Jean Imbs

Isabelle Mejean

Nicolas Coeurdacier

Jonathan Eaton

ELASTICITY OPTIMISM 1

Keywords: Elasticités du Commerce, Agrégation, Calibration, Déséquilibres Globaux, Transmission Internationale, Politique Monétaire Trade Elasticities, Aggregation, Calibration, Global Imbalances, International Transmission, International Portfolio, Monetary Policy. Classification JEL: F41, F32, F21

Les estimations de l'élasticité de substitution entre variétés domestiques et étrangères sont faibles dans les données macroéconomiques, mais nettement plus élevées dans les études sur données désagrégées. Cette divergence peut être la conséquence de l'hétérogénéité. Nous utilisons des données fines de commerce pour identifier structurellement les élasticités de substitution pour les biens américains. Nous construisons un modèle d'équilibre partiel pour agréger ces élasticités. Cet exercice nous permet de comparer les élasticités agrégées qui imposent une homogénéité inter-sectorielle et celles qui tiennent compte de l'hétérogénéité. Pour les premières, nous obtenons des valeurs similaires à ce que trouvent les études sur données macroéconomiques. Quand on tient compte de l'hétérogénéité, les élasticités sont plus de deux fois supérieures, jusqu'à 7. Ce paramètre est crucial dans la calibration de la plupart des modèles d'économie internationale. Nous discutons comment la correction que nous proposons affecte un certain nombre de résultats de la littérature, concernant notamment la dynamique des balances courantes, la transmission internationale des chocs, les choix internationaux de portefeuille, et la politique monétaire optimale.

Introduction

The substitutability between domestic and foreign goods is central to most calibrated models in international economics. Depending on the value assigned to the parameter, the predictions of virtually any calibration exercise with an international dimension change quantitatively, sometimes dramatically. The parameter value is key in models that seek to quantify the magnitude of a change in international prices consistent with a rebalancing of external balances as in [START_REF] Obstfeld | Global Current Account Imbalances and Exchange Rate Adjustments[END_REF]. It is also a crucial element to the mechanisms that underpin the international diffusion of shocks, as in [START_REF] Kose | Can the standard international business cycle model explain the relation between trade and comovement?[END_REF] or Corsetti, Dedola and Leduc (2008); or indeed to the extent of international risk sharing in [START_REF] Cole | Commodity trade and international risk sharing : How much do financial markets matter?[END_REF], and the composition of international portfolio holdings, as in [START_REF] Coeurdacier | Do trade costs in goods market lead to home bias in equities?[END_REF]. Models of international price differences have predictions that rest on the parameter value as well, as in Midrigan (forthcoming) or [START_REF] Atkeson | Pricing to Market, Trade Costs, and International Relative Prices[END_REF]. The same is true of the importance of exchange rates in the optimal conduct of monetary policy, as in Galí and Monacelli (2005). This is quite simply one of the most important parameters in international economics. Unsurprisingly, its calibrated value draws from literally decades of empirical work. Unfortunately, little consensus has emerged from the effort, except for two broad conclusions. First, finely disaggregated good-level quantities are more responsive to (international) relative prices than aggregates. Second, there are enormous differences between goods. Long time ago, [START_REF] Orcutt | Measurement of Price Elasticities in International Trade[END_REF] referred to an "elasticity pessimism", which he related to the gap between the low observed volatilities in aggregate quantities and the high volatility of international relative prices. He already conjectured that aggregates could obscure more responsive quantities at the microeconomic level. Here we ask whether this very heterogeneity may not actually be cause for optimism. We propose a correction of the elasticity of substitution between domestic and foreign macroeconomic quantities that accounts for heterogeneity at the microeconomic level.

Why should such a correction be meaningful? The robust finding that microeconomic studies uncover systematically higher estimates than aggregate data points to the possibility that aggregation matters. Suppose the price elasticity of imports varies by sector. An aggregate elasticity estimate implicitly imposes homogeneity and as a result may be biased for conventional econometric reasons. In other words, while it is the response of aggregate quantities that 1 macroeconomists are interested in, the fact they are undeniably much smoother than international prices may just be an artifact of aggregation. With heterogeneity, it does not follow that the aggregate elasticity of substitution is low. [START_REF] Orcutt | Measurement of Price Elasticities in International Trade[END_REF] reasoned that "goods with relatively low price elasticities can display the largest variation in prices and therefore exert a dominant effect on the estimated aggregate price elasticity, thereby biasing the estimate downwards." Aggregation can create a bias because the assumption that all quantities are equally substitutable and adjust homogeneously to changes in relative prices is not supported by the data. As a result there might be systematic differences between the responsiveness of aggregate traded quantities and the preference parameter it is meant to capture. It is of course the latter that should enter calibrated models. The bias we discuss matters for calibration purposes.

Our first goal is to develop a general model telling us how to properly aggregate microeconomic elasticities. We construct a measure of aggregate substitutability consistent with a representative agent choosing between aggregates of domestic and foreign quantities. At the same time, we accommodate the well documented fact that substitutability is vastly different across goods and sectors. We simulate the standard homogeneity assumption, under which elasticities are identical across sectors. This puts our theory to the test of its ability to reproduce conventional macroeconomic estimates under conventional macroeconomic assumptions.

At the same time, the model maps out an index of substitutability between domestic and foreign aggregates, and a weighted average of its disaggregated counterparts. We let theory tell us which weights must be used for aggregation.

When firm supply decisions are modeled explicitly, the observed response of quantities to prices is an imperfect reflection of substitutability, since it conflates both demand and supply driven phenomena. Traded quantities respond to altered price conditions in a way that does not only reflect consumers' preferences, but also entry decisions on the part of supplying firms, an argument recently developed in Chaney (2008). Our theory allows for this possibility. There is a discrepancy between conventional macroeconomic estimates of the elasticity of substitution, imposing equal elasticities across goods, and aggregate estimates allowing for heterogeneity.

The discrepancy persists whether firm entry decisions are permitted -or not. In fact, we are able to spell out the conditions under which firm dynamics act to reinforce an aggregation bias in conventional macroeconomic estimates of the elasticity of substitution.

Our second goal is to estimate disaggregated elasticities. To do so, we borrow from a methodology introduced by Feenstra (1994a) and recently implemented by [START_REF] Broda | Globalization and the Gains from Variety[END_REF]. In the context of our model, the approach can be used to identify the parameter of interest. Demand at a sectoral level is given by an aggregator of domestic and foreign goods varieties. Under the Armington assumption we impose, the substitutability of domestic and foreign varieties is the same as the substitutability between two foreign varieties. Domestic and foreign varieties do differ in terms of transport costs and preferences, but not substitutability.

As a consequence, we are able to identify sectoral elasticities of substitution using the observed cross-country variation in the trade flows towards a given importer. In comparison with conventional approaches, this provides estimates that are structural, and do not fall victim to the endogeneity concerns that plague any regression of (relative) quantities on (relative) prices.

The Armington assumption makes it possible to identify the substitutability between domestic and foreign goods using imported prices and quantities only. This contrasts with much of the existing empirical work, where it is the response of quantities to changes in the price of foreign goods relative to domestic varieties that is studied. It is important that we validate this identifying assumption. We do so by comparing our results to those obtained in past studies where prices are measured relative to domestic goods. At the sector level, our median estimate across manufacturing is 5.1. This is virtually identical to the estimates obtained, at the same level of aggregation and for similar activities, by [START_REF] Houthakker | Income and Price Elasticities in World Trade[END_REF] or Kreinin (1967). This similaritiy in estimates across decades suggests there is nothing special to our data or our approach relative to existing empirical work. We seem to identify the object of interest to many international economists.

In macroeconomic applications, calibration exercises typically favor values of the parameter that are inferred from aggregate estimates of imports price elasticity. This is done for lack of a consensus, often because they are construed as "plausible mid-points" to the wide range of estimates the literature has uncovered. For instance, [START_REF] Obstfeld | Global Current Account Imbalances and Exchange Rate Adjustments[END_REF] use a value of 2; [START_REF] Backus | Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?[END_REF] use 1.5. The elasticity is (crucially) unitary in [START_REF] Cole | Commodity trade and international risk sharing : How much do financial markets matter?[END_REF], 0.9 in [START_REF] Heathcote | The International Diversification Puzzle Is Not As Bad As You Think[END_REF], 1.5 in [START_REF] Chari | Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates?[END_REF], and set between 0.6 and 2 in [START_REF] Coeurdacier | International Portfolios with Supply, Demand and Redistributive Shocks[END_REF]. When all elasticities are forced to be equal across sectors, our approach generates aggregate estimates between 2.5 and 3 for the US. This is within the ballpark of the calibrated values used in the macroeconomic literature, and within the range of estimates surveyed in [START_REF] Goldstein | Income and price effect in foreign trade[END_REF]. With heterogeneity however, aggregate elasticity estimates more than double, with values in the US up to 6 or 7. This reflects the well known fact that microeconomic elasticities tend to be high on average, and also quite dispersed. It is also the result of an appropriate, theory-implied weighting of disaggregated estimates. Does the correction matter economically? We discuss some illustrations in areas as diverse as the resolution of global imbalances, the dynamics of the trade balance, the international transmission of shocks, international risk sharing, portfolio choice, models of the real exchange rate and optimal monetary policy. In short, the parameter is central to most of international macroeconomics. Whether it is in fact 1 or 7 does make a quantitative difference, and calibrated equilibrium responses often change sizably. In some instances, the correction is also relevant qualitatively: the international response to productivity shocks, the justification for a home bias in portfolio investment or the Pareto-ranking of different macroeconomic policies can all alter profoundly with an elasticity substantially above one.

The impact of heterogeneity is significant and robust. Differences between standard and corrected elasticity estimates are statistically significant at conventional confidence levels, and they prevail for a variety of alternative measures or econometric procedures. Our conclusions withstand controls for common components in prices and quantities, and a battery of alternative data sources used to aggregate up microeconomics estimates. Point estimates are minimally affected, and heterogeneity always retains an economically and statistically significant impact.

In what follows, we first present the model used to guide the aggregation of industry specific elasticities. Section 3 discusses the identification of sector specific parameters, their aggregation and the data involved. We also describe our adaptation of Feenstra's approach. Section 4 reports our results, and document their relevance in recent standard models of international trade in goods and assets. Section 5 concludes.

Trade Elasticities: Practice ahead of Theory

We open with a summary of the empirical literature concerned with estimating trade elasticities to infer substitutability. We discuss common practice, and how end estimates tend to be larger on average but heterogeneous in disaggregated data. We then lay out the theory we use to map out conventional elasticity estimates with the utility parameter of interest in macroeconomic calibration exercises. The model tells us how to compute aggregate elasticities in ways that may or may not allow for heterogeneity at the good's or sector's level. It is constructed to also accommodate firm entry decisions.

