Is Foreign Technological Advance Harmful in the Melitz Model?

Abstract

Foreign technological advance unambiguously reduces home welfare in a popular variant of the Melitz (Econometrica 71(6):1695–1725, 2003) model that assumes the presence of a costlessly traded homogeneous (outside) good (Demidova in Int Econ Rev 49(4):1437–1462, 2008). The present paper shows that this result is sensitive to the presence of the outside good and is, in fact, reversed in its absence: foreign technological advance always improves home welfare in the Melitz model without the outside good. Improvement in home welfare occurs via changes in the numbers and prices of domestic and imported varieties. For quantitative analysis of welfare effects, we calibrate an international trade model for the United States and its major trading partners. US is found to gain less from foreign technological improvements than its trading partners from US improvements. In either case, the magnitude of gains is modest.

Appendix

1.1 Sections 2.2–2.4

1.1.1 Derivation of Relations (1)–(4)

Let p _ij (z) and r _ij (z) denote, respectively, the price set and the revenue generated by a firm in country i with productivity z in country j’s market. Given the consumer demand based on Dixit-Stiglitz preferences, the optimal price and the corresponding revenue equal

$$ {p}_{ij}(z)=\frac{w_i{\tau}_{ij}}{\rho z},\kern1em {r}_{ij}(z)=\frac{p_{ij}{(z)}^{1-\sigma }{w}_j{L}_j}{{P_j}^{1-\sigma }},\kern1em i,j=1,2, $$

(A1)

where ${P_j}^{1-\sigma }={\displaystyle \sum_{i=1,2}\frac{M_i^e}{\delta_i}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty }{p}_{ij}{(z)}^{1-\sigma }d{G}_i(z)}}$ is country j’s price index. Since the price is a constant markup over marginal cost, variable profits (of a firm in country i selling in country j) equal r _ij (z)/σ. Thus the break-even condition for cut-off levels is

$$ \frac{r_{ij}\left({z}_{ij}^{\ast}\right)}{\sigma }={w}_i{f}_{ij},\kern1em i,j=1,2. $$

(A2)

Using (A1) and (A2), we can readily drive (1).

The free entry condition implies that

$$ \frac{1}{\delta_i}{\displaystyle \sum_{j=1,2}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\left(\frac{r_{ij}(z)}{\sigma }-{w}_i{f}_{ij}\right)}}\kern0.5em d{G}_i(z)={w}_i{F}_i,\kern1em i=1,2, $$

(A3)

where the LHS represents expected profits for entrants. Since $\frac{r_{ij}(z)}{r_{ij}\left({z}_{ij}^{*}\right)}={\left(\frac{z}{z_{ij}^{*}}\right)}^{\sigma -1}$ from (A1), we can use (A2) to derive $\frac{r_{ij}(z)}{\sigma }={w}_i{f}_{ij}{\left(\frac{z}{z_{ij}^{*}}\right)}^{\sigma -1}$. Using this condition, we can determine

$$ {\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\left(\frac{r_{ij}(z)}{\sigma }-{w}_i{f}_{ij}\right)}\kern0.5em d{G}_i(z)={w}_i{f}_{ij}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\left\{{\left(z/{z}_{ij}^{*}\right)}^{\sigma -1}-1\right\}}\kern0.5em d{G}_i(z)={w}_i{f}_{ij}{J}_i\left({z}_{ij}^{*}\right),\kern0.5em i,j=1,2. $$

(A4)

Using (A3) and (A4), we obtain (2). For use below, we also have

$$ \begin{array}{l}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{r_{ij}(z)}{\sigma }}\kern0.5em d{G}_i\left({z}_{ij}^{*}\right)={w}_i{f}_{ij}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\left\{{\left(z/{z}_{ij}^{*}\right)}^{\sigma -1}-1+1\right\}}\kern0.5em d{G}_i(z)\\ {}\kern7.5em ={w}_i{f}_{ij}\left[{J}_i\left({z}_{ij}^{*}\right)+1-{G}_i\left({z}_{ij}^{*}\right)\right],\kern1em i,j=1,2.\end{array} $$

