Douglass B. Lee, Jr. and Mark W. Burris1
An elasticity summarizes a large amount of information in a single number, including the choices of consumers with different preferences, with different options, which change over time. The elasticity concept normalizes for the measurement scales (e.g., pounds or kilograms, dollars or pesos) and price levels (to the degree that the demand curve is constant elasticity), but other factors are simply averaged over the consumers being observed. Moreover, although the price elasticity of travel demand is frequently mentioned in discussion as if it were a single datum, the many empirical estimates often measure conceptually different things.
The review and synthesis presented in this appendix was conducted for the purpose of establishing values for use in the HERS model to represent the short-run generalized-price elasticity of demand on a given highway section, and to estimate the long-run share parameter used in the model for estimating induced demand, as described in Appendix B, "Induced Traffic and Induced Demand." For the 2001 Conditions and Performance report to Congress, the values selected were −1.0 for short-run elasticity, and −0.6 for the long-run elasticity supplement, giving a full long-run elasticity of −1.6.
As an empirical measure, an elasticity is a microeconomic aggregate: it summarizes demand in a specific market at a point in time, at or near the prevailing price and quantity. A market could be a highway facility, a corridor, or an entire region. A point in time might be a peak hour, or a daily average. The basic concept can be represented as an arc elasticity between two demand points,
where v0 = initial traffic volume and p0 = initial price. Elasticity can be thought of as a measure of slope normalized for the arbitrary measurement scales of p and v, and, indeed, the form to the right of the second equal sign is a slope multiplied by the initial demand point. If the derivative of the demand curve is substituted for the slope, the elasticity is instantaneous at the given point, rather than over an arc. For exposition, the arc form will be used, but the principles apply to either form.
Because an elasticity is simply a summary of the behavior of a particular group of consumers for some period of time, there is no reason to believe that the same elasticity measure will have the same value at another time or place. At a different time point on the same facility, a different group of users may respond differently to a change in price, because they have different incomes, demographic characteristics, and different purposes. Additionally, prices in related markets (e.g., parallel facilities) may be different. If different facilities are being compared, all of the above may be different as well as differing substitute alternatives (routes, carpool and transit options, destinations, schedule options). Different days, seasons, regions, and forms of "price" all limit the transferability of an elasticity measured in one context to another context.
Hence - unlike the speed of light or the age of a rock - there is no underlying true number waiting to be discovered. Similar circumstances are likely to exhibit roughly similar elasticities, but the most important characteristics of these circumstances need to be made explicit.
The basic economic model of exchange reconciles supply and demand in a market, using price. The "price" is the market value of the resources given up by the buyer and received by the seller, accomplished in modern markets by means of some form of money that both parties agree to use as representing valuable resources. The market of interest in the present context is highway travel, measured as vehicle miles of travel (VMT). A single market has a single price, so a highway market could be a street length between intersections at a specific time of day. The concept of a market can be applied more broadly, however, to consist of a facility whose demand is averaged over the day, or a network of facilities. Elasticities will generally be larger (elastic) the more alternatives (route shift, time of day shift, add or forego trip) are available, and smaller (inelastic) the more broadly the market is defined (e.g., region versus facility). For present purposes, the focus is on a single facility, and its daily VMT.
If the price measure were limited to the money price paid by the user to obtain the services of the highway, then such characteristics as travel time, pavement roughness, risk of accident, scenery, and curves and grades would be attributes of the service. Such a money price (assuming the user had access to a vehicle) would be a multi-part price consisting of vehicle registration fee, drivers' license fee, excise taxes on fuel and tires, and tolls.
This formulation of the highway market has limited usefulness for several reasons. Primarily, the fees paid are a small part of the total cost to the user of highway travel, and attributes of the good (such as travel time) dominate the choice rather than price. Because travel is a derived demand, at least some of the attributes can be thought of as part of the cost to be minimized, including time and operating costs as well as user fees.
An alternative formulation is to treat some of the attributes as disutilities, and translate them into a dollar price. Operation of the vehicle, travel delay, and tolls are thus all costs to the user, or components of the price. In practice, the only way to estimate the demand elasticity of highway travel with respect to generalized price is to build up total travel demand elasticity from elasticities of the components of user costs.
If the price of a service is made up of several components, then the elasticity of demand may be broken into portions representing the components (or their elasticities). The relationship between the component elasticities and the overall elasticity can be viewed in several ways, the simplest being as arbitrary divisions of the price, the more complex way being as a set of input factors that must be purchased separately.
Suppose the price for a particular good is $2.40 (1st component) plus $1.60 (2nd component); then the price of the good is simply $4.00. Assuming that the price elasticity for this good is a known (say −0.8), then a change in its price (say −$0.80) implies a specific increase in the quantity demanded (up 16%). This example is shown in the top row and righthand column of Table C-1.
|1st component||2nd component||Overall|
|price||$ 2.40||$ 1.60||$ 4.00|
|change in price||$ (0.80)||-||$ (0.80)|
|% change in price||−33%||0||−20%|
|share of price||60%||40%||100%|
|% change in quantity||16%||0%||16%|
From the standpoint of one component (say, the 1st), this price change is a different percentage (−33%) than from the standpoint of overall price (−20%). When the elasticity of demand with respect to the price component is measured (or observed), it is smaller (−33%/+16% = −0.48) than the overall elasticity because the relative price change is magnified whereas the quantity change is the same.
If the share of the component in the total price is also known, then the overall price elasticity can be calculated (−0.80 = −0.48/0.60) from the component elasticity (or the reverse, as is done in Table C-1). If the price change is applied to the other component, or spread between them, the component elasticities and price shares are unaffected, and the percent change in quantity will sum to the overall change.
Thus elasticities can be observed empirically for components and the overall elasticity inferred, or the overall elasticity can be "decomposed" into separate elasticities for each component. Price components are necessarily stated in common numeraire units ($), and the elasticity calculated for each component always implies the same overall price elasticity. Empirical estimates, of course, may contain noise or error and therefore not give consistent overall elasticity values.
The above example is trivial in its application, but provides a framework for incorporating more realistic complexity. In such a model, the separate price components can be viewed as inputs (vehicle, fuel, time, risk, highway space) that are purchased (car payments, excise taxes, tolls) and integrated by the producer (the highway user) to yield an output (vehicle miles of travel). This process sounds a bit like a supply function, but actually it is still a demand function: the user expresses her/his willingness to pay by buying the ingredients and spending the time. This perspective, however, raises some problems with respect to elasticities:
The first problem will be handled by converting everything to dollars, the second problem will be addressed by including a shrinkage factor estimated for the specific component, and the third will be ignored.