Practice

In a conventional framework, the substitutability between foreign and domestic goods maps directly with the price elasticity of imports. Estimating the latter, or trade elasticities in general, is an old business in economics. Early contributions used simple regression analysis to evaluate the response of aggregate imported quantities to changes in international relative prices. [START_REF] Houthakker | Income and Price Elasticities in World Trade[END_REF] for instance, estimate import elasticities between -0.5 and -1.5 depending on the economy considered.1 The decades that followed saw a flurry of refinements, purporting to alleviate the econometric issues that go with having prices as independent variables. Thus, [START_REF] Marquez | Bilateral Trade Elasticities[END_REF] implements a frequency domain estimator, [START_REF] Gagnon | Productive capacity, product varieties, and the elasticities approach to the trade balance[END_REF] instruments import prices using the real exchange rate, and [START_REF] Hooper | Trade elasticities for G-7 countries[END_REF] use co-integration techniques. In most instances, estimates of the import price elasticity based on macroeconomic data are found to be weakly negative, not always significant, and rarely larger than 2 in absolute value. The conventional inference in macroeconomics is to deduce that the elasticity of substitution between aggregate bundles of domestic and foreign goods is close to zero as well.

In contrast, the results obtained from disaggregated information are centered around higher averages, and substantial heterogeneity. For instance, [START_REF] Blonigen | Explaining Armington: What Determines Substitutability Between Home and Foreign Goods?[END_REF] document substitution elasticities between zero and 3.52 in 146 US sectors. Microeconomic studies also open the door to instrumentation strategies where changes in international relative prices can be ascribed to events, such as trade liberalizations, whose magnitude and timing are arguably exogenous to each market's circumstances. Thus, using NAFTA and detailed information on good specific tariff changes, [START_REF] Head | Increasing Returns versus National Product Differentiation as an Explanation for the Pattern of U.S.-Canada Trade[END_REF] find disaggregated substitution elasticities between 8 and 12, while [START_REF] Romalis | NAFTA's and CUSFTA's Impact on International Trade[END_REF] documents estimates between 4 and 13. In the words of [START_REF] Anderson | Trade Costs[END_REF], "overall the literature leads us to conclude that the elasticity is likely to be in the range of 5 to 10" in disaggregated data (p.716).

In short, there is every indication that elasticities are higher on average at a disaggregated level. This comes as no surprise. More than fifty years ago, [START_REF] Orcutt | Measurement of Price Elasticities in International Trade[END_REF] already remarked that "it is widely recognized that the demand schedule for the product of an individual producer has, in general, far greater price elasticity than the aggregate demand schedule for the entire output of the product" (page 118). The importance of aggregation in the context of trade or substitution elasticities has a straightforward, economic intuition -one that seems to find support in the data. For instance, estimates in [START_REF] Hummels | Towards a Geography of Trade Costs[END_REF] rise from 4.79 in one-digit SITC data to 8.26 for four-digit SITC data. [START_REF] Ruhl | Solving the Elasticity Puzzle in International Economics[END_REF] proposes an interpretation. Cross-sectional elasticity estimates are naturally higher, for they embed long run phenomena, such as firm dynamics and the associated adjustments in the quantities produced. Time series data, in contrast, tend to focus on high or medium frequency developments, and may overlook most entry or exit decisions. But in practice, disaggregated datasets tend to be cross-sectional, whereas aggregate ones have information over time. The apparent importance of aggregation may therefore be an artefact of data availability and differences in econometric methodologies. The parameters estimated in micro-and macro-economic studies are in fact fundamentally different, since in practice they do not capture the extensive margin to the same extent.

In what follows, we are careful to accommodate this possibility. Both our corrected and conventional elasticity estimates arise from the same dataset. Both are affected by a putative extensive margin to exactly the same extent. In fact, Feenstra (1994a) discusses how his empirical approach can accommodate time-varying number of firms in each exporting economy, an argument we clarify when we describe the estimation. In short, our homogeneous and heterogeneous estimators are conceptually similar, and they use the same dimension in the same data. [START_REF] Ruhl | Solving the Elasticity Puzzle in International Economics[END_REF] argument cannot explain the discrepancies we identify in structural estimates of the elasticity of substitution.

Theory

Consumption in the domestic economy is an aggregate of imperfectly substitutable goods k = 1, ..., K. Utility is given by

C = k∈K C α k k α α k k
where α k denotes an exogenous preference parameter. Consumption in each sector is derived from a continuum of varieties of good k, that may be imported or not, as in

C k = i∈I, i =d (β ki C ki ) σ k -1 σ k + (β kd C kd ) σ k -1 σ k σ k σ k -1
where i ∈ I indexes varieties and d is the domestic variety of good k. Crucially, the elasticity of substitution σ k is specific to each industry, and assumed identical across all varieties, imported or not. β ki lets preferences vary exogenously across varieties, reflecting for instance differences in quality or home bias in consumption. The sectors that verify β ki = 0 for all i = d will effectively be non-traded. This structure of demand is classic in international economics. The key assumption for our purposes is equal substitutability between two varieties, no matter their origin. Introducing the assumption is largely what opened the door to the New Trade literature, pioneered by [START_REF] Krugman | Scale Economies, Product Differentiation, and the Pattern of Trade[END_REF], and laid the foundation for the more recent models of trade with heterogeneous firms, starting with [START_REF] Melitz | The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity[END_REF].

We allow for heterogeneity in the continuum of firms that produce each variety i ∈ I in each sector k. We have:

C ki = f c ρ k -1 ρ k kif df ρ k ρ k -1
where f indexes the range of firms that are active in country i and sector k. The range of active firms may differ endogenously across countries and sectors. ρ k denotes the elasticity of substitution between the varieties produced by firms in sector k, which we assume identical across all countries i. We further assume ρ k > σ k for all k. The varieties produced by two firms from the same country are more substitutable than those produced by firms located in two different countries.

The representative maximizing agent chooses her consumption allocation keeping in mind that all imported varieties incur a transport cost τ ki , i ∈ I, i = d.2 Utility maximization implies that demand for variety i in each sector k is given by

C ki = β σ k -1 ki P ki P k -σ k α k P P k C (1)
with

P ki = f p 1-ρ k kif df 1 1-ρ k P k = i∈I, i =d P ki β ki 1-σ k + P kd β kd 1-σ k 1 1-σ k P = k∈K P α k k
The demand addressed to each firm is given by:

c kif = p kif P ki -ρ k C ki (2)
where p kif is the (cost, insurance, freight) local currency price charged by firm f producing good k in country i. The paper's empirics are performed at the level of good k, allowing for different varieties across exporters i. The firm-level dimension f is introduced so that the theory-implied estimates we develop can accommodate firm entry dynamics.

We now ask our model how the estimated response of aggregate quantities to changes in aggregate international relative prices is affected depending on whether heterogeneity in σ k is permitted. For this to be a meaningful experiment in a model with multilateral trade at the industry level, we consider disturbances to international relative prices of a specific kind. First, we focus on changes in all relative prices, across all sectors k. This assumes away reallocation of demand across industries, with relative prices changing by the same amount in all sectors.

It is relative quantities whose responses may be heterogeneous, which in turn may obscure aggregate estimates.

Second, we focus on uniform shocks to the international price of domestic goods, across all exporters i in I. This assumes away reallocation of demand across source exporting economies, with relative prices changing identically in all markets. We do this for practical reasons, so that the multilateral dimension of the model collapses into a two-country version, and we can interpret our estimate as capturing the substitutability between composite goods in the domestic economy and in the rest of the world.3 A natural candidate is a domestic shock to relative production costs, driven for instance by a productivity disturbance that is not reflected in wages. It will change the international price of domestic goods by an identical amount across all sectors k and exporters i. As a short-hand, we label the shock a "domestic wage" disturbance w d . An increase in w d represents a real appreciation driven by a positive shift in relative domestic costs and ultimately prices across all sectors.

Consider the definition of an aggregate elasticity of substitution σ between bundles of domestic and foreign goods. By definition,

σ = 1 + ∂ ln k i =d P ki C ki -∂ ln P kd C kd ∂ ln w d
The elasticity of substitution captures the relative response of demand for domestic or foreign bundles of goods. Demand is expressed in nominal terms because virtually all trade data are expressed in value, especially at a disaggregated level. Since the driving force to the shift in relative prices is aggregate, the difference between the elasticity of substitution arising from volume or value data is simply 1.

Using equation (1) and its counterpart for the domestic variety, simple algebra implies

σ -1 = k i =d n ki (1 -σ k ) ∂ ln P ki ∂ ln w d - k n kd (1 -σ k ) ∂ ln P kd ∂ ln w d - k (n k -n kd ) (1 -σ k ) ∂ ln P k ∂ ln w d (3)
with

n ki ≡ P ki C ki k∈K i =d P ki C ki n kd ≡ P kd C kd k∈K P kd C kd n k = i =d n ki
The aggregate elasticity of substitution has two components. The first two terms correspond to appropriately weighted averages of the responses of individual price indices to the cost shock, at the variety level i or d. The third term captures the responses of the sector price index to the cost shock.

No Firm Entry

We clarify the separate influences of the issues raised by aggregation across all sectors k in K, and firm dynamics within each sector k. In particular, we first focus on the former and shut down firm entry. Since we are interested in long run estimates, in each sector k the representative domestic producer modifies her price according to the change in costs, ∂ ln P kd ∂ ln w d = 1. Since we assume away any changes in market structure, foreign producers do not respond to changes in domestic costs, and ∂ ln P ki ∂ ln w d = 0. As a result, the response of the sector price index is given by

∂ ln P k ∂ ln w d = ∂ ln P k ∂ ln P kd ∂ ln P kd ∂ ln w d = P kd C kd P k C k = 1 -w M k where w M k = i =d P ki C ki P k C k
is the share of imports in total expenditures on good k. Equation (3) simplifies into

σ N oF irm = k n kd σ k + k (n k -n kd ) (σ k -1) 1 -w M k (4)
The aggregate elasticity σ N oF irm contains two terms. First, a weighted average of industryspecific elasticities, with weights corresponding to the importance of sector k in overall domestic expenditures. The second term reflects the response of the industry specific price index P k .

Since by assumption the relative price of good k changes identically across all source economies i = d, the composition of the ideal price index in sector k changes significantly in response to the shock considered. We label σ N oF irm a total elasticity, i.e. one that takes the response of price indices into account. In contrast, a partial elasticity assumes aggregate price responses away. 4 We note the difference between partial and total elasticities is likely small. The second term in equation ( 4) is one order of magnitude smaller than the first. In fact, with similar sector allocation of expenditures for domestic and foreign goods, n k n kd , and partial and total elasticities are virtually identical.

Focus for now on partial elasticities and suppose ∂ ln P k ∂ ln w d = 0. The aggregate substitutability between foreign and domestic varieties is given by a weighted average of each industry's corresponding preference parameter. Strictly speaking, this weighted average is the direct equivalent of the aggregate elasticity of substitution, although disaggregated data are necessary to calculate it. It captures the direct effect of a shock to w d on P kd , and the resulting immediate change in C kd visible from equation (1). In macroeconomic data, traded quantities are summed up to the country level, and the econometrician asks how these aggregates respond to changes in international prices. In other words, she assumes σ k = σ for all k, which goes against most available empirical evidence. But equation ( 4) tells us the constraint may well be innocuous, if n kd and σ k happen to be uncorrelated. On the other hand, if substitutable varieties tend to form a large share of domestic expenditures, the unconstrained elasticity σ N oF irm will take larger values than its constrained counterpart, σNoFirm . The direction and magnitude of an aggregation bias depends therefore on the cross-sector correlation between expenditures shares n kd and goods' substitutability σ k .

