Financial crisis spread, economic growth and unemployment: a mathematical model

Calvin Tadmon; Eric Rostand Njike Tchaptchet

doi:10.1515/snde-2021-0081

Article

Financial crisis spread, economic growth and unemployment: a mathematical model

Calvin Tadmon and Eric Rostand Njike Tchaptchet

Published/Copyright: April 21, 2022

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 2

Abstract

The unemployment is the main channel through which the economic and financial crises influence the social development. In this paper, we propose a mathematical model to study the interactions between financial crisis spread, economic growth and unemployment. We also solve an optimal control problem focusing on the minimization, at the lowest cost, of the adverse effects of the financial crisis. The analysis of the model leads us to two equilibria: (1) a stress free equilibrium, where the economy and the employment are optimal, and (2) a stressed equilibrium. We obtain a theoretical confirmation of Okun’s law and a formula to compute the minimum reservation wage in terms of model parameters. Numerical simulations are performed to illustrate the theoretical results obtained.

Keywords: economic growth; financial crisis spread; mathematical modeling; numerical simulations; Okun’s law; unemployment

Corresponding author: Calvin Tadmon, Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon, E-mail: tadmoncalvin@gmail.com

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare that they have no conflict of interest.

Proof

Proposition 1

Let t ≥ 0.

From the first equation of (21), we have

d ( t ) = d ( 0 ) exp ∫ 0 t ( θ 0 k σ ( s ) − g ) ( 1 − d ( s ) ) − ξ − b d s ≥ 0 .

From the second equation of (21), we have

τ ( t ) = τ ( 0 ) exp ∫ 0 t ϕ ( u ) d u > 0 ,

where ϕ ( u ) = − α α + μ ( ρ + η + δ ) − λ 0 1 − d ( u ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α τ − ( 1 − α ) μ α ( u ) .

From the third equation of (21), we have

k ( t ) = k ( 0 ) exp ∫ 0 t λ 0 1 − d ( u ) ε k α − 1 ( u ) τ 1 − α ( u ) − ( δ + ρ + η ) d u > 0 .

The positivity of h(t) can easily be proved. From the relation h(t) + d(t) = 1, we deduce that 0 ≤ d(t) ≤ 1.

From the first equation of (21) and the positivity of τ, we have

( 1 − α ) μ α τ ( 1 − α ) μ α − 1 ( t ) τ ̇ ( t ) = − ( 1 − α ) μ α + μ ( ρ + η + δ ) τ ( 1 − α ) μ α − λ 0 ( 1 − d ( t ) ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α .

That is

d d t τ ( 1 − α ) μ α ( t ) = − ( 1 − α ) μ α + μ ( ρ + η + δ ) τ ( 1 − α ) μ α − λ 0 ( 1 − d ( t ) ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α < − ( 1 − α ) μ α + μ ( ρ + η + δ ) τ ( 1 − α ) μ α + λ 0 ( 1 − α ) μ α + μ ( 1 − α ) ( 1 − ν ) β 1 − α α .

Therefore,

τ ( 1 − α ) μ α ( t ) ≤ λ 0 ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α + τ ( 1 − α ) μ α ( 0 ) − λ 0 ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α e − ( 1 − α ) μ ( ρ + η + δ ) α + μ t .

If τ ( 0 ) ≤ λ 0 ρ + η + δ α ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ , then

lim sup t → ∞ τ ( 1 − α ) μ α ( t ) = λ 0 ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α

and

(31) τ ( t ) ≤ λ 0 ρ + η + δ α ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ .

From the second equation of (21) and the positivity of k, we have

( 1 − α ) k − α ( t ) k ̇ ( t ) = ( 1 − α ) λ 0 ( 1 − d ( t ) ) ε τ 1 − α ( t ) − ( 1 − α ) ( ρ + η + δ ) k 1 − α ( t ) .

That is,

(32) d d t k 1 − α ( t ) = ( 1 − α ) λ 0 ( 1 − d ( t ) ) ε τ 1 − α − ( 1 − α ) ( ρ + η + δ ) k 1 − α ( t ) .

