When are urban growth boundaries not second-best policies to congestion tolls?

Pines and Sadka proved that a not-too-stringent urban growth boundary is a second-best policy to congestion tolls when traffic congestion is unpriced, by assuming that all jobs are exogenously located at one urban center (monocentric city) [D. Pines, E. Sadka, Zoning, first-best, second-best and third-best criteria for allocating land to roads, Journal of Urban Economics 17 (1985) 167-183]. The result is also implied by Kanemoto [Y. Kanemoto, Cost-benefit analysis and the second-best land use for transportation, Journal of Urban Economics 4 (1977) 483-503] and Arnott [R. Arnott, Unpriced transport congestion, Journal of Economic Theory 21 (1979) 294-316]. Brueckner extrapolated this narrow theoretical result to real cities [J. Brueckner, Urban sprawl: Diagnosis and remedies, International Regional Science Review 23 (2000) 160-179]. We show that if there is no cross-commuting between city and suburb, first-best efficient tolls on traffic can reduce congestion and total travel cost by shifting worker-residents from the city to the suburbs, causing urban expansion. Then, planned urban boundaries of any stringency are not a second-best policy because they induce people to relocate to more congested areas. With cross-commuting, boundaries of any stringency can be inefficient even when tolls shrink cities, as boundaries do little but tolls do a lot to reduce inefficient suburb-to-city commuting. We also show that when the urban radius is limited by a natural boundary, then growth boundaries of any stringency are inefficient.