# A REFORMULATION OF THE CONCEPT OF RETURNS TO SCALE

Anselme Njocke

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Anselme Njocke. A REFORMULATION OF THE CONCEPT OF RETURNS TO SCALE. Finance India, 2017, XXXI (1). [https://hal-05361256v1](https://hal-05361256v1)

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# A REFORMULATION OF THE CONCEPT OF RETURNS TO SCALE

Anselme NJOCKÈ<fnref n="1" />

## ABSTRACT

We propose the *gh-returns to scale* concept which generalizes the traditional *returns to scale* concept. This new concept takes into account variations of all the inputs in unspecified proportions, in the evolution of production, contrary to the *returns to scale* concept which takes into account, only equiproportionate variations of all the inputs which lead to the implementation of production. It will appear that, when characterizing (or determining) the returns to scale (constant, increasing or decreasing), homogeneity of degree of a (production) function no longer plays the central role (except the traditional case where all the factors vary in the same proportions) it was once attributed as the nature of returns to scale can be evoked, with this new approach of unspecified variations of the proportions, whatever the degree of homogeneity of a (production) function. It will also appear that, if the producer wishes or if he is constrained to be located in a precise zone of the gh-returns to scale (for example, that of increasing gh-returns to scale, because of limited outlets or a badly filled order book), the change of its size or scale in the long run, (in case of an investment which requires capital deepening synonymous with the increase in the proportion of a factor (capital) to the detriment of the reduction in the proportion of another factor (labor)), should not be made in a hazardous way, but must rather obey specific conditions (or constraints) characterizing such a zone of the gh-returns to scale.

KEY WORDS: Returns to scale, gh-returns to scale, marginal returns, diagonal transformation, generalized homogeneous production function, homogeneous production function, homogeneity of degree, homotheticity (homothetic transformation), diagonal transformation, reference indicator, numeraire proportion, relative proportions vector.

JEL Classification Number: D24

<footnote n="1">¹ Université de Pau et des Pays de l'Adour, Faculté de Droit, d'Economie et de Gestion, Centre d'analyse théorique et de traitement des données (CATT), Avenue du Doyen Poplawski B.P. 1633 -64016, Pau Cedex, France, e-mail : anselme.njocke@univ-pau.fr.</footnote>
# I. Introduction

Generally, the notion of returns to scale is associated with the notion of a homogenous production function of degree $\alpha$ defined from a set of $n$-tuples of real numbers into the set of real positive numbers.

When browsing the economic literature, the question which is very often put is that of knowing how production varies following a variation of all the factors of production in the same proportion (it is thus referred to as the equiproportionate changes of all the factors of production). The output can increase: a) more than the proportional increase of the inputs (increasing returns to scale), b) less than the proportional increase of the inputs (decreasing returns to scale), c) in the same proportion as the increase of the inputs (constant returns to scale).

The notion of returns to scale seems to have a fairly restrictive definition in so far as practically when a company wishes to change its scale of production, i.e. its size, it simultaneously changes all the factors necessary to reach the expected level of output. Changing those factors of production depends on different types of constraints. The constraints can be economic (the relative prices of the factors), technical (the type of technology used owing to the varying indivisible nature of the goods, for example). They can result from the very nature of the output offered on the market: take for example a product with a guarantee of origin like Champagne, of the selfsame region of guaranteed origin, the increase of the soil factor in this region will be proportionately more difficult to achieve than the other factors because of the constraints of quality of the vines as well as the soil.

Because of these constraints, there is therefore a priori no reason for these factors to move or vary in the same proportion to meet the level of production required by the producer.

While having all the factors of production vary in unspecified proportions, the producer will face an increase of his output based on the various increases in inputs. We
propose a new returns to scale concept which would take into account strictly positive variations of all the factors in unspecified proportions and that we call *gh-returns to scale*.

And at the same time, the returns to scale concept associated to the homogenous production function concept does not seem suitable any more to characterize the way in which the production (the output) evolves (more than the proportional increase of the inputs, less than the proportional increase of the inputs, in the same proportion as the increase of inputs).

In this paper we shall refer to a generalized homogeneous production function concept with which we associate this new return to scale concept (which will take into account strictly positive variations of all the factors in unspecified proportions).

In this new formulation of returns to scale (gh-returns to scale), the reference indicator concept will play a central role in the characterization of the nature of the returns to scale. This reference indicator being defined as a summary or a digest of the proportions in which the inputs (factors of productions) are modulated when the producer wishes to modify his vector of outputs. In this work, for the sake of simplicity, this vector of outputs will be assimilated to a single output.

The degree of homogeneity of a (production) function will no longer appear as a central element (or a determinant) in the characterization of the nature of returns to scale. Constancy, increase and decrease of returns to scale can, completely, be evoked whatever the degree of homogeneity of the function. The constancy of returns to scale is no longer exclusively associated with a degree of homogeneity equal to the unit: this constancy could be associated with a degree of homogeneity higher or lower than the unit. The same applies to increase of returns to scale which will not exclusively be associated any more with a degree of homogeneity higher than the unit, and decrease of returns to scale which will not exclusively be associated any more with a degree of homogeneity lower than the unit.
One of the interests of this work concerns a suitable combination of the factors (a more important increase in a factor compared to another, or an increase in a factor accompanied by the reduction of another factor) if the producer wishes, or if he is constrained to be located in a precise zone of returns to scale (for example, the zone of increasing gh- returns to scale, because of limited outlets or a badly filled order book). An inappropriate reduction of manpower in a company for the sake of the capital factor cannot be compatible with an increase of returns to scale within the aforementioned company (see the numerical example given in paragraph 4.1.2.2.3, table 3.6, notice 4).

It is to be stressed that, in this unspecified variation of inputs (factors), the situation in which one maintains constant the proportion of use of a factor while modifying (upwards or downwards) the proportion of use of the other factor does not refer to the returns to scale concept (which relates to the long period) but rather to the marginal returns concept (which relates to the short period).

In a second section, we present the gh-returns to scale concept as well as the various concepts related to this gh-returns to scale concept; in the third section, we will explicitly establish the form of the gh-returns to scale concept in the case of some production functions such as the generalized Cobb-Douglas production function and the generalized Walters constant-elasticity of substitution production function. In the fourth section, we propose some numerical examples of gh-returns to scale for the Cobb-Douglas production function, and for the C.E.S. production function within the restricted framework of two factors of production. An easy reading (or interpretation) of the results is made at the end of each table as for only the Cobb-Douglas production function (reading of the results being similar with regard to the C.E.S. production function). Still in this fourth section, the question of the choice of a “suitable” reference indicator in the characterization of the nature of gh-returns to scale is tackled: we show that a change of reference indicator in the definition of gh-returns to scale
betokens nothing else than a change of measurement, the nature and the classification of gh-
returns to scale is not affected in any way.

II. Definition of the concepts related to the gh-returns to scale concept

**2.1. Definition 1 : the production possibility set and the production function**

Let $x = (x_1, \dots, x_i, \dots, x_n)$ be an input vector ($x$ is a point in a Euclidian $n$-space) and
$Y = (y_1, \dots, y_i, \dots, y_l)$ an output vector $Y = (y_1, \dots, y_i, \dots, y_l)$ (supposing that the producer can
implement $l$ outputs). The production possibility set consists of all the possible combinations
of outputs and inputs which are technically possible. Let $f$ be the border of the production
possibility set; this border $f$ is also called the production function and it is composed of the
whole of the efficient combinations of inputs and outputs. For the sake of simplicity, we will
consider that there exists only one output. The vector of outputs $Y = (y_1, \dots, y_i, \dots, y_l)$ will
thus be reduced to only one output which one will note $y$.

So, $y = f(x_1, \dots, x_i, \dots, x_n)$ stands for the traditional neo-classic single-output production function. (1)

**2.2. Definition 2 : homotheticity**

Homotheticity is an affine transformation by which the input vector
$x = (x_1, \dots, x_i, \dots, x_n)$ is multiplied by the same real number $\lambda$.

Thus, one would move from vector $x = (x_1, \dots, x_i, \dots, x_n)$ to vector
$$ \lambda x = \lambda(x_1, \dots, x_i, \dots, x_n) = (\lambda x_1, \dots, \lambda x_i, \dots, \lambda x_n). \quad (2) $$

**2.3. Definition 3 : the homogeneous production function of degree $\alpha$**

The production function is homogeneous of degree $\alpha \neq 0$ if and only if:
$$ \lambda^{\alpha} y = f(\lambda x_1, \dots, \lambda x_i, \dots, \lambda x_n), \text{ with } \lambda > 0 \quad (3) $$

In other words, that means that by replicating the technical installations and the inputs, the production could either be replicated on the same scale ($\alpha = 1$), or replicated on a smaller scale ($\alpha < 1$) or replicated on a larger scale ($\alpha > 1$).

## 2.4. Definition 4 : the traditional concept of returns to scale

One calls "returns to scale" the ratio:

$$ r(\lambda, x_1, \dots, x_i, \dots, x_n) = \frac{f(\lambda x_1, \dots, \lambda x_i, \dots, \lambda x_n)}{\lambda f(x_1, \dots, x_i, \dots, x_n)} \quad (4) $$

## 2.5. Definition 5 : the diagonal transformation

Let $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ be the modulation vector of the vector of inputs $x = (x_1, \dots, x_i, \dots, x_n)$. Each $\lambda_i$ represents the proportion in which each input $x_i$ varies.

$$ \text{Let } \Lambda = \begin{pmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & \lambda_i & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{pmatrix} \text{ be the diagonal matrix.} $$

The diagonal transformation is the product of the diagonal matrix $\Lambda$ and the transpose of the vector $x = (x_1, \dots, x_i, \dots, x_n)$, noted $x^t$.

$$ \text{That is to say: } \Lambda x^t = (\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n) \quad (5) $$

The diagonal transformation is the operation by which, in a Euclidian $n$-space, one would pass from vector $x = (x_1, \dots, x_i, \dots, x_n)$ to vector $\tilde{x} = (\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n)$.

It will clearly appear that homotheticity is a particular case of diagonal transformation.
## 2.6. Definition 6 : the generalized homogeneous production function concept

Let $\Lambda x^t = (\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n)$ be the preceding diagonal transformation.

Let us consider the class of production functions which satisfy the following property:

There exists a function $g: R^n \to R$ such that for all vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$,
$$f(\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n) = g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n) f(x) \text{ with } \lambda_i > 0 \text{ for all } i \quad (6)$$

It is noted that the scalar $g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ modulates the scale in which the output $y$ is produced. We will say that the production function $f$ is a generalized<fnref n="2" /> homogeneous production function if $f$ satisfies the property (6) above.

## 2.7. Definition 7 : the reference indicator

The components of vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ can be summarized or condensed through a scalar which we will note $\tilde{\lambda}$. We will call this scalar $\tilde{\lambda}$ the reference indicator<fnref n="3" />.

Vis-a-vis this statistical data which results from many observations, one often tests the need for summarizing these data points by a single measure of central tendency (or a summarized-value or called reference indicator), and that gives us an idea (however approximate) about the central tendency data of those data points.

It proves that there exists an infinity of manner of defining this reference indicator (this measure of central tendency), as it is the case for example with regard to the statistical average or the statistical mean of the data sample.

In a general way, the average of the data sample, noted $\bar{X}$, is defined by the relation $\tilde{\lambda} = h(\bar{X}) = \sum_{i=1}^{n} f_i h(\lambda_i)$

where $f_i = \frac{n_i}{n}$ and $\sum_{i=1}^{n} f_i = 1$, and where $h$ is a monotonic function (i.e. either always increasing, or always

<footnote n="2">² The function is generalized homogeneous in the sense that the factors of production vary in unspecified proportions.</footnote>

<footnote n="3">³ Let us consider the statistical variable noted $X$ and let $\lambda_i$ be a value taken by the statistical variable $X, i = 1, 2, \dots, n$. That is to say the data sample $(\lambda_1, \lambda_2, \dots, \lambda_i, \dots, \lambda_n)$. One will note $n_i$ the weighting associated to the value $\lambda_i$, and $n = \sum_{i=1}^n n_i$ the number of terms of the data sample.</footnote>
It should be noted that the scalar $g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ which modulates the scale in which the output $y$ is produced, can be expressed in terms of reference indicator $\tilde{\lambda}$ via a monotonically increasing function $v$ such as :

decreasing). In other words, the order of the images respect the order of the antecedents for an increasing function, on the other hand the order of the images goes contrary to the order of the antecedents for the decreasing function. So, there exists an infinity of averages in so far as there exists an infinity of such functions.

One finds some usual averages for certain functions $h$.

* If the function $h$ is such that $h(\lambda) = \lambda$, then it is question of arithmetic mean, noted $\tilde{\lambda} = \bar{\lambda}$, and such that: $\tilde{\lambda} = \bar{\lambda} = \sum_{i=1}^{n} f_i \lambda_i$.

* If the function $h$ is such that $h(\lambda) = \frac{1}{\lambda}$, then it is question of harmonic mean, noted $\tilde{\lambda} = H$, and such that: $\frac{1}{\tilde{\lambda}} = \frac{1}{H} = \sum_{i=1}^{n} \frac{f_i}{\lambda_i}$.

* If the function $h$ is such that $h(\lambda) = \ln \lambda$, then it is question of geometric mean, noted $\tilde{\lambda} = G$, and such that: $\ln \tilde{\lambda} = \ln G = \sum_{i=1}^{n} f_i \ln \lambda_i \Leftrightarrow \tilde{\lambda} = G = e^{\left(\frac{n}{\sum f_i \ln \lambda_i}\right)} \Leftrightarrow \tilde{\lambda} = G = \sqrt[n]{\prod_{i=1}^{n} \lambda_i} = \prod_{i=1}^{n} (\lambda_i)^{\frac{1}{n}}$.

* If the function $h$ is such that $h(\lambda) = \lambda^2$, then it is question of quadratic mean, noted $\tilde{\lambda} = Q$, and such that: $\tilde{\lambda}^2 = Q^2 = \sum_{i=1}^{n} f_i \lambda_i^2$.

It should be noted that : $H < G < \tilde{\lambda} < Q$.

It will appear, in paragraphs 3.1* and 3.2*, that any average $(\lambda_i = \lambda = \tilde{\lambda} = H = G = Q = \tilde{\lambda}, \forall i)$ is the reference indicator $\tilde{\lambda}$ implicitly retained in the definition of the homogeneous production function of degree $\alpha (\neq 0)$ where all the factors vary in the same proportion $\lambda$.

Although being able to define such infinity of summarized-value (here averages), it is very difficult to agree on what would be the best summarized-value. Yule and Kendall (1950) had proposed a certain number of properties (or of conditions) which should verify one summarized-value so that it can be regarded as “good” summarized-value (to be defined in an objective way (i.e. independent of the observer who calculates it), to depend on all the observations, to have a clear significance and in particular for the general public, to be simple to handle, to easily lend itself to algebraic treatment, not to be very sensitive to sampling fluctuations). And among all previous summarized-values, the arithmetic mean is the one which satisfies more Yule and Kendall's properties, hence the attraction of its use.
$$ \tilde{\lambda} = v(g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n)) \qquad (7) $$

This reference indicator will make it possible to know if the production (output) was duplicated on a higher, lower or identical size compared to this reference indicator, when one varies the factors of production in unspecified proportions

## 2.8. Definition 8 : numeraire proportion and the relative proportions vector

Let us consider the vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$. One will call numeraire proportion, noted $\lambda_u > 0$, the proportion from which all the other proportions are expressed. It thus gives

a relative proportions vector $\hat{\theta} = \left( \left( \gamma_1 = \frac{\lambda_1}{\lambda_u} \right), \left( \gamma_2 = \frac{\lambda_2}{\lambda_u} \right), \dots, \left( \gamma_i = \frac{\lambda_i}{\lambda_u} \right), \dots, \left( \gamma_n = \frac{\lambda_n}{\lambda_u} \right) \right)$.

The vector $\hat{\theta}$ can be written in the condensed form $\hat{\theta} = (\gamma_i)_{i=1,...,n}$, with

$$ \gamma_i = \frac{\lambda_i}{\lambda_u} > 0 \Leftrightarrow \lambda_i = \lambda_u \gamma_i, \forall i=1,2,...,i,...,n \qquad (8) $$

## 2.9. Definition 9 : the gh-returns to scale concept

One calls gh-returns to scale the ratio:

$$ r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{f(\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} = \frac{g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n) f(x_1, \dots, x_i, \dots, x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} \\ = \frac{f((\lambda_u \gamma_1) x_1, \dots, (\lambda_u \gamma_i) x_i, \dots, (\lambda_u \gamma_n) x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} \qquad (9) $$

## 2.9*. Définition 9* : the marginal returns concept

If at least one of the factors is maintained constant during the variation of the other factors in unspecified proportions, the suitable concept will no longer be that of gh-returns to scale (which refers to the long period); the suitable concept will be that of marginal returns (which refers to the short period).
Let us consider the vector of inputs $x = (x_1, ..., x_i, ..., x_n)$. Let us fix all the components of this vector $x$, except the component of the input $i$, that is to say $x_j = \bar{x}_j \quad \forall j \neq i$ and $j = 1, ..., n$. The vector $x$ can thus be written in the form : $x = (\bar{x}_1, ..., \bar{x}_i, ..., \bar{x}_n)$.

One calls marginal returns, the ratio noted $r^i(\lambda_i, x_i)$, which derives from the expression of gh-returns to scale, such as :

$$r(\lambda_i, \bar{x}_1, ..., \bar{x}_i, ..., \bar{x}_n) = \frac{f(\bar{x}_1, ..., \bar{x}_i, ..., \bar{x}_n)}{\lambda_i f(\bar{x}_1, ..., \bar{x}_i, ..., \bar{x}_n)} = \frac{f(\lambda_i x_i)}{\lambda_i f(x_i)} = r^i(\lambda_i, x_i) \quad (9*)$$

## 2.10. Definition 10 : the constant gh-returns to scale

There are constant gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, ..., x_i, ..., x_n) = 1 \quad (10)$$

In other words, there are constant gh-returns to scale if: $g(\lambda_1, ..., \lambda_i, ..., \lambda_n) = \tilde{\lambda}$.

## 2.10*. Definition 10* : the constant marginal returns concept

There are constant marginal returns if:

$$r^i(\lambda_i, x_i) = \frac{f(\lambda_i x_i)}{\lambda_i f(x_i)} = 1 \quad (10*)$$

That corresponds to the phase where the slope of marginal product ($f'(x_i) = \frac{dy}{dx_i} > 0$)

for input $i$ is nil ($f''(x_i) = \frac{d^2y}{dx_i^2} = 0$): in other words the slope $f'(x_i) = \frac{dy}{dx_i}$ is constant when $x_i$ increases.
## 2.11. Definition 11: increasing gh-returns to scale

There are increasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) > 1 \quad (11)$$

In others words, there are increasing gh-returns to scale if: $g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n) > \tilde{\lambda}$.

## 2.11*. Definition 11*: the increasing marginal returns concept

There are increasing marginal returns if:

$$r^i(\lambda_i, x_i) = \frac{f(\lambda_i x_i)}{\lambda_i f(x_i)} > 1 \quad (11*)$$

That corresponds to the phase where the slope of marginal product ($f'(x_i) = \frac{dy}{dx_i} > 0$)

for input $i$ is positive ($f''(x_i) = \frac{d^2y}{dx_i^2} > 0$): in other words the slope $f'(x_i) = \frac{dy}{dx_i}$ increases when $x_i$ increases.

## 2.12. Definition 12: decreasing gh-returns to scale

It will be said that the gh-returns to scale are decreasing if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) < 1 \quad (12)$$

In other words, the gh-returns to scale are decreasing if: $g(\lambda_1, \dots, \lambda_i, \dots, \lambda_n) < \tilde{\lambda}$.

## 2.12*. Definition 12*: the decreasing marginal returns concept

There are decreasing marginal returns if:

$$r^i(\lambda_i, x_i) = \frac{f(\lambda_i x_i)}{\lambda_i f(x_i)} < 1 \quad (12*)$$
That corresponds to the phase where the slope of marginal product ($f'(x_i) = \frac{dy}{dx_i} > 0$) for input i is negative ($f''(x_i) = \frac{d^2y}{dx_i^2} < 0$): in other words the slope $f'(x_i) = \frac{dy}{dx_i}$ decreases when $x_i$ increases.

# III. The expression of the gh-returns to scale concept in the case of some production functions

## 3.1. The expression of the gh-returns to scale concept in the case of the generalized Cobb-Douglas production function

The generalized Cobb-Douglas production function is written:

$$y = f(x_1, \dots, x_i, \dots, x_n) = A \prod_{i=1}^{n} x_i^{\alpha_i}, \text{ with } \alpha_i \in ]0,1[ \qquad (13)$$

The gh-returns to scale is such that:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{f(\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} = \frac{A \prod_{i=1}^{n} \lambda_i^{\alpha_i} x_i^{\alpha_i}}{\tilde{\lambda} A \prod_{i=1}^{n} x_i^{\alpha_i}} = \frac{\prod_{i=1}^{n} \lambda_i^{\alpha_i}}{\tilde{\lambda}} \qquad (14)$$

By expressing each of the components $\lambda_i$ of vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ in terms of the numeraire proportion $\lambda_u$, i.e. $\lambda_i = \lambda_u \gamma_i$, $\forall i = 1, 2, \dots, n$ (previous relation (8)), the relation (14) becomes :

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{\prod_{i=1}^{n} \lambda_i^{\alpha_i}}{\tilde{\lambda}} = \frac{\prod_{i=1}^{n} (\lambda_u \gamma_i)^{\alpha_i}}{\tilde{\lambda}} = \frac{\prod_{i=1}^{n} (\gamma_i)^{\alpha_i} (\lambda_u)^{\alpha_i}}{\tilde{\lambda}} = \frac{\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i}}{\tilde{\lambda}}.$$
$$ \text{That is to say thus the ratio : } r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) \left( (\lambda_u)^{\frac{n}{\sum \alpha_i}} \right)^{i=1} \left( (\tilde{\lambda})^{-1} \right) \quad (14^*) $$

Relation (14) or relation (14*) is the new formulation of the gh-returns to scale concept in the case of the generalized Cobb-Douglas production function.

