# Trade, Growth, and Product Innovation

Carlos Góes
UC San Diego<EMAIL_ADDRESS>I am particularly grateful for guidance and
supervision from Marc Muendler, Fabian Trottner, Fabian Eckert, Valerie Ramey,
Kyle Handley, and Johannes Wieland. I benefited from helpful discussions with
Jim Hamilton, Titan Alon, Ana Maria Santacreu, Thomas Sampson, Ezra Oberfield,
Costas Arkolakis, Andrés Rodriguez-Claire, Treb Allen, Mayara Félix, Ernest
Liu, Fernando Parro, Erhan Artuc, Chris Tonetti, Juan Herreño, Rafael Dix-
Carneiro, AV Ludovice, and Raymond Riezman as well as with seminar
participants at multiple conferences. All errors and omissions are solely my
responsibility.

(This version: June 3, 2024
First version: September 29, 2023
For the most recent version click here.)

###### Abstract

Can trade integration induce product innovation? I document that countries
that joined the European Union (EU) started producing more product varieties,
investing more in R&D, and trading more compared to candidate countries that
did not join at a given horizon. Additionally, I show that a plausibly
exogenous increase in market access increases the probability of a given
country starting production of and exporting a given product. To rationalize
this reduced-form evidence, I propose a new quantitative framework that
integrates the forces of specialization and market size. This is a dynamic
general equilibrium model of frictional trade and endogenous growth with
arbitrarily many asymmetric countries that nests the Eaton-Kortum model of
trade and the Romer growth model as special cases. The key result is an
analytical expression to decompose gains from trade into dynamic and static
components. In this framework, the product innovation growth rate increases
with higher market access. Finally, a quantitative version of the model
suggests that: (a) the EU enlargement increased its long-run yearly growth
rate by about 0.10pp; and (b) dynamic gains can account for between 65-90% of
total welfare gains from trade.

Keywords: Trade; Growth; Market Access; Extensive Margin.

JEL Codes: F12, F14, F43, O30

##  1 Introduction

Over the last decades, the trade literature converged to a broad consensus
regarding how to summarize the static gains from trade. But there is no
similar consensus on how to measure dynamic gains from trade111For a
comprehensive review of the literature and the different mechanisms that link
trade, growth, and innovation, see the paper by [38]. In this paper, I address
this topic by examining the mechanisms through which trade integration can
induce product innovation. Economic theory presents conflicting viewpoints
regarding this question. Canonical trade theory typically suggests that
increased economic integration should cause countries to produce a smaller
range of produced goods222In the class of Ricardian models, this follows
naturally: as a country opens up to trade, it specializes in a smaller set of
goods. But this also happens in the class of Melitz models. As a country opens
up to trade, due to the selection effect, the least productive firms of each
country exit the market, which results in a smaller range of firms (or,
equivalently, goods) in either market. This result holds with asymmetric
populations and symmetric productivity distributions or even with asymmetric
productivity distributions, as long as the countries are not too dissimilar
—for the latter see [21].. Models that emphasize growth and innovation, such
as those common in macroeconomics, often emphasize the role of market size for
having an incentive to innovate and produce a large range of goods333This is
true of a very large class of endogenous growth models in macroeconomics, both
with and without scale effects. See, for instance, Chapter 13 of [1]..

This paper integrates these two traditions by conceiving a global marketplace
in which the economic forces of specialization and market access are jointly
operating and developing tractable and intuitive ways of modeling them in a
dynamic framework fit for policy evaluation. First, I show that after large
events of trade integration —the expansion waves of the European Union (EU)
—the countries that joined the EU started producing more product varieties,
investing more in research and development (R&D), and trading more compared to
candidate countries that did not join at a given horizon. Additionally, I show
that a plausibly exogenous increase in market access increases the probability
of a given country starting production of and exporting a given product. These
facts are all suggestive of a dynamic market access effect. Second, to
rationalize this reduced-form evidence, I propose a new dynamic general
equilibrium model of frictional trade and endogenous growth with arbitrarily
many asymmetric countries that nests the [23] model of trade and the [44]
growth model as special cases. Third, I provide analytical expressions
decomposing gains from trade into dynamic and static components; growth and
welfare into “Romer” and “Eaton-Kortum” parts; and show analytically that the
product innovation and R&D growth rates increase with higher market access,
which is consistent the facts I used as motivation for the model. Lastly, I
use a numerical version of the model to estimate the welfare effects of 2004
enlargement of the EU, in this framework: (a) the enlargement increased its
long-run yearly growth rate by about 0.10pp; and (b) dynamic gains can account
for between 65-90% of total welfare gains from trade.

My focus on product innovation stems from two key reasons, one theoretical and
one empirical. From a theoretical standpoint, the new product margin can have
large welfare implications. Empirically, around trade liberalization episodes,
the bulk of trade creation comes from the extensive margin444For the former,
[43] has shown that in a simple trade model, adding extensive margin can make
welfare costs of a 10% tariff increase from 1% to 20%. For the latter, [36]
provide an extensive documentation of the empirical facts..

The paper starts by documenting a set of facts related to the Eastwards
enlargement of the EU. Compared to countries that selected into being
candidates of the EU but were not yet members, New Member States (NMS)
started: (a) producing more product varieties; (b) spending more on private
R&D per capita; and (c) having larger trading values.

Later, in order to go beyond correlational analysis, I exploit the fact that,
once NMS join the EU, they not only have preferential access to the European
market, but they also have to adhere to the Common Commercial Policy of the
European Union. NMS have immediate preferential access to third-party markets
via pre-existing trade agreements between the EU and these third-party
markets.

Importantly, the NMS did not get to negotiate the tariff variation that they
face —these were only a byproduct of the EU accession process. In this
context, through an event-study design, a plausibly exogenous increase in
market access leads to a higher probability of initiating production and
exporting a given product —i.e, leads to product innovation in the extensive
margin.

I develop a dynamic general equilibrium model that is consistent with both the
stylized facts and the market access mechanism to rationalize this reduced-
form evidence. Like much of the trade and growth literature, the model
presented in this paper incorporates forward-looking dynamics. However, unlike
much of the literature, it shies away from stylized simplifications, such as
symmetric countries or two-country cases. It encompasses an arbitrary number
of asymmetric countries, costly trade, and is fit for counterfactual
quantitative exercises. Therefore, it fits neatly into the tradition of
quantitative trade models in international trade or policy counterfactuals
using dynamic stochastic general equilibrium models in macroeconomics.555In
the trade literature, this is the modern world of “trade theory with numbers”
[20]. In the macroeconomics literature, this is the use of macro models as
“the leading tool” for assessing the effect of policy changes in “an open and
transparent manner.” [19].

In the model, in each source country, there are producers of final goods
varieties that combine labor and intermediate goods using a constant returns
to scale technology. They source differentiated intermediate varieties from
foreign countries. These countries differ in their product spaces: some
countries have a measure of intermediate goods that are larger than others.

Intermediate goods are non-rival in the same spirit as in the endogenous
growth literature. As new varieties are invented, they can be simultaneously
and immediately sourced by final goods producers everywhere, inducing
increasing returns.

In this framework, international trade induces substitutability across non-
rival goods. Trade also implies that the measure of intermediate varieties
that effectively diffuses to each country will be a price-weighted average of
the measure of varieties imported from all trade partners. If there are no
trade costs, all countries will share the same effective measure of varieties.
Conversely, in autarky, each country will only take advantage of its own
varieties.

At each destination, final good varieties from every source are aggregated
into a final composite good with some probability. Those actually sourced for
aggregation will be only the lowest-cost varieties at each destination. Prices
depend on productivities, taken to be the realization of a random variable.

The final composite good is used for household consumption and as an input for
the production of intermediate varieties and research & development (R&D).
Once a new blueprint is invented, each intermediate goods producer has
perpetual rights over the production of its variety. They produce under
monopolistic competition and set prices optimal prices accordingly through
market-specific price discrimination. Since this model embeds an input-output
structure, the optimal monopolist prices will depend on the price of the final
composite good at the intermediate source countries.

Forward-looking households use equity markets to invest their savings in the
R&D of new goods. For each unit of the final good invested in a new R&D
project, there is a risky return on investment with probability determined by
a Poisson process. At an aggregate level: domestic households hold a balanced
portfolio of infinitely many small firms, such that they face no idiosyncratic
risk; savings equal investment; investment flows determine the growth of
varieties; and a non-arbitrage condition connects the real interest rate (the
asset market) to real returns on R&D (the equity market).

Over the balanced growth path (BGP), I prove that the equilibrium will be
characterized by a stable distribution of income and measures of varieties,
the real interest rates will equalize across countries, and all countries will
grow at the same rate. Even though there are no international capital markets,
trade acts as a vehicle that will integrate R&D stocks and returns. Countries
with larger labor forces will have larger equilibrium measures of varieties,
which is a fact also observed in the cross-section of countries in the data.

Exploiting the linearity of income in the measure of varieties, I derive an
analytical decomposition for the BGP growth rate across labor and capital
income shares of GDP, whose elements can be further interpreted as “Romer” and
“Eaton-Kortum” components, giving intuitive meaning to the results. The Eaton-
Kortum component of growth is very much Ricardian, i.e., related to
technology, while the Romerian is related to market access, both domestically
and internationally.

Another contribution of the paper is to provide a formula for welfare gains
from trade that decomposes welfare into static and dynamic components. The
welfare formula subsumes the static results of [5] into a dynamic framework.
Like the growth formula, the static component of welfare also has Ricardian
and Romerian margins, with the Romerian margin augmenting the Ricardian one
through an extensive margin. One of the technical contributions of the model
is a tractable way of integrating a new product margin into the Eaton-Kortum
framework, which is one of the workhorse models in the international trade
literature and lacks such a margin.

By comparing the static and dynamic components of welfare, the model clarifies
that they work through different mechanisms, rationalizing the two forces of
market access and specialization. The reason is that the former operates on
households as consumers and the latter on households as producers and
investors

An additional theoretical insight lies in accounting for market access as an
avenue for growth and product innovation. Increased market access is related
to a higher steady-state equilibrium product innovation growth rate. This
finding highlights the positive impact of trade integration on fostering
product innovation and is consistent with the reduced-form evidence presented
in the beginning of the paper.

The final contribution is to set up and calibrate a quantitative version of
the model that solves for the endogenous balanced growth path of the model
with an experiment of asymmetric country groups and costly trade. I then use
this framework and apply trade cost shocks to replicate the policy scenario of
the 2004 Eastwards enlargement of the European Union.

The outcome of the numerical exercise is a set of results and decompositions
of both static and dynamic welfare as a result of the EU enlargement. This
toolkit suggests that: (a) the EU enlargement increased its long-run yearly
growth rate by about 0.10pp; (b) the share of “Eaton-Kortum” share in static
gains from trade can vary widely across countries, being as large as 90% for
some countries and as small as 10% for some other countries; (c) dynamic gains
can account for a large share of total welfare gains from trade; and (d) the
share of dynamic gains also varies across countries, ranging from 65-90% of
total welfare gains from trade.

#### Related Literature

This paper adds to the theoretical literature on trade and growth —and in
particular to trade and product innovation. The literature can be traced back
to the seminal paper by [44]. While Romer does not develop a full model, he
mentions in the paper that a natural extension of his model “pertain to its
implications for growth, trade, and research.”666This is in section VII of
[44]. Extensions of the Romer model of endogenous growth of product innovation
to a two-country framework were later done by [41] and [42] as well as [28],
in a very similar framework. I extend the Romer growth model to a multiple
asymmetric country framework and combine it with a modern quantitative
Ricardian trade model of [23].

The model is also related to the work by [2], who proposed a model with
Armington trade that features an AK-model of trade and growth with a stable
distribution of income over the balanced growth path. While groundbreaking,
they restrict their analysis to the costless trade case, while in this paper
trade costs can be positive with much more heterogeneity across countries.

Since modeling the complete state space of dynamics and countries is
nontrivial, most of the trade and growth literature has to make compromises.
Part of the literature simplifies by assuming a world of symmetric countries
([39]; [45]) or a two-country world ([24]; [32]; [31]). Another part, while
adding the heterogeneity to the cross-section, rules out forward-looking
dynamics and models growth as some external diffusion process ([13], [14]). My
model departs from most of the literature by having both asymmetric countries
and forward-looking dynamics in a theoretical and quantitative framework.

As will be clear in the next section, it is a “true macro model” combined with
a “true trade model.” In this sense, it is more similar to the very recent
models of [46] and [37]. However, unlike mine, the latter is a model of
convergence rather than a model of long-run growth and the former is a model
of firm-productivity growth rather than product innovation.

My paper makes two sets of contributions to the empirical literature. First,
it documents a collection of facts using production-and-trade data around the
enlargement episodes of the European Union. This first part of the analysis is
more akin to papers like [33], [10], [36], and [6], which provide noncausal
documentation of novel stylized facts regarding the extensive margin. But the
paper also goes beyond that, using plausibly exogenous variation in an event-
study design using a very detailed source-destination-product-year dataset. In
doing so, it relates more papers like [26], [8], [4], and [40], which estimate
well-identified empirical effects regarding product innovation.

The paper is organized as follows. Section 2 introduces the empirical evidence
that motivates the work, first summarizing some stylizing facts and then
providing some causal evidence on the relationship between market access and
product innovation. Later, Section 3 lays down the theory, introduces the
model, defines the equilibrium, and states the main results regarding the
existence of the balanced growth path and the equilibrium growth rate being
decreasing in trade costs. Afterward, Section 4 uses a numerical version of
the model to estimate the dynamic welfare effects of the 2004 enlargement of
the European Union and decomposes them using the key results of the previous
section. Finally, I conclude by trying to relate the main takeaways to the
general literature and where the main advances were.

##  2 Empirical Evidence

This section describes the evidence related to international trade and product
innovation in the context of the different enlargement waves of the European
Union (EU). First, it describes the data. Then it presents some stylized facts
comparing new member states (NMS) of the EU relative to candidate countries.
Finally, it uses an event-study approach to isolate some plausibly exogenous
variation of trade costs on the probability of initiating production of a new
product.

#### Data sources

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. Data are really high-
quality and coverage error is estimated to be below 10%. These data allows one
to create a time-series of product counts for products actually produced in
each member state and candidate countries of the European Union.

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. Bilateral trade flow data
comes from UNCOMTRADE.

I matched all of these to the production data using Eurostat’s concordance
between Prodcom product-codes, which are finer than Harmonized System (HS)
6-digit level, to a HS-6 digit level. WITS and UNCOMTRADE data come natively
at an HS-6 digit level. Combined, these constitute a novel production-and-
trade matched database.

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.

Further details on the data used, data matching, and data construction are on
Appendix D.

#### Institutional Context

Throughout this paper, the institutional setting will be expansion of the
European Union —or its enlargement, how it is typically called in EU language.
The enlargement happened in different waves as the EU included more members
from the six original members that created the group in 1957, as shown in
Figure 1.

I use this setting in three ways. First, I exploit the staggered nature of the
EU expansion to summarize some stylized facts regarding some key statistics of
New Member States (NMS) of the EU relative to candidate countries who are not
yet members of the EU at a given horizon.

Then, focusing on the largest expansion wave in 2004, I show that that the
adoption of the EU’s Common Commercial Policy induced a plausibly exogenous
variation in market access between 9 NMS that joined the EU in 2004777While
ten countries joined the EU in 2004, I do not have product-level PRODCOM data
for Malta. and 12 third-party countries with previously existing trade
agreements with the EU. Finally, in the numerical quantification exercise, I
will also use the enlargement waves as natural country groups to estimate the
welfare effects of the 2004 enlargement.

Figure 1: Institutional Context: Timeline of European Union Enlargement. The
EU enlargement comes in waves. The future cohorts serve as comparison groups,
for some time, to previous cohorts. The largest expansion wave is 2004.

#### Stylized Facts

My stylized facts compare two groups: countries that became new members of the
European Union relative to countries that self-selected into becoming
candidates for EU membership but were not yet members at a given time horizon,
exploiting the staggered nature of the enlargement. Here, one can think of
countries that became EU members as individual members of a “treatment group”
and candidate countries that applied for EU membership but had not yet become
members by that time as individual members of a “control group.” Of course,
since treatment assignment, in this case, is not random, this is not actually
a true experiment.

