1. science
2. mathematics
3. statistics

Measuring Openness to Trade
Michael E. Waugh
New York University and NBER
B. Ravikumar
Federal Reserve Bank of St. Louis and Arizona State University
October 21, 2015
ABSTRACT ————————————————————————————————————
This paper derives a new measure of openness that quantifies the potential gains to trade as
a simple function of data. Using a standard, multi-country trade model we measure a country’s potential welfare gain from the world moving to a frictionless trade regime. A country’s
openness-potential depends upon only three statistics: the county’s home trade share, its income level, and the trade elasticity. Quantitatively, we find that the world—in welfare terms—is
effectively in autarky and that a substantial portion of possible gains arise from setting the trade
barriers in all countries to the levels of developed countries.
—————————————————————————————————————————-
Please check the hyper link in the title for the most recent version. Email: [email protected]; [email protected].
1. Introduction
How open is a country to trade? How large are the welfare gains from trade? These questions
have been typically answered by computing the welfare cost of autarky—the change in real
income due to a change from the observed equilibrium to autarky. Arkolakis, Costinot, and
Rodriguez-Clare (2012) show that this calculation in a large class of models takes a simple form:
a country’s home trade share (i.e., one minus the country’s import penetration ratio) taken to
the power of the inverse of the trade elasticity.
In this paper, we deliver a measure of openness that quantifies the potential gains to trade. We
define a country’s openness potential as the the change in real income due to a change from the
observed equilibrium to the efficient, frictionless trade equilibrium. Within a standard model
of trade, we show that a country’s potential takes a simple form: it is proportional to
λii
Yi
1
1+θ
(1)
where λii is country i’s observed home trade share, Yi is country i’s observed real GDP, θ is the
trade elasticity.1 Relative to the standard welfare cost of autarky calculation, the key feature
of our measure is that it encodes how a country’s potential depends upon its technology and
endowments as summarized by its observed GDP. This distinction is important because two
countries may have the same welfare cost of autarky, but vastly different potentials. Empirically, this is the pattern in the data. The welfare cost of autarky is similar across countries, but
poor countries have the largest potential to gain from trade.
We derive our measure by embedding the multi-country trade model developed by Eaton and
Kortum (2002) into a neoclassical growth model. Each country is endowed with a stock of capital and a labor force. Both factors are immobile across countries. There is a continuum of tradable goods. The distribution of productivity over the continuum belongs to the Fréchet family
with countries differing in the centering parameter, but having a common shape parameter.
International trade is subject to barriers in the form of iceberg costs. All markets are competitive. The amount of trade between any two countries, in equilibrium, depends on wages, rental
rates, and trade costs.
Our openness potential roughly describes how much can each country gain by moving from a
current world with trade costs to a frictionless world. Operationally, this involves computing
the change in GDP per worker from its current observed level to a new level in the frictionless
world.
To measure a country’s openness potential, we proceed in two steps. We first provide a closed1
The constant of proportionality is not-country specific.
2
form expression for a country’s real GDP per worker, no matter what the trade costs are. This
expression depends upon the country’s home trade share, technology parameter, labor and capital endowments, and two non-country-specific parameters—the trade elasticity and capital’s
share in production. This expression shows how a country with a larger technology parameter
or capital endowment per worker will be relatively richer.
We can use the above expression for GDP per worker in the frictionless world and the fact that
the home trade share is a function of country-specific endowments and technology parameter
and the non-country-specific parameters. To solve for each country’s home trade share in the
frictionless world we need to know the country-specific technology parameters.
In the second step, we measure the country-specific technology parameters. This measurement
is easily done using a modified development accounting approach based on the neoclassical
growth model (see, e.g. Hall and Jones (1999) or Caselli (2005)). The combination of development accounting plus the characterization of country’s real GDP per worker in the frictionless
world allows us to eliminate the endogenous trade variables and infer a country’s opennesspotential.
Our approach yields the surprising result that a country’s openness-potential is completely
summarized by its observed home trade share and the level of GDP as shown in (1). That is,
details such as its capital stock, labor endowment, technology, etc. are not necessary in our
calculation of the openness-potential.
The important qualitative prediction of our measure is how a country’s potential depends on its
current level of income. As (1) makes clear, holding all else constant, a country whose observed
real GDP is relatively low has relatively more to gain from trade. In other words, the poorest
countries have the most openness-potential.
This prediction of our measure highlights our contribution relative to the standard welfare cost
of autarky calculation. The welfare cost of autarky calculation implies that if a rich country
and a poor country have the same home trade share, then the rich and the poor experience the
same loss from a move to autarky i.e., the welfare gain is the same across countries. Missing
from this inference, however, is the fact that the poor country has more potential to gain from
trade. Thus, while the rich and the poor country have the same welfare cost of autarky, the poor
country is a more closed economy in comparison to its potential relative to the rich country.
We quantify our measure using data on trade shares, endowment, and GDP directly from the
Penn World Tables. We find two important results. First, the world is effectively in autarky.
That is the distance between the observed equilibrium and autarky is small relative to the large
potential gains. Second, trade has large cross-country distributional consequences. Because
all countries are relatively close to autarky, but low technology/poor countries have the most
potential to gain, which implies that the potential gains can eliminate cross-country income
3
inequality.
Frictionless trade is obviously an extreme case, so we complement our theoretical results with
a quantitative assessment of the gains associated with plausible changes in trade costs. We
keep our quantitative work simple with a relatively parsimonious description of the barriers
to trade—one trade cost per country. This structure of trade costs allows us to calibrate our
model without any more data than that available in the Penn World Table. And, it provides one
parameter per country to succinctly summarize the trade frictions.
With our calibrated trade frictions, we ask how much of the potential can be achieved if the
world had the same trade cost as that of the U.S. The key result from this exercise is that while
it delivers about one-fourth of the overall gains, on average, this counterfactual delivers nearly
all the reduction in cross-country income inequality that the frictionless trade economy would.
2. Model
We outline the environment of the multi-country Ricardian model of trade introduced by EK.
We consider a world with N countries, where each country has a tradable final-goods sector.
There is a continuum of tradable goods indexed by j ∈ [0, 1].
Within each country i, there is a representative consumer of size Li . This consumer supplies
labor inelastically in the domestic labor market and also owns physical capital Ki that is inelastically supplied to the domestic capital market. This consumer also enjoys the consumption of
a CES bundle of final tradable goods with elasticity of substitution ρ > 1:
Ui =
Z
1
xi (j)
ρ−1
ρ
0
dj
ρ
ρ−1
.
