Assortative Matching of Exporters and Importers

INTERNATIONAL trade mostly takes the form of firm-to-firm transactions in which firms seek and compete for capable buyers and suppliers globally. A case example is Boeing's 787 Dreamliner team that comprises the most capable suppliers from all over the world. Trade research in the last two decades has revealed the huge heterogeneity in the capability of exporters and importers (e.g., their productivity and product quality). Thus, the way in which heterogeneous exporters and importers match along the supply chains may determine the aggregate capability of the industry and the welfare.

This paper examines how exporters and importers match based on their capability by investigating the change in such exporter-importer matching during trade liberalization. From Mexico's customs administrative records, we construct a matched exporter-importer data set for Mexican textile and apparel exports to the United States from 2004 to 2007. The Mexico–United States textile and apparel trade is particularly suitable for our purpose. First, Mexico and the United States are large trading partners with each other, and so trade between them includes numerous heterogeneous exporters and importers.¹ Second, the two countries' textile and apparel trade experienced large-scale liberalization. In 2005 the United States removed quotas on textile and apparel imports at the end of the Multifiber Arrangement (MFA). Mexican products already had quota-free access to the U.S. market under the North American Free Trade Agreement (NAFTA), and so the MFA's end effectively removed protection for Mexican products in the U.S. market and forced them to compete with imports from third countries, principally China. The liberalization varied across products substantially and was arguably exogenous because the liberalization schedule was decided at the GATT Uruguay Round (1986–1994) when China's export growth was not expected.

The MFA's end substantially changed the partnerships between Mexican exporters and U.S. importers. Mexican exports to the United States decreased by an extensive margin (stopping exports) and intensive margin (reducing export values). The intensive margin adjustment involved substantial partner switching, often including the exporter's largest main partners. Main partner switching accounted for more than 50% of the intensive margin and caused a more than 230% excess reallocation of exports across U.S. buyers beyond the intensive margin. As we explain in section II, this prevalence of main partner switching in trade liberalization was at odds with anonymous market models (e.g., neoclassical models, oligopoly models), love-of-variety models (the Krugman-Melitz model), and some recent exporter-importer matching models (e.g., Bernard et al., 2018) that combine the love-of-variety model and fixed costs of matching.

Motivated by this new fact, we develop a many-to-many matching model of exporters and importers in an intermediate good market in which partner switching is the principal margin of adjustment. The model combines Sattinger's (1979) frictionless assignment model of a continuum of agents, Melitz's (2003) standard heterogeneous firm trade model, and Bernard et al.'s (2011) multiple-product firm trade model. The model consists of final producers (importers) in the United States and suppliers (exporters) in Mexico and China. Final producers produce multiple products, and suppliers own multiple production lines. A final producer's variety-level capability depends on its firm-level capability and idiosyncratic capability, whereas a supplier's production-line-level capability depends on its firm-level capability and idiosyncratic capability. A final variety matches a production line one-to-one, resulting in the many-to-many matching of final producers and suppliers. The Beckerian PAM of varieties and production lines arises as a stable equilibrium when a variety's capability and production's capability are complements.

The model predicts that the MFA's end induced systematic partner switching that led to efficient buyer-supplier matching and improved consumer welfare. As empirically documented by Khandelwal et al. (2013), at the MFA's end, Chinese suppliers at various capability levels entered the U.S. market. Their entry lowered the capability ranking of each Mexican supplier in the market. Therefore, to achieve PAM, Mexican exporters switched to U.S. importers with lower capability, whereas U.S. importers switched to Mexican exporters with higher capability. We call these types of partner switching “partner downgrading” and “partner upgrading,” respectively. Allowing capable Chinese suppliers to match with capable U.S. final producers, this rematching achieved PAM in the global market, which improved aggregate capability and consumer welfare. By contrast, in an anonymous market in which matching is independent of capability, rematching should not occur in a systematic way or result in an efficiency gain.

We take the model's predictions on partner switching to data. Guided by the theory, we estimate the rankings of firm-level capability of Mexican exporters and U.S. importers by the rankings of their 2004 preliberalization product trade with their main partners. We then compare the partner-switching patterns between liberalized products (the treatment group) and other textile and apparel products (the control group) within Harmonized System (HS) two-digit industries. We find the partner-switching patterns to be consistent with PAM. First, U.S. importers upgrade their Mexican partners more often in the treatment group than in the control group. At the same time, Mexican exporters downgrade their U.S. partners more often in the treatment group than in the control group. Second, among firms that switch their main partners, the capability rankings of new partners are positively correlated with those of old partners. Together, these findings provide strong support for PAM and reject independent random matching. Furthermore, we confirm the model's predictions on firm exit and the number of partners. First, the capability cutoff for Mexican exporters increases. Second, U.S. importers and Mexican exporters decrease their number of partners.

