Abstract
Most trade models featuring heterogeneous firms assume a Pareto productivity distribution, on the basis that it provides a reasonable representation of the data and because of its analytical tractability. However, recent work shows that the characteristics of the productivity distribution crucially affect the estimated gains from trade. This paper thoroughly compares the gains from trade obtained under three different productivity distributions (Pareto, lognormal, and Weibull) and investigates their policy implications. We find that both the magnitude of the welfare gains and the relative importance of the fixed versus variable trade costs change significantly. Hence, relying blindly on a single distribution is dangerous when performing trade policy analysis.
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Notes
A few papers have explored the impact of modifying the demand (preference) rather than the supply (productivity distribution) side of recent trade models. This is done either by means of a generalization of the standard CES demand system (as in Behrens et al. 2014, who nest the baseline CES as a special case) or by an outright departure from CES (as in Mrázová and Neary 2014). The bottom line is that GFT are much harder to pin down. Although we find this line of research extremely interesting, we do not pursue this approach in the present paper.
More specifically, the “macro” calibration by Arkolakis et al. (2012) requires the two models to have the same trade elasticity with respect to trade costs, and the same domestic trade share, whereas the “micro” approach taken by Melitz and Redding (2015) only changes the degree of heterogeneity in the models, taking the homogeneous case as a limit (degenerate) case of the more general heterogeneous firm specification. See Melitz and Redding (2015) for more details on the two approaches.
In the standard monopolistic competition cum CES preferences that represents the backbone of Melitz-type models, firm size and firm productivity are closely related, see Sect. 2 below.
The main methodological issue has to do with the common practice of binning the data before fitting a distribution. Virkar and Clauset (2014) forcefully show that this is an important source of bias.
Gabaix (2009) provides a comprehensive review of models leading to a Pareto distribution; the lognormal is generated under a process of proportionate growth à la Gibrat (1931); finally, Growiec (2013, p. 2337) shows that, if production consists of many complementary stages, “the Weibull should approximate the true productivity distribution better than anything else”.
On the other hand, if \(\tau (f_x/f_d)^{1/(\varepsilon -1)}\le 1\), then all firms export and \(\phi _x^T=\phi _d^T\).
See Eqs. (13) and (26) in the Web Appendix of Melitz and Redding (2015).
The data on firm export are collected by French customs and have been accessed through the secure data access center CASD.
Head et al. (2014) note that the theoretical link between sales and productivity only holds for individual export markets, not total exports (although the lognormal distribution provides a good fit to total sales by firm as well). Belgium is among the most popular destinations for French firms.
As pointed out by Malevergne et al. (2011), even though the tail behaviors of the Pareto and lognormal distributions are qualitatively different (with the Pareto being heavier: the Pareto belongs to the Frećhet Maximum Domain of Attraction, while the lognormal is in the Gumbel Maximum Domain of Attraction), when \(\sigma ^2\) is large the lognormal tail becomes essentially indistinguishable from the Pareto one. Thus, when \(\sigma ^2\) increases, both the dispersion and the tail heaviness increase, and the tail becomes more and more similar to the Pareto one. As for the Weibull, its tail gets heavy as k decreases; in particular, it is commonly considered an heavy-tailed distribution when \(k<1\) (Embrechts et al. 1997). The variance gets large as well when k decreases, so that a smaller k implies both a larger dispersion and a fatter tail.
The GFT are unaffected by the mean of the lognormal as well as the scale parameters of the Pareto and Weibull. We set \(\phi _{min} = 1\) for the Pareto (as Melitz and Redding 2015), \(\mu = 0\) and \(\lambda = 0.71\) (these two values are consistent with each other) for the lognormal and the Weibull.
We thank an anonymous referee for pointing this out.
For the sake of simplicity we only discuss results obtained assuming an entry rate equal to 0.55%. Results for entry equal to 50% are available upon request.
Behrens et al. (2012, footnote 8) note that “estimation results for \(\varepsilon\) depend both on the level of aggregation and the estimation method, and vary widely. For example, Hanson (2005) using aggregate U.S. data, obtains about 7 with non-linear least squares and about 2 with GMM. Estimates in Hummels (1999) vary from 2 to 5.26. Using extremely disaggregated data, Broda and Weinstein (2006) estimate several thousand elasticities of substitution, which range, depending on the industry and the level of aggregation, from 1.3 (telecommunication equipments) to 22.1 (crude oil).”
The only cost parameter that does not change is the cost of serving the domestic market \(f_d=1\), which acts as a reference point throughout the paper and does not affect GFT.
A recent contribution by Breinlich and Cuñat (2016) shows that a workhorse heterogeneous-firm model à la Melitz (2003) severely underestimates the gains from NAFTA, unless it is extended to allow for within-firm productivity increases. We are aware that the quantitative evaluation performed in this section therefore represents a rough approximation. On the other hand, as long as within-firm productivity improvements do not affect the shape of the distribution, the extension advocated by Breinlich and Cuñat should not impact the comparison of the model results under different distributional assumptions, which is the focus of our work.
It is worth noting that the final share of exporting firms (for a given distribution) and the initial calibrated value of \(f_x\) are both invariant to \(\varepsilon\) and therefore these channels are not operating. Indeed, the interplay between \(\tau\), \(\varepsilon\) and the dispersion parameter of each distribution (being it \(\sigma\), \(\alpha\) or k) keeps \(f_x\) constant (again, for a given distribution) when \(\varepsilon\) increases. This behavior is not limited to the Pareto case, where one can derive the result analytically, but holds also for the Weibull and the lognormal distributions, although in these cases we have to rely on numerical results.
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Acknowledgements
The authors blame each other for any remaining mistake. They nevertheless agree to thank participants at the ETSG 2014 in Munich and the 2014 ISGEP meeting in Stockholm, as well as Mauro Caselli and Thierry Mayer for insightful comments on an earlier draft of the paper. Finally, the authors thank the Associate Editor, Pao-Li Chang, and two anonymous referees for their valuable suggestions.
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Appendix: Distribution fitting
Appendix: Distribution fitting
See Table 7.
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Bee, M., Schiavo, S. Powerless: gains from trade when firm productivity is not Pareto distributed. Rev World Econ 154, 15–45 (2018). https://doi.org/10.1007/s10290-017-0295-z
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DOI: https://doi.org/10.1007/s10290-017-0295-z