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,
Benefit=Q1Q2APC-MSCdQ ,
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)
Benefit=Q1Q2APCASCdQ+Q1Q2ASCdQ-Q1Q2dTSCdQdQ
Benefit=Q1Q2ASCdQ -TSC2+TSC1+Q1Q2APCASCdQ
where TSC is total social cost.

Since TSC=ASC×Q , the total cost terms, can be written as -ASC2Q2+ASC1Q1 .

The term Q1Q2ASCdQ is the area under the average cost curve between Q1 and Q2. It can be approximated by ASC2+ASC1Q2-Q1/2 .

The last term, the resource correction, can be approximated as APCASCQ2-Q1 where APC=APC1+APC2/2 and ASC=ASC1+ASC2/2 .

Combining the terms:
Benefit=ASC2+ASC1Q2-Q1/2+-ASC2Q2+ASC1Q1+APCASCQ2-Q1
which simplifies to ASC1-ASC2Q2+Q1/2 +APCASCQ2-Q1 .

Assuming ASC1-ASC2=APC1-APC2 , the benefit can be expressed as:
Benefit=APC1-APC2Q2+Q1/2 +APCASCQ2-Q1
(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, APC1-APC2<0, the formula will give a negative result, reflecting the increase in net social cost due to greater congestion.