### 3.1*. The traditional expression of the returns to scale concept in the case of the generalized Cobb-Douglas production function

In the case where $\lambda_i = \lambda_u \gamma_i = \lambda_u = \tilde{\lambda} = \lambda, \forall i = 1, 2, \dots, i, \dots, n$, i.e. in the case where the reference indicator $\tilde{\lambda}$ is any average$^4$, one finds the traditional expression of the returns to scale concept:

$$ r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) \left( (\lambda_u)^{\frac{n}{\sum \alpha_i}} \right)^{i=1} \left( (\tilde{\lambda})^{-1} = (\tilde{\lambda}) \right) \left( (\lambda_u)^{\frac{n}{\sum \alpha_i}} \right)^{i=1} = \lambda \quad (15) $$

In relation (15), the term $\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right)$ disappears from relation (14*), in so far as, in the traditional case, one takes into account only equiproportionate variations of all the factors $(\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i = 1, 2, \dots, i, \dots, n)$.

#### 3.1.1. Conditions (rules) describing the nature of gh-returns to scale in the case of the generalized Cobb-Douglas production function

<footnote>$^4$ The arithmetic mean ($\bar{\lambda} = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), the geometric mean ($G = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), the harmonic mean ($H = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), the quadratic mean ($Q = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), etc.</footnote>
In order to determine the nature of returns to scale, it is advisable to compare the ratio $r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ with the unit.

### 3.1.1.1. The condition (rule) describing the constancy of gh-returns to scale

**Condition (Rule) I:**

Given a vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ of modulation of inputs (and so given a relative proportions vector $\hat{\theta} = ((\gamma_1 = \frac{\lambda_1}{\lambda_u}), \dots, (\gamma_i = \frac{\lambda_i}{\lambda_u}), \dots, (\gamma_n = \frac{\lambda_n}{\lambda_u}))$, where $\lambda_u$ stands for the numeraire), given $\tilde{\lambda}$ the reference indicator retained, the generalized Cobb-Douglas production function $y = f(x_1, \dots, x_i, \dots, x_n) = A \prod_{i=1}^{n} x_i^{\alpha_i}$ exhibits constant gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i}}{\tilde{\lambda}} = 1.$$

$$\text{In other words: } \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} = \tilde{\lambda} \Leftrightarrow \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} = \tilde{\lambda} (\lambda_u)^{-\sum_{i=1}^{n} \alpha_i} \quad (16)$$

### 3.1.1.1*. The traditional condition (rule) describing the constancy of returns to scale

**Condition (Rule) I\*:**

In the traditional case, relation (15), there are constant returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = (\tilde{\lambda})^{\left( \frac{\sum \alpha_i}{n} - 1 \right)} = 1.$$

In other words :
$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = (\tilde{\lambda})^{\left(\sum_{i=1}^{n} \alpha_i - 1\right)} = 1 \Leftrightarrow (\tilde{\lambda})^{\left(\sum_{i=1}^{n} \alpha_i - 1\right)} = (\tilde{\lambda})^0 \Leftrightarrow \sum_{i=1}^{n} \alpha_i - 1 = 0 \Leftrightarrow \sum_{i=1}^{n} \alpha_i = 1$$

### 3.1.1.2. The condition (rule) describing the increase of gh-returns to scale

**Condition (Rule) II:**

Given a vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ of modulation of inputs (and so given a relative proportions vector $\hat{\theta} = ((\gamma_1 = \frac{\lambda_1}{\lambda_u}), \dots, (\gamma_i = \frac{\lambda_i}{\lambda_u}), \dots, (\gamma_n = \frac{\lambda_n}{\lambda_u}))$, where $\lambda_u$ stands for the numeraire), given $\tilde{\lambda}$ the reference indicator retained, the generalized Cobb-Douglas production function $y = f(x_1, \dots, x_i, \dots, x_n) = A \prod_{i=1}^{n} x_i^{\alpha_i}$ exhibits increasing gh-returns to scale

if: $r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{\left(\prod_{i=1}^{n} (\gamma_i)^{\alpha_i}\right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i}}{\tilde{\lambda}} > 1$.

In other words :

$$\left(\prod_{i=1}^{n} (\gamma_i)^{\alpha_i}\right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} > \tilde{\lambda} \Leftrightarrow \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} > \tilde{\lambda} (\lambda_u)^{-\sum_{i=1}^{n} \alpha_i} \quad (17)$$

### 3.1.1.2*. The traditional condition (rule) describing the increase of returns to scale

**Condition (Rule) II\*:**

In the traditional case, relation (15), there are increasing returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = (\tilde{\lambda})^{\left(\sum_{i=1}^{n} \alpha_i - 1\right)} > 1.$$

In other words :
$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = (\tilde{\lambda})^{\left(\frac{\sum \alpha_i - 1}{i=1} \right)} > 1 \Leftrightarrow (\tilde{\lambda})^{\left(\frac{\sum \alpha_i - 1}{i=1} \right)} > (\tilde{\lambda})^0 \Leftrightarrow \sum_{i=1}^{n} \alpha_i - 1 > 0 \Leftrightarrow \sum_{i=1}^{n} \alpha_i > 1$$

### 3.1.1.3. The condition (rule) describing the decrease of gh-returns to scale

**Condition (Rule) III:**

Given a vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ of modulation of inputs (and so given a relative proportions vector $\hat{\theta} = ((\gamma_1 = \frac{\lambda_1}{\lambda_u}), \dots, (\gamma_i = \frac{\lambda_i}{\lambda_u}), \dots, (\gamma_n = \frac{\lambda_n}{\lambda_u}))$, where $\lambda_u$ stands for the numeraire), given $\tilde{\lambda}$ the reference indicator retained, the generalized Cobb-Douglas production function $y = f(x_1, \dots, x_i, \dots, x_n) = A \prod_{i=1}^{n} x_i^{\alpha_i}$ exhibits decreasing gh-returns to scale

$$\text{if: } r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i}}{\tilde{\lambda}} < 1.$$

$$\text{In other words : } \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} < \tilde{\lambda} \Leftrightarrow \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} < \tilde{\lambda} (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} \quad (18)$$

### 3.1.1.3*. The traditional condition (rule) describing the decrease of returns to scale

**Condition (Rule) III\*:**

In the traditional case, relation (15), there are decreasing returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = (\tilde{\lambda})^{\left(\frac{\sum \alpha_i - 1}{i=1} \right)} < 1.$$

In other words :
$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = (\tilde{\lambda})^{\binom{n}{i=1} \alpha_i - 1} < 1 \Leftrightarrow (\tilde{\lambda})^{\binom{n}{i=1} \alpha_i - 1} < (\tilde{\lambda})^0 \Leftrightarrow \sum_{i=1}^{n} \alpha_i - 1 < 0 \Leftrightarrow \sum_{i=1}^{n} \alpha_i < 1$$

Of the three previous specific conditions (I*, II* and III*), it appears that, in the traditional case, the focus was exclusively on the sum<fnref n="5" /> of elasticities of production compared to each factor to characterize the nature of the returns to scale; and rightly so in so far as the influence of the multiplicative factor $\left(\prod_{i=1}^{n} (\gamma_i)^{\alpha_i}\right)$ disappears from relation (14*) since $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1$ for any input i.

### 3.1.2. Synoptic table summarizing the gh-returns to scale with the generalized Cobb-Douglas production function

$$y = f(x) = A \prod_{i=1}^{n} x_i^{\alpha_i}$$

<caption>**Table 1 :** Nature of the gh-returns to scale with the generalized Cobb-Douglas production function</caption>

$$y = f(x_1, \dots, x_i, \dots, x_n) = A \prod_{i=1}^{n} x_i^{\alpha_i}$$

<table><thead><tr><th colspan="4">Nature of the gh-returns to scale</th></tr><tr><th>n &Sigma; &alpha;<sub>i</sub><br>i=1</th><th>Constant gh-returns to<br>scale<br>r(&lambda;&#771;, x<sub>1</sub>,..., x<sub>i</sub>,..., x<sub>n</sub>) = 1</th><th>Increasing gh-returns to<br>scale<br>r(&lambda;&#771;, x<sub>1</sub>,..., x<sub>i</sub>,..., x<sub>n</sub>) > 1</th><th>Decreasing gh-returns to<br>scale<br>r(&lambda;&#771;, x<sub>1</sub>,..., x<sub>i</sub>,..., x<sub>n</sub>) < 1</th></tr></thead><tbody><tr><td>n &Sigma; &alpha;<sub>i</sub> = 1<br>i=1</td><td>(n &Sigma; &alpha;<sub>i</sub>)<sup>n</sup> (n &Sigma; &alpha;<sub>i</sub>)<sup>i=1</sup> = &lambda;&#771;<br>(a)<br>This equation contains the<br>traditional case of</td><td>(n &Sigma; &alpha;<sub>i</sub>)<sup>n</sup> (n &Sigma; &alpha;<sub>i</sub>)<sup>i=1</sup> > &lambda;&#771;<br>(d)</td><td>(n &Sigma; &alpha;<sub>i</sub>)<sup>n</sup> (n &Sigma; &alpha;<sub>i</sub>)<sup>i=1</sup> < &lambda;&#771;<br>(g)</td></tr></tbody></table>

<footnote n="5">⁵ Constant returns to scale $\binom{n}{i=1} \alpha_i = 1$, increasing returns to scale $\binom{n}{i=1} \alpha_i > 1$ and decreasing returns to scale $\binom{n}{i=1} \alpha_i < 1$.</footnote>
<table><tr><td></td><td>constant returns to scale<br>when &gamma;<sub>i</sub> = 1, &forall;i</td><td></td><td></td></tr><tr><td>n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub> &gt; 1</td><td>(n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) (n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) = &lambda;&#771;<br>(b)</td><td>(n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) (n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) &gt; &lambda;&#771;<br>(e)<br>This inequality contains<br>the traditional case of<br>increasing returns to scale<br>when &gamma;<sub>i</sub> = 1, &forall;i</td><td>(n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) (n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) &lt; &lambda;&#771;<br>(h)</td></tr><tr><td>n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub> &lt; 1</td><td>(n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) (n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) = &lambda;&#771;<br>(c)</td><td>(n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) (n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) &gt; &lambda;&#771;<br>(f)</td><td>(n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) (n &Sigma;<sub>i=1</sub> &alpha;<sub>i</sub>) &lt; &lambda;&#771;<br>(i)<br>This inequality contains<br>the traditional case of<br>decreasing returns to scale<br>when &gamma;<sub>i</sub> = 1, &forall;i</td></tr></table>

The equations and the inequalities (a), (b), (c), (d), (e), (f), (g), (h) and (i) in table 1 above lead to a whole set of solutions some of which contain the traditional solutions of constant returns to scale (a), increasing returns to scale (e) and decreasing returns to scale (i).

- Equations (a), (b) and (c), reflect all the situations of constant returns to scale,

through the connection $(\prod_{i=1}^{n}(\gamma_i)^{\alpha_i})(\lambda_u)^{\sum_{i=1}^{n}\alpha_i} = \tilde{\lambda}$, between the reference indicator $\tilde{\lambda}$ and the ratios $\gamma_i = \frac{\lambda_i}{\lambda_u}$ ($i=1,...,n$) where $\lambda_u$ stands for the numeraire proportion. The equation (a) leads to a set of solutions containing the traditional solution of constant returns to scale when $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i$; in other words when all the factors vary in the same proportion $\tilde{\lambda}$.
- Inequalities (d), (e) and (f), reflect all the situations of increasing returns to scale, through the connection $\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} > \tilde{\lambda}$, between the reference indicator $\tilde{\lambda}$ and the ratios $\gamma_i = \frac{\lambda_i}{\lambda_u} (i=1,...,n)$ where $\lambda_u$ stands for the numeraire proportion. The inequality (e) leads to a set of solutions containing the traditional solution of increasing returns to scale when $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i$; in other words when all the factors vary in the same proportion $\tilde{\lambda}$.

- Inequalities (g), (h) and (i), reflect all the situations of decreasing returns to scale, through the connection $\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} < \tilde{\lambda}$, between the reference indicator $\tilde{\lambda}$ and the ratios $\gamma_i = \frac{\lambda_i}{\lambda_u} (i=1,...,n)$ where $\lambda_u$ stands for the numeraire proportion. The inequality (i) leads to a set of solutions containing the traditional solution of decreasing returns to scale when $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i$; in other words when all the factors vary in the same proportion $\tilde{\lambda}$.

## 3.2. The expression of the gh-returns to scale concept in the case of the generalized C.E.S. production function or the generalized constant-elasticity-of-substitution production function
The generalized<fnref n="6" /> C.E.S. production function in the case of several (n) inputs has for expression :

$$
\left\{
\begin{array}{l}
y = f(x_1, \dots, x_i, \dots, x_n) = A \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}} \\
\alpha_i \in ]0,1[ \quad \forall i, \text{ the parameter of int density of the input } x_i \\
A > 0: \text{ the parameter of efficiency} \\
\rho = \frac{1-\sigma}{\sigma} : \text{ the parameter of substitution} \\
k: \text{ the degree of homogeneity of the production function } f \\
\sigma: \text{ the elasticity of substitution (which does not appear in the production function)}
\end{array}
\right.
\qquad (19)
$$

The gh-returns to scale is such that:

$$
\begin{align*}
f(\lambda x_1, \lambda x_2) &= A \left[ \alpha_1 (\lambda x_1)^{-\rho} + \alpha_2 (\lambda x_2)^{-\rho} \right]^{-\frac{k}{\rho}} \\
&= (\lambda^{-\rho})^{-\frac{k}{\rho}} A \left[ \alpha_1 (x_1)^{-\rho} + \alpha_2 (x_2)^{-\rho} \right]^{-\frac{k}{\rho}} \\
&= \lambda^k A \left[ \alpha_1 (x_1)^{-\rho} + \alpha_2 (x_2)^{-\rho} \right]^{-\frac{k}{\rho}} \\
&= \lambda^k f(\lambda x_1, \lambda x_2) = \lambda^k y
\end{align*}
$$

The Walters C.E.S. production function generalizes the C.E.S production function of Arrow, Minhas and Solow which is such that $y = f(x_1, x_2) = A[\alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho}]^{-\frac{1}{\rho}}$ and which is homogeneous of degree 1 (degree $k$ is always equal to unit in Arrow, Minhas and Solow).

The elasticity of substitution $\sigma$ is defined as the relationship between the relative variation of two given inputs and the relative variation of the rate of technological substitution between those two inputs along the isoquant (i.e. a given level of output obtained by a set of efficient combinations of inputs $x_1$ and $x_2$). The inequality $\sigma > 1$ translates the idea that by maintaining one of the inputs constant, the production can increase by increasing the quantities of the other input. On the other hand the inequality $\sigma < 1$ reflects the opposite situation in this sense that it will be difficult, by maintaining one of the inputs, to increase the production by increasing the quantities of the other input (because of the relative scarcity of the constant factor). The Leontief functions are C.E.S. production functions when $\sigma = 0$. The Cobb-Douglas production functions are C.E.S. production functions when $\sigma = 1$.

<footnote n="6">⁶ The generalized C.E.S. production function, in the case of two inputs $x_1$ and $x_2$, is expressed as follows: $y = f(x_1, x_2) = A[\alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho}]^{-\frac{k}{\rho}}$ with $\alpha_i \in ]0,1[ \quad \forall i=1,2$. This generalized C.E.S. production function is homogeneous of degree $k$ since:</footnote>
$$
r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{f(\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} = \frac{\left[ \sum_{i=1}^{n} A \alpha_i (\lambda_i x_i)^{-\rho} \right]^{-\frac{k}{\rho}}}{\tilde{\lambda} \left[ \sum_{i=1}^{n} A \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} = \frac{\left[ \sum_{i=1}^{n} \alpha_i (\lambda_i x_i)^{-\rho} \right]^{-\frac{k}{\rho}}}{\tilde{\lambda} \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} \quad (20)
$$

By expressing each of the components $\lambda_i$ of vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ in terms of
the numeraire proportion $\lambda_u$, i.e. $\lambda_i = \lambda_u \gamma_i, \forall i = 1, 2, \dots, i, \dots, n$ (previous relation (8)), the
relation (20) becomes :

$$
r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{\left[ \sum_{i=1}^{n} \alpha_i (\lambda_i x_i)^{-\rho} \right]^{-\frac{k}{\rho}}}{\tilde{\lambda} \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} = \frac{\left[ \sum_{i=1}^{n} \alpha_i (\lambda_u \gamma_i x_i)^{-\rho} \right]^{-\frac{k}{\rho}}}{\tilde{\lambda} \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} = \lambda_u^k \tilde{\lambda} (-1) \frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}
$$

$$
That is to say thus the ratio : 
$$
r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda_u^k \tilde{\lambda}^{(-1)} \frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} \quad (20*)
$$

Relation (20) or relation (20*) is the new formulation of the gh-returns to scale
concept in the case of a the generalized C.E.S. production function.

3.2*. The traditional expression of the returns to scale concept in the case of the
generalized C.E.S. production function

In the case where $\lambda_i = \lambda_u \gamma_i = \lambda_u = \tilde{\lambda} = \lambda, \forall i = 1, 2, \dots, i, \dots, n$, i.e. in the case where the
reference indicator $\tilde{\lambda}$ is any average$^7$, the traditional expression of the returns to scale
concept is as follows:

<footnote n="7">⁷ The arithmetic mean ($\bar{\lambda} = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), the geometric mean ($G = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), the harmonic mean ($H = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), the quadratic mean ($Q = \tilde{\lambda} = \lambda_i = \lambda, \forall i$), etc.</footnote>
$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda_u^k \tilde{\lambda}^{(-1)} \frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)} \quad (21)$$

In relation (21), the term $\frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}$ disappears from relation (20*), in so far as, in the traditional case, only equiproportionate variations of all the factors are taken into account ($\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i = 1, 2, \dots, n$).

### 3.2.1. Conditions (rules) describing the nature of the gh-returns to scale in the case of the generalized C.E.S. production function

In order to determine the nature of the returns to scale, it is advisable to compare the ratio $r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ with the unit.

#### 3.2.1.1. The condition (rule) describing the constancy of the gh-returns to scale

Condition (Rule) IV:

Given a vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ of modulation of inputs (and so given a relative proportions vector $\hat{\theta} = ((\gamma_1 = \frac{\lambda_1}{\lambda_u}), \dots, (\gamma_i = \frac{\lambda_i}{\lambda_u}), \dots, (\gamma_n = \frac{\lambda_n}{\lambda_u}))$, where $\lambda_u$ stands for the numeraire), given $\tilde{\lambda}$ the reference indicator retained, the generalized C.E.S. production
function $y = f(x_1, \dots, x_i, \dots, x_n) = A \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}$ exhibits constant gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda_u^k \tilde{\lambda}^{(-1)} \frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} = 1.$$

$$\text{In other words : } \lambda_u^k \tilde{\lambda}^{(-1)} = \frac{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}} \quad (22)$$

### 3.2.1.1*. The traditional condition (rule) describing the constancy of returns to scale

Condition (Rule) IV* :

In the traditional case, let us recall that, since $\lambda_i = \lambda_u = \tilde{\lambda} = \lambda, \forall i = 1, 2, \dots, i, \dots, n$, the expression of gh-returns to scale is such that (relation 21) :

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)}$$

There are constant returns to scale if: $r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda^{(k-1)} = 1$.

$$\text{In other words : } r(\tilde{\lambda}, x_1, x_2) = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)} = 1 \Leftrightarrow \lambda^{(k-1)} = \lambda^0 \Leftrightarrow (k-1) = 0$$

### 3.2.1.2. The condition (rule) describing the increase of gh-returns to scale

Condition (Rule) V :

Given a vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ of modulation of inputs (and so given a relative proportions vector $\hat{\theta} = \left( \left( \gamma_1 = \frac{\lambda_1}{\lambda_u} \right), \dots, \left( \gamma_i = \frac{\lambda_i}{\lambda_u} \right), \dots, \left( \gamma_n = \frac{\lambda_n}{\lambda_u} \right) \right)$, where $\lambda_u$ stands for the
numeraire), given $\tilde{\lambda}$ the reference indicator retained, the generalized C.E.S. production

function $y = f(x_1, \dots, x_i, \dots, x_n) = A \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}$ exhibits increasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda_u^k \tilde{\lambda}^{(-1)} \frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} > 1.$$

$$\text{In other words : } \lambda_u^k \tilde{\lambda}^{(-1)} > \frac{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}} \quad (23)$$

### 3.2.1.2*. The traditional condition (rule) describing the increase of returns to scale

Condition (Rule) V*:

In the traditional case, let us recall that, since $\lambda_i = \lambda_u = \tilde{\lambda} = \lambda, \forall i = 1, 2, \dots, i, \dots, n$, the expression of gh-returns to scale is such that (relation 21):

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)}$$

There are increasing returns to scale if: $r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda^{(k-1)} > 1$.

$$\text{In other words : } r(\tilde{\lambda}, x_1, x_2) = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)} > 1 \Leftrightarrow \lambda^{(k-1)} > \lambda^0 \Leftrightarrow (k-1) > 0$$

### 3.2.1.3. The condition (rule) describing the decrease of gh-returns to scale

Condition (Rule) VI:
Given a vector $\theta = (\lambda_1, \dots, \lambda_i, \dots, \lambda_n)$ of modulation of inputs (and so given a relative proportions vector $\hat{\theta} = ((\gamma_1 = \frac{\lambda_1}{\lambda_u}), \dots, (\gamma_i = \frac{\lambda_i}{\lambda_u}), \dots, (\gamma_n = \frac{\lambda_n}{\lambda_u}))$, where $\lambda_u$ stands for the numeraire), given $\tilde{\lambda}$ the reference indicator retained, the generalized C.E.S. production function $y = f(x_1, \dots, x_i, \dots, x_n) = A \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}$ exhibits decreasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda_u^k \tilde{\lambda}^{(-1)} \frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} < 1.$$

$$\text{In other words : } \lambda_u^k \tilde{\lambda}^{(-1)} < \frac{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}} \quad (24)$$

### 3.2.1.3*. The traditional condition (rule) describing the decrease of returns to scale

**Condition (Rule) VI\*:**

In the traditional case, let us recall that, since $\lambda_i = \lambda_u = \tilde{\lambda} = \lambda, \forall i = 1, 2, \dots, i, \dots, n$, the expression of gh-returns to scale is such that (relation 21):

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)}$$

There are decreasing returns to scale if: $r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \lambda^{(k-1)} < 1$.

