A Dynamic Economic Model with Discrete Time and Consumer Sentiment

Abstract

The paper describes a dynamical economic model with discrete time and consumer sentiment in the deterministic and stochastic cases. We seek to demonstrate that consumer sentiment may create fluctuations in the economical activities. The model possesses a flip bifurcation and a Neimark-Sacker bifurcation, after which the stable state is replaced by a (quasi) periodic motion. We associate the difference stochastic equation to the model by randomizing the control parameter 𝑑 and by adding one stochastic control. Numerical simulations are made for the deterministic and stochastic models, for different values of the control parameter 𝑑.

1. Introduction

The economic empirical evidence [1–3] suggests that consumer sentiment influences household expenditure and thus confirms Keynes' assumption that consumer “attitudes” and "animalic spirit" may cause fluctuations in the economic activity. On the other hand, Dobrescu and Opris [4, 5] analyzed the bifurcation aspects in a discrete-delay Kaldor model of business cycle, which corresponds to a system of equations with discrete time and delay. Following these studies, we develop a dynamic economic model in which the agents' consumption expenditures depend on their sentiment. As particular cases, the model contains the Hick-Samulson [6], Puu [7], and Keynes [3] models as well as the model from [3]. The model possesses a flip and Neimark-Sacker bifurcation, if the autonomous consumption is variable.

The implications of the stochastic noise on the economic process are studied. The stochastic difference equation with noise terms is scaled appropriately to account for intrinsic as well as extrinsic fluctuations. Under the influence of noise the difference equation behaves qualitatively different compared to its deterministic counterpart.

The paper is organized as follows. In Section 2, we describe the dynamic model with discrete time using investment, consumption, sentiment, and saving functions. For different values of the model parameters we obtain well-known dynamic models (Hick-Samuelson, Keynes, Pu). In Section 3, we analyze the behavior of the dynamic system in the fixed point's neighborhood for the associated map. We establish asymptotic stability conditions for the flip and Neimark-Sacker bifurcations. In both the cases of flip and Neimark-Sacker bifurcations, the normal forms are described in Section 4. Using the QR method, the algorithm for determining the Lyapunov exponents is presented in Section 5. In Section 6, a stochastic model with multiplicative noise is associated to the deterministic model. These equations are obtained by randomizing one parameter of the deterministic equation or by adding one stochastic control. Finally, the numerical simulations are done for the deterministic and stochastic equations. The obtained simulations show major changes between the deterministic and stochastic cases. The analysis of the present model proves its complexity and allows the description of the different scenarios which depend on autonomous consumption.

2. The Mathematical Model with Discrete Time and Consumer Sentiment

Let 𝑦(𝑡), 𝑡∈𝑁 be the income at time step 𝑡 and let

(1)the investment function 𝐼(𝑡), 𝑡∈ℕ, be given by 𝐼(𝑡)=𝑣(𝑦(𝑡−1)−𝑦(𝑡−2))−𝑤(𝑦(𝑡−1)−𝑦(𝑡−2))³,𝑣>0,𝑤≥0; (2.1)(2)the consumption function 𝐶(𝑡), 𝑡∈ℕ, be given by 𝐶(𝑡)=𝑎+𝑦(𝑡−1)(𝑏+𝑐𝑆(𝑡−1)),𝑎≥0,𝑏>0,𝑐≥0, (2.2) where 𝑆(𝑡),𝑡∈ℕ, is the sentiment function given by 1𝑆(𝑡)=1+𝜀exp(𝑦(𝑡−1)−𝑦(𝑡)),𝜀∈[0,1]; (2.3)(3)the saving function 𝐸(𝑡), 𝑡∈ℕ, be given by 𝐸(𝑡)=𝑑(𝑦(𝑡−2)−𝑦(𝑡−1))+𝑚𝑆(𝑡−1),𝑑≥0,𝑚≥0. (2.4) The mathematical model is described by the relation: 𝑦(𝑡)=𝐼(𝑡)+𝐶(𝑡)+𝐸(𝑡). (2.5)

From (2.1), (2.2), (2.3), (2.4), and (2.5) the mathematical model with discrete time and consumer sentiment is given by 𝑦(𝑡)=𝑎+𝑑𝑦(𝑡−2)+(𝑏−𝑑)𝑦(𝑡−1)+𝑣(𝑦(𝑡−1)−𝑦(𝑡−2))−𝑤(𝑦(𝑡−1)−𝑦(𝑡−2))³+𝑐𝑦(𝑡−1)+𝑚1+𝜀exp(𝑦(𝑡−2)−𝑦(𝑡−1)),𝑡∈ℕ. (2.6)

