J. Foncel and M. Ivaldi

supply side and the market equilibrium cannot be neglected for estimating aggregate industry models. Goldberg (1995) notices that the use of micro data for fitting a demand model alleviates the question of the endogeneity of prices. First, an individual consumer cannot alter prices. Second, micro data (i.e., data collected at the individual level) should permit to envision a high degree of product differentiation and to seize the heterogeneity of consumer behaviors. Hence, supply side considerations can be postponed until one turns to the study of the industry equilibrium. Precisely, an aggregate demand can be consistently recovered from the estimated individual demands as micro-data allow us to obtain the distribution of individual characteristics, and so to monitor the aggregation process. The only critical task left to the econometrician is to achieve a correct estimation of individual demands. In the case of durable goods, the analysis can be performed by means of discrete choice models, for which the nested multinomial logit model is an adequate tool. It is simple to estimate and it escapes from the effect of the assumption of independence of irrelevant alternatives (IIA). Indeed, if this assumption is maintained within each nest, it is relaxed in between. However, this approach can be criticized on different grounds. First, there is some adhockery in choosing the different nests, even if they correspond to common sense or expert assessment. Second, this logit-type model remains generally based on linear indirect utility, which is restrictive. Third, in consumer surveys, the observed discrete choice is often completed by an information on the usage of the durable good. For instance, one observes both the type of housing ownership and the size of dwelling, the type of car and the average annual mileage, the choice of an heating system and the power consumption, etc. Separating the two choices, the discrete choice on the durable product and the continuous choice on its usage, lies on some separability assumptions that are sometimes unrealistic. Dubbin and McFadden (1984) for the case of electric appliance, and Goldberg (1998) for the case of automobile, specify a nonlinear indirect utility function that allows recovering discrete and continuous choices. However these authors focus mainly on the continuous choice. Consider now the telecommunications demand. It involves access to the network and telephone usage that are clearly interdependent and cannot be separated. Even if a consumer does not make calls, he or she may be willing to have access just for the option value of a call in case of emergency.1 Note that access to the network requires a device or an equipment, here a telephone which is a durable good. Now, many examples show that traffic and equipment allowing for network access are interrelated in the telecommunications industry. For instance, because of the overwhelming success of Internet, producers of microcomputers are introducing simpler machines just equipped with the function of accessing and searching on the web. The emerging information economy requires network equipment capable to offer access to a large range of services and to transfer various types of information like voice, data, or images. In part, habit formation and new usage may be the spring of technical changes on equipment. More generally, the consumption pattern of a consumer informs us on his private valuation for the equipment. Understanding the relationship between product choice and usage is at the heart of business and marketing strategies, like product introduction, positioning or pricing strategies. By several specific features, the household telephone equipment (HTE) market is a very stimulating field of investigation. This market is characterized by a high degree of both horizontal and vertical differentiation of products. Marketing studies usually identify four distinct segments on the French market: one-block phone, two-block phones, phones with

1 See Wolak (1993) on modeling telecommunications demand.
answering devices, and cordless phones. Within a segment, products are likely to be horizontally differentiated. In this context, providing realistic elasticities of substitution is a sensitive task. Moreover, as in other European countries, the French HTE market presents two further dispositions which raise interesting economic questions. First, during the eighties, this market has evolved from a monopoly, the historic telecommunications operator France Telecom, renting few telephone sets to a free-entry market where few firms (the majors like Philips, Matra, Alcatel, and some smaller companies) sell a large variety of telephone models. Note that the incumbent not only keeps renting but also sells many different types of equipment. The convergence to a new equilibrium as well as the effect of ownership type (renting or buying) must receive some attention, which we do in the empirical part of our study. A survey made in 1992 on a representative sample of French households allows us to observe the choices of telephone models, the telephone bill and several socio-demographic variables on the individuals. Another marketing database provides the technical characteristics and the price ranges of different telephone models. The number of models of telephone we consider is about 160. However, since a household can enjoy more than one phone at home, the number of combinations of models that it faces, is huge (more than 12500 if we restrict attention to symmetric tuples). To cope with the dimension of the choice set, we apply a procedure introduced by McFadden (1978) which consists of drawing a subset of alternatives and performing the maximum likelihood on this subset after having corrected the choice probabilities accordingly. Finally, fitting a structural model, i.e., a model derived from a direct utility function, to micro data for analyzing a market of differentiated durable goods turns out to be a fruitful method. First, we are able to deal with the classical problems of the empirical IO literature on differentiated product markets, i.e., price endogeneity, dimensionality of product sets, and correctness of substitution patterns. Second, we provide an estimate of product shares in the whole stock of telephones as the survey bears on the equipment of households and not on recent acquisitions. From a marketing point of view, this is a helpful information since stock shares measure the effective penetration of a brand over its lifetime. Third, considering that the market solution is approximated as a Nash equilibrium, we derive markups at the product level. The next section is devoted to the specification of a demand model for HTE, taking into account the specific aspects of the HTE market. The econometric model and its estimation are presented in section 3. Empirical results are discussed in section 4 and the analysis of the market equilibrium is performed in section 5. Section 6 concludes and proposes a research agenda.