(A5)

Let M _ij denote the mass of successful entrants in country i selling in country j. The aggregate firm revenue equals ${\displaystyle \sum_{j=1,2}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{M_{ij}{r}_{ij}(z)}{1-{G}_i\left({z}_{ij}^{*}\right)}\kern0.5em }}d{G}_i(z)$. Free entry implies that aggregate revenue equals total wage income. Noting that $\left[1-{G}_i\left({z}_{ij}^{*}\right)\right]{M}_i^e={\delta}_i{M}_{ij}$ in steady state equilibrium, we can express this condition as

$$ \frac{\sigma {M}_i^e}{\delta_i}{\displaystyle \sum_{j=1,2}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{r_{ij}(z)}{\sigma }}}\kern0.5em d{G}_i(z)={w}_i{L}_i,\kern1em i=1,2. $$

(A6)

Use (A5) and (A6) to get (3). As each country’s exports equal aggregate firm revenue from sales abroad, balanced trade in steady state equilibrium implies that$\frac{\sigma {M}_1^e}{\delta_1}{\displaystyle {\int}_{z_{12}^{*}}^{\infty}\frac{r_{12}(z)}{\sigma}\kern0.5em }d{G}_1(z)=\frac{\sigma {M}_2^e}{\delta_2}{\displaystyle {\int}_{z_{21}^{*}}^{\infty}\frac{r_{21}(z)}{\sigma}\kern0.5em }d{G}_2(z)$. This condition and (A5) give (4).

1.1.2 Shifts in the CC and TB Curves

First consider the CC relation (10). For this relation, we have $\frac{\partial RHS}{\partial {w}_1}={\tau}_{12}{\left[\frac{f_{12}}{f_{22}}\right]}^{1/\left(\sigma -1\right)}{\left({w}_1\right)}^{1/\rho}\left\{\frac{\partial {h}_{22}}{\partial {z}_{21}^{*}}\frac{\partial {h}_{21}}{\partial {w}_1}+{z}_{22}^{*}\left(1/\rho \right){\left({w}_1\right)}^{-1}\right\}>0$, since $\frac{\partial {h}_{22}}{\partial {z}_{21}^{*}}<0$, and $\frac{\partial {h}_{21}}{\partial {w}_1}<0$. Also,$\frac{\partial RHS}{\partial {b}_2}={\tau}_{12}{\left[\frac{f_{12}}{f_{22}}\right]}^{1/\left(\sigma -1\right)}{\left({w}_1\right)}^{1/\rho}\frac{\partial {h}_{22}}{\partial {b}_2}>0$,$as\ \frac{\partial {h}_{22}}{\partial {b}_2}>0$,and$\frac{\partial RHS}{\partial {\theta}_2}={\tau}_{12}{\left[\frac{f_{12}}{f_{22}}\right]}^{1/\left(\sigma -1\right)}{\left({w}_1\right)}^{1/\rho}\frac{\partial {h}_{22}}{\partial {\theta}_2}<0,as\ \frac{\partial {h}_{22}}{\partial {\theta}_2}<0$.¹⁶ If ${z}_{12}^{*}$ is held constant, RHS of (10) does not change, and w ₁must fall as b ₂ increases or θ ₂ decreases. Thus an improvement in country 2’s productivity shifts the CC curve down (or to the right as it slopes upwards).

