1
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model 1
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Estimating Car Price Elasticity
Using an Inverse Product
Differentiation Logit Model
Jinghai Huo, Rubal Dua, and
Prateek Bansal
December 2023
Doi: 10.30573/KS--2023-MP05
2
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
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This publication is also available in Arabic.
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3
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Since the seminal work of Berry, Levinsohn, and Pakes (1995), random coefcient logit (RCL) has
become the workhorse model for estimating demand elasticities in markets with differentiated
products using aggregated sales data. While the ability to represent exible substitution patterns
makes RCL a preferable model, its estimation is computationally challenging due to the numerical inversion
of the demand function. The recently proposed inverse product differentiation logit (IPDL) addresses
these computational challenges by directly specifying the inverse demand function and representing
exible substitution patterns through nonhierarchical product segmentation in multiple dimensions. Unlike
the two-stage simulation-based estimation of RCL, IPDL requires the estimation of a traditional linear
instrumental variable (IV) regression model. In theory, IPDL appears to be an attractive alternative to RCL,
but its potential has not yet been explored in empirical studies. We present the rst application of IPDL
in understanding the demand for passenger cars in China using provincial-level sales data. Our results
indicate that the elasticity estimates of IPDL and RCL are not signicantly different, i.e., that IPDL can
capture substitution patterns in a similar manner as can RCL. The estimation of IPDL takes less than a
second on a regular computer (i.e., it is approximately 500 times faster than RCL). Overall, the exibility
and computational efciency of IPDL makes it a workhorse model for demand estimation using market-level
aggregated sales data.
Key Points
4
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Estimating price elasticity in markets with
differentiated products is crucial for evaluating
the effect of innovations and economic
policies. For instance, elasticity estimates can be
instrumental in simulating the impact of purchase
price subsidies on consumer demand for electric
vehicles. Berry (1994) proposed a demand model
for differentiated products under an additive random
utility maximization (ARUM) framework, which has
become popular for three reasons. First, the model
can be estimated using widely available data on
product attributes and market-level sales (instead
of individual-level choices). Second, endogeneity in
market prices can be addressed. Third, the model
can handle many differentiated products. The
central idea of Berry’s model (1994) is to obtain the
mean utility levels (across consumers) by inverting
the market share function in the rst step and
subsequently estimating the relationship between
mean utility levels and product characteristics
(including price) in the second step. Endogeneity
in market prices is addressed through instrumental
variable (IV) regression in the second step.
The framework of Berry (1994) can accommodate
several utility specications, depending on the need
to account for various product substitution patterns.
The most simplistic utility specication, with no
consumer heterogeneity and independent and
identically distributed error (i.i.d.) term, leads to the
multinomial logit (MNL) model. Due to a closed-form
inverse demand function, MNL can be estimated
using linear IV regression. MNL estimation is
computationally efcient and straightforward, but
its cross-price elasticity estimates are unrealistic
because substitution patterns are driven by the
product’s popularity/market share rather than by
their similarity (Nevo 2000). This property is also
known as independence from irrelevant alternatives
(IIA). For instance, if Hyundai and Mercedes have
the same market shares, then MNL cross-price
elasticity estimates suggest that an increase in
the price of BMW leads to an equal increase in
the demand for Hyundai and Mercedes. This
result is counterintuitive, as one would expect
more gains in Mercedes’s market shares because
its characteristics are similar to those of BMW.
Increasingly exible substitution patterns can be
incorporated in Berry’s (1994) framework either by
introducing consumer heterogeneity in the utility
or through product segmentation (i.e., a correlation
between unobserved utilities across different
product dimensions, such as body type and brand
origin in the automobile sector).
After the seminal paper by Berry, Levinsohn,
and Pakes (1995) in the context of automobile
market demand, most empirical demand models
rely on consumer heterogeneity to specify a rich
substitution pattern between products and obtain
realistic elasticity estimates. The resulting utility
specication leads to the random coefcient logit
(RCL) model. Due to the unavailability of a closed-
form inverse demand function, the estimation
of RCL incurs numerical and computational
issues due to the involvement of Monte Carlo
simulation. Interested readers can refer to Conlon
and Gortmaker (2020) to learn more about the
associated estimation challenges and best practices
in RCL implementation.
In contrast, product segmentation in one dimension
leads to the traditional nested logit (NL) model
(Cardell 1997). Unlike RCL, NL ensures fast
estimation due to the closed-form inverse demand
function. However, within-group substitution
patterns are still governed by market shares
(instead of product similarity) in NL. In other words,
IIA still holds within each nest. In multilevel NL, a
hierarchical structure is sometimes difcult to dene
(Fosgerau, Monardo, and De Palma 2022). To
obtain a more exible and nonhierarchical structure,
1. Introduction
5
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model 5
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
alternate generalized extreme value (GEV) kernel
error distributions can be adopted (McFadden
1978). However, none of these extensions of
NL has a closed-form inverse demand function.
