It’s not even “no consequences to anything”, but “the people dying and suffering are doing so outside my bubble”

Interpretation of the graphs.

Graph 1: We still see the same result. When the prices of an economy are at those predicted by the LTV, the income of every sector shrinks to 0, leading to perfect economic reproduction. However, we see that many economies have economic reproduction even without LTV prices. I have a hypothesis for this. Some of the randomly generated economies in the simulation are “disconnected”, meaning that the different industries don’t buy and sell to each other. In this case, the effect of prices of one industry on another are minimum, so the prices stop mattering much.

Graph 2: Same as graph 1, but the shape of the curve is different. Not really sure what to say about this

Graph 3: I found it very interesting that no matter how much I tried to increase the wages (at one point, I had a wage basket 2 times bigger than what the economy could actually produce on its own), the trade balance remained stubbornly positive for the overwhelming majority of the data points.

This could happen because the sectors were reorganizing themselves to exploit comparative advantage, even though I never coded them to do this!

Say the people of the country were consuming 1 million tons of grain, and 100,000 cars every time step. Producing a car takes 1 person-year, and producing a ton of grain takes 0.1 person years. This level of consumption would then require 2 million person-years of labor (1 million for the cars, 1 million for the grains).

Even if there were only 1.5 million people in the economy, they could, for example, spend all their labor producing cars. So they would make 150,000 cars and export 50,000 cars. If the price of the cars is much higher than the price of grains, they could just exchange the cars for enough grains while still maintaining a trade surplus.

This was one of the most surprising results I saw from this model.

Graph 4: This here was to test an assumption that many economists make about the economy. They assume that the profit rates of industries equalise over time. However, in my simulation at least, this never happens. There is like an invisible floor to how low the differences in profit rates can get.

I will be taking requests if someone wants me to generate data. I can change the number of sectors, the amount of wages. I can try different price seeking strategies, etc.

Also, I never thought I’d reach the “post your research annonymously on Hexbear” stage of my academic career.

’ %%%%%%%%%%

time = 100;
n = 10;
N = 100000;
connectivity = (2*n)^0.5; %The average number of intermediate commodities that go into making a commodity
threshold = connectivity/n;
e_l = 0.025;                 %proportionality rate at which hirings change per timestep
e_p = 0.025;                 %proportionality rate at which prices can change per timestep

Data = zeros([5 N*time]); %Pre allocating data matrix. Necessary to speed up simulation
Data_final = zeros([5 N]); %Pre allocating data matrix. Holds data on final time steps of each economy
w = 0.5; %Percentage of national production that the economy aims to give to labor

%%%%%%%%%%LOOP

for i = 1:N

    %Generate random workforce distribution between sectors
    L = rand([n time])*0.998 + 0.001;
    L(:,1) = L(:,1)./sum(L(:,1));    %Normalise the population to 1

    %Randomly generate direct labor use
    l = rand([n 1])*0.998 + 0.001;
    
    %Technical matrix:
    A = rand([n n]);
    A = A.*(A<=threshold);
    a = (eye(n)-A)\eye(n); %Storing the productivity matrix so it doesn't have to be recalculated over and over

    while sum(sum(a<0))>0     %If a has negative components, regenerate the economy and try again
        A = rand([n n]);
        A = A.*(A<=threshold);
        a = (eye(n)-A)\eye(n);
    end

    %LTV prices calculation
    LTV = sum(a.*l)';

    %Consumption
    basket = rand([n 1]);
    basket = w*basket./(sum(basket.*LTV)); %Consumption is scaled so that it can be in theory satisfied by the work of half the workforce

    %net production
    %o = zeros([n time]);

    %net income of sectors + agregate measures (pre-allocation)
    M = zeros([n time]);
    trade_balance = zeros([1 time]);
    profit_var = zeros([1 time]);
    
    %Randomised prices are generated for starting timestep (pre-allocation)
    P = zeros([n time]);

    %P(:,1) = rand([n 1]);             %randomly generates a set of prices

    
    %P(:,1) = (eye(n) - A - Cw)\rand([n 1]);
    P(:,1) = rand([n 1]);

    for k = 1:time
        
        if k>1
            hirings = e_l*(M(:,k-1))/sum(basket.*P(:,k-1));  %New Hirings are in proportion to the income available divided by wages
            L(:,k) = L(:,k-1) + hirings;   
            L(:,k) = L(:,k).*(L(:,k)>=(0.001/n)) + (L(:,k)<(0.001/n))*(0.01/n);   %This puts a floor on the size of sectors. Helps prevent the code from exploding.
            P(:,k) = P(:,k-1).*(1 - e_p*(hirings./L(:,k-1)));  %If the size of a sector doubles, the price decreases by e_p percent (from competititon)
            
            L(:,k) = L(:,k)./sum(L(:,k));    
        end

        P(:,k) = P(:,k).*((P(:,k)>=(0.001))) + (P(:,k)<(0.001))*(0.01);   %This puts a floor on the price. Helps prevent the code from exploding.