The TB relation (11) can be written as $\frac{w_1{L}_1}{L_2}=\frac{\varPhi_1}{\varPhi_2}$, where ${\varPhi}_1=1+\frac{f_{11}}{f_{12}}{\left(\frac{z_{12}^{*}}{h_{11}\left({z}_{12}^{*},{b}_1,{\theta}_1\right)}\right)}^{\theta_1}$, and ${\varPhi}_2=1+\frac{f_{22}}{f_{21}}{\left(\frac{h_{21}\left({w}_1,{h}_{11}\left({z}_{12}^{*},{b}_1,{\theta}_1\right)\right)}{h_{22}\left({h}_{21}\left({w}_1,{h}_{11}\left({z}_{12}^{*},{b}_1,{\theta}_1\right)\right);{b}_2,{\theta}_2\right)}\right)}^{\theta_2}$. Since $\frac{\partial {h}_{11}}{\partial {z}_{12}^{*}}<0$,$\frac{\partial {h}_{22}}{\partial {z}_{21}^{*}}<0$, and $\frac{\partial {h}_{21}}{\partial {z}_{11}^{*}}>0$, $\frac{\partial {\varPhi}_1}{\partial {z}_{12}^{*}}=\frac{\theta_1{f}_{11}}{f_{12}}{\left(\frac{z_{12}^{*}}{z_{11}^{*}}\right)}^{\theta_1-1}\left(\frac{z_{11}^{*}-{z}_{12}^{*}\partial {h}_{11}/\partial {z}_{12}^{*}}{{\left({z}_{11}^{*}\right)}^2}\right)>0$, and $\frac{\partial {\varPhi}_2}{\partial {z}_{12}^{*}}=\frac{\theta_2{f}_{22}}{f_{21}}{\left(\frac{z_{21}^{*}}{z_{22}^{*}}\right)}^{\theta_2-1}\left({z}_{22}^{*}\frac{\partial {h}_{21}}{\partial {z}_{11}^{*}}\frac{\partial {h}_{11}}{\partial {z}_{12}^{*}}-{z}_{21}^{*}\frac{\partial {h}_{22}}{\partial {z}_{21}^{*}}\frac{\partial {h}_{21}}{\partial {z}_{11}^{*}}\frac{\partial {h}_{11}}{\partial {z}_{12}^{*}}\right){\left({z}_{22}^{*}\right)}^{-2}<0$. We also have $\frac{\partial {\varPhi}_1}{\partial {b}_2}=0$, $\frac{\partial {\varPhi}_2}{\partial {b}_2}=\frac{\theta_2{f}_{22}}{f_{21}}{\left(\frac{z_{21}^{*}}{z_{22}^{*}}\right)}^{\theta_2-1}\left(\frac{-{z}_{21}^{*}\partial {h}_{22}/\partial {b}_2}{{\left({z}_{22}^{*}\right)}^2}\right)<0$, $\frac{\partial {\varPhi}_1}{\partial {\theta}_2}=0$ and $\frac{\partial {\varPhi}_2}{\partial {\theta}_2}=\frac{f_{22}}{f_{21}}{\left(\frac{z_{21}^{*}}{z_{22}^{*}}\right)}^{\theta_2}\left\{\left(\frac{-{z}_{21}^{*}\partial {h}_{22}/\partial {\theta}_2}{{\left({z}_{22}^{*}\right)}^2}\right)\frac{\theta_2{z}_{22}^{*}}{z_{21}^{*}}+ \ln \left(\frac{z_{21}^{*}}{z_{22}^{*}}\right)\right\}>0$, as $\frac{\partial {h}_{22}}{\partial {b}_2}>0$, $\frac{\partial {h}_{22}}{\partial {\theta}_2}<0$ and $\ln \left(\frac{z_{21}^{*}}{z_{22}^{*}}\right)>0$. It follows that for the TB relation, $\frac{\partial RHS}{\partial {z}_{12}^{*}}>0$, $\frac{\partial RHS}{\partial {b}_2}>0$ and $\frac{\partial RHS}{\partial {\theta}_2}<0$. If w ₁ is held constant, the RHS of the TB relation remains unchanged, and an increase in b ₂ or a decrease in θ ₂ must be offset by a decrease in ${z}_{12}^{*}$ . Thus the TB curve shifts leftwards in response to productivity improvement in country 2.

