Goods-Market Frictions and International Trade

K.7 Goods-Market Frictions and International Trade Krolikowski, Pawel M. and Andrew H. McCallum Please cite paper as: Krolikowski, Pawel M. and Andrew H. McCallum (2017). Goods-Market Frictions and International Trade. International Finance Discussion Papers 1207. https://doi.org/10.17016/IFDP.2017.1207 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1207 May 2017

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1207 May 2017 Goods-Market Frictions and International Trade Pawel M. Krolikowski and Andrew H. McCallum NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at https://www.federalreserve.gov/econres/ifdp/. This paper can be downloaded without charge from Social Science Research Network electronic library at http://www.sssrn.com.

Goods-Market Frictions and ∗ International Trade Pawel M. Krolikowski † Andrew H. McCallum ‡ First draft: September 18, 2013 This draft: May 23, 2017 Abstract We present a tractable framework that embeds goods-market frictions in a general equilibrium dynamic model with heterogeneous exporters and identical importers. These frictions arise because it is time consuming and expensive for exporters and importers to meet. We show that search frictions lead to an endogenous fraction of unmatched exporters, alter the gains from trade, endogenize entry costs, and imply that the competitive equilibrium does not generally result in the socially optimal number of searching firms. Finally, ignoring search frictions results in biased estimates of the effect of tariffs on trade flows. JEL codes: D83, F12. Keywords: Search, trade, goods, frictions, information ∗We thank Treb Allen, Pedro Amaral, Daniel Carroll, Alan Deardorff, Michael Elsby, Bruce Fallick, Jeremy Fox, Aspen Gorry, Gordon Hanson, Chris House, Andrei Levchenko, Roberto Pinheiro, James Rauch, Jagadeesh Sivadasan, Dmitriy Stolyarov, and Christian vom Lehn for helpful advice. We also thank seminar participants from the 2017 AEA meetings, Federal Reserve Bank of Cleveland, the Federal Reserve Board, the Center for Economic Studies at the Census Bureau, the University of Michigan, Utah State University, and Brigham Young University. Because this project began as McCallum’s third dissertation chapter, we also thank the many Michigan students who provided helpful comments in its earliest stages. Jessica C. Liu, Tessa A. Morrison, and Victoria Perez-Zetune provided excellent research assistance. Krolikowski and McCallum acknowledge generous financial support from the Roosa Dissertation Fellowship in Monetary Economics and the Federal Reserve Board Dissertation Internship, respectively. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the U.S. Census Bureau, the Board of Governors, the Federal Reserve Bank of Cleveland or of any other person associated with the Federal Reserve System. All results have been reviewed to ensure that no confidential information is disclosed. Any errors are our own. †Federal Reserve Bank of Cleveland, 1455 E 6th St, Cleveland, OH 44114, pawel.krolikowski@clev.frb.org ‡Board of Governors of the Federal Reserve System, Washington, DC 20551, andrew.h.mccallum@frb.gov 1

1 Introduction Locating and building connections with overseas buyers is a prevalent and costly barrier to exporting. Surveying firms in the United Kingdom, Kneller and Pisu (2011) find that ‘‘identifying the first contact’’ and ‘‘establishing initial dialogue’’ are more common obstacles to exporting than ‘‘language barriers,’’ ‘‘cultural differences,’’ or ‘‘dealing with legal, financial and tax regulations overseas.’’ Moreover, exporting firms pursue a number of costly activities to overcome these barriers. Eaton, Eslava, Jinkins, Krizan, and Tybout (2014) report that the four most expensive costs for Colombian exporters (in order) are maintaining foreign sales offices, sending sales representatives abroad, researching potential foreign buyers, and establishing a web presence. Likewise, Allen (2014) shows that it is costly for agricultural producers to gather information about market conditions across regions in the Philippines and that these information barriers alter trade flows. In this paper, we formalize this costly search process for international partners as a goods-market friction between importing retailers and exporting producers. Our search theoretic framework is motivated by a literature that documents the important role these frictions play in input markets and most closely resembles Pissarides (2000, Ch. 1). We embed these frictions into a general equilibrium model with heterogeneous producers and identical retailers in the style of Hopenhayn (1992) and Melitz (2003). Combining these two canonical models provides a tractable framework that has rich implications for individual trading relationships and economic aggregates. This framework is also consistent with the empirical relevance of international intermediaries that move goods from producers to final consumers, as documented by Bernard, Jensen, Redding, and Schott (2010) and Ahn, Khandelwal, and Wei (2011). To obtain general results about the macroeconomy, we first analyze a closed version of the model. Our main application, however, considers an open economy, as searching across international borders may be particularly expensive. 2

We begin by pointing out that any search model leads to a concept of unmatched agents in the aggregate because there always exists a mass of agents that are actively looking for partners but have yet to match. In our model, the ‘‘unmatched rate’’ of producers is the fraction of producers that are actively looking for retailers; it is analogous to the unemployment rate of workers in labor economics. This feature sets our work apart from papers in the style of Melitz (2003), in which every firm that chooses to export finds a buyer, but our model nests the standard trade model when we remove the search friction. The unmatched rate of producers is important because the associated product varieties cannot be consumed and are therefore absent from the indirect utility (welfare) function, price index, and other aggregates. We show that the absence of these varieties has first-order implications for welfare responses because any shock that changes the unmatched rate will affect welfare through this new channel. This result generalizes the findings of Arkolakis, Costinot, and Rodr´ıguez-Clare (2012). The unmatched rate depends endogenously on a search-market sufficient statistic called ‘‘market tightness,’’ which is defined as the ratio of searching retailers to searching producers. Market tightness determines the expected number of contacts between producers and retailers each instant of time, also known as the ‘‘contact rate.’’ In concurrent related work, Eaton et al. (2014) and Eaton, Jinkins, Tybout, and Xu (2016) also have endogenous contact rates but in the context of a more complex search model that includes, among other features, many-to-many matches and learning about foreign markets. Our model presents a more stylized view of the search process and, with our simplifications, admits straightforward aggregation. The tractability these simplifications afford allows us to analytically show how changes to exogenous variables affect endogenous market tightness, contact rates, the unmatched rate of producers, and welfare. In particular, if unmatched rates are exogenous, then the response of welfare to any foreign shock remains the same as in Arkolakis et al. (2012). 3

We also find that, with endogenous market tightness, goods-market frictions change the elasticity of domestic consumption shares to iceberg trade costs. We argue that the elasticity in our model with market tightness is always at least as negative as the elasticity in a model without goods-market frictions because search frictions magnify the response of trade to changes in iceberg costs. Unsurprisingly, goods-market frictions reduce aggregate trade flows relative to a model without them, but they do so through both prices and the fraction of unmatched producers. In particular, the equilibrium import price paid by the retailer and negotiated with the exporting producer is lower than the final sales price paid by consumers. This ensures that the importer can at least cover their search costs and is consistent with the empirical findings of Berger, Faust, Rogers, and Steverson (2012). This pricing approach, among other model features, is similar to that of Drozd and Nosal (2012), who use a trade model with search frictions to account for several pricing puzzles of international macroeconomics. These lower import prices in our framework reduce the value of imports relative to typical models of trade, which provide aggregate imports evaluated at final sales prices. The unmatched rate also reduces aggregate imports because a fraction of producers are not matched to importing retailers, thereby reducing the number of varieties traded in equilibrium. Benguria (2015) has a similar feature, and, as in that paper, we show that the goods-market friction provides a micro foundation for the cost associated with entering foreign markets. Moreover, our model predictions about the extensive margin differ from standard trade models because, for example, some highly-productive firms are endogenously unmatched. In this way, we provide a search theoretic explanation for what Armenter and Koren (2014) refer to as ‘‘balls-and-bins’’ facts about the extensive margin of trade. In particular, our model plainly shows how search frictions affect the fraction of firms that export. We point out that the quantity traded (intensive margin) between matched retailers and producers in our model is the same as in a model without search because the two parties 4

still seek to maximize profits earned from consumers. Our model is tractable enough that, without assuming particular functional forms or parameter values, we can compute comparative statics and provide novel search-theoretic micro foundations for several empirical approaches used by previous authors. For example, our simple closed-form gravity equation is consistent with estimates from Rauch and Trindade (2002) that a larger ethnic Chinese population within a country reduces goods-market frictions and increases bilateral trade flows between Southeast Asian countries. Similarly, Portes and Rey (2005) find that bilateral telephone traffic and the number of bank subsidiaries, which may proxy for search costs, improve information flow between countries, significantly increasing trade flows in a standard gravity equation regression. Resembling Rauch (1999), our derivations provide a justification for including distance and measures of language similarity when estimating empirical gravity equations. The tractability of the model also allows us to show that the competitive equilibrium with goods-market frictions, endogenous market tightness, and our negotiating protocol is inefficient relative to the social optimum. This inefficiency arises because the competitive equilibrium features two externalities that work in opposite directions. On the one hand, individual producers (retailers) ignore the positive market thickness externality they create for retailers (producers) when they search. On the other hand, individual producers (retailers) ignore the negative congestion externality they impose on other producers (retailers) by searching. As a result, the competitive equilibrium may have too many or two few producers (retailers) searching relative to the socially optimal level. To our knowledge, this observation has not been made before in the context of international trade. Moreover, the well-known Hosios (1990) condition from the labor search literature is, in general, insufficient to guarantee efficiency in the competitive equilibrium of our model. We briefly mention other important empirical and theoretical work related to goods-market frictions. Specifically, Rauch (1996) uses a model of search to explain multiple 5

phenomena, including the existence of large Japanese intermediate general trading companies that match producers and consumers without producing any goods themselves. Cadot, Iacovone, Pierola, and Rauch (2011); Egan and Mody (1992); Monarch and Schmidt-Eisenlohr (2015); and Rauch and Watson (2003) discuss search frictions as they relate to foreign-market entry and importer–exporter relationships. Heise (2016) highlights the importance of firm-to-firm relationships. Goods-market frictions have also recently been used outside of international trade to help explain aggregate dynamics. For example, Petrosky-Nadeau and Wasmer (2015) investigate the ability of goods- and credit-market frictions to generate persistence in labor market responses to productivity shocks, while Bai, R´ıos-Rull, and Storesletten (2012) develop a dynamic stochastic general equilibrium (DSGE) model that features search frictions between consumers and producers. Gourio and Rudanko (2014) study how goods-market frictions affect firm-level variables like investment, profits, and sales, and den Haan (2013) assesses how goods-market frictions can explain inventory dynamics over the business cycle. We think that goods-market frictions may be informative for the dynamics of aggregate trade, although this is a topic we leave to future research. The remainder of the paper is organized as follows. Section 2 introduces the closed-economy model. Section 3 presents comparative statics and analyzes the efficiency of the competitive equilibrium. Section 4 opens the economy to trade. Specifically, section 4.3 analyzes the welfare implications and derives the gravity equation implied by our framework. Lastly, section 5 presents a discussion of further research. 6

2 Searching for business affiliates 2.1 Model introduction The work on search and matching by Diamond (1982), Mortensen (1986), and Pissarides (1985) underpins the process by which producers and retailers find one another. Our model is formulated in continuous time and we focus on steady-state implications, thereby ignoring any transition dynamics. We are motivated by the facts summarized in Bartelsman and Doms (2000) and Syverson (2011): Even within similar industries, firms exhibit persistent differences in measured productivity. We incorporate this firm heterogeneity as in Hopenhayn (1992) and Melitz (2003) and index producers by their productivity, ϕ. This permanent productivity is exogenously given and known to producers. Retailers are ex-ante homogeneous but ex-post differentiated, depending on the productivity of the producing partner with whom they match. Producers and retailers make optimal decisions to search by using backward induction knowing that upon meeting, they will bargain continuously over the price, n(ϕ), and the quantity, q(ϕ), of the good to be exchanged. When bargaining, both parties know that retailers alone can access consumers and that they can sell the product to these consumers at a potentially different final price, p(ϕ). Business relationships end at an exogenous rate, λ, that is the same for all matches, regardless of the producer’s productivity level. In the remainder of this section, we focus on a closed-economy setting to derive results under general conditions. In section 4, we present an open-economy version of the model with particular functional form assumptions and derive closed-form expressions for key objects of interest, like the gravity equation, trade elasticity, and the welfare response to foreign shocks. 7

2.2 Matching function The matching function, denoted by m(uNx,vNm), gives the flow number of relationships formed at any moment in time as a function of the stock number of unmatched producers, uNx, and unmatched retailers, vNm. Nx and Nm represent the total mass of producing and retailing firms, respectively, that exist regardless of their match status. The fraction of producers looking for retailers is u, while the fraction of producers that choose not to search, and therefore remain idle, is denoted by i. The fraction of retailers that are searching for producing firms is v. We assume that the matching function is homogeneous of degree one, strictly increasing in both arguments, and concave. Due to the homogeneity assumption on the matching function, market tightness, κ = vNm/uNx, which is the ratio of the mass of searching retailers to the mass of producers, is sufficient to determine contact rates on both sides of the market. We use continuous time Poisson processes to model the random matching of retailers and producers. As such, the contact rate defines the average number of counterparty meetings during one unit of time. The rate at which retailers contact producers, χ(κ), is the number of matches in a period over the number of searching retailers: m(uNx,vNm) (cid:18) uNx (cid:19) (cid:18) 1 (cid:19) χ(κ) = = m ,1 = m ,1 . vNm vNm κ Similarly, the average rate at which producers contact retailers, κχ(κ), is the number of matches in a period over the number of searching producers: m(uNx,vNm) (cid:18) vNm(cid:19) κχ(κ) = = m 1, = m(1,κ). uNx uNx Another feature of Poisson processes is that the inverse of the contact rate is the average units of time between counterparty meetings. For example, if the units of time in the model were years, then retailers would wait an average of 1/χ(κ) years before meeting a producer. 8

Appendix C.1 contains more details. The matching function is strictly increasing in both arguments, which captures the fact that more matches are made if there are either more searching retailers or more searching producers. It also means that, because of congestion effects, the retailers’ contact rate falls with market tightness (χ(cid:48)(κ) < 0) and, because of market thickness effects, the producers’ contact rate rises with market tightness (dκχ(κ)/dκ > 0). We will refer to increasing κ as a ‘‘tighter’’ market and decreasing κ as a ‘‘looser’’ market. Market tightness is defined from the perspective of the producer, so when the market is tight, it is easy for a producer to find a retailer. Also, it will be useful to denote the elasticity of the matching function with respect to dln(m(uNx,vNm)) the number of unmatched producers, uNx, as = η(κ). This elasticity is dln(uNx) also the (negative of the) elasticity of the retailers’ finding rate with respect to market κχ(cid:48)(κ) tightness, η(κ) ≡ − , because of the homogeneity of the matching function. The χ(κ) elasticity captures the fact that when the number of searching retailers increases 1 percent more than the number of searching producers, the expected number of contacts that a retailer makes within each time period declines by η percent because of congestion. 2.3 Producers The value of a producer with productivity ϕ being matched to a retailer, X(ϕ), can be summarized by a value function in continuous time: rX(ϕ) = nq −t(q,ϕ,w,τ)−f +λ(U (ϕ)−X(ϕ)), (1) This asset equation states that the flow return at the risk-free rate, r, from the value of producing must equal the flow payoff plus the expected capital gain from operating as an exporting producer. Each producer is indexed by exogenous productivity, ϕ, with 9

cumulative density function G(ϕ) and probability density function g(ϕ), which are defined over ϕ. The flow payoff consists of nq, the revenue obtained from selling q units of the good at price n to the retailer, less the variable, t(q,ϕ,w,τ), and fixed cost of production, f. Here t(·) is an arbitrary variable cost function that depends on the level of output, q, exogenous productivity, ϕ, and input prices, w, as well as an iceberg trade cost, τ. The last term in equation (1) captures the event of a dissolution of the match, which occurs at exogenous rate λ and leads to a capital loss of U (ϕ)−X(ϕ). Here U (ϕ) represents the producer’s value of operating without a retailer and will be described in more detail shortly. In writing equation (1), we explicitly write the value X(ϕ) as a function of the producer’s productivity, ϕ. We do this to remind the reader that each endogenous term is a function of this productivity, but we conserve on notation by omitting this argument from the negotiated price, n, and traded quantity, q. Although in principle producers could circumvent retailers and contact final consumers directly, we avoid this possibility by assuming that the net value of matching with a retailer is always greater than the net value of forming a relationship directly with a final consumer. This approach is similar to earlier work by Wong and Wright (2014) who assume that a middleman is necessary rather than deriving the conditions under which this is the case. There are a few potential ways to motivate this assumption. One is that retailers specialize in reaching and serving customers and therefore have lower costs of doing so. A second is that households maximize utility in a second stage and procure the quantity desired to consume in a searching first stage so that consumers are allowed to contact producers directly. With this approach, we suspect that many of our key results about the implications of search for international trade in section 4 would go through. The value that an unmatched producing firm gets from looking for a retail partner without being in a business relationship, U (ϕ), satisfies rU (ϕ) = −l+κχ(κ)(X(ϕ)−U (ϕ)−s). (2) 10

The flow search cost, l, is what the producer pays when looking for a retailer; it captures the costs we highlighted in the introduction — namely, maintaining foreign sales offices, sending sales representatives abroad, researching potential foreign buyers, and establishing a web presence. The second term captures the expected capital gain, in which κχ(κ) is the endogenous rate at which producing firms contact retailers, and s is the sunk cost of starting up the relationship. As discussed above, market tightness is denoted by κ. The producing firm also has the option of remaining idle and not expending resources to look for a retailer. For the producer, the value of not searching, I(ϕ), satisfies rI(ϕ) = h. (3) Producers can always choose this outside option and not search for retailers. Idle firms in this context are analogous to workers who are out of the labor force. Choosing to remain idle provides the flow payoff, h. The value to a producer of remaining idle can be interpreted, for example, as the value of the stream of payments after liquidation or the flow payoff from home production if these firms are viewed as entrepreneurs. Producing firms will choose to remain idle if the value of searching, U (ϕ), is less than the value of remaining idle, I(ϕ). Generally, producers desire to match with a retailer and reach consumers, but doing so comes at the cost of searching. As such, some producers with low productivity, ϕ, may not find it optimal to engage in search. Importantly, if l is large enough relative to h, remaining idle may be optimal for many producers. 2.4 Retailers All retailers are ex-ante identical but have values that vary ex-post only because producers are heterogeneous. The value of a retailing firm being in a business relationship with a 11

producer of productivity ϕ, is defined by the asset equation, rM (ϕ) = p(q,Y,P)q −nq +λ(V −M (ϕ)). (4) The flow payoff from being in a relationship is the revenue generated by selling q units of the product to a representative consumer at a final sales price, p, determined by their inverse demand curve p(q,Y,P), less the cost of acquiring these goods from producers at negotiated price n. Consumer demand is a function of income, Y, and all goods prices, P. To highlight other aspects of the search process, we assume the retailer does not use the product as an input in another stage of production but only facilitates the match between producers and consumers. When we solve the model in section 2.5 (specifically, section 2.5.4), we show that including an additional intermediate input does not substantively affect our main conclusions. In the event that the relationship undergoes an exogenous separation, the retailing firm loses the capital value of being matched, V −M (ϕ). The value of being an unmatched retailer, V, satisfies (cid:90) rV = −c+χ(κ) [max{V,M (ϕ)}−V]dG(ϕ). (5) The retailer needs to pay a flow cost, c, to search for a producing affiliate. At Poisson rate χ(κ), the retailing firm meets a producer of unknown productivity. Because the retailer does not know the productivity of the producer it will meet, it takes the expectation over all productivities it might encounter when computing the expected continuation value of searching. As a result, the value, V, is not a function of the producer’s productivity, ϕ, but rather a function of the expected payoff. We assume that upon meeting, but before consummating a match, the retailer learns the productivity of the producer. In this sense, matches are inspection goods or search goods, as opposed to experience goods. Upon meeting, and depending on the producer’s productivity, ϕ, the retailer chooses between 12

matching with that producer, which generates value M (ϕ), and continuing the search, which is associated with value V. Hence, the capital gain to a retailer from meeting a producer with productivity ϕ can be expressed as max{V,M (ϕ)}−V, in which the retailer maximizes over accepting a match and continuing to search. We assume free entry into retailing so that in equilibrium, the value of being unmatched, V, is driven to zero. The ability to expand retail shelf space or post a product online until it is no longer valuable to do so provides an intuitive basis for this assumption. Because retailers do not generate any revenue or incur any cost before entering the state of searching, V, we also assume that, unlike their producing counterparts’, their value of remaining idle is zero. Hence, on the retailer’s side of the market, searching and not searching both have a zero value in equilibrium, allowing retailers simply to compare M (ϕ) and V. 2.5 Solution to the search model 2.5.1 Overview The retailing and producing firms use backward induction to maximize their value. The second stage is the solution that results from bargaining over price and quantity after a retailer and producer meet and decide to match. We solve this bargaining problem in sections 2.5.2 through 2.5.4. In the first stage, the retailer and the producer take as given the solution to this second-stage bargaining problem and this optimal behavior defines conditions under which they will consummate a match upon meeting and whether they will search for a business partner in the first place. Because producers are heterogeneous, the decision depends on productivity. Section 2.5.5 shows that there is a minimum productivity threshold, akin to the entry condition defined in Melitz (2003), that makes searching worthwhile. In sections 2.5.6 and 2.5.7, we present the two conditions that jointly characterize retailers’ decisions to search and define market-tightness and the negotiated price. Finally, in steady state, there 13

exists a set of unmatched producers that are actively looking for a retail partner and unmatched retailers that are actively looking for a producer. We define these concepts in section 2.5.8. 2.5.2 Continuous Nash bargaining over price and quantity Upon meeting, the retailer and producer bargain over the negotiated price and quantity simultaneously. We assume that these objects are determined by the generalized Nash bargaining solution, which, as shown by Nash (1950) and Osborne and Rubinstein (1990), is equivalent to maximizing the following Nash product: max[X(ϕ)−U (ϕ)]β[M (ϕ)−V]1−β,0 ≤ β < 1, (6) q,n in which β is the producer’s bargaining power. The total surplus created by a match, which is the value of the relationship to the retailer and the producer less their outside options, is S(ϕ) = M (ϕ)−V +X(ϕ)−U (ϕ). In appendix A.1, we show this to be pq −t(q)−f +l+sκχ(κ) S(ϕ) = . (7) r+λ+βκχ(κ) In appendix A.1, we also derive the value of a relationship, R(ϕ), in terms of model primitives. 2.5.3 Bargaining over price Producers and retailers bargaining over the negotiated price, n(ϕ), results in equations that divide the total surplus created by a match between the parties according to X(ϕ)−U (ϕ) = βS(ϕ) (8) M (ϕ)−V = (1−β)S(ϕ). 14

