_{1}

This article studies the ratio of the rates of profit and growth, in a growing economy, as a function of the average productivity of capital. It is shown that, if the savings rate and also the distribution of income between wage and profit are constant, the ratio mentioned remains constant or increases if the average productivity of capital respectively does not change or changes at a steady rate, whether it increases or decreases. If the change is repeated throughout a sufficiently large number of production cycles, the first rate grows above the second, even if in the initial situation the second rate is higher than the first. The result is the same if the savings rate and the rate of change of the average productivity of capital fluctuate within certain limits over a sufficiently large number of production cycles. In each case, the number of cycles required depends on the initial situation and the magnitude of the changes in both variables. These conclusions are compatible with the relevant historical data for economic variables involved. For this reason, they help to explain why, as a general rule, in a modern economy the rate of profit is higher than the growth rate.

Piketty ( [

This article complements the works just mentioned by studying the ratio between the rates of profit and growth in a growing economy as a function of the average productivity of capital (APK), a topic not previously treated in the specialized literature, as far as I know. To isolate this relation from the effects of changes in the distribution of income over the two rates, I consider a succession of production cycles where such distribution is the same in all cycles. The main result states that if the saving rate is constant, the ratio of the rates of profit and growth remains constant or increases if the APK respectively does not change or changes at a steady rate, whether it increases or decreases. The increase in the ratio is due to that, by increasing or reducing the APK, both profit and growth rate respectively increase or decrease but, in the first case, the profit rate increases in a greater proportion and, in the second one, decreases in a smaller proportion than the growth rate. If the change is repeated throughout a sufficiently large number of production cycles, the first rate grows above the second, even if in the initial situation the second rate is higher than the first. The result is the same if the savings rate and the rate of change of the APK fluctuate within certain limits over a sufficiently large number of production cycles. In each case, the number of cycles required depends on the initial situation and the magnitude of the changes in both variables. By means of examples, it is shown that the conclusions reached are compatible with the relevant historical data for economic variables involved. For this reason, they help to explain why, as a general rule, in a modern economy the rate of profit is higher than the growth rate. However, to know to what extent these conclusions allow to explain the difference between the two rates in particular economies and historical periods, it is necessary to perform econometric studies for each case.

In addition to this introduction, the text contains four sections and one Appendix. 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 production processes starting at 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

Let

These definitions together with Equation (5) allow writing, for each

For each

In each period t, the transactions take place at two different calendar dates. Those corresponding to the beginning of the production process are made in the afternoon of day t−1 and those corresponding to its end are made in the morning of day t.

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 t, it is possible to represent the set of goods used in production and the set of goods that constitute the net product respectively by vectors

and

Then, the capital/income ratio of period

Þ

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

For the purposes of this paper it is not necessary to choose a particular unit for measuring prices. However, it is worth mentioning that adopting for this task the whole product of the first period of production permits to relate some of the macroeconomic variables just defined with the growth rates of the different production periods. Indeed, for every

On the other hand, multiplying both sides of Equation (7) by

Equations (13) and (14) 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.

For each t,

Þ

When

Equations (19) and (20) 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

Equations (12) and (20) imply that:

For every

I also adopt the following proposition:

Hypothesis 1. In each date t, the set of households consume the goods defined by vector

As a result, for every

I will refer to

Equations (25) and (26) imply that:

which means that, from the second period, the growth rate of capital is equal to the product of the maximum profit rate and the fraction of national income devoted to saving which correspond to the preceding period. Since the last two variables are not necessarily the same in all periods, the growth rate of capital can vary from one period to another.

Furthermore, Equation (21) for period

Equations (27) and (28) allow drawing the following conclusion:

Proposition 1. 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 fraction of national income which corresponds to profit is respectively, greater than, equal to, or less than the fraction of national income destined to savings.

It is important to add that, for each

In the next section, I adopt a hypothesis about the technology employed that guaranties the satisfaction of this condition.

For each

Þ

It should be noted that, adding one unit on each side of Equation (20) yields:

Equations (31) and (33) imply that:

As a result, the APK and the maximum profit rate both increase or diminish in the same extent and in the same sense although these variations do not represent the same percentage for the two variables. Furthermore, substituting

On the other hand, I assume that, for every

where

Substituting

Þ

According to (iv) from Theorem 4.B.1 by Takayama ( [

Þ

Equation (41) also implies that:

Substituting the term between brackets in the left-hand side and also the right-hand side in this equation by their respective equivalences in accordance with Equation (31) yields:

which means that

Substituting in the penultimate equation

On the other hand, substituting

Þ

Therefore, the rate of output growth, or growth rate of the economy from period

Þ

Þ

The preceding analyses allow drawing the following conclusion.

Proposition 2. The economic growth rate is equal to the sum plus the product of the growth rates of capital and of the APK.

Substituting

Multiplying both numerator and denominator of the right-hand side of this equation by

On the other hand, Equation (20) which corresponds to the period

Þ

Þ

Replacing the first term and the term between brackets of the second term on the right-hand side of Equation (57) respectively by the left-hand side of Equations (58) and (60) yields:

Multiplying and dividing the right-hand side of this equation by

Now, substituting

Since, by hypothesis,

Replacing the first factor between brackets on the right-hand side of Equation (63) by the left-hand side of this equation result in:

Therefore, it is possible to formulate the following conclusion.

