## wp18259

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### Estimates of the intensive margin elasticity (IME)
- Preferred IME estimate: 0.4 based on inclusion of origin-year and destination-year fixed effects.
- Intensive margin accounts for approximately 40% of the variation in total exports across country pairs.
- Extensive margin accounts for approximately 60% of the variation.
- Origin-year and destination-year fixed effects alone absorb at most 59 percent of the variation in bilateral trade flows (from R-squared of OLS regression of lnXijt on origin-year and destination-year fixed effects), implying a large share of variation comes from forces behind the estimated IME.

### Robustness and sample variation
- Extended-sample IME (all country pairs) with origin-year and destination-year fixed effects: 0.58 (Panel B of Table 2).
- Preferred specification (origin-year and destination-year fixed effects):
  - IME = 0.38 among origin-destination pairs with at least 100 exporting firms (core sample).
  - IME = 0.52 among all origin-destination pairs (extended sample).
- Excluding firms with annual exports below $1,000: IME estimates change only slightly.
- Instrumenting total exports with leads and/or lags to address measurement error yields instrumented IMEs very close to OLS IME.

### Industry-level and product-level findings
- IME increases with industry disaggregation.
- HS 6-digit, core sample, with origin-year-industry and destination-year-industry fixed effects: IME = 0.51.
- Extended sample IME disaggregated at HS2 product level: IME ≈ 0.52.
- Restricting to HS 2-digit industries with low shares of firms exporting via intermediaries: IME ≈ 0.53.
- Demeaned origin-industry-destination-year plots address concern about industry composition differences.

### IME by percentiles and role of export superstars
- Estimation method: for each origin-destination-year, firms sorted into export-value percentiles; estimate IMEpct from lnxpctij = FEoi + FEdj + αpct lnXij + εij.
- Overall IME (horizontal benchmark): 0.4.
- IME for highest percentile: 0.5.
- IMEpct rises from 0.2 at the 50th percentile to 0.3 at the 80th percentile.
- Conclusion: positive overall IME is not driven exclusively by export superstars; IME increases monotonically across exporter size distribution.

### IME for multi-product firms and decomposition
- Definitions preserved exactly:
  - mij = average number of products exported from i to j by firms exporting from i to j.
  - x p ij ≡ xij / mij = average exports per product per firm exporting from i to j.
  - IMEp ≡ cov(ln ̃xpij, ln ̃Xij) / var(ln ̃Xij).
  - IME = IMEp + cov(ln ̃mij, ln ̃Xij) / var(ln ̃Xij).
- Table 5 (core sample) result: most of the IME is explained by systematic variation in average exports per product per firm rather than by average number of products exported per firm.

### Correlation between intensive and extensive margins, and relation with distance
- Positive and significant correlation between average exports per firm and number of exporting firms: 0.25 (standard error 0.01), after removing origin-year and destination-year effects.
- Table 6: elasticities with respect to log distance (controlling for origin-year and destination-year fixed effects) — margins decline with distance:
  - x_ij: 0.123*** (col 1), -0.280*** (col 2) — standard errors [0.0150], [0.0130]
  - N_ij: -0.416*** (col 1), -1.010*** (col 2) — standard errors [0.0134], [0.0128]
  - x_pij: 0.288*** (col 1), -0.071*** (col 2) — standard errors [0.0158], [0.0146]
  - m_ij: -0.165*** (col 1), -0.209*** (col 2) — standard errors [0.0059], [0.0051]

### Relation to previous empirical results
- EKK (France) find average exports per firm increase with destination market size with elasticity 1/3; regression of market size against estimated destination fixed effects in current data implies elasticity = 0.19.
- Bernard et al. (2009) find IME_p ≈ 0.23; authors find IME_p ≈ 0.29.
- Regressing ln x_pij on ln dist_ij yields positive coefficient with only origin and year fixed effects; adding destination fixed effects makes coefficient negative and significant.
- Empirical IME reported in tables and figures: 0.4 or higher.

### Melitz-Pareto model — setup and intensive-margin implications
- Key assumptions and expressions (preserved exactly):
  - Labor L_i; wage w_i. CES preferences, σ > 1. Firms measure N_i; productivity φ Pareto with shape θ > σ−1 and scale b_i: Pr(φ ≤ φ0) = G_i(φ0) = 1 − (φ0/b_i)^(−θ). Fixed trade costs F_ij and iceberg trade costs τ_ij.
  - Firm sales x_ij(φ) = A_j ( ̄σ w_i τ_ij φ )^(1−σ), A_j ≡ P_j^(1−σ) w_j L_j, ̄σ ≡ σ/(σ−1).
  - Aggregate exports and number of exporters (assuming φ*_ij > b_i):
    - X_ij = ( θ/(θ−(σ−1)) ) A_j (w_i τ_ij)^(1−σ) b_i^θ N_i (φ*_ij)^(σ−θ−1)
    - N_ij = b_i^θ N_i (φ*_ij)^(−θ)
  - Intensive margin x_ij ≡ X_ij / N_ij = ( θσ/(θ−(σ−1)) ) F_ij.
  - Log-linearized:
    - ln N_ij = μ_N,o_i + μ_N,d_j − θ ln ̃τ_ij − ̄θ ln ̃F_ij, where ̄θ ≡ θ/(σ−1).
    - lnx_ij = μ_x,o_i + μ_x,d_j + ln ̃F_ij.
- Model-implied IME (definition used: IME = cov(ln ̃x_ij, ln ̃X_ij) / var(ln ̃X_ij)):
  - IME = −( ̄θ − 1 ) var(ln ̃F_ij) − θ cov(ln ̃τ_ij, ln ̃F_ij)
    divided by var( −θ ln ̃τ_ij − ( ̄θ − 1 ) ln ̃F_ij ).
- Formal observations:
  1. If var(ln ̃F_ij) = 0 then IME = 0.
  2. If IME > 0 then corr(ln ̃F_ij, ln ̃τ_ij) < 0.
  3. cov(ln ̃x_ij, ln ̃dist_ij) / var(ln ̃dist_ij) = cov(ln ̃F_ij, ln ̃dist_ij) / var(ln ̃dist_ij).
  4. IME_pct = IME for all percentiles pct (Pareto implies constant percentile-specific IME).

### Melitz-Pareto implications versus data
- Under var(ln ̃F_ij) = 0, model predicts IME = 0 and EME = 1, in contrast to data showing IME ≥ 0.4.
- To match observed positive IME, must allow var(ln ̃F_ij) > 0 and have negative correlation between fixed and variable trade costs.
- Backing out model-implied trade costs (calibration θ = 5, σ = 5, so ̄θ = 1.25) yields:
  - corr(model-implied fixed, variable trade costs) = −0.786 (standard error = 0.007).
  - Model-implied fixed trade costs elasticity w.r.t. distance = −0.28 (standard error [0.0140]).
  - Model-implied variable trade costs elasticity w.r.t. distance = 0.272 (standard error [0.0046]).
- These implications (fixed costs falling with distance and strong negative correlation between fixed and variable trade costs) are puzzling and lack microfoundations in existing models.
- Pareto predicts constant IME across exporter-size percentiles, contradicting empirical monotonic increase.

### Multi-product extension (Bernard, Redding, Schott (2011) type) — implications
- Firms produce continuum of products λ ∈ [0,1]; productivity φ and product-specific λ drawn from Pareto distributions with shapes θ_f and θ_p and impose θ_f > θ_p > σ−1. Define ̄θ ≡ θ_f/(σ−1) and χ ≡ θ_f/θ_p.
- Log-linear relationships:
  - ln X_ij = μ_X,o_i + μ_X,d_j − θ_f ln ̃τ_ij − ( θ_f/(σ−1) − θ_f/θ_p ) ln ̃f_ij − ( θ_f/θ_p − 1 ) ln ̃F_ij.
  - lnx_pij ≡ ln X_ij − ln M_ij = μ_xp,o_i + μ_xp,d_j + ln ̃f_ij.
  - lnx_ij ≡ ln X_ij − ln N_ij = μ_xf,o_i + μ_xf,d_j + ln ̃F_ij.
- IME formulas (multi-product):
  - IME = [ −(χ − 1) var(ln ̃F_ij) + ( ̄θ − χ ) cov(ln ̃f_ij, ln ̃F_ij) + θ cov(ln ̃F_ij, ln ̃τ_ij) ] / var(ln ̃X_ij).
  - IME_p = [ −( ̄θ − χ ) var(ln ̃f_ij) + (χ − 1) cov(ln ̃f_ij, ln ̃F_ij) + θ cov(ln ̃f_ij, ln ̃τ_ij) ] / var(ln ̃X_ij).
- Observations:
  - If var(ln ̃f_ij) = 0 then IME_p = 0.
  - To obtain IME > 0 or IME_p > 0 requires negative covariances between fixed and product-level or variable costs or both.
  - Empirical product-level fixed trade costs implied by the Pareto multi-product model decrease with distance with elasticity = −0.071 (Table 7).