(2)
To produce quantity xi (j) in country i, a firm employs a Cobb-Douglas production function
combining capital and labor with factor shares α and 1 − α and productivity zi (j). Country i’s
productivity for good j is, in turn, the realization of a random variable (drawn independently
for each j) from its country-specific Fréchet probability distribution:
Fi (zi ) = exp(−Ti zi−θ ).
(3)
The country-specific parameter Ti > 0 governs the location of the distribution; higher values
of it imply that a high productivity draw for any good j is more likely. The parameter θ > 1 is
common across countries and, if higher, generates less variability in productivity across goods
in each country.
Having drawn a particular productivity level, a perfectly competitive firm from country i in4
curs a marginal cost to produce good j of riα wi1−α /zi (j), where wi is the wage rate and ri is
the rental rate of capital in country i. Shipping the good to a destination n requires a per-unit
iceberg trade cost of τni > 1 for n 6= i, with τii = 1. We assume that cross-border arbitrage
forces effective geographic barriers to obey the triangle inequality: For any three countries
i, k, n, τni ≤ τnk τki .
Below, we describe equilibrium prices, trade flows, aggregate output, and the welfare cost of
autarky.
Prices. Perfect competition implies that the price of good j from country i to destination n to
be equal to the marginal cost of production and delivery:
pni (j) =
τni riα wi1−α
.
zi (j)
(4)
So, consumers in destination n would pay pni (j), should they decide to buy good j from i.
Consumers purchase good j from the least-cost supplier; thus, the actual price consumers in n
pay for good j is the minimum price across all sources k:
pn (j) = min
k=1,...,N
pnk (j) .
(5)
The pricing rule and the productivity distribution allow us to obtain the following CES exact
price index for each destination n:
Pn =
−1
γΦn θ
where
Φn =
" N
X
k=1
In the above equation, γ = Γ
such that θ > ρ − 1.
θ+1−ρ
θ
1
1−ρ
#
Tk (τnk wk )−θ .
(6)
is the Gamma function, and parameters are restricted
Trade Flows. To calculate trade flows between countries, let Xn be country n’s expenditure
on final goods, of which Xni is spent on goods from country i. Since there is a continuum of
goods, the fraction of income spent on imports from i, Xni /Xn , can be shown to be equivalent to
the probability that country i is the least-cost supplier to country n given the joint distribution
of productivity levels, prices, and trade costs for any good j. In the equations below, we do
not include proportionality constants that are not country-specific since such constants will not
enter into our welfare gain calculations or our measure of openness.
The expression for the share of expenditures that n spends on goods from i or, as we will call
5
it, the trade share, λni , is:
Ti (τni riα wi1−α )−θ
Xni
= PN
.
λni :=
α 1−α −θ
Xn
)
k=1 Tk (τnk rk wk
(7)
Expressions (6) and (7) allow us to relate trade shares to trade costs and the price indices of each
trading partner via the following equation:
λni
=
λii
Pi τni
Pn
−θ
,
(8)
where λii is country i’s expenditure share on goods from country i, or its home trade share.
While well known, this point is worth reiterating: expression (8) is not particular to EK’s model.
Several popular models of international trade relate trade shares, prices and trade costs in the
same exact manner. These models include Anderson (1979), Krugman (1980), Bernard, Eaton,
Jensen, and Kortum (2003), and Melitz (2003), when parametrized as in Chaney (2008).
Aggregate GDP per Worker. A feature of this model (and other trade models) is that it has a
convenient representation of real gross domestic product (GDP) per worker that is similar to a
standard one-sector growth model with a total factor productivity term and capital-labor ratio
raised to a power term. The key difference is that measured TFP is endogenous and depends
on the country’s home trade share. This simple connection between trade and the tools of
development accounting is something that we will exploit continually in this paper.
To arrive at this representation, a couple of steps are needed. First, with competitive factor
markets, the rental rate on capital is pinned down by the following relationship between the
α
wi ki−1 , where k is the aggregate capital to labor
wage rate and the capital to labor ratio: ri = 1−α
ratio. Second, combining this relationship with (7) yields an expression for each country’s home
trade share.
λii = PN
ki−α wi
−θ
Ti
−α
−θ
k=1 Tk (τnk ki wi )
.
(9)
Third, a rearrangement of (9) provides an expression for the real wage
−1
1
wi
= Tiθ λiiθ kiα ,
Pi
(10)
in which wages, deflated by the aggregate price index, are a function of each country’s technology parameter, its home trade share and its capital-labor ratio. Abstracting from the role of
capital, this is the same expression discussed extensively in Arkolakis, Costinot, and RodriguezClare (2012) that relates home trade shares and the real wage and, hence, the welfare cost of
6
autarky.
Finally, market clearing conditions allow us to connect the real wage in (10) with real GDP per
worker. Balanced trade and equating production of the aggregate commodity with total factor
payments gives
yi =
w i ri k i
+
.
Pi
Pi
(11)
And then using (10) and the observation above that the wage-rental ratio is proportional to the
capital-labor ratio gives
yi = Ai kiα ,
1
where
−1
Ai = Tiθ λiiθ .
(12)
Real GDP per worker in the model is expressed in the exact same way as in the standard onesector growth model with a TFP term and capital-labor ratio raised to a power term. The one
difference is that measured TFP contains an endogenous trade factor, λ−θ
ii , and an exogenous
θ
domestic factor, Ti .
Several comments are in order regarding (12). First, the balanced trade assumption may seem
strong, but it’s not. A similar expression, but with an additive term representing the difference
between the balanced and imbalanced trade equation, is easily derivable. For now we simply
abstract from this distinction.
Second, given that the expressions for trade shares are not specific to the EK model, this implies
there is nothing unique about (12) and its association with the EK model. Many trade models
share the same “gravity equation” (i.e. equation 7), so inverting the home trade share (as we
did when going from (9) to (10)) delivers the same expression for real wage irrespective of the
micro-details of trade. In some ways, this observation is the essence of the isomorphism result
in Arkolakis, Costinot, and Rodriguez-Clare (2012).
The expression in (12) is useful for its close connection to the use of the (closed-economy)
growth model in accounting exercises such as Hall and Jones (1999) and Caselli (2005) to measure TFP.2 This connection provides a method to identify a country’s technology parameter.
Identifying a country’s technology parameter is important, because the set of technology parameters and observed endowments are sufficient to completely characterize the distribution
of income per worker in the frictionless economy.
Welfare Cost of Autarky. Given the representation of Real GDP per worker in (12), it is straightforward to compute the change in real income due to a change from an observed equilibrium
2
This “openness-adjusted” measure of total factor productivity has been noted before in Waugh (2010) and in
Finicelli, Pagano, and Sbracia (2013).
