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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
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Page 1: Automobile Prices in Market Equilibrium with …economics.yale.edu/sites/default/files/ddf_auto_discr.pdfAutomobile Prices in Market Equilibrium with Unobserved Price Discrimination

Automobile Prices in Market Equilibrium withUnobserved 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, PhilippeGagnepain, Gautam Gowrisankaran, Alessandro Iaria, John Morrow, Kathleen Nosal, Mario Pagliero,Philipp Schmidt-Dengler, Michelle Sovinsky, Hidenori Takahashi, Yuya Takahashi, Frank Verboven, NaokiWakamori and the participants of various seminars and conferences for their valuable comments. We alsothank Pierre-Louis Debar and Julien Mollet from the CCFA for providing us with the data. Isis Durrmeyeracknowledges 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]

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1 Introduction

The standard aggregate-level estimation of demand and supply models of differentiatedproducts relies on the observation of the market shares and the characteristics of theproducts, in particular prices (see Berry 1994). Because of price discrimination and pricenegotiation, transaction prices for an identical product may differ from one individual toanother. Sellers can practice third degree price discrimination according to observabledemographic characteristics such as age, gender or city of residence. There may also beroom for individual negotiation: the sellers are willing to offer discounts to the consumersthat aggressively negotiate the prices. Automobiles, furniture, kitchens or mobile phonecontracts are examples for which there is either documented or anecdotal evidence thatconsumers 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 betweenproducers and retailers. Producers are required to edit general terms and conditions ofsale. These conditions are then the starting point for individual negotiation with eachretailer.

In all these cases, precise data on transaction prices may be hard to obtain. One typicallyobserves either transaction prices on a small sample issued from a survey, or only postedprices on a large sample. In the first case, price discrimination can be studied but policyexercises cannot be performed. With large data, on the other hand, policy simulationsare 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 resultsin 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 withunobserved discounts. Our rationale for the existence of discounts over the posted pricesis that discounts allow firms to price discriminate between heterogeneous consumers andthus extract more surplus than they would with a uniform price.2 Sellers edit only one

1In France, it is also commonly admitted that negotiation is possible when purchasing a new car. Anarticle published in October 2011 by Le Figaro, which is the second largest French national newspaper,suggests that discounts up to 26% can be obtained.

2In 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 occursfor each firm at equilibrium.

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price, namely the posted price, since it is usually legally forbidden to price discriminatebetween consumers and difficult to implement, but in practice the transaction prices differfrom one individual to another. We suppose that sellers use observable characteristicsof 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 consumersthan the econometrician. This assumption may be problematic in settings where there arefew buyers, such as vertical relationships between producers and retailers. But in marketswhere sellers do not know the buyers before the transaction, it seems plausible to assumethat price discrimination is based only on a few easily observable characteristics, such assex, age and the city of residence.

We therefore extend the random coefficient discrete choice model of demand popularized byBLP to allow for unobserved price discrimination. BLP exploit the exogeneity of observedproduct characteristics, apart from price, to yield moment conditions involving the param-eters of interest. Their method does not apply directly to our framework, yet, because themoment conditions are not valid anymore if we replace the unobserved transaction priceswith posted prices. Instead, we rely on structural assumption from the supply side, andreplace the unobserved prices by their expression stemming from the first-order conditionof profit maximization.

These first-order conditions have identifying power under two assumptions. First, themarginal cost of a product is supposed to be identical for all buyers. This amounts toneglecting 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 sellingis 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. Inour application, we suppose that the prices posted by the sellers correspond to the highestdiscriminatory price, so that some consumers actually pay these observed posted prices. Inother words, we normalize the optimal discount for one group of consumers to be zero. Suchan assumption is necessary since otherwise, we could shift all discounts by an arbitraryconstant. It is also consistent with empirical evidence reported by several surveys, suchas the one made by Cetelem in 2012 in France (L’Automobile en Europe: 5 Leviers pourRebondir 2013).

The ideas underlying our approach can be applied to various settings where the informationon prices is limited, with different assumptions on observed prices. This is the case inparticular if we observe the average transaction prices paid by all consumers. In this setting,

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there is a direct link between observed and transaction prices and no further assumption onprices is needed. Our methodology could also be implemented in a setting where demandis 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 supermarketchains in different municipalities. Two recent papers have used similar ideas in differentsettings. Miller & Osborne (2014) adopt a similar methodology to analyze the cementindustry. They only observe average price and the total quantity of cement purchased inthe US. They allow as well for price discrimination across US counties. They computethe optimal prices and quantities for each location using the equilibrium conditions. Thenthey 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 unobserveddifferences 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 onother 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 forautomobile has always been estimated with posted prices when transaction prices areunobserved. As mentioned before, however, there is evidence of price discrimination inthis market. We rely on an exhaustive dataset recording all the registrations of newcars bought by households in France between 2003 and 2008. Apart from detailed carattributes, some buyers characteristics are provided. We observe in particular age andexpected income (namely, the median income of people in the same age class living in thesame municipality). As these characteristics are easily observed by sellers and presumablystrong determinants of purchases, we suppose that they are used to price discriminate.

Our results suggest that price discrimination is significant in France. The average discountis estimated to be 10.5% of the posted price. The distribution of estimated discountsspreads mostly between 0 and 25% depending on the car purchased and demographiccharacteristics. As expected, age and income are negatively correlated to the value ofdiscount. Overall, our results are in line with evidence on discounts in France (see inparticular L’Automobile en Europe: 5 Leviers pour Rebondir 2013) and with the literatureon price discrimination in the automobile market (see, e.g. Harless & Hoffer 2002, Langer2012). The magnitude of discounts obtained is also comparable to estimates obtained withsurvey data and anecdotal evidence found in specialized magazines or on internet. Finally,

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we show that ignoring price discrimination and using list prices as if they corresponded tothe transaction prices, as is usually done, would slightly overestimate the price sensitivityparameters but always overestimate the marginal costs of products.

We also study the effect of price discrimination on car manufacturers profits and consumerssurplus. This question is particularly relevant because theoretical predictions are unclearand depend on market conditions. Holmes (1989) and Corts (1998) show that in thecompetitive 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) andCowan (2012) derive sufficient conditions on demand for price discrimination to increasesocial 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, theoverall industry profit would be reduced but some firms would be better off. The gainsfrom price discrimination appear to be larger for luxury brands and French brands thathave large market shares. On the consumer side, there are winners and losers, as thetheory predicts, but price discrimination is moderately welfare enhancing at the aggregatelevel. Price discrimination carries out monetary redistribution from the older and richerpurchasers to the younger, low-income earners.

Though we do not explicitly model bargaining, our paper is related to the recent theoreticalliterature that considers hybrid models of bargaining in which sellers post a sticker price andoffer the possibility to bargain for discounts. This strategy might be profitable for sellerswhen consumers have heterogeneous bargaining costs or are imperfectly informed on theirability to bargain (see Gill & Thanassoulis 2009, 2013). Our model can be interpreted asa bargaining model in which all the bargaining power is given to the seller. Structuralmodels of demand and supply where prices are set by a bargaining process have beenrecently developed and estimated in specific industries, generally in business to businessmarkets 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 forcoronary stents. Gowrisankaran et al. (2014) also develop and estimate a structural modelof bargaining to analyze mergers between hospitals.

However, there are few empirical papers that analyze bargaining in business to consumersmarkets. A recent paper by Jindal & Newberry (2014) develops a structural model ofdemand where buyers are able to negotiate but have a bargaining cost. They estimateboth the bargaining power and the distribution of bargaining costs using individual dataon refrigerator transactions. However, their framework is very different from ours since

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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 thatincorporates unobserved negotiation between sellers and buyers to describe the second-hand car market. He estimates the model using posted prices only, as we do here. Hisidentification strategy is very different from ours and relies on the existence of dealers thatcommit not to negotiate with potential buyers. With such data at hand, he estimatesthe demand parameters together with the unobserved discounts offered by dealers thatallow for negotiation. As opposed to our methodology, he cannot identify model-specificdiscounts but rather obtains an average discount at the dealer level.

The paper is organized as follows. The second section presents the theoretical model andsection 3 explains how to estimate the model with unobserved transaction prices. Section4 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 exceptthat the demand arises from a finite number of heterogeneous groups of consumers. Firmsare supposed to observe the group of each consumer, as well as their corresponding pref-erences, such as their average price sensitivity. They then price discriminate among thesegroups, in order to take advantage of the heterogeneity in preferences.

Specifically, heterogeneous consumers are supposed to be segmented in nD groups of con-sumers, and we denote by d the group of consumer i. As in the standard BLP model, weallow consumers to be heterogeneous within a group, but assume sellers are not able todiscriminate based on this heterogeneity. Each consumer chooses either to purchase oneof the J products or not to buy any, which corresponds to the outside option denoted by0. As usual, each product is assimilated to the bundle of its characteristics. Consumersmaximize their utility, and the utility of choosing j is assumed to be a linear function ofproduct characteristics:

Udij = X ′jβ

di + αdi p

dj + ξdj + εdij,

where Xj corresponds to the vector of observed characteristics and ξdj represents the val-uation 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

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to face the same transaction price pdj . This is crucial, but not more restrictive than theassumption that ξdj is common to all individuals with characteristics d. This was shownby Berry & Haile (2014) to be necessary for identifying demand models nonparametricallyfrom aggregated data. As typical in the literature, the idiosyncratic error terms εdij areextreme-value distributed.

We make the usual parametric assumption about the intra-group heterogeneity, i.e. thatindividual parameters can be decomposed linearly into a mean, an individual deviationfrom the mean and a deviation related to individual characteristics:

βdi = βd0 + πX,d0 Ei + ΣX,d0 ζXi

αdi = αd0 + πp,d0 Ei + Σp,d0 ζpi ,

where Ei denotes demographic characteristics that are unobserved by the firm for eachpurchaser but whose distribution is common knowledge. ζi = (ζXi , ζ

pi ) 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 fromthis mean:

Udij = δdj (p

dj ) + µdj (Ei, ζi, p

dj ) + εdij,

withδdj (p

dj ) = X ′jβ

d0 + αd0p

dj + ξdj

andµdj (Ei, ζi, p

dj ) = Xj

(πX,d0 Ei + σX,d0 ζXi

)+ pdj

(πp,d0 Ei + σp,d0 ζpi

).

