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 values of the “rebound effect” for travel 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 motor vehicles (passenger cars and light trucks), 

caused in turn by regulations or technological developments. I briefly discuss the literature, then 

summarize empirical estimates done at University of California at Irvine in collaboration with 

Kurt Van Dender. Finally I present updated empirical estimates and derive their implications for 

the rebound effect in the time frame 2000-2030. 

The literature review and empirical methodology are described more fully in a published article 

(Small and Van Dender 2007a), and even more fully in the working paper from which the 

published article was derived (Small and Van Dender 2007b). The empirical estimates have been 

updated subsequently, mainly by adding three new years of data; these updated results are 

presented here and also summarized in a forthcoming short publication (Small and Van Dender 

2007c). The projections are my own, although they use a methodology adapted from one used in 

an earlier report to the California Air Resources Board (Small and Van Dender 2005). 

1. Background and definitions

1.1 Determinants of motor-vehicle travel 

The rebound effect is simply a statement of the 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 transportation, and its price to the user includes the cost of fuel. If 

the amount of service is measured as vehicle-miles traveled (VMT), then the component of price 

accounted for by fuel cost, here called “fuel cost per mile” PM, is equal to the price of fuel Pf

(e.g. stated in $/gallon) divided by fuel efficiency E (e.g. stated in vehicle-miles/gallon): 

PM = Pf / E . (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 summarized as a ratio of the percent change in 

demand, ΔM/M, to the percent change in price causing it, ΔPM/PM, 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 ΔPM is very small so that the ratio become a 

derivative. Therefore I define the elasticity of vehicle-miles traveled with respect to cost per mile 

as follows: 

M

M

M PM dP

dM

M

P = ⋅ , ε (2) 

where the derivative dM/dPM is simply the limit of ΔM/ΔPM as ΔPM becomes very small. An 

equivalent way to write this is in terms of the natural logarithms of M and PM, 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 when 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

M PM ∂

∂

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 the 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 constraints set by regulation. Therefore 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 system 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 PM (as already described), the size of the vehicle fleet, V, 

and various socio-demographic characteristics including income. The second dependent variable 

is V, which is a function of several things: a price index PV of new vehicles, the amount of travel 
3

M (since new vehicles are purchased in large part to supply desired travel), the price of travel PM, 

and other characteristics. Finally, the third dependent variable, fuel efficiency E (or its inverse), 

is presumed to be chosen based on a combination of motives including the wish to conserve on 

the cost of traveling M miles, the need to meet various regulations on fuel efficiency and/or 

emissions, and tradeoffs with vehicle performance; in our empirical work E is assumed to be a 

function of M, price of fuel PF, a variable measuring the tightness during any given year of the 

US federal Corporate Average Fuel Economy (CAFE) regulations, and other characteristics. 

Thus, our system accounts for the effects of a change in regulations through two pathways. I 

illustrate for an increase in fuel-efficiency standards. First, the regulations increase the average 

fuel economy of the fleet, and that in turn reduces the cost per mile of travel, PM, through 

equation (1); this directly changes the amount of travel because of downward-sloping demand as 

just discussed. Second, the size of the vehicle fleet 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, PV, 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 would work in the opposite direction 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 ultimate effect of both pathways on VMT, we use the symbol Mˆ to 

designate the combined effect, and designate its elasticity with respect to cost per mile as Mˆ ,PM ε , 

reserving the symbol M ,PM ε for the changes operating through the first pathway only. Small and 

Van Dender (2007a) show that these quantities are related by: 

 

1

 In principle the effect of changes in average new-vehicle price could be analyzed 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 PV, 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 using 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

M PM M V V PM

M PM

, ,

, , ,

, ˆ 1 ε ε

ε ε ε

ε −

+ = (4) 

where εM,V denotes the direct elasticity of travel with respect to vehicle fleet, εV,M denotes the 

direct elasticity of vehicle fleet with respect to amount of travel, and εV,PM denotes the elasticity 

of vehicle fleet with respect to cost per mile of travel. All the quantities on the right-hand side of 

