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Average income, income inequality and export unit
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Hélène Latzer, Florian Mayneris

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Hélène Latzer, Florian Mayneris. Average income, income inequality and export unit values. 2018. halshs-01901256

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Average income, income inequality
and export unit values

Hélène LATZER, Florian MAYNERIS

2018.31

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Average income, income inequality and export unit values *

Hélène Latzer† Florian Mayneris‡

October 9, 2018

**Abstract**

This paper analyses the relationship between a country's income distribution and its exports' unit values. Using bilateral export flows, we not only confirm the positive relationship between a country's average income and the quality of its exports, but further identify a heterogeneous impact of income inequality: we find a greater income spread to be beneficial for an exporter's unit values in the case of poor countries only. These results are robust to the inclusion of controls for other determinants of export unit values, as well as to the use of alternative measures of income inequality and of the quality index. We finally show that this heterogeneous impact of income inequality along the average income dimension is consistent with models emphasizing the role of the composition of local demand in determining the comparative advantage of countries in terms of quality.

**Keywords:** Income distribution, Export Unit Values, Product quality, Trade, Home market effect.

**JEL classification:** F12, L15, O15.

<footnote>*We are grateful to many colleagues and conference participants for very useful discussions on various versions of this paper. We also thank the Belgian French-speaking Community (Convention ARC 09/14-019 on “Geographical mobility of factors”) for financial support.</footnote>

<footnote>†Université Paris 1-Sorbonne, CNRS (France) and CEREC, Université Saint-Louis (Belgium). E-mail: helene.latzer@univ-paris1.fr.</footnote>

<footnote>‡Université du Québec à Montréal (Canada). E-mail: mayneris.florian@uqam.ca.</footnote>
# 1 Introduction

Unit values are generally considered as a proxy for quality. As a consequence, the investigation of unit values' determinants has attracted a lot of attention over the past fifteen years, since it brings insights on what drives the position of a country (or a firm) along the quality ladder.

The existing literature already explored the impact of several exporter characteristics on a country's exported quality, such as technology and/or relative factor abundance (see, among others, Schott, 2004; Fieler, 2011a,b), credit constraints (Lugovskyy and Choi, 2018) or average income and geography through a home market effect for quality (Lugovskyy and Skiba, 2015; Dingel, 2017). This paper goes further by investigating the relationship between a country's income distribution, i.e. both average income and income inequality, and the unit value of its exports. Using bilateral export flows at the 6-digit product level for around 100 countries, we confirm that unit values increase significantly with the average income of the exporting country. We then highlight on the other hand a heterogeneous relationship between unit values and income inequality along the income per capita dimension: controlling for average income, a greater income spread is positively related to export unit values in poor countries only, and negatively related to export unit values in richer ones.

Those results are robust to the inclusion of several other determinants of a country's export unit values, including the exporter's level of human capital, aggregate TFP, measures of trade openness and revealed comparative advantage, as well as the bilateral distance between the exporting and the importing country (so as to control for a potential Alchian-Allen effect). We also proceed to several robustness checks, using alternative measures of income inequality and of the quality index. More precisely, our results hold when using the share of income accruing to the top 10% and to the top 20% of the population as alternatives to the Gini index; they also come through when using a country-sector quality measure developed by Feenstra and Romalis (2014) instead of export unit values as a proxy for the quality of exports.

We then show that the highly non-linear relationship between inequality and export unit values that we identify in our empirical investigation is consistent with theoretical models where the composition of domestic demand drives the specialisation of countries in terms of quality. This literature, along with a conjecture first formulated by Linder (1961), postulates an impact of the composition of aggregate demand on the trade patterns of vertically differentiated varieties, emphasizing the existence of a "vertical home market effect" (Fajgelbaum et al., 2011). The essence of the mechanism is simple: similarly to what had been identified by the economic geography literature in the case of trade in horizontally differentiated varieties, production is expected to follow demand in presence of economies of scale and positive trade costs. The only difference is that when preferences are non-homothetic, income distribution (i.e. average income and income inequality), and not only total income, is affecting the relative size of domestic demand for high- and low-
quality varieties, and thus influences the specialisation of countries in terms of quality. So
as to illustrate this mechanism, we provide a model of intra-industrial trade in vertically
differentiated varieties that displays such a vertical home market effect,<fnref n="1" /> determining in
particular the conditions under which the quality content of exports is positively linked to
the level of income inequality in the exporting country. The key intuition is as follows:
for a given average income, an increase in inequality means both more rich and more poor
consumers; the net effect on the overall demand for quality is then a priori undetermined
and depends on how the demand for high-quality varieties evolves along income for both
income groups. It is then possible to demonstrate that the quality content of production
and exports increases unambiguously with income inequality *only* when the income share
devoted to high-quality goods is increasing and convex along income for both rich and poor
consumers; this condition is respected only for low enough values of average income (i.e.
in the case of poor countries).<fnref n="2" />

Our paper is related to several strands of the literature. As already mentioned above,
it first contributes to the literature on the determinants of a country's export unit values.
Among numerous contributions, Schott (2004), Verhoogen (2008) and Fieler (2011a,b) have
exemplified the impact of exporters' production technology and relative factor abundance;
Hummels and Skiba (2004) and Lugovskyy and Skiba (2016) discuss the effect of distance
between the trading partners on the quality content of trade, while Lugovskyy and Choi
(2018) investigate the impact of credit constraints on export prices. Finally, Dingel (2017)
uses plant-level data on manufacturing shipments to identify a home-market effect driving
the specialisation of U.S. cities in terms of quality; he shows that high-income locations
export high-quality products and that the home-market effect explains as much of this
relationship as differences in factor usage. Our paper confirms this positive relationship
between average income and the quality content of exports, and provides a further inves-
tigation of the role of income inequality (controlling for average income).

Our paper is also related to the empirical and theoretical literature on the demand-
based determinants of the quality content of trade flows. Hallak (2006) shows that richer
countries tend to import higher-price/higher-quality varieties from high income countries,
while Choi et al. (2009) show that countries displaying similar income distributions tend
to exhibit similar distributions of import prices. Bekkers et al. (2012) as well as Flach and
Janeba (2017) investigate the impact of a country's income inequality level on its import
unit values. In this paper, we instead focus on the export side and deal with the impact
of a country's income distribution on the quality of its *exports*.

The rest of the paper is structured as follows. Our data and empirical results are
presented in sections 2 and 3. We then provide a theoretical framework to rationalize these
facts in section 4 while section 5 concludes.

<footnote n="1">¹In our model, such an effect is obtained assuming non-homothetic preferences and love-for-variety at the consumer's level *a la* Hallak (2006).</footnote>

<footnote n="2">²Such a result is also in line with the predictions obtained by Fajgelbaum et al. (2011) regarding the impact of inequality on the quality content of exports in a unit consumption framework.</footnote>
# 2 Data and empirical strategy

We present in this section the data we use and the empirical strategy we follow to describe the relationship between a country's income distribution and the unit value of the goods it exports.

## 2.1 Data

To conduct our empirical analysis, we need information on countries's exports at a detailed level of the product nomenclature, on their average income and income inequality, and on various other country-level characteristics that could be correlated with both their income distribution and the unit value of their exports.

For information on trade flows, we use the Comtrade data for the years 2006, 2008, 2010 and 2012.<fnref n="3" /> The data records all bilateral trade flows at the 6-digit level of the HS nomenclature, in value (dollars) and in volume (tons). As commonly done in the literature, we clean the data and drop those flows for which the declared value is below 10,000 dollars or whose information on value or quantity is 0 or missing. Moreover, it is well known that information on unit values can be particularly noisy. For a given product and a given year, we thus drop all the flows whose unit value is below the 5th percentile or above the 95th percentile observed across all the flows in the data for that product and that year. Regarding income distribution in the exporting country, we use the data from the World Development Indicators of the World Bank. Population and total GDP allow us to compute the GDP per capita of each country. We complement this information with three measures of income inequality: the Gini index and the share of income accruing to the top 10% and the top 20% of the population in terms of income. Other characteristics of the exporting countries are taken from the Penn World Table, in particular country-level TFP and average human capital of the population (measured by an index based on years of schooling and returns to education). Finally, information on distance between countries is taken from a CEPII database.<fnref n="4" />

Note that the data on income inequality notoriously suffers from many missing observations. To limit this issue, we smoothe all the country-level information by computing moving averages of all the variables across three years (the current year and the years before and after).<fnref n="5" /> In the end, 99 exporting countries are present for at least one year in our regression sample.<fnref n="6" />

<footnote n="3">³We do not keep all the years because of the very high number of observations. Since all our regressions exploit cross-sectional variations in the data, this is not an issue for the estimation.</footnote>

<footnote n="4">⁴"dist_cepii" database, available online at http://www.cepii.fr/francgraph/bdd/distances.htm.</footnote>

<footnote n="5">⁵For example, in our data, the Gini index is available for 74 countries only in 2006. If we take the average Gini index for the year 2005, 2006 and 2007, the information becomes available for 106 countries.</footnote>

<footnote n="6">⁶See Table 1 in Appendix for the entire list.</footnote>
## 2.2 Estimated equation

We want to relate the unit value of exports to the income distribution in the exporting country. We propose two types of regressions. We first run the analysis at the exporter-importer-product-year level and estimate the following equation:

$$uv_{xmpt} = \alpha X_{x(p)t} + \beta Z_{xmt} + \mu_{mpt} + \epsilon_{xmpt}$$

where $uv_{xmpt}$ is the log unit value of exports of product $p$ by country $x$ to country $m$ at time $t$, $X_{x(p)t}$ is a vector of exporter(-product)-year characteristics, $Z_{xmt}$ is a set of bilateral determinants of unit values, $\mu_{mpt}$ is an importer-product-year fixed effect and $\epsilon_{xmpt}$ is the error term. Given the presence of the fixed effect $\mu_{mpt}$, the effect of the other variables is estimated by comparing, for a given importer and a given product in a given year, the unit values of the varieties coming from different source countries.

The vector $X_{x(p)t}$ contains the following variables: log income per capita, log population, a measure of income inequality, the level of human capital, aggregate TFP and a measure of trade openness in the exporting country. We also introduce the share of product $p$ in the exports of country $x$ in year $t$ as a measure of specialization (and thus revealed comparative advantage) of country $x$ in the production and exports of product $p$.

