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Federal Reserve Bank of Chicago

The Global Diffusion of Ideas
Francisco J. Buera and Ezra Oberfield

December 2015
WP 2016-13

The Global Diffusion of Ideas∗
Francisco J. Buera

Ezra Oberfield

Federal Reserve Bank of Chicago

Princeton University

December 22, 2015

Abstract
We provide a tractable theory of innovation and technology diffusion to explore the
role of international trade in the process of development. We model innovation and
diffusion as a process involving the combination of new ideas with insights from other
industries or countries. We provide conditions under which each country’s equilibrium
frontier of knowledge converges to a Frechet distribution, and derive a system of differential equations describing the evolution of the scale parameters of these distributions,
i.e., countries’ stocks of knowledge. In particular, the growth of a country’s stock of
knowledge depends only on its trade shares and the stocks of knowledge of its trading
partners. We use the framework to quantify the contribution of bilateral trade costs
to cross-sectional TFP differences, long-run changes in TFP, and individual post-war
growth miracles.
∗

We are grateful to Costas Arkolakis, Arnaud Costinot, Sam Kortum, Bob Lucas, Erzo Luttmer, Marc
Melitz, Jesse Perla, Andres Rodriguez-Clare, Esteban Rossi-Hansberg, Chris Tonetti, Jeff Thurk, and Mike
Waugh. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve
Bank of Chicago or the Federal Reserve System. All mistakes are our own.

Economic miracles are characterized by protracted growth in productivity, per-capita
income, and increases in trade flows. The experiences of South Korea in the postwar period
and the recent performance of China are prominent examples. These experiences suggest an
important role played by openness in the process of development.1 Yet quantitative trade
models relying on standard static mechanisms imply relatively small gains from openness,
and therefore cannot account for growth miracles.2 These findings call for alternative channels through which openness can affect development. In this paper we present and analyze
a model of an alternative mechanism: the impact of openness on the creation and diffusion
of best practices across countries.3
We model innovation and diffusion as a process involving the combination of new ideas
with insights from other industries and countries. Insights occur randomly and result from
local interactions among producers. In our theory openness affects the creation and diffusion
of ideas by determining the distribution from which producers draw their insights. Our
theory is flexible enough to incorporate different channels through which ideas may diffuse
across countries. We focus on two main channels: (i) insights are drawn from those that
sell goods to a country, (ii) insights are drawn from technologies used domestically. In our
model, openness to trade affects the quality of the insights drawn by producers because it
determines the set of sellers to a country and the set of technologies used domestically.
In this context, we provide conditions under which the distribution of productivity among
producers within each country always converges to a Frechet distribution, no matter how
trade barriers shape individual producers’ local interactions. As a consequence, the state of
knowledge within a country can be summarized by the level of this distribution, which we call
the country’s stock of knowledge. The model is thus compatible with the Eaton and Kortum
1
Sachs and Warner (1995), Dollar (1992), Ben-David (1993), Coe and Helpman (1995), and Frankel and
Romer (1999) suggest a strong relationship between openness and growth, although Rodriguez and Rodrik
(2001) subsequently argued that many estimates in the literature suffered from econometric issues including
omitted variables, endogeneity, and lack of robustness. More recent contributions to the literature have
developed strategies to overcome some of these issues. To estimate the impact of trade on growth, Feyrer
(2009a,b) studies the natural experiments of the decade-long closing of the Suez Canal in the 1970’s and
the long run decline in the cost of shipping goods by air, each of which had larger impacts on some pairs
of countries than others, and Pascali (2014) studies the introduction of the steamship which affected some
trade routes more than others. See also Lucas (2009b) and Wacziarg and Welch (2008), and Donaldson
(2015) for a review of the literature.
2
See Connolly and Yi (2015) for a quantification of the role of trade on Korean’s growth miracle. Atkeson
and Burstein (2010) also find relatively small effects in a model with innovation.
3
Parente and Prescott (1994) and Klenow and Rodriguez-Clare (2005) argue that without some form
of international spillovers or externalities, growth models have difficulty accounting for several facts about
growth and development. Each argue that these facts can be explained by catchup growth to a world
frontier of knowledge, an idea that goes back to at least Nelson and Phelps (1966). Comin and Hobijn
(2010) document large cross-country differences in the speed with which frontier technologies are adopted
and Comin et al. (2012) show that the speed of diffusion declines with distance.

(2002) machinery which has been useful in studying trade flows in an environment with many
asymmetric countries. We show that the change in a country’s stock of knowledge can be
characterized in terms of only its trade shares, its trading partners’ stocks of knowledge, and
parameters. This both yields qualitative insights and enables us to use actual trade flows to
discipline the role of trade and geography in shaping idea flows and growth.
Starting from autarky, opening to trade results in a higher temporary growth rate, and
permanently higher level, of the stock of knowledge, as producers are exposed to more
productive ideas. We separate the gains from trade into static and dynamic components.
The static component consists of the gains from increased specialization and comparative
advantage, whereas the dynamic component are the gains that operate through the flow of
ideas.
We first explore an environment in which producers in a country gain insights from those
that sell goods to the country, following Alvarez et al. (2014). With this specification of
learning, the dynamic gains from reducing trade barriers are qualitatively different from the
static gains. The dynamic gains are largest for countries that are relatively closed, whereas
the static gains are largest for countries that are already relatively open. For a country
with high trade barriers, the marginal import tends to be made by a foreign producer with
high productivity. While the high trade costs imply that the static gains from trade remain
relatively small, the insights drawn from these marginal producers tend to be of high quality.
In contrast, for a country close to free trade, the reduction in trade costs leads to large inframarginal static gains from trade, but the insights drawn from the marginal producers are
likely to have lower productivity and generate lower quality ideas.
Our model nests, at two extremes, a simple version of the Kortum (1997) model of pure
innovation and one closely related to the Alvarez et al. (2008, 2014) model of pure diffusion.
We span these two extremes by varying a single parameter, β, which we label the strength
of diffusion. β measures the contribution of insights from others to the productivity of new
ideas. One striking observation is that, for either of these two extremes, if a moderately open
country lowers its trade costs, the resulting dynamic gains from trade are relatively small,
whereas when β is in an intermediate range, the dynamic gains are larger. When β is small
so that insights from others are relatively unimportant, it follows immediately that dynamic
gains tend to be small. When β is larger, insights from others are more central. However,
in the limiting model as β approaches the extreme of one, a country accrues almost all of
the dynamic gains from trade as long as it is not in autarky.4 A moderately open country is
4

In the environment studied by Alvarez et al. (2014), both the steady state growth rate and the mass
in the right tail of countries’ productivity distributions are proportional to the number of countries not in
autarky; trade costs have no other impact on these objects. The environment we study here with β % 1 has
similar properties albeit in level differences rather than growth rate differences.

much better off than it would be in autarky, but further reductions in trade costs have little
impact. As a consequence, it is only when β is in an intermediate range that the dynamic
gains from trade are both sizable and would result from reductions in trade costs in the
empirically relevant range.
We also explore a second channel, that individuals may draw insights from others that
produce domestically, following Sampson (2015) and Perla et al. (2015). In this setting, lower
trade barriers increase domestic competition and improve the distribution of productivity
among those that continue to produce domestically, raising the quality of insights manager
might draw from. Under this specification of learning, we show that the long-run dynamic
gains from trade simply amplify the static gains. When β is larger so that insights from
others contribute more to the productivity of new ideas, the static gains from trade are
amplified more and the dynamic gains are more sizable.
We next use our model to study the dynamics of a trade liberalization. In a world that
is generally open, if a single closed country opens to trade, it will experience an instantaneous jump in real income, a mechanism that has been well-studied in the trade literature.
Following that jump, this country’s stock of knowledge will gradually improve as the liberalization leads to an improvement in the composition of insights drawn by its producers.
Here, the speed of convergence depends on the nature of learning process. Convergence is
faster if insights are drawn from goods that are sold to the country, as opening to trade
allows producers to draw insight from the relatively productive foreign producers. In contrast, if insights are drawn from technologies that are used locally, the country’s stock of
knowledge grows more slowly. In that case, a trade liberalization leads to better selection
of the domestic producers, but those domestic producers have low productivity relative to
foreign firms.
To explore the ability of the theory to account for the evolution of the world distribution
of productivity, we specify a quantitative version of the model that includes non-traded
goods and intermediate inputs, and equipped labor with capital and education, and use it
to study the ability of the theory to account for cross-country differences in TFP in 1962
and its evolution between 1962 and 2000. Following Waugh (2010), we use panel data on
trade flows and relative prices to calibrate the evolution of bilateral trade costs, and take
the evolution of population, physical and human capital, i.e., equipped labor, from the data.
Given the evolution of trade costs and equipped labor, our model predicts the evolution of
each country’s TFP.
The predicted relationship between trade and TFP depends on the value of β, the strength
of diffusion, which indexes the contribution of insights drawn from others to the productivity
of new ideas. While we provide a simple heroic strategy to calibrate this parameter, our main
3

approach is to simulate the model for various alternative values and explore how well the
model can quantitatively account for cross-country income differences and the evolution of
countries’ productivity over time.
In line with the theoretical results, the role of the trade to account for both the dispersion
of TFP and the dispersion of TFP growth is highest for intermediate values of the diffusion
parameter, β. There are several ways one might measure the contribution of changes in
trade barriers to changes in TFP. When insights are drawn from sellers, we find that, across
measures, the contribution of trade is up to three times as large when the model allows
for dynamic gains from trade. The quantitative model is particularly capable of explaining
much of the evolution of TFP in growth miracles, accounting for over a third of the TFP
growth in China, South Korea and Taiwan.
Literature Review Our work builds on a large literature modeling innovation and diffusion of technologies as a stochastic process, starting from the earlier work of Jovanovic
and Rob (1989), Jovanovic and MacDonald (1994), Kortum (1997), and recent contributions
by Alvarez et al. (2008), Lucas (2009a) and Luttmer (2012).5 We are particularly related
to recent applications of these frameworks to study the connection between trade and the
diffusion of ideas (Lucas, 2009b; Alvarez et al., 2014; Perla et al., 2015; Sampson, 2015).
In our model, the productivity of new ideas combines both insights from others and
an original component.6 As discussed earlier, our theory captures the models in Kortum
(1997) and Alvarez et al. (2008, 2014) as special, and we argue, quantitatively less promising
cases. In Kortum (1997) there is no diffusion of ideas and thus no dynamic gains from trade.
In Alvarez et al. (2014) when trade barriers are finite, changes in trade barriers have no
impact on the tail of the distribution of productivity, and therefore the model has more
limited success in providing a quantitative theory of the level and transitional dynamics
of productivity. In addition, for the intermediate cases that are the focus of our analysis,
β ∈ [0, 1), the frontier of knowledge converges to a Frechet distribution.7 This allows us use
5

Lucas and Moll (2014) and Jesse and Tonetti (2014) extends these models by endogenizing search effort.
The main text abstracts from search effort, but Appendix D studies how trade barriers affect incentives
to innovate. Following Bernard et al. (2003) we focus on a decentralization in which producers engage in
Bertrand competition, and each producer earns profit on sales to any destination to which that producer is
the lowest-cost provider of a good. Motivated by the potential for profit, producers hire labor to generate new
ideas. In this environment, we extend the result of Eaton and Kortum (2001) that on any balanced growth
path, each country’s research effort is independent of trade barriers. Chiu et al. (2011) study information
issues in the transfer of ideas, a dimension that we abstract from.
6
See König et al. (2012) and Benhabib et al. (2014) for models in which individuals can choose either to
imitate or to innovate.
7
Alvarez et al. (2014) study the case with β = 1 and deterministic arrival of ideas. In their model the
limiting distribution of productivities is only Frechet in the extreme cases of autarky and costless trade
among symmetric countries. With Poisson arrival of ideas the limiting distribution is log-logistic for these

the machinery of Eaton and Kortum (2002), enabling us to quantify the role of both trade
barriers and geography in the flow of ideas.
Eaton and Kortum (1999) also build a model of the diffusion of ideas across countries in
which the distribution of productivities in each country is Frechet, and where the evolution
of the scale parameter of the Frechet distribution in each country is governed by a system
of differential equations. In their work insights are drawn from the distribution of potential
producers in each country, according to exogenous diffusion rates which are estimated to be
country-pair specific, although countries are assumed to be in autarky otherwise. Therefore,
changes in trade do not affect the diffusion of ideas.
Our works relates to a large literature studying the connection between trade and growth,
including the early contributions by Grossman and Helpman (1991) and Rivera-Batiz and
Romer (1991). The one that is closest to ours is Grossman and Helpman (1991). They
consider a small open economy in which technology is transferred from the rest of the world
as an external effect, and the pace of technology transfer is assumed to depend on the volume of trade. Our model incorporates this channel along with several others and embeds
the mechanism in a quantitative framework. In addition, our paper relates to a large empirical literature providing evidence on the relationship between openness and diffusion of
technologies. Our reduced form evidence is reminiscent of the early evidence discussed in
Coe and Helpman (1995) and Coe et al. (1997) about the importance of knowledge spillovers
through trade. See Keller (2009) for a recent review of this empirical literature, considering
alternative channels, including trade and FDI.
The model shares some features with Oberfield (2013) which models the formation of
supply chains and the economy’s input-output architecture. In that model, entrepreneurs
discover methods of producing their goods using other entrepreneurs’ goods as inputs.8

Idea Diffusion with a General Source Distribution

We begin with a description of technology diffusion in a single country given a general source
distribution. The source distribution describes the set of insights that producers might
access. In the specific examples that we explore later in the paper, the source distribution
will depend on the profiles of productivity across all countries in the world, but in this section
we assume only that it satisfies weak tail properties. Given these assumptions, we show that
extreme cases. In our model the limiting distribution is Frechet for any β ∈ [0, 1) and any configuration of
trade costs.
8
Here, the evolution of the distribution of marginal costs depends on a differential equation summarizing
the history of insights that were drawn. In Oberfield (2013), the distribution of marginal costs is the solution
to a fixed point problem, as each producer’s marginal cost depends on her potential suppliers’ marginal costs.

the equilibrium distribution of productivity within an economy is Frechet, and derive a
differential equation describing the evolution of the scale parameter of this distribution.
We consider an economy with a continuum of goods s ∈ [0, 1]. For each good, there are
m producers. We will later study an environment in which the producers engage in Bertrand
competition, so that (barring ties) at most one of these producers will actively produce. A
producer is characterized by her productivity, q. A producer of good s with productivity q
has access to a labor-only, linear technology
y(s) = ql(s),

(1)

where l(s) is the labor input and y(s) is output of good s. The state of technology in the
economy is described by the function Mt (q), the fraction of producers with knowledge no
greater than q. We call Mt the distribution of knowledge at t.
The economy’s productivity depends on the frontier of knowledge. The frontier of knowledge is characterized by the function
F̃t (q) ≡ Mt (q)m .
F̃t (q) is the probability that none of the m producers of a good have productivity better
than q.
We now describe the dynamics of the distribution of knowledge. We model diffusion as
a process involving the random interaction among producers of different goods or countries.
We assume each producer draws insights from others stochastically at rate αt . However there
is randomness in the adaptation of that insight. More formally, when an insight arrives to
a producer with productivity q, the producer learns an idea with random productivity zq 0β
and adopts the idea if zq 0β > q. The productivity of the idea has two components. There
is an insight drawn from another producer, q 0 , which is drawn from the source distribution
G̃t (q 0 ). The second component z is an original contribution that is drawn from an exogenous
distribution with CDF H(z). We refer to H(z) as the exogenous distribution of ideas.9
This process captures the fact that interactions with more productive individuals tend to
lead to more useful insights, but it also allows for randomness in the adaptation of others’
techniques to alternative uses. The latter is captured by the random variable z. An alternative interpretation of the model is that z represents an innovator’s “original” random idea,
which is combined with random insights obtained from other technologies.10
9

From the perspective of this section, both G̃t (q) and H(z) are exogenous. The distinction between these
distributions will become clear once we consider specific examples of source distributions, in which the source
distribution will be an endogenous function of countries’ frontiers of knowledge.
10
If β = 0 our framework simplifies to a version of the model in Kortum (1997) with exogenous search

Given the distribution of knowledge at time t, Mt (q), the source distribution, G̃t (q 0 ), and
the exogenous distribution of ideas, H(z), the distribution of knowledge at time t + ∆ is

∞

Mt+∆ (q) = Mt (q) (1 − αt ∆) + αt ∆

H q/x

dG̃t (x)

The first term on the right hand side, Mt (q), is the distribution of knowledge at time t,
which gives the fraction of producers with productivity less than q. The second term is the
probability that a producer did not have an insight between time t and t + ∆ that raised her
productivity above q. This can happen if no insight arrived in an interval of time ∆, an event
with probability 1 − αt ∆, or if at least one insight arrived but none resulted in a technique

R∞
with productivity greater than q, an event that occurs with probability 0 H q/xβ dG̃t (x).
Rearranging and taking the limit as ∆ → 0 we obtain
Mt+∆ (q) − Mt (q)
d
ln Mt (q) = lim
= −αt
∆→0
dt
∆Mt (q)

∞

1 − H q/xβ

dG̃t (x).

With this, we can derive an equation describing the frontier of knowledge. Since F̃t (q) =
Mt (q)m , the change in the frontier of knowledge evolves as:
d
ln F̃t (q) = −mαt
dt

∞

1 − H q/xβ

dG̃t (x).

