# 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-AC0500OR22725A Fuel Economy Regulatory Analysis Model (FERAM) by David L. Greene, et al

35 CONTENTS

1 BACKGR OUND................................................................................................................... .................................1

2 PROPOSED MODE L STRUCTURE ..................................................................................................... ...............1 2.1 CALCULATING UTILITIES................................................................................................... ....................6 2.2 MANUFACTURERS’ USE OF TECHNOLOGY AND PRICING .............................................................6 2.3 FUEL ECONOMY PE RFORMANCE 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 VEHI CLE 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 TRADA BLE CREDITS........................................................................................................ ......................16 5.2 IMPL EMENTATION .......................................................................................................... .......................18

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

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

APPENDIX B: VALUING HORSEPOWER................................................................................................ .............28A 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 therefor e 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 ever y 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 st rategies 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 in crease fuel economy. Not only conventional gasoline technologies, but hybrid gasoline-elect ric 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, s ikjt , 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, ∆ x ikjt , and the consequences for price, ∆ p ikjt , including adjustments to vehicle prices manufacturers may make to produce a higher fuel economy sales distribution, δ ijkt , and performance ∆ h ikjt . These decision variables are chosen to maximize consumers’ welfare give n 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 thes e 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 fu el economy (unitless and >=0) ∆ x = fractional change in fuel consumption (unitless and <=0) ∆ h = potential fractional change in fuel cons umption 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 P TC = price of a tradable credit P V = price of a vehicle P f = price of fuel per gallon p = probability an alternative is chosen from a set of choices Exogenous Data S = total vehicle sales
x ijko = 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 B k = coefficient of vehicle price in consumers’ utility function for vehicles in class k
a ijk = calculated make/model/configuration intercept for consumers’ utility function b ijk = coefficient of ∆ y in technology-price function
c ijk = coefficient of ∆ y 2 in technology price function e = base of the Naperian logarithms γ k

= annual rate of decline in vehicle use M ok = annual miles per year fo r a new vehicle in class k R k = discount rate for future fuel savings L k = 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
n jk

= the number of configurations made by manufacturer j in class k
n k = the number of vehicles in class k n = the number of vehicle classes
n m = 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 prob lem. Fuel economy standards can be represented as the constrained optimization problem shown in equation 1. The obj ective function is the pres ent 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 standa rds 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.



# ( ) 

# ( )

n k n j t x x s FC specific class er manufactur OR n j t x x s FC specific er manufactur OR t x x s FC t s p h x CS r S Maximize

m n

i ikjt ikj ikjt jkt m n

k n

i ikjt ikjo ikjt jt n

j n

k n

i ikjt ikjo ikjt t n

j n

k n

i ijk ijk ijk ijk t T

t t t

jk jk m jk m jk



## , 1 , , 1 , 3 ) 1 ( / , 1 , 3 ) 1 ( 3 ) 1 ( :

. . , , , 1

1 1 1 1 11 1 11 3



## = = ≥ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ + ≥ − = ≥ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ + ≥ − ≥ ∀ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ + ≥ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ ∆ ∆ ∆ +



# ∑ ∑∑ ∑ ∑∑ ∑ ∑∑ ∑

= = = = == = == = 

## δ

(1) The change in consumers’ surplus ( ∆ CS) induced by a fuel economy regulation is summed across all manufacturers, j=1, n m , and all makes and models of class k offered by each manufacturer, i=1, n mk , and all vehicle classes sold by a manufacturer, k=1, n k . The change in consumers’ surplus captures the value to consumers of changes in fuel savings, horsepow er, 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, δ ijk , 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, P TC , 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 assume d 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, z ik (2) a constant term represen ting the average value of the unmeasured attributes of vehicle i, A ik , and, (3) an unobserved random term reflecting factors specific to the l th individual, ε ik l . Here z represents the se veral 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 appr oximately 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 jiktl ijkt k k ijk ijktl 

### z w B a u 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 z ijk = P ijk that is the price of the vehicle, w kP =1, so that the coefficien t of price, or price slope, for vehicle class k is B k . If ε ijkt l is assumed to follow a type I extreme value distri bution across consumers for th e vehicles in class k, then the probability p ij|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
k ij k I i IJ



### 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 dolla rs, 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 U B A U

B A

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 n k 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 p ik = p i|k p k . The NMNL model of vehicle choice provides a closed form measure of consumer s’ surplus that can be directly inserted into the objective function. Let U k be the average utility of a vehicle in the kth vehicle class (nest) before fuel economy changes occur and U k * be the average utility afterwards. The change in consumers’ surplus in year t is given by equation 6. The calculation of U kt 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 U k * , 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 U k without the superscript asterisk re presents utility in the absence of new fuel economy regulations. This consumers’ surp lus measure is an averag e of the changes in the measured attributes of the vehicles. Di vided 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 U ijk ).

k k n

j n

i U

k k k 

### U B A e B B A U 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.

ijk ijktl 

### a u = (8) These base utilities can be used in equation 3 to calcu late 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 no n-zero values. At the very least, there will be some changes in fuel consumption rates ( ∆ xx o ) and in prices ∆ P V .

