Social conflict and wage inequality

Abstract

This paper studies the relationship between social conflict and skilled–unskilled wage inequality through the three-sector general equilibrium approach. In the basic model without the urban unskilled minimum wage, we find that when the government enhances the degree of controlling social conflict, the skilled–unskilled wage inequality will be narrowed down (resp. widened) if the urban skilled sector is more capital intensive (resp. labor intensive) than the urban unskilled sector. The extended models address the issue under different economic structures or different types of social conflict. In the extended model with the urban unskilled minimum wage, we find that the skilled–unskilled inequality will be widened when the degree of controlling social conflict is increased. In other extended models, we find that the above obtained results are still robust.

Appendices

Dynamic adjustment process of the basic model

The dynamic adjustment process is based on the idea of the excess demand function, including the Marshallian quantity adjustment process and the Walrasian price adjustment process. Our method is in line with Beladi et al. (2008). According to the excess demand functions of the basic model, the differential equations can be constructed as:

$$\begin{aligned} \dot{{X}}= & {} d_{1} \left( {p_{X} -a_{SX} w_{S} +a_{KX} r} \right) ,\end{aligned}$$

(A1)

$$\begin{aligned} \dot{{Y}}= & {} d_{2} \left( {p_{Y} -a_{UY} w_{U} +a_{KY} r} \right) ,\end{aligned}$$

(A2)

$$\begin{aligned} \dot{{Z}}= & {} d_{3} \left( {1-a_{UZ} w_{U} +a_{TZ} \tau } \right) ,\end{aligned}$$

(A3)

$$\begin{aligned} \dot{{L}}_{A}= & {} d_{4} \left( {\frac{A\left( {\alpha ,L_{A} } \right) }{L_{A} }\left( {w_{S} \bar{{L}}_{S} +w_{U} \left( {\bar{{L}}_{U} -L_{A} } \right) +r\bar{{K}}+\tau \bar{{T}}} \right) -\left( {1-A\left( {\alpha ,L_{A} } \right) } \right) w_{U} } \right) , \nonumber \\ \end{aligned}$$

(A4)

$$\begin{aligned} \dot{{r}}= & {} d_{5} \left( {a_{KX} X+a_{KY} Y-\bar{{K}}} \right) ,\end{aligned}$$

(A5)

$$\begin{aligned} \dot{{\tau }}= & {} d_{6} \left( {a_{TZ} Z-\bar{{T}}} \right) , \end{aligned}$$

(A6)

$$\begin{aligned} \dot{{w}}_{S}= & {} d_{7} \left( {a_{SX} X-\bar{{L}}_{S} }\right) , \end{aligned}$$

(A7)

$$\begin{aligned} \dot{{w}}_{U}= & {} d_{8} \left( {a_{UY} Y+a_{UZ} Z-\left( {\bar{{L}}_{U} -L_{A} } \right) } \right) , \end{aligned}$$

(A8)

where $d_{i} >0 \quad \left( {i=1,2,...,8} \right)$ represents the speed of adjustment. The notation “.” denotes the differentiation with respect to time (e.g., $\dot{{X}}=\frac{dX}{dt})$. The determinant of the Jacobian matrix of Eqs. (A1)–(A8) (denoted as $\Phi _{1} )$ can be written as:

$$\begin{aligned} \Phi _{1} =H\left| \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0&{} 0&{} 0&{} 0&{} {-\theta _{KX} }&{} 0&{} 0&{} {-\theta _{SX} } \\ 0&{} 0&{} 0&{} {-\theta _{UY} }&{} {-\theta _{KY} }&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} {-\theta _{UZ} }&{} 0&{} {-\theta _{TZ} }&{} 0&{} 0 \\ {\theta _{X} }&{} {\theta _{Y} }&{} {\theta _{Z} }&{} {-1}&{} 0&{} 0&{} C&{} 0 \\ {\lambda _{KX} }&{} {\lambda _{KY} }&{} 0&{} {\lambda _{KY} \theta _{UY} \sigma _{Y} }&{} E&{} 0&{} 0&{} {\lambda _{KX} \theta _{SX} \sigma _{X} } \\ 0&{} 0&{} 1&{} {\theta _{UZ} \sigma _{Z} }&{} 0&{} {-\theta _{UZ} \sigma _{Z} }&{} 0&{} 0 \\ 1&{} 0&{} 0&{} 0&{} {\theta _{KX} \sigma _{X} }&{} 0&{} 0&{} {-\theta _{KX} \sigma _{X} } \\ 0&{} {\lambda _{UY} }&{} {\lambda _{UZ} }&{} F&{} {\lambda _{UY} \theta _{KY} \sigma _{Y} }&{} {\lambda _{UZ} \theta _{TZ} \sigma _{Z} }&{} {\lambda _{UA} }&{} 0 \\ \end{array} \right| =H\Delta _{1}, \end{aligned}$$

where $H=\frac{\prod \nolimits _{i=1}^8 {d_{i} \left( {1-A\left( {\alpha ,L_{A} } \right) } \right) p_{X} p_{Y} \left( {\bar{{L}}_{U} -L_{A} } \right) \bar{{L}}_{S} \bar{{K}}\bar{{T}}} }{w_{S} r\tau XYZL_{A} }>0$.

