Proposition 2 — a coordinating target rebate helps the retailer, and the manufacturer iff $\rho^C \ge 2\rho^D$

Consider one manufacturer and one retailer. Let $a > 0$ be the marginal effort cost, $\beta > 0$ the expected number of false failures at the minimum effort level, and $M_m = m + \delta_m(w - c) > 0$, $R_r = r + \delta_r(p - w) > 0$ the manufacturer's and the retailer's profits of avoiding one false failure return, in the interesting case $(M_m + R_r)\beta > a$. Suppose that at effort $\rho$ the number of false failures is $X(\rho) \sim \mathrm{Uniform}(0, 2\beta/\rho)$. Write

for the coordinated and the decentralized effort.

Let $(u, T)$ be a target rebate contract (the retailer receives $u$ for every false failure below the target $T$) with $u > 0$, $T > 0$ and $T < 2\beta/\rho^C$, which coordinates the supply chain: $\rho^C$ maximizes the retailer's expected profit $\pi_R(\rho \mid T, u)$ of Eq. (6) over efforts $\rho \ge 1$. Then

1. the contract makes the retailer better off:

2. the contract makes the manufacturer better off if and only if the coordinated effort is at least twice the decentralized effort:

Here $\pi_R(\rho^D)$ and $\pi_M(\rho^D)$ are the no-contract profits (3) and (5), and $\pi_M(\rho \mid T, u)$ is the manufacturer's profit (7) under the contract. The result says that a coordinating target rebate is a Pareto improvement over no contract exactly when the retailer's own incentive to exert effort is weak relative to the supply chain's.

Formalization Note The paper's interesting case is printed as $(m + r)\beta > a$; the condition equivalent to $\rho^C > 1$, which the proof uses, is $(M_m + R_r)\beta > a$, and that is the hypothesis here. The side condition $T < 2\beta/\rho^C$ is the paper's §3.1 requirement $T < 2\beta/\rho$ at the coordinated effort. The expectation in (6)–(7) is the true uniform expectation for every effort.