Value of Time and Elasticity of Portuguese Freeway Users: Insights from Analysis of Survey Data

Abstract

Value of time (VOT) is a crucial aspect of travel demand modeling. VOT impacts most mobility projects and the evaluations therein. It has been noted to be influenced by multiple factors, mainly related to individuals’ demographics and trips’ characteristics. This paper presents the results of a survey conducted among users of the Portuguese national freeway network, aiming to derive insights about their travel choice mechanism for testing mobility projects, in particular dynamic pricing strategies. Particular attention is dedicated to the value they attribute to travel time by analyzing willingness-to-pay for avoiding congestion and saving travel time. An elaborate questionnaire survey was distributed through online survey campaigns between March and June of 2021, eliciting 163 valid responses. Even after a stratification process, results revealed that VOT is generally lower than our original expectations; various statistical distributions were tested to fit the empirical data, the best performing ones were selected and the results were compared with a previous survey-based VOT study. We finally measured the elasticity of the freeway demand and of the whole demand for transportation, which confirmed the generally low willingness to pay for less congested travels.

1. Introduction

VOT’s analysis is considered useful for several reasons: it is a fundamental component of travel demand modeling, it explains some aspects of human behavior that are of interest in economics, and it is basic for decision making in transportation policy [1]. VOT has been observed to depend on a number of heterogeneous factors, with variations depending upon trip-related aspects, such as trip purpose, time of day and congestion levels, and individuals’ demographics, as income, gender, and family status, among others [2]. According to a recent literature review of model-based dynamic toll pricing [3], logit models constitute the most popular method to represent drivers’ choices between transportation alternatives in dynamic toll pricing studies. The ratio between cost and time parameters in logit models can be interpreted as a representative VOT value. However, logit models do not explicitly address VOT heterogeneity in the population [4] and they attribute differences in drivers’ choices to an error associated with the perception of the utility of an alternative instead of attributing it to the variability in their VOTs [5]. In [6], two main approaches to model VOT heterogeneity are described: a discrete set of VOTs associated with several distinct user classes or a continuously distributed VOT. The former method is adopted, for instance, in [7], where a discrete distribution of users’ VOT is employed, while various continuous distribution functions are considered in the literature for drivers’ VOT: a log-normal distribution is proposed in [8]; Weibull distribution is chosen for fitting in [9]; Ref. [4] adopts a simplified variant of the Burr distribution, and [10] assumes a Gaussian drivers’ VOT probability distribution.

The distribution of VOT can be estimated either based on drivers’ stated preference survey data or by observing traffic demand changes (for example, through vehicles counts) concerning the posted toll amount. However, the latter approach is feasible only when a dynamic toll pricing strategy is implemented [8]. For a more complete review of the literature on driver behavior models for model-based dynamic toll pricing, we refer the reader to [3]. Our objective is to develop a driver behavior model based on a VOT distribution that reflects the choice mechanisms of the Portuguese national freeway network users to be used for testing dynamic pricing strategies.

Currently, only flat tolls are implemented on the Portuguese freeways; most of the tolls are differentiated by vehicle class and distance-based, with the value depending on the entrance and exit point of the vehicle’s path on the freeway, while fewer tolling schemes are characterized by toll prices depending only on the vehicle class and on the tollbooth location, not being based on an origin–destination path. All the toll prices of the freeways managed by Brisa, the main freeway manager, can be consulted in [11]. An option to pay for unlimited journeys within a 3-day period, called 3 Day Virtual Card, is available only to foreign registered vehicles and just to circulate in freeways with exclusively electronic toll systems; moreover, the card cannot be purchased more than 6 times each year per each number plate [12]. The focus of this study is on Portuguese national freeway network users, so we decided to discard the influence of this unlimited trips option, which is available only to foreign occasional visitors.

Since there are no dynamic tolls implementations in Portugal, we resort to the stated preference survey method to estimate the VOT distribution of the local drivers. This paper presents some insights from the results of a survey we have been conducting during the first months of 2021 among users of the Portuguese national freeway network, aiming to derive a VOT distribution and transport demand elasticity. In Section 2, we describe the survey structure and we briefly outline the dissemination activity; in Section 3, we detail the stratification process we implemented after recognizing that the collected sample was not reflective of the actual Portuguese population; in Section 4, we refer to the results of the exploratory data analysis we conducted on the sample after stratification; in Section 5, various statistical distributions are tested for fitting of the empirical data, and we select the best performing ones; in Section 6, we compare our results with a previous survey-based VOT study; in Section 7, we analyze the elasticity of the freeway demand and of the whole demand for transportation; finally, Section 8 concludes the paper.