This property of GEV-based models removes
all the advantages of the use of product
segmentation specication over consumer
heterogeneity to specify a rich substitution pattern
between products.
To address the limitations of NL and GEV-based
models, Fosgerau, Monardo, and De Palma
(2022) propose inverse product differentiation logit
(IPDL) by directly specifying the inverse demand
function. IPDL has three mean features. First, IPDL
allows for nonhierarchical product segmentation
in multiple overlapping dimensions to capture
exible substitution patterns. Second, unlike GEV-
based models and RCL, the estimation of IPDL is
as simple as that of NL or MNL because it has a
closed-form inverse demand function. Third, IPDL
can also identify the degree of complementarity
between products conditional on the availability
of exogenous shocks. In summary, IPDL claims
to dene the tradeoff between computational
tractability and exibility. Thus, IPDL provides a
computationally efcient and fully exible framework
to model the demand for differentiated products
using market-level sales data.
Other than for the demand estimation for ready-
to-eat cereals in the original paper by Fosgerau,
Monardo, and De Palma (2022), the potential of
IPDL has not been explored in empirical studies.
This research note applies IPDL to understand the
demand for cars in the Chinese automobile market
and benchmarks it against MNL, NL, and RCL
in terms of recovering substitution patterns and
computational time. The results indicate that IPDL
has the potential to become a workhorse model
in estimating demand elasticity using market-level
aggregated sales data.
The remainder of this research is organized as
follows. Section 2 describes the mathematical
details of the existing demand models and highlights
how IPDL is formulated differently than are the
existing models. We also discuss model estimation,
elasticity, and crucial modeling details. Section 3
discusses the data and details of the empirical study
(e.g., market size and IVs). Section 4 compares the
elasticity estimates of MNL, NL, RCL, and IPDL.
The nal section summarizes the key takeaways
and some avenues for future research.
1. Introduction
6
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
We consider a linear utility specication in
which the utility of consumer i ∈ {1, 2,..., N }
due to choosing product j ∈ {0, 1, 2,..., J }
in market t ∈ {1, 2,..., T } is as follows:
Uijt =
x
x
j
j
β
β
−
α
pjt +
ξ
jt +
ε
ijt =
δ
jt +
ε
ijt
,
(1)
where j = 0 is the outside good, xj is a (1 × M) vector
of attributes for product j,
b
is the marginal utility
of xj, pjt is the price of product j in market t,
x
jt is
the unobserved product characteristics (averaged
across consumers), and
a
> 0 is the marginal
utility of income.
d
jt is the mean utility level (across
consumers), and
e
ijt captures all the observed
(interaction between demographic and product
characteristics) and unobserved factors that are
responsible for consumer preference heterogeneity.
Different structural representations of
e
ijt and
corresponding distributional assumptions lead to
different model specications (and substitution
patterns), such as MNL, NL, and RCL. We present
RCL in Section 2.2, but interested readers can refer
to Nevo (2000) for a detailed discussion on the
specication of
e
ijt.
Following Berry (1994), we relate the observed
market shares with those predicted by the model
(i.e., demand function):
sjt =
sj
(
δ
δ
t
t
;
ϕ
ϕ
)
(2)
where
ϕ
is a vector of the parameters associated
with
e
. Note that the demand function for MNL
is
s
j
MNL
(
δ
δ
t
t
;
ϕ
ϕ
)
=
exp(
δ
jt
)/ exp(
δ
kt
).
k
=0
J
∑ Thus, it is
worth noting that
ϕ
is a null set for MNL.
ϕ
contains
nest-specic correlation parameters in NL and
parameters associated with the mixing distribution of
consumer heterogeneity in RCL.
By normalizing
d
0t to 0 in all markets and assuming
the invertibility of the demand function, the inverse
demand function maps the observed market shares
st to the mean utility levels
d
jt:
δ
jt =
x
j
j
β
β
−
α
pjt +
ξ
jt =
sj
−
1(
s
t
t
;
ϕ
ϕ
)
(3)
Standard ARUM models specify the demand
function and then invert it numerically in the
estimation. This strategy is feasible for models like
MNL and NL because sj
–1 (.) has a closed-form
expression, and thus, their estimation requires
solving a linear IV regression. However, sj
–1 (.) and
Equation (14) are not in closed form for RCL, which
is the workhorse demand model. RCL estimation is
much more complex and computationally expensive
than are the other types of estimation due to its
numerical inversion of the demand function (see
Section 2.2 for details).
2.1 Inverse Product
Differentiation Logit (IPDL)
In contrast to traditional ARUM models, IPDL
directly species a closed-form inverse demand
function and thus obviates the need for numerical
inversion in the estimation (Fosgerau, Monardo, and
De Palma 2022):
sj
−
1(
st;
ϕ
ϕ
)
=
ln
Gj
(
st;
ϕ
ϕ
)
+ct=
xj
β
β
−
α
pjt +
ξ
jt
,
(4)
where the vector function G = {G0, G1, ..., GJ+1} is
invertible as a function of st. The exible substitution
patterns are governed by product segmentation in
multiple dimensions. Before specifying Gj (st;
ϕ
), we
discuss the product segmentation details.