Here the producer receives β of the total surplus, while the retailer receives the remainder. As such, we refer to this expression as the ‘‘surplus sharing rule.’’ We have relegated the details regarding the derivation of equation (8) to appendix A.2.1. We also point out that the reasoning behind the restriction that β < 1 in equation (6) is evident in equation (8). Retailing firms have no incentive to search if β = 1, as they get none of the resulting match surplus and therefore cannot recoup search costs c > 0. Any solution to the model with c > 0 and positive trade between retailers and producers also requires β < 1. Appendix A.2.1 provides greater detail. 2.5.4 Bargaining over quantity In appendix A.2.2, we show that bargaining over quantity, q, together with equation (8), yields ∂ ∂ p(q,Y,P)+ p(q,Y,P)q = t(q,ϕ,w,τ). (9) ∂q ∂q The quantity exchanged within matches, q, equates marginal revenue obtained by the retailer with the marginal production cost. This quantity is the same that we would get from a model without search frictions and therefore implies that adding search (as we do) does not change the quantity exchanged within matches. The quantity depends on consumers’ demand curve p(q,Y,P), the pricing power of retailers, and the production cost function t(q,ϕ,w,τ). We also show in appendix A.3 that including an additional input in the retailer’s production function does not change this result. Nevertheless, although the quantity exchanged does not depend on search frictions, these frictions do affect the mass of matches formed. We turn to this topic in the next section. 2.5.5 Productivity thresholds Given the outcome of the bargaining stage in the previous section, we next define conditions when retailers and producers will consummate a match upon meeting and whether they will 15

search for a business partner in the first place. Because producers differ by productivity, these conditions lead to two productivity thresholds. The lower threshold, denoted by ϕ, makes the producer and retailer indifferent between ¯ (cid:0) (cid:1) (cid:0) (cid:1) consummating a relationship upon contact and continuing to search, X ϕ −U ϕ = 0. ¯ ¯ Equation (8) implies that an identical productivity threshold could be obtained by considering the lowest productivity that would leave a retailer indifferent between matching (cid:0) (cid:1) and continuing to search for a different producer, namely M ϕ −V = 0. ¯ The upper threshold, denoted by ϕ¯, makes the producer indifferent between searching and remaining idle, so U (ϕ¯)−I(ϕ¯) = 0. By using equations (2) and (3), this can be rewritten as l+h X(ϕ¯)−U (ϕ¯) = +s. (10) κχ(κ) The equation states that the productivity necessary to induce a producer to search for a retailer equates the capital gain from forming a relationship with the expected cost of matching with a retailer. This cost includes the opportunity cost of remaining idle, h, the search cost, l, the rate κχ(κ) at which producers meet retailers, and the sunk cost of forming a relationship, s. We derive both thresholds in appendix A.4 and show that ϕ¯ > ϕ, if l+h+κχ(κ)s > 0. ¯ Intuitively, if a retailer and producer make contact, then search costs have already been paid and productivities ϕ > ϕ result in the match being consummated. Relative to producers who ¯ have already contacted retailers, producers that have not made contact with a retailer yet must incur additional search costs; this further discourages some lower-productivity producers from searching in the first place. As a result, the productivity required to engage in search exceeds the productivity required to consummate a match, that is, ϕ¯ > ϕ. ¯ We can translate equation (10) into an expression akin to the entry condition defined in Melitz (2003) even though the retailer earns pq revenue from the consumer and the producer pays the production cost. Our condition defines a threshold productivity that ensures that 16

total flow profits cover what we call the ‘‘effective entry cost,’’ which is the fixed cost of production, f, and the (appropriately discounted) flow cost of searching for a partner, l, the opportunity cost of remaining idle, h, and the sunk cost of starting up a business relationship, s. Definition 1. The effective entry cost, F (κ), is defined as (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) r+λ r+λ r+λ F (κ) ≡ f + l+ 1+ h+ s. βκχ(κ) βκχ(κ) β Proposition 1. The binding threshold productivity, ϕ¯, is determined by the implicit function π(ϕ¯,Y,P) = F (κ), (11) in which π(ϕ¯,Y,P) ≡ p(q(ϕ¯),Y,P)q(ϕ¯)−t(q(ϕ¯),w,ϕ¯,τ). Proof. See appendix A.4.2. This proposition implies that the effective entry cost, and therefore the threshold productivity, depends endogenously on the producers’ finding rate κχ(κ). Remember that κ = vNm/uNx so that as the number of searching retailers increases (or searching producers decreases), it becomes easier for a producer to meet a retailer. Intuitively, higher κ reduces the time spent searching and, along with it, the effective entry cost. Related to this, if the producers’ finding rate is exogenous, Proposition 1 provides a novel micro-level interpretation of the effective entry cost, but this cost remains a combination of exogenous parameters. Benguria (2015) makes a closely related point. Another innovation of our model is that the opportunity cost of remaining idle, h, is an important determinant of the productivity threshold and the fraction of active producers. As pointed out by Armenter and Koren (2014), the fraction of exporting firms is an important moment for parameter identification and one that has been exploited by Eaton et al. (2014) 17

and Eaton et al. (2016), among others. Allowing for the possibility that producers optimally choose not to search could change the estimates in these important papers. Another novel implication of our model is that the bargaining power, β, and the match destruction rate, λ, are determinants of the effective entry cost. Finally, Proposition 1 nests the conditions defining the threshold productivity in many trade models. In particular, with l = 0, h = −l, and s = 0, we recover the equation defining the productivity threshold in Melitz (2003). We present evidence in appendix A.4.3 on why, even though in this case it amounts to the same result, the standard trade model features h = −l instead of what one might think is the intuitive value of h = 0. In that appendix, we also relate Proposition 1 to expressions in other standard frameworks. 2.5.6 Retailer search Here we specify the conditions under which unmatched retailers search in order to match with producers. Using equation (5) together with our assumption of free entry into the market of unmatched retailers, V = 0, implies that (cid:90) c = M (ϕ)dG(ϕ). (12) χ(κ) ϕ¯ This equation defines the equilibrium market tightness, κ, that equates the expected cost of being an unmatched retailer, on the left, with the expected benefit from matching, on the right. In defining equation (12), we removed the maximum over V and M (ϕ) from equation (5) and simply integrated from the threshold productivity level defined by equation (10). This simplification is possible as long as M(ϕ) is strictly increasing in ϕ so that the ex-post value of being matched is strictly increasing in the producer’s productivity. In appendix A.5, we provide further details. It is worth emphasizing that equation (12) does not inform the binding productivity threshold ϕ¯, which is solely determined by Proposition 1. Details regarding this issue are in section 2.5.5. 18

To get intuition from equation (12), notice that as the expected benefit (the right-hand side) from retailing rises, free entry implies that retailers enter the search market, which raises market tightness, κ = vNm/uNx, reduces the rate at which searching retailers contact searching producers, χ(κ), and increases retailers expected cost of search (the left-hand side). Hence, free entry ensures that V = 0 at all times and that κ always satisfies equation (12). Equation (12) highlights that the retailers’ cost of searching for producers, c, along with our assumption of free entry into retailing is at the heart of our model. If searching for producers was free (c = 0) but matching was associated with positive expected payoff, then free entry would lead to an infinite number of retailers in the economy driving the producer’s finding rate to infinity and relieving the search friction. Proposition 2. With free entry into retailer search, market tightness, κ, is finite if and only if the retailers’ search cost, c, is positive. Proof. See appendix A.6. Free entry also interacts with assumptions for how firms of both types come into existence. We describe in detail those assumptions in appendix A.7, showing in appendix A.7.1 that for retailers, free entry into search implies free entry into existence. We also consider the alternative assumption of free entry into search for producers and show that it yields additional restrictions on equilibrium market tightness. Appendix A.7.2 includes further details. We find our baseline approach of setting V = 0 to be a natural starting point, but other approaches lead to similar effects of search frictions, and the major implications of our paper remain the same. 2.5.7 Negotiated price curve Along with the condition for market tightness in equation (12), there is an expression that determines the negotiated price, n, at which producers and retailers exchange the good after a match has been consummated. 19

Proposition 3. The negotiated price, n, at which the producer sells his good to the retailer satisfies t(q,ϕ,w,τ)+f −l−κχ(κ)s n = [1−γ]p(q,Y,P)+γ , (13) q (r+λ)(1−β) in which γ ≡ ∈ [0,1]. r+λ+βκχ(κ) Proof. Use V = 0, equations (1) and (2) along with the surplus sharing rule defined by equation (8). See appendix A.8 for a derivation of equation (13) and appendix A.9 for a proof that γ ∈ [0,1]. We remind the reader that equation (13) is a function of the producer’s productivity, ϕ, but we have not written it as such to conserve on notation. The equilibrium negotiated price is a convex combination of the final sales price and the average total production cost less the producer’s search costs. A price outside of this range would be unsustainable. The highest negotiated price, n, that the retailer is willing to pay is the final sales price, p(q,Y,P), and the lowest negotiated price that the producer is willing to accept is the average total production cost, (t(q,ϕ,w,τ)+f)/q, net of the cost of looking for a retailer, l, and the expected sunk cost, κχ(κ)s. The search costs of the producer, l and s, enter negatively in equation (13) because they erode the producer’s bargaining position and thereby allow the retailer to negotiate a lower transaction price. The negotiated price also depends on the bargaining power and the finding rate of the producer. As the producer gains all the bargaining power (β → 1), then γ → 0 and n → p, so the producer takes all the profits from the business relationship. Similarly, if producers find retailers immediately (no search friction) so that the finding rate κχ(κ) → ∞ and the sunk cost, s, is set to zero, then the negotiated price also converges to the final sales price, n → p. We provide details in appendix A.8. Importantly, the case in which n → p recovers the standard trade model, as there is, in effect, no intermediate retailer; producers can be seen as selling their goods directly to the final consumer at the price p(q,Y,P). 20

2.5.8 Matching in equilibrium In steady state, there exists a set of unmatched producers that are actively looking for a retail partner and unmatched retailers that are actively looking for a producer. These steady-state fractions of unmatched retailers and producers correspond to frictional unemployment and unfilled vacancies in the labor literature and will be positive as long as the finding rates are finite and the separation rate is non-zero. The mass of producers that are matched to retailers and selling their products is (1−u−i)Nx in which a fraction u are unmatched and actively searching for retailers and a fraction i choose not to search and therefore remain idle. To determine the steady-state fraction of unmatched producers, it is useful to think about the flow into and out of the unmatched producer state. In particular, in any given instant, (1−u−i)Nx matched producers separate exogenously at rate λ. Consequently, the inflow into the unmatched state is λ(1−u−i)Nx. Flows out of this state are κχ(κ)uNx because uNx producers find matches at rate κχ(κ). In steady state, the inflows must equal the outflows; after re-arranging, we get u λ = . (14) 1−i λ+κχ(κ) The fraction of idle producers, i, that choose not to search is defined by the steady-state threshold, ϕ¯, and the exogenous distribution of productivity: (cid:90) ϕ¯ i = dG(ϕ) = G(ϕ¯), (15) 1 in which we have assumed that G(ϕ) is defined over [1,+∞). The fraction of producers that are active, 1−i, corresponds to the labor force participation rate in the labor literature. While u is the fraction of producers that are unmatched, u/(1−i) is the fraction of active producers that are unmatched and is equivalent to the labor unemployment rate, which is 21

characterized as the fraction of the labor force that is actively searching for a job. From the definition of market tightness in this economy, κ = vNm/uNx, we can infer the fraction of unmatched retailers as v = κuNx/Nm. We assume that every matched producer must have one, and only one, retailer as their counterpart. Doing so implies the mass of matched producers and retailers must be equal in steady state: (1−u−i)Nx = (1−v)Nm. (16) 2.6 Aggregate resource constraint The aggregate resource constraint in this economy can be expressed using either the income or expenditure approach to aggregate accounting, as shown in appendix A.10. Typically, models of international trade highlight the income perspective. We find it more natural to focus on the expenditure approach: (cid:18) (cid:19) (cid:90) 1−u−i Y = Nx p(ϕ)q(ϕ)dG(ϕ) 1−i ϕ¯ (cid:124) (cid:123)(cid:122) (cid:125) Aggregateconsumption(C) +κuNxc+uNx(l+sκχ(κ))+(1−u−i)Nxf +κv−1uNxe +Nxe m x (17) (cid:124) (cid:123)(cid:122) (cid:125) Aggregateinvestment(I) (cid:18) (cid:19) (cid:90) 1−u−i + Nx (τ −1)t(ϕ)dG(ϕ). 1−i ϕ¯ (cid:124) (cid:123)(cid:122) (cid:125) Governmentexpenditure(G) Consumption expenditure, C, is total resources devoted to consumption evaluated at final consumer prices. Investment expenditure, I, is resources devoted to creating retailer-producer relationships, paying for the per-period fixed costs of goods production, and funding the creation of firms. Here e and e are the sunk, one-time, fixed costs paid by m x retailers and producers, respectively, to come into existence. In section 2.5.6, we mention that free entry into retailing implies that e must be zero (appendix A.7). m 22

The mass of producers that are matched to retailers and selling their products is (1−u−i)Nx. Producers that are idle or searching for retailers but are currently not in a business relationship do not contribute to aggregate output, consumption, or prices. The integral term times (1−i)−1 captures conditional average sales of producers that have productivity above the cutoff necessary to match. Another way to see that all aggregate variables must be scaled in this way is to compute the mass of matched producers (cid:18) (cid:19) 1−u−i Nx (cid:82)∞ dG(ϕ) = (1−u−i)Nx. 1−i ϕ¯ To account for all resources in the economy, we assume the government spends the tax/tariff revenue, which it acquires by proportionally taxing variable production cost at rate τ ≥ 1. This tax will enter the final sales price, p(ϕ), and the equilibrium quantity traded within a match, q(ϕ). According to equation (17) the government buys labor from the household but does not use the labor to produce anything. As such, those resources are wasted. We make this assumption in the closed economy because it follows the international trade treatment of iceberg trade costs in which resources simply ‘‘melt away’’ in transit. Conversely, if instead of being wasted, tax revenues were refunded lump sum to the household, then the government expenditure term would be superfluous. In that case, taxes would still have a distortionary effect but would not reduce total resources. We could also, of course, set the tax rate to zero (τ = 1) to ensure no tax distortions and that G = 0. Net exports, NX, do not appear in equation (17) because they are trivially zero in the closed economy and, when we turn to the open-economy setting in section 4, we impose balanced trade. We also treat total payments to idle producers, (1−i)Nxh, as balanced lump-sum transfers. They enter negatively in the expenditure approach as a lump-sum tax and enter positively as an additional lump-sum expenditure. As such, these cancel out on the expenditure side of the accounting identity. 23

Total resources are given by the value of the labor endowment, defined by Y = wL, (18) in which L is the exogenous size of the economy and w is the equilibrium wage. It will often be convenient to normalize the nominal, w = 1, or the real wage, w/P = 1. 2.7 The representative consumer We assume that a representative consumer maximizes utility by choosing quantities of the goods indexed by ϕ, resulting in welfare (indirect utility) equal to W (C,P) = argmaxU(q) (19) q s.t. C = Y −I −G−NX, in which consumption expenditure C equals total resources Y minus non-consumption expenditures given by equation (17). We have used the (inverse) Marshallian demand that solves this utility maximization problem denoted as p(q) extensively above. As long as preferences are homothetic (and can therefore be represented by a utility function that is homogeneous of degree one), welfare is the ratio of consumption expenditure to the price index, W (C,P) = C/P = (Y −I −G−NX)/P, as shown in appendix C.2. The price index is defined as the minimum expenditure on consumption needed to obtain a utility level of one. Without explicit functional forms for preferences, we define it as (cid:18) 1−u−i (cid:19) (cid:90) ∞ P = argmin Nx p(ϕ)q(ϕ)dG(ϕ) (20) q 1−i ϕ¯ s.t. U (q) = 1. 24

3 Equilibrium and its properties 3.1 Defining the equilibrium in the closed economy The equilibrium consists of six equations in six endogenous variables. Definition 2. A steady-state competitive equilibrium consists of six endogenous variables q, ϕ¯, κ, n, p, and P, which jointly solve six equations given by (9), (11), (12), (13), (19), and (20), given exogenous parameters β, c, f, h, l, λ, r, s, τ, exogenous productivity distribution, G(ϕ), and exogenous resource endowment, L. We elaborate on this definition in more detail in appendix A.11. 3.2 Closed-economy comparative statics In this section, we study how key equilibrium objects, like ϕ¯ and κ, respond to changes in key exogenous parameters. As we are particularly interested in the role of search frictions, we consider how endogenous variables respond to changes in the flow search costs of retailers, c, and the flow search costs of producers, l. To connect with the trade literature, we also analyze the equilibrium effects of a change in the iceberg trade cost, τ. To make definitive statements, we assume that the flow payoff from being idle is equal to (negative) the producer’s search cost (h = −l) and that the marginal cost of production is constant (horizontal supply curve) so that the final sales price, p, does not move with exogenous parameters, except for τ, which enters the cost function directly. Proposition 4. When the retailer’s flow search cost, c, rises, the productivity threshold necessary to search, ϕ¯, the price index, P, and quantity traded within a match, q, also rise. As c rises, market tightness, κ, and the negotiated price, n, fall. Proof. See appendix A.12.1. 25

These results are intuitive. For example, raising the cost of retailers’ search, c, makes finding a producer more expensive and therefore reduces retailer entry relative to unmatched producers, thereby reducing market tightness, κ. Lower market tightness means that the effective entry cost in equation (11) rises, increasing the threshold productivity, ϕ¯, so that only more productive firms decide to search in the first place. Due to lower market tightness, the producers’ finding rate, κχ(κ), also falls, which means that the producers’ threat point in the bargaining game deteriorates, which reduces their negotiated price, n. Because of lower market tightness and lower producer participation, the ideal price index, P, rises. Because we assume that goods are not perfect substitutes, this increase in the price index means that the quantity traded within a match, q, also rises. By assumption, the final sales price, p, does not move with c. Proposition 5. When the producer’s flow search cost, l, rises, the productivity threshold necessary to search, ϕ¯, and market tightness, κ, also rise. Proof. See appendix A.12.2. From equation (7), we know that as producers’ search cost, l, rises, match surplus increases, which also means that the value of retailing, M (ϕ), rises, inducing retailer entry and thereby raising market tightness, κ. Because of increased costs, the productivity threshold necessary for producers to search, ϕ¯, rises. On the one hand, the higher market tightness serves to reduce the unmatched producer rate and lower the ideal price index, P, but on the other hand, the higher threshold productivity, ϕ¯, serves to raise the price index. In general, the response of the ideal price index to changes in l, therefore, cannot be signed. As such, the response of the quantity traded within a match, q, and the negotiated price, n, also cannot be signed. As before, by assumption, the final sales price, p, does not respond to changes in l. Proposition 6. When the iceberg trade cost, τ, rises, the productivity threshold necessary 26

to search, ϕ¯, the price index, P, the final sales price, p, and the quantity traded within a match, q, also rise. As τ rises, market tightness, κ, and the negotiated price, n, fall. Proof. See appendix A.12.3. Again, the results are intuitive. As the variable costs of production rise, the surplus from a match falls and therefore the value of retailing falls, which reduces retailer entry, thereby reducing market tightness, κ. As before, this increase in market tightness raises the threshold productivity, ϕ¯, and reduces the negotiated price, n. Rising variable costs increase marginal costs and therefore increase the final sales price, p, which increases the aggregate price index, P. As goods are not perfect substitutes, this increase in the ideal price index increases the quantity traded within a match, q. 3.3 Efficiency of the competitive equilibrium The decision to search made by individual retailers and producers imposes externalities on other retailers and producers. The externalities arise because an additional producer (retailer) in the search market decreases (increases) market tightness, κ = vNm/uNx. Individual producers (retailers) ignore the positive ‘‘market thickness’’ externality they create for retailers (producers) when they search and affect κ. Likewise, individual producers (retailers) also ignore the negative ‘‘congestion’’ externality they impose on other producers (retailers) by searching. As a result, the competitive equilibrium may have too many or too few producers (retailers) searching. In a simpler setting than ours, Hosios (1990) shows that with continuous Nash bargaining, the ‘‘market thickness’’ and ‘‘congestion’’ externalities are just balanced in the competitive economy when the bargaining parameter, β, is equal to the elasticity of the matching function with respect to the number of unemployed workers, η(κ). We characterize a similar condition in our model by finding the socially optimal market tightness, κ. 27

In general, the socially optimal market tightness maximizes welfare subject to the equations of motion for the number of producers, uNx, the aggregate resource constraint, and the fact that u is constrained by an initial transversality condition to start at the steady-state value. In order to simplify the problem, we assume that the number of producers that take a draw from the productivity distribution is proportional to the size of the economy, Y. This assumption implies that N ˙x = 0. Our normalization that the real wage w/P = 1 ensures that real output is constant at Y/P = wL/P. We set G = 0 by assuming that there are no taxes, τ = 1, or that tax revenue is rebated lump sum back to the consumer. After making these assumptions and substituting the resource constraint into the objective function, the welfare maximization problem becomes (cid:90) ∞(cid:20) (1+ζ)e (cid:21) max x −κuc−u(l+sκχ(κ))−(1−u−i)f −e e−rtdt x κ ζ 0 s.t. (21) u˙ = λ(1−u)−uκχ(κ) λ(1−i) u(0) = . λ+κχ(κ) Solving this problem results in the socially optimal κ. The current value Hamiltonian, first order conditions, and steps used to derive the solution are included in appendix A.13. Proposition 7. A sufficient condition for the competitive economy to obtain the social planner’s solution is η(κ) = β and β = 1, which is not a feasible value of β. More generally, if β < 1, in order for the competitive equilibrium and the social planner’s problem to result in the same κ, it must be that η(κ) = β and two expressions, which are in general not equal to each other, must be equivalent. As a result, in general, the competitive equilibrium is inefficient under the Nash bargaining protocol. Proof. See appendix A.13. Given the work by Rogerson, Shimer, and Wright (2005), it is not surprising that the 28

competitive equilibrium in a model with random search and Nash bargaining is not, in general, socially optimal. However, we find that the standard Hosios (1990) condition (η(κ) = β) is not sufficient for efficiency in our framework. The sufficient condition in our environment ensures that the producer receives the entire surplus from the match. We find this result intuitive because the retailer in our framework is merely an intermediary and does not provide any additional services (utility) to the final consumer. Moreover, it is solely the consumption of producers’ goods that determines final consumer welfare. As a result, the social planner wants to give the retailer none of the surplus but wants there to be sufficiently many retailers so that producers make contacts instantaneously. Given a matching function that has constant elasticity with respect to the number of searching producers, the latter occurs when η(κ) = 1 so that the number of searching retailers does not matter for the number of matches. The former implies that β = 1. Aside from this amendment of the Hosios condition, we think the idea that the competitive equilibrium may be inefficient is novel in the trade context. Simply put, it implies that there can be too many (or too few) unmatched producers relative to unmatched retailers in the competitive equilibrium. 4 Open economy 4.1 Open economy introduction Up to this point, the economy has been closed to international trade. Before opening to trade, we need to outline how markets interact. We assume searching or matching in one segmented market does not affect the costs of searching across markets. In particular, there are no economies of scale in a particular market for individual producers and retailers from being active in any other market. Opening to trade is relatively simple given this setup because we can consider the behavior of individual firms in each market separately. We define each segmented market using a do subscript to denote destination–origin country 29

pairs. For example, product market tightness between destination country d and origin country o is denoted by κ and captures the number of producers and retailers in the do do product market. Because every producer in every market is a monopolist in their variety, the only interaction among agents in our economy is via aggregate variables such as output, the price index, and search market tightness. Additionally, we assume all agents ignore their individual effects on these aggregate variables when making optimal decisions. 4.2 Functional form assumptions In this section, for simplicity and clarity, we present our functional form assumptions, which will allow us to obtain closed-form expressions for key objects of interest. 4.2.1 Consumer preferences We assume the representative consumer in destination market d has Cobb-Douglas utility over a homogeneous and freely traded good, q , and a second good, q , that is a constant 1 2 elasticity of substitution (CES) aggregate of differentiated varieties from all origins. The two goods are combined with exponents 1−α and α, according to (cid:34) (cid:35) ασ (cid:88) O (cid:18) 1−u −i (cid:19) (cid:90) ∞ σ σ−1 U (q) = q1−α dk dk Nx qσ−1 (ϕ)dG(ϕ) . (22) d 1 1−i k dk dk ϕ¯ k=1 dk The differentiated goods are substitutable with constant elasticity, σ > 1, across varieties and destinations. 30