We let the dependence in pdj be explicit for reasons that will become clear below. Becauseof the logistic assumption on the εdij, the aggregate market share sdj (pd) of good j fordemographic group d satisfies, when prices are set to pd = (pd1, ..., p

dJ),

sdj (pd) =

∫sdj (e, u, p

d)dP dE,ζ(e, u), (1)

where P dE,ζ is the distribution of (E, ζ) for group d and

sdj (e, u, pd) =

exp(δdj (p

dj ) + µdj (e, u, p

dj ))∑J

k=0 exp(δdk(p

dk) + µdk(e, u, p

dk))

Now, we consider a Nash-Bertrand competition setting where firms are able to price dis-

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criminate by setting different prices to each of the nD consumers groups. Letting Jf denotethe set of products sold by firm f , the profit of f when the vector of all prices for group dis pd satisfies

Πf = M

nD∑d=1

P (D = d)∑j∈Jf

sdj (pd)×

(pdj − cdj

),

where P (D = d) is the fraction of the group of consumers d, sdj (pd) is the market share ofproduct j for group d when prices are equal to pd and M is the total number of potentialconsumers. 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

)−1sdf , (2)

where pdf , cdf and sdf are respectively the equilibrium transaction prices, marginal costs andobserved market shares vectors for firm f . Ωd

f is the matrix of typical (i, j) term equal to−∂sdj/∂pi. Prices are optimally set by the firms making the traditional arbitrage betweenincreasing 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 foreach group separately. If a group is particularly price sensitive, the monopoly seller offersa low price and is still able to extract a large surplus from the less price sensitive group bysetting a higher price for this group. In a competitive setting, this effect is mitigated bythe fact that, for a given group of consumer, the competition among sellers is reinforced.

3 Inference

3.1 GMM estimation of the model

We now turn to inference on this model. We assume that the econometrician observes themarket shares sdj corresponding to each consumer group but not the discriminatory pricespdj paid by consumers. We do assume, on the other hand, that the econometrician observesthe posted prices.

First, let us recall the standard case where the true prices are observed. Let

θd0 = (βd0 , αd0, π

X,d0 ,ΣX,d

0 , πp,d0 ,Σp,d0 )

denote the true vector of parameters for group d. The standard approach for identifica-tion and estimation of θd0, initiated by BLP, is to use the exogeneity of Zj, which includes

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the characteristics Xj and other instruments (typically, function of characteristics of otherproducts or cost shifters) to derive moment conditions involving θd0. The exogeneity con-dition takes the form

E[Zjξ

dj

]= 0. (3)

The idea is then to use the link between ξdj and the true parameters θd0 through Equation(1). Specifically, we know from Berry (1994) that for any given vector θd, Equation (1),where θd0 is replaced by θd, defines a bijection between market shares and mean utilities ofproducts δdj . Hence, we can define δdj (sd, pd; θd), where sd = (sd1, ..., s

dJ) denotes the vector of

observed market shares. Once δdj (sd, pd; θd) is obtained, the vector ξdj (pd; θd) of unobservedcharacteristics corresponding to θd and rationalizing the market shares follows easily since

ξdj (pd; θd) = δdj (s

d, pd; θd)−Xjβd − αdpdj .

The moment conditions used to identify and estimate θd0 are then

E[Zjξ

dj (p

d; θd0)]

= 0. (4)

Now let us turn to the case where the true prices are unobserved. First, remark that whenobserved prices are different from the true prices (for example when posted prices are usedinstead of transaction prices), the former approach is not valid in general. To see this,consider the simple logit model, where πX,d0 ,ΣX,d

0 , πX,d0 and ΣX,d0 are known to be zero. In

this case δdj (sd, pd; θd) takes the simple form

δdj (sd, pd; θd) = ln sdj − ln sd0

and does not depend on pd. In this context, using posted prices p instead of the true pricesamounts to relying on

ξdj (p; θd) = ln sdj − ln sd0 −Xjβ

d − αdpj,

instead of relying on ξdj (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 andof the cost shifters. If, for instance, a group of consumer values particularly the horse-power of automobiles, powerful cars will be priced higher for this group, and pj−pdj will benegatively correlated with horsepower. Because horsepower is one of the instruments, wehave E[Zj(pj − pdj )] 6= 0, and E

[Zjξ

dj (p; θ

d0)]is no longer equal to zero. In the general ran-

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dom coefficient model, this issue also arises but in addition to that, δdj (sd, pd; θd) generallydepends on pd. Thus, Zj is also correlated with δdj (sd, pd; θd)− δdj (sd, pdj ; θd).

Instead of simply replacing pd by p, we use the supply model and reasonable identify-ing conditions on marginal costs and posted prices to recover the transaction prices. Tooperationalize 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...nDpdj .

Assumption 1 amounts to neglecting differences in the costs of selling to different consumersin 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 thegoods. Assumption 2 supposes that firms post the highest discriminatory price and thenoffer some discounts according to observable characteristics of buyers in order to reachoptimal discriminatory prices. In other words, we reinforce the very mild condition thatpj ≥ pdj for all d by assuming that for each product j, there is a group dj, called thepivot group hereafter, that pays the posted price, pdjj = pj. This assumption is necessarysince otherwise, we could shift all discounts by an arbitrary constant. It is also in linewith empirical evidence on the automobile market (for France, see, e.g., L’Automobile enEurope: 5 Leviers pour Rebondir 2013). Note, however, that the pivot group is neithersupposed to be known ex ante nor constant across different products. We also consideralternative 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 transac-tion prices pdj (θ) that rationalize the market shares and the supply-side model. Precisely,Equation (2) and Assumptions 1-2 imply that

pj = cj + maxd=1...nD

[(Ωdf

)−1sdf

]j

, (5)

where [.]j indicates that we consider the j-th line of the vector only. Then, the discrimi-natory prices satisfy

pdj = pj − maxd=1...nD

[(Ωdf

)−1sdf

]j

+[(

Ωdf

)−1sdf

]j. (6)

Now, under the logit model, ∂sdj/∂pdj = −αdsdj (1 − sdj ) and ∂sdj/∂pdj′ = −αdsdjsdj′ . As a

result, Ωdf is a function of observed market shares and of α = (α1, ..., αnD) only. In turn,

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pdj can then be expressed simply as a function of α, using Equation (6). Denoting it bypdj (α), we obtain, using ξdj (p; θd) = ln sdj − ln sd0 −Xjβ

d − αdpdj (α), the moment equations

E

[Zj

(lnsdjsd0−Xjβ

d − αdpdj (α)

)]= 0.

Compared to the logit model with observed prices, the only difference is that we haveto compute pdj (α) using (6). For a given α, βd can be easily obtained by two-stage leastsquares, 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 essentiallysimilar but an additional issue arises. Equation (6) shows that for a given parameter θ,the discriminatory prices are identified up to Ωd

f . Now, taking the derivative of the marketshare function (Equation (1)) with respect to the price pdj yields:

∂sdj∂pdj

(pd) =

∫ (αd0 + πp,d0 e+ Σp,d

0 up)sdj (e, u, p

d)(1− sdj (e, u, pd))dP dE,ζ(e, u) (7)

We obtain a similar expression for ∂sdj/∂pdl (pd). These expressions show that Ωdf only

depends on the parameters θd0, on the vector of prices pd and on δd = (δd1 , ..., δdJ), through

sdj (e, u, pd). We emphasize this dependence by denoting it Ωd

f (θd0, p

d, δd). Besides, for a set ofprices pd, we can obtain by inverting the market share system the vector δd 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 = (δ1, . . . , δnD) andp = (p1, . . . , pnD) denote respectively the full vector of transaction prices. We supposehereafter 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 =

∫exp

(δdj + µdj (e, u, p

dj ))∑J

k=0 exp(δdk + µdk(e, u, p

dk))dP d

E,ζ(e, u), (8)

pdj = pj − maxd=1...nD

[(Ωdf (θ

d, pd, δd))−1

sdf

]j

+[(

Ωdf (θ

d, pd, δd))−1

sdf

]j. (9)

This assumption is satisfied in the special case where there is no unobserved heterogeneityon price sensitivity, so that αdi = αd0. In such a case, µdj (e, u, pdj ) does not depend on thetransaction price pdj anymore. As a result, the right-hand side of Equation (8) definingmarket shares only depends on δd. By the result of Berry (1994), there is a unique δd

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which solves this system. Turning to the price equation, Ωdf (θ

d, pd, δd) does not depend, forthe same reason, on pd. Therefore, the right-hand side of Equation (9) does not depend onpd, and there is indeed a unique pdj satisfying this system of equations. We conjecture thatthis result remains true at least for models where the heterogeneity coefficients on prices,πp,d and Σp,d are relatively small. Finally, Assumption 3 is related but not equivalent tothe uniqueness of the Nash-Bertrand equilibrium in prices. Even if the pivot groups wereknown, in which case we would identify directly the marginal costs and therefore wouldhave to solve for the prices in the standard supply-side first-order conditions, the equationswould still differ from those of the Nash-Bertrand equilibrium.

Under Assumption 3, we can apply the GMM to identify and estimate θ0 = (θ10, ..., θnD0 ).

Let δdj (s, θ) and pdj (s, θ) denote the mean utility and price of product d when market sharesand the vector of parameters are respectively equal to s and θ. Let also

MdJ (θ) =

1

J

J∑j=1

Zj(δdj (s, θ)−Xjβ

d − αdpdj (s, θ))

denote the empirical counterpart of the moment conditions corresponding to Equation (4).Let MJ(θ) = (M1

J(θ)′, ...,MnDJ (θ)′)′ and define

QJ(θ) = MJ(θ)′WJMJ(θ),

where WJ is a positive definite matrix. Our GMM estimator of θ0 is then

θ = arg minθQJ(θ). (10)

As in the standard BLP model, it is possible to include moments corresponding to thesupply side by imposing some additional structure on marginal costs. Let Xs be the vectorof cost shifters. Xs

j may be different fromXj but typically share some common components.We may suppose for instance that the marginal costs are log-linear:

ln(cj) = Xsj γ + ωj, (11)

where ωj stands for the unobserved cost shock. This shock is supposed to satisfy E[Zsjωj] =

0, where Zsj 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

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to s and a given vector of parameter θ. Specifically, by Equation (5),

cj(s, θ) = pj − maxd=1...nD

[(Ωdf (θ

d, pd(s, θ), δd(s, θ)))−1

sdf

]j.

We then obtain ωj(s, θ, γ) simply by

ωj(s, θ, γ) = ln (cj(s, θ))−Xsj γ.

The supply-side moment conditions are then

M sJ(θ, γ) =

1

J

J∑j=1

Zsj

[ln (cj(s, θ))−Xs

j γ]

Then we can proceed as previously, simply replacing MJ(θ) by MJ(θ, γ) = (M1J(θ)′, ...,

MnDJ (θ)′,M s

J(θ, γ))′.

