§3.1, Eq. (10): the coordinating target $T(u)$, admissible iff $u > m + \delta_m(w-c)$

Let $a > 0$, $\beta > 0$, $M_m > 0$, $R_r > 0$ with $(M_m + R_r)\beta > a$, and let $X(\rho) \sim \mathrm{Uniform}(0, 2\beta/\rho)$. For a target rebate contract with $u > 0$ and $T > 0$ the following are equivalent:

1. $(u, T)$ coordinates the supply chain (the coordinated effort $\rho^C$ maximizes $\pi_R(\cdot \mid T, u)$ over $\rho \ge 1$) and satisfies $T < 2\beta/\rho^C$;
2. $T$ is given by

and $u > M_m = m + \delta_m(w - c)$.

In words: for each rebate $u$ there is exactly one admissible coordinating target, and it exists precisely when the rebate exceeds the manufacturer's own profit of avoiding a false failure return. In particular coordinating contracts with $T < 2\beta/\rho^C$ exist, so the hypotheses of Proposition 2 can be met.