A first manuscript version of this paper, entitled âNotes pour l'estimation du nombre d'utilisateur de la CAM (Carte Autobus-Métro) dans le cadre du mod`ele ...
On the Demand for and Value of Monthly Transit Passes within the Framework of the DEMTEC Transit Ridership Demand Model by Marc Gaudry1 and Andr´e Arbic2
1
D´epartement de sciences e´ conomiques et Centre de recherche sur les transports, Universit´e de Montr´eal, C.P. 6128, succ. Centre-ville, Montr´eal, Canada H3C 3J7 2 Soci´ et´e de transport de la Communaut´e Urbaine de Montr´eal, Service de d´eveloppement, 800, de la Gaucheti`ere ouest, bureau F-1200, C.P. 2000, Montr´eal, Canada H5A 1J6
A first manuscript version of this paper, entitled “Notes pour l’estimation du nombre d’utilisateur de la CAM (Carte Autobus-M´etro) dans le cadre du mod`ele DEMTEC” was written in 1989. Agora Jules Dupuit — Publication AJD-21 2nd Draft, April 1996
1. Introduction: unobserved trips made with passes Many transit authorities have introduced over time monthly transit passes permitting wide privileges, such as unlimited use of the transit system, in parallel with the conventional transportation modes of payment, or “titles”, consisting of cash and tickets. A major inconvenience of this practice, unless it is combined with sophisticated fare and trip measurement systems, is that the transit company looses accurate day-to-day measurement of the use of the network and has to resort to occasional and expensive surveys to measure the use of passes. We want to examine the implications of this practice for the Montreal Urban Community Transit Commission (MUCTC) transit system, both in terms of the ability to understand its own ridership and in terms of the potential strategies available to price the transit passes.
2. Specific context: the DEMTEC ridership model It is convenient to discuss the problem by stating that the MUCTC has a demand model called DEMTEC (Gaudry 1973, 1975a, 1978), which considers two markets, adults and schoolchildren, and formulates for each market a demand function
D = f (P ;T ;W ;Y ;A ;ETC ) t
t
t
t
t
t
t
(1)
D is explained by prices P (transit fare, car ownership and other prices), service levels T (of transit and competing modes), weather W (rain, snow, etc), income Y , final and intermediate activities A (employment, shopping, special events, ...) and other variables ETC to account for the heterogeneity of months and various administrative decisions that affect where monthly ridership
the measurement of the dependent variable. Over the years, improvements of the specification have demonstrated the accuracy of the model. The econometric specification combines linear regression with multiple-order autocorrelation of the residuals. Further work with Box-Cox 1
transformations (Dagenais, Gaudry and Liem, 1987) demonstrated that the linear form was an excellent approximation. For each market, the dependent variable consist of the sum of cash and ticket passengers, and some variables included in
ETC effect corrections for trips made using a
small and limited number of schoolpasses. The model performed very well (with series starting in 1956) until the 1980’s, when the systematic use of monthly passes in both markets resulted in the loss of crucial information on the actual members of trips and raised the question whether separate equations should be added to account for monthly pass sales. As the addition of equations often makes it difficult to use all transit fares in all equations, because of colinearity, we want to suggest an alternative. For simplicity, we assume without loss of generality that there are only two modes of payment, tickets
y1 and monthly passes y2 ,
but that ridership with passes is not systematically observed, and that there is only one market, say for trips by adults.
3. A representative consumer A. Direct or quasi-direct demand format? As only one equation is used for each market, it is assumed that there is a representative consumer in that market deciding whether to make the trips and which transportation title should be bought. If there were not interest in the actual number of trips taken, but only in explaining how many titles of each kind are sold, one approach could be to define a mode-of-payment model similar to a mode choice model, using as explanatory variables the characteristics of all alternatives. The denominator of the resulting logit or dogit mode-of-payment model (Gaudry and Wills, 1979) could then be used to define of-payment, and
U
U , an expected maximum utility of these modes-
appear as an explanatory variable in (1). However, the dependent variable
would then also pertain to titles. The product of the two models would yield the monthly demand 2
for titles of each category, a structure that we have called the Quasi-Direct-Format (Liem and Gaudry, 1994). An alternate approach is to assume that there is an expected number of trips per month, ni , for which the consumer is indifferent between tickets and monthly passes and which induces him to switch between the titles according to whether his expected number of trips is smaller or greater than ni . This means that one could estimate this trigger trip number ni , to which corresponds a fare per trip F2 =ni , at which the consumer is indifferent between the pass fare F2 and the ticket fare
F1 . As monthly passes require a significant disbursement, involve a risk of loss and expire at the end of the month, it is expected that ni will exceed the break even number of trips nb
= F2=F1 .
