Which demand systems can be generated by discrete choice
TL;DR: This work provides a simple necessary and sufficient condition for when a multiproduct demand system can be generated from a discrete choice model with unit demands.
About: This article is published in Journal of Economic Theory.The article was published on 2015-07-01 and is currently open access. It has received 18 citations till now. The article focuses on the topics: Discrete choice.
Summary (2 min read)
Jump to: [1 Introduction] – [2 A preliminary result] – [3 Which demand systems are consistent with discrete] – [3.1 Necessity] – [3.2 Sufficiency] – [4 Applications and extensions] and [5 Conclusion]
1 Introduction
- In markets for durable goods such as cars or refrigerators, each consumer who makes a purchase typically buys one unit of one of the products on offer (or buys nothing).
- With a single product, any downward-sloping aggregate demand function can be generated by a population of unit-demand consumers–the demand function can simply be interpreted as the fraction of consumers who are willing to pay the specified price for their unit.
- The analysis in the present paper sets those contributions in a wider context.
- The main section then derives necessary and sufficient conditions for the total demand function to be consistent with discrete choice, which are then illustrated by way of some applications and extensions.
2 A preliminary result
- (1) Given the demand system q(p), define Q(p) ≡ ∑ni=1 qi(p) to be the total quantity of all products demanded with the price vector p. A result which is useful in the “sufficiency” part of the following analysis, and perhaps of interest in its own right, is the following.
- (Typically, demand is not differentiable at choke prices which make a product’s demand fall to zero.).
- Lemma 1, which is true regardless of whether demand is consistent with discrete choice, implies that the total demand function Q(·) summarises all information about the demands for individual products, which can be recovered from total demand via the procedure (2).4.
3 Which demand systems are consistent with discrete
- The authors wish to understand which restrictions on q(p) are implied if this demand system can be generated by the simplest discrete choice model.
- By “discrete choice model” the authors mean, first, that any individual consumer wishes to buy a single unit of one product (or to buy nothing).
- 5As the authors discuss and illustrate in section 3.3 there are settings where consumers buy one unit of one product if they buy at all, but where (3) is not satsified (e.g., because of search or transactions costs).
- One can just subtract v0 from each vi to return to their set-up with a deterministic outside option of zero.
- The demand for product i, qi(p), is then the measure of consumers who satisfy (3).
3.1 Necessity
- Any demand system arising out of the procedure (3) involves gross substitutes (i.e., crossprice effects are non-negative), since the right-hand side of (3) decreases with pj.
- Consider a two-product demand system where qi(p1, p2) = ai−bipi+cpj .
- (6) Since total demand Q satisfies (4), the following necessary conditions on Q are immediate: Proposition 1 Suppose that the demand system q(p) is consistent with discrete choice.
- Proposition 1(ii) implies results derived in earlier papers.
- In sum, any demand system based on a representative consumer with homothetic preferences is not consistent with discrete choice, due to its behaviour when prices are close to zero.11.
3.2 Sufficiency
- In broad terms, how the necessary conditions outlined in Proposition 1 are also sufficient for the demand system to be consistent with a discrete choice framework.the authors.
- Since the authors consider only non-negative prices, formula (4) for the candidate CDF for underlying valuations is also defined only on the non-negative orthant Rn+.
- One could adjust the argument to make the extended density continuous, if desired.
- Note that any smooth demand system which has no cross-price effects satisfies the conditions of Proposition 2, although the corresponding density g is zero throughout the positive orthant Rn+. Proposition 2 applies to demand systems which are differentiable throughout Rn+, and characterized valid total demand functions in terms of the mixed partial derivatives.
4 Applications and extensions
- The authors now consider some examples and extensions of the discrete choice model, and related examples that do not accord with it.
- For their purposes it suffices that this condition holds for k ≤ n.).
- Suppose that a consumer who investigates product 2 must pay a positive search cost to revisit product 1. system induced by this extended discrete choice model is consistent with another basic discrete choice model in which consumers buy at most one product.
