An empirical measure of factor intensity


when there are many factors and many products




Henry Thompson


Auburn University





Factor intensity is a two dimensional concept with no clear meaning when there are numerous factors of production and numerous outputs.The present paper considers the potential application of mean weighted factor intensity, a cardinal ranking across products for each factor of production.If output is measured as valued added, mean weighted intensity also ranks factors for each product.Mean weighted factor intensity successfully anticipates the comparative static elasticities of a high dimensional factor proportions model of production and trade, and may prove useful in predicting long term export potential.



A shortcoming of the factor proportions theory of production and trade is that the fundamental concept of factor intensity cannot be directly applied to data with various factors of production and numerous products.Factors can include labor skill groups, natural resources, energy inputs, and capital vintages or locations.Products can include manufacturing industries, disaggregated service industries, and agricultural crop data.Potential model simulations simply do not relate to factor intensity in high dimensional applications.††††

Proprieties of high dimensional models of production and trade in Uzawa (1964), Chang (1974), Ethier (1974), Jones and Scheinkman (1977), Takayama (1982), Choi (2004), and Thompson (2004) fall short of interpreting factor intensity.Empirical tests of factor proportions theory typically use two inputs due in part to the lack of a definition of factor intensity beyond two dimensional data.In short, a powerful concept falls short in applications with any degree of detail.

The present note defines factor intensity relative to the mean of each input across industries, generating a bilateral ranking for each factor and product.While there are no necessary links between mean weighted factor intensity and theoretical properties of production models, the metric may prove a useful empirical guide.It successfully anticipates the comparative static properties of an applied general equilibrium model of production and trade in the literature.Factor winners and losers due to trade policy may be anticipated with a relatively straightforward examination of factor intensities without having to estimate production functions or rely on model simulations under various industrial structures.


Factor intensity

Let aij be the input of factor i per unit of product j.With two factors and two products the ratio of inputs across products generates the intensity ranking a11/a21 > a12/a22.This two dimensional measure can be extended to any number of products.In the 2x3 model, the intensity ranking is a11/a21 > a12/a22 > a13/a23 with industry 1 using factor 1 intensively, industry 3 using factor 2 intensively, and industry 2 in the middle.The 2xn small open economy is over-determined but assumptions can be relaxed to create a tractable model.

In the 2x2 model, factor 1 has a higher opportunity cost in product 1 in the converted intensity condition a11/a12 > a21/a22 and adding factors extends this opportunity cost ranking.In the 3x2 model, a11/a12 > a21/a22 > a31/a32 and factor 1 is intensive in industry 1, factor 3 in industry 2, and factor 2 in the middle.Whether a country exports the product using its most abundant factor most intensively depends on factor intensity as well as substitution as developed by Ruffin (1981), Jones and Easton (1983), and Thompson (1985).

With as few as three factors and three products, there is no factor intensity ranking.In the 3x3 model, industry 1 might use factor 1 most intensively relative to industry 2 but least intensively relative to industry 3.The proposed mean weighted factor intensity MWFI provides a ranking for high dimensional models.



The mean weighted factor input factor i in product j is as mij ≡ aij/mi where mi = Sjaij/n is the mean input of factor i across the n products.Comparing this intensity across products with the same units of output (tons for instance) is straightforward but the typical data involves outputs with different physical units.Following applied production analysis, define a unit of output as the amount worth one unit of numeraire.Output is then value added and the mean weighted factor intensity MWFI can be compared across products. If mij > 1 > mih industry j uses factor i more intensively than the average industry, and industry h less intensively.The ratio mij/mih = aij/aih indicates the opportunity cost of product j in terms of product h.If mij/mih > mkj/mkh industry j has a higher opportunity cost than industry h in factor i relative to factor k.

Comparing factors across an industry mij > 1 > mkj implies industry j uses factor i intensively relative to its average input and factor k less intensively.Comparison across factors rescales the underlying factor intensity ranking mij/mkj = (aij/akj)(mi/mk).If mij/mkj > mih/mkh factor i is intensive in industry j relative to factor k in industry h.

