Proportionality Assumption in Factor Content of Trade
The concept of factor content of trade, coined by Jaroslav Vanek describes a proposition in which countries are net exporters of services relatively intensive in factors that they are endowed with. The Heckscher-Ohlin theorem lacked scalability, so Vanek shifted the reference from products to amounts of factor-services that are embodied in goods traded. In this model, both goods and factor prices are equalized which allows the evaluations of interactions between endowments, production, absorption and trade, helping to confirm trade theories and to address national trade policies.
From Vanek’s proposition, country factor content of trade can be predicted using endowments and shares of world consumption as follows:
With endowment data being sparsely available, and trade largely consisting of trade in intermediates, Reimer (2006) and Trefler and Zhu (2010) demonstrated that Vanek’s direct relation between country 𝑖’s endowment and country 𝑖’s portion of world consumption considering the world endowment can be expressed as:
Leontief’s input-output technique facilitates the exercise. The technique offers a framework that aggregates coefficients of intermediary products required in production. These coefficients track flows between sectors by arranging inflows and outflows in input-output matrices. The only drawback is the data requirements for the Leontief input-output technique. In the case of factor content of trade, as flows are domestic and international, domestic and imported input-output data is used to fill in the matrices. Domestic input-output data is widely available as national statistical agencies report them. Unfortunately, few countries report imported input-output data. As a workaround, researchers use a standard proportionality assumption by disaggregating data that is available.
The proportionality assumption involves allocating imported products proportionally to all destination sectors and final demand according to domestic demand by using bilateral trade vectors and the imported input-output matrices. For example, if 30% of wood products in the United States are imported from Canada, it is inferred that 30% of the pulp used to make Dunder Mifflin’s paper comes from Canadian wood (only a familiar illustrative example as the proportionality assumption is used at the sector level and not the firm level).
The complexity of the Leontief input-output matrix grows exponentially as countries and industries are added. Keeping it’s square shape, all corresponding inputs will have their counterpart in outputs. To illustrate this relationship, we will be using 3 sectors and 3 countries, where matrix B takes the following shape:
To help in the visualization, the following matrix B shows domestic input-output matrices on the diagonal where i = j:
Matrices on the diagonal are filled using data available from statistical agencies. The other values, referred to as imported input-output data, use the proportionality assumption, where imported products, per sector, are distributed proportionally to destinations sectors according to domestic demand. This is defined mathematically by:
Once matrix B is populated with values from the domestic and imported input-output tables, we then use direct factor requirements data to generate a matrix B that holds coefficients instead of raw values. The coefficients are based on the column sums of matrix B and column sums of the factor demand matrix for each input country-sector pair, jh. The new matrix B with coefficients can then be inverted using (I-B)^-1. Factor content of trade can then be computed using Reimer’s and Trefler and Zhu’s formulation.
