Long and Short Run of Trade Elasticities: Boehm, Levchenko & Pandalai-Nayar (2023)

What this is. This is a machine-distilled skeleton of the paper. Read the original at doi.org/10.1257/aer.20210225 to replicate or extend.

TL;DR

Boehm, Levchenko, and Pandalai-Nayar estimate the trade elasticity at every time horizon from 0 to 10 years. Using variation in MFN (most-favored-nation) tariffs across minor trading partners as a plausibly exogenous instrument, and local projections (Jordà 2005) to trace the full time path, they find an elasticity of -0.76 one year after a tariff shock, converging to approximately -2 after 7-10 years. Conventional log-levels OLS gravity estimates (-3.7 to -7.0) are biased by omitted bilateral taste and trade-cost shocks correlated with tariffs; controlling for bilateral unobservables sharply reduces the estimates. The lower long-run elasticity implies that the welfare-relevant trade elasticity is about -1, and applying the Arkolakis, Costinot, and Rodriguez-Clare (2012) gains-from-trade formula shows welfare gains five to six times larger than under the conventional value of -5.

Core results

#	Result	Locator	Magnitude as reported
R1	Short-run (h=1) trade elasticity, preferred baseline IV	Figure 2, p. 876; Table 3 col 1, p. 885	-0.76 (se=0.11)
R2	Long-run (h=10) trade elasticity, preferred baseline IV	Figure 2, p. 876; Table 3 col 1, p. 885	-2.12 (se=0.32)
R3	Log-levels OLS with multilateral resistance FE only	Table 1 cols 1-2, p. 880	-3.70 (se=0.02) to -6.96 (se=0.05)
R4	Log-levels OLS with bilateral product FE (importer-exporter-HS4)	Table 1 col 3, p. 880; Table 2 col 6, p. 882	-1.04 (se=0.02)
R5	Welfare gains from trade, ACR formula, welfare-relevant theta=-1	Figure 7, p. 901; text p. 900	US: 5.27%; world median (64 countries): 22.9%
R6	Sectoral long-run elasticity range, 11 HS sections	Figure 3, pp. 877-878	-0.75 to -5 (median years 7-10)
R7	Five-year baseline IV elasticity	Table 1 cols 6-7, p. 880	-1.17 (se=0.11) without bilateral FE/pretrends; -1.24 (se=0.19) baseline IV
R8	Persistence of the MFN tariff shock	Figure 1, p. 875	About 80% of the initial tariff shock remains after 5 years and 75% after 10 years
R9	All-data/all-tariffs 2SLS elasticity, using initial tariff changes and retaining major partners	Figure 2, p. 876; Table 1 col. 5, p. 880	-0.46 at h=1; -0.62 at h=10; five-year estimate -0.47 (se=0.03)
R10	Trade-flow pretrend check for anticipation	Figure 1, p. 875	No evidence of pretrends in trade flows, including without lagged-change controls
R11	Extensive-margin elasticities under two zero-flow samples	Table 3 cols 7-8, p. 885	h=10: -0.94 (se=0.15) with all available zeros; -1.64 (se=0.18) with only zero-to-positive or positive-to-zero flows
R12	Quantity response	Table 4 col. 4, p. 888	h=10: -1.76 (se=0.41)
R13	Unit-value response	Table 4 col. 5, p. 888	h=10: -0.08 (se=0.22), statistically insignificant
R14	Baseline elasticity across lag, sample, and clustering robustness checks	Table 3 cols 2-6, p. 885	h=10: no lags -1.46 (se=0.19); five lags -2.55 (se=1.02); FE50 -1.76 (se=0.37); two-way clustering -2.12 (se=0.33); constant sample -1.82 (se=0.54)
R15	Log-level elasticity as bilateral controls become finer	Table 2 cols 2-6, p. 882	-2.11 (se=0.02) with gravity controls; -1.39 (se=0.02) with country-pair FE; -1.21 (se=0.03) with country-pair-HS2 FE; -1.17 (se=0.02) with country-pair-HS3 FE; -1.04 (se=0.02) with country-pair-HS4 FE

Overall. The preferred IV estimates of the trade elasticity are -0.76 in the short run, falling to about -2 in the long run. It takes 7-10 years for estimates to stabilize. The welfare-relevant elasticity (adding 1 to the tariff-exclusive estimate to account for tariff-inclusive spending) is about -1, implying gains from trade five to six times larger than under the conventional elasticity of -5. Controlling for bilateral unobservables is the single most important factor distinguishing the paper’s estimates from conventional ones.

