Goodwin Model with Clustering Workers' Skills in Indonesian Economic Cycle

The economic model which deals with the economic cycle is Goodwin's Model. It presents the relationship between employment rate and wage shares. In this study, the modification model was made taking into three types of workers, namely high-, medium-, and low-skilled workers. Studies of model are conducted by determining the equilibrium point and its stability analysis. Futhermore, a numerical simulation is given to see which model satisfy the ideal of Goodwin‘s model cycle prediction by using Indonesian data from 2000 to 2020. At the end, an investigation into the effects of reducing the wage gap between the three types of workers was conducted. The results showed two equilibrium points, namely The Equilibrium Point without Employment Rate and The Wages Share (T1) and the Existence Equilibrium Point of Employment Rate and Wages Share (T2). T1 achieves a stable node locally condition when Scσ < δ + φ + n while T2 reaches a center locally condition when Scσ > δ + φ + n. The simulation showed Goodwin's model of lowskilled workers produced the ideal of Goodwin model cycle whereas Goodwin's model of high-, medium-skilled workers, and the entire economy (capitalist) didn’t produce the ideal of Goodwin model cycle. Eventually the effects of reducing the wage gap makes the economy unstable.

achieve wage determination in appropriate with the conditions of a country and the capabilities of the company, so it is necessary to determine the minimum wage that concerns the fulfillment of the Needs of Decent Living (NDL). The government sets wages in accordance with the NDL, by observing the production power and economic development [9]. Wage determination must be adjusted to the types of skills of workers. In general, workers' skills are divided into high-and low-skilled workers. To make adjustments and approaches to real phenomena, the research is needed on workers who have skills between high and low-skilled workers, namely medium-skilled workers. So this study was conducted on the skills of high, medium, and low-skilled workers and investigated its impact in the Indonesian economic cycle.
A mathematical model that predicts the existence of an economic cycle and shows fluctuations in an economic state is Goodwin's Class Struggle Model or better known as the Goodwin Model [10]. This model is widely studied from time to time, especially in its application in economic problems [11]. This model continues to develop and modify over the time. Goodwin's model only considers the employment rates and wage share, so Grasselli and Lima [12] developed their research by analyzing Goodwin's model by including debt-financed investments and this model was called the Keen model. Furthermore, Goodwin's model and Keen's model were analyzed for Minsky's financial instability hypothesis model. Sasaki et al., [13] developed a macro dynamic model taking into account two types of labor (regular and non-regular workers) and investigating how the expansion of the wage gap between regular and non-regular workers affected the economy and analyzed how the economic cycle was affected by the introduction of the minimum wage. Sasaki and Asada [14] extended Goodwin's model by considering two types of workers: low and high-skilled workers at the employment rate and the wages share. Sasaki and Asada [14] investigated how the introduction of minimum wage share affected employment rates and the wage share as well as the effect of reducing the wage gap between low and high-skilled workers using 1989-2018 Japanese economic data. The research showed the introduction of minimum wage share reduced the amplitude of fluctuations and stabilized the economy. The effect of reducing the wage gap between low-and high-skilled workers increases the amplitude of fluctuations of the wage share and employment rates.
In this study, a model modification was made that refers to Sasaki and Asada [14] by adding the type of workers to three types of workers, namely high-skilled, mediumskilled and low-skilled workers. Furthermore, the analysis is carried out by determining the fixed point of the model along with its stability and numerical simulation by applying Indonesian economic data in 2000-2020 [15] in the case of the Indonesian economic cycle to see which model satisfy the ideal of Goodwin model cycle predictions. At the end, an investigation into the effects of reducing the wage gap between high-skilled, medium-skilled, and low-skilled workers was conducted to see if the reducing the wage gap could stabilize the economic cycle.