2. Survey Structure and Dissemination

A questionnaire survey was designed so as to determine drivers’ preferences for certain factors related to travel time and travel cost. In particular, the intent of the survey was to capture the drivers’ perceived sense of time and decision mechanisms through 19 questions in total (Q1 to Q19). The first part of the survey (Q1 through Q6) solicited the respondent’s demographics (gender, age, education, job, weekly hours of work, monthly income). The second part (Q7 through Q11) explored the respondent’s mobility habits (residency location, work location, commuter trip duration, possession of driver’s license). The third part (Q12 through Q17) determined the user pattern and experience on the Portuguese national freeway network. Finally, the last group of questions (Q18 and Q19) was designed to assess the respondent’s willingness to pay various fee levels across different potential time savings and to estimate elasticity to toll pricing. A complete list of the topics investigated by each question is available in

Table 1. Topics of the survey questions.

Part	Question	Topic
Part 1: demographics	Q1	gender
	Q2	age
	Q3	education
	Q4	job
	Q5	weekly hours of work
	Q6	monthly income
Part 2: mobility habits	Q7	municipality of residency
	Q8	freguesia (parish) of residency
	Q9	municipality of work/study
	Q10	commuter trip duration
	Q11	possession of driver’s license
Part 3: freeway experience	Q12	commuting on freeways
	Q13	frequency of use of freeways
	Q14	toll of typical trip
	Q15	duration of typical trip
	Q16	presence of congestion in typical trip
	Q17	quantification of congestion in typical trip
Part 4: VOT and elasticity	Q18	VOT
	Q19	elasticity

. Some questions were asked with the aim of including the information in the following analysis, but at the end some of the collected information resulted to be superfluous or too weak and the analyses were limited to the most significant data.

We made the sample size calculations based on a preliminary hypothesis on the functional form of the VOT distribution. Specifically, a Gaussian distribution was assumed, with probability density function $f\left(\xi \right)$, with $\xi$ indicating VOT, as in Equation (1):

$$f\left(\xi \right)=\frac{1}{\sigma ·\sqrt{2·\pi }}·\exp \left[-\frac{1}{2}·{\left(\frac{\xi -\mu }{\sigma }\right)}^2\right]$$

(1)

For the mean of the distribution, $\mu$, we adopt a monetary actualization to 2020 of the VOT indicated for Portugal for car commuting trips in an urban congested environment [13]. We performed the actualization through an interactive online application provided by the Portuguese Statistics National Institute (Instituto Nacional de Estatística, IP–Portugal, henceforth referred to as INE-IP) based on the historical evolution of the Consumer Price Index (CPI).

$$\mu =7.83€$$

(2)

To determine the standard distribution, we examine studies related to VOT. Several studies indicate that the VOT of commute trips is on average approximately half of the wage rate [9]. To build the Gaussian distribution, we hypothesize that the 90th percentile of the distribution (corresponding to a standard Z-score of 1.282) is equal to the average hourly wage rate. Ref. [14] indicated for Portugal an average hourly labor cost of 15.7€, with 19.8% of non-wage costs for year 2020. Thus, our estimation of average hourly wage in Portugal for the year 2020, called $W_{2020}$, is:

$$W_{2020}=0.802·15.7=12.59€$$

(3)

Considering the symmetry of the Gaussian distribution and the correspondence of the average VOT with a value greater than half of the estimated hourly wage rate, our hypothesis also allows us to build a distribution that associates just to a relatively small group of users a negative VOT. This hypothesis allows us to compute the standard deviation to give to the distribution:

$$\sigma =\frac{W_{2020}-\mu }{1.282}=3.71€$$

(4)

Our preliminary hypothesis on the functional form of the VOT distribution is thus defined by Equation (1), with parameters determined in Equations (2) and (4). It is now possible to obtain an estimate of the target sample size ($N_{target}$) after fixing the desired confidence level and an acceptable level of error for the sample mean, with the following formula:

$$N_{target}=\frac{Z^2·{\sigma }^2}{E^2}$$

(5)

where Z is the standard Z-score associated to the desired level of confidence and E is the acceptable margin of error, which determines a confidence interval that is $2E$ units in width. Therefore, with the desired confidence level of 90% (corresponding to $Z=1.64$) and an acceptable level of error of 0.50€, the target sample size was 150 units.