To the best of our knowledge, detecting Beckerian PAM by capability in this way is a novel approach to addressing the endogeneity problem in the conventional approach. When matching matters for a firm's performance, most firm characteristics observable in typical production and customs data (e.g., inputs, outputs, and productivity measures) may reflect partners' unobserved capability as well as the firm's own capability. Therefore, the simple correlation of those characteristics across matches may suffer from endogeneity.² Instead, our approach utilizes the MFA's end as an exogenous negative shock on the capability ranking of Mexican exporters.

As matched exporter-importer data become available to researchers, the last decade since 2010 saw the burgeoning literature on buyer-supplier relationships in international trade.³ Our paper contributes to a strand of this literature studying exporter-importer matching. Rauch (1996), Casella and Rauch (2002), and Rauch and Trindade (2003) pioneered the theoretical literature by using the assignment model of symmetric firms, whereas our model features firm heterogeneity in capability as in Melitz (2003). Antras et al. (2006) analyzed offshoring as the PAM of managers and workers across countries. The assignment model captures two distinctive features in exporter-importer relationships. First, trading with high-capability firms improves a firm's performance, but the opportunity to trade with them is scarce and something that firms compete for. This view echoes with recent evidence that trading with high-capability foreign firms improves a local firm's performance through various channels.⁴ Second, buyer-supplier matching is an allocation of scarce trading opportunities. Thus, trade liberalization induces partner switching to achieve a globally efficient matching. We provide the first evidence for this matching mechanism.

Bernard et al. (2018) recently developed another approach combining match-level fixed costs and the love-of-variety (CES) production function.⁵ A buyer and a supplier are matched when the match surplus exceeds the match-level fixed costs. As the match surplus monotonically increases in the buyer's capability and the supplier's, all the matches are realized except those between low-capability firms.⁶ Thus, the model can predict the negative degree assortativity reported by Blum et al. (2010), Bernard et al. (2018), and others that a buyer's number of partners is negatively correlated with the average number of firms to which the buyer's partners sell.

Our finding of PAM can be compatible with negative degree assortativity both theoretically and empirically. In appendix D we present a two-tier model of exporter-importer matching that unifies Bernard et al.'s (2018) model and ours to predict negative degree assortativity for the firm-level matching and PAM for the product-level matching. In the model a buyer (e.g., a car maker) has a love-of-variety production function with respect to intermediate goods and decides whether to make or buy each intermediate good (e.g., tires, seats), considering the match surplus and match-level fixed costs, as in Bernard et al. (2018). For each intermediate good (e.g., a set of four tires), a buyer matches a supplier following PAM as in our model. Our data confirm the model's prediction by finding that negative degree assortativity holds when a match is defined at the firm level, but becomes weaker and statistically insignificant when a match is defined at the product level.

Another important strand of the literature studies the dynamics of an exporter's and importer's partner choice in a steady-state environment. Macchiavello (2010) introduced reputation building in an assignment model to explain an exporter's partner upgrading over time. Eaton et al. (2014) and Eaton et al. (2015) developed models incorporating search and learning frictions in partner acquisitions.⁷ Eaton et al. (2016) modeled random meeting and competition among multiple buyers and suppliers. Monarch (2021) estimated partner-switching costs in a dynamic discrete choice model. Heise (2020) documented the dependence of exchange rate pass-through on the age of trade relationships.

Benguria (2021) and Dragusanu (2014) documented positive correlations between the size and productivity measures of exporters and importers in France-Colombia trade and India–United States trade, respectively. Our model featuring Beckerian PAM also predicts these findings. Benguria (2021) and Dragusanu (2014) developed search effort models of the Stigler (1961) type to explain their findings by a different mechanism: a high-productivity exporter spends greater search efforts finding a high-productivity importer. Their models, however, do not explain Mexican exporters' partner downgrading at the MFA's end. In their models, search costs are sunk and importers are willing to trade with all exporters. Thus, Mexican exporters should continue to trade with preliberalization U.S. partners instead of downgrading partners by paying additional search costs.

Another related literature investigates nonanonymous contracts in given exporter-importer relationships, using matched exporter-importer data. Macchiavello and Morjaria (2015) examined the surplus of long-term relationships relative to anonymous spot trade. Cajal-Grossi et al. (forthcoming) found greater markups in long-term relational trade than spot trade. Bernard and Dhingra (2019) studied firms' relationship investment to avoid inefficiency in spot trade. Ignatenko (2019) reports exporters' price discriminations across importers. Our paper complements this literature by showing exporters match importers in an nonanonymous way too.

