Trade and Interdependence in International Networks

Transcription

1 Trade and Interdependence in International Networks François de Soyres Toulouse School of Economics July, 2015 Very Preliminary - Do Not Circulate Abstract This paper analyses the relationship between international trade flows and business cycle synchronization across countries. First, I show that the empirical association between trade and GDP comovement is stronger than what previous studies suggested, with a special role for trade in intermediate inputs and for the extensive margin of trade. Motivated by this fact, I then build a model of international trade in intermediates with many countries and heterogeneous firms. Value chains are segmented across countries and input-output linkages give rise to a strong interdependency between firms. International propagation of technological shocks depends on the world-wide network of firms and is strongly impacted by adjustments along the extensive margin. Quantitative explorations show that the model generates a relationship between trade and comovement that is one order of magnitude larger than previous studies and that it features a strong amplification of other sources of GDP correlations. Keywords: International Trade, International Business Cycle Comovement, Networks, Input- Output Linkages JEL Classification Numbers: F12, F17, F4, F62, L22 I am indebted to my advisor Thomas Chaney for his invaluable guidance throughout the various stages of this project. I am also grateful to the TSE macro group and in particular Patrick Fève, Simon Fuchs, Christian Hellwig, Marti Mestieri, Alban Moura, Franck Portier, Constance de Soyres, Shekhar Tomar and Robert Ulbricht for helpful discussions. For their comments, I also thank Manuel Amador, Julian di Giovanni, Oleg Itskhoki and seminar participants in Toulouse and Barcelona. All errors are mine. Contact: Toulouse School of Economics, 21 Allée de Brienne, Toulouse, France. francois.de.soyres@gmail.com. 1

2 1 Introduction Over the past four decades, the ratio of international trade flows over GDP between the main OECD countries has more than doubled while the import content of export increased by more than 50% 1, suggesting a strong internationalization of supply chains. At the same time, the average GDP correlation in the OECD went from less than 10% to more than 70%. This paper addresses the relationship between those phenomenon both empirically and theoretically. Empirics Since the seminal paper by Frankel and Rose 1998), a large empirical literature studied the link between the synchronization of economic activity and international trade flows, showing that trade proximity, an index defined as total trade flows between two countries divided by the sum of their GDP, is an important and robust determinant of GDP correlation. 2 Most of those studies explore the cross-section relationship between trade and GDP comovement and find that countries that trade more with each other experience a higher degree of output synchronization. An issue with using a single cross-section dataset is that one cannot control for country pair fixed effects. To take a simple example of why this could be important, imagine that TFP shocks are more correlated between countries that are nearby rather than far away. Since trade is also stronger between close by countries, a cross-section analysis could mistakenly conclude that higher trade leads to higher output synchronization even though the main force could be TFP correlation. I refine previous analysis by constructing a panel dataset where each country pair appears four times, one for each of the four 10-years time window ranging from 1969 to I then perform fixed effect regressions on this dataset, allowing me to control for country pair fixed effects that can be correlated with trade proximity. 3 If TFP correlation is negatively related to distance, note that adding country-pair fixed effects to a pooled OLS analysis would also yield to biased results since those fixed effects would themselves be correlated with the trade intensity variable. While the current literature usually finds that a doubling of trade proximity computed with total trade flows, both intermediate and final goods) is associated with an increase in GDP correlation of about 0.06, this paper suggests that the effect is more than three times that number, with a doubling of trade proximity being associated with an increase of 0.22 point of output correlation. Furthermore, I make use of disaggregated trade data to disentangle the effect of trade on output synchronization along two dimensions. First, I isolate the trade flows in final goods from the flows in intermediate inputs and regress GDP comovement on indexes of trade proximity in final and 1 See the OECD STAN Input-Output database for more information on this number 2 Among many others, see Imbs 2004), Baxter and Kouparitsas 2005), Kose and Yi 2006), Calderon, Chong, and Stein 2007), Inklaar, Jong-A-Pin, and Haan 2008), Di Giovanni and Levchenko 2010), Ng 2010), Liao and Santacreu 2015),... 3 In a fixed effect regression, identification comes from within country-pair variation rather than cross-country differences. 2

3 intermediate goods. Results show that trade in intermediate inputs captures all the explanatory power, whereas trade in final good is found to be insignificant. Second, using the Hummels and Klenow 2005) decomposition, I construct the extensive and the intensive margins of trade and again perform the regressions on those indexes. This analysis establishes that a higher degree of business cycle correlation is associated with an increase in the range of goods traded and is not associated with an increase in the quantity traded for a given set of goods. In other words, the extensive margin explains most of the GDP correlation, a striking result considering that the extensive margin accounts for only a fourth of the variability in total trade. 4 Theory Motivated by those facts, I propose a model of international trade in inputs with many countries and heterogeneous firms in which technological shocks propagate across countries through the whole network of input-output linkages. Production is performed by a continuum of heterogeneous firms combining in a Cobb-Douglas fashion labor and a nested CES aggregate of intermediate inputs bought from other firms from their home country as well as from abroad. Based on their expected profit, firms choose the set of countries they serve if any). In this context, a firm s marginal cost depends on the number and on the productivity of its suppliers, giving rise to a strong interdependency in the pricing as well as in the export participation decisions. By computing a closed form expression of the elasticity of domestic GDP with respect to a technological shock abroad, I show that the key elements in determining the ability of the model to feature international propagation are 1) the structure of the worldwide network of input-output linkages and 2) the degree of heterogeneity in firms productivity within each country, which governs the change in the set of firms serving each markets due to a shock abroad. The first point is related to Kose and Yi 2006) who argued that a two-country setting was inappropriate when studying international propagation because those models tend to grossly exaggerate the impact that a typical country has on its trading partners. In this paper, I show that in a world with input-output linkages, international propagation runs through the whole network of firms all around the world and that one must model the whole economy to take those effects into account. Hence, the extent to which a shock in country k can affect the GDP in country k does not only depend on the strength of the economic ties between those two countries, but also on their relative centrality in the network. To illustrate this point, I construct a fictitious economy where all country-pairs have the same index of trade proximity but have different quantitative impact on one-another due to the asymmetry of the network of linkages. This result also shows why one cannot assess the impact of trade on economic synchronization by just using a single cross-section dataset. 4 This result was already studied in Liao and Santacreu 2015) which emphasizes the role of the extensive margin. Compare to them, I am adding the panel dimension by performing fixed effect regression which alows me to control for fixed effects that can be correlated with trade intensity. 3

4 The second element - the importance of firms heterogeneity and the extensive margin adjustments - suggests that classic international real business cycle IRBC) models with representative firms tend to underestimate the extent to which technological shocks to one country can affect its trading partners. By introducing firm heterogeneity, the model features a quantitatively important transmission channel. Quantitative Analysis In the last part of the paper, I precisely calibrate the model to 14 OECD countries and a composite rest-of-the-world and assess its ability to replicate the strong relationship between trade in inputs and GDP co-movement. By exogenously varying country pair weights in firms production functions and feeding the exact same sequence of technological shocks in each configuration, I create a panel dataset where the trade proximity in intermediates ranges from - 10% to +10% of the one observed during the 1999 to 2008 period. Fixed effect regressions on this simulated dataset shows that the model is able to replicate between 60% and 88% of the tradecomovement slope depending on the correlation imposed on the TFP shocks. With point estimates between and 0.203, this is a 10 fold improvement compare to previous studies. Several parameters have a key influence on this result. First, as predicted by the theory, firms productivity distribution is an important factor of shock propagation through the extensive margin adjustment of firms and its effect on the unit cost of production in every country. The calibration of the heterogeneity of firms productivity draws is subtle in a stylized model, and several robustness checks are performed. Another important element is the structure of technological shocks that affect each country, in particular its variance-covariance matrix. In the baseline simulation, I use uncorrelated shocks across countries so that international trade is the only channel through which GDP synchronization could appear. Setting the volatility of those shocks so that the standard deviation of GDP in every country matches the one observed in the data, the model matches 60% of the trade-comovement slope but only 16% of the level 5. Introducing 20% of correlation between technological shocks allows to improve the model s performance along both dimensions level and slope ) with an average correlation of 44% and a slope of While the fact that using correlated shocks rises the average GDP correlation is tautological, the fact that it also increases the effect of trade on output comovement in the fixed-effect regression is appealing. This result suggests that exogenously induced correlation in the shocks structure creates a larger impact of trade proximity on output comovement. In other words: correlation of shocks gets amplified by trade linkages. Finally, I illustrate the importance of firm heterogeneity by simulating a version of the model with the same input-output linkages but with a single representative firm in each country. Keeping 5 The average country-pair correlation is 11.5% in the simulated economy while it is above 70% in the data. Using the data on the same 14 countries as in the simulation omitting the rest-of-the-world ), the average correlation for the time window 1999 to 2008 is 72.32%, ranging from 25.9% for Australia-Italy) to 94.8% for Austria-Spain) 4

5 the same calibration strategy, the performance of this model are greatly reduced, a result in line with Johnson 2014) which argued that trade in intermediate inputs alone could not solve the Trade- Comovement Puzzle. Relationship to the literature If the empirical association between bilateral trade and comovement has long been known, the underlying economic mechanisms leading to this relationship are still unclear. Using the workhorse IRBC with three countries, Kose and Yi 2006) have shown that the model can explain at most 10% of the slope between trade and correlation, leading to what they called the trade-comovement puzzle. Since then, many papers have refined the puzzle, highlighting different ingredients that could bridge the gap between the data and the predictions of classic models. Burstein, Kurz and Tesar 2008) show that allowing for production sharing among countries can deliver tighter business cycle synchronization if the elasticity of substitution between home and foreign intermediate inputs is very low 6. Arkolakis & Ramanarayanan 2009) analyse the impact of vertical specialization on the relationship between trade and business cycle synchronization. In their Ricardian model with perfect competition, they do not generate significant dependence of business cycle synchronization on trade intensity, but show that the introduction of price distortions that react to foreign economic conditions allows their model to reach a trade-comovement slope of 0.03, about a third of the of the slope estimated in the data in Kose & Yi 2006). Incorporating trade in inputs in an otherwise standard IRBC, Johnson 2014) shows that the puzzle cannot be solved by adding the direct propagation due to the international segmentation of supply chains only. Calibrating his model with 22 countries using the WIOD database, he finds the puzzle alive and well with the aggregate trade-comovement correlations for real value added and gross output being at most 10-20% the size of the observed correlations. In this paper, I add firm heterogeneity and monopolistic competition in every country and argue that it is a key element for the model to deliver quantitative results in line with the data. Liao and Santacreu 2015) build on Ghironi & Melitz 2005) and Alessandria & Choi 2007) to develop a two-country IBC with endogenous labor supply, capital accumulation and trade in intermediate varieties. It shows that trade in intermediate varieties lead to an endogenous correlation of measured TFP 7 across trading partners. Compare to this paper, I add a production link between importers and importers which creates a strong interdependency in firms pricing end export decisions. Furthermore, I extend the quantitative analysis from two countries to a whole network of countries and show that country-pair fixed effects due to the position in the network are quantitatively important. 6 In their benchmark simulations, the authors take the value of 0.05 for this elasticity. 7 Defined as the Solow residual at the country s level 5