Thus three relationships are central to estimating total demand elasticities from component elasticities: the component's own-price elasticity, the correspondence of a change in the price of the component to a change in the price of travel, and the expansion from the component to the total elasticity.
If X is a component of the price of travel, and we observe its own price elasticity, then
where eX is the demand price elasticity of good X, ΔqX is the change in the quantity of X that is consumed (e.g., gallons of fuel), ΔpX is the change in the price of the good (e.g., the price of gasoline at the pump), pX is the initial price of the good, qX is the initial quantity of the good, and ΔqX/qX is the percent change in the quantity of good X. This is a relationship between the price of the component and its consumption, not the consumption of the overall good of which X is a component.
The latter relationship - between the price of X and the quantity demanded of another good - is a cross-elasticity. Higher fuel prices, for example, are partly absorbed in improved fuel mileage, so that the percentage reduction in fuel consumed is greater than the percentage reduction in VMT. The cross-elasticity of total travel with respect to the price of a component is
where eT,X = elasticity of travel demand with respect to a change in the price of X. The term between the equal signs is simply the definition of a cross-elasticity, and the right-hand side indicates that the new elasticity can be obtained from the component elasticity using a factor for the percent by which travel changes for a given percent change in consumption of the component. Stated more simply, then,
|eT,X = σ × eX, σ ≤ 1|
where σ = a shrinkage factor representing the share of a reduction (or change) in consumption of X that consists of reduction in travel. The extent of this "shrinkage" between the component price elasticity and travel demand elasticity depends upon the component, and the possibilities for economizing on the component other than by reducing travel. If σ = 1 then the component and travel are necessarily consumed in fixed proportions.
If the elasticity of VMT with respect to a part of the price is known, then the elasticity of total travel demand is simply an expansion from the part to the whole,
where eT = implied demand elasticity for travel (the overall good), pT is the price of travel, and the component elasticity eX is substituted using Equation C.4. The bottom of the right-hand side is the share of the component in the total price of travel. For example, if the elasticity of fuel consumption with respect to its own price is −0.25, and the shrinkage factor is 0.6 (from increases in fuel efficiency), then the elasticity of travel with respect to fuel price is −0.15. If fuel is only 20 percent of the cost of travel, yet has an elasticity with respect to total travel of −0.15, then the implied generalized-price demand elasticity is −0.75 with respect to the total price of travel.
Considerable empirical evidence can be found in the literature, and this information can be re-interpreted with respect to generalized price elasticity.
Although empirical studies are mainly oriented toward changes in one component of price, these studies can be extrapolated to the full generalized price. Once a total price elasticity is determined, then that value can be adjusted to apply to a specific context such as a highway section. This process is represented in Figure C-1. For use in the 2001 Conditions and Performance report, the section-level effects are assumed to cancel each other, on average, and the overall price elasticity is used for sections. The evidence and analysis presented here pertain primarily to passenger travel, although freight movement can be expected to respond in similar ways.
It is important to remember that the analysis concerns vehicle-trips, not person-trips. Although persons make the decisions for vehicles, vehicle trips are more readily observed, and the price typically applies to vehicle travel rather than to person trips.
The methodological strategy for moving from information about the components of user cost or "price" to generalized travel demand elasticity is represented in Figure C-2 and described below:
These steps are explained and implemented in more detail below.
Seven categories of user cost are listed in Table C-2. They are intended to be nonoverlapping and exhaustive. Fortunately, this set of categories is generally consistent with various estimates of user costs. The purpose of these categories is to be able to combine them in subsets that provide alternative measures of the "price," to distinguish fixed from variable costs as a means for defining the relevant costs, to match with empirical estimates of costs, and to match up with empirical elasticity estimates.
|Fuel||gasoline, diesel fuel, or other fuel consumed by motor vehicles, including taxes|
|Maintenance||oil, parts, periodic maintenance, unscheduled maintenance, tires, excise taxes|
|Accidents and Insurance||costs of accidents (internal), insurance administration and profit|
|Vehicle Wear and Ownership||wear and tear, additional depreciation, financing, sales and excise taxes|
|Tolls and Fees||tolls, registration fees, license fees|
|Parking||cost of parking to the user at work, shopping or other|
|Travel Time||dollar value of time spent in traveling|
Estimates of national averages for the cost components of highway travel are provided in Table C-3. All are intended to cover internal costs borne by users, omitting externalities, since elasticities necessarily must be based on internal costs. Four sources are presented, each of which offers a different orientation:
The sources and methods for tabulating the various price components are described in the notes to the table.
|(1) Fuel - BoE assumes $1.15 for fuel (including excise taxes) and 19.7 miles per gallon average fuel economy for all passenger vehicles (FHWA 1996 Highway Statistics); Delucchi value is sum of fuel costs, oil company producer surplus, and fuel taxes (Delucchi, 1998), divided by total 1991 US annual VMT of 2,172 billion (Highway Statistics); Runzheimer values are extracted by applying their 1998 percentage distribution of costs for and intermediate car to their estimates of annual fixed ($6,934) plus operating costs ($2,240) of a Ford Taurus (Runzheimer, 1997, 1998); FHWA values are for an intermediate sedan, including fuel taxes, at $1.196 per gallon (Jack Faucett Associates, 1991).|
|(2) Maintenance - BoE based on assumed value of $1,000 per year for oil, tires, parts, and maintenance, and an average annual mileage for passenger vehicles in 1996 of 11,492 (Highway Statistics); Delucchi estimates national expenditures on maintenance, including in-house government and private fleet maintenance, and sales taxes, but excluding external property damage, divided by national VMT (see Fuel); Runzheimer and FHWA are same as for fuel.|
|(3) Accidents and Insurance - BoE assumes $1,000 per year per vehicle for insurance and accidents, divided by average mileage (see Fuel); Delucchi's estimates include insurance administration, accidents paid by users, and pain and suffering "inflicted on oneself," but not external costs; Runzheimer and FHWA same as for Fuel.|
|(4) Wear and Ownership - BoE assumes a capital cost of $12,000 over 5 years, and average passenger car mileage (see Fuel); Delucchi estimates private ownership costs, excluding sales tax, divided by total US VMT (see Fuel); Runzheimer and FHWA same as Fuel.|
|(5) Tolls and Fees - BoE takes 1996 total toll payments nationally of $4 billion (Highway Statistics) divided by 1996 US VMT of 2,360 (Highway Statistics); Delucchi omits user fees as transfers, so a rate of 1.5 cents per VMT is inserted; Runzheimer estimates registration fees only; FHWA includes parking with tolls, but no adjustment is made here.|
|(6) Parking - BoE assumes $1 per day per vehicle for 250 days per year, over 11,492 annual miles (see Maintenance); Delucchi's values combine paid private parking and public parking; Runzheimer provides only a residual "Other" category; FHWA includes tolls with parking.|
|(7) Time - BoE uses 60% of US DOT (1997) "personal" wage rate of $17 and 1.2 persons per vehicle at an average speed of 40 mph; Delucchi values time in three categories - paid time that is delay, paid uncongested time, and unpaid time whether delay or not; neither Runzheimer nor FHWA include time costs, so the Delucchi value is used.|
|(3) Accidents and Insurance||0.087||0.133||0.070||0.070|
|(4) Wear and Ownership||0.125||0.142||0.248||0.127|
|(5) Tolls, Fees||0.002||0.015||0.003||0.009|
|Total Variable w/ Time||0.538||0.620||0.495||0.490|
Several criteria might be considered for defining total price:
Generalized price is used in benefit-cost evaluation of improvement projects for estimating induced demand and associated user costs, especially congestion; such a combined elasticity, however, may be inaccurate because the user does not give each component its dollar-value weight. Pragmatically, the choice of which price to use is also affected by (a) how other elasticities have been measured empirically, for comparison, and (b) how elasticity is used in the model or analytic procedure into which the parameter is inserted.5 With proper interpretation, different measured elasticities can be used to estimate a generalized price elasticity for modeling that is defined differently from the empirical sources.
