The Rebound Effect from Fuel Efficiency Standards: Measurement and Projection to 2030

Kenneth A. Small Department of Economics University of California at Irvine

June 12, 2009

This report discusses empirical va lues of the “rebound effect” for tr avel in passenger vehicles in the United States. The rebound effect refers to effects on amount of travel that arise from changes in the fuel efficiency for light-duty mo tor vehicles (passenger cars and light trucks), caused in turn by regulations or technological developments. I brie fly discuss the literature, then summarize empirical estimates done at University of California at Irvine in collaboration with Kurt Van Dender. Finally I present updated empiri cal estimates and derive their implications for the rebound effect in the time frame 2000-2030.

The literature review and empirical methodology ar e described more fully in a published article (Small and Van Dender 2007a), and even more fu lly in the working paper from which the published article was derived (Small and Van De nder 2007b). The empirical estimates have been updated subsequently, mainly by adding three ne w years of data; these updated results are presented here and also summarized in a fort hcoming short publication (Small and Van Dender 2007c). The projections are my own, although they use a methodol ogy adapted from one used in an earlier report to the Ca lifornia Air Resources Boar d (Small and Van Dender 2005).

1. Background and definitions

1.1 Determinants of motor-vehicle travel

The rebound effect is simply a statement of th e near-universal economic principle of downwardsloping demand: when the price of a good or service increases, people purchase less of it. In this case the service is passenger trans portation, and its price to the user includes the cost of fuel. If the amount of service is measured as vehicle-mile s traveled (VMT), then the component of price accounted for by fuel cost, here called “fuel cost per mile” P M , is equal to the price of fuel P f ( e.g. stated in $/gallon) di vided by fuel efficiency E ( e.g. stated in vehicle-miles/gallon):

E P P f

M / = . (1)2 Thus if fuel efficiency E is increased, fuel cost per mile decreases, and since this is part of the price paid by consumers to drive, they will increase their VMT. See Greening, Greene and Difiglio (2000) for a more extended discussion.

The responsiveness of demand to price is often su mmarized as a ratio of the percent change in demand, Δ M / M , to the percent change in price causing it, Δ P M

/ P

M , where M designates VMT in mathematical equations and Δ designates a change in a quantity. A ratio such as this is called an elasticity , usually defined for the situation when Δ P M

is very small so that the ratio become a derivative. Therefore I define the elasticity of vehi cle-miles traveled with respect to cost per mile as follows:

M M PM M dP dM
M P

⋅ =

, ε (2)

where the derivative dM / dP M is simply the limit of Δ M / Δ P M

as Δ P M

becomes very small. An equivalent way to write this is in terms of the natural logarithms of M and P M , which we denote by lower-case letters vma and pm , respectively. (The notation vma stands for “vehicle-miles per adult”, which is how we define M in our empirical work.) Of course the equation for vma contains other variables besides pm , and these are held constant wh en considering the effects of

pm ; this makes the derivative in (2) a partial derivative, denoted using the symbol ∂ . The elasticity written in this form is:

) ( ) (

, pm vma

PM M ∂ ∂

= ε , (3)

which could be a single coefficient in the equation for vma or, if pm enters in more than one way, a combination of several coefficients.

One of the confusing aspects of th e literature is that few studies have accounted for the fact that fuel efficiency E is not simply mandated, but chosen jointly by consumers and motor-vehicle manufacturers, within certain c onstraints set by regulation. Ther efore one might ask the meaning of considering a change in E as though it could simply be set by fiat. In our empirical work, Van Dender and I meet this challenge by defining a syst em of three simultaneously determined travelrelated quantities, each applying to a state. The first dependent variable is VMT, written mathematically as M ; it is a function of P M

(as already described), th e size of the vehicle fleet, V
, and various socio-demographic characteristics in cluding income. The second dependent variable is V , which is a function of se veral things: a price index P V of new vehicles, the amount of travel3 M (since new vehicles are purchased in large part to supply desire d travel), the price of travel P M , and other characteristics. Finally, the th ird dependent variable, fuel efficiency E (or its inverse), is presumed to be chosen based on a combinati on of motives including the wish to conserve on the cost of traveling M miles, the need to meet various re gulations on fuel ef ficiency and/or emissions, and tradeoffs with vehicle performance; in our empirical work E is assumed to be a function of M , price of fuel P F , a variable measuring the tightne ss during any given year of the US federal Corporate Average Fuel Economy (C AFE) regulations, and other characteristics.

Thus, our system accounts for the effects of a ch ange in regulations th rough two pathways. I illustrate for an increase in fuel-efficiency standa rds. First, the regulations increase the average fuel economy of the fleet, and that in turn reduces the cost per mile of travel, P M , through equation (1); this directly changes the amount of travel because of downward-sloping demand as just discussed. Second, the size of the vehicle fl eet is increased because vehicles are now more useful; this change in vehicle fleet size further affects M . We estimate a simultaneous-equations model of M
, V , and E that fully accounts for these effects; empirically, we find that the first path is by far the dominant one, so that one could ignore the second path as an approximation.

Our model, through the influence of fuel price on fuel efficiency, implicitly incorporates some changes in the relative prices of vehicles of different sizes and types. However, the description just given of the effects of regulations assumes that the average price of new vehicles, P V , is held fixed. Of course, the full effect of a regulation would also include any change in this average price on new-vehicle sales. In many cases this w ould work in the opposite di rection to the effect just noted. Furthermore, changes in new-vehicle sales would also change scrappage rates and the price structure of used vehicles of different ages . These effects are not usually considered part of the “rebound effect”, and are not discussed here; 1 but they would be important to consider as part of the full effects of a regulatory change.

In order to distinguish the ul timate effect of both pathways on VMT, we use the symbol M ˆ to designate the combined effect, and designate its elasticity with respect to cost per mile as PM M
, ˆ ε , reserving the symbol PM M
, ε for the changes operating through the first pathway only. Small and Van Dender (2007a) show that th ese quantities are related by:

1 In principle the effect of changes in average new-vehicle price could be anal yzed using the results of the vehiclefleet equation in our model, if we had information about how much new-vehicle prices would increase, since that equation includes the variable P V , which is an index of nationwide new-car prices. However, our model does not estimate the coefficient of new-vehicle price very precisely , because there is little variation in that variable (none across states); so we would have less confidence in us ing it for that purpose. Probably a better approach for analyzing effects on vehicle purchases is to consider the entire range of vehicle sizes and models and how consumers shift between them.4 M V V M PM V V M PM M
PM M

, , , , ,

, ˆ 1 ε ε ε ε ε ε

− +

= (4)

where ε M,V denotes the direct elas ticity of travel with respect to vehicle fleet, ε V,M denotes the direct elasticity of vehicle fleet w ith respect to amount of travel, and ε V,PM denotes the elasticity of vehicle fleet with respect to co st per mile of travel. All the qua ntities on the right-hand side of (4) are measured directly as coefficients, or comb inations of coefficients , of the three equations in our model.

1.2 Definition of the rebound effect: short-run and long-run

While terminology differs among authors, PM M
, ˆ ε is conceptually what most writers have meant when discussing the rebound effect. To summarize: it measures the ratio of the responsiveness of travelers to the change in fuel efficiency resu lting from regulations (w ith both expressed in percentage terms), while recognizing that the cha nge in fuel efficiency is not directly set by regulations but rather results from a complex interactive pro cess. This responsiveness accounts for both the direct effect of fuel efficiency on the cost of using a given vehicle, and the indirect effect on travel through changes in the number of vehicles purchased, but all the while holding average new-vehicle prices constant.

Because the elasticity PM M
, ˆ ε is expected to negative, it is convenient to express the rebound

effect b S as a number that is normally positive:

PM M S b

, ˆ ε

− = (5)

It is also common to e xpress the rebound effect as a percentage rather than a fraction. Thus, if

PM M
, ˆ ε =-0.2, we say the rebound effect is 20%.

Our empirical system also accounts for the slowne ss with which changes can occur, especially changes in the vehicle fleet size and average efficiency, which re quire purchases and retirements of vehicles. It is able to do th is because we observed a location (a state or District of Columbia) every year for 39 years – making our data set a cross-sectional time series , sometimes also called

a panel data set . Slow adjustment is accounted for by assu ming that each of the three variables explained by our model ( M
, V , and E ) depends not only on the factors already mentioned, but also on the previous year’s value of that same quantity (called a lagged value of that variable). This is equivalent to assuming that there is a desired level of M
, V , or E , and that any deviation5 between this desired level and the level attained in the previous year is diminished in one year by a fraction (1- α ), where α is the coefficient of the lagged value of the variable. We allow α to differ across the three equations. This slow adjust ment process means that the short-run response (that occurring in the same year) is smaller th an the long-run response. Continuing to use the
notation PM M
, ˆ ε for the elasticity determined within this system, it is now a short-run elasticity because the long-run response is accounted fo r elsewhere in the equation (through the lagged variables). We represent th e corresponding short-run and long-run rebound effects as b S and b L

, respectively. They are ap proximately related by:

m PM M

m S L b b α ε

α

− −

=

− ≅ 1 1 , ˆ (6)

where α m is the coefficient of the lagged depende nt variable in the equation explaining vma . A more precise relationship accounts for the fact that in the three-equa tion system, the lagged values in more than one equation can a ffect the long-run response; specifically: 2

vm mv v m v mv v PM M L b α α α α β α α ε

− − − − − ⋅ −

= ) 1 )( 1 ( ) 1 ( 2
, (7)

where α v is the coefficient of the lagged dependent variable in the equation explaining the logarithm of vehicle stock, α mv is the coefficient of vehicle stock in the equation explaining vma ,

α

vm is the coefficient of vma in the equation explaining vehicle stock, and v 2 β is the coefficient
of pm in the equation explaining vehi cle stock. With our empirical results, the approximation in (6) is a very close one. 3 In addition to accounting for la gged values within the system determining our dependent variables, our empiri cal system accounts for th e possibility that the error terms in each equation are co rrelated over time. That is, for any given state, the unknown random factors affecting a dependent variable may have some elements that are the same year after year. Most of these common factors are accounted for by a “fix ed effects” specification, in which a distinct constant term is estimated fo r every state instead of just one for the entire system. 4

2 See Small and Van Dender (2007a), equation (7).

3 This is because α mv

, v 2 β , and α vm are all quite small. When these three coefficients are small, the term (1- α v ) cancels in the numerator and denominator of equation (7).