In other words : $r(\tilde{\lambda}, x_1, x_2) = \tilde{\lambda}^{(k-1)} = \lambda^{(k-1)} < 1 \Leftrightarrow \lambda^{(k-1)} < \lambda^0 \Leftrightarrow (k-1) < 0$.
Of the three previous specific conditions (IV*, V* and VI*), it appears that, in the
traditional case, the focus was exclusively on the homogeneous degree<fnref n="8" /> of the production
function to characterize the nature of the returns to scale; and rightly so in so far as the

influence of the multiplicative factor $\frac{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}$ disappears from relation (20*)

since $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1$ for any input $i$.

### 3.2.2. Synoptic table summarizing the gh-returns to scale for the generalized C.E.S.

production function $y = f(x_1, \dots, x_i, \dots, x_n) = A \left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}$

<table><caption>Table 2 : Nature of gh-returns to scale with the generalized C.E.S. production function</caption><thead><tr><th rowspan="2"></th><th colspan="3">Nature of the gh-returns to scale</th></tr><tr><th>Constant gh-returns to scale<br>$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = 1$</th><th>Increasing gh-returns to scale<br>$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) > 1$</th><th>Decreasing gh-returns to scale<br>$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) < 1$</th></tr></thead><tbody><tr><td>$(k-1)=0$</td><td>$\lambda_u^k \tilde{\lambda}(-1) = \frac{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}$<br>(a*)<br>This equation contains the<br>traditional case of constant returns<br>to scale when $\gamma_i = 1, \forall i$</td><td>$\lambda_u^k \tilde{\lambda}(-1) > \frac{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}$<br>(d*)</td><td>$\lambda_u^k \tilde{\lambda}(-1) < \frac{\left[ \sum_{i=1}^{n} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{n} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}$<br>(g*)</td></tr></tbody></table>

<footnote n="8">⁸ Constant returns to scale ($k = 1$), increasing returns to scale ($k > 1$) and decreasing returns to scale ($k < 1$).</footnote>
<table><tr><td>(k-1)>0</td><td>λ<sub>u</sub><sup>k</sup>λ̃(-1) = <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">[<div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">∑<sub>i=1</sub><sup>n</sup> α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ</div><div style="padding: 0 5px;">ρ</div></div>]<sup>-k/ρ</sup></div><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[∑<sub>i=1</sub><sup>n</sup> γ<sub>i</sub><sup>-</sup>ρ(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)]<sup>-k/ρ</sup></div><div style="padding: 0 5px;">ρ</div></div><br>(b*)</td><td>λ<sub>u</sub><sup>k</sup>λ̃(-1) > <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[<div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">∑<sub>i=1</sub><sup>n</sup> α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ</div><div style="padding: 0 5px;">ρ</div></div>]<sup>-k/ρ</sup></div><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[∑<sub>i=1</sub><sup>n</sup> γ<sub>i</sub><sup>-</sup>ρ(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)]<sup>-k/ρ</sup></div><div style="padding: 0 5px;">ρ</div></div><br>(e*)<br>This inequality contains the<br>traditional case of increasing<br>returns to scale when γ<sub>i</sub> = 1, ∀i</td><td>λ<sub>u</sub><sup>k</sup>λ̃(-1) < <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[<div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">∑<sub>i=1</sub><sup>n</sup> α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ</div><div style="padding: 0 5px;">ρ</div></div>]<sup>-k/ρ</sup></div><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[∑<sub>i=1</sub><sup>n</sup> γ<sub>i</sub><sup>-</sup>ρ(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)]<sup>-k/ρ</sup></div><div style="padding: 0 5px;">ρ</div></div><br>(h*)</td></tr><tr><td>(k-1)<0</td><td>λ<sub>u</sub><sup>k</sup>λ̃(-1) = <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[<div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">∑<sub>i=1</sub><sup>n</sup> α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ</div><div style="padding: 0 5px;">ρ</div></div>]<sup>-k/ρ</sup></div><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[∑<sub>i=1</sub><sup>n</sup> γ<sub>i</sub><sup>-</sup>ρ(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)]<sup>-k/ρ</sup></div><div style="padding: 0 5px;">ρ</div></div><br>(c*).</td><td>λ<sub>u</sub><sup>k</sup>λ̃(-1) > <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[<div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">∑<sub>i=1</sub><sup>n</sup> α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ</div><div style="padding: 0 5px;">ρ</div></div>]<sup>-k/ρ</sup></div><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[∑<sub>i=1</sub><sup>n</sup> γ<sub>i</sub><sup>-</sup>ρ(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)]<sup>-k/ρ</sup></div><div style="padding: 0 5px;">ρ</div></div><br>(f*)</td><td>λ<sub>u</sub><sup>k</sup>λ̃(-1) < <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[<div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">∑<sub>i=1</sub><sup>n</sup> α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ</div><div style="padding: 0 5px;">ρ</div></div>]<sup>-k/ρ</sup></div><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="padding: 0 5px;">[∑<sub>i=1</sub><sup>n</sup> γ<sub>i</sub><sup>-</sup>ρ(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)]<sup>-k/ρ</sup></div><div style="padding: 0 5px;">ρ</div></div><br>(i*)<br>This inequality contains the<br>traditional case of decreasing<br>returns to scale when γ<sub>i</sub> = 1, ∀i</td></tr></table>

The equations and the inequalities (a*), (b*), (c*), (d*), (e*), (f*), (g*), (h*) and (i*) in table 2 above lead to a whole set of solutions some of which contain the traditional solutions of constant returns to scale (a*), increasing returns to scale (e*) and decreasing returns to scale (i*).

- Equations (a*), (b*), and (c*), reflect all the situations of constant returns to scale, through the connection $λ_u^k \tilde{\lambda}(-1) [\sum_{i=1}^n \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} = [\sum_{i=1}^n \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}}$, between the reference indicator $\tilde{\lambda}$ and the ratios $γ_i = \frac{\lambda_i}{\lambda_u}$ ($i=1,...,n$) where $λ_u$ stands for the numeraire proportion. The equation (a*) leads to a set of solutions containing the traditional solution of constant return to scale when $γ_i = \frac{\lambda_i}{\lambda_u} = 1, ∀i$; in other words when all the factors vary in the same proportion $\tilde{\lambda}$.
- Inequalities (*d*'), (*e*') and (*f*'), reflect all the situations of increasing returns to scale, through the connection $\lambda_u^k \tilde{\lambda}^{(-1)} \left[ \sum_{i=1}^n \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} > \left[ \sum_{i=1}^n \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}$, between the reference indicator $\tilde{\lambda}$ and the ratios $\gamma_i = \frac{\lambda_i}{\lambda_u}$ ($i=1,...,n$) where $\lambda_u$ stands for the numeraire proportion. The inequality (*e*') leads to a set of solutions containing the traditional solution of increasing returns to scale when $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i$ ; in other words when all the factors vary in the same proportion $\tilde{\lambda}$.

- Inequalities (*g*'), (*h*') and (*i*'), reflect all the situations of decreasing returns to scale, through the connection $\lambda_u^k \tilde{\lambda}^{(-1)} \left[ \sum_{i=1}^n \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} < \left[ \sum_{i=1}^n \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}$, between the reference indicator $\tilde{\lambda}$ and the ratios $\gamma_i = \frac{\lambda_i}{\lambda_u}$ ($i=1,...,n$) where $\lambda_u$ stands for the numeraire proportion. The inequality (*i*') leads to a set of solutions containing the traditional solution of decreasing returns to scale when $\gamma_i = \frac{\lambda_i}{\lambda_u} = 1, \forall i$ ; in other words when all the factors vary in the same proportion $\tilde{\lambda}$.

IV. Numerical examples of gh-returns to scale in the case of the Cobb-Douglas production function and the C.E.S. production function within the restricted framework of two factors of production $x_1$ and $x_2$
**4.1. Numerical examples of gh-returns to scale in the case of the Cobb-Douglas production function**

$y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}$

**4.1.1. The gh-returns to scale in the case of the Cobb-Douglas production function**

$y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}$ with an unspecified reference indicator $\tilde{\lambda}$

*Reminder:* Let $\theta = (\lambda_1, \lambda_2)$ be the vector of modulation of vector of inputs $(x_1, x_2)$. Let $\hat{\theta} = (\gamma_1, \gamma_2)$ be the vector of relative proportions with $\lambda_1$ as a numeraire proportion. The expression (relation 14*) of gh-returns to scale, with an unspecified reference indicator $\tilde{\lambda}$, is such that :

$$r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{\left(\prod_{i=1}^{2} (\gamma_i)^{\alpha_i}\right) (\lambda_1)^{\sum_{i=1}^{2} \alpha_i}}{\tilde{\lambda}} = \left(\prod_{i=1}^{2} (\gamma_i)^{\alpha_i}\right) \left(\lambda_1\right)^{\sum_{i=1}^{2} \alpha_i} (\tilde{\lambda})^{-1}.$$

**4.1.2. The gh-returns to scale in the case of the Cobb-Douglas production function**

$y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}$ with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$ as a reference indicator.

Since $\lambda_1$ is the numeraire proportion, the arithmetic mean can be expressed as :

$$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_1 = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i.$$ 
The expression of gh-returns to scale is thus such as :

$$r(\lambda_1, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{\left(\prod_{i=1}^{2} (\gamma_i)^{\alpha_i}\right) (\lambda_1)^{\sum_{i=1}^{2} \alpha_i}}{\tilde{\lambda}} = 2 \frac{\left(\prod_{i=1}^{2} (\gamma_i)^{\alpha_i}\right) (\lambda_1)^{\sum_{i=1}^{2} \alpha_i - 1}}{\sum_{i=1}^{2} \gamma_i}.$$
#### 4.1.2.1. The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$, $\alpha_1 + \alpha_2 < 1$ and $\alpha_1 + \alpha_2 > 1$

There are constant gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, x_2) = 1 \Leftrightarrow 2(\lambda_1)^{\sum \alpha_i - 1} \left( \prod_{i=1}^{2} (\gamma_i)^{\alpha_i} \right) - \left( \sum_{i=1}^{2} \gamma_i \right) = 0.$$

#### 4.1.2.1.1. The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and $\forall \lambda_1 > 0$ (numeraire)

With $\alpha_1 + \alpha_2 = 1$, the solutions $\gamma_2$ to the equation⁹ $2(\gamma_2)^{\alpha_2} - \gamma_2 - 1 = 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table:

⁹ Solving the (particular type of) polynomial equation: $2x^{\alpha} - x - 1 = 0$ with $\alpha \in ]0,1[$ and $x > 0$.

Generally way for $\alpha = \frac{1}{n}$ with $n > 1$, while assuming $X = x^n$, the equation $2x^{\alpha} - x - 1 = 0$ comes back to a polynomial equation $2X - X^n - 1 = -(X - 1) \left( X^{n-1} + X^{n-2} + \dots + X^2 + X - 1 \right) = 0$. of degree $(n-1)$ one cannot factorize

* For $n=2$, that is to say $\alpha = \frac{1}{2}$, while assuming $X = x^2 = \sqrt{x}$, one is reduced to the polynomial equation $2X - X^2 - 1 = -(X - 1)^2 = -(\sqrt{x} - 1)^2 = 0$ with as only solution $\{x=1\}$.

* For $n=3$, that is to say $\alpha = \frac{1}{3}$, while assuming $X = x^3$, one is reduced to the polynomial equation $2X - X^3 - 1 = -(X - 1)(X^2 + X - 1) = 0$. The polynomial equation $X^2 + X - 1 = 0$ has two solutions: $X = \frac{-1 \pm \sqrt{5}}{2}$. However $X = \frac{-1 - \sqrt{5}}{2}$ is negative. So solution adopted is
<table><caption>Table 3: The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$</caption><tr><td>Some values<br>$\alpha_2 \in ]0,1[$</td><td>Equation : $2\gamma_2^{\alpha_2} - \gamma_2 - 1 = 0$</td><td>Link between $\lambda_1$ and $\lambda_2$ :<br>$\gamma_2 = \frac{\lambda_2}{\lambda_1} \Leftrightarrow \lambda_2 = \gamma_2\lambda_1, \forall \lambda_1$</td></tr><tr><td>$\alpha_2 = \frac{1}{2} \Rightarrow \alpha_1 = \frac{1}{2}$</td><td>$\frac{1}{2\gamma_2^2 - \gamma_2 - 1} = 0$<br>Solution :<br>$\{\gamma_2 = 1\}$</td><td>Equiproportionate variations<br>(traditional case) :<br>$\gamma_2 = 1 \Rightarrow \lambda_2 = \lambda_1$</td></tr><tr><td>$\alpha_2 = \frac{9}{10} \Rightarrow \alpha_1 = \frac{1}{10}$</td><td>$\frac{9}{2\gamma_2^{10} - \gamma_2 - 1} = 0$<br>Solutions :<br>$\{\gamma_2 = 1\}, \{\gamma_2 \approx 1013,955\}$</td><td>Equiproportionate variations<br>(traditional case) :<br>$\gamma_2 = 1 \Rightarrow \lambda_2 = \lambda_1$<br>Unspecified proportions :<br>$\gamma_2 \approx 1013,955 \Rightarrow \lambda_2 \approx 1013,955\lambda_1$</td></tr></table>

Reading (or interpretation) of the results :

$$X = x^3 = \frac{-1+\sqrt{5}}{2} \Leftrightarrow x = \left(\frac{-1+\sqrt{5}}{2}\right)^3 = -2+\sqrt{5}.$$ The equation $2x^3 - x - 1 = 0$ thus has two solutions : $\{x=1\}$ and $\left\{x = \left(\frac{-1+\sqrt{5}}{2}\right)^3 = -2+\sqrt{5} \approx 0,23607\right\}$.

* For $\alpha = \frac{1}{4}$, the equation $2x^4 - x - 1 = 0$ has two solutions : $\{x=1\}$ and $\left\{x = \frac{2}{3}\sqrt{17+3\sqrt{33}} - \frac{4}{3\sqrt{17+3\sqrt{33}}} - \frac{5}{3} \approx 0,0874\right\}$.

* For $\alpha = \frac{2}{5}$, the equation $2x^5 - x - 1 = 0$ has two solutions : $\{x=1\}$ and $\{x \approx 0,43947\}$.

* For $\alpha = \frac{2}{3}$, the equation $2x^3 - x - 1 = 0$ has two solutions : $\{x=1\}$ and $x = 2+\sqrt{5} \approx 4,2361$.

* For $\alpha = \frac{3}{4}$, the equation $2x^4 - x - 1 = 0$ has two solutions : $\{x=1\}$ and $\left\{x = \frac{2}{3}\sqrt{199+3\sqrt{33}} - \frac{68}{3\sqrt{199+3\sqrt{33}}} - \frac{11}{3} \approx 11,445\right\}$.

* For $\alpha = \frac{9}{10}$, the equation $2x^{10} - x - 1 = 0$ has two solutions : $\{x=1\}$ and $\{x \approx 1013,955\}$.
Let us consider the last line of this table (the reading is similar for the penultimate line)

* With the solution $\gamma_2 = 1$ (that is to say $\lambda_2 = \lambda_1$ (the traditional case)), the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.1}$ is said to exhibit constant gh-returns to scale, with the arithmetic mean as a reference indicator $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \lambda_1 = \lambda_2 = \lambda$.

**Proof:** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$

$$r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.1}}{\lambda (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1} (x_1^{0.9} x_2^{0.1})}{\lambda (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1}}{\lambda} = \frac{\lambda}{\lambda} = 1 \blacksquare$$

* With the solution $\gamma_2 \approx 1013,955$ (that is to say $\lambda_2 \approx 1013,955\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.1}$ is said to exhibit constant gh-returns to scale, with $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1+\gamma_2) \approx \frac{\lambda_1}{2}(1+1013,955) = \lambda_1 \frac{1014,955}{2} = 507,4775\lambda_1$.

**Proof:** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$

$$r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.1}}{507,4775\lambda_1 (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1} (x_1^{0.9} x_2^{0.1})}{507,4775\lambda_1 (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1}}{507,4775\lambda_1} \blacksquare \\ = \frac{\lambda_1^{0.9} (1013,955\lambda_1)^{0.1}}{507,4775\lambda_1} = \frac{\lambda_1 (1013,955)^{0.1}}{507,4775\lambda_1} = \frac{(1013,955)^{0.1}}{507,4775} = \frac{507,4775}{507,4775} = 1$$

#### 4.1.2.1.2. The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 < 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)

With $\alpha_1 + \alpha_2 < 1$, the solutions $\gamma_2$ to the equation $2(\lambda_1)^{\frac{2}{\sum \alpha_i} - 1} (\gamma_2)^{\alpha_2 - \gamma_2 - 1} = 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table:
<table><caption>Table 4: The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 < 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Some values of α<sub>1</sub><br>and α<sub>2</sub></th><th>Equation :</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> :</th></tr></thead><tbody><tr><td rowspan="2">α<sub>1</sub> = 2/10 et<br>α<sub>2</sub> = 7/10</td><td>2/∑α<sub>i</sub> - 1<br>2(λ<sub>1</sub>)<sup>i=1</sup> (γ<sub>2</sub>)<sup>α<sub>2</sub> - γ<sub>2</sub> - 1</sup> = 0</td><td>γ<sub>2</sub> = λ<sub>2</sub>/λ<sub>1</sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, λ<sub>1</sub> = 2</td></tr><tr><td>2<sup>0,9</sup>(γ<sub>2</sub>)<sup>0,7</sup> - γ<sub>2</sub> - 1 = 0<br>Solutions :</br>{γ<sub>2</sub> ≈ 1,654}, {γ<sub>2</sub> ≈ 3,347}</td><td>Unspecified proportions :</br>γ<sub>2</sub> ≈ 1,654 ⇒ λ<sub>2</sub> ≈ 1,654λ<sub>1</sub><br>or<br>γ<sub>2</sub> ≈ 3,347 ⇒ λ<sub>2</sub> ≈ 3,347λ<sub>1</sub></td></tr></tbody></table>

Reading (or interpretation) of the results :

Let us consider the last line of this table.

* With the solution $\gamma_2 \approx 1,654$ (that is to say $\lambda_2 \approx 1,654\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0,2} x_2^{0,7}$ is said to exhibit constant gh-returns to scale, with $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1+\gamma_2) \approx \frac{\lambda_1}{2}(1+1,654) = \lambda_1 \frac{2,654}{2} = 1,327\lambda_1$.

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$

$$ 
\begin{aligned}
r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,2} (\lambda_2 x_2)^{0,7}}{1,327 \lambda_1 (x_1^{0,9} x_2^{0,1})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7} (x_1^{0,2} x_2^{0,7})}{1,327 \lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7}}{1,327 \lambda_1} \\
&= \frac{\lambda_1^{0,2} (1,654 \lambda_1)^{0,7}}{1,327 \lambda_1} = \frac{\lambda_1^{0,9} (1,654)^{0,7}}{1,327 \lambda_1} = \lambda_1^{-0,1} \frac{(1,654)^{0,7}}{1,327} = 2^{-0,1} \frac{(1,654)^{0,7}}{1,327} \quad \blacksquare \\
&= \frac{(1,654)^{0,7}}{2^{0,1}(1,327)} = \frac{1,422246434}{1,422243385} \approx 1 \text{ (because of the succession of round-offs)}
\end{aligned}
$$

* With the solution $\gamma_2 \approx 3,347$ (that is to say $\lambda_2 \approx 3,347\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0,2} x_2^{0,7}$ is said to exhibit constant gh-returns to scale, with $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1+\gamma_2) \approx \frac{\lambda_1}{2}(1+3,347) = \lambda_1 \frac{4,347}{2} = 2,1735\lambda_1$.

