Innovation and Production in the Global Economy
Online Appendix
Costas Arkolakis
Natalia Ramondo
Andrés Rodrı́guez-Clare
Princeton, Yale and NBER
UC-San Diego and NBER
UC Berkeley and NBER
Stephen Yeaple
Penn State and NBER
2017
In this Online Appendix we discuss a number of results that appear or are discussed
in the main paper. We also incorporate a number of robustness checks. We make use and
cite various definitions and equations from the main paper, which, for brevity, we do not
reintroduce here but cite accordingly.
1
Proof Lemma 2
We first define lk =
Lk
,t
∑j Lj i
∑ j L j Tje
∑k Lk
≡ Tie , and t ≡
= ∑j lj tj.
Lemma 2 Consider a world with no worker mobility, κ → 1, where Ai = A, and
(θ +1)(θσ−σ+1)
( θ − σ +1)
ti
t
<
∀i, and assume ρ → 1 for all i. The ratio of the real wage under frictionless
trade and MP to the real wage under free trade and no MP, Wi ≡ Wi∗ /Wi , is given by the
expression:
(Wi )θ =
[(1 − η ) t + ηti ]υ t1−υ
θ/(1+θ )
ti
1/(1+θ )
∑k tk
,
(O.1)
lk
where υ = θ/ (σ − 1) − 1.
p
Proof: Notice that κ → 1, Li = Lie = Li , and Mi = Li / f e . Since we focus on frictionless
trade, we have
(
Ψin = Tie
∑
k
ψiln =
h
h
p
p
Tk (γik wk )−θ
p
Tie Tl
p −θ
γil wl
1
i
1
1− ρ
/Ψi
i
)1− ρ
1
1− ρ
≡ Ψi ,
≡ ψil ,
(O.2)
(O.3)
and
LΨ
Mi Ψ i
= i i ≡ λiE .
∑ Mj Ψ j
∑ Lj Ψj
E
λin
=
T
Using the definition of λln
and imposing free trade we have
−(1−ρ)/θ

p

wn = 
T
λnn
∑i
p −θ
Tie Tn γin
/Ψin
1/(1−ρ)
E
λin
.


E
= λiE , we can write this expression as
Using equation (O.2) and λin
−(1−ρ)/θ

p

wn = 
λnT
p 1/(1−ρ)
Tn
∑i
−θ
Tie γin
/Ψi
1/(1−ρ)
λiE
.


(O.4)
We will use this expression and consider separately the two cases: zero MP costs and infinite
MP costs.
Frictionless trade but no MP. With frictionless trade and infinite MP costs we have λnT = λnE ,
p
p
and Ψn = Tie Tn (wn )−θ so that expression (O.4) yields

p
wn = 
−(1−ρ)/θ
λnT
1/(1−ρ)
p
Tne Tn /Ψn
λnE
p
p
,

whereas equation (24) becomes
λlE
=
λlT
p
!−θ
=
Ll Tle Tl (wl )−θ
,
(O.5)
λlT
λiT
/
p p
p.
Li Tl Ti Ll Tle Tl
(O.6)
p
p
∑ j L j Tje Tj (w j )−θ
which gives us
wi
=
p
wl
No MP implies Xl = Yl , and equation (15) implies
wl Ll = (1 − η ) λlT ∑ Xn ,
p p
n
and thus
p
wl
p
wi
=
λlT λiT
/ .
L l Li
2
Combined with (O.6) this equation yields
p
!1/(1+θ )
p
wl
Li Ml Tl Tle
p
Ll Mi Ti Tie
=
p
wi
.
(O.7)
Further, combining (O.5), (O.6), and(O.7) yields the following closed form solution for
wages:
p
p
wi
Tie Ti
=
!1/(1+θ )
.
p
Tne Tn
(O.8)
Using expression ( A.24), with κ → 1, and noting that with no MP we have Xnln = 0
T
except for l = n, λnn
= Xnnn /Xn , Xn = Yn , and rn = η, the expression for welfare is
Wn = ζ
Fn
Ln
σ −1− θ
θ ( σ −1)
Tne Ln / f
e 1/θ
Xnnn
Xn
− 1
θ
σ −1− θ
( 1 − η ) θ ( σ −1) .
Using expression (O.5), we can write
Wn = ζ
Fn
Ln
σ −1− θ
θ ( σ −1)
(1 − η )
p
p
f e Tn (wn )−θ
p
p
∑ j L j Tje Tj (w j )−θ
σ −1− θ
θ ( σ −1)
!− 1
θ
.
We can now substitute (O.8) to finally obtain

Wn = ζ
Fn
Ln
σ −1− θ
θ ( σ −1)
σ −1− θ
( 1 − η ) θ ( σ −1)
e
−
1
1+ θ
p
1
1+ θ
e
Tj
 ∑ j L j Tj


θ
p 1
f e ( Tne )− 1+θ Tn 1+θ
1
θ

 .

(O.9)
Frictionless trade and MP. Using (8) and (10) together with the definition of ψiln and imposing zero MP costs we get
λlT =
∑
k
But now we have Ψi = Ψ ≡
Tie
∑j
p
p
Tl (wl )−θ
Ψk
p
p
Tj (w j )−θ
λiE =
!
1
1− ρ
1
1− ρ
1− ρ
Li Tie
∑ j L j Tje
3
Mk Ψ k
.
∑ j Mj Ψ j
, hence
,
(O.10)
and
p
p
Tl (wl )−θ
λlT =
p
1−1 ρ
p
∑k Tk (wk )−θ
1−1 ρ .
(O.11)
Therefore, relative trade shares are
λlT
λiT
p 1/(1−ρ)
(wl )−θ/(1−ρ)
p 1/(1−ρ)
(wi )−θ/(1−ρ)
Tl
=
Ti
∑l Xnln
Xn ,
E
Using (7) and noting that λnn
≡
p
p
.
(O.12)
E
E
from the definition of λin
, and recalling that λnn
=
λnE , then ( A.24) can be rewritten as
Wn∗
=ζ
Since ψiln =
Fn
Ln
p −θ
p
Tl (wl
p
)
σ −1− θ θ ( σ −1)
1/θ
1
1− ρ
p
∑k ( Tk (wk )−θ )
Wn∗ = ζ
Tne Ln
fe
1
1− ρ
Fn
Ln
≡ ψn , λnE =
σ −1− θ
θ ( σ −1)
(ψn )
(ψnnn )−
Ln Tne
∑ j L j Tje
1− ρ
θ
− 1
θ
σ −1− θ
( 1 − r n ) θ ( σ −1)
p 1− ρ
p
and Tl = Ã1−ρ Ll
∑ j L j Tje
1− ρ
− θ
λnE
! 1θ
fe
we have
σ −1− θ
( 1 − r n ) θ ( σ −1) .
(O.13)
We want to find the expression for Wn∗ when ρ → 1. We first conjecture that under this limit
wages equalize and we a) derive an expression for the last parenthetical term of the welfare
expression; b) show that ψn tends to a constant, which is finite and bounded away from zero;
and then c) show that the wage equalization conjecture is true. Combining these three results,
the limit of the expression (O.13) as ρ → 1 is
Wn∗
=ζ
Fn
Ln
σ −1− θ
∑ j L j Tje
θ ( σ −1)
! 1θ fe
(1 − η ) t
(1 − η )t + ηtn
a) First show that wages equalize as ρ → 1 and show that
1 − rn =
(1 − η ) t
.
(1 − η )t + ηtn
From the free entry condition, we have
E
wne Len = η ∑ λik
Xk
k
4
σ −1− θ
θ ( σ −1)
.
(O.14)
E
Using λnk
=
Ln Tne
∑ j L j Tje
and multiplying by Xn we have rn = η
1 − rn = 1 − η
Ln Tne ∑k Xk
∑ j L j Tje Xn
so
Ln Tne ∑k Xk
.
∑ j L j Tje Xn
(O.15)
From the current account balance condition (17) combined with the labor market clearing
condition (15), we have
E
wn Ln + η ∑ λin
Xk = Xn ,
p p
k
E
and given that λin
= λiE =
Li Tie
∑ j L j Tje
, we obtain
p p
wn Ln
Ln Tne
Xn
+η
=
.
e
∑ k Xk
∑ k Xk
∑ j L j Tj
(O.16)
p p
Substituting this expression into (O.15) and using ∑k wk Lk = (1 − η ) ∑k Xk , we obtain
1 − rn = 1 − η
Ln Tne
∑ j L j Tje (1 − η )
1
p p
wn Ln
p p
∑k wk Lk
+η
Ln Tne
∑ j L j Tje
.
p
Finally, imposing wage equalization, Ln = Ln and then reorganizing the resulting expression
using the definitions of t, tn and ln yields
1 − rn =
(1 − η ) t
.
(1 − η ) t + ηtn
completing the derivation.
b) Now we want to show that under the condition in the proposition in this limit equilibrium all countries have, ψl > 0. To show that we compute the limit
p
lim ψl = lim
ρ →1
p
Tl (wl )−θ
1−1 ρ
1
p
p
∑k Tk (wk )−θ 1−ρ
1
= lim
1
ρ →1
p
p
Tk (wk )−θ 1−ρ
∑ k T p (w p )−θ
ρ →1
l
=
l
lim λlT .
Thus, we simply need to construct the trade shares in the case of wage equalization with
5
ρ → 1. The equilibrium conditions in a frictionless equilibrium are the current account
balance,
p p
Xi = w i L i + η
Ln Tne
∑ j L j Tje
∑ Xn ,
n
and labor market clearing,
θ−σ+1
p p
Xi + (1 − 1/σ) λiT ∑ Xn = wi Li ,
σθ
n
p 1− ρ
p
with λiT given by (O.11), with Ti = Ã1−ρ Li
. Adding up across the current account
balance conditions implies
1
∑ k Xk = 1 − η ∑ k w k L k .
p p
Combining the current account balance with labor market clearing and using this last result
together with the expression for λiT implies
θ−σ+1
σθ
p p
wi L i
Ln Tne
η
+
e
∑ j L j Tj 1 − η
∑
p
!
p p
w L
k k k
+ (1 − 1/σ)
p
Li (wi )−θ/(1−ρ)
p
p
∑k Lk (wk )−θ/(1−ρ)
p
1 1
ln (wn )−θ/(1−ρ)
η θ − σ + 1 Ln Tne
+
=
=⇒
e 1 − η ∑k lk (w p )−θ/(1−ρ)
1−η
σθ
∑ j L j Tje σ
k
1
1−η
θ−σ+1
1−
σθ
∑ k wk Lk
p p
p p
wn L n
p p.
∑k wk L k
(O.17)
This system, together with a normalization for wages, gives us a system of N non-linear
equations in N unknowns. Since wages are equalized in the limit as ρ → 1, we can let
p
wi = 1, and we have from (O.17) that
λnT
p
ln (wn )−θ/(1−ρ)
θ−σ+1
θ − σ + 1 ln t n
e (1 − η ) 1 −
e
= lim
ln − σ
η
,
p −θ/(1−ρ) = σ
σθ
σθ
t
ρ →1 ∑ k l k ( w )
k
and in order for that to be positive we need to assume that
tn
(σθ − σ + 1)
(1 + θ ) > .
t
( θ − σ + 1)
(O.18)
This is the condition required in the proposition for interior solution. Notice that, around
symmetry, tn = t, since the left-hand side of this equation is always strictly greater than 1.
c) In the last step, we want to show that in the limit as ρ → 1 equilibrium wages are
equalized. Equation (O.17) can be rewritten as
6
p p
= wi L i
ai /li +
where