4 This is one of two common specifications for panel data, the other being “random effects.” A hypothesis test known as a Hausman test soundly rejects random effects in favor of fixed effects for this data set.6 Empirically, the effects of lagged dependent variables are difficult to distinguish from those of autocorrelation, a problem plagui ng earlier studies investigating changes over time. We are able to distinguish them because of the long time peri od (39 years) covered by our panel data set.

There are many ways besides those considered here that regulations on fuel efficiency or related quantities might affect travel. As already noted , such regulations might raise vehicle prices, which would affect the vehicle fl eet size and thus, indirectly, th e amount of travel. Regulations may affect fuel prices through the impact of aggregate demand for fuel on petroleum markets. They may influence technological developments, th ereby affecting the cost s and performance of future vehicles. A broader analysis of the effect s of fuel efficiency on travel might want to account for such factors, but they are outside the realm of the “r ebound effect” as I define it here and, I believe, as most researchers have used th e term. However, it is worth pointing out that if regulations raise the pric e of new vehicles, then the response to that price rise would tend to offset somewhat the rebound effect by curtailing the number of vehi cles available to travelers.

2. Prior Literature

This section of the report is taken directly from the review by Small and Van Dender (2007b), with slight adaptations.

Some empirical studies of the rebound effect have used aggregate time-series data. Greene (1992) uses annual U.S. data for 1957-1989 to estim ate the rebound effect at 5 to 15% both in the short and long run, with a best estimate of 12.7%. According to Greene, failing to account for autocorrelation – which he estimates at 0.74 – resu lts in spurious measurements of lagged values and to the erroneous conclusion that long-run effects are larger than short-run effects. 5 Greene also presents evidence that the fuel-cost-per-mil e elasticity declines over time, consistent with the effect of income just discussed; but the ev idence has only marginal statistical significance.

Jones (1993) re-examines Greene’s data, addi ng observations for 1990 and focusing on modelselection issues in time-series an alysis. He finds that although Greene’s autoregressive model is statistically valid, so are alte rnative specifications, notably those including lagged dependent variables. The latter produce long-run estimates of the rebound effect that substantially exceed the short-run estimates (roughly 31% vs. 11%). 6 Schimek (1996) uses data from a still longer

5 Another study that found autocorrelation is that by Blair, Kaserman, and Tepel (1984). They obtain a rebound effect of 30%, based on monthly data from Florida from 1967 through 1976. They do not estimate models with lagged variables.

6 These figures are from the linear lagged dependent variable model (model III in Table 1). Estimates for the loglinear model are nearly identical.7 time period and finds an even smaller short-r un but a similarly large long-run rebound effect (29%). 7 Schimek accounts for federal CAFE regul ations by including a time trend for years since 1978; he also includes du mmy variables for the years 19 74 and 1979, when gasoline-price controls were in effect, resulting in queues a nd sporadic rationing at service stations. These controls reduce the extent of au tocorrelation in the residuals.

These aggregate studies highlight the possible importance of lagge d dependent variables (inertia) for sorting out short-run and long-run effects. But they do not settle the issue because they have trouble disentangling the presen ce of a lagged dependent vari able from the presence of autocorrelation. Their estimates of these dynamic properties are especially sensitive to the time period considered and to the tr eatment of CAFE regulations.

Another type of study relies on poo led cross-sectional time-series data at a smaller geographical level of aggregation. Haughton and Sarkar (1996) c onstruct such a data set for the 50 U.S. States and the District of Columbia, from 1970 to 1991. Fuel prices vary by state, primarily but not exclusively because of different rates of fuel ta x, providing an additional opportunity to observe the effects of fuel price on travel. The author s estimate equations both for VMT per driver and for fuel intensity (the inverse of fuel efficien cy), obtaining a rebound effect of about 16% in the short run and 22% in the long run. 8 Here, autocorrelation and the effects of a lagged dependent variable are measured with sufficient precision to distinguish them; they obtain a statistically significant coefficient on the lagged dependent variable, im plying a substantial difference between long and short run. Tackling yet anot her dynamic issue, Haughton and Sarkar find that fuel efficiency is unaffected by the current price of gasoline unless that price exceeds its historical peak – a kind of hysteresis. In that equation, CAFE is taken into account through a variable measuring the difference between the lega l minimum in a given year and the actual fuel efficiency in 1975. However, that variable is so strongly correlat ed with the historical maximum real price of gasoline that th ey omit it in most specificati ons, casting doubt on whether the resulting estimates, especially of hysteresis, control adequately for the CAFE regulation.

It appears that the confounding of dynamics with effects of the CAFE regulation is a limiting factor in many studies. There is no agreement on how to control for CAFE, and results seem sensitive to the choice. This is partly because the standards were imposed at about the same time that a major increase in fuel prices occurred. But it is also because the control variables used are not constructed from an explicit theory of how CAFE worked. We attempt to remedy this in our empirical work.

7 These figures are his preferred results, from Schimek (1996), p. 87, Table 3, model (3).

8 This paragraph is based on models E and F in their Table 1, p. 115. Their variable, “real price of gasoline per mile,” is evidently the same as fuel cost per mile.8 There are also some studies measuri ng the rebound effect using micro data, i.e. data from individual vehicle owners. These studies show a wider disparity of results than those based on aggregate data, covering a range from zero to about 90%. Two recent such studies use a cross section for a single year. West (2004), using the 1997 Consumer E xpenditure Survey, estimates a rebound effect that diminishes strongly with income (across consumers) but is 87% on average, much higher than most studies. By contrast , Pickrell and Schimek (1999), using 1995 crosssectional data from the National Personal Transp ortation Survey (NTPS), obtain a rebound effect of just 4%. 9 There are a number of reasons to be cau tious about these results. West obtains an extremely low income-elasticity for travel, name ly 0.02, in the theoretically preferred model which accounts for endogeneity between vehicle- type choice and vehicle use. Pickrell and Schimek’s results are sensitive to whether or not they include residential density as an explanatory variable, apparently be cause residential density is collinear with fuel price. We think the value of cross-sectional micro data for a single year is limited by the fact that measured fuel prices vary only across states, a nd those variations are correlat ed with unobserved factors that also influence VMT – factors such as residential density, congestion, and market penetration of imports. In our work, we eliminate the spurious effects of such cross-sectional correlations by using a fixed-effects specification, i.e. by including a dummy variable for each state.

Two recent studies use micro data covering severa l different years, thereby taking advantage of additional variation in fuel price and other variables. Goldberg (1998) estimates the rebound effect using the Consumer Expenditure Survey for the years 1984-1990, as part of a larger equation system that also predicts automobile sales and prices. When estimated by Ordinary Least Squares (OLS), her usage equation implie s a rebound effect (both short- and long-run, because the equation lacks a lagged variable) of about 20%. 10 Greene, Kahn, and Gibson (1999) use micro data from the Residential Trans portation Energy Consumption Survey and its predecessor, for six different years between 1979 and 1994. Their usage equation is part of a simultaneous system including vehi cle type choice and actual fuel price paid by the individual. They estimate the rebound effect at 23% for all households (short- and long-run assumed identical), with a range from 17% for thre e-vehicle households to 28% for one-vehicle households.

9 Their model 3, with odometer readings as dependent va riable. They actually measure the elasticity of VMT with respect to gasoline price, ε M
, PF , which is equal to ε M
, PM as defined here.

10 When estimated using instrumental variables to account for the endogeneity of vehicle type, Goldberg’s estimate diminishes to essentially zero. But in that model the variab les representing vehicle type attain huge yet statistically insignificant coefficients (see her Table I), casting doubt in our minds on the ability of the data set to adequately account for simultaneity.9 Several micro studies estimate model systems in which vehicle choice and usage are chosen simultaneously, thereby accounting for th e endogeneity of fuel efficiency. 11 Mannering (1986) explicitly addresses the bias re sulting from such endogeneity, fi nding it to be large, although in the direction opposite to what we expect: his estimate of | ε M,PM | becomes considerably greater when endogeneity is taken into account. 12

Thus prior literature shows that aggregate estimates of the rebound effect, especially of the longrun effect, are sensitive to specification — in pa rticular to the treatment of time patterns and CAFE standards. Disaggregate studies tend to produce a greater range of estimates; but those that exploit both cross-sectiona l and temporal variation are mo re consistent, finding a long-run rebound effect in the neighborhood of 20-25 percent. These results parallel those of three more comprehensive reviews, which report rebound estima tes from numerous studies with means of 10-16 percent for short-run and 26-31 percent for long-run rebound effect. 13 Overall, we would regard long-run estimates of anywhere between 20 and 30 percent as compatible with previous studies, but we see less consensus on short-run estimates.

