EK) together with trade and price data in (3.1). Call this θEK
.
• Note: this is done for only the EU countries for which we have price data.
2. Estimate gravity equation using the specification employed in SW:
1
s
Dij
Diis
(As B s )− θs
V s τ s −1
θs
j
=
1
(As B s )− θs
ij
Pis
s
Vis τii
Pis
−1
θs
λsj
=
λsi
Vjs τijs
Vis
−1
θs
λsj
λsi
Hence, the log version can be written as:
ln
s
Dij
Diis
= ln
Vjs
−1
θs
−1
1
λsj − ln (Vis ) θs λsi − s ln τijs
θ
and can be estimated with fixed effects as follows:
ln
s
Dij
Diis
= Tjs − Tis −
1
s
ln
τ
,
ij
θs
(6.3)
where
Tjs = ln
Vjs
−1
θs
λsj
and
−1
Tis = ln (Vis ) θs λsi
and
ln τijs = distI + brdr
+ tblkG +
|{z} + lang
| {z }
| {z }
|{z}
distance
border
language
trade block
srcs
|{z}i
+ εsij .
source effect
Since there are zero-trade observations in trade data, we use poisson pseudo maximum likelihood
(PPML) estimation as advocated in Silva-Tenreyo (2006).
3. SW show that the inverse of the marginal cost of production (multiplied by B s ) in sector s of country
i, which is
usi =
1
s
zis (xs )θ Vis
is distributed according to:
Mis
where
Tis
= ln
−1
(Vis ) θs
λsi
(usi )
= exp
s
− (exp (Tis )) (usi )−1/θG
is the country-fixed effect estimated above.
• This can also be done by using the “inverse transform method”. The idea is that
probability draws from the F rechet (exp (Tis ) , 1/θs ) can be transformed into random
draws from a standard uniform distribution. If u has standard uniform distribution
then the inverse of the marginal cost is given by
log(u)
− exp (Tis )
−θs
.
We adopt this method in the code. This is in line with SW.
s
4. Therefore, for a given θ, say, θG
, we can use source dummies (Tis ’s) in the gravity equation to estimate
source marginal costs.
5. Using trade cost i.e., τijs and the inverse of the marginal cost of production multiplied by B s (i.e., usi ),
we can figure out “possible” destination prices and select the minimum price for each destination:
n
o
s
psi (xs ) = B s min Vjs zjs (xs )θ τijs
j
These are the simulated equilibrium prices. We allow for 50, 000 possible total goods in each sector and
think ourselves as randomly drawing good prices from these pools.
6. Given the simulated equilibrium prices of psi (xs )’s, the price Pis of the sector-level composite index Cis
can be simulated as follows:
Pis
Z
1
=
psi
s 1−η s
(x )
dx
s
1
1−η
s
,
0
where we use η s = 2 following SW. The expenditure of country i on good x imported from country j is
simply given by:
psi (xs )qis (xs )
=
psi (xs )
Pis
1−ηs
Xis ,
where Xis is the total expenditure by country i on sector s goods, i.e., Xis = Pis Cis . Adding this
expenditure across all goods imported by i from j, and then dividing both sides by Xis gives us the
simulated trade share:
Xijs
s
c
Dij = s =
Xi
Z
Ωij
psi (xs )
Pis
1−ηs
dxs ,
where Ωij is the set of goods imported by country i from country j, and we use η s = 2 as in SW.
cs /D
cs . We
7. We calculate the trade shares normalized by importing country’s own trade share, i.e., D
ij
ii
take the logarithm of these normalized trade shares and add the residuals from the gravity equation
(with replacement in each simulation). This, thus, gives us the log normalized trade shares with errors.
s
s
Denote these by log Dij
/Diis . Take the exponential of this to get Dij
/Diis . These are the normalized
simulated equilibrium trade shares. Finally, we unwind these normalized trade shares into levels to get
s
Dij
. To do that, we use that fact that for an importing country i, the sum of its trade shares across
all suppliers j = i, . . . , N is one, i.e.,
PN
j=1
Dij = 1. So, it is implied that
PN
j=1
s
/Diis = 1/Diis .
Dij
s
Accordingly, we simply divide the normalized trade shares Dij
/Diis by 1/Diis , and that gives us the level
s
trade share Dij
which, importantly, incorporates the residuals from the gravity equation.
8. Using simulated trade (incorporating the residuals) and simulated prices, estimate θs using MM estimator (as in EK) according to:
s Dij
log
Ds
| {z jj }
Simulated Trade Data
s s
Pj τij
1
= − s log
θS
Ps
{z i }
|
Simulated Price Data
which we call as θSs . We repeat this exercise for 1, 000 times.15
• This is done for only the EU countries for which we have price data.
s
s
and
, that minimizes the weighted distance between θEK
9. Within 1, 000 simulated θSs ’s, we search for θG
the average θSs :
"
1000
s
θEK
s
θSM
M = arg min
s
θG
1 X s
−
θ
1000 s=1 S
!
1000
s
W θEK
1 X s
−
θ
1000 s=1 S
!#
where W is the continuously updated weighting matrix defined as:
1000
1 X s
θ
W=
1000 s=1 S
"
1000
s
θEK
1 X s
−
θ
1000 s=1 S
!
1000
s
θEK
1 X s
−
θ
1000 s=1 S
!#
We also used an alternative W definitions such as (i) the one used by EKK based on bootstrapping, (ii)
an alternative version of W above which is
 1000
1000
1000
P s
P
P s
1
1
1
s
s
1000
θEK − 1000
θS − 1000
θEK − 1000
θS
1 X s
s=1
s=1 s=1