Sectoral price indices also respond to macroeconomic shocks. Consider a positive shock to domestic costs, driving up domestic prices and therefore P k , to an extent that increases with the share of domestic varieties, 1 -w M k . The response of the price index affects the relative prices of both domestic and imported varieties. It acts to dilute the initial upwards change in P kd P k , which dampens measured total elasticity. But it also drives a negative response in P ki P k , with opposite end effects on measured total elasticity.

Estimates of the elasticity of substitution obtained from aggregate data provide potentially biased values of the true preference parameter that reflects the aggregate substitutability between domestic and foreign varieties. The bias arises because differences in good-specific substitutability are assumed away. This is true of partial and total elasticity, and we later report results for both concepts, correcting for heterogeneity or not. In what follows, we obtain structural estimates of σ k (and its counterpart constrained to homogeneity), and infer the corresponding, theory-implied (partial or total) values for σ N oF irm and σNoFirm . We will show that the implied values for σNoFirm are in line with classic results from the macroeconomic empirical literature, whereas the implied values for σ N oF irm are not.

Allowing for Firm Entry

We now reintroduce firm heterogeneity in the definition of the aggregate elasticity of substitution, given in equation (3). Exporter's prices P ki and P kd are now indices, reflective of the continuum of firms active in sector k across countries. This range will vary endogenously in response to changes in international relative prices, and this response itself may differ across sectors and exporters because of differences in transport costs, assumptions on firms distribution, or heterogeneity in substitutability parameters. The argument is similar to Chaney (2008). In Appendix A, we go through what is now the conventional model of supply decisions made by heterogeneous firms in each sector and each country. As we describe in the Appendix, the price charged in country i and sector k is given by

P ki = M ki 1 -G ki ( φki ) ∞ φki p kif (ϕ) 1-ρ k dG ki (ϕ) 1 1-ρ k (5)
where M ki is the mass of exporting firms active in sector k and country i. G ki (ϕ) denotes the cumulative distribution function for firms' productivity in sector k and country i, and φki is an (endogenous) productivity cut-off level, above which firms are active on export markets. Equation ( 5) clearly implies that ∂ ln P ki ∂ ln w d may vary across sectors or countries, with end effects on the definition of the aggregate elasticity given in equation ( 3). Price indices can now react to domestic cost shocks because market structure changes endogenously. In particular, the response of prices now has three components. First the response of individual firm prices to the domestic shock, ∂ ln P kif ∂ ln w d . This is zero for i in I, i = d and 1 for i = d, since it reflects the pure intensive margin of adjustment. Second, the response of the productivity cut-off, φki , as firms enter markets in response to shifting relative prices. And third, the adjustment in the potential number of firms serving the market. These last two terms capture the extensive margin of price adjustments, set to zero in the previous section.

In the short run, the mass of firms in country i and sector k is usually taken as given, and the third effect is assumed equal to zero. In this case, the extensive margin works through the response of the productivity cut-off for a constant mass of potential exporters. This is the avenue followed in Chaney (2008), who shows the response of traded quantities to changes in costs is magnified by the extensive margin. Given our concern with long run elasticities, we depart from this approach and focus instead on the long run impact of changes in relative prices, when the potential number of exporters is endogenous.

When firms productivity follow a Pareto distribution, we show in the Appendix that the productivity cut-off φki depends only on exogenous parameters in a long run equilibrium where M ki adjusts. The extensive margin works only through the number of potential entrants. In the Appendix, we derive an expression for M ki , which can be used to compute

σ F irm = σ N oF irm + k n kd (σ k -1) σ k ρ k -σ k + k (n k -n kd ) (σ k -1)(1 -w M k ) σ k ρ k -σ k (6)
The aggregate elasticity of substitution continues to be given by an adequately weighted average of disaggregated elasticities σ k . An aggregation bias continues therefore to prevail in σ F irm if n kd and σ k are correlated across sectors, exactly as it did when we ignored any extensive margin.

But there are two additional summation terms. The first one is positive under our assumptions and for σ k > 1, both of which are empirically plausible. This positive correction is a direct consequence of the supply responses across all markets to a shift in relative costs. The estimate is higher because it now encapsulates the relative responses of quantities produced. This is reminiscent of [START_REF] Ruhl | Solving the Elasticity Puzzle in International Economics[END_REF] or Chaney (2008). The second correction can take either sign, depending on the relative magnitude of n k and n kd . But it is small in magnitude, especially in a symmetric setup where n k is close to n kd . We note this correction corresponds to the response in price indices at the sector level, and thus to the difference between a partial and a total elasticity.

Our main purpose in this section is to ascertain that supply responses across all markets do not alter the existence of an aggregation issue in estimates of the aggregate elasticity of substitution. In other words, the most important question here is whether the discrepancy between the estimate constrained to homogeneity, σFirm , and its unconstrained counterpart, σ F irm , differs sizably from that between σNoFirm and σ N oF irm . We noted in the previous section that the difference between σNoFirm and σ N oF irm increases in the cross-sector correlation bewteen n kd and σ k . For positive values of this correlation, a model-implied weighted average of σ k will take larger values than its counterpart imposing homogeneity σ k = σ for all k.

Equation ( 6) suggests the discrepancy will be even larger with firm entry. Under plausible conditions, a given positive value for the correlation between n kd and σ k implies larger values for σ F irm -σFirm than for σ N oF irm -σNoFirm . In other words, the theory implied bias with firm entry is larger than its counterpart assuming supply responses away. To see this, consider equation ( 6):

σ F irm -σFirm = σ N oF irm -σNoFirm + k n kd (σ k -1) σ k ρ k -σ k -(σ -1) σ ρ -σ + k (n k -n kd ) (1 -w M k ) (σ k -1) σ k ρ k -σ k -(σ -1) σ ρ -σ
where σ and ρ are homogeneous elasticities of substitution, constrained to be identical across all sectors. Under mild conditions, the first summation increases in the correlation between n kd and σ k . 5 The second summation, in turn, may take either sign, but it is one order of magnitude smaller, and corresponds to a bias in the relative responses in price indices. It will in fact be zero for symmetric models where the allocation of expenditures is identical for domestic and imported goods.

We later introduce an empirical approach to estimate σ k (and its constrained counterpart σ). For lack of cross-country firm-level data, we cannot obtain any estimates of ρ k or indeed ρ.

We are therefore only able to calculate a theory-implied aggregation bias without firm entry, σ N oF irm -σNoFirm . In our data, n kd and σ k are positively correlated so that the bias will also take positive values. Under these conditions, we just showed that σ F irm -σFirm is greater than σ N oF irm -σNoFirm . This suggests the aggregation bias we document -on the basis of our estimates of σ k -is a lower bound. It is in fact a conservative under-assessment of the bias that would obtain if an econometrician estimated directly σFirm using aggregate data inclusive of firm entry decisions.

The Price Elasticity of Imports

In most of the literature, the elasticity of substitution is inferred from the price elasticity of imports, at any level of aggregation. Since we want to validate our assumptions by comparing our results to conventional estimates (at the micro or macroeconomic level), it is important that we verify the bias we discuss continues to prevail in estimates of the price elasticity of imports.

Fortunately, the exercise follows arguments that are analogous to what was just discussed. The conventional approach to identifying the price elasticity of imports consists in estimating the response of imported quantities to changes in the relative price of imports (a negative number).6

In the model, this is equivalent to

η = ∂ ln [ k i P ki C ki ] -∂ ln w d
Demand continues to be given by equation (1). In Appendix B, we derive the following expression for η N oF irm :

η N oF irm = 1 -σ N oF irm + k n kd (σ k -1)w M k - k α k 1 -w M k (7)
The elasticity contains three terms. First, (one minus) the elasticity of substitution σ N oF irm .

The second and third summations reflect the responses of industry specific and aggregate price indices, P k and P . They correspond to the difference between partial and total elasticities. We note they are likely to be an order of magnitude smaller than

σ N oF irm , since n k , α k , w M k < 1, but σ k > 1.
The partial price elasticity of imports equals 1k n k σ k . Imposing homogeneity in σ k , it simply equals 1 -σ, where σ is the homogeneous elasticity of substitution between domestic and foreign varieties in all sectors. Constraining all elasticities to be the same, the partial price elasticity of imports and the elasticity of substitution are linearly related, a conventional result. With positive cross-sector correlation between n k and σ k , estimates of the (partial) price elasticity of imports that introduce heterogeneity will be larger in absolute value. In other words, the possibility that estimates based on macroeconomic data should be biased continue to prevail for the price elasticity of imports. This helps us relate our results with the literature. First, the bias we discuss prevails in import elasticities as well since σ N oF irm > σNoFirm implies that η N oF irm < ηNoFirm . This can explain why macroeconomic estimates of η are smaller in absolute value than those arising from disaggregated data. Second, using our structural estimates of σ k in equation ( 7).gives us an estimate of η N oF irm . We cannot calculate η F irm , the price elasticity of imports accounting for firm entry, since we do not have estimates of ρ k . But by analogy with the previous section, our results suggest that η F irm -ηFirm < η N oF irm -ηNoFirm . This means the bias we can measure is an understatement of what would obtain from aggregate data inclusive of firm entry choices. In what follows we therefore limit ourselves to computing (partial and total versions of) η N oF irm , imposing homogeneity or not. We compare these results with conventional estimates of η at both the macro and micro levels obtained from the literature. Finding similar values will suggest the Armington assumption is well supported by the data.

Identification

We review how the methodology in Feenstra (1994a) is adapted to our purposes. We first discuss the econometrics involved in estimating σ k for all sectors k in the US economy. We emphasize how we accommodate common effects across all sectors and measurement error. We then turn to the estimation of σ, a measure of elasticity constrained to be identical across sectors. We close with a description of our data.

Microeconomic Estimates

We identify the substitutability between domestic and foreign varieties using the observed crosssection of traded quantities and prices across exporters to one destination. This is afforded by the crucial assumption of an Armington aggregator between varieties of each good, irrespective of their origin. The assumption is what makes it possible to use Feenstra's (1994a) methodology in the present context, even though we do not observe prices or quantities of domestically produced varieties. We now describe our implementation of his methodology, but keep the development concise and focused on the modifications we introduce.

Demand is given in equation ( 1), which after rearranging writes:

C kit = P kit P kt 1-σ k β σ k -1 kit P kt C kt P kit
where t is a time index. Feenstra (1994a) or [START_REF] Broda | Globalization and the Gains from Variety[END_REF] impose a simple supply structure, and assume

P kit = τ kit exp(υ kit )C ω k kit
where υ kit denotes a technological shock that can take different values across sectors and exporters, τ kit is a trading cost and ω k is the inverse of the price elasticity of supply in sector k. 7 This assumes production decisions are taken on the basis of the price net of transport costs, and labeled in local currency. The potential aggregate effects of the nominal exchange rate are soaked up by the shock υ kit . It will therefore be important to implement appropriate econometric tools to control for any potential common effects in our estimated system. 8 In Appendix C we show how this expression for supply can arise from the model with firm entry developed in the previous sections.