The rest of this paper is organized as follows. Section II explains our data and documents new facts on partner switching during liberalization. Section III presents our model and derives predictions. Section IV describes our empirical strategy. Section V presents the main results and robustness checks. Section VI provides concluding remarks. The online appendix provides the calculations, proofs, data construction, extended models, robustness checks, and additional analyses rejecting alternative explanations of our results.

The MFA and its successor, the Agreement on Textiles and Clothing, are agreements about the quotas on textile and apparel imports among GATT and WTO countries. At the GATT Uruguay Round (1986–1994), the United States (together with Canada, the European Union, and Norway) promised to abolish the quotas in four steps (in 1995, 1998, 2002, and 2005). The MFA's end in 2005 was the largest liberalization, in which liberalized products constituted 49% of imports in 1990.

Three facts (taken from previous studies) about the consequences resulting from the MFA's end motivate our analysis.

By 2003 Mexico already had tariff- and quota-free access to the U.S. market through NAFTA. With the MFA's end, Mexico lost its advantage over third-country exporters and faced increased competition from Chinese exporters in the U.S. market, as the right panel of figure 1 shows.⁹ Although the two groups were stable and almost identical before 2005, the exports in the treatment group significantly declined thereafter.

From Chinese customs transaction data, Khandelwal et al. (2013) decomposed the increases in Chinese exports to the United States in liberalized products after the removal of the quota into the intensive and extensive margins. Increases in Chinese exports were mostly driven by the entry of new exporters that had not previously exported products. These new exporters have different capability levels than those of incumbent exporters, with many more capable than incumbents.¹⁰

From Mexico's customs administrative records, we construct a matched exporter-importer data set from June 2004 to December 2011 for Mexican textile and apparel exports (covering HS50 to HS63) to the United States. For each match of a Mexican exporter and a U.S. importer, the data set contains the following information: exporter ID, importer ID, HS six-digit product code, annual shipment value (USD), quantity and unit, an indicator of a duty-free processing reexport program (Maquiladora/IMMEX), and other information.

We assign the exporter ID and importer ID throughout the data set. The exporter ID is the tax number unique to each firm in Mexico. Assigning importer IDs to U.S. firms is challenging. Although the customs records report the name, address, and employment identification number (EIN) of the U.S. importer for each transaction, none of these can uniquely identify a firm because it can use multiple names or change names, own multiple plants or establishments, or change tax numbers. Furthermore, a firm's name and address may be written in multiple ways and suffer from typographical errors. Therefore, simply counting combinations of names, addresses, and EINs would wrongly assign more than one ID to one U.S. importer.

We therefore assign the importer ID by applying a series of record linkage techniques.¹¹ First, we prepare a list of name variations such as fictitious names, previous names, and name abbreviations, a list of addresses of company branches/subsidiaries, and a list of EINs from Orbis by Bureau van Dijk, which covers 20 million company branches, subsidiaries, and headquarters in the United States. Second, the address format is standardized using software certified by the U.S. Postal Service. Third, we match the lists from Orbis to each of the linking variables (name, address, EIN) in the customs data by fuzzy matching. Two types of errors can occur in fuzzy matching: “false matching” (matching records that should not be matched) and “false unmatching” (not matching records that should be matched). The criteria for fuzzy matching are chosen to minimize false unmatching because false matching is easier to identify by manual checks. Fourth, binary-matched records are aggregated into clusters so that each record matches another record in that cluster. Then, we manually check each cluster and remove falsely matched records. A resulting cluster represents a firm and receives an importer ID. Appendix B explains the data construction process in detail.

Data cleansing drops some observations. First, the data set covers only observations from June to December in 2004, and so we drop the observations from January to May in other years to make the information in each year comparable. We obtain similar results when January-May observations are included. Second, although importer information is reported for most normal trade transactions, it is sometimes missing for processing trade transactions under the Maquiladora/IMMEX program in which exporters do not have to report an importer for each shipment.¹² We drop exporters that do not report the importer information for most transactions. To address the potential selection issues caused by this action, we distinguish normal trade and processing trade in the analyses below and conduct weighted regressions in appendix B.4.

