Reducing traffic externalities by multiple-cordon pricing

Abstract

The second-best congestion pricing schemes including common optimum, one cordon, and multiple cordons schemes are compared with the first-best optimum pricing scheme. A cross-subsidy effect exists in these second-best pricing models. However, the scheme with more cordons will diminish the cross-subsidy and approach an efficient and equitable outcome. The relative efficiency of a cordon pricing scheme for the case of Taipei metropolis is very high. One single cordon yields excellent performance of 93% relative efficiency. There might be some factors causing the good results: the uncongested traffic condition, the linear unit distance cost in traffic flow forming a nonlinear cost function, and the trip demands with continuous space and the same destination (the central business district) in the network.

Appendices

Appendix 1

The following boundary conditions are used for determining the values of the six unknowns above:

• (1) The condition with the CBD (x = 0)

  When the location is the CBD, the following relations hold:

  $$p\left( {q_{1}^{**} \left( 0 \right)} \right) = C_{1} \left( 0 \right) = 0,$$

  (15a)
  $$p^{{\prime }} \frac{{dq_{1}^{**} \left( 0 \right)}}{dx} - t\left( {Q_{1} \left( 0 \right)} \right) = 0.$$

  (15b)

• (2) The condition with the first cordon location ($x = x_{m1}$)

  When the location is the first cordon location, combining (11a) and (11b) with the trip cost condition,$C_{1} \left( {x_{m1} } \right) = C_{2} (x_{m1} )$, yields:

  $$p\left( {q_{1}^{**} \left( {x_{m1} } \right)} \right) = p\left( {q_{2}^{**} \left( {x_{m1} } \right)} \right) - \tau_{1} .$$

  (16)

• (3) The condition with the second cordon location (x =  $x_{m2}$)

  When the location is the second cordon location, combining (11b) and (11c) with the trip cost condition, $C_{2} \left( {x_{m2} } \right) = C_{3} (x_{m2} )$, yields:

  $$p\left( {q_{2}^{**} \left( {x_{m2} } \right)} \right) = p\left( {q_{3}^{**} \left( {x_{m2} } \right)} \right) - \tau_{2} .$$

  (17)

• (4) The condition with city boundary (x = B)

  When the location is at the city boundary, the condition that marginal benefit equals the trip cost and the derivative for this condition must hold:

  $$p\left( {q_{3}^{**} (B)} \right) = C_{3} \left( B \right) + \tau_{1} + \tau_{2} ,$$

  (18a)
  $$p^{\prime } \frac{{dq_{3}^{**} (B)}}{dx} - t\left( {Q_{3} \left( B \right)} \right) = 0.$$

  (18b)

From the above conditions, (15a–18b), the six unknown constants are determined as follows:

$${\uplambda }_{1} = \frac{{ - 2{\text{f}}/{\upalpha } + 2{\text{ae}}^{{ - {{\upalpha B}}}} + {\uptau }_{1} \left( {{\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{\text{m1}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}}_{\text{m1}} } \right)}} } \right) + {\uptau }_{2} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{\text{m2}} } \right)}} - {\text{e}}^{{ - {{\upalpha (B}} - {\text{x}}_{\text{m2}} )}} )}}{{2{\text{b}}({\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} )}},$$

$${\uplambda }_{2} = \frac{{2{\text{f}}/{\upalpha } + 2{\text{ae}}^{{{{\upalpha B}}}} - {\uptau }_{1} \left( {{\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right) - {\uptau }_{2} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }({\text{B}} - {\text{x}}_{{{\text{m}}2}} )}} )}}{{2{\text{b}}({\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} )}},$$

$${\uplambda }_{3} = \frac{{ - 2{\text{f}}/{\upalpha } + 2{\text{ae}}^{{ - {{\upalpha B}}}} - {\uptau }_{1} ({\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}}_{\text{m1}} } \right)}} + {\text{e}}^{{ - {\upalpha }\left( {{\text{B}} + {\text{x}}_{\text{m1}} } \right)}} ) + {\uptau }_{2} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{\text{m2}} } \right)}} - {\text{e}}^{{ - {\upalpha }({\text{B}} - {\text{x}}_{\text{m2}} )}} )}}{{2{\text{b}}({\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} )}},$$

$${\uplambda }_{4} = \frac{{2{\text{f}}/{\upalpha } + 2{\text{ae}}^{{ - {{\upalpha B}}}} - {\uptau }_{1} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} ) - {\uptau }_{2} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }({\text{B}} - {\text{x}}_{{{\text{m}}2}} )}} )}}{{2{\text{b}}({\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} )}},$$

$${\uplambda }_{5} = \frac{{ - 2{\text{f}}/{\upalpha } + 2{\text{ae}}^{{ - {{\upalpha B}}}} - {\uptau }_{1} ({\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{ - {\upalpha }\left( {{\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} ) - {\uptau }_{2} ({\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }({\text{B}} + {\text{x}}_{{{\text{m}}2}} )}} )}}{{2{\text{b}}({\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} )}},$$

$${\uplambda }_{6} = \frac{{2{\text{f}}/{\upalpha } + 2{\text{ae}}^{{ - {{\upalpha B}}}} - {\uptau }_{1} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} ) - {\uptau }_{2} ({\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{{\upalpha }({\text{B}} + {\text{x}}_{{{\text{m}}2}} )}} )}}{{2{\text{b}}({\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} )}}.$$

