Support and opposition to a Pigovian tax: Road pricing with reference-dependent preferences

Journal of Urban Economics 99 (2017) 31–47

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Journal of Urban Economics journal homepage: www.elsevier.com/locate/jue

Support and opposition to a Pigovian tax: Road pricing with reference-dependent preferencesR Bruno De Borger a,∗, Amihai Glazer b a b

Department of Economics, University of Antwerp, Belgium Department of Economics, University of California Irvine, USA

a r t i c l e

i n f o

Article history: Received 26 February 2016 Revised 8 December 2016 Available online 19 January 2017 Keywords: Road pricing Loss aversion Lobbying Political economy

a b s t r a c t Loss aversion can affect support and opposition to Pigovian taxes to reduce externalities. This paper studies road pricing with reference-dependent preferences, modeled by a linear gain-loss utility function. Given this specification, we find that the socially optimal road toll is smaller than the optimal toll in the absence of reference dependence, and it declines in the degree of loss aversion. Loss aversion can also explain the empirical observation that support for road pricing is lower before than after its introduction. We further show that loss aversion may increase or reduce lobbying efforts by driver organizations against the introduction of tolling. It will increase lobbying if a high toll is proposed but drivers initially believe that the probability that it will be introduced is small. Lastly, loss aversion unambiguously reduces lobbying by organizations of non-drivers (representing, for example, environmentalists or public transport users) in favor of the introduction of a toll. © 2017 Elsevier Inc. All rights reserved.

1. Introduction This paper studies the effects of reference-dependent preferences on support and opposition to a Pigovian tax on an externality. Although the model applies more generally, the particular policy studied is road pricing to reduce congestion. We assume that people have reference-dependent preferences characterized by loss aversion, implying that they give more weight to losses than to gains of equal size (Kahneman and Tversky, 1979; Tversky and Kahneman, 1991). We capture this idea by specifying a linear gainloss utility function and explore the implications of loss aversion for the political economy of road pricing policies. Several studies emphasize the importance of referencedependent preferences and loss aversion for pricing and taxation policies. For example, Alm et al. (1992) and Dhami and al-Nowaihi (2007) argue that such preferences may explain why people pay

R We thank the Flemish Science Foundation FWO-V for financial support (grant G02.1041.13N). We are grateful to Robin Lindsey for detailed written comments on a previous version of the paper, to Mogens Fosgerau and Jonas Eliasson for useful discussions, and to Maria Börjesson for providing the data for the numerical example. We also thank Marcelo Arbex and other participants at the conference of the International Institute of Public Finance (IIPF, Lake Tahoe, August 2016) for suggestions. The comments of two referees and an editor of this journal much improved the paper. None of these people is responsible for remaining mistakes. ∗ Corresponding author. E-mail addresses: [email protected] (B. De Borger), [email protected] (A. Glazer).

http://dx.doi.org/10.1016/j.jue.2016.12.003 0094-1190/© 2017 Elsevier Inc. All rights reserved.

taxes and why we do not observe more tax evasion. Herweg and Mierendorff (2013) show the relevance of loss aversion for the optimal design of two-part tariffs. In an empirical study, Engström et al. (2015) find that loss aversion affects tax compliance, suggesting that compliance will increase and auditing costs will be reduced, if preliminary taxes are calibrated so that most taxpayers receive refunds. Alesina and Passarelli (2015) explore the implications of loss aversion in politics. However, despite these observations, a survey of the applicability of prospect theory notes that public economics is one of a few areas where more research on loss aversion may be highly relevant (Barberis, 2013). The effect of loss aversion on congestion pricing is interesting for several reasons.1 First, despite widespread support from economists, congestion pricing is rare. The cities of London, Stockholm and Milan are well known examples of successful introduction of some form of pricing, but the list of cities and countries where proposals to implement such pricing were voted down is much longer.2 Interestingly, however, in the few cases where some form of road pricing was introduced, people have reacted more

1 We consider road pricing on an existing road, not a toll to raise revenue to construct a new road. The former may hurt drivers, the second may benefit all drivers. 2 In Edinburgh, Birmingham and Manchester, road tolls were opposed by an overwhelming majority in local referenda. Although Mayor Bloomberg of New York strongly favored tolling, in 2008 the New York State Assembly ultimately decided not to vote on a proposal to introduce road tolls. In Belgium and the Netherlands, road pricing has been on the agenda for decades, but implementation has been repeatedly postponed. In the UK, an online petition against road pricing in

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B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

favorably after its introduction than before. The difference is especially well documented for the Swedish city of Stockholm (see, for example, Eliasson et al., 2009; Winslott-Hiselius et al., 2009). Just before the introduction of a trial congestion toll in Stockholm in 2005, only 36% of poll respondents supported the toll. After the trial started, support increased to 52%.3 The trial ended in July 2006, and was followed by a referendum in September. Excluding blank votes, 53% of Stockholm citizens voted to keep the charges. A poll in December 2007, after the toll was permanently adopted, indicates that 74% supported the toll; this is more than a 20% increase. A similar pattern of attitude changes appeared in London, which introduced congestion tolls in the central city in 2003. Transport for London reports at regular intervals the public attitude on the London congestion toll. Before the start of London’s congestion pricing system in late 2002, 40% rejected congestion charging, while 40% supported it. After introduction of the charging system in 2003 only 25%–30% rejected congestion pricing, whereas 50–60% were in favor. More recently, these same Transport for London surveys showed that more than 70% of Londoners said the system was effective and twice as many supported the charge as opposed it (Naparstek, 2007).4 Lastly, a study of attitudes to congestion pricing in Göteborg before and after congestion charges were introduced in January 2013 also finds that attitudes towards the charges became more favorable after they were introduced, and that the effect occurs because of a status quo bias (Börjesson et al., 2016). Many arguments have been put forward to explain the lack of popular and political support for welfare-improving road pricing, and for the change in attitudes towards pricing after its introduction. Potential explanations include uncertainty about the costs of switching modes, political uncertainty about the use of the toll revenues, attitude structures and framing, unexpectedly large benefits of road pricing, and cognitive dissonance (see, among many others, Schade and Baum, 2007; De Borger and Proost, 2012; Börjesson et al., 2016). Reference dependence and loss aversion were mentioned as possible explanations by several authors, including Hess et al. (2008) and Börjesson et al. (2016). Surprisingly, however, with one exception (Lindsey, 2011, see below) the theoretical literature neglects the effects of loss aversion on attitudes towards road pricing. A second reason why economists are interested in the effect of loss aversion on congestion pricing is that the announcement of the potential introduction of road pricing has generated intense lobbying in several countries. The interests of road users are often defended by well-organized driver organizations such as the American Automobile Association in the US, and similar organizations in most European countries. Moreover, some countries also have influential ‘anti-car’ lobbies; for example, environmentalists, public transport users, and bikers have organizations that defend their interests. The result was that, although most lobbying opposed road pricing, some groups favored its introduction.5 The incentives to

20 06-20 07 attracted more than 1.8 million signatures (http://news.bbc.co.uk/2/hi/ uk_news/6381279.stm). 3 The media image also changed. The percentage of related newspaper articles with a positive angle increased from 3 % in the autumn of 2005 to 42% in the spring of 2006; the share of negative newspaper articles was almost halved from 39 % to 22%. 4 A comparison of attitudes in Stockholm, Helsinki and Lyon concluded that the higher support in Stockholm appeared because it had experienced congestion pricing, while the others had not (see Hamilton, 2012). 5 Urban areas saw lobbying by retailer organizations as well as residents (see the evidence provided in De Borger and Russo, 2015). Road pricing on highways generated intense lobbying by driver organizations (see the German experience). In Belgium, the day after the media discussed the possibility that road pricing may be introduced, both driver organizations and individual drivers responded fiercely. Representatives of the main driver organizations went on national radio and television to argue that road pricing would be extremely unfair and unacceptable, and that it

lobby, and the effect of loss aversion on lobbying efforts by such organizations, have not been examined in the literature. Lastly, although strong empirical evidence on the effect of loss aversion on road pricing is lacking (not surprisingly, given that congestion pricing is rare), substantial evidence indicates that loss aversion affects both individuals’ and firms’ transport decisions, including their responses to changes in monetary costs and in travel times. For example, Hess et al. (2008) find clear evidence of an asymmetrical response to gains and losses relative to the reference, where the degree of asymmetry varies across attributes and population segments. De Borger and Fosgerau (2008) and Hjörth and Fosgerau (2011) explicitly study loss aversion with respect to travel time and money, finding significant loss aversion in both dimensions. Hjörth and Fosgerau (2011) further find that loss aversion depends on how well the reference is established; moreover, it depends, among others, on age and education. Masiero and Hensher (2010) analyze a freight transport experiment and find asymmetries in responses and declining sensitivity over time. Loss aversion raises several questions about the political economy of pricing. How does reference dependence and loss aversion affect socially optimal tolls? How does loss aversion affect votes by potential road users, where voting is either for or against an arbitrary (possibly not socially optimal) toll? How does the reference point affect consumers’ attitudes towards road pricing? Does loss aversion contribute to the widespread lobbying against road pricing that is observed in some countries? The model developed in this paper sheds light on these and related questions. The analysis in this paper relates to several strands of literature. By studying how loss aversion affects the attitudes of users and non-users towards Pigovian taxes, it adds to the literature, referred to before, on the effect of loss aversion for tax policies. More specifically, our analysis complements Lindsey (2011).6 Our focus differs from Lindsey (2011) in several respects: (i) We do not look at state-contingent prices. Instead, we look at the implications of reference-dependence when the reference is defined in terms of a particular pricing regime; (ii) We look not only at the effect of loss aversion on socially optimal tolls, but also study the political economy of road pricing; (iii) We study the effect of loss aversion on lobbying by special interest groups such as driver organizations. Our paper further relates to the literature on the political economy of transport pricing (see, e.g., Borck and Wrede, 2005; Brueckner and Selod, 2006; De Borger and Proost, 2012). This literature neglects the possible effect of reference-dependent preferences on behavior and consumer attitudes. Note that, although we do not formally consider distributional issues, loss aversion may also be highly relevant to study the distributional effects of pricing policies (see, e.g., Mayeres and Proost, 2001; West, 2004; van den Berg and Verhoef, 2011). In a broader perspective, our paper also relates to the literature on lobbying and on rent seeking. A theoretical analysis of rent seeking under loss aversion is provided in Cornes and Hartley (2012), who find that loss aversion reduces aggregate lobbying. However, our examination of lobbying under loss aversion differs in two main ways: (i) They consider an exogenously fixed rent that will be assigned to one of the players. They therefore have everyone expect a gain from the availability of the rent. In contrast, a road toll can directly hurt some individuals; (ii) We deshould not be introduced. Individual action consisted, for example, in protests on a website set up for opponents to register, attracting an enormous number of participants in just one day. A counter-campaign by supporters of tolling attracted much fewer participants. 6 Lindsey (2011) models state-dependent road pricing with reference-dependent preferences and shows that uncertainty with respect to the toll to be paid may explain the absence of state-dependent tolls observed in reality. Demand or supply shocks affect the capacity and the service quality of the road system, and the model studies optimal road pricing contingent on different possible states.

