# Non-linearity of Utility Function and Value of Travel Time Savings: Empirical Analysis of Inter-regional Non-business Travel Mode Choice of Japan

## Non-linearity of Utility Function and Value of Travel Time Savings: Empirical Analysis of Inter-regional Non-business Travel Mode Choice of Japan

### Authors

Hironori Kato, University of Tokyo, JP

### Description

We examine the variation of value of travel time saving over travel time. We first examine it with the comparative static method and then analyze it empirically with the data of inter-regional non-business travel mode choice of Japan.

### Abstract

1. Introduction
The willingness to pay (WTP) for travel time saving is called as the Value of Travel Time Saving (VTTS) and it is widely used for the economic evaluation of transport investment. Although the constant values are often used in practical transport planning, the constancy of the VTTS is derived simply from an assumption of the linear utility function. When using the non-linear utility function, we can derive the non-constant VTTS with respect to travel time. As for the variation of value of time over travel time, De Lapparent et al. (2002) formulate the Box-Cox Logit model and estimate the value of travel time with the empirical data in Paris. They conclude the WTP is neutral to effective travel time variation. However, the WTP derived from their model seems to be not the VTTS but the value of time as a resource (VTR). As Jiang and Morikawa (2004) present theoretically, the VTR should increase as the travel time increases under some specific conditions, but the VTTS may not increase. In addition, their empirical results could be highly influenced by the variation of wage rate with respect to travel time because they use the urban travel data. Thus we first derive the VTTS defined by DeSerpa (1971) from the discrete choice model with the non-linear utility function, and then examine the VTTS over travel time with the empirical data of inter-regional travel.

2. Derivation of the VTTS with the non-linear utility function
First, we formulate a time allocation model based on DeSerpa (1971). The model assumes an individual gains the utility from the leisure time, the travel time and the consumption of a composite good under the constraints of available time and budget with a constraint of minimum travel time. As we consider the tourism behavior on a non-work day, we assume the work time is given and fixed. Next, we derive the VTTS from the model and analyze the VTTS over travel time theoretically with the comparative static analysis. Then, we derive the indirect utility function under the condition of discrete choice of travel mode (Truong and Hensher, 1985). Finally, we approximate the utility function with the first-order, the second-order and the third-order Taylor expansions with respect to travel time and travel cost based on Blayac and Causse (2001), and derive the approximated VTTSs from them.

3. Empirical analysis on the variation of the VTTS over travel time
We use the data of the Inter-regional Travel Survey 2000 in Japan to estimate the Multinominal Logit model with the non-linear utility functions. The data contains the records of individual travel mode choice among four inter-regional travel modes: rail, air transport, automobile and bus. We made a subset of travel data by eliminating the business travel.
The results of empirical analysis show:
- If the utility function has the constant marginal utility w.r.t. income, the VTTS is estimated as 150-200 yen/minute. On the other hand, if the utility functions have the non-constant marginal utility w.r.t. income, the VTTSs are estimated as 10-50 yen/minute for automobile users, as 20-80 yen/minutes for rail users and as 50-200 yen/minute for airplane users, respectively.
- As the average wage rate in Japan is 37.6 yen/minute in 2002, the estimated VTTSs with the constant marginal utility w.r.t. income seem too high, hile the estimated VTTSs with the non-constant marginal utility seem in a quite reasonable range.
- In the cases of using the utility functions with the constant marginal utility w.r.t. income, the VTTS decreases monotonically as the travel time increases. On the other hand, in the case of using the non-linear utility functions, the VTTS of automobile users decreases monotonically as the travel time increases, while the VTTSs of other transport modes users are neutral to the travel time variation.
We discuss the reasons why the VTTS decreases as the travel time increases. First, the VTR may decrease as the travel time increases, (1) because the marginal utility w.r.t. leisure time for tourism may increase as the leisure time increases, or (2) because the individuals may relax their constraint of time budget as the travel time increases. Second, the value of time as a commodity (VTC) may decrease as the travel time increases, (1) because the marginal dis-utility with respect to travel time may decrease as the travel time increases, or (2) because the travelers may shift their choice on travel service from with higher marginal dis-utility w.r.t. travel time to lower marginal dis-utility w.r.t. travel time. In addition, we discuss further reasons why the VTTS over travel time of automobile is different from that of other transport modes.

4. Conclusions
We have examined the variation of the VTTS over travel time with the empirical data of inter-regional travel mode choice in Japan. We present the possibility of decrease of the VTTS over the travel time and discuss the possible reasons for it.

REFERENCES
Blayac, T. and Causse, A. (2001) Value of travel time: a theoretical legitimization of some nonlinear representative utility in discrete choice models, Transportation Research, Vol.35B, pp.391-400.
De Lapparent, M., de Palma, A. And Fontan, C. (2002) Non-linearities in the valuation of time estimates, Proceedings of European Transport Conference (CD-ROM).
De Serpa, A.C. (1971) A Theory of The Economics of Time, The Economic Journal, Vol.81, No.324, pp.828-846.
Jiang, M. and Morikawa, T. (2004) Theoretical analysis on the variation of value of travel time savings, Transportation Research Part A, Vol.38, pp.551-571.
Truong, T. P. and Hensher ,D. A. (1985) Measurement of travel times values and opportunity cost from a discrete-choice model, The Economic Journal, Vol. 95, pp.438-451.

#### Publisher

Association for European Transport