Abstract generated by ChatGPT¶

This work develops a probabilistic formulation of the tendency for the labor content of commodities to decline over time. Let the specific price of a commodity basket be defined as $\Psi = M/L$ , where $M$ denotes its monetary price and $L$ the amount of embodied labor. Comparing an initial basket $B_0$ with a cheaper basket $B_1$ , the analysis shows that if the monetary price decreases, the probability that the embodied labor also decreases is greater than one half. Under mild statistical assumptions, it follows that both the probability and the expected value of labor reduction are positive. By iterating this process across successive periods, the model predicts a long-run tendency toward declining labor requirements in commodity production. Monte Carlo simulations confirm the analytical results and illustrate the statistical regularity of the phenomenon. More broadly, the paper proposes an econophysical interpretation of economic dynamics, emphasizing probabilistic laws and emergent macroeconomic behavior instead of representative-agent equilibrium models.

Probabilistic Law of the Decrease in Labor Content¶

Let us work with the content of the appendix of the book How Labor Powers the Global Economy, specifically Appendix B.4 (Probabilistic Law of the Decline of Labor Content). To do so, let us begin by recalling some definitions:

Usable: Any object or service of use to human beings can be called a usable.
Gift of nature: Usables for which no input of human labor is used, directly or indirectly, to make them available. Examples of such usables are sunlight, wind, etc.
Product: A usable that does require direct or indirect input of labor for its creation. The kinds of direct inputs that go into a product are one of the following three: gifts of nature, labor, and previously produced products. Previously created products themselves require the same three kinds of direct input; by repeating this decomposition several times over, we are left with only two kinds of ultimate inputs: gifts of nature and labor.
Commodities: A special class of usables that presuppose highly abstract notions such as ownership and enforceable entitlement. A commodity is realized in a transaction of exchange or sale between a seller and a buyer. For a usable to become a commodity, therefore, it must be owned by the seller. While both products and commodities are usables, they are by no means identical categories; they merely overlap.
Commodity-product: The overlap between the two categories—products that are also commodities.
Money: A special kind of commodity—it is universal in the sense of being acceptable in exchange for any commodity.
Price: A sum of money exchanged for the transfer of ownership of the commodity.
Rent: The price of an entitlement to the use—often for a definite period of time—of some usable.
Loan of money: A special case of renting; the rent paid in this case is interest, whose rate is measured as a percentage (of the total sum lent) divided by a unit of time.
Labor power: The capacity to perform concrete labor tasks; it is a rather unique product. It requires both gifts of nature and labor as inputs. The latter includes the provision of education, training, shelter, as well as consumable products used to sustain the worker. But unlike any other product, labor power also produces an ultimate input: labor itself. In addition, labor power, being a component of an entire personality, unlike almost all major products, is “produced” outside the direct control of private firms, usually within the intimate family sphere. This is an important asymmetry that assigns to labor power a central role in the process of economic reproduction.

Let us now work with commodity-products, that is, goods produced by workers and bought and sold at various prices on the market. We want to investigate how price relates to labor time.

Over a period of time such as a year, a large quantity of products is bought and sold on the market. We may construct a list of all transactions within a given period of time and region as $\left\{1,2,3,\ldots\right\}$ . These products are produced and exchanged under many different circumstances. Now suppose a transaction involving a product with labor content $L_{i}$ and an amount of money $M_{i}$ . We may then define a price rate for this individual product as $\Psi_{i}=M_{i}/L_{i}$ , measured in dollars per hour, and we shall call this quantity the specific price.

Since prices are of central importance both for firms that produce and purchase commodities and for households that consume products, we are interested in characterizing prices across all transactions. Since each transaction involves a quantity of labor input, every transaction is the purchase and sale of a definite quantity of embodied labor.

Probability of Value Decline¶

Consider that we have a basket of commodity-products $B$ that may be mapped by a vector $\boldsymbol{b}$ $(B\Leftrightarrow\boldsymbol{b})$ . Then, considering that the labor embodied in $B$ is given by $L\left(B\right)$ , likewise $M\left(B\right)$ indicates the monetary price of this basket. From this, we define the specific price of something as the relation of dollars per hour associated with that thing, that is, the specific price of the basket would be:

\overline{\Psi}\left(B\right)=\frac{M\left(B\right)}{L\left(B\right)}\rightarrow M\left(B\right)=\overline{\Psi}\left(B\right)L\left(B\right)

(1)

Let us consider that we have a basket $B_{0}$ and then find a cheaper basket $B_{1}$ . The new basket provides a cheapening factor:

C=\frac{M\left(B_{0}\right)}{M\left(B_{1}\right)}>1

(2)

Manipulating, we obtain:

C=\frac{\overline{\Psi}\left(B_{0}\right)L\left(B_{0}\right)}{\overline{\Psi}\left(B_{1}\right)L\left(B_{1}\right)}\qquad\rightarrow C\frac{\overline{\Psi}\left(B_{1}\right)}{\overline{\Psi}\left(B_{0}\right)}=\frac{L\left(B_{0}\right)}{L\left(B_{1}\right)}

(3)

Now, if we consider something similar with respect to the embodied labor:

D=\frac{L\left(B_{0}\right)}{L\left(B_{1}\right)}=C\frac{\overline{\Psi}\left(B_{1}\right)}{\overline{\Psi}\left(B_{0}\right)}

(4)

Then, taking the logarithm:

\log\left(D\right)=\left[\log\left(C\right)+\log\overline{\Psi}\left(B_{1}\right)\right]-\left[\log\overline{\Psi}\left(B_{0}\right)\right]

(5)

We therefore want to know the probability that $D>1$ , or equivalently, that $\log\left(D\right)>0$ . This is the probability that the sum in the first bracket is greater than the sum in the second, that is:

P\left(D>1\right)=P\left(\left(\log C+\log\overline{\Psi}_{1}\right)>\left(\log\overline{\Psi}_{0}\right)\right)

(6)

Since $C>1$ , it follows that $\log C>0$ , that is, it will always add positively and contribute to making this term greater than $\log\overline{\Psi}_{0}$ . Therefore, the probability must be greater when containing this term than without it. In other words:

P\left(\left(\log C+\log\overline{\Psi}_{1}\right)>\left(\log\overline{\Psi}_{0}\right)\right) > P\left(\left(\log\overline{\Psi}_{1}\right)>\left(\log\overline{\Psi}_{0}\right)\right)

(7)

P\left(D>1\right) > P\left(\left(\log\overline{\Psi}_{1}\right)>\left(\log\overline{\Psi}_{0}\right)\right)

(8)

Now, considering that $\overline{\Psi}_{1}$ and $\overline{\Psi}_{0}$ are random values drawn from the same distribution, then the probability that $P\left(X>Y\right)=\frac{1}{2}$ . That is:

P\left(D>1\right)>50\%

(9)

In other words, there is a probabilistic tendency for the embodied labor to decrease when the price falls. We assume that the values $\overline{\Psi}(B_{0})$ and $\overline{\Psi}(B_{1})$ constitute independent realizations from the same distribution $\overline{\Psi}$ , even under the condition $M(B_{0})>M(B_{1})$ .

