Overstated product sustainability: real cases and a game-theoretical analysis

Abstract

In this paper, we examine the impact of manufacturers’ dishonest reports of product sustainability. We first investigate the Volkswagen emissions scandal and Wal-Mart fake organic food scandal in terms of the overstatement of product sustainability. Based on these two cases, we construct a two-echelon supply chain model consisting of one manufacturer and one retailer. The supply chain members can use inspections or deferred payments in trading. We find that the inspection and deferred payment mechanisms do not always help reduce the overstatements of sustainability. With the inspection mechanism, when the basic demand is higher (lower) than the threshold, inspections can (cannot) effectively reduce the levels of overstatement. With the deferred payment mechanism, when the manufacturer’s expected proportion of payments is sufficiently small (large), the deferred payments can (cannot) help reduce the levels of overstatement. Moreover, we find that when overstatements are difficult (easy) to identify by the retailer, the deferred payment (inspection) mechanism is the more effective means for the retailer to reduce overstatements. Furthermore, our numerical results indicate that when a third party organization perceives a low probability of product overstatement, the manufacturer and the retailer use neither inspections nor the deferred payment mechanism. In contrast, when a third party organization perceives a high probability of product overstatement, the inspection mechanism is more beneficial to the manufacturer, while the deferred payment is more beneficial to the retailer. The managerial implications of our findings are discussed.

Appendix—Proofs

1.1 Proof of Theorem 1

Recall the optimal overstated level in the inspection mechanism is:

$$ s_{I}^{ * } = \left( {\frac{{(1 - \theta )(1 - \lambda )\varepsilon b - (1 - \lambda )\beta \varepsilon (w_{o} - \theta v) + ((1 - \lambda )w_{o} + \lambda v - c)((1 - \theta )\alpha - \beta \varepsilon )}}{{2(1 - \theta )\eta { + 2}(1 - \lambda )\beta \varepsilon^{2} - 2(1 - \theta )(1 - \lambda )\varepsilon \alpha }}} \right)^{ + } . $$

As $(1 - \theta )\eta \ge \varepsilon ((1 - \theta )\alpha - \beta \varepsilon )$, i.e. denominator is positive, we can rearrange the numerator and yield that if $\lambda \le \frac{{(1 - \theta )\varepsilon b + ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - c) - \beta \varepsilon (w_{o} - \theta v)}}{{(1 - \theta )\varepsilon b + ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - v) - \beta \varepsilon (w_{o} - \theta v)}}$, then $s_{I}^{ * } \le 0$. In other words, the manufacturer will not overstate when $\lambda \le \frac{{(1 - \theta )\varepsilon b + ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - c) - \beta \varepsilon (w_{o} - \theta v)}}{{(1 - \theta )\varepsilon b + ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - v) - \beta \varepsilon (w_{o} - \theta v)}}$, vice verse. □

1.2 Proof of Theorem 2

Recall the optimal overstated levels in the benchmark and inspection mechanism are as follows.

$$ s_{B}^{ * } = \left( {\frac{{(1 - \theta )\varepsilon b - \varepsilon \beta (w_{o} - \theta v) + (w_{o} - c)((1 - \theta )\alpha - \beta \varepsilon )}}{{2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon }}} \right)^{ + } ,\,\,{\text{and}} $$

$$ s_{I}^{ * } = \left( {\frac{{(1 - \theta )(1 - \lambda )\varepsilon b - (1 - \lambda )\beta \varepsilon (w_{o} - \theta v) + ((1 - \lambda )w_{o} + \lambda v - c)((1 - \theta )\alpha - \beta \varepsilon )}}{{2(1 - \theta )\eta { + 2}(1 - \lambda )\beta \varepsilon^{2} - 2(1 - \theta )(1 - \lambda )\varepsilon \alpha }}} \right)^{ + } . $$

Let $s_{B}^{ * } = \left( {\frac{{(1 - \theta )\varepsilon b - \varepsilon \beta (w_{o} - \theta v) + (w_{o} - c)((1 - \theta )\alpha - \beta \varepsilon )}}{{2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon }}} \right)^{ + } = \frac{K}{L} \ge 0$, we can rewrite $s_{I}^{ * }$ to,

$$ \begin{aligned} s_{I}^{ * } & = \left( {\frac{{(1 - \theta )\varepsilon b - \beta \varepsilon (w_{o} - \theta v) + (w_{o} - c)((1 - \theta )\alpha - \beta \varepsilon ) - [(1 - \theta )\varepsilon b - \beta \varepsilon (w_{o} - \theta v) + ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - v)]\lambda }}{{2(1 - \theta )\eta { + 2}\beta \varepsilon^{2} - 2(1 - \theta )\varepsilon \alpha - [2\beta \varepsilon^{2} - 2(1 - \theta )\varepsilon \alpha ]\lambda }}} \right)^{ + } \\ & { = }\left( {\frac{K - [K + ((1 - \theta )\alpha - \beta \varepsilon )(c - v)]\lambda }{L - [L - 2(1 - \theta )\eta ]\lambda }} \right)^{ + } \\ \end{aligned} $$