We administered the survey exclusively online and disseminated it via e-mail to user lists at Universidade de Lisboa and the authors’ local personal network. At the same time, the link was advertised via social media posting on Facebook, Twitter and Instagram by Instituto Superior Técnico. At the end of the dissemination period we also tried Facebook paid advertisements without success. Dissemination started on 8 March 2021 and responses were collected until 30 June 2021.

The total number of responses reached 277. However, of those, 97 were only partially completed and were thus removed. Of the remaining 180, 12 respondents claimed that they do not possess a driver’s license in Q11 and five declared that they never use the Portuguese freeway network as drivers in Q13. As a result, they were removed from the sample, as illustrated in

Table 2. Classification of the collected responses.

Responses
total	277
partially completed	97
no driver’s license	12
no freeway users	5
valid sample	163

. Thus, the sample size of valid responses resulted in being 163 units, sufficiently higher than the estimated target sample size.

Despite the genericity of the chosen dissemination channels, which did not specifically target freeway users, more than half of the respondents declared to use the national freeway network for their daily commute in Q12 and according to the answers to Q13, about a quarter of the respondents drive on Portuguese freeways only “sometimes per year” and only five respondents, as just referred, declared never to use the network.

Given the use of dissemination channels mainly related to the university, the valid sample revealed not to represent the distribution of the actual Portuguese population. Therefore, we implemented a stratification process, described in detail in the next Section.

3. Stratification

As referred in Section 2, various demographics were collected for each respondent with the primary objective of operating an ex-post stratification of the survey results. For this purpose, the survey’s questions and answers were structured with a perfect correspondence with the population profiling emerging from statistical data publicly available on the website of INE-IP. Literature suggests a particular relation between VOT and income level: as referred by [9], the VOT of commute trips is in average approximately half of the wage rate [2,15], and VOT is known to increase with the income level [16,17,18,19]. Even though the sample that we were able to reach is not very large, it includes members of each of the income classes individuated by the classification of the employed workforce by net monthly income presented by INE-IP. Among the respondents, some individuals belong to inactive population, as retired people and students without income, and there are members of the unemployed workforce. INE-IP data also reveal other categories that were entirely not reached by the survey, as domestic workers and self-employee. Due to the small proportions associated with these categories compared with the reached ones, we decided to exclude them from our study of the VOT. For each of the considered categories, we compared the relative proportions in the INE-IP data ($p_{G,INE}$) (Source: Instituto Nacional de Estatística (INE), IP–Portugal. Consulted tables: População residente com idade entre 16 e 89 anos (Série 2021-N°) por Local de residência (NUTS-2013), Condição perante o trabalho e Condição perante o trabalho (auto classificação); Trimestral. População inativa com 16 e mais anos de idade (Série 2021-N°) por Local de residência (NUTS-2013), Sexo e Condição perante o trabalho (Inactivo); Trimestral. População empregada por conta de outrem (Série 2021-N°) por Local de residência (NUTS-2013), Sexo, Sector de actividade económica (CAE Rev. 3) e Rendimento líquido mensal; Trimestral.) and the survey’s results ($p_{G,survey}$), recognizing that the survey was not reflective of the distribution of the Portuguese population. Thus, we used the INE-IP data as a basis to formulate the stratification. We note that one of the most significant differences between the collected sample and the Portuguese population is the low representativity of the group with net monthly income between 600 and 900€, which is the largest group in the population and includes people receiving the national minimum wage. (The national gross minimum salary of Portugal was fixed to 655€ for 2021 [20]. Workers receive 14 salaries, thus minimum salary receivers have a gross average monthly salary of 775.8€ subject to 11% social security deductions.)

Once we identified the desired sample strata relative sizes with INE-IP proportions, we operated the stratification by attributing weights to the answers according to the individual’s group. The weight $w_G$ for individuals of group G was calculated as

$$w_G=\frac{p_{G,INE}}{p_{G,survey}}·\frac{1}{N_{tot}}$$

(6)

where $N_{tot}$ is the total number of valid responses. The classification of the respondents into groups and the attribution of weights are resumed by

Table 3. Classification of the responses into groups based on the answers to Q1, Q4 and Q6. For each group, we report the number of respondents ($N_G$), the relative proportions in our survey ($p_{G,survey}$), the relative proportions in the INE-IP data ($p_{G,INE}$) and the attributed weights ($w_G$).

