$\frac{w_{s}(t)}{P_{s}(t)}=M_{s}(t)\times\alpha^{1-\eta}\times\left(\frac{T_{s}}{\lambda^{F}_{ss}(t)}\right)^{\frac{1}{\theta(1-\alpha)}}\times\left(\lambda^{M}_{ss}(t)\right)^{-1}\equiv
M_{s}(t)\times\mathcal{R}^{w}_{s}(t)$ (B.6)

Similarly, I can write aggregate profits as a linear function of $M_{s}(t)$:

$\displaystyle\Pi_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\lambda^{M}_{sd}(t)P_{d}(t)Y_{d}(t)$
$\displaystyle\Pi_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\frac{M_{s}(t)\left(\alpha^{-1}\tau_{sd}P_{s}(t)\right)^{1-\eta}}{(P_{d}^{M}(t))^{1-\eta}}P_{d}(t)Y_{d}(t)$

Therefore:

$\frac{\Pi_{s}(t)}{P_{s}(t)}=M_{s}(t)\times\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\left(\frac{\alpha^{-1}\tau_{sd}P_{s}(t)}{P_{d}^{M}(t)}\right)^{1-\eta}\frac{P_{d}(t)Y_{d}(t)}{P_{s}(t)}\equiv
M_{s}(t)\times\mathcal{R}^{\pi}_{s}(t)$ (B.7)

The budget constraint then is:

$\displaystyle C_{s}(t)+I_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{w_{s}(t)}{P_{s}(t)}L_{s}+\frac{\Pi_{s}(t)}{P_{s}(t)}$
(B.8) $\displaystyle=$ $\displaystyle
M_{s}(t)\times\left[\alpha^{1-\eta}\left(\frac{T_{s}}{\lambda^{F}_{ss}(t)}\right)^{\frac{1}{\theta(1-\alpha)}}\left(\lambda^{M}_{ss}(t)\right)^{-1}L_{s}+\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\left(\frac{\alpha^{-1}\tau_{sd}P_{s}(t)}{P_{d}^{M}(t)}\right)^{1-\eta}\frac{P_{d}(t)Y_{d}(t)}{P_{s}(t)}\right]$
$\displaystyle=$ $\displaystyle
M_{s}(t)\times\left[\mathcal{R}^{w}_{s}(t)+\mathcal{R}^{\pi}_{s}(t)\right]=M_{s}(t)\times\mathcal{R}_{s}(t)$

###  B.8 Balanced Growth Path

#### Autarky

#### Proof of Proposition (1)

###### Proof.

Without loss of generality, choose an arbitrary country $s\in\boldsymbol{K}$.
Since this world economy is under autarky, evaluate (2) replacing for the real
interest rate using equations (11) and (13) and taking the limit
$\tau_{sd}\to\infty(\forall s\neq d)$. By assumption (1), $\tau_{ss}=1(\forall
s)$. Therefore, (2) collapses to:

$g_{s}^{\text{autarky}}=\frac{\alpha\cdot\psi}{\eta}\cdot\frac{Y_{s}(t^{*})}{M_{s}(t^{*})}-\rho$
(B.9)

for a BGP inclusive of each period $t\geq t^{*}$.

The next step in the proof is to show that
$g_{M_{s}}=g_{Y_{s}}=g_{C_{s}}=g_{w_{s}}=g_{A_{s}}=g_{s}^{\text{autarky}}$.
Since real wages, real profits, assets, and real output are linear functions
of $M_{s}(t)$ in each period, it follows that
$g_{M_{s}}=g_{Y_{s}}=g_{w_{s}}=g_{A_{s}}$. Since, with log preferences,
consumption is a constant fraction of output, $g_{C_{s}}=g_{Y_{s}}$. Since the
choice of $s$ was arbitrary, this holds for any $s\in\boldsymbol{K}$.

To show uniqueness, one needs to solve for growth rate in terms of parameters.
In order to do so, a few intermediate steps are necessary. First, note that
one can express the demand for intermediates as:

$\bar{x}_{ss}(t,\omega)\equiv x_{ss}(t,\omega,\nu)=\left[\alpha
z_{s}(\omega)p_{ss}(t,\omega)\right]^{\frac{1}{1-\alpha}}\cdot\ell_{s}(t,\omega)$

which, in turn, implies that the optimal price of intermediate varieties is
$p_{ss}(t,\omega,\nu)=\frac{1}{\alpha}$ and I can rewrite the production
function of the final goods producer as:

$\displaystyle y_{s}(\omega)$ $\displaystyle=$ $\displaystyle
z_{s}(\omega)\ell_{s}(t,\omega)^{1-\alpha}\left(\frac{1}{\alpha}\int_{0}^{M_{s}(t)}[\bar{x}_{ss}(t,\omega)]^{\alpha}d\nu\right)$
$\displaystyle=$ $\displaystyle
z_{s}(\omega)\ell_{s}(t,\omega)^{1-\alpha}\left(\frac{1}{\alpha}\int_{0}^{M_{s}(t)}\left[\left[\alpha
z_{s}(\omega)p_{ss}(t,\omega)\right]^{\frac{1}{1-\alpha}}\cdot\ell_{s}(t,\omega)\right]^{\alpha}d\nu\right)$
$\displaystyle=$
$\displaystyle\left[z_{s}(\omega)\right]^{\frac{1}{1-\alpha}}\cdot\left[\alpha\cdot
p_{ss}(t,\omega)\right]^{\frac{\alpha}{1-\alpha}}\cdot\ell_{s}(t,\omega)\cdot\frac{1}{\alpha}\cdot
M_{s}(t)$

Replacing for $p_{ss}(t,\omega)$ using the assumption of pricing under perfect
competition:

$\displaystyle y_{s}(\omega)$ $\displaystyle=$
$\displaystyle\left[z_{s}(\omega)\right]^{\frac{1}{1-\alpha}}\cdot\left[\alpha\frac{w_{s}(t)^{1-\alpha}\alpha^{-\alpha(1-\eta)}M_{s}^{\alpha}}{\alpha\cdot
z_{s}(\omega)}\right]^{\frac{\alpha}{1-\alpha}}\cdot\ell_{s}(t,\omega)\cdot\frac{1}{\alpha}\cdot
M_{s}(t)$ $\displaystyle=$ $\displaystyle\alpha^{-(1-\alpha)}\cdot
z_{s}(\omega)\cdot w_{s}(t)^{\alpha}\cdot
M_{s}(t)^{1-\alpha}\cdot\ell_{s}(t,\omega)$

By GDP expenditure clearing, total expenditure is equal wages plus profits:

$Y_{s}(t)=w_{s}(t)L_{s}+\frac{\alpha}{\eta}Y_{s}(t)\implies\frac{1-\alpha}{\eta}Y_{s}(t)=w_{s}(t)L_{s}\implies
Y_{s}(t)=w_{s}(t)L_{s}$

where the last equation states that, in the last equation, GDP is labor income
because labor is the only factor of income in this economy. Hence, value added
is equal to labor income.

Integrating the production function over $\omega$ and using the fact above
gives us:

$\displaystyle
Y_{s}(t)=\left[\int_{0}^{1}z_{s}(\omega)\ell_{s}(t,\omega)^{\frac{\sigma-1}{\sigma}}d\omega\right]^{\frac{\sigma}{\sigma-1}}\cdot\alpha^{-(1-\alpha)}\cdot
w_{s}(t)^{\alpha}\cdot M_{s}(t)^{1-\alpha}=L_{s}w_{s}(t)$

solving for $w_{s}(t)$:

$w_{s}(t)=\left(\left[\int_{0}^{\infty}z\ell_{s}(t,z)^{\frac{\sigma-1}{\sigma}}dF_{s}(z)\right]^{\frac{\sigma}{\sigma-1}}\right)^{\frac{1}{1-\alpha}}\cdot\alpha^{-1}\cdot
M_{s}(t)L_{s}^{-\frac{1}{1-\alpha}}$

The term in the integral denotes the joint product of productivity and labor
allocation across firms. In aggregate terms, since both the distribution of
productivity and the population are fixed for every $t$; and relative wages
are fixed along the BGP, this term will be constant.