Using (31), (32) becomes

(33) d d t k 1 − α ( t ) ≤ ( 1 − α ) λ 0 ( 1 − d ( t ) ) ε λ 0 ρ + η + δ α μ ( 1 − α ) ( 1 − ν ) β 1 μ − ( 1 − α ) ( ρ + η + δ ) k 1 − α ( t ) .

Since d(t) ≤ 1, (33) becomes

d d t k 1 − α ( t ) ≤ ( 1 − α ) λ 0 λ 0 ρ + η + δ α μ ( 1 − α ) ( 1 − ν ) β 1 − α μ − ( 1 − α ) ( ρ + η + δ ) k 1 − α ( t ) .

That is,

k 1 − α ( t ) ≤ λ 0 ρ + η + δ α + μ μ ( 1 − α ) ( 1 − ν ) β 1 − α μ + k ( 0 ) 1 − α − λ 0 ρ + η + δ α + μ μ ( 1 − α ) ( 1 − ν ) β 1 − α μ e − ( 1 − α ) ( ρ + η + δ ) t .

If k ( 0 ) ≤ λ 0 ρ + η + δ ( α + μ ) ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ , then

lim sup t → ∞ k 1 − α ( t ) = λ 0 ρ + η + δ α + μ μ ( 1 − α ) ( 1 − ν ) β 1 − α μ

and k ( t ) ≤ λ 0 ρ + η + δ ( α + μ ) ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ .□

Proof

Proposition 2

Consider the function Φ defined from 0 , ∞ × ( 0 , ∞ ) 2 to 0 , ∞ × ( 0 , ∞ ) 2 by

Φ ( d , τ , k ) = d ( θ 0 k σ ( t ) − g ) ( 1 − d ) − ξ − b − α α + μ ρ + η + δ − λ 0 ( 1 − d ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α τ − ( 1 − α ) μ α τ λ 0 ( 1 − d ) ε k α τ 1 − α − ( δ + ρ + η ) k .

Φ is a continuously differentiable function on 0 , ∞ × ( 0 , ∞ ) 2 ; hence it is locally Lipschitz. From the Cauchy–Lipschitz Theorem (Liao, Wang, and Yu 2007), problem (21) admits a unique local solution. From Proposition 1, the solution is contained in a compact subset of R 3 . Therefore it is globally defined. □

Proof

Proposition 3

We assume that β ≥ ( 1 − α ) ( 1 − ν ) λ 0 ρ + η + δ α ( 1 − α ) .

Let t ≥ 0. From Proposition 1, we have

(34) τ ( t ) ≤ λ 0 ρ + η + δ α ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ .

Therefore, if β ≥ ( 1 − α ) ( 1 − ν ) λ 0 ρ + η + δ α ( 1 − α ) , then

λ 0 ρ + η + δ α ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ ≤ 1

and τ(t) ≤ 1. □

Proof

Proposition 4

A point (d, τ, k) is an equilibrium of (21), if it is a solution of the following system

(35) d θ 0 k σ − g ( 1 − d ) − ξ − b = 0 , − α α + μ ρ + η + δ − λ 0 ( 1 − d ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α τ − ( 1 − α ) μ α τ = 0 , λ 0 ( 1 − d ) ε k α τ 1 − α − ( δ + ρ + η ) k = 0 .

From the second and third equations of (35), we obtain

(36) τ = λ 0 ρ + η + δ α ( 1 − α ) μ ( 1 − α ) ( 1 − ν ) β 1 μ ( 1 − d ) α ε ( 1 − α ) μ

and

(37) k = λ 0 ρ + η + δ α + μ μ ( 1 − α ) ( 1 − α ) ( 1 − ν ) β 1 μ ( 1 − d ) ε ( α + μ ) μ ( 1 − α ) .

