Stochastic Analysis of a Nonlinear Business Cycle Model with Correlated Random Income Disturbance

The economic cycle has always been an important feature of the evolution of an economic system. In the presence ofmany uncertain factors, it appears in the manner of very complex nonlinearity and randomness. Based on the theory of stochastic nonlinear dynamics, a nonlinear economic cycle model with correlated random income disturbance is established. The probability density evolution of the nonlinear economic cycle model under random disturbance is numerically analyzed by using a path integration method. The analysis shows the high saving rate reduces the investment and improves the probabilities of low income and low income change rate. In order to achieve a higher income, the saving rate should be controlled to some reasonable small value. The nonlinearity of the economic cycle model increases the probabilities of high income and high income change rate, which can lead to the increase of income in a probabilistic sense.The increase of random income interference enhances the uncertainty of income. Meanwhile, the increase of correlated random income disturbance can lead to a nonsymmetric distribution of the probability distributions of income and income change rate. In such cases, the income is more difficult to forecast and control.


Introduction
An economic cycle model is an effective mathematical tool to predict and control the development of economy.The economy usually undergoes a fluctuation process by the spiral way of prosperity, recession, depression, and recovery, which indicates that there exist some important rules of economic activities to be explored.This phenomenon attracts much attention from relevant scholars to use economic cycle models to explore the inherent law of circular economy change and its control factors.The corresponding effective measures are proposed to avoid severe economic shocks with a negative impact on economic activity in the market.
From the perspectives of mathematics and mechanics, an economic cycle model can be treated as a linear or nonlinear dynamical system based on the considered problems.Therefore, some experts and scholars have introduced the concept of nonlinear dynamical system to establish economic cycle models in order to study the evolution law and control factors of the complex economic cycle.Usually a constant, periodic function or random function is used to describe the spontaneous function to simulate the driving factors of a complex economic cycle.In 1940, Kaldor established a nonlinear economic cycle model based on the Keynes's income expenditure model theory and assumed that both savings and investment were nonlinear functions of income level [1].Chang and Smyth based on the Kalor model used two-dimensional differential equations and developed a continuous nonlinear dynamical economic cycle model system of income and capital [2].In 1951, Goodwin developed Hicks's idea on the consumption function and proposed a nonlinear economic cycle model, including consumption function, investment function, and its identity [3].Puu and Sushko introduced a nonlinear induced investment function of exponential quadratic form in the economic cycle model and they developed the Samuelson Hicks economic cycle model to study the nonlinear dynamic phenomena of the economic cycle model [4].Chian et al. adopted the van der Pol differential equation to model the nonlinear dynamics of business cycles under the action of a periodic exogenous force [5].Matsumoto and Szidarovszky developed a nonlinear multiplier-accelerator model with investment and consumption delays [6].Yu and Peng proposed a Kaldor-Kalecki model with both discrete and distributed delays and they used the method of multiple scales to analyze its stability and bifurcation [7].Riad et al. studied the dynamical response of a delayed business cycle model with general investment function [8].Naimzada and Pecora introduced a nonlinearity into the investment function and they studied the dynamical response of a multiplier-accelerator model [9].In fact, quite a few uncertainties exist in the macro-economy, which has attracted many researchers to develop several dynamic stochastic models to investigate the impact of randomness on macroeconomic variables [10].One important issue is to look for ways to quantify the impact of these uncertainties.Based on Goodwin's model and Puu's business cycle model, Li and Li studied the dynamic response of the stochastic nonlinear economic cycle model when the sum of the spontaneous investment is the periodic force and the Gauss white noise [11].Wingrove and Davis carried out a linear-control analysis to study the dynamical response and stability of a business cycle involving random disturbances [12].Li and Li studied the dynamical response of a nonlinear business cycle model under Poisson white noise excitation [13].Lin et al. studied the stability of a business cycle model with fractional derivative under narrow-band random excitation [14].Greenblatt developed a capitalist economic system considering the reciprocal interaction between investment, capacity utilization, and their time derivatives [15].Orlando and Zimatore examined the applicability of the recurrence quantification analysis for business cycles [16].They mentioned that, due to the presence of the randomness, nonlinearity, and nonstationarity in business cycles, some methods might fail.Bashkirtseva et al. adopted a stochastic sensitivity function technique to study the behavior of a Kaldor-type model of the business cycle with external additive and internal parametric disturbances [17].
On the other hand, the response of nonlinear dynamical systems under random disturbance is usually evaluated by its statistical moments and probability density function (PDF) [18].The PDF of nonlinear systems under Gaussian white noise is governed by the Fokker-Planck-Kolmogorov (FPK) equation.Due to the complicated form of the FPK equation, the exact stationary solutions are obtained in some special cases.Most work has to rely on numerical methods, representatively such as finite element method [19,20], weighted residual method [21], finite difference method [22], cell mapping method [23], and path integration method [24].Particularly, the path integration method has attracted much attention because of its conventional implementation and high accuracy.The path integration method can provide satisfactory solutions not only for the evolution of probability density but also for the tail of probability density.Wehner and Wolfer are the pioneers to systematically develop the path integration method [24].After that, many researchers devoted their efforts to advance the path integration method on its development and application.Naess et al. introduced a splines representation technique in implementing a path integration method for offshore structures [25,26].Yu et al. developed a new path integration procedure based on Gauss-Legendre scheme.The values of the probability density were evaluated at Gauss integral points [27].Dimentberg et al. applied the path integration method of Naess et al. [25,26] to obtain the response PDF of vibro-impact nonlinear systems [28].Chai et al. applied a path integration method to study nonlinear ship rolling in random beam seas [29].Di Paola and Bucher adopted a path integration method to solve the PDF of nonlinear system with ideal or physical barriers [30].Bucher et al. extended a path integration method to the first-passage problem for nonlinear systems under -stable Lévy white noises [31].Baravalle et al. employ a path integration method to the Hodgkin-Huxley model describing the neuronal dynamics [32].
To sum up, for the study on the nonlinear economic cycle model, the accurate estimation on its response is a critical issue.The accurate evaluation on the behavior of the nonlinear economic cycle model under different complicated situations is still a changeling topic in the field of mathematics and mechanics.Therefore, this paper is devoted to the analysis on the stochastic response of a nonlinear business cycle model with correlated random income disturbance.First, a new model is formulated by combining Goodwin's model and Puu's nonlinear economic cycle model.Gauss white noise is taken as a function of spontaneous function model for the new business cycle model.In the new model, the random income parametric interference is considered by the means of two correlated Gaussian white noises.After that, the path integration method in [27] is extended to studying the probability density of economic indicators for the nonlinear economic cycle.The probability density with spontaneous evolution function is analyzed in detail.