Following [3], note that all goods enter symmetrically in the definition of
the aggregate final good and they differ only by their productivity level.
Therefore, one can express the BGP growth rate of the economy fully in terms
of exogenous objects:

$g_{s}^{\text{autarky}}=\frac{\psi}{\eta}\cdot\left(\left[\int_{0}^{\infty}z\ell_{s}(t,z)^{\frac{\sigma-1}{\sigma}}dF_{s}(z)\right]^{\frac{\sigma}{\sigma-1}}\right)^{\frac{1}{1-\alpha}}\cdot
L_{s}^{-\frac{\alpha}{1-\alpha}}-\rho$ (B.10)

Since neither the productivity distribution $F_{s}(z)$ nor the demand
functions $\ell_{s}(t^{*},z)$ will change along the BGP and all other terms in
the growth rate are parameters, this pins down the uniqueness of the BGP under
autarky, which completes the proof. ∎

#### Zero gravity

#### Proof of Proposition (2)

###### Proof.

Without loss of generality, choose an arbitrary country $s\in\boldsymbol{K}$.
Since this world economy is under zero gravity, evaluate (2) replacing for the
real interest rate using equations (11) and (13) and evaluating
$\tau_{sd}=1(\forall s,d)$.Therefore, (2) collapses to:

$g_{s}^{\text{zero gravity}}=\left[\frac{\alpha\cdot\psi}{\eta\cdot
P_{s}(t^{*})}\cdot\frac{\sum_{k\in\boldsymbol{K}}Y_{k}(t^{*})}{\sum_{k\in\boldsymbol{K}}M_{k}(t^{*})}-\rho\right]$
(B.11)

for a BGP inclusive of each period $t\geq t^{*}$. Since there are no trade
costs, the law of one price holds, and $P_{s}(t^{*})=P_{d}(t^{*})\equiv
P(t^{*})$ for every $s,d\in\boldsymbol{K}$. Choosing $P(t^{*})$ to be
numéraire of this economy shows that the growth rate will follow the stated
equation.

Since the choice of the $s$ of arbitrary and the expression in the right-hand
side of the equation is equal for every $s\in\boldsymbol{K}$, it follows that
the $g_{s}^{\text{zero gravity}}=g^{\text{zero gravity}}$ for all
$s\in\boldsymbol{K}$, which shows that the growth rate must be common across
all countries. Furthermore, since
$Y_{k}(t^{*})=M_{k}(t^{*})\mathcal{R}_{k}(t^{*})$ and the fact that $g_{s}$
must be constant along a BGP,
$\frac{\sum_{k\in\boldsymbol{K}}Y_{k}(t^{*})}{\sum_{k\in\boldsymbol{K}}M_{k}(t^{*})}$
will only be homogeneous of degree zero in
$[M_{n}(t^{*})]_{n\in\boldsymbol{K}}$ if
$\mathcal{R}_{k}(t^{*})=\mathcal{R}(t^{*})$.

With log preferences, households will consume a constant fraction $(1-\rho)$
of their income and invest a fraction $\rho$. The non-arbitrage condition
shows that real interest rate and returns to R&D equalize globally along the
BGP:

$\frac{r_{s}(t^{*})}{P_{s}(t^{*})}=\frac{\psi\pi_{s}(t^{*},\nu)}{P_{s}(t^{*})}=\frac{\psi\Pi_{s}(t^{*})}{M_{s}(t^{*})P_{s}(t^{*})}=\frac{\psi}{M_{s}(t^{*})}M_{s}(t^{*})\times\mathcal{R}^{\pi}_{s}(t^{*})=\psi\rho\mathcal{R}(t^{*})$

The next step in the proof is to show that
$g_{M_{s}}=g_{Y_{s}}=g_{C_{s}}=g_{w_{s}}=g_{A_{s}}=g_{s}^{\text{zero
gravity}}$. Since real wages, real profits, assets, and real output are linear
functions of $M_{s}(t)$ in each period, it follows that
$g_{M_{s}}=g_{Y_{s}}=g_{w_{s}}=g_{A_{s}}$. Since, with log preferences,
consumption is a constant fraction of output, $g_{C_{s}}=g_{Y_{s}}$. Since the
choice of $s$ was arbitrary, this holds for any $s\in\boldsymbol{K}$.

For uniqueness, one needs to show that the cross-sectional equilibrium is
unique. Start from equation (15). Evaluating it under zero gravity and noting
that prices of final goods and intermediate goods equalize in that situation
results in:

$P_{s}(t^{*})Y_{s}(t^{*})=\sum_{d\in\boldsymbol{K}}\left[(1-\alpha)\frac{T_{s}w_{s}(t^{*})^{-(1-\alpha)\theta}}{\sum_{k\in\boldsymbol{K}}T_{k}w_{k}(t^{*})^{-(1-\alpha)\theta}}+\frac{\alpha}{\eta}\frac{M_{s}(t^{*})}{\sum_{k\in\boldsymbol{K}}M_{k}(t^{*})}\right]P_{d}(t^{*})Y_{d}(t^{*})$

Recall that $P_{s}(t^{*})Y_{s}(t^{*})=w_{s}(t^{*})L_{s}+\Pi_{s}(t^{*})$ and
note that, under zero gravity,
$\Pi_{s}(t^{*})=\frac{\alpha}{\eta}\frac{M_{s}(t^{*})}{\sum_{k\in\boldsymbol{K}}M_{k}(t^{*})}P_{d}(t^{*})Y_{d}(t^{*})$.
So, given $M_{s}(t^{*})$ the expenditure determination system becomes a simple
system in wages:

$w_{s}(t^{*})L_{s}=\sum_{d\in\boldsymbol{K}}\left[(1-\alpha)\frac{T_{s}w_{s}(t^{*})^{-(1-\alpha)\theta}}{\sum_{k\in\boldsymbol{K}}T_{k}w_{k}(t^{*})^{-(1-\alpha)\theta}}\right]w_{d}(t^{*})L_{d}$

Define the excess demand function:

$Z_{s}(\textbf{w},t)\equiv\frac{1}{w_{s}(t^{*})}\left(\sum_{d\in\boldsymbol{K}}\left[(1-\alpha)\frac{T_{s}w_{s}(t^{*})^{-(1-\alpha)\theta}}{\sum_{k\in\boldsymbol{K}}T_{k}w_{k}(t^{*})^{-(1-\alpha)\theta}}\right]w_{d}(t^{*})L_{d}-w_{s}(t^{*})L_{s}\right)$

and note:

$\frac{\partial Z_{s}(\textbf{w},t^{*})}{\partial
w_{d}(t^{*})}=\frac{1}{w_{s}(t^{*})}(1-\alpha)\lambda^{F}_{sd}(t^{*})\left(L_{d}+\frac{\lambda^{F}_{dd}(t^{*})}{w_{d}(t^{*})}\right)>0$

which shows that it satisfies the gross substitution property and the cross-
section equilibrium is unique. Therefore, the BGP under zero gravity will be
unique.