Perceived price is an attempt to identify the components of price that are consciously recognized by the user, as a basis for predicting user behavior. Whether making this intermediate variable (perception) explicit adds anything to predictive accuracy is doubtful, and, in any event, it does not provide much guidance for which costs to include. Consumers tend to respond, as a group, to attributes and magnitudes that have some significant impact on their well-being, whether consciously perceived or not.
This means that the user response depends upon what decisions are at stake. If ownership is not in question, then only variable costs may be considered, and more so in the short run. Four alternative definitions for total price are given in Table C-3, ranging from short-run monetary costs to full long-run costs including travel time. The distinction between variable and fixed is not clean: vehicle wear is variable, while ownership is fixed. Insurance is typically paid annually and has a large fixed component, although most exposure to risk is from operation.
5. The HERS model uses a generalized price including time, operating costs, accidents, and user fees, but not parking; demand is aggregated over peak and off-peak (therefore no time diversion) and applies to a single facility (therefore the elasticity allows for route diversion).
The several definitions of total cost yield a range of component shares, shown in Table C-4. Obviously, the range of values is quite large, depending more upon which total is used than upon the source of data. With travel time being roughly half the cost, its inclusion makes a significant difference.
|Component||Low Share||High Share|
|Accidents and Insurance||7%||37%|
|Vehicle Wear and Ownership||18%||54%|
|Tolls and Fees||0%||10%|
The number of categories drops when considering which components are suitable for empirical estimation, and more so when actual estimates are tabulated. The cost of maintenance is difficult to keep track of, and either controlled or natural experiments are hard to imagine; no such studies were found. User responses to the risk of an accident based on equipment such as air bags have been used to estimate users' implicit valuation of life, but users cannot be observed reducing either their travel or their rate of accidents in response to changes in risk. Tolls, fees, and parking are clearly candidates, but apply to very specific circumstances. Direct estimates of time elasticities have been made, and indirect estimates can be derived from changes in traffic induced by changes in capacity.
Income elasticities are treated as exogenous for purposes of estimating travel demand elasticities, by assuming that price changes are not large enough or general enough to result in a significant change in income for the average traveler.
The size of the shrinkage factor (σ in Equation C.4) for fuel can be seen in Goodwin's (1992) review of elasticity estimates. Table C-5 summarizes his result for studies based on fuel price changes. The numbers are juxtaposed to permit comparison of elasticities of fuel consumption (average of over 100 separate empirical values) versus elasticities of travel (about a dozen numbers), stratified by whether a time series model or a cross-sectional model was used and whether the intent was short term, long term, or ambiguous.
Long term elasticity estimates tend to be at least twice as large as short term values, as might be expected. For fuel, the shrinkage from fuel consumption to travel consumption seems to be about 0.5 to 0.9, meaning that half to ninety percent of the reduction in fuel expenditure is the result of less travel.
|Method||Fuel or Travel||Short Run||Long Run||Uncertain|
|Source: Goodwin (1992).|
For other price components, the shrinkage factor is more speculative. Increases in insurance and vehicle ownership costs might result in fewer vehicles but more mileage per vehicle, with the latter less than fully offsetting the former. Deterioration in pavement quality increases wear and tear, reduces fuel mileage, and reduces speeds. Increases in tolls directly affect the cost per vehicle mile, but, depending upon how the tolls are graduated, could alter the mix of vehicles and the time-of-day distribution.
The largest user response is likely to come from those users for whom the price change is relatively largest. A fuel price or tax increase will affect long trips and vehicles with low fuel efficiency; insurance costs deter ownership in urban areas thereby shifting the geographic distribution of vehicles; high parking costs deter short trips more than long ones; high ownership or insurance costs deter vehicles with low annual utilization. Explicit shrinkage factors are not estimated for these components.
Some responses occur within days or weeks, while others may take five or ten years to reach equilibrium. To usefully interpret an empirical elasticity estimate, the time dimension must be known. If the statistical measure for an empirical estimate includes all VMT or other changes that occurred within a year of the price change, then a short-run elasticity has been estimated. Longer lag periods for the same price changes yield longer run elasticities, but separating behavioral responses from background variation becomes increasingly difficult.
Several studies reviewed by Cairns, et al. (1998) reveal the degree to which individuals change their travel patterns on a daily basis. Two studies, summarized in Table C-6, are illustrative. Of a group of commuters passing a given point on a road on a given day, 60% could be found a week later within the same 2-hour block; the rest were doing something slightly or completely different. A similar study with a longer time lag showed a smaller share doing the same thing. Other studies reinforce the same conclusion: individual travel variability is high on a day-to-day basis, and more so over longer time spans. This is without any significant changes in either exogenous or endogenous factors.
One implication is that - if stability is much higher in the aggregate than in the microscopic - attempting to predict individual travel behavior is less fruitful than using aggregate elasticities for endogenous changes. Another is that, with so many individuals making changes within a short time span, the responses of travelers to changed conditions is likely to be rapid. There is not a lot of inertia in travel patterns. According to Cairns, et al. (1998), roughly 50% of the response to a change takes place within 1 to 3 years, and 90% within 5 to 10 years. Hence, long-run elasticities tend to be about twice as large as 1-year or short-run elasticities.
