metadata: authors: 0: author: Fally,Thibault
1: author: Hillberry,Russell Henry
count: World
docna: 0: docna: A Coasian model of international production chains
docty: Policy Research Working Paper
subtopic: Labor Markets,International Trade and Trade Rules,Trade and Services,Rural Labor Markets
teratopic: International Economics and Trade
abstracts: cdata!: International supply chains require the
            coordination of numerous activities across multiple
            countries and firms. This paper develops a theoretical model
            of supply chains in which the measure of tasks completed
            within a firm is determined by parameters that define
            transaction costs and the cost of coordinating more
            activities within the firm. The structural parameters that
            govern these costs explain variation in supply chain length
            as well as cross-country variation in
            gross-output-to-value-added ratios. The structural
            parameters are linked to comparative advantage along and
            across supply chains. The paper provides an analytical
            treatment of trade and welfare responses to trade cost
            change in a simple two-country model. To explore the
            model's implications in a richer setting, the model is
            calibrated to match key observables in East Asia, and the
            calibrated model is used to evaluate implications of changes
            in model parameters for trade, welfare, the length of supply
            chains, and countries' relative position within them.
repnme: repnme: A Coasian model of international production chains
display_title: A Coasian model of international
            production chains
listing_relative_url: /research/2015/10/25124943/coasian-model-international-production-chains
topicv3: Labor Markets m1326488 855,Trade m1326924 698,Trade in Services m1326928 780,Labor Supply m1504055 2751,Labor Demand m1583073 4422
publishtoextweb_dt: 2015-10-06T00:00:00Z
historic_topic: International Economics and Trade
no_of_pages: 71
author_keys: 0: author_key: 1308466
1: author_key: 1228415
model_extractions: WPS7434 Policy Research Working Paper 7435 A Coasian Model of International Production Chains _Thibault Fally Russell Hillberry_ Development Research Group Trade and International Integration Team October 2015
Policy Research Working Paper 7435

# **Abstract**

International supply chains require the coordination of numerous activities across multiple countries and firms. This paper develops a theoretical model of supply chains in which the measure of tasks completed within a firm is determined by parameters that define transaction costs and the cost of coordinating more activities within the firm. The structural parameters that govern these costs explain variation in supply chain length as well as cross-country variation in gross-output-to-value-added ratios. The structural parameters are linked to comparative advantage along and across supply chains. The paper provides an analytical treatment of trade and welfare responses to trade cost change in a simple two-country model. To explore the model's implications in a richer setting, the model is calibrated to match key observables in East Asia, and the calibrated model is used to evaluate implications of changes in model parameters for trade, welfare, the length of supply chains, and countries’ relative position within them.

This paper is a product of the Trade and International Integration Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at rhillberry@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# A Coasian Model of International Production Chains

Thibault Fally and Russell Hillberry UC-Berkeley ARE and World Bank<sup>_∗_</sup>

**Keywords** : Fragmentation of production, Transaction costs, Trade in intermediate goods, Boundary of the firm

## **JEL Classification** : F10, L23

> _∗_ We would like to thank Pol Antr`as, Arnaud Costinot, Ben Faber, Rob Feenstra, Tomoo Kikuchi, Aart Kraay, Phil Luck, Ezra Oberfield, Andres Rodriguez-Clare, Ina Simonovska and seminar participants at UC Davis, UC Berkeley, Dartmouth College, Federal Reserve Bank of Philadelphia, Federal Reserve Board of Governors, Drexel University, Columbia University, John Hopkins University, UC Santa Cruz, Barcelona GSE, CEPR/CAGE/ECARES Conference on Global Fragmentation of Production, West Coast Trade Workshop at SIEPR/Stanford, NBER ITI Spring meeting, and the SED meetings for helpful comments and discussions. Research for this paper has been supported in part by the governments of Norway, Sweden, and the United Kingdom through the Multidonor Trust Fund for Trade and Development. Contact: fally@berkeley.edu, Department of Agricultural and Resource Economics, University of California, Berkeley, CA 94720-3310, USA.
# **1 Introduction**

The nature of international trade changed dramatically in recent decades, as vertically integrated production processes spread across international borders, increasing trade in parts and components along the way.<sup>1</sup> This phenomenon raises a number of important questions for economic policy: How large are the gains from international fragmentation, and how are they distributed across countries? How do changes in trade costs affect trade flows and the distribution of value added across countries? How has China’s entry into the world trading system affected the international fragmentation of production? Answers to these questions require quantitative models that can represent the complexities of international production chains in a tractable form.

Recent evidence has documented substantial variation in supply chain length.<sup>2</sup> Even within chains, firms vary in their contribution to value added.<sup>3</sup> The length of supply chains and the degree to which they are internationalized are difficult to separate from decisions that determine firm scope. The Ford Model T, for example, was produced in a single plant, while the production of modern day automobiles can involve a myriad of heterogeneous suppliers scattered across multiple countries. International fragmentation is limited, ultimately, by the extent of fragmentation at the firm level. Yet the literature lacks a unified treatment that can explain endogenous firm boundaries within chains, formalize endogenous chain lengths, and determine comparative advantage within and across chains.

We offer a framework that accomplishes these goals. In the model, an optimal allocation of tasks determines jointly the scope of sequentially-arranged firms of varying size, the length of chains and the sequence of countries in production.<sup>4</sup> We calibrate the model using key moments from input-output tables on East Asia and the United States.<sup>5</sup> Our focus on East Asia reflects the importance of international fragmentation in that region. Based on our calibration, we are able to quantify the impact on intermediate and final goods trade, fragmentation and welfare, of changes in: 1) international trade costs, 2) productivity in China, 3) transaction costs in China, and 4) a reduction in bilateral trade costs between the US and China.

> 1Baldwin (2012) surveys these developments and provides insights into how they should affect our thinking about the economics of international trade and trade policy.

> 2See Antr`as et al. (2012) and Fally (2012).

> 3See Kraemer et al. (2011), who illustrate the distribution of value added for Apple iPhones and iPads. 4Our framework is perfectly competitive and therefore silent about the distinction between firms and plants. We shall use the term ‘firms’ throughout, but one could reinterpret everything in terms of ‘plants’ without affecting the results.

> 5Such tables have been used to quantify the extent of fragmentation and the allocation of value across countries. Johnson (2014) surveys a series of papers that use such tables to calculate value added trade. Koopman et al. (2010) use Chinese tables to calculate the domestic content of China’s exports. Antras et al. (2012) derive indices of supply chain length from input-output tables.

2
A central theoretical contribution is the development of a tractable framework in which supply chain length is endogenous. Supply chains vary in the number of firms involved because of endogenous differences in firm scope. Following Coase (1937), firm scope is determined by a marginal tradeoff between the cost of coordinating activities inside the firm and the costs of conducting market transactions. Going further, we formalize the marginal tradeoffs facing a chain of firms in a shadow market for tasks rather than focusing on the boundaries of a single firm.<sup>6</sup> Within countries, outcomes are driven by two key parameters: one that governs coordination costs within the firm and one that summarizes domestic transaction costs. Our continuous representation of a firm allows us to derive strong and transparent links between structural parameters and observables in the data. Specifically, the gross-output-to-value-added ratio at any point in the chain is equal to the ratio of the Coasian parameters that govern coordination costs and transaction costs, respectively.

We allow the Coasian parameters to vary across countries and develop implications for international trade. Vertical specialization in our model is tightly related to firm scope. In equilibrium, firm scope decreases as we move upstream along the chain. In turn, this pattern affects the sorting of countries along the chain. Within a given chain, the most upstream countries are those in which it is most difficult to coordinate multiple tasks within the firm. Transaction costs affect absolute, not comparative, advantage within the chain, but transaction costs have an indirect effect on countries’ average position in chains. Countries with high transaction costs are more likely to participate in chains for which the country has low coordination costs, which means that such countries tend to be positioned downstream.

We examine the effect of trade costs on trade and fragmentation in this setting. As expected, we find that opening to trade tends to increase the extent of fragmentation along several dimensions. The channels, however, are not trivial. Trade affects fragmentation at all stages and decreases firm scope even for firms that do not directly offshore production but are related to firms that do. The reduction in firm scope along the chain is associated with a decrease in average costs, especially downstream, and contributes to the reduction in final goods prices. We derive analytical results that reveal transparent links between the impact of fragmentation on firm scope, final goods prices and a shadow cost of tasks that governs firm scope along the chain.

As trade costs decrease, countries tend to move downstream along chains and to enter new chains. We illustrate our finding in a partial-equilibrium setting, holding the set of participating countries and their labor cost constant, and in a two-country general equilibrium setting. In

> 6Our framework builds on that of Kikuchi et al. (2014) who model a Coasian supply chain in a one-country partial equilibrium setting, assuming discrete firms. We develop a continuous firm treatment that facilitates tractable analytical solutions and calibration and extend it to a multi-country general equilibrium setting.

3
the latter, we also use our framework to examine the response of trade flows to trade costs, both in gross flows and value added content, and the welfare gains from trade. We compare our results to a single-stage Eaton and Kortum (2002) model: the gains from trade are relatively larger in countries that tend to specialize downstream and smaller in countries that tend to specialize upstream.

In order to explore the quantitative implications of our framework, we calibrate a numerical version of our model to match key features of input-output relationships in East Asia. This exercise relies on international input-output tables produced by IDE-JETRO. The data cover the US and nine East Asian countries. This region is interesting because production fragmentation there has grown quickly and is highly prevalent. The IDE-JETRO data are unique in that they track flows in four dimensions: from the making industry in the origin country to the using industry in the destination country.<sup>7</sup> To illustrate our findings, we adapt recently developed quantitative measures of firm position (i.e. upstreamness) so that they track the average number of international borders crossed, rather than plant boundaries as in the original. Our calculations indicate increasing international fragmentation over time, especially in key industries like electronics.

While the model has rich implications for trade and the fragmentation of production, its relative parsimony is useful for the purpose of calibration. We calibrate our model by targeting key moments such as GDP per capita, value added, countries’ average position in international supply chains and gross-output-to-value-added ratios. All these moments imply large crosscountry differences in productivity, transaction costs and coordination costs.

We use the calibrated model to conduct counterfactual exercises regarding changes in key structural parameters. We first examine what happens when cross-border trade costs decrease by 10%. In this counter-factual simulation, we find larger gains from trade than predicted by Arkolakis, Costinot and Rodrigues-Clare (2012)’s formula (based on imported final goods), especially in downstream countries. We also examine the response in terms of the VAX ratio (the value-added content of exports) as defined by Johnson and Noguera (2012). We find that a decrease in trade costs leads to a decrease in the VAX ratio for most countries, which can be interpreted as an increase in cross-border fragmentation. In subsequent counterfactual exercises, we simulate a 10% increase in productivity in China and a 10% decrease in Chinese transaction costs. Both the productivity and transaction cost shocks produce similar changes in Chinese and other countries’ welfare, but the shocks have different implications for the organization of international production. The productivity shock causes China to move downstream while the rest of the world moves upstream. The reduction in Chinese transaction costs causes a relative

> 7Other data that report such figures, like the World Input-Output Database (WIOD), impute these values assuming proportional treatments.

4
move upstream by China. Reducing transaction costs in China also lengthens global supply chains. This is entirely consistent with the model’s qualitative predictions, but the calibration teaches us that the shock to Chinese transaction costs has quantitative effects outside of China. Finally, we simulate a 10% reduction in trade costs between China and the US to investigate the degree to which fragmentation magnifies trade responses. At our benchmark level of trade costs there is little evidence of magnification, but it does appear when we calibrate to lower levels of trade costs.

**Relationship to the literature:** The paper contributes to the literature in two broad ways: 1) we develop a model that formalizes a role for firms (as distinct from tasks) in a sequential, multi-country general equilibrium with endogenous chain length, 2) we calibrate our model and provide quantitative implications using input-output tables for East-Asian production. We discuss the literature surrounding each of these contributions in turn.

**1. Models of production chains.** An important question in the literature on international production chains is the spatial organization of production across countries. We contribute to this literature in two ways by endogenizing the extent of fragmentation across firms and by endogenizing the relative position of countries along the chain.

In recent work, Costinot et al (2012) derive an explicitly sequential multi-country model in which mistakes can occur with given probability and these mistakes destroy all accumulated value. They show that countries with relatively high probabilities of mistakes are situated upstream. The intuition for this result broadly follows Kremer (1993), that higher rates of mistakes do less damage if they occur upstream. The Costinot et al (2012) framework has no implications for the extent of fragmentation across firms and the allocation of tasks across Instead, we formalize the firm’s internalization decision and endogenize the range of firms involved in the chain. The motivation for this follows Coase (1937), and our mathematical framework is inspired by Kikuchi et al (2014), who show how Coase’s insights can be applied to production chains. Kikuchi et al (2014) solve their model in a sequential partial equilibrium setting, and employ discrete firms. We adapt their framework to a continuum of firms in a multi-country setting where countries differ in key parameters governing transaction costs and diseconomies of scope.

As in Costinot et al (2012), we examine how countries specialize along the chain, but the patterns of specialization are now driven by interactions between firm scope, transaction costs and ad-valorem trade costs affecting cross-border transactions. In addition, we offer explicit links between the Coasian structural parameters in our model and empirical objects that can be 5
observed or constructed from input-output tables. These links make calibration of the model relatively straightforward compared to other models in the literature.

Several models fix the number of production stages by assumption (Krugman and Venables 1996, Hillberry and Hummels 2002, Yi 2003, 2010, Johnson and Moxnes 2013). The focus of this literature is often the geographic location of each production stage, relative to the other(s), and so a finite and countable number of stages is useful for analytical purposes.<sup>8</sup> Relative to our work, these models avoid the question of the allocation of activities or tasks across stages, and focus on the extensive margin of completing a specific stage in a certain location.

These models are also silent about why some countries specialize upstream while others are downstream. In Yi (2010) and in Johnson and Moxnes (2013), for example, the specialization of countries along the chain is driven by exogenous productivity shocks and trade costs. This literature makes important insights about non-linear responses of trade to trade costs and differences between gross and VA trade. Our model also contains these forces, but we introduce intra-firm coordination costs and inter-firm transaction costs as additional sources of crosscountry heterogeneity. One goal of our paper is to understand the robustness of these insights to the richer theoretical structure we offer, where the extent of fragmentation and the specialization of countries along the chain are endogenous.

**2. Quantitative implications.** We contribute to a recent quantitative literature on value chains by using new indexes to calibrate our model of cross-border fragmentation and examine the effect of trade costs on the organization of production chains, trade and welfare.

Our quantification exercise relies on input-output matrices that we exploit in a new way. Input-output matrices and direct requirement coefficients are traditionally taken as an exogenous recipe that is essentially determined by technology. Instead, we argue that input-output matrices reflect transactions in intermediate goods between firms that are themselves endogenous economic outcomes. We show that these tables can be informative about the position of firms within supply chains that link firms both within national borders and across them. Unlike previous papers, our theory determines the allocation of tasks across firms and the length of production chains endogenously, and can thus shed some light on equilibrium input-output relationships when fragmentation is endogenous, both across and within countries.

Under the assumption that IO tables effectively summarize plant-to-plant movements for a representative firm in each industry, matrix algebra can be used to calculate, for each industry in the table, two numerical values: i) a measure of the industry’s “distance” from final demand (where distance is a count of the number of plant boundaries that will be crossed prior to final

> 8Antr`as and Chor (2013) offer a different perspective by taking the location and length of production chains as exogenous but examining the optimal allocation of ownership along the chain.

6
consumption) and ii) the average number of stages embodied in an industry’s production.<sup>9</sup> We also examine more traditional indexes of fragmentation such as gross-output-to-value-added ratios and the share of intermediate goods in trade. We show that, within our framework, we can map each of these indexes to structural parameters and key summary statistics of our model. These mappings are useful when we calibrate the model to data on interregional input-output relationships in East Asia.

A key purpose of this exercise is to offer a model comparison vis-a-vis other papers in the literature. A prominent literature has emphasized that intermediate goods trade magnifies the effect of trade costs on trade. Yi (2010) and Johnson and Moxnes (2013) focus on the response of trade to trade cost shocks, whereas Krugman and Venables (1996), Hillberry and Hummels (2002), Yi (2010) and Johnson and Noguera (2014) link the spatial clustering of activities to trade costs and intermediate goods trade.<sup>10</sup> Clustering also occurs in our model, with sequential activities locating so as to avoid trade costs. Our calibrated model can be used to investigate the response of trade to trade cost shocks, as in Johnson and Moxnes (2013) or Yi (2010).

We also contribute to the recent literature on the welfare implications of trade cost change. Arkolakis, Costinot and Rodriguez-Clare (2012) show that a broad class of models imply the same response of welfare to trade costs, provided that the models are calibrated to generate the same trade response to trade cost change. Costinot and Rodriguez-Clare (2014) and Melitz and Redding (2014) show that welfare effects are magnified when intermediate goods trade is involved. Like other papers in the literature, these presume an explicit input-output relationship that governs supply chain length, in contrast to the endogenous length in our model. The Armington framework used in these papers also precludes movement along the extensive margin (in terms of countries involved in supply chains), while our theory allows this. Our calibrated model implies larger gains than in standard trade models like those described by Arkolakis et al (2012), especially for countries that tend to be downstream, but smaller gains than Costinot and Rodriguez-Clare (2014) or Melitz and Redding (2014).