According to the Routh–Hurwitz theorem, the local stability can be achieved if $\Phi _{1} >0$, which implies that $\Delta _{1} >0$. In order to make our economic system locally stable, we need $\Delta _{1} >0$.

Dynamic adjustment process of the model in Sect. 3.1

Replace (A2) and (A8) with the following differential equations, respectively:

$$\begin{aligned} \dot{{Y}}= & {} d_{2} \left( {p_{Y} -a_{UY} \bar{{w}}_{U} +a_{KY} r} \right) ,\\ \dot{{w}}_{U}= & {} d_{8} \left( {a_{UY} Y\left( {1+\lambda } \right) +a_{UZ} Z-\left( {\bar{{L}}_{U} -L_{A} } \right) } \right) . \end{aligned}$$

Under similar demonstrations in Appendix 1, we need $\Delta _{2} >0$ in order to make our economic system in Sect. 3.1 locally stable.

Dynamic adjustment process of the model with full employment in Sect. 3.2

Replace (A4), (A7), and (A8) with the following differential equations, respectively:

$$\begin{aligned}&\dot{{L}}_{A} =d_{4} \left( {\frac{A\left( {\alpha ,L_{A} } \right) }{L_{A} }\left( {w_{S} \left( {\bar{{L}}_{S} \!-\!L_{A} } \right) \!+\!w_{U} \bar{{L}}_{U} \!+\!r\bar{{K}}\!+\!\tau \bar{{T}}} \right) \!-\!\left( {1-A\left( {\alpha ,L_{A} } \right) } \right) w_{S} } \right) ,\nonumber \\ \end{aligned}$$

(A9)

$$\begin{aligned}&\dot{{w}}_{S} =d_{7} \left( {a_{SX} X-\left( {\bar{{L}}_{S} -L_{A} } \right) } \right) , \end{aligned}$$

(A10)

$$\begin{aligned}&\dot{{w}}_{U} =d_{8} \left( {a_{UY} Y+a_{UZ} Z-\bar{{L}}_{U} } \right) . \end{aligned}$$

(A11)

The determinant of the Jacobian matrix of Eqs. (A1)–(A3), (A5), (A6), and (A9)–(A11) (denoted as $\Phi _{2} )$ can be written as:

$$\begin{aligned} \Phi _{2} =I\left| \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0&{} 0&{} 0&{} {-\theta _{SX} }&{} {-\theta _{KX} }&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} {-\theta _{KY} }&{} 0&{} 0&{} {-\theta _{UY} } \\ 0&{} 0&{} 0&{} 0&{} 0&{} {-\theta _{TZ} }&{} 0&{} {-\theta _{UZ} } \\ {\theta _{X} }&{} {\theta _{Y} }&{} {\theta _{Z} }&{} {-1}&{} 0&{} 0&{} C&{} 0 \\ {\lambda _{KX} }&{} {\lambda _{KY} }&{} 0&{} {\lambda _{KX} \theta _{SX} \sigma _{X} }&{} E&{} 0&{} 0&{} {\lambda _{KY} \theta _{UY} \sigma _{Y} } \\ 0&{} 0&{} 1&{} 0&{} 0&{} {-\theta _{UZ} \sigma _{Z} }&{} 0&{} {\theta _{UZ} \sigma _{Z} } \\ 0&{} {\lambda _{UY} }&{} {\lambda _{UZ} }&{} 0&{} {\lambda _{UY} \theta _{KY} \sigma _{Y} }&{} {\lambda _{UZ} \theta _{TZ} \sigma _{Z} }&{} 0&{} F \\ {\lambda _{SX} }&{} 0&{} 0&{} {-\lambda _{SX} \theta _{KX} \sigma _{X} }&{} {\lambda _{SX} \theta _{KX} \sigma _{X} }&{} 0&{} {\lambda _{SA} }&{} 0 \\ \end{array} \right| =I\Delta _{3} , \end{aligned}$$

where $I=\frac{\prod \limits _{i=1}^8 {d_{i} \left( {1-A\left( {\alpha ,L_{A} } \right) } \right) p_{X} p_{Y} \bar{{L}}_{U} \left( {\bar{{L}}_{S} -L_{A} } \right) \bar{{K}}\bar{{T}}} }{w_{U} r\tau XYZL_{A} }>0$.

According to the Routh–Hurwitz theorem, the local stability can be achieved if $\Phi _{2} >0$, which implies that $\Delta _{3} >0$. Thus in order to make our economic system locally stable, we need $\Delta _{3} >0$.