7
to autarky, i.e., the welfare cost of autarky. In autarky, a country purchases all goods from itself
implying that λii equals one. Since technology and endowments are fixed, the change in real
income change from an observed equilibrium to autarky is
−1
λiiθ .
(13)
which is the same formula derived in Arkolakis, Costinot, and Rodriguez-Clare (2012).
The power of (13) is its simplicity. Give a researcher two statistics—the home trade share and
the trade elasticity—and one can compute the welfare cost of autarky. The weakness is that
it treats all countries in the same way. That is, conditional on an observed level of trade, the
distance in welfare terms is the same for all countries—rich or poor.
Missing from this inference is the role that technology and endowments play in determining a
country’s possibilities. What we mean by this is that a country’s “distance to the efficient frontier” will differ depending upon the country’s technology and endowments, even conditional
on an observed level of trade. Understanding the possibilities is important because while two
countries may have the same welfare cost of autarky, one may be more or less closed in comparison to its potential.
The next section derives a measure of openness-potential and shows how to measure it using
easily available data.
3. Openness Potential
This section describes several results that are useful in deriving a metric of a country’s openness potential. We first describe a country’s income level in the frictionless economy. We then
combine this measure with a development accounting approach which allows us to measure a
country’s potential gain as a function of observable data. Finally, we derive a openness metric
which computes a country’s distance between autarky and the frictionless trade equilibrium.
3.1. Openness Metrics
Several observations allow us to compute a country’s income level in frictionless trade. First,
in frictionless trade, a country’s home trade share equals its nominal GDP as a share of world
GDP. Second, frictionless trade implies that price differences are equalized and, thus, nominal
GDP is the same as real GDP. A country’s home trade share is
Li y FT
λFii T = PN i FT ,
k=1 Lk yk
8
(14)
where yiFT is a country’s real GDP per worker in frictionless trade. Combining (14) with the
general representation of real GDP per worker in (12) provides a method to solve out for a
country’s income level in frictionless trade. Proposition 1 summarizes this result.
Proposition 1 (GDP Per Worker in a Frictionless Economy.) GDP per worker in the frictionless
trade economy is
yiFT = Ω(L, T, k)
Ti
Li
1
1+θ
αθ
ki1+θ ,
(15)
where
Ω(L, T, k) =
N
X
Lk
k=1
Tk
Lk
1
1+θ
αθ
1+θ
kk
! θ1
.
(16)
Proposition 1 says much about a country’s position in the income distribution in a frictionless economy. A country with a larger technology parameter or capital endowment will have
larger GDP per worker in a frictionless economy. This is intuitive—a country’s position in the
distribution of income will reflect its advantages in technology or capital.
A country’s labor endowment also affects its level of GDP per worker in a frictionless economy. Holding all else fixed, equation (15) implies that smaller countries will have higher levels
of output per worker relative to larger countries. Thus, this model (and other trade models)
displays a peculiar form of weak scale effect. This scale effect is peculiar in the sense that it
works in the opposite direction of weak scale effects in endogenous growth models with larger
countries experiencing higher income levels. Alvarez and Lucas (2007) suggest one way to kill
these scale effects by assuming that the labor force is proportional to the technology parameter. The key problem with this solution is that it implies a counterfactual relationship between
measures of TFP from (12) and labor endowments. Ramondo, Rodrı́guez-Clare, and Saborı́oRodrı́guez (2012) provide an alternative solution by modeling the structure of trade within a
country.
The Ω term in (16) summarizes the effects of all other countries on the income level of a country.
Because Ω is common to all countries, it does not affect a country’s relative position income
level—it simple scales everything up or down. The inner term of Ω is a weighted sum of the
country-specific terms in (15). This has an implication for the covariance between size and
endowments and technology and their impact on the levels of GDP in the frictionless economy.
If the rich countries in frictionless economy are big countries, then the level effects will be larger
than if the rich countries are small countries.
9
Finally, a country’s potential income level is measurable from readily available data, e.g., the
Penn World Table. Equation (8) and development accounting procedures provide estimates of
a country’s technology parameter. Specifically, we measure the technology parameter T as
Ti = yi ki−α
θ
λii .
(17)
The term in the first set of brackets is the standard “Solow residual.” Take the Solow residual
to the power θ and scale it by a country’s home trade share and one has measured a country’s technology parameter. Substitution of (17) into (15) provides a country’s income level in a
frictionless world as a function of observable data. The potential change in real income a country could experience in the frictionless trade economy directly follows from this expression.
Proposition 2 summarizes this result.
Proposition 2 (Measuring Openness Potential) A country’s level of GDP per worker in the frictionless trade economy is
yiFT
λii
= Ω̃(Y, λ)
Yi
1
1+θ
yi
(18)
with
Ω̃(Y, λ) =
N
X
Yk
k=1
λkk
Ykk
! 1θ
1
1+θ
.
(19)
The change in real income from a country’s observed income level to the frictionless trade economy is
1
1+θ
yiFT
λii
.
= Ω̃(Y, λ)
yi
Yi
(20)
Equation (18) expresses a country’s income level in the frictionless economy as a function of
the country’s statistics today: its home trade share, total GDP, and GDP per worker. A richer
country with less trade or smaller size will have larger income level in the frictionless economy.
A interesting feature of (18) is how capital and labor endowments cancel out of the equation
with only the home trade share and GDP mattering. This says that the reason why a country is
rich or poor does not matter, what matters is just that it is rich or poor. Thus, the importance of
a country’s endowments is completely summarized by the country’s income level.3
3
The Ω̃ term also takes an interesting form as well. The inner part of Ω̃ is a GDP weighted sum of each country’s
1
1+θ
specific gain term λYiii
.
10
Dividing a country’s potential income level by to its current income level determines the potential gains in (20). Equation (20) provides a welfare metric regarding how far a country is
from the frontier as a function of several statistics. The key insight from (20) is the dependence
of a country’s distance to the frontier on its current income level. Holding fixed the volume of
trade, the poorer a country is the more potential it has.
A simple back of the envelope calculation reveals the potential that trade has to reduce income
inequality. Assume that home trade shares and labor endowments are the same across countries. Equation (18) implies that trade has the potential to reduce the variance in log income
2
θ
. Quantitatively, with a θ of four, trade has the potential to reduce income
by a factor 1+θ
inequality by 36 percent.
Using our openness measure (20) and the welfare cost of autarky in (13), we construct a summary statistic Λi that summarizes a country’s distance between autarky and the frictionless
trade. Proposition 3 summarizes the result.
Proposition 3 (Openness Metric) A country’s distance between autarky and frictionless trade is
1/θ
Λi =
1 − λii
,
1
1+θ
1/θ
λii
− λii
Ω̃(Y, λ) Yi
(21)
where this statistic has the following properties: (i) it lies between zero and one, (ii) equals zero in autarky,
(iii) equals one in frictionless trade.
