Labor market with demography: A mathematical analysis

Njike-Tchaptchet Eric Rostand<sup>a,*</sup>

<sup>a</sup>Committed Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Dschang P.O Box 67, Cameroon

**Abstract**

In this work we propose a system of nonlinear differential equations to analyze the relationship between the labor market and the demography. We obtain a unique positive equilibrium and prove its local asymptotic stability. Furthermore, under some condition on system's parameters, we prove the global asymptotic stability of the equilibrium by using a Lyapunov function. However, the condition obtained, suggests that the global stability is not economically desirable. We propose and solve an optimal control problem aiming to minimize at lowest cost the number of unemployed individuals by maintaining a higher level of labor market tightness. We also analyze the impact of some parameters on two labor market indicators. Particularly, it seems that increasing the legal working age has positive impact on the employment in short-term whereas increasing the natural birth rate has negative impact on the employment in the short term and a positive impact in the medium term. We provide numerical simulations to sustain the theoretical results.

Keywords: Mathematical modeling, Differential Equations, Unemployment,
Lyapunov function, Demography, Optimal control

MSC Classification: 34D05, 34D20, 34D23, 91B39, 91B74

**1. Introduction**

The unemployment is a complex phenomenon, resulting from a multiplicity of factors,
the interaction of which can vary over time and from country to another. Unemployment
can have dramatic consequences on society. Apart from financial misery, it causes a disin-
tegration of social ties for many unemployed individuals. Responsible governments work to
put in place policies to reduce unemployment in the short, medium and long term.
Unemployment also generates a significant cost for society. In fact, the wealth that could

<footnote>*Corresponding author
Email address: njikeeric@gmail.com (Njike-Tchaptchet Eric Rostand)</footnote>
have been produced by these unemployed people is a loss of income. Additional spending
to curb the effects of unemployment on health could have been used to improve the quality
of life of populations, in particular by investing in infrastructure. A better understanding
of the unemployment problem requires mastery of demography. The demographic growth
brings with it the need for health structures, schools and other infrastructures. This can
generate an increased demand for labor on the one hand and a possible increase in the active
population on the other.

There exists in literature many mathematical modeling approaches for the unemployment
problem. The efficiency wage approach for which the key assumption is that there is a ben-
efit as well as a cost to a firm for paying a higher wage. This approach has been proposed
among others by [9], [38], [33]. The implicit contract approach [[1], [8], [34], [14], [27], [6],
[28], [29], [35]], for which the main assumption is the existence of long-term relationships be-
tween firms and workers. The search and matching approach. Here the basic idea is that of
frictions in the labor market, making it costly (time-consuming) for workers to find firms and
vice versa. This approach has been proposed among others by [27], [30], [22], [23], [24], [21],
[3], [13], [32], [5], [4], [15], [26], [7], [36], [25], [18], [20], [19], [37]. For all the approaches pro-
posed in literature, none of them account for demography by simultaneously distinguishing
inactive individuals, unemployed individuals actively searching a job, employed individuals
and available vacancies. The contribution to the literature of this paper is to use the search
and matching approach to propose such model in order to shed light and improve our un-
derstanding of the thorny problem of unemployment. The optimal control problem proposed
with the objective of minimizing at lowest cost the number of unemployed by insuring the
ease to find a job can allow to decision makers to take appropriate actions to curd the effect
of unemployment.

This paper is organized as follows: in Section 2, we formulate the mathematical model using a
system of nonlinear ordinary differential equations. Section 3 is devoted to the mathematical
analysis of the model. In Section 4, we propose and solve an optimal control problem. Section
5 is devoted to numerical simulations to sustain theoretical results obtained. In Section 6,
we conclude the paper.

## 2. Model formulation

In this Section, we built the mathematical model under some appropriate assumptions.
## 2.1. The demography

For each time $t$ we denote by $N(t)$ the number of individuals in the entire population. To study the dynamics of $N(t)$, we make the following assumptions:

i) The repartition of individuals on the living space is homogeneous.

ii) There is no immigration or emigration.

iii) The birth rate $f$ is a nonlinear function of $N$ given by

$$f(N) = aN\left(1 - \frac{N}{A}\right), \qquad (1)$$

where $a \in (0, 1)$ is the natural birth rate and $A$ is a positive real number.

iv) The rate of change of $N(t)$ per unit of time is given by the relation

$$\dot{N}(t) = aN(t)\left(1 - \frac{N(t)}{A}\right) - \mu N(t), \qquad (2)$$

where $\mu \in (0, 1)$ is the death rate. $\frac{1}{\mu}$ is the expected lifetime.

In the sequel, we will assume that the natural birth rate is greater than the death rate. That is $a > \mu$.

## 2.2. The labor market

We assume that the entire population can be divided in two categories: i) the inactive individuals who are out of the labor market, ii) the labor force. The inactive part of the population contains the inactive individuals who belonged to the labor force but are either retired or have decided to permanently stop looking for a job and the inactive individuals under the legal working age. We denote by $I(t)$ and $J(t)$ the number of individuals who are definitely out of the labor market and the number of inactive individuals under the legal working age, at time $t$, respectively.

The labor force contains unemployed individuals that actively search a job and the employed individuals. We denote by $U(t)$ and $E(t)$ the number of unemployed and employed individuals, at time $t$, respectively. The labor market has two side: the supply and demand of labor. The supply of labor is the labor force and the demand of labor is the number of available vacancies provided by government and private sector. We denote by $V(t)$ the number of available vacancies at time $t$.

To model the labor market, we use the search and matching approach which suggests that unemployed individuals and available vacancies are heterogeneous. Furthermore, the stocks
of unemployed individuals and vacancies yield a flow of meetings between unemployed and re-cruiters. We assume that the number of meetings per unit of time is a function $g(U(t), V(t))$. This function proxies the complicated process of employer recruitment, worker search and mutual evaluation. In this paper, we assume that

$$g(U(t), V(t)) = U(t)V(t). \qquad (3)$$

The flow of meeting lead to a set of recruitment. We denote by $k \in (0, 1)$ the probability to be recruited after a meeting between an unemployed individual and an available vacancy. We make the following additional assumptions:

1. The rate of change per unit of time of the number of inactive individuals who are definitely out of the labor market depends on three factors:

* The number $mU(t)$ of unemployed individuals who decided to definitely abandon the active search of job. $m \in (0, 1)$.

* The number $rE(t)$ of retired workers. $r \in (0, 1)$ and $\frac{1}{r}$ is the average number of years of work required to be eligible for retirement benefits.

* The number of death $\mu I(t)$.

The rate of change of $I(t)$ per unit of time is summarized as follows:

$$\dot{I}(t) = mU(t) + rE(t) - \mu I(t). \qquad (4)$$

* The rate of change per unit of time of the number of individuals under the legal age of working depends on three factors:

- The number of newborns in the entire population $aN(t)\left(1 - \frac{N(t)}{A}\right)$.

- The number of death $\mu J(t)$.

- The number of $bJ(t)$ of inactive individuals who have the legal working age. Here, we assume that, once at the legal working age, individuals systematically have access to the labor market as unemployed. But, some of them can decide to not actively search a job.

The rate of change of $J(t)$ per unit of time is summarized as follows:

$$\dot{J}(t) = aN(t)\left(1 - \frac{N(t)}{A}\right) - (\mu + b)J(t). \qquad (5)$$

2. The rate of change per unit of time of the number of unemployed individuals that actively search a job depends on six factors:
* The number $bJ(t)$ of inactive individuals who have the legal working age.

* The number $mU(t)$ of unemployed individuals who decided to definitely abandon the active search of job.

* The number $\beta E(t)$ of individuals who lose their job as the result of job cuts due for example to economic reasons or the end of a contract. $\beta \in (0, 1)$.

* The number $kU(t)V(t)$ of individuals who success in the process of job matching.

* The number of death $\mu U(t)$.

* The number $\gamma U(t)$ of unemployed individuals who set up as self-employers. $\gamma \in (0, 1)$ is the self-employment rate.

The total change per unit of time of the number of unemployed individuals who actively search a job is summarized as follows:

$$ \dot{U}(t) = bJ(t) + \beta E(t) - kU(t)V(t) - (m + \mu + \gamma)U(t). \quad (6) $$

3. The rate of change per unit of time of the number of workers depends on five factors:

* The number $kU(t)V(t)$ of new hired individuals after the process of job matching.

* The number $\gamma U(t)$ of unemployed individuals who set up as self-employers.

* The number $\beta E(t)$ of individuals who lose their job as the result of job cuts due for example to economic reasons or the end of a contract.

* The number of death $\mu E(t)$.

* The number of retired $rE(t)$.

The total change per unit of time of the number of workers is summarized as follows:

$$ \dot{E}(t) = \gamma U(t) + kU(t)V(t) - (\beta + \mu + r)E(t). \quad (7) $$

4. The rate of change per unit of time of the number of available vacancies depends on six factors:

* The number $\sigma_0 U(t)$ of new vacancies created by private companies already in operation and the government. $\sigma_0 \in (0, 1)$.

* The number $\alpha \gamma U(t)$ of vacancies created by new self-employed people. These workstations are necessary for the start of the activities of the new companies. $\alpha > 0$ is a modification parameter.

* The number $\mu E(t)$ of dead workers leaving their positions vacant.
• The number $rE(t)$ of retired workers leaving their positions vacant.

• The number $\delta V(t)$ of vacancies that are canceled due to economic reasons.

• The number $kU(t)V(t)$ of vacancies filled after the matching process.