∎

#### General case

#### Proof of Proposition (3)

###### Proof.

Without loss of generality, choose an arbitrary country $s\in\boldsymbol{K}$.
From (18), real GDP is a linear function of $M_{s}(t)$:

$C_{s}(t)+I_{s}(t)=\frac{w_{s}(t)}{P_{s}(t)}L_{s}+\frac{\Pi_{s}(t)}{P_{s}(t)}=M_{s}(t)\times\mathcal{R}_{s}(t)$

Over the BGP, with log preferences, consumption is a constant fraction of GDP:
$C_{s}(t^{*})=(1-\rho)M_{s}(t)\times\mathcal{R}_{s}(t).$ From the Poisson
arrival process,
$g_{M_{s}}=\frac{\dot{M}_{s}(t^{*})}{M_{s}(t)}=\psi\rho\frac{I_{s}(t^{*})}{M_{s}(t^{*})}$.
Since trade is balanced, $I_{s}(t^{*})=\frac{\rho}{1-\rho}C_{s}(t^{*})$ and
varieties grow at the following rate:

$\displaystyle g_{M_{s}}$ $\displaystyle=$
$\displaystyle\psi\rho\left[\alpha^{1-\eta}\left(\frac{T_{s}}{\lambda^{F}_{ss}(t^{*})}\right)^{\frac{1}{\theta(1-\alpha)}}\left(\lambda^{M}_{ss}(t^{*})\right)^{-1}L_{s}+\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t^{*})\frac{P_{d}(t^{*})Y_{d}(t^{*})}{P_{s}(t^{*})M_{s}(t^{*})}\right]$

The following statements are true:

  1. 1.

$\lambda^{F}_{ss}(t^{*}),\lambda^{M}_{ss}(t^{*})$ are homogeneous of degree
zero in $\\{M_{n}(t^{*})\\}$;

  2. 2.

$\left(\frac{P^{M}_{s}(t^{*})}{P_{d}^{M}(t^{*})}\right),\left(\frac{P_{d}(t^{*})}{P_{s}(t^{*})}\right)$
are homogeneous of degree zero in $\\{M_{n}(t^{*})\\}$;

  3. 3.

$\frac{Y_{d}(t^{*})}{M_{s}(t^{*})}=\frac{M_{s}(t^{*})\times\mathcal{R}_{s}(t^{*})}{M_{s}(t^{*})}$
is homogeneous of degree zero in $\\{M_{n}(t^{*})\\}$ if and only if
$\mathcal{R}_{s}(t^{*})$ is homogeneous of degree zero in $\\{M_{n}(t)\\}$ for
all $s\in\boldsymbol{K}$.

Therefore, for $g_{M_{s}}$ to be consistent with a BGP it must also be
homogeneous of degree zero in $[M_{n}(t^{*})]_{n\in\boldsymbol{K}}$. As a
result, if $g_{M_{s}}$ is consistent with a BGP, $\mathcal{R}_{s}(t^{*})$ must
be homogeneous of degree zero in $[M_{n}(t^{*})]_{n\in\boldsymbol{K}}$ for all
$s\in\boldsymbol{K}$. As a result, it must be that varieties grow at the same
rate across countries, which implies that
$\mathcal{R}_{s}(t^{*})=\mathcal{R}(t^{*})$.

With log preferences, households will consume a constant fraction $(1-\rho)$
of their income and invest a fraction $\rho$. The non-arbitrage condition
shows that real interest rate and returns to R&D equalize globally along the
BGP:

$\frac{r_{s}(t^{*})}{P_{s}(t^{*})}=\frac{\psi\pi_{s}(t^{*},\nu)}{P_{s}(t^{*})}=\frac{\psi\Pi_{s}(t^{*})}{M_{s}(t^{*})P_{s}(t^{*})}=\frac{\psi}{M_{s}(t^{*})}M_{s}(t^{*})\times\mathcal{R}^{\pi}_{s}(t^{*})=\psi\rho\mathcal{R}(t^{*})$

The next step in the proof is to show that
$g_{M_{s}}=g_{Y_{s}}=g_{C_{s}}=g_{w_{s}}=g_{A_{s}}=g_{s}$. Since real wages,
real profits, assets, and real output are linear functions of $M_{s}(t)$ in
each period, it follows that $g_{M_{s}}=g_{Y_{s}}=g_{w_{s}}=g_{A_{s}}$. Since,
with log preferences, consumption is a constant fraction of output,
$g_{C_{s}}=g_{Y_{s}}$. Since the choice of $s$ was arbitrary, this holds for
any $s\in\boldsymbol{K}$.

∎

#### Changes in trade costs

#### Proof of Proposition 4

###### Proof.

The equilibrium growth rate of varieties:

$\displaystyle g_{M_{s}}$ $\displaystyle=$
$\displaystyle\psi\rho\left[\left(\frac{T_{s}}{\lambda^{F}_{ss}(t^{*})}\right)^{\frac{1}{\theta(1-\alpha)}}\left(\lambda^{M}_{ss}(t^{*})\right)^{-1}L_{s}+\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t^{*})\frac{P_{d}(t^{*})Y_{d}(t^{*})}{P_{s}(t^{*})M_{s}(t^{*})}\right]$

Recall that:

$\displaystyle\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t^{*})$
$\displaystyle=$
$\displaystyle\sum_{d\in\boldsymbol{K}}\frac{M_{s}(\tau_{sd}P_{s}(t^{*}))^{1-\eta}}{\sum_{k^{\prime}\in\boldsymbol{K}}M_{k^{\prime}}(\tau_{k^{\prime}d}P_{k^{\prime}}(t^{*}))^{1-\eta}}\frac{P_{d}(t^{*})Y_{d}(t^{*})}{P_{s}(t^{*})M_{d}(t^{*})}$

Since these economies are symmetric, then:
$P_{s}(t^{*})=P_{s^{\prime}}(t^{*})$, $w_{s}(t^{*})=w_{s^{\prime}}(t^{*})$,
$M_{s}(t^{*})=M_{s^{\prime}}(t^{*})$ for every $s,s^{\prime}$ and
$\tau_{sd}=\tau$ for every ${sd}$. Evaluated with symmetric economies, the
expression above becomes:

$\displaystyle\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t^{*})$
$\displaystyle=$
$\displaystyle\frac{(N-1)\tau^{1-\eta}}{[1+(N-1)\tau^{1-\eta}]}+\frac{1}{[1+(N-1)\tau^{1-\eta}]}=1$

Therefore, denoting $P_{s}(t^{*})=P(t^{*})$, $M_{s}(t^{*})=M(t^{*})$ and
noting that $Y_{d}(t^{*})=M_{d}(t^{*})\times\mathcal{R}(t^{*})$ the growth
rate becomes to:

$\displaystyle g^{*}$ $\displaystyle=$
$\displaystyle\psi\rho\left[\left(T_{s}\right)^{\frac{1}{\theta(1-\alpha)}}\left(\frac{1}{1+(N-1)\tau^{1-\theta}}\right)^{-\frac{1}{\theta(1-\alpha)}}\left(\frac{1}{1+(N-1)\tau^{1-\eta}}\right)^{-1}L_{s}+\frac{\alpha}{\eta}\mathcal{R}\right]$

Then, take the derivative of $g^{*}$ wrt $\tau$:

$\displaystyle\frac{\partial g^{*}}{\partial\tau}$ $\displaystyle=$
$\displaystyle\psi\rho\left(T_{s}\right)^{\frac{1}{\theta(1-\alpha)}}\left(\frac{1}{1+(N-1)\tau^{1-\theta}}\right)^{-\frac{1}{\theta(1-\alpha)}}\left(\frac{1}{1+(N-1)\tau^{1-\eta}}\right)^{-1}L_{s}\times$
$\displaystyle\Bigg{(}\frac{(1-\eta)\tau^{-\eta}}{1+(N-1)\tau^{1-\eta}}-\frac{\theta\tau^{-\theta-1}}{1+(N-1)\tau^{-\theta}}\Bigg{)}<0$

which is negative because $(1-\eta)<0$ and every other term in the parenthesis
is positive.