Economic Cycle Model with Correlated Random Income Disturbance
According to Goodwin's economic cycle model and Puu's nonlinear business cycle model [3,4], the investment function can be expressed as where   is the current investment; I 0t is the spontaneous investment; and ] is the capital output ratio and generally ]>0.  ,  -1 and  -2 are current income, pre-income, and prior income, respectively.Puu proposed the use of linear-pluscubic form of the deference between pre-income and prior income.This can simulate the countercyclical investment of the government on public infrastructure to resist the economic depression and benefit from lower investment during the period of great decline.
On the other hand, the consumption function is expressed as [4] where   is the current consumption;  0 is the spontaneous consumption;  is the added saving rate, 0 ≤  ≤ 1; and  is the prior savings rate, 0 ≤  ≤ 1.In a closed economic system, income equals the sum of investment and consumption where  * () =  0 +  0 denotes the sum of spontaneous investment and it is only a time variable and does not change with the subscript of  t .In this paper, the difference equation is converted into differential equation, and the dynamic response of the economic cycle can be formulated in a difference form The corresponding continuous differential equation is developed instead of the above difference equation to describe its dynamic response The reason is that the difference equation and differential equation can be used to study an identical dynamical system from different perspective.They both can equivalently describe the behaviors of the dynamical system.This study uses its differential equation in such a case that some continuous mathematical methods can easily solve this problem.
In addition, the spontaneous function  * (t) usually has three forms: constant, periodic function, or random function.Different forms of spontaneous functions can be used to simulate the effects of different economic environments.When stochastic function is used to simulate the government intervention, war, natural disasters, and other factors, the economic system has randomness and uncertainty.This paper considers that the spontaneous function  * (t) has two correlated Gaussian white noise random function forms, in which a Gaussian white noise random function acts on the revenue .Therefore, this paper studies the differential equation of nonlinear economic cycle model with correlated random income disturbance where and  is the income random interference coefficient.It is also noted that  1 (t) and  2 (t) are correlated.