To gain tractability, we make the following assumption:
Assumption 1
i. The exogenous distribution of ideas has a Pareto right tail with exponent θ, so that
limz→∞ 1−H(z)
= 1.
z −θ
ii. β ∈ [0, 1).
iii. At each t, limq→∞ q βθ [1 − G̃t (q)] = 0.
The first part of the assumption assumes that the right tail of the exogenous distribution of
ideas is regularly varying.11 We also assume that the strength of diffusion, β is strictly less
than one, introducing diminishing returns into the quality of insights one draws. For this
section we make one additional assumption: the source distribution G̃t has a sufficiently thin
intensity. The framework also nests the model of diffusion in Alvarez et al. (2008) with stochastic arrival of
ideas if β = 1, H is degenerate, and G̃t = F̃t .
11
The restriction that the limit is equal to 1 rather than some other positive number is without loss of
generality; we can always choose units so that the limit is one.

tail. In later sections when we endogenize the source distribution, this assumption will be
replaced by an analogous assumption on the right tail of the initial distribution of knowledge,
limq→∞ q βθ [1 − M0 (q)] = 0. For example, a bounded initial distribution of knowledge would
satisfy this assumption.
We will study economies where the number of producers for each good is large. As such,
it will be convenient to study how the frontier of knowledge
evolves
the
when normalized
by

1
1
number of producers for each good. Define Ft (q) ≡ F̃t m (1−β)θ q and Gt (q) ≡ G̃t m (1−β)θ q
Proposition 1 If Assumption 1 holds, then in the limit as m → ∞, the frontier of knowledge
evolves as:
Z ∞
d ln Ft (q)
−θ
xβθ dGt (x)
= −αt q
dt
0
Rt
R∞
Motivated by the previous proposition, we define λt ≡ −∞ ατ 0 xβθ dGτ (x)dτ . With
this, one can show that the economy’s frontier of knowledge converges asymptotically to a
Frechet distribution.
1/θ

−θ

Corollary 2 If Assumption 1 holds and limt→∞ λt = ∞, then limt→∞ Ft (λt q) = e−q .
−θ

Proof. Solving the differential equation gives Ft (q) = F0 (q)e−(λt −λ0 )q . Evaluating this at
−1 −θ
1/θ
1/θ
1/θ
λt q gives Ft (λt q) = F0 (λt q)e−(λt −λ0 )λt q . This implies that, asymptotically,
−θ
1/θ
limt→∞ Ft (λt q) = e−q
Thus, the distribution of productivities in this economy is asymptotically Frechet and
the dynamics of the scale parameter is governed by the differential equation
Z

∞

xβθ dGt (x).

λ̇t = αt

(2)

We call λt the stock of knowledge.
In the rest of the paper we analyze alternative models for the source distribution Gt . A
simple example that illustrates basic features of more general cases is Gt (q) = Ft (q). This
corresponds to the case in which diffusion opportunities are randomly drawn from the set
of domestic best practices across all goods. In a closed economy this set equals the set of
domestic producers and sellers. In this case equation (2) becomes
λ̇t = αt Γ(1 − β)λβt
R∞
where Γ(u) = 0 xu−1 e−x dx is the Gamma function. Growth in the long-run is obtained in
this framework if the arrival rate of insight grows over time, αt = α0 eγt . In this case, the
scale of the Frechet distribution λt grows asymptotically at the rate γ/(1−β), and per-capita
8

GDP grows at the rate γ/[(1 − β)θ]. In general, the evolution of the de-trended stock of
knowledge λ̂t = λt eγ/(1−β)t can be summarized in terms of the de-trended arrival of ideas
α̂t = αt eγt
˙
λ̂t = α̂t Γ(1 − β)λ̂βt −

γ
λ̂t ,
1−β

and on a balanced growth path on which α̂ is constant, the de-trended stock of ideas is

α̂(1 − β)
λ̂ =
Γ(1 − β)
γ

1
1−β

International Trade

Consider a world in which n economies interact through trade and ideas diffuse through
the contact of domestic managers with those who sell goods to the country as well as with
those that produce within the country. Given the results from the previous section, the static
trade theory is given by the standard Ricardian model in Eaton and Kortum (2002), Bernard
et al. (2003), and Alvarez and Lucas (2007), which we briefly introduce before deriving the
equations which characterize the evolution of countries’ knowledge in the world economy.
In each country, consumers have identical preferences over a continuum of goods. We use
ci (s) to denote the consumption of a representative household in i of good s ∈ [0, 1]. Utility
is given by u(Ci ), where the the consumption aggregate is
Z

ci (s)

Ci =

ε−1
ε

ε/(ε−1)
ds

so goods enter symmetrically and exchangeably. We assume that ε−1 < θ, which guarantees
the price level is finite. Let pi (s) be the price of good s in i, so that i’s ideal price index is
1
hR
i 1−ε
1
1−ε
Pi = 0 pi (s) ds
. Letting Xi denote i’s total expenditure, i’s consumption of good s
−ε

is ci (s) = pPi (s)
1−ε Xi .
i
In each country, individual goods can be manufactured by many producers, each using
a labor-only, linear technology (1). As discussed in the previous section, provided countries
share the same exogenous distribution of ideas H(z), the frontier of productivity in each
country is described by a Frechet distribution with curvature θ and a country-specific scale
−θ
λi , Fi (q) = e−λi q . Transportation costs are given by the standard “iceberg” assumption,
where κij denotes the units that must be shipped from country j to deliver a unit of a good
to country i, with κii = 1 and κij ≥ 1.

We now briefly present the basic equations that summarize the static trade equilibrium
given the vector of scale parameters λ = (λ1 , ..., λn ). Because the expressions for price indices,
trade shares, and profit are identical to Bernard et al. (2003), we relegate the derivation of
these expressions to Appendix B.
Given the isoelastic demand, if a producer had no direct competitors, it would set a price
ε
over marginal cost. Producers engage in Bertrand competition. This
with a markup of ε−1
means that lowest cost provider of a good to a country will either use this markup or, if
necessary, set a limit price to just undercut the next-lowest-cost provider of the good.
Let wi denote the wage in country i. For a producer with productivity q in country j, the
w κ
cost of providing one unit of the good in country i is jq ij . The price of good s in country i
is determined as follows. Suppose that country j’s best and second best producers of good
s have productivities qj1 (s) and qj2 (s).12 The country that can provide good s to i at the
lowest cost is given by
wj κij
arg min
qj1 (s)
j
If the lowest-cost-provider of good s for i is a producer from country k, the price of good s
in i is

ε wk κik wk κik
wj κij
pi (s) = min
,
, min
ε − 1 qk1 (s) qk2 (s) j6=k qj1 (s)
That is, the price is either the monopolist’s price or else it equals the cost of the next-lowestcost provider of the good; the latter is either the second best producer of good s in country
k or the best producer in one of the other countries.13
In Appendix B, we show that, in equilibrium, i’s price index is

Pi = B

(
X

)−1/θ
−θ

λj (wj κij )

where B is a constant.14
Let Sij ⊆ [0, 1] be the set of goods for which a producer in j is the lowest-cost-provider
for country i. Let πij denote the share of country i’s expenditure that is spent on goods from
12

In Appendix A we show that the joint distribution of the productivities of the best and second best
producers in a country is Ft12 (q1 , q2 ) = [1 + log Ft (q1 ) − log Ft (q2 )] Ft (q2 ),
q1 ≥ q2 .
13
Note that we have assumed for simplicity that neither consumers nor workers internalize that their
consumption or production decisions may affect the insights they may draw, and thus prices do not reflect
the possibility that idea flows may result from the production or consumption of the good. This assumption
is not innocuous;
in general,
prices depend
on how much
each agent internalizes.

−θ
−θ

ε
ε
ε−1
14 1−ε
1 − ε−1
+ ε−1
Γ 1 − ε−1
.
B
= 1− θ
θ

country j so that πij =
share is

R
s∈Sij

(pi (s)/Pi )1−ε ds. In Appendix B, we show that the expenditure
λj (wj κij )−θ
.
πij = Pn
−θ
k=1 λk (wk κik )

A static equilibrium is given by a profile of wages w = (w1 , ..., wn ) such that labor market
clears in all countries. The static equilibrium will depend on whether trade is balanced and
where profit from producers is spent. For now, we take each country’s expenditure as given
and solve for the equilibrium as a function of these expenditures.
Labor in j is used to produce goods for all destinations. To deliver one unit of good
s ∈ Sij to i, the producer in j uses κij /qj1 (s) units of labor. Thus the labor market clearing
constraint for country j is
XZ

Lj =

s∈Sij

κij
ci (s)ds.
qj1 (s)

Similarly, the total profit earned by producers in j can be written as
Πj =

XZ
i

s∈Sij

wj κij
pi (s) −
qj1 (s)

ci (s)ds.

In Appendix B, we show that these can be expressed as
wj Lj =

θ X
πij Xi
θ+1 i

and
Πj =

1 X
πij Xi
θ+1 i

Under the natural assumption that trade is balanced and that all profit from domestic
producers is spent domestically, then Xi = wi Li +Πi and the labor market clearing conditions
can be expressed as
X
wj Lj =
πij wi Li
i

As a simple benchmark, in the case of same-sized countries and costless trade (Li = Lj
wF T
and κij = 1 for all i,j), the stocks of knowledge completely determine relative wages, wiF T =
i0
1
1
1+θ
1+θ
FT
π
λ
λi
, and trade shares, πijF T = λ j0
.
λi 0
j
ij 0
Given the static equilibria, we next solve for the evolution of the profile of scale parameters

λ = (λ1 , ..., λn ) by specializing (2) for alternative assumptions about source distributions. We
consider source distributions that encompass two cases: (i) domestic producers learn from
sellers to the country, (ii) domestic producers learn from other producers in the country.

2.1

Learning from Sellers

Following the framework introduced in Section 1, we model the evolution of technologies as
the outcome of a process where managers combine “own ideas” with random insights from
technologies in other sectors or countries. We first consider the case in which insights are
drawn from sellers to the country. In particular, we assume that insights are randomly and
uniformly drawn from the distribution of productivity among all managers that sell goods
to a country.15 In this case, the source distribution is given by
Gi (q) =

GSi (q)

≡

s∈Sij |qj (s)<q

As we show in Appendix C, after specializing equation (2) to this source distribution, the
evolution of the scale of the Frechet distribution, i.e., the stock of ideas, is described by
Z
λ̇it = αit

∞

xβθ dGSi (q)

= Γ(1 − β)αit

X
j

πij

λj
πij

β
(3)

where Γ(·) is the Gamma function. That is, the evolution of the stock of knowledge is close
to a weighted sum of trading partners’ stocks of knowledge, where the weights are given by
expenditure shares.
Equation (3) shows that trade shapes how a country learns in two ways. Trade gives a
country access to the ideas of sellers from other countries. In addition, trade barriers affect
which managers are able to sell goods to a country. Trade leads to tougher competition, so
that there is more selection among the producers from which insights are drawn. Starting
from autarky, lower trade barriers make it less likely that low productivity domestic producers
can compete with high productivity foreign producers. The subsequent insights drawn from
these high productivity foreign producers will be better quality than those drawn from the
15

For the case of learning from sellers, the assumption that insights are drawn uniformly from all sellers to
the country is not central. Alternative assumptions, e.g., insights are randomly drawn from the distribution
of sellers’ productivity in proportion either to consumption of each good or to expenditure on each good,
give the same law of motion for the each country’s stock of knowledge up to a constant. See Appendix C.1.1.

low productivity domestic producers.16 Higher trade barriers, on the other hand, lead to
more selection among foreign managers into selling goods to country i. In fact, the less
a foreign country sells to country i, the stronger selection is among its producers. The
average quality of insights drawn from j is given by (λj /πij )β , where λj /πij is an average
of productivity among sellers from j to i. Holding fixed j’s stock of knowledge, a smaller
πij reflects more selection into selling goods to i, which means that the insights drawn from
sellers from j are likely to be higher quality insights.
Nevertheless, the overall quality of insights is not necessarily maximized in the case of
free trade. To optimize the quality of insights a country must bias its trade toward those
countries with more knowledge. In particular, in the short run the growth of country i’s
stock of knowledge is maximized when its expenditure shares are proportional to its trading
partners’ stocks of knowledge.17
λj
πij
=
.
πij 0
λj 0

(4)

In equilibrium, on the other hand, country i’s expenditure shares will satisfy
πij
λj (wj κij )−θ
=
.
πij 0
λj 0 (wj 0 κij 0 )−θ

(5)

Notice that (4) and (5) coincide only if differences in trade costs perfectly offset differences in
trading partners’ wages. Suppose, for example, that trade costs are symmetric. If a country
spends equally on imports from two trading partners, one with a high wage and one with
a low wage, the country would improve the quality of its insights by tilting trade toward
the trading partner with the higher wage. Intuitively, the marginal seller in the high wage
country is more productive–and would generate higher quality insights–than the marginal
seller in the low wage country, as the former must overcome the high wage.18
As discussed before, to obtain growth in the long-run we assume that the arrival rates
of insights grow over time, in which case it is convenient to analyze the evolution of the
16

This mechanism is emphasized by Alvarez et al. (2014).
P 1−β β
P
This is the solution to max{πij } j πij
λj subject to j πij = 1.
18
To be clear, iceberg trade costs are not tariffs (which both distort trade costs and provide revenue), so
the preceding argument does not show that the distorting trade represents optimal policy. However, if the
shadow value of a higher stock of knowledge is positive, a planner that maximizes the present value of a small
open economy’s real income and can set country-specific tariffs would generically set tariffs that are non-zero
and not uniform across countries. Of course, whether free trade is optimal depends on what individuals
are able to internalize; we have assumed that consumers do not internalize that their consumption decisions
affect the quality of insights drawn by managers.
17

detrended stock of knowledge λ̂it = λit e− 1−β t
n

X 1−β β
˙
λ̂it =Γ(1 − β)α̂it
πij λ̂jt −
j=1

γ
λ̂it ,
1−β

(6)

On a balanced growth path where the arrival rate of insights grows at rate γ, the de-trended
stock of knowledge solves the system of non-linear equations
n
X
(1 − β)α̂i
λ̂i =
Γ(1 − β)
πij1−β λ̂βj .
γ
j=1

2.2

(7)

Learning from Producers

Another natural source of ideas is the interaction of technology managers with other domestic
producers, or workers employed by these producers. In this section we consider the case in
which the insights are drawn uniformly from the distribution of productivity among domestic
managers that are actively producing.19 We consider only the case in which trade costs satisfy
the triangle inequality κjk < κji κik , ∀i, j, k such that i 6= j 6= k 6= i. In this case, any manager
that exports also sells domestically.20 The source distribution is
R
Gi (q) = GPi (q) =

s∈Sii |qi1 ≤q

R
s∈Sii

As we show in Appendix C specializing equation (2) to this source distribution, the evolution
of a country’s stock of knowledge is described by
Z

∞

xβθ dGPi (q)
0
β
λi
= Γ(1 − β)αit
πii

λ̇it = αit

Thus, the source distribution of country i is a function of the share of its expenditure on
domestic goods and the domestic stock of knowledge, λi .
How does trade alter a country’s stock of knowledge? In autarky, insights are drawn
from all domestic producers, including very unproductive ones. As a country opens up to
19

When insights are drawn from domestic producers, the assumption that insights are drawn uniformly,
instead of in proportion to the labor used in the production of each good, is more important. See Appendix C.1.1 for a characterization of the dynamics of the stock of ideas under alternative assumptions.
20
To see this, suppose that there were a variety s such that i exports to j and k exports to i. This means
w κjk
w κ
κik
κii
that qii(s)ji ≤ qkk (s)
and wqkk (s)
≤ wqii(s)
. Since κii = 1, these imply that κji κik ≤ κjk , a violation of the
triangle inequality and thus a contradiction.

trade the set of domestic producers improves as the unproductive technologies are selected
out. This raises the quality of insights drawn and increases the growth rate of the stock of
knowledge.21 have the additional feature that producers that drop out may upgrade their
technology by imitating others.
As before, the evolution of a country’s detrended stock of knowledge λ̂it = λit e−γ/(1−β)t
is given by
˙
λ̂it =Γ(1 − β)α̂it

λ̂it
πii

!β
−

γ
λ̂it ,
(1 − β)

(8)

and, on a balanced growth path, these solve the following system of non-linear equations

λ̂i

(1 − β)α̂i
= Γ(1 − β)
γ
1

−

λ̂i
πii

!β

∝ α̂i1−β πii 1−β .

2.3

(9)

Other Specifications of Learning

Variety
An implication of the learning from producers specification is that the rate of increase of a
country’s stock of knowledge grows without bound as the share of its expenditure spent on
domestic goods shrinks to zero. In that case, only the most productive managers would be
able to sell goods domestically, so the insights drawn from these firms would be very high
quality. This causes the frontier of knowledge to increase at a faster rate. Because the arrival
rate of ideas is independent of the mass of producers actively producing, low productivity
firms simply crowd out high quality insights.
An alternative is that a manager would gain more and better insights if she were exposed
to wider variety of production techniques. Suppose that ideas arrive in proportion to the
mass of techniques a manager is exposed to. When insights are drawn from sellers, trade
has no impact on the mass of good consumed, and hence on the variety of sellers one may
draw insight from. On the other hand, when insights are drawn from producers, ideas arrive
in proportion to the mass of domestic producers that are actively producing. This implies
21

This mechanism is emphasized by Sampson (2015) and Perla et al. (2015). Perla et al. (2015)

that a country’s the stock of knowledge evolves as22

λ̇it = Γ(1 − β)αit πii

λi
πii

β
(10)

In contrast to the baseline specification, increased trade–a lower πii –would lower the growth
rate of a country’s stock of knowledge because of the loss of variety in learning. On a
balanced growth path, the detrended stock of knowledge is
1/(1−β)

λ̂i ∝ α̂i

πii .

