## 1. Power Law in Firm Size, All Firms ## Source details **Canonical URL:** [1. Power Law in Firm Size, All Firms](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10109.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10109.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10109.pdf.json) --- ### Theoretical framework and role of the power law exponent ζ - Definition: Pr(X>x)=Cx−ζ where C and ζ are constants. - Power laws appear in city size, income, firm size, sectoral trade flows. - ζ is crucial for economic implications (macroeconomic fluctuations, regulation of entry, executive compensation). - Prior estimates: Axtell (2001) reports ζ in the range 0.996 to 1.059 with standard errors between 0.054 and 0.064. - Melitz-Pareto framework: Pareto distribution of firm productivities generates a power law in firm size. - International trade implications: - Larger, more productive firms are more likely to export and serve more markets. - Predicted consequence: estimated ζ among exporting firms is lower (flatter, more fat-tailed) than among non-exporting firms; ignoring trade biases inference about deep parameters. ### Empirical strategy and dataset - Data: exhaustive mandatory reporting of French firms’ income statements for 2006 with breakdown of domestic sales and exports. - Raw universe: - Total firms: 2,182,571. - Exporters: 194,444 (roughly 9%). - Final estimation sample after dropping non-tradeable sectors and applying a minimum sales cutoff: - Firms: 157,084. - Exporters: 67,078 (42.7%). - Average exporter total sales: Euro 24.2 million. - Average non-exporter total sales: Euro 6.4 million. - Lower sales cutoff: - Minimum sales cutoff selected: Euro750,000 of annual sales. - Dropped (very small) firms account for 7.7% of total sales. - Robustness: results robust to cutoffs as low as Euro100,000. - Sector exclusions: non-tradeable industries dropped if total exports < 5% of total sales (list provided in source). - Estimation methods for ζ: 1. CDF (Axtell 2001): ln(Pr(S_i > s)) = ln(C) − ζ ln(s). 2. PDF / bin-frequency estimator: f(s) = C ζ s^{−(ζ+1)}. 3. Log-rank estimator (Gabaix and Ibragimov 2008): ln(Rank_i − 1/2) = Constant + ζ̂_LR ln S_i + ε_i; standard error of ζ̂_LR is |ζ̂_LR|(N/2)^{−1/2}. - The three estimators give similar results in practice. ### Analytical predictions (model implications) - Domestic demand and pricing yield domestic sales D_i = M_n × B_i with B_i ≡ a_i^{1−ε}. - If productivity 1/a ∼ Pareto(b, θ), then in autarky S_i ∼ Pareto(b^{ε−1} M_n, θ/(ε−1)) and ζ = θ/(ε−1). - Two trade mechanisms that flatten the empirical size distribution: 1. Entry into progressively more foreign markets (more productive firms serve more markets) → series of rightward shifts in the CDF. 2. Stochastic firm-level fixed export costs κ_i → selection into exporting and modified distributions for exports and total sales. - With a composite foreign market characterized by φ ≡ M^*_m / M_n = τ_{mn}^{1−ε} (Y_m/Y_n) (P_n/P_m)^{1−ε}, export decision and resulting total sales S_i follow threshold rules (see equations (5) and (6) in source). - Proposition 1 (with domestic sales Pareto) gives closed-form density adjustments for domestic sales p_D(x), export-sales p_X(x), and total sales p_S(x) showing departures from a pure Pareto for total sales. - Implication: estimating a simple power law on total sales will generally not recover underlying ζ; two corrective estimation approaches are: - Estimate ζ using only non-exporting firms. - Estimate ζ using domestic sales only. ### Main empirical findings (pooled and subsample results) - Pooled (all firms, total sales): - Power law provides a very good fit: R2’s are 99% or above. - Point estimates of ζ comparable to existing studies (Axtell for the U.S.). - Exporters vs non-exporters: - For every estimation method, ζ among exporting firms is lower than ζ for non-exporting firms (exporters’ size distribution is more fat-tailed). - Differences are highly statistically significant. - Full-sample coefficient lies between exporter and non-exporter coefficients. - Fit remains good in subsamples: R2 range 0.972 to 0.999 across subsamples. - Fit is slightly worse for exporters than non-exporters (R2 systematically lower for exporters). - Nonlinearity and saturation among exporters: - As the lower sales cutoff for exporters increases, estimated