Road pricing in London attracts a great deal of interest. A challenging aspect of the London scheme is that congestion tax revenue is used to upgrade public transit networks. Although Parry and Bento (2001) show that the total social surplus would increase if congestion tax revenues are used to cut labor taxes, political difficulties exist in implementing revenue-recycling between congestion taxes and labor taxes. Given such political difficulties, the London scheme seems to be very attractive. In this paper, we develop a model that can expressly deal with the London-type revenue-recycling between congestion tax and investment in public transit, and analyze its effects.congestion tax

Yukihiro Kidokoro

English

Research Papers in Economics

London−type congestion tax with revenue−recycling 
Yukihiro Kidokoro
Center of Spatial Information Science, University of Tokyo
Abstract
Road pricing in London attracts a great deal of interest. A challenging aspect of the London
scheme is that congestion tax revenue is used to upgrade public transit networks. Although
Parry and Bento (2001) show that the total social surplus would increase if congestion tax
revenues are used to cut labor taxes, political difficulties exist in implementing
revenue−recycling between congestion taxes and labor taxes. Given such political
difficulties, the London scheme seems to be very attractive. In this paper, we develop a
model that can expressly deal with the London−type revenue−recycling between congestion
tax and investment in public transit, and analyze its effects.
I thank Kurt Van Dender for helpful suggestions and discussions. The first draft of this paper was written while I was visiting
the Department of Economics at the University of California, Irvine, and I thank the faculty members and staff there for their
hospitality. I remain solely responsible for any remaining errors and omissions.
Citation: Kidokoro, Yukihiro, (2005) "London−type congestion tax with revenue−recycling." Economics Bulletin, Vol. 18, No.
1 pp. 1−6
Submitted: March 9, 2005.  Accepted: June 16, 2005.
URL: http://www.economicsbulletin.com/2005/volume18/EB−05R40001A.pdf  1
1. Introduction 
 
Road pricing in London attracts a great deal of interest.    A challenging aspect of the London 
scheme is that revenue from congestion tax is used to upgrade public transit networks.  
Recently, Parry and Bento (2001) showed that the total social surplus would increase if 
revenue from congestion tax is used to cut labor taxes.  However, political difficulties exist 
in implementing the scheme advanced by Parry and Bento (2001).  First, a congestion tax 
would usually be a “local” tax, which is collected at the regional level, such as a city, 
although labor taxes would be a “national” tax, collected at a national level.    Such an actual 
hierarchical system of tax collection would make it difficult to implement the 
revenue-recycling scheme between congestion taxes and labor taxes.  Second, even if the 
authorities solve the tax administration problem, the issue of residents’ support for a 
congestion tax remains.  In order to get support from residents who pay congestion taxes, 
the congestion tax scheme must be understood by them.  In this regard, it would be much 
easier to use congestion tax revenue to improve the public transit system than to use the 
revenue to decrease labor taxes. 
Considering such practical difficulties to implement revenue-recycling between 
congestion taxes and labor taxes, the London scheme seems to be very attractive.    However 
no simple economic model currently exists to analyze the London congestion tax scheme.  
Thus, in this paper we develop a model that can deal explicitly with the London-type 
congestion tax with revenue-recycling and analyze its effects. 
    The rest of the paper is organized as follows.  In Section 2, we build a model.  In 
Section 3, we analyze the effects of the London-type congestion tax scheme with 
revenue-recycling on both transport demand and the social surplus to derive the optimal 
congestion tax.  In Section 4, we conclude present analysis and consider possible future 
extensions. 
 
