Mode-valued differences of in-vehicle travel time Savings

Abstract

The value of in-vehicle travel time savings (VT) estimated from mode-choice models has been sometimes found to be higher for private car than for public transportation. This mode-valued variation may seem paradoxical, if public transportation (especially the bus) is perceived as less pleasant than the private car, and because mode-valued differences in the VT cannot be attributed to self-selection. This article describes two alternative microeconomic explanations for this empirical finding. The first follows from noticing that the marginal consumption of goods may depend on travel time, but differently for each mode. A marginal reduction in travel time induces marginal savings in the consumption of goods like fuel or oil, but those marginal savings are perceived by the user only when conditioning on the use of the car. This effect can be explicitly accounted with the inclusion of technical constraints relating goods consumption and time assignment in the microeconomic framework of the VT. The second explanation follows from noticing that the activity schedule does not need to be the same conditional on the use of each mode. Since the car is usually faster and more accessible, a schedule constructed conditional on the use of the car could be more complex, justifying higher values of time as a resource for that mode. The article finishes illustrating the proposed explanations with an example and then summarizing the contributions of this research and proposing lines for further investigation.

Appendix: derivation of Eqs. (11) and (12)

Deriving the lagrangian $\ell$ in Eq. (10) with respect to the leisure activities “being at home” and “being at the coffee shop”

$$ \frac{\partial \ell }{{\partial T_{Home} }} = \alpha_{H} \rho T_{Home}^{\rho - 1} - \mu = 0 \Rightarrow \alpha_{H} \rho T_{Home}^{\rho - 1} = \mu $$

$$ \frac{\partial \ell }{{\partial T_{CS} }} = \alpha_{CS} \rho T_{CS}^{\rho - 1} - \mu = 0 \Rightarrow \alpha_{CS} \rho T_{CS}^{\rho - 1} = \mu $$

and thus

$$ \alpha_{CS} \rho T_{CS}^{\rho - 1} = \alpha_{H} \rho T_{Home}^{\rho - 1} \Rightarrow T_{CS}^{\rho - 1} = \frac{{\alpha_{H} }}{{\alpha_{CS} }}T_{Home}^{\rho - 1} \Rightarrow T_{CS}^{{}} = \left( {\frac{{\alpha_{H} }}{{\alpha_{CS} }}} \right)^{{\frac{1}{\rho - 1}}} T_{{Home^{{}} }}^{{}} . $$

Then, considering that $T_{TravelC} = t_{IV}$ and that $T_{Home}^{{}} + T_{CS}^{{}} + T_{TravelC} = \tau - \bar{\tau }$, it results that $\tau - \bar{\tau } - t_{IV} = T_{{Home^{{}} }}^{{}} + \left( {\frac{{\alpha_{H} }}{{\alpha_{CS} }}} \right)^{{\frac{1}{\rho - 1}}} T_{{Home^{{}} }}^{{}}$, arriving at Eq. (11)

$$ T_{Home} = \frac{{\tau - \bar{\tau } - t_{IV} }}{{1 + \left( {\frac{{\alpha_{H} }}{{\alpha_{CS} }}} \right)^{{\frac{1}{\rho - 1}}} }};\;T_{CS} = \frac{{\tau - \bar{\tau } - t_{IV} }}{{1 + \left( {\frac{{\alpha_{CS} }}{{\alpha_{H} }}} \right)^{{\frac{1}{\rho - 1}}} }} $$

from which $T_{Home}$ and $T_{CS}$ can be calculated given that $\tau - \bar{\tau },t_{IV} ,\alpha_{CS} ,\alpha_{H} ,\rho$ are known.

Regarding, Eq. (12) first, as it was shown before, $\mu = \alpha_{H} \rho T_{Home}^{\rho - 1} \;$ is obtained by deriving $\ell$ with respect to T _Home . $\frac{\partial U}{{\partial T_{TravelC} }} = \alpha_{TC} \rho T_{TravelC}^{\rho - 1} \;$ is obtained deriving the utility by $T_{TravelC}$. $F = \gamma T_{TravelC} \;$ is assumed to hold. $G = I - \bar{G} - P_{F} F - c_{IV} - c_{CS} \;\quad$ is obtained re-arranging terms of the budget constraint. $\psi = \lambda P_{F} - \alpha_{F} \rho F_{{}}^{\rho - 1} \;$ is obtained by deriving $\ell$ with respect to F. $\lambda = \alpha_{G} \rho G^{\rho - 1} \;$ is obtained by deriving $\ell$ with respect to G. Finally, $\kappa = \mu - \frac{\partial U}{{\partial T_{TravelC} }} + \gamma \psi \;\quad$ is obtained deriving $\ell$ with respect to T _TravelC .