Path Integration Method
This paper uses a path integration method to study the nonlinear economic cycle model expressed by (6).The stationary and nonstationary solutions of the income probability density of the economic cycle model can be studied.The strategy on controlling the income disturbance is proposed to realize the control on the nonlinear economic cycle model.The path integration method is used to discretize the continuous integral in space and time, and then the integral is replaced by the sum of integral path [27].When the probability density functions of income and income change rate of nonlinear economic cycle model are evaluated, the short-time Gaussian transition probability density between adjacent times is assumed and the probability density functions gradually evolve from the initial probability density with the increase of integration time [23].By this way, the probability density functions with time are obtained for income and income change rate.
Let () be the -dimensional state vector and its probability density function is as follows: where p(⋅) is the joint probability density function.
The Gauss-Legendre integral is used to discretize (7), and the path integration method uses the Gauss-Legendre integral and the short-time Gauss transition probability to evaluate the probability density functions.The two-dimensional expression is where  and  are the subinterval numbers along  and  axis, respectively.Δ  and Δ  are the lengths of each subinterval, respectively.( r ,   ) is the Gauss integral point coordinate.For the subintervals (  ,   ) and (  ,   ), when two Gauss integral points are taken, the point coordinates are as follows: The time step  =   − −1 and it is assumed that the transition probability density function has a Gauss distribution [23], i.e.,

𝑞 (𝑥
where The first and second moments of the model variable are obtained by using the Gauss truncation moment method.First, the moment equation of the model is where M= x    (i,  = 0, 1, 2).According to ( 13) and ( 14), (15) are formulated on the corresponding first and second moment equations where   = E[    ]; ṁ denotes the derivative of   with time.
Equations ( 15) contain some higher-order moments than second and these equations cannot be closed and be solved directly.Therefore, the Gaussian truncation moment method (assuming that the cumulants of higher order than second are zero) is used to express the higher-order terms with the first and second moments, i.e.,  03 = −2 Equations ( 16) are substituted into (15) and the Gaussian truncation moment method is used to obtain the closed equations.The initial conditions of the closed equations are set up, and the four-order Runge-Kutta method can be used to obtain the first and second moments.The transition probability density function can be obtained by using (11).
Then, the Gauss-Legendre integral solution ( 8) is used to obtain the joint probability density and marginal probability density of the income and income change rate for the nonlinear economic cycle model.

Numerical Analysis
In order to study the characteristics of the income evolution of the nonlinear economic cycle model with stochastic income disturbance and without the loss of the generality, a set of parameters is selected for numerical analysis as shown in Table 1.
In (6), other parameters are given as =0.1, 1 =0.2,  2 =0.2; the initial probability distribution is assumed as follows: where It should be noted that this study considers the stochastic response of an economic cycle model from a mathematical perspective.Therefore, the values in numerical analysis are purely numeric.The current results provide some guidelines for relevant experts when they study some practical strategy or measurements.In a practical economic problem, the details of control variable and control strategy can be determined given the real economic data of a region or a country.