Compared to the estimation of the standard BLP model, estimating our model in practiceraises two challenges. First, we have to optimize over a larger space than in the BLPsetting. In the standard BLP model where we observe the market share of j for each groupd but true prices are observed or supposed to be equal to posted prices, we could optimizeonly on θd (abstracting from supply-side conditions), for each group separately. We evenonly need to optimize over (αd0, π

X,d0 ,ΣX,d

0 , πp,d0 ,Σp,d0 ), because we can easily concentrate the

objective function on βd, by running ordinary least squares of the δdj on (Xj). In our case,we cannot estimate θd separately from θd

′ , for d′ 6= d, because θd′ matters for determiningpdj (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 theprices. Therefore, estimating the model is computationally more costly. We describe indetails our algorithm in section 4 and show that this optimization problem remains feasiblein 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 pricesensitivity within a group of consumers, so that αdi = αd0. This assumption may be rea-sonable in particular if we have a fine segmentation of consumers. In this case, we stillhave to optimize over θ = (θ1, ..., θnD). On the other hand, solving the system defined byEquations (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 Ωd

f does

3On the other hand, and as in the BLP model, we can concentrate the objective function on βd.

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not depend on pd. Thus, the computational cost is significantly reduced compared to thegeneral model. Another alternative is to rely on the logit or nested logit models. In thesimple logit model, we have seen above that the matrix Ωd

f only depends on (α1, ..., αnD).In the nested logit, it also depends on the parameters (σ1, ..., σnD) that drive substitutionswithin nests. But at the end, we also obtain a quite simple nonlinear optimization over(α1, σ1, ..., αnD , σnD) only.

3.2 Extensions

3.2.1 Other functional forms on price effects

We have assumed up to now, following the common practice, that indirect utilities dependlinearly on disposable income, namely on αi(yi − pj), where yi denotes the income beforemaking one’s choice. αiyi can then be removed, as being constant across alternatives.To incorporate, for example, credit constraints as in BLP, the indirect utility may ratherdepend on αi ln(yi − pj). Let us suppose, more generally, that the utility depends ondisposable income through q(yi − pj, αi) where q is known by the econometrician whileαi|Di = d ∼ N (αd, σ2d

α ) with (αd, σ2dα ) 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 ), withyi being one component of Ei. Then Equations (8) and (9) remain unchanged, the onlydifference being that the terms entering into Ωd

f do not satisfy Equation (7). But other thanthat, 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 whenprice discriminating the buyer. Gender and race may be important examples. It is stillpossible to apply our methodology as long as instruments for such variables are available.Specifically, suppose that we observe a discrete variable D such that (i) (ζi, ε

dij) ⊥⊥ D

and (ii) the matrix P which typical (d, d) term is the probability of belonging to group dconditional on observing d, P (D = d|D = d) has rank nD. Condition (i) is an exclusionrestriction which imposes that consumers do not differ systematically in their taste acrosscategories of D, once we control for D. Condition (ii) is similar to the standard relevancecondition in IV models and imposes that D is, basically, related to D. Let Yi denote the

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product choice of consumer i. Under the first condition, we have

P (Yi = j|Di = d) =

nD∑d=1

P(Di = d|Di = d

)P (Yi = j|Di = d, Di = d)

=

nD∑d=1

P(Di = d|Di = d

)sdj .

Then, letting sj = (s1j , ..., snDj )′, sj = (P (Yi = j|Di = 1), ..., P (Yi = j|Di = nD))′, we have,

for all j = 1...J ,sj = Psj

Because P has rank nD, this equation in sj admits a unique solution. This implies that sjis 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 observesthe buyers’ professions while sellers price discriminate based on buyers’ incomes. In thiscontext, we observe market shares of products by professional activity. The rank conditionmeans that we know the probability of belonging to an income class conditional on theprofessional activity. From this probability matrix, we are able to compute market sharesof products by income class. The exclusion restriction imposes that the differences inpreferences 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 consumersshould pay list prices, which are observed. Second, the marginal costs should be identicalfor all groups. Another crucial assumption concerns the nature of competition on themarket, which we assume to be Bertrand competition. We believe that these conditionsare realistic in many settings. In some cases, however, alternative conditions may be morenatural. Our method still applies if these alternative conditions allow us to recover themarginal costs of each product, for a given value of the parameter vector. Once we obtainthese 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 ofcompetition.

A simple example is when we observe, through survey data for instance, the price paid by

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at least one group for each product. Then, instead of using Equation (6), we can rely on

pdj = pdjj −

[(Ωdjf

)−1sdjf

]j

+[(

Ωdf

)−1sdf

]j,

where dj denotes the group for which the price of j is observed.

Similarly, suppose that we observe the average price pmj =∑nD

d=1 sdjpdj paid by all consumers

for each product, through, for example, aggregated data on sales from the firms. Then wereplace Equation (6) by

pdj = pmj −nD∑d′=1

sd′

j

[(Ωd′

f

)−1sd′

f

]j

+[(

Ωdf

)−1sdf

]j.

3.2.4 Alternative supply-side models

For a given a set of demand parameters θd, we expressed the corresponding transactionprices pd(θd) using supply-side conditions. Once these transaction prices are recovered, wecan use the standard BLP method to compute ξdj (pd(θd); θd) and then the moment condi-tions E[Zjξ

dj (p

d(θd); θd)], to check whether they are equal to zero or not. The assumptionabout the nature of competition on the market is therefore more crucial in our model thanin the standard BLP approach. Because pdj is unobserved and depends on the behavior offirms, it is impossible to estimate the demand without making assumptions on the supplyside in our setting. We do not see this as a strong limitation, however, because the supplyside 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 ourmethodology also applies to other supply-side models.

First, the identification strategy holds when there is collusion between sellers. Only theterm Ωd

f is modified to take into account the fact that the prices of all products are setby 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 forthe other groups. Our methodology also applies when the supply-side model incorporatesthe vertical relations between producers and retailers. For instance the papers by Brenkers& Verboven (2006), Mortimer (2008) and Bonnet & Dubois (2010) develop and estimatestructural models of demand and supply including vertical contracting between producersand retailers. Our methodology can still be applied for such models. Moreover, only thetype 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

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Equation (5) we can recover the marginal cost crj of retailer r by

crj(s, θ) = pj − maxd=1...nD

[(Ωdf (θ

d, pd(s, θ), δd(s, θ)))−1

sdf

]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 estimatorin practice and present some simulation results. First, note that we do not rely on theminimization program with equilibrium constraints (MPEC) approach suggested by Dubéet al. (2012) because the gradient and hessian of the constraints cannot be obtained ana-lytically easily in our model. Rather, we use the standard approach where for each valueof 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 p, ordraw a vector of initial discounts.

2. Given the current vector of transaction prices pd, compute δd = δ(sd, pd; θd). We canuse 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, thefirst 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 δd. Foreach value of transaction prices, we have to invert the market share equation to solve forthe mean utility vectors δd. We use for that purpose the contraction mapping proposedby BLP. If the computational cost is larger than for the BLP estimator, it is possible toparallelize this market share inversion as well as the computation of the mark-up terms((Ωd

f )−1sdf ), 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 theinner price-loop and by updating initial values of prices across iterations of the outer loopthat involves the parameters θ.

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In the simulations below and in the application, we use the following specifications forcomputing the GMM estimator. First, to approximate the aggregated market shares, weuse Halton normal draws for each demographic group and market (300 in the simulationsand 1,000 in the application) and the line search algorithm for minimization. Our initialvalues for the price sensitivity parameters are the estimates obtained with the simple logitmodel, while we use random draws from a uniform U [−1/2, 1/2] distribution for the randomcoefficient. As suggested by Dubé et al. (2012) and Knittel & Metaxoglou (2014), we seta tight tolerance (10−12) to compute the mean utilities and the prices, while the tolerancelevels are 10−5 for the parameters and 10−3 for the objective function. Finally, as suggestedby Knittel & Metaxoglou (2014), we carefully investigate potential convergence issues byusing different starting values and selecting the estimates that yield the lowest value of theobjective function.

To investigate the performance of our estimator and whether the algorithm produces re-liable results, we perform a Monte-Carlo simulation. We construct 50 different data setsfor T = 25 markets, J = 24 products and D = 4 demographic groups. For each marketand product, we construct the vectors of observed characteristics Xjt = (1, X1jt), unob-served characteristics ξdjt, observed cost shifters Wjt = (W1jt,W2jt,W3jt) and unobservedcost 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] andWjt follows a trivari-ate uniform distribution. ξdjt and ωjt are independent draws from the normal distributionN (0, 0.1). The parameters of preferences are summarized in Table 1. Groups of consumersare heterogeneous with respect to their average valuation of product attributes and theprice sensitivity. Group 1 is the less price sensitive group and has the highest utility ofholding 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 theexogenous characteristics (the valuation is set to 1.5 versus 2 for all the other groups). Asin our application, the unobserved heterogeneity parameters (σ) are identical for the fourdemographic groups. Finally, we assume that the market is composed by 4 firms, each ofthem producing 6 products. Once we solve for prices and market shares (sdjt, p

djt)d=1,2,3,4,

we define for each product the posted price pj as the maximal price across demographicgroups. We use for estimation the instruments Zjt = (Xjt,Wjt).

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Proportion Intercept X1 PriceGroup 1 0.3 -1 1.5 -1.5Group 2 0.2 -1 2 -2.5Group 3 0.3 -0.5 2 -2Group 4 0.2 -0.5 2 -3Random coeff. 0.5 0.4

Table 1: Parameters of preferences for the simulations

We compare the estimates of the price discrimination model with the standard modelassuming uniform pricing. Specifically, we assume in the latter case that the supply-sidefirst-order conditions are

pj = cj +[(Ωf )

−1 sf]j, (12)

where Ωf is the matrix of typical (i, j) term equal to −∂sj/∂pi and sj =∑D

d=1 P (D = d)sdj .These first-order conditions correspond to the maximization of profits under the constraintthat 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 discriminationaccurately estimates both the demand supply parameters. The pivot groups are exactlyguessed and the estimated discounts are very close to the true underlying discounts. Onthe opposite, the performances of the uniform pricing model are not as good, leading ingeneral 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 modelthan for the true model with price discrimination. The parameters of the intercept appearto be especially sensitive to misspecification. On the supply side, it is interesting to notethat apart from the parameter of the intercept, the cost equation is well estimated underthe two alternative models. The GMM objective function value is, however, much lowerfor 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 thanthe GMM estimator of the standard BLP model. On average, it is around 3.5 times slowerthan the standard uniform pricing model. However, the number of iterations is roughlythe same and the estimation time remains decent because it is possible to parallelize thecomputationally intensive part of the estimation algorithm.