Targeted Learning
If managers can glean better insights from more productive producers, one might think they
might focus their attention so that insights are drawn disproportionately from those that are
more productive.
Suppose that G̃ represents the distribution of productivity among those from whom a
manager may draw insight. We assume now that the manager can target better insights
by overweighting individual insights. Specifically, the individual can choose a schedule of
arrival rates that accompany each potential insight. Let α̂ (x) be the arrival of insights
from producers with productivity x. The manager chooses {α̂(x)} subject to the constraint
i φ−1
hR
φ
φ
∞
φ−1
≤ α for some φ > 1.23
α̂(x) dG̃(x)
0
R∞
In this case, a country’s stock of knowledge evolves as24 λ̇t = 0 α̂t (x)xβθ dGt (x). An
individual that wants to learn as quickly as possible will thus choose the schedule {α̂(x)} to
solve
Z
φ−1
Z ∞
φ
φ
βθ
max
α̂(x)x dG(x)
subject to
α̂(x) φ−1 dG(x)
≤α
{α̂(x)}

Optimal behavior implies that the change in a country’s stock of knowledge is
Z
λ̇ = α

βθ φ

φ1
dG(x)

In an Eaton-Kortum framework, πii is both the share of i’s spending on domestic goods and the fraction
of varieties produced domestically. Thus in the baseline, the arrival rate is αit , whereas in (10) the arrival
rate is αit πii .
23
The case of φ = 1 would correspond to the baseline model, in which case the constraint could be written
as supx α̂(x) ≤ α.
24
More formally, with the functional form assumptions, in
as m → ∞, the law of motion for F
the limit

R∞
R∞
−θ
1/θ
βθ
will be d ln Ft (q) = 0 α̂t (x)x dG(x) so that as t → ∞, Ft λt q → e−q with λ̇t = 0 α̂t (x)xβθ dGt (x).

With learning from producers, the change in a country’s stock of knowledge is
"
λ̇i = Γ(1 − βφ)αi

λi
πii

βφ # φ1

= Γ(1 − βφ)αi

λi
πii

For each specification, learning is faster when learning is more targeted (Γ(1 − βφ) is increasing in φ).25
With learning from sellers, the change in a country’s stock of knowledge is
"
λ̇i = Γ(1 − βφ)αi

πij

λj
πij

βφ #1/φ

As we show in Appendix F, an environment with a higher φ is quantitatively very similar to
an environment with higher β.26

Gains from Trade

As in other gravity models, a country’s real income and welfare can be summarized by its
stock of knowledge (or some other measure of aggregate productivity), its expenditure share
on domestic goods, and the trade elasticity:
wi
∝
yi ≡
Pi

λi
πii

1/θ
(11)

In our model gains from trade have a static and dynamic component. The static component,
holding each country’s stock of knowledge fixed, is the familiar gains from trade in standard
Ricardian models, e.g., Eaton and Kortum (2002).27 The dynamic gains from trade are the
ones that operate through the effect of trade on the flow of ideas.
In this section we consider several simple examples that illustrate the determinants of
the static and dynamic gains from trade, both in the short and long run. We first consider
an example of a world with symmetric countries. We study both the consequences of a
simultaneous change in common trade barriers as well as the case of a single deviant country
that is more isolated than the rest of the world. We also study how a small open economy
25

Conditioning on a value of α, learning is faster. However, conditioning on a calibration target, the faster
learning plays no role because α would be recalibrated to absorb the change.
26
Note that for either specification, a finite growth rate of the stock of knowledge requires φ < 1/β. This
limits how directly a manager can target the producers with the highest productivity.
27
See Arkolakis et al. (2012) for other examples.

responds when its trade barriers change, a case that admits an analytical characterization.28

3.1

Gains from Trade in a Symmetric Economy

Consider a world with n symmetric countries in which there is a common iceberg cost κ of
shipping a good across any border. In a symmetric world, the share of a country’s expenditure
1
on domestic goods is πii = 1+(n−1)κ
−θ , while the share of its expenditure on imports from
1−πii
each trading partner is n−1 . Specializing either equation (7) or equation (9), each country’s
de-trended stock of knowledge on a balanced growth path is
1
"
1

1−β
1−β # 1−β
1
−
π
(1 − β)α̂
ii
πii1−β + (n − 1)
Sellers : λ̂i =
Γ(1 − β)
. (12)
γ
n−1
1
1−β

− β
(1 − β)α̂
Γ(1 − β)
πii 1−β
(13)
Producers : λ̂i =
γ

The de-trended real per-capita income is obtained by substituting these into equation (11)
"
Sellers : ŷi ∝ 1 + (n − 1)β
−

1 − πii
πii

1 1
1−β # 1−β
θ

(14)

Producers : ŷi ∝ πii 1−β θ

(15)

Using these equations, we can ask how a change in trade costs would impact countries’
real incomes. It is instructive to compare three cases: the static case in which β = 0, learning
from sellers, and learning from producers. For each, we can summarize how countries’ real
incomes change with trade patterns by looking at the elasticity of real income with respect
to the share of expenditures spent on domestic goods.29
Holding a country’s stock of knowledge fixed, the change in real income arising from a
28

In Appendix G we explore an example with a richer geography in which trade barriers generate a coreperiphery structure. We examine how the dynamic gains from trade determines the gap in income between
core and periphery countries.
29
A convenient feature of the symmetric example is that, since every country has the same stock of
1
knowledge and the same wage, the share of a country’s expenditure on domestic goods is 1+(n−1)κ
−θ . Thus
the change in κ causes the same change in trade shares whether stocks of knowledge are held fixed, insights
are drawn from sellers, or insights are drawn from producers. In a world with asymmetric countries, the
a change in trade barriers (κ) would cause different changes in trade shares in each version of the model.
However, as we will show below, the overarching message—that when insights are drawn from producers the
dynamic gains amplify the static gains whereas when insights are drawn from sellers the dynamics gains are
largest when countries are close to autarky—will remain.

changing trade barriers depends only on the trade elasticity:
d ln yi
d ln πii

=−
λi fixed

1
θ

For each of the two specifications of learning, we can summarize the elasticity of real income
to the domestic expenditure share:
Sellers :

d ln ŷi
= − 1−β
dπii
π
ii

Producers :

1−πii β
n−1

d ln ŷi
1 1
=−
d ln πii
1−βθ

1
+ (1 − πii ) θ

(16)
(17)

Both the static and dynamic gains depend on the curvature of the productivity distribution,
θ; a higher θ corresponds to thinner right tails. With higher θ, there are fewer highly
productive producers abroad whose goods can be imported, and there are fewer highly
productive producers from whom insights may be drawn. The novel parameter determining
the gains from trade is β. The parameter β controls the importance of insights from others in
the quality of new ideas and hence the extent of technological spillovers associated with trade.
With higher β, insights from others are more important, and therefore, more is gained by
being exposed to more productive producers. This can be seen most clearly be comparing
autarky to costless trade. Equations (14) and (15) reveal that for either specification of
learning, the ratio of real income under costless trade (πii = 1/n) to real income under
1 1
autarky (πii = 1) is n 1−β θ . In the limit as β goes to 1, the gains from trade relative to
autarky grow arbitrarily large.30
For several values of β, the top panels of Figure 1 illustrate the common value of each
country’s stock of knowledge relative to its level under free trade. The bottom panels show
the corresponding real income per capita. The left (right) panels focus on the specification
of learning in which insights are drawn from sellers (domestic producers). As a benchmark,
the dotted line represents β = 0, which corresponds to the static trade model of Eaton
and Kortum (2002). As trade costs rise, countries become more closed and their stocks of
knowledge decline. When β is larger, the dynamic gains from trade are larger.
When insights are drawn from domestic producers, the gains from trade simply amplify
the static gains. The diffusion parameter β determines the strength of the amplification.
One way of interpreting equation (17) is that the diffusion of ideas causes the static gains
30

These limiting cases are close to the models analyzed by Alvarez et al. (2014), Sampson (2015), and
Perla et al. (2015). When β = 1, the steady state gains from moving from autarky to free trade are infinite
because integration raises the growth rate of the economy. In contrast, for any β < 1, integration raises the
level of incomes but leaves the growth rate unchanged.

Lerning from Sellers
Stock of Ideas, 6 1/3

0.8

0.6

0.4

0.4
-

0.2
0

=0
= 0.1
= 0.5
= 0.9

0.5
openness, 1-:ii

0.8

0.6

0.4

0.2

0.5
openness, 1-:ii

=0
= 0.1
= 0.5
= 0.9

0.5
openness, 1-:ii

per-capita income

0.8

0.2

per-capita income

Learning from Producers
Stock of Ideas, 6 1/3

0.5
openness, 1-:ii

Figure 1: Gain from Reducing common trade barriers
Note: This figure shows each country’s stock of ideas and per capita income relative to
their values under costless trade.

from trade to compound itself. The expression for the static and dynamic gains from trade
shares features with an analogous expression in a static world in which production uses
intermediate inputs.31
When insights are drawn from sellers, the dynamic gains from reducing trade barriers
are qualitatively different from the static gains. The dynamic gains are largest when the
world is relatively closed, whereas the static gains are largest when the world is relatively
open. This can be seen from the left panels of Figure 1 but also by inspecting the limiting
values of (16). As the world becomes more open, the total gains from reducing trade barriers
corresponds to the static gains, limπii →1/n ddlnlnπŷiii = − 1θ . In contrast, as the world becomes
more closed, the marginal dynamic gains grow arbitrarily large, limπii →1 ddlnlnπŷiii = −∞. Put
31

In a world with roundabout production, a decline in trade costs reduces the costs of production, lowering
the cost of intermediate inputs, which lowers the cost of production further, etc. Here, when trade costs
decline, producers draw better insights from others, raising stocks of knowledge, and this improves the quality
of insights others draw, etc. The parameter β gives the contribution of an insight to a new idea, just as the
share of intermediate goods measures the contribution of the cost of intermediate inputs to marginal cost.

differently, when the economy is relatively open, the total gains from reducing trade barriers
are composed mostly of the static gains, whereas when the world is relatively closed, the
total gains are composed mostly of the dynamic gains.
To understand this, consider a country close to autarky. If trade costs decline, the
marginal import tends to be made by a foreign producer with high productivity. While the
high trade costs imply that the static gains from trade remain relatively small, the insights
drawn from this marginal producer tends to be of high quality. In contrast, for a country
close to free trade, the reduction in trade costs leads to large infra-marginal static gains from
trade, but the insights drawn from the marginal producers are likely to be lower quality.
In contrast, when insights are drawn from domestic producers, the dynamic gains from
reducing trade barriers are largest when the world is already relatively open, as shown clearly
in the bottom right panel.

3.2

Asymmetric Economies

What is the fate of a single country that is more open than others? Or one that is closed
off from world trade? This section studies gains from trade in an asymmetric world in two
simple ways. We first describe how trade costs affect real income of a small open economy.
We then return to example of a symmetric world discussed in the previous section but with
a single “deviant” country that is more isolated.
Consider first a small open economy that is small in the sense its actions do not impact
other countries’ stocks of knowledge, real wages, or expenditures. Let i be the small open
economy, and suppose that all trade costs take the form of κij = κκ̃ij and κji = κκ̃ji for
j 6= i. To a first order the long-run impact of a change in κ on country i’s real income is
Sellers:
Producers:

d log yi
= − 1−Ω β
i
d log κ

1 + 2θ

πii +θ(1+πii )
1−Ωi (1−β)+β(1−πii )

d log yi
1 + 2θ
= −
π
ii )
d log κ
(1 − β) ii +θ(1+π
+1
1−πii

π 1−β λβ

where Ωi ≡ P iiπ1−βiλβ .
j ij
j
When insights are drawn from sellers, the term Ωi is the share of the growth in i’s stock of
knowledge that is associated with purchasing goods from i. One implication is that, holding
fixed πii , the response of real income to a decline in trade costs is larger when Ωi is smaller.
In words, this means that, among small open economies with the same trade shares, the
response of real income to trade will be larger when the country relies more on others for
growth in its stock of knowledge. For example, a country with a low stock of knowledge will
21

rely more on others for good quality insights. When such a country reduces trade barriers,
the impact on income is larger, a manifestation of catch-up growth.
A second implication of the claim is that for both specifications of learning, the dynamic
gains from trade are always weakly positive. The static gains can be can be found by
evaluating either expression at β = 0.
Finally, when insights are drawn from sellers, in the limiting model as β approaches one,
the dynamic gains from lower trade barriers approaches zero.32 This may seem puzzling; as
the contribution of insights from others in the productivity of new ideas becomes larger and
the model approaches one of pure diffusion, the dynamic gains from trade become relatively
unimportant.
To resolve this, it will be useful to plot the the income of the single deviant country for
various values of β. Consider now an asymmetric version of the model with n − 1 open
countries, i = 1, ..., n − 1, and a single deviant economy, i = n. The n − 1 open countries
can freely trade among themselves, i.e., κij = 1, i, j < n, but trade to and from the deviant
economy incurs transportation cost, i.e., κnj = κjn = κn ≥ 1, j < n.33
The top panels of Figure 2 show how the deviant country’s stock of knowledge changes
with the degree of openness. The x-axis measures openness as the fraction of a country’s
spending on imports, 1 − πii . On the y-axis we report the stock of knowledge relative to
what it would be if trade were costless (κn = 1). The bottom panels show the corresponding
real incomes. The different lines correspond to alternative values of β, which controls the
importance of insights from others. The solid line shows the effect of openness in the case
with no spillovers, β = 0, which corresponds to the standard static trade theory of Eaton
and Kortum (2002). The other two curves show cases with positive technological spillovers.
The left panels correspond to learning from sellers while the right panels show learning from
domestic producers.
Across balanced growth paths, as the deviant economy becomes more isolated its stock
of knowledge contracts relative to that of the balanced growth path of n economies engaging
in costless trade. As discussed earlier, trade costs have effects on per-capita income beyond
the static gains from trade.
Figure 2 replicates the curious feature that if the deviant economy is moderately open,
32

To see this, simply evaluate the expression at β = 0 and β = 1 and note that β = 0 implies Ωi = πii .
In the numerical examples that follow, we consider a world with n = 50 economies with symmetric
populations, so that each country is of the size of Canada or South Korea. We set θ = 4, the curvature of the
Frechet distribution, which also equals the tail of the distribution of exogenous ideas. This value is in the
range consistent with estimates of trade elasticities. See Simonovska and Waugh (2014), and the references
therein. Given a value of β, the growth rate of the arrival rate of ideas is calibrated so that on the balanced
γ
growth path each country’s TFP grows at 1%, (1−β)θ
= 0.01. The parameter α̂ is normalized so that in the
case of costless trade, κn = 1, the de-trended stock of ideas equals 1.
33

1.2

Learning from Sellers, 6 1/3
n

Learning from Producers, 6 1/3
n
1.2

0.8

0.6

0.4

0.4
-=0
- = 0.5
- = 0.9

0.2
0
0.01

0.1

0.2
0
0.01

0.5 1

0.8

0.6

0.4

0.2

0
0.01

0.1
0.5 1
deviant openness, 1-:nn

0.1

0.5 1

0.1
0.5 1
deviant openness, 1-:nn

Figure 2: The Stock of Knowledge and Per Capita Income of the Deviant Economy.
Note: The figure plots a deviant country’s stock of knowledge (top panels) and percapita income (bottom panels) relative to what it would be under costless trade for the
cases when insights are drawn from sellers (left panels) and domestic producers (right
panels). Each curve is continuous, and the dot along the left axis is the value in autarky,
the limit as πii → 1.

the gains from lowering trade costs are small if the model is close to one of pure innovation
(β = 0) or close to one of pure diffusion (β = 1). Those two models differ however, in
the gains relative to autarky. For β = 0.9, when a country moves from autarky to only
slightly open, the dynamic gains from trade are quite large. But any subsequent lowering of
trade costs has a relatively small impact on the country’s stock of knowledge. It is only for
intermediate values of β that lowering trade barriers would have a larger dynamic impact
on a country’s stock of knowledge for a wide range of trade shares.
Why are the dynamic gains from trade concentrated near autarky when β is close to
one? The reason is the concavity generated by β in combining insights from others with
the exogenous components of ideas. When β is large, the difference between a high and low
quality insight is magnified, and a country’s growth depends much more heavily on insights
from the most productive producers. When a country is only slightly open, it is already
23

importing goods from most of the highest productivity foreign producers. Indeed, as β → 1
as long as the deviant country is even slightly open, its stock of knowledge is as high as it
would be under costless trade.34 This feature of the model will be especially important for
understanding our quantitative results when we study the implications of actual changes in
trade volumes.