In this paper, to avoid the potential biases of the Two-Way Fixed Effects
(TWFE) estimator in summarizing the data, I adopt the Callaway-Sant’Anna (CS)
estimator888[27] shows that the TWFE estimator is a weighted average of all
possible two period-two group comparisons and that, as emphasized by [12], it
is biased if treatment effects are heterogeneous. [47] proposed a new
estimator that accommodates treatment effect heterogeneity, which was later
generalized by [16].. In a nutshell, CS calculates group-specific treatment
effects by: (a) comparing the treated group with either the not-yet-treated
groups or the never-treated groups; and then (b) aggregating them into an
average treatment on the treated given a specific set of weights. This
estimator is consistent even if true treatment effects are heterogeneous.

Therefore, even if the objective is to simply summarize the data rather than
to make causal claims, one would still want to avoid making “forbidden
comparisons.” The CS estimator, in this case, will simply recover the average
difference in outcomes for NMS relative to countries that are candidate
countries but are not yet members, at different horizons around EU enlargement
events. In Appendix D, I formal description of the CS estimator used here.

Given the weighting scheme of the CS estimator, in the estimates reported
below, the event that will have the largest weight will be the 2004 EU
expansion, which enlarged membership by ten countries, but only nine are
observed in PRODCOM data. The other episodes of expansion – Bulgaria and
Romania, in 2007; and Croatia, in 2013 – influence the estimates with
proportional weights for the horizons in which data is available. It is
important to highlight that throughout the sample, there is readily available
data for candidate countries that never became EU members, which serve as a
natural comparison group.

Here, I run the staggered difference-in-differences event study regressions
for a set of variables, using similar models. First, using the measure of
produced varieties as the dependent variable. Then with log of real private
research and development expenditures and the log of real value of yearly
trade as dependent variables.

The frame of reference is to take these variables as aggregate macro moments.
Relative to a candidate country that did not fully integrate its economy with
the European Union and did not have preferential access to the trade partners
of the EU, what happens, on average, to these variables in New Member States?

As shown in Figure 2, fifteen years after membership, the expected
differential increase in varieties is $306$, or about $17\%$ relative to the
year of membership999The average treatment on the treated is $306.23$ and the
conditional average number of produced varieties in treatment year zero is
$1804.6$.. The differential effect seems to cumulatively increase after the
year of membership.

The effects regarding private R&D, shown in Figure 3 show a clear break in
trend in the differential averages around the year of membership. Fifteen
years after the expected differential growth in private R&D expenditures is
about 60%.

Finally, the results relative to trade also show a differential growth, as
illustrated by Figure 4. There are no signs of pre-treatment trends and, seven
years after membership, the expected differential growth in the value of
yearly trade is about 50%.

Figure 2: Staggered difference-in-differences: Measure of Varieties. X-axis:
years around EU enlargement event. Y-axis: in number of produced varieties.
This plot shows the estimated coefficients $\theta(t)$ time-specific average
treatment on the treated coefficient described by equation (D.2) at the and
aggregate level. The bars around the red line denote 95% bootstrapped standard
errors. Figure 3: Staggered difference-in-differences: Log of Private Research
and Development Expenditures Per Capita. X-axis: years around EU enlargement
event. Y-axis: in log value private yearly R&D expenditures per capita
(thousand euro). This plot shows the estimated coefficients $\theta(t)$ time-
specific average treatment on the treated coefficient described by equation
(D.2) at the aggregate level. The bars around the red line denote 95%
bootstrapped standard errors. Figure 4: Staggered difference-in-differences:
Log of Real Value of Yearly Trade. X-axis: years around EU enlargement event.
Y-axis: in log real value of yearly trade (thousand US Dollars). This plot
shows the estimated coefficients $\theta(t)$ time-specific average treatment
on the treated coefficient described by equation (D.2) at the aggregate level.
The bars around the red line denote 95% bootstrapped standard errors.

#### Causal Evidence

Once NMS joins the EU, they not only have preferential access to the European
market, but they also have to adhere to the Common Commercial Policy of the
European Union. This means that these countries have immediate preferential
access to third-party markets via previously existing trade agreements between
the EU and these third parties. Furthermore, since these trade agreements
previously existed, while the NMS had immediate access to them, they did not
get to negotiate the tariff variation that they faced —these were only a
byproduct of the EU accession process.

Figure 5 illustrates how this happened in a specific example: the Free Trade
Agreement between the EU and Mexico. The EU joined a FTA with Mexico in 2000,
but the NMS only joined the EU in 2004, so in 2004 the latter immediately
adhered to these previously negotiated tariff schedule. The product-level
bilateral tariff variation $\Delta\tau_{sdip,2004}$ which was a by-product of
the EU accession process is my measure of the market accession shock.

Preferential TariffsNMSEU2000: FTA EU-MX enters into
force2004$\Delta\tau_{sdip,2004}$MFN Tariffs Figure 5: Event Study Design,
Constructing the Trade Shock: I use the fact that when the NMS joined the EU
in 2004, they had immediate preferential access to third-party markets via
previously signed EU trade agreements which the NMS did not get to negotiate.
The product-level bilateral tariff variation $\Delta\tau_{sdip,2004}$ which
was a by-product of the EU accession process is my measure of the market
accession shock. In the example above, the EU joined a FTA with Mexico in
2000, but the NMS only joined the EU in 2004, so in 2004 they immediately
adhered to these previously negotiated tariff schedule.

I focus on the largest wave of enlargement was in 2004. The source of
variation is at the source-country $\times$ destination-country $\times$
HS-6-code product level. 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. In each year, there are about 300
thousand observations.101010Leading up to 2004 there were no large changes in
bilateral tariffs between NMS and third parties but between 2003 and 2004
there was a median drop of about 2.5 percentage points. In the years after the
enlargement, there was also not a large change in the distribution of
bilateral tariff rates, which is shown in Figure 15 in Appendix D. Figure 16
in Appendix D plots the distribution of $\Delta\tau_{sdip,2004}$, excluding
the zero-valued observations.

I estimate a sequence of cross-sectional local-projection linear probability
models, which estimate what is the marginal effect of a decrease 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 $h$. These types of cross-sectional event studies with
local projections can be interpreted as differences in differences with
continuous treatments111111See [17] and [22]..

This strategy takes seriously the assertion in [7] that countries engage
endogenously in free trade agreements (FTAs). 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. For further details on the
methodology, see Appendix D.

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.

Figure 6: 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.

#### Summary of Empirical Evidence

I have documented the following novel facts: as New Member States go through a
large trade integration event with the European Union, they start producing
new product varieties; investing more in research and development; and trading
more in real values relative to candidate countries that are not yet members
at a given horizon. Furthermore, I have shown that as they are exposed to a
plausibly exogenous new market accession shock, as it happened due to the
idiosyncrasies of the Common Commercial Policy of the EU, they increase their
probability of starting production of a new product variety, which suggests
that market access increases the rate of product innovation.

To rationalize this reduced-form evidence, I develop a dynamic general
equilibrium model that is consistent with both the stylized facts and the
market access mechanism. This is what I do in the following section.

##  3 Theory

Here I present a dynamic multi-country model of the world economy with
intertemporal optimization, investment in research and development, and trade
in final and intermediate goods. In this economy, time is continuous with
$t\in\mathcal{T}\equiv[0,\infty)$ and countries indexed by
$s\in\boldsymbol{K}\equiv\\{1,\ \cdots,N\\}$.

Every country has the ability to produce final goods $\omega\in[0,1]$.
However, they differ in their ability to produce non-rival intermediate inputs
$\nu\in[0,M_{s}(t)]$, where the upper bound of the interval $M_{s}(t)$ defines
the product space of a particular country. Intermediate goods are non-rival in
the same spirit as in the endogenous growth literature: new blueprints can be
simultaneously used by multiple producers at the same time, inducing
increasing returns to scale121212See [34] and [35] for extensive reviews..

As intermediate goods are invented, trade acts as a mechanism that diffuses
new blueprints: producers expand their production function by sourcing newly
minted inputs from around the world. Exporters are monopolists in their
intermediate varieties and therefore have the incentive to invest in the
development of new varieties, thereby propelling growth. Therefore,
international trade will work as a vehicle that integrates global research and
development stocks and induces growth-rate convergence over the balanced
growth path.

My goal is to make this model easily accessible and recognizable for someone
who is familiar with either modern trade theory or modern growth theory. This
model will recover, as special cases, the [23] model of trade and the [44]
model of growth. Some functional form assumptions will be such that this
nesting is clear.

###  3.1 Demand

In each country $s\in\boldsymbol{K}$, there is a representative household the
maximizes its lifetime utility according to:

$\displaystyle\max_{C_{s}(t)}$ $\displaystyle\int_{0}^{\infty}\exp\\{-\rho
t\\}\log\left(C_{s}(t)\right)dt$ $\displaystyle s.t.\quad
P_{s}(t)I_{s}(t)+P_{s}(t)C_{s}(t)$ $\displaystyle=$ $\displaystyle
r_{s}(t)A_{s}(t)+w_{s}(t)L_{s}$

where $P_{s}(t)C_{s}(t)$ are aggregate consumption good prices and quantities
in country $s$; $I_{s}(t)$ are instantaneous investment flows; and
$w_{s}(t),r_{s}(t)$ are wages and interest rates. At any instant, the state of
asset holdings is simply the cumulative investment flows:
$A_{s}(t)\equiv\int_{0}^{t}I(s)ds$131313This, of course, implies that one can
write investments as $I_{s}(t)=\dot{A}_{s}(t)$, which clarifies the optimal
control problem at hand..

Households choose a sequence of consumption quantities for the aggregate good,
satisfying the Euler Equation:

$\frac{\dot{C}_{s}(t)}{C_{s}(t)}=\frac{r_{s}(t)}{P_{s}(t)}-\rho$ (2)

###  3.2 Production and Trade in Varieties

There are three kinds of producers in each country: those who produce
varieties of the final good, those who produce varieties of intermediate
goods, and those who invest in research and development. This section will
focus on the two first ones.

#### Final Goods Producers.

In each country, a local assembler for the final composite good $Y_{d}(t)$ who
operates under perfect competition uses the least expensive variety
$\omega\in[0,1]$ available at $d\in\boldsymbol{K}$ with the following
technology:

$Y_{d}(t)=\Big{[}\int_{0}^{1}y_{d}(t,\omega)^{\frac{\sigma-1}{\sigma}}d\omega\Big{]}^{\frac{\sigma}{\sigma-1}}$

where $\sigma>1$ is a constant elasticity of substitution across sourced
varieties $\omega$. Under these assumptions, the ideal price index of the
final good satisfies $P_{s}(t)$ satisfies:

$P_{s}(t)=\Big{[}\int_{0}^{1}p_{s}(t,\omega)^{1-\sigma}d\omega\Big{]}^{\frac{1}{1-\sigma}}$

A producer of each variety $\omega\in[0,1]$ of the final good is endowed with
a constant returns to scale technology that combines labor and intermediate
inputs $\nu\in[0,M_{s}(t)]$ coming from multiple countries
$k\in\boldsymbol{K}$:

$y_{s}(t,\omega)=z_{s}(t,\omega)[\ell_{s}(t,\omega)]^{1-\alpha}\left(\frac{1}{\alpha}\sum_{k\in\boldsymbol{K}}\int_{0}^{M_{k}(t)}[x_{ks}(t,\omega,\nu)]^{\alpha}d\nu\right)$
(3)

where $z_{s}(t,\omega)$ is total factor productivity; $\ell_{s}(t,\omega)$ is
factor demand for labor for variety $\omega\in[0,1]$ located in country $s$;
and $x_{ks}(t,\omega,\nu)$ is the demand for a intermediate good of variety
$\nu\in[0,M_{k}(t)]$ sourced from country $k$ for production as an input of a
final good in country $s$.

Non-rival intermediate goods varieties are differentiated across countries: an
input $\nu\in[0,M_{k}(t)]$ is different from $\nu\in[0,M_{n}(t)]$, even if it
is indexed by the same symbol. For instance, the first one may be a twelve-
core computer chip from Estonia while the second one may be a large language
model from Malta. Additionally, note that countries differ in their ability to
produce intermediate goods, which is denoted by the upper bound of the
integral $M_{k}(t)$. Optimal demand for an intermediate good satisfies:

$x_{ks}(t,\omega,\nu)=\left[\frac{p_{ks}(t,\omega,\nu)}{p_{ss}(t,\omega)}\right]^{-\frac{1}{1-\alpha}}\cdot\ell_{s}(t,\omega)\cdot
z_{s}(t,\omega)^{\frac{1}{1-\alpha}}$ (4)

#### Intermediate Goods Producers.

Each intermediate goods producer in country $s$ has perpetual rights over the
production of each variety $\nu\in[0,M_{s}(t)]$. They are endowed with a
linear technology that transforms one unit of the final good into one unit of
the intermediate good.

######  Assumption 1 (Trade Costs).

Trade is subject to iceberg trade costs, which implies that shipping a final
or intermediate good variety from source region $s$ to a consumer in region
$d$ requires producing $\tau_{sd}\geq 1$, where $\tau_{dd}=1$ and
$\tau_{sd}=\tau_{ds}$ for all $s,d\in\boldsymbol{K}$.

Given assumption (1), intermediate goods producers face heterogeneous marginal
costs and set optimal prices accordingly through market-specific price
discrimination. They take marginal costs and demand curves as given and choose
optimal prices to maximize profits, with the optimal price being a mark-up
over marginal costs for every variety $\nu$ and $\omega$:

$p^{M}_{ks}(t)=\frac{\tau_{ks}P_{k}(t)}{\alpha}\qquad\forall\omega\in[0,1],\quad\nu\in[0,M_{k}(t)]$

Note that this is the standard result of profit maximization under
monopolistic competition with two variations. First, as in most trade models,
prices are differentiated by destination and are inclusive of trade costs
$\tau_{ks}$. Second, since intermediate goods use one unit of the final good
at the origin country $k$ to produce one unit of the intermediate good, its
marginal cost is $P_{k}(t)$. Optimal monopolist prices being independent of
variety $\nu$ imply that demand for is symmetric:

$\bar{x}_{ks}(t,\omega)\equiv\left[z_{s}(\omega)\cdot\frac{p_{ss}(t,\omega)}{p^{M}_{ks}(t)}\right]^{\frac{1}{1-\alpha}}\cdot\ell_{s}(t,\omega)\qquad\forall\nu\in[0,M_{k}(t)]$

Given the result above, rewrite the final goods firm maximization problem in
the following way:

$\max_{\ell_{s}(t,\omega)}\frac{1}{\alpha}[p_{ss}(t,\omega)\cdot
z_{s}(t,\omega)]^{\frac{1}{1-\alpha}}\cdot\tilde{M}_{s}(t)\cdot\ell_{s}(t,\omega)-\ell_{s}(t,\omega)w_{s}(t)$
(5)

which comes from substituting for $\bar{x}_{ks}(t)$ and defining:

$\underbrace{\tilde{M}_{s}(t)}_{\begin{subarray}{c}\text{effective}\\\
\text{measure of}\\\ \text{input
varieties}\end{subarray}}\equiv\sum_{k\in\boldsymbol{K}}\underbrace{M_{k}(t)}_{\begin{subarray}{c}\text{measure
of}\\\ \text{varieties}\\\ \text{in each
}k\end{subarray}}\cdot\left(\underbrace{p^{M}_{ks}(t)}_{\begin{subarray}{c}\text{optimal
monopolist}\\\ \text{price from }k\text{ to
}s\end{subarray}}\right)^{1-\eta}\qquad\text{where }\eta=\frac{1}{1-\alpha}$
(6)

The effective measure of input varieties is a key object in this model that
captures the diffusion of non-rival intermediate goods to country $s$. It
measures input varieties sourced from each country weighted by marginal cost.
The first term on the right-hand side captures heterogeneity in the source-
country measure of varieties since final goods producers are sourcing
intermediate varieties internationally. The second term captures the
substitutability across intermediate goods, controlled by the elasticity of
substitution $\eta$. The exponent $1-\eta<0$ down-weights the relative
importance of intermediate goods coming from source countries $k$ with
relatively more expensive intermediate inputs.