Job (Q4)	Gender (Q1)	Income (Q6)	${\mathit{N}}_{\mathit{G}}$	${\mathit{p}}_{\mathit{G},\mathit{s}\mathit{u}\mathit{r}\mathit{v}\mathit{e}\mathit{y}}$	${\mathit{p}}_{\mathit{G},\mathit{I}\mathit{N}\mathit{E}}$	${\mathit{w}}_{\mathit{G}}$
unemployed	all	all	1	0.61%	5.07%	0.0507
retired	all	all	4	2.45%	30.64%	0.0766
students	male	0–310	10	6.13%	5.55%	0.0055
	female	0–310	6	3.68%	5.71%	0.0095
employees	all	0–310	13	7.98%	1.22%	0.0009
	all	310–600	6	3.68%	2.50%	0.0042
	male	600–900	4	2.45%	7.65%	0.0191
	female	600–900	6	3.68%	11.08%	0.0185
	male	900–1200	18	11.04%	5.45%	0.0030
	female	900–1200	11	6.75%	5.88%	0.0053
	male	1200–1800	17	10.43%	6.56%	0.0039
	female	1200–1800	3	1.84%	5.76%	0.0192
	male	1800–2500	24	14.72%	2.76%	0.0012
	female	1800–2500	8	4.91%	2.03%	0.0025
	all	2500–3000	17	10.43%	0.87%	0.0005
	all	3000+	15	9.20%	1.26%	0.0008
all	all	all	163	100%	100%

.

Defined the stratification weights as in Table 3, the next section is dedicated to introduce the approaches we followed for the VOT study and to expose the results of exploratory data analysis.

4. Exploratory Data Analysis

In this section, we will introduce the approaches we followed for the study of VOT and expose the results of exploratory data analysis. All the considerations regarding VOT are based on the responses to Q18, where respondents were asked to indicate their willingness to pay in exchange for various potential time savings, as shown in Figure 1.

Answers to Q18 were converted into €/h units by dividing the chosen money values by the midpoint of the corresponding time interval, expressed in hours. Thus, for each respondent, ten values of time were collected. To study the VOT distribution from the collected data, we considered three different approaches:

• Average VOT (${\xi }_{av}$): we calculated the average of the ten values expressed by each respondent and then assumed it as the respondent’s VOT.
• Delay VOT (${\xi }_d$): also in this case, we have a single VOT for each respondent, determined this time by the money value declared by the respondent for the time interval corresponding to the delay experienced by the respondent as declared in Q17. Nevertheless, we decided to open an exception for respondents who declare to experience congestion-related delays of less than 5 min because of the unrealistic values generated by the minimum choice in Q18. For these respondents, we imposed Delay VOT to be equal to Average VOT.
• All VOT (${\xi }_{all}$): all the VOTs expressed by each respondent were considered separate data points for the analysis, except those corresponding to the extreme time intervals (0–5 min and 90+ min) that were excluded for generating distortion.

Some classical descriptive statistics of the VOT calculated through each of the three approaches on the sample before stratification are available in

Table 4. Some classical descriptive statistics of the VOT calculated through each of the three approaches on the sample before stratification (in €).

Statistics	Average VOT	Delay VOT	All VOT
min	2.37	0.67	0.40
max	34.40	34.40	150
median	3.56	3.75	2.40
mean	4.99	4.78	4.07
estimated st. dev.	4.62	4.30	6.69
estimated skewness	3.98	3.88	11.07
estimated kurtosis	22.66	22.82	203.63

. The third method is the one characterized by the highest variability.

Histograms of the survey responses before and after stratification for each of the three individuated approaches are shown in Figure 2 and Figure 3.

We can observe that the histogram of the Average VOT after stratification shows a peak for the lowest monetary range, followed by a pronounced decrease and ended by stabilization around meager proportions for VOT greater than 7 €/h. No responses show an Average VOT lower than 2 €/h; in fact, the minimum identified is 2.37€. The histogram of the Delay VOT shows a peak for the values between 2 and 3 €/h, but some respondents evaluated their delay interval as less than 2 €/h, as indicated by the columns corresponding to the first two ranges. For this approach, the stratification produced a less regular distribution than the original survey data. Finally, we can also observe a decreasing pattern in the third approach, with some significant exceptions (for instance the range between 3 and 4 €/h).

The mean values of the three defined VOT (Average, Delay and All) for each of the groups individuated in Section 3, plus the groups of male and female respondents and the group of all, are displayed in

Table 5. Mean values of Average VOT (${\overline{\xi }}_{av}$), Delay VOT (${\overline{\xi }}_d$) and All VOT (${\overline{\xi }}_{all}$) for each of the groups individuated in Section 3, plus the groups of male and female respondents and the group of all the respondents.