ζ for exporters becomes larger in absolute value and converges to the non-exporter ζ (empirical convergence to 1.029 reported). - R2 improves with higher cutoffs: from 0.97 to above 0.99 for larger firm subsamples. - Interpretation: exporting behavior induces deviations from a precise power law for smaller exporters; largest exporters align with domestic ζ. ### Sector-level evidence and heterogeneity - Sector-level results are consistent with firm-level mechanisms and not driven by compositional effects. - In each of 25 tradeable sectors, |ζ_non-exporters| exceeds |ζ_exporters|. - In 22 out of 25 tradeable sectors, the difference is statistically significant. - Aggregate difference in Table 2: 0.29. Mean sector-level difference: 0.35. - Sectors more open to trade show larger deviations: - Simple bivariate correlation between sector openness (exports/total sales) and percentage difference (ζ_dom − ζ_total)/ζ_dom: 0.92. - Correlation between sector openness and (ζ_nonex − ζ_total)/ζ_nonex: 0.72. - Selected sector-level log-rank-log-size estimates (Table A2; sign conventions preserved in source): - Agriculture, Forestry, and Fishing: All Sales ζ = -1.233 (Std. Error 0.028, R2 0.997); Domestic Sales ζ = -1.292 (Std. Error 0.029, R2 0.998); No. of firms 3,949; Exports/Sales 0.092 - Food Products: All Sales ζ = -0.661 (Std. Error 0.011, R2 0.962); Domestic Sales ζ = -0.691 (Std. Error 0.012, R2 0.964); No. of firms 7,031; Exports/Sales 0.164 - Non-electrical Machinery: All Sales ζ = -0.893 (Std. Error 0.014, R2 0.989); Domestic Sales ζ = -1.007 (Std. Error 0.015, R2 0.986); No. of firms 8,722; Exports/Sales 0.347 - Electrical Machinery: All Sales ζ = -0.798 (Std. Error 0.022, R2 0.995); Domestic Sales ζ = -0.931 (Std. Error 0.026, R2 0.996); No. of firms 2,573; Exports/Sales 0.464 - Wholesale Trade, Intermediaries: All Sales ζ = -0.896 (Std. Error 0.009, R2 0.980); Domestic Sales ζ = -0.903 (Std. Error 0.009, R2 0.979); No. of firms 19,164; Exports/Sales 0.122 ### Quantitative bias magnitudes and diagnostics - Representative economy-wide estimates (selected values reported in source): - Table 1 (All Firms, total sales): - ζ (CDF): 1.017 - ζ (PDF): 1.019 - ζ (ln(Rank-0.5)): 0.825 - R2 (CDF): 0.990; R2 (PDF): 0.998; R2 (ln(Rank-0.5)): 0.991 - No. of firms: 157,084 - Table 2 (Exporters vs Non-Exporters, ln(Rank-0.5) highlights): - Exporters ζ: 0.738 - Non-Exporters ζ: 1.029 - t-stat for difference (ln(Rank-0.5) coefficients): 46.24** (significant at the 1% level) - No. of firms: Exporters 67,078; Non-Exporters 90,006 - Table 4 (All Firms, domestic sales only): - ζ (CDF): 1.048 - ζ (PDF): 1.055 - ζ (ln(Rank-0.5)): 0.869 - R2 (CDF): 0.992; R2 (PDF): 0.998; R2 (ln(Rank-0.5)): 0.992 - No. of firms: 157,084 - Bias magnitudes (comparison of conventional all-firm estimates with non-exporter “true” estimates): - For the CDF and PDF estimators, Table 1 coefficients for all firms are about 3.6% off from non-exporter estimates. - For the log-rank-log-size estimator, the difference is larger: 20%. - Figures and diagnostics: - Plots show distinct slopes for exporters vs non-exporters, stability of exporter coefficient across increasing lower cutoffs, and positive relationships between sector openness and power law deviations; OLS fits shown in figures referenced in source. ### Empirical implications and recommended estimation practice - Do not estimate the Pareto exponent ζ on total firm sales without accounting for exporting behavior; exporting systematically biases estimates downward (flattens distribution). - Two simple, theory-consistent corrections: 1. Estimate ζ using only non-exporting firms (if exporter indicator available). 2. Estimate ζ using domestic sales for all firms (exclude export sales from firm size). - Practical guidance: - Apply a lower sales cutoff (e.g., Euro750,000 used here; robust to Euro100,000). - When analyzing exporter samples, progressively raise the lower cutoff; convergence of exporter ζ toward domestic ζ and improving R2 supports exporter-induced deviations. - Use both corrections where possible to check consistency; in this analysis the two corrections yield similar estimates. - Caveat: At the economy-wide level in this dataset corrected coefficients do not differ much from unadjusted ones (ballpark estimates), but bias from exporting behavior may be larger in smaller countries, developing countries, or particular industries. *Source: _wp10109 - 1. Power Law in Firm Size, All Firms; Section III; Appendix Table A2 (excerpt) from the provided IMF PDF content.