This object also makes explicit how the model nests both the Eaton-Kortum
model of trade and the Romer growth model. If $\alpha\to 0$, then there is no
intermediate sector. The technology $\eqref{eq: production-function-main}$
collapses into a linear production function as in [23]. If the world is in
autarky —i.e., if $\tau_{sd}\to\infty$ for all $d\neq s$ —then, after setting
$P_{s}(t)=1$ as the numéraire of the home economy,
$\tilde{M}_{s}(t)=\alpha^{\frac{\alpha}{1-\alpha}}M_{s}$ and the final goods
technology becomes linear in labor with an extensive margin $M_{s}$, as in
[44]141414For a more detailed description of the nesting, see Appendix B.10..

Even outside the limiting cases expressed above, one should observe that the
final goods producer’s technology in this model is related to those in both
the Eaton-Kortum and the Romer models. It is equivalent to a simple Eaton-
Kortum model that uses the final good as an intermediate input with an added
extensive margin shifter $\tilde{M}_{s}(t)$. It also is related to the
technology of the Romer model, which is linear in labor (or human capital),
except that in this model the measure of varieties component is a weighted
average of inputs coming from domestic and international suppliers.

Note that with a slight redefinition, one can also interpret a transformation
of $\tilde{M}_{s}(t)$ as the price of a composite basket of intermediate
goods, as it is standard in many models that resort to a constant elasticity
of substitution:

$P^{M}_{s}(t)\equiv\tilde{M}_{s}(t)^{\frac{1}{1-\eta}}=\left(\sum_{k\in\boldsymbol{K}}M_{k}(t)\cdot
p^{M}_{ks}(t)^{1-\eta}\right)^{\frac{1}{1-\eta}}$ (7)

which is how, due to notational convenience, this object will appear
throughout the rest of the paper. Define the value in final goods in country
$d\in\boldsymbol{K}$ to be $P_{d}(t)Y_{d}(t)$. Then, using the definitions
above and the properties of C.E.S. and Cobb-Douglas, intermediate goods sales
by country $s$ in country $d$ equal:

$\displaystyle M_{s}(t)p^{M}_{sd}(t)x_{sd}(t)=\alpha\cdot
M_{s}\left(\frac{p^{M}_{sd}(t)}{P^{M}_{d}(t)}\right)^{1-\eta}\cdot
P_{d}(t)Y_{d}(t)=\alpha\cdot\lambda_{sd}^{M}(t)\cdot P_{d}(t)Y_{d}(t)$

The last equation follows from defining intermediate trade shares
$\lambda_{sd}^{M}(t)\equiv\frac{M_{s}p_{sd}^{M}(t)^{1-\eta}}{\sum_{k\in\boldsymbol{K}}M_{k}p_{kd}^{M}(t)^{1-\eta}}$.

In the standard [44] model, assemblers source intermediate goods exclusively
from domestic suppliers. One important implication of that assumption is
symmetry: the price of all intermediate goods will be the same. Conversely, in
this framework, when sourcing intermediate goods from multiple countries
$k\in\boldsymbol{K}$, the prices of these goods will no longer be necessarily
the same.

Economically, it is this lack of symmetry in prices that will induce
substitutability across varieties sourced from different countries, which is
reflected in the composite price of intermediate goods above. The more
dissimilar countries are in terms of their relative unit costs, the more
substitution across intermediate goods will occur. Hence, accommodating
asymmetric countries in a dynamic framework will be an important feature of
this model.

#### Trade in final goods.

The factory gate price $p_{ss}(t,\omega)$ for a variety has three components:
the unit production cost $w_{s}(t)$, the price of intermediate goods
$P^{M}_{s}(t)$, and a producer-specific productivity $z_{s}(t,\omega)$.
Destination prices also include iceberg trade costs. Under perfect
competition, consumers in country $d$ choose the lowest price variety $\omega$
available at the domestic market:

$p_{d}(t,\omega)=\min_{s\in\boldsymbol{K}}\left\\{p_{sd}(t,\omega)\right\\}=\min_{s\in\boldsymbol{K}}\left\\{\tau_{sd}p_{ss}(t,\omega)\right\\}=\min_{s\in\boldsymbol{K}}\left\\{\frac{P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha}\tau_{sd}}{z_{s}(t,\omega)}\right\\}$
(8)

######  Assumption 2 (Productivity draws).

Following [23], assume that $z_{s}(t,\omega)$ is an iid random variable drawn
from a market-specific Fréchet distribution

$F_{s}(t)(z)=\exp\\!\left\\{-T_{s}\,z^{-\theta}\right\\}.$

where $T_{s}$ is the the scale parameter and $\theta$ is the shape parameter.

Given assumption (2), both prices and demanded quantities (which are functions
of productivity draws) are also random variables. By the law of large numbers,
the share of varieties sourced from $s$ to $d$ equals151515Since there are
infinitely many varieties $\omega$ and productivities are iid random
variables, by the law of large numbers, the share of varieties sourced from
$s$ to $d$ converges almost surely to the probability of sourcing a specific
variety from $s$ to $d$.:

$\lambda^{F}_{sd}(t)\equiv\frac{E^{F}_{sd}(t)}{E^{F}_{d}(t)}=\frac{T_{s}(P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha}\tau_{sd})^{-\theta}}{\sum_{n\in\boldsymbol{K}}{T_{n}(P^{M}_{n}(t)^{\alpha}w_{n}(t)^{1-\alpha}\tau_{nd})^{-\theta}}}$
(9)

where $E^{F}_{sd}(t)$ denotes the expenditure on final goods going from
country $s$ to country $d$; $E^{F}_{d}(t)$ denotes total expenditure on final
goods in country $d$.

###  3.3 Research and Development

The research sector creates new varieties of the intermediate good. One can
think of this sector as investing in the invention of new machines, which
result in new blueprints. These firms use $\psi$ units of the final good as
inputs to research and development (R&D), but success is not guaranteed.

######  Assumption 3 (Research and Development Process).

The success rate of R&D follows a Poisson process with flow arrival rate equal
to $\psi I_{s}(t)dt$, where $I_{s}(t)$ is the research input, measured in
units of the final good per time unit.

Once researching firms invent a new machine, they hold perpetual monopoly
rights over the new variety $\nu$. They can either set up their own shop to
produce and enjoy the profits of producing such variety at the market or,
alternatively, they can sell the rights to this patent to an intermediate
variety producer. In either case, domestic households, that finance the
invention of new varieties through capital markets, will collect the profits.

The economic value of a new variety is the present value of producing the new
varieties and selling them as intermediate inputs to final goods producers,
which is, at period $t$:

$V_{s}(t,\nu)=\int_{t}^{\infty}\exp\left\\{-\int_{t}^{\tau}\frac{r_{s}(k)}{P_{s}(k)}dk\right\\}\pi_{s}(\tau,\nu)d\tau$
(10)

where $\pi_{s}(\tau,\nu)$ is the flow profit per variety per unit of time.
Research firms will only invest if the expected return of their investment is
positive, that is $\psi V_{s}(t,\nu)I_{s}(t,\nu)-P_{s}(t)I_{s}(t,\nu)\geq 0$.
With free entry, this condition holds with equality and in equilibrium it pins
down the value of each variety: $V_{s}(t,\nu)=P_{s}(t)/\psi$.

Since the only asset market in this economy is the domestic equity market,
domestic households save by funding investments in new varieties through a
balanced portfolio of infinitely many small firms, such that they face no
idiosyncratic risk. At the aggregate level, then, $\dot{M}_{s}(t)=\psi
I_{s}(t)$, where $I_{k}(t)$ is the level of aggregate investment in the
domestic economy measured in units of the final good. The value of aggregate
assets is simply the value of all invented varieties
$P_{s}(t)A_{s}(t)=M_{s}(t)V_{s}(t)$ and, since the arrival rate of ideas is
constant, the total stock of assets is at any instant a function of the total
measure of varieties $A_{s}(t)=M_{s}(t)/\psi$.

Taking the derivative of both sides of (10) with respect to time and noting
that both $V_{s}(t,\nu)$ and $\pi_{s}(\tau,\nu)$ are independent of $\nu$ pins
down the real interest rate in this economy. The result is a non-arbitrage
condition relating returns on assets to returns on R&D:

$\frac{r_{s}(t)}{P_{s}(t)}=\underbrace{\frac{\psi\cdot\pi_{s}(t,\nu)}{P_{s}(t)}}_{\text{flow
dividend rate}}+\underbrace{\frac{\dot{P}_{s}(t)}{P_{s}(t)}}_{\text{capital
gains}}$ (11)

###  3.4 Market Clearing and Equilibrium

#### Factor Market Clearing

Let $Y_{d}(t)$ denote the total output of the final good and
$X_{d}(t),I_{d}(t)$ denote the use of the final good as inputs for the
production of intermediate inputs and R&D, respectively. Then total output in
the final good for a given country must satisfy:

$Y_{d}(t)=C_{d}(t)+I_{d}(t)+X_{d}(t)$ (12)

where $I_{d}(t)$ and $C_{d}(t)$ are pinned down by the dynamic problem,
described below, and $X_{d}(t)$ can be expressed as a function of aggregate
demand in all destinations.

#### Expenditure Determination

Flow aggregate profits $\Pi_{s}(t)\equiv\int_{0}^{M_{s}(t)}\pi(t,\nu)d\nu$ are
a constant fraction of revenue:

$\Pi_{s}(t)=\frac{\alpha}{\eta}\cdot\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t)\cdot
P_{d}(t)Y_{d}(t)$ (13)

On the expenditure side, GDP of each destination country $s\in\boldsymbol{K}$
country will be exhausted as the combination of the total expenditures of
labor and capital income:

$P_{s}(t)Y_{s}(t)=w_{s}(t)L_{s}+\Pi_{s}(t)$ (14)

From the income side, nominal GDP must equal the sum of total flow payments
received domestically and from the rest of the world:

$P_{s}(t)Y_{s}(t)=\sum_{d\in\boldsymbol{K}}\left[(1-\alpha)\lambda_{sd}^{F}(t)+\frac{\alpha}{\eta}\lambda_{sd}^{M}(t)\right]P_{d}(t)Y_{d}(t)$
(15)

#### Trade Balance

Since savings equals investment and there is no access to international
capital markets in this economy, GDP accounting requires that trade balances
in each country and net exports are equal to zero at any instant:

$\displaystyle\sum_{d\neq
s\in\boldsymbol{K}}\lambda_{sd}^{F}(t)P_{d}(t)Y_{d}(t)+\alpha\sum_{d\neq
s\in\boldsymbol{K}}\lambda_{sd}^{M}(t)\left[\sum_{k^{\prime}\in\boldsymbol{K}}\lambda_{dk^{\prime}}^{F}(t)P_{k^{\prime}}(t)Y_{k^{\prime}}(t)\right]=$
$\displaystyle[1-\lambda_{ss}^{F}(t)]P_{s}(t)Y_{s}(t)+\alpha[1-\lambda_{ss}^{M}(t)]\left[\sum_{k^{\prime}\in\boldsymbol{K}}\lambda_{dk^{\prime}}^{F}(t)P_{k^{\prime}}(t)Y_{k^{\prime}}(t)\right]$
(16)

#### Dynamic Equilibrium

The dynamics in each of the countries of this world economy are governed by
the following system of differential equations:

$\displaystyle\dot{C}_{s}(t)$ $\displaystyle=$
$\displaystyle\left[\frac{r_{s}(t)}{P_{s}(t)}-\rho\right]C_{s}(t)$ (17)
$\displaystyle\dot{M}_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{r_{s}(t)}{P_{s}(t)}M_{s}(t)+\psi\frac{w_{s}(t)}{P_{s}(t)}L_{s}-\psi
C_{s}(t)$

As it is clear from the system above, the dynamics of the model are
essentially neoclassical. However, since openness to trade impacts the cross-
sectional distribution of wages and prices, it will also impact the path of
consumption product measures over time.

The first equation —the Euler Equation —states that the household in a country
$s\in\boldsymbol{K}$ will choose an upward-sloping consumption path if the
real interest rate is greater than the rate of time preference. The higher
this gap, the more a household will be willing to defer current consumption
and take advantage of higher returns in the asset and R&D markets.

The second equation is less obvious to interpret in its current form, but it
states that the growth in the product measure in each country is proportional
to the net investment rate. Since expected profits of new varieties are always
positive, the net investment rate is also always positive, which means that
new varieties are always created, inducing growth in this model.

A more explicit way to observe the net investment rate is by writing the
second equation in its equivalent asset representation. Since $M_{s}(t)=\psi
A_{s}(t)$, then:

$\dot{A}_{s}(t)=I_{s}(t)=\underbrace{\frac{r_{s}(t)}{P_{s}(t)}A_{s}(t)}_{\text{real
capital income}}+\underbrace{\frac{w_{s}(t)}{P_{s}(t)}L_{s}}_{\text{real labor
income}}-\underbrace{C_{s}(t)}_{\text{real consumption}}$

which, along with the discussion regarding the non-arbitrage condition in the
previous section, helps clarify that asset markets and varieties markets are
two sides of the same coin.

The assumption of log preferences substantially simplifies the dynamic
problem. In Appendix B.2, I show that instantaneous consumption is always
well-defined as a constant fraction of lifetime wealth:

$C_{s}(t)=\rho\left[\underbrace{A_{s}(t)}_{\text{wealth at
}t}+\underbrace{\int_{t}^{\infty}\frac{w_{s}(\tau)}{P_{s}(\tau)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(\tau)\cdot\tau\right\\}d\tau}_{\text{PV
of future labor income}}\right]$

where
$\bar{r}_{s}(\tau)\equiv\frac{1}{\tau}\int_{t}^{\tau}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu$
is the average real interest rate between periods $t$ and $\tau$. Once there
is an explicit solution for consumption at every $t$, the differential
equation for $\dot{M}_{s}(t)$ becomes autonomous, and also has an explicit
solution as a function of the path of prices.

It also shows that there is a unique initial choice of consumption that is
consistent with the optimal choices described by (17) and the transversality
condition. Since the other conditions to satisfy the Maximum Principle are
satisfied, this is equivalent to showing that the solution to the dynamic
problem is unique.

######  Definition 1 (Dynamic Equilibrium).

The dynamic equilibrium of the world economy is defined by a collection of
paths of consumption quantities, assets stocks, and profit flows
$[C_{s}(t),A_{s}(t),\Pi_{s}(t)]$; paths of final goods varieties output
quantities $[y_{s}(t,\omega)]$; paths of intermediate goods varieties output
quantities $[x_{ks}(t,\omega,\nu)]$; paths of prices
$[w_{s}(t),r_{s}(t),P_{s}(t)$, $p_{ss}(t,\omega),p_{sk}(t,\omega,\nu)]$; and a
vector of fundamentals
$(\theta,\sigma,\boldsymbol{T},\boldsymbol{\tau})^{\prime}$ where
$\boldsymbol{T}\equiv\\{T_{s}\\}$ is a collection of location parameters of
the Fréchet distribution and $\boldsymbol{\tau}\equiv[\tau_{sd}]$ is a matrix
of trade costs, such that: (a) households maximize utility given the path for
prices; (b) final goods firms maximize profits given the path for prices; (c)
intermediate goods firms choose prices to maximize profits given demand
functions and final goods prices; (d) trade balances; and (e) factors and
goods markets clear.