The Effect of Low Saving Rate.
The stationary probability density of the income and income change rate of nonlinear economic cycle model under correlated random income disturbance are obtained by the path integration method.In case 1, the low saving rate is considered.The integral interval is [-4,4] × [-2,2], and the number of each division is 40.The time step dt=1.0 and it is observed that when T=100 ×dt, the PDF of the nonlinear economic cycle model becomes stationary.Therefore, the stationary PDF is taken as T=100 ×dt.
Firstly, according to the transition probability density (11), the selection of time step is made as 1.The response point In Figure 1, the transition probability density is obtained by the short-time Gauss transition probability approximation (TPDF) and Monte Carlo simulation (MCS), respectively.The sample of MCS is 1×10 5 for the transition probability approximation.The comparison shows that the transition probability densities obtained with the two methods are in good agreement.Therefore, the short-time Gauss transition probability density with the time step of 1 can be used for the path integration method.This good agreement between TPDF and MCS also can be observed in the following cases.Secondly, based on the short-time Gaussian transition probability density and Gauss-Legendre integral with the time step of 1, the path integration method is used to evaluate the stationary probability density of the income and income change rate of the nonlinear economic cycle model as shown in Figure 2. When the total length of time is T=100 ×dt (i.e., 100), the probability density obtained by the path integration method becomes stable as described above.The further increase of the duration time does not significantly change the value of the stationary probability density.Therefore, it can be concluded that the probability densities of the income and income change rate of the nonlinear economic cycle model become stationary at T=100. Figure 2(a) shows that the stationary probability densities obtained with the path integration method and Monte Carlo simulation of 2×10 7 samples agree well.As shown in Figure 2(b), the probability density is compared using the base-10 logarithmic value in order to evaluate the tail behavior.Figures 2(a) and 2(b) show that the tails of the probability density are also in good agreement with MCS.From the stationary probability density distribution of economic indicators, it can be seen that, under random interference, the nonlinear economic cycle model has the largest probability at the origin and it is symmetric about the origin.The growth and decrease of the same amplitude have the same probability density value, which shows that random interference can significantly affect the economic development.That is, it not only can promote the economic development, but also may have a negative impact on the economy.The random interference factors in the economic cycle should be paid enough attention to.

The Effect of High Saving Rate.
Next, the saving rate  is increased from 0.1 to 0.5 in this case as listed in Table 1.The integral interval is [-2,2] × [-2,2], and the number of each division is 40.The time step dt=1.0 and the stationary PDF is obtained when the time duration T=100 ×dt.
As shown in Figures 3(a) and 3(c), the PDFs of income and income change rate agree well with the simulation results, respectively.As the tail behavior is concerned, the good agreement is also achieved between the path integration method and Monte Carlo simulation when comparison is made in Figures 3(b) and 3(d).In this case, the saving rate is increased and the investment is made with less financial input.Figure 3(a) shows that the peak PDF of income increases from about 0.35 in Figure 2(a) to 1.0.The PDFs of low income are large around the origin.This means that low income more likely happens when the saving rate is increased.If the high income is expected, the saving rate should be decreased to some extent.On the other hand, Figure 3(c) also shows that the peak PDF of income change rate differs a lot from the one in Figure 2(c).This indicates that the low income change rate also likely occurs due to the increase of the saving rate.In a word, the saving rate significantly affects the PDFs of income and income change rate.The high saving rate reduces the investment and improves the probability of   low income and low income change rate.In order to achieve a higher income, the saving rate should be controlled to some reasonable small value.

The Effect of Increased Disturbance and Correlation.
Finally, the effect of increased disturbance and correlation is considered as =0.2 and =0.5.The integral interval is , and the number of each division is 40.The time step dt=1.0 and the stationary PDF is obtained when the time duration T=100 ×dt, which is the same as the former cases.Figure 4 shows the stationary probability density distribution of income and income change rate after the increase of random income disturbance.The stationary probability density distributions obtained by the path integration method and Monte Carlo simulation method are still in good agreement, which shows that the stochastic analysis of the nonlinear economic cycle model under the random income disturbance is effective.Comparison between Figures 3(a) and 4(a) shows that the increase of random income interference slightly decreases the peak PDF of income at the origin, which means that the increase of random income interference enhances the uncertainty of income.The income is more difficult to forecast and control.At the same time, compared with the effect of saving rate, the small random income interference cannot largely affect income and income change rate.The saving rate plays an important role in the nonlinear economic cycle model.

