Quantitative Techniques for Competition and Antitrust Analysis_11 pptx

478 9. Demand System Estimation and cross-price elasticities. In practice, therefore, MNL models are quite good for learning about the characteristics which tend to be associated with high or low levels of market shares, but we recommend strongly against using MNL models in situa- tions where we must learn about substitution patterns (e.g., for merger simulation). There have been a number of responses to the problems the literature has identified with MNL and we explore some of those responses in the rest of this section. The important lesson from MNL and the property of independence of irrelevant alterna- tives (IIA) is not that the MNL is a hopeless model (though that is probably true), but rather that we can use the IIA property to our advantage; since the MNL makes unreasonable predictions about what will happen to market shares following entry, if we observe what happens to market shares following entry, we will be able to use data to reject MNL models and identify parameters in richer discrete choice models. Furthermore, the literature has grown from MNL models and many of its tools are most simply explained in that context. For example, in the next section we explore the introduction of unobserved product characteristics in the context of the MNL models but we shall see later that the basic techniques for analyzing models with unobserved product characteristics can be used in far richer discrete choice models. 9.2.4.3 Introducing Unobserved Product Characteristics in MNL Models A famous, possibly true, marketing story is that the first car which introduced a cupholder experienced dramatically high sales—customers thought it was a great novel idea. Economists working with data from the time period, however, probably would not have had a variable in their data set called “cupholder”—it would have been a product characteristic driving sales, differentiating the product, which would be observed by customers but unobserved by our analyst. Such a situation must be common. As a result Bultez and Naert (1975), Nakanishi and Cooper (1974), Berry (1994), and Berry et al. (1995) have each argued that we should introduce an unobserved product characteristic into our econometric demand models. Following Berry (1994), denote the unobserved product characteristic  j so that the conditional indirect utility function that an individual gets from a given product j is v ij DNv j C " ij D x 0 j ˇ ˛p j C  j C " ij ; where x j is a vector of the observed product characteristics,  j is the unobserved product characteristic (known to the consumer but not to the economist), and con- sumer types are represented by " i D ." i0 ;" i1 ;:::;" iJ /. In general, there may be many elements comprising  j but the class of models which have been developed all aggregate unobserved product characteristics into one. The parameters of the model which we must estimate are ˛ and ˇ. The basic MNL model attempts to force observed product characteristics to explain all of the variation in observed market shares, which they generally cannot. Instead of 9.2. Demand System Estimation: Discrete Choice Models 479 estimating a model s j D s j .p;xI˛; ˇ/ C Error j ;jD 0;:::;J; where an error term is “tagged on” to each equation in the demand system, the new model gives an explicit interpretation to the error term and integrates it fully into the consumer’s behavioral model, s j D s j .p;x;I˛; ˇ/. Of course, just introducing an unobserved product characteristic does not get you very far. In particular, there is a clear potential problem with introducing unobserved product characteristics in that the term enters in a nonlinear way—it is not obvious how to run a regression in such cases. Fortunately, Nakanishi and Cooper (1974) and Berry (1994) have shown that we can recover the unobserved product characteristics from every product in the MNL model. Berry et al. (1995) then extend the “we can recover the error terms” result to a far wider set of models. To see how, define the vector of common (across individuals) utilities with the common utility of the outside good normalized to zero Nv D .0; Nv 1 ;:::; Nv J /. Suppose we choose Nv to make the MNL model’s predicted market shares exactly match the actual market shares so that s j .p; Nv/ D s j for j D 1;:::;J: Since Nv D .0; Nv 1 ;:::; Nv J /,wehaveJ equations like the one specified above with J unknowns. If the J equations match the predicted and actual market share of all markets, then the market share of the outside good will also match s 0 .p;y; Nv/ D s 0 since the actual and predicted market shares must add to one. 40 Taking logs of the market share equations gives us an equivalent system with J equations of the form: ln s j .p ; Nv/ D ln s j for j D 1;:::;J; where at a solution we will also have ln s 0 .p; Nv/ D ln s 0 . Recalling the normalization condition Nv 0 D 0 so that expfNv 0 gD1, we can write s j .p; Nv/ D expfNv j g 1 C P K kD1 expfNv k g D s 0 .p; Nv/ expfNv j g; so that ln s j .p; Nv/ D ln s 0 .p; Nv/ CNv j for j D 1;:::;J: 40 In the continuous choice demand model context, we studied the constraint imposed by “adding up”: that total expenditure shares must add to one. In a differentiated product demand system, we get a similar “adding up” condition which enforces the condition that market shares add to one J X j D0 s j D J X j D0 s j .p;x;I ˛; ˇ/ D 1: As a result of this condition we will, as before, be able to drop one equation from our analysis and study the system of J equations. Generally, in the differentiated product context, the equation for the outside good is dropped from the system of equations to be estimated.We impose the normalization that v 0 D 0, which in turn can be generated in part by the assumption that  0 D 0. 