§3.1, Eq. (9): first-order condition for the retailer's optimal effort under a target rebate

Let $a > 0$, $\beta > 0$, $M_m > 0$, $R_r > 0$ be as in the model, and let $(u, T)$ be a target rebate contract with $u > 0$ and $T > 0$, with $X(\rho) \sim \mathrm{Uniform}(0, 2\beta/\rho)$. Suppose an effort $\rho^* > 1$ maximizes the retailer's expected profit $\rho \mapsto \pi_R(\rho \mid T, u)$ over $\rho \ge 1$, and $T < 2\beta/\rho^*$. Then

Equation (9) links $u$, $T$ and the induced effort; substituting $\rho^* = \rho^C$ gives the coordinating contracts (10) and all the profit identities of the appendix.

Formalization Note The paper's "the value $\rho^*$ that maximizes (8)" is formalized as a maximizer of the true profit (6) over $\rho \ge 1$; the hypotheses $\rho^* > 1$ (an interior maximizer) and $T < 2\beta/\rho^*$ (so that (8) is the profit near $\rho^*$) are the conditions under which the paper's first-order condition applies.