3.3

Trade Liberalization

We now study how a country’s stock of knowledge and real income evolve when it opens
to trade. Does the country experience a period of protracted growth or does it converge
relatively quickly?
Consider a world economy that starts with n − 1 open economies and a single deviant
economy that are on a balanced growth path. Figure 3 shows the evolution of the real income
in the initially deviant economy following a trade liberalization. The left panel shows an
example in which the deviant country is initially in autarky and the n − 1 open economies
trade costlessly. For each of the two learning specifications, the figure traces out the real
income in the deviant country.35 The paths of the (de-trended) stocks of knowledge solve the
differential equations in (6) and (8), depending on whether insights are drawn from sellers
or producers.36 On impact real income jumps as it would in a static model. Over time, the
deviant country’s stock of knowledge improves. When insights are drawn from sellers, real
income converges more quickly to the steady state; a trade liberalization gives immediate
access to insights from goods sold by high productivity foreign producers. In contrast, when
insights are drawn from domestic producers, the insights are initially low quality, although
they become more selected, and only gradually improve as the country’s stock of knowledge
increases.37
34

Alvarez et al. (2014) analyzed an economy similar to the limit point β = 1. In particular, their Proposition 7 and 8 show that the behavior of the tail of the distribution of productivity is independent of trade
costs, as long as they are finite.
35
We use this extreme example of a liberalization from autarky to costless trade because these are two
special cases in which both specifications of learning predict the same stocks of knowledge. This makes it
easier to contrast the speed of convergence across the two specifications.
36
We set β = 0.5. The rest of the parameters follow the calibration in footnote 33.
37
We can get a more general version of this result for a small open economy in a world with arbitrary trade
barriers. Log-linearizing around a balanced growth path, let y̌i denote the log deviation of i’s detrended real
income from its long run value and let variables with no decoration denote their long run values. The speed
of convergence of a small open economy is

Ωi − πii
β
d
log y̌i = −γ 1 −
+
(1 − Ωi )
Sellers :
dt
1 + θ (1 + πii ) 1 − β

d
β
1 − πii
Producers :
log y̌i = −γ 1 +
dt
1 − β 1 + θ (1 + πii )

The right panel of Figure 3 shows a more empirically relevant example of a world where
trade costs are such that imports comprise 5% of expenditures for the deviant country and
20% of expenditures for n − 1 open countries. At time zero, trade costs for the deviant
country fall enough so that its import share eventually rises to 20%. Each curve shows real
income relative to the new symmetric balanced growth path when insights are drawn from
sellers. In line with previous results, for intermediate values of β, the total increase in income
is larger.
In addition, the change in the deviant country’s stock of knowledge leads to a protracted
transition as the dynamic gains from trade are slowly realized. Ten years after the liberalization, only around 50% of the dynamic gains from trade have been realized.
1

Autarky to Costless Trade

0.8

1-: n = 0.05 ! 0.20

1
0.95

0.6
0.9
0.4
0.2
0

from sellers
from producers

20
years

0.85
0.8

- = 0.20
- = 0.50
- = 0.80

20
years

Figure 3: Dynamics Following a Trade Liberalization.
Note: This figure shows the evolution of real income for a deviant country whose trade
barriers suddenly fall. The left panel compares the predictions of each specification of
learning when the deviant country moves from autarky costless trade. The right panel
studies learning from sellers and compares the predictions for several values of β when
trade costs fall enough to shift the deviant country’s import share from 5% to 20%. Each
curve shows real income relative to the new symmetric balanced growth path.

Quantitative Exploration

We now explore the ability of the theory to account for the evolution of the distribution of
productivity (TFP) across countries in the post-war period. With this in mind, we extend the
where Ωi ≡

π 1−β λβ
P ii 1−βi β
λj
j πij

is the share of i’s insights drawn from i. From these expressions, one can infer both

that convergence is faster when diffusion is more important (β is larger) and that the speed of convergence
does not depend on αi . Convergence is faster with learning from sellers unless Ωi is significantly larger than
than πii , a case in which i’s stock of knowledge is much larger than that of its trading partners.

simple trade model introduced in Section 2 to incorporate intermediate inputs, non-traded
goods, and a broader notion of labor which we refer to as equipped labor. In addition, to
simplify the exposition, we focus on the case in which insights are drawn from sellers to a
market. This version of the model has particularly rich testable implications and, as we show
in this section, it provides a promising quantitative theory of dynamic gains from trade.38

4.1

Extended Trade Model

Suppose now that a producer of good s in each country i with productivity q has access to
a constant returns to scale technology combining an intermediate input aggregate (x) and
equipped labor (l)
yi (s) =

η η (1

1
qxi (s)η li (s)1−η
− η)1−η

All goods use the intermediate good aggregate, or equivalently, the same bundle of intermediate inputs. The intermediate input aggregate is produced using the same technology as
the consumption aggregate, so that the market clearing condition for intermediate inputs for
i is
Z

Z
xi (s)ds =

χi (s)

1−1/ε

ε/(ε−1)
ds

where χi (s) denotes the amount of good s used in the production of the intermediate input aggregate. Equipped labor L is produced with an aggregate Cobb-Douglas technology
requiring capital and efficiency units of labor
Z
Li =

li (s)ds =

ζ ζ (1

1
K ζ (hi L̃i )1−ζ .
− ζ)1−ζ i

In our quantitative exercises we take an exogenous path of aggregate physical and human
capital, Ki and hi , from the data. We thus abstract from modeling the accumulation of
these factors, and hold them constant in counterfactuals.39
38

In Appendix F we present results for three alternative cases. We extend the learning from sellers
specification to include targeted and study two cases in which insights are drawn from domestic producers,
either uniformly or in proportions to labor used.
39
Implicitly, we are assuming that individual technologies are
y(s) =

1
qx(s)η [k(s)ζ (hi l(s))1−ζ ]1−η
η η (1 − η)1−η ζ (1−η)ζ (1 − ζ)(1−η)(1−ζ)

and that investment can be produced with the same technology as the consumption and intermediate input
aggregates.

In addition to the iceberg transportation costs κij , we assume that a fraction 1 − µ of
the goods are non-tradable, i.e., this subset of the goods face infinite transportation costs.
The main effect of introducing non-traded goods is that in the extended model the value of
the elasticity of substitution ε affects equilibrium allocations. In Appendix E we present the
expressions for price indices, trade shares, and the evolution of stocks of knowledge for this
version of the model.

4.2

Calibration

We need to calibrate seven common parameters, (θ, η, ζ, µ, γ, ε, β), and two sets of parameters
that are country and time specific, the matrix of transportation costs Kt = [κint ] and the
vector of arrival rates αt = (α1t , ..., αnt ).
We set θ = 4. This value is in the range consistent with estimates of trade elasticities.40
We set η = 0.5 and ζ = 1/3 to match the intermediate share in gross production and the
labor share of value added. We consider a share of tradable goods µ = 0.5. We set ε = 1,
but note that alternative values do not affect the results significantly.
Following the strategy in Waugh (2010), we show in Appendix E that given values for θ,
µ, and ε as well as data on bilateral trade shares over time, the iceberg cost of shipping a
tradable good to country i from country j at time t is

κijt =

pit
pjt

1 − πiit Zit
πijt 1 − Zit

θ1



 (1 − µ) +
(1 − µ) +

where Zit solves41

− ε−1
µZit θ
− ε−1
µZjt θ

1
 ε−1



1− ε−1
θ

πiit =

(1 − µ) + µZit
(1 − µ) +

− ε−1
µZit θ

To operationalize this equation, we use bilateral trade data for 1962-2000 from Feenstra et al.
(2005) and data on real GDP and the price index from PWT 8.0.42
To assign values to the vector of arrival rates α̂t = (α̂1t , ..., α̂nt ) we proceed in two steps.
Given the evolution of trade flows summarized by Zit , we compute, in each year, the stocks
40

See Simonovska and Waugh (2014) and the references therein.
Zi can be interpreted as the share of i’s expenditure on tradable goods spent on domestic tradables,
−θ
(pη w1−η ) λi
−1/θ
equal to P iη i1−η
. Zi
is also the price index of non-tradables relative to that of tradable goods
−θ
p
w
κ
λ
(
)
ij
j
j
j j
in i.
42
In particular, we measure real GDP using real GDP at constant national prices (rgdpna). We scale the
real GDP series for each country so that its value in 1962 coincides with the real GDP measure given by the
output-side real GDP at chained PPPs (rgdpo). We measure the price index using the price level of cgdpo
(pl gdpo), where cgdpo is the output-side real GDP at current PPPs.
41

of knowledge needed to match each country’s TFP using
h

λ̂it ∝ (1 − µ) +

− ε−1
µZit θ

(1−η)θ
−θ
i ε−1
wit
Pit

This is a generalization of equation (11) for the model with intermediate inputs and nontraded goods. We measure TFP in the data as a standard Solow residual using real GDP,
physical capital (K), employment (emp), and average human capital (h) from the PWT 8.0,
i.e., T F P = real GDP/[K 1/3 · (emp · h)2/3 ].43
Given the evolution of trade flows and stocks of knowledge as well as values for β and γ,
we back out sequences of arrival rates of ideas using the law of motion of stocks of knowledge

λ̂it+1 ∝ α̂it





(1 − µ) λ̂βit + µ Zit1−β λ̂βit +



X
j6=i

πijt
1−πiit
1−Zit

!1−β



γ
λ̂βjt  −
λ̂
 1 − β it

(18)

This is a discrete time generalization of (3) for the extended model. The sequence of arrival
rates of ideas are the residuals needed to explain the evolution of TFP between 1962 and 2000
given the dynamics of trade costs. For some counterfactuals we hold country-specific arrival
rates of ideas constant at their initial levels. When doing this, we assume that countries were
on a balanced growth path in 1962, and assign to each country the arrival rate of ideas α̂i,0
to exactly match the stocks of knowledge in 1962 when specializing (18) to a steady state,
i.e., λ̂it+1 = λ̂it = λ̂i1962 .
We are left with two parameters to calibrate: the strength of the diffusion, β, and the
growth rate of the arrival rate of ideas, γ. Rather than taking a strong stand on the value
of the diffusion parameter, β, we explore how well the model can quantitatively account
for cross-country income differences and the evolution of countries’ productivity over time
for alternative values of β ∈ (0, 1). That being said, it is useful to discuss a simple (albeit
heroic) strategy to calibrate this parameter. On a balanced growth path the growth rate of
productivity is (1/θ)(λ̇/λ) = γ/(θ(1 − β)). Identifying the growth rate of the arrival of ideas
with the average growth rate of population in the US between 1962 and 2000, γ = 0.01,
43

In any calibration of the model, we must take a stand on how to apportion a country’s TFP into a stock of
knowledge, which may generate idea flows, and other factors, such as allocational efficiency, that are unlikely
to diffuse across borders. Our baseline calibration assumes that physical and human capital differences are
unlikely to diffuse across borders, but that after controlling for those, all residual TFP differences are due
to differences in the stocks of knowledge and trade barriers. In Appendix F we consider an alternative
calibration strategy. We project log T F P onto on R&D intensity, the log of the human capital stock and the
log of an import-weighted average of trading partners’ TFP. We assign the residual TFP from this regression
to a neutral productivity terms affecting the units of equipped labor and not the stock of knowledge, and
choose stocks of knowledge to match predicted TFP from the regression. There we show that the results in
this section are robust to the alternative calibration strategy.

assuming that the growth rate of productivity on a balanced growth path equals the average
growth rate for the US between 1962 and 2000 (0.8% per year), the value of θ = 4 implies
an approximate value for the diffusion parameter β = 0.7.44 When we consider alternative
value of the diffusion parameter β, we recalibrate γ to match an average growth rate of TFP
in the US of 0.8 percent.

4.3

Sample Selection

The sample of countries in our quantitative analysis consists of a balanced panel of countries
that is obtained by merging the PWT 8.0 with the NBER-UN dataset on bilateral trade flows
from 1962 to 2000. We further restrict this sample to those countries with a population above
1 million in 1962 and oil rents that are smaller than 20% of GDP in 2000. We exclude Hong
Kong, Panama and Singapore, as these are countries where re-exports play a very large role.
The final sample consists of 65 countries.45

4.4

Reduced Form Evidence

Before discussing the results from the calibrated model we present suggestive reduced form
evidence of the mechanisms emphasized by the theory. We start by discussing cross-sectional
evidence in 1962, the first year of our sample.
Given the arrival rate of ideas in a country, the theory predicts that the main drivers of
a country’s TFP are its openness and the TFP of its trading partners. The first panel of
Figure 4 shows that countries that are less open (high πii ) tend to have lower TFP, although
this relationship is not statistically significant. The second panel shows the relationship
between the TFP of a country’s trading partners and its own TFP. In particular, for each
country
we compute an import weighted average of a country’s trading partners’ TFP:
P
j6=i πij T F Pj
. The figure shows that countries with more productive trading partners tend to
1−πii
be (statistically significantly) more productive.
44
This discussion ignores the impact that changes in world openness had on the TFP growth of the US.
If we take this into account, the diffusion parameter that is consistent with the observed growth of TFP in
the US between 1962 and 2000 is approximately β = 0.6.
45
Argentina, Australia, Austria, Belgium-Luxemburg (we consider the sum of the two countries, as the
UN-NBER trade data is reported only for the sum), Bolivia, Brazil, Cameroon, Canada, Chile, China,
Colombia, Costa Rica, Cote d‘Ivoire, Denmark, Dominican Republic, Ecuador, Egypt, Finland, France,
Germany, Ghana, Greece, Guatemala, Honduras, India, Indonesia, Ireland, Israel, Italy, Jamaica, Japan,
Jordan, Kenya, South Korea, Malaysia, Mali, Mexico, Morocco, Mozambique, Netherlands, New Zealand,
Niger, Norway, Pakistan, Paraguay, Peru, Philippines, Portugal, Senegal, South Africa, Spain, Sri Lanka,
Sweden, Switzerland, Syria, Taiwan, Tanzania, Thailand, Tunisia, Turkey, Uganda, United Kingdom, United
States, Uruguay, and Zambia.

log TFP

0.5

−0.5

−1

−1.5
−2

−1.5
slope = −1.38
(1.24)

−2.5
−0.2 −0.15 −0.1 −0.05
log πii

slope = 1.87
(0.60)

−2
0

−2.5
−0.4

−0.3

−0.2

−0.1

log(πi,−i TFP−i/(1−πii))

Figure 4: Cross-sectional TFP differences in 1962
Note: The first panel shows the cross-sectional relationship between (lack of) openness,
as measured by countries’ expenditure shares on domestic goods, and TFP. The right
panel shows the cross-sectional relationship between the each country’s TFP and its
exposure to other high TFP trading partners, as measured by an import-weighted average
of trading partners’ TFP. In each panel we report the slope of the regression line and its
standard error in parenthesis.

Over time, among the many factors that would alter a country’s productivity, the model
emphasizes changes in openness, changing exposure to trading partners, and changes in
trading partners’ productivity. Figure 5 shows some simple reduced form patterns in the
data.
The first panel shows the relationship between changes in openness and changes in TFP.
Consistent with the model, countries that increased expenditures on imports tended to have
(statistically significantly) larger increases in TFP.
The second panel shows the association between the change in countries composition of
expenditures and TFP growth. For each country, we compute the changes in exposure to
P
trading partners with high initial TFP. Specifically, for country i we compute j (πij2000 −
πij1962 )T F Pj1962 . Consistent with the theory, there is a clear pattern that countries that
increased import exposure to trading partners with high initial productivity saw (statistically
significantly) larger increases in TFP.
The third panel shows that countries whose trading partners became more productive
tended to see increases in TFP. While this relationship is consistent with the model, it is
fairly weak and statistically insignificant.

∆ log TFP

1.5

0.5

−0.5

−1
−1.5

slope = −2.26
(0.69)

−0.2 −0.1

∆ log πii

−1
0

0.1

slope = 11.99
(2.87)

−1.5
−0.04−0.02

0.02 0.04

(∆ Π) TFP0

−1
−1.5
0.2

slope = 0.30
(0.67)

0.4

Π0 ∆ log TFP

0.6

Figure 5: Openness and Changes in TFP, 1962-2000
Note: The first panel shows the cross-sectional relationship between changes in countries’
TFP and changes in (lack of) openness, as measured by the change in expenditure share
on domestic goods. The second panel shows the cross-sectional relationship between
changes in countries’ TFP and changes in countries’ exposure to trading partners who
had high TFP in 1962, where exposure is an import-weighted average. The third panel
shows the cross-sectional relationship between changes in countries’ TFP and changes in
trading partners’ TFPs, weighted by expenditure shares in 1962. In each panel we report
the slope of the regression line and its standard error in parenthesis.