Rescaling unit inputs has no effect on the comparative static effects of changing prices and endowments on factor prices and outputs in the general equilibrium production model.The total endowment vi of factor i would be rescaled to vi/mi and factor prices wi rescaled to miwi.Competitive pricing conditions pj = Σiaijwi remain consistent with exogenous world prices pj.Full employment conditions vi = Σjaijxj include rescaled endowments and unit inputs.Factor shares, industry shares, and substitution elasticities are unaffected by the rescaling.

In even models with the same number of factors and products, outputs are uniquely determined given factor endowments.In uneven models such as the specific factors model, the pattern of production is not determined by endowments.Nevertheless, the present MWFI may prove useful in data exploration and applications.

Collect the mean weighted factor inputs with r factors and n products into the factor intensity matrix Frxn.The following section examines how well Frxn anticipates the comparative static results of an applied factor proportions model.Suppose there are c countries and define matrix Bnxc as the mean weighted factor abundance matrix.An empirical test of factor content theory would involve the empirical relationship between Frxn, Bnxc, and the Xnxc matrix of net exports across countries.


An application of MWFI

The 9x3 model of the US economy in Thompson (1990) provides a glimpse into potential application of MWFI.The first columns in Table 1 for the three sectors agriculture A, manufacturing M, and services S are the derived factor shares qij = aijwi/pj for capital and eight Census skill groups of labor.Output is value added with a unit of output defined by pj = 1.Capital shares are residuals of value added after labor shares.Mean weighted factor shares equal mean weighted factor intensities since 3qij/Sjqij = 3aijwi/Sjaijwi = 3aij/Sjaij = mij.

†††††† The largest labor factor shares are operators in manufacturing at .286 and professionals in services at .269.Other large labor shares are technical/sales labor in services .211, crafts in manufacturing .167, and resource labor in agriculture .139.The large residual capital share in agriculture .576 implicitly includes land input.


Table 1.1980 US Factor Intensities


†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Agriculture†††††† †††††††††††††††††††Manufacturing†††††††††††††††††††††††††††††† Services†††††††††††††††††††††††††††††††††††††† Input ratios†††


































































































































































Each input has two sets of ratios across the three sectors.Input ratios in services relative to manufactures are S/M ≡ aiS/aiM = θiSiM with the skilled wage wi equal across sectors and pA = pM = 1.This input ratio S/M and the service/agriculture intensity S/A ≡ aiS/aiA are the last two columns for each factor in Table 1.†††

Service labor, technical labor, and professional labor are the most intensive inputs in services relative to both other sectors but beyond those three inputs the rankings are not similar.Among the other inputs, capital is used intensively in services relative to manufactures but not relative to agriculture.The opposite is true for operators.For all inputs the correlation between the two rankings is 0.07 and excluding the three intensive inputs the correlation is -0.40.

MWFI aij is reported in the second column of Table 1 for each sector.Reading down columns compares factors for the sector and yields the same ordinal ranking as factor shares.Agriculture uses capital more intensively than any type of labor except resource labor.Manufactures use operators most intensively.Services use service labor and technical/sales labor.

Capital is more than twice as intensive in agriculture as in the other two sectors.Resource labor is virtually specific to agriculture.The service sector uses professional labor about twice as intensively as manufacturing which uses it about twice as intensively as agriculture.The service sector uses technical and service labor more intensively than the other two sectors.Manufacturing uses crafts, operators, and handlers intensively.

Operators are about four times as intensive in manufacturing relative to services and seven times as intensive relative to agriculture.Transport, technical, and professional labor all have about average intensity in manufacturing, which uses every type of labor except service and resource labor more intensively than capital.The service sector uses service, technical, and professional labor the most intensively.Transport labor has the least intensity variation and is close to average intensity.The simple intensity scaling illustrates the potential usefulness of the mean intensity measure.

Factor shares are a misleading guide to factor intensity in this example.Reading down the qim column for manufacturing, capital appears more intensive than handlers but reading down the mean intensity aim column capital is used only 42% (0.61/1.56) as intensively as handlers.There are other such examples.

