Hello again, I’m going through your code and taking some notes so I am getting back to you in sections. I plan on putting everything in one post, but I thought I could comment on this particular point separately in the meantime. Also, apologies if I come across as if I am speaking down you. I recognize you are acquainted with these ideas and I’m trying to be clear to avoid any confusion for both of us and anyone else reading these comments!

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The way that you calculated labor values here can work, as long as you are multiplying the labor coefficients across columns in a row when doing the element-wise multiplication. If done correctly, you are correct in that you are essentially calculating

v = v A + l = l (I - A)^-1

which is the total labor required to produce a unit net product.

I have an example here to make my point:

1. Here is my labor coefficient vector, l

2. Define Leon as the Leontief inverse matrix (I-A)^-1

MatLab trips me up with the \ operator, so I just take the inverse explicitly and define it as Leon to avoid any confusion.

This will be a little different form your approach where you are taking Leon and then matrix-multiplying by unit vectors (a matrix of unit vectors i.e. I) to perform the sum. Here, I make the sum more explicit to step through the calculation.

3. Calculate the total labor it takes to produce a unit of net output, i.e. a (standard) labor-value.

Since I am not familiar with MatLab I am not claiming you are doing this the correct or incorrect way - you can determine this since you know MatLab better than I do - but I wanted to show you a possible wrong way to calculate v depending on how you do the element-wise multiplication.

Incorrect Way If the labor coefficient value l_i is multiplied to the values in the i-th column of the Leon matrix and you sum the values of each column (sum across rows for column i) as shown below:

Then you will be getting a vector that doesn’t correctly trace the labor inputs of each sector.

You would be accidentally calculating

(I - A)^-1 l

i.e. you would be defining v_i as Leon_i,1 l₁ + Leon_i,2 l₂ + …

instead of correctly calculating it as

Correct Way

l (I - A)^-1

i.e. v_i = l₁Leon_1,i + l₂ Leon_2,i+ …

The order of the subscripts helps keep this straight, since the embodied labor in net product i is the sum of labor going from sector 1 to sector i plus labor going from sector 2 to sector i plus …, etc.

As long as your method is doing the element-wise multiplication correctly then it will work. Here are the examples I have continued:

Here, as long as the first element of l is being multiplied to the first row of Leon, and the second element of l by the second row of Leon and etc. then when you sum the columns

you get the correct calculation of the labor value, which the above shows.

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My apologies for the earlier misunderstanding. I see that you are not aggregating