#### Homogeneity of Income in Equilibrium

One of the key properties of this model is that real income, real wages, and
real profits are a function of the measure of varieties $M_{s}(t)$ and of
terms that are homogeneous of degree zero in the distribution of the measure
of varieties $\\{M_{k}(t)\\}_{k\in\textbf{K}}$. Note that for real aggregate
labor and aggregate capital income, respectively, can be expressed as:

$\displaystyle\frac{w_{s}(t)L_{s}}{P_{s}(t)}$ $\displaystyle=$ $\displaystyle
M_{s}(t)\cdot\left(\frac{T_{s}}{\lambda^{F}_{ss}(t)}\right)^{\frac{1}{\theta(1-\alpha)}}\cdot\left(\lambda^{M}_{ss}(t)\right)^{-1}\cdot
L_{s}\equiv M_{s}(t)\times\mathcal{R}^{w}_{s}(t)$
$\displaystyle\frac{\Pi_{s}(t)}{P_{s}(t)}$ $\displaystyle=$ $\displaystyle
M_{s}(t)\cdot\frac{\alpha}{\eta}\sum_{d\in\boldsymbol{K}}\frac{\left(\tau_{sd}P_{s}(t)\right)^{1-\eta}}{\sum_{k^{\prime}\in\boldsymbol{K}}M_{k^{\prime}}\left(\tau_{{k^{\prime}}d}P_{k^{\prime}}(t)\right)^{1-\eta}}\frac{P_{d}(t)Y_{d}(t)}{P_{s}(t)}\equiv
M_{s}(t)\times\mathcal{R}_{s}^{\pi}(t)$

which, of course, means that Real GDP is also a function of $M_{s}(t)$ times a
term that is homogeneous of degree zero in the distribution of the measure of
varieties:

$Y_{s}(t)=\frac{w_{s}(t)L_{s}}{P_{s}(t)}+\frac{\Pi_{s}(t)}{P_{s}(t)}=M_{s}(t)\times\left[\mathcal{R}^{w}_{s}(t)+\mathcal{R}^{\pi}_{s}(t)\right]\equiv
M_{s}(t)\times\mathcal{R}_{s}(t)$ (18)

This property is important because it is the mechanism that induces increasing
returns to scale in this model. It will also be important to characterize the
existence of the Balanced Growth Path (BGP).

Intuitively, even though $\mathcal{R}_{s}(t)$ is a complicated function, the
fact that real income is a function of $M_{s}(t)$ times a term that is
homogeneous of degree zero in $\\{M_{k}(t)\\}_{k\in\textbf{K}}$ means that
this model falls within the broader class of AK-models in macroeconomics, a
property that is inherited from the Romer side of production. Many of the
well-behaved properties of AK-models about long-run growth will carry through
to this model.

Furthermore, as it will become clear below, trade will influence both the
returns to idea creation ($\mathcal{R}_{s}$) as well as the stock of ideas
($M_{s}$). But it is important to bear in mind that some of the mechanisms
behind growth in this economy are increasing returns to scale exemplified by
the linearity of the production function.

###  3.5 Balanced Growth Path

#### Autarky

Under autarky, which is a special case in which trade costs are prohibitively
high such that countries are isolated as single-country economies, the BGP
exists and is unique for each individual economy.

######  Proposition 1 (Growth rates under autarky).

If $\tau_{sd}\to\infty$ for all $s\neq d$, then growth rates in real
consumption $g_{s}^{\text{autarky}}$ in every country $s\in\boldsymbol{K}$ are
proportional to domestic market size:

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

###### Proof.

Appendix B.8. ∎

Intuitively, Proposition (1) characterizes the BGP in a collection of closed
AK economies with expanding varieties each of them as in the original [44]
model. Growth happens endogenously in each of the countries as households
invest in the equity market to fund new intermediate varieties. However, the
mass of non-rival goods available for production will be completely different
across different countries, since final good producers only have access to
domestic intermediate inputs and are therefore less productive than they would
be if they were trading internationally. Similarly, in general, BGPs will be
characterized by different growth rates. Note that growth rates
$g_{s}^{\text{autarky}}$ are indeed constant because
$\frac{Y_{s}(t^{*})}{M_{s}(t^{*})}=\frac{M_{s}(t^{*})\times\mathcal{R}_{s}(t^{*})}{M_{s}(t^{*})}$
is homogeneous of degree zero in $M_{s}(t^{*})$ for each $s\in\boldsymbol{K}$.

#### Zero gravity

Now move on to characterize the equilibrium growth rates under the polar
opposite case: zero gravity. This is one in which trade is costless and even
geographical barriers are nonexistent. The term comes from [23].

######  Proposition 2 (BGP under zero gravity).

If $\tau_{sd}=1$ for all $(s,d)$, then there is a unique world equilibrium
growth rate $g^{\text{zero gravity}}$ that satisfies:

$g^{\text{zero
gravity}}=\frac{\alpha\cdot\psi}{\eta}\cdot\frac{\sum_{d\in\boldsymbol{K}}Y_{d}(t^{*})}{\sum_{d\in\boldsymbol{K}}M_{d}(t^{*})}-\rho$
(19)

###### Proof.

Appendix B.8 ∎

By comparing $g^{\text{zero gravity}}$ and $g_{s}^{autarky}$, it is
immediately clear that while the latter is proportional to domestic value
added per variety $Y_{s}(t^{*})/M_{s}(t^{*})$, the former is proportional to
global value added per variety
$\sum_{d\in\boldsymbol{K}}Y_{d}(t^{*})/\sum_{d\in\boldsymbol{K}}M_{d}(t^{*})$.
Intuitively, under zero gravity, growth happens as if the world were a single
integrated Romer economy.

It is clear that the growth rate must be common under zero gravity because the
expression on the right-hand side of (19) is the same for each country. Since
$g^{\text{zero gravity}}$ must be a constant, each element in the sum
$\frac{Y_{d}(t^{*})}{\sum_{d\in\boldsymbol{K}}M_{d}(t^{*})}=\frac{M_{d}(t^{*})\times\mathcal{R}_{d}(t^{*})}{\sum_{d\in\boldsymbol{K}}M_{d}(t^{*})}$
must be homogeneous of degree zero in $[M_{k}(t^{*})]_{k\in\boldsymbol{K}}$.
This, in turn, implies that returns equalize and
$\mathcal{R}_{d}(t^{*})=\mathcal{R}(t^{*})$.

In the absence of trade costs, the world economy is fully integrated in terms
of final goods varieties suppliers and the law of one price holds in the final
good. As the final good serves as an input for intermediate varieties, the
price of intermediate varieties equalizes globally. A corollary is that the
effective measure of input varieties $\tilde{M}_{s}(t^{*})$ also equalizes
globally, indicating that non-rival inputs fully diffuse across the world.

Note, however, that income levels need not be the same in this world economy.
In fact, those countries that have a higher relative wage at the start of the
BGP will have a higher wage relative forever. Therefore, under zero gravity,
this model features a stable global distribution of income as in the model of
Armington trade and capital accumulation-driven growth of [2].

Using the linearity of income in equilibrium, the growth rate over the BGP be
decomposed into the following expression, which relates how growth affects
labor and capital income, respectively:

$\small g^{\text{zero
gravity}}=\psi\rho\left[T_{s}^{\frac{1}{\theta(1-\alpha)}}\cdot\left(\frac{w_{s}(t^{*})^{\theta(1-\alpha)}}{\sum_{k\in\boldsymbol{K}}w_{k}(t^{*})^{\theta(1-\alpha)}}\right)^{-\frac{1}{\theta(1-\alpha)}}\cdot
L_{s}\cdot\left(\frac{M_{s}(t^{*})}{\sum_{k\in\boldsymbol{K}}M_{k}(t^{*})}\right)^{-1}+\frac{\alpha}{\eta}\frac{\sum_{d\in\boldsymbol{K}}Y_{d}(t^{*})}{\sum_{d\in\boldsymbol{K}}M_{d}(t^{*})}\right]$

The second term within the square brackets, which is related to capital
income, shows that profits per variety equalize under zero gravity, with every
country having the same level of market access due to the absence of trade
frictions and equalization of global prices. The first term within the
brackets relates to labor income and shows how wages $w_{s}(t^{*})$ and the
measure of varieties $M_{s}(t^{*})$, which are endogenous objects, interact
with parameters such as the labor force size $L_{s}$ and technology $T_{s}$.

Since growth rates must be equal for all countries, those countries with
better technology and higher labor forces will have a proportionately higher
real wage and a higher share in the global measure of varieties. The
relationship between the size of a country’s labor force and the measure of
varieties can also be observed in the data, as seen in Figure 7.

Figure 7: Measures of Variety and Labor Force. Long Run Averages between
2000-2020 for the cross-country correlation between the Size of the Labor
Force ($ln$ of Employment) and the Measure of Produced Varieties. Data come
from the Penn World Tables 10.10 and Prodcom, respectively.

#### Costly but finite trade

I now arrive at the more realistic case of a BGP of positive but finite trade
costs.

######  Proposition 3 (Balanced growth with costly trade).

Given a vector of fundamentals
$(\theta,\sigma,\boldsymbol{T},\boldsymbol{\tau})$, if
$\tau_{sd}\in(1,\infty)$ for all $s\neq d$, there exists a balanced growth
path world equilibrium growth rate satisfying:

$g_{s}=\psi\rho\left[\underbrace{\left(\frac{T_{s}}{\lambda^{F}_{ss}(t^{*})}\right)^{\frac{1}{\theta(1-\alpha)}}}_{\text{Eaton-
Kortum}}\times\underbrace{\alpha^{1-\eta}\cdot\frac{L_{s}}{\lambda^{M}_{ss}(t^{*})}}_{\begin{subarray}{c}\text{Romer}\\\
\text{Domestic}\end{subarray}}+\underbrace{\frac{\alpha}{\eta}\cdot\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t^{*})\cdot\frac{P_{d}(t^{*})Y_{d}(t^{*})}{P_{s}(t^{*})M_{s}(t^{*})}}_{\begin{subarray}{c}\text{Romer}\\\
\text{Global}\end{subarray}}\right]$ (20)

and where $g_{s}=g_{s^{\prime}}(\forall s,s^{\prime}\in\boldsymbol{K})$.
Furthermore, the growth rate can be decomposed into “Eaton-Kortum” and “Romer”
components.

###### Proof.

Appendix B.8. ∎

Like in the previous subsection, equation (20) shows how growth affects both
components of GDP. One can further interpret these components and relate them
to the two canonical models that are the building blocks of this framework.

I termed the capital income part, which is the second term within the square
brackets, the Romer Global component. The reason is that this component is
increasing in each country’s market access —i.e., proportional to each
country’s share of the global intermediate goods market $(\lambda_{sd}^{M})$.
Intuitively, profits will be related to a firm’s sales and to markets all over
the world and, therefore, to its market share in each of those destination
markets. This component is also decreasing in the price of the final good
$(P_{d})$ in the source country since due to the input-output structure
embedded in the lab-equipment version of the Romer model the price of the
final good is the marginal cost of R&D investment.

Real labor income can be partitioned into two components: an Eaton-Kortum
component and a Romer Domestic component. The Eaton-Kortum component is very
much Ricardian: real labor income improves with technological improvements
($T_{s}$) and as the domestic trade share in final ($\lambda^{F}_{ss}$)
decreases, consistent with the Ricardian intuition that specialization leads
to gains from trade.

The Romer Domestic component incorporates domestic market size effects, by
integrating the size of the domestic labor force $L_{s}$. But it also adjusts
for the diffusion of differentiated intermediate goods, which is embedded in
the summary statistic of domestic intermediate trade share
($\lambda^{M}_{ss}$)161616To see that, note that:
$\lambda^{M}_{ss}(t^{*})=M_{s}(t^{*})\left(\frac{p_{ss}^{M}(t^{*})}{P_{s}^{M}(t^{*})}\right)^{1-\eta}=\frac{M_{s}(t^{*})}{\tilde{M}_{s}(t^{*})}p_{ss}^{M}(t^{*})^{1-\eta}$..

The different components of (20) make it clear why a BGP requires common
growth rates. Both $\lambda_{ss}^{M}(t^{*})$ and $\lambda_{ss}^{F}(t^{*})$ are
homogeneous of degree zero in $[M_{k}(t^{*})]_{k\in\boldsymbol{K}}$ and prices
$P_{s}(t^{*})$, $P_{d}(t^{*})$ must be constant along a BGP. For a BGP,
$g_{s}$ must be a constant, and therefore
$\frac{Y_{d}(t^{*})}{M_{s}(t^{*})}=\frac{M_{d}(t^{*})\times\mathcal{R}_{d}(t)}{M_{s}(t^{*})}$
must be homogeneous of degree zero in $[M_{k}(t^{*})]_{k\in\boldsymbol{K}}$.
This can only happen if $\mathcal{R}_{d}(t^{*})$ is homogeneous of degree zero
in each $d$, which implies that $\mathcal{R}_{d}(t^{*})=\mathcal{R}(t^{*})$.

Intuitively, along the BGP, the real interest rate equalizes globally171717One
way to see that is through the Euler equation. Since the Euler Equation
governs the growth rate in consumption and over the BGP $g_{c}=g_{M}$ for all
countries, a corollary is that the real interest rate must equalize globally..
Even though there are no international equity markets, the fact that
households can invest in new varieties through equity markets and earn
expected profits that are linked to exports means that trade acts as a vehicle
to integrate international R&D and equity markets. In a balanced growth
equilibrium, then, prices and the endogenous distribution of the measure of
varieties $[M_{s}(t^{*})]_{s\in\boldsymbol{K}}$ will adjust to make sure that
returns and, therefore, growth rates equalize.

#### How are growth rates related to market access?

After characterizing the existence of the BGP, one can turn to the discussion
of what happens to the equilibrium growth rates after there is a change in
trade costs. Here, these trade costs are directly related to market access,
since the mechanism that propels growth is the incentive to have equity claims
in the profits of variety exporters. The growth rate is a general equilibrium
object that depends on the whole distribution of prices across countries and
periods. Therefore, characterizing changes to it is not a trivial task.

Nonetheless, in order to connect the theory to the empirical and quantitative
analysis, to gain some intuition, first consider what happens to the long-run
equilibrium growth rate after a permanent change in trade costs in a world of
symmetric countries. While that is an important restriction, it allows for a
closed-formed intuitive solution: in that case, $g^{*}$ can be shown to
unambiguously increase in the long run after an episode of trade
liberalization.

######  Proposition 4 (Effects of changes in trade costs over the long run in
symmetric economies).

Suppose there exist a collection of symmetric economies that grow over the BGP
with costly trade with trade costs $\tau>1$. Then $\frac{\partial
g^{*}}{\partial\tau}<0$.

###### Proof.

Appendix B.8 ∎

In this model, the long-run growth rate will change after a permanent change
in trade costs if there is a change in the effective market size, represented
by how much of the global market exporters can tap into —that is why foreign
aggregate demand $\sum_{k}P_{k}(t^{*})Y_{k}(t^{*})$ is modulated by
intermediate trade share $\sum_{k}\lambda^{M}_{sd}(t^{*})$ in the profit
formula.

In a symmetric world, it can be shown in closed form that real profits
increase when trade costs go down. The intuition translates to numerical
exercises with asymmetric countries. As an example plot in Figure 8 shows how
these results look like in a numerical exercise with two asymmetric countries.

Figure 8: Long-run growth in two asymmetric economies as a function of
changing trade costs. Results from a numerical simulation of the equilibrium
growth rate $g^{*}$ of two asymmetric economies that differ in their
populations but are otherwise equal. Parameters are the following:
$L=[1,1.03]$,
$\sigma=0.77,\theta=2.12,\alpha=1/3,\rho=0.03,T=[1,1],\psi=2.46$.

###  3.6 Welfare

With log preferences, at any moment, consumption over the BGP is a fraction of
assets plus real labor income. Since such consumption flow grows at a constant
rate $g^{*}$ and the measure of products is simply a linear transformation of
assets, as shown in Appendix B.9, welfare along the BGP can be decomposed
between a product measure component, a real income component, and a growth
component.