4.5

Explaining the Dynamics of TFP

Motivated by the reduced form evidence, this section studies the ability of the model to
account for the evolution of productivity over time. We begin our quantitative analysis
by comparing the static and dynamic gains from changes in trade costs. As discussed in
Section 4.2, we use expenditure shares to back out the evolution of bilateral iceberg trade
costs over time. Under the stark assumption that, given trade costs, each country was on
its balanced growth path in 1962, we can find the set of country-specific arrival rates {α̂i }
that perfectly match the cross-section of productivity. Finally, we ask: if the arrival rates
of ideas had remained constant over time and only trade costs changed, what would each
country’s TFP be in 2000?
Figure 6 compares the predicted change in TFP from the model to that of the data for
two calibrations of the diffusion parameter, β. Each point represents a country, and each
panel contains a regression line through the observations and a dashed 45-degree line. If the
model perfectly predicted each country’s TFP growth, each dot would be on the (dashed) 45
degree line. The red regression line in each panel provides a simple measure of the average
ability of the theory to account for cross country differences in TFP growth. The first panel
shows the predicted changes in TFP when β = 0 so that there are no dynamic gains from
31

" log TFP | , t=, 0

0.6

-=0

0.6

0.4

0.2

-0.2
-1

" log TFP, data

- = 0.7

" log TFP, data

Figure 6: Trade and the TFP Dynamics, 1962-2000
Note: Each panel plots countries’ actual changes in TFP against the predicted change in
TFP of the model if only trade costs change. We compute this counterfactual under the
assumptions that the arrival rates are heterogenous across countries, that each country
was on its balanced growth path in 1962, and that arrival rates have remained constant
since 1962. The first panel assumes that β = 0. The second panel assumes β = 0.7. In
addition, each figure plots a dashed 45-degree line and a red regression line.

trade. The model predicts only small changes in TFP, consistent with small static gains
from trade. In the second panel β is set to 0.7, the value implied by the simple calibration
discussed at the end of Section 4.2. In this panel the regression line is more upward sloping,
indicating a stronger relationship between the predicted and actual changes in TFP.
To get a sense of what is driving the model’s predictions, we can decompose the predicted
TFP changes into various components. Each of the top panels in Figure 7 displays the
changes in countries’ TFP on the x-axis and some measure of the model’s predicted changes
in TFP when β = 0.7 and insights are drawn from sellers on the y-axis. In the model,
each country’s TFP is a function of its stock of knowledge and its expenditure on domestic
goods, T F P (λi , πii ). In turn, each country’s stock of knowledge is a function of others’
stocks of knowledge and import shares, λ ({λjt }j6=i,t≥0 , {Πt }t≥0 ), where Π = {πij }i,j=1,...,N
is the matrix of trade shares. The top left panel shows the static effects of changes in
trade costs, d ln T F P (λit (λ−i0 , Π0 ), πiit ) = ln T F P (λi0 , πiit ) − ln T F P (λi0 , πii0 ), where each
country’s stock of knowledge is held fixed at its initial level. Countries that saw an increase
in the trade share tended to increase TFP more. The top middle and right panels show the
contribution of the two drivers of dynamic gains from trade. The top middle panel holds
fixed trading partner’s stocks of knowledge, but allows the dynamic gains from trade through
changing exposure to different trading partners. The top right panel holds fixed trade shares
32

" log TFP

-0.1
-1

0
1
dlog(TFP), data

ii0
0

0.2

-it

0.2

0.3

0.1

0.3

dlog(TFP(6 (6 ,& ),: ))

0.2

dlog(TFP(6it (6-i0 ,&t ),:ii0 ))

dlog(TFP(6i0 ,:iit ))

0.3

0
-0.1
-1

0
1
dlog(TFP), data

0.1
0
-0.1
-1

0.5

slope = -0.85
(0.11)

-0.5
-0.3 -0.2 -0.1

" log :ii

-0.5
-0.04

slope = 3.55
(0.49)

(" &) TFP0

0.04

-0.5

0
1
dlog(TFP), data

slope = -0.59
(1.19)

0.05

0.1

&0 " log TFP

Figure 7: Decomposition of Changes in TFP and Reduced Form Relationship between Openness and TFP Growth in Model Generated Data, 1962-2000
Note: The top left panel plots changes in TFP against predicted changes in TFP .
The top panels plot actual changes in TFP against the various components of predicted
changes in TFP, under the assumption that learning is from sellers, β = 0.7, and the
arrival rate of ideas is kept at its 1962 value, αit = αi0 . The top left panel hold fixed
all countries stocks of knowledge. The other top panels allow each country’s stock of
knowledge to evolve, but hold fixed initial trade shares in computing TFP. In the top
middle panel, learning is such that each country’s trading partners’ stocks of knowledge
are held fixed at their initial levels, but trade shares evolve. In the top right panel,
learning is such that trading partners’ stocks of knowledge evolve but initial trade shares
are held fixed at their initial levels. The bottom panels reproduce the reduced form
evidence in Figure 5, using model generated data when β = 0.7, and the arrival rate of
ideas is kept at its 1962 value, αit = αi0 .

but allows the dynamic gains from trade through changes in trading partners’ productivities.
Consistent with Figure 5, changes in exposure to trading partners plays an important role,
but changes in trading partners productivities plays almost no role.
The bottom panels of Figure 7 present the reduced form relationship between openness
and changes in TFP, using model generated data when β = 0.7, and the arrival rate of
ideas is kept at its 1962 value, αit = αi0 . The model reproduces qualitatively the patterns
reported in Figure 5. As was the case in the data, changes in own trade share (bottomleft panel) and changes in exposure to high-TFP trading partners (bottom-middle panel)
correlate significantly with changes in TFP, while changes in trading partners’ TFP is not

significantly associated with TFP growth. Interestingly, the magnitude of the coefficients
in the model generated data are smaller than in the actual data. This is true even though
in our model the gains from trade are substantially higher those of a static trade model.
Through the lens of our model, this suggests that part of relationship between openness and
growth is accounted for by the correlation between changes trade cost and changes in the
arrival rate of ideas; countries whose TFP rose due to changes in trade barriers also tended
to increase research intensity.
World Growth

0.5

Cross-Sectional Variance

0.5

Fraction explained by trade

, varying
, constant

0.4

0.3

0.2

0.1

0.5

,
,
,
,

varying, incl. cov.
constant, incl. cov.
varying, excl. cov.
constant, excl. cov.

0.5

Figure 8: The contribution of changes in trade costs to changes in TFP
Note: This figure reports the fraction of TFP growth accounted for by trade costs, for
various values of β, according to two decompositions. In both panels, the solid lines
correspond to (19) in which the contribution of trade is evaluated holding the arrival
rates constant; the dashed lines to correspond to (20) in which the contribution of trade
is evaluated at the evolving arrival rates that are consistent with data. The left panel
reports the fraction of total growth in TFP accounted for by changes in trade costs. The
right panel reports the fraction of variance in TFP growth rates accounted for by changes
in trade costs. The lines with square markers report exclude the covariance between the
contribution from trade and the contribution from changing arrival rates of ideas; the
lines without markers include the covariance. In all cases, insights are drawn from sellers.

Figure 8 shows a more systematic assessment of how the strength of diffusion alters the
explanatory power of trade in the model. For each β we compute two counterfactuals to
assess the contribution of trade to changes in TFP. Each provides a different way of dividing
changes in each country’s TFP into a contribution from changes in trade barriers and a
contribution from changes in the arrival rates of ideas.46 First, we compute how countries’
46

Of course, changes in trade costs may themselves affect incentives to innovate and hence the arrival of
ideas. In Appendix D, we extend the result of Eaton and Kortum (2001) that on any balanced growth path,
each country’s research effort is independent of trade barriers (although research effort may vary due to other
things, such as taxes). Thus over long periods of time, treating the arrival rate of ideas as independent of
trade costs may be a good approximation.

TFP would have evolved if trade costs evolved as they do in data but each country’s arrival
rate of ideas remained fixed at its 1962 level. The predicted change in TFP is the contribution
of trade and the residual is the contribution from changes in arrival rates.47 The second
counterfactual computes the changes in TFP if the arrival rates of ideas evolved as they do
in the data but the trade costs remained fixed at their 1962 levels. The predicted changes
in TFP are the contribution of changes in the arrival rates and the contribution of trade are
the residuals.48
ln

T F Pi (αt , κt )
T F Pi (αt , κt )
T F Pi (α0 , κt )
+ ln
= ln
T F Pi (α0 , κ0 )
T F Pi (α0 , κ0 )
T F Pi (α0 , κt )
{z
} |
{z
}
|
cont. from trade

T F Pi (αt , κt )
=
T F Pi (α0 , κ0 )

cont. from arrival rates

T F Pi (αt , κ0 )
T F Pi (αt , κt )
ln
+ ln
T F Pi (α0 , κ0 )
T F Pi (αt , κ0 )
|
{z
} |
{z
}

cont. from arrival rates

(19)

(20)

cont. from trade

We can summarize the role of trade in a few different ways. We first compute the
fraction of changes in TFP growth accounted for by contributions from trade and from
contributions from changes in the arrival rate of ideas.49 The solid line corresponds to the
first counterfactual in which the contribution of trade is evaluated at the initial arrival rates,
and the dashed line corresponds to the second counterfactual in which contribution of trade
is evaluated at the actual arrival rates.
According to this decomposition, both counterfactuals indicate that the static gains from
trade (β = 0) account for roughly eight percent of the growth in TFP from 1962-2000. With
β > 0, changes in trade costs are more important. The contribution trade is highest if
β = 0.7, a setting in which a quarter of the increases in TFP are accounted for by changes
in trade costs.
While the model predicts that changes in trade barriers can account for a significant
fraction of TFP growth, it is possible that model assigns growth to the wrong countries.
To address this, the right panel of Figure 8 shows the fraction of variation in TFP growth
47

This is in some ways analogous to dividing changes in nominal GDP into changes in a price index and
changes in a quantity index. If the price index is a Lespeyres index then the quantity index is a Paasche
index and vice versa.
48
We use here the shorthand that αt and κt represent a sequence of the vector of arrival rates and matrices
of trade costs that are required to match the data, and α0 and κ0 are the initial values. Thus T F Pi (α0 , κt )
represents the TFP of country i in a counterfactual in which the arrival rates of technologies are held fixed
at their initial levels but trade costs evolve as they do in the data. By construction T F Pi (αt , κt ) equals
country i’s TFP in 2000 and T F Pi (α0 , κ0 ) is what i’s TFP would have been had the world remained on the
balanced growth path since 1962 – the vector of TFP in 2000 would be a scalar multiple of the vector of
1962 TFPs.
P
contribution from trade
49
. It is thus a weighted average of the
For each of the two counterfactuals, this is i P T F Pi (αt ,κt )
i

T F Pi (α0 ,κ0 )

fraction of each country’s TFP growth due to trade, where countries are weighted by their TFP growth.

rates accounted for trade costs. The variance of TFP growth can be decomposed into three
components, the variance of contributions of changes in trade costs, the variance of the
contributions of changes in arrival rates, and twice the covariance of the two. The figure
plots four lines. The two solid lines correspond to the decomposition in (19) in which the
contribution of trade is evaluated holding the arrival rates of ideas fixed at their initial levels.
The two dashed lines correspond to (20) in which the contribution of trade is evaluated
allowing the arrival rates to evolve as they must to explain the data. The lines that are
marked with squares represent the fraction of variance of TFP growth rates accounted for
by the variance of the contributions from trade. The lines without markers add in the
covariance between the two contributions.
Three lessons emerge. First, the ability of the model to account for TFP changes is
greatest for intermediate values of the diffusion parameter, β. As highlighted when discussing
Figure 2, for β close to 1 a country’s stock of ideas depends much more heavily on insights
from the most productive producers, so that even countries close to autarky have accrued
most of the dynamic gains from trade. Consequently when β is close to 1, the model does
not predict much dispersion in TFP growth among countries that are moderately open.
Second, the covariance terms are also quite large; countries whose TFP rose most saw
increase stemming from trade but also from increasing the arrival ideas. This is consistent
with the notion that some countries reformed along many margins, which both increased
trade and increased R&D. Including this covariance, changes in trade cost can account for
more than a third of the variation of changes in TFP (when β is roughly 0.6).
Third, when the contribution of trade is evaluated at the arrival rates drawn from data,
trade accounts for more of the variance of TFP changes (either including or excluding the
covariance). This happens because changes in trade costs and in the arrival rates of ideas
are complementary. Intuitively, improvements in the quality of insights matter more when
the arrival rate of these insights is greater.

4.6

Growth Miracles

To illustrate the fit of the model more concretely, we show how the model predicts changes in
TFP during several growth miracles. We begin by comparing the implied evolution of TFP
in South Korea and the US. South Korea is a particularly interesting example as it is one
of the most successful growth miracles in the post-war period, and a country that became
most integrated with the rest of the world, as inferred from the behavior of trade flows. The
U.S. economy provides a natural benchmark developed economy.50
50

In Table 1 of the Appendix we present summary statistics for each country in our sample.

1.6

Korea

1.6

1.5

1.4

1.3

1.2

1.1

0.9

0.8
1960

1980

0.8
1960

2000

US
data
-=0
- = 0.6
- = 0.9

1980

2000

Figure 9: Openness and the Evolution of TFP: South Korea and the US
Note: This figure plots the changes in TFP for South Korea (top panels) and the US
(bottom panels) under the specification of learning insights from sellers (left panels) and
learning from producers (right panels). In each panel, we plot the actual change in TFP
and changes in TFP generated by the model for various values of β. In all cases, TFP is
detrended by the average growth rate of TFP in the US.

Figure 9 explores the implied dynamics of TFP under various assumptions. As in Figure
6, we assume that arrival rates of ideas are heterogenous, that, given trade costs, each country
was on its balanced growth path in 1962, and that arrival rates have remained constant since
1962. Figure 9 shows the evolution of TFP for South Korea (left panel) and the US (right
panel) for this case. The solid line shows the evolution of TFP in the data, de-trended by the
average growth of TFP in the U.S. The other lines correspond to simulations using alternative
values of the diffusion parameters β. The case of β = 0 (dotted line) gives the dynamics of
TFP implied by a standard Ricardian trade model, e.g., the dynamics quantified by Connolly
and Yi (2015). The other two lines illustrate the dynamic gains from trade implied by the
model.
Two clear messages stem from this figure. First, for a wide range of values of the diffusion
parameter the dynamic model accounts for a substantial fraction of the TFP dynamics of
South Korea. This is particularly true when considering intermediate values of the diffusion
parameters β. Recall from Figure 2 that for an economy that is moderately open, dynamic
gains from trade are non-monotonic in β. Second, the right panel shows that changes in
the dynamic gains from trade identified by the model are less relevant for understanding the
growth experience of a developed country close to its balanced growth path.
Figure 10 shows the evolution of TFP for a larger set of Asian countries that experienced
high growth in the post-war period. For each country, the solid line is the data, while the
dotted line is the model with β = 0 when trade costs are adjusted, but the arrival rates of
37

China
2

Korea

Data

5 , , , -=0
t

5t , , 0 , -=0.7

1.5

data - 50 , , t , -=0.7

1.5

1
1960

1
1970

1980

1990

2000

1960

1970

Taiwan
2

1.5

1
1970

1980

1990

2000

1990

2000

Thailand

1960

1980

1990

2000

1960

1970

1980

Figure 10: Growth Miracles
Note: In all cases, TFP is detrended by the average growth rate of TFP in the US.

ideas is held as in the 1962 values. The dashed line shows the evolution of TFP for the simple
calibration of the diffusion parameter, β = 0.7. For some countries such as South Korea and
China, the diffusion of ideas due to trade explains a substantial fraction of TFP growth.
For others, such as Thailand changes in trade costs account for a smaller, but significant,
fraction of TFP growth. Finally, the dashed dotted line shows the evolution of TFP netted
of the contribution of changes in the arrival rate of ideas. The gap between this and the data
is the contribution of trade as measured by (20). The fact that this second measure of the
contribution of trade tends to be larger suggests strong complementarities between changes
in trade costs and in the arrival rates of ideas, consistent with the results in Figure 8.

4.7

Explaining the Initial Distribution of TFP

We next assess the role of trade barriers in accounting for the initial cross-country TFP
differences. To this end, we use as a baseline the extreme assumption that the arrival rate
of ideas is the same in each country, α̂i = α̂. Given the calibrated trade costs and a value of
β, we solve for the balanced growth path of the model. In this case, trade is the only force
driving TFP differences.
Figure 11 compares the implied distribution of TFP on the balanced growth path of

log(TFP), model

0.5

- =0

0.5

-0.5
-1

- = 0.7

0
log(TFP), data

-0.5
-1

0
log(TFP), data

Figure 11: Openness and the Distribution of TFP in 1962
Note: Each panel plots countries actual TFP in the 1962 against the predicted TFP of
the model under the assumption that the arrival rate of ideas is uniform across countries.
The first panel assumes that β = 0. The second considers β = 0.7 and the case in which
insights are drawn from sellers. In addition, each figure plots a dashed 45-degree line
and a red regression line.

the model to the actual distribution of TFP in 1962, the first year in our sample. Each
dot represents a country. The first panel shows the case of β = 0, so that there is no
cross-country diffusion of ideas and differences in countries’ TFP represent only the static
Ricardian gains from trade. As the panel shows, these static gains generate only a small
amount of cross-country differences in productivity.
The second panel assumes that β = 0.7 so that the cross-country TFP differences represent both the static and dynamic gains from trade. The model generates more variation
in TFP across countries. The positive slope of the red regression line implies a positive
correlation between the model’s predictions and the data.
We next assess more systematically how the strength of diffusion affects the ability of
the model to account for cross-country TFP differences in Figure 12. For each model, we
divide variation in TFP into a contribution from trade and a contribution from arrival rates
of ideas. Given trade costs, we compute for the vector of arrival rates {αi } so that the world
would be on a balanced growth path. Given these, we can compute the κ̄, a number so that
if each bilateral iceberg trade cost were κ̄, the volume of world trade would be unchanged.
The contribution of trade to cross-sectional TFP differences is the variation that comes from
the counterfactual of changing trade costs from κij to κ̄.
Similar to Section 4.5, we can do this in two ways. In (21) the contribution of trade is
evaluated at common arrival rates ᾱ, while in (22) it is evaluated at the country specific
arrival rates.

T F Pi (αi , κij )
T F Pi (ᾱ, κij )
T F Pi (αi , κij )
= ln
+ ln
T F Pi (ᾱ, κ̄)
T F Pi (ᾱ, κ̄)
T F Pi (ᾱ, κij )
|
{z
} |
{z
}
cont. from trade

T F Pi (αi , κij )
=
T F Pi (ᾱ, κ)

cont. from arrival rates

T F Pi (αi , κ̄)
T F Pi (αi , κij )
ln
+ ln
T F Pi (ᾱ, κ̄)
T F Pi (αi , κij )
|
{z
}
{z
}
|

cont. from arrival rates

(21)

(22)

cont. from trade

Each panel of Figure 12 has four lines. The two solid lines correspond to the decomposition in (21) in which the contribution of trade is evaluated holding the arrival rates of ideas
fixed at a common level ᾱ. The two dashed lines correspond to (22) in which the contribution of trade is evaluated using country-specific arrival rates. The lines that are marked
with squares represent the fraction of cross-sectional variance of TFP accounted for by the
variance of the contributions from trade. The lines without markers add in the covariance
between the two types of contributions. The left and right panels illustrate the ability of the
theory to account for the cross section variance in 1962 and 2000, respectively.

