Tolls, Earmarking, and Optimal Road Capacity

ABSTRACT

A classic question in transportation economics is whether optimal road capacity is greater with or without congestion tolls. This article addresses the question under two complications. One is that tolls are levied not only to relieve congestion but also to generate revenues. The other is that toll revenues may be earmarked. Under most plausible assumptions, capacity is found to be greater without tolls. The main exception is when the marginal cost of public funds is high, and revenues are earmarked to the toll road so that capacity investments are effectively “protected” from competing demands for scarce public funds.

Key Words:

• congestion pricing
• earmarking
• road investments
• toll financing

ACKNOWLEDGMENTS

Financial support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. I am indebted to Erik Verhoef for very useful comments.

Notes

¹Small and Verhoef (2007, pp. 172–5) review empirical estimates of the magnitude of latent (or induced) demand.

²On some facilities, part of the revenues is also spent on bus service on the same right of way.

³The MCPF is loosely defined as one plus the marginal excess burden of taxation. A more precise definition is “the efficiency cost of raising one unit of tax revenue, given that the tax revenue is spent on a public good that does not affect the consumption of taxed commodities” (Proost et al., 2007, p. 66). In the case of transport infrastructure investments, consumption of taxed commodities such as those delivered by truck is, in fact, usually affected. This complication is ignored here.

⁴This assumes that if public transit is fully funded from other taxes before tolling is introduced, funding from other taxes is reduced by R after tolling begins. This violates the spirit of earmarking—if not also the legal requirement—that earmarked revenues provide a net addition to funding. If this constraint is binding, then T _P  = 0: a case considered next.

⁵The case R > P∗ violates Bös's (2000) definition of earmarking according to which earmarked revenues contribute only part of the revenues required to fund a public good. But if revenues are large, and the public good is narrowly defined, this assumption may be violated. Allegedly this was the case for the congestion-charging scheme proposed for Edinburgh in which revenues would have been earmarked to finance a small set of local transport projects (Laird et al., 2007).

⁶If the public transit service on which toll revenues are spent is a substitute for driving on the toll road, then tolling is likely to boost public transit demand and P∗. This complication is ignored. So is the possibility that subsidies to public transport are captured by service suppliers or labor unions (Button, 2006).

⁷ The qualification in footnote 4 applies to internal earmarking as well.

⁸Section 2 of Lindsey (2008) also uses this reduced form approach to estimate the benefits of a cordon toll for the City of Montréal, but with capacity treated as given. The current paper also differs in distinguishing between m _R and m _T rather than (implicitly) assuming they are the same. Furthermore, Lindsey (2008) uses linear demand and user cost functions whereas the specific model and numerical example in Sections 3 and 4 use constant elasticity functions.

⁹A two-step toll is defined here to be one with a base or off-peak toll (generally positive) and a higher peak level as in the illustrative example of Appendix 1.

¹⁰Equation (11) corresponds with equation (5.16) in Small and Verhoef (2007) for the case of a flat toll with no variable revenues.

¹¹Equation (14) corresponds with equation (5.17) in Small and Verhoef (2007) for the case of a flat toll with no variable revenues.

¹²A proof that m _R  = m _K for internal earmarking is provided in Appendix 2.

¹³As explained in Appendix 1, the efficiency gain from a fine toll that eliminates queuing is a fraction χ/(1 + χ) of d(F/ K)^χ. More generally, parameter Γ will depend on χ for any given time structure of the toll. For ease of exposition, this dependence is ignored here although it is incorporated into one of the scenarios of the sensitivity analysis in Section 4.

¹⁴In the limit as η → 0, it is necessary to assume m _R  = 1 as otherwise the optimal toll is undefined.

¹⁵See Table 1 in Wilson (1983). To understand why, note that the critical elasticity of −1 as per equation (22) obtains when all trip costs are variable. With a > 0, the partial elasticity of demand with respect to the variable cost is smaller in magnitude than the full price elasticity. Consequently, a higher (absolute) full price elasticity is required to reach the critical value of the partial elasticity.

¹⁶Varying Γ within the range [0.5, 1] does not materially alter the results.

^a Parameters δ, γ, and d recalibrated to maintain target capacity, flow, and trip price in no-toll regime.

^b Parameter γ recalibrated to maintain target capacity, flow, and trip price in no-toll regime.

^c Parameters γ and d recalibrated to maintain target capacity, flow, and trip price in no-toll regime.

^d Base (i.e., off-peak) toll.

Source: Author's calculation.

¹⁸This shift in roles is noted by Oum and Trethway (1988). The sensitivity analysis reported in Table 3 was terminated at m _T  = 1.75 because the optimal toll is unbounded for higher values.

¹⁷ The calibration procedure is explained in Appendix 4.

¹⁹Large-scale road networks can also show an approximate linear relationship between aggregate demand and trip costs (De Palma and Marchal, 1999).

²⁰See Small and Verhoef (2007, §4.2.5 and pp. 177–8) for a discussion and some empirical estimates of MCPF.