${\footnotesize\int_{t^{*}}^{\infty}\exp\\{-\rho(t-t^{*})\\}\log\left(\exp\\{g^{*}t\\}C_{s}(t^{*})\right)dt=\underbrace{\underbrace{\log\left(\frac{1}{\psi}M_{s}(t^{*})\right)}_{\text{product
measure}}}_{\text{transitional}}+\underbrace{\underbrace{\frac{1}{\rho}\log\left(\frac{w_{s}(t^{*})L_{s}}{P_{s}(t^{*})}\right)}_{\text{real
income}}}_{\text{static}}+\underbrace{\underbrace{\frac{g^{*}}{\rho^{2}}}_{\text{growth}}}_{\text{dynamic}}}$
(21)

Consider what happens to welfare after a change in trade costs from
$\boldsymbol{\tau}$ to $\boldsymbol{\tau}+d\boldsymbol{\tau}$, as in [5]
—hereinafter ACR. In this dynamic setting, to make a comparison to the static
framework, I need to compare what happens across the two BGPs, comparing the
preserved value of discounted lifetime utility across the beginning of the two
initial equilibria. For that, let me introduce some notation: suppose $t^{*}$
is the initial period of the original BGP; $t^{**}$ is the first period of the
final BGP and let $\hat{x}\equiv x(t^{**})/x(t^{*})$.

Then, relative level changes in the first component of welfare across two BGPs
can be expressed as $\log\left(\widehat{M}_{s}\right)$. Changes in the
equilibrium product measure will depend on whether the measure of varieties in
country $s$ expands or contracts, relative to the distribution of varieties
across countries, across BGPs. There is no general prediction in the model
regarding the direction of this effect.

Countries that have started with a measure of varieties above optimal
(relative to other countries) will see a shift in exports (and therefore R&D
expenditures) towards other countries and will see their measures of varieties
shrink. The opposite is true for countries that started with a measure of
varieties below optimal.

Importantly, however, this first component will not compound over time, as
highlighted by the fact is not multiplied by the factor $\rho^{-1}$ or
$\rho^{-2}$. This means that it will only change the (relative) income level
that a given country arrives with at the BGP and it will have no impact going
forward. Therefore, this is a transitional effect of welfare. For most
reasonable calibrations of $\rho$, the transitional effect will have a very
small weight on total welfare changes.

The second component will be familiar to most trade economists. It looks like
the traditional static welfare formula in ACR. In the same spirit as ACR, one
can also write the static welfare component in changes:

$\frac{1}{\rho}\log\left(\widehat{\frac{w_{s}}{P_{s}}}\right)=\frac{1}{\rho}\log\underbrace{\left(\widehat{\lambda_{dd}^{F}}^{-\frac{1}{(1-\alpha)\theta}}\right)}_{\text{Eaton-
Kortum}}+\frac{1}{\rho\eta}\log\underbrace{\left(\sum_{k\in\boldsymbol{K}}\mu_{k}\cdot\widehat{M}_{k}\cdot\left(\frac{\widehat{p}^{M}_{kd}}{\widehat{P}_{s}}\right)^{1-\eta}\right)}_{\text{Romer}}$
(22)

where
$\mu_{k}\equiv\frac{M_{k}(t^{*})\cdot\left(\frac{p^{M}_{ks}(t^{*})}{P_{s}(t^{*})}\right)^{1-\eta}}{\sum_{k\in\boldsymbol{K}}M_{k}(t^{*})\cdot\left(\frac{p^{M}_{ks}(t^{*})}{P_{s}(t^{*})}\right)^{1-\eta}}$.

This component preserves the standard feature that changes in consumer welfare
are decreasing in changes in domestic trade share
$\widehat{\lambda_{ss}^{F}}$181818The elasticity of this effect is
$-\frac{1}{(1-\alpha)\theta}$ rather than the standard $-\frac{1}{\theta}$ due
to the input-output structure of the model.. This captures the Ricardian
intuition of the model: at the margin, there are static gains from
specialization in this model.

Like the growth formula, the static component of welfare also has Eaton-Kortum
and Romer components. Here, the Romer component impacts welfare by augmenting
Ricardian gains through an extensive margin. It is represented by the weighted
change in the measure of varieties, accounting for previous weights $\mu_{k}$,
changes in the measures of varieties in each country $k\in\boldsymbol{K}$
across equilibria $\widehat{M}_{k}$, and changes in the prices of foreign
intermediate goods relative to the domestic consumer price index
$\left(\frac{p^{M}_{ks}(t^{*})}{P_{s}(t^{*})}\right)$ at the domestic market.

Note that this welfare impact from product innovation resembles how the change
in the measure of varieties shows up in the ACR formula in Melitz-type models.
This highlights that the nested structure of production featured in this model
effectively adds an extensive margin to the Eaton-Kortum framework.

While they do not compound over time, both of these effects have an impact in
every period over the BGP as it is made clear by it being multiplied by the
factor $\rho^{-1}$. For that reason, these can be understood as a level effect
or static effect of welfare.

The third and last component is the common growth rate $g^{*}$. Importantly,
since it compounds the BGP level of consumption, it is multiplied by a factor
$\rho^{-2}$ rather than $\rho^{-1}$ and it will in general have a larger
weight on welfare. This is a metric of dynamic gains from trade, which is a
growth effect of welfare.

Changes in the growth component of welfare will be defined as the change in
the growth rage: $\frac{g^{**}-g^{*}}{\rho^{2}}.$ Since growth rates equalize
along the BGP, changes in the growth component of welfare will also be shared
across all countries. However, since the other components will differ, the
share of the dynamic component of welfare in total gains from trade will
therefore be different across countries.

The discount rate $\rho$ will have an important role in attributing weights
across the dynamic, static components, and transitional components of welfare.
Intuitively, the lower the $\rho$, the more patient the agent is, and the more
relevant the dynamic component of welfare will become.

#### Welfare as buyers and as sellers

Comparing the dynamic and static components of welfare yields important
insights regarding the economic mechanisms behind this model. In fact, the
forces of specialization and innovation are reflected in these two components.

To see that, note that, as made clear by (22), since $1-\eta<1$ country $s$’s
static welfare is decreasing in the price of foreign intermediate goods. The
static welfare formula captures the effect of $s$ as final producers and
consumers. As $s$ purchases more foreign intermediate varieties for a cheaper
price, it becomes more productive by increasing its effective measure of
varieties $\tilde{M}_{s}(t)$. The other side of the coin is that it decreases
the local price index $P_{s}(t)$, which directly benefits consumers and
increases welfare.

By contrast, the growth component of welfare is increasing in the price of
foreign intermediate goods coming from $k$, in each destination markets $d$,
relative to the price of intermediate goods from the source country $s$ at
those same destination market:

$\displaystyle
g_{s}\propto\frac{\alpha}{\eta}\cdot\sum_{d\in\boldsymbol{K}}P_{d}(t)Y_{d}(t)\left[\sum_{k\in\boldsymbol{K}}M_{k}(t)\left(\frac{p_{kd}^{M}(t)}{p_{sd}^{M}(t)}\right)^{1-\eta}\right]^{-1}$

The intuition for this contrast is quite straightforward and underscores the
different underlying economic mechanisms of the model. The Romerian part of
growth captures the effect of $d$ as forward-looking investors in the R&D
market and intermediate good producers. Since the intermediate goods are
substitutes, all else equal, demand for intermediate goods from $s$ and
maximized profits are higher when the price of intermediate goods of foreign
competitors from third-party countries $k$ relative to domestic producers from
$s$ at each destination market $d$ is higher.

Intuitively, the growth effect captures that, from a seller’s perspective, the
domestically produced and exported intermediate variety $s$ is more attractive
and competitive when foreign varieties $k$ are more expensive. Conversely, the
static effect captures that, from a buyer’s perspective, when foreign
varieties $k$ relative to one’s domestic purchasing power at $d$ are more
expensive, the domestic consumer is worse off. Both channels are economically
sensible and the model captures both mechanisms.

Along the BGP, prices and measures of varieties will adjust to make sure that
growth rates equalize such that $g_{s}=g_{s^{\prime}}$ for every
$s,s^{\prime}\in\boldsymbol{K}$. While the economic mechanisms are still
operating under the hood and take over if there is any shock that drives the
system off the BGP, these different effects will wash out once differences in
prices, measures of varieties and wages endogenously adjust towards a BGP.

##  4 Quantification and Policy Exercise

This section describes a numerical quantification of the model, which solves
for three endogenous objects along the Balanced Growth Path: (a) the
distribution of wages; (b) the distribution of Measures of Varieties; and (c)
the common equilibrium growth rate. I calibrate the model to EU-15 countries
and the New Member States (NMS) that joined in the 2004 expansion.

To simplify the exercise, I group these countries into six sets: corresponding
to the six waves of the expansion of the European Union of to 2004191919This
simplification is just a matter of computational tractability. As described in
Appendix E, each guess of my solution algorithm solves for a static version of
an Eaton-Kortum model with input-output linkages, which itself has multiple
steps for solutions. So the problem grows quite fast in complexity in the
number of countries. Improving the solution algorithms for this new class of
dynamic models is a fruitful avenue of future research.. The country groups
are asymmetric both in terms of labor force and productivity. The groups are:

  1. 1.

1957: Belgium, France, Germany, Italy, Luxembourg, and the Netherlands —the
original members;

  2. 2.

1973: Denmark, Ireland, and the United Kingdom;

  3. 3.

1981: Greece;

  4. 4.

1986: Portugal and Spain;

  5. 5.

1995: Austria, Finland, and Sweden;

  6. 6.

2004: Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Slovakia,
and Slovia —the New Member States (NMS).

The solution method is straightforward. I calibrate the model to a baseline
scenario and then change iceberg trade costs to induce a trade liberalization
shock. By comparing the endogenous equilibria along the Balanced Growth Path
of these two scenarios, which include distributions of the measures of
varieties $[M_{s}(t^{*})]_{s\in\boldsymbol{K}}$ and wages
$[w_{s}(t^{*})]_{s\in\boldsymbol{K}}$ as well as a common equilibrium growth
rate $g^{*}$, I can infer the welfare consequences of a change in this
parameter along the BGP.

#### Model Calibration

My estimates of the short-term ($\sigma$) and long-term ($\theta$)
elasticities of trade come from [11], which are $\sigma=0.76$ and
$\theta=2.12$, respectively. The results are not very sensitive to using a
$\theta=4.0$. The vector of labor force $\\{L_{s}\\}$ comes from Penn World
Tables. The share of intermediate goods $\alpha=0.36$ is set to equal the
average share of intermediate goods in the sample of countries between
2000-2003 from the World Input-Output Database.

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). As shown in Appendix E, I can write trade costs as:

$\tau_{sd}=\left(\frac{E^{F}_{sd}(t^{*})}{E^{F}_{dd}(t^{*})}\cdot\frac{E^{F}_{ds}(t^{*})}{E^{F}_{ss}(t^{*})}\right)^{-\frac{1}{2\theta(1-\alpha)}}$
(23)

where each flow $E^{F}_{sd}(t^{*})$ defined to be an average between
$2000-2003$. The data on bilateral expenditure values $E^{F}_{sd}(t^{*})$
comes from the World Input-Output Database.

Figure 9 plots the change in trade costs before and after the 2004 enlargement
of the European Union, calculated from an average of for the years 2000-2003
for the immediate ”before” period and an average for the years 2004-2007 for
the immediate ”after” period. Calculations confirm that there were large
reductions ($-15-20\%$) in trade costs between NMS and the Western European
countries during this period, which is consistent with bilateral tariff data
between the NMS and Western Europe from TRAINS and the WTO202020See the
Appendices from [15] for a detailed description of the data.. Changes in trade
costs across the other groups have been comparatively small except for one
calculated increase in trade costs between Greece (g1981) and Austria,
Finland, and Sweden (g1995).

Figure 9: Changes in Trade Costs Before and After 2004 EU Enlargement (in
percentage terms). This matrix shows the bilateral changes in trade costs,
calculated using the method inferred from equation (23), before and after the
2004 EU Enlargement. The before period is an average for the years 2000-2003
and the after period is an average for the years 2004-2007. Underlying data
comes from the World Input-Output Database.

This is important because these changes in trade costs will act as the main
shock across calibrations of BGPs in my numerical exercise. It is relevant
that the key driver of changes across equilibria is the enlargement of the EU.

The location parameter of the Fréchet distribution $\\{T_{s}\\}$ and $\psi$
are free parameters that I vary to match the distribution of wages and the
average growth rate of the EU-15 countries in the $1989-2003$ period —i.e.,
fifteen years prior to the 2004 expansion of the European Union. The rationale
is that I am calibrating this model to BGP growth rate and the EU-15 countries
were very likely closer to the BGP than the transition economies of Eastern
Europe, so it is reasonable to match the model to their growth rate.

#### Model Validation

To validate the model, there are some untargeted moments one can look at.
First, compare the relative change in real wages across the two BGPs. The
predicted changes in the distribution of wages across equilibria in the model
can be compared with the relative income growth of each country group around
the EU enlargement.

Since the wages distribution is only pinned down up up to the distribution, if
real wages of a given group take a larger share of the distribution in the
later BGP, an implication is that it must have grown faster than average
between those periods. To compare the data with the model, a natural
comparison is to use GDP per capita growth rate net of the average of the EU,
which yields a income that is normalized for the periods of 1998-2003 and
2005-2010. The way to interpret the data is to see whether or not each group’s
income per capita grew faster (slower) than average across these periods.

Here, one can see that the model in fact matches the data quite correctly. It
predicts relative a catch-up of the New Member States (g2004). The model
predicts that real wages in NMS would grow about $5.1\%$ faster than the
average of the Western European countries, which is very close to observed in
the data ($5\%$). As seen in Figure 10, most of the other country groups also
fall very close to the 45-degree line, suggesting that the model’s predictions
are reasonable.

Figure 10: Model Validation: Changes in Real Wages, Relative to the Average.
In the model, the distribution of wages
$\lambda_{w}\cdot[w_{s}(t^{*})]_{s\in\boldsymbol{K}}$ is normalized with a
choice of $\lambda_{w}$ such that
$\sum_{s\in\boldsymbol{K}}w_{s}(t^{*})L_{s}=1$. What is shown in the chart is
the percentage change across equilibria
$\frac{w_{s}(t^{**})/P_{s}(t^{**})}{w_{s}(t^{*})/P_{s}(t^{*})}-1$, where
$L_{s}$ is assumed to be fixed. In the data, for consistency, I calculated
annual GDP per capita then subtracted it from the average of the group for the
periods of 1998-2003 and 2005-2010. I then calculated changes and plotted the
data. Data comes from the Penn World Tables 10.01.

One exception is the 1981 wave, for which the model substantially over-
predicts relative real income growth. The reason being that such a wave
consists of a single country: Greece. And the aftermath of the EU enlargement
$2005-2010$ includes the first years of the Greek debt crisis. Naturally, the
model cannot anticipate the negative shocks of the deep recession of the late
2010s in Greece.

Second, compare the (endogenous) distribution of the number of produced
varieties in the model to the distribution of the number of produced varieties
in the data for the 2000-2003 period. Once again, the observations fall mostly
along the 45-degree line, suggesting that the model does a good job in
replicating the empirical distribution. As one exception, now the model
underestimates the actual share of total produced varieties in the NMS
(g2004).

Figure 11: Model Validation: Distribution in the Number of Produced Varieties
Across Regions. In the model, the distribution of measures of varieties
$\lambda_{M}\cdot[M_{s}(t^{*})]_{s\in\boldsymbol{K}}$ is normalized with a
choice of $\lambda_{M}$ such that $\sum_{s\in\boldsymbol{K}}M_{s}(t^{*})=1$.
For consistency in the comparison, what I show in the data bars are the
relative shares of each country group in the total universe of the product
measure, or: $M_{s}(t)/\sum_{s^{\prime}\in\boldsymbol{K}}M_{s^{\prime}}(t)$.
This assumes, as in the model, that product varieties in the data are
differentiated across countries, so the global product space is
$\sum_{s^{\prime}\in\boldsymbol{K}}M_{s^{\prime}}(t)$. Data comes from Prodcom
(Eurostat) and are averages for the 2000-2003 period.

Finally, compare the changes in trade shares across equilibria. The model
captures changes in trade shares really well, as shown in (12). Trade expands
particularly in exports from NMS to Western European countries, which is
captured by the upper quadrant observations that lie close to the 45-degree
line. Here, the exceptions are the trade flows from Western Europe towards the
NMS. The model predicts a symmetrical response in terms of trade expansion,
while in reality, the gains were much more a relative market access from the
NMS into the EU market than the other way around.