−θ/(1−ρ)

p
wi
−(1−ρ)/θ 
p
∑k lk (wk )−θ/(1−ρ)
p
= bwi .
(O.19)
θ−σ+1
θ − σ + 1 tii
e (1 − η ) 1 −
e
η and b ≡ σ
.
ai ≡ σ
σθ
t
σθ
Assumption (O.18) is then
0 < b − ti /li .
p
p
Since, given the normalization of one wage to one, max wv ≥ 1, so that letting j = arg maxv wv ,
p
we then have b max wv − a j /l j > 0, and (O.19) implies
p
max wv
p
∑k lk (wk )−θ/(1−ρ)
Note that
lim
ρ →1
h
∑
p
−(1−ρ)/θ
= b max wv − a j /l j
i
p −θ/(1−ρ) −(1−ρ)/θ
l
(
w
)
k
k k
−(1−ρ)/θ
.
(O.20)
p
= min wk ,
and thus the left-hand side of (O.20) when we take the limit is,




p
max wv
p
max wv
.
= lim
lim −(
1
−
ρ
/θ
)
p
 ρ→1 min wvp
ρ →1 
∑k lk (wk )−θ/(1−ρ)
In addition, taking the limit on the right-hand side of (O.20)
lim
ρ →1
n
p
b max wv − a j /l j
−(1−ρ)/θ o
p
=1,
p
since b max wv − a j /l j is bounded away from zero and must be b max wv − a j /l j ≤ 1 since
the left-hand-side of (O.20) is always greater or equal to 1. Hence, taking limits of (O.20) we
have
p
lim
ρ →1
max wv
p
min wv
!
= 1.
which means that wages equalize.
We have completed the derivations of the two analytical equations for no MP and fric-
7
tionless MP. Combining equations (O.9) and (O.14) we obtain

Wn
Wn
Wnθ
1
θ
1
p 1+
θ
− 1+θ θ
σ −1− θ
Tn
∑ j L j Tje 
 ( Tne )
θ ( σ −1)
t


= 
=⇒
− 1 1 
(1 − η )t + ηti
e
1+ θ
p 1+ θ
e
Tj
∑ j L j Tj
1

1
θ
1− ρ 1+ θ
σ −1− θ
Ln
t


θ ( σ −1)
t

= 
=⇒
1 
 θ
(1 − η )t + ηti
− 1+1 θ
1− ρ 1+ θ
1+ θ
tn ∑ j l j t j
Lj
=
((1 − η )t + ηti )υ t1−υ
− 1+1 θ
θ
tn1+θ ∑ j l j t j
which is expression (O.1), with υ ≡ θ/ (σ − 1) − 1 where notice that under symmetry Wn =
1. This last derivation completes the proof of the Lemma.
2
Gains Under Unilateral MP liberalization
Proposition O.1 Consider a two-country world with perfect worker mobility (i.e., κ → ∞) and
with frictionless trade. Assume that A1 = A2 , T1e = T2e , and L1 = L2 and that countries are not
fully specialized in innovation or production. Let γ∗ ≡ (2θ − 1)(1−ρ)/θ and assume that γ21 < γ∗ ,
γ12 ≤ γ21 . Then country 1 gains when γ21 increases.
Proof: We will derive an expression for the real wage in country 1, w1 /P1 , and then use
p
p
that expression to prove part (ii). Notice that given that ω ≡ w1 /w2 we can normalize the
p
p
wage of country 2, w2 = 1, so that ω = w1 . Also, frictionless trade implies that the price
index is the same across the two countries, P1 = P2 ≡ P.
We first compute the price index. Irrespective of MP costs, under frictionless trade profits
p
p
in country i are ηL w1 + w2 λiE , where L ≡ L1 = L2 . But frictionless trade also implies that
λiE =
Mi Ψ i
,
M1 Ψ1 + M2 Ψ2
hence free entry requires
p
wi = η
L
Ψi
p
p
w1 + w2
.
e
f
M1 Ψ1 + M2 Ψ2
8
Adding up wages we get
p
p
w1 + w2 = η
L
Ψ1 + Ψ2
p
p
w1 + w2
,
e
f
M1 Ψ1 + M2 Ψ2
and hence
M1 Ψ1 + M2 Ψ2 = η
L
( Ψ1 + Ψ2 ) .
fe
(O.21)
Using (O.21) and the definition of the price index, equation (A.6), we have that
#−1/θ
" ( θ − σ +1) / (1− σ )
F
L
P = ζ −1
η e
(Ψ1 + Ψ2 )−1/θ .
L
f
(O.22)
By the definition of Ψi we have
h
Ψ1 + Ψ2 = T ω
− 1−θ ρ
+ (γ12 )
− 1−θ ρ
i 1− ρ
h
+ T (ωγ21 )
− 1−θ ρ
+1
i 1− ρ
.
(O.23)
Using the definition of ω = Ψ1 /Ψ2 we can show that ∂ ln ω/∂ ln γ1 > 0 (see part i). Also,
manipulating (O.23), we get
h
i 1− ρ
h
i 1− ρ
− θ
− θ
− θ
Ψ1 + Ψ2 = T ω 1−ρ + (γ12 ) 1−ρ
+ T (ωγ21 ) 1−ρ + 1
=
i 1− ρ
h
i 1− ρ
h
− 1−θ ρ
− 1−θ ρ
− 1−θ ρ
− 1−θ ρ
+ (γ12 )
+T ω
+ (γ12 )
/ω
= T ω
h
i 1− ρ
1
− θ
− θ
= T ω 1−ρ + (γ12 ) 1−ρ
1+
ω
(O.24)
(O.25)
We can now write the real wage of country 1, using (O.22), as
w1
=
P
ζ
−1
ω
i
h
i 1− ρ −1/θ
− 1−θ ρ
− 1−θ ρ
F ( θ − σ +1) / (1− σ ) L
η fe
T ω
+ (γ12 )
1+
L
h
h
ω
− 1−θ ρ
ω
θ
1− ρ
=
ζ
h
−1
1+ω
=
ζ −1
h
h
θ
1− ρ
+ω
θ
1− ρ
(γ12 )
− 1−θ ρ
F ( θ − σ +1) / (1− σ )
L
(γ12 )
− 1−θ ρ
i 1− ρ F ( θ − σ +1) / (1− σ )
L
η fLe
η
L
fe
i 1− ρ i−1/θ
1+
i−1/θ
1
ω
1+
T −1/θ
1/θ
T −1/θ
9
1
ω
1/θ
1
ω
−1/θ
Thus, changes in the welfare in country 1 are given by
h
∂
p
∂ w1 /P
=
∂γ1
1+ω
θ
1− ρ
(γ12 )
− 1−θ ρ
i 1− ρ 1+
1
ω
∂ω
∂ω
.
∂γ1
Since, from part (i) we know that ∂ω/∂γ1 > 0, it suffices to show that the first term is positive.
But the sign of this first term is determined by the sign of
θ
θ
ω 1−ρ (γ12 )
1+ω
θ
1− ρ
− 1−θ ρ
(γ12 )
− 1−θ ρ
−
1
.
ω+1
(O.26)
Notice that (O.26) is increasing in ω, thus if we show that is positive for γ12 = γ21 =⇒
ω = 1 it will be satisfied for any γ21 > γ12 =⇒ ω > 1. We will show that if γ2 satisfies a
p
∂(w /P)
certain condition, this expression is positive and thus ∂γ1
> 0 . Notice that for (O.26) to
21
be positive we need
θ
θω 1−ρ (γ12 )
ω
θ
1− ρ
− 1−θ ρ
θ
(ω + 1) > 1 + ω 1−ρ (γ12 )
[θ (ω + 1) − 1] > (γ12 )
− 1−θ ρ
=⇒
θ
1− ρ
For γ12 = γ21 =⇒ ω = 1 and as long as γ12 < γ∗ ,
γ∗ ≡ (2θ − 1)(1−ρ)/θ .
the inequality is satisfied, proving the second part of the proposition; QED.
3
Anti Home Market Effects in a Two-Sector Trade Model
Consider two countries the Home and the Foreign, denoted by H and F, respectively. Labor
is the only factor of production and L H < L F . There are two goods and consumers in the
two countries have symmetric Cobb-Douglas preferences over the goods, with α of total
expenditure devoted to the ek-good, and a share 1 − α of expenditure devoted to the k-good.
Each country can potentially produce both goods. The ek good, is produced under perfect
competition and is composed of a continuum of varieties ω ek ∈ [0, 1] that are aggregated
in a CES fashion with an elasticity of substitution σek > 1. These varieties are produced
in country i = H, F with a linear technology and productivity zi (ω ). There is an iceberg
cost of shipping the ek-good τ in ≥ 1, for i 6= n and τ ii = 1. Productivity is drawn from
10
a Fréchet distribution of the form Fi = e Ti z−θ , with Ti > 0 and θ > 1. We assume that
TH /L H = TF /L F = 1. The technology in this sector is identical to the technology postulated
in the perfect competition setup of Eaton & Kortum (2002).
The k-good is produced using a continuum of differentiated varieties, aggregated CES
with elasticity of substitution σk > 1. Each variety in country i = H, F is produced with
a linear production function using labor and with identical productivity across countries.
In order to produce a firm has to incur a fixed cost of entry f e . There is an iceberg cost of
shipping the k-good, γin ≥ 1, for i 6= n, and γii = 1. Firms are homogeneous and compete
monopolistically and there is free entry into this sector. The (endogenous) number of firms
is denoted by Mi , i = H, F. The technology in this sector is identical to the technology
postulated in the monopolistic competition setup of Krugman (1980).
The equilibrium is defined as firm entry, Mi , and wages, wi , for the two countries i = H, F
such that labor markets clear and free entry drives profits to zero in both countries.
We prove the following Lemma:
Lemma O.1 Assume that γ HF = γ FH = 1, τ HF = τ FH > 1, and L H > L F . Then the smaller
country specializes in the Krugman sector.
Proof: First, notice the free entry condition implies that the equilibrium entry in the Krugman sector is given by
Mi =
ri
L , i = H, F.
σk f e i
(O.27)
Thus, the market shares for each of the countries in the two sectors can be written as
ek
λin
=
k
λin
=
Ti (wi τ in )−θ
∑k Tk (wk τ kn )
−θ
Mi (wi γin )1−σ
∑k Mk (wk γkn )
1− σ
=
=
Li (wi τ in )−θ
∑k Lk (wk τ kn )
−θ
,
ri Li (wi γin )1−σ
∑k rk Lk (wk γkn )
1− σ
.
Using those expressions we now have to solve for equilibrium wages, wi , and entry ri , using
the three of the four labor market clearing conditions for the two sectors and one wage normalization, say w H = 1. Therefore, the equilibrium conditions for r H , r L and w H and w F are
w H = 1 and
w F (1 − r F ) L F = α ∑ λek
Fn wn Ln ,
(O.28)
k
wi ri Li = (1 − α) ∑ λin
wn Ln for i = H, F.
(O.29)
n
n
11
We can now prove the lemma. We start by proving that wages equalize in both countries
when γin = 1. Using the equilibrium condition for the k-sector,
( wi ) σ = (1 − α )
∑n wn L n
for i = H, F ,
1− σ
∑k rk Lk ( wk )
which implies that w H = w F . Imposing the normalization and using the equation above we
obtain a condition
∑ r k L k = (1 − α ) ∑ L n ,
(O.30)
n
k
which combined with (O.28) can be used, in turn, to solve for ri , i = H, F, and thus
rF = 1 − α ∑
n
(τ Fn )−θ Ln
∑k Lk (τ kn )
−θ
.
(O.31)
We substitute (O.31) into (O.30) to obtain
rH LH +
1−α∑
n
(τ Fn )−θ Ln
∑k Lk (τ kn )
LF
r H = (1 − α ) + α
LH
LF
r H = (1 − α ) + α
LH
−1 +
!
−θ
L F = (1 − α) ∑ Ln =⇒
1
∑k
Lk
LF
(τ kF )−θ
− LLHF (τ HF )−θ
1+
LH
LF
(O.32)
n
(τ HF )−θ
+
+
(τ FH )−θ
∑k
Lk
LH
LF
LH
(O.33)
(τ kH )−θ
!
(τ FH )−θ
1+
!
(τ FH )−θ
.
(O.34)
We show that L H > L F implies r H < 1 − α, under symmetry of trade costs. Notice that this
term is negative iff
(τ FH )
−θ
LH
1+
(τ HF )−θ
LF
LH
LF
−θ
−θ
<
1+
=⇒
(τ HF )
(τ FH )
LF
LH
LH
<
(τ HF )−θ + (τ HF )−θ (τ FH )−θ =⇒
LF
L
(τ FH )−θ + H (τ HF )−θ (τ FH )−θ
LF
−θ
−θ
−θ
τ
τ
−
τ
(
)
(
)
(
)
HF
FH
HF
LH
> 1,
L F (τ )−θ (τ )−θ − (τ )−θ
HF
FH
FH
and under symmetry of trade costs, the last parenthetical term is negative iff L H > L F , proving the result. It is straightforward to prove that r F > 1 − α using equation (O.30). QED
12
4
Plant-Level Fixed Location Costs: The case with ρ = 0
In this Section we discuss the incorporation of plant-level fixed production cost under the
special case with ρ = 0. While we maintain the rest of the assumptions of the model, we assume that for firms from i to open an affiliate in l there is a fixed cost ϕl in units of production
labor of country l. For convenience, we write this cost as ϕl = υl Fl .
We show that fixed costs of opening a plant can be incorporated into the special case of
our model with ρ = 0 in such a way that the resulting extension is isomorphic to the existing
model without plant-level fixed costs.
As noted in the discussion of the multivariate Pareto distribution, for this parameterization an entrant from a country i will draw a single country where it can produce with the
p
p
probability of this country being l is given by Tl / ∑k Tk and the productivity level given by
a Pareto distribution with shape parameter θ. Additionally, we consider only equilibria in
which the following restriction is satisfied:
p
(1 + υ l )
wl Fl /Xl
p
wn Fn /Xn
<
τ ln Pl
Pn
σ −1
for all i, l and n 6= l.
(O.35)
Note that under symmetry, this condition becomes 1 + υ < τ σ−1 . Hence, this condition
naturally extends a similar condition in Helpman, Melitz, and Yeaple (2004) to the case of
asymmetric countries.
Given these assumptions, we first show that the equilibrium is such that all affiliates in
l sell in l and that there is a positive measure of those affiliates that do not sell in n for all
n 6= l – that is, there is selection of foreign affiliates into all export markets. We begin by
defining the variable profits for a firm from i with productivity zl producing in l and selling
to n, which are given by
1
e iln (zl ) =
π
σ
e ξ iln
σ
zl
1− σ
Pnσ−1 Xn .
T , are
The productivity cutoff for a domestic firm to export to country n, denoted here by zln
p
T ) = w F . From the definition of variable profits, the cutoff is
e lln (zln
defined implicitly by π
n n
given by
p
T
zln
=
σwn Fn
Xn
!1/(σ−1)
e ξ lln
σ
.
Pn
e iln (zl ) = γ1il−σ π
e lln (zl ), the
Because the variable profits of any foreign firm from i 6= l satisfy π
cutoff productivity for all firms from i that have a plant in country l to sell to country n is
T ≡ γ zT .
ziln
il ln
13
Now, define π il (zl ) to be the profits net of marketing costs (but gross of fixed investment
costs) for a firm from i with productivity zl producing in l,
π il (zl ) =
∑ max
n
p
e iln (zl ) − wn Fn , 0 .
π
The MP productivity cutoffs for firms from i to open an affiliate in country l, zilMP , are defined
implicitly by
p
π il (zilMP ) = wl ϕl .
We can now prove the following lemma that establishes that there exists affiliates that do not
export:
T for all i, l and n 6 = l.
Lemma O.2 If condition (O.35) is satisfied, then zilMP < ziln
Proof. Consider a firm from i in l with productivity
p
1/(σ−1) σ
e wl
zil∗ = γil σwl ( Fl + ϕl ) /Xl
.
Pl
p
p
e ill (zil∗ ) − wl Fl = wl ϕl by construction, this firm would make zero profits, and so
Because π
would break even, if it sells only in market l. Now consider the additional profits that could
be earned by selling in market n 6= l:
1
p
e ξ iln /zil∗ )1−σ Pnσ−1 Xn − wn Fn
(σ
σ


