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This paper studies, within a growth model, some effects of the inequality between the profit and growth rates on the reproduction of economic elites. To this end, it considers as functions of the capital/income ratio the relations between, on the one hand, the economic growth rate and, on the other hand, the growth rates of capital and of national income. Based on this, it shows that when the income of a particular socio-economic stratum increases with respect to the national income, the lower limit for the growth rate of the first income depends almost exclusively on the variations of the capital/income ratio and of the average productivity of labor, while the employment growth rate plays a secondary role. Moreover, the paper distinguishes between three categories of renter and establishes sufficient conditions for the reproduction of each one of them. It points out that the third category, which comprises those renter dynasties whose share in the national capital stock increases with each generation, constitutes a quasi-feudal development within capitalist societies.

One of the salient features of a modern economy, as demonstrated by Piketty ( [

Building on the arguments offered by Piketty in favor of his thesis, this paper adds clarifications developed within the framework of the growth model introduced in Benítez [

There are two contributions of this paper that may be of particular importance. The first is that it shows the key part played by the capital and national income growth rates for the concentration of income when the two rates are formulated as functions of the capital/income ratio and the economic growth rate. This brings a slightly different perspective to the analysis developed in Piketty [

Regarding the first contribution, two conclusions relating some of the main variables of the model are particularly salient. First, the national income growth rate is greater than, equal to or less than the economic growth rate if the capital/income ratio respectively, decreases, stays constant or increases between two successive production periods. Second, the economic growth rate is equal to the sum plus the product of the growth rates of employment and of the average productivity of labor. These specifications are important, on the one hand, because the first one helps to explain certain aspect of income concentration as function of the capital/income ratio, such as the extension and the intensity of concentration for a given level of the economic growth rate. On the other hand, they allow establishing that the concentration of income in the higher income strata is limited almost exclusively by variations of the capital/income ratio and of the average productivity of labor, while the employment growth rate plays only a secondary role.

Regarding the second contribution, the paper underscores the difference between the concentration of income in favor of the capital owners as a class and that which favors the groups of higher incomes, referred to in the paper as economic elites. It shows that each type of concentration may occur without the other which is relevant because their meaning can also be different. Indeed, due to the fact that each consumer may own a share of capital, the first type does not exclude that workers receive part of the profit and, for this reason, income inequality depends on the distribution of capital ownership among consumers. If this distribution is particularly unequal, the second type of concentration takes place, propitiating the existence of renters and also of dynasties of renters whose share in the ownership of the national capital stock increases with each generation, which constitutes a feudal-style development within capitalist societies.

In addition to this introduction, the text contains four sections. Section 2 presents the basic model of Sraffa [

In this section, I present the basic model of Sraffa and the growth path studied here.

I consider a succession of annual production processes starting on dates

For each couple

Making

It should be added that one of the constraints of the model presented here, the fact that includes only those goods that produce all the goods, may be overcome using the Leontief’s closed model once the adaptations required are introduced. Indeed, as Benítez [

For each couple

has a unique solution

For

while, for

where, for each

and also that the vector of quantities produced in any given production cycle is a multiple of the vector produced in the first cycle.

Let

It follows from Equation (7) that, for each

Using the notation just introduced it is possible to write Equation (5) as follows:

On the other hand, for each

Starting from the second year, each production process uses the same set of means of productions as the previous one plus the part of the net product of that process that was not consumed. For this to be possible, I assume that consumers save a fraction of the net product of each year and also that, for every pair

In each period

In this section, I present the definitions and some properties of the main variables considered in this study.

It follows from Equation (3) that, for every

and

Then, the capital/income ratio of period

Therefore, this ratio is independent of the distribution of income and depends only on the technique of the period considered.

Prices will be measured with the value of the whole product of the first period of production, which permits to relate some macroeconomic variables with the growth rates of the different production periods. Indeed, for every

On the other hand, multiplying both sides of Equation (9) by

The last two equations imply that:

This result and the definitions of capital and national income presented above imply respectively that:

and

Due to the fact that these formulas are independent of changes in relative prices taking place in the different production periods, they facilitate comparing capital and income pertaining to those periods, as shown in the next section.