Mathematical Model
In this article, mathematical models are formulated based on assumptions below: 1. Goodwin's model is reviewed based on the skill of workers that are high-, medium-, and low-skilled workers and the entire economy (capitalist), where the entire economy (capitalist) is the sum of all workers' skills or representing workers as a whole 2. The measurement of workers' skills is based on the last level of education completed 3. High-skilled workers are workers who have the highest level of education completed is Diploma I/II/III/Academy/University (workers aged 19 years and older) 4. Medium-skilled workers are workers who have the highest level of education that is completed is high school/vocational school (workers aged 17 years and older) 5. Low-skilled workers are workers who have the highest level of education that is completed are junior high school/MTs (workers aged 15 years and older) 6. There are only two factors of production: labor and capital 7. We suppose have an economy in which low-skilled workers, medium-skilled workers, high-skilled workers, and capitalists coexist 8. Firms produce a single good used for both consumption and investment using low-, medium-, and high-skilled workers, (as a labor) and capital stock according to the following Leontief production function 9. Labor productivity of both groups increases at the same constant rate ( ) 10. Firms are assumed to adopt cost-minimizing behaviors 11. The labor supply ( ) grows at a constant rate ( ) 12. The growth rate of the real wage rate of high-skilled workers ( ) is an increasing function of the employment rate of high-skilled workers 13. The real wage rate of high-skilled workers is more than that of medium-skilled workers by a constant factor and the real wage rate of medium-skilled workers is more than that of low-skilled workers by a constant factor. 14. The low-skilled workers ( ), medium-skilled workers ( ), and high-skilled workers ( ) save their wage incomes at constant rates, respectively, and that capitalists ( ) save their profit incomes at a constant rate 15. Saving rates < < < 16. All the parameters are positive.
Based on the assumptions above, the dynamics of the Goodwin model with three types of worker can be represented in the following set of differential equations divided based on the types of each skill.
Goodwin models of high-skilled workers: Goodwin models of medium-skilled workers: Goodwin models of low-skilled workers: Goodwin model of the entire economy (capitalist): Description: : The employment rate of high-skilled workers : The wage share of high-skilled workers : The employment rate of medium-skilled workers : The wage share of medium-skilled workers : The employment rate of low-skilled workers : The wage share of low-skilled workers : The employment rate of the entire economy (capitalists) : The wage share of the entire economy (capitalists) The description of all parameters presented in Table 5. ) has a good meaning in economically because the employment rates and the wage share of high-skilled workers has positive value. The determination of stability first lineing the equations (1) and (2). The Jacobian matrix is obtained:

Equilibrium Points and it's Stability Analysis of Goodwin's Model with Three Types of Workers
Where: The stability of the equilibrium point studied by local condition. It is seen from the eigenvalue that generated by substitute the equilibrium point without employment rate and the wages share 1 ( , ) and the existence equilibrium point of employment rate and wages share 2 ( * , * ) into the Jacobian matrix ( , ) in equation (9), the eigen value of the the equilibrium point 1 ( , ) follows: To satisfy the stability, all the eigenvalues must be negative. Thus the stability of equilibrium point of 1 ( , ) is a stable node locally condition if it satisfy < + + . And the eigen value of the the equilibrium point 2 ( * , * ) follows: From the eigen value obtained, the stability of equilibrium point of 2 ( * , * ) is a center locally condition when > + + . The stability of the equilibrium point without employment rate and the wages share 1 ( , ) and the existence equilibrium point of employment rate and wages share 2 ( * , * ) based on the eigenvalues obtained results such as in Table 1.  (3) and (4) is obtained by solving equations ̇= 0 and ̇= 0 . The results showed two equilibrium points, namely the equilibrium point without employment rate and the wages share ( 1 ) where 1 ( , ) = (0,0) and the existence equilibrium point of employment rate and wages share ( 2 ) where 2 ( * , * ) = ). The equilibrium point without employment rate and the wages share 1 ( , ) = (0,0) in the economy has no meaning because the employment rates ( ) and the wages share of medium-skilled workers ( ) are both worth zero. While the existence equilibrium point of employment rate and wages share 2 ( * , * ) = ( ( + ) ̅ , ) has a good meaning in economically because the employment rates and the wage share of medium-skilled workers has positive value.