4.2.2 The ideal price index Given these preferences, the price index for differentiated goods from origin country o to destination country d is given by (cid:20)(cid:18) 1−u −i (cid:19) (cid:90) ∞ (cid:21) 1− 1 σ P = do do Nx p (ϕ)1−σdG(ϕ) , (23) do 1−i o do do ϕ¯ do in which pd (ϕ) is the domestic final sales price for each variety. The total price index for do differentiated goods in country d is given by (cid:34) (cid:35)1/(1−σ) (cid:34) (cid:35) 1 (cid:88) O (cid:88) O (cid:18) 1−u −i (cid:19) (cid:90) ∞ 1−σ P = P1−σ = dk dk Nx pd (ϕ)1−σdG(ϕ) , (24) d dk 1−i k dk dk ϕ¯ k=1 k=1 dk and the overall price index with both homogeneous and differentiated goods is (cid:18) p (cid:19)1−α(cid:18) P (cid:19)α 1 d Ξ = . More details are provided in appendix B.1. d 1−α α 4.2.3 Production structure We will use the familiar variable cost function for producers with productivity ϕ given by t(q ,ϕ,w ) = q w τ ϕ−1. Here w is the competitive wage in the exporting (origin) do o do o do o country, τ ≥ 1 is a parameter capturing one plus the iceberg transport cost between do destination d and origin o. Total production cost is t(q ,ϕ,w )+f in which f is the do o do do fixed cost of production. This cost function implies a constant-returns-to-scale production function in which labor is the only input. The firm that produces variety q has efficiency ϕ do and marginal cost equal to w τ ϕ−1. o do 4.2.4 Productivity distribution Productivity is Pareto distributed with cumulative density function G[ϕ˜ < ϕ] = 1−ϕ−θ over [1,+∞) so the probability density function is g(ϕ) = θϕ−θ−1. We assume that θ > σ−1 so that aggregate variables determined by the integral (cid:82)∞ zσ−1dG(z) are bounded. ϕ¯ 31

4.2.5 Optimal final sales price and quantity traded Total resources in country d, as discussed in section 2.6, are the value of the labor endowment, Y = w L , in which w is the wage, and L is the endowment in labor units. d d d d d Given this income and the ideal price index, P , consumer preferences imply that the d demand for variety ϕ in the differentiated goods sector is q (ϕ) = p (ϕ)−σαY Pσ−1. The do do d d quantity traded is determined by negotiation between importers and exporters and ensures that marginal revenue equals marginal cost as determined by equation (9). If retailers are monopolistic in their variety, equation (9), together with our functional form assumptions on the demand curve, q , and the cost function, t(q ,ϕ,w ), imply that do do o the final is p (ϕ) = µw τ ϕ−1 in which µ = σ/(σ −1). The price charged for the imported do o do good in the domestic market takes the standard markup times marginal cost form. Using the demand curve and domestic optimal price implies the imported quantity q (ϕ) = (µw τ ϕ−1) −σ αY Pσ−1. We present the details of these derivations in appendix do o do d d B.2. 4.2.6 Productivity threshold Flow profits from a match for a producer that has productivity ϕ are π (ϕ) = p (ϕ)q (ϕ)−t (ϕ). With these functional form assumptions, profits become do do do do α (cid:18) µw τ (cid:19)1−σ π (ϕ) = o do Y ϕσ−1 −f , with details in appendix B.2. Using this, the do d do σ P d expression for the cutoff from equation (11) becomes (cid:16)σ(cid:17) 1 (cid:18) w τ (cid:19) 1 1 ϕ¯ = µ σ−1 o do Y 1−σFσ−1, (25) do α P d do d in which (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) F = f + l + 1+ h + s . (26) do do do do do βκ χ(κ ) βκ χ(κ ) β do do do do 32

Notice that equation (25) is identical to equation (7) in Chaney (2008) except for our effective entry cost, F , which differs from the fixed entry cost given there. do 4.3 Analysis of the open economy 4.3.1 Real income In this section, we discuss how adding search frictions changes the response of real income to foreign shocks. We relate this to Arkolakis et al. (2012), who show that, in many different trade models, real income changes can be summarized by two sufficient statistics: the change in the domestic consumption share in response to a shock and the elasticity of trade with respect to variable trade costs. Our expression includes the change in the fraction of matched producers. Definition 3. Define a foreign shock in country d as a change from (L,λ,r,f,c,l,s,τ) to (L(cid:48),λ(cid:48),r(cid:48),f(cid:48),c(cid:48),l(cid:48),s(cid:48),τ(cid:48)) such that (L ,λ ,r ,f ,c ,l ,s ,τ ) = (L(cid:48),λ(cid:48),r(cid:48),f(cid:48),c(cid:48),l(cid:48),s(cid:48),τ(cid:48)). d d d d d d d d d d d d d d d d Proposition 8. Suppose the functional form restrictions in section 4.2 hold and all consumption is devoted to differentiated goods so that α = 1. Then the change in real income associated with any foreign shock in country d can be computed as W ˆ = λ ˆ− θ 1 F ˆθ 1− σ− 1 1 (cid:18) 1− (cid:92) u dd (cid:19)θ 1 , (27) d dd dd 1−i dd in which xˆ ≡ x(cid:48)/x denotes the change in any variable x between the initial and the new equilibrium, λ ≡ C /Y is the share of country d’s total expenditure that is devoted to do do d goods from origin country o, θ governs the productivity distribution, and F is the effective dd cost of entering the home market. Proof. See appendix B.3. 33

ˆ Equation (27) states that the change in real income in country d, W , is a function of the d changes in the share of domestic expenditure, λ , the effective entry cost, F , and the dd dd (cid:18) (cid:19) u dd matched rate, 1− . As a result, unlike in Arkolakis et al. (2012), changes in the 1−i dd trade share are not sufficient to capture changes in real income. Moreover, if we think that trade liberalization increases the domestic matched rate of producers, our framework would predict larger gains from trade than a framework without goods-market frictions. 4.3.2 Trade elasticity In their work, Arkolakis et al. (2012) show that in the Melitz (2003) model, the trade elasticity with respect to a change in the iceberg trade costs is given by    (1−σ)+ψ +(ψ −ψ ) ∂ln(ϕ¯ dd ) if d(cid:48) = d   do do dd ∂ln(τ )   do ∂ln(C /C )  do dd = εACRdd(cid:48) = , (28) ∂ln(τ ) o d(cid:48)o     ∂ln(ϕ¯ )     (ψ do −ψ dd ) ∂ln(τ dd ) if d(cid:48) (cid:54)= d d(cid:48)o ∂ln(Ψ ) in which ψ = do ≥ 0 and Ψ = (cid:82)∞ ϕσ−1dG(ϕ). This expression is the same as do ∂ln(ϕ¯ do ) do ϕ¯ do the expression in Arkolakis et al. (2012) except that ψ ≤ 0 while γ ≥ 0 because we define do ij our model in terms of productivity, ϕ, while they define theirs in terms of marginal cost. In our context, the trade elasticity turns out to be similar to equation (28) but in general has many additional endogenous terms. We provide the full expression in appendix B.4 and present a version under simplifying assumptions here. Proposition 9. The elasticity of consumption shares to iceberg trade costs in our model 34

with goods-market frictions is given by  ∂ln(C /C )  ε + A o (cid:18) CR 1 d − u d d (cid:48) i o do (cid:19)(cid:18) ∂ln ∂ κ ln do ( χ τ d ( o κ ) do ) (cid:19) − (cid:18) 1− u d i d dd (cid:19)(cid:18) ∂ln ∂ κ ln dd ( χ τ d ( o κ ) dd ) (cid:19) if d(cid:48) =d do dd = , (29) ∂ln(τ ) d(cid:48)o  ε + A o (cid:18) CRd u d d (cid:48) o (cid:19)(cid:18) ∂lnκ do χ(κ do ) (cid:19) − (cid:18) u dd (cid:19)(cid:18) ∂lnκ dd χ(κ dd ) (cid:19) if d(cid:48) (cid:54)=d 1−i ∂ln(τ ) 1−i ∂ln(τ ) do d(cid:48)o dd d(cid:48)o in which we assume that the number of producers in d and o do not change with tariff changes, ∂ln(Nx)/∂ln(τ ) = ∂ln(Nx)/∂ln(τ ) = 0 and we assume that l = 0 = h so d d(cid:48)o o d(cid:48)o that F and F are parameters. dd do Proof. See appendix B.4. Our trade elasticity depends on the usual trade elasticity but also on the fraction of unmatched producers and the elasticity of the producers’ finding rate in the do product market. Using equation (12), we know that raising tariffs, τ , reduces the value of do importing, M (ϕ), and therefore reduces market tightness, κ , and the producers’ finding do ∂lnκ χ(κ ) do do rate, κ χ(κ ). This comparative static implies that < 0. We suspect that, do do ∂ln(τ ) do ∂lnκ χ(κ ) dd dd because of protectionism, > 0. In other words, raising tariffs in the do market ∂ln(τ ) do makes the domestic market more attractive, encouraging domestic retailer entry and thus raising domestic market tightness, κ , and the domestic producers’ finding rate. Because dd both dd and do unmatched rates of producers are weakly positive, the trade elasticity in our model is at least as negative as the trade elasticity in the model of Arkolakis et al. (2012). 4.3.3 The gravity equation The gravity structure in our model, albeit more complicated, is similar to the gravity structure common to many trade models. To show this, we begin with the definition of total 35

imports (free on board) by destination d from origin o in the differentiated goods sector (cid:18) u (cid:19) (cid:90) ∞ I = 1− do Nx n (ϕ)q (ϕ)dG(ϕ). (30) do 1−i o do do do ϕ¯ do Performing the required integration in equation (30) gives the following proposition. Proposition 10. The gravity equation in our model is: I = (cid:18) 1− u do (cid:19) (1−b(σ,θ,γ ,δ ,F ))α (cid:18) Y o Y d (cid:19)(cid:18) w o τ do (cid:19)−θ F −( σ− θ 1 −1) , (31) do 1−i do do do Y ρ do do d u do in which the fraction of matched exporters 1− and the bundle of search frictions 1−i do 1−b(σ,θ,γ ,δ ,F ) are each weakly in the unit interval. As such, in a model with search do do do frictions, trade is reduced to a fraction of the value that would be obtained without search frictions. Proof. See appendix B.5. The main message is clear: Search frictions have a first order effect on the level of total imports. Although it is not surprising that adding search frictions reduces trade flows, it is important to realize that instead of simply raising the expected entry cost, F , faced by do firms, they reduce imports to a fraction of the value that would be obtained without search frictions. Investigating aggregate trade flows in greater detail shows that they reduce trade flows in three ways. First, search frictions give rise to a fraction of unmatched exporters, (cid:18) (cid:19) u do 1− . From equation (14) we know that this fraction is less than one, if we ignore 1−i do the possibility of corner solutions, such as c = 0 (as in Proposition 2) or λ = 0. Second, trade flows are diminished because imports are computed using import prices (as opposed to final sales prices) and these negotiated import prices are lower than final sales prices. This is consistent with work by Berger et al. (2012), who use prices for the same product to document that retail prices are 50 to 70 percent higher than at the dock. These lower import 36

prices lead to the bundle of search friction parameters 1−b(σ,θ,γ ,δ ,F ), which is less do do do than one, as shown in appendix B.5. Third, search frictions reduce imports because of the negative exponent on the effective entry cost, F , which is increasing in search frictions, as do shown in equation (26). Even if imports are measured at final sales prices, as they are in the typical gravity equation, search frictions have a significant effect on trade flows. By evaluating equation (31) at p (ϕ) instead of n (ϕ) and using our functional form assumptions, we compute do do imports at final sales prices in appendix B.5 as C = (cid:18) 1− u do (cid:19) α (cid:18) Y o Y d (cid:19)(cid:18) w o τ do (cid:19)−θ F −( σ− θ 1 −1) . (32) do 1−i Y ρ do do d By definition, this provides consumption expenditure in destination d on differentiated goods produced in origin o. Even without the wedge between final and import prices reflected in (cid:18) (cid:19) u do b(·), search frictions lead to a mass of inactive and searching producers 1− , 1−i do which lowers imports. Search frictions also enter effective entry cost, F , as before. do Consumption expenditure must equal imports plus the period profits of matched importers, C = I +Π . Combining equations (30) and (32) gives total period profits do do do accruing to importers in matched relationships: Π = b(σ,θ,γ ,δ ,F )α (cid:18) Y o Y d (cid:19)(cid:18) w o τ do (cid:19)−θ F −( σ− θ 1 −1) . (33) do do do do Y ρ do d We could also obtain this quantity if we integrate profit to each variety p (ϕ)q (ϕ)−n (ϕ)q (ϕ) over all imported varieties. Despite the value of posting a do do do do vacancy being driven to zero by free entry, flow profits are always positive as long as the retailers’ search cost, c, is positive, so importers can recoup the costs expended while searching. 37

4.3.4 Implications for estimation The fact that introducing search frictions into a model of trade results in a scalar times the typical gravity equation has a few interesting implications for estimation. First, if the fraction of matched exporters and the bundle of search friction parameters do not vary by destination–origin pairs, then their effect on trade would be lost in the constant term of a gravity regression. In this case, while estimates of the other coefficients in the model would be unbiased, search frictions could be a pervasive feature of international trade but would not be identifiable using the gravity equation. Second, if the fraction of unmatched producers and bundle of search frictions vary by importer–exporter pair, they may provide an additional rationale for why language, currency, common legal origin, historical colonial ties or other variables often included in gravity equations have an effect on aggregate trade flows. In particular, Rauch and Trindade (2002) argue populations of ethnic Chinese within a country facilitate the flow of information, provide matching and referral services, and otherwise reduce informal barriers to trade. Their empirical specification matches the gravity equation with search that we have derived here if the destination–origin search frictions are a function of the ethnically Chinese population. Third, any gravity regression that does not include adequate proxies for search frictions would suffer from omitted variable bias. In particular, suppose that a researcher omits search frictions, as measured by the matched rate, and estimates the following equation: lnI = A +B +β lnτ +ν , (34) do d o 0 do do in which I are the imports from origin o to destination d, A is an importer-specific term, do d B is an exporter-specific term, β is the partial elasticity of bilateral imports with respect to o 0 variable trade costs, and ν is an error term. Econometric theory suggests that the omitted do u do ˆ variable, Z = 1− , will introduce bias into the ordinary least squares estimate of β , do 0 1−i do 38

according to the well-known formula: E[β ˆ |X] = β +ρ(τ ,Z )ρ(I ,Z ), (35) 0 0 do do do do in which X is a vector of all right-hand-side variables and ρ(X,Y) is the correlation between X and Y. We know that ρ(τ ,Z ) < 0 (higher variable trade costs, τ , raise the do do do threshold productivity, ϕ¯ , increasing the fraction of idle firms, i , and lowering the do do matched rate) and ρ(I ,Z ) > 0 (increasing the matched rate increases trade flows) so the do do sign of the bias is negative. Proposition 11. Omitting the matched rate from a standard gravity equation implies that the estimate of trade elasticity with respect to variable trade costs is more negative than if one included the matched rate in the estimating equation. Proof. This result follows from the discussion in the text. 5 Conclusion Although the international trade literature has recently made substantial progress in modeling and estimating the costs and attributes of micro-level relationship formation in Heise (2016), Eaton et al. (2014), and Monarch and Schmidt-Eisenlohr (2015), among others, less is know about how these costs affect aggregate quantities in a general equilibrium framework. To improve understanding on this score, we have combined canonical models from search and trade to present a rich and tractable framework for studying the importance of costly relationship formation for economic aggregates. This framework shares the central tenet of any search model: In equilibrium, there exists a mass of unmatched agents who are actively looking for partners. This simple observation leads to profound implications because the product varieties associated with these unmatched agents (producers) cannot be 39

consumed and are therefore absent from indirect utility (welfare), the price index, trade flows, and other aggregates. Specifically, we show that unmatched varieties have first-order implications for welfare responses because any shock that changes the unmatched rate will affect welfare through this new channel. As such, we generalize the findings of Arkolakis et al. (2012) to include search frictions and take up the task of deriving aggregate welfare implications in a model of search and trade that more complex models, such as Chaney (2014) and Eaton et al. (2016), left to future work. In addition to these welfare results, we show that the competitive equilibrium in our search framework will typically have too many or too few searching retailers and producers, which implies that the competitive equilibrium, in general, is not socially optimal. Not surprisingly, adding search friction also reduces aggregate trade flows, but we point out that it does so in two ways. First, search frictions introduce a wedge between import and final-sales prices. Second, they change the mass of varieties traded, because some producers are unmatched. Both of these effects are evident in the closed-form gravity equation derived from our model. Beyond aggregate predictions, our framework also provides a search-theoretic foundation for micro features such as the up-front cost firms face when entering foreign markets. The interpretation of these costs are that they include the search, fixed, sunk, and opportunity costs of serving a foreign market and that they can depend on the rate at which counterparts are contacted. Admittedly, to obtain this amount of analytical simplicity, our framework ignores some of the details of microeconomic trading relationships, including many-to-many matches, learning about product quality, and endogenous search intensity. Following the pattern for how search models were developed in labor and monetary economics, we recommend future work incorporate these features into our tractable framework to derive additional implications for the aggregate economy and to better match micro data. 40

First, the model can be extended to incorporate endogenous separations, in the spirit of Mortensen and Pissarides (1994), so that larger, more productive firms are in more stable trading relationships. Second, we have focused on the steady state of the model, but the framework is a dynamic one and could be extended to include the transition path after a relevant exogenous shock. This direction would dovetail nicely with Chaney (2014) and would go a long way to providing a dynamic, continuous-time model that admits easy aggregation and retains the basic features of Melitz (2003). Third, as in most models of search, it would be interesting to see if other matching and bargaining protocols support the results of the present paper. Locating and building connections with overseas buyers is a prevalent and costly barrier to exporting. Our tractable setting provides a promising baseline for analyzing the implications of these costs and for future research considering search and trade. 41

References Ahn, J., A. K. Khandelwal, and S.-J. Wei (2011): ‘‘The Role of Intermediaries in Facilitating Trade,’’ Journal of International Economics, 84, pp. 73–85. Allen, T. (2014): ‘‘Information Frictions in Trade,’’ Econometrica, 82, pp. 2041–2083. Arkolakis, C., A. Costinot, and A. Rodr´ıguez-Clare (2012): ‘‘New Trade Models, Same Old Gains?’’ American Economic Review, 102, pp. 303–07. Armenter, R. and M. Koren (2014): ‘‘A Balls-and-Bins Model of Trade,’’ American Economic Review, 104, pp. 2127–51. Bai, Y., J.-V. R´ıos-Rull, and K. Storesletten (2012): ‘‘Demand Shocks as Productivity Shocks,’’ Mimeo, University of Minnesota. Available at http://www.sas.upenn.edu/~vr0j/papers/brs_Sept_27_2012.pdf. Bartelsman, E. J. and M. Doms (2000): ‘‘Understanding Productivity: Lessons from Longitudinal Microdata,’’ Journal of Economic Literature, 38, pp. 569–594. Benguria, F. (2015): ‘‘The Matching and Sorting of Exporting and Importing Firms: Theory and Evidence,’’ Mimeo, University of Kentucky. Available at http://felipebenguria.weebly.com/. Berger, D., J. Faust, J. H. Rogers, and K. Steverson (2012): ‘‘Border Prices and Retail Prices,’’ Journal of International Economics, 88, pp. 62–73. Bernard, A. B., J. B. Jensen, S. J. Redding, and P. K. Schott (2010): ‘‘Wholesalers and Retailers in US Trade,’’ American Economic Review, 100, pp. 487–493. Cadot, O., L. Iacovone, D. Pierola, and F. Rauch (2011): ‘‘Success and Failure of African Exporters,’’ Center for Economic Performace Working Paper 1054. Available at http://cep.lse.ac.uk/pubs/download/dp1054.pdf. Chaney, T. (2008): ‘‘Distorted Gravity: The Intensive and Extensive Margins of International Trade,’’ American Economic Review, 98, pp. 1707–21. --------- (2014): ‘‘The Network Structure of International Trade,’’ American Economic Review, 104, 3600--3634. den Haan, W. J. (2013): ‘‘Inventories and the Role of Goods-Market Frictions for Business Cycles,’’ CFM discussion paper series, CFM-DP2014-2. Available at http://eprints.lse.ac.uk/58231/. Diamond, P. A. (1982): ‘‘Aggregate Demand Management in Search Equilibrium,’’ Journal of Political Economy, 90, pp. 881–894. Drozd, L. A. and J. B. Nosal (2012): ‘‘Understanding International Prices: Customers as Capital,’’ American Economic Review, 102, pp. 364–395. 42

Eaton, J., M. Eslava, D. Jinkins, C. J. Krizan, and J. Tybout (2014): ‘‘A Search and Learning Model of Export Dynamics,’’ Mimeo, Pennsylvania State University. Available at http://www.davidjinkins.com/docs/EEJKT_02_26_2014.pdf. Eaton, J., D. Jinkins, J. Tybout, and D. Xu (2016): ‘‘Two-sided Search in International Markets,’’ Mimeo, Pennsylvania State University. Available at http://www.ucdenver.edu/academics/colleges/CLAS/Departments/economics/ Seminars/Documents/ejtx_draft_v14.pdf. Egan, M. L. and A. Mody (1992): ‘‘Buyer–Seller Links in Export Development,’’ World Development, 20, pp. 321–334. Gourio, F. and L. Rudanko (2014): ‘‘Customer Capital,’’ Review of Economic Studies, 81, pp. 1102–1136. Heise, S. (2016): ‘‘Firm-to-Firm Relationships and Price Rigidity Theory and Evidence,’’ Working Paper 6226, CESifo. Hopenhayn, H. A. (1992): ‘‘Entry, Exit, and Firm Dynamics in Long Run Equilibrium,’’ Econometrica, 60, pp. 1127–1150. Hosios, A. J. (1990): ‘‘On the Efficiency of Matching and Related Models of Search and Unemployment,’’ Review of Economic Studies, 57, pp. 279–298. Kneller, R. and M. Pisu (2011): ‘‘Barriers to Exporting: What are They and Who do They Matter To?’’ World Economy, 34, pp. 893–930. Kudoh, N. and M. Sasaki (2011): ‘‘Employment and Hours of Work,’’ European Economic Review, 55, pp. 176–192. Melitz, M. J. (2003): ‘‘The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity,’’ Econometrica, 71, pp. 1695–1725. Monarch, R. and T. Schmidt-Eisenlohr (2015): ‘‘Learning and the Value of Trade Relationships,’’ Available at http://dx.doi.org/10.2139/ssrn.2594035. Mortensen, D. T. (1986): ‘‘Job Search and Labor Market Analysis,’’ in Handbook of Labor Economics, ed. by O. C. Ashenfelter and R. Layard, Elsevier Science, vol. 2, pp. 849–919. Mortensen, D. T. and C. A. Pissarides (1994): ‘‘Job Creation and Job Destruction in the Theory of Unemployment,’’ Review of Economic Studies, 61, pp. 397–415. Nash, Jr., J. F. (1950): ‘‘The Bargaining Problem,’’ Econometrica, 18, pp. 155–162. Osborne, M. J. and A. Rubinstein (1990): Bargaining and Markets, London: Academic Press. 43