WA =
θ
1000
1000
1000
P s
P
P s
1000 s=1 S 
1
1
1
s
s
×
θEK − 1000
θS − 1000
θEK − 1000
θS
s=1
s=1




s=1
and (iii) the identity matrix; however, the results were very close to each other. Currently, we are using
s
s
the benchmark W defined above. The selected θG
is the SMM estimate of θs , which we denote by θSM
M.
15
Note that according to
log
s
Pjs τij
s
Pi
=
max {rij (xs )}
,
H
Ps
s
s
[rij (x )] /H
j=1
n
o
s s
Pj τij
θs s
where B s ’s as in psi (xs ) = B s min Vjs zjs (xs ) τij
cancel each other out while calculating log P
; therefore, we don’t
s
j
need to know B s ’s while estimating θs ’s.
i
Following Eaton, Kortum, and Kramarz (2011) and SW, we calculate standard errors using a bootstrap
technique, taking into account both sampling error and simulation error. In particular, we proceed as follows:
1. Using the fitted values and residuals in the gravity equation of (3.3), resample residuals with replacement
and generate a new set of data using the fitted values. This is very similar to Step 5 in SMM estimation,
above.
2. For each resampling b, with the generated data set, estimate θs using MM estimator (as in EK) together
with trade and price data in (3.1). Call this θbs .
3. To account for simulation error, set a new seed to generate a new set of model-generated moments; i.e.,
s
follow Steps 2-7 for SMM estimation above to estimate θb,SM
M for each bootstrap b.
s
4. Repeat this exercise 25 times and compute the estimated standard error of the estimate of θSM
M as
follows:
# 12
25
X
1
0
s
s
s
s
S.E. (θSM
θs
− θSM
θb,SM
M) =
M
M − θSM M
25 b=1 b,SM M
"
s
where θb,SM
M is a vector with the size of (25 × 1).
One Sector Eaton-Kortum Model
Here we present the version of our model with one traded goods sector (T ). There is continuum of tradable
goods xi ∈ [0, 1] in the traded goods sector of country i, and each good is produced by combining labor and
intermediate inputs through a Cobb-Douglas production technology.
1−β

Y
qi (xi ) = zi (x)−θ [li (x)]β 
Cim (x)ξ
T,m

m={T,N }
The traded goods sector’s composite good is given by
CiT =
Z
q i (x)
where
η−1
η
N
Y
f (x) = f (z) =
i=1
η
η−1
f (x) dx
!
λi
exp −
,
N
X
!
λi zi
i=1
Given that the individual goods can be bought from domestic or foreign producers, the price of good x in
country i is
n
o
θ
pi (x) = min BVj zj (xj ) τij
j
(6.4)
where Vi , the unit cost of production, is given by
1−β

ξ T,m
Y
Vi = [wi ]β 
(Pim )
,

(6.5)
m={T,N }
and
(1−β)

Y
B = (β)−β (1 − β)−(1−β) 
ξ T,m
−ξT,m
.

m={T,N }
Following the steps as outlined in the multi-sector model, the expression for price of traded goods sector
composite is given by
PiT =
1
Z
1−η
pi (x)
1
1−η
dx
0
N
X
= AB
!−θ
(6.6)
ψij
j=1
where
ψij = (Vj τij )
and A =
∞
R
−1
θ
λj ,
(6.7)
1
1−η
uθ(1−η) exp (−u) du
is a Gamma function.
0
The non-traded sector’s homogenous good is produced as follows:

N γ
CiN = AN
li 
i
Y
CiN,m
φN,m
1−γ
,

m={T,N }
The price of the non-traded good is
PiN = E
[wi ]γ
hQ
m φ
m={T,N } (Pi )
N,m
i1−γ
,
AN
i
where
(1−γ)

