Math supplement to “Capitalist Equilibrium?”

This is, more or less, the argument set out by Karl Marx in Das Kapital, vol. 2.

Department one produces constant capital (c) for both departments. (Different capitals within Department 1 sell and buy c from each other while all of the capitalists in Department 2 must buy c from someone in Department 1) Department two produces consumer goods for both departments. Workers buy consumer goods with their wages (v) and capitalists out of their surplus (s). So if the entire output of Department one is equal to the demand for constant capital from both Departments, then there is no overproduction of c. And if the entire output of Department two is equal to the wages (v) and profits (s) of both Departments, there is no overproduction of consumer goods. The system is in equilibrium. Marx proceeds to lay create a numerical example of such a system in balance. It looks like this:

Department 1: 4000c + 1000v  + 1000s = 6000 units of c
Department 2: 2000c +  500v     + 500s  = 3000 units of consumer goods

Total demand for c = 6000
Total demand for consumer goods= 3000

These are arbitrary figures. The point is just to establish a hypothetical example of a system in balance. The numbers stand not for amount of commodities or dollar prices but for units of labor time embodied in commodities. (We abstract from money at this point in the equation because we are assuming that the addition of this factor in the equation would only create destabilization and not aid in our goal of establishing the theoretical possibility of equilibrium.)

But don’t get too excited about the fact that our model is in balance because we haven’t accounted for growth yet. The whole point of building this model is to figure out what reinvestment strategies will allow for balanced growth. So let’s change the model to allow for this “expanded reproduction”. This is a different model, one in which there is as surplus of c created by Department 1.

Department 1: 4000c + 1000v + 1000s = 6000 units of c
Department 2: 1,500c + 750v +  750s   =  3000 units of consumer goods

Total demand for c= 5,500
Total demand for consumer goods= 3500

So we have an imbalance between departments. Department 1 has a surplus of 500 units, while Department 2 appears to have 500 too few units to meet the demand for consumer goods. But this assumes that Department 1 actually plans to spend its surplus on consumer goods. It doesn’t. Instead it plans to reinvest its surplus of 500s in production. In order not to change the ratio of c to v we invest 400 of the surplus in c and 100 in v.  By doing so we get this:

Department 1: 4000c + 400c +1000v + 100v + 500s = 6000 units

This leaves an extra 100v of consumer goods to produce while still requiring another 100c of surplus to be mopped up. If Department 2 steps and reinvests some of its surplus we can take care of the problem:

Department 1: 4000c + 400c +1000v + 100v + 500s = 6000 units
Department 2: 1,500c + 100c + 750v + 50v + 600s = 3000

Total demand for c= 6000 units
Total demand for consumer goods= 3000

We have achieved balance merely by redistributing surplus invested at strict proportions. Notice that we have no growth yet in the economy. But see what happens in the next period:

Department 1: 4400c  +1100v + 1100s = 6,600
Department 2: 1,600c + 800 + 800s     = 3,200

Total demand for c= 6000
Total v+s= 3,800

The new investments have created a surplus of machines again as well as a surplus of v+s. But if we reinvest half of 1s (550) again…

Department 1: 4400c+ 440c + 1100v+110c + 550s= 6600

And if we adjust Department 2′s reinvestment to mop of the surplus again:

Department 1:  4400c+ 440c + 1100v+110c + 550= 6600
Department 2:  1,600c+160c + 800v + 80v   + 560s= 3,200

Total demand for c= 6,600
Total v+s= 3,200

The reinvestment has mopped up the surplus. In the next period…

Department 1: 4840c + 1210v + 1210s = 7260
Department 2: 1760c +  880v +  880s    = 3520

Total demand for c= 6,600
Total v+s= 4180

If we continue to reinvest at these same proportions (Department 1 reinvesting half of its surplus in c and v at a 4:1 ratio and Department 2 investing a three tenths of its profit in c and v at a 3:1 ratio) we find that this abstract model can continue to grow at a rate of 10% per period with no more overproduction and hence no crisis. This is a model for capitalism in equilibrium.