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$
$$
\begin{align*}
r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,2} (\lambda_2 x_2)^{0,7}}{2,1735 \lambda_1 (x_1^{0,9} x_2^{0,1})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7} (x_1^{0,2} x_2^{0,7})}{2,1735 \lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7}}{2,1735 \lambda_1} \\
&= \frac{\lambda_1^{0,2} (3,347 \lambda_1)^{0,7}}{2,1735 \lambda_1} = \frac{\lambda_1^{0,9} (3,347)^{0,7}}{2,1735 \lambda_1} = \lambda_1^{-0,1} \frac{(3,347)^{0,7}}{2,1735} = 2^{-0,1} \frac{(3,347)^{0,7}}{2,1735} \quad \blacksquare \\
&= \frac{(3,347)^{0,7}}{2^{0,1} (2,1735)} = \frac{2,329480067}{2,329499621} \approx 1 \text{ (because of the succession of round-offs)}
\end{align*}
$$

**Remark :** It thus appears that there are constant returns to scale, with the arithmetic mean

$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_1 = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time the sum of

elasticities of the Cobb-Douglas production function is lower than the unit $(\alpha_1 + \alpha_2 < 1)$

4.1.2.1.3. The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 > 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and
$\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value$^{10}$ could have been set)

With $\alpha_1 + \alpha_2 > 1$, the solutions $\gamma_2$ to the equation $2(\lambda_1)^{\sum \alpha_i - 1} (\gamma_2)^{\alpha_2 - \gamma_2 - 1} = 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<caption>Table 5 : The constancy of gh-returns to scale with $\alpha_1 + \alpha_2 > 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption>

<footnote>$^{10}$ With $\lambda_1 = 3$, the equation $(2\lambda_1)^{i=1}$ ($\gamma_2)^{\alpha_2 - \gamma_2 - 1} = 0$ becomes :
- for $\alpha_2 = \frac{2}{10} = 0,2$ and $\alpha_1 = \frac{9}{10} = 0,9$: $2 \times 3^{0,1} (\gamma_2)^{0,2} - \gamma_2 - 1 = 0$, with a solution $\gamma_2 = 0,0199$ or $\gamma_2 = 1,3812$
- for $\alpha_2 = \frac{3}{10} = 0,3$ and $\alpha_1 = \frac{9}{10} = 0,9$: $2 \times 3^{0,2} (\gamma_2)^{0,3} - \gamma_2 - 1 = 0$, with a solution $\gamma_2 = 0,0575$ or $\gamma_2 = 2,1226$
- for $\alpha_2 = \frac{4}{10} = 0,4$ and $\alpha_1 = \frac{9}{10} = 0,9$: $2 \times 3^{0,3} (\gamma_2)^{0,4} - \gamma_2 - 1 = 0$, with a solution $\gamma_2 = 0,0975$ or $\gamma_2 = 3,6857$
- for $\alpha_2 = \frac{5}{10} = 0,5$ and $\alpha_1 = \frac{9}{10} = 0,9$: $2 \times 3^{0,4} (\gamma_2)^{0,5} - \gamma_2 - 1 = 0$, with a solution $\gamma_2 = 0,13334$ or $\gamma_2 = 7,4996$
- for $\alpha_2 = \frac{6}{10} = 0,6$ and $\alpha_1 = \frac{9}{10} = 0,9$: $2 \times 3^{0,5} (\gamma_2)^{0,6} - \gamma_2 - 1 = 0$, with a solution $\gamma_2 = 0,16192$ or $\gamma_2 = 19,7387$
- for $\alpha_2 = \frac{7}{10} = 0,7$ and $\alpha_1 = \frac{9}{10} = 0,9$: $2 \times 3^{0,6} (\gamma_2)^{0,7} - \gamma_2 - 1 = 0$, with a solution $\gamma_2 = 0,18453$ or $\gamma_2 = 87,3362$</footnote>
<table><thead><tr><th>Some values of α₁ and α₂</th><th>Equation :</th><th>Link between λ₁ and λ₂ :</th></tr></thead><tbody><tr><td></td><td>2(λ₁)<sup>2</sup> / (Σαᵢ - 1) + (γ₂)<sup>α₂</sup> - γ₂ - 1 = 0</td><td>γ₂ = λ₂ / λ₁ ⇔ λ₂ = γ₂λ₁, λ₁ = 2</td></tr><tr><td>α₁ = 9/10 = 0,9<br>and<br>α₂ = 5/10 = 0,5</td><td>2<sup>1,4</sup>(γ₂)<sup>0,5</sup> - γ₂ - 1 = 0<br>Solutions :</td><td>Unspecified proportions :</td></tr><tr><td></td><td>{γ₂ ≈ 0,2103}, {γ₂ ≈ 4,7541}</td><td>γ₂ ≈ 0,2103 ⇒ λ₂ ≈ 0,2103λ₁<br>or<br>γ₂ ≈ 4,7541 ⇒ λ₂ ≈ 4,7541λ₁</td></tr><tr><td>α₁ = 9/10 = 0,9<br>and<br>α₂ = 7/10 = 0,7</td><td>2<sup>1,6</sup>(γ₂)<sup>0,7</sup> - γ₂ - 1 = 0<br>Solutions :</td><td>Unspecified proportions :</td></tr><tr><td></td><td>{γ₂ ≈ 0,2975}, {γ₂ ≈ 36,8774}</td><td>γ₂ ≈ 0,2975 ⇒ λ₂ ≈ 0,2975λ₁<br>or<br>γ₂ ≈ 36,8774 ⇒ λ₂ ≈ 36,8774λ₁</td></tr></tbody></table>

Reading (or interpretation) of the results :

Let us consider the last line of this table.

* With the solution $γ₂ \approx 0,2975$ (that is to say $λ₂ \approx 0,2975λ₁$), the production function

$y = f(x₁, x₂) = x₁^{0,9} x₂^{0,7}$ is said to exhibit constant gh-returns to scale, with

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_{i} = \frac{\lambda_{1}}{2}(1 + \gamma_{2}) \approx \frac{\lambda_{1}}{2}(1 + 0,2975) = \lambda_{1} \frac{1,2975}{2} = 0,64875\lambda_{1}. $$

**Proof :** let us check that $r(\tilde{\lambda}, x₁, x₂) = \frac{f(\lambda₁x₁, \lambda₂x₂)}{\tilde{\lambda}f(x₁, x₂)} = 1$

$$ \begin{align*} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,9} (\lambda_2 x_2)^{0,7}}{0,64875 \lambda_1 (x_1^{0,9} x_2^{0,7})} = \frac{\lambda_1^{0,9} \lambda_2^{0,7} (x_1^{0,9} x_2^{0,7})}{0,64875 \lambda_1 (x_1^{0,9} x_2^{0,7})} = \frac{\lambda_1^{0,9} \lambda_2^{0,7}}{0,64875 \lambda_1} \\ &= \frac{\lambda_1^{0,9} (0,2975 \lambda_1)^{0,7}}{0,64875 \lambda_1} = \frac{\lambda_1^{1,6} (0,2975)^{0,7}}{0,64875 \lambda_1} = \lambda_1^{0,6} \frac{(0,2975)^{0,7}}{0,64875} = 2^{0,6} \frac{(0,2975)^{0,7}}{0,64875} \blacksquare \\ &= \frac{0,648722374}{0,64875} \approx 1 \text{ (because of the succession of round-offs)} \end{align*} $$

* With the solution $γ₂ \approx 36,8774$ (that is to say $λ₂ \approx 36,8774λ₁$), the production

$y = f(x₁, x₂) = x₁^{0,9} x₂^{0,7}$ is said to exhibit constant gh-returns to scale, with

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_{i} = \frac{\lambda_{1}}{2}(1 + \gamma_{2}) \approx \frac{\lambda_{1}}{2}(1 + 36,8774) = \lambda_{1} \frac{37,8774}{2} = 18,9387\lambda_{1}. $$
**Proof:** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$

$$
\begin{align*}
r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} \\
&= \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.7}}{18,9387 \lambda_1 (x_1^{0.9} x_2^{0.7})} \\
&= \frac{\lambda_1^{0.9} \lambda_2^{0.7} (x_1^{0.9} x_2^{0.7})}{18,9387 \lambda_1 (x_1^{0.9} x_2^{0.7})} \\
&= \frac{\lambda_1^{0.9} (36,8774 \lambda_1)^{0.7}}{18,9387 \lambda_1} \\
&= \frac{\lambda_1^{1.6} (36,8774)^{0.7}}{18,9387 \lambda_1} \\
&= \lambda_1^{0.6} \frac{(36,8774)^{0.7}}{18,9387} \\
&= 2^{0.6} \frac{(36,8774)^{0.7}}{18,9387} \quad \blacksquare \\
&= \frac{18,93870168}{18,9387} \approx 1 \quad (\text{because of the succession of round-offs})
\end{align*}
$$

**Remark :** It thus appears that there are constant returns to scale, with the arithmetic mean

$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_1 = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time the sum of elasticities of the Cobb-Douglas production function is higher than the unit $(\alpha_1 + \alpha_2 > 1)$.

**General conclusion (result) on the constancy of returns to scale:**

$\forall \sum_{i=1}^{2} \alpha_i \in ]0,2[$, in other words, whatever the degree of homogeneity (the sum of elasticities $\sum_{i=1}^{2} \alpha_i$) of the Cobb-Douglas production function $y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}$, there are constant returns to scale ($r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$) if, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, the components of the relative vector of modulation $\hat{\theta} = (\gamma_1 = \frac{\lambda_1}{\lambda_1}, \gamma_2 = \frac{\lambda_2}{\lambda_1})$, generated by the numeraire proportion $\lambda_1$, are bound by the relation $2(\lambda_1)^{\sum_{i=1}^{2} \alpha_i - 1} \left( \prod_{i=1}^{2} (\gamma_i)^{\alpha_i} \right) - \left( \sum_{i=1}^{2} \gamma_i \right)^{\sum_{i=1}^{2} \alpha_i} = 0$.
#### 4.1.2.2. The increase of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$, $\alpha_1 + \alpha_2 < 1$ and $\alpha_1 + \alpha_2 > 1$

There are increasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, x_2) > 1 \Leftrightarrow 2(\lambda_1)^{\sum \alpha_i - 1} \left( \prod_{i=1}^{2} (\gamma_i)^{\alpha_i} \right) - \left( \sum_{i=1}^{2} \gamma_i \right) > 0.$$

##### 4.1.2.2.1. The increase of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and $\forall \lambda_1 > 0$ (numeraire)

With $\alpha_1 + \alpha_2 = 1$, the solutions $\gamma_2$ to the inequality $2(\gamma_2)^{\alpha_2} - \gamma_2 - 1 > 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table:

<table><caption>**Table 6 :** The increase of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Values of<br>$\alpha_2 \in ]0,1[$</th><th>Inequality : $2\gamma_2^{\alpha_2} - \gamma_2 - 1 > 0$</th><th>Link between $\lambda_1$ and $\lambda_2$ :<br>$\gamma_2 = \frac{\lambda_2}{\lambda_1} \Leftrightarrow \lambda_2 = \gamma_2\lambda_1, \forall \lambda_1$</th></tr></thead><tbody><tr><td>$\alpha_2 = \frac{1}{2} \Rightarrow \alpha_1 = \frac{1}{2}$</td><td>$\frac{1}{2\gamma_2^2 - \gamma_2 - 1 > 0}$<br>Solution : $\emptyset$, i.e. the empty set</td><td>No link</td></tr><tr><td>$\alpha_2 = \frac{9}{10} \Rightarrow \alpha_1 = \frac{1}{10}$</td><td>$\frac{9}{2\gamma_2^{10} - \gamma_2 - 1 > 0}$<br>Solutions : $\gamma_2 \in ]1;1013,955[$</td><td>Unspecified proportions :<br>$\lambda_2 = \gamma_2\lambda_1$</td></tr></tbody></table>

Reading (or interpretation) of the results:

Let us consider the last line of this table.

* Case : $\gamma_2 \in ]1;1013,955[$: With the solution $\gamma_2 = 9$, for example, (that is to say $\lambda_2 = 9\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0,1}x_2^{0,9}$ is said to exhibit increasing gh-returns to scale, with the arithmetic mean as a reference indicator
$$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(\gamma_1 + \gamma_2) = \frac{\lambda_1}{2}(1 + \gamma_2) = 5\lambda_1.$$
**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} > 1$

$$r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,1} (\lambda_2 x_2)^{0,9}}{5 \lambda_1 (x_1^{0,1} x_2^{0,9})} = \frac{\lambda_1^{0,1} \lambda_2^{0,9} (x_1^{0,1} x_2^{0,9})}{5 \lambda_1 (x_1^{0,1} x_2^{0,9})} = \frac{\lambda_1^{0,1} (9 \lambda_1)^{0,9}}{5 \lambda_1} = \frac{9^{0,9}}{5} \approx 1,45 > 1 \blacksquare$$

**Remark :** It thus appears that there are increasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_1 = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time the sum of elasticities of the Cobb-Douglas production function is equal to the unit $(\alpha_1 + \alpha_2 = 1)$.

**4.1.2.2.2. The increase of gh-returns to scale with $\alpha_1 + \alpha_2 < 1$, $\alpha_1, \alpha_2 \in ]0,1[, \text{ and}**

**$\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)**

With $\alpha_1 + \alpha_2 < 1$, the solutions $\gamma_2$ to the inequality $2(\lambda_1)^{\frac{2}{\sum \alpha_i - 1}} (\gamma_2)^{\alpha_2 - \gamma_2 - 1} > 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<table><caption>**Table 7 : The increase of gh-returns to scale with $\alpha_1 + \alpha_2 < 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$**</caption><thead><tr><th>Some values of α<sub>1</sub><br>and α<sub>2</sub></th><th>Inequality :</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> :</th></tr></thead><tbody><tr><td rowspan="2">α<sub>1</sub> = <sup>2</sup>&frasl;<sub>10</sub> and<br>α<sub>2</sub> = <sup>7</sup>&frasl;<sub>10</sub></td><td>2<sup>0,9</sup>(γ<sub>2</sub>)<sup>0,7</sup> - γ<sub>2</sub> - 1 > 0</td><td>γ<sub>2</sub> = <sup>λ<sub>2</sub></sup>&frasl;<sub>λ<sub>1</sub></sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, λ<sub>1</sub> = 2</td></tr><tr><td>Solutions : γ<sub>2</sub> ∈ ]1,654;3,347[</td><td>Unspecified proportions :</td></tr></tbody></table>

Reading (or interpretation) of the results :

Let us consider the last line of this table.

* Case : $\gamma_2 \in ]1,654;3,347[$ : With the solution $\gamma_2 = 3$, for example, (that is to say $\lambda_2 = 3\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0,2} x_2^{0,7}$ is said to exhibit
increasing gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} (\gamma_1 + \gamma_2) = \frac{\lambda_1}{2} (1 + \gamma_2) = 2\lambda_1. $$

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} > 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,2} (\lambda_2 x_2)^{0,7}}{2\lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7} (x_1^{0,2} x_2^{0,7})}{2\lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,9} (3\lambda_1)^{0,7}}{2\lambda_1} \\ &= \lambda_1^{0,6} \frac{3^{0,7}}{2} = 2^{0,6} \frac{3^{0,7}}{2} = \frac{3^{0,7}}{2^{0,4}} \approx 1,635 > 1 \end{aligned} $$

**Remark :** It thus appears that there are increasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time the sum of elasticities of the Cobb-Douglas production function is lower than the unit $(\alpha_1 + \alpha_2 < 1)$.

**4.1.2.2.3. The increase of gh-returns to scale with $\alpha_1 + \alpha_2 > 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)**

With $\alpha_1 + \alpha_2 > 1$, the solutions $\gamma_2$ to the inequality $2(\lambda_1)^{\sum \alpha_i - 1} (\gamma_2^{\alpha_2} - \gamma_2 - 1 > 0)$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<table><caption>**Table 8 : The increase of gh-returns to scale with $\alpha_1 + \alpha_2 > 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$**</caption><thead><tr><th>Some values of α<sub>1</sub><br>and α<sub>2</sub></th><th>Inequality :</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> :</th></tr></thead><tbody><tr><td>α<sub>1</sub> = <sup>9</sup>&frasl;<sub>10</sub> = 0,9 and<br>α<sub>2</sub> = <sup>2</sup>&frasl;<sub>10</sub> = 0,5</td><td>2<sup>1,4</sup>(γ<sub>2</sub>)<sup>0,5</sup> - γ<sub>2</sub> - 1 > 0<br>Solutions : γ<sub>2</sub> ∈ ]0,2103;4,7541[</td><td>γ<sub>2</sub> = <sup>λ<sub>2</sub></sup>&frasl;<sub>λ<sub>1</sub></sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, λ<sub>1</sub> = 2<br>Unspecified proportions :</br>λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub> with γ<sub>2</sub> ≠ 1<br>Equiproportionate variations<br>(traditional case):<br>λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub> = λ<sub>1</sub> with γ<sub>2</sub> = 1</td></tr></tbody></table>
<table><tr><td>α₁ = <sup>9</sup>&frasl;<sub>10</sub> = 0,9 and<br>α₂ = <sup>2</sup>&frasl;<sub>10</sub> = 0,7</td><td>2<sup>1,6</sup>(γ₂)<sup>0,7</sup> - γ₂ - 1 > 0<br>Solutions : γ₂ ∈ ]0,2975;36,8774[</td><td>Unspecified proportions:<br>λ₂ = γ₂λ₁ with γ₂ ≠ 1<br>Equiproportionate variations<br>(traditional case):<br>λ₂ = γ₂λ₁ = λ₁ with γ₂ = 1</td></tr></table><p>Note : ∀α₁, α₂ ∈ ]0,1[ with α₁ + α₂ > 1, γ₂ = 1 is the solution to the inequality</p><p>2(λ₁)<sup>2</sup> &sum;<sub>i=1</sub><sup>i=1</sup> αᵢ - 1<br>(γ₂)<sup>α₂</sup> - γ₂ - 1 > 0.</p>

Reading (or interpretation) of the results :

Let us consider the penultimate line of this table.

* Case : $γ_2 \in ]0,2975;36,8774[$ : With the solution $γ_2 = \frac{1}{2}$, for example, (that is to say

$λ_2 = \frac{1}{2}λ_1$), the production function $y = f(x_1, x_2) = x_1^{0,9} x_2^{0,7}$ is said to exhibit increasing gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} (\gamma_1 + \gamma_2) = \frac{\lambda_1}{2} (1 + \gamma_2) = \frac{3}{4} \lambda_1 . $$

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} > 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,9} (\lambda_2 x_2)^{0,7}}{\frac{3}{4} \lambda_1 (x_1^{0,9} x_2^{0,7})} = \frac{\lambda_1^{0,9} \lambda_2^{0,7} (x_1^{0,9} x_2^{0,7})}{\frac{3}{4} \lambda_1 (x_1^{0,9} x_2^{0,7})} = \frac{\lambda_1^{0,9} \left(\frac{1}{2} \lambda_1\right)^{0,7}}{\frac{3}{4} \lambda_1} \\ &= \lambda_1^{0,6} \frac{\left(\frac{1}{2}\right)^{0,7}}{\frac{3}{4}} = \lambda_1^{0,6} \frac{\frac{1}{2^{0,7}}}{\frac{3}{4}} = \lambda_1^{0,6} \frac{1}{2^{0,7}} \times \frac{4}{3} = 2^{0,6} \frac{1}{2^{0,7}} \times \frac{4}{3} = 2^{-0,1} \frac{2^{2}}{3} = \frac{2^{1,9}}{3} \approx 1,244 > 1 \end{aligned} $$

**Notice 1 :** The traditional case of increasing returns to scale (when all the factors are duplicated in the same proportion $γ_2 = 1 \Rightarrow λ_2 = λ_1$) really seems a typical case of increasing gh-returns to scale. Indeed, for $\tilde{\lambda} = λ_2 = λ_1 = λ$, one has well,

$$ r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda x_1)^{0,9} (\lambda x_2)^{0,7}}{\lambda (x_1^{0,9} x_2^{0,7})} = \frac{\lambda^{0,9} \lambda^{0,7} (x_1^{0,9} x_2^{0,7})}{\lambda (x_1^{0,9} x_2^{0,7})} = \lambda^{0,6} = 2^{0,6} \approx 1,1557 > 1. $$
**Notice 2:** In the case of solution $\gamma_2 = \frac{1}{2} \Rightarrow \lambda_2 = \frac{1}{2}\lambda_1$, since by hypothesis $\lambda_1 = 2$ (as said before, any other value $\lambda_1 > 0$ could have been set), we get $\lambda_2 = \frac{1}{2} \times 2 = 1$: That means that the unspecified variation of the factors corresponds here to the case where factor $x_2$ remains fixed ($\lambda_2 = 1$) while factor $x_1$ is doubled ($\lambda_1 = 2$). It is therefore the examination of the marginal returns of factor $x_1$ (factor $x_1$ being the only one to vary: the passage from $\lambda_1 = 1$ to $\lambda_1 = 2$), which is indeed a specific case of gh-returns to scale.

**Notice 3:** A solution ($\gamma_2 = \frac{1}{3} \in ]0,2975;36,8774[$ is such as $\lambda_2 = \frac{1}{3}\lambda_1$. Thus, since by hypothesis $\lambda_1 = 2$, we get $\lambda_2 = \frac{2}{3} \approx 0,67$: that means that in the unspecified variation of the factors, we are in the case where factor $x_2$ sees its use being amputated by a third ($\frac{1}{3}$) when factor $x_1$ is doubled (since $\lambda_1 = 2$). It is therefore a deepening capital investment or an investment of the opposite type, since factor $x_2$ is partly sacrificed to the profit of the factor $x_1$.

**Notice 4:** If the producer does not make the wise choice of solution $\gamma_2 \in ]0,2975;36,8774[$, corresponding to the increase of returns to scale, when it plans to reduce one of the factors to the benefit of another, but rather the choice of a hazardous solution ($\gamma_2 = \frac{1}{10} \notin ]0,2975;36,8774[$, then it will certainly be located in the zone of decreasing gh-returns to scale (the gh-returns to scale $r_a(\tilde{\lambda}, x_1, x_2)$ not being more higher than the unit). Indeed $\gamma_2 = \frac{1}{10} \Rightarrow \lambda_2 = \frac{1}{10}\lambda_1$. Since by hypothesis $\lambda_1 = 2$, we get $\lambda_2 = \frac{2}{10} = 0,2$: in other words, if the producer cuts down factor $x_2$ of eight tenth ($\frac{8}{10} = 0,8$) when he doubles factor
$x_1$ ($\lambda_1 = 2$), with the arithmetic mean as a reference indicator

$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} (1 + \gamma_2) = \frac{(1+0,1)}{2} \lambda_1 = 0,55 \lambda_1$, the gh-returns to scale $r_a(\tilde{\lambda}, x_1, x_2)$ is such

that:

$$r_a(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,9} (\lambda_2 x_2)^{0,7}}{0,55 \lambda_1 (x_1^{0,9} x_2^{0,7})} = \frac{\lambda_1^{0,9} \lambda_2^{0,7} (x_1^{0,9} x_2^{0,7})}{0,55 \lambda_1 (x_1^{0,9} x_2^{0,7})} = \frac{\lambda_1^{0,9} (0,1 \lambda_1)^{0,7}}{0,55 \lambda_1} = \lambda_1^{0,6} \frac{(0,1)^{0,7}}{0,55} \\ = 2^{0,6} \frac{(0,1)^{0,7}}{0,55} = 0,5498 < 1$$

It thus follows that an inappropriate modification of the capital intensity on behalf of the producer or the entrepreneur can lead him to the zone of decreasing gh-returns to scale and not in the zone of increasing gh-returns to scale.

