Welfare Analysis for Discrete Choice: Some New Results

Transcription

1 Welfare Analysis for Discrete Choice: Some New Results Debopam Bhattacharya University of Oxford May 6,. Abstract We develop methods of empirical welfare-analysis for multinomial choice, which require no restriction on unobserved preference heterogeneity across consumers. We present results on nonparametric identi cation of Hicksian welfare distributions from choice-probabilities in three practically important scenarios, viz., (i) simultaneous price-change of multiple alternatives, e.g., following rm-mergers, (ii) introduction of new goods, and (iii) multinomial choice with nonexclusive options. A key insight is that identi cation of preference-parameters, although su - cient, is not necessary for welfare-analysis. The methods are illustrated with school/work choice data on Indian teenagers. Welfare-analyses in presence of income-endogeneity, market-level data and interval-reporting of income are brie y discussed. Welfare analysis lies at the heart of economic policy evaluations. The economic impacts of taxes and subsidies, introduction of new goods, rm-mergers, changes in trade-policy etc. are all evaluated by policy-makers primarily in terms of their potential impact on consumer welfare. The conceptual cornerstones of such welfare-analysis are the notions of compensating and equivalent variation (CV and EV, henceforth) originating with Hicks, 946. In this approach, welfare e ects are measured in terms of the income adjustment necessary for maintaining consumers utility levels. The popularity of these "money-metric" welfare measures stems from their basis in fundamental economic principles of constrained utility maximization and their representation in monetary units which facilitates cost-bene t analysis. In real-life applications, the source of welfare-relevant information are micro-level data recording consumers purchase behavior at di erent prices and income. But CV/EV calculations require empirical measurement of utilities which cannot be directly observed in any such dataset. So the I am grateful to Phil Haile, and seminar participants at the University of Wisconsin for useful feedback and suggestions. All errors are mine.

2 key methodological challenge is to infer welfare-relevant features of utilities from observable purchase behavior. An important consideration in any such exercise is the presence of unobserved heterogeneity in individual preferences. For the same price change and at the same income, unobserved heterogeneity causes di erent individuals to undergo a di erent amount of welfare-change. Because the underlying heterogeneity is unobservable, one cannot in general learn the individuallevel welfare changes, and can only hope to infer features of the welfare distribution, such as the average welfare-change and its variance across subpopulations. This paper concerns empirical welfare analysis in the common real-life setting of discrete choice, allowing for completely unrestricted preference heterogeneity across consumers. Previous researchers analyzing this problem worked under highly restrictive and arbitrary assumptions on preferences and/or unobserved heterogeneity. These include assumptions of quasilinear preferences implying absence of income e ects (c.f., Domencich and McFadden, 97, Small and Rosen 9, Trajtenberg, 99), and parametrically speci ed utility functions and heterogeneity distributions (Herriges and Kling, 999, Dagsvik and Karlstrom, ). Apart from standard concerns about model mis-speci cation leading to false conclusions, these parametric approaches do not clarify whether the empirical conclusions arise from the functional form assumptions or they are more fundamental in the sense that choice probabilities contain all the welfare-relevant information, with parametric computations being simply a convenient approximation. More recently, Berry and Haile (4) have developed methods for empirical demand and welfare analysis with market-level (as opposed to individual) data by relaxing a few of the restrictive assumptions commonly imposed in the existing empirical IO literature. The rst fully nonparametric approach to empirical welfare analysis for discrete choice, using individual data, was recently developed by Bhattacharya, (DB, henceforth). This paper showed that for a discrete good with mutually exclusive alternatives, the marginal distributions of the equivalent/compensating variation (EV/CV) resulting from ceteris paribus price-change of an alternative can be expressed as closed-form transformations of choice probabilities. Surprisingly, this result requires neither prior knowledge nor identi ability of the functional form of utilities or the dimension (and hence the distribution) of unobserved preference heterogeneity. The present paper complements DB, and makes three contributions. First, it presents pointidenti cation results for distributions of individual welfare change resulting from simultaneous price change of several alternatives; this includes situations where some price changes are negative, some zero and some positive. A key observation is that welfare e ects of simultaneous change in multiple prices cannot be obtained by iterating the single price-change result of DB. This is because

3 the de nition of welfare-distribution corresponding to simultaneous change of several prices di ers from the one that arises from sequential ceteris paribus changes, owing to the presence of unobserved heterogeneity. Multiple price-changes are likely consequences of a single price change in price-elastic markets. For example, a price-cut in one inter-city coach service is likely to trigger price-cuts for competing coach services; a school-tuition subsidy in an area with child labor is likely to raise children s wages in response, and a merger between rms providing di erent telecommunication services would potentially raise prices of all products concerned (c.f., Hausman, ). In the traditional parametric approach used in the empirical IO literature, one typically assumes (i) absence of income e ects, (ii) Type extreme valued errors, and (iii) linear index form of utility di erences, which result in the so-called log-sum formula (c.f., Train, 9) for welfare comparisons, based on evaluating the index at di erent price vectors. This formula, though convenient, is based on strong, unsubstantiated assumptions and potentially leads to erroneous substantive conclusions. In the present paper, we show that one can obtain correct welfare estimates without making any assumptions on unobserved heterogeneity and functional form of utilities, and the resulting expressions are both easy to compute, and continue to hold even when income e ects are non-negligible, as is likely for bigger purchases like durables or health insurance plans. Our second set of results concern welfare e ects resulting from introduction of a new alternative to consumers choice set. Hausman, 996 and Nevo, had previously investigated this problem in the context of breakfast cereal brands, Petrin, considered new products in the automobile market and Trajtenberg, 99 studied innovations in CT scanners. We consider welfare evaluation of new products when individual purchase data in the post-introduction period are available. Indeed, existing methods based on the virtual price (c.f., Willig et al, 99, and Hausman, ) do not allow for unobserved heterogeneity in preferences, and so fundamentally new results are required for welfare analysis in presence of such heterogeneity. The third part of this paper considers multinomial choice with non-exclusive alternatives. For example, if a cable-tv company o ering a sports package and a movie package raises product-prices, then the resulting welfare calculations requires new results beyond DB because the packages are not exclusive alternatives for a potential consumer. In the present paper, we show that for multiple non-exclusive discrete goods, the welfare distribution resulting from a single price-change can be point-identi ed but that for multiple price-changes cannot, unlike the case of multinomial choice The impact of such simultaneous price-changes on consumer welfare is the key consideration for regulatory agencies, such as the US Federal Trade Commission and the UK Competition and Markets Authority, in their decision on whether to allow the merger (c.f., Willig et al, 99).