Expected Value of Decline¶

In other words, it is assumed that the selection of baskets by monetary price does not introduce systematic bias into the distribution of the specific price $\overline{\Psi}(B)=\frac{M(B)}{L(B)}$ .

Regarding expected values, we have:

\left\langle \log\left(D\right)\right\rangle = \left\langle \log\left(C\right)\right\rangle + \left\langle \log\overline{\Psi}\left(B_{1}\right)\right\rangle - \left\langle \log\overline{\Psi}\left(B_{0}\right)\right\rangle

(10)

Since the last two terms have the same expected value, given that $\overline{\Psi}(B_{0})$ and $\overline{\Psi}(B_{1})$ are independent realizations from the same distribution, it follows that:

\left\langle \log\left(D\right)\right\rangle = \left\langle \log\left(C\right)\right\rangle >0

(11)

since $C>1$ . Now, using the Jensen inequality for a concave function $g\left(x\right)$ , we have that $\left\langle g\left(X\right)\right\rangle \leq g\left(\left\langle X\right\rangle \right)$ . Thus, taking $g\left(x\right)=\log\left(x\right)$ , we obtain:

0< \left\langle \log\left(D\right)\right\rangle \leq \log\left(\left\langle D\right\rangle \right)

(12)

That is, for this to be possible we must have $\log\left(\left\langle D\right\rangle \right)>0$ , which requires that $\left\langle D\right\rangle >1$ . Therefore, there is also an average tendency for embodied labor to decline alongside the reduction in monetary price.

Long-Term Tendency¶

Now, if we replace it with a basket 2, we would have at a second moment:

D_{2}=\frac{L\left(B_{1}\right)}{L\left(B_{2}\right)}

(13)

Or, considering the process from the very beginning:

D^{*}_{2} = \frac{L\left(B_{0}\right)}{L\left(B_{2}\right)} = \frac{L\left(B_{0}\right)}{L\left(B_{1}\right)} \frac{L\left(B_{1}\right)}{L\left(B_{2}\right)} = D_{1}D_{2}

(14)

Generalizing, we may then write:

D^{*}_{m} = \prod^{m}_{i=1}D_{i} = \frac{L\left(B_{0}\right)}{L\left(B_{m}\right)}

(15)

Taking the logarithm:

\log\left(D^{*}_{m}\right) = \sum^{m}_{i=1}\log D_{i}

(16)

Now, considering that all $D_{i}$ are identically distributed, that is, drawn from the same distribution, then for any $i$ we have $\left\langle \log D_{i}\right\rangle =c>0$ , since we have already shown that $\left\langle \log\left(D\right)\right\rangle >0$ . Therefore, by the linearity of expectation $\left(\left\langle a+b\right\rangle =\left\langle a\right\rangle +\left\langle b\right\rangle \right)$ , we may write:

\left\langle \log\left(D^{*}_{m}\right)\right\rangle = mc >0

(17)

Here we must make clear that $D^{*}_{m}$ is not equal to $D_{i}$ . For example, as seen before, $D^{*}_{2}\neq D_{2}$ . Since all $D_{i}$ are identically distributed, it is as though each $D_{i}$ were a random step (but with $p>0.5$ of moving forward), while $D^{*}_{i}$ represents the accumulated distance after several steps. If at each step we expect to advance by $\left\langle D_{i}\right\rangle$ , then after $n$ steps we expect to have traveled $\left\langle D^{*}_{n}\right\rangle$ . By analogy, it is as if, after observing many trajectories of $m$ steps, our walker had advanced on average by $mc$ .

Now, looking at a single trajectory, by the Law of Large Numbers:

\frac{1}{m}\log(D^{*}_{m})\to c

(18)

Or equivalently:

\log(D^{*}_{m})\approx mc

(19)

That is, a sufficiently long single trajectory tends to exhibit behavior close to the average behavior obtained over the set of all possible trajectories. As a result, the probability that the embodied labor decreases approaches 1.

To test this, a code was developed. Using a normal distribution with mean 100 and standard deviation 10, and performing 10⁶ tests, we drew a $\Psi_{i}$ and an $L_{i}$ from the distribution for each of the two commodities, then calculated $M_{i}=\Psi_{i}L_{i}$ . We were able to verify that in approximately $75\%$ of the cases in which the price falls, the embodied labor also falls. In this case, we also obtained on average a cheapening rate of embodied labor of $\left\langle D_{i}\right\rangle \approx1.09$ .

To test whether, for $m\rightarrow\infty$ , we have $\log(D^{*}_{m})\approx\left\langle \log(D^{*}_{m})\right\rangle$ , we ran 10 simulations with 10⁶ exchanges and calculated the ratio between the average over all these simulations and the result obtained from a single execution:

\frac{\left\langle \log(D^{*}_{m})\right\rangle }{\log(D^{*}_{m})}\approx1.00

(20)

# @title
import numpy as np

N = 1000000

count = 0
Ds = []
P=[]
M=[]
L=[]
logs=[]
for _ in range(N):

    # preço específico (sempre positivo)
    psi1 = np.random.normal(loc=100, scale=10)
    psi2 = np.random.normal(loc=100, scale=10)

    # trabalho contido
    L1 = np.random.normal(loc=100, scale=10)
    L2 = np.random.normal(loc=100, scale=10)

    # preços monetários
    M1 = psi1 * L1
    M2 = psi2 * L2

    M.append(M1)
    M.append(M2)
    P.append(psi1)
    P.append(psi2)
    L.append(L1)
    L.append(L2)

    # definir D usando a cesta mais cara e a mais barata
    if M1 > M2:
        D = L1 / L2
    else:
        D = L2 / L1

    Ds.append(D)
    logs.append(np.log(D))

    if D > 1:
        count += 1

Ds = np.array(Ds)
logs = np.array(logs)
print(f"Podemos notar que :\nP(D>1) = {count / N:.2f} > 0.50")
print(f"  <D>  = {np.mean(Ds):.2f} > 1.00")

Podemos notar que :
P(D>1) = 0.75 > 0.50
  <D>  = 1.09 > 1.00

# @title
 import numpy as np

# -----------------------------
# parâmetros
# -----------------------------

num_traj = 10   # número de trajetórias
m = 1000000     # número de trocas por trajetória

traj_logs = []

# -----------------------------
# gerando trajetórias
# -----------------------------

for _ in range(num_traj):

    logDm = 0
    L=0
    for _ in range(m+1):

        # preço específico
        psi1 = np.random.normal(loc=100, scale=10)
        psi2 = np.random.normal(loc=100, scale=10)

        # trabalho contido
        L1 = np.random.normal(loc=100, scale=10)
        L2 = np.random.normal(loc=100, scale=10)

        # preços monetários
        M1 = psi1 * L1
        M2 = psi2 * L2

        # troca para cesta mais barata
        if M1 > M2:
            D = L1 / L2
        else:
            D = L2 / L1

        # acumulando log(D_i)
        logDm += np.log(D)

    traj_logs.append(logDm)

traj_logs = np.array(traj_logs)