 
2. The Model 
 
Consider the following situation.  A representative consumer who lives in the surrounding 
area to the CBD demands the composite consumer good  z, and the transport services to the 
CBD, 
1 x  and 
2 x .  
1 x  is road transport for which congestion tax is levied, and 
2 x  is 
public transit that receives congestion tax revenue in the form of additional investment.  
This setup would be realistic provided that the transport authority wants to divert traffic from 
road transport to public transit due to a lower social marginal cost of the latter.  However, 
our argument is applicable if 
1 x  and 
2 x  are substitutable roads in a road network and the 
transport authority finances the investment costs of 
2 x  using the toll revenues from 
1 x .  
The essential point is that 
1 x  is taxed and 
2 x  receives the tax revenue in the form of 
investment.  In this paper, both 
1 x  and 
2 x  are congestion-prone.  For the sake of 
simplicity, traffic from the CBD to the surrounding area of the CBD, that is the reverse 
direction of traffic, is ignored. 
The utility function of a representative consumer is:   
12 (,) Uz u x x =+ ,                ( 1 )  
which is assumed to be strictly concave.    The quasi-linear utility function of (1) implies that 
income effects are ignored.  This simplifying assumption would be justified if transport’s 
expenditure share is low.  The price of the composite consumer good,  z, is normalized to 
unity.  The generalized prices of 
1 x  and 
2 x , which include time costs, are 
1 p  and 
2 p .    2
The budget constraint is:   
11 22 yzp x p x =+ +                 ( 2 )  
where  y   is full income.    In this paper, labor hours are assumed to be fixed.    This implies 
that labor market distortions are ignored. 
  A  representative  consumer  maximizes  his  or her utility, (1), subject to the budget 
constraint,  (2).  The  maximization  yields the following first order conditions:   
1
1
x up = ,                   ( 3 )  
2
2
x up = ,                   ( 4 )  
where the subscripts denote partial derivatives throughout the paper.  From (3) and (4), we 
derive the demand functions of 
112 (,) x pp and 
212 (,) x pp, which satisfy 
12
12
21xx
dx dx
u
dp dp
== − .                ( 5 )  
In this paper, 
1 x  and 
2 x   are assumed to be substitutes and consequently, 
12
12
21 0
xx
dx dx
u
dp dp
== −> , which implies  12 0
xx u < .  For  our  purpose, 
1 x  and 
2 x  need  not  be 
perfect substitutes for which Wardrop (1952) equilibrium applies. 
    The authorities (the city of London or Tokyo, for instance) impose congestion tax, τ , 
for road transport, 
1 x .    The price of road transport is then 
11 1 () p cx τ =+ ,                  ( 6 )  
where 
11 () cx  is the generalized price of road transport net of congestion tax and 
11 ()0 cx ′ > . 
    The price of public transit (or the subsidized transport service), 
2 x , is 
22 2 (, ) p cxI = ,                 ( 7 )  
where 
22 (, ) cxI  is the generalized price of public transit and  I  is monetized investment.  
We assume that the congestion of public transit causes delays, that is,  2
2 0
x c ≥  and 
investment relieves the delays to lower its generalized costs, that is, 
2 0 I c < .  We assume 
there is no possibility of a congestion tax for public transit. 
    Congestion tax revenue is denoted by  R , which is assumed to be used as the investment 
for public transit,  I .    Thus, the following equation holds 
1 I Rx τ == .                 ( 8 )  
 
 
3. Analysis 
 
3.1  The effects of the congestion tax on transport with revenue-recycling 
 
In Subsection 3-1, we examine the effects of congestion tax, τ , on transport demand, 
1 x  
and 
2 x  with  revenue-recycling.  Totally  differentiating and rearranging (3), (4), and (6)-(8), 
we obtain 
1() x τ  and 
2() x τ , which satisfies: 
{} 22 2 12
1
1 221 () 0 I xx x xx
dx
Du c c u x
dτ
− =− − <             ( 9 )    3
{ } 12 11
2
1 21 1 () I xx xx
dx
Duc uc x
d
τ
τ
− ⎡ ⎤ ′ =− + −+ ⎢ ⎥ ⎣ ⎦
         ( 1 0 )  
where 
11 22 12 12
11 22 12 22 2 2 11 12
11 2
21 2 2 2
() () ( )
() ( ) 0
I xx xx xx xx
I xx xx xx xx x x xx xx
Du c u c uu c
uu u cu c c u c u
τ
τ
′′ =− − − −
′ =− − − − + >
 
from the strict concavity of utility function, 
1 0 c ′ > ,  2
2 0
x c > , 
2 0 I c < , and  12 0
xx u < . 
    We begin by looking at 
1 dx
dτ
.  From (9), we know that a change in τ  yields two 
effects.  Congestion tax for road transport makes its generalized price relatively higher than 
that for public transit.  The demand for road transport decreases by this effect, which is 
represented by the term of  22 2
1 2 () 0
xx x Du c
− − <  in (9).  This effect is caused by the 
congestion tax on 
1 x  itself; the effect exists whether a revenue-recycling mechanism is 
adopted or not.  A revenue-recycling scheme increases public transit investment, which 
lowers public transit’s generalized price.  This effect also works to make the generalized 
price of road transport relatively higher and consequently, the demand for road transport is 
further decreased.  This effect through revenue-recycling is represented by the term of 
12
1 21 () 0 I xx Dc u x
− −<  in (9).  Since both effects work to decrease road transport, we know 
that a congestion tax with revenue-recycling unambiguously decreases road transport 
demand.   
Next, we look at 
2 dx
dτ
.    Congestion tax has an effect to increase public transit demand, 
because it makes the generalized price of public transit relatively lower than that of road 
transport.  This effect is represented by the term of  12
1() 0
xx Du
− − >  in (10).  This effect 
stems from the congestion tax on 
1 x  itself and exists whether a revenue-recycling 
mechanism is adopted or not.  The first effect of revenue-recycling is represented by 
{ } 11
1 21 1 () 0 I xx Dc u c x
− ′ −>  in (10).  Revenue-recycling increases public transit investment 
and lowers its generalized price.  Thus, public transit demand is increased by this effect.  
However, the second effect of revenue-recycling, 
1 2 () 0 I Dc τ
− <  in (10), makes the sign of 
2 dx
dτ
 ambiguous.  An increase in congestion tax decreases the demand for road transport, 
and reduces the revenues from congestion tax ceteris paribus.  This second effect of 
revenue-recycling decreases investment in public transit and raises its generalized price.  
Thus, this second effect works to counteract the first effect of revenue-recycling.  Whether 
the first effect outweighs the second effect or not depends on the size of τ  and 
11
11 ()
xx cux ′ − .  Usually the second effect would be minor and the first effect would be 
stronger than the second; the net effect of revenue recycling works to increase public transit 
demand.  In this case, the overall effect of an increase in the congestion tax on 
2 x  is 
positive, that is, 
2
0
dx
dτ
> . 
 