Petrosky-Nadeau, N. and E. Wasmer (2015): ‘‘Macroeconomic Dynamics in a Model of Goods, Labor, and Credit Market Frictions,’’ Journal of Monetary Economics, 72, pp. 97–113. Pissarides, C. A. (1985): ‘‘Short-Run Equilibrium Dynamics of Unemployment, Vacancies, and Real Wages,’’ American Economic Review, 75, pp. 676–690. --------- (2000): Equilibrium Unemployment Theory, The MIT Press, second ed. Portes, R. and H. Rey (2005): ‘‘The Determinants of Cross-Border Equity Flows,’’ Journal of International Economics, 65, pp. 269–296. Rauch, J. E. (1996): ‘‘Trade and Search: Social Capital, Sogo Shosha, and Spillovers,’’ NBER Working Papers 5618, National Bureau of Economic Research, Inc. --------- (1999): ‘‘Networks Versus Markets in International Trade,’’ Journal of International Economics, 48, pp. 7–35. Rauch, J. E. and V. Trindade (2002): ‘‘Ethnic Chinese Networks In International Trade,’’ Review of Economics and Statistics, 84, pp. 116–130. Rauch, J. E. and J. Watson (2003): ‘‘Starting Small in an Unfamiliar Environment,’’ International Journal of Industrial Organization, 21, pp. 1021–1042. Rogerson, R., R. Shimer, and R. Wright (2005): ‘‘Search-Theoretic Models of the Labor Market: A Survey,’’ Journal of Economic Literature, 43, pp. 959–988. Ross, S. M. (1995): Stochastic Processes, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., 2 ed. Syverson, C. (2011): ‘‘What Determines Productivity?’’ Journal of Economic Literature, 49, pp. 326–365. Wong, Y. and R. Wright (2014): ‘‘Buyers, Sellers and Middlemen: Variations on Search-Theoretic Themes,’’ International Economic Review, 55, pp. 375–397. 44

A Search model appendix A.1 The surplus, value, and expected duration of a relationship Denote the joint surplus accruing to both sides of a match as S(ϕ). The bargain will divide this surplus such that the value of being a retailer equals M (ϕ)−V = (1−β)S(ϕ) and the value of being a producer is X(ϕ)−U (ϕ) = βS(ϕ), in which β is the producer’s bargaining power. Using the value functions presented in the main text (1), (2), (4), and (5), we can write the surplus equation as pq −t(q)−f +l+sκχ(κ) S(ϕ) = . r+λ+βκχ(κ) The surplus created by a match is the appropriately discounted flow profit, with the search cost l and the sunk cost s also entering the surplus equation because being matched avoids paying these costs. There are three things to notice here. First, the surplus from a match is a function of productivity. We show in appendix A.5 that matches that include a more productive exporting firm lead to greater surplus, that is, S(cid:48)(ϕ) > 0. Second, the value of the relationship will fluctuate over the business cycle as shocks hit the economy and change the finding rate κχ(κ). Finally, surplus is greater than or equal to zero when pq −t(q)−f +l+sκχ(κ) ≥ 0. Specifically, at the binding productivity cutoff we can use equation (10) and the surplus sharing rule to write l+h βS(ϕ¯) = +s, κχ(κ) which, in order for surplus to be positive, puts a restriction on the parameter choices and the equilibrium value of market tightness, κ. With the definition of surplus in hand, the value of a matched relationship, (cid:18) (cid:19) r+κχ(κ)β l R(ϕ) = X(ϕ)+M (ϕ), can be expressed as R(ϕ) = S(ϕ) − . The value r r of the relationship to the producer is, of course, X(ϕ) and to the retailer M(ϕ). The value of a relationship in product markets has been of recent interest in Monarch and Schmidt-Eisenlohr (2015) and Heise (2016). Relationships are destroyed at Poisson rate λ in the model, which implies the average duration of each match is 1/λ. Because the destruction rate is exogenous and does not vary in our model, the average duration of each match is constant. A.2 Solving the Nash bargain For this section, it will be helpful to note that equations (1) and (4) imply that 45

nq −t(q)−f +λU (ϕ) X(ϕ) = (36) r+λ and p(q)q −nq M (ϕ) = . (37) r+λ A.2.1 Bargaining over the price Take equation (6), log and differentiate with respect to the price n and rearrange to get q −q β +(1−β) = 0, (38) X(ϕ)−U (ϕ) M (ϕ)−V which implies the simple surplus sharing rule: The retailer receives β of the total surplus from the trading relationship, S(ϕ) = M (ϕ)−V +X(ϕ)−U (ϕ). The producer receives the rest of the surplus, (1−β)S(ϕ). In section 2.5.3 of the main text, we point out the restriction that β < 1 in equation (6) is evident in equation (8), which results from equation (38). Retailing firms have no incentive to search if β = 1 because they get none of the resulting match surplus and therefore cannot recoup search costs c > 0. Any solution to the model with c > 0 and positive trade between retailers and producers also requires β < 1. This result can be shown explicitly by using equations (4), (5), and (13) together with β = 1 to show that for productivity, ϕ, levels above the reservation productivity, ϕ¯, (defined in section 2.5.5), the retailing firm has no incentive to search. Finally, we do not need to calculate the partial derivative with respect to U (ϕ) or V (ϕ) because the individual firms are too small to influence aggregate values. Hence, when they meet, the firms bargain over the negotiated price-taking behavior in the rest of the economy as given. In particular, the outside option of the firms does not vary with the individual’s bargaining problem. A.2.2 Bargaining over the quantity Take equation (6), log and differentiate with respect to the quantity q to get 1 1 β (n−p(cid:48)(q))+(1−β) (p(q)+p(cid:48)(q)q −n) = 0, (39) X(ϕ)−U (ϕ) M (ϕ)−V in which we compute the partials of X(ϕ) and M (ϕ) using equations (36) and (37). Now, notice that equation (38) implies that X(ϕ)−U (ϕ) = β (M (ϕ)−V), and plugging this 1−β into equation (39) and rearranging slightly gives p(q)+p(cid:48)(q)q = t(cid:48)(q). (40) 46

This expression says that the quantity produced and traded is pinned down by equating marginal revenue in the domestic market with marginal production cost in the foreign country. This restriction is the same as what we get from a model without search and therefore implies that adding search does not change the quantity traded. The profit maximization implied by this equation is crucial: Despite being separate entities, the retailer and the producer decide to set marginal revenue equal to marginal cost. The result follows because of the simple sharing rule, the maximization of joint surplus, and the trivial role of the retailer. To maximize surplus, the parties choose to equate marginal revenue and marginal cost. A.3 Retailer production function In this section, we show that including another input for the retailer does not affect the conclusions of this paper under some weak additional assumptions. With an additional input, the value of being in a relationship for a retailer changes to rM(ϕ) = p(f(q,k))f(q,k)−nq −n k −λ(M(ϕ)−V), (41) k in which retailer combines the input, denoted by k, with the input purchased from the producer, q, according to production function f(q,k) for the final good sold to consumers. The price of the additional input, n , is determined outside of the search model and is taken k as given by the retailer. With this new Bellman equation, logging and differentiating the Nash product in equation (6) with respect to p gives the same surplus sharing (8) rule as before. The first order condition of equation (6) with respect to q, however, becomes n−t(cid:48)(q) p(cid:48)(f(q,h))f (q,h)f(q,h)+p(f(q,h))f (q,h)−n q q β +(1−β) = 0. (42) X(ϕ)−U(ϕ) M(ϕ)−V Combining this with the surplus sharing rule (8) yields an expression similar to equation (9): p(cid:48)(f(q,h))f (q,h)f(q,h)+p(f(q,h))f (q,h) = t(cid:48)(q). (43) q q This equation states that retailers and producers will negotiate to trade a quantity of q that ensures the marginal revenue equals marginal cost. As the price of input k is taken as given, the firm chooses the optimal level of the input, k∗, so that f (q,k∗) = n . Strict concavity k k of the function f(q,k) is sufficient to ensure that f (q,k) is invertible. Making this k assumption gives f−1(q,n) = k∗ which can be substituted into equation (43) to get one k equation in one unknown, q. The quantity traded depends on n , the price of the other k input, but search frictions still do not enter the equation (43). The result in the main text, that optimal q is determined by the condition that ensures marginal revenue from q equals the marginal cost of producing q, remains intact. 47

A.4 Solving for the productivity thresholds A.4.1 Solving for the lowest productivity threshold First, let’s solve for an expression for X(ϕ)−U(ϕ) by plugging in equations (1) and (2): rX(ϕ)−rU(ϕ) = nq −t(q)−f −λ(X(ϕ)−U(ϕ))+l−κχ(κ)(X(ϕ)−U(ϕ)−s) = nq −t(q)−f +l+κχ(κ)s−(λ+κχ(κ))(X(ϕ)−U(ϕ)) ⇒ (r+λ+κχ(κ))(X(ϕ)−U(ϕ)) = nq −t(q)−f +l+κχ(κ)s (44) nq −t(q)−f +l+κχ(κ)s ⇒ X(ϕ)−U(ϕ) = . r+λ+κχ(κ) Now plug this expression into the definition of ϕ from the main text to get ¯ nq −t(q)−f +l+κχ(κ)s = 0 r+λ+κχ(κ) ⇒ nq −t(q)−f +l+κχ(κ)s = 0. By using the fact that X(cid:48)(ϕ)−U(cid:48)(ϕ) > 0 from above we can state that this threshold is unique. l+h We can be sure that for any positive cost of forming a relationship, +s, if and κχ(κ) (cid:0) (cid:1) (cid:0) (cid:1) only if l+h+κχ(κ)s > 0, the expression X(ϕ¯)−U (ϕ¯) exceeds X ϕ −U ϕ . This result ¯ ¯ implies that as long as X(ϕ)−U (ϕ) is increasing in ϕ, then ϕ¯ > ϕ. In appendix A.5, we ¯ show the very general conditions under which X(ϕ)−U (ϕ) is increasing in ϕ. The binding productivity threshold defining the mass of producers that have retail partners is the greater of these two and hence ϕ¯. In other words, the productivity necessary to induce a producer to search for a retail partner is greater than the productivity necessary to consummate a match after meeting a retailer due to the costs that are incurred while searching. Similarly, the productivity necessary to form a match is greater than the productivity to maintain one already in place. 48

A.4.2 Proof of Proposition 1: Solving for the binding productivity threshold Using equation (44) in the definition of ϕ¯ in equation (10) yields nq −t(q)−f +l+κχ(κ)s l+h = +s r+λ+κχ(κ) κχ(κ) sκχ(κ)+l+h ⇒ nq −t(q)−f +l+κχ(κ)s = (r+λ+κχ(κ)) κχ(κ) sκχ(κ)+l+h ⇒ nq −t(q)−f +l+κχ(κ)s = (r+λ) +sκχ(κ)+l+h κχ(κ) sκχ(κ)+l+h ⇒ nq −t(q)−f = (r+λ) +h κχ(κ) l+h ⇒ nq −t(q)−f −(r+λ) −h = (r+λ)s κχ(κ) l+h ⇒ nq −t(q)−f = (r+λ)s+(r+λ) +h. κχ(κ) Now, plug in for the equilibrium import price, n, from equation (13), to get (cid:18) (cid:19) (r+λ) (r+λ) (1−γ)p(q)q +γ(t(q)−f −l−κχ(κ)s)−t(q)−f = (r+λ)s+ l+ 1+ h. κχ(κ) κχ(κ) which can be rearranged to obtain (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) (r+λ) (r+λ) p(q)q −t(q)−f = (1−γ)−1 (r+λ+γκχ(κ))s+ γ + l+ 1+ h . κχ(κ) κχ(κ) Further simplification of the terms with γ imply that (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) p(q)q −t(q) = f + l+ 1+ h+ s. βκχ(κ) βκχ(κ) β which is the expression in the main text. A.4.3 Comparing our productivity threshold to previous models (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) Defining F (κ) ≡ f + l+ 1+ h+ s in this framework would βκχ(κ) βκχ(κ) β allow us to replace the fixed cost in the standard models with F (κ) from here. The key thing to remember when working with the other quantities of our model is we want to work with them in terms of the cutoff and not in terms of these fundamental frictions just yet. Another interesting comparison is to Eaton et al. (2014). That framework includes a flow search cost, l, but did not have a sunk costs s or any idle state. If we set h = 0, we are implicitly including an idle state because the producer will have a zero value for being in the idle state but have a negative flow cost, −l, for being in the searching state because that 49

state requires a payment each period of l > 0. In other words, because the producer cannot opt out of searching we must set the flow of the idle state to h = −l instead of what one might think is the intuitive value of that state h = 0. Making this assumption together with s = 0 provides: (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) p(q)q −t(q) = f + l+ 1+ h+ s βκχ(κ) βκχ(κ) β = f −l. This result is the very reason why Eaton et al. (2014) must have that f > l. Notice that we recover the standard model when we make these assumptions together with l = 0: (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) p(q)q −t(q) = f + l+ 1+ h+ s βκχ(κ) βκχ(κ) β = f. Another interesting way to remove just the search friction, l, from the model is to set the (r+λ) finding rate κχ(κ) → ∞ so that → 0 βκχ(κ) (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) p(q)q −t(q) = f + l+ 1+ h+ s βκχ(κ) βκχ(κ) β (r+λ) = f +h+ s. β The interpretation of the fixed cost includes the sunk cost, bargaining power of the producer, and the flow from the outside option, even if one finds a partner immediately. If bargaining power differs by market, for example, the bundle of entry costs will as well. A.5 The value of importing is strictly increasing in productivity Here we show that the value of importing, M(ϕ), is strictly increasing with the producer’s productivity level, ϕ. This fact allows us to replace the integral of the max over V and M(ϕ) (equation 5) with the integral of M(ϕ) from the productivity threshold, ϕ¯ (equation 12). Starting with equation (4) and V = 0 obtain (r+λ)M (ϕ) = pq −nq = pq −[1−γ]pq −γ(t(q)+f −l−κχ(κ)s) = γpq −γ(t(q)+f)+γ(l+κχ(κ)s) = γ(pq −t(q)−f)+γ(l+κχ(κ)s). (r+λ)(1−β) Remember that γ ≡ . It is clear from the integral in the import relationship r+λ+βκχ(κ) creation equation (12) that neither the finding rate for retailers, χ(κ), nor the tightness, κ, 50

∂[p(q)q −t(q)−f] are functions of the productivity, ϕ. Given this, M(cid:48)(ϕ) and will have ∂ϕ the same sign. As long as flow profits without search frictions are strictly increasing in productivity, M(cid:48)(ϕ) > 0. Using the specific functional forms for t(q)+f used above, as well as the equilibrium values for n, p, and q, we can derive this result explicitly. In this case, (cid:18) 1 (cid:19)(cid:18) µ−σ (cid:19) M(ϕ) = γ (w τ )1−σαY Pσ−1ϕσ−1 −γ f +γ (l +κ χ(κ )s ). do o do d do do do do do do do r+λ σ −1 Therefore the derivative is (cid:18) (cid:19) ∂M (ϕ) 1 = γ µ−σ(w τ )1−σαY Pσ−1ϕσ−2. do o do d ∂ϕ r+λ which is always positive. As long as M(cid:48)(ϕ) > 0, we can demonstrate the way in which many other important quantities depend on the producer’s productivity level, ϕ. From the surplus sharing rule (38) can be rewritten as βM (ϕ) = (1−β)(X(ϕ)−U (ϕ)), (45) We know that in equilibrium, because M(cid:48)(ϕ) > 0, it must be that X(cid:48)(ϕ)−U(cid:48)(ϕ) > 0. Differentiating both sides of equation (2) gives rU(cid:48)(ϕ) = κχ(κ)(X(cid:48)(ϕ)−U(cid:48)(ϕ)) > 0. We can combine these facts to show X(cid:48)(ϕ) > U(cid:48)(ϕ) > 0. Using the definition of the joint surplus of a match S(ϕ) = X(ϕ)+M (ϕ)−U (ϕ)−V we get S(cid:48)(ϕ) > 0. Likewise, the value of a relationship, R(ϕ) = X(ϕ)+M (ϕ), has R(cid:48)(ϕ) > 0. A.6 Proof of Proposition 2: Market tightness and the cost of search Let’s first prove that κ < ∞ if c > 0. To do this, let’s prove the contrapositive: assume that c = 0 and show that κ = ∞. Rearrange equation (12) slightly to get (cid:90) 0 = c = χ(κ) M (ϕ)dG(ϕ). ϕ¯ We have shown than M (ϕ¯) ≥ 0 for any consummated match in equilibrium (Nash bargaining together with appendix A.4) and M(cid:48)(ϕ) > 0 (appendix A.5). Therefore we know (cid:82) that M (ϕ)dG(ϕ) > 0. As such, χ(κ) must be zero. Because χ(cid:48)(κ) < 0 this is true if and ϕ¯ only if κ = ∞. To prove that if c > 0 then κ < ∞, let’s use equation (12) again. In particular, because (cid:82) c > 0 it must mean that χ(κ) M (ϕ)dG(ϕ) > 0. As before, we know that ϕ¯ (cid:82) M (ϕ)dG(ϕ) > 0 so it must be that χ(κ) > 0 as well, which is true if and only if κ < ∞. ϕ¯ 51

A.7 Producer and retailer existence A.7.1 Retailing firms Free entry implies that the ex-ante expected value from entering for a potential retailer equals the expected cost of entering. Assume for a moment that the potential retailers consider the value of becoming a retailer as defined by E . This value is characterized by m the following Bellman equation rE = −e +(V −E ). (46) m m m The potential retailer could sell the value E and invest the proceeds at the interest rate r m getting flow payoff rE forever after. Alternatively, they could pay a cost e to become a m m retailer, at which point they will begin in the state of having a vacancy with value V (with certainty) and give up the value of being a potential retailer E . Free entry into becoming a m retailer implies that E = 0 in equilibrium so that m 0 = −e +V m e = V. m Hence, free entry into vacancies V = 0 implies e = 0 and we cannot have a sunk cost m for entry into retailing. In other words, free entry into the search market along with assuming that one must post a vacancy before matching implies free entry into retailing. Free entry into the search market subsumes free entry into retailing and so we only have one condition defined by free entry on the retailing side given by equation (12) and restated here (cid:90) c = M (ϕ)dG(ϕ). χ(κ) ϕ¯ Remember this states that product vacancies continue being created until the expected cost (cid:82) of being an unmatched retailer, c/χ(κ), equals the expected benefit M (ϕ)dG(ϕ). ϕ¯ Because each potential retailer must post a product vacancy before forming a match, the expected cost of becoming a retailer (entering as a retailer) is the same as the expected cost of being an unmatched retailer. Likewise, the expected benefit of posting a vacancy and the expected benefit of becoming a retailer are also the same because we assume retailers must post a vacancy before matching. A.7.2 Producing firms Similar to the entry decision of retailers, the value of entry for producers, E , is defined by x (cid:90) rE = −e + max{I(ϕ),U (ϕ)}dG(ϕ)−E x x x (cid:90) ϕ¯ (cid:90) ∞ = −e + I(ϕ)dG(ϕ)+ U (ϕ)dG(ϕ)−E . x x 1 ϕ¯ 52

We assume that the potential producer must transit through the unmatched state before forming a match. After paying e and taking a productivity draw ϕ, the potential producer x loses the value E with certainty and, depending on the drawn productivity, chooses x between searching for a retailer and getting value U (ϕ) or remaining idle and getting value I(ϕ). If we assumed free entry into production, we would get E = 0 and that x (cid:90) ϕ¯ (cid:90) ∞ e = I(ϕ)dG(ϕ)+ U (ϕ)dG(ϕ). (47) x 1 ϕ¯ which ensures the expected value of taking a productivity draw equals the expected cost. Free entry into production, therefore, imposes another restriction on the equilibrium. We can use the facts that X(ϕ)−U (ϕ) = (1−β)S(ϕ) and that M (ϕ) = βS(ϕ) to write (cid:18) (cid:19) 1−β X(ϕ)−U (ϕ) = M (ϕ). Applying this to equation (2) gives β (cid:18)(cid:18) (cid:19) (cid:19) 1−β rU (ϕ) = −l+κχ(κ) M (ϕ)−s . β Computing the relevant integrals in equation (47) gives (cid:90) ∞ (cid:18) l+sκχ(κ) (cid:19) κχ(κ) (cid:18) 1−β (cid:19)(cid:90) ∞ U (ϕ)dG(ϕ) = − (1−G(ϕ¯))+ M (ϕ)dG(ϕ). r r β ϕ¯ ϕ¯ Likewise, from (3) we have (cid:90) ϕ¯ h I(ϕ)dG(ϕ) = G(ϕ¯). r 1 Combining these with equation (47) gives h (cid:18) l+sκχ(κ) (cid:19) κχ(κ) (cid:18) 1−β (cid:19)(cid:90) ∞ e = G(ϕ¯)− (1−G(ϕ¯))+ M (ϕ)dG(ϕ¯). (48) x r r r β ϕ¯ which is the restriction that free entry into production for producers would place on equilibrium market tightness κ. From equation (47), we can see that free entry into search for producers would require (cid:82)∞ U (ϕ)dG(ϕ) = 0, in which case we would be left with ϕ¯ h e = G(ϕ¯). (49) x r which still places a restriction on market tightness because ϕ¯ from Proposition (11) includes κ. Finally, we note that simultaneous combinations of free entry on both sides of the market are possible. Combining free entry into both existence and search for retailers from equation 53

(12) with free entry into existence for producers from equation (48) gives (cid:18) (cid:19) (cid:18) (cid:19) h l+sκχ(κ) cκ 1−β e = G(ϕ¯)− (1−G(ϕ¯))+ . x r r r β Likewise, allowing for free entry into both existence and search for retailers and producers would give the following system that defines κ (cid:90) c = M (ϕ)dG(ϕ) χ(κ) ϕ¯ h e = G(ϕ¯). x r A.8 Proof of Proposition 3: Solving for the equilibrium negotiated price Start with equation (39) and rearrange considerably: p(q)q −nq nq −t(q)−f +l+κχ(κ)s β = (1−β) r+λ r+λ+κχ(κ) nq = p(q)q(1−γ)+γ[t(q)+f −l−κχ(κ)s]. in which (r+λ)(1−β) γ = . r+λ+βκχ(κ) We show that γ ∈ [0,1] in appendix A.9. (r+λ)(1−β) The limit of γ when the finding rate κχ(κ) → ∞ is simply γ ≡ → 0. r+λ+βκχ(κ) More complicated is the limit of γκχ(κ) as κχ(κ) → ∞. First rewrite the expression as (r+λ)(1−β) γκχ(κ) = κχ(κ). r+λ+βκχ(κ) Dividing the top and bottom of this expression by κχ(κ) yields (r+λ)(1−β) γκχ(κ) = . r+λ +β κχ(κ) 54

