The Inefficiency of Marginal cost pricing on roads


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1 The Inefficiency of Marginal cost pricing on roas Sofia GrahnVoornevel Sweish National Roa an Transport Research Institute VTI CTS Working Paper 4:6 stract The economic principle of roa pricing is that a roa toll shoul eual the marginal cost impose y an aitional user, since this will lea to efficient use of the transport facility. However, when the roa is use y traffic oth from the roa proviing region as well as y traffic from another region, the supplie roa stanar is likely to e too low, since the consumer surplus of the users from outsie the region is not taken into account. This can e solve y letting an authority level higher than the roa supplier use taxes an earmarke transactions to raise the roa stanar. (In Europe we see this one in the Trans European Network). To o this the higher authority nees very etaile information aout the roa an the users on local level. Further raising taxes an transactions also involve costs, that can e sustantial. nother prolem is that transactions of this type it is har to separate from other political interferance. This paper analyzes how a limite toll on top of the marginal cost can serve the purpose of of solving this prolem locally, without involving a higher authority. Keywors: Marginal cost pricing, congestion, roa uality JEL Coes: R 4, R 48 Centre for Transport Stuies SE 44 Stockholm Sween
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3 The ine ciency of Marginal cost pricing on roas So a GrahnVoornevel Sweish National Roa an Transport Research Institute Center for Transport Stuies, Stockholm, Sween P.O. ox Stockholm, 5 Sween so Septemer, 4 stract The economic principle of roa pricing is that a roa toll shoul eual the marginal cost impose y an aitional user, since this will lea to e cient use of the transport facility. However, when the roa is use y tra c oth from the roa proviing region as well as y tra c from another region, the supplie roa stanar is likely to e to low, since the consumer surplus of the users from outsie the region is not taken into account. This can e solve y letting an authority level higher than the roa supplier use taxes an earmarke transactions to raise the roa stanar. (In Europe we see this one in the Trans European Network). To o this the higher authority nees very etaile information aout the roa an the users on local level. Further raising taxes an transactions also involve costs, that can e sustantial. nother prolem is that transactions of this type is har to separate from other poitical interferance. This paper analyzes how a limite toll on top of the marginal cost can serve the purpose of of solving this prolem locally, without involving a higher authority. Introuction Roa pricing is a growing issue in transport research. In Europe roa pricing has evelope along two i erent lines; tolls to control congestion in uran areas (for instance Stockholm an Lonon) an marginal cost pricing in interuran areas (so far solely focuse on heavy goos vehicles (HVF) introuce in Switzerlan, ustria, Germany an others) The main part of the roa pricing literature has focuse on the possiility to use tolls in orer to control roa congestion (for a short overview see Linsey ).
4 For tolls not use to control congestion the economic principle is that a toll shoul eual the marginal cost impose y an aitional user, since this will lea to e cient use of the transport facility in uestion. This is also the pricing policy that is spreaing in Europe for interuran roas. However the situation is not as trivial as it souns. Quite often a roa is not only use y tra c from the region supplying an nancing the roa, ut also from other regions. The roa supplying region will therefore not consier the welfare of all users when eciing uality an capacity of the roa. Further, the users from outsie the roa proviing region can not contriute to increase uality an capacity even if this woul e in line with their preferences. nother prolem with roa tolls is the sustantial pulic opposition. lthough when it comes to tolls in orer to control congestion, experience show that the opposition uite often change to support after the tolling reform is introuce. Naturally this implies that there is a prolem introucing roa tolls, since politicians have to make ecisions against the will of the majority (see for instance Schae an aum (7) an De orger an Proost ). The reason for a marginal cost policy even when there are users from outsie the region trivial. The existing roa will e e