The Effect of Strong Nonlinearity with Low Saving Rate.
In case 4, the capital output ratio ] is taken as 0.5 compared with case 1.The strong nonlinearity of the economic cycle model is further studied.The numerical results show that the integral interval is [-4,4] × [-2,2], and the number of each division is 50.The time step dt=0.5 and it is observed that when T=100 ×dt.Under such settings, the stationary PDFs of income and income change rate can be obtained accurately.The good agreement with the simulation still can be observed in this case, which is also observed in the other two cases.Therefore, the PDF distributions of income and income change rate are discussed in detail in the following cases.Figure 2(a) shows that the PDF of income at origin is larger than 0.35 in Figure 2(a) with weak nonlinearity, whereas the one in Figure 5 manner can be observed in the case of income change rate in Figures 2(c) and 5(c).That is, the increase of nonlinearity of the economic cycle model significantly decreases the magnitudes of the peak probabilities of income and income change rate.That is to say, the nonlinearity of the economic cycle model increases the PDF of income and income change rate far from the origin.In a practical situation, the nonlinearity, simulating the countercyclical investment of the government on public infrastructure to resist the economic depression and benefit from lower investment during the period of great decline, can lead to the increase of income in a probabilistic sense.However, due to the symmetrical shape of the PDF distribution, the nonlinearity also leads to the increase of the PDF of negative income, which should be avoided.

The Effect of Strong Nonlinearity with High Saving Rate.
In this case, the saving rate increases from 0.1 in case 4 to 0.5 in case 5.The effect of strong nonlinearity with high saving rate is studied herein.The comparison between Figures 5 and  6 shows the effect of high saving rate.The numerical settings are the same as case 5, which is also employed in the last case.The peak of the PDFs of income and income change rate are much larger than those of case 5, correspondingly.The peak of PDFs of income is above 0.8 in Figure 6(a) and below 0.35 in Figure 5(a).The case of income change rate is similar.Therefore, the increase of saving rate can lead to the high probability of income and income change rate at origin.The PDFs far from the origin significantly decrease.This indicates that low income more likely happens when the saving rate is increased, which is similar to case 2. The strong nonlinearity of the economic cycle model cannot change the impact pattern of saving rate.

Conclusions
This paper combines Goodwin's model and Puu's nonlinear economic cycle model, takes the Gaussian white noise as a function of spontaneous function model, and considers random income disturbance.The path integration method is adopted to study the stationary PDF of the nonlinear economic cycle.The study shows that the obtained PDFs of the nonlinear economic cycle agree well with the simulation results, even in the tail region.In numerical analysis, the

Figure 1 :
Figure 1: Comparison on the TPDFs of income and income change rate.
Logarithmic PDF of income change rate

Figure 2 :
Figure 2: Comparison on the PDFs of income and income change rate in case 1.

Figure 3 :
Figure 3: Comparison on the PDFs of income and income change rate in case 2.

Figure 4 :
Figure 4: Comparison on the PDFs of income and income change rate in case 3.

Figure 5 :
Figure 5: Comparison on the PDFs of income and income change rate in case 4.

4. 6 .
The Effect of Full Correlation.In the last case, the full correction of random income disturbance is considered by setting =0.2 and =1.0.Comparison between Figures6 and 7shows that the differences on the globe PDFs of income and income change rate are insignificant.This is due to the small magnitude of .However, the tail regions of the PDFs of income and income change rates are significantly different as shown in Figures6(b), 6(d), 7(b), and 7(d), correspondingly.Furthermore, the nonsymmetric distributions can be formulated in the tail regions in Figures 7(b) and 7(d).These observations indicate that the increase of correlated random income disturbance can lead to a nonsymmetric distribution of the PDFs of income and income change rate.

Table 1 :
Parameter settings in numerical analysis.