Singular value decomposition of a technology matrix

• Published date: 01 March 2014
• DOI: http://doi.org/10.1111/ijet.12026
• Date: 01 March 2014
• Author: Eric O'N. Fisher

doi: 10.1111/ijet.12026

Singular value decomposition of a technology matrix

Eric O’N. Fisher∗

This paper is the first application of the singular value decomposition (SVD) in general equi-

librium theory. Every technology matrix can be decomposed into three parts: a definition of

composite commodities; a definition of compositefactors; and a simple map of composite factor

prices into composite goods prices. This technique gives an orthogonal decomposition of the

price space into two complementary subspaces: vectors that generate the price cone; and a basis

that describe the flats on the production possibility frontier. This decomposition can be used

easily to compute Rybczynski effects.

Key wor ds singular value decomposition, international trade, Rybczynski, Stolper–Samuelson

JEL classification F1, D5

Accepted 24 October 2013

1 Introduction

A technology matrix is a description of the direct and indirect factor inputs that minimize unit costs,

given local factor prices. If there are constant returns toscale, these requirements are independent of

the quantities of output. Let wbe the vector of local factor costs. Then a technology is a mapping

w→A(w).

It is customary to write this rule as an n×fmatrix whose element aij (w) is the cost-minimizing

unit input requirement of factor jin the production of good i. In the textbook case with two goods

and two factors, this rule has dimension 4, but in most empirical applications, the number of goods is

near n≈60 and the number of factors is near f≈5, so the mapping has dimension approximately

nf ≈300. In this paper, I will present a novel technique that summarizes the local information in a

technology matrix succinctly.

This technique is used routinely in the transmission of images by computers. Here is a simple

visual application that makes my point. The first image is a file of 2,800 kb (Figure 1), and the second

is of size 90 kb (Figure 2). Eventhough the compressed file is 97 percent smaller than the original, the

image is quite readily recognizable as the same economist; the Philadelphia Museum of Art and the

spring foliage around the Schuylkill are still clear in the picture with lower resolution.If it is costly to

∗Orfalea College of Business, California Polytechnic State University, San Luis Obispo, USA. Email: efisher@calpoly.edu

I would like to thank an anonymous referee, Tanner Starbard and seminar participants at the Econometric Society

Australasian Meetings at the University ofSydney and the Midwest International Economics Group at the University of

Michigan for comments on an earlier draft. I owe a great debt of gratitude to Bill Ethier,whose scholarship has inspired

me, whose conversations have shaped this manuscript, and whose friendship survived a star-crossedt rip to the summit

of Mt. Fuji.

International Journal of Economic Theory 10 (2014) 37–52 © IAET 37

International Journal of Economic Theory

SVD of a technology matrix Eric O’N. Fisher

Figure 1 The full file with 2,800 kb.

transmit information or if the original file consists of a true depiction and some noise, then it pays

to separate the wheat from the chaff.

What does it mean to compress the information inherent in a technology matrix? Consider a

matrix with n=61 goods and f=5factors:

A(w)=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

12345

10 20 30 40 50

.

.

..

.

..

.

..

.

..

.

.

590 1,080 1,770 2,360 2,950

600 1,200 1,800 2,400 3,001

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

It has dimension 305, a size typical of technology matrices in empirical work. The small “error” in

the lower right corner gives this matrix full rank 2, but it is obvious that every sector is using factors

in fixed proportions

vT=(1,2,3,4,5)

and that the sectors differ by the inverses of total factor productivities

u=(1,10,20,...,600)T.

The singular value decomposition transmits all the information inherent in this matrix—

including the minor “error”—by four vectors consisting of 2(n+f)=2×(5 +61) =132

38 International Journal of Economic Theory 10 (2014) 37–52 © IAET