Automobile Prices in Market Equilibrium with
Unobserved Price Discrimination∗
Xavier D’Haultfœuille†
Isis Durrmeyer‡
Philippe Février§
December 24, 2014
Abstract
In markets where sellers are able to price discriminate, or the buyers to bargain,
individuals receive discounts over the posted prices that are usually not observed
by the econometrician. This paper considers the structural estimation of a demand
and supply model à la Berry et al. (1995) when only posted prices are observed.
We consider that heterogeneous discounts occur due to price discrimination by firms
on observable characteristics of consumers. Within this framework, identification is
achieved by assuming that the marginal costs of producing and selling the goods do
not depend on the characteristics of the buyers. We also require a condition relating
the posted prices to the prices actually paid. For instance, we can assume that at
least one group of individuals pays the posted prices. Under these two conditions,
the demand and supply parameters, as well as the exact discounts corresponding to
each type of consumers, can be identified. We apply our methodology to estimate the
demand and supply in the new automobile market in France. Results suggest that
discounting arising from price discrimination is important. The average discount is
estimated to be 10.5%, with large variation depending on the buyers’ characteristics
and cars’ specifications. Our results are in line with discounts generally observed in
European and American automobile markets.
∗
We would like to thank Steven Berry, Juan Esteban Carranza, Rob Clark, Pierre Dubois, Philippe
Gagnepain, Gautam Gowrisankaran, Alessandro Iaria, John Morrow, Kathleen Nosal, Mario Pagliero,
Philipp Schmidt-Dengler, Michelle Sovinsky, Hidenori Takahashi, Yuya Takahashi, Frank Verboven, Naoki
Wakamori and the participants of various seminars and conferences for their valuable comments. We also
thank Pierre-Louis Debar and Julien Mollet from the CCFA for providing us with the data. Isis Durrmeyer
acknowledges financial support from the Deutsche Forschungsgemeinschaft through SFB-TR 15.
†
CREST. E-mail: [email protected]
‡
University of Mannheim. E-mail: [email protected]
§
CREST. E-mail: [email protected]
1
1
Introduction
The standard aggregate-level estimation of demand and supply models of differentiated
products relies on the observation of the market shares and the characteristics of the
products, in particular prices (see Berry 1994). Because of price discrimination and price
negotiation, transaction prices for an identical product may differ from one individual to
another. Sellers can practice third degree price discrimination according to observable
demographic characteristics such as age, gender or city of residence. There may also be
room for individual negotiation: the sellers are willing to offer discounts to the consumers
that aggressively negotiate the prices. Automobiles, furniture, kitchens or mobile phone
contracts are examples for which there is either documented or anecdotal evidence that
consumers receive some discounts (on new automobiles, see, e.g. Ayres & Siegelman 1995,
Goldberg 1996, Harless & Hoffer 2002, Morton et al. 2003, Langer 2012, Chandra et al.
2013).1 Loans and insured mortgages have also proved to be negotiable (see Charles et al.
2008, Allen et al. 2014). Such phenomena also exist in vertical relationships between
producers and retailers. Producers are required to edit general terms and conditions of
sale. These conditions are then the starting point for individual negotiation with each
retailer.
In all these cases, precise data on transaction prices may be hard to obtain. One typically
observes either transaction prices on a small sample issued from a survey, or only posted
prices on a large sample. In the first case, price discrimination can be studied but policy
exercises cannot be performed. With large data, on the other hand, policy simulations
are usually done without taking the issue of limited observation of prices into account.
Because the instrumental variables approach used in Berry et al. (1995, henceforth, BLP)
to control for price endogeneity does not solve this nonclassical measurement error problem
(namely, observing posted prices instead of transaction prices), ignoring it generally results
in an inconsistent estimation of the structural parameters and biases in policy exercises.
This paper proposes a method to estimate a structural demand and supply model with
unobserved discounts. Our rationale for the existence of discounts over the posted prices
is that discounts allow firms to price discriminate between heterogeneous consumers and
thus extract more surplus than they would with a uniform price.2 Sellers edit only one
1
In France, it is also commonly admitted that negotiation is possible when purchasing a new car. An
article published in October 2011 by Le Figaro, which is the second largest French national newspaper,
suggests that discounts up to 26% can be obtained.
2
In some cases, all profits could be higher if all firms did not price discriminate (see, e.g. Holmes 1989,
Corts 1998). But without any commitment devise against price discrimination, price discrimination occurs
for each firm at equilibrium.
2
price, namely the posted price, since it is usually legally forbidden to price discriminate
between consumers and difficult to implement, but in practice the transaction prices differ
from one individual to another. We suppose that sellers use observable characteristics
of the buyers to price discriminate and set an optimal discount over the posted price.
Importantly, we assume that the sellers do not have more information about consumers
than the econometrician. This assumption may be problematic in settings where there are
few buyers, such as vertical relationships between producers and retailers. But in markets
where sellers do not know the buyers before the transaction, it seems plausible to assume
that price discrimination is based only on a few easily observable characteristics, such as
sex, age and the city of residence.
We therefore extend the random coefficient discrete choice model of demand popularized by
BLP to allow for unobserved price discrimination. BLP exploit the exogeneity of observed
product characteristics, apart from price, to yield moment conditions involving the parameters of interest. Their method does not apply directly to our framework, yet, because the
moment conditions are not valid anymore if we replace the unobserved transaction prices
with posted prices. Instead, we rely on structural assumption from the supply side, and
replace the unobserved prices by their expression stemming from the first-order condition
of profit maximization.
These first-order conditions have identifying power under two assumptions. First, the
marginal cost of a product is supposed to be identical for all buyers. This amounts to
neglecting differences in selling costs to different consumers in the total cost of a product.
This assumption is likely to be satisfied in many markets, such as the automobile market,
where the major part of the marginal cost is production, not sale, and the cost of selling
is probably not very different from one consumer to another. The second condition states,
basically, that there is a known relationship between observed and transaction prices. In
our application, we suppose that the prices posted by the sellers correspond to the highest
discriminatory price, so that some consumers actually pay these observed posted prices. In
other words, we normalize the optimal discount for one group of consumers to be zero. Such
an assumption is necessary since otherwise, we could shift all discounts by an arbitrary
constant. It is also consistent with empirical evidence reported by several surveys, such
as the one made by Cetelem in 2012 in France (L’Automobile en Europe: 5 Leviers pour
Rebondir 2013).
The ideas underlying our approach can be applied to various settings where the information
on prices is limited, with different assumptions on observed prices. This is the case in
particular if we observe the average transaction prices paid by all consumers. In this setting,
3
there is a direct link between observed and transaction prices and no further assumption on
prices is needed. Our methodology could also be implemented in a setting where demand
is observed in different markets, while the prices are observed only in a subset of markets,
as could be the case for example with the automobile market in Europe or supermarket
chains in different municipalities. Two recent papers have used similar ideas in different
settings. Miller & Osborne (2014) adopt a similar methodology to analyze the cement
industry. They only observe average price and the total quantity of cement purchased in
the US. They allow as well for price discrimination across US counties. They compute
the optimal prices and quantities for each location using the equilibrium conditions. Then
they compare the corresponding average prices and total quantities with the observed ones.
Apart from this similarity, their model and estimation strategy is very different from ours.
In particular, due to data limitations, they cannot account for observed and unobserved
differences in preferences across counties. Also related is the paper by Dubois & Lasio
(2014), which estimates marginal costs when observed prices are regulated and, therefore,
no longer related to the marginal cost. They use the first-order conditions of the firms on
other countries that do not regulate the prices of the same drug. Contrary to us, however,
they do not use the first-order conditions of the firms to identify the demand model.
We apply our method to the French market of new cars. Up to now, the demand for
automobile has always been estimated with posted prices when transaction prices are
unobserved. As mentioned before, however, there is evidence of price discrimination in
this market. We rely on an exhaustive dataset recording all the registrations of new
cars bought by households in France between 2003 and 2008. Apart from detailed car
attributes, some buyers characteristics are provided. We observe in particular age and
expected income (namely, the median income of people in the same age class living in the
same municipality). As these characteristics are easily observed by sellers and presumably
strong determinants of purchases, we suppose that they are used to price discriminate.
Our results suggest that price discrimination is significant in France. The average discount
is estimated to be 10.5% of the posted price. The distribution of estimated discounts
spreads mostly between 0 and 25% depending on the car purchased and demographic
characteristics. As expected, age and income are negatively correlated to the value of
discount. Overall, our results are in line with evidence on discounts in France (see in
particular L’Automobile en Europe: 5 Leviers pour Rebondir 2013) and with the literature
on price discrimination in the automobile market (see, e.g. Harless & Hoffer 2002, Langer
2012). The magnitude of discounts obtained is also comparable to estimates obtained with
survey data and anecdotal evidence found in specialized magazines or on internet. Finally,
4
we show that ignoring price discrimination and using list prices as if they corresponded to
the transaction prices, as is usually done, would slightly overestimate the price sensitivity
parameters but always overestimate the marginal costs of products.
We also study the effect of price discrimination on car manufacturers profits and consumers
surplus. This question is particularly relevant because theoretical predictions are unclear
and depend on market conditions. Holmes (1989) and Corts (1998) show that in the
competitive framework, third degree price discrimination can induce profit loss for firms,
depending on the degree of competition and the demand shape. Aguirre et al. (2010) and
Cowan (2012) derive sufficient conditions on demand for price discrimination to increase
social welfare and consumer surplus, but these conditions may not be satisfied in practice.
We show, in our application, that if all firms could commit not to price discriminate, the
overall industry profit would be reduced but some firms would be better off. The gains
from price discrimination appear to be larger for luxury brands and French brands that
have large market shares. On the consumer side, there are winners and losers, as the
theory predicts, but price discrimination is moderately welfare enhancing at the aggregate
level. Price discrimination carries out monetary redistribution from the older and richer
purchasers to the younger, low-income earners.
Though we do not explicitly model bargaining, our paper is related to the recent theoretical
literature that considers hybrid models of bargaining in which sellers post a sticker price and
offer the possibility to bargain for discounts. This strategy might be profitable for sellers
when consumers have heterogeneous bargaining costs or are imperfectly informed on their
ability to bargain (see Gill & Thanassoulis 2009, 2013). Our model can be interpreted as
a bargaining model in which all the bargaining power is given to the seller. Structural
models of demand and supply where prices are set by a bargaining process have been
recently developed and estimated in specific industries, generally in business to business
markets where there are few identifiable actors. Crawford & Yurukoglu (2012), for instance,
estimate a structural model of bargaining between television stations and cable operators,
whereas Grennan (2013) analyzes price discrimination and bargaining in the market for
coronary stents. Gowrisankaran et al. (2014) also develop and estimate a structural model
of bargaining to analyze mergers between hospitals.
However, there are few empirical papers that analyze bargaining in business to consumers
markets. A recent paper by Jindal & Newberry (2014) develops a structural model of
demand where buyers are able to negotiate but have a bargaining cost. They estimate
both the bargaining power and the distribution of bargaining costs using individual data
on refrigerator transactions. However, their framework is very different from ours since
5
they omit competition and they do observe the transaction prices at the individual level.
Last but not least, the paper by Huang (2012) develops a structural model of demand that
incorporates unobserved negotiation between sellers and buyers to describe the secondhand car market. He estimates the model using posted prices only, as we do here. His
identification strategy is very different from ours and relies on the existence of dealers that
commit not to negotiate with potential buyers. With such data at hand, he estimates
the demand parameters together with the unobserved discounts offered by dealers that
allow for negotiation. As opposed to our methodology, he cannot identify model-specific
discounts but rather obtains an average discount at the dealer level.
The paper is organized as follows. The second section presents the theoretical model and
section 3 explains how to estimate the model with unobserved transaction prices. Section
4 describes our estimation algorithm and presents the results of Monte-Carlo simulations.
The application on the French new car market is developed in the fifth part of the paper.
We conclude in Section 6.
2
Theoretical model
We first present our theoretical model. The approach is identical to the BLP model except
that the demand arises from a finite number of heterogeneous groups of consumers. Firms
are supposed to observe the group of each consumer, as well as their corresponding preferences, such as their average price sensitivity. They then price discriminate among these
groups, in order to take advantage of the heterogeneity in preferences.
