The Economics of Innovation, Knowledge Diffusion, and Globalization
Summary and Keywords
A recent body of literature on quantitative general equilibrium models links the creation and diffusion of knowledge and technology to openness to international trade and to the activity of multinational firms. The unifying theme of this literature is methodological: productivities are Fréchet random variables and arise from Poisson innovation and diffusion processes for ideas. The main advantage of this modeling strategy is that it delivers closed-form solutions for key endogenous variables that have a direct counterpart in the data (e.g., prices, trade flows). This tractability makes the connection between theory and data transparent, helps clarify the determinants of the gains from openness, and facilitates the calculation of counterfactual equilibria.
Motivating Empirical Evidence
A classic question in international economics has always been if—and how—openness increases the wealth of nations. A large part of the literature seeks to establish an empirical link between income and openness, while, simultaneously, a growing theoretical literature explores mechanisms through which openness increases income.
This article reviews the literature on quantitative general equilibrium models, spurred by Eaton and Kortum (2002) (EK), that link innovation and knowledge diffusion to international trade and the activity of multinational firms—henceforth, multinational production (MP).
The unifying theme of this review is methodological. All the models reviewed are models that treat productivity as random draws from a Fréchet distribution. This treatment of productivities has been the transformative contribution of EK.
Given this organizing methodology, it is natural to start by analyzing growth models of innovation that generate Fréchet production possibilities frontiers. The article first presents extensions of the EK model of trade that incorporate technology innovation through semi-endogenous growth models (as in Jones, 1995; Kortum, 1997; Jones, 2005), and international technology diffusion (as in Eaton & Kortum, 1999, 2001). EK-type models in which trade is the channel for international diffusion, as in Buera and Oberfield (2016), are also reviewed. Extensions of the EK model of trade in which the channel for international technology diffusion is multinational production (MP), as introduced by Ramondo and Rodríguez-Clare (2013), are analyzed next. The article finalizes reviewing the research in Lind and Ramondo (2018), which generalizes the EK model of trade to capture spatial correlation of technologies. This generalization encompasses most of the EK extensions in the literature and links to the innovation and diffusion models that generate Fréchet technology frontiers.
Different theories have been (fully and partially) motivated by the empirical evidence on international technology diffusion and the role of international trade and MP as channels for technology transfer across countries. Most of the studies focus on outcomes, such as productivity and income, in receiving countries.
First, a large literature documents the link between growth and openness to trade. Cross-country evidence suggests that more open economies have higher income levels and also they grow faster. Early evidence in Sachs and Warner (1995) showed that a group of eight always-open economies have consistently higher growth rates in the period 1965 to 1990 than 40 always-closed economies.1 Alcala and Ciccone (2004) took into account the endogeneity of trade openness and estimated that the elasticity of productivity with respect to real openness is around 1.2, after controlling for quality of institutions, country size, and geography.2 More recent studies try and establish a causal link between openness and growth by resorting to natural experiments. In that regard, Feyrer (2009) used the closure of the Suez Canal during the Six-Day War to establish the effect of distance on trade flows and of trade on income. Effects are large: A 10% decrease in ocean distance results in a 5% increase in trade, and one more dollar of trade increases the income by about 25 cents. Finally, Pavcnik (2002) uses detailed firm-level data to document the effects of trade liberalization on industry productivity and turnover (see Harrison & Rodríguez-Clare, 2010, for a survey).
Second, a similarly large empirical literature documents how openness to foreign firms impacts receiving countries (see Harrison & Rodríguez-Clare, 2010; Keller, 2010; Antrás & Yeaple, 2014, for surveys). The basic premise is that MP is a channel through which new technology diffuses to firms in developing countries. The primary focus, as for trade flows, has been on the effect of Foreign Direct Investment (FDI) on the productivity of the receiving industries and domestic firms. Using firm-level data, the typical empirical tests revolve around measuring: the advantage of multinational firms, relative to domestic firms, in terms of firm outcomes such as productivity and revenues; and whether more exposure to foreign presence increases the productivity of domestic firms (“spillover”).3 The firm-level evidence robustly finds that multinational firms—and their foreign affiliates—perform better than domestic firms in several dimensions. This result provides support to the hypothesis that multinational firms not only have better technologies, but they transfer them to their affiliates abroad, thus constituting a direct positive effect on the receiving industry. In contrast, the evidence on spillovers is mixed; robust findings are mostly related to vertical, not horizontal, linkages with domestic firms (see Javorcik, 2004).
Finally, the gains from globalization can materialize through direct international technology diffusion flows that allow domestic firms to use foreign technologies. This flow is difficult to measure. As noticed in Ramondo, Rodríguez-Clare, and Saborío-Rodríguez (2016), “Except for the small part that happens through licensing, technology diffusion does not leave a paper trail that can be used to directly measure the value of production done in a country by domestic firms using foreign technologies.” Some indirect evidence shows that international technology diffusion is responsible for most of the productivity growth in OECD countries, except for the United States (see Eaton & Kortum, 1996, 1999; Keller, 2004). Comin and Hobijn (2004) undertook a more direct effort to measure international technology diffusion and adoption patterns across countries and link them with income convergence. They build an impressive dataset that spans two centuries, covers as many as 166 countries—starting with 23 industrialized economies—and between 15 and 20 technologies belonging to textile production, steel manufacturing, communications, information technology, transportation, and electricity. In this and subsequent papers (Comin & Hobijn, 2010; Comin, Dmitriev, & Rossi-Hansberg, 2013; Comin & Mestieri, 2018), the authors document the characteristics and extent of international diffusion and adoption. The salient facts are that most technologies originate in advanced economies, get adopted there first, and subsequently diffuse to countries that lag economically. Newer technologies get adopted faster than older technologies, and adoption rates are heterogeneous across countries, even though adoption lags between rich and poor countries are narrower than in the 19th and beginning of the 21st century. The differences in adoption rates account for at least 25% of per capita income differences, and the degree of openness to trade is among the most important determinants of the speed of adoption. Finally, technology diffusion follows a spatial pattern where nearby countries adopt one another’s technology more rapidly.
The Ricardian Model of Trade
This section presents the workhorse model of trade introduced by EK. Consider a global economy consisting of countries. Countries produce and trade in a continuum of product varieties . Consumers in all countries have identical constant-elasticity-of-substitution (CES) preferences with elasticity of substitution . Expenditure by destination on variety is where is the cost of the variety in terms of numeraire, is the price level in country , and is the country’s total expenditure.
