Environmental disparities in an urban area, rural–urban migration, and urban unemployment

Abstract

The environment is not always uniform for all of the residents of a specific region. This study explicitly incorporates this aspect into a general equilibrium model of rural–urban migration with urban unemployment. Pollution from the urban manufacturing sector is assumed to have a negative externality on urban residents’ utility. Moreover, this study assumes that in an urban area, the effects of pollution on employed and unemployed workers are different. Thus, disparities in the environment develop for the residents within that area. Rural–urban migration is assumed to occur according to a rural–urban difference in expected utilities. This study performs a comparative static analysis. It is shown that, in an urban area, a decrease in the effect of pollution on an employee (unemployed worker) increases his/her utility; however, it decreases an unemployed worker’s (employee’s) utility.

Appendix: stability

Consider the following dynamic systems:

$$\dot{E}^{e} = \bar{E} - E^{e} - \alpha \lambda F^{M} \left( {L_{M} , K_{M} } \right) \equiv \phi \left( {E^{e} , E^{u} } \right) ,$$

(32)

$$\dot{E}^{u} = \bar{E} - E^{u} - \beta \lambda F^{M} \left( {L_{M} , K_{M} } \right) \equiv \varPsi \left( {E^{e} , E^{u} } \right) .$$

(33)

For given $E^{e}$ and $E^{u}$, six Eqs. (3), (4), (7), (8), (9), and (23) determine six endogenous variables: $L_{M}$, $L_{U}$, $L_{A}$, $K_{M}$, $K_{A}$, and $w$. Necessary and sufficient condition for the linear approximation system of (32) and (33) to be globally stable is that the trace of $S$ (${\text{tr }} S$) is negative and the determinant of matrix $S$ (det $S$) is positive, where $S \equiv \left[ {\begin{array}{*{20}c} {\partial \phi /\partial E^{e} } & {\partial \phi /\partial E^{u} } \\ {\partial \varPsi /\partial E^{e} } & {\partial \varPsi /\partial E^{u} } \\ \end{array} } \right]$. By differentiating (3), (4), (7), (8), (9), and (23) to investigate the signs of ${\text{tr }} S$ and det $S$, we obtain

$$\left[ {\begin{array}{*{20}c} {pF_{LL}^{M} } & 0 & 0 & {pF_{LK}^{M} } & 0 & 0 \\ 0 & 0 & {F_{LL}^{A} } & 0 & {F_{LK}^{A} } & { - 1} \\ {pF_{KL}^{M} } & 0 & { - F_{KL}^{A} } & {pF_{KK}^{M} } & {-F_{KK}^{A} } & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ x & y & 0 & 0 & 0 & { - V_{w}^{a} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {dL_{M} } \\ {dL_{U} } \\ {dL_{A} } \\ {dK_{M} } \\ {dK_{A} } \\ {dw} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ { - g} \\ \end{array} } \right]dE^{e} + \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ { - h} \\ \end{array} } \right]dE^{u} .$$

(34)

We denote the determinant of the square matrix of (34) as $J$. We then have

$$J = pF_{LK}^{M} F_{KL}^{A} \left( {V^{e} - V^{u} } \right)L_{A} \left[ {K_{M} /\left( {L_{M} + L_{U} } \right) - K_{A} /L_{A} } \right]/\left( {L_{M} + L_{U} } \right)K_{A} .$$

Thus, the following holds:

$$J > ( < )0 \Leftrightarrow K_{M} /\left( {L_{M} + L_{U} } \right) > ( < )K_{A} /L_{A} .$$

From (34), we obtain

$$dL_{M} /dE^{e} = gpF_{LK}^{M} F_{KL}^{A} /J ,$$

(35)

$$dK_{M} /dE^{e} = - gpF_{LL}^{M} F_{KL}^{A} /J ,$$

(36)

$$dL_{M} /dE^{u} = hpF_{LK}^{M} F_{KL}^{A} /J ,$$

(37)

$$dK_{M} /dE^{u} = - hpF_{LL}^{M} F_{KL}^{A} /J .$$

(38)

Using (35)–(38), we have

$${\text{tr }} S = \partial \phi /\partial E^{e} + \partial \varPsi /\partial E^{u}$$

$$\begin{aligned} = - 1 - \alpha \lambda \left( {F_{L}^{M} \partial L_{M} /\partial E^{e} + F_{K}^{M} \partial K_{M} /\partial E^{e} } \right) - 1 - \beta \lambda \left( {F_{L}^{M} \partial L_{M} /\partial E^{u} + F_{K}^{M} \partial K_{M} /\partial E^{u} } \right) \hfill \\ = {\kern 1pt} {\kern 1pt} - 2 - \lambda pF_{KL}^{A} m\left( {\alpha g + \beta h} \right)/J. \hfill \\ \end{aligned}$$

Thus, the following holds:

$${\text{tr }} S<0 \Leftarrow J> 0 \Leftrightarrow K_{M} /\left( {L_{M} + L_{U} } \right) > K_{A} /L_{A} .$$

(39)

Moreover, we obtain

$${\text{det }} S = \left( {\partial \phi /\partial E^{e} } \right)\left( {\partial \varPsi /\partial E^{u} } \right) - \left( {\partial \phi /\partial E^{u} } \right)\left( {\partial \varPsi /\partial E^{e} } \right)$$

$$= \left[ { - 1 - \alpha \lambda \left( {F_{L}^{M} \partial L_{M} /\partial E^{e} + F_{K}^{M} \partial K_{M} /\partial E^{e} } \right)} \right]\left[ { - 1 - \beta \lambda \left( {F_{L}^{M} \partial L_{M} /\partial E^{u} + F_{K}^{M} \partial K_{M} /\partial E^{u} } \right)} \right]$$

$$- \left[ { - \alpha \lambda \left( {F_{L}^{M} \partial L_{M} /\partial E^{u} + F_{K}^{M} \partial K_{M} /\partial E^{u} } \right)} \right]\left[ { - \beta \lambda \left( {F_{L}^{M} \partial L_{M} /\partial E^{e} + F_{K}^{M} \partial K_{M} /\partial E^{e} } \right)} \right]$$

$$= 1 + \lambda pF_{KL}^{A} m\left( {\alpha g + \beta h} \right)/J .$$

Therefore, the following holds:

$${ \det }{\kern 1pt} S > 0 \Leftarrow J > 0 \Leftrightarrow K_{M} /\left( {L_{M} + L_{U} } \right) > K_{A} /L_{A} .$$

(40)

From (39) and (40), if the urban area is more capital abundant than the rural area, that is, $K_{M} /\left( {L_{M} + L_{U} } \right) > K_{A} /L_{A}$, then the linear approximation system of (32) and (33) is globally stable.