If the suburban toll revenue falls short of the suburban road
cost, then by (39) O 1 40, which means that a subsidy of O 1
must be paid to each core resident. Then, by (40) the core’s
ADLR is exhausted by financing: (i) the public transit invest-
ment in the core, k; (ii) the shortfall in the tolling of suburban
roads; (iii) the subsidy to the core’s residents. The intuition is
that since the tolls fall short of the Pigouvian level, subsidiz-
ing residence location in the core compensates for the
incompleteness in tolling and helps to further reduce sub-
urban congestion.
If the suburban toll revenue exceeds the road cost, then by
(39) O 1 o0 and a tax O 1
must be levied on each core
resident. In this case, therefore, the core’s ADLR would be
supplemented by the location tax revenue from the core as
well as the suburban toll surplus and all three funding
sources would be exhausted to finance the public transit
investment. The intuition is that since the suburban tolls
exceed their Pigouvian level, taxing location in the core
compensates for the over reduction of congestion in the
suburbs.
It is now helpful to check how the marginal utilities may be
equalized in the case of a couple of specific indirect utility
functions. First, suppose that indirect utility is linear in disposable
income with constant marginal utility of income a: V iðR i ,m iÞ ¼
v iðR iÞþam i . In this case, the marginal utilities are y ¼
@V 1
@m 1
¼
@V 2
@m 2
¼ a. Since this condition is always satisfied, the instrument
O 1 is redundant because transferring public income between core
and suburb does not improve the value of welfare, and the road
toll can be set at its Pigouvian value t ¼ n 2 t 0
ðn 2 ,AÞ. Then, the toll