3. Data and specification for this report

Our data set is a cross-sectional time series fo r 50 US states plus District of Columbia, each measured annually for years 1966-2004. Variables are constructed from pu blic sources, mainly the US Federal Highway Administration, US Census Bureau, and US Energy Information Administration. Data sources and a fuller de scription are given in Small and Van Dender (2007a,b).

I now list the primary variables used in our statistical esti mation. As already noted, we simultaneously estimate three equations, each of which explains one dependent variable. All the dependent variables, and many others as well, are measured as natural logarithms. Variables starting with lower case letters are logarithms of the variable described. All monetary variables are real (i.e. inflation-adjusted).

Dependent Variables

11 Examples include Train (1986), Hensher et al. (1992), Goldberg (1998), and West (2004).

12 This could occur if people who drive a lot spend more time in stop-and-go traffic, or if they invest more heavily in fuel-consuming amenities like air cond itioning or stronger acceleration.

13 De Jong and Gunn (2001), Table 2; Graham and Glaister (2002), p. 23; Goodwin, Dargay, and Hanly (2004), Tables 3, 4. The National Research Council (2002, p. 19), without distinguishing short from long run, quotes 10-20 percent.10 M : Vehicle miles traveled (VMT) divided by a dult population, by state and year (logarithm:

vma , for “v ehicle-m iles per a dult”).
V : Vehicle stock divided by adult population (logarithm: vehstock ). 1/

E : Fuel intensity, F / M , where F is highway use of gasoline (logarithm: fint ).

Independent Variables other than CAFE
P

M : Fuel cost per mile, P F / E. Its logarithm is denoted pm ≡ ln( P F )–ln( E ) ≡ pf+fint . For convenience in interpreting in teraction variables based on pm , we have normalized it by subtracting its mean over the sample.
P V : Index of real new vehicle prices (1987=100) (logarithm: pv ). 14

P F : Price of gasoline, deflated by consumer price index (1987=1.00) (cents per gallon). Variable pf is its logarithm normalized by subtracting the sample mean. Other : See Small and Van Dender (2007a), Appendix A. All three equations include time trends to proxy for unmeasured trends su ch as residential dispersion, other driving costs, lifestyle changes, and technology. As described below, in equation (8), the se t of variables denoted
X

M

includes the variable ( pm ) 2 and interactions between normalized pm and two other normalized variables: log real per capita income ( inc ) and fraction urbanized ( Urban ).

Variable to Measure CAFE Regulation ( R E ) We define a variable measuring the tightness of CAFE regulation, starting in 1978, based on the difference between the mandated efficiency of ne w passenger vehicles and the efficiency that would be chosen in the absence of regulation. The variable becomes zero when CAFE is not binding or when it is not in effect . In our system, this variable he lps explain the efficiency of new passenger vehicles, while the lagged dependent vari able in the fuel-intensity equation captures the inertia due to slow turnover of the vehicle fleet.

The calculation proceeds in four steps, descri bed more fully in Small and Van Dender (2007a), Appendix B. First, we estimated a reduced-for m equation explaining log fuel intensity from 1966-1977, prior to CAFE regulations. Next, this e quation is interpreted as a partial adjustment model, so that the coefficient of lagged fuel inte nsity enables us to form a predicted desired fuel intensity for each state in each year, including ye ars after 1977. Third, for a given year, we averaged desired fuel intensity (in levels, weight ed by vehicle-miles travel ed) across states to get a national desired average fuel intensity. Finally, we compared the reciprocal of this desired nationwide fuel intensity to the minimum effi ciency mandated under CAFE in a given year (averaged between cars and light trucks using VMT weights, and corrected for the difference between factory tests and real-w orld driving). The variable cafe is defined as the difference

14 We include new-car prices in the second equation as indi cators of the capital cost of owning a car. We exclude used-car prices because they are likely to be endog enous; also reliable data by state are unavailable.11 between the logarithms of mandated and desired fuel efficiency, truncated below at zero. The resulting variable suggests that the CAFE st andard has been binding throughout its time of application, but that its tightne ss rose dramatically during its fi rst six years and then gradually diminished until it was just barely binding after 2000. 15 This pattern is obviously quite different from a trend starting at 1978 and from the CAFE st andard itself, both of which have been used as a variable in VMT equati ons by other researchers.

Implicit in the definition of our regulatory variable is a view of the CAFE regulations as exerting a force on every state toward greate r fuel efficiency of its fleet, regardless of the desired fuel efficiency in that particular state. Our reason fo r adopting this view is that the CAFE standard applies to the nationwide fleet average for each manufacturer; the manufacturer therefore has an incentive to use pricing or other means to improve fuel efficiency everywhere, not just where it is low.

State population data Several variables of our specification, including the first two endogenous variables, make use of data on adult or total state population. Such da ta are published by the U.S. Census Bureau as midyear population estimates; they use demographic information at the stat e level to update the most recent census count, taken in years endi ng with zero. However, these estimates do not always match the subsequent census count, and the Census Bureau does not update them to create a consistent series. As a result, the publis hed series contains many instances of implausible jumps in the years of the census count. For our preferred specification, we apply a correction assuming that the actual census counts taken ever y ten years are accurate, and that the error in estimating population between them grows linearly over that ten-ye ar time interval. We believe this approach is better than using the publishe d estimates because it makes use of Census year data that were not available at the time the pub lished estimates were constructed (namely, data from the subsequent census count). See Small and Van Dender (2007b) for details.

Appendix A shows summary statistics for the data used in our main specification.

4. Results of the Empirical Analysis

A major limitation of the previous literature is its inability to determ ine whether or not the rebound effect has changed over time. Theoreti cal arguments, especially by Greene (1992), suggest that it should. Basically, the argument is that the respon siveness to the fuel cost of driving will be larger if that fuel cost is a larg er proportion of the total cost of driving. If initial fuel cost is high, that increases the proportion; but if the perceived valu e of time spent in the

15 See Small and Van Dender (2007a), Fig. 1.12 vehicle is high, either because of congestion (closely related to ur banization) or because of a high value of time (closely related to income), that decreases the proportion. Thus we expect the rebound effect to increase with increasing initial fuel cost, and decrease with increasing income and urbanization. On the few occasions when such factors are even discussed, most analysts have presumed that income is the dominant one a nd therefore have hypothesi zed a decline in the rebound effect over time, due to rising real income s. Previously used data sets, however, have covered too short a time span to test any of these arguments satisfactorily. 16

With the longer time span of the data set compiled by Small and Van Dender (2007a), and the even longer one used here (39 years), we have a much better opportunity to see such changes. In Section 4.1, I explore the questi on in a simple way, by estimati ng the same model, postulating a constant rebound effect, on data fr om different time periods. As we shall see, the data reveal a substantial diminution in the rebound effect in the most recent 15 years. In Section 4.2, I explore the causes of this diminution. Theoretical argumen ts just noted suggest that likely causes are changes in income, fuel costs, and urbanization; each of these factors is entered in the model in such a way that the rebound effect can vary with them (rather than varying over time in an unexplained manner). We find that it does indeed vary with all three, in exactly the manner predicted by theory: th e rebound effect (measured as a positiv e number) declines with increasing income and urbanization, and increases with increasing fuel cost.

To understand better why changes in income and fu el costs may be important to behavior, Figure 1 shows three categories of the short-run costs of driving and how they are likely to progress over coming decades. The values are for 2005, take n from Small and Verhoef (2007, Table 3.3) and updated to 2007 prices. Small and Verhoef compile figures from various sources to estimate the average private cost per mile of an urban commuting trip by automobile. The values placed by travelers on travel time and unr eliability are taken from statistical literature examining how people are willing to trade off t hose factors against money. I have then projected fuel costs per mile into the future, using the Energy Informa tion Administration’s projec tions for fuel prices and fuel efficiency in their reference scenario (US EIA 2009). I have projected the values of travel time and unreliability into the future by assuming that the amounts of time and unreliability are unchanged (a conservative assumption given trends toward increased congestion) while the values of time and unreliability increase with rising per capita real income

16 A recent exception is two studies by Wadud, Graham and Noland (2007a, 2007b) using time-series cross sections of individual households from the US Consumer Expenditure Survey. Cross-sectionally, they find a U-shaped pattern of the absolute value of the price elasticity of fuel consumption, taking values of 0.35 for the lowest income quintile, falling to 0.20 for the middle, and rising again to 0.29 for the highest (2007b, Table 2). But when they hold other variables constant while allowing income to vary both cross-sectionally and over time (1997-2002), they obtain a nearly steady, though sm all, decline of the absolute value of elastic ity with income, from 0.51 in the lowest two income quintiles to 0.40 in the highest.13 according to an elasticity of 0.8, a recommendation of Mackie et al. (2003) based on many studies of how value of time depends on in come (Small and Verhoef 2007, Section 2.6.5).