2. A CONTINUOUS-DISCRETE CHOICE MODEL OF TELEPHONE DEMAND 2.1. Notations and definitions Consider a household (or a consumer) who simultaneously chooses a telephone equipment in a set E of alternatives which are precisely defined below, and a level of telecommunications usage which is a continuous variable. This consumer is assumed to be connected to the network and to own at least one phone.2
2 As about ninety nine percent of French households are connected to the network, there is no need to consider an outside
alternative composed with no telephone. In a standard discrete choice model, omitting the outside alternative implies the unrealistic feature that aggregate demand for all brands or products remain unchanged due to a general increase in price.
An alternative or a choice is an equipment composed with one or two brands of telephone.3 An equipment is denoted by ( j, k ) , j or k being a model of telephone. Note that, without loss of generality, an equipment composed of one phone is treated as a two-phones bundle, with one of the two telephones being the nil phone. A symmetry condition on the structure of equipment is needed as we do not observe which component of the equipment (i.e., j or k) has been bought first. Hence, the model does not differentiate between alternative ( j, k ) and alternative (k, j ). (See however in Section 4 how the price of an alternative is defined.) Finally, the set E contains all mutually exclusive alternatives ( j, k ) comprising one or two phones. Consumer ns preferences are translated into a conditional direct utility function, which provides the utility level of using a level x of telecommunications and a level z of numraire, conditional to holding the alternative ( j, k ). A modified version of the Blackburn utility function, discussed by Hanemann (1984) and applied by Hobson and Spady (1988) for analyzing the demand of telephone usage, serves to specify this conditional utility function as: U njk = U (x, z , jk , n , njk , njk ) = x (1 + ln n ln x ) + x jk + hn z + njk + njk , (1)
index of equipment ( j, k ) that depends on observable attributes of each component of alternative ( j, k ) , njk is a term aimed at measuring the individual valuation for quality, i.e., relating the taste of individual n for equipment

where is a non-negative parameter of scale to be estimated, n is a heterogeneity index (supposed to be strictly positive) which can be measured through observable characteristics of household n, hn is the marginal utility of numraire specific to household n, jk is a quality

( j, k )

to observable variates, and njk is a
random term which is not observed by the econometrician. Note that the quasi-linearity is a fairly admissible assumption as telephone call patterns should not depend on the consumption structure of other goods. The household faces the budget constraint px + z = y n F jk , where p is the price of a call unit,4 F jk is the cost of equipment household income over the period under study.
Here, the possibility of combining brands of telephone to set up an equipment avoids a similar drawback. In response to a general price increase, the consumer could shift to an equipment composed of only one phone, when he/she used to own a two-pieces equipment. 3 Restricting attention to equipment that has less than two telephones is not a strong assumption as the percent of households having more than two telephones is very low. 4 Because of the tariff structure practised by the French operator, the cost of each call depends on the calling area and the time-of-use. In fact, this tariff structure provides exchange rates between calls of different types. All calls performed by a customer are aggregated in terms of a measurement unit of duration associated with a particular call, i.e., a specific duration for a call on a particular calling area at a particular time of the day. The price p is the price for one unit of this type of call. Note that, because of this tariff structure, the price per minute is an endogenous variable. In other terms, the average expenditure on telephone calls is specific to the individual and depends on his calling pattern. In this context, the demand model we present could be understood as the second step of the a full consumer program or as the result of a twostage budgeting analysis. In a first stage, the consumer decides for the total expenditure, px, on telephone usage and on the composite good. The other stage is devoted to the allocation of this telephone budget to different types of calls and, to the