0.6

Cross-Sectional Variance, 1962

0.6

0.5

0.4

0.3

0.2

0.1

0.5

Cross-Sectional Variance, 2000

,
,
,
,

varying, incl. cov.
constant, incl. cov.
varying, excl. cov.
constant, excl. cov.

0.5

Figure 12: Mean squared error of predicted TFP
Note: This figure reports the fraction of the cross-sectional variance in log TFP in
1962 accounted for by trade, for various β, according to two decompositions. The solid
lines correspond to (21) in which the contribution of trade is evaluated at a common
arrival rate; the dashed lines to correspond to (22) in which the contribution of trade is
evaluated at country-specific arrival rates. The lines with square markers report exclude
the covariance between the contribution from trade and the contribution from variation
in arrival rates of ideas; the lines without markers include the covariance. In all cases,
insights are drawn from sellers.

When β = 0 so that there is no diffusion of ideas, the model accounts from roughly 2% to
8% of the variation, consistent with the first panel of Figure 11. As we consider specifications
40

for which the strength of diffusion is larger, the model accounts for more of the variation
in TFP. Again, while the ability of the model to account for the initial differences in TFP
initially rises with the strength of diffusion, it is greatest for cases with intermediate values
of the diffusion parameter, β. For a large range of values of β, however, variation in trade
barriers accounts on average for up to 45% of the cross-sectional dispersion in TFP.51

Conclusion

In this paper we have provided a tractable theory of the cross-country diffusion of ideas
across countries and a quantitative assessment of the role of trade in the transmission of
knowledge. We found that when the model is specified so that the strength of diffusion is
at an intermediate level, the model predicts a stronger response of TFP to changes in trade
barriers than if the model were specified at either extreme of pure innovation or of pure
diffusion. We show quantitatively that ability of trade barriers to account for changes in
TFP from 1962-2000 is up to three times as large when the model allows for dynamics gains
from trade.
The analysis points to critical importance of the strength of diffusion, β. While we
provided one crude strategy to calibrate β, a more robust strategy would make better use of
the variation in trade costs identified by Feyrer (2009b,a).
Of course we omitted many channels that may complement or offset the role of trade in
the diffusion of ideas. Chief among these are FDI and purposeful imitation. The productivity
spillovers from trade are modeled as an external effect, which likely reflects how some but not
all ideas diffuse. In addition we have abstracted from variation across industries. Knowledge
from one industry may be more useful in generating ideas to be used in the same industry
than in other industries. In light of this, our quantitative results assessing the role of openness
should be viewed as a first step rather than the final word.

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sample, e.g., Niger, were to face the trade cost of open developed economies, e.g., Belgium. Appendix G.2
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Technology Diffusion

Lemma 3 Under Assumption 1, there is a K < ∞ such that for all z,

1−H(z)
z −θ

≤K

Proof. Choose δ > 0 arbitrarily. Since limz→∞ 1−H(z)
= 1, there is a z ∗ such that z > z ∗
z −θ
θ
∗ θ
∗ θ
implies 1−H(z)
< 1 + δ. For z < z ∗n
, we have that
z −θ
o z [1 − H(z)] ≤ (z ) [1 − H(z)] ≤ (z ) .
Thus for any z, 1−H(z)
≤ K ≡ max 1 + δ, (z ∗ )θ
z −θ
Claim 4 Suppose that Assumption 1 holds. Then in the limit as m → ∞, the frontier of
knowledge evolves as:
Z ∞
d ln Ft (q)
−θ
xβθ dGt (x)
= −αt q
dt
0
1

Proof. Evaluating the law of motion at m (1−β)θ q and using the change of variables w =
1
m− (1−β)θ x we get

1
∂
(1−β)θ
ln F̃t m
q = −mαt
∂t

∞

Z
0

∞

Z
= −mαt
0

m (1−β)θ q
1−H
xβ



1
(1−β)θ

dG̃t (x)


1
q 

 m
(1−β)θ w
m
d
G̃
1
−
H


β  t
1
m (1−β)θ w

1
1
Since Ft (q) ≡ F̃t m (1−β)θ q , and Gt ≡ G̃t m (1−β)θ q , this is

∂ ln Ft (q)
= −mαt
∂t



∞

1
(1−β)θ



q 

 m
1 − H 
β  dGt (w)
1
0
(1−β)θ
w
m
"
#
Z ∞
1/θ
−β
1
−
H
m
qw
= −αt q −θ
wβθ dGt (w)
−θ
1/θ
−β
[m qw ]
0

We want to take a limit as m → ∞. To do this, we must show that we can take the limit
inside the integral. By Lemma 3, there is a K < ∞ such that for any z, 1−H(z)
≤ K. Further,
z −θ
R∞
given the assumptions on the tail of Gt , the integral 0 Kwβθ dG̃t (w) is finite. Thus we can
take the limit inside the integral using the dominated convergence theorem to get
∂ ln Ft (q)
= −αt q −θ
∂t

Z
0

∞

wβθ dGt (w)

In the model, managers engage in Bertrand competition. In that environment, an important object is the joint distribution of the productivities of the best and second best
producers of a good. We denote the CDF of this joint distribution as F̃t12 (q1 , q2 ), which, for
q1 ≥ q2 , equals52
F̃t12 (q1 , q2 ) = Mt (q2 )m + m [Mt (q2 ) − Mt (q1 )] Mt (q2 )m−1 .
Since the frontier of knowledge at t satisfies F̃t (q) = Mt (q)m , the joint distribution can be
written as

1/m
12
F̃t (q1 , q2 ) = 1 + m
F̃t (q1 )/F̃t (q2 )
− 1 F̃t (q2 ),
q 1 ≥ q2 .
Normalizing this joint distribution by the number of producers,

1
1
Ft12 (q1 , q2 ) ≡ F̃t12 m (1−β)θ q1 , m (1−β)θ q2 ,
taking the limit as m → ∞ gives the following corollary:
Claim 5 Under Assumption 1,
lim Ft12 (q1 , q2 ) = [1 + log Ft (q1 ) − log Ft (q2 )] Ft (q2 ),

m→∞

B
B.1

q1 ≥ q2

Trade
Equilibrium

This section derives expressions for price indices, trade shares, and market clearing conditions
that determine equilibrium wages. The total expenditure in i is Xi . Throughout this section,

−θ
we maintain that Fi12 (q1 , q2 ) = 1 + λi q1−θ − λi q2−θ e−λi q .
For a variety s ∈ Sij (produced in j and exported to i) that is produced with producw κ
tivity q, the equilibrium price in i is pi (s) = jq ij , the expenditure on consumption in i is

1−ε

1−ε
pi (s)
pi (s)
1
X
,
consumption
is
Xi , and the labor used in j to produce variety
i
Pi
pi (s)
Pi

1−ε
κ /qj1 (s) pi (s)
s for i is ijpi (s)
Xi .
Pi
λ (w κ )−θ

j ij
Define πij ≡ P jλ (w
−θ . We will eventually show this is the share of i’s total expendik κik )
k k
ture that is spent on goods from j. We begin with a lemma which will be useful in deriving

Intuitively, there are two ways the best and second best productivities can be no greater than q1 and q2
respectively. Either none of the productivities are greater than q2 , or one of the m draws is between q1 and
q2 and none of the remaining m − 1 are greater than q2 .

a number of results.
Lemma 6 Suppose τ1 and τ2 satisfy 0 ≤ τ1 < 1 and 0 ≤ τ1 + τ2 < 1. Then
# τ2

Z
qj1 (s)

τ1 θ

pi (s)

−τ2 θ

ds = B̃(τ1 , τ2 )

s∈Sij

−θ

λk (wk κik )

πij

n
where B̃(τ1 , τ2 ) ≡ 1 −

τ2
1−τ1

−θ(1−τ1 )
ε
ε−1

λj
πij

τ1

Γ (1 − τ1 − τ2 )

Proof. We begin by defining the measure Fij to satisfy
q2

wk κik x wk κik x
Fij (q1 , q2 ) =
,
Fj12 (dx, x)
wj κij wj κij
0 k6=j

Z q1 Y
wk κik q2 wk κik q2
12
+
Fk
,
Fj12 (dx, q2 )
w
κ
w
κ
j ij
j ij
q2 k6=j
Z

Fk12

(23)

Fij (q1 , q2 ) is the fraction of varieties that i purchases from j with productivity no greater
w κ
than q1 and second best provider of the good to i has marginal cost no smaller than jq2 ij .
There are two terms in the sum. The first term integrates over goods where j’s lowest-cost
producer has productivity no greater than q2 , and the second over goods where j’s lowest
2
cost producer has productivity between q1 and q2 . The corresponding density ∂q∂1 ∂q2 Fij (q1 , q2 )
will be useful because it is the measure of firms in j with productivity q that are the lowest
cost providers to i and for which the next-lowest-cost provider has marginal cost wj κij /q2 .
We first show that

− 1 λj q−θ
Fij (q1 , q2 ) = πij + λj q2−θ − q1−θ e πij 2
The first term of equation (23) can be written as
Z
0

Y
k6=j

Fk12

wk κik x wk κik x
,
wj κij wj κij

Fj12 (dx, x)

−

k6=j

λk

wk κik
wj κij

−θ

x−θ

−θ

θλj x−θ−1 e−λj x dx

0
−
λj (wj κij )−θ
= P
e
−θ
k λk (wk κik )
λ

− π j q2−θ

= πij e

k λk

wk κik
wj κij

−θ

q2−θ

The second term is
Z

Fk12

k6=i

wk κik q2 wk κik q2
,
wj κij
wj κij

Fj12

−

(dx, q2 ) = e

k6=j

λk

wk κik
wj κij

−θ

q2−θ

−θ

θλj x−θ−1 e−λj q2 dx

= e

−

λk

−θ

wk κik
wj κij

q2−θ

λj q2−θ − q1−θ

−θ

− π j q2−θ

= e

λj q2−θ − q1

Together, these give the expression for Fij , so the joint density is

− 1 λj q−θ
∂2
1
Fij (q1 , q2 ) =
θλj q1−θ−1 θλj q2−θ−1 e πij 2
∂q1 ∂q2
πij
R
We next turn to the integral s∈Sij qj1 (s)θτ1 pi (s)−θτ2 ds. Since the price of good s is set
ε
at either a markup of ε−1
over marginal cost or at the cost of the next lowest cost provider,
this integral equals
Z

∞

q1θτ1

ε wj κij
wj κij
,
q2 ε − 1 q1

−θτ2

∂ 2 Fij (q1 , q2 )
dq1 dq2
∂q1 ∂q2

ε wj κij
wj κij
,
q2 ε − 1 q1

−θτ2

− 1 λj q−θ
1
θλj q1−θ−1 θλj q2−θ−1 e πij 2 dq1 dq2
πij

min

q2
∞

∞

q1θτ1

=
0

min

λj −θ
q
πij 1

Using the change of variables x1 =
(wj κij )−θτ2 πij

Define B̃(τ1 , τ2 ) ≡

λj
πij

R ∞ R x2
0

τ1 +τ2 Z

∞

qj1 (s)

−θτ2

pi (s)

(
1
x−τ
min x2 ,
1

n
1
x−τ
x2 ,
min
1
θτ1

and x2 =

θ
ε
ε−1

o−τ2

λj −θ
q
πij 2

, this becomes

ε
ε−1

λj (wj κij )−θ
P
−θ ,
k λk (wk κik )

−θτ2

ds = B(τ1 , τ2 ) (wj κij )

we have (wj κij )−θτ2

λj
πij

e−x2 dx1 dx2

e−x2 dx1 dx2 , so that the integral is

πij

s∈Sij

Using πij =

)−τ2

τ2

λj
πij

τ1 +τ2

−θ
k λk (wk κik )

i τ2

. Finally we

complete the proof by providing an expression for B̃(τ1 , τ2 ):
Z

∞

B̃ (τ1 , τ2 ) =

1
min x2 ,
x−τ
1

∞

=
0

ε
( ε−1
) x2
Z ∞ Z ( ε )−θ x2
ε−1

x1−τ1

(

1−τ
∞ x2 1

−

ε
ε−1

)−τ2

θ
x1

e−x2 dx1 dx2

1 −τ2 −x2
dx1 dx2
x−τ
1 x2 e

−θ

+
Z

(

−θ
ε
ε−1

ε
ε−1

)−τ2

θ
x1

e−x2 dx1 dx2

1−τ1

x2−τ2 e−x2 dx2
0

−θ 1−τ1 −τ2
ε

−θτ2 Z ∞
x2
ε−1
ε
+
e−x2 dx2
ε−1
1 − τ1 − τ2
0

−θ(1−τ1 ) Z
−θ(1−τ
)
Z
1
ε
ε
∞
∞
1 − ε−1
x21−τ1 −τ2 e−x2 dx2 + ε−1
x21−τ1 −τ2 e−x2 dx2
=
1 − τ1
1
−
τ
−
τ
1
2
0
0
(
−θ(1−τ1 )
−θ(1−τ1 ) )
ε
ε
1 − ε−1
=
+ ε−1
Γ (2 − τ1 − τ2 )
1 − τ1
1 − τ1 − τ2
(

−θ(1−τ1 ) )
τ2
τ2
ε
+
=
1−
Γ (1 − τ1 − τ2 )
1 − τ1 1 − τ1 ε − 1

1 − τ1

where the final equality uses the fact that for any x, Γ(x + 1) = xΓ(x).
We first use this lemma to provide expressions for the price index in i and the share of
i’s expenditure on goods from j.
Claim 7 The price index for i satisfies
#− θ1

1 "
ε − 1 1−ε X
λk (wk κik )−θ
Pi = B̃ 0,
θ
k
n

−θ
ε
where B ≡
1 − ε−1
1
−
+
θ
ε−1
share of i’s expenditure on goods from j.

−θ
ε
ε−1

Γ 1−

Proof. The price aggregate of goods provided to i by j is
this equals

ε−1
θ

. πij =

R
s∈Sij

λ (w κ )−θ
P j j ij
−θ
k λk (wk κik )

pi (s)1−ε ds. Using Lemma 6,

# ε−1

"X
θ
ε
−
1
−θ
1−ε
pi (s) ds = B̃ 0,
λk (wk κik )
πij
θ
s∈Sij
k

is the

The price index for i therefore satisfies

Pi1−ε =

# ε−1
"X

θ
ε
−
1
λk (wk κik )−θ
pi (s)1−ε ds = B̃ 0,
θ
s∈Sij
k

XZ
j

and i’s expenditure share on goods from j is
R
s∈Sij

pi (s)1−ε ds
Pi1−ε

= πij

We next turn to the market clearing conditions.
Claim 8 Country j’s expenditure on labor is

θ
θ+1

πij Xi .

i
Proof. i’s consumption of good s is pi (s)−ε PX1−ε
. If j is the lowest-cost provider to i, then
i
κij
j’s expenditure on labor per unit delivered is wj qj1 (s) . The total expenditure on labor in j to
R
w κij
i
produce goods for i is then s∈Sij qj1j (s)
pi (s)−ε PX1−ε
ds. Using Lemma 6, the total expenditure
i
on labor in j is thus

XZ
i

s∈Sij

Z
X
Xi
wj κij
−ε Xi
pi (s)
ds =
wj κij 1−ε
qj1 (s)−1 pi (s)−ε ds
qj1 (s)
Pi1−ε
P
s∈Sij
i
i
"
# θε

− θ1
1 ε X
Xi X
λj
−θ
= B̃ − ,
λk (wk κik )
wj κij 1−ε
πij
θ θ
πij
Pi
i
k

The result follows from B̃ − 1θ , θε =
−1
B̃ 0, ε−1
.
θ

θ
B̃
θ+1

0, ε−1
and
θ

wj κij
Pi1−ε

−θ
k λk (wk κik )

i θε

λj
πij

− θ1

Source Distributions

This appendix derives expressions for the source distributions under various specifications.
We begin by describing learning from sellers.

