To each year corresponds a particular index t such that $t=1,2,\cdots$ For each t, $L_t\left(w\right)$ is the labor used, $Q_t\left(w\right)$ the quantity produced, $K_t\left(w\right)$ the quantity consumed, and $Y_t\left(w\right)$ the net product ( $Q_t\left(w\right)-K_t\left(w\right)$) obtained producing the single good in the year t. Therefore, the average labor productivity in that year and in the period t₁ − t₂, which runs from the beginning of the year t₁ to the end of the year t₂ are, respectively:

$AL{P}_t\left(w\right)=\frac{Y_t\left(w\right)}{L_t\left(w\right)}$ (1)

$AL{P}_{t_1-t_2}\left(w\right)=\frac{{\displaystyle {\sum }_{t=t_1}^{t_2}{Y}_t\left(w\right)}}{{\displaystyle {\sum }_{t=t_1}^{t_2}{L}_t\left(w\right)}}.$ (2)

Let ${\varkappa }_{t_1-t_2}\ge 0$ be the increase in average labor productivity over the period t₁ − t₂ above $AL{P}_{t_1}\left(1\right)$ that is attributable to the zero-profit level. That is, the increase that could have occurred if the profit level had been zero during the reference period. We measure the quantities of the good using as a unit of measurement the average net product per labor unit corresponding to the zero level of profit. For this reason, we have:

$AL{P}_{t_1}\left(1\right)+{\varkappa }_{t_1-t_2}=1.$ (3)

It should be noted that, although it is not specified on the right side, on both sides of this equation we have a quantity of the good per unit of labor.

To calculate the real wage, we notice that the total quantity of the only good that can be purchased by the wage-earners in a given year t or in the period t₁ − t₂ results from multiplying by $w$ the numerator in the right side of Equations (1) and (2), respectively. To obtain the real wage in each case we divide the product by the corresponding quantity of labor. From this conclusion, it follows that in a given year t:

$s_t\left(w\right)=wAL{P}_t\left(w\right)$ (4)

and, in the period from t₁ to t₂:

$s_{t_1-t_2}\left(w\right)=wAL{P}_{t_1-t_2}\left(w\right).$ (5)

Since the subject of our study is the relation between the growth of profit and that of the average labor productivity, we shall assume that for every couple $\left(w_1,w_2\right)\in \left]0,1\right]$ such that $w_1<w_2$ the following two propositions are true:

${\displaystyle {\sum }_{t=t_1}^{t_2}{Y}_t\left(w_1\right)}\ge {\displaystyle {\sum }_{t=t_1}^{t_2}{Y}_t\left(w_2\right)}$ (6)

$AL{P}_{t_1-t_2}\left(w_1\right)\ge AL{P}_{t_1-t_2}\left(w_2\right).$ (7)

Hence, profit in real terms and average labor productivity are both non-decreasing functions of profit. There are several sets of sufficient conditions for Propositions (6) and (7) to be true that are more or less realistic. For the sake of simplicity, we adopt the following six assumptions of which the first three and the last two are true all along the reference period:

(I) Regardless of the wage level both single good and labor markets are always in equilibrium.

(II) Production plans are chosen following the criterion of profit maximization.

(III) The propensity to consume of the society is a non-increasing function of both income and profit.

(IV) For every $t>t_1$, the investment in the year t ( $K_t\left(w\right)-K_{t-1}\left(w\right)$) is equal to the savings in the year t − 1.

(V) Net product and employment are both monotonous increasing functions of capital. However, as capital increases, the corresponding percentage change in net product is greater than that which takes place in employment.

(VI) There is no increasing marginal productivity of labor.

The following four comments may be useful for a better understanding of the model.

First, with respect to Assumption (I), it is important to remember that the real wage cannot fall below a certain limit without the supply of labor decreasing enough to cause a diminishing of the net product, so inequality (6), as a rule, ceases to be valid for real wage levels that are below that limit. Nevertheless, our assumption that there is no such limit does not substantially affect the main results of this study, as argued in Remark 1 at the end of the next section.

Second, changes in production processes taking place before the reference period are not considered. For this reason, the capital stock of the first year is the same for all levels of income distribution. Therefore, if the employment level is also the same, average labor productivity in this year is constant and independent of income distribution. On the other hand, if the employment level is a decreasing function of the wage, it follows from Assumption (VI) that average labor productivity is never greater than when the wage equals one.