        %Calculate gross output of industries
        O = L(:,k)./l;

        P(:,k) = P(:,k)./sum(O.*P(:,k));%Normalises these prices so that total economy wide revenue is always 1

        Cw = basket * l';
        profit_var(:,k) = var(((eye(n) - A - Cw)*P(:,k))./P(:,k));

        %Calculate net production
        o = O - A*O;       %Net production can be negative. We will assume the existence of imports
                                %negative net production will show up as
                                %negative sales (the external market is
                                %selling to the economy)

        %Inter-industry sales
        R = O.*P(:,k);  %Market value of gross production by sector
        C = A' .*O*P(:,k); %Costs of inputs to production by sector
        
        %Industry to market sales
        S = o.*P(:,k); %Sales to consumers by sector
        Y = sum(S); %Total industry income from market sales
                    %Under balanced conditions, this income would be
                    %exactly matched by industry outflows to consumers
                    %(wages + dividends)

                    %Here it is assumed that the industry pays enough in
                    %(wages + dividends) to afford a fixed basket of 
                    % consumption.

                    %Any leftover income is the trade balance

        trade_balance(k) = sum((o - basket).*P(:,k));

        W = L(:,k).*(sum(basket.*P(:,k))); %Wages paid out vector by industry
        
        M(:,k) = R - C - W; %Net Income by industry
        
        M_per_worker = (1/n)*M./L(:,k); %I want to see if this givees any interesting results

        %Accounting identities
        % Y = sum(W) + trade_balance
        %Y = sum(R - C)

    end 
    %%%%%%%%%%%%%%Computing more time steps%%%%%%%%%%%%%%%%%%%%%%%%
    
    
    %%%%%%%%%%%%%Processing data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    %poopoo = LTV./LTV(1,:); 
    %Peepee = P./P(1,:); 

    ratios = P./LTV;
    ratios = log(ratios);
    ratios = ratios - mean(ratios);
    specific_price = sum(abs(ratios))/n;

    M = sum(abs(M));
    M_per_worker = sum(abs(M_per_worker));
    
    %specific_price = sum(abs(log(Peepee./poopoo)))/(n-1);

    % trade_balance; trade_intensity
    Data(:,(1+ (i-1)*time ):(i*time)) = [specific_price; M; M_per_worker; trade_balance; profit_var];
    Data_final(:,(1+ (i-1) ):(i)) = [specific_price(time); M(time); M_per_worker(time); trade_balance(time); (profit_var(time)).^0.5];

end
%%%%%%%%%%LOOP end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

resolution = 1001;
scale = 1;

ptsy = linspace(0, 1, resolution);
ptsx = linspace(0, scale, resolution);
%ptsx = linspace(-0.1, 5, 1001);
%H = log(histcounts2(Data_final(2,:), Data_final(1,:), pts, pts));
H = log(histcounts2(Data(2,:), Data(1,:), ptsy, ptsx));
imagesc(ptsx, ptsy, H);
axis xy;
set(gca, 'XLim', ptsx([1 end]), 'YLim', ptsy([1 end]), 'YDir', 'normal');
colormap copper
a=colorbar;
a.Label.String = "Density of simulation outcomes [natural log scale]";
xlabel {Deviation from LTV prices [natural log scale]}
ylabel {Deviation from reproduction [linear scale]}
title {Absolute sector income vs LTV pricing}
exportgraphics(gcf,"repro_inv_M10.png","Resolution",600);

figure 
ptsy = linspace(0, 1, resolution);
ptsx = linspace(0, scale, resolution);
%H = log(histcounts2(Data_final(3,:), Data_final(1,:), pts, pts));
H = log(histcounts2(Data(3,:), Data(1,:), ptsy, ptsx));
imagesc(ptsx, ptsy, H);
axis xy;
set(gca, 'XLim', ptsx([1 end]), 'YLim', ptsy([1 end]), 'YDir', 'normal');
colormap copper
a=colorbar;
a.Label.String = "Density of simulation outcomes [natural log scale]";
xlabel {Deviation from LTV prices [natural log scale]}
ylabel {Deviation from reproduction (scaled by employment) [linear scale]}
title {Per worker sector income vs LTV pricing}
exportgraphics(gcf,"repro_inv_Mw10.png","Resolution",600);

figure 

ptsy = linspace(-1, 1, resolution);
ptsx = linspace(0, scale, resolution);
%H = log(histcounts2(Data_final(4,:), Data_final(1,:), pts, pts));
H = log(histcounts2(Data(4,:), Data(1,:), ptsy, ptsx));
imagesc(ptsx, ptsy, H);
axis xy;
set(gca, 'XLim', ptsx([1 end]), 'YLim', ptsy([1 end]), 'YDir', 'normal');
colormap copper
a=colorbar;
a.Label.String = "Density of simulation outcomes [natural log scale]";
xlabel {Deviation from LTV prices [natural log scale]}
ylabel {Trade balance [linear scale]}
title {Trade balance vs LTV pricing}
exportgraphics(gcf,"repro_inv_T10.png","Resolution",600);

