(19), p. 15 — the manufacturer's unique best response is e* = p(1 − (1 − β)p)/(2(1 − α)k)

Consider Setting (2B) with R&D cost scale $k>0$ and consortium price $p\in(0,1)$. For every subsidy policy $(\alpha,\beta)\in[0,1)^2$, the manufacturer's profit $e\,p\,[1-(1-\beta)p]-(1-\alpha)ke^2$ has exactly one maximizer over $e>0$, namely

This is the follower's response in the Stackelberg game; every statement about the government's problem evaluates welfare and spending at this efficacy.

Formalization Note The best-response set is the set of maximizers over $e>0$, and the theorem says it equals the singleton $\{e^*\}$, i.e. both existence and uniqueness. The hypotheses $k>0$ and $0<p<1$ are the paper's standing assumptions made explicit: with $p\ge1$ and $\beta=0$ the bracket $1-(1-\beta)p$ is not positive and no maximizer with $e>0$ exists. The page says "Equation (10)"; the manufacturer's objective is (9).