Now use this to derive the limit (r+λ)(1−β) lim γκχ(κ) = lim r+λ κχ(κ)→∞ κχ(κ)→∞ +β κχ(κ) (r+λ)(1−β) = . β This can be used to derive limit of the negotiated price, n, as κχ(κ) → ∞: (cid:20) (cid:21) t(q,ϕ,w,τ)+f −l−κχ(κ)s lim n = lim [1−γ]p(q,Y,P)+γ κχ(κ)→∞ κχ(κ)→∞ q (cid:20) (cid:20) (cid:21) (cid:21) t(q,ϕ,w,τ)+f −l γκχ(κ)s = lim p(q,Y,P)−γp(q,Y,P)+γ − κχ(κ)→∞ q q = p(q,Y,P)−p(q,Y,P) lim γ κχ(κ)→∞ (cid:20) (cid:21) t(q,ϕ,w,τ)+f −l s + lim γ − lim γκχ(κ) q κχ(κ)→∞ q κχ(κ)→∞ (cid:20) (cid:21) t(q,ϕ,w,τ)+f −l s(r+λ)(1−β) = p(q,Y,P)−p(q,Y,P)·0+ ·0− q qβ s(r+λ)(1−β) = p(q,Y,P)− . qβ The negotiated price is the final sales price, less the amount required to compensate the producer for the sunk cost to start up the business relationship. Notice that if s = 0 then the negotiated price would be the final sales price as in standard trade models. A.9 Bounding the search friction Recall the definition (r+λ)(1−β) γ ≡ . r+λ+βκχ(κ) Here we show that γ ∈ [0,1]. First, because all parameters are positive, γ ≥ 0. The lower bound, γ = 0, is reached only when β = 1 and c = 0 simultaneously. Second, prove that γ ≤ 1 by contradiction. Assuming γ > 1 implies that 0 > βκχ(κ), which is a contradiction, as β ≥ 0 and κχ(κ) ≥ 0. A.10 Closed economy accounting We first work through the expenditure approach and then turn to the income approach in the closed-economy framework. 55

A.10.1 Expenditure approach The expenditure approach to accounting allows total resources in the economy, Y, to be allocated to consumption, C, investing in searching and creating firms, I, and funding the government, G. Y = C +I +G. Consumption expenditure is (cid:18) (cid:19) (cid:90) 1−v C = Nm p(ϕ)q(ϕ)dG(ϕ). 1−i ϕ¯ Investments in searching to create retailer–producer relationships, pay for period fixed goods production costs, and fund entry is I = vNmwc+uNx(wl+wsκχ(κ))+(1−u−i)Nxwf +Nmwe +Nxwe . m x Government is assumed to tax production cost at a proportional rate τ ≥ 1 and levy a lump-sum tax on household’s consumption to fund payments to idle firms. (cid:18) (cid:19) (cid:90) 1−u−i G = Nx (τ −1)t(q)dG(ϕ)+(1−i)Nxwh−(1−i)Nxwh 1−i ϕ¯ (cid:18) (cid:19) (cid:90) 1−u−i = Nx (τ −1)t(q)dG(ϕ). 1−i ϕ¯ To account for all resources in the economy, we assume the government spends the tax/tariff revenue raised by proportionally taxing variable production cost at rate τ ≥ 1, which enters equilibrium value p(ϕ)q(ϕ). According to this equation the government buys labor from the household but does not use the labor to produce anything. As such, those resources are wasted. We make this assumption in the closed economy because it follows the international trade treatment of iceberg trade costs in which resources simply ‘‘melt away’’ in transit. We could assume that tax revenues are refunded lump sum to the household in order to ensure that G = 0. In that case, taxes would still have a distortionary effect on production costs but would not reduce available resources. Combining the terms defined above through the expenditure accounting identity gives (cid:18) (cid:19) (cid:90) 1−v Y = Nm p(ϕ)q(ϕ)dG(ϕ) 1−i ϕ¯ + vNmwc+uNx(wl+wsκχ(κ))+(1−u−i)Nxwf + Nmwe +Nxwe m x (cid:18) (cid:19) (cid:90) 1−u−i + Nx (τ −1)t(q)dG(ϕ). 1−i ϕ¯ To simplify this expression, we assume that every matched producer must have one, and only one, retailer as their counterpart. Doing so implies the mass of matched producers and 56

retailers must be equal, (1−u−i)Nx = (1−v)Nm. We also use the definition of κ to substitute for Nm = κv−1uNx leading to the expression in the main text (cid:18) (cid:19) (cid:90) 1−u−i Y = Nx p(ϕ)q(ϕ)dG(ϕ) 1−i ϕ¯ + κuNxwc+uNx(wl+wsκχ(κ))+(1−u−i)Nxwf + κv−1uNxwe +Nxwe m x (cid:18) (cid:19) (cid:90) 1−u−i + Nx (τ −1)t(q)dG(ϕ). 1−i ϕ¯ We will get the same expression if we instead use the income approach to national accounting. A.10.2 Income approach Total income in the economy is the sum of variable operating profits, Π, income earned by variable factors used to produce goods, Φ , payments made in the search for business p relationships and to cover fixed production costs, Φ , and income to the factors that create i retailing and producing firms Φ , along with government income, Φ : e g Y = Π+Φ +Φ +Φ +Φ . p i e g We define aggregate operating profits as revenue, R, less variable input costs of production, Φ , so that Π = R−Φ . Profits are zero when production is zero. Excluding costs that do p p not vary with output from aggregate profit implies the economy faces aggregate fixed costs. For example, factors in our economy must be paid income Φ +Φ without production. If i e instead Π = R−Φ −Φ −Φ so that costs invariant to the level of output were included in p i e operating profits, income to fixed factors would enter the accounting identity as a transfer instead of as a fixed cost. Aggregate operating profits are the sum of retailer and producer profits, Π = Π +Π in m x which the sum is defined as: (cid:18) (cid:19) (cid:90) (cid:18) (cid:19) (cid:90) 1−v 1−u−i Π = Nm (p(q)q −nq)dG(ϕ)+ Nx (nq −t(q))dG(ϕ). 1−i 1−i ϕ¯ ϕ¯ Retailer profits include the revenue of selling the good to final consumers less the cost paid for the good to the producer. The latter is the producer’s revenue and the producer pays t(q) variable cost to produce q units of the good. Variable factors of production are paid (cid:18) (cid:19) (cid:90) 1−u−i Φ = Nx t(q)dG(ϕ). p 1−i ϕ¯ because each producer in a relationship pays variable production cost t(q). The search costs invested by unmatched retailers and producers in the hope of forming 57

relationships and the investment made before production can occur is given by Φ = vNmwc+uNx(wl+wsκχ(κ))+(1−u−i)Nxwf. i In order to come into existence, potential retailers and producers must pay an entry cost, e , and cost, e , in which both are measured in labor units m x Φ = Nmwe +Nxwe . e m x Finally, the government levies an ‘‘iceberg’’ tariff on the variable production cost of each good that is exchanged between retailers and producers. The tariff means that τ ≥ 1 units must be produced in order to ensure one unit of the good is exchanged. Total government income is therefore (cid:18) (cid:19) (cid:90) 1−u−i Φ = Nx (τ −1)t(q)dG(ϕ). g 1−i ϕ¯ Using the income approach our model yields (cid:18) (cid:19) (cid:90) (cid:18) (cid:19) (cid:90) 1−v 1−u−i Y = Nm (p(q)q −nq)dG(ϕ)+ Nx (nq −t(q))dG(ϕ) 1−i 1−i ϕ¯ ϕ¯ (cid:18) (cid:19) (cid:90) 1−u−i + Nx t(q)dG(ϕ) 1−i ϕ¯ + vNmwc+uNx(wl+wsκχ(κ))+(1−u−i)Nxwf + Nmwe +Nxwe m x (cid:18) (cid:19) (cid:90) 1−u−i + Nx (τ −1)t(q)dG(ϕ). 1−i ϕ¯ As in the expenditure approach, to simplify this expression we must assume one-to-one matches so that (1−u−i)Nx = (1−v)Nm. We again use the definition of κ to substitute for Nm = κv−1uNx. Using these two expressions above and canceling terms yields (cid:18) (cid:19) (cid:90) 1−u−i Y = Nx p(q)qdG(ϕ) 1−i ϕ¯ + κuNxwc+uNx(wl+wsκχ(κ))+(1−u−i)Nxwf + κv−1uNxwe +Nxwe m x (cid:18) (cid:19) (cid:90) 1−u−i + Nx (τ −1)t(q)dG(ϕ). 1−i ϕ¯ This expression is precisely the one we got with the expenditure approach. In summary, our model gives a resource constraint that is consistent with both income and expenditure approaches. 58

A.11 Equilibrium equations for closed economy First consider the fully general definition of the equilibrium: Definition 4. A steady-state competitive equilibrium consists of eight endogenous variables p, q, n, κ, ϕ¯, Nx, Nm, w, which jointly solve eight equations given by (9), (11), (12), (13), (16), (17), (18), (19) given exogenous parameters β, c, f, h, l, λ, r, s, τ, exogenous productivity distribution, G(ϕ), and exogenous resource endowment, L. Definition 5. The equilibrium is comprised of three prices p, n, w, and five quantities q, κ, ϕ¯, Nx, Nm. These eight endogenous variables jointly solve 8 equations. 1. Firms maximize profits (a) Equation (9) ensures retailers and producers maximize total profits within each match. (b) Equation (11) ensures that producers and retailers only form matches for varieties that will give positive profits. (c) Equation (12) ensures retailers maximize expected profits from creating vacancies by continuing to do so until the expected cost to create a vacancy equals the expected benefit. (d) Equation (13) ensures retailers and producers divide total profits within each match in a way that maximizes their own individual profits. 2. Markets clear (a) Equation (17) clears the market for creating firms. (b) Equation (16) clears the matched relationship market by ensuring the mass of matched retailers equals the mass of matched producers. (c) Equation (18) ensures the labor market clears. 3. Consumers maximize utility (a) Equation (19) ensures consumers maximize utility. In the closed economy, a steady-state competitive equilibrium consists of six endogenous variables p, q, n, κ, ϕ¯, and P, and six equations. A representative consumer maximizes utility given aggregate income, Y, and the ideal price index, P, as in equation (19), which defines the demand curve p(q). Bargaining over q between the producer and retailer yields equation (9) and defines q as a function of w, P, and Y p(q)+p(cid:48)(q)q = t(cid:48)(q), in which P and Y enter consumer demand and w enters the cost function. 59

Three equations are needed to solve jointly for ϕ¯, κ, and p. Equation (11) defines ϕ¯, the reservation productivity for producer participation in search, as a function of p(q), q, and κ (cid:18) (cid:19) (cid:18) (cid:19) (r+λ) (r+λ) (r+λ) p(q)q −t(q) = f + l+ 1+ h+ s. βκχ(κ) βκχ(κ) β As a reminder, q = q(ϕ¯) and κ = κ(ϕ¯), which highlights that this equation implicitly defines ϕ¯. Equation (12), which comes from free entry into retailing, is an implicit function of κ and defines κ as a function of p(q), q, n, and ϕ¯. (cid:90) c = M (ϕ)dG(ϕ). χ(κ) ϕ¯ Equation (13), a result of bargaining between retailers and producers over the trading price, defines n as a function of p(q), q, and κ t(q)+f −l−κχ(κ)s n(ϕ) = [1−γ]p(q)+γ . q The aggregate price index is defined by (cid:20)(cid:18) 1−u−i (cid:19) (cid:90) ∞ (cid:21) 1− 1 σ P = Nx p(ϕ)1−σdG(ϕ) . 1−i ϕ¯ Notice that this is really just a function of κ, Nx, ϕ¯, and p(ϕ). It should be clear that the rest of the equilibrium objects can be derived from these endogenous variables. Equation (15) defines i as a function of ϕ¯ (cid:90) ϕ¯ i = dG(ϕ) = G(ϕ¯). 1 Equation (14) defines u as a function of ϕ¯, via i and κ λ(1−i) u = . λ+κχ(κ) The vacancy rate is defined by κuNx v = , Nm which is a function of Nm, Nx, κ, and u. We assume as in Chaney (2008) that the number of producers is proportional to aggregate output and this pins down Nx. In particular, Nx = Y/(1+π˜), in which π˜ is the share of profits from domestic retailers and producers rebated back to domestic consumers. In turn, the number of retailers, Nm, is given by one-to-one matches and the equilibrium condition (1−v)Nm = (1−u−i)Nx, which states that every retailer must have a producing partner. In the closed economy we normalize the price of labor, w, to one so that 60

labor is the numeraire. The measure of workers in the economy, L, is given exogenously. Aggregate output, Y, is given by wL. A.12 Comparative statics in the closed economy A.12.1 Proof of Proposition 4 Here are our six equations: (cid:90) c F1 : − M (κ,p,q,w,ϕ,Y;Ω)dG(ϕ) = 0 χ(κ) ϕ¯ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) r+λ r+λ r+λ F2 : π(q,p,P,w,ϕ¯,Y;Ω)−f − l− 1+ h− s = 0 βκχ(κ) βκχ(κ) β F3 : p−p(q,Y,P) = 0 F4 : −P +P (u,i,N ,ϕ¯,p(q,P,Y)) = 0 x F5 : n(κ,p(q,P,Y),q,γ,w | Ω)q(Y,P)−[1−γ]p(q,Y,P)q(Y,P) −γ(t(q,w | ϕ,τ)+f −l−κχ(κ)s) = 0 F6 : q = q(Y,P). Take the derivative of F1 with respect to c and group the partials to get (cid:34) (cid:35) c (cid:90) ∂M ∂κ ∂ϕ¯ ∂P (cid:90) ∂M ∂q 1 − χ(cid:48)(κ)+ dG(ϕ) +M(κ,p,q,w,ϕ¯,Y;Ω)g(ϕ¯) − dG(ϕ) = − , χ(κ)2 ∂κ ∂c ∂c ∂c ∂q ∂P χ(κ) ϕ¯ ϕ¯ ∂p in which we assume that = 0. This assumption implies that the supply curve for the ∂q good is flat. This assumption is true in our trade application, as the final sales price is simply a markup over marginal cost and the marginal cost is constant. Taking the derivative of F2 with respect to c gives: ∂π(q,p,P,w,ϕ¯,Y;Ω)∂q ∂π(q,p,P,w,ϕ¯,Y;Ω)∂p ∂π(q,p,P,w,ϕ¯,Y;Ω)∂P + + + ∂q ∂c ∂p ∂c ∂P ∂c (cid:18) (cid:19) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂ϕ¯ r+λ ∂κ + (1−η(κ))(h+l) = 0. ∂ϕ¯ ∂c βκ2χ(κ) ∂c Taking the derivative of F3 with respect to c gives: ∂p ∂p∂q ∂p ∂P − − = 0, ∂c ∂q ∂c ∂P ∂c 61

and as in the simple case, equilibrium p is just markup over marginal cost, we have that ∂p ∂p = 0 and = 0, so we are left with ∂q ∂P ∂p = 0. ∂c Taking the derivative of F4 with respect to c gives: ∂P ∂P ∂u∂κ ∂P ∂i ∂ϕ¯ ∂P ∂ϕ¯ ∂P ∂p∂q ∂P ∂p − + + + + + = 0, ∂c ∂u ∂κ ∂c ∂i ∂ϕ¯ ∂c ∂ϕ¯ ∂c ∂p ∂q ∂c ∂p ∂c ∂p and as the equilibrium price p is just a markup over marginal cost, we have that = 0 and ∂q ∂p = 0, so we end up with ∂P (cid:18) (cid:19) ∂P ∂u∂κ ∂P ∂i ∂P ∂ϕ¯ ∂P + + − = 0. ∂u ∂κ ∂c ∂i ∂ϕ¯ ∂ϕ¯ ∂c ∂c Taking the derivative of F5 with respect to c gives: (cid:20) (cid:21) ∂n ∂q ∂p ∂q ∂γ∂κ ∂γ∂κ ∂κ q+n −(1−γ) q+p + pq− (t+f −l−κχ(κ)s)+γχ(κ)(1−η(κ))s = 0 ∂c ∂c ∂c ∂c ∂κ ∂c ∂κ ∂c ∂c (cid:18) (cid:19) ∂γ ∂γ ∂κ ∂p ∂n ∂q pq− (t+f −l−κχ(κ)s)+γχ(κ)(1−η(κ))s −(1−γ)q + q+(n−(1−γ)p) = 0 ∂κ ∂κ ∂c ∂c ∂c ∂c (cid:18) (cid:19) ∂γ ∂κ ∂p ∂n ∂q (pq−(t+f −l−κχ(κ)s))+γχ(κ)(1−η(κ))s −(1−γ)q +q +(n−(1−γ)p) = 0. ∂κ ∂c ∂c ∂c ∂c Taking the derivative of F6 with respect to c gives: ∂q ∂q ∂P − = 0 ∂c ∂P ∂c 62

The system therefore looks like this  1  −  χ(κ)   0     0  =   0    0  0 ∂κ − (cid:34) c χ(cid:48)(κ)+(cid:82) ∂M dG(ϕ) (cid:35) M(κ,p,q,w,ϕ¯,Y;Ω)g(ϕ¯) 0 (cid:82) ∂M(κ,p,q,w,ϕ,Y;Ω) ∂q dG(ϕ) 0 0  ∂c χ(κ)2 ϕ¯ ∂κ ϕ¯ ∂q ∂P      ∂ ∂ ϕ c ¯        (cid:32) r+λ (cid:33) (1−η(κ))(h+l) ∂π(q,p,P,w,ϕ¯,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω) 0 ∂π(q,p,P,w,ϕ¯,Y;Ω)     ∂p βκ2χ(κ) ∂ϕ¯ ∂p ∂P ∂q       ∂ ∂ P c      ∂P 0 ∂u ∂P ∂i 0 ∂P 1 0 0 0   .   ∂c       ∂u ∂κ ∂i ∂ϕ¯ + ∂ϕ¯ 0 −1 0 0    ∂n∂γ    ∂c      ∂κ (pq−(t+f−l−κχ(κ)s))+γχ(κ)(1−η(κ))s 0 −(1−γ)q 0 q (n−(1−γ)p)    ∂q ∂q  0 0 0 − 0 1 ∂c ∂P By Cramer’s rule we know that the ith element of the partials vector will be determined by the determinant of the left-hand-side (LHS) matrix and the determinant of the matrix formed by replacing the ith column of the LHS matrix with the right-hand-side vector. We omit many of the details in this appendix about how to obtain and sign these determinants but details are available from the authors upon request. 63

A.12.2 Proof of Proposition 5: Comparative statics with respect to producer search cost The negotiated price does not interact with the four equations. Hence, we add the negotiated price to the system of equations and find its derivative with respect to l without redoing the derivatives for all the other equations. (cid:90) c F1 : − M (κ,p,q,w,ϕ,Y;Ω)dG(ϕ) = 0 χ(κ) ϕ¯ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) r+λ r+λ r+λ (cid:0) ¯ (cid:1) F2 : π q,p,P,w, ¯,Y;Ωϕ −f − l− 1+ h− s = 0 βκχ(κ) βκχ(κ) β F3 : p−p(q,Y,P) = 0 F4 : −P +P (u,i,N ,ϕ¯,p(q,P,Y)) = 0 x F5 : n(κ,p(q,P,Y),q,γ,w | Ω)q−[1−γ]p(q,Y,P)q−γ(t(q,w | ϕ,τ)+f −l−κχ(κ)s) = 0 F6 : q = q(Y,P), (r+λ)(1−β) pq −t−f +l+sκχ(κ) in which γ = and M (κ,p,q,w,ϕ,Y;Ω) = (1−β) . r+λ+βκχ(κ) r+λ+βκχ(κ) Differentiate F1 with respect to l: c ∂κ (cid:26) ∂ϕ¯ (cid:90) ∂M ∂M ∂κ ∂M ∂p ∂M ∂q ∂P (cid:27) − χ(cid:48)(κ) − − M(κ,p,q,w,ϕ,Y;Ω)g(ϕ¯)+ + + + dG(ϕ) = 0 χ(κ)2 ∂l ∂l ϕ¯ ∂l ∂κ ∂l ∂p ∂l ∂q ∂P ∂l c ∂κ (cid:26) ∂ϕ¯ (cid:90) ∂M ∂κ ∂M ∂M ∂q ∂P (cid:27) (cid:90) ∂M − χ(cid:48)(κ) − − M(κ,p,q,w,ϕ,Y;Ω)g(ϕ¯)+ + 0+ dG(ϕ) = dG(ϕ) χ(κ)2 ∂l ∂l ϕ¯ ∂κ ∂l ∂p ∂q ∂P ∂l ϕ¯ ∂l c ∂κ ∂ϕ¯ (cid:90) ∂M ∂κ ∂M ∂q ∂P (cid:90) ∂M − χ(cid:48)(κ) + M(κ,p,q,w,ϕ,Y;Ω)g(ϕ¯)− + dG(ϕ) = dG(ϕ) χ(κ)2 ∂l ∂l ϕ¯ ∂κ ∂l ∂q ∂P ∂l ϕ¯ ∂l (cid:20) c (cid:90) ∂M (cid:21)∂κ ∂ϕ¯ ∂P (cid:90) ∂M ∂q (cid:90) ∂M − χ(cid:48)(κ)+ dG(ϕ) + M(κ,p,q,w,ϕ¯,Y;Ω)g(ϕ¯)− dG(ϕ) = dG(ϕ). χ(κ)2 ϕ¯ ∂κ ∂l ∂l ∂l ϕ¯ ∂q ∂P ϕ¯ ∂l Differentiate F2 with respect to to l: 64

∂π(q,p,P,w,ϕ¯,Y;Ω)∂q ∂π(q,p,P,w,ϕ¯,Y;Ω)∂p ∂π(q,p,P,w,ϕ¯,Y;Ω)∂P + + ∂q ∂l ∂p ∂l ∂P ∂l (cid:18) (cid:19) (cid:18) (cid:19) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂ϕ¯ r+λ ∂κ r+λ + + (1−η(κ))(h+l) = ∂ϕ¯ ∂l βκ2χ(κ) ∂l βκχ(κ) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂q ∂π(q,p,P,w,ϕ¯,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂P + 0+ ∂q ∂l ∂p ∂P ∂l (cid:18) (cid:19) (cid:18) (cid:19) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂ϕ¯ r+λ ∂κ r+λ + + (1−η(κ))(h+l) = ∂ϕ¯ ∂l βκ2χ(κ) ∂l βκχ(κ) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂q ∂π(q,p,P,w,ϕ¯,Y;Ω)∂P + ∂q ∂l ∂P ∂l (cid:18) (cid:19) (cid:18) (cid:19) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂ϕ¯ r+λ ∂κ r+λ + + (1−η(κ))(h+l) = . ∂ϕ¯ ∂l βκ2χ(κ) ∂l βκχ(κ) Differentiate F3 with respect to to l: ∂p ∂p∂q ∂p ∂P − − = 0 ∂l ∂q ∂l ∂P ∂l and as in the simple case, equilibrium p is just markup over marginal cost, we have that ∂p ∂p = 0 and = 0, so we are left with ∂q ∂P ∂p = 0. ∂l Differentiate F4 with respect to to l: ∂P ∂P ∂u∂κ ∂P ∂i ∂ϕ¯ ∂P ∂ϕ¯ ∂P ∂p∂q ∂P ∂p − + + + + + = 0, ∂l ∂u ∂κ ∂l ∂i ∂ϕ¯ ∂l ∂ϕ¯ ∂l ∂p ∂q ∂l ∂p ∂l ∂p and as the equilibrium price p is just a markup over marginal cost, we have that = 0 and ∂q ∂p = 0, so we end up with ∂P ∂P ∂P ∂u∂κ ∂P ∂i ∂ϕ¯ ∂P ∂ϕ¯ − + + + = 0 ∂l ∂u ∂κ ∂l ∂i ∂ϕ¯ ∂l ∂ϕ¯ ∂l (cid:18) (cid:19) ∂P ∂u∂κ ∂P ∂i ∂P ∂ϕ¯ ∂P + + − = 0. ∂u ∂κ ∂l ∂i ∂ϕ¯ ∂ϕ¯ ∂l ∂l Differentiate F5 with respect to to l: 65

