Universal Gravity * Costas Arkolakis Yale and NBER. First Version: August 2014 This Version: April Abstract

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1 Universal Gravity * Treb Allen Dartmouth and NBER Costas Arkolakis Yale and NBER Yuta Takahashi Northwestern First Version: August 2014 This Version: April 2017 Abstract This paper studies the theoretical properties and counterfactual predictions of a large class of general equilibrium trade and economic geography models. We begin by presenting a framework that combines aggregate factor supply and demand functions with market clearing conditions. We prove that existence, uniqueness and given observed trade flows the counterfactual predictions of any model within this framework depend only on the demand and supply elasticities (the gravity constants ). We propose a new strategy to estimate these gravity constants using an instrumental variables approach that relies on the general equilibrium structure of the model. Finally, we use these estimates to compute the impact of a trade war between US and China. * We thank Andy Atkeson, David Atkin, Lorenzo Caliendo, Arnaud Costinot, Jonathan Dingel, Dave Donaldson, Jonathan Eaton, Pablo Fajgelbaum, John Geanakoplos, Penny Goldberg, Sam Kortum, Xiangliang Li, Giovanni Maggi, Kiminori Matsuyama, Francesc Ortega, Ralph Ossa, Nina Pavcnik, Steve Redding, Andres Rodriguez-Clare, Bob Staiger, Chris Tonetti, our editor Harald Uhlig, and four anonymous referees for excellent comments and suggestions. A Matlab toolkit which is the companion to this paper is available on Allen s website. All errors are our own. 1

2 1 Introduction Over the past fifteen years, there has been a quantitative revolution in spatial economics. The proliferation of general equilibrium gravity models incorporating flexible linkages across many locations now gives researchers the ability to conduct a rich set of real world policy analyses. However, the complex general equilibrium interactions and the variegated assumptions underpinning different models has resulted in our understanding of the models properties to lag behind. As a result, many important questions remain either partially or fully unresolved, including: When does an equilibrium exists and when is it unique? Do different models imply different counterfactual policy implications? In this paper, we characterize the theoretical and empirical properties common to a large class of gravity models. We first provide a universal gravity framework combining aggregate factor demand and supply functions with standard market clearing conditions that incorporates many workhorse trade and economic geography models. 1 We show that existence and uniqueness of the equilibria of all models under the auspices of our framework can be characterized solely based on their factor demand and supply elasticities (the gravity constants ). Moreover, the counterfactual predictions of these models can be expressed solely as a function of the gravity constants and observed data. Hence, the key theoretical properties and counterfactual predictions of all gravity models depend ultimately on the value of two parameters the elasticities of supply and demand. We show how these gravity constants can be estimated using an instrumental variables approach that relies on the general equilibrium structure of the model. Finally, we use these estimates to compute the impact of a trade war between US and China. To construct our framework, we consider a representative economy in which an aggregate factor is traded across many locations subject to the following six economic conditions: 1) iceberg type bilateral trade frictions; 2) a constant elasticity of substitution (CES) aggregate factor demand function; 3) a CES aggregate factor supply function; 4) factor market clearing; 5) balanced trade; and 6) a choice of the numeraire. Any model in which the equilibrium can be represented in a way that satisfies these conditions is said to be contained within the universal gravity framework. 2 Moreover, these conditions impose sufficient struc- 1 Examples of gravity trade models included in our framework are perfect competition models such as Anderson (1979), Anderson and Van Wincoop (2003), Eaton and Kortum (2002), Caliendo and Parro (2010) monopolistic competition models such as Krugman (1980), Melitz (2003) as specified by Chaney (2008), Arkolakis, Demidova, Klenow, and Rodríguez-Clare (2008), Di Giovanni and Levchenko (2008), Dekle, Eaton, and Kortum (2008), and the Bertrand competition model of Bernard, Eaton, Jensen, and Kortum (2003). Economic geography models incorporated in our framework include Allen and Arkolakis (2014) and Redding (2016). 2 See Table 1 for the mapping from work-horse trade and economic geography models to the universal gravity framework. 2

3 ture to completely characterize all general equilibrium interactions. It turns out that the demand elasticity from condition C.2 and the supply elasticity from condition C.3 play a particularly important role in this characterization. We first provide sufficient conditions for the existence and uniqueness of the equilibrium of the model that depend solely on the gravity constants. Existence occurs everywhere except for a knife-edge constellation of parameters (corresponding e.g. to Leontief preferences in an Armington trade model or when agglomeration forces are just strong enough to create a black hole equilibrium in an economic geography model). An equilibrium is unique as long as demand is decreasing and supply is increasing (or vice versa and both elasticities are greater than one in magnitude). Multiplicity may occur if demand and supply are both decreasing (for example, in an economic geography model if agglomeration forces are sufficiently strong) or if demand and supply are both increasing (for example, in a trade model if goods are complementary). We also show that these sufficient conditions can be extended further if trade costs are quasi symmetric a common assumption in the literature. We then examine how a shock to bilateral trade frictions affects equilibrium trade shares, income shares, and real factor prices. To do so, we derive an analytical expression for the counterfactual elasticities of these endogenous variables to changes in all bilateral trade frictions. To provide economic intuition for this expression, we show that it can be written as series of terms expressing how a shock propagates through the trading network, e.g. the direct effect of a shock, the effect of the shock on all locations trading partners, the effect on all locations trading partners trading partners, etc. Importantly, we show that this expression depends only on observed trade flows and the gravity constants, demonstrating that conditional on these two model parameters, the macro-economic implications for all gravity models are the same. We proceed by estimating the gravity constants using a novel procedure that can be applied to any model contained within the universal gravity framework. We show that the supply and demand elasticities can be recovered from a gravity regression of normalized bilateral trade flows on relative own expenditure share and relative incomes, respectively, as long as there exists instruments that are uncorrelated with unobserved supply shifters (e.g. productivity). To construct such instruments, we rely on the general equilibrium structure of the model by calculating the equilibrium own expenditure shares and incomes of a hypothetical world where no such unobserved supply shifters exist and bilateral trade frictions are only a function of distance. Using this procedure, we estimate a demand elasticity in line with previous estimates from the trade literature (e.g. Simonovska and Waugh (2014)) and a supply elasticity that is larger than is typically (implicitly) calibrated to in trade models but appears reasonable given estimates from the economic geography literature. 3