The total change per unit of time of the number of available vacancies is summarized as follows:

$$ \dot{V}(t) = (\sigma_0 + \alpha\gamma)U(t) + (r + \mu)E(t) - \delta V(t) - kU(t)V(t) \quad (8) $$

In this paper, we use two main indicators to study the labor market:

• The unemployment rate $\tau$ defined as the percentage of people in the labor force who are unemployed. For a given time $t$, it is given by the relation

$$ \tau(t) = \frac{U(t)}{U(t) + E(t)}. \quad (9) $$

• The labor market tightness $\xi$. The labor market is said to be tight when the imbalance between labor supply and labor demand manifests itself in an abundance of job opportunities along with scarcity of workers available and willing to take those jobs. For a given time $t$, it is given by the relation

$$ \xi(t) = \frac{V(t)}{U(t)}. \quad (10) $$

Putting together the equations (2), (4), (5), (6), (7) and (8), we obtain the following system of ordinary differential equations modeling the labor market with demography.

$$ \left\{ \begin{aligned} \dot{N}(t) &= aN(t)\left(1 - \frac{N(t)}{A}\right) - \mu N(t), \\ \dot{I}(t) &= mU(t) + rE(t) - \mu I(t), \\ \dot{J}(t) &= aN(t)\left(1 - \frac{N(t)}{A}\right) - (\mu + b)J(t), \\ \dot{U}(t) &= bJ(t) + \beta E(t) - kU(t)V(t) - (m + \mu + \gamma)U(t), \\ \dot{E}(t) &= \gamma U(t) + kU(t)V(t) - (\beta + \mu + r)E(t), \\ \dot{V}(t) &= (\sigma_0 + \alpha\gamma)U(t) + (r + \mu)E(t) - \delta V(t) - kU(t)V(t). \end{aligned} \right. \quad (11) $$

Since the number of individuals in the entire population $N(t)$ satisfies:

$$ N(t) = I(t) + J(t) + U(t) + E(t), \quad (12) $$
the system (11) can be rewritten as

$$
\left\{
\begin{aligned}
\dot{I}(t) &= mU(t) + rE(t) - \mu I(t), \\
\dot{J}(t) &= a \left( I(t) + J(t) + U(t) + E(t) \right) \left( 1 - \frac{I(t)+J(t)+U(t)+E(t)}{A} \right) - (\mu + b)J(t), \\
\dot{U}(t) &= bJ(t) + \beta E(t) - kU(t)V(t) - (m + \mu + \gamma)U(t), \\
\dot{E}(t) &= \gamma U(t) + kU(t)V(t) - (\beta + \mu + r)E(t), \\
\dot{V}(t) &= (\sigma_0 + \alpha\gamma)U(t) + (r + \mu)E(t) - \delta V(t) - kU(t)V(t).
\end{aligned}
\right.
\qquad (13)
$$

3. Mathematical analysis of the model

In this Section, we investigate the main properties of the system (13).

3.1. Positivity, boundedness and global existence of solutions

The following result is about the positivity, the boundedness and the global definition of
a solution to (13).

(14)

**Theorem 3.1.** Let $I_0, J_0, U_0, E_0$ and $V_0$ be given initial condition such that $I_0 > 0, J_0 > 0, U_0 > 0, E_0 > 0$ and $V_0 > 0$.

i) The Cauchy problem

$$
\left\{
\begin{aligned}
\dot{I}(t) &= mU(t) + rE(t) - \mu I(t), \\
\dot{J}(t) &= a \left( I(t) + J(t) + U(t) + E(t) \right) \left( 1 - \frac{I(t)+J(t)+U(t)+E(t)}{A} \right) - (\mu + b)J(t), \\
\dot{U}(t) &= bJ(t) + \beta E(t) - kU(t)V(t) - (m + \mu + \gamma)U(t), \\
\dot{E}(t) &= \gamma U(t) + kU(t)V(t) - (\beta + \mu + r)E(t), \\
\dot{V}(t) &= (\sigma_0 + \alpha\gamma)U(t) + (r + \mu)E(t) - \delta V(t) - kU(t)V(t), \\
(I(0), J(0), U(0), E(0), V(0)) &= (I_0, J_0, U_0, E_0, V_0),
\end{aligned}
\right.
\qquad (15)
$$

admits a unique continuous solution $(I(t), J(t), U(t), E(t), V(t))$ defined for all $t \in [0, T)$, where $T \in (0, \infty)$. Furthermore, we have $I(t) > 0, J(t) > 0, U(t) > 0, E(t) > 0$ and $V(t) > 0$ for all $t \in [0, T)$.

ii) The set Ω defined by

$$
\Omega = \left\{ (I, J, U, E, V) \in \mathbb{R}^5, I, J, U, E > 0, 0 < I+J+U+E \le \frac{(a-\mu)A}{a}, 0 < V < \frac{(\sigma+r+\mu)(a-\mu)A}{a\delta} \right\} \quad (16)
$$

is positively invariant for the system (13). Here $\sigma = \sigma_0 + \alpha\gamma$.
iii) The solution $(I(t), J(t), U(t), E(t), V(t))$ is defined for all $t \ge 0$.

The following result is useful to prove the positivity of solution to (15).

**Lemma 3.1.** Let $N_0 > 0$ be the initial number of individuals in the entire population.

Then,

$$N(t) > 0, \quad \forall t \ge 0.$$

Furthermore, if $N_0 \le \frac{(a-\mu)A}{a}$, then

$$N(t) \le \frac{(a - \mu)A}{a}, \quad \forall t > 0.$$

*Proof.* Let $N_0 > 0$. The equation (2) can be rewritten as

$$\dot{N}(t) = (a - \mu)N(t) \left(1 - \frac{aN(t)}{(a - \mu)A}\right).$$

That is,

$$N(t) = \frac{(a - \mu)A}{a \left( 1 + \left( \frac{(a - \mu)A}{a N_0} - 1 \right) \exp \left( -(a - \mu)t \right) \right)}, \quad \forall t \ge 0.$$

Since $N_0 \le \frac{(a-\mu)A}{a}$, then

$$\limsup_{t \to \infty} N(t) \le \frac{(a - \mu)A}{a}. \qquad (17)$$

That is, $N(t) \le \frac{(a-\mu)A}{a}$ for all $t \ge 0$. □

In the sequel, we assume that $N_0 \le \frac{(a-\mu)A}{a}$.

*Proof.* Let $I_0, J_0, U_0, E_0$ and $V_0$ be given initial condition such that $I_0 > 0, J_0 > 0, U_0 > 0, E_0 > 0$ and $V_0 > 0$.

i) The system (13) can be rewritten as follows:

$$\dot{X}(t) = \Phi(X(t)), \qquad (18)$$

where

$$\Phi(I, J, U, E, V) = \begin{pmatrix} mU + rE - \mu I \\ a(I + J + U + E)\left(1 - \frac{I + J + U + E}{A}\right) - (\mu + b)J \\ bJ + \beta E - kUV - (m + \mu + \gamma)U \\ \gamma U + kUV - (\beta + \mu + r)E \\ (\sigma_0 + \alpha\gamma)U + (r + \mu)E - \delta V - kUV \end{pmatrix}.$$

The function $\Phi$ is continuously differentiable on $\mathbb{R}^5$ and therefore, is locally Lipschitz.

Using the Cauchy-Lipschitz Theorem [17], the Cauchy problem (15) admits a unique
local solution $(I(t), J(t), U(t), E(t), V(t))$. That is, there exists a positive real number $T$ such that $(I(t), J(t), U(t), E(t), V(t))$ is defined for all $t \in [0, T)$.

Now, we prove the positivity of the solution $I(t), J(t), U(t), E(t)$, and $V(t)$ for all $t \in [0, T)$.

Let prove the positivity of $J(t)$ for all $t \in [0, T)$.

From Lemma 3.1, the second equation of (13) becomes

$$ \dot{J}(t) \geq \mu N(t) - (\mu + b)J(t) > -(\mu + b)J(t). \quad (19) $$

Using the comparison Theorem [16], we deduce from the inequality (19)

$$ J(t) > J_0 \exp(-(\mu + b)t) > 0, \quad \forall t \in [0, T). \quad (20) $$

To prove the positivity of $U(t)$, we assume that there exists some $t \in (0, T)$ such that $U(t) < 0$. Let $t_1 \in (0, T)$ be the first time such that $U(t_1) < 0$. Hence, as $U$ is a continuous function of time, there exists $t_2 \in (0, t_1)$ such that

$$ U(t_2) = 0, \quad J(t_2) > 0, \quad E(t_2) \geq 0, \quad \text{and} \quad V(t_2) \geq 0. $$

Using the third equation of the system (13), we obtain for $t = t_2$

$$ \dot{U}(t_2) = bJ(t_2) + \beta E(t_2) > 0. \quad (21) $$

The relation (21) implies that the function $U : t \mapsto U(t)$ cannot assume negative values. The positivity of $I(t), E(t)$ and $V(t)$ all $t \in [0, T)$ are proven similarly.

ii) Adding the first fourth equations of the system (13), we obtain the equation of the dynamics of $N(t)$ given by the equation (2).

If $I_0 + J_0 + U_0 + E_0 = N_0 \le \frac{(a-\mu)A}{a}$, then from Lemma 3.1, $\limsup_{t \to \infty} N(t) \le \frac{(a-\mu)A}{a}$.

Therefore, as $N(t) = I(t) + J(t) + U(t) + E(t)$ and $I(t), J(t), U(t), E(t)$ are positive real numbers, we have

$$ I(t) < \frac{(a - \mu)A}{a}, \quad J(t) < \frac{(a - \mu)A}{a}, \quad U(t) < \frac{(a - \mu)A}{a}, \quad E(t) < \frac{(a - \mu)A}{a}, \quad \text{for all } t \ge 0. \quad (22) $$

From the fifth equation of system (13) and the inequalities (22), we have

$$ \dot{V} < ((\sigma_0 + \alpha\gamma) + r + \mu) \frac{(a - \mu)A}{a} - \delta V. \quad (23) $$

If, $V_0 < ((\sigma_0 + \alpha\gamma) + r + \mu) \frac{(a-\mu)A}{a\delta}$, using the comparison Theorem [16], we obtain from the inequality (23)

$$ \limsup_{t \to \infty} V(t) < ((\sigma_0 + \alpha\gamma) + r + \mu) \frac{(a - \mu)A}{a\delta}. $$
That is, $V(t) < ((\sigma_0 + \alpha\gamma) + r + \mu) \frac{(a-\mu)A}{a\delta}$ for all $t \ge 0$.

iii) From ii) the solution $(I(t), J(t), U(t), E(t), V(t))$ is contained in a compact subset of $\mathbb{R}^5$ and therefore is globally defined.