In practice, the approach uses expenditure shares to alleviate measurement error in unit values, following [START_REF] Kemp | Error of Measurements and Bias in Estimates of Import Demand Parameters[END_REF]. We define s kit = P kit C kit P kt C kt and rewrite demand as

s kit = P kit P kt 1-σ k β σ k -1 kit
We do not observe domestically produced consumption. In addition, prices are measured Free on Board. We introduce tilded variables to denote the observed counterparts to theory-implied prices and quantities. We observe Pkit ≡ P kit /τ kit . The empirical market shares are therefore given by

skit ≡ Pkit C kit i =d Pkit C kit = s kit τ kit 1 + P kdt C kdt i =d Pkit C kit ≡ s kit τ kit µ kt
Taking logarithms, it is straightforward to rewrite demand as

∆ ln skit = (1 -σ k )∆ ln Pkit + Φ kt + ε kit (8)
with Φ kt ≡ (σ k -1)∆ ln P kt + ∆ ln µ kt , a time-varying intercept common across all varieties, and ε kit ≡ (σ k -1)∆ ln β kit -σ k ∆ ln τ kit an error term that captures random trade cost and taste shocks, via changes in τ kit and β kit . Feenstra (1994a) shows this implies the demand system is robust to quality changes in variety i of good k -or indeed to time-varying number of firms producing good k in country i. The estimation is robust to the presence of an extensive margin within exporting economies.

After rearranging, substituting in log-linearized supply yields

∆ ln Pkit = Ψ kt + ω k 1 + ω k σ k ε kit + δ kit (9) with Ψ kt ≡ ω k 1+ω k σ k Φ kt + ∆ ln i ( Pkit C kit )
a time-varying factor common across varieties, which subsumes sector specific prices and quantities. δ kit ≡ 1 1+ω k σ k ∆υ kit is an error term encapsulating movements in the exchange rate or aggregate technological developments in country i and sector k.

Under standard assumptions on taste shocks β kit and technology shocks υ kit , it is possible to identify the system formed by equations ( 8) and ( 9). Identification rests on the cross-section of exporters i to the domestic economy, and is achieved in relative terms with respect to a reference country r.9 We follow Feenstra (1994a) and summarize the information contained in the system with the following estimable regression

Y kit = θ 1k X 1kit + θ 2k X 2kit + u kit ( 10
)
where

Y kit = (∆ ln Pkit -∆ ln Pkrt ) 2 , X 1kit = (∆ ln skit -∆ ln skrt ) 2 , X 2kit = (∆ ln skit - ∆ ln skrt )(∆ ln Pkit -∆ ln Pkrt ) and u kit = (ε kit -ε krt ) (δ kit -δ krt ) (σ k -1)(1+ω k ) 1+ω k σ k
. Estimates of equation ( 10) map directly with the parameters of interest, since

θ 1k = ω k (σ k -1)(1 + ω k ) θ 2k = ω k σ k -2ω k -1 (σ k -1)(1 + ω k )
Equation ( 10) still suffers from an endogeneity issue. We follow Feenstra (1994a) and instrument the regressors with country-sector specific fixed effects, and correct the estimation for heteroskedasticity across exporters i. As in Feenstra, identification is therefore based on the cross-sectional dimension of equation ( 10). We also include an intercept to account for the measurement error arising from using unit values to approximate prices. Given the origin of potential measurement error, we let it prevail at the most granular level afforded by our data.10

The system summarized by equation ( 10) can accommodate developments that are specific to each sector k. But in macroeconomic applications where the universe of economic activities that form Gross Domestic Product is considered, it is important to allow for more general, aggregate influences. Aggregate technology shocks for instance, or movements in the nominal exchange rate, presumably affect prices and quantities jointly in all sectors. If it were a shock in the exporting economy, that would correspond to a common component of υ kit across all k.

We allow for such correlated effects in as general and parsimonious a manner as possible. We implement a correction suggested by [START_REF] Pesaran | Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure[END_REF] to purge all "Common Correlated Effects" (CCE) from sector level data, and estimate

Y kit = θ 0 + θ 1k X1ki + θ 2k X2ki + θ 3k X 1it + θ 4k X 2it + u kit ( 11
)
where the intercept allows for HS6-specific measurement error, hatted variables are the instrumented versions of X 1kit and X 2kit , and X 1it and X 2it control for the time-varying component of Y kit that is common across all sectors. In particular, following [START_REF] Pesaran | Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure[END_REF], X 1it and X 2it are the cross-sector arithmetic averages of X 1kit and X 2kit .

Armed with consistent (and sector-specific) estimates of θ 1k and θ 2k , it is straightforward to infer elasticities. In particular, the model implies

σk = 1 + θ2k + ∆ k 2 θ1k if θ1k > 0 and θ1k + θ2k < 1 σk = 1 + θ2k -∆ k 2 θ1k if θ1k < 0 and θ1k + θ2k > 1 with ∆ k = θ2
2k + 4 θ1k . Appendix D details how these are also used to infer standard deviations around these point estimates.

As is apparent, there are combinations of estimates in equation ( 11) that do not correspond to any theoretically consistent estimates of σk . This is a problem we encounter in our data, as [START_REF] Broda | Globalization and the Gains from Variety[END_REF] did. We follow their approach, and use a search algorithm that minimizes the sum of squared residuals in equation ( 11) over the intervals of admissible values of the supply and demand elasticities. We use this approach whenever direct estimates of θ 1k and θ 2k cannot be used to infer σk . Whenever CCE are included, we hold constant the estimates of θ 0 , θ 3k and θ 4k obtained from the direct instrumental variable regression, and search the combination of values for σ k and ω k that minimizes the sum of squared residuals in equation ( 11). The corresponding standard errors are obtained via bootstrapping of the procedure using 1,000 repetitions.

Macroeconomic Estimates

We invoke equation (4) to aggregate adequately the estimates of σk just obtained. Our purpose is to compare these results to what is obtained when sectoral elasticities are constrained to be homogeneous, as they would in conventional regression analysis based on macroeconomic data.

To do so, we impose σ k = σ and modify equation ( 11) into

Y kit = θ 0 + θ 1 X1ki + θ 2 X2ki + θ 3 X 1it + θ 4 X 2it + u kit (12)
We maintain the assumption of a HS6-specific intercept, to continue to accommodate the possibility that measurement error varies at the disaggregated level. Not doing so will conflate two potential sources of bias, and the one we are pursuing relates only to the estimates of θ 1 and θ 2 . These are now constrained to equality across all sectors k. Identification continues to rest on the cross-section of exporters i, but equation ( 12) is now estimated on the pooled dataset formed by observations on all sectors. It is noteworthy that identification in equations ( 11) and ( 12) rests in practice on the same dimension of the same dataset. It is therefore difficult to ascribe the discrepancy we find to a difference in the extent of an extensive margin. Whether our data (and procedure) capture or not firm dynamics, they do so equally in both estimations.

In the presence of heterogeneity in σ k , equation ( 12) suffers from a conventional aggregation bias, of an econometric nature, distinct from and non-competing with the argument developed in the previous sections. The bias arises because imposing the constraint that θ jk = θ j for all k and j = 1, 2, 3, 4 in equation ( 11) will impose homogeneity in coefficients that theory implies are heterogeneous across sectors. As in [START_REF] Pesaran | Estimating Long Run Relationships from Dynamic Heterogeneous Panels[END_REF], ignored heterogeneity enters the residuals in equation ( 12), and creates a potential bias in the estimates of θ j , j = 1, 2, 3, 4. The direction of this bias is ambiguous, and depends on the correlation between the heterogeneity in the residuals of equation ( 12) and its regressors. A careful analysis of this possibility is beyond the scope of this paper. But we note that, as a result, empirical estimates of the constrained elasticity of substitution σ may well be away from a simple average of its sectoral counterparts.

We continue to allow for the possibility that aggregate shocks in any country i should affect all sectors simultaneously, and include adequately modified CCE terms. The instrumentation and correction for heteroskedasticity are also modified accordingly. In particular, country-specific effects are used as instruments across the whole panel of sectors. Armed with estimates of θ 1 and θ 2 , it is easy to obtain a value for the constrained elasticity of substitution σ. Our model then implies conventional macroeconomic estimates of the elasticity of substitution are given by σNoFirm = σ + (σ -1)

k (n k -n kd ) 1 -w M k
with a standard error given by a first-order approximation, as detailed in Appendix D. σNoFirm is the constrained total elasticity, and σ is a constrained partial elasticity.

Data

The Armington assumption requires that imports from different countries be imperfectly substitutable varieties. The hypothesis is increasingly palatable as the granularity of the data augments. We choose to use disaggregated, multilateral trade data from the Base Analytique du Commerce International (BACI), released by the Centre d'Etudes Prospectives et d'Informations Internationales (CEPII), and available at the 6-digit level of the harmonized system (HS6). The data cover around 5,000 products over the 1996-2004 period for a large cross-section of countries. The database describes bilateral trade at the sectoral level, building on the United Nations ComTrade database with some added effort put in the harmonization of trade flows on the basis of both import and export declarations. The improvement acts to limit measurement error.

Unlike Feenstra (1994a) or [START_REF] Broda | Globalization and the Gains from Variety[END_REF], we do not estimate elasticities at the most disaggregated level in the main body of the text, but rather partition our data into 56 ISIC (Revision 3) industries where we implement our methodology. The constrained estimation given by equation ( 12) is effectively estimated on a panel of HS6 sectors, and our partition of HS6 sectors into 56 ISIC categories is entirely innocuous here. Homogeneity is imposed across all sectors, and two HS6 goods have to be equally substitutable whether or not they belong to the same ISIC sector. The partition is only relevant for the unconstrained estimation, and is performed for lack of detailed information on n k , n kd , α k or w M k at such high level of granularity. It corresponds to the assumption that all HS6 goods are equally substitutable within an ISIC category, but not between. This does presumably assume some heterogeneity away, and possibly creates a bias as a result. We conjecture that heterogeneity between ISIC industries is more sizable, and thus creates more of a bias. We do however perform some robustness in section 4.4, using all HS6 goods in the unconstrained estimation. But to do so we have to maintain some rather stringent assumptions on the values for n k , n kd , α k or w M k .

The approach adapted from Feenstra (1994a) requires relatively little information on traded flows. To estimate equation ( 11) we only need measures of Pkit and the expenditure shares skit .

As is conventional, we use unit values to approximate bilateral prices, and divide values of We subject our data to sampling with a view to limiting the role of extreme outliers. These are notoriously frequent in approaches making use of unit values to approximate prices. For instance, tonnage is not always appropriate to capture the traded volumes of all HS6 goods, which can instill artificial (massive) volatility in the resulting time series on prices. In each sector, we exclude annual variations in prices and market shares that exceed five times the median value. In addition, we impose a minimum of 20 exporters for each HS6 good over the whole observed time period. The cross-section of exporters is what ultimately achieves identification. Measurement error may prevail in estimates of σk that are based on too few exporters, which would translate into biased values for (unconstrained) aggregate elasticities.

We require that at least 20 exporters be present to alleviate this concern. Our data ultimately represent 73 percent of the total value of US imports, across 56 ISIC sectors.

In the model, n k and n kd depend directly on the import share w M k and the expenditure share α k . In particular, we have

n k = α k w M k k α k w M k and n kd = α k 1 -w M k k α k (1 -w M k )
Calibration is therefore only needed for α k and w M k . In the main body of the text, we consider the following data sources. The expenditure shares α k are obtained from the OECD STAN dataset, as the 1997 ratio of sectoral absorption (value added and imports net of exports) relative to the aggregate across sectors. The import shares w M k are computed from the US input/output (IO) tables, available in the ISIC (Revison 3) nomenclature, as the 1997 ratio of imports over domestic gross output. Values for n k and n kd are calculated accordingly. 12In section 4.4, we verify our results do not depend on this specific choice of data sources.