∎

###  B.9 Welfare

Recall that $C_{s}(t^{*})$ can be expressed as a constant fraction of total
lifetime wealth:

$C_{s}(t^{*})=\rho\left[A_{s}(t^{*})+\int_{t^{*}}^{\infty}\frac{w_{s}(\tau)}{P_{s}(\tau)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(\tau)\cdot\tau\right\\}d\tau\right]$

where $\bar{r}_{s}=\frac{1}{\tau}\int_{t^{*}}^{\tau}r_{s}(t)dt$ is the average
interest rate between $t^{*}$ and $\tau$. Since this holds along the BGP,
$\frac{w_{s}(\tau)}{P_{s}(\tau)}=\frac{\exp\\{(\tau-t^{*})g_{w_{s}}\\}w_{s}(t^{*})}{P_{s}(t^{*})}$.
Furthermore, since $\frac{r_{s}(t^{*})}{P_{s}(t^{*})}$ is constant along the
BGP, $\bar{r}_{s}(\tau)=\frac{r_{s}(t^{*})}{P_{s}(t^{*})}$ for all $\tau\geq
t^{*}$. Replacing those above results in:

$\displaystyle C_{s}(t^{*})$ $\displaystyle=$
$\displaystyle\rho\left[A_{s}(t^{*})+\frac{w_{s}(t^{*})}{P_{s}(t^{*})}L_{s}\int_{t^{*}}^{\infty}\cdot\exp\left\\{-\left(\frac{r_{s}(t^{*})}{P_{s}(t^{*})}-g_{w_{s}}\right)\cdot(\tau-t^{*})\right\\}d\tau\right]$
$\displaystyle=$
$\displaystyle\rho\left[A_{s}(t^{*})+\frac{w_{s}(t^{*})}{P_{s}(t^{*})}\frac{L_{s}}{\frac{r_{s}(t^{*})}{P_{s}(t^{*})}-g_{w_{s}}}\right]$
$\displaystyle=$ $\displaystyle\rho
A_{s}(t^{*})+\rho\frac{w_{s}(t^{*})}{P_{s}(t^{*})}\frac{L_{s}}{\frac{r_{s}(t^{*})}{P_{s}(t^{*})}-g_{w_{s}}}$

Since $g_{w_{s}}=g_{C_{s}}$ and
$g_{C_{s}}=\frac{r_{s}(t^{*})}{P_{s}(t^{*})}-\rho$,
$\frac{r_{s}(t^{*})}{P_{s}(t^{*})}-g_{w_{s}}=\rho$. Hence, over the BGP, real
consumption is a fraction of assets plus real labor income:

$C_{s}(t^{*})=\rho A_{s}(t^{*})+\frac{w_{s}(t^{*})L_{s}}{P_{s}(t^{*})}$

Welfare over the BGP is:

$\displaystyle\int_{t^{*}}^{\infty}\exp\\{-\rho(t-t^{*})\\}\log\left(\exp\\{g^{*}t\\}C_{s}(t^{*})\right)dt$
$\displaystyle=$
$\displaystyle\int_{t^{*}}^{\infty}\exp\\{-\rho(t-t^{*})\\}\log\left(C_{s}(t^{*})\right)dt$
$\displaystyle+$
$\displaystyle\int_{t^{*}}^{\infty}\exp\\{-\rho(t-t^{*})\\}g^{*}tdt$
$\displaystyle=$
$\displaystyle\frac{\log\left(C_{s}(t^{*})\right)}{\rho}+\frac{g^{*}}{\rho^{2}}$
$\displaystyle=$
$\displaystyle\log\left(A_{s}(t^{*})\right)+\frac{1}{\rho}\log\left(\frac{w_{s}(t^{*})L_{s}}{P_{s}(t^{*})}\right)+\frac{g^{*}}{\rho^{2}}$

Finally, using the fact that $\psi A_{s}(t^{*})=M_{s}(t^{*})$, I can write:

$\int_{t^{*}}^{\infty}\exp\\{-\rho(t-t^{*})\\}\log\left(\exp\\{g^{*}t\\}C_{s}(t^{*})\right)dt=\log\left(\frac{1}{\psi}M_{s}(t^{*})\right)+\frac{1}{\rho}\log\left(\frac{w_{s}(t^{*})L_{s}}{P_{s}(t^{*})}\right)+\frac{g^{*}}{\rho^{2}}$

#### Static welfare

For real labor income, start from equation (9) evaluated at $s=d$ and use the
fact that, as shown in equation (B.5) of Appendix B.5,

$P_{s}(t)=\gamma\cdot\left[\sum_{n\in\boldsymbol{K}}{T_{n}(w_{n}(t)^{1-\alpha}P_{n}^{M}(t)^{\alpha}\tau_{nd})^{-\theta}}\right]^{-\frac{1}{\theta}}$

where
$\gamma\equiv\Gamma\left(\frac{\theta+1-\sigma}{\theta}\right)^{\frac{1}{1-\sigma}}$.
Then, own trade share in a given country can be represented by:

$\lambda^{F}_{dd}(t)=\gamma^{\theta}\cdot\frac{T_{d}(w_{d}(t)^{1-\alpha}(P_{d}(t)^{M})^{\alpha})^{-\theta}}{[P_{d}(t)]^{-\theta}}$

Solving for $\frac{w_{d}(t)}{P_{d}(t)}$ delivers:

$\displaystyle\frac{w_{d}(t)}{P_{d}(t)}$ $\displaystyle=$
$\displaystyle\gamma^{\frac{1}{1-\alpha}}\lambda_{dd}(t)^{-\frac{1}{(1-\alpha)\theta}}T_{d}^{\frac{1}{(1-\alpha)\theta}}\left(\frac{P_{d}^{M}(t)}{P_{d}(t)}\right)^{-\alpha}$

Replacing for the definition of
$P^{M}_{d}(t)=\left[\sum_{k\in\boldsymbol{K}}M_{k}\left(\frac{\tau_{kd}P_{k}(t)}{\alpha}\right)^{-\frac{\alpha}{1-\alpha}}\right]^{-\frac{1-\alpha}{\alpha}}$
results in:

$\displaystyle\frac{w_{d}(t)}{P_{d}(t)}$ $\displaystyle=$
$\displaystyle\gamma^{\frac{1}{1-\alpha}}\lambda_{dd}(t)^{-\frac{1}{(1-\alpha)\theta}}T_{d}^{\frac{1}{(1-\alpha)\theta}}\left[\sum_{k\in\boldsymbol{K}}M_{k}\left(\frac{\tau_{kd}P_{k}(t)}{\alpha
P_{d}(t)}\right)^{-\frac{\alpha}{1-\alpha}}\right]^{1-\alpha}$

Consider what happens to welfare after a change in trade costs from
$\boldsymbol{\tau}$ to $\boldsymbol{\tau}+d\boldsymbol{\tau}$, as in [5]. In
this dynamic setting, to compare the static component of welfare, I need to
compare what happens across the two BGPs, comparing the two initial
equilibria. Suppose $t^{*}$ is the initial period of the original BGP while
$t^{**}$ is the first period of the final BGP. To fit this framework to the
general trade literature, I will compare the static component of these BGP as
if they happened in the same period, and compound the difference over time.