3.2  The effects of the congestion tax on the social surplus with revenue-recycling   4
 
In Subsection 3-2, we examine the effects of the congestion tax,  τ , with revenue-recycling 
on the social surplus.  From (1), (2), and (6)-(8), the total social surplus, SS , can be 
formulated as 
11 1 1 1 22 2 2 2
12 111 22 12 1
() (, )
(,) () (, ) .
SS U p x c x x p x c x I x I
yu x x c xx cx xx x τ τ
=+ − + − −
=+ − − −
      ( 1 1 )  
Totally differentiating (11) with respect to  τ  yields 
2
1 111 22 2 2 2
11 1 22 22 2
(( ) )
() () ( 1 ) ,
I x
I
dSS
cxxx cx x c x I I
d
p MC x p MC x c x I
ττ τ τ
τ ττ
τ
τ
′ =− − − −
=− +− + − −
         ( 1 2 )  
where 
111 1 MCcc x ′ ≡+  and  2
222 2
x MCcc x ≡+ . 
    The first term of (12), 
11 1 () p MC xτ − , represents a change in the social surplus that 
congestion tax causes through a change in the demand for road transport.  If roads are 
heavily congested and, consequently, road transport’s generalized price is lower than its 
marginal cost, this term is positive from (9).  The second term of (12), 
22 2 () p MC xτ −  
shows a change in the social surplus that a congestion tax produces via a change in the 
demand for public transit.  If public transit is not heavily congested and consequently, its 
generalized price is higher than its marginal cost and 
2 0 xτ > , this term is also positive.  
(Remember that 
2 0 xτ >  holds true in usual cases where the net effect of revenue recycling 
works to increase public transit demand.)  The first and second terms of (12) imply that an 
increase in congestion tax is welfare-improving as long as road transport congestion is heavy 
and public transit congestion is light.   
    The third term of (12), 
22 (1 ) I cx I τ −−, represents the effect caused by revenue-recycling.   
We call this effect the “revenue-recycling effect” after literature on “double dividend” of 
environmental taxes.  Denoting the elasticity of road transport demand with respect to 
congestion tax by 
1
1
x
x
ττ
ε ≡− ,  Iτ  can be rewritten as 
1(1 ) Ix τ ε = − .  Thus, we obtain 
0 Iτ >  if  1 ε < .    It would be reasonable to assume  1 ε <  throughout  the  paper.  Thus,  the 
revenue-recycling effect is positive and works to increase the social surplus as long as 
22 1 I cx −> , which means that the social benefit from investment in public transit outweighs its 
investment cost.  Obviously, revenue-recycling will not always work to increase the social 
surplus; for example, if 
22 1 I cx −< , revenue-recycling works to lower the social surplus.  
This is because a lump-sum transfer is socially preferred if the social benefit from investment 
in public transit is smaller than its investment cost.   
 