Equation (18) for period

Dividing term by term Equation (18) by Equation (20), results in:

Substituting

Furthermore, Equation (14) implies that:

Equation (26) corresponding to period

Dividing term by term Equation (27) by Equation (26) and simplifying results in:

Equations ((23) and (28)) taken together imply that:

For each

Thus, we can formulate the following conclusion:

Proposition 1. The capital growth rate from period

The next example allows having an idea of the order of magnitude of the first factor between parentheses on the right hand side of Equation (30).

Example 1. According to Graphic 3.6 by Piketty ( [

constant increase, the average value of capital in that 60 years period was

Therefore, due to the increase in the capital/income ratio the capital growth rate exceeded the economic growth rate in a percentage equal to

Equation (19) for period

Dividing term by term Equation (19) by Equation (32), results in:

Substituting

Furthermore, Equation (14) implies that:

Equation (38) corresponding to period

Dividing term by term Equation (39) by Equation (38) and simplifying, results in:

Substituting the first factor in the right-hand side of Equation (35) by the left-hand side of Equation (40) gives:

For each

Thus, we can formulate the following conclusion:

Proposition 2. The national income growth rate from period

Finally, Equations ((41) and (42)) taken together imply that:

The next example allows having an idea of the order of magnitude of the first factor between parentheses on the right hand side of Equation (42).

Example 2. Substituting

Therefore, due to the increase in the capital/income ratio the economic growth rate exceeded the national income growth rate in a percentage equal to

For every

When

Equations ((46) and (47)) taken together imply that:

According to this equation, the profit rate is equal to the maximum profit rate multiplied by the fraction of national income which corresponds to profits. It should be added that, for each

According to system (10) for each

Equations ((17) and (49)) taken together imply that the average productivity of labor

I assume that, for each

where

and

Substituting

Now, substituting in Equation (53)

Þ

Multiplying and dividing the right hand side of this equation by

Now, substituting the numerator and the term between square brackets in the right-hand side of this equation by their respective equivalences in Equations ((6) and (8)) yields:

Equations ((49), (50) and (59)) taken together imply that:

This means that is the rate of variation of the average productivity of labor from period

Þ

For each

Equations ((8), (62) and (63)) imply that:

Þ

The preceding analysis enables us to draw the following conclusion.

Proposition 3. The economic growth rate is equal to the sum plus the product of the growth rates of employment and of the

Equations ((14) and (47)) taken together imply that:

In turn, this result and Equation (48) imply that:

Þ

Þ

Equation (69) corresponding to period

Dividing term by term Equation (69) by Equation (70) results in:

Þ

The preceding analysis enables us to draw the following conclusion.

Proposition 4. Given two successive periods of production, for each level of the profit rate common to both periods, the fraction of national income corresponding to wages in the first is greater than, equal to or less than that which corresponds to the second if the capital/income ratio of the first is, respectively, less than, equal to or greater than the second.

Due to the fact that the capital/income ratio and the average productivity of capital change in opposed sense (see Benítez [

Example 3. It follows from

Therefore, due to the increase in the capital/income ratio, the wage share decreased

For each

To study the effects of changes in the average productivities of capital and labor on the wage unit, it is useful to substitute in Equation (74) variables

Equations ((42) and (64)) taken together imply that:

Substituting the sum

This result and Equation (74) corresponding to period

Therefore, the wage unit in period will be greater than, equal to or less than in period

is respectively, greater than, equal to or less than one. If the capital/income ratio increases in the second period, the first two factors in the product (80) are less than one for each level of the profit rate common to periods

If, on the other hand, the capital/income ratio decreases, the first two factors of function (80) are greater than one for each level of the profit rate common to periods

The next example allows having an idea of the order of magnitude of product (80).

