Pricing transport networks with fixed residential location

Regional Science and Urban Economics 34 (2004) 289 – 307 www.elsevier.com/locate/econbase

Pricing transport networks with fixed residential location Kurt Van Dender * Department of Economics, University of California, Irvine, CA 92697-5100, USA Received 17 April 2001; accepted 21 October 2002 Available online 8 July 2003

Abstract For a congestible static traffic network which is used by different households, the conditions for optimal link tolls and destination charges are analysed for the case where not all links or destinations in the network can be tolled (partial network pricing). It is shown that differences in transport costs across households lead to unequal treatment of equal households at the social welfare optimum. Constraints on network pricing imply complex deviations from marginal social cost pricing because of network interactions. Parking charges rival second-best link tolls under some conditions. When no differentiated lump sum transfers are available, the optimal link tolls will deviate from marginal external congestion costs, even in the absence of network pricing constraints. Stylised examples are used to illustrate the interactions. D 2003 Elsevier B.V. All rights reserved. JEL classification: H23; R10; R41; R48 Keywords: Congestion; Road transport; Pricing; Networks

1. Introduction A number of recent studies deal with the problem of determining optimal congestion tolls in transport networks, under the second-best restriction that not all links in the network can be subject to a toll (partial network tolling). The basic trade-off between efficient route choice and trip demand decisions in the case of one origin– destination pair connected by two links, is analysed in, amongst others, McDonald (1995), Braid (1996), Verhoef et al. (1996), Liu and McDonald (1999), Arnott et al. (1998) and de Palma and Lindsey (2000). * Tel.: +1-949-824-9698; fax: +1-949-824-2182. E-mail address: [email protected] (K. Van Dender). 0166-0462/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0166-0462(03)00057-7

290

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

Second-best schemes are seen to be potentially effective for reallocating traffic flows, but taxing one link produces much smaller welfare gains than taxing both. This result is confirmed for a wide range of models and parameter values, with the qualification that the benefits from second-best schemes are revised upward when account is taken of heterogeneity in marginal values of time savings (Verhoef and Small, 1999; Small and Yan, 2001). Verhoef (2002) extends the analysis to networks with any number of links and origin – destination pairs, finding that second-best link tolls depend on the pricing conditions on other links, as well as on network and demand interactions. The optimal toll may be above or below external congestion costs on the link, and an interior second-best toll configuration is not necessarily unique nor is one guaranteed to exist.1 A first-best solution is to set all link tolls equal to marginal external costs on the link. The second-best partial network tolling problem has attracted attention in the transportation science literature, with a focus on the design of algorithms for finding the optimal link tolls under tolling restrictions. The difficulty here is the presence of discontinuities in a bilevel problem (cf., amongst others, Labbe´ et al., 1998; Transportation Research B (special issue), 2001; Ferris and Kanzow, 2002). Hearn and Ramana (1998) and Hearn and Yildirim (2002) show that multiple toll configurations may support the first-best outcome when transport demand is elastic, and study how a first-best configuration with the minimal number of toll booths can be found. Shepherd and Sumalee (1999) find that derivativebased approaches for finding the second-best solution to the partial network tolling problem may fail because of multiple optima and changes in the set of used paths (implying discontinuities). This paper generalises the network pricing problem from an economic point of view, by treating it as an optimal commodity tax problem. The network is simultaneously used by multiple households, who derive utility from trips to various destinations which may be substitutes. Households’ residential location is fixed. The extensions allow taking income effects into account, as well as possibly different social welfare weights for different households. The role of income effects is clarified by considering alternative toll revenue redistribution schemes. The impact of social welfare weights is assessed for the case where households have identical normative welfare weights but different private marginal utilities of income. It is shown that the Mirrlees paradox (Mirrlees, 1972) affects the network tolls when lump sum transfers cannot be differentiated across households residing at different locations. Furthermore, the analytical expressions for link tolls and destination charges (parking charges) are compared. The theoretical analysis is in Section 2, Section 3 provides examples, and Section 4 concludes.

2. Theory Section 2.1 defines the network and its use. Section 2.2 discusses the optimal network tolls. 1 Both non-uniqueness and non-existence would require an untolled link subject to much stronger congestion than the tolled links.

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

291

2.1. Consumer equilibrium in a static transport network Denote a static transport network by a graph GðN; AÞ; where N is a set of nodes and A a set of links. Each node can be reached from every other node through at least one sequence of links (a path). Links are congestible when time costs ðca Þ are increasing in traffic flow on the link (fa , in passenger car units per unit of time): ðAca =Afa uca VÞz 0; baaA:2 The network technology is simplified by assuming Aca =Afb ¼ 0; ba p baA; without loss of generality. Consider Ni identical consumers at each trip origin i ¼ 1; . . . ; I in the network (iaN; bi). The trip origin coincides with the fixed residential location. Utility functions (1) are defined over a composite nume´raire good xi and over transport commodities qri; j , where j ¼ 1; . . . ; J is a trip destination (jaN; i p j; bj). Trip origins and destinations are connected by paths raPi;j :
 Ui ¼ Ui xi ; qri;j ; bi ¼ 1; . . . ; I

where

j ¼ 1; . . . ; J

and

r a Pi;j

ð1Þ

A path represents a route for a given transport mode, or it refers to different transport modes. Link flow is the sum of demands for all paths that use the link.3 fa ¼

XX X i

j

dai;j;r Ni qri; j ; ba

ð2Þ

raPi;j

The indicator variable dai;j;r equals one when link a belongs to path r, and zero otherwise. The utility function allows substitution between destinations. In practice, this depends on the trip motive and on the time horizon. Shopping destinations may be substitutes in the short run, while commuting destinations (employment locations) probably only exhibit significant substitutability in the long run. Each household faces a budget constraint (3). Exogenous income Yi , the lump sum transfer Ti, transport time costs ca and tolls ta are in generalised terms, implying a constant value of marginal time savings or losses. The price of a commodity qri;j is the sum of time costs and tolls incurred on all network links which are used to reach destination j, starting from origin i and using path r.

