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Online Appendix: Measured Aggregate Gains from
International Trade
Ariel Burstein
Javier Cravino
UCLA and NBER
University of Michigan
March 3, 2014
In this online appendix we derive additional results discussed in the paper. In the
first section, we include two extensions of the model with perfect competition of Section
3. First, we incorporate intermediate inputs into the production function. Second, we
consider the case in which one country operates the trading technologies of all countries.
For each of these two extensions, we derive real GDP from the production side and from
the expenditure side, and show the equivalence between real GDP and real consumption
at the world level. In deriving our second extension, we consider the case (discussed in
Footnote 16) where the trading technology is operated by intermediaries rather than by
the producers themselves.
In the second section we show that our assumption in equation (19) of the paper
that aggregate profits represent a constant share of total revenues (Pit = ki Yit , where
´
Yit = Ân W pint qint dMit ), is satisfied in the remaining two special cases of our model
int
described in Section 4. In deriving these results, we show that under the assumptions of
Colloraries 2 and 3, in response to changes in variable trade costs the mass of entering
firms is constant across steady states.
1
General Production Technologies with Perfect Competition
1.1
Allowing for intermediate inputs
In this section we extend our model in Section 3 to allow for intermediate inputs. We
allow for a general specification in which each good z can be used both for consumption
1
and as an intermediate input. We denote by qcni (z) and qm
ni ( z ) the quantities of good z
from country n used in country i for consumption and as intermediate inputs. Market
clearing implies qni (z)=qcni (z) + qm
ni ( z ).
1.1.1
Real GDP, production side
We start by deriving expression (8) in the paper. The profit maximization problem is given
by:
Pit =
max
{yin (z) , xi (z)}
subject to
ˆ
 pint (z) yin (z) /tint
Wi n
Wit xi (z) dz
Â
n
ˆ
Wn
dnit pnit (z) qm
nit ( z ) dz
(1)
 yin (z) , qmnit (z) , xi (z)
2 Yi (z) for all z.
Current-dollar GDP is given by:
GDPit =
Â
n
ˆ
Wi
pint (z) yint (z) /tint dz
Â
n
ˆ
Wn
pnit (z) qm
nit ( z ) dz.
Real GDP from the production side is calculated using the double deflation method,
which consist in deflating output and intermediate inputs using different price indexes.
In particular, we calculate real GDP as:
RGDPit =
Ân
´
Wi
pint (z) yint (z) /tint dz
PPIit
Ân
´
Wn
pnit (z) qm
nit ( z ) dz
MPIit
.
Where MPIit denotes the input price index in country i. To a first order approximation,
the change in RGDP is given by:
dlogRGDPit =
pint0 (z) yint0 (z) /tint0
(dlog ( pint (z) yint (z) /tint ) dlogPPIit ) dz
GDPit0
n Wi
ˆ p (z) qm (z) ⇣
⌘
nit0
nit0
m
p
z
q
z
dlogMPI
(2)
(
)
(
)
nit0
it dz.
 W
nit0
GDPit0
n
n
Â
ˆ
The change in the producer price index is given by equation (7) in the paper. The change
in the input price index is given by:
dlogMPIit ⌘
Â
n
ˆ
Wn
m
lint
(z) dlogpnit (z) dz.
2
where
m
lint
0
(z) =
pnit (z)qm
nit0 ( z )
´ 0
.
Ân Wn pnit0 (z)qm
nit ( z )
0
Substituting dlogPPIit and dlogMPIit in expression
(2), and following the steps used in Proposition 1 in the paper, we can write the change in
aggregate productivity as:
Wit0 Xit0
dlogAit = dlogRGDPit
dlogXit
GDPit0
ˆ
= Â lint0
lint0 (z) (dlogpint (z)
Wi
n
+
Â
⇥
1
dnit0
⇤
n
ˆ
pnit0 (z) qm
nit0 ( z )
Wn
GDPit0
dlogtint
dlog p̄int (z)) dz
(3)
dlogqm
nit0 ( z ) dz.
Note that in the absence of tariffs on intermediate inputs, equation (3) coincides with
expression (8) in the paper.
1.1.2
Real GDP, expenditure side
We now derive the equivalence between real GDP from the production and the expenditure side. To a first order approximation, the change in real GDP from the expenditure
side is given by,
GDPitE0 dlogRGDPitE
=Â
n
ˆ
Â
ˆ
pint0 (z) qint0 (z) (dlog ( pint (z) yint (z) /dlogtint )
Â
ˆ
pnit0 (z) qnit0 (z) (dlog ( pnit (z) qnit (z))
+
dnit0 pnit0 (z) qcnit0 (z) (dlog (dnit pnit (z) qcnit (z))
Wi
n 6 =i Wn
n 6 =i Wn
dlogCPIit ) dz
dlogEPIit ) dz
dlogIPI it ) dz.