Assume that the production function for varieties presents constant returns to scale in labor and depends on the origin country where the good gets produced. For each , output satisfies(1)
where is the amount of labor used to produce variety at origin and is the marginal product of labor—referred as productivity.
Suppose that to ship goods from country to there is an iceberg-type cost : Firms from need to ship units of the good for one unit to arrive to . Additionally, the usual triangular inequality is satisfied, for any , , and , , and trade is frictionless within a country, .
The marginal cost to deliver good to destination from origin is then(2)
where is the nominal wage in country . As in the original model, we assume perfect competition so that prices are equal to unit costs, . Good is provided to country by the cheapest supplier,(3)
Modeling productivity as a random draw across varieties captures heterogeneity in production possibilities. EK assumed that productivity is a Fréchet random variable, independent both across the continuum of goods and across exporters. The joint distribution of productivity across countries is(4)
where is the scale parameter for origin and is a common shape parameter. A higher scale parameter means that a country has on average better productivity draws (i.e., larger ‘s increases the probability of larger values of ), capturing aggregate productivity of country (i.e., its absolute advantage). A higher shape parameter means less heterogeneity in productivity draws across the continuum of goods (within each origin), whereby capturing relative productivities differences (i.e., comparative advantage).
Under independent Fréchet with a common shape parameter, tractability is achieved thanks to a property known as max stability: The Fréchet distribution is preserved under the maximum operator,(5)
This max-stability property is key for deriving equilibrium prices and bilateral trade shares in closed form.
In particular, the joint distribution of productivity determines the joint distribution of potential import prices. Using equation (2), the distribution of potential import prices into a destination country is given by(6)
where . For each origin , the marginal distribution of prices, , is a Weibull distribution with scale parameter and shape parameter .
Using equation (6), the distribution of prices actually paid by consumers in is(7)
The scale is and the shape is . The shape parameter indicates the price dispersion across the continuum of goods, while the scale summarizes the overall level of prices in the destination market. This variable indicates how easy it is to access market because it determines the overall price index in the destination market.
Using equation (7), the CES price index in country is calculated as the following expected value(8)
where and is the Gamma function. This expression states that countries that are easy to access have lower , and therefore, a lower price index; in autarky (i.e., for all ), prices are the highest and ; and in a frictionless world (i.e., , for all ), the law of one price holds with , for all .
Given the distribution of prices, we can also solve for the fraction of goods that destination imports from source . Because draws across goods are i.i.d. and there is a continuum of goods, by the law of large numbers, the fraction of goods imported from into is equal to the probability that country is the lowest-cost producer of good to destination ,(9)
The lower bilateral trade costs , and the higher source productivity relative to other sources, the larger the fraction of goods that country imports into .
The probability in equation (9) coincides with the share of expenditure of on goods from due to a key feature of the EK model: The distribution of prices among goods sold in a destination is equal to the distribution of prices among goods sold by any source to the destination. This result is a consequence of the max-stability property of the Fréchet distribution, which implies that the conditional and unconditional distribution of the maximum (minimum) are identical. This translates into identical distribution of prices in unconditional and conditional on a source being the lowest-cost producer for ,
Origins with better comparative advantage (i.e., lower trade costs, better productivity, or lower labor costs) in serving sell more goods to exactly to the point where the distribution of prices of their goods sold to is the same as the overall price distribution in .
Total expenditure in country devoted to goods from country is then given by(10)
Expenditure is a CES import demand system with elasticity of substitution . As a result, in the words of Arkolakis et al. (2012), it constitutes a gravity system since it can be expressed as a log-linear function of two set of country effects and a bilateral term,(11)
where and .(12)
The share of goods that a country imports is referred as a country’s openness. This equation states that real per-capita income for a country is given by its degree of openness and its average productivity level. The same parameter, , governs the influence of both openness and average productivity on welfare.
In autarky, the real wage is simply(13)
so that the gains from trade for country , defined as the change in real wages from autarky to the actual equilibrium, is(14)
The gains from trade are positively related to the degree of openness.
This expression, together with equation (12), makes clear that the parameter is key in several dimension: it captures the degree of productivity dispersion across goods, equals the elasticity of trade expenditure to trade costs, determines the sensitivity of the gains from trade to openness and the sensitivity of income to average productivity in a country. For instance, a lower indicates more heterogeneity, stronger comparative advantage forces, stronger effects of openness and average productivity on income.
The next section turns to innovation and diffusion models that generate production possibility frontiers that are Fréchet.
Innovation and Diffusion in the Ricardian Model of Trade
This section reviews the literature that adds knowledge creation and diffusion to the EK model of trade. First, it shows how the literature has gone about generating production possibility frontiers that are Fréchet, following Kortum (1997). Second, it presents an EK-type model of trade based on Buera and Oberfield (2016) in which international diffusion of knowledge occurs through trade. Finally, it analyzes the extension of the EK model of trade to multinational production (MP) in Ramondo and Rodríguez-Clare (2013) in which diffusion of knowledge occurs through the activities of multinational firms.
Fréchet as a Result of Innovation
The previous section showed that due to the max-stability property, the Fréchet distribution is the “right” distribution to solve Ricardian-type models in closed form. This is because the key feature of these models is a head-to-head comparison of unit costs across producers of the same good for destination market . But what economic justification underlies the Fréchet assumption?
The basic idea was introduced in Kortum (1997). He shows that if the production technology is determined by the best “idea,” or blueprint, and if ideas become available according to a Poisson process, then after a sufficiently long period productivity can be approximated by an extreme value distribution.4
For illustration purposes, suppose that the number of ideas available for producing a good is . Assume that the productivity of each idea is drawn i.i.d. from a distribution , referred to as the exogenous distribution of ideas. Consider the productivity of the most efficient idea: . Then the distribution of the maximum is .5
One important insight from Kortum (1997) is that the limiting distribution belongs to the extreme value family if the number of ideas available is itself a Poisson random variable with average number of innovations —so that . In this case, the productivity distribution of the best idea has a closed form,
for any such that , and otherwise.
A Fréchet distribution arises if the distribution of productivity across innovations has a Pareto right tail, as specified in Assumption 1.
Assumption 1 (Pareto-Tailed Exogenous Distribution of Ideas). There exists such that
Under this assumption, for sufficiently large , the productivity distribution is approximately Fréchet, as in equation (4),
As the scale parameter of the Poisson process gets large, most goods have many innovations, and the distribution converges to an extreme value distribution. The specific limiting distribution depends on the properties of the innovation distribution, . Under Assumption 1, the limiting distribution is Fréchet.