6 The idea that trade disruption can result in significant decrease in efficiency for firms relying heavily on foreign inputs has been studied by Gopinath and Neiman 2014). Focusing on the Argentinian crisis with a detailed dataset at the firm level, they show that the aggregate productivity drop following trade disruption can be significant. They then build a model with an exogenous cost of changing the number of firm s suppliers and are able to replicate the data, showing the importance of within firms dynamics to understand aggregate productivity. Finally, the role of firms heterogeneity in international business cycles has been pioneered by Ghironi & Melitz 2005) and the business cycle implication of firms heterogeneity is studied in Fattal-Jaef & Lopez 2014). The rest of the paper is organized as follow: the second section studies empirically the relationship between trade proximity and output synchronization and highlights the important role of trade in intermediate inputs. The third section proposes a model of international trade in intermediate goods with many countries. In the fourth section, I explain my calibration strategy and the quantitative results are presented in section five. Section six concludes. 2 Empirical Relationship In this section, I update and refine the initial Frankel and Rose 1998) regressions using a sample of 20 OECD countries 8 from 1969Q1 to 2008Q4. Using fixed effect regressions to control for unobserved effects that can be correlated with the index of trade proximity, my results are threefold. First, the empirical association between business cycle synchronization and international trade is stronger than found in previous studies. Second, there is a very strong role of international trade in intermediate goods in explaining GDP comovement, while the explanatory power of trade in final good is found not significant. Third, higher GDP comovement is associated with an increase in the set of goods traded the extensive margin) while it is not associated with an increase in the quantity of goods traded, for a given set of goods the intensive margin). I first describe the data, then I explain the econometric strategy and finally I present the results in details. I use quarterly data on real GDP from the OECD database which I transform using Hodrick and Prescott filtering to capture the business cycle frequencies. In the appendix, I show that my results are robust to other types of filtering, including log difference and band pass filtering that capture the medium term frequencies as suggested by Comin & Gertler 2006)). Trade data come from the NBER-UN world trade database provided by the Center for International Data CID). It features bilateral trade flows at the 4-digit level of disaggregation SITC Rev. 4). Such a high level of disaggregation allows me to deepen the anlysis by studying the impact of trade along two dimensions: the end usage of goods final consumption or intermediate inputs) as well as the margins 8 The list of countries can be found in appendix 6

7 of trade. In order to disentangle the influence of trade flows in inputs from the final goods, I follow Feenstra and Jensen 2012) and transform the SITC code into End-Use categorization. The End-use codes are used by the Bureau of Economic Analysis BEA) to allocate goods to their final use, and are similar to the Broad Economic Categories of the United Nations Statistics Division. It allows me separate products between final and intermediate goods. Following Hummels & Klenow 2005) as well as use the Feenstra & Markusen 1994), I then construct the Extensive and Intensive margins of trade between countries j and m using the Restof-the-World as a reference country k. The extensive margin EM) is defined as a weighted count of varieties exported from j to m relative to those exported from k to m. If all categories are of equal importance and the reference country k exports all categories to m, then the extensive margin is simply the fraction of categories in which j exports to m. More generally, categories are weighted by their importance in k s exports to m. The corresponding intensive margin IM) is defined as the ratio of nominal shipments from j to m and from k to m in a common set of goods. With this construction, the product of both margins of trade between j and m is equal to the ratio of total trade flows between j and m to total trade flows from the reference country k to m, which is usually denoted as OT. Formally, the margins are defined as: p kmi q kmi i I jm Extensive Margin EM jm = p kmi q kmi i I p jmi q jmi i I jm Intensive Margin IM jm = p kmi q kmi i I jm Trade Ratio OT jm = X j m X k m = EM jm IM jm Where I jm is the set of observable categories in which j has a positive shipment to m and I is the set of all categories exported by the reference country which is supposed to be the universe of all categories. Since those measures are not symmetric within every country-pair, I define for a given country pair i, j) as the sum of the margins from i to j and the margin from j to i, which are then averaged over the time window. The extent to which countries have correlated output can be influenced by many factors, including international trade, correlated shocks, financial linkages, monetary policies, etc... In order to separate the effect of trade linkages from other elements, I construct a panel dataset by creating four periods of ten years each. In every time window, I compute GDP correlation for all country pairs as well as 7

8 trade indexes as defined by the sum of import and export between two countries divided by the sum of their GDP. Index are constructed using total trade flows, trade in final goods as well as trade in intermediate inputs. The index relative to a given time window is the average of all yearly indexes. Using panel data allows me to control for country-pair specific factors that are not observables. Moreover, because those other factors correlation of shocks, financial linkages, common monetary policy,...) can be themselves correlated with the index of trade proximity in the cross section, using a simple pooled OLS with country-pair fixed effect would yield biased results. To overcome this issue, I use Fixed-Effect regressions and estimate the following three equations: I) II) III) corryit HP, Yjt HP ) = α I + β T logtotal ijt ) + ɛ I,ijt corryit HP, Yjt HP ) = α II + β I logintermediate ijt ) + β F logfinal ijt ) + ɛ II,ijt corryit HP, Yjt HP ) = α III + β EM logem ijt ) + β IM logim ijt ) + ɛ III,ijt In table 1, the first three columns show the results of pooled OLS without controlling for countrypair fixed effects while the last three present the results of the Fixed-Effect regressions. The point estimate in column I A matches closely the estimations found in the literature for the role of trade proximity using total trade flows, implying that a doubling of the median index is associated with an increase of about 0.06 of output correlation. However, the estimation using the fixed effect method, as presented in column I B is more than three times larger, with a doubling of the medial index leading to an increase of 0.22 point of output correlation. Moreover, looking at columns II A and II B, one can separately identify the effect of trade in final and intermediate goods as well as the importance of country-pair specific fixed effects. The results highlight a specific role for trade in intermediate inputs. According to the pooled OLS estimation, a doubling of the median index of trade proximity in intermediate goods is associated with an increase of of the median correlation. Moreover, when using Fixed-Effect regressions, the coefficients imply that a doubling of trade proximity in intermediate goods results in an increase of 0.16 point of correlation, about twice the effect estimated with pooled OLS. It is important to remember that in Fixed-Effect regressions coefficients are identified by within-firms variations, implying that the increase in correlation due to a change in trade proximity has to be interpreted as being relative to each country-pair average. This result imply a large and important role for trade in inputs in the explanation of GDP comovement but also points toward significant country-pair specific factors among which one could think of the correlation of underlying shocks or financial linkages for example. The theory presented below will also shed light on those fixed effect by showing the importance of the relative location of the country pair compare to the whole network of input-output linkages. Finally, columns III A and III B use the Hummels & Klenow 2005) decomposition to disentangle the influence of the intensive and the extensive margins of trade on GDP synchronization. While 8

9 the results do not significantly differ between the pooled OLS and the Fixed Effect regressions, both point toward the same message: only the extensive margin of trade matters for the. This result is particularly striking given that most of the variation in trade is explained by variations along the intensive margin. Indeed, performing a Shapley value decomposition of OT on the intensive and extensive margins, one can show that only one fourth of the total variance in OT is explained by the variation of the extensive margin. Put simply: even though EM does not vary too much compare to IM), its variation are strongly correlated with the variation of output comovement. Pooled OLS FE Regressions I A ) II A ) III A ) I B ) II B ) III B ) logtotal) 0.096*** 0.315*** 11.18) 15.04) logintermediate) 0.121*** 0.231*** 6.22) 9.01) logfinal) ) 0.67) logem) 0.249*** 0.246*** 8.91) 6.27) logim) ) 0.45) N 760 Note: The dependent variable is correlation of HP filtered GDP. Total is the sum of bilateral trade flows divided by the sum of GDPs. Final and Intermediate are similar but using only trade in final and intermediate goods. EM and IM are the Extensive and the Intensive margins of trade as defined in Hummels & Klenow. t stat. in parentheses, *** means p < Table 1: Trade-Comovement analysis for a sample of 20 OECD countries By using fixed effect regressions, I assumed that something within each country-pair may impact or bias output synchronization. One might also consider that the variation across country-pairs are assumed to be random and uncorrelated with the GDP comovement or the trade proximity indexes, in which case a random effect treatment would be a better fit. To discriminate between fixed or random effects, I run a Hausman specification test where the null hypothesis is that the preferred model is random effects against the fixed effects. This tests whether the error terms ɛ ijt 9

10 are correlated with the regressors, with the null hypothesis being they are not. Results display a significant difference p < 0.001), indicating that the two models are different enough to reject the null hypothesis, and hence to reject the random effects in favor of the fixed effects model. Finally, I performed several robustness checks which results can be found in appendix. Firstly, I changed the filtering technique used to extract the business cycle fluctuations of output and used log first-difference as well as the Baxter and King band pass filter. The first method removes any exponential growth trend while the second keeps explicitly medium term fluctuations between 32 and 200 quarters as suggested by Comin & Gertler 2006). Secondly, in order to test if the results were robust to a change in the definition of the four time windows, I created two twenty-years time windows from 1969Q1 to 2008Q4 and performed the same analysis as above. 9 In all those cases, the results presented here hold with the trade proximity in intermediates and the extensive margin capturing all the statistically significance leaving respectively trade in final goods and the intensive margin of trade with no impact. lastly, it is interesting to test for cross-sectional correlation between trade proximity and output synchronization, as has been done in previous studies. Such an exercise does not allow for country-pair fixed effects to be captured, and the results must be interpreted with caution. Running the same regression on each of the four time windows reveals contrasted results for different periods. 3 A model of International Trade in Inputs 3.1 Setup Motivated by the empirical section and in particular the importance of trade in intermediates and the extensive margin of trade, I build a model of international trade in inputs with heterogeneous firms and assess its ability to replicate the data. I consider an environment with N countries indexed by k with respective size L k and aggregate productivity level Z k. In each country, there is a representative agent with preferences over leisure and the set of available goods Ω k described by U k = Ω k q σ 1 σ i σ σ 1 ψ k L 1+ν 1 + ν where ψ k is a scaling parameter controlling the size of the country, ν is the inverse of the Frisch elasticity of labor supply and σ the elasticity of substitution between final goods. Production is performed by a continuum of heterogeneous firms combining in a Cobb-Douglas fashion labor l k and intermediate inputs I k bought from other firms from their home country as well as from abroad. 9 Note that with only two time windows, fixed effect regression is equivalent to first differences 10