4 among exclusive options. Taken together, these three results provide new theoretical insights into welfare analysis in discrete choice situations, as well as providing practitioners with useful empirical tools for evaluating policy changes in real-life settings. The rest of the paper is organized as follows. In Section, we discuss multiple price changes, in Section we analyze welfare e ects of introducing a new alternative, and in section we consider discrete choice among non-exclusive options. In section 4, we provide a brief discussion of how to implement these results in practice, and how to circumvent common data problems such as income endogeneity, unavailability of individual demand data and interval reporting of income. An empirical illustration is presented in section, using the well-known Herriges and Kling (999) data on choice of mode of shing in Southern California. All proofs are collected in the appendix. Multiple Price-Changes Set-up: Individual income is denoted by Y ; individual heterogeneity is denoted by with marginal CDF F (). Individual utility from choosing alternative j is U j (Y P j ; ), j = ; :::; J. De ne the structural choice probabilities evaluated at price vector p and income y, denoted fq j (p; y)g, j = ; :::; J, as Z q j (p; y) = U j (y p j ; ) > max J6=j fu J (y p J ; )g df (). Assumption Assume that for each and for each j = ; :::; J, the utility function U j (; ) is continuous and strictly increasing. Consider arbitrary price changes from p (p ; p ; :::; p J ) to p (p ; p ; :::; p J ). Assume without loss of generality that p J p J p J ; p J ; ::: p p. () That is, label the alternative with the smallest price change (the smallest could be a negative number, representing a fall in price) alternative, the next smallest as alternative and so on. This result is unrelated to identi cation failure for ordered choice, shown in DB, since the alternatives in the non-exclusive case need not be ordered. 4

5 De ne EV as the solution S to the equation max fu (y p ; ) ; U (y p ; ) ; :::; U J (y p J ; )g = max fu (y S p ; ) ; U (y p S; ) ; :::; U J (y p J S; )g. () Similarly, de ne CV as the solution S to the equation max fu (y + S p ; ) ; U (y + S p ; ) ; :::; U J (y + S p J ; )g = max fu (y p ; ) ; U (y p ; ) ; :::; U J (y p J ; )g. () Under assumption, the solutions to the equations () and () are unique. For instance, if S and S satisfy () and S > S, then max fu (y S p ; ) ; U (y p S; ) ; :::; U J (y p J S; )g = max fu (y p ; ) ; U (y p ; ) ; :::; U J (y p J ; )g = max U y S p ; ; U y p S ; ; :::; U J y p J S ; > max fu (y S p ; ) ; U (y p S; ) ; :::; U J (y p J S; )g, by assumption. This is a contradiction. Corresponding to the multiple price-changes, one can attempt to de ne the resulting change in average consumer surplus via the line-integral CS = Z JX q j (p; y) dp j, L j= where L denotes a path from p to p. The negative sign stems from the fact that rise in price leads to a loss in consumer surplus. In the set-up described above, we will establish two results, as follows.. For arbitrary price changes from p j to p j for j = ; :::; J, allowing for the possibility that some price changes are negative, some positive and some zero, individual level compensating and equivalent variations, and consequently average EV and CV, are path independent but average consumer surplus is not.. For arbitrary price changes from p j to p j for j = ; :::; J, the marginal distributions of individual-level EV and CV are point-identi ed and closed-form expressions for their CDFs can be obtained as functionals of the structural choice probabilities.

6 . Path Independence Results It is well known that for continuous choice with multiple prices changing simultaneously, the Marshallian Consumer Surplus is generically path-dependent but the Hicksian CV and EV are not (c.f., Tirole, 9, page ). Our rst result establishes that the same conclusion is also true for discrete choice. Theorem Consider the multinomial choice set-up with J alternatives. Consider a price change from p (p ; p ; :::; p J ) to p (p ; p ; :::; p J ). Assume without loss of generality that p J p J p J ; p J ; ::: p p. Under assumption, the individual level compensating and equivalent variations are path independent. (Proof in Appendix) Marshallian Consumer Surplus: It can be shown that for discrete choice with multiple price changes, the average CS is path dependent. In the appendix, we demonstrate path dependence of the consumer surplus for the case with alternatives (J = ). Thus it is no longer true that the average CS is identical to the average EV, as found in DB for the case of a ceteris paribus price increase of a single alternative.. Welfare Distributions for Multiple Price-Changes We begin by noting that one cannot iterate the single price change result of DB to obtain the welfare distribution for simultaneous changes in the prices of multiple alternatives. To see this, consider a choice among three alternatives (J = ), and suppose that price of alternative changes from p to p, and that of from p to p, and price of is unchanged at p. Suppose we try to calculate the overall CV by starting with, say, price change of alternative, followed by from p to p (the order in which we do this does not matter, by path independence). Suppose the CVs 6

