The elasticities of passenger transport demand in the Northeast Corridor

Research in Transportation Economics 78 (2019) 100759

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Research paper

The elasticities of passenger transport demand in the Northeast Corridor ~ uela Romana Ignacio Escan Loyola University, Andalusia, Spain

A R T I C L E I N F O

A B S T R A C T

JEL classification: D12 R41 C13 C32 R11

The objective of this study is to estimate the values for compensated and uncompensated price elasticities and income elasticities for the various relevant modes of passenger transport between cities along the Northeast Corridor of the United States. The theoretical assumptions of the theory of utility maximization by a rational consumer are a necessary condition, a Rotterdam demand model is applied due to several relevant advantages, and the seemingly unrelated regression method is used. The problem of aggregation in the Rotterdam model will also be considered. I will approximate the data sets for the various modes of transport, reconstructing the quantities and prices. The quantification of demand elasticities is strongly relevant in order to understand substitution and income effects on the demand for transport modes. However, efficient multimodal transport is crucial to the economic and social life of such vast metropolitan areas. In general, the relatively successful estimation of this demand model, and the information it provides, shapes a proposal for an understanding of transport in the so-called mega-regions, which are increasing in size and global relevance.

Keywords: Demand Maximization Elasticity Transport Megaregion

1. Introduction and aim In this paper I conduct research into the demand for intercity modes of transport inside the Northeast Corridor (NEC) of the United States: plane, train, car and bus. The aim is to estimate compensated and un­ compensated price elasticities and income elasticity. The final objective is to provide a general model for the estimation of the elasticities of transport demand through an understanding of how the different modes of transport interact inside a mega-region. This produces a very inter­ esting result, since mega-regions are presumably increasing in social and economic global importance. I estimate a Rotterdam demand model by the seemingly unrelated regression (SUR) method, using as necessary conditions the theoretical assumptions of the theory of rational maximization. This demand model is one of the most widely used (Clements & Gao, 2015). The other is the “almost ideal demand system” (AIDS). Both are flexible at the local level, in the sense that they presumably do not have conditions imposed on the possible elasticities in a point (Barnett & Seck, 2008, p. 2). Both are easy to estimate and can be used to test theoretical restrictions (Faroque, 2008, p. 4). The Rotterdam model is chosen for its connection to theory and for taking first differences of all its variables, thus avoiding a spurious regression. Scientific literature does not appear to contain anything close to an overall multi-equation estimation for demand elasticities, prices and income, therefore imposing restrictions from rational behaviour in the

various NEC modes of transport. In fact, the Rotterdam demand model, as a tool for analysis and comprehension, appears to not have been applied to passengers transport operations (Liu and He 2016 reject its application). Moreover, following Oum, Waters, and Yong (1990, p.10) and Oum, Waters, and Yong (1992, pp. 152–153), the estimation must control the presence of intermodal competition, since the absence of this could affect its own price elasticities estimates. This paper seems to offer relevant results: imposing theoretical and intermodal conditions for the estimation, applying the Rotterdam model to the NEC and generally to mega-regions). The second section is dedicated to the description of the NEC and a mega-region. Section 3 and annex 1 and 2 discuss the Rotterdam model and the SUR method. Sections 4 and 5, and the annex 3, include data and results. Finally, the reached conclusions are in the sixth section. 2. The Northeast Corridor and the mega-regions The NEC is an economic and social entity. It links Boston, Provi­ dence, New York, Philadelphia, Baltimore and Washington DC, as well as the nearby urban and rural areas. The NEC traverses some 457 miles of railroad on its longest journey between Boston and Washington DC. It passes through four large metropolitan urban areas: Boston, New York, Philadelphia and Washington DC. These areas overlap as specialised interdependent entities (Beiler, McNeil, Ames, & Gayley, 2013; Oswald, Gayley, Ames, & McNeil, 2009, p. 1). The NEC is “North America’s

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.retrec.2019.100759 Received 25 February 2019; Received in revised form 6 August 2019; Accepted 15 October 2019 Available online 29 October 2019 0739-8859/© 2019 Elsevier Ltd. All rights reserved.