Specifically, heterogeneous consumers are supposed to be segmented in nD groups of consumers, and we denote by d the group of consumer i. As in the standard BLP model, we
allow consumers to be heterogeneous within a group, but assume sellers are not able to
discriminate based on this heterogeneity. Each consumer chooses either to purchase one
of the J products or not to buy any, which corresponds to the outside option denoted by
0. As usual, each product is assimilated to the bundle of its characteristics. Consumers
maximize their utility, and the utility of choosing j is assumed to be a linear function of
product characteristics:
Uijd = Xj0 βid + αid pdj + ξjd + dij ,
where Xj corresponds to the vector of observed characteristics and ξjd represents the valuation of unobserved characteristics. pdj is the price set by the seller for the category d
and is not observed by the econometrician. Consumers with characteristics d are supposed
6
to face the same transaction price pdj . This is crucial, but not more restrictive than the
assumption that ξjd is common to all individuals with characteristics d. This was shown
by Berry & Haile (2014) to be necessary for identifying demand models nonparametrically
from aggregated data. As typical in the literature, the idiosyncratic error terms dij are
extreme-value distributed.
We make the usual parametric assumption about the intra-group heterogeneity, i.e. that
individual parameters can be decomposed linearly into a mean, an individual deviation
from the mean and a deviation related to individual characteristics:
(
X
βid = β0d + π0X,d Ei + ΣX,d
0 ζi
p
αid = α0d + π0p,d Ei + Σp,d
0 ζi ,
where Ei denotes demographic characteristics that are unobserved by the firm for each
purchaser but whose distribution is common knowledge. ζi = (ζiX , ζip ) is a random vector
with a specified distribution such as the standard multivariate normal distribution.
The utility function can be expressed as a mean utility and an individual deviation from
this mean:
Uijd = δjd (pdj ) + µdj (Ei , ζi , pdj ) + dij ,
with
δjd (pdj ) = Xj0 β0d + α0d pdj + ξjd
and
µdj (Ei , ζi , pdj ) = Xj π0X,d Ei + σ0X,d ζiX + pdj π0p,d Ei + σ0p,d ζip .
We let the dependence in pdj be explicit for reasons that will become clear below. Because
of the logistic assumption on the dij , the aggregate market share sdj (pd ) of good j for
demographic group d satisfies, when prices are set to pd = (pd1 , ..., pdJ ),
sdj (pd )
Z
d
sdj (e, u, pd )dPE,ζ
(e, u),
=
(1)
d
where PE,ζ
is the distribution of (E, ζ) for group d and
exp δjd (pdj ) + µdj (e, u, pdj )
sdj (e, u, pd ) = PJ
k=0
exp δkd (pdk ) + µdk (e, u, pdk )
Now, we consider a Nash-Bertrand competition setting where firms are able to price dis7
criminate by setting different prices to each of the nD consumers groups. Letting Jf denote
the set of products sold by firm f , the profit of f when the vector of all prices for group d
is pd satisfies
Πf = M
nD
X
d=1
P (D = d)
X
sdj (pd ) × pdj − cdj ,
j∈Jf
where P (D = d) is the fraction of the group of consumers d, sdj (pd ) is the market share of
product j for group d when prices are equal to pd and M is the total number of potential
consumers. cdj is the marginal cost of the product j for group d.
The first-order condition for the profit maximization for group d yields
pdf = cdf + Ωdf
−1
sdf ,
(2)
where pdf , cdf and sdf are respectively the equilibrium transaction prices, marginal costs and
observed market shares vectors for firm f . Ωdf is the matrix of typical (i, j) term equal to
−∂sdj /∂pi . Prices are optimally set by the firms making the traditional arbitrage between
increasing prices and lowering sales. When a monopoly seller is able to price discriminate,
it is less constrained than with a uniform pricing strategy since this arbitrage is made for
each group separately. If a group is particularly price sensitive, the monopoly seller offers
a low price and is still able to extract a large surplus from the less price sensitive group by
setting a higher price for this group. In a competitive setting, this effect is mitigated by
the fact that, for a given group of consumer, the competition among sellers is reinforced.
3
3.1
Inference
GMM estimation of the model
We now turn to inference on this model. We assume that the econometrician observes the
market shares sdj corresponding to each consumer group but not the discriminatory prices
pdj paid by consumers. We do assume, on the other hand, that the econometrician observes
the posted prices.
First, let us recall the standard case where the true prices are observed. Let
p,d
p,d
θ0d = (β0d , α0d , π0X,d , ΣX,d
0 , π0 , Σ0 )
denote the true vector of parameters for group d. The standard approach for identification and estimation of θ0d , initiated by BLP, is to use the exogeneity of Zj , which includes
8
the characteristics Xj and other instruments (typically, function of characteristics of other
products or cost shifters) to derive moment conditions involving θ0d . The exogeneity condition takes the form
E Zj ξjd = 0.
(3)
The idea is then to use the link between ξjd and the true parameters θ0d through Equation
(1). Specifically, we know from Berry (1994) that for any given vector θd , Equation (1),
where θ0d is replaced by θd , defines a bijection between market shares and mean utilities of
products δjd . Hence, we can define δjd (sd , pd ; θd ), where sd = (sd1 , ..., sdJ ) denotes the vector of
observed market shares. Once δjd (sd , pd ; θd ) is obtained, the vector ξjd (pd ; θd ) of unobserved
characteristics corresponding to θd and rationalizing the market shares follows easily since
ξjd (pd ; θd ) = δjd (sd , pd ; θd ) − Xj β d − αd pdj .
The moment conditions used to identify and estimate θ0d are then
E Zj ξjd (pd ; θ0d ) = 0.
(4)
Now let us turn to the case where the true prices are unobserved. First, remark that when
observed prices are different from the true prices (for example when posted prices are used
instead of transaction prices), the former approach is not valid in general. To see this,
X,d
consider the simple logit model, where π0X,d , ΣX,d
and ΣX,d
are known to be zero. In
0 , π0
0
this case δjd (sd , pd ; θd ) takes the simple form
δjd (sd , pd ; θd ) = ln sdj − ln sd0
and does not depend on pd . In this context, using posted prices p instead of the true prices
amounts to relying on
ξjd (p; θd ) = ln sdj − ln sd0 − Xj β d − αd pj ,
instead of relying on ξjd (pd ; θd ). The problem comes from the fact that pj − pdj is not a
classical measurement error. The true price depends on the characteristics of the good and
of the cost shifters. If, for instance, a group of consumer values particularly the horsepower of automobiles, powerful cars will be priced higher for this group, and pj − pdj will be
negatively correlated with horsepower. Because horsepower is one of the instruments, we
have E[Zj (pj − pdj )] 6= 0, and E Zj ξjd (p; θ0d ) is no longer equal to zero. In the general ran-
9
dom coefficient model, this issue also arises but in addition to that, δjd (sd , pd ; θd ) generally
depends on pd . Thus, Zj is also correlated with δjd (sd , pd ; θd ) − δjd (sd , pdj ; θd ).
Instead of simply replacing pd by p, we use the supply model and reasonable identifying conditions on marginal costs and posted prices to recover the transaction prices. To
operationalize this idea, we impose the two following assumptions.
Assumption 1. (Constant marginal costs across consumers) For all d and j, cdj = cj .
Assumption 2. (Posted prices as maximal prices) For all j, pj = maxd=1...nD pdj .
Assumption 1 amounts to neglecting differences in the costs of selling to different consumers
in the total cost of a product. This assumption is likely to be satisfied in many settings,
such as the automobile market, where most costs stem from producing, not selling the
goods. Assumption 2 supposes that firms post the highest discriminatory price and then
offer some discounts according to observable characteristics of buyers in order to reach
optimal discriminatory prices. In other words, we reinforce the very mild condition that
pj ≥ pdj for all d by assuming that for each product j, there is a group dj , called the
d
pivot group hereafter, that pays the posted price, pj j = pj . This assumption is necessary
since otherwise, we could shift all discounts by an arbitrary constant. It is also in line
with empirical evidence on the automobile market (for France, see, e.g., L’Automobile en
Europe: 5 Leviers pour Rebondir 2013). Note, however, that the pivot group is neither
supposed to be known ex ante nor constant across different products. We also consider
alternative conditions to Assumption 2 below.
Let us first present our method in the simple case of the logit model. As explained above,
the idea is to compute, for a given value of the parameter θ = (θ1 , ..., θnD ), the transaction prices pdj (θ) that rationalize the market shares and the supply-side model. Precisely,
Equation (2) and Assumptions 1-2 imply that
pj = cj + max
˜
d=1...n
D
˜ −1
Ωdf
˜
sdf
,
(5)
j
where [.]j indicates that we consider the j-th line of the vector only. Then, the discriminatory prices satisfy
pdj
= pj − max
˜
d=1...n
D
−1
d˜
Ωf
˜
sdf
+
j
h
−1
Ωdf
sdf
i
.
(6)
j
Now, under the logit model, ∂sdj /∂pdj = −αd sdj (1 − sdj ) and ∂sdj /∂pdj0 = −αd sdj sdj0 . As a
result, Ωdf is a function of observed market shares and of α = (α1 , ..., αnD ) only. In turn,
10
pdj can then be expressed simply as a function of α, using Equation (6). Denoting it by
pdj (α), we obtain, using ξjd (p; θd ) = ln sdj − ln sd0 − Xj β d − αd pdj (α), the moment equations
"
E Zj
!#
sdj
ln d − Xj β d − αd pdj (α)
= 0.
s0
Compared to the logit model with observed prices, the only difference is that we have
to compute pdj (α) using (6). For a given α, β d can be easily obtained by two-stage least
squares, as usually. But we still have to solve a nonlinear optimization over α ∈ RnD .
Let us turn to the general random coefficient model, for which the method is essentially
similar but an additional issue arises. Equation (6) shows that for a given parameter θ,
the discriminatory prices are identified up to Ωdf . Now, taking the derivative of the market
share function (Equation (1)) with respect to the price pdj yields:
∂sdj d
(p ) =
∂pdj
Z p
d
α0d + π0p,d e + Σp,d
u
sdj (e, u, pd )(1 − sdj (e, u, pd ))dPE,ζ
(e, u)
0
(7)
We obtain a similar expression for ∂sdj /∂pdl (pd ). These expressions show that Ωdf only
depends on the parameters θ0d , on the vector of prices pd and on δ d = (δ1d , ..., δJd ), through
sdj (e, u, pd ). We emphasize this dependence by denoting it Ωdf (θ0d , pd , δ d ). Besides, for a set of
prices pd , we can obtain by inverting the market share system the vector δ d of mean utilities.
Hence, to obtain the discriminatory prices for a given vector of parameter θ = (θ1 , ..., θnD ),
we need to solve a system of non-linear equations in (δ, p), where δ = (δ 1 , . . . , δ nD ) and
p = (p1 , . . . , pnD ) denote respectively the full vector of transaction prices. We suppose
hereafter that this system of equations admits a unique solution.
Assumption 3. (Uniqueness of (δ, p)) For any θ and vector of market shares s = (s1 , ..., snD ),
there is a unique (δ, p) satisfying, for all j ∈ {1, ..., J} and d ∈ {1, ..., nD },
sdj
pdj
exp δjd + µdj (e, u, pdj )
Z
=
PJ
= pj −
δkd
d
(e, u),
dPE,ζ
+ µdk (e, u, pdk )
k=0 exp
˜ ˜ ˜ ˜ −1
max
Ωdf (θd , pd , δ d )
˜
d=1...n
D
˜
sdf
+
j
h
Ωdf (θd , pd , δ d )
(8)
−1
sdf
i
.
(9)
j
This assumption is satisfied in the special case where there is no unobserved heterogeneity
on price sensitivity, so that αid = α0d . In such a case, µdj (e, u, pdj ) does not depend on the
transaction price pdj anymore. As a result, the right-hand side of Equation (8) defining
market shares only depends on δ d . By the result of Berry (1994), there is a unique δ d
11
which solves this system. Turning to the price equation, Ωdf (θd , pd , δ d ) does not depend, for
the same reason, on pd . Therefore, the right-hand side of Equation (9) does not depend on
pd , and there is indeed a unique pdj satisfying this system of equations. We conjecture that
this result remains true at least for models where the heterogeneity coefficients on prices,
π p,d and Σp,d are relatively small. Finally, Assumption 3 is related but not equivalent to
the uniqueness of the Nash-Bertrand equilibrium in prices. Even if the pivot groups were
known, in which case we would identify directly the marginal costs and therefore would
have to solve for the prices in the standard supply-side first-order conditions, the equations
would still differ from those of the Nash-Bertrand equilibrium.