We discuss four alternatives. First, we compute w M k directly from the BACI dataset used in our main estimation, rather than the IO tables, normalized by a measure of domestic output taken from the OECD STAN data. But we continue to compute both n k and n kd on the basis of their model-implied values. Second, the IO tables provide enough information to directly compute n k , rather than on the basis of a model-implied formula. In our second variant, we do so, and therefore use IO tables to calibrate both w M k and n k . But n kd continues to be computed according to the model, since we do not have information on domestic production. Our third variant combines both insights. We infer w M k from the BACI and STAN dataset, but now also use BACI to calibrate n k . Finally, we return to our original data sources in our fourth variant, get α k from STAN and w M k from the IO tables. But now, we compute sectoral absorption on the basis of gross output rather than value added.

Results and Relevance

We first review the microeconomic estimates, obtained across 56 ISIC sectors, and relate them with existing evidence. We then aggregate them, first preserving heterogeneity. We compare the results with our estimation imposing homogeneity -using the same data sources to perform the aggregation. We discuss the discrepancy, and in particular whether its magnitude is significant economically. We argue the corrected estimates we uncover change dramatically the quantitative and qualitative predictions of a vast range of international macroeconomic models. We close with some robustness.

Microeconomic Results

Figure 1 reports sectoral estimates of σk for 56 ISIC sectors. On average, σk is equal to 6.7, with values ranging from 28 to 3.1 and a standard deviation of 4.9. The median value is only 5.1, reflecting a skewed distribution of elasticities: only 5 out of 56 estimates are above 10.

How do our results compare with existing studies of the substitutability between foreign and domestic varieties, at similar aggregation levels? If anything, a median value of 5.1 lies at the low end of the range of estimates obtained in the empirical trade literature. [START_REF] Romalis | NAFTA's and CUSFTA's Impact on International Trade[END_REF] finds elasticities of substitution between 4 and 13 at the HS6 level. [START_REF] Head | Increasing Returns versus National Product Differentiation as an Explanation for the Pattern of U.S.-Canada Trade[END_REF] find values between 7.9 and 11.4 at the 3-digit SIC level. [START_REF] Hanson | Market Potential, Increasing Returns and Geographic Concentration[END_REF] finds estimates between 4.9 and 7.6 using data at the US county level. A common denominator across these studies is their focus on disaggregated, microeconomic information on traded quantities and/or tariffs. In particular, they report cross-sector averages of microeconomic estimates, rather than estimates based on macroeconomic data. In theory, the parameters we estimate are comparable with the values obtained in the conventional approach regressing imported quantities on relative prices. The relative price of imports is typically measured with respect to domestically produced varieties, as in [START_REF] Houthakker | Income and Price Elasticities in World Trade[END_REF] or Kreinin (1967). The (partial) price elasticity of imports is then given by 1-σk .

These elasticities were the object of a vast literature spread over the 1960s and 1970s, with two differences. First, in most instances, the results put forward focused on short-run elasticity estimates, typically for reasons of data availability especially at a disaggregated level. In contrast our estimates correspond to long run elasticities, since the identification is in cross-section.

Second, the data used were more coarse, focusing on just a few sectors. Still, in what follows we strive to ensure our disaggregated estimates are consistent with the existing estimates of import elasticities.

Houthakker and Magee (1969) report in their Table 6 a long run price elasticity in manufactures estimated at -4.05. This is virtually identical to the median value we obtain across our 56 manufacturing sectors, equal to -4.1 (= 1 -5.1). Kreinin (1967) documents similar estimates, with an elasticity for manufactures equal to -4.71. It is remarkable that such different data sources, coverages and methodologies should yield strikingly similar median estimates. Our assumptions -and in particular the Armington aggregator -find comfort in this convergence of results.

The data in [START_REF] Houthakker | Income and Price Elasticities in World Trade[END_REF] and Kreinin (1967) are much coarser than ours, but they also discuss the relative magnitudes of elasticity estimates across the categories they observe. Manufactures have the higher estimates, followed by semi-manufactures and crude foods and materials. Similar relative rankings come out of the survey in Goldstein and Khan (1985). They summarize their Table 4.4 commenting that "the price elasticity of demand for manufactures is significantly larger than that for non-manufactures. Within non-manufactures, price elasticities for raw materials appear to be larger than those for food and beverages" (pages 1084-1085). A precise mapping is difficult given the differences in granularity, but the ranking is roughly prevalent in our results as well. There are exceptions, but our highest estimates concern finished manufactures, such as aircrafts, TVs, telephones, photo instruments, footwear, motor vehicles or office machinery. At the other end of the spectrum, we find relatively low elasticities for dairy, wood, food, beverages and semi-manufactures like wires or metal products.

Mapping our most disaggregated, sector-specific estimates with the literature becomes quickly difficult, once again because of data availability as of 20 or 30 years ago. In fact, not many papers have attempted to estimate sector-specific price elasticities of imports, say at the two or three digit level of aggregation. We were able to identify two exceptions. [START_REF] Stone | Price Elasticities of Demand for Imports and Exports: Industry Estimates for the U.S., the E.E.C. and Japan[END_REF] presents US estimates at the two digit level. On the whole, his estimates are lower than ours, but that can simply reflect his focus on short run elasticities. A few examples may nevertheless help illustrate the relative similarities in our results. For "Inorganic Chemicals",

Stone estimates an import price elasticity of -3.40, as against -3.60 for "Other Chemicals" in Figure 1. He finds -2.22, -2.32, and -3.71 in "Rubber Products", "Plastic Materials and Articles" and "Dyeing, Tanning and Coloring Agents", as against -4.1, -3.1 and -5.9 in "Rubber

Products", "Manufactures of Plastic Products" and "Tanning and Dressing of Leather" in [START_REF] Shiells | Errors in Import-Demand Estimates Based upon Unit-Value Indexes[END_REF] estimates long run elasticities at the three digit SITC level, but only for 12 US sectors. Once again, an accurate mapping is impossible in most cases. Interestingly however, his estimate in "Newsprint" is -3.6, indistinguishable from our value of -3.4 for "Publishing".

He also finds -3.5 in "Steel Plate and Sheet", relatively close to the estimate of -5.2 we find for "Manufacturing of Basic Iron and Steel", even though sector definitions are different. The discrepancies become even less substantial when taking into account Shiells' relatively large standard errors.

This comparison exercise is not meant to suggest we reproduce exactly sector-specific results that were obtained several decades ago in totally different data using drastically different methodologies. Rather, we seek to ascertain the identification strategy we follow is not fundamentally falsified. In particular, the Armington assumption is what exonerates us from having to observe any characteristics of domestically produced goods. With the Armington aggregator, the observed prices and quantities of imports originating from a cross-section of countries are sufficient to identify the elasticity of interest, between domestic and foreign varieties.

From this point of view, it is reassuring that our mean and median estimates should be strikingly close to seminal, fundamental contributions to the literature on imports price elasticities.

Virtually all the papers there do make use of domestic prices in their estimations: import prices are evaluated relative to their domestically produced counterpart. That we should find similar results without any information on domestic prices vindicates the Armington assumption. The few punctual comparisons we report at the level of individual sectors do, as well.

There is of course an obvious comparison absent from our analysis so far. We have implemented a variant of the methodology introduced by Feenstra (1994a), just as [START_REF] Broda | Globalization and the Gains from Variety[END_REF] have. Our objectives are fundamentally different, as are some of our identifying restrictions and some of the corrections we introduce. Still, Broda and Weinstein estimate the universe of substitution elasticities in disaggregated US data, just as we do. Given the similarities in methodologies, it is not surprising that our estimates should be similar, and they are. Their mean estimate at the three digit level is 4.0, with a standard deviation of 7.9.

"Petroleum Oils and Oils from Bituminous Minerals, Crude", "Aircraft and Associated Equipment" or "Fuel Wood" are sectors with relatively high elasticities, whereas "Lighting Fixtures", "Radio-Broadcast Receivers" or 'Motorcycles and Cycles" all rank towards the bottom of their list.

But the comparison is not especially informative in terms of validating our assumptions.

What is key here is the Armington assumption that the substitutability between two foreign varieties should equal that between domestic and foreign varieties. If this is true in the data, we can infer directly the price elasticity of imports. Since the latter has been the object of a vast empirical literature, it is with it that we have striven to compare our results.

Macroeconomic Results

We now turn to macroeconomic estimates of the elasticity of substitution implied by the values in Figure 1. To do so, we apply the aggregation procedure spelled out in the previous sections.

Our main point concerns the difference in estimates of η N oF irm and σ N oF irm where σ k = σ as against those where σ k is left unconstrained. Since we have the bias thus induced will increase in the correlations between σ k and n k . The second summation has second order effects only. In our data, the correlation is mildly positive. This suggests constraining all elasticities to be the same will act to lower estimates of both σ N oF irm and η N oF irm .

σ N oF irm = k n kd σ k + k (n k -n kd ) (σ k -1) 1 -w M k
Table 1 reports estimates of both aggregate elasticities. We first report estimates of η N oF irm , the total price elasticity of imports, when we impose that σ k be equal across all sectors. Our point estimate suggests a value for the parameter of -1.98. A confidence interval at standard significance levels implies values ranging roughly from -1.6 to -2.3. This is at the high end of the range of values obtained in conventional estimates of the elasticity based on macroeconomic data. For instance, [START_REF] Goldstein | Income and price effect in foreign trade[END_REF] claim that " [START_REF] Harberger | Some Evidence on the International Price Mechanism[END_REF] judgment of 25 years ago that the price elasticity of import demand for a typical country lies in or above the range of -0.5 to -1.0 still seems on the mark". In their Table 4.1, they report estimates for the US between -1.03 and -1.76, an interval that is not significantly distinct from the one we estimate.

Once again, this vindicates the assumption of an Armington aggregator. We obtain an aggregate estimate with nothing but import prices, that is not significantly different from one obtained on the basis of relative prices computed with domestic price indices. More importantly, -1.98 is consistent with the choices made in the vast majority of calibration exercises in international macroeconomics. We refer to 1 -η N oF irm as the natural proxy for aggregate substitutability, because macroeconomic calibrations would typically infer the elasticity of sub-stitution from the price elasticity of imports, estimated on the basis of macroeconomic data. It is therefore the right comparison with the literature. Here, the implied elasticity of substitution is between two and three, which includes the ranges of values used in, say, [START_REF] Obstfeld | Global Current Account Imbalances and Exchange Rate Adjustments[END_REF] or [START_REF] Backus | Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?[END_REF]. Note that in theory, it is only the partial version of σ N oF irm that can be inferred as 1 -η N oF irm . The theoretical relation bewteen total elasticities is more complex, as illustrated in equation ( 7).

Constrained estimates stand in contrast with the value of η N oF irm obtained when σ k is left unconstrained across sectors. As shown in the Table, the parameter jumps to -4.5, with standard errors that guarantee a significant difference at conventional confidence levels. This more than doubles the corresponding value for the estimated elasticity of substitution, at 5.5 -again, on the basis of a direct inference on σ N oF irm as given by 1 -η N oF irm . The right panel of the Table reports the theoretical values for σ N oF irm , which we find is in fact around 7. We argue this is the value that should enter the utility of a representative agent with heterogeneous preferences across sectors. Given the overwhelming evidence that substitutability is heterogeneous across goods or sectors, we contend a value around 7 is preferable from a calibration standpoint. Such a high value characterizes adequately the average substitutability of a representative agent who has heterogeneous preferences across goods.

