Appendix A Benefit where the related market is a congested road - technical proof

This appendix provides a technical proof of the alternative measure of benefit on related infrastructure referred to in footnote in section 7.2.

The exact benefit area is negative the gap between the marginal social cost (MSC) and average perceived cost (APC) curves between Q1 and Q2, 

$\text{Benefit}={\int }_{Q_1}^{Q_2}\left(\text{APC}-\text{MSC}\right)d\mathrm{Q }$

$$which is consistent with the formula in the text of section 7.2 except for replacing average social cost (assumed constant in the formula in section 7.2) with marginal social cost. (See Harberger 1972, pp. 262-3)

$\text{Benefit}={\int }_{Q_1}^{Q_2}\left(\text{APC}–\text{ASC}\right)dQ+{\int }_{Q_1}^{Q_2}\text{ASC}dQ-{\int }_{Q_1}^{Q_2}\frac{d\text{TSC}}{dQ}dQ$
$\text{Benefit}={\int }_{Q_1}^{Q_2}\text{ASC}dQ -{TSC}_2+{TSC}_1+{\int }_{Q_1}^{Q_2}\left(\text{APC}–\text{ASC}\right)dQ$

where TSC is total social cost.

Since $\text{TSC}=\text{ASC}\times Q$, the total cost terms, can be written as ${-\text{ASC}}_2{Q}_2+{\text{ASC}}_1{Q}_{1.}$

The term ${\int }_{Q_1}^{Q_2}\text{ASC}dQ$ is the area under the average cost curve between Q1 and Q2. It can be approximated by $\left({\text{ASC}}_2+{\text{ASC}}_1\right)\left(Q_2-Q_1\right)/2.$

The last term, the resource correction, can be approximated as $\left(\text{APC}–\text{ASC}\right)\left(Q_2-Q_1\right)$ where $\text{APC}=\left({\text{APC}}_1+{\text{APC}}_2\right)/2$ and $\text{ASC}=\left({\text{ASC}}_1+{\text{ASC}}_2\right)/2.$

Combining the terms:
$\text{Benefit}=\left({\text{ASC}}_2+{\text{ASC}}_1\right)\left(Q_2-Q_1\right)/2+{-\text{ASC}}_2{Q}_2+{\text{ASC}}_1{Q}_1+\left(\text{APC}–\text{ASC}\right)\left(Q_2-Q_1\right)$ which simplifies to $\left({\text{ASC}}_1-{\text{ASC}}_2\right)\left(Q_2+Q_1\right)/2 +\left(\text{APC}–\text{ASC}\right)\left(Q_2-Q_{1.}\right)$

Assuming ${\text{ASC}}_1-{\text{ASC}}_2={\text{APC}}_1-{\text{APC}}_2$ , the benefit can be expressed as:
$\text{Benefit}=\left({\text{APC}}_1-{\text{APC}}_2\right)\left(Q_2+Q_1\right)/2 +\left(\text{APC}–\text{ASC}\right)\left(Q_2-Q_1\right)$ (Neuberger 1971, p. 56 has this formula without the resource correction.)

Note that if the cost change is positive, as would occur on an upstream or downstream road where the demand curve shifts rightward, ${\text{APC}}_1-{\text{APC}}_2<0$, the formula will give a negative result, reflecting the increase in net social cost due to greater congestion.