In line with previous evidence, we expect the log income per capita to be positively related to unit values since richer countries have been shown to have a comparative advantage in high-quality and thus high-price varieties (due to both supply- and demand- side mechanisms, as emphasized, e.g., in Schott, 2004; Dingel, 2017). The prior is less clear for population. Fajgelbaum et al. (2011) show that an increase in population increases disproportionately the number of varieties that are more horizontally differentiated. Since we can reasonably think that high-quality varieties are more differentiated than low-quality ones, this argument points at a possible positive correlation between export unit values and population size. On the other hand, Desmet and Parente (2010) show that bigger markets exhibit lower markups and consequently bigger firms, which favors process innovation. This could lead, all else equal, to lower prices in bigger countries. Given these conflicting theoretical insights, we have no prior on the empirical correlation between unit values and population. The same applies to human capital and TFP. Indeed, most models of trade with vertical differentiation generally assume that a more educated workforce and/or more productive firms allow for a more efficient production and thus cheaper varieties, as well as for the production of higher-quality and thus more expensive varieties (see, e.g. Verhoogen, 2008; Hallak and Sivadasan, 2013; Eckel et al., 2015). Finally, we introduce the ratio of total exports plus imports over GDP as a measure of trade openness. This control has two main purposes. First, trade openness might directly affect the quality content of exports. Indeed, trade models with firm heterogeneity in terms of quality highlight a selection mechanism where high-productivity/high-quality firms are the most likely to export. In such models, the quality produced by the marginal exporter decreases as trade becomes easier (Crozet et al., 2012; Johnson, 2012); this selection mechanism then generates a negative
correlation between trade openness and the average quality of the export basket. Second, trade openness might not only affect the quality content of exports but also earnings inequality both across and within firms, which could then distort the correlation we capture between income inequality and export unit values. This effect channels through: i) an increase in the wages paid by internationally active (and thus more profitable) firms as compared to domestic ones (Amiti and Davis, 2012; Helpman et al., 2017), ii) a skill-biased technology and quality upgrading, the firms in low- and middle-income countries needing to adapt their processes and the quality content of their production so as to serve the richer countries when they open to trade (Verhoogen, 2008; Bustos, 2011; Bas, 2012; Iacovone and Javorcik, 2012).

The set of bilateral determinants of unit values $Z_{xmt}$ only contains the bilateral distance between the exporting and the importing country. It is now well established that unit values of exports (both at the country-product and firm-product level) are positively related to the bilateral distance between the trading countries (Hummels and Skiba, 2004; Baldwin and Harrigan, 2011; Bastos and Silva, 2010; Martin, 2012; Manova and Zhang, 2012). This feature is in line with the so called Alchian-Allen effects where high quality varieties tend to be relatively more exported to distant countries. This effect might arise due to the selection of high-quality firms in more difficult destinations (Johnson, 2012), or to a demand-side mechanism when part of the trade costs are additive and thus relatively less important for more expensive varieties (for a thorough discussion of the Alchian-Allen effect and its relationship with export unit values, see Lugovskyy and Skiba, 2016). We thus expect the coefficient on distance to be positive in our regressions.

We present in Table 2 in Appendix A a correlation table for all the variables we take into account in our empirical analysis. The variable of interest in these regressions will be the proxy for income inequality. We will show that our results are robust to the use of three alternative measures of income inequality: the Gini index and the share of income in the hands of the top 10% and top 20% of the population in the income distribution.

Besides this analysis of the determinants of export unit values at the exporter-importer-product-year level, we replicate the analysis at the exporter-product-year level. Indeed, if we assume that countries produce and export only one type of quality or export the same quality mix to all the destination countries (which is of course an extreme assumption), keeping the importer dimension “artificially” duplicates the observations. To avoid this issue, we first regress the log unit value $uv_{xmpt}$ on exporter-product-year and importer-product-year fixed effects, as well as on the log bilateral distance between the exporting and the importing countries; we then retrieve the exporter-product-year fixed effect and use it as a dependent variable in a regression similar to equation 2.2. The only difference is that the log distance is removed from the regressors and the fixed effect $\mu_{mpt}$ is replaced by a fixed effect in the product-year dimension. This amounts to comparing for a given product in a given year the average export unit value of the various countries exporting that product (net of the effect of distance and importing country characteristics).

Finally, our dependent variable is exporter(-importer-)product-year specific, while our
variables of interest are exporter-year specific. According to Moulton (1990), standard-
errors of the coefficients on exporter-year characteristics might consequently be downward-
biased. To correct for this, we cluster all the regressions at the exporter-year level.

3 Results

In this section, we first present some descriptive statistics on average income and income inequality in the exporting countries present in our sample. We then detail our baseline results and provide several additional robustness checks.

3.1 Descriptive statistics

Table 1 shows there is a great heterogeneity in our sample both in terms of average income and income inequality. While average income is equal to 18990$, the median is much lower at 9220$ over the period under study. This shows that the distribution is very much right skewed, which is reflected in a relatively high standard deviation as compared to the mean (and thus a high coefficient of variation). Based on the categories established by the World Bank in 2006 (first year in our sample), there are 19 low-, 53 middle- and 27 high- income countries in our sample. Regarding income inequality, the heterogeneity is also high even though less massive than the one observed for average income (the coefficient of variation is respectively equal to 23.5%, 23.3% and 17.4% for the Gini index and the share of income accruing to the top 10% and the top 20%).

<table><caption>Table 1: Average income and income inequality in the estimation sample</caption>
  <thead>
    <tr>
      <th></th>
      <th>Mean</th>
      <th>Med</th>
      <th>Min</th>
      <th>Max</th>
      <th>Sd</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Average income</td>
      <td>18991.18</td>
      <td>9219.21</td>
      <td>162.62</td>
      <td>112028.1</td>
      <td>22191.01</td>
    </tr>
    <tr>
      <td>Gini</td>
      <td>37.90</td>
      <td>35.5</td>
      <td>24.3</td>
      <td>64.8</td>
      <td>8.92</td>
    </tr>
    <tr>
      <td>Share of top 10%</td>
      <td>29.77</td>
      <td>27.37</td>
      <td>20.37</td>
      <td>54.2</td>
      <td>6.95</td>
    </tr>
    <tr>
      <td>Share of top 20%</td>
      <td>44.89</td>
      <td>42.5</td>
      <td>15.8</td>
      <td>71</td>
      <td>7.82</td>
    </tr>
  </tbody>
</table>

Figure 1 plots our three measures of income inequality against the log of average income. The picture is very similar for the three measures:<fnref n="7" /> there is a slight negative correlation between average income and income inequality (however never bigger than 0.35 in absolute value in the data), but overall, for any level of income inequality, we observe countries with very different levels of average income. We thus clearly have enough variation in the data to disentangle the relationship between export unit values and these two dimensions of the income distribution in the exporting country.

<footnote n="7">⁷Which is not surprising since the pairwise correlation between any two of the three measures is above 94% (unreported computations available upon request).</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 1: Average income and income inequality in 2006</figcaption></figure>

## 3.2 Export prices and income distribution within the exporting country

Table 2 displays the results when using bilateral unit values as the dependent variable and the Gini index as the measure of income inequality. All regressions include importer-product-year fixed effects. In column (1), we only include the income distribution in the exporting country on the top of the fixed effects; income per capita is positively and significantly correlated with export unit values, while inequality is on the opposite slightly negatively correlated with export prices. We add several other export characteristics in column (2), and our results are all in line with the intuition or previous literature. Bigger countries tend to export cheaper varieties, in line with the theoretical intuition in Desmet and Parente (2010). More educated countries export on the opposite more expensive, and thus certainly higher quality varieties. Aggregate TFP is positively but not significantly correlated with export unit values. Our measure of trade openness is significantly negatively correlated with export unit values, which lends support to the idea that greater openness to trade allows low price/low quality producers to enter the export markets (Feenstra and Romalis, 2014). Specialization is not significantly related to export unit values. This is due to the fact that countries with a comparative advantage in a given product are certainly both more efficient for a given quality-level, which allows them to sell their varieties at a lower price, but also more able to produce and export high-quality varieties, which plays in the opposite direction in terms of export unit values. Finally, more expensive varieties tend to be shipped to more distant countries, which is coherent with the extensive literature on the Alchian-Allen effect we already cited. Controlling for all of these covariates, average income is still positively and very robustly associated with export unit values, but the
<table><caption>Table 2: Bilateral export unit values and exporter characteristics</caption><thead><tr><th></th><th colspan="4">Ln uv<sub>xmpt</sub></th></tr><tr><th></th><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th></tr></thead><tbody><tr><td>Ln GDP per cap.<sub>xt</sub></td><td>0.182<sup>a</sup><br>(0.019)</td><td>0.132<sup>a</sup><br>(0.016)</td><td>0.372<sup>a</sup><br>(0.063)</td><td>0.201<sup>a</sup><br>(0.026)</td></tr><tr><td>Gini<sub>xt</sub></td><td>-0.005<sup>b</sup><br>(0.002)</td><td>-0.005<sup>c</sup><br>(0.003)</td><td>0.062<sup>a</sup><br>(0.017)</td><td>0.014<sup>a</sup><br>(0.004)</td></tr><tr><td>Gini<sub>xt</sub> × Ln GDP per cap.<sub>xt</sub></td><td></td><td></td><td>-0.007<sup>a</sup><br>(0.002)</td><td></td></tr><tr><td>Gini<sub>xt</sub> × Middle inc. country<sub>x</sub></td><td></td><td></td><td></td><td>-0.014<sup>a</sup><br>(0.003)</td></tr><tr><td>Gini<sub>xt</sub> × High inc. country<sub>x</sub></td><td></td><td></td><td></td><td>-0.016<sup>a</sup><br>(0.003)</td></tr><tr><td>Ln Population<sub>xt</sub></td><td></td><td>-0.031<sup>b</sup><br>(0.014)</td><td>-0.035<sup>b</sup><br>(0.015)</td><td>-0.048<sup>a</sup><br>(0.013)</td></tr><tr><td>Human capital index<sub>xt</sub></td><td></td><td>0.094<sup>b</sup><br>(0.038)</td><td>0.130<sup>a</sup><br>(0.033)</td><td>0.109<sup>a</sup><br>(0.038)</td></tr><tr><td>Aggregate TFP<sub>xt</sub></td><td></td><td>0.299<br>(0.352)</td><td>0.414<br>(0.356)</td><td>0.121<br>(0.323)</td></tr><tr><td>Trade openness<sub>xt</sub></td><td></td><td>-0.077<sup>a</sup><br>(0.027)</td><td>-0.107<sup>a</sup><br>(0.027)</td><td>-0.063<sup>b</sup><br>(0.025)</td></tr><tr><td>Share of product <i>p</i> in total exports of <i>x</i></td><td></td><td>0.123<br>(0.322)</td><td>0.135<br>(0.323)</td><td>0.058<br>(0.330)</td></tr><tr><td>Ln Bilateral distance<sub>xm</sub></td><td></td><td>0.059<sup>a</sup><br>(0.005)</td><td>0.060<sup>a</sup><br>(0.005)</td><td>0.062<sup>a</sup><br>(0.005)</td></tr><tr><td>Importer-Product (HS 6-diti)-Year fixed effects</td><td>Yes</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Observations</td><td>10,018,400</td><td>10,018,400</td><td>10,018,400</td><td>10,018,400</td></tr><tr><td>R-squared</td><td>0.862</td><td>0.863</td><td>0.864</td><td>0.864</td></tr></tbody></table>

Standard errors clustered at the exporter-year level.

a p<0.01, b p<0.05, c p<0.1

coefficient is lower as compared to column (1). The coefficient on the Gini index barely changes, but it becomes significant at the 10% level only.