□

## 3.2. The existence and stability of stationary solutions

In what follows, we investigate the existence and stability of stationary solutions to (13).

### 3.2.1. The existence of equilibria

The following result is about the existence of equilibria for system (13).

**Theorem 3.2.** The system (13) admits a unique equilibrium $P_* = (I_*, J_*, U_*, E_*, V_*)$ in $\Omega$ given by:

$$I_* = \frac{(a - \mu)Ab(m(\beta + \mu + r) + r(kV_* + \gamma))}{a(b + \mu)((kV_* + \gamma + m + \mu)(r + \mu) + \beta(m + \mu))}, \quad (24)$$

$$J_* = \frac{(a - \mu)\mu A}{a(b + \mu)}, \quad (25)$$

$$U_* = \frac{(\beta + \mu + r)(a - \mu)Ab\mu}{a(b + \mu)((kV_* + \gamma + m + \mu)(r + \mu) + \beta(m + \mu))}, \quad (26)$$

$$E_* = \frac{(a - \mu)Ab\mu(kV_* + \gamma)}{a(b + \mu)((kV_* + \gamma + m + \mu)(r + \mu) + \beta(m + \mu))}, \quad (27)$$

and

$$V_* = \frac{-\delta a(b + \mu)((m + \mu)(\beta + \mu + r) + \gamma(r + \mu)) + k\beta(a - \mu)Ab\mu + \sqrt{\Delta}}{2k\delta a(r + \mu)(b + \mu)}, \quad (28)$$

with

$$\Delta = \left( \delta a(b + \mu)((m + \mu)(\beta + \mu + r) + \gamma(r + \mu)) + k\beta(a - \mu)Ab\mu \right)^2 + 4k\delta a(r + \mu)(b + \mu) \left( (a - \mu)Ab\mu(\beta(\sigma_0 + \alpha\gamma) + (\gamma + (\sigma_0 + \alpha\gamma))(r + \mu)) \right).$$

*Proof.* Let $t \ge 0$. A point $(I(t), J(t), U(t), E(t), V(t))$ is a stationary solution to system (13), if it satisfies the relation

$$\dot{I}(t) = \dot{J}(t) = \dot{U}(t) = \dot{E}(t) = \dot{V}(t) = 0. \quad (29)$$
That is,

$$ egin{cases} mU(t) + rE(t) - \mu I(t) = 0, \\ a(I(t) + J(t) + U(t) + E(t)) \left( 1 - \frac{I(t)+J(t)+U(t)+E(t)}{A} \right) - (\mu + b)J(t) = 0, \\ bJ(t) + \beta E(t) - kU(t)V(t) - (m + \mu + \gamma)U(t) = 0, \\ \gamma U(t) + kU(t)V(t) - (\beta + \mu + r)E(t) = 0, \\ (\sigma_0 + \alpha\gamma)U(t) + (r + \mu)E(t) - \delta V(t) - kU(t)V(t) = 0. \end{cases} \quad (30) $$

Adding the first four equations of (30), we obtain the relation

$$ a(I(t)+J(t)+U(t)+E(t)) \left( 1 - \frac{I(t)+J(t)+U(t)+E(t)}{A} \right) - \mu(I(t)+J(t)+U(t)+E(t)). \quad (31) $$

That is,

$$ I + J + U + E = 0, \quad \text{or} \quad I + J + U + E = \frac{(a - \mu)A}{a}. $$

For $I + J + U + E = 0$, the positivity of $I(t)$, $J(t)$, $U(t)$ and $E(t)$ leads to

$$ I = J = U = E = 0. $$

This is not economically meaningful. Therefore

$$ I + J + U + E = \frac{(a - \mu)A}{a}. \quad (32) $$

Using the second equation of (30) and the relation (32), we obtain

$$ J = \frac{(a - \mu)\mu A}{a(b + \mu)}. \quad (33) $$

Using the first, the fourth and the fifth equation of (30) and after some computations, we obtain

$$ egin{cases} I = \frac{(a-\mu)Ab(m(\beta+\mu+r)+r(kV+\gamma))}{a(b+\mu)((kV+\gamma+m+\mu)(r+\mu)+\beta(m+\mu))}, \\ U = \frac{(\beta+\mu+r)(a-\mu)Ab\mu}{a(b+\mu)((kV+\gamma+m+\mu)(r+\mu)+\beta(m+\mu))}, \\ E = \frac{(a-\mu)Ab\mu(kV+\gamma)}{a(b+\mu)((kV+\gamma+m+\mu)(r+\mu)+\beta(m+\mu))}. \end{cases} \quad (34) $$

With $V$ satisfying the following second order polynomial equation (35)

$$ e_2 V^2 + e_1 V + e_0 = 0, \quad (35) $$

where

$$ e_0 = (a - \mu)Ab\mu(\beta(\sigma_0 + \alpha\gamma) + (\gamma + (\sigma_0 + \alpha\gamma))(r + \mu)), $$

$$ e_1 = -\delta a(b + \mu)((m + \mu)(\beta + \mu + r) + \gamma(r + \mu)) + k\beta(a - \mu)Ab\mu, \text{ and} $$

$$ e_2 = -k\delta a(r + \mu)(b + \mu). \quad \square $$
The discriminant $\Delta$ of the polynomial $e_2V^2 + e_1V + e_0$ is a positive real number. That is, the equation (35) admits two real numbers as solutions. Furthermore the product $\frac{e_0}{e_2}$ and the sum $-\frac{e_1}{e_2}$ of these solutions are negative real numbers. After some computations, the appropriate value of $V$ is:

$$V = \frac{-\delta a(b + \mu)((m + \mu)(\beta + \mu + r) + \gamma(r + \mu)) + k\beta(a - \mu)Ab\mu + \sqrt{\Delta}}{2k\delta a(r + \mu)(b + \mu)},$$

with

$$\Delta = \left( \delta a(b + \mu)((m + \mu)(\beta + \mu + r) + \gamma(r + \mu)) + k\beta(a - \mu)Ab\mu \right)^2 + 4k\delta a(r + \mu)(b + \mu) \left( (a - \mu)Ab\mu(\beta(\sigma_0 + \alpha\gamma) + (\gamma + (\sigma_0 + \alpha\gamma))(r + \mu)) \right). (36)$$

### 3.2.2. The stability of equilibrium

The following result is about the local stability of the unique positive equilibrium.

**Theorem 3.3.** The unique equilibrium $P_* = (I_*, J_*, U_*, E_*, V_*)$ is locally asymptotically stable.

*Proof.* Linearizing the function $\Phi(I, J, U, E, V)$ around the equilibrium $P_*$, we obtain the following Jacobian matrix

$$J(P_*) = \begin{pmatrix} -\mu & 0 & m & r & 0 \\ -a + 2\mu & -(a - \mu) - b & -a + 2\mu & -a + 2\mu & 0 \\ 0 & b & -kV_* - (\gamma + \mu + m) & \beta & -kU_* \\ 0 & 0 & \gamma + kV_* & -(\beta + r + \mu) & kU_* \\ 0 & 0 & (\sigma_0 + \alpha\gamma) - kV_* & r + \mu & -\delta - kU_* \end{pmatrix}. \quad (37)$$

The characteristics equation of the matrix $J(P_*)$ is the polynomial $Q(\lambda)$ given by

$$Q(\lambda) = \left(\lambda + \mu\right) \left(\lambda + (a - \mu + b)\mu\right) \left(\lambda + \mu\right) \left((a - \mu)((kV_* + \gamma + \mu + m) + b(m + \mu)) + b(kV_* + \gamma)\right) \left(\lambda^2 + c_1\lambda + c_0\right), \quad (38)$$

where,

$$c_1 = (a - \mu)(b + \mu)((r + \mu)(kV_* + \gamma + m + \mu) + \beta(m + \mu)) + (a - \mu + b)\mu((\gamma + (\sigma_0 + \alpha\gamma))kU_* + \delta(kV_* + \gamma)) + (\delta + kU_*)(a - \mu)(m + \mu)(b + \mu),$$

$$c_0 = kU_*(a - \mu)(m + \mu)(b + \mu)(kV_*\beta\mu(a - \mu + b) + (\sigma_0 + \alpha\gamma)(a - \mu)b(r + \mu)) + \beta(m + \mu)\left((a - \mu + b)\mu((\gamma + (\sigma_0 + \alpha\gamma))kU_* + \delta(kV_* + \gamma)) + (\delta + kU_*)(a - \mu)(m + \mu)(b + \mu)\right)$$
$$+(a - \mu)(b + \mu)(r + \mu) \times \left[ \begin{aligned} & \gamma\mu \left( (kV_* + \gamma + m + \mu)(\delta(a - \mu + b) + (a - \mu)kU_*) + (kV_* + \gamma)kU_*b \right) \\ & + (\sigma_0 + \alpha\gamma)kU_*\mu \left( (a - \mu)(kV_* + \gamma + m + \mu) + b(kV_* + \gamma) \right) + (kV_* + \gamma + m + \mu)\delta kV_*\mu(a - \mu + b) \\ & + (m + \mu)(\delta(b + \mu)(kV_* + \gamma + m + \mu) + bkU_*(kV_* + \gamma)) \end{aligned} \right].$$

Solving the equation $Q(\lambda) = 0$, we obtain

$$\lambda = -\mu < 0, \quad \lambda = -(a-\mu+b)\mu < 0, \quad \lambda = -\mu \left( (a-\mu)((kV_*+\gamma+\mu+m)+b(m+\mu)) + b(kV_*+\gamma) \right) < 0$$

or

$$\lambda^2 + c_1\lambda + c_0 = 0. \quad (39)$$

Since $c_1 > 0$ and $c_0 > 0$, the equation (39) admits two solutions with negative real parts.