> 9The first is described as “distance to final demand” in Fally (2012) and “upstreamness” in Antras et al (2012). The second is developed in Fally (2012) and computed using the BEA input-output tables for the US. Using the highly disaggregated US table, Fally (2012) finds that the two indicators are not correlated, and that the indicators do not appear to be especially sensitive to aggregation concerns. Note also that Fally (2012) and Antras et al (2012) compute these indexes using input-output table for a single country. Here we extend these indicators to multi-rational input-output tables.

> 10More recently, Kee and Tang (2013), Bernard et al. (2014) and Antr`as et al. (2014) have used firm-level data to examine both intensive and extensive margins in import decisions. Firm-level data, however, do not allow a full consideration of supply chains over several countries. Multi-country input-output tables are more suitable for exercises like ours.

7
# **2 Model setup**

We develop a model where the production of each variety of final good requires a continuum of tasks and firms organized across countries. We describe, in turn, consumers’ preferences in final goods, tasks and firms involved in the production of each good, the forces shaping firm scope and firm entry along the chain, differences between varieties and the labor market.

**Preferences:** Consumers have identical Cobb-Douglas preferences over varieties of final goods indexed by _ω_ :

where _q_<sup>_F_</sup> ( _ω_ ) denotes quantities of final goods. As in Eaton and Kortum (2002), all countries have access to the same set of product varieties _ω_ but at different prices.

**Tasks and firms along the chain:** In order to produce the final good of variety _ω_ , a range [0 _,_ 1] of tasks must be performed sequentially. These tasks may be performed across different and countries.

Firms are arranged sequentially along the chain to produce each good _ω_ . A chain is specific to each variety _ω_ of the final good and the location of final producers. On each chain, we assume that there is a continuum of firms indexed by _f_ . Firms may be located in different countries. For each chain, we rank countries along the chain and index by _i_ ( _n, ω_ ) the _n_<sup>_th_</sup> country, _i_ (1 _, ω_ ) being the most downstream country and _i_ ( _N, ω_ ) being the most upstream country along the chain.

We denote by _Fn_ ( _ω_ ) the range of firms involved in the chain in the _n_<sup>_th_</sup> country _i_ ( _n_ ). An elementary firm _df_ performs a range _snf_ ( _ω_ ) of tasks. Both the range of firms _Fn_ ( _ω_ ) and firm scope _snf_ ( _ω_ ) are endogenous, but the range of tasks performed across all firms must sum up to one to obtain a final good: Denoting _Sn_ ( _ω_ ) = � _fF_ =0 _n_ ( _ω_ ) _snf_ ( _ω_ ) _df_ the total range of tasks to be performed in country _n_ , the last constraint can be rewritten: for all chains _ω_ .

**Coordination costs:** There are costs and benefits to fragmenting production across firms and countries. Fragmentation across firms reduces total costs because of diseconomies of scope. As firms need to manage employees across different tasks and perform tasks that are away from

8
their core competencies, unit costs increase with the scope of the firm. We will refer to these costs as “coordination costs” that occur within the Formally, we assume that an elementary firm _df_ in country _i_ requires one unit of intermediate goods and _ci_ ( _s, ω_ ) _df_ units of labor which is a function of firm scope _s_ . The cost of labor is _wi_ in country _i_ and labor is the only production input besides intermediate goods. We assume that _ci_ is convex in firm scope _s_ , thus generating gains from fragmentation across firms.

In particular, we specify the following labor requirements: where _t_ is the distance from the first task and _ai_ ( _ω_ ) and _θi_ ( _ω_ ) are specific parameters for each country _i_ for variety _ω_ .<sup>11</sup> The marginal cost of performing additional tasks within the firm increases with _t_ . This follows recent work on the division of labor (Chaney and Ossa, 2013), and in this context represents the productivity loss associated with movement away from the firm’s core competencies. _θi_ ( _ω_ ) parameterizes “coordination costs” and governs the convexity of the cost function. The higher is _θi_ ( _ω_ ), the greater the increase in costs when firms need to manage a larger range of tasks.<sup>12</sup> After integrating and multiplying by the cost of a unit of labor in country _i_ , the cost function appears as:

**Transaction costs:** Fragmenting production across firms incurs transaction costs. We model transaction costs like iceberg transport costs in standard trade models. More specifically, a transaction in country _i_ with an elementary firm _df_ involves losing a fraction _γidf_ of the good.

Within each country, quantities thus follow a simple evolution depending on transaction costs _γi_ and the position on the chain _f_ . As we go upstream, quantities increase exponentially with the number of firms _f_ to cross along the chain:

> 11 _ai_ ( _ω_ ) and _θi_ ( _ω_ ) are constant along the chain (for a given country).

> 12Note that we assume diseconomies of scope but constant returns to scale in production. This differs from Chaney and Ossa (2013) and more closely follows Kikuchi et al (2014). In keeping with Kikuchi et al (2014), this framework allows us to examine patterns of fragmentation across firms while keeping a perfectly-competitive framework where the competitive allocation of tasks across firms is optimal.

9
Since part of the production is lost when transactions occur, upstream firms must produce larger quantities. The necessary increase in quantities is starker when transaction costs are high and when the chain is more fragmented.

In a similar fashion, a _cross-border_ transaction between two consecutive countries _i_ = _i_ ( _n_ ) and _j_ = _i_ ( _n_ +1) along the chain involves an iceberg trade cost _τ >_ 1 such that: where _qj,_ 0( _ω_ ) denotes the quantities produced by the most downstream plant in the upstream country _j_ and _qi,F_ ( _ω_ ) denotes quantities produced by the next plant, i.e. the most upstream plant in the next country along the chain going downstream. For simplicity, we assume away geographical elements other than borders and impose a common border cost. Here, quantities also increase exponentially as we cross borders along the chain.

**Market structure:** In addition to assuming constant returns to scale in production, we assume perfect competition as in typical Ricardian frameworks. In this setup, the market equilibrium and the optimal allocation correspond to the social optimum.<sup>13</sup>

**Prices along the chain:** The price of intermediate goods is thus equal to their unit cost of production. Here, this cost accounts for all transaction costs incurred along the chain (going upstream) and the labor costs incurred by each firm. Within country borders, the price of intermediate goods satisfies the following differential equation which describes its evolution along the chain:

where _ci_ ( _sfi_ ) denotes the cost of performing a range _sfi_ of tasks at stage _f_ in country _i_ as specified above. This equation is close to Costinot, et al. (2012) and also features increasing intermediate goods prices as we go downstream. A key difference, however, is that the labor share is endogenous since _snf_ is endogenous and thus not simply driven by differences in input prices along the chain. In particular, the cost of inputs per unit of labor is no longer necessarily larger for downstream firms. Many of the results in Costinot et al (2012) are driven by this feature and thus no longer hold in our framework.

Across borders, the price is simply multiplied by the international trade cost _τ_ :

> 13While there are decreasing return to scale in terms of firm scope, there are constant returns to scale in production in terms of quantities. The equilibrium under perfect competition corresponds to the social optimum. This insight follows Kikuchi et al (2014) generalized to a multi-country setting with heterogenous costs and a continuum of

10
for cross-border transactions from the most downstream plant in _j_ to the most upstream plant in _i_ .

**Industry heterogeneity:** While the previous assumptions are sufficient to generate interesting patterns of specialization along a particular chain, we still need to specify how chains vary across varieties. Following Eaton and Kortum (2002), we assume that labor efficiency is a random variable drawn independently across varieties and countries. Specifically, we assume that the labor cost parameter _ai_ ( _ω_ ) is drawn from a Frechet distribution as in Eaton and Kortum (2002). For each country _i_ , the cumulative distribution function for _ai_ is:

where _Ti_ parameterizes the country average productivity and where _ξ_ is inversely related to productivity dispersion.<sup>14</sup> Note that _ai_ ( _ω_ ) is thus constant along the chain for a specific country and variety _ω_ . Unlike Yi (2003, 2010), Rodriguez-Clare (2010) and Johnson and Moxnes (2013), our framework does not require _ai_ ( _ω_ ) to differ across tasks along the chain to generate trade in intermediate goods. Another component of the cost function is _θi_ ( _ω_ ). We will explore different settings. In section 4.2 we simply consider two countries _U_ and _D_ : one where _θU_ ( _ω_ ) = _θU_ across all varieties, and another country with _θD_ ( _ω_ ) = _θD < θU_ across all varieties. In sections 4.3 and 5 (the calibration exercise), we allow _θi_ ( _ω_ ) to vary across countries and varieties. Specifically, we assume that _θi_ ( _ω_ ) is log-normally distributed with a country-level shifter _θ_<sup> ̄</sup> _i_ and a common standard deviation.

**Labor supply:** Finally, to close the model, we assume that workers are homogenous and perfectly mobile within each country, with an inelastic supply of labor _Li_ in country _i_ . By Walras’ law, trade is balanced.

# **3 Partial equilibrium: optimal organization of chains**

In this section, we take wages _wi_ as given and focus on the optimal fragmentation and location of production for a specific chain corresponding to a final good variety _ω_ . For the sake of presentation, we drop the index _ω_ . The reader should keep in mind, however, that the optimal fragmentation and allocation of value across firms, as well as costs parameters _ai_ and _θi_ , are all specific to each variety of final good _ω_ .

For a given chain, we can formulate the equilibrium as the solution to a social planning problem. Given our assumption of perfect competition and constant returns to scale, prices

> 14Parameter _ξ_ corresponds to the notation _θ_ in Eaton and Kortum (2002).

11
equal unit costs and the competitive equilibrium corresponds to the social optimum.

Let us denote by _i_ ( _n_ ) the ranking of countries along the chain, with _i_ (1) being the most downstream country and _i_ ( _N_ ) the most upstream country, assuming that _N_ countries are involved in the chain. One should keep in mind that the ranking of countries is an equilibrium outcome that we will characterize subsequently. In partial equilibrium (i.e. for a given variety, taking wages as given), the optimal organization of chains minimizes the price of the final good. Hence equilibrium can be summarized by the following optimization problem: _under the constraints:_ where _N_ is the optimal number of countries involved in the chain and _Pn ≡ p_ 0 _,n_ denotes the price at the most downstream stage in country _i_ ( _n_ ) at the _n_<sup>_th_</sup> position. Recall that exponential terms e<sup>_γi_(</sup><sup>_n_)</sup><sup>_f_</sup> reflect the evolution of quantity requirements along the chain as described in equation (5). The transaction cost parameter _γi_ ( _n_ ) and the cost function _ci_ ( _n_ )( _s_ ) are indexed by _i_ ( _n_ ) because they depend on which country _i_ ( _n_ ) is at the _n_<sup>_th_</sup> position upstream. As an abuse of notation, _PN_ +1 refers to the price of the most upstream good and is set to zero.<sup>15</sup> Chains are optimized along several dimensions. Choice variables include firm scope _si,f_ , the range of firms _Fi_ in the chain in each country and the range of tasks _Si_ to be performed in each country _i_ . The ranking of countries along the chain, from most downstream _i_ (1) to most upstream _i_ ( _N_ ), is itself an endogenous outcome described below.<sup>16</sup>

# **3.1 Fragmentation of production within countries**

Before turning to the cross-border organization of chains, we focus on optimal fragmentation within each country. The optimization problem described in (10) can be formulated as a nested

> 15Alternatively, we could set an exogenous price _PN_ +1 = _p_  ̄ of the most upstream good reflecting the price of primary commodity such as oil and minerals available from an outside economy in exchange for final goods.

> 16We can either interpret this optimization from the point of view of the most downstream country, in which case _i_ (1) is fixed, or we can optimize from the point of view of the consumer, in which case the final good price needs to be multiplied by trade costs _τ_ if the optimum location _i_ (1) is different from the location of the final consumers.

12
optimization problem. In the inner nest, firms in a specific country _i_ are organized to minimize the price of goods exported by country _i_ conditional on the overall range of tasks _Si_ to be performed in _i_ and _Pi_<sup>_M_</sup> the price of intermediate goods imported by country _i_ . The outer nest allocates stages across countries to minimize the cost of producing the final good. Here we focus on the inner nest that optimizes within country _i_ .

The price _Pi_ can be expressed as the solution of the following optimization: under the constraint: To examine the optimal allocation of tasks across firms and the optimal range of firms, it is _Fi_ useful to introduce the Lagrange multiplier _λi_ associated with the constraint �0<sup>_sfdf_=</sup><sup>_Si_.</sup> The first-order conditions of this planning program are: These conditions help us solve for firm scope ( _sfi_ ) and the number of firms involved in the chain ( _Fi_ ). Both _sfi_ and _Fi_ depend on _λi_ , the shadow cost of a task.

Equation (13) defines a shadow market for tasks. All firms in the chain provide a measure of tasks _sif_ such that their marginal cost of tasks equals the shadow price of a task, _λi_ . In this way, the conditions that determine the scope of individual firms also define the allocation of tasks across firms that minimizes the cost of producing a measure of tasks _Si_ in country _i_ .<sup>17</sup> Condition (13) offers an additional insight about the relationship between firm heterogeneity and relative position along the chain. A move upstream (i.e. towards higher index _f_ ) increases required quantities _e_<sup>_γif_</sup> , which must be balanced by a reduction in the marginal cost _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_sif_).</sup> Hence, with convex costs, condition (13) implies that more upstream firms have smaller firm scope _sif_ and provide less value added. We can be more explicit about this using our parameterization: _c_<sup>_′_</sup> _i_<sup>=</sup><sup>_aisθ_</sup> _if_<sup>_i_,whichimpliesthatfirmscopeislog-linearinupstreamness</sup><sup>_f_:</sup> 17 _′ ′_ Our Lagrangian formulation in (13) generalizes the condition _δc_ ( _sf_ +1) = _c_ ( _sf_ ) in Kikuchi et al (2014) that links the marginal costs of tasks between (discrete) firms _f_ that neighbor one another in the chain.

13
From a broader perspective, _λi_ also links firm scope decisions across countries, a relationship we develop further in the following section of the paper. For those relationships it is helpful to recognize that _λi_ =<sup>_<u>∂</u>_</sup> _∂SP_<sup>�</sup> _ii_<sup>.</sup> In an appendix we solve for _sif_ and _Fi_ as a function of _λi_ . We apply these in turn to the _Fi_ constraint �0<sup>_sifdf_=</sup><sup>_Si_andderiveanexplicitsolutionfortheshadowcostoffragmentation.</sup> _λi_ increases with all cost parameters _ai_ , _θi_ and _γi_ , with the price of intermediate goods _Pi_<sup>_M_</sup> and with the range of tasks to be performed _Si_ .

Having solved for the shadow cost of fragmentation, we can now solve for the price of the last-stage goods _Pi_ , the extent of fragmentation _Fi_ in country _i_ and firm scope _sif_ across all firms _f_ within the country. We also examine the (endogenous) intensity in intermediate goods at each stage.

**Firm scope:** The model is tractable enough to solve for firm scope _sif_ all along the chain. Firm scope _si,Fi_ for the most upstream firm is:

while the most downstream firm has scope : Using expression (15), scope at intermediate positions corresponds to: log _sif_ = _−_<sup>_<u>γ</u>_</sup> _θi_<sup>_<u>i</u>f_+log</sup><sup>_si,_0.</sup> Note again that firms are _ex ante_ homogenous but end up with different firm scope due to their position on the chain. The difference<sup>_<u>γ</u>_</sup> _θi_<sup>_<u>i</u>Si_between the scope of the most downstream and</sup> upstream firms in country _i_ is illustrative of this within-country heterogeneity in firm scope. Heterogeneity is rising in _Si_ because more tasks produced in country i implies more firms, and thus more room for heterogeneity, conditional on _θi_ and _γi_ . Larger values of transaction costs _γi_ imply more heterogeneity in firm scope because upstream firms must reduce _sif_ relatively more to satisfy equation (13). Larger values of _θi_ imply that scope remains more uniform across Of further interest is the relationship between firm scope and the price of intermediate goods relative to labor costs _aPiiw_<sup>_M_</sup> _i_<sup>.</sup> The scope of both the most upstream and downstream 14
firms are rising in this ratio. The intuition is that when the price of intermediates is relatively high, the cost of outsourcing is relatively higher and firms will choose to add more value inhouse. Conversely, when labor costs are high, firms will produce relatively few stages before outsourcing to upstream firms.

**Length of the chain:** The number (mass) of firms involved sequentially in production is a key measure of fragmentation of the chain. Here, since the range of tasks performed by each firm is endogenous, the length of the chain also becomes endogenous and is no longer proportional to _Si_ . For a given price _Pi_<sup>_M_</sup> of imported intermediate goods and range _Si_ of tasks to be performed, the mass of sequential suppliers is:

The mass of suppliers depends negatively on the price of intermediate goods because more expensive components make transactions more costly and lead to more fragmentation. The number of suppliers also depends negatively on transaction costs and positively on _θi_ , the parameter for diseconomies of scope.

**Aggregate price:** After solving for firm scope _sif_ and the number of firms _Fi_ (see appendix), we find that the price of the most downstream good in country _i_ , i.e. the solution of the minimization program (11), is:

expressed as a function of the synthetic parameter _Ai_ : This _Ai_ depends on exogenous country-specific parameters _θi_ , _ai_ and _γi_ , and reflects the effective labor productivity in country _i_ . Note that, conditional on _Ai_ , prices no longer depend on transaction costs _γi_ . The price mimics a CES cost function with two inputs: imported intermediate goods and labor, where the weight for labor depends on the range of tasks, productivity, transaction costs and coordination costs. The apparent elasticity of substitution is _θi_ +1. When coordination costs _θi_ are larger, production has to be more fragmented and there is a larger amount of production lost in transaction costs. These costs are larger when the price of intermediate goods _Pi_<sup>_M_</sup> is high.