4 Finally, we use the estimated gravity constants along with the expression for comparative statics to evaluate the effect of a trade war between the U.S. and China on the real expenditure of all countries in the world. Given our large estimated supply elasticity, we find modest declines in real factor prices but large declines in total real expenditure. Third country effects are also substantial, with important trading partners of China (e.g. Vietnam and Japan) and the U.S. (e.g. Canada and Mexico) being especially adversely affected. This paper is related to a number of strands of literature in the fields of international trade, economic geography, and general equilibrium theory. There is a small but growing literature examining the structure of general equilibrium models of trade and economic geography. In particular, Arkolakis, Costinot, and Rodríguez-Clare (2012) provide conditions under which a gravity model yields a closed form expression for changes in welfare as a function of changes in openness, while the recent paper Adao, Costinot, and Donaldson (2017) show how to conduct counterfactual predictions in neo-classical trade models without imposing gravity. In contrast, our paper incorporates models with elastic aggregate factor supply curves, thereby allowing analysis of economic geography models and trade models with intermediate round-about production. In terms of the theoretical properties of the equilibrium, Alvarez and Lucas (2007) use the gross substitutes property to establish sufficient conditions for uniqueness for gravity trade models. We instead generalize results from the study of nonlinear integral equations (see e.g. Karlin and Nirenberg (1967); Zabreyko, Koshelev, Krasnosel skii, Mikhlin, Rakovshchik, and Stetsenko (1975); Polyanin and Manzhirov (2008)) to systems of nonlinear integral equations. As a result, the sufficient conditions we provide are strictly weaker. In particular, our conditions allows the supply elasticity to be larger in magnitude than the demand elasticity (in which case gross substitutes may not hold), which is what we find when we estimate the elasticities. In previous work, Allen and Arkolakis (2014) provide sufficient conditions for existence and uniqueness for economic geography models. Unlike those results, our conditions do not require symmetric trade costs nor do we require finite trade costs between all locations. Unlike both Alvarez and Lucas (2007) and Allen and Arkolakis (2014), our theoretical results cover both trade and economic geography models simultaneously. Our analytical characterization of the counterfactual predictions is related to the exact hat algebra methodology pioneered by Dekle, Eaton, and Kortum (2008) and extended in Costinot and Rodriguez-Clare (2013) (and many others). Unlike that approach, we characterize the elasticity of endogenous variables to trade shocks (i.e. we examine local shocks instead of global shocks). There are several advantages of the local approach: first, all possible counterfactuals can be calculated simultaneously through a single matrix inversion; second, our analytical characterization will hold even if there are multiple equilibria (and 4

5 the exact hat approach is ill-defined); third, the local analytical expression admits a simple economic interpretation as a shock propagating through the trading network. (In this regard, our paper is related to the recent working paper by Bosker and Westbrock (2016) which examines how shocks propagate through global production networks). Our estimation strategy uses equilibrium income and own expenditure shares from a hypothetical economy as instruments to identify the demand and supply elasticities. Following Eaton and Kortum (2002), we use the fixed effects of a gravity equation as the dependent variable in an instrumental variables regression (although we use the regression to estimate the supply elasticity along with the demand elasticity). One advantage of our approach is the simplicity of calculating our instruments using bilateral distances and observed geographic variables; in this regard, we owe credit to Frankel and Romer (1999) who instrument for trade with geography (albeit not in a general equilibrium context). The idea of using the general equilibrium structure of the gravity model to recover key parameters is originally due to Anderson and Van Wincoop (2003). Following this, several papers have sought to improve the typical gravity equation estimation by accounting for equilibrium conditions. For example, Anderson and Yotov (2010) pursues an estimation strategy imposing that the equilibrium adding up constraints of the multilateral resistance terms are satisfied, whereas Fally (2015) proposes the use of a Poisson Pseudo-Maximum- Likelihood estimator whose fixed effects ensure that such constraints are satisfied, and Egger and Nigai (2015) develops a two-step model consistent approach that overcomes bias arising from general equilibrium forces and unobserved trade costs. Unlike these papers, here our focus is on recovering the demand and supply elasticities rather than estimating trade cost coefficients in a model consistent manner. Recent work by Anderson, Larch, and Yotov (2016) explores the relationship between trade and growth examined by Frankel and Romer (1999) in a structural context. They recover the demand (trade) elasticity from a regression of income on a multilateral resistance term, where endogeneity concerns are addressed by calculating multilateral resistance based on international linkages only. Our estimation strategy, in contrast, recovers both the demand and supply elasticities from a gravity regression and overcomes endogeneity concerns using an instrumental variables approach based on the general equilibrium structure of the model. Finally, we should note that the brief literature review above is by no means complete and refer the interested reader to the excellent review articles by Baldwin and Taglioni (2006), Head and Mayer (2013), Costinot and Rodriguez-Clare (2013) and Redding and Rossi-Hansberg (2017), where the latter two focus especially on quantitative spatial models. The remainder of the paper is organized as follows. In the next section, we present the universal framework and discuss how it nests existing general equilibrium gravity models. In 5