Therefore the Jacobian matrix possesses five eigenvalues with negative real parts and the equilibrium $P_*$ is locally asymptotically stable. $\square$

The following result is about the global stability of the equilibrium $P_*$.

**Theorem 3.4.** If the condition

$$(r+u)E_* \geq \left(1 + \frac{kV_* + \gamma(1+\alpha) + \sigma_0}{m+\mu}\right)kU_*V_* \quad (40)$$

is satisfied, then the unique positive equilibrium $P_*$ is globally asymptotically stable.

The following lemmas will be useful in the proof of Theorem 3.4.

**Lemma 3.2.** For all initial condition $(I_0, J_0, U_0, E_0, V_0) \in \Omega$, we have

$$\lim_{t \to \infty} \left( I(t) + J(t) + U(t) + E(t) \right) = \frac{(a - \mu)A}{a}. \quad (41)$$

*Proof.* Let $N_0 = I_0 + J_0 + U_0 + E_0$ such that $(I_0, J_0, U_0, E_0, V_0) \in \Omega$. We recall that the dynamics of $N(t) = I(t) + J(t) + U(t) + E(t)$ for all $t \ge 0$ and satisfies the relation (2). The function $L_0$ defined by

$$L_0(N) = \frac{1}{2} \left( N - \frac{(a - \mu)A}{a} \right)^2 \quad (42)$$

is a Lyapunov function for the dynamics of $N(t)$.

In fact, deriving the function $L_0$ along a solution to (2) leads to

$$\frac{d}{dt} L_0(t) = -\frac{aN(t)}{A} \left( N(t) - \frac{(a - \mu)A}{a} \right)^2 \le 0. \quad (43)$$

Moreover, $\frac{d}{dt} L_0(t) = 0$ holds if and only if $N = \frac{(a-\mu)A}{a}$. By the LaSalle's Principle [12], it follows that every solution $N(t)$ of equation (2) with initial condition $N_0 = I_0 + J_0 + U_0 + E_0$, such that $(I_0, J_0, U_0, E_0, V_0) \in \Omega$, approaches the value $N_* = \frac{(a-\mu)A}{a}$ as $t \to \infty$. $\square$
**Lemma 3.3.** Given initial condition $(I_0, J_0, U_0, E_0, V_0) \in \Omega$, each solution $(I(t), J(t), U(t), E(t), V(t))$ to system (13) is such that

i)

$$ \lim_{t \to \infty} J(t) = J_* = \frac{(a - \mu)\mu A}{a(b + \mu)}. \qquad (44) $$

ii)

$$ \liminf_{t \to \infty} U(t) \geq \frac{b\delta(a - \mu)\mu A}{(b + \mu)((\gamma + m + \mu)\delta a + kA((\sigma_0 + \alpha\gamma) + r + \mu)(a - \mu))}. \qquad (45) $$

In the sequel, we denote $\liminf_{t \to \infty} U(t) = U_\infty$.

*Proof.* Let $(I_0, J_0, U_0, E_0, V_0) \in \Omega$ be given initial condition.

i) From relation (41) of Lemma 3.2, for all $\epsilon > 0$ sufficiently small, there exists $t_1 > 0$ such that for all $t > t_1$, we have

$$ \frac{(a - \mu)\mu A}{a} - \epsilon \leq aN(t)\left(1 - \frac{N(t)}{A}\right) \leq \frac{(a - \mu)\mu A}{a} + \epsilon. \qquad (46) $$

Using the second equation of (13), we deduce that for all $t > t_1$,

$$ \frac{(a - \mu)\mu A}{a} - \epsilon - (\mu + b)J(t) \leq \dot{J}(t) \leq \frac{(a - \mu)\mu A}{a} + \epsilon - (\mu + b)J(t). \qquad (47) $$

Using the comparison Theorem [16] and taking the limit as $t$ tends to infinity, on the left and the right inequalities of (47), we get

$$ \frac{(a - \mu)\mu A}{a(b + \mu)} - \frac{\epsilon}{(b + \mu)} \leq \lim_{t \to \infty} J(t) \leq \frac{(a - \mu)\mu A}{a(b + \mu)} + \frac{\epsilon}{(b + \mu)}. \qquad (48) $$

From the arbitrariness of $\epsilon$, the relation (48) leads to

$$ \lim_{t \to \infty} J(t) = \frac{(a - \mu)\mu A}{a(b + \mu)}. $$

ii) From i), $\lim_{t \to \infty} J(t) = \frac{(a - \mu)\mu A}{a(b + \mu)}$. Therefore, for all $\epsilon' > 0$ sufficiently small, there exists $t_2 > 0$ such that for all $t > t_2$ sufficiently large,

$$ \frac{(a - \mu)\mu A}{a(b + \mu)} - \epsilon' \leq J(t) \leq \frac{(a - \mu)\mu A}{a(b + \mu)} + \epsilon'. $$

Using the third equation of (13), we obtain the inequality

$$ \dot{U}(t) \geq \frac{b(a - \mu)\mu A}{a(b + \mu)} - b\epsilon' - \left(\gamma + m + \mu + \frac{k((\sigma_0 + \alpha\gamma) + r + \mu)(a - \mu)A}{\delta a}\right)U(t). \qquad (49) $$
Using once more the comparison Theorem [16], with the inequality(49), we obtain for all $t > t_2$ sufficiently large

$$ U(t) \geq U_0 e^{-(\gamma+m+\mu+\frac{k((\sigma_0+\alpha\gamma)+r+\mu)(a-\mu)A}{\delta a})t} + C_0 \left(1 - \exp\left(-(\gamma+m+\mu+\frac{k((\sigma_0+\alpha\gamma)+r+\mu)(a-\mu)A}{\delta a})t\right)\right) - C_1 \epsilon' \left(1 - \exp\left(-(\gamma+m+\mu+\frac{k((\sigma_0+\alpha\gamma)+r+\mu)(a-\mu)A}{\delta a})t\right)\right), \quad (50) $$

where,

$$ C_0 = \frac{b\delta(a-\mu)\mu A}{(b+\mu)((\gamma+m+\mu)\delta a + kA((\sigma_0+\alpha\gamma)+r+\mu)(a-\mu))} \quad \text{and} $$

$$ C_1 = \frac{b\delta a}{((\gamma+m+\mu)\delta a + kA((\sigma_0+\alpha\gamma)+r+\mu)(a-\mu))}. $$

Taking the limit as $t$ tends to infinity in the above inequality and using the arbitrariness of $\epsilon'$, we obtain

$$ \liminf_{t \to \infty} U(t) \geq \frac{b\delta(a-\mu)\mu A}{(b+\mu)((\gamma+m+\mu)\delta a + kA((\sigma_0+\alpha\gamma)+r+\mu)(a-\mu))} \quad \square $$

We give now the proof of Theorem 3.4.

*Proof.* We consider the following function defined in $\Omega$ with values in $\mathbb{R}_+$

$$ L_2 = \frac{1}{2}(N - N_*)^2 + (J - J_* \ln J) + (U - U_* \ln U) + \alpha_1(E - E_* \ln E) + \alpha_2(V - V_* \ln V), \quad (51) $$

where $\alpha_1$ and $\alpha_2$ are positive real numbers to be determined.

We recall that $N = I + J + E + U$ and $N_* = I_* + J_* + U_* + E_*$.

Deriving $L_2$ along a trajectory of a solution to (13), we obtain:

$$ \begin{aligned} \frac{dL_2}{dt} &= (N - N_*)\dot{N} + \left(1 - \frac{J_*}{J}\right)\dot{J} + \left(1 - \frac{U_*}{U}\right)\dot{U} + \alpha_1\left(1 - \frac{E_*}{E}\right)\dot{E} + \alpha_2\left(1 - \frac{V_*}{V}\right)\dot{V} \\ &= -\frac{aN}{A}(N - N_*)^2 + \left(1 - \frac{J_*}{J}\right)\left(aN\left(1 - \frac{N}{A}\right) - (\mu + b)J\right) \\ &\quad + \left(1 - \frac{U_*}{U}\right)\left[bJ - bJ_*\frac{U_*}{U} + \beta E_*\left(\frac{E}{E_*} - \frac{U}{U_*}\right) - kU_*V_*\left(\frac{UV}{U_*V_*} - \frac{U}{U_*}\right)\right] \\ &\quad + \alpha_1\left(1 - \frac{E_*}{E}\right)\left[\gamma U_*\left(\frac{U}{U_*} - \frac{E}{E_*}\right) + kU_*V_*\left(\frac{UV}{U_*V_*} - \frac{U}{U_*}\right)\right] \\ &\quad + \alpha_2\left(1 - \frac{V_*}{V}\right)\left[(\sigma_0 + \alpha\gamma)U_*\left(\frac{U}{U_*} - \frac{V}{V_*}\right) + (r + \mu)E_*\left(\frac{E}{E_*} - \frac{V}{V_*}\right)\right]. \end{aligned} \quad (52) $$

Let $\epsilon > 0$ sufficiently small.