Type of Travel:
|Leeds 1 Week commuting||Southampton 4 months regular trips|
|Source: Cairns, et al. (1998).|
|a travelled the same route by car within the same 2-hour time period.|
|No trip/different trip||6%||25%|
The results of the above process are displayed in Table C-7, showing empirical estimates of component travel elasticities, along with their implied short- and long-run total travel demand elasticities. The range of possible values is wide, extending from −0.22 to −3.7 for short-run demand and −0.57 to −5.1 for long-run demand. The most plausible numbers, however, lie in the −0.5 to −1.0 range for short-run demand, and −1.0 to −2.0 for long-run demand. These elasticities apply to vehicle travel, not person travel, which can be considerably less elastic and still be consistent with these vehicle elasticities due to changes in vehicle occupancy and other adaptations.6 The "high" values come from using the full generalized price as the base in Equation C.5., while the "low" values come from using subsets such as variable costs.
Numerous empirical studies have estimated the price elasticity of gasoline, and a few have measured the travel elasticity with respect to fuel price. The review and summary by Goodwin described above reflects the results of these studies, and subsequent studies have tended to confirm his conclusions. Thus a value of −0.16 for short-run travel impacts of fuel price changes and −0.33 for long-run impacts are used, with Equation C.5, in Table C-7.
6. Cairns, et al. (1998) provide an illuminating list of examples that illustrate the many ways in which individuals and households can satisfy their travel requirements while reducing vehicle miles of travel.
|User Cost Component||Component Elasticities||Implicit Total Travel Elasticities|
|Wear and Ownership||−0.12||−0.31||−0.22||−0.57||−0.6||−1.7|
Holding exogenous factors constant, an increase in the real price of vehicles of the same quality causes a reduction in the purchases of vehicles, especially new ones. The most likely behavior response is to defer purchase of a new or better vehicle, and keep using the old one. If, however, the response is measured in the aggregate as total vehicle ownership, then fewer vehicles means less VMT, offset by the extent to which vehicles are shared.
Dargay (1998) compared several ownership and operation elasticities between the UK and France, including price elasticity and income elasticity. Converting the price elasticities to VMT elasticities, using a shrinkage factor of 0.9 and the values from Table C-4 in Equation C.5, provides the results shown in Table C-7. The source of imprecision in applying ownership elasticities is the uncertain share of total cost per VMT comprised by ownership and wear-and-tear costs.
Elasticities in response to toll variations are difficult to impute because the quality of travel, in terms of delay and total travel time, often changes when the toll changes. First consider the case of a change in a flat rate toll, for example, a toll increase of $2 to $3 over the entire day. In this instance, the traveler must consider trade-offs between paying the higher toll and:
Therefore, considerations include a trade-off between a longer trip or paying the increased toll (value of time) and the trade-off between money and travel (price elasticity). Many empirical estimates of flat rate toll price elasticities in the U.S. have found elasticities generally ranging from −0.1 to −0.31.7
Next, the case of variable toll rates, where the toll varies by time of day or congestion level, is examined. In this case, tolls are generally greater during periods of peak traffic demand. Assuming congested conditions during the peak period, travelers must consider a trade-off between their preferred time of travel or traveling at an alternate time but paying a reduced toll (time elasticity) plus experiencing reduced delay (value of time).
The empirical evidence regarding driver response to variable tolls is limited and subject to considerable variation based on the specific opportunities available to avoid the toll. For example, some facilities that offer a variable toll are situated in the median of a free route and, therefore, altering travel route to avoid the toll is relatively easy. Other facilities offer discounts or free trips to high occupancy vehicles, increasing the benefit of sharing a ride over and above simply splitting the toll between vehicle occupants. Therefore, it was not surprising that elasticities found in the cases with variable toll rates had a wider variation that flat toll rates and were generally larger. They typically ranged from −0.02 to −1.0 (see Table C-8).
|Lee County (Cape Coral and Midpoint Bridges)||−0.02 to −0.36||Cain, Burris, and Pendyala (2001)|
|California (SR 91)||−0.9 to −1.0
−0.7 to −0.8
|Sullivan, 2000; Jan, Small, and Sullivan (2002)|
|California (I-15)||−0.02 to −0.16
−0.34 to −0.42
|Dahlgren (1999); San Diego Association of Governments (1999)|
|Singapore (City Center)||−0.25||Menon, Lam, and Fan (1993)|
|France (A-1)||−0.16 to −0.28||Small and Gomez-Ibanez (1997)|
|Houston (Hardy Toll Road)||−0.4 to −0.8||Spock (1998)|
Since this effort is focused on changes in VMT and not on the time of travel, the elasticities found for flat rate toll are used in developing a total travel demand elasticity. Because most trips are not tolled, national averages of tolls per VMT are not useful. Of trips that are tolled, $1.25 might be about average, yielding a share of total costs on a 15-mile trip ranging from 10-30%. This range is used in Table C-7 instead of the range shown in Table C-4 on page C-11.
Shoup (1994), and Willson and Shoup (1990) review more than a dozen studies of parking pricing, including their own as well as Shoup and Pickrell (1980). From these studies it is possible to extract five case studies that provide sufficient data to construct ordinary price elasticities. In all of these examples, the price of parking was zero for the base alternative in the comparison, so the calculations in Table C-9 base the elasticity estimates on an assumed total price for travel, rather than for parking alone. The elasticity magnitudes are large even when the price change is measured against only a small share of long-run cost.
7. Wuestefeld and Regan (1981); Harvey (1994); Wilbur Smith Associates (1995); Gifford and Talkington (1996); Samuel (2000). Samuel uses a phased-in toll increase on the Ohio Turnpike to make a back-of-the envelope calculation of trip elasticity of −0.23 and VMT elasticity of −0.15, in his Toll Roads Newsletter for February, 2000.
|Trip Ratea||Parking Priceb||Total Price Elasticityc|
|a Autos driven per 100 employees.|
|b Price in dollars per month.|
|c Full vehicle "backward" arc (low price to high) price elasticity based on 35-mile round trip average for LA reported in Willson and Shoup (1990), at an average user cost of $.22 per VMT for variable costs only (=$163 per month for the Low estimate) or $.70 for all costs including time but excluding parking (=$515 for the High estimate). Backward elasticities are lower than midpoint or forward arc elasticities for downward-sloping demand curves.|
|Before/After Case Studies:|
|Mid Wilshire, LA||48||30||0||58||−1.05||−3.33|
|Warner Center, LA||92||64||0||30||−1.65||−5.23|
|Ottawa CBD, Canada||94||80||0||30||−0.81||−2.56|
|With/Without Case Studies:|
|Century City, LA||39||32||0||23||−1.27||−4.02|
|Civic Center, LA||78||50||0||30||−1.95||−6.16|
In the before/after case studies, the price of parking changed at a particular work site, and the behavior responses were tracked for up to a year after parking became priced. Some of the employers had ridesharing incentive programs, which were ineffective so long as parking was free. These examples are interpreted as representative of short-run demand elasticities. The with/without case studies compare similar work sites, one priced and one not. These are interpreted here as long-run elasticities, on the rationale that commuters had sufficient time to make long run adjustments. These elasticities would be higher if all employee parking were priced, because more people at more sites would be seeking ridesharing or transit arrangements.