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

rive the effects of the policy, and of lobbying against the policy, on different groups (drivers, non-drivers). As we will show, these features of our model make the effect of loss aversion on lobbying ambiguous. Experimental work on contests finds evidence for loss aversion, and evidence that subjects consider the reference point as depending on expectations of outcomes given their own efforts (see Gill and Prowse, 2012).7 To simplify our analysis we suppose, as does much of the literature, that the reference point is exogenously given. But, unlike the evidence provided in Gill and Prowse (2012), we consider situations in which a person may suffer a loss if he takes no action. We consider consumer decisions whether to commute by car, both in the absence and in the presence of a toll. In the latter case, we specify reference-dependence and loss aversion using a linear gain-loss utility function (Köszegi and Rabin, 2006; Lindsey, 2011). We initially treat the zero-toll situation as the reference. It is assumed that people evaluate gains and losses under road pricing in terms of two dimensions: the net utility of making the trip by car and the income increase associated with the redistributed toll revenues. Using this setup, we show several results. Some follow rather directly from the concept of loss aversion itself, others are much less obvious. First, we show that the socially optimal toll under loss aversion is smaller than the socially optimal toll in its absence, and the toll declines in the degree of loss aversion. The intuition is that, given the setup of the model, loss aversion makes it less desirable to move away from the reference point, so that the optimal policy is mitigated. Loss aversion therefore generates status quo bias in the sense that the optimal toll ‘remains closer’ to the reference; it bridges only part of the gap between the status quo and the social optimum in the absence of loss aversion. Second, our model explains the observation that support for road pricing is lower before than after its introduction. Third, we find that loss aversion may increase or reduce lobbying efforts by driver organizations against the introduction of tolling. It is more likely to increase lobbying efforts if the reference probability of toll introduction is small, a high toll is proposed, and lobbying strongly affects the probability that the toll will not be introduced. Lastly, loss aversion reduces lobbying by non-drivers in favor of the introduction of a toll. The remainder of this paper has the following structure. The first section presents the assumptions underlying our analysis and introduces the structure of the model. Section 2 derives and interprets the socially optimal toll under loss aversion. We then turn to the effect of loss aversion on the voting behavior of drivers and non-drivers in Section 3, and we study the implications of changes in the reference situation for voting outcomes. Section 4 analyses the effects of loss aversion on lobbying by drivers and non-driver organizations. A final section summarizes our main conclusions. 2. Assumptions To specifically focus on reference dependence and to isolate the implications of loss aversion from other phenomena (for example, risk aversion under uncertainty), the model assumes perfect information and no uncertainty.8 We do introduce uncertainty in

7 Gill and Stone (2010) study tournaments with loss averse players where the reference point depends on relative effort. 8 One contribution of our model is to show that many people may oppose a congestion toll even under perfect information and no risk. Furthermore, even if risk aversion can explain why, under imperfect information, voters would oppose the imposition of a new toll, risk aversion does not explain why after its introduction there is a bias to continue levying the toll. For after the toll is imposed, voters know the effects of the toll. The effects may be better or worse than what they had expected, and so in cases where the effects were worse, voters should favor abolishing the toll. We, in contrast, show a consistent bias caused by loss aversion to favor any change in policy that was adopted.

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the analysis of lobbying below, see Section 4. In what follows, we first present the setting in the standard case without referencedependence, then we look at the model under reference dependence. 2.1. The standard case: no reference-dependence We use the setup described in De Borger and Proost (2012). The model reflects commuting decisions, and in the basic version assumes all commuters travel by car. Introducing alternative modes (for example, public transport) is straightforward and does not affect the results with respect to the effects of loss aversion. Each potential road user makes at most one trip. There are M potential road users who are ranked in declining order of willingness to pay. A road user indexed n has utility

Y + Max {a − bn − P, 0}, where Y is income and P is the generalized price of road use. In the absence of tolls, this is given by the average cost function

P = d + cN. In this expression, N is the number of users. The marginal external congestion cost is cN, average social cost is d + 2cN . All people for whom using the road yields higher utility than not doing so will make the road trip. Solving for equilibrium road use at a zero toll gives

N0 =

a−d . b+c

Similarly, for an arbitrarily given toll τ equilibrium road use solves a − bN − (d + cN + τ ) = 0 for N; we find

Nτ =

a−d−τ . b+c

For comparison we describe the implications of this setting for the socially optimal road toll, and for potential road users’ voting behavior with respect to the introduction of a toll (see De Borger and Proost, 2012 for details). First, the socially optimal toll equals the marginal external cost: τ ∗ = cN∗ . It then follows that the number of road users is

N∗ =

a−d . b + 2c

Second, who supports an arbitrary (not necessarily optimal) toll

τ ? Suppose toll revenues are redistributed lump-sum to all votτ

N ers; each person then gets τ M of the revenues, where M is the number of voters. People who continue to drive (n in the range (0, Nτ )) are worse off; people who did not drive and continue to not drive after the toll is imposed (n in the range (N0 , M)) are better off (because they share in the toll revenues). Somewhere in the range (Nτ , N0 ) there is a cutoff value N such that all individuals n to the left (n < N ) oppose the toll; all those to the right (N < n) favor the toll. People in this range are affected in three ways: they gain because they get redistributed revenues, they lose the value of the trip they no longer make, and they save the time cost they no longer have to incur. The total gain (which may be negative) for person n in this range is

−(a − bn ) + (d + cN 0 ) +

τ Nτ M

.

Setting this expression equal to zero and solving for n shows that all people with n < N oppose road pricing, where N is

N = N0 −

τ Nτ bM

.

For future reference, note that opposition to road pricing (a larger N ) increases with the number of initial drivers and declines

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B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

the net utility of driving when the toll remains zero. Note that: Fig. 1. Relation between traffic levels (M: number of voters; N0 : traffic level in the absence of the toll; Nτ : traffic level in the presence of the toll; N : cutoff value such that all voters n < N are worse off under the toll).

with toll revenues (because toll revenues are distributed to the population). Very mild conditions imply that Nτ < N’ < N0 .9 Note that N defines for every toll level the cutoff such that people with a willingness to pay index n < N oppose the toll. A majority will favor the toll if N  < M 2 . Note that, in case the toll is socially optimal, the cutoff value is

N = N0 −

c (N ∗ ) . bM 2

2.2. Reference-dependent preferences Consider now the case where people evaluate gains and losses separately, according to a gain-loss utility function. Utility is standard utility plus a gain-loss correction involving loss aversion, see below. To fix ideas, assume that everyone treats a zero toll as the reference. This reference may apply if people expect no toll, and the introduction of tolls surprises them. The reference traffic level people expect is the traffic level associated with a zero toll N0 . Of course, the assumption of zero tolls at the reference is a strong one. As will be shown below, however, most of our results are robust with respect to alternative assumptions about the reference situation. Consider a person indexed n who is a road user at a zero toll and who continues to drive if the toll is positive (hence, n is in the range (0, Nτ )). We normalize his income Y at zero without loss of generality. Note that a person who continues to drive is affected in three ways by the introduction of the toll: he suffers less congestion due to the decline in traffic, he pays the toll, and he receives his share of the redistributed toll revenues. In principle, drivers may evaluate gains and losses separately for each of these three components. However, to avoid complicating the model and without losing essential insights, we restrict gain-loss utility to two components: the toll makes a driver lose the net utility of driving, and he gains his share of the redistributed toll revenues. The distinction we make may be interpreted as a particular type of mental accounting, whereby people think differently about the net utility of driving and the receipt of redistributed toll revenues. This mental accounting seems plausible, not only because the utility of driving and toll revenues received are fundamentally different reasons for changes in well-being, but also because they are experienced at different moments in time. Specifically, borrowing the linear formulation from Köszegi and Rabin (2006) and Lindsey (2011), we define reference dependent utility as:

uτ ( n ) + d

τ Nτ  M

(2)

The superscript ‘d’ refers to a person who drives; the subscript ‘τ ’ refers to a positive toll, and ‘0 refers to a zero toll. In (1), the parameter η > 0 captures gain-loss utility; λ > 1 measures the degree of loss aversion. Gains are valued at η, losses at λη > η. The specification of reference-dependent utility in (1) consists of three terms. The first term is ‘standard’ utility when the toll is positive: the net willingness to pay for the road trip plus the user’s share in toll revenues. The second term is gain-loss utility applied to the change in net willingness to pay. Note that the toll necessarily exceeds the time gain, so that for continuing drivers

udτ (n ) − ud0 (n ) < 0.

A graphical illustration of the relation between the different traffic levels is in Fig. 1.



udτ (n ) = a − bn − (d + cN τ + τ ) ud0 (n ) = a − bn − (d + cN 0 ).



+ λη uτ (n ) − d

ud0

  τ Nτ  (n ) + η −0 . M

(1)

In this expression, udτ (n ) captures the net utility of driving (the willingness to pay minus the cost of a trip inclusive of any toll paid) of this road user when the toll is positive; the term ud0 (n ) is 9 The condition is bM − (a − d − τ ) > 0.This is a rather weak condition. A sufficient (but by no means necessary) condition is that the number of voters (potential road users) M exceeds the number of road users at a zero generalized price. In that case bM − (a − d − τ ) = d + τ > 0.