Why the discrepancy? Does this not mean that aggregate quantities should be measurably more responsive to aggregate relative prices? [START_REF] Orcutt | Measurement of Price Elasticities in International Trade[END_REF] already speculated an explanation, reported by Goldstein and Khan (1985). On page 1070, they explain that, "in aggregate trade equations, goods with relatively low price elasticities can display the largest variation in prices and therefore exert a dominant effect on the estimated aggregate price elasticity, thereby biasing the estimate downwards." We view the results in Table 1 as a confirmation of this decade-old conjecture. With heterogeneity, the estimated response of aggregate quantities is not indicative of the true average elasticity of substitution.

It is useful to check the aggregation bias we document continues to prevail in estimates of the partial import elasticity. We report its constrained and unconstrained values in Table 1.

As is patent, a bias continues to prevail, with a constrained estimate at -2.7, jumping to -6.5

when sector specific elasticities are permitted. The adjustment of price indices does not explain our results away. But as expected, it tends to dilute measured elasticity, as partial import elasticities are systematically higher.

Relevance

Is the correction we document relevant in economic terms? We now discuss the quantitative and qualitative consequences of our corrected estimates. The most straightforward implication concerns models directly dealing with the responsiveness of traded quantities to relative prices, and in particular the resolution of global imbalances. Most prominently, [START_REF] Obstfeld | Global Current Account Imbalances and Exchange Rate Adjustments[END_REF] use a calibrated model to argue a reversal of the US current account is compatible with a 30% depreciation of the real exchange rate. The calibration sets substitutability at 2. In a slightly simplified two-country version, we obtained depreciation rates of 22 and 21% for values of the parameter of 5 and 7, respectively, down from 31% with an elasticity of 2. 13 The parameter is quantitatively important, and shaves off one third of the "required" depreciation, almost all the way to the 19.3% that obtains for an elasticity of 100. This is true even though

Obstfeld and Rogoff's calibration gives prominence to another parameter, the elasticity of substitution between traded and non-traded goods, important in this instance because the US is a largely closed economy. Still, the effects are sizeable and probably important in terms of welfare as well. [START_REF] Cole | Commodity trade and international risk sharing : How much do financial markets matter?[END_REF] showed the endogenous response of the terms of trade can deliver perfect insurance against country-specific shocks when the elasticity of substitution between domestic and foreign goods is exactly unitary. The result is meant as an illustrative special case of a powerful mechanism. Still, models of international portfolio holdings will have drastically different qualitative predictions depending on which side of one the parameter lies. For instance, [START_REF] Heathcote | The International Diversification Puzzle Is Not As Bad As You Think[END_REF] show that, with complete markets, complementarity between domestic and foreign goods can generate a home equity bias. A positive domestic productivity shock will increase the relative return on domestic stocks as long as the terms of trade do not respond too strongly: there will be portfolio home bias for values of the elasticity of substitution below one. A contrario, in [START_REF] Coeurdacier | Do trade costs in goods market lead to home bias in equities?[END_REF], domestic consumers choose to hold foreign assets to insure against shocks to domestic consumption, provided the terms of trade respond 13 We are grateful to Cedric Tille for graciously giving us the simulation code.

strongly enough in response to real shocks. This happens for values of the elasticity above one. Both papers then move on to introduce labor income risk and endogenous production, or incomplete markets. Our purpose here is not to settle the question of the origins of an equity home bias. Rather, we stress the fact our estimated aggregate elasticity is substantially above one has far-reaching implications in models of international portfolio choice. [START_REF] Atkeson | Pricing to Market, Trade Costs, and International Relative Prices[END_REF] propose to explain observed deviations from PPP in a model with trade costs, imperfect competition and variable markups. Variables profit margins require that the elasticity of substitution between the (foreign and domestic) varieties that form a sector be larger than the substitutability across sectors. In their model, markups then depend on market shares, and can therefore fluctuate over time. In their calibration, the elasticity equals 10; deviations from PPP virtually disappear for an alternative value set at 3.

It is perhaps not surprising that models of the real exchange rate should have predictions that depend on the substitutability between domestic and foreign goods. A direct corollary is that the policy consequences of international price differences will also depend crucially on the parameter. Presumably, the relevance of the exchange rate in the monetary policy rule developed in Galí and Monacelli (2005) depends on the substitutability between foreign and domestic varieties. Galí and Monacelli focus on unitary elasticity, so the result is not directly apparent there. But [START_REF] Benigno | Optimal Monetary and Fiscal Policy for a Small Open Economy[END_REF] introduce a generalization of their model, and their conclusions point to that direction. As in [START_REF] Cole | Commodity trade and international risk sharing : How much do financial markets matter?[END_REF] with unitary elasticity, a marginal reduction in the utility value of output is accompanied by an exactly offsetting reduction in the utility value of consumption. This insulates the economy from terms of trade movements. With non unitary elasticity however, policy shocks that affect the terms of trade also affect welfare, in a way that crucially depends on whether the calibrated parameter is above or below one. The fact that we find an aggregate estimate substantially above one must therefore have important policy implications.

The elasticity of substitution of a representative agent living in a one-sector model should be aggregated adequately on the basis of the microeconomic elasticities that we know are heterogeneous. We have shown estimating that average on the basis of macroeconomic data can be misleading. Our recommendation is that σ N oF irm should be preferred to σNoFirm in calibrating one-sector theoretical economies if one wants to capture the fact that σ k are heterogeneous in the data. We close with a simple exercise to validate this claim. We construct a two-sector version of a classical model in international economics, due to [START_REF] Backus | Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?[END_REF] [BKK], where the sources of heterogeneity are sector-specific elasticities of substitution between domestic and foreign goods, along with α k and w M k . We calibrate this version of the model using our sectoral results, and simulate a J-curve from it. We then compare this prediction with the conventional one-sector model. We ask what value of the (single) elasticity of substitution in the one sector model reproduces the J-curve that is implied by the calibrated two-sector version. We expect an adequately weighted average of the calibrated values of σ k to come closest to the dynamics implied by the multi-sector version. Since the workings of the model are well known, we leave a detailed description of the details to Appendix E. We compare the relative performances of three models. First, a two-sector version of BKK, calibrated on our data and our results. In particular, we choose (σ 1 ; σ 2 ) = (4.8; 12.9), which corresponds to below-and above-mean averages of σ k . The second model uses the arithmetic average of σ 1 and σ 2 in a conventional one-sector version of BKK. This version is meant to capture a macroeconomic calibration fully ignorant of heterogeneity issues.14 Finally, we calibrate a one-sector version of BKK using a weighted average of σ 1 and σ 2 , k=1,2 n kd σ k , consistent with the allowance for heterogeneity we have argued matters quantitatively. In all three models, we calibrate α k and w M k using our data. Figure 2 reports the J-curves implied by the three models. As is patent, the one-sector version of BKK that best matches the dynamics of the trade balance implied by the two-sector model is one that accounts for heterogeneity in the manner that we have described in this paper.

Stability

This section verifies the robustness of our results in three dimensions. First, we ascertain our results do not depend on a particular choice of data source in computing α k and w M k . Second, we investigate the importance of "Common Correlated Effects" in obtaining estimates of σ k .

Third, we relax our assumption that elasticities of substitution be identical across the HS6 categories regrouped in each ISIC sector. Instead, like [START_REF] Broda | Globalization and the Gains from Variety[END_REF] we estimate a value of σ k for each HS6 category. We discuss the necessary shortcuts this requires in terms of aggregation.

Table 2 compares the constrained and unconstrained values of the total elasticities σ N oF irm and η N oF irm using different weighting vectors. The data sources and computations behind the four alternative variants we present in the Table are discussed in Section 3.3. The first line repeats the results implied by the benchmark weights we have used so far. Across the four variants, constrained estimates of η N oF irm range around -2 and are not significantly distinguishable from conventional estimates, for instance in [START_REF] Goldstein | Income and price effect in foreign trade[END_REF]. Unconstrained estimates reach -5. The bias continues to be quantitatively important across these four alternatives.

The inclusion of Common Correlated Effects in the estimation of σ k is justified by our interest in the macroeconomic implications of the microeconomic values we obtain. After all, the quantities traded at sector level, and their prices, do presumably respond to common, aggregate, Note: Benchmark: w M k using imports and output from IO tables, α k using STAN sectoral interior demand. Variant 1: w M k using imports from BACI and output from STAN, α k using STAN sectoral interior demand. Variant 2: n k and w M k using imports and output from IO tables, α k using STAN sectoral interior demand. Variant 3: n k and w M k using imports from BACI and output from STAN, α k using STAN sectoral interior demand. Variant 4: w M k using imports and output from IO tables, α k using STAN sectoral interior demand (absorption in terms of value added). macroeconomic influences. When estimating the "true", sector-specific substitutability between domestic and foreign varieties, one want to ascertain one is not capturing aggregate dynamics.

This would amount to double-counting at the time of aggregation. Does this matter in our estimations? Table 3 provides a mixed answer, using again our benchmark weights. Without CCE, the constrained estimate of η increases slightly, to -2.17, whereas the unconstrained estimate decreases slightly, to -4.08. These changes are not strongly significant relative to our benchmark results, and they do not alter the conclusion of a significant heterogeneity bias. But they nevertheless suggest the introduction of a CCE term in equation ( 11) is not innocuous.

Finally, we relax our assumption that the substitutability between two HS6 categories be identical within each ISIC sector. In other words, we allow for heterogeneity in σ k even within each ISIC sector. Like [START_REF] Broda | Globalization and the Gains from Variety[END_REF], we estimate an elasticity of substitution for each HS6 sector, and then use equation ( 4) to aggregate them at the macroeconomic level.

This raises the question of what values for α k and w M k to use: we do not observe any direct information on any of these weights at such a refined aggregation level. We choose to impose similar values of the weights for all HS6 categories that belong to one ISIC sector. Clearly, this assumes away some possible source of a heterogeneity bias, but there is simply no alternative.

But we know choosing other values for α k and w M k do not matter at the ISIC level. At least, this whether the heterogeneity in estimates of σ k within each ISIC category can be such that our conclusions are altered.

Naturally, the constrained estimates of η N oF irm continue to be identical, for instance at -2.17 without a CCE corrective term. After all, this is an estimation that constrains all coefficients to be identical, within and between ISIC categories. The difference arises for unconstrained estimates. We estimated values for σ k in 4,021 HS6 categories, and aggregated them using our benchmark (ISIC) weights. We obtained a value of -5.23 for the unconstrained elasticity.

Computing standard error bands across this point estimate is not tractable, but the aggregation bias appears to be even stronger. We conclude ignoring heterogeneity within ISIC sectors is not important to our conclusions.