* ### 1. Power Law in Firm Size, All Firms................................................................... 18 ### 1. Power Law in Firm Size, All Firms ### Theoretical framework and role of the power law exponent ζ - A distributional power law is defined as: Pr(X>x)=Cx−ζ where Pr(X>x) is the probability that a random variable X is greater than x, and C and ζ are constants. - Power laws arise across contexts including city size (Zipf 1949), income (Champernowne 1953), firm size (Axtell 2001), and sectoral trade flows (Hinloopen and van Marrewijk 2006, Easterly, Reshef and Schwenkenberg 2009). - The precise value of the power law exponent ζ is important for economic implications (e.g., macroeconomic fluctuations, regulation of entry, executive compensation). - Prior estimates for the firm size power law exponent: Axtell (2001) reports a range between 0.996 and 1.059 with standard errors between 0.054 and 0.064. - The Melitz (2003) model with a Pareto distribution of firm productivities (Melitz-Pareto framework) delivers a power law in firm size. - In the presence of international trade, the Melitz-Pareto model predicts the power law exponent is not constant across exporting and non-exporting firms because: - Larger firms are more likely to export. - More productive firms serve more markets. - Consequence: estimated power law exponent among exporting firms is expected to be lower (in absolute value) than among non-exporting firms; ignoring international trade can bias inference about deep parameters. ### Empirical strategy and dataset - Empirical evaluation uses data on production and exports for a large sample of French firms. - Comparison is made between: - The full sample that includes all firms. - Subsamples of exporting firms and non-exporting firms. - Alternative estimation approaches consistent with heterogeneous firms models. ### Key empirical findings - In the full sample (all firms), the power law in firm size closely matches Axtell (2001) results for U.S. firms: the estimated power law exponent and fit are nearly identical to those prior findings. - When separating firms: - The estimated power law coefficient for exporting firms is consistently lower than in the full sample. - The estimated power law coefficient for non-exporting firms is consistently higher than in the full sample. - These differences are present across all estimators and are highly statistically significant. - Additional empirical regularities consistent with the theoretical prediction: - The power law exponent for exporting firms converges to the power law exponent for domestic firms as the sample is restricted to larger and larger exporters. - At the disaggregated industry (sector) level, power law coefficients show the same pattern. - Differences between power law coefficients are larger in sectors that are more open to trade. ### Implications for estimation and model calibration - Empirical power law estimates are used to pin down crucial parameters in calibrated heterogeneous firms models (e.g., Helpman, Melitz and Yeaple 2004; Chaney 2008; Buera and Shin 2008; di Giovanni and Levchenko 2009a). - Quantitative results can be sensitive to the precise value of ζ: - Di Giovanni and Levchenko (2009b) show welfare gains from reductions in fixed costs are an order of magnitude lower, and gains from reductions in variable costs an order of magnitude higher, in a model calibrated to Zipf’s Law compared to the counterfactual case in which ζ=2. - Two alternative estimation approaches consistent with the canonical heterogeneous firms model of trade are proposed: - Use a sample of only non-exporting firms to estimate the power law parameter. - Use only domestic sales to estimate the power law parameter. - These approaches are simple to implement and do not require additional modeling or estimation techniques; they rely on appropriate modification of the sample or variables used in estimation. ### Relation to existing literature - Eaton, Kortum and Kramarz (2008) estimate parameters of a multi-country heterogeneous firms model using Simulated Method of Moments with French firm data. - The paper’s