The third term applies gain-loss utility to the change in toll revenues. Apart from assuming linearity, our specification (1) makes two additional strong and related assumptions. First, we apply the same gain-loss parameters to changes in the net willingness to pay for driving and to the change in revenue from tolling (see Lindsey, 2011 for discussion). Second, we apply the same gain-loss parameters to the time and money components of the net willingness to pay for driving. Although there is some empirical evidence that time and money losses are evaluated differently (De Borger and Fosgerau, 2008), allowing this in our model greatly complicates the analysis, and is unlikely to affect the main insights.10 Some initial drivers will stop driving after the toll is adopted. To determine who will continue to drive, we compare reference dependent utility of an initial driver when he continues to drive under the toll (see (1)) with reference-dependent utility of an initial driver who no longer drives under a positive toll. For an initial driver who avoids driving (superscript ‘nd’ for ‘non-driver’) after the toll is imposed (n is in the range (Nτ , N0 )), referencedependent utility is



und τ (n ) +

τ Nτ  M





d + λη und τ ( n ) − u0 ( n ) + η

 τ Nτ M



−0 .

For a non-driver und τ (n ) = 0, so that the expression reduces to





−λη ud0 (n ) + (1 + η )

 τ Nτ  M

.

(3)

Comparing (3) to (1) and using (2) then shows that the traffic level when the toll is imposed is

Nτ =

a−d−τ . b+c

(4)

Lastly, for non-drivers that remain non-drivers (n lies in the range (N0 , M)) reference dependent utility is

(1 + η )

 τ Nτ  M

.

(5)

3. Socially optimal road toll under loss aversion We are interested in the socially optimal toll in the presence of reference-dependence, initially assuming voters treat the zero toll 10 Under the assumption of separate evaluation of changes in trip time and in the toll paid, all people with index n < Nτ would suffer a loss due to the toll payment, and no one would suffer time losses. Continuing drivers would enjoy a time saving due to reduced congestion, and people no longer driving after the introduction of the toll avoid the time loss of the trip they no longer make. As loss aversion only affects losses, the results would be driven by the losses due to toll payments. We experimented with a model version that assumed people evaluate gains and losses separately for the time gain, the toll paid, and the toll revenue received. The model became analytically very cumbersome but, as far as we could see, it did not affect the main insights derived in this paper.

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

as the reference. We discuss the robustness of our results with respect to this and other simplifying assumptions at the end of this section. We further assume throughout that toll revenues are redistributed lump-sum to all voters, road users and non-users alike. Aggregate welfare in the presence of the toll is defined as follows. First, total reference-dependent utility of all continuing drivers, using (1), is



Nτ



 τ Nτ 

  (1 + λη ) udτ (n ) − λη ud0 (n ) + (1 + η ) dn.

M

n=0

(6)

For initial drivers who no longer drive under the positive toll total utility is (see (3))





N0





−λη ud0 (n ) + (1 + η )

n=N τ

 τ Nτ  M

dn.

(7)

Continuing non drivers have total utility (see (5))





M n=N 0

(1 + η )

 τ Nτ 

M − N0 dn = (1 + η )(τ N τ )

M

M

 .

(8)

Total social welfare is defined as the sum of reference dependent utility of all individuals (note that toll revenues have been redistributed to voters; this has been taken into account in the definition of standard utility, see above). Summing (6)–(8) and rearranging yields11

SW = (1 + λη ) −λη



N0 n=0



Nτ

n=0





 udτ (n ) dn 

ud0 (n ) dn + (1 + η )(τ N τ ).

(a − d ){b[η (1 − λ )] + c[1 + η]} . b[1 + η (2 − λ )] + 2c[1 + η]

(9)

(10)

The denominator of (10) is positive by the second-order condition (see Appendix A). Substituting (10) in (4) and rearranging gives the socially optimal traffic level as

Nτ =

a−d−τ (a − d )(1 + η ) = . b+c b[1 + η (2 − λ )] + 2c[1 + η]

(11)

If there is no gain-loss utility (so η = 0) or if there is gain-loss utility but no loss aversion (η > 0, λ = 1), then expressions (10) and (11) reduce to the socially optimal toll and traffic level rules mentioned above.12 Loss aversion necessarily implies a smaller socially optimal toll. To see this, rewrite (10) as

τ =c

(a − d ) (a − d )[bη (b + c )(1 − λ )] + . (b + 2c ) (b + 2c )[b(1 + η (2 − λ )) + 2c(1 + η )]

The first term on the right-hand side is the expression for the socially optimal toll in the absence of reference-dependence (see footnote 11 above). The second term on the right-hand side is necessarily negative if there is loss aversion, so that

τ
11

Table 1 Toll and traffic volumes under three tolling scenarios. Toll (τ ) Zero toll Optimal toll (no loss aversion) Optimal toll (loss aversion)

Number of road users (N)

c (a−d ) b+2c

a−d b+c a−d b+2c

(a−d ){b[η (1−λ )]+c[1+η]} b[1+η (2−λ )]+2c[1+η]

(a−d )(1+η ) b[1+η (2−λ )]+2c[1+η]

0

Hence, loss aversion implies a smaller socially optimal toll.13 It creates a bias towards preserving the status quo which, given our setup, is the zero-toll situation. Loss aversion makes it less desirable to move away from the reference point, so that the optimal policy is mitigated. If the reference is the status quo, as we assumed, loss aversion can therefore be seen as leading to status quo bias. To see the intuition for the qualitative result that loss aversion reduces the socially optimal toll, rewrite the social welfare function (9) as:

 SW =

Nτ n=0

+λη





udτ (n ) dn + (τ N τ )



Nτ n=0



uτ ( n ) − d

ud0

  (n ) d n −

N0 Nτ



ud0

  (n ) dn +η (τ Nτ ). (9 )

Note that the second integral term is independent of the toll to be derived, and hence can be ignored in the optimization problem. Appendix A derives the socially optimal toll under referencedependence that results from maximizing the reference dependent social welfare function (9). The socially optimal toll is

τ=

35

(a − d ) = τ ∗. ( b + 2c )

Observe that if there is no gain loss utility we would have  Nτ SW = n=0 {udτ (n )}dn + τ N τ . (a−d ) τ = (a−d ) . These are the socially optimal 12 ; N τ = a−d− Indeed, we have τ = cb+2 c b+c b+2c expressions in the absence of reference-dependence, see Section 1.

The first two terms give social welfare in the absence of loss aversion. The remaining three terms are specifically due to gainloss utility: continuing drivers suffer utility losses compared to the reference (the toll payment exceeds the time gain), and initial drivers that give up driving due to the toll lose the utility associated with driving in the reference; lastly, there is a gain in toll revenues. If there were no gain-loss utility, (9 ) reduces to standard social welfare, yielding the standard social optimum. With loss aversion, however, drivers lose net utility. The higher is loss aversion, the more the social planner will try to avoid these losses. To do so, the toll is reduced. Differentiating (10) with respect to the gain-loss and loss aversion parameters yield:

∂τ −bη (a − d )(b + c )(1 + η ) = <0 ∂λ {b[1 + η (2 − λ )] + 2c[1 + η]}2 ∂τ b(a − d )(b + c )(1 − λ ) = ≤ 0. ∂η {b[1 + η (2 − λ )] + 2c[1 + η]}2

(12)

Not surprisingly in view of our earlier findings, the socially optimal toll declines in the degree of loss aversion. It depends on the degree of gain-loss utility only if there is loss aversion (remember that λ ≥ 1, where equality implies the absence of loss aversion). In that case it also declines in the gain-loss utility parameter η. For purposes of comparison, Table 1 summarizes the expressions for the toll and the resulting traffic level under three scenarios: a zero toll, the social optimum in the absence of loss aversion, and the social optimum under loss aversion. For λ = 1the last two scenarios are identical. We summarize with: Proposition 1. Assume all people have reference-dependent preferences as given by (1). The socially optimal toll under loss aversion

13 The toll can be written as a weighted average of the zero and socially optimal toll in the absence of reference-dependence. The weight of the optimal toll is a complex function of all model parameters and it declines in the loss aversion parameter: more loss aversion brings the optimal toll under loss aversion closer to the zero toll of the reference.

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B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

is smaller than the socially optimal toll in its absence, and it declines in the degree of loss aversion. If loss aversion is substantial and policy makers believe they should capture reference dependence in the design of policy, the proposition implies that optimal tolls may be much lower than in the absence of loss aversion.14 As an illustration of the potential effects of loss aversion for the socially optimal toll we develop a numerical example, calibrated using aggregate data for Stockholm. This city offers a particularly appealing example. It is one of the few cities where some form of road pricing has actually been introduced. Moreover, the whole process is remarkably well documented, and many studies analyzed the implications of the toll (see, among others, Börjesson et al., 2012; Hamilton, 2012; Börjesson and Eliasson, 2014; Eliasson, 2014; Börjesson and Kristofferson, 2014). Using aggregate data from these published studies together with information provided by one of the authors, we calibrate the parameters of the simple model used in this paper.15 We then apply the model for a reasonable range of reference-dependence parameters to investigate the effect of loss aversion on the socially optimal toll, and compare the optimal toll to the average toll implemented. To calibrate the model, we need aggregate data on traffic levels before and after the introduction of the toll, the toll level, trip length, travel times before and after toll introduction and the associated time gains following the reduction in congestion, values of time, etc. A brief overview of the calibration process follows. Traffic levels before and after the toll was introduced was approximately given by 447,60 0 and 352,0 0 0, respectively. Average trip length was estimated at 13 km. Taking into account the characteristics of the network (some roads have intersections, traffic lights, etc.), we assume a free-flow speed of 60 km/h, so that the travel time at free-flow speed is 13 min per trip. The value of time for Stockholm car drivers was estimated as 12.1 euro per hour, or 0.2017 euro per minute (see Börjesson and Eliasson, 2014). Using information in Eliasson (2014, Fig. 5, inner city inward morning peak) we estimate the pre-toll and post toll travel times at 2.28 times the travel time at free-flow speed and 2.02 times the travel time at free-flow speed, respectively. This results in the average travel time pre-toll and post-toll of, respectively, 2.28 ∗ 13 = 29.64 min and 2.02 ∗ 13 = 26.26 min. These figures are in line with the time gain due to the toll, estimated on average at 3.36 min per trip (see Börjesson and Kristofferson, 2014). Using a monetary cost per kilometer of 0.15 euro (see Börjesson et al., 2012) and adding the monetary and time cost per trip, we find the generalized cost in the pre-toll situation as 7.925 euro per trip. The average toll was 1.28 euro per trip (Börjesson et al., 2012). Note that the Stockholm system distinguishes three toll levels according to different periods of the day; the average of 1.28 euro is a weighted average over time periods. Using this information we calculate the generalized cost at the post-toll situation as 8.525 euro per trip. Finally, using the calculated generalized cost in the pre- and post- toll situation, the average toll level, and the estimated vol14 In theory, the socially optimal toll calculated using (10) may be negative. However, the setup of the model assumed a positive toll: drivers were assumed to lose in terms of road utility, and it was assumed that positive toll revenues were available for redistribution. In practice, therefore, the socially optimal toll (10) expression applies only to non-negative optimal tolls. A zero toll is optimal if loss aversion is so large that for any positive toll the evaluation of losses to drivers exceeds the perceived gains of redistributed toll revenues to all individuals. We ignore negative tolls. Note that these would require a different setup: they imply that drivers gain and that all individuals incur a monetary loss (they finance the subsidies to drivers). This necessitates adapting (1), (3) and the social welfare function (9) to reflect that road utility increases and toll revenues are negative. 15 We are grateful to Maria Börjesson for providing some missing information in private communication.