E = γ −γ (1 − γ)−(1−γ) 
(6.8)
Y
φN,m
−φN,m

m={T,N }
The utility of the representative household in each country is given by
Ui = Yi
.
where Yi , the final consumption good, is a Cobb-Douglas aggregator of the sectoral composite goods.
Y
Yi =
Cif,m
δf,m
,
m={T,N }
The price of the final good, therefore, is given by
Y
Pi =
δ f,m
−δf,m
(Pim )δ
f,m
.
(6.9)
m={T,N }
The market clearing conditions are as follows:
Z
1
0
|
Z
|0
lis (xs ) dxs +liN ≤ 1 , i = 1, . . . , N ,
{z
}
liT
1
CiT,s (xs ) dxs +CiN,s + Cif,s ≤ Cis , i = 1, . . . , N , s = {T, N } .
{z
}
CiT,s
Finally, the share of country j in country i’s total expenditure on traded goods is:
XijT
ψij
.
Dij = T = πij = PN
Xi
n=1 ψin
Using (6.6), we can rewrite this expression as:
− θ1
Dij = (AB)
Vj τij
PiT
−1
θ
λj .
(6.10)
To estimate the gravity equation we sum up the trade flows for a country pair across all sectors to arrive
at the aggregate trade flows for the single traded goods sector. Thus, now the gravity equation is estimated
as follows:
ln
Dij
Dii
= Tj − Ti −
1
ln (τij )
θ
(6.11)
where trade cost specification is unchanged, except that the source country effect is not sector-specific:
+ tblkG +
ln τij = distI + brdr
|{z} + lang
| {z }
| {z }
|{z}
distance
border
language
trade block
src
|{z}i
+ εij ,
(6.12)
source effect
To estimate θ, we pool good of different sectors together to belong to the single traded goods sector, and
then employ the bilateral aggregate trade flows with these pooled prices to carry out the SMM estimation as
explained in the multi-sector model. The key equations are:
log
Dij
Djj
log
PjT τij
PiT
1
= − log
θ
where
!
=
PjT τij
PiT
!
,
(6.13)
max {rij (x)}
.
H
P
[rij (x)] /H
(6.14)
j=1
The solution methodology works in the same manner as for the multi-sector model
Step 1 - Estimate trade costs, τij , and country dummies, Ti , using the gravity equation - (6.11) and (6.12).
Step 2 - Compute price of traded goods sector composite using
N
X
PiT = AB
!−θ
eTj τij
− θ1
.
j=1
Step 3 - Taking Li and Dij from the data, solve for per capita expenditures, Xis for s ∈ {T, N }, as a function
of wage, wi , and use the balanced trade condition to solve for wi .
Li Xis
f,s
N,s
= δ wi Li + (1 − γ) φ
Li XiN
+ (1 − β) ξ
T,s
N
X
Lj XjT Dji ,
j=1
N
X
Lj XjT Dji = Li XiT .
j=1
To solve for labor allocations - liT and liN - use
Li wi liT
=β
N
X
Lj XjT Dji ,
j=1
Li wi liN = γLi XiN .
Step 4 - Lastly, the productivity parameter for country i, λi , is obtained using the relationship
1
Ωi = (Vi )− θ λi = eTi ,
(6.15)
where Vi is the factor cost, given by (6.5).
1−β

