A distance measure of factor abundance

with many factors and many countries

 

 

 

Myeongjoo Kang

Auburn University

kangmye@auburn.edu

 

 

Mostafa Malki
University of Texas – Brownsville

mostafa.malki@utb.edu

 

 

 

Farhad Rassekh

University of Hartford

rassekh@mail.hartford.edu

 

 

Henry Thompson

Auburn University

thomph1@auburn.edu

 

 

 

 

Factor abundance is a bilateral concept in factor proportions trade theory that has no definition when there are many countries and various factors of production.  The present paper proposes a general definition, the Euclidean distance to the intersection of abundance rays with unit hyperplanes.  “Distance factor abundance” is compared with other measures using a data set from the literature.

 

 

 

 

Factor proportions trade theory is based on the idea that differences in relative endowments of productive factors across countries explain patterns of production and trade.  Countries would export products using their abundant factor or factors intensively. A difficulty with empirical application has been that the theory defines factor abundance for two countries and two factors while the empirical investigator faces many countries and many factors.  The present paper proposes a general definition that produces a unique ranking of countries for each factor and collapses to the two dimensional definition.

Empirical studies of factor proportions theory ideally would (but do not) include independent measures of factor abundance and factor intensity, as pointed out by Bowen, Leamer, and Sveikauskas (1987) and Leamer (1994).  There is no measure of factor abundance, for instance, in the classic studies of Leontief (1953) and Baldwin (1971).  Horiba (1974) develops a method to apply bilateral concepts to many countries but does not provide a general definition of factor abundance.  There are two measures of factor abundance in the applied trade literature, share abundance based on the theory of Vanek (1968) and world abundance developed by Leamer (1980).  The present paper introduces a “distance” measure and compares it to the other two using the data set of Trefler (1995). 

Distance factor abundance is defined as the Euclidean distance from the unit value of a factor to the intersection of a factor abundance ray with the unit factor hyperplane.  For each factor, countries are ranked by distance abundance resulting in a measure that can be used in direct estimations or applications of factor proportions theory.  The ultimate empirical issue is the extent to which countries tend to export the products that use their “abundant” factors “intensively” at least in some senses with many countries and various productive factors.

 

1.  Factor Abundance Measures in the Literature

Factor abundance is clearly defined for two countries or two productive factors.  Let vij represent the endowment of factor i in country j.  With two countries and two factors, country 1 (2) is abundant in factor 1 (2) if

(1)            v11/v21 > v12/v22.                                                                                                                                                                                        

Free trade would imply equal product prices across the two countries.  If there are identical neoclassical production functions, factor price equalization (FPE) follows and their input ratios would be identical.  With homothetic production, country 1 would produce a higher ratio of product 1 to 2.  Countries with identical homothetic utility functions would consume products in the same ratio and export the product using their abundant factor intensively. 

Vanek (1968) extends the two dimensional factor content model to any number of factors.  Country 1 is more abundant in factor m and country 2 is more abundant in factor n if

(2)            …> vm1/vm2 >…> vn1/vn2 >                                                                                                                                                                  

Factor abundance can similarly be defined when there are many countries and two factors. 

Share abundance rests on the special set of assumptions in Vanek (1968) including FPE.  Let sk be the share of country k in world income, sk = yk/yw.  Given free trade and identical homothetic preferences, country k would have to consume a share of the world output xjw of each product j according to its income share, cjk = skxjw.  Country k is share abundant in factor i if its endowment of factor i relative to the world is greater than its share of world income, vik/viw > sk or

(3)            vik  -  skviw > 0.                                                                                                     

Trefler (1995) calculates share abundance with data on nine factors and 33 countries and furnished the present data set.   The less developed countries in the sample have very small income shares due to low wages and cheap nontraded products and as a result are share abundant in many factors. 

Leamer (1980) develops the world abundance measure.  Country k is world abundant in factor m relative to factor n if

(4)            vmk/vnk > vmw/vnw.                                                                                           

World abundance is equivalent to a ranking of world endowment shares as in vmk/vmw > vnk/vnw.   Thompson (1999) shows that share and world abundance are identical to (1) when there are two countries and two factors but with more countries or factors they are weaker than the definition in (1).  With many countries and factors, world abundance might hold between pairs of countries and factors when (1) does not.  Share abundance is the weakest condition and might hold between pairs of countries and factors when world abundance does not. 

