Urban Structure and Environmental Externalities

Abstract

The objective of this paper is to analyze policy design for air pollution management in the spatial context of urban development. We base our analysis on the paper of Ogawa and Fujita (Reg Sci Urban Econ 12:161–196, 1982), which offers a proper theoretical framework of non-monocentric urban land use using static microeconomic theory where the city structure is endogenous. First, we show that when households internalize industrial pollution in their residential location choice, spatialization within the city is reinforced. This impacts directly the emissions of greenhouse gases from commuting. Then, we analyze policy instruments in order to achieve optimal land use pattern when the policy maker has to manage both industrial and commuting related polluting emissions, that interact through the land market.

Appendices

Appendix 1

The density, locational potential and environmental quality functions in the incompletly mixed case are:

• In the integrated district:

  $$\begin{aligned} h(x)= & {} \frac{L_b}{S_b+S_hL_b}, \quad b(x)= \frac{1}{S_b+S_hL_b} \\ E(x)= & {} \bar{E} - e M + \left\{ \frac{\eta }{S_b}\left( f_{2I}^2 - f_{1I}^2\right) + \frac{\eta }{S_b+S_hL_b}\left( f_{1I}^2+x^2\right) \right\} \\ F(x)= & {} \alpha M - \left\{ \frac{\tau }{S_b}\left( f_{2I}^2 - f_{1I}^2\right) + \frac{\tau }{S_b+S_hL_b}\left( f_{1I}^2+x^2\right) \right\} \\ \end{aligned}$$

• In the business district:

  $$\begin{aligned} h(x)= & {} 0 , \quad b(x)=\frac{1}{S_b}\\ E(x)= & {} \bar{E} - e M + \left\{ \frac{\eta }{S_b}\left( f_{2I}^2 - 2f_1x + x^2\right) + \frac{2\eta }{S_b+S_hL_b}f_1x\right\} \\ F(x)= & {} \alpha M - \left\{ \frac{\tau }{S_b}\left( f_{2I}^2 - 2f_{1I}x + x^2\right) + \frac{2\tau }{S_b+S_hL_b}f_{1I}x\right\} \\ \end{aligned}$$

• In the residential area:

  $$\begin{aligned} h(x)= & {} \frac{1}{S_h}, \quad b(x)=0 \\ E(x)= & {} \bar{E} - e M + \left\{ \frac{2\eta }{S_b}\left( f_{2I} - f_{1I}\right) x + \frac{2\eta }{S_b+S_hL_b}f_{1I}x\right\} \\ F(x)= & {} \alpha M - \left\{ \frac{2\tau }{S_b}\left( f_{2I} - f_{1I}\right) x + \frac{2\tau }{S_b+S_hL_b}f_{1I}x\right\} \\ \end{aligned}$$

Appendix 2

As in the case of a completely mixed urban configuration, from (4), (6) and (22a), we obtain the wage profile W(x) in the integrated district. On the residential area, the wage profile is a linear function of distance. To summarize, the wage profile in the city is given by:

$$\begin{aligned} W(x) = \left\{ \begin{array}{ll} \frac{S_hF(x) +S_b\left( U^* - \gamma \ln S_h\right) - S_b E(x)}{S_b+S_hL_b} &{} \text {if} \quad x \in [0,f_{1I}] \\ W(f_{1I})-t(x-f_{1I})&{} \text {if} \quad x \in [f_{1I},f_{3I}] \end{array} \right. \end{aligned}$$

(34)

Using (4) and the second part of (34), we can compute the value of W(x) at $f_{1I}$ depending on the level of equilibrium utility $U^*$. Knowing that this value of $W(f_{1I})$ must be equal to the value in the first part of (34), we can determine the equilibrium utility level $U^*$ as a function of $f_{1I}$ and $f_{3I}$:

$$\begin{aligned} U_I^*&= \frac{F(f_{1I}) }{L_b} + \frac{S_b\left( E\left( f_{3I}\right) -E\left( f_{1I}\right) \right) - \left( S_b+S_hL_b\right) \left( S_hR_a+t\left( f_{3I}-f_{1I}\right) \right) }{S_hL_b} + \gamma \ln S_h + Eq\left( f_{3I}\right) \end{aligned}$$

(35)