### Figure 1: 

## Costs of Driving

0 25 50 75 100

2005 2015 2025Cents Per Mile 

### Other Costs

Travel Time and Unreliability Fuel Cost

4.1. Variation by Time Period

This section presents the results of estimating a relatively simple versi on of the three-equation system described earlier. In this version, the variable pm (the logarithm of fu el cost per mile) is simply included in the equation explaining vma (the logarithm of vehicle-miles traveled per adult). Its coefficient, the “structural elasticity,” is the elasticity of VMT with respect to fuel cost per mile, holding vehicle fleet constant. Accounti ng for how the vehicle fleet also varies with fuel cost, and how lagged adjustment create s differences between short-run and long-run responses, we get the short- and long-run rebound effects from equations (4), (5), and (7).

In order to see whether the re bound effect changes over time, I car ry out this estimation on two subsamples: 1966-1989 and 1990-2004. Table 1 show s the estimation resu lts. Specifically, it shows the structural elasticity PM M
, ε and its long-run counterpart, ) 1 /(

, m PM M α ε

− . As described14 earlier, these are nearly identical (except for the minus sign) to the short-run and long-run rebound effects, and their values come immediatel y from the estimated results. The table shows that the short-run rebound effect falls by nearly 40 percent be tween these two time periods. The long-run value falls by even more, by nearly two- thirds. (The reason they do not fall by the same proportion is because the lagged coefficient α m is estimated to be smaller in the later period, indicating a smaller difference betwee n short-run and long-run responses.)

Table 1. Structural Elastici ty of VMT with respect to fuel cost per mile, estimated on subsamples 1966-1989 1990-2004 Short Run -0.0482 -0.0292 (0.0066) (0.0086)

Long Run -0.2107 -0.0773 (0.0309) (0.0225) (Standard errors are in parentheses.)

4.2. Variation by Income, Fuel Cost, and Urbanization

As described in Section 1.1, in our three-equati on system the rebound effect is a combination of elasticities of the three distin ct equations (known as “structural equations”). Because of the relative sizes of these elasticities, it is approximated by one: namely ε M,PM , giving the effect of fuel cost per mile in the structural equation for vehicle-miles traveled per adult. In our notation, using lower-case names for variables that are expr essed in natural logarithms, that elasticity is given by equation (3), i.e. ε M,PM = ∂ ( vma )/ ∂ ( pm ).

In the previous subsection, fuel cost per m ile was entered just as a single variable ( pm in logarithmic terms) in the equation for vehicle-miles ( vma ). The elasticity was just its coefficient, which we may call β pm . But it is easy to specify the equation for vma so that pm appears not only as a single variable, but also interacted with ot her variables including itself. We define three such variables: pm ⋅ inc
, pm ⋅ pm ≡ pm 2 , and pm ⋅ Urban , where inc is the logarithm of per capita real
income and Urban is the fraction of state population that is urbanized; I call their coefficients β 1 , β

2 , and β 3 . Then the derivative in (3) consists of four terms:

Urban pm inc pm vma

pm PM M ⋅ + ⋅ + ⋅ + = ∂ ∂

= 3 2 1
, 2 ) ( ) ( β β β β ε . (8)15 (The factor 2 in this equation is a consequence of properties of the derivative of the quadratic function ( pm ) 2 .) Inserting (8) into e quations (4) and (7) for th e short- and long-run rebound effects implies that those rebound effects also depend on inc
, pm , and Urban

.
In order to facilitate interpretation of coefficients, we “normalize” the values of inc
, pm , and Urban by subtracting from each variable its mean value over our entire data set. This has no effect on the coefficients except to change the co nstant terms in the equations; but it means that even in this expanded specification, where elasti cities vary across the samples, the coefficient β

pm still gives the estimated elasticity ε M,PM at the point where each vari able is equal to its mean value in our data set – as can be seen by setting the three normalized variables in (8) to zero. This is especially convenient because the short-r un and long-run rebound effects are approximately

- ε

M,PM and - ε M,PM /(1- α m ), respectively, where α m is coefficient of lagged vma in the vma equation. Thus, one can see the approximate value of the estimated rebound effect, under average conditions over the sample period, just by looking at - β pm and α m

.
The model is estimated using the maximum-lik elihood simultaneous-equations estimator in Eviews 5 (Quantitative Micro Software 2004). Tec hnical details are provided in Small and Van Dender (2007a). The full results of estimating th is model are presented in Appendix A; the most important coefficients are summarized here in Table 2.16 Table 2. Selected Results of Full Model with Interactions Equation and Variable Coefficient Symbol Coefficient Estimate Standard Error Equation for vma :

pm β pm -0.0407 0.0042

pm * inc β 1 0.0696 0.0132

pm 2 β 2 -0.0169 0.0064

pm * Urban β 3 0.0255 0.0100
inc 0.1044 0.0134 lagged vma α m 0.7980 0.0120
Equation for fint :

pf + vma -0.0297 0.0064 cafe -0.0882 0.0110 lagged fint α f 0.8450 0.0127

Notes:

vma = logarithm of vehicle-miles traveled per adult

pm = logarithm of fuel cost per mile (normalized)
inc = logarithm of income per capita Urban = fraction of population living in urban areas fint = logarithm of fuel intensity, i.e. log (1/ E ) where E = fuel efficiency pf

= logarithm of fuel price cafe = variable reflecting how far the CAFE standard is above the desired fuel efficiency based on other variables (Small and Van Dender 2007a, Section 3.3.3) pf+vma is log (price of fuel * vehicle- miles traveled), representing the logarithm of the incremental annual fu el cost of a unit change in fuel intensity; thus it may be interpreted as the logarithm of the “price” to the user of higher fuel intensity.17 All coefficients shown in Table 2 easily pass the conven tional test of statis tical significance, having estimates more than twice the standard deviation of those estimates. The coefficient α m of lagged vma shows that the long-run effect of any variable on VMT is about 1/(1- α m )=4.95 times larger than the corresponding short-run effect.

The coefficient of inc confirms the conventiona l expectation that vehicl e-miles traveled rises with rising income: the income-elasticity is approx imately 0.1 in the short run and 0.5 in the long run. Both annual fuel costs and CAFE standard s are shown to be impor tant determinants of average fleet fuel efficiency – which responds to changes with an even longer lag, causing the long-run effects of these variables to be 1/(1- α f )=6.45 times larger than the corresponding shortrun effects. Another way to interpret this is that each year, fleet turnover and/or changes in driving patterns are able to close (1- α f )=15.5% of the gap between the fuel intensity desired this year (on the basis of variable in the model) and that achieve d by the previous year’s fleet.

Our primary interest is in the first four coe fficients shown in Tabl e 2. The short-run rebound effect for average conditions in this sample is approximately - β pm =0.0407, i.e. 4.07%, while the long-run rebound is 4.95 times this value, or 20.1%. Furthermore, the coefficients β 1 –

β

3 for the three interacted variables involving pm show that the magnitude of the rebound effect, given approximately by the negative of equation (8), d eclines with increasing income and urbanization and increases with increasing fuel cost of driving.

To get a better idea of the magnitude of this dependence, I show in Table 3 the estimated elasticities, computed more precisel y using equations (4), (5), and (7), at two different sets of values for the explanatory variables inc
, pm , and Urban ; one set consists of the average values over our sample and the other consists of the av erage values over the la st five years of the sample. Under average conditions ov er our entire sample period, the measured rebound effect is 4.1% short run and 21.0% long run. However, th ese values are found to fall dramatically when we consider conditions that prevailed in 2000-2004: over th ose years the rebound effect on average is just 1.1% short run and 5.7% long run.

The decline in the rebound effect portrayed in Table 3 is consiste nt with the findings of Section 4.1; but now we are able to fine-tune those findings by compu ting them for a narrower time period. 17 Furthermore, now we can quantify the sour ces of this decline in the rebound effect.

17 This decline in sensitivity applies equally to the sensitivity of travel to fuel price – a topic of great interest to policy in general although no t specifically to the “rebound effect.” This is because the fuel cost of driving, P M , is defined as the ratio of fuel price to fuel efficiency, P F / E . Therefore the elasticity of VMT with respect to fuel price is just the negative of the elasticity of VMT with respect to fu el efficiency. By contrast w ith VMT, fuel intensity has a sensitivity with respect to fuel price, elasticity approximately -0.33, that changed very little over our sample period. (We explored specifications in which it could change with income, much as the VMT elas ticity does, but we found no effect.) With sensitivity of travel declining but that of fuel intensity much less so, the overall elasticity of fuel18 Doing so, we find that about three-fifths of the decline is due to rising real incomes over the time period. The other two-fifths is mostly due to fall ing real fuel costs of operating cars; this fall in turn is about two-thirds caused by a rise in fuel efficiency and one-third caused by a decline in real fuel prices. 18

Table 3. Estimated Elasticities

A verage values (real 2006 $):

Per capita income ($/year) Fuel price ($/gal) Calculated elasticities: Short run Long run Short run Long run

Vehicle-miles traveled -0.041 -0.210 -0.011 -0.057 Fuel intensity -0.035 -0.193 -0.031 -0.191 Fuel consumption -0.074 -0.363 -0.041 -0.237 Rebound effect (%) 4.1% 21.0% 1.1% 5.7% Source: Small and Van Dender (2007c), estimated using full data set. 1.91 1.69

Columns labeled "2000-2004" use average income and fuel cost for the last five years of the sample period. Elasticities are with respect to fuel cost per mile for Vehicle-miles traveled, and fuel price for other quantities.