Yi þ Ti ¼ xi þ

XX X j

r

! dai;j;r ðca

þ ta Þqri;j ; bi

ð3Þ

a

Using ki as the multiplier for the budget constraint, the first order conditions for maximising (1) subject to (3) and non-negativity constraints, are given by Eqs. (4) and 2

Zero derivatives allow for the introduction of virtual links, which may represent flow independent costs (e.g. parking costs, waiting times, or some types of taxes). 3 A conversion from trip demand to flow in terms of passenger car units is required when transport modes with different occupancy rates or different passenger car equivalents are considered. I abstract from this for reasons of clarity.

292

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

(5). Households neglect the congestion externality. When paths are perfect substitutes, condition (5) implies that the prices on all paths raPi; j which carry positive flow are equal and not larger than prices on all paths saPi;j that do not carry any flow (6). The Wardropian network equilibrium conditions (Wardrop, 1952) thus are a special case of the present specification. For simplicity, I further abstract from the complementarity condition by considering paths with positive flow only.4 AUi ¼ ki ; bi Axi

ð4Þ

! X AUi a ¼ ki di;j;r ðca þ ta Þ ; bi; bj; r:qri; j > 0 Aqri;j a

ð5Þ

X a

! dai;j;r ðca

þ ta Þ

¼

X

! dbi;j;s ðcb

þ tb Þ

if qri; j > 0; qsi; j > 0

ð6Þ

b

The indirect utility function is given by: Vi ðca þ ta ; a ¼ 1; . . . ; A; Yi þ Ti Þ; bi

ð7Þ

Congestion is not an argument of the utility function, so marginal external congestion costs—which on the link level equal cV a fa —are valued at the marginal utility of income. 2.2. Optimal congestion pricing in a static transport network Section 2.2.1 presents a general condition for optimal link tolls, allowing for partial network taxation and for differences in households’ welfare weights. Its properties are clarified by considering special cases. First, in Section 2.2.2, location-specific lump sum transfers are introduced and the focus is on the impact of network pricing constraints. Second, Section 2.2.3 uses a utilitarian social welfare function to show the impact on link tolls of ruling out location-dependent lump sum transfers. 2.2.1. A general expression for link tolls I simplify by normalising the number of households at each location to 1. Restrictions on link tolls are indicated by ja, equal to one when a link toll (of any value) is possible and zero otherwise. W, social welfare, is maximised subject to R, the government revenue requirement, using available link taxes and lump sum transfers Ti as instruments. Using l as the multiplier for the government budget constraint, programme produces (8) first order conditions (9). Substituting Eq. (10), which gives the indirect utility effect of a marginal link toll change, in all conditions (9) where ja ¼ 1, using the definition of link flow, and 4

This simplification is acceptable for the characterisation of an equilibrium, in which the set of used paths can be treated as given. When analysing the transition between equilibria, the set of used paths is not fixed in general, and is in fact likely to change (Van Dender, 2001).

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

293

defining bi ¼ AW =AVi ki ; bi; allows rewriting the first order condition for each ta as in (11).5 ! ! X XXX X a r I ¼ W ðVi ; i ¼ 1; . . . ; I Þ þ l di;j;r ja ta qi;j  Ti  R a

i

j

r

i

ð8Þ

X AW AVi XXX X Afb þl dai;j;r qri;j þ j b tb AV At Ata i a j r i i b

! ¼ 0; ba : ja ¼ 1

XX XX X AVi Afb ¼ ki dai;j;r qri;j þ dbi;j;r cVb qri;j Ata Ata r j r j b

ð9Þ

!! ; bi; a : ja ¼ 1 ð10Þ

ta

¼

! X b  l  X X XXX ca þ ta i a di;r;j dai;r;j cVa qri;j þ l e r r j j i i  Afb X X X X
Ata b r  jb tb  di;r;j cVb qi;j ; ba : ja ¼ 1 Afa r j i bpa Ata ! XXX A dai;r;j qri;j ca þ ta Afa ca þ ta r j i where e ¼ ¼ Ata fa Ata fa

ð11Þ

Expression (11) is the first order condition for ta . Note that ta also appears in the conditions for all other link tolls. This interdependence is further discussed in Section 2.2.2. Here I focus on the interpretation of the general first order condition. It says that the optimal link toll consists of three components. First, the toll on link a is determined by revenue raising concerns (the stringency of the government requirement R), through the deviation between the marginal social value of 5 Expression (11) is an implicit expression for the link toll. As is usual, explicit solutions cannot be found. Note also that the expression is in terms of uncompensated demand derivatives, and that the feedback effects of congestion are implicitly taken into account. A more detailed analysis would distinguish between substitution and income effects, and would separate out the feedback effect of congestion. Also, instead of measuring the social marginal utility of income for household i, the net social marginal utility of income could be used (which takes account of the marginal tax paid out of that extra unit of income). I do not provide this level of detail, in order to not make inaccessible an already complex expression. A detailed optimal tax analysis of congestion tolls in a nonspatial context can be found in Mayeres and Proost, 1997.