Because parking is free to the user for 99% of all trips (over 90% of urban work trips), the average share of parking in the cost of travel to the user is not a valid base value for these elasticities. As Shoup (1994) states,
"It is important to remember that the elasticity estimates [average 0.15] refer to commuter response to changes in only the parking price of their trip and are therefore smaller than the elasticity of demand with respect to changes in the full price of automobile trips."(p. 159)
He gives other reasons why these estimates are low, including the likely availability of cheaper parking nearby and the inelasticity of work trip demand.8
8. Harvey (1994) offers examples from San Francisco and Boston airport parking, which are consistent with Shoup's summary if Harvey's elasticities measure the number of vehicles parking. This does not necessarily equate, however, to the same percentage reduction in VMT, because some of the deterred parking is shifted to taxi trips.
Several studies have tabulated traffic volumes subsequent to an increase in capacity, or occasionally in response to a decrease in capacity or change in travel time. For those based on change in capacity, the measure of elasticity is
where %Δcap = percent change in capacity, with capacity measured in lane miles. Hansen et al. (1993) estimate this elasticity for eighteen highway sections in California, and include controls for trend VMT. To transform this measure into a price elasticity requires substituting a price measure for the capacity measure, such that
|eT =||%ΔVMT||=||%ΔVMT||×||%Δcap||= eT,capacity ×||%Δcap|
i.e., a conversion factor is needed from the Hansen elasticity to a price elasticity, consisting of the ratio of an increase in capacity to its corresponding reduction in price. Taking time as the only component affected, the question is what are the time savings from a given added capacity? Most of Hansen's expansions are from four to six lanes or six to eight lanes; if it is assumed that two lanes in the same direction are congested, and that adding a third will increase average speed from 40 to 60 mph9 for at least a few years, then a 50% capacity increase is equivalent to a 33% time savings (neither the value of time nor occupancy affect this result), for a conversion factor of −1.5. Thus Hansen's low or short-run value of about 0.25 becomes −0.375 and his high or long-run value of 0.45 becomes −0.675.
Cohen (1995) reviews several time-travel elasticity studies, whose results are somewhat erratic, but generally consistent with the above. Often, some types of induced traffic are counted (e.g., new travel by users already in the market) and others omitted (e.g., route diversions). Unlike other components of user price, time cannot be economized by sharing the cost among additional vehicle occupants. Therefore, elasticity with respect to time cost should be lower than for the components whose cost can be shared among passengers.
The total price elasticity reflects the change in vehicle miles of travel in response to the generalized price of travel. For application to the specific context of evaluating improvement alternatives for a section of highway, the elasticity for price changes on the section requires several adjustments. In this context, the price is the generalized price per vehicle mile of travel, and the quantity is the volume of vehicle travel on the facility, per hour or per day.10
9. This speed change implies an average savings of 0.50 minutes per VMT. Using the HERS (Chapter 6) delay equations for expressways, which model average daily delay per VMT as a function of AADT/c(apacity), a 50% increase in capacity at an AADT/c of 15 (fairly high) results in delay savings of 0.855 minutes per mile, whereas an initial AADT/c of 12 yields a savings of 0.255 per mile.
10. A possibly superior way to calculate section elasticity is to base it on trips over the section rather than VMT. This automatically incorporates the length of the section. In HERS, the elasticity is applied per VMT but adjusted for section length.
Any price component that applies to the vehicle and can be divided among occupants creates an incentive to increase occupancy; such monetary components include tolls, excise taxes, wear and ownership costs, parking, and operating costs except for some portion of accidents.11 The remaining non-divisible non-monetary price components are time and collective accident costs. Thus empirical evidence should show higher observed elasticities for monetary components than non-monetary, other things being equal. A strong pattern is not evident from Table C-7, although at the high end the time elasticity is lower than all but wear-ownership.12
Conceptually, the overall total price elasticity has two components, one derived from the divisible portion of the price, the other from the indivisible portion. Using Figure C-3, the elasticity at a point p0, v0 can be represented as the slope of a demand curve.13 The less elastic demand is labeled Dnd (although we are dealing with point elasticities, the explanation is easier if we imagine them as arc elasticities).
If all price components were indivisible, there would be no incentive for increased occupancy, so elasticity would be relatively low; for a given price change from p0 to p1, the additional traffic would be from v0 to vnd. Alternatively, if all price components were monetized to the vehicle and divisible among occupants, the applicable demand curve would be Dd and the induced traffic would be from v0 to vd. More generally, where the components are mixed, the demand curve is a weighted average, represented by the dashed demand Dactual. The end result at point b can be decomposed into a movement along Dnd for the non-divisible portion, plus a movement along line ab (parallel to Dd) for the divisible portion. Hence the incremental volume from v0 to vactual is comprised of a portion Δvnd from non-divisible price components, plus a portion Δvd from the divisible components. The share of movement along Dnd versus Dd is determined by the shares of price components, Δpnd and Δpd.
11. Insurance rates are often higher for carpool drivers, but the component is probably shareable in the same way operating costs can be shared. Damages to the vehicle and other property can be divided, but all occupants are exposed to risk of personal injury and fatality if the vehicle is involved in a crash.
12. An increase in the money share of the generalized price - assuming a mix of travelers with differing values of time - will shift the mix of travelers toward those with higher time values. We do not know, however, if the demand elasticity of high income travelers with respect to the generalized price is higher or lower than that of low income travelers.
In elasticity terms, the actual or combined elasticity is simply a weighted average of the two component elasticities,
|e = ηend+(1 ∠ η)ed|
where η = share of generalized price that consists of non-divisible components, end = elasticity for non-divisible components, ed = elasticity for divisible components, and e = combined or "actual" elasticity.