Under Assumption 3, we can apply the GMM to identify and estimate θ0 = (θ01 , ..., θ0nD ).
Let δjd (s, θ) and pdj (s, θ) denote the mean utility and price of product d when market shares
and the vector of parameters are respectively equal to s and θ. Let also
MJd (θ)
J
1X
=
Zj δjd (s, θ) − Xj β d − αd pdj (s, θ)
J j=1
denote the empirical counterpart of the moment conditions corresponding to Equation (4).
Let MJ (θ) = (MJ1 (θ)0 , ..., MJnD (θ)0 )0 and define
QJ (θ) = MJ (θ)0 WJ MJ (θ),
where WJ is a positive definite matrix. Our GMM estimator of θ0 is then
θb = arg min QJ (θ).
θ
(10)
As in the standard BLP model, it is possible to include moments corresponding to the
supply side by imposing some additional structure on marginal costs. Let X s be the vector
of cost shifters. Xjs may be different from Xj but typically share some common components.
We may suppose for instance that the marginal costs are log-linear:
ln(cj ) = Xjs γ + ωj ,
(11)
where ωj stands for the unobserved cost shock. This shock is supposed to satisfy E[Zjs ωj ] =
0, where Zjs denotes a vector of instruments for the supply side. As for the demand, we
construct the moment conditions by first recovering the marginal cost cj (s, θ) associated
12
to s and a given vector of parameter θ. Specifically, by Equation (5),
cj (s, θ) = pj − max
d=1...nD
h
−1 d i
Ωdf (θd , pd (s, θ), δ d (s, θ))
sf .
j
We then obtain ωj (s, θ, γ) simply by
ωj (s, θ, γ) = ln (cj (s, θ)) − Xjs γ.
The supply-side moment conditions are then
MJs (θ, γ) =
J
1 X s
Zj ln (cj (s, θ)) − Xjs γ
J j=1
Then we can proceed as previously, simply replacing MJ (θ) by MJ (θ, γ) = (MJ1 (θ)0 , ...,
MJnD (θ)0 , MJs (θ, γ))0 .
Compared to the estimation of the standard BLP model, estimating our model in practice
raises two challenges. First, we have to optimize over a larger space than in the BLP
setting. In the standard BLP model where we observe the market share of j for each group
d but true prices are observed or supposed to be equal to posted prices, we could optimize
only on θd (abstracting from supply-side conditions), for each group separately. We even
p,d
p,d
only need to optimize over (α0d , π0X,d , ΣX,d
0 , π0 , Σ0 ), because we can easily concentrate the
objective function on β d , by running ordinary least squares of the δjd on (Xj ). In our case,
0
0
we cannot estimate θd separately from θd , for d0 6= d, because θd matters for determining
pdj (s, θ) (see Equation (6)).3 Second, for each θ, we need to solve not only Equation (8),
but also simultaneously Equation (9), in order to obtain both the mean utilities and the
prices. Therefore, estimating the model is computationally more costly. We describe in
details our algorithm in section 4 and show that this optimization problem remains feasible
in a reasonable amount of time.
We can also reduce the computational cost by considering restricted versions of the model.
In particular, things are significantly simpler when assuming no heterogeneity on price
sensitivity within a group of consumers, so that αid = α0d . This assumption may be reasonable in particular if we have a fine segmentation of consumers. In this case, we still
have to optimize over θ = (θ1 , ..., θnD ). On the other hand, solving the system defined by
Equations (8)-(9) is easy. Equation (8) reduces to the standard inversion of market shares,
while Equation (9) provides an explicit expression for transaction prices, since Ωdf does
3
On the other hand, and as in the BLP model, we can concentrate the objective function on β d .
13
not depend on pd . Thus, the computational cost is significantly reduced compared to the
general model. Another alternative is to rely on the logit or nested logit models. In the
simple logit model, we have seen above that the matrix Ωdf only depends on (α1 , ..., αnD ).
In the nested logit, it also depends on the parameters (σ 1 , ..., σ nD ) that drive substitutions
within nests. But at the end, we also obtain a quite simple nonlinear optimization over
(α1 , σ 1 , ..., αnD , σ nD ) only.
3.2
3.2.1
Extensions
Other functional forms on price effects
We have assumed up to now, following the common practice, that indirect utilities depend
linearly on disposable income, namely on αi (yi − pj ), where yi denotes the income before
making one’s choice. αi yi can then be removed, as being constant across alternatives.
To incorporate, for example, credit constraints as in BLP, the indirect utility may rather
depend on αi ln(yi − pj ). Let us suppose, more generally, that the utility depends on
disposable income through q(yi − pj , αi ) where q is known by the econometrician while
αi |Di = d ∼ N (αd , σα2d ) with (αd , σα2d ) unknown. Our methodology also applies to this
setting. In such a case, one has to include entirely q(yi − pj , αi ) into µdj (Ei , ζi , pdj ), with
yi being one component of Ei . Then Equations (8) and (9) remain unchanged, the only
difference being that the terms entering into Ωdf do not satisfy Equation (7). But other than
that, the construction of the moment conditions follows exactly the same methodology.
3.2.2
Discrimination based on unobserved characteristics
The econometrician may not have access to all information available to the seller when
price discriminating the buyer. Gender and race may be important examples. It is still
possible to apply our methodology as long as instruments for such variables are available.
e such that (i) (ζi , εdij ) ⊥⊥ D
e
Specifically, suppose that we observe a discrete variable D
e term is the probability of belonging to group d
and (ii) the matrix P which typical (d, d)
˜ P (D = d|D
e has rank nD . Condition (i) is an exclusion
e = d)
conditional on observing d,
restriction which imposes that consumers do not differ systematically in their taste across
e once we control for D. Condition (ii) is similar to the standard relevance
categories of D,
e is, basically, related to D. Let Yi denote the
condition in IV models and imposes that D
14
product choice of consumer i. Under the first condition, we have
e =
e i = d)
P (Yi = j|D
nD
X
e i = de P (Yi = j|Di = d, D
e i = d)
P Di = d|D
d=1
=
nD
X
e i = de sdj .
P Di = d|D
d=1
e i = 1), ..., P (Yi = j|D
e i = n e ))0 , we have,
Then, letting sj = (s1j , ..., snj D )0 , esj = (P (Yi = j|D
D
for all j = 1...J,
esj = Psj
Because P has rank nD , this equation in sj admits a unique solution. This implies that sj
is identified. We can then apply the methodology above, using these market shares.
As an example of this IV approach, consider a scenario where the econometrician observes
the buyers’ professions while sellers price discriminate based on buyers’ incomes. In this
context, we observe market shares of products by professional activity. The rank condition
means that we know the probability of belonging to an income class conditional on the
professional activity. From this probability matrix, we are able to compute market shares
of products by income class. The exclusion restriction imposes that the differences in
preferences across professional activities only reflect the differences across income classes.
3.2.3
Alternative conditions on costs and prices
Our methodology relies crucially on two conditions. First, one of the group of consumers
should pay list prices, which are observed. Second, the marginal costs should be identical
for all groups. Another crucial assumption concerns the nature of competition on the
market, which we assume to be Bertrand competition. We believe that these conditions
are realistic in many settings. In some cases, however, alternative conditions may be more
natural. Our method still applies if these alternative conditions allow us to recover the
marginal costs of each product, for a given value of the parameter vector. Once we obtain
these marginal costs, we can compute the transaction prices for each consumer group,
using the first-order conditions associated to the profit maximization, given the nature of
competition.
A simple example is when we observe, through survey data for instance, the price paid by
15
at least one group for each product. Then, instead of using Equation (6), we can rely on
pdj
=
d
pj j
−
d
Ωfj
−1
d
sf j
+
h
Ωdf
−1
sdf
i
j
,
j
where dj denotes the group for which the price of j is observed.
PnD d d
Similarly, suppose that we observe the average price pm
j =
d=1 sj pj paid by all consumers
for each product, through, for example, aggregated data on sales from the firms. Then we
replace Equation (6) by
pdj
=
pm
j
−
nD
X
0
sdj
0
Ωdf
−1
+
j
d0 =1
3.2.4
0
sdf
h
−1
Ωdf
sdf
i
.
j
Alternative supply-side models
For a given a set of demand parameters θd , we expressed the corresponding transaction
prices pd (θd ) using supply-side conditions. Once these transaction prices are recovered, we
can use the standard BLP method to compute ξjd (pd (θd ); θd ) and then the moment conditions E[Zj ξjd (pd (θd ); θd )], to check whether they are equal to zero or not. The assumption
about the nature of competition on the market is therefore more crucial in our model than
in the standard BLP approach. Because pdj is unobserved and depends on the behavior of
firms, it is impossible to estimate the demand without making assumptions on the supply
side in our setting. We do not see this as a strong limitation, however, because the supply
side is usually modelled, as it is crucial to perform counter-factual analysis. Following BLP,
we have assumed up to now that firms are involved in a price competition game. But our
methodology also applies to other supply-side models.
First, the identification strategy holds when there is collusion between sellers. Only the
term Ωdf is modified to take into account the fact that the prices of all products are set
by the same decision-maker. Thus, our identification argument is valid in this framework.
Given the parameter value of θ, we can identify the pivot group and compute the prices for
the other groups. Our methodology also applies when the supply-side model incorporates
the vertical relations between producers and retailers. For instance the papers by Brenkers
& Verboven (2006), Mortimer (2008) and Bonnet & Dubois (2010) develop and estimate
structural models of demand and supply including vertical contracting between producers
and retailers. Our methodology can still be applied for such models. Moreover, only the
type of competition in the downstream market matters for recovering transaction prices.
For example, if we assume a Nash-Bertrand equilibrium in the downstream market, using
16
Equation (5) we can recover the marginal cost crj of retailer r by
crj (s, θ) = pj − max
h
d=1...nD
−1 d i
Ωdf (θd , pd (s, θ), δ d (s, θ))
sf .
j
Then, as before, it is possible to compute the discriminatory prices using Equation (6).
4
Estimation algorithm and simulations
In this section, we provide additional discussion on how to compute our GMM estimator
in practice and present some simulation results. First, note that we do not rely on the
minimization program with equilibrium constraints (MPEC) approach suggested by Dubé
et al. (2012) because the gradient and hessian of the constraints cannot be obtained analytically easily in our model. Rather, we use the standard approach where for each value
of θ, we solve for the system of non-linear equations given by (8)-(9). For that purpose,
we use the following iterative procedure:
1. Start from initial values for pd for all groups, use for example the posted price p̄, or
draw a vector of initial discounts.
2. Given the current vector of transaction prices pd , compute δ d = δ(sd , pd ; θd ). We can
use for that purpose the contraction mapping suggested by BLP.
3. Given the current vector of mean utilities, compute the corresponding matrix Ωdf and
update the transaction prices, using Equation (6).
4. Iterate 2 and 3 until convergence of prices.
The construction of the moment conditions therefore involves two nested inner loops, the
first one, the price-loop searches over the vector of prices for every demographic group pd .
Inside the price-loop, we have the delta-loop that searches over the mean utilities δ d . For
each value of transaction prices, we have to invert the market share equation to solve for
the mean utility vectors δ d . We use for that purpose the contraction mapping proposed
by BLP. If the computational cost is larger than for the BLP estimator, it is possible to
parallelize this market share inversion as well as the computation of the mark-up terms
((Ωdf )−1 sdf ), as they are independent across markets and demographic groups. We can also
save time by updating the initial values for the mean utilities after each iteration of the
inner price-loop and by updating initial values of prices across iterations of the outer loop
that involves the parameters θ.
17
In the simulations below and in the application, we use the following specifications for
computing the GMM estimator. First, to approximate the aggregated market shares, we
use Halton normal draws for each demographic group and market (300 in the simulations
and 1,000 in the application) and the line search algorithm for minimization. Our initial
values for the price sensitivity parameters are the estimates obtained with the simple logit
model, while we use random draws from a uniform U [−1/2, 1/2] distribution for the random
coefficient. As suggested by Dubé et al. (2012) and Knittel & Metaxoglou (2014), we set
a tight tolerance (10−12 ) to compute the mean utilities and the prices, while the tolerance
levels are 10−5 for the parameters and 10−3 for the objective function. Finally, as suggested
by Knittel & Metaxoglou (2014), we carefully investigate potential convergence issues by
using different starting values and selecting the estimates that yield the lowest value of the
objective function.