11 Firms productivity is the product of an idiosyncratic part ϕ and a country specific part Z k. In each country, the intermediate input index I k takes a nested CES form allowing me to separate between the micro elasticity of substitution σ between firms from the same country and the macro Armington) elasticity ρ that measures the substitutability of different country-specific bundles. The production function writes: Y k ϕ) with I k ϕ) = = Z k ϕ I 1 β k ϕ) l β k ϕ) α k k ρ 1 1 ) ρ ρ Mk,k k σ and M k,k = Ω k,k m σ 1 σ i ) ρ ρ 1 σ 1 where M k,k is a country-pair specific CES bundle of all varieties produced in country k that are exported to country k and used for production, α k k ) is the share of country k in the production process of country k with k α k k ) = 1 and Ω k,k is the endogenous set of firms based in k and exporting to k. For later use, it is useful to define notations for the ideal price indexes dual to the two layers of the nested CES aggregation. I denote by P k,k the price of the country-pair specific bundle M k,k and IP k the price if the intermediate input bundle I k. The unit cost of the Cobb Douglas bundle aggregating I k and l k is P B k and represents the price of the basic production factor in country k. The exact expressions of those objects are classic and can be found in appendix for reference. In all countries, the distribution of productivity is Pareto with shape parameter γ k and density gϕ) = γ k ϕ γ k 1. For simplicity and in line with the empirical results showing that only trade in intermediate inputs is statistically significant in explaining GDP comovement, I restrict trade to be only between firms and not with final consumers. In order to be allowed to sell its variety to a country j, a firm from country i must pay a fixed cost f ij in unit of the input bundle ) as well as a variable iceberg) cost τ ij. Based on their expected profit, firms choose which countries they enter if any), affecting both the level of competition and the marginal cost of all firms in the country. As will be clear below, profits are strictly increasing with productivity ϕ so that export decision in equilibrium is defined by a simple country-pair specific threshold ϕ k,k above which firms from k find it profitable to pay the fixed cost f kk and serve country k. Finally, there is an exogenous mass of potential but not actual) entrants M k which I assume to be proportional to the size of labor force L k This is the same assumption than in Chaney 2008). Arkolakis, Demidova, Klenow and Rodriguez-Clare 2008) showed that it can be micro-founded with a free entry condition imposing that the average profits over all firms in country k is equal to a fixed cost f E,k of developing a variety in country k.) if all entry fixed entry costs are paid 11

12 3.2 Equilibrium In this section, I present the key conditions that characterize the equilibrium of the model, leaving to the appendix the precise derivations. Introducing X k the aggregate consumers revenue in k and S k the total firms spendings including fixed costs payments that are denominated in unit of the production bundle) in country k respectively, one can show that the total demand faced by firm ϕ is given by ) pk,k ϕ) σ X k qϕ) = + P k P k k ) pk,k ϕ) σ ) ρ Pk,k α k k)1 β)s k P k,k IP k IP k 1) where p k,k ϕ) is the price charged by a firm from country k, with productivity ϕ, when selling in country k and the summation is done over all markets that are served by a firm with productivity ϕ. I assume that firms are monopolists within their variety. Classically, they choose their price at a constant markup over marginal cost and the markup depends on the price elasticity of demand. In this case, the only elasticity that is relevant to firms pricing is the micro one σ), capturing the fact that firms compete primarily with other firms coming from their home country and do not internalize the effect of their own pricing on their country-specific price index. 11 The marginal cost of a firm with productivity ϕ in country k is P B k /Z k ϕ) and its optimal price is given by: σ P B k p k,k = τ k,k σ 1 Z k ϕ 2) Unlike in the canonical Krugman 1980) or Melitz 2003) models of international trade, one cannot solve for prices independently for each firm. Through P B k, the price charged by firm ϕ in country k depends on the prices charged by all firms supplying country k both domestic and foreign) which in turn depend on the prices charged by their suppliers and so on and so forth. The input-output linkages across firms creates a link between the pricing strategies of all firms and one needs to solve for all those prices at once. Doing so requires solving for all country-pair specific price indexes P k,k. The definition of price indexes gives rise to a simple relationship between the price of the country k specific bundle at home, M k,k, and its counterpart in country k, M k,k: P k,k = τ kk ϕk,k ϕ k,k ) σ γ k 1 1 σ P k 3) Intuitively, the ratio between the price of a country specific bundle in two different markets depends in unit of the destination production factor. 11 With a finite number of firms, both elasticities σ and ρ would appear in the pricing strategy. In such a case, every firm would take into account the fact that its own price has an impact on the unit cost of the corresponding country-specific bundle. Therefore, when decreasing its price a firm would attract more demand compare to firms from its own country but also increase the share of total demand that goes to every other firms from the its country. 12

13 on the relative iceberg costs as well as the relative thresholds. Using this relation in the definition of price indexes in every country yields a system of N equations with the same number of unknowns which define the price indexes: P 1 ρ k = µ k α k k ) k τ k k ϕk,k ϕ k,k ) ) σ γk 1 1 ρ P k 1 β, k = 1,..., N 4) with µ k constants depending on thresholds and parameters. 12 show that this system admits a unique non negative solution. 13 For given thresholds ϕ k,k, one can Turning now to the determination of export strategies, the thresholds above which firms from country k optimally decide to pay the fixed cost and serve market k are simply given by the threshold system: P B k π k,k ϕ k,k ) = f k,k for all k and k 5) Z k where π k,k ϕ) is the variable profit earned by a firm with productivity ϕ in market k. I assume that the fixed cost f k,k is paid in unit of the basic production factor in country k deflated by the aggregate level of productivity, as is the case in Ghironi and Melitz 2005) or Johnson 2014) for example. The model is then closed by the labor and the good market clearing conditions. The former, which characterize the wage rate w k and the labor soze L k in every country, is simply the intersection of labor demand w k L k = βs k and labor supply ψl ν = w P. The latter, however, requires a few more steps. First, I define R k as the total sales of firms from country k made on all markets. Trade being allowed only in intermediate goods, revenues in foreign countries come from other firms spending while domestic revenues also include consumers spendings. Then, total revenues writes R k = X k + [ k Pk,k IP k ) 1 ρ α k k)1 β)s k ] This formula has a simple interpretation: firms in country k receive revenues from their final good sales to their home consumers for a total amount of X k ) as well as from sales as intermediate goods on all markets. In every country k, firms allocate a constant fraction 1 β of all their spendings to intermediate inputs, which is then scaled by the weight α k k) representing the importance of 12 1 σ 1 ρ µ k = γ σ γ k 1 kϕ k,k M γ k σ 1) k ) 1 σ σ w β k 1 σ 1 β β.1 β) 1 β Z k 13 Following Kennan 2001) and denoting Y k = P 1 ρ k and Y the associated N 1 vector, it suffices to show that the system is of the form Y = fy ) with f : R N R N a vector function which is strictly concave with respect to each argument, which is obvious as long as 0 < β < 1. On a more conceptual level, this argument stresses the importance of decreasing return to scale with respect to intermediate inputs in order to ensure unicity of the equilibrium. 6) 13

14 country k in the production process of country k. Finally, since country k specific bundle in k is in competition with other country specific bundles in the input market, total revenues of k-firms when selling in k also depend on the ratio of P k,k to IP k to a power reflecting the relevant the price elasticity, in this the macro Armington) one ρ. Computing revenues in all countries requires an expression for consumers spendings. With a fixed mass of potential entrants, firms make profits that are then redistributed to consumers, so that X k = w k L k + Π k. An expression of Π k can be found using a property first noted by Eaton and Kortum 2005) and used for example in DiGiovanni and Levchenko 2012) according to which total profit in country k are proportional to total revenues. Lemma 1 : Total profit in country k are proportional to total revenues: Π k = σ 1 γσ R k 7) Proof: see Appendix. Using this expression, together with the fact that total labor payment w k L k is a fraction β of firms variable spendings, the good market clearing condition can be written in compact form as IN W T P )) }{{} =M S 1. S N = 0 R N 8) ) 1 ρ where W the weighting matrix defined as W ij = α i j), P a matrix defined by P ij = IP i and is the element-wise Hadamard) product. To gain intuitions, one can note that the matrix P scales the weights α k k) in order to account for the competition across country-specific bundles. If the Armington elasticity ρ is above unity country specific bundles are substitutes) then a country i which is able to charge low prices in some market j a low P i,j ) will attract a higher share of total expenditures from all firms in this country. Classically, this effect completely disappears in the case of a Cobb-Douglas aggregation of country specific bundles, because in such a case the spending shares are fixed. Finally, one can note that the solutions of this system form a one dimensional vector space, revealing that, as in any general equilibrium model, one needs to set one price in order to fully characterize the decisions of all agents. Setting w 1 = 1 provides a unique solution for all variables by solving together the price system 4), the threshold system 5), the Spending system 8) and the labor market equilibrium conditions. The equations above completely describe the equilibrium for Pi,j 14

15 given levels of technology. In this context GDP is defined as total consumer spendings in real terms: GDP k = X k P k 9) This definition insures that GDP is expressed in the same unit in every country, namely in units of utility. 3.3 Closed form solution in a simplified case In order to investigate the mechanisms driving the propagation of shocks across countries in the model, let us study a special case with ρ = 1 and inelastic labor supply. The goal of this section is to compute the elasticity of GDP in country i with respect to a technology shock in country 1 for every country i: η GDPi,Z 1 = loggdp i) logz 1 ) Inspecting the spending system 8) reveals that when ρ = 1 the matrix P disappears, which simply reflects the fact that in a Cobb-Douglas production function, the shares of total spending allocated to each production factor is fixed and independent of their price. As a result, one can solve for all spendings using only the system 8) and the choice of numeraire, independently of the technology level in every country. Moreover, using lemma 1 we have a simple relationship between revenues ) and spendings, S k = 1 σ 1 γ k σ R k which further implies that total consumers spending X k does not depend either on the technology level. The relationship between technology shocks and GDP works only through the variations in the price indexes P k. In appendix, I compute the elasticity of all endogenous variable with respect to technological shocks. This process leads to the closed-form formula in lemma 2. Lemma 2 : In the Cobb-Douglas ρ = 1) and fixed labor supply case, the elasticity of every GDP with respect to a technology shock in country 1 is given by: η GDP1,Z 1. η GDPN,Z 1 1 = I N 1 β)w T ) ) with W the weighting matrix defined above and T a Transmission matrix 14, function of γ k and σ. 14 T = diagλ 1,..., Λ N ), with Λ k = 1 σ+ σ 1)2 γ k σ 1) 15