figure 

ptsy = linspace(0, 2.5, resolution);
ptsx = linspace(0, scale, resolution);
%H = log(histcounts2(Data_final(5,:), Data_final(1,:), pts, pts));
H = log(histcounts2(Data(5,:), Data(1,:), ptsy, ptsx));
imagesc(ptsx, ptsy, H);
axis xy;
set(gca, 'XLim', ptsx([1 end]), 'YLim', ptsy([1 end]), 'YDir', 'normal');
colormap copper
a=colorbar;
a.Label.String = "Density of simulation outcomes [natural log scale]";
xlabel {Deviation from LTV prices [natural log scale]}
ylabel {STD of profitability rates of sectors}
title {Profit STD vs LTV pricing}
exportgraphics(gcf,"repro_inv_p10.png","Resolution",600);

’

The graphs:

Each graph has 10 million points, There are 100 points (1 for each time step) for 100,000 economies

Graph 1: The average deviation of sectorial income (Revenue - Costs - Wages) from reproduction vs average deviation of prices from LTV.

Graph 2: Graph 2, but the incomes of each sector are given per worker

Graph 3: The trade balance of each economy at each time step

Graph 4: The variability of profit rates between different sectors

I don’t know if I should post them here or on lemmygrad (a place I don’t go to often enough). If you have any suggestions for a community let me know.

Posting it on the free chat should be fine. Stuff gets traction there. Also, you could try posting in Chapo then linking to the lemmygrad. I think that could boost the engagement numbers and get lots of community interaction.

which can be rewritten as …

The very last part of your comment is very interesting. At some point in my coding, I literally did randomly generate prices using that exact same equation. The vector pi was randomly generated (and so was A+). I didn’t use that pricing generation strategy because it caused the deviation of the prices to clump around the average for large sector numbers. But, that was my incomplete model. I think I can try and see how my finished model reacts to these prices.

For the model in this post, there is no assumption of equal profit rates, no structual mechansim for enforcing equalizafion of r, so the profit is simply a residual.

Not exactly. It’s more so that in this model, there is no class other than the working class. There is also no mechanism for profit rate equalization because this model doesn’t have time steps. The only purpose of the model is to investigate the relationship between prices and economic stability.

I do have a model that I have just about finished coding that incorporates the idea of time, movement, wage rates and external trade. The post for that is in the works.

Maybe the third model in the series model will have profits and a capitalist class in it. The second version is still a massive upgrade, so I want to release it first.

Profit equalization is also something I don’t plan on assuming. The agents in my 3rd model will try to maximize profits, and maybe profit equalization might emerge.

Although, the sectors in the 2nd model do hire more workers when the sector’s income is high, which is something like a profit maximization behaviour. Of course, the sectors in my 2nd model can’t choose prices (it wouldn’t make sense, since all gross outputs are fully sold), so it’s not true profit maximization behavior. The sectors in my models (so far) are also not individual firms, which limits their agency as agents.

Working under the assumption that the net product is the net consumption of workers, i.e. n = c (also, don’t worry about using a standard notation - there isn’t really much of one. I am using a mix of Ian Wright’s, Pasinetti’s and my own ) the line S = o.*P; is calculating the below, correct?

Yup, 100% correct

Works out because you couldn’t element-wise multiply a n x 1 array L with a 1 x mag array Y.

This is indeed a valid operation in Matlab, also equivalent to L*Y (I could have clearer here)

Thanks for taking the time to discuss this!

No problem

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.

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.

Having a power plant that literally solves the energy problem is worth pretty much anything. Especially for a country like China, which has to import huge amounts of oil through geopolitically risky routes.

True. I see a store, and I loose all control.

Must, buy, groceries

Looking forward to the 10,000 year reign of the burger emperor, his golden throne (the toilet) fueled by the souls of a 1000 immigrants being sacrificed a day.

All of those sound interesting, I will check them out.

also love to hear about any updates!

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).

7. 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.

8. 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).

unit cost = A^T p

This is what I have done in C (except I also multiplied by gross output yo get total costs per sector)

If you wanted an aggregate quantity across all sectors, this would be the p q - you have already calculated this as R.

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.

And perhaps this is what you’ve done, but just in an aggregate way.

my shitty code is causing people to think I aggregated everything even though everything is disagregated.

6. uh, I haven’t really read staffa or anything, but my approach for computing ltv prices was

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.

1. correct

2. correct

3. 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

4. yes

5. 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

I have some comments, questions, and possible suggestions.

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 😴