A Fuel Economy Regulatory Analysis Model (FERAM) 

For the Energy Information Administration 

David L. Greene 

Oak Ridge National Laboratory 

2360 Cherahala Boulevard 

Knoxville, Tennessee 37932 

K.G. Duleep 

Doug Elliott 

Energy & Environmental Analysis, Inc. 

1655 North Fort Meyer Drive, Suite 600 

Arlington, VA 22209 

Sanjana Ahmad 

University of Tennessee 

2360 Cherahala Boulevard 

Knoxville, Tennessee 37932 

April 28, 2005 

Prepared by the 

OAK RIDGE NATIONAL LABORATORY 

P.O. BOX 2008 

Oak Ridge, Tennessee 37830 

managed by 

UT-Battelle, LLC 

for the 

U.S. DEPARTMENT OF ENERGY 

under contract No. DE-AC0500OR22725 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

35 

CONTENTS 

1 BACKGROUND....................................................................................................................................................1 

2 PROPOSED MODEL STRUCTURE ....................................................................................................................1 

 2.1 CALCULATING UTILITIES.......................................................................................................................6 

 2.2 MANUFACTURERS’ USE OF TECHNOLOGY AND PRICING .............................................................6 

 2.3 FUEL ECONOMY PERFORMANCE TRADE-OFF...................................................................................8 

 2.4 PRICING, SALESMIX AND FUEL ECONOMY .......................................................................................9 

3 VEHICLE CHOICE MODEL STRUCTURE........................................................................................................9 

 3.1 CALIBRATION AND INTRODUCTION OF DIESEL AND HYBRID 

 VEHICLES .................................................................................................................................................10 

 3.2 CALIBRATION OF VEHICLE AND CLASS INTERCEPTS ..................................................................10 

 3.3 CALIBRATION OF VEHICLE AND CLASS CHOICE PRICE SLOPES................................................11 

 3.4 CALIBRATION OF COEEFICIENT OF THE CHANGE IN FUEL 

 CONSUMPTION........................................................................................................................................12 

4 REGULATORY STRUCTURES.........................................................................................................................13 

 5 IMPLEMENTATION OF DECISION VARIABLES AND CONSTRAINTS WITH 

 DISCRETE REDESIGN INTERVALS ...............................................................................................................14 

 5.1 TRADABLE CREDITS..............................................................................................................................16 

 5.2 IMPLEMENTATION .................................................................................................................................18 

REFERENCES ............................................................................................................................................................18 

APPEDIX A: ESTIMATING VEHICLE-SPECIFIC FUEL ECONOMY COST CURVES......................................20 

APPENDIX B: VALUING HORSEPOWER.............................................................................................................28 

A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

1. BACKGROUND 

The Energy Information Administration (EIA) requires a model for analyzing the impacts of alternative 

fuel economy regulations. The model is to be used to analyze changes to the current CAFE regulatory 

system under consideration by NHTSA. Reforms considered could include attribute-based standards, 

tradable fuel economy credits, modifications to vehicle classifications (e.g., passenger car vs. light truck), 

changes to the “two fleet rule,” and uniform percentage increase standards. A model in which varied forms 

of standards can be formulated and tested is therefore required. The model should also be able to analyze 

the impacts of including class IIb (>8,500 lbs GVW) light trucks in the regulatory framework. 

It is essential that the model be able to calculate impacts on individual manufacturers. Thus, the model 

must include sufficient detail not only to identify every major manufacturer, but also to represent its market 

position in terms of the market segments and product designs it sells. The model is not expected to predict 

changes in manufacturers’ marketing strategies nor to predict mergers or market entries or exits. This has 

important implications for interpreting model predictions. Impacts on consumers must also be represented, 

and the model must also be able to estimate overall economic costs and benefits in a partial equilibrium 

framework. 

A key factor will be the use of technologies to increase fuel economy. Not only conventional gasoline 

technologies, but hybrid gasoline-electric and diesel power trains must be included. Manufacturers must 

also be free to use pricing strategies to shift sales among vehicles with differing levels of fuel economy in 

order to meet regulatory requirements. 

The model proposed below is capable of addressing all of the issues outlined above in a rigorous 

framework based on microeconomic theory. In implementing and testing the model, however, we have 

found that in its more complex forms it is at or beyond the capabilities of the IMSL non-linear optimization 

software. This has not only slowed progress in implementation and testing but raised the question of 

whether the problem, as currently posed, is inherently too complex. 

2. PROPOSED MODEL STRUCTURE 

The proposed modeling method represents the following key components as endogenous variables: (1) 

market shares, sikjt , of individual make/model/configurations of light-duty vehicles based on prices and 

attributes related to or affected by fuel economy, and (2) manufacturers’ decisions about changes in fuel 

consumption levels, ∆xikjt , and the consequences for price, ∆pikjt, including adjustments to vehicle prices 

manufacturers may make to produce a higher fuel economy sales distribution, δijkt , and performance ∆hikjt . 

These decision variables are chosen to maximize consumers’ welfare given fuel prices, the costs of fuel 

economy technology, and other parameters, in the face of various levels and forms of fuel economy 

standards. Manufacturers’ decisions to introduce new product lines for diesel and hybrid vehicles will be 

specified exogenously, but consumers’ demand for these products will be endogenous. Table 1 provides a 

list of all variables and their definitions. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

38 

Table 1. Variable Definitions 

Decision Variables 

∆y = fractional increase in fuel economy (unitless and >=0) 

∆x = fractional change in fuel consumption (unitless and <=0) 

∆h = potential fractional change in fuel consumption used to raise horsepower (unitless) 

 δ = change in vehicle price not caused by use of fuel economy technology (not 

 implemented in first version of model) 

Other Endogenous Variables

∆p = change in vehicle price caused by use of fuel economy technology 

 s = market share of a vehicle 

 S = annual vehicle sales in units 

∆CS = change in consumers surplus resulting from changes in decision variables 

 TC = number of tradable credits per vehicle 

 PTC = price of a tradable credit 

 PV = price of a vehicle 

 Pf = price of fuel per gallon 

 p = probability an alternative is chosen from a set of choices 

Exogenous Data 

 S = total vehicle sales 

 xijko = base vehicle fuel consumption in gallons per mile 

 y = base year fuel economy in miles per gallon 

 R = fuel consumption constraint level 

 K = vehicle weight in kilograms 

 H = vehicle footprint equal to track width times wheelbase 

Parameters 

 B = coefficient of inclusive value (average generalized price) for choice among the the k 

 vehicle classes 

 Bk = coefficient of vehicle price in consumers’ utility function for vehicles in class k 

 aijk = calculated make/model/configuration intercept for consumers’ utility function 

 bijk = coefficient of ∆y in technology-price function 

 cijk = coefficient of ∆y2

 in technology price function 

 e = base of the Naperian logarithms 

γk = annual rate of decline in vehicle use 

 Mok = annual miles per year for a new vehicle in class k 

 Rk = discount rate for future fuel savings 

 Lk = vehicle lifetime in years 

Indexes 

 i indexes vehicle make/model/configurations within a vehicle class and manufacturer 

 k indexes vehicle classes 

 j indexes manufacturers 

 t indexes calendar years 

 njk = the number of configurations made by manufacturer j in class k 

 nk = the number of vehicles in class k 

 n = the number of vehicle classes 

 nm = the number of manufacturers

 T = final year of the analysis (2030) 

The fundamental method is mathematical optimization, subject to constraints. It is similar in form to the 

model developed by Greene and Hopson (2003). Manufacturers are assumed to maximize the value of 

their products to consumers, subject to two types of constraints: (1) equations describing technology (i.e., 

the trade-offs between fuel economy and vehicle price and between fuel economy and vehicle 

performance) which are embedded in the consumers’ utility functions via the price of increased fuel 

economy (decreased fuel consumption x), and (2) fuel economy regulations represented as mathematical 

constraints on the optimization problem. Fuel economy standards can be represented as the constrained 

optimization problem shown in equation 1. The objective function is the present value of consumers’ 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

39 

surplus from all vehicles sold in all future years. The fuel economy standards are constraints on the 

maximization of consumers’ surplus. The simple form in equation 1 does not represent tradable fuel 

economy credits, which are taken up separately below. 