!1− σ
p
e γil wl τ ln
σ
Xn
p
− 1
= wn Fn 
p
∗
zil Pn
σwn Fn
1− σ
σ −1 !
p
w
F
+
ϕ
/X
(
)
τ
P
τ
P
l
l
p
ln l
ln l
l
l
= wn Fn
−
< 0,
p
Pn
Pn
wn Fn /Xn
p
e iln (zil∗ ) − wn Fln =
π
where the inequality at the end follows from the assumption in the lemma. This implies that
∑ max
n
p
p
p
e iln (zil∗ ) − wn Fn , 0 = π
e ill (zil∗ ) − wl Fl = wl ϕl
π
and hence
zilMP
=
zil∗
=
p
γil σwl
p
1/(σ−1) σ
e wl
.
( Fl + ϕl ) /Xl
Pl
T then follows directly from the definition of z T and the assumption in the
Finally, zilMP < ziln
iln
lemma.
14
The next step is to solve for price indices as a function of entry levels and wages. Integrating using the Pareto distribution of entrants and substituting for the cutoffs for exporting, the
price indices must satisfy
Pn1−σ =
∑ ∑ Mi Til
i l 6=n
+ ∑ Mi Tin
i
1− σ

p
σwn Fn
!1/(σ−1)
 σ − θ −1
p
e γin wn τ ln 
σ
θ

σ−θ−1
Xn
Pn

 σ − θ −1
#1/(σ−1)
"
p
p
e
σw
F
σ
γ
w
θ
p 1− σ
n n
in n 

e γin wn
σ
(1 + υ n )
σ−θ−1
Xn
Pn
p
e γil wl τ ln
σ
Note that we have substituted ϕl = υl Fl to obtain this expression. Letting µll = 1 and µln = 0
if l 6= n, we can solve for Pn , which can be written compactly as
Pn−θ =
θ
θ − ( σ − 1)
p
σwn Fn
Xn
!1−θ/(σ−1)
∑ Mi Til
i,l
p
e γil wl τ ln
σ
−θ
(1 + µln υn )1−θ/(σ−1) .
(O.36)
Now, integrating over the sales of the individual exporters that originate from i and sell to n
from country l, we obtain
e ξ iln )
Xiln = Mi Til (σ
1− σ
Pnσ−1 Xn
θ
θ − ( σ − 1)
and
e ξ ill )
Xill = Mi Til (σ
1− σ
Plσ−1 Xl
θ
θ − ( σ − 1)
T
γil zln
σ − θ −1
γil zllMP
for n 6= l
σ − θ −1
.
Substituting for the price index using (O.36), we have
Xiln
Mi Til (ξ iln )−θ (1 + µln υn )1−θ/(σ−1)
=
Xn
−θ
p
1−θ/(σ−1)
(1 + µkn υn )
∑ j,k Mk Tjk γ jkl wk τ kn
for all n and l.
Finally, the equilibrium (with ∆ = 0) is a set of firms, production worker wages, and
innovation worker wages for each country, M, w p , we such that the market for innovation
labor clears,
η ∑ Xiln = wie Lie ,
n,l
15
(O.37)
and the market for production labor clears,
p p
wl Ll
1
1
= ∑ Xiln + 1 − η −
Xl .
e i,n
e
σ
σ
(O.38)
Note that these two equations are the same as in the case without fixed costs up to the calculation of Xiln , which we now address.1
p
To link the equilibrium without fixed costs to the one with fixed costs, let τ ln and Tl be
the parameters for trade costs and productivity in the model without fixed costs and denote
with tildes the ones in the world with fixed costs. Then, in the world without fixed costs, we
have
−θ
p
γil wl τ ln
=
− θ Xn ,
p
p
e
∑ j,k Mk Tj Tk γ jkl wk τ kn
p
Xiln
Mi Tie Tl
and in the world with fixed costs, we have
−θ
p
γil wl τ̃ ln
(1 + µln υn )1−θ/(σ−1)
=
Xn .
−θ
1−θ/(σ−1)
e T̃ p γ w p τ̃
M
T
1
+
µ
υ
(
)
∑ j,k k j k
jkl k kn
kn n
p
Xiln
Mi Tie T̃l
Thus, if we set
p
p
T̃l ≡ Tl
and
τ̃ ln ≡
τ ln
(1 + µln υn )1/(σ−1)−1/θ
then the two equilibria yield the same (M, w p , we ). Note that
τ̃ ll ≡ τ ll = 1 for all l
and
τ̃ ln ≡ τ ln (1 + υn )1/(σ−1)−1/θ > 1 for l 6= n,
where the inequality is strict for all υn > 0 since θ > σ − 1 by assumption.
Note that prices are also the same in the two equilibria. Prices in the equilibrium without
1 One may have thought that we needed to subtract M w ϕ from η
∑n,l Xiln , but recall that ηXill are the
il l l
profits net of marketing costs for firms from i producing in l and selling domestically in l. When the cutoff is
p
p
p
determined by wl ( Fl + ϕl ) rather than wl Fl , then this means that ηXill is already net of wl ( Fl + ϕl ), and hence
it should not be subtracted again.
16
fixed costs are given by
Pn−θ =
θ
θ − ( σ − 1)
p
w Fn
σ n
Xn
!1−θ/(σ−1)
∑ Mi Tie Tl
p
i,l
p
e γil wl τ ln
σ
−θ
,
while in the equilibrium with fixed costs we would have
Pn−θ =
θ
θ − ( σ − 1)
p
w Fn
σ n
Xn
!1−θ/(σ−1)
∑ Mi Tie T̃l
p
i,l
p
e γil wl τ̃ ln
σ
−θ
(1 + µln υ)1−θ/(σ−1) .
Because (M, w p , we ) are the same and given the definition of the variables with tildes, then
prices must also be the same. Finally, the gravity relationships of the two models are also the
same as long as “own country” pairs are excluded.
Finally, because our counterfactuals entail moving τ 0 s or γ0 s by a certain percent change,
the counterfactual implications of a percent change in the τ 0 s (with no transformation of γ0 s)
are the same if we lived in a world with fixed investment costs as long as the key inequality
(O.35) holds.
5
Isomorphism
In this Section we present a formal isomorphism of our model where the firm has a multivariate Pareto productivity to one where each firm’s productivity in a location is the product of
a core productivity and a location-specific efficiency shock as in Tintelnot (2017). The results
are independent from the assumptions on labor mobility across sectors.
5.1
Environment
The basic environment of the model with location-specific productivity shocks is the same as
in the main paper. The difference is that a firm’s productivity is the product of two random
variables: a “core productivity” parameter φ and a vector of location specific productivity
adjustment parameters, z = (z1 , z2 , ..., z N ). A firm with productivity variables φ and z producing in country l has labor productivity φ × zl .
These assumptions imply that a firm with productivity φ and z producing in location l to
p
serve market n has a unit cost given by wl τ ln / (φzl ). Such a firm chooses to serve country n
17
from the cheapest production location l, charging a price
pin = min { piln }
σ
min
σ−1 l
(
p
γil wl τ ln
φzl
)
.
We assume that φ is drawn from a Pareto distribution,
κ
bi
φ ∼ Fi (φ) = 1 −
,
φ
where κ + 1 − σ > 0, while zl for firms from i are drawn i.i.d from a Fréchet distribution with
parameters θ and Til ,
−θ
zl ∼ e−Til z .
p
We again assume that firms incur a destination-specific fixed cost wn Fin in order to have
the possibility of serving market n. In addition, we assume that these fixed costs are paid
before the vector of location-specific efficiency shocks z is observed, but knowing the firm’s
core productivity φ. We assume that firms have to decide ex-ante how many markets they
might end up serving; once, the vector z is observed, the set of possible markets to serve
is given. Hence, firms choose to pay the entry cost to the destination market n if expected
profits are larger than entry costs. Firms make this calculation market-by-market.
5.2
Firm’s Problem
∗ for which
The entry decision into each market n gives us a threshold productivity level φin
p
∗ pay the fixed cost w F and serve n and firms with
firms from country i with φ ≥ φin
n in
∗ do not. To derive φ∗ , imagine first that a firm from i with productivity vector ( φ, z )
φ < φin
in
is forced to supply a country n from l. The revenues of this firm would be ( Pn /piln )σ−1 Xn .
Let viln ≡
σ −1
σ φpiln .
Expected profits (gross of the fixed marketing cost) for such a firm are
Pnσ−1
σ
σ−1
1− σ
−σ
φσ−1 ( Xn /σ) E(v1iln
).
18
Letting G (z; T ) ≡ 1 − e−Tz , then we know that viln is distributed according to G (viln ; (γil wl τ ln )−θ Til ).
θ
Next, notice that for every c ≥ 1 we have
c
)]
[ E(viln
1/c
=
"Z
∞
0
c
v dG (v;
σ
γ wτ
σ − 1 il l ln
#1/c
−θ
=Γ
Til )
θ+c
θ
1/c h
(γil wl τ ln )
−θ
Til
i−1/θ
,
(O.39)
where we denote γc = Γ
θ +c
θ
1/c
.
This result implies that expected profits in this case are
−σ
Pnσ−1 φσ−1 ( Xn /σ ) E(v1iln
) = Pnσ−1 φσ−1 ( Xn /σ)γ̃
where
γ̃ =
σ
σ−1
h
p
γil wl τ ln
−θ
Til
i(σ−1)/θ
,
1− σ θ+1−σ
Γ
.
θ
In fact firms in i that are not forced to supply a country n from l, they will choose the production location that minimizes delivery cost, hence
"
E
max Pnσ−1
l
σ
σ−1