Neither of the "pure" (all non-divisible or all money) demand curves can be observed directly, and although there may be enough variation in the distribution η to extract the two elasticities econometrically, whatever evidence has been acquired has not been studied in this way.14 If it can be assumed that end and ed are proportionally related, then
|ed = (1 + x)end|
where x = fraction by which the more elastic divisible-price demand is greater in magnitude than the non-divisible demand.
In responding to a change in the generalized price of highway travel, the total price elasticity incorporates changes in vehicle trips due to changes in mode or occupancy, change in destination or trip chaining that alter trip length, and adding or deleting person-trips (it omits route and temporal diversion). Using Table C-3 and averaging values, the share of existing price that is monetary is about one-half.15 The portion of total elasticity that is due to occupancy is unknown, but might likely be around one third, for a value of x of 0.5. This is enough information to calculate the two divisible/non-divisible elasticities, given an initial overall elasticity, as shown in Table C-10.
If the share of the price that consists of divisible components remains fixed, nothing is gained by this decomposition. If, however, specific sections are priced differently from the overall average, or the intent is to test a different pricing policy, then the component elasticities can be recalculated based on price components that pertain to the particular section. The example shown in Table C-10 might represent a shift to congestion pricing, such that delay is almost eliminated but the money price is much higher. The result is a large decline in the non-divisible costs, but only a modest increase in average elasticity.
14. Average vehicle occupancies vary by trip purpose from barely one to over three, as evidenced by FHWA (1992) or MTC (1998); the effect on average occupancy of current variations in the divisible money price is likely to be much smaller in magnitude, thus hard to detect empirically.
Most traffic on a given section of highway can be assumed to be using more than that section to accomplish its trips. If, for example, the price for the section being evaluated will decrease by 20% if the improvement is implemented, and the section constitutes 60% of the complete vehicle trip, then the price for the trip will decline by 20% times 60%, or 12%. If total elasticity for the vehicle trip is −1.0, the elasticity for the section is 60% of the total, or −0.6; the 20% price decrease will thus result in a 12% reduction in volume, same as if the price change were calculated for the full trip and applied to the −1.0 elasticity. Specifically,
|es = e||Ls|
where es = section elasticity, e = total price elasticity, adjusted for occupancy, Ls = section length, and LT = average trip length for vehicles using the section.
|total vehicle price elasticity (base elasticity)||−1.00|
|share of base price that is nondivisible (%) (η)||50%|
|elasticity increment for divisible price (%) (x)||50%|
|elasticity for nondivisible price components (end)||−0.8|
|elasticity for divisible price (ed)||−1.2|
|normal travel time||0.30|
|nondivisible portion of price (η)||0.37|
|base elasticity adjusted for divisible/nondivisible components||−1.05|
If the section is longer than the average trip length, then the multiplier should be one, assuming that the price change on the section applies to the full trip for the average traveler on the section. Individual trips that use only part of the section would show a lower elasticity (approaching zero), but there would be several or many of them in each average daily VMT across the entire section, giving the same result as if a single trip used the entire section.
For most of the components of the generalized price of vehicle travel, route diversion is not an available substitute (e.g., fuel price, operating cost). For the parking studies, employee parking at a substitute destination is not feasible. Toll elasticities of major facilities such as the Golden Gate Bridge incorporate very few alternative route possibilities, and time elasticities based on facility expansion allow for some indeterminate amount of diversion in the empirical values. Thus the elasticity for a single facility or section will be higher than the value derived from empirical estimates for total price elasticity, because travelers can divert to parallel routes.
The magnitude of an elasticity is greater if there are many close substitutes. A route diversion elasticity will be high if there are alternative routes that are only slightly "worse" in the sense of having a higher generalized price. For example, a simplified representation of route diversion is shown in Figure C-4. The section being considered for improved is S0. In the base case, the preferred path for a trip between A and B uses the section S1, but an alternative route uses Sa1, S0, and Sa2. The alternate might become preferred if S0 were improved to the extent that the total trip price were lower via the alternative route, i.e., if
|pa1 + p0 + pa2 < p1|
where pi = generalized price on section Si. Sections Sa1 and Sa2 are "access" sections that would be used to get to and from the improved section. For a given primary section of interest, S0, there might be many trip ends such as A and B that do not use S0 as their preferred route, but might choose to do so if the price on S0 were lower, e.g., faster.
The demand elasticity on S0 with respect to route diversion only depends upon the relative volumes of traffic on the two routes (as well as other alternate routes) and the difference in the price between them. If v represents the volume on all routes such as S1 that do not use S0 but could, stated as a ratio to the volume on S0, then
where s = share of corridor traffic not using S0 for which the price of S0 is r% greater than their preferred route, σ = the access share of the alternative route that incorporates S0, and V0 is the volume on S0. The numerator is the percent change in volume on S0 from a change in price of on S0, and the denominator is the percent change in the price of the alternate route from a change of r% on S0.
None of these parameters is readily observable, although they could be extracted from traffic assignment simulations. As an example, parameter values have been entered into Table C-11 and the diversion elasticity calculated. There exists some distribution of the share of traffic in the corridor according to the percentage by which the price of the preferred path differs from the path that incorporates S0, and an example distribution is shown in its cumulative form. Because its shape is unknown, it is possible (and likely) that the elasticity on S0 varies by the magnitude of the percentage price change on S0, i.e., the diversion elasticity is not a constant. In general, it might be expected that network redundancy is higher in urban areas, implying that route diversion elasticities are higher (in absolute magnitude) in urban areas.
|ROTC = trips in the rest of the corridor that do not use the primary section but could.|
|% by which route with section S0 is worse than preferred path (r)||0.05||0.1||0.2||0.4||0.5|
|cumulative share of ROTC within % of trip cost (urban) (s)||0.10||0.25||0.50||0.85||0.95|
|ratio of ROTC volume to section AADT (trips) (v)||0.60|
|access share of price (a)||0.50|
|percent increase on section S0||0.06||0.15||0.30||0.51||0.57|
The diversion elasticity can be added to the previous total elasticity that has been adjusted for section length and occupancy. Because the section length relative to the total trip length is corrected for separately, section length does not need to be considered for route diversion.16 Also, because speed-volume or delay functions account for the increase in price from congestion, the capacity of S0 does not need to be considered in diversion.
A price component that applies only to the peak should have a higher elasticity than that for the total price as estimated from the empirical evidence, considering only the peak period, because some travel will be diverted to the off-peak. If diurnal demand periods are modeled separately (i.e., two or more demand periods per day), then the interrelationships among the demand periods must be explicitly modeled. This is more than just a problem of elasticities.17 Of narrower interest here is whether peak-offpeak price differentiation alters the overall - or daily - elasticity. The bottleneck model provides an example of a substitution of price for queuing delay, without any change in overall traffic, but the model does not include the option of changing the total number of trips.