umes reported above, we obtain the following calibrated linear demand and cost functions:

P = 10.7365 − 0.0063 N GC = 4.7396 + 0.0071 N + τ . To put the calibrated parameters in perspective, note that they imply that in the absence of loss aversion the socially optimal toll is

τ∗ = c

(a − d ) = 2.08 ( b + 2c )

euro per trip. As mentioned above, the average toll paid in Stockholm was 1.28. We calculate the socially optimal toll for different values of the reference dependence and loss aversion parameters η, λ. Although the specific gain-loss utility function assumed in this paper is often used in theoretical models, there are no direct empirical estimates of η, λ available; the available studies use a variety of different specifications to capture reference dependence. Some evidence on the importance of loss aversion is available, however. For example, Tversky and Kahneman (1992) report that, on average, people value losses 2.25 times more than equivalent gains. Lindsey (2011) mentions that values of about 2 are reasonable, although many studies found smaller values. Furthermore, De Borger and Fosgerau (2008) estimate loss aversion specifically using data on transport decisions; their results (see Model 3; this assumes equal parameters for money and time) imply a loss aversion parameter of about 2.7. Given the absence of direct estimates of η and the range of loss aversion parameters suggested in the empirical literature, we calculate the socially optimal toll for three different values of η and for values of the loss aversion parameter λ ranging between 1, the case of zero loss aversion, and 4. We report the results graphically, see Fig. 2. To put the reported values for the socially optimal toll in perspective, remember that the current average toll equals 1.28 euro per trip. It follows from Fig. 2 that loss aversion strongly affects optimal tolls. In the absence of loss aversion, the socially optimal toll is 2.08, as calculated before. It declines strongly in the parameters capturing reference-dependence. For example, holding η fixed at 0.5, assuming λ = 2 yields a socially optimal toll of 1.63, as compared to 2.08 in the absence of loss aversion. An increase in the parameter η capturing the evaluation of gains further reduces the toll to approximately 1.25 (see the parameter combination η = 1, λ = 2), close to the current average toll of 1.28. Further, note that at high values of both reference-dependence parameters the toll declines to (less than) zero: a zero toll avoids the losses in road utility that drivers suffer; if these are highly valued a zero toll may be optimal. The example just described is based on highly aggregate data and, consistent with our theoretical setting, linear demand and cost functions. Although these may be quite accurate in the neighborhood of the observed volume and generalized price information, accuracy may be substantially less at much higher or lower traffic volumes.16 Moreover, average cost and demand levels hide substantial variation on different links of the network. Therefore, although the example illustrates the potential importance of loss aversion for tolling, it is no more than a stylized exercise. To conclude this section, it is useful to briefly reflect on the robustness of our findings in Proposition 1. First, instead of drivers 16 Note that d = 4.7386 is a crude estimate of the generalized cost at zero traffic and toll levels, under the assumption of a linear generalized cost function. Calculating this cost independently – using the monetary cost per trip, the value of time and the free flow travel time as reported above – yields 1.95 + 0.2017 ∗ 13 = 4.565. This suggests that our linear approximation provides a decent estimate of the intercept of the generalized cost function.

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

37

2.5

Opmal toll

2 1.5 ɳ=0.1 ɳ=0.5

1

ɳ=1 0.5 0

0

1

2

3

4

5

loss aversion (λ) Fig. 2. The effect of loss aversion on the socially optimal toll (η, λ are the gain-loss parameters, see the main text).

treating a zero toll as the reference, they may believe that a positive toll will be introduced and treat this positive toll as the reference. Appendix B shows that, as long as the reference toll is less than the socially optimal toll, such a positive reference toll does not change the expression for the socially optimal toll under loss aversion (Expression (10)).17 Given that a reference toll smaller than socially optimal is the most realistic assumption – noting the current absence of tolling in most major cities – one implication of the previous finding is that loss aversion also matters in the long-run. Suppose people adapt the reference toll over time, then as long as this reference is less than socially optimal, the socially optimal toll under loss aversion is still given by (10).18 In the much less plausible case where people expect a very high toll (above socially optimal), loss aversion leads the socially optimal toll to be higher than it would be in its absence, see Appendix B. This is to be expected: as mentioned before, loss aversion generates a status quo bias, attenuating policies towards the reference. Second, we assumed a particular linear gain-loss utility function to capture reference dependence. The linearity assumption is often used and, although it may affect the strength of the effect identified, it is unlikely to affect its sign. More importantly, our utility specification combined loss aversion with a particular type of mental accounting, in the sense that drivers separately evaluate changes in net road utility and redistributed revenues. This assumption simplifies the analytical derivations and generates very transparent results, though it is clear that different formulations of loss aversion will yield different rules for the optimal toll. Again, however, it is unlikely to invalidate the finding summarized in Proposition 1. 19

17

But do note that changing the reference does affect optimal social welfare, see (9) and (9 ). The reason is that losses relative to the reference matter for referencedependent utility and social welfare. 18 Suppose people expect that a positive toll will be introduced with some probability and adjust the zero toll reference accordingly. With a minor qualification, as described in Appendix B, this change to the basic setting also does not affect our main result. Appendix B also shows that the results continue to hold in a 2-period model. 19 Although we believe our specification is quite plausible, we developed an alternative setting of the model in which people value gains and losses not for road utility and revenues separately, but for changes in overall well-being (road utility plus redistributed revenue). This much complicated the technical analysis: the firstorder conditions implied a polynomial of the third degree in the toll, so that no explicit solution for the optimal toll could be derived. However, under plausible assumptions, one does obtain that loss aversion reduces the socially optimal toll.

Third, we used road pricing to cope with congestion as the driving example, but Proposition 1 also holds if the policy was a Pigovian tax to deal with pollution. Suppose that, instead of congestion, we consider pollution. The absence of congestion can be captured by setting c = 0. Suppose a pollution externality exists and that the marginal external cost is a constant e per unit of N (say, per kilometer driven). If the government maximizes social welfare under reference-dependence (capturing total road utility and total external pollution costs) the optimal tax under loss aversion is now:

τ=

(a − d )[η (1 − λ )] + e(1 + η ) . [1 + η (2 − λ )]

In the absence of loss aversion (λ = 1), the socially optimal toll equals the marginal external pollution cost e. And as before, one finds that loss aversion reduces the socially optimal toll. 4. Effects of loss aversion on the politics of road tolls This section studies some implications of loss aversion for the political outcomes on road pricing, assuming simple majority voting. We proceed in several steps. First, assuming that a zero toll is the reference, we analyze the effect of loss aversion on votes for or against an arbitrary positive toll (which need not be socially optimal). Lastly, we study how the reference point affects voting for road pricing. As will be shown, loss aversion can explain the widespread observation, mentioned in the introduction, that more people oppose road pricing before than after its implementation. 4.1. Loss aversion and voting behavior This subsection analyzes the effect of loss aversion for voting: will more or fewer people support a toll than in the absence of loss aversion? Voters who never drive are clearly better off with any positive toll. Moreover, under mild assumptions initial road users who continue to drive under the toll are worse off, people who are road users in the reference but avoid driving after introduction of the toll can be better or worse off. To see this, note that for an initial driver, by definition reference-dependent utility for the individual at Nτ is the same whether or not he continues to drive. Let us then, as we did before in the case of standard preferences, determine the cutoff value N of the index n such that all voters for whom n < N oppose the toll, and all for whom n > N favor the toll. To determine the cutoff, note that a person no longer

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Fig. 3. The effect of the reference point (case 1: zero toll is the reference; case 2: positive toll is the reference).

driving (n in the range (Nτ , N0 )) has reference-dependent utility of

−ληud0 + (1 + η )

τ Nτ

(13)

M

where, as before, his utility in the reference situation is:

ud0 = a − bn − (d + cN 0 ).

(14)

Subtracting (14) from (13) and solving for n yields the cutoff N . Using the definition of N0 yields

N = N0 −

 1 + η  τ Nτ . 1 + λη bM

(15)

As before, we can show that under a weak condition Nτ < N’ < 0 20 N . A majority favors the toll if N  < M 2. Because loss aversion requires that λ > 1, it follows from (15) that reference dependence with loss aversion means that more people oppose road pricing, so that getting a majority in favor will be more difficult. We have: Proposition 2. Assume all voters have reference-dependent preferences as given by (1). Loss aversion leads more people to oppose the introduction of a toll. This result holds for arbitrary positive toll levels. 4.2. The effect of the reference point In the few cases where some form of road pricing has been introduced, people view it more favorably after its introduction than before. Although the literature mentions loss aversion as a possible explanation for this phenomenon, its role has not been formally studied (for a review of arguments put forward in the literature, see Börjesson et al., 2016). It is straightforward to do so in the model of this paper. One way of looking at the problem of change in attitudes ex ante versus ex post is to compare (i) voter behavior with respect to the introduction of a toll, assuming a zero toll is the reference, and (ii) voter behavior with respect to the removal of an existing toll, assuming the tolled situation is treated as the reference. Our earlier analysis focused on the first problem; in what follows, we study the second one. Assume that a toll is in place and that voters expect it to stay, so that they treat the tolled situation as the reference. The analysis is analogous to that of Section 3.1. We therefore relegate the technical derivations to Appendix C. There we determine the cutoff value N that separates voters in favor from those against removal of the toll. We find that the cutoff is

(1 + λη ) N  = N 0 −

τ Nτ

(1 + η ) bM

.