7 corresponding to the two price changes are denoted by S and S respectively. Then max fu (y p ; ) ; U (y p ; ) ; U (y p ; )g = max fu (y + S p ; ) ; U (y + S p ; ) ; U (y + S p ; )g 9 + S + S p >< {z } ; A ; + S + S p {z } ; A ; >= =S, overall CV S = max. + S + S {z } S p ; A Then using theorem of DB, we can get the marginal distribution of S but we cannot get the marginal distribution of S + S because the price of both alternative and have changed between lines and of the previous display and so theorem of DB does not apply. Secondly, because we cannot calculate the value of the CV S for an individual (we can only calculate its distribution across all individuals), we cannot apply theorem of DB to calculate even the marginal distribution of S, since the income at which we could potentially apply the theorem depends on S which is unknown, unlike y which is a xed known constant. Thus, a new result is required for welfare analysis corresponding to simultaneous changes in multiple prices, and it is given by the following theorem. >; Theorem Consider the multinomial choice set-up with J alternatives. Consider a price change from p (p ; p ; :::; p J ) to p (p ; p ; :::; p J ). Assume without loss of generality that p J p J p J ; p J ; ::: p p. Under assumption the marginal distribution of individual EV evaluated at income y is given by = Pr S EV a, if a < p >< p, P j k= q k (p ; :::; p j ; p j+; + a; :::; p J + a; y), if a [p j p j ; p j+; p j+; ), j J, (4), if a p J p J, while that of individual CV is given by = Pr S CV a >< P j k= q p ; :::; p j ; p j+; + a; :::; p J + a; y + a, if a < p p, A, if p j p j a < p j+; p j+;, j J, (), if a p J p J, 7

8 Proof in appendix. For example, in the three alternative case, the above results specialize to Pr S EV a >< = while for CV it is given by Pr S CV a >< =, if a < p p, q (p ; p + a; p + a; y), if p p a < p p, 4 q (p ; p ; p + a; y) +q (p ; p ; p + a; y), if p p a < p p,, if a p p,, if a < p p, q (p ; p + a; p + a; y + a), if p p a < p p, 4 q (p ; p ; p + a; y + a) +q (p ; p ; p + a; y + a), if p p a < p p,, if a p p. (6) (7) The following discussion applies to the general result in theorem. But it is easier to interpret them in the context of the -alternative illustration above. First, note that the distributional results cover positive, zero and negative price-changes. For example, in the -alternative case, if p p = p p < p p, then (6) and (7) can be rewritten as (letting p = p = p ) Pr S EV a = Pr S CV a = >< ><, if a < p p, q (p a; p ; p ; y a), if p p a <, q (p ; p ; p + a; y) + q (p ; p ; p + a; y), if a < p p,, if a p p., if a < p p, q (p a; p ; p ; y), if p p a <, 4 q (p ; p ; p + a; y + a) +q (p ; p ; p + a; y + a), if a < p p,, if a p p.. ()

9 This is because adding the same constant to all price arguments and income does not change the choice probability, e.g., q (p ; p + a; p + a; y) = Pr [U (y p ; ) max fu (y p a; ) ; U (y p a; )g] = Pr [U (y a (p a) ; ) max fu (y a p ; ) ; U (y a p ; )g] = q (p a; p ; p ; y a), Second, if there is an outside option j, with zero price before and after the price change, then that possibility is already incorporated in the above set-up simply set p j = p j = in the above speci cation. The EV result can be understood intuitively as follows. First note that lowering income at the initial situation by an amount smaller than p p will make everyone s utility from consuming any of the three alternatives at the nal price and initial income lower, relative to the utilities of consuming them at the initial price and reduced income. Therefore, no one will have an EV less than p p. Now consider those individuals whose compensation is at most a where a satis es p p a < p p. Since y a p is strictly larger than y p, no such individual s choice in the nal situation can be alternative and, by the same logic, alternative. So their reference utility must be U (y p ; ). If the EV is a, then taking out income a in the initial situation results in indirect utility exactly equal to the reference utility. And thus EV is no greater than a means that if we take out an amount greater than a at the low price initial situation, then the resulting indirect utility will fall short of the reference utility level, viz., U (y p ; ). Therefore, Pr S EV a < = Pr 4U (y p ; ) max : 9 U (y p a; ) ; =. U (y p a; ) ; U (y p a; ) ; But because a p p, the compensated utility of consuming alternative at income lowered by a cannot be the indirect utility because the reference utility must exceed it. Therefore, Pr S EV a = Pr [U (y p ; ) max fu (y p a; ) ; U (y p a; )g]. The case where compensation is at most a with p p a < p p, the reference utility by a similar logic is either U (y p ; ) or U (y p ; ). If the former, then we must have that U (y p ; ) > U (y p ; ) and take out an amount greater than a at the low price 9