294

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

private income, bi , and of government revenue, l (first bracketed term). bi measures the marginal contribution to social welfare of a unit of income given to household i ; distributional concerns matter through the inter-household differences between the bi . The first term is zero when the government revenue requirement is zero, lump sum transfers are available, and distributional concerns are unimportant. More specifically, when there is no government revenue requirement, the only function of the first term is to redistribute income between consumers. However, when lump sum transfers can be used, they should be used for this purpose as they are non-distortionary (as is assumed in Section 2.2.2). Also, when distributional concerns are absent from the welfare function, no redistribution is required. But when lump sum transfers are not available and distributional issues matter, then each link tax will be used to redistribute. The extent of its redistributional function is determined by a trade-off between the social welfare weights of link users and the elasticity of demand for the link (as in the standard Ramsey tax rule, the tax is decreasing in the elasticity of demand; see Atkinson and Stiglitz, 1980, for this standard optimal tax result). Second, the toll is used for charging the marginal external congestion cost on link a (second term, Pigouvian taxation). Third, ta is affected by the network-wide gap between taxes and marginal external congestion costs. In case this gap is zero, i.e. when all links are taxed at marginal external congestion cost, the term disappears. It is clear that taxing all links at marginal external congestion cost is (a) not feasible when not all links can be taxed, and (b) not optimal when revenue raising and distributional issues affect the link tolls. The next sections clarify these issues by considering special cases. First, when optimal lump sum transfers are available, link tolls are affected by network pricing constraints, while distributional issues are addressed through the transfers (Section 2.2.2). Second, the separation of tax functions no longer holds when optimal transfers are ruled out. Section 2.2.3 shows the specific nature of the redistribution problem in the present context. 2.2.2. Optimal network pricing with individualised lump sum transfers The location-specific lump sum transfer is used to equalise social marginal utilities of income (12). When the normative component of the social welfare weight AW =AVi is equal across consumers, this requires equality of consumers marginal private utilities of income. The next section discusses the impact of that requirement in detail. bi ¼ lZ

AW l ¼ ; bi AVi ki

ð12Þ

0P 1 Afb X B bpaðcVb fb  jb tb Þ Atz C ta ¼ cVa fa þ @ A; ba : ja ¼ 1; b; zaA Afa

P ta ¼ cVa fa þ bpa

ð13Þ

Atz

z:jz ¼1

cVb fb Afa Ata

Afb Ata

ð14Þ

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

295

Eq. (13) is the optimal link toll condition for the case where lump sum transfers are available. It is obtained from considering the first order conditions for all possible link tolls.6 Observe that when ja ¼ 1 for all a, setting all link tolls equal to marginal external costs on the link (ta ¼ cVa fa) satisfies the optimality conditions.7 With full network pricing, the link tolls optimise trip demand and path choice simultaneously. With partial network pricing, indirect effects matter and trip demand and path choice effects are conflicting to some extent. The impact of link pricing constraints can be seen from the case where ja ¼ 1 for one link in the network only, i.e. only one link can be taxed. Eq. (13) then reduces to (14), which says that the link toll is the sum of marginal external congestion costs on the link and a correction term. This correction term specifies how the optimal tax deviates from the marginal external congestion cost on the tolled link. Its denominator is negative. It scales the indirect effect of ta on congestion on other links (the numerator) by the marginal effect on link a: if setting ta above or below external costs on link a has large effects on the flow on a, the indirect effect on other links becomes less important in determining ta . The numerator will be large when substitution to paths and origin– destination pairs that do not use link a is easy, or when trip demand is very elastic. When substitution effects are large, the correction term is small and the tax is close to the marginal external congestion cost on the tolled link: substitution is encouraged only in as far as it follows from internalising the externality on link a. When trip demand is very elastic, price changes have large effects on consumer welfare. These effects are to be avoided when they do not directly reflect the internalisation of external congestion costs. In as far as ta moves traffic to other parts of the network, the correction term’s numerator measures to which extent congestion in the rest of the network is affected. The numerator may take either sign, depending on the extent to which flow on other links goes up or down after a marginal change in ta. When the numerator is small, the correction term is small and ta is close to c aVfa. This is the case when congestion on other links is limited, either because these links are not strongly congestible or because link loads are small. This would indicate strong concentration of congestion in the network. The effect of ta on flow on other links is more difficult to qualitatively assess. Tolling link a could strongly decrease overall transport demand, when it is a link that is shared by many paths (e.g. a bridge providing access to a downtown area, for which no substitutes are available). More in general, the toll on link a could decrease demand for some part of the network, while increasing it for others (e.g. downtown access from regions that do not require crossing the bridge), and the sign of the net effect is not known in general. When much traffic is diverted to the rest of the network by changing ta, this tends to reduce the correction factor (it becomes more negative) in as far as the rest of the network is

6 The expression is equivalent to expression (11) in Verhoef (2002), except that it is derived within an indirect utility approach, so that consumer equilibrium is implicitly taken into account. 7 Actually, the condition is that taxes on each path for each consumer and each destination equal marginal external costs. In some networks this can be achieved without marginal social cost pricing on each link, but in a non-trivial network with a large number of links, paths and consumers, the existence of equivalent systems becomes less likely. It is therefore safe to conclude that in general the combination of marginal social cost pricing and optimal transfers is the only solution to the first best network pricing problem.