The difference between the break-even number of trips nb and the trigger or indifference number 2
ni implies that a discount equal to F2 =nb 0 F2 =ni
3
=
2
0
F2= nb 0 ni
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has to be granted for
consumers to purchase monthly passes.
B. Estimation of the trigger usage within a direct demand format
In view of the demonstrated robustness of the DEMTEC model, both in terms of stability (functional form, elasticities, etc) and as a precise tool of analysis and forecasting for over 15 years at the MUCTC (Gaudry, 1975b, Arbic et al., 1979), we shall consider that the model is “given” and focus on the conditional estimation of ni . Regrouping other explanatory variables than the fare, the formulation can be 0
y1 + n y2 i
1
= f F1 + Fn2i 3
+
X k
k X k + u
(2)
where it is clear that a grid search can be performed to estimate ni because, for a given value
X X + u;
of ni , one may write
y 3 = f [Fn3 ] + i
k
(3)
k
k
and the coefficients can be either the same as before the introduction of monthly passes or (with additional assumptions) reestimated. Note also that, if the monthly pass market differed significantly from the representative consumer market, a distinct coefficient for each of the fares could be identified in (2); however, in view of the colinearity between different transit fares, such a two-dimensional search may not be profitable. C. The value of actual usage of monthly passes Although the actual trip rate na is only occasionally observed through special surveys, it is known to be much larger than the break-even trip rate nb . Whether one assumes the existence of one or two representative consumers (i.e. of one or two f coefficients in (2)), many users would be willing to pay a price for the privilege of unlimited use. The value of this gift is the difference between then effective fare Fn3a implied by their use rate na and the fare Fn3i implied by their indifference rate ni . This can be construed as a measure of consumer surplus, as shown in the shaded area of Figure 1. Figure 1 Value of monthly pass privilege
Fare
Fn*i
⋅
⋅
Fn*a
⋅ na
nb ni 4
Trips
Clearly, some individuals who expect to make very large numbers of trips could have individual trigger rates ni that would be lower than the break-even rate nb . However, this is unlikely to be the case for the average user, unless the price of passes is extremely low relative to that of tickets. In this case one could find an empirical value of ni smaller than the break-even value nb . In such a case, the proper measure of consumer surplus would take no account of the area between Fn3b and Fn3i . REFERENCES 1. Arbic, A., Boucher, C. et M. Gaudry, “Pr´evisions de l’achalandage de la C.T.C.U.M. en 1979 et 1980”, Rapport technique #G.11.79, Service de la planification, C.T.C.U.M., 18 p., 1979. 2. Dagenais, M., Gaudry, M. and T.C. Liem, “Urban Travel Demand: the Impact of BoxCox Transformation with Nonspherical Residual Errors”, Transportation Research B 21, 443–477, 1987. 3. Gaudry, M., “The Demand for Public Transit in Montreal and its Implications for Transportation Planning and Cost-Benefit Analysis”, Ph.D. Thesis, Economics Department, Princeton University, and University Microfilms, Ann Harbor, Michigan, 188 p., 1973a. 4. Gaudry, M., “An Aggregate Time-Series Analysis of Urban Transit Demand: the Montreal Case”, Transportation Research 9, 4, 249–258, 1975a. 5. Gaudry, M., “Quelques apports du mod`ele DEMTEC a` l’explication de l’achalandage pass´e et futur de la C.T.C.U.M.”, Rapport technique #G.1.75, Service de la planification, C.T.C.U.M., 21 p., 1975b. 6. Gaudry, M., “Seemingly Unrelated Static and Dynamic Urban Travel Demands”, Transportation Research 12, 3, 195–211, 1978. 7. Gaudry, M. and M. Wills, “Testing the Dogit Model with Aggregate Time Series and Cross Sectional Travel Data”, Transportation Research B 13, 2, 155–166, 1979. 5
8. Liem, T. and M. Gaudry, “QDF: Quasi-Direct Format used to Combine Total and Mode Choice Results to Obtain Modal Elasticities and Diversion Rates” Publication #982, Centre de recherche sur les transports, Universit´e de Montr´eal, 18 p., June 1994.
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