E.R. Ignacio

Research in Transportation Economics 78 (2019) 100759

busiest and most complex rail corridor” (Regional Plan Association, 2013, p. 1). It is a metropolitan area that in 2012 produced 20% of the USA’s GDP and held 18% of its population within 2% of the territory (Regional Plan Association, 2013, p.1). The NEC is demarcated by transport-based criteria: “The boundaries (…) are based on three criteria. First, the portion of I-95 between Boston and Washington, D.C. is used as the primary point of reference. Second, the outer boundary is drawn to include the space within 50 miles of the I-95 in all directions. Finally, in cases in which part of a metropolitan planning organization (MPO) is located within a 50-mile radius of I-95 [and part] beyond, all of the area covered by the MPO is included in [the] boundaries of the NEC” (Beauchamp and Warren, 2009, p. 4). Its connectivity is not defined solely by the motorway, inasmuch as: “the initial regional structure and transportation “spine” of Megalopolis was established by the passenger rail system that dates back to the late nineteenth century” (Oswald, 2007, p. 1). The transport of passengers and freight within the area is the key factor behind the internal inter­ connectedness of the Corridor: by train, road or air transport. Without efficient and economical transport, the NEC would not exist. NEC rail infrastructures are shared by numerous services, with a density of some 2300 trains running each day (GAO, 2016, pp. 16–67). Acela Express (since December 2000) and Northeast Regional (since 1995) offer train transport between cities. Both are electrified services owned by Amtrak. In 2014, they transported 11.6 million passengers, which represented 38% of all Amtrak passengers nationwide (Archila, 2013; GAO, 2016, pp. 16–67). Acela is faster and more costly. Acela takes three and a half hours (231 miles) from NY to Boston, while Northeast Regional takes about 4 h 10 min (2016). Various alternative NEC road routes exist between the large cities, as well as a dense network connecting these cities with other major cities and rural areas. The I-95 is the main connecting motorway and runs about parallel to the railroad within the NEC. Motorway travel accounts for 72,000 vehicular trips per day, registering peaks of 300,000 on some days (the I-95). Road use has constantly been on the increase: “The growth rate in the demand for transportation is exceeding the ability of the highway system to expand at a rate to handle the growth” (McNeil, Oswald, & Ames, 2010, p. 7). Finally, bus services connect both city centres and, also, outskirts and suburbs. O’Toole (2011, p. 680) emphasises the new bus service fea­ tures: online sale of tickets, passenger departure and arrival other than at bus stations (at locations nearest to the point requested), routes with fewer stops to reduce times, and better bus amenities. These features would also be adopted by more traditional operators, and they have been taking place since the beginning of the considered period. No substantial air, road and rail infrastructural changes have been introduced and hence speeds, journeys and frequencies have remained unvaried in the NEC between 2004 and 2016. For example, the number of miles of interstate motorway in Delaware1 or Maryland2 did not change (2002/2011, 2000/2016). This enables us to consider this period as homogeneous with respect to transport and the estimation. This model and its estimation could help us to understand demand for the modes of transportation in the mega-regions, which continue to increase in size and importance. Krugman (1991) emphasised the importance of two factors: economies of large-scale production and lower transportation costs. “The way the space economy is organized depends on the mutual interactions between mobility costs and scale economies” (Thisse, 2009, p. 30). It is calculated that in 2050, ¾ of the national population and income will be concentrated in the mega-regions of the US (Zhang & Zhang, 2013). A mega-region is an extensive, interconnected network of metropolitan centres and sur­ rounding service areas (Ross, 2011), brought about by the externalities

of human capital which tend to create concentrations of production and innovations (Florida, Gulden, & Mellander, 2008). Transport in­ frastructures and services are the main concern (Siemens & Hazel, 2010). “Improved highways and new freight rail will be absolutely essential” (U.N. Agenda 21, 2013, p. 100). Efficiency and competitive­ ness will depend on transport, which in turn depends on investment and innovations (U.N. Agenda 21, 2013). Elasticities are crucial to an un­ derstanding of this sector. Transport modes can compete with or complement one another (Albalate, Bel, & Fageda, 2015, p. 7). There is vast scientific literature, and cooperation or competition varies from market to market (Sun, Zhang, & Wandelt, 2017, p. 1). 3. Methods. The Rotterdam model and its estimation Barnett (2001), He and Liu (2016) and others propose the require­ ment of demand appraisal as a function subject to theoretical re­ strictions, as deduced from the theory of maximization by a rational consumer. The results of the estimation of a function, in the absence of the said theoretical conditions, could be biased (He & Liu, 2016, p. 1 and 13) and would not satisfy the secondary conditions of optimisation and duality theory of the consumer (Barnett, 2001). Therefore, this estima­ tion starts with the requirement to fulfil all the hypotheses of the theory of consumer maximization. Moreover, the estimation must control the presence of intermodal competition, since the absence of this could affect its own price elasticities estimates (Oum et al., 1990). This paper is based on the Rotterdam demand model. It was created by Barten (1964) and Theil (1965) and is characterised by its departure point of demand functions, differentiating them completely and using the theory of utility maximization to set restrictions (Clements & Gao, 2015). It falls under a theoretical perspective that is not based on the assumption of any function of utility. Consequently, this model must be consistent with the various functions of utility (Clements & Gao, 2015, p.1). What the model does, in conclusion, is this: “infinitesimal changes are replaced with finite changes and the model is parameterized by taking the marginal shares and Slutsky coefficients to be constants” (Clements & Gao, 2015, p.13). It is Barten (1964) who takes the share of expenditure and elasticities as fixed, and replaces them with sample means. Theil (1965) multiplies by the share in expenditure on the goods in demand, and then resets the parameters of the equation. The disad­ vantages would be related to the constant nature of the share in expenditure on each good, or set of goods considered, and the assumed fixed nature of the coefficients. This parameterization is necessary for arriving at a model applicable to the data: “Note that there is not strong a priori reason why (…) (parameters) should be held constant. Never­ theless, some decision must be taken” (Deaton, 1974, p. 344). The variables to the right of the equations are assumed to be exog­ enous. Prices, quantities and budget shares are the considered relevant factors. Sun et al. (2017, p.8) suggest the inclusion of travel time, among other things, as a critical factor in competition between transport modes. They also point out that elasticity values are influenced by the socio­ economic context of each area. Other characteristics have been considered by scientific literature in estimating travel demand, such as frequency, quality of service and schedule reliability (departures and arrivals). This paper follows the suggestion put forward by Deaton (1974) that it is reasonable to expect the combined effects of these omitted factors to have a stochastic structure. Their aggregate effect would be reflected in the error term of the equations, which would follow the assumptions of linear regression. However, adds Deaton (1974), it would be unreasonable to expect the median of its combined effect to be zero, and consequently it is advisable to include a constant term in the regression equations and to verify its statistical relevance. This would serve to remove any bias. However, the estimation of the Rotterdam model on the NEC indicates that this fixed coefficient does not enhance the results and in fact introduces incorrect signs. The Rotterdam model, as do other demand systems of equations,