Theory / model

The paper develops a partial equilibrium (PE) dynamic model of sluggish adjustment to trade cost shocks (§V.A, pp. 890-897), nesting dynamic versions of the Krugman (1980), Melitz (2003), and Arkolakis (2010) models. Trade in period t is:

$X_t = p_t^* q_t n_t,$

where $p_t^*$ is the exporters’ price exclusive of tariffs, $q_t$ is quantity per unit mass, and $n_t$ is the mass of active exporters. Crucially, $p_t^*$ and $q_t$ adjust instantaneously to tariff changes, while $n_t$ is predetermined by one period: it captures the entry and investment decisions made last period.

The value of exporting $v_t$ and the mass dynamics are governed by equations (8)-(9), p. 891. Their forward and backward solutions are equations (10)-(11), p. 892:

v_t = \frac{1}{1+r}\,E_t\bigl[\pi_{t+1} + (1-\delta)v_{t+1}\bigr], \tag{8}

n_t = n_{t-1}(1-\delta) + G(v_{t-1}), \tag{9}

where $\delta$ is the exogenous exit rate and $G(\cdot)$ is an increasing function mapping the value of exporting to new entrants. Solving (8) forward and (9) backward yields analytical expressions for $v_t$ and $n_t$ in terms of the tariff path.

$$v_t = \frac{1}{1+r} E_t\left[\sum_{k=0}^{\infty}\left(\frac{1-\delta}{1+r}\right)^k \pi(\tau_{t+k+1})\right]. \tag{10}$$ $$n_t = \sum_{\ell=0}^{t-1}(1-\delta)^\ell G(v_{t-1-\ell}) + (1-\delta)^t n_0. \tag{11}$$

The four component elasticities are defined in equation (7), p. 891:

$$\eta_{q,p}=\frac{\partial\ln q}{\partial\ln p^x},\quad \eta_{q,\tau}=\frac{\partial\ln q}{\partial\ln\tau},\quad \eta_{p,\tau}=\frac{\partial\ln p^x}{\partial\ln\tau},\quad \eta_{\pi,\tau}=\frac{\partial\ln\pi}{\partial\ln\tau}. \tag{7}$$

The short-run trade elasticity, where $n_t$ is fixed (eq. 12, p. 892), is:

\varepsilon^0 := \frac{d\ln X_{t_0}}{d\ln\tau_{t_0}}=(1+\eta_{q,p})\eta_{p,\tau}+\eta_{q,\tau}. \tag{12}

The long-run elasticity adds the endogenous adjustment of $n$ to its steady-state value (eq. 13, p. 892):

\varepsilon := \varepsilon^0 + \chi\eta_{\pi,\tau}, \tag{13}

where $\chi := g(v)v/G(v) > 0$ captures the elasticity of the mass of exporters with respect to the value of exporting. Because $\eta_{\pi,\tau} < 0$ and $\chi > 0$, the long-run elasticity is strictly larger in absolute value than the short-run elasticity. In the CES-monopolistic competition version, $\varepsilon^0 = -\sigma$ and $\varepsilon = -\sigma(1 + \chi)$.

The impulse-response decomposition and horizon elasticity (eqs. 14-15, p. 893) are:

$$\frac{d\ln X_{t_0+h}}{d\ln\tau_{t_0}}=\varepsilon^0\frac{d\ln\tau_{t_0+h}}{d\ln\tau_{t_0}}+\frac{d\ln n_{t_0+h}}{d\ln\tau_{t_0}}. \tag{14}$$ $$\varepsilon^h=\frac{\frac{d\ln X_{t_0+h}}{d\ln\tau_{t_0}}}{\frac{d\ln\tau_{t_0+h}}{d\ln\tau_{t_0}}}=\varepsilon^0+\frac{\frac{d\ln n_{t_0+h}}{d\ln\tau_{t_0}}}{\frac{d\ln\tau_{t_0+h}}{d\ln\tau_{t_0}}}. \tag{15}$$

Proposition 1 expresses the mass response for an arbitrary tariff path (eq. 16, p. 893):

$$\frac{d\ln n_{t_0+h}}{d\ln\tau_{t_0}}=\chi\eta_{\pi,\tau}\frac{\delta+r}{1+r}\delta\sum_{k=0}^{h-1}(1-\delta)^{h-1-k}E_{t_0+k}\left[\sum_{\ell=0}^{\infty}\left(\frac{1-\delta}{1+r}\right)^\ell\frac{d\ln\tau_{t_0+k+\ell+1}}{d\ln\tau_{t_0}}\right]. \tag{16}$$

