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.
Introduction
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 a_{ij}
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 a_{11}/a_{21} > a_{12}/a_{22}. This two dimensional measure can be extended
to any number of products. In the 2x3
model, the intensity ranking is a_{11}/a_{21} > a_{12}/a_{22}
> a_{13}/a_{23} 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 overdetermined
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 a_{11}/a_{12} > a_{21}/a_{22} and
adding factors extends this opportunity cost ranking. In the 3x2 model, a_{11}/a_{12}
> a_{21}/a_{22} > a_{31}/a_{32} 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.
MWFI
The mean weighted
factor input factor i in product j is as m_{ij} ≡ a_{ij}/m_{i}
where m_{i} = S_{j}a_{ij}/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 m_{ij} > 1 > m_{ih} industry j uses factor i more
intensively than the average industry, and industry h less intensively. The ratio m_{ij}/m_{ih} = a_{ij}/a_{ih}
indicates the opportunity cost of product j in terms of product h. If m_{ij}/m_{ih} > m_{kj}/m_{kh} industry j has a higher opportunity
cost than industry h in factor i relative to factor k.
Comparing factors
across an industry m_{ij} > 1 > m_{kj} 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 m_{ij}/m_{kj} = (a_{ij}/a_{kj})(m_{i}/m_{k}). If m_{ij}/m_{kj} > m_{ih}/m_{kh} 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 v_{i }of
factor i would be rescaled to v_{i}/m_{i}
and factor prices w_{i} rescaled to m_{i}w_{i}. Competitive pricing conditions p_{j}
= Σ_{i}a_{ij}w_{i} remain consistent with
exogenous world prices p_{j}. Full
employment conditions v_{i} = Σ_{j}a_{ij}x_{j}
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 F_{rxn}. The following
section examines how well F_{rxn} anticipates the comparative static
results of an applied factor proportions model.
Suppose there are c countries and define matrix B_{nxc} as the
mean weighted factor abundance matrix.
An empirical test of factor content theory would involve the empirical
relationship between F_{rxn}, B_{nxc}, and the X_{nxc}
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 q_{ij} = a_{ij}w_{i}/p_{j}
for capital and eight Census skill groups of labor. Output is value added with a unit of output
defined by p_{j} = 1. Capital
shares are residuals of value added after labor shares. Mean weighted factor shares equal mean
weighted factor intensities since 3q_{ij}/S_{j}q_{ij} = 3a_{ij}w_{i}/S_{j}a_{ij}w_{i} = 3a_{ij}/S_{j}a_{ij} = m_{ij}.
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

q_{iA} 
m_{iA} 
w_{iA} 

q_{iM} 
m_{iM} 
w_{iM} 

q_{iS} 
m_{iS} 
w_{iS} 

a_{iSM} 
a_{iSA} 
Capital 
.576 
1.65 
0.32 

.214 
0.61 
0.45 

.261 
0.75 
1.13 

1.22 
0.45 
Professional 
.059 
0.37 
0.18 

.148 
0.93 
0.42 

.269 
1.69 
1.60 

1.82 
4.56 
Tech/Sales 
.033 
0.28 
0.17 

0.12 
0.96 
0.60 

.211 
1.76 
1.78 

1.83 
6.39 
Service 
.003 
0.18 
0.15 

.008 
0.47 
0.75 

.041 
2.41 
1.90 

5.12 
13.7 
Resource 
.139 
2.90 
5.94 

.002 
0.04 
0.16 

.002 
0.04 
5.11 

1.00 
0.01 
Crafts 
.086 
0.74 
0.04 

.167 
1.44 
1.31 

.096 
0.83 
0.36 

0.57 
1.12 
Operators 
.045 
0.34 
0.17 

.286 
2.17 
3.44 

.067 
0.51 
2.27 

0.23 
1.49 
Transport 
.030 
0.94 
0.01 

.031 
0.97 
0.77 

.036 
1.13 
0.22 

1.16 
1.20 
Handlers 
.008 
0.44 
0.25 

.028 
1.56 
0.28 

.018 
1.00 
0.47 

0.64 
2.25 
Each input has
two sets of ratios across the three sectors.
Input ratios in services relative to manufactures are S/M ≡ a_{iS}/a_{iM}
= θ_{iS}/θ_{iM} with the skilled wage w_{i}
equal across sectors and p_{A} = p_{M} = 1. This input ratio S/M and the
service/agriculture intensity S/A ≡ a_{iS}/a_{iA} 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 a_{ij} 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 q_{im} column for manufacturing, capital
appears more intensive than handlers but reading down the mean intensity a_{im} column capital is used only 42%
(0.61/1.56) as intensively as handlers.
There are other such examples.
The MWFI
anticipate the StolperSamuelson (1941) dw_{i}/dp_{j}
elasticities reported in the third columns for each sector in Table 1. The model uses estimates of translog
production functions across states but these dw_{i}/dp_{j}
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 24 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.
Conclusion
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.
References
Chang, Winston ((1979) Some theorems of trade and general equilibrium
with many goods and factors, Econometrica
47, 70926.
Chipman, John (1966) A Survey of the Theory of International Trade:
Part 3, The Modern Theory, Econometrica
34, 1876.
Choi, Kwan (2004) Implications of many industries in the HeckscherOhlin
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, 199206.
Jones, Ron and Stephen Easton (1983) Factor intensities and factor
substitution in general equilibrium, Journal
of International Economics 15, 90935.
Jones, Ron and José Scheinkman (1977) The relevance of the two sector
production model in trade theory, Journal
of Political Economy 85, 90935.
Ruffin, Roy (1981) Trade and factor movements with three factors and two
goods, Economics Letters 7, 17782.
Stolper, Wolfgang and Paul Samuelson (1941) Protection and real wage, Review of Economic Studies 8, 5873.
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, 138.
Thompson, Henry (1985) Complementarity in a simple general equilibrium
production model, Canadian Journal of
Economics 18, 61621.
Thompson, Henry (1990) Simulating a multifactor general equilibrium
model of production and trade, International
Economic Journal 4, 2134.
Thompson, Henry (2004) Robustness of the StolperSamuelson 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, 21620.