Diana Andrada

Cyrille Piatecki

An overview on the non-newtonian calculus and its potential applications to economics

Keywords: MR Subject Classification: 91B02, 91B62. Keywords: non-newtonian calculus, growth, geometric calculus, bigeometric calculus. Digital Object Identifier(

Until now, non-newtonian calculus, multiplicative calculus in particular, has been presented as a curiosity and is nearly ignored for the social scientists field. In this paper, after a brief presentation of this calculus, we try to show how it could be used to re-explore from another perspective classical economic theory, more particularly the economic growth and in the maximum likelihood method from statistics.

§1 Introduction

The development of the occidental science has been an incredible success through the introduction of the newtonian calculus. One could say that until the XXth century when discontinuous methods became available, without the derivative nothing of great could have been accomplished.

In the social science, for a very long time, the derivative has been the center of nearly all the analysis. But in an unexpected way, it appears that we can develop another approach which seems to threat more realistically growth phenomenon which are involved in the model of economic growth.

The change of paradigm is to consider that the variations are more naturally taken into account if the deviations are measured by ratios instead of differences. But even if Galileo has discussed briefly such an opportunity, it is not until 1972 that Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] imagine a non-newtonian calculus.

In the last twenty years, a number of reviews of the non-Newtonian calculus have appeared, scattered in various journals, primarily of pedagogical scope.

We think that the works of Bashirov Bashirov et al. [START_REF] Bashirov | Multiplicative calculus and its applications[END_REF] and Özyapici et al. [START_REF] Özyapici | On multiplicative and Volterra minimization methods[END_REF] has open a new era where non-newtonian calculus, from a mere curiosity, will become an operational calculus. We have shown elsewhere Filip and Piatecki [START_REF] Filip | A non-newtonian examination of the theory of exogenous economic growth[END_REF] how calculus has been a locked-in and how Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF], surfing on Volterra [START_REF] Volterra | Sui fondamenti della teoria delle equazioni differenziali lineari[END_REF], has begun to unlock it. There have been two main reasons to this state of fact: the first one is that non-mathematician researchers are very conservative in what concern the mathematical instruments they use and don't understand why they should explore other kinds of mathematics, since the classical methods are well fited for what they expect. The second one is that, in the particular case of the non-newtonian calculus, as there is a gateway between calculus and non-newtonian calculus, they doesn't find any reason to adopt another type of calculus.

Nevertheless, inside any social science modelization, there is always an equilibrium problem, based on a balance equation. Since the end of the 6 th century, following Luca Paccioli, book accounting are universally based on an additive algebra. But, as Ellerman [START_REF] Ellerman | Double-entry accounting : The mathematical formulation and generalization[END_REF] has clearly proved, one could also defined a multiplicative bookkeeping. Now, if in the standard accounting system resources are kept through positive numbers and deficit through negative ones, giving to zero the balance role in an additive system, in a multiplicative bookkeeping, the balance is given by one, resources by numbers greater than one, deficit by numbers lower than one, in a multiplicative system.

So, if one adopts a multiplicative bookkeeping system, multiplicative calculus becomes the only way to coherently manage models, since in multiplicative calculus, as in multiplicative accounting system, one plays the role devoted to zero in the standard system. §2 Glimpses on the non-newtonian calculus 2.1 Definition of the ⋆ derivative Definition 2.1. Let f : R → R + be a positive function. The ⋆ derivative of the function f is given by:

f ⋆ (t) = lim h→0 f (t + h) f (t) 1 h
In fact, we have :

f ⋆ (t) = lim h→0 f (t+h) f (t) 1 h = lim h→0 1 + f (t+h)-f (t) f (t) ( f (t) f (t+h)-f (t) . f (t+h)-f (t) h . 1 f (t) ) = lim h→0 1 + f (t+h)-f (t) f (t) ( f (t) f (t+h)-f (t) ) ( f (t+h)-f (t) h . 1 f (t) ) = e ḟ (t) f (t) = e (ln • f ) ′ (t)
for (ln •f )(t) = ln(f (t)). If the second order derivative of f at t exists, then by substitution, one will find that :

f ⋆⋆ (t) = e (ln •f ⋆ ) ′ (t) = e (ln •f ) ′′ (t)
We can see that (ln • f ) ′′ exists because f ′′ (t) exists. If we repeat n times this procedure, we can conclude that if f (t) is a positive function and f (n) (t) exists,

f ⋆(n) (t) = e (ln • f ) (n) (t)
We must note that this formula includes the case n = 0 as well because :

f (t) = e (ln •f )(t)
It must signaled that in multiplicative calculus ⋆ differentiability imply continuity but continuity doesn't imply ⋆ differentiability. Secondly the ⋆ derivative of a positive constant function is 1 and to fix some results one has the Table 1.

In Stanley [START_REF] Stanley | A multiplicative calculus[END_REF] is stated that if a function f (t) is differentiable at t, then is also ⋆ differentiable at t, and the converse affirmation is true in the following terms: if a positive function f (t) is ⋆ differentiable at t, and if f * (t) = 0, then it is also differentiable at t.

We have:

f * (t) = e f ′ (t) f (t) f ′ (t) = f (t) ln f * (t) ln(f * (t)) = ln ′ (f (t))
The commutative diagram presented in Figure 1. As we has already signaled, for a complete account of the product calculus we must consult Bashirov et al. [START_REF] Bashirov | Multiplicative calculus and its applications[END_REF] but it may be useful to begin by Stanley [START_REF] Stanley | A multiplicative calculus[END_REF].

g ln ln(g) g ′ g g * exp * d dt
Stanley [START_REF] Stanley | A multiplicative calculus[END_REF]. For more details see Filip and Curt [START_REF] Filip | Méthodes quantitatives en économie[END_REF].

a level curve. Note that if the function f is an utility function, then a level curve is called an indifference curve.

The slope of the line that is tangent to the level curve f (x, y) = c at a particular point is given by the derivative y ′ (x). This derivative is the rate of change of y with respect to x on the level curve and hence is approximately the amount by which the y coordinate of a point on the level curve changes when the x coordinate is increased by 1. Since

y ′ (x) = lim ∆x→0 ∆y ∆x ≈ ∆y ∆x then ∆y ≈ y ′ (x) ∆x
which gives us the change in y needed to compensate a small change ∆x in x so that the value of the function f remain unchanged.

Recall the ⋆ derivative formula

f ⋆ (x) = lim h→0 f (x + h) f (x) 1 h
and consider ∆x = x 2 -x 1 the variation of the independent variable, and

∆ * y = y 2 y 1 = y(x2) y(x1)
the variation of the independent variable. It follows that

y * (x) = lim ∆x→0 (∆ * y) 1 ∆x
and we consider that

y ⋆ (x) ≈ (∆ ⋆ y) 1 ∆x
. By taking the ∆x power of this expression we obtain

∆ ⋆ y ≈ [y ⋆ (x)] ∆x ∆ ⋆ y ≈ e y ′ (x) y(x) ∆x
which gives

y 2 ≈ y 1 e y ′ (x) y(x) ∆x .
We can reach a similar result if we analyse the ⋆ Euler's method for solving a differential equation, as it can be seen in Campbell [START_REF] Campbell | Multiplicative calculus and student projects[END_REF]. Let consider the first-order differential equation with initial conditions (the Cauchy problem)

   y ′ (t) = f (t, y) y(t 0 ) = y 0
Using the limit definition y ′ (t) = lim ∆t→0 y(t + ∆t) -y(t) ∆t and considering small ∆t, we can write y(t + ∆t) ≈ y(t) + y ′ (t)∆t This approximation naturally induces the idea of the Euler's method which uses an iterative process in order to produce approximate values for the unknown function f (t) as follows:

   t n+1 = t n + ∆t a n+1 = a n + f (t n , a n )∆t
for n = 0, 1, 2, ... . A two-variable discrete dynamical system was obtained.

If we consider now the ⋆ calculus, the approximation which uses the ⋆ derivative has the form: One can remark that the Euler's method gives the exact solutions only in the case of linear functions, which implies that differential equations of the form y ′ (t) = m can be exactly solved.

y(t +
In Campbell [START_REF] Campbell | Multiplicative calculus and student projects[END_REF] is stated that ⋆ Euler's method gives exact solution to the differential equations of the form:

   y ′ (t) = ky y(0) = y 0 over the time interval 0 ≤ t ≤ T ,
where k is a constant, which is the case of exponential functions. By choosing a number of N iterations and set ∆t = T N and set initial conditions t 0 = 0 and a 0 = y 0 , then iterate as follows:

   t n+1 = t n + ∆t a n+1 = a n e kyn yn ∆t = a n e k∆t
It can be seen that these iterations are repeated multiplications by the constant e k∆t . This implies that the closed form of this recurrence relation is a N = a 0 e k∆t N = y 0 e kT which is the exact solution.

The connection between Euler's method and ⋆ Euler's method can be obtained by using Taylor series of the function e x e x = 1 + x 1! + x 2 2! + ...

+ x n n! + ... a n+1 = a n e f (tn,an ) an ∆t = a n 1 + f (t n , a n ) a n ∆t + 1 2! f (t n , a n ) a n ∆t 2 + ... = a n + f (t n , a n )∆t + 1 2! [f (t n , a n )] 2 a n (∆t) 2 + ...
from which we can see that Euler's method is simply a truncation of ⋆ Euler's method. In addition, when a n > 0 we get an answer which is greater than the answer from the Euler's method and when a n < 0 the answer is less than the Euler's method answer. This is connected by socalled the concavity problem. This problem appears as well if we study the linear approximation versus the exponential approximation, as can be seen in Campbell [START_REF] Campbell | Multiplicative calculus and student projects[END_REF].

