While Marx himself has always emphasized the qualitative side of his theory, most of his critics seem to be focusing on the quantitative side instead. It is frequently asserted that Marx's theory is not mathematically consistent for various reasons or that actual prices in Capitalism have no relation to labor whatsoever. I thought it would be helpful to do a mathematical formulation of the theory starting with how it appears directly in developed Capitalism - the gravitation of prices towards prices of production.
In classical theory, the gravitation towards prices of production has been a widely accepted conjecture. In modern economics this conjecture is not universally accepted. For the purposes of the following formulation, I will explicitly assume that this tendency holds. But such an assumption is not without basis. Various mathematical models exist that demonstrate such convergence with only standard assumptions. Unfortunately, the most extensive such models seem to be locked behind a substantial paywall. For this reason I'm going to reference this paper. While it does make a somewhat controversial assumption of no fixed capital, it must be noted that there are models that work without such an assumption.
Given this hypothesis, the long-term competitive equilibrium prices can be represented by this equation:
P = (1 + r)(LW + AP) = (1 + <rate of profit>)(<wages> + <cost of means of production expended>)
I'm using a slightly different notation that the one in the paper to make it easier to present on Reddit. I'm also formulating it slightly differently to better present my point. Here:
P - nx1 vector of long-term competitive equilibrium prices, which is assumed to also be the vector of prices of production.
r - the general rate of profit(which is the result of the equalization of the rate of profit between industries). This rate is risk-adjusted.
A - the input matrix nxn, which represents the quantities of input goods necessary to produce one unit of the corresponding output good.
L - the direct SNLT matrix nxm, which represents the socially-necessary duration of various kinds of labor directly employed, for example in hours to produce one unit of a good. Socially-necessary basically means that we average the total duration of labor employed in an industry over the total quantity of goods sold.
W - the wage-rate vector mx1, which represents all the different wages paid depending on the type of labor performed, for example per hour
So like this:
-- -- -- -- -- -- ---
p1 | l11, l12, l13, ..., l1m | w1 | a11, a12, a13, ..., a1n | p1
p2 | l21, l22, l23, ..., l2m | w2 | a21, a22, a23, ..., a2n | p2
p3=(1+r)*(| l31, l32, l33, ..., l3m | * w3 + | a31, a32, a33, ..., a3n | * p3)
.. | ..., ..., ..., ..., ... | ... | ..., ..., ..., ..., ... | ...
pn | ln1, ln2, ln3, ..., lnm | wn | an1, an2, an3, ..., ann | pn
-- -- -- -- -- -- ---
(My best attempt at a matrix representation on Reddit)
We assume a static equilibrium merely to simplify our presentation. Thus, we assume A,L,W,P,r do not change between the production cycles. In reality the equilibrium is dynamic, and so to model this dynamic equilibrium we would simply replace the `AP` component with `AP'`, where P' denotes the previous equilibrium price vector. This assumption does not affect the conclusions drawn from our formulation, merely simplifies the math a bit.
Deconstructing the prices of production
Let's investigate an arbitrary equilibrium price px out of the equilibrium price vector P
px = (1+r)(LxW + AxP),
Where Lx is the 1xm vector of direct SNLT and Ax is the 1xn vector of direct input goods, both corresponding to the xth rows in their respective matrices L and A, which correspond to the production requirements of the xth good.
This equation can be recursively expanded by substituting (1 + r)(AP + LW) for P
px = (1+r)(LxW + Ax(L'W + (1+r)*A*(L''W + (1+r)*A*... ) ) )
In order to keep track of direct and indirect labor employed, have added the ' annotation to denote that this L-matrix is the SNLT matrix from the previous production cycle. So L' is labor employed one production cycle ago, L'' is labor employed 2 production cycles ago etc
px = (1+r)(LxW + Ax( (1+r)L'W + (1+r)^2AL''W + ...) )
Since all of the components of px are linear in the direct or indirect labor-time matrices Lx, L', L''... , then it follows that if we were to multiply all elements of Lx, L', L'' ... by an arbitrary value q, then P would be multiplied by q as well. In other words, a proportional increase in the total required SNLT will result in a corresponding increase in the equilibrium price, ceteris paribus. For the following segment of the formulation we will assume that all transactions take place at the equilibrium price, we will consider the remaining possibilities later.
Profit, wages and class conflict
Once a sale is realized, px becomes a real, fixed sum of money that constitutes the revenue. Some of this money is necessarily spent on replacing the means of production, some goes towards the workers' wages and some the capitalist keeps as profit. Each component part of px corresponds to a labor performed directly or indirectly, and consequently px follows the "proportionality rule" established above.
vx = LxW
This is the part that goes towards replacing the wages. We call this part variable capital
sx = r(LxW + Ax((1+r)L'W + (1+r)^2AL''W + ...)) = r(vx + cx)
This is the profit of the capitalist
cx = Ax((1+r)L'W + (1+r)^2AL''W + ...)
This is the part of the money that necessarily goes towards replacing/maintaining the spent means of production(actually replacing the wages of the workers who made them and realizing the profit of the capitalists who provisioned them). We call this part constant capital
The rate of profit specific to industry x can be expressed as:
rx = sx/(vx+cx)
As a result of the equalization of the rates of profit, rx=r is forced. The proportion in which the revenue is divided between profit and wages can therefore be expressed as:
sx/vx = r(vx+cx)/vx = r(1+kx)
Where kx=cx/vx is the Value Composition of Capital(VCC) in industry x
Because r is an external variable not determined by our model(exogeneous), it can assume many different values independent of all the other variables. This means that the proportion in which this fixed amount of revenue is distributed between profit and wages can vary. Marxists assume r to be socially-determined(politically, culturally, etc). This points to a clear conflict of interests between the workers and the capitalists, as the proportion between wages and profit is subject to political struggle.