(4) are measured directly as coefficients, or combinations 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, Mˆ ,PM 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 resulting from regulations (with both expressed in 

percentage terms), while recognizing that the change in fuel efficiency is not directly set by 

regulations but rather results from a complex interactive process. 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 Mˆ ,PM ε is expected to negative, it is convenient to express the rebound 

effect bS

 as a number that is normally positive: 

M PM

S b , = −ε ˆ (5) 

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

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

Our empirical system also accounts for the slowness with which changes can occur, especially 

changes in the vehicle fleet size and average efficiency, which require purchases and retirements 

of vehicles. It is able to do this 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 assuming 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 deviation 
5

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 adjustment process means that the short-run response 

(that occurring in the same year) is smaller than the long-run response. Continuing to use the 

notation Mˆ ,PM ε for the elasticity determined within this system, it is now a short-run elasticity 

because the long-run response is accounted for elsewhere in the equation (through the lagged 

variables). We represent the corresponding short-run and long-run rebound effects as bS

 and bL, 

respectively. They are approximately related by: 

 m

M PM

m

S

L b b

α

ε

α −

− = − ≅

1 1

, ˆ

 (6) 

where αm is the coefficient of the lagged dependent variable in the equation explaining vma. A 

more precise relationship accounts for the fact that in the three-equation system, the lagged 

values in more than one equation can affect the long-run response; specifically:2

m v mv vm

v mv v

L M PM b

α α α α

ε α α β

− − −

− ⋅ − − = (1 )(1 )

(1 ) , 2 (7) 

where αv

 is the coefficient of the lagged dependent variable in the equation explaining the 

logarithm of vehicle stock, αmv is the coefficient of vehicle stock in the equation explaining vma, 

αvm is the coefficient of vma in the equation explaining vehicle stock, and v β 2 is the coefficient 

of pm in the equation explaining vehicle stock. With our empirical results, the approximation in 

(6) is a very close one.3

 In addition to accounting for lagged values within the system 

determining our dependent variables, our empirical system accounts for the possibility that the 

error terms in each equation are correlated 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 “fixed effects” specification, in 

which a distinct constant term is estimated for every state instead of just one for the entire 

system.4

 

 

2

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

3

 This is because αmv, v β2 , and αvm are all quite small. When these three coefficients are small, the term (1-αv) 

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 plaguing earlier studies investigating changes over time. We are able 

to distinguish them because of the long time period (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 fleet size and thus, indirectly, the amount of travel. Regulations 

may affect fuel prices through the impact of aggregate demand for fuel on petroleum markets. 

They may influence technological developments, thereby affecting the costs and performance of 

future vehicles. A broader analysis of the effects of fuel efficiency on travel might want to 

account for such factors, but they are outside the realm of the “rebound effect” as I define it here 

and, I believe, as most researchers have used the term. However, it is worth pointing out that if 

regulations raise the price of new vehicles, then the response to that price rise would tend to 

offset somewhat the rebound effect by curtailing the number of vehicles 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 estimate 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 – results 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-mile elasticity declines over time, consistent with 

the effect of income just discussed; but the evidence has only marginal statistical significance. 

Jones (1993) re-examines Greene’s data, adding observations for 1990 and focusing on modelselection issues in time-series analysis. He finds that although Greene’s autoregressive model is 

statistically valid, so are alternative 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-run but a similarly large long-run rebound effect 

(29%).7

 Schimek accounts for federal CAFE regulations by including a time trend for years 

since 1978; he also includes dummy variables for the years 1974 and 1979, when gasoline-price 

controls were in effect, resulting in queues and sporadic rationing at service stations. These 

controls reduce the extent of autocorrelation in the residuals. 

These aggregate studies highlight the possible importance of lagged 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 presence of a lagged dependent variable from the presence of 

autocorrelation. Their estimates of these dynamic properties are especially sensitive to the time 

period considered and to the treatment of CAFE regulations. 