6 Section 3, we present the theoretical results for existence and uniqueness. In Section 4, we present the results concerning the counterfactual predictions of the model. In Section 5, we estimate the gravity constants. In Section 6 we calculate the effects of a U.S. - China trade war. Section 7 concludes. 2 A universal gravity framework Before turning to the universal gravity framework, we present two variants of the simple Armington gravity model to provide a concrete example of the type of models that fall within our framework. Suppose there are N locations, where in what follows we define the set S {1,..., N}, each producing a a differentiated good. The only factor is labor, where we denote the allocation of labor in location i S as L i and assume the total world labor endowment is i S L i = L. Shipping the good from i S to final destination j incurs an iceberg trade friction, where τ ij 1 units must be shipped in order for one unit to arrive. Consumers have CES preferences with elasticity of substitution σ 0. In the first trade variant of the model, suppose that the labor supplied to a location is perfectly inelastic and exogenous, as in Anderson (1979) and Anderson and Van Wincoop (2003). Suppose too that there is roundabout production, as in Eaton and Kortum (2002), that combines labor and an intermediate input in a Cobb-Douglas fashion. Thus, the production function of the differentiated variety is Q i = (A i L i ) ζ I 1 ζ i, with ζ (0, 1] the labor share, A i is the labor productivity in location i S and the intermediate input I i ( k S ( Q I ki ) σ 1 σ ) σ σ 1, and the unit cost of production in location i is simply p i = (w i /A i ) ζ P 1 ζ i, where w i is the wage. In the second economic geography variant of the model, we suppose instead that the labor supplied to a location is perfectly elastic so that welfare is equalized across locations, as in Allen and Arkolakis (2014). 3 We denote by u i the amenity of living in location i S and further assume that productivity and amenities are subject to spillovers: A i = ĀiL a i and u i = ū i L b i. In this variant of the model, the quantity of the differentiated variety produced is 1+a Q i = ĀiLi and the unit cost of production is p i = w i / ( ) Ā i L a i. 4 In both variants of the model, consumers CES preferences for the goods from each location yields a gravity equation for the expenditure in location j on the differentiated 3 In addition, this formulation incorporates many prominent economic geography models, e.g. Helpman (1998); Donaldson and Hornbeck (2012); Redding (2016). 4 Since labor is inelastic, it is trivial to add externalities into the trade variant of the model. It is also straightforward to add round-about production into the economic geography variant of the model (see Table 1); we omit to do so here to keep our illustrative examples as simple as possible. 6

7 variety from location i: X ij = (p i τ ij ) 1 σ P σ 1 j E j for all j where E j is expenditure in location j and P j ( k S (p jτ kj ) 1 σ) 1 1 σ price index. is the Dixit-Stiglitz More subtly, both variants of the model also exhibit a supply curve for the quantity of the differentiated variety produced in each location. In the trade variant of the model despite the labor supply being perfectly inelastic we can use the fact that a constant share of revenue is paid to both workers and intermediates to write the supply of the composite factor as a CES function of the real factor payments: Q i = ( ) 1 ζ p ζ A i L i i P i ( ) 1 ζ k S A p ζ kl k k P k Q, where Q i S Q i. Similarly, in the economic geography variant of the model despite the labor supply being perfectly elastic we can use the welfare equalization condition to write: Q i = Ā b 1 a+b i ū 1+a a+b i k S Ā b 1 a+b k ū 1+a a+b k ( ) 1+a p a+b i P i ( ) 1+a p a+b k P k Finally, in both variants, we close the model by requiring that the payment to the factors of production is equation to total sales (market clearing), which in turn is equal to total expenditure (balanced trade), i.e.: p i Q i = j S X ij = j S X ji. Q. Substituting in the CES demand (gravity) and supply equations into the market clearing and balanced trade conditions yields the following identical system of equilibrium equations for both variants of the model: ( ) ψ p 1+φ pi i C i = ( ) ψ τ φ ij P φ j P p pj jc j i S (1) i P j j S P φ i = j S τ φ ji p φ j i S, (2) 7