 

2.  A Distance Measure of Factor Abundance

Factor abundance is a ratio that can be treated as a distance.  The proposed definition is based on the distance from the unit value of a factor to the intersection of an abundance ray with its unit hyperplane.  Figure 1 illustrates the distance measure with two countries and two factors.  Rays m and n represent endowments of two countries.  Consider the unit value of factor 1, v1j = 1, and its intersections with the two abundance rays.  Country m is abundant in factor 1 since the distance d1m = v2m/v1m to the unit axis from ray m is less than the distance d1n = v2n/v1n from ray n.  Any number of countries can be ranked by their abundance in factor 1.  

The model with three factors illustrates the generality of distance abundance.  Figure 2 pictures the distance measure for country m.  On the left, the three factors are measured along axes v1m, v2m, and v3m.  Consider the v1m = 1 plane and its intersection with endowment ray m at point M.  The right side of Figure 2 is the v1m = 1 plane with its origin at point 1 and endowment point M.  By the Pythagorean theorem, the Euclidean distance from point M to the origin 1 is

(5)            d1m = ((n2m/n1m)2 + (n3m/n1m)2)1/2.                                                                     

Across any number of factors, the Euclidean distance for factor 1 in country k would be

(1)            d1k = ((v2k/v1k)2 + … + (vrk/v1k)2)1/2.                                                                           

To standardize by an arbitrary factor h, divide endowments of each factor i in country k by vhk to find the distance abundance,

(2)            dhk = (Siąh(vik/vhk)2)1/2.                                                                                        

This procedure produces a ranking of countries for each factor.  Distance abundance collapses to (1) in the two dimensional model.  The next step for theory would be to establish a link between distance abundance and production but there are no necessary links between factor endowments and production with more than two factors even in the special situation of universal, identical, homothetic, constant returns production functions.  As an example, Thompson (1985) uncovers 7 possible Rybczynski comparative static sign patterns of endowments differences on outputs in the model with only 3 factors and 2 products.  If production is not necessarily linked to factor endowments in a unique manner, neither is trade.  Measuring factor abundance might appear pointless without such necessary theoretical links but applied trade theorists face the challenge of applying the concepts of factor proportions theory.  The empirical issue is the extent to which factor abundance, at least under some interpretation, explains trade. 

 

3.  A Comparison of Factor Abundance Measures

This section compares the three measures of factor abundance using the data set of Trefler (1995).  All abundance measures are scaled to a maximum of 1 and a minimum of 0 for comparison.  The factors of production are listed in Table 1.

Table 2 reports share abundance.  Due to its large share of world income, the US has the lowest share abundance of all factors except capital and manufacturing labor.  At the other extreme, the LDCs have small shares of world income and high share abundances for a number of factors.   For instance, Bangladesh, Columbia, Indonesia, Pakistan, Sri Lanka, Thailand, and Yugoslavia have higher than average abundance in professional labor.  Capital, clerical, and manufacturing labor share abundances are skewed right with most countries below the mean and a thin long tail of countries with above average abundance.  The other factors are skewed left with most countries above the mean and a long tail of scarce countries.  All distributions except manufacturing labor are highly leptokurtic with narrow high peaks and many countries close to the mean.   The coefficient of variation is the standard deviation relative to the mean and it indicates a very high degree of dispersion for capital and manufacturing labor.  The least dispersed endowments are cropland and pastureland. 

Table 3 reports world abundance, each country’s share of the world endowments of each factor.  In direct contrast with share abundance, the US has the highest world abundance of every factor except agricultural labor and pastureland.  The US has the largest portion of world capital, other countries having an average of about 10% of the US level and Japan the closest at 55.8%.  In contrast to share abundance, the LDCs generally have low world abundances except for agricultural labor, cropland, and pastureland.  World abundances are all skewed to the right with most countries below the mean and a long tail of abundant countries, in contrast to the left skews of most of the share abundances.  All of the distributions except pastureland are highly leptokurtic with narrow high peaks and many countries close to the mean.  Coefficients of variation indicate a higher degree of variation than share abundance except for capital, clerical, and manufacturing labor.  The least variation occurs with cropland.  Share and world abundance generate very different measures for particular factors.  Comparing pastureland, for instance, there is a higher mean, less dispersion, more countries above the mean, and more countries close to the mean with share abundance. 