### Granularity (finite firm numbers) — theory and simulation
- Discrete-firm equations (exact):
  - lnN_ij = μ_{N,o i} + μ_{N,d j} − θ ln ̃τ_ij − ̄θ ln ̃F_ij + ξ_ij
  - lnx_ij = μ_{x,o i} + μ_{x,d j} + ln ̃F_ij + ε_ij
- IME with granularity (equation 23) includes var(ε_ij) and covariance terms (COV) and can be positive even with cov(ln ̃F_ij, ln ̃τ_ij) > 0 if var(ε_ij) is large relative to −COV.
- Empirical tests and simulations:
  - Accounting for granularity reduces magnitude of negative distance elasticity of fixed trade costs but estimates remain negative.
  - Simulation (Table 9) for var(̃F_ij) = 0 and different ̄θ and sales correlation:
    - corr(αjφ, αkφ) = 0:
      - ̃θ = 2.4 → 0.005
      - ̃θ = 1.25 → 0.133
      - ̃θ = 10 → 0.333
    - corr(αjφ, αkφ) = 1:
      - ̃θ = 2.4 → 0.001
      - ̃θ = 1.25 → 0.036
      - ̃θ = 10 → 0.103
  - For ̄θ = 2.4 and no demand shocks (perfect correlation) simulated IME = 0.001.
  - For ̄θ = 1 and no correlation simulated IME = 0.33.
- Conclusion: granularity alone is not a plausible explanation for observed positive IME given empirical trade elasticities and percentile IME patterns.

### Melitz-lognormal model — theory (continuum approximation)
- Assume firm productivity distribution lognormal with parameters μ_{φ,i} and σ_φ. Define h(x) ≡ Φ′(x) / Φ(x) and ̄σ_φ ≡ (σ−1) σ_φ.
- Key identities:
  - x_ij / x_ij(φ^*_ij) = h[−(ln φ^*_ij − μ_{φ,i})/σ_φ] / h[−(ln φ^*_ij − μ_{φ,i})/σ_φ + ̄σ_φ]
  - Using 1 − G_i(φ^*_ij) = N_ij / N_i define Ω(N_ij / N_i) so:
    - x_ij / x_ij(φ^*_ij) = Ω(N_ij / N_i) = h(Φ^{-1}(N_ij / N_i)) / h(Φ^{-1}(N_ij / N_i) + ̄σ_φ).
  - ln x_ij = μ_{x,o i} + μ_{x,d j} + ln Ω(N_ij / N_i) (equation 28).
  - ln ̃F_ij = δ_{F,o i} + δ_{F,d j} + ln x_ij − ln Ω(N_ij / N_i) (equation 29).
  - (σ−1) ln ̃τ_ij = δ_{τ,o i} + δ_{τ,d j} − ln x_ij + ln Ω_i(N_ij / N_i) + ̄σ_φ Φ^{-1}(1 − N_ij / N_i) (equation 30).
- Theoretical implication: under lognormal productivity, ̃φ(φ^*)/φ^* decreases in φ^*, so a decline in τ_ij increases average exports per firm — positive IME can arise even with var(̃F_ij) = 0.

### Full Melitz-lognormal model — specification and identification
- Enriched model: joint lognormal for ln φ, ln α_1,..., ln α_J, ln f_1,..., ln f_J per origin i with structured covariance allowing correlations.
- Latent sales Z_{ij} ≡ D_{ij} + ln α_j + (σ−1) ln φ; observed X_{ij} = Z_{ij} if ln σ + ln f_{ij} ≤ Z_{ij}, otherwise non-export.
- Z_i = (Z_{i1},...,Z_{iJ}) ∼ N(d_i, Σ_i) with covariance elements ̄σ^2_{φ,i} + σ^2_{α,i} on diagonal and ̄σ^2_{φ,i} off-diagonal.
- Estimation strategy:
  - Use firm-level data for fifteen destinations for year 2007 across 39 origins (2 dropped for convergence).
  - Likelihood constructed from density of observed sales realizations; log-likelihood sum over firms.
  - Origin-specific parameter vector Θ_i includes {d_{ij}, ̄μ_{f,ij}}_{i,j}, ̄σ_{φ,i}, σ_{α,i}, σ_{f,i}, ρ_i.
  - Use Chernozhukov and Hong (2003) MCMC Metropolis-Hastings algorithm for estimation.

### Simple Melitz-lognormal results (continuum approximation)
- QQ-estimates yield conservative ̄σ_φ = 4.02 (minimum among estimates; corresponds to largest 25% subsample).
- For ̄σ_φ = 4.02, implied IME = 0.28 (close to data).
- Most contribution to IME comes from right tail of exporter size distribution.
- Model-implied variable and fixed trade costs increase with distance (Table 11 and Figure 11).
- Under lognormal, correlation between model-implied variable and fixed trade costs is closer to zero (−0.3) versus Pareto (−0.8), but still negative.

### Estimated full Melitz-lognormal model — fit and parameters
- Data: EDD and China for 2007, 39 origins (2 dropped), 15 destinations per origin, set σ = 5.
- Fit diagnostics:
  - Density of standardized log sales: estimated model closely fits pooled data.
  - Predicted deviations from strict market hierarchy close to data.
  - Correlation of export values across destinations: model implies mostly positive correlation; data shows more dispersion.
- Median parameter estimates across 37 origins (Table 12):
  - Median ̄σ_{φ,i} = 3.18
  - Median σ_{α,i} = 2.67
  - Median σ_{f,i} = 2.39
  - Median ρ_i = 0.50
- IME implied by estimated model:
  - Simulation of one million firms per origin yields IME implied = 0.63.
  - Using same sample of 37 origins and 4 destinations for 2007, data IME = 0.67 (95% CI [0.61, 0.73]); full model IME = 0.63 (95% CI [0.59, 0.67]).
  - IME by percentile from model closely matches data pattern.
- Trade-cost elasticities with distance (Table 14; medians and CIs):
  - corr( ̃F ij, ̃τ ij): -0.31 (95% CI [-0.45, -0.1])
  - Distance elasticity, Fixed costs: 0.31 (95% CI [0.18, 0.41])
  - Distance elasticity, Variable costs: 0.34 (95% CI [0.30, 0.37])
- Note: under the estimated full lognormal model both variable and fixed trade costs are strongly increasing in distance (positive elasticities).

### Counterfactual analysis — exact hat algebra extension and applications
- Extended Dekle et al. (2008) “exact hat algebra” to generalized Melitz with arbitrary productivity distribution and firm-level shocks; define normalized variables using H_{ij}(·), n_{ij} ≡ N_{ij}/N_i and h_{ij} ≡ H_{ij}(n_{ij}).
- Hat-change system (equations 42–48) solves for hat changes of endogenous variables given exogenous changes in ˆτ_{ij} and ˆ∆_j.
- Counterfactual exercises (subset of 37 origins for tractability):
  1. Uniform reductions in international trade costs: ˆτ_{ij} ∈ {0.99, 0.95, 0.9, 0.75} for i ≠ j; ˆτ_{ii} = 1.
  2. Estimate trade elasticity θ_{ˆτ} by OLS from counterfactuals; obtained θ_{0.99} = 4.5, θ_{0.95} = 4.49, θ_{0.9} = 4.47, θ_{0.75} = 4.43.
  3. Compute Melitz-Pareto counterfactuals using these θ values.
- Main findings:
  - Welfare changes from uniform liberalization: Melitz-Pareto and Melitz-lognormal yield very similar welfare changes across origins; average trade elasticity differences small.
  - Bilateral trade flows: Melitz-Pareto can substantially over- or underpredict changes relative to Melitz-lognormal depending on local elasticities.
  - Asymmetric shocks (25% reduction only to each origin’s largest destination): Melitz-lognormal yields smaller welfare gains than Melitz-Pareto (affected pairs have low trade elasticities); differences in trade-flow changes larger.
  - ACR welfare formula approximates welfare changes well in estimated Melitz-lognormal model because bilateral trade elasticity varies little across counterfactuals (three quarters of bilateral elasticities lie between 4 and 4.8).