8 where in the trade variant of the model ψ 1 ζ and C ζ i A i L i in the economic geography variant of the model ψ 1+a and C a+b i Ā b 1 a+b i ū 1+a a+b i, and in both models φ σ 1. This example highlights the close relationship between trade and geography models and suggests the need for a unified analysis of the properties of such spatial gravity models. In what follows, we present a framework comprising six simple economic conditions about the trade of an aggregate representative factor between many locations. We show that the equilibrium of any model for which a representative factor can be constructed that satisfies these conditions can be represented by the solution to equations (1) and (2). We assume that the world comprises a set S {1,..., N} locations and is endowed with a quantity Q R ++ of a single representative factor. 5 We define the factor price in each location as {p i } i S R N + the factor supplied to each location as {Q i } i S R N +. We say an economy is contained within the universal gravity framework if a representative factor can be constructed that satisfies the following six economic conditions: Condition 1. The price faced by location j for a factor from i can be written as: p ij = p i τ ij, (3) where, as above, {τ ij } i,j S S R ++ are bilateral trade frictions. 6 Condition 2. (CES Factor Demand). There exists a scalar φ R such that the aggregate expenditure function in location j S can be written as: E j (p, W j ) = ( i S p φ ij ) 1 φ W j, (4) where W j is the welfare of the representative owner of the aggregate factor. By Shephard s lemma, Condition C.2 implies that aggregate expenditure in location j S on the factor from i S can be written as: X ij = p φ ij P φ j E j, (5) ( ) 1 where P j k S p φ φ kj is a CES aggregator of all factor prices faced by location j. This demand side condition is equivalent to the gravity demand structure in Anderson and 5 The choice of a finite number of locations is not necessary for the results that follow, but it saves on notation and avoids several thorny technical issues, and is consistent with the majority of the trade literature. See Allen and Arkolakis (2014) for the characterization of the model with a continuum of locations and the comparison with the discrete location counterpart. The choice of Q does not affect the equilibrium factor prices, factor price indices, trade shares, or income shares. Although the value of Q does affect the scale of Q i, without loss of generality one can choose units so that Q is equal to one. 6 R ++ is defined as R ++ { }. If τ ij =, then there is no trade between i and j. 8

9 Van Wincoop (2004), Condition R3 in Arkolakis, Costinot, and Rodríguez-Clare (2012) and the CES factor demand specification considered in Adao, Costinot, and Donaldson (2017). In what follows, we refer to the scalar φ as the aggregate demand elasticity, although we note that it is often referred to as the trade elasticity in the literature. Unlike these papers, however, we allow for the factor supply in a location to be potentially endogenous by specifying the following supply-side equation: Condition 3. (CES Factor Supply) There exists constants {C i } R N ++ and a scalar ψ R such that factor supply in each location i S can be written as: Q i = C i ( p i k S C k P i ) ψ ( p k P k ) ψ Q (6) In what follows, we refer to the constants C i as a supply shifters and the scalar ψ as the aggregate supply elasticity. We also refer to the parameter pair (φ, ψ) as the gravity constants. Finally, to close the model, we impose two standard market clearing conditions and choose our numeraire: Condition 4. (Factor market clearing). For all i S, Q i = j S τ ijq ij. Note that my multiplying both sides of Condition C.4 by the factor price we have that income is equal to total sales, i.e. Y i p i Q i = j S X ij. Condition 5. (Balanced trade). For all i S, E i = p i Q i. Balanced trade is the standard assumption in (static) gravity models, despite trade imbalances being a common occurrence empirically. When we combine the general equilibrium structure of the model with data to characterize the counterfactual implications of gravity models, we relax Condition C.5 to allow for (exogenous) trade deficits. Our final condition is a normalization. Note that the system of equations defined by conditions C.1-C.5 already implies a version of Walras law. 7 As a result, we make the following assumption to pin down the equilibrium level of nominal total trade flows. 7 To see this note that summing these two equations over all i N and equating them we obtain i N j X ij = i N j X ji. By the definition of gross world income being total trade across all markets we obtain trade balance for the Nth location which implies Walras law. 9

10 Condition 6. World income equals to one: Y i = 1. (7) i In what follows, for any given gravity constants {φ, ψ}, supply shifters, {C i } i S, aggregate factor supply Q, and bilateral trade frictions {τ ij } i,j S S, we define a regular equilibrium of the universal gravity framework as a set of factor prices {p i } i S R N ++ and price indices {P i } i S R N ++ that satisfy conditions C.2-C.6. Note that given factor prices and price indices, factor quantities can be immediately recovered from equation 6. By combining conditions C.2 and C.5, we can construct the indirect utility function of an owner of one unit of aggregate factor in location i: W j = p j P j. (8) We refer to this welfare measure as the real factor price in what follows. 8 We say that any gravity model for which a representative factor can be constructed that satisfies conditions C.1-C.6 in contained within the universal gravity framework. As Table 1 summarizes, many well-known trade and economic geography models are contained within the universal gravity framework. On the demand side, it is well known (see e.g. Arkolakis, Costinot, Donaldson, and Rodríguez-Clare (2012) and Adao, Costinot, and Donaldson (2017)) that many trade models imply an aggregate CES factor demand system as specified in condition C.2. 9 For example, in the Armington perfect competition model, a CES demand combined with linear production functions implies φ = σ 1, in the Eaton and Kortum (2002) model, a Ricardian model with endogenous comparative advantage across goods and Frechet distributed productivities across sectors with elasticity θ implies that φ = θ. Similarly, a class of monopolistic models with CES or non-ces demand, linear production function, and Pareto distributed productivities with elasticity θ, summarized in Arkolakis, Costinot, Donaldson, and Rodríguez-Clare (2012) also implies φ = θ. Economic geography models delivering gravity equations for trade flows such as Allen and Arkolakis (2014) and 8 It is important to note that the real factor price may not correspond to the concept of welfare in a particular model. For example, in the Armington trade model with intermediates discussed above, the real factor price is equal to the real wage per efficiency unit of labor to the power of the labor share, i.e. ( ) p i 1 w ζ, P i = i A i P i whereas in the Armington economic geography model, the real factor price is equal to the p real wage per efficiency unit of labor, i.e. i P i = 1 w i A i P i. From condition C.6, total real expenditure is related to ( ) 1+ψ ( ) the real factor price through the supply elasticity, as E i /P i = κc pi i P i, where κ Q/ k S C pk ψ. k P k 9 The class of trade models considered by Arkolakis, Costinot, and Rodríguez-Clare (2012) (under their CES demand assumption R3 ) are a strict subset of the models which fall within the universal gravity framework, corresponding to the case of ψ = 0. 10