ciently use, an the region proviing the roa oes not have to carry any costs for users from outsie the region, nor can they exploit this tra c. However, this is precisely the cause of a welfare loss, since this will lea to a stanar of the roa that is not e cient consiering all tra c. This is not a new thought. The solution has een that a ecision level higher than the region, proviing the roa, give earmarke nancing to this region in orer to raise uality. This is for instance the case with the Trans European Network (TEN). However, this means that this higher ecision level must have very etaile information of all roas an their use. Further it means that political ecisions such as supporting a certain region, via infrastructure investments, is har to separate from the transaction to ajust roa stanar with respect to tra c from outsie the region. Further it is uestionale if the etour via a higher authority level is e cient. There is a large risk that a signi cant chunk of the resources are lost in this process. The EU nticorruption Report (4) claim that corruption costs the European Economy illions per year. This is an implication that it might e a goo iea not to involve more levels than is strictly necessary in hanling transactions, an to make these transactions as transparent as possile, oth in orer to save resources ut also to increase acceptance for a region to contriute to a facility their memers use in another region. The purpose of this paper is to investigate how this coul e one y aing a limite toll on top of the marginal cost. When a toll is ae on top of the marginal cost, this means that the welfare function, of the region proviing the roa, inirectly will take the consumer surplus of users from the other region into account via the tollincome. This woul e a more transparent way of hanling the prolem. Further the apparatus to collect such a toll alreay exists ue to the marginal cost pricing.
5 However, since tra c from i erent regions are not necessarily willing to pay eually much for raising roa stanar, the toll has to e set so that all tra c ene t. Therefore it is esirale that the principle for how to restrict tolls an the use of tollincome is mae on a higher authority level that take an interest in the welfare of all involve regions. The uestions that nees to e answere is therefore what restrictions nee to e set for the size of the toll an the use of the toll income. simple moel with two regions; region an region is use. Region provies a roa which is use y tra c from oth an. The two regions in the moel can e interprete as for instance communities or nations. This is a very general setting applicale to various situations, such as ajacent communities with a commuting population, or nations with highways with international tra c. The paper analyzes an compare three i erent toll systems. Marginal cost pricing a toll on top on marginal cost pricing, where toll incomes can e use freely a toll on top of marginal cost pricing, where toll incomes are use to increase uality an capacity In the rst part of the paper it is assume that there is no congestion. The roa supplying region sets the uality. In the secon part uality is assume to e xe. The roa su ers from congestion, an region choose capacity. Policy implications are given in the last section. Moel with no congestion, region ecies uality In this rst version of the moel, it is assume that there is no congestion. Region is proviing a roa use y tra c from oth region an region. The variale to ecie for region is with what uality to provie the roa. The tra c volumes are enote y the strictly positive functions x an x. The inverse eman functions are given y p i a i i x i where the coe cients a i an i are strictly positive real numers. The reason to chose an inverse linear eman function is to simplify the further analysis. 3
6 . Usage charge with marginal cost The supplier of the roa, region, is allowe to charge the users for their marginal cost of their usage. The marginal cost is assume either to e xe, or to epen on the uality with which the roa is provie. The generalize user cost functions are given y g i i + i for i f; g where i an i are assume to e strictly positive real numers. If the marginal cost is assume to e xe, this cost is inclue in i : If the marginal cost is epenent on the uality, we assume that it is inverse proportional to the uality, in which case it is inclue in i : In euilirium the generalize prices euals the generalize user cost, thus a i i x i i + i From euation () the tra c volumes x an x can e etermine. x i i a i i i for i f; g () The welfare of region, proviing the roa, consists of the consumer surplus for users from this region R x p (x) x; minus the user cost x i g i ; minus the costs for proviing the roa with the chosen uality. For simplicity let the cost of proviing the roa with uality e. The welfare of region is now given y () W Z x p (x) x x g x Rememer that p i g i in euilirium. The welfare of region consists of the consumer surplus, for users from this region, minus the user cost. Thus Z x W p (x) x x x g (3) Region will choose the uality in orer to maximize its welfare. Therefore, to n the welfaremaximizing uality, the welfare function of region nees to e analyze. 4