Let $\hat{x}\equiv x(t^{*}*)/x(t^{*})$. Then cumulative changes in static
welfare are:

$\frac{1}{\rho}\log\left(\widehat{\frac{w_{s}(t^{**})}{P_{s}(t^{**})}}\right)=\frac{1}{\rho}\log\left(\widehat{\lambda_{dd}^{F}}(t^{**})^{-\frac{1}{(1-\alpha)\theta}}\right)+\frac{1}{\rho\eta}\log\left(\sum_{k\in\boldsymbol{K}}\mu_{k}(t^{*})\widehat{M}_{k}(t^{**})\cdot\left(\frac{\widehat{\tau}_{kd}\widehat{P}_{k}(t^{**})}{\widehat{P}_{d}(t^{**})}\right)^{1-\eta}\right)$

where
$\mu_{k}(t)\equiv\frac{M_{k}(t)\cdot\left(\frac{\tau_{kd}P_{k}(t)}{P_{d}(t)}\right)^{1-\eta}}{\sum_{k\in\boldsymbol{K}}M_{k}(t)\cdot\left(\frac{\tau_{kd}P_{k}(t)}{P_{d}(t)}\right)^{1-\eta}}$

###  B.10 Nesting of Romer and Eaton-Kortum

In this subsection, I will briefly describe how to recover the canonical [44]
and [23] models from the framework described above.

#### Eaton-Kortum

Setting $\alpha=0$ implies that the value of new varieties is zero since the
demand for and profits of varieties is also zero. Therefore, $I_{s}(t)=0$ and
$A_{s}(t)=0$ for all $t$ and $s$. While the Eaton-Kortum model is a static
model, here it can be thought of as an infinite sequence of static models with
no intertemporal decision, since there are no longer asset markets that permit
households to save:

$\displaystyle\max_{C_{s}(t),c_{s}(t,\omega)_{\omega\in[0,1]}}$
$\displaystyle\int_{0}^{\infty}\exp\\{-\rho t\\}\log\left(C_{s}(t)\right)dt$
$\displaystyle s.t.\quad P_{s}(t)C_{s}(t)$ $\displaystyle=$ $\displaystyle
w_{s}(t)L_{s}$ $\displaystyle C_{s}(t)$ $\displaystyle=$
$\displaystyle\Big{[}\int_{0}^{1}c_{s}(t,\omega)^{\frac{\sigma-1}{\sigma}}d\omega\Big{]}^{\frac{\sigma}{\sigma-1}}$
$\displaystyle P_{s}(t)C_{s}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}p_{s}(t,\omega)c_{s}(t,\omega)d\omega$

Furthermore, since $\alpha=0$, the intermediate and research and development
sectors disappear. The problem of the final goods producer becomes:

$\max_{\ell_{s}(t,\omega)}p_{ss}(t,\omega)\cdot
z_{s}(t,\omega)\cdot\ell_{s}(t,\omega)-\ell_{s}(t,\omega)w_{s}(t)$

which is identical to the one in the standard Eaton-Kortum model. Equilibrium
will take the form of a system of labor market determination equations that
solve for $N$ wages using trade expenditure shares.

#### Romer

Setting $\tau_{sd}\to\infty$ for $s\neq d$ implies trade costs are
prohibitively high internationally, such that varieties of both final goods
and intermediate goods become sold only locally. Normalizing the price of the
domestic final good to be the numéraire in each country, I write the dynamic
household problem as:

$\displaystyle\max_{C_{s}(t),c_{s}(t,\omega)_{\omega\in[0,1]}}$
$\displaystyle\int_{0}^{\infty}\exp\\{-\rho t\\}\log\left(C_{s}(t)\right)dt$
$\displaystyle s.t.\quad I_{s}(t)=\dot{A}(t)$ $\displaystyle=$ $\displaystyle
r_{s}(t)A_{s}(t)+w_{s}(t)L_{s}-C_{s}(t)$ $\displaystyle C_{s}(t)$
$\displaystyle=$
$\displaystyle\Big{[}\int_{0}^{1}c_{s}(t,\omega)^{\frac{\sigma-1}{\sigma}}d\omega\Big{]}^{\frac{\sigma}{\sigma-1}}$

Furthermore, redefine assumption (2) in the following terms:

######  Assumption 4 (Productivity draws to recover Romer).

To recover the Romer model as a special case of the general model, I need to
specify productivity terms $z_{s}(\omega)$ which are homogeneous across firms
in each country. In order to do so, redefine the cumulative distribution
function $F_{s}(t)(z)$ of the baseline case to be one of a degenerate random
variable with a point mass concentrated at a certain scalar for each country.
Formally:

$F_{s}(t)(z)=\begin{cases}0\text{ for }z<T_{s}\\\ 1\text{ for }z\geq
T_{s}\end{cases}$

Using the symmetry assumption above, the numéraire normalization and the
unavailability of foreign intermediate goods in the domestic market, the final
goods assembler technology becomes:

$y_{s}(t,\omega)=T_{s}[\ell_{s}(t,\omega)]^{1-\alpha}\left(\frac{1}{\alpha}\int_{0}^{M_{s}(t)}[x_{ss}(t,\omega,\nu)]^{\alpha}d\nu\right)$

which is identical to the single-country Romer model. Profits and demand per
variety $\nu\in[0,M_{s}(t)]$ will be constant and growth will be driven by the
domestic R&D sector. Equilibrium will take the following form: labor markets
will clear; total final goods produced being equal to total final goods used
for consumption, intermediate production; and R&D production; and optimized
household optimal dynamics will be described by an Euler equation and an
asset/measure accumulation equation.

##  Appendix C Qualitative evidence: life among product innovators

As an initial exploratory part of this research, I conducted a qualitative
survey of managers in firms of New Member States. I first collected a list of
notable firms from publicly available sources, restricted the sample to those
who were active for at least two years before the time their respective
countries joined the European Union, and then crawled through their English-
language websites to collect the publicly available contact information. I
sent the questionnaire below to 221 firms.

My goal was to assess if the description of the world that macro theorists set
forth aligns with the practical intuitions of entrepreneurs. And it turns out
that, at among the group of managers that responded to my email, they do. I
will highlight two illustrative cases in the text.

For instance, the dynamic mechanism that propels growth, as I have described
in the theory section is that increased access to foreign markets increases
expected profits, thereby increasing the incentive to invest in research and
development. This is entirely consistent with the description of the facts by
one Czech biotech entrepreneur:

> “Once we joined the EU […] this allowed us to increase our exports and fund
> our own genetic programmes.” —CEO of a Czech Biotech company

In their comments, they went on to specify the importance of having access of
not only to the European market itself, but also third party markets. They
mentioned that after the Czech Republic joined the EU, his firm had immediate
access to the standards for labeling and certification in existing trade
agreements between the EU and third parties, which facilitated their firm’s
exports. These kinds of non-tariff barriers are typically considered part of
trade costs $\tau$ in most trade models.

In this firm’s particular case, product innovation came through the invention
of breeding of new varieties of farm animals, that were then commercialized.
But we see a similar story in a very different market: alcoholic beverages:

> “In 2004, we first started producing the ultra-luxury variation of our
> signature vodka, which became a popular export product […] and later started
> production of 18 new products.” —Spokesperson of a Latvian liquor
> manufacturer

In this case, the firm reported having used the European market’s exports as a
platform for global expansion. For context, 2004 marks the year Latvia
accessed the EU —and also the year that this manufacturer decided to expand
its product line by introducing the ultra-luxury versions of its signature
product, which they claimed was adequate to the Western European market.

Once again, this is qualitatively consistent with the theoretical mechanism
proposed in the model, with market access likely inducing product innovation.
Of course, these individual experiences are not necessarily representative of
a large universe of firms, which is why in the next two sections I will
perform a detailed quantitative exploration of the data, first detailing some
stylized facts, then going into causal inference. Nonetheless, the type of
qualitative evidence presented here is useful to show to that the big picture
is consistent with the individual experiences.