Example 4. According to

Thus, notwithstanding the decrease in the wage share, due to the increase of the average productivity of labor, the wage unit increased by

In this section, I introduce a model of a Bank centralizing transactions among economic agents and also define some related concepts.

In every production period, there is a single company in each industrial branch, to which corresponds the index of the good produced in the branch. For each

There is a Bank in which, at noon of the first date of each period, are found deposited all the goods of the economy, which belong entirely to the consumers. Each consumer

In the evening of each date

In the morning of each date

On each date

I will represent with

For each pair

and the total income is:

Thus, for each pair

and, for each pair

For each

Equation (86) allows us to show that, when there is an increase in the wage unit and a concomitant decrease of the profit rate, the income of those agents whose income depends to a greater extent on labor will increase and vice versa (see Equations (48) and (74)). Therefore, given the distribution of labor and capital among consumers, the income distribution will vary depending on the rate of profit, and for each level of the same, such distribution is determined unequivocally. There is therefore, for each

For each pair

The propensity to save for the set of consumers is obtained dividing the sum of saving by the total income, resulting in:

Substituting the second factor in the right-hand side of this equation by the left-hand side of Equation (89) yields:

In this manner, the propensity to save of the economy is the sum of the individual propensities, each one weighted by the fraction of the national income belonging to the corresponding consumer.

Let

implies that, if the propensity to save is the same for all consumers, the collective propensity to save is equal to the propensity to save of each consumer. On the other hand, if

Multiplying Equation (86) by

Adding the first term on the right-hand side from the

Substituting the first term of this function by the sum

Hence:

In this way, the difference between profit and savings is equal to the difference between the amount of profits destined to consumption and the amount of wages destined to savings. This result and Proposition 1 of Benítez [

Proposition 5. Given two successive production periods, the profit rate of the first period is greater than, equal to, or less than the capital growth rate of the second if in the first period the amount of profit destined to consumption is, respectively, greater than, equal to, or less than the amount of wages destined to savings.

In this section, I establish some conditions for the concentration of income to take place both among individuals and among social strata as well.

For each

The wage part of a consumer income can vary from a period to the next by a change in the amount of work done, a change in the wage unit, or both. However, I assume that for each couple

This is justified by the simplification introduced in the analysis and, additionally, by the following reasons. On the one hand, the average amount of work performed by an employee under normal circumstances varies little from a period to the next, and the variation in salary must also be small since, as I will suppose from now on, the profit rate is the same in periods

Thus, making

Now, the difference between

Equations ((108) and (109)) taken together imply that:

Therefore:

In this section, I study the concentration of income within a given cohort

The sum

It is important to note that assuming that the first

On the other hand, for each

This fraction indicates the part of the income of cohort

Substituting

Replacing the first factor on the right-hand side of the last equation with its equivalent in accordance with definition (89) gives:

Dividing by

which leads to the following conclusion.

Proposition 6. The fraction of the income of cohort

I will say that a concentration of income takes place within a cohort from date

Dividing term by term Equation (111) by Equation (43) results in:

Substituting the left-hand side of this equation by its equivalence in Equation (89) and simplifying the numerator on the right-hand side, we can write this equation in the following form:

In this way, replacing the first factor on the right-hand side of the last equation with its equivalent according to Equation (89) gets:

This result allows drawing the following conclusion.

Proposition 7. For each

I will say that a concentration of income takes place within two successive cohorts from date

It follows from Equations ((89), (114) and (115)) taken together that every pair

Let

Let

Since employment is growing at the rate

the earlier date, due to this

On the other hand, replacing

The first term of the sum in parenthesis on the right-hand side of this equation is equal to

In accordance with the foregoing, it is possible to write the right-hand side of Equation (130) as follows:

Substituting

Substituting now

It should be noted that the second factor between square brackets on the right-hand side of this equation approaches one when the difference between the average earnings of the two groups of consumers, the one with

Example 5. If the average income of the consumer belonging to stratum b already on date

Therefore, if

Unlike what happens with the concentration of income in favor of individual consumers, the concentration of income in favor of a particular stratum

In this section, I study the effect of saving on the existence of a class of renters under the assumption that the propensity to consume has the properties indicated by Keynes ( [

I assume that the per capita income

Hypothesis 1. There is a function

It follows from the above that the inverse function exists

According to these rules, for each

On the other hand, for each

I call renter an individual who possesses a capital reporting a profit that allows him to live comfortably without having to participate in the production process. It should be added that this does not imply that the individual does not work but only that he has the possibility to refrain from working thanks to his share in the ownership of capital. It is useful to distinguish the following three types of renters.