Let us consider that the modeler assumes product
segmentation in D overlapping dimensions. In the
empirical application, we segment passenger cars
across body type, powertrain, and brand origin
(D = 3). Each dimension has a xed number of
nests, which are exogenous and market invariant.
2. Demand Models
7
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model 7
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Each product belongs to only one nest in each
dimension. For instance, we have three nests—
combustion, hybrid electric, and battery or plug-in
electric—in the powertrain dimension. Under this
product segmentation, ln Gj (st;
ϕ
) in IPDL is dened
as follows:
ln
Gj
(
st;
ϕ
ϕ
)
=
1
−
d=
1
D
∑
ρ
d
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
ln
sjt +
d=
1
D
∑
ρ
d
ln
k∈Gd
(
j
)
∑skt
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ (5)
where
ϕ
= {ρ1, ρ2, ..., ρD} and
Gd
(j
)
is a set
of products that are grouped with product j in
dimension d. Note that IPDL is ARUM consistent
only if
ρ
d<
1
d
=1
D
∑
and ρd > 0 for all dimensions.
These conditions are similar to those for NL. In
fact, IPDL simplies to the traditional one-level
NL when D = 1 and to MNL when there is no
product segmentation. Similar to NL, a higher
ρd value implies that products in the same nest
are more similar in dimension d than in other
dimensions.
Whereas the IIA property holds within each nest in
NL, it holds for products of the same type in IPDL. In
IPDL, a group of products are considered as being
of the same type if they belong to the same nest
in all dimensions. Assuming that the outside good
is the only product of its type, ln G0 (st;
ϕ
) = ln s0t.
Thus, through the use of Equations (4) and (5), the
IPDL boils down to the following:
ln
sjt
sot
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟=xj
β
β
−
α
pjt +
d=
1
D
∑
ρ
d
ln
sjt
k∈Gdj
( )
∑skt
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
+
ξ
jt
,
(6)
Equation (6) is linear in terms of the parameters,
and the parametrization of IPDL depends only
on the number of dimensions (not on the number
of products). Thus, similar to MNL and NL, IPDL
estimation requires solving a linear IV regression.
However, apart from price, log-share sums are
also endogenous. Therefore, at least D + 1 IVs are
required to address the endogeneity bias in the
estimation.
Conditional on the availability of the price
derivative of demand, the elasticity calculations are
straightforward. Fosgerau, Monardo, and De Palma
(2022) provide the following expression for the price
derivative of demand for each market in IPDL:
∂
∂
s
∂
∂
p= −∝
([
J
ln
G
(
s
)]
−
1
−ssT
),
(7)
where entry in cell ij of Jacobian
J
InG(s) is as follows:
∂
ln
Gi
∂sj
=
1
−
ρ
d
d=1
D
∑
si
1
{
i=j
}
+
d=
1
D
∑
ρ
d
k∈Gdi
( )
∑sk
1
{
j∈Gd
(
i
)},
(8)
where 1{.} is an indicator function and the market
subscript is suppressed. If we set D = 1, then we
obtain the price derivatives for NL.
2.2 Random Coefcient Logit
(RCL)
Following Nevo (2000), we discuss the specication
and estimation of RCL. The model incorporates
unobservable taste heterogeneity through the
inclusion of random coefcients. The indirect utility
of consumer i from purchasing product j in market t
is as follows:
Uijt =xj
β
β
i−
α
ipjt +
ξ
jt +
ε
ijt =
δ
jt +
µ
ijt +
ε
ijt
,
(9)
where all notations are identical to those in Equation
(1). Individual-specic
b
i and
a
i are assumed to be
distributed as follows:
α
i
β
β
i
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟=
α
β
β
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟+
1
!
0
" # "
0
!
1
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
M+
1
ϑ
i,1
!
ϑ
i
,
M+
1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
, (10)
2. Demand Models
8
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
where
ϑ
i
,
m
~
N
(0,
σ
m
2)
and the mean utility in
Equation (9) is as follows:
δ
jt =xj
β
β
−
α
pjt +
ξ
jt
.
(11)
The consumer-specic portion of utility is
µ
ijt =[−pjt ,xj].
ϑ
i,1
!
ϑ
i
,
M+
1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
. (12)
The probability of consumer i choosing product j in
market t, sijt, is
sijt =
exp(
δ
jt +
µ
ijt
)
exp(
δ
kt +
µ
ikt
)
k−
0
J
∑
. (13)
Then, the aggregate market share of product j, sjt,
can be obtained by summing the probabilities of
individuals’ choices:
sjt =∫
exp(
δ
jt +
µ
ijt
)
exp(
δ
kt +
µ
ikt
)
k−
0
J
∑dG
(
α
i
,
β
β
i
),
(14)
where G(αi,
β
i) is a multivariate normal distribution.