1.1.3 Proof of Proposition 1

To prove proposition 1, we show that in the new equilibrium with a higher b ₂ or a lower θ ₂, ${z}_{12}^{*}$ must decrease because a different outcome would lead to contradiction. First, note that (5) and (6) imply that $\frac{z_{21}^{*}}{z_{22}^{*}}={\tau}_{12}{\tau}_{21}{\left(\frac{f_{12}{f}_{21}}{f_{11}{f}_{22}}\right)}^{1/\left(\sigma -1\right)}\frac{z_{11}^{*}}{z_{12}^{*}}$ . Using this relation to substitute for $\frac{z_{21}^{*}}{z_{22}^{*}}$ in the TB relation, we get

$$ {w}_1=\frac{L_2}{L_1}\frac{1+\frac{f_{11}}{f_{12}}{\left(\frac{z_{12}^{*}}{z_{11}^{*}}\right)}^{\theta_1}}{1+\frac{f_{22}}{f_{21}}\left[{\left\{{\tau}_{12}{\tau}_{21}{\left(\frac{f_{12}{f}_{21}}{f_{11}{f}_{22}}\right)}^{1/\left(\sigma -1\right)}{\left(\frac{z_{12}^{*}}{z_{11}^{*}}\right)}^{-1}\right\}}^{\theta_2}\right]}. $$

(A7)

Now suppose that ${z}_{12}^{*}$ increases or does not change in the new equilibrium. In this case, ${z}_{11}^{*}$ decreases or is unchanged according to (7), and the ratio ${z}_{12}^{*}/{z}_{11}^{*}$ increases or remains the same. It follows that the RHS of (A7) would increases or not change. This result clearly holds in the case of an increase in b ₂. To see that it also holds in the case of a decrease in θ ₂, note that the expression in the square bracket equals ${\left(\frac{z_{21}^{*}}{z_{22}^{*}}\right)}^{\theta_2}$ and a lower θ ₂ would further decrease this expression as $\frac{z_{21}^{*}}{z_{22}^{*}}>1$. The above result that the RHS of (7) increases or does not change, however, is not possible because the LHS of (A7) must decrease as w ₁ falls in the new equilibrium. Thus an increase in b ₂ or a decrease in θ ₂ would cause a decrease in ${z}_{12}^{*}$, and hence an improvement in country 1’s welfare.

1.2 Section 2.5

1.2.1 Derivation of (13) and (14)

Using (A1), we have ${P_j}^{1-\sigma }={\displaystyle \sum_{i=1,2}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{M_{ij}{p}_{ij}{(z)}^{1-\sigma }}{1-{G}_i\left({z}_{ij}^{*}\right)}d{G}_i(z)}}={\displaystyle \sum_{i=1,2}{M}_{ij}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty }{\left(\frac{w_i{\tau}_{ij}}{\rho z}\right)}^{1-\sigma}\frac{1}{1-{G}_i\left({z}_{ij}^{*}\right)}d{G}_i(z)}}$. Letting ${\tilde{z}}_{ij} \equiv {\left({\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{z^{\sigma -1}}{1-{G}_i\left({z}_{ij}^{*}\right)}d{G}_i(z)}\right)}^{\frac{1}{\sigma -1}}$ we readily obtain (13). Next, totally differentiating (13) and simplifying, we get $\frac{d{P}_j}{P_j}=\frac{1}{P_j}{\displaystyle \sum_{i=1,2}{M}_{ij}{\left(\frac{w_i{\tau}_{ij}}{\rho {\tilde{z}}_{ij}}\right)}^{1-\sigma}\left(\frac{d{w}_i}{w_i}-\frac{d{\tilde{z}}_{ij}}{{\tilde{z}}_{ij}}-\frac{1}{\sigma -1}\frac{d{M}_{ij}}{M_{ij}}\right)}$, which can be expressed as (14) with ${s}_{ij}={M}_{ij}{\left(\frac{w_i{\tau}_{ij}}{\rho {\tilde{z}}_{ij}}\right)}^{1-\sigma}\div {\displaystyle \sum_{i=1,2}{M}_{ij}{\left(\frac{w_i{\tau}_{ij}}{\rho {\tilde{z}}_{ij}}\right)}^{1-\sigma }}$. Note that the share of country i’s sales in country j’s market equals ${M}_{ij}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{r_{ij}{(z)}^{1-\sigma }}{1-{G}_i\left({z}_{ij}^{*}\right)}d{G}_i(z)}\div {\displaystyle \sum_{i=1,2}{M}_{ij}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{r_{ij}{(z)}^{1-\sigma }}{1-{G}_i\left({z}_{ij}^{*}\right)}d{G}_i(z)}}$, which simplifies to the above expression for s _ij [using (A1)].