7 W x ( ) a a (4) The rst orer conition gives that the welfare optimizing toll satis es W a a a 3 3 a + (5) This is a cuic function, an it is well known that such a function has three roots of which either two or no root is complex. Even though we therefore know that this function has at least one real root it is not possile to express this solution, in the general case, without using complex numers. This expression will therefore e of no help in the further analysis of this paper. However it is possile to etermine some properties of the welfare maximizing root. y assumption x ; x ; W ; W ; an are strictly positive. Inirectly this means that we have assume that there is a positive root yieling x ; W > : Lemma ssume that there is a > such that maximizes the function W x, with x a >. Then is the largest root of the cuic function 3 a ) + ( : Proof. The rst orer conition for a maximum of W is given y euation (5) as) as aove, i.e. the cuic function 3 a ) + ( : Thus the three roots to this cuic function are the extreme points of the function W : It is trivial to see that this function is continuos everywhere except in the point where the function has an incontinuity such that lim W () 5
8 , an lim + W () : Further the function has two asymptotes; lim W () + a an lim W () + a : Since lim an lim W () + a it follows from continuity that there is at least one local minimum where <. Since lim +, it is trivial to see that W is positivefor values close to zero. Further gives W an x a a a ; meaning that the function W cross the xaxis at least once etween an : Since we have assume that there is a > with x > it follows a from continuity that the function W has a local minimum in the interval ;. From this we can euce that the that maximizes the function W is larger than the values of the other two local extremepoints, thus is the largest vaue satisfying the rst orer conition given y the cuic function, meaning that is the largest root of this function. Even though we o not have the exact expression for the uality, the result aove is enough to make it possile to o a comparison etween the e ects of marginal cost pricing an a toll on top of marginal cost pricing.. Usage charge with a toll on top of marginal cost Now assume that region, that provies the roa, is allowe to charge a xe toll > on top of the marginal cost: Region oes not have to invest the toll incomes in the roa. This means that the general user cost function is given y g i i + i + In euilirium the generalize user cost euals the generalize price; i + i + ai The tra c ows can now e expresse y: x i i a i i i i x i for i f; g For simplicity lets enote a a The welfare of region is given y the consumer surplus minus the user cost plus the toll incomes minus the roa provier costs. 6
9 Z x W p (x) x x g + x + x x + x + x + ( + + ) + ( + + ( + ) ) + ( + ) + + ( + ) + ( + ) + ( ) + (6) The welfare of region is given y the consumer surplus minus the user cost Z x W p (x) x x x g (7) The roa proviing region sets the uality to maximize its welfare. The rst orer conition for the welfare maximizing is given y erivating euation (6). W (8) Lets multiply (8) y 3 7
10 (9) Let e the uality maximizing W : Looking at function W an its conition for extreme points given y the cuic function (9), it is easy to see that we can use the same reasoning as in Lemma. Thus t is the largest root to euation (9). Theorem The uality will increase when is allowe to charge a toll on top of the marginal cost: Proof. In orer to show that > lets compare the rst orer conitions for maximizing the welfare of region with an without the toll (euation (5) an (9) respectively): y assumption an are larger than zero, an we know from Lemma that these are the largest roots to euation (5) an (9) respectively. The i erence etween euation (5) an (9) is a positive term. From this it follows that satisfying euation (5) has to e larger than satisfying euation (9). Hence > : The welfare of region can e written as W W + (x ) + ( ) + When the welfare function of region is written on this form, it is trivial to see that in the case of only local tra c (x ), a toll will reuce the welfare of 8