The determination of stability first lineing the equations (3) and (4). The Jacobian matrix is obtained: Where: The stability of the equilibrium point studied by local condition. It is seen from the eigenvalue that generated by substitute the equilibrium point without employment rate and the wages share 1 ( , ) and the existence equilibrium point of employment rate and wages share 2 ( * , * ) into the Jacobian matrix ( , ) in equation (10), the eigen value of the the equilibrium point 1 ( , ) follows: To satisfy the stability, all the eigenvalues must be negative. Thus the stability of equilibrium point of 1 ( , ) is a stable node locally condition if it satisfy < + + . And the eigen value of the the equilibrium point 2 ( * , * ) follows: 12 = ± √( + )( − − − ) From the eigen value obtained, the stability of equilibrium point of 2 ( * , * ) center locally condition when > + + . So that the stability of the equilibrium point without employment rate and the wages share 1 ( , ) and the existence equilibrium point of employment rate and wages share 2 ( * , * ) based on the eigenvalues obtained results such as in Table 2.  (5) and (6) is obtained by solving equations ̇= 0 and ̇= 0 . The results showed two equilibrium points, namely the equilibrium point without employment rate and the wages share ( 1 ) where 1 ( , ) = (0,0) and the existence equilibrium point of employment rate and wages share ( 2 ) where 2 ( * , * ) = ). The equilibrium point without employment rate and the wages share 1 ( , ) = (0,0) in the economy has no meaning because the employment rates ( ) and the wages share of low-skilled workers ( ) are both worth zero. While the existence equilibrium point of employment rate and wages ) has a good meaning in economically because the employment rates and the wage share of low-skilled workers has positive value. The determination of stability first lineing the equations (5) and (6). The Jacobian matrix is obtained: Where: The stability of the equilibrium point studied by local condition. It is seen from the eigenvalue that generated by substitute the equilibrium point without employment rate and the wages share 1 ( , ) and the existence equilibrium point of employment rate and wages share 2 ( * , * ) into the Jacobian matrix ( , ) in equation (11), the eigen value of the the equilibrium point 1 ( , ) follows: To satisfy the stability, all the eigenvalues must be negative. Thus the stability of equilibrium point of 1 ( , ) is a stable node locally condition if it satisfy < + + . And the eigen value of the the equilibrium point 2 ( * , * ) follows: 12 = ± √( + )( − − − ) From the eigen value obtained, the stability of equilibrium point of 2 ( * , * ) center locally condition when > + + . So that the stability of the equilibrium point without employment rate and the wages share 1 ( , )and the existence equilibrium point of employment rate and wages share 2 ( * , * ) based on the eigenvalues obtained results such as in Table 3.  (7) and (8)  ) has a good meaning in economically because the employment rates and the wage share of the entire economy (capitalists) has positive value. The determination of stability first lineing the equations (7) and (8). The Jacobian matrix is obtained: Where: The stability of the equilibrium point studied by local condition. It is seen from the eigenvalue that generated by substitute the equilibrium point without employment rate and the wages share 1 ( , ) and the existence equilibrium point of employment rate and wages share 2 ( * , * ) into the Jacobian matrix ( , ) in equation (12), the eigen value of the the equilibrium point 1 ( , )follows: 1 = − − − ⋁ 2 = − − To satisfy the stability, all the eigenvalues must be negative. Thus the stability of equilibrium point of 1 ( , )is a stable node locally condition if it satisfy < + + . And the eigen value of the the equilibrium point 2 ( * , * ) follows: 12 = ± √( + )( − − − ) From the eigen value obtained, the stability of equilibrium point of 2 ( * , * ) center locally condition when > + + . So that the stability of the equilibrium point without employment rate and the wages share 1 ( , )and the existence equilibrium point of employment rate and wages share 2 ( * , * ) based on the eigenvalues obtained results such as in Table 4.

