§3.1 — the profit jump at $p=\bar c$ (corrected: demand conditions)

At the market price the platform's profit jumps by a positive amount:

Hence $\pi_m<\sup_{p>0}\pi_N(p)$, and every maximizer of $\pi_N$ over prices $p>0$ lies in $(0,\bar c]$.

So, in the non-scalable scenario, an optimal price always attracts the sellers.

Differences from the source:

• correction: D > 0, D′ < 0, D″ ≤ 0, D(0) = ∞ on outputs q ≥ 0 (p.4) → outputs in J = (0, q̄), the signs on J, and (1 − α)D(q) = p solvable in J for every p ∈ (0, c̄]; reason: the printed conditions are unsatisfiable: a positive concave function on (0, ∞) cannot be strictly decreasing, and a concave function on (0, b) cannot tend to +∞ at 0; solvability of the first-order condition is the only use of D(0) = ∞
• generalization: α a commission rate, Assumption 2 → 0 ≤ α < 1, and of Assumption 2 only f < F; reason: the argument uses no more, and low-type sellers are absent at μ = 1; every source instance, with α ∈ (0, 1), is covered
• encoding: c ∈ (c̲, c̄) with no sign on c̲ (p.6) → 0 < c̲; reason: (1 − α)D(q(c̲)) = c̲ with D > 0 and α < 1 forces c̲ > 0, so no source instance is excluded
• reading: its profit jumps → the size of the jump, its sign, and the consequence for maximizers of π_N; reason: the source's claim made quantitative; the last clause is what p.8 uses when it reads p* against c̄