### Conclusions — empirical and modeling implications
- Empirical evidence (EDD + China) indicates at least 40% of variation in exports occurs along the intensive margin (preferred estimate 0.4).
- Findings robust across samples, firm-size exclusions, and industry disaggregation.
- IME rises with exporter percentile.
- Pareto-based Melitz models reconcile positive IME only with implausible patterns: negatively correlated fixed and variable trade costs and fixed costs decreasing with distance; Pareto also fails to generate rising IME across percentiles.
- Multiproduct structure and granularity do not resolve these puzzles.
- Lognormal productivity model:
  - Resolves puzzles: yields positive IME, IME rising with exporter percentile, and model-implied fixed trade costs increasing with distance.
  - Full Melitz-lognormal estimated by likelihood fits EDD data well and reproduces IME magnitude and percentile pattern.
- Counterfactual implications:
  - For uniform trade liberalization, Melitz-Pareto approximates welfare gains well because average trade elasticities change little under the estimated lognormal model.
  - Melitz-Pareto can mispredict bilateral trade-flow changes and asymmetric liberalizations produce larger model differences.
- Broader implication: choice between Pareto and lognormal productivity distributions matters for dynamics and spillovers — e.g., Pareto implies more entry of marginal exporters after liberalization than lognormal, affecting dynamic gains from learning and growth mechanisms.

*Source: wp18259 (IMF working paper PDF content).*

### 0.46 in the core sample that we will use for the analysis in the next two sections.  Our preferred

### wp18259 - 0.46 in the core sample that we will use for the analysis in the next two sections.  Our preferred

### Estimates of the intensive margin elasticity (IME)
- Preferred IME estimate: 0.4 based on inclusion of origin-year and destination-year fixed effects (as in Figure 1).
- In this estimate:
  - Intensive margin accounts for approximately 40% of the variation in total exports across country pairs.
  - Extensive margin accounts for approximately 60% of the variation.
- Origin-year and destination-year fixed effects alone absorb at most 59 percent of the variation in bilateral trade flows (from R-squared of OLS regression of lnXijt on origin-year and destination-year fixed effects), implying a large share of variation comes from forces behind the estimated IME.

### Robustness of IME estimates
- Extended-sample IME including all country pairs (even those with less than 100 exporting firms) reaches 0.58 when origin-year and destination-year fixed effects are included (Panel B of Table 2).
- Preferred specification with origin-year and destination-year fixed effects:
  - IME = 0.38 among origin-destination pairs with at least 100 exporting firms (core sample).
  - IME = 0.52 among all origin-destination pairs (extended sample).
- Excluding firms whose annual exports fell below $1,000 in any year: corresponding IME estimates change only slightly (Table 3 for core sample; Online Appendix Table I4 for extended sample).
- Instrumenting total exports with leads and/or lags to address serially uncorrelated classical measurement error yields instrumented IMEs very close to OLS IME (Online Appendix Table I5).

### Industry-level and product-level findings
- IME increases when moving to industry-level data.
- At HS 6-digit (lowest aggregation) for the core sample with origin-year-industry and destination-year-industry fixed effects:
  - IME = 0.51.
- Extended sample IME disaggregated at HS2 product level: IME close to 0.52 (Online Appendix Table I6).
- Concern that country differences in industry composition combine with industry differences in average exports per firm is addressed by plotting demeaned intensive and extensive margins at origin-industry-destination-year level using HS 2-digit industries (Figure 2).
- Presence of large trading firms: restricting to HS 2-digit industries with low shares of firms exporting via intermediaries yields IME ≈ 0.53 (Online Appendix Table I7).

### IME by percentiles and role of export superstars
- Method: for each origin-destination-year, distribute exporting firms into percentiles by export value and estimate IMEpct from lnxpctij = FEoi + FEdj + αpct lnXij + εij.
- Overall IME (horizontal line in Figure 3): 0.4.
- IME for highest percentile: 0.5.
- IMEpct rises from 0.2 at the 50th percentile to 0.3 at the 80th percentile.
- Conclusion: positive overall IME not driven exclusively by export superstars; IME increases monotonically across exporter size distribution.

### IME for multi-product firms and decomposition
- Definitions:
  - mij = average number of products exported from i to j by firms exporting from i to j.
  - x p ij ≡ xij / mij = average exports per product per firm exporting from i to j.
  - IMEp ≡ cov(ln ̃xpij, ln ̃Xij) / var(ln ̃Xij).
  - IME = IMEp + cov(ln ̃mij, ln ̃Xij) / var(ln ̃Xij).
- Table 5 (core sample) shows most of the IME is explained by systematic variation in average exports per product per firm rather than by average number of products exported per firm.

### Correlation between intensive and extensive margins, and relation with distance
- Positive and significant correlation between average exports per firm and number of exporting firms: 0.25 (standard error 0.01), after removing origin-year and destination-year effects.
- Table 6: elasticities of average exports per firm, number of firms, average number of products exported per firm, and average exports per product per firm with respect to log distance are all negative and significant when controlling for origin-year and destination-year fixed effects — these margins decline with distance between trade partners.

### Relation to previous empirical results
- Comparison to EKK (France): EKK find average exports per firm increase with destination market size with elasticity 1/3.
  - Regression of market size against estimated destination fixed effects in current data implies elasticity = 0.19.
- On multi-destination productivity (EKK): using x il|j and most popular destination l*(i) to proxy domestic-market productivity:
  - Plot of log x il*(i)|j / x il*(i)|l*(i) against log(Nij / Nil*(i)) for all i and j (core sample) shows firms that sell in more markets have higher sales in origin country’s most popular destination market, consistent with EKK.
- Comparison to EKS: EKS report average exports per firm similar across four origins; running analogous regression (pooling years, including year fixed effects) in current dataset finds origin fixed effects do vary significantly across countries (coefficient of variation in estimated origin fixed effects ranges from...).

### Summary takeaway: the IME in the EDD
- Intensive margin elasticity is positive and significant, both statistically and economically.
- Robust to inclusion of various fixed effects, sample definitions, exclusion of small firms, and industry disaggregation.
- IME is positive and monotonically increasing across exporter size distribution.
- Systematic cross-country-pair variation of average exports per firm is primarily driven by average exports per product per firm.

*Source: wp18259 - 0.46 in the core sample that we will use for the analysis in the next two sections.  Our preferred*

### 0.81 to 2.56, depending on the sample used) and are higher for countries with higher GDP per

### wp18259 - 0.81 to 2.56, depending on the sample used) and are higher for countries with higher GDP per

### Empirical regularities on the intensive margin
- Regression of lnx_ij on origin-year and destination-year fixed effects yields R-squared = 0.65 when only country pairs with N_ij >100 are considered and R-squared = 0.37 when all country pairs are considered.
- Bernard et al. (2009) find IME_p ≈ 0.23; the authors find IME_p ≈ 0.29.
- Tables and figures referenced in the text show an IME of 0.4 or higher in the data.
- Regressing ln x_pij on ln dist_ij with only origin and year fixed effects (no destination fixed effects) yields a positive and significant coefficient; adding destination fixed effects makes the coefficient negative and significant.
- Empirical finding: average exports per firm fall with distance (see Table 6).

*Key numeric values and empirical magnitudes preserved exactly*
- R-squared: 0.65 and 0.37.
- IME_p data comparison: 0.23 versus 0.29.
- Empirical IME: 0.4 or higher.