Numerical Simulation of Goodwin Model Modification with Three Types of Workers
Numerical simulations are used to show whether Goodwin's model of three types of workers satisfy the ideal of Goodwin's model cycle. Simulation on Goodwin model with three types of workers conducted by substituting parameter values that have been estimated (using ordinary least squares) and determined based on Indonesian economic data from 2000 to 2020. The data obtained from website of Badan Pusat Statistik Indonesia [16]. Before estimating the parameters, the step that must be done is to find out which conditions in the data from 2000 to 2020 that satisfy the Phillips Curve theory in Goodwin's model which is satisfying the conditions of the inverse relationship between the unemployment rate and the wage growth rate.  Table 5. The initial value of Goodwin model of high-skilled workers ( and ), Goodwin model of medium-skilled workers ( and ), Goodwin model of low-skilled workers ( and ), Goodwin model of the entire economy (capitalists) ( and ) are 0.2. The simulation results using parameter values on Table 5 and an initial value of 0.2 are obtained as follows:  Figure 1 shows that the employment rate and wage share of the high-skilled workers increase at the beginning of the cycle followed by a decrease towards stability then fluctuate again every 20 years. The trivial equilibrium point 1 ( , ) = (0,0) is unstable (saddle) with eigenvalues 1 = 6.95 and 2 = −1.53 and 2 ( * , * ) = (0.11, 0.37) is center with eigen values 0. + 3.26 , 0. − 3.26 in accordance with the results of stability analysis in Table 1. Figure 3 and Figure Figure 9 shows that employment rate and the wage share of the low-skilled workers increase at the beginning of the cycle followed by a decrease to stability then fluctuates again every 20 years. The trivial equilibrium point 1 ( , ) = (0,0) is unstable (saddle) with eigenvalues 1 = 6.95 and 2 = −1.53 and 2 ( * , * ) = (0.16, 0.22) is center with eigen values 0. + 3.26 , 0. − 3.26 in accordance with the results of stability analysis in Table 3. Figure 11 and Figure 12 show the cycle of dynamics between employment rate and the wage share of the low-skilled workers period 2010-2014. The resulting cycle is similar to the period 2001-2004 but the fluctuations occur every 25 years. This shows  Figure 13 shows that the employment rate decreases at the beginning of the cycle followed by a rise and fluctuates again every 25 years. While the wage share begins with an increase and then decreases to stability and fluctuates again every 25 years. The trivial equilibrium point 1 ( , ) = (0,0) is unstable (saddle) with eigenvalues 1 = 6.95 and 2 = −1.53 and 2 ( * , * ) = (0.58, 1.20) is center with eigen values 0. + 3.26 , 0. − 3.26 in accordance with the results of stability analysis in Table 3. Figure 15 and Figure 16 show the cycle of dynamics between employment rate and

The Effect of Reducing the Wage Gap Between Three Types of Workers
In this section, an investigation into the effects of reducing the wage gap between the three types of workers by reducing the value of wage gap parameters between high-and medium-skilled workers ( ) and the parameters of the wage gap between medium-and low-skilled workers ( ) at each period by 10%, 20% and 30%.   Figure 18 show the cycle of dynamics between employment rate and the wage share of the high-skilled workers with a reduction in the wage gap between the three types of workers. The results showed the greater the decrease in the value of the and the more shifting the orbit downwards and minimized the cycle. This indicates that the smaller value of the and then the economic cycle becomes unstable. Goodwin Model of Medium-Skilled Workers with Reduced The Value of Parameters and Figure 19 and Figure 20 show the cycle of dynamics between employment rate and the wage share of the medium-skilled workers with a reduction in the wage gap between the three types of workers. Figure 19 shows the reduction value of and as 10% and 30% maximizing the cycle and a 20% reduction minimizing the cycle but not shifting the orbit. This indicates that the smaller the value of the and then the economic cycle formed is not fixed but the cycle formed is stable because it does not cause significant cyclical changes from the previous model. Figure 20 shows the reduction value of and as 10% and 20% maximizing the cycle and a 30% reduction minimizing the cycle but not shifting the orbit. This indicates that the smaller the value of the and then the economic cycle formed is also not fixed but the cycle formed is stable because it does not cause significant cyclical changes from the previous model. Of the two cycles formed, the more ideal cycle is in the period 2001-2004 because it produces a cycle change that is not significant compared to the period 2010-2014.  Figure 22 show the cycle of dynamics between employment rate and the wage share of the low-skilled workers with a reduction in the wage gap between the three types of workers. Figure 21 and Figure 22 show the greater the decrease in the value of the and further shifting the orbit upwards and enlarging the cycle. This indicates that the reduction of the value of and makes good changes characterized by an enlarged cycle, that indicates a large economic recovery process but cannot stabilize the economy due to very significant differences with the previous model and hence the economic cycle becomes unstable.