#
1− σ
φ
σ −1
( Xn /σ)(v1iln−σ )
=
Pnσ−1
σ
σ−1
1− σ
φσ−1 ( Xn /σ) E(min viln )1−σ .
l
However, vin ≡ minl viln is distributed according to G (vin ; Ψin ), where
Ψin ≡
∑
j
−θ
p
γij w j τ jn
Tij .
(O.40)
We finally have that the expected profits we are interested in are
Pnσ−1 φσ−1 ( Xn /σ) Bin ,
where
(σ−1)/θ
Bin ≡ E(min viln )1−σ = γ1−σ Ψin
l
∗
This definition implies that Pnσ−1 φin
σ −1
"
∗
φin
= σ̃
.
(O.41)
p
( Xn /σ) Bin = wn Fin , and hence
p
wn Fin
Xn
1
Bin
#
1
σ −1
1
.
Pn
(O.42)
Given that the expected marginal cost of producing and shipping the good from country i to
19
n, using production location l is
)!
p
γil wl τ ln
E min σ̃
∗
zl φin
l
(
)! "
# 1
p
p
1− σ
γil wl τ ln
wn Fin 1
E min
σ̃ σ
Pn
zl
Xn Bin
l
# 1
"
p
1− σ
wn Fin 1
−1/θ
Ψ1/θ
Ψin σ̃ σ
in Pn
Xn γ 1− σ
"
# 1
p
1− σ
1
w
F
n
in
σ
−
1
γ 1 ( γ 1− σ )
σ̃ σ
Pn .
Xn
(
c∗n =
=
=
=
where we have made use of definition (O.43). Notice that this expected cutoff cost, is the
same as the realized cutoff cost, up to a constant, in the main paper.
We can also calculate the share of firms from i serving in n that choose to do so from
p
location l, ϕiln . Since viln is distributed G (vin ; (γil wl τ ln )−θ Til ) and l = arg minl viln , then
standard results with the Fréchet distribution imply that ϕiln (which is also the share of sales
by firms in i in n that is produced in location l) imply that
p
ψiln
5.3
γ w τ ln
= il l
Ψin
−θ
Til
.
(O.43)
Price index
To calculate the price index Pn , note that since piln =
σ
σ−1 viln /φ
and the measure of firms
from country i with φ is Mi dFi (φ) then
Pn1−σ
=
σ
σ−1
1− σ
∑ Mi
Z ∞
∗
φin
1− σ
E min viln
l
φσ−1 dFi (φ)
and hence
(σ−1)/θ κ
bi
Pn1−σ = γ1 ∑ Mi Ψin
∗ −(κ +1−σ)
,
(φin
)
where
c1 ≡ γ̃
κ
.
κ+1−σ
Plugging in from (O.42) we get (after some simplifications)
20
p
Pn = c2
wn
Xn
!(κ +1−σ)/κ (σ−1) "
−(κ +1−σ)/(σ−1)
∑ biκ Mi Fin
i
#−1/κ
Ψκ/θ
in
,
(O.44)
where
c2 ≡ c1 −1/κ (σ/γ̃)(κ +1−σ)/κ (σ−1) .
5.4
Market Shares
We will construct and make use of three market shares that are relevant for our model and
potentially empirically relevant. To introduce these shares, let Xiln denote the sales to market
n by firms originating in country i that are produced in country l. Notice that total spending
E
T
and total output are given by Xn = ∑i,l Xiln and Yl = ∑i,n Xiln . The share λin
, λln
, λilM , can be
derived using the formulas (8), (10), (11).
We can use standard arguments about price indices in this kind of environment to get
−(κ +1−σ)/(σ−1)
E
λin
=
Mi biκ Fin
Ψκ/θ
in
−(κ +1−σ)/(σ−1)
∑ j M j bκj Fjn
Ψκ/θ
jn
.
(O.45)
In relation to the main paper, this expression is ‘distorted’ through the term Fin which, of
course, cancels out if Fin = Fn . Notice that the term Ψin is slightly different from that paper,
but very closely related.
To get trade shares, recall that firms from i that sell in n produce a share ψiln in country n
(equation 6). Hence the import share by n from l is the weighted average of these production
E
shares across i weighed by the λin
,
T
λln
=
∑ ψiln λinE .
(O.46)
i
To get MP shares, note that the value of goods produced in l by firms from i to be delivered
E
E
to n is ψiln λin
Xn , hence the total value of goods produced in l by firms from i is ∑n ψiln λin
Xn ,
and so
λilM =
E
Xn
∑n ψiln λin
.
E
∑ j,n ψ jln λ jn Xn
Thus, these expression are all as in the main paper.
21
(O.47)
5.5
Profits
Let us now compute total profits made by firms from i from their production in country l,
E
Xn /σ.
Πil . We know that total variable profits made by firms from i in country l are ∑n ψiln λin
What are the fixed costs paid by these firms? The measure of firms in country i that serve
∗ −κ , hence
country n through location l is ψiln Mi biκ φin
Πil =
∑
n
E
∗ −κ p
ψiln λin
Xn /σ − ψiln Mi biκ (φin
) wn Fin .
(O.48)
To proceed, note that from (O.42) and (O.41) we have
biκ
∗ −κ
(φin
)
biκ (γ̃)κ/(σ−1) Ψκ/θ
in
=
!κ ,
h p i σ−1 1
wn Fin
1
σ Xn
Pn
whereas from (O.44) and (8) we have
p
wn
Xn
Pn−κ = c2−κ
!−(κ +1−σ)/(σ−1)
−(κ +1−σ)/(σ−1)
Mi biκ Fin
E
Ψκ/θ
in /λin ,
hence
biκ
∗ −κ
(φin
)
= ψiln Mi h
biκ γ2κ (γ̃)κ/(σ−1) Ψκ/θ
in
p
σ wXn Fnin
i σ−κ 1 = c2κ (γ̃/σ)κ/(σ−1)
p
wn
Xn
−(κ +1−σ)/(σ−1)
−(κ +1−σ)/(σ−1)
Mi biκ Fin
E
Ψκ/θ
in /λin
E
λin
Xn
p
Mi wn Fin
Using expression (O.48) and the above definitions we have that
Πil =
∑ ψiln λinE Xn /σ − ∑ ψiln c2κ (γ̃/σ)κ/(σ−1)
n
= (1/σ − c2κ (γ̃/σ)
n
κ/(σ−1)
E
λin
Xn p
wn Fin
p
wn Fin
E
) ∑ ψiln λin
Xn
n
so
Πil = ηλilM Yl ,
where
η≡
σ−1
,
σκ
22
(O.49)
with the interesting result the realized share of profits is constant but depends on κ.
5.6
Equilibrium
p
Wages, spending, and number of firms, wi , Xi and Mi solve the exact same system of equaE
tion in Section 2.3 in the paper, provided that λin
and ψiln are the same. Notice that the
occupational choice model does not interfere with the rest of the derivations in the model as
p
the equilibrium equations are the same given wi , Xi and Mi , which in turn are fully determined by the equations above. Given that we proceed to characterize welfare in the model.
5.7
Welfare
To look at welfare, we start with
T
λnn
p
Using Ψin ≡ ∑l wl /τ ln
−θ
p −υ
wn
=∑
p
i ∑k wk τ kn
!−1/θ
T
λnn
p
(σ−1)/θ
E
λin
=
γbiκ Mi Ψin
p 1/(σ−1)
∗ = σ wn Fin
Plugging in for φin
Xn B
in
∗
φin
−(κ +1−σ)
Pn1−σ
1
Pn
(σ−1)/θ
and Bin = ΓΨin
c1−κ biκ Mi Ψκ/θ
in
p
wn Fin
Xn
1/θ
c1 bn−1 Mn−1/κ Ψ−
nn
we then get
−(κ +1−σ)/(σ−1)
Pnκ +1−σ
E
λin
hence
Pn =
.
E
Tin /Ψin
∑i λin
We also have
=
Tik
E
λin
.
Til this implies
wn =
Pn1−σ
Tin
−θ
p
wn Fnn
Xn
23
!(κ +1−σ)/κ (σ−1)
E
λnn
1/κ
and real wage is
p
wn
Pn
=
T
λnn
−1/θ E
λnn
−1/κ c1
p
wn Fnn
Xn
E
Tin Ψnn /Ψin
∑i λin
(κ +1−σ)/κ (σ−1)
1/θ
bn Mn1/κ
Let εiln ≡ Xiln / ∑ j X jln , then we can show that
Ψnn
∑ Tin Ψin πin =
i
E
Tnn λnn
,
εnnn
so finally we have
Xn
=
Pn
( Ln /(1 − η ))
κ +1− σ 1
σ −1 κ +1
Xn
Yn
κ +1− σ 1 +1 σ −1
κ
T
λnn
c1 ( Fnn )
−1/θ
−(κ +1−σ)
σ −1
1/θ −1/κ
E
(εnnn )−1/θ λnn
( Tnn )1/θ
−1/κ
bn Mn1/κ ,
(O.50)
so the gains from openness are
GO =
=
Xn
Yn
Xn
Yn
κ +1− σ 1 + 1 σ −1
κ
T
λnn
κ +1− σ 1 + 1 σ −1
κ
−1/θ
Xnnn
Xn
(εnnn )
−1/θ −1/θ
∑l Xnln
Xn
E
λnn
1/θ −1/κ
1/θ −1/κ
(O.51)
T
E
where we used the definitions of λnn
, εnnn , and λnn
. This expression is the same as the main
paper as long as 1/θ = (1 − ρ) /θ̃ and 1/κ − 1/θ = ρ/θ̃ so that
1/κ = 1/θ̃, 1/θ = (1 − ρ) /θ̃ .
Notice that the case of ρ = 0 in the main paper corresponds to the case κ = θ here.
5.8
Formally Connecting the Models
We finally formally connect this model to the model in the main text. The variables with
tildes correspond to the variables of the model in the main text.
i) Parameters γ, τ, T and L are the same,
ii) We set Fin = Fn = F̃n
iii) Set 1/κ = 1/θ̃, 1/θ = (1 − ρ̃) /θ̃ =⇒ θ̃ = κ,
24
θ̃
1−ρ̃
=θ
iv) Set wages, wi = w̃i , spending, Xi = X̃i , and entry, Mi = M̃i , the same.
E
E
v) Given those, we can define Ψin = Ψ̃in , ψiln = ψ̃iln and λin
= λ̃in since
Ψin ≡
∑
γij w j τ jn
−θ
Tij =
j
∑
γ̃ij w̃ j τ̃ jn
−
θ̃
1− ρ
j
1/(1−ρ̃)
Tij = Ψ̃in
,
and
−(κ +1−σ)/(σ−1)
E
λin
=
=
Mi biκ Fin
Ψκ/θ
in
−(κ +1−σ)/(σ−1)
∑ j M j bκj Fjn
M̃i b̃iκ Ψ̃in
Ψκ/θ
jn
E
∑ j M̃ j b̃κj Ψ̃in
= λ̃in
and also
ψiln =
=
(γil wl τ ln )−θ Til
Ψin
(γ̃il w̃l τ̃ ln )−θ̃/(1−ρ̃) T̃il
1/(1−ρ̃)
Ψ̃in
= ψ̃iln
completing the isomorphism of all the variables in the two models. Notice that these derivaT
M
T
tions also imply λln
= λ̃ln and λilM = λ̃il .
vi) We finally show that the price index is the same up to a constant. Using (O.44) we can
write the price index as
Pn = γ2
wn
Xn
(κ +1−σ)/κ (σ−1) "
−(κ +1−σ)/(σ−1)
Mn bnκ Fnn
Ψκ/θ
nn
#−1/κ
E
λnn
and using the definitions above
Pn = c
= c