# --------------------------------
# grandezas principais
# --------------------------------

# média ensemble
ensemble_mean = np.mean(traj_logs)

# trajetória típica
traj_typical = traj_logs[0]

# --------------------------------
# resultados
# --------------------------------


print(f"\nRazão entre <log(Dm)> e log(Dm) típico: {(ensemble_mean/traj_typical):.2f}")


Razão entre <log(Dm)> e log(Dm) típico: 1.00

What this section seeks to demonstrate, therefore, is that there exists a probabilistic tendency for cheaper commodities to contain less embodied labor. Defining the specific price as $\overline{\Psi}(B)=\frac{M(B)}{L(B)}$ and comparing an initial basket $B_{0}$ with a new cheaper basket $B_{1}$ , we define $D=\frac{L(B_{0})}{L(B_{1})}$ , where $D>1$ means that the new basket contains less labor. Since the monetary cheapening factor $C=\frac{M(B_{0})}{M(B_{1})}>1$ enters positively into the inequality, it follows that $P(D>1)>50\%$ , assuming that the specific prices $\overline{\Psi}(B_{0})$ and $\overline{\Psi}(B_{1})$ are independent realizations from the same distribution. Furthermore, we showed that the expected value also favors the reduction of embodied labor, $\langle D\rangle>1$ , indicating an average tendency toward a decrease in embodied labor associated with the fall in monetary price. Finally, by successively iterating this process, we obtain

D^{*}_{m}=\frac{L(B_{0})}{L(B_{m})}

(21)

and, by the Law of Large Numbers,

\log(D^{*}_{m})\approx mc,\quad(c>0),

(22)

which implies that, in the long run, a typical trajectory tends almost certainly toward the continuous reduction of embodied labor.

The idea is that although embodied labor $L$ and the specific price $\Psi$ may initially be assumed to be independent random variables, this does not mean that they remain independent after conditioning on the observed monetary price. The central point is that the monetary price is defined by:

M=\Psi L

(23)

Thus, when we specifically select a cheaper basket, we are imposing the condition $M_{1}<M_{0}$ , that is, we are conditioning on the product of the two variables. This conditioning induces statistical dependence between $L$ and $\Psi$ , even if they were originally independent. The reduction in the monetary price must result from at least one of the following possibilities: a) a reduction in $\Psi$ ; b) a reduction in $L$ ; or c) a simultaneous reduction in both. Since we assume that $\Psi$ has no systematic tendency between the compared baskets, it follows that part of the reduction in price tends probabilistically to fall upon $L$ . Therefore, the result

P(D>1)>\frac{1}{2}

(24)

emerges not because there exists a direct causal relationship between embodied labor and the specific price, but because both enter multiplicatively into the determination of the monetary price.

Standard economic theory is based on rather rigid assumptions regarding the behavior of individual agents in the economy. It is assumed that agents adopt optimal actions with respect to various economic response functions, often visualized as ‘‘curves’’ that shift or intersect. The collective result of optimization by agents is a supposed equilibrium state in which opposing economic forces balance one another until disturbed by ‘‘external’’ factors. Frequently, microeconomic assumptions are transferred into macroeconomic modeling through a ‘‘representative’’ agent who optimizes on behalf of all individual agents.

In contrast, the probabilistic framework possesses radically different microeconomic foundations (indeed, almost indifferent ones!). Very few assumptions are made about individual agents, who are regarded as only weakly coordinated at the local level within a capitalist market economy. They operate in diverse ways, yet remain subject to certain fundamental economic constraints. If certain structural variables are considered to be in equilibrium, this is a statistical equilibrium, where even within such a state the microeconomic configurations remain subject to incessant change.

A central concept in standard economic theory is that of the ``production function.‘’ It supposedly specifies the technological relationship between the quantities of productive inputs and the outputs generated by a given firm. In practice, this relationship is quantified in monetary terms, such that $Y=M(B)$ is the value of the basket $B$ produced by the firm and is considered to be determined by a production function:

Y=f\left(K,T\right)

(25)

where $K$ is the value of capital and $T$ is the amount of direct labor time employed. The most commonly used production function takes the following form (the Cobb--Douglas production function):

Y=A\cdot K^{\beta}\cdot T^{1-\beta}

(26)

where $A$ is a supposedly ‘‘technical’’ coefficient and $\beta$ is a parameter between 0 and 1. Through the lens of the production function, labor and capital become substitutable ‘‘factors of production.’’

It also conforms to the principles of income distribution in standard economic theory: each production agent receives an amount of wealth corresponding to what that agent creates. Capital and labor are here viewed as factors making independent ‘‘contributions’’ to production.

Although widely used, the production function is filled with deep theoretical problems: a) the individual production functions of all firms in the economy do not aggregate easily into a single one; b) there is overwhelming empirical evidence that average wages are above the theoretical ‘‘marginal product of labor’’ in most capitalist economies. Thus, the production function serves more as an exercise in curve fitting than as a modeling of underlying laws that would provide genuine predictive power.

In contrast, the probabilistic framework does not rely on any assumptions that firms possess well-behaved production functions. Starting from a few basic postulates, it generates theoretically consistent and empirically testable claims.

We have already discussed how to calculate a measure of the embodied labor contained in a basket of goods, and we now show a probabilistic connection between market price and the law of decline of embodied labor, which is considered a central dynamic force within the capitalist economy.

Upper Limit¶

Considering $\mathbb{L}$ as the labor added during the year, then the embodied labor contained in the basket representing the accumulated material wealth of a region $B$ may be approximated, for example, by:

L\left(B\right)\simeq\alpha\cdot\mathbb{L}

(27)

There is therefore a limit to $\delta$ (the rate of decline of embodied labor), since otherwise all material wealth could be produced by a very small quantity of labor. Insofar as such assets were accessible on the market, they would rapidly cease to remain concentrated, and the dependence of households on wages paid by capitalist firms would soon disappear.

Let us investigate what this limit is. Consider changes in wealth $B$ from a year $t$ to a year $t+1$ . Part of this wealth is destroyed or consumed, so that only a remaining wealth $B'$ persists. That is, part of the embodied labor contained in the initial stock, $L_{t}\left(B\right)$ , is transformed into $L_{t+1}\left(B'\right)$ .

In addition, new products are produced and accumulated each year, so that a portion of $\mathbb{L}$ is accumulated. This portion has as its upper limit the part of the surplus not consumed by the labor force, $\left(1-\omega\right)$ , where $\omega$ is the fraction of embodied labor consumed by the labor force as remuneration. We are simplifying here to the scenario in which the labor added remains unchanged from one year to the next.

If material wealth does not decline, we may express accumulation as:

L\left(B_{t+1}\right)\geq L\left(B_{t}\right)

(28)

L_{t+1}\left(B'\right)+\left(1-\omega\right)\mathbb{L} \geq L_{t}\left(B\right)

(29)

We may then write:

$L_{t+1}\left(B'\right)=\left(1-\delta\right)L_{t}\left(B'\right)$ , that is, the embodied labor in $B'$ undergoes a decline of $\delta$ from one year to the next.
$L_{t}\left(B'\right)=\left(1-d\right)L_{t}\left(B\right)$ , where $d$ is the rate of material decline, that is, the material fraction of $B$ that was consumed.