This openness metric is useful for putting into perspective the relative magnitude of the welfare
cost of autarky and openness potential in (20). For example, a country’s welfare cost of autarky
may appear to be large, but (21) places this number in context relative to its openness potential.
So while a country may have a relatively high welfare cost of autarky, it may still be relatively
far from frictionless trade. Proposition 3 provides an openness metric that encodes this feature.
3.2. Quantifying the Gain From Trade
The key feature of (20) and (21) is that all the data required to compute the potential gains and
a country’s relative openness metric is GDP data, home trade shares, and the trade elasticity.
Below we discuss the country-specific measures the trade elasticity that we use.
3.2.A. Cross-country Data and the Trade Elasticity
Output per worker, the number of workers are standard measures used in development and
growth accounting exercises. In our computations, we use the expenditure side of output measure because this measure of real GDP treats trade balances with how we treat them in the
11
1000
COM
Percent Increase in GDP Per Worker
900
800
LBR
GNB
GMB
CAF
700
400
300
200
100
0
1/128
COD
DJI
BTN
VCT
MDV
LCA
BLZ
SLE
TGO
BMU
MNE
LSO
RWA
SWZ
MRT
MWINER
KGZ
BRB
SUR
LAO
GIN BEN
COG MNG
GNQ
ATG
TJK
MDA
FJI
MLI
BFA
ARM
NAM
MDG
ZMB
MOZ
GEO
TCD
ALB MUSMKD GAB
BRN
BHS
PRY
KHM
SEN
BWATTO
MAC
HND
NPL
UGA
BOL
MLT
PAN
URY
CMR
AZE
TKM
ZWE
TZA
BIH
CYP
ISL
JAM
CIV
ETH
BHR
KEN GHA AGOSDN YEM
JOR
CRI LVA LBN
DOM
GTM
OMN
SRB
SYR
LKA
ECU
TUN
LTU
UZB
BGR
HRV
IRQ BLR
KWT
SVN
MAR
NZL
KAZ
PER
EST
BGD
SVK
CHL
ROU
VEN
NOR
VNMNGA
ISR
FIN
IRL
COL
PRT
EGY
PHL UKR
DNK
PAK
CZE GRC
ZAFARG HUN
SWE
CHE
AUT
POL
SAU
MYS
THA
IDN
IRN
AUS LUX
TUR
TWN
BRA
KOR
MEX
ESP
CAN
RUS
NLD
ITA
FRA
GBR
IND
JPN
DEU
CHN
USA
BDI
600
500
CPV
1/64
1/32
1/16
1/8
1/4
Log GDP Per Worker (Data, USA = 1)
1/2
1
QAT
2
Figure 1: Openness Potential Versus Output Per Worker Data
model (i.e., where exports and imports are deflated together and not separately as in production side measures of GDP). See, for example, the discussions in Feenstra, Heston, Timmer, and
Deng (2004) and Waugh (2010) for a more detailed explanation.
We measure a country’s home trade share λii , as one minus a country’s ratio of imports to GDP
at current prices. A complicating issue with our approach is that imports are largely intermediates and measured in gross terms, while GDP is a value added measure. One approach to
correct this mismatch is to “gross up” GDP by a multiplier that represents intermediates share
in value added. If this multiplier is constant across countries, the only effect would be on the
level of potential gains and not their distributional consequences. Another alternative would
be to explicitly model intermediates directly as in Eaton and Kortum (2002), Alvarez and Lucas
(2007), or Waugh (2010) and construct measures of trade penetration using gross production
rather than value added. Measures of home trade shares using gross production would only
affect the level and the distributional consequences if they varied in a systematic way across
countries.
We focus on the year 2005 which is the the benchmark year for the PWT 8.1. We drop any
countries with missing data. We also drop the few countries (e.g. Belgium, Panama, etc.) that
have an imports to GDP ratio which is larger than one. This leaves 160 countries in the sample.
As a baseline we set θ equal to four. This is consistent with the estimates from Simonovska
12
Table 3: Openness Potential
Data
Frictionless
Autarky
Relative Openness
Mean
0.33
1.28
0.27
0.03
Standard Deviation
1.20
1.04
1.16
0.92
90/10 Ratio
25.2
14.8
22.6
11.8
Note: The first three columns report statistics for Real GDP per worker in the data and the model.
The last column reports the openness metric in (21.
and Waugh (2014). Simonovska and Waugh (2014) also provides an extensive discussion on
other estimates from the literature. A short summary is that a variety of different methods
and estimation approaches point to a plausible range of between three and five. As discussed
above, the potential reduction in income inequality decreases with a larger θ.
3.2.B. Openness Potential
Figure 1 plots a countries openness potential in (20) (in percent change terms) versus the logarithm of country’s observed of output per worker. Below we make several observations.
First, the change in the level of income for all countries is large. To provide one measure, Table
3 shows that the average income level in the data relative to the frictionless economy is nearly
a factor of four. Eye-balling the overall level of gains in Figure 1 reveals a similar observation.
Second, poor countries have substantially more potential than rich countries. Figure 1 shows
the negative relationship between the potential gains and a country’s observed income level.
This relationship implies that trade has a large potential to increase income inequality. Table
3 reports summary measures of income inequality: the standard deviation of log income per
worker and the ninety-ten percentile ratio. Both these metrics of cross-country income inequality decline substantially.
The reason the potential gains from trade are larger for poor countries is because they do not
trade more than rich countries. In fact, poor countries trade systematically less then rich countries.Iin the context of (20), this observation implies that the potential gains from trade will be
larger the poorer a country is.