The parameters from (2.6) have the following economic interpretations. The parameter 𝑎 represents the autonomous expenditures. The parameter 𝑑 is the control, 𝑑∈[0,1], and it characterizes a part of the difference between the incomes obtained at two time steps 𝑡−2 and 𝑡−1, which is used for consumption or saving in the time step 𝑡. The parameter 𝑐, 𝑐∈[0,1], is the trend towards consumption. The parameter 𝑚, 𝑚∈[0,1], is the trend towards the saving. The parameter 𝑏, 𝑏∈(0,1), represents the consumer's reaction against the increase or decrease of his income. When the income (strongly) decreases, the consumer becomes pessimistic and consumes 0<𝑏<1 of his income. When the income (strongly) increases, the consumer becomes optimistic and consumes 𝑏<𝑏+𝑐<1 of his income. Note that Souleles [8] finds, in fact, that higher consumer confidence is correlated with less saving and increases in relation to expected future resources. The parameters 𝑣 and 𝑤, 𝑣>0, 𝑤≥0 describe the investment function. If 𝑤=0, the investment function is linear. The parameter 𝜀, 𝜀∈[0,1], describes a family of the sentiment functions.

For different values of the model parameters, we obtain the following classical models:

(1)for 𝑎=0,𝑏=1−𝑠,𝑑=0,𝑚=0,𝜀=0,𝑠∈(0,1) from (2.6) we obtain the Hick-Samuelson model [6]: 𝑦(𝑡)=(1+𝑣−𝑠)𝑦(𝑡−1)−𝑣𝑦(𝑡−2),𝑣>0,𝑠∈(0,1); (2.7)(2)for 𝑣=𝑤=0,𝜀=0,𝑚=0, (2.6) gives us the Keynes model [6]: 𝑦(𝑡)=𝑑(𝑦(𝑡−2)−𝑦(𝑡−1))+𝑎+𝑏𝑦(𝑡−1); (2.8)(3)for 𝑣=𝑤=0,𝜀=1, (2.6) leads to the model from [4]: 𝑦(𝑡)=𝑑(𝑦(𝑡−2)−𝑦(𝑡−1))+𝑎+𝑏𝑦(𝑡−1)+𝑐𝑦(𝑡−1)+𝑚1+exp(𝑦(𝑡−2)−𝑦(𝑡−1)),𝑡∈ℕ; (2.9)(4)for 𝑎=0,𝑏=𝑠−1,𝑑=0,𝑚=0,𝑤=1+𝑣,𝜀=0, from (2.6) we get the Puu model [7]: 𝑦(𝑡)=𝑣(𝑦(𝑡−1)−𝑦(𝑡−2))−(1+𝑣)(𝑦(𝑡−1)−𝑦(𝑡−2))³−(1−𝑠)𝑦(𝑡−1). (2.10)

3. The Dynamic Behavior of the Model (2.6)

Using the method from Kusnetsov [4, 6, 9], we will analyze the system (2.6), considering the parameter 𝑎 as bifurcation parameter. The associated map of (2.6) is 𝐹∶ℝ²→ℝ² given by 𝐹𝑦𝑧=𝑎+(𝑏−𝑑+𝑣)𝑦+(𝑑−𝑣)𝑧−𝑤(𝑦−𝑧)³+(𝑐𝑦+𝑚)𝑦1+𝜀exp(𝑧−𝑦). (3.1)

Using the methods from [5, 6, 9], the map (3.1) has the following properties.