Table 4 reports the proposed distance measure, inverted and rescaled for comparison with 1 the most abundant and 0 the least abundant country.  Singapore is the most abundant country in capital as well as in more skilled labor groups due to its lack of cropland and pastureland.  The US is relatively scarce in capital and most types of labor due to its abundant pastureland.  Japan has higher than average distance abundance for capital and all types of labor.  Countries with above average capital abundances are Singapore, Hong Kong, Japan, the Netherlands, and Belgium.  Distance abundance measures are skewed to the right with very high peaks and most countries below the mean, similar to world abundance.  Coefficients of variation are relatively low except for agricultural labor, cropland, and pastureland.  Capital and most types of labor are highly leptokurtic with high peaks while agricultural labor, cropland, and pastureland are more evenly spread across countries.

Table 5 reports correlations between the three abundance measures.  Notation is S for share, W for world, and D for distance abundance.  Correlations are typically small and over half are negative.  Only 4 of the 27 correlations are greater than 0.5.  World abundance is negatively correlated with distance abundance for most factors.  The three measures are most consistent for agricultural labor.  Clearly these abundance measures measure different things and tests of factor proportions theory would vary depending on the measure utilized.

Table 6 reports the signs of US factor contents from Trefler (1995) with positive (negative) signs indicating net factor exports (imports).  Factor content calculations based on US input coefficients as in this data set are not necessarily appropriate for other countries.  The US is a net importer of capital and all types of labor except agricultural, and is a net exporter of cropland and pastureland.  Table 6 also reports “signs” of the three US abundance measures.  The sign of the original share abundance measure is reported.  For world and distance abundance, the US is compared to world means with a positive (negative) sign indicating above (below) average abundance. 

Abundance measures completely agree only for the scarcity of agricultural labor but the US is a net exporter of agricultural labor due to its capital intensive agricultural production.  Distance scarcities of capital and manufacturing labor correctly predict those net imports in direct contrast to the other two measures.  For pastureland, only share abundance has the wrong prediction.  Both share and distance abundance correctly anticipate net imports of professional, clerical, sales, and service labor.  World abundance anticipates US factor content for only 2 of the 9 factors and share abundance for 4 of the 9.  Distance abundance is correct for 7 of the 9 factors, missing only agricultural labor and cropland due to the capital intensive production techniques of US agriculture. 

 

5.  Conclusion

Bilateral concepts of factor proportions theory fall short empirically when facing data with many countries and various factors of production.  Share abundance is questionable in practice because it relies on a special set of theoretical assumptions including factor price equalization.  Having both developed and developing countries in the same sample is especially troublesome when one applying share abundance.  World abundance produces more sensible rankings but is biased due to country size.  The European Union, for instance, is more world abundant in every factor than any of its individual countries. 

Empirical economists can adopt one of the two following approaches.  In the first approach, they formulate a model as close to the specifications of the theory as possible and estimate the model using real world data to determine the applicability (not the accuracy) of the theory.  This is the approach adopted by Rassekh and Thompson (1997) for the Stolper-Samuelson theorem.  The second approach involves searching for model specifications that best explain observed trade.  Davis and Weinstein (1998) adopt this approach in explaining the factor content of capital and labor.  Both approaches are scientifically valid.

Empirical international economists using either approach want a reliable independent measure of factor abundance.  The proposed distance abundance measure is a step in the direction of a more complete realization of the empirical scope of factor proportions trade theory.

 

References

Baldwin, Robert (1971) “Determinants of the Commodity Structure of U.S. Trade,” American Economic Review 61.

 

Bowen, Harry P., Edward E. Leamer, and Leo Sveikauskas (1987)“Multicountry, Multifactor Tests of the Heckscher-Ohlin-Vanek Model,” American Economic Review 77,

791-809.