Third, it follows from the conclusions of the second comment and Assumption (III) that the amount of savings in the first year is a non-decreasing function of profit. Then, according to Assumption (IV), $K_2\left(w\right)$ is also a non-decreasing function of profit. This result taken together with Assumption (V) implies that both $Y_2\left(w\right)$ and $AL{P}_2\left(w\right)$ are non-decreasing functions of profit. Upon this basis, a similar reasoning allows us to conclude successively that $Y_t\left(w\right)$ and $AL{P}_t\left(w\right)$ are non-decreasing functions of profit for every $t>2$. In turn, these conclusions permit us to prove Propositions (6) and (7).

Fourth, if the propensity to consume is 100% and constant the equality holds in both Propositions (6) and (7). The inequality holds if the propensity to consume is a decreasing function of both income and profit.

To calculate $w_{t_1-t_2}\ast$, it is useful to remember what has been said above regarding

the right side of Equation (10). On the basis of this we can see that $w_{t_1-t_2}\ast$ is the wage for which the decrease in the purchasing power of the first term (with respect to $w=1$) is equal to the increase in the real wage concomitant with this decrease due to the increase in the average labor productivity. Therefore, $w_{t_1-t_2}\ast$ satisfies the following equation:

$1-w_{t_1-t_2}\ast =w_{t_1-t_2}\ast {\mu }_{t_1-t_2}\left(1-w_{t_1-t_2}\ast \right).$ (16)

Dividing the two sides of the equation by $1-w_{t_1-t_2}\ast$ yields:

$1=w_{t_1-t_2}\ast {\mu }_{t_1-t_2}$ (17)

⇒

$w_{t_1-t_2}\ast =\frac{1}{{\mu }_{t_1-t_2}}.$ (18)

This result shows that $w_{t_1-t_2}\ast$ is a monotonic decreasing function of ${\mu }_{t_1-t_2}$ which tends to 1 when ${\mu }_{t_1-t_2}$ tends to 1 and tends to 0 when ${\mu }_{t_1-t_2}$ tends to $+\infty$.

To calculate $w_{t_1-t_2}\ast \ast$, we equal the right side of Equation (13) to zero:

$1+{\mu }_{t_1-t_2}\left(1-2w_{t_1-t_2}\ast \ast \right)=0$ (19)

⇒

$1+{\mu }_{t_1-t_2}-2{\mu }_{t_1-t_2}{w}_{t_1-t_2}\ast \ast =0$ (20)

⇒

$2w_{t_1-t_2}\ast \ast {\mu }_{t_1-t_2}=1+{\mu }_{t_1-t_2}$ (21)

⇒

$w_{t_1-t_2}\ast \ast =\frac{1+{\mu }_{t_1-t_2}}{2{\mu }_{t_1-t_2}}.$ (22)

This result shows us that $w_{t_1-t_2}\ast \ast$ is a monotonic decreasing function of ${\mu }_{t_1-t_2}$, which tends to 1 when ${\mu }_{t_1-t_2}$ tends to 1 and tends to $\frac{1}{2}$ when ${\mu }_{t_1-t_2}$ tends to $+\infty$. Thus, when wage-earners receive the optimal wage, as the productivity/profit rate increases, the fraction of income corresponding to profit increases by tending to $\frac{1}{2}$.

$AL{P}_{t_1-t_2}\ast \ast$ is the value of the average labor productivity corresponding to the optimal wage. Therefore:

$AL{P}_{t_1-t_2}\ast \ast =ALP\left(w_{t_1-t_2}\ast \ast \right).$ (23)

To calculate this value, we substitute $w$ in the right side of Equation (9) with the right side of Equation (22):

$AL{P}_{t_1-t_2}\ast \ast =1+{\mu }_{t_1-t_2}\left(1-\frac{1+{\mu }_{t_1-t_2}}{2{\mu }_{t_1-t_2}}\right)$ (24)

$=1+{\mu }_{t_1-t_2}\left(\frac{2{\mu }_{t_1-t_2}}{2{\mu }_{t_1-t_2}}-\frac{1+{\mu }_{t_1-t_2}}{2{\mu }_{t_1-t_2}}\right)$ (25)

$=1+\frac{2{\mu }_{t_1-t_2}-1-{\mu }_{t_1-t_2}}{2}$ (26)

$=\frac{2+{\mu }_{t_1-t_2}-1}{2}$ (27)

$=\frac{{\mu }_{t_1-t_2}+1}{2}.$ (28)

This result shows us that $AL{P}_{t_1-t_2}\ast \ast$ is an increasing monotonic function of ${\mu }_{t_1-t_2}$ which tends to be equal to half the value of this variable as it grows.