Eaton and Kortum (2001) developed a model of trade based on this result for the special case of Pareto innovations: , for .6 Researchers in country draw ideas independently across countries at a Poisson rate with ideas distributed uniformly across goods. The distribution of innovations across varieties is then independent across countries and Poisson. Applying the previous result, if as , the distribution of productivity within each country is asymptotically Fréchet and independent across countries.7
Building on the idea in Kortum (1997) for a closed economy, Eaton and Kortum (1999) present a model in which ideas diffuse across countries and in each country the distribution of productivities is Fréchet. In this model, however, the international diffusion of ideas occurs at exogenous rates, and countries are otherwise in autarky. That is, international trade is not a channel through which ideas get diffused, or diffusion accelerates; international trade and international diffusion are substitutes.8
However, as documented by Comin et al. (2013), international diffusion follows a spatial pattern that may be linked to trade and other international flows such as the activity of multinational firms (as implied by the empirical evidence in Keller & Yeaple, 2013). The next two sections present models in which diffusion occurs through trade and multinational activity, respectively.
Diffusion Through Trade
This section focuses on a recent paper by Buera and Oberfield (2016), which incorporates innovation and international diffusion into an EK-type model, but, crucially, international diffusion happens because of international trade.9 Their model belongs to the class of models that generate a Fréchet production possibility frontier for each country. Moreover, the model delivers a simple system of differential equations for the evolution of the country’s stock of knowledge—the scale parameters of the Fréchet distributions.
Assume that for each good there are competitive firms within each country.10 For each firm and time , the production function for good is the same as in equation (1) and marginal cost is the same as in equation (2). Perfect competition implies that in each country the firm with the lowest marginal cost produces good . The frontier of knowledge in for producing is then given by the productivity of the most efficient firm, .
Provided that is independent across within , the frontier distribution of productivity is
This distribution is the result of taking the maximum of i.i.d. random variables. As explained in the previous section, if the distribution converges as (with an appropriate normalization), then it must converge to an extreme value distribution.
Assume that firms within origin draw insights independently at a Poisson rate . These insights may come from firms worldwide. The distribution of productivity from which ideas are drawn, the source distribution, is . To start, this distribution is taken as exogenous. When an idea arrives to a firm, it has productivity . The component is drawn from the source distribution and captures learning from others. The parameter controls the degree of diminishing returns in the adoption of ideas. The component is drawn from a time-invariant distribution , the exogenous distribution of ideas, and represents randomness in adapting the idea. A firm with original productivity adopts the idea if it increases its productivity, . As a result, for a small time increment , the distribution of productivity is given by
This distribution is given by the old distribution of productivity multiplied by the share of varieties for which productivity did not change due to diffusion. This share equals the share of varieties for which no new insight arrived, , plus the share of varieties for which new insights arrived, , but none of them were an improvement over previous productivity. The quantity is the chance that an insight does not improve the firm’s productivity.
As a consequence, the frontier distribution evolves as
The evolution of the frontier distribution in any given origin depends on the rate at which ideas diffuse to firms and the source distribution from which firms draw ideas.
We get a Fréchet frontier distribution for productivity in each origin by considering the limiting case of this economy as . We need the two following assumptions in addition to Assumption 1.
Assumption 2 (Diminishing Returns to Learning). .
Assumption 3 (Thin-Tail Source Distribution). , .
The assumption of diminishing returns implies that the model generates semi-endogenous growth. The second assumption is a regularity assumption that ensures that growth in the frontier distribution of productivity remains finite at each point in time.11 Later, when the source distribution is endogenized, Assumption 3 is replaced with an analogous restriction on the initial frontier distribution in each country.
Similar to the approach in the previous section, the frontier distribution of productivity is scaled by , and defined as , and . Changing variables yields
As , due to the dominated convergence theorem, the limiting case is given by
For any fixed and initial , this equation is a differential equation with solution
One can immediately see that if the initial knowledge frontier is Fréchet with scale and shape , then the distribution at is also Fréchet with the same shape, . Additionally, the evolution of the scale parameter reflects the Poisson rate at which ideas diffuse into country and the source distribution of productivity from which ideas are drawn,(15)
This model of knowledge diffusion implies that at any point in time productivity is distributed independently Fréchet across countries with a common shape parameter. As a result, the static trade theory is identical to the EK model of trade.
To examine the implications of the static trade equilibrium for the dynamics of knowledge diffusion, one needs to endogenize the source distribution of productivity. In what follows, the focus is on a specification in which domestic firms can learn—and adapt ideas—from those firms in the global economy that sell goods in . Following the assumption in Alvarez, Buera, and Lucas (2014), the source distribution of productivity in country is given by(16)
The set includes varieties produced in and sold in (i.e., is the lowest-cost supplier to ) at time . International trade matters for the evolution of the frontier productivity distribution because access to new ideas depends on the set of firms (foreign and domestic) that sell in the domestic market.
Given the endogeneity of the source distribution, Assumption 3 has to be replaced by the following condition on the initial frontier of knowledge in each country.
Assumption 4 (Thin-Tail Initial Distribution). , .
Buera and Oberfield (2016) showed that in the case of an endogenous source distribution as in equation (16), Assumption 3 is the result of this assumption on the initial distribution, and, hence, the results derived for an exogenous source distribution go through. In particular, the law of motion for the scale parameter in equation (15) specializes to
where is the Gamma function. Aggregate productivity growth in country depends on the exogenous rate at which domestic firms draw ideas, , as well as an expenditure-weighted average of the average productivity among those in exporting into —the quantity —to the power. Trade influences the diffusion of knowledge through two channels. International competition increases productivity growth because trade allows for more productive foreign firms to sell in the domestic market. When a country lowers its trade barriers, low-productivity domestic producers lose market share to high-productivity foreign producers (expenditure shares shift toward countries with higher ). As a result, freer trade improves the average productivity of the set of firms selling in the domestic market, which means faster learning and higher growth. The second channel is an offsetting selection effect: with lower trade barriers, many low productivity foreign firms can compete in the domestic market. As trade barriers on foreign country fall, the expenditure share increases and the average productivity of firms from sell in falls. This selection toward low-productivity foreign firms dampens learning and growth.
An important implication of these two offsetting effects is that growth is not necessarily maximized at free trade. Rather, growth is maximized when trade is biased toward those foreign countries with better technology than the domestic country. Buera and Oberfield (2016) showed that this maximum is achieved by choosing relative trade barriers proportional to wage differences across trading partners. Intuitively, average productivity is high among firms in countries with high wages. Hence, tilting trade barriers to increase trade with high wage countries will lead to more rapid diffusion of knowledge and higher growth.
