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.

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

Topics: Discrete choice (63%)

### 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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Munich Personal RePEc Archive
Which demand systems can be generated
by discrete choice?
Armstrong, Mark and Vickers, John
Department of Economics, University of Oxford
April 2015
Online at https://mpra.ub.uni-muenchen.de/63439/
MPRA Paper No. 63439, posted 04 Apr 2015 06:02 UTC

Which demand systems can be generated by discrete
choice?
Mark Armstrong John Vickers
April 2015
Abstract
We provide a simple necessary and su¢cient condition for when a multiproduct
demand system can be generated from a discrete choice model with unit demands.
Keywords: Discrete choice, unit demand, multiproduct demand functions.
1 Introduction
In a variety of economic settings the decision problem facing agents is one of discrete
choice. For example, 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 o¤er
i
is a consumer’s valuation for product i and p
i
is its price, then the
rational consumer will buy the product with the best value for money given her preferences,
i.e., the highest (v
i
p
i
) if that is positive, and will otherwise buy nothing. By specifying
a probability distribution for the vector of valuations within the population of consumers,
one can derive aggregate multi-product demand as a function of the vector of prices. Such
a demand system necessarily involves products being substitutes, but otherwise appears to
permit rich possibilities of behaviour.
1
Both authors at All Souls College, University of Oxford. Thanks for helpful comments are due to two
referees, as well as to Simon Anderson, Sonia Ja¤e, Howard Smith and Glen Weyl. Contact information
for corresponding author: john.vickers@economics.ox.ac.uk, tel: +44 (0)1865 279300.
1
For example, Hotelling (1932, section 2) provides an early example of a discrete choice demand system.
This example exhibits Edgeworth’s Paradox, in which an increase in the unit cost of a product (as a result
of imposing a new tax, say) causes a multiproduct monopolist to reduce all of its prices.
1

In this paper we investigate which aggregate demand functions have discrete choice
micro-foundations. With a single product, any (bounded) 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 speci…ed price for their unit. With more than one product, though, the answer is less
obvious. We show that discrete choice foundations for an aggregate demand system (which
is bounded and exhibits the usual Slutsky symmetry property) exist if and only if all mixed
partial derivatives (with respect to prices) of the total quantity demanded are negative.
Thus there is a simple test for whether a given demand system is consistent with discrete
choice.
Early contributions to the theory and econometrics of discrete choice are surveyed by
McFadden (1980), who developed the modern economics of discrete choice analysis in a
variety of applications including choices of education and residential location. Relation-
ships between discrete choice models and demand systems for di¤erentiated products are
explored in chapter 3 (and elsewhere) of the classic analysis by Anderson, de Palma and
Thisse (1992). In particular, their Theorem 3.1 states necessary and su¢cient properties
of demand functions that ensure these demands are consistent with discrete choice. Their
result presumes that consumers must buy one option or another, so that total demand
always sums to one. In most situations of interest, however, consumers have, and use, the
option to buy nothing, and we provide a result in the same spirit as Anderson et al., but
which allows for this. Indeed, the way that total demand varies with prices is the key to
our analysis.
More recently, Ja¤e and Weyl (2010) show how a linear demand system cannot be gen-
erated from (continuous) discrete choice foundations when there are at least two products
and buyers can consume an outside option.
2
Ja¤e and Kominers (2012) extend this analysis
to show how (continuous) discrete choice cannot induce a demand system where demand
for each product is additively separable in its own price. The analysis in the present paper
sets those contributions in a wider context.
2
Strictly speaking, they show that linear demand does not have discrete choice foundations where the
valuations are continuously distributed (so a density exists). In section 3.2 we show how linear demand
is often consistent with a discrete choice model in which the support of valuations does not have full
dimension.
2

The next section states a preliminary result, which is not speci…c to discrete choice, that
individual product demands can be derived from the total demand function. The main
section then derives necessary and su¢cient 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
Suppose there are n products, with associated price vector p = (p
1
; :::; p
n
), where the
aggregate demand for product i = 1; :::; n is given by q
i
(p) 0. We only consider prices
in the non-negative orthant R
n
+
, and we assume quasi-linear preferences, so that demand
q
i
is the derivative of an indirect utility function CS(p): q
i
(p) @CS(p)=@p
i
, where
CS() is convex and decreasing in p. For simplicity, suppose that demand functions are
di¤erentiable, in which case we have Slutsky symmetry:
@q
i
(p)
@p
j
@q
j
(p)
@p
i
for i 6= j . (1)
Given the demand system q(p), de…ne Q(p)
P
n
i=1
q
i
(p) to be the total quantity of all
products demanded with the price vector p. We make the innocuous assumptions that
Q(0) > 0 and that Q(p) ! 0 as all prices p
i
simultaneously tend to in…nity.
A result which is useful in the “su¢ciency” part of the following analysis, and perhaps
of interest in its own right, is the following.
3
Lemma 1 Suppose the demand system satis…es (1). Then the demand for product i, q
i
(p),
satis…es
q
i
(p) =
Z
1
0
@
@p
i
Q(p
1
+ t; :::; p
n
+ t)dt ; (2)
where Q
P
i
q
i
is total demand.
Proof. We need to show that
q
i
(p) =
Z
1
0
@
@p
i
Q(p
1
+ t; :::; p
n
+ t)dt =
Z
1
0
n
X
j=1
@q
j
@p
i
(p
1
+ t; :::; p
n
+ t)dt :
3
Expression (2) remains valid if Q is continuous and piecewise-di¤erentiable. (Typically, demand is not
di¤erentiable at choke prices which make a product’s demand fall to zero.)
3

But (1) implies that the right-hand side above is equal to
Z
1
0
n
X
j=1
@q
i
@p
j
(p
1
+ t; :::; p
n
+ t)dt =
Z
1
0
d
dt
q
i
(p
1
+ t; :::; p
n
+ t)dt = q
i
(p)
as required.
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
choice?
We 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” we mean,
…rst, that any individual consumer wishes to buy a single unit of one product (or to buy
nothing). In particular, a consumer gains no extra utility from buying more than one
product or from buying more than one unit of a product. Speci…cally and furthermore
5
,
the discrete choice model assumes that a consumer has a valuation v
i
for a unit of product i
(where valuations can be negative), where the vector of valuations v = (v
1
; :::; v
n
) is drawn
from a joint cumulative distribution function (CDF), denoted G(v), and if she makes a
purchase she buys the product which o¤ers the greatest net surplus v
i
p
i
nothing she obtains a deterministic pay of zero.
6
Faced with price vector p, the type-v
consumer in this discrete choice problem will therefore
choose product i if v
i
p
i
max
j6=i
f0; v
j
p
j
g : (3)
4
For instance, if total demand is additively separable in prices, it follows from (2) that demand for a
particular product depends only on its own price. If total demand depends only on the sum of prices, so
does the demand for each product.
5
As we 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 satsi…ed (e.g., because of search or transactions costs).
Such settings do not come within the discrete choice model as we have de…ned it.
6
The following analysis applies equally to the situation where the consumer’s outside option, say v
0
,
is stochastic, and a consumer buys product with the highest value of (v
i
p
i
) provided this is above v
0
.
However, one can just subtract v
0
from each v
i
of zero.
4

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