11 Redding (2016) also satisfy condition C.2. As discussed in the example above, labor mobility across locations generates a CES factor supply satisfying condition C.3, with a supply elasticity of ψ = 1+a. In this case, the labor a+b supply elasticity depends on the strength of the agglomeration / dispersion forces summarized by a + b. Assuming a > 1, if dispersion forces dominate (a + b < 0), the aggregate supply elasticity is positive, whereas when agglomeration forces dominate (a + b > 0), the aggregate supply elasticity is negative. As we will see in the next section, a negative aggregate supply elasticity admits the possibility of multiple equilibria. What is perhaps more surprising is that trade models incorporating round-about trade with intermediates goods also exhibit CES factor supply, even though workers are immobile across locations. As discussed in the example above, the labor supply elasticity is ψ = 1 ζ ζ and hence positive and increasing in the share of intermediates in the production In the next two sections, we show that any trade and economic geography models sharing the same gravity constants will also share the same theoretical properties and counterfactual implications. What types of trade and economic geography models are not contained within the universal gravity framework? Conditions C.2 and C.3 are violated by models that do not exhibit constant demand and supply elasticities, which include Novy (2010), Head, Mayer, and Thoenig (2014), Melitz and Redding (2014), Fajgelbaum and Khandelwal (2013) and Adao, Costinot, and Donaldson (2017). Models which multiple factors of production combined in more general production functions than Cobb-Douglas typically cannot admit a single representative aggregate factor and hence are also not contained within the universal gravity framework (although the tools developed below can often be extended to analyze such models depending on the particular functional forms). Condition C.5 is violated both by dynamic models in which the trade deficits are endogenously determined and by models incorporating additional sources of revenue (like tariffs); hence these models are not contained within the universal gravity framework. However, we show in Online Appendix B.4 how the results below can be applied to a trade model with tariffs. 3 Theoretical properties of the Universal Gravity Framework It is straightforward to show that conditions C.1-C.5 can be combined to write the equilibrium prices and price indices of any model contained within the universal gravity framework as the solution to equations (1) and (2) above. Together with the normalization in condition C.6, 11

12 equations (1) and (2) are sufficient to characterize the equilibrium of the universal gravity framework. 10 Before proceeding, we impose two mild conditions on bilateral trade frictions {τ ij } : Assumption 1. The following parameter restrictions hold: i) τ ii < for all i S. ii) The graph of the matrix of trade costs {τ ij } i,j S S is strongly connected. The first part of the assumption imposes strictly positive diagonal elements of the matrix of bilateral trade costs. The assumption guarantees that quantities and prices are positive for all countries. The second part of the assumption strong connectivity requires that there is a sequential path of finite bilateral trade costs that can link any two locations i and j for any i j. This condition has been applied previously in general equilibrium analysis as a condition for existence in McKenzie (1959, 1961), Arrow, Hahn, et al. (1971), invertibitility by Cheng (1985); Berry, Gandhi, and Haile (2013), and uniqueness by Allen (2012). In our case strong connectivity guarantees that all countries have positive production. We mention briefly (but do not need to assume) a third condition. We say that trade costs are quasi-symmetric if for any i, j S S, we can write τ ij = τ ij τ A i τj B, where τ ij = τ ji and τi A and τi B are arbitrary positive vectors. Quasi-symmetry is a common assumption in the literature (see for example Anderson and Van Wincoop (2003), Eaton and Kortum (2002), Waugh (2010), Allen and Arkolakis (2014)), and we prove in Online Appendix B.1 that Conditions C.1, C.2, C.4, and C.5 together imply that the origin and destination-specific terms in the bilateral trade flow expression are equal up to scale, i.e. p φ i = p 1+ψ i P φ ψ i C i, which in turn implies that equilibrium trade flows will be symmetric, i.e. X ij = X ji for all i, j S S. Intuitively, the only way the trade can be balanced when trade costs are quasi-symmetric is to make trade flows bilaterally balanced. As a result, equations (1) and 10 Mathematically, we can in turn express equations (1) and (2) as the following system of two sets of equations and two sets of unknowns, γ i, δ i, B i γ α i δ β i = j S K ij γ i δ j i S B i γ α i δ β i = j S K ji γ j δ i i S, φ φ ψ ( ) where K ij τ φ ij, B i Ci, α 1+ψ ψ φ, β ψ ψ φ, γ i p φ pj ψand i, δ i = C j P j we normalize i S B iγi αδβ i = 1. The analysis of Karlin and Nirenberg (1967) and Zabreyko, Koshelev, Krasnosel skii, Mikhlin, Rakovshchik, and Stetsenko (1975) focused on such mathematical systems with one set of equations and one sets of unknowns. In what follows, we generalize their techniques to extend the analysis to the two sets of equations with two unknowns. 12