**Notice 5 :** In an anecdotic way, another problem could be that certain public policy aids vis-à-vis companies as regards recruiting or alleviation of unemployment. Sometimes, these policies are failed because of a bad diagnosis: the Public authorities may not correctly have identified the true returns zone where the company which they wish to help finds itself. A bad diagnosis can bring to suggest erroneous solutions.

If a company is in the zone of decreasing (marginal) returns, where arise difficulties such as that of the fixity of a factor, or that of wear and/or obsolescence of a factor (one thinks primarily of the machines, but that can as well relate to labor), then such assistance measures can be effective.

On the other hand, if a company is (because of limited outlets or a badly filled order book) in the zone (or phase) of increasing (marginal) returns (i.e. where for example a worker has a positive slope of marginal product, in other words where the worker is increasingly powerful), it may not find it beneficial to hire a worker who does not match such a marginal product).

This company is actually confronted with a problem of plethoric manpower: it will refuse such a help, if it is rational, especially when it is accompanied by constraints. And it will
accept this help (like manna from heaven) to improve its results if it is accompanied by no constraint.

**General conclusion (result) on the increase of returns to scale:**

$\forall \sum_{i=1}^{2} \alpha_i \in ]0,2[$, in other words, whatever the degree of homogeneity (the sum of elasticities $\sum_{i=1}^{2} \alpha_i$) of the Cobb-Douglas production function $y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}$, there are increasing gh-returns to scale ($r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} > 1$) if, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, the components of the relative vector of modulation $\hat{\theta} = (\gamma_1 = \frac{\lambda_1}{\lambda_1}, \gamma_2 = \frac{\lambda_2}{\lambda_1})$, generated by the numeraire proportion $\lambda_1$, are bound by the relation $2(\lambda_1)^{\sum \alpha_i - 1} (\prod_{i=1}^{2} (\gamma_i)^{\alpha_i}) - (\sum_{i=1}^{2} \gamma_i) > 0$.

**4.1.2.3. The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$, $\alpha_1 + \alpha_2 < 1$ and $\alpha_1 + \alpha_2 > 1**

There are decreasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, x_2) < 1 \Leftrightarrow 2(\lambda_1)^{\sum \alpha_i - 1} \left( \prod_{i=1}^{2} (\gamma_i)^{\alpha_i} \right) - \left( \sum_{i=1}^{2} \gamma_i \right) < 0$$
**4.1.2.3.1. The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$, $\alpha_1, \alpha_2 \in ]0,1[, \text{ and } \forall \lambda_1 > 0$**

*(numeraire)*

With $\alpha_1 + \alpha_2 = 1$, the solutions $\gamma_2$ to the inequality $2(\gamma_2)^{\alpha_2} - \gamma_2 - 1 < 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<table><caption>**Table 9 : The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 = 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$**</caption><thead><tr><th>Some values of<br>$\alpha_2 \in ]0,1[$</th><th>Inequality : $2\gamma_2^{\alpha_2} - \gamma_2 - 1 < 0$</th><th>Link between $\lambda_1$ and $\lambda_2$ :<br>$\gamma_2 = \frac{\lambda_2}{\lambda_1} \Leftrightarrow \lambda_2 = \gamma_2\lambda_1, \forall \lambda_1$</th></tr></thead><tbody><tr><td>$\alpha_2 = \frac{1}{2} \Rightarrow \alpha_1 = \frac{1}{2}$</td><td>$\frac{1}{2\gamma_2^2 - \gamma_2 - 1} < 0$<br>Solutions : $\gamma_2 \in ]0,1[ \cup ]1;+\infty[$</td><td>Unspecified proportions :<br>$\lambda_2 = \gamma_2\lambda_1$</td></tr><tr><td>$\alpha_2 = \frac{9}{10} \Rightarrow \alpha_1 = \frac{1}{10}$</td><td>$\frac{9}{2\gamma_2^{10} - \gamma_2 - 1} < 0$<br>Solutions : $\gamma_2 \in ]0,1[ \cup ]1013,955;+\infty[$</td><td>Unspecified proportions :<br>$\lambda_2 = \gamma_2\lambda_1$</td></tr></tbody></table>

Reading (or interpretation) of the results :

Let us consider the last line of this table.

* Case 1 : $\gamma_2 \in ]1013,955;+\infty[$ : With the solution $\gamma_2 = 1999$, for example, (that is to say $\lambda_2 = 1999\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0,1}x_2^{0,9}$ is said to exhibit decreasing gh-returns to scale, with the arithmetic mean as a reference indicator
$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1 + \gamma_2) = \frac{\lambda_1}{2}(1 + 1999) = 1000\lambda_1. $$

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,1} (\lambda_2 x_2)^{0,9}}{1000 \lambda_1 (x_1^{0,1} x_2^{0,9})} = \frac{\lambda_1^{0,1} \lambda_2^{0,9} (x_1^{0,1} x_2^{0,9})}{1000 \lambda_1 (x_1^{0,1} x_2^{0,9})} = \frac{\lambda_1^{0,1} \lambda_2^{0,9}}{1000 \lambda_1} \\ &= \frac{\lambda_1^{0,1} (1999 \lambda_1)^{0,9}}{1000 \lambda_1} = \frac{\lambda_1^{0,9} (1999 \lambda_1)^{0,9}}{1000 \lambda_1} = \frac{1999^{0,9}}{1000} \approx 0,9348 < 1 \quad \blacksquare \end{aligned} $$
* Case 2: $\gamma_2 \in ]0,1[$: With the solution $\gamma_2 = 0,5$, for example, (that is to say $\lambda_2 = 0,5\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0,1}x_2^{0,9}$ is said to exhibit decreasing gh-returns to scale, with the arithmetic mean as a reference indicator
$$ \tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i = \frac{\lambda_1}{2}(1+\gamma_2) = \frac{\lambda_1}{2}(1+0,5) = \frac{3}{4}\lambda_1. $$

**Proof:** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,1} (\lambda_2 x_2)^{0,9}}{\frac{3}{4} \lambda_1 (x_1^{0,1} x_2^{0,9})} = \frac{\lambda_1^{0,1} \lambda_2^{0,9} (x_1^{0,1} x_2^{0,9})}{\frac{3}{4} \lambda_1 (x_1^{0,1} x_2^{0,9})} = \frac{\lambda_1^{0,1} \lambda_2^{0,9}}{\frac{3}{4} \lambda_1} \\ &= \frac{4}{3} \frac{\lambda_1^{0,1} (0,5\lambda_1)^{0,9}}{\lambda_1} = \frac{4}{3} \frac{\lambda_1 0,5^{0,9}}{\lambda_1} = \frac{4}{3} \times 0,5^{0,9} \approx 0,7145 < 1 \end{aligned} $$

**Remark:** It thus appears that there are decreasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i = \frac{1}{2}\sum_{i=1}^{2}\gamma_i\lambda_1 = \frac{\lambda_1}{2}\sum_{i=1}^{2}\gamma_i$ as a reference indicator, while at the same time the sum of elasticities of the Cobb-Douglas production function is equal to the unit $(\alpha_1 + \alpha_2 = 1)$.

#### 4.1.2.3.2. The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 < 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)

With $\alpha_1 + \alpha_2 < 1$, the solutions $\gamma_2$ to the inequality $2(\lambda_1)^{i=1}^{\frac{2}{\sum \alpha_i} - 1} (\gamma_2)^{\alpha_2 - \gamma_2 - 1} < 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table:

<table><caption>**Table 10 :** The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 < 1$ and $\tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i$.</caption><thead><tr><th>Some values of<br>α₁ and α₂</th><th>Inequality :<br>2(λ₁)<sup>2</sup>(γ₂)<sup>α₂ - γ₂ - 1</sup></th><th>Link between λ₁ and λ₂ :<br>γ₂ = λ₂/λ₁ ⇔ λ₂ = γ₂λ₁, λ₁ = 2</th></tr></thead><tbody><tr><td></td><td></td><td></td></tr></tbody></table>
<table><tr><td>α₁ = 2/10 and<br>α₂ = 7/10</td><td>2<sup>0.9</sup>(γ₂)<sup>0.7</sup> - γ₂ - 1 &lt; 0<br>Solutions: γ₂ ∈ ]0;1,654[ ∪ ]3,347;+∞[</td><td>Unspecified proportions :<br>λ₂ = γ₂λ₁ with γ₂ ≠ 1<br>Equiproportionate variations<br>(traditional case):<br>λ₂ = γ₂λ₁ = λ₁ with γ₂ = 1</td></tr></table>

Reading (or interpretation) of the results :

Let us consider the last line of this table.

* Case 1 : $γ_2 \in ]3,347;+\infty[$ : With the solution $γ_2 = 5$, for example, (that is to say $λ_2 = 5λ_1$), the production function $y = f(x_1, x_2) = x_1^{0,2}x_2^{0,7}$ is said to exhibit decreasing gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1 + \gamma_2) = \frac{\lambda_1}{2}(1 + 5) = 3\lambda_1. $$

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,2} (\lambda_2 x_2)^{0,7}}{3\lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7} (x_1^{0,2} x_2^{0,7})}{3\lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7}}{3\lambda_1} \\ &= \frac{\lambda_1^{0,2} (5\lambda_1)^{0,7}}{3\lambda_1} = \frac{5^{0,7} \lambda_1^{0,9}}{3\lambda_1} = \frac{5^{0,7} \lambda_1^{-0,1}}{3} = \frac{5^{0,7}}{3 \times 2^{0,1}} \approx 0,9595 < 1 \quad \blacksquare \end{aligned} $$

* Case 2 : $γ_2 \in ]0;1,654[$ : With the solution $γ_2 = 1,5$, for example, (that is to say $λ_2 = 1,5λ_1$), the production function $y = f(x_1, x_2) = x_1^{0,2}x_2^{0,7}$ is said to exhibit decreasing gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1 + \gamma_2) = \frac{\lambda_1}{2}(1 + 1,5) = 1,25\lambda_1. $$

**Proof :** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$
$$
\begin{aligned}
r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,2} (\lambda_2 x_2)^{0,7}}{1,25 \lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7} (x_1^{0,2} x_2^{0,7})}{1,25 \lambda_1 (x_1^{0,2} x_2^{0,7})} = \frac{\lambda_1^{0,2} \lambda_2^{0,7}}{1,25 \lambda_1} \\
&= \frac{\lambda_1^{0,2} (1,5 \lambda_1)^{0,7}}{1,25 \lambda_1} = \frac{1,5^{0,7} \lambda_1^{0,9}}{1,25 \lambda_1} = \frac{1,5^{0,7} \lambda_1^{-0,1}}{1,25} = \frac{1,5^{0,7}}{1,25 \times 2^{0,1}} \approx 0,9514 < 1
\end{aligned}
$$

**Notice :** The solution $\left( \gamma_2 = \frac{\lambda_2}{\lambda_1} = 1 \right) \in ]0;1,654[$, i.e. $\lambda_1 = \lambda_2$, corresponds to the traditional case of decreasing returns to scale ($\alpha_1 + \alpha_2 < 1$) where all the factors are duplicated in the same proportion. Indeed, for $\tilde{\lambda} = \lambda_2 = \lambda_1 = \lambda$, one has well:

$$r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda x_1)^{0,2} (\lambda x_2)^{0,7}}{\lambda (x_1^{0,2} x_2^{0,7})} = \frac{\lambda^{0,2} \lambda^{0,7} (x_1^{0,2} x_2^{0,7})}{\lambda (x_1^{0,2} x_2^{0,7})} = \lambda^{-0,1} = 2^{-0,1} \approx 0,933 < 1.$$

**4.1.2.3.3. The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 > 1$, $\alpha_1, \alpha_2 \in ]0,1[$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)**

With $\alpha_1 + \alpha_2 > 1$, the solutions $\gamma_2$ to the inequality $2(\lambda_1)^{i=1} \quad (\gamma_2)^{\alpha_2} - \gamma_2 - 1 < 0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<table><caption>**Table 11 :** The decrease of gh-returns to scale with $\alpha_1 + \alpha_2 > 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Some values of α₁<br>and α₂</th><th>Inequality :<br>2/∑αᵢ - 1<br>2(λ₁)<sup>i=1</sup> λ₂<sup>α₂ - γ₂ - 1</sup> < 0</th><th>Link between λ₁ and λ₂ :<br>γ₂ = λ₂/λ₁ ⇔ λ₂ = γ₂λ₁, λ₁ = 2</th></tr></thead><tbody><tr><td>α₁ = 9/10 = 0,9 and<br>α₂ = 2/10 = 0,5</td><td>2<sup>1,4</sup>(γ₂)<sup>0,5</sup> - γ₂ - 1 < 0<br>Solutions :<br>γ₂ ∈ ]0;0,2103[ ∪ ]4,7541;+∞[</td><td>Unspecified proportions :<br>λ₂ = γ₂λ₁</td></tr><tr><td>α₁ = 9/10 = 0,9 and<br>α₂ = 2/10 = 0,7</td><td>2<sup>1,6</sup>(γ₂)<sup>0,7</sup> - γ₂ - 1 < 0<br>Solutions :<br>γ₂ ∈ ]0;0,2975[ ∪ ]36,8774;+∞[</td><td>Unspecified proportions :<br>λ₂ = γ₂λ₁</td></tr></tbody></table>

Reading (or interpretation) of the results :
Let us consider the last line of this table.

* Case 1: $\gamma_2 \in [36, 8774+\infty[$: With the solution $\gamma_2 = 39$, for example, (that is to say $\lambda_2 = 39\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.7}$ is said to exhibit decreasing gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1 + \gamma_2) = \frac{\lambda_1}{2}(1 + 39) = 20\lambda_1. $$

**Proof:** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.7}}{20 \lambda_1 (x_1^{0.9} x_2^{0.7})} = \frac{\lambda_1^{0.9} \lambda_2^{0.7} (x_1^{0.9} x_2^{0.7})}{20 \lambda_1 (x_1^{0.9} x_2^{0.7})} = \frac{\lambda_1^{0.9} (39 \lambda_1)^{0.7}}{20 \lambda_1} \\ &= \lambda_1^{0.6} \frac{(39)^{0.7}}{20} = 2^{0.6} \frac{(39)^{0.7}}{20} \approx 0.9847 < 1 \end{aligned} $$

* Case 2: $\gamma_2 \in [0, 2975[$: With the solution $\gamma_2 = 0, 1$, for example, (that is to say $\lambda_2 = 0, 1\lambda_1$), the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.7}$ is said to exhibit decreasing gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1 + \gamma_2) \approx \frac{\lambda_1}{2}(1 + 0,1) = 0,55\lambda_1. $$

**Proof:** let us check that $r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$

$$ \begin{aligned} r(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.7}}{0.55 \lambda_1 (x_1^{0.9} x_2^{0.7})} = \frac{\lambda_1^{0.9} \lambda_2^{0.7} (x_1^{0.9} x_2^{0.7})}{0.55 \lambda_1 (x_1^{0.9} x_2^{0.7})} = \frac{\lambda_1^{0.9} \lambda_2^{0.7}}{0.55 \lambda_1} \\ &= \frac{\lambda_1^{0.9} (0.1 \lambda_1)^{0.7}}{0.55 \lambda_1} = \frac{0.1^{0.7} \lambda_1^{1.6}}{0.55 \lambda_1} = \frac{0.1^{0.7} \lambda_1^{0.6}}{0.55} = \frac{0.1^{0.7} \times 2^{0.6}}{0.55} \approx 0.5498 < 1 \end{aligned} $$

**Remark:** It thus appears that there are decreasing returns to scale, with the arithmetic mean

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i $$ as a reference indicator, while at the same time the sum of elasticities of the Cobb-Douglas production function is higher than the unit $(\alpha_1 + \alpha_2 > 1)$.
**General conclusion (result) on the decrease of returns to scale:**

$\forall \sum_{i=1}^{2} \alpha_i \in ]0,2[$, in other words, whatever the degree of homogeneity (the sum of elasticities $\sum_{i=1}^{2} \alpha_i$) of the Cobb-Douglas production function $y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}$, there are decreasing gh-returns to scale ($r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$) if, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, the components of the relative vector of modulation $\hat{\theta} = (\gamma_1 = \frac{\lambda_1}{\lambda_1}, \gamma_2 = \frac{\lambda_2}{\lambda_1})$, generated by the numeraire proportion $\lambda_1$, are bound by the relation $2(\lambda_1)^{\sum_{i=1}^{2} \alpha_i - 1} (\prod_{i=1}^{2} (\gamma_i)^{\alpha_i}) - (\sum_{i=1}^{2} \gamma_i) < 0$.

**4.2. Numerical examples of gh-returns to scale in the case of the generalized C.E.S. production function**

$$y = f(x_1, x_2) = A[\alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho}]^{-\frac{k}{\rho}}$$

**4.2.1. Gh-returns to scale in the case of the generalized C.E.S. production function**

$$y = f(x_1, x_2) = A[\alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho}]^{-\frac{k}{\rho}} \quad \text{with an unspecified reference indicator } \tilde{\lambda}$$

*Reminder:* Let $\theta = (\lambda_1, \lambda_2)$ be the vector of modulation of vector of inputs $(x_1, x_2)$. Let $\hat{\theta} = (\gamma_1, \gamma_2)$ be the vector of relative proportions with $\lambda_1$ as numeraire proportion. The
expression (relation 20*) of gh-returns to scale, with an unspecified reference indicator $\tilde{\lambda}$, is

$$ \text{such that : } r(\tilde{\lambda}, x_1, x_2) = \lambda_1^k \tilde{\lambda}^{(-1)} \frac{\left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}. $$

### 4.2.2. The gh-returns to scale in the case of the generalized C.E.S. production<fnref n="11" /> function

$$ y = f(x_1, x_2) = A \left[ \alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho} \right]^{-\frac{k}{\rho}} \quad \text{with arithmetic mean } \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i \quad \text{as a reference indicator}^{12} $$

Since $\lambda_1$ is the numeraire proportion, the arithmetic mean can be expressed as :

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_1 = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i. \quad \text{The expression of gh-returns to scale is thus as follows :} $$

$$ r(\tilde{\lambda}, x_1, x_2) = \frac{\lambda_1^k}{\tilde{\lambda}} \frac{\left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}} = 2 \frac{\lambda_1^{k-1}}{\sum_{i=1}^{2} \gamma_i} \frac{\left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}}}{\left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}}}. $$

#### 4.2.2.1. The constancy of gh-returns to scale with $k=1, k<1$ and $k>1$

There are constant gh-returns to scale if:

<footnote n="11">¹¹ The reader will be spared the reading of results after each table, as was the case with the generalized Cobb-Douglas production function.</footnote>

<footnote n="12">¹² The analysis of gh-returns to scale with the geometric mean as a reference indicator will not be broached in so far as the conclusions for the classification of gh-returns to scale are not affected (there is only a change of scale: see paragraph 4.3.1.)</footnote>
$$
r(\tilde{\lambda}, x_1, x_2) = 1 \Leftrightarrow \left( \sum_{i=1}^{2} \gamma_i \right) \left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}} - 2 \lambda_1^{k-1} \left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} = 0
$$

4.2.2.1.1. The constancy of gh-returns to scale with $k=1$ and $\forall \lambda_1 > 0$ (numeraire)

With $k=1, \forall \alpha_1, \alpha_2 \in ]0;1[, \forall \rho, \forall x_i (i=1,2), \gamma_2=1$ is the solution to the equation

$$
\left(\sum_{i=1}^{2} \gamma_i\right) \left[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2 \lambda_1^{k-1} \left[\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} = 0.
$$

An illustration of the solution $\gamma_2 = 1$, for some values of $\alpha_2 \in ]0,1[$, with $\rho = -0,5$,
and $x_1 = x_2 = 1$, is given in the following table :

<table><caption>Table 12 : The constancy of gh-returns to scale with $k=1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption>
  <thead>
    <tr>
      <th>Some values of<br>α₁, α₂ ∈ ]0;1[<br>with k = 1</th>
      <th>Equation :<br>(1 + γ₂)[α₁ + α₁]² - 2[α₁ + α₂γ₂⁰,⁵]² = 0</th>
      <th>Link between λ₁ and λ₂ :<br>γ₂ = λ₂/λ₁ ⇔ λ₂ = γ₂λ₁, ∀λ₁ > 0</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>α₁ = 5/10 = 0,5</td>
      <td>γ₂⁰,⁵ - 0,5γ₂ - 0,5 = 1/2(√γ₂ - 1)² = 0</td>
      <td rowspan="2">Equiproportionate variations<br>(traditional case) :<br>γ₂ = 1 ⇒ λ₂ = λ₁</td>
    </tr>
    <tr>
      <td>α₂ = 5/10 = 0,5</td>
      <td>Solution : {γ₂ = 1}</td>
    </tr>
    <tr>
      <td>α₁ = 8/10 = 0,8</td>
      <td>1,92γ₂⁰,⁵ - 1,24γ₂ - 0,68 = 0</td>
      <td rowspan="2">Equiproportionate variations<br>(traditional case) :<br>γ₂ = 1 ⇒ λ₂ = λ₁</td>
    </tr>
    <tr>
      <td>α₂ = 6/10 = 0,6</td>
      <td>Solution : {γ₂ = 1}</td>
    </tr>
  </tbody>
</table>