Proposition 3.1. (i) If (1+𝜀)(1−𝑏)−𝑐>0, then, for the map (3.1), the fixed point with the positive components is 𝐸₀(𝑦₀,𝑧₀), where 𝑦₀=𝑝₁𝑎+𝑝₂,𝑧₀=𝑦₀, (3.2)𝑝₁=(1+𝜀)(1+𝜀)(1−𝑏)−𝑐,𝑝₂=𝑚(1+𝜀)(1−𝑏)−𝑐. (3.3)
(ii) The Jacobi matrix of the map 𝐹 in 𝐸₀ is given by 𝑎𝐴=¹¹𝑎₁₂10, (3.4) where: 𝑎₁₁=𝑝₃𝑎−𝑝₄+𝑝₅, 𝑎₁₂=−𝑝₃𝑎+𝑝₄, 𝑝₃=𝜀𝑐(1+𝜀)²𝑝₁,𝑝₄=𝑑−𝑣−𝜀𝑚(1+𝜀)²−𝜀𝑐(1+𝜀)²𝑝₂,𝑝₅𝑐=𝑏+(1+𝜀). (3.5)
(iii) The characteristic equation of matrix 𝐴 is given by 𝜆²−𝑎₁₁𝜆−𝑎₁₂=0. (3.6)
(iv) If the model parameters 𝑑,𝑣,𝜀,𝑏,𝑐,𝑚 satisfy the following inequality: (1+𝑑−𝑣)(1+𝜀)((1+𝜀)(1−𝑏)−𝑐)−𝑚𝜀(1−𝑏)>0, (3.7) then, for (3.6), the roots have the modulus less than 1, if and only if 𝑎∈(𝑎₁,𝑎₂), where 𝑎₁=2𝑝₄−𝑝₅−12𝑝₃,𝑎₂=1+𝑝₄𝑝₃. (3.8)
(v) If the model parameters 𝑑, 𝑣, 𝜀, 𝑏, 𝑐, 𝑚 satisfy the inequality (3.7) and 𝑎=𝑎₁, then, one of equation (3.6)'s roots is −1, while the other one has the modulus less than 1.
(vi) If the model parameters 𝑑,𝑣,𝜀,𝑏,𝑐,𝑚 satisfy the inequality (3.7) and 𝑎=𝑎₂, then, (3.6) has the roots 𝜇₁(𝑎)=𝜇(𝑎),𝜇₂(𝑎)=𝜇(𝑎), where |𝜇(𝑎)|=1.

Using [7] and Proposition 3.1, with respect to parameter 𝑎, the asymptotic stability conditions of the fixed point, the conditions for the existence of the flip and Neimark-Sacker bifurcations are presented in the following.

Proposition 3.2. (i) If (1+𝜀)(1−𝑏)−𝑐>0, the inequality (3.7) holds and 2𝑝₄−𝑝₅−1>0, then for 𝑎∈(𝑎₁,𝑎₂) the fixed point 𝐸₀ is asymptotically stable. If (1+𝜀)(1−𝑏)−𝑐>0, the inequality (3.7) holds, and 2𝑝₄−𝑝₅−1<0, then for 𝑎∈(0,𝑎₂) the fixed point 𝐸₀ is asymptotically stable.
(ii) If (1+𝜀)(1−𝑏)−𝑐>0, the inequality (3.7) holds, and 2𝑝₄−𝑝₅−1>0, then 𝑎=𝑎₁ is a flip bifurcation and 𝑎=𝑎₂ is a Neimark-Sacker bifurcation.
(iii) If (1+𝜀)(1−𝑏)−𝑐>0, the inequality (3.7) holds and 2𝑝₄−𝑝₅−1<0, then 𝑎=𝑎₂ is a Neimark-Sacker bifurcation.

4. The Normal Form for Flip and Neimark-Sacker Bifurcations

In this section, we describe the normal form in the neighborhood of the fixed point 𝐸₀, for the cases 𝑎=𝑎₁ and 𝑎=𝑎₂.

We consider the transformation: 𝑢₁=𝑦−𝑦₀,𝑢₂=𝑧−𝑧₀, (4.1) where 𝑦₀,𝑧₀ are given by (3.2). With respect to (4.1), the map (3.1) is 𝐺∶ℝ²→ℝ², where 𝐺(𝑢₁,𝑢₂𝑔)=¹(𝑢₁,𝑢₂)𝑔₂(𝑢₁,𝑢₂)𝑔₁,(𝑢₁,𝑢₂𝑟)=−1+𝜀+(𝑏−𝑑+𝑣)𝑢₁+(𝑑−𝑣)𝑢₂−𝑤(𝑢₁−𝑢₂)³+𝑐𝑢₁+𝑟1+𝜀exp(𝑢₂−𝑢₁),𝑔₂(𝑢₁,𝑢₂)=𝑢₁,𝑟=𝑐𝑦₀+𝑚. (4.2)(4.3) The map (4.2) has 𝑂(0,0) as fixed point.

We consider

𝑎₁₁=𝜕𝑔₁𝜕𝑢₁(0,0),𝑎₁₂=𝜕𝑔₁𝜕𝑢₂(0,0),𝑙₂₀=𝜕²𝑔₁𝜕𝑢²₁(0,0),𝑙₁₁=𝜕²𝑔₁𝜕𝑢₁𝜕𝑢₂𝑙(0,0),₀₂=𝜕²𝑔₁𝜕𝑢²₂(0,0),𝑙₃₀=𝜕³𝑔₁𝜕𝑢³₁(0,0),𝑙₂₁=𝜕³𝑔₁𝜕𝑢²₁𝜕𝑢₂(0,0),𝑙₁₂=𝜕³𝑔₁𝜕𝑢₁𝜕𝑢²₂𝑙(0,0),₀₃=𝜕³𝑔₁𝜕𝑢³₁(0,0). (4.4)