Table 1 reports the summary statistics for the product-level and firm-level matching. A product-level match occurs if an importer and an exporter trade in a particular product, whereas a firm-level match occurs if an importer and an exporter trade in at least one product. Columns (a) and (b) in table 1 report the mean and median of the product-level matching.¹³ The first four rows show that eleven to fifteen exporters and fifteen to twenty importers exist in an average product market, but the majority of firms trade with only one partner.¹⁴ Rows (5) and (6) show that even for firms that trade with multiple partners, more than 70% of their trade occurs with their single main partner.¹⁵

Our new finding is that exporters and importers actively switch partners during liberalization. Panel A in table 2 reports the changes in Mexican textile and apparel exports to the United States between 2004 and 2007 by incumbent exporters in 2004 separately for liberalized products (quota-bound) and other products (quota-free). The changes in total exports in column 1 are decomposed into the extensive margin in column 2 by exiters that stopped exporting by 2007 and intensive margin in column 3 by continuing exporters in 2007.¹⁶ The intensive margin in column 3 is further decomposed into three margins of partner changes: Partner Staying in column 4 expresses the changes in exports to continuing buyers that import from the exporter in both 2004 and 2007, Partner Adding in column 5 expresses those to new buyers in 2007 that did not import from the exporter in 2004, and Partner Dropping in column 6 expresses those to dropped partners that imported from the exporter in 2004 but not in 2007. The parentheses in columns 5 and 6 report the share of export changes by Partner Switchers that simultaneously add and drop partners. These high shares imply that most partner changes are in fact partner switching. Column 7 reports the excess reallocation of partners, that is, $|(5)|+|(6)|-|(5)+(6)|$⁠.

As table 1 suggests, the switching of main partners plays a major role in the adjustment. In panel C in table 2, the intensive margin in column 1, which is column 3 in panel A, is decomposed according to main partner's involvement: export changes not involving main partners in column 2, exports to continuing main partners in 2004 and 2007 in column 3, those to new main buyers in 2007 that were not main buyers in 2004 in column 4, and those to dropped main buyers that were main buyers in 2004 but not in 2007 in column 5. Column 6 reports the excess reallocation associated with main partners, that is, $|(4)|+|(5)|-|(4)+(5)|$⁠.

Table 2 shows that in liberalized industries, main partner switching [columns 4+5] accounted for 54% of the intensive margin [column 1] and caused a more than 230% excess reallocation of exports across U.S. buyers beyond the intensive margin. This prevalence of main partner switching is at odds with anonymous market models (perfectly competitive and oligopoly models) and love-of-variety models (the Krugman-Melitz model) including some production networks models (e.g., Bernard et al., 2018). First, as we show in section III, in anonymous markets where firms are indifferent about partners, partner changes should be minimized to save partner-switching costs. Exporters may either add or drop buyers but should not switch among surviving buyers; that is, the excess reallocations in panels A and C should be zero. Second, as appendix D shows, in models combining the love-of-variety model and match-specific fixed costs, firms add and drop marginally important partners rather than main partners. Thus, the large main partner excess export reallocation in panel C is puzzling to these models.

The decompositions in panels A and C show the overall importance of partner switching. To examine the impact of liberalization at the disaggregated level, we regress each margin of the HS six-digit product-level exports on the dummy variable of quota liberalization (the Binding dummy) with the HS two-digit fixed effects. Panels B and D in table 2 report the estimated coefficients. The large and statistically significant coefficients in columns 5–7 in panel B and columns 4–6 in panel D confirm the significant roles of partner switching.

This section develops an exporter-importer matching model in which partner switching is the principal margin of adjustment. Section IIIA sets up the model for the case of one-to-one matching, and section IIIB derives the main insights about partner switching in trade liberalization. Section IIIC introduces many-to-many matching and derives predictions that we take to the data.

The model includes three types of a continuum of firms, namely, U.S. final producers, Mexican suppliers, and Chinese suppliers.¹⁷ U.S. final producers may be retailers or wholesalers. The model has two stages. In Stage 1 a U.S. final producer matches with a supplier from either Mexico or China to form a team that produces one variety of differentiated final goods. Suppliers tailor intermediate goods and transact them only within the team. Firms match under perfect information, and each firm joins only one team. This one-to-one frictionless matching model is the simplest model predicting PAM. Introducing search frictions does not change the qualitative predictions that we take to the data.¹⁸ In Stage 2 teams compete in the U.S. final good market under monopolistic competition.

The team's capability $θ=θ(x,y)$ is increasing in the final producer's capability $x$ and supplier's capability $y$ in that team, that is, $θ1≡∂θ(x,y)/∂x>0$ and $θ2≡∂θ(x,y)/∂y>0$⁠. One finds a fixed mass $MU$ of final producers in the United States, $MM$ of suppliers in Mexico, and $MC$ of suppliers in China. The cumulative distribution function (CDF) for U.S. final producers' capability is $F(x)$ with support $[xmin,xmax]$⁠. For simplicity, a Chinese supplier is a perfect substitute for a Mexican supplier of the same capability. The capability of Mexican and Chinese suppliers follows an identical distribution with the CDF $G(y)$ and support $[ymin,ymax]$⁠.¹⁹

Production technology is of the Leontief type. When a team with capability $θ$ produces $q$ units of final goods, the team supplier produces $q$ units of intermediate goods at costs $cyθβq+fy$⁠; then the final producer assembles these intermediate goods into final goods at costs $cxθβq+fx$⁠, where $ci$ and $fi$ are positive constants (⁠$i=x,y$⁠). The team's total costs are $c(θ,q)=cθβq+f,$ where $c≡cx+cy$ and $f≡fx+fy$⁠. The externalities within teams make firms' marginal costs dependent on both their partner's capability and their own capability.²⁰ For simplicity, we assume that the firm's marginal costs depend on the team's capability. The team's capability $θ$ shifts both demand and marginal costs depending on $α$ and $β$⁠. Therefore, $θ$ may represent productivity (e.g., Melitz, 2003) and/or quality (e.g., Verhoogen, 2008; Baldwin & Harrigan, 2011).

