Which of the following is not a factor affecting the size of the trade area?

18.1 Introduction

This chapter discusses trade elasticities – the response of traded quantities to changes in prices of tradable goods. While the results of computable general equilibrium [CGE] experiments depend upon a number of inputs, trade elasticities are of particular interest because they significantly impact upon the modeled effects of policy experiments on trade patterns, welfare and factor returns, among other important phenomena.

It is common when calibrating CGE models to select trade elasticities from “the literature” while selecting other [taste and technology] parameters to allow the theory to replicate the data.1 Curiously, there is no clear consensus on which elasticities to use. Major trade-focused CGE models draw elasticities from many different econometric studies. These econometric studies use very different data samples, response horizons and estimating techniques, and arrive at elasticities as much as an order of magnitude different from each other. This raises the critical question: which elasticities are “right?” Or at least, which are right for the particular modeling application at hand?

As a starting point for thinking about these issues, Figure 18.1 presents a simple partial equilibrium diagram in which the price and quantity of traded goods depends on export supply and import demand. Using this diagram, we can think through the effects of a policy experiment such as raising a tariff on foreign goods.

To fix ideas, consider a parsimonious representation of import demand in which quantities imported depend on prices in the foreign country [F], inclusive of tariffs and real expenditures in the home country [H]:

[18.1]lnqF=lnEH−σlnpF[1+τ].

The key parameter is σ, which can be thought of as a reduced form measuring the elasticity of import quantities with respect to import prices, but is more commonly given a structural interpretation. For example, in many common CGE frameworks, this demand function arises from a constant elasticity of substitution [CES] cost or utility function in which buyers regard home and foreign varieties as imperfect substitutes. This is known as the Armington assumption [and σ is sometimes referred to as the Armington parameter or Armington elasticity], although very similar formulations arise in other common modeling frameworks such as monopolistic competition.

The parameterization of σ has important quantitative implications for a number of variables that are of interest to economists and policy makers alike, and we highlight these in Section 18.2. The most direct and explicit link is via import quantities. In Figure 18.1, a rise in tariff rates shifts the export supply curve upwards along the import demand curve. Here, the elasticity of import demand effectively summarizes the first-order response of traded quantities to changes in trade cost changes. These first-order effects, as summarized in Equation [18.2] imply that doubling the trade elasticity will double the response in measured quantities.2

In Section 18.3 we survey the literature estimating import demand elasticities. We highlight important differences across the econometric literature in the price shocks observed, the time horizon over which responses are measured, the comparison set of countries and the level of aggregation. Estimates of σ vary considerably, and we provide a lengthy discussion of why these estimates vary and which are appropriate in different circumstances.

A recurring theme throughout the chapter is the difficulty of separating supply and demand parameters. Ideally, one would observe movements in export supply induced by policy shocks in the manner described in Figure 18.1. Unfortunately, data experiments of this sort are somewhat rare and so many early studies exploit time-series variation in foreign prices pF in Equation [18.1]. Since prices are jointly determined by supply and demand this raises a critical issue of identification: are these time-series studies observing shocks to supply and identifying the elasticity of import demand or are they observing shocks to demand and observing the elasticity of export supply, or some combination of the two? In more recent econometric papers we survey, price variation is driven by shocks to tariffs or transportation costs in precisely the manner described in Figure 18.1. This sort of estimation procedure provides a reasonably close match to thought experiments typically contemplated in CGE trade liberalization exercises and also allows the econometrician to better control for shocks to demand.

In Section 18.4 we turn to the literature on estimating the elasticity of export supply. Single-country CGE trade models do not provide an explicit modeling of production and demand in the rest of the world. Instead, they may parameterize a country’s exports to the rest of the world [and the supply of imports into that country from the rest of the world] in a reduced form way. These approaches have a weak connection between the underlying supply-side details that give rise to an export supply curve as in Figure 18.1 and we discuss the reduced form econometric work used to parameterize it.