Another type of study relies on pooled cross-sectional time-series data at a smaller geographical 

level of aggregation. Haughton and Sarkar (1996) construct 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 tax, providing an additional opportunity to observe 

the effects of fuel price on travel. The authors estimate equations both for VMT per driver and 

for fuel intensity (the inverse of fuel efficiency), 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, implying a substantial difference 

between long and short run. Tackling yet another 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 legal minimum in a given year and the actual fuel 

efficiency in 1975. However, that variable is so strongly correlated with the historical maximum 

real price of gasoline that they omit it in most specifications, 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 measuring 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 Expenditure 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 Transportation Survey (NTPS), obtain a rebound effect 

of just 4%.9

 There are a number of reasons to be cautious about these results. West obtains an 

extremely low income-elasticity for travel, namely 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 because 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, and those variations are correlated 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 several 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 implies 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 Transportation Energy Consumption Survey and its 

predecessor, for six different years between 1979 and 1994. Their usage equation is part of a 

simultaneous system including vehicle 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 three-vehicle households to 28% for one-vehicle 

households. 

 

9

 Their model 3, with odometer readings as dependent variable. They actually measure the elasticity of VMT with 

respect to gasoline price, εM,PF, which is equal to εM,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 variables 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 the endogeneity of fuel efficiency.11 Mannering (1986) 

explicitly addresses the bias resulting from such endogeneity, finding it to be large, although in 

the direction opposite to what we expect: his estimate of |εM,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 particular 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-sectional and temporal variation are more 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 estimates 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 for 50 US states plus District of Columbia, each 

measured annually for years 1966-2004. Variables are constructed from public sources, mainly 

the US Federal Highway Administration, US Census Bureau, and US Energy Information 

Administration. Data sources and a fuller description are given in Small and Van Dender 

(2007a,b). 

I now list the primary variables used in our statistical estimation. 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 conditioning 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 adult population, by state and year (logarithm: 

vma, for “vehicle-miles per adult”). 

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

PM: Fuel cost per mile, PF/E. Its logarithm is denoted pm ≡ ln(PF)–ln(E) ≡ pf+fint. For 

convenience in interpreting interaction variables based on pm, we have normalized it by 

subtracting its mean over the sample. 

PV: Index of real new vehicle prices (1987=100) (logarithm: pv). 14

PF: 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 such as residential dispersion, other driving costs, lifestyle 

changes, and technology. As described below, in equation (8), the set of variables denoted 

XM 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 (RE)

We define a variable measuring the tightness of CAFE regulation, starting in 1978, based on the 

difference between the mandated efficiency of new 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 helps explain the efficiency of new 

passenger vehicles, while the lagged dependent variable in the fuel-intensity equation captures 

the inertia due to slow turnover of the vehicle fleet. 

The calculation proceeds in four steps, described more fully in Small and Van Dender (2007a), 

Appendix B. First, we estimated a reduced-form equation explaining log fuel intensity from 

1966-1977, prior to CAFE regulations. Next, this equation is interpreted as a partial adjustment 

model, so that the coefficient of lagged fuel intensity enables us to form a predicted desired fuel 

intensity for each state in each year, including years after 1977. Third, for a given year, we 

averaged desired fuel intensity (in levels, weighted by vehicle-miles traveled) across states to get 

a national desired average fuel intensity. Finally, we compared the reciprocal of this desired 

nationwide fuel intensity to the minimum efficiency 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-world driving). The variable cafe is defined as the difference 

 