Firms choose their partners and decide how to split team profits, taking $A$ as given. Profit schedules, $πxx$ and $πyy$⁠, and matching functions, $mxx$ and $my(y)$⁠, characterize equilibrium matching.²¹ A final producer with capability $x$ matches with a supplier having capability $mxx$ and receives the residual profit $πxx$ after paying profits $πy(mxx)$ to the partner. $my(y)$ is the inverse function of $mx(x)$⁠, where $mx(my(y))=y$⁠.

The denominator in equation (5) is positive from the second-order condition, and so the sign of $θ12$ is the same as the sign of $mx'(x)$⁠, namely, the sign of sorting in stable matching (e.g., Becker, 1973). For simplicity, we consider three cases in which the sign of $θ12$ is constant for all $x$ and $y$⁠: (1) Case C (Complement) $θ12>0$⁠, (2) Case I (Independent) $θ12=0$⁠, and (3) Case S (Substitute) $θ12<0$⁠.²³ In Case C we have PAM (⁠$mx'(x)>0$⁠): high-capability firms match with high-capability firms, whereas low-capability firms match with low-capability firms. In Case S we have negative assortative matching (⁠$mx'(x)<0$⁠): high-capability firms match with low-capability firms. In Case I we cannot determine a matching pattern (i.e., $mx[x]$ cannot be defined as a function) because each firm is indifferent about partner capability. Therefore, we assume that matching is random and independent of capability in Case I.

Case I is a useful benchmark because it nests two important classes of standard models. The first is anonymous market models in which each firm is indifferent about partner capability. The second is heterogeneous firm trade models with one-sided heterogeneity in which firm heterogeneity exists either among exporters (⁠$θ1=θ12=0$⁠) or among importers (⁠$θ2=θ12=0$⁠). In the following we focus on Case C and Case I in the main text, and we examine Case S in appendix A.3.

This section analyzes the effect of the MFA's end on matching. Motivated by fact 3 shown in section II that new Chinese entrants had different levels of capability, we model the event as an increase in the mass of Chinese suppliers (⁠$dMC>0$⁠). We assume that a firm changes its partner only if it strictly prefers the new match over the current match. We denote the variables and functions before the MFA's end by “B” (before) and variables after the MFA's end by “A” (after).

Figure 2B shows how matching changes from $mxB(x)$ to $mxA(x)$ for the given capability $x$⁠. Area $A$ expresses U.S. importers with capability higher than $x$⁠. They initially match with suppliers in areas $B+C$ that have higher capability than $mxB(x)$⁠. After the MFA's end, the original matches become unstable because some U.S. importers are willing to switch to the new entrants. In the new matching, final producers in area $A$ match with suppliers in areas $B+D$ that have higher capability than $mxA(x)$⁠. A U.S. final producer with capability $x$ switches its main partner from one with capability $mxB(x)$ to one with higher capability, namely,  $mxA(x)$⁠. We call this change “partner upgrading” by U.S. final producers. This in turn implies “partner downgrading” by Mexican suppliers. Mexican suppliers with capability $mxA(x)$ match with final producers with strictly higher capability than $x$ before the MFA's end. Not all Mexican suppliers can match with new partners, however, and those with low capability exit the market, as proven in appendix A.2.

Figure 2C shows that the MFA's end increases the supplier's cutoff from $yLB$ to $yLA$⁠, as proven in appendix A.2. Whether $xL$ increases or decreases is generally ambiguous, and so the figure depicts the case in which $xL$ unchanged. As low-capability suppliers in Area C exit, U.S. importers that matched with them switch to new Chinese suppliers in Area D. Other firms do not change their partners, although they change the price and quantity of goods traded. Firms are indifferent about their partners as long as those partners have a capability level above the cutoffs.

The MFA's end causes two adjustments. First, new Chinese suppliers with high capability enter the market, and Mexican suppliers with low capability exit. This replacement effect occurs in both Cases C and I, and it corresponds to the extensive margin adjustment in table 2. Second, incumbent firms rematch. This rematching effect occurs only in Case C and corresponds to the partner excess reallocation in table 2.