To formalize this argument, consider the slope coefficient from a projection of (20) on GDP per
worker data. The sign of this slope coefficient corresponds with correlation coefficient between
13
50
MLT
45
ISL
Percent Increase in GDP Per Worker
SUR
40
FJI
JAM
VCT
35
NLD
HUN
BHS
SVN
LCA
30
SVK
LSO
JOR
BLZ
25
BMU
CYP
AUT
DNK
CHE
BHR
LTU CZE
SWE
BIH
IRL
MYS
DEU
LVA
FIN
CPV
DJI
HRV ISR
CRI
BRB
15
TTO PRT
TWN
SWZ
MRT
BGR
CAN
MDV
NZL
HND
GTM
LBR
MDA
MKD
MNE
MUS
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ESP
FRA
NAM
THA
MAC
MOZ
ITA
LBN
10
GRC
SEN
ROU
CIV
POL KOR
MAR
BLR
SRB MEX
ALB
TUN
COMVNM
NOR
CHL
GINZMB
GNQ
STP COG PHL
QAT
TUR
AUS
OMN
BWA
PRY
GEO
ARM
SAU
MLI
URY
AGO
MWI
MNG
ECU
BTN
USA BRN
UKRIRQ
PAN
KHM
TGO
KAZZAF
DOM
KEN
5
AZE
SLEGNB
GHA
MDG
KGZ
JPN
GMB
SYR
BOL
LKA
CODBDI
YEM
KWT
LAO
ETH NER
TZA
TJK
BFA
CHN
PER VEN
BEN
IDN
CMR
COL
ARG GAB
NGA SDN
BGD
BRA
CAF
ZWE
EGY
RWA UGA
TKM RUS IRN
IND PAK
NPL
UZB
TCD
0
1/128
1/64
1/32
1/16
1/8
1/4
1/2
1
2
Log GDP Per Worker (Data, USA = 1)
20
1/128
Figure 2: Welfare Cost of Autarky vs GDP Per Worker Data
(20) GDP per worker data or
Corr
1
log
1+θ
λii
Yi
, log yi .
(22)
A simple way for this correlation to be negative is if home trade shares are largely uncorrelated
with real GDP. A way for this correlation to take a large negative sign is if home trade shares
are negatively correlated with real GDP. The latter case is the pattern in the data. The correlation between log home trade shares and log gdp per worker is −0.39 and statistically different
from zero. This correlation can be seen in Figure 2 since the welfare cost of autarky is just an
inverse function of the home trade share. Given that gdp per worker is strongly correlated with
total GDP, these observations imply that poor countries have more to gain from trade than rich
countries.
3.2.C. Welfare Cost of Autarky
Figure 2 provides a point of comparison. It plots the welfare cost of autarky as measured by
(13). The third column of Table 3 reports summary statistics. Several observations.
First, rich countries have larger welfare costs of autarky. Consistent with the discussion above,
because home trade shares are negatively correlated with income level, this implies the welfare
14
cost of autarky is positively correlated with income level. Rich countries trade more and, thus,
have the most to loose from closing to trade. This pattern has the implication that international
trade is modestly increasing cross-country income inequality. The inequality statistics in Table
3 illustrate this point with the standard deviation and ninety-ten ratio decreasing in the closed
economy.
The losses from autarky, however, are very modest. For example, the average loss across all
countries is 13 percent. Moreover, these loses become tiny when compared to the potential
gains seen in Figure 1. This observation suggests that relative to the potential gains, the world
is quite close to autarky. Our openness metric in 21 formalizes this observation.
3.2.D. Openness Metric
Our openness metric in (21) maps a country’s observed position into a value between zero
and one. If a country has the value near one, then the country is close the frictionless trade
benchmark, if a country has the value near zero it is close to autarky.
Figure 3(a) and 3(b) both plot the same openness metric, but on with different scales; Figure
3(a) plots the whole scale between zero and one and Figure 3(b) zooms in.
Figure 3(a) shows the world is effectively in autarky. Many countries lie near zero with notable
exceptions being Germany and the Netherlands. The summary statistics in Table 3 quantify this
observation with the average openness level being 0.03 or 97 percent away from the frictionless
trade benchmark.
Figure 3(a) and 3(b) show that rich countries are relatively more open. This is not because they
have large welfare costs of autarky, but because their welfare cost of autarky relative to their
potential is large. The US is a good example of this pattern. The US is not exceptional in the
size of its welfare cost of autarky—yet the US is among the top ten most open countries in the
world by our openness metric. The insight is that the cost of autarky for the US is not large, but
relative to its potential, it is a very open country.
To generalize the observation, note that the correlation between log GDP per worker and the
welfare cost of autarky is 0.38. The correlation of log GDP per worker and our relative openness
metric is fifty percent larger at 0.57. This implies that rich countries are systematically closer to
their potential than poor countries, and, hence measured to be more open countries.
4. Openness and Trade Frictions
This section asks several follow up questions. First, how do these openness measures relate to
the underlying frictions that impede trade? Second, what are plausible potential gains and how
do they relate to the frictionless gains? The difficulty in asking these questions is that we must
15
1
Opennes Metric (0 = Autarky, 1 = Frictionless)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
NLD
DEU
EST
HUN
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AUT USA
CHE
DNK
GBR
MYS
SWE
ATG
SVKSVN
CZEMLT
FRA
CAN
ISL
TWN
ESP
ITA
IRL
JAM
FIN
KOR
ISR
LTU
BHS
PRT
FJITHAJOR
CYP
SUR
MEX
BHR
POL
CHN
HRV
JPN
BIH
LVA
GRC
NZL
TUR
BGR
CRI
LSO
BMU
AUS
VCT
LCA
ROU
GTM
SAU
VNM
NOR
TTO
BLZ
MAR
PHL
CHL
BLR
HND
LBN
ZAF
TUN
BRB
SRB
MKD
MUS
UKR
CIV
MAC
MRT
SWZ
MDA
CPV
NAM
DJI
SEN
BRA
MOZ
IDN
OMN
MNE
KAZ
ALB
IND
ECU
RUS
IRQ
ZMB
AGO
VEN
DOM
IRNGAB
LBRMWI
ARG
BWA
GNQ
URY
PRY
GIN
COL
KEN
SYR
KWT
LKAEGY
GEO
NGA
PERMDV
PAN
GHA
ARM
COG
AZE
KHM
BRN QAT
MLI
YEM
BGD
PAK
MNG
BOL
MDG
TZA
COD BDIETH
COM
CMR
TGO
KGZ
SDN
BTN
SLE
NER
BFA
LAO
TJK
UGA
BEN
STP
ZWE
GMB
GNB
TKM
NPL
UZB
CAF
RWA
TCD
0
1/128
1/64
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1/8
1/4
1/2
1
2
Log GDP Per Worker (Data, USA = 1)
(a) Openness Metric Λi vs GDP Per Worker Data
0.2
Opennes Metric (0 = Autarky, 1 = Frictionless)
0.18
NLD
0.16
DEU
0.14
EST
0.12
HUN
0.1
AUT
CHE
USA
GBR
MYS
SWE
SVN DNK
ATG
SVK
CZE
0.08
FRA
CAN
ISL
TWN
ESP
MLT
ITA
IRL
JAM
FIN
0.06
KOR
ISR
LTU
BHS
PRT
FJITHAJOR
CYP
SUR
MEX
BHR
CHN
HRV JPN
BIH POL
LVA
0.04
GRC
NZL
TUR
BGR
CRI
LSO
BMU
AUS
VCT
LCA
ROU
GTM
SAUNOR
VNM
TTO
BLZ
MAR
PHL
CHL
BLR
HND
LBN BRB
ZAF
TUN
SRB
MKD
MUS
UKR
CIV
MAC
MRT
SWZ
MDA
QAT
0.02
CPV
NAM
DJI
SEN IND
BRA
MOZ
IDN
OMN
MNE
KAZ
ALB
ECU
RUS
MDV
IRQ
ZMB
AGOPRY
VEN
DOM
IRN
LBR
ARG
GNQ
URY
GIN
COL
SYR
KWT
LKAEGY
GEO
PER BWA
PAN
GHANGA
ARM
COGAZE
KHM
BRN
MLIKEN
YEM
BGD
PAK
MNG
BOL
MDG
TZA
COD ETHMWINER
COM
CMR
TGO
KGZ
SDN
BTN
SLE
LAO STP
GAB
TJK
UGA
BEN
ZWE
GMB
GNB
BDI
TKM
NPL
UZB
CAF
RWABFA
TCD
0
1/128
1/64
1/32
1/16
1/8
1/4
1/2
1
2
Log GDP Per Worker (Data, USA = 1)
(b) Zoomed In
Figure 3: Openness Metric Λi vs GDP Per Worker Data
16
now take an explicit stand on the underlying frictions.