( j, k ) ,

and yn is the
exists and is unique for the given discrete choice ( j, k ). On the other hand, the optimal demand * of the composite commodity, z njk , is positive provided that income is large enough compared to the value of consumption. Assuming that household n selects an alternative optimal conditional level of telephone consumption is given by
* xnjk = n exp ( jk hn p ).
The utility function (1) is strictly quasi-concave, continuous and smooth with respect to * x and z. On the one hand, it ensures that the conditional optimal level of telephone usage, xnjk ,
Note that the telephone demand is always positive as long as n is positive. Individual heterogeneity enters the demand in two ways. First, due to the nonmonotonicity of preferences, the telephone consumption can reach a saturation level, which is specific to the consumer, a realistic hypothesis in the case of telephone. This saturation level, i.e., the total usage of telephone services that an individual is able to consume when the price of telephone calls is zero, is given by n exp ( jk ) , a function of the heterogeneity index. Second, another source of heterogeneity is introduced through the marginal utility of numraire, namely hn (also called marginal utility of income herein). It is realistic that individuals differ in their valuations of the exchange rate between the utility levels provided by the telephone usage and the numraire (i.e., French Francs). It allows us to discriminate among the different consumption levels in terms of income levels, which turns out to be empirically adapted to our context. (See section 4 paragraph 3.) The optimal discrete choice can now be defined. Inserting equations (2) and (3) into equation (1) yields the conditional indirect utility

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level of its telephone bill over two successive months of 1992.13 It also reports a lot of sociodemographic variables, such as the number of individuals in the household, the job position of the family head, the population density of the area where the household lives or the ownership of a teletext terminal, and the income class. The second database14 describes all the 134 models of telephone marketed in the survey year, including their prices, their attributes and in particular their type. In 1992, marketing analysts considered four market segments corresponding to four different types of telephone: One-block telephone (referenced by the acronym OB) for which dialling buttons are located on the listening part; two-blocks telephone (TB); cordless telephone (CD); answering telephone (AW). The telephone type is used as an attribute in the empirical model. Other attributes are for instance, the number of memories, the presence or not of a speaker or a screen to show the numbers of incoming or outgoing calls. Further informations about sales by brand and type of telephone from 1988 (after deregulation) to 1992 are also provided. For each of these years, the set of models and their attributes (including price) are reported. Variables used in the empirical model are listed in Appendix 3 while descriptive statistics on individual and product variables are gathered in Appendix 4. 4.2. Specification Now we specify the heterogeneity index n , the marginal utility of numraire hn , the quality index jk , the cost of equipment Fjk , and the private valuation of equipment njk , in terms of the observable variables. The heterogeneity index As this index is individual-specific and must be positive always, a classical specification is to set
Q n = exp c0 + cq vnq , q =1
where (cq )q =0 ,.,Q are parameters to be estimated and vnq is the vector of observable individual characteristics q of household n. The selection of relevant socio-demographic variables is an empirical issue. The marginal utility of income In order to differentiate the marginal utility of numraire among individuals, we set hn = mi yin ,

i =1 I

where i = 1,. , I indexes income classes (increasing with i), yin is a dummy variable which takes value equal to one if the household's income belongs to class i or value zero otherwise,
13 If d denotes the telephone bill (expressed in French francs) of household n during two months, the annual level of n
usage x n is approximated by computing 6d n p with p=0.73 FF. We assume that any seasonal variation is included in the measurement error of the equation of telephone consumption. 14 The marketing institute GFK, Paris, has provided this database.

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- 14 -

Table 1: Estimation of model parameters
Model parameters Heterogeneity index, Variable acronyms Constant cqs URBA1 URBA2 URBA3 NUMB SES1 SES2 SES3 MINIT Estimates 6.9106 0.1946 0.1540 0.0821 0.0259 0.2095 0.1828 0.0974 0.2436 0.0025 0.0021 0.0021 0.0017 0.0017 0.0017 0.0021 0.0019 0.0000 0.0001 0.0002 0.0001 0.0001 2.2947 0.2546 3.1976 France Telecom dgs Philips Alcatel Matra d5 d6 Other parameters 0.2568 0.1410 0.0592 0.1811 0.6387 -0.3570 0.8150 0.5390 470.9400 Standard errors 0.2341 0.0437 0.0355 0.0424 0.0102 0.0535 0.0439 0.0418 0.0352 0.0007 0.0006 0.0006 0.0006 0.0004 0.0004 0.0004 0.0004 0.0000 0.0000 0.0001 0.0001 0.0001 0.9467 0.1107 1.1682 0.0320 0.0292 0.0314 0.0245 0.0552 0.0880 0.0827 0.0091 107.5300 t-ratios 29.520 4.452 4.335 1.937 2.525 3.918 4.168 2.328 6.913 3.710 3.329 3.308 2.812 4.341 4.459 4.769 4.220 1.784 2.270 2.006 0.957 2.236 2.424 2.298 2.737 8.022 4.835 1.886 7.400 11.570 -4.068 9.856 59.080 4.379
Marginal utility of income, h INC1 mis INC2 INC3 INC4 Quality index, OB als TB CD AW MEM AMPL AFFIC NMD VOL 3