(4)
The term in the first line is the change in nominal expenditures, deflated by the CPI. The
second and third lines denote exports deflated by the EPI and imports deflated by the IPI
respectively.
We can write the consumer price index as:
Eit0 d log CPIit =
Â
n
ˆ
Wi
dnit0 pnit0 (z) qcnit0 (z) (dlogdnit (z) + dlogpnit (z)) dz
(5)
The log change in the export price index (EPI) can be written as,
Â
ˆ
n 6 =i Wn
pint0 (z) qint0 (z) dlogEPIit =
Â
ˆ
n 6 =i Wn
3
pint0 (z) qint0 (z) dlog p̄int (z) dz,
(6)
and the log change in the import price index (IPI) can be written as:
Â
ˆ
n 6 =i Wn
pnit0 (z) qnit0 (z) dlogIPIit =
Â
ˆ
n 6 =i Wn
pnit0 (z) qnit0 (z) dlogpnit (z) dz.
(7)
Substituting (5), (6) and (7) into equation (4), and using qni (z)=qcni (z) + qm
ni ( z ) we obtain:
GDPitE0 dlogRGDPitE
Â
=
ˆ
n 6 =i Wi
+Â
n
ˆ
Â
ˆ
+
Wn
n 6 =i Wn
⇥
dnit0
⇤
1 pnit0 (z) qcnit0 (z) dlogqcnit (z)
pint0 (z) qint0 (z) dlogyint (z)
pint0 (z) qint0 (z) (dlogpint (z)
Â
n
ˆ
Wn
m
pnit0 (z) qm
nit0 ( z ) dlogqnit ( z ) dz
dlogtint
dlog p̄nit (z)) dz.
This can be written as:
dlogRGDPitE
1.1.3
1
1
=
dlogRGDP
+
it
GDPitE0
GDPitE0
Â
ˆ
n 6 =i Wi
⇥
dnit0
⇤
1 pnit0 (z) qcnit0 (z) dlogqcnit (z) .
World-level equivalence between real consumption and Real GDP
From the proof of Proposition 4 in the paper, the equivalence between world real GDP and
world real consumption holds if the world import price index equals the world export
price index. This follows directly from equations (6) and (7) for the case in which each
country operates its own trading technology. The case in which one country operates the
trading technologies of all the other countries is presented below.
1.2
Trading technologies concentrated in one country
We now consider the case in which there is a sector in country is that operates the trading
technologies of every country. The section also shows how to extend our results when the
trading technology is operated by intermediaries instead of the good producers themselves.
4
1.2.1
Real GDP, production side
Current-dollar GDP in country is is given by:
GDPis t =
Â
n
ˆ
Wis
+Â Â
pis nt (z) /tis nt yis nt (z) dz
ˆ
i n 6 =i Wi
pint (z) yint (z) /tint dz
ÂÂ
ˆ
i n 6 =i Wi
pint (z) /tint yint (z) dz.
(8)
The term in the first line denotes value added of goods producers from country is . The
two terms in the second line capture value added in the sector that operates the trading technology. This sector purchases the production from each source country i that is
bound to foreign countries n 6= i at prices pint (z) /tint , and resells a fraction 1/tint of this
production in each destination country n at prices pint (z).
Computing real GDP from the production side implies using the double deflation
method to deflate value added the sector operating the trading technology. That is:
RGDPis t
=(
RGDPis t 1
Ân
´
Wis
pis nt (z) /tis nt yis nt (z) dz
g
PPIis t
Âi  n 6 =i
´
Wi
+
Âi  n 6 =i
pint (z) /tint yint (z) dz
MPIiss t
´
Wi
)/GDPis t
pint (z) yint (z) /tint dz
PPIiss t
1.
g
Here, PPIis t denotes the producer price index of the goods sector in country i, while PPIiss t
and MPIiss t denote the producer and the input price index of the sector operating the
trading technology. Taking a first order approximation around time t = t0 and using the
fact that all deflators are normalized to equal 1 at date t0 we obtain:
dlogRGDPis t =
 lis nt0
ˆ
+
Âi  n 6 =i
´
n
Wis
Wi
⇣
lis nt0 (z) dlogpis nt (z)
pint0 (z) yint0 (z) /tint0 dz
GDPis t0
dlogtis nt + dlogyis nt (z)
dlogMPIiss t
dlogPPIiss t .