10 initial situation, will make the resulting indirect utility fall short of the reference utility level, viz., U (y p ; ). Thus Pr S EV a < = Pr 4U (y p ; ) max : U (y p ; ) ; U (y p a; ) ; U (y p a; ) ; U (y p a; ) = Pr [U (y p ; ) max fu (y p ; ) ; U (y p a; )g] = q (p ; p ; p + a; y). 9 = ; Interchanging the roles of U (y p ; ) and U (y p ; ) yields q (p ; p ; p + a; y). Similarly, the CV result can be understood as follows. The smallest price rise is p p and so a compensation of less than p p at the post-change situation will reduce the utility of consuming each alternative relative to the utility of consuming it in the original situation and therefore cannot restore anyone s utility to the original level. Hence the probability mass is zero below p p. Next, consider p p a < p p. Suppose the price of alternatives and increase by a and we compensate an individual by giving him an extra a of money. This will enable him to reach the original utility level of consuming alternatives or. Suppose the individual still chooses to buy alternative. This means that his utility from buying at higher price with higher income y + a exceeds his original utilities from buying alternatives and. Since p a p, his utility of buying at price p with income y + a (weakly) also exceeds his original utility from buying at the lower price. Thus his utility of buying at price p with income y + a exceeds his utilities from buying either of the three alternatives at income y and the original price. Such an individual can therefore be made as happy as in the original situation by paying a compensation less than a. Thus Pr S CV a equals the probability of buying alternative at price p and income y + a when the prices of the alternatives and are p + a and p + a, respectively. Finally, consider the case p p a < p p. Now suppose the price of alternative goes up to p + a and we compensate an individual by raising his income by a. He can therefore consume alternative and enjoy the same utility as in the original situation. Suppose he still prefers not to buy. Then his utility from buying either or at higher income y + a and price p or p must exceed his utility from buying in the original situation. Since p p p p a, his utility in the eventual situation also exceeds that from buying or in the original situation. Such a person can therefore be made equally well-o by paying a compensation (weakly) lower than a. That is, Pr S CV a equals the probability of not buying alternative at prices (p ; p ; p + a) and income y + a.

11 Finally, note that in the above results, the CDFs are non-decreasing because of assumption. For instance, for the EV, we need that q (p ; p + a; p + a; y) q p ; p ; p + a ; y (9) whenever a < p p a. This holds because U (y p ; ) max fu (y p a; ) ; U (y p a; )g ) U (y p ; ) max fu (y p ; ) ; U (y p a; )g ) U (y p ; ) max U (y p ; ) ; U y p a ;, since y p a y p and y p a y p a, respectively. Consequently, the C.D.F. of EV in (6) is non-decreasing. For the CV, we require q (p ; p + a; p + a; y + a) q p ; p + a ; p + a ; y + a () whenever p p a < a. This is true because the LHS is the probability of the event U (y + a p ; ) max fu (y p ; ) ; U (y p ; )g ) U y + a p ; max fu (y p ; ) ; U (y p ; )g, since a > a, 9 () U y + a p ; < U (y + a (p + a ) ; ) ; = max : U (y + a (p + a ) ; ) ;, whose probability is the RHS of the previous display. Note that inequalities (9) and () can be interpreted as Slutsky/Revealed Preference inequalities for continuous choice. Introduction of New Alternative Consider a setting of multinomial choice among exclusive alternatives f; :::; Jg. Suppose a new alternative J + is introduced subsequently into the choice set, which expands the choice alternatives facing each consumer and may also a ect the prices of other, pre-existing alternatives. Assume, as in Hausman (), that we have data on a cross-section of individual choices in the nal, i.e., post-introduction situation. We wish to evaluate the impact of this new alternative on individual welfare. Toward that end, consider an individual with initial income y and unobserved heterogeneity whose utility from consuming alternative j at price p j is given by U j (y p j ; ). The problem is to nd the distribution of welfare arising from introduction of the new alternative J + as varies

12 across individuals. Suppose from an initial price vector (p ; :::; p J ), the eventual price vector becomes (p ; :::; p J ; p J+ ), with p J+ denoting the price of the new alternative. De ne the EV as the income compensation in the initial situation necessary to attain the eventual utility. Formally, EV is the solution S to the equation max fu (y p ; ) ; :::; U J (y p J ; ) ; U J+ (y p J+ ; )g = max fu (y + S p ; ) ; :::; U J (y + S p J ; )g. Analogously, the CV is de ned as the income reduction in the nal situation necessary to attain the initial utility, i.e., max fu (y S p ; ) ; :::; U J (y S p J ; ) ; U J+ (y S p J+ ; )g = max fu (y p ; ) ; :::; U J (y p J ; )g. De ne the eventual (i.e., latter period) choice probabilities, for k = ; :::; J +, as q k (p ; :::; p J+ ; y) = Pr U k (y p k ; ) max j (y jf;:::;j+gnfkg p j ; ). () Finally, for later use, de ne the constant a (p ; :::; p J ; p J+ ; y) as the smallest a satisfying q ; :::; p {z J ; p } J+ + a; y + aa =, () p and the constant a (p ; :::; p J ; p J+ ; y) as the smallest a satisfying Recall that q J+ q J+ (p ; p J+ + a; y + a) = ; :::; p {z J ; p } J+ + a; ya =. () p U J+ (y p J+ ; ) max U j (y + a p j ; ) jf;:::;jg q J+ (p ; p J+ + a; y) = Pr U J+ (y p J+ a; ) max j (y jf;:::;jg p j ; ) Under strict monotonicity of the U j (; ) functions, it follows that q J+ (p ; p J+ + a; y + a) and q J+ (p ; p J+ + a; y) are non-increasing in a, so that a (p ; p J+ ; y) and a (p ; p J+ ; y) are wellde ned. At a =, the RHS of both displays above give the probability of choosing the new alternative with the prices of all other alternatives xed at their original level. We will assume that p J+ is low enough that this probability is strictly positive. That is, if the price of the other options do not change, then the price p J+ of the new alternative that we consider for calculating.

