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



More intriguing information

1. Strategic Investment and Market Integration
2. The Making of Cultural Policy: A European Perspective
3. Regional specialisation in a transition country - Hungary
4. The name is absent
5. The Institutional Determinants of Bilateral Trade Patterns
6. The name is absent
7. The name is absent
8. Optimal Private and Public Harvesting under Spatial and Temporal Interdependence
9. The name is absent
10. Does Presenting Patients’ BMI Increase Documentation of Obesity?