15
**Labor vs. imported intermediate goods demand:** Each unit of the final-stage good produced in country _i_ also generates a demand _e_<sup>_γiFi_</sup> for the most upstream intermediate goods, i.e. intermediate goods imported from the next country in the chain. In terms of value rather than quantities, we obtain that the share of imported inputs in the total cost of production in country _i_ is:

Using this expression, we can retrieve the demand for local labor in country _i_ . The share of local demand in the production of country _i_ has a simple interpretation: it corresponds to the value-added content of exports for country _i_ in that chain. As with the price of the produced good, this expression mimics a CES cost function. The share of labor (one minus the above expression) depends positively on the range of tasks to be performed as well as the price of intermediate goods. The elasticity of substitution between imported inputs and local labor is in turn endogenously determined by diseconomies of scope at the firm level.

**Gross-output-to-value-added ratio:** We define gross output as: _GOi_ = �0 _Fi_<sup>_pfieγifdf_by</sup> integrating the value of all transactions along the chain, while total value added by country _i_ corresponds to: _V Ai_ = �0 _Fi_<sup>_ci_(</sup><sup>_sfi_)</sup><sup>_eγifdf_.Theratioofthesetwovariableshasausefulempirical</sup> counterpart since it is readily available in typical input-output tables provided by statistical agencies. Here, we find that the GO-VA ratio equals:

Strikingly, this result also holds at the firm level. To be more precise, the ratio of price to cost at each stage is constant and equal to: We can interpret this ratio as an index of fragmentation at the firm level. In particular, this ratio reflects the two key forces present in our model: stronger diseconomies of scope (coordination costs) _θi_ lead to more fragmentation while larger transaction costs _γi_ lead to less fragmentation. As seen in equations (15) and (18), this ratio also dictates the difference in scope between upstream and downstream firms.

The relationship between the structural parameters and summary measures of fragmentation are summarized in the following lemma: 16
_–_ **Lemma 1** _Production fragmentation within countries captured either by the GO/VA ratio – or by the range Fi of firms involved in the chain increases with coordination costs θi and decreases with transaction costs γi. In particular, the GO/VA ratio equals γ_<sup>_<u>θi</u>_</sup> _i_<sup>_._</sup>

**Free entry and cost decomposition:** We can also use (24) to better understand the link between our model and perfect competition, as well as the nature of the growth in output prices along the chain. Perfect competition implies that firms’ average and marginal costs of producing a stage will be equalized along the chain. If average costs exceed marginal costs firms can reduce costs by expanding their scope. If marginal costs exceed average costs there will be entry and firms will reduce their average scope. Applying (24) and (13), we can equate average and marginal cost for firm _f_ , and link these to the shadow cost (per unit of quantity):

It is also useful to decompose the sources of costs in the left-hand-side term of (25). Average cost has two components: labor costs associated with producing tasks inside the firm and transaction costs linked to shipments between firms. A decomposition exercise highlights the central role of the coordination cost parameter _θi_ , and will be useful in a later discussion of comparative advantage. Using (24), we solve for changes in average cost as we move along the implicit price function.

The contribution of input prices to total cost growth is solely a function of _θ_ . The share of labor costs is, by implication: _θi_ <u>1+1</u><sup>.Anotableoutcomeinthiscalculationistheabsenceof</sup> a role for _γi_ in the decomposition of cost growth, which arises because firms react to higher values of _γi_ by bringing more stages inside the firm. We revisit this issue when we describe comparative advantage within the supply chain.

# **3.2 Cross-border fragmentation**

Now that we have described the allocation of tasks along the chain within borders, we turn to the optimal allocation of tasks and firms across borders. In particular, we need to characterize the ordering of countries _i_ ( _n_ ) on the chain, with _i_ (1) being the most downstream and _i_ ( _N_ ) the most upstream country.<sup>18</sup>

> 18Recall that we drop for now the variety subscript _ω_ while most parameters vary across varieties.

17
Given the optimal fragmentation of production across firms in each country _i_ = _i_ ( _n_ ), summarized by the price function from equation (20), _P_<sup>�</sup> _i_ ( _S, P_<sup>_M_</sup> ), the optimal global value chain corresponds to the following minimization program: under the constraints: and where the function _P_<sup>�</sup> _i_ ( _S, P_<sup>_M_</sup> ) is the solution of the optimization described in equation (20) in the previous section.

For a given sequence of countries _i_ ( _n_ ), we can go quite far in characterizing prices, ranges of tasks completed and labor demand along the chain. First, it is useful to explicitly express the Lagrangian: The Lagrange multipliers associated with price equations correspond to quantities required for each unit of final good. To be more precise, _qn_ correspond to quantities _qi_ ( _n_ ) _,_ 0 _/qi_ (1) _,_ 0 required at the most downstream task performed in the _n_<sup>_th_</sup> country _i_ ( _n_ ) per unit of final good _qi_ (1) _,_ 0. The first-order condition _∂P∂nL_ +1<sup>= 0isequivalenttoimposing</sup><sup>_qn_+1=</sup><sup>_τqneγi_(</sup><sup>_n_)</sup><sup>_Fn_(usingtheprice</sup> derivative described in equation 22).

The first-order condition _∂S∂Ln_<sup>=0reflectstheoptimalallocationoftasksacrosscountries.</sup> At the optimum, the marginal cost of completing another task should be equalized across all countries on the chain, up to quantities _qn_ produced by country _n_ : _λG_ = _qn ∂∂SP_<sup>�</sup> _i_ <u>(</u> _nn_ <u>)</u><sup>=</sup><sup>_qnλn_.This</sup> implies: for any pair of countries _i_ and _j_ along the chain, where _λi_ is the shadow cost of fragmentation within country _i_ (per unit of goods exported by the country). The tight links between the Lagrange multipliers in successive countries serves to link the shadow cost of stages across markets.

Since a move upstream along the chain increases quantities (because of transaction costs and cross-border trade costs), the shadow cost _λi_ ( _n_ ) _> λi_ ( _n_ +1) must decrease. Concretely, a first implication is that firm scope tends to decrease as we go upstream, not just within countries 18
but also across countries. The F.O.C. in _Si_ implies the following expression which generalizes equation (13) across countries along the chain: where _qne_<sup>_γi_(</sup><sup>_n_)</sup><sup>_f_</sup> corresponds to the quantities of intermediate goods required for each unit of final good. Since the latter increases with upstreamness, we obtain that firm scope _snf_ would be smaller if a country _i_ = _i_ ( _n_ ) specializes upstream than if it specializes downstream. Therefore, a country with large within-firm coordination costs would have a relatively larger cost downstream than upstream compared to a country with low coordination costs.<sup>19</sup> This feature has important implications for the sorting of countries along the chain. Because firm scope can be smaller upstream, diseconomies of scope have a smaller impact on upstream stages than downstream stages. Hence, we should then expect countries with high- _θ_ to specialize upstream while low- _θ_ countries tend to specialize downstream. Formally, we can confirm this insight by examining second-order conditions of the optimization problem described in Equation (27), which yields the following Proposition:

**Proposition 1** _Let us denote by i_ ( _n_ ) _the ranking of countries involved in the same production chain, i_ (1) _being the most downstream and i_ ( _N_ ) _the most upstream country. In equilibrium, the relative position of countries along the chain is fully determined by coordination costs θi; countries with smaller coordination costs specialize downstream:_

Proposition 1 describes comparative advantage within a supply chain, conditional on a country’s participation in the chain. Two implications are of primary interest here: the central role of _θi_ in determining within-chain comparative advantage, but also the absence of a role for the transaction cost parameter _γi_ . The lack of a role for _γi_ would seem to run counter to Costinot et al. (2011), where cross-country differences in the rates of mistakes in production drive comparative advantage within the chain. The closest counterpart in our model to the mistakes in Costinot et al. (2011) is the _γi_ parameter.<sup>20</sup>

> 19Note, however, that firm scope does not necessarily increase across countries as we go upstream since firm scope also depends on transaction costs. We find that transaction-costs-adjusted firm scope<sup>_s_</sup> _γ_<sup>_i_</sup> _i_<sup>_<u>f</u>_decreaseswith</sup> a move upstream, but firm scope is not necessarily lower in a country specializing upstream than in a country specializing downstream.

> 20Costinot et al. (2011) offer cross-country differences in contract enforcement as a rationale for differences in the rates of mistakes. Here, the parameter most closely related to contract enforcement is clearly _γi_ .

19
What our model shares with Costinot et al. (2011) is the dependence of comparative advantage on the parameter that determines the convexity of the price function as value accumulates along the chain. In both models, prices are increasing at an increasing rate as the measure of completed tasks increases and, in both models, countries with less convex price schedules tend to specialize downstream. In Costinot et al. (2011), convexity arises because the losses due to mistakes increase with the value of the good along the chain, while, in our model, coordination costs are the key source of convexity.

Another way to see this is to exploit the insights in Costinot (2009), who links comparative advantage to the mathematics of log super-modularity. The accumulation of value added along the chain insures that the cost of intermediate goods is rising along the chain. This means that if production costs are log-supermodular in a parameter and input prices, then countries that have low values of that parameter will locate downstream. Using previous results on cost decomposition (equation 26), we find for any country _i_ : In Costinot et al. (2011), _sfi_ is fixed, so average cost is log supermodular in _pfi_ and _γi_ . The middle term _wici_ ( _sγfiip_ )+ _<u>f</u> i γipfi_<sup>wouldbeincreasingin</sup><sup>_γi_iffirmscopewerefixed.Thisimpliesthat</sup> transaction costs _γi_ have the least impact on average cost when _pfi_ is low (i.e. early in the chain), so that lower transaction costs create a comparative advantage in downstream tasks.

In contrast, _γi_ does not appear in (31), and thus does not affect comparative advantage within the chain. The simple explanation is that in our model, firm scope is endogenous to changes in _γi_ ; in countries with larger transaction costs, firms will endogenously increase firm scope to mitigate the role of higher transaction costs. These endogenous responses nullify the role that transaction cost would otherwise play if firm scope were exogenous. Instead, _θi_ plays a singular role in determining countries’ positions within the chain. As shown in equation (31), countries with higher coordination costs _θi_ should specialize upstream to mitigate the effect of input prices on value added.<sup>21</sup>

**Equilibrium allocation of tasks across countries:** Given the ranking of countries described in Proposition 1, we now describe the range of tasks performed by each one. Using marginal conditions imposed by the optimization problem, we can also determine prices and firm scope along the chain depending on wages and relative productivity. Specifically, the first-order

> 21A related implication is that there will be no international fragmentation without cross-country variation in _θi_ . Also, Proposition 1 implies that there is no back-and-forth trade along a specific chain in equilibrium, except when a final good is shipped back to be consumed in an upstream country.

20
conditions determine the c.i.f. price between consecutive countries _i_ ( _n_ ) and _i_ ( _n_ + 1). Firstorder conditions between three consecutive countries _i_ ( _n −_ 1), _i_ ( _n_ ) and _i_ ( _n_ + 1) then yield the range of tasks performed in _i_ ( _n_ ). Denoting _An_ , _wn_ and _θn_ the productivity, wages and the coordination cost parameter in the _n_<sup>_th_</sup> country _i_ ( _n_ ) along the chain, we obtain: where _τ_ is the trade cost between any two countries.

Given the range of tasks performed in country _i_ ( _n_ ), it is interesting to derive the share of local labor in exports to the next country in the chain. Because ours is a single-factor model, this corresponds to the value added locally in the exports of country _i_ ( _n_ ), a key statistic for economic policy.<sup>22</sup> Here, we find that the demand for labor (in value) in country _i_ ( _n_ ) per dollar of good exported to the next country _i_ ( _n −_ 1) in the chain is: Intuitively, the share of local value added in exports is higher when the relative labor cost is lower, as lower labor costs allows the country to serve as the low cost location for a larger measure of stages. This operates through margins that are both up- and down-stream. A lower labor cost makes country _i_ ( _n_ ) more competitive at the margin than the previous upstream country _i_ ( _n_ + 1) as well as the next country _i_ ( _n −_ 1) downstream.

The effect of trade costs on this statistic also operates through two channels: higher trade costs reduce the contribution of country _i_ ( _n_ ) in the downstream country _i_ ( _n −_ 1) operations, but they also reduce the upstream country’s contribution. For a country in the middle of the chain, trade costs have a positive effect on local labor content only if there are stronger complementarities with downstream rather than upstream countries, i.e. when the differences in _θn_ are larger with the downstream country than with the upstream country: _θn − θn−_ 1 _> θn_ +1 _− θn_ .

Conditional on the set of countries participating (with _θn_ increasing with _n_ along the chain), we can go further and obtain a simple expression for the price of final goods (i.e. price of downstream goods in country 1) as a function of costs parameters _A_ , _θ_ and wages _w_ . Conditional on the set of countries, we can also derive simple expressions for the share of labor costs from a specific country.

> 22For example Koopman et al. (2010) investigate the share of domestic value added in China’s exports.

21
**Lemma 2** _Conditional on the set of countries n_ = 1 _,_ 2 _, etc. participating (with θn increasing with n along the chain), the price of the final good is:_

_where_ Θ( **_wA_** _, τ_ ) _<_ 1 _captures the gains from fragmentation for the chain:_ _Moreover, country i_ ( _n_ ) _’s contribution to each dollar of final good being produced is equal to:_ In the expression for the final good price above, the first term ( _θ_ 1 _<u>A</u>_ +1)1 _<u>w</u>_<sup>_θ_1+1</sup> <u>1</u><sup>isthecostof</sup> production in country 1 if there is no possibility to fragment production across countries, while the second term Θ( **wA** _, τ_ ) is the price reduction obtained from fragmenting production across countries. We can verify that this term increases with trade costs. It also increases with labor requirements _A_ in each upstream country.

Reductions in trade costs allow chains to reorganize some of the tasks abroad, which in turn has an effect on all other firms along the chain. Equation (30) shows that the marginal cost of increasing firm scope has to be equalized across all stages. A decrease in trade costs leading to a decrease in the final good’s price also lead to a decrease in firm scope at other stages. The price of the final good is itself tightly linked to the shadow cost of fragmentation: As expressed with the Θ term, there is a tight connection between the gains from fragmentation (which reduces the final good’s price) and the shadow cost of fragmentation. Any increase in fragmentation and decrease in the final good price follows: _d_ log _λG_ = _θ_ 1 _<u>θ</u>_ +11<sup>_d_log Θ.</sup> A change in trade costs and wages along the chain has implications for firm scope everywhere on the chain. Each firm equalizes the cost of the marginal task and the shadow cost _λG_ of performing the task somewhere else. Hence, a change in the shadow cost of fragmentation _λG_ has implications for firm scope everywhere along the chain. In particular, the marginal cost of increasing firm scope in the most downstream firm in the most downstream country, 22
_w_ 1 _c_<sup>_′_</sup> ( _s_ 1 _,f_ =0) is equal to _λG_ , which implies: <u>1</u> Hence: _d_ log _s_ 1 _,f_ =0 = _θ_ 1+1<sup>_d_log Θ,whichformalizeshowachangeinfragmentationandtrade</sup> costs (changes in Θ) affects firm scope for the last firm in the chain, the one that produces the finished good.

Note, however, that the scope of the average firm in an upstream country _i_ ( _n_ ) (with _n >_ 1) does not decrease with trade costs. As trade costs decrease, a country moves downstream where firms tend to be larger. Upstream firms, which tend to be smaller in scope, exit or relocate. More specifically, we find that both the size of the most downstream and the most upstream firm within a country increase as trade costs decrease: Proposition 2 below summarizes the effect of trade costs on a chain in partial equilibrium (exogenous wages) for a given set of countries involved in the chain:

# **Proposition 2** _Holding wages constant, a decrease in cross-border trade costs leads to:_

_i) a decrease in the price of the final good;_ _ii) an increase in the value share of imported inputs at any stage of the chain; iii) an increase in the range of tasks being offshored;_ _iv) a decrease in the shadow cost of fragmentation λG;_ _v) a decrease in firm scope sn,f at a given stage f ;_ _vi) an increase in average firm scope in upstream countries n >_ 1 _._ While the ranking of countries along the chain (from downstream to upstream stages) is dictated by the ranking in _θi_ (Proposition 1), it is more difficult to characterize the participation of a specific country in the chain. Expression (32) for _Si_ can be used to obtain a necessary condition for _Si >_ 0, but cannot be used to derive a sufficient condition for country _i_ to participate in the chain. Moreover, the reader should keep in mind that we have dropped the variety subscript _ω_ to simplify the notation, but the costs parameters _Ai_ and _θi_ are assumed to be specific to a particular variety of final good _ω_ . Hence, the organization of the chains across firms and countries is specific to each variety and country of final destination.

23
We address this problem below in two special cases with simple analytical expressions: first, as in Costinot et al (2012), with symmetric chains and frictionless trade; second, in a two-country case with trade costs and heterogeneous chains. We also examine numerically a ten-country case calibrated using input-output data (Section 5). Using expressions (34) and (35) from Lemma 2, we can dramatically reduce the complexity of the numerical problem and reformulate the problem into a simpler linear programming problem that allows us to solve for large economies with a large number of final goods.