The elasticity of demand for alternative j due to the
change in the price of alternative k is as follows:
η
jkt =∂sjt pkt
∂pkt sjt
=
−pjt
sjt
∫
α
isijt
(1
−sijt
)
dP
ϑ
ϑ
(
ϑ
ϑ
)
,
if
j=k
pkt
sjt
∫
α
isijt sikt dP
ϑ
ϑ
(
ϑ
ϑ
)
,
if
j≠k
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
,
(15)
where P
ϑϑ
(
ϑ
ϑ
)
is a multivariate (1 × [M + 1]) normal
distribution with zero mean. In the absence of
a closed-form inverse demand function, RCL is
estimated using a two-step iterative process. In
the rst step, for the given parameter values, we
use the contraction mapping method to solve for
those mean utility levels (
d
jt) that minimize the
differences between the estimated and observed
market shares. This step involves the numerical
computation of integrals presented in Equation (14).
Using the mean utility level and mean marginal
utilities, residuals are obtained. These residuals
are further interacted with IVs to form the objective
function of the generalized method of moments
(GMM), which is minimized using the Nelder–Mead
simplex algorithm in the second step. This two-step
process is repeated until convergence is achieved.
We use the MATLAB code provided by Nevo (2000)
for RCL estimation.
2. Demand Models
9
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
To understand the demand for passenger
cars in China, we use the provincial-level
sales data of passenger cars in 2017. The
sales data and characteristics of products (including
price) are provided by JATO, a leader in automotive
business intelligence solutions. The original data
have more than 1,000 make-model combinations.
However, after removing those make-models with
nationwide sales below 500, we end up with 584
make-models. Since a make-model can have
variants with different powertrains and body types,
we dene a product based on make, model, body
type, and powertrain. Thus, the Chinese automobile
market has 651 unique passenger cars. The nal
sales data consist of 24.4 million cars, which is
close to the number of passenger cars sold in
China (i.e., 24.7 million) in 2017 (Statista 2020a).
We have data for 651 products (J = 651) across 31
provinces (T = 31), and thus, the linear IV regression
in Equation (6) should ideally have 31 × 651 = 20,181
observations. However, since some provinces
have no sales of some products, we have 19,061
observations. Similar to Berry, Levinsohn, and Pakes
(1995), apart from the market-specic price, we
consider length x width and power/weight as proxies
for car size and performance, respectively. In the
estimation, we also include market-specic xed
effects. While estimating linear IV regressions, we do
not put any constraints on nesting parameters ρd.
In NL, we consider product segmentation across
only body type, but in IPDL, we consider three
dimensions—body type (6 nests ranging from
mini to multipurpose vehicle (MPV)), powertrain
(3 nests: combustion, hybrid electric, and plug-in/
battery electric), and make-origin (2 nests: Chinese
and non-Chinese). Following Berry, Levinsohn,
and Pakes (1995), we assume that the household
population of a province is its potential market size
(Statista 2020b). Subsequently, we compute the
share of outside goods and other products using
Equation (16).
s
0
t=
HHPOP
t
−
j=1
J
∑Sales jt
HHPOP
t
;
sjt =
Sales jt
HHPOP
t
,
(16)
where HHPOPt is the household population in
province t and Salesjt is the sales of product
j in province t. We create IVs as suggested
by Berry, Levinsohn, and Pakes (1995) and
Fosgerau, Monardo, and De Palma (2022). For
each dimension, we take the average of the
characteristics of other products in the same nest.
Given that we have two product characteristics
related to size and performance, we have two
IVs in each dimension. In RCL, we use 200
quasirandom draws from the standard normal
distribution to numerically integrate the market
share functions.
3. Empirical Study
10
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Table 1 presents the parameter estimates of
MNL, NL, RCL, and IPDL under different
specications of xed effects and IVs. Since
including the make-origin dimension in IPDL makes it
inconsistent with the ARUM approach (i.e.,
ρ
d>
1
d=
1
3
∑),
we consider only body type and powertrain as two
dimensions in the nal specication of IPDL.
Across all models, the signs of parameters are as
expected: negative for price and positive for size and
performance-related attributes. The differences in
the marginal utilities of the price between the base
model (i.e., without IV) and specications 1 and 2 in
MNL indicate that addressing endogeneity is crucial
in market-level demand models. The F-statistic
values corresponding to all endogenous variables
are much higher than 10 in all specications,
indicating that the IVs satisfy the relevance condition.
A comparison of the estimates in specications 1
and 2 suggests that province-specic xed effects
have a noticeable impact only on one of the nesting
parameters in IPDL and the standard deviations
of random parameters in RCL. In addition, Table 1
shows that while RCL estimation takes approximately
9 minutes (owing to numerical simulation and
iterative estimation), all other considered models,
including IPDL, take less than 1 second.