Definition 1. A renter is a consumer who owns a capital reporting an annual profit equal to or greater than

Definition 2. A renter dynasty is a sequence of generations of a family lineage in which each member of the family belonging to these generations inherits a capital that, during the period between the granting of the heritage in two successive generations, increases enough so that the heirs of the following generation are also renters.

Definition 3. A quasi-feudal renter is a member of a dynasty of renters where each member bequeaths to each one of his descendants a fraction of the total capital that is at least equal to the one he himself received from his predecessors.

Now, I will proceed to calculate

A) Calculation of

B) Calculation of

Then, to belong to a dynasty of renters, a consumer

In order to simplify, I assume that the rate of saving is equal to

Þ

Þ

Conditions ((137) and (138)) taken together imply that:

This result and inequality (144) taken together imply that there can be dynastic renters only if:

Assuming that this inequality is met, the amounts of annual income satisfying condition (144) are characterized by the following inequality:

Accordingly, to satisfy condition (141) it is enough for consumer

We thus reach the following conclusion. If inequality (146) is satisfied, then:

The next example allows forming an idea of the order of magnitude of the variables considered.

Example 6. Let

Substituting the corresponding values in inequality (144) gives:

Þ

Therefore, to satisfy condition (141) it is enough owning a capital sufficiently large to live with

Þ

Þ

Substituting in the right hand side of this equation

This equation indicates a sufficient annual income. The corresponding capital is:

Equations ((139) and (159)) taken together imply that

Hence, in order to belong to a dynasty of renters it is enough to inherit the amount of capital just indicated.

C) Calculation of

Þ

To simplify, I assume that the propensity to save

Þ

Þ

This condition and inequality (145) taken together imply that there can be quasi-feudal renters only if:

On the other hand, according to condition (140) we have

Proposition 8. If the rates of profit of capital growth and of individual savings are constant, there can be quasi-feudal renters only if the amount of profit destined to consumption is greater than the amount of wages destined to savings every year or the period considered.

Assuming that condition (167) is satisfied, it follows from inequality (166) that the next condition is also required:

The amounts of annual income satisfying this condition are characterized by the following inequality:

In order to satisfy this condition, it is enough for consumer

Thus, we reach the following conclusion. If inequality (167) is satisfied, then:

Those that inherit an amount of capital equal to or greater than

The next example allows forming an idea of the order of magnitude of the variables considered.

Example 7. It follows from the data presented in Example 1 that, if the capital growth rate in the French economy had been constant during the period 1950-2010, the following equation must be satisfied:

Þ

Substituting in the right-hand side of inequality (167) by the right-hand side of Equation (174) and also substituting

This result, taken together with the fact that the average annual profit rate in the French economy during the period considered was equal to

Þ

Therefore, to satisfy condition (162) it is enough owning a capital sufficiently large to live with

This equation indicates a sufficient annual income. The corresponding capital is:

Equations ((161) and (182)) taken together imply that

Hence, in order to be a quasi-feudal renter it is enough to inherit the amount of capital just indicated.