Multicountry CGE trade models do provide explicit modeling of production and demand worldwide, and so do not parameterize export supply in this reduced form way. While these models are primarily interested in import demand elasticities, the identification issue just discussed requires the econometrician to account for supply. We next discuss a literature that estimates systems of export supply and import demand in order to get proper identification of each. While this literature has primarily been mined for import demand elasticities, it provides a potentially useful source of export supply elasticities.

To make progress on the export supply front it is necessary to move away from reduced forms and provide a parameterization that is closely linked to theory. We discuss developments in the literature on trade with heterogeneous firms that provide such a link. These developments are useful on one dimension and challenging on another – the possibility of within industry heterogeneity calls into question the identification of demand parameters used throughout a large literature, they suggest that econometricians are actually, and only, estimating supply responses!

Finally, in Section 18.5 we discuss structural estimation as a possible way forward. We step the reader through a progression of papers in order to show the assumptions under which import demand and export supply parameters can be extracted from available data. None of these approaches are “magic bullets.” Our discussion instead highlights the point that these papers differ primarily in what they hold fixed, or what external parameters they bring to bear in order to extract residual parameters from the data.

Ultimately, the interpretation of trade responses is model dependent. CGE practitioners should come away from this survey with a sense of where the elasticities “in the literature” come from, how they are identified and what they purport to measure. We do not ultimately pronounce upon the question of which estimates provide the “right” elasticities. Rather we hope to inform the choice of trade response parameters by informing practitioners about the nature of the assumptions econometricians have undertaken in order to move from the theory to the data to resulting estimates.

7.5.3 Behavioral parameters

It is common practice for regional CGE models to use elasticities borrowed from their national counterparts [e.g. Jones and Whalley, 1989; Madden, 1990]. While this is a reasonable approach in the case of most types of elasticities, frequent concern is expressed in the case of the inter-regional trade Armington elasticities [e.g. Partridge and Rickman, 2010]. It is commonplace for regional CGE modelers to undertake sensitivity analysis on these latter elasticities [Turner, 2009].

In the case of many countries, there is a dearth of the regional data required to undertake econometric estimates of inter-regional trade elasticities. There have been, however, some studies [e.g. Bilgic et al., 2002; Ha et al., 2010] that econometrically estimate inter-regional Armington elasticities for the US where commodity flow surveys are undertaken by the Bureau of Transportation Statistics at 5-year intervals. Ha et al. compare their results for inter-regional elasticities with Bilgic et al.’s, and with three studies estimating Armington estimates in international trade.73 The comparison is made for agriculture, mining and seven manufacturing commodities, and it suggests, in general, that inter-regional and international elasticities are within the same broad order of magnitude. They give no support to the often-held idea that international trade elasticities form a lower bound for the corresponding inter-regional import elasticities.

Certainly, it is an area calling for further econometric work. Examination of the movement in inter-regional twist parameters and price movements from historical multiregional CGE simulations might reveal evidence as to whether the inter-regional import elasticity estimates currently being used are reasonable. However, on the face of it, the use of inter-regional import elasticities equal to or close to their international counterparts would seem reasonable.

3.1.3 Trade Elasticity and Trade Policy

As we conclude the topic of the effects of trade policy on trade flows and gradually move toward an analysis of its effects on the gains from trade, one more observation is necessary. Recent work on the gains from trade [Arkolakis et al., 2010] has highlighted the importance of the reduced-form trade elasticity in computing the aggregate gains from trade. Given that the trade elasticity relates—by its very definition—changes in trade flows to changes in trade costs, exploiting observable changes in trade policy [ie, tariff reductions] seems an obvious way to credibly estimate it. What trade elasticity estimates do changes in trade policy imply?