We develop the function 𝐺(𝑢),𝑢=(𝑢₁,𝑢₂)^𝑇 in the Taylor series until the third order and obtain 1𝐺(𝑢)=𝐴𝑢+21𝐵(𝑢,𝑢)+2𝐷(𝑢,𝑢,𝑢)+𝑂(|𝑢|⁴), (4.5) where 𝐴 is given by (3.4) and 𝐵(𝑢,𝑢)=(𝐵¹(𝑢,𝑢),0)^𝑇,𝐷(𝑢,𝑢,𝑢)=(𝐷¹(𝑢,𝑢,𝑢),0)^𝑇,𝐵¹(𝑢,𝑢)=𝑢^𝑇𝑙₂₀𝑙₁₁𝑙₁₁𝑙₀₂𝐷𝑢,₁(𝑢,𝑢,𝑢)=𝑢^𝑇(𝑢₁𝑙₃₀𝑙₂₁𝑙₂₁𝑙₁₂+𝑢₂𝑙₂₁𝑙₁₂𝑙₁₂𝑙₀₃)𝑢. (4.6)(4.7)

For 𝑎=𝑎₁, given by (3.8) with the condition (v) from Proposition 3.1, we have the following.

Proposition 4.1. (i) The eigenvector 𝑞∈ℝ², given by 𝐴𝑞=−𝑞, has the components: 𝑞₁=1,𝑞₂=−1. (4.8)
(ii) The eigenvector ℎ∈ℝ², given by ℎ^𝑇𝐴=−ℎ^𝑇, has the components: ℎ₁=11+𝑎₁₂,ℎ₂𝑎=−¹²1+𝑎₁₂. (4.9)
The relation ⟨𝑞,ℎ⟩=1 holds.
(iii) The normal form of the map (3.1) on the center manifold in 𝑂(0,0) is given by 1𝜂→−𝜂+6𝜈𝜂³+𝑂(𝜂⁴), (4.10) where 𝜈=(1/(1+𝑎₁₂))(𝑙₃₀−3𝑙₂₁+3𝑙₁₂−𝑙₀₃)+(3/(1−𝑎₁₁−𝑎₁₂))(𝑙₂₀−2𝑙₁₁+𝑙₀₂)².

The proof results from straight calculus using the formula (2.6): 𝜈=⟨𝑟,𝐷(𝑞,𝑞,𝑞)⟩+3𝐵(𝑞,(𝐼−𝐴)⁻¹𝐵(𝑞,𝑞)),𝐼=1001. (4.11)

For 𝑎=𝑎₂, given by (3.8) with the condition (vi) from Proposition 3.1, one has the following.

Proposition 4.2. (i) The eigenvector 𝑞∈ℂ², given by 𝐴𝑞=𝜇₁𝑞, where 𝜇₁ is the eigenvalue of the matrix 𝐴, has the components: 𝑞₁=1,𝑞₂=𝜇₂=𝜇₁. (4.12)
(ii) The eigenvector ℎ∈ℂ², given by ℎ^𝑇𝐴=𝜇₂ℎ^𝑇, where 𝜇₂=𝜇₁, has the components: ℎ₁=11+𝜇²₁𝑎₁₂,ℎ₂=𝜇₁𝑎₁₂1+𝜇²₁𝑎₁₂. (4.13)
The relation ⟨𝑞,ℎ⟩=1 holds.

Using (4.6) and (4.8) one has 𝐵¹(𝑞,𝑞)=𝑙₂₀+2𝑙₁₁𝜇₂+𝑙₀₂𝜇²₂,𝐵¹(𝑞,𝑞)=𝑙₂₀+𝑙₁₁(𝜇₁+𝜇₂)+𝑙₀₂𝜇₁𝜇₂,𝐵¹(𝑞,𝑞)=𝑙₂₀+2𝑙₁₁𝜇₁+𝑙₀₂𝜇²₁. (4.14)