### The basic Melitz-Pareto model — setup and key theoretical expressions
- Setup and assumptions:
  - Labor in fixed supply L_i; wage w_i.
  - CES preferences with elasticity of substitution σ > 1.
  - Continuum of single-product firms measure N_i; firm productivity φ Pareto with shape θ > σ−1 and scale b_i: Pr(φ ≤ φ0) = G_i(φ0) = 1 − (φ0/b_i)^(−θ).
  - Fixed trade costs F_ij and iceberg trade costs τ_ij.
- Firm sales in destination j: x_ij(φ) = A_j ( ̄σ w_i τ_ij φ )^(1−σ), where A_j ≡ P_j^(1−σ) w_j L_j and ̄σ ≡ σ/(σ−1).
- Productivity cutoff φ*_ij defined by x_ij(φ*_ij) = σ F_ij.
- Aggregate exports and number of exporters (assuming φ*_ij > b_i):
  - X_ij = ( θ/(θ−(σ−1)) ) A_j (w_i τ_ij)^(1−σ) b_i^θ N_i (φ*_ij)^(σ−θ−1)
  - N_ij = b_i^θ N_i (φ*_ij)^(−θ)
- Intensive margin (average exports per firm):
  - x_ij ≡ X_ij / N_ij = ( θσ/(θ−(σ−1)) ) F_ij
- Log-linearized equations (after decomposing trade costs into origin, destination, and pair-specific components):
  - ln N_ij = μ_N,o_i + μ_N,d_j − θ ln ̃τ_ij − ̄θ ln ̃F_ij, where ̄θ ≡ θ/(σ−1).
  - lnx_ij = μ_x,o_i + μ_x,d_j + ln ̃F_ij.

### Intensive margin elasticity (IME) implications in the basic model
- Definition used: IME = cov(ln ̃x_ij, ln ̃X_ij) / var(ln ̃X_ij).
- Model-implied IME formula:
  - IME = −( ̄θ − 1 ) var(ln ̃F_ij) − θ cov(ln ̃τ_ij, ln ̃F_ij)
    divided by var( −θ ln ̃τ_ij − ( ̄θ − 1 ) ln ̃F_ij ).
- Four formal observations:
  1. If var(ln ̃F_ij) = 0 then IME = 0.
  2. If IME > 0 then corr(ln ̃F_ij, ln ̃τ_ij) < 0 (i.e., negative correlation between pair-specific fixed and variable trade costs, ignoring origin/destination fixed costs).
  3. cov(ln ̃x_ij, ln ̃dist_ij) / var(ln ̃dist_ij) = cov(ln ̃F_ij, ln ̃dist_ij) / var(ln ̃dist_ij) — fixed trade costs elasticity w.r.t. distance equals the empirical distance elasticity of average exports per firm.
  4. IME_pct = IME for all percentiles pct of the exporter size distribution (Pareto implies constant percentile-specific IME).

### Model implications versus data (basic Melitz-Pareto)
- Under the assumption that fixed trade costs vary only by origin and destination (var(ln ̃F_ij) = 0), the model predicts IME = 0 and EME = 1, which sharply contrasts the data showing IME ≥ 0.4.
- To reconcile the model with the observed positive IME, one must abandon var(ln ̃F_ij) = 0. Observation 2 then requires a negative correlation between model-implied fixed and variable trade costs.
- Backing out model-implied trade costs using:
  - ln ̃F_ij = δ_F,o_i + δ_F,d_j + lnx_ij
  - θ ln ̃τ_ij = δ_τ,o_i + δ_τ,d_j − ̄θ lnx_ij − ln N_ij
  - Calibration used: θ = 5 and σ = 5, implying ̄θ = 1.25.
- Empirical consequences of computed model-implied trade costs:
  - corr(model-implied fixed, variable trade costs) = −0.786 (standard error = 0.007).
  - Model-implied fixed trade costs decrease with distance; model-implied variable trade costs increase with distance.
  - Distance elasticities reported:
    - Fixed trade costs elasticity w.r.t. distance = −0.28.
    - Variable trade costs elasticity w.r.t. distance = 0.272.
  - These implications (fixed costs falling with distance and strong negative correlation between fixed and variable trade costs) are labeled puzzling and lack microfoundations in existing models.
- Constant IME across exporter-size percentiles predicted by the model contradicts empirical variation across percentiles.

### Multi-product extension (Bernard, Redding, Schott (2011) type)
- Extension features:
  - Firms produce a continuum of products λ ∈ [0,1]; productivity φ common across products and product-specific component λ.
  - φ drawn from Pareto G_f with shape θ_f; λ drawn from Pareto G_p with shape θ_p; impose θ_f > θ_p > σ−1 so θ_f > θ_p and ̄θ ≡ θ_f/(σ−1) and χ ≡ θ_f/θ_p.
  - Firms pay firm-level fixed cost F_ij to enter market j and product-level fixed cost f_ij per product; variable trade costs τ_ij.
- Key log-linear relationships (after Pareto assumptions and decomposition):
  - ln X_ij = μ_X,o_i + μ_X,d_j − θ_f ln ̃τ_ij − ( θ_f/(σ−1) − θ_f/θ_p ) ln ̃f_ij − ( θ_f/θ_p − 1 ) ln ̃F_ij.
  - lnx_pij ≡ ln X_ij − ln M_ij = μ_xp,o_i + μ_xp,d_j + ln ̃f_ij.
  - lnx_ij ≡ ln X_ij − ln N_ij = μ_xf,o_i + μ_xf,d_j + ln ̃F_ij.
- IME expressions in multi-product model:
  - IME = [ −(χ − 1) var(ln ̃F_ij) + ( ̄θ − χ ) cov(ln ̃f_ij, ln ̃F_ij) + θ cov(ln ̃F_ij, ln ̃τ_ij) ] / var(ln ̃X_ij).
  - IME_p = [ −( ̄θ − χ ) var(ln ̃f_ij) + (χ − 1) cov(ln ̃f_ij, ln ̃F_ij) + θ cov(ln ̃f_ij, ln ̃τ_ij) ] / var(ln ̃X_ij).
- Observations and implications:
  - Observation 5: If var(ln ̃f_ij) = 0 then IME_p = 0.
  - Observation 6: If IME > 0 then either cov(ln ̃f_ij, ln ̃F_ij) < 0 or cov(ln ̃F_ij, ln ̃τ_ij) < 0 (or both).
  - Observation 7: If IME_p > 0 then either cov(ln ̃f_ij, ln ̃F_ij) < 0 or cov(ln ̃f_ij, ln ̃τ_ij) < 0 (or both).
  - Observation 8 (product-level distance relation): cov(ln ̃x_pij, ln ̃dist_ij) / var(ln ̃dist_ij) = cov(ln ̃f_ij, ln ̃dist_ij) / var(ln ̃dist_ij).
- Empirical implications maintained:
  - To match a positive IME_p in the data, multi-product model requires var(ln ̃f_ij) > 0.
  - The model still implies negatively correlated fixed trade cost components or negative covariances with variable trade costs.
  - Model-implied product-level fixed trade costs decrease with distance with elasticity = −0.071 (third column of Table 7, Figure 7).

### Firm-level demand and fixed-cost shocks (EKK extension) — robustness and limits
- EKK allow log-normal firm-level destination-specific demand and fixed-cost shocks; under their assumptions, equations (7) and (8) remain valid up to constants capturing net effects of shocks, so Observations 1–3 continue to hold in their environment.
- Critical technical note: If productivity is Pareto, adding log-normal demand or fixed-cost shocks generally breaks equations (7) and (8) unless one takes the limiting construction used by EKK (loosely: scale parameter of the Pareto converges to zero while the measure of firms goes to infinity). In that limit the simple aggregate expressions are recovered.
- Interpretation and consequence:
  - The EKK construction shows one route to reconcile richer firm-level shocks with the aggregate Melitz-Pareto predictions, but the core implications (need for pair-specific variation in fixed trade costs, negative correlation between fixed and variable costs to obtain positive IME, and downward-sloping fixed costs with distance) persist unless Pareto and limiting assumptions are relaxed.

*Italic source attribution: Extracted from wp18259 (IMF working paper PDF content).*

### Section 3.1, if for alli,jand all possible(α

### wp18259 - Section 3.1, if for alli,jand all possible(αj,fj)we haveφ∗ij(αj,fj)> bi, we can easily show that we still have IME= 0.