Goodwin Model of Low-Skilled Workers with Reduced The Value of Parameters and
Goodwin Model of The Entire Economy (Capitalists) with Reduced The Value of Parameters and Figure 23 and Figure 24 show the cycle of dynamics between employment rate and the wage share of the entire economy (capitalists) with a reduction in the wage gap between the three types of workers. Figure 23 and Figure 24 show the greater decrease the value of the and minimizing the cycle. This indicates that the smaller the value of the and then the economic cycle becomes unstable.
Numerical simulation results of the period 2001-2004 and the period 2010-2014 on each type of worker's skill showed when the values of and decreased by 10%, 20%, and 30% resulting in an unstable economic cycle in high-and low-skilled workers and the entire (capitalist) economy. The reduction in the value of and appears to be desired by low-skilled workers but not for economic stability because the resulting cycle shifts the orbit upwards and enlarges the cycle which means this reduction in gap makes good changes characterized by the grinding of the orbit upwards and the enlarging of the cycle. While in medium-skilled workers the resulting cycle stabilizes the economy. So it can be concluded that reducing the wage gap cannot stabilize the economy as a whole. The wage gap must be determined according to each worker's ability to produce a stable economic cycle.

CONCLUSIONS
Goodwin's model is constructed based on economic variables that can describe the dynamics between employment opportunity rates and wages share. Goodwin's model is modified by adding consideration of the type of workers from two types of workers, namely high and low-skilled workers into three types of workers, namely high, medium and low-skilled workers. The results showed two equilibrium points, namely The Equilibrium Point without Employment Rate and The Wages Share ( 1 ) and the Existence Equilibrium Point of Employment Rate and Wages Share ( 2 ). 1 achieves a stable node condition when < + + while 2 reaches a stable center condition when > + + . Based on Indonesia's economic data from 2000 to 2020, conditions were obtained that satisfy the Phillips curve theory in the Goodwin Model are the period 2001-2004 and the period 2010-2014. Numerical simulations showed that Goodwin's model with three types of workers fluctuated periodically. From the estimation results, low-skilled workers produce ideal Goodwin model cycle while Goodwin's model of high-, medium-skilled workers and the entire economy (capitalist) does not produce ideal Goodwin model cycle. Furthermore, gap reduction is carried out to see the effect on each type of worker's skills in economic stability. Numerical simulation results show when the values of and decrease by 10%, 20%, and 30% resulting in an unstable economic cycle in high-skilled, low-skilled workers and the entire economy (capitalists). Reductions in the value of and appear to be desirable by low-skilled workers but not for economic stability. And for medium-skilled workers, the resulting cycle can stabilize the economy. Thus, the wage gap cannot stabilize the overall economy and must be determined according to each worker's ability to produce a stable economic cycle.