Then:

L_{t+1}\left(B'\right) = \left(1-\delta\right)L_{t}\left(B'\right) = \left(1-\delta\right)\left(1-d\right)L_{t}\left(B\right)

(30)

Substituting:

\left(1-\delta\right)\left(1-d\right)L_{t}\left(B\right) + \left(1-\omega\right)\mathbb{L} \geq L_{t}\left(B\right)

(31)

\left(1-d-\delta+\delta d\right)\alpha\mathbb{L} + \left(1-\omega\right)\mathbb{L} \geq \alpha\mathbb{L}

(32)

\left(1-d\right)\alpha - \left(1-d\right)\alpha\delta + \left(1-\omega\right) \geq \alpha

(33)

-\left(1-d\right)\delta \geq 1-\frac{\left(1-\omega\right)}{\alpha}-\left(1-d\right)

(34)

-\delta \geq \frac{1}{\left(1-d\right)} - \frac{\left(1-\omega\right)}{\alpha\left(1-d\right)} -1

(35)

-\delta \geq -\frac{\left(-\alpha+1-\omega\right)}{\alpha\left(1-d\right)} -1

(36)

\delta \leq 1+ \frac{\left(-\omega-\alpha+1\right)}{\alpha\left(1-d\right)}

(37)

\delta \leq 1- \frac{\omega+\alpha-1}{\alpha\left(1-d\right)}

(38)

Or equivalently:

\boxed{ \delta \leq 1-\frac{\omega+\alpha-1}{\alpha\left(1-d\right)} }

(39)

Empirical data indicate that $\alpha\approx5$ , that is, accumulated material wealth is approximately five times the labor added to the economy during the year. Taking an estimated material depreciation rate of $5\%$ to $8\%$ , let us adopt $d=0.05$ , together with a wage share on the order of $\omega=0.5$ . Then $\delta\leq5.3\%$ .

That is, if the rate of reduction of embodied labor in commodities exceeds approximately $5\%$ per year, the labor content of material wealth will tend to decline persistently. This implies a growing difficulty for the reproduction of capitalist accumulation, since the system depends on living labor as the source of value and surplus-value.

Paradoxically, this reduction in labor content may represent an advance from a technical and human point of view, since it means that society becomes capable of producing increasing quantities of material wealth with ever less necessary labor. However, if productivity grows too rapidly, capitalism comes to require an increasingly smaller quantity of workers to reproduce the material basis of society, thereby weakening the very foundation of capital valorization.

The available alternatives for increasing the limit $\delta$ are to reduce $\omega$ , that is, to reduce the quantity of embodied labor allocated to workers, thereby intensifying exploitation. Reducing $d$ faces a limit, since even for $d=0$ , in our case we obtain an upper limit of only $10\%$ . And if we look at accumulation $\alpha$ , increasing the limit requires reducing accumulated wealth itself. But here we encounter the contradiction: in order to ensure that material wealth does not decline persistently, it becomes necessary to reduce accumulated wealth.

Testing this numerically, for the constants specified above, when $\delta=5\%<\delta_{lim}$ , material wealth measured in embodied labor grew by $1.64\%$ , whereas for $\delta=5.5\%>\delta_{lim}$ it declined by $1.45\%$ .

# @title
LB = 5  #Trabalho contido na riqueza inicial
LB0=LB
a=5     #Multiplo do trabalho adicionado
L=LB/a  #Trabalho adicional no ano
w=0.5   #wageshare
d=0.05  #Taxa de depreciação material
g=0.050  #Taxa de decréscimo do trabalho contido
N=10    #anos
for i in range(N):
  LB1=LB*(1-d) #O que resta depois do decréscimo material
  LB2=LB1*(1-g) #A diminuição no trabalho contido
  LB3=LB2+(1-w)*L
  LB=LB3


print(f"\nApós {(N):d} anos, a riqueza material cresceu {(100*(LB/LB0-1)):.2f}%")


Após 10 anos, a riqueza material cresceu 1.64%

Productivity¶

The standard way of measuring productivity is as the ratio between monetary value added (that is, the sale price minus the cost of the inputs used in production) and the quantity of labor directly employed in the production of the basket $B$ .

Considering the basket of the economy’s net product $B_{out}$ , we may write:

P=\frac{M(B_{out})}{L(B_{out})}

(40)

where $M(B_{out})$ represents the monetary value added and $L(B_{out})$ represents the labor directly employed in the production of this basket.

To track changes in productivity over time, a reference basket $B_{ref}$ is used, whose prices are rescaled in real terms (that is, adjusted for inflation), such that:

M_{t+1}(B_{ref})\approx M_{t}(B_{ref})

(41)

Defining $\delta$ as the rate of decline of labor content, we have:

\delta = 1-\frac{L_{t+1}(B_{ref})}{L_{t}(B_{ref})}

(42)

= 1-\frac{M_{t+1}/P_{t+1}}{M_{t}/P_{t}}

(43)

\approx 1-\frac{P_{t}}{P_{t+1}}

(44)

Then:

\delta \approx \frac{P_{t+1}-P_{t}}{P_{t+1}}

(45)

That is, by disregarding inflation through the use of real prices, we treat the monetary value added of the net product basket as approximately constant over time. In this case, increases in productivity imply reductions in the quantity of labor required to produce that basket.

In other words, by the very traditional definition of productivity as the ratio between monetary value added and the labor directly expended in production, if monetary value added remains approximately constant, then an increase in productivity necessarily corresponds to a reduction in the labor embodied in the production of those goods.

The most relevant aspect is not merely the algebraic identity between productivity and labor content, but rather the fact that the rate of reduction of labor content corresponds approximately to the relative variation in productivity.

This discussion regarding the decline in labor content also helps to explain how an increase in exploitation may occur even without a reduction in nominal wages. We may imagine a scenario without inflation in which a worker receives a minimum wage sufficient to purchase a basic basket of goods necessary for survival.

With technical progress, the quantity of labor required to produce this basket tends to decline. If the nominal wage remains constant and the monetary prices of the basket do not fall in the short run, then the worker will continue purchasing the same physical basket, yet this basket will represent a smaller quantity of socially necessary labor.

One might object that, as previously discussed, reductions in labor content tend to produce reductions in prices. However, this does not necessarily occur immediately. In the short run, prices may remain relatively rigid or be offset by inflation and by changes in relative prices throughout the economy.

For example, other components of the cost of living may become relatively more expensive. Suppose a second basket containing twice as many labor hours. If technical progress reduces the labor content of the basic basket while its monetary price remains stable, then the relative difference between the labor contents of the two baskets increases. This may be reflected in an increase in their relative price.

Thus, even if certain industrial commodities significantly reduce their labor content, other components of the cost of living may become relatively more expensive, especially in sectors in which human labor remains central.

In this way, the nominal wage may remain unchanged while the worker comes to control a smaller share of the total social labor produced in the economy.