What are the gains from trade after accounting for trade-induced knowledge diffusion? Since the import demand system is CES, the results of Arkolakis, Costinot, and Rodriguez-Clare (2012) apply: A country’s real wage, at each time , is given by the expression in equation (12), and the static gains from trade (holding fixed ) are as in equation (14).12
To gain intuition for the dynamic gains from trade, it is useful to consider the case of a symmetric world economy with cross-country trade barriers and within country trade barriers equal to one. In this case, each country’s self-trade share is constant, , and the import trade share with each partner is . Additionally, in order to incorporate exogenous growth of the global economy, assume that the arrival rate of ideas grows over time at rate . Let denote de-trended productivity and, analogously, . On a balanced growth path (i.e., constant ) the evolution of aggregate (de-trended) productivity is given by
where . This result implies that along a balanced growth path aggregate (de-trended) productivity is constant,
The dynamic gains from trade, which incorporate the difference in productivity relative to autarky along a balanced-growth path (BGP), are then13
While the first term represents the static gains from trade, as in Arkolakis et al. (2012), the second term captures the dynamic effect of knowledge diffusion and learning. When there is no learning ( ), the model collapses to EK, and the dynamic gains from trade coincide with the static gains from trade. For , the increase in aggregate productivity depends on the degree of openness of the economy: As a country opens up to trade, it gains opportunities to learn from foreign firms, and the productivity of their domestic firms increases. As the force of diminishing returns in learning weakens ( increases), the effect is magnified, and the dynamic part of the gains from trade becomes more and more important.
Buera and Oberfield (2016) consider an alternative way of endogenizing the source distribution of productivity: Domestic firms learn from other domestic producers rather than from worldwide sellers.14 In this case, the source distribution of productivity in country is specified as(17)
where is the set of varieties produced and sold in at time . Following the derivations in Buera and Oberfield (2016) and specializing equation (15) to the source distribution in equation (17) delivers
Trade increases a country’s stock of knowledge because as a country opens up to trade, unproductive technologies are selected out, and more productive domestic producers survive. This raises the average quality of idea draws and increases the growth rate of the stock of knowledge.
As in the case in which learning occurs through global sellers, the dynamic gains from trade consider not only the static gains from trade but also the gains coming from the changes in the stock of knowledge due to diffusion. In a symmetric world, the de-trended aggregate productivity in a balance growth path is simply
so that the gains from trade in a balance growth path are
As for the case of a source distribution in which domestic firm learn from worldwide sellers, with no learning , the dynamic and static gains from trade coincide. For , learning opportunities increase with openness due to the selection effect on unproductive domestic producers, with a larger effect as increases.
The model developed in Buera and Oberfield (2016) constitutes a parsimonious, yet quantitative, theory in which diffusion and international trade interact to shape the gains from openness. As with all the models reviewed in this article, the Fréchet productivity distribution is at the center of their analysis.
Diffusion Through Multinational Production
Multinational production (MP) is arguably the most important channel through which firms choose to serve foreign consumers.15 As documented by UNCTAD (2017), in 2016, world sales of foreign affiliates of multinational firms were almost twice as high as world exports, with an increase over the 1990–2010 of a factor of more than six against a factor of almost four for exports. Additionally, according to the Bureau of Economic Analysis (BEA), foreign affiliates of U.S. MNEs accounted, in 2009, for 75% of U.S. sales to foreign customers; the remaining 25% was made of U.S. exports.
Multinational firms also are, not surprisingly, the top innovators in the world. UNCTAD (2005) estimates that multinational firms constituted close to half of world R&D expenditure, and at least two-thirds of business R&D expenditures at the beginning of the 2000s. The link between R&D and MP activity is also documented by Arkolakis, Ramondo, Rodriguez-Clare, and Yeaple (2018), which showed that the most innovative OECD economies, measured by R&D expenditures in manufacturing relative to local value added, are also the ones with net outward MP flows (measured by the sales of foreign affiliates of home firm minus the sales of foreign multinational affiliates at home).16
Facts documented by Bilir (2014) using detailed firm-level data suggest that multinational firms are indeed a channel for international technology transfer: Intellectual property rights have a positive effect on the location decision of multinational firms and more so if they operate in sectors in which the scope for imitation is higher. Further evidence is provided in Bilir and Morales (2018). Using a long panel of U.S. multinational firms from the BEA, they document that innovation at a parent company substantially increases the efficiency of affiliates abroad: The average U.S. multinational firm realizes over a quarter of the gains from innovation abroad.
Both the macro and micro evidence suggests that MP is an important channel through which countries exchange not only goods, but also ideas and technologies, and, as such, it may lead to large gains from international technology sharing.
The idea of MP as a channel for international technology diffusion was present in early work by Markusen (1984): His concept of “knowledge capital” entailed the idea that this type of capital was freely shared within different units of the firm, at home and abroad. Building on the idea of non-rivalry, more recent papers develop and quantify models in which international technology diffusion occurs through MP. Using a neoclassical growth model, McGrattan and Prescott (2009) pair MP with the transfer (and reproduction) of firm-specific technology capital in foreign countries.17 They calculate (steady state) gains from MP (from autarky) of 27% of real consumption, for a small country, and of 1% for a large country. Ramondo (2014) used an EK-type framework in which MP is paired with the (costly) transfer of home technologies abroad. She calculates gains from MP (from autarky) of almost 9% of real per capita income. Irarrazabal, Moxnes, and Opromolla (2013) extend the proximity-concentration tradeoff framework in Helpman, Melitz, and Yeaple (2004) to incorporate intrafirm transfers of firm-specific inputs, both tangible and intangible. Using firm-level data for Norway, they estimate that the share of a parent’s input costs in the affiliate total costs is substantial (9/10). Similarly, Shikher (2012) and Ramondo and Rodríguez-Clare (2013) also pair MP with the international transfer of technologies for production abroad. Their model incorporates both trade and MP into an EK-type model with the goal of quantifying the gains from trade and MP. In particular, Ramondo and Rodríguez-Clare (2013) extend the probabilistic representation of technologies introduced by EK, and show that their framework is consistent with growth models that generate a Fréchet production possibility frontier.18
Next, we present a simplified version of the model in Ramondo and Rodríguez-Clare (2013). We abstract from non-tradable goods, intermediate tradable goods, an input-output structure, and the shipment of inputs from the home country to the country of the affiliate. These are all important components to be taken into account when quantifying the model; this section just aims to illustrate how MP can be incorporated into the canonical Ricardian model as a channel for technology diffusion.
