Proof of Proposition 1, p. 35 — π₁(z^N, b) − π^N_1 = 2(b − z^N)F(z^N), positive for b > z^N

Let $(F,f)$ be a monotone-hazard-rate distribution on $[0,1]$ and $z^N$ the benchmark marginal consumer of Eq. (1). For data sharing on $[z^N,b]$, firm 1's profit is $\pi_1(z^N,b)=\int_0^{z^N}\bigl((2b-1)+(1-2x)\bigr)\,dF(x)$. Then for every real $b$,

and this difference is strictly positive whenever $b>z^N$.

Sharing data on firm 2's customer base $[z^N,b]$ costs firm 1 no market share, and raises the price it can charge each of its consumers by $2(b-z^N)$; this identity is the firm-1 half of Proposition 1.