Following the steps used in Proposition 1 in the paper, the first line equals
g
⌘
g
dlogPPIis t dz
(9)
Wis t0 Xis t0
GDPis t0 dlogXis t ,
since prices in PPIis t do not include changes in trade costs tis nt . The second line captures
the difference between input prices and final prices in the operating the trading technology. Noting that final prices in the PPIiss t equal pint (z), and that input prices in the MPIits
5
equal pint (z) /tint , we obtain:
dlogMPIiss t
s
dlogPPIist
=
pint (z) qint0 (z) dlogtint dz
´ 0
.
Âi Ân W pint0 (z) qint0 (z)
Âi  n 6 =i
´
Wi
(10)
i
Substituting into equation (8) in the paper we can write the change in aggregate productivity as:
GDPit0
dlogAis t = Â
lint0 dlogtint .
GDPis t0 nÂ
i
6 =i
1.2.2
Real GDP, expenditure side
We now show the relation in Proposition 2 between changes in real GDP from the production and the expenditure side. We assume that tariffs dnis t only apply to goods that
imported to be consumed in country is (and not to goods that are imported by intermediaries in country is to be resold in other countries). GDP from the expenditure side for
country is is given by:
GDPiEs t
=Â
n
ˆ
Wn
ÂÂ
dnis t pnis t (z) qnis t (z) dz + Â
Â
i n6=i,n6=is
ˆ
i 6 =is n 6 =i Wn
ˆ
Wi
pint (z) yint (z) /tint dz.
pint (z) /tint yint (z) dz.
The first terms in this expression denotes current-dollar expenditures in country is ,
inclusive of tariffs. The second term denotes exports from country is , which equal the
foreign sales from each source country i, except those bound to country is . The last term
denotes imports of is , which equal expenditures in foreign goods in every country n,
excluding expenditure in goods produced in county is . To a first-order approximation,
the change in real GDP from the expenditure side is:
GDPiEs t0 dlogRGDPiEs t
=Â
n
+Â
ˆ
dnis t0 pnis t0 (z) qnits0 (z) (dlog (dnis t pnis t (z) qnis t (z)) dlogCPI is t ) dz
ˆ
pint0 (z) qint0 (z) (dlog ( pint (z) qint (z)) dlogEPI is t ) dz
Â
Wn
i n6=i,n6=is
ÂÂ
ˆ
i 6 =is n 6 =i Wn
Wi
pint0 (z) qint0 (z) (dlog ( pint (z) qint (z))
dlogIPI is t ) dz.
(11)
We re-write the log-change in the consumer price index in country is given by equation
6
(13) in the paper:
d log CPIis t =
Â
n
ˆ
Wis
dnis t0 pnis t0 (z) qnis t0 (z)
(dlogdnis t + dlogpnis t (z)) dz.
E i s t0
(12)
All export prices for country is include trade costs. The log change in the EPI is:
dlogEPI is t =
pint0 (z) qint0 (z) dlogpint (z) dz
´
.
Âi Ân6=i,n6=is W pint0 (z) qint0 (z) dz
Âi Ân6=i,n6=is
´
Wi
(13)
i
In contrast, import prices do not include trade costs. The log change in the IPI is,
d log IPIit =
Âi 6 =is  n 6 =i
´
pint0 (z) qint0 (z) (dlogpint (z) dlogtint ) dz
´
.
Âi6=is Ân6=i Wi pint0 (z) qint0 (z) dz
Wi
(14)
Substituting (8), (12), (13) and (14) in expression (11) we can write:
GDPiEs t0 dlogRGDPiEs t
=Â
n
ˆ
dnis t0 pnis t0 (z) qnis t0 (z) dlogqnis t (z) dz
ˆ
pint0 (z) qint0 (z) dlogqint (z) dz
Â
Wn
+Â
Wi
i n6=i,n6=is
ˆ
ÂÂ
i 6 =is n 6 =i Wn
pint0 (z) qint0 (z) (dlogqint (z) + dlogtint ) dz.
