Jump to content
Sign in to follow this  
Freeman

Eigenvalue and Eigenvector problem

Recommended Posts

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 :confused: If some bright young feller would help, that would be fantastic.

Share this post


Link to post
Share on other sites
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.

Share this post


Link to post
Share on other sites

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:

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

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

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

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.

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
Sign in to follow this  

×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.