The effect of complex models of externalities on estimated optimal tolls

Abstract

Transport externalities such as costs of emissions and accidents are increasingly being used within appraisal and optimisation frameworks alongside the more traditional congestion analysis to set optimal transport policies. Models of externalities and costs of externalities may be implemented by a simple constant cost per vehicle-km approach or by more complex flow and speed dependent approaches. This paper investigates the impact of using both simple and more complex models of CO₂ emissions and cost of accidents on the optimal toll for car use and upon resulting welfare levels. The approach adopted is to use a single link model with a technical approach to the representation of the speed-flow relationship as this reflects common modelling practice. It is shown that using a more complex model of CO₂ emitted increases the optimal toll significantly compared to using a fixed cost approach while reducing CO₂ emitted only marginally. A number of accident models are used and the impact on tolls is shown to depend upon the assumptions made. Where speed effects are included in the accident model, accident costs can increase compared to the no toll equilibrium and so tolls should in this case be reduced compared to the congestion optimal toll. Finally it is shown that the effect of adding variable CO₂ emission models along with a fixed cost per vehicle-km for accidents can increase the optimal toll by 44% while increasing the true welfare gained by only 8%. The results clearly demonstrate that model assumptions for externalities can have a significant impact on the resulting policies and in the case of accidents the policies can be reversed.

Appendix A: Derivation of constants b and b 1 from the MARS model

In MARS the speed-flow relationship used is that by Singh (1999) and is of the following form:

$$ V=\frac{V_f }{1+\gamma (DF)^{\alpha }} $$

(A1)

where V is speed in km/h; V _f is free-flow speed (km/h); α, γ are parameters taken as 4 and 0.15 respectively (Based on data from 1964 Highway Capacity Manual); DF is the demand factor.

The demand factor is defined further as follows:

$$ DF=DF_0 \left( {\frac{T}{T_0 }} \right) $$

(A2)

where DF ₀ is the demand factor in the base case and T and T ₀ are the demand in trips in the do-something and base case respectively. The initial demand factor DF ₀ is calibrated to the initial average speed (35 km/h) and the free-flow speed (50 km/h) as follows:

$$ DF_0 =\root{\alpha}\of{\frac{V_f -V}{\gamma V}}=\root{4}\of{\frac{50-35}{0.15(35)}}=1.3 $$

(A3)

Thus we can say that average speed V is related to trips as follows:

$$ V=\frac{V_f}{1+\gamma (DF)^{\alpha}}=\frac{V_f}{1+\gamma \left( {DF_0 \left( {\frac{T}{T_0 }} \right)} \right)^{\alpha }}=\frac{1}{\frac{1}{V_f }+b_1 T^{4}} $$

(A4)

Thus we have:

$$ \frac{dV}{dT}=\frac{-4b_1 T^{3}}{\left( {\frac{1}{V_f }+b_1 T^{4}} \right)^{2}}=-4b_1 T^{3}V^{2} $$

(A5)

As supply curves are more traditionally written as costs which increase with flow we can invert (A1) to give the time per/km, t, as:

$$ t=\frac{1+\gamma (DF)^{\alpha }}{V_f }=\frac{1}{V_f }+b_1 T^{4} $$

(A6)

where b ₁ is a constant

$$ b_1 =\frac{\gamma (DF_0 )^{\alpha }}{V_f T_0^\alpha }=\frac{0.15(1.3)^{4}}{50(1212414)^{4}}=3.9654E^{-27} $$

(A7)

The constant b used in Eq. 1 in the main body of the paper is calculated by converting from time per km in hours to time per average trip in minutes by multiplying by 60 times the average trip length i.e.

b = 60*14.46*3.9654E⁻²⁷ = 3.4404E⁻²⁴