Linear approximation versus exponential approximation

As we have shown in Filip and Piatecki [START_REF] Filip | A non-newtonian examination of the theory of exogenous economic growth[END_REF] and it can be found as well in Stanley [START_REF] Stanley | A multiplicative calculus[END_REF], linear functions of the form x(t) = at + b has constant derivative and x(t + 1) = x(t) + x ′ (t). In addition, if any two differential functions x(t) and y(t) differ by a constant, then x ′ (t) = y ′ (t). Furthermore, a function x(t) differentiable at a point t = a has a linear approximation L near a defined as follows:

L(t) = x(a) + x ′ (t)(t -a)
On the other side, exponential functions of the form x(t) = ab t , with b > 0, has constant ⋆ derivative and x(t + 1) = x ⋆ (t) x(t). Moreover, the following two properties are true for all positive differential functions:

1. If one positive function is a constant multiple of another, i.e. x(t) = c y(t), they have the same ⋆ derivative. This is the central idea of the multiplicative rate of change : if one function is a constant multiple of another, their multiplicative rate of change is the same (as the ⋆ derivative quantifies the multiplicative rate of change ) that is, the two functions change by the same factor over any given interval [t, t + ∆t]. In other words, for any functions x(t) and y(t) = c x(t), if we have x(t + ∆t) = G x(t) for some G, then y(t + ∆t) = G y(t) for the same factor G (G is the multiplicative rate of change ).

2. Any positive function x(t) differentiable at a point t = a has an exponential approximation E near a defined as follows:

Stanley [15] E(t) = x(a)x ⋆ (a) (t-a) .
In Stanley [START_REF] Stanley | A multiplicative calculus[END_REF], it is stated that the graph of the exponential approximation E "hugs" the graph of the function x(t) and furthermore, the graph of E is a curve that coincides with the graph of x at a point a in these three ways:

• its value: E(a) = x(a) • its ⋆ derivative: E ⋆ (a) = x ⋆ (a)
• its additive derivative (its slope):

E ′ (a) = x ′ (a)
In order to illustrate linear and exponential approximation in Stanley [START_REF] Stanley | A multiplicative calculus[END_REF]aut]Stanley, D. is taken the function x(t) = 1 t and a point a = t = 2. Using the rules for ⋆ derivative presented in Table 1 we have

x(t) = 1 t , x ′ (t) = - 1 t 2 , x ⋆ = e -1 t x(2) = 1 2 , x ′ (t) = - 1 4 , x ⋆ = e -1 2
Linear approximation at t = 2 is

L(t) = x(2) + x ′ (2)(t -2) = 1 2 - 1 4 (t -2) = 1 - 1 4 t Exponential approximation at t = 2 is E(t) = x(2) x ⋆ (2) (t-2) = 1 2 e -1 2 (t-2) = 1 2 e -t 2 +1
For this example, the exponential approximation is a closer match that the linear one, as it can be seen in the Figure 2. The problem is that the ⋆ derivative is an exponential function which is a convex function and it cannot approximate closely any concave function. For instance (see Stanley [START_REF] Stanley | A multiplicative calculus[END_REF]aut]Stanley, D.) the exponential approximation to concave function sin t at t = π 6 is far worse than the linear one, for the function 1 -sin t its exponential approximation is fairly good and has about the same degree of closeness of match as the linear one.

0 2 t E(t) x(t) L(t)
In Stanley [START_REF] Stanley | A multiplicative calculus[END_REF]aut]Stanley, D. the ⋆ mean value theorem versus the mean value theorem, the ⋆ Rolle's theorem versus the Rolle's theorem and ⋆ Taylor products versus Taylor series are presented.

For the function x(t) = 1 1+e -t at t = 1 second order polynomial and exponential approximations are computed and better results are obtained in the exponential approximation case.

Recall the function x(t) = sin t and a = π 6 . Computing E(t), E 2 (t), E 3 (t), E 4 (t) and E 5 (t) -Coco [START_REF] Coco | Multiplicative Calculus[END_REF] -where

E(t) = x(a) [x ⋆ (a)] t-a = sin π 6 e cot π 6 = 1 2 e √ 3(t-π 6 ) E 2 (t) = x(a) [x ⋆ (a)] t-a [x ⋆⋆ (a)] (t-a) 2 2! = E(t) e -1 sin 2 π 6 1 2 (t-π 6 ) 2 = = E(t) e -2(t-π 6 ) 2 = 1 2 e √ 3(t-π 6 ) e -2(t-π 6 ) 2 E 3 (t) = x(a) [x ⋆ (a)] t-a [x ⋆⋆ (a)] (t-a) 2 2! x ⋆(3) (a) (t-a) 3 3! = = E 2 (t) e 8 √ 3 1 6 (t-π 6 ) 3 E 4 (t) = E 3 (t) e -80 1 24 (t-π 6 ) 4 E 5 (t) = E 4 (t) e 352 √ 3 1 120 (t-π 6 ) 5
and illustrating by graphics we can see that by increasing the order of the exponential approximation we obtain increasingly better results. One can remark that the multiplicative calculus can be extended as follows: if ϕ is a bijective function define § derivative and § integral by -Coco [START_REF] Coco | Multiplicative Calculus[END_REF]:

f § = ϕ ϕ -1 • f ′ (x) (we can see that f ⋆ (x) = e (ln •|f |) ′ (x) ) b a f (x)d § x = ϕ (ϕ -1 • f )(x)dx (we can see that b a f (x) dx = e b a (ln •|f |)(x)dx ).

Geometric calculus and bigeometric calculus

As we have mentioned, the multiplicative calculus or ⋆ calculus was called by Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] geometric calculus in order to emphasize that changes in function arguments are measured by differences, while changes in values are measured by ratios. The entire theory works only for positive functions. This is why it is mandatory not to work with standard (additive) accounting since, in case of deficit, we should have negative numbers, while with multiplicative accounting, all numbers are greater than zero. In addition, a positive function which is continuous and has the same geometric change over any two intervals of equal classical extent (i.e. x(t2) x(t1

) = x(t4) x(t3) if t 2 -t 1 = t 4 -t 3 and t 1 ≤ t 2 , t 3 ≤ t 4 ) is called geometrically-uniform function.
The name geometric calculus comes from at least two reasons. The first one is the property of a geometrically-uniform function that each arithmetic progression of arguments implies geometric progression of corresponding values. The second one must be started by recalling the definition of the geometric average of n positive numbers v 1 , v 2 , ..., v n , that is the positive number (v 1 v 2 ...v n ) 1 n . In Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] and Grossman [START_REF] Grossman | Bigeometric Calculus: A System with a Scale-Free Derivative[END_REF], is explained how a simply algebraic identity let them to construct the geometric calculus as follows:

Consider any n points (a 1 , v 1 ),(a 2 , v 2 )...,(a n , v n ), where a 1 < a 2 < ... < a n and a 2 -a 1 = a 3 -a 2 = ... = a n -a n-1 = k. Thus, k(n -1) = a n -a 1 . Connect the n points by line segments, of which are n -1. The arithmetic average of their classical slopes equals the classical slope of the line segment containing the end points (a 1 , v 1 ) and (a n , v n ), that is:

v 2 -v 1 k + v 3 -v 2 k + ... + vn-vn-1 k n -1 = v n -v 1 a n -a 1
If we assume that the v i values are positive, the following identity is obvious:

v 2 v 1 1 k v 3 v 2 1 k ... v n v n-1 1 k 1 n-1 = v n v 1 1 an-a 1
The left side of this equality is the geometric average of the n -1 numbers vi v i-1 1 k which they imagined to be the slopes of a new kind.

Speaking about the representation of a geometrically-uniform function x on a semi-log paper (logarithmically scaled on the ordinate axis), the result is a straight line whose classical slope equals the natural logarithm of the geometric slope of x.

One could ask what happens if both changes in arguments and values are measured by ratios? The theory developed by Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] and Grossman [START_REF] Grossman | Bigeometric Calculus: A System with a Scale-Free Derivative[END_REF] works for functions with positive arguments and positive values and it is called bigeometric calculus.

A positive function which is continuous and has the same geometric change over any two positive intervals of equal geometric extent (i.e. x(t 2 )

x(t 1) = x(t 4 )

x(t 3 ) if t 2 t1 = t 4 t3 and 0 < t 1 ≤ t 2 , 0 < t 3 ≤ t 4 ) is called bigeometrically-uniform function. The characteristic of a bigeometricallyuniform function is that each geometric progression of arguments implies geometric progression of values.

Speaking about the representation of a bigeometrically-uniform function x on a log-log paper (logarithmically scaled on both abscise and ordinate axis), the result is a straight line whose classical slope equals the natural logarithm of the bigeometric slope of x.

In the bigeometric calculus, the bigeometric derivative is defined as being lim t→a x(t) x(a)

1 ln t-ln a
and, by making an analogy with the geometric derivative or ⋆ derivative we propose the notation x ⋆ (a ⋆ ) for the bigeometric derivative of the function f on a.

It can be proved that x ′ (a) and x ⋆ (a ⋆ ) coexist and the relationship between them is

x ⋆ (a ⋆ ) = e ax ′ (a)
x(a) . The expression ax ′ (a)

x(a) is called by economists the elasticity of x at a, while Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] and Grossman [START_REF] Grossman | Bigeometric Calculus: A System with a Scale-Free Derivative[END_REF] called x ⋆ (a ⋆ ) the resiliency of x at a. We can see that the elasticity equals the natural logarithm of resiliency.