16 Proof: see Appendix. This expression is reminiscent of what can be found in Cobb-Douglas network models such as Acemoglu et al 2012) for example, with an additional effect coming from firm heterogeneity and the extensive margin adjustments captured by the matrix T. From a purely theoretical point of view, this means that virtually any level of propagation could be obtained providing that set the values of σ and γ i accordingly. 15 From a quantitative point of view, it means that the calibration of those two parameters will be an important for the model s ability to generate international propagation. The computations leading to the expressions of the elasticity in this lemma are greatly simplified by the fixed labor supply assumption. In the general model, however, this constitute an important amplification channel through two effects. First, as in many macro models, a positive productivity shock in any country contributes to the decrease of prices all over the world and hence an increase in real wage. This triggers a increase in labor supply that amplifies the benefits of the shock in terms of output. 16 In addition to this classic effect, there is a second channel going through the change in the mass of active firms in every country. With the assumption that the mass of potential entrepreneurs is fixed and proportional to the labor size, an increase in labor supply results in a similar increase in the mass of potential entrants. Whether the mass of actual producing firms goes up or down in any country k will also be determined by the changes in the thresholds ϕ ik for all i which in turns crucially depends on the value of the Armington elasticity ρ. In the Cobb-Douglas case where country specific bundles are complements and the expenditure shares are fixed, it is easy to see that a positive technological shock will result in a decrease of all entry thresholds in every market. Putting pieces together, a positive shock triggers at the same time more potential entrepreneurs and a decrease of the entry threshold in every market. As a result, the new structure of input-output linkages amplifies the benefits of the shock. Illustration of the Network Effect Equation 10) shows that the impact that a given country i has on its trading partner j not only depends on how intensively those country directly trade with one another, but also on the shape of the overall network of linkages. In order to illustrate this simple point, I created four different economies with six countries and computed the elasticity of GDP in countries number 1 and 3 with respect to a shock in country 2. In every configuration, I keep constant the trade proximity between 15 One can see that for a fixed σ, Λγ k ) is a strictly increasing function. When γ k σ 1, Λ k 0 and when γ k +, Λ k 1/σ. in other words, when σ 0, Λ k could take any real value providing one chooses γ k adequately. This, of course, means that the values of γ k and σ must be chosen in order to match precise moments in the data in order to discipline the model s prediction. 16 With GHH preferences, there are no wealth effect in the labor supply decision, hence a positive technological shocks always result in an increase in labor supply. 16

17 1 and 2 on the one hand and between 2 and 3 on the other hand, changing only the configuration of the network. In figure 3.3, I show the value of the elasticities η GDP 1,Z2 and η GDP 3,Z2 normalized by their value in the complete network. One can see that those elasticities differ importantly in different network, implying that for a given value of trade proximity, the position of a given country pair within the worldwide network is a key element of the level of interdependence between countries. A direct consequence of this result is that one cannot properly estimate the impact of trade on business cycle comovement using a single cross section. In the example above, different the same trade proximity leads to different elasticities, showing that one must control for the position of each country pair when assessing the impact of trade proximity. This illustrates the importance of using panel data and controling for country-pair fixed effect, which is what was done in the empirical section where I used fixed effect regression. Discussion In the simplified case as well as in the general model, the quantitative effect of the change in the mass of firms in every market after a technological shock depends on the size distribution of firms. My model, as the very vast majority in the trade literature and due to the fact that the only degree of heterogeneity among firms is their productivity, implies a strict hierarchy of exporting markets and the decision to export to a specific country is captured by a simple threshold equilibrium. In 17

18 the data, if exporters are on average bigger and more efficient than non exporters, there is a very large overlap between their distribution. Looking at the firm size distribution in French data shows that there is no strict ordering between firms that export and those who don t. 17 Consequently, the true extensive margin is probably due to changes in the exporting status not only of firms near a single clear-cut threshold, but all along the size distribution. In addition, the single threshold prediction coupled with a fat tail distribution implies that a very small number of exporters can account for a realistic value of trade flows. 18 If the right tail of the firm size distribution is well approximated by a power law with exponent -1 Zipf s Law) 19, this approximation is patently at odds when considering the whole distribution, not only the right tail see Stanley et al 1995)). For those reasons, calibrating the slope of the Pareto distribution is not trivial: when matching the actual fat tail observed in the data, I would underestimate strongly the extensive margin adjustments that occur all along the size distribution. I discuss this issue in the next section, together with the whole calibration strategy. 4 Calibration In the remaining of the paper, I perform a quantitative exercise and assess the model s ability to match the strong empirical relationship between trade proximity in intermediate input and output synchronization. The model is calibrated to 14 countries and a composite rest-of-the-world for the time period 1999 to I then vary the weighting matrix W from -10% to +10% of its actual value and simulate the model for 5,000 periods in each configuration and compute all bilateral correlations and average trade proximity for every country pair. Fixed effect regressions on this simulated dataset shows that the model is able to replicate 62% of the trade-comovement slope, a 10 times improvement from previous studies such as Kose and Yi 2006) or Johnson 2014). Moreover, using correlated shocks improve the model s performance and the implied slope reaches In what follows, I explain in detail my calibration strategy while the results are gathered in the next section. For a simulation with N countries, there are 3 N 2 + N + 6 parameters to set. For each countrypair i, j), there are the shares α i j), the iceberg trade cost τ ij and the fixed cost f i,j, then for every country i we have the scaling parameter ψ i and finally the set of common parameters: β for the 17 In appendix, I plot the distribution of French firms and separate between exporters and non exporters. 18 More precisely, when calibrating the fixed access cost to the Doing Business Indicator database as in DiGiovanni and Levchenko and the weighting matrix W to match the exact trade flows across all country pairs, the model implies that less than 4% of all firms are exporters when in the data it is close to 20%. 19 Since Axtell 2001), many researchers have provided evidence that Zipf s law is a good approximation of the right tail of the firms size distribution interestingly, power laws in general and Zipf s law in particular apply to many other phenomenon, see Gabaix 2014)). 18

19 labor share in the production function, ν for the elasticity of labor supply, γ for the distribution of productivity draws, σ for the within country micro) elasticity of substitution across varieties, ρ for the macro) elasticity of substitution of country-specific bundles and the ratio of the mass of potential entrepreneurs to the labor size. My calibration is a mixture of estimations from micro data taken from the literature as well as re-estimated) and a matching of macro moments. The goal is to match exactly the relative GDP across all country pairs as well as the trade proximity in intermediate goods in order to give a reasonable account of the ability of the model to generate a strong link between trade and output synchronization despite the fact that typical trade flows between two given countries are very low compare to their GDPs. 20 Finally, as a robustness exercise, and in order to illustrate Kose & Yi 2006) s point in the context of my model, I also simulated the model with only two countries and taking the US and Canada which features one of the highest proximity index. With only those two countries, the calibrated model delivers a very high slope of with uncorrelated shocks, and introducing a third rest-of-the-world composite generates a slope of Those numbers, much higher than the slope implied in the full model with 14 countries and a composite rest-of-the-world, confirm the Kose & Yi 2006) s insight that one cannot quantitatively study the trade-comovement puzzle without modelling the whole economy. From micro data The parameter β is chosen to be 0.5 as is used in Jones 2011) and documented in Acemoglu et al 2012) and implying a share of intermediate goods of 50%. The inverse) elasticity of labor supply ν is 1 as is used in Johnson 2014), Flaaen et al 2015) and many others. 21 The ratio of potential entrepreneurs M and the labor size L is taken to be 0.1 and is in line with the ratio of total number of firms divided by the population in the US, taking into account that not all potential entrepreneurs enter the economy in equilibrium. The variable iceberg) trade costs are taken from the ESCAP World Bank: International Trade Costs Database 22. This database features symmetric bilateral trade costs in its wider sense, including not only international transport costs and tariffs but also other trade cost components discussed in Anderson and van Wincoop 2004). 20 Using the same 14 countries as in the simulations not including the rest-of-the-world), the average trade proximity in intermediate goods that is: total trade flows in intermediate inputs divided by the sum of GDP) across all country pairs is about 0.01% of GDP for the time period 1999 to 2008, with a value of about half a percent for US-Canada and for France-Germany. 21 The calibration of this parameter is subject to numerous debates between micro- and macro-economists. Estimates with micro data usually yields a low Frish elasticity of labor supply Chetty et al 2011) review many work and suggest that macro models with representative agents should use a Frish elasticity of 0.75) while the macro models traditionally take higher values. 22 See at 19