−(κ +1−σ)/(σ−1)
κ
 M̃n b̃n F̃n

w̃n
X̃n
(κ +1−σ)/κ (σ−1)
w̃n
X̃n
(κ +1−σ)/κ (σ−1) "
1/(1−ρ̃) (1−ρ̃)
Ψ̃in


E
λ̃nn
−(κ +1−σ)/(σ−1)
M̃n b̃nκ F̃n
E
Ψ̃in
−1/θ̃
#−1/θ̃
= P̃nn
λ̃nn
E
E
given that λnn
= λ̃nn , so that this would imply Pn = P̃n .
vii) The final step is to show that the models solve the same equilibrium conditions. Given
25
the above definitions and the discussion in Subsection 5.6 this holds completing the derivation of the isomorphism between the two setups.
6
Process Innovation
Process innovation involves a conscious effort on the part of firms to lower their marginal
cost of production, but like all innovation involves uncertainty. Suppose that when firms
enter, in our model, each entrant in country i can augment its Tie by a proportion ai . The cost
of this possibility is aiα /α in terms of home labor.
From the firm’s perspective, the relevant concern is how process innovation will affect its
expected costs of serving foreign markets. Adapting (A.2), we have

"
N
Pr (Ci1n ≥ ci1n , ..., Ciln = ciln , ..., CiNn ≥ ciNn ) = θ  ∑ ( Tie ai ) Tk
p
k =1
p
−θ
( Tie ai ) Tl ξ iln
1
1− ρ
ξ ikn
cikn
−θ # 1−1 ρ
−ρ

