Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5192968, 11 pages

https://doi.org/10.1155/2017/5192968

## Optimal Fare, Vacancy Rate, and Subsidies under Log-Linear Demand with the Consideration of Externalities for a Cruising Taxi Market

Department of Tourism, Aletheia University, New Taipei City 25103, Taiwan

Correspondence should be addressed to Chun-Hsiao Chu

Received 25 June 2016; Accepted 5 December 2016; Published 15 January 2017

Academic Editor: Chaudry M. Khalique

Copyright © 2017 Chun-Hsiao Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Externality is an important issue for formulating the regulation policy of a taxi market. However, this issue is rarely taken into account in the current policy-making process, and it has not been adequately explored in prior research. This study extends the model proposed by Chang and Chu in 2009 with the aim of exploring the effect of externality on the optimization of the regulation policy of a cruising taxi market. A closed-form solution for optimizing the fare, vacancy rate, and subsidy of the market is derived. The results show that when the externality of taxi trips is taken into consideration, the optimal vacancy rate should be lower and the subsidy should be higher than they are under current conditions where externality is not considered. The results of the sensitivity analysis on the occupied and vacant distance indicate that the relation of the vacant distance to the marginal external cost is more sensitive than the occupied distance. The result of the sensitivity analysis on the subsidy shows the existence of a negative relationship between the marginal external cost and the optimal subsidy.

#### 1. Introduction

The taxi is a mode of paratransit in the urban public transit system. It is a popular mode of transportation in urban areas because of its embedded features of convenience, speediness, privacy, comfort, long hours of operation, and door-to-door delivery with no need to pay parking fees. If the fare of a taxi service is reasonable and the quality is good enough, using the service of taxis is attractive to private vehicle users. This may also result in a reduction in the external cost incurred by using private vehicles (e.g., environmental pollution, congestion, and fuel consumption) and enhance the efficiency and safety of the transportation system in urban areas.

A cruising taxi means a taxi that is continually driving about looking for business. When a client is found, the taxi is defined as having business. Although intelligent dispatching is now being used in the taxi business, the drivers of vacant taxis still cruise the streets to find business opportunities. Cruising taxis always exist and are perhaps the most basic mode of operation in the taxi industry.

Due to the characteristics of taxi operation, taxi trips have both positive and negative external effects. When a taxi is occupied by passengers, it may create positive external effect because it reduces the use of private vehicles and the external cost incurred. However, when a cruising taxi carries no passengers, it may cause the same negative external effects as a private vehicle. Externality in the taxi market is an important issue which is worth noting and being researched. However, it is often overlooked by authorities and researchers alike.

As a kind of public transportation system, the taxi business is often regulated and supervised by public authorities in most cities around the world. The regulation policies include quantity (considering the vacant and occupied distance in the market), fare, and subsidy/taxation. Ideally, the externality of the taxi business should be taken into account when the regulation policy is being discussed by the authorities because, if not considered, the most effective allocation of public resources may not be achieved.

In the studies related to taxi regulation, the log-linear demand function has often been specified to describe the market demand of taxi trips because this specification has the advantages of allowing the demand elasticity to be obtained directly from the formula and its parameters are very easily calibrated. However, the common restriction of the conventional log-linear demand function is that the price elasticity has to be restricted to be less than −1. Hence, Chang and Chu [1] introduced the concept of “maximum social willingness-to-pay (MSWTP)” to deal with the restriction. However, in their model, the concept of externality, which is an important but often overlooked by authorities and researchers issue, was not taken into account.

In this study an extended model based on Chang and Chu’s one is proposed and solved to analyze the effect of externality on the optimization of the fare, vacancy rate, and subsidy of the taxi cruising market in the first-best environment where taxi trips need to be subsidized [2].

#### 2. Literature Review

The taxi industry is highly regulated. The regulation policies include quantity, fare, and subsidy/taxation. Hence, the optimization of vacancy rate, fare, and subsidies is an important issue in the policy-making process.

In 1972, Douglas [3] proposed an abstract model to explore the effect of different fares and characteristics of equilibrium for the cruising taxi market. In Douglas’s study, demand is a function of fare and service quality, which is evaluated by the users’ waiting time. Ghahraman et al. [4] discussed a time- and space-based fare structure for the taxi industry. Ardekani et al. [5] suggested that certain factors need to be considered in the establishment of fare structure such as fuel cost, driver salary, operation costs, and reasonable profit. Chang and Tu [6] proposed a time-based fare scheme that took 12 cost factors into account in the cost-sharing procedure. They also divided the 12 cost factors into 3 different types, namely, fixed cost, variable cost, and mixed cost, in accordance with their correlations to travel time and distance. Chang and Sun [7] proposed a concept for a flexible initial fare and suggested that the authorities set a range so that the taxi operators can flexibly adjust the initial fare to reflect changing demands. To realize the concept of the flexible initial fare, Chang and Chu [8] further developed a discrete dynamic control model and solved the problem of optimal control rate of the initial fare to match with the changing demand and maintain the vacancy rate on a predetermined level.