Rate of Accumulation¶

If the labor produced during the year is $\mathbb{L}$ , and a fraction $\alpha$ is reinvested in order to generate new capital, and we denote $\mathbb{K}=L\left(B_{cap}\right)$ as the labor embodied in capital, then the rate of profit over the total capital stock, in terms of embodied labor, may be approximately written as:

R=\frac{\alpha_{max}\mathbb{L}}{\mathbb{K}}

(46)

That is, a ratio between all surplus labor appropriated by capitalists and the capital stock, where $\alpha_{max}=1-\omega$ , since $\omega$ is the fraction appropriated by workers. Evidently, $\alpha_{max}$ also represents the maximum fraction of added labor that could be reinvested by capitalists, such that:

0\leq\alpha\leq\alpha_{max}

(47)

In the following year, the capital stock then consists of part of the labor produced in the previous year that was converted into capital, together with what remains of the previous year’s stock after discounting both the material depreciation of capital $(d)$ and the reduction in the labor content of commodities caused by productivity growth $(\delta)$ .

\mathbb{K}_{t+1} = \alpha\mathbb{L}_{t} + \left(1-\delta\right)\left(1-d\right)\mathbb{K}_{t}

(48)

And if we use $l$ to denote the growth of labor added annually, we have:

\mathbb{L}_{t+1} = \left(1+l\right)\mathbb{L}_{t}

(49)

Then

R_{t+1} = \alpha_{max}\frac{\mathbb{L}_{t+1}}{\mathbb{K}_{t+1}}

(50)

= \alpha_{max} \frac{\left(1+l\right)\mathbb{L}_{t}} {\alpha\mathbb{L}_{t}+\left(1-\delta\right)\left(1-d\right)\mathbb{K}_{t}}

(51)

= \left( \alpha_{max}\frac{\mathbb{L}_{t}}{\mathbb{K}_{t}} \right) \frac{\left(1+l\right)} { \frac{\alpha}{\alpha_{max}} \left( \alpha_{max}\frac{\mathbb{L}_{t}}{\mathbb{K}_{t}} \right) + \left(1-\delta\right)\left(1-d\right) }

(52)

= R_{t} \frac{\left(1+l\right)} {AR_{t}+B}

(53)

where we defined

A=\frac{\alpha}{\alpha_{max}}, \qquad B=\left(1-\delta\right)\left(1-d\right).

(54)

Now, taking the inverse $x_{t}=\frac{1}{R_{t}}$ :

\frac{1}{x_{t+1}} = \frac{1}{x_{t}} \frac{\left(1+l\right)} {\frac{A}{x_{t}}+B}

(55)

A+Bx_{t} = x_{t+1}\left(1+l\right)

(56)

Or equivalently:

x_{t+1} = \frac{A}{\left(1+l\right)} + \frac{B}{\left(1+l\right)}x_{t} = a+bx_{t}

(57)

The equilibrium solution is therefore obtained when $x_{t+1}=x_{t}=x^{*}$ :

x^{*}=a+bx^{*}

(58)

That is, the equilibrium point is:

x^{*}\left(1-b\right)=a \rightarrow x^{*}=\frac{a}{\left(1-b\right)}

(59)

The distance of a term $x_{t}$ from equilibrium may then be defined by:

y_{t}=x_{t}-x^{*}

(60)

and, since $x^{*}=a+bx^{*}$ :

x_{t+1}=a+bx_{t}

(61)

y_{t+1}+x^{*} = a+by_{t}+bx^{*}

(62)

y_{t+1}+x^{*} = \left(a+bx^{*}\right)+by_{t}

(63)

y_{t+1}=by_{t}

(64)

It is then easy to see that for $t=0$ we have:

y_{1}=by_{0},

(65)

for $t=1$ :

y_{2}=by_{1}=b^{2}y_{0},

(66)

and, in general:

y_{t}=b^{t}y_{0}.

(67)

Returning then to:

y_{t}=x_{t}-x^{*}

(68)

we obtain:

b^{t}y_{0}=x_{t}-x^{*}

(69)

Rearranging, and recalling that $y_{t}=x_{t}-x^{*}$ :

x_{t} = x^{*}+b^{t}y_{0} = x^{*}+b^{t}\left(x_{0}-x^{*}\right)

(70)

Therefore, if $\left|b\right|< 1$ , then for $t\rightarrow\infty$ we have:

x_{t}\rightarrow x^{*}.

(71)

That is, the system is stable if:

\left|b\right| = \left| \frac{B}{\left(1+l\right)} \right| = \left| \frac{\left(1-\delta\right)\left(1-d\right)} {\left(1+l\right)} \right| < 1

(72)

And the equilibrium point is

x^{*} = \frac{a}{\left(1-b\right)}

(73)

= \frac{\frac{\alpha}{\alpha_{max}}}{\left(1+l\right)} \frac{1} { \left( 1- \frac{\left(1-\delta\right)\left(1-d\right)} {\left(1+l\right)} \right) }

(74)

= \frac{\alpha}{\alpha_{max}} \frac{1} { \left( \left(1+l\right) - \left(1-\delta\right)\left(1-d\right) \right) }

(75)

= \frac{\alpha}{\alpha_{max}} \frac{1} { \left( \left(1+l\right) - \left(1-d-\delta+d\delta\right) \right) }

(76)

= \frac{\alpha}{\alpha_{max}} \frac{1} { \left( l+d+\delta-d\delta \right) }

(77)

Considering $d\delta\approx0$ :

R \approx \frac{\alpha_{max}}{\alpha} \left( l+d+\delta \right)

(78)

That is, in the stationary state, the greater the fraction of surplus reinvested $\alpha$ in the accumulation of capital $B_{cap}$ , the lower the average rate of profit tends to be.

Specific Price of a Sufficiently Large Basket¶

The book argues that, for sufficiently large and heterogeneous baskets, the average specific price tends approximately toward the same value. This implies that

\frac{M\left(B_{i}\right)}{M\left(B_{j}\right)} \approx \frac{L\left(B_{i}\right)}{L\left(B_{j}\right)},

(79)

that is, the ratio between the monetary prices of two large baskets tends to follow the ratio between their labor contents. Under this approximation, the average rate of profit calculated in monetary terms is approximately equal to the rate of profit calculated in terms of embodied labor.

For this condition to hold, it is necessary to assume that the specific prices of individual commodities,

\Psi_{i}=\frac{M_{i}}{L_{i}},

(80)

are drawn from the same relatively stable statistical distribution. Thus, by the Law of Large Numbers, the average specific prices of large heterogeneous baskets tend toward the same expected value.

That is:

\overline{\Psi}\left(B\right) = \sum^{m}_{j=1}w_{j}\Psi_{j}

(81)

where the weights

w_{j}=\frac{L_{j}}{\sum_{k}L_{k}}

(82)

are determined by the labor content of the basket. Then:

\overline{\Psi}\left(B\right) =\sum^{m}_{j=1}\frac{L_{j}}{\sum_{k}L_{k}}\Psi_{j}=\frac{\sum_{j}L_{j}\frac{M_{j}}{L_{j}}}{\sum_{k}L_{k}}=\frac{\sum^{m}_{j}M_{j}}{\sum_{k}L_{k}}=\frac{M\left(B\right)}{L\left(B\right)}

(83)

By the Law of Large Numbers, if the basket contains a sufficiently large and heterogeneous quantity of commodities, the average specific price of the basket is expected to converge toward the theoretical mean of the distribution. To test this, a code was written constructing two baskets with 10⁶ commodities drawn from a normal distribution with mean 100 and standard deviation 10. The ratio between the specific prices of both baskets was found to be approximately 1.00.

