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

C2 Π is strictly submodular and Π₁ (x, y) < 0 and Π₂ (x, y) > 0

C3 Π(x, x) < Π(y, y) if x>y

C4 ∣∏ι(x, x)∣ > ∣∏2(x, x)|, ∀x ∈ [0, c]

C5 f⁰ (x) ≥ 0 and f(0) ≥ 0

C6 f0 (0) < -β∏2 (c, c) -∏1(c,c) and f0(c) > -(1 -β)∏1 (0, 0)

The overall payoff of firm 1,F(x, y), defined as in (1), is given by the dif-
ference between its second stage profit and first stage R&D cost. The payoff of
firm 2 is G(y, x) by symmetry.

U(x, y)=β∏ (c - x, c - x)+(1- β) ∏ (c - x, c - y) - f (x)      (6)

L(x,y)=β∏(c - y,c- y)+(1- β)∏(c - x,c - y) - f (x)      (7)

We can easily check that assumptions A1, A2 and A3 indeed hold in or-
der to apply Theorem 3.1. U and L are continuous and differentiable because
they result from the sum of continuous and differentiable functions. U (x, x)=
L(x, x), ∀x ∈ [0,c],soF and G are continuous. A1 can be checked by using
the cross-partial test and the fact that ∏(x, y) is submodular (Assumption C1).
Also, Using C2, and

U1 (x, x)=- [∏1 (c - x, c - x)+β∏2 (c - x, c - x)] - f⁰ (x)       (8)

L₁ (x, x)=- (1 - β) ∏₁ (c - x, c - x) - f⁰(x)                     (9)

we obtain that U1 (x, x) >L1 (x, x) , therefore A2 is verified.

From Theorem 3.1, we can conclude that payoff functions for both players
are submodular and thus reaction curves are downward sloping and there exists
at least one PSNE. Moreover the reaction curves do not intersect the diagonal
and by C6, U1 (0, 0) > 0 and L1 (c, c) < 0, so there is no symmetric equilibrium
in x ∈ [0, c]. The uniqueness of a pair of asymmetric PSNEs is shown by
imposing conditions that secure that the reaction curves are contractions, so
that Theorem 3.2 can be applied. Similarly, extra conditions are needed to

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