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True Discrimination BLP Uniform BLPMean Bias RMSE Mean Bias RMSE

Price sensitivityGroup 1 -1.5 -1.5 0 0.05 -1.46 0.04 0.16Group 2 -2.5 -2.5 0 0.09 -2.36 0.14 0.27Group 3 -2 -2 0 0.06 -1.89 0.11 0.2Group 4 -3 -3 0 0.1 -2.83 0.17 0.31sigma 0.4 0.4 0 0.04 0.35 -0.05 0.16InterceptGroup 1 -1 -1 0 0.1 -1.08 -0.08 0.4Group 2 -1 -0.99 0.01 0.15 -0.51 0.49 0.68Group 3 -0.5 -0.5 0 0.11 -0.32 0.18 0.45Group 4 -0.5 -0.5 0 0.15 0.32 0.82 0.94Exogenous characteristicGroup 1 1.5 1.5 0 0.06 1.46 -0.04 0.21Group 2 2 2 0 0.08 1.96 -0.04 0.33Group 3 2 2 0 0.06 1.96 -0.04 0.22Group 4 2 2 0 0.12 1.98 -0.02 0.42sigma 0.5 0.49 -0.01 0.09 0.46 -0.04 0.33Marginal cost equationIntercept 0.7 0.7 0 0.04 0.86 0.16 0.17X1 0.7 0.7 0 0.02 0.73 0.03 0.04W1 1 1 0 0.02 0.99 -0.01 0.03W2 1 1 0 0.02 0.99 -0.01 0.03W3 1 1 0 0.02 0.99 -0.01 0.03Average discount (in %)Group 1 0.03 0.03Group 2 10.90 10.90Group 3 7.13 7.12Group 4 13.77 13.76Frequency pivot (in %)Group 1 98.7 98.6Group 2 0 0Group 3 1.3 1.4Group 4 0 0Mean objective function value 0.16 1.19% replications converging 100 100Number of iterations 457 466Time (sec) 551 167

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 uniformpricing model is the true one. We use the same values of parameters of demand and supplyexcept that prices are no longer group-specific but optimally set by firms given the globaldemand that arises from the heterogeneous groups of consumers. Table 3 summarizes theestimation results over the 50 replications. As expected, the bias of our GMM estimatoris larger than the standard BLP GMM estimator. The price sensitivity parameters areunderestimated and the parameters of the intercept estimates corresponding to the unob-served discrimination model exhibit a large bias. The model with unobserved discountsyields positive and significant discount, which is natural since the values of discounts arepinned 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 withuniform pricing.

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True Uniform BLP Discrimination BLPMean Bias RMSE Mean Bias RMSE

Price sensitivityGroup 1 -1.5 -1.48 0.02 0.07 -1.35 0.15 0.21Group 2 -2.5 -2.48 0.02 0.1 -2.27 0.23 0.36Group 3 -2 -1.98 0.02 0.07 -1.85 0.15 0.25Group 4 -3 -2.98 0.02 0.1 -2.83 0.17 0.33sigma 0.4 0.39 -0.01 0.06 0.24 -0.16 0.22InterceptGroup 1 -1 -1.05 -0.05 0.19 -1.4 -0.4 0.57Group 2 -1 -1.05 -0.05 0.22 -2.24 -1.24 1.36Group 3 -0.5 -0.55 -0.05 0.18 -1.28 -0.78 0.9Group 4 -0.5 -0.54 -0.04 0.17 -1.99 -1.49 1.6Exogenous characteristicGroup 1 1.5 1.5 0 0.04 1.5 0 0.11Group 2 2 2.01 0.01 0.06 1.96 -0.04 0.22Group 3 2 2 0 0.04 1.97 -0.03 0.15Group 4 2 2 0 0.09 1.9 -0.1 0.29sigma 0.5 0.48 -0.02 0.09 0.39 -0.11 0.27Marginal cost equationIntercept 0.7 0.69 -0.01 0.04 0.45 -0.25 0.27X1 0.7 0.7 0 0.02 0.7 0 0.03W1 1 1 0 0.02 1.07 0.07 0.08W2 1 1 0 0.02 1.08 0.08 0.09W3 1 1.01 0.01 0.02 1.08 0.08 0.09Average discount (in %)Group 1 0.01Group 2 10.9Group 3 7.2Group 4 13.9Frequency pivot (in %)Group 1 99.53Group 2 0Group 3 0.47Group 4 0Mean objective function value 0.22 1.08% replications converging 100 98Number of iterations 464 493Time (sec) 170 638

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 theunicity condition of Assumption 3. For that purpose, we compute the value of the transac-tion prices for 50 different initial values of prices and the true value of the parameters usingthe algorithm detailed above. Under Assumption 3, we should expect to obtain the sametransaction prices for each of these initial values, whenever the algorithm converges. More-over, 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 al-gorithm always converged to the true value of the transaction prices. Besides, convergenceoccurs very quickly. We computed, at each iteration of the price-loop, the average andmaximal absolute differences between the true prices and those obtained by the algorithm,across all products. We then averaged these average and maximal absolute differences overthe 50 initial draws. The results, displayed in Table 4, show that the sequence of vectorsof prices converges very quickly to the true vector.

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Iteration 1 2 3 4 5Average 1.28 0.052 0.0014 4.2×10−5 1.8 ×10−6

Maximal 3.92 0.29 0.011 6.1 ×10−4 4.1×10−5

Lecture notes: “average” (resp. “Maximal”) is the average (resp. maxi-mal) absolute differences between the true prices and those obtained bythe algorithm across all products. The figures are average over the 50 sim-ulations. 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 unobserveddiscounts in the new automobile industry, using a dataset from the association of Frenchautomobile manufacturers (CCFA, Comité des Constructeurs Français d’Automobiles) thatrecords all the registrations of new cars purchased by households in France between 2003and 2008. Each year, we observe a sample of about one million vehicles. For each registra-tion, the following attributes of the car are reported: brand, model, fuel energy, car-bodystyle, number of doors, horsepower, CO2 emissions, cylinder capacity and weight. Thesecharacteristics 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 onlyobserve here posted prices that come from manufacturers catalogs.

We now turn to the construction of the consumer groups that are used by firms to pricediscriminate. Apart from car attributes, the date of the registration and some character-istics of the owner are provided in the CCFA database : municipality of residence andage. The age (or the age class) is presumably a strong determinant of purchase, and iseasily observed by a seller even if he does not know the buyer before the transaction. Wetherefore assume that these characteristics are used by the automobile makers to pricediscriminate. The income is also likely to affect preferences for different car attributes andprice sensitivity. The income is, however, likely to be unobserved by the seller but insteadinferred from the municipality the buyer lives in and the age class. We compute a pre-dictor of buyer’s income, namely the median household income in his age class and in hismunicipality using data from the French national institute of statistics (Insee).4 It seems

4There are over 36,000 municipalities in France. Note also that Paris, Lyon and Marseille, the three

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reasonable to assume that the seller does not have a far better prediction of the buyer’sincome in such anonymous market, where buyers and sellers do not know each other beforethe transaction. It is crucial for our approach that buyers cannot lie about their individualcharacteristics, and in our application it implies that buyers do not make geographicalarbitrage, i.e. buy the car in another municipality where discounts are higher. We believethat this assumption is reasonable since buyers have high incentive to buy a new car at aclose dealer to minimize transportation costs and take advantage of the after-sale servicesand guarantees. We thus define groups of buyers by interacting three age classes and twoincome classes.5 We choose the common thresholds of 40 and 60 for the age classes, and27,000 euros per year as the threshold for income. This amount corresponds roughly tothe median yearly income in France in 2008.6

Group FrequencyAge < 40, income <27,000 15.7%Age < 40, income ≥ 27,000 11.5%Age ∈ [40,59], income <27,000 16.3%Age ∈ [40,59], income ≥ 27,000 22.3%Age ≥ 60, income <27,000 20.8%Age ≥ 60, income ≥ 27,000 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 (itis likely that firms discriminate along several dimensions) and accuracy of the observedproportion of sales sdj as estimators of the true market shares sdj . The six groups that weconsider are large enough to avoid in most cases the problem of zero sales (see Table 17in Appendix A.2 for the fraction of products with null market shares). Moreover, ratherthan discarding those products, we replace the proportion of sales by a predictor of sdj that

minimizes the asymptotic bias, namely sdj =ndj+0.5

Nd , ndj denoting the number of sales ofproduct j in group d and Nd the number of potential buyers with characteristics d (seeAppendix 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 medianincome across municipalities is large.

5We do not observe owners’ gender in our database. Even if this information was available, it would behard to use since the owner and the buyer can be different persons. Furthermore, many couples are likelyto buy their car together.

6We estimate our model with alternate thresholds. The results, which are overall very similar, areavailable upon request.

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estimator has been proposed by Gandhi et al. (2013). We show in Appendix A.2 that ourresults 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 of3205 products for the six years is obtained. Table 6 presents the average characteristics ofnew cars purchased for each group of consumers. We find significant heterogeneity acrossthese groups. On average, the medium age, high income class purchases more expensivevehicles. They also choose larger and more powerful cars. Young purchasers are moreinterested in smaller cars (lighter and with three doors) whereas station-wagons are morepopular among the medium age class. The highest age group purchases lighter vehiclesthan 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 19,803 6.2 5.7 1,182 19.0% 9.7%A < 40, I ≥ 27, 000 20,911 6.5 6.0 1,221 16.8% 12.9%A ∈ [40,59], I <27,000 21,521 6.5 6.1 1,231 14.3% 12.7%A ∈ [40,59], I ≥ 27, 000 21,739 6.8 6.2 1,236 14.8% 13.1%A ≥ 60, I <27,000 20,117 6.9 5.9 1,194 11.4% 8.9%A ≥ 60, I ≥ 27, 000 20,831 7.0 6.0 1,219 10.9% 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, HPstands 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 thedistribution sector is not modeled in this application. We make the traditional assumptionthat manufacturers have only exclusive dealers and are perfectly integrated. As discussedin the previous section, adding vertical relations between manufacturers and dealers wouldbe possible as long as the competition on the downstream market still implies a Nash-Bertrand equilibrium. We also suppose that prices are set at the national level, which isconsistent with the fact that listed prices are set by manufacturers at such a level. Withsufficient 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 ofpricing strategy and competition intensity (see, e.g., Nurski & Verboven 2012). Due to alack of such available data, we abstract from these issues afterwards.

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5.2 Parameter estimates and comparison with the standard model

We first present the estimations of different models. We estimate the nested logit modelwith and without unobserved price discrimination. We also estimate the standard BLPmodel with uniform pricing, where the first-order conditions of the supply side is (12), andthe BLP model with unobserved price discrimination. In models with uniform pricing, weassume that sellers do not price discriminate and that the posted prices, which correspondto the transaction prices, are optimal given the heterogeneous preferences of the differentgroups of consumers. For all specifications, we control for the main characteristics of thecars such as horsepower, weight and the cost of driving 100 kilometers in the demandfunction. 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 forall demographic groups. For the two random coefficient models, we allow for unobservedheterogeneity of preferences inside groups of consumers in terms of price, fuel cost and forthe utility of buying a new car, represented by the intercept. To obtain more accurateresults, we constrain the heterogeneity parameters to be identical for all demographicgroups. In the marginal cost equation, we use horsepower, fuel consumption (in liters for100 kilometers) and weight as cost-shifters. We also introduce brand dummies to control formanufacturer’s specific unobserved quality of cars. Finally, the nested logit model requiresa segmentation of the market. We take, as in the literature, a segmentation according tothe 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 momentconditions stemming from the demand and from the marginal cost equation (Equation(11)).7 The implementation of the estimation follows the method described in Section4. We also verify that Assumption 3 is satisfied at the estimated value of parametersby applying the test performed in the simulation analysis and find that after drawingseveral initial values of transaction prices, the algorithm always converges to the same valueof estimated transaction prices. In addition to exogenous characteristics we include thefollowing 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 thecar weight multiplied by a composite price index that aims at approximating the averageinput price.8 The other instruments are close to those suggested by BLP. We include the

7We also estimated the models without using the moment conditions stemming from the marginal costsequation. The results are very similar.

