Karim Kilani - Academia.edu (original) (raw)

Papers by Karim Kilani

Luce’s Independence from Irrelevant Alternatives (IIA) choice axiom states that the ratio of choi... more Luce’s Independence from Irrelevant Alternatives (IIA) choice axiom states that the ratio of choice probabilities of two given alternatives out of any set of alternatives remains unchanged. The complete decomposition axiom arises as a natural extension of the IIA axiom when the axiomatization is extended from the simple best choice to ranking probabilities. This axiom states that the event that a ranking of some alternatives occur is independent from the fact that another alternative will precede all those alternatives. In this case, the choice probabilities also have the IIA property (logit form) and the ranking probabilities can be expressed as a product of logit choice probabilities. This paper deals with the axiomatization of ranking probabilities. We consider instead of the "complete" IIA and decomposition axioms weaker conditions: "partial" IIA and decomposition axioms. We also consider the "reverse" partial IIA and partial decomposition axioms, t...

We study the descriptive and the normative consequences of price and/or other attributes changes ... more We study the descriptive and the normative consequences of price and/or other attributes changes in additive random utility models. We …rst derive expressions for the transition choice probabilities associated to these changes. A closed-form formula is obtained for the logit. We then use these expressions to compute the cumulative distribution functions of the compensating variation conditional on ex-ante and/or ex-post choices. The unconditional distribution is also provided. The conditional moments of the compensating variation are obtained as a one-dimensional integral of the transition choice probabilities. This framework allows us to derive a stochastic version of Shephard's lemma, which relates the expected conditional compensating variation and the transition choice probabilities. We compute the compensating variation for a simple binary linear in income choice model and show that the information on the transitions leads to better estimates of the compensating variation than those obtained when only ex-ante or ex-post information on individual choices is observed. For the additive in income logit, we compute the conditional distribution of compensating variation, which generalizes the logsum formula. Finally, we derive a new welfare formula for the disaggregated version of the representative consumer CES model.

We show that the number of individuals selecting their worst alternatives within a …nite set of a... more We show that the number of individuals selecting their worst alternatives within a …nite set of alternatives can be written as an alternating sum of the number of individuals having their best choice within subset of alternatives. The identities are then applied to random utility models, including the multinomial logit model, the mixed logit model and the disaggregated version of the CES representative consumer model. Finally, we show that better estimates are obtained if respondents are asked to reveal their worst instead of their best choices.

The paper considers models for best, worst and best-worst choice probabilities, that use a single... more The paper considers models for best, worst and best-worst choice probabilities, that use a single common set of random utilities. Choice probabilities are derived for two distributions of the random terms: i.i.d. extreme value, i.e. Logit, and multivariate normal, i.e. Probit. In Logit, best, worst and best-worst choice probabilities have a closed form. In Probit, worst choice probabilities are simply obtained from best choice probabilities by changing the sign of the systematic utilities. Strict log-concavity of the likelihood, with respect to the coefficients of the systematic utilities, holds, under a mild necessary and sufficient condition of absence of perfect multicollinearity in the matrix of alternative and individual characteristics, for best, worst and best-worst choice probabilities in Logit, and for best and worst choice probabilities in Probit. The assumption of substitutability between best and worst choices is tested with data on mode choice, collected for the assessm...

Discrete choice models have been used to describe imperfect competition between …rms selling hori... more Discrete choice models have been used to describe imperfect competition between …rms selling horizontally di¤erentiated products. In all theoretical models, the indirect utility function is assumed to be linear in income so that there is no income e¤ect. We consider here a situation in which income enters nonlinearly into the indirect utility function. We propose a correct (hicksian) measure of consumer surplus based on a willingness to pay principle. In order to grantee the existence of a price equilibrium, match values are assumed logconcavilly distributed. Using a correct measure of welfare, we extent the results of Anderson, de Palma and Nesterov to the case where income e¤ects are involved. We proof that under these general assumptions, overentry prevails. Our …ndings, which extend the conventional discrete choice oligopoly approach provide various guidelines for empirical research.

Discrete choice models have been used to describe imperfect competition between firms selling hor... more Discrete choice models have been used to describe imperfect competition between firms selling horizontally differentiated products. In all theoretical models, the indirect utility function is assumed to be linear in income so that there is no income effect. We consider here a situation in which income enters nonlinearly into the indirect utility function. We propose a correct (hicksian) measure of consumer surplus based on a willingness to pay principle. In order to grantee the existence of a price equilibrium, match values are assumed logconcavilly distributed. Using a correct measure of welfare, we extent the results of Anderson, de Palma and Nesterov to the case where income effects are involved. We proof that under these general assumptions, overentry prevails. Our findings, which extend the conventional discrete choice oligopoly approach provide various guidelines for empirical research.