296

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

congestible. Since trip demand generally is not very elastic, and since paths are perfect substitutes and most networks will allow at least some alternative paths for an origin – destination pair, the correction term can be expected to be a large negative number: network interactions usually imply optimal tolls that are well below marginal external congestion costs on the tolled link. The general second best tax expression in (13) has a similar interpretation to (14). One difference is that account should be taken of the deviation between taxes and marginal external costs on all other links, instead of taking only congestion costs into account. The second difference is that the marginal flow effects of all possible link tolls on all links matter for the determination of the optimal ta . To the extent that an optimal second-best link toll distorts the assignment of traffic flow over the network, alternative second-best instruments become potentially attractive. A destination charge (parking charge), for example, directly affects trip demand, but not path choice. Expressions (15) and (16) give the welfare programme and the condition for optimal partial parking charges. jj here indicates whether a tax at destination j is possible (jj ¼ 1) or not (jj ¼ 0). Note that parking charges here are used only to indirectly control congestion, not for parking-specific issues as such.

I ¼ W ðVi ; i ¼ 1; . . . ; I Þ þ l

XXX i

0

 PPP P

B XB B i tm ¼ B B k:jk ¼1@

r

a

j

r

j

tm ¼

i

r

j

a

! Ti

ð15Þ



1 Aqri; j jj tj At C þ k C i jpm r C C ð16Þ r P P Aqi;m C A Atk

dai;j;r cVa qri; j

X X Aqri;m Atm r i



X i

Af dai; j;r cVa qri; j Ata k

i

XXX X

jj tj qri; j

P P P

r

Afa Atm

! ð17Þ

For ease of interpretation, first consider the case where a parking charge is possible at one destination only, as in (17). The parking charge is used to internalise congestion on the whole network, in as far as it manages to reduce flow. The flow reduction will be smaller than the direct demand reduction for destination m suggests, as travel costs for other destinations decrease. The direct reduction is taken into account in the denominator, so that the parking charge at m will be used less for reducing network congestion as it decreases

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

297

Fig. 1. A two serial links network with two household locations and one destination.

demand for m more. As long as the charge at m reduces network flow, it will be positive. Next, when more destinations can be taxed (16), the parking charge at m decreases to the extent that the other taxes reduce network flow. This says that the flow reduction becomes more evenly spread over the network as more destinations are charged for (in as far as demand elasticities do not strongly differ between origin –destination pairs). Will partial parking charges outperform partial link tolls? In case path choices before taxes are not very inefficient, and in case reducing demand does not make them inefficient, parking charges may well rival link tolls. In case trip demand is not price elastic, link tolls are more attractive. The question is an empirical one, and is further addressed in Section 3.3. Note that parking charges and link tolls are not necessarily substitutes in a second-best setting. This issue is not further studied here. Calthrop et al. (2000), in a non-spatial analysis, suggest that a combination of parking charges and link tolls is a sensible secondbest pricing strategy. 2.2.3. Optimal network pricing with restricted lump sum transfers Now assume that appropriate lump sum transfers are not available, so that link tolls depend on all tax functions. Of specific interest here is the problem of ‘unequal treatment of equals’, also called the Mirrlees paradox (Mirrlees, 1972). The paradox follows from the spatial nature of the problem: households with identical preferences face different transport costs for identical destinations, so that their marginal utilities of income differ at identical utility levels. This can be seen in a simple network consisting of serial links a and b, which connect the residence of two identical households to a common destination D, as in Fig. 1. Demand for D is D1 and D2 . Both links are congestible, and link b is shared by both households. The flow on link a is D1, the flow on link b equals D1 þ D2. By assumption, incomes in the no-toll equilibrium are such that the utilities are equal. This implies a higher income for household 1 than for household 2, and a higher marginal utility of income for household 2 than for household 1.8 Assume a utilitarian welfare function, and assume that congestion tax revenues are redistributed lump sum according to a predetermined share s for household 1 and ð1  sÞ for household 2. The government then solves programme (18): the sum of (indirect) utilities is maximised, subject to the constraint that each household receives a transfer Ti , i ¼ 1; 2 which is financed out of an exogenous share s of toll revenues. The Lagrangean multipliers for these constraints are ci ; i ¼ 1; 2. The govern8 Note that the direction of the Mirrlees paradox is reversed with respect to Mirrlees’ (1972) discussion. His model is set in the context of a monocentric city model with a land market. Households living further out of the CBD face lower land prices, so that their marginal utility of income is higher. Here, the transport cost differences drive the differences in the marginal utility of income, while land markets are abstracted from. The direction of the paradox consequently is reversed.

298

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

ment’s problem is equivalent to maximising the sum of indirect utilities after substitution of the expression for the transfer into the indirect utility functions. As the private and marginal social utilities of income are equal, ki ¼ ci, bi. Note however that because of the Mirrlees paradox, the marginal utilities of income in general differ across households. Since transport costs are higher for household 1, c1 < c2 . I denote this by c1 ¼ ac2, with 0 < a < 1. I ¼ V1 ðca þ ta ; cb þ tb ; Y1 þ T1 Þ þ V2 ðcb þ tb ; Y2 þ T2 Þ þ c1 ðsðja ta D1 þ jb tb ðD1 þ D2 ÞÞ  T1 Þ þ c2 ðð1  sÞðja ta D1 þ jb tb ðD1 þ D2 ÞÞ  T2 Þ