8Specifically, we use a weighted average of steel, aluminium and plastic prices taken in January. Theweights we use are equal to 0.77, 0.11 and 0.12, respectively, reflecting the relative importance of each ofthese inputs in car manufacturing.

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sum of continuous exogenous characteristics (namely weight, horsepower and fuel cost)of other brands’ products. We also consider the sums of these characteristics over otherbrands’ products of the same segment, supposed to be closer substitutes. Finally, weinclude the sums of these characteristics of the other products of the brand belonging tothe same segment.9

The results for the different models are presented in Table 7. Columns (1) and (2) presentthe estimation results for the nested logit specifications, which abstract from individualheterogeneity, while Columns (3) and (4) display the estimation results for the randomcoefficient models. The estimated parameters are generally similar for the nested logitand the random coefficient models. Note that for two groups (the old purchasers with lowand high income), we obtain negative intra-segment correlation, which is absurd since thisparameter should belong to [0, 1]. Thus, we constrain these two parameters to be equalto zero in the estimation, which amounts to consider the logit specification for these twogroups of consumers. The random coefficient models imply higher price sensitivities thanthe nested logit models and significant within-group individual heterogeneity. We obtaina standard deviation of 1.12 for the model with uniform pricing and 0.95 for the modelwith price discrimination. We thus discuss in more detail the results for the models withrandom coefficient while the results of the nested logit specification serve as a benchmarkto check the general credibility of the models with individual unobserved heterogeneity.

9Armstrong (2014) has recently shown that such instruments could be weak when the number ofproducts 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 purposethe F-statistic of the joint nullity of the coefficients of these instruments in the linear regression of priceson the characteristics and these instruments. We obtain F ' 24.1, which is far above the threshold of10 suggested by Staiger & Stock (1997) and usually used to detect weak instruments. This is thereforereassuring on the identification of the model and the validity of inference here.

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Nested-logit Random CoefficientsUniform Price discrimination Uniform Price discrimination

Parameter Std-err Parameter Std-err Parameter Std-err Parameter Std-errPrice sensitivityAge < 40, I = L -2.69∗∗ 0.18 -2.70∗∗ 0.185 -4.51∗∗ 0.278 -4.73∗∗ 0.271Age < 40, I = H -2.55∗∗ 0.173 -2.55∗∗ 0.176 -4.27∗∗ 0.265 -4.55∗∗ 0.278Age ∈ [40,59], I = L -2.24∗∗ 0.174 -2.24∗∗ 0.177 -3.87∗∗ 0.265 -4.17∗∗ 0.269Age ∈ [40,59], I = H -2.16∗∗ 0.169 -2.16∗∗ 0.178 -3.68∗∗ 0.25 -3.87∗∗ 0.254Age ≤ 60, I = L -1.95∗∗ 0.179 -1.98∗∗ 0.185 -3.75∗∗ 0.288 -4.07∗∗ 0.287Age ≤ 60, I = H -1.79∗∗ 0.172 -1.88∗∗ 0.105 -3.51∗∗ 0.257 -2.87∗∗ 0.249Std. dev. (σp) 1.12∗∗ 0.081 0.95∗∗ 0.081Intra-segment correlationAge < 40, I = L 0.17∗ 0.068 0.17∗ 0.073Age < 40, I = H 0.29∗∗ 0.07 0.3∗∗ 0.072Age ∈ [40,59], I = L 0.22∗∗ 0.066 0.22∗∗ 0.069Age ∈ [40,59], I = H 0.29∗∗ 0.074 0.29∗∗ 0.08Age ≤ 60, I = L 0 0Age ≤ 60, I = H 0 0InterceptAge < 40, I = L -6.49∗∗ 0.536 -7.09∗∗ 0.515 -5.52∗∗ 0.49 -6.42∗∗ 0.524Age < 40, I = H -6.43∗∗ 0.527 -7.05∗∗ 0.488 -6.35∗∗ 0.501 -7.17∗∗ 0.533Age ∈ [40,59], I = L -6.92∗∗ 0.493 -7.29∗∗ 0.457 -6.21∗∗ 0.504 -7.01∗∗ 0.529Age ∈ [40,59], I = H -6.46∗∗ 0.54 -6.89∗∗ 0.496 -6.47∗∗ 0.459 -7.06∗∗ 0.497Age ≤ 60, I = L -7.86∗∗ 0.248 -7.92∗∗ 0.296 -5.86∗∗ 0.47 -6.66∗∗ 0.501Age ≤ 60, I = H -8.2∗∗ 0.242 -8.23∗∗ 0.238 -6.39∗∗ 0.487 -6.45∗∗ 0.496Std. dev (σx) 0.35 1.465 0.39 1.335Fuel costAge < 40, I = L -6.03∗∗ 0.339 -6.03∗∗ 0.349 -6.00∗∗ 0.234 -5.43∗∗ 0.238Age < 40, I = H -4.86∗∗ 0.297 -4.84∗∗ 0.302 -5.09∗∗ 0.249 -4.7∗∗ 0.244Age ∈ [40,59], I = L -5.06∗∗ 0.3 -5.04∗∗ 0.306 -5.2∗∗ 0.233 -4.9∗∗ 0.225Age ∈ [40,59], I = H -4.13∗∗ 0.272 -4.12∗∗ 0.278 -4.21∗∗ 0.221 -4∗∗ 0.214Age ≤ 60, I = L -4.17∗∗ 0.244 -4.19∗∗ 0.248 -3.56∗∗ 0.225 -3.42∗∗ 0.212Age ≤ 60, I = H -3.52∗∗ 0.235 -3.62∗∗ 0.199 -2.77∗∗ 0.218 -2.55∗∗ 0.161Std. dev (σx) 0.09 0.128 0.23† 0.117HorsepowerAge < 40, I = L 5.8∗∗ 0.501 5.8∗∗ 0.514 3.7∗∗ 0.441 2.54∗∗ 0.412Age < 40, I = H 5.21∗∗ 0.476 5.2∗∗ 0.482 3.1∗∗ 0.455 2.17∗∗ 0.414Age ∈ [40,59], I = L 4.31∗∗ 0.474 4.3∗∗ 0.482 2.15∗∗ 0.439 1.72∗∗ 0.381Age ∈ [40,59], I = H 4∗∗ 0.462 3.97∗∗ 0.486 1.7∗∗ 0.408 1.29∗∗ 0.356Age ≤ 60, I = L 2.93∗∗ 0.488 2.99∗∗ 0.502 1.22∗∗ 0.424 1.03∗∗ 0.354Age ≤ 60, I = H 2.49∗∗ 0.467 2.73∗∗ 0.322 0.77† 0.402 0.19 0.17WeightAge < 40, I = L 4.15∗∗ 0.393 4.17∗∗ 0.396 5.74∗∗ 0.339 6.53∗∗ 0.371Age < 40, I = H 4.05∗∗ 0.357 4.06∗∗ 0.357 5.82∗∗ 0.339 6.71∗∗ 0.383Age ∈ [40,59], I = L 4.2∗∗ 0.353 4.21∗∗ 0.353 5.66∗∗ 0.323 6.54∗∗ 0.355Age ∈ [40,59], I = H 3.88∗∗ 0.342 3.89∗∗ 0.34 5.56∗∗ 0.315 6.2∗∗ 0.358Age ≤ 60, I = L 3.53∗∗ 0.342 3.58∗∗ 0.349 4.49∗∗ 0.316 5.47∗∗ 0.359Age ≤ 60, I = H 3.56∗∗ 0.332 3.69∗∗ 0.256 4.57∗∗ 0.309 3.92∗∗ 0.213Three doorsAge < 40, I = L -0.09 0.118 -0.09 0.12 0.10 0.118 0.17 0.12Age < 40, I = H -0.26∗ 0.104 -0.26∗ 0.105 -0.05 0.117 0.00 0.119Age ∈ [40,59], I = L -0.22∗ 0.105 -0.22∗ 0.105 -0.04 0.114 -0.03 0.115Age ∈ [40,59], I = H -0.36∗∗ 0.098 -0.35∗∗ 0.1 -0.2† 0.115 -0.18 0.116Age ≤ 60, I = L -0.6∗∗ 0.117 -0.61∗∗ 0.117 -0.52∗∗ 0.112 -0.53∗∗ 0.112Age ≤ 60, I = H -0.65∗∗ 0.114 -0.67∗∗ 0.111 -0.59∗∗ 0.11 -0.5∗∗ 0.104Station-wagonAge < 40, I = L -0.59∗∗ 0.086 -0.59∗∗ 0.088 -0.75∗∗ 0.081 -0.75∗∗ 0.083Age < 40, I = H -0.42∗∗ 0.074 -0.42∗∗ 0.075 -0.61∗∗ 0.08 -0.62∗∗ 0.083Age ∈ [40,59], I = L -0.45∗∗ 0.084 -0.45∗∗ 0.085 -0.64∗∗ 0.079 -0.66∗∗ 0.081Age ∈ [40,59], I = H -0.46∗∗ 0.084 -0.46∗∗ 0.086 -0.71∗∗ 0.081 -0.71∗∗ 0.084Age ≤ 60, I = L -0.7∗∗ 0.078 -0.7∗∗ 0.078 -0.73∗∗ 0.078 -0.75∗∗ 0.082Age ≤ 60, I = H -0.67∗∗ 0.077 -0.69∗∗ 0.076 -0.72∗∗ 0.079 -0.64∗∗ 0.076Marginal cost equationIntercept -0.25∗∗ 0.048 -0.47∗∗ 0.059 -0.05 0.03 -0.19∗∗ 0.03Horsepower 0.49∗∗ 0.026 0.5∗∗ 0.028 0.43∗∗ 0.028 0.28∗∗ 0.028Fuel inefficiency -2.69∗∗ 0.323 -2.49∗∗ 0.347 -2.03∗∗ 0.307 -0.83∗∗ 0.307break -0.07∗∗ 0.012 -0.09∗∗ 0.014 -0.07∗∗ 0.011 -0.07∗∗ 0.011Three doors -0.05∗∗ 0.007 -0.05∗∗ 0.009 -0.04∗∗ 0.007 -0.04∗∗ 0.007Weight × average input price 0.13∗∗ 0.004 0.15∗∗ 0.005 1.01∗∗ 0.028 1.1∗∗ 0.028Value of objective function 4,220 4,236 2,400 1,794Number of observations 22,435 22,435 22,435 22,435Significance levels : † : 10% ∗ : 5% ∗∗ : 1%

Table 7: Parameter estimates for the four specifications.