Job (Q4)	Sex (Q1)	Income (Q6)	${\mathit{N}}_{\mathit{G}}$	${\overline{\mathit{\xi }}}_{\mathit{av}}$	${\overline{\mathit{\xi }}}_{\mathit{d}}$	${\overline{\mathit{\xi }}}_{\mathit{all}}$
unemployed	all	all	1	2.37	2.31	1.42
retired	all	all	4	3.15	3.15	2.29
students	male	0–310	10	3.62	3.39	2.74
	female	0–310	6	3.06	3.07	2.19
employees	all	0–310	13	4.52	4.11	3.33
	all	310–600	6	7.84	8.81	6.81
	male	600–900	4	2.86	2.14	1.93
	female	600–900	6	3.68	4.03	2.71
	male	900–1200	18	6.41	6.04	5.41
	female	900–1200	11	4.35	3.95	3.70
	male	1200–1800	17	4.79	4.37	4.03
	female	1200–1800	3	3.08	3.50	2.22
	male	1800–2500	24	6.51	5.94	5.45
	female	1800–2500	8	4.72	4.45	4.08
	all	2500–3000	17	4.25	4.37	3.53
	all	3000+	15	5.62	5.51	4.55
all	male	all	103	5.71	5.36	4.73
all	female	all	60	3.76	3.76	2.95
all	all	all	163	4.99	4.78	4.07

. Curiously, the highest values correspond to the group of respondents with monthly income between 310 and 600€, and men seem to have on average higher VOTs than women.

In the next section, after going through the main steps of the adopted fitting process, various statistical distributions are tested for fitting of the empirical data after stratification, and we select the best performing ones.

5. Fitting

This section will select parametric functions to efficiently describe the distributions of VOT obtained after the stratification process, referring to each of the three approaches described in Section 4 for VOT estimation. For this purpose, as a preliminary step, besides the plots displayed in Figure 2 and Figure 3, we also examine the empirical distributions’ skewness and kurtosis, which are descriptive statistics linked to their third and fourth moments. In particular, “a non-zero skewness reveals a lack of symmetry of the empirical distribution, while the kurtosis value quantifies the weight of tails in comparison to the normal distribution for which the kurtosis equals 3” [21], and these characteristics may be very useful to guide the choice of the most appropriate parametric distributions. Besides their values, reported in Table 4, an effective representation of the mentioned descriptive statistics is given by the Cullen and Frey graph, which has the square of skewness on the x-axis and kurtosis on the y-axis. The graph includes one point corresponding to the empirical distribution of the collected sample and bootstrapped values deriving from random resampling. These are compared with the values for some common distributions to help the choice of distributions to fit to data. For some distributions (as normal, uniform, logistic and exponential), there is only one possible value for the skewness and the kurtosis: these distributions are represented by a single point on the plot. Normal, uniform and logistic distributions are characterized by symmetry. Thus they are represented by points with zero skewness. For other distributions, areas of possible values are represented, consisting of lines (as for gamma and lognormal distributions) or larger areas (as for beta distribution). Weibull is close to gamma and lognormal lines. The Cullen and Frey graphs for the three approaches in this paper, obtained through the descdist function of R with boot = 1000, are displayed in Figure 4. Both the observation and the bootstrapped values are positioned rather far from the points representing symmetric distributions for all the approaches. Therefore, we exclude these from the candidate distributions for fitting, concentrating on the other possibilities.

Specifically, we considered exponential, beta, log-normal, gamma, Weibull and Burr distributions. Exponential is a mono-parametric function, Burr is defined by three parameters, whereas all the others have two parameters. The expressions of the probability density function (PDF) and the cumulative density function (CDF) and a list of the parameters of each of the distributions tested for fitting are available in

Table 6. Probability density function (PDF), cumulative density function (CDF) and list of parameters of each of the distributions tested for fitting. Used symbols: $\xi$ = VOT, $B(·)$ = beta function, $I_{\xi }(\alpha ,\beta )$ = regularized incomplete beta function, $\mathrm{erf}(·)$ = error function, $\mathsf{\Gamma }(·)$ = gamma function, and $\gamma (·)$ = lower incomplete gamma function.

