Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

3.2 Applications

In this section we present examples of economic models that constitute spe-
cial cases of the general framework developed above. While the assumptions
validating Theorem 3.1 might at first appear somewhat special, they are sat-
isfied in several a priori unrelated studies that have established endogenous
heterogeneity in strategic settings. There are also some studies where asym-
metric equilibria are produced via a mechanism similar to our Theorem 3.1,
without being a special case in a formal sense. Going over some of these ex-
amples illustrates the unifying character of our results and allows us to provide
some contextual interpretations of endogenous heterogeneity, or our version of
symmetry-breaking.

3.2.1 R&D investment

The first example is based on the model by Amir and Wooders (2000). Two a
priori identical firms with initial unit cost c are engaged in a two stage game
of R&D investment and production. In the first stage, autonomous cost reduc-
tions x and y for firms 1 and 2, respectively, are chosen. The novel feature of
this study is that spillovers are postulated to flow only from the more R&D
active firm to the rival, but not vice versa. The effective (post-spillover) cost
reductions X and Y when x ≥ y are given by:

{x with probability β

(5)
y with probability 1 - β

Second stage product market competition, be it Cournot or Bertrand, is assumed
to have a unique PSNE with equilibrium payoffs given by Π :[0,c]² → R.¹⁰
Π(x, y ) is the payoff of the firm whose unit cost is the first argument. f :
[0, c] → R is a known R&D cost schedule. Amir and Wooders (2000) assume
the following:

C1 Π and f are twice continuously differentiable

¹⁰This is a standard assumption in the literature.

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