1.2.2 Variety and Price Effects in the Paper’s Model

Using (1) and the Pareto distribution assumption, we get

$$ {\widehat{\tilde{z}}}_{ij}={\widehat{\xi}}_i+{\widehat{z}}_{ij}^{*}={\widehat{\xi}}_i+\frac{\sigma }{\sigma -1}{\widehat{w}}_i-{\widehat{P}}_j-\frac{1}{\sigma -1}{\widehat{w}}_j,\kern1em i,j=1,2, $$

(A8)

where ${\xi}_i \equiv {\left(\frac{\theta_i}{\theta_i-\sigma +1}\right)}^{\frac{1}{\sigma -1}}$. In this case, for j = 1, w ₂ = 1 and ${\widehat{\xi}}_1=0$, using (A8) to determine ${\displaystyle \sum_{i=1,2}{s}_{ij}{\widehat{\tilde{z}}}_{ij}}$ and substituting this value in (14), we derive the variety channel effect as $\frac{s_{11}}{\sigma -1}{\widehat{M}}_{11}+\frac{s_{21}}{\sigma -1}{\widehat{M}}_{21}={s}_{21}\left(\frac{1}{\sigma -1}{\widehat{w}}_1-{\widehat{\xi}}_2\right)$. Also, using (6), we have

$$ {\widehat{\tilde{z}}}_{21}={\widehat{\xi}}_2+{\widehat{z}}_{21}^{*}={\widehat{\xi}}_2-\frac{\sigma }{\sigma -1}{\widehat{w}}_1+{\widehat{z}}_{11}^{*}. $$

(A9)

Using (A9), we can express the price channel effect as ${s}_{11}{\widehat{\tilde{z}}}_{11}+{s}_{21}\left({\widehat{w}}_1+{\widehat{\tilde{z}}}_{21}\right)={\widehat{z}}_{11}^{*}+\left[{s}_{21}\left({\widehat{\xi}}_2-\frac{1}{\sigma -1}{\widehat{w}}_1\right)\right]$. The price channel now contains an additional term in square brackets, which exactly offsets the variety effect.

1.2.3 Truncated Pareto Distribution

Let b _Li and b _Hi be the lower and higher bounds for the truncated Pareto distribution for country i. Then the country’s CDF is given by ${G}_i(z)=\frac{1-{b_{Li}}^{\theta_i}{z}^{-{\theta}_i}}{1-{b_{Li}}^{\theta_i}{b_{Hi}}^{-{\theta}_i}}$. For this case, the relation between average productivity and the cut-off level can be derived as ${\tilde{z}}_{ij}={\left\{\left(\frac{\theta_i}{\theta_i-\sigma +1}\right)\left[\frac{1}{{\left({z}_{ij}^{*}\right)}^{-{\theta}_i}-{\left({b}_{Hi}\right)}^{-{\theta}_i}}\right]\left[{\left({z}_{ij}^{*}\right)}^{-{\theta}_i+\sigma -1}-{\left({b}_{Hi}\right)}^{-{\theta}_i+\sigma -1}\right]\right\}}^{\frac{1}{\sigma -1}}$. Totally differentiating this relation and (1) and using the resulting expressions, we can express

$$ {\widehat{\tilde{z}}}_{ij}={\alpha}_{ij}{\widehat{\theta}}_i+{\beta}_{ij}{\widehat{z}}_{ij}^{*}={\alpha}_{ij}{\widehat{\theta}}_i+{\beta}_{ij}\left(\frac{\sigma }{\sigma -1}{\widehat{w}}_i-{\widehat{P}}_j-\frac{1}{\sigma -1}{\widehat{w}}_j\right), $$