We show that the rematching effect in Case C is a new mechanism of gains from trade that did not exist in standard trade models nested in Case I (perfectly competitive models and Krugman-Melitz models with one-sided heterogeneity). We consider a hypothetical “no-rematching” equilibrium at which firms switch partners only if their current partners exit the market, and we denote variables in this equilibrium by “NR.” The following proposition compares the price indices across the three cases (the proof is in appendix A.2).

The effect of liberalization on the price index $PB-PA$ can be decomposed into the replacement effect $PB-PNR$ and rematching effect $PNR-PA$⁠. The gain from the replacement effect is well known in the heterogeneous firm trade literature. In Case C the rematching effect creates an additional consumer gain. The proof applies a classic theorem in matching theory that stable matching maximizes the aggregate payoff, $AΘ-Mf$⁠, for the given $A$ (Koopmans & Beckmann, 1957; Shapley & Shubik, 1971; Gretsky et al., 1992) and proves that aggregate capability increases as $ΘA>ΘNR>ΘB$⁠.²⁴ In other words, trade liberalization improves consumer welfare by improving global buyer-supplier matching and aggregate capability.

Proposition 1 also implies that a preferential trade agreement can create inefficient “matching diversion.” High-capability U.S. final producers are diverted to match with low-capability Mexican suppliers instead of high-capability Chinese suppliers.²⁵

This section introduces many-to-many matching in an intermediate good market. A final producer produces multiple product varieties, and a supplier owns multiple production lines. Matching occurs between varieties and production lines, resulting in many-to-many matching.

A supplier owns multiple production lines. Each line specializes in a particular variety. A supplier with firm capability $y$ owns $n(y)$ production lines and can match with at most $n(y)$ buyers. One reason for such buyer capacities is a manager's span of control. A supplier requires a manager's resource to collaborate with each buyer. We assume that $n(y)$ is weakly increasing in $y$⁠.

The production line $k$ of supplier $j$ with firm capability $yj$ has line capability $υjk=yj+ɛjk$, where $ɛjk$ is i.i.d. idiosyncratic capability with $Eɛjk=0$ and support $[ɛmin,ɛmax]$⁠. $yj$ and $ɛjk$ are independent. Their marginal densities are $gy(y)$ and $gɛ(ɛ)$⁠, respectively, which are common for both Mexican and Chinese suppliers. We assume that $fη(η)$ and $gɛ(ɛ)$ are log concave.²⁶ In other respects, the model has the same structure as the one in section IIIA.

Because the equilibrium conditions remain the same as in section IIIA the effects of the MFA's end on the matching functions, capability cutoffs, and price indices are qualitatively the same as those in section IIIB. Let $Pt$ (⁠$t∈{A,B,NR}$⁠) be the product-level price indices. Then the following lemma holds with essentially the same proofs as in section IIIB.

We derive the model's predictions of firm-to-firm matching that we take to the data. Our data on Mexico–United States trade record only partner switching by firms engaging in Mexico–United States trade both before and after the MFA's end. We call these firms U.S. continuing importers and Mexican continuing exporters.

We examine a firm's main partner choice because of its importance in our dataset. First, consider Case C. Let $χi*≡xi+maxsηis$ be the highest product capability of final producer $i$⁠. The mean firm capability of final producer $i$'s main partner is $y¯t(χi*)≡E[y|y+ɛ=mχt(χi*)]$ for $t∈{A,B}$⁠. Similarly, the mean firm capability of supplier $j$'s main partner is $x¯t(υj*)≡E[x|x+η=mυt(υj*)]$ for $t∈{A,B}$⁠, where $υj*≡yj+maxkɛjk$⁠. A final producer $i$ upgrades its main partner if $y¯A(χi*)>y¯B(χi*)$⁠, and downgrades if $y¯A(χi*)<y¯B(χi*)$⁠. Similarly, a supplier $j$ upgrades its main partner if $x¯A(υj*)>x¯B(υj*)$⁠, and downgrades if $x¯A(υj*)<x¯B(υj*)$⁠.

As shown in appendix A.2.2, the log concavity of $fη(η)$ and $gɛ(ɛ)$ implies that $E[x|x+η=mυ(υj*)]$ increases in $mυ(υj*)$ and that $E[y|y+ɛ=mχ(χi*)]$ increases in $mχ(χi*)$⁠. Therefore, from Lemma 1, U.S. continuing importers upgrade Mexican main partners, whereas Mexican continuing exporters downgrade U.S. main partners. Another testable implication is that the relative ranking of main partner's firm capability preserves. For each pair of final producers $i$ and $j$⁠, if $y¯B(χi*)>y¯B(χj*)$⁠, then $y¯A(χi*)>y¯A(χj*)$ holds; similarly, for each pair of suppliers $k$ and $h$⁠, if $x¯B(υk*)>x¯B(υh*)$⁠, then $x¯A(υk*)>x¯A(υh*)$ holds. That is, the ranking of new partners' firm capability is positively correlated with the ranking of that of old partners.