For this work we stop here with reviewing non-newtonian calculi, even if in Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] other kind of calculi are presented and developed. §3 Applications and further research directions 3.1 ⋆ calculus or geometric calculus

Galilean ratios

In the book by Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] is presented a problem which comes from Renaissance during Galileo, when two estimates, i.e. 10 and 1000 , were proposed for the price of a horse and the question was which estimate, if any, deviates less from the true value of 100? The first approach was to measure the deviations by differences and because of this, the estimate of 10 appears to be closer to the true value. Galileo proposed that the deviation should be measured by ratios and from this, both estimates deviated equally form the true value. From this story we can state as it was shown in Filip and Piatecki [START_REF] Filip | A non-newtonian examination of the theory of exogenous economic growth[END_REF] that growth phenomenon could be better modelled by using ⋆ derivative.

Personal finance application

In the same book by Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF], a more sophisticated problem is presented in order to make a comparison between classical calculus and ⋆ calculus: at time t 1 a person invested x(t 1 ) dollars with a promoter who guarantees that at a certain subsequent of time t 2 , the value of the investment would be x(t 2 ) dollars. In event that the investor should desire to withdraw at any other time t ∈ (t 1 , t 2 ), it was agreed that the value of the investment increases continuously and uniformly. The question is how much, i.e. x(t), would the investor be entitled to at time t? Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] presented two solutions which are both reasonable.

The first one is based on the assumption that the function x increases by equal amounts in equal times. This implies (because of the continuous increasing of the investment) that the growth has to be linear and, by considering the equation of a straight line which pass through the points (t 1 , x(t 1 )) and (t 2 , x(t 2 )) which is

x(t) -x(t 1 ) t -t 1 = x(t 2 ) -x(t 1 ) t 2 -t 1 we have x(t) = x(t 1 ) + x(t 2 ) -x(t 1 ) t 2 -t 1 (t -t 1 ).
Notice that the ratio x(t2)-x(t1) t2-t1

represents the classical slope of the straight line which pass through the points (t 1 , x(t 1 )) and (t 2 , x(t 2 )).

The second solution is based on the assumption that if the value of the investment increases uniformly we may reasonable consider that it increases by equal percents in equal times. From this we can prove that the growth has to be exponential, i.e.

x(t) x(t 1 )

1 t-t 1 = x(t 2 ) x(t 1 ) 1 t 2 -t 1 which gives x(t) = x(t 1 )
x(t 2 )

x(t 1 )

1 t 2 -t 1 t-t1
The expression x(t 2 )

x(t 1) 1 t 2 -t 1 is, when t 2 → t 1 , the ⋆ derivative x ⋆ (t 1 ).

Relative derivative

Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] spoke about the relative change of a positive function x over the interval [t 1 , t 2 ], i.e. x(t2)-x(t1)

x(t1) which equalized by 1 gives the geometric change of the function x over the interval [t 1 , t 2 ], i.e. x(t2) x(t1) . The relative slope of a geometrically-uniform function is its relative change over any interval of classical extent 1, from where if the interval is [ t 1 , t 2 ] we have x(t2) x(t1) 1 t 2 -t 1 as relative slope. The example given by the mentioned authors is one from securities analysis which is called compound growth rate. They supposed that if one paid 64 dollars for a share of stock, three years later the price of the share will be 216 dollars. The compound (annual) growth rate of the price over the time interval [0, 3] is given by: 216 64

1 3-0 -1 = 0.5 = 50%
The significance of this result is that if an original investment of 64 dollars increases 50% in each of the three years, the final value would be 216 dollars, which is true if we compute directly 64 + 64 2 + 96 2 + 144 2 = 216. They introduced the notion of relative derivative of a positive function x as being the limit:

lim t→a x(t) x(a) 1 t-a -1
if it exists and is greater than -1. They also state that the relative derivative of x at a coexists with, and equals 1 less than, the geometric derivative ( ⋆ derivative) of x at a.

Non-newtonian Euler's formula for homogeneous function

As an example of application of the non-newtonian calculus, we have shown, in the paper Filip and Piatecki [START_REF] Filip | A non-newtonian examination of the theory of exogenous economic growth[END_REF], what happen to the celebrated Euler's formula for homogeneous functions. This approach was also developed by Córdova-Lepe [START_REF] Córdova-Lepe | The multiplicative derivative as a measure of elasticity in economics[END_REF]. But it's results was given for another definition of multiplicative derivative for positive functions :

Qf (x 0 ) = lim h→1 f (x 0 h) f (x 0 ) 1 ln(h) , f :]0, ∞[→]0, ∞[, x 0 ∈]0, ∞[, if this limit exists.
If we take an homogeneous function of degree r, which means

f (µx 1 , . . . , µx n ) = µ r f (x 1 , . . . , x n ) we have rf (x 1 , . . . , x n ) = x 1 f ′ x1 (x 1 , . . . , x n ) + • • • + x n f ′ xn (x 1 , . . . , x n ) which is equivalent with r = x 1 f ′ x1 (x 1 , . . . , x n ) f (x 1 , . . . , x n ) + • • • + x n f ′ xn (x 1 , . . . , x n ) f (x 1 , . . . , x n )
and by exponentiation, we obtain :

e r = e f ′ x 1 f x1 • • • e f ′ xn f x n
We introduce the ⋆ partial derivative of f as being

f ⋆ x 1 = e f ′ x 1 f
by making a parallel with the definition of the ⋆ derivative of f :

f ⋆ = e f ′ f
We obtain the non-newtonian Euler's formula for homogeneous functions:

f ⋆ x 1 x1 • • • f ⋆ x n xn = e r
In the case of the Cobb-Douglas function, we have :

F (K, L) = AK α L β
the non-newtonian Euler's formula is :

e α K K e β L L = e r ⇐⇒ e α+β = e r
in such a way that if there are constant return to scale (the function is homogeneous of degree one), α + β = 1 = r.

Growth model

The neoclassical growth model. In the paper Filip and Piatecki [START_REF] Filip | A non-newtonian examination of the theory of exogenous economic growth[END_REF] we have considered the Solow-Swan exogenous growth model and we have try to reconstruct the model.

If there is a model know by all macro-economists, it is certainly this one, but it is necessary to remember how it works. We have :

                   I(t) = S(t)
Equilibrium of the good & services market

S(t) = sY (t) Saving function Y (t) = F (K(t), L(t))
Production function homogeneous of first degree K(t) = I(t) -δK(t)

The increase in the capital is equal to the investment less the obsolescence

L(t)/L(t) = n
The constant of growth of the labor force for I the investment, S the savings, Y the production, K the capital, L the labor force, δ the rate of obsolescence of the capital and n the constant rate of growth of the labor force. We know that if we define k(t) as the capital by headi.e. : k(t) = K(t)/L(t) -, we will find that his accumulation is given by the celebrated ordinary differential equation :

k(t) = sf (k(t)) -(n + δ)k(t), k(0) = k 0
which can be further developed in postulating for instance that the production function is a Cobb-Douglas one Y (t) = AK(t) α L(t) 1-α . This gives Bernoulli first order ODE :

k(t) = sAk α (t) -(n + δ)k(t), k(0) = k 0
whose solution is given by :

k(t) = k(0) 1-α - s δ + n e -(1-α)(δ+n)t + s δ + n 1 1-α
Since the production fonction is homogeneous of degree one, one knows, by one of the most used Euler's theorems, that if the representative enterprise remunerate its factors to the marginal product (F K = r/p and F L = w/p), the product is completely affected between the two type of costs, i.e.:

Y = F K K + F L L = r p K + w p L
It is know that this conduct to the fact that the profit by head π = Π/L could be written :

π = pf (k)
-rk -w which on its turn implies for a maximising firm that f ′ (k) = r/p and that π = 0 or equivalently

f (k) = r p k + w
All those information are subsumed in the Figure 3. We must also stress that in the long term when all capital adjustment has been done, that is to say that, in equilibrium, we have :

k(t) = 0 ⇐⇒ sf (k) = (n + δ)k
We must apologize to present here what is elementary text book economics, but we think that, in order to contrast our results with the standard approach this is mandatory. which says simply that savings sf (k) must equal the capital taken into account the constant growth of the labor force and to replace the obsolete one. In the steady state, output per worker is constant but total output increases at the rate n than the labor force. One must notice that obviously we obtain the same results if we decide to use the ⋆ derivative to calculate marginal productivity. This is for the newtonian Solow-Swan model. The problem one face now is to construct the non-newtonian equivalent approach. In the Figure 4, we can see that if savings are greater than (n + δ)k, the capital by workers will tend to rise and, vice versa, if savings are lower than (n + δ)k, it will tend to fall. Now suppose that with Phelps [START_REF] Phelps | Second essay on the golden rule of accumulation[END_REF] -see also Burmeister and Dobell [3] -, we want to see if, in the stationary state, there is a better saving rate among all the feasible saving rates. That is to say, we are searching for a saving rate which maximize the consumption by worker in the steady state, i.e. :

sf (k) > (n + δ)k sf (k) < (n + δ)k k(t) k(t) → → → → → ← ← ← ←
ŝ = argmax {s} {c = (1 -s)f (k)|sf (k) = (n + δ)k}
From the constraint it comes that s = (n + δ)k/f (k) in such a way that the consumption by worker is given by :

c = f (k) -(n + δ)k =⇒ f ′ (k) = (n + δ) =⇒ ŝ = f ′ (k)k f (k)
In other word, we learned that, if we ignore initial conditions and simply chose the collective saving rate associated with the steady growth path we must equal it to the share of the profit f ′ (k)k in the product.

The non-newtonian Solow-Swan model. In order to show how non-newtonian calculus can be applied to keep Solow-Swan results for the exogenous growth, we are obliged to adapt the equations of the model in such a way that the rate of growth of the capital-labor ratio be described exactly by the same equation that the celebrated one. As it will be shown, only the differential equations must be changed.

