Linear algebra:Input output matrices>

[oddiseas]

Homework Statement

I am trying to make sense of some of the terminology.
Basically i get that each industry in an input matrix is dependent on each of those industries to some specified extent in order for production.

Is the output what each sector produces before we take into acount the consumption of each matrix? and is the production what we have left after consumption?

I will post two sample questions that i have been working on.

1)If the gross production for this economy is $10 million of tourism, $15 million of trans-
portation, and $20 million of services, what is the total value of the inputs consumed by
each sector during the production process?

Is the gross production what we have after consumption.Ie Cx=total consumption, and thus total production +consumption = total output?

2)If the total outputs of the tourism, transportation, and services sectors are $70 million, $50
million, and $60 million, respectively, what is the net production of each sector?

So in this part they have used the term output and net production,Now what is the output, is it everything produced "prior" to consumption of each sector? and what is the difference between net production and gross production.The book i have has like 1 page on input output, so i couild use some help, because i want to make sure i am interpreting everything correctly.

(i have not posted the consumption matrix because i am trying to get a better ubderstanding of the terminology and what it means, solving is pretty straightforward)

Homework Equations

The Attempt at a Solution

 

[Mark44]

I'll take a stab at this, but you should also verify what I say against what your book says.

The input and output refer to vectors that are associated with your consumption matrix. An input vector represents the amounts of money fed into a consumption matrix in regard to some number of industries, and an output vector represents the money produced by each of these same industries. The output vector would represent (I believe) the gross production of these industries. The net production would be a vector of the gross production minus the cost to produce those amounts.

You haven't told us anything about the production matrix, so I'll make some assumptions. In the first problem, you mention 3 industry sectors, so I'll assume that the matrix is 3 x 3.

You're basically looking at the matrix equation Ax = y, where A is the matrix, x is the inputs, and y is the outputs.

For the first problem, you are supposed to solve for x in the equation Ax = <10, 15, 20>^T. Hopefully, you know something about matrices, so that should suggest something you can do to solve for x, the inputs.

 

[oddiseas]

Hi, thanks for the reply.It did help me.

Can anyone help me verify this. I basically spent awhole day studying only this topic and i think i almost have it.

b)If the gross production for this economy is $10 million of tourism, $15 million of trans-portation, and $20 million of services, what is the total value of the inputs consumed by each sector during the production process?

So the total production will be what is produced prior to taking into acount "consumption". Thus A.p
where p is the production vector gives us the consumption "from each sectors production"
But the question is worded as what is consumed "by each sector' not what value of the production is consumed which would imply the total consumption from each three sectors from a given production output sector. So what i did in this case was represent the consumptions in matrix form and sum each column for the total amount of the "total productio: that each sector consumed.

 

[Mark44]

Without seeing the whole problem, I can't say for sure, but the p vector in the matrix product Ap is, I believe, the input vector, not the production vector. The total production would be the vector Ap.