(16)

Algebra shows that Nτ < N” < N0 only if loss aversion is mild. In the case of much loss aversion we may have N” < Nτ < N0 . To interpret (16), note that in the absence of loss aversion the number of people opposing the introduction of road pricing equals

20

Algebra shows that bM − (a − d − τ ) > 0 is again a sufficient condition.

the number of people favoring eliminating an existing toll; we have

N  = N  = N 0 −

τ Nτ bM

.

Second, comparing N to N (see (15) and (16)) shows that in the presence of loss aversion N ’ < N . More people would oppose the introduction of the toll (when the reference is a zero toll) than would favor its removal once it was introduced (when the reference is the tolled situation).21 The situation is depicted graphically on Fig. 3. We have the following proposition. Proposition 3. Let voters have reference-dependent preferences as captured by (1). Loss aversion implies that a. Fewer people favor removal of an existing toll (treated as the reference). b. The number of people who favor removing the toll (once the toll is treated as the reference) is smaller than the number of people who oppose introducing the toll (when the zero toll case is the reference). Hence, support for road pricing is lower before than after its introduction. c. If the reference shifts from a zero toll to a positive toll, introducing a toll and then removing it makes people worse off. Using the parameters calibrated for the Stockholm congestion charge, we calculated the implications of reference dependence and loss aversion for N and N .22 These give, respectively, the number of people against the toll ex ante and the number of people against removal of the toll after it is introduced. Fig. 4 summarizes the results for two levels of the toll: τ = 1 and τ = 2. The former is close to the current average toll, the latter is close to the socially optimal toll absent loss aversion. The figure is based on η = 1 throughout (many more detailed results are available on request). The results show that loss aversion raises the number of people against the toll ex ante (N ), but it reduces the number of people in favor of toll removal ex post (N ). In the absence of loss aversion (λ = 1), both numbers are equal. It also follows from Fig. 4 that the level of the toll reduces the number of people opposing introduction of the toll. The reason is, as noted before, that opposition to the toll declines when toll revenues increase, because revenues are redistributed to the population. 5. How loss aversion affects lobbying Individuals or groups who would suffer from road pricing often fiercely lobby against it. Similarly, individuals or groups who 21 Note that a corollary of previous results is that moving from a zero to a positive toll and back leaves all individuals worse off. The reason is that losses have a larger weight than gains. For a formal proof, see Appendix C. 22 Of course, the numerical results depend to some extent on M, the total number of voters assumed (see (15) and (21)). Stockholm is a city with some 90 0,0 0 0 inhabitants (Eliasson, 2014), and we assumed M = 600 (M is expressed in thousands of voters). Note that the number of people that were allowed to vote in the Stockholm referendum played a crucial role in determining the outcome.

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

39

Fig. 4. Reference dependence and voting on road tolls (N : number of people against the toll before its introduction; N : number of people in favor of removing the toll once it has been introduced).

Fig. 5. Timing of events.

would benefit from road pricing may lobby for its introduction. This section examines lobbying by special interests groups under loss aversion. We distinguish between (i) organizations that represent drivers and that, therefore, may lobby against road tolls, and (ii) organizations that may lobby for tolls (organizations protecting the environment, supporters of public transport, biking federations, etc.). A previous theoretical analysis of rent seeking under loss aversion finds that loss aversion reduces aggregate lobbying (Cornes and Hartley, 2012). However, as shown below, our model predicts that the effect of loss aversion on lobbying is ambiguous. The reason for the difference is that the standard analysis of rent seeking considers an exogenously fixed rent that will be assigned to one of the players so that, aside from spending by the loser on lobbying, a person can be no worse off than in the status quo. Not so in our model: a person who does nothing may still lose from the introduction or non-introduction of a policy.

5.1. Assumptions about lobbying Lobbying, especially by opponents of a policy, is typically undertaken only if the relevant organizations or their members believe that the probability that, absent lobbying, the policy will be adopted is sufficiently high. To make the analysis realistic, we therefore extend the setting of the basic model considered before. Specifically, we consider three periods, as depicted in Fig. 5. The initial period (period 0) has a zero toll. In period 1 consumers realize that with positive probability a toll may be adopted; they adapt their reference utility accordingly. Given this new reference, the lobbying organization decides on lobbying efforts. We discuss the different periods sequentially. Period 0: We start from a reference which has a zero toll. This reflects the idea that at some point in time tolls neither existed

nor were they seriously considered by policy makers. The initial reference utility is ud0 . Period 1: In period 1, voters recognize that with some probability a toll may be adopted. For example, this could reflect the period when policy makers and the media discuss the potential introduction of road pricing. This may be interpreted by consumers that with some probability π 0 the toll will remain zero, but with some probability (1 − π 0 ) that a toll may be imposed. Consumers adapt the reference taking into account this information. Specifically, we assume consumers treat expected reference-dependent utility as the new reference in period 1; we denote this updated reference as R1 . For example, the new reference in period 1 for a person who drove when the toll was zero and who continues to drive under the toll is

R1 =



   π 0 ud0 + (1 − π 0 ) udτ + λη udτ − ud0  τ Nτ   τ Nτ + (1 − π 0 ) +η −0 . M

(17)

M

Algebra shows that:23

 ∂ R1 = (1 − π 0 )η udτ − ud0 < 0 ∂λ  ∂ R1 d τ Nτ = u0 − udτ (1 + λη ) − ( 1 + η ) > 0. 0 M ∂π Increased loss aversion reduces reference-dependent utility in period 1. The reason is that loss aversion puts a larger weight on the decreased utility from driving (udτ < ud0 ) than on the gain from redistributed toll revenue. The effect of the perceived probability π 0 that the toll will remain zero is to raise reference-dependent utility. It will be instructive for the analysis that follows to split the new reference-dependent utility R1 , as given by (17), into its ‘road’ utility and ‘toll revenue’ component. The reason is that our specification of the gain-loss utility function (1) implies that people evaluate gains and losses differently, and drivers may be better off in one dimension but worse off in the other. We define the ‘road’ utility component in (17) as







R1 (u ) = π 0 ud0 + (1 − π 0 ) udτ + λη udτ − ud0



.

(18)

23 The second inequality is shown using the definitions of the different utilities and working out the resulting expression.

40

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

The ‘toll revenue’ component is

R1 (rev ) = (1 − π 0 )(1 + η )

 τ Nτ  M

.

(19)

Of course, R1 = R1 (u) + R1 (rev), see (17)–(19). The change in road utility between the new and the initial zero toll reference for continuing drivers is





R1 (u ) − ud0 = (1 − π 0 ) udτ − ud0 (1 + λη ) < 0.

(20)

The new reference in period 1 is necessarily worse than the initial zero toll reference, as there now is a positive probability of tolling. Surprisingly, the relation between road utility at the new reference and road utility if the toll were imposed with certainty is ambiguous; it is given by:



R1 (u ) − udτ = ud0 − udτ



 π 0 − (1 − π 0 )λη .

(21)

This implies that

R1 (u ) − udτ ≥ 0

⇔

π0 ≥

R1 (u ) − udτ < 0

⇔

π0 <

λη

1 + λη

λη

1 + λη

.

(22)

To interpret the above inequalities, note that if π 0 is close to 1, people are confident the toll will remain zero, so that their new reference utility (ignoring redistributed toll revenues) remains close to that of the zero-toll case. If the toll were introduced, people are taken by surprise and their ‘standard’ utility is well below the new reference utility. However, if π 0 is small and loss aversion substantial, drivers believe that the toll will most likely be imposed, making them worse off. Large loss aversion then implies that they adjust their reference-dependent utility in period 1 so far downward that it makes them feel worse off than were the toll introduced but there was no loss aversion: the adjusted reference is actually worse than standard utility with the toll. Lastly, note that the ‘toll revenue component’ R1 (rev) in the reference of period 1, as given by (19), may be larger or smaller than the revenue per person if the toll were imposed with certainty. This depends on the reference probability π 0 and on η, the parameter capturing how consumers value a revenue gain relative to the reference. We have

(1 − π 0 )(1 + η )

 τ Nτ  M

−

τ Nτ M

>0

⇔

π0 <

η

1+η

.

(23)

The relation between the reference probability π 0 and the parameters η, λ, as established in (22) and (23), has important implications for the analysis below. Period 2: Period 2 can see lobbying against the introduction of the toll. In principle, drivers may act individually; alternatively, they may contribute to a driver organization that coordinates lobbying activities. Under individual lobbying a free-rider problem appears whereby drivers may contribute little, because the benefits of individual efforts are largely captured by others. We therefore focus on organized lobbying by, say, a driver organization. We assume that all drivers Nτ belong to the lobbying organization. Moreover, it is assumed that the organization determines spending on lobbying L (paid out of general membership revenues; we assume revenues are sufficient to finance lobbying) so as to maximize the total expected reference-dependent utility of all members, net of lobbying costs.

Fig. 6. Three cases corresponding to the reference probability.

The horizontal axis shows the value of π 0 , or consumers’ subjective probability that the toll remains zero. In what follows, we will show that the effect of loss aversion on lobbying expenditures is ambiguous, but there is a clear pattern in terms of Fig. 6. At sufficiently low values of the reference probability π 0 that the toll will remain zero (so in the initial reference people expect that the toll is likely to be introduced), loss aversion reduces lobbying. Instead, at sufficiently high values of π 0 more loss aversion increases lobbying against the toll. Note that the analysis below focuses on the economic intuition, relegating many technical details to the appendix. Appendix D shows that the first two cases captured on Fig. 6 yield the same and quite intuitive result, viz., that more loss aversion reduces lobbying against the toll. Here we just point at the intuition. The relatively strong belief that the toll will be adopted (π 0 is small) implies that drivers’ ‘road’ utility in the new reference of period 1 is low: they adapt the reference to reflect the high probability of toll introduction. As a consequence, the new reference ‘road’ utility associated with making the trip by car in period 1 is lower than standard road utility with a zero toll (ud0 ), and it is also below standard utility if the toll were introduced λη (given by udτ ). Indeed, as long as π 0 < 1+ λη we showed above (see 22) that:

ud0 − R1 (u ) > 0, udτ − R1 (u ) > 0.