Distribution	PDF $\mathit{f}\left(\mathit{\xi }\right)$	CDF $\mathit{F}\left(\mathit{\xi }\right)$	Parameters
exponential	$\theta e^{-\theta \xi }$	$1-e^{-\theta \xi }$	$\theta$
beta	$\frac{{\xi }^{\alpha -1}{(1-\xi )}^{\beta -1}}{B(\alpha ,\beta )}$	$I_{\xi }(\alpha ,\beta )$	$\alpha ,\beta$
log-normal	$\frac{e^{-\frac{{(ln\xi -\mu )}^2}{2{\sigma }^2}}}{\sqrt{2\pi }\sigma \xi }$	$\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{ln\xi -\mu }{\sqrt{2}\sigma }\right)$	$\mu ,\sigma$
gamma	$\frac{{\beta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}{\xi }^{\alpha -1}{e}^{-\beta \xi }$	$\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\gamma (\alpha ,\beta \xi )$	$\alpha ,\beta$
weibull	$\frac{c}{\theta }{\left(\frac{\xi }{\theta }\right)}^{c-1}{e}^{-{(\xi /\theta )}^c}$	$1-e^{-{(\xi /\theta )}^c}$	$c,\theta$
burr	$\frac{\alpha \beta c^{\alpha }{\xi }^{\beta -1}}{{(c+{\xi }^{\beta })}^{\alpha -1}}$	$1-{\left(\frac{c}{c+{\xi }^{\beta }}\right)}^{\alpha }$	$\alpha ,\beta ,c$

.

We classified the weighted data into unitary ranges for each of the three approaches of VOT definition. The adopted fitting method consisted in optimizing the functional parameters of each distribution by minimizing the absolute error between the empirical distribution function and the parametric one through Excel’s solver, where the absolute error was estimated by summing up the absolute errors between the fitting CDF and the empirical CDF calculated for the midpoints of each unitary range of VOT values. The results of the fitting process are resumed in

Table 7. Results of the fitting process. For each of the considered distributions, we report the values of the parameters obtained through the fitting process, the absolute error compared to the empirical CDF, the mean and the median.

Approach	Distribution	Parameters	Error	Mean (€)	Median (€)
Average VOT $\left({\xi }_{av}\right)$	empirical	-	-	3.82	2.80
	exponential	$\theta =0.32$	13.17	3.09	2.14
	beta	$\alpha =1.29$	0.24	2.75	2.12
		$\beta =15.14$
	log-normal	$\mu =1.29$	0.14	3.07	2.40
		$\sigma =0.70$
	gamma	$\alpha =1.53$	0.22	0.82	2.23
		$\beta =1.84$
	weibull	$\theta =2.83$	0.23	2.74	2.02
		$c=1.09$
	burr	$\alpha =0.81$	0.11	3.22	2.39
		$\beta =2.82$
		$c=8.57$
Delay VOT $\left({\xi }_d\right)$	empirical	-	-	3.74	2.87
	exponential	$\theta =0.35$	0.50	2.83	1.96
	beta	$\alpha =1.62$	0.24	2.86	2.36
		$\beta =18.21$
	log-normal	$\mu =0.84$	0.21	3.05	2.33
		$\sigma =0.74$
	gamma	$\alpha =1.77$	0.34	1.09	2.38
		$\beta =1.63$
	weibull	$\theta =3.12$	0.39	2.85	2.39
		$c=1.38$
	burr	$\alpha =0.78$	0.19	3.22	2.39
		$\beta =2.89$
		$c=8.73$
All VOT $\left({\xi }_{all}\right)$	empirical	-	-	2.93	2.31
	exponential	$\theta =0.1$	19.86	10	6.93
	beta	$\alpha =1.09$	0.37	0.53	0.37
		$\beta =70.12$
	log-normal	$\mu =0.44$	0.17	2.43	1.56
		$\sigma =0.94$
	gamma	$\alpha =1.10$	0.36	0.52	1.65
		$\beta =2.09$
	weibull	$\theta =2.34$	0.37	2.28	1.65
		$c=1.06$
	burr	$\alpha =1.13$	0.15	2.55	1.61
		$\beta =1.82$
		$c=2.81$

and Figure 5, Figure 6 and Figure 7.

The described fitting process did not work properly for the exponential distribution in Average VOT and All VOT cases, leaving a high value of the absolute error and indicating that it is not the proper function to describe these VOT distributions synthetically. On the other hand, for all the three approaches, Burr distribution results as the one with the smallest associated absolute error and with mean values that are the closest to the empirical ones. Furthermore, the median value of the Burr distribution is also the closest to the empirical one for Delay VOT and is near to the closest for the other approaches. Finally, concerning the two-parameters distributions, log-normal is the one which performed best for our stratified sample, both for error and mean and limited to Average VOT, also for median.

In the next section, we compare our results with the ones of a previous survey-based VOT study.