We then test for the conjecture that the weak significance of the Gini index hides a heterogeneous relationship between export unit values and income inequality along average income. This is confirmed by our results in column (3), where we introduce an interaction term between the Gini index and income per capita. The coefficient on the Gini is now positive while its interaction with income is negative and significant: income inequality is positively associated with export unit values in poor countries only, which is further emphasized in column (4), where we interact the Gini index with dummies identifying middle- and high-income countries.<fnref n="8" />

We then replicate the analysis using as the dependent variable the average unit value of exports at the exporting country-product-year level (computed as described in section 2.2). The results presented in Table 3 are remarkably stable, both qualitatively and also for most of the variables quantitatively (only the proxy for specialization is now negatively and significantly related to unit values). Regarding the link between export unit values and income distribution, the main message holds: richer countries export more expensive

<footnote n="8">⁸Poor countries are thus the excluded category, so that the coefficient on the Gini index alone corresponds to the coefficient for these low-income countries.</footnote>
<table><caption>Table 3: Country-product export unit values and exporter characteristics</caption><thead><tr><th></th><th colspan="4">Ln uv<sub>xpt</sub></th></tr><tr><th></th><th>(1)</th><th>(2)</th><th>(3)</th><th>(4)</th></tr></thead><tbody><tr><td>Ln GDP per cap.<sub>xt</sub></td><td>0.150<sup>a</sup><br>(0.012)</td><td>0.129<sup>a</sup><br>(0.009)</td><td>0.293<sup>a</sup><br>(0.038)</td><td>0.175<sup>a</sup><br>(0.017)</td></tr><tr><td>Gini<sub>xt</sub></td><td>-0.005<sup>a</sup><br>(0.001)</td><td>-0.003<sup>c</sup><br>(0.002)</td><td>0.042<sup>a</sup><br>(0.010)</td><td>0.008<sup>a</sup><br>(0.003)</td></tr><tr><td>Gini<sub>xt</sub> × Ln GDP per cap.<sub>xt</sub></td><td></td><td></td><td>-0.005<sup>a</sup><br>(0.001)</td><td></td></tr><tr><td>Gini<sub>xt</sub> × Middle inc. country<sub>x</sub></td><td></td><td></td><td></td><td>-0.009<sup>a</sup><br>(0.001)</td></tr><tr><td>Gini<sub>xt</sub> × High inc. country<sub>x</sub></td><td></td><td></td><td></td><td>-0.011<sup>a</sup><br>(0.002)</td></tr><tr><td>Ln Population<sub>xt</sub></td><td></td><td>-0.039<sup>a</sup><br>(0.008)</td><td>-0.033<sup>a</sup><br>(0.008)</td><td>-0.040<sup>a</sup><br>(0.007)</td></tr><tr><td>Human capital index<sub>xt</sub></td><td></td><td>0.027<br>(0.033)</td><td>0.053<sup>c</sup><br>(0.030)</td><td>0.046<br>(0.035)</td></tr><tr><td>Aggregate TFP<sub>xt</sub></td><td></td><td>0.201<br>(0.178)</td><td>0.276<br>(0.171)</td><td>0.138<br>(0.160)</td></tr><tr><td>Trade openness<sub>xt</sub></td><td></td><td>-0.028<sup>b</sup><br>(0.013)</td><td>-0.034<sup>a</sup><br>(0.013)</td><td>-0.030<sup>b</sup><br>(0.013)</td></tr><tr><td>Share of product <i>p</i> in total exports of <i>x</i></td><td></td><td>-0.570<sup>a</sup><br>(0.213)</td><td>-0.618<sup>a</sup><br>(0.216)</td><td>-0.687<sup>a</sup><br>(0.206)</td></tr><tr><td>Product (HS 6-diti)-Year fixed effects</td><td>Yes</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Observations</td><td>530,308</td><td>530,308</td><td>530,308</td><td>530,308</td></tr><tr><td>R-squared</td><td>0.152</td><td>0.158</td><td>0.161</td><td>0.164</td></tr></tbody></table>

Standard errors clustered at the exporter-year level.

a p<0.01, b p<0.05, c p<0.1

varieties, while income inequality is positively related to unit values, but in poor countries only. For the rest of the empirical analysis, we stick to exporter-product-year regressions.

## 3.3 Quantitative assessment

To get a sense of how much average income and inequality contribute to the variations in export unit values observed across countries, we use the regression coefficients presented in Table 3 and compute the percentage change in unit values associated with a one standard deviation increase in both variables. We find that for the average country in our sample in terms of average income and Gini index, a one standard deviation increase in average income is associated with a 15.7% increase in export unit values, while a one standard deviation increase in the Gini coefficient is associated with a 1.4% decrease in unit values. However, the contribution of income inequality varies a lot across countries, from -9.6% for the average rich country to +8.9% for the average poor country in our sample.<fnref n="9" />

Average income is thus the main dimension of income distribution to drive export unit values; however, an important message of this quantification exercise is that income inequality is clearly not a negligible dimension for both rich and poor countries.

<footnote n="9">⁹The picture is qualitatively similar when using the share of income accruing to the top 10% and the top 20% as measures of income inequality.</footnote>
<table><caption>Table 4: Country-product export unit values and exporter characteristics - Various measures of income inequality</caption><thead><tr><th></th><th colspan="3">Ln uv<sub>xpt</sub></th></tr><tr><th></th><th>(1)</th><th>(2)</th><th>(3)</th></tr></thead><tbody><tr><td>Ln GDP per cap.<sub>xt</sub></td><td>0.293<sup>a</sup><br>(0.038)</td><td>0.318<sup>a</sup><br>(0.042)</td><td>0.433<sup>a</sup><br>(0.060)</td></tr><tr><td>Gini<sub>xt</sub></td><td>0.042<sup>a</sup><br>(0.010)</td><td></td><td></td></tr><tr><td>Gini<sub>xt</sub> × Ln GDP per cap.<sub>xt</sub></td><td>-0.005<sup>a</sup><br>(0.001)</td><td></td><td></td></tr><tr><td>Share of inc. owned by top 10</td><td></td><td>0.063<sup>a</sup><br>(0.014)</td><td></td></tr><tr><td>Share of inc. owned by top 10 × GDP per cap. (Ln)</td><td></td><td>-0.007<sup>a</sup><br>(0.002)</td><td></td></tr><tr><td>Share of inc. owned by top 20</td><td></td><td></td><td>0.066<sup>a</sup><br>(0.013)</td></tr><tr><td>Share of inc. owned by top 20 × GDP per cap. (Ln)</td><td></td><td></td><td>-0.007<sup>a</sup><br>(0.001)</td></tr><tr><td>Ln Population<sub>xt</sub></td><td>-0.033<sup>a</sup><br>(0.008)</td><td>-0.034<sup>a</sup><br>(0.008)</td><td>-0.032<sup>a</sup><br>(0.008)</td></tr><tr><td>Human capital index<sub>xt</sub></td><td>0.053<sup>c</sup><br>(0.030)</td><td>0.062<sup>b</sup><br>(0.030)</td><td>0.070<sup>b</sup><br>(0.029)</td></tr><tr><td>Aggregate TFP<sub>xt</sub></td><td>0.276<br>(0.171)</td><td>0.298<sup>c</sup><br>(0.169)</td><td>0.345<sup>b</sup><br>(0.163)</td></tr><tr><td>Trade openness<sub>xt</sub></td><td>-0.034<sup>a</sup><br>(0.013)</td><td>-0.027<sup>b</sup><br>(0.013)</td><td>-0.027<sup>b</sup><br>(0.013)</td></tr><tr><td>Share of product <i>p</i> in total exports of <i>x</i></td><td>-0.618<sup>a</sup><br>(0.216)</td><td>-0.628<sup>a</sup><br>(0.218)</td><td>-0.632<sup>a</sup><br>(0.219)</td></tr><tr><td>Product (HS 6-diti)-Year fixed effects</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Observations</td><td>530,308</td><td>530,308</td><td>530,308</td></tr><tr><td>R-squared</td><td>0.161</td><td>0.160</td><td>0.161</td></tr></tbody></table>

Standard errors clustered at the exporter-year level.

<sup>a</sup> p<0.01, <sup>b</sup> p<0.05, <sup>c</sup> p<0.1

## 3.4 Robustness checks

In this section, we show that the heterogeneous relationship between unit values and income inequality along average income is robust to several checks.