1

“State Data Delaware”, 2002 and 2011. Maryland Department of Transportation, State Highway Administration, 2000 and 2016. 2

2

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Research in Transportation Economics 78 (2019) 100759

faces the problem of aggregation. In general, “the empirical importance of possible distortions caused by aggregation errors is largely unknown” (Barten, 1977, p. 36). As concluded by Varian: “The consumer theory places no restriction on aggregate behaviour in general” (1992, page 153). However, as Barten (1977) adds, the alternative is to estimate without using theoretical conditions, which means disregarding theory (p. 36). Positive confirmation of the maximising theory of the rational consumer only has particular value, but it is considered necessary. McFadden (1964) analyses the Rotterdam model by taking the funda­ mental equation of the differential specification of demand, which is typical of this model, applying the conditions required by rational consumer maximization, and integrating them to see the underlying demand function, and from there the utility function. He arrives at the following conclusion: “The only system of demand functions which is consistent with the differential specification globally (…) is a system of Cobb-Douglas demand functions, derived by maximization of log-additive utility” (p.1). Yoshihara (1969), among others, adds that, to escape from this limitation, it would be necessary to deny the integra­ bility of the model and, therefore, it would lack theoretical basis. Barnett (1979) emphasises the idea that “integrability of any model at the aggregate level [is obtained] only if a community utility function exists” (p.109), but this is “extremely restrictive” (p.110). Barnett (Barnett (1979)) imposes weak assumptions and then takes the probability limits of Slutsky equations as the number of consumers increases. The first assumption is that each consumer’s instantaneous income at the time is “sampled randomly from an infinite population of potential income paths” (Barnett (1979), p. 111). The second assumption concerns the finiteness of the first two moments of the macro and micro factors implied inside the functions (Barnett (1979), p.112). The conclusion is that: “No other model has been shown to have such an attractive connection with theory after aggregation over consumers under weak assumptions” (Barnett y Serletis, 2009, p. 77). The Rotterdam demand model presents a series of advantages. First, “the Rotterdam model has the virtue of making the regressors station­ ary” (Lewbel & Ng, 2005, p. 487) because it is a model with first-difference variables. Therefore, it follows Granger and Newbold (1974): “we recommend taking first differences of all variables that appear to be highly autocorrelated. Once more, this may not completely remove the problem but should considerably improve the interpret­ ability of the coefficients” (p.118). Second, it is most closely connected to the theory after adding consumers on the basis of weak assumptions (Barnett & Serletis, 2009, p. 77). Lastly, the Rotterdam model has a well-behaved error structure (Barnett & Serletis, 2008). Empirical rejection of He and Liu (2016) is, in my view, incorrect because it would be based on a disregard of the fact that the matrix is singular (Clements and Gao, 2014; Conniffe & Hegarty, 1980, p. 103; Deaton & Muellbauer, 1980, p. 321, p.7). The model (Barnett & Serletis, 2009; Clements & Gao, 2015) starts from a Marshallian demand for good. It takes logarithms (log) and ap­ plies the total differential to this equation. The Rotterdam demand equation, in this version of absolute, non-deflated prices, states that the change in the log for the consumed quantity of the good, weighted by share in expenditure, is equal to the change in actual income (approxi­ mated by the change in the log of one index of quantity consumed), multiplied by elasticity in relation to income and by this share in expenditure, plus the sum of the changes in the logs of the prices of the various goods multiplied by their respective compensated price elas­ ticities, again multiplied by this share. It is a linear model. Equations in annex 1. Schematically, the theoretical restrictions are as follows (annex 1):

� Symmetry: the cross-derivatives of the Hicks demand function are equal. � Negativity and curvature, deduced from the concavity of the expenditure function. The Hessian matrix of second partial de­ rivatives of the cost function is a negative semidefinite. Own-price elasticity for the Hicks demand is not positive. � Positivity: that the cost function is non-negative. � Monotonicity. The consumer always prefers more of the goods if possible (a condition emphasised by Barnett, 2001). Aggregation, homogeneity and symmetry are set as conditions in the estimation. Positivity and monotonicity must be observed in the esti­ mation. Negativity and curvature must be proven from the Hessian matrix values. The model to be estimated has three equations, where i, j ¼ 1,2,3. The dependent variable in the first equation is x1 (air passengers), in the second x2 (train passengers) and in the third x3 (a synthetic index of car and bus passengers), subject to the restrictions set out. The estimation is performed with and without constants. ui is the error term, under the assumptions of white noise. Equation (7) is the main component of the system: X ω*it Dxit ¼ ki þ θi DQt þ π ij Dpjt þ uit (7) j