ð18Þ

From the first order condition for ta I get (19), which describes the condition for the optimal ta for a given level of tb . Similar expressions can be found for tb.  ta ¼

   a a D
1 cVa D1  1  AD1 as  s þ 1 as  s þ 1 Ata

  aD1 þ D2  jb tb  cVb as  s þ 1

ð19Þ

The first term on the right hand side of (19) states that a fraction of marginal external congestion costs on link a is charged (the Pigouvian component). The second term relates to demand effects, and it gets a weight of one minus the weight of the Pigouvian component (the Ramsey – Mirrlees component, where ‘Ramsey’ refers to its dependence on the price elasticity of demand, and ‘Mirrlees’ to the fact that it appears because of transport cost differences). The weight of the first (second) term is positive and increasing (decreasing) in a and in s. A higher a implies an increase of the relative social welfare contribution of household 1, and causes the tax to be closer to the marginal external cost. An increase in s means that more of the tax revenue goes to household 1, the household with higher transport costs. This is says that congestion tax revenues can be used less easily to exploit the Mirrlees paradox. Consequently, the tax is closer to a Pigouvian tax. The third term on the right of (19) is the network interaction component. It shows that ta is corrected for the deviation of tb from the social welfare value of congestion on link b (D1 is weighted by a). This term increases with a and s, so that using ta as an indirect way of addressing congestion on link b becomes more attractive as the social welfare contribution of household 1 increases. This stands to reason, as here only household 1 uses both links. Therefore, the larger the social welfare impact of household 1, the more the tax system is directed towards internalising the externalities affecting household 1. The indirect effects on household 2 receive less attention. Notice that ta differs from marginal external costs on link a as long as a < 1, i.e. as long as transport cost differences exist, irrespective of whether link b can be taxed or not. When

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

299

a ¼ 1, the Ramsey –Mirrlees component reduces to zero, and ta equals marginal external congestion costs on link a in case both links can be taxed (a zero network interaction term). As long as the Pigouvian component differs from zero, the network interaction component is different from zero as well. Therefore, in general, all three tax components are different from zero, also when all links in the network can be optimally tolled. The reason is the interaction between the Mirrlees paradox and network taxes. This interaction disappears only when optimal lump sum transfers are possible or when all households share the same location: when lump sum transfers are available, they will be used to redistribute income instead of the link tolls; when all households share the same location, their marginal utilities of income are equal and there is no reason to redistribute (as long as the welfare function is utilitarian).

3. Examples 3.1. Properties of the network examples Two examples illustrate the theoretical analysis. They are stylised and not meant to generate definitive policy conclusions. Data are derived from realistic network cost and demand data (obtained from a network model for the city of Namur, Belgium), in order to obtain reasonable orders of magnitude for the model parameters.9 Section 3.2 investigates the importance of the Mirrlees paradox under various tax redistribution assumptions, using a two link serial network. Section 3.3 assumes that individualised lump sum transfers compensate each household exactly for the amount of congestion taxes it pays, while the households do not take this connection between the lump sum transfer and the link taxes into account (if they did, congestion tolls would have no effect). This compensation scheme neutralises the Mirrlees paradox as much as possible, as it rules out the use of toll revenues to redistribute income to households with higher utilities of income. The Mirrlees effect is not completely neutralised, as it will still affect the toll levels. Nevertheless, the assumption allows focusing on the impact of network pricing restrictions in a three link network. In both examples household preferences are represented by nested CES functions,  defined over a composite commodity and over trip destinations: Ui ¼ Ui xi ; Ti qi;j ; bi; where Ti is the sub-utility function for transport commodities. The elasticities of substitution are chosen to obtain a price elasticity of transport demand of 0:3 in the reference equilibrium (in line with, for example, Small, 1992). Social welfare is the sum of household utilities. Finally, taxes are bounded below at zero in the applications. 3.2. Example 1: the Mirrlees paradox in a two link serial network The first example illustrates the impact of inter-household differences in transport costs on optimal network prices. The user cost functions, the reference demands for

9

See http://aris.ss.uci.edu/econ/personnel/VanDender/VanDender.html for detailed background information.

300

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

Table 1 Cost and demand characteristics of the two link network (reference equilibrium) Aggregate demand (trips) 1–D 2–D

2583 1089

Costs (Eurocent) and flow (vehicles)

Link a Link b

Intercept

Slope

Flow

Link travel cost

20 45

0.01 0.01

2583 3672

45.83 81.72

the representative consumers at each origin, the link flows and the user costs are in Table 1. In the reference equilibrium, the utility of households at both locations is equal by construction. This implies that per capita income at location 2 is 4.2% lower than at location 1, in the assumption that the transport expenditures contained in the example represent 10% of total expenditures. The remaining 90% is spent on a composite nume´raire commodity. When consumption transfers between households are allowed, the full exploitation of transport cost differences implies that income from households at location 1 is transfered to consumers at location 2 until the social marginal utility of income is equalised across locations. Here this implies transferring all income to households at location 2. Less extreme results are obtained when only congestion tax revenues can be redistributed. When household 1 receives all the congestion tax revenues, welfare increases by 0.2%. The gain linearly increases to 0.4% when household 2 receives all the revenues. The social welfare gains from optimal transport prices are low because of limited congestion. Of course, the utility of household 2 is increasing in the tax revenue share when both links are taxed. The utility of household 1 drops below the reference level, once more than 12% of congestion tax revenues goes to household 2. The impact of the redistribution scheme on

Fig. 2. Taxes/marginal external congestion costs, with full and partial network pricing.