θ/(1−ρ)−1
ciln
×
.
Assuming as before that marketing fixed costs are such that firms do not operate on the
support of the distribution, we still have
Pr arg min Cikn = l ∩ min Cikn = c
k
= Pr (Ci1n ≥ c, ..., Ciln = c, ..., CiNn ≥ c)
k
= θ (Ψin ai )
ρ
− 1− ρ
p
−θ
( Tie ai ) Tl ξ iln
1
1− ρ
c θ −1
= ai ψiln Ψin θcθ −1 ,
where Ψin and ψiln continue to be given by the expressions in the text. Given this linearity,
(A.3) becomes
Pr arg min Cikn = l ∩ min Cikn ≤
k
and hence,
Pr min Cikn ≤
k
k
c∗n
c∗n
= ai ψiln Ψin (c∗n )θ
= ∑ ai ψikn Ψin (c∗n )θ = ai Ψin (c∗n )θ .
k
The expected sales of the firm from i through l to n that has invested in process innovation
26
of ai are
e
E ( xiln ) = ai ψiln Ψin σ
1− σ
Xn Pnσ−1
Z c∗
n
0
θcθ −σ dc,
e 1− σ θ
σ
ψiln Ψin Xn Pnσ−1 (c∗n )θ −σ+1 ,
= ai
θ−σ+1
and the expected marketing costs are
Pr min Cikn ≤
k
c∗n
wn Fn = wn Fn ∑ ai ψikn Ψin (c∗n )θ ,
p
p
=
k
p
ai wn Fn Ψin
(c∗n )θ .
Thus, the total expected profits (net of innovation costs) of serving n for a firm from i choosing process innovation level n are
e 1− σ
σ
θ
p
∗ θ
Xn Pnσ−1 (c∗n )1−σ − wn Fn
Eπ in ( ai ) = ai Ψin (cn )
θ−σ+1 σ
σ−θ 1
σ−1
Xn Pnσ−1 σ
p
= ai
Ψ
wn Fn
p
1− σ
θ − σ + 1 in
wn Fn σ
e
!
Summing over all n and subtracting off the process innovation cost and the entry cost, the
total expected profits of innovation level ai
( ai )α
e
EΠi ( ai ) = ∑
+ fi
α
n
σ−θ 1
α
σ−1
Xn Pnσ−1 σ
p
e ( ai )
e
= ai ∑
wn Fn − wi
Ψin
+ fi .
p
α
wn Fn σ
e 1− σ
n θ−σ+1
Eπ in ( ai ) − wie
The first-order condition for the choice of the optimal level of process innovation is
σ−1
∑ θ − σ + 1 Ψin
n
Xn Pnσ−1 σ
p
wn Fn σ
e 1− σ
σ−θ 1
p
wn Fn = wie ( ai )α−1 .
Substituting the first order condition into the zero expected profit condition yields
( ai )α =
α
f e.
α−1 i
27
To complete the analysis, we can define the country innovation parameter as
ee = T e
T
i
i
α
fe
α−1 i
1/α
,
(O.52)
and the country entry fixed cost parameter as
( ai )α
e
e
fi =
+ f ie ,
α
α
1
α
e
=
f
+ f ie .
α α−1 i
(O.53)
This expression shows that total innovation costs are the sum of the process innovation costs
(first term) and product innovation costs (second term). All the derivations in the paper are
therefore consistent with a model in which firms can choose the level of process innovation
at the time of entry with the parameters Tie and f ie adjusted to reflect the bundling of process
and product innovation.
7
7.1
Gravity: Alternative Estimations
Poisson PML estimation
By estimating the restricted and unrestricted gravity equations via Ordinary Least Square
(OLS), we assume that any source of variation in the error term, εln , is orthogonal to the
independent variables. This is a strong assumption, but one that is routinely made in gravity
equation estimations. However, if εln is heteroscedastic, OLS estimates may be biased (see
Silva & Tenreyro (2006)). To study the effect of this bias, we re-estimate both the restricted
and unrestricted gravity equations using the Poisson pseudo-maximum-likelihood (PML)
estimator proposed by Silva & Tenreyro (2006). We report the coefficient estimates in Table 1.
7.2
Instrumental-variable estimation
We instrument trade costs with distance dummies, border, and language dummies, as well
as an index of the quality of a country’s infrastructure from the World Competitiveness Report.
Trade costs are the sum of freight costs (in logs), calculated from cif/fob U.S. imports and
then assumed to be symmetric for U.S. exports, and inward and outward tariffs combined
as log(1 + til /100) × (1 + tli /100). As dependent variable, we use a Head-Ries measures of
trade flows: for restricted gravity, log( Xili × Xiil ) − log( Xill × Xiii ) with i = USA; and for
28
Table 1: Restricted and unrestricted gravity, PPML estimator.
Tariff
Distance Dummies
D2
D3
D4
D5
D6
Other Gravity Controls
Self Border Lang. Chi-sq.
Restricted
-8.4
(2.6)
-0.4
(0.3)
-1.7
(0.4)
-0.4
(0.2)
-1.6
(0.4)
-1.2 2.2
(0.6) (0.5)
0.3
(0.4)
-0.2
(0.2)
6006
Unrestricted
-5.4
(1.7)
-0.6
(0.1)
-0.6
(0.3)
-0.8
(0.1)
-1.7
(0.2)
-1.3 2.8
(0.2) (0.2)
0.8
(0.2)
-0.2
(0.2)
34975
Notes: Robust standard errors in parenthesis.
Table 2: Restricted and unrestricted gravity, instrumental variables (2SLS).
OLS
Restricted Unrestricted
Trade costs
Constant
Observations
R-sq.
IV
Restricted Unrestricted
(1)
-11.8
(1.02)
-5.5
(1.10)
(2)
-6.9
(3.10)
-6.9
(1.00)
(3)
-14.38
(4.98)
-4.69
(1.42)
(4)
-9.74
(5.22)
-5.99
(1.59)
45
0.21
45
0.10
45
0.20
45
0.08
Notes: The dependent variable for the restricted gravity equation is log( Xili × Xiil ) − log( Xill × Xiii ) with i = USA, while
for the unrestricted gravity equation is log( Xli × Xil ) − log( Xll × Xii ). The variable ”trade costs” is the sum of freight
costs (in logs), and inward and outward tariffs combined as log(1 + til /100) × (1 + tli /100) with til being the tariff rate
for goods from i to l. Trade costs are instrumented in columns 3 and 4 using distance dummies, self, border, and language
dummies, as well as an index of quality of infrastructure. Robust standard errors in parenthesis.
unrestricted gravity, log( Xli × Xil ) − log( Xll × Xii ). We choose to use these indices, rather
than the trade flows (in logs) directly, because the data on bilateral freight costs are available
only for the United States, impeding the inclusion of two sets of country fixed effects in the
gravity equations. Table 2 presents the results.
7.3
Estimation using other tariff-weighting schemes
We estimate the restricted and unrestricted gravity equations using tariff measures that are
calculated using as the common weights the value of global trade in that industry divided by
the value of total global trade. The underlying tariff data is from the World Integrated Trade
Solution (WITS) data provided by the World Bank and calculated at the H.S. six-digit level
29
Table 3: Restricted and unrestricted gravity, alternative tariff weights. OLS
Tariff
Distance Dummies
D2
D3
D4
D5
D6
Other Gravity Controls
Self Border Lang. R-sq.
Restricted
-12.8
(-3.5)
-0.5
(0.5)
-0.4
(0.6)
-1.4
(0.6)
-2.9
(0.5)
-2.7 2.1
(0.6) (0.7)
0.2
(0.6)
0.6
(0.3)
0.84
Unrestricted
-5.6
(1.9)
-0.7
(0.3)
-0.7
(0.3)
-0.7
(0.3)
-2.0
(0.3)
-1.7 3.3
(0.3) (0.4)
1.0
(0.3)
0.0
(0.1)
0.89
Notes: Robust standard errors in parenthesis.
for the year closest to 1999 for which data were available. The elasticity estimates, obtained
via OLS, are shown in Table 3.
8
Additional Tables
30
Table 4: Aggregate Variables. Data and Model.
Xidata
Ximodel
L̄i
Yidata
Yimodel
ridata
rimodel
∆i /Xidata
Australia
0.042
Austria
0.024
Benelux
0.078
Brazil
0.093
Canada
0.079
China
0.303
Cyprus
0.001
Denmark
0.014
Spain
0.089
Finland
0.017
France
0.152
United Kingdom 0.154
Germany
0.263
Greece
0.016
Hungary
0.010
Ireland
0.011
Italy
0.172
Japan
0.598
Korea
0.105
Mexico
0.081
Poland
0.024
Portugal
0.018
Romania
0.007
Sweden
0.030
Turkey
0.042
United States
1.000
0.024
0.020
0.056
0.102
0.073
0.292
0.001
0.014
0.066
0.010
0.139
0.149
0.267
0.010
0.017
0.005
0.103
0.511
0.117
0.077
0.023
0.020
0.006
0.020
0.053
1.000
0.053
0.028
0.077
0.184
0.096
1.501
0.002
0.024
0.084
0.019
0.179
0.183
0.382
0.013
0.035
0.011
0.121
0.523
0.200
0.266
0.126
0.033
0.065
0.031
0.109
1.000
0.04
0.02
0.09
0.10
0.09
0.33
0.00
0.02
0.09
0.02
0.16
0.16
0.30
0.01
0.01
0.02
0.19
0.66
0.12
0.08
0.02
0.02
0.01
0.03
0.04
1.00
0.03
0.02
0.07
0.10
0.08
0.32
0.00
0.01
0.07
0.01
0.15
0.15
0.30
0.01
0.02
0.01
0.11
0.56
0.13
0.08
0.02
0.02
0.01
0.02
0.05
1.00
0.137
0.140
0.212
0.140
0.130
0.139
0.158
0.192
0.133
0.203
0.174
0.160
0.176
0.151
0.045
0.092
0.150
0.182
0.160
0.127
0.115
0.084
0.149
0.169
0.150
0.179
0.128
0.140
0.213
0.145
0.130
0.140
0.157
0.193
0.135
0.201
0.175
0.160
0.178
0.156
0.046
0.090
0.150
0.180
0.163
0.130
0.116
0.087
0.158
0.166
0.159
0.180
0.151
0.101
-0.189
0.058
-0.012
0.003
0.392
-0.057
0.086
-0.188
-0.028
0.044
-0.073
0.271
0.196
-0.275
-0.018
-0.053
-0.031
0.103
0.128
0.182
0.090
-0.104
0.052
0.034
Average
0.12
0.21
0.14
0.13
0.15
0.15
0.03