( ) ( )

FC s x x t j n k n

OR manufacturer class specific

FC s x x t j n

OR manufacturer specific

st FC s x x t

CS x h p

r

S Maximize

m

n

i

jkt ikjt ikj ikjt

m

n

k

n

i

jt ikjt ikjo ikjt

n

j

n

k

n

i

t ikjt ikjo ikjt

n

j

n

k

n

i

t ijk ijk ijk ijk

T

t

t

t

jk

jk

m jk

m jk

(1 ) 3, 1, , 1,

/

(1 ) 3, 1,

. .: (1 ) 3

, , , 1

1

1 1

1 11

3 1 11

⎥ ∀ ≥ = =

⎦

⎤ ⎢

⎣

⎡

≥ + ∆

−

⎥ ∀ ≥ =

⎦

⎤ ⎢

⎣

⎡

≥ + ∆

−

∀ ≥

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

≥ + ∆

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∆ ∆ ∆ ∆

+

∑

∑∑

∑ ∑∑

∑ ∑ ∑∑

=

= =

= ==

= = ==

δ

(1) 

The change in consumers’ surplus (∆CS) induced by a fuel economy regulation is summed across all 

manufacturers, j=1, nm , and all makes and models of class k offered by each manufacturer, i=1, nmk , and 

all vehicle classes sold by a manufacturer, k=1, nk . The change in consumers’ surplus captures the value to 

consumers of changes in fuel savings, horsepower, and price increases due to change made by 

manufacturers to meet a particular set of standards. The objective function also includes any changes in 

vehicle prices, δijk , manufacturers elect to make for the purpose of shifting sales in order to meet the 

standard (the potential to change prices independently of fuel economy will not be included in version 1 of 

the model).7

 Finally, if credit trading is allowed, the cost of credits purchased must be included. This is 

equal to the price per credit, PTC , times the tradable credits purchased per vehicle, TC, times the number of 

vehicles sold by manufacturer j. It can be demonstrated that solving this problem is equivalent to a 

competitive market solution, provided that the technology constraints reflect cost-minimizing behavior on 

the part of manufacturers. 

Consumers’ vehicle choices will be represented by a random utility model, the Nested Multinomial Logit 

(NMNL) model. In the NMNL framework, it is assumed that the utility (or general index of value) of a 

particular vehicle i, in vehicle class k, to the l

th consumer (manufacturer subscripts are omitted for 

simplicity) can be represented as a sum of three components: (1) a function of measurable attributes such as 

price, fuel economy, and horsepower, zik (2) a constant term representing the average value of the 

unmeasured attributes of vehicle i, Aik , and, (3) an unobserved random term reflecting factors specific to 

the lth individual, εikl . Here z represents the several variables that determine the desirability of different 

vehicles in the eyes of consumers. In forecast years these variables will be the ∆x’s, ∆P’s, ∆h’s, and δ’s 

chosen so as to optimize consumers’ surplus. In the base year all the z variables are combined into a 

make/model/configuration-specific constant term. Calibration of these constants is explained in the 

appendix. 

If it is further assumed that the utility function is approximately linear in these factors, it can be written as 

follows. 

 

7

 Adding these variables would double or more than double the number of decision variables. Unless 

software far more powerful than the IMSL nonlinear optimization routines can be found, this does not 

appear to be feasible. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

40 

ijktl ijk k k ijkt jiktl u = a + B ∑w z + ε

λ

λ λ

(2) 

The index scripted l is used here as a temporary index over measured variables while l is used as a 

temporary subscript representing individual car buyers. For the zijk = Pijk that is the price of the vehicle, wkP

=1, so that the coefficient of price, or price slope, for vehicle class k is Bk . 

If εijktl is assumed to follow a type I extreme value distribution across consumers for the vehicles in class k, 

then the probability pij|k , that a consumer will choose vehicle I made by manufacturer j, given that the 

consumer will select a vehicle from class k, is given by the multinomial logit equation. 

∑∑= =

= m

j

n

i

u

u

ij k k

I

IJ i

e

e

p

1 1

|

(3) 

The index IJ is used here to specify a particular vehicle (i=I) made by a particular manufacturer (j=J) in 

class k. In calibrating the NMNL model, the base year intercepts (a) include the base year values of all the 

z variables. The optimization problem will calculate changes in those variables. 

 

The probability that a consumer will select from class k is a logit function of the expected utility of class k, 

U ̄k (Ubarsubk). The expected (or average) utility, in dollars, of vehicles in class k depends on the utility of 

each vehicle in class k according to equation 4. 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∑=

k

ik

n

i

u

k

k e

B

U

1

ln 1

(4) 

Note that here B is subscripted by the class index k because it is the price slope for the choice among 

vehicles in class k. The probability that a consumer will choose a vehicle from class k is then given by the 

following logit function. 

∑=

+

+

= n

k

A BU

A BU

K

k k

K K

e

e

p

1

(5) 

Here the index K is used to indicate a particular vehicle class, k is used for summation over all vehicle 

classes and nk is the number of vehicle classes. The probability that vehicle i in class k will be chosen 

(which is equivalent to its expected market share) is given by the product of the class and conditional 

vehicle choice probabilities pik = pi|k pk . 

The NMNL model of vehicle choice provides a closed form measure of consumers’ surplus that can be 

directly inserted into the objective function. Let Uk be the average utility of a vehicle in the kth vehicle 

class (nest) before fuel economy changes occur and Uk* be the average utility afterwards. The change in 

consumers’ surplus in year t is given by equation 6. The calculation of Ukt follows in equation (7). 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

41 

⎪

⎪

⎭

⎪

⎪

⎬

⎫

⎪

⎪

⎩

⎪

⎪

⎨

⎧

∆ = −

∑

∑

=

=

n

k

U

n

k

U

t

kt

kt

e

e

B

CS

1

1

*

ln 1

(6) 

In equation 6, the superscript asterisk on Uk

*

 , indicates the value of the utility function after fuel economy 

changes have been made in response to future fuel economy regulations, and after new diesel and hybrid 

products have been introduced, while Uk without the superscript asterisk represents utility in the absence of 

new fuel economy regulations. This consumers’ surplus measure is an average of the changes in the 

measured attributes of the vehicles. Divided by B, which has units of utils/$, ∆CS will be in dollars. 

The utility of class k is a function of the utilities of all vehicles in class k (all the Uijk ). 

k k

n

j

n

i

U

k

k k e A BU

B

U A B m jk

ijk = + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = + ∑∑= = 1 1

ln 1

(7) 

2.1 CALCULATING UTILITIES 

Because all the z variables in equation 2 are changes in vehicle attributes from the base year, they are all 

zero in the base year. Since the expected value of the error terms is zero, the utility in the base year equals 

the intercept terms. 

uijktl = aijk

(8) 

These base utilities can be used in equation 3 to calculate make/model/configuration shares within a class 

(equivalent to the probability of choosing a specific make/model/configuration given that a vehicle will be 

chosen from class k). 