13 welfare e ects is one where there is positive demand for the new product. This is intuitive because producers have no incentive to sell the product at a price where its demand is zero. This also means that a and a are strictly positive. Continue to assume that p J; p J; ::: p p. We now state the theorem describing the marginal distribution of EV and CV resulting from introduction of the new alternative. Theorem Assume that for each j = ; :::; J +, U j (; ) is continuous and strictly increasing. Let q k (; :::; ; y) be as de ned in () and suppose that p J+ is small enough that q J+ (p ; p J+ ; y) > so that a and a, as de ned in () and (), are both strictly positive. Then = Pr (EV a) >< P j k= k p ; :::; p j ; p j+; + a; :::; p J + a; p J+ + a; y + a, if a < p p, A, if p j p j a < p j+; p j+;, j < J, q J+ (p ; :::; p J ; p J+ + a; y + a), if p J p J a. On the other hand, = Pr (CV a) >< P j k= k p ; :::; p j ; p j+; + a; :::; p J + a; p J+ + a; y, if a < p p, A, if p j p j a < p j+; p j+;, j < J, q J+ (p ; :::; p J ; p J+ + a; y), if p J p J a. When a p J p J p J. p J, the CDF at a for EV (resp., CV) will be if a (resp., a), is smaller than Note that in order to calculate the probabilities appearing in theorem, we need to observe adequate cross-sectional variation in the price of the new alternative, over and above those of the existing alternatives and income, in the post-change period. Corollary If introduction of the new alternative has no e ect on prices of the other alternatives, then p j = p j for all j = ; :::; J, and the above results simplify to, if a <, >< Pr (EV a) = q J+ (p ; :::; p J ; p J+ + a; y + a), if a < a,, if a a. (4)

14 >< Pr (CV a) =, if a <, q J+ (p ; :::; p J ; p J+ + a; y), if a < a,, if a a. () These corollaries have clear intuitive interpretation. First, take the result (). Consider an individual with income y and tastes for whom the reservation price for the new alternative is p J+ + r J+ (p ; p J+ ; y; ), meaning that U J+ y p J+ r J+ (p ; p J+ ; y; ) = max fu (y p ; ) ; :::; U J (y p J ; )g. Thus, from a situation where he has access to the new good, faces prices (p ; p J+ ) and has income y, if we reduce his income by rj+ (p ; p J+ ; y; ), then he will enjoy the same utility as he was enjoying before introduction of the new good. Thus his CV is rj+ (p ; p J+ ; y; ). Therefore, the CDF of CV at a equals the fraction of people whose reservation price for the new good is no larger than p J+ + a. This fraction is equal to the fraction of people not buying the new good at price p J+ + a and is thus given by q J+ (p ; p J+ + a; y). Next, consider the result (4). Recall that EV is de ned as the solution S to max fu (y p ; ) ; :::; U J (y p J ; ) ; U J+ (y p J+ ; )g = max fu (y + S p ; ) ; :::; U J (y + S p J ; )g. Observe that any individual who does not consume the new product in the post-change situation su ers no welfare change relative to the initial situation when the product was unavailable. Therefore, we only need to consider those individuals who consume the new product in the post-change situation. So the EV is positive and less than a for those individuals who, with the additional income a, would enjoy a higher utility than what they are currently getting from consuming the new alternative. Since buying the new alternative at price p J+ with income y yields the same utility as buying it at price p J+ + a with income y + a, the probability of EV being less than a equals the probability of not buying the new alternative at price p J+ + a when income is y + a. Corollary If introduction of the new alternative has no e ect on prices of the other alternatives, then the average values of individual welfare change are given by E (EV ) = E(CV ) = Z a Z a q J+ (p ; :::; p J ; p J+ + a; y + a) da, q J+ (p ; :::; p J ; p J+ + a; y). 4

15 Remark Note that p J+ + a may be interpreted as the "virtual" price of the new alternative in the pre-introduction period, and correspondingly a can be interpreted as the virtual price change for the J + th alternative. The concept of the "virtual price" was rst discussed in Rothbarth, and later used by Hausman,, as the price that sets compensated demand for the new product equal to zero. Hausman, discussed measurement of the EV resulting from introduction of a new good using the idea of a virtual price, but did not incorporate unobserved preference heterogeneity directly in his analysis. His expressions are therefore di erent from those stated above. Remark A popular "intuition" regarding the virtual price is that we can perform demand analysis "as if" the new alternative was available originally at the virtual price. It is important to note that for the purpose of welfare analysis, this intuition is only partly correct. For instance, in the expressions for the CV distribution in theorem, the change in (virtual) price from p J+ + a to p J+ plays a very di erent role from changes in the true prices of pre-existing alternatives, viz., p p ; :::p J p J. In particular, the CDF is not a ected by where a lies within p p ::: p J p J, except determining whether the CDF reaches at p J p J. An intuitive explanation of this di erence is as follows. The CDF of the CV is the fraction of individuals whose welfare gain from introduction of the new alternative is small enough, i.e., whose indirect utility in the original situation was large enough. This indirect utility could not have been be attained for any from consumption of alternative J +, since it was unavailable in the original situation, unlike alternatives ; :::; J, any one of which could have provided the original indirect utility with positive probability. Non-exclusive Discrete Choice We now change the set-up described above to allow for non-exclusive choice. Accordingly, assume that there are two binary choices which are non-exclusive among themselves. For example, suppose choice for a household is whether to subscribe to a sports package o ered by a cable TV network which costs P and choice is whether to subscribe to a movie package which costs P. A household then has four exclusive options fg ; fg ; f; g ; none with respective utilities U (Y P ; ), U (Y P ; ), U (Y P P ; ) and U (Y; ), respectively. Consider the CV corresponding to a rise in the price of the sports package from p to p with the price of the movie package xed at p. The CV evaluated at income Y = y is the solution S to