480 9. Demand System Estimation So that our J equations become ln s j D ln s 0 .p; Nv/ CNv j for j D 1;:::;J: At a solution we know that ln s 0 .p; Nv/ D ln s 0 so that we know a solution must have the form Nv j D ln s j  ln s 0 ; where the shares on the right-hand side are observed data. Thus the mean utilities Nv D .0; Nv 1 ;:::; Nv J / that exactly solve the market share equations s j .p; Nv/ D s j are just Nv j D ln s j  ln s 0 for j D 1;:::;J; where s 0 D 1  P J kD1 s k . This formula states that, for the MNL model, we only need information about the levels of market shares to figure out what the utility levels of the models must be in order to rationalize those market shares. The mean utility vector Nv is uniquely determined by the observed market shares. This allows us to write and estimate the linear equation with now “observed” level of utility as the dependent variable: ln s j  ln s 0 D x j ˇ ˛p j C  j : Note that this formulation of the model provides a simple linear-in-the-parameters regression model to estimate, a familiar activity. The prices p j and product char- acteristics x j are observed, the parameters to be estimated are ˛ and ˇ and the error term is the unobserved product characteristic  j . Since this is a simple linear equation we can use all of our familiar techniques upon it, including instrumental variable techniques. For the avoidance of doubt, note that the market shares in this equation are volume market shares (or equivalently here number of purchasers, since in this model only one inside good can be chosen per person). In addition, the market shares must be calculated as a proportion of the total potential market S including the set of people who choose the outside good. The appropriate way to calculate the total potential market can be a matter of controversy, depending on the setting. In the new car market it may be reasonable to assume that the largest potential market is for each person of driving age to buy a new car. In breakfast cereals it may be reasonable to assume that at most all people in the country will eat one portion of cereal a day, so, for example, no one eats bacon and eggs for breakfast if the price of cereal is sufficiently low and the quality sufficiently high. Obviously, such propositions are not uncontroversial: some people own two cars and some people eat two bowls of cereal a day. It may sometimes be possible to estimate the market size S , though few academic articles have managed to. More frequently, it is a very good idea to test the sensitivity of estimation results to whatever assumption has been made. Table 9.5 presents results from Berry et al. (1995). Specifically, in the first column they report an OLS estimation of the logit demand specification and in the second 9.2. Demand System Estimation: Discrete Choice Models 481 Table 9.5. Estimation of the demand for cars. OLS logit IV logit OLS Variable demand demand ln.price/ on w Constant 10.068 9.273 1.882 (0.253) (0.493) (0.119) HP/weight a 0.121 1.965 0.520 (0.277) (0.909) (0.035) Air 0.035 1.289 0.680 (0.073) (0.248) (0.019) MP$ a 0.263 0.052 — (0.043) (0.086) — MPG a — — 0.471 — — (0.049) Size 2.431 2.355 0.125 (0.125) (0.247) (0.063) Trend — — 0.013 — — (0.002) Price 0.089 0.216 — (0.004) (0.123) Number of inelastic 1,494 22 n.a. demands (˙2 SEs) (1,429–1,617) (7–101) R 2 0.387 n.a. 0.656 Notes: The standard errors are reported in parentheses. a The continuous product characteristics—horsepower/weight, size, and fuel efficiency (miles per dollar or miles per gallon)—enter the demand equations in levels but enter the column 3 price regression in natural logs. Source: Table III from Berry et al. (1995). Columns 1 and 2 report MNL demand estimates obtained using (1) OLS and (2) IV. Column 3 reports a regression of the price of car j on the characteristics of car j , sometimes called a “hedonic” pricing regression. If a market were perfectly competitive, then price would equal marginal cost and the final regression would tell us about the determinants of cost in this market. column the instrumental variable (IV) estimation. Note in particular that the move from OLS to IV estimation moves the price coefficient downward. This is exactly as we would expect if price were “endogenous”—if it is positively correlated with the error term in the regression. Such a situation will