We denoted by 𝑔₂₀=𝑣₁𝐵¹(𝑞,𝑞),𝑔₁₁=𝑣₁𝐵¹(𝑞,𝑞),𝑔₀₂=𝑣₁𝐵¹(𝑞,ℎ𝑞),₁₂₀=(1−𝑣₁−𝑣₁)𝑔₂₀,ℎ¹₁₁=(1−𝑣₁−𝑣₁)𝑔₁₁,ℎ¹₀₂=(1−𝑣₁−𝑣₁)𝑔₀₂,ℎ²₂₀=−(𝑣₁𝜇₂+𝑣₁𝜇₁)𝐵¹(𝑞,𝑞),ℎ²₁₁=−(𝑣₁𝜇₂+𝑣₁𝜇₁)𝐵¹(𝑞,ℎ𝑞),₂₀₂=−(𝑣₁𝜇₂+𝑣₁𝜇₁)𝐵¹(𝑞,𝑤𝑞),₂₀=(𝜇²₁𝐼−𝐴)⁻¹ℎ¹₂₀ℎ²₂₀,𝑤₁₁=(𝐼−𝐴)⁻¹ℎ¹₁₁ℎ²₁₁,𝑤₀₂=(𝜇²₂−𝐴)⁻¹ℎ¹₀₂ℎ²₀₂, (4.15)(4.16) where 𝐼=(¹⁰₀₁), 𝐴 is given by (3.4), and 𝑟₂₀=𝐵¹(𝑞,𝑤₂₀),𝑟₁₁=𝐵¹(𝑞,𝑤₁₁𝐷),₀=𝐷¹(𝑞,𝑞,𝑞)=𝑙₃₀+(𝜇₁+2𝜇₂)𝑙₂₁+𝜇₂(2𝜇₁+𝜇₂)𝑙₁₂+𝜇₁𝜇²₂𝑙₀₃,𝑔₂₁=𝑣₂(𝑟₂₀+2𝑟₁₁+𝐷₀). (4.17)

Using the normal form for the Neimark-Sacker bifurcation of the dynamic systems with discrete time [6] and (4.15), (4.16), and (4.17), we obtain the following.

Proposition 4.3. (i) The solution of the system (2.6) in the neighborhood of the fixed point (𝑦₀,𝑧₀)∈ℝ^𝑛 is given by 𝑦(𝑡)=𝑦₀+𝜇₂𝑥(𝑡)+𝜇₁1𝑥(𝑡)+2𝑤²₂₀𝑥(𝑡)²+𝑤²₁₁𝑥(𝑡)1𝑥(𝑡)+2𝑤²₀₂𝑥(𝑡)²,𝑦(𝑡−1)=𝑧(𝑡)=𝑧₀+𝑥(𝑡)+1𝑥(𝑡)+2𝑤¹₂₀𝑥(𝑡)²+𝑤¹₁₁𝑥(𝑡)1𝑥(𝑡)+2𝑤¹₀₂𝑥(𝑡)², (4.18) where 𝑥(𝑡)∈ℂ is the solution of the following equation: 𝑥(𝑡+1)=𝜇₁1𝑥(𝑡)+2𝑔₂₀𝑥(𝑡)²+𝑔₁₁𝑥(𝑡)1𝑥(𝑡)+2𝑔₀₂𝑥(𝑡)²+12𝑔₂₁𝑥(𝑡)²𝑥(𝑡). (4.19)
(ii) A complex variable transformation exists so that (4.18) becomes 𝜔(𝑡+1)=𝜇₁𝜔(𝑡)+𝐿_𝑐𝜔²_𝑡𝜔(𝑡)+𝑂(|𝜔(𝑡)|⁴), (4.20) where 𝐿_𝑐=𝑔₂₀𝑔₁₁(𝜇₂−3−2𝜇₁)2(𝜇²₁−𝜇₁)(𝜇₂+−1)|𝑔₁₁|²(1−𝜇₁)+|𝑔₂₀|²2(𝜇²₁−𝜇₂)+𝑔₂₁2 (4.21) is the Lyapunov coefficient.
(iii) If 𝑙₀=ℝ𝑒(𝑒^−𝑖𝜃𝐿_𝑐)<0, where 𝜃=arg(𝜇₁), then in the neighborhood of the fixed point (𝑦₀,𝑧₀) there is a stable limit cycle.