The model inherits the main features of the EK Ricardian model of trade, but it is extended so that both trade and MP are two possible ways of reaching foreign consumers. As in models of “horizontal” FDI (e.g., Helpman et al., 2004) trade and MP are competing ways to serve a foreign market. Additionally, foreign affiliates of multinational firms can serve markets other than where they produce through exports (i.e., export-platform FDI).
There are countries and a continuum of tradable goods produced under constant returns to scale with only labor. Knowledge transfers through MP take the following form. Firms in country have a technology to produce each good at home and in each foreign country. These technologies are described by the vector . MP occurs when a country produces in another country with productivity . If for all , and for all , then the model reverts to the EK model of trade.
The productivity vector for each good is a random variable drawn independently across goods and countries from the following distribution,(18)
where and . The distribution in equation (18) is called a symmetric multivariate -Fréchet distribution for reasons that will become clear in the next section.19 The scale and shape parameters are, respectively, and , and have the same interpretation as in the baseline trade model: Lower means more heterogeneous productivity draws and hence stronger comparative advantage forces; higher means better productivity draws on average for firms from producing anywhere. The parameter determines the degree of correlation among productivity draws in : If , productivity draws are uncorrelated across production locations, while as productivity draws are perfectly correlated across locations of productions for a good —that is, , for all .
Assume that trade is subject to iceberg-type costs: units of any good must be shipped from country for one unit to arrive in country , with for all and for all . Similarly, MP from country with production in incurs a productivity loss modeled as an iceberg-type bilateral MP cost, , with , for all . These costs are meant to capture the various costs of technology transfer incurred by multinational firms when they operate in a different production location other than their country of origin. Finally, following Head and Mayer (2018), assume that MP from country selling in destination market incurs a loss also modeled as an iceberg-type bilateral cost, , with . As they argue, this loss is meant to capture marketing costs incurred because the headquarters of the firm is different from the market where consumers are located.20
The marginal cost of delivering good to destination produced in location with technologies originated in country is(19)
where . Good is delivered to consumers in : through trade if ; through horizontal FDI if ; through export-platform FDI if ; and through vertical FDI if .21 Domestic firms will be the ones serving market in good when .
Under perfect competition, unit costs are equal to prices, and the lowest-cost producer will deliver good to market ,(20)
Given that the symmetric -Fréchet distribution has the max-stability property—for reasons explained in the next section—the same reasoning as for the EK model of trade can be applied and similar results derived.
The expenditure share on goods produced in for destination with technology from is(21)
where , , and the price index in is(22)
where with the Gamma function. The first term on the right-hand side of equation (21) is the share of expenditure that country allocates to goods produced with country ‘s technology using labor in country , while the second term is the overall share of goods produced using ‘s technology.
The aggregate expenditure share on goods produced in for —that is, the bilateral trade share—is , while the expenditure share on goods from produced in . That is, the bilateral MP share—is .
The questions we ask are: How does trade and diffusion through MP interact to shape the gains from openness? Does trade complement or substitute diffusion forces that act through MP? Ramondo and Rodríguez-Clare (2013) defined the concepts of substitution and complementarity through the relation between the implied gains from trade (MP) in the model with both flows, and the implied gains from trade (MP) implied by a model with only trade (MP). The gains from trade (MP) in the full model, denoted by ( ), are given by the change in the real wage from a situation without trade (MP), but MP (trade), to a situation with both trade and MP. The gains from trade (MP) coming from a model with only trade (MP), denoted by ( ), are given by the change in the real wage from isolation to an equilibrium with only trade (MP) but in which trade (MP) flows are the same as in the model with both trade and MP (i.e., the observed flows). Then: If , trade is MP complement; if trade is MP substitute; and if trade is MP independent. Analogously, if , MP is trade complement; if , MP is trade substitute; and if , MP is trade independent.
Even the simplified model presented here delivers rich patterns of complementarity and substitutability between the two modes of openness. The key parameter is the parameter indicating correlation in productivity across production locations, . As Ramondo and Rodríguez-Clare (2013) showed, the special case of independence, , is particularly illuminating.
First, when , trade and MP shares collapse, respectively, to
Assuming that , both trade and MP shares satisfy gravity equations,
where , , , and .
Second, under independence, the gains from trade are exactly equal to the gains from trade coming from a model with only trade as in Arkolakis et al. (2012),(23)
and trade is MP independent. Independent productivity draws across production locations basically make the tradeoff between trade and MP irrelevant as alternative ways of serving a foreign market.22 MP, however, is not trade independent. The gains from diffusion through MP for a country that it is already open to trade are given by(24)
where is the domestic trade share in the equilibrium with trade but not MP. Because the gains from MP coming from an MP-only model are given by the first term on the right-hand side of equation (24), the expression in equation (24) implies that MP can be either trade substitute, trade complement, or trade independent. The size of trade costs, countries, and aggregate productivity interacts to shape the decision of serving markets through trade, MP, and export-platform MP, and determines the relation between and .
For the case of , Ramondo and Rodríguez-Clare (2013) provided special cases to illustrate the different forces that determine whether trade and MP are substitutes or complements of each other. For instance, they are able to show analytically that, with symmetric countries, substitution forces dominate: Trade is an MP substitute, , and MP is a trade substitute, . The calibrated version of their model, which matches the observed bilateral trade and MP data, implies that the gains from trade can be twice as high as the gains calculated in trade-only models, suggesting that diffusion of technologies through MP complements trade flows; in contrast, MP is a mild trade substitute.23
There is an extensive literature that builds and quantifies models of trade and MP using static EK-type approaches. For instance, Alviarez (2018) extends the model in Ramondo and Rodríguez-Clare (2013) to many sectors and shows that the gains from MP are three times higher than in the model with one sector. Arkolakis et al. (2018) extended Ramondo and Rodríguez-Clare (2013) to allow for heterogeneous production firms á la Melitz and creation of firms through free entry—what they call innovation. They use their framework to quantify the gains and losses of globalization for innovation and production workers. Tintelnot (2017) incorporated fixed costs of production into a framework similar to Arkolakis et al. (2018). Solving a complex permutational problem, he calibrates the model and evaluates the role of fixed costs in generating export platforms and its consequences on the gains from MP. Fan (2017) extended Arkolakis et al. (2018) to further incorporate the possibility of offshore R&D: Multinational firms established in a given country match with researchers around the world to develop new blueprints and then engage in production—possible offshore—and exports. That is, in his framework, the home country of a multinational firm can be different from the place where they decide to carry innovation, and in turn, different from the production sites and consumer markets. When he quantifies the model, he finds that offshore R&D is an important channel shaping the gains from globalization. Finally, Sun (2018) used the framework in Arkolakis et al. (2018) to add a firm’s technology choice: MP not only implies the transfer of Hicks-neutral productivity but also the transfer of technologies of (varying) factor intensity. In a model with both capital and labor as factors of production, he shows that MP, by coming from capital-abundant countries and transferring capital-intensive technologies to its host countries, changes the aggregate demand of capital relative to labor, and hence, contributes to the decline in the labor share observed in several countries.