For a one-time tariff shock followed by a constant tariff change, the transitional elasticity is (eq. 17, p. 894):

$$\varepsilon^h=\varepsilon^0+\chi\eta_{\pi,\tau}\left[1-(1-\delta)^h\right],\qquad h\geq 1. \tag{17}$$

For an AR(1) tariff change, the finite-horizon expression is (eq. 18, p. 895):

$$\varepsilon^h=\varepsilon^0+\chi\eta_{\pi,\tau}\frac{(\delta+r)\delta}{[1+r-(1-\delta)\rho](1-\frac{1-\delta}{\rho})}\left[1-\left(\frac{1-\delta}{\rho}\right)^h\right]. \tag{18}$$

Proposition 2 (p. 894) establishes that $\lim_{h\to\infty}\varepsilon^h = \varepsilon$ as long as the tariff shock is not fully mean-reverting, validating the horizon-10 estimates as long-run estimates.

Proposition 3 (p. 895) shows that the model delivers the local-projections estimating equation (2) up to first order. The importer-product-time and exporter-product-time fixed effects absorb weighted averages of past, present, and expected future demand and supply shifters: dynamic analogues of the Anderson and van Wincoop (2003) multilateral resistance terms.

The model is calibrated with $\sigma = 1.1$, $\chi = 0.82$, $\delta = 0.25$, $r = 0.03$ to match the empirical time path of elasticities (Figure 5, p. 897). Convergence to the long run is geometric at rate $\delta$, taking approximately a decade.

Method

The horizon-h trade elasticity $\varepsilon^h$ is estimated by combining local projections (Jordà 2005, builds-on) with a WTO MFN instrumental variable. The key innovation relative to conventional gravity estimation is that a separate regression is run at each horizon $h = 0, 1, \ldots, 10$, tracing the full impulse-response function of trade to tariff shocks without imposing a parametric dynamic model.

The initial local-projection regression relates h-period trade changes to the one-period tariff shock (eq. 2, p. 866):

$$\Delta_h \ln X_{i,j,p,t} = \beta^{X,h} \Delta_0 \ln \tau_{i,j,p,t} + \delta^{d,X,h}_{i,p,t} + \delta^{s,X,h}_{j,p,t} + \delta^{b,X,h}_{i,j,p} + u^{X,h}_{i,j,p,t}. \tag{2}$$

The fixed effects are importer-product-time, exporter-product-time, and importer-exporter-product. The panel is annual; standard errors are clustered by country-pair-product. Equation (3) estimates the tariff impulse response with the same fixed effects, while equation (4) estimates the combined IV specification below.

The horizon-h trade elasticity is defined (eq. 1, p. 865) as:

\varepsilon^h := \frac{\Delta_h \ln X_{i,j,p,t}}{\Delta_h \ln \tau_{i,j,p,t}}, \tag{1}

where $i$ indexes the importing country, $j$ the exporting country, $p$ the product, $t$ time, and $\Delta_h x_t := x_{t+h} - x_{t-1}$ is the h-period change. The long-run elasticity is $\varepsilon = \lim_{h\to\infty}\varepsilon^h$.

Instrument. To address the endogeneity of tariffs, the paper exploits the WTO’s MFN principle: when an importing country changes its applied MFN tariff on a product, all WTO partners trading on MFN terms experience that change. Minor trading partners (not among the top-10 exporters of product p to importer i) are unlikely to have driven the tariff change. The baseline instrument (eq. 5, p. 870) is:

\Delta_0\ln\tau^{\text{instr}}_{i,j,p,t} = \mathbf{1}\!\left\{\tau_{i,j,p,t} = \tau^{\text{appliedMFN}}_{i,j,p,t}\right\} \times \mathbf{1}\!\left\{\tau_{i,j,p,t-1} = \tau^{\text{appliedMFN}}_{i,j,p,t-1}\right\} \times \left(\ln\tau^{\text{appliedMFN}}_{i,j,p,t} - \ln\tau^{\text{appliedMFN}}_{i,j,p,t-1}\right), \tag{5}

retaining only observations where exporter j is not a top-10 trading partner of importer i (in total trade or in product p). Countries in preferential trade agreements (PTAs) with the importer serve as the control group: their applied tariffs differ from MFN rates so they do not experience the MFN tariff change. The instrument is equivalent to an instrumented difference-in-differences comparing minor MFN partners (treated) to PTA partners (control).