The inclusive value for a class is just the logsum of the exponentiated intercept terms. 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∑∑= =

m k

ik

n

j

n

i

a

k

k e

B

U

1 1

ln 1

(9) 

The class utilities are given by equation 7. These can be used in equation 5 to calculate class shares. The 

final individual make/model/configuration shares are the product of equations 3 and 5. 

In future years, the changes in attributes will have non-zero values. At the very least, there will be some 

changes in fuel consumption rates (∆xxo) and in prices ∆PV . 

ijkt ijk k kFC oijkt ijkt Bk PVijkt u = a + B w (x ∆x ) + ∆

(10) 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

42 

The shares of make/model/configurations within class k in year t can then be computed using equation 3. 

Equation 4 can be used to calculate the class inclusive values and equation 5 to estimate the new class 

shares in year t. The individual vehicle market shares are again the product of the conditional 

make/model/vehicle shares (equation 3) and the class shares (equation 5). 

2.2 MANUFACTURERS’ USE OF TECHNOLOGY AND PRICING 

Cost minimization, subject to other vehicle attributes remaining constant, can be insured by using a cost 

function for fuel economy improvements that orders the implementation of technologies in terms of costeffectiveness. Let ∆x be a relative decrease in fuel consumption (a reduction in gallons per mile or liters 

per 100 km and therefore <0) and ∆p be the resulting increase in vehicle price. Greene and Ahmad (2003) 

have shown that when technologies are ranked in descending order of cost-effectiveness (∆x /∆p), cost as a 

function of relative change in fuel economy can be very closely approximated by a quadratic function. 

Fitting quadratic price functions to both relative changes in miles per gallon (>0) and gallons per mile (<0) 

produced consistently good fits in mpg space but unacceptable fits in gpm space. The fitted curves in gpm 

space often exhibited negative costs for reducing fuel consumption in the range of 5% to 15%, a result 

inconsistent with the data. Fortunately, relative changes in miles per gallon can be easily transformed into 

equivalent changes in gallons per mile by equation 11. 

1

1

1

1 1

1 1

1

1 1

− ∆ +∆ =

= −

−

∆ =

= − − ∆ =

= =

y

x

mpg

mpg

mpg

mpg mpg x

mpg

mpg

mpg

mpg mpg y

y mpg

x

o

o

o

o o

o

(11) 

Quadratic functions have excellent mathematical properties for use in constrained optimization problems. 

2 ∆p = b∆y + c∆y

(12) 

In equation 12, b and c are constants to be estimated, ∆y is the fractional change in fuel economy relative to 

the base year fuel economy rate, and ∆p is defined in terms of retail price equivalent, including full 

overhead and return to capital. 

Industry average data for calibrating equation 12 at a vehicle class level are available from a number of 

sources, including EEA and the National Academy of Sciences’ CAFE effectiveness study. EEA has also 

developed data for calibrating passenger car versus light truck fuel economy v. price functions at a 

manufacturer-specific level for the largest manufacturers. A higher degree of disaggregation is needed for 

the purposes of this study, we believe, to increase the validity of manufacturer-specific impacts. In 

addition, separate cost curves will be developed for gasoline, diesel and hybrid vehicles. 

Price change versus relative change in fuel economy curves are specific to each and every vehicle 

configuration (make, model, engine, transmission), since each may contain a different use of fuel economy 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

43 

technologies and reflect different design decisions. Detailed data on the use of fuel economy technologies 

by each vehicle was developed by Energy and Environmental Analysis, Inc. (Duleep, 2005) for this 

purpose. 

2.3 FUEL ECONOMY PERFORMANCE TRADE-OFF 

Manufacturers can also trade-off lower fuel consumption (FC) for greater horsepower (HP), or other 

attributes such as weight, or alter vehicle prices to change their sales distributions.8

 The use of fuel 

economy technology to meet alternative design goals is handled by allowing the optimization model to 

allocate potential fuel economy improvement to these uses. If we interpret ∆y as the result of the total 

change in use of fuel economy technology (which could be measured in units of potential fractional change 

in fuel economy), then the sum of the alternative uses must be exactly equal to ∆y. 

FC HP ∆y = ∆y + ∆y

(13) 

The total cost of these uses of fuel economy technology will be the cost of ∆y as measured by equation 14. 

( ) ( )

2

FC HP FC HP ∆p = b ∆y + ∆y + c ∆y + ∆y

(14) 

The two components of ∆y, namely ∆yFC and ∆yHP , enter the consumer’s utility function in differently. ∆y 

is transformed into a fractional change in the rate of fuel consumption, ∆x, and (1/(1+∆y))xo = (1+∆x)xo , 

will be multiplied by BkwkFC , where Bk is the price slope for class k and wkFC is the discounted present 

dollar value of a 1 unit change in gallons per mile for vehicles in class k. 

∆yHP , however, is not a change in horsepower but rather a fractional change in potential fuel consumption 

that must be transformed into a change in horsepower before being entered into the consumers’ utility 

function. Let η be the elasticity of fuel economy with respect to engine horsepower. The new horsepower 

level after the application of potential fuel economy technology to increase horsepower instead of fuel 

economy, HP, is given in terms of the initial horsepower level and ∆y in equation 15. 

( )

[ ]η

η

1

1

1

o HP

o

o

o o

HP HP y

mpg

y mpg

HP

HP

mpg

mpg

= + ∆

+ ∆ = ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

(15) 

For the elasticity of fuel consumption with respect to horsepower (weight assumed constant) we use η = 

0.38 (EEA, 1993, p. 6-8. With engine efficiency constant, engine power is proportional to engine size). 

If we can now specify a function to provide a dollar value of horsepower as a function of the level of 

horsepower, the value of this additional horsepower to the consumer can be entered into the consumers’ 

utility function and its impact on consumers’ surplus calculated. 

2.4 PRICING, SALESMIX AND FUEL ECONOMY 

In the face of tighter fuel economy regulations, it is possible that manufacturers would not only respond by 

using technology to increase fuel economy at increased cost, but also by raising the prices of their lower 

fuel economy vehicles and, or subsidizing their more efficient vehicles. The final decision variable for 

manufacturers is a price subsidy or surcharge that could be applied to each vehicle, δijk , to change the sales 

 

8

 This capability was also not implemented in version 1, once again due to limitations on the number of 

decision variables the IMSL non-linear optimization software can handle. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

44 

mix in order to help meet fuel economy regulations. The net cost of these subsidies and surcharges must 

therefore be subtracted from the objective function. This feature will not be implemented in the initial 

version of the model. 

3. VEHICLE CHOICE MODEL STRUCTURE 

A nested multinomial logit model will be used to represent consumers’ choices among light-duty vehicles. 

Vehicles will be grouped into classes (nests) of similar vehicles (e.g., small SUVs, midsize passenger cars, 

minivans, etc.). The nesting structure allows a more realistic representation of own and cross price 

elasticities. In general, the price sensitivity of choices among vehicles within a nest is greater than the price 

sensitivity of choices across vehicle classes. One possible nesting structure is shown in Figure 1. In it, a 

makes, models and configurations of small cars, for example, are in one group. All passenger cars (Small, 

Compact, Midsize-Large) are grouped into another nest. Standard passenger vehicles, cars and light trucks, 

are grouped into yet another nest. It is very likely that the structure shown in figure 1 will be modified for 

this model based on recent evidence indicating that cargo vehicles should be separated from passenger 

vehicles at the highest nesting level. 