First, our results are not sensitive to the income inequality measure we use. We propose two alternative proxies: the share of income accruing to the top 10% as well as to the top 20% of the population. For the sake of brevity, we only reproduce in Table 4 the regression with the interaction term between average income and inequality. Whatever the measure of income inequality, the non-linear relationship is there. Quite interestingly, the estimated level of income above which income inequality stops being positively related to export unit values is quite similar across the three specifications. In log, it is comprised between 8.5 and 9, i.e. between 5,200 and 8,400$ approximately.<fnref n="10" />

If export unit values are a proxy for quality, we can expect the relationship between income distribution and unit values to be less intense for products that are less differentiated. We use the elasticities of substitution estimated by Imbs and Mejean (2015) as a

<footnote n="10">¹⁰Given the specification, this income threshold is equal (in log) to $\frac{\text{coef. inc. ineq.}}{\text{coef. inc. ineq.} \times \text{avg inc.}}$. Based on our estimations, the smallest value is obtained for the Gini index, and the biggest one for the share of income accruing to the top 20% of the population.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 2: Marginal effect of the Gini index</figcaption></figure>

measure of product differentiation. We define as non-differentiated products those whose elasticity of substitution is above 6.5.<fnref n="11" /> We estimate the regression in column (1) of Table 3 for these highly substitutable products. We do the same for the products defined as (strictly) homogeneous in the Rauch classification. Based on the coefficients we obtain, we can compute the marginal effect of the Gini index at each income level. Figure 2 presents the estimated marginal effect of the Gini index for all the products, highly elastic ones and (Rauch-based) homogeneous ones. At each income level, the marginal effect of income inequality is lower in absolute value for highly elastic and homogeneous products than for the entire sample of products; moreover, the slope of the marginal effect is lower in absolute value for these less differentiated products. We provide in Appendix A the same graphs computed with the share of income owned by the top 10% and the top 20% respectively: the picture we obtain is very much the same. This shows that income distribution relates to export unit values for the products that are more vertically differentiated, corroborating the interpretation in terms of quality.

Still, we know that the correlation between unit values and quality is not perfect, other factors such as the production costs or the exchange rate determining export prices. This is why several procedures have been developed to recover quality measures from trade data (Khandelwal, 2010; Hallak and Schott, 2011; Khandelwal et al., 2013). Khandelwal (2010) in particular shows that the shorter the product quality ladder, the lower the correlation between unit values and his quality index. The derivation of these indexes mostly relies on

<footnote n="11">¹¹6.5 corresponds to the 9th decile in the distribution of the elasticities estimated by Imbs and Mejean (2015).</footnote>
the demand side, following the intuition that conditional on price, varieties with a higher market share must be higher-quality varieties. Feenstra and Romalis (2014) propose a framework with a richer supply side (where the so-called Alchian-Allen effect is taken into account as well as the selection of high-productivity/high-quality firms into exporting) and where the demand side allows for non-homothetic preferences. They perform the analysis at the SITC 4-digit level. Their estimated country-sector quality indices are available online. We use them for the period 2006-2011 (the year 2012 is not available) as a dependent variable to reproduce the analysis of section 3.2.

For the purpose of brevity, we directly reproduce the specification with all the controls and the interaction term between average income and inequality. As can be seen in Table 3.4, all the results go through: income per capita in the exporting country is positively related to the quality of exports, while the effect of income inequality is highly non linear, positive and significant in poor countries only.

Among the controls, human capital and aggregate TFP are positively related to the quality of exports (even though the coefficient is not significant for TFP). Quite interestingly, it seems that the countries that are more open to trade tend to export lower quality varieties. This is actually well in line with the prediction in Feenstra and Romalis (2014): when trade becomes easier, more firms enter the markets, and the marginal exporter tends to export lower quality varieties.

Finally, we can note that given the coefficients we obtain on income inequality and its interaction with the log GDP per capita, the average income above which quality and income inequality are not positively related anymore is estimated to lie between 10,000 and 11,000$, i.e. a bit above the estimated value when using export unit values as the dependent variable;<fnref n="12" /> it is still in the range of the average income values we observe in middle-income countries.

## 3.5 Discussion

All of the results presented above highlight a robust non-linear relationship between average unit value/quality content of exports and income inequality in the exporting country. Controlling for average income, income inequality is positively related to the quality content of exports in poor countries only. Ideally, to push further the causal inference on the relationship between income distribution and the quality content of exports, one would like to exploit exogenous variations of average income and income inequality. However, variables that effect average income and income inequality without affecting directly the quality of production are not easy to find.

Still, we believe we control for a great deal of possible supply-side confounding factors: the skills of the workforce (through the human capital index), aggregate productivity (through TFP) and product-specific comparative advantage (through the specialization index). We also control for trade openness of the exporting country, so that we are confi-

<footnote n="12">¹²Note however that the results are not directly comparable since the product nomenclature and level of aggregation is different across both exercises.</footnote>
<table><caption>Table 5: Quality of exports and exporter characteristics</caption><thead><tr><th></th><th colspan="3">Feenstra-Romalis quality index<sub>xpt</sub></th></tr><tr><th></th><th>(1)</th><th>(2)</th><th>(3)</th></tr></thead><tbody><tr><td>Ln GDP per cap.<sub>xt</sub></td><td>0.191<sup>a</sup><br>(0.050)</td><td>0.227<sup>a</sup><br>(0.051)</td><td>0.307<sup>a</sup><br>(0.074)</td></tr><tr><td>Gini<sub>xt</sub></td><td>0.031<sup>b</sup><br>(0.013)</td><td></td><td></td></tr><tr><td>Gini<sub>xt</sub> × Ln GDP per cap.<sub>xt</sub></td><td>-0.003<sup>b</sup><br>(0.001)</td><td></td><td></td></tr><tr><td>Share of inc. owned by top 10%<sub>xt</sub></td><td></td><td>0.051<sup>a</sup><br>(0.016)</td><td></td></tr><tr><td>Share of inc. owned by top 10%<sub>xt</sub> × Ln GDP per cap.<sub>xt</sub></td><td></td><td>-0.005<sup>a</sup><br>(0.002)</td><td></td></tr><tr><td>Share of inc. owned by top 20%<sub>xt</sub></td><td></td><td></td><td>0.050<sup>a</sup><br>(0.015)</td></tr><tr><td>Share of inc. owned by top 20%<sub>xt</sub> × Ln GDP per cap.<sub>xt</sub></td><td></td><td></td><td>-0.005<sup>a</sup><br>(0.002)</td></tr><tr><td>Ln Population<sub>xt</sub></td><td>-0.072<sup>a</sup><br>(0.009)</td><td>-0.071<sup>a</sup><br>(0.009)</td><td>-0.070<sup>a</sup><br>(0.009)</td></tr><tr><td>Human capital index<sub>xt</sub></td><td>0.154<sup>a</sup><br>(0.037)</td><td>0.159<sup>a</sup><br>(0.037)</td><td>0.162<sup>a</sup><br>(0.037)</td></tr><tr><td>Aggregate TFP<sub>xt</sub></td><td>0.366<br>(0.258)</td><td>0.365<br>(0.255)</td><td>0.367<br>(0.254)</td></tr><tr><td>Trade openness<sub>xt</sub></td><td>-0.354<sup>a</sup><br>(0.033)</td><td>-0.350<sup>a</sup><br>(0.033)</td><td>-0.352<sup>a</sup><br>(0.033)</td></tr><tr><td>Share of product in total exports<sub>xpt</sub></td><td>-0.282<br>(0.507)</td><td>-0.293<br>(0.506)</td><td>-0.290<br>(0.506)</td></tr><tr><td>Product (SITC 4-digit)-Year fixed effects</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Observations</td><td>236,008</td><td>236,008</td><td>236,008</td></tr><tr><td>R-squared</td><td>0.180</td><td>0.180</td><td>0.180</td></tr></tbody></table>

Standard errors clustered at the exporter-year level.

<footnote>a p<0.01, b p<0.05, c p<0.1</footnote>
dent in the fact that the correlation we capture between income inequality and the quality content of exports is not driven by the effect of trade openness on quality upgrading and then *earnings* inequality.

Regarding this latter aspect, there are several reasons why we can reasonably think that our estimations are not plagued by reverse causality. First, the theoretical and empirical literature that emphasized the effect of globalization on quality upgrading and inequality considers *earnings* inequality. However, in our data, income inequality measures are based on the total disposable income of households, calculated by adding together the personal income received by all the household members plus the income received at the household level. It encompasses earnings from work, but also private income from investment and property, transfers between households and all social transfers received in cash including old-age and pensions. Therefore, our measures of income and income inequality goes beyond wage and wage inequality. Actually, based on data from the International Labor Organization for 95 developed and developing economies, we find that the median ratio of the white collar to the blue collar earnings is equal to 2.04.<fnref n="13" /> As a comparison, the median ratio of the average income of people in the top 10% to the average income of the rest of the population is much higher in our sample, equal to 3.39 (2.96 for the ratio of the top 20%). The figures are even more striking if we consider the income ratio of the top to bottom 10% and 20%, whose median value is respectively equal to 10.38 and 3.59. This is not surprising since as emphasized by Atkinson et al. (2011), “aggregate economic growth per capita and Gini inequality indexes are sensitive to excluding or including top incomes”. They show that top incomes play a key role in the evolution of inequality in the past decades, the evolution of top incomes themselves being mainly driven by top managers’ and CEOs’ wages in some countries, and by capital income in other countries (in Scandinavia in particular). In addition, Philippon and Reshef (2013) point at the role of the financial sector in the evolution of wages and inequality. Consequently, in our data, wages and wage inequality in industries that export goods are certainly not the main drivers of our measures of income distribution; the risk of reverse causation between the quality content of exports and our measure of income inequality is thus limited.

# 4 Non-homothetic preferences and quality of exports - a theoretical framework

To rationalize our empirical findings, we now present a theoretical framework that replicates the main empirical regularities we identified. We propose a demand-based explanation for the relationship between income distribution and the quality content of exports. The results we obtain are highly reminiscent of those obtained by Fajgelbaum et al. (2011); the two frameworks however differ in the way the non-homotheticity of consumers' preferences

<footnote n="13">¹³We consider as white collars the managers, the professionals and the technicians and associate professionals. We define as blue collars the craft and related trades workers, the plant and machine operators and assemblers, and the elementary occupations.</footnote>
(which is necessary so as to obtain an impact of income distribution on the composition of aggregate demand) is modelled.<fnref n="14" /> In particular, our alternative framework enables us to derive conditions under which income inequality positively affects the quality of exports that are intuitively related to the shape of the Engel curve for high-quality varieties.

We model international trade between a domestic (D) and a foreign (F) country. Each country features a two-class society with $N_r$ ($r = D, F$) consumers differing in their effective labor endowment, and hence belonging either to a poor (P) or a rich (R) class. The extent of inequality within each economy is determined by the share $\beta_r$ of poor consumers within the population and by the distribution of the aggregate amount of effective labor supply $L$ available in the economy.<fnref n="15" /> $\theta_r \in (0, 1)$ is defined as the ratio of a poor consumer's labor supply $l_{Pr}$ relative to the average per-capita labor supply $L/N_r$: $\theta_r = \frac{l_{Pr}}{L/N_r}$. As $\theta_r$ gets closer to 1, the level of inequality within the economy $r$ diminishes. Given $\theta_r$, it is possible to compute the labor supply of respectively a poor and a rich consumer in country $r$ as $l_{Pr} = \theta_r \frac{L}{N_r}$ and $l_{Rr} = \frac{1-\beta_r \theta_r L}{1-\beta_r N_r}$. In this framework, a mean-preserving increase in the level of inequality corresponds to a decrease in $\theta_r$, while an increase in the average income, leaving the level of inequality unchanged, corresponds to a decrease in $N_r$.