Equipment cost, F

Equipment valuation,
Note: Standard errors are obtained from the consistent-heteroskedastic matrix of White
Comparing our joint model with a simpler model omitting the continuous choice would also be of interest. However, the discrete part of our model is not identifiable from the data. Note that, as the estimated value of is relatively small, the continuous part of the model is meaningful. Moreover, as the estimated value of is large, the variance of the s tends to become small, so that the discrete choice model is informative. What do the other results teach us on the microeconomic behavior of households? First, the effects of socio-demographic variables on telephone consumption agree with the common intuition. The higher is the population density in the area where the household lives, the lengthier is the duration of telephone usage. This result should be interpreted as an indicator of the presence of a network effect. Similarly, the larger the size of household, the higher the telephone usage. Note that white collars have a higher consumption than blue collars. Finally, the ownership of a teletex terminal also increases consumption. Second, as expected, for each individual, the marginal utility of numraire is positive and decreasing with the class of income level. Moreover, individual price elasticities of telephone consumption (i.e., ln x n ln p = hn p ) are negative and increasing in absolute

t Pnjk1 in equations (19) and (20). Finally, an estimate of sales D j of telephone j Ct* during
period t can be readily obtained by computing D j = S tj S tj1. (21)

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Table 2: Telephone equipment of French households: Estimated stock shares in percent
Market segment Firms France Telecom Matra Philips Alcatel Modulophone Comoc HPF Tfal Radialva Dialatron All firms One-block OB 0.62 6.66 6.34 4.03 2.05 0.62 0.98 1.01 0.00 0.10 22.41 Two-block TB 28.60 14.70 3.01 5.72 5.39 4.16 1.88 0.75 0.62 0.00 64.83 Cordless CD 2.90 3.02 1.37 0.85 0.18 Answering AW 1.27 0.95 1.44 0.53 0.26 All segments 33.39 25.33 12.15 11.13 7.87 4.78 2.86 1.77 0.62 0.10 100.00

0.00 8.30

Notes: - An empty cell means that the firm is not present on the corresponding market segment. - The stock does include rented telephones
Table 3: Telephone equipment of French households: Observed stock shares in percent
Market segment Firms France Telecom Matra Philips Alcatel Modulophone Comoc HPF Tfal Radialva Dialatron All firms One-block OB 2.72 8.34 4.58 4.00 1.24 2.54 0.72 1.21 0.00 0.71 26.07 Two-block TB 27.81 10.90 3.35 4.31 2.56 2.20 0.97 0.59 0.40 0.24 53.32 Cordless CD 4.00 4.13 3.09 1.80 0.21 Answering AW 1.16 1.33 3.06 1.19 0.47 All segments 35.69 24.70 14.08 11.30 4.49 4.74 1.69 1.80 0.40 1.11 100.00

0.16 13.40

Note: All figures are obtained from the marketing institute GFK.
Table 4: Telephone equipment of French households: Estimated market shares in percent
Market segment Firms France Telecom Matra Philips Alcatel Modulophone Comoc HPF Tfal Radialva Dialatron All firms One-block OB 4.20 6.21 5.97 4.44 0.89 0.40 0.71 0.85 0.67 24.35 Two-block TB 25.58 8.27 2.81 1.91 2.76 2.57 2.77 0.42 0.16 0.00 47.26 Cordless CD 7.11 8.54 2.45 2.14 0.22 Answering AW 1.99 1.77 2.57 0.91 0.70 All segments 38.88 24.79 13.80 9.40 4.57 2.98 3.47 1.27 0.16 0.67 100.00
Table 5: Telephone equipment of French households: Observed market shares in percent
Market segment Firms France Telecom Matra Philips Alcatel Modulophone Comoc HPF Tfal Radialva Dialatron All firms One-block OB 7.92 7.50 4.22 3.18 0.21 2.14 0.21 0.88 1.90 28.16 Two-block TB 23.73 10.29 2.39 4.02 0.82 1.71 0.29 0.64 0.20 0.66 44.75 Cordless CD 5.73 4.79 4.24 2.46 0.55 Answering AW 3.25 2.26 2.65 1.08 0.08 All segments 40.63 24.84 13.50 10.74 1.66 3.85 0.50 1.52 0.20 2.56 100.00