advantage is simplicity: alternative estimation strategies that explicitly account for international trade lower barriers to obtaining reliable power law estimates and can be applied easily in many contexts. - The qualitative mechanisms highlighted apply beyond Pareto distributions to other underlying distributions of firm size. _Italic: Source: _wp10109 - 1. Power Law in Firm Size, All Firms (excerpt) from the provided IMF PDF content._ ### Section III describes the dataset used in the analysis and the methodology for estimating ### _wp10109 - Section III describes the dataset used in the analysis and the methodology for estimating ### Theoretical framework and analytical predictions - Consumers maximize ∫_{J_n} c_{ni}^{ε−1/ε} di subject to ∫_{J_n} p_{ni} c_{ni} di = Y_n. Demand for variety i in country n: - c_{ni} = Y_n P_n^{1−ε} p_{ni}^{−ε}, where P_n = [∫_{J_n} p_{ni}^{1−ε} di]^{1/(1−ε)}. - Firms have marginal cost a_i measured in input-bundle units; input-bundle price in country n is ω_n. - Profit-maximizing price: p_{ni} = (ε/(ε−1)) ω_n a_i. Domestic sales D_i: - D_i = Y_n P_n^{1−ε} (ε/(ε−1) ω_n a_i)^{1−ε} = M_n × B_i, - where M_n ≡ Y_n P_n^{1−ε} (ε/(ε−1) ω_n)^{1−ε} and B_i ≡ a_i^{1−ε}. - Variable profits are proportional to D_i: π_VD(a_i) = D_i/ε. Firm operates domestically iff D_i/ε ≥ f, defining D = εf and cutoff a_nn = (M_n/(εf))^{1/(ε−1)}. - Exporting to country m requires per-firm fixed cost κ_{mni} and iceberg cost τ_{mn} > 1. Export sales expressed as M^*_m B_i where M^*_m = Y_m P_m^{1−ε} (ε/(ε−1) τ_{mn} ω_n)^{1−ε}. - Exporting condition: M^*_m B_i/ε ≥ κ_{mni}. - Power law specification for firm sales: - Pr(S_i > s) = C s^{−ζ}. (equation (3)) - If productivity 1/a ∼ Pareto(b, θ), then in autarky with S_i = D_i = M_n B_i: - Pr(S_i > s) = (b^{ε−1} M_n)^{θ/(ε−1)} s^{−θ/(ε−1)}. - Hence ζ = θ/(ε−1) and S_i ∼ Pareto(b^{ε−1} M_n, θ/(ε−1)). (equation (4)) - Two trade mechanisms that flatten (make more right-skewed) the empirical size distribution: 1. Entry into progressively more foreign markets: more productive firms serve more markets → series of parallel rightward shifts in the CDF of firm size. 2. Stochastic firm-level fixed export costs κ_i: leads to selection into exporting and modified distributions for exports and total sales. - Export probability function formulation for a single composite foreign market: - Define φ ≡ M^*_m / M_n = τ_{mn}^{1−ε} (Y_m/Y_n) (P_n/P_m)^{1−ε}. - Export decision: φ D_i^{ε} ≥ κ_i. Let H(x) = Pr(κ_i ≤ φ x^{ε}). - Export sales X_i = 0 with probability 1−H(D_i), and X_i = φ D_i with probability H(D_i). - Total sales S_i = D_i if φ D_i^{ε} < κ_i (probability 1−H(D_i)), and S_i = (1+φ) D_i if φ D_i^{ε} ≥ κ_i (probability H(D_i)). (equations (5) and (6)) - Proposition 1 (density forms when underlying domestic sales are Pareto): - p_D(x) = k x^{−ζ−1} 1_{x>D} - p_X(x) = K x^{−ζ−1} H(x/φ) 1_{x>φD} - p_S(x) = k x^{−ζ−1} [1−H(x) + H(x/(1+φ))(1+φ)^{ζ}] 1_{x>(1+φ)D} + k x^{−ζ−1} 1_{D s)) = ln(C) − ζ ln(s). Estimate by regressing ln empirical exceedance probability on ln(s) using a log-equidistant s grid. 2. PDF / bin-frequency estimator: f(s) = C ζ s^{−(ζ+1)}; bin on log scale, regress frequency on s (geometric mean of bin endpoints). Coefficient estimates −(ζ+1). 3. Log-rank estimator (Gabaix and Ibragimov 2008): ln(Rank_i − 1/2) = Constant + ζ̂_LR ln S_i + ε_i. Standard error of ζ̂_LR is |ζ̂_LR|(N/2)^{−1/2}. The three estimators give similar results in practice. ### Main empirical results (Section IV) - Replication of pooled results (all firms, ignoring exporting behavior): - Power law provides a very good fit: R^2’s are 99% or above. - Point estimates of ζ comparable to existing studies (e.g., Axtell for the U.S.). (Figure 4 referenced.) - Key result: exporting vs non-exporting firm size distributions - Estimates in Table 2 (separate estimations for exporters and non-exporters) show: - For every estimation method, the exponent ζ among exporting firms is lower than that for non-exporting firms. - Interpretation: exporting firms’ size distribution is more fat-tailed (flatter CDF) than non-exporting firms. - Differences are highly statistically significant. Column (7) reports the t-statistic comparing coefficients (columns (5) and (6)) and finds significant differences. - Additional observations: - Full-sample coefficient lies between exporter