The utility of a type $i$ ($i = P, R$) consumer living in country $r$ is assumed to be of the form:

$$U_{ir} = M_{ir}^{\mu} A_{ir}^{1-\mu} \quad (1)$$

with $M_{ir}$ being an index of consumption of the varieties of a both vertically- and horizontally differentiated good, and $A_{ir}$ being the consumed quantity of a homogenous good. The homogenous good is priced competitively, freely traded, and produced with unit efficient labor requirement, therefore implying that wages equalize across countries and can be normalized to 1. With $n = n_D + n_F$ being the total number of varieties of the differentiated good being produced (i.e. both domestic and foreign), we define $M_{ir}$ as:

$$M_{ir} = \left[ \int_{0}^{n} \left( \gamma_{k}^{\phi_{i}} c_{ir}(k) \right)^{\frac{\sigma-1}{\sigma}} dk \right]^{\frac{\sigma}{\sigma-1}}, \quad \sigma \in (1, +\infty) \quad (2)$$

where $\gamma_k$ and $c_{ir}(k)$ are respectively the quality and the quantity of a variety $k$ consumed by a type $i$ consumer living in country $r$, $\sigma$ is the elasticity of substitution between any two varieties, and $\phi_i$ is a type-specific taste parameter that determines the intensity of preferences for product quality. Along Hallak (2006) and Lugovskyy and Skiba (2015), we assume that $\phi_i$ is a positive function of individual income $l_i$, i.e. that richer households value quality more.

<footnote n="14">¹⁴Indeed, as it will be exposed below, our model features love-for-variety at the individual level, while Fajgelbaum et al. (2011) assume heterogeneous consumers and unit consumption. Such a result equivalence between a framework featuring heterogeneous consumers and unit consumption and models with love-for-variety at the individual level is reminiscent of the one demonstrated by Anderson et al. (1992) in a horizontal framework.</footnote>

<footnote n="15">¹⁵Since we want to neutralize any supply-based variation in the quality mix being produced and exported, we assume that both countries have the same fixed amount of overall labor supply $L$.</footnote>
Consumers use two-stage budgeting. A type i consumer living in country r will devote a share $\mu$ of its overall income $l_{ir}$ to the consumption of the differentiated good; she will then spend the following amount of those expenses $\mu l_{ir}$ on a given variety k:

$$p_r(k)c_{ir}(k) = \left( \frac{\left( \frac{p_r(k)}{\gamma_k^{\phi_i}} \right)^{1-\sigma}}{\int_0^n \left( \frac{p_r(k)}{\gamma_k^{\phi_i}} \right)^{1-\sigma} dk} \right) \mu l_{ir} \quad (3)$$

with $p_r(k)$ being the price charged for the variety k in country r. Assuming there exists only two possible qualities for each variety, i.e. high quality $\gamma_H$ and low quality $\gamma_L$ ($\gamma_H > \gamma_L$), using (3) and introducing specific consumption indices $C_{ir}^L$ and $C_{ir}^H$ for low- and high-quality variety bundles,<fnref n="16" /> the share $s_j(l_{ir})$ of those expenses $\mu l_{ir}$ devoted to varieties of quality j ($j = H, L$) is:

$$s_j(l_{ir}) = \frac{P_{rj} C_{ir}^j}{\mu l_{ir}} = \frac{\left( \frac{P_{rj}}{\gamma_j^{\phi_i}} \right)^{1-\sigma}}{\left( \frac{P_{rL}}{\gamma_L^{\phi_i}} \right)^{1-\sigma} + \left( \frac{P_{rH}}{\gamma_H^{\phi_i}} \right)^{1-\sigma}} \quad (4)$$

with $P_{rj} = [\int_0^{n_{rj}} p_{rj}(k)^{1-\sigma} dk + \int_0^{n_{sj}} p_{rj}^s(k)^{1-\sigma} dk]^{\frac{1}{1-\sigma}}$ ($r, s = D, F, r \neq s$) being a quality and country-specific price index, and $p_{rj}^s$ being the price of a good of quality j produced in country s and sold in country r ($p_{rj}$ being the mill price).

Focusing on the share devoted to high-quality goods, we have the following properties:

**Proposition 1 (Properties of the expenditure share devoted to high-quality varieties in the case of a quality-augmented CES utility specification):** For a given set of prices ($P_{rH}, P_{rL}$), we have the following properties.

**Property 1 (P1):** The average propensity to consume high-quality varieties increases along income: $\frac{\partial s_H(l_{ir})}{\partial l_{ir}} > 0$.

**Property 2 (P2):** For levels of income $l_{ir}$ for which we have $s_H(l_{ir}) < s_L(l_{ir})$, the share of expenditures devoted to the consumption of high-quality varieties is convex along income: $\frac{\partial^2 s_H(l_{ir})}{\partial l_{ir}^2} > 0$.

**Proof.** See Appendix B.

Hence, high-quality varieties are goods whose share in a given consumer's consumption basket increases along individual income. Moreover, as long as the share of high-quality varieties is lower than the share of low-quality varieties

<footnote n="16">¹⁶ $C_{ir}^j = (n_{Dj}(\gamma_j^{\phi_i} c_{ijr}^D)^{\frac{\sigma-1}{\sigma}} + n_{Fj}(\gamma_j^{\phi_i} c_{ijr}^F)^{\frac{\sigma-1}{\sigma}})^{\frac{\sigma}{\sigma-1}}$ with $c_{ijr}^r$ and $n_{rj}$ denoting respectively the consumption for a variety of quality j ($j = H, L$) produced in country r by a type i consumer and the number of firms producing quality j within country r.</footnote>
in the consumption basket, the share of high-quality varieties increases along income in a convex way.

Firms compete monopolistically. In the quality segment j, producing a quantity $x_j(k)$ of variety k requires $f_j + a_j x_j(k)$ units of labor, with $f_j$ and $a_j$ being respectively the fixed and marginal labor requirements for quality j.<fnref n="17" /> We impose $a_H > a_L$, in line with the idea that high-quality varieties are more expensive to produce (see, for example Kugler and Verhoogen, 2012), and assume free entry in each segment of the market.<fnref n="18" /> Our model features “iceberg” trade costs: in order to export to country r ($r \in \{D, F\}$) one unit of quality j's output manufactured in country s, a firm must ship $\tau_j \ge 1$ units. Finally, firms fully pass on their shipping costs to their foreign customers: one unit of variety k of quality j manufactured in country s is sold to consumers of country r at price $p_{sj}^r(k) = \tau p_{sj}$, where $p_{sj}$ is the mill price. Denoting by $D_{rj} = \beta_r N_r C_{Pr}^j + (1-\beta) N_r C_{Rr}^j$ the total demand in country r for all varieties of quality j (both domestically- and foreign produced), (3) yields the following expression for the demand $d_{rj}$ devoted to a variety of quality j produced in country r: $d_{rj} = p_{rj}^{-\sigma} (P_{rj}^{\sigma} D_{rj} + \tau^{1-\sigma} P_{sj}^{\sigma} D_{sj})$. The resolution of the firm's profit maximization problem within each country and quality segment is similar to the benchmark monopolistic competition model, and yields the following standard mill price and break-even output:

$$p_{rj} = \frac{\sigma}{\sigma - 1} a_j, \quad d_{rj} = \frac{f_j(\sigma - 1)}{a_j} \qquad (5)$$

The price index in country r for quality j can then be re-expressed as:

$$P_{rj} = (n_{rj} + \tau^{1-\sigma} n_{sj})^{\frac{1}{1-\sigma}} \frac{\sigma}{\sigma - 1} a_j \qquad (6)$$

It is then convenient to introduce along Fajgelbaum et al. (2011) the notion of “effective competitors” of quality j present on the domestic market r: $\tilde{n}_{rj} = n_{rj} + \tau^{1-\sigma} n_{sj}$. The intuition behind the concept is straightforward: while love for variety guarantees that for each quality j, each consumer in each country will devote a non-null part of its overall expenses to every available variety (both domestic and foreign), the market penetration of foreign varieties is discounted by a factor $\tau^{1-\sigma}$, capturing the fact that the price charged for foreign varieties bears the burden of shipping costs. Substituting for (6) in $d_{rj}$, we then get:

$$d_{rj} = \tilde{n}_{rj}^{\frac{\sigma}{1-\sigma}} D_{rj} + \tau^{1-\sigma} \tilde{n}_{sj}^{\frac{\sigma}{1-\sigma}} D_{sj} \qquad (7)$$

Equating demand and supply within each country and quality segment and using the fact that $d_{Dj} = d_{Fj} = d_j$ (i.e. domestic demand faced by a producer of a quality j variety is the same in both countries), equations (4), (5) and (7) yield the following four equilibrium

<footnote n="17">¹⁷Since we want to neutralize any supply-side determinant of a country's vertical specialization, we assume that those costs are similar across countries.</footnote>

<footnote n="18">¹⁸We assume that firms are mono-variety in our set-up: a single firm cannot enter both quality segments of the market.</footnote>
conditions:

$$ \frac{f_j \sigma}{\mu L} = \frac{(1 + \tau^{1-\sigma})(\beta_r \theta_r s_j(l_{Pr}) + (1 - \beta_r \theta_r) s_j(l_{Rr}))}{\tilde{n}_{rj}}, \quad j = H, L, r = D, F \quad (8) $$

**Proposition 2 (Existence and uniqueness of the equilibrium with trade):** For given income distribution parameters $\beta_r$, $N_r$, $L_r$ and $\theta_r$ ($r = D, F$), there exists a unique positive solution to the system of four equations defined by (8), determining the distribution of effective firms across countries and sectors ($\tilde{n}_{DL}, \tilde{n}_{DH}, \tilde{n}_{FL}, \tilde{n}_{FH}$).