Country Name
0.13
Note: Expenditure Xidata is from the WIOD, an average over 1996-2001, for manufacturing; output Yidata is calculated
using the expenditure data and bilateral trade shares in manufacturing, from the WIOD, an average over 1996-2001,
T data data
as Yidata = ∑n (λin
) Xn ; labor L̄i is equipped labor from Klenow & Rodrı́guez-Clare (2005), an average for the
nineties, adjusted by the share of manufacturing employment from UNIDO; the trade and MP deficits are calculated
as ∆i = Xidata − Yidata (1/e
σ) + Xidata (1 + θ − σ )/(σθ ) + η ∑l (λilM )data Yldata , where the bilateral MP shares are from Ramondo
et al. (2015), an average over 1996-2001; and the innovation share is calculated as ridata = 1 − (Yidata (1/e
σ ) + Xidata (1 + θ −
data
model
model
model
σ )/(σθ ))/( Xi − ∆i ). The variables X
,Y
, and r
are as implied by the calibrated model with trade and MP
imbalances. Variables X, Y, and L̄ are relative to the United States.
31
Table 5: Gains from Openness, Trade, and MP. Baseline calibration.
GO, overall
GO, direct
GO, indirect
GT
GMP
(1)
(2)
(3)
(4)
(5)
Australia
Austria
Benelux
Brazil
Canada
China
Cyprus
Denmark
Spain
Finland
France
United Kingdom
Germany
Greece
Hungary
Ireland
Italy
Japan
Korea
Mexico
Poland
Portugal
Romania
Sweden
Turkey
United States
1.207
1.346
1.602
1.038
1.490
1.034
1.372
1.324
1.112
1.291
1.190
1.267
1.182
1.130
1.440
1.895
1.111
1.051
1.049
1.167
1.133
1.265
1.139
1.395
1.058
1.098
1.422
1.383
1.529
1.069
1.572
1.074
1.372
1.280
1.160
1.245
1.177
1.281
1.171
1.140
1.730
2.182
1.135
1.035
1.056
1.224
1.209
1.405
1.150
1.403
1.068
1.076
0.848
0.973
1.048
0.970
0.949
0.963
1.000
1.034
0.959
1.037
1.011
0.989
1.009
0.992
0.833
0.868
0.978
1.015
0.993
0.953
0.937
0.900
0.990
0.994
0.990
1.020
0.924
1.033
1.097
0.987
1.056
0.978
1.370
1.104
0.996
1.074
1.040
1.029
1.025
1.071
0.948
0.990
1.014
1.026
1.011
1.018
0.988
0.968
1.076
1.052
1.026
1.032
1.117
1.110
1.278
1.003
1.068
1.000
1.004
1.071
1.019
1.093
1.073
1.130
1.102
1.000
1.163
1.277
1.015
1.027
1.012
1.010
1.014
1.067
0.998
1.125
0.995
1.053
Average
1.246
1.290
0.971
1.036
1.070
Note: The gains from openness refer to changes in real expenditure between autarky and the calibrated equilibrium.
The direct and indirect effects refer to the first and second terms, respectively, on the right-hand side of (27). The gains
from trade (MP) refer to changes in real expenditure between an equilibrium with only MP (trade) and the calibrated
equilibrium with both trade and MP. Changes are with respect to the baseline calibrated equilibrium without trade and
MP imbalances, ∆ = 0.
32
Table 6: MP Liberalization. Baseline calibration.
% change in:
innovation share
r
real expenditure
X/P
real production wage
w p /P
real innovation wage
we /P
Australia
Austria
Benelux
Brazil
Canada
China
Cyprus
Denmark
Spain
Finland
France
United Kingdom
Germany
Greece
Hungary
Ireland
Italy
Japan
Korea
Mexico
Poland
Portugal
Romania
Sweden
Turkey
United States
-11.49
-0.52
10.63
-4.14
-3.06
-5.33
0.00
2.33
-5.94
2.88
1.02
2.16
1.29
-1.85
-14.80
7.52
-3.70
0.99
-1.02
-8.26
-9.40
-9.72
-2.65
2.51
-2.14
0.75
3.40
3.00
5.61
0.26
3.03
0.26
0.06
2.30
0.93
2.81
2.24
3.36
2.71
0.11
3.76
4.94
0.83
0.72
0.49
0.81
0.95
2.64
0.02
3.63
-0.01
1.32
3.83
3.05
4.17
0.61
3.27
0.70
0.06
2.01
1.41
2.45
2.13
3.14
2.57
0.28
4.24
4.60
1.16
0.61
0.59
1.45
1.60
3.18
0.27
3.38
0.20
1.23
-2.72
2.73
11.08
-1.84
1.44
-2.45
0.06
3.49
-2.11
4.28
2.76
4.46
3.37
-0.82
-4.23
8.81
-1.05
1.22
-0.03
-3.44
-3.91
-2.47
-1.32
4.93
-1.08
1.70
Average
-2.00
1.93
2.01
0.88
Note: MP liberalization refers to a five-percent decrease in all MP costs with respect to the baseline calibrated values. The
variable we is the wage per efficiency unit in the innovation sector, while w p is the wage per efficiency unit in the production
sector. Percentage changes are with respect to the baseline calibrated equilibrium without trade and MP imbalances, ∆ = 0.
33
Table 7: The Rise of the East. Baseline calibration.
% change in:
r
China in autarky
w p /P we /P X/P
MP liberalization into China
r
w p /P we /P X/P
Frictionless MP into CHN from US
r
w p /P we /P
X/P
Australia
Austria
Benelux
Brazil
Canada
China
Cyprus
Denmark
Spain
Finland
France
United Kingdom
Germany
Greece
Hungary
Ireland
Italy
Japan
Korea
Mexico
Poland
Portugal
Romania
Sweden
Turkey
United States
-0.57
-1.02
-1.04
0.07
-2.26
19.55
0.00
-1.18
0.19
-2.98
-0.60
-0.58
-1.57
-0.03
0.06
-0.79
-0.37
-1.81
-1.18
0.34
0.24
-0.51
-0.03
-1.13
0.02
-2.53
-0.47
-0.80
-0.51
-0.18
-0.48
-4.84
-1.57
-1.24
-0.13
-0.56
-0.67
-0.50
-0.28
-0.35
-0.22
-0.33
-0.57
-0.57
-0.27
-0.14
-0.09
-0.30
-0.29
-0.45
-0.22
-0.48
-0.77
-1.40
-1.16
-0.14
-1.78
5.74
-1.57
-1.96
-0.02
-2.40
-1.03
-0.84
-1.23
-0.36
-0.19
-0.76
-0.79
-1.67
-0.98
0.06
0.05
-0.58
-0.31
-1.12
-0.21
-2.03
-0.49
-0.89
-0.64
-0.17
-0.65
-3.29
-1.57
-1.38
-0.12
-0.91
-0.73
-0.56
-0.44
-0.35
-0.22
-0.37
-0.60
-0.76
-0.39
-0.11
-0.07
-0.33
-0.29
-0.55
-0.22
-0.76
0.67
-0.30
0.95
-0.11
1.16
-11.95
0.00
0.22
-0.10
1.68
0.32
0.48
1.00
-0.07
-0.31
0.58
-0.04
1.21
0.68
-0.25
-0.10
-0.18
-0.05
0.86
-0.06
1.34
0.10
0.01
0.10
0.01
0.14
1.73
-0.08
-0.03
0.02
0.17
0.02
0.06
0.13
-0.02
0.08
0.12
0.00
0.13
0.09
0.03
0.03
0.03
-0.02
0.10
0.00
0.16
0.45
-0.17
0.70
-0.06
0.80
-5.45
-0.08
0.11
-0.03
1.22
0.21
0.35
0.73
-0.07
-0.09
0.43
-0.03
0.86
0.50
-0.11
-0.03
-0.08
-0.05
0.61
-0.04
0.98
0.12
-0.02
0.22
0.00
0.22
0.75
-0.08
0.00
0.02
0.38
0.06
0.11
0.23
-0.03
0.07
0.14
-0.01
0.26
0.16
0.01
0.02
0.02
-0.03
0.18
-0.01
0.31
-4.40
3.22
-0.81
-0.89
-10.63
-92.25
0.00
0.69
-0.03
-3.73
-0.77
-1.32
-4.66
0.09
2.03
-5.56
0.56
-0.86
-1.60
-3.36
0.72
0.51
-0.07
-2.34
0.04
21.87
0.64
0.67
0.20
0.10
0.87
43.88
0.75
1.31
0.22
-0.05
0.44
0.33
-0.02
0.10
0.17
0.92
0.54
0.09
-0.06
0.59
0.08
0.10
0.04
0.24
0.18
2.27
-1.75
2.56
-0.31
-0.43
-5.38
-62.65
0.75
1.75
0.20
-2.37
-0.03
-0.46
-2.85
0.16
1.24
-2.16
0.87
-0.43
-1.02
-1.36
0.49
0.38
0.00
-1.16
0.20
15.75
0.48
0.95
0.09
0.02
0.09
34.20
0.75
1.40
0.22
-0.50
0.36
0.21
-0.51
0.11
0.23
0.68
0.59
0.00
-0.21
0.33
0.13
0.13
0.03
0.02
0.18
4.85
Average
0.01
-0.64
-0.67
-0.65
-0.09
0.12
0.06
0.12
-3.98
2.10
-2.23
1.72
Note: China in autarky refers to the counterfactual scenario in which trade and MP costs from/to China are set to infinity;
MP liberalization into China refers to the counterfactual scenario in which MP costs into China are decreased by ten
percent; frictionless MP into China from USA refers to the counterfactual scenario in which MP costs from the United
States into China are set to one. The variables are: real expenditure, X/P; innovation share, r; real wage in production,
w p /P; and real wage in innovation, we /P. Percentage changes are with respect to the baseline calibrated equilibrium
without trade and MP imbalances, ∆ = 0.
34
Table 8: The Fall of the West. Baseline calibration.
% change in:
Australia
Austria
Benelux
Brazil
Canada
China
Cyprus
Denmark
Spain
Finland
France
United Kingdom
Germany
Greece
Hungary
Ireland
Italy
Japan
Korea
Mexico
Poland
Portugal
Romania
Sweden
Turkey
United States
Average
Rise in MP barriers from US
r
w p /P we /P X/P
r
”Brexit I”
w p /P we /P
X/P
r
”Brexit II”
w p /P we /P
X/P
26.55
5.72
9.31
6.10
35.30
6.20
0.00
5.73
3.73
6.60
8.78
15.68
9.25
1.78
6.86
35.56
4.78
3.95
4.11
19.08
0.83
2.26
0.16
9.08
0.77
-16.67
-5.44
-0.04
-2.32
-0.74
-5.00
-0.67
-0.31
0.00
-0.16
-0.20
-0.68
-3.06
-0.82
-0.15
-0.69
-7.08
-0.23
-0.27
-0.41
-2.23
0.22
-0.01
-0.02
-1.33
-0.04
-1.29
7.40
3.29
3.36
2.78
13.51
2.88
-0.31
3.54
1.99
3.88
4.57
5.84
4.69
0.91
2.88
9.84
2.56
2.12
2.02
8.28
0.69
1.23
0.07
3.97
0.41
-11.52
-4.53
0.45
-1.14
-0.22
-2.41
-0.17
-0.31
0.69
0.14
0.61
0.26
-1.60
0.16
0.02
-0.48
-5.66
0.19
0.16
-0.01
-0.77
0.28
0.11
0.00
-0.45
0.03
-3.07
0.09
-0.15
0.02
0.02
0.01
0.00
0.00
0.01
-0.45
-0.22
0.06
-0.63
0.13
0.03
-1.18
0.03
0.04
0.02
0.01
0.00
-0.34
-0.45
-0.01
-0.51
0.01
0.06
0.01
-0.04
-0.55
0.01
0.00
0.00
-1.42
-0.05
-0.18
-0.17
-0.01