Some have suggested that the taxi industry should be subsidized. Teal and Berglund [9] argued that subsidizing a taxi-pool may be an alternative for the taxi industry in the US. Arnott [2] suggested that taxi travel should be subsidized and that the subsidy should cover the shadow cost of taxi idle time in the first-best environment. Frankena and Pautler [10] reviewed the regulation policies for four taxicab market segments (namely, cruising cabs, cabs that wait at stands, radio-dispatched cabs, and cabs that provide contract services) and discussed market imperfections including externalities of congestion and pollution that might occur in the taxicab service market. Their study showed that some imperfections indeed provide theoretical rationale for regulation of the cruising taxicab market. However, they also admit that economists have not developed a model of the cruising cab segment that determines a unique equilibrium fare and service combination.

To regulate the number of taxis in the market is also an important area of concern. Some studies have focused on this subject area. Beesley and Glaister [11] considered that the taxi demand is a function of fare and waiting time and discussed the variation of taxi demand and waiting time when the taxi fare is changed. Schaller [12] studied the elasticity of price and the service availability by using econometric methods. Wong and Yang [13], Yang and Wong [14], Wong et al. [15], and Yang et al. [16] developed a series of taxi optimization models using mathematical programming methods based on the O-D matrix and the network of the service area. By materializing Douglas’ research, Chang and Huang [17] proposed an optimal vacancy rate and fare model to maximize the social welfare for a cruising taxi market with a break-even constraint. However, under Chang and Huang’s model, the consumers’ surplus would not exist when the price elasticity is greater than −1. To deal with the problem of inexistence of the consumer’s surplus in Chang and Huang’s model, Chang and Chu [1] further introduced the concept of “maximum social willingness-to-pay (MSWTP)” and obtained the optimum of fare, vacancy rate, and subsidy for cases in which the absolute value of price elasticity is less than 1.

Based on the above, it can be found that the issue of externality of the vacancy rate, fare, and subsidy of the taxi industry has rarely been presented in the prior research related to the optimization of taxi regulation. As a type of public transportation, the taxi business could cause positive and/or negative external effects on social welfare. This study thus aims at exploring the effect of externality on the optimization of vacancy rate, fare, and subsidy of the taxi industry and attempts to develop a model accordingly.

#### 3. Analytical Model

An analytical model based on the model proposed by Chang and Chu in 2009 is discussed in this section for the optimization of the fare, vacancy rate, and subsidy with the consideration of externality.

A review of previous research is carried out first. The externality function of the cruising taxi industry derived in this study will also be discussed. We then derive and solve the optimization problem for the fare, vacancy rate, and subsidy with the consideration of externality.

##### 3.1. Models in Previous Studies

In 1972, Douglas presented an abstract model for a cruising taxi market and gave an example with a log-linear delay (waiting time) function.

In Douglas’ study, the demand was assumed to be a function of the fare and waiting time. The demand of the taxi market can be written aswhere is the total demand of the taxi market, is the taxi fare, is the waiting time of the taxi, and .

Further, the waiting time is a function of the density of vacant taxis and can be expressed aswhere is the density of vacant taxis.

Regarding the supply, Douglas assumed the supply of taxicabs and drivers to be perfectly elastic, given as a cost per unit taxi service, . It is also assumed that the cost of operating cruising taxicabs is independent of the division of operation time between “vacant” and “occupied.” Hence, the total cost can be given byUnder these settings, equilibrium occurs at or

By materializing Douglas’ research, Chang and Huang [17] proposed an optimum vacancy rate and fare model that maximized the social welfare for a cruising market with the break-even constraint. Chang and Huang’s study follows the setting of cost structure and the relationship between the density of vacant taxis and the waiting time in Douglas’ research and specified a log-linear function as the demand function.

In Chang and Huang’s study, the demand function was expressed aswhere is daily occupied mileage in the market (km/day), is total daily vacancy mileage (km/day),* P* is fare rate of taxis (dollars/taxi-day),* w* is waiting time of taxi passengers, is price elasticity of taxi demand, is vacant mileage elasticity of waiting time, is waiting time elasticity of taxi demand, is constant term of the demand function, and is constant term of the waiting time function.

Given Chang and Huang’s settings, the consumer surplus () is yielded as follows:

Equation (10) shows that the consumers’ surplus does not exist when the price elasticity () lies within the closed interval of under the log-linear demand since is an improper integral and is its improper point.

To deal with the problem of the nonexistent consumer surplus, Chang and Chu further introduced the concept of the “maximum social willingness-to-pay” [1]. By the introduction of the “maximum social willingness-to-pay,” the improper point from the CS could be shifted to become a negative point. The concept of the “maximum social willingness-to-pay” can be briefly explained as follows.

Assume that is a demand function that satisfies

Equation (11) shows that is a conventional log-linear demand function whose price elasticity is and (12) means that is a divergent improper integral with an improper point .

Equation (12) is an improper integral since the maximum willingness-to-pay is not considered in (11). Hence, it is assumed that there exists a maximum social willingness-to-pay () so that if and that the shape of the demand is fixed. Geometrically, the introducing of the maximum social willingness-to-pay means the demand curve shifts to the left, say , as shown in Figure 1.