C.1

Learning from Sellers

Here we characterize the learning process when insights are drawn uniformly from sellers. If
producers are equally likely to learn from all active sellers, the source distribution is
P R
j

Gi (q) =

{s∈S |q (s)≤q}
P Rij j1
j s∈Sij ds

R∞

The change in i’s stock of knowledge depends on

P R
j

βθ

q dGi (q) =

s∈S

P Rij

qj1 (s)βθ ds

s∈Sij

λj
πij

. Using

Lemma 6, this is
Z
0

C.1.1

∞

B̃(β, 0) X
πij
q dGi (q) =
B̃(0, 0) j
βθ

λj
πij

β
= Γ(1 − β)

πij

β
(24)

Alternative Weights of Sellers

Here we explore two alternative processes by which individuals can learn from sellers. In
the baseline, individuals are equally likely to learn from all active sellers, independently of
how much of the seller’s variety they consume. In the first alternative, insights are drawn
from sellers in proportion to the expenditure on the sellers good. In the second, insights
are drawn in proportion to consumption of each sellers’ goods. In each case, the speed of
learning is the same as our baseline (equation (24)) up to a constant.
Learning from Sellers in Proportion to Expenditure
Here we characterize the learning process when insights are drawn from sellers in proportion
to the expenditure on each seller’s good. Consider a variety that can be produced in j at
productivity q. Since the share of i’s expenditure on good s is (pi (s)/Pi )1−ε , the source
distribution is
XZ
Gi (q) =
(pi (s)/Pi )1−ε ds
j

{s∈Sij |qj1 (s)≤q}

The change in i’s stock of knowledge depends on
Z

∞
βθ

q dGi (q) =
0

XZ
j

qj1 (s)βθ (pi (s)/Pi )1−ε ds

s∈Sij

Using Lemma 6, this is
Z
0

∞

# ε−1

"X
β
θ
ε
−
1
λj
−θ
βθ
q dGi (q) =
πij
λk (wk κik )
1−ε B̃ β,
θ
πij
Pi
j
k

X λj β
B̃ β, ε−1
θ

πij
=
πij
B̃ 0, ε−1
θ
j
1

Learning from Sellers in Proportion to Consumption
i
. If producers learn in proportion to coni’s consumption of goods s is ci (s) = pi (s)−ε PX1−ε
i
sumption, then the source distribution is

P R
j

Gi (q) =

{s∈Sij |qj1 (s)≤q}

i
pi (s)−ε PX1−ε
ds
i

i
p (s)−ε PX1−ε
ds
{s∈Sij } i

P R
j

The change in i’s stock of knowledge depends on
Z

qj1 (s)βθ pi (s)−ε ds

P R

∞
βθ

q dGi (q) =
0

s∈Sij

P R
j

s∈Sij

pi (s)−ε ds

Using Lemma 6, this is
Z
0

C.2

∞

B̃ β,

ε
θ

−θ

i θε

λj
πij

πij
k λk (wk κik )
iε
h
P
P
−θ θ
ε
B̃
0,
λ
(w
κ
)
πij
k ik
j
k k
θ

β
B̃ β, θε X
λj

=
π
ij
πij
B̃ 0, θε j

q βθ dGi (q) =

Learning from Producers

Here we briefly describe the learning process in which insights are equally likely to be drawn
from all active domestic producers. As discussed in the text, we consider only the case in
which trade costs satisfy the triangle inequality which implies that, all producers
that export
R
s∈Sii |qi1 ≤q ds
also sell domestically. As a consequence, the source distribution is Gi (q) = R
. The
s∈Sii ds
change in i’s stock of knowledge depends on
Z

∞

q βθ dGi (q) =

s∈Sii

q βθ ds

R
s∈Sii

Using Lemma 6, this is
Z

B̃(β, 0)πii

∞
βθ

q dGi (q) =
0

C.2.1

λi
πii

B̃(0, 0)πii

= Γ(1 − β)

λi
πii

Alternative Weights of Producers

Here we characterize the learning process when insights are drawn from domestic producers
in proportion to labor used in production. For each s ∈ Sij , the fraction of j’s labor used
κ
i
to produce the good is L1j qj1ij(s) ci (s) with ci (s) = pi (s)−ε PX1−ε
. Summing over all destinations,
i
the source distribution would then be
XZ
Xi
1 κij
pi (s)−ε 1−ε ds
Gj (q) =
Pi
s∈Sij |qj1 (s)≤q Lj qj1 (s)
i
The change in j’s stock of knowledge depends on
∞

βθ

q dGj (q) =
0

XZ
i

q βθ

s∈Sij

Xi
1 κij
pi (s)−ε 1−ε ds
Lj qj1 (s)
Pi

Using Lemma 6, this is
Z
0

∞

# θε

β− θ1
"X
X κij Xi
1 ε
λj
−θ
βθ
q dGj (q) =
λk (wk κik )
πij
1−ε B̃ β − ,
Lj Pi
θ θ
πij
j
k

Using the expressions for Pi and πij from above, this becomes
Z
0

∞

β
B̃ β − 1θ , θε
1 X
λj

q dGj (q) =
π
X
ij
i
wj Lj j
πij
B̃ 0, ε−1
θ
βθ

Research

This section endogenizes the arrival rate of ideas, broadly following Rivera-Batiz and Romer
(1991) and Eaton and Kortum (2001). Labor can engage in two types of activities, production
and research. The production sector is described by Section 2. In the research sector,
entrepreneurs generate ideas by hiring labor. The labor resource constraint in country i at
t is thus
LPit + LR
it = Lit

where LPit is the labor used in production and LR
it is the labor used in research. We will show
that on a balanced growth path, the fraction of labor engaged in research is independent of
trade barriers.
We assume that there is a mass of managers and each manager is, in principle, capable of
producing all varieties s ∈ [0, 1]. Each manager is characterized by a profile of productivities
q(s) with which she can produce the various goods. We assume that if an individual manager
employs l units of labor in research, then ideas arrive independently for each variety at rate
α̃l. Thus the arrival of goods is uniform across goods; research effort is not directed at
particular goods.53
Suppose also that the entrepreneurs behave as if there is a tax Ti on profit. This may be
an actual tax, or it may stand in for other distortions (as in Parente and Prescott (1994)).
Let Vit is the expected pretax value of a single idea generated in i at t. Each manager chooses
a research intensity l to maximize α̃l(1 − Ti )Vit − wit l. For research to be interior, it must
be that
α̃Vit = wit
We next compute the expected pretax value of an idea, Vit . In the next subsection, we
prove the following intermediate step: if Πiτ is total flow of profit earned by entrepreneurs
in i at time τ , then the flow of profit earned in i at time τ from ideas generated between t
λ −λ
and t0 (with t < t0 < τ ) is itλ0 iτ it Πiτ . The basic idea is that, among ideas on the frontier at
time τ , knowing the time at which the idea was generated does not provide any additional
information about the quality of the idea.54
Taking the limit as t0 → t implies that the flow of profit at τ from ideas generated at the
instant t is λλ̇iτit Πiτ . As a consequence, the present value of revenue from ideas generated in i
at instant t is
Z ∞
Pit λ̇it
e−ρ(τ −t)
Πiτ dτ
Piτ λiτ
t
where e−ρ(τ −t) PPiτit is the real discount factor between t and τ . The cumulative arrival of ideas
R
at t is α̃LR
it , so that the total pretax value of the ideas generated at t is α̃Lit Vit . We thus
have
Z ∞
Pit λ̇it
R
α̃Lit Vit =
e−ρ(τ −t)
Πiτ dτ
Piτ λiτ
t
53

We could just as easily have assumed that each manager is capable of producing a subset of the goods
with positive measure, and call this subset an industry. The part of the assumption that is crucial is that
the research effort is uniform across varieties.
54
Both the arrival rate of ideas and the source distribution at time t affect the probability that the idea is
on the frontier at τ ≥ t and the unconditional distribution of the idea’s productivity, these have no impact
on the conditional distribution of productivity conditioning on being on the frontier. This is a useful and
well known property of extreme value distributions, see Eaton and Kortum (1999).

The optimal choice of research intensity implies α̃(1 − Ti )Vit = wit . In addition, as we
show in the claim below, profit among all entrepreneurs is proportional to the wage bill in
w LP
production, Πiτ = iτθ iτ . Together these imply that the optimal research intensity satisfies
wit LR
it

Z
= (1 − Ti )

∞

e−ρ(τ −t)

Letting rit =

LR
it
Lit

Pit λ̇it wiτ LPiτ
dτ
Piτ λiτ θ

be the fraction of labor engaged in research, this can be rearranged as
1 − Ti
rit =
θ

∞

e−ρ(τ −t)

Pit λ̇it (1 − riτ ) wiτ Liτ
dτ
Piτ λiτ
wit Lit

Finally, using wit /Pit ∝ (λit /πiit )1/θ , this can be written as
1 − Ti
rit =
θ

∞

−ρ(τ −t)

λ̇it (λiτ /πiiτ )1/θ Liτ
(1 − riτ )
dτ
λiτ (λit /πiit )1/θ Lit

If labor grows at rate γ so that Liτ = Lit eγ(τ −t) , then there is balanced growth path with
γ
rit = ri , λit = e 1−β t λi , πiit = πii . Plugging these in gives
1 − Ti
ri =
θ

∞

e
t

−ρ(τ −t)

(1 − ri )

γ
1−β
γ
(τ −t)
1−β

1/θ
γ
eγ(τ −t) dτ
e 1−β (τ −t)

Integrating and rearranging gives a simple characterization of the fraction of the labor force
engaged in research:
1 − Ti
ri
i
= h
(25)
1 − ri
θ (1 − β) ρ + β − 1
γ

Equation 25 implies that on a balanced growth path, the fraction of labor engaged in research
is independent of both trade barriers and the cross-country distribution of knowledge. The
only thing that alters research effort are distortions on the payoff to innovation. This aligns
with results of Eaton and Kortum (2001), Atkeson and Burstein (2010), and the knowledge
specification of Rivera-Batiz and Romer (1991) with flows of only goods, all of which imply
that integration has little impact on R&D effort.
It is important to keep in mind that, in this context, integration still has an impact on
a country’s stock of knowledge. Even if a country’s R&D effort does not change, integration
could lead to larger increases in a country’s stock of knowledge if new ideas are based on
better insights.55
55

See also Rivera-Batiz and Romer (1991) and Baldwin and Robert-Nicoud (2008).

Finally, we define
α̃
rit Lit
m
where rit is defined in equation (25) and depends on country-specific distortion to R&D
effort.
αit =

D.1

Research and Profit

In this section we prove the following claim:
Claim 9 If Πiτ is total flow of profit earned by entrepreneurs in i at time τ , then the flow
of profit earned in i at time τ from ideas generated between t and t0 (with t < t0 < τ ) is:
λit0 − λit
Πiτ
λiτ
(t,t0 ]

Proof. For v1 ≤ v2 , let Ṽjiτ (v1 , v2 ) be the probability that at time τ , the lowest cost
technique to provide a variety to j was discovered by a manager in i between times t and t0 ,
that the marginal cost of that lowest-cost technique is no lower than v1 , and that marginal
cost of the next-lowest-cost supplieris not lower than v2 . Just as in Appendix B, we will
1
1
(t,t0 ]
(t,t0 ]
define Vjiτ (v1 , v2 ) = limm→∞ Ṽjiτ m− (1−β)θ v1 , m− (1−β)θ v2 .
(t,t0 ]

Let Πiτ be profit from all techniques drawn in i between t and t0 . Thus total profit in

−ε
(−∞,τ ]
ε
. Defining p (v1 , v2 ) ≡ min v2 , ε−1
i at τ is Πiτ
v1
, we can compute each of these by
summing over profit form sales to each destiantion j:
(t,t0 ]
Πiτ

XZ
j

(−∞,τ ]

Πiτ

∞

XZ
j

∞

(t,t0 ]

[p (v1 , v2 ) − v1 ] p (v1 , v2 )−ε Pjε−1 Xj Vjiτ (dv1 , dv2 )

v1
∞

[p (v1 , v2 ) − v1 ] p (v1 , v2 )−ε Pjε−1 Xj Vjiτ

(−∞,τ ]

(dv1 , dv2 )

v1
(t,t0 ]

−λ

(−∞,τ ]

We will show below that Vjiτ (v1 , v2 ) = itλ0 iτ it Vjiτ
(v1 , v2 ). It will follow immediately
that
λit0 − λit (−∞,τ ]
(t,t0 ]
Πiτ =
Πiτ
λiτ
(t,t0 ]

(t,t0 ]

We now compute Vjiτ (v1 , v2 ). For each of the m managers in i, let Mi (q) be the
probability that the no technique drawn between t and t0 delivers productivity better than
(t,t0 ]
(t,t0 ]
q. Similarly, define F̃i
(q) ≡ Mi (q)m be the probability that none of the m managers
(t,t0 ]
drew a technique with productivity better than q between t and t0 . Finally, let Fi
(q) =

(t,t0 ]

limm→∞ F̃i

(t,t0 ]

1 1
m 1−β θ q . We have

Ṽjiτ (v1 , v2 ) =















R ∞ hQ

wk κjk
(−∞,τ ] wi κji m−1
(−∞,t] wi κji
Mi
Mi
x
x
x
v2
(t0 ,τ ] wi κji
(t,t0 ] wi κji
Mi
dMi
x
x m−1
i

R v2 h Q
(−∞,τ ] wk κjk
(−∞,τ ] wi κji
(−∞,t] wi κji
F̃
M
+m v1
M
i
i
k
k6=i
v2
v2
x
(t,t0 ] wi κji
(t0 ,τ ] wi κji
dMi
Mi
x
x

(−∞,τ ]

k6=i F̃k

(t,t0 ]















The expression for Ṽjiτ (v1 , v2 ) contains two terms. The first represents the probability that
the best technique to serve j delivers marginal cost greater than v2 and was drawn by a
(t,t0 ] wi κji
measures the
manager in i between t and t0 . For each of the m managers in i, dMi
x
0
likelihood that the managers best draw between t and t delivered marginal cost x ∈ [v2 , ∞],
(−∞,t] wi κji
(t0 ,τ ] wi κji
M
Mi
is the probability that the manager had no better draws, and
x
hQi
i x
(−∞,τ ] wk κjk
(−∞,τ ] wi κji m−1
Mi
is the probability that no other manager from any
k6=i F̃k
x
x
destination would be able to provide the good at marginal cost lower than x. The second
terms represents the probability that the best technique to serve j delivers marginal cost
between v1 and v2 and was drawn by a manager in i between t and t0 , and that no other
manager can deliver the variety with marginal cost lower than v2 .
(−∞,τ ]
(−∞,τ ]
(−∞,τ ]
(t,t0 ]
(t0 ,τ ]
(−∞,t]
(q)m and
(q) = Mi
(q), F̃i
(q) Mi (q) = F̃i
(q) Mi
Using Mi
0
0
(t,t ]
(t,t ]
(q) = Mi (q)m , this can be written as
F̃i

(t,t0 ]

Ṽjiτ (v1 , v2 ) =










1

i dF̃ (t,t0 ]

w κ

( ix ji )
w κ
v2
( ix ji )
1

R v2 hQ (−∞,τ ] wk κjk i F̃i(−∞,τ ] ( wixκji ) m dF̃i(t,t0 ] ( wixκji )

+ v1
(−∞,τ ] wi κji
k F̃k
(t,t0 ] wi κji
v2
F̃i
F̃i
( x )
v2
R ∞ hQ

wk κjk
x

(−∞,τ ]
k F̃k

i
(t,t0 ]
F̃i

Evaluating this at m− 1−β θ v1 and m− 1−β θ v2 and taking a limit as m → ∞ gives






i dF (t,t0 ]

w κ

( ix ji )
i
0
(t,t ] wi κji
v2
Fi
(t,t0 ]
( x )
Vjiτ (v1 , v2 ) =
h

i
(t,t0 ] wi κji
R
Q

dFi
( x )
v2
(−∞,τ ] wk κjk


 + v1
k Fk
(t,t0 ] wi κji
v2
Fi
( x )
R ∞ hQ

wk κjk
x

(−∞,τ ]
k Fk

(t,t0 ]

Finally, following the logic of Section 1, we have Fi
(t,t0 ]

dFi

(t,t0 ]

wi κji
x
wi κji
x






(q) = e−(λit0 −λit )q , so that

(−∞,τ ]

λit0 − λit dFi
=
(−∞,τ ]
λiτ
Fi
58

−θ







wi κji
x
wi κji
x











We thus have
(t,t0 ]

Vjiτ (v1 , v2 ) =

λit0 − λit (−∞,τ ]
Vjiτ
(v1 , v2 )
λiτ

which completes the proof.

Quantitative Model

This appendix derives expressions for the price index, expenditure shares, and the law of
motion of the stock of knowledge for the extended model discussed in Section 4, incorporating
non-tradable goods, intermediate inputs, and equipped labor. The price index satisfies

∝ (1 − µ)
p1−ε
i

1−η −θ

pηi wi

λi

i− 1−ε
θ

"
+µ

n
X

pηj wj1−η κij

−θ

#− 1−ε
θ
λj

j=1

and the share of i’s spending on non-tradable goods is

πiN T

h
−θ i ε−1
θ
(1 − µ) pηi wi1−η
λi
=
h
hP
i ε−1

−θ i ε−1
θ
θ
η 1−η
η 1−η −θ
(1 − µ) pi wi
+µ
λi
λk
k pk wk κik
−θ

(pηj wj1−η κij ) λj
denote the share of i’s tradable spending spent on tradable
−θ
η 1−η
κik ) λk
k (pk wk
goods from j. The fraction country i’s total expenditure on goods from country j 6= i is
πij = (1 − πiN T )Zij . The fraction of country i’s total expenditure spent on its own goods is
given by the sum of the non-tradable and tradable shares πii = πiN T + (1 − π N T )Zii . The
evolution of i’s stock of knowledge when learning is from sellers is
Let Zij ≡

λ̇i ∝ (1 −

µ)λβi

+µ

X
j

Zij

λj
Zij

The evolution of the stock of knowledge when learning is uniformly from domestic producers
is
β
(1 − µ)λβi + µZii Zλiii
λ̇i ∝
(1 − µ) + µZii
The market clearing conditions are the same as in the baseline model once labor is reinterpreted as equipped labor.

Quantitative Exploration

This section presents three additional quantitative exercises: (i) We consider an alternative
calibration strategy for the arrival rate of ideas, using data on R&D intensity; (ii) we explore
the extension with targeted learning discussed in Section 2.3 for the case in which insights
are drawn from sellers; (iii) we present results for learning from producers, under the baseline
assumptions that insights are drawn uniformly from domestic producers or the alternative
in which insights are drawn in proportion to labor used.

