Economics Letters 70 (2001) 69–72
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Measuring diversity in product space: the missing axiom
*
Daniel M. Bernhofen
Department of Economics, Clark University, Worcester, MA 01610-1477, USA
Received 25 January 2000; accepted 15 July 2000
Abstract
In contrast to the standard monopolistic competition models of product differentiation, an n-good quadratic utility specification allows for asymmetries in the degree of substitutability among products. However, in order to guarantee the existence of a market equilibrium, these asymmetries need to be constrained to fulfil the ‘transitivity of closeness’ axiom. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Quadratic utility; Metric product space
JEL classification: D11; L11
1. Introduction
In a recent paper in this journal, Gans and Hill (1997) examined an alternative approach to the measurement of product diversity in contrast to the monopolistic competition models, where diversity is a function of the number of varieties available to consumers. Inspired by the work of Weitzman (1992), the authors state a set of axioms for bilateral distances between products and specify a utility function, which they claim incorporate these distance measures in a meaningful way. In particular, the utility function is defined over n products and is characterized by n(n21) / 2 ‘distance’ parameters,
n(n21 ) / 2
with the permissible parameter space being equal to [0,1] .
This paper revisits Gans and Hill (1997) and illustrates, in the context of a simple example, that the permissible parameter space needs to be reduced significantly in order to guarantee the existence of a market equilibrium. In particular, it postulates that the distance function should fulfil the ‘transitivity of closeness’ axiom, which guarantees an ‘ordering’ in a product space. From a topological point of view, this means that an economically meaningful product space must be a metric space.
*Tel: 11-508-793-7185; fax: 11-508-793-8849.
E-mail address: [email protected] (D.M. Bernhofen).
70 D.M. Bernhofen / Economics Letters 70 (2001) 69 –72
2. Problem formulation
Consider a set X of potential products and let x , . . . ,x be n elements from this set. In the context1 n of a representative consumer model, one can then define a utility function U as follows:
n
U(x , . . . ,x ,s )5
O
v(x )2O
s x x 1M (1)1 n ij i ij i j
i51 i,j
2
wherev(x)5x2x / 2 and M is money spent on outside goods. The above utility function, which is a
simple reformulation of Eq. (2) in (Gans and Hill, 1997; p. 148), allows for a convenient economic n
interpretation. The first term,o v(x ), is the sum of n subutility functionsv(.) and can be viewed as
i51 i
1
capturing the taste for variety of the representative consumer. The second term, o s x x , is the
i,j ij i j sum of n(n21) / 2 interaction terms.
The main insight of Gans and Hill (1997) is that the above utility function is flexible enough to allow for n(n21) / 2 interaction parameterss that are specific to each pair of products (x ,x ), where
ij i j
0#s #1. Consequently, thes can be used to define a bilateral distance function: d(x ,x )512s .
ij ij i j ij
Since the distance function d(.) is non-negative and –by construction– also symmetric, the authors
2
claim that it is a well-defined distance measure between two goods.
3. Inconsistent parameter configuration: a simple example
Gans and Hill (1997) allow the n(n21) / 2 parameters s in the utility function (1) to take any
ij
arbitrary values in the interval [0,1]. Hence, they postulate that the parameter space is equal to n(n21 ) / 2
S5[0,1] . The authors proceed then by showing that it is always possible to estimate the
distance parameterss from observed price and quantity data, as long as one assumes that the data are
ij
generated by the utility maximization of the representative consumer. However, this argumentation ignores the production side of the economy. I provide an example that demonstrates that the parameter space needs to be significantly reduced in order to guarantee that a production equilibrium actually exists.
Assume now that there are 3 goods with the following parameter configuration: s 51, s 50
12 13
and s 50.5. This implies that good 1 and good 2 are perfect substitutes (i.e. d 50), good 2 and
23 12
good 3 are independent (i.e. d 51) and good 2 and good 3 are imperfect substitutes (i.e. d 50.5).
13 23
The representative consumer maximizes the utility function (1), subject to its budget constraint
3
M05M1oj51 p x . If one assumes that the consumer’s income Mj j 0 is sufficiently large, the first-order conditions, U 5p , ( j51,2,3) yield the following inverse demand schedules:
j j
The subutility function v is characterized by diminishing marginal utility for a good (i.e.v9.0 and v0,0 for x,1). i 2
D.M. Bernhofen / Economics Letters 70 (2001) 69 –72 71
Let us assume now that each good is produced by a single firm and that marginal production costs
3
are equal to zero. Since goods 1 and 2 are perfect substitutes, the two goods must have the same price in equilibrium, i.e. p5p 5p . From (2) and (3) it follows that x 50, which implies that producers 1
1 2 3
and 2 face the following inverse market demand curve: p512x 2x . Through a straightforward
1 2
calculation, it can be shown that the Cournot–Nash equilibrium quantities are given by x 5x 51 / 3.
1 2
However, if one applies these values into Eq. (4), one obtains that p 55 / 6, which contradicts the fact
3
that x 50. Hence, for the chosen parameter configuration s 51, s 50, and s 50.5 a market
3 12 13 23
equilibrium doesn’t exist.
4. The ‘transitivity of closeness axiom’
The above example has demonstrated that the interaction parameterss in the above utility function
ij
can’t take any arbitrary values between 0 and 1. In particular, the chosen distances lead to an ‘inconsistent’ market structure. However, in order to obtain sensible distances in the product space, one needs to impose some ordering among the products. This can be accomplished by requiring the distance function to fulfil the transitivity of closeness property; that is, if good i is close to good j and good j is close to good k, then good j must also be close to good k.
Formally, the set of permissible parameter values s needs to be restrained such that the
ij
corresponding bilateral distances d(x ,x )512s fulfil the following axiom:
i j ij
Triangle inequality: d(x ,x )#d(x ,x )1d(x ,x ) for all i, j and k. (5)
i k i j j k
The triangle inequality guarantees the transitivity of closeness property and implies that the product
4
space is a metric space.
To illustrate that the triangle inequality will guarantee the appropriate ordering in product space, let us look at the example from the previous section. Assuming that s 51 ands 50, we can use the
12 23
triangle inequality to infer the ‘permissible’ values for the parameter s In particular, we obtain
13.
15d(x ,x )5d(x ,x )1d(x ,x )5(12s ). Solving this inequality for s , one obtains s #0.
2 3 1 3 1 2 13 13 13
Hence, the only permissible value for s is 0.
13
Consequently, if goods 1 and 2 are perfect substitutes and goods 1 and 3 are independent, a metric product space would require that goods 2 and 3 are also independent. Under such a parameter configuration firms 1 and 2 engage in duopolistic competition and produce x 5x 51 / 3. Since the
1 2
representative consumer perceives good 3 as independent from the other two goods, firm 3 acts as a monopolist and produces x 51 / 2.
3
3
It is an implicit assumption that positive fixed costs prevent other firms from entering the market. 4
72 D.M. Bernhofen / Economics Letters 70 (2001) 69 –72
References
Gans, J., Hill, R., 1997. Measuring product diversity. Economics Letters 55 (1), 145–150. Mendelson, B., 1990. Introduction to Topology. Dover Publications, New York.