In Case I, no systematic partner change occurs. No U.S. continuing importers or Mexican continuing exporters change main partners. The firm capability ranking of new partners is independent of the ranking of old partners. In summary, we establish the following proposition.

We derive the model's predictions of firm exit and the number of partners that holds in both Cases C and I. First, the firm capability cutoff for Mexican suppliers $yL=υL-ɛmax$ increases. Second, from equation (9), the number of partners $NM(x)$ and $nS(y)$ decrease.

To test the predictions in Propositions 2 and 3, we estimate the ranking of firm capability as follows. Let $I(x)$ be the mean imports of the intermediate good by U.S. importers with firm capability $x$ from the main partners and let $X(y)$ be the mean exports by Mexican exporters with firm capability $y$ to the main partners. The following lemma holds from the monotonic relationship between firm capability and within-match trade (the proof is in appendix A.2.3).

For each HS six-digit product, we rank all the U.S. importer and all the Mexican exporters, using their imports and exports of the product from their main partner in 2004, respectively. We use these rankings using 2004 data throughout our sample period (2004–2007) during which the ranking is stable.²⁹ Section VD presents the results using alternative rankings.

We first create three variables using these rankings for each product $g$ in country $c$⁠: (1) firm $i$'s own ranking, $OwnRankigc$⁠, (2) the ranking of the firm's main partner of product $g$ in 2004, $OldPartnerRankigc$⁠, and (3) the ranking of the firm's main partner of product $g$ in 2007, $NewPartnerRankigc$⁠. We choose 2004–2007 as the sample period to avoid potential confounding from the impact of the 2008 financial crisis on Mexican exports. These rankings are standardized using the number of firms to fall into the range of [0, 1]. Smaller rankings indicate higher capability (e.g., first ranking means the best). $OldPartnerRankigc$ differs from $NewPartnerRankigc$ if and only if the firm switches its main partner during 2004–2007. Finally, the partner upgrading dummy $Upigsc$ equals one if $NewPartnerRankigsc<OldPartnerRankigsc$⁠, and the partner downgrading dummy $Downigsc$ equals one if $NewPartnerRankigsc>OldPartnerRankigsc$⁠.

The coefficients of interest $βUc$ and $βDc$ in equation (10) are identified by comparing the treatment and control groups within HS two-digit sectors. The treatment is the removal of binding quotas on Chinese exports to the United States. The coefficients estimate the removal's impact on the probability of partner upgrading and downgrading, respectively. The HS two-digit fixed effects control for basic product characteristics such as textile and apparel and knit and woven.

Prediction C1 for PAM states that at the MFA's end, all the continuing U.S. importers upgrade their main partners, whereas all the continuing Mexican exporters downgrade. Although the frictionless matching model predicts that all the firms will change their partners, in reality, other factors such as transaction costs are likely to prevent some from making such a change, at least in the short run. Accordingly, we reformulate prediction C1 as follows: U.S. importers' partner upgrading and Mexican exporters' partner downgrading will occur more frequently in the treatment group than in the control group, which corresponds to $βUUS>0$⁠, $βDUS=βUMex=0$⁠, and $βDMex>0$ in equation (10).

Prediction I1 for independent matching states that at the MFA's end, no continuing U.S. importer and Mexican exporter would change their partners. In reality, some idiosyncratic shocks appearing as error terms in equation (10) could induce partner changes. Thus, we reformulate prediction I1 as follows: no difference should exist in the probability of partner changes in any direction between the treatment and control groups, which corresponds to $βUUS=βDUS=βUMex=βDMex=0$ in equation (10).

Our regression equation (10) does not suffer from the endogeneity problem that existed in the conventional correlation approach to detecting PAM that regresses an exporter's characteristics on those of an importer. For instance, the cross-sectional regression of an exporter's rank on an importer's rank could produce a mechanical positive correlation regardless of the sign of sorting.³¹ We use firm characteristics (trade volume) only to construct the outcome variables on the left-hand side. Any discrepancy between the true capability ranking and trade ranking should appear in the error terms $ɛUigsc$ and $ɛDigsc$⁠, which might reflect the capability of the firm and its partners, and other unobservable firm and product characteristics. However, as long as the binding dummy is uncorrelated with these unobservables, $βUc$ and $βDc$ are consistently estimated.³²

Another advantage of equation (10) is controlling for the various unobservable determinants of a firm's partner rankings. First, idiosyncratic shocks to demand and cost may change firm capability and generate partner switching. As long as these shocks appearing as error terms in equation (10) are uncorrelated to the MFA liberalization, they should not bias our estimates. Second, the dependent variables are constructed from time differences in partner rankings. Time differencing controls for all the time-invariant firm-specific determinants of the level of partner rankings.