4.2.2.1.2. The constancy of gh-returns to scale with $k < 1$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)

With some values $k \in ]0;1[$, the solutions $\gamma_2$ to the equation

$$
\left(\sum_{i=1}^{2} \gamma_i\right) \left[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2 \lambda_1^{k-1} \left[\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} = 0 \text{ are given in the following table:}
$$
<table><caption>Table 13 : The constancy of gh-returns to scale with $k < 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_{i}$.</caption><thead><tr><th>Some values of k ∈ ]0;1[ with<br>α₁ = <sup>7</sup>&frasl;<sub>10</sub> = 0,7<br>α₂ = <sup>2</sup>&frasl;<sub>10</sub> = 0,2<br>ρ = -0,5<br>x₁ = x₂ = 1</th><th>Equation :<br><span style="font-size: 2em; vertical-align: middle;">[</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><sup>2</sup>&frasl;<sub>∑<sub>i=1</sub><sup>γi</sup></sub><span style="display: block;">[</span><span style="display: block;">2</span><span style="display: inline-block; vertical-align: middle; text-align: center;"><sup>2</sup>&frasl;<sub>∑<sub>i=1</sub><sup>αix<sub>i</sub><sup>-</sup>ρ</sup></sub><span style="display: block;">[</span><span style="display: block;">k</span><span style="display: inline-block; vertical-align: middle; text-align: center;">- ρ</span><span style="display: inline-block; vertical-align: middle; text-align: center;">- 2λ<sub>1</sub><sup>k-1</sup></span><span style="display: inline-block; vertical-align: middle; text-align: center;"><sup>2</sup>&frasl;<sub>∑<sub>i=1</sub><sup>γ<sub>i</sub><sup>-</sup>ρ</sup></sub><span style="display: block;">[</span><span style="display: block;">2</span><span style="display: inline-block; vertical-align: middle; text-align: center;">- ρ</span><span style="display: inline-block; vertical-align: middle; text-align: center;">(α<sub>i</sub>x<sub>i</sub><sup>-</sup>ρ)</sup></span><span style="display: inline-block; vertical-align: middle; text-align: center;"><sup>k</sup>&frasl;<sub>ρ</sub></span><span style="display: inline-block; vertical-align: middle; text-align: center;">= 0</span></span><span style="font-size: 2em; vertical-align: middle;">]</span><span style="font-size: 2em; vertical-align: middle;">]</span></th><th>Link between λ₁ and λ₂ :<br>γ₂ = <sup>λ₂</sup>&frasl;<sub>λ₁</sub> ⇔ λ₂ = γ₂λ₁, λ₁ = 2</th></tr></thead><tbody><tr><td>k = 0,5</td><td>2<sup>0,5</sup>(0,7 + 0,2 × γ<sub>2</sub><sup>0,5</sup>) - 0,9γ₂ - 0,9 = 0<br>Solution<fnref n="13" /> : {γ₂ ≈ 0,26028}</td><td>Unspecified proportions :<br>γ₂ ≈ 0,26028⇒ λ₂ ≈ 0,26028λ₁</td></tr><tr><td>k = 0,7</td><td>2<sup>0,7</sup>(0,7 + 0,2 × γ<sub>2</sub><sup>0,5</sup>)<sup>1,4</sup> - (0,9)<sup>1,4</sup>γ₂ - (0,9)<sup>1,4</sup> = 0<br>Solution : {γ₂ ≈ 0,46663}</td><td>Unspecified proportions :<br>γ₂ ≈ 0,46663⇒ λ₂ ≈ 0,46663λ₁</td></tr></tbody></table>

**Remark :** It thus appears that there are constant returns to scale, with the arithmetic mean

$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_{i} = \frac{1}{2} \sum_{i=1}^{2} \gamma_{i} \lambda_{i} = \frac{\lambda_{1}}{2} \sum_{i=1}^{2} \gamma_{i}$ as a reference indicator, while at the same time

homogeneity of degree of the C.E.S. production function is lower than the unit ($k < 1$).

Indeed:

$$2^{0,5}(0,7+0,2 \times \gamma_2^{0,5}) - 0,9\gamma_2 - 0,9 = 0 \Leftrightarrow 0,7\sqrt{2} + 0,2\sqrt{2\gamma_2} - 0,9\gamma_2 - 0,9 = 0 \Leftrightarrow \frac{7}{10}\sqrt{2} + \frac{2\sqrt{2}}{10}\sqrt{\gamma_2} - \frac{9}{10}\gamma_2 - \frac{9}{10} = 0 \\ \Leftrightarrow -9\gamma_2 + 2\sqrt{2\gamma_2} + 7\sqrt{2} - 9 = 0$$

While assuming $\Gamma = \sqrt{\gamma_2}$, the equation $-9\gamma_2 + 2\sqrt{2\gamma_2} + 7\sqrt{2} - 9 = 0$ comes back to the equation $-9\Gamma^2 + 2\sqrt{2\Gamma} + 7\sqrt{2} - 9 = 0$. The discriminant of the equation $-9\Gamma^2 + 2\sqrt{2\Gamma} + 7\sqrt{2} - 9 = 0$ is such that: $\Delta = 8 + 36(7\sqrt{2} - 9) = 4(63\sqrt{2} - 79) > 0$. The equation $-9\Gamma^2 + 2\sqrt{2\Gamma} + 7\sqrt{2} - 9 = 0$ thus has two solutions: $\Gamma = \frac{-2\sqrt{2} \pm \sqrt{4(63\sqrt{2} - 79)}}{-18} = \frac{\sqrt{2} \pm \sqrt{(63\sqrt{2} - 79)}}{9}$. However $\Gamma = \frac{\sqrt{2} - \sqrt{(63\sqrt{2} - 79)}}{9}$ is negative. So the solution adopted is: $\Gamma = \frac{\sqrt{2} + \sqrt{(63\sqrt{2} - 79)}}{9}$. The equation $2^{0,5}(0,7+0,2 \times \gamma_2^{0,5}) - 0,9\gamma_2 - 0,9 = 0$ thus has one solution:

$$\gamma_2 = \left( \frac{\sqrt{2} + \sqrt{(63\sqrt{2} - 79)}}{9} \right)^2 = \frac{2 + 2\sqrt{(126\sqrt{2} - 158)} + 63\sqrt{2} - 79}{81} = \frac{63\sqrt{2} - 77 + 2\sqrt{(126\sqrt{2} - 158)}}{81} \approx 0,26028.$$

<footnote n="13">¹³ One can get the exact expression of the solution to the equation $2^{0,5}(0,7+0,2 \times \gamma_2^{0,5}) - 0,9\gamma_2 - 0,9 = 0$.</footnote>
4.2.2.1.3. The constancy of gh-returns to scale with $k > 1$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)

With some values $k \in ]1; +\infty[$, the solutions $\gamma_2$ to the equation
$$ \left( \sum_{i=1}^{2} \gamma_i \right) \left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} \left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} = 0 $$ are given in the following table:

<table><caption>Table 14 : The constancy of gh-returns to scale with $k > 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Some values of k ∈ ]1; +∞[ with<br>α₁ = <sup>5</sup>&frasl;<sub>10</sub> = 0,5<br>α₂ = <sup>5</sup>&frasl;<sub>10</sub> = 0,5<br>ρ = -0,5<br>x₁ = x₂ = 1</th><th>Equation :<br>$$ \left(\sum_{i=1}^{2} \gamma_i\right) \left[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} \left[\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} = 0 $$</th><th>Link between λ₁ and λ₂ :<br>γ₂ = <sup>λ₂</sup>&frasl;<sub>λ₁</sub> ⇔ λ₂ = γ₂λ₁, λ₁ = 2</th></tr></thead><tbody><tr><td>k = 2</td><td>2²(0,5 + 0,5×γ₂<sup>0,5</sup>)⁴ - γ₂ - 1 = 0<br>Solution : {γ₂ ≈ 0,24342}</td><td>Unspecified proportions :<br>γ₂ ≈ 0,24342 ⇒ λ₂ ≈ 0,24342λ₁</td></tr><tr><td>k = 8</td><td>2⁸(0,5 + 0,5×γ₂<sup>0,5</sup>)¹⁶ - γ₂ - 1 = 0<br>Solution : {γ₂ ≈ 0,18425}</td><td>Unspecified proportions :<br>γ₂ ≈ 0,18425 ⇒ λ₂ ≈ 0,18425λ₁</td></tr></tbody></table>

**Remark :** It thus appears that there are constant returns to scale, with the arithmetic mean
$\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time
homogeneity of degree of the C.E.S. production function is higher than the unit ($k > 1$).

**General conclusion (result) on the constancy of returns to scale:**

∀k > 0, in other words, whatever the degree of homogeneity of the generalized C.E.S. production function $y = f(x₁, x₂) = A[\alpha₁x₁^{-ρ} + \alpha₂x₂^{-ρ}]^{-\frac{k}{\rho}}$, there are constant returns to scale
($r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = 1$) if, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, the components of the relative vector of modulation $\hat{\theta} = (\gamma_1 = \frac{\lambda_1}{\lambda_1}, \gamma_2 = \frac{\lambda_2}{\lambda_1})$, generated by the numeraire proportion $\lambda_1$, are bound by the relation $(\sum_{i=1}^{2} \gamma_i) [\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} [\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} = 0$.

### 4.2.2.2. The increase of gh-returns to scale with $k=1$, $k<1$ and $k>1$.

There are increasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, x_2) > 1 \Leftrightarrow \left( \sum_{i=1}^{2} \gamma_i \left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} \left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} > 0$$

#### 4.2.2.2.1. The increase of gh-returns to scale with $k=1$, $\forall \lambda_1 > 0$ (numeraire)

With $k=1$, $\alpha_1, \alpha_2 \in ]0;1[$, $\rho = -0,5$, $x_1 = x_2 = 1$, the solutions $\gamma_2$ to the inequality

$(\sum_{i=1}^{2} \gamma_i) [\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} [\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} > 0$ are given in the following table:

<table><caption>**Table 15 :** The increase of gh-returns to scale with $k=1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Some values of α<sub>1</sub>, α<sub>2</sub> ∈ ]0;1[ with k = 1</th><th>Inequality : (1 + γ<sub>2</sub>)[α<sub>1</sub> + α<sub>1</sub>]<sup>2</sup> - 2[α<sub>1</sub> + α<sub>2</sub>γ<sub>2</sub><sup>0,5</sup>]<sup>2</sup> > 0</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> : γ<sub>2</sub> = λ<sub>2</sub>/λ<sub>1</sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, ∀λ<sub>1</sub> > 0</th></tr></thead><tbody><tr><td>α<sub>1</sub> = 5/10 = 0,5</td><td>γ<sub>2</sub><sup>0,5</sup> - 0,5γ<sub>2</sub> - 0,5 > 0</td><td rowspan="2">Unspecified proportions : λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub></td></tr><tr><td>α<sub>2</sub> = 5/10 = 0,5</td><td>Solutions : γ<sub>2</sub> ∈ ]1;+∞[</td></tr></tbody></table>
<table><tr><td>α₁ = 8/10 = 0,8</td><td>1,92γ₂<sup>0,5</sup> - 1,24γ₂ - 0,68 > 0</td><td rowspan="2">Unspecified proportions :<br/>λ₂ = γ₂λ₁</td></tr><tr><td>α₂ = 6/10 = 0,6</td><td>Solutions : γ₂ ∈ ]1;+∞[</td></tr></table>

**Remark :** It thus appears that there are increasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time homogeneity of degree of the C.E.S. production function is equal to the unit ($k = 1$).

**4.2.2.2.2. The increase of gh-returns to scale with $k < 1$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)**

With some values $k \in ]0;1[$, the solutions $\gamma_2$ to the inequality $(\sum_{i=1}^{2} \gamma_i)[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1}[\sum_{i=1}^{2} \gamma_i^{-\rho}(\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} > 0$ are given in the following table:

<table><caption>**Table 16 : The increase of gh-returns to scale with $k < 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$**</caption><tr><td>Some values of $k \in ]0;1[$ with<br/>α₁ = 7/10 = 0,7,<br/>α₂ = 2/10 = 0,2,<br/>ρ = -0,5, and<br/>x₁ = x₂ = 1</td><td>Inequality :<br/>$(\sum_{i=1}^{2} \gamma_i)[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1}[\sum_{i=1}^{2} \gamma_i^{-\rho}(\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} > 0$</td><td>Link between $\lambda_1$ and $\lambda_2$ :<br/>γ₂ = $\frac{\lambda_2}{\lambda_1}$ ⇔ λ₂ = γ₂λ₁, λ₁ = 2</td></tr><tr><td>k = 0,5</td><td>2<sup>0,5</sup>(0,7 + 0,2×γ<sub>2</sub><sup>0,5</sup>) - 0,9γ₂ - 0,9 > 0<br/>Solutions : γ₂ ∈ ]0;0,26028[</td><td>Unspecified proportions :<br/>λ₂ = γ₂λ₁</td></tr><tr><td>k = 0,7</td><td>2<sup>0,7</sup>(0,7 + 0,2×γ<sub>2</sub><sup>0,5</sup>)<sup>1,4</sup> - (0,9)<sup>1,4</sup>γ₂ - (0,9)<sup>1,4</sup> > 0<br/>Solutions : γ₂ ∈ ]0;0,46663[</td><td>Unspecified proportions :<br/>λ₂ = γ₂λ₁</td></tr></table>
**Remark :** It thus appears that there are increasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i = \frac{1}{2}\sum_{i=1}^{2}\gamma_i\lambda_1 = \frac{\lambda_1}{2}\sum_{i=1}^{2}\gamma_i$ as a reference indicator, while at the same time homogeneity of degree of the C.E.S. production function is lower than the unit ($k < 1$).

**4.2.2.2.3. The increase of gh-returns to scale with $k > 1$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)**

With some values $k \in ]1;+\infty[$, the solutions $\gamma_2$ to the inequality
$$ \left(\sum_{i=1}^{2}\gamma_i\right)\left[\sum_{i=1}^{2}\alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1}\left[\sum_{i=1}^{2}\gamma_i^{-\rho}(\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} > 0 $$
are given in the following table:

<table><caption>**Table 17 : The increase of gh-returns to scale with $k > 1$ and $\tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i$**</caption><thead><tr><th>Some values of k ∈ ]1;+∞[ with α<sub>1</sub> = <sup>5</sup>&frasl;<sub>10</sub> = 0,5, α<sub>2</sub> = <sup>5</sup>&frasl;<sub>10</sub> = 0,5, ρ = -0,5, and x<sub>1</sub> = x<sub>2</sub> = 1</th><th>Inequality : <br> $\left(\sum_{i=1}^{2}\gamma_i\right)\left[\sum_{i=1}^{2}\alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1}\left[\sum_{i=1}^{2}\gamma_i^{-\rho}(\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} > 0$</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> : <br> γ<sub>2</sub> = <sup>λ<sub>2</sub></sup>&frasl;<sub>λ<sub>1</sub></sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, λ<sub>1</sub> = 2</th></tr></thead><tbody><tr><td>k = 2</td><td>2<sup>2</sup>(0,5 + 0,5×γ<sub>2</sub><sup>0,5</sup>)<sup>4</sup> - γ<sub>2</sub> - 1 > 0<br> Solutions : γ<sub>2</sub> ∈ ]0;0,24342[</td><td>Unspecified proportions :<br> λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub></td></tr><tr><td>k = 8</td><td>2<sup>8</sup>(0,5 + 0,5×γ<sub>2</sub><sup>0,5</sup>)<sup>16</sup> - γ<sub>2</sub> - 1 > 0<br> Solutions : γ<sub>2</sub> ∈ ]0;0,18425[</td><td>Unspecified proportions :<br> λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub></td></tr></tbody></table>

**General conclusion (result) on the increase of returns to scale:**

∀k > 0, in other words, whatever the degree of homogeneity of the generalized C.E.S. production function $y = f(x_1, x_2) = A[\alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho}]^{-\frac{k}{\rho}}$, there are increasing gh-returns to
scale ($r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} > 1$) if, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, the components of the relative vector of modulation $\hat{\theta} = (\gamma_1 = \frac{\lambda_1}{\lambda_1}, \gamma_2 = \frac{\lambda_2}{\lambda_1})$, generated by the numeraire proportion $\lambda_1$, are bound by the relation $(\sum_{i=1}^{2} \gamma_i) [\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} [\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} > 0$.

### 4.2.2.3. The decrease of gh-returns to scale with $k=1$, $k<1$ and $k>1$.

There are decreasing gh-returns to scale if:

$$r(\tilde{\lambda}, x_1, x_2) < 1 \Leftrightarrow \left(\sum_{i=1}^{2} \gamma_i\right) \left[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} \left[\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} < 0$$

#### 4.2.2.3.1. The decrease of gh-returns to scale with $k=1$, $\forall \lambda_1 > 0$ (numeraire)

With $k=1$, $\alpha_1, \alpha_2 \in ]0;1[$, $\rho = -0,5$, $x_1 = x_2 = 1$, the solutions $\gamma_2$ to the inequality

$(\sum_{i=1}^{2} \gamma_i) [\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} [\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} < 0$ are given in the following table:

<table><caption>**Table 18 :** The decrease of gh-returns to scale with $k=1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Some values of α<sub>1</sub>, α<sub>2</sub> ∈ ]0;1[ with k = 1</th><th>Inequality : (1+γ<sub>2</sub>)[α<sub>1</sub> + α<sub>1</sub>]<sup>2</sup> - 2[α<sub>1</sub> + α<sub>2</sub>γ<sub>2</sub><sup>0,5</sup>]<sup>2</sup> < 0</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> : γ<sub>2</sub> = λ<sub>2</sub>/λ<sub>1</sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, ∀λ<sub>1</sub> > 0</th></tr></thead><tbody><tr><td>α<sub>1</sub> = 5/10 = 0,5</td><td>γ<sub>2</sub><sup>0,5</sup> - 0,5γ<sub>2</sub> - 0,5 < 0</td><td rowspan="2">Unspecified proportions : λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub></td></tr><tr><td>α<sub>2</sub> = 5/10 = 0,5</td><td>Solutions : γ<sub>2</sub> ∈ ]0;1[</td></tr><tr><td>α<sub>1</sub> = 4/10 = 0,4</td><td>0,8γ<sub>2</sub><sup>0,5</sup> - 0,31γ<sub>2</sub> - 0,49 < 0</td><td>Unspecified proportions :</td></tr></tbody></table>
<table><tr><td>α<sub>2</sub> = <sup>5</sup>&frasl;<sub>10</sub> = 0,5</td><td>Solutions : γ<sub>2</sub> ∈ ]0;1[</td><td>λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub></td></tr><tr><td>α<sub>1</sub> = <sup>8</sup>&frasl;<sub>10</sub> = 0,8</td><td>1,92γ<sub>2</sub><sup>0,5</sup> - 1,24γ<sub>2</sub> - 0,68 &lt; 0</td><td rowspan="2">Unspecified proportions :<br/>λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub></td></tr><tr><td>α<sub>2</sub> = <sup>6</sup>&frasl;<sub>10</sub> = 0,6</td><td>Solutions : γ<sub>2</sub> ∈ ]0;1[</td></tr></table>

**Remark :** It thus appears that there are decreasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{1}{2} \sum_{i=1}^{2} \gamma_i \lambda_i = \frac{\lambda_1}{2} \sum_{i=1}^{2} \gamma_i$ as a reference indicator, while at the same time homogeneity of degree of the C.E.S. production function is equal to the unit ($k=1$).