arise when firms know more about their product than we have data about and price the product accordingly. In terms of our opening example, a car which introduces the feature of cupholder will see high sales and the firm selling it may wish to increase its price to take advantage of high or inelastic demand. If so, then the unobserved product characteristic (our error term) and price will be correlated. We have mentioned previously that the multinomial logit model, even with the introduction of an unobserved product characteristic, imposes severe and unde- sirable structure on own- and cross-price elasticities. To see that result, recall 482 9. Demand System Estimation that ln s j .p;x;/ DNv j .p;x;/  ln.s 0 .p;x;// DNv j .p j ;x j ; j /  ln  1 C J X kD1 expfv k .p k ;x k ; k /g à ; where Nv j D x j ˇ ˛p j C  j . Differentiating, it follows that @ ln s j .p ;x;/ @ ln p k D˛p k s k .p;x;/ D˛p k s k for j ¤ k; @ ln s j .p;x;/ @ ln p j D˛p j .1  s j .p;x;// D˛p j .1  s j /; where the latter equalities follow when we evaluate the elasticities at a point where predicted and actual market shares match. This means that all own- and cross-price elasticities between any pair of products j and k are entirely determined by one parameter ˛, the market share of the good whose price changed and also the price of that good. Most strikingly, substitution patterns do notdepend on how good substitutesgoods j and k really are, for example, whether they have similar product characteristics. Because of the inflexible and unrealistic structure that the MNL model imposes on the preferences, they probably should never be used in merger simulation exercises or in any other exercise where the pattern of substitution plays a central role in informing decision makers about appropriate policy. Despite all of the comments above, the MNL model does remain tremendously useful in allowing analysts a simple way of exploring which product characteristics play an important role in determining the levels of market shares. However, it is often the departures from the simple MNL model that are most informative. For example, it can be informative to include rival characteristics in product j ’s payoff since that may inform us when close rival products drive down each individual product’s market share because each product cannibalizes the demand for the other. Indeed, it is precisely such patterns in the data that richer models will use to generate more realistic substitution patterns than those implied by models such as MNL with IIA. The observation is useful generally, but it also provides the basis of the formal specification tests for the MNL proposed by Hausman and McFadden (1984). 9.2.5 Extending the Multinomial Logit Model In this section we follow the literature in extending the MNL model to allow for additional dimensions of consumer heterogeneity. To illustrate the process, we bring together the MNL model with the Hotelling model and also the vertical product differentiation model. 9.2. Demand System Estimation: Discrete Choice Models 483 Specifically, suppose that the conditional indirect utility function can be defined as v j .z j ;L j ;p j ; j ; i ;L i ;" ij / D  i z j  tg.d.L i ;L j //  ˛p j C  j C " ij ; where the term z j is a quality characteristic where all consumers agree that all else equal more is better than less—a vertical source of product differentiation. Additionally, products are available in different locations L j and depending on the consumer’s location L i the travel cost may be small or large—a horizontal source of product differentiation. Finally, we suppose that consumers have an intrinsic preference for particular products as in the multinomial logit. The consumer type in this model is  D ." i0 ;" i1 ;:::;" iJ ;L i ; i /, where " ik represents the idiosyncratic preference of consumer i for product k, L i indicates the individual’s taste for the horizontal product characteristic, and  i represents his or her willingness to pay for the vertical product characteristic. As usual, aggregate demand is simply the sum of individual demands, x j .z;L;p;I i ;L i ;" i0 ;:::;" iJ /; over the set of all consumer types. In the first instance, that sum involves a .J C3/- dimensional integral involving (J C 1) dimensions for the epsilons plus 1 each for the location and vertical tastes L i ,  i . Thus aggregate demand is D j .s;L;p;/ D • ";L; x j .z;L;p;I i ;L i ;" i0 ;:::;" iJ /f ";L; .";L i ;IÂ/d" dL i d D Z  Z L expfz k  tg.d.L j ;L i //  ˛p j g P K kD1 expfz k  tg.d.L k ;L i //  ˛p k g f L; .L i ;/dL i d: For any given L i ,  i , the model is exactly an MNL model. Thus we can use the MNL formula to perform the integration over the .J C 1/ dimensions of consumer het- erogeneity arising from the epsilons. Doing so means that the resulting integration problem becomes in this instance just two dimensional, which is a relatively straight- forward activity that can be accomplished using numerical integration techniques such as simulation. 