4.1. Calibration
Calibrating trade models of this type present the challenge in that there are a large number
of trade costs and technology parameters to discipline. In fact, this observation illustrates the
value added of the welfare cost of calculations in 13, 20, or 21—they are model consistent, yet
they do not require explicit stands on (potentially) hard to infer parameters.
Our objective is to have a relatively parsimonious description of the barriers to trade, and add
as little extra data that we have worked with so far. To achieve these objectives we reduce the
parameter space such that each country faces only one trade cost, τi , to import from all other
countries. And this one number will succinctly summarize the frictions that each country face
to trade.
To calibrate these trade frictions, we use the following procedure. First, we infer the technology
parameters using our development accounting equation in (8). Unlike our results in Proposition
2, we must use data on capital stocks and capital shares to invert the technology parameters.
Recall, that this was not necessary before as a country’s total GDP was a sufficient statistic for
the role that endowments play.
Given the model’s structure resulting in equation (12), we want α to be consistent with the
exercises in the income accounting literature. To do so, we set α equal to 1/3. Gollin (2002)
provides an argument for setting α equal to 1/3 by calculating labor’s share for a wide crosssection of countries and finding it to be around 2/3 with no systematic variation across income
levels. Given this value, we then use the capital stock measures for the year 2005 from the PWT
8.1.
Given the technology parameters, we then chose τi s to such that the model in equilibrium
exactly fits each country’s observed home trade share. The Appendix describes the algorithm
used. This procedure results that the equilibrium of the model exactly match the observed
distribution of output per worker across countries and each country’s home trade share. This
also implies that the frictionless trade equilibrium of our calibrated economy will be the same
as the results presented above.
Calibrated Trade Costs. Table 2 presents statistics summarizing the calibrated trade costs.
Figure 4 plots each country’s calibrated trade cost versus its income level. There are several
important observations to make.
First, an important aspect of the calibrated trade costs are that they are large. The mean and the
median trade cost are about seven and six and a half. The trade elasticity plays an important
role in determining the size. With a larger θ, the inferred trade costs would be smaller.
17
Table 2: Calibrated Trade Costs
Mean τ
Median τ
All Countries
6.99
6.59
Rich
4.92
4.64
Poor
9.06
8.44
Note: Rich is the set of countries above the median GDP
per worker. Poor is the set below.
Second, the trade costs are substantially larger for poorer countries. Table 2 illustrates this point
by reporting the trade costs for those “poor” countries which are defined by those countries
below median GDP per worker; “rich” countries are those above the median. By this demarcation, poor countries have trade costs that are almost twice as large as those in rich countries.
Figure 4 further illustrates this point with a strong, downward sloping relationship between
the calibrated trade costs and income level.
The intuition for both these two observations lies in Section 3.2. Relative to the frictionless
benchmark the observed levels of trade are small. Thus, to reconcile the small levels of trade,
the model needs large frictions. Moreover, poor countries, have the most to gain relative to
frictionless trade—thus, one infers that poor countries require even larger frictions to reconcile
their relatively low levels of trade. The next observation makes this connection between our
openness metric and the trade costs event tighter.
Third, the calibrated trade costs correlate highly with our relative openness metric in (21). In
fact, they move almost one-for-one with each other. Figure 5(a) illustrates this observation
by plotting the log of our relative openness measure in (21) versus the log of the calibrated
trade costs. Figure 5(a) illustrates very clearly that country’s with measures of higher relative
openness have low trade costs and vice versa. The correlation between these two measures is
−0.94.
As a point of comparison, Figure 5(b) plots the welfare cost of autarky in (13), versus the calibrated trade costs. The calibrated trade costs are more modestly correlated with the welfare
cost of autarky with a correlation coefficient of −0.48. Similar to the discussion above, the US
is a good example. The US is measured to have a low welfare cost of autarky, yet it also has
very low trade costs. This incongruence disappears when using our relative openness metric:
the US is one of the most open countries and it has some of the lowest trade costs.
A concern with these claims is that these correlation are spurious or mechanical, e.g., relative
18
16
Log Trade Cost
8
4
2
1
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GNB
GMB
COM
NPL
BTN
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UZB
NER
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GAB
LBRMWI
MNG
MDG
CMR COG
SDN
MDV
MLI
TZA
BOL
ARM
ETH
CPV GEO DJI
BRN
GIN
KHM
MNEGNQ
AZE YEM
PAN
PRY
BWA
ZMB
GHA
URY
SWZ
KEN
ALB
MRT
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EGY
BLZ
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NAM
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DOM
MOZ
BRB
LCA
PER
AGO PAK
SEN NGA
MAC KWT
MKD
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ECU
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COL MUS
OMN
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ARG
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IRN
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UKR
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CHL
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CYP NOR
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ROU
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MLT ISL
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JAM JOR
SAU
LTU
GRC
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TUR
POL
ISR
PRT
FIN IRL
SVNJPN
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MEX SVK
CHN
KORDNK
CZE
SWE
EST
TWN
MYS
ESP
ITA
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AUT
CAN
HUN
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CAF
RWA
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1/8
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Log GDP Per Worker (Data, USA = 1)
1/2
1
QAT
2
Figure 4: Trade Costs
openness is correlated with GDP as are trade costs etc. This is not the case. For example, after
conditioning on income level, the correlation between relative openness and the trade costs is
−0.88. In contrast, after conditioning on income level, the correlation between the welfare cost
of autarky and the trade costs declines to −0.32.