To investigate the performance of our estimator and whether the algorithm produces reliable results, we perform a Monte-Carlo simulation. We construct 50 different data sets
for T = 25 markets, J = 24 products and D = 4 demographic groups. For each market
and product, we construct the vectors of observed characteristics Xjt = (1, X1jt ), unobd
served characteristics ξjt
, observed cost shifters Wjt = (W1jt , W2jt , W3jt ) and unobserved
cost shifters ωjt . The marginal cost of j then satisfies
cj = 0.7 + 0.7X1jt + W1jt + W2jt + W3jt + ωjt .
We suppose that X1jt is drawn from a uniform distribution U [1, 2] and Wjt follows a trivarid
ate uniform distribution. ξjt
and ωjt are independent draws from the normal distribution
N (0, 0.1). The parameters of preferences are summarized in Table 1. Groups of consumers
are heterogeneous with respect to their average valuation of product attributes and the
price sensitivity. Group 1 is the less price sensitive group and has the highest utility of
holding a car, so it is likely to be the pivot group in the model with price discrimination.
To decrease its chance to be pivot, we assumed that Group 1 has a lower valuation of the
exogenous characteristics (the valuation is set to 1.5 versus 2 for all the other groups). As
in our application, the unobserved heterogeneity parameters (σ) are identical for the four
demographic groups. Finally, we assume that the market is composed by 4 firms, each of
them producing 6 products. Once we solve for prices and market shares (sdjt , pdjt )d=1,2,3,4 ,
we define for each product the posted price p̄j as the maximal price across demographic
groups. We use for estimation the instruments Zjt = (Xjt , Wjt ).
18
Group 1
Group 2
Group 3
Group 4
Random coeff.
Proportion
0.3
0.2
0.3
0.2
Intercept
-1
-1
-0.5
-0.5
X1
1.5
2
2
2
0.5
Price
-1.5
-2.5
-2
-3
0.4
Table 1: Parameters of preferences for the simulations
We compare the estimates of the price discrimination model with the standard model
assuming uniform pricing. Specifically, we assume in the latter case that the supply-side
first-order conditions are
pj = cj + (Ωf )−1 sf j ,
where Ωf is the matrix of typical (i, j) term equal to −∂sj /∂pi and sj =
(12)
PD
d=1
P (D = d)sdj .
These first-order conditions correspond to the maximization of profits under the constraint
that all groups of consumers pay the posted price. The results are displayed in Table 2.
We observe that the GMM estimator corresponding to the model with price discrimination
accurately estimates both the demand supply parameters. The pivot groups are exactly
guessed and the estimated discounts are very close to the true underlying discounts. On
the opposite, the performances of the uniform pricing model are not as good, leading in
general to an underestimation of the price sensitivity parameters. For all the parameters,
the root mean squared errors (RMSE) appear to be higher for the uniform pricing model
than for the true model with price discrimination. The parameters of the intercept appear
to be especially sensitive to misspecification. On the supply side, it is interesting to note
that apart from the parameter of the intercept, the cost equation is well estimated under
the two alternative models. The GMM objective function value is, however, much lower
for the model with unobserved price discrimination than for the uniform pricing model.
For both models the estimation algorithm converged for every replication. As expected,
the GMM estimator corresponding to our model is more computationally intensive than
the GMM estimator of the standard BLP model. On average, it is around 3.5 times slower
than the standard uniform pricing model. However, the number of iterations is roughly
the same and the estimation time remains decent because it is possible to parallelize the
computationally intensive part of the estimation algorithm.
19
True
Price sensitivity
Group 1
-1.5
Group 2
-2.5
Group 3
-2
Group 4
-3
sigma
0.4
Intercept
Group 1
-1
Group 2
-1
Group 3
-0.5
Group 4
-0.5
Exogenous characteristic
Group 1
1.5
Group 2
2
Group 3
2
Group 4
2
sigma
0.5
Marginal cost equation
Intercept
0.7
X1
0.7
W1
1
W2
1
W3
1
Average discount (in %)
Group 1
0.03
Group 2
10.90
Group 3
7.13
Group 4
13.77
Frequency pivot (in %)
Group 1
98.7
Group 2
0
Group 3
1.3
Group 4
0
Mean objective function value
% replications converging
Number of iterations
Time (sec)
Discrimination BLP
Mean
Bias
RMSE
Uniform BLP
Mean
Bias
RMSE
-1.5
-2.5
-2
-3
0.4
0
0
0
0
0
0.05
0.09
0.06
0.1
0.04
-1.46
-2.36
-1.89
-2.83
0.35
0.04
0.14
0.11
0.17
-0.05
0.16
0.27
0.2
0.31
0.16
-1
-0.99
-0.5
-0.5
0
0.01
0
0
0.1
0.15
0.11
0.15
-1.08
-0.51
-0.32
0.32
-0.08
0.49
0.18
0.82
0.4
0.68
0.45
0.94
1.5
2
2
2
0.49
0
0
0
0
-0.01
0.06
0.08
0.06
0.12
0.09
1.46
1.96
1.96
1.98
0.46
-0.04
-0.04
-0.04
-0.02
-0.04
0.21
0.33
0.22
0.42
0.33
0.7
0.7
1
1
1
0
0
0
0
0
0.04
0.02
0.02
0.02
0.02
0.86
0.73
0.99
0.99
0.99
0.16
0.03
-0.01
-0.01
-0.01
0.17
0.04
0.03
0.03
0.03
0.03
10.90
7.12
13.76
98.6
0
1.4
0
0.16
100
457
551
1.19
100
466
167
Table 2: Simulation results when the true model is the model of price discrimination
Furthermore, we also compare the uniform pricing model with our model when the uniform
pricing model is the true one. We use the same values of parameters of demand and supply
except that prices are no longer group-specific but optimally set by firms given the global
demand that arises from the heterogeneous groups of consumers. Table 3 summarizes the
estimation results over the 50 replications. As expected, the bias of our GMM estimator
is larger than the standard BLP GMM estimator. The price sensitivity parameters are
underestimated and the parameters of the intercept estimates corresponding to the unobserved discrimination model exhibit a large bias. The model with unobserved discounts
yields positive and significant discount, which is natural since the values of discounts are
pinned down by the differences in price sensitivities across demographic groups. As before,
the value of the objective function is lower under the true model, namely the model with
uniform pricing.
20
Price sensitivity
Group 1
Group 2
Group 3
Group 4
sigma
Intercept
Group 1
Group 2
Group 3
Group 4
Exogenous characteristic
Group 1
Group 2
Group 3
Group 4
sigma
Marginal cost equation
Intercept
X1
W1
W2
W3
Average discount (in %)
Group 1
Group 2
Group 3
Group 4
Frequency pivot (in %)
Group 1
Group 2
Group 3
Group 4
Mean objective function value
% replications converging
Number of iterations
Time (sec)
True
Uniform BLP
Mean
Bias
RMSE
Discrimination BLP
Mean
Bias
RMSE
-1.5
-2.5
-2
-3
0.4
-1.48
-2.48
-1.98
-2.98
0.39
0.02
0.02
0.02
0.02
-0.01
0.07
0.1
0.07
0.1
0.06
-1.35
-2.27
-1.85
-2.83
0.24
0.15
0.23
0.15
0.17
-0.16
0.21
0.36
0.25
0.33
0.22
-1
-1
-0.5
-0.5
-1.05
-1.05
-0.55
-0.54
-0.05
-0.05
-0.05
-0.04
0.19
0.22
0.18
0.17
-1.4
-2.24
-1.28
-1.99
-0.4
-1.24
-0.78
-1.49
0.57
1.36
0.9
1.6
1.5
2
2
2
0.5
1.5
2.01
2
2
0.48
0
0.01
0
0
-0.02
0.04
0.06
0.04
0.09
0.09
1.5
1.96
1.97
1.9
0.39
0
-0.04
-0.03
-0.1
-0.11
0.11
0.22
0.15
0.29
0.27
0.7
0.7
1
1
1
0.69
0.7
1
1
1.01
-0.01
0
0
0
0.01
0.04
0.02
0.02
0.02
0.02
0.45
0.7
1.07
1.08
1.08
-0.25
0
0.07
0.08
0.08
0.27
0.03
0.08
0.09
0.09
0.01
10.9
7.2
13.9
0.22
100
464
170
99.53
0
0.47
0
1.08
98
493
638
Table 3: Simulation results when the true model is the standard BLP model
Finally, using the DGP with unobserved price discrimination, we perform a check of the
unicity condition of Assumption 3. For that purpose, we compute the value of the transaction prices for 50 different initial values of prices and the true value of the parameters using
the algorithm detailed above. Under Assumption 3, we should expect to obtain the same
transaction prices for each of these initial values, whenever the algorithm converges. Moreover, these transaction prices should correspond to the true transaction prices of the model.
We draw initial values of transaction prices equal to R × pj , where R ∼ U [0.25, 1]. The algorithm always converged to the true value of the transaction prices. Besides, convergence
occurs very quickly. We computed, at each iteration of the price-loop, the average and
maximal absolute differences between the true prices and those obtained by the algorithm,
across all products. We then averaged these average and maximal absolute differences over
the 50 initial draws. The results, displayed in Table 4, show that the sequence of vectors
of prices converges very quickly to the true vector.
21
Iteration
Average
Maximal
1
1.28
3.92
2
0.052
0.29
3
0.0014
0.011
4
4.2×10−5
6.1 ×10−4
5
1.8 ×10−6
4.1×10−5
Lecture notes: “average” (resp. “Maximal”) is the average (resp. maximal) absolute differences between the true prices and those obtained by
the algorithm across all products. The figures are average over the 50 simulations. The average true price here is 3.87, with a range of [2.07; 5.80].
Table 4: Average and maximal price difference across iterations
5
Application to the French new car market
5.1
Description of the data
We apply our methodology to estimate demand and supply together with unobserved
discounts in the new automobile industry, using a dataset from the association of French
automobile manufacturers (CCFA, Comité des Constructeurs Français d’Automobiles) that
records all the registrations of new cars purchased by households in France between 2003
and 2008. Each year, we observe a sample of about one million vehicles. For each registration, the following attributes of the car are reported: brand, model, fuel energy, car-body
style, number of doors, horsepower, CO2 emissions, cylinder capacity and weight. These
characteristics have been complemented with fuel prices to compute the cost of driving
(in euros for 100 kilometers). Automobile sellers are well known to price discriminate,
negotiate or to offer discounts to close the deal. But as in our theoretical model, we only
observe here posted prices that come from manufacturers catalogs.
We now turn to the construction of the consumer groups that are used by firms to price
discriminate. Apart from car attributes, the date of the registration and some characteristics of the owner are provided in the CCFA database : municipality of residence and
age. The age (or the age class) is presumably a strong determinant of purchase, and is
easily observed by a seller even if he does not know the buyer before the transaction. We
therefore assume that these characteristics are used by the automobile makers to price
discriminate. The income is also likely to affect preferences for different car attributes and
price sensitivity. The income is, however, likely to be unobserved by the seller but instead
inferred from the municipality the buyer lives in and the age class. We compute a predictor of buyer’s income, namely the median household income in his age class and in his
municipality using data from the French national institute of statistics (Insee).4 It seems
4
There are over 36,000 municipalities in France. Note also that Paris, Lyon and Marseille, the three
22
reasonable to assume that the seller does not have a far better prediction of the buyer’s
income in such anonymous market, where buyers and sellers do not know each other before
the transaction. It is crucial for our approach that buyers cannot lie about their individual
characteristics, and in our application it implies that buyers do not make geographical
arbitrage, i.e. buy the car in another municipality where discounts are higher. We believe
that this assumption is reasonable since buyers have high incentive to buy a new car at a
close dealer to minimize transportation costs and take advantage of the after-sale services
and guarantees. We thus define groups of buyers by interacting three age classes and two
income classes.5 We choose the common thresholds of 40 and 60 for the age classes, and
27,000 euros per year as the threshold for income. This amount corresponds roughly to
the median yearly income in France in 2008.6
Group
Age < 40, income <27,000
Age < 40, income ≥ 27,000
Age ∈ [40,59], income <27,000
Age ∈ [40,59], income ≥ 27,000
Age ≥ 60, income <27,000
Age ≥ 60, income ≥ 27,000
Frequency
15.7%
11.5%
16.3%
22.3%
20.8%
13.2%
Table 5: Definition of the groups of consumers and frequency
As usual, when defining the groups of consumers, we face a trade-off between realism (it
is likely that firms discriminate along several dimensions) and accuracy of the observed
proportion of sales sbdj as estimators of the true market shares sdj . The six groups that we
consider are large enough to avoid in most cases the problem of zero sales (see Table 17
in Appendix A.2 for the fraction of products with null market shares). Moreover, rather
than discarding those products, we replace the proportion of sales by a predictor of sdj that
minimizes the asymptotic bias, namely sbdj =
ndj +0.5
,
Nd
ndj denoting the number of sales of
product j in group d and N d the number of potential buyers with characteristics d (see
Appendix A.2 for details). Note that another simple correction of the basic market shares
largest cities, are split into smaller units (“arrondissement”). As a result, the heterogeneity in the median
income across municipalities is large.
