1. Regression results from the average treatment effect estimation
Note: Superscripts ***, ** and * represent significance levels of 0.1, 0.05 and 0.01 respectively.
Source: Author s own calculation
2. Deriving the Margins
Following from Hummels and Klenow (2005), consider there are h = 1,...,H countries in the world. Each country produces many types of goods and all of them can be exported. Assume that the set of goods produced and exported by country h at time t is
the quantity of good i is
and thus the vector of each type of good produced in country h at time t is
There are some goods that common to countries h and j, which we name as common goods set, It. Next we further assume the production function in country h follows a CES transformation function with aggregate resources
then the total outputs of country h equals,
where ω > 0 is the elasticity of transformation between goods.
The ratio of two CES functions across two countries h and j at time t is the product of the price index of the common goods,
to both countries
multiplied by the terms that indicate the revenue share of the non-common goods.
are weights calculated using the revenue shares of countries h and j .
The weights sum to unity over the common goods set It.
Now the revenue shares of the non-common goods are,
Note that the revenue shares are always calculated relative to the common goods set, It . Unless
we always have
As long as there are goods not in the common goods set that country h produces,
This means if country h produces certain goods that country j does not produce, then
will be strictly less than 1. In other words, the higher is
(the total value of the non-common goods country h produces), the lower is the ratio of the revenue shares
Therefore, the inverse of
represents the relative export variety of country h to country j.
Now if we add the world F as reference, then
represents all the varieties that the world produces at time t and considering the pair-wise comparison between country h and country j, we have the extensive margin of the exports from country h to country j equals to,
Estimated extensive margin of exports is within the range of 0 to 1, and its value can be interpreted as how much of the exports by country h to country j can be explained by export varieties. It can also be taken as a weighted count of country h's product set relative to the world's product set.
Applying similar logic, we derive the intensive margin of the exports from country h to country j as,
The estimated intensive margin of exports is also within the range of 0 to 1, and its value can be interpreted as how much of the exports by country h to country j can be explained by the volume.
The product of the extensive and intensive margins equals to country h's share of exports in country j,
Essentially, the above equals to the amount of exports from country h to country j divided by the total imports by country j, i.e. the import share of country h in country j.
3. Theoretical Set-up of Arkolakis et al. (2009)
Consider a traditional Armington (1969) model, there are i = 1,...,n countries, each produces a differentiated good using labour. The labour supply is inelastic and given by Li . There is a representative agent in each country that has the following Dixit-Stigliz utility function,
where qij is the quantity of country j's good consumed by country i and
is the elasticity of substitution between goods. The associated price index in country j is then given by,
is country i's wage and
are the trade costs between country i and country j.
When the Dixit-Stigliz utility equation is maximized with respect to the price index, the total imports from country i is then equal to,
being country j's total expenditure and
is the partial elasticity of relative imports with respect to the variable trade costs (i.e. the trade elasticity). Also, trade is balanced as
Assume that there is a shock that affects foreign labour endowment,
and trade cost,
but not those of country j's. The change in real income is then,
where λij is the country j's share of expenditure on goods from country i. By simple manipulation, we can see that the change in relative imports would be,
Combining the last two equations together, i.e. substituting the relative imports equation into the equation for change in real income, we have the change in real income as,
This gives rise to the final expression for changes in income,
The above expression equals to the change of income in the initial equilibrium and the new equilibrium. The interpretation of this result is straight forward; change in real income is dependent on terms of trade changes which in turn are dependent on changes of relative import demand. Thus, the system can be reduced to an expression with two sufficient statistics, λ and σ. Arkolakis et al. (2009) also demonstrate that under various assumptions on preferences, technologies of production and market structures, this expression still holds.
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