Figure 12: Model Validation: Changes in Trade Shares. The model object is
plotted is $\hat{\lambda}^{F}_{sd}$: the change in the final sector trade
share. In the data, this is total trade shares renormalized to account for the
fact that there is no rest-of-the-world in the sample. The before and after
periods are 1998-2003 and 2005-2010, respectively.

#### Results

The main result of this exercise relates to the theoretical welfare
decomposition in equation (21). One can compare two paths of consumption along
the BGP and decompose them into:

$\displaystyle\int_{\tau}^{\infty}\exp\\{-\rho(t-\tau)\\}\left[\log\left(\exp\\{g^{**}t\\}C_{s}(t^{**},\tau)\right)-\log\left(\exp\\{g^{*}t\\}C_{s}(t^{*},\tau)\right)\right]dt=$
$\displaystyle\underbrace{\log\left(\hat{M}_{s}\right)}_{\text{transitional}}+\underbrace{\frac{1}{\rho}\log\left(\widehat{\frac{w_{s}}{P_{s}}}\right)}_{\text{static}}+\underbrace{\frac{g^{**}-g^{*}}{\rho^{2}}}_{\text{dynamic}}$

where $C_{s}(t^{*},\tau),C_{s}(t^{**},\tau)$ are the paths of consumption
along the original and new BGPs, respectively.

For all countries, the transitional component is negligible. They never
contribute with more than $0.03\%$ of total absolute value of welfare, in the
largest case.

Static gains from trade can be as large as 5-6% of domestic income in the case
of NMS (g2004) or Greece (g1981) or even negative or close to zero in the case
of the Western European countries such as Portugal and Spain (g1986). In the
case of Greece and the NMS, they account for 38% and 32% of total welfare
gains from trade, respectively.

These changes can be further decomposed into “Eaton-Kortum” and ”Romer” parts
of static welfare using equation (22). Results in Figure 13 show a wide
variation of the Ricardian component share in total changes in static welfare.
While for most country groups that share is about 10% of total changes in
static welfare, for Portugal and Spain (g1986) it accounts for nearly 25% of
static welfare changes while for Greece (g1981) the Ricardian share accounts
for more than 90% of changes in total welfare.

Figure 13: Static Welfare Decomposition. Static Welfare Decomposition Across
its Eaton-Kortum and Romer Components, according to equation (22).

Finally, the main numerical outcome of the exercise is the differences in
growth rates across BGPs $g^{**}-g^{*}$. In the current calibration, the trade
liberalization embedded in the 2004 enlargement of the European Union induced
the EU long-run yearly growth rate to increase $0.10pp$. One implication is
that the dynamic part of welfare accounts for the most of gains from trade for
all countries. Therefore, not accounting for this channel ignores the majority
of gains from trade.

However, the share of total welfare gains it accounts for varies across
country groups. According to this model, in the current parametrization, the
share of dynamic gains in total welfare gains is between $65\%$ and $90\%$.
This is in line with estimates from [32] ($78\%$) and [39] ($85\%$).

However, in this model, the change in the equilibrium measure of varieties
(and hence the real wage) between one BGP and the other can actually decrease,
which implies a negative static welfare share. Therefore, for some countries,
such as Portugal and Spain (g1986) the share of dynamic welfare in total
welfare is larger than 100%. These decompositions are in Figure 14.

Figure 14: Total Welfare Decomposition. Welfare Decomposition Across its
Transitional Static and Dynamic Components, according to equation (21).

In monetary terms, a back-of-the-envelope calculation suggests an additional
$0.10\%$ yearly growth rate to the aggregate GDP of the Western European plus
the New Member States since the year of accession —that is, between 2004 and
2023 —would have induced an additional current production level of
approximately $\$332$ billion in the continent, which accounts for $2.0\%$ of
the total level of production of the European Union.

##  5 Conclusion

I focus on the long-lasting question of the relationship between trade and
growth and, in particular, trade and product innovation. I make several
contributions: theoretical, empirical, and quantitative.

On the theoretical front, my main contribution is a new framework that
reconciles the forces of specialization and market size, rationalizes foreign
market access as a rationale for growth in a dynamic framework, and provides
an analytical formula for dynamic gains from trade. In all of those points, I
maintain active dialogues with the literature, such as nesting the Eaton-
Kortum model of trade and Romer growth model as special cases of my model and
subsuming the ACR static welfare formula in my dynamic welfare formula.

In my empirical work, I rely on the eastward expansion of the European Union
and document several new facts that are consistent with the mechanisms of my
model. Compared to countries that selected into becoming candidates but had
not joined at given horizon, countries started producing more product
varieties, investing more in R&D, and trading more.

I go beyond these facts and exploit plausibly exogenous variation to show that
a plausibly exogenous increase in market access leads to a higher probability
of initiating production and exporting a given product, which is consistent
with the main mechanism of the theoretical model.

Finally, I solve for a quantitative model and replicate the 2004 expansion of
the European Union in the computer. The results of the simulation imply that:
(a) the EU expansion increased its long-run yearly growth rate by about
0.10pp; and (b) dynamic gains from trade account for somewhere between 65-90%
of total welfare gains from trade.

This paper points to the fact that dynamic gains from trade are likely too
large to be ignored. The big generalizable takeaway is that the previous
literature has largely underestimated gains from trade, perhaps by as much as
one order of magnitude. Advancing on this agenda, perhaps by understanding the
transition dynamics, is a fruitful avenue of future research.

## References

  * [1] Daron Acemoglu  “Introduction to Modern Economic Growth” Google-Books-ID: DsPH5fWNdrsC  Princeton University Press, 2008
  * [2] Daron Acemoglu and Jaume Ventura  “The World Income Distribution*”  In _The Quarterly Journal of Economics_ 117.2, 2002, pp. 659–694  DOI: 10.1162/003355302753650355
  * [3] Fernando Alvarez and Robert E. Lucas  “General equilibrium analysis of the Eaton–Kortum model of international trade”  In _Journal of Monetary Economics_ 54.6, 2007, pp. 1726–1768  DOI: 10.1016/j.jmoneco.2006.07.006
  * [4] David Argente, Salome Baslandze, Douglas Hanley and Sara Moreira  “Patents to Products: Product Innovation and Firm Dynamics”  In _SSRN Electronic Journal_ , 2020  DOI: 10.2139/ssrn.3577811
  * [5] Costas Arkolakis, Arnaud Costinot and Andrés Rodríguez-Clare  “New Trade Models, Same Old Gains?”  In _American Economic Review_ 102.1, 2012, pp. 94–130  DOI: 10.1257/aer.102.1.94
  * [6] Costas Arkolakis, Sharat Ganapati and Marc-Andreas Muendler  “The Extensive Margin of Exporting Products: A Firm-level Analysis”, 2020  DOI: 10.3386/w16641
  * [7] Scott L. Baier and Jeffrey H. Bergstrand  “Do free trade agreements actually increase members’ international trade?”  In _Journal of International Economics_ 71.1, 2007, pp. 72–95  DOI: 10.1016/j.jinteco.2006.02.005
  * [8] Maria Bas  “Input-trade liberalization and firm export decisions: Evidence from Argentina”  In _Journal of Development Economics_ 97.2, 2012, pp. 481–493  DOI: 10.1016/j.jdeveco.2011.05.010
  * [9] A.. Bernard, S.. Redding and P.. Schott  “Multiproduct Firms and Trade Liberalization”  In _The Quarterly Journal of Economics_ 126.3, 2011, pp. 1271–1318  DOI: 10.1093/qje/qjr021
  * [10] Andrew B Bernard, J. Jensen, Stephen J Redding and Peter K Schott  “The Margins of US Trade”  In _American Economic Review_ 99.2, 2009, pp. 487–493  DOI: 10.1257/aer.99.2.487
  * [11] Christoph E. Boehm, Andrei A. Levchenko and Nitya Pandalai-Nayar  “The Long and Short (Run) of Trade Elasticities”, Working Paper Series  National Bureau of Economic Research, 2020  DOI: 10.3386/w27064
  * [12] Kirill Borusyak, Xavier Jaravel and Jann Spiess  “Revisiting Event Study Designs: Robust and Efficient Estimation” arXiv:2108.12419 [econ]  arXiv, 2022  URL: http://arxiv.org/abs/2108.12419
  * [13] Francisco J. Buera and Ezra Oberfield  “The Global Diffusion of Ideas”  In _Econometrica_ 88.1, 2020, pp. 83–114  DOI: https://doi.org/10.3982/ECTA14044
  * [14] Jie Cai, Nan Li and Ana Maria Santacreu  “Knowledge Diffusion, Trade, and Innovation across Countries and Sectors”  In _American Economic Journal: Macroeconomics_ 14.1, 2022, pp. 104–145  DOI: 10.1257/mac.20200084
  * [15] Lorenzo Caliendo, Luca David Opromolla, Fernando Parro and Alessandro Sforza  “Goods and Factor Market Integration: A Quantitative Assessment of the EU Enlargement” Publisher: The University of Chicago Press  In _Journal of Political Economy_ 129.12, 2021, pp. 3491–3545  DOI: 10.1086/716560
  * [16] Brantly Callaway and Pedro H.. Sant’Anna  “Difference-in-Differences with multiple time periods”  In _Journal of Econometrics_ 225.2, Themed Issue: Treatment Effect 1, 2021, pp. 200–230  DOI: 10.1016/j.jeconom.2020.12.001
  * [17] Gabriel Chodorow-Reich  “Geographic Cross-Sectional Fiscal Spending Multipliers: What Have We Learned?”  In _American Economic Journal: Economic Policy_ 11.2, 2019, pp. 1–34  DOI: 10.1257/pol.20160465
  * [18] Gabriel Chodorow-Reich  “Regional data in macroeconomics: Some advice for practitioners”  In _Journal of Economic Dynamics and Control_ 115, St. Louis Fed -JEDC-SCG-SNB-UniBern Conference, titled ”Disaggregate Data and Macroeconomic Models”, 2020, pp. 103875  DOI: 10.1016/j.jedc.2020.103875
  * [19] Lawrence J. Christiano, Martin S. Eichenbaum and Mathias Trabandt  “On DSGE Models”  In _Journal of Economic Perspectives_ 32.3, 2018, pp. 113–140  DOI: 10.1257/jep.32.3.113
  * [20] Arnaud Costinot and Andrés Rodríguez-Clare  “Trade Theory with Numbers: Quantifying the Consequences of Globalization”  In _Handbook of International Economics_ 4 Elsevier, 2014, pp. 197–261  DOI: 10.1016/B978-0-444-54314-1.00004-5
  * [21] Svetlana Demidova  “Productivity Improvements and Falling Trade Costs: Boon or Bane?” Publisher: [Economics Department of the University of Pennsylvania, Wiley, Institute of Social and Economic Research, Osaka University]  In _International Economic Review_ 49.4, 2008, pp. 1437–1462  URL: https://www.jstor.org/stable/20486843
  * [22] Arindrajit Dube, Daniele Girardi, Òscar Jordà and Alan M. Taylor  “A Local Projections Approach to Difference-in-Differences Event Studies”, Working Paper Series  National Bureau of Economic Research, 2023  DOI: 10.3386/w31184
  * [23] Jonathan Eaton and Samuel Kortum  “Technology, Geography, and Trade”  In _Econometrica_ 70.5, 2002, pp. 1741–1779  DOI: https://doi.org/10.1111/1468-0262.00352
  * [24] Jonathan Eaton and Samuel Kortum  “Innovation, Diffusion, and Trade”, 2006, pp. w12385  DOI: 10.3386/w12385
  * [25] Jonathan Eaton, Samuel Kortum, Brent Neiman and John Romalis  “Trade and the Global Recession”  In _American Economic Review_ 106.11, 2016, pp. 3401–3438  DOI: 10.1257/aer.20101557
  * [26] P.. Goldberg, A.. Khandelwal, N. Pavcnik and P. Topalova  “Imported Intermediate Inputs and Domestic Product Growth: Evidence from India”  In _The Quarterly Journal of Economics_ 125.4, 2010, pp. 1727–1767  DOI: 10.1162/qjec.2010.125.4.1727
  * [27] Andrew Goodman-Bacon  “Difference-in-differences with variation in treatment timing”  In _Journal of Econometrics_ 225.2, Themed Issue: Treatment Effect 1, 2021, pp. 254–277  DOI: 10.1016/j.jeconom.2021.03.014
  * [28] Gene M. Grossman and Elhanan Helpman  “Comparative Advantage and Long-Run Growth” Publisher: American Economic Association  In _The American Economic Review_ 80.4, 1990, pp. 796–815  URL: https://www.jstor.org/stable/2006708
  * [29] Keith Head and Thierry Mayer  “Gravity Equations: Workhorse,Toolkit, and Cookbook”  In _Handbook of International Economics_ 4 Elsevier, 2014, pp. 131–195  DOI: 10.1016/B978-0-444-54314-1.00003-3
  * [30] Keith Head and John Ries  “Increasing Returns versus National Product Differentiation as an Explanation for the Pattern of U.S.-Canada Trade” Publisher: American Economic Association  In _The American Economic Review_ 91.4, 2001, pp. 858–876  URL: https://www.jstor.org/stable/2677816
  * [31] Elhanan Helpman  “Foreign Competition and Innovation”, Working Paper Series  National Bureau of Economic Research, 2023  DOI: 10.3386/w31840
  * [32] Wen-Tai Hsu, Raymond Riezman and Ping Wang  “Innovation, Growth, and Dynamic Gains from Trade”, 2019, pp. w26470  DOI: 10.3386/w26470
  * [33] David Hummels and Peter J Klenow  “The Variety and Quality of a Nation’s Exports”  In _American Economic Review_ 95.3, 2005, pp. 704–723  DOI: 10.1257/0002828054201396
  * [34] Charles I. Jones  “Growth and Ideas”  In _Handbook of Economic Growth_ 1 Elsevier, 2005, pp. 1063–1111  DOI: 10.1016/S1574-0684(05)01016-6
  * [35] Charles I. Jones  “Paul Romer: Ideas, Nonrivalry, and Endogenous Growth”  In _The Scandinavian Journal of Economics_ 121.3, 2019, pp. 859–883  DOI: 10.1111/sjoe.12370
  * [36] Timothy Kehoe and Kim J. Ruhl  “How Important Is the New Goods Margin in International Trade?”, 2013  DOI: 10.1086/670272
  * [37] Benny Kleinman, Ernest Liu, Stephen J Redding and Motohiro Yogo  “Neoclassical Growth in an Interdependent World”, 2023
  * [38] Marc Melitz and Stephen Redding  “Trade and Innovation”, 2021, pp. w28945  DOI: 10.3386/w28945
  * [39] Jesse Perla, Christopher Tonetti and Michael E. Waugh  “Equilibrium Technology Diffusion, Trade, and Growth”, 2015  DOI: 10.3386/w20881
  * [40] Swapnika Rachapalli  “Learning between Buyers and Sellers Along the Global Value Chain”, 2021, pp. 81  URL: https://cowles.yale.edu/sites/default/files/2022-11/rachapalli-learningbetweenbuyersandsellersalongtheglobalvaluechain-learning-between-buyers-and.pdf
  * [41] Luis A. Rivera-Batiz and Paul M. Romer  “Economic Integration and Endogenous Growth” Publisher: Oxford University Press  In _The Quarterly Journal of Economics_ 106.2, 1991, pp. 531–555  DOI: 10.2307/2937946
  * [42] Luis A. Rivera-Batiz and Paul M. Romer  “International trade with endogenous technological change”  In _European Economic Review_ 35.4, 1991, pp. 971–1001  DOI: 10.1016/0014-2921(91)90048-N
  * [43] Paul Romer  “New goods, old theory, and the welfare costs of trade restrictions”  In _Journal of Development Economics_ 43.1, 1994, pp. 5–38  DOI: 10.1016/0304-3878(94)90021-3
  * [44] Paul M. Romer  “Endogenous Technological Change”  In _The Journal of Political Economy_ 98.5,, 1990, pp. S71–S102  URL: http://www.jstor.org/stable/2937632
  * [45] Thomas Sampson  “Dynamic Selection: An Idea Flows Theory of Entry, Trade, and Growth *”  In _The Quarterly Journal of Economics_ 131.1, 2016, pp. 315–380  DOI: 10.1093/qje/qjv032
  * [46] Thomas Sampson  “Technology Gaps, Trade and Income”  In _American Economic Review_ VOL. 113.2, 2023, pp. 66
  * [47] Liyang Sun and Sarah Abraham  “Estimating dynamic treatment effects in event studies with heterogeneous treatment effects”  In _Journal of Econometrics_ 225.2, Themed Issue: Treatment Effect 1, 2021, pp. 175–199  DOI: 10.1016/j.jeconom.2020.09.006