Figure 1. Nested Multinomial Logit Model Structure and Approximate Price Elasticities 

(shown in bold numbers) 

 -1 

 -2 

 -3 -3 

Lux Sport Sportscar Lux Car Lux Trk 

 -3 

 

-4 -4 -3 

Small Comp Mid-Lg Small Van Small SUV Large SUV Sm P-U Lg P-U Van 

 -5 or more 

Economic theory requires that sensitivities of choices to price are greatest at the lowest nesting levels and 

decrease at more aggregate levels. The elasticity of overall demand for light-duty vehicles shown in Figure 

1 is -1. The exact nesting shown in figure 1 is intended to illustrate the structure of an NMNL model; the 

structure implemented may differ based on the best available evidence. 

3.1 CALIBRATION AND INTRODUCTION OF DIESEL AND HYBRID VEHICLES 

The model will be calibrated to base year 2003 light-duty vehicle sales at the configuration level. 

Calibration consists of two steps. First, average attribute values will be derived, using a methodology 

described in Greene (2001) and also used in the current NEMS alternative fuel vehicles model. Next, 

constant terms will be calibrated using the equations for nested multinomial logit models presented in 

Greene et al. (2004), Appendix A.6. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

45 

For calibration, NHTSA’s light-duty vehicle CAFE data base, augmented by EEA to include relevant 

attributes such as vehicle prices and horsepower, will be used. At this time, we assume that similar data 

will be provided by NHTSA for manufacturers’ production plans in 2008 and 2012. If so, we will create 

data bases for intervening years by explicitly deleting and adding product lines according to the best 

available information and our judgment. The model will be calibrated to the vehicle attributes and sales 

data for each and every year. Having data at this level of detail will make it unnecessary to forecast the 

introductions and retirements of product lines. It will also make it unnecessary to predict the introductions 

of diesel and hybrid vehicles. 

3.2 CALIBRATION OF VEHICLE AND CLASS INTERCEPTS 

Vehicle and class intercepts can be calculated from the base year market shares of individual 

make/model/configurations and vehicle classes. 

Let Sik = sales in base year of vehicle i in class k. 

Let n = the number of classes (k = 1,n) 

∑∑= =

= n

k

n

i

ik

ik

ik k

S

S

s

1 1

(16) 

Let aIk = the intercept term for a specific vehicle configuration i=I, in class k. 

∑=

= − k n

i

ik

k

Ik Ik s

n

a s

1

ln( ) 1 ln( )

(17) 

Define the inclusive value, IV, of class k as 

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∑=

nk

i

ik

k

k a

B

IV B

1

ln exp( ) 1

(18) 

Calculate class intercepts, Ak 

For an arbitrarily chosen class, say class 1, 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥ −

⎦

⎤ ⎢

⎣

⎡ = − + ∑ ∑ = =

n

k

k

n

k

k IV

s

s

n

A IV

1 1 1

1 1 ln 1

(19) 

For the remaining classes, 

k

k

j IV IV

s

s A A + − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − 1

1

1 ln

(20) 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

46 

For a check, see that the class shares calculated using the intercepts match the base year shares. 

∑=

+

+ = n

k

k k

k k

k

A IV

A IV

s

1

exp( )

exp( )

(21) 

The B and Bk terms must be known in advance in order to calibrate the class and vehicle intercepts. 

Calculation of these coefficients is explained below. 

3.3 CALIBRATION OF VEHICLE AND CLASS CHOICE PRICE SLOPES 

Both the class level choice and vehicle make/model/configuration choice price slopes can be calculated 

from price elasticities, if they are available. In general, price elasticity is a function of price slope, B, at a 

particular market share, s, and price, PV. 

V

V

s P

B

B s P

(1 )

(1 )

− =

= −

β

β

(22) 

If the Bk are price slopes for choices within class k and B is the price slope for choices among the k classes, 

then the NMNL model requires that B > Bk for all k. Since both the Bk’s and B are less than zero, this 

implies that absolute values of the Bk’s must be greater than the absolute value of B. In other words, 

choices among vehicles within classes must be more sensitive to price than the choice among classes of 

vehicles. 

For B, a reasonable choice for β is approximately -2.0. The median share for 11 vehicle classes would be 

about 0.1. If the average price of a vehicle were $25,000, this would imply the following value for B. 

0.0000888

(1 0.1)25000

2 = − −

− B =

(23) 

Reasonable price elasticity values for calculating the Bk‘s range from about -4 to -8. Economy vehicles are 

likely to be more price elastic, luxury vehicles less so. Class specific price slopes should be calculated 

using the median share of a make/model/configuration in that class and the average (or median) price of a 

vehicle in that class. 

3.4 CALIBRATION OF COEFFICIENT OF THE CHANGE IN FUEL CONSUMPTION 

The fractional change in fuel consumption multiplied by the base year fuel consumption rate gives a change 

in fuel consumption in gallons per mile. The coefficient of this variable in the utility function for a 

particular make/model/configuration of vehicle is the product of the class-specific price slope and a 

weighting factor representing the dollar value of a one unit change in gallons per mile. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

47 

( ) k kFC FC o B w ∆x x

(24) 

Bk is the vehicle class k price slope of the utility function and has units of utils/dollar. The fuel 

consumption weight wkFC transforms a change in gallons per mile into dollars; it is the dollar value (in retail 

price equivalent) of a 1 gallon per mile change in fuel consumption. If we assume economically rational 

consumers who discount future fuel savings to present value, wkFC can be directly computed from a few key 

items of data: 1) the lifetime of a vehicle, L, 2) rates of vehicle use over time, M(t), 3) the price of fuel, pf , 

and, 4) the consumer’s discount rate, r. 

The value of a change in fuel consumption is the product of fuel savings per mile, total miles, and the price 

of fuel. Because the fuel savings will accrue over the life of the vehicle as it is used, they should be 

discounted to present value in order to be comparable to changes in vehicle price. We assume that vehicle 

use decreases exponentially over time with decay rate γ and that the initial rate of use is Mo . The rate of 

use in year t is given by equation 27. 

t

o M t M eγ ( ) =

(25) 

The present value of discounted future fuel savings, assuming a discount rate of r, is given by equation 28. 

P represents the price of fuel which varies over time but is here assumed to be constant (as if the consumer 

had constant expectations about fuel prices). Reasonable values for the key parameters of equation 28 are 

as follows: Mo = 15,000, γ = 0.04, r = 0.06. Different parameters may be specified for each vehicle class to 

produce a different value of fuel economy for each class. The subscript, k is omitted from the equations for 

simplicity. 

 

[ ] FC FC o

f o r L

FC o

L

t

L

t

r t

FC o f o

t rt

FC o f o e w x x r

P M

PV x x P t M e e dt x x P M e dt x x = ∆

⎭

⎬

⎫

⎩

⎨

⎧ − + =∆

⎭

⎬

⎫

⎩

⎨

⎧ = ∆ = ∆ − +

= =

− − − +

∫ ∫ ( )

0 0

( ) 1

( ) ( ) γ γ γ

γ

(26) 

Equation 28 also shows that the appropriate weight for fuel consumption is the term in brackets. Since 

vehicle usage rates, lifetimes and possibly discount rates may vary across the k vehicle classes, it is 

possible to use equation 28 to compute different weights for each vehicle class. Since w translates a given 

change in fuel consumption into its present value in dollars, BkwkFC translates a given ∆xxo into a change in 

utils, and is therefore the utility function coefficient of ∆xxo. 