´
Adding and subtracting Ân6=is Wn pnis t0 (z) qnits0 (z) dlogqnis t (z) dz/GDPiEs t0
´
Ân6=is Wi pis nt0 (z) qints0 (z) dlogyis nt (z) dz/GDPiEs t0 we obtain:
GDPiEs t0 dlogRGDPiEs t
=Â
n
+
Â
ˆ
n 6 =is
Wi
ˆ
pis nt0 (z) qints0 (z) dlogyis nt (z) dz
Wn
+Â Â
ˆ
ÂÂ
ˆ
⇥
⇤
1 pnis t0 (z) qnis t0 (z) dlogqnis t (z) dz
dnis t0
i n 6 =i Wi
i n 6 =i Wn
pint0 (z) qint0 (z) dlogqint (z) dz
pint0 (z) qint0 (z) (dlogqint (z) + dlogtint ) dz.
7
and
Re-arranging, using equations (9) and (10), we can write:
dlogRGDPiEs t
 n 6 =is
GDPis t0
=
dlogRGDP
+
i
t
s
GDPiEs t0
´
Wn
⇥
dnis t0
⇤
1 pnis t0 (z) qnis t0 (z) dlogqnis t (z) dz
GDPiEs t0
(15)
which coincides with expression (11) in the paper.
1.2.3
World level real consumption and real GDP
We now show the equivalence between real consumption and real GDP from the expenditure side at the world level. As in Proposition 4 in the paper, we need to show that, to a
first approximation, the change in the world’s import price index equals the world export
price index. The change in the world import price index is given by:
 Importsit0 dlogIPI it =  Importsit0 dlogIPI it + Importsis t0 dlogIPI is t
i
i 6 =is
=Â
Â
ˆ
n i 6=is ,i 6=n Wn
+
ÂÂ
ˆ
n 6 =is n 6 =i Wn
pnit (z) ynit (z) /tnit dlogpnit (z) dz
pnit (z) ynit (z) /tnit (dlogpnit (z)
dlogtnit ) dz.
(16)
The change in the export price index is given by:
 Exportsit0 dlogEPI it =  Exportsit0 dlogEPI it + Exportsis t0 dlogEPI is t
i
i 6 =is
=
 Â
ˆ
n6=is ,n6=i Wn
+Â
Â
n i 6=n,i 6=is
ˆ
pnit0 (z) ynit0 (z) /tnit (dlogpnit (z)
Wn
dlogtnit ) dz
pnit0 (z) ynit0 (z) /tnit dlogpnit (z) dz.
8
(17)
.
Expressions (16) and (17) coincide. From expression equation (11) in the paper and (15),
the change in world GDP is given by:
E
dlogRGDPwt
=
+
+
1
Ewt0
1
Ewt0
1
Ewt0
  Exportsint0 dlogtint dz
i
n
 Wit0 Xit0 dlogXit
i
ÂÂ
ˆ
i n 6 =i Wi
⇥
dnit0
⇤
1 pnit0 (z) qnit0 (z) dlogqnit (z) dz.
which coincides with expression (14) in the paper.
2
Proofs and Derivations: Monopolistic competition, heterogeneous firms and quality
2.1
Deriving the share of profits in total revenues
We show that our assumption in equation (19) of the paper that aggregate profits repre´
sent a constant share of total revenues (Pit = ki Yit , where Yit = Ân W pint qint dMit ), is
int
satisfied in the remaining two special cases of our model described in Section 4. In the
first case, there are no fixed costs of selling into individual countries so that all firms sell
in each country. In the second case, there are positive fixed costs of selling in individual
countries (incurred in either the exporting or importing country) and productivities are
Pareto distributed. We derive equation (19) for the general case in which a fraction 1 f
of these fixed costs are incurred in the exporting country and a fraction f of these fixed
costs are incurred in the importing country. We also allow for a fraction 1 y of the innovation costs to be incurred in the exporting country and a fraction y to be incurred in the
importing country. The baseline model in the body of the paper assumes f = y = 0. We
also show that, in these two cases, the mass of firms is unchanged following a trade liberalization. Remember that in the third special case described in Section 4, when r ! 0, it is
straightforward to show that the free entry condition implies that ki = 0 in steady-state.
We start by re-writing some preliminary equations of the model to make this section
self-contained. First, aggregate profits in country i in period t net of fixed labor costs,
innovation costs, and entry costs are:
Pit = Yit
p
Wit Lit
Â
n
1 f f
Wit Wnt f int
ˆ
Wint
dMit
9
Â
n
ˆ
1 y
Wint
Wit
y
Wnt hint (z) dMit
Wit f Ei MEit ,
Aggregate revenues Yit = Ân Yint are proportional to variable labor payments:
Yit =
r
p
1
r
Wit Lit .