The MWFI anticipate the Stolper-Samuelson (1941) dwi/dpj elasticities reported in the third columns for each sector in Table 1.The model uses estimates of translog production functions across states but these dwi/dpj elasticities are insensitive to a wide range of factor substitution depending almost entirely on factor shares.An increase of 1% in the price of manufactures raises the operator wage over 3% and lowers the service worker wage by just under 1%.Figures 2-4 also present these dw/dp elasticities.

The correlation of the comparative static dw/dp elasticity vector with factor shares across all three sectors is only 0.32 while its correlation with mean weighted factor intensities is 0.74.The correlation between factor shares and mean weighted factor intensities of 0.42 indicates the difference in the two measures.

Due to reciprocity in the comparative static results, mean weighted factor intensities also anticipate the effects of changing factor supplies on outputs.Countries more abundant in a factor are expected to produce and export more of the products using that factor intensively, at least given identical homothetic preferences and no transport costs.With many factors and countries, a similar mean weighted measure of factor abundance can be formulated.Whether the MWFI and abundance anticipate the direction of trade is an empirical issue.††††



Mean weighted factor intensities provide a metric to anticipate and interpret general equilibrium properties of high dimensional factor proportions models of production and trade.They are comparable across products for each factor and across factors for each product.In contrast, factor shares are comparable only across factors for each product and are apparently misleading guides to theoretical predictions.

Examination of the empirical links between mean weighted factor intensities and trade would provide a test of the relevance of factor content theory.Mean weighted factor abundance matrices can be used alongside mean weighted factor intensity matrices given data for various factors of production, products, and countries.An empirical test of factor content theory would involve the empirical relationship between the factor abundance matrix, the factor intensity matrices, and the matrix of net exports across countries.

As a forward looking application, consider the potential effect of liberalized trade inside the evolving Free Trade Area of the Americas FTAA.The 34 countries in FTAA will increase trade in hundreds of manufactured goods, services, and natural resource products classified with the NAICS system.There is associated input data on labor, energy, and residual capital, and there is some data for skilled labor groups.A comparison of mean weighted intensities across industries and mean weighted abundance across countries would predict which countries will export which products.Countries with above average abundance in a factor might be expected to export products with above average intensity in that factor.Labor groups with below average abundance could expect falling wages.Such projections would avoid estimation of cost or production functions in detailed production models.Policymakers could use the projections to help make a decision about whether to invest in a container port at a particular location, or whether to alter tax rates in anticipation of the income redistribution that will follow trade.




Chang, Winston ((1979) Some theorems of trade and general equilibrium with many goods and factors, Econometrica 47, 709-26.


Chipman, John (1966) A Survey of the Theory of International Trade: Part 3, The Modern Theory, Econometrica 34, 18-76.


Choi, Kwan (2004) Implications of many industries in the Heckscher-Ohlin trade model,
Handbook of International Trade, Volume 1, Blackwell.


Ethier, Wilfred (1974) Some of the theorems of international trade with many goods and factors, Journal of International Economics 6, 199-206.


Jones, Ron and Stephen Easton (1983) Factor intensities and factor substitution in general equilibrium, Journal of International Economics 15, 909-35.


Jones, Ron and Josť Scheinkman (1977) The relevance of the two sector production model in trade theory, Journal of Political Economy 85, 909-35.


Ruffin, Roy (1981) Trade and factor movements with three factors and two goods, Economics Letters 7, 177-82.


Stolper, Wolfgang and Paul Samuelson (1941) Protection and real wage, Review of Economic Studies 8, 58-73.


Takayama, Akira (1982) On theorems of general competitive equilibrium of production and trade - A survey of recent developments in the theory of international trade, Keio Economic Studies 9, 1-38.


Thompson, Henry (1985) Complementarity in a simple general equilibrium production model, Canadian Journal of Economics 18, 616-21.


Thompson, Henry (1990) Simulating a multifactor general equilibrium model of production and trade, International Economic Journal 4, 21-34.


Thompson, Henry (2004) Robustness of the Stolper-Samuelson price link, Handbook of International Trade, Volume 1, Blackwell.


Uzawa, H. (1964) Duality principles in the theory of cost and production, International Economic Review 5, 216-20.