If the price varies over the time of day according to the level of demand, then overall delay will be reduced (relative to no differentiation) by redistributing traffic from congested periods to time periods with lower v/c ratios. This suggests that total traffic levels will be higher because the average price is lower. If, however, we hold average price constant as well as the share that is monetary, it is not obvious that the demand elasticity is changed by altering the price between peak and offpeak.
Two effects can be separated: first, the change in volumes between a differentiated price and a constant price; and second, whether the elasticity is different for total demand if the price is differentiated or constant.
16. A very short section might have negligible impact on generating new trips but still have a large diversion impact. If the section were isolated, then the access portions of the diversion would neutralize any diversion, but if the access were easy (e.g., turn right on one block rather than the next), the amount of diversion could be large. The benefits from this diversion might be small in absolute terms, but so might the costs. Incremental consumer surplus would be made up almost entirely of diverted (rather than new) trips.
Because the share of the price that is paid in money is recognized in the occupancy factor, for purposes of peak spreading the price can be treated as a single dimension. Empirical evidence on peak spreading applies to a context in which peak and offpeak prices are differentiated only by the amount of delay. If we imagine a case with the same demand curves but constant price across all (both) periods, and the same average price, then more demand will occur in the peak and less in the offpeak. If elasticities differ between the two periods, then total volume will be higher if peak elasticity is greater then offpeak, and lower if the reverse.
If we then ask what difference this makes for total price elasticity, the simplest answer is that it makes no difference if the two elasticities are the same. If we use constant elasticity demand curves, located at different volume points but having the same elasticity, then the overall elasticity is not affected by the differentiation of price between the two periods, for the same average price.
If the elasticities are different between the two periods, then the amount of differentiation between peak and offpeak prices may cause the aggregate elasticity to be greater or lesser than the case where the price is constant, but the effect seems to be relatively small.18
Despite the widely varying orientations, data sources, and scope of applicable empirical studies, and the fact that none was attempting to estimate generalized-price travel demand elasticity, the results are roughly consistent. Users respond to changes in any of the components of travel cost that are measurable, and the response starts immediately and continues over many years.
Taking the short run to be approximately a year or less, vehicle demand-price elasticity tends to fall in the range of −0.5 to −1.0, with −0.7 to −0.8 being the most likely for typical conditions. The long run may occur over twenty years, but five years is enough to cover most of the effects. Long run elasticities are about twice as high as short run, with a range of about −1.0 to −2.0. Response to variable and obvious money costs such as parking and fuel show higher elasticities than for fixed and more hidden costs. These elasticities apply to vehicle-trips, not person-trips.
These total price elasticities can be applied to specific highway sections, adjusting for changes in the share of divisible price components, the section length, and possibilities for route diversion. The elasticity values are somewhat uncertain, due to the fact that every context is different, but the uncertainty could be reduced with focused data collection and research.
Cain, A., M. W. Burris, and R. M. Pendyala, "The Impact of Variable Pricing on the Temporal Distribution of Travel Demand," Transportation Research Record, 1747 (2001), pp. 36-43.
Cairns, Sally, Carmen Hass-Klau, and Phil Goodwin, "Traffic Impact of Highway Capacity Reductions: Assessment of the Evidence," prepared for London Transport, London, UK: Landor, March 1998.
Cambridge Systematics, Inc., "Time-of-Day Modeling Procedures: State-of-the-Practice, State-of-the-Art," prepared for FHWA, Washington, DC: US DOT/BTS, February 1997.
Chan, Yupo, and F.L. Ou, "A Tabulation of Demand Elasticities for Urban Travel Forecasting," paper for TRB, Washington, DC, January 1978.
Cohen, Harry S., "Review of Empirical Studies of Induced Traffic," in Transportation Research Board (ed.), Expanding Metropolitan Highways: Implications for Air Quality and Energy Use, Special Report 245, pp. 295-309, Washington, DC: National Academy Press, 1995.
Dahl, Carol A., and Thomas Sterner, "Analyzing Gasoline Demand Elasticities," Energy Economics (1991), pp. 203-210.
Dahl, Carol A., and Thomas Sterner, "A Survey of Econometric Gasoline Demand Elasticities," International Journal of Energy Systems, 11, 2 (1991).
Dahlgren, Joy, "High Occupancy Vehicle/Toll Lanes: How Do They Operate and Where Do They Make Sense?," California PATH Working Paper, Berkeley, CA: University of California, 1999.
Dargay, Joyce M., "Demand Elasticities: A Comment," Journal of Transport Economics and Policy, 27, 1 (1993), pp. 87-90.
Dargay, Joyce M., "Estimation of Dynamic Car Ownership Model for France Using a Pseudo-Panel Approach," prepared for INRETS, London: ESRC Transport Studies Unit, University College London, January 1998.
Dargay, Joyce M., and Dermot Gately, "The Demand for Transportation Fuels," Transportation Research (B), 31B, 1 (1997).
Dehgani, Youssef, and Antti Talvitie, "Model Specification, Modal Aggregation, and Market Segmentation in Mode-Choice Models: Some Empirical Evidence," Transportation Research Record, 775 (1980), pp. 28-34.
DeJong, John G. J., "Estimation of Gasoline Price Elasticities for New Jersey," Transportation Research Record, 812 (1981), pp. 64-67.
Delucchi, Mark A., "The Annualized Social Cost of Motor-Vehicle Use in the U. S., 1990-1991: Summary of Theory, Methods, and Results," Vol. 1, of 20, The Annualized Social Cost of Motor-Vehicle Use in the United States, based on 1990-1991 Data, Davis, CA: Institute of Transportation Studies, University of California, June 1997.
Federal Highway Administration, "1990 Nationwide Personal Transportation Survey: Summary of Trends," Washington, DC: US DOT/FHWA, March 1992.
Federal Highway Administration, "Cost of Owning and Operating Automobiles and Vans 1984," Washington, DC: FHWA, 1984.
Federal Highway Administration, "Highway Statistics 1996," Washington, DC: US DOT/FHWA, November 1997.
Gifford, Jonathan L., and Scott W. Talkington, "Demand Elasticity Under Time- Varying Prices: Case Study of Day-of-Week Varying Tolls on Golden Gate Bridge," Transportation Research Record, 1558 (1996), pp. 55-59.
Goodwin, Phil B., "Empirical Evidence on Induced Traffic," Transportation, 23, 1 (1996), pp. 35-54.