# **4 General equilibrium and aggregation**

In this section we investigate the properties of a general equilibrium version of the model. Initially we assume cross country variation in _θi_ , but no heterogeneity across chains ( _ω_ ) and no trade costs. In this setting, we prove the existence and uniqueness of the solution to the general equilibrium system. In order to investigate the implications of trade costs we develop an analytically tractable two-country version of the model in subsection 4.2 and provide analytical solutions to describe the effects of trade costs on trade and on welfare. In subsection 4.3, we discuss the general equilibrium setting that we take to the calibration in section 5. In subsection 4.4, we show that the model can be easily extended to include a distinct assembly sector.

## **4.1 Equilibrium with frictionless trade and homogenous goods**

In the section 3 we derived model implications under the partial equilibrium conditions of fixed wages and incomes. We now examine the existence and uniqueness of equilibrium. We focus here on the case with no trade costs ( _τ_ = 1) and homogenous productivity across varieties. Also, a natural assumption is that no country shares the same coordination cost parameter ( _θi_ = _θj_ if _i_ = _j_ ) to avoid any indeterminacy. In this case, we can prove that the model features a unique equilibrium:

### **Proposition 3** _With frictionless trade, homogeneous productivity:_

- _i) the organization of production is symmetrical across all chains, i.e.: all countries participate in all varieties, and their position along the chain and contribution to value added is identical across varieties;_

- _ii) there exists a unique equilibrium such that each country’s labor market clears, with a fixed labor supply and labor demand computed as in equation (35)._

24
The proof of this proposition is provided in an appendix. As in the Costinot et al (2011), cross-country differences in the convexity parameter (in our case _θi_ ) ensures a unique allocation of tasks across countries in the free trade general equilibrium. As discussed above, our model distinguishes itself from Costinot et al (2011) in its ability to explain endogenous chain lengths as measured by F, and in its links to data elements such as the GOVA ratio. Moreover, note that our model offers more flexibility as it does not impose any relationship between wages and the position along supply chains.

However, the free-trade model presented here (and its counterpart in Costinot et al., 2011) retains predictions that make it unsuitable for quantitative analysis. Contrary to what we observe in international trade data, both models predict that every country but one will produce (and export) only intermediate goods. In all but the country producing final goods, both models imply that imports constitute 100% of consumption, and both models predict that two-way trade in goods occurs only within the final two countries in the chain. These free-trade models also predict extremely long chains of production, with every country participating in every chain. This implies, in turn, extremely large measures of upstreamness (or other measures of chain length), which is inconsistent with evidence that we present below. Clearly a richer model structure is needed. We introduce trade costs and heterogeneity into our framework in order to build a richer theory of supply chain trade. We begin by relaxing the free trade assumption in a two-country model that allows analytical links between trade costs, trade flows and welfare.

# **4.2 A two-country case**

In this subsection, we reintroduce trade costs _τ >_ 1 as well as heterogeneity in productivity as in Eaton and Kortum (2002).

**Two-country setting:** We consider only two countries: country _D_ and country _U_ . Country _D_ has a parameter value _θD_ for coordination costs, while country _U_ has a parameter value _θU_ . To justify these country names, we assume that _θU > θD_ .

As specified in equation (9) in section 2, labor efficiency _aD_ ( _ω_ ) and _aU_ ( _ω_ ) are distributed Frechet with coefficient _TD_ and _TU_ respectively for countries _D_ and _U_ . We make no assumption about the relative ranking of _TD_ and _TU_ . We also make no assumption about relative transaction costs _γD_ and _γU_ for countries _D_ and _U_ . Also, we normalize _wD_ = 1. As demonstrated in the previous sections (equation 21), it is useful to instead define an _θD θU_ adjusted labor costs parameter _AD_ ( _ω_ ) = _aD_ ( _ω_ ) � _γD_<sup>_<u>θD</u>_</sup> _θD_<sup><u>+1</u></sup> � and _AU_ ( _ω_ ) = _aU_ ( _ω_ ) � _γU_<sup>_<u>θU</u>_</sup> _θU_<sup><u>+1</u></sup> � . The effect of transaction costs is equivalent to a shift in labor productivity. The resulting _AD_ ( _ω_ ) 25
and _AU_ ( _ω_ ) parameters also follow a Frechet distribution with adjusted shift parameters:<sup>23</sup> Following Dornbusch et al. (1977), we rank varieties _ω_ between 0 and 1 and specify the following relative cost: where _A_ ( _ω_ ) is defined as the relative labor requirement in country _U_ . This ordering implies that _U_ has a comparative advantage in low- _ω_ chains while country _D_ has a comparative advantage in high- _ω_ chains. For the sake of exposition, we normalize _AD_ ( _ω_ ) to unity. It is otherwise equivalent to redefine all prices as relative to _AD_ ( _ω_ ).

**Definition of an equilibrium:** The optimal organization of chains is described by the optimization problem in (10). In addition, labor demand must equal labor supply in each country.

**Sourcing patterns:** As shown in Proposition 1, the ranking _θD < θU_ determines relative position on the chain. A chain that involves the two countries necessarily features country _U_ specializing upstream and country _D_ specializing downstream. Some chains may also involve country _U_ only. However, when country _D_ produces the final good, we find that country _U_ is also involved in the chain, at least for some of the most upstream tasks.

When country _D_ produces the final good (with country _U_ involved in upstream tasks), the price of the final good in _D_ is: Consumers in _U_ can also import these goods at a price _τPD_ ( _ω_ ). When country _U_ produces the entire range of tasks, the price of final goods in _U_ is: while consumers in country _D_ can also import these goods for a price _τPU_ ( _ω_ ).

Given the patterns of labor costs across varieties, the ratio of prices<sup>_P_</sup> _P_<sup>_<u>D</u>_</sup> _U_ (<sup><u>(</u></sup> _ω_<sup>_ω_</sup> )<sup><u>)</u>strictlyincreases</sup> with _ω_ . For each final destination _X ∈{D, U }_ , there is a unique threshold _ωX_<sup>_∗_forwhichthe</sup>

> 23Our parameter _ξ_ is the same as the dispersion parameter _θ_ in Eaton and Kortum (2002).

26
two prices are equal. These thresholds _ωD_<sup>_∗_and</sup><sup>_ω_</sup> _U_<sup>_∗_areimplicitlydefinedby:24</sup> As in Dornbusch, Fisher and Samuelson (1977), these cutoffs _ωD_<sup>_∗_and</sup><sup>_ω_</sup> _U_<sup>_∗_correspondtothe</sup> goods for which consumers (resp. in _D_ and _U_ ) are indifferent between purchasing locally or importing.

Figure 1: Final goods prices depending on source and destination countries <!-- Start of picture text --> Final good price<br>(relative to price if good P<br>entirely produced in D) DU<br>P<br>UU<br>P<br>UD<br>Gains from<br>τ<br>offshoring<br>P<br>DD<br>τ<br>Produced<br>τ<br>in U only Production sharing when Production sharing,<br>final destination is D all final goods<br>imported from D<br>1<br>* *<br>0 ωD ωU Varieties<br><!-- End of picture text --> These results are illustrated in Figure 1. Figure 1 plots the price of final goods for each sourcing strategy. The threshold _ωD_<sup>_∗_isdefinedbytheintersectionbetweenthecurvesfor</sup><sup>_PD_</sup> and _τPU_ (prices for consumers in _D_ while for final goods purchased from _D_ and _U_ ), while _ωU_<sup>_∗_isdefinedbytheintersectionbetweenthecurvesfor</sup><sup>_τPD_and</sup><sup>_PU_.Consumersin</sup><sup>_D_thus</sup> purchase goods _ω ∈_ [ _ωD_<sup>_∗,_1]locally(withupstreamtasksbeingoffshoredto</sup><sup>_U_)andconsumers</sup> in _U_ purchase goods _ω ∈_ [0 _, ωU_<sup>_∗_]intheirowncountry.</sup> The effect of fragmentation on prices is reflected by the upward slope of the price schedule _PD_ and _τPD_ (when the final goods is purchased from _D_ ) which would have been flat without cross-border fragmentation (i.e. no offshoring of upstream tasks in _U_ ). As described below,

> 24The solution for _ωD∗_<sup>hasananalyticalexpression:</sup><sup>_ω_</sup> _D_<sup>_∗_=</sup> _T_  ̃ _DT_  ̃+  ̃ _UTτU_<sup>_−_</sup> _τ_<sup>_ξ_</sup> _−wξU_<sup>_−_</sup> _w_<sup>_ξ_</sup> _U_<sup>_−_</sup><sup><u>(</u></sup><sup>_θξD_(</sup><sup>_θ_+1</sup><sup>_D_+1))</sup><sup>_−_(</sup><sup>_θU−_(</sup><sup>_−θUθD−θD_)</sup><sup>_ξ_)</sup><sup>_ξ_.</sup>

27
this generates a stronger effect of trade costs on _ωU_<sup>_∗_thaninasituationwithnofragmentation.</sup> Since the production of final goods in _D_ relies on country _U_ to perform upstream tasks, the demand for labor does not only relate to the demand in final goods but also depends on intermediate goods. Using the results from Lemma 2, the demand for labor in _U_ for each dollar of final goods produced in _D_ (at a price _PD_ ) equals: Wages at equilibrium are determined by a labor market clearing condition or, equivalently, a trade balance condition (see equation 59 in appendix). Based on this, we can show that there is a unique equilibrium in the wage _wU_ of country _U_ relative to country _D_ , and that _τwU_ decreases as trade costs _τ_ decrease.<sup>25</sup>

**Effect of trade costs on final goods trade:** We examine trade in final goods before intermediate goods and the value added content of trade. The share of final goods consumed in _D_ that are imported from _U_ corresponds to the threshold _ωD_<sup>_∗_,while the share of goods consumed</sup> by _U_ and imported from _D_ equals 1 _− ωU_<sup>_∗_.We find that the elasticity of final good imports with</sup> respect to trade costs depends on the final destination of consumption and more specifically on the local labor content of the exporter. For country _D_ , this elasticity is the same as in Eaton and Kortum (2002), i.e. _ξ_ in our notation:

essentially because the exporter (country _U_ ) does not rely on imported goods for its exports: from country _D_ ’s perspective, a 1% increase in trade costs from _U_ is exactly compensated by a 1% decrease in labor costs in _U_ .

Country _D_ , however, relies on imports from _U_ to produce goods that it exports back to _U_ . As in Yi (2010), this back and forth trade generates a higher trade elasticity. There are two reasons for that. When trade costs increase by 1%, it affects the price of imported goods by _U_ from _D_ by more than 1% since the production of these goods in _D_ relies itself on goods imported in _U_ (double penalty). The second reason is that country _D_ would need to decrease its labor cost by more than 1% to offset a 1% increase in the price of its exports since its labor only contributes to a fraction of the value of the good. Combining these two effects, the

> 25While we can get clear-cut results in terms of trade-costs-adjusted wages _wU τ_ , the effect of trade costs on _wU_ itself is ambiguous like in a typical Ricardian trade model: the effect of trade on the relatively wage of country _U_ tends to be larger if _U_ is smaller.

28
elasticity of imports in _U_ to trade costs equals: _wU lU_ <u>(</u> _ωU_<sup>_∗_</sup><sup><u>)</u></sup> where _PD_ ( _ωU_<sup>_<u>∗</u>_)</sup> is the foreign labor content of production in country _D_ for the marginal variety _ωU_<sup>_∗_importedbycountry</sup><sup>_U_(seeequation44).Thenumerator,whichisgreaterthanunity,</sup> reflects the effect of trade costs on the price after the double penalty. The denominator, which is less than unity, corresponds to the share of _D_ ’s labor costs in total costs. The lower the trade costs, the higher the trade elasticity. Because lower trade costs leads to more fragmentation, the foreign labor content for the marginal variety increases. When trade becomes frictionless, the foreign labor content for this marginal variety converges to unity and the trade elasticity _ε_<sup>_F_</sup> _U_<sup>goestoinfinity.</sup>

**Vertical specialization and the value-added content of trade:** Some of the goods are produced in two locations and some of them are even re-exported to consumers in other countries. There are not many combinations with a two-country model but we can still provide predictions about how the model fits with some of indexes used to describe the extent of fragmentation across countries. We focus here on Johnson and Noguera (2012a)’s “VAX ratio”, which is itself a generalization of Hummels, Ishii and Yi (2001)’s vertical specialization index. The VAX ratio for an exporter is the ratio of the value-added content of exports and gross exports. In turn, the value-added content of exports corresponds to the value added to the production of goods eventually consumed by foreign consumers. A decrease in the VAX ratio reflects an increase in fragmentation across borders, characterized by a growing difference between trade measured in gross flows and the value added by each exporter (Johnson 2014).<sup>26</sup>

For country _D_ , the VAX ratio is smaller than unity because country _D_ relies on imported intermediate goods to produce the goods that are then exported: where 1 _− ωU_<sup>_∗_istheshareofimportedgoodsbyconsumersin</sup><sup>_U_andwhere</sup> _wPUDlU_ ( _ω_ <u>(</u> _ω_ ) <u>)</u> is the share of foreign labor in the production of variety _ω_ in country _D_ . As trade costs decrease, we show that the foreign labor content increases and therefore that the VAX ratio for country _D_ decreases.

> 26As noted in Fally (2012), the inverse of the VAX ratio for the world can also be interpreted as the embodied number of border crossings.

29
For country _U_ , all exported goods rely on domestic labor. Hence, the index developed by Hummels et al. (2001) would be equal to one for _U_ . The VAX ratio, however, is again lower than unity and is equal to country _D_ ’s VAX ratio. For country _U_ , value-added exports only account for the export of intermediate goods that are embodied in final consumption in _D_ , while gross exports also include the exports of intermediate goods that are embodied in the final goods that are re-exported back to _U_ . We show that this back-and-forth trade grows faster than other trade flows as trade costs decrease, which implies that the VAX ratio for country _U_ decreases as trade costs decrease.

We summarize these results on the trade cost elasticity and VAX ratio in the following Proposition:

**Proposition 4** _The effect of trade costs on trade is such that:_

_i) The elasticity of trade in final goods to trade costs is higher than without fragmentation; ii) This elasticity is larger when trade costs are smaller;_ _iii) The VAX ratio (value-added content of trade) decreases as trade costs decrease._ The results of Proposition 4 are intuitive and supported by recent empirical evidence. In particular, Johnson and Noguera (2013, 2012b) use multi-country input-output tables to show that the VAX ratio has decreased over the past decades and that the bilateral VAX ratio depends positively on bilateral trade costs.

**Gains from trade:** How does trade affect welfare in upstream and downstream countries? A key policy question is whether a country is affected differently depending on its position in international production chains. To examine this question, we derive an exact expression for the price index and the gains from trade relative to autarky and compare it to standard models without cross-border fragmentation of production. In particular, we use the formula developed by Arkolakis et al. (2012) as a benchmark.

For country _D_ , which tends to specialize downstream, the change in the real wage compared to autarky is: 1 where _PD_ = exp ��0<sup>log</sup><sup>_PD_(</sup><sup>_ω_)</sup><sup>_dω_</sup> � is the price index (the nominal wage _wD_ is normalized to unity in country _D_ ). The first term<sup><u>1</u></sup> _ξ_<sup>log (1</sup><sup>_−ω_</sup> _D_<sup>_∗_)correspondstotheArkolakisetal(2012)</sup> formula based on final demand trade: the log of the gains from trade are proportional to the log of the domestic content of consumption, where the proportionality coefficient is the inverse of the trade elasticity _ξ_ . For country _D_ , expression (45) indicates that the Arkolakis et al 30
(2012) formula underestimates the gains from trade, because we must adjust for the foreign _wU lU_ <u>(</u> _ω_ <u>)</u> labor content in the production of final goods by _D_ : _PD_ ( _ω_ )<sup>.</sup> Conversely, we find that the Arkolakis et al (2012) formula based on final goods overstates the gains from trade for the upstream country _U_ . Specifically, we find that the difference in real wage compared to autarky equals: While these results are shown here only for a two-country case, our counterfactual simulations in Section 5 suggest that this insight holds more generally. Conditional on final goods trade, our results suggest that gains from trade tend to be underestimated for downstream (and generally richer) countries and overstated for upstream (and often poorer) countries. We calibrate our model and compute gains from trade by using input-output tables and information on domestic and foreign labor content which, as shown above, are crucial to obtain a more adequate measure of the gains from trade when production is fragmented across borders.

**Fragmentation and firm scope along the chain:** Finally, we examine the effect of trade on fragmentation and firm scope along the chain within each country.

A key determinant of firm scope along the chain is the Lagrange multiplier associated with the total range of tasks, i.e. the shadow cost of fragmentiation _λG_ ( _ω_ ) (which varies across varieties). At optimum, this shadow cost depends negatively on the relative wage in _U_ adjusted for trade costs _τ_ : (for final goods produced in _D_ ). Using Lemma 2, we obtain that the shadow cost of fragmentation decreases with trade cost, which lead to more fragmentation.

