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Appendix A: Proof of Proposition 2

To verify the comparative statics of the system (4)-(5)-(7) with respect to s, let us
use the definitions where β =(n - 2)h(x) + h(z) and β_H ≡ (n - 1)h(x) to rewrite
it in terms of the three unknown variables x, z and β_H :

Π1 [x, h(z) - h(x)+βH, 0] = 0

Π1^H[z,βH,s]=0                        (19)

Π [x, h(z) - h(x)+βH, 0] = F

The second equation provides an implicit relationship z = z(βH, s) with dz/двн =
—ΠH>∕ΠH and ∂z∕∂s = — ΠH3∕ΠH > 0. Substituting this expression we obtain a
system of two equations in two unknowns, x and βH :

Π1 [x, h(z(βH, s)) — h(x) + βH , 0] = 0

Π [x, h(z(βH,s)) — h(x) + βH, 0] = F

Totally differentiating the system we have:

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