Conclusion

The elasticity of substitution between domestic and foreign varieties is central in international economics. But no clear consensus has emerged from a vast empirical literature seeking to pin down the parameter, except for one essential disagreement. On average, microeconomic data tend to imply substantially higher values than macroeconomic aggregates. The point estimates are also much more heterogeneous. We propose that this heterogeneity is the reason why aggregate results are close to zero. We compute structural estimates of aggregate substitutability allowing or not for heterogeneity at the sectoral level. We find that imposing homogeneity is enough to obtain aggregate estimates in line with the macroeconomic evidence, even using a disaggregated dataset. Allowing for heterogeneity results in an aggregate parameter value of up to 7. This discrepancy validates the conjecture of an aggregation bias in elasticity estimates that goes back at least to [START_REF] Orcutt | Measurement of Price Elasticities in International Trade[END_REF]. Such high parameter values change dramatically the conclusions of calibrated models in areas of international economics as varied as the international transmission of shocks, global imbalances, international risk sharing, portfolio choice and optimal monetary policy.

A Heterogeneous Supply

Here we present the now classic model of production with heterogeneous firms making entry decisions, inspired from [START_REF] Melitz | The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity[END_REF]. The theory is written for a given sector k located in country i, and indexes are dropped to facilitate the exposition. All prices expressed in the exporter's currency are denoted with an asterisk * . Firms in each sector k and country i are monopolistically competitive, produce differentiated varieties and are heterogeneous in terms of productivity. They decide whether to pay a fixed cost f e prior to knowing their own productivity, which allows them to export to foreign markets. Once the cost is sunk, productivity ϕ is drawn from a distribution with probability density function g(ϕ), and an associated cumulated distribution G(ϕ). Productivity is revealed, and the firm decides whether to produce for the foreign market, with a technology featuring constant marginal costs and a fixed overhead export cost f . Both costs are expressed in terms of the unique composite factor of production, whose price is denoted with w * . Finally, exporting involves an iceberg cost τ , potentially specific to each sector and country. Note that for i ∈ I this setup departs slightly from the conventional [START_REF] Melitz | The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity[END_REF] model, where the decision to produce for the domestic market predates that to export. We assume there are fixed costs involved in gathering information on export markets, that are quite separate from those involved in a purely domestic production decision. 15 Total costs in exporter's currency are given by

T C * (ϕ) = w * f + w * τ c(ϕ) ϕ
where c(ϕ) is the equilibrium demand addressed to the firm by the foreign market. Individual costs are decreasing in ϕ, as firms with higher productivity can produce the same amount of output with fewer workers. Firms face a probability δ of exogenous exit in each period. Under these assumptions, the decision to enter export markets and pay the sunk cost f e is governed by the following maximization of the (probability) discounted value of profits

v(ϕ) = max 0, ∞ t=s (1 -δ) t-s π * (ϕ) = max 0, 1 δ π * (ϕ) (A.1)
15 See for instance Segura-Cayuela and Villarubia (2008).

where profits, expressed in the exporter's currency, are given by

π * (ϕ) = p * (ϕ) - τ w * ϕ c -w * f
There is a unique threshold productivity φ such that v(ϕ) > 0 if and only if ϕ > φ. Firms that are productive enough remain in the market after having drawn their productivity and produce in every period. Their export price is set by maximizing profits under a demand constraint given in the body of the paper by equation ( 2), i.e.

p * (ϕ) = ρ ρ -1 τ w * ϕ (A.2)
As [START_REF] Melitz | The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity[END_REF], we introduce a weighted average productivity measure for firms active in

the market φ = 1 1 -G( φ) ∞ φ ϕ ρ-1 g(ϕ)dϕ 1 ρ-1
Then define average profits π * :

π * ≡ 1 1 -G( φ) ∞ φ π * (ϕ)g(ϕ)dϕ = 1 1 -G( φ) 1 ρ c p E ρ ρ ρ -1 τ w * 1-ρ ∞ φ ϕ ρ-1 g(ϕ)dϕ -w * f
Since the threshold firm makes no profit, average profits simplify into

π * = w * f φ φ( φ) 1-ρ -1 (A.3) Equation (A.3) is a "Zero Cutoff Profit" (
* = w * f κ ρ -κ + 1 - 1
The ZCP condition characterizes a short run equilibrium, where the threshold productivity φ is determined in function of the model's parameters. It is a short run equilibrium in that it takes firms' location decisions, and thus the potential number of firms active in the market, as given.

A long run equilibrium obtains when a free entry condition holds, that ensures the expected discounted value of profits for a potential entrant equals the fixed cost of entry in the export market. The condition will pin down both threshold productivity and the number of firms active in the export market. Rearranging the condition

∞ 0 v(ϕ)g(ϕ)dϕ = f e implies π * = δf e 1 -G( φ)
When productivity is distributed according to a Pareto distribution, the Free Entry (FE) con-

dition becomes π * = δw * f e φ b κ
The ZCP and FE conditions define a system of two equations in two unknowns, π * and φ.

In the particular case of a Pareto distribution, the two conditions give an expression for the productivity cut-off φ as a function of exogenous parameters: .4) shows that, once adjustments in the mass of entering firms are accounted for, the productivity cut-off is independent from cost shocks. In the long run, firms relocate in response to price changes. When productivity is distributed according to a Pareto-distribution, the threshold productivity φ remains unchanged. The adjustment along the extensive margin happens exclusively via firm relocation and adjustment in the mass of potential exporters. This differs from Chaney (2008) where the total mass of potential entrants is taken as given.

φ = b f δf e 1/κ κ ρ -κ + 1 -1 1/κ (A.4) Equation (A
Finally, the mass M of firms is obtained using the definition of the sectoral price index

p = M 1 -G( φ) ∞ φ (p * (ϕ)E) 1-ρ g(ϕ)dϕ 1 1-ρ
where E is the bilateral exchange rate, expressed in terms of the importer's currency. Using optimal pricing, this implies

p 1-ρ = M ρ ρ -1 τ Ew * 1-ρ φρ-1
Substituting the expression into the definition for average profits implies

π * = 1 ρ pc EM -w * f
Using the ZCP condition finally gives

M = ρ -κ + 1 κ 1 ρ 1 w * f pc E
We now derive how prices respond to the cost shock. Since we identify separately domestic and foreign entities, we re-introduce sector and country indices. Foreign price indices P ki , i ∈ I, i = d, can now react to domestic cost shocks because market structure responds endogenously.

For all i in I, i = d, we have

∂ ln P ki ∂ ln w d = 0 + 1 1 -ρ k p kif ( φki ) P ki 1-ρ k M ki g( φki ) φki ∂ ln φki ∂ ln w d + 1 1 -ρ k ∂ ln M ki ∂ ln w d (A.5)
while the response of the domestic price index is slightly different, Using equilibrium values for pc, straightforward algebra implies Equations (A.7), (A.8) and (A.9) form a system that can be solved for the response of the price index P k to a domestic cost shock, allowing for long run firm entry decisions in both the domestic and all foreign markets. This writes

∂ ln P kd ∂ ln w d = 1 + 1 1 -ρ k p kdf ( φkd ) P kd 1-ρ k M kd g( φkd ) φkd ∂ ln φkd ∂ ln w d + 1 1 -ρ k ∂ ln M kd ∂ ln w d (A .
∂
∂ ln P k ∂ ln w d = ρ k ρ k -1 1 -w M k - 1 ρ k -1 ∂ ln P ∂ ln w d
which completes the derivation in the text.

B The Price Elasticity of Imports

The price elasticity of imports is defined as Using the definitions of all price indices, we obtain

η N oF irm = ∂ ln [ k i P ki C ki ] -∂
η N oF irm = k n k (1 -σ k )(1 -w M k ) - ∂ ln P ∂ ln w d = 1 - k n k σ k + k n k (σ k -1)w M k - k α k 1 -w M k
which is the expression in the text.

C Derivation of the Supply Curve

Omitting once again indexes for sector k and country i, the definition of the sector price index implies

p 1-ρ = M ρ ρ -1 τ w * E 1-ρ 1 1 -G( φ) ∞ φ ϕ ρ-1 dϕ = M ρ ρ -1 τ w * E 1-ρ φ( φ) ρ-1
Now in the long run we have

M = φ φ( φ) ρ-1 1 ρ 1 w * f pc E Substituting M out gives p = φ 1-ρ ρ (w * ρf E) 1 ρ c -1 ρ ρ ρ -1 τ w * E 1-ρ
Now recognize that φ is constant, and define exp(υ kit ) = (w * ρf E)

1 ρ ρ ρ-1 w * 1-ρ
(τ E) -ρ to obtain the expression in the text. Note that we have assumed constant returns to scale in our theory, but the effective estimation procedure allows for non-constant returns to scale, i.e for non negative values of ω.

D Estimated Variances

The variance of σk is computed using the second-order moments of θ1k and θ2k and a first-order approximation of σ k around its true value: Using the same reasoning, the first-order approximation of the aggregate elasticity around its estimated value gives:

σ k =
σ N oF irm = σNoFirm - k∈K [n k + (n k -n kd )(1 -w M k )](σ k -σk )
The variance of σNoFirm is then defined as:

V ar(σ N oF irm ) ≡ E(σ N oF irm -σNoFirm ) 2 ⇒ V ar(σ N oF irm ) =

k∈K [n k + (n k -n kd )(1 -w M k )] 2 V ar(σ k ) + k∈K k =k [n k + (n k -n k d )(1 -w M k )][n k + (n k -n kd )(1 -w M k )]Cov(σ k , σk )
Since we control for common correlated effects in the estimation of the σ k s, Cov(σ k , σk ) is effectively zero, and the estimated variance is given by

V ar(σ N oF irm ) = k∈K [n k + (n k -n kd )(1 -w M k )] 2 V ar(σ k )
In the constrained case, the same reasoning gives

V ar( σNoFirm ) = k∈K [n k + (n k -n kd )(1 -w M k )] 2 V ar(σ)
The variance of the constrained and unconstrained import elasticities are given by:

V ar(η N oF irm ) = where U = log c + γ (1-nt) 1-η 1-η and c it (n it ) denote aggregate consumption (hours worked). Aggregate consumption is a Cobb-Douglas function of sector-specific consumption

c i,t = c a i,t α i c b i,t 1-α i α α i i (1 -α i ) 1-α i
where c a i,t and c b i,t are the consumption baskets of good a and b, and α i is the share of sector a in nominal aggregate consumption. The same structure prevails for aggregate investment:

i i,t = i a i,t α i i b i,t 1-α i α α i i (1 -α i ) 1-α i
Sectoral output is produced with capital k and labor n following a Cobb-Douglas function:

y k i,t = z i,t k k i,t θ n k i,t 1-θ , i = 1, 2, k = a, b
The quantity y k i,t denotes country i's production of good k, in units of the local good. In equilibrium, it is equal to domestic sales c ik i,t +i ik i,t plus exports c i k i,t +i i k i,t . The vector z t = (z 1,t , z 2,t ) is a stochastic shock to productivity. Importantly, productivity shocks are assumed symmetric across sectors. The cross-sector symmetry assures that, in each country, producer prices are homogenous. In what follows, domestic prices are normalized to unity and the relative price of foreign goods is denoted P .