20 As in DiGiovanni and Levchenko 2012), fixed access costs are computed from the Doing Business Indicators. 23 In particular, I measure the relative entry fixed costs in domestic markets by using the information on the amount of time required to set up a business in the country relative to the US. 24 If according to the Doing Business Indicators database, in country i it takes 10 times longer to register a business than in the U.S., then f i,i = 10 f US,US. I normalize the lowest entry fixed cost so that no entry threshold lies below the lower bound o the productivity distribution, which is taken to be one in every country. To measure the fixed costs associated with entry in a foreign market, I use the Trading Across Borders module of the Doing Business Indicators. I choose the number of days it takes to import to a specific country, using the same normalization as for the domestic entry cost. This approach means that the fixed cost associated with trade from France to the US is the same as the one from Germany to the US. One must keep in mind, however, that the iceberg variable cost will differ. In the benchmark simulations, I choose the macro Armington) elasticity ρ to be equal to unity while the micro elasticity σ is equal to 4. There are many papers estimating those elasticities for intermediate or final goods. Saito 2004) provides estimations from 0.24 to 3.5 for the Armington elasticity 25 and Anderson and van Wincoop 2004) report available estimates for the micro elasticity in the range of 3 to 10. Following Bernard, Eaton, Jensen, and Kortum 2003), Ghironi and Melitz 2005) choose a micro elasticity of 3.8. Recently, papers such as Barrot and Sauvagnat 2015) or Boehm, Flaaen and Pandalai-Nayar 2015) argue that firms ability to substitute can be very low. There is also a theoretical convenience to use ρ = 1, as it allows the model to take the same form as other network models such as Acemoglu 2012), Bigio and La O 2015) and many others. Matching of macro moments I jointly set the country size parameters ψ i ) i=1,...,n as well as the shares α i j) the matrix W ) in order to match all countries relative GDP and all relative trade flows in real terms. More precisely, I normalize the real GDP of the composite rest-of-the-world to 100 and set all other real GDPs so that the ratio of their real GDP to the one of the rest-of-the-world in the simulated economy matches exactly its counterpart in the data for the time window 1999 to My targets are 23 The World Bank s Doing Business Initiative collected data on regulations regarding obtaining licenses, registering property, hiring workers, getting credit, and more. See trading-across-borders and 24 As argued in DIGiovanni and Levchenko 2012), using the time taken to open a business is a good indicator because it measures entry costs either in dollars or in units of per capita income, because in the model f i,i is a quantity of inputs rather than value. 25 Feenstra et al 2014) studies the macro and micro elasticities for final goods and reports estimates between and 4.08 for the Armington elasticity. They find that for half of goods the macro elasticity is significantly lower than the micro elasticity, even when they are estimated at the same level of disaggregation. 20

21 Parameter Value Counterpart β 0.5 Labor share see Jones2011) and Acemoglu et al 2012)) σ 4 Micro Elasticity of substitution see Anderson and van Wincoop 2004)) ν 1 Frisch Elasticity M/L 0.1 Mass of plants over working population τ ij [1-3] Iceberg trade cost From ESCAP - World Bank f ij [1-10] Fixed trade cost Doing Business Indicators Table 2: Parameter fixed using micro data then N real GDP targets as well as N 2 directed trade flows over GDP), with an equal number of parameters to match N 2 + N) if one considers the coefficients in the weighting matrix W and the scaling vector ψ i ) i=1,...,n. As a remark, it is interesting to note that it is necessary to jointly set the values of those parameters, as a country s GDP is influenced not only by the size of its labor force, chiefly governed by ψ i, but also by its productive structure which is captured by the whole W matrix. The choice of γ, governing the heterogeneity of the productivity draws, is key for the model to deliver a sizeable extensive margin adjustment and thus for the internal propagation mechanism. A classic target for this parameter is the right tail of the firm-size distribution implying a value such that the slope of the size distribution in a log-log scale is close to one Zipf Law) as in the data. 26 This would impose a value of approximately γ = σ 0.4 = 3.6 here. However, as argued above, such a stylized model delivers some realistic predictions as well as some simplifications of the reality. In particular, the simple threshold equilibrium that governs export decision leads to the fact that all adjustments along the extensive margin concern only firms in the neighbourhood of this threshold. On the contrary, data feature much more heterogeneity and the empirical literature suggests a high degree of churning in exports. 27 In appendix, I plot the size distribution of French firms, distinguishing exporters from non exporters. If the distribution of exporters first order stochastically dominates the one relative to non exporters, one can also see the large overlap between those densities, suggesting that firms opt in and out of export markets all along the size distribution and not only close to a single threshold. For this reason, calibrating the Pareto shape only to the right tail of the firm distribution might impede the model s internal propagation properties and lead to downward biased quantitative results. While it is not easy to find a suitable calibration target for this parameter, I choose to set the parameter γ in order to match the volatility of the number of products exported from France across all destinations present in the simulations. In the model, each variety is produced and sold by a different firm, which is not true in the data. hence, matching the 26 For papers using this calibration targets, see for example Chaney 2008) or many others. 27 See for example Lawless 2009) for evidence using Irish firms, Eaton et al 2007) for Columbian firms or Chaney 2014) for French firms 21

22 variance of the number of exported products is more meaningful that matching the variance of the number of exporting firms. Using French data, I compute the number of products exported to every country. After taking the logarithm to remove any level effect, 28 I then apply the HP filter to isolate the business cycle frequency fluctuations and take the standard deviation. For the model to perform well in matching the proportional) movements in the number of exported products, I set γ = In the sensitivity analysis, I present the value of the target for other choices of γ. Finally, I need to calibrate the variance-covariance matrix for the country-level TFP shocks Z i ) i=1,...,n. The volatility of those shocks, but also their correlation across countries plays an important role in the results. In order not to overestimate the impact of idiosyncratic shocks, I chose to set their volatility so that the model can replicate the volatility of output de-trended using HP filtering). This allows me to generate fluctuations in the simulated economy that are similar to those observed in the data. Concerning the degree of correlation, I will perform two distinct exercises that yields interesting results. In a first set of calibration, I set the off diagonal element of the covariance matrix to zero, so that the correlation in output that the model generates is solely to trade linkages across countries. 30 The results show that such an economy features 11.5% of output correlation, and that the trade-comovement slope is 60% the one computed in the data. Then, in a second exercise, I feed the model with a sequence of correlated shocks, setting all correlations to 20% for the sake of the example). In those simulations, the average level of correlation across output is obviously larger and reaches 44%, but more interestingly the trade-comovement slope is also increased and reaches 88% of the data. I discuss in detail those results in the next section. Procedure The calibration exercise insuring that the model matches the selected targets is done in two steps. Once the parameters estimated using micro data are fixed, I numerically solve the model at the steady state, meaning for a fixed value of the aggregate part of the TFP fixed and equal to one in every country. With the results, I compute the relative GDPs for all country-pairs as well as the trade flows and compare them to the data. If the moments do not match, I update the values of the shares α i j)) i,j=1,...,n and the scaling parameters ψ i ) i=1,...,n and iterate the process until 28 I choose not to match the fraction of exported products for two reasons. First, computing the fraction of exported products to all products produced in France would imply that all products are exportable, which is not the case. Secondly, I could try to match the fraction of exporting firms in the French data not products), which is around 20% of all firms. In order to match this number would require either an extremely low fixed cost significantly lower than the fixed cost associated with entry in the domestic market) or an unreasonable value of gamma γ > 100). 29 In appendix, I present the value of the volatility of the fraction of French firms or products, which is the same in the model) that are active in the export market for different values of γ, showing that my choice of γ = 9 is consistent with the magnitude of changes in firms export status. 30 Obviously, simulating uncorrelated shocks to those countries when they are in autarky would lead to uncorrelated output. 22

23 the model delivers exactly the targeted moments. Secondly, once all parameters are fixed but the shock structure, I feed the model with a sequence of shocks and evaluate the volatility of output for every country. If those volatility do not match the data, I update the variance-covariance matrix and iterate the process until the standard deviations of outputs in the simulated economy matches exactly their counterpart in the data. This method insures that the model is a meaningful approximation of the true economy in terms of relative country size, their trade flows and the fluctuations they experience. 5 Model fit and quantitative results The goal of this section is to assess the ability of the model to replicate the strong empirical relationship between trade proximity in intermediate inputs and the synchronization of output. The results of several quantitative explorations yield two interesting messages: firstly, when using a sequence of uncorrelated shocks, the model is able to replicate a trade-comovement slope of more than 62% of the value in the data), a 10 times improvement from previous studies. However, in this case, the level of synchronization is only 11.5% the one observed in the data. Secondly, introducing exogenously some correlation in country-specific technological shocks improves the model s performance along both dimensions. With 20% of correlations in the TFP shocks, the model generates 44% of output correlation and a trade-comovement slope of This suggests that the model is able to endogenously amplify an exogenous source of output comovement. The procedure presented in the last section yields values for all parameters so that the model economy matches the data for the period 1999 to From this reference point, I vary the shares α i j) recorded in the weighting matrix W ) from -10% to +10% and create 7 versions of the economy. For each value of W, I feed the economy with the same sequence of 5,000 shocks and record the correlation of HP-filtered GDP as well as the average index of trade proximity. This gives rise to a panel data set in which, for every value of W, I have 14 13/2 = 91 observations In the first set of simulations, the shocks are uncorrelated across countries so that international trade is the only source of output synchronization. To dig further into the model s internal propagation channels, I create a sequence of TFP shocks with 20% of correlation among all country pairs. This exploration gives rise to interesting results that are discussed below. In the last part of this section, I also present a sensibility analysis in which I vary the values of σ and γ. Trade-Comovement Slope Simulations using different values for W allow me to generate a panel dataset in which country pairs appear for different values of their respective shares. Using fixed effect regressions, I use the 23

24 variations in trade proximity to. The identification in the simulated economy is cleaner than the one in the data, since I used exactly the same sequence of shocks for all values of the matrix W. Hence, the different output comovement for a given country-pair are solely due to the variation of shares and this method allows me to precisely identify the effect of trade proximity on output synchronization. With uncorrelated shocks, I find a trade-comovement slope of 0.144, meaning that a doubling of trade proximity is associated with an increase of 0.1 of the output correlation relative to each country-pair average. In the case of correlated shocks, the slope rises to implying that mutliplying by two the index of trade proximity results in an additional correlation of This shows a very interesting feature of the model: when shocks are correlated, this exogenous correlation is propagated through the network of trading partners. To refine this property of the model, I then turn to the analysis of the level of correlation for both sequence of shocks. Data Uncorrelated Shocks Correlated Shocks logintermediate) 0.231*** 0.144*** 0.203*** 9.01) 22.05) 41.51) N Note: The dependent variable is correlation of HP filtered GDP. Intermediate is the sum of bilateral trade flows in intermediate goods divided by the sum of GDPs. t stat. in parentheses, *** means p < Table 3: Trade-Comovement Slope in the Data and in the Simulations Comovement Level With uncorrelated shocks, international trade is the only channel through which output can be synchronized across countries. In the baseline simulations, the simulated economy is able to achieve an average output correlation of 11.5%, which is significantly lower than the 72.3% observed in the data, suggesting that output comovement is not only the result of trade linkages but also other factors such as correlated shocks or financial linkages. Introducing an exogenous motive for GDP synchronization via a sequence of correlated shocks improves dramatically this result and provides interesting insights. When I impose 20% of correlation among all TFP shocks, the model is reaches an average output synchronization of 44%, meaning that the additional correlation due to endogenous propagation of shocks is = 24% which is larger than what is obtained in the case of uncorrelated shocks. Together with the increase in the slope, those facts suggest international trade alone cannot account for the very high degree of output comovement in the data, and that the model possesses the ability to propagate and amplify exogenous sources of output synchronization. 24