In what concern the increase of the capital the non-newtonian equation must be the following:

K ⋆ (t) = I(t) K(t) δ
As weird as can look this equation it is no more than the rate of growth of the capital written in terms of logarithm as it is shown by the fact that

K ⋆ = e K K = I K δ ⇐⇒ ˙ ln K = K K = ln I -δ ln K
This equation is, in fact the equation of the capital increase in the newtonian model, but which is not written in level.

In what concern the population growth, the Malthus equation becomes

L ⋆ = e n
which, in fact, is nothing more than the standard equation, since according to the rule of the ⋆ calculus :

L ⋆ = e L
L = e n So now, the non-newtonian Solow-Swan model is:

                       I(t) = S(t)
Equilibrium of the good & services market

S(t) = sY (t) Saving function Y (t) = F (K(t), L(t))
Production function homogeneous of first degree

K ⋆ (t) = I(t) K(t) δ
The increase in the capital equals the ratio between investment and obsolescence L ⋆ (t) = e n Growth of the labor force With some simple algebraic manipulations we can show that from this model we can derived the Solow-Swan capital-labor ratio growth equation.

k ⋆ = K L ⋆ = K ⋆ L ⋆ = e K K L ⋆ = e I(t)-δK(t) K(t)
e n = = e

I(t) L(t) -δ K(t) L(t) K(t) L(t)
e n = e

I(t) K(t) -δk k e n = e sf -δk k e n = = e sf -δk-nk k = e sf -(n+δ)k k = e k k ⇒ k = sf -(n + δ)k
We can see that by re-writing the model by considering the ⋆ derivative in the differential equations the classical model is conserved.

Maximum likelihood method versus the ⋆ maximum likelihood method

With the help of the ⋆ derivative, we are able to avoid the logarithmization of likelihood function in classical statistics. In the classical method, the random characteristic X, which has the PDF f (x; λ 1 , λ 2 , ..., λ s ) depending on unknown the parameters λ 1 , λ 2 , ..., λ s , is investigated. Let consider X 1 , X 2 , ..., X n a sample of n independent and identically distributed (iid) observations which follow the same probability law as the random characteristic X. It follows that the PDF of the random vector (X 1 , X 2 , ..., X n ) is the function called the likelihood given by expression:

L(x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ s ) = n k=1 f (x; λ 1 , λ 2 , ..., λ s )
The estimators λi = λi (X 1 , X 2 , ..., X n ) are called maximum likelihood estimators of the parameters λ 1 , λ 2 , ..., λ s if they realize the likelihood maximum.

The maximum likelihood estimators are determined by solving the equations system:

∂L(x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ s ) ∂λ i = 0, i = 1, s
which is usually replaced by the system (easier to solve):

∂ ln L(x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ s ) ∂λ i = 0, i = 1, s
obtained by considering the optimization problem for the natural logarithm of the likelihood ln

L(x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ s ) = n k=1 ln f (x k ; λ 1 , λ 2 , ..., λ s )
This calculus artifice is due to the calculus complexity or impossibility to calculate the partial derivatives of the likelihood.

Thanks to the product ⋆ derivative rule, i.e. (f g) ⋆ (x) = f ⋆ (x)g ⋆ (x), we are now able to avoid this problem, if we ⋆ differentiate the likelihood function L(x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ s ) = f (x 1 ; λ 1 , λ 2 , ..., λ s )...f (x n ; λ 1 , λ 2 , ..., λ s ) as follows: which conduct to the maximum likelihood system:

L ⋆ λi (x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ s ) = f ⋆ λi (x 1 ; λ 1 , λ 2 , ..., λ s )...f ⋆ λi (x n ; λ 1 , λ 2 , ..., λ s ) = e f ′ λ i (x 1 ;λ 1 ,λ 2
f ′ λ i (x 1 ; λ 1 , λ 2 , ..., λ s ) f (x 1 ; λ 1 , λ 2 , ..., λ s ) + ... + f ′ λ i (x n ; λ 1 , λ 2 , ..., λ s ) f (x n ; λ 1 , λ 2 , ..., λ s ) = 0, i = 1, s
equivalent with the system obtained in the classical maximum likelihood method. Matched with the ⋆ Newton-Raphson method developped by Özyapici et al. [START_REF] Özyapici | On multiplicative and Volterra minimization methods[END_REF], which is known to be faster than the standard Newton-Raphson method, we expect that the implementation of the proposed ⋆ maximum likelihood method improve the speed and the accuracy of the search for the maximum likelihood estimator.

Bigeometric calculus

Resiliency

As we have shown a relationship between the elasticity of a function x at a and the resiliency of x at a is the following: elasticity =ln(resiliency ) where elasticity = ax ′ (a)

x(a) and resiliency = x ⋆ (a ⋆ ) = e ax ′ (a)

x(a) .

Suggested applications of resiliency

Grossman and Katz [START_REF] Grossman | Non-Newtonian Calculus[END_REF] and Grossman [START_REF] Grossman | Bigeometric Calculus: A System with a Scale-Free Derivative[END_REF] supposed that:

• psychophysicists could find some interests in bigeometric calculus because of one of their basic laws which can be stated as follows: "the resiliency of the stimulus-sensation function is constant", and they continued: "that constant is determined by the nature of the stimulus".

• biologists could use the bigeometric calculus for a fundamental law of growth as follows: "if f is the function relating the size of one organ to the size of any other given organ in the same body at the same instant, then, within certain time limits, the resiliency of f is constant".

• physicists who preferred not to settle on specific units of time and distance could assert that the bigeometric speed of an object falling freely to the earth is constant.

Bigeometric derivative

In Grossman [START_REF] Grossman | Bigeometric Calculus: A System with a Scale-Free Derivative[END_REF] is studied the case of a bipositive function when each of its points ( x, y) is changed to (px, qy), where p and q are positive constants, so a change of scales has been made in the function. It could be proved that the bigeometric derivative is independent of the scales used for function arguments and values and this fact is considered to be as a point of interest for scientists who wish to express laws in scale-free form. §4 Conclusion For nearly thirty years, non-newtonian calculus stays confidential because of the conservative position of social scientists and perhaps a lack of information on the fact that it could help to acquired new insight on classical subjects, or solve directly some problems which could only reached by approximations.

Until now, there is a lot of papers which describe this calculus as Stanley [START_REF] Stanley | A multiplicative calculus[END_REF], Spivey [START_REF] Spivey | A product calculus[END_REF], Bashirov et al. [START_REF] Bashirov | Multiplicative calculus and its applications[END_REF], Bashirov et al. [START_REF] Bashirov | On modelling with multiplicative differential equations[END_REF], but economic applications are rare and doesn't get away from the simple justification of the technics.

In this paper, we have tried to show that it has some very interesting potentials that we have begin to study.

Derivative Rules

Examples

f (t) f ⋆ (t) (cx) ⋆ (t) = (x) ⋆ (t

Figure 1 :

 1 Figure 1: Relationship of g ′ and g *

Figure 2 :

 2 Figure 2: Linear versus exponential approximation to x(t) = 1 t at the point t = 2

Figure 3 :

 3 Figure 3: The repartition of the product in the Solow-Swan model

Figure 4 :

 4 Figure 4: Phase diagram of capital accumulation in the Solow model

  ,...,λs ) f (x 1 ;λ 1 ,λ 2 ,...,λs ) ... e

		f ′ λ i f (xn;λ 1 ,λ 2 ,...,λs ) (xn;λ 1 ,λ 2 ,...,λs )
	= e	f ′ λ i f (x 1 ;λ 1 ,λ 2 ,...,λs ) +...+ (x 1 ;λ 1 ,λ 2 ,...,λs )	f ′ λ i f (xn;λ 1 ,λ 2 ,...,λs ) (xn;λ 1 ,λ 2 ,...,λs )
	= 1
	= e 0

Table 1 :

 1 Some examples of product calculus

	)	C	1

⋆ (t) = f ⋆ (t)y ⋆ (t) ⋆ (t) = f ⋆ (t) y(t) f (t) y ′ (t) Ct e 1 t Chain rule (x • y) ⋆ (t) = f ⋆ (y(t)) y ′ (t) αt + β e ⋆ (t) = f ⋆ (t) f (t)

f (t)+y(t) y ⋆ (t)


Cyrille Piatecki

Leo

Study choices in an evolutionary game

Keywords: Labor Economics, Evolutionary Game Theory, Strategic Interactions

 contribution, the accumulation of human capital has been developed only within the framework of individual choices based upon wage expectations. For over forty years, this approach showed itself fruitful. However, with the rarefaction of jobs during periods of crisis, it seems light to explain most of the decisions regarding human capital accumulation. Indeed, because nobody can be sure to obtain an employment which will yield a positive return to education, the decision to acquire further education can only be apprehended as a strategic decision taken in a situation of imperfect information. This individual decision will depend on the answer to the following simple question: With the level of studies which I intend to acquire, will I have bigger chances to obtain a job than my classmates who stopped their studies at a lower level?

Introduction

In economics, many topics should be considered from the complexity point of view.

First of all, while it took approximately four hundred years to establish a utility theory from the early works by Edme Mariotte2 -see Mariotte (1717) -, thirty years were sufficient to make the building vacillate on its basis.

Among other sciences, Behavioral Psychology and Neurosiences question the very existence of a utility function derived from preferences and used to construct practical decisions. The assumption that human behavior can generate rational decisions3 has been denied by new experimental evidences from the recent access to the brain through magnetic resonance imaging (MRI) and molecular biology. Thanks to the MRI, it is now proven that decisions can not be taken without reference to emotions and that we are sensitive to chemical substances which alter profoundly our behavior and decisions -see [START_REF] Damásio | Descartes' Error: Emotion, Reason, and the Human Brain[END_REF], [START_REF] Jevons | The Theory of Political Economy[END_REF].