41 Berry et al. (1995) show that even in this kind of context we canfollow an approach similar to that taken to analyze the MNL model. We discuss their model below, but before doing so we describe the nested logit specification, which is a less flexible but more tractable alternative popular among some antitrust practitioners. 41 For an introductory discussion in this context, see Davis (2000). For computer programs and a good technical discussion, see Press et al. (2007). For a classic text, see Silverman (1989). For the econometric theory underlying estimation when using simulation estimators, see Pakes and Pollard (1989), McFadden (1989), and Andrews (1994). 484 9. Demand System Estimation Rigids Tractors Outside good Truck models Truck models Figure 9.5. A model for the demand for trucks. Source: Ivaldi and Verboven (2005). 9.2.6 The Nested Multinomial Logit Model The nested multinomial logit (NMNL) model is a somewhat more flexible structure than the MNL model and yet retains its tractability. 42 It is based on the assumption that consumers each choose a product in stages. The concept is very similar to the nested model we studied by Hausman et al. (1994) for the demand for beer. In each case, consumers first choose a broad category of products and then a specific product within that category. Hausman et al. estimated their model using different regressions for each stage. In contrast, the NMNL model allows us to estimate the demand for the final products in a single estimation. Ivaldi and Verboven (2005) apply this methodology in their analysis of a case from the European merger jurisdiction, the proposed Volvo–Scania merger. 43 The product overlap of concern involved the sale of trucks generally and heavy trucks in particular since the commission found that heavy trucks constituted a relevant market. The authors suggest that the heavy trucks market can be segmented into two groups involving (1) rigid trucks (“integrated” trucks, from which no semi-trailer can be detached) and (2) tractor trucks, which are detachable. A third group is specified for the outside good. Figure 9.5 describes the nesting structure they adopt. The NMNL model itself can be motivated in a number of ways. Motivation method 1. McFadden (1978) initially motivated the NMNL model by assuming that consumers undertook a two-stage decision-making process. At the first stage he suggested they decide which broad category (group) of goods g D 1;:::;Gto buy from and then, at the second stage, they choose between goods within that group. Each of the groups consists of a set of products and all products are in only one group. The groups are mutually exclusive and exhaustive collections of products. 42 The link between consumer theory and discrete choice models is discussed in McFadden (1981) and for the NMNL model, in particular, see also Verboven (1996). 43 Case no. COMP/M. 1672. Their exercise is described in chapter 8. 9.2. Demand System Estimation: Discrete Choice Models 485 Motivation method 2. Cardell (1997) (see also Berry 1994) provide an alternative way to motivate the NMNL model as a random coefficient model with a conditional indirect utility function defined as v ij D K X lD1 x jl ˇ il C  j C & ig C .1  /" ij for product j in group g; v i0 D & i0 C .1  /" i0 for the outside good; where x jl is the lth observed product characteristics of product j ,  j are the unob- served product characteristics, & ig is the consumer preference for product group g, and " ij is the idiosyncratic preference of the individual for product j . For reasons we describe below, since for every individual any products in group g get the same value of & ig , which in turn depends on , the parameter  introduces a correlation in all consumers’tastes across products within a group. Consumers with a high taste for group g, a large & ig , will tend to substitute for other products in that group when the price of a good in group g goes up. The consumer type in a model with G pre-specified groups is  i D .& i1 ;:::;& iG ;" i0 ;" i1 ;:::;" iJ /: Cardell (1997) showed that for given  ,if& ig are independent with " ij having a type I extreme value distribution, then the expression & ig C.1/" ij will also have a type I extreme value distribution if and only if & ig has a particular type I extreme value distribution. 44 Cardell (1997) also showed that the required distribution of & ig depends on the parameter  so that some authors prefer to write & ig ./ and & ig ./ C .1  /" ij . The parameter  is restricted to be between zero and one. As  approaches zero the model approaches the usual MNL model and the correlation between goods in a given group becomes zero. On the other hand, as  increases to one, so does the relative weight on & ig and hence correlation between tastes for goods within a group. Motivation method 3: the MEV class of models. A third way to motivate the NMNL model is to consider it a special case of McFadden’s (1978) generalized extreme-value (GEV) class of models (which is probably more appropriately called the multivariate extreme-value (MEV) class of models since the statistics com- munity use GEV to mean a generalization of the univariate extreme value distri- bution). That model effectively relaxes the independence assumptions across the tastes ." i0 ;:::;" iJ / embodied in the MNL model. The basic bottom line is that the MEV class of models assumes that the joint distribution of consumer types can be expressed as F." i0 ;:::;" iJ I/ D exp.H.e " i0 ;:::;e " iJ I//; 44 As Cardell describes, his result is analogous to the more familiar result that if "  N.0;  2 1 / and " and v are independent, then " C v  N.0;  2 1 C  2 2 / if and only if v  N.0;  2 2 /. 