That said, having the theoretical restrictions verified is one condition, but it does not guarantee that the neoclassical theory of utility maxi­ mization can be satisfied, due to the fact that it is estimated with noninfinitesimal data and assumed to have constant parameters over time (Deaton, 1974). The Rotterdam model assumes that the variables to the right of the equation are exogenous. Estimation by ordinary least squares (OLS), or by the seemingly unrelated regression (SUR) method, is preferable to the instrumental variable or the two-stage least squares methods. Zellner’s (1962) SUR method is used. The two objectives in using the SUR method are to set and/or test restrictions for the parameters in the different equations and improve efficiency in the estimation (Zellner, 1962, p. 363). However, in the Rotterdam model, all the exogenous variables are the same in all the estimated equations. The OLS residuals are orthog­ onal to the exogenous variables, which are now the same in every equation, complying with the OLS condition. Therefore, the SUR method is only used to impose and test cross-equation theoretical restrictions over the system of demand equations. Mathematical equations for the SUR method can be found in Annex 2. The model assumes that the conditions of linear regression are satisfied and that the inference is valid for linear estimators. The error terms of the equations are not correlated, not heteroscedastic and have a normal distribution. In order to avoid a singular covariance matrix, and due to the equality between total expenditure and the sum of the prices multiplied by quantities (Barten, 1968, 1977), two of the three equations in the system are estimated. Barten demonstrated that the estimation did not vary, irrespective of the hidden equation. The coefficients of the equa­ tion not included can be recalculated from the restrictions. 4. Data The Acela train service commenced in late 2000. However, the years 2001 and 2002 were affected by the September terrorist attacks, together with the gradual introduction of the Acela service, and for these reasons it seems reasonable to begin the estimation in 2003. As the firstdifferences in the Rotterdam model are taken, the annual data ultimately covers 2004 to 2016. Between 2003 and 2013, the passengers and prices data set for the Acela and Northeast Regional services can be found in Archilla T�ellez’s (2013) Table 2.4, “Performance of Acela Express and Northeast Regional (NR) FY 2003–2012”. To complete the set, the data

� All of the income is spent. The sum of the change in expenditure for the different goods must be equal to the change in total expenditure or income (Barten, 1969). � Zero-degree homogeneity. Multiplying prices and income in the same proportion does not change demand decisions. 3

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Research in Transportation Economics 78 (2019) 100759

can be found in the “Monthly Performance Report” (Amtrak) and in the surveys published by the Rail Passengers Association. Data on air passenger transport is as follows: since 1993, the Bureau of Transportation Statistics has been carrying out random monthly surveys on 10% of all tickets sold in the United States, prices and quantities (U.S. DB1A data, U.S. domestic). It gives the average price of the journey for each route, weighted by the total number of passengers on each route. The routes are: New York-Washington DC, Boston-New York, Boston-Philadelphia and Boston-Washington DC. In regard to the data for car and bus, there is no direct estimation of the number of transported persons. “Accurate passenger traffic statistics are not available for the intercity bus sector due to the fact that no federal government agency compiles and audits such statistics” (Schwieterman et al., 2011, p. 6). It is necessary to use the vehicle count for the various state-run road services at points on the motorway linking the Corridor, i.e. locations that can reasonably be expected to have an approximate count of vehicles that travel between the cities. Several series of data were compared to give consistency to the estimation. Of all the sets available, I have chosen the following:

For a “Small sedan”, in 2008, the gasoline cost would be 13.08 cts./mile, whereas the overall cost of owning and driving it would increase to 35 cts./mile, from 15.29 cts./mile to 46 cts./mile for a “Medium sedan”, from 17.25 cts./mile to 58 cts./mile for a “Large sedan” (American Automobile Association 2008; in Litman, 2009, p. 31, VTPI, p. 5.1–6). Estimation is pursued for both cases. The set of prices for bus use comes from the Table 3–18 “Average Passenger Fares (Current dollars) Class I bus, intercity” (Bureau of Trans­ portation Statistics). As it does not cover the entire time series, I am completing and extrapolating it from the increases in index numbers, calculated from diesel prices ((Energy Information Administration US) 2018). Finally, a passenger-weighted average is calculated from car and bus prices. There are two alternative series: WCBP when the total cost of cars is considered and MCBP when only petrol prices are. Expenditure statistics are found in Table 1100 “Quintiles of income before taxes: Average annual expenditures and characteristics, Consumer Expenditure Survey” (U.S. Bureau of Labor Statistics BLS). Additional data are given in Table 1800 “Region of residence: Average annual ex­ penditures and characteristics” (U.S. BLS). Table 1100 distinguishes be­ tween consumer expenditure on “Gasoline and motor oil” and “Gasoline on out-of-town trips”, and “Parking fees” and “Parking fees on out-oftown trips”. Moreover, this Table provides data about expenditure on: “Airline fares”, “Intercity bus fares”, “Intercity train fares” and “Taxi fares and limousine services on trips”. However, the expenditures on “Vehicle purchases (net outlay)” and “Other vehicles expenses” do not include this distinction about out-of-town trips. I calculate the per­ centage of expenditures on out-of-town trips in relation to expenditures on the entire related set (e.g. Gasoline and others on out-of-trips as a % of Gasoline and others). I then multiply this value by the different ex­ penditures on cars.

