Journal ArticleDOI

# Which demand systems can be generated by discrete choice

01 Jul 2015-Journal of Economic Theory (Academic Press)-Vol. 158, pp 293-307
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.

### 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.

Did you find this useful? Give us your feedback

##### Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the authors discuss situations in which consumers search through their options in a deliberate order, in contrast to more familiar models with random search, and show how ordered search can be reformulated as a simpler discrete choice problem without search frictions.
Abstract: The paper discusses situations in which consumers search through their options in a deliberate order, in contrast to more familiar models with random search. Topics include: network effects (consumers may be better off following the same search order as other consumers); the use of price and non-price advertising to direct search; the impact of consumers starting a new search with their previous supplier; the incentive sellers have to merge or co-locate with other sellers; and the incentive a seller can have to raise its own search cost. I also show how ordered search can be reformulated as a simpler discrete choice problem without search frictions.

83 citations

Journal ArticleDOI
TL;DR: This paper provides a thorough exploration of the microeconomic foundations for the multi-variate linear demand function for differentiated products, which is widely used in industrial organization and finds that strict concavity of the quadratic utility function is critical for the demand system to be well defined.

61 citations

Journal ArticleDOI
TL;DR: It is shown that all tractable versions of the model used in practice are (almost) identical and have a mean-variance structure.

30 citations

### Cites background or methods from "Which demand systems can be generat..."

• ...Armstrong and Vickers (2015) generalize Jaffe and Weyl and show, in particular, that linear demand can be consistent with a discrete choice model in which the support of valuations does not have full dimension....

[...]

• ...Prominent examples are Deneckere and Davidson (1985),8 Vives (1985),9 Shaked and Sutton (1990), Bagwell and Ramey (1991), Shaffer (1991), Dobson and Waterson (1997), and Sutton (1997). In, Motta (2004), influential book “Competition Policy: Theory and Practice”, the Levitan and Shubik’s model is used (in particular in chapter 5 on horizontal mergers) to illustrate some properties with a closed-form model. Among the articles relying on QQUM, there is a literature on comparing prices, quantities, profits, welfare, between Bertrand and Cournot competition. Levitan and Shubik themselves have compared prices when all goods are substitutes, see Levitan and Shubik (1967b) page 7, but this strand of literature really started with Singh and Vives (1984) and Vives (1985), the main reference remaining Amir and Jin (2001)....

[...]

• ...Prominent examples are Deneckere and Davidson (1985),8 Vives (1985),9 Shaked and Sutton (1990), Bagwell and Ramey (1991), Shaffer (1991), Dobson and Waterson (1997), and Sutton (1997)....

[...]

• ...Prominent examples are Deneckere and Davidson (1985),8 Vives (1985),9 Shaked and Sutton (1990), Bagwell and Ramey (1991), Shaffer (1991), Dobson and Waterson (1997), and Sutton (1997). In, Motta (2004), influential book “Competition Policy: Theory and Practice”, the Levitan and Shubik’s model is used (in particular in chapter 5 on horizontal mergers) to illustrate some properties with a closed-form model....

[...]

Journal ArticleDOI
TL;DR: In this article, the authors show that a firm can use the purchase price and the fine imposed on detected payment evaders to discriminate between unobservable consumer types, and they illustrate with data from fare dodging on public transportation.
Abstract: This paper shows that a firm can use the purchase price and the fine imposed on detected payment evaders to discriminate between unobservable consumer types. Assuming that consumers self‐select into regular buyers and payment evaders, we show that the firm typically engages in second‐degree price discrimination in which the purchase price exceeds the expected fine. In addition, we find that higher fines do not necessarily reduce payment evasion. We illustrate with data from fare dodging on public transportation.

11 citations

### Cites background from "Which demand systems can be generat..."

• ...We first establish that the partial derivatives of consumer surplus are linked to demands—a standard property (Armstrong and Vickers [2015])....

[...]

Journal ArticleDOI
TL;DR: A supply chain is constructed to investigate the reason why it is uncommon for the traditional retailer to function as the pre-warehouse of the online retailer, and an intrinsic rationale for theTraditional retailer not to supply the common product is identified, even if he is unambiguously cost advantageous vis-a-vis the third-party suppliers.
Abstract: We construct a supply chain to investigate the reason why it is uncommon for the traditional retailer to function as the pre-warehouse of the online retailer. The product only offered by the tradit...

8 citations

##### References
More filters
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....

[...]

• ...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....

[...]

Journal ArticleDOI

786 citations

Journal ArticleDOI
TL;DR: A review of recent developments in econometric demand analysis which may be of interest in market research is given in this article, with particular attention given to models which yield tree structures of similarities between alternatives.
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.

751 citations

Journal ArticleDOI
TL;DR: In this paper, a new model was developed to study competition between print and online newspapers and the welfare impact of charging positive online prices was investigated, using new micro data from Washington, DC.
Abstract: Many important economic questions hinge on the extent to which new goods either crowd out or complement consumption of existing products. Recent methods for studying new goods rule out complementarity by assumption, so their applicability to these questions has been limited. I develop a new model that relaxes this restriction, and use it to study competition between print and online newspapers. Using new micro data from Washington, DC, I estimate the relationship between the print and online papers in demand, the welfare impact of the online paper's introduction, and the expected impact of charging positive online prices.

346 citations

Book ChapterDOI
TL;DR: In this article, the authors show that a tax imposed on the seller of a monopolized article may lead to an actual lowering of the price to the buyer, and that the result of a tax is to cheapen the untaxed as well as the taxed commodities.
Abstract: That a tax imposed on the seller of a monopolized article may lead to an actual lowering of the price to the buyer has been shown by F. Y. Edgeworth.2 His example was of a railway supplying two classes of passenger service at different prices and, unhindered by governmental interference, setting its rates so as to make its own profit a maximum. When the company is compelled to pay a tax on each first-class ticket, it finds it profitable, in Edgeworth’s example, to reduce rates on both classes of accommodations. Regarding this paradoxical conclusion, Professor Seligman writes:3 The mathematics which can show that the result of a tax is to cheapen the untaxed as well as the taxed commodities will surely be a grateful boon to the perplexed and weary secretaries of the treasury and ministers of finance throughout the world!

317 citations