From Lemma 3.2 and Lemma 3.3, there exists $\tilde{T} > 0$ such that for all $t > \tilde{T}$ large enough, we get

$$ F_1 - C(\epsilon)\epsilon < \frac{dL_2}{dt} < F_1 + C(\epsilon)\epsilon, \quad (53) $$
where $C(\epsilon) = \frac{2J_*}{J_* - \epsilon} + b\left(1 + \frac{U_*}{U_\infty}\right)$ and

$$F_1 = -\frac{aN}{A}(N-N_*)^2 - \frac{b+\mu}{J}(J-J_*)^2 + \left(1-\frac{U_*}{U}\right)\left[bJ_*\left(1-\frac{U_*}{U}\right) + \beta E_*\left(\frac{E}{E_*}-\frac{U_*}{U}\right) - kU_*V_*\left(\frac{UV}{U_*V_*}-\frac{U_*}{U}\right)\right] \\ +\alpha_1\left(1-\frac{E_*}{E}\right)\left[\gamma U_*\left(\frac{U_*}{U}-\frac{E_*}{E}\right) + kU_*V_*\left(\frac{UV}{U_*V_*}-\frac{E_*}{E}\right)\right] \\ +\alpha_2\left(1-\frac{V_*}{V}\right)\left[(\sigma_0+\alpha\gamma)U_*\left(\frac{U_*}{U}-\frac{V_*}{V}\right) + (r+\mu)E_*\left(\frac{E}{E_*}-\frac{V_*}{V}\right) - kU_*V_*\left(\frac{UV}{U_*V_*}-\frac{V_*}{V}\right)\right].$$

Choosing $\alpha_1 = \frac{m+\mu}{\gamma(1+\alpha)+\sigma_0+kV_*}$ and $\alpha_2 = \frac{kV_*+\gamma+m+\mu+(\sigma_0+\alpha\gamma)}{kV_*+\gamma(1+\alpha)+\sigma_0}$, for $t > \tilde{T}$ large enough, the relation (53) becomes

$$F_2 - C(\epsilon)\epsilon < \frac{dL_2}{dt} < F_2 + C(\epsilon)\epsilon, \quad (54)$$

with

$$F_2 = -\frac{aN}{A}(N-N_*)^2 - \frac{b+\mu}{J}(J-J_*)^2 + \beta E_* \left( 2 - \frac{EU_*}{E_*U} - \frac{UE_*}{U_*E} \right) \\ + \left( r + \mu + \frac{\beta(m+\mu)}{kV_* + \gamma(1+\alpha) + \sigma_0} \right) E_* \left( 3 - \frac{E}{E_*} - \frac{U_*}{U} - \frac{UE_*}{U_*E} \right) \\ + \frac{(m+\mu)(\sigma_0 + \alpha\gamma)U_*}{kV_* + \gamma(1+\alpha) + \sigma_0} \left( 3 - \frac{V}{V_*} - \frac{U_*}{U} - \frac{UV_*}{U_*V} \right) \\ + \left( 1 + \frac{m+\mu}{kV_* + \gamma(1+\alpha) + \sigma_0} \right) kU_*V_* \left( 3 - \frac{U_*}{U} - \frac{UVE_*}{U_*V_*E} - \frac{EV_*}{E_*V} \right) \\ + \left( \frac{(m+\mu)(r+\mu)E_* - kU_*V_* (kV_* + \gamma(1+\alpha) + m + \mu + \sigma_0)}{kV_* + \gamma(1+\alpha) + \sigma_0} \right) \left( 4 - \frac{V}{V_*} - \frac{U_*}{U} - \frac{EV_*}{E_*V} - \frac{UE_*}{U_*E} \right).$$

If $(r+u)E_* \ge (1+\frac{kV_*+\gamma(1+\alpha)+\sigma_0}{m+\mu})kU_*V_*$, using the arithmetic-geometric mean, we get $F_2 \le 0$ for $t$ sufficiently large. From the arbitrariness of $\epsilon$, the relation (54) leads to

$$\frac{dL_2}{dt} = F_2 \le 0. \quad (55)$$

Furthermore, $P_*$ is the unique point in $\Omega$ such that $F_2 = 0$. Therefore, from LaSalle's Principle [12], it follows that $P_*$ is globally asymptotically stable.

This complete the proof of Theorem 3.4. $\square$

<figure><img src="image_1.png" /></figure>

**Remark 3.1.** From the global stability of $P_*$, we obtain the long-term behavior of the state variables necessary to study the unemployment. Therefore, for each initial values $\tau_0$ and $\xi_0$ of the unemployment rate and the labor market market tightness, if the condition (40) is satisfied, then

$$\lim_{t \to \infty} \tau(t) = \tau_* \quad \text{and} \quad \lim_{t \to \infty} \xi(t) = \xi_*.$$
However, the condition (40) entails $\delta V_* > (\sigma_0 + \alpha\gamma)U_*$. That is, at equilibrium, the number of closed vacancies is greater than the number of new vacancies created. This is not a beneficial situation for the economy. More favorable conditions can be obtained by compromising the global stability $((r+u)E_* < (1 + \frac{kV_*+\gamma(1+\alpha)+\sigma_0}{m+\mu})kU_*V_*)$. For example, policymakers can put in place mechanisms to encourage self-employment (by increasing the value of $\gamma$) and improve the success rate in the matching process between available vacancies and job seekers (by increasing the value of $k$).

### 3.3. Impact of some parameters on the dynamics of the unemployment rate and the labor market tightness

In what follows, we will use the local sensitivity analysis to study the change on the behaviors of the unemployment rate and the labor market tightness with respect to some parameters.

For a given parameter $p$, we estimate its impact on the unemployment rate $\tau$ and the labor market tightness $\xi$ by studying the dynamics of the quantities $\frac{\partial \tau}{\partial p}$ and $\frac{\partial \xi}{\partial p}$.

We have

$$ \frac{\partial \tau}{\partial p} = \frac{\frac{\partial U}{\partial p}E - \frac{\partial E}{\partial p}U}{(U + E)^2} \quad \text{and} \quad \frac{\partial \xi}{\partial p} = \frac{\frac{\partial V}{\partial p}U - \frac{\partial U}{\partial p}V}{U^2}. \qquad (56) $$

To investigate the impact of the parameter $p$, we need to study the behaviors of $\frac{\partial U}{\partial p}$, $\frac{\partial E}{\partial p}$ and $\frac{\partial V}{\partial p}$. In the sequel, we will denote by $Z_p^I(t)$, $Z_p^J(t)$, $Z_p^U(t)$, $Z_p^E(t)$ and $Z_p^V(t)$ the partial derivatives of $I(t)$, $J(t)$, $U(t)$, $E(t)$ and $V(t)$ with respect to $p$, respectively. Here, $(I(t), J(t), U(t), E(t), V(t))$ is a solution to (13).

To obtain $\frac{\partial U}{\partial p}$, $\frac{\partial E}{\partial p}$ and $\frac{\partial V}{\partial p}$, it suffices to solve the following Cauchy problem

$$ \left\{ \begin{aligned} \dot{I} &= mU + rE - \mu I, \\ \dot{J} &= aN\left(1 - \frac{N}{A}\right) - (\mu + b)J, \\ \dot{U} &= bJ + \beta E - kUV - (m + \mu + \gamma)U, \\ \dot{E} &= \gamma U + kUV - (\beta + \mu + r)E, \\ \dot{V} &= (\sigma_0 + \alpha\gamma)U + (r + \mu)E - \delta V - kUV, \\ \dot{Z}_p^I &= \frac{\partial f_1(X,p)}{\partial p} + \frac{\partial f_1(X,p)}{\partial I} Z_p^I + \frac{\partial f_1(X,p)}{\partial J} Z_p^J + \frac{\partial f_1(X,p)}{\partial U} Z_p^U + \frac{\partial f_1(X,p)}{\partial E} Z_p^E + \frac{\partial f_1(X,p)}{\partial V} Z_p^V, \\ \dot{Z}_p^J &= \frac{\partial f_2(X,p)}{\partial p} + \frac{\partial f_2(X,p)}{\partial I} Z_p^I + \frac{\partial f_2(X,p)}{\partial J} Z_p^J + \frac{\partial f_2(X,p)}{\partial U} Z_p^U + \frac{\partial f_2(X,p)}{\partial E} Z_p^E + \frac{\partial f_2(X,p)}{\partial V} Z_p^V, \\ \dot{Z}_p^U &= \frac{\partial f_3(X,p)}{\partial p} + \frac{\partial f_3(X,p)}{\partial I} Z_p^I + \frac{\partial f_3(X,p)}{\partial J} Z_p^J + \frac{\partial f_3(X,p)}{\partial U} Z_p^U + \frac{\partial f_3(X,p)}{\partial E} Z_p^E + \frac{\partial f_3(X,p)}{\partial V} Z_p^V, \\ \dot{Z}_p^E &= \frac{\partial f_4(X,p)}{\partial p} + \frac{\partial f_4(X,p)}{\partial I} Z_p^I + \frac{\partial f_4(X,p)}{\partial J} Z_p^J + \frac{\partial f_4(X,p)}{\partial U} Z_p^U + \frac{\partial f_4(X,p)}{\partial E} Z_p^E + \frac{\partial f_4(X,p)}{\partial V} Z_p^V, \\ \dot{Z}_p^V &= \frac{\partial f_5(X,p)}{\partial p} + \frac{\partial f_5(X,p)}{\partial I} Z_p^I + \frac{\partial f_5(X,p)}{\partial J} Z_p^J + \frac{\partial f_5(X,p)}{\partial U} Z_p^U + \frac{\partial f_5(X,p)}{\partial E} Z_p^E + \frac{\partial f_5(X,p)}{\partial V} Z_p^V, \\ (I(0), J(0), U(0), E(0), V(0), Z_p^I(0), Z_p^J(0), Z_p^U(0), Z_p^E(0), Z_p^V(0)) &= (I_0, J_0, U_0, E_0, V_0, 0, 0, 0, 0, 0), \end{aligned} \right. \qquad (57) $$
where

$$X = (I, J, U, E, V),$$

$$f_1(X, p) = mU + rE - \mu I,$$

$$f_2(X, p) = a(I + J + U + E)\left(1 - \frac{I + J + U + E}{A}\right) - (\mu + b)J,$$

$$f_3(X, p) = bJ + \beta E - kUV - (m + \mu + \gamma)U,$$

$$f_4(X, p) = \gamma U + kUV - (\beta + \mu + r)E, \text{ and}$$

$$f_5(X, p) = (\sigma_0 + \alpha\gamma)U + (r + \mu)E - \delta V - kUV.$$

The first fifth equations of (57), are independent of $Z_p^I(t)$, $Z_p^J(t)$, $Z_p^U(t)$, $Z_p^E(t)$ and $Z_p^V(t)$.