Two common themes in all these papers are that MP is a channel for international technology diffusion and that the modeling strategy is based on the EK Ricardian model of trade with a Fréchet productivity distribution. By having flexible general equilibrium models that capture the rich heterogeneity of the data, all of these papers are able to draw quantitative conclusions through counterfactual exercises on the effects of MP on their host and home countries.
New Directions for the Ricardian Model of Trade
The EK model of trade assumes that productivities are independent Fréchet random variables across countries with a common shape. This assumption is restrictive in several ways: First, it does not allow for spatial correlation in technology, as the empirical evidence seems to suggest Comin et al., 2013; Keller & Yeaple, 2013); second, as pointed out by Arkolakis et al. (2012), it restricts trade shares to the CES class, generating symmetric substitution patterns (i.e., a change in unit costs has the same effect on the aggregate price index of the importer regardless of the identity of the exporter); and finally, it does not include non-CES extensions of the EK model of trade (e.g., many sectors, multinational production, global value chains). The appeal of this assumption, however, has been that it generates max-stability—the distribution of the maximum is the same up to scale—which, in turn, is the key property that makes the EK model tractable.
Lind and Ramondo (2018) generalized the distribution of productivities in EK to a multivariate -Fréchet distribution, allowing for an arbitrary spatial correlation structure of productivity while maintaining the key max-stability property necessary for tractability of the EK Ricardian model. This generalization also encompasses various extensions of the EK model in the literature to non-CES setups. Moreover, it naturally links to the models of innovation and diffusion that generate Fréchet production possibility frontiers.24
Productivity as a Max-Stable Multivariate Fréchet Random Variable
This section characterizes multivariate max-stable Fréchet distributions and their properties, following the results from Lind and Ramondo (2018).
Definition 1. (Multivariate -Frìhet distribution). A random vector, , has a multivariate -Fréchet distribution if for any with the random variable has a Fréchet distribution with shape parameter . In this case, the marginal distributions are Fréchet with (common) shape parameter and, for each , satisfy(25)
for some scale parameter .
This definition implies that a multivariate -Fréchet distribution is max stable—up to scale the distribution of the max is the same as the distribution of the marginals. It includes as special cases the distribution in EK which is independent -Fréchet, and the symmetric -Fréchet distribution in Ramondo and Rodríguez-Clare (2013).
The class of multivariate -Fréchet random variables puts minimal restrictions on dependence, but maintains the property of max stability.25 The joint distribution of a multivariate -Fréchet random variable can be characterized by a correlation function as defined next.
Definition 2. (Correlation Function). is a correlation function if: (1) Normalization: ; (2) Homogeneity: is homogeneous of degree one; (3) Unboundedness: as for any ; and (4) Differentiability: The mixed partial derivatives of exist and are continuous up to order , with the ‘th partial derivative of with respect to distinct arguments non-negative if is odd and non-positive if is even.
A correlation function is closely related to a max-stable copula and adds a normalization restriction to the social surplus function in GEV discrete choice models (McFadden, 1978). This normalization allows us to distinguish between absolute advantage—captured by scale parameters—and comparative advantage, which is captured by a correlation function.
The joint distribution of any multivariate -Fréchet random vector, , can always be parameterized in terms of the scale parameters of the marginal distributions, for , and a correlation function with the properties in Definition 226,(26)
The restrictions defining a correlation function ensure that equation (26) characterizes a valid multivariate Type II extreme value (Fréchet) distribution.
Importantly, max-stability arises from the characterization in equation (26) and the homogeneity property of the correlation function. The maximum of a multivariate -Fréchet random vector is -Fréchet,(27)
with scale parameter given by and shape parameter given by . The correlation function acts as an aggregator, returning the scale parameter of the maximum when evaluated at the scale parameters of the marginal distributions.
Correlation functions reflect comparative advantage because they measure relative productivity levels across varieties and across origin countries within the same destination market. Comparing the case of independence in equation (5) with equation (27), it is clear that the EK model has an additive correlation function. But an additive correlation function imposes a strong assumption, namely that comparative advantages across countries are symmetric. By allowing for heterogeneity in correlation, this generalization of the EK model allows for heterogeneity in comparative advantage. Moreover, the conditional and unconditional distributions of the maximum are identical, . This property, as in EK, implies that expenditure shares simply reflect the probability of importing from an origin country.27
The next question is: What are the implications for trade shares and the gains from trade of using a multivariate -Fréchet distribution with arbitrary correlation across exporters?
The joint distribution of productivity determines the joint distribution of potential import prices into market ,
Given the distribution of potential import prices, a country imports each variety from the cheapest source. As for the Ricardian model under independence, the max-stability property, together with the previous characterization of the potential import price distribution, leads to closed-form results for trade shares and the price index, respectively,(28)
where , , and(29)
with and the Gamma function.
The formula for the expenditure share in equation (28) takes the same form as a choice probability in generalized extreme value (GEV) discrete choice models (McFadden, 1978), with the import price index taking the place of choice-specific utility. Accordingly, the implied import demand system, , is referred as a GEV import demand system. In turn, the price level in each destination market is determined by the correlation function, , which can be interpreted as an aggregator that defines the welfare-relevant price index. In analogy to the discrete choice literature, welfare calculations depend crucially on the specification of this function.
The direct consequence of this result is that a Ricardian model with a multivariate -Fréchet productivity distribution as in equation (26) is equivalent to any model that generates a generalized extreme value (GEV) import demand system as in equation (28). Since expenditure shares do not depend on overall expenditure, the GEV class is homothetic. It also satisfies the gross substitutability property (implying equilibrium uniqueness) and belongs to the large class of invertible demand systems.