###  C.1 Qualitative Questionnaire

  1. 1.

After your country joined the European Union, did your company:

     * •

start producing more products/services or varieties;

     * •

start producing fewer products/services or varieties; or

     * •

keep producing about the same number of products/services or varieties?

  2. 2.

If your company changed the number of products/services or varieties after EU
accession, how was the change implemented and what were the results? Please
include any important information or relevant anecdotes.

  3. 3.

If your company changed the number of products/services or product/service
varieties after EU accession, was the decision primarily motivated by access
to new technologies/imports, access to new markets/exports, or both? Explain.

  4. 4.

After your country joined the European Union, did your company:

     * •

stay in the same industry;

     * •

expanded to another industry; or

     * •

move completely to a new industry?

  5. 5.

If your company expanded to another industry or moved to a new industry.
Please explain whether the change was related to your country’s EU accession.

##  Appendix D Data and Empirical Appendix

###  D.1 Extensive Description of the Data

#### Production data

Production data comes from Eurostat’s Prodcom (Production Communautaire),
which is an annual full coverage survey of the European mining, quarry and
manufacturing sectors, reporting the value of production of 4,000+ different
product-lines of EU members and candidate countries. Prodcom reports, for each
product line, country, and year, the value (in euros) and volume (in kg,
$m^{2}$, number of items, etc.) of production. Product lines follow the
Statistical Classification of Products by Activity in the EU (CPA).

The target population of the full coverage sample is every enterprise that
manufactures some good in the Prodcom List. Data quality is good for member
countries since European Law212121“PRODCOM statistics are compiled under the
legal basis provided by Council Regulation (EEC) NO 3924/1991 of 19 December
1991 and by Commission Regulation (EC) No 0912/2004 of 29 April 2004
implementing the Council Regulation (EEC) No 3924/91 on the establishment of a
Community survey of industrial production. Additionally, a Commission
Regulation updating the PRODCOM classification is available annually since
2003.” mandates National Statistical Institutes to collect enterprise-level
information on the value and volume of production covering at least 90% of
national production in each NACE class, defined as the first four digits of
each product code. In practice, reporting goes beyond this minimum threshold
and, according to Eurostat, the coverage error is estimated to be below 10%.

Let $n,i,p,t$ index countries, sectors, products, and periods, respectively;
and denote $Y_{inpt}$ as the market value of production of product $p$222222To
construct sector codes, I use Eurostat concordances to map Prodcom product
codes to Harmonized System (HS) product codes. I then used the respective HS-2
division codes as sector codes.. The set of varieties produced in each sector
is $\mathcal{M}_{nit}=\\{k\mathrel{\mathop{\mathchar 58\relax}}Y_{nikt}>0\\}$.
The measure of varieties is simply the cardinality of the set of produced
varieties
$M_{nit}=|\mathcal{M}_{nit}|=\sum_{k}\mathbb{1}_{\\{k\mathrel{\mathop{\mathchar
58\relax}}Y_{nikt}>0\\}}$. The overall measure over varieties produced in a
country is, then: $M_{nt}=\sum_{i}M_{nit}$. These measures can be directly
calculated from Prodcom’s table.

Oftentimes, the value of production is labeled as confidential information by
the National Statistical Institute, particularly in cases in which production
is concentrated on a few enterprises. In those cases, while the value and
volume are not publicly available, Eurostat reports this number as
confidential, which still allows one to infer that $Y_{nikt}>0$ for that
particular variety $k$, implying that the variety is produced.

Typically, production information at the variety level is not available, which
pushed researchers to use product-level trade data instead. Some exceptions
include [26] and [40], who use firm-product links from the Indian Survey of
Manufacturers; [9], who use US Manufacturing Censuses firm-product data.

#### Tariff and trade flow data

Bilateral tariff data come from WITS (World Integrated Trade Solution Trade
Stats). It consolidates tariff data from the UNCTAD’s Trade Analysis
Information System (TRAINS) as well as from the WTO.

To construct effective tariff rates, one starts from baseline tables of most
favored nation tariffs at the source-country $\times$ destination-country
$\times$ HS6-code $\times$. Then, one superimposes every bilateral product
level preferential tariff available in the WITS database on each of these
tables. Furthermore, whenever there are gaps between two identical bilateral
preferential tariffs, one fills in those gaps. The result is a dataset of
effectively applied tariff rates.

Bilateral trade flow data comes from UNCOMTRADE. These data, which are widely
used in research, come natively in a source-country $\times$ destination-
country $\times$ HS-6 product-code $\times$ year format, which makes it
readily compatible with the tariff data mentioned above.

Let $s,d,i,p,t$ index source countries, destination countries, sectors,
products, and periods, respectively; and denote $X_{sdipt}$ as the market
value of bilateral trade of product $p$.

The set of traded varieties in each sector is
$\mathcal{X}_{nit}=\\{k\mathrel{\mathop{\mathchar 58\relax}}X_{sdikt}>0\\}$.
Analogously as with production, one can observe the total number of traded
varieties $\sum_{k}\mathbb{1}_{\\{k\mathrel{\mathop{\mathchar
58\relax}}X_{nikt}>0\\}}$. To make sure these are comparable to PRODCOM’s
codes, whenever possible, I used concordances and restricted the set of goods
to create a dataset that matched both trade and production.

#### Other data

I also collected data on (a) the dates of accession of new member states to
the European Union; (b) trade agreements existent and entered into force
between the European Union and third parties before 2004; and (c) expenditure
in private research & development expenditures per capita. The first two come
from hand collecting documents and tables from the European Commission’s
official websites while the latter comes from Eurostat.

###  D.2 Formal Description of the Callaway & Sant’Anna Estimator

Formally, let a “treatment” group $g$ be defined as being treated for all
periods $t\geq g$. Note that, since the EU enlargement happened simultaneously
for more than one country, there is more than one country $n$ for each
$g_{n}=g$. If some country cluster is in group $g$, then $G_{nt}=g$ $(\forall
t)$. If it is never treated, it is in the control group, and then
$G_{nt}=\infty$ $(\forall t)$.

The parameter of interest is the average treatment on the treated for a given
treatment group $g$ and horizon $t$, i.e.:

$ATT(g,t)=\mathbb{E}[M_{nt}(g)-M_{nt}(0)|G_{nt}=g]$ (D.1)

where $M_{nt}(g)$ is the potential outcome of country $n$ at period $t$ if
treated at period $g$; $M_{nt}(0)$ is the potential outcome country $n$ at
period $t$ if untreated; $X_{n{g-1}}$ are pre-treatment time-invariant
covariates; and $G_{nt}=g$ is a group indicator.

Note that the $ATT(g,t)$ is group and period-specific. It can be recovered
under assumptions similar to the standard difference-in-differences framework:
parallel trends and no-anticipation232323Formally, parallel trends is the
assumption that potential outcomes evolve almost surely equally to the
untreated group:
$\mathbb{E}[M_{nt}(0)-M_{n{t-1}}(0)|G_{nt}=g]=\mathbb{E}[M_{nt}(0)-M_{n{t-1}}(0)|G_{nt}>g]$
for all $t\geq g$. No anticipation means that potential outcomes for a treated
group are equal to the untreated group for any date before the treatment
—i.e., for all $t<g$,
$\mathbb{E}[M_{nt}(g)|G_{nt}=1]=\mathbb{E}[M_{nt}(0)|G_{nt}=1]$ almost
surely.. The next step is to summarize the ATT across groups by appropriately
weighting the results as:

$\theta(t)=\sum_{g}\mathbb{1}\\{g\leq t\\}w_{gt}ATT(g,t)$ (D.2)

for some weights $w_{gt}$. [16] propose the weights
$w_{gt}=P(G_{nt}=g|G_{nt}\leq t)$, which is the share of country clusters from
group $g\geq t$ out of all country clusters being treated at time $t$.