Perhaps surprisingly, estimates of the trade elasticity based on actual trade policy changes are scarce, and the few that exist are all over the place. As discussed in Hillberry and Hummels [2013] in their review of the trade elasticity parameters used in the literature, the “trade elasticity” is usually estimated either based on cross-sectional [cross-country and cross-industry] variation of trade costs other than trade policy barriers or based on time series variation stemming from exchange rate fluctuations. Studies that rely on cross-sectional variation are often labeled “micro” studies and yield high values for the trade elasticity [around five or higher]. Studies that rely on time-series variation are often identified as “macro” studies and yield low estimates for the trade elasticity, around one or lower. A standard explanation for these divergent results is that cross-sectional studies identify long-run effects corresponding to different steady states associated with different trade costs, while studies based on time-series variation capture only the short-run effects of changing trade costs. Economic agents have time to adjust in the long run so the long-run trade elasticity is larger than the short-run elasticity. While this explanation is appealing, it abstracts from the fact that the two types of studies rely on very different sources of variation, so that the different estimates could instead be due to these different sources of variation. Indicatively, Shapiro [2014] relies on panel data in order to estimate the trade elasticity. The use of panel data implies that his elasticity estimate should be best thought of as a short-run one; yet, his results are closer to the ones obtained by cross-sectional studies because he relies on similar sources of variation.

This review does not examine work on estimation of the trade elasticity, but given the central role that trade elasticity plays in a number of trade models and in welfare analysis, it is surprising that trade policy has not been exploited to a larger extent to identify this crucial parameter. To our knowledge, the only exceptions to this pattern are the work by Yi [2003]—who however calculated the trade elasticity implied by tariff reductions only to subsequently denounce it as implausible—and the estimates provided in recent work by Caliendo and Parro [2015]. Caliendo and Parro estimate sectoral trade elasticities based on the import tariff reductions associated with NAFTA. The estimates displayed in table 1 of their paper display substantial heterogeneity, with trade elasticities ranging from 0.37 to 51.8! The authors reject the null hypothesis of a common elasticity across sectors. The heterogeneity of the estimates suggests that trade elasticity estimates may vary by sector, time, and country. This makes careful empirical work that exploits trade policy variation in order to identify the trade elasticity/ies more important. The fact that a key parameter in the trade literature is so rarely estimated based on trade policy variation speaks to the secondary role assigned to trade policy.

12.3.2 Systematic sensitivity analysis

Based on the foregoing discussion, it is clear that we are unlikely to have access to a fully validated, global CGE model in the near future. A more modest goal is to provide sufficient robustness checks to assure policy makers that key findings are not simply a function of certain arbitrary [or worse yet, strategically selected] parameter settings. This leads to the topic of systematic sensitivity analysis [SSA] – a tool that has been widely employed in the GTAP community to explore the sensitivity of model results to parametric uncertainty. The basic idea is to sample from a set of parameter distributions, each time re-solving the model and saving the results. After completion of the SSA, the user can compute standard statistics – most commonly the mean and variance of model results, thereupon providing model consumers with appropriately constructed confidence intervals. Thus, it should be possible to say, for example: “Given the overall structure of the GTAP model, we are 95% confident that this policy will improve regional welfare.” Of course, the only sources of uncertainty are the parameters and the policy shocks, which can be varied separately and/or together according to a specified covariance matrix [Horridge and Pearson, 2011].

There is now a long history of CGE studies utilizing SSA [for early contributions to this literature, see Pagan and Shannon, 1987; Wigle, 1991; Harrison and Vinod, 1992; Harrison et al., 1993]. The fundamental problem with SSA in a large-scale CGE model is that the individual model solutions can be quite time-consuming, even with modern computational facilities. This may preclude solving the model 10,000 times, as might be desirable from the point of view of Monte Carlo Analysis. Fortunately, other methods have been developed that appear to perform quite well in the context of standard CGE models. In particular, DeVuyst and Preckel [1997] show that a modest number of solutions via Gaussian Quadrature can approximate the true mean and variance of model results quite well for a simple, global CGE model. The Gaussian Quadrature approach has been tailored for use in GTAP through a series of GTAP Technical Papers [Arndt, 1996; Pearson and Arndt, 2000; Horridge and Pearson, 2011]. Indeed, the tools for implementing SSA in RunGTAP make it difficult for any author to excuse themselves from providing such robustness checks on their results and journal reviewers are increasingly insisting on SSA as part of the peer-review process.