At the same time, with the emergence of a new type of society where information is huge and plethoric, we have begun to understand that the brain is, in some aspect, a wired electrical network with limited transmission speed and limited capacity to treat information. Furthermore, since Poincaré, but more effectively since Edward Lorenz, we know that linearity, through which one believes to understand the most essential structure of our universe, is misleading and that a very small departure from it can produce some processes that are truly indistinguishable from randomness -Poincaré (1890), [START_REF] Lorenz | The problem of deducing the climate from the governing equations[END_REF] and also [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF].

At last, one begins to understand that strategic interactions, which are mainly described in the setting of standard game theory may not bee understood from rationality itself.

Nevertheless, despite all these new evidences, many analysts are globally satisfied with their inherited instrument and are not yet ready to change paradigm. For instance, the core of the modelling of economic markets has not been fundamentally changed in thirty years.

For instance, let us have a look at labor economics. Since the development of the Human Capital Theory -Schultz (1961), [START_REF] Becker | Human Capital[END_REF] -, economists have concentrated their attention essentially on the main advantage that the education confers to individuals: an increase in their income.

The mechanics highlighted under this original human capital theory allowed, in a standard neo-classic framework, to propose an alternative explanation to the theory of compensatory differences that was first advanced by Adam Smith and in which individuals were considered as indistinguishable.

Two research areas have caught the attention and the energy of economists. On the one hand, following Mincer's works -see [START_REF] Mincer | Schooling Experience and Earnings[END_REF] -, an effort has been made to estimate empirically education returns through some earning functions. That approach enables to estimate a return curve for each educational level and to make international comparisons. On the other hand, more recently, human capital has been introduced in models of endogenous growth.

Following [START_REF] Lucas | On the mecanics of economic development[END_REF], macroeconomists have favored the stock of human capital over technology to explain growth, and the differences of rates in its accumulation to explain the persistence of international differences in growth rates. In particular, because of the externalities generated by its human capital, for a given level of qualification, a worker is more productive in a country already strongly endowed with human capital. Therefore, he is better paid. This mechanism could explain the strong migratory pressures of the South towards the North.

Without questioning the legitimacy of the Beckerian individualistic approach of human capital accumulation, it seems, nevertheless, important to reinterpret the contemporary logics of its accumulation in the context of jobs competition. In the Beckerian logic, the rise of the level of education appears as a real solution against unemployment, from the individual point of view, because the level of education is positively correlated with the speed of transition towards employment. This transition advantage is also assumed by the filter theory developed by [START_REF] Arrow | Higher education as a filter[END_REF] and the signal theory developed by [START_REF] Spence | Job marker signaling[END_REF].

Other things being equal, considering a homogeneous population, this transition advantage given by education is expected to motivate all individuals to apply for further education and training. Hence, the ultimate consequence of this dynamic process is the deletion of the number of nongraduates.

In practice, we observe a conflicting logic which substitutes itself to the Beckerian individualistic approach as soon as such a situation is reached or, at least, as soon as a critical threshold of graduates is exceeded. Indeed, the diploma competitive advantage tends to disappear with an increase in the number of its graduates. Outside a conflicting approach of the human capital accumulation, we can only predict a desertion of the educational sphere: education does not generate any more a return on investment which can be evaluated in comparison to an investment in stocks.

However, from the empirical point of view, it is rare to observe parents preferring to donate the cost of their education to their children in the form of a financial endowment because its seems to them higher and less risky than that of the studies that their children undertake. Indeed, in spite of its depreciation, human capital can become the entry key to active life. So, on the contrary to the predictions of the Beckerian model, the model of the conflicting logic seems to predict a wild race towards the accumulation, the non-accumulation appearing as an almost insuperable handicap.

However, if a model which rests on a conflicting logic ends in an extreme distribution -either every player graduates or nobody graduates -, it must be rejected as empirically not relevant because, as far as one can observe, in populations where the individual characteristics are distinguished with difficulty, the coexistence of the two logics of accumulation of human capital -i.e.: the maximalist logic of the acquisition of the highest diploma and the minimalist logic of the acquisition of the lowest diploma.

This paper proposes to examine the evolution of the graduate population within the framework of the theory of evolutionary games. For every generation, for a given unemployment rate, the percentage of graduates is predetermined by their expected remuneration relative to the previous generation. The graduates, in a context where the diploma gives essentially a priority access to jobs, are all the stronger as there are few qualified people. Consequently and a priori, we should expect that the global educational level does not stop to grow excluding the possibility of a mixed balance, that is a balance of population in which coexists both categories of agents. However, when the trained population is numerous, the high training costs decrease the relative advantage conferred by the studies and can incite to disinvest in education.

The problem hereby underlined is a real problem that all developed countries seem to encounter. For instance, in France, for the year 2000, it has been revealed that one third of the youth undergo a downgrading. Even if the apex of the phenomena has been observed between 1986 and 1995 for the graduates at the Baccalauréat (High School exit examination) level, and between 2001 and 2004, at the BAC+2 level (two-year post-secondary or tertiary level degree), due to the overall decrease in employment, even the higher-level graduates got heavily impacted. The chances for a newly graduate to get a position as an executive fell from 85% to 70%. Only 26% of the youth have a perfectly matched job according to their formation, in level and in speciality. At the same time, we still observe that the higher the level of the diploma, the higher the smoother the transition towards the first job.

In the first section, we consider the static evolutionary game and its dynamics when the matching of the young job-seekers is realized pairwise. The second section considers explicitly the effect of the unemployment growth on the population dynamics. The third section studies the formation of the long-term balances when irrational mutations appear. The fourth and last section is devoted to the commandability of the education system4 .

Pair Matching dynamics

Here we study more particularly a game in which the players have two strategies:

① undertake short studies -strategy (C); ② undertake long studies -strategy (L);

At first, we are interested in two players that compete for a job, both of them assigning to it a common present value V. The agent having adopted the L strategy will be given priority access to the job compared to the agent having adopted the C strategy. This preference for the most qualified refers to the human capital theory -under its shape filter or under its shape signal. Because our focus is on the advantage conferred by education, we suppose that the level of the remuneration V is independent from the educational level.

Labor markets are segmented and it is notable that the multiplication of the graduates at every level of study entailed a toughening of the dualism -for a definition of the dualism, we can consult5 , for example, [START_REF] Taubman | Segmented labor markets[END_REF]. Young people that are excluded from markets, which access conditions correspond exactly to their diplomas, are thus brought to look for an outlet on a market for which they are overqualified. This downgrading search creates an additional bottleneck for their companions who had acquired the diploma required to enter on the secondary market.

We can unwind this reasoning up to the markets which require no specific training and which are, in turn, assailed by young graduates. This logic leads to a more or less important eviction of the non-graduates of the labor market 6 . Naturally, later in their active life, some individuals will manage to return to their desired market, but they will certainly represent an exception to the rule of the strict dualism. So, on an isolated market, we can expect to observe applications emanating from young graduates possessing the minimum level of required diplomas -see the figure 1. A rather reliable indicator of these observations would be established by the ratio between the level of education of the candidates at public service job competitions and the education level required by the examination. We observe that a large share of the candidates have a must higher level of education and qualification than required by the examination. As a consequence, these overqualified candidates systematically evict the least endowed without benefiting of a more generous salary scale. On the other hand, on the analytical point of view, we can imagine that the status of the labor market is such that companies are in a position to monopolize a rent or to make sure of the honesty of their employees. This point was developed in particular by [START_REF] Shapiro | Equilibrium unemployment as a worker discipline device[END_REF].

Returning to the model, let us assume that initially all players were used to short studies. As ex ante none of the players is able of determining the stemming from the overcoming conflict, by adopting Laplace's insufficient reason principle, it is possible to suppose that they anticipate to each have a 1/2 probability to monopolize the desired job. The couple of payoffs in a conflict between two individuals having adopted the short strategy is thus: (0.5V, 0.5V).

Let us suppose that after a punctual mutation7 , a fraction 1ǫ of the population decides to opt for a new strategy consisting of a cycle of longer studies. These players can be perceived as predators by all those players who have decided not to change strategy. Indeed, we suppose that it is of common knowledge that, if a long player is accidentally mated to a short player, the first one will obtain the employment and will be paid V, while the second will have to pursue its job search. However, we suppose that, compared to strategy C (short studies), the long studies yield a fixed additional cost C. The couple of payoffs bound to such a confrontation is then: (V -C, 0).

Obviously, a mutant can be mated with an other mutant. As previously, in this case, they will obtain the couple of remuneration (0, 5(V -C); 0, 5(V -C)). Presented under normal form, the bi-matrix of the payoffs, associated with this game, can then been spell as in figure 2.

C L C V 2 , V 2 (0, V -C) L (V -C, 0) V-C 2 , V-C 2 Fig. 2

. Payoff matrix for the game

In what follows we will postulate that 0 ≤ C ≤ V ≤ 2C. This hypothesis is as fundamental as natural. First of all, we imagine badly a negative cost. Then, because a cost lower than the present value of the income of the promised job is a necessary and sufficient condition to ensure that some individuals will wish to bear it. And finally, one has to impose that V ≥ 2C. Indeed, let us suppose the opposite: V > 2C. In this case, (V -C)/C which is the ex post rate of return of longer studies is lower than the unit. Now, if the return was greater than 100%, there would be no student to not undertake longer studies.