NL and IPDL are ARUM consistent because the
nesting parameters are positive, and their sum
is below 1. As expected, a higher estimate of the
nesting parameter for body type implies that cars
with the same body type are much closer substitutes
than are cars with the same powertrain. The RCL
results indicate that unobserved taste heterogeneity
is prevalent in all three attributes in specication 1,
but the standard deviation of marginal utility is not
signicantly different from zero, at a 0.05 signicance
level, after the inclusion of provincial-level xed effects.
Since marginal utilities are difcult to compare across
models, we compute own and cross-price elasticities
for specication 2 of all models. For brevity, we
average the price elasticities across markets and
present them in Table 2 after aggregating across
body type. We also aggregate averaged price
elasticities (across markets) across the top ve
automobile makes (brands) in Table 3. Whereas the
marginal utility of price and the nesting parameter
for body type are not very different between NL and
IPDL, the differences in their own and cross-price
elasticities are substantial. Tables 2 and 3 show
that the absolute values of the own price elasticity
estimates for RCL and IPDL are much higher than
those for NL and MNL, but the own price elasticity
estimates for IPDL and RCL are close to one another.
The pattern of cross-price elasticities in Table 2
illustrates the importance of product segmentation.
The rows of the cross-price elasticity matrix are
the same in MNL due to IIA. We observe a similar
trend in NL, but the diagonals are different due to
segmentation across body type. However, we do not
see any specic pattern in the cross-price elasticity
matrix of IPDL and RCL due to segmentation in the
powertrain dimension (in addition to body type) and
accounting for consumer preference heterogeneity.
We see a similar trend of cross-price elasticity for
MNL in Table 3 as that in Table 2, but the trend is
different for NL between Tables 2 and 3. The trend
in NL elasticities is as expected because the IIA in
NL holds only within products of the same body type
but not within products of the same make. Tables 2
and 3 also show that while the price elasticity
estimates of IPDL and RCL are similar, the cross-
price elasticities of both models appear different.
To further explore whether these differences
between the RCL and IPDL models in cross-price
elasticity are signicant, we take 100 draws from
the asymptotic sampling distribution of model
parameters and compute cross-price elasticity
values for each draw. We sort 100 elasticity values
in increasing order and use the 3rd and 97th
4. Results
11
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model 11
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Table 1. Parameter estimates of choice models.
Specications Multinomial logit
(MNL)
Nested logit
(NL)
Random coefcient logit (RCL) Inverse product
differentiation
logit (IPDL)
Base 1 2 1 2 1 2 1 2
Mean Mean Mean Mean Mean Mean SD Mean SD Mean Mean
Purchase price −1.77
(0.07)
−5.60
(0.25)
−5.84
(0.25)
−4.07
(0.17)
− 4.13
(0.17)
−22.39
(0.64)
4.82
(3.87)
−22.00
(0.59)
1.85
(1.58)
−4.54
(0.18)
−4.28
(0.17)
Length × Width 1.25
(0.18)
3.23
(0.25)
3.37
(0.25)
3.90
(0.17)
4.10
(0.17)
5.87
(1.12)
3.18
(3.56)
6.30
(0.87)
5.99
(0.85)
4.25
(0.17)
4.22
(0.17)
Power/Weight 0.90
(0.08)
3.65
(0.19)
3.82
(0.19)
2.49
(0.13)
2.53
(0.12)
4.62
(1.16)
1.16
(0.95)
4.51
(1.60)
0.98
(0.35)
2.94
(0.14)
2.66
(0.13)
ln
sjt
k∈Gbody
(
j
)
∑skt
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
0.62
(0.03)
0.68
(0.03)
0.62
(0.03)
0.69
(0.03)
ln
sjt
k∈Gpowertrain
(
j
)
∑skt
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
0.23
(0.04)
0.07
(0.03)
F-statistic (First stage)
Purchase price 389.7 391.8 389.7 391.8 389.7 391.8 389.7 391.8 391.8 389.7
ln
sjt
k∈Gbody
(
j
)
∑skt
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
147.1 153.2 147.1 153.2
ln
sjt
k∈Gpowertrain
(
j
)
∑skt
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
97.6 143.3
Provincial-level xed effects Yes No Yes No Yes No Yes No Yes
Instrumental variables (IVs) No Yes Yes Yes Yes Yes Yes Yes Yes
Estimation time (Seconds) 0.367 0.367 0.367 0.367 0.367 530.161 571.241 0.367 0.367
Notes: Numbers in parentheses are standard errors.
observations as the lower and upper bounds of the
95% condence interval (CI), respectively. These
95% CIs of cross-price elasticity estimates for IPDL
and RCL are presented in Table 4 for the top ve
brands. Since RCL has two levels of heterogeneity
(standard error and taste heterogeneity) and much
higher standard errors of the parameters, its CIs are
much broader than are those of IPDL. Many CIs of
4. Results
12
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Table 2. Comparison of price elasticity estimates across car body type.