### Conditions giving IME = 0
- If for all i, j and all possible (αj, fj) we have φ∗ij(αj, fj) > bi, then IME = 0.
- Notation and expressions preserved exactly: φ∗ij(αj,fj) > bi; IME= 0.

### Implications of lognormal shocks
- If αj and fj are lognormally distributed, then for bi > 0 for all i there must be a positive mass of firms for which φ∗ij(αj,fj) < bi.
- For those firms with φ∗ij(αj,fj) < bi, there would be a positive intensive margin elasticity (positive IME for that subset).

### EKK approach and its avoidance of positive IME
- EKK essentially avoid the positive-mass issue by taking the limit with bi → 0 for all i.

### Trade-off of extending Melitz-Pareto to include lognormal demand and fixed-cost shocks
- In principle, a Melitz model with Pareto distributed productivity but extended to allow for log-normally distributed demand and fixed-cost shocks could match the positive IME observed in the data.
- However, such an extended model would lose convenient features of the canonical Melitz-Pareto model:
  - the sales distribution is not distributed Pareto,
  - the trade elasticity is not common across country pairs and fixed,
  - the gains from trade are not given by the ACR formula.

### Authors' approach and motivations
- The paper moves to a model where productivity as well as destination-specific demand and fixed-cost shocks are lognormally distributed.
- Advantage highlighted: such a model is computationally tractable and amenable to likelihood estimation methods.

*Source: wp18259 - Section 3.1, if for alli,jand all possible(αj,fj)we haveφ∗ij(αj,fj)> bi, we can easily show that we still have IME= 0.*

### Section 4.

### Section 4.

### Granularity (extension of Melitz-Pareto)
- Model setup: discrete and finite number of firms with equations (21) and (22):
  - lnN_ij = μ_{N,o i} + μ_{N,d j} − θ ln ̃τ_ij − ̄θ ln ̃F_ij + ξ_ij
  - lnx_ij = μ_{x,o i} + μ_{x,d j} + ln ̃F_ij + ε_ij
- Intensive margin elasticity (IME) expression with granularity (equation 23):
  - IME = −( ̄θ − 1) var(ln ̃F_ij) − θ cov(ln ̃τ_ij, ln ̃F_ij) + var(ε_ij) + COV
    var(−θ ln ̃τ_ij − ( ̄θ − 1) ln ̃F_ij + ε_ij + ξ_ij)
  - COV ≡ cov(ln ̃F_ij + ε_ij, ξ_ij) + cov(ln ̃F_ij + ln ̃τ_ij, ε_ij)
- Key insight:
  - If var(ε_ij) is large relative to −COV, IME could be > 0 even with cov(ln ̃F_ij, ln ̃τ_ij) > 0 — in theory granularity could explain positive IME without implausible fixed-cost patterns.
- Empirical tests (summary; details in Online Appendix B):
  - Estimation of distance elasticities of model-implied firm- and product-level fixed trade costs accounting for granularity:
    - Distance elasticities are significantly lower than estimates ignoring granularity (Table 7 vs Table 8) but remain negative — model-implied fixed trade costs are decreasing with distance.
    - Conclusion: granularity does not eliminate the puzzle that model-implied fixed trade costs decrease with distance.
  - Simulation exercise (Table 9):
    - Simulate exports of N_ij firms per country pair, assuming var(̃F_ij) = 0 to isolate granularity.
    - Consider two extreme cases of firm sales correlation across destinations: perfect correlation (no demand shocks) and sales independent across destinations (all heterogeneity from destination-specific demand shocks).
    - Simulate for three values of ̄θ:
      - ̄θ = 2.4 (estimate using Eaton et al. (2011) procedure)
      - ̄θ = 1.25 (inferred from θ = 5 and σ = 5)
      - ̄θ = 1 (Zipf’s Law)
    - Findings:
      - Simulated IME decreases with ̄θ.
      - Simulated IME is highest when productivity is less correlated across destinations.
      - For ̄θ = 2.4 and no demand shocks (perfect correlation) simulated IME = 0.001.
      - For ̄θ = 1 and no correlation between demand shocks and productivity simulated IME = 0.33.
    - Interpretation:
      - The ̄θ = 1, uncorrelated-sales case (simulated IME = 0.33) is extreme and implausible given empirical trade elasticities; it implies superstar firms drive almost all of the positive IME.
      - Percentile analysis: simulated IME is almost zero for small percentiles and high for a small number of top percentiles; fails to reproduce data pattern where IME increases steadily across percentiles with a spike at the top percentile.
  - Overall conclusion: granularity is not a plausible explanation for the positive estimated IME in the data.

### The Intensive Margin in the Melitz-Lognormal Model — Theory
- Departure: assume firm productivity distribution is lognormal rather than Pareto.
- Key formulae and definitions:
  - Ratio of average to minimum exports per firm (equation 24):
    - x_ij / x_ij(φ^*_ij) = ( ̃φ_i(φ^*_ij) / φ^*_ij)^{σ−1}
  - For lognormal g_i(φ) with parameters μ_{φ,i} and σ_φ:
    - G_i(φ) = Φ((ln φ − μ_{φ,i}) / σ_φ)
    - Define h(x) ≡ Φ′(x) / Φ(x)
    - ( ̃φ_i(φ^*_ij) / φ^*_ij)^{σ−1} = h[−(ln φ^*_ij − μ_{φ,i})/σ_φ] / h[−(ln φ^*_ij − μ_{φ,i})/σ_φ + ̄σ_φ], where ̄σ_φ ≡ (σ−1) σ_φ (equation 26)
  - Using 1 − G_i(φ^*_ij) = N_ij / N_i, define Ω(N_ij / N_i) so:
    - x_ij / x_ij(φ^*_ij) = Ω(N_ij / N_i) = h(Φ^{-1}(N_ij / N_i)) / h(Φ^{-1}(N_ij / N_i) + ̄σ_φ) (equation 27)
  - Model-implied average exports per firm (in logs) (equation 28):
    - ln x_ij = μ_{x,o i} + μ_{x,d j} + ln Ω(N_ij / N_i)
  - Model-implied log fixed costs and variable costs:
    - ln ̃F_ij = δ_{F,o i} + δ_{F,d j} + ln x_ij − ln Ω(N_ij / N_i) (equation 29)
    - (σ−1) ln ̃τ_ij = δ_{τ,o i} + δ_{τ,d j} − ln x_ij + ln Ω_i(N_ij / N_i) + ̄σ_φ Φ^{-1}(1 − N_ij / N_i) (equation 30)
- Theoretical implication:
  - Under lognormal productivity, ̃φ(φ^*)/φ^* is decreasing in φ^*, hence a decline in τ_ij increases average exports per firm — yielding a positive IME even when var(̃F_ij) = 0.
- Data inputs and estimation notes:
  - Need values for ̄σ_φ and N_i for each country; use Bento and Restuccia (2015) data to estimate N_i (with imputation for missing values projecting log number of firms on log population; elasticity = 0.945 in their BR dataset).
  - Use QQ-estimation (Head et al. (2014)) to obtain σ_φ and μ_{φ,i} (Online Appendix C).