An important class of models within the GEV class is the CES model. This model is generated by an additive correlation function, and entails expenditure shares of the form
The GEV class, however, is much larger than the CES class and accommodates many non-CES Ricardian trade models commonly used in the literature.28
The MP model in Ramondo and Rodríguez-Clare (2013) and the multisector EK model in Caliendo and Parro (2015) belong to the GEV class. To illustrate, consider versions of these models without an input-output loop. They are both generated by a correlation function that takes a cross-nested CES form,(30)
where , with , and refers to the country of origin of MP and to sectors, alternately. The import demand system implied by this correlation function is(31)
Gains From Trade
As for the EK model under independence, the gains from trade stemming from a GEV import demand system can be easily calculated. The calculation entails a simple correction to the formula for gains in Arkolakis et al. (2012). Using a multivariate -Fréchet distribution with correlation across exporters adds, however, an important channel for the gains from trade: Countries gain more if they trade with partners that have more dissimilar technology. The implied GEV import demand system also allows for richer patterns of substitution across exporters than CES: By having , for , the effect of a change in the unit cost on the aggregate importer price index can vary substantially depending on the exporter.(32)
Solving for the real wage yields(33)
where . Under autarky, . Hence, the gains from trade from autarky are given by(34)
For CES, and we are back to the formula from the gains from trade in Arkolakis et al. (2012)—presented in equation (14)—that only depends on the self-trade share of a country. For GEV, countries with the same self-trade share can have very different gains depending on the way their technology is correlated with their trading partners.30
The next section shows how a multivariate -Fréchet distribution with arbitrary correlation function can be generated from an underlying innovation and adoption process.
Generating Max-Stable Multivariate Fréchet Distributions
The next result provides a tool to build multivariate max-stable Fréchet distributions from Poisson processes. Based on primitive assumptions on technology, this result establishes necessary and sufficient conditions for productivity to be distributed multivariate -Fréchet. By doing so, it links productivity distributions with flexible correlation patterns to underlying micro-foundations while maintaining the max-stability property that has proved key in the literature inspired by EK.
Theorem 1 in Lind and Ramondo (2018) establishes that productivity, , is multivariate -Fréchet if and only if it satisfies the following three assumptions.
Assumption 5 (Innovation Decomposition). There exists and, for each , a countable set of global innovations, , with global productivity and spatial applicability satisfying(35)
Assumption 6 (Independence). is independent of and i.i.d. over and with .
Assumption 7 (Poisson Innovations). The collection consists of the points of a Poisson process with intensity measure , and is i.i.d. over .
Assumption 5 states that productivity is the result of countries adopting the best global innovation (i.e., a blueprint) to produce a good , based on each country’s ability to apply the innovation. For a given good , each innovation has a global productivity component, , and an origin-destination specific spatial applicability component, . Global productivity measures the fundamental efficiency of the production technique and is identical across all origins and destinations. In turn, spatial applicability captures origin-destination specific factors that determine the efficiency of the technique when adopted at origin to deliver goods to destination .
The key aspect of Assumption 6 is that it does not impose independence of applicability across origin countries; instead, it allows for arbitrary patterns of spatial correlation. Moreover, the assumption does not impose any particular probability distribution for applicability; this distribution can belong to any family provided that first moments exist.
Finally, Assumption 7 states that the global productivity component, , follows a non-homogenous Poisson process over ‘s, for each . This assumption implies that the number of innovations with productivity satisfying is a Poisson random variable. Among those innovations with productivity above , is distributed Pareto with lower bound and shape .
The equivalence between a structure for technology that satisfies Assumptions 5 to 7 and multivariate -Fréchet productivity is a consequence of the spectral representation theorem for max-stable processes (De Haan, 1984; Penrose, 1992; Schlather, 2002).31
Theorem 1 (Innovation Representation). Productivity, , is multivariate -Fréchet if and only if it satisfies Assumptions 5, 6, and 7. In this case, the joint productivity distribution is as in equation (26), with the scale parameters given by , for , and correlation function(36)
Proof. See Lind and Ramondo (2018).
This characterization of productivity establishes that -Fréchet-distributed productivity can always be interpreted as arising from the adoption of technologies and that patterns of adoption depend on the ability of exporters to apply global innovations. Intuitively, both absolute advantage (the scale parameters) and comparative advantage (the correlation function) are the result of the ability of exporters to adopt technological innovations. In fact, Theorem 1 provides a method to compute scale parameters and correlation functions: They are simply the first moments of spatial applicability and the expected value of the maximum of (scaled) spatial applicability.
The following example illustrates how to use this theorem to construct multivariate -Fréchet distributions. Assume that the spatial applicability of individual technologies, is independently distributed Fréchet over , with scale , shape , and the gamma function.32 The scale parameters of productivity equal the first moments of spatial applicability: (using properties of the Fréchet distribution). To derive the correlation function, use properties of the Fréchet random variables to compute the expectation over the maximum of (scaled) spatial applicability in equation (36). First, is distributed -Fréchet with scale . Due to independence and max-stability, the maximum over is also -Fréchet and its scale is the sum of the underlying scale parameters. Computing the expectation of the maximum yields(37)
and the joint distribution of productivity is
The correlation function in equation (37) takes the form of a CES aggregator. The coefficient measures the degree of correlation, which arises from dispersion in spatial applicability, and is controlled by the shape parameter . As , dispersion in applicability is high, and . Intuitively, when applicability becomes very fat tailed, it dominates the contribution of the common global component of productivity. In this limiting case, productivity is independent across exporters and the correlation function is additive due to the assumption that applicability is independent across countries. In contrast, as , dispersion in applicability becomes negligible and . In this case, applicability becomes deterministic and heterogeneity in productivity is entirely determined by the global component, . As a result, productivity becomes perfectly correlated across countries.
This example provides intuition for how Theorem 1 generates varying degrees of correlation in productivity from underlying assumptions on the applicability of technologies across the globe. High dispersion in spatial applicability dampens the importance of the common global component of productivity and reduces correlation, while the opposite is true when dispersion in spatial applicability is low.
The result in Theorem 1 from Lind and Ramondo (2018) constitutes a tool to build max-stable copulae from underlying Poisson processes. The appeal of this approach is that it leads to closed-form results while also flexibly incorporating rich heterogeneity. This flexibility is particularly important when building quantitative models. As such, this tool can be applied to tractably model correlation in a variety of contexts—from growth models of innovation and diffusion to selection-type models of migration, occupation, and commuting decisions. More generally, this tool is useful for answering questions throughout economics where technology plays a central role.