486 9. Demand System Estimation where H.r 0 ;:::;r J I/ is a possibly parametric function (hence the inclusion of parameters ) with some well-defined properties (e.g., homogeneity of some pos- itive degree in the vector of arguments). We have already mentioned that the stan- dard MNL model has distribution function F." ij / D exp.e " ij / so that under independence the multivariate distribution of consumer types is F." i0 ;:::;" iJ I/ D F." i0 /F ." i1 / F." iJ / D exp   J X j D0 e " ij à : In that case the MNL corresponds to the simple summation function H.r 0 ;:::;r J I/ D J X j D0 r j : The “one-level” NMNL model developed by McFadden (1978) corresponds to a choice of function H.r 1 ;:::;r J I/ D G X gD1  J X j 2= g r 1=.1/ j à 1 ; where = g denotes the set of products placed into group g,  D , and the distribution function is evaluated at r j D e " ij . The outside good will often be put into its own group. Davis (2006b) discusses this approach to understanding the discrete choice literature and also proposes a new member of the MEV class of discrete choice models which can be used to estimate discrete choice models which have far less restrictive substitution patterns. Whichever method is used to motivate the NMNL model, specifying the groups appropriately is absolutely vital for the results one will obtain. The groups must be specified before proceeding to estimate the model, and the choice of groups will have implications for which goods the model predicts will be better substitutes for one another. Recall that the parameter  controls the correlation in tastes between goods within a group. Company information on market segments or consumer surveys may be helpful in establishing which products are likely to be “closer” substitutes and therefore form distinct market segments that can be associated with a particular group. Following the earlier literature, Berry (1994) shows that in a manner very similar to that used for the MNL model the NMNL model can also be estimated using a regression equation linear in the parameters that can be estimated with instrumental variables (see Bultez and Naert 1975; Nakanishi and Cooper 1974). In particular, we have ln s j  ln s 0 D K X lD1 x jl ˇ l C  ln s j jg C  j ; 9.2. Demand System Estimation: Discrete Choice Models 487 where s j jg is the market share of product j among those purchased in group g.If q j denotes the volume of sales of product j , then s j jg D q j = P j 2= g q j . The use of instrumental variables is likely to be essential when using this regression equation since there will be a clear correlation between the error term  j and the conditional market shares s j jg . Verboven and Brenkers (2006) suggest allowing the parameter of the model controlling the within-group taste correlation to be group-specific so that H.r 1 ;:::;r J I/ D G X gD1  J X j 2= g r 1=.1 g / j à 1 g : In that case, they show that Berry’s regression can be estimated similarly by estimating G group-specific taste parameters, ln s j  ln s 0 D K X lD1 x jl ˇ l C  g ln s j jg C  j : The additional taste parameters will help free-up substitution patterns across goods within each group since they are no longer constrained to be the same across groups. However, even this model will suffer from similar problems as MNL when examining substitution across groups. 9.2.7 Random Coefficient Models Economists studying discrete choice demand systems have used consumer hetero- geneity to generate models with better properties than either pure MNL or even NMNL models. These approaches have been taken with both aggregate data and also consumer-level data. We focus primarily on approaches with aggregate-level data but note that the models are identical, although their method of estimation typi- cally is not. 45 In the aggregate demand literature, the first random coefficient models were estimated by Boyd and Mellman (1980) and Cardell and Dunbar (1980) using data from the U.S. car industry. Those authors did not incorporate an unobserved product characteristic into their model. The modern variant of the random coeffi- cient model for aggregate data was developed in Berry et al. (1995) and through their initials (Berry, Levinsohn, and Pakes) is often referred to as the “BLP” model. In principle, random coefficients can provide us with very flexible models that put few constraints on the substitution patterns in demand. If the models place few con- straints on substitution patterns, then in an ideal world with enough data we will be able to use that data to learn about the true substitution patterns. Because the utility is expressed in terms of product characteristics and not in terms of products, the number of parameters to be estimated does not increase exponentially with the number of products in the market as in the case of the AIDS 45 See Davis (2000) and the references therein for more on the connections between the two types of discrete choice models. [...]