� Delaware, “Highway Statistics”, data on annual vehicle miles travelled. � Maryland, “HISD Reports”, data on Interstate 95 (the I-95). There are alternatives such as the US 1 or the US 29, but the data on the I-95 give us an approximate picture of the annual evolution. It is set on annual vehicle miles travelled, with functional classification.3 There are two data-collection points. These sets are comprehensive for the whole period and their high correlation indicates an evolution in the same direction. Several ad­ justments are made: dividing the mean journey by motorway on the NEC and estimating the share corresponding to cars, as well as that consisting of buses. This is done from Table 1–40, “U.S. Passenger-Miles (Millions)” (Bureau of Transportation Statistics. National Transportation Statistics.) 2017, which gives an overall quantification of nationwide passenger-miles in the United States and a classification of vehicles from which the percentage of traveller-miles completing the journey by car4 and by bus can be established. Lastly, the sets are multiplied by the number of travellers on average per vehicle; travellers by car: 1.54 (“Average vehicle occupancy, car”, Office of Energy Efficiency and Renewable Energy); travellers by bus: 15.1712 (mean for NEC zones in the set “Annual Average Vehicle Occupancy Factors per Cars, Buses and Trucks for PHED Metrics”, buses, FHWA, from the National Transit Database). Two series are used: DELHQ containing car and bus passengers from Delaware AVMT statistics and AVEHQ from the average AVMT (Dela­ ware and Maryland). The prices for car use come from Table 3–17, “Average Cost of Owning and Operating an Automobile, assuming 15,000 Vehicle-Miles per Year” (Bureau of Transportation Statistics, from the American Automobile Association). This gives an estimation of cost per mile. The prices for petrol and other oil products can be found in Tables 9.4 and 9.7 (U.S. Department of Energy, EIA, Monthly Energy Review) and in Table 5.3 (US Department of Transportation, FHWA, Highway Statistics). These costs are multiplied by the mean distance of the NEC journey and divided by the mean number of travellers per vehicle. There is doubt as to whether to enter the overall cost of buying and maintaining a car, prorated by miles travelled. Dealing with inter-city travel can be un­ dertaken either by considering as a marginal cost for each additional mile the entire prorated cost of the purchase, maintenance, insurance and others, or a marginal cost determined solely by the price of petrol.

5. Results Group 1: NEC air passenger travel. Group 2: NEC train passenger travel. Group 3: weighted index of NEC car and bus passenger travel. Aggregation, homogeneity and symmetry are set as restrictions in the estimation. Monotonicity, positivity and curvature must be proven. Average shares in total expenditure over the period considered: Air transport (ω1) 0.4000; train (ω2) 0.0211; car and bus (ω3) 0.5789. The estimation of coefficients is shown in Tables 1.1 and 1.2. Elas­ ticities are in Tables 2.1 and 2.2. Alternative elasticities are in Table 3. The estimation has been obtained by using R programming language (R Foundation for Statistical Computing) and the software package is “systemfit” (Henningsen & Hamann, 2007). Estimation with fixed parameters gives incorrect signs. Positivity and Table 1.1 Parameters estimates of demand for modes of passenger transport in the NEC. Rotterdam model, SUR estimation. 2004–2016a. ω*it Dxit ¼ ki þ θi DQt þ

P

πij Dpjt þ uit

i, j ¼ 1, 2, 3. Air transport (1), train (2), car and bus (3). Option 1. Car fares: WCBP (total cost). Road passengers: DELHQ (Delaware AVMT). Marginal budget share

Slutsky coefficients

Transport Mode

θi

πi1

πi2

πi3

Air transport (i ¼ 1).

0.6168 (0.1051) (5.8666) 0.0015 (0.0239) (0.0642) 0.3817 (0.1047) (3.6408)

0.0690 (0.0293) (-2.3560) 0.0046 (0.0153) (0.2982) 0.0644 (0.0247) (2.6125)

0.0046 (0.0153) (0.2982) 0.0079 (0.0142) (-0.5609) 0.0034 (0.0048) (0.7073)

0.0644 (0.0247) (2.6125) 0.0034 (0.0048) (0.7073) 0.0678 (0.0247) (-2.7563)

Train (i ¼ 2). Car/bus (i ¼ 3).

3

11 – INTERSTATE. 4 “Passenger cars” up to 2006, “Light duty vehicle, short wheel base” from 2007.

a

4

Standard errors and t-Student values in parentheses.

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Research in Transportation Economics 78 (2019) 100759

Table 1.2 Parameters estimates of demand for modes of passenger transport in the NEC. Rotterdam model, SUR estimation. 2004–2016a. ω*it Dxit ¼ ki þ θi DQt þ

P

Table 2.2 Elasticities estimates. Demand for modes of passenger transport in the NEC. Rotterdam model, SUR estimation. 2004–2016. Option 2. Car fares: WCBP (total cost). Road passengers: AVEHQ (average AVMT).

πij Dpjt þ uit

i, j ¼ 1, 2, 3. Air transport (1), train (2), car and bus (3).

i,j ¼ 1, 2, 3. Air transport (1), train (2), car and bus (3).

Option 2. Car fares: WCBP (total cost). Road passengers: AVEHQ (average AVMT).

Hicksian price elasticities (ηij)

Marginal budget share

Slutsky coefficients

Transport Mode

θi

πi1

πi2

πi3

Air transport (i ¼ 1).