∂n ∂q (cid:20)∂p ∂q(cid:21) ∂γ∂κ ∂γ∂κ ∂κ q+n −(1−γ) q+p + pq− (t+f−l−κχ(κ)s)+γχ(κ)(1−η(κ))s +γ = 0 ∂l ∂l ∂l ∂l ∂κ ∂l ∂κ ∂l ∂l (cid:18)∂γ ∂γ (cid:19)∂κ ∂p ∂n ∂q pq− (t+f−l−κχ(κ)s)+γχ(κ)(1−η(κ))s −(1−γ)q + q+(n−(1−γ)p) = −γ ∂κ ∂κ ∂l ∂l ∂l ∂l (cid:18)∂γ (cid:19)∂κ ∂p ∂n ∂q (pq−(t+f−l−κχ(κ)s))+γχ(κ)(1−η(κ))s −(1−γ)q +q +(n−(1−γ)p) = −γ. ∂κ ∂l ∂l ∂l ∂l Differentiate F6 with respect to to l: ∂q ∂q ∂P − = 0. ∂l ∂P ∂l 66

Hence, the system looks like (cid:82) ∂M  dG(ϕ)   ϕ (cid:18) ¯ ∂ r l +λ (cid:19)      βκχ(κ)   =  0     0     −γ  0 ∂κ − (cid:20) c χ(cid:48)(κ)+ (cid:82) ∂M dG(ϕ) (cid:21) M(κ,p,q,w,ϕ¯,Y;Ω)g(ϕ¯) 0 (cid:82) ∂M ∂q dG(ϕ) 0 0   ∂l  χ(κ)2 ϕ¯ ∂κ ϕ¯ ∂q ∂P    ∂ϕ¯    (cid:18) r+λ (cid:19) ∂π(q,p,P,w,ϕ¯,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω)    ∂ ∂ p l       βκ2χ(κ) (1−η(κ))(h+l) ∂ϕ¯ 0 ∂P 0 ∂q      0 0 1 0 0 0   ∂ ∂ P l     ∂P ∂u (cid:18)∂P ∂i ∂P(cid:19)   .   + 0 −1 0 0    ∂ ∂ n l      ∂γ ∂u ∂κ ∂i ∂ϕ¯ ∂ϕ¯      (pq−(t+f−l−κχ(κ)s))+γχ(κ)(1−η(κ))s 0 −(1−γ)q 0 q (n−(1−γ)p)   ∂l ∂κ  ∂q ∂q  0 0 0 − 0 1 ∂l ∂P 67

A.12.3 Proof of Proposition 6: Comparative statics with respect to iceberg trade cost We start with the system of equations (cid:90) c F1 : − M (κ,p,q,w,ϕ,Y;Ω)dG(ϕ) = 0 χ(κ) ϕ¯ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) r+λ r+λ r+λ F2 : π(q,p,P,w,ϕ¯,τ,Y;Ω)−f − l− 1+ h− s = 0 βκχ(κ) βκχ(κ) β F3 : −P +P (u,i,N ,ϕ¯,p(q,P,Y)) = 0 x F4 : n(κ,p(q,P,Y),q,γ,w | Ω)q(Y,P)−[1−γ]p(q,Y,P)q(Y,P) −γ(t(q,w | ϕ,τ)+f −l−κχ(κ)s) = 0 F5 : q = q(Y,P) F6 : p−p(q,Y,P) = 0. in which pq −t(τ)−f +l+sκχ(κ) M (κ,p,q,w,ϕ,Y;Ω) = . r+λ+βκχ(κ) The negotiated price, n, does not enter the other equilibrium equations so it largely does not change the analysis with three equations. Also, the final sales price is a markup over marginal cost so it does not interact with any of the other endogenous variables. We know, ∂p however, that > 0. We will use this in the analysis below without using F6 above. ∂τ Take the derivative of F1 with respect to τ − c χ(cid:48)(κ) ∂κ − (cid:40) − ∂ϕ¯ M(κ,p,q,w,ϕ,Y;Ω)g(ϕ¯)+ (cid:90) ∂M + ∂M ∂κ + ∂M ∂q ∂P + ∂M ∂p dG(ϕ) (cid:41) = 0 χ(κ)2 ∂τ ∂τ ϕ¯ ∂τ ∂κ ∂τ ∂q ∂P ∂τ ∂p ∂τ ∴− c χ(cid:48)(κ) ∂κ − ∂κ(cid:90) ∂M dG(ϕ)− ∂P (cid:90) ∂M ∂q dG(ϕ)+ ∂ϕ¯ M(κ,p,q,w,ϕ,Y;Ω)g(ϕ¯) = (cid:90) ∂M + ∂M ∂p dG(ϕ) χ(κ)2 ∂τ ∂τ ϕ¯ ∂κ ∂τ ϕ¯ ∂q ∂P ∂τ ϕ¯ ∂τ ∂p ∂τ − (cid:32) c χ(cid:48)(κ)+ (cid:90) ∂M dG(ϕ) (cid:33)∂κ +M(κ,p,q,w,ϕ,Y;Ω)g(ϕ¯) ∂ϕ¯ − ∂P (cid:90) ∂M ∂q dG(ϕ) = (cid:90) ∂M + ∂M ∂p dG(ϕ). χ(κ)2 ϕ¯ ∂κ ∂τ ∂τ ∂τ ϕ¯ ∂q ∂P ϕ¯ ∂τ ∂p ∂τ ∂M Note: There is no term because M does not depend directly on P. Also, we cannot ∂P ∂p pull out out of the integral on the right hand side (RHS) because p = p(w,τ,ϕ) depends ∂τ on productivity. Taking the derivative of F2 with respect to τ gives 68

∂π(q,p,P,w,ϕ¯,τ,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂q ∂π(q,p,P,w,ϕ¯,Y;Ω)∂p ∂π(q,p,P,w,ϕ¯,τ,Y;Ω)∂ϕ¯ + + + ∂τ ∂q ∂τ ∂p ∂τ ∂ϕ¯ ∂τ ∂π(q,p,P,w,ϕ¯,τ,Y;Ω)∂P (cid:18) r+λ (cid:19) ∂κ + + (1−η(κ))(h+l) =0 ∂P ∂τ βκ2χ(κ) ∂τ ∂π(q,p,P,w,ϕ¯,τ,Y;Ω)∂ϕ¯ ∂π(q,p,P,w,ϕ¯,τ,Y;Ω)∂P (cid:18) r+λ (cid:19) ∂κ ∂π(q,p,P,w,ϕ¯,Y;Ω)∂q + + (1−η(κ))(h+l) + ∂ϕ¯ ∂τ ∂P ∂τ βκ2χ(κ) ∂τ ∂q ∂c ∂π(q,p,P,w,ϕ¯,τ,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂p =− − , ∂τ ∂p ∂τ ∂π(q,p,P,w,ϕ¯,Y;Ω) and we have the term because profits depend directly on τ through ∂q ∂π(q,p,P,w,ϕ¯,Y;Ω)∂p the cost function and we treat as a constant in the system. ∂p ∂τ Taking the derivative of F3 with respect to τ gives ∂P ∂P ∂u∂κ ∂P ∂i ∂ϕ¯ ∂P ∂ϕ¯ ∂P ∂p − + + + + = 0 ∂τ ∂u ∂κ∂τ ∂i ∂ϕ¯∂τ ∂ϕ¯ ∂τ ∂p ∂τ (cid:18) (cid:19) ∂P ∂u∂κ ∂P ∂i ∂P ∂ϕ¯ ∂P ∂P ∂p + + − = − . ∂u ∂κ∂τ ∂i ∂ϕ¯ ∂ϕ¯ ∂τ ∂τ ∂p ∂τ Taking the derivative of F4 with respect to τ gives (cid:20) (cid:21) ∂n ∂q ∂p ∂q ∂γ∂κ ∂γ∂κ ∂κ q+n −(1−γ) q+p + pq− (t+f −l−κχ(κ)s)+γχ(κ)(1−η(κ))s = 0 ∂τ ∂τ ∂τ ∂τ ∂κ∂τ ∂κ∂τ ∂τ (cid:18) (cid:19) ∂γ ∂γ ∂κ ∂p ∂n ∂q pq− (t+f −l−κχ(κ)s)+γχ(κ)(1−η(κ))s −(1−γ)q + q+(n−(1−γ)p) = 0 ∂κ ∂κ ∂τ ∂τ ∂τ ∂τ (cid:18) (cid:19) ∂γ ∂κ ∂n ∂q ∂p (pq−(t+f −l−κχ(κ)s))+γχ(κ)(1−η(κ))s +q +(n−(1−γ)p) =(1−γ)q . ∂κ ∂τ ∂τ ∂τ ∂τ Taking the derivative of F5 with respect to τ gives ∂q ∂q ∂P − = 0. ∂τ ∂P ∂τ 69

The system therefore looks like this  (cid:82) ∂M ∂M ∂p  + dG(ϕ) ϕ¯ ∂τ ∂p ∂τ    ∂π(q,p,P,w,ϕ¯,τ,Y;Ω) ∂π(q,p,P,w,ϕ¯,Y;Ω)∂p − −   ∂τ ∂p ∂τ    ∂P ∂p =  −   ∂p ∂τ   ∂p     (1−γ)q  ∂τ 0  (cid:32) (cid:33)  ∂κ − c χ(cid:48)(κ)+ (cid:82) ∂M dG(ϕ) M(κ,p,q,w,ϕ¯,Y;Ω)g(ϕ¯) (cid:82) ∂M ∂q dG(ϕ) 0 0  χ(κ)2 ϕ¯ ∂κ ϕ¯ ∂q ∂P             ∂ ∂ ∂ ∂ ∂ ∂ ∂ P ϕ n τ τ τ τ ¯                        (cid:18) ∂γ [pq−(t (cid:18) + β f κ r 2 − + χ l ( λ − κ) κ (cid:19) χ ∂ ∂ ( P u ( 1 κ − ∂ ∂ ) κ u s η )] (κ + )) γ ( χ h ( + κ) l ( ) 1−η(κ))s (cid:19) ∂π( (cid:18) q, ∂ ∂ p P i ,P ∂ ∂ , ϕ ∂ i ¯ w 0 ϕ¯ , + ϕ¯, ∂ ∂ τ P ϕ¯ ,Y (cid:19) ;Ω) ∂π(q,p,P, ∂ − w 0 P 1 ,ϕ¯,τ,Y;Ω) 0 0 q ∂π( ( q n ,p − ,P (1 ∂ , 0 w q − , γ ϕ¯ ) , p Y ) ;Ω)             . ∂q  ∂κ   ∂q  ∂τ 0 0 − 0 1 ∂P By Cramer’s rule we know that the ith element of the partials vector will be determined by the determinant of the LHS matrix and the determinant of the matrix formed by replacing the ith column of the LHS matrix with the RHS vector. We omit many of the details in this appendix about how to obtain and sign these determinants but details are available from the authors upon request. 70

A.13 Proof of Proposition 7: Comparing the competitive economy and the social planner’s problem In general, the social planner (SP) maximizes welfare subject to the equations of motion for u and Nx and the aggregate resource constraint in which we initiate the economy at the steady-state unemployment rate and the steady-state number of producers (cid:90) ∞ C max e−rtdt κ P 0 s.t. (uN ˙ x) = u˙Nx +uN ˙x (cid:16) (cid:17) N ˙x = d u˙,u,N ˙x,Nx,κ λ(1−i) u(0) = λ+κχ(κ) Nx(0) = Nx Y = C +I +G. (cid:16) (cid:17) Here d u˙,u,N ˙x,Nx,κ is some function characterizing the dynamics of Nx. The predetermined variables u and Nx are constrained by initial conditions to start at their steady-state values. We assume that the economy is endowed with a number of producers given by Nxe = ζL, with ζ ∈ (0,∞), and L given exogenously. These assumptions imply x that N ˙x = 0 and removes the equation of motion for Nx and the corresponding initial condition. We set G = 0 by assuming that there are no taxes, τ = 1, or that tax revenue is rebated lump sum back to the consumer. Along with this assumption, the additional endowment means that Y w(1+ζ)e x = . Nx ζ As such, after rearranging the resource constraint and dividing through by Nx and P, we can write C Y I = − NxP NxP NxP w(1+ζ)e κuwc−u(wl+wsκχ(κ))−(1−u−i)wf −κv−1uwe −we x m x = − . ζP P Here we have written the parameters c, l, s, f, e , and e in labor units instead of dollars, m x as presented in the main text. There is no issue with this; we could have rewritten the whole model in terms of labor units and we would obtain this expression for real consumption. We normalize the real wage to one so that w/P = 1, which yields C (1+ζ)e = x −κuc−u(l+sκχ(κ))−(1−u−i)f −κv−1ue −e . NxP ζ m x 71

We show in appendix A.7 that e = 0. Furthermore, scaling the Hamiltonian by a constant, m Nx, makes no difference to the solution of the optimal control problem, so we can maximize C/(NxP) instead of C/P as long as we divide our constraint by Nx as well: (uN ˙ x) = u˙Nx +uN ˙x = u˙Nx = λ(1−u−i)Nx −κχ(κ)Nx (uN ˙ x) = u˙ = λ(1−u−i)−κχ(κ). Nx After making these assumptions and substituting the resource constraint into the objective function, the SP’s problem becomes (cid:90) ∞(cid:20) (1+ζ)e (cid:21) max x −κuc−u(l+sκχ(κ))−(1−u−i)f −e e−rtdt x κ ζ 0 s.t. u˙ = λ(1−u)−uκχ(κ) λ(1−i) u(0) = . λ+κχ(κ) The current value Hamiltonian associated with this problem is (1+ζ)e x H = −κuc−u(l+sκχ(κ))−(1−u−i)f −e +ν[λ(1−u)−uκχ(κ)]. x ζ The first order conditions that solve this problem are ∂ϕ¯ H : −uc−us(κχ(cid:48)(κ)+χ(κ))+g(ϕ¯) f −νu(χ(κ)+κχ(cid:48)(κ)) = 0 κ ∂κ ∂ϕ¯ H : −κc−(l+sκχ(κ))+f +g(ϕ¯) f −νλ−νκχ(κ) = −ν˙ +rν. u ∂u with transversality condition lim e−rtνu = 0. t→∞ Solving H = 0 for ν gives κ −c g(ϕ¯) ∂ϕ¯ ν = −s+ f, χ(κ)(1−η(κ)) uχ(κ)(1−η(κ))∂κ in which we define the elasticity of the matching function with respect to κ as κχ(cid:48)(κ) −η(κ) ≡ so that χ(κ)+κχ(cid:48)(κ) = χ(κ)(1−η(κ)). Differentiating ν with respect χ(κ) 72

to time gives (λ(1−u)−uκχ(κ)) ∂ϕ¯ ν˙ = − g(ϕ¯) f, u2χ(κ)(1−η(κ)) ∂κ ∂ϕ¯ ∂ϕ¯ in which κ˙ = 0 ensures that = κ˙ = 0. Notice that we substitute the equation of ∂t ∂κ motion for u in place of u˙. Using ν and ν˙ and many simplifying steps, we can write the first order condition H = −ν˙ +rν as u  ∂ϕ¯  [λ+ur]g(ϕ¯) f c = (1−η(κ))  ∂κ −s(r+λ+κχ(κ))−g(ϕ¯) ∂ϕ¯ f−f+l+sκχ(κ). (50) χ(κ) r+λ+η(κ)κχ(κ)u2χ(κ)(1−η(κ)) ∂u  The comparable expression from the competitive equilibrium (CE) economy is (cid:90) c (1−β) = (p(q)q −t(q)−f +l+sκχ(κ))dG(ϕ). (51) χ(κ) r+λ+βκχ(κ) ϕ¯ It is often the case that extreme constellations of parameters are required for the CE in search models to be efficient; Kudoh and Sasaki (2011) is one such example. Using equations (50) and (51), when c > 0, it appears that sufficient conditions for the CE and SP to result in the same κ are β = η(κ) = 1, which imposes that the matching function exhibits constant elasticity equal to β and that foreign producers receive all match surplus. In this extreme case, however, the retailing importer has no incentive to be in a relationship with a foreign producer. A retailer must have some bargaining power, β < 1, to recoup search cost c > 0. These apparent sufficient conditions lead to a contradiction and are therefore not sufficient. The case in which a retailing importer has some bargaining power, β < 1, the CE and the SP’s coincide when equation (50) equals equation (51). It is evident that they are not generally equal, but we can consider a few special cases. If f = l = s = 0, then setting equation (51) equal to (50) provides (cid:90) (p(q)q −t(q))dG(ϕ) = 0, ϕ¯ which is never true as long as variable profits are increasing in productivity. If f = s = 0, then it would have to be that β = η(κ) and (cid:90) (p(q)q −t(q)+l)dG(ϕ) = l, ϕ¯ which defines a knife-edge requirement for the endogenous quantities in the model. If f = 0, then we need that β = η(κ) and (cid:90) (p(q)q −t(q)+l+sκχ(κ))dG(ϕ) = l−s(r+λ), ϕ¯ 73

which also defines a knife-edge requirement for the endogenous quantities in the model. ∂ϕ¯ ∂ϕ¯ If l = h = 0, then = 0. If it is also the case that = 0 and β = η(κ), then ∂κ ∂u efficiency requires (cid:90) (p(q)q −t(q)+f +sκχ(κ))dG(ϕ) = −f −s(r+λ), ϕ¯ which can never hold, as p(ϕ¯)q(ϕ¯)−t(ϕ¯) ≥ 0. 74

B Appendix for the open economy B.1 The open economy ideal price index The ideal price index for differentiated goods in the destination/domestic market will take a form similar to the index in Chaney (2008): (cid:34) (cid:35) 1 O 1−σ (cid:88) P = P1−σ , d dk k=1 in which the price index for goods trade between destination d and origin o is given by (cid:20)(cid:18) 1−u −i (cid:19) (cid:90) ∞ (cid:21) 1− 1 σ P = do do Nx p (ϕ)1−σdG(ϕ) . do 1−i o do do ϕ¯ do Combining these two expression gives (cid:34) (cid:35) 1 (cid:88) O (cid:18) 1−u −i (cid:19) (cid:90) ∞ 1−σ P = dk dk Nx p (ϕ)1−σdG(ϕ) . d 1−i k k dk ϕ¯ k=1 dk The utility function from equation (22) includes both a homogeneous and differentiated (cid:18) p (cid:19)1−α(cid:18) P (cid:19)α 1 d good so that the overall price index is given by Ξ = . The ideal price d 1−α α index relies on the final sales price and not the price negotiated between retailer and producers. Following Chaney (2008), we assume that the number of producers in the origin market that take a draw from the productivity distribution is proportional to the size of the economy, Y . The basic intuition behind this is that larger economies have a larger stock of o potential entrepreneurs. To make this explicit, we denote the total mass of potential entrants as Nx = ξ Y , in which the proportionality constant ξ ∈ [0,∞) captures exogenous o o o o structural factors that affect the number of potential entrants in country k. Among others, these could include such factors as literacy levels and attitudes toward entrepreneurship. Because the number of producers is fixed, they make profits that are then split between retailers and producers. We assume that a global mutual fund collects profits from retailers and producers and redistributes them as π dividends per share to each worker who owns w o shares. Total income in each country is labor income w L plus dividends income, o o Y = (1+π)w L . With these assumptions, the number of producers is o o o Y Y Nx = ξ (1+π)w L . It will be useful to write this as Nx = o , which we simplify o o o o o (1+π) Y by setting ξ = 1. o Another perspective on these conditions is to view the economy as being endowed with labor of value w L and the number of producers, the value of which is proportional to the o o labor endowment, Nxex = ζ w L , in which ζ ∈ [0,∞) and ex ≥ 0 is the setup cost to o o o o o o o 75

ζ create each producer. We can rewrite this expression as Nx = o w L , which matches the o ex o o o ζ o discussion above when = ξ (1+π). ex o o With our functional form assumptions, demand for a variety, ϕ, in the differentiated αY goods sector is q (ϕ) = p (ϕ)−σ d . Given this demand, monopolistic competition and do do P1−σ d constant returns-to-scale production imply retailers set optimal final sales prices of p (ϕ) = µw τ ϕ−1. Using this and the number of producers allows us to compute the do o do integral as (cid:32) (cid:33)1/(1−σ) Y (cid:88) O (cid:18) 1−u −i (cid:19) Y θϕ¯σ−θ−1 P = dk dk o (µw τ )1−σ dk , d k dk (1+π) 1−i Y θ−σ +1 dk k=1 θϕ¯σ−θ−1 in which (cid:82)∞ zσ−1dG(z) = . Now, using the productivity thresholds, ϕ¯ , defined ϕ¯ dk θ−σ +1 dk earlier we can write the price index as 1− 1 P = λ ×Y θ σ−1 ×ρ , d 2 d d in which (cid:32) (cid:33)−1 ρ ≡ (cid:88) O Y k (cid:18) κ dk χ(κ dk ) (cid:19) (w τ )−θF −[ σ− θ 1 −1] θ , d Y λ +κ χ(κ ) k dk dk dk dk dk k=1 and (cid:18) θ (cid:19)− θ 1 (cid:16)σ(cid:17) 1 −1 (cid:18) Y (cid:19)− θ 1 λ ≡ σ−1 θ µ . 2 θ−(σ −1) α 1+π B.2 Functional form implications Total income to workers in country d is Y = w L +w L π, in which w is the wage, L is d d d d d d d labor endowment, and π is the dividend per share paid by the global mutual fund. See Chaney (2008) for details regarding the global mutual fund. Given the consumer preferences in equation (22), income, and the ideal price index P , the inverse demand for each variety d within each sector is p(q) = (cid:0) αY d P d σ−1 (cid:1) σ 1 q− σ 1 . The price charged in the domestic market is defined by the marginal cost equals marginal revenue expression (9) from above. Given the functional form assumptions we have made for the inverse demand curve and cost functions, this becomes w τ o do p = µ , ϕ 76