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If product j is new, D j = S tj. Of course equation (21) applies only to products that are sold in period t. Note that for all products sold in 1992, our estimated demands turn out to be positive. The model predicts a volume of total sales up to 2.88 millions of telephones. This number should be compared to the estimation of 3.06 millions telephones sold in 1992, according to GFK. Table 4 provides the 1992 estimated market shares as percent of total sales by market segment and firm, while observed shares are given in Table 5. Again we observe that the model behaves quite well. These simulations permit also to analyze the renting side of the market. In 1992, the stock of rented telephones has increased slightly by 2.3 percent from a stock of 17.25 millions in 1991. However, as the estimated proportion of rented telephones decreases from 50.86 percent in 1991 to 47.43 in 1992, our estimate agrees with the facts as reported by marketing experts. 5.2. Price elasticities As explained in Section 2, the structure of the utility function does not impose unrealistic substitution patterns of aggregate demands. This can be easily checked by a glance at Table 6 that displays the own and cross-price elasticities of demand for some models of telephone. The elasticity of product js demand with respect to the 1992 price of product q is

of its products given the prices of competitors.20 Note that the set
of all products and their attributes are exogenous. Assume that the marginal cost of product j can be written mc j = mc j + j where mc j is the deterministic part of marginal cost (which is a function of product attributes and parameters) and where the component j is not observable for the econometrician and probably correlated with unobserved quality. Firm g maximizes expected profits, i.e., solves Max E ( f j mc j + j ) D j ( f ) + j , ( f j )j g j g
where D j ( f ) is the estimated demand for product j, which depends on the vector f of product prices (and on all other exogenous variables that are omitted for convenience), and j is a random term relative to product j, related to the non observable attributes of this product, in particular the unobservable perceived quality. This last term is introduced to represent what it is not explained by the model. Recall that, when prices are correlated with unobservable attributes of products (i.e., with a term like j ) on the demand side, there is a potential problem of endogeneity, which may prevent to find a correct solution to the first order conditions associated with the program defined by equation (22). Neglecting this endogeneity problem has statistical implications when one estimates a system of demand and supply with aggregate data. As already noticed by Goldberg (1995), this problem is alleviated when demand and supply are estimated sequentially on micro data as here. Indeed, because of our specification, our estimated demands account correctly for the perceived quality of products by the consumers. Hence it is reasonable to assume that j is not correlated with the unobservable perceived quality, and hence not correlated with j (or that the correlation between j and j is negligible). Then, a good approximation of the program solved by firm g is

mc j ) D j ( f ).

20 The operator France Telecom has an asymmetric role in this game. We assume here that its choice concerning the price
of telecommunications is not related to the competition on the differentiated product market. Its strategy can be analyzed conditionally to the given tariff of telecommunications.

- 20 -

This leads to the first order condition21

Dj ( f ) +

fr mcr

Dr f fj

( ) =0,
Define by a J J matrix whose element (i.j) is such that ij = Di f f j , with J the number of products. Let E be the J J matrix with generic element (i.j) such that ij = 1 if products i and j are produced by the same firm and 0 otherwise. In matrix notations, equations (24) is written D( f ) + ('. )( f mc ) = 0. (25)

where the operator. defines the element-by-element matrix multiplication and mc is the vector of marginal costs. Assuming that '. is invertible, the vector of markups evaluated at the observed prices is obtained as