5
We do not observe owners’ gender in our database. Even if this information was available, it would be
hard to use since the owner and the buyer can be different persons. Furthermore, many couples are likely
to buy their car together.
6
We estimate our model with alternate thresholds. The results, which are overall very similar, are
available upon request.
23
estimator has been proposed by Gandhi et al. (2013). We show in Appendix A.2 that our
results are robust to the choice of the market shares correction.
We define a product as a brand, model, segment, car-body style and fuel type. A total of
3205 products for the six years is obtained. Table 6 presents the average characteristics of
new cars purchased for each group of consumers. We find significant heterogeneity across
these groups. On average, the medium age, high income class purchases more expensive
vehicles. They also choose larger and more powerful cars. Young purchasers are more
interested in smaller cars (lighter and with three doors) whereas station-wagons are more
popular among the medium age class. The highest age group purchases lighter vehicles
than medium age classes, but these vehicles are on average less fuel efficient.
Consumer group
Price
Fuel cost
HP
Weight
Three doors
Station wagon
A < 40, I <27,000
A < 40, I ≥ 27, 000
19,803
20,911
6.2
6.5
5.7
6.0
1,182
1,221
19.0%
16.8%
9.7%
12.9%
A ∈ [40,59], I <27,000
A ∈ [40,59], I ≥ 27, 000
21,521
21,739
6.5
6.8
6.1
6.2
1,231
1,236
14.3%
14.8%
12.7%
13.1%
A ≥ 60, I <27,000
A ≥ 60, I ≥ 27, 000
20,117
20,831
6.9
7.0
5.9
6.0
1,194
1,219
11.4%
10.9%
8.9%
10.5%
Lecture notes : A represents the age class and I the income class. Prices are in constant
(2008) euros, fuel cost is the cost of driving 100 kilometers, in constant (2008) euros, HP
stands for horsepower, weight is in kilograms.
Table 6: Average characteristics of new cars purchased across groups of consumers
The dataset does not contain any information on the distribution network, and thus the
distribution sector is not modeled in this application. We make the traditional assumption
that manufacturers have only exclusive dealers and are perfectly integrated. As discussed
in the previous section, adding vertical relations between manufacturers and dealers would
be possible as long as the competition on the downstream market still implies a NashBertrand equilibrium. We also suppose that prices are set at the national level, which is
consistent with the fact that listed prices are set by manufacturers at such a level. With
sufficient observations on sales at the dealer level, and individual characteristics of dealers
(location and brands offered), we would be able to take into account heterogeneity of
pricing strategy and competition intensity (see, e.g., Nurski & Verboven 2012). Due to a
lack of such available data, we abstract from these issues afterwards.
24
5.2
Parameter estimates and comparison with the standard model
We first present the estimations of different models. We estimate the nested logit model
with and without unobserved price discrimination. We also estimate the standard BLP
model with uniform pricing, where the first-order conditions of the supply side is (12), and
the BLP model with unobserved price discrimination. In models with uniform pricing, we
assume that sellers do not price discriminate and that the posted prices, which correspond
to the transaction prices, are optimal given the heterogeneous preferences of the different
groups of consumers. For all specifications, we control for the main characteristics of the
cars such as horsepower, weight and the cost of driving 100 kilometers in the demand
function. We also introduce dummies for station-wagon car-body style and three doors.
Finally, we introduce year and brand dummies that are constrained to be identical for
all demographic groups. For the two random coefficient models, we allow for unobserved
heterogeneity of preferences inside groups of consumers in terms of price, fuel cost and for
the utility of buying a new car, represented by the intercept. To obtain more accurate
results, we constrain the heterogeneity parameters to be identical for all demographic
groups. In the marginal cost equation, we use horsepower, fuel consumption (in liters for
100 kilometers) and weight as cost-shifters. We also introduce brand dummies to control for
manufacturer’s specific unobserved quality of cars. Finally, the nested logit model requires
a segmentation of the market. We take, as in the literature, a segmentation according to
the main use of the car. See Appendix A.3 for more details.
All the models are estimated using the GMM approach, relying both on the moment
conditions stemming from the demand and from the marginal cost equation (Equation
(11)).7 The implementation of the estimation follows the method described in Section
4. We also verify that Assumption 3 is satisfied at the estimated value of parameters
by applying the test performed in the simulation analysis and find that after drawing
several initial values of transaction prices, the algorithm always converges to the same value
of estimated transaction prices. In addition to exogenous characteristics we include the
following instruments. The first is the number of kilometers per fuel liter (“fuel inefficiency”
in Table 7 below), which replaces fuel cost in the marginal cost equation. The second is the
car weight multiplied by a composite price index that aims at approximating the average
input price.8 The other instruments are close to those suggested by BLP. We include the
7
We also estimated the models without using the moment conditions stemming from the marginal costs
equation. The results are very similar.
8
Specifically, we use a weighted average of steel, aluminium and plastic prices taken in January. The
weights we use are equal to 0.77, 0.11 and 0.12, respectively, reflecting the relative importance of each of
these inputs in car manufacturing.
25
sum of continuous exogenous characteristics (namely weight, horsepower and fuel cost)
of other brands’ products. We also consider the sums of these characteristics over other
brands’ products of the same segment, supposed to be closer substitutes. Finally, we
include the sums of these characteristics of the other products of the brand belonging to
the same segment.9
The results for the different models are presented in Table 7. Columns (1) and (2) present
the estimation results for the nested logit specifications, which abstract from individual
heterogeneity, while Columns (3) and (4) display the estimation results for the random
coefficient models. The estimated parameters are generally similar for the nested logit
and the random coefficient models. Note that for two groups (the old purchasers with low
and high income), we obtain negative intra-segment correlation, which is absurd since this
parameter should belong to [0, 1]. Thus, we constrain these two parameters to be equal
to zero in the estimation, which amounts to consider the logit specification for these two
groups of consumers. The random coefficient models imply higher price sensitivities than
the nested logit models and significant within-group individual heterogeneity. We obtain
a standard deviation of 1.12 for the model with uniform pricing and 0.95 for the model
with price discrimination. We thus discuss in more detail the results for the models with
random coefficient while the results of the nested logit specification serve as a benchmark
to check the general credibility of the models with individual unobserved heterogeneity.
9
Armstrong (2014) has recently shown that such instruments could be weak when the number of
products is large. Note however that identification is secured here by the inclusion of the cost shifters.
Nonetheless, we checked that the instruments are indeed relevant for prices. We use for that purpose
the F-statistic of the joint nullity of the coefficients of these instruments in the linear regression of prices
on the characteristics and these instruments. We obtain F ' 24.1, which is far above the threshold of
10 suggested by Staiger & Stock (1997) and usually used to detect weak instruments. This is therefore
reassuring on the identification of the model and the validity of inference here.
26
Price sensitivity
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Std. dev. (σ p )
Intra-segment correlation
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Intercept
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Std. dev (σ x )
Fuel cost
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Std. dev (σ x )
Horsepower
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Weight
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Three doors
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Station-wagon
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Marginal cost equation
Intercept
Horsepower
Fuel inefficiency
break
Three doors
Weight × average input price
Value of objective function
Number of observations
Significance levels :
Nested-logit
Uniform
Price discrimination
Parameter
Std-err
Parameter
Std-err
Random Coefficients
Uniform
Price discrimination
Parameter Std-err
Parameter Std-err
-2.69∗∗
-2.55∗∗
-2.24∗∗
-2.16∗∗
-1.95∗∗
-1.79∗∗
0.18
0.173
0.174
0.169
0.179
0.172
-2.70∗∗
-2.55∗∗
-2.24∗∗
-2.16∗∗
-1.98∗∗
-1.88∗∗
0.185
0.176
0.177
0.178
0.185
0.105
-4.51∗∗
-4.27∗∗
-3.87∗∗
-3.68∗∗
-3.75∗∗
-3.51∗∗
1.12∗∗
0.278
0.265
0.265
0.25
0.288
0.257
0.081
-4.73∗∗
-4.55∗∗
-4.17∗∗
-3.87∗∗
-4.07∗∗
-2.87∗∗
0.95∗∗
0.271
0.278
0.269
0.254
0.287
0.249
0.081
0.17∗
0.29∗∗
0.22∗∗
0.29∗∗
0
0
0.068
0.07
0.066
0.074
0.17∗
0.3∗∗
0.22∗∗
0.29∗∗
0
0
0.073
0.072
0.069
0.08
-6.49∗∗
-6.43∗∗
-6.92∗∗
-6.46∗∗
-7.86∗∗
-8.2∗∗
0.536
0.527
0.493
0.54
0.248
0.242
-7.09∗∗
-7.05∗∗
-7.29∗∗
-6.89∗∗
-7.92∗∗
-8.23∗∗
0.515
0.488
0.457
0.496
0.296
0.238
-5.52∗∗
-6.35∗∗
-6.21∗∗
-6.47∗∗
-5.86∗∗
-6.39∗∗
0.35
0.49
0.501
0.504
0.459
0.47
0.487
1.465
-6.42∗∗
-7.17∗∗
-7.01∗∗
-7.06∗∗
-6.66∗∗
-6.45∗∗
0.39
0.524
0.533
0.529
0.497
0.501
0.496
1.335
-6.03∗∗
-4.86∗∗
-5.06∗∗
-4.13∗∗
-4.17∗∗
-3.52∗∗
0.339
0.297
0.3
0.272
0.244
0.235
-6.03∗∗
-4.84∗∗
-5.04∗∗
-4.12∗∗
-4.19∗∗
-3.62∗∗
0.349
0.302
0.306
0.278
0.248
0.199
-6.00∗∗
-5.09∗∗
-5.2∗∗
-4.21∗∗
-3.56∗∗
-2.77∗∗
0.09
0.234
0.249
0.233
0.221
0.225
0.218
0.128
-5.43∗∗
-4.7∗∗
-4.9∗∗
-4∗∗
-3.42∗∗
-2.55∗∗
0.23†
0.238
0.244
0.225
0.214
0.212
0.161
0.117
5.8∗∗
5.21∗∗
4.31∗∗
4∗∗
2.93∗∗
2.49∗∗
0.501
0.476
0.474
0.462
0.488
0.467
5.8∗∗
5.2∗∗
4.3∗∗
3.97∗∗
2.99∗∗
2.73∗∗
0.514
0.482
0.482
0.486
0.502
0.322
3.7∗∗
3.1∗∗
2.15∗∗
1.7∗∗
1.22∗∗
0.77†
0.441
0.455
0.439
0.408
0.424
0.402
2.54∗∗
2.17∗∗
1.72∗∗
1.29∗∗
1.03∗∗
0.19
0.412
0.414
0.381
0.356
0.354
0.17
4.15∗∗
4.05∗∗
4.2∗∗
3.88∗∗
3.53∗∗
3.56∗∗
0.393
0.357
0.353
0.342
0.342
0.332
4.17∗∗
4.06∗∗
4.21∗∗
3.89∗∗
3.58∗∗
3.69∗∗
0.396
0.357
0.353
0.34
0.349
0.256
5.74∗∗
5.82∗∗
5.66∗∗
5.56∗∗
4.49∗∗
4.57∗∗
0.339
0.339
0.323
0.315
0.316
0.309
6.53∗∗
6.71∗∗
6.54∗∗
6.2∗∗
5.47∗∗
3.92∗∗
0.371
0.383
0.355
0.358
0.359
0.213
-0.09
-0.26∗
-0.22∗
-0.36∗∗
-0.6∗∗
-0.65∗∗
0.118
0.104
0.105
0.098
0.117
0.114
-0.09
-0.26∗
-0.22∗
-0.35∗∗
-0.61∗∗
-0.67∗∗
0.12
0.105
0.105
0.1
0.117
0.111
0.10
-0.05
-0.04
-0.2†
-0.52∗∗
-0.59∗∗
0.118
0.117
0.114
0.115
0.112
0.11
0.17
0.00
-0.03
-0.18
-0.53∗∗
-0.5∗∗
0.12
0.119
0.115
0.116
0.112
0.104
-0.59∗∗
-0.42∗∗
-0.45∗∗
-0.46∗∗
-0.7∗∗
-0.67∗∗
0.086
0.074
0.084
0.084
0.078
0.077
-0.59∗∗
-0.42∗∗
-0.45∗∗
-0.46∗∗
-0.7∗∗
-0.69∗∗
0.088
0.075
0.085
0.086
0.078
0.076
-0.75∗∗
-0.61∗∗
-0.64∗∗
-0.71∗∗
-0.73∗∗
-0.72∗∗
0.081
0.08
0.079
0.081
0.078
0.079
-0.75∗∗
-0.62∗∗
-0.66∗∗
-0.71∗∗
-0.75∗∗
-0.64∗∗
0.083
0.083
0.081
0.084
0.082
0.076
-0.25∗∗
0.49∗∗
-2.69∗∗
-0.07∗∗
-0.05∗∗
0.13∗∗
0.048
0.026
0.323
0.012
0.007
0.004
-0.47∗∗
0.5∗∗
-2.49∗∗
-0.09∗∗
-0.05∗∗
0.15∗∗
0.059
0.028
0.347
0.014
0.009
0.005
-0.05
0.43∗∗
-2.03∗∗
-0.07∗∗
-0.04∗∗
1.01∗∗
0.03
0.028
0.307
0.011
0.007
0.028
-0.19∗∗
0.28∗∗
-0.83∗∗
-0.07∗∗
-0.04∗∗
1.1∗∗
0.03
0.028
0.307
0.011
0.007
0.028
† : 10%
4,220
22,435
∗ : 5%
4,236
22,435
2,400
22,435
∗∗ : 1%
Table 7: Parameter estimates for the four specifications.