### Full Melitz-lognormal model — richer specification and identification
- Enriched model features:
  - Joint lognormal distribution for ln φ, ln α_1,..., ln α_J, ln f_1,..., ln f_J per origin i (multivariate normal with origin-specific means and a structured covariance allowing correlation between demand shocks and fixed costs).
  - Latent sales variable Z_{ij} ≡ D_{ij} + ln α_j + (σ−1) ln φ, where D_{ij} ≡ ln[A_j (w_i τ_ij)^{1−σ}], and observed sales X_{ij} = Z_{ij} if ln σ + ln f_{ij} ≤ Z_{ij}, otherwise non-export.
  - Z_i = (Z_{i1},...,Z_{iJ}) ∼ N(d_i, Σ_i) with covariance elements ̄σ^2_{φ,i} + σ^2_{α,i} on diagonal and ̄σ^2_{φ,i} off-diagonal (equation 32).
- Estimation strategy:
  - Use firm-level data for fifteen destinations (USA, Germany, Japan, France and the 11 largest destinations by exports value for each origin) for the year 2007 across 39 origins (drop 2 origins due to convergence issues).
  - Construct likelihood by computing density g_{X_i1,...,X_iJ}(x_{i1},...,x_{iJ}) for observed sales realizations; log-likelihood is sum over firms (equation 33).
  - Origin-specific parameter vector Θ_i includes {d_{ij}, ̄μ_{f,ij}}_{i,j}, ̄σ_{φ,i}, σ_{α,i}, σ_{f,i}, ρ_i (equation 34), where ̄μ_{f,ij} = ln σ + μ_{f,ij} and ρ_i = σ_{αf,i} / (σ_{α,i} σ_{f,i}).
  - Use Chernozhukov and Hong (2003) MCMC Metropolis-Hastings algorithm to generate chains Θ^{(n)}_i and take average ̄Θ as consistent estimator (method handles nonconcavity and high-dimensional parameter vector).
- Identification intuition:
  - d_{ij} and ̄μ_{f,ij} identified from export flows and number of exporters across pairs.
  - ̄σ_{φ,i} + σ_{α,i} from variance of firm sales within pairs.
  - Relative magnitude of σ_{φ,i} versus σ_{α,i} from cross-destination correlation of firm sales.
  - σ_{f,i} from left-tail shape of sales density (selection weights derived from fixed cost distribution); ρ_i from prevalence of small firms and joint distribution of demand shocks and fixed costs.

### Simple Melitz-lognormal results (continuum approximation)
- QQ-estimates of ̄σ_φ (Online Appendix Table I9) across three samples (full, largest 50%, largest 25% firms) yield a conservative choice:
  - Adopt ̄σ_φ = 4.02 (minimum among estimates; corresponds to largest 25% subsample).
- Findings for simple model with lognormal productivity:
  - A lognormal distribution allows positive IME even with a continuum of firms.
  - For ̄σ_φ = 4.02, implied IME = 0.28 (close to data).
  - Most contribution to IME comes from right tail of exporter size distribution (Figure 10).
  - Model-implied variable and fixed trade costs are increasing with distance (Table 11 and Figure 11) — in contrast to Pareto results where fixed costs decrease with distance.
  - Under lognormal, correlation between model-implied variable and fixed trade costs is closer to zero (−0.3) versus Pareto (−0.8), but still negative.

### Estimated full Melitz-lognormal model — fit and parameter estimates
- Data and estimation specifics:
  - Firm-level EDD and China data for 2007, 39 origins (2 dropped for convergence), 15 destinations per origin, set σ = 5.
  - For China, use random sample of 5% of exporters (computational tractability).
- Model fit diagnostics:
  - Density of standardized firm-level log sales: estimated model closely fits data pooled across origins and destinations (Figure 12).
  - Deviations from strict market hierarchy: share of firms selling only to less popular destinations predicted by model is close to data (Figure 13).
  - Correlation of export values across two destinations among top three: model implies mostly positive correlation driven by productivity shocks; data shows more dispersion (Figure 14).
- Variance-covariance parameter estimates (Table 12; medians across 37 origins):
  - Median ̄σ_{φ,i} = 3.18
  - Median σ_{α,i} = 2.67
  - Median σ_{f,i} = 2.39
  - Median ρ_i = 0.50
  - Parameters are precisely estimated for each origin but vary across origins; in general positive correlation between demand and fixed cost shocks, though some origins show negative correlation.
- IME implications from estimated model:
  - Simulation: draw one million firms per origin, compute average sales with selection, multiply by observed N_ij to get total exports.
  - IME implied by estimated model = 0.63.
    - This is higher than preferred IME estimate of 0.4 from Section 2, but sample differences explain much of gap.
    - Using same sample of 37 origins and 4 destinations for 2007, estimated IME = 0.67 (standard error 0.03), statistically indistinguishable from model-implied 0.63.
  - IME by percentile: pattern across percentiles from model is remarkably close to data (Figure 15).
- Trade cost elasticities with distance (Table 14):
  - Both variable and fixed trade costs are strongly increasing in distance in the estimated full lognormal model.
  - Despite improvement, still a negative correlation between fixed and variable trade costs remains.

### Counterfactual analysis — exact hat algebra extension and applications
- Extension of Dekle et al. (2008) “exact hat algebra” to generalized Melitz with arbitrary productivity distribution and firm-level demand and fixed-cost shocks:
  - Define normalized variables and selection boundary using H_{ij}(·) with n_{ij} ≡ N_{ij}/N_i and h_{ij} ≡ H_{ij}(n_{ij}).
  - Equilibrium conditions expressed in levels (equations 35–41) and transformed to hat-change system (equations 42–48).
  - System solves for hat changes of endogenous variables {ˆh_{ij}, ˆλ_{ij}, ˆw_i, ˆN_i, ˆP_j, ˆP_{ij}, ˆX_j} given exogenous ˆτ_{ij} and ˆ∆_j and known functions g_{ij}, known data λ_{ij}, h_{ij}, X_j, Y_i.
- Counterfactuals computed for a subset: 12 Latin American countries + China (37 origins estimated), for tractability.
  - Construct h_{ij} from data N_{ij} and N_i; construct X_{ii} = manufacturing value-added / 0.25; set N_{ii} = N_i.
- Exercise steps:
  1. Apply uniform reductions in international trade costs: ˆτ_{ij} ∈ {0.99, 0.95, 0.9, 0.75} for i ≠ j; ˆτ_{ii} = 1. Compute counterfactuals in estimated full Melitz-lognormal model.
  2. Estimate trade elasticity θ_{ˆτ} by OLS regression ln ˆX_{ij} = γ^o_i + γ^d_j − θ ln ˆτ_{ij} + ζ_{ij} (equation 49) using the counterfactual results for each ˆτ.
     - Use θ_{0.99} (local approximation) and θ_{ˆτ} (average elasticity) as Pareto shape parameters to compute counterfactuals in Melitz-Pareto model.
     - Obtained values: θ_{0.99} = 4.5; θ_{0.95} = 4.49; θ_{0.9} = 4.47; θ_{0.75} = 4.43.
  3. Compute counterfactual implications in Melitz-Pareto model with these θ values.
- Main counterfactual findings:
  - Welfare comparisons (Figure 16):
    - ˆW^{LN}_i − 1 (lognormal) versus ˆW^{P}_i − 1 (Pareto) show very similar welfare changes across origins for uniform shocks, even when Pareto uses local θ_{0.99}.
    - Reason: average trade elasticity implied by regression changes little across equilibria in estimated lognormal model (θ_{0.99} = 4.5 vs θ_{0.75} = 4.43).
    - Distribution of local trade elasticities in lognormal model ranges from 4 to 6.9 (std dev 0.59); higher elasticities associated with low n_{ij} (Online Appendix G).
    - For uniform shocks, average elasticity matters most; Pareto is a good approximation for welfare gains from uniform liberalization.
  - Bilateral trade flows (Figure 17):
    - Melitz-Pareto can substantially over- or underpredict changes in bilateral trade flows relative to Melitz-lognormal depending on local trade elasticities; higher lognormal elasticities produce larger changes in trade flows than Pareto.
  - Asymmetric shock case:
    - Extreme asymmetric shock: for each origin i, reduce τ only for export to destination with highest N_{il} (ˆτ_{ij} = 0.25 if j = arg max_l N_{il}, else 1).
    - Expectation: Melitz-lognormal delivers smaller welfare gains than Melitz-Pareto because affected pairs have low trade elasticities.
    - Confirmed in Figure 18: lognormal yields smaller welfare gains, but differences are small.
    - Greater differences between models in changes to trade flows in asymmetric shocks (Figure 19).
  - Comparison with Melitz and Redding (2015):
    - Melitz and Redding find ACR ex-post formula performs poorly for welfare evaluation when trade elasticity varies a lot (truncated Pareto example).
    - Here, ACR formula approximates welfare changes well in estimated Melitz-lognormal model because trade elasticity varies little across counterfactuals (three quarters of bilateral elasticities lie between 4 and 4.8). Reproducing Melitz and Redding results requires parameter values far from estimated ones (Appendix H).