5. The Lyapunov Exponents

If 𝑎₁>0, then for 𝑎∈(0,𝑎₁) or 𝑎∈(𝑎₂,1) the system (2.6) has a complex behavior and it can be established by computing the Lyapunov exponents. We will use the decomposed Jacobi matrix of map (3.1) into a product of an orthogonal matrix Q and an uppertriangular matrix R with positive diagonal elements (called QR algorithm [6]). The determination of the Lyapunov exponents can be obtained by solving the following system: 𝑦(𝑡+1)=𝑎+(𝑏−𝑑+𝑣)𝑦(𝑡)+(𝑑−𝑣)𝑧(𝑡)−𝑤(𝑦(𝑡)−𝑧(𝑡))³+𝑐𝑦(𝑡)+𝑚,1+exp(𝑧(𝑡)−𝑦(𝑡))𝑧(𝑡+1)=𝑦(𝑡),𝑥(𝑡+1)=arctan(−cos𝑥(𝑡)𝑓₁₁cos𝑥(𝑡)−𝑓₁₂sin𝑥(𝑡)),𝜆(𝑡+1)=𝜆(𝑡)+ln(|(𝑓₁₁−tan𝑥(𝑡+1))cos𝑥(𝑡)cos𝑥(𝑡+1)−𝑓₁₂sin𝑥(𝑡)cos𝑥(𝑡+1)|),𝜇(𝑡+1)=𝜇(𝑡)+ln(|(𝑓₁₁−tan𝑥(𝑡+1)+1)sin𝑥(𝑡)cos𝑥(𝑡+1)+𝑓₁₂cos𝑥(𝑡)cos𝑥(𝑡+1)|), (5.1) with 𝑓₁₁=𝜕𝑓₁𝜕𝑦(𝑦(𝑡),𝑧(𝑡))=𝑏−𝑑+𝑣−3𝑤(𝑦(𝑡)−𝑧(𝑡))²+𝑐+𝜀(𝑐+𝑚+𝑐𝑦(𝑡))exp(𝑧(𝑡)−𝑦(𝑡))(1+𝜀exp(𝑧(𝑡)−𝑦(𝑡)))²,𝑓₁₂=𝜕𝑓₁𝜕𝑧(𝑦(𝑡),𝑧(𝑡))=𝑑−𝑣+3𝑤(𝑦(𝑡)−𝑧(𝑡))²−𝜀(𝑚+𝑐𝑦(𝑡))exp(𝑧(𝑡)−𝑦(𝑡))(1+𝜀exp(𝑧(𝑡)−𝑦(𝑡)))². (5.2) The Lyapunov exponents are 𝐿₁=lim_𝑡→∞𝜆(𝑡)𝑡,𝐿₂=lim_𝑡→∞𝜇(𝑡)𝑡. (5.3) If one of the two exponents is positive, the system has a chaotic behavior.

6. The Stochastic Difference Equation Associated to Difference Equation (2.6)

Let (Ω,𝐹,{ℱ_𝑡}_𝑡∈ℕ,𝑃) be a filtered probability space with stochastic basis and the filtration {ℱ_𝑡}_𝑡∈ℕ. Let {𝜉(𝑡)}_{𝑡∈𝐼ℕ} be a real valued independent random variable on (Ω,ℱ,{ℱ_𝑡}_𝑡∈ℕ,𝑃) with 𝔼(𝜉(𝑡))=0 and 𝔼(𝜉(𝑡))²<∞.

The stochastic difference equation associated to the difference equation (2.6) is given by 𝑦_𝑡+1=𝑎+𝑑𝑦(𝑡−1)+(𝑏−𝑑)𝑦(𝑡)+𝑣(𝑦(𝑡)−𝑦(𝑡−1))−𝑤(𝑦(𝑡)−𝑦(𝑡−1))³+𝑐𝑦(𝑡)+𝑚1+𝜀exp(𝑦(𝑡−1)−𝑦(𝑡))+𝑔(𝑦(𝑡),𝑦(𝑡−1))𝜉(𝑡). (6.1) The function 𝑔(𝑦(𝑡),𝑦(𝑡−1)) is the contribution of the fluctuations while it is under certain circumstances it is given by 𝑔(𝑦(𝑡))=𝛼(𝑦(𝑡)−𝑦₀),𝛼≥0 (6.2) or 𝑔(𝑦(𝑡−1),𝑦(𝑡))=𝛼(𝑦(𝑡−1)−𝑦(𝑡)),𝛼≥0. (6.3) From (6.1) and (6.2) the difference equation with multiplicative noise associated to the difference equation (2.6) is given by 𝑦(𝑡+1)=𝑎+𝑑𝑧(𝑡)+(𝑏−𝑑)𝑦(𝑡)+𝑣(𝑦(𝑡)−𝑧(𝑡))−𝑤(𝑦(𝑡)−𝑧(𝑡))³+𝑐𝑦(𝑡)+𝑚1+𝜀exp(𝑧(𝑡)−𝑦(𝑡))+𝛼(𝑦(𝑡)−𝑦₀)𝜉(𝑡),𝑧(𝑡+1)=𝑦(𝑡), (6.4) where 𝑦₀ is the fixed point of map (2.6).