The game as a perfect information game in normal form

In pure strategies, under the hypothesis V < 2 C, there are two Nash- equilibria in this game : ENSP 1 = (C, C) and ENSP 2 = (L, L). Both equilibria are symmetric -i.e. : both players must choose the same strategy -, which in itself is not so astonishing since the players are indistinguishable. To those pure strategy equilibria one must add a mixed strategy equilibria. One finally has :

ENSM 1 = (0, 0), ENSM 2 = V -C C , V -C C et ENSM 3 = (1, 1)
The figure 3 give the three equilibria as intersections of the players' reaction correspondences.

p q V-C C V-C C p = R 1 (q)
Pure Strategy Nash equilibrium Mixed Strategy Nash equilibrium

y = R j (x)          = 1 if x ∈ V-C C , 1 ∈ [0, 1] if x = V-C C = 0 if x ∈ 0, V-C C p = R 1 (q) q = R 2 (p)
Fig. 3. The Svastika for Nash equilibria

In front of the embarrassment induced by the multiplicity of the Nash equilibria, game theorists have developed a number of procedures aiming at extracting a single equilibrium, which will most probably be played by the player8 . The first seducing approach consists in postulating that the players are going to notice that one of the two pure equilibria is socially more favorable than the other one, in the sense that the expected payoff is higher there -here, it is about the (C, C) which gives a payoff equal to V/2 which is greater to the payoff of (V -C)/2 associated to thee second pure strategies equilibrium. (C, C) is called the Pareto-dominant equilibrium.

A standard approach of the selection of the multiple equilibria was proposed by [START_REF] Harsanyi | A General Theory of Equilibrium Selection in Games[END_REF]. This approach to equilibrium selec-tion is known as the risk dominance. This approach of equilibrium selection presents the interest of a double interpretation. On the one hand, a player can realize an experiment in which he tries to guess the strategy adopted by the opponent with whom he will be mated -this is a game theory interpretation. On the other hand, he can, ex ante, try to conceive the characteristics of the distribution of the players between both strategies as members of the population to which they belong -this is a sociological interpretation.

Let us begin by replacing us in a frame where only pure strategies could be elected and let us endorse the clothes of one for the two matched players.

If he thinks that his opponent will choose the C strategy with probability q and the L strategy with probability 1 -q, his expected utility is :

EU 1 (s) =    (1 -q) V 2 if s = L q V 2 + V-C 2 if s = C
The player will prefer to adopt the C strategy -the L strategy -over the L strategy -the C strategy -, if :

EU(C) < (>) EU(L)
Writing q, the q value which makes the player indifferent between both strategies, we find that :

q = C V + C
Now, the player's choice depends of his mental scheme which drives him to postulate a q specific value. However it's clear that if 1 -q ≥ q -or, more simply, if q ≥ 1/2 -, which is the present case since C ≤ V, the opponent will take a greater risk in postulating that the player will choose the C strategy over the L one. As this is a symmetric game what is valuable for one player is also valuable for the other. So one will say with Harsanyi and Selten that the (L, L) equilibrium risk dominate the (C, C) equilibrium if q > 1 -q, which means that there is a greater likelihood that both players coordinate on the (L, L) equilibrium than on the (C, C) one. In this game, according to the risk dominance concept, the players should be incited to adopt a coordination over long studies.

The static evolutionary game

The application of traditional game theory to the duration of education raises two problems:

① On the one hand, we cannot consider two isolated players, because the game is played by a very large number of players simultaneously. ② On the other hand, the players have to make decisions without knowing which type of player they will later be mated with -an C player or an L player. Now, the yield of a strategy ex ante depends on the percentage of players which endorse it in the population. In particular, even if there are C players in the population, a player can be interested in being an L player because the probability to be mated to an C player is low and because, by choosing to be an L player, he does not have to support the additional cost of the extra time of studies.

We begin by studying the pure strategies game. Let us note ǫ, the percentage of players who use the C strategy. In such a way, the expected payoff of a C player in a mixed population is :

EU(C|L) = ǫU(C, C) + (1 -ǫ)U(C, L)
where U(X, Y) is the utility of a X player matched with a L player. In the same way, one can write :

EU(L|L) = ǫU(C, C) + (1 -ǫ)U(C, L) = ǫ(V -C) + (1 -ǫ) V -C 2 = 1 + ǫ 2 (V -C)
In the evolutionary game terminology, a strategy X is evolutionary stable if :

U(X, Y) > U(Y, Y)
for every other Y strategy. This means that X offers a better fit 9 to the game than every other Y. [START_REF] Maynard-Smith | The theory of games and the evolution of animal conflicts[END_REF] has shown that X is evolutionary stable 10 if the two following conditions are satisfied :

① A Nash Equilibrium Condition : U(X, X) > U(X, Y).
9 In the sense of greater expected utility. 10 One speaks of an Evolutionary Stable Strategy -ESS.

② A stability Condition

: if U(X, X) = U(X, Y) then U(X, Y) > U(Y, Y).
It is obvious that the pure strategies Nash equilibria are ESS, which here was predictable as far as the game is symmetric. Consequently, by neglecting the fact that they are immersed in a larger population than that of both matched players, and as far as they manage to coordinate on an equilibrium -for example the risk dominating equilibrium -, the players protect themselves against the invasion of mutant players, who would have adopted the other strategy. Now, in the same way as in the traditional interpretation, what happens if one allows mixed strategies? If one redefines the utilities in terms of mixed strategies, that is:

EU(p|q) = ǫ p ⊤ U p + (1 -ǫ) p ⊤ Uq
where p = [p, 1 -p] ⊤ is the mixed strategy of the first player, q = [q, 1q] ⊤ the mixed strategy of the second one and U is defined as earlier. Evaluating in the same way EU(q|p), we can define an Evolutionary Stable Mixed Strategy -ESMS -in the same way as for the ESS. In that case the Maynard-Smith theorem gives that p is and ESMS if two conditions are verified:

① A Nash Equilibrium Condition : p ⊤ U p > p ⊤ Uq. ② A stability Condition : if p ⊤ U p = p ⊤ Uq then p ⊤ Uq > q ⊤ Uq.
Now for exactly the same reason as in the pure strategy case, the three Nash equilibria are SMES.

Population dynamics

The direct ESS11 or SMES approach are clearly insufficient. Indeed, unlike the classic theory of antagonistic games, evolutionary game theory does not postulate any form of rationality from the players. Generally, it considers that they inherit their behavior. On the labor market, and in an unstable environment, the only way of making a decision more or less justified, is, from one period to the next, when the information is available, to observe the relative fit of a strategy with regard to an other one. Therefore, on a macroscopic plan, one shall notice an increasing evolution on behalf of the population which adopts the strategy C if that strategy gives, on average, a satisfactory payoff in the matching with the upper strategy L.

If students can observe that the C strategy can offer a greater utility than the average utility associated with a population divided out of ǫ(t) percent of their own type and 1ǫ(t) percent of the other, then in all likelihood, ǫ(t) is going to rise. This leads to a Malthusian population dynamics of the C players called replication dynamics. This dynamics is describe by the following differential equation 12 :

ǫ(t) ǫ(t) = EU(C) -[ǫEU(C) + (1 -ǫ(t))EU(L)] = C 2 (1 -ǫ(t))(ǫ(t) -ǫ)
with 0 < ǫ = (V -C)/C < 1. This dynamics 13 to translate the idea according to which the C players' population growth rate is proportional to their relative payoff in a mixed population. Now, the average payoff in a mixed population is defined by the sum of the product between payoffs and probabilities of the realisation of a specific matching. This replication dynamics may be also written as :

ǫ(t) = ǫ(t)[1 -ǫ(t)][EU(C) -EU(L)]
which gives also the following interpretation : the increase of the C players population is proportionate to the expected utility differential gap between both strategies, conditioned by the emergence of a mixed matching 14 .

Define the h(ǫ(t)) function as :

h(ǫ(t)) = C 2 ǫ(t)(1 -ǫ(t))(ǫ(t) -ǫ)
which leads to rewrite the replication dynamics as :

ǫ(t) = h(ǫ(t))
12 Only a small bunch of studies was interested in the justification of the recourse to the replication dynamics in the economic environment. Nevertheless, we can quote [START_REF] Weibull | Evolutionary Game Theory[END_REF], who shows how the dynamics of replication can appear as approximation for imitation behavior. 13 To obtain EU(C) and EU(L) one needs only to substitute ǫ(t) to qcf supra.

14 It would be possible to generalize the dynamics by the use of an increasing function of the right side of the replication dynamics, but this does not change the qualitative behavior of the dynamics.

The h(t) function possesses three fixed points: ǫ 0 = 0, which is the case for the degenerated population composed exclusively of L players; ǫ 1 = ǫ, which corresponds to a mixed population; and ǫ 2 = 1, which is the case for the degenerated population composed exclusively of C players. These three fixed points are the three SMES . As the derivative of h

(ǫ) is h ǫ (ǫ) = (C/2)[-3ǫ 2 + 2 (1 + ǫ)ǫ -ǫ],
it is very easy to show that ǫ 0 and ǫ 1 are locally stable equilibria15 when ǫ 2 is locally unstable, the C strategy is completely evinced in favor of the L strategy. From a formal point of view, we can develop a symmetric argumentation 16 . So there are two attraction manifolds for the locally stable equilibria -see figure 4: Thus, according to its initial distributionǫ(0) ∈ B 0 , ǫ(0) = ǫ 0 or ǫ(0) = ǫ 1 -, the population converges towards the Pareto-dominant equi- librium, remains in the internal equilibrium -if by chance it was yet at this equilibrium -, or converges towards the risk dominant equilibrium.