The functional form of F(x) and E(x) allow us to say that F(x) is strictly concave on BD and linear on RA and E(x) is convex on BD and linear on RA. Then, we conclude that R(x) will be concave on BD and linear on RA, so the rest of the land market conditions are equivalent to:

$$\begin{aligned} R(x)= & {} \Phi ^*(x) = \Psi ^*(x) \quad \text {at} \quad x = f_{1I}, f_{2I} \\ R(x)= & {} \Psi ^*(x) = R_a \quad \text {at} \quad x=f_{3I} \end{aligned}$$

Appendix 3

To compare the aggregate commuting distances in the monocentric and incompletely mixed cases, we plug the definition of $f_{1M}$, $f_{2M}$, $f_{1I}$, $f_{2I}$ and $f_{3I}$ in equation (15) and (29):

$$\begin{aligned} D_M-D_I=\frac{1}{16}\frac{S_bS_h^2\left( L_bS_h+S_b\right) }{L_b\left( S_b\eta +S_h\tau \right) ^3} \left[ 2tL_b\left( L_bS_h+S_b\right) -N\left( S_b\eta +S_h\tau \right) \right] D_1 \end{aligned}$$

(36)

where $D_1$ is a second-order equation in t, positive over positive t:

$$\begin{aligned} D_1= & {} 4tL_b\left( L_bS_h+S\right) \left[ tL_b\left( L_bS_h+S_b\right) -N\left( S_b\eta +S_h\tau \right) \right] \nonumber \\&+\,\left( S_b\eta +S_h\tau \right) \left[ 7N^2-6tL_b\left( L_bS_h+S_b\right) \right] \end{aligned}$$

(37)

Hence, $D_M-D_I$ is of the sign of the expression in brackets, which is positive in an incompletly mixed equilibrium.

Appendix 4

In the monocentric case, the regulator maximises SW $_M=N U^*_M - \gamma _c$ TDC $_M -\gamma _f$ TDF $_M$, with respect to a and T, where TDC $_M=\frac{1}{2}kS_b f_{2M} (f_{2M}-f_{1M})^2$, TDF $_M= (e-a)M(f_{2M}-f_{1M})-\frac{\eta }{S_b}\left[ \frac{f_{2M}^3}{3} + f_{1M}^2f_{2M}-\frac{4f_{1M}^3}{3}\right]$ and the utility is derived from $\Psi (f_{2M})=R_a$, from which we can express $W(0)=S_hR_a+tf_{2M}+U^*-E(f_{2M})-\gamma \ln S_h$; plugging it into $\Psi (f_{1M})=\Phi (f_{1M})$ we obtain $U^*_M$ is:

$$\begin{aligned} U^*_M=\frac{S_b}{S_hL_b+S_b}\left[ -t\left( f_{2M}-f_{1M}\right) -S_hR_a+E\left( f_{2M}\right) +\gamma \ln S_h + \frac{S_b}{S_hL_b+S_b} U_1 \right] \end{aligned}$$

with $U_1 = \left( \frac{F(f_{1M})-C(a)}{S_b}-\frac{E(f_{1M})+\gamma \ln S_h}{S_h}\right)$.

In the completely mixed case, setting $\Psi (f_{1C})=\Phi (f_{1C})=R_a$ we obtain:

$$\begin{aligned} U^*_{C}=E(F_1)+\gamma \ln S_h -\frac{S_h}{S_b} F(f_{1C})+ \frac{S_b}{S_hL_b+S_b} \left( \frac{F(f_{1C})-C(a)}{S_b}-R_a\right) \end{aligned}$$

Furthermore, TDC $_C=0$ and TDC $_F=(e-a)Mf_{1C}-\frac{4}{3}\frac{\eta }{S_hL_b+S_b} f_{1C}^3$.

In the incompletely mixed case, the regulator maximises $SW_I=$ NU $^*_I - \gamma _c$ TDC $_I -\gamma _f$ TDF $_I$ with respect to T and a. The utility level and the new wage profile are determined in Appendix 2. The damage terms are: TDC $_I = k D_I(T) = k\cdot \frac{1}{2} S_b (f_{3I}-f_{1I}) (f_{3I}-f_{2I})^2$ and TDF $_I = \int _X\int _Y [e - \eta |x-y|]b(y)dydx$:

$$\begin{aligned} {\textit{TDF}}_I= & {} f_{1I} \left[ M(e-a)-\frac{\eta }{S_b}\left( f_{2I}^2-f_{1I}^2\right) -\frac{4}{3}\frac{\eta f_{1I}^2}{S_hL_b+S_b} \right] \\&+ \frac{M\left( f_{3I}-f_{2I}\right) }{S_h}\left[ e-a - \frac{\eta }{2}(f2+f3)\right] \end{aligned}$$