16 the equation < max : < = max : U (y + S; ) ; U (y + S p ; ) ; U (y + S p ; ) ; U (y + S p p ; ) ; 9 U (y; ) ; U (y p ; ) ; = U (y p ; ) ; U (y p p ; ) ;. (6) 9 = The following proposition (proof in appendix) establishes that the marginal distribution of CV corresponding to a single price change is point-identi ed. The intuition for this result is as follows. Suppose the price P changes from p to p with the price P xed at p. Group option fg and ( \ ) together and options fg and fg together. De ne " def = (p ; ) V A (y p ; ") def = max fu (y p ; ) ; U (y p p ; )g, V B (y; ") def = max fu (y; ) ; U (y p ; )g. Correspondingly (6) becomes max fv A (y + S p ; ") ; V B (y + S; ")g = max fv A (y p ; ") ; V B (y; ")g. (7) Assumption implies that V A (; ") and V B (; ") are continuous and strictly increasing for each ". Now we can apply theorem of DB for binary choice to this problem and get the marginal distribution of the compensating variation. For example, for a < p gives p, theorem of DB Pr S CV a = Pr [V B (y + a; ") V A (y + a (p + a) ; ")] max fu (y + a; ) ; U (y + a p ; )g 9 = Pr 6 < U 4 (y + a (p + a) ; ) ; = 7 max : U (y + a (p + a) p ; ) ; = q (p + a; p ; y + a) + q (p + a; p ; y + a). The key property enabling us to write (6) as the binary choice CV (7) is that P is being held xed at p ; if P also varied across individuals, then the distribution of " would vary beyond the variation of and the binary formulation would no longer be applicable, as we will see below. We summarize the above discussion for a single price-change into a formal proposition. 6

17 Proposition Under assumption, the C.D.F. of CV for a single price change of P from p to p with the price P xed at p, is given by Proof in appendix. Pr S CV a >< =, if a <, 4 q (p + a; p ; y + a) +q (p + a; p ; y + a), if a p p., if a < p p, Now consider a simultaneous rise in p and p from (p ; p ) to (p ; p ). Assume that < p p < p p. Then the CV is de ned via 9 < U (y + S; ) ; U (y + S p ; ) ; = max : U (y + S p ; ) ; U (y + S p p ; ) ; Now, = max fu (y; ) ; U (y p ; ) ; U (y p ; ) ; U (y p p ; )g. () Pr [S = p p ] >< = Pr 6 4 U (y p ; ) max < = Pr 4U (y p ; ) max : 9 U (y; ) ; U (y p ; ) ; U (y p p ; ) ; >= U (y + p p ; ) ; U (y + p p p ; ) ; 7 U (y p p ; ) >; 9 U (y p ; ) ; U (y p p ; ) ; =. (9) U (y + p p ; ) ; In order for this to be expressible as a choice probability, we need a combination of alternative speci c prices p ; p and income y such that y = y + p p y p = y p y p = y p y p p = y p p. But adding the rst and fourth equation in the previous display we get y p p = y + p p p () 7

18 whereas adding the nd and rd equation gives y p p = y p p, () Now, equations () and () can hold simultaneously if and only if p = p which reduces to the single price change case. Therefore, if both price changes are positive, we cannot express (9) as a choice probability and therefore we cannot obtain the probability that the CV is p p. But for any generic probability distribution on, the probability in equation (9) will be positive. We summarize this conclusion as a proposition. Proposition In a discrete choice setting with non-exclusive choices, the marginal distribution of the CV resulting from simultaneous change in the prices of multiple alternatives is not pointidenti ed. One can nonetheless bound the probability in (9). In particular, an upper bound can be obtained via 9 U >< (y p ; ) ; U (y p p ; ) ; >= Pr 6 4 U (y p ; ) max 7 + p p ; A {z } >; Pr [U (y p ; ) max fu (y p ; ) ; U (y p p ; ) ; U (y; )g] = q (p ; p ; y), whereas a lower bound is given by 9 < U (y p ; ) ; U (y p p ; ) ; = Pr 4U (y p ; ) max : U (y + p p ; ) ; >< + p p p Pr 6 4 U (y p ; ) max {z } ; A ; U (y p p ; ) ; U (y + p p ; ) U (y + p p (p + p p ) ; ) 9 U = Pr >< (y + p p p ; ) ; >= 6 max U 4 (y + p p (p + p p ) p ; ) ; 7 U (y + p p ; ) >; = q (p + p p ; p ; y + p p ). 9 >= 7 >;