Own Cross
123456
Multinomial logit (MNL)
1−1.63 7. 8 0 E - 0 5 2.40E-05 8.60E-05 8.20E-05 2.20E-05 3.90E-05
2−0.53 7. 8 0 E- 0 5 2.40E-05 8.60E-05 8.20E-05 2.20E-05 3.90E-05
3−4 7. 8 0 E- 0 5 2.40E-05 8.60E-05 8.20E-05 2.20E-05 3.90E-05
4−0.78 7. 8 0 E - 0 5 2.40E-05 8.60E-05 8.20E-05 2.20E-05 3.90E-05
5−0.7 7. 8 0 E - 0 5 2.40E-05 8.60E-05 8.20E-05 2.20E-05 3.90E-05
6−1.87 7. 8 0 E- 0 5 2.40E-05 8.60E-05 8.20E-05 2.20E-05 3.90E-05
Nested logit (NL)
1−3.63 6 .10 E- 0 3 2.30E-05 9.60E-05 5.90E-05 2.70E-05 5.50E-05
2−1.17 7. 8 0 E- 0 5 8.90E-03 9.60E-05 5.90E-05 2.70E-05 5.50E-05
3−8.8 7. 8 0 E- 0 5 2.30E-05 8.90E-02 5.90E-05 2.70E-05 5.50E-05
4−1.74 7. 8 0 E - 0 5 2.30E-05 9.60E-05 7.10 E- 0 3 2.70E-05 5.50E-05
5−1.43 7. 8 0 E - 0 5 2.30E-05 9.60E-05 5.90E-05 5.90E-02 5.50E-05
6−4 7. 8 0 E- 0 5 2.30E-05 9.60E-05 5.90E-05 2.70E-05 1.20E-01
Random coefcient logit (RCL)
1−5.04 3 .10 E- 0 3 8.10 E- 0 4 3.60E-03 2.40E-03 4.00E-04 2.30E-03
2−1.84 2.30E-03 8 .10 E- 0 4 2.00E-03 2.30E-03 4.10 E- 0 4 1.50E-03
3−10.53 3.60E-03 5.80E-04 5.60E-03 1.90E-03 3.30E-04 2.80E-03
4−2.79 2.80E-03 8.80E-04 2.70E-03 2.50E-03 4.20E-04 1.90E-03
5−1.88 1.40E- 03 5.00E-04 1.30E-03 1.30E-03 3.40E-04 9.00E-04
6−5.9 3.50E-03 7. 8 0 E- 0 4 4.40E-03 2.40E-03 4.00E-04 2.70E-03
Inverse product differentiation logit (IPDL)
1−4.94 9.10 E- 0 3 7. 4 0 E- 0 5 2.20E-04 2 .10 E- 0 4 4.40E-04 1.10 E- 0 4
2−1.59 2 .10 E- 0 4 1.30E-02 2.20E-04 2.00E-04 7.30E-04 1.10 E- 0 4
3−11.9 4 2.30E-04 8.80E-05 1.30E-01 2.70E-04 7.20E-04 9.50E-05
4−2.36 2.30E-04 9.70E-05 2.30E-04 1.10E- 0 2 8.90E-04 1.00E-04
5−1.93 2.50E-04 7. 6 0 E- 0 5 2.60E-04 2.40E-04 8.80E-02 1.00E-04
6−5.42 2.10 E- 0 4 7. 9 0 E- 0 5 2.30E-04 1.80E-04 3.10 E- 0 4 1.80E-01
Notes:
1: SUV/Upper medium.
2: Small/Mini multipurpose vehicle (MPV).
3: Sports/Executive/Luxury.
4: Lower medium/Medium MPV.
5: Mini.
6: Full size MPV/Small commercial.
RCL encompass the CIs of IPDL (with some IPDL
CIs being slightly above or below those of RCL). The
results thus indicate that RCL and IPDL can capture
substitution patterns similarly in this study.
4. Results
13
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model 13
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Table 3. Comparison of price elasticity estimates for the top ve brands.