$s_{t_1-t_2}\ast \ast$ is the real wage that corresponds to the optimal wage. Therefore:

$s_{t_1-t_2}\ast \ast =s\left(w_{t_1-t_2}\ast \ast \right).$ (29)

We can calculate $s_{t_1-t_2}\ast \ast$ by substituting $w$ and $AL{P}_{t_1-t_2}\left(w\right)$ into the right side of Equation (5) with the right side of Equation (22) and the right side of Equation (28), respectively:

$s_{t_1-t_2}\ast \ast =\left(\frac{1+{\mu }_{t_1-t_2}}{2{\mu }_{t_1-t_2}}\right)\left(\frac{{\mu }_{t_1-t_2}+1}{2}\right)$ (30)

$=\frac{1}{4}\left[\frac{1+2{\mu }_{t_1-t_2}+{\mu }_{t_1-t_2}^2}{{\mu }_{t_1-t_2}}\right].$ (31)

It follows from this formula that $s_{t_1-t_2}\ast \ast$ is an increasing monotonic function of ${\mu }_{t_1-t_2}$ which tends to be equal to a quarter of the value of this variable as it grows.

Here, we calculate the average labor productivity over a period $t_1-t_2$ by knowing the average annual growth rates of net product (g₁) and employment (g₂) over that period. The net product in the first year is $Y_{t_1}\left(w\right)$ units, in the second year it is $Y_{t_1}\left(w\right)\left(1+g_1\right)$ units, in the third year $Y_{t_1}\left(w\right){\left(1+g_1\right)}^2$ units and so on. Likewise, the amount of work used in the first year is $L_{t_1}\left(w\right)$ units, in the second year it is $L_{t_1}\left(w\right)\left(1+g_2\right)$ units, in the third year $L_{t_1}\left(w\right){\left(1+g_2\right)}^2$ units and so on. The average labor productivity over the period is the quotient of the sum of the first sequence of quantities divided by the sum of the second sequence:

$AL{P}_{t_1-t_2}\left(w\right)=\frac{Y_{t_1}\left(w\right)\left[1+\left(1+g_1\right)+{\left(1+g_1\right)}^2+\cdots +{\left(1+g_1\right)}^{t_2-t_1}\right]}{L_{t_1}\left(w\right)\left[1+\left(1+g_2\right)+{\left(1+g_2\right)}^2+\cdots +{\left(1+g_2\right)}^{t_2-t_1}\right]}.$ (32)

$AL{P}_{t_1-t_2}\left(w\right)=\left[\frac{Y_{t_1}\left(w\right)}{L_{t_1}\left(w\right)}\right]\left[\frac{1+\left(1+g_1\right)+{\left(1+g_1\right)}^2+\cdots +{\left(1+g_1\right)}^{t_2-t_1}}{1+\left(1+g_2\right)+{\left(1+g_2\right)}^2+\cdots +{\left(1+g_2\right)}^{t_2-t_1}}\right]$ (33)

$=\left[AL{P}_{t_1}\left(w\right)\right]\left[\frac{1+\left(1+g_1\right)+{\left(1+g_1\right)}^2+\cdots +{\left(1+g_1\right)}^{t_2-t_1}}{1+\left(1+g_2\right)+{\left(1+g_2\right)}^2+\cdots +{\left(1+g_2\right)}^{t_2-t_1}}\right].$ (34)

It is possible to observe that both numerator and denominator in the second factor of the left side of this equation are sums of geometric progressions whose first term is 1 and whose common ratios are $\left(1+g_1\right)$ and $\left(1+g_2\right)$, respectively. Therefore, applying the formula for this type of sums, we can write the last equation as follows:

$AL{P}_{t_1-t_2}\left(w\right)=\left[AL{P}_{t_1}\left(w\right)\right]\frac{\left[\frac{{\left(1+g_1\right)}^{t_2-t_1+1}-1}{\left(1+g_1\right)-1}\right]}{\left[\frac{{\left(1+g_2\right)}^{t_2-t_1+1}-1}{\left(1+g_2\right)-1}\right]}.$ (35)

Dividing the two sides of this equation by $AL{P}_{t_1}\left(w\right)$, we get the following formula:

$AL{P}_{t_1-t_2}\left(w\right)=\frac{\left[\frac{{\left(1+g_1\right)}^{t_2-t_1+1}-1}{\left(1+g_1\right)-1}\right]}{\left[\frac{{\left(1+g_2\right)}^{t_2-t_1+1}-1}{\left(1+g_2\right)-1}\right]},$ (36)

in which the average labor productivity during the reference period appears as a multiple of the average labor productivity in the first year of the period.