13 (2) simplify to a single set of equilibrium equations, which allows allows us to relax the conditions on the following theorem regarding existence and uniqueness: Theorem 1. Consider any model contained within the universal gravity framework satisfying Assumption 1. Then: i) If 1 + ψ + φ 0, then there exists a regular equilibrium. ii) If {φ 0; ψ 0} or {φ 1; ψ 1} (or, if trade costs are quasi-symmetric and either { φ 1 2 ; ψ 1 2} or { φ 1 2 ; ψ 1 2} ) then the equilibrium is unique. Proof. See Appendix A.1. Theorem 1 deals with a number of challenges that arise in the general equilibrium analysis of spatial models as well as their empirical implementation. A key advantage of Theorem 1 is that despite the large dimensionality of the parameter space (N supply shifters {C i } and N 2 trade costs {τ ij }), the conditions are only stated in terms of the two gravity constants. Of course, since we provide sufficient conditions, there may be certain parameter constellations such as particular geographies of trade costs where uniqueness may still occur even if the conditions of Theorem 1 are not satisfied. 11,12 The sufficient conditions for existence and uniqueness from Theorem 1 are illustrated in Figure 1. In the case of existence, standard existence theorems (see e.g. Mas-Colell, Whinston, and Green (1995)) guarantee existence for endowment economies when preferences are strictly convex. This is also true in the universal gravity framework: existence may fail only when 1 + ψ + φ 0, which corresponds to the Armington trade model (without intermediate goods) where σ = 0, i.e. with non-convex Leontief preferences. Moreover, in the economic geography example above, a regular equilibrium does not exist in the knifeedge case where σ = 1+a, as agglomeration forces lead to the concentration of all economic a+b activity in one location (see Allen and Arkolakis (2014)). The conditions for uniqueness are intuitive: the equilibrium is unique as long as demand is downward sloping (i.e. φ > 0) and supply is upward sloping (i.e. ψ > 0). Uniqueness 11 Alvarez and Lucas (2007) provide an alternative approach based on the gross substitute property to provide conditions for uniqueness of the Eaton and Kortum (2002) model. In Online Appendix B.3, we show that the gross substitutes property directly applied to our system may fail if the supply elasticity ψ is larger in magnitude than the demand elasticity φ, i.e. in ranges ψ > φ 0 or ψ < φ 1. Theorem 1 provides strictly weaker sufficient conditions in that regard. Such parameter constellations are consistent with economic geography models with weak dispersion forces or trade models with large intermediate goods shares. Importantly, in Section 5, we estimate that ψ > φ > 0 empirically. 12 Theorem 1 generalizes Theorem 2 of Allen and Arkolakis (2014) in three ways: 1) it allows for asymmetric trade costs; 2) it allows for infinite trade costs between certain locations; and 3) it applies to a larger class of general equilibrium spatial model, including notably trade models with inelastic labor supplies (i.e. models in which ψ = 0). Theorem 1 also provides a theoretical innovation, as it shows how to extend the mathematical argument of Karlin and Nirenberg (1967) to systems of non-linear integral equations like equations (1) and (2). 13

14 is also guaranteed where supply is downward sloping and demand is upward sloping and both elasticities have magnitudes greater than one, although such parameter constellations are less economically meaningful. Multiplicity may arise in cases when supply and demand are both upward sloping (which occurs e.g. in trade models when goods are complements) or when supply and demand are both downward sloping (which occurs e.g. in economic geography models when agglomeration forces are stronger than dispersion forces, as discussed by Krugman (1991)). 13 Finally, quasi-symmetric trade costs allow us to extend the range of gravity constants for which uniqueness is guaranteed, but do not qualitatively change the intuition for the results. 4 Counterfactual predictions of the universal gravity framework In this section, we turn from the theoretical properties of the universal gravity framework to how the framework can be used to make predictions of how policy shocks alter equilibrium factor prices, price indices, and relative factor supplies (and hence trade shares, income shares and real factor prices) in each location. 14 It turns out that the answer to this question depends only on observed trade flows and the value of the two gravity constants. Define X to be the (observed) N N trade flow matrix whose i, j th element is X ij, Y is the N N diagonal income matrix whose i th diagonal element is Y i, and E is the N N diagonal income matrix whose i th diagonal element is E i. 15 as follows: A ( (1 + ψ + φ) Y (1 + ψ) X ψy (φ ψ) X φx T With a slight abuse of notation, let A 1 kl A. 16 Define the 2N 2N matrix A φe denote the kl th element of the (pseudo) inverse of 13 Such examples of multiplicity are easy to construct - for example, consider an Armington world of two identical locations separated by trade costs. With strong demand complementarity (in an Armington trade model) or agglomeration forces (in an Armington economic geography model), three equilibria are possible: identical economic output in both locations, higher economic output in location 1, or higher economic output in location In what follows, we focus on the policy shocks that alter bilateral trade frictions {τ ij } i,j S. In Online Appendix B.4, we show how one can apply similar tools to characterize the theoretical properties and conduct counterfactuals in trade models with tariffs. 15 In what follows (apart from part (iii) of Theorem 2), we do not assume that Condition C.5 holds in the data, i.e. that income is necessarily equal to expenditure; rather, we allow for income and expenditure to differ by a location-specific scalar, i.e. we allow for (exogenous) deficits. 16 Because of homogeneity of degree 0, we can without loss of generality normalize one price; moreover, from Walras law, if 2N 1 equilibrium conditions hold, then the last equation holds as well. As a result, ). 14