1,794
22,435
The two random coefficients models produce similar price sensitivities except for the group
of old with high income. This group is the least price sensitive group and turns out to
be always pivot in the model with unobserved price discrimination. Specifically, the price
sensitivity of the pivot group is overestimated with uniform pricing compared to the model
with unobserved price discrimination. The price sensitivity decreases with both age and
income, leaving the young with low income the more price sensitive group. The parameters
of the intercept are negative, reflecting the fact that the major part of consumers choose
the outside option, namely not to buy a car or buy one on the second-hand market.
The heterogeneity of this parameter across groups does not follow a clear pattern. As
expected, consumers display a preference for horsepower, but how much they value it differ
substantially across groups. Young consumers have a high valuation for the engine power
while the eldest care less about this attribute. As expected, all groups of consumers dislike
large fuel expenses. The parameters of sensitivity to the fuel cost are consistent with the
parameters of sensitivity to the car price. The old purchasers with high income appear to
be also the less sensitive to the cost of driving while the more sensitive consumers are also
the young and middle-age groups with a low income. As weight is a proxy for the size and
the space of the car, it is positively valued by all the consumers. Three doors and stationwagon vehicles are negatively valuated, reflecting that most of the consumers buy sedan
or hatchback cars with five doors (four doors plus the trunk). Finally, the cost equation
parameters have the predicted signs. The marginal cost of production is increasing in the
horsepower and in the proxy of inputs cost while it appears costly to produce fuel efficient
cars.
We obtain a lower value of the objective function for the model with price discrimination
than for the model with uniform pricing (1,794 versus 2,400). In line with the simulation
results, and though it seems difficult to construct a formal statistical test based on these
values,10 we see this as evidence in favor of unobserved price discrimination in our context. Note also that if qualitatively similar, the results we obtain with the two models
exhibit some quantitative differences. This is especially the case for the pivot group, for
which the effects of price, fuel cost and horsepower are lower in magnitude under the price
discrimination model.
10
The test of Rivers & Vuong (2002) is sometimes conducted in the literature (see Jaumandreu & Moral
2006, Bonnet & Dubois 2010, Ferrari & Verboven 2012). The main issue in applying such a test in our
context is to obtain a consistent estimator of the standard errors of the difference between these two
objective functions, or any other statistics, under the null hypothesis. The problem is that both models
may be wrong under the null hypothesis of the test. In such a case, the residuals (ξjd (pdj , θ0d ))j=1...J that
we obtain under each models are not independent to each other, and the dependence between them is
unknown. Thus, neither the standard GMM formula based on independence, nor the standard bootstrap,
allow one to compute standard errors in a consistent way.
28
To understand what these differences imply, we compare the price elasticities and the markup rates under the two random coefficient models. The results for the nested logit models,
as well as other results on these models, are displayed in Appendix A.1. Table 8 focuses on
price elasticities implied by the uniform pricing model and the price discrimination model.
Price elasticities are, in absolute terms, higher for the model with uniform pricing mainly
because the prices are overestimated. In the uniform pricing model we find average price
elasticities varying from -3.7 to -6.3, which are in line with those obtained by BLP, who
report elasticities between -3.5 and -6.5, but below those of Langer (2012) who finds, using
transaction prices, a range between -6.4 to -17.8. These elasticities imply mark-ups that
are around 20% in both models, with, as we could expect, substantial heterogeneity across
groups in the price discrimination model. The average mark-up for the group of young,
low-income consumers is around 17.6%, contrasting with the 29% the firms obtain for the
old and high-income group.
Group of consumers
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Average
Discrimination BLP
-4.73
-4.55
-4.17
-3.87
-4.07
-2.87
-4.03
Uniform BLP
-6.25
-6.25
-5.79
-5.41
-5.38
-3.71
-5.46
Table 8: Comparison of average price elasticities under the uniform pricing and unobserved
price discrimination models.
Figure 1 displays the distribution of the differences in estimated marginal costs between
the two models. We compute here the relative difference (ĉu − ĉd )/ĉd . The costs are
always overestimated in the uniform pricing model, with an average difference of 10.9%
and differences that exceed 20% for 3.2% of the products. These differences stems from the
fact that, in the uniform pricing model, the marginal cost is deduced from the difference
between the posted price and the average mark-up. In contrast, in the price discrimination
model, the marginal cost is equal to the difference between the posted price and the markup of the pivot group. This mark-up is higher than the average mark-up estimated in the
standard model, resulting in lower marginal cost. Ultimately, the errors in the estimation
of marginal costs translate into errors in counterfactual simulation exercises.
29
0.18
0.16
0.14
Frequency
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
Relative cost difference (in %)
25
30
Figure 1: Distribution of the relative difference between estimated marginal costs.
5.3
Analysis of the discounts
Table 9 presents the average discount for each demographic group estimated using the
model with unobserved price discrimination. We compute average discounts weighted by
actual sales in each group but also using the same weighting scheme for all groups of
consumers, namely, the overall product market shares (“basket-weighted” method). This
allows us to eliminate the potential group-specific demand composition effect. The results
with both weighting methods are similar. As expected, the pattern on average discounts
across groups is similar to the one on price elasticity. The estimated pivot group (the group
assumed to be paying the posted price) is identical for all the products and corresponds to
the group with the lowest price elasticity. These are the 13.2% of the population over 60
year old with income over 27,000 euros. On average, the sales-weighted discount is 10.5%,
with a large heterogeneity across consumers. Around 25% receive a discount greater or
equal to 12.9%. Clearly, income and age are both important determinants of the discount
obtained. On average, young purchasers with a low income pay 14.4% less than the posted
price, while young, high income buyers get an average discount of 13.6%. These percentages
represent a gross gain of around 2,800 euros. Middle age consumers get smaller discounts
(12.2% for the low income group and 10.6% for the high income group). Finally, while
old, low income, individuals receive an average discount of 11.3%, old, high income, buyers
30
receive no discount since they constitute the pivot group for all products.
Group of consumers
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Average
Average discount
(in % of posted price)
Sales-weighted Basket-weighted
14.36
14.49
13.64
13.84
12.16
12.07
10.63
10.58
11.31
11.27
0
0
10.53
10.54
Average gross discount
(in euros)
Sales-weighted
Basket-weighted
2,805
3,015
2,863
2,891
2,659
2,546
2,391
2,251
2,276
2,387
0
0
2,210
2,219
Reading notes: the “basket-weighted” discounts are obtained by using the same artificial basket of cars
for all groups.
Table 9: Average discounts by groups of consumers
These figures average vehicle specific discounts. Our methodology allows us to analyze
further the heterogeneity across car models, since we can estimate a discount value for each
model and demographic group. Figure 5.3 displays the resulting distribution of discounts
across products. The corresponding average discount, averaged by product rather than by
consumers, is equal to 10.6%, with substantial heterogeneity. For 10% of the products,
the discount is smaller than 7.6%, while for the 10% most discounted cars, the rebate is
larger than 13%, and it even exceeds 34.3% for 1% of the fleet. To understand better the
source of this heterogeneity, we regress these discounts on the characteristics of the cars.
The results are displayed in Table 10. Discounts increase with posted price and horsepower
but decrease with weight and fuel cost. These results reflect both the differences in sales
between consumer groups (e.g. products mostly sold to the pivot group tend to have
a small average discount) and differences in the pricing strategy. Results with baskedweighted discounts are however similar, showing in particular that it is profitable for firms
to offer larger discounts for their most expensive cars.
31
0.1
Frequency
0.08
0.06
0.04
0.02
0
0
5
10
15
Average discount by product (in %)
20
25
Figure 2: Distribution of estimated discounts across products
Variable
Intercept
Posted price
Horsepower
Fuel cost
Weight
Three doors
Station wagon
R2
Parameter
13.3∗∗
3.27∗∗
2.65∗∗
-3.33∗∗
-8.44∗∗
1.16∗∗
0.85∗∗
0.51
Std-err
0.35
0.11
0.41
0.33
0.34
0.18
0.14
Table 10: Regression of average product discount on cars characteristics
How do our estimates compare with other evidence of discounts? First, to the best of
our knowledge, there is no comprehensive and reliable data on transaction prices in the
French automobile market. However, a recent survey conducted by the French credit
company, Cetelem (L’Automobile en Europe: 5 Leviers pour Rebondir 2013), provides a
useful benchmark. First, it reveals that in 2012, 87% of the purchasers benefited from a
discount from their car dealers, which is exactly what we estimate with our model (86.8%).
Interestingly, a quarter of them also indicate that they did not even need to negotiate to
obtain a rebate, which may be seen as evidence of price discrimination rather than a true
32
bargaining process. Further, for 68% of individuals who indicated that they negotiated the
car price, the average discount was around 11%. This result is very close to our average on
the whole population, and also comparable to the average discount we obtain on individuals
below 60 year old (12.4%), who also represent around two third of the whole population.
We were unable to find precise statistics on the dispersion of discounts, but we can report
some anecdotal evidence. For example, when searching online using the keywords “how
much discount for new car ” (in French), the first website listed states that “discounts are
generally between 5% and 20%”.11 The fourth website associated to the same key words
search is a forum asking the question of how much discount one can expect to obtain on the
purchase of a new car. One reply states that discount do not exceed 20%, while another
mentions an average discount of 6%.12 Our estimations are overall consistent with these
figures.
A recent study by Kaul et al. (2012) investigates the effect of the scrapping policy on the
magnitude of discounts in Germany, using data collected to a sample of dealers. When
excluding demonstration cars and sales to employees, which are typically much more discounted, they obtain an average discount of 14%. This magnitude is consistent, though
somewhat higher, with our estimate. Their study focuses on the period 2007-2010, which
corresponds to the beginning of the economic crisis. If posted prices did not adjust immediately, it is likely that car dealers reacted to this adverse economic climate by reducing
their margins and increasing the discounts. The assumption that the posted price is equal
to the transaction price for one group may also explain part of this difference. Specifically,
we re-estimated our model imposing a discount of 4% instead of 0% for the pivot group.
We obtained an average discount of 14%, fully consistent with the one observed by Kaul
et al. (2012). In their regression analysis, they also find a positive link between discounts
and posted prices, which is in line with the results displayed in Table 10.
In 2000, the UK Competition Commission investigated the competitiveness of the UK new
car market and gathered data on average discounts by brand and segment (New cars: A
report on the supply of new motor cars within the UK 2000). The dataset is very reliable
since it was collected directly from dealers. The report reveals that the average discount
lies between 7.5% and 8%, also broadly in line with our estimated average discount. Once
more, the difference may stem from differences between the two markets and the periods
under consideration. This report also refers to a consumer survey conducted in 1995 asking
11
See http://www.choisir-sa-voiture.com/concessionnaire/meilleur-prix-voiture.php. We
performed this search in November 2014 using Google search engine.