6. Comparison with a Previous Survey-Based VOT Study

After the exploratory data analysis exposed in Section 4 and the fitting process described in Section 5, this section is dedicated to comparing our results with the VOT distribution function built by [9] for the users of the A1 highway in Lebanon. Since the study was conducted at the end of 2014, to compare the results, we converted the money values of the midpoints of the intervals individuated in [9] from Lebanese Lira (LL) into Euros considering the exchange rate of 31 December 2014 (1€ = 1647.30 LL) and actualizing it to 2019 by considering both inflation in Lebanon (8.85% in total, according to data from the Word Bank) and Portugal (3.85% in total, according to data from the Word Bank). The year 2019 was chosen for comparison because afterwards Lebanon entered a period of crisis because of the COVID-19 Pandemic and the 2020 Beirut explosion, registering a 84.86% inflation rate in 2020, while Portugal registered a negligible deflation rate between 2019 and 2020 (−0.012%) [22]. Then, we applied to these values our best performing distribution for the approach All VOT because this approach was the most similar to the one followed in [9]. The comparison between the graphs of our Burr CDF for All VOT and the one in [9], in Figure 8, clearly reveals that all the quartiles of our distribution are located on lower values than the ones of the Lebanese survey, indicating, in general a lower willingness to pay emerging from our survey’s results.

Since the Lebanese study was conducted on a situation where the untolled alternative was a highly congested roadway, one possible explanation for the lower willingness to pay found in this study could be deriving from the fact that many respondents of our survey (57%) were facing low-congestion conditions in their typical trip.

Concerning mean values, the one in the Lebanese survey (almost 8000 LL = 4.86€) is slightly lower than our empirical pre-stratification one and amounts to around 60% of the average hourly wage of the respondents, consistently with literature evidence, which indicates values around 50%; the mean values of our empirical distributions as well as our fitting options fall in between 20% and 40% of the value of the average wage rate found for Portugal, which amounts to about $12.59€$ for 2020, as calculated in Equation (3). This fact suggests a lower willingness to pay in comparison to our original expectations deriving from literature.

These findings suggest not to impose very high tolls in a dynamic toll pricing strategy for a Portuguese freeway, in order to not exclude too many drivers from the freeway. Moreover, the results suggest that a little toll increase can have a major impact on the drivers’ reactions.

The next section is dedicated to our estimation of the elasticity of the freeway demand and of the whole demand for transportation from survey data.

7. Elasticity

Defined elasticity as the measurement of the percentage change of one economic variable in response to a change in another [23], toll price elasticity of transport demand should be measured by analyzing the reaction of the users to a change in the toll price of a specific tolled infrastructure. However, in this survey, we did not target the users of a particular infrastructure. Thus, we observe a high variability of the toll prices paid by the respondents for their typical travel on freeways. The distribution of the toll values declared by the respondents in Q14 is displayed in Figure 9.

Q14 was the only free text question of the survey, while all the others were single-choice questions. A large number of respondents declared to pay no toll. In contrast, reported toll values greater than 100€ seem quite unrealistic because there are no regular routes with such high tolls in the Portuguese national freeway network [11]. Considering these as typing errors committed when answering Q14, we assumed that the fact does not affect the credibility of the other answers of a respondent.

The last question of the survey, Q19, was designed to get an idea of the price elasticity of the surveyed population by asking the respondents which choice they would make if the toll price associated with their typical journey on freeways doubled with the guarantee of better traffic conditions.

Four possible choices are contemplated in Q19, as displayed in Figure 10: the respondents could declare that they would continue to use their private vehicles in the same freeway (choice 1), that would use their private vehicles in an alternative untolled route (choice 2), that would switch to public transportation solutions (choice 3), or that would consider a change in trip’s destination or residency location (choice 4).

These four alternatives are meant to investigate four different types of elasticity: from the first, we could estimate the price elasticity of the freeway demand; the second and the third deal with cross-price elasticities of untolled road and public transportation demands, respectively; and the fourth should capture the price elasticity of the whole demand for transportation.

The frequencies associated to each possible choice of Q19, resumed in

Table 8. Absolute frequencies and percentages associated to each choice of Q19 in valid responses.

Choice	Frequency	Percentage
1	58	36%
2	78	48%
3	15	9%
4	12	7%
total	163	100%

, show that almost half of the respondents (48%) declared the intention to switch to an alternative untolled route in case of doubling of the toll, about one third (36%) would continue to use the same freeway and relatively lower percentages of respondents are ready to switch to public transportation or to change house or work location in consequence of a toll raise.