**Proof.** See Appendix B.

This result concerning the number of effective firms within each country does however not guarantee that we will observe trade of the differentiated good at the equilibrium. Indeed, we have the following expression for $n_{rj}$, i.e. the number of local firms producing varieties of quality j within country r:

$$ n_{rj} = \frac{\tilde{n}_{rj} - \tau^{1-\sigma} \tilde{n}_{sj}}{1 - \tau^{2(1-\sigma)}}, \quad r \neq s, j = H, L, r, s = D, F \quad (9) $$

which entails the following condition for $n_{rj}$ to be positive, i.e. to have partial specialization of both countries:

$$ \tau^{1-\sigma} < \frac{\tilde{n}_{rj}}{\tilde{n}_{sj}} < \frac{1}{\tau^{1-\sigma}}, \quad r \neq s, j = H, L, r, s = D, F \quad (10) $$

Condition (10) is scarcely respected for low levels of transport costs, i.e. $\tau$ very close to 1, but always met for high enough values of $\tau$.<fnref n="19" /> From now on, we hence assume the transport costs $\tau$ are sufficiently high to guarantee that both countries produce and export the two qualities, i.e. that $n_{rj} > 0$ for $j = H, L$ and $r = D, F$.

**Proposition 3 (Impact of the average income and the level of inequality on the average quality of the export bundle):**

For given income distribution parameters $\theta_D$, $N_D$, $N_F$ and $\theta_F$ and for high enough transport costs $\tau$, we have the following comparative statics along $N_D$ and $\theta_D$:

(i) An increase in average income within country D (i.e. a decrease in $N_D$) generates an increase in the average quality of country D's export bundle: $\frac{\partial n_{DH}}{\partial N_D} < 0$, $\frac{\partial n_{DL}}{\partial N_D} > 0$.

(ii) Provided we have $s_H(l_{iD}) < s_L(l_{iD})$ for both $i = P, R$, a mean-preserving spread of income within country D (i.e. a decrease in $\theta_D$) generates an increase in the average quality of country D's export bundle: $\frac{\partial n_{DH}}{\partial \theta_D} < 0$, $\frac{\partial n_{DL}}{\partial \theta_D} > 0$.

**Proof.** See Appendix B.

**Proposition 3** implies that domestic income distribution has an impact on the quality mix being exported to trading partners. This result is the vertical

<footnote n="19">¹⁹For low values of $\tau$, condition (10) is respected when countries D and F are relatively similar in terms of average income $\frac{L_r}{N_r}$ and efficient labor size $L_r$.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 3: Heterogenous impact of inequality along the average income dimension</figcaption></figure>

translation of the classic horizontal “home-market effect” identified by Krugman (1980): a bigger market for varieties of a given quality *j* ensures the possibility to serve more consumers with sales that do not bear shipping costs, generating the entry of a greater number of producers of quality *j* and resulting in a shift in the quality level of exports.

Part (i) of Proposition 3 states that the average quality of the export bundle increases along the average income of consumers. This result is straightforward: since the share of overall consumption devoted to high-quality goods increases along income, an increase of average income leads to an increase in the size of the market for high-quality varieties. Such a demand shift raises the relative profitability of high-quality varieties, leaving the possibility for a higher number of firms to enter the market: $\tilde{n}_{DH}$ increases, leading to an increase (resp. decrease) in $n_{DH}$ (resp. $n_{DL}$) and driving the exported quality mix upwards.

Part (ii) of Proposition 3 states that inequality has a positive impact on the exported quality mix provided both $l_{PD}$ and $l_{RD}$ are below the income threshold for which $s_H(l_{iD})$ becomes greater than $s_L(l_{iD})$, i.e. if the evolution of the income share devoted to high-quality varieties is convex along income for both rich and poor consumers (cf property (P2) of the consumers’ preference system). This result is intuitively less straightforward, since mean-preserving variations in the spread of income impact in opposite ways the consumption of high-quality varieties of the poor and the rich: $\frac{\partial s_H(l_{PD})}{\partial \theta_D} > 0$, while $\frac{\partial s_H(l_{RD})}{\partial \theta_D} < 0$. As it can be seen from Figure 3, the properties of concavity/convexity of the evolution of a consumer’s income share devoted to high-quality varieties following an increase in her income are then essential so as to grasp the mechanism at work. When the income share devoted to high-quality varieties increases in a convex way for both poor and rich consumers within the economy (cf graph on the left of Figure 3), the marginal increase of rich consumers’ demand for high-quality varieties following an increase in inequality is more important than the marginal decrease of poor consumers’ demand. Moreover, an increase in inequality gives more weight to rich consumers in total income. This leads to an overall increase in aggregate demand for high-quality varieties. As exemplified in Figure 3, the
convexity of the Engel curve for high-quality varieties for both rich and poor consumers is more likely to be observed in poorer country. This property fails to follow through when the average income in the economy is high enough for $s_H(l_{RD})$ to increase in a concave way following a positive income shock (cf graph on the right of Figure 3).

As already mentioned at the beginning of this section, those results regarding the effect of income and inequality on a country's quality mix are in line with those obtained by Fajgelbaum et al. (2011).<fnref n="20" /> The nature of the adjustment of aggregate demand for high- and low-quality varieties is however different in the two models. In our model featuring love-for-variety at the individual level, it derives from changes in the quantity of each quality consumed at the individual level; in their model featuring heterogeneous consumers and unit consumption, it stems from changes in the number of people choosing a variety of a given quality.

# 5 Conclusion

In this paper, we have shown that income distribution in a given country is significantly related to the quality content of its exports. If average income is non-ambiguously and strongly related to export unit values, the message is more subtle for income inequality: controlling for income, income inequality is associated with a higher quality content of exports for poor countries only, while the opposite is true for rich countries. This non-linear pattern is robust to the inclusion of many controls and to several robustness checks. It can be rationalized thanks to a demand-based explanation: when the demand for high-quality varieties increases with income in a convex way, which is more likely to happen in poor countries, higher income inequality increases the aggregate demand for high-quality varieties. In the presence of increasing returns to scale and trade costs, this translates into a greater specialization of countries in the production and exports of high-quality varieties through a “vertical home-market effect”.

We believe we are the first to highlight empirically this pattern. Given the growing academic and policy interest for the determinants and the consequences of income inequality, we think this is a valuable contribution that could be extended in several dimensions. First, progress could be made in terms of causal inference by looking for exogenous variations in income distribution that could be related to changes in the quality content of production and exports. Second, the effect of income distribution on the composition of the aggregate demand might not only matter for the quality content of exports, but also for other outcomes such as the demand for (and thus public expenses on) healthcare and education for example.

<footnote n="20">²⁰Indeed, the convexity property of the evolution of the income share devoted to high-quality varieties (needed so as to guarantee a positive impact of the inequality level on the average quality of the export bundle) is similar in the two models. In Fajgelbaum et al. (2011)'s unit consumption model, it implies that a majority of any income class purchases low-quality goods; in our model featuring love-for-variety at the individual level, this property is similarly verified for countries in which both rich and poor consumers devote a greater share of their income to low-quality varieties.</footnote>
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<figure><img src="image_8.png" /><figcaption>Figure 1: Marginal effect of the Top 10% share of income</figcaption></figure>

# APPENDIX

## 1 Appendix A: Marginal effect of income inequality and average income
<table><caption>Table 1: List of exporting countries in the estimation sample</caption><thead><tr><td>ISO3 code</td><td>Country</td></tr></thead><tbody><tr><td>AUS</td><td>Australia</td></tr><tr><td>AUT</td><td>Austria</td></tr><tr><td>BDI</td><td>Burundi</td></tr><tr><td>BEL</td><td>Belgium</td></tr><tr><td>BEN</td><td>Benin</td></tr><tr><td>BFA</td><td>Burkina Faso</td></tr><tr><td>BGR</td><td>Bulgaria</td></tr><tr><td>BOL</td><td>Bolivia</td></tr><tr><td>BRA</td><td>Brazil</td></tr><tr><td>BWA</td><td>Botswana</td></tr><tr><td>CAF</td><td>Central African Republic</td></tr><tr><td>CAN</td><td>Canada</td></tr><tr><td>CHE</td><td>Switzerland</td></tr><tr><td>CHL</td><td>Chile</td></tr><tr><td>CHN</td><td>China</td></tr><tr><td>CIV</td><td>Cote d'Ivoire</td></tr><tr><td>CMR</td><td>Cameroon</td></tr><tr><td>COL</td><td>Colombia</td></tr><tr><td>CRI</td><td>Costa Rica</td></tr><tr><td>CYP</td><td>Cyprus</td></tr><tr><td>CZE</td><td>Czech Republic</td></tr><tr><td>DEU</td><td>Germany</td></tr><tr><td>DNK</td><td>Denmark</td></tr><tr><td>DOM</td><td>Dominican Republic</td></tr><tr><td>ECU</td><td>Ecuador</td></tr><tr><td>EGY</td><td>Egypt, Arab Rep.</td></tr><tr><td>ESP</td><td>Spain</td></tr><tr><td>EST</td><td>Estonia</td></tr><tr><td>FIN</td><td>Finland</td></tr><tr><td>FJI</td><td>Fiji</td></tr><tr><td>FRA</td><td>France</td></tr><tr><td>GAB</td><td>Gabon</td></tr><tr><td>GBR</td><td>United Kingdom</td></tr><tr><td>GRC</td><td>Greece</td></tr><tr><td>GTM</td><td>Guatemala</td></tr><tr><td>HND</td><td>Honduras</td></tr><tr><td>HRV</td><td>Croatia</td></tr><tr><td>HUN</td><td>Hungary</td></tr><tr><td>IDN</td><td>Indonesia</td></tr><tr><td>IND</td><td>India</td></tr><tr><td>IRL</td><td>Ireland</td></tr><tr><td>IRN</td><td>Iran, Islamic Rep.</td></tr><tr><td>IRQ</td><td>Iraq</td></tr><tr><td>ISL</td><td>Iceland</td></tr><tr><td>ISR</td><td>Israel</td></tr><tr><td>ITA</td><td>Italy</td></tr><tr><td>JOR</td><td>Jordan</td></tr><tr><td>JPN</td><td>Japan</td></tr><tr><td>KAZ</td><td>Kazakhstan</td></tr><tr><td>KEN</td><td>Kenya</td></tr><tr><td>KGZ</td><td>Kyrgyz Republic</td></tr><tr><td>KOR</td><td>Korea, Rep.</td></tr><tr><td>LKA</td><td>Sri Lanka</td></tr><tr><td>LSO</td><td>Lesotho</td></tr><tr><td>LTU</td><td>Lithuania</td></tr><tr><td>LUX</td><td>Luxembourg</td></tr><tr><td>LVA</td><td>Latvia</td></tr><tr><td>MAR</td><td>Morocco</td></tr><tr><td>MDA</td><td>Moldova</td></tr><tr><td>MEX</td><td>Mexico</td></tr><tr><td>MLT</td><td>Malta</td></tr><tr><td>MNG</td><td>Mongolia</td></tr><tr><td>MOZ</td><td>Mozambique</td></tr><tr><td>MRT</td><td>Mauritania</td></tr><tr><td>MUS</td><td>Mauritius</td></tr><tr><td>MYS</td><td>Malaysia</td></tr><tr><td>NAM</td><td>Namibia</td></tr><tr><td>NER</td><td>Niger</td></tr><tr><td>NGA</td><td>Nigeria</td></tr><tr><td>NIC</td><td>Nicaragua</td></tr><tr><td>NLD</td><td>Netherlands</td></tr><tr><td>NOR</td><td>Norway</td></tr><tr><td>PAN</td><td>Panama</td></tr><tr><td>PER</td><td>Peru</td></tr><tr><td>PHL</td><td>Philippines</td></tr><tr><td>POL</td><td>Poland</td></tr><tr><td>PRT</td><td>Portugal</td></tr><tr><td>PRY</td><td>Paraguay</td></tr><tr><td>ROU</td><td>Romania</td></tr><tr><td>RUS</td><td>Russian Federation</td></tr><tr><td>RWA</td><td>Rwanda</td></tr><tr><td>SDN</td><td>Sudan</td></tr><tr><td>SEN</td><td>Senegal</td></tr><tr><td>SVK</td><td>Slovak Republic</td></tr><tr><td>SVN</td><td>Slovenia</td></tr><tr><td>SWE</td><td>Sweden</td></tr><tr><td>TGO</td><td>Togo</td></tr><tr><td>THA</td><td>Thailand</td></tr><tr><td>TUN</td><td>Tunisia</td></tr><tr><td>TUR</td><td>Turkey</td></tr><tr><td>TZA</td><td>Tanzania</td></tr><tr><td>UKR</td><td>Ukraine</td></tr><tr><td>URY</td><td>Uruguay</td></tr><tr><td>USA</td><td>United States</td></tr><tr><td>VEN</td><td>Venezuela, RB</td></tr><tr><td>ZAF</td><td>South Africa</td></tr><tr><td>ZWE</td><td>Zimbabwe</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>Figure 2: Marginal effect of the Top 20% share of income</figcaption></figure>