Appendix 2

$$\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}1}} }} = \frac{{ - {\uptau }_{1} {\upalpha }\left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0, \quad for\;0 \le {\text{x}} \le {\text{x}}_{\text{m1}} ,$$

(19a)

$$\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}1}} }} = \frac{{ - {\uptau }_{1} {\upalpha }\left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0,\quad for\;{\text{x}}_{{{\text{m}}1}} \le {\text{x}} \le {\text{x}}_{{{\text{m}}2}} ,$$

(19b)

$$\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}1}} }} = \frac{{ - {\uptau }_{1} {\upalpha }\left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0,\quad for\;{\text{x}}_{{{\text{m}}2}} \le {\text{x}} \le {\text{B}},$$

(19c)

$$\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}2}} }} = \frac{{ - {\uptau }_{2} {\upalpha }\left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0, \quad for\;0 \le {\text{x}} \le {\text{x}}_{{{\text{m}}1}} ,$$

(20a)

$$\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}2}} }} = \frac{{ - {\uptau }_{2} {\upalpha }\left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0, \quad for\;{\text{x}}_{{{\text{m}}1}} \le {\text{x}} \le {\text{x}}_{{{\text{m}}2}} ,$$

(20b)

$$\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{\text{m2}} }} = \frac{{ - {\uptau }_{ 2} {\upalpha }\left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{\text{m2}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{\text{m2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{\text{m2}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{\text{m2}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0,\quad for\;x_{\text{m2}} \le {\text{x}} \le {\text{B}}$$

(20c)

$$\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{1} }} = \frac{{\left( {{\text{e}}^{{ - {\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} - {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} > 0,\quad for\;0 \le {\text{x}} \le {\text{x}}_{{{\text{m}}1}} ,$$

(21a)

$$\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{1} }} = \frac{{ - \left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0, \quad for\;{\text{x}}_{{{\text{m}}1}} \le {\text{x}} \le {\text{x}}_{{{\text{m}}2}} ,$$

(21b)

$$\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{1} }} = \frac{{ - \left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}1}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}1}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0, \quad for\;{\text{x}}_{{{\text{m}}2}} \le {\text{x}} \le {\text{B,}}$$

(21c)

$$\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{2} }} = \frac{{\left( {{\text{e}}^{{ - {\upalpha }\left( {{\text{B}} + {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{B}} + {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} > 0,\quad for\;0 \le {\text{x}} \le {\text{x}}_{{{\text{m}}1}} ,$$

(22a)

$$\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{2} }} = \frac{{\left( {{\text{e}}^{{ - {\upalpha }\left( {{\text{B}} + {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{ - {\upalpha }\left( {{\text{B}} - {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( {{\text{B}} + {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} - {\text{e}}^{{{\upalpha }\left( {{\text{B}} - {\text{x}} - {\text{x}}_{{{\text{m}}2}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} > 0,\quad for\;{\text{x}}_{{{\text{m}}1}} \le {\text{x}} \le {\text{x}}_{{{\text{m}}2}} ,$$

(22b)

$$\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{2} }} = \frac{{ - \left( {{\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{ - {\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} + {\text{x}}_{{{\text{m}}2}} } \right)}} + {\text{e}}^{{{\upalpha }\left( { - {\text{x}} + {\text{B}} - {\text{x}}_{{{\text{m}}2}} } \right)}} } \right)}}{{2{\text{b}}\left( {{\text{e}}^{{{{\upalpha B}}}} + {\text{e}}^{{ - {{\upalpha B}}}} } \right)}} < 0, \quad for\;{\text{x}}_{{{\text{m}}2}} \le {\text{x}} \le {\text{B}} .$$

(22c)

Appendix 3

Substituting trip in (13a–13c) into (14), the first-order conditions are:

$$\mathop \int \limits_{0}^{{{\text{q}}_{1}^{**} ({\text{x}}_{{{\text{m}}1}} )}} {\text{p}}\left( {\text{q}} \right){\text{dq}} - {\text{C}}_{1} \left( {{\text{x}}_{{{\text{m}}1}} } \right){\text{q}}_{1}^{**} \left( {{\text{x}}_{{{\text{m}}1}} } \right) - \left[ {\mathop \int \limits_{0}^{{{\text{q}}_{2}^{**} ({\text{x}}_{{{\text{m}}1}} )}} {\text{p}}\left( {\text{q}} \right){\text{dq}} - {\text{C}}_{2} \left( {{\text{x}}_{{{\text{m}}1}} } \right){\text{q}}_{2}^{**} \left( {{\text{x}}_{{{\text{m}}1}} } \right)} \right] + \mathop \int \limits_{0}^{{{\text{x}}_{{{\text{m}}1}} }} \left[ {{\text{p}}\left( {{\text{q}}_{1}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{1} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}1}} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}1}} }}^{{{\text{x}}_{{{\text{m}}2}} }} \left[ {{\text{p}}\left( {{\text{q}}_{2}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{2} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}1}} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}2}} }}^{\text{B}} [{\text{p}}\left( {{\text{q}}_{3}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{3} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)]\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}1}} }}{\text{dx}} = 0,$$