- 20 Specifically, suppose that all consumers have the same demand for a given product, and each consumer has demand xi(pi) if she buys product i with price pi.
5 Conclusion
- Propositions 1 and 2 together show that, assuming that total demand is differentiable and bounded, the necessary and sufficient condition for consistency with the discrete choice model is that all mixed partial derivatives of total demand be non-positive.
- (More fundamentally, without requiring differentiability the condition is that 1−Q exhibits the required properties of a joint CDF.).
- The authors have focused on the basic discrete choice model where each consumer buys one unit of one product, specifically the product with highest (vi − pi), or else nothing.
- The authors have also focused on those situations in which linear prices are used.
- When facing unit-demand consumers, a seller can never benefit from the use of two-part tariffs, nonlinear pricing or bundling, while if the seller faced a single consumer with the same aggregate demand it will usually prefer to use a two-part tariff instead of linear prices.
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TL;DR: In this paper, the authors analyzed a market in which consumers buy horizontally differentiated products without knowing their exact valuations and their shopping strategy is similar to the search strategy of a consumer in a sequential costly search model.
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Cites methods from "Which demand systems can be generat..."
...Alternatively, one could apply the discrete choice modelling, as in Armstrong and Vickers (2015), Armstrong (2016) and Choi et al. (2016), which I use to prove the existence of a symmetric equilibrium in the proof of Proposition 1....
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TL;DR: In this article, the authors present an alternative microeconomic foundation by deriving the same demand system directly from a population of heterogeneous buyers, which can be easily adapted to different demand specifications.
Abstract: We critically assess the representative consumer model that forms the foundation of a well-known class of linear oligopoly demand structures. It is argued that this approach has several limitations. We present an alternative microeconomic foundation by deriving the same demand system directly from a population of heterogeneous buyers. Our approach can be easily adapted to different demand specifications.
6 citations
Posted Content•
TL;DR: It is shown that the optimal policy for this search and discovery problem is fully characterized by tractable reservation values and it is proved that a predetermined index fully specifies the purchase decision of a consumer following the optimal search policy.
Abstract: This paper studies a search problem where a consumer is initially aware of only a few products. At every point in time, the consumer then decides between searching among alternatives he is already aware of and discovering more products. I show that the optimal policy for this search and discovery problem is fully characterized by tractable reservation values. Moreover, I prove that a predetermined index fully specifies the purchase decision of a consumer following the optimal search policy. Finally, a comparison highlights differences to classical random and directed search.
5 citations
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...…of the number of available alternatives, 2Choi et al. (2018) introduced the name and noted that “Our eventual purchase theorem was anticipated by Armstrong and Vickers (2015) and has been independently discovered by Armstrong (2017) and Kleinberg et al. (2017).” and decrease as more products…...
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TL;DR: In this paper , a random subset of traders who buy cocoa from farmers in Sierra Leone illustrates this idea by interpreting treatment-control differences in prices and quantities purchased from farmers through a model of Cournot competition reveals differentiation between traders is low.
Abstract: Interference across competing firms in RCTs can be informative about market structure. An experiment that subsidizes a random subset of traders who buy cocoa from farmers in Sierra Leone illustrates this idea. Interpreting treatment-control differences in prices and quantities purchased from farmers through a model of Cournot competition reveals differentiation between traders is low. Combining this result with quasi-experimental variation in world prices shows that the number of traders competing is 50 percent higher than the number operating in a village. Own-price and cross-price supply elasticities are high. Farmers face a competitive market in this first stage of the value chain. (JEL L13, L14, O13, Q12, Q13)
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TL;DR: In this paper, a model that links the diffusion process of a technological innovation and the dissemination of a social norm is presented, where the market is linked to the process of norm adoption via psychological forces, endogenizing the rates of adoption and abandonment.
