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?