① a first attraction manifold B 0 = [0, ǫ[ for ǫ 0 . ② a second attraction manifold B 1 =]ǫ, 1] for ǫ 1 . 1 ← ← ← ← ← ← ← ← ← → → → → → →→ → → ǫ 0 ǫ 1 ǫ 2 h(ǫ) hǫ (ǫ) B 0 B 1 B 0 = ǫ 0 attraction manifold B 1 = ǫ 1 attraction manifold
We can propose a simple explanation for this result. For an initial population such as ǫ(0) < ǫ, the players are matched with numerous players who, on average, have made long studies rather than short. The cost of longer studies is more than compensated by the low probability to be mated with an other long studies player. The expected relative advantage of a strategy over another determines its adoption.

Hence, at the final stage of the dynamics, the C strategy is completely evinced in favor of the L one. From a formal point of view, one can develop a symmetric argumentation17 for the case where ǫ(0) > ǫ.

However, this symmetry is only visible. Indeed, if we suppose that, initially, only the short studies existed and that a spontaneous mutation has moved a small percentage of players towards the L strategy, it is not very likely, that this percentage makes tip over the initial condition to the attraction manifold of the other SMES, i.e. to the long studies attraction manifold.

While the risk dominance argumentation is more convincing than the Pareto-dominance argumentation, in the final equilibrium choice decided by the population of players, it seems that the replication dynamics pleads clearly in favor of the eviction of long studies from the education system, unless, it is possible to produce an initial mutation which drives the initial population in the other attraction manifold -i.e. : which moves ǫ(0) over ǫ. A classic argument is then to suggest the appearance of behavior determined by a rationality still weakened by the number of agents.

Population dynamics with continuous mutations

As pointed out in [START_REF] Orlean | De la stabilité évolutionniste à la stabilité stochastique[END_REF], this dynamics leaves aside an important aspect of the evolutionary logic, namely the constant presence of mutational factors. Indeed, two elements can overlap to explain that the moving decision of a number of players is not directly driven by the relative advantage of a strategy. First of all, there is always a certain percentage of the population which has a very restricted access to information -in the replication dynamics, the access to information is supposed free and without cost for the players. Similarly, a certain percentage of the population is likely to decide to undertake or not to undertake long studies only based on its interest for the studied fields.

Then, the State, for reasons which run from the adequacy of the population to jobs which require an accumulation of human capital always more important for the prolonged, to the young people withdrawal of the labor market in period of under-employment, to make migrate sub-populations having adopted a particular strategy to an other one in a permanent way. So, a second dynamic system is build upon the first one, and this new system does not answer the arguments of the replication dynamics.

This dynamics translates into the fact that, over a lapse of time of length h, a percentage m of the players go from the L strategy to the C strategy and a percentage l goes from the C strategy to the L strategy. In t + h, the percentage of the population that has adopted the C strategy is then determined by the recursive equation:

ǫ(t + h) = [1 -mhǫ(t)]
probability not to migrate for a C player

ǫ(t) + [lhǫ(t)] probability to migrate for a L player (1 -ǫ(t)) C ǫ(t) L ǫ(t) mhǫ(t) l -lhǫ(t) l -mhǫ(t) lh(1 -ǫ(t))
Fig.

The continuous mutations population dynamics

By making h stretch to dt, one finally obtains:

ǫ(t) = [l -(m + l)ǫ(t)]ǫ(t) = k(ǫ(t))
This dynamics has two fixed points ǫ 0 = 0 and ǫ 4 = l/(m + l). As k ǫ (ǫ) = l -2(m + l)ǫ, ǫ 0 is locally unstable, unless l ≡ 0, and ǫ 4 is locally sta- ble. ǫ 1 = 1 is no more an equilibrium. It could be under the restriction m ≡ 0, in which case ǫ 4 = ǫ 1 = 1. This has a very simple explanation : if at each period of time some C players migrate to become L players, it is impossible that the final population-state be fixed in an all C players population. This dynamics, which could be called a rational extreme dynamics18 , in opposition to the replication dynamics, insists on the central role of an equilibrium where C players and L players coexist -see the figure 6.

Let us consider the general population dynamics corresponding to the juxtaposition of the two types of behavior: the strategic behavior and the predetermined behavior. It takes the form :

ǫ(t) = h(ǫ(t)) + k(ǫ(t)) = C 2 ǫ(t)(1 -ǫ(t))(ǫ(t) -ǫ) + ([l -(m + l)ǫ(t)]ǫ(t)) = ǫ(t) C 2 (1 + ǫ)ǫ(t) -ǫ + ǫ 2 (t) + [l -(m + l)ǫ(t)] = H(ǫ(t)|m, l) (dm1) ← ← ← ← ← ← → → → → → → → ǫ 0 ǫ 4 k(ǫ(t)) 1 0

Fig. 6. The extreme arational dynamics

Here the mutations remind a trembling hand process -see [START_REF] Orlean | De la stabilité évolutionniste à la stabilité stochastique[END_REF]. They constitute a permanent and autonomous source of variability inside the labor market system.

Then one can also envision a new population dynamics in which, α percent of the population share a quasi-rational behavior, when the other 1α has adopted an irrational one. In this case, the dynamics becomes:

ǫ(t) = αh(ǫ(t)) + (1 -α)k(ǫ(t)) = α C 2 ǫ(t)(1 -ǫ(t))(ǫ(t) -ǫ) + (1 -α) ([l -(m + l)ǫ(t))ǫ(t)] = K(ǫ(t)|m, l) (dm2)
It is particularly noticeable that both dynamics are not reducible. Indeed, in the dynamics (drnl), the irrational dynamics appears as stacked over the replication dynamics, whereas in the dynamics (dm2) they work in parallel. In this case, for one unit value of α, we get back to the replication dynamics and for α = 0, we find the irrational dynamics. However, in spite of these differences, the two dynamics approach the same phenomenon in an increased complexity with regard to both elementary dynamics.

In what follows, we will adopt the subsequent strategy: we will pose α = 1, m = l = 0 from the replicator dynamics, and study the deformations induced on it from small variations of the parameters.

The (dm1) dynamics

Let us note that the (dml) dynamics is identical to the replicator dynamics for m = l = 0 because: H(ǫ(t)|0, 0) = h(ǫ(t)). As the presence of two new parameters fairly complicates the problem, let us begin by assuming that l ≡ 0. In such a way, we start with three equilibria: ǫ 0 =0, ǫ 1 and ǫ 2 .

l ≡ 0

In this case, let us leave from the replication dynamics to study the way a tiny variation apply on it. An equilibrium of the (dml) dynamics is characterized by: H(ǫ(t)|0, 0) = 0. Let us suppose that the value of m increases from m = 0.

The equilibria move in the neighborhood of ǫ i -i = 1, 2, 4 -in agreement with the equation:

H ǫ (ǫ(t)|0, 0)dǫ + H m (ǫ(t)|0, 0)dm = 0 ⇐⇒ dǫ(t) dm m=0 = - H m (ǫ i |0, 0) H ǫ (ǫ i |0, 0)
Now:

H m (ǫ i |0, 0) = -ǫ i H ǫ (ǫ i |0, 0) = C 2 -3ǫ 2 i + 2(1 -ǫ)ǫ i -ǫ
In consequence of what, a small variation of m induces -the results are shown in the (a)-quadrant of the figure 7 -:

① an invariance of ǫ 0 since in that case H m (ǫ i |0, 0) = 0. ② an increase of ǫ 2 = ǫ since :

dǫ dm m=0 = ǫ (C/2)ǫ(1 -ǫ) > 0 ③ a decrease of ǫ 2 = 1, which is driven inside [0, 1] since : dǫ dm m=0 = -ǫ (C/2)ǫ(1 -ǫ) < 0
So, a small variation of m in the neighborhood of m = 0 when l = 0, eliminates the all in C equilibrium, in which all the players become C playersi.e. : eliminates the Pareto-dominant equilibrium. In fact, the disappeared equilibrium swerves inward, which gives now two internal equilibrium, the new one being stable and at the heart of its attraction manifold.

As long as m remains smaller than m = (C/2)[(1ǫ)/2] 2 , nothing changes. Then, at m = m, there is a drastic qualitative change19 because, even if ǫ 0 remains stable, both interior equilibria merge to become a saddle point equilibrium -noted ǫ 5 -, which will be reached from every starting point ǫ(0) > ǫ 5 but not from a starting point ǫ(0) < ǫ 5 -see the (b) quadrant in figure 6. Finally, if m > m, ǫ 0 remains an equilibrium. The (d) quadrant in figure 6 -called the equilibrium bifurcation diagram -summarizes the qualitative properties of the equilibria. The main lesson induced by the perturbation of the replicator dynamics, with the exclusive irrational flight from the C players to the L players, is that, when m rises, the attraction manifold of the all L players equilibrium, is the emergence of a mixed stable equilibrium which may vanish with a further increase of m beyond m.

← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → → → → → → → → ← ← ← ← ǫ ǫ 0 1 ǫ ǫ 4 ǫ ′ m = 0 0 < m ≪ 1 (a) ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ǫ ǫ 0 1 ǫ ǫ 4 ǫ ′ m = 0 m = m < 1 (b) ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ǫ ǫ 0 1 ǫ ǫ 4 ǫ ′ m = 0 m = m < 1 (c) ǫ 0 ǫ 2 ǫ 1 1 2 equilibria 1 Equilibrium 1 equilibrium (d)
In this scenario, the risk dominant equilibrium tends to win over the Pareto equilibrium. One has :

① 3 equilibria for m ∈ [0, m[, ② 2 equilibria for m = m, ③ 1 equilibrium for m ∈] m, 1].
Furthermore, we can observe that it is really the irrational dynamics -a pure mutation dynamics -, which pollutes the replication dynamics because the subset of m value for which one has a stable mixed equilibrium is very small. This tends to confirm that the replication dynamics plays the central role in this scenario.

m ≡ 0 Now that the scenarii analysis tools have been set up, we will proceed with a graphic analysis based on figure 7. This time, a small variation of l induces the immediate disappearance of the equilibrium ǫ 0 = 0. In that case, it could not be any more the case of the convergence towards a state where nobody chose to become a C player. ǫ 2 remains a stable equilibrium in all the declinations of this situation. At first, two internal balances coexist, the equilibrium ǫ 5 arisen from the shift of the defunct ǫ 0 being stable, while the equilibrium ǫ ′ , born from the ǫ equilibrium remains unstable. But, when l reaches the limit value l = (C/2)(ǫ/2) 2 , there is a fork which compacts both internal equilibria in a unique saddle point, which immediately disappears. Only one equilibrium remains. One has :

① 3 equilibria for l ∈ [0, l[, ② 2 equilibria for l = l, ③ 1 equilibrium for l ∈] l, 1].
Here again, it is the irrational dynamics which pollutes the replication dynamics because, when m rises, ǫ and its siblings -interior equilibria generated by the m induced sliding of ǫ, change only at the margin in their qualitative behavior.