The actual basket of goods that the worker gets for his wage can increase despite a rise in profit due to increases in productivity, so it might appear as if the interests of the workers and the capitalists can be reconciled. But such changes render the worker increasingly more replaceable, throw ever greater masses of people out on the streets and take away their means of subsistence, so the conflict is not resolved and merely takes on a different form.
Redistribution of profit between industries
In the previous section we've derived the profit of the capitalist sx
sx = r(vx + cx) = r(LxW + Ax((1+r)L'W + (1+r)^2AL''W + ...))
This profit is the sole property of the capitalist who provisioned the good. But notice that there are terms that correspond to indirect labor-time of workers that the capitalist didn't hire (L',L'' ...). It is therefore possible for the capitalist to realize a profit that corresponds to the labor of workers who made the input goods, that the capitalist didn't even directly hire. We call this phenomenon the redistribution of profit
Conversely, because the revenue from the same labor exists as profit across many industries where the labor wasn't even directly employed, then what appears in the equation as the profit from L', L'' ... is in fact only one part of the total profit corresponding to this labor. To know the total profit we'd have to add up all the profit corresponding to this labor across industries.
How is this possible? In order to understand this phenomenon, let's imagine an industry α, where the redistribution of profit does not take place, and as a result all profit corresponds exactly to the labor employed directly, sα = σαLαW = σαvα. Here σα denotes the rate of exploitation, or the ratio of the total generated profit to variable capital employed. But at the same time the equalization of the rate of profit forces rα=r, so sα = r(vα+cα). Both can only hold simultaneously if
σαvα = r(vα+cα)
σα = r(vα+cα)/vα
σα/(1 + kα) = r
σx/(1 + kx) is a theoretical pre-equalization rate of profit in industry x, rx*. The extra terms therefore don't appear if the pre-equalization rate of profit is already equal to the general rate of profit.
If rx* > r, then the profit from direct labor in industry x is obtained at a higher rate than the general rate of profit. Noticing the higher returns, the capitalists are compelled to invest in this industry. Consequently, more capital is diverted to this industry, increasing supply and lowering prices until rx = r is restored. Vice versa if rx* < r, then the prices are increased until rx = r again. The capitalist competition between industries forces the equalization in the rates of profit.
The industries where the pre-equalization rate of profit is higher than the general rate of profit will therefore experience deflated prices, and vice versa for industries with a lower pre-equalization rate of profit. The pre-equalization rate of profit on the other hand depends on the Variable Composition of Capital kx, rx* = σx/(1+kx). Companies with a higher than average VCC will therefore experience extra profits that are related to the labor of workers employed in other industries which compensate for the low profit obtained from direct labor. Companies with low VCC on the other hand will experience a relative loss in profits. In this way profits are redistributed between industries to force the general rate of profit.
That's how a high-tech low-labor industry can still experience the same rate of profit as a low-tech high-labor industry, or even higher under certain conditions. The extra profit comes from the fact that high-VCC industries can sell their outputs at inflated prices, while low-VCC industries have to sell their outputs at deflated prices. But inflated/deflated in relation to what?
Social product and value
If we sum up all the prices of final goods sold in some period, excluding credit, we will obtain the total net social product expressed in money. Let Qf be the 1xn row-vector of quantities of final goods sold. Then the net social product can be expressed as:
QfP = Qf(1+r)(LW + (1+r)AL'W + (1+r)^2A^2L''W + ...).
Thus the net social product follows the same "proportionality rule" to SNLT as the prices of production, since it simply consists of a sum of prices of production.
This social product is divided between the workers and capitalists as wages and profit respectively. But considered as a whole, this product represents the total realized value of society. All component parts of this product correspond to some labor, past or present, which is embodied in a particular good. The product however is not distributed between industries according to labor embodied but in the first place based on the principle forced by the equalization of the rates of profit. The total net social product created across industries by the sale of a good is it's value in monetary form.
It is in relation to this value that the production prices of goods are inflated or deflated. By selling above value, one company records a gain relative to selling the good at it's value; but another company records a relative loss of the same magnitude due to having bought above value. Vice versa if the sale takes place below value. The sum total of the social revenue is not affected by this outcome. Only new value can increase it.
The investigation of what constitutes the nature of value and it's origin does not belong here but in a separate text focusing on the qualitative side of the theory.
The disequilibrium case
Until now we've assumed that all transactions occur at the equilibrium. In such a case all of our conclusions immediately follow. Let's now investigate the remaining cases of transactions taking place above or below the equilibrium.
By selling above the equilibrium(prices of production) one person records a gain relative to selling at the equilibrium, but by buying above the equilibrium another person records a relative loss of the same magnitude. Vice versa if the transaction takes place below the equilibrium. So the relative loss becomes someone else's relative gain. But the loss can not come from nowhere and must instead come either from wages or from profit. It is therefore a loss in wages or in profit relative to the wages and profit obtained at the equilibrium, but the sum total of value remains unaffected.
Just like with the effects of the equalization of the rate of profit, the disequilibrium case does not, by itself, affect the total social product, and thus can't create new value. In certain cases disequilibrium can prevent the value of a good from becoming realized in exchange, in this way reducing the net social product. But it can never increase it.