# Eigenvalue and Eigenvector problem

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My friend asked me to help him on his math for his economics, but I got rather lost since I do not know too much about Linear Algebra. His problem was that he had an economy (this is, for those who are curious, from Piero Sraffa) such that the inputs equal the outputs. The economy was:

280 qr. Corn + 12.t. iron --> 400 qr. corn

120 qr Corn + 8 t. iron --> 20 t. iron

The solution to me was simple, just subtract from both sides the corresponding units to derive the exchange values of 10 qr. Corn = 1 t. iron.

That much was correct, but I think by coincidence. Where my friend and I got lost was with a surplus. Suppose the economy became

280 qr. Corn + 12.t. iron --> 575 qr. corn

120 qr Corn + 8 t. iron --> 20 t. iron

I thought "Aha! Use an Eigenvalue...or eignevector...or some eigenthing!" However, that's the only thing I've heard of in Linear Algebra.

I do know that Piero Sraffa (that madman who came up with this Linear nightmare) said to use a rate of profit scalar, r, and multiply both sides by "1+r". The problem alas was that he never stated how to find the thing!

I tried mathworld, but all I got was that I think I need some sort of right eigenvector. I'm more lost than found If some bright young feller would help, that would be fantastic.

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My friend asked me to help him on his math for his economics' date=' but I got rather lost since I do not know too much about Linear Algebra. His problem was that he had an economy (this is, for those who are curious, from Piero Sraffa) such that the inputs equal the outputs. The economy was:

280 qr. Corn + 12.t. iron --> 400 qr. corn

120 qr Corn + 8 t. iron --> 20 t. iron

The solution to me was simple, just subtract from both sides the corresponding units to derive the exchange values of 10 qr. Corn = 1 t. iron.[/quote']

presumably these aren't equations so it makes no sense to talk about subtracting things from each side, and if they are equations then they make no sense.

Nevermind...

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Unfortunately, I'm not familiar with the language of economics, but it seems to me that I understand the meaning of the problem.

When economy is balanced, we have the constant exchange rate: 10 qr corn to 1t iron. This is obtained by solving the system of equations:

280x+12y=400x

120x+8y=20y

These two equations are linearly dependent and only the relation 10x=y may be recovered.

Transferring RHS to LHS we have:

-120x+12y=0

120x-12y=0

We can write it in the matrix notation:

$\left( \begin{array}{cc}-120 & 12\\ 120 & -12 \end{array} \right) \left( \begin{array}{c} x\\y \end{array} \right)=0$

Thix matrix has the eigenvalues 0 and -11, while the value 0 corresponds to the eigenvector (1,10). So I guess, that the balanced economy corresponds to the eigenvalue 0 and then the exchange rate assumes values 10 to 1.

Now for the surplus conditions we have

$\left( \begin{array}{cc}-295 & 12\\ 120 & -12 \end{array} \right) \left( \begin{array}{c}x\\y \end{array} \right)=0$

This matrix has eigenvalues -300 and -7, while the latter corresponding to the exchange rate 1 to 24, so I guess that (1,24) is the "equilibrium distribution" and the eigenvalue -7 characterizes something like the rate of convergence to the equilibrium (maybe this is the "profit rate" you have mentioned).

I'm pretty sure that not everything I've written is actually correct, but, perhaps it will push you into the right direction.

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