and non-exporter coefficients (consistent with aggregation). - Fit remains good in subsamples: R^2 range 0.972 to 0.999 across subsamples. - Fit of power law is slightly worse for exporters than non-exporters (R^2 systematically lower for exporters). - Nonlinearity and saturation among exporters - Theory predicts that for sufficiently large exporting firms, estimated power law exponent should converge back to ζ (the domestic-sales exponent), due to saturation (all big firms export or fixed exporting costs bounded). - Empirical test: re-estimate power law on exporting firms while increasing the lower sales cutoff. - As cutoff increases, estimated exponent for exporters becomes larger in absolute value, converging to 1.029 (the non-exporting firms coefficient). - Fit (R^2) improves with higher cutoffs: from 0.97 to above 0.99 for larger firm subsamples. - This pattern supports the theoretical mechanisms: exporting behavior induces deviations from a precise power law for smaller exporters, but the power law holds better among the largest exporters. ### Sector-level evidence (robustness and heterogeneity) - Interpreting model at sector level yields similar predictions; results are not driven by compositional effects. - Table 3 (sector-level estimates, log-rank-log-size): - In each of 25 tradeable sectors, the absolute value of ζ for non-exporting firms exceeds that for exporting firms. - In 22 out of 25 tradeable sectors, the difference is statistically significant. - Aggregate difference in Table 2: 0.29. Mean sector-level difference: 0.35 (slightly larger). - Additional auxiliary prediction: - Sectors more open to trade should show greater deviations induced by exporting behavior. - Figure 7(a) plots percentage difference (ζ_dom − ζ_total)/ζ_dom against sector openness (exports/total sales); OLS fit reported. (Underlying coefficients and openness ratios documented in the source.) ### Empirical implications and recommended estimation practice - When calibrating heterogeneous-firm trade models or estimating underlying Pareto productivity parameters: - Do not estimate the Pareto exponent ζ on total firm sales without accounting for exporting behavior; exporting systematically biases estimates downward (flattens distribution). - Preferable estimation strategies: - Estimate ζ using only non-exporting firms. - Estimate ζ using domestic-sales data only (i.e., exclude export sales from firm size). - Practical guidance from the paper’s evidence: - Apply a lower sales cutoff (e.g., Euro750,000 was used here; results robust down to Euro100,000) because power laws tend to hold only above a minimum firm size. - When exploring exporter samples, examine sensitivity by progressively raising the lower cutoff—convergence toward the domestic ζ and improving R^2 is evidence of exporter-induced deviations. *Italic source: _wp10109 - Section III describes the dataset used in the analysis and the methodology for estimating (PDF chapter/section).* ### Appendix Table A2. ### Appendix Table A2. ### Relationship between sector openness and power law deviations - The more open the sector, the greater the difference between the conventional power law coefficient (estimated on total sales) and the power law coefficient estimated on domestic sales only. - Simple bivariate correlation between those two variables: 0.92. - Correlation between sector openness and the difference between the conventional power law coefficient and the coefficient estimated on non-exporting firms only (ζ_nonex − ζ_total)/ζ_nonex: 0.72. - Conclusion: International trade changes the distribution of firm size in systematic and predictable ways; sector openness is strongly associated with deviations in estimated power law exponents. ### Corrections and quantitative implications - Theory links the power law exponent in firm size, ζ, to θ/(ε−1) where θ is the parameter in the distribution of productivities and ε is the elasticity of substitution between goods. - Estimating ζ on all firms (ignoring exporting behavior) does not generally recover θ/(ε−1). - Two theory-consistent corrections to recover θ/(ε−1): - Restrict the estimation sample to non-exporting firms only; in that sample ζ corresponds to θ/(ε−1). - Fit a power law on domestic sales for all firms (exporters and non-exporters); domestic sales obey the power law with exponent