This observation is important because is tells us that the simple statistic in (21) summarizes
well the underlying frictions that a country faces (in addition to measuring how open a country
is). In contrast, the welfare cost of autarky is not a good summary statistic of the underlying
frictions in the economy.
4.2. Counterfactual: Toward the US Trade Cost
We now perform several exercises to understand plausible, intermediate levels of trade costs.
The first exercise endows all countries with the same trade costs that the US faces (which is
approximately 2.25) and then computes the new equilibrium. To be clear, this is a simultaneous
change and general equilibrium effects are taken into account as a new equilibrium is computed. The idea behind this exercise is to use the US trade costs a plausible benchmark without
altering the technological considerations for moving goods across space.
Table ?? reports the summary statistics. The move to the US trade costs gives rise to large gains.
19
1/2
1/4
Log Relative Openness
1/8
LUX
NLD
DEU
1/16
1/32
1/64
1/128
1/256
EST
HUN
AUT
CHE
USAGBR MYS
DNK
SWE
SVN
ATG
SVK
CZE
FRA
CAN
ISL
TWN IRL
ESP
MLT
ITA
JAM
KORFIN
ISR
BHS
PRT LTU
JOR
FJISUR
THA
CYP
MEX
BHR
CHN
HRV
JPNPOL GRC
BIH
LVA
NZL
TUR
BGR
LSO
BMU
AUS ROU CRI
VCT
LCA
GTMTTO
SAU
VNM
NOR
BLZ
MAR
PHL
CHL
BLRLBN
HND BRB
ZAF
TUN
MUS
UKR SRB
CIVMKD
MAC
MRT
SWZ DJI
MDA
CPV
NAM
SEN
BRA
MOZ
IDN QAT
OMN
MNEMDV
ALB
IND
ECU
RUS KAZ
IRQ
ZMB
AGO
VEN
DOM
IRN
ARG
BWA
GNQ
URY
PRY
GIN LBR
COL
KEN
SYR
KWT
LKA
GEO
NGA
PER
PAN
GHA
ARM
COG
AZE
KHM
BRN
MLI
BGDYEM
PAK
MNG
EGY
MWI
BOL
ETH
MDG
TZA
COD
CMR
TGO
KGZ
SDN
BTNCOM
SLE
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ZWEBEN
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GNB STP
BDI
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UZBNPL
CAF
RWA
1/512
TCD
2
4
8
16
Log Trade Cost
(a) Openness Metric Λi vs Trade Cost τi
2
ATG
Log Welfare Cost of Autarky
EST
MLT
ISL
1.41
NLD
DEU
1
2
HUN
JAM
SVN
SUR
FJI
BHS
VCT
LCA
BMU
LSO
BLZ
SVK
CYP
AUT
JOR
DNK
CHE
BHR
LTU
CZE
SWE
BIH
MYS IRL
LVA
FIN
CPV
DJI
ISR
HRV CRI
BRB
TTO
PRT
TWN
SWZ
MRT
BGR
CAN
MDVLBR
NZL GTM HND
MDA
MKD
MUS
GBR
ESP
FRA
NAM MNE
MOZ
ITA KORTHA
LBNCIVMAC
ROU MAR
POL GRC
BLR
SRB SENALB
TUN
VNM
COM
NOR
MEX TUR PHL
ZMB
CHL
GIN
GNQ
STP
QAT
AUS
COG
OMN
BWA
PRY
GEO
ARM
SAU ZAF
MLI
URY
AGO
MWI TGO
MNG
ECU
BTN GMB
USA
UKR KAZ
BRN
PAN
KHM
DOM
IRQ
KEN
AZE
SLE
GHA
MDG
GNB
KGZ
JPN
SYR
BOL
LKA
COD
NER
YEM
KWT
LAO
ETH
TZA
TJK
VEN
BFA
CHN
PER
BDI
BEN
IDN
CMR
GAB
COL
ARG
NGA
BGD
BRA
UGA
SDN
CAF
ZWE
RUSIRN PAK
EGY
TKM
IND
NPL RWA
UZB
TCD
4
8
Log Trade Cost
(b) Welfare Cost of Autarky vs Trade Cost τi
Figure 5: Openness Metrics and Trade Costs
20
16
Table 3: Counterfactual τ s
Data
US τ
τ =1
yi
Openness Λi
yi
Openness Λi
yi
Openness Λi
Mean
0.33
0.03
0.54
0.25
1.28
1.0
Standard Deviation
1.20
0.92
1.05
0.21
1.04
—
90/10 Ratio
25.2
11.8
14.5
1.78
14.8
—
Note:
The average percent increase is almost 100 percent. Moreover, there is substantial heterogeneity, with the gains largely accruing to the poorest countries. Figure 6(a) illustrates this point by
plotting the percent gain versus gdp per worker data. The second column of Table ?? summarizes this finding by showing that the standard deviation in income and the 90/10 ratio decline
by large amounts.
The third row of Table ?? provides a point of comparison by reporting the same statistics in
the frictionless trade economy. The most important (and surprising) result is that that crosscountry inequality is nearly the same as in the US trade cost economy. In other words, the
US trade friction delivers all the reduction in inequality that the frictionless trade equilibrium
delivers.
The key difference between row two and row three of Table ?? is the level of the gains. The difference between going from a trade friction of 2.2 to frictionless trade amounts to an additional
300 percentage point increase in the gains from trade.
The difference between levels versus inequality reduction can be seen in Figure 6(b). Figure 6(b)
plots the results from both the US trade friction economy and the frictionless trade economy
with the same axis. The key thing to note is that the slope between output per worker and the
gains are nearly the same. The only difference is the intercept or level.