σ in which µ = . Notice the price charged for the imported good in the domestic market σ −1 takes the standard markup over marginal cost form. Using the demand curve and domestic optimal price implies the imported quantity (cid:18) w τ (cid:19)−σ q = µ o do αY Pσ−1. ϕ d d We will use a general form for the production cost function w τ o do t(q ,ϕ,w ,| τ )−f = q +f , do o do do do do ϕ in which w is the wage in the exporting (origin) country, τ ≥ 1 is a parameter capturing o do one plus the iceberg transport cost between the domestic destination d and origin o, and f do is the corresponding fixed cost of production for the export market in units of the numeraire. w τ o do The firm that produces variety q has efficiency ϕ and marginal cost equal to . ϕ p(q) = (cid:0) αY d P d σ−1 (cid:1) σ 1 q− σ 1 (52) The price charged in the domestic market is defined by the marginal cost equals marginal revenue expression (9) from above. Given the functional form assumptions we have made for the inverse demand curve and cost functions, this becomes w τ o do p(ϕ) = µ , (53) ϕ σ in which µ = . Notice the price charged for the imported good in the domestic market σ −1 takes the standard markup over marginal cost form. Using the demand curve and domestic optimal price implies the imported quantity (cid:18) w τ (cid:19)−σ q(ϕ) = µ o do αY Pσ−1. ϕ d d Using the functional forms for the production cost function, consumer preferences, and the price index from equation (24), we know that final sales of each variety is (cid:18) w τ (cid:19)1−σ p(ϕ)q(ϕ) = µ o do αY Pσ−1. ϕ d d Putting this together with the definition of flow profits, π (ϕ) = p(ϕ)q(ϕ)−t (ϕ), we do do get that (cid:18) w τ (cid:19)1−σ π (ϕ) = µ o do αY Pσ−1 −αw τ (µw τ )−σY Pσ−1ϕσ−1 −f, do ϕ d d o do o do d d 77

which simplifies to α (cid:18) µw τ (cid:19)1−σ π (ϕ) = o do Y ϕσ−1 −f. do d σ P d B.3 Proof of proposition 8: Changes in real income We prove proposition 8 assuming monopolistic competition and following steps similar to those used to prove proposition 1 in Arkolakis et al. (2012). Because, with the exception of the search frictions, the functional form assumptions detailed in section 4, are the same as in Arkolakis et al. (2012), we can related the differentiated goods price index in our model from section 4.2.2 to the price index equation (A22) in Arkolakis et al. (2012, p. 123) (cid:18) (cid:19) 1−u −i P1−σ = do do (cid:0) PACR (cid:1)1−σ , (54) do 1−i do do in which (cid:0) PACR (cid:1)1−σ = Nx(µw τ )1−σΨ , and it will be useful to define the important do o o do do one-sided moment Ψ ≡ (cid:82)∞ zσ−1dG(z). Also define the elasticity of this integral with do ϕ¯ do ∂ln(Ψ ) do respect to the cutoff as ψ ≡ . A sufficient condition for ψ ≤ 0 is σ > 1. The do do ∂ln(ϕ¯ ) do overall price index with both homogeneous and differentiated goods is (cid:18) p (cid:19)1−α(cid:18) P (cid:19)α 1 d Ξ = , d 1−α α in which p is the price of the freely trade homogeneous good and α is the share of 1 consumption devoted to the differentiated goods bundle. Lastly, it will be very useful to denote the total derivative of the log of a variable x, as dlnx = ln(x(cid:48)/x) = ln(xˆ) and so exp(dlnx) = xˆ. B.3.1 Step 1: Small changes in per capita real income satisfy dln(W ) = −αdln(P ). (55) d d w d Per capital real income in country d is given by W ≡ , in which Ξ is the overall price d d Ξ d index and w is the wage. Using the price index gives d (cid:18) p (cid:19)α−1(cid:18) P (cid:19)−α 1 d W = w . d d 1−α α Normalizing w pα−1 = (1−α)α−1, then taking logs and totally differentiating gives equation d 1 (55). Throughout we will impose labor market clearing, which implies all labor is employed so that Y = w L . d d d 78

B.3.2 Step 2: Small changes in the consumer price index satisfy O (cid:20) (cid:18) (cid:19) (cid:88) λ 1−u −i dk do do dlnP = dln d α(1−σ +α−1ψ ) 1−i d do k=1 (cid:21) ψ dln(F ) + (1−σ +ψ )(dlnw +dlnτ )+dlnNx + do do , (56) do o do o σ −1 in which ψ is defined above and ψ = (cid:80)O λ ψ . do d k=1 dk dk Equation (56) is analogous to equation (A33) in Arkolakis et al. (2012, p. 125) other than a typo in the first multiplicative term in which they should have γ instead of γ . j ij When the utility function has only differentiated goods (α = 1) and there are no search frictions (u = 0), equations (A33) and (56) are the same. The sign on ψ and ψ differ do d do between the models because ours is defined in terms of productivity, while theirs is defined in terms of costs. We derive equation (56) by starting with total consumption in destination country d for the differentiated goods bundle from origin country o by integrating over all varieties at final prices. Because CES preferences define the differentiated goods aggregate given in equation (22), this integral is the value of CES demand for the bundle of country o products (cid:18) 1−u −i (cid:19) (cid:90) ∞ P1−σY C = do do Nx p (ϕ)q (ϕ)dG(ϕ) = α do d . do 1−i o do do P1−σ do ϕ¯ do d Define the consumption to income ratio (which in our model is different from the observed import share) as λ do C P1−σY P1−σ λ ≡ do = α do d = α do . do Y P1−σY P1−σ d d d d We will work with P1−σ using the definition of the price index for the differentiated good in d the destination market d, given by (cid:34) (cid:35) 1 O 1−σ (cid:88) P = P1−σ . d dk k=1 Take the log of this expression to get (1−σ)lnP = ln (cid:80)O P1−σ and then totally d k=1 dk differentiate to get (cid:88) O P1−σ dP1−σ (1−σ)dlnP = dk dk . d P1−σ P1−σ k=1 d dk 79

λ 1 do Rearrange λ to get = and then use this to simplify do αP1−σ P1−σ do d O (cid:88) λ (1−σ)dlnP = dk dlnP1−σ. d α dk k=1 Taking logs of equation (54) and totally differentiating gives (cid:18) (cid:19) 1−u −i dlnP1−σ = dln do do +dln (cid:0) PACR (cid:1)1−σ . do 1−i do do Employing our functional form assumptions, which gives (cid:0) PACR (cid:1)1−σ = Nx(µw τ )1−σΨ , do o o do do we can derive dln (cid:0) PACR (cid:1)1−σ = dlnNx +(1−σ)(dlnw +dlnτ )+ψ dln(ϕ¯ ), do o o do do do in which we use the chain rule to get dlnΨ = ψ dln(ϕ¯ ). do do do Putting these parts together gives O (cid:20) (cid:18) (cid:19) (cid:21) (1−σ)dlnP = (cid:88)λ dk dln 1−u do −i do +(1−σ)(dlnw +dlnτ )+dlnNx+ψ dln(ϕ¯ ) . (57) d α 1−i o do o do do do k=1 If we set α = 1 and u = 0, we match equation (A34) in Arkolakis et al. (2012, p. 125). do Next, take the log and total derivative of the cutoff expression from equation (25) (cid:18) (cid:19) 1 dln(ϕ¯ ) = −dln(P )+dln(τ )+ dln(F )+dln(w ), (58) do d do do o σ −1 which is the analog to equation (A36) in Arkolakis et al. (2012, p. 126). There are a few differences between equation (58) here and their equation (A36). First, the signs are reversed because we define everything in terms of productivity, while they use costs. Second, their term ξ captures the fixed cost of entry like our term F (see equation (A27) on page ij do 124). And while their term ρ allows for some foreign labor to be used to enter a foreign ij country, we do not. Making the same restriction in their model would require setting h = 1 ij and hence ρ = 1. ij Combining equations (57) and (58) gives equation (56). 80

B.3.3 Step 3: Small changes in the consumer price index satisfy O (cid:18) (cid:19) (cid:18) (cid:19) (cid:88) dln(λ )−dln(λ ) ψ −ψ do dd dd d dlnP = λ + dln(ϕ¯ ) d dk α(1−σ +α−1ψ ) α(1−σ +α−1ψ ) dd d d k=1 dlnNx + d α(1−σ +α−1ψ ) d ψ dln(F ) d dd + (σ −1)α(1−σ +α−1ψ ) d (cid:18) (cid:19) 1−u −i dd dd dln 1−i + dd . (59) α(1−σ +α−1ψ ) d If α = 1, u = 0, and F is a constant, then (59) becomes equation (A37) of Arkolakis do dd et al. (2012, p. 126). P1−σ λ P1−σ Start again with the consumption to income ratio λ = α do and form do = do . do P1−σ λ P1−σ d dd dd λ do Substitute into the ratio our functional form assumptions for the price index, take logs, λ dd and then totally differentiate to get dln(λ )−dln(λ ) = (1−σ)(dlnw +dlnτ )+ψ dln(ϕ¯ )−ψ dln(ϕ¯ ) do dd o do do do dd dd + dlnNx −dlnNx o d (cid:18) (cid:19) (cid:18) (cid:19) 1−u −i 1−u −i do do dd dd + dln −dln . (60) 1−i 1−i do dd In obtaining this expression we have simplified terms by recalling that we are considering a foreign shock so that dlnτ = 0 and that we have made the normalization dd w pα−1 = (1−α)α−1. We also assume that the homogeneous good price remains fixed so d 1 that dlnw = 0. We have ordered the terms as presented in Arkolakis et al. (2012, p. 126) d to make the comparison easy. We can derive two cutoff expressions dln(F ) do dln(ϕ¯ ) = −dln(P )+dln(τ )+ +dln(w ), do d do o σ −1 and also dln(F ) dd dln(ϕ¯ ) = −dln(P )+ , dd d σ −1 in which we again impose that τ = 1 and the normalization w pα−1 = (1−α)α−1. dd d 1 81

Combining these two cutoff expressions gives dln(F ) dln(F ) do dd dln(ϕ¯ ) = dln(ϕ¯ )+dln(w )+dln(τ )+ − , (61) do dd o do σ −1 σ −1 which is akin to the last equation of Arkolakis et al. (2012, p. 126) with the exception that they have a typo because the equal sign should be a minus sign. In our model, it is not necessarily the case that dln(F ) = 0 in response to a foreign shock because the effective dd entry cost, F , is an endogenous variable and not a parameter. dd Combine expression (61) with dln(λ )−dln(λ ) to get do dd dln(λ )−dln(λ ) = (1−σ +ψ )(dlnw +dlnτ ) do dd do o do (cid:18) (cid:19) dln(F ) dln(F ) do dd + ψ − (62) do σ −1 σ −1 + (ψ −ψ )dln(ϕ¯ )+dlnNx −dlnNx do dd dd o d (cid:18) (cid:19) (cid:18) (cid:19) 1−u −i 1−u −i do do dd dd + dln −dln . 1−i 1−i do dd Equation (62) is analogous to equation (A38) in Arkolakis et al. (2012, p. 127) which has a typo because α∗ should be α∗ . Substituting equation (62) into equation (56) and ij jj performing algebra gives equation (59). B.3.4 Step 4: Small changes in the consumer price index satisfy −dln(λ ) dlnNx dlnP = dd + d d α(1−σ +α−1(σ −1−θ)) α(1−σ +α−1(σ −1−θ)) (σ −1−θ)dln(F ) dd + (σ −1)α(1−σ +α−1(σ −1−θ)) (cid:18) (cid:19) 1−u −i dd dd dln 1−i + dd . (63) α(1−σ +α−1(σ −1−θ)) We depart somewhat from the approach taken in step 4 of Arkolakis et al. (2012, p. 127) in simplifying equation (59) to derive equation (63). They invoke macro-level restriction number 3, ‘‘R3: The import demand system is such that for any importer j and any pair of exporters i (cid:54)= j and i(cid:48) (cid:54)= j, εii(cid:48) = ε < 0 if i = i(cid:48), and zero otherwise.’’ As they describe on j page 103, this restriction imposes symmetry on the elasticity of the consumption ratio to changes in variable trade costs. That elasticity in our model in general is given by equation (69) and need not be symmetric across countries. A sufficient condition to derive equation (63), however, is that productivity distributions and consumer preferences are symmetric. For now, we impose those restrictions in the following steps but could likely relax them in future work. The term we need to consider from equation (59) is ψ = (cid:80)O λ ψ , which is the d k=1 dk dk 82

consumption share weighted average of the elasticity of the moment of the productivity dln(Ψ ) do distribution, in which ψ = . In section 4.2, we present functional forms for do dln(ϕ¯ ) do preferences and productivity. In particular, productivity ϕ ∈ [1,+∞) is Pareto distributed with CDF G[ϕ˜ < ϕ] = 1−ϕ−θ and PDF g(ϕ) = θϕ−θ−1 where, as usual, θ > σ −1 in order θϕ¯σ−θ−1 to close the model. With this distribution, the moment Ψ = do and the elasticity do θ−σ +1 ϕ¯σ θϕ¯−θ−1 ψ = − do do = −(θ−σ +1). Notice that the restriction that θ > σ −1 ensures do Ψ do ψ < 0 and ψ < 0. Also notice that ψ = ψ and the term we are actually interested in do d do dd becomes O (cid:88) ψ = λ ψ = σ −1−θ, (64) d dk do k=1 because by definition consumption shares 1 = (cid:80)O λ Substituting equation (64) into (59) k=1 dk and also using the fact that Euler’s Homogeneous Function Theorem gives (cid:80)O λ dln(λ ) = 0 provides (63). k=1 dk do B.3.5 Step 5: Small changes in the number of exporters satisfy −dln(λ ) dd dlnP = d α(1−σ +α−1(σ −1−θ)) (σ −1−θ)dln(F ) dd + (σ −1)α(1−σ +α−1(σ −1−θ)) (cid:18) (cid:19) 1−u −i dd dd dln 1−i + dd . (65) α(1−σ +α−1(σ −1−θ)) Restricted entry immediately gives dlnNx = 0 and free entry with total profits proportional d to output also gives dlnNx = 0. d B.3.6 Combining steps 1 to 5 into the final expression Combining equation (55) with equation (65) provides the change in real income in response to a foreign shock in our model 83

dln(λ ) dd dln(W ) = d (1−σ +α−1(σ −1−θ)) (σ −1−θ)dln(F ) dd − (σ −1)(1−σ +α−1(σ −1−θ)) (cid:18) (cid:19) 1−u −i dd dd dln 1−i − dd . (66) (1−σ +α−1(σ −1−θ)) The terms (1−σ +α−1(σ −1−θ)) ≤ 0 and 1−σ +θ ≤ 0. These inequalities imply that an increase (decrease) in the domestic consumption share lowers (raises) welfare, an increase (decrease) in the effective entry cost lowers (raises) welfare, and an increase (decrease) in the matched rate raises (lowers) welfare. Assume we only have differentiated goods (α = 1) so that (1−σ +α−1(σ −1−θ)) = −θ then equation (66) becomes (cid:18) (cid:19) 1−u −i dd dd dln dln(λ ) (σ −1−θ)dln(F ) 1−i dlnW = − dd + dd + dd . d θ θ(σ −1) θ Integrating this gives W ˆ = λ ˆ− θ 1 F ˆθ 1− σ− 1 1 (cid:18) 1− (cid:92) u dd (cid:19)θ 1 , (67) d dd dd 1−i dd (σ −1−θ) 1 1 in which = − ≤ 0. If effective entry costs are a parameter so that θ(σ −1) θ σ −1 u dd dln(F ) = 0 and we assume 1− ≡ 1, then dd 1−i dd ˆ ˆ−1 W = λ θ. d dd This is the expression implied by the steps in Arkolakis et al. (2012) if we use the functional form assumptions we made in section 4.2. B.4 Proof of Proposition 9: Consumption elasticity B.4.1 Relating price indexes Start with the functional form assumptions detailed in section 4. Because, with the exception of the search frictions, these functional form assumptions are the same as in Arkolakis et al. (2012) we can related the price index in our model given in section 4.2.2 to 84

the price index equation (A22) in Arkolakis et al. (2012, p. 123) (cid:0) PACR (cid:1)1−σ = Nx(µw τ )1−σΨ , do o o do do in which it will be useful to define Ψ = (cid:82)∞ ϕσ−1dG(ϕ) and the elasticity of this integral do ϕ¯ do ∂ln(Ψ ) do with respect to the cutoff ψ = ≤ 0 a sufficient condition for which is σ > 1. do ∂ln(ϕ¯ ) do B.4.2 Demand for a country’s bundle of goods We can derive total consumption in destination country d for the goods bundle from origin country o by integrating over all varieties at final prices. Because we have CES preferences, this integral is the value of CES demand for the bundle of country o products (cid:18) 1−u −i (cid:19) (cid:90) ∞ P1−σY C = do do Nx p (ϕ)q (ϕ)dG(ϕ) = α do d . do 1−i o do do P1−σ do ϕ¯ do d Define the consumption to income ratio (which we note in our model is different from the observed trade share) as λ do C P1−σY P1−σ λ = do = α do d = α do . do Y P1−σY P1−σ d d d d We can also form relative consumption ratios which is equivalent to Arkolakis et al. (2012) equation (21), page 110 and is just the ratio of the price indexes raised to a power λ C P1−σ do = do = do . λ C P1−σ dd dd dd Using the country specific price indexes given above we have (cid:18) (cid:19) 1−u −i do do (cid:0) PACR (cid:1)1−σ P1−σ 1−i do do = do , (cid:18) (cid:19) P1−σ 1−u −i dd dd dd (PACR) 1−σ 1−i dd dd in which we used the definition of PACR . Taking the log of relative consumption ratios do therefore gives ln (cid:18) C do (cid:19) =ln (cid:16)(cid:0) PACR(cid:1)1−σ (cid:17) −ln (cid:16)(cid:0) PACR(cid:1)1−σ (cid:17) +ln (cid:18) 1−u do −i do (cid:19) −ln (cid:18) 1−u dd −i dd (cid:19) . (68) C do dd 1−i 1−i dd do dd 85

B.4.3 Derivative of consumption ratio with respect to tariffs The goal is to derive two derivatives. The first is the direct effect of a change in the tariffs τ on the consumption ratio do ∂ln(C /C ) do dd = εdd. ∂ln(τ ) o do The second is the indirect effect, which documents how changing tariffs between a third country d(cid:48) and the origin o changes relative consumption in country d ∂ln(C /C ) do dd = εdd(cid:48). ∂ln(τ ) o d(cid:48)o B.4.4 Direct effect of tariff changes (d(cid:48) = d case) ∂ln(C /C ) We begin by deriving do dd = εdd in the most general form and then apply the ∂ln(τ ) o do assumptions made by Arkolakis et al. (2012). With that said, two assumptions that we will ∂ln(w ) ∂ln(w ) d o make from the start are that = 0 and = 0. The wage in the origin ∂ln(τ ) ∂ln(τ ) do do economy w is a function of the τ but only through general equilibrium. Another argument o do for how we can ignore how tariff changes affect the equilibrium wage is to normalize the w o relative wage = 1. w d B.4.5 First and second terms of equation (68) (d(cid:48) = d case) Differentiating and simplifying the first term of equation (68) gives ∂ (cid:16) (cid:17) ∂ln(Nx) ∂ln(ϕ¯ ) ln (cid:0) PACR (cid:1)1−σ = o +(1−σ)+ψ do , ∂ln(τ ) do ∂ln(τ ) do ∂ln(τ ) do do do and similarly ∂ (cid:16) (cid:17) ∂ln(Nx) ∂ln(ϕ¯ ) ln (cid:0) PACR (cid:1)1−σ = d +ψ dd . ∂ln(τ ) dd ∂ln(τ ) dd ∂ln(τ ) do do do Combining these gives ∂ ln (cid:16) (cid:0) PACR(cid:1)1−σ (cid:17) − ∂ ln (cid:16) (cid:0) PACR(cid:1)1−σ (cid:17) = (1−σ)+ψ ∂ln(ϕ¯ do ) −ψ ∂ln(ϕ¯ dd ) ∂ln(τ ) do ∂ln(τ ) dd do ∂ln(τ ) dd ∂ln(τ ) do do do do ∂ln(Nx) ∂ln(Nx) + o − d . ∂ln(τ ) ∂ln(τ ) do do The elasticities of the cutoffs ϕ¯ and ϕ¯ are related because changing tariff τ changes the do dd do price index P which changes the cutoff ϕ¯ . We can derive this relationship by d dd differentiating the explicit expression for the cutoff given in equation (25) 86

∂ln(ϕ¯ ) ∂ln(ϕ¯ ) do dd = 1+ ∂ln(τ ) ∂ln(τ ) do do (cid:18) (cid:19)(cid:20) (cid:21) 1 ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) do do do dd dd dd + − . σ −1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do do dd dd do Substituting this into the elasticity of the general expression for the ratio of relative price indexes and simplifying gives ∂ ln (cid:16) (cid:0) PACR(cid:1)1−σ (cid:17) − ∂ ln (cid:16) (cid:0) PACR(cid:1)1−σ (cid:17) = (1−σ)+ψ +(ψ −ψ ) ∂ln(ϕ¯ dd ) ∂ln(τ ) do ∂ln(τ ) dd do do dd ∂ln(τ ) do do do (cid:18) (cid:19)(cid:20) (cid:21) ψ ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) do do do do dd dd dd + − σ−1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do do dd dd do ∂ln(Nx) ∂ln(Nx) + o − d . ∂ln(τ ) ∂ln(τ ) do do The first line is the same as equation (21) in ACR (2012) except that ψ ≤ 0 in our case, do while γ ≥ 0 in the ACR expressions because we define our model in terms of productivity, ij ϕ, while they define theirs in terms of marginal cost. B.4.6 Elasticity of destination–origin market underutilization rate Next, we calculate the elasticity of destination–origin market underutilization rate. Because 1−u −i κ χ(κ ) do do do do we are studying a steady state, we use the definition = to derive 1−i λ+κ χ(κ ) do do do (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) ∂ 1−u −i u ∂lnκ χ(κ ) do do do do do ln = , ∂ln(τ ) 1−i 1−i ∂ln(τ ) do do do do in which we used the chain rule to write (cid:18) (cid:19) ∂lnκ χ(κ ) ∂lnκ χ(κ )∂κ χ(κ ) 1 ∂κ χ(κ ) do do do do do do do do = = . ∂ln(τ ) ∂κ χ(κ ) ∂ln(τ ) κ χ(κ ) ∂ln(τ ) do do do do do do do B.4.7 Elasticity of destination–destination market underutilization rate Calculating the elasticity of destination–destination market underutilization rate with 1−u −i κ χ(κ ) dd dd dd dd respect to τ also relies on the definition of = . The steps to do 1−i λ+κ χ(κ ) dd dd dd derive this will be identical to the ones we took in calculating the destination–origin market underutilization rate with only the sub-indexes changing. The final derivative is (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) ∂ 1−u −i u ∂lnκ χ(κ ) dd dd dd dd dd ln = , ∂ln(τ ) 1−i 1−i ∂ln(τ ) do dd dd do 87

in which we used the chain rule again to calculate (cid:18) (cid:19) ∂lnκ χ(κ ) ∂lnκ χ(κ )∂κ χ(κ ) 1 ∂κ χ(κ ) dd dd dd dd dd dd dd dd = = . ∂ln(τ ) ∂κ χ(κ ) ∂ln(τ ) κ χ(κ ) ∂ln(τ ) do dd dd do dd dd do B.4.8 General expression for d(cid:48) = d case ∂ln(w ) d Here we try to write the most general possible expression only assuming that = 0 ∂ln(τ ) do ∂ln(w ) o and = 0. Combining the general ACR term expression with the elasticity of the ∂ln(τ ) do finding rate with respect to tariffs gives ∂ (cid:18) C (cid:19) ∂ln(ϕ¯ ) ∂ln(Nx) ∂ln(Nx) ln do = (1−σ)+ψ +(ψ −ψ ) dd + o − d ∂ln(τ ) C do do dd ∂ln(τ ) ∂ln(τ ) ∂ln(τ ) do dd do do do (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) u ∂lnκ χ(κ ) u ∂lnκ χ(κ ) + do do do − dd dd dd 1−i ∂ln(τ ) 1−i ∂ln(τ ) do do dd do (cid:18) (cid:19)(cid:20) (cid:21) ψ ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) + do do do do − dd dd dd . σ−1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do do dd dd do B.4.9 Indirect effect of tariff changes (d(cid:48) (cid:54)= d case) The second derivative is the indirect effect which documents how changing tariffs between a third country d(cid:48) and the origin o changes relative consumption in country d ∂ln(C /C ) do dd = εdd(cid:48). ∂ln(τ ) o d(cid:48)o B.4.10 First and second terms of equation (68) (d(cid:48) (cid:54)= d case) Following the general pattern above we first derive the change in the ACR price indexes as ∂ (cid:16) (cid:17) ∂ln(Nx) ∂ln(ϕ¯ ) ln (cid:0) PACR (cid:1)1−σ = o +ψ do , ∂ln(τ ) do ∂ln(τ ) do ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o and similarly ∂ (cid:16) (cid:17) ∂ln(Nx) ∂ln(ϕ¯ ) ln (cid:0) PACR (cid:1)1−σ = d +ψ dd . ∂ln(τ ) dd ∂ln(τ ) dd ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o 88