... conduct, and market outcomes In principle, estimating market demand functions for homogeneous products is the easiest activity for an applied economist as there is only one demand equation to estimate and it depends on only one price variable (and any demand shifters such as income) Still, one must be careful to understand the drivers of variation in the observed data and doing so will involve understanding... between the individualproduct taste random vector " and the individual’s income and tastes for characteristics We also assume the multinomial logit distribution for " allows us to express the individual demand for product j given the individual’s tastes for characteristics and MNL income, which we have denoted sij p; x; I yi ; ˇ1i ; : : : ; ˇiK / Computing aggregate demand then “only” requires the K C... potential motivations for imposing vertical restraints or perhaps even vertically integrating As we shall see, many of the possible motivations for vertical restraints are likely to fit well with the market generating good outcomes for consumers On the other hand, vertical restraints and integration can generate harm to consumers and therefore form a legitimate focus of attention for a competition authority... about the likelihood of foreclosure and consumer harm rather than detailed quantitative analysis There have, however, been some interesting attempts at empirical estimation of the effect of vertical practices and vertical integration and we explore many of them in this chapter Moreover, since the trend in the legal 10.1 Rationales for Vertical Restraints and Integration 503 standard for evaluation is toward... markups will be much higher for high-end BMWs and Lexuses than for low-end Mazdas and Fords 9.3 Demand Estimation in Merger Analysis The above introduction to the common models used for demand system estimation has hopefully served at least to illustrate that estimating demands, although an essential part of many quantification exercises, is quite a complex and even optimistic task An analyst is faced... survey data Demand estimation is another tool in the economists’ toolbox—but one that is sometimes easy to physically implement and yet difficult to use well If demand estimation produces unrealistic demand elasticities, one must revise the specification of the demand model Assuming that the demand estimation is correctly specified and that proper instruments are being used, one must check for other sources... as big as the linear demand model and the logit model showed an increase in price 50% higher than the linear demand model These results reflect the fact that the greater the curvature of the demand curve, the lower the price elasticity of demand as prices increase (think about moving upward and leftward along an inverse demand curve that is either steeply or shallowly curved) and the greater the incentives... of the factors involved and the difficulty of even predicting the direction of potential effects 10.1 Rationales for Vertical Restraints and Integration Vertical restraints can take on many forms.1 Some take the form of price restraints and impose conditions on the price that the downstream firm can charge for the good that is purchased from the upstream firm while others take the form of nonprice restraints... provided by Rey and Tirole (2005) See also the special edition of the Journal of Industrial Economics, September 1991, edited by John Vickers and Mike Waterson (see Vickers and Waterson (1991) and, of course, Church (2004)) Finally, the interested reader may like to refer to Dobson and Waterson (1996) 2 For the source of this example and a detailed examination of it, see Asker (2005) 504 10 Quantitative. .. other hand, aggregation can sometimes reduce the effect of measurement error.54 The possibility 53 Short-run elasticities of demand can be far greater than long-run elasticities of demand (or vice versa) depending on the context For the recent literature, see the overview by Hendel and Nevo (2004) For a more technical dynamic model of consumer choice with inventories, see Hendel and Nevo (2006a,b) For . be much higher for high-end BMWs and Lexuses than for low-end Mazdas and Fords. 9.3 Demand Estimation in Merger Analysis The above introduction to the common models used for demand system estimation has. Pakes and Pollard (1989), McFadden (1989), and Andrews (1994). 484 9. Demand System Estimation Rigids Tractors Outside good Truck models Truck models Figure 9.5. A model for the demand for. taste random vector " and the individual’s income and tastes for characteris- tics. We also assume the multinomial logit distribution for " allows us to express the individual demand for