We can also express firm scope at all stages along the chain as a function of relative wages. In country _D_ , firm scope at stage _f_ is given by: A decrease in trade costs leads to a decrease in trade-cost adjusted wage _τwU_ and therefore a decrease in firm scope in _D_ at each stage. Moreover, the overall range of tasks _SD_ ( _ω_ ) performed in country _D_ decreases when trade costs decrease: 31
which also implies that the range of tasks performed by country _U_ : _SU_ ( _ω_ ) = 1 _−SD_ ( _ω_ ) increases when trade costs decrease. Hence both countries tend to perform more downstream tasks as trade costs decrease (i.e. “move up the value chain”).

While firm scope decreases for each firm in each country, note that average firm scope in a country does not decrease, due to the reallocation of tasks across countries. Since both countries _D_ and _U_ tend to move downstream, each country only keep its largest firms (downstream firms) when trade costs decrease. Hence, average firm size in each country actually _increases_ once we account for this margin of adjustment along the chain (see appendix for mode details). These results are illustrated in Figure 2.<sup>27</sup> Proposition 5 below also summarizes the key results on fragmentation in this two-country setting:

**Proposition 5** _In general equilibrium, a decrease in trade costs τ leads to: i) For each variety, a decrease in the shadow cost of fragmentation λG; ii) At any given stage, a smaller firm scope;_

_iii) A greater share of inputs from U embodied in final goods produced in D; iv) A larger share of imports of final goods._ Figure 2: Effect of a decrease in trade costs on firm scope for a given chain <!-- Start of picture text --> Firm scope<br>s<br>Effect of a decrease<br>in trade costs<br>(in blue)<br>s<br>D,0<br>Tasks<br>in D<br>sD,FD<br>s Tasks<br>U,0<br>in U<br>sU,FU s<br>0 SD SU 1 Tasks<br>(downstream) (upstream)<br><!-- End of picture text -->

- 27Instead of expressing firm scope as a function of upstreamness _f_ (in terms of firms), we plot firm scope as

- a function of the position on the range of tasks (ordered from 0 to 1) which yields linear relationships.

32
# **4.3 General equilibrium and aggregation: general case**

In the quantitative section, section 5, we examine a situation with 10 countries involved in general equilibrium. Several comments are in order to describe the difference between the two-country case above and the calibrated multi-country model.

First, we change our assumptions about _θi_ ( _ω_ ). Empirical evidence suggests large heterogeneity in a country’s position on supply chains. As shown in the data, all countries export a mix of final goods and intermediate goods. According to Proposition 1, this would not happen if _θi_ ( _ω_ ) is constant across all varieties. We extend the approach taken by Eaton and Kortum (2002) and specify that _θi_ ( _ω_ ) varies across varieties and is the realization of a random variable. We assume in the calibration section that it is log-normally distributed, with i.i.d. shocks across varieties within countries. There are however systematic differences that can be captured here by differences in countries’ average _θ_<sup> ̄</sup> _i_ .

Second, this heterogeneity in _θi_ ( _ω_ ) opens up an indirect role for cross-country variation in _γi_ . As noted above, transaction costs play no direct role for comparative advantage within chains, but now they can play an indirect role. As shown in (21) the effect of _γi_ is subsumed in the summary measure of productivity _Ai_ . As such, _γi_ affects country _i_ ’s absolute advantage for a given chain, but not its comparative advantage. But _Ai_ itself is log-supermodular in _γi_ and _θi_ ( _ω_ ) for each variety _ω_ . The implication is that countries with high transaction costs will tend to have especially high production costs in varieties with idiosyncratically large _θi_ ( _ω_ ). As a result, these countries will more often successfully enter chains _ω_ for which they have low values of _θi_ ( _ω_ ). Using proposition 1, this implies that high transaction cost countries will tend to participate in relatively downstream stages in those chains in which they do participate. Hence, in the calibrated model, both _θi_ and _γi_ affect a countries’ average position within chains.

# **4.4 Model extension with assembly**

One of the most visible manifestations of production fragmentation in East Asia is pure assembly activity. Often, firms engaged in pure assembly use more than one component, import most of their components, employ relatively low wage labor and add relatively little value in the assembly process. We can extend our model to address these phenomena with a treatment in which the completed continuum of tasks merely produces a component, rather than a final good that is ready for consumption. The component is tradeable, and components are assembled using labor in a capstone stage of production. There is trade in each variety of the final good. In this way we transform our model of ‘snakes’ into one of ‘spiders’, using the terminology of Baldwin and Venables (2010). To formalize matters let _P_ 1 _ik_ ( _ω_ ) represent the price in _i_ of the component _k_ used in variety _ω_ , inclusive of trade costs. The cost function used to produce the 33
final good in region j is then: where _ηk_ (<sup>�</sup> _k_<sup>_η_</sup> _k_<sup>= 1)representsaconstantCobb-Douglascostshareofcomponentk,</sup><sup>_β_repre-</sup> _−_ sents the share of labor in assembly, and 1 _β_ the share of components in the cost of assembly. We assume the technology of assembly is the same across all commodities and regions. The price in region i of the final good, _Pi_<sup>_F_(</sup><sup>_ω_),istheminimumdeliveredpriceaccountingfortrade</sup> costs _τ_ when the good is imported. We describe the calibration of this model in section 5.5.

# **5 Quantitative analysis**

## **5.1 Data**

Our main sources of data are the Asian input-output tables developed by IDE-JETRO. These tables provide information on gross output, value-added, and (most importantly) input purchases by product, parent industry (downstream industry), source country and destination country. For instance, the data report the amount of metals purchased from China by the auto industry in Japan. These 4-dimensional input-output tables, are, as far as we know, the only tables that track international transactions directly, rather than imputing them from trade flows. This is an exceptional data set for investigating the organization and evolution of international production fragmentation in a region of the world where fragmentation is an important feature of international trading relationships.

The dataset covers 9 Asian countries and the US.<sup>28</sup> Our analysis mostly focuses on the year 2000 (the most recent year available), but we also compare our results to IDE-JETRO data from 1975 and 1990. This period marks a time in which the region began to emerge as an important location for internationally fragmented production (see Baldwin and Lopez-Gonzalez (2012) for example).

Information on input purchases and production is disaggregated at the 76-sector level in 2000. For the sake of comparison to previous input-output tables (1975 and 1990), we construct a more aggregated 46-sector classification to obtain harmonized product categories across years. The sector classification is far more detailed for manufacturing goods and commodities than services (among the 46 sectors, only 5 of them are service industries). We thus mostly restrict our attention to tradable goods: commodities and manufacturing goods.

> 28The countries in the data base are the US, Japan, China, Taiwan, Korea, Singapore, Malaysia, Thailand, Indonesia and the Philippines.

34
The information provided in these tables goes beyond a simple aggregation of country-level input-output tables. Besides the harmonization of product categories, input flows by parent and source countries are estimated by means of additional surveys and thus deviate from the proportionality assumption<sup>29</sup> which, according to Puzello (2012), is rejected in these data. This constitutes an important advantage of using the IDE-JETRO input-output compared to previous attempts at constructing input-output tables based on the proportionality assumption (as in Johnson and Noguera 2012, for example).

# **5.2 Measuring the position along the chain: indexes** _N_ **,** _N_<sup>_∗_</sup> **,** _D_ **&** _D_<sup>_∗_</sup>

To better understand the degree of fragmentation in vertical production chains we adopt four indexes that generalize the two indexes proposed in Fally (2012) and applied there to US data. The indexes are designed to describe industries’ position in vertical production chains by exploiting information about relationships in the input-output table. The ‘D’ index measures an industry’s weighted average distance to final demand, where distance is a measured by the apparent number of plants visited by the industry’s output before reaching consumers.<sup>30</sup> The ‘N’ index calculates, for each industry, the number of stages that are embodied in each industry’s production. These two calculations are distinct for each industry, and in the US data there is only a weak correlation between them.

**Distance to final demand or “upstreamness”:** We turn to a formal representation of the two indices. Consider a variable _Dik_ , which is intended to measure the distance of a product _k_ from final demand. Some part of product _k_ ’s sales will be intermediate trade purchased by downstream industries, so the industry in question’s distance measure will depend upon which industries buy its output, and in turn how far those downstream industries are from final demand. Because an industry’s sales go to several industries, which will vary in their respective measures of D, the industry measure must be weighted, and it must also be defined recursively. Let _Dik_ indicate the distance measure in region _i_ for product _k_ . We define _Dik_ as:

where _φikjl_ denotes the amount of output from product _k_ in country _i_ that is used to produce

> 29The proportionality assumption is made to construct input purchase by source country and parent industry when only partial information is provided. For instance, traditional country-level input-output tables describe how much steel is used by the auto industry in each country. Using trade flow data (which describe how much steel is imported from a particular country), previous international input-output tables have been constructed by allocating the use of input across source countries on a proportional basis.

> 30The measure is equivalent to the ‘upstreamness’ measure in Antras, et al. (2012).

35
one dollar’s worth of product in sector _l_ in country _j_ . The entire system of equations that includes a _Dik_ for each industry and country can be solved to produce a measure for each sector-country pair.

As shown in Antras, et al. (2012), this index can be interpreted as the average number of stages that goods cross before reaching final consumers. Using the input-output matrix, we can decompose the different trajectories taken by the good across and within industries. Each trajectory is associated with a specific number of transactions across or within industries. Index _D_ would then correspond to the average number of transactions weighted by the fraction of output corresponding to each trajectory.

Notice that _Dik_ does not only rely on inter-industry linkages but also depends on the extent of fragmentation within each industry. If an industry’s production is partly used an input by other firms in the industry (e.g. electronic parts used as parts for other electronic parts within the same country), the coefficient _φikik_ would be strictly positive and would contribute to a higher index _Dik_ (since it would also correspond to a higher number of transactions).<sup>31</sup> How to interpret ‘D’ in the model? First, suppose that products _k_ correspond to stages _f_ . When _f_ is strictly positive, i.e. when it does not refer to the most dowstream stage in the _n_<sup>_th_</sup> country _i_ ( _n_ ), then all sales are made to the next plant _f − df_ in the chain: If _f_ = 0 and _i_ ( _n_ ) is not the most downstream country _i_ (1), then all sales go towards the most upstream firm in the next country in the chain. After integrating, we obtain that the model counterpart of _Dif_ corresponds to the total range of firms located downstream: summing across all downstream countries _i_ ( _n_<sup>_′_</sup> ) with _n_<sup>_′_</sup> _< n_ .

In terms of the model, we can also interpret _Di,k_ , for any country _i_ , as a semi-elasticity of

> 31One may argue that the industry classifications are too aggregated and create biases in computing _Dik_ and _Nik_ compared to what would be obtained with more precise data. Fally (2012) examines the aggregation properties of indexes _D_ and _N_ and shows that aggregating industries does not much affect the average of _D_ and _N_ across industries.

36
required quantities w.r.t. to transaction costs.<sup>32</sup> Formally, _Dif_ corresponds to:

**Embodied stages:** The _Nik_ index captures a weighted average of the number of plants involved sequentially in the production of good _k_ in country _i_ . It is defined recursively by:

where _μikjl_ denotes the amount of input _l_ from country _j_ used to produce one dollar of product _k_ in country _i_ . This is a single equation, but, as with the D index, the system of equations can be solved to produce a measure of N for each sector-country pair. As shown in Fally (2012),<sup>33</sup> this index can also be expressed as a weighted average of the number of stages required to produce good _k_ in country _i_ , weighted by how much each stage of production contributes to the final value of that good.

We can also interpret _N_ in light of our theoretical framework. In the model, the amount of input purchased by other firms corresponds to the price of the good minus the labor cost incurred at each stage, i.e.: _wicip_ <u>(</u> _sifif_ <u>)</u> _df_ , where _pif_ denotes the price of the good in country _i_ at stage _f_ . The model counterpart of index _N_ would thus correspond to the following recursive with a similar equation when the chain crosses a border. The solution to this differential equation equals the average of the number of production stages required to produce a good at stage _f_ in country _i_ .

There is a strong connection between the _N_ and _D_ index. Since the number of stages between firm _f_<sup>_′_</sup> in the _m_<sup>_th_</sup> country _i_ ( _m_ ) and firm _f_ in the _n_<sup>_th_</sup> country _i_ ( _n_ ) corresponds to _Di_ ( _m_ ) _,f ′ − Di_ ( _n_ ) _,f_ , we obtain formally: where the integral is taken across all upstream firms either in _i_ ( _n_ ) at a more upstream stage

> 32In a model with a discrete number of firms instead of a continuum of suppliers, transaction costs would correspond to _δ_ = _e_<sup>_γ_</sup> (as in Kikuchi et al 2014) and _D_ would correspond to the elasticity of quantities w.r.t transaction costs instead of the semi-elasticity.

> 33See Proposition 1 in Fally (2012).

37
_f_<sup>_′_</sup> _> f_ or in more upstream countries _i_ ( _m_ ) with _m > n_ , and where the price _pif_ can be itself re-expressed as the sum of all costs incurred in upstream stages, adjusting for quantities: _qi_ ( _n_ ) _,f pi_ ( _n_ ) _,f_ = �( _m,f_<sup>_′_</sup> ) _>_ ( _n,f_ )<sup>_qi_(</sup><sup>_m_)</sup><sup>_,f′ci_(</sup><sup>_m_)(</sup><sup>_si_(</sup><sup>_m_)</sup><sup>_,f′_).</sup> The connection between the two indexes _N_ and _D_ is clearest if we look at the most downstream stage. For the most downstream country _i_ = 1 and the most downstream firm _f_ = 0 in the country, index _N_ corresponds to a weighted average of _D_ : with the price _pi_ =1 _,f_ =0 =<sup>�</sup> _j_ � _fF_<sup>_′_</sup> _j_ =0<sup>_qif′cj_(</sup><sup>_sjf′_)</sup><sup>_df ′_beingthesumofallupstreamcosts.</sup> As for _D_ , we can also use the model to interpret _Nik_ as a semi-elasticity of w.r.t. to transaction costs, looking at prices instead of quantities. Formally, _Nif_ corresponds to:<sup>34</sup>

**Borders vs. stages:** The _N_ and _D_ calculations are independent of whether or not the shipments in question cross international borders. Since the organization of international production networks is a key question for this literature, and for this study, we propose alternative forms of the indexes in Fally (2012). Using analogous methods to those used in the calculation of _D_ and _N_ we ask how many international borders are crossed for each region-industry pair’s outputs and inputs. D* measures the average number of national borders crossed before final consumption. N* measures the weighted average number of production stages that are embodied in the output of an industry.

Formally, and with most of the variables defined above. _Xik_ represents exports by sector _k_ in _i_ , _Mik_ represents imports, and _Yik_ is sector gross output.

In our model, the equivalent is _Dif_<sup>_∗_reflecting the number of country borders to cross before</sup>

> 34Based on the results in Fally (2012), we should also note that _Nif_ equals the aggregate gross-output-tovalue-added ratio across upstream activities. It can therefore be interpreted as a weighted average of _γ_<sup>_<u>θi</u>_</sup> _i_<sup>across</sup> upstream activities.

38
<!-- Start of picture text --> 3<br>2.5<br>2<br>1.5<br>1 m Di<br>m Di*<br>05<br>ie)<br>yn o o =] © oO Cc Cc vo cig<br>v £ aw c a7) ov o o 5 vn<br>= s g 2 = iS) ZB o 2 =)<br>2 3 S i 7 S<br>= 3 = s - r=<br>z 5 B<br><!-- End of picture text --> <!-- Start of picture text --> 3<br>2.5<br>2<br>1.5<br>1 mDi<br>m Di*<br>05<br>ie)<br>n o o =] © oO Cc Cc o og<br>v £ D c a7) ov o o = vw<br>=2s g32s= S ZBi7o 2&=)<br>== no]= <=- S e aoo<br><!-- End of picture text -->
for defining weights, including value added- or export-weighting for D and output-weighting for N. As argued in Fally (2012), a natural weight for the upstreamness index _D_ is value added, and final production by sector-country for index _N_ . We do the same for _N_<sup>_∗_</sup> and _D_<sup>_∗_</sup> .

In order to best capture the relative position of countries on international production chains, our preferred aggregate is an export-weighted average of index _Dik_ . This export-weighted average is the statistic calculated in Antr`as et al. (2012) to document countries’ comparative advantage along production chains. Formally, we define _DXi_ by country with the following: where _Xik_ represents country i’s exports of product k.<sup>35</sup>

**Descriptive statistics:** We calculate these indices using the IDE-JETRO data from 1975, 1990 and 2000. In order to present this information succinctly, we offer two aggregates, _Di_ and _Di_<sup>_∗_,where the aggregates are calculated as averages weighted by sector</sup><sup>_k_value added.We</sup> report the aggregate measures below in graphical form in order to show cross-country variation and within country variation over time.

In part A) of Figure 3, we see the results for all tradable goods. There is some variation in the levels and trends of the _D_ across countries. While the cross-country variation is, in some cases, interesting in its own right, the key takeaway is the absence of a consistent story in the simple measures that are neutral with respect to national borders. By contrast, one can see a general pattern of growth in the _D_<sup>_∗_</sup> measures (upstreamness in border crossings), which is partially driven by decreases in trade costs but may also reflect a movement upstream. Both indexes increase in 2000 for most countries (with Japan and the US as notable exceptions), which suggests that the countries in question moved into upstream positions within internationalizing production chains.

