Here is the lemmygrad post I made it at (don’t wanna have to copy everything over).
Please give the post lots of 
, the post would really appreciate it
Don’t be afraid to ask questions.
Here is the lemmygrad post I made it at (don’t wanna have to copy everything over).
Please give the post lots of , the post would really appreciate it
Don’t be afraid to ask questions.
I have some comments, questions, and possible suggestions. I think this is really great, and simulations are incredibly fun and I love digging into them. Some of my questions are because I know Python but not MatLab, so I may need to check I understand what your code is doing. Other comments may be due to differences in how quantities are calculated (particularly how the labor-value and the costs were calculated), and your final comparison of aggregated quantities instead of sector quantities. Keep in mind that I don’t mean any comment to sound aggressive, and it is possible that I am misunderstanding the code or the concept that the calculation represents.
Because I don’t know MatLab well, I’m going to go through the sections of the code and we can confirm my understanding of each part.
1.) You make N economic simulations. For each iterative economic simulation, i, you:
2.) First, generate a random net product vector n for the global economy (you call it o in the simulation).
3.) Then you randomly generate an input output table A.
4.) Then you calculate the gross product vector q, or as you call it O. This is calculated via q = (I-A)-1 n. It took me a while to realize that “\” is MatLab’s way of doing a matrix inverse followed by a multiplication. So A \ b is MatLab’s way of calculating A-1b, correct?
5.) You calculate the gross labor use as L=l q. This is element wise multiplication, or equivalently, the dot product. I wasn’t sure why you normalized the net product and the gross labor, though.
I added in the explict checks. It turns out, a huge number of the matrices being produced were non-productive. Instead of trying to keep generating matrices, I made a different fix (which makes the technical matrices more realistic, so win-win)
I made it so that the average number of entries in each row of the technical matrix is (2*n)^0.5
This means that as the economy grows larger, the matrices grow sparser. This makes productive matrices much more likely (at which point, I just have a check which makes it so that non-productive matrices are regenerated).
Curiously, this change doesn’t have that big of an effect on the outcome. I’ve verified. The model simply handles negative net production and treats it like purchasing commodities from the external market (so something like imports). Still, I have removed it for now.
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 pi / vi for each sector i.
Assume 1 unit of net output for a commodity
Calculate how much gross production would required for each sector in that case
Element wise multiply this with direct labor use to get needed labor from each sector
Add up all of the labors from all the sectors.
In formal terms, for sector 1
(I-A)^-1 * [1;0] = g = Gross product for 1 unit of sector 1
Then
sum(g.*l)
The “(eye(n)-A)\eye(n))” computes “g” for every sector at once (the output is [g1 g2 g3 g4 …])
I believe this is equivalent to the equation you have also provided, except your equation involves fewer steps.
Also, just as a context thing, matlab, for some reason sums matrices down columns by default. So the output the ltv prices equation is a row vector of the summed labors (which I transpose using the apostrophe symbol ')
Indeed I am pretty sure that my code actually is using prices and values for each sector separately, otherwise the code should be giving me an error regarding the dimensionality of the code.
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!
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:
Here is my labor coefficient vector, l
Define
Leonas the Leontief inverse matrix (I-A)-1MatLab trips me up with the
\operator, so I just take the inverse explicitly and define it asLeonto avoid any confusion.This will be a little different form your approach where you are taking
Leonand 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.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 li is multiplied to the values in the i-th column of the
Leonmatrix 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 vi as
Leoni,1 l1 +Leoni,2 l2 + …instead of correctly calculating it as
Correct Way
l (I - A)-1
i.e. vi = l1
Leon1,i + l2Leon2,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 ofLeonand etc. then when you sum the columnsyou get the correct calculation of the labor value, which the above shows.
My apologies for the earlier misunderstanding. I see that you are not aggregating
Matlab element wise multiplication indeed works in the correct way that you described.
sum(Leon.*l)
is equivalent to
l*Leon
which is why MatLab’s “sum” function by default sums down columns rather than by rows. The sum(Leon.*l) notation keeps things explicit (helps me in coding consistently), but the MatLab compiler knows how to optimize these things.