Proposition 3. If the distribution of income between wage and profit in two successive periods of production is the same, the profit rate decreases, remains constant or increases in the second period with respect to the first if the APK of the second is, respectively, less than, equal to, or greater than the APK of the first. In the first and in the third case, the absolute value of the rate at which the rate of profit varies is greater than the one corresponding to the rate of variation of the APK and the difference between these two figures will be lower the greater the maximum profit rate previously to the change of productivity.

In this section, I study different aspects of the ratio between the rates of profit and growth as a function of the APK. In the first subsection, I establish for

Equation (62) implies that:

Now, substituting the sum between brackets in the right-hand side of Equation (65) by the left-hand side of Equation (66) result in:

On the other hand,

Dividing Equation (67) term to term by the last equation yields:

Hence, the ratio of the rates of profit and growth in period t is greater than, equal to, or less than in period

Equation (66) implies that:

Furthermore, Equation (61) for the period

Substituting in the right-hand side of Equation (70) the rate

Doing the same substitution in Equation (27) results in:

Now substituting

Similarly, Equation (27) for the period

while Equation (53) for the same period is:

Substituting in this equation

Equations (72), (74) and (77) allow writing the following conclusion:

The right-hand side of this equation is the factor which, multiplied by the ratio of the rates of profit and growth of a production period

Furthermore, for each

Therefore,

In this Section and in the next one, I assume the following propositions.

Hypothesis 2. In each production period, the same fraction of national income is saved.

Hypothesis 3. In each production period, the APK varies at the same rate.

Let

Regarding the magnitude of h in this equation, it is important to note that there is no upper limit for its positive values. However, negative values are limited in accordance with the following proposition.

Lemma 1. For each

Proof. See Appendix A.2.

For each_{t}_{,min} indicates a range of possible values of h determined by the given values of s and R. Thus, it usually changes from one period to another. The following proposition relates changes in the APK and the sum

Theorem 1. The sum _{t}_{,min}.

Proof. See Appendix A.3.

The preceding analyses allow the drawing of the following conclusions.

Proposition 4. Given two successive production periods in a growing economy in which the distribution of income between wage and profit, the rate of savings and the rate of variation of the APK are constant, the ratio of the rates of profit and growth of the second period is equal to or greater than the ratio of the first period if the rate of variation of the APK is respectively equal to or different from zero but greater than h_{t}_{,min}.

As can be noted in Equations (65) and (53) when the APK increases or decreases both the profit and the growth rate respectively increase or decrease. This observation and Proposition 4 imply that, in the first case, the profit rate increases in a greater proportion and, in the second one, decreases in a smaller proportion than the growth rate.

In the examples, I use certain data and formulas for the sole purpose of indicating the order of magnitude of a particular variable, as in the case presented below related to the sum

Example 1. According to

values yields

Equation (49) that, if the annual rate of change of the APK had been steady between the first two dates and also between the last two, then

maximum profit rates corresponding to the beginning and the end of the period, then, for the first period

Hence,

Equation (83) shows that both maximum profit and growth rates increased from period

Proceeding in analog form with data from the period 1950-2010 results in:

Hence,

The penultimate equation shows that both maximum profit and growth rates decreased from period

It follows from Equations (69) and (79) that, given two dates

Substituting in the penultimate equation the quotient

Þ

Therefore, in order to satisfy the inequality:

It is enough that:

Regarding the magnitude of

Lemma 2. When

Proof. See Appendix A.4.

It follows from this lemma that, in some cases, inequality (96) may not be true for any t or be true only for a t extremely high if R, the quotient

big enough. However, the data offered by Piketty indicate historical levels of the variables involved much lower than those required in those two cases, as shown below.

Example 2. Substituting in Equation (92) the data from Example 1, results in:

Substituting in the right-hand side of Equation (98) the factor within brackets by the right-hand side of Equation (100) gives:

Therefore, under the assumptions adopted, the ratio of the rate of profit and the rate of growth in the French economy increased 28.4823% during the period 1910-2010 due to variations of the APK.

It can be noted in Equation (78) that the quotient

Theorem 2. The sum

Therefore, provided that the saving rate decreases with respect to its value in the previous period, the sum

Example 3. In the case of period 1910-1950 discussed in Example 1, it follows from Equations (78), (81) and (83) that, for the sum

Þ

Þ

Hence,

for the ratio of the profit and growth rates to decrease, it is enough that the savings rate increases at a rate greater than 0.3891% from one period to the next. In this regard, it should be noted that the historical series on the value of the savings rate present significant differences in the magnitude of this variable in different countries and also between different historical periods in the same country. However, in the last 70 years, for the USA, Germany, Japan, France, United Kingdom, Italy, Canada, and Australia the savings rate fluctuates in each country around the corresponding national average value, with increases and decreases of the order of one percentage point from one year to the next one (see Piketty and Zucman [

For a

Theorem 3. The sum

Proof. See Appendix A.5.