4. REGULATORY STRUCTURES 

A fuel economy standard is comprised of three elements: 

1. A vector of target values defined by a metric, levels, and timing 

2. A constraint structure specifying which vehicles and manufacturers must meet which target 

values, and 

3. Credit trading mechanisms that allow constraints to be met by banking and borrowing of fuel 

economy credits and define the markets for those credits. 

Alternative metrics will include, but need not be limited to, the following three: 

1. Corporate Average Fuel Economy constraints 

2. Uniform Percentage Increase constraints 

3. Attribute-based constraints 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

48 

a. Footprint-based (wheelbase or length × track width) 

b. Weight-based 

For example, it is also possible to define a fuel economy metric based on interior volume and cargo 

capacity. 

Out of the enormous number of possible constraint structures, five will be programmed in the model. 

Others can be added by model users, as desired. 

1. Industry-wide 

2. Manufacturer 

3. Manufacturer & Vehicle Class (incl. passenger car vs. light truck) 

4. Manufacturer and Weight Formula 

5. Manufacturer and Footprint Formula 

Plotkin, Greene and Duleep (2002) tested a weight-based constraint using a regression formula relating 

vehicle weight to fuel consumption. The FERAM model, however, will not predict how manufacturers 

might game attribute-based standards by altering vehicle attributes. This would require a separate 

engineering and design analyses. It is also possible to define weight class and footprint class formulas. 

This will be left for future development. 

The constraint equations shown below are written in terms of fuel consumption (1/MPG) rather than in fuel 

economy (MPG) for mathematical simplicity. A linear fuel consumption constraint is equivalent to a 

harmonic mean constraint on fleet average miles per gallon. 

Industry-wide constraint (there is one constraint for every year from t=3 to T). 

∑∑∑ = ==

≥ m k n

j

n

k

n

i

t ijkt ijkt F s x

1 11

(27) 

Manufacturer-specific constraints (there are nm constraints for each year, t=3 to T). 

∑∑= =

≥

n

k

n

i

jt ijkt ijkt

k

F s x

1 1

(28) 

The manufacturer-specific targets can be defined in many ways. If they are defined as percentage 

decreases from each manufacturer’s base year average fuel consumption rate, this defines a Uniform 

Percentage Increase fuel economy standard. 

If all manufacturers must meet the same fuel consumption target, Fjt = Ft , equation 30 becomes a CAFE 

standard. Equation 30 can also be readily modified to specify different fuel consumption requirements for 

different vehicle types. 

Manufacturer-Vehicle Class Standard (there are potentially nm × n constraints for each year). 

∑=

≥

nk

i

jkt ijkt ijkt F s x

1

(29) 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

49 

Equation 31 defines a separate standard for each of the n vehicle classes. It is also possible to define 

standards for sets of classes. In particular, a passenger car versus light truck standard will also be included 

in the set of default constraint structures. 

5. IMPLEMENTATION OF DECISION VARIABLES AND CONSTRAINTS WITH DISCRETE 

REDESIGN INTERVALS

Decision variables are defined for each year, but only for those vehicles that can be redesigned in that year. 

If the first year of the fuel economy constraint is t=3, then the set of decision variables is ∆xijk3 , ∆hijk3 , δijk3

. The fuel economy constraint for year 3 recognizes that those vehicles scheduled for redesign in year 3 are 

affected by the ∆xijk3 , but the rest of the vehicles are not. In all equations below we assume ∆x < 0, ∆h > 

0, and δ > 0. An industry-wide constraint for year 3 is shown in equation 32. 

∑∑ ∑ ∑ ( ) == = + =

⎥

⎦

⎤ ⎢

⎣

⎡

≥ + ∆ + m km km

km

n

j

n

k

n

i

n

i n

ijk ijko ijk ijk ijko FC s x x s x

11 1 1

3 3 3 3

3

3

1

(30) 

The number of vehicles in class k, sold by manufacturer j that are slated to be redesigned in year 3 is 

denoted by the rather cumbersome notation, nkm3 . 

The cost of the changes in year 3 also reflects only those vehicles that can be redesigned but includes the 

costs of both fuel consumption and horsepower changes. Note that ∆y not ∆x determines the cost of 

increasing fuel economy or decreasing fuel consumption. This is because the quadratic curve provides an 

empirically good fit to relative increases in fuel economy but not to relative decreases in fuel consumption. 

∆x is computed from ∆y using equation 11. 

[ ( ) ( ) ] 2

ijk 3 ijk 3 ijk ijk 3 ijk 3 ijk ijk 3 hijk 3 P = δ + b ∆y + ∆h + c ∆ ∆y + ∆

(31) 

The initial vehicle price is reflected in the calibrated constant term for each vehicle and so only the change 

in price need be considered. 

The utility functions and consumers’ surplus change for year 3 includes changes in fuel consumption and 

horsepower but, again, only for those vehicles subject to redesign. Manufacturers may change the prices of 

all vehicles at any time, so δ’s are included for all vehicles (not included in first version of model). For 

vehicles that can be redesigned in year 3 the utility function is the following. 

( ( ) ( ) ) ( ) 3 3

2

ijk 3 ijk k jik 3 ijk ijk 3 ijk 3 ijk ijk 3 ijk 3 ( k kFC ) ijko 1 ijk akHP hijk u = A + B δ + b ∆y + ∆h + c ∆y + ∆h + B w x + ∆x + ∆

(32) 

For vehicles not eligible for redesign in year 3, the utility function depends only on the δ’s (which are not 

included in version 1 of the model). 

uijk3 = Aijk + Bkδ ijk3

(33) 

Note that there is a δ for every vehicle in every year of fuel economy standards. Since this creates 

thousands of decision variables, the first model version implemented will not include the δ’s to allow the 

more important features of the model to be tested with far fewer decision variables. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

50 

In the fourth year, vehicles that were redesigned in year 3 now have their fuel economy and horsepower 

fixed at the year 3 levels, but vehicles eligible for redesign in year 4 can be changed. All other vehicles 

remain at their base year fuel economy and horsepower levels. The fuel consumption constraint is the 

following. 

∑∑ ∑ ∑ () () ∑ == = = + +

+

=

⎥

⎦

⎤ ⎢

⎣

⎡

≥ + ∆ + + ∆ + m km km

m m

km km

m

n

j

n

k

n

i

n

i n n

ijk ijko

n n

i n

ijk ijko ijk ijk ijko ijk FC s x x s x x s x

11 1 1

4 4 3 4 4 4

3

3 4

3 4

3

1 1

(34) 

The utility functions for vehicles redesigned in year 3 remains the same as for year 3, except that the 

pricing change decision variable is now subscripted for year 4. The utility functions for vehicles eligible 

for redesign in year for are updated. The utility functions for all other vehicles not yet redesigned are 

affected only by the δ’s. 

Analogous definitions of constraints and utility functions apply through year 10. In year 11, vehicles 

redesigned in year 3 are again eligible for redesign. By this time, all other vehicles will already have been 

redesigned. The ∆x’s and ∆h’s for year three are replaced by ∆x’s and ∆h’s for year 11 for these vehicles, 

while the others remain the same. 