# @title
import numpy as np
N=1000000
c1=0;c2=0;T1=0;T2=0
for x in range(N):
  P1=np.random.normal(loc=100, scale=10)
  P2=np.random.normal(loc=100, scale=10)
  L1=np.random.normal(loc=100, scale=10)
  L2=np.random.normal(loc=100, scale=10)
  c1=c1+P1*L1
  T1=T1+L1
  c2=c2+P2*L2
  T2=T2+L2
c1=c1/T1
c2=c2/T2
print(f"\nAmos os preços específicos convergem com uma razão de {(c1/c2):.2f}")


Amos os preços específicos convergem com uma razão de 1.00

Accumulation Limit¶

It is often said that capitalism is an economic engine of development because it requires the continuous development of the productive forces. We may verify this in the fact that the material accumulation that a capitalist is capable of possessing is limited by annual production.

That is, if we denote accumulated material wealth by $\mathbb{W}$ , then we expect something of the form:

\mathbb{W}\propto\mathbb{L}

(84)

Let us now calculate this constant. Considering that a fraction $\alpha$ of the labor produced in one year, $\mathbb{L}$ , is transformed into wealth, together with the same factors of material depreciation $d$ and reduction in labor content $\delta$ :

\mathbb{W}_{t+1} = \alpha\mathbb{L} + \left(1-\delta\right)\left(1-d\right)\mathbb{W}_{t}

(85)

Then:

\frac{\mathbb{W}_{t+1}}{\mathbb{L}_{t+1}} = \frac{ \alpha\mathbb{L} + \left(1-\delta\right)\left(1-d\right)\mathbb{W}_{t} }{ \left(1+l\right)\mathbb{L}_{t} }

(86)

= \frac{\alpha}{1+l} + \frac{ \left(1-\delta\right)\left(1-d\right) }{ \left(1+l\right) } \frac{\mathbb{W}_{t}}{\mathbb{L}_{t}}

(87)

x_{t+1}=a+bx_{t}

(88)

We obtain the same result as before, except now:

a=\frac{\alpha}{1+l}

(89)

and

b=\frac{\left(1-\delta\right)\left(1-d\right)}{\left(1+l\right)}.

(90)

In addition:

x=\frac{\mathbb{W}}{\mathbb{L}}.

(91)

Therefore:

x^{*} = \frac{a}{\left(1-b\right)}

(92)

= \frac{ \frac{\alpha}{\left(1+l\right)} }{ \left( 1- \frac{ \left(1-\delta\right)\left(1-d\right) }{ \left(1+l\right) } \right) }

(93)

= \frac{ \alpha }{ \left( \left(1+l\right) - \left(1-\delta\right)\left(1-d\right) \right) }

(94)

\approx \frac{ \alpha }{ \left( \left(1+l\right) - \left(1-d-\delta\right) \right) }

(95)

\approx \frac{\alpha}{\left(l+d+\delta\right)}

(96)

That is:

\mathbb{W} \approx \left[ \frac{\alpha}{\left(l+d+\delta\right)} \right] \mathbb{L}

(97)

The transformation problem¶

This chapter is inspired by Chapter 6 of the book Laws of Chaos: Probabilistic Approach to Political Economy . The chapter is entitled The Dissolution of the Transformation Problem and deals, as its title suggests, with the transformation problem. Here, the transformation problem should be understood as “showing how an equal average rate of profit can be achieved not only without violating the law of value, but through it.”

The term “transformation problem” can also be reinterpreted in a broader sense, namely, as the problem of elucidating the systematic connections between value categories (or labor-content categories), on the one hand, and price categories, on the other.

Let us begin by understanding Marx’s theory of prices. We start from the idea of ideal prices, that is, prices that would prevail in a hypothetical state of equilibrium. Under this conception of prices, market prices are said to fluctuate around these ideal prices and to tend to converge toward them.

We begin with the model in which the labor content $\lambda\left(C\right)$ of a commodity $C$ is related to its ideal price $\overline{\pi}\left(C\right)$ through the following equation:

\frac{\overline{\pi}\left(C\right)}{\lambda\left(C\right)}=\psi_{0}

(98)

Where $\psi_{0}$ is a universal constant for all commodities. We now need to make a few distinctions.

Labor power is the worker’s capacity to perform labor. Thus, the amount of labor power that an employer purchases corresponds to the length of time for which this capacity to work is rented, for example, 8 hours per day. It is important to make it clear that labor power is itself a commodity. However, it has a particularity when discussing its price and value.

Workers must consume a minimum quantity of commodities in order to reproduce their labor power, that is, to remain capable of working and surviving. These commodities include food, housing, transportation, clothing, and so on, although the exact composition of this bundle depends on historical and social conditions.

The value (labor content) of labor power corresponds to the socially necessary labor embodied in this bundle of commodities required for the worker’s reproduction. Equivalently, the price of labor power corresponds to the price of this bundle of commodities. Naturally, the numerical values depend on the time unit under consideration, such as weekly, monthly, or annual consumption.

Let us assume, for example, that $\overline{\pi}_{0}$ is the ideal price of labor power—for instance, the weekly wage—equal to the price of a standard bundle of commodities that a worker (and their family) must consume each week under given social conditions in order to remain able to work for another week. Likewise, $\lambda_{0}$ is the labor content of the same bundle, that is, the amount of labor required to produce it. If we work with a weekly time unit, then we must have $\lambda_{0} < 1$ ; that is, the bundle of commodities consumed by a worker during one week must require less than one week of labor to produce. We therefore have

\psi_{0}=\frac{\overline{\pi}_{0}}{\lambda_{0}}.

(99)

Let us now consider a firm that hires $n$ workers for one week. Since we are using one week as the unit of time, the firm has hired $n$ units of labor power. Naturally, if the price is $\overline{\pi}_{0}$ per unit of labor power, then the weekly labor cost is $n\overline{\pi}_{0}$ .

Furthermore, during the week, the workers produce a bundle of commodities $\boldsymbol{C}$ , so that the firm’s revenue is $\overline{\pi}\left(\boldsymbol{C}\right)$ .

During the production process, the firm also consumes a bundle of commodities $\boldsymbol{I}$ , representing an additional cost of $\overline{\pi}\left(\boldsymbol{I}\right)$ . Therefore, the weekly profit is given by

l=\overline{\pi}\left(\boldsymbol{C}\right)-\left[\overline{\pi}\left(\boldsymbol{I}\right)+n\overline{\pi}_{0}\right].

(100)

We can calculate the rate of profit by taking the ratio of profit to the firm’s total capital $\boldsymbol{K}$ . Thus, we obtain

r=\frac{\overline{\pi}\left(\boldsymbol{C}\right)-\left[\overline{\pi}\left(\boldsymbol{I}\right)+n\overline{\pi}_{0}\right]}{\overline{\pi}\left(\boldsymbol{K}\right)}.