Equation (6), p. 870, states the sample exclusion: drop an observation if the applied MFN tariff binds in both periods and exporter j is a major trading partner at either date, measured in aggregate trade or product p. The baseline uses the top ten partners:

$$\mathbf{1}\{\tau_{i,j,p,t}=\tau^{\text{appliedMFN}}_{i,j,p,t}\}\,\mathbf{1}\{\tau_{i,j,p,t-1}=\tau^{\text{appliedMFN}}_{i,j,p,t-1}\}\left(\mathbf{1}\{j \text{ major in aggregate at }t-1\}+\mathbf{1}\{j \text{ major in product }p\text{ at }t-1\}+\mathbf{1}\{j \text{ major in aggregate at }t\}+\mathbf{1}\{j \text{ major in product }p\text{ at }t\}\right)>0. \tag{6}$$

Standard errors are clustered at the country-pair-product level throughout; first-stage F-statistics exceed 10 at all horizons (Online Appendix Table B2).

Empirical specifications

The combined specification estimated at each horizon h is (eq. 4, p. 867):

\Delta_h \ln X_{i,j,p,t} = \beta^h \Delta_h \ln\tau_{i,j,p,t} + \delta^{d,h}_{i,p,t} + \delta^{s,h}_{j,p,t} + \delta^{b,h}_{i,j,p} + u^h_{i,j,p,t}, \tag{4}

where $\delta^{d,h}_{i,p,t}$ is an importer-HS4-year fixed effect, $\delta^{s,h}_{j,p,t}$ is an exporter-HS4-year fixed effect, and $\delta^{b,h}_{i,j,p}$ is a source-destination-product fixed effect (absorbing bilateral trends in trade). All specifications include one lag of log changes in tariffs and trade as pretrend controls. When $\Delta_0\ln\tau^{\text{instr}}_{i,j,p,t}$ instruments for $\Delta_h\ln\tau_{i,j,p,t}$, the IV estimator $\hat{\beta}^h$ identifies $\varepsilon^h$.

To account for tariff autocorrelation, the paper also runs a complementary local projection of the tariff change (eq. 3, p. 867):

\Delta_h \ln\tau_{i,j,p,t} = \beta^h_\tau \Delta_0\ln\tau_{i,j,p,t} + \delta^{d,\tau,h}_{i,p,t} + \delta^{s,\tau,h}_{j,p,t} + \delta^{b,\tau,h}_{i,j,p} + u^{\tau,h}_{i,j,p,t}, \tag{3}

with the trade elasticity recovered as $\varepsilon^h = \hat{\beta}^h_X / \hat{\beta}^h_\tau$. This is important because tariff changes are autocorrelated in the data: about 80% of the initial shock survives at 5 years and 75% at 10 years (Figure 1 panel A, p. 875). Failing to account for this autocorrelation would cause h-period differences to conflate elasticities across horizons.

Bias diagnostics (Table 1, p. 880). A log-levels OLS specification assuming all tariff variation is exogenous, similar to Head and Ries (2001), yields coefficients of -3.70 to -6.96. Conventional static gravity in log-levels without bilateral effects:

$\ln X_{i,j,p,t} = \beta\ln\tau_{i,j,p,t} + \delta^d_{i,p,t} + \delta^s_{j,p,t} + u_{i,j,p,t}$

yields $\hat{\beta} = -6.96$ (col 2) or $-3.70$ (col 1 without multilateral resistance). Adding bilateral product fixed effects $\delta^b_{i,j,p}$ in log-levels (Table 2, p. 882) yields $-1.04$, demonstrating that controlling for bilateral unobservables is the dominant force pushing estimates toward the paper’s preferred IV values.

The 5-year IV baseline is $-1.24$ (Table 3 col 1, p. 885) and the 10-year baseline is $-2.12$, consistent across a wide range of robustness checks: alternative pretrend lags (Table 3 cols 2-3), alternative clustering (col 5), constant sample (col 6), and extensive margin specifications (cols 7-8).