 

Davis, Donald and David Weinstein (1998) “An Account of Global FactorTrade,” NBER Report, available online at www.nber.org

 

Horiba, Yutaka (1974) “General Equilibrium and the Heckscher-Ohlin Theory of Trade: The Multicountry Case,” International Economic Review 15, 440-49.

 

Leamer, Edward E. (1994) “Testing Trade Theory,” in David Greenaway and L. Alan Winters (ed.) Surveys in International Trade, Cambridge: Basil Blackwell, 1994.

 

Leamer, Edward E. (1980) “The Leontief Paradox Reconsidered,” Journal of Political Economy 88, 495-503.

 

Leontief, Wassily W. (1953) “Domestic Production and Foreign Trade: The American Capital Position Re-Examined,” Proceedings of the American Philosophical Society,

97, 332-49.

 

Thompson, Henry (1985) “Complementarity in a Simple General Equilibrium Production Model,” Canadian Journal of Economics, 616-21.

 

Thompson, Henry (1997) “International Differences in Production Functions and Factor Price Equalization,” Keio Economic Studies, 1997, 43-54.

 

Thompson, Henry (1999) “Definitions of Factor Abundance and the Factor Content of Trade,” Open Economies Review 10, 385-93.

 

Trefler, Daniel (1995) “The Case of the Missing Trade and Other Mysteries,” American Economic Review 85, 1029-46.

 

Vanek, Jaroslav (1968) “The Factor Proportions Theory: The N-Factor Case,” Kyklos 21, 749-56.

 

 

 

Table 1.  Factors of production

K = capital                            

P = professional/technical labor            

C = clerical labor                           

S = sales labor             

R = service labor                            

A = agricultural labor

M = manufacturing labor    

T = cropland                

U = pastureland

 

Table 2.  Share Abundance

 

 

        K

       P

        C

        S

       R

      A

      M

       T

       U

Austria

.038

.687

.179

.457

.906

.631

.053

.980

.980

Bangladesh

.001

.814

.267

.671

1.00

.818

.256

.997

.997

Belgium

.041

.717

.182

.448

.891

.625

.048

.974

.974

Canada

.146

.756

.314

.439

.852

.589

.123

.912

.911

Columbia

.022

.713

.182

.486

.934

.657

.078

.990

.990

Denmark

.021

.709

.155

.451

.910

.633

.023

.985

.985

Finland

.024

.712

.121

.451

.906

.635

.028

.986

.986

France

.297

.740

.409

.391

.776

.548

.298

.829

.829

Greece

.024

.712

.153

.466

.911

.645

.054

.989

.989

Hong Kong

.012

.681

.159

.466

.921

.636

.068

.992

.992

Indonesia

.111

.953

.522

1.00

.994

1.00

.651

.976

.976

Ireland

.008

.702

.135

.461

.914

.641

.018

.996

.996

Israel

.013

.721

.146

.454

.910

.636

.008

.993

.993

Italy

.268

.760

.409

.435

.824

.584

.398

.876

.875

Japan

.600

.337

1.00

.689

.608

.482

.993

.652

.651

Netherlands

.060

.749

.188

.449

.882

.618

.049

.961

.961

New Zland

.010

.700

.146

.458

.911

.639

.016

.994

.994

Norway

.026

.709

.100

.452

.905

.633

.020

.986

.986

Pakistan

.017

.855

.276

.602

.951

.782

.374

.992

.992

Panama

.000

.695

.131

.458

.919

.643

.000

     1.00

1.00

Portugal

.019

.721

.227

.473

.926

.646

.081

.994

.994

Singapore

.011

.682

.137

.460

.912

.637

.015

.995

.995

Spain

.107

.717

.263

.480

.893

.628

.260

.947

.947

Sri Lanka

.008

.740

.187

.482

.927

.664

.065

1.00

1.00

Sweden

.031

.805

.133

.446

.896

.624

.039

.972

.972

Switzland

.050

.663

.141

.438

.886

.623

.030

.970

.970

Thailand

.028

.806

.176

.579

.933

.850

.165

.989

.989

Trinidad

.001

.688

.126

.457

.916

.641

.001

.999

.999

UK

.179

1.00

.465

.378

.831

.543

.359

.849

.849

USA

1.00

.000

.000

.000

.000

.000

1.00

.000

.000

Uruguay

.003

.704

.151

.464

.923

.642

.012

.999

1.00

W. Germany

.335

.456

.448

.400

.735

.534

.418

.803

.803

Yugoslavia

.061

.829

.283

.474

.916

.661

.171

.984

.983

mean

 