0.7468 (0.1167) (6.3990) 0.0052 (0.0221) (0.2369) 0.2479 (0.1142) (2.1689)

0.0561 (0.0274) (-2.0456) 0.0074 (0.0123) (0.6031) 0.0487 (0.0238) (2.0461)

0.0074 (0.0123) (0.6031) 0.0104 (0.0111) (-0.9379) 0.0030 (0.0039) (0.7692)

0.0487 (0.0238) (2.0461) 0.0030 (0.0039) (0.7692) 0.0517 (0.0234) (-2.2193)

Train (i ¼ 2). Car/bus (i ¼ 3).

a

ηi2

ηi3 0.1610 0.1598 0.1171

εi2

εi3

εi1

0.7893 0.1867 0.1524

0.0212 0.3771 0.0081

σi1

0.4312 0.5395 0.2782

σi2

0.5395 17.7654 0.2760

0.7317 0.1178 0.4988

σi3

0.2782 0.2760 0.2023

Income elasticities (ξi) Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

ξ1 ξ2 ξ3

0.8870 0.2513 0.0873

0.0210 0.4993 0.0038

0.9593 0.0004 0.3373

σi1

σi2

0.3505 0.8760 0.2102

0.8760 23.3714 0.2483

ξ1 ξ2 ξ3

1.8672 0.2476 0.4283

σi3

0.2102 0.2483 0.1543

intercity passenger transport on the NEC complies with the necessary theoretical restrictions. Secondly, this estimation gives us an acceptable approximation to the actual value of the demand elasticities for air and car/bus transport modes on the NEC, in relation to income and their prices. Thirdly, the quantification of demand for train passenger trans­ port does not show the required statistical relevance. In order to recognise the reasons for this statistical non-relevance, we would need more and more precise information about the different modes inside the NEC. “One way of interpreting lack of statistical significance is that further information might change one’s decision recommendations” (Gelman & Stern, 2006, p. 329). Without forgetting the non-significance of the parameters for train transport, all the modes of transport seem to have negative Hicksian elasticities for their price and the prices of the other modes seem posi­ tive. They are, therefore, net substitutes. However, the coefficients denote that these elasticities are low and the price substitution effects are relatively weak. Air transport seems to be a luxury good, while train and car/bus would be necessary goods. The Marshallian price elasticity is negative for the specific prices of each mode of transport. The value is low and the effect inelastic. The demand for plane and car/bus show that both of them are grossly complementary to one another and to the train service because the income effect predominates that of the substitution. The transport of passengers by train seems to be gross substitute for other services. The Allen-Uzawa elasticities indicate that there is a low substitution between the different modes of transport in the NEC.

Allen-Uzawa elasticities of substitution (σιϕ) Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

εi3

Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

Marshallian price elasticities (εij) Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

εi2

εi1

Income elasticities (ξi)

Hicksian price elasticities (ηij) 0.0114 0.3756 0.0058

0.1217 0.1437 0.0893

Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

i,j ¼ 1, 2, 3. Air transport (1), train (2), car and bus (3).

0.1724 0.2158 0.1113

ηi3

0.0185 0.4941 0.0052

Allen-Uzawa elasticities of substitution (σιϕ)

Option 1. Car fares: WCBP (total cost). Road passengers: DELHQ (Delaware AVMT).

ηi1

ηi2

0.1402 0.3504 0.0841

Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

Table 2.1 Elasticities estimates. Demand for modes of passenger transport in the NEC. Rotterdam model, SUR estimation. 2004–2016.

Transport Mode

ηi1

Marshallian price elasticities (εij)

Standard errors and t-Student values in parentheses.

Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

1.5422 0.0726 0.6593

monotonicity are satisfied: quantities and expenditure shares are nonnegative. Negativity and curvature conditions require the Hessian ma­ trix to be a negative semidefinite. First, the Hessian matrix is singular due to the theoretical restrictions of the Rotterdam model, its determi­ nant is zero and one eigenvalue is also zero. Second, the determinant of the 1 � 1 top left submatrix of the Hessian matrix is: D1 < 0, the deter­ minant of the 2 � 2 top left submatrix is: D2 > 0. The eigenvalues are (0.0000, 0.0119, 0.1328) and (0.0000, 0.0155, 0.1027). The re­ quirements are fulfilled. Therefore, the theoretical conditions seem to be satisfied. Estimated elasticities from using gasoline prices can be found in annex 3. The demands for air travel and car/bus services have significant coefficients for θ1 and θ3 (approx. 99% significance) and π11, π13, π31 and π33 (approx. 95%). There is a lack of significance of the coefficients associated with the prices of train travel and with the regression equa­ tion for the number of train passengers. Consequently, the estimation of a Rotterdam system of demand for

6. Conclusions The precedent empirical quantification allows us to affirm the following conclusions in relation to demand for passenger transport between cities in the NEC for the most relevant modes (air, train or road). First, the estimation of the price and income elasticities of demand must depend on two necessary conditions: the restrictions of the theory of rational maximization of consumers’ behaviour and the simultaneous estimation for all relevant modes. Second, the Rotterdam demand model is a theoretical system that can be applied to this estimation problem with a series of advantages, allowing quantification of these elasticities. The estimation method is 5

E.R. Ignacio

Research in Transportation Economics 78 (2019) 100759

the SUR. The two previous points are a methodological proposal in order to understand the transport problem in the mega-regions. Third, the shares in expenditure on car/bus and air transport are high in terms of the overall transport expenditure considered. Fourth, all the income elasticities are positive: all the transport modes are normal goods. The income elasticity for air transport is high and shows that it is a luxury good. The demand for train and car/bus would be relatively inelastic with respect to rent. Strong economic growth would generate a much higher pressure of demand on air transport and this would necessitate enlargement of air service in­ frastructures and capacity, thereby avoiding congestion problems. Similarly, an economic recession would impact more decisively, in proportion, on air travel than on other modes. Fifth, all modes of transport have negative demand price elasticities, whether compensated and uncompensated, although their values are less than 1. What is not surprising is that “transportation is a derived demand, it tends to be inelastic” (Oum et al., 1990, p. 8). The Mar­ shallian elasticity of air transport is, however, in absolute value, superior to that of the other modes. Sixth, all the Hicksian elasticities show consistent signs of being competing services, all the modes of transport being net substitutes. However, these elasticities are low and the price substitution effects are relatively weak. The demand for plane and car/bus transport shows both of them to be grossly complementary to one another and to the train service because the income effect predominates that of the substitution.