Variables Own Cross
Volkswagen Honda Toyota Geely Buick
Multinomial logit (MNL)
Volkswagen −1.41 2.90E-04 9.30E-05 5.20E-05 1.00E-04 3.10 E- 0 4
Honda −1.11 2.90E-04 9.30E-05 5.20E-05 1.00E-04 3.10 E- 0 4
Toyota −1.95 2.90E-04 9.30E-05 5.20E-05 1.00E-04 3.10 E- 0 4
Geely −0.54 2.90E-04 9.30E-05 5.20E-05 1.00E-04 3.10 E- 0 4
Buick −1.27 2.90E-04 9.30E-05 5.20E-05 1.00E-04 3.10 E- 0 4
Nested logit (NL)
Volkswagen −3.13 8.30E-03 5.00E-03 5.10 E- 0 3 3.10E- 0 3 1.30E-02
Honda −2.47 7. 4 0E - 0 3 5.10 E- 0 3 6.20E-03 2.50E-03 1.30E-02
Toyota −4.28 7. 9 0 E- 0 3 5.60E-03 4.40E-03 2.50E-03 1.40E-02
Geely −1.19 9.20E-03 4.00E-03 4.80E-03 2.90E-03 4.80E-03
Buick −2.66 8.00E-03 5.00E-03 5.40E-03 2.60E-03 5.00E-03
Random coefcient logit (RCL)
Volkswagen −4.87 8.70E-03 4.80E-03 5.00E-03 3.00E-03 7. 8 0 E- 0 3
Honda −3.92 8.80E-03 4.90E-03 4.80E-03 3.20E-03 7. 9 0 E- 0 3
Toyota − 6.10 8.70E-03 4.70E-03 5.20E-03 2.80E-03 7. 8 0 E- 0 3
Geely −2.10 8 .10 E- 0 3 4.60E-03 4.20E-03 3.30E-03 7. 0 0 E- 0 3
Buick −4.35 9.10 E- 0 3 5.00E-03 5.00E-03 3.20E-03 8 .10 E- 0 3
Inverse product differentiation logit (IPDL)
Volkswagen −4.25 1.23E-02 7. 39 E- 0 3 7.47 E- 0 3 4.85E-03 1.89E-02
Honda −3.34 1.05E-02 8.22E-03 1.22E-02 3.43E-03 1.86E-02
Toyota −5.80 1.14 E- 0 2 9.14 E- 0 3 7.8 7E- 0 3 3.59E-03 2 .12E- 0 2
Geely −1.61 1.37E-02 5.88E-03 7.07 E- 0 3 4.33E-03 7. 2 3 E- 0 3
Buick −3.60 1.12 E- 0 2 8 .17E- 0 3 1.03E-02 4.01E-03 7. 01E - 03
4. Results
14
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Table 4. Condence intervals (CIs) for cross-price elasticities across the top ve brands.
Brand Own
elasticity
Volkswagen Honda Toyota Geely Buick
Random
coefcient logit
(RCL)
Volkswagen –4.87 8.70E-03 4.80E-03 5.00E-03 3.00E-03 7.80E- 03
[2.21E-03, 2.05E-02] [1.20E-03, 1.04E-02] [1.34E-03, 1.39E-02] [6.03E-04, 7.28E-03] [2.07E-03, 1.69E-02]
Honda –3.92 8.80E-03 4.90E-03 4.80E-03 3.20E-03 7.9 0E- 03
[2.12E-03, 2.01E-02] [1.17E-03, 1.09E-02] [1.26E- 03, 1.25E-02] [6.22E-04, 7.74E-03] [1.94E-03, 1.70E-02]
Toyota – 6 .10 8.70E-03 4.70E-03 5.20E-03 2.80E-03 7.80E- 03
[2.29E-03, 2.23E-02] [1.27E-03, 1.08E-02] [1.28E-03, 1.42E-02] [6.03E-04, 6.78E-03] [2.25E-03, 1.81E-02]
Geely – 2.10 8.10 E- 0 3 4.60E-03 4.20E-03 3.30E-03 7. 0 0 E- 0 3
[1.67E-03, 2.11E-02] [9.60E-04, 1.12E-02] [8.87E-04, 1.12E-02] [5.71E-04, 9.15E-03] [1.50E-03, 1.65E-02]
Buick –4.35 9.10 E- 0 3 5.00E-03 5.00E-03 3.20E-03 8.10 E- 0 3
[2.15E-03, 2.08E-02] [1.16E- 03, 1.12E- 02] [1.29E-03, 1.30E-02] [6.11E-04, 7.65E-03] [1.95E-03, 1.74E-02]
Inverse product
differentiation
logit (IPDL)
Volkswagen –4.25 1.23E-02 7. 39 E- 0 3 7.47E- 0 3 4.85E-03 1.89E-02
[1.22E-02, 1.25E-02] [ 7. 2 9 E- 03 , 7.47 E- 03] [ 7. 37E- 03 , 7.5 6E- 0 3] [4.79E-03, 4.91E-03] [1.87E-02, 1.92E-02]
Honda –3.34 1.05E-02 8.22E-03 1.22E-02 3.43E-03 1.86E-02
[1.03E-02, 1.06E-02] [8.11E-03, 8.32E-03] [1.20E-02, 1.23E-02] [3.38E-03, 3.47E-03] [1.83E-02, 1.88E-02]
Toyota –5.80 1.14 E- 0 2 9.14 E- 0 3 7.87E- 0 3 3.59E-03 2.12E-02
[1.12E- 02, 1.15E- 02] [9.02E-03, 9.25E-03] [7.76E-03, 7.97E-03] [3.54E-03, 3.63E-03] [2.09E-02, 2.14E-02]
Geely –1.61 1.37E-02 5.88E-03 7.07E- 0 3 4.33E-03 7. 2 3 E - 0 3
[1.35E-02, 1.38E- 02] [5.80E-03, 5.95E-03] [6.97E-03, 7.15E-03] [4.27E-03, 4.38E-03] [7.13 E - 03 , 7. 31E- 0 3]
Buick –3.60 1.12 E- 0 2 8.17 E- 0 3 1.03E-02 4.01E-03 7. 01E- 0 3
[1.10E- 02, 1.13E- 02] [8.06E-03, 8.27E-03] [1.02E-02, 1.05E-02] [3.95E-03, 4.05E-03] [6.91E-03, 7.09E-03]
Notes: CIs in bold do not match between the RCL and IPDL models.