Prediction C2 predicts $γc>0$⁠, and prediction I2 predicts $γc=0$⁠.

Two additional points need to be mentioned. First, if we run equation (11) only for firms that do not change partners, then $γc$ equals one by construction. To avoid this mechanical correlation, we estimate equation (11) only for firms that change partners. Second, the regression equation (11) combines both the treatment and the control groups since prediction C2 should hold for both groups in Case C.³³

For surviving exporters in the final year of period $r$⁠, the minimum of their exports of product $g$ in the initial year of period $r$ proxies for the capability cutoff, $ExportCutoffgsr$⁠. Since importer information is unavailable before 2004, we use Mexican exporters' product exports as the capability proxy, which is highly correlated with exports to the main partners in the 2004–2007 data. $Afterr$ is an indicator of whether period $r$ is 2004–2007, $λs$ represents the HS two-digit-level fixed effects, and $uigsc$ are the error terms.

We use the difference-in-difference specification to test the predictions about the cutoff changes. In equation (12), the cutoff increase in prediction E1 implies $δ2>0$ as the coefficient of interest. On the contrary, $δ1$ estimates the difference in the levels of the cutoffs between the liberalized and nonliberalized products. We perform a placebo check of no difference in the prior trends in the cutoffs by estimating equation equation (12) for the two preliberalization periods (1998–2001 and 2001–2004).

where $Δ#Partnersigsc$ is the changes in the number of firm $i$'s partners in product $g$ during 2004–2007, $λs$ represents the HS two-digit-level fixed effects, and $ɛigsc$ are the error terms. Prediction E2 implies $ζ1Mex<0$ and $ζ1US<0$⁠.

Panel A in table 3 examines partner changes during 2004–2007 using linear probability models.³⁶ The columns with odd numbers report the estimates of $βdc$ (⁠$c=US,Mex$ and $d=U,D$⁠) from the baseline regressions equation (10). We find that $βUUS$ in column 1 and $βDMex$ in column 7 are positive and statistically significant, while $βDUS$ in column 3 and $βUMex$ in column 5 are close to and not statistically different from zero. These signs of $βdc$ support Case C and reject Case I. The removal of binding quotas from Chinese exports increased the probability of U.S. importers' partner upgrading by 5.2 percentage points and the probability of Mexican exporters' partner downgrading by 12.7 percentage points.³⁷ These effects are quantitatively large compared with the sample averages of $UpigsUS$ and $DownigsMex$⁠, which are three and fifteen percentage points, respectively.³⁸

Panel B in table 3 examines partner changes in the later periods of 2007–2011 and 2009–2011 to check our assumption that both the treatment and the control groups exhibit similar partner change patterns if the treatment is absent.⁴⁰ For each period, we reconstruct the capability rankings based on trade in the new initial years and re-create the upgrading and downgrading dummies. If the transition from the old to the new equilibrium was largely completed by 2007, we should observe no difference in partner changes between the two groups. Small and insignificant estimates for $βUUS$ and $βDMex$ in 2007–2011 and 2009–2011 support our assumption.⁴¹

We conduct numerous robustness checks in appendix E.3. First, we include as additional controls several product-level and firm-product-level characteristics that statistically differ between the treatment and control groups.⁴² Second, we conduct three exercises to address potential within-firm interactions in firms trading multiple products and firms with multiple partners. We add the number of products that a firm trades and its interaction with the binding dummy, address the case that the main partner switching occurs within initial partners, and distinguishing firms that had a single partner and those that had multiple partners. Finally, we adopt alternative variable definitions. We define partner switching using rank bins, define quota binding under alternative criteria, and use alternative year windows. Our results are robust to all of these alternatives.⁴³

Table 4 reports the tests of prediction E1. Column 1 reports the baseline specification of product-level regression equation (12), and column 2 includes as additional control variables the product characteristics for the initial year in each period and their interactions with the after dummy. These controls, when available, are the same as in footnote 42.⁴⁴ The estimates of the positive and significant $δ2$ confirm the prediction that the MFA's end increased the capability cutoff for Mexican exporters. Column 5 reports the baseline specification of firm-level regression (13), and column 6 includes the product characteristic variables and their interactions with the after dummy. The estimates of the positive and significant $δ2$ confirm that the MFA's end increased their exit probability for a given capability level. In addition, the negative estimates of $δ4$ and $δ4+δ5$ confirm that small exporters are more likely to exit the market.