(b) 2000-2004, short run: 0.007 for "Vehicle- miles traveled", 0.020 for "Fuel efficiency" and "Fuel consumption". (c) Long run: 5-6 times as large as for short run. 1966-2004 2000-2004

Columns labeled "1966-2004" use average income and fuel cost for the entire sample period.

Standard errors are approximately as follows:

(a) 1966-2004, short run: 0.004 for "Vehicle- miles traveled", 0.020 for "Fuel efficiency" and "Fuel consumption". 26,506 33,669

As indicated in Table 3, the rebound effect is numerically equal to the elasticity of vehicle-miles traveled with respect to fuel price, except for the convention that the latter is expressed as a positive percentage rather than as a negative fr action. The underlying reason for this is the theoretical assumption that VMT responds to fuel price and fuel efficiency only through their ratio, which is how we have defi ned fuel cost per mile. (This a ssumption is almost universal in

consumption becomes increasingly dominated by changes in fuel efficiency. This observation would be especially relevant to policies aimed at changing fuel prices.

18 Stated in constant 2005 dollars, average fuel prices were $1.50 in 2000-04 compared to $1.85 over the longer time period 1966-2004. This serves as a useful reminder that real fuel prices declined precipitously from 1981 to 1986, as well as more slowly from 1966 to 1973; these declines mo re than offset the price ri ses during the 1970s and from 1999 to 2004.19 this literature.) The elasticity of VMT with resp ect to fuel price is shown in the top row of the part of Table 3 labeled “calculate d elasticities”. The next row s hows the effect of fuel price on fuel intensity (the inverse of fuel efficiency). Together, these two elastici ties determine the price elasticity of gasoline consumption, shown in the third row; but the latter is not just the sum of others due to an interaction between them. 19 One way to look at this interaction is that a VMT reduction of x% in these circumstances reduces fuel consumption by somewhat less than x% because it is operating on a fleet with improved fuel efficiency. Taking this into account, Figure 2 shows the price elasticit y of gasoline, both for the full sample and for the last five years of the sample, and how it is decomposed into changes in VMT and changes in fuel intensity. 20



### Figure 2

0.041 

### Price Elasticity of Gasoline - Components

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Short Run Long Run Short Run Long Run

1966-2004 2000-2004 Vehicle-Miles Traveled Fuel Intensity

0.074 0

. 363

0.041 0.237

19 The precise relationship is PF E PF E PM M PF F

, ~

, ~

, ˆ

, ) 1 ( ε ε ε ε

− − ⋅ = , where PM M
, ˆ ε and PF E
, ~ ε

− are the elasticities reported in the first two panels of Table 4. This equatio n is derived by USDOE (1996, p. 5-11) and Small and Van Dender (2005, eqn. 6). In Small and Van Dender (2007a), the factor ) 1 (

, ~ PF E ε

− was inadvertently omitted when calculating elasticities of fuel consumption, causing those elasticities to be slightly overstated: see Small and Van Dender (2007b) for the correct values.

20 The VMT and fuel intensity components of Figure 2 are the terms ) 1 (

, ~

, ˆ PF E PM M ε ε

− ⋅ and PF E
, ~ ε

− , respectively.20 5. Implications of the Empiri cal Analysis: Projections to 2030

By distinguishing the causes of the observed de cline in the rebound effect, we are in a position to consider how the rebound effect is likely to cha nge in the future. By inserting projected values for real per capita income, real fuel costs of driv ing, and urbanization into our model, we obtain a projection for the rebound effect.

There are two important qualifications to such an exercise. First, like any empirical equation, there is statistical uncertainty in the estimated parameters, and this uncertainty may become magnified in situations where one or more variab les in the model takes large values. Second, the farther the projected variables de part from the observed range over which the model is fit, the more likely it is that the specific functional fo rm assumed by the model will inadequately portray projected behavior. In our case, the main concer n is with real per capita income, which has a strong influence on the rebound effect and which is e xpected to grow to values much higher than those observed over the sample period 1966-2004.

The first problem can, in principle, be handl ed by computing confidence bands around the projected values. The second, howev er, is inherent in any attemp t to project beyond the range of past experience. Therefore, it must be recogni zed that one can have le ss confidence in projected values than in averages such as those reported in Table 3.

In our case, there is one part icular problem concerning the functional form, but it can be ameliorated. The linear form of equation (8), whic h calculates the structural elasticity of travel with respect to fuel cost per mile, does not constr ain this elasticity to be negative, as theory predicts it should be. This is not a problem for estimation because the predicted value is negative for all but 5.1% of our data poi nts; and for those 5.1%, it may be regarded as differing from zero within normal error ranges. But when projected to high incomes, the elasticity will be predicted to be positive ( i.e. , the rebound effect negative) , contrary to theory. A lthough error bands might still assure us that theoretically appropriate valu es are within the range of statistical uncertainty, the projected values could not be considered th e most likely beliefs abou t the rebound effect in the future.

One solution to this problem would be to c onstrain the projected re bound effect to be zero whenever the equation predicts a negative rebound eff ect. This would be statistically justified, in the sense that zero is the most likely value given th e theoretical constraint. However, I want to be more conservative in asserting how small the reb ound effect may be in the future, and for this reason I have adopted an auxiliary adjustment for th e projections. The effect of this adjustment is to make the projected rebound effects larger than would be the case if I strictly followed the statistical analysis.21 I have therefore fit an aux iliary equation explaining the ne gative of the elasticity ε M,PM , as predicted within our data set by equation (8), as a function of income, fuel cost per mile, and urbanization, but using a functional form that guarantees the rebound effect remains positive. 21 The auxiliary equation is a declini ng exponential approximation to (8):

) 2 exp( ˆ 3 2 1 0
, μ ε + ⋅ + ⋅ + ⋅ − = Urban b P b Income b R M PM M (9)

where Income
, P M , and Urban are their actual values (not lo garithms and not normalized), and μ represents approximation e rror. The symbol ^ over εis a reminder that the values on the lefthand side of (9) are the values predicted from our main empirical system, through equation (8). The factor 2 is just for mnemoni c convenience, making coefficients b 1 , b 2 , and b 3 analogous to β

1 , β 2 , and β 3 in (8).

We can also write (9) as:

μ ε + ⋅ + ⋅ + ⋅ + = − Urban b P b Income b R M PM M 3 2 1 0
, 2 ln ) ˆ ln( . (10)

where again ln denotes the natural logarithm. Equa tion (10) is in a convenient form to estimate, which I do by ordinary least squares. Before estimating (10), I delete the smallest 10% and largest 10% of the 1,989 values of PM M
, ˆ ε in the data set, in order to reduce the impact of extreme values; this also eliminates any negative valu es, which are incompatible with (9) and (10).

The results of this estimation are shown in Tabl e 4 and illustrated in Fi gure 3, which shows how the negative of the structural elasticity (approximately the reb ound effect) varies with income when the other two variables – fuel cost per mile and urbanization – are held at their 2000-2004 average values. For convenience, income has been converted to 2007 dollars in this figure and in subsequent tables. 22 The figure shows that for per capita incomes up to about $55,000, both the original estimated equation and the auxiliary equation are within the 95% confidence bands implied by statistical uncertainty in our coefficients. It also illustrates the manner in which our

21 Ideally it would be better to have used this functional form in the original estimation, but for numerical reasons we were unsuccessful in an earlier project in obtaining estimates this way; I believe the reason for this is that it was too difficult for the statistical routine to fit such a highly nonlinear function while also accounting for simultaneity, lagged dependent variables, and autocorrelation.

22 This affects only how the data are portrayed in graphs and tables, not the actual calculations: all calculations are done using income stated in constant 1987 dollars, consistent with its definition in both the original model estimated and the approximation to it shown in Table 5.22 auxiliary procedure projects positive rebound effect s in situations where the statistical model alone would predict that they are negative.

Table 4. Auxiliary regression results Dependent variable: predicted structural elasticity ( PM M
, ˆ ε

− )

Variable Coefficient

name Coefficient estimate Standard

error
constant ln R 0 -1.4789 0.0512
Income (thousands $) b 1 -0.1706 0.0027 2* P M b 2 0.0826 0.0015 Urban b 3 -0.7961 0.0340

Standard error of regression 0.233 R-squared 0.895

Note: Income and P M

are in measured at 1987 price levels: Income in 1000s of constant 1987 dollars, P M

in constant 1987 cents/veh-mile. The factor 2 multiplying P M

is for convenience of interpretation, making (10) analogous to (9). These choices of units are arbitrary and have no effect on m easured or projected rebound effects.