-0.47
0.00
-0.01
-0.12
-0.02
0.01
0.00
0.01
0.00
-0.09
-0.10
-0.02
-0.34
0.02
0.01
0.05
-0.12
-0.53
0.03
0.01
0.01
-1.42
-0.04
-0.44
-0.30
0.03
-0.84
0.08
0.00
-0.74
0.00
0.03
0.02
0.02
0.00
-0.28
-0.35
-0.02
-0.64
0.03
0.04
0.01
-0.05
-0.54
0.01
0.00
0.00
-1.42
-0.05
-0.22
-0.20
0.00
-0.53
0.01
-0.01
-0.15
-0.02
0.01
0.01
0.01
0.00
-0.11
-0.12
-0.02
-0.38
0.02
0.01
0.14
0.26
-0.33
0.01
0.10
0.01
0.00
0.04
-0.14
-0.26
0.05
-2.35
-0.07
0.11
-0.48
-1.44
0.19
0.07
0.02
0.00
0.08
0.92
0.08
-0.73
0.03
0.28
0.01
-0.07
-0.54
0.02
0.02
0.01
-1.37
-0.16
-0.18
-0.31
-0.14
-1.35
-0.23
-0.05
-0.21
-0.47
0.02
0.01
0.02
0.00
-0.12
-0.50
-0.04
-0.48
0.04
0.02
0.08
0.08
-0.75
0.03
0.08
0.01
-1.37
-0.13
-0.26
-0.47
-0.11
-2.73
-0.28
0.02
-0.46
-1.25
0.13
0.05
0.03
0.00
-0.08
0.00
0.01
-0.92
0.06
0.19
0.01
-0.05
-0.58
0.02
0.03
0.01
-1.37
-0.15
-0.19
-0.34
-0.13
-1.57
-0.24
-0.04
-0.22
-0.53
0.04
0.02
0.02
0.00
-0.12
-0.45
-0.03
-0.55
0.04
0.05
8.13
-1.27
3.11
-0.68
-0.13
-0.13
-0.21
-0.14
-0.13
-0.23
-0.31
-0.24
Note: Rise in MP barriers from U.S. refers to an increase in γil of 20 percent, for i = US, i 6= l. ”Brexit I” refers to an increase
in τ il of five percent, for i = GBR, l ∈ EU, and i 6= l, ”Brexit II” refers to an increase in τ il and γil of five percent each, for
i = GBR, l ∈ EU, and i 6= l. The variables are: real expenditure, X/P; innovation share, r; real wage in production, w p /P;
and real wage in innovation, we /P. Percentage changes are with respect to the baseline calibrated equilibrium without
trade and MP imbalances, ∆ = 0.
35
Table 9: Gains from Openness, Trade, and MP. Calibration with ρ = 0.
r
GO, overall
GO, direct
GO, indirect
GT
GMP
(1)
(2)
(3)
(4)
(5)
(6)
Australia
0.093
Austria
0.128
Benelux
0.184
Brazil
0.127
Canada
0.116
China
0.125
Cyprus
0.150
Denmark
0.165
Spain
0.121
Finland
0.175
France
0.155
United Kingdom 0.143
Germany
0.156
Greece
0.141
Hungary
0.042
Ireland
0.076
Italy
0.135
Japan
0.161
Korea
0.144
Mexico
0.115
Poland
0.105
Portugal
0.077
Romania
0.136
Sweden
0.148
Turkey
0.136
United States
0.164
1.080
1.156
1.378
1.026
1.264
1.023
1.188
1.170
1.068
1.190
1.117
1.161
1.128
1.077
1.156
1.314
1.074
1.040
1.030
1.101
1.070
1.114
1.084
1.215
1.043
1.086
1.176
1.193
1.320
1.060
1.327
1.060
1.188
1.147
1.112
1.153
1.110
1.172
1.119
1.090
1.407
1.476
1.097
1.026
1.038
1.157
1.142
1.247
1.106
1.218
1.062
1.067
0.918
0.970
1.044
0.968
0.953
0.966
1.000
1.020
0.960
1.032
1.007
0.991
1.008
0.988
0.822
0.890
0.979
1.014
0.992
0.952
0.937
0.893
0.980
0.997
0.982
1.018
0.969
1.055
1.135
0.987
1.117
0.985
1.188
1.129
1.009
1.113
1.056
1.046
1.044
1.057
0.949
1.062
1.031
1.027
1.012
1.038
1.004
0.977
1.074
1.098
1.021
1.041
1.044
1.068
1.256
1.008
1.087
1.007
1.007
1.059
1.021
1.104
1.067
1.104
1.091
1.009
1.087
1.144
1.023
1.029
1.010
1.016
1.016
1.045
1.005
1.107
1.005
1.063
Average
1.129
1.164
0.972
1.047
1.057
0.131
Note: The gains from openness refer to changes in real expenditure between autarky and the calibrated equilibrium.
The direct and indirect effects refer to the first and second terms, respectively, on the right-hand side of (27). The gains
from trade (MP) refer to changes in real expenditure between an equilibrium with only MP (trade) and the calibrated
equilibrium with both trade and MP. Changes are with respect to the baseline calibrated equilibrium without trade and
MP imbalances, ∆ = 0.
36
Table 10: MP Liberalization. Calibration with ρ = 0.
% change in:
innovation share
r
real expenditure
X/P
real production wage
w p /P
real innovation wage
we /P
(1)
(2)
(3)
(4)
Australia
Austria
Benelux
Brazil
Canada
China
Cyprus
Denmark
Spain
Finland
France
United Kingdom
Germany
Greece
Hungary
Ireland
Italy
Japan
Korea
Mexico
Poland
Portugal
Romania
Sweden
Turkey
United States
-11.54
-3.74
10.13
-4.93
-3.25
-5.33
0.00
1.42
-6.08
2.63
0.26
0.65
0.68
-1.94
-26.19
-4.96
-3.38
1.18
-1.41
-7.46
-10.29
-15.68
-2.95
0.65
-2.85
1.23
1.45
1.66
5.27
0.28
2.32
0.26
0.17
1.28
0.60
2.17
1.52
2.51
2.04
0.24
2.90
3.57
0.59
0.65
0.26
0.53
0.63
1.61
0.15
2.48
0.14
1.36
2.05
1.94
4.07
0.64
2.54
0.64
0.17
1.14
1.02
1.89
1.50
2.45
1.98
0.40
3.48
3.78
0.85
0.53
0.37
1.02
1.24
2.27
0.38
2.42
0.36
1.24
-4.58
-0.26
10.47
-2.23
0.65
-2.45
0.17
2.00
-2.50
3.50
1.65
2.84
2.39
-0.74
-11.60
0.96
-1.13
1.24
-0.45
-3.29
-4.69
-6.70
-1.34
2.81
-1.30
1.98
Average
-3.58
1.41
1.55
-0.48
Note: MP liberalization refers to a five-percent decrease in all MP costs with respect to the baseline calibrated values. The
variable we is the wage per efficiency unit in the innovation sector, while w p is the wage per efficiency unit in the production
sector. Percentage changes are with respect to the baseline calibrated equilibrium without trade and MP imbalances, ∆ = 0.
37
Table 11: Gains from Openness, Trade, and MP. Calibration with κ = 5.
r
GO, overall
GO, direct
GO, indirect
GT
GMP
(1)
(2)
(3)
(4)
(5)
(6)
Australia
0.055
Austria
0.150
Benelux
0.197
Brazil
0.145
Canada
0.128
China
0.139
Cyprus
0.167
Denmark
0.194
Spain
0.137
Finland
0.193
France
0.175
United Kingdom 0.158
Germany
0.172
Greece
0.161
Hungary
0.067
Ireland
0.065
Italy
0.151
Japan
0.178
Korea
0.162
Mexico
0.133
Poland
0.122
Portugal
0.102
Romania
0.160
Sweden
0.160
Turkey
0.160
United States
0.183
1.213
1.345
1.597
1.037
1.487
1.033
1.373
1.323
1.110
1.288
1.190
1.267
1.181
1.130
1.384
1.873
1.110
1.051
1.049
1.165
1.128
1.250
1.139
1.394
1.058
1.098
1.520
1.375
1.538
1.069
1.576
1.074
1.373
1.278
1.159
1.246
1.176
1.282
1.172
1.139
1.668
2.273
1.135
1.036
1.056
1.223
1.206
1.388
1.150
1.407
1.068
1.075
0.798
0.978
1.039
0.970
0.945
0.962
1.000
1.035
0.958
1.033
1.012
0.988
1.007
0.992
0.830
0.824
0.978
1.015
0.994
0.952
0.935
0.901
0.990
0.991
0.991
1.021
0.940 1.123
1.042 1.109
1.081 1.273
0.989 1.002
1.071 1.066
0.980 0.999
1.372 1.004
1.099 1.070
1.000 1.018
1.065 1.090
1.040 1.073
1.032 1.130
1.023 1.101
1.073 1.000
0.982 1.122
1.019 1.264
1.017 1.014
1.025 1.026
1.012 1.012
1.026 1.008
0.996 1.010
0.986 1.055
1.078 0.998
1.054 71.124
1.027 0.995
1.030 1.053
Average
1.24
1.29
0.97
1.04
0.15
1.07
Note: The gains from openness refer to changes in real expenditure between autarky and the calibrated equilibrium.
The direct and indirect effects refer to the first and second terms, respectively, on the right-hand side of (27). The gains
from trade (MP) refer to changes in real expenditure between an equilibrium with only MP (trade) and the calibrated
equilibrium with both trade and MP. Changes are with respect to the baseline calibrated equilibrium without trade and
MP imbalances, ∆ = 0.
38
Table 12: MP Liberalization. Calibrations with κ = 5.
% change in:
innovation share
r
real expenditure
X/P
real production wage
w p /P
real innovation wage
we /P
(1)
(2)
(3)
(4)
Australia
Austria
Benelux
Brazil
Canada
China
Cyprus
Denmark
Spain
Finland
France
United Kingdom
Germany
Greece
Hungary
Ireland
Italy
Japan
Korea
Mexico
Poland
Portugal
Romania
Sweden
Turkey
United States
-16.59
-1.12
19.29
-4.51
-6.11
-5.91
0.00
2.80
-7.13
3.10
0.80
2.24
0.90
-1.96
-20.66
13.28
-4.79
0.92
-1.14
-9.65
-11.36
-12.30
-2.65
2.04
-2.22
0.90
3.91
2.98
5.75
0.21
2.99
0.20
-0.04
2.30
0.85
2.83
2.25
3.36
2.78
0.10
3.62
5.05
0.78
0.71
0.48
0.73
0.87
2.53
0.01
3.65
-0.03
1.34
4.11
3.02
4.73
0.36
3.17
0.39
-0.04
2.17
1.07
2.67
2.22
3.27
2.74
0.18
3.92
4.85
0.95
0.67
0.52
1.03
1.19
2.81
0.11
3.57
0.06
1.30
0.20
2.75
9.55
-0.71
1.70
-1.01
-0.04
2.87
-0.63
3.46
2.42
3.82
2.96
-0.29
-1.07
7.70
-0.20
0.90
0.25
-1.29
-1.53
-0.13
-0.53
4.07
-0.47
1.52
Average
-2.38
1.93
1.96
1.39
Note: MP liberalization refers to a five-percent decrease in all MP costs with respect to the baseline calibrated values. The
variable we is the wage per efficiency unit in the innovation sector, while w p is the wage per efficiency unit in the production
sector. Percentage changes are with respect to the baseline calibrated equilibrium without trade and MP imbalances, ∆ = 0.
39
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40