**4.2.2.3.2. The decrease of gh-returns to scale with $k<1$, and $\lambda_1$ fixed ($\lambda_1=2$ will be taken: any other value $\lambda_1>0$ could have been set)**

With some values $k \in ]0;1[$, the solutions $\gamma_2$ to the inequality $(\sum_{i=1}^{2} \gamma_i)[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1}[\sum_{i=1}^{2} \gamma_i^{-\rho}(\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} < 0$ are given in the following table:

<table><caption>**Table 19 : The decrease of gh-returns to scale with $k<1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$**</caption><tr><td>Some values of $k \in ]0;1[$ with $\alpha_1 = \frac{5}{10} = 0,5$, $\alpha_2 = \frac{5}{10} = 0,5$, $\rho = -0,5$, and $x_1 = x_2 = 1$</td><td>Inequality : $\left[\sum_{i=1}^{2} \gamma_i\right] \left[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} \left[\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} < 0$</td><td>Link between $\lambda_1$ and $\lambda_2$ : $\gamma_2 = \frac{\lambda_2}{\lambda_1} \Leftrightarrow \lambda_2 = \gamma_2 \lambda_1$</td></tr><tr><td>$k = 0,5$</td><td>2<sup>0,5</sup>(0,7 + 0,2 × γ<sub>2</sub><sup>0,5</sup>) - 0,9γ<sub>2</sub> - 0,9 < 0</td><td>Equiproportionate variations (traditional case): $\lambda_2 = \gamma_2 \lambda_1$ with $\gamma_2 = 1$</td></tr><tr><td></td><td>Solutions : $\gamma_2 \in ]0,26028;+\infty[$</td><td>Unspecified proportions : $\lambda_2 = \gamma_2 \lambda_1$ with $\gamma_2 \neq 1$</td></tr><tr><td>$k = 0,7$</td><td>2<sup>0,7</sup>(0,7 + 0,2 × γ<sub>2</sub><sup>0,5</sup>)<sup>1,4</sup> - (0,9)<sup>1,4</sup>γ<sub>2</sub> - (0,9)<sup>1,4</sup> < 0</td><td>Equiproportionate variations (traditional case): $\lambda_2 = \gamma_2 \lambda_1$ with $\gamma_2 = 1$</td></tr><tr><td></td><td>Solutions : $\gamma_2 \in ]0,46663;+\infty[$</td><td>Unspecified proportions : $\lambda_2 = \gamma_2 \lambda_1$ with $\gamma_2 = 1$</td></tr></table>
$\lambda_2 = \gamma_2 \lambda_1$ with $\gamma_2 \neq 1$

Note : $\forall k \in ]0;1[, \gamma_2 = 1$ is the solution to the inequality :

$$ \left( \sum_{i=1}^{2} \gamma_i \right) \left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}} - 2 \lambda_1^{k-1} \left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} < 0. \text{ The solution } \gamma_2 = \frac{\lambda_2}{\lambda_1} = 1, \text{ i.e.} $$

$\lambda_1 = \lambda_2$, corresponds to the traditional case of decreasing returns to scale ($k < 1$) where all the factors are duplicated in the same proportion

#### 4.2.2.3.3. The decrease of gh-returns to scale with $k > 1$, and $\lambda_1$ fixed ($\lambda_1 = 2$ will be taken: any other value $\lambda_1 > 0$ could have been set)

With some values $k \in ]1;+\infty[$, the solutions $\gamma_2$ to the inequality

$$ \left( \sum_{i=1}^{2} \gamma_i \right) \left[ \sum_{i=1}^{2} \alpha_i x_i^{-\rho} \right]^{-\frac{k}{\rho}} - 2 \lambda_1^{k-1} \left[ \sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho}) \right]^{-\frac{k}{\rho}} < 0 \text{ are given in the following table:} $$

<table><caption>**Table 20 :** The decrease of gh-returns to scale with $k > 1$ and $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$.</caption><thead><tr><th>Some values of k ∈ ]1;+∞[ with α<sub>1</sub> = 5/10 = 0,5, α<sub>2</sub> = 5/10 = 0,5, ρ = -0,5, and x<sub>1</sub> = x<sub>2</sub> = 1</th><th>Inequality : <br> $\left(\sum_{i=1}^{2} \gamma_i\right) \left[\sum_{i=1}^{2} \alpha_i x_i^{-\rho}\right]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1} \left[\sum_{i=1}^{2} \gamma_i^{-\rho} (\alpha_i x_i^{-\rho})\right]^{-\frac{k}{\rho}} < 0$</th><th>Link between λ<sub>1</sub> and λ<sub>2</sub> : <br> γ<sub>2</sub> = λ<sub>2</sub>/λ<sub>1</sub> ⇔ λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub>, λ<sub>1</sub> = 2</th></tr></thead><tbody><tr><td>k = 2</td><td>2<sup>2</sup>(0,5 + 0,5×γ<sub>2</sub><sup>0,5</sup>)<sup>4</sup> - γ<sub>2</sub> - 1 < 0<br> Solutions : γ<sub>2</sub> ∈ ]0,24342;+∞[</td><td>Equiproportionate variations (traditional case):<br> λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub> with γ<sub>2</sub> = 1<br> Unspecified proportions :<br> λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub> with γ<sub>2</sub> ≠ 1</td></tr><tr><td>k = 8</td><td>2<sup>8</sup>(0,5 + 0,5×γ<sub>2</sub><sup>0,5</sup>)<sup>16</sup> - γ<sub>2</sub> - 1 < 0<br> Solutions : γ<sub>2</sub> ∈ ]0,18425;+∞[</td><td>Equiproportionate variations (traditional case):<br> λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub> with γ<sub>2</sub> = 1<br> Unspecified proportions :<br> λ<sub>2</sub> = γ<sub>2</sub>λ<sub>1</sub> with γ<sub>2</sub> ≠ 1</td></tr></tbody></table>
**Remark:** It thus appears that there are decreasing returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i = \frac{1}{2}\sum_{i=1}^{2}\gamma_i\lambda_i = \frac{\lambda_1}{2}\sum_{i=1}^{2}\gamma_i$ as a reference indicator, while at the same time homogeneity of degree of the C.E.S. production function is higher than the unit ($k > 1$).

**General conclusion (result) on the decrease of returns to scale:**

$\forall k > 0$, in other words, whatever the degree of homogeneity of the generalized C.E.S. production function $y = f(x_1, x_2) = A[\alpha_1 x_1^{-\rho} + \alpha_2 x_2^{-\rho}]^{-\frac{k}{\rho}}$, there are decreasing gh-returns to scale ($r(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} < 1$) if, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2}\sum_{i=1}^{2}\lambda_i = \frac{\lambda_1}{2}\sum_{i=1}^{2}\gamma_i$ as a reference indicator, the components of the relative vector of modulation $\hat{\theta} = (\gamma_1 = \frac{\lambda_1}{\lambda_1}, \gamma_2 = \frac{\lambda_2}{\lambda_1})$, generated by the numeraire proportion $\lambda_1$, are bound by the relation $(\sum_{i=1}^{2}\gamma_i)[\sum_{i=1}^{2}\alpha_i x_i^{-\rho}]^{-\frac{k}{\rho}} - 2\lambda_1^{k-1}[\sum_{i=1}^{2}\gamma_i^{-\rho}(\alpha_i x_i^{-\rho})]^{-\frac{k}{\rho}} < 0$.

### 4.3. Gh-returns to scale, geometric mean versus arithmetic mean as reference indicator !

This discussion could naturally have related to the arithmetic mean and any other average (mean); and also between two averages (reference indicators) others than the arithmetic mean and the geometric mean.

#### 4.3.1.: The relation between gh-returns to scale expressed in terms of geometric mean as a reference indicator and gh-returns to scale expressed in terms of arithmetic mean as a reference indicator
Let us suppose (relation 8) that $\lambda_u$ is the numeraire, in other words $\gamma_i = \frac{\lambda_i}{\lambda_u} > 0$, i.e.

$$ \lambda_i = \lambda_u \gamma_i, \forall i=1,2,...,i,...,n. $$

* The expression of the arithmetic mean is such that:

$\tilde{\lambda} = \sum_{i=1}^{n} f_i \lambda_i = \frac{1}{n} \sum_{i=1}^{n} \lambda_i = \frac{\lambda_u}{n} \sum_{i=1}^{n} \gamma_i$ with $f_i = \frac{1}{n}$ and $\sum_{i=1}^{n} f_i = 1$ (it is supposed, to simplify, that each proportion $\lambda_i$ has an identical weight (or the same importance) in the calculation of the reference indicator).

* The expression of the geometric mean is such that:

$$ \tilde{\lambda} = \sqrt[n]{\prod_{i=1}^{n} \lambda_i} = \sqrt[n]{\left(\lambda_i\right)^n} = \prod_{i=1}^{n} (\lambda_i)^{f_i} = \prod_{i=1}^{n} (\lambda_u \gamma_i)^{\frac{1}{n}} = \lambda_u \prod_{i=1}^{n} (\gamma_i)^{\frac{1}{n}}, \text{ with } f_i = \frac{1}{n} \text{ and } \sum_{i=1}^{n} f_i = 1. $$

Initially, we will highlight the connection (or the relation) which exists between gh-returns to scale expressed in terms of geometric mean, and gh-returns to scale expressed in terms of arithmetic mean. This relation will be established only for the Cobb-Douglas production function; a relation of the same type can be established, in a similar way, with regard to the C.E.S. production function.

Typically, the gh-returns to scale concept, for an unspecified reference indicator $\tilde{\lambda}$, is given by the expression:

$$ r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \frac{f(\lambda_1 x_1, \dots, \lambda_i x_i, \dots, \lambda_n x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} = \frac{f((\lambda_u \gamma_1) x_1, \dots, (\lambda_u \gamma_i) x_i, \dots, (\lambda_u \gamma_n) x_n)}{\tilde{\lambda} f(x_1, \dots, x_i, \dots, x_n)} $$

In the case of the generalized Cobb-Douglas production function

$y = f(x) = A \prod_{i=1}^{n} x_i^{\alpha_i}$, we know that the expression of gh-returns to scale is such that:

$$ r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} \tilde{\lambda}^{-1} \text{ (expression 14*)} $$
* If $\tilde{\lambda}$ is the arithmetic mean, i.e. if $\tilde{\lambda} = \sum_{i=1}^{n} f_i \lambda_i = \frac{1}{n} \sum_{i=1}^{n} \lambda_i = \frac{\lambda_u}{n} \sum_{i=1}^{n} \gamma_i$, then the expression of gh-returns to scale, noted $r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$, is such that:

$$r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = n \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) \left( \sum_{i=1}^{n} \gamma_i \right)^{-1} \lambda_u \left( \sum_{i=1}^{n} \alpha_i - 1 \right) \quad (14*a)$$

* If $\tilde{\lambda}$ is the geometric mean, i.e. if $\tilde{\lambda} = \lambda_u \prod_{i=1}^{n} (\gamma_i)^{\alpha_i}$, then the expression of gh-returns to scale (14*), noted $r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$, is such that:

$$r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) \left( \prod_{i=1}^{n} (\gamma_i)^{\frac{1}{n}} \right)^{-1} \lambda_u \left( \sum_{i=1}^{n} \alpha_i - 1 \right) \quad (14*b)$$

* The ratio between $r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ and $r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ is such that:

$$\frac{r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)}{r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)} = \frac{\left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) \left( \prod_{i=1}^{n} (\gamma_i)^{\frac{1}{n}} \right)^{-1} \lambda_u \left( \sum_{i=1}^{n} \alpha_i - 1 \right)}{n \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) \left( \sum_{i=1}^{n} \gamma_i \right)^{-1} \lambda_u \left( \sum_{i=1}^{n} \alpha_i - 1 \right)} = \frac{1}{n} \frac{\left( \sum_{i=1}^{n} \gamma_i \right)}{\left( \prod_{i=1}^{n} (\gamma_i)^{\frac{1}{n}} \right)} \\ \Leftrightarrow \\ r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \eta r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) \text{ with } \eta = \frac{1}{n} \frac{\left( \sum_{i=1}^{n} \gamma_i \right)}{\left( \prod_{i=1}^{n} (\gamma_i)^{\frac{1}{n}} \right)} > 0$$

* The relation $r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \eta r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ thus represents only a change of scale when one changes reference indicators, in the characterization of
the nature of gh-returns to scale. All occurs as if instead of evaluating the
temperature in degrees Celsius, one preferred to evaluate it in Fahrenheit, or
Kelvin, or Rankine, or Réaumur, etc.

* Hence the following equivalence, for the characterization of the nature of gh-returns to scale:

$$
\begin{equation}
\left\{
\begin{aligned}
r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) &= \eta r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = 1 \\
r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) &= \eta r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) > 1 \\
r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) &= \eta r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) < 1
\end{aligned}
\right.
\iff
\left\{
\begin{aligned}
r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) &= \frac{1}{\eta} \\
r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) &> \frac{1}{\eta} \\
r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) &< \frac{1}{\eta}
\end{aligned}
\right.
\end{equation}
$$

* Thus, when one retains the geometric mean as a reference indicator, the comparison of $r_g(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ to the unit and the comparison of $r_a(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n)$ to a proportion of the unit, i.e. to the scalar $\frac{1}{\eta}$, lead to the same conclusions as for the nature of gh-returns to scale.

* A change of reference indicator in the definition of the gh-returns to scale concept anything else represents only one change of measurement, which thus does not deteriorate in any way nature or classification of gh-returns to scale. And it was to be anticipated in light of the expression of gh-returns to scale

$$r(\tilde{\lambda}, x_1, \dots, x_i, \dots, x_n) = \left( \prod_{i=1}^{n} (\gamma_i)^{\alpha_i} \right) (\lambda_u)^{\sum_{i=1}^{n} \alpha_i} \tilde{\lambda}^{-1} \quad \text{which to some extent confers, a dumb character to the reference indicator } \tilde{\lambda}.$$
### 4.3.2.: Numerical examples of the relation between gh-returns to scale ($r_g$) calculated with the geometric mean and gh-returns to scale ($r_a$) calculated with the arithmetic mean in the case of the Cobb-Douglas production function

#### 4.3.2.1.: The relation between gh-returns to scale $r_g$, and gh-returns to scale $r_a$ in the case of constant gh-returns to scale

##### 4.3.2.1.1.: The gh-returns to scale $r_a$ calculated with the arithmetic mean

Let us take to simplify, the last line of table 3, concerning the constant gh-returns to scale for a Cobb-Douglas production function, and the arithmetic mean

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_{i} = \frac{\lambda_{1}}{2}(1 + \gamma_{2}) $$
as a reference indicator.

<caption>Table 3: The constancy of gh-returns to scale for a Cobb-Douglas production function</caption>

$$ y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}, \alpha_1 + \alpha_2 = 1, \text{ with the arithmetic mean } \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i $$

<table><thead><tr><th>Values of α₂ ∈ ]0,1[</th><th>Equation : 2γ₂<sup>α₂</sup> - γ₂ - 1 = 0</th><th>Link between λ₁ and λ₂ :<br>γ₂ = λ₂/λ₁ ⇔ λ₂ = γ₂λ₁, ∀λ₁</th></tr></thead><tbody><tr><td>α₂ = 9/10 ⇒ α₁ = 1/10</td><td>9/(2γ₂<sup>10</sup> - γ₂ - 1 = 0)</td><td>Equiproportionate variations<br>(traditional case) :<br>γ₂ = 1 ⇒ λ₂ = λ₁<br>Unspecified proportions :<br>γ₂ ≈ 1013,955 ⇒ λ₂ ≈ 1013,955λ₁</td></tr></tbody></table>

* Case 1: With the solution $γ₂ = 1$, that is to say $λ₂ = λ₁$ (the traditional case), we showed that the production function $y = f(x₁, x₂) = x₁^{0.9} x₂^{0.1}$ exhibited constant gh-returns to scale, with the arithmetic mean as a reference indicator

$$ \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_{i} = \lambda_{1} = \lambda_{2} = \lambda $$
since:
$$r_a(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.1}}{\lambda (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1} (x_1^{0.9} x_2^{0.1})}{\lambda (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1}}{\lambda} = \frac{\lambda}{\lambda} = 1$$

* Case 2: In the same way, with the solution $\gamma_2 \approx 1013,955$, that is to say $\lambda_2 \approx 1013,955\lambda_1$ (unspecified proportions), we showed that the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.1}$ exhibited constant gh-returns to scale, with $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^2 \lambda_i = \frac{\lambda_1}{2}(1+\gamma_2) \approx \frac{\lambda_1}{2}(1+1013,955) = \lambda_1 \frac{1014,955}{2} = 507,4775\lambda_1$ since:

$$r_a(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.1}}{507,4775 \lambda_1 (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1} (x_1^{0.9} x_2^{0.1})}{507,4775 \lambda_1 (x_1^{0.9} x_2^{0.1})} = \frac{\lambda_1^{0.9} \lambda_2^{0.1}}{507,4775 \lambda_1} \\ = \frac{\lambda_1^{0.9} (1013,955 \lambda_1)^{0.1}}{507,4775 \lambda_1} = \frac{\lambda_1 (1013,955)^{0.1}}{507,4775 \lambda_1} = \frac{(1013,955)^{0.1}}{507,4775} = \frac{507,4775}{507,4775} = 1$$

#### 4.3.2.1.2.: The gh-returns to scale $r_g$ calculated with the geometric mean

The geometric mean $\tilde{\lambda} = \prod_{i=1}^{2} (\lambda_i)^{\frac{1}{2}}$ as a reference indicator, with $\lambda_1$ being the numeraire proportion, is such as : $\tilde{\lambda} = \prod_{i=1}^{2} (\gamma_i \lambda_1)^{\frac{1}{2}} = \lambda_1 (\gamma_2)^{\frac{1}{2}}$.

Thus the expression gh-returns to scale (relation 14*) becomes as follows :

$$r(\tilde{\lambda}, x_1, x_2) = \left( \prod_{i=1}^{2} (\gamma_i)^{(\alpha_i - \frac{1}{2})} \right) (\lambda_1)^{\sum \alpha_i - 1} = \gamma_1 (\alpha_1 - 0.5) \times \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{\sum \alpha_i - 1} = \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{\sum \alpha_i - 1}$$

The condition characterizing constant gh-returns to scale is thus such as :

$$r(\tilde{\lambda}, x_1, x_2) = \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{\frac{2}{\sum \alpha_i - 1}} = 1.$$
The solutions $\gamma_2$ to the equation $\gamma_2^{(\alpha_2-0,5)}(\lambda_1)^{i=1} -1=0$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<caption>**Table 3\* : The constancy of gh-returns to scale for a Cobb-Douglas production function**</caption>

$$y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}, \text{ with } \alpha_1 + \alpha_2 = 1 \text{ and the geometric mean } \tilde{\lambda} = \prod_{i=1}^{2} (\lambda_i)^{\frac{1}{\alpha_i}}$$

<table><thead><tr><th>Values of<br>α₂ ∈ ]0,1[<br>with α₁ + α₂ = 1</th><th>Equation :<br>γ₂(α₂-0,5)(λ₁)<sup>i=1</sup> -1 = 0<br>⇔<br>γ₂(α₂-0,5)-1 = 0</th><th>Link between λ₁ and λ₂ :<br>γ₂ = λ₂/λ₁ ⇔ λ₂ = γ₂λ₁, ∀λ₁</th></tr></thead><tbody><tr><td>α₁ = α₂ = 0,5</td><td>Solutions :<br>∀γ₂ = k ∈ ℝ, and solutions adopted<br>are: ∀γ₂ = k ∈ ℝ*, i.e. γ₂ > 0</td><td>Equiproportionate variations<br>(traditional case) :<br>λ₂ = γ₂λ₁ = λ₁ with γ₂ = 1<br>Unspecified proportions :<br>λ₂ = γ₂λ₁ for γ₂ ≠ 1</td></tr><tr><td>α₂ = 0,9</td><td>Solution :<br>γ₂(α₂-0,5)-1 = 0 ⇔ γ₂(0,9-0,5)-1 = 0<br>⇔ γ₂<sup>0,4</sup> -1 = 0 ⇔ γ₂ = 1</td><td>Equiproportionate variations<br>(traditional case) :<br>γ₂ = 1 ⇒ λ₂ = λ₁</td></tr></tbody><tfoot><tr><td colspan="3">Note : ∀α₂ ∈ ]0,1[ with α₁ + α₂ = 1, γ₂ = 1 is the solution to the equation<br>γ₂(α₂-0,5)(λ₁)<sup>i=1</sup> -1 = 0.</td></tr></tfoot></table>

* With the solution $\gamma_2=1$, that is to say $\lambda_2 = \lambda_1$ (i.e. the traditional case), the production function $y = f(x_1, x_2) = x_1^{0,9} x_2^{0,1}$ is said to exhibit constant gh-returns to scale, with geometric mean $\tilde{\lambda} = \lambda_1 \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{\alpha_i}} = \lambda_1 (\gamma_2)^{\frac{1}{\alpha_1}} = \lambda_1 = \lambda_2 = \lambda$ since:

$$r_g(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0,9} (\lambda_2 x_2)^{0,1}}{\lambda (x_1^{0,9} x_2^{0,1})} = \frac{\lambda_1^{0,9} \lambda_2^{0,1} (x_1^{0,9} x_2^{0,1})}{\lambda (x_1^{0,9} x_2^{0,1})} = \frac{\lambda_1^{0,9} \lambda_2^{0,1}}{\lambda} = \frac{\lambda}{\lambda} = 1$$
#### 4.3.2.1.3.: The relation between the gh-returns to scale $r_g$, and the gh-returns to scale $r_a$ in the case of constant gh-returns to scale

The relation $r_g(\tilde{\lambda}, x_1, x_2) = \eta r_a(\tilde{\lambda}, x_1, x_2)$ with $\eta = \frac{\sum_{i=1}^{2} \gamma_i}{\prod_{i=1}^{2} (\gamma_i)^2} > 0$ is verified

since for $\gamma_2 = 1$, one has:

$$ \left\{ \eta = \frac{\sum_{i=1}^{2} \gamma_i}{\prod_{i=1}^{2} (\gamma_i)^2} = \frac{\gamma_1 + \gamma_2}{(\gamma_1)^2 \times (\gamma_2)^2} = \frac{2}{2} = 1; \text{ and } \begin{array}{l} r_g(\tilde{\lambda}, x_1, x_2) = 1 \\ r_a(\tilde{\lambda}, x_1, x_2) = 1 \end{array} \right\} \Rightarrow \frac{r_g(\tilde{\lambda}, x_1, x_2)}{r_a(\tilde{\lambda}, x_1, x_2)} = 1 $$

#### 4.3.2.2.: The relation between gh-returns to scale $r_g$, and gh-returns to scale $r_a$ in the case of increasing returns to scale

#### 4.3.2.2.1.: The gh-returns to scale $r_a$ calculated with the arithmetic mean

Let us take, to simplify, the first line and the penultimate line of table 8, concerning the increasing gh-returns to scale for a Cobb-Douglas production function, and the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1 + \gamma_2)$ as the reference indicator.