The transport of passengers by train seems to be gross substitute to the other services. The impact of movements in the price of road passenger transport is reflected in the other modes through the effect on the aggregated expenditure. Negative impact of a rising road pricing on air transport demand stands out owing to the fact that this transport demand is elastic in regard to income. Finally, air transport would be key when the economy and, accord­ ingly, income is experiencing intense growth, although trains would be able to sustain a very limited replacement demand away from planes. Moreover, road transport does not have clear substitutes, being inelastic in the face of own-price or income, grossly complementary with air transport service, and with a near zero replacement by train service. In this sense, the possibility noted by McNeil et al. (2010) may because for concern: “The growth rate in the demand for transportation is exceeding the ability of the highway system to expand at a rate to handle the growth” (p.7). An analysis of investments for preventing congestion points, as well as for increasing mobility between the modes of trans­ port, seems to be a matter of urgency. Acknowledgements I thank Professors.Genoveva Mill� an V� azquez de la Torre and Leonor M. P�erez Naranjo for their support. I thank Taylor J. Wilson, economist, Bureau of Labor Statistics, for his help to access data. And I have received the valuable comments of three anonymous referees.

Annex 1. The Rotterdam model (Barnett & Serletis, 2009; Clements & Gao, 2015) starts from the neoclassical consumer theory: taking a utility function, subject to the budget constraint, obtaining first-order conditions for utility maximization, and arriving at the Marshallian demand for good i. The Rotterdam model comes from the total differential of those first-order conditions of the utility problem (Barnett & Serletis, 2009, pp. 63–65). In addition, the Slutsky equation is introduced and log are applied. Certain decisions are taken about parameterization. The equations of the model are as follows, from Barnett and Serletis (2009). The differential Rotterdam demand system in relative prices, equation (1), being i ¼ 1, 2, …, n (Barnett and Serletis (2009), p. 68): X � (1) ωi d lnxi ¼ θi d lnQ þ vij d lnpj d ln Pf j

Where x are quantities, p are prices, ωi is budget share of i, θi is marginal budget share of the i use of money income (θi ¼ ωi ηiy ¼ piyxi P λp p u coefficient of the j relative price ( vij ¼ φθi ¼ ∂∂yλ yλ piyxi ∂∂xyi xyi ¼ i yj ij ), y is income, λ is the Lagrange coefficient.

∂ xi y ∂y xi ),

vij is the

j

dlnPf is the Frisch price index: it uses marginal shares as weights, instead of budget shares, used by the Divisia price index. And dlnQ is the change in real income (applying a Divisia volume index). X d lnPf ¼ θj d lnpj (2) j

X d ln Q ¼ d ln y

(3)

ωj d lnxj

d ln P ¼ J

The Rotterdam demand system in absolute prices is shown in equation (4), i ¼ 1, 2, …, n (Barnett and Serletis (2009), p. 69): X ωi dlnxi ¼ θi dlnQ þ πij d lnpj

(4)

j

ηiy ¼ IncomeElasticity ¼

θi

(5)

ωi

ηij * ¼ CompensatedPriceElasticity ¼

π ij ωi

(6)

πij is the Slutsky coefficient: πij ¼ vij

φθi θj . The Rotterdam model’s parameterization of these equations is as follows:

� Infinitesimal changes are now finite changes, from t-1 to t. � The coefficients are constants. 6

E.R. Ignacio

� ω*it ¼ ωit þ2ωit

Research in Transportation Economics 78 (2019) 100759 1

(Barten, 1969).

Therefore, the absolute prices equation to be estimated is (i, j ¼ 1, 2, 3; t ¼ 1, 2, …, T): X ω*it Dxit ¼ ki þ θi DQt þ πij Dpjt þ uit

(7)

j

Where Dzt ¼ lnzt

lnzt

1.

Now: DQt ¼

P3

* j¼1 ωjt Dxjt .

θi and πij are constants.

Mathematical theoretical restrictions are as follows. All of the budget is spent (adding up):

3 X

(8)

θi ¼ 1 i¼1

Zero-degree homogeneity (demand does not move if all prices and expenditure change proportionally): 3 X

(9)

π ij ¼ 0

j¼1

The cross-derivatives of the Hicks demand function are equal, since utility must be constant through demand compensation: (10)

π ij ¼ πji

Negativity and curvature. The Slutsky matrix [πij] is negative semidefinite and the compensated own-price effect must be non-positive (C is the cost function): H¼

∂2 Cðp; uÞ �0 ∂pi ∂pj

(11)

∂hi ðu; pÞ �0 ∂ pi

(12)

Therefore, all the eigenvalues of the [πij] matrix are less than or equal to 0. Positivity: that the cost function is positive. Monotonicity: that the derivatives of the cost function are non-negative, expenditure must be

monotonically increasing in prices. Since Shepard’s Lemma determines that: hi ðp;uÞ ¼ ∂Cðp;uÞ ∂pi , this can be checked by testing whether the quantities and expenditure shares are non-negative. Therefore, the ω must be positive. Aggregation, homogeneity and symmetry are set as conditions in the esti­ mation. Positivity and monotonicity must be observed in the estimation. Negativity and curvature must be proven from the [πij] matrix values. Annex 2. SUR estimation: the starting point is m linear regressions (Zellner, 1962, p. 349): Yi ¼ Xi βi þ ui