### Figure 3


### Rebound Effect Projections

-0.04 0.00 0.04 0.08 0.12 0.16

0 20 40 60 Real Income (thousands of const 2007$/yr)-Elasticity Estimated eqn. on full data set (with 95% conf bands)

Exponential fit to 80% of data Range covered by middle 80% of data

Min. income in data Max. Income in data (Fuel cost per mile & Urbanization at 2000-04 averages)23 Based on this estimate of equation (10) , we can then project values for PM M
, ˆ ε

− in years 2005- 2030, given actual and anticipated va lues of income, fuel cost pe r mile, and urbanization. (Fuel cost per mile is equal to fuel price divided by fuel efficiency.) For years beginning 2006, each of these variables, except urbaniza tion, is taken from the detailed tables in the Annual Energy Outlook (AEO) (US EIA 2009). For per capita income , the values are calculated for each year from the AEO macroeconomic tables, then adjust ed for the difference between our measure (real 2007 dollars using a consumer price index deflat or) and the measure used in the AEO (chainweighted year-2000 dollars). 23 For average fuel price and fuel efficiency, which together determine fuel cost per mile, I show results from two alternate scenarios: the AEO baseline projection (“Reference Case”) and one that a ssumes higher price growth (“High Price Case”). 24 Urbanization, which plays only a small role, is assumed to grow slowly from 83.3% in 2005 to 89.7% in 2030. 25 Once - ε M
, PMis calculated for each year for each scenario, using (9), the shortand long-run rebound effects are calcula ted from (4), (5), and (7).

The results of these projections are shown in Ta ble 5. The table starts with calculations using observed values for variables in years 2000-2008. In the Reference Case the rebound effect in those years is calculated as 0.8%–1.2% for the sh ort run and 4.0–5.8% for th e long run. (It rises, rather than falls, from 2002 to 2008 due to the ra pid run-up of fuel pric es in those years.) 26 The projections show these rebound eff ects falling gradually over most of the projection period, with the long-run rebound effect taking values of 3.9% in 2015 and 1.5% in 2025 under the Reference Scenario. The values are somewhat higher in the High-Price Scenario, but still falling to 5% in 2016 and to 1.4% by 2030.

23 That is, in projecting income, fleet fu el efficiency, and average fuel price, I apply the EIA actual (for 2006-08) and projected (for 2009-2030) growth rates to our 2005 values, in order to avoid spurious shifts due to our use of a different data source than EIA for our estimations and histor ical values. In addition to the slight difference in price index, the EIA values are for real disposable personal income per capita, which differs only very slightly (in terms of recent growth rates) from the measure used in our empirical work, which is real personal income per capita. I have calculated the EIA values from US EI A (2009), “Year-by-Year Reference Ca se Tables” and “Year-by-Year High Price Case Tables”, Table 20 (“Macroeconomic Indicators”), by dividing Real Disposable Personal Income (row 2116) by Population with Armed Forces Overseas (row 2104). I then calculate the annual growth rate between each successive pair of years, starting 2008-2009, and apply that ra te to the historical values found for the series used in our estimation, starting in 2006.

24 See US EIA (2009), “Year-by-Year Re ference Case Tables (2006-2030),” Tabl e 3, and High Price Case Tables (2006-2030), Table 3, row 1361, “Motor Gasoline”. Prices are projected to grow in real (inflation-adjusted) terms by an average of 1.4% per year over the period 2007-2030 in the Reference Case, and by 2.9% per year in the High Price Case (see last column in the row for “Motor Gasoline” in each table).

25 This is based on our data on urbanization along with the projected growth rates for US urbanization of the United Nations Department of Economic and Social Affairs, Population Division, World Urbanization Prospects: The 2005 Revision , Data Table A.2 ( http://www.un.org/esa/population/publications/WUP2005/2005wup.htm ).

26 The values shown here for 2000-2004 are slightly lower than the values of 1.1% and 5.7% shown for 2000-2004 in Table 3, mainly because of the approximation error visible in Figure 3 at incomes in the range $35,000–36,000.24 Year Income Fuel Price Fuel Price (2007$) (2007$/gal) Short Long Dynamic (2007$/gal) Short Long Dynamic Run Run (start 2010) Run Run (start 2010)

2000 35,933 1.84 0.9 4.4 1.84 0.9 4.4 2001 35,781 1.64 0.8 4.1 1.64 0.8 4.1 2002 35,492 1.52 0.8 4.0 1.52 0.8 4.0 2003 35,458 1.70 0.9 4.4 1.70 0.9 4.4 2004 36,321 1.98 0.9 4.5 1.98 0.9 4.5 2005 36,611 2.41 1.0 5.1 2.41 1.0 5.1 2006 37,373 2.65 1.0 5.1 2.65 1.0 5.1 2007 38,065 2.78 1.0 5.1 2.78 1.0 5.0 2008 38,258 3.16 1.2 5.8 3.16 1.2 5.7 2009 38,909 2.30 0.8 3.8 2.25 0.7 3.7 2010 38,565 2.86 1.0 4.8 1.0 3.04 1.0 5.2 1.0 2011 39,186 3.05 1.0 4.8 1.7 3.31 1.1 5.4 1.9 2012 40,030 3.20 0.9 4.7 2.3 3.58 1.1 5.4 2.6 2013 40,815 3.29 0.9 4.4 2.8 3.86 1.1 5.4 3.2 2014 41,807 3.41 0.8 4.1 3.0 4.18 1.1 5.4 3.6 2015 42,669 3.50 0.8 3.9 3.2 4.43 1.1 5.3 4.0 2016 43,419 3.54 0.7 3.6 3.3 4.59 1.0 5.0 4.2 2017 44,126 3.57 0.7 3.3 3.3 4.77 1.0 4.8 4.3 2018 45,000 3.59 0.6 3.0 3.3 4.87 0.9 4.4 4.4 2019 45,966 3.61 0.5 2.7 3.2 4.93 0.8 4.0 4.3 2020 46,778 3.63 0.5 2.4 3.0 4.97 0.7 3.6 4.2 2021 47,524 3.66 0.4 2.2 2.9 4.97 0.6 3.2 4.0 2022 48,307 3.69 0.4 2.0 2.7 5.01 0.6 2.9 3.8 2023 49,148 3.68 0.4 1.8 2.6 5.07 0.5 2.7 3.6 2024 50,010 3.71 0.3 1.7 2.4 5.07 0.5 2.4 3.4 2025 50,894 3.74 0.3 1.5 2.3 5.12 0.4 2.2 3.2 2026 51,787 3.77 0.3 1.4 2.1 5.18 0.4 2.0 3.0 2027 52,605 3.80 0.3 1.3 2.0 5.23 0.4 1.8 2.8 2028 53,399 3.85 0.2 1.2 1.8 5.29 0.3 1.7 2.6 2029 54,139 3.87 0.2 1.1 1.7 5.34 0.3 1.6 2.4 2030 54,848 3.91 0.2 1.0 1.6 5.40 0.3 1.4 2.2 Table 5. Projected Rebound Effect, US Average, 2000-2030 Assumed growth in real per capita income: AEO 2009

High fuel-price scenario Reference fuel-price scenario -----Rebound Effect (%)----- -----Rebound Effect (%)-----

The projected values in Table 5 for the the EIA Reference Scenario are depicted in Figure 4.25 

### Figure 4

Projected Rebound Effect

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

2005 2010 2015 2020 2025 2030Rebound Effect (%) Short Run

Long Run

Table 5 also shows two columns, one for each scenario, entitled “Dynamic Rebound Effect.” This uses the projection methodology to answer the question of how th e actual effects on VMT would play out over time if a regulation changing average fuel efficiency went into effect in year 2010 and remained unchanged thereafter. In the fi rst year (2010), the dynamic rebound effect is simply the short-run rebound effect. In subsequent years, two counteracting forces are at work: consumers gradually adjust their travel patter ns toward that predicted by the long-run rebound effect (hence raising their VMT response), but the long-run rebound effect also diminishes over time (hence lowering their VMT response). In the re ference scenario, the response peaks in years 2016-18 at a value of 3.3%, then gradually falls, tracking the long-run reb ound effect with a lag. In the high-price scenario, the response peaks in year 2018, at a value of 4.4%.26 All of the rebound effects shown in this report describe the respons e to regulations of just those vehicles subject to the regulation. 27 Since most realistic regulations apply only to new vehicles, the response of aggregate VMT to regulations is much slower ev en than depicted by the dynamic rebound effect.

The low rebound effect projected in this report depends strongly on the future growth of per capita income. As a sensitivity analysis, I project it also using an alternative growth rate equal to the 1.18% annual rate seen in one of the lowe r-growth periods of recent history, namely 1990- 2005. 28 In that case, the rebound eff ect declines less rapidly; the full results are shown in Appendix B (Table B1).

Appendix B also shows the results of using the inco me, fuel price, and fuel efficiency price paths in the Updated AEO 2009 scenario issued in Ap ril 2009. The latter include s the effects of the major stimulus package passed just previously (t he American Recovery and Reinvestment Act) as well as revised economic pr ojections. A comparison of the rebound effects, calculated under the two alternate projections (original and upda ted AEO), is shown in Table B2. The Updated scenario has slightly lower incomes but substant ially lower fuel prices over the near term (2009- 2015), resulting on net in lower rebou nd effects. In later years, th e two fuel-price projections are nearly identical but the Updated income projections are slightly higher, resulting also in lower rebound effects.

Table 6 shows a comparison of selected rebound-eff ect calculations for five scenarios: two from the original AEO, two with lower income growth, and one with Updated AEO.