19 The key conclusion from the above discussion is that in a discrete choice setting when alternatives are non-exclusive, the distributions of individual welfare change for a single ceteris paribus price rise are identi ed, but for simultaneous increase in several alternatives prices, they are not. This is in contrast to the multinomial case with exclusive alternatives where welfare distributions are identi ed for both single (shown in DB) and multiple (shown above) price changes. An intuition for the result is that in the non-exclusive case, there is a systematic relationship between the prices of the di erent (composite) options, so that in a dataset we can never observe independent price variations across the various options, no matter how many price-income combinations are observed in the data. This restriction in observable variation leads to the failure of point-identi cation. 4 Implementation The identi cation results reported above are fully nonparametric in that no functional form assumptions are required to derive them. When implementing these results in practical applications, one can therefore estimate conditional choice probabilities nonparametrically, e.g., using kernel or series regressions and use these estimates to calculate welfare distributions. For instance, recall the -alternative case, leading to (), where we have, say, p p < = p p < p p. Now, for any random variable X with CDF F () and nite mean, the expectation satis es E (X) = thus the mean CV from () is given by Z Z F (x) dx + p p q (p a; p ; p ; y) da + Z Z p p ( F (x)) dx; q (p ; p ; p + a; y + a) da, () where q j (p ; p ; p ; y) denotes the choice probability of alternative j when the price of the three alternatives are (p ; p ; p ) and income is y. If the dataset is of modest size, so that kernel regressions are imprecise, then one can alternatively use a parametric approximation to the choice probabilities. At the end of the appendix, we provide illustrative calculations for a parametric speci cation of choice-probabilities, viz., McFadden and Train s mixed logit in the three alternative case given by eqn. () above. 4. Some Practical Issues We now discuss some practical data problems which may arise in the context of implementing the above results on real datasets. 9

20 Endogeneity: In some applications, observed income may be endogenous with respect to individual choice. An obvious example is when omitted variables, such as unrecorded education level, can both determine individual choice and be correlated with income. As noted in DB (c.f., Remark, page 6), under endogeneity, one can de ne welfare distributions either unconditionally, or conditionally on income. In either case, the observed choice probabilities can potentially di er from the structural choice probabilities which appear in the welfare distributions, and controlfunction type methods are required to estimate them. An interesting point, not previously noted, is that the distribution of income-conditioned CV is a ected by endogeneity, whereas that of the EV is not. To see why, recall the three alternative case discussed above, and de ne the conditionalon-income structural choice probabilities as qj c p ; p ; p ; y ; y Z = U j y p j ; max U k y p k ; df (jy), k6=j where F (jy) denotes the distribution of the unobserved heterogeneity for individuals whose current income is y. Now, for a real number a, satisfying p p a < p p, it is easy to see that similar to equations (6) and (7), the distributions of EV at a, evaluated at income y, conditional on current income being y, are given by Pr S EV ajinc = y = q c (p ; p + a; p + a; y; y), while for CV it is given by Pr S CV ajinc = y = q c (p ; p + a; p + a; y + a; y), Now, q c (p ; p + a; p + a; y; y), by de nition, is the fraction of individuals currently at income y who would choose alternative at prices (p ; p + a; p + a), had their income been y. But this is directly observable in the data, and therefore, no corrections are required owing to endogeneity. However, q c (p ; p + a; p + a; y + a; y) is the fraction of individuals currently at income y who would choose alternative at prices (p ; p + a; p + a), had their income been y+a. This fraction is counterfactual and not directly estimable because the distribution of is likely to be di erent across people with income y +a relative to those with income y, due to endogeneity. To summarize, if the objective of welfare analysis is to predict the EV distribution resulting from price changes at the current income, then endogeneity of income is irrelevant to the analysis. This suggests that if exogeneity of income is suspect and no obvious instrument or control function is available, then a researcher can still perform meaningful welfare analysis based on the EV distribution at current income.

21 Market-level data: In some situations, researchers may have access only to market level aggregate data. If they also know the marginal distribution of incomes within each market, e.g., from Census extracts, then the two datasets may be combined, together with a parametric speci cation of choice probabilities, to yield useful moment conditions. Speci cally, suppose that there are M markets indexed by m. In market m, we observe the J price vector p m for the J alternatives and the market share s mj of each alternative. Suppose we also have individual income data y m ; :::; y mnm for the mth market, from the census, for each m = ; :::; M. Then we have JM equations in dim unknowns, given by s mj = n m n m X i= q j (p m ; y mi ; ), j = ; :::; J, m = ; :::; M, where denotes the parameters entering a parametric speci cation of q j (p; y). For example, for the mixed logit model, (; ) will denote the mean and covariance matrix of the random coe cients on the J prices and income, leading to dim = J + + (J+)(J+) = (J+)(J+4), which needs to be smaller than M. Once these equations are solved to yield estimates of, one can plug the ^s into () and (6) to estimate mean and quantiles of individual welfare e ects at a hypothetical income y. Interval-Income: In many consumer datasets, including scanner-data such as Nielsen s, individual income is recorded in intervals, e.g. K-K, K-4K etc., in order to produce higher response rates. Then the researcher observes not the true income Y but lower and upper limits y l and y u s.t. y l Y y u w.p.. Since income is a key ingredient of the choice probability, fundamental modi cations are required for calculating welfare distributions when income is intervalmeasured. Speci cally, consider binary choice (i.e., J = in section above with one outside good alternative with zero price change), and let P denote the price of alternative. If the good in question is normal, then the conditional choice probability of alternative, q (p; ) would be increasing for each p. Then (p; y l ; y u ) def Pr (choose jp = p; Y [y l ; y u ]) [q (p; y {z } l ) ; q (p; y u )]. observed For a parametric approximation, e.g., probit, q (p; y) = + p p + I y, the above inequalities would represent a set of inequalities in, given by + p p + I y l p; y l ; y u + p p + I y u, where ; y l ; y u can be nonparametrically estimated using individuals whose income lies between a speci c pair (y l ; y u ). Strict monotonicity of () permits us to express the above inequalities as