###  D.3 Further Details on Causal Inference

Since the largest wave of enlargement was in 2004, in this analysis I will
focus exclusively on that wave. The source of variation is at the source-
country $\times$ destination-country $\times$ HS-code product level. In each
year, there are about 300 thousand observations. Figure 15 shows the
interquartile range of bilateral product-level tariff rates between NMS and
the set of countries that had concluded trade agreements with the EU prior to
2004.

It shows that there is not much change in tariffs leading up to membership and
then a median drop of about 2.5 percentage points between 2003 and 2004. In
the years immediately after membership, there is also not a large change in
the distribution of bilateral tariff rates. There are some changes after 2007,
possibly because some future provisions in trade agreements kick in.

Figure 15: Distribution of Tariff Changes Over Time: Interquartile Range
Bilateral HS6-Product-Level Tariff Rates Between New Member States (2004 EU
Enlargement) and Set of Countries that Concluded Trade Agreements with EU
prior to 2004. Data were constructed from WITS Preferential and MFN databases.
Figure 16: Tariff Shock: Distribution of the (non-zero) observations of the
changes in Bilateral HS6-Product-Level Tariff Rates Between New Member States
(2004 EU Enlargement) and Set of Countries that Concluded Trade Agreements
with EU prior to 2004. Data were constructed from WITS Preferential and MFN
databases.

The metric of the tariff shock change is simply
$\Delta\tau_{sdip,2004}\equiv(\tau_{sdip,2004}-\tau_{sdip,2003})$, which is
the change in the level of effectively applied bilateral tariffs at the
product level between 2003 and 2004. Figure 16 plots the distribution of
$\Delta\tau_{sdip,2004}$, excluding the zero-valued observations. The average
$\Delta\tau_{sdip,2004}$ is $2.14\%$ and the standard deviation is $12\%$.

I estimate a sequence of cross-sectional local-projection linear probability
models, which estimate what is the marginal effect of an increase in the
tariffs on exports of a given product $p$, conditional on that country $s$ not
producing that particular product before joining the EU in 2003. The fact the
data is highly granular permits me to exploit within $industry\times
source\times destination\times horizon$ (across product) variation.

Formally, I estimate the following equation:

$\displaystyle P\left(X_{sdip,h}>0\middle|Y_{s\cdot ip,2003}=0\right)$
$\displaystyle=$
$\displaystyle\alpha_{h}+\beta_{h}\cdot\Delta\tau_{sdip,2004}+\gamma_{sdi,h}+\nu_{sdip,h}$
$\displaystyle\text{for }h\in\\{2000,\cdots,2010\\}$

where $X_{sdip,h}$ is the market value of exports between country $s$ and
country $d$ of product $p$ of industry $i$ at horizon $h$; $Y_{s\cdot
ip,2003}$ is the market value of production in country $s$ of product $p$ of
industry $i$ in $2003$; $\alpha_{h}$ are horizon (time) fixed-effects;
$\gamma_{sdi,h}$ are $source\times destination\times industry$ interactions
fixed-effects for each $h$.

Note that, since these are local projections, the right-hand side
coefficients, the regressor $\tau_{sdip,2004}$ is fixed for all horizons, and
the coefficients $\beta_{h}$ change. As initially argued by [18] and later
formalized by [22], these types of cross-sectional event studies with local
projections can be interpreted as differences in differences with continuous
treatments. If consistently estimated, the estimated coefficients $\beta_{h}$,
then, are simply the average treatment on treated compared to the potential
outcomes of not being treated, normalized to a treatment of intensity of one
unit.

This strategy takes the assertion in [7] (henceforth B&B) that countries
engage endogenously in free trade agreements (FTAs) and one needs to look for
a plausibly exogenous source of variation to check whether or not FTA
“actually increase members’ international trade” seriously. Here, I rely on
their strategy of running dynamic panels with fixed effects to control for
unobserved heterogeneity.

Importantly, while they estimate their models at the aggregate country level
with $source\times destination\times period$ fixed effects, I have enough
variability and data availability to estimate it at the product-level adding
$industry\times source\times destination\times period$ fixed effects. Hence,
this approach adds granularity to B&B’s strategy, thereby controlling for more
unobserved heterogeneity.

The identification assumption is that conditional on the very saturated fixed
effects that this model includes, the unobserved components $\nu_{sdip,h}$ are
uncorrelated with the change in tariffs $\Delta\tau_{sdip,2004}$. Intuitively,
the identification is robust to a NMS (say, Poland’s) policymakers
endogenously targeting EU accession to have preferential access to a third-
party’s (say, Mexico’s) car industry (relative to other industries and
countries), but not if they want to have preferential access to compact cars
relative to SUVs in Mexico.

The identification strategy is plausible. In general, neither lobbyists of
industry trade groups nor trade negotiations work in such a disaggregated
product-level setting. Typically, lobbyists consolidate the interests of the
producers of many products under the same umbrella and try to influence
negotiations. Similarly, even when governments are negotiating tariffs
schedule changes —which was not the case in this particular case —these
negotiations typically also happen in blocs, with governments exchanging
positions in some products for others. Hence, the fact that this is a highly
disaggregated dataset at the product level adds a lot of strength to the
identification strategy.

As shown in Figure 6, an increase in market access by 1 percentage point
increases the probability of starting to produce and export a given product by
about 1 percent by 2010. To benchmark this result, it is about one-third of
the conditional mean $\mathbb{E}[X_{sdip,h}>0|X_{s\cdot
ip,h}>0,h>2003]=2.9\%$. There are no signs of a pre-existing trend before
2004: both the magnitude of the coefficients and the standard errors are very
small before the treatment date.

The related set of continuation regressions, is very similar to the model
estimated in equation (2), except that now it conditions in initial production
being active:

$\displaystyle P\left(X_{sdip,h}>0\middle|Y_{s\cdot ip,2003}=1\right)$
$\displaystyle=$
$\displaystyle\alpha_{h}+\beta_{h}\cdot\Delta\tau_{sdip,2004}+\gamma_{sdi,h}+\nu_{sdip,h}$
$\displaystyle\text{for }h\in\\{2000,\cdots,2010\\}$

In this case, there are no effects observed on the extensive margin. When
countries already have the ability to produce a given product, additional
market access produces very noisy results in the extensive margin. The
coefficients are large and bounce between positive and negative and the
confidence bands are even larger. One potential explanation is that the
countries possibly already had market access before 2004, as illustrated by
the positive (albeit insignificant results) for 2000-03, since they already
had the production capacity. It is possible that most of the effects
concentrate on the intensive margin, something that futures iteration of this
paper would need to check.

Figure 17: Entry Regressions. This plot shows the coefficients $\beta_{h}$ of
the local projection linear probability models specified in equation (2). Each
year is a different cross-sectional regression with approximately 300 thousand
observations. The whiskers show 95% confidence intervals with robust standard
errors clustered at the source-destination-industry level. Figure 18:
Continuation Regressions. This plot shows the coefficients $\beta_{h}$ of the
local projection linear probability models specified in equation (D.3). Each
year is a different cross-sectional regression with approximately 300 thousand
observations. The whiskers show 95% confidence intervals with robust standard
errors clustered at the source-destination-industry level.

##  Appendix E Computational Appendix

This computation appendix explains how I solve for the BGP growth rate.

  1. 1.