<table><caption>Table 2: Correlation matrix between explanatory variables</caption><thead><tr><th></th><th scope="col">Ln GDP per cap.</th><th scope="col">Gini index</th><th scope="col">Ln Pop.</th><th scope="col">Trade openness</th><th scope="col">Human capital index</th><th scope="col">Aggregate TFP</th></tr></thead><tbody><tr><th scope="row">Ln GDP per cap.</th><td>1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><th scope="row">Gini index</th><td>-.45</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><th scope="row">Ln Pop.</th><td>-.12</td><td>.19</td><td>1</td><td></td><td></td><td></td></tr><tr><th scope="row">Trade openness</th><td>.04</td><td>-.26</td><td>-.45</td><td>1</td><td></td><td></td></tr><tr><th scope="row">Human capital index</th><td>.78</td><td>-.52</td><td>-.11</td><td>.24</td><td>1</td><td></td></tr><tr><th scope="row">Aggregate TFP</th><td>.20</td><td>-.08</td><td>-.07</td><td>-.01</td><td>.01</td><td>1</td></tr></tbody></table>
## 2 Appendix B: Proofs of the theoretical model

### Proof of Proposition 1

We have the following expressions for the different derivatives w.r.t. income $l_i$ considered in Proposition 1:

$$ 
\begin{aligned}
\frac{\partial s_H(l_i)}{\partial l_i} &= \frac{\partial \phi_i}{\partial l_i} \frac{\partial s_H(l_i)}{\partial \phi_i} = \frac{\partial \phi_i}{\partial l_i} (\sigma - 1) s_H(l_i) s_L(l_i) (\ln(\gamma_H) - \ln(\gamma_L)) \\
\frac{\partial^2 s_H(l_i)}{\partial l_i^2} &= (\sigma - 1)(\ln(\gamma_H) - \ln(\gamma_L)) s_H(l_i) s_L(l_i) \left[ \frac{\partial^2 \phi_i}{\partial l_i^2} + \left(\frac{\partial \phi_i}{\partial l_i}\right)^2 (\sigma - 1)(\ln(\gamma_H) - \ln(\gamma_L))(s_L(l_i) - s_H(l_i)) \right]
\end{aligned}
$$

Since $\gamma_H > \gamma_L$ and $\phi_i$ increases along $l_i$, we have unambiguously $\frac{\partial s_H(l_i)}{\partial l_i} > 0 \forall l_i > 0$. The sign of the second derivative depends on the sign of $s_L(l_i) - s_H(l_i)$ and $\frac{\partial^2 \phi_i}{\partial l_i^2}$: provided we have $s_H(l_i) < s_L(l_i)$ and the relationship between income and taste for quality is linear or convex, we hence have $\frac{\partial^2 s_H(l_i)}{\partial l_i^2} > 0$. This ends the proof. □

### 2.1 Proof of Proposition 2

Using (4) and (5), it is possible to reformulate the share $s_j(l_{ir})$ devoted to the consumption of varieties of quality j of a type i consumer living in country r as:

$$ s_j(l_{ir}) = \frac{a_j^{1-\sigma} \tilde{n}_{rj} \gamma_j^{\phi_i(\sigma-1)}}{a_H^{1-\sigma} \tilde{n}_{rH} \gamma_H^{\phi_i(\sigma-1)} + a_L^{1-\sigma} \tilde{n}_{rL} \gamma_L^{\phi_i(\sigma-1)}} $$

The equilibrium conditions featured in (8) represent the possible combinations for numbers of low- and high-quality effective producers consistent with market clearing and zero profits in the two market segments in both countries. More precisely, for a given country r we have:

$$ \frac{f_L \sigma}{\mu L_r} = \frac{(1 + \tau^{1-\sigma})(\beta_r \theta_r s_L(l_{Pr}) + (1 - \beta_r \theta_r) s_L(l_{Rr}))}{\tilde{n}_{rL}} \quad (11) $$

$$ \frac{f_H \sigma}{\mu L_r} = \frac{(1 + \tau^{1-\sigma})(\beta_r \theta_r s_H(l_{Pr}) + (1 - \beta_r \theta_r) s_H(l_{Rr}))}{\tilde{n}_{rH}} \quad (12) $$

(11) and (12) yield two implicit functions $\tilde{n}_{rH} = \psi^L(\tilde{n}_{rL})$ and $\tilde{n}_{rH} = \psi^H(\tilde{n}_{rL})$. $\psi^L$ and $\psi^H$ are implicitly defined by writing (11) and (12) as $L(\tilde{n}_{rH}, \tilde{n}_{rL}) = 0$ and $H(\tilde{n}_{rH}, \tilde{n}_{rL}) = 0$ with:

$$ 
\begin{aligned}
L(\cdot) &= -\frac{f_L \sigma}{(1 + \tau^{1-\sigma}) \mu L_r} + \frac{\beta_r \theta_r s_L(l_{Pr})}{\tilde{n}_{rL}} + \frac{(1 - \beta_r \theta_r) s_L(l_{Rr})}{\tilde{n}_{rL}} \\
H(\cdot) &= -\frac{f_H \sigma}{(1 + \tau^{1-\sigma}) \mu L_r} + \frac{\beta_r \theta_r s_H(l_{Pr})}{\tilde{n}_{rH}} + \frac{(1 - \beta_r \theta_r) s_H(l_{Rr})}{\tilde{n}_{rH}}
\end{aligned}
$$

$\psi_L$ and $\psi_H$ can be represented as downward-sloping curves in the $(\tilde{n}_{rH}, \tilde{n}_{rL})$ plane (respectively $LL$ and $HH$ in figure 3), since an increase in the number of competitors in one quality segment necessarily leads to a decrease in the number of competitors in the other segment in order to preserve profitability. More precisely, we have $\tilde{n}_{rL} \to \frac{f_L \sigma}{(1+\tau^{1-\sigma})\mu L}$ as $\tilde{n}_{rH} \to 0$ and $\tilde{n}_{rL} \to 0$ as $\tilde{n}_{rH} \to \infty$ in (11), while we have $\tilde{n}_{rH} \to \frac{f_H \sigma}{(1+\tau^{1-\sigma})\mu L}$ as $\tilde{n}_{rL} \to 0$ and $\tilde{n}_{rH} \to 0$ as $\tilde{n}_{rL} \to \infty$ in (12). The two curves hence necessarily intersect in the positive quadrant, i.e. there exists a positive equilibrium with $\tilde{n}_{rH} > 0$ and $\tilde{n}_{rL} > 0$.
<figure><img src="image_10.png" /><figcaption>Figure 3: HH and LL in the $(\tilde{n}_{rH}, \tilde{n}_{rL})$ plane</figcaption></figure>

Such an equilibrium is unique if $LL$ is always steeper than $HH$, i.e. if $\frac{\partial \psi_H}{\partial \tilde{n}_{rL}} < \frac{\partial \psi_H}{\partial \tilde{n}_{rL}} \forall \tilde{n}_{rL} > 0$. Using the implicit function theorem, this amounts to showing that we have $\frac{\partial L}{\partial \tilde{n}_{rL}} \frac{\partial H}{\partial \tilde{n}_{rH}} - \frac{\partial H}{\partial \tilde{n}_{rL}} \frac{\partial L}{\partial \tilde{n}_{rH}} > 0$. We note that $\frac{\partial s_j(l_i)}{\partial \tilde{n}_{rj}} = \frac{1}{\tilde{n}_{rj}} s_j(l_i)s_{-j}(l_i)$, and use the following notations to simplify the demonstration:

$$ 
\begin{aligned}
E[s_j] &= \beta_r \theta_r s_j(l_{Pr}) + (1 - \beta_r \theta_r) s_j(l_{Rr}) \\
E[s_H s_L] &= \beta_r \theta_r s_L(l_{Pr}) s_H(l_{Pr}) + (1 - \beta_r \theta_r) s_L(l_{Rr}) s_H(l_{Rr})
\end{aligned}
$$