It is worth mentioning that Condition (167) is not always met. However, in an economy where individual capitals obtain different profit rates, even if condition (167) is not satisfied by the average rate of profit of the economy it may be satisfied by the profit rate of some individual capitals Furthermore, if the capital/income ratio decreases during the period considered, as occurred for instance in the French economy during the years 1910-1950 (see Benítez [

The three categories of renters can include the same set of individuals although typically this does not happen. A renter may not be able to inherit to his descendants amounts of capital large enough for them to be also renters. A dynastic renter may belong to a renter dynasty where the proportion of total capital own by each of the members decreases from one generation to the following one. A quasi-feudal renter may lose his status either temporarily or definitely if changes in the profit rate of his individual capital stock, in the capital growth rate of the economy or in both variables are not favorable enough. For these reasons, the duration of a renter dynasty may vary widely.

Substituting the sum

Regarding profits, we can infer from this equation two different effects on the concentration of income caused by an increase of the capital/income ratio while the other variables in the quotient between square brackets remain constant. The first one can be called extensive due to the growth of the percentage of the national capital stock whose profit is benefited by an increase with respect to the sum of income. Indeed, with the decrease in the national income growth rate, the participation in the total income increases not only for the individual profits that already did previously to this decrease, but also for the profits of some other capitals growing at smaller rates. The second can be called intensive because the individual profits that already increased their participation in the total income now grow in a proportion greater that before.

If instead, the capital/income ratio decreases, two effects take place opposed to those just mentioned. On the one hand, it diminishes the percentage of the national capital stock whose profit increases with respect to the national income. This is due to the fact that, because of the increase of the national income growth rate, some individual profits that previously grew or kept constant now decrease with respect to the national income. On the other hand, it reduces the rate of income concentration because profits that still increase their participation in the total income do it now in a smaller proportion than before.

Therefore, when the capital/income ratio varies, the increase in the capital share and the concentration of income in favor of the economic elites do not occur always together. It is important to note that the latter may occur both if this ratio increases as if it decreases, although both the extent and the intensity of the process of concentration are favored in the first case and decreased in the second. Furthermore, when the capital/income ratio increases, the growth rates of capital and of national income increase and decrease respectively. For these reasons, an increase of the capital/income ratio favors the formation of renter dynasties but hinders that of quasi-feudal dynasties and vice versa.

Finally, it is important to add that, while in the model studied here the profit rate is the same for all capitals, it follows from formula (125) that the concentration of income is an increasing function of the profit rate obtained by each individual capital. Hence, as already indicated, for that concentration to take place it is not required for the average profit rate of an economy to be higher than the growth rate of its national income. It is enough that one individual capital obtains a profit rate sufficiently greater than the mentioned growth rate.

Excluding the particular case in which all consumers receive an equal income, given a situation chosen randomly, in each cycle of production a concentration of income takes place within the corresponding cohort and, starting from a certain degree of inequality also is produced a concentration of income between two successive cohorts. This last development does not require for the average profit rate of an economy to be higher than the growth rate of its national income. It is enough that for at least one individual capital, the product of the corresponding profit and savings rates be above the national income growth rate, which tends to favor the growth of larger capitals given that they tend to get higher profit rates. Furthermore, consumers whose incomes are growing at a faster rate than the national income, also come to increase their propensity to save. Thus, in a succession of production periods there may be a persistent concentration of income in favor of the same consumers.

When capital property is sufficiently concentrated, some individuals can live comfortably without participating in the production process, their expenses being covered with their profits. A social class consisting of renters is thus formed and, within this, renter dynasties can be established. Moreover, some dynasties of particularly wealthy renters give a feudal bias to the development of capitalist societies since their participation in the ownership of the capital stock of the society increases with each generation.

It should be recalled that these conclusions refer to the particular model studied in this article, in which the forecasts of all agents are met, and therefore do not include the causes that can disrupt the uninterrupted growth of some individual capitals considered here. For this reason, they describe possible trends within capitalist economies whose presence in real economies must be verified in each case, which represents a considerable task for future research.

I am grateful to two anonymous referees for helpful comments and suggestions.

Benítez Sánchez, A. (2016) Piketty’s Inequality between the Profit and Growth Rates and Its Implications for the Reproduction of Economic Elites. Theoretical Economics Letters, 6, 1363- 1392. http://dx.doi.org/10.4236/tel.2016.66125