The electronics sector is particularly interesting over this period. Complex international production chains are, anecdotally, an important phenomenon in East Asian manufacturing. This is even more notably so within the electronics sector. Moreover, there has been important growth in the region’s trade in electronics, which constituted only 8% of Asian exports in 1975, and 34% in 2000. In part B) of Figure 3, we report results of our calculations for the electronics sector only, by country and by year.

> 35While a VA-weighted average of _D∗_ can also capture the relative position on international supply chains, an advantage of the measure _DXi_ over _D_<sup>_∗_</sup> is that it is less sensitive to the country’s level of openness. For instance, we may be concerned that Singapore has high values of _D_<sup>_∗_</sup> mainly because of low trade costs rather than being relatively more upstream. As our sole interest is in countries’ relative position along the chain, a measure that conflates position with openness is problematic. The index _DXi_ puts more emphasis on the composition of exports than openness.

40
These measures indicate quite a bit more heterogeneity in the levels of both indexes, _D_ and _D_<sup>_∗_</sup> . Production chains appear to be somewhat shorter in the U.S. than in Asia. Moreover there is sharp upward movement for most indexes and for most countries. One explanation would be that this is due to electronics becoming much more important as inputs into production of other sectors (i.e. automobiles), but this upward trend also reflects a specialization of Asian countries into relatively upstream stages within the electronics industry.

**Correlation between upstreamness and value-added content:** The model predicts that: i) countries with high coordination costs _θ_ should specialize upstream while countries with low _θ_ should specialize downstream; ii) the value-added-to-gross-output ratio decreases with _θ_ since it equals the ratio of _θ_ and transaction costs _γ_ . By combining these two predictions, we should observe a negative correlation between upstreamness and the value-added-to-gross-output ratio.

Figure 4: Value-added-to-gross-output ratio as a function of upstreamness _Dik_ <!-- Start of picture text --> 1 2 3 4<br>Upstreamness<br>95% CI Fitted values<br>.4<br>.35<br>.3<br>.25<br><!-- End of picture text --> Figure 4 shows that we indeed observe such a negative correlation between upstreamness and value added content.<sup>36</sup> This finding supports the prediction that firm scope tends to be smaller upstream than downstream. In Table 1, we regress the value-added-to-gross-output ratio on upstreamness to find a significant and negative correlation between the two, whether we include no fixed effect (column 1), country fixed effects (column 2), industry fixed effects (column 3) or both (column 4). Consistent with the model, the correlation is the strongest

> 36Figure 4 covers manufacturing industries only, trimming observations with upstreamness above 4.

41
when most of the variation is driven by cross-country variation, i.e. when we include industry fixed effects but no country fixed effects.

Table 1: Correlation between upstreamness _Dik_ and the value-added-to-gross-output ratio

||(1)|(2)<br>(3)|(4)|
|---|---|---|---|
|Dependent var.:||VA/GO||
|Upstreamness|-0.025<br>[0.012]<sup>_∗∗_</sup>|-0.060<br>-0.019<br>[0.008]<sup>_∗∗_</sup><br>[0.008]<sup>_∗_</sup>|-0.028<br>[0.008]<sup>_∗_</sup>|
|Industry FE|No|Yes<br>No|Yes|
|Country FE|No|No<br>Yes|Yes|
|R2|0.106|0.429<br>0.152|0.542|
|N Obs.|478|478<br>478|478|

_Notes_ : Notes: OLS regression with robust s.e.; manufacturing industries, trimming observations with upstreamness above 4;<sup>_∗_</sup> significant at 5%;<sup>_∗∗_</sup> significant at 1%.

# **5.3 Calibration**

Our calibration exercise focuses on the 10 countries that are covered by the IDE-JETRO inputoutput tables. The general equilibrium model described above is calibrated so as to reproduce key features of the data. The parsimony of the model allows us to consider only a small number of parameters to calibrate, those listed in the left column of Table 2.

Thanks to Lemma 2 and the analytical results described in section 3.2, we can reduce the optimization problem described in equation (10) to a linear programming problem for each chain. Numerical simulations are performed in Matlab. We approximate a continuum of varieties by assuming 1,000,000 different final goods.

We now describe each calibrated parameter and its targeted moment, as described respectively in the left and right columns of Table 2.

**Labor supply:** Each country is endowed with an exogenous supply of factors. In the benchmark case, we consider only one factor of production: labor. For each country, we choose the labor force _Li_ to match aggregate value-added in tradeable goods sectors (i.e. excluding services) divided by the cost of labor (proxied by income per capita in our benchmark simulation).

**Labor productivity:** There is a tight link between wages and labor productivity in country _−_<sup><u>1</u></sup> _ξ i_ . As in Eaton and Kortum (2002), average labor productivity is driven by _A_<sup> ̄</sup> _i ≡ Ti_ where _Ti_ is a shift parameter for the Frechet distribution of _Ai_ ( _ω_ ) in country _i_ (equation 9). Given

42
the connection between _A_<sup> ̄</sup> _i_ and wages, we use data on per capita income as a proxy for labor cost to be matched by the model. In turn, the dispersion parameter _ξ_ is calibrated using the main estimate of Simonovska and Waugh (2014). Data on per capita income is obtained from the Penn World Tables for the year 2000.

Conditional on wages and calibrated parameters, we can compute labor demand for each country. Equality between aggregate labor demand and labor supply is attained by adjusting labor productivity. Of course, productivity is tighly related to wages. As shown in Table 2, there is almost a log-linear (downward-sloping) relationship between wages _wi_ and _A_<sup> ̄</sup> _i_ in our benchmark calibration.

**Coordination costs:** As shown in Proposition 1, the coordination cost parameter _θi_ is a key determinant of the position of a country along the chain, downstream or upstream. A country tends to export final goods when coordination costs are low and export intermediate goods when these costs are higher. Since all countries export a mix of final and intermediate goods, we assume that _θi_ is heterogeneous across varieties _ω_ , as discussed in section 4.3. We assume that it is log-normally distributed. In calibration countries are allowed a different shift parameter _θ_<sup> ̄</sup> _i_ and a different standard deviation _σiθ_ . We use _DX_ as the primary moment to calibrate _θ_<sup> ̄</sup> _i_ and _σiθ_ to fit countries’ intermediate share in total exports. While the correlation of the values in Table 2 is weak (0.11), the prediction of the model is confirmed: countries that are more upstream have higher average values of _θ_<sup> ̄</sup> _i_ . Empirically we note than most of the countries in the model have values of DX near the center of the distribution. Only Indonesia and China stand out as countries that are, respectively, relatively upstream and downstream.

**Transaction costs:** Another key parameter of the model is _γi_ , the cost of transactions between two firms. This cost is assumed to be positive even for transactions that occur within borders. Transaction costs are difficult to estimate in practice, but our model indicates that the grossoutput-to-value-added ratio equals the ratio of coordination and transaction costs parameters _γθii_<sup>andthuscanbeusedtoretrieveanestimateof</sup><sup>_γi_onceweknow</sup><sup>_θ_ ̄</sup><sup>_i_(onaverage).Results</sup> are provided in Table 2. As a credibility check we compare our results to plausible realworld counterparts of the _γi_ parameter using the Doing Business Database (World Bank). Reassuringly, we find expected correlations of our calibrated _γi_ variables with the costs of enforcing a contract claim (0.84), the time to enforce contracts (0.29), and the recovery rate in insolvency proceedings (-0.51).<sup>37</sup>

We note that Singapore has unusually low transaction costs in the calibration, which is consistent with its high gross output to value added ratio. Singapore’s value of _θ_<sup> ̄</sup> _i_ is also relatively

> 37We could regress _γi_ on each type of transaction cost (contract vs. bankruptcy, etc.) but with only 10 countries this regression would be meaningless.

43
low. Singapore often appears as a special case in the simulations that follow. In addition to its small size, it is likely that the low values of _θ_<sup> ̄</sup> _i_ and _γi_ are responsible for the particular behavior of Singapore in our counterfactual analysis. At the other end of the spectrum, Indonesia has the higher transaction costs, which is consistent with the Doing Business indicators and recent literature (e.g. Barron and Olken, 2009).

**Trade costs:** In addition to transaction costs that are also incurred within borders, crossborder transactions face an additional burden. The nature and size of the extra cost affecting international transactions is the matter of an extensive debate in the trade literature (transport costs, asymmetric information, marketing cost, technical barriers, or cultural differences). There is however a consensus that these costs are large and have large effect on cross-border trade. While distance usually plays a key role in explaining the pattern of international trade (see Disdier and Head, 2008), distance is not as crucial for trade among Asian countries.<sup>38</sup> We therefore assume that cross-border trade costs are uniform across all country pairs in our sample. We fit the trade cost parameter by asking the model to replicate the global ratio of trade to output.

**Simulation results on other moments:** Before turning to the counterfactual results we briefly describe the benchmark equilibrium. By construction, our model is able to reproduce key indicators of fragmentation such as the gross-output-to-value-added ratio and values of _D_<sup>_∗_</sup> . Alternatively, we can examine how the fitted model fares in terms of other indexes such as indexes _D_ , _N_ and _N_<sup>_∗_</sup> described above.

Table 3 compares indexes from the model vs. data. In broad terms the magnitudes are consistent with the data, even though they are constructed in very different ways. These indexes from the data are computed at the industry level then averaged across industries. Index _N_ in the benchmark calibration is computed for the most downstream firm in the chain while index _D_ is a weighted average across firms weighted by value-added at each stage (same for _D_<sup>_∗_</sup> ).

Indexes _D_ and _N_ tend to be too high in our calibrated model (too much fragmentation), but they are strongly correlated with their data counterpart (86% correlation for _N_ and 44% correlation for _D_ ). With the exception of Singapore, the richest and poorest countries tend to be more downstream (lower index _D_ ) and have a smaller number of embodied stages (index _N_ ). With the exception of Singapore, middle-income countries in our sample such as Taiwan and Malaysia end up with the highest indexes _D_ and _N_ both in the calibrated model and the data.

> 38In a standard gravity equation of trade using the IDE-JETRO data, the coefficient for distance is not significant while the estimate border effect is large and significant, economically and statistically. Of course, this is a small sample of only 10 countries.

44
Table 2: Parameter choice and moments to match

|_Parameters:_<br><sup>1</sup>|||_Moments to match:_|||
|---|---|---|---|---|---|
|Average <sup> ̄</sup>_Ai_ = <sup> ̃</sup>_T _<sup>_−_</sup><br>_ξ_|USA|1.000|GDP per capita|USA|35,080|
|by country<br>|SGP|4.768|(PWT)|SGP|32,808|
|(relative to the US)|JPN|1.634||JPN|26,721|
||TWN|4.977||TWN|21,891|
||KOR|4.521||KOR|17,208|
||MYS|14.249||MYS|7,917|
||THA|17.885||THA|5,178|
||IDN|17.657||IDN|2,549|
||CHN|25.049||CHN|2,442|
||PHL|41.881||PHL|2,210|
|Labor supply|USA|53,551|Total value-added|USA|1,878.6|
|in tradeable goods|JPN|41,665|in tradeable goods|JPN|1,113.3|
|(x1000 workers)|SGP|735|(in $M)|SGP|24.1|
||TWN|3,889||TWN|85.1|
||KOR|10,491||KOR|180.5|
||MYS|5,637||MYS|44.6|
||THA|10,410||THA|53.9|
||IDN|36,585||IDN|93.3|
||CHN|266,707||CHN|651.3|
||PHL|13,618||PHL|30.1|
|Average coordination costs <sup> ̄</sup>_θi_|USA|1.161|DX Index|USA|2.555|
|by country|SGP|0.590|(Export weighted)|SGP|2.659|
||JPN|1.029||JPN|2.375|
||TWN|0.692||TWN|2.613|
||KOR|0.825||KOR|2.637|
||MYS|0.695||MYS|2.590|
||THA|0.742||THA|2.360|
||IDN|1.173||IDN|3.002|
||CHN|0.809||CHN|1.973|
||PHL|0.790||PHL|2.508|
|Transaction costs _γi_|USA|0.487|aggregate GO / VA ratio|USA|2.666|
|by country|SGP|0.244||SGP|4.676|
||JPN|0.442||JPN|2.773|
||TWN|0.285||TWN|3.951|
||KOR|0.358||KOR|3.359|
||THA|0.325||THA|3.452|
||MYS|0.383||MYS|3.101|
||IDN|0.740||IDN|2.035|
||CHN|0.343||CHN|3.110|
||PHL|0.464||PHL|2.629|
|Simple average border cost|All|50%|Trade/output ratio|All|26%|

45
Table 3: Fragmentation indexes: model vs. data

|Index|D||D*||N||N*||M sha|re|
|---|---|---|---|---|---|---|---|---|---|---|
||Data|Model|Data|Model|Data|Model|Data|Model|Data|Model|
|USA|2.252|2.680|0.208|0.121|2.422|2.804|0.316|0.089|0.657|0.387|
|SGP|2.558|4.254|1.062|0.234|2.656|4.228|0.535|0.143|0.672|0.231|
|JPN|2.370|2.683|0.278|0.132|2.364|2.894|0.182|0.133|0.558|0.317|
|TWN|2.567|3.778|0.699|0.227|2.542|3.832|0.423|0.173|0.644|0.258|
|KOR|2.537|3.256|0.503|0.240|2.536|3.329|0.358|0.174|0.632|0.310|
|MYS|2.755|3.471|0.912|0.264|2.515|3.449|0.501|0.155|0.704|0.272|
|THA|2.373|2.951|0.591|0.252|2.443|3.001|0.400|0.164|0.593|0.265|
|IDN|2.509|2.893|0.521|0.524|2.109|2.234|0.244|0.104|0.823|0.523|
|CHN|2.707|2.785|0.278|0.101|2.620|3.160|0.183|0.189|0.472|0.197|
|PHL|2.006|2.859|0.483|0.329|2.216|2.692|0.307|0.137|0.677|0.315|
|Correl.|with data|0.443||0.321||0.860||0.159||0.726|

# **5.4 Counterfactual simulations**

East-Asian economies have been the setting for tremendous changes in recent decades. Arguably the most significant changes are the increased fragmentation of production, China’s opening to international trade and its subsequent rapid economic growth. In our theory these phenomena can certainly be related, as China’s opening to trade could have facilitated fragmentation along chains in which it is now involved. Rapid economic growth may be associated with trade-related increases in productivity, but multi-factor productivity growth not specifically related to trade might also have been important. With a calibrated model at hand, we can now examine various counterfactual simulations to study how structural changes would affect economic outcomes such as output, trade, welfare and the fragmentation of production.

We see at least four experiments that would provide interesting insight into the reorganization of supply chains in Asia:

- Counterfactual 1): Trade costs have fallen significantly over the past decades and their reduction is cited as the most likely source of the increased fragmentation of production in Asia. Trade costs may decline even further in the near future, as there is still room to improve trade agreements, especially on a multi-lateral basis (Baldwin, 2008). We model this structural change with a 10% reduction in cross-border trade costs.

- Counterfactual 2): Arguably the most dramatic change in the Asian economy over the past two decades has been the very high rates of growth in the Chinese economy, along with its opening to trade. With GDP growth rates reaching 10%, the Chinese economy is inducing very large changes in how production in Asia is organized. To examine the

46
role of China in light of our model, we shock the productivity parameter there. We first simulate a 10% productivity increase in China. Formally, this corresponds to a 10% <u>1</u> increase in _TCHNξ_<sup>.</sup>

- Counterfactual 3): We simulate a reduction in the transaction costs _γCHN_ for China. This scenario could be used to understand growing transparency in contractual disputes, for example. Reduced transaction costs should encourage relatively more domestic outsourcing in China, and raise the share of domestic value added in production/exports.<sup>39</sup>

- Counterfactual 4): Finally, we consider a bilateral trade cost reduction. This allows us to offer a local estimate of the trade elasticity. In particular we are interested in a quantitative evaluation of the claim in Proposition _4i_ , that the elasticity of final goods trade to trade cost changes is larger in the presence of fragmentation. In order to do this we reduce trade costs between China and the US.

# **5.4.1 Reduction in trade costs**

Our first scenario is a 10% reduction in international trade costs _τ_ . These results are reported in Table 4. In addition to the statistics already introduced we report the intermediate share of exports “M share.” The distribution of welfare gains across countries (column 2) suggests that countries with larger economies benefit least. This likely reflects the fact that these larger economies are less exposed to changes in international trade costs. Compared to the Arkolakis et al (2012) formula, gains from trade with international production chains tend to be larger. Moreover, our simulations indicate that downstream countries gain relatively more than upstream countries, relative to the ACR formula (we find a _−_ 0 _._ 54 correlation between index DX and the ratio of gains from trade: simulated vs. ACR).

The general increase in _D_<sup>_∗_</sup> and _N_<sup>_∗_</sup> is consistent with more international fragmentation. On average, the reduction in international trade costs means that production crosses more borders, both upstream of countries’ production _N_<sup>_∗_</sup> and downstream of it _D_<sup>_∗_</sup> . Greater international fragmentation can also be seen in the decrease in the VAX ratio (lower value-added content in trade). Interestingly, a falling intermediate share of trade indicates that final goods trade is more sensitive than intermediate goods trade. A consequence is that countries tend to move downstream as trade costs decrease, as reflected in the negative change in export-weighted upstreamness, _DX_ . This overall decrease in upstreamness is consistent with the changes described in Fally (2012, Figure 4).

> 39Kee and Tang (2013) and Li and Liu (2014) provide evidence that the share of domestic value added in Chinese exports has been growing over time.