4. Results
15
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
We present the rst application of the IPDL
model in estimating the demand for
passenger cars in the automobile market.
The modeling and results provide three main
insights in favor of IPDL over the other logit models.
First, IPDL can nearly mimic the RCL model via two-
dimensional market segmentation (i.e., body type
and power train). Second, the demand elasticity
estimates of IPDL are more certain than are those
of RCL. Third, since IPDL is as easy to estimate as
is the two-stage least squares (2SLS) regression,
its estimation takes less than 1 second on a regular
computer (i.e., it is approximately 500 times faster
than RCL). Unlike RCL, IPDL does not suffer from
computational and numerical issues.
We also identify two challenges, similar to those of
NL, in the empirical application of IPDL. First, the
modeler must specify the product segmentation
dimensions. There is neither an elegant way to
select these dimensions nor any guidance on
the required number of dimensions to extract the
maximum possible information from the data.
Second, to ensure the consistency of IPDL with
the utility-maximization framework, constraints on
the nesting parameters must be satised. These
constraints make the segmentation search even
more difcult. Thus, the quality of the results
produced by IPDL depends on the modeler’s
understanding of the context to correctly specify
the nesting structure. However, the challenge in
segmentation should not be seen as a bottleneck in
IPDL applications because several segmentations
can be tested quickly. Notably, specication
search challenges manifest differently in RCL,
such as those involving specications for mixing
distribution and for whether or not to consider taste
heterogeneity for a specic attribute.
Overall, the exibility and computational efciency
of IPDL can make it a workhorse model for demand
estimation using market-level aggregated sales
data. At the least, researchers can leverage the
computational efciency of IPDL to search the
specication, followed by running the RCL model
for the nal specication (i.e., saving themselves
from having to deal with the numerical issues
and computational efforts required of RCL
and expediting the model-building process).
Nevertheless, additional empirical studies are
needed to validate the generalizability of the ndings
presented in this study.
5. Conclusions
16
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
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References
17
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
Notes
18
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
About the Authors
Jinghai Huo
Jinghai Huo is pursuing a Master of Engineering (MEng) degree at the National
University of Singapore, specializing in the study of consumer preferences
for energy-efcient technologies such as electric vehicles and e-scooters. His
ongoing projects revolve around incorporating eye-tracking data to improve the
effectiveness of choice modeling.
Rubal Dua
Rubal Dua, a research fellow at KAPSARC, is actively engaged in exploring
emerging challenges and priorities in sectors that critically inuence
transportation and infrastructure, viewed through the lens of energy economics,
policy, and sustainability. He holds a Ph.D. from KAUST, Saudi Arabia, an
M.Sc. from the University of Pennsylvania, USA and a B.Tech. from the Indian
Institute of Technology (IIT), Roorkee.
Prateek Bansal
Prateek Bansal is an assistant professor at the National University of
Singapore, working primarily on Bayesian machine learning methods and
causal inference models with applications to transport systems. He holds a
Ph.D. from Cornell University, an M.Sc. degree from The University of Texas at
Austin, and a B.Tech. from the Indian Institute of Technology (IIT) Delhi.
19
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
About the Project
Promoting the adoption of energy efcient vehicles has become a key policy imperative in both
developed and developing countries. Understanding the impacts of various factors on adoption
rates forms the backbone of KAPSARC’s efforts in the light-duty vehicle demand eld. These
factors include (i) consumer-related factors—demographics, behavioral, and psychographics;
(ii) regulatory factors—policies, incentives, rebates, and perks; and (iii) geotemporal factors—
weather, infrastructure and network effects. Our team is currently developing models at different
levels: microlevel models using large-scale data comprising new car buyers’ proles, and
macrolevel models using aggregated adoption data to understand and project the effects of various
factors affecting the adoption rate of energy efcient vehicles.
20
Estimating Car Price Elasticity Using an Inverse Product Differentiation Logit Model
www.kapsarc.org
