Design of Sustainable Cordon Toll Pricing Schemes in a Monocentric City

Abstract

This paper addresses the design issue of sustainable cordon toll pricing schemes in a monocentric city. An analytical model that maximizes the total social welfare of urban system is first proposed for simultaneous optimization of the cordon toll location and charge level. The solution properties of the model with/without considering traffic congestion and/or environmental effects are explored and compared analytically. The proposed model is then extended to explicitly incorporate the effects of subsidizing the retrofit of old vehicles on reduction in average vehicle emissions. The optimal subsidy scheme for maximizing the social welfare of the system is also determined. Finally, a numerical example is given to illustrate the model applications. Insightful findings are reported on the interrelationships among cordon toll scheme, traffic congestion and environmental effects, urban population distribution, and subsidy scheme as well as their implications in practice.

Appendix A: Derivation of Equation (26)

Substituting Eq. (25) into Eq. (4), one obtains

$$ u(x)=\left({c}_v+\beta {t}_0\right)\left(x-{x}_c\right)+{c}_f. $$

(A.1)

With the assumption that the traffic congestion effects can be ignored (i.e. t( ⋅ ) = t ₀), from Eqs. (13)–(15), the air pollution cost can be expressed as

$$ \widehat{C}(x)=\xi \left(x-{x}_c\right), $$

(A.2)

where ξ is a constant, which is given by

$$ \xi ={\rho}_1\gamma {t}_0 \exp \left(\frac{\rho_2}{t_0}\right). $$

(A.3)

As u(x) does not contain x _m , q _i (x) (i = 1, 2) does not also contain x _m in terms of Eqs. (2) and (3). Therefore, we have

$$ \frac{\partial {q}_i(x)}{\partial {x}_m}=0,i=1,2. $$

(A.4)

Substituting (A.4) into (23), we have

$$ \frac{\partial \varPhi }{\partial {x}_m}=\pi {\alpha}_p\tau {x}_mm\left({x}_m\right)\left(\tau -2\widehat{C}\left({x}_m\right)\right)=0. $$

(A.5)

As πα _p τx _m m(x _m ) > 0 always holds, we obtain

$$ \tau -2\widehat{C}\left({x}_m\right)=0. $$

(A.6)

This means that $\tau =2\widehat{C}\left({x}_m\right)$ holds.

On the other hand, as u(x) does not contain τ, from Eq. (2) we have

$$ \left\{\begin{array}{l}\frac{\partial {q}_1(x)}{\partial \tau }=0,\\ {}\frac{\partial {q}_2(x)}{\partial \tau }=-m(x){\alpha}_p\frac{\partial {p}_2(x)}{\partial \tau }=-m(x){\alpha}_p.\end{array}\right. $$

(A.7)

When the traffic congestion effects can be ignored, we have $\frac{\partial \widehat{C}(x)}{\partial {q}_i(x)}=0$ in terms of Eq. (A.2). Therefore, from Eq. (24), we have

$$ \frac{\partial \varPhi }{\partial \tau }={\displaystyle {\int}_{x_m}^B2\pi x{\alpha}_pm(x)\left(\widehat{C}(x)-\tau \right) dx}=0. $$

(A.8)

Substituting Eqs. (1), (A.2) and (A.6) into Eq. (A.8), we have

$$ \begin{array}{c}\hfill -5a{x}_m^4+\left(8b+4a{x}_c\right){x}_m^3-6b{x}_c{x}_m^2+\left(8a{B}^3-12b{B}^2\right){x}_m\hfill \\ {}\hfill +\left(4b{B}^3+6{x}_cb{B}^2-3a{B}^4-4a{x}_c{B}^3\right)=0.\kern5em \hfill \end{array} $$

(A.9)

Equation (A.9) can further be written as

$$ \left({x}_m-B\right)\left({\omega}_3{x}_m^3+{\omega}_2{x}_m^2+{\omega}_1{x}_m+{\omega}_0\right)=0, $$

(A.10)

where ω ₃, ω ₂, ω ₁, and ω ₀ can be given by Eq. (27). This completes the derivation of Eq. (26).