47
Table 4: Counterfactual 1): 10% decrease in border trade costs

|(10 x change)|Welfare|1<br>_ξ_ <sup>log</sup><sup>_πF_</sup><br>_ii_|DX|D|D*|N|N*|M share|VAX|
|---|---|---|---|---|---|---|---|---|---|
|USA|0.094|0.070|-0.517|-0.083|0.057|-0.051|0.036|-0.121|-0.073|
|SGP|0.464|0.426|-0.598|-0.335|-0.029|-0.160|-0.005|-0.080|-0.076|
|JPN|0.139|0.094|-0.546|-0.086|0.052|-0.022|0.048|-0.116|-0.074|
|TWN|0.381|0.307|-0.624|-0.083|0.001|-0.007|-0.020|-0.092|-0.075|
|KOR|0.313|0.236|-0.627|-0.027|0.032|-0.014|-0.001|-0.108|-0.072|
|MYS|0.427|0.373|-0.553|-0.161|-0.013|-0.065|-0.016|-0.092|-0.076|
|THA|0.411|0.343|-0.508|-0.132|-0.008|-0.065|-0.014|-0.098|-0.078|
|IDN|0.407|0.298|-0.458|-0.038|0.025|-0.050|0.006|-0.136|-0.056|
|CHN|0.188|0.110|-0.475|-0.059|0.024|0.041|0.050|-0.087|-0.077|
|PHL|0.455|0.405|-0.485|-0.204|-0.023|-0.101|-0.003|-0.109|-0.074|

# **5.4.2 Increasing productivity in China**

<u>1</u> Table 5 reports results from a 10% shock to productivity in China, _TCHNξ_<sup>.Thisisauniform</sup> shock that improves productivity at all points in the chain. Our interest is in seeing how such shocks affect China’s relative position in chains, and the degree to which such shocks spill over into other countries.

Table 5: Counterfactual 2): 10% increase in productivity in China

|(10 x change)|Welfare|1<br>_ξ_ <sup>log</sup><sup>_πF_</sup><br>_ii_|DX|D|D*|N|N*|M share|VAX|
|---|---|---|---|---|---|---|---|---|---|
|**CHN**|0.968|-0.047|-0.289|-0.190|-0.052|-0.112|-0.009|-0.047|-0.002|
|USA|0.014|0.011|0.188|0.052|0.028|0.012|0.002|0.046|0.003|
|SGP|0.036|0.040|0.191|0.154|0.029|0.079|-0.012|0.026|0.005|
|JPN|0.015|0.012|0.164|0.066|0.027|0.019|-0.001|0.038|0.004|
|TWN|0.030|0.029|0.174|0.146|0.029|0.065|-0.012|0.027|0.005|
|KOR|0.025|0.023|0.173|0.135|0.033|0.049|-0.010|0.031|0.005|
|MYS|0.035|0.037|0.148|0.143|0.026|0.051|-0.013|0.022|0.005|
|THA|0.032|0.034|0.140|0.135|0.030|0.041|-0.013|0.026|0.004|
|IDN|0.040|0.034|0.176|0.188|0.049|0.039|-0.010|0.034|0.005|
|PHL|0.038|0.039|0.145|0.149|0.031|0.039|-0.012|0.024|0.005|

The welfare changes reported in column 2 show that the vast majority of the welfare gains accrue to China, which sees a nearly 10% increase in welfare from a 10% productivity shock. The gains elsewhere are limited, and reasonably similar across countries. The most notable changes have to do with changes in relative position in supply chains. Changes in China’s _DX_ , _D_ and _D_<sup>_∗_</sup> indices indicate that the technology shock moves Chinese production significantly 48
closer to final demand, while the other countries move upstream. China’s move downstream can also be seen in columns 8 (M share), which shows a reduction in the intermediate share of China’s exports.

# **5.4.3 Reduced transaction costs in China**

The welfare effects of a 10% reduction in transaction costs in China are similar to those that were calculated for the Chinese productivity shock. China’s welfare rises by almost 10%. The welfare gains of the countries outside of China are, however, somewhat larger than in the case of the Chinese productivity shock. In contrast with the previous counterfactual, the shock to internal transaction costs moves China’s production upstream, as indicated by the movements in _D_<sup>_∗_</sup> , _D_ and _N_<sup>_∗_</sup> . Similarly, the increase in _N_ reflects greater fragmentation of production within China. Other countries also move upstream because falling Chinese transaction costs lead to longer chains. In broad terms, a move upstream by China is consistent with the evidence presented in Kee and Tang (2013), who find that Chinese exporters have been shifting their purchases of inputs from foreign to domestic sources.

Table 6: Counterfactual 3): 10% decrease in Chinese transaction costs _γi_

|(10 x change)|Welfare|DX|D|D*|N|N*|M share|VAX|
|---|---|---|---|---|---|---|---|---|
|**CHN**|0.953|1.609|3.983|0.033|3.490|-0.027|0.119|0.055|
|USA|0.026|0.341|0.098|0.025|0.143|0.016|0.030|0.008|
|SGP|0.050|0.187|0.166|0.016|0.116|-0.003|0.013|0.002|
|JPN|0.027|0.256|0.096|0.022|0.164|0.015|0.025|0.005|
|TWN|0.042|0.195|0.159|0.019|0.153|0.003|0.016|0.002|
|KOR|0.038|0.210|0.161|0.020|0.163|0.007|0.015|0.002|
|MYS|0.047|0.192|0.179|0.017|0.125|-0.001|0.013|0.002|
|THA|0.045|0.182|0.169|0.019|0.133|0.002|0.013|0.001|
|IDN|0.048|0.372|0.363|0.033|0.107|0.001|0.017|0.003|
|PHL|0.050|0.186|0.194|0.011|0.099|0.001|0.003|-0.001|

# **5.4.4 Reduced trade costs between China and the US**

As noted above and in Yi (2010), international fragmentation raises the elasticity of trade to trade costs. In order to explore the quantitative magnitude of this effect, we shock trade costs for a single country pair (US and China) and measure trade responses. We compare results in our model to those in a version of the same model, except that no trade in intermediates is allowed in the comparison model. Both models are calibrated in the same manner. We conduct one set of exercises using our baseline calibration, and one that does the same for a lower level 49
Table 7: Counterfactual 4): 10% bilateral decrease in US-China trade costs Counterfactual 4A: Baseline calibration with _τ_ = 1 _._ 5

|I|mporter-exporter|With cr|oss-border|No|
|---|---|---|---|---|
||pair|fragm|etation|fragmentation|
|||_All trade_|_Final goods_|_Final goods_|
|∆log_πni_|USA-CHN|0.138|0.151|0.119|
||CHN-USA|0.087|0.120|0.090|
|Trade cost elasticities|USA-CHN|4.399|4.828|3.893|
||CHN-USA|3.609|4.071|3.962|

Counterfactual 4B: Starting from twice lower trade costs, _τ_ = 1 _._ 25

||Importer-exporter|With cr|oss-border|No|
|---|---|---|---|---|
||pair|fragm|etation|fragmentation|
|||_All trade_|_Final goods_|_Final goods_|
|∆log_πni_|USA-CHN|0.095|0.101|0.066|
||CHN-USA|0.053|0.066|0.040|
|Trade cost elasticities|USA-CHN|5.273|5.601|3.794|
||CHN-USA|4.265|3.997|3.895|

of trade costs. In each exercise we track two outcomes in our model (total trade and trade in final goods alone), and track final goods trade in the model with no trade in intermediates. We report the percentage change in trade flows in each direction, and a trade elasticity measured as ∆log _π_<sup>_π_</sup> _<u>nn</u>_<sup>_<u>ni</u>_We hold wages fixed in these simulations because this partial elasticity is conceptually</sup> ∆log _τni_<sup>.</sup> closer to empirical estimates of the trade elasticity than its general equilibrium counterpart.

In the baseline calibration we find somewhat larger trade responses in final goods when fragmentation is allowed. This is consistent qualitatively with the Proposition 4, which demonstrated magnification of the final goods trade elasticity in a two-country version of the model. When we look at trade responses using trade that includes intermediates, the change in the calculated trade elasticity is ambiguous. These results show that the qualitative predictions of magnification in the theory have relatively little quantitative importance at existing levels of implied trade costs.

We impose lower levels of trade costs ( _τ_ = 1 _._ 25), recalibrate as before and again simulate reductions in _τ_ . In this case, magnification through fragmentation is quantitatively more important. In both directions the elasticity of total trade and trade in final goods to changing trade costs are larger in the fragmentation model than in the model without fragmentation.

50
This is also consistent with Proposition 4, which showed larger trade responses at lower levels of trade costs. These exercises are similar to those conducted in Johnson and Moxnes (2013), who also found relatively small magnification effects at current levels of trade but larger effects when trade costs were lower.

# **5.5 Calibrated model with assembly**

As an extension in section 4.4 we added a capstone assembly sector to the model. Our quantitative investigation of this extension focuses on the implied changes in the parameterization of the model that occur in this new structure. We treat assembly as a discrete additional stage and rework our indices ( _D_ , _DX_ , _N_ , etc.) to incorporate it. As before, we calibrate coordination costs, transaction costs, productivity and trade costs ( _τ_ = 1 _._ 36) so that the model replicates the data for upstreamness (variable _DXi_ ), gross-output-to-value-added ratios, wages and the overall trade-to-gross-output ratio. In addition, we assume that each final good has two components with equal cost shares in assembly ( _ηik_ = 0 _._ 5) and we assume that labor costs account for 10 percent of assembly costs ( _β_ = 0 _._ 1). Conditional on our estimate of trade costs, this value is chosen to be sufficiently low so that no country imports both components used to assemble a variety and then re-exports the final good.

Table 8: Calibration with Assembly

||Baseli|ne calibr|ation|Assem|bly calibr|ation||
|---|---|---|---|---|---|---|---|
|Country| ̄_Ai_| ̄_θi_|_γi_| ̄_Ai_| ̄_θi_|_γi_|_X_<sup>_asm_</sup><br>_i_<br>�<br>_i _<sup>_Xasm_</sup><br>_i_|
|US|1.000|1.172|0.490|1.000|0.893|0.511|0.000|
|Singapore|4.783|0.619|0.262|5.122|0.436|0.202|0.000|
|Japan|1.627|1.044|0.446|1.708|0.731|0.430|0.000|
|Taiwan|4.933|0.703|0.289|5.142|0.533|0.254|0.000|
|Korea|4.442|0.812|0.359|4.535|0.631|0.345|0.009|
|Malaysia|14.585|0.731|0.333|15.018|0.572|0.318|0.001|
|Thailand|17.993|0.761|0.393|16.563|0.659|0.461|0.019|
|Indonesia|17.850|1.159|0.726|3.519|2.236|2.671|0.050|
|China|25.432|0.793|0.339|14.228|1.121|0.691|0.884|
|Philippines|41.306|0.807|0.480|14.095|1.402|1.551|0.036|

The structural parameters from the calibrated assembly model are reported in Table 8 along with each country’s exports of assembled goods ( _Xi_<sup>_asm_</sup> ) as a share of the world total. Crosscountry homogeneity of labor productivity in the assembly activity means that the assembly 51
model naturally locates assembly activities in low-wage countries. Since assembly is the most downstream sector, fitting the _DXi_ in the new structure means that _θ_<sup> ̄</sup> _i_ ’s must fall in rich countries and rise in poor countries to offset the location of an additional downstream sector in the poor countries. Thus, rich countries are more downstream within production chains while poor countries are more upstream than in the baseline calibration. Calibrated transaction costs remain broadly similar for the rich countries, but _γi_ rises substantially in the poor countries in the new calibration in order to offset the increase in _θ_<sup> ̄</sup> _i_ . In this calibration both transaction costs and coordination costs are higher in poor countries than in rich countries. One implication is that there is less need for the productivity distribution to generate wage differences. The calibrated values of poor countries’ _A_<sup> ̄</sup> _i_ ’s fall as a result. The final column shows that only the poor countries export assembled goods in this calibration, with China playing a dominant role.

# **6 Concluding remarks**

Recent empirical work has documented sizable differences in supply chain length. In this paper we attempt to explain such variation in an integrated framework that links the internalization decisions of firms within a supply chain to the organization of the chain across countries. We develop a continuous firm representation of the optimal organization of a multi-country supply chain, with an endogenous allocation of tasks across firms and countries. We derive formal and intuitive representations of the gains from fragmentation within a chain and relate these to the implicit price of tasks and the price of the final good.

In this Coasian setting, we show that the same parameters that shape the boundaries of firms also determine comparative advantage within international supply chains. Conditional on participation in a supply chain, the lower a country’s coordination costs the more downstream it will be. Low within-firm coordination costs also imply an ability to host larger firms. By contrast, countries with high transaction costs will tend to participate downstream because their disadvantages can only be offset in chains for which they have low within-firm-coordination costs and larger firms.

In order to link the model to the prominent literature on the welfare gains of trade, we use a conventional Ricardian framework to produce a general equilibrium model with multiple chains, and with exogenous productivity shocks _across_ chains. We derive implications for trade elasticities and welfare, relative to standard theoretical benchmarks (Arkolakis et al 2012). Among a number of theoretical results we show that the elasticity of final goods trade to trade costs is larger in the presence of fragmentation. Relative to the Arkolakis et al (2012) formula without fragmentation, we show that welfare effects are smaller for upstream countries and larger for downstream countries in the presence of fragmentation.

52
To illustrate the quantitative implications of the model, we conduct a calibration exercise with counterfactual simulations. In our model, the Coasian structural parameters determine the gross-output-to-value added ratio, a fact that facilitates calibration. We shock international trade costs and find numerical evidence that is consistent with our theoretical results. We find that shocks to Chinese productivity and to the coordination cost parameter in China to highlight different implications for welfare, spillover to other countries and specialization along production chains. In order to better tie the model to available data, we develop an extension of the calibrated model with an explicit representation of assembly activities to better formalize the participation of China in international production chains.

# **References**

- **Antr`as, Pol and Davin Chor** , “Organizing the global value chain,” _Econometrica_ , 2013, _81_ (6), 2127–2204.

- **, , Thibault Fally, and Russell Hillberry** , “Measuring the Upstreamness of Production and Trade Flows,” _American Economic Review Papers and Proceedings_ , 2012, _102_ (3), 412– 416.

- **, Teresa C Fort, and Felix Tintelnot** , “The Margins of Global Sourcing: Theory and Evidence from US Firms,” _Working paper_ , 2014.

- **Arkolakis, Costas, Arnaud Costinot, and Andres Rodriguez-Clare** , “New Trade Models, Same Old Gains?,” _American Economic Review_ , 2012, _102_ (1), 94–130.

- **Baldwin, Richard** , “Global supply chains: Why they emerged, why they matter, and where they are going,” _C.E.P.R. Discussion Papers_ , August 2012, _9103._

- **and Anthony Venables** , “Relocating the Value Chain: Offshoring and Agglomeration in

- the Global Economy,” _NBER Working Papers_ , 2010, _1661._

- **and Javier Lopez-Gonzalez** , “Supply-chain Trade: A portrait of global patterns and

- several testable hypotheses,” _Working Paper_ , December 2012, pp. 1–57.

- **Baldwin, Richard E.** , “Managing The Noodle Bowl: The Fragility Of East Asian Regionalism,” _The Singapore Economic Review (SER)_ , 2008, _53_ (03), 449–478.

- **Bernard, Andrew B, Andreas Moxnes, and Karen Helene Ulltveit-Moe** , “Two-Sided Heterogeneity and Trade,” _NBER Working Paper no. 20136_ , 2014.

53
- **Chaney, Thomas and Ralph Ossa** , “Market size, division of labor, and firm productivity,” _Journal of International Economics_ , 2013, _90_ (1), 177–180.

- **Coase, Ronald H.** , “The Nature of the Firm,” _Economica_ , 1937, _4_ (16), 386–405.

- **Costinot, Arnaud** , “An Elementary Theory of Comparative Advantage,” _Econometrica_ , 2009, _77_ (4), 1165–1192.

- **and Andres Rodriguez-Clare** , “Trade Theory with Numbers: Quantifying the Conse-

- quences of Globalization,” _Handbook of International Economics_ , 2014, _4_ , 197.

- **, Jonathan Vogel, and Su Wang** , “An Elementary Theory of Global Supply Chains,” _NBER working paper no. 16936_ , 2011.

- **Disdier, Anne-Celia and Keith Head** , “The Puzzling Persistence of the Distance Effect on Bilateral Trade,” _The Review of Economics and Statistics_ , 2008, _90_ (1), 37–48.

- **Dornbusch, Rudiger, Stanley Fischer, and Paul Anthony Samuelson** , “Comparative advantage, trade, and payments in a Ricardian model with a continuum of goods,” _The American Economic Review_ , 1977, pp. 823–839.

- **Eaton, Jonathan and Samuel Kortum** , “Technology, Geography, and Trade,” _Econometrica_ , 2002, _70_ , 1741–1780.

- **Fally, Thibault** , “Production Staging: Measurement and Evidence,” _Working paper_ , 2012.

- **Hillberry, Russell and David Hummels** , “Explaining home bias in consumption: The role of intermediate input trade,” _NBER working paper no. 9020_ , 2002, pp. 1–33.

- **Hummels, David, Jun Ishii, and Kei-Mu Yi** , “The nature and growth of vertical specialization in world trade,” _Journal of international Economics_ , 2001, _54_ (1), 75–96.