21
450
STP
Percent Increase in GDP Per Worker
400
350
COM
300
GNB
GMB
LBR
250
CPV
CAF
BDI
200
150
COD
100
50
0
1/128
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BTN
VCT
MDV
LCA
BLZ
SLE
TGO
BMU
MNE
RWA
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SWZ
MRT
MWINER
KGZ
BRB
SUR
MNG
LAO
GIN BEN
COG
GNQ
TJK
ATG
MDA
MLI
BFA
ARM
FJI
NAM
MDG
ZMB
TCD
MOZ
GEO
ALB MUSMKD GAB
BRN
BHS
PRY
KHM
BWATTO
SEN
MAC
HND
NPL
UGA
BOL
PAN
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MLT
CMR
TKM
AZE
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TZA
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CYP
ETH
ISL
JAM
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BHR
CRI
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SRB
LKA ECUBGR
QAT
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TUN
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IRQ
BLR
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KWT
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SVN
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NOR
VNMNGA
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COL
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FIN IRL
PAK
PHL UKR
GRC
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CZE DNK
SAU
POL
IRN
IDN
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THA
TUR
HUN
CHE
AUT
MYS
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NLD USALUX
DEU
1/64
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Log GDP Per Worker (Data, USA = 1)
(a) Counterfactual: US τ
1
Opennes Metric (0 = Autarky, 1 = Frictionless)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
VCT ATG
COM STP CPV
LCA
LBRCAF
DJI
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GNB
BMU
SUR
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MDV
LSO
BTN
FJI SWZ
BDI
MNEGNQ BHS
SLE
TGO
MRT
MLT ISL
BRB
RWA
MWI
NER
MDA MNG
KGZ
GIN
COG
JAMNAM
LAO
BEN
TJK
ESTGAB
MUS
MLI
ARM
MKD
LUXBRN
MOZ MDG
CYP MAC
BFA KHM
ZMB
ALB
GEO
TTO
HND
COD
BIH
SEN
PRY
JOR
BWA
BHR
TCD
LVA
PAN
URY
NPL KEN GHA
BOL
CIV
CRI
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SVN
CMR
TZA UGA
LTU
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ZWE
GTM
ETH
LBN
AGOSDN YEM
SRB
HRV
OMN
BGR
SVK
TUN
ECU
LKA SYR
BLR
NZL
IRQDOM
MAR
UZB
HUN
IRL
FIN
KAZ
ISR
DNK
PER VEN
CZE GRC
CHL PRT
BGD
VNMNGA
AUTNOR KWT
CHE
SWE
COLROU
PHL UKR
EGY
PAK
NLD
ZAFARGMYS
POL
THA
SAU
TWN
AUS
TUR
IDN
IRN
CAN
KOR
ESP
MEX
BRA
ITA
FRA
RUS
GBR
DEU
IND
JPN
CHN
USA
QAT
0.1
0
1/128
1/64
1/32
1/16
1/8
Log Trade Cost
1/4
1/2
(b) Counterfactual: US τ vs. Frictionless Trade
Figure 6: Change in Output Per Worker
22
1
2
5. Conclusion
23
References
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AND
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AND
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F INICELLI , A., P. PAGANO , AND M. S BRACIA (2013): “Ricardian selection,” Journal of International Economics, 89(1), 96–109.
G OLLIN , D. (2002): “Getting Income Shares Right,” Journal of Political Economy, 110(2), 458–474.
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25
Appendix
Below, we walk through the derivation of some of the relationships that we exploit. What we
want to do is find an expression for a country’s real GDP per worker in frictionless trade.
First, we want walk through a balanced trade condition. In general the balanced trade condition is
Li (wi + ri ki ) =
N
X
Lk (wk + rk kk )λki
(23)
k=1
which says that total income payments in country i must equal the total purchases of i’s goods
made by all countries. These country-specific purchases equal the income in country k times
the expenditure share that country k spends on goods from country i, i.e. λki . Note the right
hand side of (23) includes purchases of country i from itself.
In frictionless trade, all countries purchase the same amount from country i. That is, λki = λii .
This then simplifies the equation to be
Li (wi + ri ki ) = λi
N
X
Lk (wk + rk kk )
(24)
k=1
which states that the share of goods all countries purchase from country i must equal country
is share in world GDP:
Li (wi + ri ki )
.
λii = PN
k=1 Lk (wi + ri ki )
(25)
Then the final observation is to note that (up to some non-country specific scale factor), real
GDP per worker corresponds with nominal GDP per worker in frictionless trade. Thus we
have
Li y FT
λFii T = PN i FT .
k=1 Lk yk
(26)
Then to solve for real GDP in frictionless trade we substitute this expression into 12, giving
yiF T = Ti
yiFT
=
1
θ
N
X
k=1
Li yiFT
PN
FT
k=1 Lk yk
Lk ykFT
! −1
θ
1
! 1+θ
26
Ti
Li
−1
λiiθ kiα ,
1
1+θ
(27)
αθ
ki1+θ .
(28)
The final issue is that in the first bracket terms are endogenous variables, not primitives. We
can substitute them out in the following manner. First define
N
X
Ω=
Lk ykFT
1
! 1+θ
1
1+θ
ki1+θ .
k=1
δi =
Ti
Li
(29)
αθ
(30)
So that Ω is the non-country specific component and δi is the country specific component of
real GDP. Now we will take logs and repeatedly substitute this expression into (28). Doing and
walking through each of the steps gives. . .
log yiF T
log yiF T
N
X
1
=
log
Lk Ωδi
1+θ
k=1
!
+ log δi
N
X
1
1
log Ω +
log
Lk δi
=
1+θ
1+θ
k=1
!
+ log δi
N
X
!
(32)
N
X
1
Lk δi
+
log
1+θ
k=1
1
1+θ
2
log Ω +
log yiF T =
1
1+θ
n
N
X 1 n
X
Lk δi
log Ω +
log
1
+
θ
n=1
k=1
log yiF T
1
1+θ
n
N
X
1
log Ω +
log
Lk δi
1+θ
k=1
log yiF T =
1
1+θ
...
=
log yiF T =
log yiF T
1
log
1+θ
N
X
Lk δi
k=1
N
X
1
= log
Lk δi
θ
k=1
!
!
2
(31)
log
Lk δi
k=1
1
1 + log δi
1 − 1+θ
+ log δi
!
!
+ log δi
X 1 n
+ log δi
1+θ
n=0
!
+ log δi
(33)
(34)
(35)
(36)
(37)
(38)
27
Then piecing everything together yields
yiFT =
N
X
θ
1+θ
1
1+θ
Lk Tk
αθ
1+θ
kk
k=1
! θ1 Ti
Li
1
1+θ
αθ
ki1+θ .
(39)
Algorithm for Calibrating the Model
The strategy is we want to pick T s and one τ per country to exactly replicate the distribution
of income per worker and home trade shares λii . Below we describe the procedure to calibrate
the model in this way.
Step 1. We first use our development accounting results to recover the T s. So
Ti =
yi
kiα
θ
λii .
(40)
This is straightforward.
Step 2. With the assumption of one importing trade barrier per country, the home trade share
can be expressed as
′
λii = 1 −
Φ (w, τ, T, k)
τi−θ i
Φi (w, τ, T, k)
(41)
′
where Φi is Φi minus the term for country i in the summation. This equation provides a map′
ping from observed trade, two endogenous objects Φi and Φ, into the unobserved trade friction.
What this allows us to do is to set up a system of equations to find the τi that satisfies the relationship above subject to the equilibrium constraints, i.e. that wages are consistent with market
clearing.
28