Sectoral consumption and investment, c k i,t and i k i,t are composites of foreign and domestic goods:

c k i,t = β k i c i k i,t σ k -1 σ k + (1 -β k i )c ik it σ k -1 σ k σ k σ k -1 i k i,t = β k i i i k i,t σ k -1 σ k + (1 -β k i )i ik it σ k -1 σ k σ k σ k -1
The elasticity of substitution between foreign and domestic varieties σ k is sector-specific. The weights β i k are related to the share of imports in the sectoral consumption of good k. In the calibration, they are assumed symmetric across countries but potentially different across sectors. To calibrate these, we use the sectoral data in our estimation of the aggregate substitution elasticity. In terms of the model presented in Section 2, α i , the share of sector a in nominal consumption, is directly related to α k . In a symmetric steady state with P = 1, the β k i parameters are linked with the w M k parameters according to

β k i = w M k 1 -w M k 1 1-σ k + 1 -1
As before, n kd can be expressed as:

n kd = 1 -w M k α k k (1 -w M k ) α k

  bilateral trade flows by their volume. In BACI, values are denominated in USD and are Free On Board. 11 Quantities are in tons. The empirical model described in section 3.1 is not sensitive to the currency denomination of trade data, nor to the treatment of trade costs, as both are passed into the residuals. Expenditure shares are measured as skit = Pkit C kit i =d Pkit C kit .

Figure 1 :

 1 Figure 1: Microeconomic estimates of the elasticity of substitution

  Figure 1. Keeping in mind ours are estimates of long run elasticitites, these values lie in similar ballparks.

Figure 2 :

 2 Figure 2: The J-curve in a two-sector BKK model

  ZCP) condition. It delineates a relation between average profits π * and the productivity cut-off φ. In particular, when the distribution of productivity follows a Pareto distribution, G(ϕ) = 1 -

  6) because each domestic firm prices at a fixed markup above its marginal costs w d . The first term in both equations measures the response of individual firm prices to the domestic shock, ∂ ln P kif ∂ ln w d . This is zero for i in I, i = d and 1 for i = d, since it reflects the pure intensive margin of adjustment. The second term captures the response of the productivity cut-off, φki , and the third term corresponds to the adjustment in the potential number of firms serving the market. In the long run, and under a Pareto distribution, equation (A.4) implies ∂ ln φki ∂ ln w d = ∂ ln φkd ∂ ln w d = 0.

  -Sector Version of BKK 16Two countries i = 1, 2 face aggregate productivity shocks. Each country is inhabited by a large number of identical agents and labor is internationally immobile. Our main departure from BKK is that each country produces two goods, a and b. Preferences of the representative agent in country i are characterized by utility functions of the formE 0 ∞ t=0 β t U (c it , 1 -n it )

Table

  our calibration. Our only deviation from the classical BKK model pertains to the elasticities of substitution at the sector level. In particular, the two new parameters in the multi-sector version of BKK pertain to the calibration of the Armington and Cobb-Douglas aggregators for consumption.

Table 1 :

 1 Estimation with common correlated effects

		Import Elasticity Substitution Elasticity
		η N oF irm	σ N oF irm
	Constrained total elasticity	-1.980 a (.175)	4.124 a (0.300)
	Constrained partial elasticity	-2.738 a (.262)	3.738 a (0.263)
	Unconstrained total elasticity	-4.508 a (.745)	7.226 a (0.962)
	Unconstrained partial elasticity	-6.553 a (1.100)	6.921 a (0.697)
	Number of sectors Number of grid searches	56 11	56 11
	Note: Standard errors in parentheses (obtained by bootstrapping for grid searched sectors), a denotes significance at
	the 1% level.

Table 2 :

 2 Variants on the weights

		Import elasticity	Substitution Elasticity
		Unconstrained Constrained Unconstrained Constrained
	Benchmark	-4.51	-1.98	7.22	4.12
	Variant 1	-5.17	-2.21	6.93	4.05
	Variant 2	-4.38	-2.08	7.36	4.02
	Variant 3	-4.60	-2.15	6.77	4.06
	Variant 4	-4.41	-2.10	7.27	4.12

Table 3 :

 3 Estimation without common correlated effects

		Import Elasticity Substitution Elasticity
		η N oF irm	σ N oF irm
	Constrained total elasticity	-2.166 a (.150)	4.442 a (.257)
	Constrained partial elasticity	-3.016 a (.225.)	4.016 a (.225)
	Unconstrained total elasticity	-4.075 a (.112)	6.584 a (0.145)
	Unconstrained partial elasticity	-5.946 a (.209)	6.321 a (0.138)
	Number of sectors Number of grid searches	56 12	56 12
	Note: Standard errors in parentheses (obtained by bootstrapping for grid searched sectors), a denotes significance
	at the 1% level.

  ln M ki ∂ ln w d = (1 -σ k ) ∂ ln P ki ∂ ln w d + (σ k -1) ∂ ln P k ∂ ln w d + ∂ ln P ∂ ln w d ln P ki ∂ ln w d is now different from zerobecause of the response of market structure in all foreign economies. Substituting in equations (A.5) and (A.6), respectively, we obtain∂ ln P ki ∂ ln w d = σ k -1 σ k -ρ kBy definition, and since P ki can now change in all countries i, we have

	and
	∂ ln M kd ∂ ln w d	= (1 -σ k )	∂ ln P kd ∂ ln w d	+ (σ k -1)	∂ ln P k ∂ ln w d	+	∂ ln P ∂ ln w d	-1
	where ∂ ∂ ln P k ∂ ln w d	+	1 σ k -ρ k	∂ ln P ∂ ln w d	(A.7)
	and
	∂ ln P kd ∂ ln w d	= -	ρ k σ k -ρ k	+	σ k -1 σ k -ρ k	∂ ln P k ∂ ln w d	+	1 σ k -ρ k	∂ ln P ∂ ln w d	(A.8)
	Notice this implies
				∂ ln P ki ∂ ln w d	=	∂ ln P kd ∂ ln w d	+	ρ k σ k -ρ k
		∂ ln P k ∂ ln w d	= w M k	∂ ln P ki ∂ ln w d	+ 1 -w M k	∂ ln P kd ∂ ln w d	(A.9)

  ln w d

	Using equilibrium traded values, this simplifies into
	η N oF irm =	k	i	n ki (σ k -1)	∂ ln P ki ∂ ln w d	+ (1 -σ k )	∂ ln P k ∂ ln w d	-	∂ ln P ∂ ln w d

Most papers in this literature also take interest in the response of quantities traded to changes in income. We do not discuss this parameter here.

We could introduce without loss of generality an additional price wedge, reflecting distribution costs that presumably affect both domestic and foreign varieties. This would merely add some notation, but no further insight. In the empirics, the price of each variety is measured Free on Board, i.e. net of both retail and transportation costs.

The second assumption is made for convenience. The intuition remains the same if we focus on a change in (all) relative prices between the domestic economy and a specific exporter i. The data needed to perform aggregation are just slightly different, and identification becomes more complicated since it relies on the crosssection of all exporters to a given destination.

The second term in equation (4) exists because of our focus on an aggregate shock in relative prices. If instead the shock considered were microeconomic in nature and focused on a specific exporter i -a change in tariff-then under standard atomistic assumptions and large I, the second term in equation (4) would disappear, and we would be left with a partial elasticity.

This will be true as long as corr (n kd , ρ k ) < ρ k σ k corr (n kd , σ k ). The condition is plausible, since the correlation between ρ k and the expenditure share of sector k is likely to be smaller than that between σ k and the expenditure share. After all, ρ k denotes a substitutability between firms within a sector, whereas σ k captures a substitutability between countries.

 6 We focus on estimates of the price elasticity of imports abstracting from firm dynamics, since we just established they do not affect our main results.

We followFeenstra (1994a) and assume all exporters have the same supply elasticity.

υ kit will also absorb any heterogeneity in the extent of the exchange rate pass-through.

In the empirics, we choose a reference country that is present in the US market during the whole observed period.

For the instrumentation to be consistent, there must be some cross-country differences in the relative variance of the demand and supply curves. For an intercept to capture measurement error, its variance must be equal across exporting countries. Of course, it may still be specific to each sector. SeeFeenstra (1994aFeenstra ( , 1994b)).

In general, trade data are collected by national customs offices in the currency of the declaring country. These data are then converted in US dollars by the United Nations, using the current nominal exchange rate.

α k and w M k do not sum to one because of non-traded sectors. Since n k and n kd both sum to unity by definition, we normalize each definition so that it is the case.

We note the arithmetic average of σ 1 and σ 2 is not necessarily an accurate proxy for σ. We have discussed in section 3.2 the possibility that macroeconomic estimates of σ should suffer from an heterogeneity bias of an econometric nature.

We are grateful to Jean-Olivier Hairault who kindly gave us his codes to the one-sector version of the BKK model.

We thank the Fondation Banque de France and the National Center of Competence in Research "Financial Valuation and Risk Management" for financial support. The National Centers of Competence in Research (NCCR) are a research instrument of the Swiss National Science Foundation. This paper was begun while Imbs was visiting the Hong Kong Institute for Monetary Research, whose hospitality is gratefully acknowledged, and while Méjean was at CEPII.

Manuf. of electric lamps & lighting

Manuf. of TV & radio receivers

Manuf. of transport equipment n.e.c.

Manuf. of other fabricated metal products

Manuf. of structural metal products

Sawmilling

& planing of wood Manuf. of insulated wire & cable Manuf. of other textiles Manuf. of products of wood Manuf. of plastics products Manuf. of other food products Manuf. of paper products Manuf. of non-metallic mineral products Manuf. of beverages Publishing Accessories for motor vehicles Manuf. of furniture Mining & quarrying n.e.c. Manuf. of other chemical Manuf. of electric motors Manuf. of grain mill products & related Production of food Manuf. of domestic appliances n.e.c. Printing & related Manuf. of special purpose machinery Manuf. of rubber products Manufacturing n.e.c. Manuf. of knitted articles Manuf. of basic precious Manuf. of elect. valves & other elect. Comp. Manuf. of basic chemicals Manuf. of man-made fibres Crops; horticulture Manuf. of general purpose machinery Manuf. of medical appliances & measuring Manuf. of accumulators & primary batteries Spinning, weaving & finishing of textiles Manuf. of other electrical equipment n.e.c. Manuf. of elect. distribution & control Manuf. of basic iron & steel Manuf. of motor vehicles Manuf. of wearing apparel Tanning & dressing of leather Farming of animals Manuf. of office & computing machinery Manuf. of glass products Manuf. of tobacco products Manuf. of footwear Manuf. of optical & photo instruments Manuf. of refined petroleum products Fishing & fish farms Manuf. Of TV, telephone & radio transmitters

Crude petroleum & natural gas

Manuf. of aircraft & spacecraft

Forestry, logging & related

How does the magnitude of our individual sectoral estimates compare with the literature?

The aggregate capital stock evolves in each country according to:

where δ is the depreciation rate. Adjustment costs for capital follow Obstfeld and Rogoff (1996):

Finally, fluctuations arise from persistent shocks to aggregate productivity:

where ε Z is distributed normally and independently over time with variance V Z . The correlation between the technology shocks, z 1 and z 2 is determined by the off-diagonal elements of A and

We can obtain national income and product accounts for each country. Aggregate GDP in country 1 in period t, in units of domestically produced goods, is y 1t = y a 1t + y b 1t . The resource contraint equates sectoral GDPs to the sum of (domestic and foreign) consumption and investment:

National output is related to expenditure components according to:

Finally, the trade balance, defined as the ratio of net exports to output, both measured in current prices, is:

and the terms of trade P t equal the sectoral marginal rate of transformation between the two varieties in country 1, evaluated at equilibrium quantities.