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The two random coefficients models produce similar price sensitivities except for the groupof old with high income. This group is the least price sensitive group and turns out tobe always pivot in the model with unobserved price discrimination. Specifically, the pricesensitivity of the pivot group is overestimated with uniform pricing compared to the modelwith unobserved price discrimination. The price sensitivity decreases with both age andincome, leaving the young with low income the more price sensitive group. The parametersof the intercept are negative, reflecting the fact that the major part of consumers choosethe 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. Asexpected, consumers display a preference for horsepower, but how much they value it differsubstantially across groups. Young consumers have a high valuation for the engine powerwhile the eldest care less about this attribute. As expected, all groups of consumers dislikelarge fuel expenses. The parameters of sensitivity to the fuel cost are consistent with theparameters of sensitivity to the car price. The old purchasers with high income appear tobe also the less sensitive to the cost of driving while the more sensitive consumers are alsothe young and middle-age groups with a low income. As weight is a proxy for the size andthe space of the car, it is positively valued by all the consumers. Three doors and station-wagon vehicles are negatively valuated, reflecting that most of the consumers buy sedanor hatchback cars with five doors (four doors plus the trunk). Finally, the cost equationparameters have the predicted signs. The marginal cost of production is increasing in thehorsepower and in the proxy of inputs cost while it appears costly to produce fuel efficientcars.

We obtain a lower value of the objective function for the model with price discriminationthan for the model with uniform pricing (1,794 versus 2,400). In line with the simulationresults, and though it seems difficult to construct a formal statistical test based on thesevalues,10 we see this as evidence in favor of unobserved price discrimination in our con-text. Note also that if qualitatively similar, the results we obtain with the two modelsexhibit some quantitative differences. This is especially the case for the pivot group, forwhich the effects of price, fuel cost and horsepower are lower in magnitude under the pricediscrimination model.

10The test of Rivers & Vuong (2002) is sometimes conducted in the literature (see Jaumandreu & Moral2006, Bonnet & Dubois 2010, Ferrari & Verboven 2012). The main issue in applying such a test in ourcontext is to obtain a consistent estimator of the standard errors of the difference between these twoobjective functions, or any other statistics, under the null hypothesis. The problem is that both modelsmay be wrong under the null hypothesis of the test. In such a case, the residuals (ξdj (p

dj , θ

d0))j=1...J that

we obtain under each models are not independent to each other, and the dependence between them isunknown. Thus, neither the standard GMM formula based on independence, nor the standard bootstrap,allow one to compute standard errors in a consistent way.

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To understand what these differences imply, we compare the price elasticities and the mark-up 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 onprice 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 mainlybecause the prices are overestimated. In the uniform pricing model we find average priceelasticities varying from -3.7 to -6.3, which are in line with those obtained by BLP, whoreport elasticities between -3.5 and -6.5, but below those of Langer (2012) who finds, usingtransaction prices, a range between -6.4 to -17.8. These elasticities imply mark-ups thatare around 20% in both models, with, as we could expect, substantial heterogeneity acrossgroups 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 theold and high-income group.

Group of consumers Discrimination BLP Uniform BLPAge < 40, I = L -4.73 -6.25Age < 40, I = H -4.55 -6.25Age ∈ [40,59], I = L -4.17 -5.79Age ∈ [40,59], I = H -3.87 -5.41Age ≤ 60, I = L -4.07 -5.38Age ≤ 60, I = H -2.87 -3.71Average -4.03 -5.46

Table 8: Comparison of average price elasticities under the uniform pricing and unobservedprice discrimination models.

Figure 1 displays the distribution of the differences in estimated marginal costs betweenthe two models. We compute here the relative difference (cu − cd)/cd. The costs arealways 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 thefact that, in the uniform pricing model, the marginal cost is deduced from the differencebetween the posted price and the average mark-up. In contrast, in the price discriminationmodel, the marginal cost is equal to the difference between the posted price and the mark-up of the pivot group. This mark-up is higher than the average mark-up estimated in thestandard model, resulting in lower marginal cost. Ultimately, the errors in the estimationof marginal costs translate into errors in counterfactual simulation exercises.

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0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Relative cost difference (in %)

Fre

quency

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 themodel with unobserved price discrimination. We compute average discounts weighted byactual sales in each group but also using the same weighting scheme for all groups ofconsumers, namely, the overall product market shares (“basket-weighted” method). Thisallows us to eliminate the potential group-specific demand composition effect. The resultswith both weighting methods are similar. As expected, the pattern on average discountsacross groups is similar to the one on price elasticity. The estimated pivot group (the groupassumed to be paying the posted price) is identical for all the products and corresponds tothe group with the lowest price elasticity. These are the 13.2% of the population over 60year 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 orequal to 12.9%. Clearly, income and age are both important determinants of the discountobtained. On average, young purchasers with a low income pay 14.4% less than the postedprice, while young, high income buyers get an average discount of 13.6%. These percentagesrepresent 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, whileold, low income, individuals receive an average discount of 11.3%, old, high income, buyers

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receive no discount since they constitute the pivot group for all products.

Average discount Average gross discount(in % of posted price) (in euros)

Group of consumers Sales-weighted Basket-weighted Sales-weighted Basket-weightedAge < 40, I = L 14.36 14.49 2,805 3,015Age < 40, I = H 13.64 13.84 2,863 2,891Age ∈ [40,59], I = L 12.16 12.07 2,659 2,546Age ∈ [40,59], I = H 10.63 10.58 2,391 2,251Age ≤ 60, I = L 11.31 11.27 2,276 2,387Age ≤ 60, I = H 0 0 0 0Average 10.53 10.54 2,210 2,219Reading notes: the “basket-weighted” discounts are obtained by using the same artificial basket of carsfor all groups.

Table 9: Average discounts by groups of consumers

These figures average vehicle specific discounts. Our methodology allows us to analyzefurther the heterogeneity across car models, since we can estimate a discount value for eachmodel and demographic group. Figure 5.3 displays the resulting distribution of discountsacross products. The corresponding average discount, averaged by product rather than byconsumers, 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 islarger than 13%, and it even exceeds 34.3% for 1% of the fleet. To understand better thesource 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 horsepowerbut decrease with weight and fuel cost. These results reflect both the differences in salesbetween consumer groups (e.g. products mostly sold to the pivot group tend to havea small average discount) and differences in the pricing strategy. Results with basked-weighted discounts are however similar, showing in particular that it is profitable for firmsto offer larger discounts for their most expensive cars.

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0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

Average discount by product (in %)

Fre

quency

Figure 2: Distribution of estimated discounts across products

Variable Parameter Std-errIntercept 13.3∗∗ 0.35Posted price 3.27∗∗ 0.11Horsepower 2.65∗∗ 0.41Fuel cost -3.33∗∗ 0.33Weight -8.44∗∗ 0.34Three doors 1.16∗∗ 0.18Station wagon 0.85∗∗ 0.14R2 0.51

Table 10: Regression of average product discount on cars characteristics

How do our estimates compare with other evidence of discounts? First, to the best ofour knowledge, there is no comprehensive and reliable data on transaction prices in theFrench automobile market. However, a recent survey conducted by the French creditcompany, Cetelem (L’Automobile en Europe: 5 Leviers pour Rebondir 2013), provides auseful benchmark. First, it reveals that in 2012, 87% of the purchasers benefited from adiscount 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 toobtain a rebate, which may be seen as evidence of price discrimination rather than a true

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bargaining process. Further, for 68% of individuals who indicated that they negotiated thecar price, the average discount was around 11%. This result is very close to our average onthe whole population, and also comparable to the average discount we obtain on individualsbelow 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 reportsome anecdotal evidence. For example, when searching online using the keywords “howmuch discount for new car ” (in French), the first website listed states that “discounts aregenerally between 5% and 20%”.11 The fourth website associated to the same key wordssearch is a forum asking the question of how much discount one can expect to obtain on thepurchase of a new car. One reply states that discount do not exceed 20%, while anothermentions an average discount of 6%.12 Our estimations are overall consistent with thesefigures.

A recent study by Kaul et al. (2012) investigates the effect of the scrapping policy on themagnitude of discounts in Germany, using data collected to a sample of dealers. Whenexcluding demonstration cars and sales to employees, which are typically much more dis-counted, they obtain an average discount of 14%. This magnitude is consistent, thoughsomewhat higher, with our estimate. Their study focuses on the period 2007-2010, whichcorresponds to the beginning of the economic crisis. If posted prices did not adjust imme-diately, it is likely that car dealers reacted to this adverse economic climate by reducingtheir margins and increasing the discounts. The assumption that the posted price is equalto 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 Kaulet al. (2012). In their regression analysis, they also find a positive link between discountsand 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 newcar market and gathered data on average discounts by brand and segment (New cars: Areport on the supply of new motor cars within the UK 2000). The dataset is very reliablesince it was collected directly from dealers. The report reveals that the average discountlies between 7.5% and 8%, also broadly in line with our estimated average discount. Oncemore, the difference may stem from differences between the two markets and the periodsunder consideration. This report also refers to a consumer survey conducted in 1995 asking

11See http://www.choisir-sa-voiture.com/concessionnaire/meilleur-prix-voiture.php. Weperformed this search in November 2014 using Google search engine.

12See http://forum.hardware.fr/hfr/Discussions/Auto-Moto/negocier-voiture-concession-sujet_15899_1.htm.

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automobile purchasers whether or not they obtained a discount over the posted price. Thissurvey reveals that 17% of purchasers paid the posted price whereas 37% bargained andobtained a discount and 29% were automatically offered a discount. This figure of 17% isclose to our estimation of 13% of the cars are sold without discount. Furthermore, the factthat some purchasers were “automatically offered a discount” corroborates our assumptionthat discounts are used as a tool to price discriminate because the posted price is notoptimal for some consumers.

A direct comparison of the distribution of discounts we obtain and evidence on the USmarket 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 availableto consumers in France. It reports in particular the negotiability, the fair purchase priceand 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 zipcodes, they also provide the distribution of transaction prices, which indicates geographicalprice dispersion in the US. Therefore, the notion of posted prices and discounts, as definedin 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 oncemore consistent with our estimated discounts.