Vijkt k ijkt oijkt kFC k ijk ijkt 

### P B x x w B a u ∆ + ∆ + = ) ( (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 inclus ive 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 costfunction 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 fitte d curves in gpm space often exhibited negative costs fo r reducing fuel consumption in th e range of 5% to 15%, a result inconsistent with the data. Fortunately, relative chan ges 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 mpg y x

o

o o o o o

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

2 

### y c y b p ∆ + ∆ = ∆ (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 calibratin g equation 12 at a vehicle class level are available from a number of sources, including EEA and the National Academy of Sc iences’ 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 econo my curves are specific to each and every vehicle configuration (make, model, engine, transmission), since each may contain a different use of fuel economyA 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 lo wer 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 improveme nt 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.

HP FC 

### 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 HP FC HP FC 

### y y c y y b p ∆ + ∆ + ∆ + ∆ = ∆ (14) The two components of ∆ y, namely ∆ y FC and ∆ y HP , 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))x o = (1+ ∆ x)x o , will be multiplied by B k w kFC , where B k is the price slope for class k and w kFC is the discounted present dollar value of a 1 unit change in gallons per mile for vehicles in class k.

∆ y HP , however, is not a change in horsepower but rather a fractional change in potential fuel consumption that must be transformed into a change in horsepow er 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 hors epower instead of fuel economy, HP, is given in terms of the initial horsepower level and ∆ y in equation 15. 

# ( )


# [ ] η η

1 

### 1 1

HP o o o

o o


### y

HP HP mpg mpg y 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, SALES MIX AND FUEL ECONOMY In the face of tighter fuel economy regulations, it is po ssible 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, δ ijk , 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 repr esent consumers’ choices among light-duty vehicles. Vehicles will be grouped into classes (nests) of simila r vehicles (e.g., small SUVs, midsize passenger cars, minivans, etc.). The nesting structure allows a mo re realistic representation of own and cross price elasticities. In general, the price sensitivity of choices among vehicles with in 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 a ttribute values will be deri ved, using a methodology described in Greene (2001) and also used in the cu rrent 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’ producti on 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 w ill 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 w ill 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 fr om the base year market shares of individual make/model/configurations and vehicle classes.

Let S ik = 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 a Ik = the intercept term for a specific vehi cle configuration i=I, in class k. 

# ∑

= 

### − = k n

i ik k Ik Ik 

### s n s a 1 

### ) ln( 1 ) ln(
(17)
Define the inclusive value, IV, of class k as



### ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

= 

# ∑

= k n

i ik k k 

### a B B IV
1 

### ) exp( ln 1

(18) Calculate class intercepts, A k For an arbitrarily chosen class, say class 1,


### ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

− ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +

− = 

# ∑ ∑

=

= n

k k n

k k 

### IV s s

n IV A
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 

### IV A IV A s

1 

### ) exp( ) exp(

(21)

The B and B k 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 genera l, price elasticity is a function of price slope, B, at a particular market share, s, and price, P V .

V V



### P s B P s B

)

1 ( ) 1 (

− = − =
β β

(22)

If the B k are price slopes for choices within class k and B is the price slope for choi ces among the k classes, then the NMNL model requires that B > B k for all k. Since both the B k ’s and B are less than zero, this implies that absolute values of the B k ’s must be greater than the absolu te 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.



### 0000888

. 0 25000 ) 1

. 0 1 ( 2

− =

− − = B
(23) Reasonable price elasticity values for calculating the B k ‘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 th e 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 

# ( )

o FC kFC k 

### x x w B ∆ (24)

B k

is the vehicle class k price slope of the utility function and has units of utils/dollar. The fuel consumption weight w kFC 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, w kFC 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, p f , and, 4) the consumer’s discount rate, r. The value of a change in fuel consum ption 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 vehicl e 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 M o . The rate of use in year t is given by equation 27.

t
o 

### e M t M γ 

### = ) ( (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: M o = 15,000, γ = 0.04, r = 0.06. Different parameters may be specified for each vehicle class to produce a different value of fuel econo my for each class. The subscript, k is omitted from the equations for simplicity.