11 , i.e. the region is etter o if the local government pays all costs for the roa rather than the travellers contriuting to these costs. However, when there is tra c from region also contriuting to the costs for the roa the situation is i erent. It is trivial to see that when the tra c ow from is large enough, will e etter o with a toll then without. The next uestion to ask is therefore whether it can e ene cial for oth region an region to introuce a toll. Theorem 3 When the uality satis es the euation 3 3 < it is possile to set a toll ; resulting in a new uality ; such that oth an ene ts from introucing this toll even though can use the pro t free. Proof. From euation (9) 3 a ) + ( we can see how epens on the toll. Lets erivate this expression with respect to the toll ; while viewing the uality as a function of the toll, i.e. () : 3 " + " 3 " # ## ( ) () Since > we know that > in the point ; this means that 3 ( ) > : Further, since > we can euce that 3 ( ) >. Hence > everywhere in the interval ( ; ). 9
12 Now lets erivate euation () with respect to. 3 ( ) 3 ( ) 4 3 ( ) ( ) 6 Since 3 3 > 3 ( ) > it is ovious that < in the interval ( ; ) : This means that even if the uality increase when the toll is raise, the proportion etween the raise in toll an the uality change ecomes less an less ene cial for the larger the toll is. For this reason it is enough to check if it is possile that a small toll is ene cial for (since if this is not the case neither will a large change). It is trivial to see from the welfare function of region with an without toll (euation (3) an (7)) W W x x that ene ts from the toll if x > x : Thus > > () Using the implicit function theorem we can now erivate the expression () in the point ;with respect to while viewing as a function of. > > This means that region ene ts from the toll if the introuction of the toll results in a raise in that is larger than : Note that in the point :
13 Now using euation () we have that > 3 ( ) > 3 3 < () Thus if () hols, region ene ts from the toll. We now nee to check criteria for when ene ts from a small toll, i.e. when, i.e. when lim : This means that we want to check when W > in the point : Therefore lets erivate W W W + W W lim " with respect to W () > + + # " + # Now lets use that : W () " #
14 From euation (5) we know that 3 Since. W () ) + ( : Thus > > is always true a small toll will always e ene cial for Hence oth region an will ene t from a toll satisfying 3 < ( ) ) + ( This shows that a small toll on top of the marginal cost will always increase the uality, even though region oes not have to invest the toll incomes in the roa.. Further, if uality is enough important for region (a large means that the users from region are very sensitive to uality changes) such a toll will e ene cial for oth regions..3 Usage charge with marginal cost plus a toll use to increase uality Lets now consier a situation where region is allowe to charge a toll ; ut has to use the toll incomes to increase the uality from (the uality in the case of marginal cost pricing) to the uality + x + x. Further for simplicity lets assume that x x x: Thus the inverse eman function is given y : p a x an the generalize user cost is given y g + + x + In euilirium the generalize price euals the generalize user cost. x + x + x + x The welfare of region an are given y the generalize user cost is now given y g + x +
15 Thus in euilirium we have that Which gives the tra c ows a x + x + x( + ) x + ( ) + Theorem 4 If the toll < toll. oth region an region will ene t from the Proof. It is trivial to see from the welfare functions that oth region an region will ene t from the toll if x > x : 3
16 Lets assume that x > x : T hus x > x + x > + x > + + x < + + x < + + x < + < < + + < + < < < < ( ) (3) From euation (5) we have that the welfare maximizing uality in the case of only marginal cost pricing satis es 3 + ( ) (4) 4