∑∑ ∑ ∑ ( ) () ( ) ∑ == = = + + +

+

=

⎥

⎦

⎤ ⎢

⎣

⎡

≥ + ∆ + + ∆ + + + ∆ m km km

m m

km km

m

n

j

n

k

n

i

n

i n n

ijk ijko ijk

n n

i n

ijk ijko ijk ijk ijko ijk FC s x x s x x s x x

11 1 ... 1

11 11 11 11 4 11 10

3

3 9

3 4

3

1 1 ... 1

(35) 

For year 12 the vehicles redesigned in year 4 will be eligible, those redesigned in all other years will remain 

the same. The utility functions are changed in a completely analogous manner and the process may 

continue with as many redesigns as desired. 

5.1 TRADABLE CREDITS 

Although, tradable fuel economy credits can take any form, the most economically appropriate 

denomination for credits is a “vehicle-gallon-per-mile” (Rubin, Leiby and Greene, 2004). Under this 

definition each credit effectively represents a right to consume a certain amount of gasoline. Multiplying 

one vehicle-gallon-per-mile credit times the average lifetime miles traveled by an automobile gives the 

average gallons consumed by an automobile over its life. In effect, the fuel efficiency credit market 

becomes a fuel use credit market. 

In a tradable credit system, manufacturers may either reduce the fuel consumption of their fleet by the 

number of vehicle-gallons-per-mile necessary to meet the standard or buy vehicle-gallon-per-mile credits in 

the credit market, or any combination of the two options. Let the number of tradable credits purchased by 

manufacturer j in period t be TCjt . If we assume the standard must be met in each period (no banking or 

borrowing of credits) then the number of credits purchased must be equal to (or greater than) the salesweighted difference between the standard and the manufacturer’s actual fleet average fuel efficiency. 

( )⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = −∑∑ + ∆ = =

n

k

n

i

jt jt t ijkt ijko ijkt

k

TC S x s x x

1 1

* 1

(36) 

In equation 38 the ∆x’s for vehicles that have not yet been redesigned will be zero. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

51 

Equation 38 becomes a new constraint for the optimization problem shown in equation 1. In addition, the 

net proceeds from the sales and purchases of credits must be included in the objective function. As defined 

in equation 38, TC>0 implies sales of credits since the manufacturer’s actual sales-weighted fuel 

consumption rate will be less than the standard. Purchases will be negative. Thus, the net proceeds 

(PTCTC) should be added to the objective function. 

( ) ( )

TC t

TC S x s x x t j n k n

OR manufacturer class specific

TC t

TC S x s x x t j n

manufacturer specific

CS x h p P TC

r

S Maximize

m

k

m

m mk

m mk

n

j

n

k

jkt

m

n

i

jkt jkt kt ikjt ikj ikjt

n

j

jt

m

n

k

n

i

jt jt t ikjt ikjo ikjt

n

j

TCt jt

n

k

n

i

t ijk ijk ijk ijk

T

t

t

t

= ∀

∀ ≥ = = ⎥

⎦

⎤ ⎢

⎣

⎡ = − + ∆

−

= ∀

∀ ≥ = ⎥

⎦

⎤ ⎢

⎣

⎡ = − + ∆

−

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∆ ∆ ∆ ∆ +

+

∑∑

∑

∑

∑∑

∑ ∑ ∑∑

= =

=

=

= =

= = ==

0

(1 ) 3, 1, , 1,

/

0

(1 ) 3, 1,

, , , 1

1 1

1

*

1

1 1

*

3 1 11

δ

(37) 

Banking and borrowing of credits over time can also be represented by formulating “bank balance” 

constraints that must hold over a moving window of years (e.g., see, Rubin, Leiby and Greene, 2004, draft). 

Equation 39, which does not allow banking and borrowing over time, requires that the balance be nonnegative for each year. Manufacturer j’s bank balance for year t, Bjt , for a window extending zb years back 

from year t and zf years forward is defined in equation 40, in which v is used as an index for the years in the 

moving window. 

∑

+

= −

=

f

b

t z

v t z

Bjt TCjv

(38) 

The new programming model with banking and borrowing is represented in equation 39. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

52 

( ) ( )

B TC t j

TC S x s x x t j n k n

OR manufacturer class specific

B TC t j

TC S x s x x t j n

manufacturer specific

CS x h p P TC

r

S Maximize

f

b

k

f

b

m mk

m mk

t z

v t z

n

k

jt jkv

m

n

i

jkt jkt kt ikjt ikj ikjt

t z

v t z

jt jv

m

n

k

n

i

jt jt t ikjt ikjo ikjt

n

j

TCt jt

n

k

n

i

t ijk ijk ijk ijk

T

t

t

t

0, ,

(1 ) 3, 1, , 1,

/

0, ,

(1 ) 3, 1,

, , , 1

1

1

*

1 1

*

3 1 11

= ≥ ∀

∀ ≥ = = ⎥

⎦

⎤ ⎢

⎣

⎡ = − + ∆

−

= ≥ ∀

∀ ≥ = ⎥

⎦

⎤ ⎢

⎣

⎡ = − + ∆

−

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∆ ∆ ∆ ∆ +

+

∑ ∑

∑

∑

∑∑

∑ ∑ ∑∑

+

= = −

=

+

= −

= =

= = ==

δ

(39) 

Many different credit trading schemes can be implemented. One interesting variation recommended by the 

NAS (2002) Committee on CAFE Standards allows manufacturers to buy credits from the government at a 

fixed price, PTC . This makes the credit price a constant rather than a decision variable. Manufacturers may 

buy as many credits as they need at the fixed price. They need not generate them by exceeding the fuel 

consumption standards. This changes the tradable credit constraints. 

 

5.2 IMPLEMENTATION 

The FERAM optimization model will be implemented in the FORTRAN programming language, making 

use of IMSL non-linear constrained optimization software. Data for the model and preliminary calculations 

will be implemented in Excel spreadsheets. Calibration of the NMNL model to base year and future year 

sales/production estimates will also be carried out in FORTRAN. Estimation of the NMNL choice model 

parameters will be implemented in a FORTRAN program. Estimation of the fuel consumption cost curves 

will also be implemented in FORTRAN. 

REFERENCES 

Duleep, K.G. 2005. “Vehicle and Fuel Economy Technology Database,” Energy and Environmental 

Analysis, Inc., Arlington, Virginia. 

(EEA) Energy and Environmental Analysis, Inc. 1993. “Documentation of Attributes of Technologies to 

Improve Automotive Fuel Economy,” Final Report, Energy and Environmental Analysis, Inc., Arlington, 

Virginia, February. 

Greene, D.L. 2001. TAFV Alternative Fuels and Vehicles Choice Model Documentation, ORNL/TM2001/134, Oak Ridge National Laboratory, Oak Ridge, Tennessee, July. 

Greene, D.L. and S. Ahmad. 2003. “A Preliminary Analysis of Tradable Fuel Economy Credits,” 

presented at the Meetings of the Transportation Research Board, Washington, DC, January. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

53 

Greene, D.L. and J.L. Hopson. 2003. “Analysis of Alternative Forms of Automotive Fuel Economy 

Standards for the United States,” Transportation Research Record 1842, pp. 20-28, Transportation 

Research Board, National Research Council, Washington, DC. 