- **Johnson, Robert and Andreas Moxnes** , “Technology, Trade Costs, and the Pattern of Trade with Multi-Stage Production,” _Working Paper_ , 2013.

- **and Guillermo Noguera** , “Accounting for intermediates: Production sharing and trade

- in value added,” _Journal of International Economics_ , 2012, _86_ (2), 224–236.

- **Johnson, Robert C.** , “Five Facts about Value-Added Exports and Implications for Macroeconomics and Trade Research,” _Journal of Economic Perspectives_ , 2014, _28_ (2), 119–42.

- **Johnson, Robert C and Guillermo Noguera** , “Proximity and production fragmentation,” _The American Economic Review_ , 2012, _102_ (3), 407–411.

54
- **and** , “Fragmentation and trade in value added over four decades,” _Working paper_ , 2013.

- **Kee, Hiau Looi and Heiwai Tang** , “Domestic Value Added in Chinese Exports: Firm-level Evidence,” _Working Paper_ , 2013.

- **Kikuchi, Tomoo, Kazuo Nishimura, and John Stachurski** , “Transaction Costs, Span of Control and Competitive Equilibrium,” _Working Paper_ , 2014.

- **Koopman, Robert, Zhi Wang, and Shang-Jin Wei** , “How much of Chinese exports is really made in China? Assessing domestic value-added when processing trade is pervasive,” _Working Paper_ , 2010.

- **Kraemer, Kenneth L., Greg Linden, and Jason Dedrick** , “Who Captures Value in the Apple iPad and iPhone?,” _Working paper_ , 2011.

- **Kremer, Michael** , “The O-Ring Theory of Economic Development,” _The Quarterly Journal of Economics_ , 1993, _108_ (3), 551–75.

- **Krugman, Paul and Anthony J Venables** , “Integration, specialization, and adjustment,” _European Economic Review_ , 1996, _40_ (3), 959–967.

- **Li, Ben G. and Yibei Liu** , “Moving up the value chain,” _Boston College Working Paper_ , 2014.

- **Melitz, Marc J. and Stephen J. Redding** , “Missing Gains from Trade?,” _American Economic Review_ , 2014, _104_ (5), 317–21.

- **Puzello, Laura** , “A proportionality assumption and measurement biases in the factor content of trade,” _Journal of International Economics_ , 2012, _87_ (1), 105–111.

- **Rodriguez-Clare, Andres** , “Offshoring in a Ricardian World,” _American Economic Journal: Macroeconomics_ , 2010, _2_ (2), 227–258.

- **Simonovska, Ina and Michael E Waugh** , “The elasticity of trade: Estimates and evidence,” _Journal of International Economics_ , 2014, _92_ (1), 34–50.

- **Yi, Kei-Mu** , “Can vertical specialization explain the growth of world trade?,” _Journal of Political Economy_ , 2003, _111_ (1), 52–102.

- , “Can Multistage Production Explain the Home Bias in Trade?,” _American Economic Review_ , 2010, _100_ (1), 364–93.

55
<!-- Start of picture text -->
[)<br>po<br>|<br>| (J<br>| 7<br>si |<br>—(—) ee<br><!-- End of picture text -->
<!-- Start of picture text --> ~(—<br>/<br>/ oe<br>yf<br>Ot<br>> (>) (—}<br>> |<br>——()~<br>[—<br>( —)<br>“t= O N ]<br><!-- End of picture text -->
**Prices along the chain** : To obtain a simple expression for the value-added-to-gross-output ratio<sup>_ci_</sup><sup><u>(</u></sup><sup>_sif_</sup><sup><u>)</u>thefirststepistocompute</sup><sup>_pif_isthepricealongthechain.</sup> _pif_<sup>,</sup>

Hence the gross-output-to-value-added ratio at the firm level is: We also obtain the same expression for the aggregate gross-output-to-value-added ratio. If we define gross output as the total value of all transactions: we obtain:

# **Proof of Proposition 1**

To simplify the exposition, we index countries by _n_ , with _n_ = 1 referring to the most downstream country and _n_ = _N_ the most upstream country. The goal is to minimize: 58
under the constraints: with: Under which condition can country _n_ be downstream from country _n_ + 1? Let us take as given the price in country _n_ + 2 and consider the following function: This function _m_ ( _x_ ) indicates by how much the price of output in _n_ will increase if we shift a measure _x_ of tasks from country _n_ to country _n_ + 1.

If we are at equilibrium, the function _m_ ( _x_ ) must be at its minimum at _x_ = 0. The first-order condition imply that _m_<sup>_′_</sup> ( _x_ ) = 0. We obtain that: must equal zero at _x_ = 0.

More importantly, to prove Proposition 1, one needs to examine the second order condition, which imposes _m_<sup>_′′_</sup> ( _x_ ) _>_ 0. If _m_<sup>_′′_</sup> ( _x_ ) were negative, _x_ = 0 would not be a local minimum and it would be more efficient to shift some tasks to either country _n_ or _n_ + 1.

As one can see in equation (55), the right-hand-side term is increasing in _x_ (i.e. _m_<sup>_′′_</sup> ( _x_ ) _>_ 0) only if the exponent<sup>_θn_</sup> _θ_<sup><u>+1</u></sup> _n_ +1<sup>_−θn_</sup> is positive. This proves that we must have _θn_ +1 _> θn_ at equilibrium.

Finally, it is not difficult to verify that two consecutive countries cannot have the same _θn_ = _θn_ +1 as long as we have non-zero trade costs _τ −_ 1 _>_ 0.

# **Other proofs for Section 3.2**

**Prices along the chain** : Using again equation (55), the first-order condition _m_<sup>_′_</sup> (0) = 0 implies:

59
<!-- Start of picture text --> Se<br>a<br>|<br>—<br>(—) (——)<br>— (——)<br>(i) sr (~~)<br>Ee (—) 5 (—)<br>ye (——) © (—)<br>(—) " x (—)<br><!-- End of picture text -->
<!-- Start of picture text -->
[—-<br>[— |<br>— (—)<br>—— & ()<br>s ()<br>a ae<br>—<br><!-- End of picture text -->
~~(yo~~ ~~<u>)</u>~~ <!-- Start of picture text --> Hy<br>()<br>(—)<br>ee<br>——(——)<br>—(—)~<br><!-- End of picture text -->
which implies that: i) the price in the final good and iv) the shadow cost _λG_ decrease when trade costs decrease. Given equation (30) we obtain that a decrease in _λG_ affects firm scope everywhere along the chain and leads to a decrease in _snf_ , conditional on the position on the chain, wages and the set of countries involved in the chain (point v). As trade costs decrease, however, countries tend to move downstream. Since firms scope is larger downstream, moving up the chain implies larger average firm scope for each country _i >_ 1 (except the most downstream one). This can be seen in expressions (38): firm scope at both end of the chain in country _i_ is a decreasing function of _τ_ (point vi in Proposition 2). Finally, we can also see above that _S_ 1 is an increasing function of trade costs (conditional on wages), which proves point iii).

# **Proof of Proposition 3: Free Trade Equilibrium**

To prove part 1, we can use our previous result that, for a given chain _ω_ , the sorting of countries is given by the ranking of _θi_ . A key implication is that each chain is divided into a finite number of segments. We structure the proof by contradiction to show that no two countries with different values of _θ_ will produce the same tasks, not even on different chains.

Suppose there exist two varieties over which two countries are involved in parallel on the same range of tasks, i.e. country _i_ products one variety while country _j_ produces another variety on a common segment of tasks [ _S,_<sup> ̄</sup> _S_<sup> ̄</sup> + ∆] with _S >_ 0.

A key implication of free trade is that prices must be identical across varieties at any point on the production chain, where location on the chain represents the the cumulative range of tasks that have been performed. Denote by _PM_ the price after completing a range _S_<sup> ̄</sup> of tasks. For any _s <_ ∆, free trade implies that: (this comes from equation 20).

For this equality to hold for all values of _s <_ ∆, all derivatives must be equal. Taking ratios of the second and first derivatives evaluated at _s_ = 0, it is easy to show that the two parameters _θi_ and _θj_ that govern the convexity of prices (with respect to additional tasks) must be equal. Hence these two countries cannot be distinct.

This proves that there is a partition of the range of tasks that is common across all varieties. Each country corresponds to a specific segment of the range of tasks [0 _,_ 1] in which it specializes for all varieties.

To prove part (ii) we characterize aggregate labor demand and equilibrium in the labor market. Denoting labor supply by _Ln_ , we can use equations (34) and (35) from Lemma 2, to obtain a simple expression for aggregate labor demand as a function of the vector of wages **w** for all countries _n >_ 1: 63
For the most upstream country _N_ , the right term involving _wN_ +1 is dropped. For the most downstream country _n_ = 1, we have: We can verify that Walras’ Law is satisfied:<sup>�</sup> _n_<sup>_w_</sup> _n_<sup>_LD_</sup> _n_<sup>(</sup><sup>**w**) =�</sup> _n_<sup>_L_</sup> _n_<sup>_w_</sup> _n_<sup>.</sup> Equilibrium on the labor market requires: for each country _n_ .

To prove existence and uniqueness of equilibrium, one would be tempted to define the excess demand function _Zn_ ( **w** ) = _L_<sup>_D_</sup> _n_<sup>(</sup><sup>**w**)</sup><sup>_−Ln_asinAlvarezandLucas(2007).Whilethis</sup> excess demand function would satisfy all sufficient conditions to prove existence using standard theorems, it does not satisfy the gross substitute property that is commonly used to prove uniqueness.

Instead, we can prove uniqueness (and existence) with a more simple recursive approach using our analytical results. We normalize wages such that: in order to simply the labor market conditions in each country.

� In a first step, using this normalization, we define a function _wN −_ 1( _wN_ ) implicitly such that: With the normalization above, this is equivalent to imposing: � � which has a unique solution _wN −_ 1 as a function of _wN_ . We can see that this function _wN −_ 1( _wN_ ) _wN −_ <u>1(</u> _wN_ <u>)</u> is continuous, monotonic and strictly increasing in _wN_ . Moreover, both the ratio<sup>�</sup> _wN_ and � _wN −_ <u>1(</u> _wN_ <u>)</u> _wN −_ 1( _wN_ ) go from 0 to +� _∞_ as _wN_ goes from 0 to + _∞_ . We also obtain that the ratio<sup>�</sup> _wN_ is increasing in _wN_ , i.e. _wN −_ 1( _wN_ ) grows faster than _wN_ .

Recursively, for any other country _n_ , we can define a continuous function _w_ � _n_ ( _wN_ ) such that: 64
� � If _w_ � _n_ ( _wN_ ) and _w_ � _wn_ +1 _n_ <u>(</u> _w_ ( _wNN_ <u>)</u> )<sup>bothincreasewith</sup><sup>_wN_,wefindthat</sup> _w_ � _n−_ 1( _wN_ ) and _ww_ � _<u>n−n</u>_ (1 _w_ <u>(</u> _wNN_ ) <u>)</u> also increase with _wN_ .

� If both _w_ � _n_ ( _wN_ ) and the ratio _w_ � _wn_ +1 _n_ <u>(</u> _w_ ( _wNN_ <u>)</u> )<sup>gofrom0to+</sup><sup>_∞_(as</sup><sup>_wN_goesfrom0to+</sup><sup>_∞_),then</sup> we also find that _w_ � _n−_ 1( _wN_ ) and the ratio _w_<sup>�</sup> _w_ � _<u>n−n</u>_ (1 _w_ <u>(</u> _wNN_ ) <u>)</u><sup>go from 0 to +</sup><sup>_∞_as</sup><sup>_wN_goes from 0 to +</sup><sup>_∞_.</sup> By recurrence, these monotonicity and limit properties apply to _w_ � _n−_ 1( _wN_ ) and _w_<sup>�</sup> _w_ � _<u>n−n</u>_ (1 _w_ <u>(</u> _wNN_ ) <u>)</u><sup>for</sup> all _n ∈_ [1 _, N −_ 1].

Finally, we are left to show that we can find _wN_ (and thus all other wages) such that the normalization is (where _w_ � _N_ ( _wN_ ) _≡ wN_ ). One can see that the left hand side is strictly increasing in _wN_ given the monotonicity properties shown above. When _wN_ approaches to zero, the left hand side has a limit zero. As _wN_ goes to infinity, the left hand side also goes to infinity. This proves existence and uniqueness of a solution _wN_<sup>_∗_suchthattheabovenormalizationequationholds.</sup> � Consequently, this also proves existence and uniqueness of wages _wn_ = _wn_ ( _wN_<sup>_∗_) such that labor</sup> markets clear.

# **Proofs for Section 4.2: Two-country case**

# **Proof of Proposition 4**

**Final good prices and consumption choices** : Using the results from above, final goods prices are:

respectively when country _D_ or _U_ produces the final stages. This leads to the following thresholds in the text: There is no analytical solution for _ωU_<sup>_∗_butitiseasytocheckthefollowingsolutionin</sup><sup>_ω_</sup> _D_<sup>_∗_:</sup> 65
We can also verify that the foreign content for the marginal variety _ωD_<sup>_∗_isequaltoone:</sup> where<sup>_w_</sup> _P_<sup>_<u>U</u>_</sup> _D_<sup>_lU_</sup> ( _ω_<sup><u>(</u></sup><sup>_ω_</sup> )<sup><u>)</u></sup> is the share of value-added from country _U_ in final goods sold by _D_ :

**General equilibrium:** We prove here that _τwU_ decreases as trade costs _τ_ decrease.

Wages at equilibrium are determined by a labor market clearing condition or, equivalently, a trade balance condition. Here, trade balance imposes: where the left-hand side correspond to exports of final goods by _D_ and the right-hand side corresponds to exports of final and intermediate goods by _U_ .

It is equivalent to a trade balance in value-added content: where<sup>_w_</sup> _P_<sup>_<u>U</u>_</sup> _D_<sup>_lU_</sup> ( _ω_<sup><u>(</u></sup><sup>_ω_</sup> )<sup><u>)</u></sup> denotes the foreign value-added content of final goods sold by _D_ (equation 58).

We arrive at a contradiction if we assume that _τwU_ increases while _τ_ decreases. The righthand-side term of expression (60) would decrease since it is a strictly decreasing function of _τwU_ : country _D_ sources less from _U_ if trade-cost-adjusted wages increase in _U_ . On the other hand the term on the left would increase because of higher income _LU wU_ , a lower import threshold _ωU_<sup>_∗_</sup> (since goods from _D_ would become relatively cheaper) and higher foreign value-added content<sup>_lU_</sup><sup><u>(</u></sup><sup>_ω_</sup><sup><u>)</u></sup> 1 _−_<sup>_w_</sup> _P_<sup>_<u>U</u>_</sup> _D_ ( _ω_ )<sup>.Henceitmustbethat</sup><sup>_τwU_decreaseswhen</sup><sup>_τ_decreases.</sup>

**Trade elasticity** : For country _D_ , it is easy to check that the elasticity is the same as in Eaton and Kortum (2002):

For country _U_ , we take the derivative of _τPD_ ( _ωU_<sup>_∗_) =</sup><sup>_PU_(</sup><sup>_ω_</sup> _U_<sup>_∗_) with respect to log</sup><sup>_τ_,which gives:</sup> In the expression above, the partial derivative<sup>_<u>∂</u>_</sup> _∂_<sup>lo</sup> log<sup><u>g</u></sup><sup>_P_</sup> _τ_<sup>_<u>U</u>_</sup> is equal to unity. The partial derivative _<u>∂∂</u>_ lologg _P τD_<sup>islowerthanoneandequalstheshareofvaluecomingfrom</sup><sup>_U_forthethresholdvariety</sup> 66
_ω_<sup>_∗_</sup> _U_<sup>:</sup> _d_ log _A_ <u>(</u> _ωU_<sup>_∗_</sup><sup><u>)</u></sup> After solving for _d_ log _τ_ , we find: The trade elasticity in final goods for country _U_ is then:

**VAX ratios:** For country _D_ , the value-added content of exports to country _U_ is:

For a given _ω_ , the term in the integral sum increases with trade-cost-adjusted wages _τwU_ , which itself increases with _τ_ (larger domestic value added share as trade costs increase). This is a direct effect. There is also a composition effect: _ωU_<sup>_∗_increaseswith</sup><sup>_τ_,sothatcountry</sup><sup>_D_only</sup> exports high-value-added goods (varieties _ω_ closer to one) when trade costs are higher. This second effect also leads to an increase in _V AXD_ when trade costs increase.

A similar intuition holds for the VAX ratio for country _U_ , as described in the text. The VAX ratio for country _U_ equal the one for country _D_ in this two-country example because we have a trade balance in gross flows as well as in the value-added content of trade.

# **Gains from trade**

**Gains from trade for country** _D_ **:** For country _D_ , the wage and the price index under autarky are normalized to zero (in log). Hence the log of the price index with trade also reflects the gains from trade. Gains from trade are given by:

with: 67
<!-- Start of picture text --> H) | |<br>i (—) J<br>f (—) | —<br>- | _<br>| ;——<br>f—— -s——<br>po)<br>||<br>|-<br><!-- End of picture text -->
<!-- Start of picture text --> (J) /<br>h (i 6 4<br>h () -<br>[| (——) J ——<br>/—-<br>- {| ———<br><!-- End of picture text -->

| ~~-~~