Here, we study both average labor productivity and real wage as functions of income distribution when ${\mu }_{t_1-t_2}=2$. Table 2 presents various wage values as well as those corresponding to the two functions. Their graphs are shown in Figure 3 .

Substituting ${\mu }_{t_1-t_2}$ with 2 into Equation (18), we get:

$w_{t_1-t_2}\ast =\frac{1}{2}.$ (37)

Therefore, the real wage is higher than 1 for every $w\in \left]\frac{1}{2},1\right[$.

To obtain the optimal wage, we substitute ${\mu }_{t_1-t_2}$ with 2 into Equation (22):

$w_{t_1-t_2}\ast \ast =\frac{1+2}{2\left(2\right)}$ (38)

$=\frac{3}{4}.$ (39)

Now, substituting ${\mu }_{t_1-t_2}$ with 2 into Equation (31) yields:

$s_{t_1-t_2}\ast \ast =\frac{1}{4}\left[\frac{1+2\left(2\right)+2^2}{2}\right]$ (40)

$=\frac{1}{4}\times \frac{9}{2}$ (41)

$=\frac{9}{8}.$ (42)

This means that with the optimal wage, the real wage is $\frac{1}{8}$ higher than when $w=1$.

Finally, in order to know the increase in the average labor productivity when $w=w\ast \ast$, we substitute ${\mu }_{t_1-t_2}$ with 2 into Equation (28). Then, we have:

$AL{P}_{t_1-t_2}\ast \ast =\frac{2+1}{2}$ (43)

$=\frac{3}{2}.$ (44)

According to this result, when the wage falls from 1 to $\frac{3}{4}$, there is also a 50% increase in the average labor productivity. The increase of $\frac{1}{2}$ unit of the good produced per labor unit increases in $\frac{1}{8}$ units the real wage and in $\frac{3}{8}$ units the profit per unit of labor.

According to Weinstock (2023) in the United States between 1949 and 2021, the average annual growth rate of labor productivity (g₃) in the non-farm business sector was 2.1%, according to FXEMPIRE (2023) the average annual rate of GDP growth (g₁) in the United States from 1947 to 2023 was 3.16% and, according to York (2023) , the average share of wages in national income from 1929 to 2023 was 69.9%. Adopting these data for the periods 1950-1985 and 1950-2020, we will roughly calculate successively the average labor productivity and the real wage for each period. With this purpose, to obtain g₂ we use the following formula:

$g_2=\frac{1+g_1}{1+g_3}-1.$ (45)

Substituting into this equation g₁ and g₃ with the corresponding values, we get:

$g_2=\frac{1+0.0316}{1+0.021}-1$ (46)

$=0.0103$ (47)

Of particular note in this example is the difference in real wage between the two periods, which is due to the cumulative nature of technological progress.

Substituting each variable in Equation (36) with its value during the period 1950-1985, we get:

$AL{P}_{1950\text{-}1985}\left(0.699\right)=\frac{\left[\frac{{1.0316}^{36}-1}{1.0316-1}\right]}{\left[\frac{{1.0103}^{36}-1}{1.0103-1}\right]}$ (48)

$=\frac{\left[\frac{2.0648}{0.0316}\right]}{\left[\frac{0.4461}{0.0103}\right]}$ (49)

$=\frac{65.3417}{43.3106}$ (50)

$=1.5086$ (51)

This result means that the average labor productivity during the period was 1.5086 times the average labor productivity of the first year of the period. Now, substituting each variable in Equation (5) with its value during the period 1950-1985 we get the real wage:

$s_{1950\text{-}1985}\left(0.699\right)=0.699\times 1.5086$ (52)

$=1.0541$ (53)

Hence, the average real wage over the period was 5.41% higher than the average labor productivity in the first year. During the reference period, average labor productivity increased 50.86% with respect to its initial value, of which 35.51% was added to wages and 15.35% to profits. These percentages result from computing $\left(1.0541-0.699\right)\times 100$ and $\left[0.5086-\left(1.0541-0.699\right)\right]\times 100$, respectively.