(A10)

where the coefficients α _ij and β _ij depend on the initial values of θ _i and ${z}_{ij}^{*}$. Using (A10) to evaluate ${\displaystyle \sum_{i=1,2}{s}_{ij}{\widehat{\tilde{z}}}_{ij}}$ and substituting this value in (14), we can determine the net effect of the variety channel for the home country (j = 1,${\widehat{w}}_2=0$, ${\widehat{\theta}}_1=0$) as $\frac{s_{11}}{\sigma -1}{\widehat{M}}_{11}+\frac{s_{21}}{\sigma -1}{\widehat{M}}_{21}=-\left(\overline{\beta}-1\right)\left({\widehat{w}}_1-{\widehat{P}}_1\right)-{s}_{21}\left(1-\frac{\sigma {\beta}_{21}}{\sigma -1}\right){\widehat{w}}_1-{s}_{21}{\alpha}_{21}{\widehat{\theta}}_2$, where $\overline{\beta}={s}_{11}{\beta}_{11}+{s}_{21}{\beta}_{21}$. Using (A10) again and noting that ${\widehat{z}}_{21}^{*}=-\frac{\sigma }{\sigma -1}{\widehat{w}}_1+{\widehat{z}}_{11}^{*}$, we can derive the net effect of the price channel as ${s}_{11}{\widehat{\tilde{z}}}_{11}+{s}_{21}\left({\widehat{w}}_1+{\widehat{\tilde{z}}}_{21}\right)={\widehat{z}}_{11}^{*}+\left[\left(\overline{\beta}-1\right)\left({\widehat{w}}_1-{\widehat{P}}_1\right)+{s}_{21}\left(1-\frac{\beta_{21}\sigma }{\sigma -1}\right){\widehat{w}}_1+{s}_{21}{\alpha}_{21}{\widehat{\theta}}_2\right]$. The term in the square brackets exactly offsets the net variety effect.

1.3 Section 3.1

1.3.1 Derivation of (19)

Country i’s share of exports in income equals $e{s}_i=\left[\frac{\sigma {M}_i^e}{\delta_i}{\displaystyle {\int}_{z_{12}^{*}}^{\infty}\frac{r_{ij}(z)}{\sigma}\kern0.5em }d{G}_i(z)\right]\div \left[\frac{\sigma {M}_i^e}{\delta_i}{\displaystyle \sum_{j=1,2}{\displaystyle {\int}_{z_{ij}^{*}}^{\infty}\frac{r_{ij}(z)}{\sigma }}}\kern0.5em d{G}_i(z)\right]$, with i ≠ j in the first square bracket. Using this condition and (A5), and noting that ${J}_i\left({z}_{ij}^{*}\right)+1-{G}_i\left({z}_{ij}^{*}\right)=\frac{\theta_i}{\theta_i-\sigma +1}{\left(\frac{b_i}{z_{ij}^{*}}\right)}^{\theta_i}$, we get (13).

1.3.2 Data

Steady-state values of the relative wage (w ₁) and the labor supply ratio (L ₁/L ₂) are measured, respectively, as weighted averages of GDP per capita (in US dollars, averaged over 2001–2010) and population (also averaged over 2001–2010) of major US trading partners relative to US levels. The trading partners group includes 15 leading US trade partners accounting for 73.4 % of US imports and 71.5 % of US exports.¹⁷ The weights used to compute averages for this group are based on the share of US trade (average of imports and exports) with each partner in the total trade with the group. The source of the data for GDP per capita and population is the World Development Indicators 2011 (from the World Bank) and for trade flows is the US Census Bureau.