(24)

The observation that drivers do not suffer losses in road utility has clear implications for loss aversion: it implies that loss aversion (parameter λ) does not affect lobbying through its effect on road utility at all. It affects lobbying spending only through two channels, viz. through the expected redistributed toll revenue and through its effect on the cost of lobbying. Intuitively then, one expects loss aversion to reduce lobbying by the driver organization for two reasons. First, loss aversion raises the perceived marginal cost of lobbying. Second, loss aversion leads the lobbying organization to avoid the loss in redistributed toll revenues in case the toll is not introduced. This further weakens the incentives to lobby, because lobbying would raise the probability that revenue losses are incurred. As a consequence, both cases 1 and 2 on Fig. 6 imply that loss aversion reduces lobbying. The most interesting and arguably the most realistic case on λη 0 Fig. 6 is the third one, where 1+ λη < π ≤ 1: in the initial reference drivers believe that there is a reasonable chance that the toll will remain zero. We showed before that in the parameter range considered here the following holds:

ud0 − R1 (u ) > 0, udτ − R1 (u ) < 0, and

τ Nτ M

− (1 − π 0 )(1 + η )

τ Nτ M

> 0.

(25)

5.2. Loss aversion and lobbying against tolling The discussion above suggests that we should distinguish between different cases, depending on the value of the parameters λ, η, π 0 . In principle, because λ ≥ 1 and noting that 1+ληλη increases

with λ, there are three cases to consider, as illustrated on Fig. 6.

Note that there is now a loss in drivers’ utility of road use compared to the reference in period 1 (as voters believe the toll is unlikely to be imposed) if the toll were actually imposed. We assume the lobbyist maximizes the total expected reference dependent utility of all members, relative to the reference in

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

period 1, net of lobbying costs. In other words, the lobby organization maximizes:

π (L, π 0 )





Nτ n=1







ud0 + η ud0 − R1 (u )

+λη 0 − (1 − π 0 )(1 + η )

   + 1 − π (L, π 0 )

+η

 τ Nτ

Nτ n=1

τ Nτ 



 τ Nτ udτ + λη udτ − R1 (u ) + M

− (1 − π 0 )(1 + η )

M −L(1 + λη ).

τ Nτ  M

dn

(26)

π 0)

In this expression, π (L, is the probability that the toll will remain zero as perceived by opponents of the toll. We assume this probability increases with lobbying expenditures L, where ∂π (L,π 0 ) > 0 captures the effectiveness of lobbying. Moreover, we ∂L

also allow this probability to depend on the reference probability

π 0 perceived by drivers, where

∂π (L,π 0 ) > 0. Lastly, observe that ∂π0

the loss incurred from lobbying expenses is assumed to be separable from gains and losses with respect to driving and toll revenues. Note from (26) that loss aversion now affects lobbying both via the road utility component and via redistributed toll revenues. If the toll is imposed, drivers suffer a loss in road utility compared to the reference of period 1. It follows from (21) that this loss increases with loss aversion; moreover, it increases with the reference probability π 0 . If the toll remains zero, there is a revenue loss compared to the period 1 reference. However, this revenue loss decreases with the probability π 0 . These effects will be helpful below. The first-order condition for an interior optimum (L > 0) can be manipulated to yield:

∂π (L, π 0 ) τ Nτ {bM[1 + η + η (λ − 1 )(K )] ∂L M (b + c ) −(a − d − τ )(1 + η )[1 + η (λ − 1 )(1 − π 0 )]} = 1 + λη

First, for a given effectiveness of lobbying ( ∂π (∂L,Lπ ) ), we pointed out above that increasing π 0 raises the road utility loss when the toll is actually introduced, but it reduces the revenue loss were the toll to remain zero. One therefore expects that at some sufficiently high value of π 0 greater loss aversion will lead to more lobbying. Increased lobbying makes it more likely that the toll remains zero, avoiding the large loss in road utility. This more than compensates for the small increase in toll revenue losses when the toll remains zero. The argument can be formalized by reconsidering (30). Holding the effectiveness of lobbying constant, an increase in the ∂ Z . Differentiating (30) reference probability unambiguously raises ∂λ shows that this effect is 0

∂π (L, π 0 ) τ Nτ ∂L M (b + c ) ×{bMη[1 + η (2λ − 1 )] + (a − d − τ )(1 + η )η} > 0. This confirms that at relatively high probabilities π 0 we may ∂ Z > 0 so that, by (29), more loss aversion increases lobbyhave ∂λ ing:

∂L > 0. ∂λ

(28)

Second, a higher π 0 may affect the effectiveness of lobbying. This may strengthen or counteract the previous effect. For example, if both drivers and the organization that represents them believe that the probability that the toll will remain zero is very high, raising that probability even further by lobbying may become difficult. A higher π 0 then reduces the marginal effect of lobbying

Not surprisingly, it follows from (27) that an interior solution is obtained when the proposed toll is high, the effectiveness of lobbying is large, and the reference probability π 0 is small. A corner solution with zero lobbying will result if the toll is low, lobbying is ineffective in reducing the probability of toll introduction, and the perceived probability that the toll will be introduced is small anyway. We are interested in the effect of loss aversion on lobbying. Denoting the first-order condition in implicit form as Z(L, λ, π 0 ) = 0 and using the implicit function theorem, this effect is: ∂Z ∂L = − ∂λ . ∂Z ∂λ ∂L

Reconsidering the first-order condition (27) for π 0 = 1, multiplying the result by η and dividing by (1 + λη) immediately implies that the expression just given is positive. It then finally follows from (29) that lobbying against the toll increases with loss aversion. To understand this result, consider what happens when π 0 increases. This has two separate effects.

(27)

where

K = π 0 − (1 − π 0 )λη.

from (28) that K = 0, so that (30) is unambiguously negative, and (29) then implies that loss aversion reduces L. This is as expected, given what we found before for cases 1 and 2. Second, evaluating (30) at the upper boundary π 0 = 1 gives

∂ Z ∂π (L ) τ Nτ = {bMη} − η. ∂λ ∂ L M (b + c )

dn

M

41

(29)

The denominator is negative by the second-order condition. Moreover, algebra shows that the numerator equals

  ∂Z ∂π (L, π 0 ) τ Nτ  = bMη (K ) − η (λ − 1 )(1 − π 0 ) ∂λ ∂L M (b + c )  −(a − d − τ )(1 + η )η (1 − π 0 ) − η. (30) In general, the sign of (30), and hence of (29), is ambiguous. It depends on parameter values, so we will use numerical analysis below to get further insight. However, some intuition can be obtained by evaluating (30) at the lower and the upper boundary of λη the parameter range considered here. First, let π 0 = 1+ λη . It then follows that loss aversion reduces lobbying. To see this, we have

( ∂ π∂ L(∂L,ππ0 ) < 0), reducing the incentives to lobby. In sum, the effect of loss aversion on lobbying is theoretically ambiguous. Expression (30) suggests that for a wide range of parameters it will be negative when both the reference probability π 0 and the proposed toll are low. If the reference probability that the toll remains zero is high, the road utility loss of drivers – in case the toll is indeed introduced – becomes more important and loss aversion will increase lobbying. Numerical analysis illustrates our findings. We report results for optimal values of L for different values of the loss aversion parameter λ and the reference probability π 0 . As before, we use the calibrated demand and cost parameters for Stockholm. As no information on lobbying spending L was available, we did not try to calibrate the probability function based on observable information. We assume that the lobbyist against the toll has rational expectations, in the sense that π 0 is the true probability the toll will remain zero in the absence of lobbying. To reflect this idea, we specify: 2

0

π (L, π 0 ) =

π 0 − (1 − exp(L )) exp(L )

⇔

1 − π (L, π 0 ) =

1 − π0 . exp(L )

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B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

This specification implies:

Proposition 4. Loss aversion and lobbying against the introduction of a toll

π ( L = 0, π 0 ) = π 0 lim π (L, π 0 ) = 1 L→∞

∂π (L, π 0 ) > 0; ∂L

∂π (L, π 0 ) > 0; ∂π0

∂ π 2 (L, π 0 ) < 0. ∂ L∂ π 0

This specification implies that lobbying is zero if drivers’ reference probability is equal to one: it does not make sense to lobby if the toll is certain to remain zero. If the reference probability that the toll remains zero is the probability perceived by the driver organization at zero lobbying (π (L = 0, π 0 ) = π 0 ), then π (L = 0, π 0 = 1) = 1. As lobbying is costly and cannot raise this probability any further, the organization will not lobby. Lastly, if the reference probability π 0 = 0 – drivers are sure the toll will be introduced – then

π (L, π 0 ) =

exp(L ) − 1 exp(L )

⇔

1 − π (L, π 0 ) =

1 exp(L )

In that case, without lobbying the toll will certainly be introduced. We summarize the numerical results graphically (more detailed tables are available from the authors). Fig. 7 gives the relation between lobbying L and loss aversion for three different values of the reference probability π 0 (0.3, 0.7 and 0.9), and for three different toll levels τ (0.5, 1 and 2). All other parameters are fixed.24 Note that one implication of the absence of information on lobbying and the specification of the probability function used is that the values we obtained for L only have a relative interpretation. However, they do allow us to investigate in what direction lobbying changes when loss aversion increases. The results confirm our earlier discussion. Higher loss aversion may increase or reduce lobbying. On the one hand, loss aversion reduces lobbying when the reference probability of a zero toll is relatively small (see the results for π 0 = 0.3). In that case ‘road utility’ losses are limited, as drivers already incorporated the high probability of toll introduction in their updated reference in period 1. However, revenue losses are large, because people thought it quite likely that the toll would be imposed. Loss aversion then reduces lobbying; this reduces the losses of foregone redistributed revenues. Note that, if the proposed toll is low, high loss aversion may well imply zero lobbying (see panel a. of Fig. 7). On the other hand, loss aversion increases lobbying if the reference probability that the toll will remain zero is fairly large (in the numerical example, for π 0 = 0.7 and 0.9). The road utility loss when the toll is imposed is now substantial, because people were not expecting that it would be introduced and adapted their reference accordingly. To avoid this loss, the lobbyist raises lobbying activities in an effort to reduce the probability of toll introduction. Lastly, comparing lobbying for different levels of the proposed toll (compare the values of L across the three panels of Fig. 7), we observe that lobbying rises at higher tolls. This is plausible, as higher tolls raise the road utility loss if the toll is introduced, as well as the foregone toll revenues when it is not. The results therefore suggest that, depending on parameter values, loss aversion may increase lobbying or reduce it. What matters is the relative size of the reference probability (reflecting beliefs about the likelihood of toll introduction) and the degree of loss aversion (reflecting how much drivers dislike unfavorable changes in toll revenues and in road utility). We summarize our findings: 24 We find that there is no lobbying (L=0) when the toll was low and the reference probability that the toll would remain zero was very high. All parameter combinations used are consistent with the assumptions underlying Case 3, as described in the text.

