S. Salop provides an instructive model of product differentiation. He asks us
to conceptualize the demand for a product group as varying along a circular
spectrum of characteristics (the model can also be thought of as a spatial
model with consumers located around a circle). Demanders are located at each
point on this circle and each demands one unit of the good. Demanders incur
costs if they must consume a product that does not precisely meet the
characteristics they prefer. As in the Hotelling model, these costs are given
by \(t x\) (where \(x\) is the "distance" of the consumer's preferred
characteristic from the characteristics being offered by the nearest supplier
and \(t\) is the cost incurred per unit distance). Initially there are \(n\) firms
each with identical cost functions given by \(T Q=/+c q,\) For simplicity we
assume also that the circle of characteristics has a circumference of
precisely 1 and that the \(n\) firms are located evenly around the circle at
intervals of \(1 / n\)
a. Each firm is free to choose its own price \((p),\) but is constrained by the
price charged by its nearest neighbor \(\left(p^{*}\right) .\) Explain why the
extent of any one firm's market \((x)\) is given by the equation
\\[
p+t x=p^{*}+t /(l / n)-x J
\\]
b. Given the pricing decision illustrated in part a, this firm sells \(q,=2 x\)
because it has a mar ket on "both sides." Calculate the profit-maximizing
price for this firm as a function of \(p^{*}, c,\) and \(t\)
c. Assuming symmetry among all firms will require that all prices are equal,
show that this results in an equilibrium in which \(p=p^{*}=c+t / n .\) Explain
this result intuitively.
d. Show that in equilibrium the profits of the typical firm in this situation
are \(T T,=t / n^{2}-f\)
e. Assuming free entry, what will be the equilibrium level of \(n\) in this
model?
f. Calculate the optimal level of differentiation in this model-defined as
that number of firms (and products) that minimizes the sum of production costs
plus demander dis tance costs. Show that this number is precisely half the
number calculated in part (e). Hence, this model suffers from "over-
differentiation." (For a further exploration of this model, see S. Salop
"Monopolistic Competition with Outside Goods," Bell Journal of Eco nomics,
Spring \(1979, \text { pp. } 141-156 .)\)