12
See http://forum.hardware.fr/hfr/Discussions/Auto-Moto/negocier-voiture-concession-sujet_
15899_1.htm.
33
automobile purchasers whether or not they obtained a discount over the posted price. This
survey reveals that 17% of purchasers paid the posted price whereas 37% bargained and
obtained a discount and 29% were automatically offered a discount. This figure of 17% is
close to our estimation of 13% of the cars are sold without discount. Furthermore, the fact
that some purchasers were “automatically offered a discount” corroborates our assumption
that discounts are used as a tool to price discriminate because the posted price is not
optimal for some consumers.
A direct comparison of the distribution of discounts we obtain and evidence on the US
market is more complicated. Rebates and negotiation are extremely common in the US,
and the popular Kelley Blue Book website provides a lot of information that is not available
to consumers in France. It reports in particular the negotiability, the fair purchase price
and the fair market range in any given area (zip code) and for almost every car model.
The price quotes are computed using weekly data on transaction prices. For the larger zip
codes, they also provide the distribution of transaction prices, which indicates geographical
price dispersion in the US. Therefore, the notion of posted prices and discounts, as defined
in our paper, are less relevant in the US. Despite these differences, Busse et al. (2012)
report that the rebates represent on average 9.8% of the transaction prices, which is once
more consistent with our estimated discounts.
Finally, few papers correlate the magnitude of discounts to individual characteristics. Harless & Hoffer (2002) and Langer (2012), in particular, conduct such an analysis on the
US market, using respectively dealer margins and a survey on transaction prices (see also
Chandra et al. 2013, for an analysis of the Canadian market, focusing on gender discrimination). They both report a negative correlation between the discounts and purchasers’
age. In the web appendix of the 2012 version of her paper, Langer documents significant
price discrimination with respect to income, the high income groups of consumers (for both
men and women) being associated with higher margins. These two results are in line with
our findings on the estimated discounts and mark-up rates.
5.4
The impact of price discrimination on firms and consumers
If third degree price discrimination is always profitable for a monopoly seller, this may
not be the case in an oligopoly, because price discrimination may reinforce competition
among firms. Under certain conditions, all firms may actually be worse off than if they
could commit to a uniform pricing strategy (Holmes 1989, Corts 1998). The effect on
consumers is also ambiguous. For a given group of consumers, some products may turn
34
to be cheaper without price discrimination. We investigate in this subsection the effect of
price discrimination on firms and consumers. We thus compute, using our estimates of the
model with price discrimination, the counterfactual prices and profits that would occur if
firms could commit to set a single price for all groups of consumers.
Profit with price
Profit without price
Gain from
Brand
discrimination (in Me) discrimination (in Me)
discrimination
Renault
645.92
618.88
4.37%
Peugeot
546.75
529.19
3.32%
Citroen
455.37
433.61
5.02%
Volkswagen
172.46
171.74
0.42%
Toyota
162.47
157.53
3.13%
Mercedes
149.48
137.35
8.83%
Ford
134.43
130.61
2.92%
Opel
106.89
106.02
0.82%
B.M.W.
104.84
99.85
5.00%
Audi
82.15
81.58
0.71%
Fiat
64.6
63.68
1.45%
Dacia
55.28
54.16
2.07%
Seat
54.21
55.8
-2.86%
Suzuki
52.75
52.46
0.55%
Nissan
49.65
48.61
2.15%
Mini
34.06
34.24
-0.54%
Honda
30.24
29.12
3.83%
Hyundai
29.09
28.55
1.91%
Skoda
22.52
22.46
0.25%
Mazda
19.22
19.04
0.94%
Kia
17.95
17.68
1.56%
Alfa Romeo
17.81
17.72
0.55%
Land Rover
15.74
15.07
4.44%
Smart
11.37
11.28
0.83%
Mitsubishi
10.08
9.83
2.49%
Porsche
9.22
7.41
24.44%
Jeep
6.89
6.78
1.67%
Chrysler
6.16
6.07
1.55%
Lancia
5.08
4.91
3.4%
Saab
4.29
4.14
3.74%
Daewoo
3.42
3.37
1.51%
Dodge
3.24
3.27
-0.98%
Jaguar
3.04
2.7
12.55%
Daihatsu
2
1.95
2.29%
Subaru
1.87
1.9
-1.58%
Ssangyong
1.81
1.85
-1.88%
Lexus
1.55
1.48
4.6%
Rover
0.05
0.05
2.22%
Total industry
3,094
2,992
3.41%
Reading notes: Profits are annual profits, for the year 2007, in millions of euros. The gains
from price discrimination represent the profit gains or losses of switching from the uniform
pricing equilibrium to the price discrimination equilibrium.
Table 11: Gains and losses from price discrimination by brand.
35
Results on firms profits are displayed in Table 11. Gains from price discrimination are
rather small but heterogeneous. We observe that if price discrimination is profitable for
most of the manufacturers, it makes 5 out of the 38 manufacturers worse off. The gains
associated to price discrimination are particularly high for brands that commercialize powerful vehicles, such as Mercedes (+8.8%), Jaguar (+12.6%) and Porsche (+24.4%). This
makes sense, given that higher prices and horsepowers are associated to higher discounts or,
put it another way, more price discrimination. Price discrimination appears to be also more
profitable than average for the major French manufacturers (+4.4%, +3.3% and +5% for
respectively Renault, Peugeot and Citroen) and more moderate for Dacia (+2.1%). The
total gain from price discrimination is rather small but significant, the industry profit
increasing by 3.4% compared to the uniform pricing equilibrium.
We also investigate the impact of price discrimination on consumers. In Table 12, we
compute the average price differences between the uniform and the discriminatory prices
for each group of consumers and report the number of products for which the discriminatory
price is lower than the uniform one (see Column 3). We also compute average surplus for
each group of consumers under the two pricing equilibria (see Table 13). For the young
groups, all products are cheaper under uniform pricing, and price discrimination makes
them save around 600 euros. The situation is more contrasted for the 40-59 and 60+ yearold group. In particular, all prices are lower under uniform pricing for the pivot group, who
would save on average the substantial amount of 2,153 euros. Overall, price discrimination
is hardly beneficial for consumers as it increases the global average individual surplus by
only 0.37%. Again, this global impact hides heterogeneous effects. The group experiencing
the highest welfare gain is the group of young consumers with low income (+3.7%), while
the pivot group is, not surprisingly, the one that suffers the most from price discrimination
(-2.8%).
#{j : pdj <
puniform
}
j
Average gains in purchases
Group of consumers
Frequency
Sales-weighted
Basket-weighted
Age < 40, I = L
15.7
3,205
627
761
Age < 40, I = H
11.5
3,205
454
637
Age ∈ [40,59], I = L
16.3
2,925
271
292
Age ∈ [40,59], I = H
22.3
1,382
6
-2.6
Age ≥ 60, I = L
20.8
2,252
180
133
Age ≥ 60, I = H
13.2
0
-2,153
-2,254
Reading notes: the third column indicates how many products (among the 3,205) have lower
prices with the price discrimination regime. The “basket-weighted” gains are obtained by
using the same artificial basket of cars for all groups.
Table 12: Gains of price discrimination for groups of consumers.
36
Age
Age
Age
Age
Age
Age
< 40, I = L
< 40, I = H
∈ [40,59], I = L
∈ [40,59], I = H
≤ 60, I = L
≤ 60, I = H
Average
Price discrimination
Uniform pricing
Gain from
discrimination (in %)
13,208
14,666
15,980
18,286
15,680
35,759
18,424
12,735
14,244
15,803
18,213
15,503
36,795
18,356
3.72
2.96
1.12
0.4
1.14
-2.82
+0.37
Table 13: Comparison of average individual surplus for the different groups of consumers
with price discrimination and uniform pricing.
6
Conclusion
This paper investigates the recurring problem of observing only posted prices instead of
transaction prices in structural models of demand and supply in markets with differentiated
products. We propose an approach that incorporates unobserved price discrimination by
firms based on observable individual characteristics. This approach requires to have data
on aggregate sales on the corresponding groups of purchasers and, as usual, characteristics
of products. We use this model to describe the French new car market where price discrimination may occur through discounts. Our results suggest significant discounting by
manufacturers which is consistent with previous studies on price dispersion, survey data
and anecdotal evidence on the magnitude of discount in the French market.
We implemented our methodology in the standard Berry et al. (1995) framework, but it can
be easily extended to other demand and supply models. It also applies when the data collected by the econometrician is unreliable or limited. We finally explained how to deal with
price discrimination based on characteristics that are unobserved by the econometrician.
37
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40
A
A.1
Appendix
Results with the nested logit specification
We present in this appendix the same results as those given in Tables 8 to 10 and Figure 5.3,
but for the nested logit. Table 14 first shows that the average price elasticities are similar
than with the random coefficient model. Under price discrimination, they range from 6.5 to -3.9, lower than the range [−4.7, −2.9] that we obtain with the random coefficient
model. Here again, older people are the less price sensitive. Perhaps surprisingly, on the
other hand, high-income individuals appear to be more price sensitive in general, both
under price discrimination and uniform pricing. The pivot group is nevertheless still the
older, high-income consumers. We also observe, as with the random coefficient model,
that the model without price discrimination slightly overestimates price elasticities and
always overestimates the marginal costs. Figure 3 displays the distribution of the relative
cost differences between the two alternative models. The average difference is 10.5%,
with substantial heterogeneity. In particular, the difference exceeds 20% for 10.3% of the
products. Turning to the discounts, we obtain again that the youngest purchasers obtain
the highest discount, though such discounts are on average smaller than with the random
coefficient model. Interestingly, the high-income groups also receive smaller discounts, in
line with the results on the random coefficient model. This shows that price sensitivity
alone does not determine the amount of the discounts. The heterogeneity in the valuation of
other characteristics such as fuel cost or horsepower also plays an important role. Finally,
we display in Figure 4 the distribution of average discounts over car models. Both the
average (6.6%) and the standard deviation (3.3%) are lower than the figures obtained with
the random coefficient model (10.6% and 4.8%, respectively), but for 10% of the fleet
discounts still exceed 11.5% (versus 13% for the random coefficient model). Finally, a
regression of the discounts on cars’ characteristics shows, as before, that large fuel costs
and heavy vehicles are associated with lower discounts, while horsepower is associated to
greater discounts. On the other hand, the price has a negative rather than positive effect on
discounts in this specification, contradicting in particular the results of Kaul et al. (2012).
41
Group of consumers
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Discrimination n. logit
Uniform n. logit
-5.64
-6.54
-5.6
-5.92
-3.91
-3.92
-5.20
-6.34
-7.38
-6.06
-6.52
-3.92
-3.73
-5.59
Average
Table 14: Comparison of average price elasticities for the nested logit models with uniform
pricing and unobserved price discrimination.
0.09
0.08
0.07
Frequency
0.06
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20
Relative cost difference (in %)
25
30
Figure 3: Distribution of the relative difference between estimated costs
Group of consumers
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Average
Average discount (in % of posted price)
Sales-weighted
Basket-weighted
12.81
13.74
9.71
10.52
1.62
0
7.86
12.45
13.89
9.95
10.75
1.57
0
7.9
(ĉu −ĉd )
.
ĉd
Average gross discount (in euros)
Sales-weighted Basket-weighted
2,249
2,516
1,801
1,951
288
0
1,431
2,262
2,523
1,808
1,952
285
0
1,435
Reading notes: the “basket-weighted” discounts are obtained by using the same artificial basket of cars
for all groups.
Table 15: Average discount by groups of consumers for the nested logit model.
42
0.08
0.07
Frequency
0.06
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
Average discount by product (in %)
20
25
Figure 4: Distribution of estimated discounts for the nested logit model.
Variable
Intercept
Posted price
Horsepower
Fuel cost
Weight
Three doors
Station wagon
R2
Parameter
15.29∗∗
-1.45∗∗
2.12∗∗
-2.16∗∗
-3.50∗∗
1.11∗∗
0.31∗∗
0.65
Std-err
0.21
0.06
0.25
0.20
0.20
0.10
0.08
Table 16: Regression of average product discount on cars characteristics
A.2
Correction for null market shares
We first display the fraction of products with null market shares, given the choice of our
groups and consumers.