25 Sensibility Analysis According to 10), the extensive margin adjustments play an important role in the international transmission of shocks in the model. When setting ρ = 1 Cobb-Douglas aggregation of countryspecific bundles), this margin is governed chiefly by two parameters: the micro elasticity of substitution σ and the degree of heterogeneity in idiosyncratic productivity draws γ. In tables 4 to 7, I present the results of the simulations when I vary both parameters. For each value of the pair σ, γ), the model is entirely re-calibrated, 31 and I perform the same analysis than in the baseline simulations. As expected, the results for both the level and the slope are strongly impacted by the choice of the σ, γ) pair. Countries are more interdependent if the micro elasticity σ is low or of the degree of heterogeneity is low low γ). The impact of σ is due to a strong love for variety effect: when σ is low, the price index dual to every country-specific bundle in a given market is very sensitive to the number of firms serving this market. Any change in the threshold ϕ i,j translates into an important change in the corresponding price index, and hence a on the real GDP. Similarly, international transmission of technological shocks is stronger when within country firms heterogeneity is lower implying a higher value of γ). Indeed, large heterogeneity leads to lower density for any given productivity ϕ and reduces the volatility of the fraction of exporters in the simulation. Intuitively, if heterogeneity is reduced to two points ϕ H and ϕ L ), then we can have situations where a small technological shock leads to half the firms dropping in or out of the export market. One can also gain intuition about this effect by considering the disproportionate weight of the biggest firms when γ is small. With large heterogeneity small γ) and a fat tail distribution, movements along the extensive margin are reduced because there are many very large and productive firms that will not change their export status for small shocks. 32 Finally, recall that this type of model requires γ > σ 1 to ensure that the firm size distribution has a finite mean. For this reason, the sensibility tables do not feature any result for the upper right cells. 31 For different values of σ and γ, I have to adapt the values of all shares α ij), scaling parameters ψ i and most importantly the volatility of the every country-specific TFP shocks Z i in order to match the relative GDPs, Trade Proximity as well as standard deviation of HP-filtered GDPs. 32 This is reminiscent of the DiGiovanni & Levchenko 2012) paper that argues that if the size distribution of firms follows Zipf s Law, then movement along the extensive margin are virtually insignificant because trade flows are almost entirely due to the biggest firms. 25

26 σ = 4.0 σ = 4.5 σ = 5.0 γ = % γ = 5 6.1% 3.2% 2.6% γ = 7 8.7% 4.2% 2.7% γ = % 4.3% 3.0% γ = 15 19% 5.2% 3.8% γ = % 6.6% 3.4% Table 4: Average GDP Correlation Uncorrelated shocks σ = 4.0 σ = 4.5 σ = 5.0 γ = % γ = % 30.6% 27.8% γ = % 30.1% 28.7% γ = 9 44% 33.3% 29.4% γ = % 32.4% 30.3% γ = % 37.3% 30.8% Table 5: Average GDP Correlation Correlated shocks 20%) σ = 4.0 σ = 4.5 σ = 5.0 γ = γ = γ = γ = γ = γ = Table 6: Trade-Comovement Slope Uncorrelated shocks σ = 4.0 σ = 4.5 σ = 5.0 γ = γ = γ = γ = γ = γ = Table 7: Trade-Comovement Slope Correlated shocks 20%) Discussion The quantitative explorations presented in tables 4 to 7 show that the model goes a long way toward replicating the empirical relationship between output synchronization and trade proximity in intermediate goods. However, one should keep in mind that the calibration of such a stylized model is subject to debates since it produces sensible predictions along many dimensions while neglecting other aspects such as the great overlapping between the size distribution of exporters and non exporters. However, if the precise numbers found in this section must be taken with caution, they clearly show that taking into account the input-output linkages between countries as well as the powerful amplification mechanism at play through the extensive margin adjustments improve significantly the ability of the model to solve the Trade-Comovement Puzzle. Kose and Yi 2006) argued that when studying the trade comovement puzzle, it was necessary to take into account at least three countries so that the model can be consistent with the fact that the typical trade proximity between two given countries is very small, any given trade partner is 26

27 small compare to the rest of the world. The model developed here extents this argument and shows that one must take into account not only a third rest-of-the-world partner but the whole network of input-output linkages. In order to illustrate Kose and Yi 2006) s point, I also simulated the model using only two countries The United-States and Canada) isolated from the rest of the world. In this simplified quantitative versions, using uncorrelated shocks leads to an output correlation of 60.7%, which is about twice the correlation obtained between the US and Canada when using the full model with 15 countries. Moreover, varying the matrix W with the same magnitude leads to an implied trade-comovement slope of with uncorrelated shocks, a increase of more than 140% compare to the value of obtained in the baseline simulation with the full model. Introducing a third country representing all other trading partners decreases the slope to 0.311, in line with Kose & Yi 2006) s message but still significantly higher than the result of the full model. The key point to take away from this exercise is the importance of international interactions between many countries in order to explain bilateral correlations. Modelling many countries and hence allowing for worldwide supply chains reorganisation after a technological shock dramatically impacts the results. As a consequence, models featuring only two countries or even a simplified third rest-ofthe-world face the risk of strongly over-estimating the impact of one country on its trading partners. As argued in the theoretical part and in line with the quantitative results, firm heterogeneity plays an important role in the transmission of shocks. I illustrate quantitatively this point by simulating a version of the model featuring the same input-output linkages across countries but with a single homogeneous firm per country. Keeping the same calibration strategy in particular, one needs to recalibrate the matrix W, the scaling parameters ψ and most importantly the variance of each country specific shock 33 ), such a model feature a slope of with uncorrelated shocks and with 20 percent of TFP correlation. Hence, without firm heterogeneity, the model accounts for less than 10% of the observed association between trade and GDP synchronization, which is in line with the results featured in Johnson 2014). Finally, one can relates this result to the debate on the role of firm heterogeneity in international trade models, as discussed by Arkolakis, Costinot and Rodriguez-Clare 2012). With input-output linkages, equation 10) and tables 4 to 7 show that the extent to which a technological shock in one country impacts its trading partners depends on the degree of heterogeneity in productivity captured by the parameter γ. Hence, not taking into account this channel would tend to greatly reduce the ability of the model to replicate the empirical association between trade and comovement. 33 as expected, when the model does not feature firm heterogeneity, the variance of the shocks needed to match the observed volatility of GDP in every country is larger. In this case, without the powerful amplification mechanism due to firm heterogeneity, the shocks must be more than 20 times more volatile in order to match the data. 27

28 6 Conclusion This paper studies the relationship between output synchronization and trade proximity both empirically and theoretically. With fixed effect regression on a panel dataset from 1969 to 2008, I show that the association between trade and GDP comovement is stronger than what cross-section studies found in the past. Using disaggregated data, I then refine the analysis and show that all the statistical significance is captured by trade flows in intermediate goods and by adjustments along the extensive margin. Motivated by those facts, I propose a model of international trade in intermediate inputs with many countries and heterogeneous firms. With round-about production, the model features international supply chains which give rise to a strong interdependency across all firms in the economy for both their pricing and their export strategies. In a simplified case, closed form solution are found for the elasticity of GDP in all country with respect to a technological shock abroad. Those results suggest that two elements are important for the ability of the model to propagate shocks internationally: 1) the whole network structure of input-output linkages and not only the bilateral relationship) and the adjustments along the extensive margin. Finally, I calibrate this model to match precisely the relative GDPs, trade proximities and standard deviation of output for 14 OECD countries and assess its ability to replicate the empirical findings. Using uncorrelated TFP shocks, the model reaches an average correlation of 11.5% and a trade-comovement slope of 60% of its true value. Exogenously imposing correlation of technological shocks, the model reaches an average correlation of 44% and a slope of 88% of its value in the data, implying the existence of a strong amplification channel. Overall, the quantitative exercise suggests that the model makes an important step toward solving the Trade Comovement Puzzle. On a broader level, the purpose of this work is to highlight the consequences of the increasing segmentation of value chains across countries. This phenomenon leads to a strong interdependency between trading partners which in turn could have an impact on the optimal design of countryspecific policies. Depending on the exact nature of the international network of input-output linkages, diversification of trading partners might not lead to a hedge against adverse shocked, but rather increase total interdependency. Finally, this work also sheds light on the nature of gains from trade. With international production chains, a productivity increase in one country is propagated to all its customers. Equivalently, reducing trade barriers between countries leads to adjustments to the unit production costs for firms engaged in trade but also for those who are linked to firms engaged in international trade at any point of their production chain. Those effect can potentially be large, as is attested by the quantitative results presented in this paper. 28

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32 A Empirical Appendix A.1 Data description I focus the empirical analysis on a subset of all OECD countries. For the evolution of GDP correlation, I use 24 countries for which I could gather quarterly data on real GDP, which gives me NN 1)/2 = 276 country pairs. Those are Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Luxembourg, Mexico, Netherlands, New-Zealand, Norway, Portugal, South-Africa, Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States. For the study of the trade-comovement, I took out four countries from the sample. I get rid of Belgium and Luxembourg because they were pooled together in the trade data until the year 2000, preventing me from using 4 time windows and perform the fixed effect regression properly. I also took out New Zealand and South Africa because their trade data contained many zeros for some time periods, 34 resulting in some country-pairs being presents only for some time windows and hence reducing the effectiveness of the fixed effect regression. This leaves me in the end with 20 countries, so 190 country pairs for each of the four time window. I use data from the OECD database VPVOBARSA which features quarterly GDP. In this database, numbers are expressed as constant 2005 prices converted with 2005 PPPs. As for the trade flows, I use two datasets. From 1968 to 1999, I use the NBER-UN world trade data updated on the 30th of January, From 2000 onward, I use the revision of those data. In both datasets, trade flows are categorized using SITC4, which represents the first 4 digits of the SITC Rev 2 categorization. I used two different ways to separate final from intermediate goods. In the first method, I translate the SITC4 codes into END USE codes using the concordance table available on the CID website. 36 The end-use codes are used by the Bureau of Economic Analysis to allocate goods to their final use, within the National Income and Product Accounts. I then label as intermediate goods the products which end-use codes correspond to the list put together by Feenstra and Jensen. As a robustness check, I also use a second method to separate final and intermediate goods, which results are available upon request and are virtually the same. In this method, I translate the initial SITC4 codes into BEC using a table mappingone into the other from the World Bank website. 37 However, this table is not complete at all and, for example, almost 10% of the total flows for the 34 To fix ideas, in 1969, while Spain was importing from 134 countries and exporting to 140 countries, South Africa was importing from only 65 and exporting to 77 countries. As for New Zealand, the numbers were 105 imports) and 127 exports). 35 This data is constructed from United Nations trade data by Robert Feenstra and Robert Lipsey. It can be downloaded, as well as the associated documentation, from the Center for International Data CID) s website at 36 See at under the directory U.S. import data - SAS and STATA. 37 See 32