##  Appendix A Timeline of EU Trade Agreements

Partner | Signed |  Provisional application | Full entry into force
---|---|---|---
Switzerland |  1972 |  | 1973
Iceland | 1992 |  | 1994
Norway | 1992 |  | 1994
Turkey | 1995 |  | 1995
Tunisia | 1995 |  | 1998
Israel |  1995 | 1996 | 2000
Mexico | 1997 |  | 2000
Morocco | 1996 |  | 2000
Jordan | 1997 |  | 2002
Egypt | 2001 |  | 2004
North Macedonia |  2001 | 2001 | 2004
South Africa | 1999 | 2000 | 2004
Chile | 2002 | 2003 | 2005

##  Appendix B Mathematical derivations

###  B.1 Optimal control problem

In the dynamic optimal control problem, the household chooses an optimal path
of $C_{s}(t)$ at every instant, taking as given prices. The problem of
choosing varieties $c_{s}(t,\omega)$ is separable and can be solved
conditional on a path for $C_{s}(t)$, such that only aggregates matter for the
dynamic path. Therefore, the current-value Hamiltonian for this problem is:

$H(t,C,L,\mu)=\log\left(C_{s}(t)\right)+\mu_{s}(t)\left[\frac{r_{s}(t)}{P_{s}(t)}A_{s}(t)+\frac{w_{s}(t)}{P_{s}(t)}L_{s}-C_{s}(t)\right]$

with optimality conditions satisfying:

$\displaystyle\frac{1}{C_{s}(t)}$ $\displaystyle=$ $\displaystyle\mu_{s}(t)$
$\displaystyle\frac{\dot{\mu}_{s}(t)}{\mu_{s}(t)}$ $\displaystyle=$
$\displaystyle\rho-\frac{r_{s}(t)}{P_{s}(t)}$

and a transversality condition:

$\lim_{t\to\infty}\left[\exp\\{-\int_{0}^{t}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu\\}P_{s}(t)A_{s}(t)\right]=0$

Taking time derivatives of the first optimality condition and then replacing
for $\frac{\dot{\mu}(t)}{\mu(t)}$ yields the Euler equation:

$\frac{\dot{C}_{s}(t)}{C_{s}(t)}=\left[\frac{r_{s}(t)}{P_{s}(t)}-\rho\right]$

###  B.2 Solution to the dynamic problem

Growth in each of the $s\in\boldsymbol{K}$ of the national economies evolve
according to the following system of differential equations:

$\displaystyle\dot{C}_{s}(t)$ $\displaystyle=$
$\displaystyle\left[\frac{r_{s}(t)}{P_{s}(t)}-\rho\right]C_{s}(t)$
$\displaystyle\dot{M}_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{r_{s}(t)}{P_{s}(t)}M_{s}(t)+\psi\frac{w_{s}(t)}{P_{s}(t)}L_{s}-\psi
C_{s}(t)$

In this section, I will first derive this system of equations, then solve it.
First, one sees that consumption evolves according to a first-order
differential equation. Let
$a(t)\equiv\left[\frac{r_{s}(t)}{P_{s}(t)}-\rho\right]$ and write the Euler
equation as:

$\dot{C}_{s}(t)=a(t)C_{s}(t)$

Multiplying both sides by the integration factor
$\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}$:

$\dot{C}_{s}(t)\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}-a(t)C_{s}(t)\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}=0$

Now, using Leibnitz lemma, note that the time derivative of
$\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}C_{s}(t)$ is
$\dot{C}_{s}(t)\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}-a(t)C_{s}(t)\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}$.
Therefore, integrating both sides with respect to time:

$\exp\\{-\int_{0}^{t}a(\tau)d\tau\\}C_{s}(t)=C(0)$

where $C(0)$ is the constant of integration. Dividing both sides by
$\exp\\{-\int_{0}^{t}a(s)ds\\}$ and replacing for $a(t)$ yields the solution
for the consumption path:

$C_{s}(t)=C(0)\exp\left\\{\int_{0}^{t}\left[\frac{r_{s}(\tau)}{P_{s}(\tau)}-\rho\right]d\tau\right\\}$

which can be rewritten as:

$C_{s}(t)=C_{s}(0)\exp\left\\{\left[\bar{r}_{s}(t)-\rho\right]t\right\\}$

where
$\bar{r}_{s}(t)\equiv\frac{1}{t}\int_{0}^{t}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu$
is the average real interest rate between periods $0$ and $t$. Now recall that
the budget constraint is:

$P_{s}(t)I_{s}(t)+P_{s}(t)C_{s}(t)=r_{s}(t)A_{s}(t)+w_{s}(t)L_{s}$ (B.1)

and that $\psi I_{s}(t)=\dot{M}_{s}(t)$ and $\psi A_{s}(t)=M_{s}(t)$.
Replacing above and solving for $\dot{M}_{s}(t)$ results in:

$\dot{M}_{s}(t)=\frac{r_{s}(t)}{P_{s}(t)}M_{s}(t)+\psi\frac{w_{s}(t)}{P_{s}(t)}L_{s}-\psi
C_{s}(t)$

which, after replacement, yields the following equation:

$\dot{M}_{s}(t)=\frac{r_{s}(t)}{P_{s}(t)}M_{s}(t)+\psi\frac{w_{s}(t)}{P_{s}(t)}L_{s}-\psi
C_{s}(0)\exp\left\\{\left[\bar{r}_{s}(t)-\rho\right]t\right\\}$

In turn, this equation has a solution satisfying:

$\displaystyle M_{s}(t)$ $\displaystyle=$ $\displaystyle
M_{s}(0)\cdot\exp\left\\{\int_{0}^{t}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu\right\\}$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\psi\frac{w_{s}(\xi)}{P_{s}(\xi)}L_{s}\cdot\exp\left\\{-\int_{0}^{\xi}\frac{r_{s}(\upsilon)}{P_{s}(\upsilon)}d\upsilon\right\\}d\xi\cdot\exp\left\\{\int_{0}^{t}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu\right\\}$
$\displaystyle-$ $\displaystyle\int_{0}^{t}\psi
C_{s}(0)\exp\left\\{\left[\bar{r}_{s}(\xi)-\rho\right]\xi\right\\}\cdot\exp\left\\{-\int_{0}^{\xi}\frac{r_{s}(\upsilon)}{P_{s}(\upsilon)}d\upsilon\right\\}d\xi\cdot\exp\left\\{\int_{0}^{t}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu\right\\}$

which, using the definition of $\bar{r}(t)$, becomes:

$\displaystyle M_{s}(t)$ $\displaystyle=$ $\displaystyle
M_{s}(0)\cdot\exp\left\\{\bar{r}(t)\cdot t\right\\}$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\psi\frac{w_{s}(\xi)}{P_{s}(\xi)}L_{s}\cdot\exp\left\\{-\bar{r}(\xi)\cdot\xi\right\\}d\xi\cdot\exp\left\\{\bar{r}(t)\cdot
t\right\\}$ $\displaystyle-$ $\displaystyle\psi
C_{s}(0)\cdot\int_{0}^{t}\exp\left\\{\left[\bar{r}_{s}(\xi)-\rho\right]\xi\right\\}\cdot\exp\left\\{-\bar{r}(\xi)\cdot\xi\right\\}d\xi\cdot\exp\left\\{\bar{r}(t)\cdot
t\right\\}$

simplifying the last integral:

$\displaystyle M_{s}(t)$ $\displaystyle=$ $\displaystyle
M_{s}(0)\cdot\exp\left\\{\bar{r}(t)\cdot t\right\\}$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\psi\frac{w_{s}(\xi)}{P_{s}(\xi)}L_{s}\cdot\exp\left\\{-\bar{r}(\xi)\cdot\xi\right\\}d\xi\cdot\exp\left\\{\bar{r}(t)\cdot
t\right\\}$ $\displaystyle-$ $\displaystyle\psi
C_{s}(0)\cdot\int_{0}^{t}\exp\left\\{-\rho\xi\right\\}d\xi\cdot\exp\left\\{\bar{r}(t)\cdot
t\right\\}$

Finally, note that both $P_{s}(t)$ and $r_{s}(t)$ are functions of wages.
Therefore, given the initial measure of varieties $M_{s}(0)$ and the wages for
all countries, which are defined at every instance through the trade
equilibrium, paths for consumption $C_{s}(t)$, varieties $M_{s}(t)$ and assets
$A_{s}(t)=1/\psi M_{s}(t)$ follow the equations above.

As a final step, one needs to pin down the starting values. $M_{s}(0)$ is
given and calibrated to reflect the technological level of country $s$. Choice
of $C_{s}(0)$, by contrast, is an endogenous object that guarantees that,
given lifetime income and the initial level of assets, consumption as governed
by the euler equation will be optimal. Start from the equation above, multiply
both sides by $\exp\left\\{-\bar{r}(t)\cdot t\right\\}$:

$\displaystyle\exp\left\\{-\bar{r}(t)\cdot t\right\\}M_{s}(t)$
$\displaystyle=$ $\displaystyle
M_{s}(0)+\int_{0}^{t}\psi\frac{w_{s}(\xi)}{P_{s}(\xi)}L_{s}\cdot\exp\left\\{-\bar{r}(\xi)\cdot\xi\right\\}d\xi$
$\displaystyle-$ $\displaystyle\psi
C_{s}(0)\cdot\int_{0}^{t}\exp\left\\{-\rho\xi\right\\}d\xi$

Now evaluate this equation taking the limit $t\to\infty$.

$\displaystyle\lim_{t\to\infty}\left(\exp\left\\{-\bar{r}(t)\cdot
t\right\\}M_{s}(t)\right)$ $\displaystyle=$ $\displaystyle
M_{s}(0)+\int_{0}^{\infty}\psi\frac{w_{s}(t)}{P_{s}(t)}L_{s}\cdot\exp\left\\{-\bar{r}(t)\cdot
t\right\\}dt$ $\displaystyle-$ $\displaystyle\psi
C_{s}(0)\cdot\int_{0}^{\infty}\exp\left\\{-\rho t\right\\}dt$

Recall that the transversality condition is:

$\lim_{t\to\infty}\left[\exp\\{-\int_{0}^{t}\frac{r_{s}(\nu)}{P_{s}(\nu)}d\nu\\}P_{s}(t)A_{s}(t)\right]=0$

which states that the value of assets cannot grow faster than the interest
rate, the standard no-Ponzi scheme condition. Using the fact that $\psi
A_{s}(t)=M_{s}(t)$, noting that prices $P_{s}(t)$ are always positive and
finite, and dividing both sides by $P_{s}(t)/\psi$, we can rewrite this as:

$\lim_{t\to\infty}\left[\exp\\{-\bar{r}_{s}(t)t\\}M_{s}(t)\right]=0$

Using the fact that $\lim_{t\to\infty}\left(\exp\left\\{-\bar{r}(t)\cdot
t\right\\}M_{s}(t)\right)=0$, we can then solve for $C_{s}(0)$ as:

$C_{s}(0)=\left[\frac{1}{\psi}M_{s}(0)+\int_{0}^{\infty}\frac{w_{s}(t)}{P_{s}(t)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(t)\cdot
t\right\\}dt\right]\cdot\left[\int_{0}^{\infty}\exp\left\\{-\rho
t\right\\}dt\right]^{-1}$

Using the fact that $\int_{0}^{\infty}\exp\left\\{-\rho
t\right\\}dt=\frac{1}{\rho}$, then:

$\displaystyle C_{s}(0)$ $\displaystyle=$
$\displaystyle\rho\left[\frac{1}{\psi}M_{s}(0)+\int_{0}^{\infty}\frac{w_{s}(t)}{P_{s}(t)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(t)\cdot
t\right\\}dt\right]$ $\displaystyle=$
$\displaystyle\rho\left[\underbrace{A_{s}(0)}_{\text{initial
wealth}}+\underbrace{\int_{0}^{\infty}\frac{w_{s}(t)}{P_{s}(t)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(t)\cdot
t\right\\}dt}_{\text{PV of real labor income}}\right]$

Therefore, at any instant $t$, consumption is proportional to lifetime wealth:

$\displaystyle C_{s}(t)$ $\displaystyle=$
$\displaystyle\rho\left[A_{s}(0)+\int_{0}^{\infty}\frac{w_{s}(\tau)}{P_{s}(\tau)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(\tau)\cdot\tau\right\\}d\tau\right]\cdot\exp\left\\{\left[\bar{r}_{s}(t)-\rho\right]t\right\\}$
$\displaystyle=$
$\displaystyle\rho\left[A_{s}(t)+\int_{t}^{\infty}\frac{w_{s}(\tau)}{P_{s}(\tau)}L_{s}\cdot\exp\left\\{-\bar{r}_{s}(\tau)\cdot\tau\right\\}d\tau\right]$

###  B.3 Final varieties producers problem

Each final goods producer chooses intermediate inputs and labor to maximize
profits according to:

$\displaystyle\max_{\ell_{s}(t,\omega),\\{x_{ks}(t,\omega,\nu)\\}}$
$\displaystyle
p_{ss}(t,\omega)z_{s}(t,\omega)[\ell_{s}(t,\omega)]^{1-\alpha}\left(\frac{1}{\alpha}\sum_{k\in\boldsymbol{K}}\int_{0}^{M_{k}(t)}[x_{ks}(t,\omega,\nu)]^{\alpha}d\nu\right)$
$\displaystyle-$ $\displaystyle
w_{s}(t)\ell_{s}(t,\omega)-\sum_{k\in\boldsymbol{K}}\int_{0}^{M_{k}(t)}p_{ks}(t,\nu)x_{ks}(t,\omega,\nu)d\nu$

There are infinitely many first order conditions for this problem: one for
each variety $\nu$ and one for labor. These satisfy:

$\displaystyle w_{s}(t)\ell_{s}(t,\omega)$ $\displaystyle=$
$\displaystyle(1-\alpha)\cdot
p_{ss}(t,\omega)z_{s}(t,\omega)[\ell_{s}(t,\omega)]^{1-\alpha}\left(\frac{1}{\alpha}\sum_{k\in\boldsymbol{K}}\int_{0}^{M_{k}(t)}[x_{ks}(t,\omega,\nu)]^{\alpha}d\nu\right)$
$\displaystyle p_{ks}(t,\nu)x_{ks}(t,\omega,\nu)$ $\displaystyle=$
$\displaystyle\alpha\cdot
p_{ss}(t,\omega)z_{s}(t,\omega)[\ell_{s}(t,\omega)]^{1-\alpha}\left(\frac{1}{\alpha}[x_{ks}(t,\omega,\nu)]^{\alpha}\right)$

Solving for $x_{ks}(t,\omega,\nu)$ yields equation (4):

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

###  B.4 Intermediate varieties producers problem

Each intermediate varieties producer holds perpetual rights over variety
$\nu$, which they sell to final goods varieties in every country
$d\in\boldsymbol{K}$. For each destination, they take demand as given and
choose prices to maximize profits at every moment:

$\max_{p_{ks}(t,\omega,\nu)}\frac{1}{\tau_{ks}}p_{ks}(t,\omega,\nu)x_{ks}(t,\omega,\nu)-P_{k}(t)x_{ks}(t,\omega,\nu)$

Replacing for $x_{ks}(t,\omega,\nu)$:

$\max_{p_{ks}(t,\omega,\nu)}\left[p_{ks}(t,\omega,\nu)-\tau_{ks}P_{k}(t)\right]\left[\frac{p_{ks}(t,\omega,\nu)}{p_{ss}(t,\omega)}\right]^{-\frac{1}{1-\alpha}}\cdot\ell_{s}(t,\omega)\cdot
z_{s}(t,\omega)^{\frac{1}{1-\alpha}}$

which, after taking the FOC and solving for $p_{ks}(t,\omega,\nu)$ yields the
optimal price as a mark-up over marginal price, which is independent of
$\omega$ or $\nu$:

$p_{ks}(t,\omega,\nu)=\frac{\tau_{ks}P_{k}(t)}{\alpha}\qquad\forall\omega\in[0,1],\quad\forall\nu\in[0,M_{s}(t)]$

Flow aggregate profits $\Pi_{s}(t)\equiv\int_{0}^{M_{s}(t)}\pi(t,\nu)d\nu$ are
a constant fraction of revenue:

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

Profits per variety $\pi_{s}(t,\nu)=\frac{1}{M_{s}(t)}\Pi_{s}(t)$ are
independent of $\nu$.