 
3.3  Optimal congestion tax with revenue-recycling 
 
Setting (12) equal to zero, the optimal congestion tax is derived as 
2
22 2
11 2 2
11
(1 ) I
x
x cx I
cx cx
xx
τ τ
ττ
τ
−− ′ =+ − .             ( 1 2 )  
    The first term of the right-hand side of (12) is the marginal congestion cost.  If the 
model was the simple first-best model, the optimal congestion tax would be equal to the 
marginal congestion cost.    Under our setup, however, there are two additional effects which   5
change the first-best result. 
    The second term of the right-hand side of (12) stems from non-optimal pricing for public 
transit, which is caused by the assumption that congestion tax is not levied on public transit.  
A congestion tax for road transport must be set to take into account the subsequent effects on 
public transit congestion.  We hereafter call the second term of the right-hand side of (15) 
“congestion spill-over effect” after Verhoef et al. (1996).  A congestion spill-over effect 
arises because the congestion tax on a certain part of the transport network transfers the 
congestion to another part of the transport network.  In our normal context where a 
congestion tax increases the demand for public transit, where 
2 0 xτ > , the congestion 
spill-over effect is negative and works to lower the optimal congestion tax.   
    The third term of the right-hand side of (12) is the effect of revenue-recycling on 
congestion.  If 
22 1 I cx −> , revenue-recycling works to increase the optimal congestion tax.  
In such a case, higher congestion tax than the first-best level is socially justified, because 
transfer via revenue recycling is socially desirable as compared with lump-sum transfer. 
    The relative magnitude of any congestion spill-over effect and revenue-recycling effect 
determines whether the optimal congestion tax is larger than the marginal congestion cost or 
not.    Two special cases are useful to obtain some insight. 
    First, in the case where public transit is not congested, that is,  2
2 0
x c = , (12) is reduced to 
22
11 1 1 1
1
(1 )
()
I cx I
cx c x x
x
τ
τ
τ
−− ′′ =− >  if 
22 1 I cx − > .       ( 1 3 )  
If there is no congestion spill-over effect, the optimal congestion tax is higher than the 
marginal congestion cost, provided revenue-recycling is socially preferable to lump-sum 
transfer. 
    Second, if there is no revenue-recycling effect, investment in public transit,  I , is fixed 
and accordingly, 
2 0 I cI τ == .    In this case, (12) is reduced to 
2
2
11 2 2 11
1 x
x
cx cx cx
x
τ
τ
τ ′ ′ =+ <,               ( 1 4 )  
(Note also that 
2 0 xτ >  unambiguously holds in the case of 
2 0 I c =  from (10)).  (14) is 
basically the same as (14) of Van Dender (2004) and demonstrates that the optimal 
congestion tax is lower than the marginal congestion cost by the congestion spill-over effect.   
If  11 22 12 xx xx xx uuu ==, that is, the demand for road transport and that for public transit are 
perfect substitutes, rearranging (14) using (9) and (10) yields 
 
11
2
21 1
11 2 2
2
xx
x
xx x
u
cx cx
cu
τ
⎛⎞ − ′ =−⎜⎟ ⎜⎟ − ⎝⎠
,             ( 1 5 )  
which corresponds to (2) of Verhoef et al. (1996).   
 
 
4. Conclusion 
 
In this paper, we analyze a congestion tax scheme with revenue recycling between road 
transport and public transit.    The main results are as follows: 
(i)    the demand for road transport, for which congestion tax is levied, decreases; 
(ii)   the demand for public transit, which receives revenues in the form of additional 
investment, is increased in usual cases where the net effect of revenue recycling works to   6
increase the demand for public transit; 
(iii) a congestion tax with revenue-recycling works to increase the total social surplus if the 
social benefits from investment in public transit exceed its investment cost; and 
(iv)   revenue-recycling works to raise the optimal congestion tax if the social benefit from 
investment in public transit is higher than its investment cost. 
    Our analysis has been conducted under the simplest possible model.  We would need 
various extensions to apply our analysis to an actual situation.  For instance, 
revenue-recycling in this paper is somewhat extreme in that all congestion tax revenue is 
applied to public transit in the form of additional investment.  In reality, some of this 
revenue would be used to invest in road transport on which a congestion tax is imposed.  
How to determine the appropriate intensity of revenue-recycling, with a given level of 
congestion tax, also remains unsolved.  These lines of research are deferred to future 
studies. 
 
References 
 
Parry, I. and A. Bento (2001) “Revenue recycling and the welfare effects of road pricing” 
Scandinavian Journal of Economics 99, 261-279. 
Parry, I. and K. Small (2002) “Does Britain or the United States have the right gasoline tax?” 
Discussion Paper 02-12, Resources for the Future. 
Van Dender K. (2004) “Pricing transport networks with fixed residential location” Regional 
Science and Urban Economics 34 289-307. 
Verhoef, E., P. Nijkamp and P. Rietveld (1996) “Second-best congestion pricing: the case of 
an untolled alternative” Journal of Urban Economics 40, 279-302. 
Wardrop, J. G. (1952) “Some Theoretical Aspects of Road Traffic Research” Proceedings of 
the Institute of Civil Engineers, Part II-1, 325-378. 
 

London-type congestion tax with revenue-recycling

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