14 We include new-car prices in the second equation as indicators of the capital cost of owning a car. We exclude 

used-car prices because they are likely to be endogenous; 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 standard has been binding throughout its time of 

application, but that its tightness rose dramatically during its first 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 standard itself, both of which have been used as 

a variable in VMT equations 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 greater fuel efficiency of its fleet, regardless of the desired fuel 

efficiency in that particular state. Our reason for 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 data are published by the U.S. Census Bureau as 

midyear population estimates; they use demographic information at the state level to update the 

most recent census count, taken in years ending 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 published 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 every ten years are accurate, and that the error in 

estimating population between them grows linearly over that ten-year time interval. We believe 

this approach is better than using the published estimates because it makes use of Census year 

data that were not available at the time the published 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 determine whether or not the 

rebound effect has changed over time. Theoretical arguments, especially by Greene (1992), 

suggest that it should. Basically, the argument is that the responsiveness to the fuel cost of 

driving will be larger if that fuel cost is a larger proportion of the total cost of driving. If initial 

fuel cost is high, that increases the proportion; but if the perceived value 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 urbanization) 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 and therefore have hypothesized a decline in the 

rebound effect over time, due to rising real incomes. 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 question in a simple way, by estimating the same model, postulating a 

constant rebound effect, on data from 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 arguments 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: the rebound effect (measured as a positive number) declines with increasing 

income and urbanization, and increases with increasing fuel cost. 

To understand better why changes in income and fuel 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, taken 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 unreliability are taken from statistical literature examining how 

people are willing to trade off those factors against money. I have then projected fuel costs per 

mile into the future, using the Energy Information Administration’s projections 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 small, decline of the absolute value of elasticity 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 income (Small and Verhoef 2007, Section 2.6.5). 

Figure 1: 

Costs of Driving

0

25

50

75

100

2005 2015 2025

Cents 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 version of the three-equation 

system described earlier. In this version, the variable pm (the logarithm of fuel 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. Accounting for how the vehicle fleet also varies with 

fuel cost, and how lagged adjustment creates 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 rebound effect changes over time, I carry out this estimation on two 

subsamples: 1966-1989 and 1990-2004. Table 1 shows the estimation results. Specifically, it 

shows the structural elasticity M ,PM ε and its long-run counterpart, /(1 ) ,

m ε M PM −α . As described 
14

earlier, these are nearly identical (except for the minus sign) to the short-run and long-run 

rebound effects, and their values come immediately from the estimated results. The table shows 

that the short-run rebound effect falls by nearly 40 percent between 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 αm is estimated to be smaller in the later period, 

indicating a smaller difference between short-run and long-run responses.) 

Table 1. Structural Elasticity 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-equation system the rebound effect is a combination of 

elasticities of the three distinct equations (known as “structural equations”). Because of the 

relative sizes of these elasticities, it is approximated by one: namely εM,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 expressed in natural logarithms, that elasticity is 

given by equation (3), i.e. εM,PM = ∂(vma)/∂(pm). 

In the previous subsection, fuel cost per mile 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 βpm. But it is easy to specify the equation for vma so that pm appears not only 

as a single variable, but also interacted with other 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 β1, 

β2, and β3. Then the derivative in (3) consists of four terms: 

inc pm Urban

pm

vma

M PM pm = + ⋅ + ⋅ + ⋅ ∂

∂ , = 1 2 2 3 ( )

( ) ε β β β β . (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 equations (4) and (7) for the 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 constant terms in the equations; but it means that 

even in this expanded specification, where elasticities vary across the samples, the coefficient 

βpm still gives the estimated elasticity εM,PM at the point where each variable 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-run and long-run rebound effects are approximately 

-εM,PM and -εM,PM /(1-αm), respectively, where αm 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 -βpm and αm. 

The model is estimated using the maximum-likelihood simultaneous-equations estimator in 

Eviews 5 (Quantitative Micro Software 2004). Technical details are provided in Small and Van 

Dender (2007a). The full results of estimating this 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 βpm -0.0407 0.0042 

 pm*inc β1 0.0696 0.0132 

 pm

2 β2 -0.0169 0.0064 

 pm*Urban β3 0.0255 0.0100 

 inc 0.1044 0.0134 

 lagged vma αm 0.7980 0.0120 

Equation for fint: 

 pf+vma -0.0297 0.0064 

 cafe -0.0882 0.0110 

 lagged fint αf 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 fuel 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 conventional test of statistical significance, 

having estimates more than twice the standard deviation of those estimates. The coefficient αm of 

lagged vma shows that the long-run effect of any variable on VMT is about 1/(1-αm)=4.95 times 

larger than the corresponding short-run effect. 