Substituting each variable in Equation (36) with its value in the period 1950-2020, we get:

$AL{P}_{1950\text{-}2020}\left(0.699\right)=\frac{\left[\frac{{1.0316}^{71}-1}{1.0316-1}\right]}{\left[\frac{{1.0103}^{71}-1}{1.0103-1}\right]}$ (54)

$=\frac{\left[\frac{8.1055}{0.0316}\right]}{\left[\frac{1.07}{0.0103}\right]}$ (55)

$=\frac{256.5031}{103.8834}$ (56)

$=2.4691$ (57)

This result means that the average labor productivity of the period was 2.4691 times the average labor productivity of the first year of the period. Now, substituting each variable in Equation (5) with its value during the period 1950-2020, we get the real wage:

$s_{1950\text{-}2020}\left(0.699\right)=0.699\times 2.4691$ (58)

$=1.7259$ (59)

Therefore, the average real wage during the period was 72.59% higher than average labor productivity in the first year. During the reference period, average labor productivity increased 146.91% with respect to its initial value, of which 102.69% was added to wages and 44.22% to profits. These percentages result from computing $\left(1.7259-0.699\right)\times 100$ and $\left[1.4691-\left(1.7259-0.699\right)\right]\times 100$, respectively.

As noted in our presentation of the WCS, the increase in average labor productivity when the profit is zero must be very small compared to what can be achieved at other levels of profit. Indeed, if there is no profit, investment loses its greatest attraction¹⁰ and, with it, the growth of markets and the division of labor lose their greatest momentum. For these reasons, for simplicity’s sake, in the example studied in this subsection we assume that when the profit is zero the increase in average labor productivity is also zero. In addition, we assume that the change in average labor productivity when the wage increases from 69.9% to 100% in the year 1950 is negligeable for our illustrative purposes. Consequently, according to Equation (3), the unit of measurement of the good is equal to the amount of that good produced on average per labor unit during the first year of the period when the wage is 69.9%. This conclusion implies that it is appropriate to employ in Equation (8) the value of $AL{P}_{t_1-t_2}\left(w\right)$ calculated by means of Equation (36).

Now, we will calculate successively the productivity/profit rate, the wage interval with real wages above one, the optimal wage and the corresponding real wage in the two periods presented in Example 2.

First, substituting each variable in Equation (8) with its value for the period 1950-1985 we calculate the productivity/profit rate:

${\mu }_{1950\text{-}1985}=\frac{1.5086-1}{1-0.699}$ (60)

$=1.6897$ (61)

Substituting ${\mu }_{t_1-t_2}$ with 1.6897 into Equation (18), we get:

$w_{1950\text{-}1985}\ast =\frac{1}{1.6897}$ (62)

$=0.5918$ (63)

Therefore, the real wage is higher than 1 for every $w\in \left]0.5918,1\right[$.

Now, we calculate the optimal wage for the period 1950-1985 by substituting ${\mu }_{t_1-t_2}$ with 1.6897 into Equation (22):

$w_{1950\text{-}1985}\ast \ast =\frac{1+1.6897}{2\left(1.6897\right)}$ (64)

$=0.7959$ (65)

Finally, we get the real wage corresponding to the optimal wage for the period 1950-1985 by substituting ${\mu }_{t_1-t_2}$ with 1.6897 into Equation (31):

$s_{1950\text{-}1985}\ast \ast =\frac{1}{4}\left[\frac{1+2\left(1.6897\right)+{1.6897}^2}{1.6897}\right]$ (66)

$=\frac{1+3.3794+2.855}{6.7588}$ (67)

$=1.0703$ (68)

Comparing this result with Equation (53), it can be seen that the real wage corresponding to the optimal wage is barely 1.53% higher than the real wage for the period.

First, substituting each variable in Equation (8) with its value for the period 1950-2020, we calculate the productivity/profit rate:

${\mu }_{1950\text{-}2020}=\frac{2.4691-1}{1-0.699}$ (69)

$=4.8807$ (70)

Substituting ${\mu }_{t_1-t_2}$ with 4.8807 into Equation (18), we get:

$w_{1950\text{-}2020}\ast =\frac{1}{4.8807}$ (71)

$=0.2048$ (72)

Therefore, the real wage is higher than 1 for every $w\in \left]0.2048,1\right[$.

Now, we calculate the optimal wage for the period 1950-2020 by substituting ${\mu }_{t_1-t_2}$ with 4.8807 into Equation (22):

$w_{1950\text{-}2020}\ast \ast =\frac{1+4.8807}{2\left(4.8807\right)}$ (73)

$=0.6024$ (74)

Finally, we get the real wage corresponding to the optimal wage for the period 1950-2020 by substituting ${\mu }_{t_1-t_2}$ with 4.8807 into Equation (31):

$s_{1950\text{-}2020}\ast \ast =\frac{1}{4}\left[\frac{1+2\left(4.8807\right)+{4.8807}^2}{4.8807}\right]$ (75)

$=\frac{1+9.7614+23.8212}{19.5228}$ (76)

$=1.7713$ (77)

Comparing this result with Equation (59), it is possible to see that the real wage that corresponds to the optimal wage is slightly higher (2.63%) than the real wage for the period.