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

301

Table 2 Effects of network pricing with exact compensationa

tA=0, tB = 0 tA*, tB* tA*, tB = 0 tA= 0, tB* a

(1) Utility index households 1

(2) Utility index households 2

(3) SW share households 1

(4) tA/mecA

(5) tB/mecB

1 1.0024 1.00015 1.0027

1 1.0018 1.0022 1.0005

0.703 0.704 0.703 0.704

0 1.083 2.492 0

0 0.963 0 1.406

The * sign indicates that a tax is optimised.

link tolls is illustrated in Fig. 2. The horizontal axis shows the share of congestion tax revenues which goes to household 2. The vertical axis gives the ratio of the congestion tolls and the marginal external congestion costs. The series ‘tA/mecA (tB=0)’ describes that ratio for link a, in case tb is restricted to equal zero, while ‘tA/mecA’ gives the same ratio when the toll on link b is simultaneously optimised. The series relating to tb are analogously defined. Clearly, taxes are affected by the Mirrlees paradox as well as by marginal congestion costs. In case both links can be taxed (full network pricing, series ‘tA/mecA’), taxes nearly are double the marginal external congestion costs on link a, when households 1 receive all the revenue. The ratio increases as more tax revenues go to household 2, because both the direct tax effects and the tax redistribution effects facilitate the exploitation of cost differences. For link b, taxes are below marginal external congestion costs except when all tax revenues go to household 2, in which case they are equal to marginal external congestion costs (in case both links can be taxed, series ‘tB/mecB’): there is no reason to tax below marginal external costs on link b, as the efficiency gains from internalisation are not counteracted by transferring income to the high cost household 1; also, there is no reason for taxing above marginal external costs, as transport cost differences can be exploited fully by the tax on link a. Consequently, a tax equal to marginal external costs is obtained on link b. The interaction between partial pricing, tax redistribution assumptions and tax levels is straightforward here. Taxing only link a indirectly decreases congestion for household 2, thereby re-enforcing the effects of the transport cost differences. The optimal tax is higher than in full network pricing, and it increases above marginal external costs at a quicker rate as the revenue share of household 2 grows (series ‘tA/mecA(tB=0)’).10 The share of household 1 in social welfare drops compared to full network pricing, and it drops below the reference level as soon as a congestion tax on link a is implemented (even if household 1 receives all revenues). When only link b can be taxed, the optimal tax is slightly above marginal external costs in case all revenues go to households 1. It is also higher than the optimal tax under full network pricing. The use of the tax on link b to internalise congestion on link a is counteracted by the effect of excessively high taxes on the utility of household 2. When more of the revenue goes to household 2, higher taxes will counteract to a lesser degree the tendency to advantage household 2, so that the tax rises above 10

When all tax revenues go to households 1, the second term in (19) equals zero.

302

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

marginal external costs at nearly the same rate as under full network pricing. Taxing link b only leads to a higher share of household 1 in social welfare. A further possibility for restricting lump sum transfers is to impose that each household is compensated by the amount of congestion taxes that it pays. This prevents use of the revenues for exploitation of the Mirrlees paradox, but the tax levels are still affected by the Ramsey – Mirrlees term. Table 2 presents the main results for this case. Despite the fact that the optimal taxes with full network pricing (row 2) are affected by the Ramsey –Mirrlees term, the share of households in social welfare is virtually unchanged. The deviation between taxes and the marginal external costs is small, see columns (4) and (5), and partial pricing has no significant effect on the distribution of aggregate welfare. Furthermore, the gains from partial pricing are nearly as high as those of efficient pricing. When the exploitation of transport cost differences is ruled out, partial taxation is an effective instrument in the two link network, because congestion is limited, so that the efficiency gains from congestion pricing are small. In summary, the illustration shows that the Mirrlees paradox leads to deviations of optimal taxes from marginal external costs. This is the case with full and with partial network pricing, and the deviations depend on the specific tax redistribution assumptions. With full network pricing, links predominantly used by households with low transport costs tend to be priced below marginal social cost, and those mostly used by households with high transport costs are priced above social cost. This tendency is stronger when the tax redistribution mechanism favours low cost households, and it is neutralised to a large extent when each household is compensated by the amount of congestion taxes it pays. Using this assumption in the next example enables us to focus on link toll restrictions. 3.3. Example 2: network interactions in a three link network Example 2 highlights the impact of link toll restrictions on the effectiveness of network congestion pricing. The network (Fig. 3), consisting of three directed links (a; b; c) and three nodes (1,2,3), is a stylised representation of high capacity roads used for through-

Fig. 3. Topology for network example 2.

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

303

traffic in Namur, during the morning peak. The idea is to test partial pricing schemes (taxes on (combinations of) links a; b and c) and optimal parking charges, by comparing their efficiency to that of the solution in which all links are priced. Note that, while in theory optimal tolls can be negative, in the application all tolls are bounded below at zero. As mentioned, it is assumed that households are compensated for the congestion taxes that they incur. Sensitivity analysis on cost and demand characteristics has been performed, but the results are omitted for reasons of brevity. Table 3 gives the reference demands for the representative consumer at each origin, and the cost function parameters. For trips from node 1 to node 3, paths ab and c can be chosen. In the reference equilibrium link c is under-used from the social point of view (see below). I assume that there is no substitution between destinations: increased costs for one destination cannot be avoided by switching to another. In contrast, the paths from node 1 to node 3 (ab and c) are perfect substitutes. Conform Eq. (6), the sum of link travel costs on a and b is equal to the cost on link c. The main results from the partial pricing combinations are summarised in Table 4. The maximal welfare gain is obtained by taxing all network links (a; b; c ) at approximately marginal external cost. The impact of the Mirrlees paradox on optimal prices is small because of the assumption of exact compensation, and because of the presence of alternative paths for traffic going from node 1 to node 3 (network interaction). The optimal prices reduce demand by f6% for all origin –destination pairs, and increase the share of link c for trips from node 1 to node 3 from 14.5 to 18%. Optimal pricing on all links achieves efficiency in terms of demand levels as well as in terms of path choice, if combined with exact compensation of households through the tax redistribution mechanism. The share of the first-best welfare improvement from partial pricing schemes ranges from 0% (link c) to 92% (links b and c). A tax on link c shifts traffic going from node 1 to node 3, to path ab, which is not desirable in terms of assignment. This is not compensated by a demand reduction (as paths are perfect substitutes), so that the optimal tax on link c is zero if no other links can be taxed (and subsidising link c is ruled out). To the contrary, taxes on links a or b improve path choice and demand levels, permitting substantial welfare gains. Taxing link b performs better than taxing link a because it