1 MK = ('. ) D( f ).

For some models of telephone, Table 7 provides the associated markup, this markup as a percent of the price and the profit (expressed in millions of French Francs) which is simply the markup times the estimated sales. At the segment level, the average ratio of markups to prices are equal to 31.46 percent for answering telephones, 34.09 percent for cordless telephones, 19.03 percent for two-blocks type and 15.27 percent for one-block type. These numbers seems high although the pattern is plausible and agrees with a common feature of differentiated products markets: The higher the ratios, the higher the quality. Concerning profits now (see last column of Table 7), note that products in segments with an intensive competition in terms of number of products (like OB and TB) are generally associated with lower level of profits than the other segments (CD and AW). While it is often observed in market studies that high markups come with small market shares, here they are combined with relatively large market shares. In our case, demand for high-quality products is rising while competition is not so fierce since the market is just experiencing deregulation and the number of sophisticated telephone models remains limited. After 1992, many new highquality models appear (like telephones with both answering and cordless functions) and prices decreased significantly. The most striking result is the very large profit generated by the two-blocks RONDO which was a very popular telephone and which was also proposed for renting from 1993. Table 8 below confirms the leadership of France Telecom, Matra and Philips in terms of profits as in terms of shares. Note that the high quality segments are very profitable (especially the cordless one).
21 Checking that second order conditions hold ensures that estimated parameters are consistent with the existence of the
equilibrium even if unicity is not guaranteed (see also Berry, Levinshon and Pakes, 1998, Feenstra and Levinshon, 1995, Petrin, 1999).

- 21 -

Table 7: Estimated markups and profits by model of telephone
Brand MATRA PHILIPS MATRA ALCATEL PHILIPS ALCATEL COMOC MATRA MODULOPHONE TEFAL DIALATON FRANCE TELECOM FRANCE TELECOM MATRA MATRA MATRA MODULOPHONE Model RIP30 TD9460C AMPLITE DAYTONR TD9230 SURFMEM DAND101 VOILE10 MP2020T COMPAC2 SCANDAS DUO RONDO ADVENT1 CONTACP TM1 DYNASTM Type AW AW CD CD CD OB OB OB OB OB OB TB TB TB TB TB TB Price 1413.84 1386.26 1450.43 1363.80 894.93 282.38 157.01 309.64 185.19 273.96 171.45 369.39 459.00 586.77 353.37 576.30 304.55 % Markup 31.34 34.90 36.55 31.20 28.20 16.88 13.11 16.13 12.82 12.84 13.09 22.73 24.36 26.71 21.86 26.83 18.53 Markup 443.10 483.80 530.13 425.51 252.37 47.67 20.58 49.94 23.74 35.18 22.44 83.96 111.81 156.73 77.25 154.62 56.43 Profits 6.73 6.95 5.88 2.39 4.45 1.19 0.24 2.33 0.68 0.49 0.24 7.68 8.70 2.35 1.38 3.51 0.17

APPENDIX 1: Proof of Proposition 1
The first task is to define the market share for any product. Denoting by s jk the market share of equipment (j,k), we have that s jk = 1 N

Pnjk ,

(A1.1)
where Pnjk is defined in equation (7). If is the number of households in the population, the total demand for equipment (j,k) is S jk = s jk. Now, taking into account the symmetry of alternatives, the total stock for product j is S j = S jj +
, where C * is the set C of all telephones to which one
adds the nil telephone. The market share of product j is then s j = S j

- 23 -

P + Pnjk njj * n =1 kC kC * . = N s + *s j'k Pnj ' j ' + *Pnj 'k j' j' n =1 j 'C kC kC s jj +

(A1.2)

Now we compute the cross price-elasticity of product j with respect of product q. We have the following expressions (for a given n) exp Wnjj 2h exp Wnqq hn exp Wnqk Pnjj kC * \{q } = 2 Fq exp Wnjk ( j '',k '' )E
= h P P + *Pnqk , n njj nqq kC

Pnjk * kC Fq Hence

= hn Pnqq + Pnqk Pnjk Pnjq . * kC * kC
N N Pnjj + Pnjk = hn Pnjj + Pnjk Pnqq + Pnqk Pnjq , Fq n =1 n =1 kC * kC * kC *
and Pnj ' j ' = hn Pnqq + Pnqk Pnj ' j ' 2 Pnqq , Fq j 'C kC * j 'C Fq So we have Pnjk * Pnjj F + kCF q n =1 q

Pnj 'k = hn ' j C kC*

Pnj 'q + Pnqq + Pnqk Pnj 'k Pnqk.