Therefore, the sum

Example 4. In the case of period 1910-1950 discussed in Example 1, it follows from Equations (78) and (81) that, for the sum

Þ

Þ

Þ

Þ

Þ

Therefore,

That is, for the ratio of the profit and growth rates to decrease, it is enough that the rate of variation of the APK grow more than 2.7109% compared to its value in the previous production cycle.

For any

Proposition 5. In a growing economy in which, for each

It may be noted in Equation (78) that, in the particular case when the savings rate is constant and

If the rate of change of the APK is constant, either to increase or to decrease, over a certain number of production cycles, at the end of these cycles result an increase in the ratio of the profit and growth rates and, respectively, an increase or a decrease in the APK of the last cycle with respect to the first. It is important to note that, in accordance with Proposition 4, it is possible to obtain the same variation in the ratio of the profit and growth rates through a succession of production cycles in which the APK at the first and the last cycle are equal. For this purpose it is enough that, throughout all cycles, the APK fluctuates properly, for example initially increasing and then decreasing, respecting the condition stated in Proposition 5, as shown below.

Example 5. Suppose that the APK from the French economy continues to decline, after 2010, at the same annual average rate of the period 1950-2010. In accordance with Equation (49) and with the data from Example 1, the number of years required (x) from 1950 so the APK descends to its level of 1910 must satisfy the equation

According to these analyses, if between two successive periods of production increases do not occur in the savings rate of the economy nor in the rate of growth of the APK which exceed certain percentages, the ratio between profit and growth rates increases. In a succession of production cycles, even if the percentages indicated are surpassed and this ratio decreases in some cycles, it is possible that these decreases are offset by what happens in other cycles, so that the average of the variations of the proportion that we are interested in is greater than zero.

In a growing economy in which remain constant the savings rate and also the distribution of income between profits and wages, when the APK varies at a steady rate, either to increase or to decrease, in each production cycle the profit rate increases with respect to the growth rate. This is due to that, by increasing or reducing the APK, both profit and growth rate respectively increase or decrease but, in the first case, the profit rate increases in a greater proportion and, in the second one, decreases in a smaller proportion than the growth rate. If such variation is repeated throughout a sufficiently large number of cycles, the accumulation of increases leads the first rate to grow above the second, even if in the initial situation the second rate is higher than the first. The same result is reached if the savings rate and the rate of change of the APK fluctuate within certain limits over a sufficiently large number of production cycles; however, if any of these variables increases surpassing the corresponding limit from one period to another, the profit rate may decrease with respect to the growth rate. In addition, the variation in the distribution of income between wage and profit affects the rate of profit and, by consequence, the ratio between the latter and the growth rate of the economy. For these reasons, to know to what extent the fluctuations of the APK explain the relationship between the rates indicated in a growing economy, it is necessary to study individual cases.

I am grateful to an anonymous referee for helpful comments and suggestions.

Benítez Sánchez, A. (2016) Piketty’s r > g Explained by Changes in the Average Productivity of Capital. Theo- retical Economics Letters, 6, 1034-1059. http://dx.doi.org/10.4236/tel.2016.65102

In the first part of this appendix I present a formulation of Equation (80) that facilitates the demonstrations of the two sections following.

A.1 A Reformulation of Equation (80)

Dividing by s all terms of the denominator of the right-hand side of Equation (80) and making operations there described, it is possible to write the equation in the following form:

Let:

Using this notation, it is possible to write Formula (115) in the following manner:

Finally, simplifying results in:

A.2 Proof of Lemma 1

In order to avoid division by zero in Equation (80), and for the economy to grow, respectively, the following three inequalities and the last two of them must be satisfied:

When

On the other hand, at the beginning of each production period

Þ

Substituting in this inequality

Þ

Substituting in this inequality

When

The lemma is satisfied making:

Remark. To demonstrate Theorem 1, it is useful to check separately that division by zero is avoided in Equation(119). For which the following inequalities must be satisfied:

When

val

ma is satisfied.

A.3 Proof of Theorem 1

As can be noted in Equation (119), if

Therefore, given that

Þ

As a result, the right-hand side of Equation (119) is greater than one. Finally, according to Lemma 1, if

allows to conclude that:

Furthermore, multiplying by h both sides of inequality (132) yields:

For this reason,

This result and inequality (130) imply that:

In turn, this result and inequality (136) imply that:

Therefore, also in this case inequality (134) is satisfied which, together with inequality (131), imply that the right-hand side of the Equation (119) is greater than one.

A.4 Proof of Lemma 2

Dividing by R each term on the right-hand side of Equation (80), gives:

Making:

it is possible to write:

Hence,

The lemma is inferred from this result and Equation (79).

A.5 Proof of Theorem 3

I will show that the derivative with respect to

Let

Deriving with respect to

Since the sign of the derivative depends only on the sign of the numerator of the above ratio, in what follows I will deal exclusively with this numerator. Multiplying it by B results in:

Multiplying this result by A gives:

=

Condition (23) implies that