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



Proof. (Lemma 8.1) Take any point (x, x) on the diagonal, belonging to the
domain of F .Forα>
0 small enough

U1 (x, x) ' U(x + α, x) - U(x, x) and L1(x, x) ' (x, x) - L(x - α, x).   (17)

Hence, from A2:

U(x + α, x) - U(x, x) > L(x, x) - L(x - α, x).            (18)

From A1 we know that

U(x + α, x - α) - U(x, x - α) U(x + α, x) - U(x, x).        (19)

Take bε> 0 such that

U(x + α, x) - U (x, x) L(x, x) - L(x - α, x)+bε.          (20)

From continuity of U(x, y) in x (for fixed y = x - α), it follows that

bε

0 < ε ≤ -, δ > 0 such that

bε

|x (x α)| = α δ and |U(x, x α) U(x α,x α)| ≤ ε ≤ —.

Moreover,

bε

0 < ε ≤ -, δ > 0 such that

bε

|x + α x| = α δ and |U(x + α,x α) U(x, x α)| ≤ ε ≤ —.

This allows us to establish that:

(U(x + α, x α) U(x, x α)) (U (x, x α) U(x α, x α)) 2ε bε

(21)

since, for |A| <εand |B| <ε,thenA B<2ε.

Rewriting (21),

U(x + α, x α) U(x, x α) U(x, x α) U(x α, x α)+bε.   (22)

33



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