Of course, none of the foregoing discussion touches on the question of how to specify the parameter distributions and these are central in determining the distributions of model results. Some authors have taken the approach of surveying the literature, treating each estimate of a given parameter as a draw from the underlying distribution [Harrison et al., 1993]. The problem with this approach is that there is a limited pool of published studies and many of them make different assumptions in their estimation approaches. In addition, the process of peer review has a tendency to lead to overly narrow parameter distributions, with so-called “unreasonable” values being ruled out a priori during the peer-review process. A more common approach to SSA is to simply specify a uniform or a triangular distribution with a lower endpoint of zero [for non-negative elasticity values]. This reassures the reader that the author is being suitable conservative by specifying a generous variance in the underlying distribution. However, none of this is really satisfactory. It would be far preferable to actually estimate the relevant parameters and the associated distributions and use these directly in the SSA.

This led Hertel et al. [2003] to undertake such an exercise in the context of their analysis of the proposed Free Trade Area of the Americas [FTAA]. Following earlier work by Hummels,11 their econometric work focuses on the estimation of a particular parameter, the elasticity of substitution among imports from different countries, which is central to any evaluation of a discriminatory trade agreement such as the FTAA. They match the data in the econometric exercise [from North and South America] to the policy experiment at hand, and employ both point estimates and the associated standard errors in a policy analysis which takes explicit account of the degree of uncertainty in the underlying parameters. In particular, they sample from the distribution of parameter values given by their econometric estimates in order to generate a distribution of model results from which they then construct confidence intervals. These authors find that imports increase in all regions of the world as a result of the FTAA, and this outcome is robust to variation in the trade elasticities. Nine of the 13 FTAA regions experience a welfare gain in which they are more than 95% confident. The authors conclude that there is great potential for combining econometric work with CGE-based policy analysis in order to produce a richer set of results that are likely to prove more satisfying to the sophisticated policy maker.

5.3 Calibrating Elasticities

The key parameters for counterfactual analysis using gravity models, like other CGE models, are elasticities. We conclude this section by discussing some of the issues arising when calibrating elasticities.

Trade Elasticities. As can already be seen from the counterfactual analysis carried in the context of the Armington model in Section 2, the single most important structural parameter in gravity models is the trade elasticity. Conditional on observed trade shares, it determines both the response of bilateral trade flows and real consumption. In more general environments such as those considered in Section 3, it remains one of the key statistics required to estimate the gains from international trade; see equations [23], [27], and [32]. So, how large are trade elasticities?

This question is an old one. It is as important for recent gravity models reviewed in this chapter as for earlier CGE models. The broad consensus in the CGE literature is, in the words of Dawkins et al. [2001], that “the quantity and quality of literature-based elasticity parameters for use in calibrated models is another Achilles’ heel of calibration.” As John Whalley notes “It is quite extraordinary how little we know about numerical values of elasticities […] In the international trade area researchers commonly use import price elasticities in the neighborhood of unity, even for small economies, even though elasticity estimates as high as nine appear in the literature.” The same pessimism can be found in the review of trade elasticities by McDaniel and Balistreri [2003]: “The estimates from the literature provide a wide range of point estimates, and little guidance on correct estimates to apply to a given commodity in a given model for a given regional aggregation. Most of the controversy surrounding the [trade] elasticities reduces to a general structural inconsistency between the econometric models used to measure the response and the simulation models used to evaluate policy.”

Part of the success of new quantitative work in international trade lies in the tight connection between the structural estimation of the trade elasticity and the underlying economic model. In their seminal work, Eaton and Kortum [2002] offer multiple ways to estimate structurally the trade elasticity using a gravity equation akin to equation [16]. Their estimates range from ε=3.60 to ε=12.86 with a preferred value of ε=8.28 when using price gaps as a measure of trade costs. While the range is wide, it remains in the range of elasticities used in earlier CGE models; see Hertel [1997]. In recent work, Simonovska [2011] refines Eaton and Kortum’s [2002] preferred estimation strategy to take into account the fact that price gaps are only lower-bounds on trade costs. Simonovska and Waugh [2011] propose a simulated method of moments to correct for the fact that trade elasticities using price gaps tend to overestimate the sensitivity of trade flows to trade costs. Their preferred estimate of ε is 4.12.