# [ ]

o FC FC L r o f

o FC L

t L

t t r o f o FC rt t
o f o FC 

### x x w e r M P x x dt e M P x x dt e e M t P x x PV ∆ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧

− + ∆ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∆ = ∆ = +

−

= = +

− − − 

# ∫ ∫ ) (

0 0 ) ( 

### 1 ) ( ) ( γ γ γ 

### γ (26) Equation 28 also shows that the appr opriate 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. Sin ce w translates a given change in fuel consumption into its present value in dollars, B k w kFC translates a given ∆ xx o into a change in utils, and is therefore the utility function coefficient of ∆ xx o .

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 includ e, but need not be limited to, the following three:

1. Corporate Average Fuel Economy constraints 2. Uniform Percentage Increase constraints 3. Attribute-based constraintsA 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 term s 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 ijkt ijkt t 

### x s F
1 11 (27) Manufacturer-specific constraints (there are n m constraints for each year, t=3 to T). 

# ∑∑

= = 

### ≥ n

k n

i ijkt ijkt jt k 

### x s F
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 consump tion rate, this defines a Uniform Percentage Increase fuel economy standard. If all manufacturers must meet the same fuel consumption target, F jt

= F t , 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 n m × n constraints for each year). 

# ∑

= 

### ≥ k n

i ijkt ijkt jkt 

### x s F
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 VARI ABLES AND CONSTRAINT S 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 ∆ x ijk3 , ∆ h ijk3 , δ ijk3 . The fuel economy constraint for year 3 recognizes th at those vehicles scheduled for redesign in year 3 are affected by the ∆ x ijk3 , 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

n i ijko ijk ijk ijko ijk 

### x s x x s FC
11 1 1 3 3 3 3 3

3 

### 1
(30)
The number of vehicles in class k, sold by manufacturer j that are slat ed to be redesigned in year 3 is denoted by the rather cumbersome notation, n km3 . The cost of the changes in year 3 also reflects only th ose 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 becau se 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 3 3 3 3 3 3 ijk ijk ijk ijk ijk ijk ijk ijk 

### h y c h y b P ∆ + ∆ ∆ + ∆ + ∆ + = δ (31) The initial vehicle price is reflected in the calibrated co nstant term for each vehicl e 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 thos e vehicles subject to redesign. Manu facturers 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 3 3 3 3 3 3 

### 1 ) ( ijk kHP ijk ijko kFC k ijk ijk ijk ijk ijk ijk jik k ijk ijk 

### h a x x w B h y c h y b B A u ∆ + ∆ + + ∆ + ∆ + ∆ + ∆ + + = δ

(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).

3 3 ijk k ijk ijk 

### B A u 

### δ + = (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 horsepow er levels. The fuel consumption constraint is the following. 

# () () ∑∑ ∑ ∑ ∑

== = + + = +

= 

### ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∆ + + ∆ + ≥ m km km

m m km km

m n

j n

k n

i n

n n i ijko ijk n n

n

i ijk ijko ijk ijk ijko ijk 

### x s x x s x x s FC
11 1 1 4 4 4 3 4 4 3

4 3 4 3

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 f unctions 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

n n i ijk ijko ijk n n

n

i ijk ijko ijk ijk ijko ijk 

### x x s x x s x x s FC
11 1 1

... 10 11 4 11 11 11 11 3

9 3 4 3

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 repr esents a right to consume a certain amount of gasoline. Multiplying one vehicle-gallon-per-mile credit times the average lif etime miles traveled by an automobile gives the average gallons consumed by an automobile over its lif e. In effect, the fuel efficiency credit market becomes a fuel use credit market. In a tradable credit system, manufact urers 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 TC jt

. If we assume the standard must be met in each period (no banking or borrowing of credits) then the numberof credits purchased must be equal to (or greater than) the salesweighted difference between the sta ndard and the manufacturer’s actual fleet average fuel efficiency. 

# ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ +

− = 

# ∑∑

= = n

k n

i ijkt ijko ijkt t jt jt k 

### x x s x S TC
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 (P TC TC) should be added to the objective function.


# ( ) 

# ( )

t TC n k n j t x x s x S TC specific class er manufactur OR t TC n j t x x s x S TC specific er manufactur TC P p h x CS r S Maximize

m k m m mk m mk

n

j n

k jkt m n

i ikjt ikj ikjt kt jkt jkt n

j jt m n

k n

i ikjt ikjo ikjt t jt jt n

j jt TCt n

k n

i ijk ijk ijk ijk t T

t t t



### ∀ = = = ≥ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ +

− = − ∀ = = ≥ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ +

− = − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∆ ∆ ∆ ∆ +



# ∑∑ ∑ ∑ ∑∑ ∑ ∑∑ ∑

= = = = = = = == =



### 0 , 1 , , 1 , 3 ) 1 ( / 0 , 1 , 3 ) 1 ( , , , 1

1 1 1 * 1 1 1 * 1 11 3 

### 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 year s (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, B jt , for a window extending z b years back from year t and z f years forward is defined in equation 40, in which v is used as an index for the years in the moving window. 

# ∑ +

− = 

### = f

b z t

z

t v jv jt 

### TC B
(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 

# ( ) 

# ( )

j t TC B n k n j t x x s x S TC specific class er manufactur OR j t TC B n j t x x s x S TC specific er manufactur TC P p h x CS r S Maximize

f

b k f

b

m mk m mk

z t

z

t v n

k jkv jt m n

i ikjt ikj ikjt kt jkt jkt z t

z

t v jv jt m n

k n

i ikjt ikjo ikjt t jt jt n

j jt TCt n

k n

i ijk ijk ijk ijk t T

t t t



## , , 0 , 1 , , 1 , 3 ) 1 ( / , , 0 , 1 , 3 ) 1 ( , , , 1

1 1 * 1 1 * 1 11 3



## ∀ ≥ = = = ≥ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ +

− = − ∀ ≥ = = ≥ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ +

− = − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∆ ∆ ∆ ∆ +



# ∑ ∑ ∑ ∑

∑∑ ∑ ∑∑ ∑

+

− = = = +

− = = = = == = 

## δ

(39) Many different credit trading schemes can be implemented. One interesting variation recommended by the NAS (2002) Committee on CAFE Standards allows manuf acturers to buy credits from the government at a fixed price, P TC . This makes the credit price a constant rath er 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 change s the tradable credit constraints.

5.2 IMPLEMENTATION The FERAM optimization model will be implemente d 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 Prelimin ary 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 Co rporate 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 Pub lic Policy, University of Maine, Or ono, 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 necessa ry 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 no t were combined with the necessary precursor technologies, to get an “aggregate” technology by su mming fuel economy improvements as well as RPEs. For every vehicle, it is possible to identify the set of t echnologies that are both app licable 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, cumula tive fuel economy improvement and cumulative retail prices increases were computed as follows: 

# ∑

= 

### ∆ = ∆ n

i i n 

### y Y
1 and 

# ∑

= 

### ∆ = ∆ n

i i n 

### p P
1 (B-2) A subroutine was used to estimate the parameters b and c of the quadratic equation, 2

n n n 

### Y c Y b P ∆ + ∆ = ∆ by linear regression. Let X by the matrix containing ∆ Y n and ∆ Y n 2 for a particular vehicle.



### ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

∆

∆ ∆ ∆ ∆ ∆

=

2 2 2 2 2 1 1

n n 

### Y Y 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 

### ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

∆

∆ ∆

=

n 

### P 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 R 2 value are calculated:



### ) ' ( ) ' ( 1 

### Y X X X b 

### * - 

### = ) ( ) (
2 2 2 

### Y n Y

n R

− − = Y Y' Y

X' b'

(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 MPGA 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 techsA 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=0A 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 a HP , or (2) one can make use of the fact that at a competitive market equilibrium, the marginal costs of increasing fuel consumption, horsepow er, 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 a h directly from the marginal cost of increasing horsepower.

h 

### a B h U B h P ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

= ∂ ⎟

⎠ ⎞ ⎜ ⎝ ⎛ ∂

= ∂ ∂ 1 1

(A-1)

Both ∆ y FC and ∆ y HP 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 ∆ y HP as a decision variable also imposes a restriction on calibrating the vehicle choice model, namely that there should have been no incentive for further increas es in power in the year in which each vehicle was designed. In other words, the marginal utility of increased horsepower mu st 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 a 2 and a 3 , respectively, then this requirement and equation 16 imply the following relationship between a 2 and a 3 .
Marginal Cost of Increasing Horsepower = Marginal Value of Additional HP

1

2 3 

### 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 consum ption. It appears to require calibrating vehicle-specific ho rsepower coefficients. Inasmuch as this formulation has not been tested before, its implementation should be considered experimental until it has been tested.