Goodwin, Phil B., "Extra Traffic Induced By Road Construction," in OECD (ed.), Infrastructure Induced Mobility, ECMT Round Table 105, Paris: OECD, 1998.
Goodwin, Phil B., "A Review of New Demand Elasticities with Special Reference to Short and Long Run Effects of Price Changes," Journal of Transport Economics and Policy, 26, 2 (1992), pp. 155-170.
Hall, Jane V., "The Role of Transport Control Measures in Jointly Reducing Congestion and Air Pollution," Journal of Transport Economics and Policy, 29, 1 (1995), pp. 93-103.
Hansen, Mark, David Gillen, Alison Dobbins, Yuanlin Huang, and M. Puvathingal, "The Air Quality Impacts of Urban Highway Capacity Expansion: Traffic Generation and Land Use Change," prepared for California Department of Transportation, Berkeley, CA: Institute of Transportation Studies, University of California, April 1993.
Harvey, Greig W., "Transportation Pricing and Travel Behavior," in Transportation Research Board (ed.), Curbing Gridlock: Peak-Period Fees To Relieve Traffic Congestion, Special Report 242, pp. 89-114, Washington, DC: National Academy Press, 1994.
Jack Faucett Associates, "Cost of Owning and Operating Automobiles and Vans 1991," prepared for Federal Highway Administration, Washington, DC: FHWA, 1992.
Luk, James, and Stephen Hepburn, "New Review of Australian Travel Demand Elasticities," Research Report ARR No. 249, Victoria, Australia: Australian Road Research Board, December 1993.
Menon, G., S. Lam, and H. Fan, "Singapore's Road Pricing System: Its Past, Present and Future," Institute of Transportation Engineers Journal, 63, 12 (1993), pp. 43-48.
Metropolitan Transportation Commission, "Travel Forecasting Assumptions '98 Summary," Oakland, CA: MTC, August 1998.
Meyer, Michael D., and Eric J. Miller, Urban Transportation Planning: A Decision-Oriented Approach, New York: McGraw-Hill, 1984.
Oum, Tae Hoon, and W. G. Waters II, "Chapter 12: Transport Demand Elasticities," in David A. Hensher and Kenneth J. Button (eds.), Handbook of Transport Modeling, pp. 197-210, Amsterdam, The Netherlands: Pergamon, 2000.
Oum, Tae Hoon, W. G. Waters II, and Jong-Say Tong, "Concepts of Price Elasticities of Transport Demand and Recent Empirical Estimates," Journal of Transport Economics and Policy, 26, 2 (1992), pp. 139-154.
Poole, Jr., Robert W., and C. Kenneth Orski, "Building a Case for HOT Lanes: A New Approach to Reducing Urban Highway Congestion," Policy Study 257, Los Angeles, CA: Reason Public Policy Institute, April 1999.
Pritchard, Tim, and Larry DeBoer, "The Effect of Taxes and Insurance Costs on Automobile Registrations in the United States," Public Finance Quarterly, 23, 3 (1995), pp. 283-304.
Runzheimer International, "Runzheimer Analyzes 1998 Car, Van, and Light Truck Costs," news release, Rochester, WI: Runzheimer, November 3 1997.
Runzheimer International, "Runzheimer Analyzes Typical Vehicle Costs: Where Does Your Automotive Dollar Go?," news release, Rochester, WI: Runzheimer, June 1 1997.
Samuel, Peter, Toll Roads Newsletter, February 2000.
San Diego Association of Governments, "Fastrak Users' Response to Toll Pricing Change: A Time Series Analysis of Time of Travel," San Diego, CA: SANDAG, 1999.
Schimek, Paul, "Gasoline and Travel Demand Models Using Time Series and Cross-Section Data from United States," Transportation Research Record, 1558 (1996), pp. 83-89.
Shoup, Donald C., "Cashing Out Employer-Paid Parking: A Precedent for Congestion Pricing?," in Transportation Research Board (ed.), Curbing Gridlock: Peak-Period Fees To Relieve Traffic Congestion, Special Report 242, pp. 152-199, Washington, DC: National Academy Press, 1994.
Shoup, Donald C., and Don H. Pickrell, "Free Parking as a Transportation Problem," prepared for US DOT, Los Angeles, CA: University of California, 1980.
Small, Kenneth A., Urban Transportation Economics, Chur, UK: Harwood Academic, 1992.
-Small, Kenneth A., and Jose A. Gomez-Ibanez, "Road Pricing for Congestion Management: The Transition from Theory to Policy," in K.J. Button and E.T. Verhoef (eds.), Road Pricing, Traffic Congestion and the Environment: Issues of Efficiency and Social Feasibility, pp. 213-246, Cheltenham, UK: Edward Elgar, 1998.
Spock, L., "Toll Practices for Highway Facilities," NCHRP Synthesis of Highway Practice 262, Washington, DC: Transportation Research Board, 1998.
Sterner, Thomas, and Carol A. Dahl, "Gasoline Demand Modeling: Theory and Application," in Thomas Sterner (ed.), International Energy Modeling, London, UK: Chapman and Hall, 1992.
Sterner, Thomas, Carol A. Dahl, and Mikael Frazen, "Gasoline Tax Policy, Carbon Emissions and the Global Environment," Journal of Transport Economics and Policy, 26, 2 (1992), pp. 109-120.
Strathman, James G., and Kenneth J. Dueker, "Transit Service, Parking Charges and Mode Choice for the Journey-to-Work: Analysis of the 1990 NPTS," Portland, OR: Center for Urban Studies, Portland State University, January 1996.
Symons, N. R., J. R. K. Standingford, and R. B. Jones, "Sensitivity of Travel Demand to Toll Charges: Experience at West Gate," Hobart, Australia: Australian Road Research Board, August 1984.
Taplin, John H.E., "Inferring Ordinary Elasticities From Choice or Mode-Split Elasticities," Journal of Transport Economics and Policy (1982).
U.S. Department of Transportation, "The Value of Saving Travel Time: Departmental Guidance for Conducting Economic Evaluations," Washington, DC: US DOT/OST, April 9 1997.
Wuestefeld, Norma H., and Edward W. Regan, "Impact of Rate Increases on Toll Facilities," Traffic Quarterly, 35 (1981), pp. 639-655.
Wilbur Smith Associates, "Potential for Variable Pricing of Toll Facilities," prepared for Federal Highway Administration, New Haven, CT: WSA, 1995.
Willson, Richard W., and Donald C. Shoup, "Parking Subsidies and Travel Choices: Assessing the Evidence," Transportation, 17, 2 (1990), pp. 141-157.