The merits of a tight connection between theory and data notwithstanding, the state of affairs remains far from ideal. An issue with simulation using earlier CGE models, such as those used in the evaluation of NAFTA, is that they require a very large number of elasticities. Many recent papers, following Eaton and Kortum [2002], side-step this issue by assuming that all goods enter utility functions through a unique CES aggregator. But empirically, there is ample evidence of significant variation in the trade elasticity across sectors; see e.g. Feenstra [1994], Broda and Weinstein [2006], and Hummels and Hillberry [2012]. If so, why should we be more confident in the counterfactual predictions of simpler gravity models that abstract from this heterogeneity? Given the heterogeneity in the trade elasticity across sectors, is the trade elasticity estimated from an aggregate gravity equation like [16] the “right” trade elasticity to calibrate a one-sector model?32

A natural way to address the previous concerns is to write down multiple-sector models, such as those considered in Section 3.3, and incorporate formally the heterogeneity in trade elasticities across sectors. But this raises new issues. As the number of elasticities that needs to be estimated increases, the precision with which each of those elasticities is estimated tends to decrease. Accordingly, results become much more sensitive to the presence of outliers. To take a concrete example, the sector-level elasticities from Caliendo and Parro [2010] used in Section 3.3 are around 8 on average. But for some sectors, like automobiles, trade elasticities are not statistically different from zero. An elasticity of zero would imply infinite gains from trade.33

Upper-Level Elasticities [I]: Substitution Across Sectors. Going from a one-sector to a multi-sector model raises another question: How large is the elasticity of substitution between sectors? All papers referenced in Section 3.3 follow what Dawkins et al. [2001] refer to as the “idiot’s law of elasticities”: all elasticities are equal to one until shown to be otherwise. How important is the assumption that upper-level utility functions are Cobb-Douglas for the predictions of multi-sector quantitative trade models?

To shed light on this question, let us make the same assumptions as in Section 3.3, except for the fact the upper-level utility function is now given by

[43]Cj=∑s=1Sβj,sCj,s[γ-1] /γγ/[γ-1],

where γ>0 denotes the upper-level elasticity of substitution between goods from different sectors; βj,s≥0 are exogenous preference parameters, which we normalize such that ∑s=1Sβj,s=1 for all j; and C j,s still denotes total consumption of the composite good s in country j. The Cobb-Douglas case studied in Section 3.3 corresponds to γ=1.

For simplicity let us focus on the case of perfect competition, δs=0 all s. Following a procedure similar to that in Section 3.1, one can show that if sector-level price indices are given by equation [20], then the welfare impact of a shock generalizes to

[44]Cˆj=∑s=1Sej,sλ ˆjj,s-1[γ-1]/εs1/[γ-1].

Gains from trade are thus given by

[45]Gj =1-∑s=1Sej,s λjj,s[γ-1]/εs1/[γ-1].

In Section 3.3, we have pointed out that multi-sector models with Cobb-Douglas preferences predict significantly larger gains than one-sector models. Using equation [45], we can now quantify the importance of the Cobb-Douglas assumption for this prediction.