Abstract: The paper presents a model that links the diffusion process of a technological innovation and the dissemination of a social norm. Our approach introduces a new dimension to the interaction between markets and social norms beyond the interplay of monetary and non-monetary incentives: the innovation of material goods as a catalyst of norm evolution. We analyze how the dissemination of a social norm may be affected by product innovation, which adds to the variation of products with respect to their level of norm compliance. The market is linked to the process of norm adoption via psychological forces, endogenizing the rates of adoption and abandonment. We derive necessary and sufficient conditions for a) a positive impact of the innovation on the level of norm adoption and b) for multiplicity of norm equilibria. In concluding, we discuss several policy implications.
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References
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Book•
16 Oct 1992
TL;DR: This important study shows that an understanding of product differentiation is crucial to understanding how modern market economies function and that differentiated markets can be analyzed using discrete choice models of consumer behavior.
Abstract: Product differentiation - in quality, packaging, design, color, and style - has an important impact on consumer choice It also provides a rich source of data that has been largely unexplored because there has been no generally accepted way to model the information available This important study shows that an understanding of product differentiation is crucial to understanding how modern market economies function and that differentiated markets can be analyzed using discrete choice models of consumer behavior It provides a valuable synthesis of existing, often highly technical work in both differentiated markets and discrete choice models and extends this work to establish a coherent theoretical underpinning for research in imperfect competition The discrete choice approach provides an ideal framework for describing the demands for differentiated products and can be used for studying most product differentiation models in the literature By introducing extra dimensions of product heterogeneity, the framework also provides richer models of firm location Discrete Choice Theory of Product Differentiation introduces students and researchers to the field, starting at the beginning and moving through to frontier research The first four chapters detail the consumer-theoretic foundations underlying choice probability systems (including an overview of the main models used in the psychological theory of choice), while the next four chapters apply the probabilistic choice approach to oligopoly models of product differentiation, product selection, and location choice The final chapter suggests various extensions of the models presented as well topics for further research
2,280 citations
"Which demand systems can be generat..." refers background or methods in this paper
...This observation is useful in relating Proposition 1 to Theorem 3.1 of Anderson et al. (1992), which was highlighted in the Introduction....
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...Thus it would appear that, with demands by assumption always adding to one, Theorem 3.1 in Anderson et al. (1992) could likewise be stated in terms of demand for a single product rather than all....
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Abstract: I understand the discipline of marketing exists to answer questions such as: "Will housepersons buy more Brand A soap if its perfume content is increased?" Traditional econometric demand analysis provides no answer. Its attention has been concentrated on consumption levels of broad commodity classes (e.g., housing services), examined using aggregate market data, with demand models constructed on the twin pillars of economic rationality and consumer sovereignty. The market researcher has understandably looked elsewhere-to psychology and survey research-for answers to his questions. Realities have forced econometric demand analysts to broaden their perspective. Public intervention in the supply of some commodities, notably in the areas of transportation, energy, and communications, have required economists to recognize the marketing considerations implicit in issues of policy. (The decision of whether to build and how to design a public This paper reviews several recent developments in econometric demand analysis which may be of interest in market research. Econometric models of probabilistic choice, suitable for forecasting choice among existing or new brands, or switching between brands, are surveyed. These models incorporate attribute descriptions of commodities, making them statistical counterparts of the Court-GrilichesLancaster theory of consumer behavior. Particular attention is given to models which yield tree structures of similarities between alternatives. Also reviewed are methods for estimating econometric models of probabilistic choice from "point-ofsale" sample surveys. * Prepared for presentation at the Conference on Interfaces between Marketing and Economics, Graduate School of Management, University of Rochester, April 7, 1978. Research was supported in part by the National Science Foundation through grant SOC75-22657 to the University of California, Berkeley. Portions of this paper were written while the author was an Irving Fisher Visiting Professor of Economics at the Cowles Foundation for Research in Economics, Yale University.
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