Here is the lemmygrad post I made it at (don't wanna have to copy everything over).
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, the post would really appreciate it
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this post was submitted on 29 Mar 2025
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chapotraphouse
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6.) Calculating the LTV “prices” is the part that confuses me the most and I have a few comments and questions on
You calculate this by first calculating (I-A)^-1^, and then perform an element-wise multiplication (this is what “.*” does in MatLab, correct?) by the labor coefficient vector l. Then you sum this together.
I am confused by this calculation as the standard equation for calculating labor values is v = v A + l = l (I-A)^-1^. Note that l is a 1 x n row vector and so v is also a row (or left) vector. Alternatively, you could calculate with column vectors and the transpose of the inverse, i.e. v = ((I-A)^-1^)^T^ l.
This gives you the amount of labor required in a vertically integrated subsystem to produce one unit net product. This is using Sraffa and Pasinetti’s work to give a more concrete theoretical understanding of what a labor-value is. It is like a total labor input per unit net product. By that definition, though, it isn’t meaningful to sum up these elements as they don’t have the same units. You would first have to multiply v by some commodity-quantity, typically the net product. Note that v n = L, this provides an alternative way to “dividing up” the economy’s social labor.
This is similar to how you couldn’t actually meaningful sum up the elements of a price vector, p. A price has the units of money per unit commodity, and each commodity would hence have a different price unit. You would first have to multiply each price by the quantity of commodities that one is purchasing in order to convert it to a common unit (money) and then you could add it up. The same logic works for the standard definition of the labor value vector.
Another reason to not sum the value vector is that it would be useful to compare each element of the value vector, i.e. each commodity’s labor value, to the emergent price in the market. But, this would require you to add a mechanism where each firm can adjust prices and I don’t believe that has been added into the model yet. At the moment you are comparing aggregate quantities, i.e. the sum of prices with the sum of values-per-commodity (which I am not confident is meaningful in your present calculation), but an improvement could be to compare each sector’s emergent price (the price required for reproduction) with the sector’s labor value. In another post I have some papers where you can read how Ian Wright’s simulations handle the price adjustment and the reallocation of labor. This can be one possible future direction to head toward. I.e. you can inspect the ratio of p~i~ / v~i~ for each sector i.
7.) Then you have a vector of randomized prices. This is an n x 1000 table, though. I am confused why there is an extra 1000 here, when the loop itself will create 1000 instances of the economy. I may be misunderstanding the intention of this step so any clarification can help.
8.) You then calculate the costs, the sales to consumers, the total wages paid out, and the net income (I am also confused why the net income is normalized)? Some of the cost calculations confuse me, so I will discuss my understanding of calculating cost and we can also check if we are converging on our understanding: Sector i will spend the following amount of money on their means of production per unit product:
unit cost~i~ = p~1~ a_~1,i~ + p~2~a_~2,i~ + …
This is the amount of money that sector i must shell out to all other sectors in order to built a unit quantity. You can think of it as the row vector of prices multiplied by the i-th column in the input-output matrix A.
This unit cost for each sector can be expressed via matrix algebra as
unit cost = p A where p is a n x 1 row vector of prices
This gives you a row vector of each sector’s unit cost.
Alternatively, you can calculate this as unit cost = A^T^ p where p is a column vector. I just prefer to use column vectors for physical quantity vectors, and row vectors for prices, labor coefficients, and values. It makes the math easier to write, and you avoid transposing. It also hints at a duality present in the system.
Since this is a unit cost, we can calculate the total cost of production for sector i as
cost~i~ = unit cost~i~ x q~i~ = (p~1~ a~1,i~ + p~2~a~2,i~ + …) q~i~
You can do this for all sectors by first calculating the unit cost vector above, so you can do p A using matrix multiplication, then you can convert these unit costs to real costs by element-wise multiplication with q. This will give you each sector’s individual costs for means of production.
If you wanted aggregate costs for the whole economy though, then you can do the matrix multiplication of p A q, but I would suggest moving from an aggregate to a sector-specific model so you can test the differences between sector prices and sector labor values. I think that would be very interesting!
The money that each sector receives would then be the element-wise multiplication of p~i~ q~i~. If you wanted an aggregate quantity across all sectors, this would be the p q - you have already calculated this as R.
Then you also need to include the wage payments for each sector. Each sector pays there workers w l~i~ q~i~ = w L~i~. In the aggregate if workers are spending all of their wage and they are the only consumers of the net product, then w L = p n. But if you wanted to find how much each sector was spending for their wages, then you have to disaggregate this. You could Introduce the wage as a new parameter. Then the total wage spend by each sector is w l~i~ q~i~ = w L~i~
Divide the working class into sectors, just as the industry is divided, and note that the i-th working class consumes a proportion of L~i~/L n of the net product. Then, by conservation of the worker’s wages with their expenditure, note that wage spent by each sector is then p~i~ n~i~ L~i~/L.
With that, you can then find the net income as you have. And perhaps this is what you’ve done, but just in an aggregate way. As I mentioned, MatLab code is a little difficult for me to understand. And also, I think disaggregating this can be a good next step!
9.) The sector income then acts as a proxy for reproducibility. If the income is negative, then the sector is not sustainable and the prices must be updated. A future direction can be to have some dynamic model where each sector can adjust its prices if it is finding that the income is negative. A problem though that you will have to address if you do move in this direction is you will have to have a mechanism for modeling prices in a semi-realistic fashion and address appropriate labor reallocation.
If a sector’s income is very high then perhaps it is because the prices are very high. But the prices won’t be high in a real market economy if the supply of the product is also very high, as competition would result in competitive firms within the sector lowering prices for market advantage. A high supply would result in a downward pressure on prices. This downward pressure causes the income to lower over time, and firms would start to move out of this sector and labor will be reallocated to other sectors. This also occurs in reverse for sectors with a high demand but low supply. This mechanism is important for the operation of the law of value.
I am very well aware that the dynamism is important for the law of value. I have tried making simulations of micro actors in the past to simulate commodity exchange, but those tended to become computationally intensive beyond what my cheap laptop could handle. (Either that, or trying model various effects to make it more accurate would start becoming like a full time job, forcing me to focus on my actual studies instead. )
This simulation is intended only to model a single time step (for now, adding more time steps comes in once I perfect the simulation for a single time step).
I was in that same boat with my micro models. It got to where I felt like I had to add X, but to add X properly it felt like I also needed to handle Y, and so on and so on. So I get the struggle! I can have a problem of not knowing when and how to set the level of abstraction and not let the perfect be the enemy of the good. I have so many scrap projects that stall out because of this. And proletariat science unfortunately has to deal with the fact that we have other jobs that take away our time.
I will give it another look now that I understand what you were doing with the 1000 prices and that you did disaggregate. It isn't bad code, it just isn't my first language!
Don't know when I will get back, but I hope do it soon as I genuinely love this sort of stuff!