43
Group
1
2
3
4
5
6
Characteristics
Age < 40, Income <27,000
Age < 40, Income ≥ 27, 000
Age ∈ [40,59], Income <27,000
Age ∈ [40,59], Income ≥ 27, 000
Age ≥ 60, Income <27,000
Age ≥ 60, Income ≥ 27, 000
Frequency of null sale
11.6%
10.3%
7.5%
4%
7.8%
7.6%
Table 17: Fraction of products with null market shares in the final sample
We now provide a rationale for the choice of our estimator sbdj =
ndj +0.5
Nd
of sdj , where ndj
denoting the number of sales of product j in group d and N d the number of potential
buyers with characteristics d. The idea is to consider simple estimators of sdj of the form
ndj +c)/N d , and fix c such that the expectation of ln((ndj +c)/N d ) is asymptotically unbiased.
The reason we are looking for such a c is that ln(sdj ) plays an important role at least in
the logit or nested logit models. With an unbiased estimator of ln(sdj ), we could estimate
consistently and as usually the demand parameters. However, in our framework where
individuals choose independently from each others, so that ndj ∼ Binomial(Nd , sdj ), it is
well-known that only polynomials of sdj of degree at most Nd can be estimated without
bias. Our aim is then to find instead an estimator that is asymptotically unbiased at the
first order.
For that purpose, we consider an asymptotic approximation where sj is small but λdj ≡
q
Nd sdj → ∞. Let Zjd = (ndj − λdj )/ λdj . A second-order Taylor expansion of (ndj + c)/N d
around sdj yields

2
q d2
sj
c
1
c
λdj ln((ndj + c)/N d ) − ln sdj = Zjd + q − d2 q Zjd + q  ,
2e
sj
λdj
λdj
λdj
where sedj is between (ndj +c)/N d and sdj . The first order term, Zjd , is asymptotically standard
normal and thus asymptotically centered. Now, considering the second-order term,

2
q nq d2
o
s
c
j
λdj
λdj ln((ndj + c)/N d ) − ln sdj − Zjd = c − d2 Zjd + q  .
2e
sj
λdj
44
Moreover,
sd2
sd2
j
j /e
P
−→ 1 and
Zjd
+
√c d
λj
2
L
−→ χ21 . Hence,
q nq o
1
L
λdj
λdj ln((ndj + c)/N d ) − ln sdj − Zjd −→ c − χ21 .
2
Choosing c = 1/2 therefore ensures that this second-order term is asymptotically centered
around 0.
To examine the robustness of the estimation results to the correction of the null shares
adopted. We re-estimate the different models using the Laplace transformation of the
market share equation used by Gandhi et al. (2013). This correction replaces the market
share by :
s̃dj =
N d sbdj + 1
.
Nd + J + 1
As Table 18 suggests, the estimation results are robust to the choice of a correction to deal
with products with null market shares. The estimated parameters are very close to each
other. As a result, subsequent results (not displayed here) on, e.g., discounts, are also close
to each other.
45
Price sensitivity
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Intra-segment correlation
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Intercept
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Fuel cost
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Horsepower
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Age < 40, I = L
Weight
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Three doors
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Station-wagon
Age < 40, I = L
Age < 40, I = H
Age ∈ [40,59], I = L
Age ∈ [40,59], I = H
Age ≤ 60, I = L
Age ≤ 60, I = H
Marginal cost equation
Intercept
Horsepower
fuel inefficiency
Three doors
Station-wagon
Weight × average price index
Our correction
Parameter
Std-err
Gandhi correction
Parameter
Std-err
-2.7∗∗
-2.55∗∗
-2.24∗∗
-2.16∗∗
-1.98∗∗
-1.88∗∗
0.185
0.176
0.177
0.178
0.185
0.105
-2.54∗∗
-2.41∗∗
-2.13∗∗
-2.03∗∗
-1.83∗∗
-1.87∗∗
0.18
0.171
0.17
0.174
0.099
0.168
0.17∗
0.3∗∗
0.22∗∗
0.29∗∗
0
0
0.073
0.072
0.069
0.08
0.08
0.21∗∗
0.16∗
0.2∗
0
0
0.071
0.072
0.068
0.081
-7.09∗∗
-7.05∗∗
-7.29∗∗
-6.89∗∗
-7.92∗∗
-8.23∗∗
0.515
0.488
0.457
0.496
0.296
0.238
-7.62∗∗
-7.49∗∗
-7.63∗∗
-7.34∗∗
-7.9∗∗
-8.31∗∗
0.506
0.48
0.45
0.501
0.224
0.273
-6.03∗∗
-4.84∗∗
-5.04∗∗
-4.12∗∗
-4.19∗∗
-3.62∗∗
0.349
0.302
0.306
0.278
0.248
0.199
-5.95∗∗
-4.81∗∗
-4.99∗∗
-4.23∗∗
-3.88∗∗
-3.51∗∗
0.319
0.282
0.286
0.274
0.187
0.224
5.8∗∗
5.2∗∗
4.3∗∗
3.97∗∗
2.99∗∗
2.73∗∗
4.17∗∗
0.514
0.482
0.482
0.486
0.502
0.322
0.396
5.46∗∗
4.91∗∗
4.07∗∗
3.7∗∗
2.71∗∗
2.84∗∗
4.15∗∗
0.498
0.467
0.463
0.478
0.301
0.454
0.38
4.17∗∗
4.06∗∗
4.21∗∗
3.89∗∗
3.58∗∗
3.69∗∗
0.396
0.357
0.353
0.34
0.349
0.256
4.15∗∗
4.01∗∗
4.13∗∗
3.91∗∗
3.29∗∗
3.59∗∗
0.38
0.344
0.34
0.338
0.241
0.321
-0.09
-0.26∗
-0.22∗
-0.35∗∗
-0.61∗∗
-0.67∗∗
0.12
0.105
0.105
0.1
0.117
0.111
-0.05
-0.22∗
-0.19†
-0.32∗∗
-0.57∗∗
-0.65∗∗
0.119
0.104
0.103
0.102
0.105
0.107
-0.59∗∗
-0.42∗∗
-0.45∗∗
-0.46∗∗
-0.7∗∗
-0.69∗∗
0.088
0.075
0.085
0.086
0.078
0.076
-0.62∗∗
-0.43∗∗
-0.47∗∗
-0.5∗∗
-0.66∗∗
-0.66∗∗
0.086
0.074
0.083
0.088
0.071
0.072
-0.47∗∗
0.5∗∗
-2.49∗∗
-0.09∗∗
-0.05∗∗
0.15∗∗
0.059
0.028
0.347
0.014
0.009
0.005
-0.49∗∗
0.5∗∗
-2.51∗∗
-0.09∗∗
-0.05∗∗
0.15∗∗
0.06
0.029
0.352
0.014
0.009
0.005
Table 18: Estimation of parameters : Nested logit model with our correction and Gandhi
et al. correction.
46
A.3
Segmentation of the market
The nested logit approach requires to define a segmentation of the market in homogeneous
groups of products. Our segmentation, based on the main use of the vehicle, is close to the
one of The European New Car Assessment Program one (Euro NCAP). Table 19 displays
the eight segments that we consider and their market shares over the period. Note in
particular that sport cars include all convertible cars as well as vehicles with a high ratio
horsepower/weight, while small multi-purpose vehicles (MPV) include small vans such as
Renault Kangoo. The entire classification is presented in Table 22.
Segment
Supermini
Executive
Small Family
Large Family
Small MPV
Large MPV
Sports
Allroad
Market shares
(in %)
45.14
1.17
17.01
8.67
17.56
1.07
5.11
4.77
Table 19: Segments and their market shares.
47
48
Cuore, Sirion, YRV
-
Fortwo, Forfour
Mito
500,
Palio,
Panda,
Punto, Seicento
Y
Fiesta, Ka,
-
Kalos, Matiz
Kalos, Matiz
Corsa
Jazz
Atos, Getz, I10
Picanto, Rio
2
Colt
Micra, Note
25, Streetwise
-
Justy
Alto, Ignis, Splash,
Swift, SX4
Aygo, IQ, Yaris
A2
Arosa, Ibiza
Fabia
Fox, Lupo, Polo
Renault
Dacia
B.M.W
Mini
Chrysler
Jeep
Dodge
Daihatsu
Mercedes
Smart
Alfa Romeo
Fiat
Jaguar
Land Rover
Volvo
Chevrolet
Daewoo
Opel
Saab
Honda
Hyundai
Kia
Lada
Mazda
Mitsubishi
Nissan
Porsche
Rover
Ssangyong
Subaru
Suzuki
Toyota
Lexus
Audi
Seat
Skoda
Volkswagen
Honda
Hyundai
Lada
Mazda
Mitsubishi
Nissan
Porsche
Rover
Ssangyong
Subaru
Suzuki
Toyota
VW Group
GM Europe
Ford
Fiat
Daihatsu
Daimler
Chrysler
B.M.W
Lancia
Ford
106,
107,1007,206,
207
Clio, Modus, Twingo
Sandero
Mini
-
Peugeot
Renault
Supermini
C1, C2, C3, Saxo
Make
Citroen
Group
PSA
Auris
A3
Cordoba, Leon
Eos, Golf, Jetta, Newbeetle
Impreza
Liana
45
-
Cee-d, Cerato
111, 112
3
Lancer
Almera, Qashqai
Civic
Accent, Coupe, I30
C30, V50
Aveo, Lacetti, Nubira
Lanos
Astra
-
-
147
Bravo, Stilo
Caliber
A-Class
Megane
Logan
1-Series
-
306, 307, 308
Small family
Xsara
Avensis, Prius
IS
A4, A5
Toledo
Octavia, Superb
Scirocco, Passat
Legacy
-
75
-
Magentis
6
Carisma
Primera
C70, S40, S60, V70
Epica, Evanda
Insigna, Signum, Vectra
9-3
Accord
Elantra, Sonata
X-Type
-
Lybra
Mondeo
156, 159, GT
Croma
Journey
C, CLK-Class
Laguna
3-Series
Sebring
-
406, 407
Large family
C5
GS, LS
A6, A8, R8
-
-
911, Boxter, Cayman
-
RX8
350Z, Maxima-Q
9-5
-
V40, S80
Evanda
Omega
S-Type, XJ, XK
-
Thesis
-
Viper
E, CL, R, S, SL, CLS,
SLR-Class
166, Brera
-
Vel Satis
5, 6, 7-Series
300C, 300M, Crossfire
-
607
Executive
C6
Celica, MR
S3, S4, S6, S8 TT
Phaeton
-
-
MX5
-
S2000
-
Corvette
Tigra, Speedster
-
Puma
Coupe, Roadster
GTV, Spider
Barchetta
Copen
SLK-Class
Z4
-
-
Sports car
-
Table 22: Segmentation of the automobile market
Corolla
Altea
Roomster
Caddy, Touran
Wagon-R
-
Carens, Soul
5, Premacy
Spacestar
Almera
Rezzo, Tacuma
Rezzo
Agila,
Combo,
Meriva, Zafira
FR-V, Stream
Matrix
Doblo, Fiorino, Idea,
Multipla
Musa
Focus,
Fusion,
T.Connect,
Tourneo
-
B-Class, Vaneo
Kangoo, Megane
PT Cruiser
-
Small MPV
Berlingo, C4, Nemo,
Xsara
Bipper, Partner
Previa
Alhambra
Sharan
-
Rodius, Stavic
Carnival
MPV
Grandis
-
Trajet
-
-
Phedra
Galaxy, S-Max
Ulysse
Viano
Espace
Voyager, G.Voyager
-
807
Large MPV
C8
CR-V, HR-V
Tucson, Santafe, Terracan
Sorento, Sportage
Niva
Outlander, Pajero
X-Trail,
Murano,
Pathfinder,
Patrol,
Terrano
Cayenne
Actyon, Korando, Kyron, Rexton
Forester, B9Tribeca
G.
Vitara,
Jimny,
Samurai, Vitara
RAV4, L.Cruiser
RX
Allroad, Q5, Q7
Tiguan, Touareg
Freelander, Defender,
Discovery, R.Rover
XC60, XC70, XC90
Captiva, Tahoe
Korando
Antara, Frontera
Kuga
Koleos
X3, X5, X6
Compass,
Cherokee,
Commander,
G.Cherokee,
Wrangler
Durango, Nitro
Terios
G, GL, GLK, MLClass
Sedici
4007
Allroad/SUV
C-Crosser