Using (6.1) and (6.3) we have 𝑦(𝑡+1)=𝑎+𝑑𝑧(𝑡)+(𝑏−𝑑)𝑦(𝑡)+𝑣(𝑦(𝑡)−𝑧(𝑡))−𝑤(𝑦(𝑡)−𝑧(𝑡))³+𝑐𝑦(𝑡)+𝑚1+𝜀exp(𝑧(𝑡)−𝑦(𝑡))+𝛼(𝑧(𝑡)−𝑦(𝑡))𝜉(𝑡),𝑧(𝑡+1)=𝑦(𝑡). (6.5) By (2.6), randomizing parameter 𝑑 equation (6.5) is obtained.

The analysis of the stochastic difference equations (6.4) and (6.5) can be done by using the method from [10–12].

In what follows, we calculate an estimation of the upper (forward 2 th moment) stability exponent (given in [12]) for the stochastic process (𝑦(𝑡),𝑧(𝑡))_𝑡∈ℕ.

Let (𝑦(𝑡),𝑧(𝑡))_𝑡∈ℕ be the process that satisfies (6.4). The system of stochastic difference equations (6.4) has the form: 𝑦(𝑡+1)=𝑦(𝑡)+𝑓₁(𝑦(𝑡),𝑧(𝑡))+𝑓₂(𝑦(𝑡))𝜉(𝑡),𝑧(𝑡+1)=𝑧(𝑡)+𝑓₃(𝑦(𝑡),𝑧(𝑡)), (6.6) where 𝑓₁(𝑦(𝑡),𝑧(𝑡))=(𝑏−𝑑+𝑣−1)𝑦(𝑡)+(𝑑−𝑣)𝑧(𝑡)−𝑤(𝑦(𝑡)−𝑧(𝑡))³+𝑐𝑦(𝑡)+𝑚,𝑓1+𝜀exp(𝑧(𝑡)−𝑦(𝑡))₂(𝑦(𝑡))=𝛼(𝑦(𝑡)−𝑦₀𝑓),₃(𝑦(𝑡),𝑧(𝑡))=𝑦(𝑡)−𝑧(𝑡). (6.7)

The upper (forward 2 th moment) stability exponent of the stochastic process (𝑦(𝑡),𝑧(𝑡))_𝑡∈ℕ is defined by [12] 𝜆₂=limsup_𝑡→∞1𝑡ln𝔼(𝑦(𝑡)²+𝑧(𝑡)²), (6.8) provided that this limit exists.

Using Theorem 2.2 [12], for (6.5), with 𝑓₁, 𝑓₂, 𝑓₃ given by (6.7) and 𝑤=0, we get the following.

Proposition 6.1. Let (𝑦(𝑡),𝑧(𝑡)) be the process which satisfies the stochastic difference equation (6.6) with 𝑤=0. Assume that, forall𝑡∈ℕ, (𝑦(𝑡),𝑧(𝑡))∈ℝ², 𝑓₁(𝑦(𝑡),𝑧(𝑡))𝑦(𝑡)+𝑓₃(𝑦(𝑡),𝑧(𝑡))𝑧(𝑡)≤𝑘₁(𝑦(𝑡)²+𝑧(𝑡)²𝑓),₁(𝑦(𝑡),𝑧(𝑡))²+𝑓₃(𝑦(𝑡),𝑧(𝑡))²≤𝑘₂(𝑦(𝑡)²+𝑧(𝑡)²𝑓),₂(𝑦(𝑡),𝑧(𝑡))²≤𝑘₃(𝑦(𝑡)²+𝑧(𝑡)²), (6.9) where 𝑘₁, 𝑘₂, 𝑘₃ are finite, deterministic, real numbers. Then 𝜆₂≤2𝑘₁+𝑘₂+𝜎²𝑘₃, (6.10) with 𝔼(𝜉(𝑡))²=𝜎².

If the parameters of the model are 𝑎=250, 𝑣=0.1, 𝑤=0, 𝑐=0.1, 𝑏=0.45, 𝜀=1, 𝑚=0.5, 𝑑=0.8, 𝜎=0.5, and 𝛼=1, then for 𝑘₁=15, 𝑘₂=350, 𝑘₃=1, the inequalities (6.9) are satisfied and 𝜆₂≤380.25.

A similar result can be obtained for (6.5).

The qualitative analysis of the difference equations (6.4) and (6.5) is more difficult than that in the deterministic case and it will be done in our next papers.

7. Numerical Simulation

The numerical simulation is done using a Maple 13 program. We consider different values for the parameters which are used in the real economic processes. We use Box-Muller method for the numerical simulation of (6.4).

For system (6.4) with 𝛼=1, 𝑎=250, 𝑣=0.1, 𝑤=0, 𝑐=0.1, 𝑏=0.45, 𝜀=1, 𝑚=0.5 and the control parameter 𝑑=0 we obtain in Figure 1 the evolution of the income in the time domain (𝑡,𝑦(𝑡)), in Figure 2 the evolution of the income in the phase space (𝑦(𝑡−1),𝑦(𝑡)), in Figure 3 the evolution of the income in the stochastic case, and in Figure 4 the evolution of the income in the phase space in the stochastic case. In the deterministic case the Lyapunov exponent is negative and the system has not a chaotic behavior.