Greene, D.L., P.D. Patterson, M. Singh and J. Li. 2004. “Feebates, Rebates and Gas-Guzzler Taxes: A 

Study of Incentives for Increased Fuel Economy,” Energy Policy, forthcoming. 

(NAS) National Research Council. 2002. Effectiveness and Impact of Corporate Average Fuel Economy 

(CAFE) Standards, National Academies Press, Washington, DC. 

Plotkin, S., D.L. Greene and K.G. Duleep. 2002. Examining the Potential for Voluntary Fuel Economy 

Standards in the United States and Canada, ANL/ES/02-5, Transportation Technology R&D Center, 

Argonne, National Laboratory, Argonne, Illinois. 

Rubin, J., P. Leiby, and D.L. Greene. 2004. “Tradable Fuel Economy Credits for Cars and Light Trucks.” 

Margaret Chase Smith Center for Public Policy, University of Maine, Orono, Maine, draft, March 4, 2004. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

54 

APPENDIX A 

ESTIMATING VEHICLE-SPECIFIC FUEL ECONOMY COST CURVES 

To estimate the impact of alternative fuel economy standards, it is necessary to predict the extent to which 

manufacturers can apply new technologies to reduce vehicle fuel consumption. This will depend not only 

on the costs and effectiveness of fuel economy technology but also on the extent to which they have 

already been incorporated in existing vehicles. In this section, a method is described that allows all three 

key factors to be represented for each individual make, model, and configuration of vehicle sold in the U.S. 

EEA has identified 49 technologies applicable for reducing fuel consumption. We assume that these 

technologies will be applied to the vehicles in the most cost effective order. The cost-effectiveness of each 

technology is measured by the ratio of its fractional improvement in fuel economy divided by its retail price 

equivalent (RPE). 

 

i

i

i

p

y

∆

∆ CE =

(B-1) 

When ranked by decreasing CE-ratio, the technologies were checked to make sure they remained in a 

feasible implementation order. Those that did not were combined with the necessary precursor 

technologies, to get an “aggregate” technology by summing fuel economy improvements as well as RPEs. 

For every vehicle, it is possible to identify the set of technologies that are both applicable and not already in 

use. Each technology was assigned one of the following index values: 1, already on the vehicle; 0, 

applicable; or N, not applicable. These indexes were used to develop a specific data set for each individual 

make, model, and configuration that included the applicable technologies, potential improvement in fuel 

economy, ∆y, and retail price equivalent, ∆p. 

A FORTRAN program was written to calculate quadratic cost curves using the data described above. The 

program estimates the coefficients for one car at a time. First, the data is sorted in descending order of 

cost-effectiveness. Next, cumulative fuel economy improvement and cumulative retail prices increases 

were computed as follows: 

∑=

∆ = ∆

n

i

n i Y y

1

 and ∑=∆ = ∆

n

i

Pn pi

1

(B-2) 

A subroutine was used to estimate the parameters b and c of the quadratic equation, 

2

n n Yn ∆P = b∆Y + c∆ by linear regression. Let X by the matrix containing ∆Yn and ∆Yn

2

 for a particular 

vehicle. 

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

∆ ∆

∆ ∆

∆ ∆

=

2

2

2 2

2

1 1

Yn Yn

Y Y

Y Y

Μ Μ

X

(B-3) 

Let Y be the vector of cumulative prices. 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

55 

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

∆

∆

∆

=

Pn

P

P

Μ

2

1

Y

(B-4) 

Let b be the vector of coefficients. 

⎥

⎦

⎤

⎢

⎣

⎡ = c

b

b

(B-5) 

Using the following formulas, the coefficients and a R2

 value are calculated: 

( ' ) ( ' ) 1 b X X * X Y - =

( )

( )

2

2

2

nY

nY R

−

− =

Y'Y

b'X'Y

(B-6) 

Curves were estimated in both gpm, ∆x, and mpg space, ∆y. The fitted curves are shown in the following 

graph. The mpg-curve produced consistently the best results and was thus used for all makes/models. 

The FORTRAN code for estimating technology/price curves is listed below. 

$(500)

$-

$500

$1,000

$1,500

$2,000

$2,500

$3,000

$3,500

$4,000

$4,500

-0.50 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00

Percent Change in Fuel Consumption (GPM)

Cumulative Cost

Costs

Quadratic in GPM

Quadratic in MPG
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

56 

! ----------------------------------- 

! MAIN PROGRAM 

! Call the various subroutines to get 

! cost curve coefficients for every car 

! in Matrix Car 

 PROGRAM OLS_Regression 

 IMPLICIT NONE 

 

 INTEGER, PARAMETER :: M=50 ! Total number of technologies 

 INTEGER, PARAMETER :: W=348 ! Total number of cars 

 DOUBLE PRECISION, DIMENSION (W,125) :: Cars ! Data set of cars 

 

 DOUBLE PRECISION, DIMENSION (M,4) :: One_Car ! Data for one car 

 DOUBLE PRECISION, DIMENSION (M,2) :: X_Vars ! LHS variables for regress 

 DOUBLE PRECISION, DIMENSION (M) :: Y_Vars ! RHS variables for regress 

 DOUBLE PRECISION, DIMENSION (2) :: B_Coeff ! Regression coefficients 

 DOUBLE PRECISION :: FC_Base ! Starting FC 

 DOUBLE PRECISION :: R_Sq1 ! R-Squared 

 

 INTEGER :: k, i, j, g ! Loop constants 

 INTEGER :: N ! Number of techs applicable to vehicle k 

 open(UNIT=1,FILE='techs_cars.txt') 

 read (1,*) ((Cars(i,k), k=1,M), i=1,W) 

 open(UNIT=2,FILE='costs_cars.txt') 

 read (2,*) ((Cars(i,k), k=M+1,2*M), i=1,W) 

 open(UNIT=3,FILE='coefficients.txt') 

 write (3,*), "COST CURVE COEFFICIENTS" 

 write (3,*), "Car # A B R-Sq#1" 

 k=1 

 DO WHILE (k .LE. W) 

 ! Put non-zero data into matrix One_Car 

 N=Cars(k,2*M) 

 i=2 

 j=1 

 DO WHILE(i .LE. M) 

 IF(Cars(k,i) .NE. 0) THEN 

 One_Car(j,1)=Cars(k,i) 

 One_Car(j,2)=Cars(k,i+M-1) 

 j=j+1 

 i=i+1 

 ELSE 

 i=i+1 

 ENDIF 

 END DO 

 

 ! Change percent to decimal 

 One_Car(1:N,1)=One_Car(1:N,1)/real(100) 

 

 CALL Rank(M,One_Car, FC_Base,N) ! Rank matrix in terms of cost-effectiveness 

 CALL Regression(M, One_Car, X_Vars, Y_Vars, B_Coeff, N, R_Sq1) ! Calculate Coefficients 

 

 ! Write calculations to file 

 write (3,FMT=10), Cars(k,1), B_Coeff(1), B_Coeff(2), R_Sq1 

10 format(1(F6.1,5X), 2(F10.4,5X), 1(F10.7,5X)) 

 

 ! Reset matrix for next vehicle 

 One_Car(1:M,1) = 0 

 One_Car(1:M,2) = 0 

 One_Car(1:M,3) = 0 

 One_Car(1:M,4) = 0 

 

 k=k+1 

 END DO 

 END PROGRAM OLS_Regression 

! ----------------------------------- 

! SUBROUTINE RANK 

! Take Matrix One_Car and rank the techs 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