33 year 1962 cannot be allocated to BEC codes. In order to improve the quality of the match, I took the list of all SITC4 codes that were still unmatched in the 1962 data the first year available in the CID data) and used the SITC to BEC table provided on the EUROSTAT website 38. This table is in PDF format and one needs to manually go through file in order to identify which of those unmatched SITC codes at the 4 digits level) can be allocated uniquely and without ambiguity to any of my two categories. As an illustration, this method allows me to take care of 97.64% of all trade flows in the year 1962 and 99.48% of all trade flows in Finally, from BEC codes, one can categorize goods into Intermediate, Capital and Final goods with a concordance table available on the UNSTAT website. 39 A.2 Evolution of GDP correlation In the graph below, I computed all country pair correlation of GDP using a sample of 24 OECD countries from 1960 to 2012 using a 10 years rolling window. Each point of this graph represents the average correlation of GDP across all country-pair in the 10 years preceding the point. The vertical red line is set at 2007Q3 and is a rough materialization of the last financial crisis. The point which is exactly at the red line is then the average GDP correlation from 1997Q3 to 2007Q3 and hence is not affected by the collapse of subprime markets and the events that followed. The graph on the left is the correlation HP filtered in order to keep the business cycle frequency, while on the right I used the Baxter and King filter to extract fluctuations from 32 to 200 quarters as suggested by Comin and Gertler 2006). A.3 Additional regression results This section provides several robustness checks for the empirical part of the paper. Changing the filtering method to extract the short or medium term fluctuation of GDP does not alter the main message that trade is an important explanatory variable for the bilateral output comovement and in particular trade in intermediate input captures most of the statistical power. In the main text, I 38 See at 39 See at 33

34 showed that it was true when using the HP filter with smoothing parameter Here, I provide evidence that the result holds when taking the log first-difference as well as the Baxter and King band pass filter with fluctuations between 32 and 200 quarters to keep the medium term fluctuations as suggested by Comin and Gertler 2006). As in the main text, I created four time windows of 10 years each, from 1969 to For every time window, I computed all bilateral output correlation as well as the average of the trade proximity index which is the sum of bilateral trade flows divided by the sum of GDP) using total trade flows first, and then differentiating between trade in intermediate goods and trade in final goods. Finally, I perform fixed effect regressions on this panel dataset, allowing me to identify separately the evolution of a variation of trade proximity and a variation in output correlation taking away the important country-pair fixed effects such as correlation of shocks or financial linkages. Alternative filtering methods Log Difference Baxter & King I) II) I) II) logtotal) 0.193*** 0.330*** 12.69) 12.52) logintermediate) 0.156*** 0.289*** 7.62) 9.45) logfinal) ) -0.95) adj. R N 760 Note: The dependent variable is correlation of filtered GDP. The first two columns use a log first-difference to remove the trend. The last two columns use a band pass filter by Baxter and King and keep the fluctuations between 32 and 200 quarters as suggested by Comin and Gertler 2006). Total is the sum of bilateral trade flows divided by the sum of GDPs. Final and Intermediate are similar but using only trade in final and intermediate goods. t stat. in parentheses, *** means p < Table 8: Trade-Comovement Slope for 20 OECD countries with other filtering methods Changing the definition of time windows 34

35 Pooled OLS FE Regressions I) II) I) II) logtotal) *** 0.287*** 10.46) 10.89) logintermediate) *** 0.164*** 3.54) 4.34) logfinal) ) 1.86) N 380 Note: In this table, I constructed 2 time windows of 20 years each. The dependent variable is correlation of HP-filtered GDP. Total is the sum of bilateral trade flows divided by the sum of GDPs. Final and Intermediate are similar but using only trade in final and intermediate goods. t stat. in parentheses, *** means p < Table 9: Trade-Comovement Slope for 20 OECD countries with only 2 time windows of 20 years each Cross Section Regressions 1969 to 1978) 1979 to 1988) 1989 to 1998) 1999 to 2008) I) II) I) II) I) II) I) II) logtotal) 0.068*** 0.075*** 0.094*** 0.054*** 4.29) 5.15) 5.09) 5.78) logintermediate) 0.090* ) 1.62) -1.85) -0.41) logfinal) ** 0.063** -0.74) 0.60) 3.61) 2.68) N 190 adj. R Note: The dependent variable is correlation of HP filtered GDP. Total is the sum of bilateral trade flows divided by the sum of GDPs. Final and Intermediate are similar but using only trade in final and intermediate goods. t stat. in parentheses, *** means p < Table 10: Cross-Section Trade-Comovement Slope for a sample of 20 OECD countries 35

36 A.4 Size distribution in France Using a detailed dataset with the universe of all French firm, I can compute the size distribution of exporters and non exporters. The graph below show that there is a very large overlap between those two distributions, which is at odds with the model s prediction. This suggests that calibrating the model to match the actual size distribution, or its right tail as it is commonly done in international trade models, might lead to an underestimation of adjustments along the extensive margin. A.5 Fluctuation in the number of exported varieties in France Using French Data, the table below presents the standard deviation of the logarithm of the number of products exporting to different destinations. Data are annual and HP filtered with smoothing parameter Destination Std. Dev of lognb of Products) Australia Austria Canada Denmark Germany Ireland Italy Mexico Netherlands Spain United Kingdom United States Average Table 11: Standard Deviation of the Number of products exported from France In order to match the magnitude of the movement along the extensive margins, I set the pa- 36

37 rameter γ so that the simulated model matches the average standard deviation in table 11. Fixing σ = 4, the values of the volatility of lognumberof exportingf irms), HP filtered are computed for different values of γ. The results are presented in table 12. Data γ = 3.5 γ = 5 γ = 7 γ = 9 γ = Table 12: Standard Deviation of the Number of products exported from France B Theoretical Appendix B.1 Equilibrium Conditions in the general CES case Price Indexes and Pricing System Denoting by Ω k,k = [ϕ k,k, + ), we can write the price of country specific bundles M k,k, the input bundles I k and the production bundles B k = I 1 β k P k,k = IP k = Ω k,k k =1,...,N l β k respectively as: p k,k ϕ) 1 σ gϕ)dϕ α k k )P 1 ρ k,k 1 1 ρ P B k = 1 β) β 1 β β IP 1 β k 1 1 σ w β k The optimal pricing strategy is p k,k = τ k,k σ σ 1 P B k Z k ϕ and the price of any firm is a function of the prices charged by every other firms in the economy through P B k. Solving for the vector of prices comes down to solving for the price indexes relative to country specific bundles, which can be done by solving P 1 ρ k = µ k α k k ) k τ k k ϕk,k ϕ k,k ) ) σ γk 1 1 ρ P k 1 β, k = 1,..., N Entry Thresholds Variable profits are a fraction 1/σ of total revenues and firms enter a market if and only if they can make a positive profit in it. 37

38 At Home π k,k ϕ k,k ) = f k,k P B k Z k ϕ k,k = ) σ P B k 1 σ 1 Z k P k σf k,k P B k Z k X k + Pk IP k ) 1 ρ αk k)1 β)s k 1 σ 1 Abroad P B k π k,k ϕ k,k ) = f k,k Z k ϕ k,k = τ kk σ σ 1 ) P B k 1 Z k P k,k Pk,k IP k P B σf k k,k Z k ) 1 ρ αk k)1 β)s k 1 σ 1 Replacing P k,k by its expression using P k, we also get γ 1+ k σ 1)).σ ρ) σ 1) ϕ 2 k,k = τ ρ 1 σ 1 kk σ σ 1 P B k Z k 1 ρ σ 1 IPk ϕ σ γ k 1 σ 1 k,k ) ρ σ ) P B ) 1 σ 1 σfk,k k σ 1 Z.P k k α k k)1 β)s k Labor Market Equilibrium w k L k = βs k and ψl ν = w P Spending System Using lemma 1, we have an expression of total firms spendings in country k. Indeed, total firms sales are R k while total profits net of fixed costs are σ 1 γ k σ R k. Hence, we can write S k = 1 σ 1 ) R k γ k σ Moreover, with the labor demand equation w k L k = βs k, we get a nice expression of total consumer s spending: X k = βs k + σ 1 γ k σ R k Plugging those two equations in the expression of revenue 6) for every country k and simplifying the 1 β) term directly leads to the spendings system 8). 38

39 B.2 Equilibrium Conditions in the Cobb Douglas case ρ = 1) When ρ = 1, the production function writes Y k ϕ) = Z k ϕ k M αkk ) k,k ) 1 β l β k with k α k k ) = 1. In order to save on notation, I denote α k k ) = α k k ) 1 β). Price Indexes and Pricing System P k,k = p k,k ϕ) 1 σ gϕ)dϕ 1 1 σ P B k = Ω k,k k αk k ) ) α k k ) Using the optimal pricing strategy p k,k = τ k,k β β σ P B k σ 1 Z k ϕ to each country specific bundle, we have the pricing system: k = µ k P 1 σ k τ k,k with µ k = γ σ γ k 1 kϕk,k γ k σ 1) M σ k σ 1 wβ k β β k ϕk,k ϕ k,k k P αkk ) k,k ) w β k with the definition of the price index relative ) ) αk σ γk 1 k ) P 1 σ k ) αk k ) αkk ) be transformed into a linear one by writing Y k = logp 1 σ k ): with λ k = log Y 1. Y N µ k k τ k,k ϕk,k ϕ k,k = I N W ) 1 1 Z k ) 1 σ. This non linear system can λ 1. λ N ) ) σ γk 1 α k k ) constants that depend on the entry thresholds in every country and every export market and W the weighting matrix which entries are the shares α i j). 39