Hence, (57) can be reduced to the following Cauchy problem

$$ 
\left\{ 
\begin{aligned}
\dot{Z}_p^I(t) &= \frac{\partial f_1(X(t),p)}{\partial p} + \frac{\partial f_1(X(t),p)}{\partial I} Z_p^I + \frac{\partial f_1(X(t),p)}{\partial J} Z_p^J + \frac{\partial f_1(X(t),p)}{\partial U} Z_p^U + \frac{\partial f_1(X(t),p)}{\partial E} Z_p^E + \frac{\partial f_1(X(t),p)}{\partial V} Z_p^V, \\
\dot{Z}_p^J(t) &= \frac{\partial f_2(X(t),p)}{\partial p} + \frac{\partial f_2(X(t),p)}{\partial I} Z_p^I + \frac{\partial f_2(X(t),p)}{\partial J} Z_p^J + \frac{\partial f_2(X(t),p)}{\partial U} Z_p^U + \frac{\partial f_2(X(t),p)}{\partial E} Z_p^E + \frac{\partial f_2(X(t),p)}{\partial V} Z_p^V, \\
\dot{Z}_p^U(t) &= \frac{\partial f_3(X(t),p)}{\partial p} + \frac{\partial f_3(X(t),p)}{\partial I} Z_p^I + \frac{\partial f_3(X(t),p)}{\partial J} Z_p^J + \frac{\partial f_3(X(t),p)}{\partial U} Z_p^U + \frac{\partial f_3(X(t),p)}{\partial E} Z_p^E + \frac{\partial f_3(X(t),p)}{\partial V} Z_p^V, \\
\dot{Z}_p^E(t) &= \frac{\partial f_4(X(t),p)}{\partial p} + \frac{\partial f_4(X(t),p)}{\partial I} Z_p^I + \frac{\partial f_4(X(t),p)}{\partial J} Z_p^J + \frac{\partial f_4(X(t),p)}{\partial U} Z_p^U + \frac{\partial f_4(X(t),p)}{\partial E} Z_p^E + \frac{\partial f_4(X(t),p)}{\partial V} Z_p^V, \\
\dot{Z}_p^V(t) &= \frac{\partial f_5(X(t),p)}{\partial p} + \frac{\partial f_5(X(t),p)}{\partial I} Z_p^I + \frac{\partial f_5(X(t),p)}{\partial J} Z_p^J + \frac{\partial f_5(X(t),p)}{\partial U} Z_p^U + \frac{\partial f_5(X(t),p)}{\partial E} Z_p^E + \frac{\partial f_5(X(t),p)}{\partial V} Z_p^V, \\
(Z_p^I(0), Z_p^J(0), Z_p^U(0), Z_p^E(0), Z_p^V(0)) &= (0, 0, 0, 0, 0),
\end{aligned}
\right.
\qquad (58) 
$$

where the coefficients are functions of $I(t)$, $J(t)$, $U(t)$, $E(t)$, $V(t)$, and $p$.

We have the following result

**Theorem 3.5.** The Cauchy problem (58) admits a unique solution defined for all $t \ge 0$.

*Proof.* The problem (58) can be written as

$$ 
\left\{ 
\begin{aligned}
\dot{Z}_p(t) &= B(t)Z_p(t) + G_p(t), \\
Z_p(0) &= (0, 0, 0, 0, 0),
\end{aligned}
\right. 
\qquad (59) 
$$

where

$$Z_p = (Z_p^I, Z_p^J, Z_p^U, Z_p^E, Z_p^V)^\top,$$

$$ B(t) = \begin{pmatrix} -\mu & 0 & m & r & 0 \\ a\left(1 - \frac{2N(t)}{A}\right) - (\mu + b) & a\left(1 - \frac{2N(t)}{A}\right) & a\left(1 - \frac{2N(t)}{A}\right) & a\left(1 - \frac{2N(t)}{A}\right) & a\left(1 - \frac{2N(t)}{A}\right) \\ 0 & b & -kV(t) - (\gamma + \mu + m) & \beta & -kU(t) \\ 0 & 0 & \gamma + kV(t) & -(\beta + r + \mu) & kU(t) \\ 0 & 0 & \sigma_0 + \alpha\gamma - kV(t) & r + \mu & -\delta - kU(t). \end{pmatrix}, $$

and

$$G_p(t) = \left( \frac{\partial f_1(X(t),p)}{\partial p}, \frac{\partial f_2(X(t),p)}{\partial p}, \frac{\partial f_3(X(t),p)}{\partial p}, \frac{\partial f_4(X(t),p)}{\partial p}, \frac{\partial f_5(X(t),p)}{\partial p} \right)^\top.$$

The coefficients of $B(t)$ and $G(t)$ are continuous functions and defined for all $t \ge 0$ (Theorem 3.1). Therefore, the Cauchy problem (59) admits a unique solution defined for all $t \ge 0$. $\square$
The impact of a, b and γ are obtained by solving the following Cauchy problems

$$ \begin{cases} \dot{Z}_a(t) = B(t)Z_a(t) + G_a(t), \\ Z_a(0) = (0, 0, 0, 0), \end{cases} \qquad (60) $$

with $G_a(t) = (0, N(t)(1 - \frac{N(t)}{A}), 0, 0, 0)$.

$$ \begin{cases} \dot{Z}_b(t) = B(t)Z_b(t) + G_b(t), \\ Z_b(0) = (0, 0, 0, 0, 0), \end{cases} \qquad (61) $$

with $G_b(t) = (0, -J(t), J(t), 0, 0)$.

$$ \begin{cases} \dot{Z}_\gamma(t) = B(t)Z_\gamma(t) + G_\gamma(t), \\ Z_\gamma(0) = (0, 0, 0, 0, 0), \end{cases} \qquad (62) $$

with $G_\gamma(t) = (0, 0, -U(t), U(t), \alpha U(t))$.

From Theorem 3.5, each of the Cauchy problems (60), (61) and (62) admits a unique solution defined for all $t \ge 0$.

## 4. An optimal control of the model

To reduce at lower cost the number of unemployed individuals while ensuring relative ease of finding work, we propose the following strategy:

* Promote the self-employment by accompanying the unemployed in the process of creating businesses.

* Improve the success rate in the search for employment.

* Maintain a high level of labor market tightness.

The strategy is settled as follows:

$$ \left\{ \begin{array}{l} \min_{\{(u,v) \in L^1[0,t_f], [\underline{u},\underline{v}] \times [\underline{v},\bar{v}]\}} \int_0^{t_f} (\eta_1 u^2(t) + \eta_2 v^2(t) + \eta_3 U(t) - \eta_4 \frac{V(t)}{U(t)}) dt \\ \text{subject to} \\ \dot{I} = mU + rE - \mu I, \\ \dot{J} = a(I + J + U + E) \left(1 - \frac{I+J+U+E}{A}\right) - (\mu + b)J, \\ \dot{U} = bJ + \beta E - kvUV - (m + \mu + \gamma)U - uU, \\ \dot{E} = \gamma U + kvUV - (\beta + \mu + r)E + uU, \\ \dot{V} = (\sigma_0 + \alpha\gamma)U + (r + \mu)E - \delta V - kvUV + \alpha uU, \\ (I(0), J(0), U(0), E(0), V(0)) = (I_0, J_0, U_0, E_0, V_0), \end{array} \right. \qquad (63) $$
where $\eta_i$, $i = 1...4$, $\underline{u}$, $\overline{u}$, $\underline{v}$, $\overline{v}$, $t_f$, $I_0$, $J_0$, $U_0$, $E_0$ and $V_0$, are positive real numbers. The integral represents the general cost of implementation of government policies to reduce the unemployment rate in the period $[0, t_f]$. The constant $\eta_1$ is the average amount of money spent per unemployed person in order to support the installation as self-employed. $\eta_2$ is the amount of money invested to implement the government policies intended to improve the rate of successful matchings between job seekers and vacancies. $\eta_3$ and $\eta_4$ represent relative importance in the cost of the control, of the number of unemployed and the level of labor market tightness, respectively. The states variables $(I, J, U, E, V)$ belongs to $AC([0, t_f]; \mathbb{R}^5)$, the set of absolutely continuous functions from $[0, t_f]$ to $\mathbb{R}^5$. The control $(u, v)$ belongs to the set $L^1([0, t_f], [\underline{u}, \overline{u}] \times [\underline{v}, \overline{v}])$ of Lebesgue integrable functions from $[0, t_f]$ to $[\underline{u}, \overline{u}] \times [\underline{v}, \overline{v}]$. For given $t \ge 0$, $u(t)$ is the fraction of unemployed having benefited to the government initiatives intended to encourage self-employment. $v(t)$ is the efficiency rate of the government policies intended to improve the rate of successful matchings between jobseekers and vacancies.