Finally, few papers correlate the magnitude of discounts to individual characteristics. Har-less & Hoffer (2002) and Langer (2012), in particular, conduct such an analysis on theUS market, using respectively dealer margins and a survey on transaction prices (see alsoChandra et al. 2013, for an analysis of the Canadian market, focusing on gender discrim-ination). 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 significantprice discrimination with respect to income, the high income groups of consumers (for bothmen and women) being associated with higher margins. These two results are in line withour 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 maynot be the case in an oligopoly, because price discrimination may reinforce competitionamong firms. Under certain conditions, all firms may actually be worse off than if theycould commit to a uniform pricing strategy (Holmes 1989, Corts 1998). The effect onconsumers is also ambiguous. For a given group of consumers, some products may turn

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to be cheaper without price discrimination. We investigate in this subsection the effect ofprice discrimination on firms and consumers. We thus compute, using our estimates of themodel with price discrimination, the counterfactual prices and profits that would occur iffirms could commit to set a single price for all groups of consumers.

Profit with price Profit without price Gain fromBrand discrimination (in Me) discrimination (in Me) discriminationRenault 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 gainsfrom price discrimination represent the profit gains or losses of switching from the uniformpricing equilibrium to the price discrimination equilibrium.

Table 11: Gains and losses from price discrimination by brand.

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Results on firms profits are displayed in Table 11. Gains from price discrimination arerather small but heterogeneous. We observe that if price discrimination is profitable formost of the manufacturers, it makes 5 out of the 38 manufacturers worse off. The gainsassociated to price discrimination are particularly high for brands that commercialize pow-erful vehicles, such as Mercedes (+8.8%), Jaguar (+12.6%) and Porsche (+24.4%). Thismakes 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 moreprofitable than average for the major French manufacturers (+4.4%, +3.3% and +5% forrespectively Renault, Peugeot and Citroen) and more moderate for Dacia (+2.1%). Thetotal gain from price discrimination is rather small but significant, the industry profitincreasing by 3.4% compared to the uniform pricing equilibrium.

We also investigate the impact of price discrimination on consumers. In Table 12, wecompute the average price differences between the uniform and the discriminatory pricesfor each group of consumers and report the number of products for which the discriminatoryprice is lower than the uniform one (see Column 3). We also compute average surplus foreach group of consumers under the two pricing equilibria (see Table 13). For the younggroups, all products are cheaper under uniform pricing, and price discrimination makesthem save around 600 euros. The situation is more contrasted for the 40-59 and 60+ year-old group. In particular, all prices are lower under uniform pricing for the pivot group, whowould save on average the substantial amount of 2,153 euros. Overall, price discriminationis hardly beneficial for consumers as it increases the global average individual surplus byonly 0.37%. Again, this global impact hides heterogeneous effects. The group experiencingthe highest welfare gain is the group of young consumers with low income (+3.7%), whilethe pivot group is, not surprisingly, the one that suffers the most from price discrimination(-2.8%).

#j : pdj < Average gains in purchasesGroup of consumers Frequency puniform

j Sales-weighted Basket-weightedAge < 40, I = L 15.7 3,205 627 761Age < 40, I = H 11.5 3,205 454 637Age ∈ [40,59], I = L 16.3 2,925 271 292Age ∈ [40,59], I = H 22.3 1,382 6 -2.6Age ≥ 60, I = L 20.8 2,252 180 133Age ≥ 60, I = H 13.2 0 -2,153 -2,254Reading notes: the third column indicates how many products (among the 3,205) have lowerprices with the price discrimination regime. The “basket-weighted” gains are obtained byusing the same artificial basket of cars for all groups.

Table 12: Gains of price discrimination for groups of consumers.

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Price discrimination Uniform pricingGain from

discrimination (in %)Age < 40, I = L 13,208 12,735 3.72Age < 40, I = H 14,666 14,244 2.96Age ∈ [40,59], I = L 15,980 15,803 1.12Age ∈ [40,59], I = H 18,286 18,213 0.4Age ≤ 60, I = L 15,680 15,503 1.14Age ≤ 60, I = H 35,759 36,795 -2.82Average 18,424 18,356 +0.37

Table 13: Comparison of average individual surplus for the different groups of consumerswith price discrimination and uniform pricing.

6 Conclusion

This paper investigates the recurring problem of observing only posted prices instead oftransaction prices in structural models of demand and supply in markets with differentiatedproducts. We propose an approach that incorporates unobserved price discrimination byfirms based on observable individual characteristics. This approach requires to have dataon aggregate sales on the corresponding groups of purchasers and, as usual, characteristicsof products. We use this model to describe the French new car market where price dis-crimination may occur through discounts. Our results suggest significant discounting bymanufacturers which is consistent with previous studies on price dispersion, survey dataand 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 canbe easily extended to other demand and supply models. It also applies when the data col-lected by the econometrician is unreliable or limited. We finally explained how to deal withprice discrimination based on characteristics that are unobserved by the econometrician.

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A Appendix

A.1 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 similarthan 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 coefficientmodel. Here again, older people are the less price sensitive. Perhaps surprisingly, on theother hand, high-income individuals appear to be more price sensitive in general, bothunder price discrimination and uniform pricing. The pivot group is nevertheless still theolder, high-income consumers. We also observe, as with the random coefficient model,that the model without price discrimination slightly overestimates price elasticities andalways overestimates the marginal costs. Figure 3 displays the distribution of the relativecost 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 theproducts. Turning to the discounts, we obtain again that the youngest purchasers obtainthe highest discount, though such discounts are on average smaller than with the randomcoefficient model. Interestingly, the high-income groups also receive smaller discounts, inline with the results on the random coefficient model. This shows that price sensitivityalone does not determine the amount of the discounts. The heterogeneity in the valuation ofother 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 theaverage (6.6%) and the standard deviation (3.3%) are lower than the figures obtained withthe random coefficient model (10.6% and 4.8%, respectively), but for 10% of the fleetdiscounts still exceed 11.5% (versus 13% for the random coefficient model). Finally, aregression of the discounts on cars’ characteristics shows, as before, that large fuel costsand heavy vehicles are associated with lower discounts, while horsepower is associated togreater discounts. On the other hand, the price has a negative rather than positive effect ondiscounts in this specification, contradicting in particular the results of Kaul et al. (2012).

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Group of consumers Discrimination n. logit Uniform n. logitAge < 40, I = L -5.64 -6.34Age < 40, I = H -6.54 -7.38Age ∈ [40,59], I = L -5.6 -6.06Age ∈ [40,59], I = H -5.92 -6.52Age ≤ 60, I = L -3.91 -3.92Age ≤ 60, I = H -3.92 -3.73Average -5.20 -5.59

Table 14: Comparison of average price elasticities for the nested logit models with uniformpricing and unobserved price discrimination.

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Relative cost difference (in %)

Fre

quency

Figure 3: Distribution of the relative difference between estimated costs (cu−cd)cd

.

Average discount (in % of posted price) Average gross discount (in euros)Group of consumers Sales-weighted Basket-weighted Sales-weighted Basket-weightedAge < 40, I = L 12.81 12.45 2,249 2,262Age < 40, I = H 13.74 13.89 2,516 2,523Age ∈ [40,59], I = L 9.71 9.95 1,801 1,808Age ∈ [40,59], I = H 10.52 10.75 1,951 1,952Age ≤ 60, I = L 1.62 1.57 288 285Age ≤ 60, I = H 0 0 0 0Average 7.86 7.9 1,431 1,435Reading notes: the “basket-weighted” discounts are obtained by using the same artificial basket of carsfor all groups.

Table 15: Average discount by groups of consumers for the nested logit model.

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0 5 10 15 20 250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Average discount by product (in %)

Fre

quency

Figure 4: Distribution of estimated discounts for the nested logit model.

Variable Parameter Std-errIntercept 15.29∗∗ 0.21Posted price -1.45∗∗ 0.06Horsepower 2.12∗∗ 0.25Fuel cost -2.16∗∗ 0.20Weight -3.50∗∗ 0.20Three doors 1.11∗∗ 0.10Station wagon 0.31∗∗ 0.08R2 0.65

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 ourgroups and consumers.

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Group Characteristics Frequency of null sale1 Age < 40, Income <27,000 11.6%2 Age < 40, Income ≥ 27, 000 10.3%3 Age ∈ [40,59], Income <27,000 7.5%4 Age ∈ [40,59], Income ≥ 27, 000 4%5 Age ≥ 60, Income <27,000 7.8%6 Age ≥ 60, Income ≥ 27, 000 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 sdj =ndj+0.5

Nd of sdj , where ndjdenoting the number of sales of product j in group d and Nd the number of potentialbuyers with characteristics d. The idea is to consider simple estimators of sdj of the formndj+c)/N

d, and fix c such that the expectation of ln((ndj+c)/Nd) is asymptotically unbiased.

The reason we are looking for such a c is that ln(sdj ) plays an important role at least inthe logit or nested logit models. With an unbiased estimator of ln(sdj ), we could estimateconsistently and as usually the demand parameters. However, in our framework whereindividuals choose independently from each others, so that ndj ∼ Binomial(Nd, s

dj ), it is

well-known that only polynomials of sdj of degree at most Nd can be estimated withoutbias. Our aim is then to find instead an estimator that is asymptotically unbiased at thefirst order.

For that purpose, we consider an asymptotic approximation where sj is small but λdj ≡Nds

dj → ∞. Let Zd

j = (ndj − λdj )/√λdj . A second-order Taylor expansion of (ndj + c)/Nd

around sdj yields

√λdj[ln((ndj + c)/Nd)− ln

(sdj)]

= Zdj +

c√λdj

−sd2j2sd2j

1√λdj

Zdj +

c√λdj

2

,

where sdj is between (ndj+c)/Nd and sdj . The first order term, Zd

j , is asymptotically standardnormal and thus asymptotically centered. Now, considering the second-order term,

√λdj

√λdj[ln((ndj + c)/Nd)− ln

(sdj)]− Zd

j

= c−

sd2j2sd2j

Zdj +

c√λdj

2

.

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Moreover, sd2j /sd2jP−→ 1 and

(Zdj + c√

λdj

)2L−→ χ2

1. Hence,

√λdj

√λdj[ln((ndj + c)/Nd)− ln

(sdj)]− Zd

j

L−→ c− 1

2χ21.

Choosing c = 1/2 therefore ensures that this second-order term is asymptotically centeredaround 0.

To examine the robustness of the estimation results to the correction of the null sharesadopted. We re-estimate the different models using the Laplace transformation of themarket share equation used by Gandhi et al. (2013). This correction replaces the marketshare by :

sdj =Ndsdj + 1

Nd + J + 1.

As Table 18 suggests, the estimation results are robust to the choice of a correction to dealwith products with null market shares. The estimated parameters are very close to eachother. As a result, subsequent results (not displayed here) on, e.g., discounts, are also closeto each other.