27 Recall that the adjustment lags entering equation (7) for the long-run rebound effect involve only changes in vehicle travel and, to a minor extent, in total fl eet size – that is, they involve lag coefficients α m and α v . The equation does not depend on lagged effects in changes in fuel intensity (coefficient α f ), which can be interpreted as gradual fleet turnover. Thus, the long-run rebound effect calculated here depends on fleet turnover only for the realization of changes in vehicle fleet size, whic h play a very minor role in equation (7) due to the small sizes of α mv , α vm , and v 2 β

.

28 This rate is calculated from per capita personal income (US BEA 2009, Table 657) divided by the Consumer Price Index, all urban consumers, all items (US Council of Economic Advisors 2009, Table B-60).27 Table 6. Long-run rebound effect, 2015 and 2025, under alternate scenarios (%) Year Income Growth Fuel price AEO Reference AEO High Price AEO Updated Reference AEO Reference 3.9 5.3 AEO Updated Ref. 3.7 2015 1.18% annual 4.3 5.9

AEO Reference 1.5 2.2 AEO Updated Ref. 1.4 2025 1.18% annual 2.2 3.2

6. Conclusions

The research reported here confirms the findings of previous studies that the long-run rebound effect, measured over a period of several decades extending back to 1966, is about 20%. I also find a short-run (one-year) rebound eff ect of 4%, which is harder to compare to previous studies because previous work contains so much vari ation depending on the treatment of dynamics and of CAFE regulations.

This research also provides strong evidence that the rebound effect became substantially lower in the period since 1990, and that probably this was due to a combination of higher real incomes, lower real fuel costs, and higher urbanization. Because time spent in travel rises with urbanization and its attendant conge stion, and the value of that tim e rises with incomes, all three of these differences tend to make fuel costs a sma ller portion of the total cost of traveling. Thus it is not surprising that people would become less se nsitive, on a percentage basis, to changes in those fuel costs.

The statistical procedures describe d here enable us to isolate the separate influences of income, fuel cost, and urbanization on the rebound effect. Then we can see how actual changes in those values probably influenced the rebound effect during any given period covered by our data. This reveals that in the years 2000-2004, the rebound eff ect was likely 5.7% in the long run and 1.1% in the short run.

Similarly, the model enables us to see how proj ected changes in those same three variables can be expected to influence the rebound effect in th e future. These quantitative predictions are less28 certain because we are projecting beyond the range of data with which the model was estimated. Nevertheless, the results strongl y suggest that inco me growth overcomes other influences, causing the rebound effect to declin e substantially in the future. With income and fuel prices changing according to the Energy Information Administration’s AEO 2009 reference scenario, the long-run rebound effect will most likely fall to 3.9% in 2015 and 1.5% in 2025, with the short-run rebound effect about one-fourth of those values. Even if income grows at the much slower rate experienced from 1990-2005, and if fu el prices rise according to the EIA “high-price scenario” — two conditions that produce th e highest rebound effects — the long-run rebound effect rises for a few years, to a high of 5.9% in 2015, then declines steadily over the period 2016-2030 to a value of 2.4% (Table B-1, right panel). Under the Updated Reference Case, the rebound effect is slightly smaller than under the original Referen ce Case, due to lower projected fuel prices in the early years and slightly higher projected incomes in the later years.29 Appendix A. Full Estimation Results of the Si multaneous Three-Equation Model of Vehicle Travel, Vehicle Stock, and Fuel Intensity, 1966-2004.

Table A1 provides summary statistics some of the main variables used in our statistical estimation. Monetary variables (prices, costs, and incomes) are shown in constant 1987 dollars; they may be multiplied by 1.72 to express them in 2005 dollars (as is done in Tables 5 and B1).

Table A1. Summary Statistics for Selected Variables

Name Definition Mean Std. Dev. Min. Max.
Vma VMT per adult 11,209 2,709 4,737 24,107 Vehstock Vehicles per adult 1.007 0.192 0.458 1.716 Fint Fuel intensity (gal/mi) 0.0603 0.0127 0.0317 0.0919
Pf Fuel price, real (cents/gal) 107.9 23.1 60.3 194.9 pf log Pf, normalized 0 0.2008 -0.5602 0.6127 Pm Fuel cost/mile, real (cents/mi) 6.625 2.292 2.782 14.205 pm log Pm, normalized 0 0.3552 -0.8059 0.8245
Income Income per capita, real 14,936 3,504 6,448 30,763 inc log Income, normalized 0 0.2350 -0.8127 0.7499 Adults/road-mile Adults per road mile 58.08 67.35 2.50 490.20 Pop/adult Population per adult 1.4108 0.0904 1.2314 1.7360 Urban Fraction of pop. in urban areas 0.7143 0.1940 0.2895 1.0000 Railpop 0.0907 0.2107 0.0000 1.0000

Pv Price of new vehicles (index) 1.040 0.210 0.703 1.493 Interest Interest rate, new-car loans (% ) 10.43 2.72 5.17 16.49 Licenses/adult Licensed drivers per adult 0.907 0.078 0.603 1.173 Fraction of pop. in metro areas served by heavy rail

Variables with capitalized names are shown here as levels ; even if it is their logarithms (lower-case names) that enter our specification. "Urban" is shown here unnormalized, but it is normalized when entering our specification.

Tables A2–A4 show the simultaneous estimates of the three indicated structural equations, performed using three-stage least squares on th e entire system. The estimation routine accounts for both the lagged dependent variab les and autocorrelation; for this reason the predicted value in any year depends on observed values for two previ ous years, hence the firs t two years of data do not enter as separate observations for the estimation. (This is w hy the number of observations is 51 states times 37 years, although there are 39 years of data.)30 Small and Van Dender (2007a,b) provide furthe r details about the estimation procedure including the choice of instrume ntal variables, which has spec ial features due to variables entering the equation non-linearly and the presence of autoco rrelation and lagged dependent variables.

Variable Coefficient Stndrd. Error t-statistic
vma(t-1) 0.7980 0.0120 66.40 vehstock 0.0253 0.0093 2.72

p m -0.0407 0.0042 -9.61
p m^2 -0.0169 0.0064 -2.66
p m*inc 0.0696 0.0132 5.28
p m*Urban 0.0255 0.0100 2.54

inc 0.1044 0.0134 7.80 adults/road-mile 0.0214 0.0044 4.81
p op/adult 0.1661 0.0396 4.20 Urban -0.1090 0.0501 -2.17 Railpop -0.0041 0.0062 -0.67 D7479 -0.0438 0.0035 -12.59 Tren d 0.0006 0.0003 2.07

constant 1.9033 0.1180 16.13 rho -0.0893 0.0219 -4.07
No. observations 1,887 Adjusted R-square d 0.9816 S.E. of regression 0.0314 Durbin-Watson stat 1.9725 Sum squared resi d 1.7927

Notes to Tables 2-4:

Estimates of fixed effects coefficients (one for each state except Wyoming) are not shown

. Variables inc
, Urban , and the constituent variables in pm are normalized by subtracting their mean value in the sample, both in the variable itself and in any interactions it takes. As a result, the coefficient of any variable in its uninteracted form gives the effect of that variable on vma at the mean values of the other variables. Bold or italic type indicates the coefficient is statistically significant at the 5% or 10% lev e Table A2. Vehicle-Miles Traveled Equation

NOTE: coefficient of variable “adults/r oad-mile” whould be preceded by a – sign.31 Variable Coefficient Stndrd. Erro r vehstock(t-1) 0.8635 0.0149 57.79

vma 0.0285 0.0172 1.66

p v 0.0240 0.0333 0.72
p m -0.0011 0.0070 -0.15

inc 0.0196 0.0166 1.18 adults/road-mile 0.0258 0.0071 3.63 Trend 0.0005 0.0007 0.63 interest 0.0070 0.0047 1.49 licenses/adult 0.0475 0.0203 2.34

constant -0.1934 0.1699 -1.14 rho -0.1144 0.0264 -4.34
No. observations 1,887 Adjusted R-squared 0.9571 S.E. of regression 0.0401 Durbin-Watson stat 2.0198 Sum squared resid 2.9332 Table A3. Vehicle Stock Equation

Variable Coefficient Stndrd. Error
fint(t-1) 0.8450 0.0127 66.65 vma+pf -0.0297 0.0064 -4.66 cafe -0.0882 0.0110 -8.03

inc -0.0040 0.0159 -0.25
p op/adult -0.0056 0.0643 -0.09 Urban -0.1081 0.057 6 -1.88 D7479 -0.0102 0.0046 -2.19

Trend66-73 0.0005 0.0010 0.45 Trend74-79 -0.0036 0.0010 -3.64 Trend80+ -0.0024 0.0004 -6.13

constant -0.1777 0.0779 -2.28 rho -0.1491 0.0221 -6.74
No. observations 1,887 Adjusted R-square d 0.9606 S.E. of regression 0.0411 Durbin-Watson stat 2.0004 Sum squared resi d 3.0898 Table A4. Fuel Intensity Equation32 Table A5 shows various elasticities calculated fr om the estimated structural equations, according to equations (4) and (7) in the text, and equati on (8), (9), and (10) in Small and Van Dender (2007b).