### Conclusions (Section 6 summary points reiterated)
- Canonical Melitz-Pareto prediction: conditional on fixed costs, variation in exports across partners should be due entirely to number of exporting firms (extensive margin); intensive margin should be zero.
- Empirical evidence using World Bank Exporter Dynamics Database + China:
  - At least 40% of variation in exports occurs along the intensive margin (preferred estimate 0.4).
  - Robust across samples (all destinations vs largest destinations, all firms vs excluding very small firms, country-pair sample size, industry disaggregation).
  - Intensive margin importance increases with exporter percentile.
- Explanations tested:
  - Pareto with country-pair variation in fixed trade costs can match intensive margin only if fixed costs are negatively correlated with distance — implausible pattern; also fails to reproduce monotone rising IME across percentiles.
  - Allowing multiproduct firms or granularity does not resolve puzzles.
- Lognormal productivity implications:
  - Moving to lognormal firm productivity resolves puzzles: positive IME overall, IME rising by exporter percentile, model-implied fixed trade costs increasing with distance.
  - Full Melitz-lognormal model estimated by likelihood methods fits EDD data well; generates IME close to data and percentile IME pattern similar to data.
- Counterfactual welfare implications:
  - Although trade elasticity is not constant in Melitz-lognormal, average trade elasticity for uniform shocks varies little in estimated model, so Melitz-Pareto provides a good approximation to welfare gains from uniform trade liberalization.
  - However, Melitz-Pareto can mispredict bilateral trade flow changes depending on local elasticities; asymmetry in liberalization can accentuate differences.
- Broader implications:
  - Choice between Pareto and lognormal productivity distributions matters for dynamics and spillovers: e.g., trade liberalization induces more entry of marginal exporters under Pareto than under lognormal, impacting dynamic gains from learning and growth mechanisms.

*Source: wp18259 - Section 4.*

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- Nigai, Sergey, “A tale of two tails: Productivity distribution and the gains from trade,” Journal of International Economics, 2017,104(C), 44–62.
- Ossa, Ralph, “Why Trade Matters After All,” Working Paper 18113, National Bureau of Economic Research May 2012.
- Perla, Jesse, Christopher Tonetti, and Michael E. Waugh, “Equilibrium Technology Diffusion, Trade, and Growth,” 2015.

### Tables and Figures

### Table 1: Core Sample of EDD countries + China, years firm-level data is available
- Countries and years (ISO3, Country name, 1st year, Last year) include:
  - ALB Albania 2004 2012
  - BFA Burkina Faso 2005 2012
  - BGD Bangladesh 2005 2013
  - BGR Bulgaria 2003 2006
  - BOL Bolivia 2006 2012
  - BWA Botswana 2003 2013
  - CHL Chile 2003 2012
  - CHN China 2003 2008
  - CIV Cote d’Ivoire 2009 2012
  - CMR Cameroon 2003 2013
  - COL Colombia 2007 2013
  - CRI Costa Rica 2003 2012
  - DOM Dominican Republic 2003 2013
  - ECU Ecuador 2003 2013
  - EGY Egypt 2006 2012
  - ETH Ethiopia 2008 2012
  - GAB Gabon 2003 2008
  - GEO Georgia 2003 2012
  - GIN Guinea 2009 2012
  - GTM Guatemala 2005 2013
  - HRV Croatia 2007 2012
  - IRN Iran 2006 2010
  - JOR Jordan 2003 2012
  - KEN Kenya 2006 2013
  - KGZ Krygyztan 2006 2012
  - KHM Cambodia 2003 2009
  - LAO Laos 2006 2010
  - LBN Lebanon 2008 2012
  - MAR Morocco 2003 2013
  - MDG Madagascar 2007 2012
  - MEX Mexico 2003 2012
  - MKD Macedonia 2003 2010
  - MMR Myanmar 2011 2013
  - MUS Mauritius 2003 2012
  - MWI Malawi 2009 2012
  - NIC Nicaragua 2003 2013
  - NPL Nepal 2011 2013
  - PAK Pakistan 2003 2010
  - PRY Paraguay 2007 2012
  - PER Peru 2003 2013
  - QOS Kosovo 2011 2013
  - ROU Romania 2005 2011
  - RWA Rwanda 2003 2012
  - THA Thailand 2012 2013
  - TZA Tanzania 2003 2012
  - UGA Uganda* 2003 2010
  - URY Uruguay 2003 2012
  - YEM Yemen 2008 2012
  - ZAF South Africa 2003 2012
  - ZMB Zambia 2003 2011
- Note: * indicates that Uganda does not have data for 2006

### Table 2: IME regressions, core sample — key coefficients and statistics
- Panel a: country pairs with N_ij ≥100
  - IM elasticity: 0.438*** (col 1), 0.459*** (col 2), 0.400*** (col 3)
  - Standard error: [0.0058], [0.0041], [0.0049]
  - R2: 0.55, 0.74, 0.85
  - Variation in lnX_ij explained by FE, %: 0.01, 0.20, 0.59
  - Observations: 7,781; 7,768; 7,324
- Panel b: all country pairs
  - IM elasticity: 0.503***, 0.530***, 0.579***
  - Standard error: [0.0018], [0.0017], [0.0022]
  - R2: 0.77, 0.81, 0.85
  - Variation in lnX_ij explained by FE, %: 0.00, 0.20, 0.50
  - Observations: 47,129; 47,129; 47,037
- Year FE: Yes
- Origin×year FE: Yes (cols 2–3)
- Destination×year FE: Yes (col 3)
- Note: Robust standard errors reported in brackets. *, **, and *** represent the 5%, 1%, and 0.1% significance levels respectively.

### Table 3: IME regression, small firms excluded, core sample — key coefficients and statistics
- Panel a: country pairs with N_ij ≥100
  - IM elasticity: 0.437***, 0.459***, 0.398***
  - Standard error: [0.0058], [0.0042], [0.0050]
  - R2: 0.54, 0.74, 0.85
  - Variation in lnX_ij explained by FE, %: 0.01, 0.19, 0.59
  - Observations: 7,698; 7,684; 7,234
- Panel b: all country pairs
  - IM elasticity: 0.497***, 0.525***, 0.573***
  - Standard error: [0.0013], [0.0013], [0.0015]
  - R2: 0.77, 0.81, 0.84
  - Variation in lnX_ij explained by FE, %: 0.00, 0.19, 0.50
  - Observations: 46,925; 46,925; 46,832
- Year FE: Yes
- Origin×year FE: Yes (cols 2–3)
- Destination×year FE: Yes (col 3)
- Note: Average and total exports per destination calculated using the sales of firms with at least $1000 to that destination. Robust standard errors reported in brackets. *, **, and *** represent the 5%, 1%, and 0.1% significance levels respectively.

### Table 4: IME regression, disaggregated within manufacturing, core sample — key coefficients and statistics
- Panel a: HS 2-digit (sample restricted to origin-destination-product cells with at least 100 exporters)
  - IM elasticity: 0.569***, 0.510***, 0.467***
  - Standard error: [0.0022], [0.0017], [0.0049]
  - Observations: 37,321; 35,621; 10,732
- Panel b: HS 4-digit
  - IM elasticity: 0.651***, 0.569***, 0.515***
  - Standard error: [0.0019], [0.0013], [0.0069]
  - Observations: 62,776; 58,516; 4,640
- Panel c: HS 6-digit
  - IM elasticity: 0.664***, 0.593***, 0.508***
  - Standard error: [0.0020], [0.0014], [0.0094]
  - Observations: 67,967; 61,501; 2,972
- Year×HS FE: Yes (col 1)
- Origin×Year×HS FE: Yes (cols 2–3)
- Destination×Year×HS FE: Yes (col 3)
- Note: Robust standard errors reported in brackets. *, **, and *** represent the 5%, 1%, and 0.1% significance levels respectively.

### Table 5: Product-level IME regression, core sample — key coefficients and statistics
- IM elasticity: 0.380*** (col 1), 0.397*** (col 2), 0.288*** (col 3)
- Standard error: [0.0070], [0.0054], [0.0073]
- R2: 0.35, 0.62, 0.78
- Variation in lnX_ij explained by FE, %: 0.01, 0.20, 0.59
- Observations: 778; 17,768; 7,324
- Year FE: Yes
- Origin×year FE: Yes (cols 2–3)
- Destination×year FE: Yes (col 3)
- Note: Average exports per product defined as average exports divided by the number of HS6 products exported by all firms from origin i to destination j in a given year. Sample restricted to origin-destination pairs with at least 100 exporters. Robust standard errors reported in brackets. *, **, and *** represent the 5%, 1%, and 0.1% significance levels respectively.