###  B.5 Trade in Final Goods

#### Trade shares

In this model, since there are infinitely many varieties in the unit interval,
the expenditure share of destination region $d\in\boldsymbol{K}$ on goods
coming from source country $s\in\boldsymbol{K}$ converge to their expected
values. Let $\lambda_{sd}(t,\omega)$ denote the probability that consumers in
region $d\in\mathcal{D}$ source variety $\omega$ from region
$s\in\mathcal{D}$. For each each $n$, let
$A_{n}^{-1}(t,\omega)\equiv\frac{\tilde{x}_{sd}(t)}{\tilde{x}_{nd}(t)}$, with
$x_{sd}(t)\equiv(P_{s}^{M}(t))^{\alpha}w_{s}(t)^{1-\alpha}\tau_{sd}$. This
probability will satisfy:

$\displaystyle\lambda_{sd}(t,\omega)$ $\displaystyle=$ $\displaystyle
Pr\Big{(}s\text{ is the lowest cost supplier of }\omega\text{ to }d\Big{)}$
(B.4) $\displaystyle=$ $\displaystyle
Pr\Big{(}\frac{\tilde{x}_{sd}(t)}{z_{s}(t,\omega)}<\min_{(n\neq
s)}\Big{\\{}\frac{\tilde{x}_{nd}(t)}{z_{n}(t,\omega)}\Big{\\}}\Big{)}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}Pr(z_{s}(t,\omega)=z)Pr(z_{n}(t,\omega)<zA_{n}(t))dz$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}f_{s}(t)(z)\Pi_{(n\neq
s)}F_{n}(t)(A_{n}z)dz$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\theta
T_{s}z^{-(1+\theta)}\exp\left\\{-\left(\sum_{n\in\boldsymbol{K}}T_{n}A_{n}(t)^{-\theta}\right)z^{-\theta}\right\\}dz$
$\displaystyle=$
$\displaystyle\frac{T_{s}\left(\tilde{x}_{sd}(t)\right)^{-\theta}}{\sum_{n\in\boldsymbol{K}}T_{n}\left(\tilde{x}_{nd}(t)\right)^{-\theta}}$
$\displaystyle=$
$\displaystyle\frac{T_{s}\left(\tilde{x}_{sd}(t)\right)^{-\theta}}{\sum_{n\in\boldsymbol{K}}T_{n}\left(\tilde{x}_{nd}(t)\right)^{-\theta}}$
$\displaystyle=$
$\displaystyle\frac{T_{s}(w_{s}(t)^{1-\alpha}P^{M}_{s}(t)^{\alpha}\tau_{sd})^{-\theta}}{\sum_{n\in\boldsymbol{K}}{T_{n}(w_{n}(t)^{1-\alpha}P^{M}_{n}(t)^{\alpha}\tau_{nd})^{-\theta}}}$

Now note that $\lambda_{sd}(t,\omega)$ is independent of $\omega$, so the
probability of sourcing each variety from $s$ to $d$ is identical. A corollary
is that aggregate expenditure trade shares of final goods from $s$ in $d$ will
be equal to the probability of sourcing an arbitrary variety from $s$ in $d$.

#### Price distributions and ideal price index

Recall that, under the assumption of perfect competition, prices equal their
marginal costs, such that the price of a variety $\omega$ produced in country
$s$ and shipped to $d$ satisfies
$p_{sd}(t,\omega)=\frac{\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha}}{z_{s}(t,\omega)}$.

Since $z_{s}(t,\omega)$ is a random variable, $p_{sd}(t,\omega)$ is also a
random variable. We can derive the distribution of prices through the
following steps. First, note that
$z_{s}(t,\omega)=\frac{\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha}}{p_{sd}(\omega)}$.
Then, note that:

$p_{sd}(t,\omega)<p=\frac{\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}^{1-\alpha}}{z}\iff
z_{s}(t,\omega)>z=\frac{\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}^{1-\alpha}}{p}$

Therefore:

$\displaystyle G_{sd}(t,\omega)(p)$ $\displaystyle=$ $\displaystyle
Pr(p_{sd}(t,\omega)<p)$ $\displaystyle=$ $\displaystyle
Pr\left(z_{s}(t,\omega)>\frac{\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}^{1-\alpha}}{p}\right)$
$\displaystyle=$ $\displaystyle
1-\exp\\{-T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t))^{-\theta}p^{\theta}\\}\qquad\forall\omega\in[0,1]$

which is the distribution of prices of any variety $\omega$ conditional on $s$
being the lowest cost supplier of such a variety to $d$. To derive the
unconditional distribution of prices at $d$, realize that:

$\displaystyle G_{n}(t,\omega)$ $\displaystyle\equiv$ $\displaystyle
Pr(p_{s}(t,\omega)<p)$ $\displaystyle=$ $\displaystyle Pr((\exists s)\text{
for which }p_{s}d(t,\omega)<p)$ $\displaystyle=$ $\displaystyle 1-Pr((\nexists
s)\text{ for which }p_{sd}(t,\omega)<p)$ $\displaystyle=$ $\displaystyle
1-\prod_{s\in\boldsymbol{K}}Pr(p_{sd}(t,\omega)>p)$ $\displaystyle=$
$\displaystyle
1-\prod_{s\in\boldsymbol{K}}\exp\\{-T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t))^{-\theta}p^{\theta}$
$\displaystyle=$ $\displaystyle
1-\exp\\{-\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t))^{-\theta}p^{\theta}\\}$

Recall that the price index is defined as:

$\displaystyle P_{d}(t)$ $\displaystyle=$
$\displaystyle\left[\int_{0}^{1}p_{d}(t,\omega)^{1-\sigma}d\omega\right]^{\frac{1}{1-\sigma}}$
$\displaystyle=$
$\displaystyle\left[\int_{0}^{\infty}p^{1-\sigma}dG_{n}(t,p)\right]^{\frac{1}{1-\sigma}}$
$\displaystyle=$ $\displaystyle\left[\int_{0}^{\infty}p^{1-\sigma}\theta
p^{\theta-1}\exp\left\\{-\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha})^{-\theta}p^{\theta}\right\\}dp\right]^{\frac{1}{1-\sigma}}$

Using a change of variables, let
$\nu\equiv\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha})^{-\theta}p^{\theta}$
and note that $d\nu=\theta
p^{\theta-1}\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha})^{-\theta}p^{\theta}dp$.
Then:

$\displaystyle P_{d}(t)$ $\displaystyle=$
$\displaystyle\left[\int_{0}^{\infty}\left(\frac{\nu}{\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha})^{-\theta}}\right)^{\frac{1-\sigma}{\theta}}\exp\left\\{-\nu\right\\}d\nu\right]^{\frac{1}{1-\sigma}}$
$\displaystyle=$
$\displaystyle\Gamma\left(\frac{\theta+1-\sigma}{\theta}\right)^{\frac{1}{1-\sigma}}\left(\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha})^{-\theta}\right)^{-\frac{1}{\theta}}$
$\displaystyle=$
$\displaystyle\Gamma\left(\frac{\theta+1-\sigma}{\theta}\right)^{\frac{1}{1-\sigma}}\left(\sum_{s\in\boldsymbol{K}}T_{s}(\tau_{sd}w_{s}(t)^{1-\alpha})^{-\theta}\left(\sum_{n\in\boldsymbol{K}}M_{n}(t)\left[\frac{\tau_{ns}P_{n}(t)}{\alpha}\right]^{-\frac{\alpha}{1-\alpha}}\right)^{(1-\alpha)\theta}\right)^{-\frac{1}{\theta}}.$

which shows that, given parameters $T_{s}.\tau_{sd}$ and the vector of state
variables $\boldsymbol{M}_{s}(t)=[M_{1}(t),\cdots,M_{N}(t)]^{\prime}$, the
closed form solution for the ideal price index $P_{d}(t)$ is a function of the
vector of wages $\boldsymbol{w}(t)=[w_{1}(t),\cdots,w_{N}(t)]^{\prime}$.

###  B.6 Market Clearing and Trade Balance

#### Market Clearing

Let $Y_{d}(t)$ denote the total output of the final good and
$X_{d}(t),I_{d}(t)$ denote the use of the final good as inputs for the
production of intermediate inputs and R&D, respectively. Then total output in
the final good for a given country must satisfy:

$Y_{d}(t)=C_{d}(t)+I_{d}(t)+X_{d}(t)$

where $I_{d}(t)$ and $C_{d}(t)$ are pinned down by the dynamic problem,
described below, and $X_{d}(t)$ can be expressed as a function of aggregate
demand in all destinations:

$X_{d}(t)\equiv\sum_{k\in\boldsymbol{K}}M_{d}(t)\cdot\left(\frac{p^{M}_{dk}(t)}{P_{k}^{M}(t)}\right)^{-\eta}\cdot\alpha\cdot\left(\frac{P^{M}_{k}(t)}{P_{k}(t)}\right)^{-1}\cdot
Y_{k}(t)$

Combining the equations, one can express aggregate output as a function of the
state variable $M_{d}(t)$, parameters, and wages (both $r_{d}(t)$ and
$P_{d}(t)$ are functions of wages in every country):

$Y_{d}(t)=I_{d}(t)+C_{d}(t)+\sum_{k\in\boldsymbol{K}}M_{d}(t)\cdot\left(\frac{p^{M}_{dk}(t)}{P_{k}^{M}(t)}\right)^{-\eta}\cdot\alpha\cdot\left(\frac{P^{M}_{k}(t)}{P_{k}(t)}\right)^{-1}\cdot
Y_{k}(t)$

#### Expenditure Determination

Flow aggregate profits $\Pi_{s}(t)\equiv\int_{0}^{M_{s}(t)}\pi(t,\nu)d\nu$ are
a constant fraction of revenue:

$\Pi_{s}(t)=\frac{\alpha}{\eta}\cdot\sum_{d\in\boldsymbol{K}}\lambda_{sd}^{M}(t)\cdot
P_{d}(t)Y_{d}(t)$

On the expenditure side, GDP of each destination country $s\in\boldsymbol{K}$
country will be exhausted as the combination of the total expenditures of
labor and capital income:

$P_{s}(t)Y_{s}(t)=w_{s}(t)L_{s}+\Pi_{s}(t)$

From the income side, nominal GDP must equal the sum of total flow payments
received domestically and from the rest of the world:

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

#### Trade Balance

Total exports are equal to:

$EX_{d}(t)=\underbrace{\sum_{d\neq
s\in\boldsymbol{K}}\lambda_{sd}^{F}(t)P_{d}(t)Y_{d}(t)}_{\text{exports in
final goods}}+\underbrace{\alpha\sum_{d\neq
s\in\boldsymbol{K}}\lambda_{sd}^{M}(t)\left[\sum_{k^{\prime}\in\boldsymbol{K}}\lambda_{dk^{\prime}}^{F}(t)P_{k^{\prime}}(t)Y_{k^{\prime}}(t)\right]}_{\text{exports
in intermediates}}$

Total imports are equal to:

$IM_{d}(t)=\underbrace{[1-\lambda_{ss}^{F}(t)]P_{s}(t)Y_{s}(t)}_{\text{imports
in final
goods}}+\underbrace{\alpha[1-\lambda_{ss}^{M}(t)]\left[\sum_{k^{\prime}\in\boldsymbol{K}}\lambda_{dk^{\prime}}^{F}(t)P_{k^{\prime}}(t)Y_{k^{\prime}}(t)\right]}_{\text{imports
in intermediates}}$

Since there are no international capital markets in this economy, trade will
be balanced at any instant. This means that:

$\displaystyle\sum_{d\neq
s\in\boldsymbol{K}}\lambda_{sd}^{F}(t)P_{d}(t)Y_{d}(t)+\alpha\sum_{d\neq
s\in\boldsymbol{K}}\lambda_{sd}^{M}(t)\left[\sum_{k^{\prime}\in\boldsymbol{K}}\lambda_{dk^{\prime}}^{F}(t)P_{k^{\prime}}(t)Y_{k^{\prime}}(t)\right]=$
$\displaystyle[1-\lambda_{ss}^{F}(t)]P_{s}(t)Y_{s}(t)+\alpha[1-\lambda_{ss}^{M}(t)]\left[\sum_{k^{\prime}\in\boldsymbol{K}}\lambda_{dk^{\prime}}^{F}(t)P_{k^{\prime}}(t)Y_{k^{\prime}}(t)\right]$

###  B.7 Homogeneity of Income in Equilibrium

The trade share for final goods is:

$\displaystyle\lambda^{F}_{sd}(t)$ $\displaystyle\equiv$
$\displaystyle\frac{E^{F}_{sd}(t)}{E^{F}_{d}(t)}=\frac{T_{s}(P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha}\tau_{sd})^{-\theta}}{\sum_{n\in\boldsymbol{K}}{T_{n}(P^{M}_{n}(t)^{\alpha}w_{n}(t)^{1-\alpha}\tau_{nd})^{-\theta}}}=\frac{T_{s}(P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha}\tau_{sd})^{-\theta}}{P_{s}(t)^{-\theta}}$

Evaluating it at $\lambda_{ss}(t)$, noting that
$\lambda^{M}_{s}(t)=\frac{M_{s}(t)(p^{M}_{ss}(t))^{1-\eta}}{(P^{M}_{s}(t))^{1-\eta}}=\frac{M_{s}(t)(\frac{1}{\alpha}P_{s}(t))^{1-\eta}}{(P^{M}_{s}(t))^{1-\eta}}$
and solving it for $P_{s}(t)$ allows me to write it linear in $M_{s}(t)$:

$\displaystyle P_{s}(t)$ $\displaystyle=$ $\displaystyle
T_{s}^{-\frac{1}{\theta}}\lambda^{F}_{ss}(t)^{\frac{1}{\theta}}(P^{M}_{s}(t)^{\alpha}w_{s}(t)^{1-\alpha})$
$\displaystyle P_{s}(t)$ $\displaystyle=$
$\displaystyle\left(\frac{T_{s}}{\lambda^{F}_{ss}(t)}\right)^{-\frac{1}{\theta}}\left(\frac{M_{s}(t)}{\lambda^{M}_{ss}(t)}\right)^{\frac{\alpha}{1-\eta}}\cdot\alpha^{-\alpha}\cdot
P_{s}(t)^{\alpha}(w_{s}(t)^{1-\alpha})$ $\displaystyle P_{s}(t)$
$\displaystyle=$
$\displaystyle\alpha^{-\frac{\alpha}{1-\alpha}}\cdot\left(\frac{T_{s}}{\lambda^{F}_{ss}(t)}\right)^{-\frac{1}{\theta(1-\alpha)}}\left(\frac{M_{s}(t)}{\lambda^{M}_{ss}(t)}\right)^{\frac{1}{1-\eta}\frac{\alpha}{1-\alpha}}\cdot
w_{s}(t)$ $\displaystyle P_{s}(t)$ $\displaystyle=$
$\displaystyle\alpha^{-\frac{\alpha}{1-\alpha}}\cdot\left(\frac{T_{s}}{\lambda^{F}_{ss}(t)}\right)^{-\frac{1}{\theta(1-\alpha)}}\frac{\lambda^{M}_{ss}(t)}{M_{s}(t)}\cdot
w_{s}(t)\qquad\left(\because\frac{\alpha}{1-\alpha}=1-\eta\right)$

which allows me to write real wages as a linear function of $M_{s}(t)$: