CONSUMER VEHICLE CHOICE MODEL DOCUMENTATION 

David L. Greene 

Changzheng Liu 

March 2012 

Prepared for 

U.S. ENVIRONMENTAL PROTECTION AGENCY 

Prepared by 

OAK RIDGE NATIONAL LABORATORY 

Oak Ridge, Tennessee 37831-6283 

Managed by 

UT-BATTELLE, LLC 

for the 

U.S. DEPARTMENT OF ENERGY 

under contract DE-AC05-00OR22725 

iii 

TABLE OF CONTENTS

LIST OF FIGURES ........................................................................................................................ v

LIST OF TABLES .......................................................................................................................... v

ACKNOWLEDGEMENT ............................................................................................................ vii

1. Introduction ................................................................................................................................ 1

1.1 Project Overview .............................................................................................................. 1

1.2 Model Usage and Results Interpretation .......................................................................... 2

1.2.1 Model Functionality and Usage ................................................................................ 2

1.2.2 Prediction Errors ....................................................................................................... 3

2. Literature Review on New Vehicle Type Choice Modeling ..................................................... 7

2.1 Aggregate Demand Models .................................................................................................. 7

2.2 Discrete Choice or Random Utility Models .......................................................................... 9

2.2.1 Multinomial Logit ....................................................................................................... 10

2.2.2 Probit and Nested Multinomial Logit ......................................................................... 11

2.2.3 Mixed Logit Model (MLM) ........................................................................................ 14

2.3 Summary Observations ....................................................................................................... 18

3. Methodology ............................................................................................................................ 21

3.1 Nesting Structure ................................................................................................................ 21

3.2 Equations............................................................................................................................. 24

3.2.1 Prelude ......................................................................................................................... 24

3.2.2 Two-level CVCM Equations ...................................................................................... 25

3.2.3 Full Scale CVCM Equations ....................................................................................... 26

3.3 Value of Fuel Economy ...................................................................................................... 28

3.4 Calibration........................................................................................................................... 29

3.4.1 Generalized Cost Coefficient Determination ............................................................... 29

3.4.2 Constant Term Calibration .......................................................................................... 35

4. Implementation and User Guide .............................................................................................. 37

4.1 User Interface ...................................................................................................................... 37

4.1.1 Input ............................................................................................................................. 37

4.1.2 Output .......................................................................................................................... 39

4.2 Interaction with OMEGA ................................................................................................... 40

References ..................................................................................................................................... 41

Appendix A: Derivation of Nested Logit Model Equations and Relevant Properties ................. 45
iv 

Appendix B: Model Sensitivity analysis...................................................................................... 49

B.1 The Distribution of Own Price Elasticities ........................................................................ 49

B.2 Sensitivity Analysis ............................................................................................................ 52
v 

LIST OF FIGURES 

Figure Page 

1. Nested Multinomial Logit Structure of Consumer Choice Model.................................... 21

2. Distribution of Elasticities ................................................................................................ 50

LIST OF TABLES 

Tables Page 

1. Demand Elasticities of Kleit's Vehicle Class Demand Model (Kleit, 2004) ...................... 7

2. Market Segment and Nameplate Own Price Elasticities Estimated by Bordley (1993) ..... 8

3. Vehicle Class Definition in the CVCM ............................................................................ 23

4. Own Price Elasticities of New Vehicle Demand in the Literature ................................... 33

5. Generalized Cost Coefficient Calibration ......................................................................... 34

6. Format of Vehicle Sheet ................................................................................................... 38

7. Structure of “GlobalParameter” Sheet .............................................................................. 39

8. List of Vehicles with Very High Elasticities (in absolute value) ...................................... 50

9. Descriptive Statistics of Elasticities .................................................................................. 51

10. Sensitivity Analysis Results .............................................................................................. 53

11. Market Shares by MPG Decile ......................................................................................... 53

12. Rebound Effect ................................................................................................................. 54

 

 
vi 

 
vii 

ACKNOWLEDGEMENT 

This document was prepared as part of a research project sponsored by the U.S. Environmental 

Protection Agency (EPA). The authors would like to express their gratitude to Michael Shelby, 

Sharyn Lie and Gloria Helfand, EPA, for the leadership and support in developing the Consumer 

Vehicle Choice Model (CVCM). The authors are grateful to Gloria Helfand and Michael Shelby 

for valuable comments on an earlier draft of this documentation. The authors also thank Ari 

Kahan and Richard Rykowski, EPA, for reviewing and testing the CVCM. We are especially 

grateful to our peer reviewers, Professor David Bunch, Professor Trudy Cameron, and Dr. 

Walter McManus, for a very thorough and helpful review of the model and documentation. Any 

remaining errors or deficiencies are the authors’ responsibility. 

 
viii 
1

1. INTRODUCTION 

1.1 PROJECT OVERVIEW

In response to the Fuel Economy and Greenhouse Gas (GHG) emissions standards, automobile 

manufacturers will need to adopt new technologies to improve the fuel economy of their vehicles 

and to reduce the overall GHG emissions of their fleets. The U.S. Environmental Protection 

Agency (EPA) has developed the Optimization Model for reducing GHGs from Automobiles 

(OMEGA) to estimate the costs and benefits of meeting GHG emission standards through 

different technology packages. However, the model does not simulate the impact that increased 

technology costs will have on vehicle sales or on consumer surplus. As the model 

documentation states, “While OMEGA incorporates functions which generally minimize the cost 

of meeting a specified carbon dioxide (CO2) target, it is not an economic simulation model which 

adjusts vehicle sales in response to the cost of the technology added to each vehicle.” 

Changes in the mix of vehicles sold, caused by the costs and benefits of added fuel economy 

technologies, could make it easier or more difficult for manufacturers to meet fuel economy and 

emissions standards, and impacts on consumer surplus could raise the costs or augment the 

benefits of the standards. Because the OMEGA model does not presently estimate such impacts, 

the EPA is investigating the feasibility of developing an adjunct to the OMEGA model to make 

such estimates. This project is an effort to develop and test a candidate model. The project 

statement of work spells out the key functional requirements for the new model. 

“ORNL shall develop a Nested Multinomial Logit (NMNL) or other appropriate model capable of 

estimating the consumer surplus impacts and the sales mix effects of GHG emission standards. The 

model will use output from the EPA’s Optimization Model for reducing Emissions of Greenhouse 

gases from Automobiles (OMEGA), including changes in retail price equivalents, changes in fuel 

economy, and changes in emissions, to estimate these impacts. ...The model will accept 

approximately 60 vehicle types, with the flexibility to function with fewer or more vehicle types, and

will use a 15 year planning horizon, matching the OMEGA parameters. It will be calibrated to 

baseline sales projection data provided by the EPA and will include a buy/no-buy option to simulate 

the possibility that consumers will choose to keep their old vehicle or to buy a used vehicle. The first 

version of the model must be completed by the spring of 2011. Additional versions may be created in 

the future, pending further discussion and negotiation between the consultant and the EPA.” 

Briefly, given changes in each vehicle’s price and fuel economy, the model 

(1) calculates impacts of standards on vehicle sales mix, and 

(2) calculates cost of standards in terms of consumer surplus. 

The initial version of the model, at least, is not intended to project market trends due to other 

factors, although this might be a fruitful area for future research and development. The goal of 

this project is to create a simple model to test the concept of incorporating market share and 

consumer surplus changes to the OMEGA model and to produce a working initial model. 
2

A research team at Oak Ridge National Laboratory (ORNL) has designed and implemented a 

Consumer Vehicle Choice Model (CVCM) for the project based on NMNL theory with a 

representative consumer. This document will detail CVCM design principles, model equations, 

parameter calibration, implementation and user guide. Specifically, Section 1.2 further explains 

CVCM functionality and intended usage and summarizes possible sources of prediction errors. 

Section 2 reviews relevant new vehicle type choice models in the literature and compares their 

merits and limitations. Then Section 3 describes NMNL equations and model calibration 

procedure. Finally, Section 4 gives instructions to use the model. 

1.2 MODEL USAGE AND RESULTS INTERPRETATION 

1.2.1 Model Functionality and Usage 

The CVCM is intended to perform specific functions as an adjunct to the EPA’s OMEGA model. 

As such, it has been designed to use the same theoretical basis and premises as the OMEGA 

model. Specifically, it has been designed to self-calibrate to the baseline vehicle sales 

distribution used by OMEGA and, given estimates for each individual vehicle of (1) changes in 

vehicle fuel economy, and (2) changes in vehicle prices, it: 

1. calculates impacts of those changes on vehicle sales and the distribution of 

vehicle sales and the resulting impact on manufacturers’ abilities to meet fuel 

economy standards and, 

2. calculates changes in consumers’ surplus as a consequence of the changes in fuel 

economy and vehicle purchase cost. 

The CVCM is not intended to be a tool for forecasting the future vehicle fleet. There is no doubt 

that, over time periods longer than a few years, vehicle designs will come and go, new vehicle 

models will be introduced and others retired, new manufacturers will enter the U.S. market, 

existing manufacturers will exit, and there will be mergers and divestitures. However, predicting 

such events is outside the scope of the CVCM. It is also likely that over future time periods 

manufacturers will introduce new types of vehicles: plug-in hybrid, battery electric, hydrogen 

fuel cell vehicles and perhaps vehicles that are not foreseen at the present time. The CVCM was 

not designed to predict consumers’ acceptance of these advanced technology vehicles. This 

capacity was left for future research and development. 

The CVCM was developed to test the concept of predicting the differential sales impacts of fuel 

economy changes together with price changes brought about by fuel economy standards. It is 

intended to produce credible estimates of such changes to determine whether they may have 

important implications for manufacturers’ abilities to meet the standards and for consumer wellbeing. Given the EPA’s need for periodic and timely analyses to support its responsibilities for 

GHG emissions and fuel economy rulemakings, the CVCM should be capable of being readily 

calibrated to new data sets and updated with new sales and fuel economy data. 

Given the intended purpose and functions of the model, it is most appropriately used for 

estimating changes in the following variables relative to the baseline values: 
3

1. Market-wide consumers’ surplus, total sales, total gross revenue, and fleet average miles 

per gallon (MPG) and GHG emissions, 

2. Sales, average MPG and GHG emissions by manufacturer, and 

3. Sales by market segment. 

The CVCM models vehicle type choice at the most complete level of detail possible, 

corresponding to the level of detail at which fuel economy measurements are made by the EPA. 

Given that the price sensitivity of consumers’ choices is greatest at the lowest1

 level of the 

NMNL nest, i.e., when vehicles are the closest substitutes, modeling at the greatest feasible level 

of detail can capture the full range of sales mix shifts. If vehicle type choice were modeled at a 

more aggregate level, the modeling process may be open to the questions about whether it misses 

important sales mix changes that would have been evident had the model operated at a greater 

level of detail. However, the CVCM prediction at the lowest level (i.e., make, model, engine and 

transmission configuration) is most sensitive to changes in input data and model parameters. 

Reporting results at this level may imply a higher degree of precision than is appropriate. Thus, 

we recommend reporting CVCM predictions at more aggregate levels. 

The model provides highly detailed results. For reasons discussed in Section 1.2.2, the model’s 

predictions are unlikely to be as precise as is suggested from the model output. The detail is 

provided for situations where the CVCM would be used iteratively with OMEGA, where the 

detail may provide advantages for model convergence. On the other hand, when final results are 

presented for consideration, false precision should be avoided. The sensitivity analyses we have 

done (see Appendix B) suggest that outputs should be presented to no more than three digits and 

perhaps only two in the case of consumers’ surplus impacts and impacts on total vehicle sales. 

1.2.2 Prediction Errors 

The aggregate, or representative consumer, NMNL model makes simplifying assumptions about 

consumer behavior. Since consumer behavior is complex, we have focused the modeling 

initially on the decisions by consumers to trade-off fuel savings for higher vehicle prices, holding 

all other vehicle attributes constant. A change in a particular vehicle’s fuel economy is translated 

into a change in price equivalent (present value dollars) based on a model or theory of how 

consumers value fuel economy. The change in the present value of future fuel costs perceived 

by consumers is added to the estimated change in the vehicle’s price ∆p . A price sensitivity 

parameter, B, translates the resulting net change in present value into a change in a utility index 

that determines a vehicle’s market share. The change in utility for the i

th vehicle in nest j, Uij ,is 

the following, where PV represents whatever function is chosen to transform a change in fuel 

economy to a change in the present value of fuel savings considered in the vehicle purchase 

decision. 

∆ = ∆ − U B p PV fuel savings ij j i ( { }) (1) 

 

1

 In this document, the higher/lower levels are referred to by their relative positions in the nested tree/nesting structure 

(Figure 1 in chap. 3) which has a buy/no-buy decision at the top/highest level and vehicle configurations (combinations of make, 

model, engine, and transmission) at the bottom/lowest level. It implies that the lower levels in the tree are more disaggregated. 
4

The NMNL model is a tool for estimating changes in market shares as a function of changes in 

the present value (in dollars) of vehicles. If the changes are small relative to the prices of the 

vehicles and if the price sensitivity parameters are reasonably accurate, the NMNL model should 

give reasonably accurate predictions. 2

 Prediction errors arise from incorrectly estimating 

changes in the utility index, caused either by errors in the estimation of the role of a change in 

fuel economy or inaccurate specification of the price sensitivity parameter, B. Such errors have 

a specific functional form in logit models. 

For illustrative purposes, consider a simple multinomial logit (MNL) model (a derivation of the 

NMNL model that begins with a specification of the simple MNL model can be found in section 

2.2.1 below).3

 The derivative of a vehicle’s market share Si with respect to a change in its utility 

index is the following: 

2

1

( )

i

j

U

i

N U i i

i i j

dS d e S S

dU dU e

=

= = −

∑

 (2) 

Since, in general, Si (which is between 0 and 1) will be approximately two orders of magnitude 

larger than 2

( )iS , the change in market share dSi

 is approximately the change in utility weighted 

by the vehicle’s market share, SidUi

. As changes in utility are propagated up the nesting 

structure (as inclusive values, or expected utility changes) this simple relationship applies at each 

step. Since a shock (error) in the utility index of vehicle i is a change in its utility, the impact of 

errors in the utility index on the predicted share is proportional to the market share of vehicle i. 

Prediction errors will be negatively correlated between alternatives within a nest. At the lowest 

level of nesting, shocks to the utilities of individual vehicles are independent and identically 

distributed, in theory. However, the errors in one vehicle’s utility index induce a change in the 

predicted shares of other vehicles that are negatively related to changes for the initial vehicle. 

The error term of a utility function directly induces a change in utility so its impact can be 

described by the derivative of the share of vehicle i with respect to a change in (shock to) the 

utility of vehicle j. 

( )

2

1

1

1

i

j j i

j

U N

i U U U

N U i j

j j j

j

S e

e e e S S

U U e

−

=

=

∂ ∂  

= = − = −   ∂ ∂   ∑

∑

 (3) 

Thus, a shock to the utility index of vehicle j induces a negative error in the prediction of the 

share of vehicle i that is proportional to the product of their shares, and prediction errors within a 

nest are negatively correlated. Because of this, errors in utility functions within a nest will tend 

to cancel, and the sum of the shares within a nest (i.e., the share of that nest) will have a smaller 

relative error than the relative errors of the individual vehicles within the nest. 

 

2

 There is reason to expect the changes in dollar value to be small relative to the vehicle’s price in that they will, in 

general, be comprised of an increase in price (>0) minus a present value of future fuel savings (also >0). 

3

 Each lowest level nest of a NMNL model is a simple multinomial logit model. 
5

In reality, prediction errors can arise from a number of simplifications in the CVCM, errors in 

parameters and errors in input data. 

1. Non-optimizing consumer behavior 

2. Aggregate NMNL model applied to heterogeneous consumers 

3. Errors in NMNL model structure 

4. Errors in NMNL parameters 

5. Omitted variables (including manufacturer pricing decisions) 

6. Inaccuracies in baseline sales data 

7. Inaccuracies in OMEGA model predictions 

8. Unanticipated technological innovations over time 

9. Changes in consumers’ behavior over time 

There is substantial evidence that consumers’ decision-making in markets for energy efficiency, 

and in particular fuel economy, may not correspond to the classical rational economic model 

(e.g., Jaffe and Stavins, 1994; Greene, 2009). A review of the econometric evidence found 

contradictory and inconclusive results (Greene, 2010). If, as Greene (2011) proposes, consumers’ 

decisions about fuel economy are best described by prospect theory of behavioral economics, 

then the theory of utility maximization that underlies random utility models like the NMNL 

model would not be the most appropriate context for evaluating consumers’ surplus impacts. 

Further theoretical and empirical research is needed to better understand how consumers' value 

fuel economy and how fuel economy and emissions standards affect consumers' surplus. 

The CVCM is a market or representative consumer NMNL. It does not explicitly represent 

differences in consumers’ preferences. The only recognition of differences in consumers’ tastes 

is in the logit formulation itself which assumes that each individual perceives a different value 

for each vehicle (e.g., Train, 1993 ch. 2). However, this representation of heterogeneity is very 

limited and, in particular, does not allow for different price sensitivities. The population of 

consumers is undoubtedly heterogeneous but it is not known how important that heterogeneity is 

to the intended purpose of the CVCM. If further research and development is undertaken, 

investigating the importance of consumer heterogeneity should be given a high priority. Explicit 

heterogeneity was not incorporated in the CVCM in order to keep the model and its calibration 

simple. 

The nesting structure used in the CVCM is similar to nesting structures used in empirically 

estimated models and in constructed models such as Bunch et al. (2011) and NERA (2009). 

Grouping vehicles by size, functionality and price is intuitive and consistent with the theoretical 

requirement that vehicles in a nest be similar with respect to unobserved attributes, i.e., be close 

substitutes. However, there is no guarantee that the nesting structure chosen is the best possible 

nesting structure. 

Price sensitivities and alternative-specific constants are the two classes of parameters of the 

CVCM. Price sensitivities are the most important because the constants are computed so that the 

model exactly predicts the baseline market shares, given the assumed price sensitivities. The 

price sensitivities have been chosen to be consistent with the estimates in the published literature 
6

and to conform to the theoretical requirement that price sensitivities increase in absolute value as 

one moves down along the nesting tree. However, the price sensitivity parameters have not been 

estimated to be consistent with a specific data set and it is always possible that an additional 

empirical analysis could yield insights missing from the existing literature. 

Numerous possible explanatory variables have been excluded from the CVCM. Indeed, the only 

variables included are the changes in price and fuel economy supplied by the OMEGA model. 

Other variables are implicitly held constant in that they are included in the baseline constant 

terms. Including factors such as income and demographic variables may be desirable in a model 

to be used for estimation over an extended time period. However, this would require predicting 

values for those variables over the same time period. A potentially important endogenous 

variable in the OMEGA/CVCM system might be internal pricing decisions by manufacturers to 

meet especially stringent (strongly binding) fuel economy and emissions standards. This is 

beyond the scope of the current CVCM project, however. 

To the extent that the baseline sales data, including the definitions of individual vehicles, differ 

from the actual market data, errors could be induced in the CVCM estimates. OMEGA is itself a 

model and thus its estimates undoubtedly contain some differences from what will occur, and 

these will also affect the accuracy of CVCM estimates. 

Over extended time periods, automotive technology will change, and may change in ways that 

cannot be foreseen at the present time. Furthermore, consumers’ preferences may also change in 

unpredictable ways. The 2002 National Research Council report on the CAFE standards and 

potential for fuel economy improvement did not foresee a successful market for hybrid vehicles 

(NRC, 2002). The emergence of minivans, SUVs, crossovers, the near disappearance of station 

wagons, and more, could not have been predicted with any certainty a long time period (e.g. 15 

years) in advance. Assessment of technological innovation and trends in consumers’ preferences 

is beyond the state-of-the-art in economic modeling and is probably best handled by scenario 

analysis. 

The CVCM was designed to estimate the impacts of changes in vehicle prices and fuel economy 

provided by the OMEGA model on consumers’ surplus and changes in vehicle sales that could 

impact manufacturers’ abilities to meet fuel economy and GHG emissions standards. It was 

developed as a first test of the potential for such estimations to contribute to improved rule 

making. The goal was to develop a simple model that could be readily calibrated and operated in 

conjunction with the OMEGA model, and that had a sound theoretical and empirical basis. 
7

2. LITERATURE REVIEW ON NEW VEHICLE TYPE CHOICE 

MODELING 

The impacts of changes in vehicle prices and fuel economy on vehicle sales and consumer 

surplus can be estimated by means of systems of demand equations and discrete choice models, 

which are reviewed in this section. The emphasis is on two types of discrete choice models –

NMNL and Mixed Logit (ML) models. 

2.1 AGGREGATE DEMAND MODELS 

Automobile demand by type of vehicle can be represented by a system of linear or non-linear 

demand equations. Kleit (2004, 2002a & b, 1990) created a market segment vehicle demand 

model that he used to evaluate the costs and benefits of CAFE standards. Kleit divided the 

market into eleven vehicle classes and four manufacturers. Demand functions, in the 2002b 

paper at least, were specified as simple linear functions of vehicle price (i.e., Q = a + bP). 

These equations can be calibrated given an initial set of prices and quantities and own price 

elasticities. Kleit estimated own price elasticities and cross price elasticities by exercising a 

proprietary model developed by GM (Table 1). Own price elasticities for the eleven vehicle 

classes ranged from -1.5 for large trucks to -4.5 for large cars. The own price elasticity for 

luxury cars is -1.7, less than those of standard cars (-2.8 to -4.5) but of the same order of 

magnitude. In general, cross price elasticities are small relative to own price elasticities. Cross 

elasticities are not symmetric because classes with high sales volumes have a greater effect on 

classes with low sales volumes than vice versa. However, there are sets of classes for which 

cross price elasticities are substantial, indicating that the vehicle types are relatively close 

substitutes. The groupings in Kleit’s model suggest that standard cars are relatively close 

substitutes, with small cars being better substitutes for medium cars than for large cars. Small 

and Large SUVs are relatively good substitutes, as are small and large (pickup) trucks. Cars and 

pickup trucks are not close substitutes, and the only vehicle that is even a weak substitute for a 

full size van is a minivan. In the discussion of discrete choice models below, such grouping 

become “nests”. 

Table 1 Demand Elasticities of Kleit's Vehicle Class Demand Model (Kleit, 2004) 

1 2 3 4 5 6 7 8 9 10 11 

1 Small Car -2.808 0.423 0.063 0.018 0.000 0.036 0.027 0.009 0.009 0.009 0.000 

2 Medium Car 0.684 -3.528 1.107 0.027 0.018 0.018 0.018 0.036 0.045 0.054 0.009 

3 Large Car 0.270 1.926 -4.500 0.027 0.216 0.009 0.054 0.018 0.063 0.054 0.009 

4 Sport Car 0.549 0.423 0.324 -2.250 0.009 0.090 0.198 0.045 0.108 0.018 0.000 

5 Luxury Car 0.045 0.405 1.062 0.009 -1.737 0.000 0.027 0.045 0.189 0.072 0.009 

6 Small Truck 0.162 0.099 0.000 0.009 0.000 -2.988 0.702 0.045 0.054 0.009 0.009 

7 Large Truck 0.063 0.072 0.018 0.009 0.000 0.234 -1.548 0.027 0.090 0.018 0.036 

8 Small SUV 0.216 0.279 0.099 0.027 0.009 0.090 0.351 -3.645 0.747 0.108 0.072 

9 Large SUV 0.117 0.243 0.171 0.018 0.018 0.054 0.387 0.414 -2.043 0.234 0.108 

10 Minivan 0.081 0.171 0.063 0.000 0.009 0.009 0.045 0.027 0.135 -2.286 0.180 

11 Van 0.027 0.036 0.009 0.009 0.000 0.009 0.054 0.036 0.072 0.387 -2.385 
8

Automobile supply was represented by assuming a short-run price elasticity of supply of +2 and 

a long-run elasticity of +4. 

Bordley4

 (1993) estimated own and cross price elasticities for 200 passenger car nameplates 

using aggregate time-series sales data by market segment plus survey data on the first and second 

choices of consumers who had just purchased a new car. The aggregate sales data allowed 

estimation of own price elasticities for seven passenger car market segments and an overall price 

elasticity of automobile demand. The survey data were used to estimate “diversion fractions” 

quantifying the propensity of purchasers of one nameplate to buy any of the others given an 

increase in its price. Bordley estimated an own price elasticity for passenger car purchases 

versus all other commodities of -1.0. Car segment elasticities ranged from -1.7 for small cars 

to -3.4 for sporty cars (Table 2). Elasticities for individual nameplates ranged from -1.7 to -8.2; 

mean values within segments ranged from -2.4 to -4.7. 

Table 2 Market Segment and Nameplate Own Price Elasticities Estimated by Bordley (1993) 

Car Class Class Elasticity Car Nameplate Elasticities 

 Minimum Average Maximum 

 

Economy -1.9 -3.3/-3.4 -4.7 -8.2/-8.1 

Small -1.7 -1.9/-1.7 -2.4 -3.1/-3.4 

Compact -2.0 -2.1/-2.2 -3.1 -4.9/-4.7 

Midsize -2.3 -2.3/-2.6 -3.3 -4.6/-4.2 

Large -3.0 -3.1/-3.5 -3.8 -4.3/-4.0 

Luxury -2.4 -3.2/-3.4 -3.7 -5.3/-4.5 

Sporty -3.4 -2.6/-3.4 -4.2 -6.5/-5.3 

Bordley’s method could be used to calibrate a system of linear nameplate demand equations, as 

was done by Kleit (1990). More complex systems including cross price elasticities can also be 

calibrated, as Bordley (1993) points out, but does not explicitly describe calibration of such a 

model. 

Austin and Dinan (2005) used the own- and cross-price elasticity matrix developed by Kleit 

(2002a) to estimate the impacts of changes in vehicle prices due to fuel economy standards on 

consumers’ demand for 10 vehicle classes. Consumer demand for a class is a linear function of 

the difference between vehicle price and the value of future fuel savings induced by the 

standards. For manufacturer i, demand for its vehicle classes is given by the following matrix 

equation, 

( ) i i i i q A p c = − (4) 

in which qi is the vector of quantities for each of the 10 vehicle classes, pi is the vector of prices, 

ci is the vector of present value of fuel economy improvements and Ai is a matrix of own- and 

cross-price elasticities. Austin and Dinan (2005) do not provide the numerical values for the 

 

4

 Bordley was employed by General Motors Research Laboratory at the time he conducted and published his study. 

Thus, there may be a relationship between Kleit’s elasticity estimates, which are based on a GM model, and Bordley’s. 
9

elasticities they used in their model, nor does Kleit (2002a), apparently because the model is 

proprietary. 

2.2 DISCRETE CHOICE OR RANDOM UTILITY MODELS 

Discrete choice models, sometimes referred to as random utility (RU) models, are by far the 

most common methodology used to mathematically model automobile demand. Baltas and 

Doyle (2001) succinctly summarize the methodology. 

“In RU models, preferences for such discrete alternatives are determined by the realization of latent 

indices of attractiveness, called product utilities. Utility maximization is the objective of the 

decision process and leads to observed choice in the sense that the consumer chooses the alternative 

for which the utility is maximal. Individual preferences depend on characteristics of the alternatives 

and the tastes of the consumer....The analyst cannot observe all the factors affecting preferences and 

the latter are treated as random variables.” (Baltas and Doyle, 2001, pp. 115). 

Since the early applications of random utility models in the 1970s (McFadden, 1973), 

formulations of RU models have proliferated. Baltas and Doyle (2001) identified fourteen 

different methods which they grouped into three fundamentally different approaches depending 

on the nature of the random utility: 

• Unobserved product heterogeneity, 

• Taste Variation (consumer heterogeneity), and 

• Choice Set Heterogeneity. 

Nearly all applications of random utility models to automobile choice fall into the first two 

groups because the availability of different types of automobiles is rarely a significant issue. 

Randomness in the simple multinomial logit model derives primarily from unobserved attributes. 

Its error term may also include unobserved variations in taste but the representation of these 

variations is limited and simplistic. The same applies to NMNL Models although their ability to 

represent randomness in unobserved attributes and tastes is much more complex. In these 

models, heterogeneity in consumers’ preferences is commonly represented by explicit functional 

relationships between product attributes and consumer characteristics. MNL models allow 

variations in consumers’ preferences to be represented by random coefficients, whose 

distributions can be inferred either from survey or market shares data. 

Which methodology is best for a given application depends not only on the richness of the 

modeling approach but on the objectives of the exercise, as well as practical constraints, 

including data and resource availability. Baltas and Doyle sum up the dilemma well. 

“Finally, a general concern relates to overall model practicality. As our discussion illustrates, recent 

developments have increased model complexity and made estimation, interpretation, and forecasting 

less straightforward. Some specifications are still rather impractical. The issue can be viewed as the

common dilemma between simplicity and flexibility. There is no universal answer to this question as 

it depends on one’s rate of exchange between the two criteria.” (Baltas and Doyle, 2001, p. 123). 
10

2.2.1 Multinomial Logit 

The first application of a multinomial logit model to automobile choice appears to be the seminal 

paper by Lave and Train (1979) which estimated a multinomial logit model of consumers’ 

choices among 10 vehicle classes using what was then a new method for analyzing qualitative 

choice behavior (McFadden, 1973). The probability of an individual consumer choosing a 

vehicle class was assumed to be a function of a vector of vehicle attributes and household 

attributes. The model formulation allowed for interaction of household and vehicle variables in a 

linear “representative” utility function. Let Xijk be the k

th variable, for the jth vehicle class and the 

i

th consumer. The representative utility function is defined as, 

1

K

ij k ijk ij

k

V X β ε

=

= + ∑ (5) 

in which the βks are fixed coefficients and the εijks are independent, identically distributed 

random variables that have extreme value distributions. The probability that consumer i will 

purchase a vehicle from class j is a multinomial logit function of the representative utilities of all 

classes. 

1

j

l

V

ij L V

l

e

P

e

=

=

∑

 (6) 

In the Lave and Train model, vehicle price was represented by price divided by household 

income and the same variable squared. The results implied both that sensitivity to price 

decreased with increasing vehicle price and that price sensitivity decreased with increasing 

income. The model was calibrated to survey data from 541 households collected in seven U.S. 

cities in 1976. 

McCarthy and Tay (1998) estimated a MNL model of consumers’ choices among 68 makes and 

models. Their objective was to test whether buyers of domestic, European and Japanese 

manufactured vehicles valued vehicle attributes in the same way. Their analysis rejected the 

hypothesis that vehicle attributes are similarly valued regardless of country of origin. They also 

noted certain “anomalies” in their coefficient estimates. For example, faster acceleration 

decreased the probability of choosing American and Japanese vehicles, while operating costs 

were an insignificant variable for makes and models of Japanese manufacture. Similar results 

have been observed in other studies and may point to an inherent difficulty in estimating random 

utility models. A key assumption of such models is that the unobserved attributes are 

uncorrelated with the observed attributes. If they are not, then biased estimates can result. 

Given the strong correlations among many observed attributes (e.g., size, price, horsepower, fuel 

economy, weight, interior volume, number of seats, etc.), the assumption that unobserved 

attributes are uncorrelated with observed attributes seems unlikely. In addition, the problem of 

defining and obtaining measures of precisely the right attributes that determine consumers’ 

choices has also been a persistent issue for random utility models. Is acceleration best measured 

by the ratio of horsepower to weight, by 0-60 mph time, or by the various measures the industry 

uses to capture the experiences of launch from a stop, intermediate speed range acceleration, 
11

passing acceleration, and responsiveness? Inaccuracies in defining and measuring attributes lead 

to errors in observed variables. Correlated omitted variables, errors in observed variables and 

correlated observed variables makes statistical inferences challenging indeed. 

Lave and Train (1979) noted two key limitations of the MNL model. First is the so-called 

Independence of Irrelevant Alternatives (IIA) property, which makes the ratio of the probabilities 

of choice of any two alternatives independent of the presence or attributes of any other 

alternatives. A related property is that all alternatives are assumed to have the same probability 

distribution of unobserved utility (i.e., ε has the same distribution for every alternative) and that 

these distributions are independent. These properties severely restrict the patterns of substitution 

the model can represent. For example, apart from the measured utility component, Lave and 

Train’s MNL model implies that a two-seater sports car is just as good a substitute for a luxury 

sedan as it is for a sporty subcompact. Note that the measured utility component in the LaveTrain model directly accounted for factors such as the number of seats, household size and 

acceleration performance. Unobserved factors might be such things as styling or image. 

Second, because automobile attributes do not vary across the population of consumers, it is not 

possible to estimate a MNL model that includes vehicle attributes and a vehicle specific constant. 

In a model estimated using household data, attributes can only be entered when interacted with 

some household characteristic. On one hand, this allows attribute values to vary across 

individuals. On the other, it imposes specific functional relationships on how attribute values 

vary that may not be supported by any theory. Thus, heterogeneity of consumers’ preferences is 

an inherent property of MNL models estimated using household data but is restricted to specific 

functional relationships chosen by the researcher. 

2.2.2 Probit and Nested Multinomial Logit 

The shortcomings of the simple MNL model, especially its IIA property, led researchers to 

explore alternative formulations that allowed greater flexibility in patterns of substitution among 

vehicles and representations of heterogeneous consumer preferences. The probit model was 

derived by relaxing the assumptions of independent, identical error distributions (see, e.g., Train, 

1993). Instead the error terms in a probit model are assumed to be jointly normally distributed. 

Instead of leading to a simple, closed form equation for the choice probabilities (like equation (6) 

for the MNL model), the probit model requires numerical integration of a series of integrals. The 

probit model’s inherent complexity, combined with the ability of a variant of the MNL model to 

overcome most of its limitations, is responsible for the very infrequent use of probit models in 

modeling automobile choice. 

The NMNL Model, a special case of the Generalized Extreme Value (GEV) model, is based on 

the premise that the full choice set can be portioned into subsets (nests) within which the IIA 

property is appropriate but across which it is not. Put another way, within a nest all vehicles are 

assumed to be equal substitutes, conditional on their observed utility. Formally, within a subset 

alternatives error terms are independent and identically distributed. Across subsets, they are not. 

Building on the notation of equations (5) and (6), the probability that a consumer will choose a 

specific make, model, engine and transmission configuration, m, given that the consumer will 

choose a vehicle in nest (class) j, is a simple MNL probability. 
12

|

1

ijm

ijl

U

im j L

U

l

e

P

e

=

=

∑

 (7) 

The probability that consumer i will choose class j is a function of the utility of attributes 

common to class j, Vj, as well as a function of the composite utility of all vehicles within class j, 

Iij. 

1

ij j ij

il l il

V I

ij V I

l

L

e

P

e

λ

λ

+

+

=

=

∑

 (8) 

The term Iij is the “inclusive value” or expected value of the utility of vehicles in set j. It is 

defined by equation (9). 

1

1

K

jk ijmk

k

M X

ij

m

I ln e

β

=

=

  ∑

=  

   

∑ (9) 

In equation (9) each nest has a different set of coefficients that map vehicle attributes into the 

utility index. In particular for this model, these coefficients differ across nests. This allows 

different degrees of substitutability for the choices within different nests. The unconditional 

probability of consumer i choosing vehicle m in class j is the following. 

P P P ijm im j ij = |

 (10) 

Another feature of the NMNL model that helps overcome the limitations of the MNL model is 

the ability to define any number of levels of nesting. A key advantage of this is that the top nest 

can represent the choice to buy or not to buy a new automobile. Thus, an NMNL market model 

can predict the impacts of changes in vehicle attributes and other factors on total vehicle sales as 

well as the type of vehicles purchased. 

The flexibility and mathematical simplicity of the NMNL model have made it the most widely 

used tool for modeling automobile choices. Goldberg (1995, 1998) estimated NMNL models of 

automobile choice in order to evaluate the impacts of fuel economy standards. In the 1995 study, 

her nests comprised (1) small cars including subcompacts and compacts, (2) luxury automobiles 

including sports cars, and (3) all other vehicles. A likelihood ratio test was used to test (and 

reject) the hypothesis that coefficient values within the three nests were equal. While such tests 

can be used to reject a nesting structure, there is no accepted methodology for identifying a 

correct nesting structure. Goldberg’s 1998 study used nine vehicle classes, within which 

consumers could choose between a foreign or domestic car. This structure was chosen to allow 

exploration of differential impacts of standards on foreign and domestic manufacturers. 

Stated preference survey data were used by Brownstone et al. (1996) to estimate a NMNL model 

of consumers’ choices among conventional and alternative fuel vehicles. The 1993 California 
13

survey asked households to choose among hypothetical vehicles that included alternative fuel 

vehicles. Stated preference methods were necessitated by the fact that very few households 

purchase or have any experience with vehicles powered by non-petroleum fuels. 

Most often, NMNL models are calibrated via statistical inference based on the vehicle choices of 

individual consumers or households. However, MNL and NMNL models can also be interpreted 

as representing the choice probabilities of a representative consumer or a population of 

consumers with diverse tastes (Anderson et al., 1988). In this interpretation, the random error 

term (ε) represents not only unobserved attributes but also unobserved variations in tastes and 

errors in perception and optimization by consumers (Madalla, 1992, p. 60). Several modelers 

have used NMNL models in this way to represent aggregate market behavior. 

Greene (1994) constructed a NMNL choice model for predicting market shares of alternative 

fuel vehicles. Rather than estimating a model based on stated preference survey data, Greene 

followed a methodology invented by Donndenlinger and Cook (1997) to infer the values of 

automobile attributes. The model coefficients were constructed by postulating how vehicle 

attributes such as range or fuel economy would be valued by consumers, deriving a coefficient 

that translates unit changes in each variable to a present dollar value and applying a multiplier to 

transform that coefficient into one that translates unit changes into the utility index. This 

multiplier is referred to as generalized cost coefficient in the remainder of this document. Greene 

reasoned that since the overwhelming majority of consumers had no first-hand experience with 

alternative fuel vehicles (e.g., battery electric vehicles, compressed natural gas vehicles, etc.) 

stated preference surveys data would likely be misleading. The model did not include a buy/nobuy decision. The first level nest included eight alternative fuel technologies. The second level 

nest comprised the choice of fuel for bi-fuel or flex-fuel vehicles. A similar model also 

constructed by Greene (2001) contained Conventional Internal Combustion Engine (ICE) 

vehicles, Dedicated Alternative Fuel vehicles (CNG and LPG), Hydrogen Fuel Cell vehicles and 

Battery Electric vehicles in the first level nest and subcategories of these vehicle technologies in 

the second. E.g., ICE was divided into conventional liquid fuel vehicles, hybrid vehicles and 

gaseous-fueled vehicles. Within the conventional liquid fuel nest, consumers chose among 

gasoline, diesel, ethanol FFVs and methanol FFVs. Within the FFV nests, consumers chose fuel 

types, e.g., gasoline or E85. To estimate price coefficients for the nests in his model, Greene 

(2001) relied on existing studies and the theoretical requirement that sensitivity to price must 

increase from the top nest to the bottom (from vehicle technology choice to fuel choice). Since 

the overall price elasticity of automobile demand is generally believed to be approximately -1.0 

(Kleit, 1990; McCarthy, 1996; Bordley, 1993) and the choice of fuel is highly but not infinitely 

elastic (approximately -10 or more: Greene, 1998, p. 228), this bounds the range of price 

sensitivity for nests in between. Although this range is an order of magnitude, with three nests 

between the top and bottom choices it provides useful information that can be used in 

conjunction with estimates from published studies to greatly reduce the uncertainty about 

coefficient values. 

Greene et al. (2005) and Greene (2009) calibrated constructed NMNL models to the market 

shares of over 800 carline/engine/transmission configurations. The data sets included every 

vehicle in the National Highway Traffic Safety Administration’s (NHTSA) model year 2000 and 

2005 fuel economy data sets, respectively, except those with annual sales below 25 units per year. 
14

Generalized cost coefficients were chosen based on the published literature and the relative value 

rule for nests described above. Vehicle-specific constants were used to insure the model exactly 

fit the base year data. The ability to calibrate the model to fit any given year’s sales data is an 

advantage for use in policy analysis where the correspondence of model estimates to real world 

experience is of value. 

Harrison et al. (2008), like Greene et al. (2005), used a constructed NMNL model to evaluate the 

benefits and costs of the 2011-2015 CAFE standards. The authors assumed a plausible nesting 

structure based on their judgments about the substitutability of different types of vehicles. The 

guiding principle is that vehicles within a nest are closer substitutes for one another than they are 

for vehicles in other nests.5

 Consumers are assumed to decide to buy or not to buy at the top nest, 

then choose among three car classes: passenger cars, pickup truck/full-size van, or 

SUV/minivans. The next level contains 14 vehicle classes based on size and price. Within these 

subclasses are non-intersecting subsets of over 200 vehicle models. Like Greene et al. (2005), 

Harrison et al. made use of the NMNL requirement that price sensitivity (price coefficients) must 

decrease in absolute value (increase in value) as one moves up the nesting tree. They began with 

a price elasticity of -1.0 for the buy/no-buy decision, and then assumed the ratios of parameters 

at each level in order to calculate price coefficients for each lower nest. Harrison et al. also 

calculated a constant term for each model, as Greene et al. did, but then regressed those constant 

terms against other vehicle attributes in an effort to infer the value of those attributes. 

2.2.3 Mixed Logit Model (MLM) 

The MLM adds to the NMNL a greater capability to include heterogeneous consumer tastes. 

The utility of vehicle m to consumer i is given by equation (11). 

1 1

K H

im m k imk ih imh im

k h

U X z δ β μ ε

= =

= + + + ∑ ∑ (11) 

In equation (11), δm represents the average utility (intercept term) of vehicle m, the Ximk are 

vehicle attributes interacted with consumer characteristics, the βk are mean coefficient values for 

these variables, the μih are individual specific random coefficients reflecting deviations of 

individual tastes from those βk for which tastes vary, and the zimh are vehicle attributes interacted 

with consumer characteristics for which tastes vary (Train and Winston, 2007). Assuming that 

the εim are independent and identically distributed and have an extreme value distribution, the 

probability that consumer i chooses vehicle m is given by the mixed logit model (the integral 

sign represents many integrals over the many probability distributions of the random variables). 

1 1

1 1

K H

m k imk ih imh k h

K H

l k ilk ih ilh k h

X z

im X z

l

e

P

e

δ β μ

δ β μ

= =

= =

+ +

+ +

∑ ∑

=

∑ ∑ ∫

∑

 (12) 

 

5

 More accurately, the vehicles are more similar with respect to their unobserved attributes. Vehicles may differ greatly 

with respect to the measured attributes that enter the utility index function yet still belong in the same nest. 
15

Train and Winston (2007) estimated a mixed logit model of vehicle choice using a random 

sample of 458 U.S. consumers who had just purchased a new model year 2000 vehicle in order 

to investigate reasons for the declining market shares of U.S. auto manufacturers. Each 

consumer’s choice set consisted of 200 makes and models. There is no closed form solution for 

estimating the parameters of the MLM. Instead, simulation was used to approximate the 

integrals for choice probabilities and the resulting log likelihood function. 

The parameters of the MLM are functions of consumer attributes and random variables. For 

example, the price coefficient in the Train and Winston model is, 

1.60 0.86 0.073

Y Y i i

ν

β

−

= − − + (13) 

The variable ν is a standard normal random variable. This adds richness to the model by 

representing varying tastes across the population. There is even some small probability of 

finding a consumer who prefers higher prices (β>0). On the other hand, the functional form is, 

to a degree, chosen a priori by the researcher, and both estimating the model and predicting with 

it are substantially more complicated but still quite feasible. Both require simulations (perhaps 

only a few hundred) and both require information about the distributions of consumer 

characteristics (available from national surveys). 

A comparison of MLM and NMNL models was made by Brownstone et al. (2000), combining 

stated and revealed preference survey data for California households. The authors observed that 

the MLM improves the fit of model to data, and indicated substantial heterogeneity of 

preferences across the population. They also noted that revealed preference (RP) data are 

essential for obtaining realistic predictions of consumers’ choices of vehicle types. However, 

they also commented on the difficulty of statistical inference using RP data. 

“RP data appear to be critical for obtaining realistic body-type choice and scaling information, but 

they are plagued by multicollinearity and difficulties with measuring vehicle attributes. SP data are

critical for obtaining information about attributes not available in the marketplace, but pure SP 

models with these data give implausible forecasts.” (Brownstone et al., 2000) 

Bento et al. (2005, 2009) estimated a mixed logit model of vehicle choice and a paired model of 

vehicle use using data from the 2001 National Household Travel Survey. They divided vehicles 

into 10 vehicle classes, 5 age categories and 7 manufacturers. The paired models not only 

estimate new vehicle choices but vehicle use, as well as aging and scrappage. Jacobsen (2010) 

used the model to assess the impacts of CAFE standards on manufacturers but did not include in 

his model the option they have to use technology to improve the fuel economy of vehicles at 

increased cost. The mean price elasticity of new vehicle demand was estimated to be -2.0, 

substantially more than the unit elasticity found in models cited above. 

Cambridge Econometrics (2008) estimated a mixed logit model of vehicle choice in the UK 

based on a survey of households who had purchased a new or less than 1-year-old vehicle during 

the years 2004 to 2007. Households identified the manufacturer, model and engine size of their 
16

vehicle, which the researchers matched to a separate data base of vehicle attributes. The survey 

asked what attributes consumers considered important to the purchase of a vehicle. Respondents 

cited many difficult to measure factors, such as reliability, safety, comfort, warranty and security. 

Estimated mean price elasticities by vehicle class ranged from -0.96 for multi-passenger vehicles, 

to -3.51 for luxury vehicles. Relatively elastic market segments included Minicars (-2.46), 

Upper Medium cars (-2.81) and Executive cars (-3.24). Less price elastic segments were 

Superminicars (-1.15), Lower Medium cars (-1.15), Sports cars (-1.79) and 4X4s (-1.75). The 

observed patterns of own- and cross- price elasticities led the researchers to comment on the 

importance of models that allow flexibility in substitution patterns. 

“We observe substitution patterns that represent a significant departure from proportional 

substitution, i.e. there is a higher level of substitution between similar models of cars.” (Cambridge 

Econometrics, 2008, p. vii) 

Mixed logit models can also be estimated using aggregate market shares, as first shown by Boyd 

and Mellman (1980) and Cardell and Dunbar (1980) and later in a seminal paper by Berry, 

Levinsohn and Pakes (BLP) (1995). BLP provided a practical method of estimating a mixed 

logit model from aggregate sales data. Prices are endogenous in the BLP model, an issue they 

addressed by means of instrumental variables comprised of the attributes of other vehicles. 

Estimates relying on instrumental variables in this context can be unreliable, as Knittel and 

Metaxoglou (2008) demonstrated using BLP’s data. Noting that the objective function in the 

BLP model is highly nonlinear and thus prone to multiple local optima, they tested 10 different 

optimization algorithms, using 50 different starting values for each. Their results call for caution 

both in interpreting parameter estimates from BLP-type models and in their use for forecasting. 

“We find that convergence may occur at a number of local extreme, at saddles and in regions of the 

objective function where first-order conditions are not satisfied. We find own- and cross-price 

elasticity estimates that differ by a factor of over 100 depending on the set of candidate parameter 

estimates.” (Knittel and Metaxoglou, 2008) 

On the other hand, other researchers, using variants of the BLP model and different estimation 

procedures, have obtained more stable results. 

Moon, Shum and Weidner (2010) extend the BLP method by adding interactive fixed effects to 

the unobserved product characteristics. The specification multiplicatively combines timespecific fixed effects with vehicle-specific fixed effects. The consumer’s utility function is, 

1 1

K R

ijt ik kjt rj rt jt ijt

k r

U X f α λ δ ε

= =

= + + + ∑ ∑ (14) 

in which α’s are coefficients measuring the marginal value of each of the K vehicle attributes X, 

whose mean value also includes the R interactive fixed effects of product j, plus δjt, represented 

by the third hand side term. The difference between this formulation and that of BLP is the 

specific structure imposed on the distribution of product-specific tastes. The final term, εijt, is the 

individual, product and time specific utility component. Note that if there are on the order of 103
17

vehicles and just a few time periods, this model has thousands of parameters. In addition, 

projecting taste heterogeneity into the future requires specifying future values for frt and δjt or 

assuming they remain constant. If these are assumed to be constant at the values of a given year 

or at average values, the heterogeneity of tastes is limited to product-specific heterogeneity. 

An advantage of the Moon et al. approach is that it explicitly represents some endogenous factors 

by means of interactive fixed effects and thereby reduces the need for instrumental variables, in 

particular, to represent price endogeneity. The authors find that, given their formulation, 

coefficient estimates produced by methods that assume prices are exogenous versus endogenous 

differ little. 

The Moon et al. method also produces price elasticities that are much higher in absolute value 

than those obtained by the standard BLP model estimation methods. This is apparently due to 

the inclusion of the fixed effect variables. They applied the method to the same data used by 

BLP (1995). Own and cross price elasticities were estimated for 23 vehicle classes. Using their 

interactive fixed effect formulation and assuming prices to be endogenous produced own price 

elasticities ranging from -7.0 for Cadillacs to -36.5 for large Mercurys. Twenty of the 23 

estimated elasticities were more price elastic than -25.0. Omitting the interactive fixed effects 

produced own price elasticity estimates ranging from -7.8 (again, for Cadillac) to -17.6 for a 

“remainder of the market” category. This time, 10 of the 23 elasticity estimates were more 

elastic than -15.0. 

In a study for the UK Department of Transport, the Economics for the Environment Consultancy 

(EFTEC) estimated a MLM model of consumers’ choices of automobiles in the UK (EFTEC, 

2008). The researchers estimated their model using the method of BLP and data on new car 

market shares for 2,190 different vehicle types registered by private households in 11 regions of 

the UK. They note that their choice set is considerably larger than that of any previous study. 

The ability to calibrate a model to such a large choice set is a consequence of the BLP estimation 

procedure. Vehicles were nested into 9 classes based on size, body style and price. Estimated 

median price elasticities ranged from -1.3, for vehicles in the SUV class with a range from -2.4 

(90th percentile) to -1.0 (10th percentile), to -5.4 for vehicles in the small-to-medium size family 

car segment with a range from -7.1 (90th) to -4.5 (10th). Sports cars also had relatively low price 

elasticities and subcompact and mini car choices were relatively price elastic. 

A number of recent studies have employed forms of the Mixed Logit model to estimate the 

relative effects of vehicle price and fuel economy or fuel costs on vehicle choice (e.g., Allcott 

and Wozny, 2009; Klier and Linn, 2008; Gramlich, 2008; Sawhill, 2008). These and other 

related studies were reviewed by Greene (2010). All used extensive, detailed data bases on 

vehicle purchases in the United States but reached very different conclusions about how 

consumers trade off vehicle price and fuel economy. Some of the differences can be attributed to 

how consumers form expectations about future fuel prices, although most models assumed static 

expectations based on the observation that fuel prices appear to follow a random walk. 

Aggregate, mixed logit type models can be used to predict market shares and estimate changes in 

consumer surplus. For example, Greene and Liu (1988) used both a random coefficient MNL 

model and Lave and Train’s (1979) model to estimate the impacts of changes in vehicle 
18

attributes related to fuel economy on the consumer surplus associated with automobiles sold in 

the United States between 1978 and 1985. The random coefficient model utilized Monte Carlo 

simulation to execute repeated draws from the vector distribution of random coefficients. 

Greene and Liu found that the estimated mean consumer surplus values were highly sensitive to 

the mean values of attributes but they did not test sensitivity of consumer surplus estimates to the 

variance of attribute values. 

2.3 SUMMARY OBSERVATIONS 

All three categories of models (aggregate demand models, NMNL, and MLM) can be used to 

estimate changes in market shares and consumer surplus due to increases in vehicle prices and 

fuel economy. Aggregate demand models, like those developed by Kleit (2002a) or Austin and 

Dinan (2005), could, in principle, produce estimates for 60 or even 800 vehicle types. Given 

own- and cross-price elasticities, calibration of such models to sales data would be 

straightforward. Estimating the price elasticity matrix, however, is a major challenge. An 800 

by 800 matrix would require 640,000 elasticity estimates and even a 60 by 60 matrix would need 

3,600 elasticity values. Bordley’s (1993) method offers a potential solution to this problem but it 

requires rarely available data on consumers’ first and second choices. Perhaps this is why it 

appears not to have been used in subsequent studies. 

The ability of mixed logit models to represent consumer heterogeneity also comes at the price of 

greater information requirements for model calibration and simulation. Mixed logit models 

require specification of not only the central tendencies of key parameters but also their variance, 

and possibly their correlations. Running a mixed logit model requires repeated randomized 

draws from the distributions of parameters. Fortunately, software is available for performing the 

necessary simulations. Calibration and updating of MLMs requires considerable effort. Survey 

based estimation methods require extensive, detailed survey data. Aggregate methods have more 

modest data requirements but the validity of the estimates by the most prevalent algorithms has 

been called into question by recent research (Knittel and Metaxoglou, 2008). In either case, 

there is presently no evidence that MLMs produce more accurate predictions than other methods. 

Should the EPA determine that vehicle choice modeling can make an important contribution to 

its regulatory analyses, it may be worthwhile to determine whether the potential benefits of using 

mixed logit models to represent consumer heterogeneity are worth the extra complexity and data 

requirements of the mixed logit model. 

NMNL models have been constructed, calibrated and used in policy analyses of fuel economy 

issues by Greene et al. (2005), Harrison et al. (2008) and Bunch et al. (2011). All three 

applications modeled vehicle choices at a fine level of detail, ranging from 200 makes and 

models to over 800 make/model/engine/transmission combinations. This high level of detail was 

considered necessary to adequately represent the changes in market shares that might result from 

fuel economy and emissions standards or fiscal policies. Given that the price sensitivity of 

consumers’ choices is greatest at the lowest level of the NMNL nest, i.e. when vehicles are the 

closest substitutes, modeling at the greatest feasible level of detail should produce a model with 

the potential to measure the full impacts of price and fuel economy changes on fleet average fuel 

economy and consumer surplus. 
19

Given a nesting structure and corresponding price coefficients, NMNL models can be quickly 

and precisely calibrated to historical or projected sales data using closed form equations. NMNL 

models are capable of accommodating the introduction, termination, or modification of product 

lines. They are not capable, however, of predicting when product lines will be introduced or 

terminated. NMNL models that must be calibrated to sales data are also not able to predict the 

sales of newly introduced vehicles, since there is no vehicle-specific constant term available for 

new products. This is a general limitation of models that include fixed effects to accurately 

predict sales shares and applies to Mixed Logit Models and other formulations, as well. 

For the purpose of developing an initial model to test the value of making such estimates, the 

NMNL method appears to be a good compromise between flexibility and simplicity. It can be 

readily calibrated with only a small amount of information about price elasticities and base year 

sales data. It allows for substantial flexibility in representing substitutions among vehicle types. 

On the other hand, it does not allow great flexibility in representing heterogeneous consumer 

preferences. This may be a fruitful area of future research and development, especially if it can 

be shown that more detailed representations of consumer tastes lead to more accurate predictions. 
20

 
21

3. METHODOLOGY 

This project constructs and calibrates a NMNL model along the line of Greene et al. (2005) and 

Bunch et al. (2011). Generalized cost coefficients are derived from the literature and NMNL 

properties. Given generalized cost coefficients, constant terms of the model are calibrated to 

baseline sales data such that the model prediction replicates baseline market share. 

3.1 NESTING STRUCTURE 

Choice alternatives in the CVCM are represented in detail, by make, model, engine and 

transmission, corresponding to the level of detail at which fuel economy measurements are made 

by the EPA. There are on the order of 1,000 choice alternatives. Individual vehicles are grouped 

into nests as in Figure 1 to allow differential substitution patterns within and between nests. The 

structure has 5 levels: Lev0 (Buy a new vehicle/Don’t buy a new vehicle), Lev1 (Passenger 

Vehicles, Cargo Vehicles and Ultra Prestige vehicles), Lev2 (vehicle types: Two Seaters, 

Prestige Cars, Standard Cars, Prestige SUVs, MiniVans, Standard SUVs, Pickup Trucks, Vans, 

and Ultra Prestige Vehicles), Lev3 (vehicle classes (see Table 3) and Lev4 (vehicle 

configurations (one configuration is defined as a combination of make, model, engine size and 

transmission type)). Define Lev0 as the highest level and Lev4 as the lowest level. Right above 

Lev0 is root node (not drawn in Figure 1), which is the origin of the nesting structure/tree. 

Figure 1 Nested Multinomial Logit Structure of Consumer Choice Model 

Note: “Standard” is synonymous with “Non-Prestige” 

The nesting structure in Figure 1 is defined according to general principles that group closer 

substitutes in a nest and ensure price sensitivity (price coefficient) and substitutability increase as 

one goes down to the bottom of the nesting structure

6

. The inclusion of the buy/no-buy option is 

necessary to predict impacts on total sales, not just the distribution of sales among makes, 

 

6

 The requirement that price sensitivity increases as one goes down to the bottom is explained in Appendix A. 
22

models and vehicle classes. Conditioning on buying a new vehicle, vehicle configurations are 

grouped according to functionality and size of vehicles and prestige/non-prestige. Thus lev1 

distinguishes between passenger vehicles, cargo vehicles, and ultra-prestige vehicles (see its 

definition in Table 3), which are least substitutable. Lev2 further divides passenger vehicles into 

Two Seaters, Prestige Cars, Standard Cars, Prestige SUVs, Standard SUVs, and MiniVans, 

acknowledging increasing substitutability among these alternatives (e.g. Standard SUVs and 

MiniVans, which are both passenger vehicles, are closer substitutes than Standard SUVs and 

Small Pickup Trucks, because Small Pickup Trucks are cargo vehicles). Cargo vehicles are 

divided into Pickup Trucks and Vans. Lev3 continue dividing some nodes in lev2 by vehicle size 

or prestige/non-prestige. 

The literature provides evidence that support our definition of nesting structure. A no-buy 

alternative is often included in previous studies (e.g., Berkovec, 1985; Berry, 1994; Berry et al., 

1995; Goldberg, 1995; NERA, 2009). It is very common to segment vehicle market by vehicle 

size, functionality, and prestige/non-prestige (e.g. Lave and Train, 1979; Berkovec and Rust, 

1985; Berkovec, 1985; Goldberg, 1995; Kleit, 2004; NERA, 2009). For example, Kleit (2004) 

classifies vehicles into small car, midsize car, large car, sports car, luxury car, small truck, large

truck, small SUV, large SUV, minivan, and van, which is consistent with our class definition 

(Table 3). Moreover, our structure has advantages over other structures in the literature: 

(1) It models vehicle market at a high level of detail, which enables the CVCM to potentially 

simulate the full range of sales mix shifts. The structure includes 5 levels, and choice 

alternatives are vehicle configurations (on the order of 1000), while the literature studies 

typically include two or three levels, and choice alternatives are vehicle size classes or 

makes/models (on the order of 200); 

(2) The passenger and cargo vehicle distinction in Lev1 is fully compatible with EPA emissions 

standards’ compliance categories for cars and trucks; 

(3) Our structure has a more thorough treatment of prestige vehicles in consideration that they 

have different price sensitivities from non-prestige vehicles. In addition to grouping prestige 

two seaters, cars and SUVs into their own nests in Lev2 and Lev3, the structure also groups 

ultra-prestige vehicles into a nest in Lev1. The special treatment of ultra-prestige vehicles is 

to recognize that these vehicles have very distinct consumer demand and thus are hardly 

ever substitutes for other inexpensive vehicles. Technically speaking, positioning ultraprestige vehicle nest in Lev1 allows us to assign a small price coefficient to these vehicles. 

The structure in Figure 1 is implemented in the CVCM by default. Future versions of the CVCM 

could support user-defined structure. Alternative structures may have impacts on sales 

predictions. Sales in the level of vehicle configurations will be most sensitive to the structure 

change. The degree of sensitivity diminishes as the prediction is targeted at more aggregate 

levels. 
23

Table 3 Vehicle Class Definition in the CVCM 

CVCM Class

No. of 

Configurations1 Corresponding EPA Class

1. Prestige2 Two-Seaters 27 Two Seaters

2. Prestige Subcompact Cars 49 Subcompact Cars, Minicompact Cars

3. Prestige Compact Cars and Small Station 

Wagons 71 Compact cars, Small Station Wagons

4. Prestige Midsize Cars and Station Wagons 66 Midsize Cars, Midsize Station Wagons

5. Prestige Large Cars 17 Large Cars

6. Two-Seater 26 Two Seaters

7. Subcompact Cars 58 Subcompact Cars, Minicompact Cars

8. Compact Cars and Small Station Wagons 82 Compact Cars, Small Station Wagons

9. Midsize Cars and Station Wagons 100 Midsize Cars, Midsize Station Wagons

10. Large Cars 29 Large Cars

11. Prestige SUVs 109 SUVs

12. Small3 SUVs 17 SUVs

13. Midsize SUVs 72 SUVs

14. large SUVs 137 SUVs

15. MiniVans 19 MiniVans

16. Cargo/Large Passenger Vans 42 Cargo Vans, Passenger Vans

17. Small Pickup Trucks 49 Small Pickup Trucks

18. Standard Pickup Trucks 67 Standard Pickup Trucks

19. Ultra Prestige Vehicles393 See the definition (note 4) below

Notes: 

(1) Number of configurations is the number of configurations which a CVCM class contains. It is not 

an attribute of the model itself, but specific to the vehicle data base to which the model is 

calibrated: a configuration is a record in the data base and a CVCM class consists of multiple 

records. 

(2) Prestige and non-prestige classes are defined by vehicle price: the prestige are vehicles whose 

prices are higher than or equal to unweighted average price in the corresponding EPA class, and 

vice versa for non-prestige vehicles; these calculations are done after ultra-prestige vehicles (see 

below) are put in a separate nest. E.g., Prestige Two-Seater class is the set of relatively expensive

vehicle configurations in EPA class of two seaters with prices higher than or equal to the 

unweighted average price of EPA two seaters. 

(3) Non-prestige SUVs are divided into small, midsize and large SUVs by vehicle’s footprint (small: 

footprint <43; midsize: 43<=footprint<46; large: footprint>=46) 

(4) Ultra Prestige class is defined as the set of vehicles whose prices are higher than or equal to 

$75,000.

 
24

3.2 EQUATIONS 

The CVCM includes a series of equations to define or calculate vehicle utilities, to calculate 

market share and sales of each vehicle configuration, and to estimate consumer surplus change 

brought by the installation of fuel economy technologies. 

3.2.1 Prelude 

We start from a review of Multinomial Logit (MNL) equations. The representative component of 

the utility expression for an alternative is defined in terms of four parts – the attributes k

x , 

attribute coefficients βk, alternative specific constant α j, and scale parameter μ . With the 

assumption that the variance of unobserved factors is distributed extreme value with variance 

2

2

6

π

μ

 (Train, 2009), the utility of alternative j for individual n is 

,

r

nj nj nj j r jr nj j p jr nj j p j nj

r r p

U V x x G βε α β ε α β ε α β ε

β

= + = + + = + + = + + ∑ ∑ (15) 

2

2 Var( ) , 2

6

nj

π

ε σ

μ

= = (16) 

where the sum G j

represents a “generalized cost” (Greene, 2001) for alternative j, β pis the 

coefficient of vehicle price attribute and the scale parameter μ is proportional to the inverse of 

the standard deviation of the error term. The choice probability of alternative j is 

exp ( ) exp exp ( )

.

exp exp ( ) exp ( )

j r jr

nj j p j r

nj

ni i r ir i p i

i i r i

x

V G

P

V x G

μ α β

μ μ α β

μ μ α β μ α β

+

+

= = =

+ +

∑

∑ ∑ ∑ ∑ (17) 

Note that the scale parameter μ and coefficients α jand β pare not separately identified and only 

the product of them can be estimated (Train, 2009). Thus in the CVCM, utility and choice 

probabilities have been expressed as 

, U A BG j j j j = + + ε (18) 

exp ( )

.

exp ( )

j j

j

i j

i

A BG

P

A BG

+

=

∑ +

 (19) 

with Aj j = μα and B = β μp

. Coefficient B is called generalized cost coefficient since it reflects 

the derivative of utility with respect to price or generalized cost. Generalized cost coefficient is 
25

proportional to scale parameter and thus inversely proportional to the standard deviation of error 

terms. Subscript n for individuals is omitted since the CVCM models the demand of a 

representative consumer. 

3.2.2 Two-level CVCM Equations 

The CVCMNMNL equations are first introduced in a simplified context with a two-level 

(vehicle configurations and vehicle classes) nested tree. Then full equations will be detailed in 

the next section. 

The CVCM assumes that fuel economy and vehicle price are the only factors changing between 

model runs, and other attributes (e.g. performance and size) remain constant. This assumption is 

consistent with the current version of OMEGA which only predicts changes in fuel economy and 

vehicle prices. Other attributes can be included if the value of the attributes can be accurately 

quantified. The average value of unmeasured vehicle attributes is represented by an alternativespecific constant term. The constant for each alternative is calibrated to match baseline sales data. 

The utility7

 for vehicle j in class k is 

( ) , U A B G A B C FS jk jk k j jk jk k jk jk jk = + + = + − + ε ε (20) 

where 

Ajk: constant term for vehicle jin class k,

Bk: generalized cost coefficient parameter for vehicles in class k, 

Cjk : incremental cost for improving fuel economy of vehicle j, and 

FS jk : the amount of fuel savings from improved fuel economy, valued by consumers when 

making purchase decisions. 

The utility function for the class k is 

ln exp ln exp[ ( )] root root

k k jk k jk k jk jk

k j k j k k

B B

U A U A A B C FS

B B ∈ ∈

= + = + + − ∑ ∑ (21) 

where Ak is constant term representing attributes shared by all alternatives in class k and Broot is 

the generalized cost coefficient for vehicle classes. Note that the log-sum term ln exp jk

j k

U

∈

∑ is 

often referred to as the “inclusive value” in the literature (e.g. Train, 2009). Choice probability 

for alternative j is 

P P P j j k k = |

, (22) 

 

7As seen in the appendix A, equation (20) only represents a component of the total utility that is unique to vehicle j. 

The utility component common to all vehicles in one class is captured by a class specific constant term. 
26

with 

|

' '

'

exp( )

exp( )

j k j

j k

j k j

j k

A B G

P

A B G

∈

+

=

∑ +

 (23) 

and 

' '

' ' '

exp[ ln exp( )]

.

exp[ ln exp( )]

root

k j k j

k j k

k

root

k j k j

k j k k

B

A A B G

B

P

B

A A B G

B

∈

∈

+ +

=

+ +

∑

∑ ∑

 (24) 

where Pj k|is the conditional probability of choosing alternative j given that an alternative in class 

k is chosen, and Pkis the marginal probability of choosing an alternative in class k. Appendix A 

will show the equivalence of the NMNL specification here to more general formulations in the 

literature. 

3.2.3 Full Scale CVCM Equations 

We could list out NMNL equations for all the five levels. But a simpler alternative is to define 

utilities and calculate choice probabilities recursively based on the notations in Daly (2001). We 

reproduce the notation here for convenience: 

• The tree function t(c) is used to define the nested logit structure: If c is a node 

in the tree, t(c) denotes the unique parent node at the higher level to which c is 

attached. For instance, Passenger Vehicle node is the parent of Standard Car 

node in Figure 1. 

• The set ALL(c) denotes the set of nodes consisting of c and all its ancestors: 

ALL(c) = {c, t(c), t(t(c)), ..., k| t(k) = root} 

• Each node c can be considered an “alternative” in its own right. Nodes in the 

bottom level are “elemental alternatives”, which are vehicle configurations. 

Nodes higher than the bottom level are viewed as “composite alternatives” that 

include all the elementary alternatives below it. 

•

Utilities of nodes are then defined by 

U A B G j j t j j = + ( ) (25) 

( )

( )

( )

ln exp t c

c

c c a c t c

c t a c

B

U A U A B U

B =

= + = + ∑ (26) 

 
27

with 

( )

1

c ln exp . a

c t a c

U U

B =

= ∑ (27) 

Equation (25) defines utilities for elementary alternatives including all vehicle configurations 

and No-Buy alternative. Equation (26) is recursive, defining the utility of node c as the 

summation of constant term Ac and generalized cost coefficient ( ( ) / B B t c c ) weighted log-sum 

term. The log-sum term is calculated over all the child nodes of c, where Ua is the utility of a 

child node a and again its utility can be expressed by equation (26). 

In particular, the utility for root node (overall composite utility for the choice set) is 

( )

1 1 ln exp ln(exp exp ) root a Buy NoBuy

root root t a root

U U U U

B B =

= = + ∑ (28) 

with 

. U A NoBuy NoBuy = (29) 

The utility function in equation (28) can be used to measure the consumer surplus change, 

consistent with Small and Rosen (1981): 

1 0

( ) ( )

1

(ln exp ln exp ) a a

root t a root t a root

CS U U

B = =

∆ = − − ∑ ∑ (30) 

where the superscripts 0 and 1 refer to before and after the change, and - Broot is marginal utility 

of income. 

The choice probability of each alternative is found by solving the following equation for Pc: 

( ) ( ) ( )

ln { ln exp }. c a b

a ALL c t b t a

P U U

∈ =

= − ∑ ∑ (31) 

Specifically, the choice probabilities of bottom level elementary alternatives are calculated as the 

product of a series of probabilities: 

| ( ) | ( ) | ... , P P P P j j c j c t c Buy root = (32) 

with 

| ( )

( ) ( )

exp

exp

c

c t c

b

t b t c

U

P

U

=

=

∑

, (33) 
28

where Pc t c | ( ) is conditional probability of choosing node c given its parent node t(c) is chosen. In 

the CVCM, the market share of a vehicle segment is equivalent to the probability of choosing the 

corresponding node. Vehicle sales then equal the product of market size and market share: 

, N M S M P c c c = = (34) 

where Nc is sales for the vehicle segment represented by node c, Sc is corresponding market share 

and M is market size, estimated by number of households. 

The key input parameters for these equations include constant terms and generalized cost 

coefficients at each level of the nesting structure, change in vehicle price due to the installation 

of fuel economy technologies, and the value of fuel economy improvement perceived by 

consumers. The derivation of constant terms and generalized cost coefficients will be described 

in Section 3.4 on model calibration. Vehicle price change is assumed to be equal to increased 

vehicle cost, a direct output of the OMEGA. The assumption and calculation of consumer value 

of fuel economy will be presented in the next section. 

3.3 VALUE OF FUEL ECONOMY 

How consumers value fuel economy improvements has very significant implications for the costs 

and benefits of fuel economy and emissions policies. The accuracy of consumer choice models 

depends much on how close the assumption of value of fuel economy resembles the reality. 

However the literature has not achieved a consensus on this subject. On one hand, economically 

rational consumers would measure the value of fuel economy by the expected discounted present 

value of fuel saved over the full life of the vehicle. On the other hand, there is evidence that very

few consumers actually make such quantitative assessments (Turrentine and Kurani, 2007). 

Greene et al. (2009) show that typical consumer loss aversion combined with the uncertainty of 

future fuel savings could lead to a significant undervaluing of future fuel savings relative to the 

expected present value. Greene (2010) concludes that econometric studies are nearly evenly 

divided about whether car buyers value fuel savings in accord with rational economic principles 

or significantly undervalue future fuel. 

Reflecting this controversy, the National Research Council (2002) fuel economy study 

considered two alternative methods of estimating fuel savings valued by consumers, full lifetime 

discounted fuel savings and a 3-year simple payback. The OMEGA has calculated fuel savings 

as the payback from the first 5 years with 3% discount. In order to be consistent with the 

OMEGA, the CVCM implemented the same calculation method by default. However, users can 

always change the parameters (r and L in equation(35)) in the input file to reflect their own 

assumptions on fuel savings calculation. 

Denote scenario 0 as the baseline scenario, with fuel economy at an initial value; denote scenario 

1 as the policy scenario, where fuel economy changes over time in response to fuel economy and 

emissions policies. Define considered Fuel savings as fuel saved that the consumer takes into 

account in the vehicle purchase decision in policy scenario relative to baseline scenario: 
29

0 1

1

1 1 1 ( ) ( ) ( )[ ]

(1 ) ( ) ( )

t L

i t

t i i

FS t P M t

r MPG t MPG t τ

τ

τ τ

η η

+

−

= +

= − −

+

∑ (35) 

where 

FSi(t): considered fuel savings of model year t vehicle i relative to its baseline configuration 

P(τ): price of fuel in year τ

M(τ − t ): annual miles traveled for a vehicle with age of τ − t

r: consumer discount rate 

η : OnRoad discount factor that discounts fuel economy (MPG) in order to reflect realworld driving conditions 

L: assumed payback period, in years. 

3.4 CALIBRATION 

Generalized cost coefficients and alternative specific constant terms are key input parameters to 

the CVCM. Generalized cost coefficients can be directly assigned, as in NERA (2009), or can be 

derived from other measures, e.g. price elasticities, as in this CVCM. Constant terms represent 

baseline utilities before any changes to vehicles. It is necessary to calibrate constant terms such 

that the CVCM prediction replicates market shares in the baseline scenario. 

3.4.1 Generalized Cost Coefficient Determination 

3.4.1.1 Methods 

Generalized cost coefficients in the CVCM have been determined based on multiple 

relationships and rules. Firstly generalized cost coefficients can be estimated from price 

elasticities according to the following relationship: 

(1 ), j j c j η = − p B S (36) 

(1 ) (1 )

j c

c

j j c c

B

p S p S

η η

= ≈

− −

 (37) 

where ηjis the own-price elasticity of demand for alternative j, j

p is the price of j, jS is j’s 

conditional market share given nest c is chosen, Bcis the generalized cost coefficient for 

alternatives in nest c, c

p is average price for alternatives in nest c , 

c

S is average conditional 

market share, and ηcis a representative value of s η j. Equation (36) is derived from the definition 

of elasticities and logit model equations (for further details, please refer to Train, 2009). Price 

elasticities can be chosen based on an evaluation of values found in the literature. 

Secondly, theoretical requirement of NMNL on generalized cost coefficients provides useful 

information for determining generalized cost coefficients. The NMNL theory requires that the 

absolute value of generalized cost coefficients must increase as one goes down to the bottom 
30

(vehicle configurations level) for the NMNL model to be consistent with utility maximization 

(see Appendix A): 

( ) ( ( )) | | | | | | ... | |, B B B B c t c t t c root ≥ ≥ ≥ ≥ (38) 

where Bt c( ) is the generalized cost coefficient associated with the parent node of c. Thus 

generalized cost coefficients at bottom level provide upper bounds (in terms of absolute value) 

and generalized cost coefficient at the root node (choice to buy a new vehicle or not) provides a 

lower bound for all other generalized cost coefficients at intermediate nodes. 

Thirdly, generalized cost coefficient of a nest has certain relationship with the price of that nest. 

We know that generalized cost coefficient is inversely proportional to the standard deviation of 

unobserved attributes in the nest (Appendix A). Prestige vehicle classes or nests may have large 

variance in unobserved attributes since consumers value these attributes very differently. Thus 

generalized cost coefficients of prestige vehicle classes or nests are lower in absolute value than 

those of non-prestige vehicle nests. This is consistent with the finding of Goldberg (1995) of 

lower generalized cost coefficients for higher-price market segments. 

We further extend this relationship with evidence from empirical studies. Disaggregate vehicle 

type choice models (e.g. Train and Winston, 2007) typically include in the utility function the 

ratio of vehicle price ( j

p ) and household income (Yn

): 

... ... .

j

jn jn j jn

n n

p

U p

Y Y

β

= + + = + + β ε ε (39) 

So generalized cost coefficient ( / β Yn

here) is inversely proportional to income. Assuming that 

income elasticity of expenditure is 1 (i.e., expenditure on vehicle purchase is approximately 

proportional to income), we conclude that generalized cost coefficients are approximately 

inversely proportional to vehicle purchase expenditure and, roughly speaking, vehicle price.8

That is, 

'

'

,

c c

c c

B p

B p

≈ (40) 

where c and c’ represent two nests, and B and p are generalized cost coefficient and average 

price respectively. The CVCM models the choice of a representative consumer and cannot 

directly incorporate income difference at the household level. Equation (40) can act as a proxy to 

represent price sensitivity variation due to household income difference. 

3.4.1.2 Calculation 

The calculation of generalized cost coefficients according to Equation (37) requires the input of 

price elasticities. Table 4 4 has summarized elasticity values from relevant literature that study 

new vehicle demand and report elasticities explicitly. Although these literature elasticities are 

valuable, it is difficult to directly use them in the CVCM due to the following reasons: (1) 

literature studies and the CVCM have different nesting structures,9 and (2) elasticities could be 

 

8

 Our intention is not to derive a definitive relationship, but to obtain a rule of thumb from empirical observations, 

which would be useful to generalized cost coefficient calibration. 

9

 Thus the categories presented in Table 4 do not correspond to the categories used in the nesting structure of the 

CVCM, but instead reflect the categories used in the cited studies. 
31

quite different from one study to another depending on dataset and model assumptions. In view 

of these difficulties, one shall cautiously utilize these literature elasticities and also consider 

other constraints (equations (38) and (40)) to determine generalized cost coefficients. In the 

following sections, we will detail how generalized cost coefficients are calculated at each level 

of Table 5 by integrating all available information. Note that the choice levels in Table 4 do not 

exactly match those in Table 5. Roughly speaking, “Choice to Buy a New Vehicle or Not” in 

Table 4 corresponds to Level 0 of Table 5; “Choice of Market Segment” in Table 4 corresponds 

to Level 3 of Table 5; “Choice of Configurations” in Table 4 corresponds to Level 4 in Table 5. 

“Choice of Make/Model” in Table 4 has no direct corresponding level in Table 5 and is an 

intermediate level between level 3 and 4 of Table 5. 

The overall price elasticity of automobile demand is set at -0.8 (Lev0 of Table 5), consistent with 

McCarthy (1996) and Levinsohn (1988). Following equation (37), the generalized cost 

coefficient for the buy/no buy decision is calculated and the value (-3.39E-05) serves as a lower 

bound (in absolute value) for all generalized cost coefficients in Table 5. 

Not many studies report elasticities at vehicle configuration level (Level 4 of Table 5). So we 

first look at the make/model level, whose elasticities values are lower bounds10 (in absolute value) 

of configuration level elasticities. Table 4 shows the average elasticity for choices among all 

individual makes and models is in the range of -2.3 to -4 (see ”Choice of Make/Model” Section 

and Row “Average elasticity” in Table 4). Elasticities for choices among makes and models in 

each market segment vary, with the range of –3.3 to -4.7 for small, medium and large size 

segments, -1.2 to -3.7 for luxury vehicles and -1.2 to -4.2 for sport vehicles. Based on these 

estimates, we assume that price elasticities at make/model level are around -4 for non-luxury cars 

(-4 is about the central value of the literature estimates) and around -2 for luxury and sport cars 

(-2 is about the central estimate). Generalized cost coefficients (usually but not always ranked in 

the same way as elasticities) at vehicle configuration level (Level 4 of Table 5) shall be larger in 

absolute value than at make/model level. Therefore the representative value of elasticities is set 

at -5.0 at vehicle configuration level for non-prestige11 cars (classes 6, 7, 8, 9, and 10 of Level 4 

in Table 5) and -3.5 for prestige cars and two seaters (classes 1, 2, 3, 4, and 5 of Level 4 in Table

5. These values are within the range of literature estimates in Table 4 (“Choice of Configuration” 

section and studies of Berry et al., 1995 and EFTEC, 2008). Generalized cost coefficients for 

classes 1-10 are calculated according to equation (37) given elasticities are known. We don’t 

have sufficient information to choose elasticities for other classes in Level 4 of Table 5 and will 

rely on equation (40) to calculate generalized cost coefficients. We select class 10 as the base 

class for non-prestige vehicles. Generalized cost coefficients of classes 12-18 are derived from 

class 10 generalized cost coefficient. Similarly for prestige vehicles, we select class 5 as the base 

class. Generalized cost coefficients of classes 11 and 19 are derived from class 5 generalized cost 

coefficient. Generalized cost coefficients in Level 4 serve as upper bounds (in absolute value) 

for other generalized cost coefficients in Table 5.

For elasticities at the vehicle class level (Level 3 of Table 5 and “Choice of Market Segment” of 

Table 4), Table 4 summarizes that own price elasticity is around -1.8 to -2.8 for small size 

 

10 Exactly speaking, generalized cost coefficients for choices among makes and models are lower bounds of 

generalized cost coefficients for choices among configurations. This relationship is approximately true for elasticities. 

11Prestige cars in Table 3 have the same definition as luxury cars in Table 4. 
32

segment, -1.3 to -3.5 for medium size segment, and -2.8 to -4.5 for large size segment. 

According to this observation, the representative value of price elasticities for choice among 

vehicle classes within Standard Car type is set at -3 (Row “Standard Car” in Level 3 of Table 5). 

Generalized cost coefficient is calculated according to equation (37). For luxury and sport cars, 

elasticities are around -1.7 to -3.5 in Table 4. Thus the representative value of price elasticities 

for choice among luxury/sport vehicle classes was initially set at -2.5 (Rows “Two Seater” and 

“Prestige Car” in Level 3 of Table 5), which is about the mean of literature estimates. However, 

calculated generalized cost coefficients based on these elasticities are larger in absolute values 

than their upper bounds and hence violate the theoretical requirement of NMNL (equation(38)). 

So price elasticities are adjusted to be -2.2 for Row “Prestige Car” and -1.3 for Row “Two Seater” 

so that12 calculated generalized cost coefficients satisfy the constraint in (38). Again we are not 

trying to choose elasticity values for Prestige SUV, Standard SUV, and Pickup of Lev3 in Table 

5. Instead, we selected Standard Car of Level 3 as the base. Generalized cost coefficients of 

Standard SUV and Pickup of Level 3 are derived from Standard Car generalized cost coefficient 

according to equation (40). 

For Levels 2 and 1 of Table 5, we don’t find relevant elasticity estimates in the literature. Thus 

the calibration of generalized cost coefficients is based on equations (38) and (40). First for 

choice of vehicle type within passenger category (Level 2-Passenger in Table 5), generalized 

cost coefficient is chosen to be -5.23e-5, which is the mean of its upper bound (Lev3-Two Seater: 

-7.08e-5) and lower bound (Lev0-RootNode: -3.38e-5). Then applying equation (40), generalized 

cost coefficient for choice of vehicle type within Cargo category (Lev2-Cargo) is calculated as -

5.23e-5. Since there are no vehicle types within Ultra Prestige category, generalized cost 

coefficient of Lev2-Ultra Prestige is copied from Lev4-Ultra Prestige. For Lev1, generalized cost 

coefficient is simply set as the mean of its upper and lower bounds (-3.92e-5 and -3.38e-5 

respectively). 

So far all generalized cost coefficients are obtained from equation (37) or from equation (40) in 

the case that elasticities are not available, with equation (38) providing upper and lower bounds. 

On the other hand, unknown elasticities can be calculated once generalized cost coefficients are 

obtained. For convenience of implementing CVCM in C#, we simply provide C# program with 

all price elasticities in an input file (with some of the price elasticities back-calculated from the 

generalized cost coefficients as described above) and calculate all generalized cost coefficients 

using equation (37). 

 

 

12 A range of elasticities satisfy this constraint. We gradually increase the initial elasticity value of -3 by 0.1. The final 

value of -2.2 (-1.3 for the case of Two Seater) in the table is the first value in this process that satisfies the constraint. 
33

Table 4 Own Price Elasticities of New Vehicle Demand in the Literature 

 

Own Price Elasticity of Demand Values Used in Calibration

Choice of Configuration

Small Berry et al. (1995): -6.4 for Mazda 323; Eftec (2008): -4.5 -5

Midsize Berry et al. (1995): -4.8 for Nissan Maxima; Eftec (2008): -5.4 -5

Large Berry et al. (1995): -4.8 for Honda Accord; Eftec (2008): -3.6 -5

Luxury Berry et al. (1995): -3.1 for Lexus LS400; Eftec (2008): -4.0 -3.5

Sport Eftec (2008): -1.6 -3.5 for two seaters

Choice of Make/Model Average 

elasticity

Bordely (1993):-3.6; Goldberg (1995): -3.3; Goldberg (1998):-3.1;

Brownstone et al. (2000):-3; Train and Winston (2007):-2.3; NERA Small Bordley (1993): -3.4; Goldberg (1995): -3.5 -4

Midsize Bordley (1993): -3.3; Goldberg (1995): -4.6; Goldberg (1996,1998):-4 -4

Large Bordley (1993): -3.8; Goldberg (1995): -4.7; Goldberg (1996,1998):-4 -4

Luxury Bordley (1993): -3.7; Goldberg (1995): -2; Goldberg(1996):-1.2; -2

Sport Bordley (1993): -4.2; Goldberg (1995): -1.4; Goldberg(1996,1998):-1.2 -2 for two seaters

Truck Goldberg (1995): -3.16

Van Goldberg (1995): -4.56

Choice of Market segment Small Bordley (1993):-1.9; Kleit (2002):-2.8; Cambridge (2008): -1.8 -3

Midsize Bordley (1993):-2.3; Kleit (2002):-3.5; Cambridge (2008): -1.3 -3

Large Bordley (1993):-3; Kleit (2002):-4.5; Cambridge (2008): -2.8 -3

Luxury Bordley (1993):-2.4; Kleit (2002):-1.7; Cambridge (2008): -3.5 -2.2

Sport Bordley (1993):-3.4; Kleit (2002):-2.3; Cambridge (2008): -1.8 -1.3 for two seaters

Truck Kleit (2002):-3 for small truck, -1.5 for large truck

SUV Kleit (2002):-3 for small suv, -2 for large suv

Van Kleit (2002):-2.4

Choice to Buy a New Veh or 

Not

ranged from -0.8 to -1

Levinsohn(1988), Kleit (1990), McCarthy (1996,1998), Goldberg (1998)

-0.8

Small Berry et al. (1995): -6.4 for Mazda 323; Eftec (2008): -4.5 -5

Midsize Berry et al. (1995): -4.8 for Nissan Maxima; Eftec (2008): -5.4 -5

Large Berry et al. (1995): -4.8 for Honda Accord; Eftec (2008): -3.6 -5

Luxury Berry et al. (1995): -3.1 for Lexus LS400; Eftec (2008): -4.0 -3.5

Sport Eftec (2008): -1.6 -3.5 for two seaters
34

Table 5 Generalized Cost Coefficient Calibration 

LEVEL 4 Choice of Make, Model, Engine Transmission C onfiguration within a Class

Class Name Sales Price Share No. MembersAve. Share1

Elasticity Slope

1 Prestige Two-Seater 79692 $50,888 0.47% 27 3.7% -3.5 -7.14E-05

2 Prestige Subcompact 276351 $41,808 1.63% 49 2.0% -3.5 -8.55E-05

3 Prestige C ompact and Small Station Wagon536024 $34,369 3.16% 71 1.4% -3.5 -1.03E-04

4 Prestige Midsize C ar and Station Wagon 727577 $42,988 4.29% 66 1.5% -3.5 -8.27E-05

5 Prestige Large 113968 $47,762 0.67% 17 5.9% -3.5 -7.79E-05

6 Two-Seater 112099 $26,656 0.66% 26 3.8% -3.5 -1.37E-04

7 Subcompact 1608947 $18,869 9.48% 58 1.7% -5.0 -2.70E-04

8 C ompact and Small Station Wagon 2392457 $17,901 14.10% 82 1.2% -5.0 -2.83E-04

9 Midsize C ar and Station Wagon 3180971 $21,132 18.75% 100 1.0% -5.0 -2.39E-04

10 Large C ar 752846 $24,217 4.44% 29 3.4% -5.0 -2.14E-04

11 Prestige SUV 1011890 $46,765 5.96% 109 0.9% -3.7 -7.95E-05

12 Small SUV 167691 $18,591 0.99% 17 5.9% -4.9 -2.79E-04

13 Midsize SUV 1082846 $24,133 6.38% 72 1.4% -5.1 -2.15E-04

14 Large SUV 2485225 $29,134 14.65% 137 0.7% -5.1 -1.78E-04

15 Minivan 801143 $28,413 4.72% 19 5.3% -4.9 -1.82E-04

16 C argo / large passenger van 84530 $25,002 0.50% 42 2.4% -5.1 -2.07E-04

17 C argo Pickup Small 353636 $20,929 2.08% 49 2.0% -5.1 -2.47E-04

18 C argo Pickup Standard 984260 $28,444 5.80% 67 1.5% -5.1 -1.82E-04

19 Ultra Prestige 214002 $94,930 1.26% 93 1.1% -3.7 -3.92E-05

TOTAL2

16966155 $27,227 100.00% 1130

LEVEL 3 Choice Among 19 Vehicle Classes within Vehicle Type

Type Name Sales Price3Share No. MembersAve. Share Elasticity Slope

1 Two-Seater 191791 $36,725 1.13% 2 50.0% -1.3 -7.08E-05

2 Prestige C ar 1653920 $40,326 9.75% 4 25.0% -2.2 -7.27E-05

3 Standard C ar 7935221 $19,992 46.77% 4 25.0% -3.0 -2.00E-04

4 Prestige SUV 1011890 $46,765 5.96% 1 100.0% na -7.95E-05

5 Standard SUV 3735762 $27,211 22.02% 3 33.3% -2.7 -1.47E-04

6 Minivan 801143 $28,413 4.72% 1 100.0% na -1.82E-04

7 C argo Van 84530 $25,002 0.50% 1 100.0% na -2.07E-04

8 Pickup 1337896 $26,457 7.89% 2 50.0% -2.0 -1.51E-04

9 Ultra Prestige 214002 $94,930 1.26% 1 100.0% na -3.92E-05

TOTAL 16966155 $27,227 100.00% 18

Level 2 Choice of Vehicle Type within Passenger or Cargo C ategories

C ategory Name Sales Price Share No. MembersAve. Share Elasticity Slope

1 Passenger 15329727 $26,362 90.35% 6 16.7% -1.1 -5.23E-05

2 C argo 1422426 $26,371 8.38% 2 50.0% -0.7 -5.23E-05

3 Ultra Prestige 214002 $94,930 1.26% 1 100.0% na -3.92E-05

Level 1 Choice of Passenger,C argo or Ultra Prestige Vehicle

Name Sales Price Share No. MembersAve. Share Elasticity Slope

Buy a new vehicle 16966155 $27,227 100.00% 3 33.3% -0.7 -3.65E-05

Level 0 Choice to Buy a New Vehicle or Not

US HHs4

Price Buy Share Elasticity Slope

Root Node 129973385 $27,227 13.05% -0.8 -3.38E-05

Note:

1)"Ave. Share" is the average of conditional shares of members in a nest. It is approximated by 1 over number of members

2)"Total" operation is not applicable to "price" column, which is sales weighted average price.

3) "Price" here reflects sales weighted average price.

4) "US HHs" is numer of households in the U.S. in base year
35

3.4.2 Constant Term Calibration 

Given generalized cost coefficients, constant terms at each level of the nesting structure are 

calibrated to baseline sales data. Baseline market share and constants have the following 

relationship for any two vehicle configurations within the same vehicle class: 

0 0

|

0 0

|

0 0

, ,

ln ln , ,

ik

jk

A

i k i

A

j k j

ik jk i j

P S e

i j k

P S e

A A S S i j k

= = ∀ ∈

=> − = − ∀ ∈

 (41) 

where superscript 0 represents baseline scenario, 0 Pi k| and 0 Pj k|are conditional probabilities of 

choosing vehicle i and j given class k has been chosen, and 0

i

S and 0

j

S are baseline market share 

of vehicles i and j. If we normalize one of the constants, e.g., A1k, to be zero, then 

0 0

1

ln ln , . A S S i k ik ik k = − ∀ ∈ (42) 

Vehicle class level constants can be derived from the following equation: 

0 0

|

0 0

|

exp[ ln( )]

, ,

exp[ ln( )]

ik

il

h A

kh

k h k k i k

l h l h A

lh

l i l

B

A e

P S B k l nest h

P S B

A e

B

∈

∈

+

= = ∀ ∈

+

∑

∑

 (43) 

where 0 Pk h|

 and 0 Pl h|are conditional probabilities of choosing vehicle class k and l given nest h

has been chosen, 0

k

S and 0

l

S are base year market shares of vehicle classes k and l in the nest h, 

and Akh and Alh are class-specific constant terms. Normalizing the first class specific constant A1h

to be zero, we get 

1

1

1 1

ln( ) ln( ) ln ln , h h A A i ik

kh k

i class i k k

B B A e e S S k nest h

B B ∈ ∈

= − + − ∀ ∈ ∑ ∑ (44) 

Again, we can use Daly (2001) notations to write a general equation. Denote c as a composite 

alternative and t(c) as its parent. The following equation holds for any two composite alternatives 

in the same level of the nesting structure: 

0

0

( )

0 0

| ( ) ( )

0 0

( ) | ( )

( )

exp[ ln( )]

, , ,such that ( ) ( )

exp[ ln( )]

a

a

t c U

c

c t c c c t a c

b t b b t b U

b

b t a b

B

A e

P S B

c b t c t b

P S B

A e

B

=

=

+

= = ∀ =

+

∑

∑

 (45) 

where 0 Pc t c | ( ) and 0 Pb t b | ( ) are conditional probabilities of choosing alternatives c and b given their 

parent has been chosen, 0

c

S and 0

b

S are base year market shares of vehicle segments represented 

by c and b, and 0 U a

 is baseline utility for an alternative a, as described by the recursive equation 

(26) with initial condition of 

0

, U A j j j = ∀ (46) 

and 
36

0

0. U A NoBuy NoBuy = = (47) 

Constant terms can be solved from equation (45) given 0

c

S , 

0

b

S , and generalized cost coefficients 

are known. The constant for the alternative of “not buying a new vehicle” is assumed to be 0. 
37

4. IMPLEMENTATION AND USER GUIDE 

The CVCM has been implemented in C# at the editor environment of Visual Studio 2010. The 

C# code reads input, calibrates the model parameters including constant terms and generalized 

cost coefficients, calculates utilities, choice probabilities, sales, and consumer surplus, and 

finally writes output to an Excel file. The C# code is distributed as a Windows installation file 

and users can install the program on the destination computers with Windows operating 

systems.13 

4.1 USER INTERFACE 

User interface of the CVCM is straightforward. The File menu has mainly two items: “Output 

Files to...” and “Open”. The “Output Files to...” item specifies the folder of output files. The 

default output folder is CVCM installation folder\output. The “Open” item selects input file. 

Input files could be located in any folder. But by default, they are in CVCM installation 

folder\input. The CVCM installation program has copied example input files in the input folder. 

Each scenario has one input file, indicated by the file name. Select and open one input file to 

read in data. Then some of the data content will be displayed in the two tables of the user 

interface and users can check if the data are correctly read. The gray car button on the upper right 

corner will turn green. Users can click on the green button and run the program. After the run is 

finished, users can then select another input file to start another run. 

The CVCM takes input on model parameters, vehicle characteristics in the baseline scenario (e.g. 

price, sales and fuel economy) and fuel economy improvement and associated incremental 

vehicle price in the policy scenario, where fuel economy changes over time in response to fuel 

economy and emissions policies. It then outputs predictions on sales and consumer surplus in the 

policy scenario. 

All input and output are in Excel Files. As described in Section 4.1.2, the user can name the 

output file via the "GlobalParameter" sheet of the input file. 

4.1.1 Input 

A list of input data and data sources is as follows. 

• Vehicle database: detailed database at vehicle configuration level. It includes vehicle 

identification information (e.g. make, model, and engine size), price, baseline sales, and 

baseline fuel economy. 

• Predictions on fuel economy and incremental price at vehicle configuration level: key 

input data, commonly obtained from OMEGA output.

• Generalized cost coefficients and alternative-specific constants: model parameters. 

Generalized cost coefficients are derived from price elasticities and NMNL properties, 

 

13 The current version of the CVCM does not work on non-windows operations systems (e.g. Mac and UNIX).

 
38

given elasticities are known from the literature. Constant terms are calibrated from 

baseline sales data and generalized cost coefficients. 

• Market size: size of consumer market, which is typically approximated by the number 

of households. Household number projection for the United States is obtained from 

U.S. Census and American Community Survey. 

• Nesting structure: default nesting structure is built in the model. In the future, users 

may be able to specify their own structure. 

• Fuel prices: used to calculate fuel savings. Source: Annual Energy Outlook (AEO) 

2010 from Energy Information Administration (EIA). 

• Annual and lifetime driving mileage for a typical car or truck: used to calculate fuel 

cost and VMT weighted GHG emissions for manufacturers. Source: consistent with 

OMEGA assumptions. 

• Emission standards and vehicle footprint: used to check manufacturer compliance with 

GHG emissions standards. This information is optional and requires linking the CVCM 

outputs to OMEGA. This linkage may be made available in future releases of the 

CVCM. At the current time, this field can be left blank. 

Sample input files in the CVCM installation folder contain all the above information. Users 

should follow the format in these files to prepare their own input. An input file has 7 data sheets, 

listed as follows. The file also contains a sheet (“InputValidation”) to validate the input. Click on 

“Validation Data” button in this sheet and error messages will prompt out if the input in the data 

sheets is not in right data type or within appropriate range. If the inputs fail the validation test,

the implicit meaning is that the model nesting structure is not consistent with the parameters. An 

error message box will pop out and instruct users to check input files. Users are not able to run 

the model with invalid inputs. 

4.1.1.1 Vehicle 

Each row in this table contains attributes (see Table 6) of a vehicle configuration. CVCM 

classes are classified based on EPA classes, according to the relationship in Table 3. Users will 

need to provide data for the columns “predicted mpg” and “incremental price” based on 

OMEGA output or other sources. 

Table 6 Format of Vehicle Sheet 

4.1.1.2 Manufacturer 

It includes a list of manufacturer names, which must be consistent with column “manufacturer” 

in “vehicle” sheet. 

4.1.1.3 Logit 

This sheet lists price elasticities at each level of the nesting structure for the purpose of model 

calibration. Users can change the values of price elasticities, but not the nesting structure. 

4.1.1.4 GlobalParameter 

The structure of “GlobalParameter” sheet is as follows: 

veh id manufacturer nameplate model CVCM class EPA class fleet type fuel type footprint

baseline price baseline mpg baseline sales predicted mpg incremental price
39

Table 7 Structure of “GlobalParameter” Sheet 

Scenario name defines the name of the output file. Payback period and discount rate are 

parameters for calculating the value of fuel economy improvement perceived by consumers. 

“OnRoad Discount” is used in fuel cost calculation to discounts EPA fuel economy (MPG) test 

value, which is displayed in fuel economy window stickers and used in the CVCM, to better 

reflect fuel economy under real-world driving conditions. Market Size data are used to calculate 

sales and calibrate logit model constants at the level of Buy/No-Buy. 

4.1.1.5 Other Sheets 

“VehicleUse” and “Fuel” sheets include parameters for calculating fuel cost. In “VehicleUse” 

sheet, Annual driving mileage of a car (truck) at certain age equals the product of VMT and 

survival rate. “Fuel” sheet simply records fuel prices with year 1 as the 1st year after redesign 

year. 

“Target” sheet specifies footprint function parameters as in 2012-2016 EPA GHG emissions 

standards final rule (Table III.B.2–1 and Table III.B.2–2, page 25409 of EPA and NHTSA, 

2010). The redesign year in the example input files is 2016. Thus parameters in this sheet reflect 

2016 emissions standard. 

4.1.2 Output 

Each run will generate an output file, with its name defined by the user (at cell B2 of Global 

Parameter sheet of the Input file). An output file consists of two sheets: raw output and aggregate 

output. 

“Raw Output” sheet first repeats the input data for the convenience of reading. The model output 

includes sales, market share, revenue (sales times the sum of vehicle price and incremental price), 

net price change (incremental price less fuel savings), and sales changes relative to the baseline 

scenario at the level of vehicle configuration. 

“Aggregate Output” sheet outputs variables at more aggregate levels, including market-wide 

consumer surplus change, total sales, industry revenue, sales weighted average fuel economy and 

CO2 emissions; manufacturer level sales, sales weighted average fuel economy and CO2

emissions; sales at the level of passenger vehicle, cargo vehicle, or ultra-prestige14, and sales at 

each vehicle class. Note that the fleet average fuel economy is calculated as a harmonic mean: 

with gpmi as the gallons per mile for vehicle i, 

∑

=

i

i i

Sales gpm

TotalSales FleetAvgFuelEcon

*

Fleet average CO2 values are calculated two ways: first, sales weighted: 

 

14 Passenger vehicle, cargo vehicle, and ultra-prestige distinction corresponds to level 1 of the nested choice structure. 

Scenario Name Payback Period Discount rate OnRoad/Tested MPG Market Size
40

∑

∑=

i

i

i

i i

Sales

Sales CO

SalesWtdCO

* 2

2

Secondly, it is also calculated as VMT-weighted, with the VMT based on the full lifetime 

undiscounted VMT of the vehicle; VMT differs by whether a vehicle is classified as a car or a 

truck for regulatory purposes: 

i i

i i

i

i i

Sales VMT

Sales CO VMT

VMTSalesWtdCO ∑

∑=

*

* 2 *

2

4.2 INTERACTION WITH OMEGA 

The CVCM could interact with the OMEGA at different degree. At this stage of model 

development, they run as two separate programs and pass information via excel files. In the 

future, the CVCM can be fully integrated into the OMEGA as one program.

15 The framework of 

interaction is as follows: 

 Step 1: Run OMEGA model. 

 Step 2: Collect data from OMEGA output and prepare input file for the CVCM. 

 Step 3: Run CVCM 

3a: Calibrate CVCM using baseline sales data and price elasticities 

3b: Calculate sales, market share and consumer surplus 

 3c: Output 

 If the convergence criteria is met (i.e. emissions standards are complied), STOP here. 

 Otherwise, Go back to step 1 

 

 

15 One possible way is to program the CVCM as a dynamic link library (dll) file and call this dll from the OMEGA. 
41

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45

APPENDIX A: DERIVATION OF NESTED LOGIT MODEL EQUATIONS 

AND RELEVANT PROPERTIES 

The primary purpose of this appendix is to provide a general form of nested logit model 

equations and demonstrate CVCM equations as one specific instance of the general form. The 

secondary purpose is to derive conditions on structural parameters in nested logit models. 

Without loss of generality, we consider a two-level nesting structure for the convenience of 

discussion, which is the most common case in the literature. Formulations for two-level models 

can be extended to multi-level cases. 

The following formulation framework is consistent with William (1977), Daly and Zachary 

(1978) and Train (2009). Let j denote an elemental alternative in the nested tree and c the upper 

level composite alternative (or nest) to which j belongs. The utility of alternative j is 

| | | , U V V V V j c j c j c j c c j c = + + = + + + ε ε ε (A-1) 

where the observed component of utility is decomposed into two parts: a part labeled Vc

 that is 

constant for all alternatives within the nest c, and a part labeled Vj c|that varies over alternatives 

within the nest. The error term j

ε is also divided into two independent components of c

ε and j c|

ε . 

The following assumptions are made. Errors j c|

ε are identically and independently distributed 

(iid) Gumbel with scale parameter μc

. Errors c

ε are distributed such that total errors j

ε are 

distributed Gumbel with scale parameter μroot . 

We first see how the above definition and assumptions imply a relationship of scale parameters

μc

and μroot . The variance of total errors is the sum of variance of two error components: 

Var( )=Var( )+Var( ) j c j c|ε ε ε , (A-2) 

which, because the variance of the Gumbel distribution is 

2

6

π

μ

with μ as the scale parameter , can 

be expressed as 

2 2

=Var( )+ .

6 6 c

root c

π π

ε

μ μ

 (A-3) 

Since Var( )

c

ε is non-negative, the above implies the following structural condition (Williams, 

1977): 

. μ μ root c ≤ (A-4) 
46

Then we compare variations of choice probability expressions. Choice probability for alternative 

j (Carrasco and Ortuzar, 2002) is 

P P P j j c c = |

, (A-5) 

with 

|

|

'|

'

exp( )

exp( )

c j c

j c

c j c

j c

V

P

V

μ

μ

∈

=

∑

 (A-6) 

and 

|

' ' | ' ' '

' ' ' ' '

exp[ ln exp( )] exp[ ]

,

exp[ ln exp( )] exp[ ]

root root

root c c j c root c c

c j c c

c

root root

root c c j c root c c

c j c c c c

V V V I

P

V V V I

μ μ

μ μ μ

μ μ

μ μ

μ μ μ

μ μ

∈

∈

+ +

= =

+ +

∑

∑ ∑ ∑

 (A-7) 

where Pj c|

is the conditional probability of choosing alternative j given that an alternative in nest 

c is chosen, Pcis the marginal probability of choosing an alternative in nest c, and I is so called 

inclusive value of nest c. 

Choice probability form in (A-5) to (A-7) is consistent with the one in Train (2009) (equations 

(4.4) and (4.5) of chapter 4) if the scale parameter μroot is normalized to 1 and μcreplaced by

1

λ k

, the so-called log-sum coefficient. Next we wish to show that CVCM formulation is a 

specific instance of this general form. Assume utility function Vj c|

takes a specific form similar to 

the one in CVCM: 

V p j c j j | = + α β (A-8) 

whereα j

is alternative specific constant and p is vehicle price or generalized cost. Then choice 

probability can be expressed as 

|

' '

'

exp[ ( )]

exp[ ( )]

c j j

j c

c j j

j c

p

P

p

μ α β

μ α β

∈

+

=

∑ +

 (A-9) 

and 
47

' ' ' '

' ' '

exp[ ln exp( ( ))]

.

exp[ ln exp( ( ))]

root

root c c j j

c j c

c

root

root c c j j

c j c c

Y p

P

Y p

μ

μ μ α β

μ

μ

μ μ α β

μ

∈

∈

+ +

=

+ +

∑

∑ ∑

 (A-10) 

On the other hand, choice probability in the CVCM is 

|

' '

'

exp( )

exp( )

j c j

j c

j c j

j c

A B G

P

A B G

∈

+

=

∑ +

 (A-11) 

and 

' '

' ' '

exp[ ln exp( )]

.

exp[ ln exp( )]

root

c j c j

c j c

c

root

c j c j

c j c c

B

A A B G

B

P

B

A A B G

B

∈

∈

+ +

=

+ +

∑

∑ ∑

 (A-12) 

If one defines 

Aj c j = μ α (A-13) 

A Y c root c = μ (A-14) 

Bc c = βμ (A-15) 

Broot root = βμ (A-16) 

and 

,

j j p G= (A-18) 

then CVCM equations are equivalent to the general form in (A-9) to (A-10). Parameters Bc

and 

Broot are called generalized cost coefficients in the CVCM, since they reflect the derivative of 

utility with respect to price (or generalized cost). Equation (A-15) and (A-16) indicate that the 

absolute value of generalized cost coefficients are proportional to scale parameters and thus 

proportional to the inverse of standard deviation of random errors. 

 
48

 
49

APPENDIX B: MODEL SENSITIVITY ANALYSIS 

Among the model’s parameters, the most important ones are price elasticities and those defining 

how consumers value fuel savings from fuel economy improvement. In this section, we examine 

the sensitivity of model results to variation of price elasticities and consumers’ evaluation of fuel 

savings. 

The input data (baseline sales and predicted fuel economy) are from an OMEGA run which 

simulates the scenario of light duty vehicles meeting 2016 EPA greenhouse gas emissions 

standards. The OMEGA results were provided by EPA in November, 2011. Sensitivity analysis 

results are specific to the data, which is why we use a realistic OMEGA data set. Since OMEGA 

tends to (but does not precisely) equalize the marginal cost of installing fuel economy 

technologies across all vehicles, we expect the impact of fuel economy improvement on sales 

mix to be small. 

We will start by describing the distribution of price elasticities and then present sensitivity 

analysis results. 

B.1 THE DISTRIBUTION OF OWN PRICE ELASTICITIES 

Generalized cost coefficients are calibrated from assumed price elasticities (see Table 5). The 

elasticities represent the average elasticities of alternatives in a nest. Each vehicle has its own 

price elasticity depending on its price and market share as well as the generalized cost coefficient 

for its nest. Once the CVCM is calibrated, the actual elasticity of an alternative can be calculated 

based on equation (36). Alternatively, we can also get actual price elasticities through simulation. 

For example, the own price elasticity of a vehicle can be obtained by calculating the relative 

sales change of the vehicle in response to 1% change in its price. We calculated own price 

elasticities of 1130 vehicle configurations using equation (36). 16 The distribution of these 

individual elasticities is shown in Figure 2. 

 

16 Equation (36) is derived in the case of simple logit models. Thus elasticities calculated using this equation for nested logit 

models (e.g. CVCM here) are only an approximation to true values. However the error is very small. 
50

Figure 2 Distribution of Own Price Elasticities

Most individual vehicle price elasticities are in the range of -6 to -3. Only 10 vehicle 

configurations have price elasticities less than -8.0, as displayed in Table 8. Those vehicles are 

either ultra-prestige vehicles with very high prices or relatively expensive cars in their classes. 

Table 8 List of Vehicles with Very High Elasticities (in absolute value) 

manufacturer nameplate model vehicle class 

generalized 

cost 

coefficient elasticity price 

Daimler MERCEDES-BENZ SLR Ultra Prestige -0.000038 -18.9 497750 

BMW ROLLS-ROYCE PHANTOM Ultra Prestige -0.000038 -15.6 409000 

BMW ROLLS-ROYCE PHANTOM EWB Ultra Prestige -0.000038 -15.4 405000 

BMW ROLLS-ROYCE 

PHANTOM 

DROPHEAD 

COUPE Ultra Prestige -0.000038 -13.0 342000 

VOLKSWAGEN BENTLEY AZURE Ultra Prestige -0.000038 -12.6 332585 

VOLKSWAGEN BENTLEY ARNAGE RL Ultra Prestige -0.000038 -10.1 266585 

VOLKSWAGEN BENTLEY ARNAGE Ultra Prestige -0.000038 -8.5 224585 

Ford FORD MUSTANG Subcompact -0.000264 -8.3 31525 

TOYOTA LEXUS IS 250 Subcompact -0.000264 -8.3 31220

TOYOTA LEXUS IS 250 Subcompact -0.000264 -8.2 31220

Mitsubishi MITSUBISHI ECLIPSE SPYDER Subcompact -0.000264 -8.0 30224 

Mitsubishi MITSUBISHI ECLIPSE SPYDER Subcompact -0.000264 -8.0 30224 

Ford VOLVO V50 FWD Compact -0.000289 -8.0 27560 

Mazda MAZDA MAZDA RX-8 Subcompact -0.000264 -8.0 30108 

Table 9 provides additional descriptive statistics for elasticities of vehicle configurations within 

each class. In particular the medians of these elasticities are comparable to elasticity inputs 

shown in Table 5.

0

10

20

30

40

50

60

70

80

90

Frequency of Elasticities
51

Table 9 Descriptive Statistics of Elasticities 

Vehicle Classes 

No. of 

vehicles Min 

1st 

Quartile Median 

3rd 

Quartile Max 

Std. 

deviation 

Prestige Two-Seater 27 -5.5 -4.2 -3.6 -3.4 -3.2 0.6 

Prestige Subcompact 54 -6.3 -4.6 -3.6 -3.2 -2.8 0.9 

Prestige Compact and Small Station 

Wagon 71 -7.1 -4.3 -3.7 -3.4 -3.0 0.8 

Prestige Midsize Car and Station Wagon 58 -6.1 -4.5 -3.9 -3.2 -2.8 0.9 

Prestige Large 17 -5.4 -3.6 -3.5 -2.9 -2.7 0.8 

Two-Seater 26 -5.9 -5.5 -4.4 -3.8 -2.0 1.1 

Subcompact 76 -8.3 -7.0 -6.0 -5.1 -3.0 1.4 

Compact and Small Station Wagon 82 -8.0 -6.9 -5.4 -4.6 -3.8 1.3 

Midsize Car and Station Wagon 85 -7.8 -6.1 -5.2 -4.6 -3.3 1.2 

Large Car 29 -6.6 -5.9 -5.5 -4.9 -4.3 0.6 

Prestige SUV 108 -6.1 -4.1 -3.6 -3.2 -2.8 0.7 

Small SUV 17 -7.3 -6.1 -5.3 -5.0 -4.5 0.8 

Midsize SUV 78 -7.0 -5.7 -5.1 -4.7 -3.4 0.7 

Large SUV 132 -6.5 -6.0 -5.4 -4.7 -2.9 0.8 

Minivan 19 -6.5 -5.3 -4.6 -4.4 -3.7 0.8 

Cargo / large passenger van 42 -6.2 -5.8 -5.7 -5.4 -4.7 0.4 

Cargo Pickup Small 49 -6.7 -5.6 -4.9 -4.5 -3.8 0.8 

Cargo Pickup Standard 67 -7.8 -5.6 -5.3 -4.6 -3.8 0.9 

Ultra Prestige 93 -18.9 -4.8 -3.6 -3.2 -2.9 2.9 

B.2 THE DISTRIBUTION OF CROSS PRICE ELASTICITIES 

We have also obtained cross price elasticities at vehicle class level through simulation. The 

demand elasticity of class k with respect to the price of class l is calculated as the relative change 

in class k sales (total sales for all vehicles in class k) given 1% price change for all vehicles in 

class l. The cross elasticities are shown in Table 10, where own price elasticities are in bold text 

and large cross elasticities are in red. The values reported in Table 10 are comparable to those in 

Table 1 in the chapter of literature review. 
52

Table 10 Price Elasticities at Vehicle Class Level 

B.3 SENSITIVITY ANALYSIS 

Eight cases are defined for the sensitivity analysis (see Table 10). Case 1 Baseline is the case to 

which other cases are compared. Cases 2-4d assume vehicles are required to improve fuel 

economy and vehicle prices are increased consequently. The fuel economy improvements and 

price increases of individual vehicle configurations are the same for all policy cases 2-4d, as 

provided by the OMEGA output dataset. Case 2 Reference uses default CVCM assumptions. It 

assumes consumers value the first 5 years of fuel savings using an annual discount rate of 3%. 

The default elasticities in Table 5 were used to calibrate the model. Varying the length of 

payback period and the discount rate generates policy cases 3a and 3b, and varying the 

elasticities generates policy cases 4a-4d. We examine the impact of these different assumptions 

on consumer surplus change, fleet average MPG and total sales. 

 

Class Name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Prestige Two-Seater 1 -3.47 0.01 0.01 0.01 0.01 0.43 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00

Prestige Subcompact 2 0.02 -2.90 0.16 0.16 0.16 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.00 0.00 0.00 0.00

Prestige Compact and 

Small Station Wagon 3 0.02 0.24 -2.25 0.24 0.24 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01

Prestige Midsize Car 

and Station Wagon 4 0.04 0.40 0.40 -2.73 0.40 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.01 0.01 0.01 0.01

Prestige Large 5 0.01 0.08 0.08 0.08 -3.33 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00

Two-Seater 6 0.36 0.00 0.00 0.00 0.00 -1.51 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Subcompact 7 0.05 0.05 0.05 0.05 0.05 0.05 -3.10 0.67 0.67 0.67 0.05 0.05 0.05 0.05 0.05 0.01 0.01 0.01 0.01

Compact and Small 

Station Wagon 8 0.06 0.06 0.06 0.06 0.06 0.06 0.72 -2.72 0.72 0.72 0.06 0.06 0.06 0.06 0.06 0.02 0.02 0.02 0.02

Midsize Car and Station 

Wagon 9 0.09 0.09 0.09 0.09 0.09 0.09 1.19 1.19 -3.01 1.19 0.09 0.09 0.09 0.09 0.09 0.03 0.03 0.03 0.03

Large Car 10 0.05 0.05 0.05 0.05 0.05 0.05 0.59 0.59 0.59 -4.24 0.05 0.05 0.05 0.05 0.05 0.01 0.01 0.01 0.01

Prestige SUV 11 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 -2.38 0.06 0.06 0.06 0.06 0.02 0.02 0.02 0.02

Small SUV 12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -2.67 0.09 0.09 0.00 0.00 0.00 0.00 0.00

Midsize SUV 13 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 1.15 -2.45 1.15 0.06 0.02 0.02 0.02 0.02

Large SUV 14 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 1.48 1.48 -3.02 0.08 0.02 0.02 0.02 0.02

Minivan 15 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 -1.52 0.01 0.01 0.01 0.01

Cargo / large passenger 16 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.25 0.05 0.05 0.00

Cargo Pickup Small 17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 -2.71 0.39 0.00

Cargo Pickup Standard 18 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.31 2.58 -1.60 0.02

Ultra Prestige 19 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 -3.84
53

Table 11 Sensitivity Analysis Results 

Case 

ID Name 

Payback 

Period 

Disc. 

Rate Elasticities 

Avg. Veh. 

Net Value 

($) 

Consumer 

Surplus 

Change 

($/veh.) 

Fleet 

MPG 

Total 

Sales 

(millions) 

1 Baseline NA NA NA 0 0 27.35 16.65 

2 Reference 5 0.03 default 1284 1227 33.67 17.29 

3a 

Lower value of fuel 

savings 2 0 default 23 17 33.66 16.66 

3b 

Higher value of fuel 

savings 15 0.3 default 4054 3591 33.65 18.64 

4a 50% lower elasticities 5 0.03 default *0.5 1250 1222 33.7 16.96 

4b 25% lower elasticities 5 0.03 default *0.75 1267 1225 33.69 17.12 

4c 

25% higher 

elasticities 5 0.03 default *1.25 1301 1230 33.65 17.45 

4d 

50% higher 

elasticities 5 0.03 default *1.50 1318 1232 33.64 17.62 

The estimated consumer surplus change is highly sensitive to how consumers are assumed to 

value fuel savings from fuel economy improvements relative to the baseline case. The greater the 

amount of fuel savings that consumers consider when buying their vehicles, the larger the 

column “average vehicle net value” is. The vehicle net value is defined as the value of fuel 

savings taken into account minus vehicle price increase. Consumer surplus is largest in case 3b, 

where consumers are assumed to take into account fuel savings over the full expected lifetime of 

a vehicle. Price elasticities have much smaller impacts on consumer surplus. The higher the price 

elasticities, the larger the consumer surplus change is. 

The impacts on total sales follow the same pattern as consumer surplus change. Total sales are 

most sensitive to the value of fuel savings perceived by consumers, with largest sales in case 3b. 

Total sales also increase when demand is more price elastic because the OMEGA data imply 

large gains in net values. The assumed price elasticity of new vehicle demand is -0.8. 

Fleet average MPG is robust to all the variables varied in the sensitivity tests. In general, the 

differences are less than one tenth of a MPG among sensitivity test cases 2-4d. Fleet average 

MPG provided by OMEGA, if weighted by baseline sales, is 33.73, which is higher than all 

sensitivity test case fleet MPGs. This suggests the existence of a very small sales mix rebound 

effect: In the OMEGA data supplied, the greatest benefits of improved fuel economy (fuel 

savings minus vehicle price increase) tend to accrue to lower fuel economy vehicles. Thus, we 

see a sales mix shift towards lower fuel economy vehicles, as shown by Table 11. The table 

displays market shares in the baseline and reference cases for each baseline MPG decile. The 

share of lower fuel economy vehicles (decile 1 and 2) increases, while the share of higher fuel 

economy vehicles (decile 9 and 10) decreases. 

Table 12 Market Shares by MPG Decile 

MPG Decile 1 2 3 4 5 6 7 8 9 10 

Baseline 5.1% 20.4% 31.2% 24.3% 13.7% 2.7% 0.6% 0.0% 0.3% 1.6% 
54

Reference 5.8% 21.2% 30.4% 24.7% 13.4% 2.4% 0.6% 0.0% 0.2% 1.3% 

The rebound effect can be quantified using the following equation 

0

'

,

'

s

s MPG MPG

MPG MPG

φ

−

=

−

 (48) 

where 

S: index for a policy case 

MPG0

: fleet MPG in the baseline case 

MPGs: fleet MPG in the policy case (average OMEGA MPG weighted by sales in the policy 

case) 

MPG’: fleet average OMEGA MPG weighted by the baseline sales. 

If MPGs is smaller than MPG’, then there is a rebound effect. Table 12 indicates that the rebound 

effect is small, in the range of 1%, and higher price elasticities tend to magnify the rebound 

effect. 

Table 13 Rebound Effect 

Case 

ID Name Fleet MPG Rebound Effect 

2 Reference 33.67 0.9% 

3a

Lower value of fuel 

savings 33.66 1.1% 

3b

Higher value of fuel 

savings 33.65 1.3% 

4a 50% lower elasticities 33.70 0.5% 

4b 25% lower elasticities 33.69 0.6% 

4c 25% higher elasticities 33.65 1.3% 

4d 50% higher elasticities 33.64 1.4% 

In summary, the sensitivity analysis results suggest that fleet MPG is robust to assumptions 

about price elasticities and the value of fuel economy perceived by consumers. Consumer surplus 

and total sales are very sensitive to perceived value of fuel economy and not very sensitive to 

variation in price elasticities at lower levels in the nesting structure.