Any measure of elasticity should consider demand both before and after the price change. Thus, it is impossible to estimate cross-price elasticities from our data, but we could obtain estimates of both price elasticity of the freeway demand and of the whole demand for transportation. Two different formulas are employed to calculate elasticities in this paper: for the group of respondents who declared a non-zero value of the toll associated with their typical travel, we used the arc elasticity, whereas for the other group, we used mid-point (or linear) arc elasticity. Arc elasticity is defined by a logarithmic formulation [24]:

$${\eta }_{arc}=\frac{log\left(q_f\right)-log\left(q_i\right)}{log\left({\pi }_f\right)-log\left({\pi }_i\right)}$$

(7)

where $q_i$ and $q_f$ represent the demand before and after the price change, respectively, and ${\pi }_i$ and ${\pi }_f$ represent the price value before and after the change, respectively. The formula in Equation (7) was employed to estimate elasticities associated with drivers who declared non-zero tolls because its denominator could easily be computed for any case of toll doubling regardless of the original value of the toll, resulting in $log2$. However, we may observe that Equation (7) does not allow the calculation in case ${\pi }_i=0$, thus for these cases, we employ the mid-point (or linear) arc elasticity, which is the only formulation allowing calculation of elasticity when the original price equals zero [24]:

$${\eta }_{linear}=\frac{(q_f-q_i)/[\frac{1}{2}·(q_f+q_i)]}{({\pi }_f-{\pi }_i)/[\frac{1}{2}·({\pi }_f+{\pi }_i)]}$$

(8)

The price elasticity of the freeway demand and of the whole demand for transportation calculated for both drivers with zero and non-zero original toll are shown in

Table 9. Price elasticity of the freeway demand and of the whole demand for transportation calculated for both drivers with zero and non-zero original toll and their weighted average.

Formula	Freeway	Transport
${\eta }_{linear}$	−1.28	−0.13
${\eta }_{arc}$	−0.92	0.00
$\overline{\eta }$	−1.23	−0.11

. This table also includes an average value of elasticity ($\overline{\eta }$), calculated as an average of the obtained values of ${\eta }_{linear}$ and ${\eta }_{arc}$, weighted by the proportion of respondents to which each formula was applied, i.e., the proportion of zero toll and non-zero toll respondents.

Since it is a derived demand, automobile transportation demand tends typically to be inelastic, i.e., with elasticity between 0 and −1 [23], meaning that a percentage change in price usually causes a proportionally smaller change in transportation demand, except for discretionary travel, as consumption or entertainment-related trips [24,25,26,27]. The estimated elasticity of the whole demand for transportation for respondents with non-zero tolls is indeed in this range, as the one of the freeway demand limiting to drivers who currently experience zero tolls. The whole demand for transportation results as perfectly inelastic, with elasticity equal to zero, for respondents declaring zero tolls. On the contrary, the freeway demand results as elastic, with elasticity less than −1, when considering respondents declaring non-zero tolls and also averagely.

8. Summary and Concluding Remarks

The main objective of this paper was to develop a driver behavior model that reflects the choice mechanisms of the users of the Portuguese national freeway network to be used for testing dynamic pricing strategies. For this purpose, we conducted a survey between March and June of 2021 among users of the Portuguese national freeway network, aiming to derive a VOT distribution and demand elasticity. The survey was administered exclusively online and totalized 163 valid responses. Since the validated sample was not reflective of the actual distribution of the Portuguese population, we carried out a stratification process based on published statistical data. For each of the approaches followed for VOT definition (Average VOT, Delay VOT and All VOT), we operated a fitting process that consisted in optimizing the functional parameters of each distribution by minimizing the absolute error between the empirical distribution function and the parametric one, concluding that the Burr distribution is the best performing one amid the tested functions both in terms of absolute error and in terms of the mean value. Our results indicate generally lower VOTs than the ones predicted from the literature. In particular, the mean values we obtain are somewhat below half of the average wage rate, and the distributions are characterized by significant portions of the users with low VOT, followed by long tails of values with small associated frequencies. The low willingness to pay for traffic condition improvement was also confirmed by a comparison with a distribution built in a previous survey-based VOT study and an elasticity estimation that suggests freeway demand to be averagely elastic, contrary to literature indications.

We can conclude that we fulfilled our original objective of developing a survey-based driver behavior model for testing dynamic pricing strategies on the Portuguese freeway network presenting a method that could possibly be followed in other countries for analogous studies. For the future, we would aim to conduct more targeted inquiries, focusing on the infrastructure where the toll pricing will be implemented to capture the choice mechanisms of local drivers and resort to local in-person interviews, which were impeded by the ongoing pandemic.