15 We now characterize the elasticity of all endogenous variables to all possible trade cost shocks: Theorem 2. Consider any model contained in the universal gravity framework. Suppose that X satisfies strong connectivity. If A has rank 2N 1, then: i) The elasticities of factor prices and factor price indices are given by: ln p l ( ) = φx ij A 1 l,i + A 1 l,n+j ln τ ij and ln P l ln τ ij = φ ( A 1 N+l,i + A 1 N+l,N+j) Xij. (9) ii) If the largest absolute value of eigenvalues of A is less than one, then A 1 has the following series expansion: A 1 = k=0 ( ) 1 D k 1+ψ+φ Y 1 0, 0 1 φ E 1 where D is given by D = ( 1+ψ 1+ψ+φ Y 1 X, I + 1+ψ+φ E 1 X T 0 ψ φ ψ 1+ψ+φ Y 1 X ). iii) If either φ ψ 1 or φ ψ 1 and condition C.5 holds, then A has rank 2 2N 1 and the absolute value of its largest eigenvalue will be less than one. Proof. See Appendix A.2. The first part of Theorem 2 states that the (local) counterfactuals of gravity models can be characterized solely as a function of the gravity constants and data. This shows that given observed trade flows and the gravity constants φ and ψ, all models contained in the universal gravity framework deliver the same counterfactual predictions in response to any shock. 17 The second part of Theorem 2 provides a simple interpretation of the counterfactuals as a shock propagating through the trade network, which we discuss below. The third part of A will have at most 2N 1 rank and A 1 can be calculated by simply eliminating one row and column of A and then calculating its inverse. The values of the eliminated row can then be determined using the normalization condition C.6. For example, if one removes the first row and column, can be chosen to ln Y i ln p1 ln τ ij ensure that i S ln τ ij = 0 so that Condition C.6 is satisfied. 17 In Online Appendix B.5, we show how the exact hat algebra pioneered by Dekle, Eaton, and Kortum (2008) and extended by Costinot and Rodriguez-Clare (2013) can be applied to any model in the universal gravity framework to calculate the effect of any (possibly large) trade shock. The key takeaway that counterfactual predictions depend only on observed data and the value of the gravity constants remains true globally. Unlike the Proposition 2, however, determining the global shocks require that the uniqueness conditions of Theorem 1 part (ii) hold to ensure the global counterfactual predictions are unique. 15

16 Theorem 2 provides sufficient conditions for the conditions necessary for parts 1 and 2 of the theorem to hold. Note, however, that because A depends only on observed data and the gravity constants, one can check immediately whether or not it has rank 2N 1 and if the absolute value of its largest eigenvalue is less than one. The second part of Theorem 2 states that the inverse matrix A 1 can be expressed as an infinite geometric series of a matrix D. It turns out that this infinite series has an intuitive economic interpretation of a shock propagating throughout a network, where the shape of the network and the strength of the network connections are determined by the observed trade flows. To see this, let us consider a shock ln τ ij and call the k th element of the series (i.e. the term including D k ) the k th degree effect of the shock. The 0 th degree term is the direct effect of the shock on locations i and j, holding prices and quantities in all other locations constant. The 1 st degree effects are the general equilibrium effects of the 0 th degree on all the trading partners of i and j, holding the prices and quantities of their trading partners constant. In a similar fashion, the k th degree effect is the impact of the (k 1) th degree effect on the trading partners, holding the prices and quantities of the trading partners trading partners constant. To get some intuition, consider first a trade model with inelastic factor supply and downward sloping aggregate factor demand, i.e. ψ = 0 and φ > 0 and suppose there is a reduction in trade frictions from i to j, i.e. ln τ ij < 0. Figure 2 depicts how a shock propagates through the trade network in such a model in aggregate factor supply and demand space for location i. The figure looks a little bit like half a Christmas tree, with each ascending branch reflecting a higher degree response to the initial trade shock and the slope of the top of each branch determined by the factor demand elasticity. In particular, the bottom branch represents the 0 th degree effect, i.e. the direct shock on location i. Because i faces higher demand from j, its factor price rises. The extent to which the factor price for i rises depends on its trade volume with j; the greater i s export share to j, the more its factor price will rise, which is why the 0 th degree horizontal shift of the demand curve depends on Y 1 X (see the green arrow in Figure 2). The extent to which this horizontal shift affects the factor price in i depends in turn on the magnitude of the demand elasticity φ. Now consider the second branch, which represents the 1 st degree effects. While Figure 2 depicts the effect on location i, the following intuition holds for any location k which trades with i and j. Location k is affected by the shock in two ways: first, k experiences higher demand from i since the income of i goes up due to the increase of the factor price; second, k experiences less demand from j since j consumes more goods for i instead of k. These two 16

17 net effects are mathematically represented as X Y ln p(0) }{{} >0 +φ X (0) ln P Y }{{}, <0 and depicted by the second green arrow in Figure 2. The equilibrium adjustment happens due to the shift of the demand curve and yields ln p (1), which is represented by the blue arrow. It is clear that the effect for k s factor price is ambiguous; if k exports a greater share to i than j, then k experiences a net positive effect from the shock, which induces higher demand and hence higher factor price for k. On the other hand, the effect on the price index for k is unambiguous, as k faces a higher price for good i, which increases the price index for k (things get more complicated when factors are mobile, see below). The higher degree effects (not shown on the figure due to space constraints) share the same intuition, although it should be emphasized that they represent the sum of all of the degree-minus-one effects on all countries trading partners. The total local counterfactual prediction (the height of the tree) is the infinite sum of the effect of the shock on all degrees of connections within the network. Next consider the case where the supply and demand elasticities are both strictly positive, i.e. ψ > 0 and φ > 0. Figure 3 depicts an example of this case. Now the Christmas tree has branches in both directions, with the slope of the top of the right branches again determined by the demand elasticity and the slope of the top of the left branches determined by the supply elasticity. As in the case that ψ = 0, the bottom branch reflects the 0 th degree shock causing the demand curve to shift right by φy 1 X ln τ. Because the supply curve is elastic, however, the aggregate factor will move toward locations where the factor price increases (or the factor price index decreases), as illustrated by the bottom light green arrow, which reflects the 1 st degree supply response. As above, the 1 st degree demand response (the second-to-the-bottom dark green branch) reflects the effect of the 0 th degree shock on all trading partners of i and j. The total 1 st degree effect is represented by the brown arrow. Again, higher degree effects are illustrated by higher branches and combine the total effect of the degree-minus-one shock on all trading partners. The total height of the tree (and its angle of repose) is determined by the infinite sum of the effect of the shock on all degrees of connections within the network. 5 Estimating the gravity constants In the previous section, we saw that any counterfactual in a gravity model can be determined solely from observed trade flow data and the value the demand and supply elasticities. In 17