Luce’s Independence from Irrelevant Alternatives (IIA) choice axiom states that the ratio of choi... more Luce’s Independence from Irrelevant Alternatives (IIA) choice axiom states that the ratio of choice probabilities of two given alternatives out of any set of alternatives remains unchanged. The complete decomposition axiom arises as a natural extension of the IIA axiom when the axiomatization is extended from the simple best choice to ranking probabilities. This axiom states that the event that a ranking of some alternatives occur is independent from the fact that another alternative will precede all those alternatives. In this case, the choice probabilities also have the IIA property (logit form) and the ranking probabilities can be expressed as a product of logit choice probabilities. This paper deals with the axiomatization of ranking probabilities. We consider instead of the "complete" IIA and decomposition axioms weaker conditions: "partial" IIA and decomposition axioms. We also consider the "reverse" partial IIA and partial decomposition axioms, t...

We study the descriptive and the normative consequences of price and/or other attributes changes ... more We study the descriptive and the normative consequences of price and/or other attributes changes in additive random utility models. We …rst derive expressions for the transition choice probabilities associated to these changes. A closed-form formula is obtained for the logit. We then use these expressions to compute the cumulative distribution functions of the compensating variation conditional on ex-ante and/or ex-post choices. The unconditional distribution is also provided. The conditional moments of the compensating variation are obtained as a one-dimensional integral of the transition choice probabilities. This framework allows us to derive a stochastic version of Shephard's lemma, which relates the expected conditional compensating variation and the transition choice probabilities. We compute the compensating variation for a simple binary linear in income choice model and show that the information on the transitions leads to better estimates of the compensating variation than those obtained when only ex-ante or ex-post information on individual choices is observed. For the additive in income logit, we compute the conditional distribution of compensating variation, which generalizes the logsum formula. Finally, we derive a new welfare formula for the disaggregated version of the representative consumer CES model.

We show that the number of individuals selecting their worst alternatives within a …nite set of a... more We show that the number of individuals selecting their worst alternatives within a …nite set of alternatives can be written as an alternating sum of the number of individuals having their best choice within subset of alternatives. The identities are then applied to random utility models, including the multinomial logit model, the mixed logit model and the disaggregated version of the CES representative consumer model. Finally, we show that better estimates are obtained if respondents are asked to reveal their worst instead of their best choices.

The paper considers models for best, worst and best-worst choice probabilities, that use a single... more The paper considers models for best, worst and best-worst choice probabilities, that use a single common set of random utilities. Choice probabilities are derived for two distributions of the random terms: i.i.d. extreme value, i.e. Logit, and multivariate normal, i.e. Probit. In Logit, best, worst and best-worst choice probabilities have a closed form. In Probit, worst choice probabilities are simply obtained from best choice probabilities by changing the sign of the systematic utilities. Strict log-concavity of the likelihood, with respect to the coefficients of the systematic utilities, holds, under a mild necessary and sufficient condition of absence of perfect multicollinearity in the matrix of alternative and individual characteristics, for best, worst and best-worst choice probabilities in Logit, and for best and worst choice probabilities in Probit. The assumption of substitutability between best and worst choices is tested with data on mode choice, collected for the assessm...

Discrete choice models have been used to describe imperfect competition between …rms selling hori... more Discrete choice models have been used to describe imperfect competition between …rms selling horizontally di¤erentiated products. In all theoretical models, the indirect utility function is assumed to be linear in income so that there is no income e¤ect. We consider here a situation in which income enters nonlinearly into the indirect utility function. We propose a correct (hicksian) measure of consumer surplus based on a willingness to pay principle. In order to grantee the existence of a price equilibrium, match values are assumed logconcavilly distributed. Using a correct measure of welfare, we extent the results of Anderson, de Palma and Nesterov to the case where income e¤ects are involved. We proof that under these general assumptions, overentry prevails. Our …ndings, which extend the conventional discrete choice oligopoly approach provide various guidelines for empirical research.

Discrete choice models have been used to describe imperfect competition between firms selling hor... more Discrete choice models have been used to describe imperfect competition between firms selling horizontally differentiated products. In all theoretical models, the indirect utility function is assumed to be linear in income so that there is no income effect. We consider here a situation in which income enters nonlinearly into the indirect utility function. We propose a correct (hicksian) measure of consumer surplus based on a willingness to pay principle. In order to grantee the existence of a price equilibrium, match values are assumed logconcavilly distributed. Using a correct measure of welfare, we extent the results of Anderson, de Palma and Nesterov to the case where income effects are involved. We proof that under these general assumptions, overentry prevails. Our findings, which extend the conventional discrete choice oligopoly approach provide various guidelines for empirical research.