Advances at the beginning of the 21st century in the international trade literature have led to quantitative models of knowledge creation and international technology diffusion, as well as their links with international flows of goods, services, and firms. This paper reviews the literature on innovation and knowledge diffusion built on a common methodological theme: Fréchet productivity distributions that arise from underlying Poisson processes. The central pieces of the analysis are the model of innovation in Kortum (1997) and the Ricardian model of trade in Eaton and Kortum (2002). As such, all the models reviewed are quantitative general equilibrium models suitable for performing counterfactual analysis, have quantitative implications for the gains from trade, and fit into the sufficient-statistic approach proposed in Arkolakis et al. (2012) and extended in Lind and Ramondo (2018).
The methodology developed in Lind and Ramondo (2018) to build max-stable copulae from underlying Poisson processes opens promising opportunities for future research. Indeed, one of the important tasks ahead is to generate max-stable Fréchet productivity distributions with arbitrary spatial correlation in dynamic settings. This may provide new insights on the determinants of living standards, as well as broaden the understanding of the effects of integrating production and research activities between countries. It can also shed light on the important question of why some countries ended up sharing, for example, similar levels of income per capita, while others do not. The methodology is also well-suited to build quantifiable frameworks that examine the consequences of technology transfers, through multinational firms, on a variety of issues, from R&D and patent policies to the increases in economic concentration that occurred between 1990 and 2010. Finally, the methodology can be applied to analyze the motivation for—and sustainability of—certain policies, such as trade wars, bilateral trade and investment treaties, and the formation of fiscal and economic unions.
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(2.) Real openness is measured as imports plus exports in exchange rate U.S. dollars relative to GDP in purchasing power parity U.S. dollars.
(3.) Ramondo (2009) and Alfaro and Chen (2018) also document the effects of foreign entry on the productivity distribution of the receiving industry through turnover of domestic firms (i.e., exit and entry). They find that multinational entry increases the exit probability of domestic firms but also increases the productivity of entrants.
(4.) There is a relatively large body of literature that models innovation and diffusion of technologies as stochastic processes in closed economy setups, starting by the early work of Jovanovic and McDonald (1994) and Jovanovic and Rob (1989), followed by Eeckhout and Jovanovic (2002), Alvarez et al. (2008), Lucas (2009), Luttmer (2012), Lucas and Moll (2014), Perla and Tonetti (2014), and Benhabib, Perla, and Tonetti (2017), among others.
(7.) The average number of innovations is given by , where is number of researchers in country at time . A sufficient condition for as is that , for all .
(8.) In the same spirit, Rodríguez-Clare (2007) incorporates diffusion into the EK model of trade. As in Eaton and Kortum (1999), trade and diffusion are substitutes and any complementarity between them is ruled out. A similar approach is taken in Ramondo and Rodríguez-Clare (2010) who further enlarge the model in Rodríguez-Clare (2007) to also incorporate MP in ways explained below.
(11.) For example, a bounded distribution would satisfy Assumption 3.
(12.) The counterfactual here is one in which a country has a self-trade share of and moves to autarky for a single moment.
(13.) These are the gains from trade coming from a unilateral move to autarky.
(15.) Following Ramondo (2014), the term “multinational production,” rather than “Foreign Direct Investment” (FDI), is used to refer to the activity of foreign affiliates (e.g., sales, employment). FDI is a financial category of the Balance of Payments and, as such, is one of the possible channels through which affiliates finance their activities abroad.
(16.) Arkolakis et al. (2018) documented a similar positive correlation for the United States across time: R&D expenditures relative to manufacturing value-added in the United States grew from 8.7% to 12.7% between 1999 and 2009, while at the same time, U.S. multinational firms increased the share of employment located in their foreign affiliates from 22% to 31%.
(17.) In their own words “A firm’s technology capital is its unique know-how from investing in research and development, brands, and organization capital.” The main characteristic of this capital is its non-rivalry.
(18.) All these papers treat technology as non-rival within the firm. In contrast, Burstein and Monge-Naranjo (2007) focused on a scarce resource that cannot be easily replicated within the firm: managerial know-how. This, in turn, shapes firm productivity. MP arises as the reallocation of managerial talent to foreign countries to gain control of foreign factors of production. They calculate output gains from removing barriers to managerial mobility of 12%.
(19.) See the Appendix in the working paper version of Ramondo and Rodríguez-Clare (2013) for an extension of Eaton and Kortum (2001) that generates a production possibility frontier that is has symmetric -Fréchet form.
(20.) Head and Mayer (2018) applied their model to the car industry. They argue that “‘deep’ integration agreements contain whole chapters that do not operate on the origin-destination path traversed by goods. Rather, topics such as harmonization of standards, protection of investments, and facilitation of temporary movement of professionals, mainly affect the flows of headquarters services to production and distribution affiliates.” Thus, these frictions, they argue, become key in evaluating the welfare gains of trade agreements such as NAFTA and the European Union.
(22.) Trade becomes an MP complement as soon as one assumes that some inputs from the home country are needed for production in the host country, and these inputs are sufficiently imperfect substitute of local inputs (see Proposition 4 in Ramondo & Rodríguez-Clare, 2013).
(23.) Their calibrated model includes tradable intermediate goods, non-tradable final goods, an input-output structure, and the shipment of inputs from the home country to the affiliate.
(25.) The restriction to a common shape is necessary for that property; general multivariate Fréchet distributions may have marginal distributions with different shape parameters, in which case the maximum, even under independence, is not distributed Fréchet.
(28.) Multisector models (Costinot, Donaldson, & Komunjer, 2012; Costinot & Rodríguez-Clare, 2014; and Caliendo & Parro, 2015; Ossa, 2015; Levchenko & Zhang, 2016; French, 2016; Lashkaripour & Lugovskyy, 2017); multinational production models (Ramondo & Rodríguez-Clare, 2013; Alviarez, 2018); global value chains models (Antras & de Gortari, 2017); and models of trade with domestic geography (Fajgelbaum & Redding, 2014; Ramondo et al., 2016; Redding, 2016).
(29.) With , the expenditure shares across sectors are Cobb-Douglas. More generally, French (2016) used a CES aggregator across sectors, but he restricts the elasticities of substitution for each sector to be the same, , for all .
(31.) In contrast to the results in Kortum (1997), in which a independent -Fréchet productivity distribution was obtained asymptotically, this result is exact and for a -Fréchet distribution with arbitrary correlation patterns.
(32.) The constant on the scale ensures that productivity remains finite as .