18 this section, we show how these gravity constants can be estimated. We use data on trade flows between countries, so for the remainder of the paper we refer to a location as a country. 5.1 Methodology We first derive an equation that shows that the relationship between three observables relative trade shares, relative incomes, and relative own expenditure shares are governed by the two gravity constants. We then show how this relationship under minor assumptions can be used as an estimating equation to recover the gravity constants. We begin by combining conditions C.1 and C.2 to express the expenditure share of country j on trade from i relative to its expenditure on its own goods as a function of the trade frictions, the factor prices in i and j, and the factor demand elasticity: X ij X jj = ( ) φ τjj p j. We then use the relationship Y i = p i Q i to re-write this expression in terms of incomes and aggregate quantities and rely on Condition 3 to write the equilibrium factor quantity as a τ ij p i function of factor prices and the factor price index: X ij X jj = ( ) ( Yj τ jj C j ) ψ p i P i τ ij ( Y i C i ) ( pj P j ) ψ φ. (10) We now define λ jj X jj /E j to be the fraction of income country j spends on its own goods (j s own expenditure share ). By again combining Conditions 1 and 2, we note j s own ( ) φ p expenditure share can be written as λ jj = τ j jj P j, which allows us to write equation (10) (in log form) as: ln X ij X jj = φ ln τ ij τ jj + φ ln Y j Y i + ψ ln λ jj λ ii φ ln C j C i + φψ ln τ jj τ ii. (11) Equation (11) shows that the demand elasticity φ is equal to the partial elasticity of trade flows to relative incomes, whereas the supply elasticity ψ is equal to the partial elasticity of trade flows to the relative own expenditure shares. Intuitively, the greater j s income relative to i (holding all else equal, especially the relative supply shifters ln C j C i ), the greater the price of the factor in j relative to i and hence the more it would want to consume of goods from i relative to goods from j; the greater the demand elasticity φ, the greater the effect of the price difference on expenditure. Conversely, because the real factor price is inversely related 18

19 to a country s own expenditure share, the greater j s own expenditure share relative to i, the lower the relative factor supply to j and hence the more j will consume from i relative to j; the larger the factor supply elasticity ψ, the more responsive factor supply will be to differences in own expenditure share. 18 Equation (11) forms the basis of our strategy for estimating the gravity elasticities φ and ψ. However, it also highlights two important challenges in estimation. First, equation (11) suggests that for any observed set of trade flows {X ij } and any assumed set of gravity elasticities {φ, ψ}, own trade frictions {τ ii }, and supply shifters {C i }, there will exist a unique set of trade frictions {τ ij } i j for which the observed trade flows are the equilibrium trade flows of the model. 19 As a result, trade flow data alone will not provide sufficient information to estimate the gravity elasticities. Second, equation (11) highlights that the gravity elasticities are partial elasticities holding the (unobserved) relative supply shifters {C i } fixed. Because both income and own expenditure shares are correlated with supply shifters through the equilibrium structure of the model, any estimation procedure must contend with this correlation between observables and unobservables. In order to address both concerns, we combine plausibly exogenous observed geographic variation with the general equilibrium structure of the model to estimate the gravity elasticities. We proceed in a two-stage procedure. 20 First, we re-write equation (11) as: ln X ij X jj = φ ln τ ij τ jj ln π i + ln π j, where ln π i φ ln Y i +ψ ln λ ii φ ln C i +φψ ln τ ii is a country-specific fixed effect. We assume relative trade costs can be written as a function of their continent of origin c, continent of destination d, and the decile of distance between the origin and destination countries, l: ln τ ij τ jj = β l cd + ε ij, where ε ij is a residual that is assumed to be independent across origin-destination pairs. The 18 In particular, it is straightforward to show that the real factor price from equation 8 can be written as p i /P i = λ 1 φ ii ; see Online Appendix B See Online Appendix B.6 for a formal proof of this result. 20 While the two step procedure we follow resembles the procedure used in Eaton and Kortum (2002) to recover the trade elasticity from observed wages, there are two important differences. First, our procedure applies to a large class of trade and economic geography models and allows us to simultaneously estimate both the demand (trade) elasticity and the supply elasticity (rather than assuming e.g. that the population of a country is exogenous and calibrating the model to a particular intermediate good share). Second, our procedure relies on the general equilibrium structure of the model to generate the identifying variation (rather than e.g. instrumenting for wages with the local labor supply, which would be inappropriate for economic geography models). 19