(101)

Naturally, we are assuming that all transactions take place at their ideal prices. Since $\psi_{0}$ is a universal constant, independent of the commodity under consideration, we can rewrite this expression in terms of labor content:

r=\frac{\lambda\left(\boldsymbol{C}\right)-\left[\lambda\left(\boldsymbol{I}\right)+n\lambda_{0}\right]}{\lambda\left(\boldsymbol{K}\right)}.

(102)

Now let us focus on the difference

p=\lambda\left(\boldsymbol{C}\right)-\lambda\left(\boldsymbol{I}\right).

(103)

We should keep in mind that this expression must yield a quantity of labor content, which in our case is measured in weeks. Since the entire production process takes place within a single week, the difference in labor content between the commodities consumed during production and the final output can only originate from the labor performed by the workers themselves. Because the production process lasts one week and employs $n$ workers throughout this period, the total labor added to production is $n$ units of labor content. Therefore, we have $p=n$ , which is precisely the amount of labor embodied in the final product. Hence,

r=\frac{n-n\lambda_{0}}{\lambda\left(\boldsymbol{K}\right)} =\frac{n}{\lambda\left(\boldsymbol{K}\right)}\left(1-\lambda_{0}\right).

(104)

Once again, we see the requirement that $\lambda_{0}< 1$ , since otherwise there would be no surplus available to be appropriated by the capitalists. Considering the economy as a whole, the average rate of profit is

r_{G}=\frac{N\left(1-\lambda_{0}\right)}{\lambda\left(\boldsymbol{K}_{G}\right)},

(105)

where $N$ is the total number of workers in the economy and $\boldsymbol{K}_{G}$ is the aggregate capital.

Looking at the economy as a whole, what differs across firms is the number of workers, $n$ , and the amount of capital, $\boldsymbol{K}$ . More specifically, the rate of profit is inversely proportional to the amount of capital per worker, $K_{n}=\frac{\lambda\left(\boldsymbol{K}\right)}{n},$ so that

r=\frac{1-\lambda_{0}}{K_{n}}.

(106)

Clearly, the profit rates $r$ can only be uniform across firms if $K_{n}$ is itself uniform, which does not appear to be a realistic assumption. Let us now rewrite

N\lambda_{0}=\lambda\left(\boldsymbol{V}\right),

(107)

simply to emphasize that $\boldsymbol{V}$ denotes the bundle of commodities consumed by workers in order to reproduce their labor power. We may then write

r_{G}=\frac{N-\lambda\left(\boldsymbol{V}\right)}{\lambda\left(\boldsymbol{K}_{G}\right)}.

(108)

Since $N$ is the total number of workers, it is also the total amount of labor (measured in weeks of labor) performed by these workers. There is an important point to emphasize here: the quantity of living labor added does not necessarily correspond to the number of physical hours worked within the firm. Because we are using the week as the fundamental unit of time, one unit of labor power corresponds to renting a worker’s capacity to work for an entire week. Thus, if a worker is hired on a weekly basis to produce a chair and delivers the finished product at the end of the week, we count one weekly unit of labor as having been embodied in the product, regardless of how the actual working hours were distributed throughout the week. In this framework, it is irrelevant whether the worker labored one hour or eight hours per day; what matters is that one week’s worth of labor power was mobilized in the production process. Therefore, if a firm hires $N$ workers on a weekly basis, the production process incorporates $N$ weekly units of living labor.

We are also working under the abstraction that workers are homogeneous with respect to their average productive capacity. Otherwise, a firm employing $n$ workers could incorporate either more or less than $n$ units of labor content, since its workers might be either more or less productive than the social average. We can therefore recover a version analogous to our first expression for $r$ , since

N=\lambda\left(\boldsymbol{C}_{G}\right)-\lambda\left(\boldsymbol{I}_{G}\right),

(109)

which gives

r_{G}=\frac{\lambda\left(\boldsymbol{C}_{G}\right)-\left[\lambda\left(\boldsymbol{I}_{G}\right)+\lambda\left(\boldsymbol{V}\right)\right]}{\lambda\left(\boldsymbol{K}_{G}\right)}.

(110)

However, we now abandon the notion of ideal prices and the deterministic formulation. Instead of assuming $\frac{\overline{\pi}\left(C\right)}{\lambda\left(C\right)}=\psi_{0},$ we adopt the specific price of a commodity bundle introduced earlier:

\overline{\Psi}\left(B\right) =\frac{M\left(B\right)}{L\left(B\right)} =\frac{\pi\left(B\right)}{\lambda\left(B\right)}.

(111)

Now consider again that the commodity bundles $\left(\boldsymbol{C}_{G},\boldsymbol{I}_{G},\boldsymbol{K}_{G},\boldsymbol{V}\right)$ each contain an extremely large number of commodities. Recall that the specific price $\Psi_{j}$ of commodity $j$ is modeled as a random variable drawn from a probability distribution common to all commodities. Furthermore, the average specific price of a bundle $\boldsymbol{A}$ is given by

\overline{\Psi}(\boldsymbol{A}) = \sum_{j} \frac{\lambda_{j}}{\sum_{k}\lambda_{k}} \Psi_{j},

(112)

where $\lambda_{j}$ denotes the labor content of commodity $j$ . Since the $\Psi_{j}$ are independent and identically distributed random variables, and since the bundles under consideration contain a very large number of commodities, it follows from the law of large numbers that

\overline{\Psi}(\boldsymbol{A}) \approx \overline{\Psi}(\boldsymbol{B}),

(113)

for any sufficiently large bundles $\boldsymbol{A}$ and $\boldsymbol{B}$ .

Therefore,

r_{G} = \frac{\pi\left(\boldsymbol{C}_{G}\right)-\left[\pi\left(\boldsymbol{I}_{G}\right)+\pi\left(\boldsymbol{V}\right)\right]} {\pi\left(\boldsymbol{K}_{G}\right)}.

(114)

However, $\pi$ no longer denotes the ideal price. Instead, it represents the actual realized price. Now, if the profit rate of firm $i$ is

r_{i} = \frac{l_{i}}{c_{i}},

(115)

where $l_{i}$ is the firm’s realized profit and $c_{i}$ is its capital, then the capital-weighted average profit rate across the entire economy is therefore

\left\langle r\right\rangle = \frac{\sum_{i}c_{i}r_{i}}{\sum_{j}c_{j}} = \frac{\sum_{i}c_{i}\frac{l_{i}}{c_{i}}}{\sum_{j}c_{j}} = \frac{\sum_{i}l_{i}}{\sum_{j}c_{j}}.

(116)

Since $r_{G}$ is precisely the ratio between the total profit and the total capital of the entire economy, it follows that

r_{G}=\left\langle r\right\rangle.

(117)

In other words, we have abandoned the hypothesis of deterministic ideal prices, according to which every commodity satisfies exactly

\frac{\overline{\pi}(C)}{\lambda(C)}=\psi_{0},

(118)

and replaced it with a probabilistic formulation in which the specific prices of commodities are treated as random variables. Nevertheless, for sufficiently large aggregates of commodities, the law of large numbers implies that individual price fluctuations offset one another, causing the ratio between price and labor content for these aggregates to converge toward a common average value, $\overline{\Psi}$ .