(13)

� E ui uj ¼ σij IT

(14)

In the form of matrices and vectors: 1 0 10 1 0 1 y1 X1 0 ⋯ 0 β1 u1 B y2 C B 0 X2 ⋯ 0 CB β2 C B u2 C B C¼B CB C þ B C @ ⋯ A @ ⋯ ⋯ ⋯ ⋯ A@ ⋯ A @ ⋯ A 0 0 ⋯ Xm ym βm um 0

Next, we have (Zellner, 0 σ11 I σ 12 I X B σ21 I σ 22 I B VðuÞ ¼ ¼@ … … σm1 I σm2 I

(15)

1962, p. 350): 1 … σ 1m I X … σ 2m I C C¼ �I … … A C … σmm I

(16)

Where � means Kronecker product. The disturbance vector is assumed to have the variance-covariance matrix Σ. I is a unit matrix of order T � T (t ¼ 1, 2, …T: j ¼ 1, 2, …, m). Variance (i ¼ j) and covariances: σij ¼ Eðuit ujt Þ. 0 11 1 1 σ I … σ 1m I X 1 @ ¼ V ðuÞ ¼ (17) … … … A σ m1 I … σ mm I The best linear unbiased estimator of the coefficients, then, is (Zellner, 1962, p. 351):

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Research in Transportation Economics 78 (2019) 100759

E.R. Ignacio

1

0

0

βest

^ 11 ’ B σ X 1 X1 1 � 1 X 1 B σ21 X ’ X � X B 2 1 ¼ X� X X� y ¼ B B … @ σm1 X ’m X1

σ 12 X ’1 X2 σ 22 X ’2 X2

… …

…

…

σm2 X ’m X2 …

m X σ1j X1’ yj C B 1B C j¼1 1m ’ C 0 B 1 σ X 1 Xm CB C est β m 1 C BX 2m ’ C B est C σ X 2 Xm CB σ2j X2’ yj C C ¼ B β2 C CB C @ … A j¼1 CB … C B AB … C est β m C σ mm X ’m Xm B m C BX mj ’ A @ σ X yj

(18)

m

j¼1

This is a generalized least squares estimator. If a normality assumption is added, it is a maximum likelihood estimator (Zellner, 1962, p. 351) “Maximum likelihood (ML) estimators are consistent, asymptotically efficient, and asymptotically normally distributed” (Barten, 1969, p. 22). Σ is unknown. The estimation of Σc, and then Σ, is performed from the variance-covariance matrix: {σij} are estimated from {u’iui’}, by iterative methods (Barten, 1969, p. 47). When all the independent variables are the same, as in the Rotterdam model, there is no covariance of error terms and no data that can be added to improve the estimations. The SUR would be equal to single-equation least-squares (OLS) estimators (Zellner, 1962, p. 351), in the absence of multi-equational restrictions. However, the SUR method is used because of those cross-equation restrictions; OLS estimation, equation by equation, is not possible. The covariance matrix of the system is singular, since there are n dependent variables and m > n equations (behavioural ones plus identity re­ strictions). One equation is deleted before the estimation procedure (Haupt & Oberhofer, 2000). Barten proved that any equation could be suppressed; the results would be the same (Barten, 1969). The omitted equation can be estimated by applying restrictions on coefficients. Annex 3. Alternative estimation of elasticities

Table 3 Alternative elasticities estimates. Demand for modes of passenger transport in the NEC. Rotterdam model, SUR estimation. 2004–2016. Option 3. Car fares: MCBP (gasoline cost). Road passengers: DELHQ (Delaware AVMT).

Option 4. Car fares: MCBP (gasoline cost). Road passengers: AVEHQ (average AVMT).

i,j ¼ 1, 2, 3. Air transport (1), train (2), car and bus (3).

i,j ¼ 1, 2, 3. Air transport (1), train (2), car and bus (3).

Hicksian price elasticities (ηij)

Hicksian price elasticities (ηij)

Transport Mode

ηi1

ηi2

ηi3

Transport Mode

ηi1

ηi2

ηi3

Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

0.1131 0.3403 0.0657

0.0180 0.4362 0.0035

0.0952 0.0959 0.0692

Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

0.0865 0.3741 0.0461

0.0198 0.4648 0.0033

0.0667 0.0907 0.0494

ei2 0.0147 0.4391 0.0103

ei3 0.8008 0.0179 0.4474

ei2 0.0199 0.4691 0.0056

ei3 1.0207 0.0265 0.2938

Marshallian price elasticities (eij) Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

ei1 0.7321 0.2864 0.1955

Marshallian price elasticities (eij) Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

Allen-Uzawa elasticities of substitution (sij) Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

si1 0.2829 0.8509 0.1644

si2 0.8509 20.6346 0.1656

Allen-Uzawa elasticities of substitution (sij) si3 0.1644 0.1656 0.1196

Transport Mode Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

Income elasticities. (xi) Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

x1 x2 x3

ei1 0.8378 0.2931 0.1227 si1 0.2163 0.9353 0.1153

si2 0.9353 21.9878 0.1567

si3 0.1153 0.1567 0.0854

Income elasticities. (xi) 1.5476 0.1348 0.6532

Air transport (i ¼ 1) Train (i ¼ 2) Car/bus (i ¼ 3)

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x1 x2 x3

1.8785 0.2024 0.4222

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