Table 3 Cost and demand characteristics of the three link network (reference equilibrium) Aggregate demand (trips) 1–2 1–3 2–3

993 1590 1089

Costs (Eurocent) and flow (vehicles) Intercept Link a Link b Link c

57 83 150

Slope

Flow

Link travel cost

0.004 0.007 0.072

2353 2449 230

66.4 100.1 166.5

304

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

Table 4 The effects of partial pricing in the three link network Reference

Optimal tax on link(s) a, b, c

a, b

a, c

b, c

0

54.4

72.8

92.1

(2) Percentage demand change per origin – destination pair 1,2 0 5.6 2.5 1.5 1,3 0 5.6 2.5 1.5 2,3 0 6.3 0.4 3.6

0 0 0

1.8 1.8 2.8

5.8 5.7 0.4

4.0 4.2 8.7

(3) Ratio of taxes over marginal external costs per link A 0 1.005 0.93 B 0 0.998 0 C 0 0.999 0

0 0 0

2.1 0 0.7

0 1.4 0.9

81.8 18.2

82.2 17.8

(1) Percentage of maximal welfare gain 0 100

a

b

46.6

53.9

(4) Distribution of demand for 1,3 over paths AB 85.5 82.0 79.2 C 14.5 18.0 20.8

c

0 0.55 0

78.9 21.1

85.5 14.5

0.21 0.45 0

78.7 21.3

affects traffic originating from both nodes 1 and 2, on the link suffering from the strongest congestion problem (link b). Taxes for all one link pricing schemes are below marginal external costs, and the demand reductions are smaller and more diverse than under complete network pricing. The network assignment is shifted towards over-usage of link c, instead of under-usage in the reference equilibrium (from the first-best point of view). Referring to Eq. (14), the example suggests that the second term on the right hand side is negative and large. For the two link pricing schemes, the welfare gain of a combined tax on the serial links a and b is much smaller than the sum of the welfare gains of a single tax on links a and b. The reverse holds for parallel links: the welfare gain of a combined tax on links a and c (b and c) is much larger than the sum of the gains from single taxes on a and c (b and c). Taxing a sequence of links does not outperform taxing one link in the sequence by a lot when the network interactions are similar. However, taxation of parallel links allows to, imperfectly, control path choice and trip demand, allowing much larger welfare gains. As in schemes ac (bc), the tax on link a (b) is used to, imperfectly, control traffic on the sequence ab, it is not surprising that taxes rise above external costs. While link tolls are thought of as being expensive to implement, parking charges at trip destinations may be cheaper. I compute the welfare effects of optimal parking charges at destination 2 and/or destination 3 (results not shown in Table 4). Intuitively it is clear that the effectiveness of parking charges depends on the contribution of excessive transport demand levels and of inefficient assignment to the overall transport inefficiency, as parking charges can not correct path choice but they can restrain demand. Since an optimal parking charge at both trip destinations realises only 14% less welfare gains than first-best congestion taxes, excess demand is more important here than the assignment inefficiencies. An optimal parking charge at destination 2 only, produces 58% of the maximal

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

305

welfare gain (67% of the gain of optimally charging for parking at both destinations). Similarly, an optimal parking charge at destination 3 only, produces 79% of the maximal welfare gain (92% of the gain of optimally charging for parking at both destinations). In other words, the performance of optimal parking charges is comparable to that of two link pricing schemes.11 Of course, demand reductions are lower for destinations where no parking charge is implemented. Demand for destination 3 increases slightly when only destination 2 is subject to the charge, because of reduced congestion on path ab. The network assignment is not improved by parking charges, as there is a switch to the overused path (ab). Because of the demand reduction, the costs of the assignment inefficiency are reduced however.

4. Concluding comments This paper presented rules for optimal link tolls and parking charges, for a general static transport network used by households whose residential location is fixed. The analysis indicates that when not all links can be tolled, link tolls will deviate strongly from marginal external congestion costs on the link, because of network interactions. In most circumstances taxes will be lower than marginal external congestion costs. When optimal individualised lump sum transfers are not available, the optimal link tolls are influenced by the Mirrlees paradox, a concept referring to the tendency of the optimal tax structure to favour households with relatively high marginal utilities of income. It was shown that optimal link tolls will deviate from marginal external congestion costs, even in the absence of link pricing constraints. The welfare potential of partial pricing systems strongly depends on the characteristics of the no-tax network assignment and on the particular links which are tolled. The comparison of partial link tolls and parking charges suggests that the contribution of improved network assignment to transport efficiency is smaller than that of restraining demand. This can be expected to hold in other networks, as long as the user assignment and the cost minimising assignment for a given level of demand are similar. Instruments like parking charges then should be considered instead of partial link pricing schemes. The model allows other applications, e.g. the assessment of link capacity changes. A further problem is that of link selection. The analysis has concentrated on the description of second-best link tolls. Alternatively, one could ask which links should tolled when tolling all of them is not possible. However, as the set of used paths depends on the tolls, the information from an observed equilibrium is not sufficient to characterise the equilibrium after the tax change. The applicability of the present analysis for the link selection problem hence is limited. Finally, I mention just one caveat. The assumption of fixed location rules out compensating reactions to transport cost changes in the land market. Endogenous location choices will affect the impact of congestion tolls, in terms of congestion and in terms of the

11 The relative performance of parking charges decreases in a network with more variation in origins, destinations and trip distances. Parking charges also do not deter through traffic (Glazer and Niskanen, 1992).