Inner loop (Prices of Final Goods). Given parameters
$\\{\theta,\psi,\alpha,\textbf{L},\textbf{T},\boldsymbol{\tau}\\}$ and guesses
for wages $\boldsymbol{w}$, measures of varieties $\boldsymbol{M}$ and some
common return $\mathcal{R}$, use the input-output structure of the model to
solve for the prices of the final goods.

$\displaystyle P_{s}(t)$ $\displaystyle=$
$\displaystyle\gamma\cdot\left[\sum_{n\in\boldsymbol{K}}T_{n}\left(P_{n}^{M}(t)^{\alpha}w_{n}(t)^{1-\alpha}\tau_{ns}\right)^{-\theta}\right]^{-\frac{1}{\theta}}$
$\displaystyle P_{s}(t)$ $\displaystyle=$
$\displaystyle\gamma\cdot\left[\sum_{n\in\boldsymbol{K}}T_{n}\left(w_{n}(t)^{1-\alpha}\tau_{ns}\right)^{-\theta}\left(\sum_{k\in\boldsymbol{K}}M_{k}(t)\left(\frac{\tau_{kn}P_{k}(t)}{\alpha}\right)^{-\frac{\alpha}{1-\alpha}}\right)^{\theta(1-\alpha)}\right]^{-\frac{1}{\theta}}$

with
$\gamma\equiv\Gamma\left(\frac{\theta+1-\sigma}{\theta}\right)^{\frac{1}{1-\sigma}}$.
The last equation makes it explicit that, given parameters, wages, and
measures of varieties, this is a system of $|N|$ equations and $|N|$ unknowns
in final goods prices. A simple grid search algorithm finds a fixed point for
final goods prices.

  2. 2.

Intermediate loop. Given parameters
$\\{\theta,\psi,\alpha,\textbf{L},\textbf{T},\boldsymbol{\tau}\\}$ and guesses
the measures of varieties $\boldsymbol{M}$, some common return $\mathcal{R}$,
and the prices from the following step, use the expenditure determination
equation to solve for final demand.

$P_{s}(t)Y_{s}(t)=\sum_{d\in\boldsymbol{K}}\left[(1-\alpha)\lambda_{sd}^{F}(t)+\frac{\alpha}{\eta}\lambda_{sd}^{M}(t)\right]P_{d}(t)Y_{d}(t)$

Update the guess for usages using a constant fraction $(1-\rho)$ of income
over the BGP and taking advantage of returns $\mathcal{R}^{g}$ and measures
$M^{g}_{k}$:

$w_{s}(t)L_{s}(t)=(1-\alpha)\sum_{k\in\boldsymbol{K}}\lambda_{sk}(t)^{F}(1-\rho)\mathcal{R}^{g}M_{k}(t)^{g}$

Re-normalize
$w_{s}(t)=\frac{w_{s}(t)}{L_{s}\cdot\sum_{k\in\boldsymbol{K}}w_{k}(t)L_{k}}$
to ensure it always maps onto a compact space, it is an operator and converges
according to the contraction mapping theorem.

  3. 3.

Outer loop (Growth rates). Given parameters
$\\{\theta,\psi,\alpha,\textbf{L},\textbf{T},\boldsymbol{\tau}\\}$, prices,
wages, and trade shares calculated in the previous steps, update the guesses
for $M_{s}^{g}$ using:

$M_{s}^{g^{\prime}}=\left(\frac{\lambda_{ss}^{F}(t^{*})}{T_{s}}\right)^{-\frac{1}{\theta(1-\alpha)}}\left(\lambda_{ss}^{M}(t^{*})\right)^{-1}L_{s}\frac{M_{s}^{g}}{\mathcal{R}^{g}}+\frac{\alpha}{\eta}\sum_{k\in\boldsymbol{K}}\left(\lambda_{sk}^{M}(t^{*})\frac{\sum_{l\in\boldsymbol{K}}\lambda_{kl}^{F}(t^{*})(1-\rho)M_{l}^{g}}{P_{s}(t^{*})M_{s}^{g}}\right)$

Again, to make sure it always maps onto a compact space, it is an operator and
converges according to the contraction mapping theorem, renormalize the
measure of varieties:
$M_{s}^{g^{\prime}}=\frac{M_{s}^{g^{\prime}}}{P_{s}(t^{*})\cdot\sum_{k\in\boldsymbol{K}}M_{s}^{g^{\prime}}P_{s}(t^{*})}$

And update the guesses for the global return rates:

$\mathcal{R}^{g^{\prime}}=\frac{1}{\sum_{k\in\boldsymbol{K}}M_{s}^{g^{\prime}}P_{s}(t^{*})}$

A test of this algorithm is, starting from a random guess, knowing that a
group of symmetric countries will eventually converge towards the same measure
of varieties within some tolerance criterion $<\varepsilon$. One numerical
illustration of this convergence is the Figure below, for a group of 4
symmetric countries, starting for a random guess, that eventually converge to
$0.25$ (the sum of the measure of varieties is normalized to sum to 1).

###  E.1 Calibration of Trade Shares

I use observed trade flows to infer trade costs. The strategy goes back to
[30]. According to the handbook chapter by [29], the index is called the Head-
Ries Index (HRI) since 2011 (when the working paper version of [25] was
published).

Expenditure in final goods is defined as:

$E^{F}_{sd}(t)=\lambda_{sd}^{F}(t)P_{d}(t)Y_{d}(t)=\frac{T_{s}\left(\tilde{M}_{s}(t)^{1-\alpha}\right)^{\theta}(w_{s}(t)^{1-\alpha}\tau_{sd})^{-\theta}}{\sum_{n=1}^{N}{T_{n}\left(\tilde{M}_{n}(t)^{1-\alpha}\right)^{\theta}(w_{n}(t)^{1-\alpha}\tau_{nd})^{-\theta}}}\cdot
P_{d}(t)Y_{d}(t)$

The ratio between $E^{F}_{sd}(t)$ and $E^{F}_{dd}(t)$ is, then:

$\frac{E^{F}_{sd}(t)}{E^{F}_{dd}(t)}=\frac{T_{s}\left(\tilde{M}_{s}(t)^{1-\alpha}\right)^{\theta}(w_{s}(t)^{1-\alpha}\tau_{sd})^{-\theta}}{T_{d}\left(\tilde{M}_{d}(t)^{1-\alpha}\right)^{\theta}(w_{d}(t)^{1-\alpha}\tau_{dd})^{-\theta}}$

Analogously, the ratio between $E^{F}_{ds}(t)$ and $E^{F}_{ss}(t)$ is:

$\frac{E^{F}_{ds}(t)}{E^{F}_{ss}(t)}=\frac{T_{s}\left(\tilde{M}_{d}(t)^{1-\alpha}\right)^{\theta}(w_{d}(t)^{1-\alpha}\tau_{ds})^{-\theta}}{T_{d}\left(\tilde{M}_{s}(t)^{1-\alpha}\right)^{\theta}(w_{s}(t)^{1-\alpha}\tau_{ss})^{-\theta}}$

Therefore:

$\frac{E^{F}_{sd}(t)}{E^{F}_{dd}(t)}\cdot\frac{E^{F}_{ds}(t)}{E^{F}_{ss}(t)}=\left(\frac{\tau_{sd}\tau_{ds}}{\tau_{ss}\tau_{dd}}\right)^{-(1-\alpha)\theta}$

Using Assumption (1), $\tau_{ss}=\tau_{dd}=1$ and $\tau_{sd}=\tau_{ds}$.
Hence, I can express the trade cost $\tau_{sd}$ as:

$\tau_{sd}=\left(\frac{E^{F}_{sd}(t)}{E^{F}_{dd}(t)}\cdot\frac{E^{F}_{ds}(t)}{E^{F}_{ss}(t)}\right)^{-\frac{1}{2\theta(1-\alpha)}}$
(F.1)