.108

.704

.239

.476

.864

.629

.187

.926

.926

variance

.042

.029

.032

.096

.029

.023

.068

.032

.032

CV

1.89

.241

.755

.298

.198

.233

1.40

.196

.197

skewness

3.04

-2.30

2.45

.584

-4.10

-1.79

1.96

-4.19

-4.18

kurtosis

12.4

10.4

10.3

10.1

20.5

12.0

6.18

21.0

21.0

 

 


Table 3.  World Abundance

 

 

K

P

C

S

R

A

M

T

U

Austria

.038

.025

.034

.033

.034

.010

.040

.009

.068

Bangladesh

.004

.040

.035

.267

.139

.478

.130

.048

.020

Belgium

.041

.040

.040

.031

.025

.004

.041

.004

.023

Canada

.139

.125

.132

.116

.116

.021

.109

.242

.800

Columbia

.023

.021

.025

.053

.056

.063

.047

.030

1.00

Denmark

.023

.026

.024

.018

.031

.006

.023

.014

.008

Finland

.025

.025

.015

.016

.020

.009

.025

.012

.005

France

.277

.219

.236

.187

.197

.059

.236

.098

.426

Greece

.025

.022

.020

.030

.022

.032

.036

.021

.175

Hong Kong

.015

.010

.018

.026

.031

.002

.042

.000

.000

Indonesia

.098

.105

.110

.702

.181

1.00

.332

.103

.398

Ireland

.011

.010

.008

.013

.007

.007

.015

.005

.162

Israel

.015

.020

.014

.010

.011

.002

.012

.002

.027

Italy

.245

.169

.188

.169

.159

.071

.260

.065

.171

Japan

.558

.317

.540

.825

.364

.202

.661

.025

.020

Netherlands

.059

.065

.055

.053

.044

.009

.048

.005

.038

New Zland

.012

.012

.013

.013

.008

.005

.015

.002

.500

Norway

.027

.024

.011

.018

.019

.005

.021

.004

.003

Pakistan

.018

.058

.042

.192

.075

.390

.189

.106

.167

Panama

.003

.004

.003

.004

.006

.005

.004

.003

.039

Portugal

.020

.018

.029

.031

.031

.024

.046

.019

.018

Singapore

.013

.006

.010

.014

.009

.001

.014

.000

.000

Spain

.101

.072

.085

.112

.094

.061

.157

.107

.357

Sri Lanka

.010

.016

.015

.032

.019

.062

.036

.011

.015

Sweden

.034

.068

.033

.032

.039

.007

.038

.016

.024

Switzerland

.049

.030

.036

.025

.027

.007

.034

.002

.054

Thailand

.029

.048

.024

.169

.055

.579

.090

.100

.010

Trinidad

.004

.002

.003

.003

.003

.001

.005

.001

.000

UK

.177

.268

.227

.140

.235

.012

.255

.037

.377

US

1.00

1.00

1.00

1.00

1.00

.095

1.00

1.00

.792

Uruguay

.005

.007

.008

.012

.013

.005

.011

.008

.454

W. Germ

.313

.171

.271

.238

.197

.070

.307

.039

.156

Yugoslavia

.057

.060

.052

.048

.041

.083

.095

.041

.213

 

mean

.105

.094

.102

.140

.100

.103

.133.

.066

.198

variance

.014

.033

.038

.058

.033

.045

.043

.031

.070

CV

.527

.520

.522

.584

.550

.484

.642

.377

.803

skewness

3.21

3.99

3.38

2.49

3.86

2.88

2.78

4.75

1.55

kurtosis

13.6

19.9

14.8

8.20

18.9

11.0

11.0

25.3

4.51


 

 

 

 

Table 4.  Distance Abundance

 

 

 

K

P

C

S

R

A

M

T

U

 