57 

 SUBROUTINE Rank(M, One_Car, FC_Base,N) 

 DOUBLE PRECISION, DIMENSION (M,4) :: One_Car ! Data for one car 

 DOUBLE PRECISION :: FC_Base ! Starting FC 

 INTEGER :: N ! Number of techs applicable to vehicle k 

 

 ! Declare local variables 

 INTEGER :: i ! Loop constants 

 INTEGER :: swaps ! Swaps made in checking order of data 

 DOUBLE PRECISION, DIMENSION (1,3):: temp 

 ! Calculate cost-effectiveness ratio 

 i=1 

 DO WHILE (i .le. N) 

 One_Car(i,3) = One_Car(i,1)/One_Car(i,2) 

 i=i+1 

 END DO 

 

 ! Sort technologies by ratio 

 swaps=5 

 DO WHILE (swaps .gt. 0) 

 swaps = 0 

 i=2 

 DO WHILE (i .le. N) 

 IF(One_Car(i-1,3) .lt. One_Car(i,3)) THEN 

 temp(1,1:3)=One_Car(i,1:3) 

 One_Car(i,1:3)=One_Car(i-1,1:3) 

 One_Car(i-1,1:3)=temp(1,1:3) 

 swaps = swaps + 1 

 END IF 

 i=i+1 

 END DO 

 END DO 

 

 ! Calculate Cumulative %Chg in MPG and Price 

 i=2 

 DO WHILE (i .le. N) 

 One_Car(i,1) = One_Car(i-1,1) + One_Car(i,1) 

 One_Car(i,2) = One_Car(i-1,2) + One_Car(i,2) 

 i=i+1 

 END DO 

 

 RETURN 

 END SUBROUTINE 

! ----------------------------------- 

! ----------------------------------- 

! SUBROUTINE REGRESSION 

! Find coefficients for cost curves 

 SUBROUTINE Regression(M, One_Car, X_Vars, Y_Vars, B_Coeff,N,R_Sq1) 

 DOUBLE PRECISION, DIMENSION (M,4) :: One_Car ! Data for one car 

 DOUBLE PRECISION, DIMENSION (M,2) :: X_Vars ! LHS variables for regr 

 DOUBLE PRECISION, DIMENSION (M) :: Y_Vars ! RHS variables for regr 

 DOUBLE PRECISION, DIMENSION (2) :: B_Coeff ! Regression coefficients 

 DOUBLE PRECISION :: R_Sq1 ! R-Squared 

 INTEGER :: N ! Number of techs applicable to vehicle k 

 

 ! Declare local variables 

 INTEGER :: i ! Loop constants 

 DOUBLE PRECISION :: Det_X ! Determinant of (X'X) 

 DOUBLE PRECISION :: Sum1, Sum2, Sum3 , Sum4, Sum5, Sum6 ! Summation 

 ! Fill X: Column1=%Chg X, Column2=%ChgX^2 and Y=Chg in Price 

 i=1 

 DO WHILE (i .le. N) 

 X_Vars(i,1) = One_Car(i,1) 

 X_Vars(i,2) = One_Car(i,1)**2 

 Y_Vars(i) = One_Car(i,2) 

 i=i+1 

 END DO 

 

 ! Calculate the determinant of (X'X) 

 i=1 

 Sum1=0 

 Sum2=0 

 Sum3=0 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

58 

 DO WHILE (i .le. N) 

 Sum1 = Sum1 + X_Vars(i,2) ! Sum of x(i) 

 Sum2 = Sum2 + X_Vars(i,2)**2 ! Sum of x(i)^2 

 Sum3 = Sum3 + X_Vars(i,1)**3 ! Sum of x(i)^3 

 i=i+1 

 END DO 

 Det_X=(Sum1*Sum2)-(Sum3**2) 

 IF(Det_X .EQ. 0) THEN ! Check determinant for singularity 

 print *, "Determinant is zero. Regression Failed." 

 B_Coeff(1)=99999 

 B_Coeff(2)=99999 

 R_Sq1=99999 

 ELSE 

 i=1 

 Sum4=0 

 Sum5=0 

 Sum6=0 

 DO WHILE (i .le. N) 

 Sum4 = Sum4 + X_Vars(i,1)*Y_Vars(i) ! Sum of x(i)*y(i) 

 Sum5 = Sum5 + X_Vars(i,2)*Y_Vars(i) ! Sum of x(i)^2*y(i) 

 Sum6 = Sum6 + Y_Vars(i)**2 ! Sum of y(i)^2 

 i=i+1 

 END DO 

 ! Calculate coefficients 

 B_Coeff(1)=((Sum2*Sum4) - (Sum3*Sum5))*(real(1)/Det_X) 

 B_Coeff(2)=((Sum1*Sum5) - (Sum3*Sum4))*(real(1)/Det_X) 

 

 ! Calculate average of Y 

 i=1 

 Avg_Y = 0 

 DO WHILE (i .le. N) 

 Avg_Y = Avg_Y + Y_Vars(i) !Sum of y(i) 

 i=i+1 

 END DO 

 Avg_Y=Avg_Y/N 

 

 R_Sq1=(B_Coeff(1)*Sum4+B_Coeff(2)*Sum5) 

 R_Sq1=R_Sq1-((N-2)*(Avg_Y**2)) 

 R_Sq1=R_Sq1/(Sum6-((N-2)*(Avg_Y**2))) 

 ENDIF 

 END SUBROUTINE 

! ----------------------------------- 
A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al 

59 

APPENDIX B 

VALUING HORSEPOWER 

There are two options: (1) based on the literature or some other source such as a special study, one can 

choose a value for aHP , or (2) one can make use of the fact that at a competitive market equilibrium, the 

marginal costs of increasing fuel consumption, horsepower, or weight must equal their marginal utilities. 

In retail price equivalents, and translating utility into dollars by dividing by the price coefficient, B , this 

provides a means of calculating ah directly from the marginal cost of increasing horsepower. 

h a

h B

U

B

h

P ⎟

⎠

⎞ ⎜

⎝

⎛ = ∂

⎟

⎠

⎞ ⎜

⎝

⎛

∂

= ∂

∂ 1

1

(A-1) 

Both ∆yFC and ∆yHP determine consumers’ surplus via the utility function of the vehicle choice model. 

Both are decision variables in the optimization problem. Equation 14 insures that an increase in power will 

require either a decrease in fuel economy or an increase in price. However, adding ∆yHP as a decision 

variable also imposes a restriction on calibrating the vehicle choice model, namely that there should have 

been no incentive for further increases in power in the year in which each vehicle was designed. In other 

words, the marginal utility of increased horsepower must have equaled the marginal cost of lost fuel 

economy that would be caused by the increase in power. If the coefficients of fuel consumption and 

horsepower are a2 and a3 , respectively, then this requirement and equation 16 imply the following 

relationship between a2 and a3 . 

Marginal Cost of Increasing Horsepower = Marginal Value of Additional HP 

1

3 2 1 1

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∆

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∆ = +

∂

∂

∂

∂ = ∂

∂

η

η

o o o

o

h

hp

x

fc

h

x

a a

h

x

x

u

h

u

(A-2) 

This restriction determines the base year marginal utility of horsepower as a function of the marginal utility 

of fuel consumption and the technical relation between horsepower and fuel consumption. It appears to 

require calibrating vehicle-specific horsepower coefficients. Inasmuch as this formulation has not been 

tested before, its implementation should be considered experimental until it has been tested.