### Table 6: Margins of trade and distance — elasticities
- Elasticity with respect to distance
  - x_ij: 0.123*** (col 1), -0.280*** (col 2)
  - Standard error: [0.0150], [0.0130]
- N_ij
  - Coefficient: -0.416*** (col 1), -1.010*** (col 2)
  - Standard error: [0.0134], [0.0128]
- x_pij
  - Coefficient: 0.288*** (col 1), -0.071*** (col 2)
  - Standard error: [0.0158], [0.0146]
- m_ij
  - Coefficient: -0.165*** (col 1), -0.209*** (col 2)
  - Standard error: [0.0059], [0.0051]
- Observations: 7,725; 7,320
- Origin×year FE: Yes (cols shown)
- Destination×year FE: Yes (col 2)
- Note: The table represents the estimated coefficients of the regression of log average exports, number of firms, average exports per product, and number of products on log distance between origins and destinations with origin×year fixed effects (column 2), origin×year and destination×year fixed effects (column 3). The data are aggregated at the year-origin-destination level for a set of origin-years listed in Table 1.

*Reference list and tables extracted from the original content unit.*

### 1.  Population-weighted distance between origins and destinations is

### 1. Population-weighted distance between origins and destinations is

### Trade costs and distance (Table 7)
- Regression of implied log fixed firm-level trade costs (column 1), log variable trade costs (column 2), and log fixed product-level trade costs (column 3) on log distance between origins and destinations.
- Calculations use equations 12 and 13 with θ = 5.
- Data aggregated at the year-origin-destination level; sample restricted to origin-destination pairs with at least 100 exporters.
- Population-weighted distance between origins and destinations taken from Mayer and Zignago (2011).
- Robust standard errors reported in brackets.
- Estimated coefficients:
  - ln ̃F ij on lndist ij: -0.280***
    - Standard error [0.0140]
  - ln ̃τ ij on lndist ij: 0.272***
    - Standard error [0.0046]
  - ln ̃f ij on lndist ij: -0.071***
    - Standard error [0.0146]
- Observations:
  - Column 1: 7,320
  - Column 2: 7,320
  - Column 3: 7,320

### Fixed trade costs distance elasticity and granularity (Table 8)
- Estimated via Poisson pseudo maximum likelihood (Online Appendix B).
- Population-weighted distance from Mayer and Zignago (2011); sample restricted to origin-destination pairs with at least 100 exporters.
- Robust standard errors in brackets.
- Fixed trade costs elasticity estimates:
  - Firm level ζ: -0.022***
    - Standard error [0.0029]
    - Observations: 7,320
  - Product level ζ: -0.007**
    - Standard error [0.0026]
    - Observations: 7,320

### IME under granularity (Table 9)
- Estimated coefficients from regression of implied log average exports on log total exports using numerical simulations (Online Appendix B).
- Sample restricted to year 2007 and origin-destination pairs with at least 100 exporters (867 observations).
- Results reported for models with differing correlation of αjφ and αkφ and different ̃θ values:
  - corr(αjφ, αkφ) = 0:
    - ̃θ = 2.4 → 0.005
    - ̃θ = 1.25 → 0.133
    - ̃θ = 10 → 0.333
  - corr(αjφ, αkφ) = 1:
    - ̃θ = 2.4 → 0.001
    - ̃θ = 1.25 → 0.036
    - ̃θ = 10 → 0.103

### Number of firms and population (Table 10)
- Regression of log number of firms (from Bento and Restuccia (2015)) on log population (World Development Indicators).
- Year fixed effects included.
- Estimates:
  - Coefficient on log population: 0.945***
    - Standard error [0.0136]
    - Observations: 468
    - Year FE: Yes
  - Alternative column: 0.944***
    - Standard error [0.0139]
    - Observations: 468
    - Year FE: Yes

### Melitz-lognormal model: trade costs and dispersion (Tables 11–14)
- Table 11: Melitz-lognormal model regressions of log fixed and log variable trade costs on log distance; sample restricted to origin-destination pairs with at least 100 exporters.
  - log fixed costs on lndist: 0.156***
    - Standard error [0.0155]
    - Observations: 77,387
  - log variable costs on lndist: 0.299***
    - Standard error [0.0051]
    - Observations: 77,387
- Table 12: Estimates of dispersion, full Melitz-lognormal model (sample includes 37 origin countries and 15 destinations per origin). Mean, median, min, max across origins:
  - ̄σφ: mean 3.32, median 3.18, min 0.93, max 5.82
  - σ α: mean 2.72, median 2.67, min 1.94, max 3.64
  - σ f: mean 2.39, median 2.39, min 1.64, max 3.11
  - ρ: mean 0.47, median 0.50, min -0.33, max 0.90
- Table 13: Implied IME in full Melitz-lognormal model (sample: 37 origins and 4 main destinations: USA, Germany, France, Japan, in 2007).
  - Data IME: 0.67
    - 95% CI [0.61, 0.73]
  - Full Melitz-lognormal model IME: 0.63
    - 95% CI [0.59, 0.67]
  - Point estimates and 95% CIs based on 1,000 simulations from Monte-Carlo Markov chain.
- Table 14: Implied trade costs in full Melitz-lognormal model (sample: 37 origins and 4 main destinations in 2007).
  - corr( ̃F ij, ̃τ ij): -0.31
    - 95% CI [-0.45, -0.1]
  - Distance elasticity:
    - Fixed costs: 0.31
      - 95% CI [0.18, 0.41]
    - Variable costs: 0.34
      - 95% CI [0.30, 0.37]

### Figures: empirical patterns and model fit (figure notes and key inputs)
- Figures use Exporter Dynamics Database (EDD) for core and extended samples; analyses often restrict to origin-destination pairs with more than 100 exporting firms.
- Model-implied calculations often use:
  - θ = 5 (Head and Mayer (2014))
  - σ = 5 (Bas et al. (2015))
  - Estimated σ φ in lognormal model: ̄σφ = 4.02 (used in some simulations)
- Selected figure notes and methodological points:
  - Figure 1: plots average size of exporters (intensive margin) and number of exporters (extensive margin) vs. log total exports demeaned by origin-year and destination-year fixed effects; Melitz-Pareto model slope shown.
  - Figure 2: industry-level analogous regressions at origin-HS 2-digit-product-destination-year level with fixed effects at origin-HS 2-digit-year and destination-HS 2-digit-year.
  - Figure 3 and Figure 15: IME for each percentile — regressions of log average exports per firm in exporter size percentiles on log total exports, demeaned by fixed effects.
  - Figure 4: destination fixed effects for log average exports per firm regressed on manufacturing absorption (manufacturing gross production plus manufacturing imports minus manufacturing exports in billions of USD), where manufacturing gross production is calculated as manufacturing value-added divided by 0.418.
  - Figure 6 and Figure 11: model-implied fixed and variable trade costs plotted against log distance (Mayer and Zignago (2011)) and demeaned by origin/destination fixed effects.
  - Figure 12: pdf of standardized log sales — comparison of pooled empirical standardized log sales, model simulated standardized log sales (1MM draws per origin-destination), and standard normal.
  - Figures 16–19: welfare gains and counterfactual changes in trade flows under variable trade costs shocks in full Melitz-lognormal model versus Melitz-Pareto model, using:
    - Dekle et al. (2008) ‘exact hat’ algebra for Melitz-Pareto changes in trade shares
    - Arkolakis et al. (2012) formula for gains from trade liberalization
    - Specific experiments: reductions in trade costs of 1%, 5%, 10%, 25%; asymmetric 25% decline in costs of exporting to biggest market.
  - Figure 17: compares differences in changes in trade flows between models for a 25% reduction in variable trade costs, across trade elasticities implied by the full Melitz-lognormal model.

*Source: wp18259 - 1. Population-weighted distance between origins and destinations is (PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18259.pdf_