### 4.1. The existence and uniqueness of optimal solution

The problem 63 can be rewritten as follows:

$$ 
\left\{
\begin{array}{l}
\displaystyle \min_{\{(u,v) \in L^1([0,t_f],[\underline{u},\overline{u}] \times [\underline{v},\overline{v}])\}} \int_0^{t_f} \mathcal{L}(X(t), \mathcal{U}(t)) dt \\
\text{subject to} \\
\dot{X}(t) = f(X(t), \mathcal{U}(t)) \\
X(0) = (I_0, J_0, U_0, E_0, V_0),
\end{array}
\right.
\qquad (64)
$$

where,

$$X = (I, J, U, E, V) \in AC([0, t_f], \mathbb{R}^5),$$

$$\mathcal{U} = (u, v) \in L^1([0, t_f], [\underline{u}, \overline{u}] \times [\underline{v}, \overline{v}]),$$

$$\mathcal{L}(X(t), \mathcal{U}(t)) = \eta_1 u^2(t) + \eta_2 v^2(t) + \eta_3 U(t) - \eta_4 \frac{V(t)}{U(t)}, \text{ and}$$

$$f(X, \mathcal{U}) = \begin{cases}
mU + rE - \mu I \\
a(I + J + U + E) \left(1 - \frac{I + J + U + E}{A}\right) - (\mu + b)J \\
bJ + \beta E - kvUV - (m + \mu + \gamma)U - uU \\
\gamma U + kvUV - (\beta + \mu + r)E + uU \\
(\sigma_0 + \alpha\gamma)U + (r + \mu)E - \delta V - kvUV + \alpha uU
\end{cases}$$
It is obvious that $\mathcal{L}$ and $f$ are continuous functions of $X$ and $\mathcal{U}$. We say that a pair $(X, \mathcal{U})$ in $AC([0, t_f], \mathbb{R}^5) \times L^1[0, t_f], [\underline{u}, \bar{u}] \times [\underline{v}, \bar{v}]$ is a feasible pair if it satisfies

$$ \left\{ \begin{array}{l} \dot{X}(t) = f(X(t), \mathcal{U}(t)), \quad a.e \ t \in [0, t_f], \\ X(0) = (I_0, J_0, U_0, E_0, V_0). \end{array} \right. \qquad (65) $$

We denote by $\mathcal{G}$ the set of all feasible pairs.

We begin by examining sufficient conditions for the existence of a solution to the optimal control problem. We refer to the conditions in Theorem III.4.1 and its corresponding Corollary in [10]. The nontrivial requirements from Fleming and Rishel's theorem are satisfied. In fact,

(H$_1$) The set of all solutions to the following system

$$ \left\{ \begin{array}{l} \dot{I} = mU + rE - \mu I, \\ \dot{J} = a(I + J + U + E) \left( 1 - \frac{I + J + U + E}{A} \right) - (\mu + b)J, \\ \dot{U} = bJ + \beta E - kvUV - (m + \mu + \gamma)U - uU, \\ \dot{E} = \gamma U + kvUV - (\beta + \mu + r)E + uU, \\ \dot{V} = (\sigma_0 + \alpha\gamma)U + (r + \mu)E - \delta V - kvUV + \alpha uU, \\ (I(0), J(0), U(0), E(0), V(0)) = (I_0, J_0, U_0, E_0, V_0), \end{array} \right. \qquad (66) $$

with control in $(u, v) \in L^1([0, t_f], [\underline{u}, \bar{u}] \times [\underline{v}, \bar{v}])$ is nonempty.

(H$_2$) The state system can be written as a linear function of the control variables with coefficients dependent on time and the state variables.

(H$_3$) The function $\mathcal{L}$ is convex with respect to $\mathcal{U}$ and additionally satisfies

$$ \mathcal{L}(X, \mathcal{U}) \geq c_0 \|\mathcal{U}\|^{c_1} - c_2, $$

where $c_0 > 0$, $c_1 > 1$, $c_2 \in \mathbb{R}$ and $\| \cdot \|$ is a norm in $\mathbb{R}^2$.

The following theorem is about the existence of optimal solution of problem (64).

**Theorem 4.1.** There exists an optimal control pair $(u^*, v^*)$ and corresponding solution $(I^*, J^*, U^*, E^*, V^*)$ to the state initial value problem (66) that minimize

$$ \int_{0}^{t_f} (\eta_1 u^2(t) + \eta_2 v^2(t) + \eta_3 U(t) - \eta_4 \frac{V(t)}{U(t)}) dt. $$

*Proof.* Since the controls $u$ and $v$ are bounded, by proceeding as in the proof of Theorem 3.1, $(H_1)$ is obtained.
Condition ($H_2$) is verified by observing the linear dependence of the state equations on controls $u$ and $v$.

To verify condition ($H_3$), note that $\mathcal{L}$ is a quadratic function of the controls and is clearly convex as function of controls. To prove the bound on the function $\mathcal{L}$, we use Theorem 3.1 and Lemma 3.3.

In fact, we have

$$ \mathcal{L}(X, \mathcal{U}) = \eta_1 u^2 + \eta_2 v^2 + \eta_3 U - \eta_4 \frac{V}{U} \geq \frac{1}{2} \min(\eta_1, \eta_2) \|(u, v)\|^2 - \frac{(\sigma_0 + \alpha(1 + \bar{u}) + r + \mu)(a - \mu)A}{a\delta U_{\infty}}, $$

where $\|(u, v)\| = \sqrt{u^2 + v^2}$ is the Euclidian norm in $\mathbb{R}^2$.

Therefore, we obtain the existence of $(u^*, v^*)$ and the corresponding optimal trajectory.

For small value of $t_f$, we obtain the uniqueness of the optimal solution [11]. □

### 4.1.1. The characterization of the optimal controls

We apply Pontryagin's Maximum Principle [31] and convert the optimization problem (63) to the problem of finding the point-wise minimum relative to $\gamma$ and $v$ of the Hamiltonian. All control minimizers in problem (63) are essentially bounded. By the Pontryagin Maximum Principle [31], we address the question of how to identify the solutions predicted by Theorem 4.1. Moreover, our optimal control problem (63) has only fixed initial conditions, with the state variables being free at the final time, that is, $I(t_f)$, $J(t_f)$, $U(t_f)$, $E(t_f)$ and $V(t_f)$ are free. This implies that abnormal minimizers [2] are not possible in our context and we can fix the cost multiplier associated with the Lagrangian to be equal to one.

The Hamiltonian of problem (63) is given by

$$ \begin{aligned} \mathcal{H}(X, \mathcal{U}, P) = & \eta_1 u^2 + \eta_2 v^2 + \eta_3 U(t) - \eta_4 \frac{V}{U} + p_1 [mU + rE - \mu I] \\ & + p_2 \left[ a(I + J + U + E) \left( 1 - \frac{I + J + U + E}{A} \right) - (\mu + b)J \right] \\ & + p_3 [bJ + \beta E - kvUV - (m + \mu + \gamma + u)U] \\ & + p_4 [(\gamma + u)U + kvUV - (\beta + \mu + r)E] \\ & + p_5 [(\sigma_0 + \alpha(\gamma + u))U + (r + \mu)E - \delta V - kvUV], \end{aligned} $$

with $P = (p_1, p_2, p_3, p_4, p_5)$.

The following theorem gives necessary optimality conditions for the optimal control problem.

**Theorem 4.2.** If $(I^*, J^*, U^*, E^*, V^*, u^*, v^*)$ is a minimizer of problem (63), then there exist
multipliers $p_1(\cdot), p_2(\cdot), p_3(\cdot), p_4(\cdot), p_5(\cdot)$ in $AC([0, t_f], \mathbb{R})$ such that

$$
\left\{
\begin{aligned}
\dot{p}_1 &= \mu p_1 - a\left(1 - \frac{2(I+J+U+E+V)}{A}\right)p_2, \\
\dot{p}_2 &= -\left(a\left(1 - \frac{2(I+J+U+E+V)}{A}\right) - (\mu + b)\right)p_2 - bp_3, \\
\dot{p}_3 &= -mp_1 - a\left(1 - \frac{2(I+J+U+E+V)}{A}\right)p_2 + (kvV + m + \mu + \gamma + u)p_3 - (kvV + \gamma + u)p_4 \\
&\quad - (\sigma_0 + \alpha(\gamma + u) - kvV)p_5 - \eta_3 - \eta_4 \frac{V}{U^2}, \\
\dot{p}_4 &= -rp_1 - a\left(1 - \frac{2(I+J+U+E+V)}{A}\right)p_2 - \beta p_3 + (\beta + \mu + r)p_4 - (r + \mu)p_5, \\
\dot{p}_5 &= kvUp_3 - kvUp_4 + (\delta + kvU)p_5 + \frac{\eta_4}{U},
\end{aligned}
\right.
\tag{67}
$$

for almost all $t \in [0, t_f]$, with transversality conditions

$$p_1(t_f) = p_2(t_f) = p_3(t_f) = p_4(t_f) = p_5(t_f) = 0. \tag{68}$$

The optimal control pair for all $t \in [0, t_f]$ is given by

$$u^*(t) = \min \left\{ \max \left\{ \underline{u}, \frac{(p_3(t) - p_4(t) - \alpha p_5(t))U^*(t)}{2\eta_1} \right\}, \overline{u} \right\}, \tag{69}$$

and

$$v^*(t) = \min \left\{ \max \left\{ \underline{v}, \frac{(p_3(t) - p_4(t) + p_5(t))kU^*(t)V^*(t)}{2\eta_2} \right\}, \overline{v} \right\}. \tag{70}$$

*Proof.* The result follows from a direct application of a version of Pontryagin's Maximum Principle for bounded controls [31].

The optimality conditions dictate that

$$\frac{\partial \mathcal{H}}{\partial u} = \frac{\partial \mathcal{H}}{\partial v} = 0$$

for the optimal pair $(\gamma^*, v^*)$ on the interior of the interval $[\underline{u}, \overline{u}] \times [\underline{v}, \overline{v}]$, and this condition is simplified in equations (69) and (70) considering the bounds $\underline{u}, \overline{u}, \underline{v}$ and $\overline{v}$. $\square$

## 5. Numerical simulations and economics interpretations

In this section, using ODE 45 routine and the shooting method implemented on MATLAB software, we now carry out numerical simulations to illustrate the theoretical results obtained. To perform numerical simulations, we made the following assumptions:

* The initial conditions $(I_0, J_0, U_0, E_0, V_0)$ are such that $I_0 = 1.5 * 10^4$, $J_0 = 10^5$, $U_0 = 1.7 * 10^4$, $E_0 = 3.5 * 10^4$, $V_0 = 5 * 10^3$.

* We assume that the expected lifetime $\frac{1}{\mu} = 90$ years. That is $\mu = 0.011$.
* The legal working age $\frac{1}{b} = 18$ years. That is $b = 0.055$.

