Eqs. (1)–(2): the coordinated profit is concave and maximized at $\rho^C$

Let $a > 0$ be the marginal effort cost, $\beta > 0$ the expected number of false failures at minimum effort, and $M_m > 0$, $R_r > 0$ the manufacturer's and retailer's profits of avoiding one false failure return. The coordinated supply chain's expected profit

is concave on $\rho > 0$, and it is maximized over $\rho > 0$ at the coordinated effort

This identifies $\rho^C$ as the effort level a coordinating contract has to induce.

Formalization Note Concavity and maximality are stated on the open half-line $\rho > 0$, where the formula is meaningful; the effort domain $\rho \ge 1$ of the paper is contained in it, and $\rho^C \ge 1$ in the interesting case (see the next milestone).