a. Loss aversion may increase or decrease lobbying efforts against the introduction of road pricing. b. Loss aversion raises lobbying efforts if (i) the reference probability of toll introduction is small (ii) a high toll is proposed (iii) lobbying is effective in raising the probability that the toll remains zero. 5.3. Loss aversion and lobbying in favor of road pricing Just as opponents may lobby against road pricing, people who would benefit from its introduction – say, public transport users or bikers – may lobby in favor of such policies. The question then is whether loss aversion reduces or increases lobbying in favor of road pricing by non-drivers? Consider an environmental organization that organizes nondrivers. Assume all non-drivers in the initial reference situation are members. The organization contemplates spending S on lobbying to support road pricing, where we assume that such lobbying raises the probability that road pricing will be introduced. As before, π is the probability after lobbying that the toll will remain

zero, and now assume that ∂π (∂S,Sπ ) < 0. Again, we assume the lobbyist has sufficient funds to finance such lobbying. The non-driver maximizes expected reference-dependent utility, where the reference is the zero toll case (where members did not drive). Noting that und = und τ = 0 the problem is to 0 0



M

Max S

N0



(1 − π (S ) )



 τ Nτ M

+η

 τ Nτ

− (1 − π 0 )

M

τ Nτ  M

−S(1 + λη ) dn. The first-order condition is

 τ Nτ  ∂π (S, π 0 )  τ Nτ − + ηπ 0 = (1 + λη ). ∂S M M

(31)

If lobbying is sufficiently effective in increasing the probability that the toll will be introduced ( ∂π (∂S,Sπ ) is sufficiently large) an interior solution will result. Note that the left-hand side of (31) is the marginal benefit of lobbying in favor of tolls (extra revenues); the right-hand side is the marginal cost. Loss aversion raises the marginal cost but does not affect the marginal benefit, so that loss aversion reduces the incentive to lobby for road pricing. 0

Proposition 5. Loss aversion implies less lobbying by organizations of non-drivers in favor of the introduction of a toll. 6. Summary and conclusions This paper studied how reference-dependent preferences and loss aversion affect people’s attitudes for or against a Pigovian tax. Although the model is more generally applicable to other Pigovian taxes, we focused on road pricing to reduce congestion. We considered potential road users with reference-dependent preferences, modeled by a specific linear gain-loss utility function characterized by loss aversion. We analyzed the implications of loss aversion on socially optimal tolls, analyzed the political economy effects by looking at the effect of loss aversion on attitudes towards a road toll, and studied the effect of loss aversion on lobbying against the introduction of tolls by driver organizations that oppose the toll. We find several results. If all voters have reference-dependent preferences and treat the zero toll as the reference, the socially optimal toll is smaller than the socially optimal toll in the absence of

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

Fig. 7. (a) The effect of loss aversion on lobbying (τ = 0.5). (b) The effect of loss aversion on lobbying (τ = 1). (c) The effect of loss aversion on lobbying (τ = 2).

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B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

reference dependence, and it declines in the degree of loss aversion. The intuition is that loss aversion attenuates the optimal policy; the optimal toll remains closer to the reference and bridges only part of the gap between the status quo and the social optimum in the absence of loss aversion. Loss aversion also implies that more people will oppose the introduction of an arbitrary toll than in the standard case. Support for road pricing is lower before than after its introduction. Loss aversion may increase or reduce lobbying efforts by driver organizations against the introduction of tolling. It is more likely to raise lobbying efforts if the reference probability of toll introduction is small, a high toll is proposed, and lobbying strongly affects the probability that the toll will be zero. Lastly, loss aversion yields less lobbying by non-drivers in favor of the introduction of a toll.

Consider the problem

Max τ

SW = (1 + λη )

Nτ

n=0

 d

uτ dn − λη



N0 n=0

+(1 + η )(τ N τ ).



d

SW = (1 + λη ) −λη



Nτ

n=0



N0

n=0

uτ =0 dn

SW = (1 + λη )

b (a − d − cNτ − τ )Nτ − (Nτ )2 b 0 2 (N ) 2





+ ( 1 + η )τ N τ .

 ∂ SW  ∂ SW ∂ Nτ + = 0. ∂τ Nτ ∂ Nτ ∂τ

 ∂ SW  = −(1 + λη )N τ + (1 + η )N τ = η (1 − λ )N τ . ∂τ Nτ

=−

 1 

{τ [η (1 − λ )] b+c +(1 + λη )[a − d − (b + 2c )N τ ]} = 0.

(A1.1)

The second-order condition requires 2





Nτ

n=0





udτ (n ) dn − λη



Nτ

re f

n=0





udτ re f (n ) dn

Noting that the second integral in SW as well as the term re f are independent of τ , the social welfare function to be maximized is – up to a constant – the same as in (9). The same socially optimal toll is found as before (see (10)). The case just described seems by far the most plausible. For completeness sake, do note other possibilities. Consider, for example, the unlikely case where the reference toll exceeds the socially optimal toll, so high that drivers would benefit from the optimal toll, and that the reference toll yields higher revenues than socially optimal. Similar arguments as before then show that social welfare is now



Nτ

n=0





udτ (n ) dn − η



+ (1 + λη )τ N τ − λη (τ re f N τ

Nτ

re f

n=0 re f





udτ re f (n ) dn

).

Algebra produces then the following optimal toll:

(a − d ){b[η (1 − λ )] + c[1 + λη]} . b[1 + η (1 − 2λ )] + 2c[1 + λη]

τ =c

1 . b+c

η ( 1 − λ )N τ −



The denominator is positive by the second-order condition. Rewriting previous expression yields

Substituting rewrites the first-order condition as

dSW = dτ

Appendix B. Robustness of Proposition 1

τ=

∂ SW = (1 + λη )[a − d − (b + 2c )N τ − τ ] + (1 + η )τ . ∂ Nτ ∂τ

The traffic level is found by using the optimal toll in the definition of the number of users given above.

SW = (1 + η )

The first term is the derivative with respect to the toll, holding the traffic level constant. The second term captures the effect of the toll on social welfare via its intermediate effect on traffic levels. We further have

∂ Nτ

(A1.4)

τ re f Nτ

Note that the second term is independent of the toll. Furthermore, the first-order condition can be written as

dSW = dτ

(a − d ){b[η (1 − λ )] + c[1 + η]} . b[1 + η (2 − λ )] + 2c[1 + η]

re f

2

− λη (a − d − cN 0 )N 0 −

τ=

+ (1 + η )τ N τ − η (τ re f N τ ).

 a − bn − (d + cN 0 ) dn + (1 + η )τ N τ .



Next, multiply (A1.1) by (b + c), then note that the number of τ , and solve for the optimal road users with the toll is N τ = a−d− b+c toll under reference dependence to find

SW = (1 + λη )

{a − bn − (d + cNτ + τ )}dn

Working out the integrals yields



(A1.3)

driver utility (in essence udτ (n ) < ud re f (n )) and the toll to be inτ troduced yields more revenue than the reference toll (just as in the case of a zero toll reference), our result remains unchanged. Indeed, going through the same steps as before, social welfare in that case is



Using earlier definitions rewrite SW as



b[1 + η (2 − λ )] + 2c[1 + η] > 0.

Consider extensions of the model to allow the reference point to differ from a zero toll. First, suppose people treat a positive toll τ ref rather than a zero toll as the reference, with traffic level re f τ re f . As long as the consumer is worse off in terms of N τ = a−d− b+c

Appendix A. Socially optimal toll



This can be worked out; it implies

d SW 1 =− [η (1 − λ )] b+c dτ 2  1  ∂ Nτ + [η (1 − λ )] + (1 + λη )(b + 2c ) < 0. b+c ∂τ (A1.2)

(a − d ) (a − d )[bη (b + c )(λ − 1 )] + . (b + 2c ) (b + 2c )[b(1 + η (1 − 2λ )) + 2c(1 + λη )]

Hence, the optimal toll under reference dependence now exceeds the socially optimal toll if there was no loss aversion.25 Intuitively, loss aversion puts more weight on the revenue loss compared to the reference, resulting in a higher optimal toll. Second, the result given by (9) still holds when we reconsider the effect of loss aversion on socially optimal road pricing in a more realistic two-period setup. Suppose that in a first period consumers expect no toll and treat the zero toll case as the reference. 25 In two other cases loss aversion does not play a role in optimizing social welfare, so that the optimal toll is the one in the absence of loss aversion, as given in Section 1. This is the case when (i) the optimal toll makes drivers worse off and yields less revenue than the reference toll, or (ii) the optimal toll makes drivers better off and it yields more revenues.

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

However, assume that in period one – because politicians and the media discussed the possibility of introducing a toll in the future – they believe that with some positive probability such a toll will indeed be introduced. Denoting the probability that, after lobbying, that toll will remain zero by π 0 and that it will be positive by (1 − π 0 ), a continuing driver then adjusts the reference in period 1. The new reference is





 π 0 ud0 (n ) + (1 − π 0 ) udτ (n ) + λη udτ (n ) − ud0 (n )  τ Nτ   τ Nτ + (1 − π 0 ) +η −0 . M

M

Similar adaptations occur for people who avoid driving under the toll, and for people that did not drive initially. The new reference utility for people who drove before the toll but do not after it is imposed is



 π (n ) + (1 − π ) −λη (n )  τ Nτ   τ Nτ + (1 − π 0 ) +η −0 . 0 d u0



0

M

ud0

(1 − π )

 τ Nτ M

+η

M

−0

SW = (1 − π )(1 + λη )





.