Austria

.030

.038

.031

.201

.022

.071

.019

.033

.982

Bangladesh

.001

.018

.009

.051

.026

1.00

.018

.425

.098

Belgium

.080

.149

.089

.050

.040

.068

.047

.050

.998

Canada

.006

.010

.006

.004

.004

.008

.003

.080

.745

Columbia

.002

.003

.002

.003

.003

.039

.002

.008

.273

Denmark

.018

.039

.022

.012

.020

.044

.111

.435

.137

Finland

.022

.043

.016

.012

.015

.075

.013

.581

.101

France

.025

.039

.025

.015

.015

.050

.013

.061

.918

Greece

.008

.013

.007

.008

.006

.093

.007

.031

.950

Hong Kong

.474

.604

.684

.752

1.00

.511

1.00

.005

.023

Indonesia

.001

0.02

.011

.054

.014

.830

.018

.068

.847

Ireland

.005

.008

.004

.005

.003

.025

.004

.008

.289

Israel

.035

.085

.036

.019

.021

.036

.016

.021

.718

Italy

.038

.050

.033

.022

.021

.101

.024

.100

.603

Japan

.230

.252

.257

.292

.125

.765

.167

.137

.177

Netherlands

.085

.180

.091

.064

.053

.120

.042

.031

.951

New Zland

.002

.003

.002

.002

.001

.006

.001

.001

.045

Norway

.067

.114

.032

.004

.039

.108

.031

.311

.178

Pakistan

.002

.011

.005

.016

.006

.357

.011

.168

.366

Panama

.005

.011

.006

.005

.009

.071

.004

.021

.704

Portugal

.012

.020

.020

.015

.015

.129

.016

.275

.222

Singapore

1.00

1.00

1.00

1.00

.633

.505

.782

.010

.055

Spain

.009

.013

.009

.009

.007

.050

.009

.080

.748

Sri Lanka

.009

.030

.016

.026

.015

.535

.020

.301

.298

Sweden

.023

.089

.026

.019

.022

.046

.016

.172

.354

Switzerland

.061

.072

.051

.026

.028

.074

.026

.011

.368

Thailand

.003

.010

.003

.016

.005

.584

.006

1.00

.024

Trinidad

.055

.061

.044

.031

.035

.116

.040

.478

.103

UK

.027

.080

.040

.018

.030

.016

.002

.026

.836

US

.007

.013

.008

.006

.006

.006

.004

.033

.983

Uruguay

.001

.002

.001

.002

.002

.007

.001

.004

.154

W. Germany

.074

.077

.072

.047

.039

.151

.044

.066

.858

Yugoslavia

.012

.024

.012

.008

.007

.157

.012

.051

1.00

 

mean

.074

.096

.081

.085

.069

.205

.077

.154

.488

variance

.035

.039

.042

.046

.038

.072

.046

.017

.131

CV

.393

.488

.393

.396

.346

.759

.359

.702

1.35

skewness

4.00

3.54

3.57

3.37

3.86

1.58

3.59

2.15

0.17

kurtosis

18.9

15.3

14.8

13.3

16.8

4.25

14.4

7.78

1.34

 

 

 Table 5.  Correlations of Abundance Measures

 

K

P

C

S

R

A

M

T

U

 

 

SW

.999

-.708

.323

.087

-.942

.548

.959

-.900

-.404

DS

-.028

-.091

.029

.061

.057

.453

-.084

.154

-.312

DW

-.030

-.088

-.033

-.012

-.089

.726

-.084

-.065

.264

 

 

 

 

Table 6.  US Factor Content and Abundances Relative to World Means

 

K

P

C

S

R

A

M

T

U

 

 

Content

-

-

-

-

-

+

-

+

+

 

S

+

-

-

-

-

-

+

-

-

W

+

+

+

+

+

-

+

+

+

D

-

-

-

-

-

-

-

-

+

 

 

 

 

 

 

 

 

   v2n/v1n

 

   v2k

 

n

 
 

m

 

  v2m/v1m

 

v1k

 

1

 
 

 

 

 

 


                            

Figure 1.  Factor abundance distance

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 


v2m/v1m

 

M

 

m

 
Figure 2.  Factor abundance with 3 factors
 

v3m/v1m

 

1

 

M

 

v3m

 

v2m

 

1

 

v3k

 

v1k

 

v2k