Rate of Surplus Value and the Tendency of the Rate of Profit to Fall¶

Returning to our initial formulation in terms of prices, let us define the ideal wage unit as $\overline{\pi}_{0}=1$ . Let $\lambda_{0}$ denote the labor content of this unit. Then, for labor power, we have

\frac{\overline{\pi}_{0}}{\lambda_{0}}=\psi_{0}.

(119)

For any other commodity bundle, it follows that

\frac{\overline{\pi}\left(C\right)}{\lambda\left(C\right)} = \psi_{0} = \frac{\overline{\pi}_{0}}{\lambda_{0}} = \frac{1}{\lambda_{0}}.

(120)

If we identify the rate of surplus value as

e_{M} = \frac{N-\lambda\left(\boldsymbol{V}\right)} {\lambda\left(\boldsymbol{V}\right)} = \frac{N-\lambda_{0}N} {\lambda_{0}N} = \frac{1-\lambda_{0}} {\lambda_{0}},

(121)

then the rate of surplus value corresponds to the ratio between the labor content of the surplus appropriated by the capitalists and the labor content of the bundle of commodities required to reproduce labor power.

Therefore,

\lambda_{0} = \frac{1}{1+e_{M}}.

(122)

Substituting this result into the previous expression, we obtain

\psi_{0} = \frac{\overline{\pi}\left(C\right)} {\lambda\left(C\right)} = \frac{1}{\lambda_{0}} = \frac{1}{\frac{1}{1+e_{M}}} = 1+e_{M}.

(123)

Connecting this with the previous discussion, in which we moved from the deterministic formulation to the probabilistic one, $\psi_{0}\rightarrow\overline{\Psi}$ , we may expect that the average specific price of a sufficiently large aggregate of commodities converges to $1+e_{M}$ . This is precisely what we shall now demonstrate.

To do so, recall that $\boldsymbol{V}$ denotes the aggregate of commodities consumed by the working population during one week, and that $\pi\left(\boldsymbol{V}\right)$ is its total price. We shall ignore savings, taxes, and similar factors, assuming that workers spend their entire income. Under this assumption, the aggregate weekly wage paid to all workers is simply $\pi\left(\boldsymbol{V}\right)$ .

Since we have chosen the average weekly wage as the monetary unit, the aggregate wage paid to $N$ workers over one week is numerically equal to $N$ . Hence,

\overline{\Psi}(\boldsymbol{V}) = \frac{\pi\left(\boldsymbol{V}\right)} {\lambda\left(\boldsymbol{V}\right)} = \frac{N} {\lambda\left(\boldsymbol{V}\right)} = 1+e_{M}.

(124)

Therefore, for any sufficiently large aggregate of commodities $\boldsymbol{A}$ , we expect that

\overline{\Psi}(\boldsymbol{A}) \approx 1+e_{M}.

(125)

In this formulation, the labor theory of value is no longer interpreted as a deterministic theory of individual prices, but rather as a statistical regularity that holds only at the aggregate level.

Furthermore, let $\lambda\left(\boldsymbol{V}\right)=v$ and $\lambda\left(\boldsymbol{K}_{G}\right)=k$ . Since $e_{M} = \frac{N-v}{v},$ and $1+e_{M} = 1+\frac{N-v}{v} = \frac{N}{v},$ , we obtain

r_{G} = \frac{N-v}{k} = \frac{N}{k} \left( \frac{v}{N} \frac{N-v}{v} \right) = \frac{1}{q_{G}} \left( \frac{e_{M}} {1+e_{M}} \right).

(126)

Since $\lim_{e_{M}\rightarrow\infty} \left( \frac{e_{M}}{1+e_{M}} \right) = 1,$ it follows that if the amount of capital per worker, $q_{G}=\frac{k}{N},$ tends to increase, that is, $q_{G}\rightarrow\infty,$ then, regardless of any increase in the rate of surplus value $e_{M}$ , the general rate of profit tends to decline:

\lim_{q_{G}\rightarrow\infty}r_{G}=0.

(127)

According to the authors of this book, however, this tendency is not supported by the available empirical evidence. Other interpretations argue that it is not the general rate of profit that tends to decline, but rather the incremental rate of profit (that is, the rate of return on new investments).

Furthermore, we may relate the quantity $q_{G}$ to the well-known organic composition of capital, $c_{0}=\frac{k}{v}.$ Since $e_{M} = \frac{N}{v}-1,$ we have $v=\frac{N}{1+e_{M}},$ and therefore

c_{0} = \frac{k}{v} = \frac{k}{N}\left(1+e_{M}\right) = \left(1+e_{M}\right)q_{G},

(128)

which implies

q_{G} = \frac{c_{0}}{1+e_{M}}.

(129)

Thus, we may rewrite the previous expression directly in terms of the organic composition of capital:

r_{G} = \frac{e_{M}}{c_{0}}.

(130)

Once again, an increase in the organic composition of capital, while holding $e_{M}$ constant, leads to a decline in the general rate of profit.

Conclusion¶

The main weakness of the first model does not appear to lie in its connection between price and value, but rather in its strictly deterministic character. Once reinterpreted probabilistically—allowing individual prices to fluctuate randomly while regularities emerge only at the aggregate level—the model becomes compatible with the empirically observed dispersion of both prices and profit rates.

The model presented in Volume III, on the other hand, retains its deterministic character and furthermore begins by assuming the uniformity of profit rates. This assumption is introduced precisely because the original deterministic model of Volume I cannot generate uniform profit rates across sectors with different capital compositions. However, in attempting to impose such uniformity, the model introduces new conceptual and mathematical difficulties.

Whereas the model in Volume I of Capital is primarily concerned with the origin of profit—which is why concepts such as value and surplus value occupy a central role—the model in Volume III shifts its attention to the distribution of profit. Although these two questions can be treated separately, as demonstrated by several theorists, there must ultimately be some connection between them. It is precisely in formulating this connection that many of the controversies associated with the so-called transformation problem arise.

It is important to bear in mind that non-Marxist approaches, being concerned only with the distribution of profit, may regard the labor theory of value as unnecessary. The Marxist approach, by contrast, seeks not only to explain the distribution of profit but also its origin. It therefore requires a framework that preserves some connection between profit, value, and surplus value. Consequently, rather than abandoning the original model in favor of a new deterministic system based on uniform profit rates, it seems more natural to reinterpret the original model directly in probabilistic terms. Under this approach, individual prices and profit rates are free to fluctuate, while the regularities predicted by the labor theory of value emerge only at the statistical and aggregate levels.

It may also be said that the authors reject the hypothesis that profit rates become equalized across individual capitals. Instead, they replace it with the concept of a general rate of profit defined as the capital-weighted average of firms’ profit rates, where the weights correspond to the magnitude of each firm’s capital. Rather than assuming a uniform profit rate across different sectors, this approach assumes the existence of a relatively stable distribution of profit rates across firms.

Automatic translation into English from my Portuguese manuscripts.