Using (36), (37) and following the same development as in Bucci et al. (2019), we obtain the result. □

Proof

Proposition 5

The Jacobian matrix J(d, τ, k) of (21) is defined by

(38) J ( d , τ , k ) = J 1 ( d , τ , k ) 0 J 2 ( d , τ , k ) J 3 ( d , τ , k ) J 4 ( d , τ , k ) 0 J 5 ( d , τ , k ) J 6 ( d , τ , k ) J 7 ( d , τ , k ) ,

where

J 1 ( d , τ , k ) = θ 0 k σ − g ( 1 − 2 d ) − ξ − b ,

J₂(d, τ, k) = θ₀σd(1 − d)k^σ−1,

J 3 ( d , τ , k ) = − λ 0 ε α α + μ ( 1 − d ) ε − 1 ( 1 − α ) ( 1 − ν ) β 1 − α α τ 1 − ( 1 − α ) μ α ,

J 4 ( d , τ , k ) = − α α + μ ρ + η + δ − 1 − ( 1 − α ) μ α λ 0 ( 1 − d ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α τ − ( 1 − α ) μ α ,

J₅(d, τ, k) = −λ₀ɛ(1 − d)^ɛ−1k^ατ^1−α,

J₆(d, τ, k) = (1 − α)λ₀(1 − d)^ɛk^ατ^−α, and

J₇(d, τ, k) = αλ₀(1 − d)^ɛk^α−1τ^1−α − (ρ + η + δ).

At Q 0 * , the eigenvalues of (38) are
λ 1 = θ 0 λ 0 ρ + η + δ σ ( α + μ ) μ ( 1 − α ) ( 1 − α ) ( 1 − ν ) β σ μ − ( ξ + g + b )
λ 2 = − ( 1 − α ) μ ( ρ + η + δ ) α + μ and λ 3 = ( α − 1 ) ( ρ + η + δ ) .

Since α < 1, it follows that λ₂ are λ₃ are negative real numbers. Therefore, for Q 0 * to be locally stable, λ₁ must be non-positive, that is

θ 0 λ 0 ρ + η + δ σ ( α + μ ) μ ( 1 − α ) ( 1 − α ) ( 1 − ν ) β σ μ ≤ ξ + g + b .

Similarly, by using the Routh–Hurwitz criterion, we obtain the local asymptotic stability of Q*.

□

Proof

Proposition 6

For given value of d = d_*, system (21) becomes

(39) τ ̇ ( t ) = − α α + μ ρ + η + δ − λ 0 ( 1 − d * ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α τ − ( 1 − α ) μ α ( t ) τ ( t ) , k ̇ ( t ) = λ 0 ( 1 − d * ) ε k α ( t ) τ 1 − α ( t ) − ( δ + ρ + η ) k ( t ) .

It is easy to verify that (τ_*, k_*) is the unique equilibrium of (39).

From the first equation of (39) and the positivity of τ, we have

( 1 − α ) μ α τ ( 1 − α ) μ α − 1 τ ̇ = − ( 1 − α ) μ α + μ ( ρ + η + δ ) τ ( 1 − α ) μ α − λ 0 ( 1 − d * ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α .

That is

d d t τ ( 1 − α ) μ α ( t ) = − ( 1 − α ) μ α + μ ( ρ + η + δ ) τ ( 1 − α ) μ α − λ 0 ( 1 − d * ) ε ( 1 − α ) ( 1 − ν ) β 1 − α α .

Therefore, for t ≥ t₀

τ ( 1 − α ) μ α ( t ) = λ 0 ( 1 − d * ) ε ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α + τ ( 1 − α ) μ α ( t 0 ) − λ 0 ( 1 − d * ) ε ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α e − ( 1 − α ) μ ( ρ + η + δ ) α + μ t .

If τ ( 1 − α ) μ α ( t 0 ) ≤ λ 0 ( 1 − d * ) ε ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α , that is τ(t₀) ≤ τ_*, then

lim sup t → ∞ τ ( 1 − α ) μ α ( t ) = λ 0 ( 1 − d * ) ε ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α

and for all t ≥ t₀, τ(t) ≤ τ_*.

If τ(t₀) ≥ τ_*, then

lim inf t → ∞ τ ( 1 − α ) μ α ( t ) = λ 0 ( 1 − d * ) ε ρ + η + δ ( 1 − α ) ( 1 − ν ) β 1 − α α

and for all t ≥ t₀, τ(t) ≥ τ_*.