* The birth rate $a = 0.09$.

* The average number of years of work required to be eligible for retirement benefits $\frac{1}{r} = 40$ years. That is $r = 0.025$.

* Table 1 gives the values of other parameters used.

<table><caption>Table 1: Values of the parameters used to perform simulations</caption><thead><tr><td>Parameter</td><td>Value</td><td>Source</td></tr></thead><tbody><tr><td>A</td><td>7 * 10<sup>5</sup></td><td>Assumed</td></tr><tr><td>k</td><td>1.08 * 10<sup>-5</sup></td><td>[25]</td></tr><tr><td>β</td><td>0.01</td><td>[25]</td></tr><tr><td>σ<sub>0</sub></td><td>0.15</td><td>Assumed</td></tr><tr><td>γ</td><td>0.2</td><td>Assumed</td></tr><tr><td>δ</td><td>0.075</td><td>[25]</td></tr><tr><td>m</td><td>0.007</td><td>Assumed</td></tr><tr><td>α</td><td>0.33</td><td>Assumed</td></tr><tr><td>u</td><td>0.05</td><td>Assumed</td></tr><tr><td>u&#773;</td><td>0.4</td><td>Assumed</td></tr><tr><td>v</td><td>1</td><td>Assumed</td></tr><tr><td>v&#773;</td><td>10</td><td>Assumed</td></tr></tbody></table>

## 5.1. The asymptotic behavior of unemployment and labor market indicators

On Figure 1, given different initial conditions, we illustrate the global asymptotic stability of the unique positive equilibrium. Figure 2 gives the long-term behavior of the unemployment rate and the labor market tightness for given different initial conditions.

## 5.2. Impact of some parameters on the dynamics of labor market indicators

In this part of the paper, we will illustrate graphically the impact of an increase of the natural birth rate, the legal age for working or the self-employment rate on the dynamics of unemployment rate and labor market tightness.
<figure><img src="image_2.png" /><figcaption>Figure 1: The global asymptotic stability of the equilibrium</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: The global asymptotic stability of the labor market indicators</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: The figure on the left gives the profile of the impact of an increase in the natural birth rate on the dynamics of the unemployment rate. The figure on the right provides an illustration of this impact.</figcaption></figure>

### 5.2.1. The impact of the natural birth rate

We evaluated the impact of the natural birth rate by solving the equation (60).

The profile of its impact on the dynamics of the labor market indicators are given by the left-hand sides of Figure 3 and Figure 4 (summarized in Table 2).

#### Interpretations:

<table><caption>Table 2: The impact of an increase of the natural birth rate on the labor market indicators</caption><thead><tr><td>Period</td><td>The sign of <sup>∂τ(t)</sup>⁄⁄∂a</td><td>The sign of <sup>∂ξ(t)</sup>⁄⁄∂a</td></tr></thead><tbody><tr><td>Short term</td><td>Positive</td><td>Negative</td></tr><tr><td>Medium term</td><td>Negative</td><td>Positive</td></tr><tr><td>Long term</td><td>Zero</td><td>Negative</td></tr></tbody></table>

*Ceteris paribus,*

* An increase in the natural birth rate has the effect of an increase in the unemployment rate in the short term and a reduction in the unemployment rate in the medium term (illustration on the right-hand side of Figure 3).
<figure><img src="image_5.png" /><figcaption>Figure 4: The figure on the left gives the profile of the impact of an increase in the natural birth rate on the dynamics of the labor market tightness. The figure on the right provides an illustration of this impact.</figcaption></figure>

* In the long term, there is no longer any effect of an increase of the natural birth rate on the unemployment rate (illustration on the right-hand side of Figure 3).

* An increase in the natural birth rate has the effect of a reduction of the labor market tightness in the short and long term (illustration on the right-hand side of Figure 4).

* In the medium term, an increase in the natural birth rate leads to an increase of the labor market tightness (illustration on the right-hand side of Figure 4).

From the above interpretations, we infer that an increase of the natural birth rate entails a difficulty to have a job in short term. However, this difficulty fades in medium term. At long term, there is not enough available vacancies but it doesn't affect the unemployment rate.

### 5.2.2. The impact of the legal age of working

We evaluated the impact of the legal age of working by solving the equation (61).

Let denote by $\omega = \frac{1}{b}$ the legal age of working. Therefore, the impacts of its increase on the labor markets indicators are given by

$$ \frac{\partial \tau(t)}{\partial \omega} = -b^2 \frac{\partial \tau(t)}{\partial b}, $$
<figure><img src="image_6.png" /><figcaption>Figure 5: The figure on the left gives the profile of the impact of the legal age of working on the dynamics of the unemployment rate. The figure on the right provides an illustration of this impact.</figcaption></figure>

and

$$ \frac{\partial \xi(t)}{\partial \omega} = -b^2 \frac{\partial \xi(t)}{\partial b}. $$

The left-hand sides of Figure 5 and Figure 6 give the profiles of its impact on the dynamics of the labor market indicators (summarized in Table 3).

Interpretations:

<table><caption>Table 3: The impact of an increase of the legal age of working on the labor market indicators</caption><thead><tr><th>Period</th><th>The sign of <sup>∂τ(t)</sup>⁄⁄<sub>∂w</sub></th><th>The sign of <sup>∂ξ(t)</sup>⁄⁄<sub>∂ω</sub></th></tr></thead><tbody><tr><td>Short term</td><td>Negative</td><td>positive</td></tr><tr><td>Medium term</td><td>Positive</td><td>Negative</td></tr><tr><td>Long term</td><td>Zero</td><td>Positive</td></tr></tbody></table>

Ceteris paribus,

* An increase of the legal age of working has the effect of a reduction in the unemployment rate in the short term and an increase in the unemployment rate in the medium term
<figure><img src="image_7.png" /><figcaption>Figure 6: The figure on the left gives the profile of the impact of the legal age of working on the dynamics of the labor market tightness. The figure on the right provides an illustration of this impact.</figcaption></figure>

(illustration on the right-hand side of Figure 5).

* In the long term, there is no longer any effect of an increase of the legal age of working on the unemployment rate (illustration on the right-hand side of Figure 5).

* An increase in the legal age of working has the effect of an increase of the labor market tightness in the short and long term (illustration on the right-hand side of Figure 6).

* In the medium term, an increase in the legal age of working leads to a reduction of the labor market tightness (illustration on the right-hand side of Figure 6).

From the above interpretations, one can infer that an increase of the legal age of working entails less difficulty to finding work in short term. However, this difficulty resurfaces in medium term. At long term, despite of the existence of vacancies, the unemployment rate is not affected.

## 6. The impact of the self-employment rate

We evaluated the impact of an increase of the self-employment rate $\gamma$ by solving the equation (62).
The profile of its impact on the dynamics of the labor market indicators are given by the left-hand sides of Figure 7 and Figure 8. It is almost always negative for the unemployment rate and almost always positive for the labor market tightness. Hence, *ceteris paribus*, an increase of the self-employment rate is always benefits for the labor market (illustrations on the right-hand sides of Figure 7 and Figure 8).

<figure><img src="image_8.png" /><figcaption>Figure 7: The figure on the left gives the profile of the impact of $\gamma$ on the dynamics of the unemployment rate. The figure on the right provides an illustration of this impact.</figcaption></figure>

## 6.1. The optimal control

On Figure 10, we have the comparison between optimal paths and uncontrolled paths of labor market indicators. It shows the impact of the controls on the unemployment rate and the level of the labor market tightness. We observe simultaneously the reduction of unemployment rate with a higher level of the labor market tightness.

## 7. Conclusion

The aim of this paper was the mathematical analysis of the labor market by taking into account the demography. We built a system of nonlinear differential equations based on some general assumptions. Our main findings are summarized as follows:
<figure><img src="image_9.png" /><figcaption>Figure 8: The figure on the left gives the profile of the impact of $\gamma$ on the dynamics of the labor market tightness. The figure on the right provides an illustration of this impact.</figcaption></figure>

* The system admits a unique positive equilibrium: we prove its local asymptotic stability and under appropriate condition we built a Lyapunov function to prove its global asymptotic stability.

* The condition obtained for the global asymptotic stability of the equilibrium is not economically desirable. In fact, it induces that at equilibrium the number of closed vacancies is greater than the number of newly created vacancies. Therefore, it seems that only the local asymptotic stability is beneficial for the economy.

* We analyze the impact of some parameters on the dynamics of two labor market indicators: the unemployment rate and the labor market tightness. We made the following observations: *Ceteris paribus*,

* An increase in the natural birth rate has the effect of an increase in the unemployment rate in the short term and a reduction in the unemployment rate in the medium term.

* In the long term, there is no longer any effect of an increase of the natural birth rate on the unemployment rate.
<figure><img src="image_10.png" /><figcaption>Figure 9: The profiles of the optimal controls</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: The impact of the controls on the labor market indicators</figcaption></figure>
* An increase in the natural birth rate has the effect of a reduction of the labor market tightness in the short and long term.

* In the medium term, an increase in the natural birth rate leads to an increase of the labor market tightness.

* An increase of the legal age of working has the effect of a reduction in the unemployment rate in the short term and an increase in the unemployment rate in the medium term.

* In the long term, there is no longer any effect of an increase of the legal age of working on the unemployment rate.

* An increase in the legal age of working has the effect of an increase of the labor market tightness in the short and long term.

* In the medium term, an increase in the legal age of working leads to a reduction of the labor market tightness.

* In order to improve the employment state, we defined a strategy of control aiming to reduce the number of unemployed individuals by insuring a relative ease to find a job.

However, we neglect in this paper the age structure of the population. May be accounting it can give more explanation of the dynamics of the unemployment rate. Also, we didn't account for the informal and formal sector and we assume an homogeneous repartition of the population. Our next challenge is to address these issues.

## Data availability statement

Our manuscript has no associated data.

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