Elasticity Short Run Long Run Elasticity of VMT with respect to fuel cost per mile: (a) At sample average -0.0407 -0.2078 (0.0042) (0.0219) At US 2000-2004 avg. (b) -0.0103 -0.0532 (0.0071) (0.0359) Elasticity of fuel intensity with respect to fuel price: At sample average -0.0286 -0.1593 (0.0062) (0.0368) At US 2000-2004 avg. (b) -0.0294 -0.1838 (0.0064) (0.0421) Elasticity of fuel consumption

w i t h respect to f ue l pr i ce: At sample average -0.0681 -0.3340 (0.0050) (0.0293) At US 2000-2004 avg. (b) -0.0394 -0.2272 (0.0075) (0.0391) Table A5. Rebound Effect and Other Price Elasticities

Notes: (a) The rebound effect is just the negative of this number (multiplied by 100 if expressed as a percent).

Asymptotic standard errors in parentheses are calculated from the covariance matrix of estimated coefficients using th e Wald test procedure for an arbitrary function of coefficients in Eviews 5. (b) Elasticities measured at the average 2000-2004 values of pm , inc , and Urban for all US.

Total fuel consumption is the product of VMT and fuel intensity, and so it might seem that the price-elasticity of fuel consumption would be th e sum of the price-elasticities of VMT and fuel intensity. But because both are endogenous, there is an interaction and the co mbined elasticity is:

PF I PF I PM M PF F

, ~

, ~

, ˆ

, ) 1 ( ε ε ε ε + + ⋅ = (11)33 where PM M
, ˆ ε and PF I
, ~ ε are the elasticities reported in the first two panels of Table A5. This equation is derived by USDOE (1996, p. 5-11) and Small and Van Dender (2007b, eqn. 10). 29

Appendix B. Projection results for alte rnate assumptions about income growth

Table B1 shows the results with all other assumpti ons the same as in Table 5, except income is projected to grow at only 1.18% per year starting in 2008. This re sults in higher rebound effects.

Table B2 shows the results with all the assu mptions of the Updated Reference Scenario (produced April 2009), and compares them with th e original Reference Scenario (same as Table 5). The Updated scenario shows slightly lower in comes but substantially lower fuel prices over the near term (2009-2015), resulting in lower re bound effects. Over the long term the fuel-price projections are nearly the same but the income projections slightly higher, resulting also in lower rebound effects.

29 Small and Van Dender (2007a) inadvertently omitted the term in parentheses in equation (11) when computing

ε

F
, PF for Table 5, and therefore overstated the magnitude of ε F
, PF ; the correct results are shown in Small and Van Dender (2007b, Table 5).34 Year Income Fuel Price Fuel Price (2007$) (2007$/gal) Short Long Dynamic (2007$/gal) Short Long Dynamic Run Run (start 2010) Run Run (start 2010)

2000 35,933 1.84 0.9 4.4 1.84 0.9 4.4 2001 35,781 1.64 0.8 4.1 1.64 0.8 4.1 2002 35,492 1.52 0.8 4.0 1.52 0.8 4.0 2003 35,458 1.70 0.9 4.4 1.70 0.9 4.4 2004 36,321 1.98 0.9 4.5 1.98 0.9 4.5 2005 36,611 2.41 1.0 5.1 2.41 1.0 5.1 2006 37,373 2.65 1.0 5.1 2.65 1.0 5.1 2007 38,065 2.78 1.0 5.1 2.78 1.0 5.0 2008 38,258 3.16 1.2 5.8 3.16 1.2 5.7 2009 38,710 2.30 0.8 3.8 2.30 0.8 3.8 2010 39,168 2.86 0.9 4.6 0.9 3.10 1.0 4.9 1.0 2011 39,631 3.05 0.9 4.6 1.7 3.39 1.0 5.1 1.8 2012 40,100 3.20 0.9 4.6 2.3 3.66 1.1 5.3 2.5 2013 40,574 3.29 0.9 4.5 2.7 3.95 1.1 5.5 3.1 2014 41,054 3.41 0.9 4.4 3.1 4.27 1.2 5.8 3.7 2015 41,539 3.50 0.9 4.3 3.3 4.53 1.2 5.9 4.1 2016 42,030 3.54 0.8 4.0 3.5 4.69 1.2 5.7 4.5 2017 42,527 3.57 0.8 3.8 3.6 4.88 1.1 5.6 4.7 2018 43,030 3.59 0.7 3.6 3.6 4.97 1.1 5.3 4.8 2019 43,539 3.61 0.7 3.3 3.6 5.04 1.0 5.0 4.9 2020 44,054 3.63 0.6 3.1 3.5 5.08 0.9 4.6 4.9 2021 44,575 3.66 0.6 2.9 3.4 5.08 0.9 4.2 4.8 2022 45,102 3.69 0.5 2.7 3.3 5.12 0.8 4.0 4.6 2023 45,635 3.68 0.5 2.5 3.2 5.18 0.7 3.7 4.5 2024 46,175 3.71 0.5 2.4 3.0 5.19 0.7 3.4 4.3 2025 46,721 3.74 0.4 2.2 2.9 5.24 0.6 3.2 4.1 2026 47,273 3.77 0.4 2.1 2.8 5.30 0.6 3.0 3.9 2027 47,832 3.80 0.4 2.0 2.6 5.34 0.6 2.8 3.7 2028 48,398 3.85 0.4 1.9 2.5 5.40 0.5 2.7 3.5 2029 48,970 3.87 0.4 1.8 2.4 5.46 0.5 2.5 3.4 2030 49,549 3.91 0.3 1.7 2.3 5.52 0.5 2.4 3.2 Table B1. Projected Rebound Effect, US Average, 2000-2030 Assumed growth in real per capita income: 1.18%/year (historic 1990-2005)

Reference fuel-price scenario High fuel-price scenario -----Rebound Effect (%)----- -----Rebound Effect (%)-----35 Year Income Fuel Fuel Price Income Fuel Fuel Pric e (2007$) Effic. (2007$/gal) Short Long Dynamic (2007$) Effic. ( 2007$/gal ) Short Long Dynamic (mi/gal) Run Run ( start 2010) (mi/gal) Run Run (start 2010)

2008 38,258 22.6 3.16 1.2 5.8 38,168 22.7 3.12 1.1 5.7 2009 38,909 22.5 2.30 0.8 3.8 38,829 22.8 1.87 0.6 3.2 2010 38,565 22.6 2.86 1.0 4.8 1.0 38,931 23.0 2.11 0.7 3.5 0.7 2011 39,186 22.8 3.05 1.0 4.8 1.7 39,143 23.2 2.47 0.8 3.9 1.3 2012 40,030 23.0 3.20 0.9 4.7 2.3 39,618 23.4 2.66 0.8 4.0 1.9 2013 40,815 23.3 3.29 0.9 4.4 2.8 39,893 23.8 2.81 0.8 4.0 2.3 2014 41,807 23.6 3.41 0.8 4.1 3.0 40,749 24.2 2.96 0.8 3.8 2.6 2015 42,669 24.0 3.50 0.8 3.9 3.2 41,593 24.7 3.12 0.7 3.7 2.8 2016 43,419 24.4 3.54 0.7 3.6 3.3 42,368 25.2 3.23 0.7 3.5 3.0 2017 44,126 24.9 3.57 0.7 3.3 3.3 43,185 25.7 3.34 0.7 3.3 3.1 2018 45,000 25.4 3.59 0.6 3.0 3.3 44,115 26.3 3.44 0.6 3.0 3.1 2019 45,966 25.9 3.61 0.5 2.7 3.2 45,148 26.8 3.51 0.5 2.7 3.0 2020 46,778 26.4 3.63 0.5 2.4 3.0 46,275 27.4 3.57 0.5 2.4 2.9 2021 47,524 27.0 3.66 0.4 2.2 2.9 47,331 27.9 3.59 0.4 2.2 2.8 2022 48,307 27.6 3.69 0.4 2.0 2.7 48,356 28.5 3.62 0.4 1.9 2.6 2023 49,148 28.1 3.68 0.4 1.8 2.6 49,427 29.0 3.64 0.3 1.7 2.5 2024 50,010 28.6 3.71 0.3 1.7 2.4 50,483 29.5 3.64 0.3 1.5 2.3 2025 50,894 29.1 3.74 0.3 1.5 2.3 51,600 30.0 3.63 0.3 1.4 2.1 2026 51,787 29.6 3.77 0.3 1.4 2.1 52,701 30.5 3.67 0.2 1.2 2.0 2027 52,605 30.0 3.80 0.3 1.3 2.0 53,786 30.9 3.66 0.2 1.1 1.8 2028 53,399 30.4 3.85 0.2 1.2 1.8 54,823 31.4 3.71 0.2 1.0 1.7 2029 54,139 30.8 3.87 0.2 1.1 1.7 55,859 31.8 3.81 0.2 0.9 1.5 2030 54,848 31.1 3.91 0.2 1.0 1.6 56,919 32.1 3.77 0.2 0.8 1.4 Original (published) version Updated version Compares the Reference Scenarios in the original published (March) and updated (April) versions Table B2. Projected Rebound Effect: AEO 2009 Original (Published) and Updated Compared

-----Rebound Effect (%)----- -----Rebound Effect (%)-----36 References

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