### Welfare Gains from Trade in Multi-Sector Models: The Role of Aggregation and Income Elasticities

**One sentence summary**: In the calculation of welfare gains from trade, the income elasticity interacts with the trade elasticity that can be estimated in a bilateral trade regression as the coefficient in front of aggregated unit prices instrumented by gravity variables.

**Abstract**

Sectoral heterogeneity has been shown to affect country-level welfare gains from trade that can be calculated by sector-specific trade elasticities and home expenditure shares. However, empirical analyses of multi-sector models are restricted to a limited number of countries and sectors, mostly due to the lack of data on sector-specific home expenditure shares. This paper first proposes a solution to this limitation by changing the way that foreign products are aggregated at the destination country. Second, when firm-level productivity differences are controlled for in a bilateral aggregate trade equation, gravity variables are shown to be perfect instruments for aggregated unit prices that enter trade regressions for the estimation of trade elasticities. Third, when the assumption of unitary income elasticity is relaxed, the trade elasticity in the calculation of welfare gains is shown to be replaced by the newly-introduced welfare elasticity, a function of trade and income elasticities. Empirical evidence is shown for the heterogeneity of these elasticity measures across countries.

**Non-technical Summary**

Welfare gains from trade (measured as costs of autarky) can be expressed by two key parameters, namely the trade elasticity and home expenditure share. When the one-sector environment is extended to a multi-sector one, due to the way that sectors are aggregated at the destination country (i.e., an upper tier aggregation of individual utility across sectors), the two key parameters are required at the sector level in order to calculate welfare gains from trade. Although the trade elasticity can be estimated for pretty much any sector and any country by using the corresponding trade and price/tariff data, home expenditure data are available only for certain aggregation of sectors in certain countries. Accordingly, in order to calculate welfare gains in a multi-sector environment, several studies focusing on the estimation of both parameters (at the sectoral level) have been restricted to a limited number of countries and a limited number of sectors. Since one-sector trade elasticity measures are biased due to sectoral heterogeneity, unbiased welfare gains from trade cannot be calculated for several countries that lack sector-level home expenditure data (although they have all the necessary trade data). Moreover, when the number of countries is limited, the estimated elasticity parameters (especially those that are common across countries) simply cannot represent global trade patterns, which would lead into biased welfare calculations.

Considering a multi-sector framework in order to have
unbiased welfare implications, this paper proposes a new aggregation approach
that does not require any sector-level home expenditure data. Instead, the
overall home expenditure share data are shown to be enough to calculate the
welfare gains from trade. This is achieved by using an Armington model, where
aggregation at any destination country is achieved across source countries at
the upper tier and across sectors at the middle tier; the lower tier aggregates
firm-level goods. Having an upper-tier aggregation across source countries
corresponds to having a unique trade elasticity measure interacting with a
unique home expenditure share in order to calculate welfare gains from trade
for any destination country. Accordingly, when the upper-tier elasticities are
estimated, welfare gains can be calculated for any country that has data for
overall home expenditure share.

On top of solving the problem of not having sector-level home expenditure data, this paper also relaxes the restrictive assumption of having unitary income elasticity in the calculation of welfare gains. In an Armington (1969) framework, it is shown that the standard trade elasticity in the case of unitary income elasticity (which is one minus the elasticity of substitution) is replaced with the country-specific income elasticity minus the elasticity of substitution (that we call welfare elasticity in this paper). It is implied that income effects counteract those of substitution effects when economies open up for trade, because consumers may simply value the utility out of home products different from that of foreign products due to the heterogeneity in income elasticity measures.

The inclusion of income elasticity in welfare gains
calculations is achieved by using implicitly additively separable nonhomothetic
constant elasticity of substitution (CES) preferences across source countries
at the upper tier of individual utility. Such an approach is essential to
separately capture the income effects in the utility function, without giving
away the standard features of having CES preferences, so that one can easily
distinguish between income and substitution effects, even in the calculation of
welfare gains.

The model is estimated by using UN Comtrade data at the six
digit Harmonized System (HS) level between 1995-2015. The estimation of the
country-specific income elasticity and the country-specific elasticity of
substitution is achieved by using data on bilateral trade (of imports measured
at the destination country) and unit prices. Since the lower tier aggregation
of individual utility is achieved across firm-level goods, firm-level
productivity differences are carefully connected to the data on unit prices and
the corresponding estimation; therefore, although firm-level data are not
utilized, firm-level productivity differences are still taken into account at the
aggregated level, without making any assumptions on their distribution.

Following the literature, the aggregation across sectors is
achieved by a Cobb-Douglas aggregation. Due to the tiers of aggregation
introduced earlier, this corresponds to having a weighted average of log unit
destination prices in the bilateral aggregate trade estimation, where the
weights are simply the expenditure weights of sectors. Nevertheless, since
firm-level productivity measures are carried over to the bilateral aggregate
trade estimation as residuals, aggregated log unit prices become endogenous. By
using the implications of the model, bilateral trade costs measured by standard
gravity variables are shown to be strong instruments for aggregated unit prices
in a Two-Stage Least Squares (TSLS) estimation. In this bilateral aggregate
trade estimation, the coefficient in front of aggregated unit prices represents
the country-specific trade elasticity (i.e., one minus the elasticity of
substitution), while the coefficient in front of log total imports (due to
having non-unitary income elasticity) represents the country-specific income
elasticity; these estimates are further used to construct country-specific
welfare elasticity estimates.

The corresponding welfare elasticity estimates have a median
value of -4.3 across countries with a range between -6.0 and -2.7 that are
highly consistent with trade elasticity estimates in the literature. Although
the estimates are comparable, the methodology introduced in this paper in a
multi-sector framework is much easier to implement, and it allows us to
calculate welfare gains in pretty much all countries that have trade data. In
order to show the contribution of this paper in a clear way, the obtained welfare
elasticity estimates are further compared to the common (across countries)
trade elasticity estimate of about -3.8 that is obtained by the very same data
set. It is shown that the difference between country-specific welfare
elasticity estimates and common trade elasticity estimate is the key to
understand the heterogeneity of welfare gains across countries for a given
measure of home expenditure share. This heterogeneity is further connected to
the trade patterns and per capita income of countries by using the implications
of the model.

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