In Figure 4.4 we plot Gj as a function of γ using equation [45] for several countries—the United States, Canada, France, Germany, and Japan—as well as the average Gj for all countries considered in Table 4.1. We see that the value of the upper-level elasticity γ—for which the existing empirical literature provides little guidance—has large effects on the magnitude of the gains from trade. As we go from the Leontief case, γ=0, to the Cobb-Douglas case, γ=1, to an upper-level elasticity equal to the average of lower-level elasticities, γ=8, average gains from trade decrease from 45% to 15% to 3%.34 The intuition is simple. If the elasticity of substitution between sectors is high, then the consequences of autarky are mitigated by consumers’ ability to substitute consumption away from the most affected sectors, i.e., those with lowest values of λjj,s1/εs, towards the least affected sectors, i.e., those with highest values of λjj,s1/εs . By the same token, however, the gains from further trade liberalization would tend to be higher with a higher γ, since consumers could more easily reallocate their consumption towards goods that experience larger price declines.35

Upper-Level Elasticities [II]: Domestic versus Foreign. Another, and perhaps deeper issue regarding gravity estimates of the trade elasticity is that they capture the elasticity of substitution between foreign sources of imports. Yet, the elasticity that one needs, for instance, for measuring the gains from trade is the elasticity of substitution between home and import goods. To see this formally, let us go back to the simple Armington model presented in Section 2.1, but let us generalize equation [1] so that

Cj=[Cjj][γ-1]/γ+ CjM[γ-1]/γγ/[γ -1],

where CjM measures total consumption of imported goods,

CjM=∑i≠jψij σ/[σ-1]Cij[σ-1]/σσ/[σ-1].

The upper-level elasticity γ>1 now represents the elasticity of substitution between the domestic good and the composite of the foreign goods, whereas the lower-level elasticity σ>0 still represents the elasticity of substitution between foreign goods. The simple Armington model corresponds to the special case, γ=σ. Under this new demand system, bilateral trade flows still satisfy a gravity-like equation:

[46]Xij=PjMPj1-γPijPjM1-σEj, for alli≠j,

where Pij=τijPii is the price of goods from country i in country j;Pj M≡[∑i≠jψij1-σ Pij1-σ]1/[1-σ] is the import price index; and Pj=[[Pjj] 1-γ+[PjM]1-γ]1/[1-γ] is the consumer price index in country j. In this more general environment, one can still rearrange bilateral trade flows as we did in equation [13] and use the cross-sectional variation in trade flows and trade costs to estimate 1-σ.

Now, like in Section 2.2, consider a small change in trade costs that affects country j. The change in the consumer price index is still given by

dln Pj=λjjdlnPjj+1-λjjdlnPjM.

But our new demand system now implies

dln1-λjj-dlnλjj=1-γdlnPjM-dlnPjj.

Following the same strategy as in Section 2.2, one can therefore show that

[47]d lnCj=dlnλjj/1-γ.

While gravity estimates can uncover the lower-level elasticity, σ, equation [47] shows that the upper-level elasticity, γ, i.e., the elasticity of substitution between domestic and foreign goods, is the relevant elasticity for welfare analysis.36

In standard gravity models, it is only the assumption of symmetric CES utility that allows researchers to go from the commonly estimated elasticity, σ, to the welfare-relevant elasticity, γ. When estimated, does the elasticity of substitution between home and import goods, γ, look similar to the elasticity of substitution between foreign goods, σ? Head and Ries [2001] suggest that the answer is yes. They measure the average of the elasticity of demand for Canadian goods in Canada relative to U.S. goods and the elasticity of demand for U.S. goods in the United States relative to Canadian goods. If all trade was U.S.–Canada trade, their estimate would therefore provide an estimate of γ. They find an average elasticity equal to 7.8, quite in line with previous gravity estimates of σ. Likewise, using the methodology of Feenstra [1994] to estimate both γ and σ, Feenstra et al. [2013] cannot reject the null that γ and σ are equal.

Factor Supply Elasticities. The multi-sector gravity models that we have reviewed assume a perfectly-elastic factor supply to each sector. Thus, except for the case with sector-level differences in factor intensities considered in Section 3.5, the aggregate production possibilities frontier [PPF] is linear. In practice one may expect factors to be imperfect substitutes across sectors. For instance some workers may have a comparative advantage in particular sectors, as in a Roy-type model, or some natural resources may be critical inputs to production in some sectors and not others. Such considerations would lead to more “curvature” in the PPF and, conditional on observed trade flows, larger gains from trade.