Nτ

n=0

+ [π 0 − λη (1 − π 0 )]



 uτ (n ) dn

n=0





Appendix C. Loss aversion and voting on toll removal of an existing toll Consider voters’ attitudes towards the potential removal of an existing toll. Suppose a toll exists and people expect it to stay; they treat the tolled situation as the reference. Consider a person indexed n who is a road user at the current toll τ > 0 and who will continue to drive if the toll is removed. His reference-dependent utility when the toll is removed is then

    τ Nτ  ud0 (n ) + η ud0 (n ) − udτ (n ) + λη 0 − . M

(A3.1)

Note that standard utility now improves as compared to the reference, whereas there is a loss in toll revenues. Now calculate the difference in reference-dependent utility when removing the toll and with the toll; this is









ud0 (n ) + η ud0 (n ) − udτ (n ) + λη 0 −



− uτ ( n ) + d

τ Nτ M

τ Nτ  M

( b + c )M

 τ Nτ  M

M

= −(1 + λη )

τ Nτ M

< 0.

(A3.2)

(A3.3)

.

(A3.4)

a−d . b+c

{(1 + η )bM − (1 + λη )(a − d − τ )}.

If loss aversion is modest and a mild assumption on M holds, this expression is positive: the group of drivers under the toll that





− und τ (n ) +







ud0 (n ) + η ud0 (n ) − und τ (n ) + λη 0 −



τ Nτ  M

τ Nτ  M

.

Working out we can rewrite this as

 τ Nτ    (1 + η ) ud0 (n ) − (1 + λη ) .

(A3.5)

M

Setting this equation in n equal to zero yields the cutoff that separates voters in favor and against removal of the toll. We find the cutoff (denoted N ) as

N  = N 0 −

(1 + λη ) τ Nτ . (1 + η ) bM

(A3.6)

Algebra shows that Nτ < N” < N0 only if loss aversion is limited. In the case of much loss aversion we may have N” < Nτ < N0 . Lastly, consider the sequence of decisions (introduce the toll, then remove the toll) for a person always using the road. The first change (introduction of the toll) implies a gain equal to referencedependent utility at the positive toll but treating zero toll as the reference) minus reference-dependent utility at a zero toll (which is the reference):



.

Using the definitions given before and working out, rewrite this expression as

τ

τ Nτ

Comparing (A3.3) and (A3.4) and solving for n then immediately shows that the number of drivers when the toll is removed is

The second term is independent of the toll; noting that (1 − π 0 ) is also unaffected by the toll it is clear that maximizing social welfare with respect to the toll gives expression (10) again.





und τ (n ) +

Then calculate for this group the difference in reference dependent utility when the toll is removed and when the toll was in place:



ud0 (n ) dn

+(1 − π 0 )(1 + η )(τ N τ ).





 τ Nτ    (1 + η ) ud0 (n ) − λη .

N0 =

d

N0

M

−

These people are necessarily worse off; they lose the toll revenues. Lastly, consider a person who does not drive under the toll (the reference). Such a person may or may drive when the toll is removed. If he drives his reference-dependent utility, noting that for a non-driver und τ (n ) = 0, is given by:

−λη

Now let the social planner, as before, maximize social welfare. This can now be written as 0

τ Nτ

−λη

Alternatively, if a non-driver at the tolled reference does not drive after removal of the toll, reference-dependent utility is:

M

 τ Nτ

also drive when the toll is removed is certainly better off without the toll.26 Now look at the group that did not drive with the toll and does not drive even when the toll is removed. Compare the difference in reference-dependent utility when removing the toll with that under the toll (which is the reference):

M

The new reference for people who do not drive is 0

45







udτ (n ) + λη udτ (n ) − ud0 (n ) + (1 + η )

 τ Nτ  M





− ud0 (n ) .

The gain when moving from a positive toll to a zero one when the positive toll is the reference is similarly









ud0 (n ) + η ud0 (n ) − udτ (n ) − λη

 τ Nτ  M



− udτ (n ) +

 τ Nτ  M

26 If loss aversion is severe these people might, in principle, be worse off. In that case, all drivers under the toll would vote against removal of the toll.

.

46

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

Simplifying and taking the sum yields



  τ Nτ  η (1 − λ ) ud0 (n ) − udτ (n ) + . M

Under loss aversion this expression is negative. A similar procedure shows that the other groups are also worse off when introducing a toll and then removing it. Appendix D. The effect of loss aversion on lobbying As argued in the main body of the paper, three cases are to be considered. Here we deal with the first two cases.

Case1 : π 0 ≤

η

1+η

This case applies when drivers fear that the toll is quite likely to be imposed. In this case, we showed above (see (23) and (24)) that

ud0 − R1 (u ) > 0, udτ − R1 (u ) > 0, τ Nτ  τ Nτ  − (1 − π 0 )(1 + η ) < 0. M M

(A4.1)

The strong belief that the toll will be adopted implies that drivers’ ‘road’ utility in the new reference of period 1 is low: they adapt the reference to reflect this high probability. Given our assumptions, the problem for the lobbying organization is to choose L so as to maximize the expected referencedependent utility of its members, treating period 1 as the reference, minus lobbying costs. It maximizes:

π (L, π 0 )



Nτ

n=1







 ud0 + η ud0 − R1 (u )

+λη 0 − (1 − π 0 )(1 + η )





 + 1 − π (L, π 0 )



τ Nτ 



M

M



M

In this expression π (L, π 0 ) is the probability that the toll will remain zero; it increases with lobbying expenditures and with the reference probability. Note from the objective function that loss aversion (parameter λ) does not affect lobbying through its effect on ‘road’ utility, but only affects perceived expected redistributed toll revenue and the perceived cost of lobbying. Because the integral boundaries are independent of lobbying, the first-order condition of maximizing expression (A4.2) is

∂π (L, π 0 ) d τ Nτ (u0 − udτ )(1 + η ) − (1 + λη ) Nτ = 1 + λη. ∂L M (A4.3) Given our ‘road’ utility specifications it follows that

ud0 − udτ =

τb . b+c

(A4.5)

Here the denominator is negative by the second-order condition for a maximum. Moreover, it follows from (A4.4) that

∂ Z ∂π (L, π 0 ) −τ Nτ = [ ( a − d − τ )η ] − η < 0. ∂λ ∂L M (b + c )

∂L < 0. ∂λ For intuition, observe that for the parameter range considered in this case, drivers adjust their reference utility of road use in period 1 downward in response to the high probability that the toll will be introduced so that, whether or not the toll is actually introduced, they suffer no further losses in road utility. Loss aversion therefore only affects the losses in redistributed toll revenues and the cost of lobbying. On the one hand, loss aversion raises the marginal cost of lobbying. On the other hand, lobbying raises the probability that the toll remains zero, and the stronger is loss aversion, the larger the value attached to the losses in expected toll revenues. Hence, loss aversion reduces lobbying.

η

Case 2 :

1+η

≤ π0 ≤

λη

1 + λη

Under these assumptions, we have

τ Nτ τ Nτ ud0 − R1 (u ) > 0, udτ − R1 (u ) > 0, − (1 − π 0 )(1 + η ) > 0. M M The lobbyist therefore maximizes

Using this result, rewrite the first-order condition (A4.3) as

∂π (L, π 0 ) τ Nτ [bM (1 + η ) − (a −d− τ )(1 + λη )] = 1 + λη. ∂L M (b + c ) (A4.4) Although we will focus on interior solutions, it is useful to understand under what conditions a corner solution with zero lobbying may be optimal. Start from a situation without loss aversion (λ = 1). In that case, expression (A4.4) reduces to

∂π (L, π 0 ) τ Nτ [bM − (a − d − τ )] = 1. ∂L M (b + c )

∂Z ∂L = − ∂λ . ∂Z ∂λ ∂L

Combining this with expression (A4.5), we find that more loss aversion reduces lobbying:

dn

 τ Nτ udτ + η udτ − R1 (u ) + M n=1    τ Nτ τ τN +λη − (1 − π 0 )(1 + η ) dn − L(1 + λη ). (A4.2) Nτ

The term between brackets on the left-hand side is positive, see above. We will therefore have an interior solution, and positive lobbying expenditure, (i) if the toll is sufficiently large and/or (ii) if, evaluated at L = 0, a marginal increase in lobbying has a sufficiently large effect on the probability that the toll remains zero. Then introduce loss aversion. Expression (A4.4) suggests that even when there would be lobbying in the absence of loss aversion, a corner solution with zero lobbying is possible when loss aversion is large. Indeed, large loss aversion implies that the toll revenue loss compared to the reference is perceived as very serious. Because lobbying raises the probability that the toll remains zero, this increases the probability that this loss will materialize. Hence, under strong loss aversion a corner solution and no lobbying (L = 0) is optimal. Then turn to interior solutions. The right-hand side of (A4.4) is independent of L but increases with loss aversion; the left-hand side decreases in L (by the second-order condition), and increases in the toll but decreases in loss aversion. This suggests that more loss aversion will reduce lobbying. This is indeed the case; it can formally be shown as follows. Writing the first-order condition (A4.4) in implicit form as Z(L, λ, π 0 ) = 0 yields the effect of loss aversion on lobbying as

π (L, π 0 )





Nτ

n=1







ud0 + η ud0 − R1 (u )

+λη 0 − (1 − π 0 )(1 + η )

   + 1 − π (L, π 0 )

+η

 τ Nτ M

Nτ

n=1





τ Nτ 

dn

M

 τ Nτ udτ + η udτ − R1 (u ) + M

− (1 − π 0 )(1 + η )

τ Nτ  M



dn − L(1 + λη ).

B. De Borger, A. Glazer / Journal of Urban Economics 99 (2017) 31–47

The first-order condition for a maximum is now



τ Nτ

∂π (L, π ) (1 + η )Nτ (ud0 − udτ ) − ∂L M = 1 + λη. 0



 1 + η (λ − 1 )(1 − π 0 )

τ b reformulate as Using ud0 − udτ = b+ c

 ∂π (L, π 0 ) τ Nτ (1 + η ) {bM − (a − d − τ ) ∂L M (b + c ) ×[1 + η (λ − 1 )(1 − π 0 )]} = 1 + λη. We now find, again denoting the first-order condition Z(L, λ) = 0 and assuming an interior solution

 ∂Z ∂π (L, π 0 ) τ Nτ (1 + η ) =− η (1 − π 0 )(a − d − τ ) − η < 0. ∂λ ∂L M (b + c ) So again ∂Z ∂L = − ∂λ < 0. ∂Z ∂λ ∂L

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