From: Robert Vienneau <rvien@see.sig.com>
Subject: Re: Eigenvalues, practical use ?
Date: Sat, 25 Nov 2000 18:49:01 -0500
Newsgroups: sci.math,sci.econ
Summary: [missing]

An use from economics:

Consider an economy which produces n goods available at the end of the
year from inputs of n goods and labor purchased at the start of the year.

Let ao(j), j = 1, 2, ..., n, be the person years of labor used in the
production of one unit of the jth good. Let a0 be a row vector with
these components.

Suppose a worker spends his wage on consumption goods denoted by the
column vector c. c(i), i = 1, 2, ..., n, is the amount of the ith good
purchased out of the wage of a worker.

Let b(i, j), i,j = 1, 2, ..., n, be amount of the ith good used as an
input in producing one unit of the jth good. Let B be the nxn matrix
with these components.

Let A be the matrix of material inputs combining B and workers'
consumption:

   A = B + c a0

where ' indicates a transpose. Let p be a row vector of prices and
r be the rate of profits. Steady state prices solve the system


   p A (1 + r) = p

Or

   p A = [1/(1+r)] p

Or

  p { A - [1/(1+r)] I } = 0.

Prices are components of the left-hand eigenvector and the rate of
profits is related to the eigenvalue.

This sort of model earned Leontief the "Nobel" prize in economics and
is widely used in indicative planning throughout the world.

Eigenvectors are also used in finding steady-state probabilities for a
Markov process, an elementary kind of stochastic process.

-- 
Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html
r           c
 v         s a           Whether strength of body or of mind, or wisdom, or
  i       m   p          virtue, are found in proportion to the power or wealth
   e     a     e         of a man is a question fit perhaps to be discussed by
    n   e       .        slaves in the hearing of their masters, but highly
     @ r         c m     unbecoming to reasonable and free men in search of
      d           o      the truth.    -- Rousseau
==============================================================================

From: Robert Vienneau <rvien@see.sig.com>
Subject: Re: Eigenvalues, practical use ?
Date: Sun, 26 Nov 2000 02:04:19 -0500
Newsgroups: sci.math,sci.econ

In article <m9_T5.1558$tj4.5136@tor-nn1.netcom.ca>, "David Lloyd-Jones" 
<icomm5@netcom.ca> wrote:

> "Robert Vienneau" <rvien@see.sig.com> wrote
 
> > This sort of model earned Leontief the "Nobel" prize in economics and
> > is widely used in indicative planning throughout the world.

> Leontief won his Nobel for sectoral input-output analysis, a neat idea, 
> well
> worked-out, but which I think nobody takes seriously any more.

Better let these people know:

  <http://www.policy.rutgers.edu/cupr/iioa/iioa.htm>

-- 
Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html
r           c
 v         s a           Whether strength of body or of mind, or wisdom, or
  i       m   p          virtue, are found in proportion to the power or wealth
   e     a     e         of a man is a question fit perhaps to be discussed by
    n   e       .        slaves in the hearing of their masters, but highly
     @ r         c m     unbecoming to reasonable and free men in search of
      d           o      the truth.    -- Rousseau