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Our correction Gandhi correctionParameter Std-err Parameter Std-err

Price sensitivityAge < 40, I = L -2.7∗∗ 0.185 -2.54∗∗ 0.18Age < 40, I = H -2.55∗∗ 0.176 -2.41∗∗ 0.171Age ∈ [40,59], I = L -2.24∗∗ 0.177 -2.13∗∗ 0.17Age ∈ [40,59], I = H -2.16∗∗ 0.178 -2.03∗∗ 0.174Age ≤ 60, I = L -1.98∗∗ 0.185 -1.83∗∗ 0.099Age ≤ 60, I = H -1.88∗∗ 0.105 -1.87∗∗ 0.168Intra-segment correlationAge < 40, I = L 0.17∗ 0.073 0.08 0.071Age < 40, I = H 0.3∗∗ 0.072 0.21∗∗ 0.072Age ∈ [40,59], I = L 0.22∗∗ 0.069 0.16∗ 0.068Age ∈ [40,59], I = H 0.29∗∗ 0.08 0.2∗ 0.081Age ≤ 60, I = L 0 0Age ≤ 60, I = H 0 0InterceptAge < 40, I = L -7.09∗∗ 0.515 -7.62∗∗ 0.506Age < 40, I = H -7.05∗∗ 0.488 -7.49∗∗ 0.48Age ∈ [40,59], I = L -7.29∗∗ 0.457 -7.63∗∗ 0.45Age ∈ [40,59], I = H -6.89∗∗ 0.496 -7.34∗∗ 0.501Age ≤ 60, I = L -7.92∗∗ 0.296 -7.9∗∗ 0.224Age ≤ 60, I = H -8.23∗∗ 0.238 -8.31∗∗ 0.273Fuel costAge < 40, I = L -6.03∗∗ 0.349 -5.95∗∗ 0.319Age < 40, I = H -4.84∗∗ 0.302 -4.81∗∗ 0.282Age ∈ [40,59], I = L -5.04∗∗ 0.306 -4.99∗∗ 0.286Age ∈ [40,59], I = H -4.12∗∗ 0.278 -4.23∗∗ 0.274Age ≤ 60, I = L -4.19∗∗ 0.248 -3.88∗∗ 0.187Age ≤ 60, I = H -3.62∗∗ 0.199 -3.51∗∗ 0.224HorsepowerAge < 40, I = L 5.8∗∗ 0.514 5.46∗∗ 0.498Age < 40, I = H 5.2∗∗ 0.482 4.91∗∗ 0.467Age ∈ [40,59], I = L 4.3∗∗ 0.482 4.07∗∗ 0.463Age ∈ [40,59], I = H 3.97∗∗ 0.486 3.7∗∗ 0.478Age ≤ 60, I = L 2.99∗∗ 0.502 2.71∗∗ 0.301Age ≤ 60, I = H 2.73∗∗ 0.322 2.84∗∗ 0.454Age < 40, I = L 4.17∗∗ 0.396 4.15∗∗ 0.38WeightAge < 40, I = L 4.17∗∗ 0.396 4.15∗∗ 0.38Age < 40, I = H 4.06∗∗ 0.357 4.01∗∗ 0.344Age ∈ [40,59], I = L 4.21∗∗ 0.353 4.13∗∗ 0.34Age ∈ [40,59], I = H 3.89∗∗ 0.34 3.91∗∗ 0.338Age ≤ 60, I = L 3.58∗∗ 0.349 3.29∗∗ 0.241Age ≤ 60, I = H 3.69∗∗ 0.256 3.59∗∗ 0.321Three doorsAge < 40, I = L -0.09 0.12 -0.05 0.119Age < 40, I = H -0.26∗ 0.105 -0.22∗ 0.104Age ∈ [40,59], I = L -0.22∗ 0.105 -0.19† 0.103Age ∈ [40,59], I = H -0.35∗∗ 0.1 -0.32∗∗ 0.102Age ≤ 60, I = L -0.61∗∗ 0.117 -0.57∗∗ 0.105Age ≤ 60, I = H -0.67∗∗ 0.111 -0.65∗∗ 0.107Station-wagonAge < 40, I = L -0.59∗∗ 0.088 -0.62∗∗ 0.086Age < 40, I = H -0.42∗∗ 0.075 -0.43∗∗ 0.074Age ∈ [40,59], I = L -0.45∗∗ 0.085 -0.47∗∗ 0.083Age ∈ [40,59], I = H -0.46∗∗ 0.086 -0.5∗∗ 0.088Age ≤ 60, I = L -0.7∗∗ 0.078 -0.66∗∗ 0.071Age ≤ 60, I = H -0.69∗∗ 0.076 -0.66∗∗ 0.072Marginal cost equationIntercept -0.47∗∗ 0.059 -0.49∗∗ 0.06Horsepower 0.5∗∗ 0.028 0.5∗∗ 0.029fuel inefficiency -2.49∗∗ 0.347 -2.51∗∗ 0.352Three doors -0.09∗∗ 0.014 -0.09∗∗ 0.014Station-wagon -0.05∗∗ 0.009 -0.05∗∗ 0.009Weight × average price index 0.15∗∗ 0.005 0.15∗∗ 0.005

Table 18: Estimation of parameters : Nested logit model with our correction and Gandhiet al. correction.

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A.3 Segmentation of the market

The nested logit approach requires to define a segmentation of the market in homogeneousgroups of products. Our segmentation, based on the main use of the vehicle, is close to theone of The European New Car Assessment Program one (Euro NCAP). Table 19 displaysthe eight segments that we consider and their market shares over the period. Note inparticular that sport cars include all convertible cars as well as vehicles with a high ratiohorsepower/weight, while small multi-purpose vehicles (MPV) include small vans such asRenault Kangoo. The entire classification is presented in Table 22.

Market sharesSegment (in %)Supermini 45.14Executive 1.17Small Family 17.01Large Family 8.67Small MPV 17.56Large MPV 1.07Sports 5.11Allroad 4.77

Table 19: Segments and their market shares.

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Table22

:Segm

entation

oftheau

tomob

ilemarket

Gro

up

Make

Supermini

Smallfamily

Largefamily

Executive

Sportscar

SmallMPV

LargeMPV

Allroad/SUV

PSA

Citro

en

C1,C2,C3,Saxo

Xsara

C5

C6

-Berlingo,C4,Nem

o,

Xsara

C8

C-C

rosser

Peugeo

t106,

107,1007,206,

207

306,307,308

406,407

607

-Bipper,Partner

807

4007

Renault

Renault

Clio,Modus,

Twingo

Megane

Laguna

Vel

Satis

-Kangoo,Megane

Espace

Koleos

Dacia

Sandero

Logan

--

--

--

B.M

.WB.M

.W-

1-Series

3-Series

5,6,7-Series

Z4

--

X3,X5,X6

Min

iMini

--

--

--

-Chrysler

Chry

sler

--

Sebring

300C,300M,Crossfire

-PT

Cruiser

Voyager,G.Voyager

-Jee

p-

--

--

--

Compass,

Chero-

kee,

Commander,

G.Cherokee,

Wran-

gler

Dodge

-Caliber

Journey

Viper

--

-Durango,Nitro

Daihatsu

Daih

ats

uCuore,Sirion,YRV

--

-Copen

--

Terios

Daim

ler

Merc

edes

-A-C

lass

C,CLK-C

lass

E,CL,R,S,SL,CLS,

SLR-C

lass

SLK-C

lass

B-C

lass,Vaneo

Viano

G,

GL,

GLK,

ML-

Class

Sm

art

Fortwo,Forfour

--

-Coupe,

Roadster

--

-Fiat

Alfa

Rom

eoMito

147

156,159,GT

166,Brera

GTV,Spider

--

-Fia

t500,

Palio,

Panda,

Punto,Seicento

Bravo,Stilo

Croma

-Barchetta

Doblo,Fiorino,Idea,

Multipla

Ulysse

Sedici

Lancia

Y-

Lybra

Thesis

-Musa

Phedra

-Ford

Ford

Fiesta,Ka,

-Mondeo

-Puma

Focus,

Fusion,

T.Connect,

Tour-

neo

Galaxy,

S-M

ax

Kuga

Jaguar

--

X-T

ype

S-T

ype,

XJ,XK

--

--

Land

Rover

--

--

--

-Freelander,Defender,

Discovery,

R.R

over

Volv

o-

C30,V50

C70,S40,S60,V70

V40,S80

--

-XC60,XC70,XC90

GM

Europe

Chevro

let

Kalos,

Matiz

Aveo,Lacetti,Nubira

Epica,Evanda

-Corvette

Rezzo,Tacuma

-Captiva,Tahoe

Daewoo

Kalos,

Matiz

Lanos

-Evanda

-Rezzo

-Korando

Opel

Corsa

Astra

Insigna,Signum,Vec-

tra

Omega

Tigra,Speedster

Agila,

Combo,

Meriva,Zafira

-Antara,Frontera

Saab

--

9-3

9-5

--

--

Honda

Honda

Jazz

Civic

Accord

-S2000

FR-V

,Stream

-CR-V

,HR-V

Hyundai

Hyundai

Atos,

Getz,

I10

Accent,

Coupe,

I30

Elantra,Sonata

--

Matrix

Trajet

Tucson,Santafe,Ter-

racan

Kia

Picanto,Rio

Cee-d,Cerato

Magentis

--

Carens,

Soul

Carnival

Sorento,Sportage

Lada

Lada

-111,112

--

--

-Niva

Mazda

Mazd

a2

36

RX8

MX5

5,Premacy

MPV

-Mitsubishi

Mitsu

bis

hi

Colt

Lancer

Carism

a-

-Spacestar

Grandis

Outlander,Pajero

Nissan

Nis

san

Micra,Note

Alm

era,Qashqai

Primera

350Z,Maxim

a-Q

-Alm

era

-X-Trail,

Murano,

Pathfinder,

Patrol,

Terrano

Porsche

Pors

che

--

-911,Boxter,

Cayman

--

-Cayenne

Rover

Rover

25,Streetw

ise

45

75

--

--

-Ssangyong

Ssa

ngyong

--

--

--

Rodius,

Stavic

Actyon,Korando,Ky-

ron,Rexton

Subaru

Suba

ruJusty

Impreza

Legacy

--

--

Forester,B9Tribeca

Suzuki

Suzu

ki

Alto,

Ignis,

Splash,

Swift,

SX4

Liana

--

-Wagon-R

-G.

Vitara,

Jim

ny,

Samurai,Vitara

Toyota

Toyota

Aygo,IQ

,Yaris

Auris

Avensis,

Prius

-Celica,MR

Corolla

Previa

RAV4,L.Cruiser

Lexus

--

ISGS,LS

--

-RX

VW

Group

Audi

A2

A3

A4,A5

A6,A8,R8

S3,S4,S6,S8TT

--

Allroad,Q5,Q7

Sea

tArosa,Ibiza

Cordoba,Leon

Toledo

--

Altea

Alhambra

-Skoda

Fabia

-Octavia,Superb

--

Roomster

--

Volk

swagen

Fox,Lupo,Polo

Eos,Golf,Jetta,New

-beetle

Scirocco,Passat

-Phaeton

Caddy,

Touran

Sharan

Tiguan,Touareg

48