We then have:

$$ 
\begin{aligned}
\frac{\partial L}{\partial \tilde{n}_{rL}} \frac{\partial H}{\partial \tilde{n}_{rH}} - \frac{\partial H}{\partial \tilde{n}_{rL}} \frac{\partial L}{\partial \tilde{n}_{rH}} &= (1/\tilde{n}_{rL}^2) (1/\tilde{n}_{rH}^2) E[s_L] E[s_H] \left( \left( \frac{E[s_H s_L]}{E[s_L]} - 1 \right) \left( \frac{E[s_H s_L]}{E[s_H]} - 1 \right) - \frac{E[s_H s_L]^2}{E[s_H] E[s_L]} \right) \\
&= (1/\tilde{n}_{rL}^2) (1/\tilde{n}_{rH}^2) E[s_L] E[s_H] \left( 1 - \frac{E[s_L s_H]}{E[s_L] E[s_H]} \right)
\end{aligned}
$$

Using the fact that $s_L(l_{ir}) = 1 - s_H(l_{ir})$, we have $E[s_L]E[s_H] = E[s_H] - E[s_H]^2$, while $E[s_H s_L] = E[s_H] - E[s_H^2]$; we hence have $\frac{E[s_L s_H]}{E[s_L]E[s_H]} < 1$, and $\frac{\partial L}{\partial \tilde{n}_{rL}} \frac{\partial H}{\partial \tilde{n}_{rH}} - \frac{\partial H}{\partial \tilde{n}_{rL}} \frac{\partial L}{\partial \tilde{n}_{rH}} > 0$. This ends the proof. $\square$

## Proof of Proposition 3

Using the implicit function theorem, the comparative statics of $\tilde{n}_{DH}$ and $\tilde{n}_{DL}$ with respect to a parameter $\eta$ ($\eta = N_D, \theta_D$) can be obtained with the formula:

$$ 
\begin{pmatrix} \frac{\partial \tilde{n}_{DH}}{\partial \eta} \\ \frac{\partial \tilde{n}_{DL}}{\partial \eta} \end{pmatrix} = - \begin{pmatrix} \frac{\partial H}{\partial \tilde{n}_{DH}} & \frac{\partial H}{\partial \tilde{n}_{DL}} \\ \frac{\partial L}{\partial \tilde{n}_{DH}} & \frac{\partial L}{\partial \tilde{n}_{DL}} \end{pmatrix}^{-1} \begin{pmatrix} \frac{\partial H}{\partial \eta} \\ \frac{\partial L}{\partial \eta} \end{pmatrix}
$$

which yields:

$$ 
\begin{pmatrix} \frac{\partial \tilde{n}_{DH}}{\partial \eta} \\ \frac{\partial \tilde{n}_{DL}}{\partial \eta} \end{pmatrix} = - \frac{1}{\frac{\partial H}{\partial \tilde{n}_{DH}} \frac{\partial L}{\partial \tilde{n}_{DL}} - \frac{\partial H}{\partial \tilde{n}_{DL}} \frac{\partial L}{\partial \tilde{n}_{DH}}} \begin{pmatrix} \frac{\partial H}{\partial \eta} \frac{\partial L}{\partial \tilde{n}_{DL}} - \frac{\partial H}{\partial \tilde{n}_{DL}} \frac{\partial L}{\partial \eta} \\ -\frac{\partial L}{\partial \tilde{n}_{DH}} \frac{\partial H}{\partial \eta} + \frac{\partial H}{\partial \tilde{n}_{DH}} \frac{\partial L}{\partial \eta} \end{pmatrix}
$$

The sign of the fraction is straightforward: considering demonstration of proposition 1, we have $-\frac{\partial H}{\partial \tilde{n}_{rH}} \frac{\partial L}{\partial \tilde{n}_{DL}} - \frac{\partial H}{\partial \tilde{n}_{DL}} \frac{\partial L}{\partial \tilde{n}_{DH}} < 0$. We are left to determine the signs of the derivatives of
H and L with respect to $\theta_D$ and $N_D$:

$$
\begin{align*}
\frac{\partial L}{\partial N_D} &= \frac{\beta_D \theta_D}{\tilde{n}_{DL}} \left( \frac{\partial s_L}{\partial l_{PD}} \frac{\partial l_{PD}}{\partial N} \right) + \frac{(1 - \beta_D \theta_D)}{\tilde{n}_{DL}} \left( \frac{\partial s_L}{\partial l_{RD}} \frac{\partial l_{RD}}{\partial N} \right) \\
\frac{\partial H}{\partial N_D} &= -\frac{\beta_D \theta_D}{\tilde{n}_{rH}} \left( \frac{\partial s_H}{\partial l_{PD}} \frac{\partial l_{PD}}{\partial N} \right) + \frac{(1 - \beta_D \theta_D)}{\tilde{n}_{rH}} \left( \frac{\partial s_H}{\partial l_{RD}} \frac{\partial l_{RD}}{\partial N} \right) \\
\frac{\partial L}{\partial \theta_D} &= \frac{\beta_D}{\tilde{n}_{DL}} (s_L(l_{PD}) - s_L(l_{RD})) + \frac{\beta_D L_D}{N_D \tilde{n}_{DL}} \left[ \theta_D \frac{\partial s_L}{\partial l_{PD}} - \frac{1 - \beta_D \theta_D}{1 - \beta_D} \frac{\partial s_L}{\partial l_{RD}} \right] \\
\frac{\partial H}{\partial \theta_D} &= \frac{\beta_D}{\tilde{n}_{rH}} (s_H(l_{PD}) - s_H(l_{RD})) + \frac{\beta_D L_D}{N_D \tilde{n}_{rH}} \left[ \theta_D \frac{\partial s_H}{\partial l_{PD}} - \frac{1 - \beta_D \theta_D}{1 - \beta_D} \frac{\partial s_H}{\partial l_{RD}} \right]
\end{align*}
$$

(i) We have $\frac{\partial l_{PD}}{\partial N_D} = -\theta_D \frac{L}{N_D^2} < 0$ and $\frac{\partial l_{RD}}{\partial N_D} = \frac{1-\beta_D\theta_D}{1-\beta_D} \frac{L}{N_D^2} < 0$. Along P1, we are further able to state that $\frac{\partial s_H(l_{iD})}{\partial l_{iD}} > 0$ and $\frac{\partial s_L(l_{iD})}{\partial l_{iD}} < 0$. We hence obtain unambiguously that $\frac{\partial L}{\partial N_D} > 0$ and $\frac{\partial H}{\partial N_D} < 0$. The implicit function theorem then entails that $\frac{\partial \tilde{n}_{rH}}{\partial N_D} < 0$ and $\frac{\partial \tilde{n}_{DL}}{\partial N_D} > 0$.

An alternative and more intuitive demonstration of part (i) of Proposition 2 can be obtained by considering a slightly modified version of the equilibrium condition (12):

$$ \frac{f_H \sigma \tilde{n}_{DH}}{\mu L_D (1 + \tau^{1-\sigma})} = \beta_D \theta_D s_H(l_{PD}) + (1 - \beta_D \theta_D) s_H(l_{RD}) \quad (13) $$

As already said, an increase in $N_D$ decreases both $l_{PD}$ and $l_{RD}$, and hence generates a decrease of both $s_H(l_{PD})$ and $s_H(l_{RD})$ (cf property P1). The RHS of condition (13) hence unambiguously decreases. Considering the concavity of $s_H(l_{iD})$ along $\tilde{n}_{DH}$ ($\frac{\partial^2 s_H(l_{iD})}{\partial \tilde{n}_{DH}^2} < 0$, cf demonstration of Proposition 1) and the fact that the LHS is linear in $\tilde{n}_{DH}$, such a decrease of the RHS cannot be compensated by an increase in $\tilde{n}_{DH}$. The LHS necessarily needs to decrease for the equality to be respected again, leading to a decrease in $\tilde{n}_{DH}$ following an increase in $N_D$.

(ii) As stated in Proposition 2, we place ourselves in the case where both $l_{RD}$ and $l_{PD}$ are under the income threshold $l^T$ beyond which we have $s_H(l^T) > s_L(l^T)$. Along P1 and since $l_{RD} > l_{PD}$, we have that $s_H(l_{PD}) - s_H(l_{RD}) < 0$ and $s_L(l_{PD}) - s_L(l_{RD}) > 0$. Along (P2), we have that $\frac{\partial s_H}{\partial l_{RD}} > \frac{\partial s_H}{\partial l_{PD}}$ and $\frac{\partial s_L}{\partial l_{RD}} < \frac{\partial s_L}{\partial l_{PD}}$. Using those properties, we can deduce $\frac{\partial L}{\partial \theta_D} > 0$ and $\frac{\partial H}{\partial \theta_D} < 0$. Considering the formula obtained with the implicit function theorem, we then obtain unambiguously that $\frac{\partial \tilde{n}_{DH}}{\partial \theta_D} < 0$ and $\frac{\partial \tilde{n}_{DL}}{\partial \theta_D} > 0$.

Adding up the equilibrium conditions in both quality segments for country D yields the following condition that needs to be met at the equilibrium:

$$ f_L \sigma \tilde{n}_{DL} + f_H \sigma \tilde{n}_{DH} = \mu L_D (1 + \tau^{1-\sigma}) \quad (14) $$

Hence, at fixed overall labor supply $L_D$, condition (14) guarantees that an increase in $\tilde{n}_{DH}$ is only possible through a decrease in $\tilde{n}_{DL}$. Furthermore, we have that:

$$ \frac{\partial n_{rj}}{\partial \tilde{n}_{rj}} > 0 \quad j = H, L, r = D, F \quad (15) $$

Those comparative statics imply that, provided that we are in an equilibrium with partial trade specialization (i.e. for high enough values of $\tau$), an increase in the number $\tilde{n}_{rj}$ of “effective” producers of a given quality $j$ in country $r$ increases the number $n_{rj}$ of domestic producers of this quality. We can hence directly interpret an increase in $\tilde{n}_{DH}$ as an increase in $n_{DH}$, and a decrease in $\tilde{n}_{DL}$ as a decrease in $n_{DL}$. In other words, an increase in $\tilde{n}_{DH}$ leads to a shift of the export mix of the domestic country D towards high a higher average quality at the equilibrium. This ends the proof. $\square$