<table><caption>**Table 8 :** The increasing gh-returns to scale for a Cobb-Douglas production function</caption><tr><td>y = f(x<sub>1</sub>, x<sub>2</sub>) = x<sub>1</sub><sup>α<sub>1</sub></sup> x<sub>2</sub><sup>α<sub>2</sub></sup>, with α<sub>1</sub> + α<sub>2</sub> > 1, and λ̃ = <sup>1</sup>/<sub>2</sub> &sum;<sub>i=1</sub><sup>2</sup> λ<sub>i</sub></td><td>Some values of α<sub>1</sub></td><td>Inequality :</td><td>Link between λ<sub>1</sub> and λ<sub>2</sub>:</td></tr></table>
### 4.3.2.2.2.: The gh-returns to scale $r_g$ calculated with the geometric mean

We know (see paragraph 4.3.2.1.2.), that with the geometric mean
$$ \tilde{\lambda} = \prod_{i=1}^{2} (\gamma_i \lambda_1)^{\frac{1}{\gamma_i}} = \lambda_1 (\gamma_2)^{\frac{1}{\gamma_2}}, $$ the expression of gh-returns to scale is such as :

$$ r(\tilde{\lambda}, x_1, x_2) = \left( \prod_{i=1}^{2} (\gamma_i)^{(\alpha_i - \frac{1}{2})} \right) (\lambda_1)^{\sum_{i=1}^{2} \alpha_i - 1} = \gamma_2^{(\alpha_2 - 0.5)} (\lambda_1)^{\sum_{i=1}^{2} \alpha_i - 1}. $$

The condition characterizing the increasing gh-returns to scale is thus such as :

$$ r(\tilde{\lambda}, x_1, x_2) = \gamma_2^{(\alpha_2 - 0.5)} (\lambda_1)^{i=1} > 1. $$

The solutions $\gamma_2$ to the inequality $\gamma_2^{(\alpha_2 - 0.5)}(\lambda_1)^{i=1} > 1$, for some values $\alpha_2 \in ]0,1[$, are given in the following table :

<caption>**Table 8\* : The increasing of gh-returns to scale for a Cobb-Douglas production function**</caption>

$$ y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}, \text{ with } \alpha_1 + \alpha_2 > 1 \text{ and the geometric mean } \tilde{\lambda} = \prod_{i=1}^{2} (\gamma_i \lambda_1)^{\frac{1}{\gamma_i}} $$

<table><tr><td rowspan="2">Some values of α₁ and α₂</td><td>Inequality :<br>γ₂<sup>(α₂-0.5)</sup>(λ₁)<sup>i=1</sup> &gt; 1<br>1 - Σαᵢ<br>i=1<br>(α₂-0.5)</td><td>Link between λ₁ and λ₂ :<br>γ₂ = λ₂/λ₁ &harr; λ₂ = γ₂λ₁, λ₁ = 2</td></tr><tr><td>Solutions : ∀γ₂ &gt; (λ₁)<br>2<sup>0.6</sup>(γ₂)<sup>0.2</sup> - 1 &gt; 0</td><td>Unspecified proportions:<br>λ₂ = γ₂λ₁ with γ₂ ≠ 1<br>Equiproportionate variations (traditional case):<br>λ₂ = γ₂λ₁ = λ₁ with γ₂ = 1</td></tr><tr><td>α₁ = 9/10 = 0.9 and<br>α₂ = 7/10 = 0.7</td><td>Solutions :<br>γ₂ ∈ [1/2³; +∞] = ]0,125; +∞[</td><td></td></tr><tr><td colspan="3">Note : ∀α₁, α₂ ∈ ]0,1[ with α₁ + α₂ > 1, γ₂ = 1 is the solution to the inequality</td></tr></table>
$$ \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{i=1} \sum_{i=1}^{2} \alpha_i > 1. $$

Let us consider the penultimate line of this table.

* Case : $\gamma_2 \in [\frac{1}{2^3}; +\infty[$ :

* With the solution $\gamma_2 = 1$, that is to say $\lambda_2 = \lambda_1$ (the traditional case), the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.1}$ is said to exhibit increased gh-returns to scale, with the geometric mean $\tilde{\lambda} = \lambda_1 \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}} = \lambda_1 (\gamma_2)^{\frac{1}{2}} = \lambda_1 = \lambda_2 = \lambda$ since:

$$ r_g(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda x_1)^{0.9} (\lambda x_2)^{0.7}}{\lambda (x_1^{0.9} x_2^{0.7})} = \frac{\lambda^{0.9} \lambda^{0.7} (x_1^{0.9} x_2^{0.7})}{\lambda (x_1^{0.9} x_2^{0.7})} = \lambda^{0.6} = 2^{0.6} \approx 1{,}1557 > 1 $$

* In the same way with the solution $\gamma_2 = \frac{1}{2} \Rightarrow \lambda_2 = \frac{1}{2} \lambda_1$ (unspecified variations), the production function $y = f(x_1, x_2) = x_1^{0.9} x_2^{0.7}$ is said to exhibit increased gh-returns to scale, with the geometric mean $\tilde{\lambda} = \lambda_1 \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}} = \lambda_1 (\gamma_2)^{\frac{1}{2}} = \lambda_1 (\frac{1}{2})^{\frac{1}{2}} = \frac{\lambda_1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \lambda_1$ since:

$$ \begin{align*} r_g(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.9} (\lambda_2 x_2)^{0.7}}{\frac{\sqrt{2}}{2} \lambda_1 (x_1^{0.9} x_2^{0.7})} = \frac{\lambda_1^{0.9} \lambda_2^{0.7} (x_1^{0.9} x_2^{0.7})}{\frac{\sqrt{2}}{2} \lambda_1 (x_1^{0.9} x_2^{0.7})} = \frac{\lambda_1^{0.9} (\frac{1}{2} \lambda_1)^{0.7}}{\frac{\sqrt{2}}{2} \lambda_1} = \lambda_1^{0.6} \frac{(\frac{1}{2})^{0.7}}{\frac{\sqrt{2}}{2}} \\ &= \lambda_1^{0.6} \frac{1}{2^{0.7}} \times \frac{2}{\sqrt{2}} = 2^{0.6} \frac{1}{2^{0.7}} \times \frac{2}{\sqrt{2}} = \frac{2^{0.9}}{\sqrt{2}} \approx 1{,}3195 > 1 \end{align*} $$

#### 4.3.2.2.3.: The relation between gh-returns to scale $r_g$, and gh-returns to scale $r_a$ in the case of increasing returns to scale
The relation $r_g(\tilde{\lambda}, x_1, x_2) = \eta r_a(\tilde{\lambda}, x_1, x_2)$ with $\eta = \frac{1}{2} \frac{\left(\sum_{i=1}^{2} \gamma_i\right)}{\left(\prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}}\right)} > 0$ is verified

since, for $\gamma_2 = \frac{1}{2}$, one has:

$$ \left\{ \begin{array}{l} \eta = \frac{1}{2} \frac{\left( \sum_{i=1}^{2} \gamma_i \right)}{\left( \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}} \right)} = \frac{1}{2} \frac{\gamma_1 + \gamma_2}{(\gamma_1)^{0,5} \times (\gamma_2)^{0,5}} = \frac{1}{2} \frac{1+0,5}{(1)^{0,5} \times (0,5)^{0,5}} = \frac{1,5}{2 \times \frac{\sqrt{2}}{2}} = \frac{3}{2\sqrt{2}} \approx 1,06066 \\ \\ r_g(\tilde{\lambda}, x_1, x_2) = \frac{2^{0,9}}{\sqrt{2}} \approx 1,3195 \\ \\ r_a(\tilde{\lambda}, x_1, x_2) = \frac{2^{1,9}}{3} \approx 1,244 \end{array} \right\} \Rightarrow \frac{r_g(\tilde{\lambda}, x_1, x_2)}{r_a(\tilde{\lambda}, x_1, x_2)} = \frac{2^{0,9}}{\sqrt{2}} \times \frac{3}{2^{1,9}} = \frac{3}{2\sqrt{2}} \approx 1,06066 $$

One can check that with the previous solution $\gamma_2 = 1$, the relation $r_g(\tilde{\lambda}, x_1, x_2) = \eta r_a(\tilde{\lambda}, x_1, x_2)$ is verified since $r_g(\tilde{\lambda}, x_1, x_2) = r_a(\tilde{\lambda}, x_1, x_2) = 1,1557$ and $\eta = 1$.

### 4.3.2.3.: The relation between gh-returns to scale $r_g$, and gh-returns to scale $r_a$ in the case of decreasing returns to scale

#### 4.3.2.3.1.: The gh-returns to scale $r_g$ calculated with the arithmetic mean

Let us consider table 10 again, concerning the decreasing gh-returns to scale for a Cobb-Douglas production function, and the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1+\gamma_2)$ as the reference indicator.
<caption>Table 10: Decreasing gh-returns to scale for a Cobb-Douglas production function</caption>

$$y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}, \text{ with } \alpha_1 + \alpha_2 < 1, \text{ and } \tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i$$

<table><thead><tr><th>Some values of<br>α₁ and α₂</th><th>Inequality :</th><th>Link between λ₁ and λ₂ :</th></tr></thead><tbody><tr><td rowspan="2">α₁ = 2/10 and<br>α₂ = 7/10</td><td>2(λ₁)<sup>2</sup> / (Σαᵢ - 1) + (γ₂)α₂ - γ₂ - 1 < 0</td><td>γ₂ = λ₂ / λ₁ ⇔ λ₂ = γ₂λ₁, λ₁ = 2</td></tr><tr><td>2<sup>0.9</sup>(γ₂)<sup>0.7</sup> - γ₂ - 1 < 0<br>Solutions: γ₂ ∈ ]0;1,654[ ∪ ]3,347;+∞[</td><td>Unspecified proportions :</td></tr><tr><td></td><td></td><td>λ₂ = γ₂λ₁ with γ₂ ≠ 1<br>Equiproportionate variations<br>(traditional case):<br>λ₂ = γ₂λ₁ = λ₁ with γ₂ = 1</td></tr></tbody></table>

* Case : $γ_2 \in ]0;1,654[$ :

* With the solution $γ_2 = 1$, that is to say $λ_2 = λ_1$ (the traditional case), we showed that the production function $y = f(x_1, x_2) = x_1^{0.2} x_2^{0.7}$ exhibited decreased gh-returns to scale, with the arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \lambda_1 = \lambda_2 = \lambda$, since:

$$r_a(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda x_1)^{0.2} (\lambda x_2)^{0.7}}{\lambda (x_1^{0.2} x_2^{0.7})} = \frac{\lambda^{0.2} \lambda^{0.7} (x_1^{0.2} x_2^{0.7})}{\lambda (x_1^{0.2} x_2^{0.7})} = \lambda^{-0.1} = 2^{-0.1} \approx 0.933 < 1$$

* With the solution $γ_2 = 1,25$, that is to say $γ_2 = 1,25 \Rightarrow λ_2 = 1,25λ_1$ (unspecified variations), we showed that the production function $y = f(x_1, x_2) = x_1^{0.2} x_2^{0.7}$ exhibited decreased gh-returns to scale, with arithmetic mean $\tilde{\lambda} = \frac{1}{2} \sum_{i=1}^{2} \lambda_i = \frac{\lambda_1}{2}(1+\gamma_2) = \frac{\lambda_1}{2}(1+1,25) = 1,125\lambda_1$, since:

$$\begin{aligned} r_a(\tilde{\lambda}, x_1, x_2) &= \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.2} (\lambda_2 x_2)^{0.7}}{1,125 \lambda_1 (x_1^{0.2} x_2^{0.7})} = \frac{\lambda_1^{0.2} \lambda_2^{0.7} (x_1^{0.2} x_2^{0.7})}{1,125 \lambda_1 (x_1^{0.2} x_2^{0.7})} = \frac{\lambda_1^{0.2} \lambda_2^{0.7}}{1,125 \lambda_1} \\ &= \frac{\lambda_1^{0.2} (1,25 \lambda_1)^{0.7}}{1,125 \lambda_1} = \frac{1,25^{0.7} \lambda_1^{0.9}}{1,125 \lambda_1} = \frac{1,25^{0.7} \lambda_1^{-0.1}}{1,125} = \frac{1,25^{0.7}}{1,125 \times 2^{0.1}} \approx 0,969575 < 1 \end{aligned}$$
#### 4.3.2.3.2.: The gh-returns to scale $r_g$ calculated with the geometric mean

We know (see paragraph 4.3.2.1.2.), that with the geometric mean

$$ \tilde{\lambda} = \prod_{i=1}^{2} (\gamma_i \lambda_1)^{\frac{1}{2}} = \lambda_1 (\gamma_2)^{\frac{1}{2}}, \text{ the expression of gh-returns to scale is such as :} $$

$$ r(\tilde{\lambda}, x_1, x_2) = \left( \prod_{i=1}^{2} (\gamma_i)^{(\alpha_i - \frac{1}{2})} \right) (\lambda_1)^{\frac{2}{\sum \alpha_i - 1}} = \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{\frac{2}{\sum \alpha_i - 1}}. $$

The condition characterizing the decreasing gh-returns to scale is thus such as :

$$ r(\tilde{\lambda}, x_1, x_2) = \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{\frac{2}{\sum \alpha_i - 1}} < 1. $$

The solutions $\gamma_2$ to the inequality $\gamma_2(\alpha_2 - 0.5)(\lambda_1)^{\frac{2}{\sum \alpha_i - 1}} < 1$, for some values $\alpha_2 \in [0,1[$, are given in the following table :

<caption>**Table 10\* : The decrease of gh-returns to scale for a Cobb-Douglas production function**</caption>

$$ y = f(x_1, x_2) = x_1^{\alpha_1} x_2^{\alpha_2}, \text{ with } \alpha_1 + \alpha_2 < 1 \text{ and the geometric mean } \tilde{\lambda} = \prod_{i=1}^{2} (\gamma_i \lambda_1)^{\frac{1}{2}} $$

<table><tr><td rowspan="2">Some values of α₁ and α₂</td><td>Inequality :</td><td>Link between λ₁ and λ₂ :</td></tr><tr><td>γ₂(α₂-0.5)(λ₁)<sup>i=1</sup> &lt; 1<br>1 - Σαᵢ<br>(α₂-0.5)<sup>i=1</sup></td><td>γ₂ = λ₂/λ₁ &harr; λ₂ = γ₂λ₁, λ₁ = 2</td></tr><tr><td>α₁ = 2/10 and<br>α₂ = 7/10</td><td>Solutions : γ₂ &lt; (λ₁)<sup>2/Σαᵢ - 1</sup><br>2⁻⁰.₁(γ₂)⁰.² - 1 &lt; 0<br>Solutions : γ₂ ∈ [0; √2] ≈ [0; 1.414]</td><td>Unspecified proportions :</td></tr><tr><td colspan="3">λ₂ = γ₂λ₁ with γ₂ ≠ 1<br>Equiproportionate variations (traditional case):<br>λ₂ = γ₂λ₁ = λ₁ with γ₂ = 1</td></tr></table>

Note : ∀α₁, α₂ ∈ [0,1[ with α₁ + α₂ < 1, γ₂ = 1 is the solution to the inequality
$$ \gamma_2 (\alpha_2 - 0.5) (\lambda_1)^{i=1}^{\sum \alpha_i - 1} < 1. $$

Let us consider the penultimate line of this table.

* Case : $\gamma_2 \in [0; \sqrt{2}]$ :

* With the solution $\gamma_2 = 1$, that is to say $\lambda_2 = \lambda_1$ (the traditional case), the production function $y = f(x_1, x_2) = x_1^{0.2} x_2^{0.7}$ is said to exhibit decreased gh-returns to scale, with the geometric mean $\tilde{\lambda} = \lambda_1 \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}} = \lambda_1 (\gamma_2)^{\frac{1}{2}} = \lambda_1 = \lambda_2 = \lambda$ since:

$$ r_g(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda x_1)^{0.2} (\lambda x_2)^{0.7}}{\lambda (x_1^{0.2} x_2^{0.7})} = \frac{\lambda^{0.2} \lambda^{0.7} (x_1^{0.2} x_2^{0.7})}{\lambda (x_1^{0.2} x_2^{0.7})} = \lambda^{-0.1} = 2^{-0.1} \approx 0.933 < 1 $$

* With the solution $\gamma_2 = 1,25$, that is to say $\gamma_2 = 1,25 \Rightarrow \lambda_2 = 1,25\lambda_1$ (unspecified variations), the production function $y = f(x_1, x_2) = x_1^{0.2} x_2^{0.7}$ is said to exhibit decreased gh-returns to scale, with the geometric mean $\tilde{\lambda} = \lambda_1 \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}} = \lambda_1 (\gamma_2)^{\frac{1}{2}} = \lambda_1 (1,25)^{\frac{1}{2}} = \lambda_1 \sqrt{1,25}$ since:

$$ r_g(\tilde{\lambda}, x_1, x_2) = \frac{f(\lambda_1 x_1, \lambda_2 x_2)}{\tilde{\lambda} f(x_1, x_2)} = \frac{(\lambda_1 x_1)^{0.2} (\lambda_2 x_2)^{0.7}}{\lambda_1 \sqrt{1,25} (x_1^{0.2} x_2^{0.7})} = \frac{\lambda_1^{0.2} \lambda_2^{0.7} (x_1^{0.2} x_2^{0.7})}{\lambda_1 \sqrt{1,25} (x_1^{0.2} x_2^{0.7})} = \frac{\lambda_1^{0.2} (1,25 \lambda_1)^{0.7}}{\lambda_1 \sqrt{1,25}} \\ = \lambda_1^{-0.1} \frac{(1,25)^{0.7}}{\sqrt{1,25}} = 2^{-0.1} \frac{(1,25)^{0.7}}{\sqrt{1,25}} = \frac{(1,25)^{0.7}}{2^{0.1} \times \sqrt{1,25}} \approx 0,9756 < 1 $$

#### 4.3.2.3.3.: The relation between gh-returns to scale $r_g$, and gh-returns to scale $r_a$ in the case of decreasing returns to scale
The relation $r_g(\tilde{\lambda}, x_1, x_2) = \eta r_a(\tilde{\lambda}, x_1, x_2)$ with $\eta = \frac{1}{2} \frac{\left(\sum_{i=1}^{2} \gamma_i\right)}{\left(\prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}}\right)} > 0$ is verified

since, for $\gamma_2 = 1,25$, one has:

$$ 
\left\{ 
\begin{aligned}
\eta &= \frac{1}{2} \frac{\left( \sum_{i=1}^{2} \gamma_i \right)}{\left( \prod_{i=1}^{2} (\gamma_i)^{\frac{1}{2}} \right)} = \frac{1}{2} \frac{\gamma_1 + \gamma_2}{(\gamma_1)^{0,5} \times (\gamma_2)^{0,5}} = \frac{1}{2} \frac{1+1,25}{(1)^{0,5} \times (1,25)^{0,5}} = \frac{2,25}{2\sqrt{1,25}} = \frac{1,125}{\sqrt{1,25}} \\
r_g(\tilde{\lambda}, x_1, x_2) &= \frac{(1,25)^{0,7}}{2^{0,1} \times \sqrt{1,25}} \approx 0,9756 \\
r_a(\tilde{\lambda}, x_1, x_2) &= \frac{1,25^{0,7}}{1,125 \times 2^{0,1}} \approx 0,969575
\end{aligned}
\right\} \Rightarrow \frac{r_g(\tilde{\lambda}, x_1, x_2)}{r_a(\tilde{\lambda}, x_1, x_2)} = \frac{(1,25)^{0,7}}{2^{0,1} \times \sqrt{1,25}} \times \frac{1,125 \times 2^{0,1}}{1,25^{0,7}} = \frac{1,125}{\sqrt{1,25}}
$$

One can check that with the previous solution $\gamma_2 = 1$, the relation $r_g(\tilde{\lambda}, x_1, x_2) = \eta r_a(\tilde{\lambda}, x_1, x_2)$ is verified since $r_g(\tilde{\lambda}, x_1, x_2) = r_a(\tilde{\lambda}, x_1, x_2) = 0,933$ and $\eta = 1$.

**Concluding remarks about the change of reference indicator:** The relation between the gh-returns to scale $r_g$, and the gh-returns to scale $r_a$ in the three cases of constant, increasing and decreasing gh-returns to scale, shows well that a change of indicator, only represented a change of scale, $(\frac{r_g(\tilde{\lambda}, x_1, x_2)}{r_a(\tilde{\lambda}, x_1, x_2)} = \eta > 0)$, in the characterization of the nature of returns to scale; what thus counts ultimately is the choice of a reference mark (arithmetic mean, geometric mean or any other reference indicator), and which this choice is carried out, to make as much may be, in manner reasoned so that such a choice can be the object of a broad consensus, if need be was!
V. Conclusion

The gh-returns to scale concept seems more general and better explains how flexible
the company could become when it wishes to modify its size. The restriction of the
company's change of size by means of a simple homotheticity of its input vector seems very
far from reality. One should rather consider the modification of the company's size by means
of a *diagonal transformation* as it is proposed in this analysis; homothetic *transformations*
being only one particular (typical) case of the *diagonal transformations*.

In this new formulation of returns to scale (gh-returns to scale), the reference indicator
played a central role in the characterization of the nature of returns to scale and in the
definition of the generalized homogeneous production function.

In the traditional definition of returns to scale (when all the factors necessary vary in
the same proportion), the reference indicator, whose existence not mentioned (or rather whose
existence is mentioned in a very implicit way), is actually only the average (arithmetic,
quadratic, harmonic, geometric, etc.) of these identical proportions. In the case of diagonal
transformations (where all the factors necessary do not vary in the same proportion) the need
for an explicit evocation of the reference indicator (whatever it is) is essential.

A change of reference indicator in the definition of the gh-returns to scale concept
represents only one scaling without a change in origin, in the characterization of the nature of
returns to scale. All occurs as if instead of evaluating the temperature in degrees Celsius, one
preferred to evaluate it in Fahrenheit, or Kelvin, or Rankine, or Reaumur, etc.

We will point out the fact that the question of the choice of a reference indicator rather
than another is a difficult question in so far as one is directly challenged by the problem of
choosing a selection criterion for the reference indicators, allowing to rule out certain
reference indicators. The selection of a better reference indicator leads to the question of
choice of the properties which a reference indicator must take on in order to decide between
these various reference indicators. In so far as the questions of standards (or good properties) hardly find a universal echo, we leave aside this question of selection of the best reference indicator, by considering that it is solved in an exogenous way, in other words that the reference indicator is imposed.

In this work, homogeneity of degree of a (production) function no longer seems a central element in the characterization of the nature of gh-returns to scale. The conditions (or the rules) describing constancy, increase or decrease of gh-returns to scale, in the case of the Cobb-Douglas production function and of the C.E.S. production function, highlight the fact that the constancy, increase and decrease of returns to scale can completely be evoked whatever the homogeneity of degree of such functions.

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