The coefficient of inc confirms the conventional expectation that vehicle-miles traveled rises 

with rising income: the income-elasticity is approximately 0.1 in the short run and 0.5 in the long 

run. Both annual fuel costs and CAFE standards are shown to be important 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-αf

)=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-αf

)=15.5% of the gap between the fuel intensity desired this 

year (on the basis of variable in the model) and that achieved by the previous year’s fleet. 

Our primary interest is in the first four coefficients shown in Table 2. The short-run rebound 

effect for average conditions in this sample is approximately -βpm=0.0407, i.e. 4.07%, while the 

long-run rebound is 4.95 times this value, or 20.1%. Furthermore, the coefficients β1–β3 for the 

three interacted variables involving pm show that the magnitude of the rebound effect, given 

approximately by the negative of equation (8), declines 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 precisely 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 average values over the last five years of the 

sample. Under average conditions over our entire sample period, the measured rebound effect is 

4.1% short run and 21.0% long run. However, these values are found to fall dramatically when 

we consider conditions that prevailed in 2000-2004: over those 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 consistent with the findings of Section 

4.1; but now we are able to fine-tune those findings by computing them for a narrower time 

period. 17 Furthermore, now we can quantify the sources 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 not specifically to the “rebound effect.” This is because the fuel cost of driving, PM, is 

defined as the ratio of fuel price to fuel efficiency, PF/E. Therefore the elasticity of VMT with respect to fuel price is 

just the negative of the elasticity of VMT with respect to fuel efficiency. By contrast with 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 elasticity does, but we found 

no effect.) With sensitivity of travel declining but that of fuel intensity much less so, the overall elasticity of fuel 
18

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 falling 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 

Average 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 fraction. 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 defined fuel cost per mile. (This assumption 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 more than offset the price rises during the 1970s and from 

1999 to 2004. 
19

this literature.) The elasticity of VMT with respect to fuel price is shown in the top row of the 

part of Table 3 labeled “calculated elasticities”. The next row shows the effect of fuel price on 

fuel intensity (the inverse of fuel efficiency). Together, these two elasticities 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 elasticity 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 F PF M PM E PF E,PF ~ ,

~ , , ˆ ε = ε ⋅(1− ε ) − ε , where Mˆ ,PM ε and E,PF −ε ~ are the elasticities 

reported in the first two panels of Table 4. This equation 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 ) , ~

E PF − 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 ) ,

~ , Mˆ PM E PF ε ⋅ − ε and E,PF −ε ~ , respectively. 
20

5. Implications of the Empirical Analysis: Projections to 2030 

By distinguishing the causes of the observed decline in the rebound effect, we are in a position to 

consider how the rebound effect is likely to change in the future. By inserting projected values 

for real per capita income, real fuel costs of driving, 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 variables in the model takes large values. Second, the 

farther the projected variables depart from the observed range over which the model is fit, the 

more likely it is that the specific functional form assumed by the model will inadequately portray 

projected behavior. In our case, the main concern is with real per capita income, which has a 

strong influence on the rebound effect and which is expected to grow to values much higher than 

those observed over the sample period 1966-2004. 

The first problem can, in principle, be handled by computing confidence bands around the 

projected values. The second, however, is inherent in any attempt to project beyond the range of 

past experience. Therefore, it must be recognized that one can have less confidence in projected 

values than in averages such as those reported in Table 3. 

In our case, there is one particular problem concerning the functional form, but it can be 

ameliorated. The linear form of equation (8), which calculates the structural elasticity of travel 

with respect to fuel cost per mile, does not constrain 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 points; 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. Although error bands might 

still assure us that theoretically appropriate values are within the range of statistical uncertainty, 

the projected values could not be considered the most likely beliefs about the rebound effect in 

the future. 