j 'C kC *

s j Fq
N Pnj ' j ' + Pnj 'k n=1 j 'C kC * N N P + Pnjk njj * F Pnj ' j ' + F j 'C n =1 kC q n =1 q 2 N Pnj ' j ' + Pnj 'k n =1 j 'C kC*
N P + Pnj 'k j nj ' j ' k* n=1 'C C

Pnj 'k j 'C * kC

This can be rewritten

- 24 -

N hn Pnjj + Pnjk Pnqq + Pnqk Pnjq s j n=1 kC * kC * = N Fq Pnj ' j ' + *Pnj 'k n =1 j 'C kC N hn n =1
Pnj 'q + Pnqq + Pnqk 1 + Pnj ' j ' + Pnj 'k Pnqq

j 'C kC *

Pnj ' j ' + Pnj 'k

n=1 j 'C

Finally:
N hn Pnjj + Pnjk Pnqq + Pnqk Pnjq s j Fq n=1 kC * kC * = Fq Fq s j N Pnjj + *Pnjk n =1 kC N hn n =1 Fq
P + Pnj 'k nj ' j ' k* C
As this expression depends on j, the result is proved. By the same token, own-price elasticities can be computed. (See Foncel, 1997.)
APPENDIX 2: Proof of Proposition 2
i) The joint probability Pr (Emn , xn , Dn ) of drawing a level of usage x n , a subset Dn and an alternative which is known to belong to the group Emn , is given by Pr U Vnjk + njk Vnjk + njk , ( j, k ) E and ( j , k ) ( j , k ) {xn } {Dn }. ( j ,k )Emn
Then, Pr (Emn , xn , Dn ) = However,

( j ,k )Emn

Pr( j, k , xn , Dn ).
Pr( j , k , xn , Dn ) = Pr (Dn j, k , xn ) Pr ( j , k , xn ).
Given the sampling procedure, the way alternatives are selected is independent of the consumption level. So we have Pr (Dn j, k , xn ) =Pr (Dn j, k ). By Bayes' theorem, Pr ( j, k , xn Dn ) = However, Pr(Dn ) = Pr (Dn j , k ) Pr ( j , k , xn ). Pr (Dn )

q y 0 , z > 0 , all z Z. Then the
expression q y 0 , z f ( y , z )dH attains its maximum at y = y 0.
( ) q(y , z ) f (y , z ) dH q(y , z ) f (y , z ) dH , and q(y , z )[ f (y , z ) f (y , z )]dH 0. This implies that there exists z Z such that f (y , z ) f (y , z ) which contradicts the assumption. From Lemma 1, the expression B (v , x, D, m; )ln B(v , x, D, m; ) dx attains its maximum
PROOF: Suppose y1 = arg max q y 0 , z f ( y , z ) dH. Then,

y z z z 0 z M 0

at = , for all (D, v ). It is just required to take an appropriate measure composed by a Lebesgue measure over x and a discrete one over m. From Lemma 2, consistency is achieved by taking the appropriate measure over D and v.

m =1 x

APPENDIX 3: Variable definitions
A3.1. Individual variables All these variables enter the heterogeneity index, . URBA1 : takes value 1 if the household lives in a city with a population size larger than 1,000,000 inhabitants and 0 otherwise. URBA2 : takes value 1 if the household lives in a city with a population size between 100,000 and 1,000,000 inhabitants and 0 otherwise. URBA3 : takes value 1 if the household lives in a city with a population size between 20,000 and 100,000 inhabitants and 0 otherwise. URBA4 : takes the value 1 if the household lives in a city with a population size lower than 100,000 inhabitants or in a rural area (the parameter relative to this variable is normalised to 0). NUMB : is the number of persons in the household. SES1 : takes value 1 if the household head is an upper level white collar. SES2 : takes value 1 if the household head is a lower level white collar. SES3 : takes value 1 if the household head is a blue collar, a farmer or a craftsman. SES4 : takes value 1 if the household head is unproductive or pensioned off (the parameter relative to this variable is normalised to 0). MINIT : takes value 1 if the household owns a teletext terminal.

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A3.2. Income variables These variables correspond to vector ( yi ) i =1,., I in the model. INC1 INC2 INC3 INC4 : takes value 1 if households annual income y is lower than FF 84,000. : takes value 1 if households annual income is between FF 84,000 and FF 165,000. : takes value 1 if households annual income is between FF 165,000 and FF 270,000. : takes value 1 if households annual income is greater than FF 270,000.

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