# 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.

Abstract: We provide a simple necessary and sufficient condition for when a multiproduct demand system can be generated from a discrete choice model with unit demands.

## Summary (2 min read)

### 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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##### Citations

64 citations

43 citations

10 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....

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...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....

[...]

9 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])....

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4 citations

### 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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##### References

2,184 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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685 citations

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