One solution to this problem would be to constrain the projected rebound effect to be zero 

whenever the equation predicts a negative rebound effect. This would be statistically justified, in 

the sense that zero is the most likely value given the theoretical constraint. However, I want to be 

more conservative in asserting how small the rebound effect may be in the future, and for this 

reason I have adopted an auxiliary adjustment for the 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 auxiliary equation explaining the negative of the elasticity εM,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 declining exponential approximation to (8): 

 ) εˆM ,PM = −R0 exp(b1 ⋅ Income + 2b2 ⋅ PM + b3 ⋅Urban + μ (9) 

where Income, PM, and Urban are their actual values (not logarithms and not normalized), and μ

represents approximation error. The symbol ^ over ε is 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 mnemonic convenience, making coefficients b1, b2, and b3 analogous to 

β1, β2, and β3 in (8). 

We can also write (9) as: 

ln( . (10) −εˆM ,PM ) = ln R0 + b1 ⋅ Income + 2b2 ⋅ PM + b3 ⋅Urban + μ

where again ln denotes the natural logarithm. Equation (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 M ,PM εˆ in the data set, in order to reduce the impact of extreme 

values; this also eliminates any negative values, which are incompatible with (9) and (10). 

The results of this estimation are shown in Table 4 and illustrated in Figure 3, which shows how 

the negative of the structural elasticity (approximately the rebound 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 effects in situations where the statistical model 

alone would predict that they are negative. 

Table 4. Auxiliary regression results 

Dependent variable: predicted structural elasticity ( M ,PM − εˆ ) 

Variable Coefficient 

name 

Coefficient 

estimate 

Standard 

error 

constant lnR0 -1.4789 0.0512 

Income (thousands $) b1 -0.1706 0.0027 

2*PM b2 0.0826 0.0015 

Urban b3 -0.7961 0.0340 

 

Standard error of regression 0.233 

R-squared 0.895 

Note: Income and PM are in measured at 1987 price levels: Income in 1000s of constant 

1987 dollars, PM in constant 1987 cents/veh-mile. The factor 2 multiplying PM is for 

convenience of interpretation, making (10) analogous to (9). These choices of units are 

arbitrary and have no effect on measured 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 M ,PM − εˆ in years 2005-

2030, given actual and anticipated values of income, fuel cost per 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 urbanization, 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 adjusted for the difference between our measure (real 

2007 dollars using a consumer price index deflator) 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 assumes 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 -εM,PM is calculated for each year for each scenario, using (9), the shortand long-run rebound effects are calculated from (4), (5), and (7). 

The results of these projections are shown in Table 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 short run and 4.0–5.8% for the long run. (It rises, 

rather than falls, from 2002 to 2008 due to the rapid run-up of fuel prices in those years.)26 The 

projections show these rebound effects 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 fuel 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 historical 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 EIA (2009), “Year-by-Year Reference Case 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 rate to the historical values found for the series used in 

our estimation, starting in 2006. 

24 See US EIA (2009), “Year-by-Year Reference Case Tables (2006-2030),” Table 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

Reference fuel-price scenario High 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 2030

Rebound 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 the 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 first 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 patterns 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 reference scenario, the response peaks in years 

2016-18 at a value of 3.3%, then gradually falls, tracking the long-run rebound 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 response 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 even 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 lower-growth periods of recent history, namely 1990-

2005.28 In that case, the rebound effect declines less rapidly; the full results are shown in 

Appendix B (Table B1). 

Appendix B also shows the results of using the income, fuel price, and fuel efficiency price paths 

in the Updated AEO 2009 scenario issued in April 2009. The latter includes the effects of the 

major stimulus package passed just previously (the American Recovery and Reinvestment Act) 

as well as revised economic projections. A comparison of the rebound effects, calculated under 

the two alternate projections (original and updated AEO), is shown in Table B2. The Updated 

scenario has slightly lower incomes but substantially lower fuel prices over the near term (2009-

2015), resulting on net in lower rebound effects. In later years, the 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-effect 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 fleet size – that is, they involve lag coefficients αm and αv

. The equation 

does not depend on lagged effects in changes in fuel intensity (coefficient αf

), 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, which play a very minor role in equation (7) due to the small sizes of αmv, αvm, and v β 2 . 