Figure 1 

(𝑡,𝑦(𝑡)) in the deterministic case for 𝑑=0.

Figure 2 

(𝑦(𝑡−1),𝑦(𝑡)) in the deterministic case for 𝑑=0.

Figure 3 

(𝑡,𝑦(𝑡)) in the stochastic case for 𝑑=0.

Figure 4 

(𝑦(𝑡−1),𝑦(𝑡)) in the stochastic case for 𝑑=0.

Comparing Figures 1 and 2 with Figures 3 and 4 we observe the solution behaving differently in the deterministic and stochastic cases.

For the control parameter 𝑑=0.6 Figure 5 displays the evolution of the income in the time domain (𝑡,𝑦(𝑡)), Figure 6 the evolution of the income in the phase space (𝑦(𝑡−1),𝑦(𝑡)), Figure 7 the evolution of the income in the stochastic case, Figure 8 the evolution of the income in the phase space in the stochastic case, and Figure 9 shows the Lyapunov exponent (𝑡,𝜆(𝑡)/𝑡).

Figure 5 

(𝑡,𝑦(𝑡)) in the deterministic case for 𝑑=0.6.

Figure 6 

(𝑦(𝑡−1),𝑦(𝑡)) in the deterministic case for 𝑑=0.6.

Figure 7 

(𝑡,𝑦(𝑡)) in the stochastic case for 𝑑=0.6.

Figure 8 

(𝑦(𝑡−1),𝑦(𝑡)) in the stochastic case for 𝑑=0.6.

Figure 9 

(𝑡,𝜆(𝑡)/𝑡) the Lyapunov exponent in the deterministic case for 𝑑=0.6.

Comparing Figures 5 and 6 with Figures 7 and 8 we observe the solution behaving differently in the deterministic and stochastic cases.

The Lyapunov exponent is positive, therefore the system has a chaotic behavior.

For the control parameter 𝑑=0.8, Figure 10 shows the evolution of the income in the time domain (𝑡,𝑦(𝑡)), Figure 11 the evolution of the income in the phase space (𝑦(𝑡−1),𝑦(𝑡)), Figure 12 the evolution of the income in the stochastic case, Figure 13 the evolution of the income in the phase space in the stochastic case, and Figure 14 shows the Lyapunov exponent (𝑡,𝜆(𝑡)/𝑡).

Figure 10 

(𝑡,𝑦(𝑡)) in the deterministic case for 𝑑=0.8.

Figure 11 

(𝑦(𝑡−1),𝑦(𝑡)) in the deterministic case for 𝑑=0.8.

Figure 12 

(𝑡,𝑦(𝑡)) in the stochastic case for 𝑑=0.8.

Figure 13 

(𝑦(𝑡−1),𝑦(𝑡)) in the stochastic case for 𝑑=0.8.

Figure 14 

(𝑡,𝜆(𝑡)/𝑡) the Lyapunov exponent in the deterministic case for 𝑑=0.8.

Comparing Figures 10 and 11 with Figures 12 and 13 we notice the solution behaving differently in the deterministic and stochastic cases.

The Lyapunov exponent is positive; therefore the system has a chaotic behavior.

Considering 𝑎 as parameter, we can obtain a Neimark-Sacker bifurcation point or a flip bifurcation point.

8. Conclusion

A dynamic model with discrete time using investment, consumption, sentiment, and saving functions has been studied. The behavior of the dynamic system in the fixed point's neighborhood for the associated map has been analyzed. We have established asymptotic stability conditions for the flip and Neimark-Sacker bifurcations. The QR method is used for determining the Lyapunov exponents and they allows us to decide whether the system has a complex behavior. Also, two stochastic models with multiplicative noise have been associated to the deterministic model. One model was obtained by adding one stochastic control and the other by randomizing the control parameter 𝑑. Using a program in Maple 13, we display the Lyapunov exponent and the evolution of the income in the time domain and the phase space, in both deterministic and stochastic cases. A qualitative analysis of the stochastic models will be done in our next papers.

Acknowledgments

The authors are grateful to Akio Matsumoto and the anonymous referee for their constructive remarks. The research was supported by the Grant with the title “The qualitative analysis and numerical simulation for some economic models which contain evasion and corruption,” The National University Research Council from Ministry of Education and Research of Romania (Grant no. 520/2009).