Combining these gives ∂ (cid:16) (cid:17) ∂ (cid:16) (cid:17) ∂ln(ϕ¯ ) ∂ln(ϕ¯ ) ln (cid:0) PACR (cid:1)1−σ − ln (cid:0) PACR (cid:1)1−σ = ψ do −ψ dd ∂ln(τ ) do ∂ln(τ ) dd do ∂ln(τ ) dd ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o d(cid:48)o ∂ln(Nx) ∂ln(Nx) + o − d . ∂ln(τ ) ∂ln(τ ) d(cid:48)o d(cid:48)o The elasticities of the cutoffs ϕ¯ and ϕ¯ with respect to τ are also related because do dd d(cid:48)o changing tariff τ changes the price index P which changes the cutoff ϕ¯ . We can derive d(cid:48)o dd this relationship by differentiating the explicit expression for the cutoff given in equation (25) (cid:18) (cid:19) ∂ln(ϕ¯ ) ∂ln(P) 1 ∂ln(F ) do do = − + ∂ln(τ ) ∂ln(τ ) σ −1 ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o and symmetrically (cid:18) (cid:19) ∂ln(ϕ¯ ) ∂ln(P ) 1 ∂ln(F ) dd d dd = − + . ∂ln(τ ) ∂ln(τ ) σ −1 ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o So the relationship between the two cutoff elasticities is ∂ln(ϕ¯ ) ∂ln(ϕ¯ ) do dd = ∂ln(τ ) ∂ln(τ ) d(cid:48)o d(cid:48)o (cid:18) (cid:19)(cid:20) (cid:21) 1 ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) do do do dd dd dd + − , σ −1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do d(cid:48)o dd dd d(cid:48)o and we use the chain rule to expand the derivatives with respect to the finding rate. Substituting the relationship between the cutoffs into the general expression for the ratio of relative prices and simplifying gives ∂ ln (cid:16)(cid:0) PACR(cid:1)1−σ (cid:17) − ∂ ln (cid:16)(cid:0) PACR(cid:1)1−σ (cid:17) =(ψ −ψ ) ∂ln(ϕ¯ dd ) + ∂ln(N o x) − ∂ln(N d x) ∂ln(τ ) do ∂ln(τ ) dd do dd ∂ln(τ ) ∂ln(τ ) ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o d(cid:48)o d(cid:48)o (cid:18) (cid:19)(cid:20) (cid:21) ψ ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) + do do do do − dd dd dd . σ−1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do d(cid:48)o dd dd d(cid:48)o B.4.11 Elasticity of destination–origin market underutilization rate We continue to follow the pattern above and calculate the elasticity of destination–origin market underutilization rate. Because we are studying a steady state, use the definition 1−u −i κ χ(κ ) do do do do = to derive 1−i λ+κ χ(κ ) do do do (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) ∂ 1−u −i u ∂lnκ χ(κ ) do do do do do ln = , ∂ln(τ ) 1−i 1−i ∂ln(τ ) d(cid:48)o do do d(cid:48)o 89

in which we used the chain rule to write ∂lnκ χ(κ ) ∂lnκ χ(κ )∂κ χ(κ ) 1 ∂κ χ(κ ) do do do do do do do do = = . ∂ln(τ ) ∂κ χ(κ ) ∂ln(τ ) κ χ(κ ) ∂ln(τ ) d(cid:48)o do do d(cid:48)o do do d(cid:48)o For the record, the elasticity of the third term boils down to (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) ∂ 1−u −i u ∂lnκ χ(κ ) do do do do do ln = . ∂ln(τ ) 1−i 1−i ∂ln(τ ) d(cid:48)o do do d(cid:48)o The thing that matters is the capacity utilization rate and the elasticity of the finding rate with respect to tariffs. B.4.12 Elasticity of destination–destination market underutilization rate The fourth term requires that we calculate (cid:18) (cid:19) ∂ 1−u −i dd dd ln . ∂ln(τ ) 1−i d(cid:48)o dd The steps are identical to the ones we took in calculating the destination–origin market underutilization rate derivative with only the sub-indexes changing. The end result is (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) ∂ 1−u −i u ∂lnκ χ(κ ) dd dd dd dd dd ln = , ∂ln(τ ) 1−i 1−i ∂ln(τ ) d(cid:48)o dd dd d(cid:48)o in which we again used the chain rule to write (cid:18) (cid:19) ∂lnκ χ(κ ) ∂lnκ χ(κ )∂κ χ(κ ) 1 ∂κ χ(κ ) dd dd dd dd dd dd dd dd = = . ∂ln(τ ) ∂κ χ(κ ) ∂ln(τ ) κ χ(κ ) ∂ln(τ ) d(cid:48)o dd dd d(cid:48)o dd dd d(cid:48)o B.4.13 General expression for d(cid:48) (cid:54)= d case ∂ln(w ) d Here we try to write the most general possible expression only assuming that = 0 ∂ln(τ ) d(cid:48)o ∂ln(w ) o and = 0. The general ACR term expression was ∂ln(τ ) d(cid:48)o ∂ ln (cid:16)(cid:0) PACR(cid:1)1−σ (cid:17) − ∂ ln (cid:16)(cid:0) PACR(cid:1)1−σ (cid:17) =(ψ −ψ ) ∂ln(ϕ¯ dd ) + ∂ln(N o x) − ∂ln(N d x) ∂ln(τ ) do ∂ln(τ ) dd do dd ∂ln(τ ) ∂ln(τ ) ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o d(cid:48)o d(cid:48)o (cid:18) (cid:19)(cid:20) (cid:21) ψ ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) + do do do do − dd dd dd . σ−1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do d(cid:48)o dd dd d(cid:48)o Combining these with the elasticity of utilization rates gives 90

∂ln(C /C ) ∂ln(ϕ¯ ) ∂ln(Nx) ∂ln(Nx) do dd = (ψ −ψ ) dd + o − d ∂ln(τ ) do dd ∂ln(τ ) ∂ln(τ ) ∂ln(τ ) d(cid:48)o d(cid:48)o d(cid:48)o d(cid:48)o (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) u ∂lnκ χ(κ ) u ∂lnκ χ(κ ) + do do do − dd dd dd 1−i ∂ln(τ ) 1−i ∂ln(τ ) do d(cid:48)o dd d(cid:48)o (cid:18) (cid:19)(cid:20) (cid:21) ψ ∂ln(F ) ∂ln(κ χ(κ )) ∂ln(F ) ∂ln(κ χ(κ )) + do do do do − dd dd dd . σ−1 ∂ln(κ χ(κ )) ∂ln(τ ) ∂ln(κ χ(κ )) ∂ln(τ ) do do d(cid:48)o dd dd d(cid:48)o B.4.14 Final expression The final expression is  ∂ln(ϕ¯ ) ∂ln(Nx) ∂ln (cid:0) Nx(cid:1) ∂ln(C do /C dd ) =εdd(cid:48) =  + + ( (cid:18) (cid:16) 1 σ ψ 1 − − d − u o 1 σ d (cid:17) i o ) d(cid:20) + o ∂ (cid:19) ψ l (cid:18) d n o ∂ ( ∂ κ + ln l d n ∂ o ( ( ψ κ F χ ln d d d ( o o o ( κ χ τ ) − d d o ( o κ ) ψ ) ) d d o ∂ d ) ) l (cid:19) n ∂ ∂ ( − κ l l n n d (cid:18) ( o ( τ χ τ 1 d d d ( o − u d o κ ) d ) d i d + d o d )) (cid:19) ∂ − l (cid:18) n ∂ ∂ (τ l l d n n o ∂ o ∂ ( κ ) l κ l n d n − d d ( d ( χ τ F χ ∂ d ( d o ( κ l d ) κ n d ) d d ( d τ ) ) d(cid:19) ) d o ) ∂ln ∂ (κ ln dd ( χ τ d ( o κ ) dd ))(cid:21) if d(cid:48)=d . (69) ∂ln(τ d(cid:48)o ) o  + + ( (cid:16) (cid:18) ψ σ ψ 1 d − d o − u o 1 d − (cid:17) i o d ψ (cid:20)o d ∂ (cid:19) d l ) (cid:18) n ∂ ∂ ∂ ( ∂ κ l l l n n l n d n ∂ o ( ( ( κ F χ l τ ϕ¯ n d d d d ( (cid:48) o ( o κ d o τ χ ) ) d ) d o ( (cid:48) + o κ )) ) d ∂ o ∂ ∂ ) l l l n (cid:19) n n ∂ ( ( ( − τ κ N l d n d (cid:18) (cid:48) o x o ( o τ ) χ ) 1 d ( (cid:48) − − u o κ d ) d i d ∂ ∂ d o d ) l l n n ) (cid:19) ( (cid:0) − τ (cid:18) N d ∂ (cid:48) d ∂ x o (cid:1) l l ) n n ∂ ∂ ( κ l κ n l d n d d ( d ( τ χ F χ d ( (cid:48) d ( o κ d κ ) d ) d d d ) ) (cid:19) ) ∂ln ∂ (κ ln dd (τ χ d ( (cid:48)o κ ) dd ))(cid:21) if d(cid:48)(cid:54)=d B.4.15 Interesting restrictions on the general elasticity We will make a few assumptions to show how the elasticity is related to models without ∂ln(Nx) ∂ln(Nx) search frictions. First, assume that o = d in which case the first additive ∂ln(τ ) ∂ln(τ ) do do term in each of the cases becomes the ACR elasticity denoted εACR,dd(cid:48). Second, assume that o ∂ln(F ) ∂ln(F ) dd do l = 0 = h so that F and F are parameters making = = 0. Together dd do ∂ln(τ ) ∂ln(τ ) do do these give ∂ln(C do /C dd ) =εdd(cid:48) =   εA o CRdd(cid:48)+ (cid:18) 1− u d i o do (cid:19)(cid:18)∂ln ∂ κ ln do ( χ τ d ( o κ ) do )(cid:19) − (cid:18) 1− u d i d dd (cid:19)(cid:18)∂ln ∂ κ ln dd ( χ τ d ( o κ ) dd )(cid:19) if d(cid:48)=d ∂ln(τ d(cid:48)o ) o εA o CRdd(cid:48)+ (cid:18) 1− u d i o do (cid:19)(cid:18)∂ln ∂ κ ln do (τ χ d ( (cid:48)o κ ) do )(cid:19) − (cid:18) 1− u d i d dd (cid:19)(cid:18)∂ln ∂ κ ln dd (τ χ d ( (cid:48)o κ ) dd )(cid:19) if d(cid:48)(cid:54)=d Signing the effect of the addition of search depends on our work using the implicit function theorem. 91

B.5 Proof of Proposition 10: Deriving the gravity equation The value of total imports will be (cid:18) u (cid:19) (cid:90) ∞ I = 1− do Nx n(ϕ)q (ϕ)dG(ϕ). do 1−i o do do ϕ¯ do We need to integrate over the varieties to get the total value of imports going into the domestic market. Demand for a variety, ϕ, in the differentiated goods sector is αY q (ϕ) = p (ϕ)−σ d . Given this demand, monopolistic competition, and constant do do P1−σ d returns-to-scale production imply producers set optimal prices of p (ϕ) = µw τ ϕ−1. For do o do notational simplicity, define B ≡ α(µw τ )−σY Pσ−1 and combine the optimal price with do o do d d the demand curve to get q (ϕ) = B ϕσ. Evaluated at final prices, the value of sales of do do each variety is p (ϕ)q (ϕ) = µw τ B ϕσ−1 and the cost to produce q (ϕ) units of this do do o do do do variety is t (ϕ)+f = w τ B ϕσ−1 +f . These expressions imply that total profits do do o do do do generated by each variety are p (ϕ)q (ϕ)−t (ϕ) = A ϕσ−1, in which it is also useful to do do do do define A = w τ B [µ−1]. Using this profits expression, the productivity cutoff is do o do do (cid:18) (cid:19) 1 F σ−1 do ϕ¯ = , in which F is given in equation (25). The value of total imports from do do A do the negotiated price curve in equation (13) is n(ϕ)q (ϕ) = [1−γ ]p (ϕ)q (ϕ)+γ [t (ϕ)+f −l −κ χ(κ )s ]. do do do do do do do do do do do Using the functional forms assumptions from above this becomes n(ϕ)q (ϕ) = (σ −γ )A ϕσ−1 −γ [−f +l +κ χ(κ )s ]. do do do do do do do do do Substituting the value of imports for a particular variety into the integral defining the value of total imports gives (cid:18) u (cid:19) (cid:90) ∞ I = 1− do Nx (σ −γ )A ϕσ−1 −γ [−f +l +κ χ(κ )s ]dG(ϕ). do 1−i o do do do do do do do do do ϕ¯ do We assume productivity, ϕ, has a Pareto distribution over [1,+∞) with cumulative density function G[ϕ˜ < ϕ] = 1−ϕ−θ and probability density function g(ϕ) = θϕ−θ−1. The Pareto parameter and the elasticity of substitution are such that θ > σ −1 which ensures that the integral (cid:82)∞ zσ−1dG(z) is bounded. Using these assumptions we can compute the integral to ϕ¯ get (cid:18) u (cid:19) (cid:20) θϕ¯σ−θ−1 (cid:21) I = 1− do Nx (σ −γ )A do −γ [−f +l +κ χ(κ )s ]ϕ¯−θ , do 1−i o do do θ−σ +1 do do do do do do do do in which we use the relevant moment of the productivity distribution θϕ¯σ−θ−1 (cid:82)∞ zσ−1dG(z) = do . Define δ ≡ f −l −κ χ(κ )s to conserve on notation, ϕ¯ do θ−σ +1 do do do do do do 92

substitute the export productivity threshold into this expression, and simplify to get (cid:18) (cid:19) (cid:20) (cid:21) I = 1− u do Nx (σ −γ ) θ +γ δ do F −( σ− θ 1 −1) Aσ− θ 1. do 1−i o do θ−σ +1 do F do do do do Next, utilize the assumption that the number of producers in the origin market is (cid:18) (cid:19) Y Y proportional to output in that market Nx = o and the definition for o 1+π Y A = µ−σα(w τ )1−σY Pσ−1[µ−1] to write do o do d d (cid:18) (cid:19)(cid:18) (cid:19) (cid:20) (cid:21) u Y Y θ δ do o do I = 1− (σ −γ ) +γ do do do 1−i 1+π Y θ−σ +1 F do do × F −( σ− θ 1 −1)(cid:0) µ−σα(w τ )1−σY Pσ−1[µ−1] (cid:1) σ− θ 1 . do o do d d We presented the price index earlier as 1− 1 P = λ ×Y θ σ−1 ×ρ . d 2 d d Substituting that into our value of imports and simplifying gives (cid:18) (cid:19)(cid:20) (cid:21) u θ δ do do I = 1− (σ −γ ) +γ do do do 1−i θ−σ +1 F do do × (cid:0) µ−σα[µ−1] (cid:1) σ− θ 1 (cid:18) Y (cid:19) λθ (cid:18) Y o Y d (cid:19)(cid:18) w o τ do (cid:19)−θ F −( σ− θ 1 −1) . 1+π 2 Y ρ do d In the price index section earlier we also define λ , which we can now substitute in here and 2 then simplify to get the final gravity equation I = (cid:18) 1− u do (cid:19)(cid:18) 1− γ do (cid:18) θ− δ do (θ−(σ −1)) (cid:19)(cid:19) α (cid:18) Y o Y d (cid:19)(cid:18) w o τ do (cid:19)−θ F −( σ− θ 1 −1) . do 1−i σθ F Y ρ do do do d We could have also evaluated the integral at the beginning of this section at final sales prices p (ϕ) instead of negotiated prices n (ϕ). From equation (13), we can see p (ϕ) = n (ϕ) do do do do if γ = 0. Setting γ = 0 then gives do do C = (cid:18) 1− u do (cid:19) α (cid:18) Y o Y d (cid:19)(cid:18) w o τ do (cid:19)−θ F −( σ− θ 1 −1) . do 1−i Y ρ do do d The utilization rate is bounded by (cid:18) (cid:19) (cid:18) (cid:19) u κ χ(κ ) do do do 1− = ∈ [0,1], 1−i λ+κ χ(κ ) do do do because the finding and destruction rates must be finite. Similarly, the bundle of search 93

parameters (cid:18) (cid:18) (cid:19)(cid:19) γ δ do do 1− θ− (θ−(σ −1)) ∈ [0,1] σθ F do Proving the second condition takes a few steps. First, prove that (cid:18) (cid:18) (cid:19)(cid:19) γ δ do do 1− θ− (θ−(σ −1)) ≤ 1. We can prove this by noting that σθ F do δ ≡ f −l −κ χ(κ )s so it must be that δ ≤ F and therefore δ F−1 ≤ 1. Also, do do do do do do do do do do the restriction that σ > 1, ensures θ−(σ −1) < θ. Together, these ensure δ do θ− (θ−(σ −1)) ≥ 0. Combining this with the fact that γ ∈ [0,1] ensures that do F do (cid:18) (cid:19) γ δ do do 1− θ− (θ−(σ −1)) ≤ 1. σθ F do (cid:18) (cid:18) (cid:19)(cid:19) γ δ do do Next, show that 1− θ− (θ−(σ −1)) ≥ 0 by showing that σθ F do (cid:18) (cid:19) γ δ γ do do do 1 ≥ θ− (θ−(σ −1)) . Because γ ∈ [0,1] and σ > 1 we know that < 1. do σθ F σ do (θ−(σ −1)) Likewise, σ > 1 ensures θ−(σ −1) < θ so that < 1. We assume above that θ (θ−(σ −1)) θ−(σ −1) > 0 in order to close the model. Together these imply that ∈ [0,1]. θ δ (θ−(σ −1)) Finally, because δ F−1 ≤ 1 we have that do ≤ 1 and we have proved the do do F θ do result. 94

C Appendix for general mathematical results C.1 Poisson process Consider a continuous time Poisson process where the number of events, n, in any time interval of length t is Poisson distributed according to (λt)n P {N (t+s)−N (s) = n} = e−λt n = 0,1,... n! in which s,t ≥ 0, N (0) = 0, and the process has independent increments. The mean number of events that occur by time t is E[N (t)] = λt. Notice that λ is defined in units of time as λ events per t. For example, if producers in our model contact 9 retailers every six months, on average, then we could recast our model measured in years with t = 1 and λ = 4.5 because λt = 9×1/2 = 4.5. Using the Poisson process above, the probability that the first event occurs after time t equals the probability no event has happened before P {t > t} = P [N (t) = 0] = e−λt, 1 in which t denotes the time between the (n−1)st and nth events so t is the time of the n 1 first event. The arrival time of the first event is an exponential random variables with parameter λ. Conversely, the probability the first event occurs between time 0 and time t is P {t ≤ t} = 1−e−λt. Because the Poisson process has independent increments, the 1 distribution of time between any two events, t , for n = 1,2,... will also be an exponential n random variables with parameter λ. The sequence of times between all events, {t ,n ≥ 1}, n also known as the sequence of inter-arrival times, will be a sequence of i.i.d. exponential random variables with parameter λ. Given this distribution, the mean time between events is 1 E[t ] = n λ For example, if producers in our model contact nine retailers every six months, on average, so that λt = 9/2, then the average time between contacts is 1/λ = 2/9 years (or about 365.25×2/9 = 81.17 days). The arrival time of the nth event, S , also called the waiting time, is the sum of the time n between preceding events n (cid:88) S = t . n i i=1 Because S is the sum of n i.i.d. exponential random variables where each has parameter λ n and the number of events n is an integer, S has an Erlang distribution with cumulative n 95

density function (cid:88) ∞ (λt)i P {S ≤ t} = P [N (t) ≥ n] = e−λt , n i! i=n and probability density function (λt)n−1 f (t) = λe−λt . (n−1)! The Erlang distribution is a special case of the gamma distribution where the gamma allows the number of events n to be any positive real number while the Erlang restricts n to be an integer. The above discussion relies heavily on (Ross, 1995, Chapter 2). C.2 Deriving aggregate welfare Here we outline the steps to show that the indirect utility function (welfare) is C/P, in which C is total consumption expenditure, n is the vector of prices for each good, and P is the ideal price index. Assume that preferences are homothetic, which is defined in MWG 3.B.6, page 45. This means that they can be represented by a utility function that is homogeneous of degree one in quantities and that the corresponding indirect utility function is linear in total consumption expenditure. We can begin with the indirect utility function and then manipulate it as follows W (p,C) = W (p,1)C W (p,e(p,u)) = W (p,1)e(p,u) u = W (p,1)e(p,u) 1 = W (p,1)e(p,1) 1 = W (p,1), e(p,1) in which the first line comes from homothetic preferences; the second line follows by plugging in for consumption expenditure C = e(p,u); the third line comes from equation (3.E.1) in MWG that says W (p,e(p,u)) = u (also known as duality); and in the fourth line we plug in for utility level u = 1. The function e(p,u) is the consumption expenditure function which solves the expenditure minimization problem. Using this result and the fact that the price index is defined as e(p,1) ≡ P we can show that 1 C W (p,C) = W (p,1)C = C = . e(p,1) P Hence, as long as preferences are homothetic, we will always get welfare equal to consumption expenditure divided by the price index, W (p,Y) = C/P. The expenditure approach to accounting can be particularly useful for computing aggregate welfare in this C Y −I −G−NX setting because, W (p,C) = = . P P 96

C.3 Moving from an index to a distribution of goods Melitz uses the following steps to move from index ω over a continuum of goods available to consume, Ω, to the distribution of productivity G(ϕ) and the measure of goods available for consumption, N. Assume that the continuum of goods Ω has measure M = |Ω|. In our model, this continuum of goods will have measure (1−u−h)N. The following steps keep the notation in Melitz’s original work. Change of variables, also known as integration by substitution, states (cid:90) i(b) (cid:90) b f(ω)dω = f(h(ϕ))h(cid:48)(ϕ)dϕ. i(a) a Choose ω = h(ϕ) = G(ϕ)M where G(ϕ)M is differentiable in ϕ then we can apply the rule from right to left to get (cid:90) ∞ ∂G(ϕ)M (cid:90) G(∞)M (cid:90) M (cid:90) f(G(ϕ)M) dϕ = f(ω)dω = f(ω)dω = f(ω)dω. ∂ϕ 0 G(0)M 0 ω∈Ω Choosing G(0) = 0 allows us to start the continuous indexing such that the upper bound is the measure of Ω and is without loss of generality. Notice that in equation (22) we integrate from ϕ¯ to ∞ and not 0 to ∞, because for all productivities below the cutoff,ϕ¯ , the s s quantity consumed is zero, so we can ignore that mass. Remember that in our context f(·) is the indexing function so the final step, (cid:90) ∞ ∂G(ϕ)M (cid:90) ∞ ∂G(ϕ)M f(ϕ) dϕ = f(G(ϕ)M) dϕ, ∂ϕ ∂ϕ 0 0 simply reassigns the indexing number G(ϕ)M to ϕ. 97