Naturally, the most realistic situation corresponds to m = 0, l = 0 and m = l. However, this situation is particularly complex and a more precise approach could be obtained in posing l = m. In fact, one can rely on intuition to assume that the values of m and l are not too much apart from each other. First of all, they are mutation rates and, by nature, they are restrained on [0, 1]. Then, if those mutations are irrational, i.e. induced by purely subjective elements without any reference to any strategy from mutants, in all likelihood one can suppose that, in a great population, we will observe very similar percentages of mutant from both populations.

m = l = k ǫ(t) = C 2 ǫ(t)(1 -ǫ(t))(ǫ(t) -ǫ) + k[1 -2ǫ(t)]
Once again, let us study the properties of this case from a purely graphical

→ → → → ← ← ← ← ← ← ← ← ← → → → → → → → → →→ → → → → → → ǫ ǫ 0 1 ǫ ǫ ′ 0 < l ≪ 1 l = 0 (a) → → → → → → → → → → → → → → → → → → → → →→ → → → → → → ǫ ǫ 0 1 ǫ ǫ ′ ǫ 5 m = 0 m = m < 1 (b) → → → → → → → → → → → → → → → → → → → → → → → → → → ǫ ǫ 0 1 ǫ ǫ 5 m = 0 m = m < 1 (c) ǫ 0 l ǫ 1 0 ǫ 2 3 equilibria 2 Equilibria 1 equilibrium (d)
Fig. 8. The mixed dynamics for m ≡ 0 point of view -see figure 8. In contrast with the two polar cases studied above, the focus in this case is concentrated on the internal mixed equilibrium. Indeed, not only the increase in k drives both extreme equilibria to their disappearance but the internal equilibrium nature is modified by the variations of k. For k < k = (C/2)(ǫ 2ǫ + 1), it is unstable, while stable for k > k. In fact, in k there is a bifurcation of the dynamics and the internal equilibrium is the only equilibrium to remain -see figure 10 for the separation hyperbola as a function of ǫ for C given -one must not forget that ǫ is a function of C and V.

Consequently, as soon as k exceeds k, the replication dynamics is absorbed by the irrational dynamics, which imposes its properties on the replication dynamics equilibrium. As far as a stable mixed equilibrium seems to better correspond to the observed situation of the labor market than any other equilibrium, the model preaches in favour of a significant pollution of the replication dynamics by the irrational dynamics. Contrary to what asserts [START_REF] Orlean | De la stabilité évolutionniste à la stabilité stochastique[END_REF], with the increase of k, the attraction manifold of the risk dominant equilibrium -but also the attraction manifold of the Paretodominant equilibrium -, if it can rise for a moment, eventually disappears to make way for a unique stable equilibrium.

This conclusion applies to the general model, i.e. to the model for which m = l. Indeed, by the methodology of the tiny variations in the neighbor- hood of m = l = 0, if the measure or the variation of m is equal to that of l, one finds the same result as the one from which we had just moved from. It would therefore be useful to explore in detail the case m = l.

→ → → → ← ← ← ← ← ← ← ← ← → → → → → → → →→ → → → ← ← ← ← ǫ ǫ 0 1 ǫ ǫ ′ 0 < k < k < 1 k = 0 (a) → → → → → → → ← ← ← ← ←←← ← ← ← ← ← ← ← ǫ ǫ 0 1 ǫ ǫ ′ ǫ 5 m = 0 m = m < 1 (b) → → → → → → ← ← ← ← ← ǫ ǫ 0 1 ǫ ǫ 6 m = 0 m = m < 1 (c) ǫ 0 l ǫ 1 0 ǫ 2 3 equilibria 1 Equilibria

The (dm2) dynamics

We can immediately observe that in the case of this representation of the pollution of the replication dynamics by the irrational dynamics, only one value of α = 1 allows to find the pareto-dominating balances and the dominant risk anew. The general case pleads for internal equilibria, even for a unique stable internal stable equilibrium when the irrational dynamics dominates the replication dynamics. Some simulations -which are not presented here -prove that the irrational dynamics dominates very quickly the replication dynamics. Consequently, the (dm2) dynamics seems to confirm the results obtained during the study of the (dm1) dynamics -see figure 11. In fact, this description is only true on the surface, because the dynamics embed a local emergence phenomenon. While one should expect that with the decrease of α the only observed behavior of the dynamics is the change of sign of its slope leading the internal equilibrium from instability to stability, we can observe for a very peculiar range of values of α the emergence of an unexpected unstable equilibrium .

Unfortunately, we can not show this emergence other than by specifying the parameter values. We have chosen ǫ = .5 and l = m = 0.1. For this specific range, a simple simulation shows that for the approximate value of their bounds, we have the emergence of an unstable internal equilibrium on ]0, ǫ[ when α ∈]0.167667, 0.285667[ -see figure 12 .

Conclusion

In this paper, one has tried to explain the persistence of a mixed equilibria composed of C players and L players. It is no way, at least in the evolutionary game formalism, a trivial affair since one is obliged to pollute the standard evolutionary game by arational behavior to generate such an equilibrium. One even shows that there is an embedded emergence phenomenon inside the mixed model. Many extensions to this approach are under study: First of all, a differential wage distinguishing job introduce according to their place in the dual hierarchy of the labor market can easily be introduced. Then one can also improve the description of the matching by introducing two distinct effects: agents can run several things at once, in that they can not receive a simple but several jobs and / or not be matched in pairs but by N-tuple. One can also introduce rigid agents who never change their mind according to the length of the study. Finally, the information available on the labor market is necessarily dated and it would be interesting to see what is the impact on the evolutionary dynamics of a delayed reaction In all the case, this approach reveal the complexity of the specific decision wether to undertake studies even in a stationary world.

Fig. 1 .

 1 Fig. 1. The graduates' downgrading process

Fig. 4 .

 4 Fig. 4. The replication dynamics

Fig. 7 .

 7 Fig. 7. The mixed dynamics for l ≡ 0

Fig. 9 .Fig. 10 .

 910 Fig. 9. The mixed dynamics for m = l = k

Fig. 11 .

 11 Fig. 11. the (dm2) dynamics

Fig. 12 .

 12 Fig. 12. Local emergence of an instable equilibrium

I would like to thank A. Clark, J.-B. Desquilbet, C. Nieradzik and C. Garrouste for their helpful comments. All remaining errors are the sole responsibility of the author.

Mariotte is the second name to appear in the appendix of[START_REF] Jevons | The Theory of Political Economy[END_REF] dedicated to mathematico-economic books, memoir, and other published writings.

This assumption originated from the great greek philosophers (essentially Aristote), followed by Daniel Bernoulli, Emmanuel Kant (for the moral point of view), Jeremy Bentham, Stanley Jevons and Von Neumann, among many others

See the survey published by the Céreq at http://www.cereq.fr/gsenew/ concours2008/cereq/G98ind/premierepage.htm.

Nevertheless, one can follow[START_REF] Piore | Dualism in the labor market : A response to uncertainty and flux. the case of france[END_REF]. According this theory, the labor market is divided in two sector : a primary which give right to better paying, promotion, more stables opportunities and a secondary which contains the poor paying unsecure and otherwise unattractive

jobs.6 It is not in our objective to explain why the non-qualified market can not be itself balanced, while we describe a balancing process by downgrading rejection of populations towards the least qualified jobs, and while the existence of an involuntary unemployment is a necessary condition for the justification of the model.

The term of mutation, borrowed from the biologist vocabulary, is here excessive. We can imagine that this phenomenon comes upstream from a human capital investment behavior or, more simply from the invasion of the labor market for the non-qualified by individuals stemming from a qualified labor market which has reached saturation, hypothesis which is defended in the body of the text.

In a finite game, every pure strategy Nash equilibrium is also a mixed strategy Nash equilibrium.

It is important to distinguish both because the first one is clearly less debated than the second. Numerous economists refuse to consider mixed strategies.

If we note x j (t) = ǫ(t)ǫ j , in the neighborhood of ǫ j , we get ẋj (t) = h ǫ (ǫ j )x j .

Since h(0) = 0 and h ǫ (0) < 0, as ǫ 2 ∈]0, 1[, h ǫ (ǫ 2 ) is, by necessity, positive.

When ǫ(0) = ǫ, the population repartition is temporally invariant.

This type of dynamics is not derived from the rationality of learning as in the evolutionary game and its associated replication dynamics. Nothing forbids the agents to base this type of behavior on another logic, even a logic of maximization.

H(ǫ|m, 0) has ǫ 0 = 0 and ǫ ± = (2 C) -1 [C(1 + ǫ) ± √ ∆] with ∆ = C 2 ǫ 2 -2 C 2 ǫ + C 2 -8Cm. One can see that, when m = m: ǫ + = ǫ -.