From the second equation of (39) and the positivity of k, we have

( 1 − α ) k − α k ̇ ( t ) = ( 1 − α ) λ 0 ( 1 − d * ) ε τ 1 − α − ( 1 − α ) ( ρ + η + δ ) k 1 − α ( t ) .

That is,

d d t k 1 − α ( t ) = ( 1 − α ) λ 0 ( 1 − d * ) ε τ 1 − α − ( 1 − α ) ( ρ + η + δ ) k 1 − α ( t ) .

Therefore, for t ≥ t₀

(40) k 1 − α ( t ) = k 1 − α ( t 0 ) e − ( 1 − α ) ( ρ + η + δ ) t + ( 1 − α ) λ 0 ( 1 − d * ) ε ∫ 0 t e ( s − t ) ( 1 − α ) ( ρ + η + δ ) τ 1 − α ( s ) d s .

If τ(t₀) ≤ τ_*, then from (40) we have the inequality
(41) k 1 − α ( t ) ≤ k 1 − α ( t 0 ) e − ( 1 − α ) ( ρ + η + δ ) t + k * 1 − α 1 − e − ( 1 − α ) ( ρ + η + δ ) t .
Additionally if k(t₀) ≤ k_*, then from (41) we get
lim sup t → ∞ k ( t ) = k *

and for all t ≥ t₀ k(t) ≤ k_*.

If τ(t₀) ≥ τ_*, then from (40) we have the inequality
(42) k 1 − α ( t ) ≥ k 1 − α ( t 0 ) e − ( 1 − α ) ( ρ + η + δ ) t + k * 1 − α 1 − e − ( 1 − α ) ( ρ + η + δ ) t .
Additionally if k(t₀) ≥ k_*, then from (42) we get
lim inf t → ∞ k ( t ) = k *
and for all t ≥ t₀, k(t) ≥ k_*.

Note from (a) and (b) that, if τ(t₀) = τ_* and k(t₀) = k_*, there is no further variations of τ and k.

Now, if τ(t₀) < τ_* and k(t₀) < k_*, then

λ 0 ( 1 − d * ) ε ρ + η + δ τ 1 − α ( t 0 ) − k 1 − α ( t 0 ) < k * 1 − α − k 1 − α ( t 0 ) < 0

and

α τ * ( 1 − α ) μ α − τ ( 1 − α ) μ α ( t 0 ) ( α + μ ) λ 0 ( 1 − d * ) ε ρ + η + δ τ 1 − α ( t 0 ) − k 1 − α ( t 0 ) k − α ( t 0 ) τ 1 − ( 1 − α ) μ α ( t 0 ) > 0 .

□

Proof

Theorem 2

Direct computations show that system (28) follows from the adjoint system of the PMP (Sethi 2018). Similarly, the equalities in (30) are directly given by the transversality conditions of the PMP. It remains to characterize the control using the minimality condition of the PMP (Sethi 2018).

The minimality condition on the set t ∈ [ 0 , t f ] : 0 < u * ( t ) < 1 is

∂ H ∂ u * = 2 γ 1 u * − p 1 d * = 0 .

Thus, on this set,

u * = p 1 d * 2 γ 1 .

If t ∈ t ∈ [ 0 , t f ] : u * = 1 , then the minimality condition is

∂ H ∂ u * = 2 γ 1 u * − p 1 d * ≤ 0 , i . e , p 1 d * 2 γ 1 ≥ 1 .

If t ∈ t ∈ [ 0 , t f ] : u * = 0 , then the minimality condition is

∂ H ∂ u * = 2 γ 1 u * − p 1 d * ≥ 0 , i . e , p 1 d * 2 α 1 ≤ 0 .

Therefore, we obtain (30). □

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2021-0081).

Received: 2021-08-19

Accepted: 2022-03-26

Published Online: 2022-04-21

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Articles in the same Issue

https://doi.org/10.1515/snde-2021-0081

Keywords for this article

economic growth; financial crisis spread; mathematical modeling; numerical simulations; Okun’s law; unemployment