Vi = [wi ]β 
Y
m={T,N }
(Pim )ξ
T,m

.
Appendix D: Other
Table 6: List of Sectors: ISIC Revision 2
ISIC Code
311
313,314
321
322
323
324
331
332
341,342
352
355
356
361
362
369
371
381
383
384
400
Sector Description
Food products
Beverages and Tobacco
Textiles
Wearing apparel, except footwear
Leather products
Footwear, except rubber or plast
Wood products, except furniture
Furniture, except metal
Paper and products and printing and publishing
Other chemicals
Rubber products
Plastic products
Pottery, china, earthenware
Glass and products
Other non-metallic mineral products
Iron and steel
Fabricated metal products
Machinery, electric
Transport equipment
Non-traded sector
Table 7: Input-Output Structure
Share of Column Sector in Row Sector’s Expenditure on Intermediates
For Traded Goods Sectors Expenditure on Intermediates - ξs,m
311
313,314
321
322
323
324
331
332
341,342
352
355
356
361
362
369
371
381
383
384
311
313,314
321
322
323
324
331
332
341,342
352
355
356
361
362
369
371
381
383
384
400
0.1092
0.1092
0.0002
0.0002
0.0002
0.0002
0.0006
0.0006
0.0641
0.0148
0.0123
0.0123
0.0046
0.0046
0.0046
0.0009
0.0413
0.0005
0.0007
0.0062
0.0062
0.1272
0.1272
0.1272
0.1272
0.0006
0.0006
0.0201
0.1546
0.0124
0.0124
0.0011
0.0011
0.0011
0.0011
0.0016
0.0011
0.0010
0.0062
0.0062
0.1272
0.1272
0.1272
0.1272
0.0006
0.0006
0.0201
0.1546
0.0124
0.0124
0.0011
0.0011
0.0011
0.0011
0.0016
0.0011
0.0010
0.0062
0.0062
0.1272
0.1272
0.1272
0.1272
0.0006
0.0006
0.0201
0.1546
0.0124
0.0124
0.0011
0.0011
0.0011
0.0011
0.0016
0.0011
0.0010
0.0062
0.0062
0.1272
0.1272
0.1272
0.1272
0.0006
0.0006
0.0201
0.1546
0.0124
0.0124
0.0011
0.0011
0.0011
0.0011
0.0016
0.0011
0.0010
0.0005
0.0005
0.0105
0.0105
0.0105
0.0105
0.1132
0.1132
0.0252
0.0345
0.0184
0.0184
0.0043
0.0043
0.0043
0.0305
0.0656
0.0022
0.0016
0.0005
0.0005
0.0105
0.0105
0.0105
0.0105
0.1132
0.1132
0.0252
0.0345
0.0184
0.0184
0.0043
0.0043
0.0043
0.0305
0.0656
0.0022
0.0016
0.0017
0.0017
0.0029
0.0029
0.0029
0.0029
0.0061
0.0061
0.4278
0.0615
0.0118
0.0118
0.0004
0.0004
0.0004
0.0013
0.0095
0.0011
0.0016
0.0065
0.0065
0.0012
0.0012
0.0012
0.0012
0.0005
0.0005
0.0415
0.3691
0.0168
0.0168
0.0016
0.0016
0.0016
0.0061
0.0195
0.0029
0.0008
0.0009
0.0009
0.0096
0.0096
0.0096
0.0096
0.0045
0.0045
0.0473
0.3888
0.0565
0.0565
0.0047
0.0047
0.0047
0.0155
0.0164
0.0054
0.0018
0.0009
0.0009
0.0096
0.0096
0.0096
0.0096
0.0045
0.0045
0.0473
0.3888
0.0565
0.0565
0.0047
0.0047
0.0047
0.0155
0.0164
0.0054
0.0018
0.0004
0.0004
0.0020
0.0020
0.0020
0.0020
0.0040
0.0040
0.0568
0.0659
0.0060
0.0060
0.0649
0.0649
0.0649
0.0106
0.0111
0.0044
0.0048
0.0004
0.0004
0.0020
0.0020
0.0020
0.0020
0.0040
0.0040
0.0568
0.0659
0.0060
0.0060
0.0649
0.0649
0.0649
0.0106
0.0111
0.0044
0.0048
0.0004
0.0004
0.0020
0.0020
0.0020
0.0020
0.0040
0.0040
0.0568
0.0659
0.0060
0.0060
0.0649
0.0649
0.0649
0.0106
0.0111
0.0044
0.0048
0.0004
0.0004
0.0003
0.0003
0.0003
0.0003
0.0023
0.0023
0.0180
0.0529
0.0035
0.0035
0.0039
0.0039
0.0039
0.2511
0.0393
0.0213
0.0178
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0030
0.0030
0.0245
0.0375
0.0155
0.0155
0.0035
0.0035
0.0035
0.2840
0.1154
0.0105
0.0057
0.0002
0.0002
0.0004
0.0004
0.0004
0.0004
0.0024
0.0024
0.0232
0.0274
0.0265
0.0265
0.0056
0.0056
0.0056
0.0469
0.0627
0.3393
0.0119
0.0001
0.0001
0.0056
0.0056
0.0056
0.0056
0.0058
0.0058
0.0106
0.0147
0.0225
0.0225
0.0037
0.0037
0.0037
0.0755
0.0447
0.0666
0.3630
0.0133
0.0133
0.0013
0.0013
0.0013
0.0013
0.0073
0.0073
0.0332
0.0251
0.0070
0.0070
0.0047
0.0047
0.0047
0.0110
0.0258
0.0241
0.0110
0.0207
0.0207
0.0037
0.0037
0.0037
0.0037
0.0051
0.0051
0.0303
0.0300
0.0052
0.0052
0.0021
0.0021
0.0021
0.0092
0.0139
0.0202
0.0397
For Traded Goods Sectors Expenditure on Intermediates - φN,m
400
311
313,314
321
322
323
324
331
332
341,342
352
355
356
361
362
369
371
381
383
384
400
0.6185
0.2700
0.2700
0.2700
0.2700
0.5212
0.5212
0.4448
0.5029
0.3484
0.3484
0.6228
0.6228
0.6228
0.5743
0.4736
0.4121
0.3345
0.7950
0.7735