MatLab simulation of labor theory of value pricing

So, recently, I made a matlab simulation/visualization for prices in a 2-sector economy. I did this to see for myself whether labor theory of value prices are the prices at which an economy can reproduce itself exactly. That is, when the unit price of each sector matches the price predicted by the LTV, then the net financial position of both sectors of the economy (and the consumers) neither improves or worsen (so nobody is going into debt to another).

Of course, this was inspired by Marx's famous reproduction scheme method, where he did something similar. However, I have better tech than he did.

I basically randomly generated a 1000 different random economies, and in each economy, I generate 1000 different prices. Then, for each price and economy, I computed the balance of payments between the 2 sectors and the consumers. I also computed the predicted LTV prices for each of the 1000 economies

These 1 million data points are plotted on a density map in the linked image. The brighter the spot on the map, the more data points fell there. And it's exponential. A 7 on the color scale has almost 3 times as many points there as a 6.

The y-axis is the net income of sector 1. If that net income is anything but 0, the economy isn't reproducing itself. The x-axis is the ratio between the actual price ratio of the sectors (randomly generated for each point) and the price ratio predicted by the LTV.

If you look at this graph, you will see 2 black regions. There are sharp black regions both horizontally and vertically emanating from the center point (the situation of reproduction and LTV prices). This means that reproduction implies LTV prices, and LTV prices imply reproduction. So the only way for a perfectly stable economy to exist is for it to employ LTV prices, and vice versa, an economy is stable if it employs LTV prices.

This is pretty much what the theory expects. What might be an interest further avenue to research is that hourglass shape. From pretty much the moment I started coding the simulation (so even during my early attempts when I was doing things wrong), I kept seeing that hourglass shape. I don't know if it is an artefact of how I did my simulation, or if it is something more fundamental.

If I could create a model for the shape of that hourglass, I could use it to predict things like

"How much can the prices in this economy deviate from LTV prices given this much growth in a sector?"

Which would be very cool.

Unfortunately, the ability to code basic shit in MatLab does not give me the ability to do advanced mathematics, so I will really need to think about this.

For anyone interested in my actual simulation approach, I have it written out in a comment (and will also provide the code)

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This is so cool!
- Wait till you see what I'm cooking up next

pancake also did some simulations about LTV a while ago, here are their posts about it: Marxist Game Theory? et Marxist Game Theory: correction.

God I know I'm a fucking nerd by the fact that I thought this was cool as hell, and could comprehend it a little (though I'm much more familiar with Python and industry specific coding, rather then economic analysis.)

        mag = 1000;
    n = 2;
    N = 1000;
    
    Data = zeros([n N*mag]);
    
    %%%%%%%%%%LOOP
    
    for i = 1:N
    
        %Net production of [MOP; MOC]
        o = rand([n 1]); %[MOP; MOC] of net production
        o = o./sum(o); %Normalise the net production
        
        %Direct Labor use for [MOP; MOC]
        l = rand([n 1]);
        
        %Technical matrix:  [MOP to MOP, MOP to MOC;
        %                    MOC to MOP, MOC to MOC]
        A = rand([n n]);
        
        %Gross production
        O = (eye(n)-A)\o;
        
        %Gross labor use
        L = O.*l;
        L = L/sum(L); %Normalising gross labor use
        
        %LTV prices calculation
        LTV = sum(((eye(n)-A)\eye(n)).*l)';
        
        %Randomised prices
        P = rand([n mag]);
        
        %Interindustry flows
        %F = A.*(O'); %Physical flows, NOT USED IN THIS SIM
        
        %Inter-industry sales
        R = O.*P;  %Market value of gross production
        C = A' .*O*P; %Costs of inputs to production
        
        %Industry to consumer sales (assuming consumers save/lose nothing)
        S = o.*P; %Sales to consumers
        Y = sum(S); %Total consumer spending = wages paid out
        W = L.*Y; %Wages paid out vector by industry
        
        M = R - C - W; %Net Income by industry
        M = mag*M./sum(abs(M),2); %Normalise net income
        
        specific_price = log((P(2,:)*LTV(1))./(LTV(2)*P(1,:))); %the x-coordinate variable
    
        Data(:,(1+ (i-1)*mag ):(i*mag)) = [specific_price; M(1,:)]; 
    
    end
    %%%%%%%%%%LOOP
    
    pts = linspace(-3, 3, 251);
    H = log(histcounts2(Data(1,:), Data(2,:), pts, pts));
    imagesc(pts, pts, H);
    axis xy;
    set(gca, 'XLim', pts([1 end]), 'YLim', pts([1 end]), 'YDir', 'normal');
    colormap copper
    a=colorbar;
    a.Label.String = "Density of simulation outcomes [natural log scale]";
    
    xlabel {Ratio of actual price to LTV price [natural log scale]}
    ylabel {Net income of sector 1 [linear scale]}
    
    saveas(gcf,"repro.png");
    `

Why does this formatting look so weird 😭

My approach in the sim was:
Randomly generate a 2-sector economy a. generate 2x1 net production vector (total product sold to consumers) b. generate 2x1 direct labor vector (direct labor needed to produce 1 unit of gross output) c. generate 2x2 technical matrix (row i column k tells how many units of product i is needed to make 1 unit of product k)
Calculate the economy a. Calculate gross production as (Identity_matrix - technical_matrix)^-1 * net_production Gross production is a 2x1 vector of the total production of both sectors. The gross products of both sectors have to be consumed as intermediate goods to support the production process (ex - some electricity is needed to make steel, and some steel is needed to make electricity). This formula for computing the gross product can be found from Paul Cockshott's Towards a new socialism b. Calculate the gross labor use for each sector by multiplying their gross production and direct labor use per product.c. Predict LTV prices for sector k by calculating how much gross production would be needed to produce 1 unit of net output (formula in 2a), then finding how much labor would be used to do that gross production (formula in 2b)
Randomly generate a 1000 pair of prices (1 per sector) and compute financial flows (so this step is repeated 1000 times for each pair of prices)a. Each sector pays the other for the product it purchased from that sector and at the price that was generated.b. Each sector sells its net output to consumers at the randomly generated prices.c. It is assumed that consumers are not going into debt, which means the total money they spending on net consumption comes from the wages (or dividends) paid by industry. Wages are paid by each industry proportionally to their gross labor usage
Repeat steps 1-3 a thousand times. This gives an overall 1 million data points from 1000 distinct economies.

There's a little bit of discussion on the hexbear post going on.

8 comments