306

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

influence of the Mirrlees paradox. The effects depend on assumptions concerning the exogeneity or endogeneity of lot sizes. Acknowledgements Earlier versions of this paper were written while the author was affiliated with the Center for Economic Studies, KULeuven, Belgium. I acknowledge support of the Belgian Federal Office for Scientific, Technical and Cultural Affairs, Sustainable Mobility Programme, contract MD/01/026. Earlier versions were presented at CES, KULeuven and at the Department of Economics, University of California, Irvine. I am grateful to participants, in particular Stef Proost, Ken Small, Erik Verhoef, Bruno De Borger and Erik Schokkaert, for their remarks. I also thank two anonymous referees for their in-depth comments. Eric Corne´lis provided data for the examples. The usual disclaimer applies.

References Transportation Research B (special issue), 2001. Vol. 35 (1), pp. 1 – 105. Arnott, R., de Palma, A., Lindsey, R., 1998. Recent developments in the bottleneck model. In: Button, K., Verhoef, E. (Eds.), Road Pricing, Traffic Congestion and the Environment. Issues of Efficiency and Social Feasibility. Edward Elgar, Massachusetts, pp. 79 – 112. Atkinson, A., Stiglitz, J., 1980. Lectures on Public Economics McGraw-Hill, New York. Braid, R.M., 1996. Peak-load pricing of a transportation route with an unpriced substitute. Journal of Urban Economics 40, 179 – 197. Calthrop, E., Proost, S., Van Dender, K., 2000. Parking policies and road pricing. Urban Studies 37 (1), 63 – 76. de Palma, A., Lindsey, R., 2000. Private toll roads: competition under various ownership regimes. Annals of Regional Science 34, 13 – 35. Ferris, M.C., Kanzow, C., 2002. Complementarity and related problems. In: Pardalos, P.M., Resende, M.G.C. (Eds.), Handbook of Applied Optimization. Oxford University Press, New York, pp. 514 – 530. Glazer, A., Niskanen, E., 1992. Parking fees and congestion. Regional Science and Urban Economics 22, 123 – 132. Hearn, D.W., Ramana, M.V., 1998. Solving congestion toll pricing models. In: Marcotte, P., Nguyen, S. (Eds.), Equilibrium and Advanced Transportation Modeling. Kluwer, Boston, Dordrecht, London, pp. 109 – 124. Hearn, D.W., Yildirim, M.B., 2002. A toll pricing framework for traffic assignment problems with elastic demand. In: Marcotte, P. (Ed.), Current Trends in Transportation and Network Analysis—Papers in Honor of Michael Florian. Kluwer, in press. Labbe´, M., Marcotte, P., Savard, G., 1998. A bilevel model of taxation and its application to optimal highway pricing. Management Science 44, 1595 – 1607. Liu, L.N., McDonald, J.F., 1999. Economic efficiency of second-best congestion pricing schemes in urban highway systems. Transportation Research Part B 33, 157 – 188. McDonald, J.F., 1995. Urban highway congestion: an analysis of second-best tolls. Transportation 22, 353 – 369. Mayeres, I., Proost, S., 1997. Optimal tax and public investment rules for congestion type of externalities. Scandinavian Journal of Economics 99, 261 – 279. Mirrlees, J.A., 1972. The optimum town. The Swedish Journal of Economics 74, 114 – 135. Shepherd, S., Sumalee, A., 1999. A genetic algorithm based approach to optimal toll level and location problems. Paper presented at the ETC Conference, Cambridge, UK. Available from http://www.personal.leeds.ac.uk/ ftraas/. Small, K., 1992. Urban Transportation Economics Harwood, Chur. Small, K., Yan, J., 2001. The value of ‘value pricing’ of roads: second-best pricing and product differentiation. Journal of Urban Economics 49, 310 – 336.

K. Van Dender / Regional Science and Urban Economics 34 (2004) 289–307

307

Van Dender, K., 2001. Assessment of transport network pricing reform: a simulation approach. In: Van Dender, K. (Ed.), Aspects of Congestion Pricing For Urban Transport, pp. 63 – 87 (PhD Thesis nr. 149, ETEW, Leuven). Verhoef, E.T., 2002. Second best congestion pricing in general static transportation networks with elastic demands. Regional Science and Urban Economics 32, 281 – 310. Verhoef, E.T., Nijkamp, P., Rietveld, P., 1996. Second-best congestion pricing: the case of an untolled alternative. Journal of Urban Economics 40, 279 – 302. Verhoef, E.T., Small, K.A., 1999. Second-best congestion pricing with heterogeneity under public and private ownership. Paper presented at the Helsinki Workshop on The Politics of Pricing, 19 – 20 August 1999, Helsinki. Wardrop, J., 1952. Some theoretical aspects of road traffic research. Proceedings of the Institute of Civil Engineers 1, 325 – 378.