17 When inserting (4) in (3) aove we have that < ( ) ( ) < ( ) ( ) < ( ) < Thus if the toll is smaller than it is true that x > x ; an oth region an ene t from introucing the toll. Oviously uality will increase when this toll is introuce. If the toll is not to large oth regions will ene t from this toll reform. 3 Moel with congestion, region ecies capacity In this version of the moel we assume that the roa in uestion is congeste. The variale for the roa proviing region is the capacity. Quality is assume to e xe. The tra c volumes are enote y the strictly positive functions x an x. ing congestion makes the moel very much more complicate to analyze. Therefore lets make the simpli cation that the tra c ows are eual, thus x x x. The inverse eman functions are given y p a x where the coe cients a an are strictly positive real numers. The roa is assume to e congeste. Further the proviing region is allowe to charge the users for their marginal cost. The congestion cost is not inclue in the marginal cost since it is not a cost that falls upon the proviing region ut on the users. The congestion cost is inverse proportional to the capacity. 3. Usage charge with marginal cost The generalize user cost function is given y the value of time times the inverse of the capacity per vehicle. g + x 5
18 For simplicity lets enote a In euilirium the generalize prices euals the generalize user cost, thus a x + x x( + ) x + The welfare of region (proviing the roa) consists of the consumer surplus of the users from region R x p (x) x; minus the user cost xg; minus the costs for proviing the roa with the chosen capacity. For simplicity let the cost of proviing the roa with capacity e. The welfare of region is therefore given y W Z x p (x) x xg an the welfare of region is analogously given y W x (5) Z x p (x) x xg x (6) The proviing region will choose the capacity in orer to maximize the welfare of the region. The rst orer conition for a maximum is given y W x W + ( + ) ( + ) ( + ) 3 ( + ) (7) We can now reason analogously to lemma an euce that the capacity that maximizes the welfare W is the largest root to euation (7). Lets enote this root y : 6
19 3. usage charge with marginal cost plus toll with free use Now assume that region is allowe to charge a toll : The generalize user cost is now given y g + x + Thus in euilirium we have that a x + x + x( + ) Which gives the tra c ows x + ( ) + Region will set the capacity in orer to maximize its welfare W + x. The rst orer conition for a maximum is given y x W x + x W x x ( + ) + x ( + ) + + ( + ) ( + ) ( ) ( + ) ( + ) ( + ) ( 4 ( ) ( + ) ) (8) Reasoning as in lemma we can euce that the maximum welfare is given y the largest root to euation (8). 7
20 Theorem 5 The capacity ecie y region will ecrease when is allowe to charge a toll Proof. In orer to show that < lets compare the rst orer conitions for maximizing the welfare of region with an without the toll :(euation (7) an (8) respectively) ( 4 ( ) ( + ) ) Since ( ) is positive we know that the expression on the right sie of euation (8) is negative. y assumption an are larger than zero, an we know from Lemma that these are the largest roots to euation (7) an (8) respectively. The i erence etween euation (7) an (8) a negative term. From this it follows that has to e smaller than. Hence < : This result might seem counter intuitive. However this result can e explaine y the fact that a toll will increase the cost for travelling also means that the eman for travelling is reuce thus reucing the nee for capacity. This result also makes it clear that region will never ene t from such a toll. For region to ene t from the toll it woul have to e true that ( ) + x > x + > + > > + ut since we know that < this can never e the case. 3.. toll use to increase capacity Now lets assume that region is allowe to charge a toll on top of the marginal cost given that the income from the toll are use to increase the capacity. The generalize user cost is now given y g + x + x + Thus in euilirium we have that a x + x + x + x( + + x ) 8
21 Which gives that x toll + +x The welfare function of an are given y W Z xtoll x toll p (x) x x toll g W Z xtoll p (x) x x toll g x toll Since the welfare of an without toll is given y euation (5) an (6) respectively W Z x x p (x) x x g W Z xl p (x) x x g x it is trivial to see that oth region an region ene ts from the toll if x toll > x : Theorem 6 If the toll < 4 (+) (+) oth region an ene ts from a toll given that the toll incomes are use to increase the capacity Proof. We want to show the criteria for when x toll > x : Thus + +x toll ( ) + > + > + + x toll > + x toll Since we have assume that x toll > x we know that if > + x > + x toll 9