θ/(ε−1). - Quantitative findings on bias magnitude (comparing conventional all-firm estimates with non-exporter “true” estimates): - For the CDF and PDF estimators, the Table 1 coefficients for all firms are only about 3.6% off from the “true” estimates based on non-exporting firms only. - For the log-rank-log-size estimator, the difference is larger: 20%. - Empirical consistency: - Table 4 (power law on domestic sales for all firms) yields coefficients higher in absolute value than Table 1 (total sales), as predicted by theory. - Coefficients in Table 4 are quite similar to the non-exporter coefficients of Table 2, indicating the two corrections deliver mutually consistent results. - Practical implication: At the economy-wide level in this dataset, corrected coefficients do not differ much from traditional unadjusted ones — suggesting exporting behavior may not introduce large systematic biases for a ballpark estimate of firm size heterogeneity. Caveat noted: effects could be larger in smaller countries, developing countries, or in specific industries. ### Key empirical estimates (representative values from reported tables) - Table 1 (Power Law in Firm Size, All Firms, total sales): - ζ (CDF): 1.017 - ζ (PDF): 1.019 - ζ (ln(Rank-0.5)): 0.825 - R2 (CDF): 0.990; R2 (PDF): 0.998; R2 (ln(Rank-0.5)): 0.991 - No. of firms: 157,084 - Table 2 (Power Laws, Exporters vs Non-Exporters, ln(Rank-0.5) column highlights): - Exporters ζ: 0.738 - Non-Exporters ζ: 1.029 - t-stat for difference (ln(Rank-0.5) coefficients): 46.24** (significant at the 1% level) - No. of firms: Exporters 67,078; Non-Exporters 90,006 - Table 4 (Power Law in Firm Size, All Firms, domestic sales only): - ζ (CDF): 1.048 - ζ (PDF): 1.055 - ζ (ln(Rank-0.5)): 0.869 - R2 (CDF): 0.992; R2 (PDF): 0.998; R2 (ln(Rank-0.5)): 0.992 - No. of firms: 157,084 - Table A2 (Power Laws by sector, All Sales vs Domestic Sales — selected sector examples, log-rank-log-size estimator): - Agriculture, Forestry, and Fishing: All Sales ζ = -1.233 (Std. Error 0.028, R2 0.997); Domestic Sales ζ = -1.292 (Std. Error 0.029, R2 0.998); No. of firms 3,949; Exports/Sales 0.092 - Food Products: All Sales ζ = -0.661 (Std. Error 0.011, R2 0.962); Domestic Sales ζ = -0.691 (Std. Error 0.012, R2 0.964); No. of firms 7,031; Exports/Sales 0.164 - Non-electrical Machinery: All Sales ζ = -0.893 (Std. Error 0.014, R2 0.989); Domestic Sales ζ = -1.007 (Std. Error 0.015, R2 0.986); No. of firms 8,722; Exports/Sales 0.347 - Electrical Machinery: All Sales ζ = -0.798 (Std. Error 0.022, R2 0.995); Domestic Sales ζ = -0.931 (Std. Error 0.026, R2 0.996); No. of firms 2,573; Exports/Sales 0.464 - Wholesale Trade, Intermediaries: All Sales ζ = -0.896 (Std. Error 0.009, R2 0.980); Domestic Sales ζ = -0.903 (Std. Error 0.009, R2 0.979); No. of firms 19,164; Exports/Sales 0.122 - Note: Table A2 reports sector-level exports/total sales ratios in the last column. ### Empirical diagnostics and figures (summary) - Figures illustrate: - Distinct slopes (power law exponents) for exporting vs non-exporting firms: exporting firms have lower power law exponents than non-exporting firms. - Panel diagnostics (Figure 6): stability of estimated coefficient for exporting firms across lower sales cutoffs and corresponding R2 evolution. - Figure 7: plots of percentage differences (ζ_dom − ζ_total)/ζ_dom and (ζ_nonex − ζ_total)/ζ_nonex against sector-level exports/total sales, showing positive relationships and clear separation between tradeable (solid dots) and non-tradeable (hollow dots) sectors; OLS fit lines shown. ### Practical recommendations for empirical work on firm-size power laws - If dataset contains explicit exporter indicator (even without export values), estimate θ/(ε−1) by restricting the sample to non-exporting firms. - Alternatively, estimate power law exponents on domestic sales for all firms (exporters and non-exporters). - Use both corrections where possible to check consistency; in this analysis the two corrections yield similar estimates. - Exercise caution when generalizing results across countries or sectors: the bias from exporting behavior may be larger in smaller countries, developing countries, or particular industries. *Source: Appendix Table A2 of the provided IMF working paper content.* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10109.pdf_