F.1

Calibrating α with R&D Data

In our main results we calibrate the evolution of the arrival rate of ideas α̂t to match the
evolution of TFP given the dynamics of trade costs. In doing this, we took the strong stand
that all residual TFP differences are due to differences in the stocks of knowledge and trade
barriers. In this section we consider an alternative calibration using information on R&D
and human capital stocks to more directly measure the stock of knowledge.
In particular, for the initial period, we project log TFP onto R&D intensity, the log of
the human capital stock, the log of the own trade share, and the log of an imported-weighted
average of trading partners’ TFP. We assign the residual initial TFP from this regression
to a neutral productivity terms affecting the units of equipped labor and not the stock
of knowledge, and choose initial stock of knowledge to match predicted TFP from these
regressions. The initial arrival rate of ideas is chosen to exactly match the initial stock of
knowledge, taking as given the initial trade flows, and assuming that the world economy was
on a balanced growth path in 1962. The knowledge-driven TFP accounts for 31 percent of
the variation of log TFP (as measured by R-squared).
To calibrate the evolution of the arrival rate of ideas in the subsequent periods we proceed
in two steps. First, we project the initial calibrated value for the arrival rate of ideas on
initial R&D intensity and the human capital stock. We then use the resulting coefficients
and the data on the evolution of R&D intensity and the human capital stock to predict the
evolution of arrival rates of ideas. The evolution of stocks of knowledge then depends on
the evolution of the arrival rates of ideas and trade costs. As before, we assign the residual
evolution of TFP to a neutral productivity terms affecting the units of equipped labor and
not the stock of knowledge. We denote the part of TFP accounted by the evolution in the
stock of knowledge the knowledge-driven TFP growth.
Figure 13 reports the fraction of knowledge-driven TFP growth accounted by trade costs.
This figure is the counterpart of Figure 8. The message in the main body of the paper is
preserved, with trade explaining a bit less of the level and a bit more of the variance of
60

World Growth

Fraction explained by trade

0.5

Cross-Sectional Variance

0.5

, varying
, constant

0.4

0.3

0.2

0.1

0.5

,
,
,
,

varying, incl. cov.
constant, incl. cov.
varying, excl. cov.
constant, excl. cov.

0.5

Figure 13: The contribution of changes in trade costs to changes in knowledge-driven TFP.
Note: This figure reports the fraction of knowledge driven TFP growth accounted for
by trade costs, for various values of β, according to two decompositions described in
Section 4.5. The left panel reports the fraction of total growth in TFP accounted for by
changes in trade costs. The right panel reports the fraction of variance in TFP growth
rates accounted for by changes in trade costs. See the note to Figure 8 for details.

knowledge-driven TFP growth.

F.2

Targeted Learning

World Growth

Fraction explained by trade

0.5

Cross-Sectional Variance

0.5

, varying
, constant

0.4

0.3

0.2

0.1

0.5

,
,
,
,

varying, incl. cov.
constant, incl. cov.
varying, excl. cov.
constant, excl. cov.

0.5

Figure 14: The contribution of changes in trade costs to changes in TFP, Targeted Learning.
Note: This figure reports the fraction of TFP growth accounted for by trade costs, for
various values of β, according to two decompositions described in Section 4.5. The left
panel reports the fraction of total growth in TFP accounted for by changes in trade costs.
The right panel reports the fraction of variance in TFP growth rates accounted for by
changes in trade costs. See the note to Figure 8 for details.

Figure 14 reports the fraction of TFP growth accounted for by trade costs for the exten61

sion of the benchmark model with targeted learning discussed in Section 2.3. In particular,
we present results for the case with φ = 1.5. Notice that now we must restrict the values
of β < 1/φ. As the figure clearly illustrates, for each value of β the results are qualitatively
very similar to an environment without targeted learning and a higher value of β.

F.3

Learning from Producers

In this section we present results when insights are drawn from producers. We present two
sets of results, under the respective assumptions that insights are drawn uniformly and in
proportions to labor used.
World Growth

0.5

Cross-Sectional Variance

0.5

Fraction explained by trade

, varying
, constant

0.4

0.3

0.2

0.1

0.5

,
,
,
,

varying, incl. cov.
constant, incl. cov.
varying, excl. cov.
constant, excl. cov.

0.5

Figure 15: The contribution of changes in trade costs to changes in TFP: Learning from
Producers, Uniformly.
Note: This figure reports the fraction of TFP growth accounted for by trade costs, for
various values of β, according to two decompositions described in Section 4.5. The left
panel reports the fraction of total growth in TFP accounted for by changes in trade costs.
The right panel reports the fraction of variance in TFP growth rates accounted for by
changes in trade costs. See the note to Figure 8 for details.

Figure 15 reports the fraction of TFP growth explained by trade costs in the specification
where insights are drawn from domestic producers uniformly (see Section 2.2). In this version
of the model diffusion provides a simple amplification mechanism for the static gains from
trade. Changes in the composition of trading partners, which was an important correlate of
TFP gains in the reduced form evidence in Figure 5, and an important driver of TFP gains
in the benchmark model with learning from sellers is absent. Therefore, it is not surprising
that the ability of trade costs to account for TFP gains is significantly muted.
Figure 16 reports the fraction of TFP growth explained by trade costs in the specification where insights are drawn from domestic producers in proportion to the labor used
(see Appendix C.2.1). This version of the model has some qualitative features similar to
62

the benchmark model with learning from sellers, and some important differences when considering values of β close to 1. In the limit as β → 1, a balanced growth path requires
P
an joint restriction on arrival rates and trade shares, αi = j πji αj . Since the the counterfactuals (which do not impose BGP) violate this restriction, they produce extreme (but
uninteresting) results.

World Growth

Fraction explained by trade

0.5

Cross-Sectional Variance

0.5

, varying
, constant

0.4

0.3

0.2

0.1

0.5

,
,
,
,

varying, incl. cov.
constant, incl. cov.
varying, excl. cov.
constant, excl. cov.

0.5

Figure 16: The contribution of changes in trade costs to changes in TFP: Learning from
Producers, in proportion to labor used.
Note: This figure reports the fraction of TFP growth accounted for by trade costs, for
various values of β, according to two decompositions described in Section 4.5. The left
panel reports the fraction of total growth in TFP accounted for by changes in trade costs.
The right panel reports the fraction of variance in TFP growth rates accounted for by
changes in trade costs. See the note to Figure 8 for details.

The Role of Geography: Examples

This section describes two examples that illustrate the role of geography in shaping income.

G.1

A Core-Periphery Economy

The interaction between geography and the diffusion of knowledge can be easily seen with
the example of an economy that takes a core-periphery structure. Suppose there are n
core countries and n periphery countries. Trade between a core country and any other
country incurs an iceberg trade cost of κ. Trade among any two periphery countries must
pass through the core, and thus incurs an iceberg cost of κ2 . All countries are otherwise
symmetric.
Figure 17 shows the real income of periphery countries relative to that of the core countries. Each curve corresponds a level of κ, and shows the ratio of real incomes for various
63

1.1

0.9

0.8

0.7
5
5
5
5
5

0.6

0.5

=1
= 1.2
=2
=5
= 10

0.2

0.4

0.6

0.8

Figure 17: A Core-Periphery Economy
Note: For various values of iceberg trade costs, this figure plots the ratio of real income
in periphery countries to real income in core countries.

values of β. Note that for each level of trade barriers, the relative income of periphery countries as β approaches one is the the same as it would be in a static trade model. Consistent
with the earlier discussion, the income gap is wider when β takes an intermediate value.
The income of core and periphery countries are similar when trade costs are either very
low (κ ≈ 1) or very high (κ % ∞); in either case, core countries effectively have no advantage.
Thus if trade costs fall steadily, income differences will initially grow and eventually shrink.

G.2

Niger in Belgium or Switzerland

To further illustrate the role of geography in determining productivity differences and the
(potential) dynamics of the model, we consider two counterfactual experiments where we
assign to Niger, one of the poorest countries in our sample, the trade costs faced by Belgium
and Switzerland, two rich countries with populations of comparable sizes.56 This exercise is
presented in Figure 18.
The left panel plots the evolution of TFP, normalized by the TFP of the US. On impact
the TFP jumps due to the static gains from trade, reflecting increased specialization and
comparative advantage. Given that Niger is a very isolated economy, compared to the more
integrated developed countries, the static gains increase Niger’s income by 5 to 10 percentage
points. Over time, as firms in Niger interact and get insights from more productive foreign
firms, TFP continues to grow. The second phase is more gradual, as it is mediated by the
random arrival of insights. The dynamic gains are large, more than doubling the static gains.
56

While Niger’s population is comparable to that of Belgium and Switzerland, 4% vs. 4% and 3% the US
population, respectively, its endowment of equipped labor is an order of magnitude lower.

0.5

TFP, Niger

0.5

0.4

0.3

0.2

0.1
0
2000

0.1

in Belgium
in Switzerland

2050

1-: ii , Niger

0
2000

2100

years

2050

2100

years

Figure 18: Transitional Dynamics after Assigning Belgium and Switzerland’s Trade Costs
to Niger.
Overall, the theory predicts that over a century the productivity of Niger would increase from
15 percent of that of the US to between 30 and 45 percent.
The right panel shows the evolution of Niger’s import share in the counterfactual experiments. The sharp increases in import share illustrates the large differences in trade costs
characterizing poor economies.

Table 1: TFP Growth by Country, 1962-2000. Data vs. Models with β = 0.7.
Data
ln
Thailand
China
Ireland
Taiwan
Japan
Korea, Republic of
Israel
Greece
Sri Lanka
Egypt
Finland
Pakistan
Belgium+Lux
Malaysia
Tunisia
Indonesia
Norway
Italy
United Kingdom
Mozambique
Brazil
Turkey
Denmark
Australia
Portugal
Cote d‘Ivoire
Austria
India
United States
Netherlands
Sweden
France
Mali
Spain
Cameroon
Ecuador
Paraguay
New Zealand

T F Pi (λt ,κt )
T F Pi (λ0 ,κ0 )

0.889
0.854
0.559
0.554
0.548
0.534
0.402
0.395
0.378
0.346
0.326
0.295
0.273
0.272
0.246
0.243
0.225
0.200
0.161
0.155
0.144
0.138
0.111
0.108
0.106
0.106
0.106
0.101
0.083
0.073
0.063
0.054
0.050
0.049
0.021
-0.004
-0.007
-0.041

Static Effect
Static+Dynamic Effect
T F Pi (λ0 ,κt )
T F Pi (λ(α0 ,κt ),κt )
(λ(αt ,κt ),κt )
ln T F Pi (λ0 ,κ0 ) ln T F Pi (λ(α0 ,κ0 ),κ0 ) ln TT FF PPii(λ(α
t ,κ0 ),κ0 )
0.116
0.049
0.146
0.094
0.000
0.048
0.016
0.025
0.022
0.003
0.036
-0.009
0.157
0.201
0.046
0.023
-0.002
0.024
0.024
0.046
0.003
0.023
0.004
0.017
0.051
-0.011
0.043
-0.009
0.023
0.027
0.029
0.037
-0.033
0.040
-0.008
0.018
0.037
0.028

0.218
0.352
0.181
0.203
0.014
0.188
0.052
0.083
0.065
0.137
0.134
0.038
0.190
0.253
0.134
0.154
0.006
0.050
0.046
0.167
0.031
0.077
-0.001
0.085
0.111
0.028
0.141
0.059
0.066
0.038
0.044
0.078
-0.045
0.079
0.013
0.049
0.060
0.123

0.253
0.573
0.255
0.277
0.053
0.300
0.081
0.032
0.064
0.160
0.152
0.062
0.149
0.231
0.106
0.205
0.057
0.044
0.020
0.125
0.029
0.039
-0.002
0.079
0.068
0.029
0.125
0.087
0.052
0.061
0.051
0.058
-0.010
0.052
0.025
0.037
0.060
0.085

Table 2: TFP Growth by Country, 1962-2000. Data vs. Models with β = 0.7
(cont’d).
Data
ln
Germany
Syria
Mexico
Zambia
Canada
Morocco
Chile
Tanzania
Guatemala
Argentina
Colombia
Switzerland
Philippines
South Africa
Uruguay
Costa Rica
Dominican Republic
Jamaica
Peru
Kenya
Honduras
Bolivia
Senegal
Uganda
Niger
Jordan
Ghana

T F Pi (λt ,κt )
T F Pi (λ0 ,κ0 )

-0.079
-0.094
-0.097
-0.101
-0.106
-0.108
-0.110
-0.128
-0.129
-0.131
-0.151
-0.170
-0.180
-0.194
-0.196
-0.219
-0.229
-0.245
-0.265
-0.271
-0.278
-0.288
-0.293
-0.329
-0.371
-0.392
-0.419

Static Effect
Static+Dynamic Effect
T F Pi (λ0 ,κt )
T F Pi (λ(α0 ,κt ),κt )
(λ(αt ,κt ),κt )
ln T F Pi (λ0 ,κ0 ) ln T F Pi (λ(α0 ,κ0 ),κ0 ) ln TT FF PPii(λ(α
t ,κ0 ),κ0 )
0.031
-0.005
0.061
-0.093
0.055
0.024
0.031
-0.002
0.005
0.001
0.006
0.010
0.061
0.004
0.017
-0.006
-0.003
-0.013
-0.030
-0.018
0.000
-0.005
0.003
-0.005
0.009
-0.017
0.004

0.054
0.035
0.098
-0.121
0.089
0.082
0.082
0.176
0.075
0.026
0.030
0.004
0.118
0.049
0.059
0.044
0.053
-0.013
-0.061
0.041
0.102
-0.007
0.032
0.283
0.085
0.053
0.005

0.042
0.043
0.073
-0.164
0.087
0.030
0.092
0.072
0.054
0.021
0.018
0.016
0.099
0.025
0.048
0.048
0.020
-0.045
-0.047
-0.004
0.073
0.006
-0.027
0.044
0.016
0.025
0.005

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Benjamin Chabot, Eric Ghysels, and Ravi Jagannathan

WP-14-27

Early Life Environment and Racial Inequality in Education and Earnings
in the United States
Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder

WP-14-28

Poor (Wo)man’s Bootstrap
Bo E. Honoré and Luojia Hu

WP-15-01

Revisiting the Role of Home Production in Life-Cycle Labor Supply
R. Jason Faberman

WP-15-02

Working Paper Series (continued)
Risk Management for Monetary Policy Near the Zero Lower Bound
Charles Evans, Jonas Fisher, François Gourio, and Spencer Krane
Estimating the Intergenerational Elasticity and Rank Association in the US:
Overcoming the Current Limitations of Tax Data
Bhashkar Mazumder

WP-15-03

WP-15-04

External and Public Debt Crises
Cristina Arellano, Andrew Atkeson, and Mark Wright

WP-15-05

The Value and Risk of Human Capital
Luca Benzoni and Olena Chyruk

WP-15-06

Simpler Bootstrap Estimation of the Asymptotic Variance of U-statistic Based Estimators
Bo E. Honoré and Luojia Hu

WP-15-07

Bad Investments and Missed Opportunities?
Postwar Capital Flows to Asia and Latin America
Lee E. Ohanian, Paulina Restrepo-Echavarria, and Mark L. J. Wright

WP-15-08

Backtesting Systemic Risk Measures During Historical Bank Runs
Christian Brownlees, Ben Chabot, Eric Ghysels, and Christopher Kurz

WP-15-09

What Does Anticipated Monetary Policy Do?
Stefania D’Amico and Thomas B. King

WP-15-10

Firm Entry and Macroeconomic Dynamics: A State-level Analysis
François Gourio, Todd Messer, and Michael Siemer

WP-16-01

Measuring Interest Rate Risk in the Life Insurance Sector: the U.S. and the U.K.
Daniel Hartley, Anna Paulson, and Richard J. Rosen

WP-16-02

Allocating Effort and Talent in Professional Labor Markets
Gadi Barlevy and Derek Neal

WP-16-03

The Life Insurance Industry and Systemic Risk: A Bond Market Perspective
Anna Paulson and Richard Rosen

WP-16-04

Forecasting Economic Activity with Mixed Frequency Bayesian VARs
Scott A. Brave, R. Andrew Butters, and Alejandro Justiniano

WP-16-05

Optimal Monetary Policy in an Open Emerging Market Economy
Tara Iyer

WP-16-06

Forward Guidance and Macroeconomic Outcomes Since the Financial Crisis
Jeffrey R. Campbell, Jonas D. M. Fisher, Alejandro Justiniano, and Leonardo Melosi

WP-16-07

Working Paper Series (continued)
Insurance in Human Capital Models with Limited Enforcement
Tom Krebs, Moritz Kuhn, and Mark Wright

WP-16-08

Accounting for Central Neighborhood Change, 1980-2010
Nathaniel Baum-Snow and Daniel Hartley

WP-16-09

The Effect of the Patient Protection and Affordable Care Act Medicaid Expansions
on Financial Wellbeing
Luojia Hu, Robert Kaestner, Bhashkar Mazumder, Sarah Miller, and Ashley Wong

WP-16-10

The Interplay Between Financial Conditions and Monetary Policy Shock
Marco Bassetto, Luca Benzoni, and Trevor Serrao

WP-16-11

Tax Credits and the Debt Position of US Households
Leslie McGranahan

WP-16-12

The Global Diffusion of Ideas
Francisco J. Buera and Ezra Oberfield

WP-16-13

Full text of Working Papers (Federal Reserve Bank of Chicago) : The Global Diffusion of Ideas, Working Paper 2016-13

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