Thanks for the response! Hopefully I’m not overloading you with questions - this is helping me understanding MatLab and answering some other questions I had. I’ve asked another question too re. the wages for whenever you have the time.
Thanks again!
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 costi = p1 a_1,i + p2a_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 = AT 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
costi = unit costi x qi = (p1 a1,i + p2a2,i + …) qi
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 pi qi. 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 li qi = w Li. 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 li qi = w Li
Divide the working class into sectors, just as the industry is divided, and note that the i-th working class consumes a proportion of Li/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 pi ni Li/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!
the 1000 different prices are there to see what happens to the reproduction condition with 1000 different prices. I am not directly calculating reproduction prices for each economy. I am just letting random guesses show me what happens to the reproduction condition at various price points.
The net income being normalised was there just to improve the visualisation. I have run the code with all sorts of parameters with and without the normalisation, and it is difficult to decide which is more useful for gaining insight.
I’ve tried generating economies with upto 100 sectors (my poor laptop), but right now, I am facing a different problem I am trying to solve (with more and more sectors, my current random price generation strategy rarely ever produces prices close to LTV. Law of large numbers and all).
This is what I have done in C (except I also multiplied by gross output yo get total costs per sector)
R is a vector denoting the revenue by sector. I think part of your misunderstanding might be from MatLab’s element-wise multiplication function, whose output can be difficult to understand.
O is a n long coming vector, P is a n long column vector, when element wise multiplied, the output is also a n long column vector.
And I do suppose that using standard notation (which I have never seen before tbh) would probably help greatly.
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.
To test Wright’s non-standard value you’ll need to introduce an exploiting class. For the purpose of the simulation you would have to add some population that consumes but does not contribute labor to production. You can have separate consumption vectors for the working class and the exploiting class. The net product n would then be sum of the worker’s and the exploiter’s consumption, n = cW + cK. Start with the exploiting class consumption = 0, and you can slowly increase it to check the impact it has on prices.
For a micro model which with no exploitation and is simple commodity production you can find his paper The Emergence of the Law of Value in a Dynamic Simple Commodity Economy. For a different micro model with exploitation (but abstracts away from different commodities and input output tables) you can check out his Implicit Microfoundations for Macroeconomics or his The Social Architecture of Capitalism, both papers have similar simulations. But value isn’t discussed in the last two papers.
For a macro model, with the emergence of value and prices corresponding to values, you can check out Convergence to Natural Prices in Simple Production for simple commodity production with no exploitation and Classical Macrodynamics and the Labor Theory of Value for a macro model with exploitation.
Also, for a theoretical discussion of the work you can either check out his thesis, The Law of Value or specific papers such as a discussion paper where he discusses his non-standard value Nonstandard Labor Values, or a more detailed paper Marx’s Transformation Problem and Pasinetti’s Vertically Integrated Subsystems
I find this field fascinating and I love simulations. Hopefully I can share something of my own one day. So I’d like to congratulate you, and also love to hear about any updates!
All of those sound interesting, I will check them out.
I would love to share them. Hopefully I have something good by the end of the week.
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!
correct
correct
I was under the impression that since matlab’s “rand” generates values between 0 and 1, all of the technical matrices should be automatically productive, but you are correct. I should add explicit checks
yes
I normalise the net product so that different economies are more comparable to one another. I am interested in the ratio of output between 2 industries, and not the scale of overall production
I normalise the gross labor use because that variable should actually be “share of workforce employed”. The name is a relic from my previous attempt at the code where I had tried to incorporate the population into the sim. I seem to have forgotten to change the name.
Matlab’s elementwise multiplication is not a dot product
Exactly what I’m looking for :)
I’m not exactly a mathematician, so I had intended the simulation to more so be a learning excercise. Feedback is appreciated. I’m going to take a look at your comments once I am done sleeping 😴
Looking forward to any discussion. And like I said, I may be misunderstanding some things!