Proposition 3 (p. 895) maps the tariff impulse-response coefficient into the horizon elasticity:

$$\beta^{X,h}=\chi\eta_{\pi,\tau}\frac{r+\delta}{1+r}\delta\sum_{k=0}^{h-1}(1-\delta)^{h-1-k}\sum_{\ell=0}^{\infty}\left(\frac{1-\delta}{1+r}\right)^\ell\beta^{\tau,k+\ell+1}+\varepsilon^0\beta^{\tau,h}.$$

The paper also estimates conventional log-level gravity with importer-product-time and exporter-product-time effects, adding bilateral gravity controls or importer-exporter and importer-exporter-HS2/HS3/HS4 effects (Tables 1-2, pp. 880-882). The log-level specification is:

$$\ln X_{i,j,p,t}=\beta\ln\tau_{i,j,p,t}+\delta^d_{i,p,t}+\delta^s_{j,p,t}+u_{i,j,p,t}.$$

Table 1 also reports five-year log-difference OLS and IV specifications. Written out from Table 1, p. 880, the fixed-window specification is:

$$\Delta_5\ln X_{i,j,p,t}=\beta\Delta_5\ln\tau_{i,j,p,t}+\delta^d_{i,p,t}+\delta^s_{j,p,t}+\delta^b_{i,j,p}+u_{i,j,p,t}.$$

The level specification with bilateral product effects, reported in Table 1 col. 3 and Table 2 col. 6 (pp. 880, 882), adds (\delta^b_{i,j,p}) to the displayed log-level form above. These specifications use positive trade flows and tariffs; the level samples have about 105 million observations, while the five-year baseline IV sample has 16.7 million. Table 3 varies pretrend lags, minimum fixed-effect cluster size, standard-error clustering, constant-sample restriction, and extensive-margin treatment; the latter replaces log changes in trade with differenced inverse-hyperbolic-sine changes, omits pretrend controls, and clusters by importer-exporter-HS4 (p. 885). Table 4 keeps equation (4)‘s baseline fixed effects and pretrends while changing partner cutoff/instrument, outcome (quantities or unit values), initial-trade weights, tariff-variation sample, or PTA control assignment (p. 888). The Table 3 extensive-margin regression, written out from its stated transformation (p. 885), is:

$$\Delta_h\operatorname{asinh}(X_{i,j,p,t})=\beta^h\Delta_h\ln\tau_{i,j,p,t}+\delta^{d,h}_{i,p,t}+\delta^{s,h}_{j,p,t}+\delta^{b,h}_{i,j,p}+u^h_{i,j,p,t},$$

with the baseline instrument for the tariff change. Standard errors are clustered by country-pair-product in Tables 1-2 and by importer-exporter-HS4 in Tables 3-4; the two-way Table 3 specification also clusters by year.

Datasets used

Dataset	Role in paper	Wiki page
BACI (CEPII version of UN COMTRADE)	Trade values and quantities at HS6 level, 183 economies, 1995-2018; main outcome variable	no page yet
UN TRAINS (UNCTAD, 1995-2018)	Applied and MFN tariff rates at HS6 level; source of the instrument and the tariff regressor	no page yet

Sample: 183 economies, 5,000+ HS6 product categories, 1995-2018 (annual). Baseline estimation sample at h=1: approximately 26 million country-pair-product-year observations. Additional gravity covariates (distance, common border, common language, colonial relationship) from the CEPII GeoDist database are used in robustness exercises.

When to read the full paper

Read the original when you need: (i) first-stage F-statistics and the full robustness matrix across all horizons (Online Appendix Tables B1-B8); (ii) the multicountry multisector general equilibrium extension and GE impulse responses of US imports to tariff shocks (Figure 6, pp. 898-899); (iii) sectoral heterogeneity results broken down by 11 HS sections and benchmarked to Ossa (2015) (Figure 3, pp. 877-878); (iv) proofs of Propositions 1-3 on the impulse-response of firm mass and the micro-foundation of the estimating equation (Online Appendix C); or (v) country-level gains-from-trade quantification at alternative elasticity values (Online Appendix Table B9).

Attribution and rights

Boehm, Christoph E., Andrei A. Levchenko, and Nitya Pandalai-Nayar. 2023. “The Long and Short (Run) of Trade Elasticities.” American Economic Review 113(4): 861-905. doi:10.1257/aer.20210225.

Replication code: Boehm, Levchenko, and Pandalai-Nayar (2023). “Replication Data for: The Long and Short (Run) of Trade Elasticities.” American Economic Association / ICPSR openICPSR. doi:10.3886/E182781V1.

This page is an LLM-distilled extract (not human-verified, not reproduced). Results are drawn from Figures 1-7 and Tables 1-4 of the published paper. Extract-only redistribution; the original is paywalled via the AEA.