This is a sequel to my previous post. The idea is the same, but I'm using better methods as was suggested in the comments.

As u/Sodium_nitride (thank you!) explained, here...

• ...I use a production matrix instead of the Cobb-Douglas function.
• ...I use capital-time instead of capital, to handle depreciation.
• ...classes consume commodities, seeking to maximize the amount consumed.

Also, I purchased the book suggested by u/davel :)

We use the following definitions:

• Labor is measured relative to A's total labor power.
• B has b labor power, assumed to be proportional to population.
• Capital-time and commodities are measured in units of what can be produced directly from 1 unit of labor.
• Labor sold is represented with w, and the salary is used to purchase capital-time s_k and commodities s_c.
• Consumption of A is c, while that of B is z*b*c, where z is the ratio of per capita consumption.

Production matrix:

        /0 0 0\
    A = |1 m 0|
        \1 n 0/

Input vectors:

          /  1 - w  \
    x_A = |k_A + s_k|
          \    0    /

          /  b + w  \
    x_B = |k_B - s_k|
          \    0    /

Demand vectors:

          / w - 1 \
    y_A = | -s_k  |
          \c - s_c/

          /  -b - w   \
    y_B = |    s_k    |
          \z*b*c + s_c/

Payoff functions:

	X_A = c
	X_B = z*c

Case 1: full equilibrium

In this case, we assume that A and B can negotiate w, s_k and s_c freely, with no party being able to obtain a better bargaining position.

The Nash equilibrium is:

    w = s_k = s_c = 0
    c = (m - n - 1)/(m - 1)
    z = 1

That is, both groups are independent and produce their own capital-time and commodities. Their consumption is directly proportional to their labor power. Effectively, there is no difference between A and B, any member of either group belongs to the same class.

Case 2: asymmetric capital ownership

Here, we set k_A = s_k = 0, so A owns no capital-time. A and B can negotiate w and s_c under the same conditions as in Case 1.

The Nash equilibrium is:

    w = 1
    s_c = c = (1/2)*(m - n - 1)/(m - 1)
    z = 2 + 1/b

As can be seen, in this case A works for B and obtains a salary. Interestingly, this salary is exactly half of what A would have obtained in Case 1. From this and z's non-dependence on m and n, we can deduce that increases in productivity scale both A's and B's earnings with the same coefficient, so it's impossible for B to force A's income to any specific minimum.

We also see that B's per capita income is higher when less people belong to the group. For a small enough group, B's total income approaches that of A, just extremely concentrated.

A plausible hypothesis here is that, if the initial situation is Case 2 but productivity is more than high enough to sustain A's needs (thanks to the inevitable scaling described before), then A would be able to eventually negotiate their way to the final equilibrium, Case 1, provided a minimally feasible way to obtain capital.

If that is the case, the (surreal, but theoretically interesting) requirements to get to the equilibrium could be summarized like this:

1. All members of A cooperate perfectly (obviously false).
2. B has no way to gain an advantage (bourgeois state in general).
3. The productive forces have developed beyond a critical point.

Further questions

• How could one verify the hypothesis above? I know how to use production matrices in a state of equilibrium, but what about transient states?
• What if individuals can freely move across groups as their economic status changes and so do their interests? I know nothing about cooperative game theory, so this could be an interesting start.
• What if members of A and/or B do not cooperate perfectly?
• What are the minimum requirements for a mechanism that could allow the cooperative result in a non-cooperative Nash equilibrium?

I'm learning game theory these days, and I've tried my hand at some problems inspired by ML theory. Here's one I found really interesting.

Let's assume the following (clearly unrealistic) situation:

1. Working class (A) and bourgeoisie (B) form perfectly cooperative coalitions.
2. They may negotiate salaries, with no class having any mechanism to obtain a better bargaining position.
3. A cooperatively owns some amount k of capital, while B owns (arbitrarily) 1. In principle, k < 1.
4. B has some labor power b, while A has (arbitrarily) 1. Again, b < 1.
5. Production follows a Cobb-Douglas function, where the sum of the output elasticities of capital and labor is 1.
6. Both classes consume the same proportion of their income and use the remainder to acquire more capital.

Therefore, the payoffs in our problem can be stated like this (where w is the labor given to B and s is the compensation given to A for this labor):

X_A = A k**α (1 - w)**β + s
X_B = A (b + w)**β - s

We would like to find how the ratio of A's capital to B's changes over time, so we compute its derivative using the quotient rule, and note that the proportion of income (p) spent on capital is the same for both classes:

R = p X_A - k p X_B

Finally, we find the Nash equilibrium and its corresponding R at each combination of k and b:

Note the attractor curve coinciding with k = 1/b. In other words, ~~if both k and b are higher than 0~~, the eventual equilibrium will be for A and B to own capital proportional to their labor power.

Edit: fixed some missing solutions near k = 0.

1 more year has passed, and I'm still tracking these numbers, albeit now posting with a different username. The upward tendency has not just continued, but even increased; now Linux is nearing 4 % market share globally and over 2 % on Steam.

My old keyboard served me well, but lately I'm having to replace a broken switch every month so I'm not sure it's worth it. It's also noisy as hell and I hate the backlighting with every piece of my heart. So here's the replacement.

I've ordered it from WASD Keyboards, hmu for the design file. Obviously Spanish layout, I chose MX Cherry Brown switches, light pastel colors to improve visibility under dim lighting, and a pattern from a Gray-Scott reaction-diffusion system to decorate special keys. I've added a few (superfluous) icons for editing operations and arrow keys for Vim, as well as part of an Aristotle quote I like, just because the spacebar felt so empty. I used the old Greek translation simply to avoid distracting myself (I can barely read even modern Greek, so this looks like an uneventful string of accented letters to me).