To take an extreme example, consider the petroleum sector. The trade elasticity, εs, for this sector estimated by Caliendo and Parro [2010] is around 70. The formula presented in Section 3.3 would therefore predict very small gains from trade in this sector. Yet, of course, one would expect many oil importing countries to face enormous losses from moving to autarky. One simple way to capture such considerations would be to go back to the multi-factor model presented in Section 3.5 and assume factors employed in the petroleum sector are specific to that sector, effectively making petroleum an endowment. To explore the quantitative importance of these type of considerations, we have recomputed the gains from trade in the multi-factor model under the assumption that all sectors are perfectly competitive and all factors are sector-specific.37 In line with the previous discussion, we find larger gains from trade under the assumption that factors are sector-specific, with the cross-country average for Gj going from 15.3% to 17.2%.

Other Elasticities: Love of Variety and Extensive Margin. We conclude by discussing the calibration of the elasticity of substitution, σ, and the extensive margin elasticity, η, introduced in equation [15] and its sector-level counterparts, equations [20], [26], and [30]. Given estimates of the trade elasticity, ε, these two elasticities are irrelevant for welfare analysis under perfect competition. Under monopolistic competition, however, we have seen that in the presence of intermediate goods, as in Section 3.4, or in multi-sector models with general CES preferences, as discussed above, the values of σ and η do matter above and beyond the value of the trade elasticity, ε. In these richer environments, the predictions of models with and without firm-level heterogeneity are different and the magnitude of the difference crucially depends on how σ and η are calibrated.

As shown in Section 3.2, the three elasticities ε,σ, and η are not independent of one another. In gravity models, ε determines the overall response of trade flows to changes in trade costs, whereas σ-1 determine their responses at the intensive margin and η=ε-σ-1 σ-1 determines their response at the extensive margin, ε-σ-1, weighted by the love of variety, σ-1. Given an estimate of the trade elasticity, ε, one therefore only needs an estimate of σ to compute η and vice versa. The most direct way to estimate σ or η is to use firm-level trade data; see e.g. Crozet and Koenig [2010] and Eaton et al. [2011]. When available, they offer a simple way to estimate the intensive margin elasticity, i.e., by how much the sales of a given set of firms respond to changes in trade costs, and the extensive margin elasticity, i.e., by how much the number of firms responds to changes in trade costs.38Eaton et al. [2011] estimate a value of 1.5 for η in a one-sector model, whereas Crozet and Koenig [2010] obtain estimates of εs and ηs for several sectors. Interestingly, the average ηs across s estimated by Crozet and Koenig [2010] is also 1.5 , with little variation across sectors.39

Balistreri and Rutherford [2012] nicely illustrate the issues inherent to the calibration of models of monopolistic competition. The authors compare the predictions of a model without firm-heterogeneity, like Krugman [1980], to those of a model with firm-heterogeneity, like Melitz [2003], in a model with identical countries, three sectors, nested CES preferences, but no inter-industry trade, ej,s=rj,s. Changes in real consumption are given by equation [22]. Using the fact that ηs=εs- σs-1σs-1, one can show that the overall scale effects, eˆj,sηs rˆj,sδs/εs , are equal to rˆj,s1/σs-1 in the two models. Thus the only difference between the predictions of the two models comes from the different values of σ s used in the calibration of the two models. In their calibration, Balistreri and Rutherford [2012] assume that εs is equal to 4.6 in both models, but that ηs is equal to 0 in the Krugman-version and 0.65 in the Melitz-version. This implies calibrated values of σs =εs/1+ηs+1 equal to 5.6 in the Krugman-version and 3.8 in the Melitz-version. This leads to stronger love of variety and, in turn, larger entry effects and gains from trade liberalization in the latter model. The question, of course, is whether one should take seriously that love of variety is much stronger than previously thought because the intensive margin is only one particular margin of adjustment of trade flows, σs-12aH-12aH. Correspondingly, the real exchange rate misalignment will take the form of over- or undervaluation, respectively.

For a large home bias in consumption, the case ϕ