22 > + x > + + ( ) + + > > + 4 > 4 4 > ( + ) > ( + ) 4 ( + ) ( + ) > For this to e possile we nee to check that 4 euation (7) we can euce that ( + ) > : From ( + ) 3 Lets multiply 4 ( + ) > with ( + ) thus Since this is always true we have that < 4 ( + ) ( + ) 3 4 ( + ) ( + ) ( + ) > > 4 Policy implications For tolls not use to control congestion the economic principle is that a toll shoul eual the marginal cost impose y an aitional user, since this will lea to e cient use of the transport facility in uestion. This is also the pricing policy that is spreaing in Europe for interuran roas. Quite often a roa is not only use y the region supplying an nancing the roa, ut also y tra c from other regions. The region supplying the roa,
23 will not consier the welfare of users from other regions, when eciing uality an capacity of the roa. The solution has een that a ecision level, higher than the region proviing the roa, give earmarke nancing to this region in orer to raise uality. However, this means that this higher ecision level must have very etaile information of all roas an their use. Further it means that political ecisions such as supporting a certain region, via infrastructure investments, is har to separate from the ajustment of uality an capacity ue to users from outsie the region proviing the roa. Further the EU nticorruption Report (4) claim that corruption costs the European Economy illions per year. This is an implication that it might e a goo iea not to involve more levels than is strictly necessary in hanling transactions, an to make these transactions as transparent as possile, in orer to save resources, ut also to make regions more willing to accept paying for a facility their memers use in a another region. more transparent way to arrange this is to use a toll on top of the marginal cost, even if transactions is more e cient in a moel without transaction costs an corruption. Further the apparatus to collect such a toll alreay exist in orer to hanle the marginal cost pricing. The purpose of this paper is to investigate how this coul e one y aing a limite toll on top of the marginal cost. When a toll is ae on top of the marginal cost, this means that the welfare function, of the region proviing the roa, inirectly will take the consumer surplus of users from the other region into account via the tollincome. However, since tra c from i erent regions are not necessarily willing to pay eually much for raising roa stanar, the toll has to e set so that all tra c ene t. Therefore it is esirale that the principle for how to restrict tolls an the use of tollincome is mae on a higher authority level that take an interest in the welfare of all involve regions. This paper has use a simple moel with two regions, an, in orer to analyze how such a toll a ects uality, capacity an welfare levels, an what restrictions nee to e set for the toll an the use of the tollincome In the case of a noncongeste roa a toll top of the marginal cost will always lea to a raise in uality even if no restriction is put on how to use the tollincome. Moreover, if uality is very important to the users, such a toll can e ene cial for oth regions. However, in general the use of toll incomes nee to e restricte to investments of the roa, in orer for oth regions to ene t from the toll. Further, since the tra c eman for he two regions are likely to e i erent the toll also nee to e restricte in orer to make certain that oth regions really ene t from paying the toll to increase uality or capacity. Such an upper level of the toll is given in the paper. The policy implication of the paper is that a correctly set toll, on top of the marginal cost, can serve the purpose of ajusting the roa stanar with respect to users from outsie the region. This is a simple way to avoi involving
24 a higher authority. There is an important uestions that this paper oes not answer, where further research is neee. Is it cost e cient to nanse the iscusse raise in roa stanar with a toll rather than with taxes an transactions? The author ns it likely that the transaction costs an lack of transparency y involving a higher ecision level motivates elegating the ajustments of the roa stanar y tolling. However to answer this uestion real ata woul nee to e analyze. 5 References De orger. an Proost S. (): " political economy moel of roa pricing", Journal of Uran Economics, Vol 7, pp EU nticorruption Report nal. (4): russels 3..4 COM (4)38  Linsey R. (): " Roa pricing an investment", Economics of Transportation, Vol, pp Schae J., an aum M. (7): "Reactance or acceptance? Reactance against the introuction of roa pricing", Transportation Research Part, Vol 4, pp The analyzis of the paper was complicate since it eals with the general case. However, when solving this kin of euations for speci c cases there are very straight forwar algorithms to e use.
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