40 Entry Thresholds In very market, entry occurs until the profit of the least productive firms is equal to the fixed cost of accessing the market. The yields At Home Abroad π k,k ϕ k,k ) = 0 ϕ k,k = π k,k ϕ k,k ) = 0 ϕ k,k = ) σ P B k 1 σ 1 Z k P k τ kk σ σ 1 σfk,k P B k Z k X k + α k k)s k ) 1 σ 1 ) ) 1 P B P B k 1 σfk,k k σ 1 Z k Z k P k,k α k k)s k Labor Market Equilibrium w k L k = βs k and ψl ν = w P Spending System In the Cobb-Douglas case, total revenue of all firms from country k can be written as R x = X x + k αk k)s k Then using lemma 1 and the labor demand equation the whole spending system can be simply written as S 1. S N = βi N + W ) T S 1. S N 1 β)i N W ) S Ạ T. }{{} =M S N = 0 R N One can easily show that the matrix M is non invertible 40 and is of rank exactly N 1, meaning that the solutions of the system is a one dimensional space. This is reassuring because it means we can normalize one price to one. I then normalize w 1 = 1 and with the labor demand equation this results is S 1 = L 1 β 40 One can easily see that summing all rows results in a column of zero. 40

41 B.3 Proof of Lemma 1 Reminder of Lemma 1 : Total profit in country k are proportional to total revenues: Proof Π k = σ 1 γσ R k First, since firms charge a constant markup σ/σ 1) over marginal cost, we know that variable profits are a fraction 1/σ of total revenues. Hence, total profits net of fixed costs for all firms in k are simply Π k = R k σ k F C k k where F C k k is the sum of fixed cost payment from all firms from country k serving market k. Then, note that total fixed cost payment for all firms in country k is F C k k = M k + ϕ k,k f kk P B k Z k γ k ϕ γ k 1 dϕ P B k = M k f kk ϕ γ k Z k,k k If k k, we can also express total revenues sales) from k to k as R k,k = M k = + ϕ k,k τ kk γ k M k γ k σ 1) σ σ 1 τ kk ) P B k 1 1 σ α k k)s k ϕ σ 1 gϕ)dϕ Z k P k,k σ σ 1 Next, using the expression for the threshold ϕ σ 1 k,k get γ k M k R k,k = And we recognize finally that R k,k = ) P B k 1 1 σ α k k)s k ϕ σ γ k 1 Z k P k,k k,k γ k σ 1) σf k,k P B k derived above as a function of P k,k ), we Z k γ k γ k σ 1) σf C k k For domestic revenues, we can show using the same steps that ϕ γ k k,k X k + R k,k = γ k γ k σ 1) σf C k k 41

42 Combining those expressions, we get F C k k = γ k σ 1) γ k k σ = γ k σ 1) γ k σ R k X k + k R k,k ) Using this expression of k F C k k in the definition of profits completes the proof. B.4 Proof of Lemma 2 Reminder of Lemma 2 : In the Cobb-Douglas ρ = 1) and fixed labor supply case, the elasticity of every GDP with respect to a technology shock in country 1 is given by: η GDP1,Z 1. η GDPN,Z 1 1 = I N W T ) 1 0. with W the weighting matrix defined above and T a Transmission matrix function of γ k and σ. Proof: In this simplified case ρ = 1 and fixed labor supply), the labor demand schedule w k L k = βs k provides a one to one mapping between total spendings S k and the wages w k. Moreover, inspecting the spending system SS) when ρ = 1 reveals that once a choice of numeraire is done that is, taking w 1 = 1 and hence fixing S 1 = L 1 /β), the vector of spendings S i ) i=1,...,n is independent of the technology level and is completely pinned down by the spending system 8). Using lemma 1 and the fact thtat labor supply is fixed, we can then show that total consumers spending X i also independent of technology level. Thus, the GDP elasticity is simply the opposite of the elasticity of the country s consumers price index. Moreover, with fixed labor supply and the assumption that the mass of potential entrepreneurs is proportional to labor size, the mass of firms M i is fixed for every country i. In the next sections, I compute elasticities of all endogenous variables step by step until I can solve for the price index elasticities. 42

43 B.4.1 Price Indexes Home Price Index at home P k Using the definitions of price indexes, we can easily show that logp k ) logz k ) = 1 + logp B k) logz k ) + γk σ 1) σ 1 ) logϕk,k ) logz k ) We can see in this formula the direct effect of lowering all prices in country k plus two other indirect effects : the propagation going through the input-output linkages in the P B k term as well as the extensive margin of entry of new firms through the ϕ k,k term. Foreign Price Index at their home P k logp k ) logz k ) = logp B k ) + logz k ) ) γk σ 1) logϕk,k ) σ 1 logz k ) The foreign price index at their home is not affected directly but only through the effects going through the input output linkages as well as the entry of new firms. Export Price indexes P i,j The price index relative to varieties from i sold on j s market is affected by the shock according to: logp i,j ) logz k ) = logp i) logz k ) + γk σ 1) σ 1 ) logϕi,j ) logz k ) logϕ ) i,i) logz k ) We can see that the effect of a technology shock on exporting price indexes depends on the widening in the range of exported goods, as measured by the second term, in the brackets. Input Bundle Price P B k Abroad Using the fact that wages are not affected by technology shocks, I can compute the elasticity of the input bundle price with respect to a technology shock at home as follow: logp B k ) logz k ) = 1 β) j [ logpj ) α k j) logz k ) + γj σ 1) σ 1 ) logϕj,k ) logz k ) logϕ )] j,j) logz k ) Putting Pieces together for a shock to Z 1 Replacing the elasticities of Input Bundle s price into the elasticities of price indexes, we get the following nice interpretable system where I use the notation η a,b to denote the elasticity of an 43

44 endogenous variable a with respect to an exogenous variable b) η P1,Z 1 1 η P1,Z 1. = β)w. η PN,Z 1. η PN,Z 1 κ 1 η ϕ1,1,z 1 +α 1 2)κ 2 η ϕ2,1,z 1 η ϕ2,2,z 1 ) α 2 1)κ 1 η ϕ1,2,z 1 η ϕ1,1,z 1 ) +κ 2 η ϕ2,2,z Where W is still the weight matrix and I used the notation κ k = γ k σ 1) σ 1. This expression shows the different effects at play in the model: The direct effect that concerns the shocked country only An indirect effect due to input-output linkages represented by the weight matrix W An entry margin effect du to the change in the mass of entrants in all markets, captured by the last term. I will now characterize precisely the effect of a technology shock to the entry thresholds. B.4.2 Thresholds Home Entry Threshold ϕ k,k at Home Using the definition of the thresholds from above and replacing logp B k) logz k ) the expression of the elasticity of the Home price index at home, we get logϕ k,k ) logz k ) = 1 σ 1 + κ k σ logp k) logz k ) 1 by its expression in 1 The scaling factor σ 1+κ k σ ) is positive while the second term is negative, meaning that a positive technology shock trigers the entry of more firms in the country, which amplifies the effect of the shock. Export Entry Threshold ϕ k,k for Home firms exporting to k Using the second definition of the export thresholds from above, we get γ k σ 1 ) logϕ k,k ) logz k ) = ) logp σ 1 ) Bk ) logz k ) 1 logϕ k,k ) + κ k logz k ) logp k) logz k ) 44

45 Moreover, replacing logp B k) logz k ) 1 by its expression we get logϕ k,k ) logz k ) = σ 1 γ k 1 + κ k σ 1 + κ k σ logp k) logz k ) Finally, using the fact that 1 + κ k = γ k σ 1, we get logϕ k,k ) logz k ) = 1 σ 1 + κ k σ logp k) logz k ) Home Entry Threshold ϕ k,k Abroad Using the definition of the thresholds from above and replacing logp B k ) logz k ) by its expression, we get logϕ k,k ) logz k ) = 1 σ 1 + κ k σ logp k ) logz k ) Export Entry Threshold ϕ k,j for Foreign firms exporting to j With the second definition of the threshold and using the expression of η ϕk,k,z k, one can show that the elasticity of the exporting threshold is proportional to the elasticity of the domestic entry threshold and that the scaling factor do not depend on the export market considered : logϕ k,j) logz k ) = 1 + κ k γ k σ 1 logϕ k,k ) logz k ) Finally, using the expression for 1 + κ k, we get logϕ k,j) logz k ) = logϕ k,k ) logz k ) = 1 σ 1 + κ k σ logp k ) logz k ) B.4.3 Final Expression Now that I have an expression for the elasticity of all thresholds as functions of the elasticities of price 1 indexes, I can gather the results. Introducing Λ k =, I define a matrix T for Transmission) σ+ σ 1)2 γ k σ 1) as T = diagλ 1,..., Λ N ). This matrix characterizes the additional propagation mechanism due to the change in the mass of firms in all markets. Then, the price index elasticities are defined by η P1,Z 1. η PN,Z 1 Finally, noting that for all i, η Pi,Z 1 = η GDPi,Z 1 1 = I N 1 β)w T ) concludes the demonstration. In order to gain intuition on this formula, a few comments are in order. First, note that in the case 45

46 of complete autarcy of all countries we have W = I N to that the elasticity for country 1 is simply η GDP1,Z 1 = 1/β Λ 1 ) whereas all other elasticities are zero. This result is reminiscent of what is found in Jones 2011) with the additional propagation mechanism due to the adjustment along the extensive margin captured by Λ 1. Interestingly, this special case highlights the fact that we need 1 β) + Λ 1 < 1 in order to get a positive own-country elasticity. This condition is necessary for the validity of 10), since it corresponds to imposing that the reason of the geometric sequence is below one Secondly, noting that Λ k =, one can see that for a fixed σ, Λγ σ+ σ 1)2 k ) is a strictly γ k σ 1) increasing function. When γ k σ 1, Λ k 0 and when γ k +, Λ k 1/σ. For a labor share β = 0.5 we can see that any value σ > 2 is sufficient to insure the validity of the condition 1 β) + Λ k < 1 for any value of γ k within the range of admissible values γ k > σ 1). B.5 Additional Results Interestingly, using correlated shocks allows the model to match the ranking of bilateral correlation: countries that tend to be more synchronize in the data are also more synchronized in the simulations, as is shown by the positive slope of the fitted line in the figure 2 Figure 1: Uncorrelated Shocks Figure 2: 20% Correlated Shocks 41 The formula 10) is the matrix analogue of summing an infinite geometric sequence. 1 β)w + T corresponds to the first order effect of the shock, 1 β)w + T ) 2 is the second order effect, etc... The total effect can then be described by the matrix I N 1 β)w T ) 1 if and only if the eigenvalues of the matrix 1 β)w + T all lie within the unit circle. In the autarcy case, this condition is insured by 1 β) + Λ 1 < 1. 46