(23a)

$$\mathop \int \limits_{0}^{{{\text{q}}_{2}^{**} ({\text{x}}_{{{\text{m}}2}} )}} {\text{p}}\left( {\text{q}} \right){\text{dq}} - {\text{C}}_{2} \left( {{\text{x}}_{{{\text{m}}2}} } \right){\text{q}}_{2}^{**} \left( {{\text{x}}_{{{\text{m}}2}} } \right) - \left[ {\mathop \int \limits_{0}^{{{\text{q}}_{3}^{**} ({\text{x}}_{{{\text{m}}2}} )}} {\text{p}}\left( {\text{q}} \right){\text{dq}} - {\text{C}}_{3} \left( {{\text{x}}_{{{\text{m}}2}} } \right){\text{q}}_{3}^{**} \left( {{\text{x}}_{{{\text{m}}2}} } \right)} \right] + \mathop \int \limits_{0}^{{{\text{x}}_{{{\text{m}}1}} }} \left[ {{\text{p}}\left( {{\text{q}}_{1}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{1} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}2}} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}1}} }}^{{{\text{x}}_{{{\text{m}}2}} }} \left[ {{\text{p}}\left( {{\text{q}}_{2}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{2} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}2}} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}2}} }}^{\text{B}} \left[ {{\text{p}}\left( {{\text{q}}_{3}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{3} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\text{x}}_{{{\text{m}}2}} }}{\text{dx}} = 0,$$

(23b)

$$\mathop \int \limits_{0}^{{{\text{x}}_{{{\text{m}}1}} }} \left[ {{\text{p}}\left( {{\text{q}}_{1}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{1} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{1} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}1}} }}^{{{\text{x}}_{{{\text{m}}2}} }} \left[ {{\text{p}}\left( {{\text{q}}_{2}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{2} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{1} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}2}} }}^{\text{B}} \left[ {{\text{p}}\left( {{\text{q}}_{3}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{3} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{1} }}{\text{dx}} = 0,$$

(23c)

$$\mathop \int \limits_{0}^{{{\text{x}}_{{{\text{m}}1}} }} \left[ {{\text{p}}\left( {{\text{q}}_{1}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{1} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{1}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{2} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}1}} }}^{{{\text{x}}_{{{\text{m}}2}} }} \left[ {{\text{p}}\left( {{\text{q}}_{2}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{2} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{2}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{2} }}{\text{dx}} + \mathop \int \limits_{{{\text{x}}_{{{\text{m}}2}} }}^{\text{B}} \left[ {{\text{p}}\left( {{\text{q}}_{3}^{**} \left( {\text{x}} \right)} \right) - {\text{C}}_{3} \left( {\text{x}} \right) - {\text{E}}\left( {\text{x}} \right)} \right]\frac{{\partial {\text{q}}_{3}^{**} \left( {\text{x}} \right)}}{{\partial {\uptau }_{2} }}{\text{dx}} = 0,$$

(23d)

$$E(x) = \left\{ \begin{gathered} \int\limits_{0}^{x} {t^{\prime } \left( {Q_{1} \left( y \right)} \right)Q_{1} \left( y \right)dy,\quad 0 \le x \le x_{{m1}} } \hfill \\ \int\limits_{0}^{{x_{{m1}} }} {t^{\prime } \left( {Q_{1} \left( y \right)} \right)Q_{1} \left( y \right)dy} + \int\limits_{{x_{{m1}} }}^{x} {t^{\prime } \left( {Q_{2} \left( y \right)} \right)Q_{2} \left( y \right)dy,\quad x_{{m1}} \le x \le x_{{m2}} } \hfill \\ \int\limits_{0}^{{x_{{m1}} }} {t^{\prime } \left( {Q_{1} \left( y \right)} \right)Q_{1} \left( y \right)dy} + \int\limits_{{x_{{m1}} }}^{{x_{{m2}} }} {t^{\prime } \left( {Q_{2} \left( y \right)} \right)Q_{2} \left( y \right)dy} + \int\limits_{{x_{{m2}} }}^{B} {t^{\prime } \left( {Q_{3} \left( y \right)} \right)Q_{3} \left( y \right)dy,\quad x_{{m2}} \le x \le B.} \hfill \\ \end{gathered} \right.$$

(24)

E(x) represents the externality incurred from an additional trip from x on all drivers using the road between 0 and x. The first two parts of (23a) represent the direct effect on total surplus caused by the outward move of the first cordon location. The other three parts of (23a) represent the indirect effect on total surplus via the change in trips on the three sections of the road by an outward move for the first cordon location. Similarly, (23b) represents the direct effect and indirect total surplus caused by an outward move for the second cordon location. In addition, (23c) and (23d) represent the indirect effect on total surplus via the change in the trips on the three sections of the road by a unit increase of toll at the first cordon location and at the second cordon location, respectively.