22 linear restrictions on the s. Accordingly, for each combination the set of s satisfying + p p + I y l p; y l ; y u, + p p + I y u p; y l ; y u. p; y l ; y u in the data, de ne B as Now consider the CV resulting from a price increase of alternative from p to p at income y. Applying () with J = and p = = p, we have that, if a <, >< Pr fcv ag = + p (p + a) + I (y + a), if a < p p,, if a p p, Then the upper and lower bounds on the CDF of the CV at a with a < p obtained by solving the linear programs () p can be max = min + p (p + a) + I (y + a), s.t. B [ p + I, where p + I comes from the monotonicity/slutsky restrictions discussed above just before section (). Plugging in these min and max value functions into eqn. () yields bounds on the distribution function of CV. An Illustration We now illustrate the above methods by estimating average welfare e ects in the context of children s school attendance in India. Our data come from a large-scale household survey, conducted by the Indian National Sample Survey Organization of India in 4- (6st round, schedule ), which focuses on employment and unemployment. Our subsample of interest consists of approximately households from across India, which have a single teenage child of school-going age (- years) and belong to one of the two dominant religious groups Hindu and Muslim. Each child faces three alternatives attend school (option ), work for a wage (option ) or neither (option ). We drop the two states of Kerala and Himachal Pradesh which typically score very high relative to the rest of India on human development indicators including children s education. For income y, we use the monthly per capita household expenditure. For potential price faced by a household It is worth noting that this problem is di erent from Manski and Tamer () whose object of interest was the set of s satisfying a set of inequality constraints (analogous to the set B here). In contrast, here the objects of interest are the min or max of known linear functionals of the s when s come from the identi able set B.

23 related to school-attendance, we use the median annual educational expense (divided by ) per school-going child across all households in the same income stratum in the village or urban block where the household resides. 4 These expenses comprise the annual school fees and study-related spending (e.g., books, private tuition etc.), which are recorded as responses to a survey question. The reason for using the median expense is that some households might spend excessively large amounts on educational resources, which does not represent the true "price" envisaged by a typical household making the choice between sending and not sending their child to school. The price of working is taken to be the negative of the average wage for children aged - in the village or urban block where the household resides. the price of staying at home is normalized to zero. All money amounts are expressed as Indian rupees per month, with Indian rupee=. US dollars in 4. Summary statistics are presented in Table. In table, we show results t-values from a multinomial logit regression of school-attendance and working on price and income and a set of household characteristics, using home-stay as the base outcome. For choice of schooling one gets a positive income e ect, negative price and household size e ect; female children are disadvantaged, as are Muslims and the disadvantaged castes relative to other Hindus. The overall t is relatively high, with McFadden s R_squared about %. We consider two policy changes. The rst is a simultaneous price change for schooling and working induced by a tuition subsidy and a potential general equilibrium increase in wages. The second is elimination of the work option with no induced change on school fees, as would result from enforcement of a child labor legislation. Policy Change : For the rst policy exercise, we consider a change from initial price of schooling of 4 and initial wage to a nal school-price of 76 and wage with the price of staying at home (i.e., the outside option) constant at. These change in prices correspond to a fall from the 7th to the th percentile for schooling and a rise from the th to the 7th percentile for child wage. We calculate the average welfare gain for this price change at the median monthly expenditure (a typical proxy for income in developing countries) y = 4 separately by 4 An advantage of the Indian NSS is that it divides each village and urban block into two income strata (labelled "second stage stratum" in the survey documentation) and samples households from both strata. In forming our measure of potential price faced by a household, we use median school expenses across households in the same income strata and the same village as the reference household. In cases where no household from a village and a speci c income stratum sends any child to school, we use the median educational expenditure in the entire village as the potential price.

24 gender for Hindus at age, household size and adult literacy rate 6%. To use our formuale for welfare change, we need to estimate the choice probabilities for schooling and work at these covariate values. We model these choice probabilities via quadratics in prices and income. This formulation should be interpreted as a sieve approximation to the underlying nonparametric choice probabilities. The alternative to the above calculations is the Small and Rosen (9) approach which ignores income-e ects, assumes Type extreme valued errors and leads to average welfare calculations via the popular "log-sum" formula, c.f., Train, 6, page. Details of calculations are provided in the appendix. We use the term "income" throughout the rest of this section to mean monthly per capita expenditure. Policy Change : The second policy change we consider is the hypothetical removal of the "work" option without any change in school tuition with the welfare being evaluated at a speci c wage. The corresponding analysis can be performed using corollary above by rst considering the reverse policy of adding the work option at a particular price and applying theorem, and then interchanging the labels of EV and CV. It is important to stress that here welfare loss refers to the immediate loss in private utilities, ignoring any external social impact of eliminating child labor. Accordingly, consider a child with characteristics given above, family income y = 4 and school-expenditure (all being at their median value in the sample) who initially could not work but then is o ered the option of working at wage. The question is what would be the average CV/EV resulting from this option. The required welfare loss measured by the EV/CV from the elimination of the work option would simply be the CV/EV gains, respectively, of introducing the work option. Using corollary, the former quantities are measured by integrating the adjusted probability of working E (EV ) = E(CV ) = Z Z q ( q ( + a; ; 4 + a) da, + a; ; 4) da. Results: In Table, we present results on mean welfare e ects of the price changes at median income, and in Figure, we plot the estimated mean welfare change at various values of income for female children. Three interesting features of the CV and EV graphs are evident from the gure: (a) welfare e ects rise with income, because at relatively higher levels of income, a larger proportion of children attend schools and gain from the fall in tuition, with the mean EV/CV reaching the entire tuition subsidy of 74 at the highest income levels, (b) the log-sum based estimates of welfare gains, do not vary with income due to assumed lack of income e ects, and are systematically di erent from the EV/CV estimates, and (c) the EV estimates are lower than CV 4