§3, p. 382: in the interesting case, $\rho^C > 1$

Let $a > 0$, $\beta > 0$, $M_m > 0$ and $R_r > 0$ be as in the model. Suppose the total potential saving of avoiding false failures exceeds the marginal cost of the minimum effort level,

Then the coordinated effort exceeds the minimum effort level:

This is the "interesting case" of the paper, in which the coordinated supply chain exerts more than the minimum effort; every later result about coordinating contracts is stated in it.

Formalization Note The paper writes the condition as $(m + r)\beta > a$. That condition does not imply $\rho^C > 1$ when $\delta_m(w - c) + \delta_r(p - w) < 0$; the condition the conclusion rests on is $(M_m + R_r)\beta > a$, which is used here.