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 

2015 AEO Updated Ref. 3.7 

1.18% annual 4.3 5.9 

AEO Reference 1.5 2.2 

2025 AEO Updated Ref. 1.4 

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 effect of 4%, which is harder to compare to previous studies 

because previous work contains so much variation 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 congestion, and the value of that time rises with incomes, all three 

of these differences tend to make fuel costs a smaller portion of the total cost of traveling. Thus it 

is not surprising that people would become less sensitive, on a percentage basis, to changes in 

those fuel costs. 

The statistical procedures described 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 effect was likely 5.7% in the long run and 1.1% 

in the short run. 

Similarly, the model enables us to see how projected changes in those same three variables can 

be expected to influence the rebound effect in the future. These quantitative predictions are less 
28

certain because we are projecting beyond the range of data with which the model was estimated. 

Nevertheless, the results strongly suggest that income growth overcomes other influences, 

causing the rebound effect to decline 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 fuel prices rise according to the EIA “high-price 

scenario” — two conditions that produce the 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 Reference 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 Simultaneous 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 the entire system. The estimation routine accounts 

for both the lagged dependent variables and autocorrelation; for this reason the predicted value in 

any year depends on observed values for two previous years, hence the first two years of data do 

not enter as separate observations for the estimation. (This is why 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 further details about the estimation procedure 

including the choice of instrumental variables, which has special features due to variables 

entering the equation non-linearly and the presence of autocorrelation 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

pm -0.0407 0.0042 -9.61

pm^2 -0.0169 0.0064 -2.66

pm*inc 0.0696 0.0132 5.28

pm*Urban 0.0255 0.0100 2.54

inc 0.1044 0.0134 7.80

adults/road-mile 0.0214 0.0044 4.81

pop/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

Trend 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-squared 0.9816

S.E. of regression 0.0314

Durbin-Watson stat 1.9725

Sum squared resid 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% leve

Table A2. Vehicle-Miles Traveled Equation

NOTE: coefficient of variable “adults/road-mile” whould be preceded by a – sign. 
31

Variable Coefficient Stndrd. Error

vehstock(t-1) 0.8635 0.0149 57.79

vma 0.0285 0.0172 1.66

pv 0.0240 0.0333 0.72

pm -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

pop/adult -0.0056 0.0643 -0.09

Urban -0.1081 0.0576 -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-squared 0.9606

S.E. of regression 0.0411

Durbin-Watson stat 2.0004

Sum squared resid 3.0898

Table A4. Fuel Intensity Equation
32

Table A5 shows various elasticities calculated from the estimated structural equations, according 

to equations (4) and (7) in the text, and equation (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

with respect to fuel price:

 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 the 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 the sum of the price-elasticities of VMT and fuel 

intensity. But because both are endogenous, there is an interaction and the combined elasticity is: 

F PF M PM I PF I ,PF ~ , ~ , , ˆ ε = ε ⋅(1+ε ) +ε (11) 
33

where Mˆ ,PM ε and I ,PF 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 alternate assumptions about income growth

Table B1 shows the results with all other assumptions the same as in Table 5, except income is 

projected to grow at only 1.18% per year starting in 2008. This results in higher rebound effects. 

Table B2 shows the results with all the assumptions of the Updated Reference Scenario 

(produced April 2009), and compares them with the original Reference Scenario (same as Table 

5). The Updated scenario shows slightly lower incomes but substantially lower fuel prices over 

the near term (2009-2015), resulting in lower rebound 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 

εF,PF for Table 5, and therefore overstated the magnitude of εF,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 Price

(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

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