(18)
If h (z; a) = gg0 h̄ (z) ag , aggregate innovation expenditures are a constant fraction of aggregate revenues:
ˆ
1
1 y y
(19)
 W Wit Wnt h (z; ain ) dMit = rg Yit .
n
int
If the trade balance is a constant share k̄i of GDP, and condition (19) in the paper holds,
aggregate revenues can be written as:
Yit =
1
1 k̄¯i
W L ,
ki (1 k̄¯i ) it it
(20)
where k̄¯i = yk̄i /(y + rg) and if there are no fixed fixed costs of selling into individual
y
(q +1 r)(g 1)
countries, and k̄¯i = k̄i /(1 + (f
+ rg
) 1 ) if there are fixed costs of selling into
rqg
individual countries and productivities are Pareto distributed.
Suppose we are on a steady-state equilibrium in which aggregate variables are constant. In steady-state, the distribution of firms is given by Mi (z) = MdEi Gi (z) (we omit
time subscripts for the reminder of this section to simplify notation). The aggregate freeentry condition (equation B.6. in the paper) in steady-state is:
(r + d) Wi f Ei MEi = Yi
p
Wi Li
Â
n
1 f f
Wi Wn
[1
Gi (z̄in )] f in
Â
n
ˆ
1 y
Win
Wi
y
Wn hin (z) dMi .
(21)
In what follows, we solve for the constant of proportionality Pi /Yit = ki = k in
steady-state under two special cases of our model. We then show that, in these two special
cases, the aggregate response to a change in variable or fixed trade costs is immediate (i.e.
there are no transition dynamics), so that k remains constant over time.
Case 1: No fixed costs
Assume that there are no fixed costs of selling in individual countries, i.e. f ii = f in = 0,
so that there is no selection. In this case, the aggregate free entry condition (21) is:
"
Wi f Ei MEi
1
=
Y
d
r+d i
p
Wi Li
Â
n
ˆ
Wint
10
1 y y
Wi Wn h (z, ain
#
(z)) dMi (z) .
and using (18) and (19):
Wi f Ei MEi =
d g 1
Y.
r + d rg i
(22)
Aggregate profits are:
Pi = Yi
=
Â
Wi Li
n
r g 1
Y,
r + d rg i
ˆ
1 y
Win
Wi
y
Wn hin (z) dMi
Wi MEi f Ei
so k = r+r d grg1 . Note that if r > 0, aggregate cross-sectional profits are positive even
though discounted profits at entry are zero.
The steady-state mass of entering firms is given by:
MEi =
d g 1
r + d rg 1
1 k̄¯i
Li
,
ki (1 k̄¯i ) f Ei
where we used (20) and (22). Hence, the mass of entrants MEi does not change in response
to permanent changes in variable or fixed trade costs. Therefore, there are no transition
dynamics to the new steady-state, and kit = k.
Finally, aggregate variable profits gross of entry costs are: Pi + Wi MEi f Ei = grg1 Yi .
Hence, with restricted entry (so that there are no costs incurred in entry), equation (19) in
the paper holds with k = grg1 .
Case 2: Pareto distributed productivities
Now assume that there are positive fixed costs of selling in individual countries and that
the distribution of entering firms Gi is Pareto with shape parameter q, i.e. Gi = 1 z q
for z
1. We also assume that the productivity cutoffs are interior, z̄in > 1. We start by
noting that aggregate fixed labor costs are proportional to aggregate revenues.
MEi
d
1 f
 Wi
n
f
Wn f in [1
Gi (z̄in )] =
q+1
q
rg 1
Y.
rg i
(23)
Using (18), (19), and (23), we can write the aggregate free entry condition (21) as:
MEi Wi f Ei =
d r 1g 1
Yi ,
r + d rq
g
11
(24)
Finally, combining (18), (23) and (24), aggregate profits are
Pi =
so ki =
r r 1g 1
r +d qr g .
r r 1g 1
Yi
r + d qr
g
The steady-state mass of entering firms is given by:
MEi =
d r 1g 1
r + d rq
g 1
1 k̄¯i
Li
,
ki (1 k̄¯i ) f Ei
where we used (20), and (24). Hence, the mass of entrants MEi does not change in response to permanent changes in variable or fixed trade costs. Therefore, there are no
transition dynamics to the new steady-state.
r 1
Finally, aggregate variable profits gross of entry costs are Pi + Wi MEi f Ei = qr g g 1 Yi .
Hence, in the model with restricted entry (in which there are no entry costs), equation
(19) holds with k = g g 1 (r 1) / (qr).
12