Suppose $s_{I}^{ * } \le s_{B}^{ * }$, we find

$$ \begin{aligned}\frac{K}{L}& \ge \frac{ - ((1 - \theta )\alpha - \beta \varepsilon )(c - v)}{2(1 - \theta )\eta } \Leftrightarrow \frac{{(1 - \theta )\varepsilon b - \varepsilon \beta (w_{o} - \theta v) + (w_{o} - c)((1 - \theta )\alpha - \beta \varepsilon )}}{{2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon }} \\&\ge \frac{ - ((1 - \theta )\alpha - \beta \varepsilon )(c - v)}{2(1 - \theta )\eta } \end{aligned}$$

$$ \Leftrightarrow b \ge \frac{{(1 - \theta )\eta [ \beta \varepsilon (w_{o} - \theta v)-((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - \theta v)] + ((1 - \theta )\alpha - \beta \varepsilon)^{2} \varepsilon (c - v)}}{{(1 - \theta )^{2} \varepsilon \eta }} $$

that is, if $\Leftrightarrow b \ge \frac{{(1 - \theta )\eta [\beta \varepsilon (w_{o} - \theta v) - ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - v)] + ((1 - \theta )\alpha - \beta \varepsilon )^{2} \varepsilon (c - v)}}{{(1 - \theta )^{2} \varepsilon \eta }}$, then $s_{I}^{ * } \le s_{B}^{ * }$; otherwise, $s_{I}^{ * } > s_{B}^{ * }$.

Based on the comparison of the overstated level, the optimal sale price in the benchmark and inspection are:

$$ r_{B}^{ * } = \frac{{(1 - \theta )(\alpha s_{B}^{ * } + b) + \beta (w_{o} + \varepsilon s_{B}^{ * } - \theta v)}}{2\beta (1 - \theta )},\,\,{\text{and}}\,\,r_{I}^{ * } = \frac{{(1 - \theta )(\alpha s_{I}^{ * } + b) + \beta (w_{o} + \varepsilon s_{I}^{ * } - \theta v)}}{2\beta (1 - \theta )}. $$

We can get that, if

$$ b \ge \frac{{(1 - \theta )\eta [\beta \varepsilon (w_{o} - \theta v) - ((1 - \theta )\alpha - \beta \varepsilon )(w_{o} - v)] + ((1 - \theta )\alpha - \beta \varepsilon )^{2} \varepsilon (c - v)}}{{(1 - \theta )^{2} \varepsilon \eta }}, $$

then $r_{I}^{ * } \le r_{B}^{ * }$; otherwise, $r_{I}^{ * } > r_{B}^{ * }$. □

1.3 Proof of Theorem 3

In order to simplify our calculation, let $N = 2\beta \varepsilon w_{o}$, $Y = (1 - \theta )\varepsilon b + \beta \varepsilon \theta v + (1 - \theta )\alpha w_{0}+ \beta \varepsilon c$, we can rewrite $s_{D}^{ * }$ to

$$ \begin{aligned} s_{D}^{ * } & = \left( {\frac{{A(1 - \theta )\varepsilon b - A\beta \varepsilon (Aw_{o} - \theta v) + ((1 - \theta )\alpha - A\beta \varepsilon )(Aw_{o} - c)}}{{2(1 - \theta )\eta + 2A^{2} \beta \varepsilon^{2} - 2A(1 - \theta )\alpha \varepsilon }}} \right)^{ + } \\ & = \left( {\frac{{ - 2A^{2} \beta \varepsilon w_{o} + A((1 - \theta )\varepsilon b + \beta \varepsilon \theta v + (1 - \theta )\alpha w_{0} + \beta \varepsilon c) -(1- \theta) \alpha c}}{{2(1 - \theta )\eta + 2A^{2} \beta \varepsilon^{2} - 2A(1 - \theta )\alpha \varepsilon }}} \right)^{ + } \\ & = \left( {\frac{{ - A^{2} N + AY - (1 - \theta )\alpha c}}{{2(1 - \theta )\eta + 2A^{2} \beta \varepsilon^{2} - 2A(1 - \theta )\alpha \varepsilon }}} \right)^{ + } . \\ \end{aligned} $$

Let $F(A) = - A^{2} N + AY - (1 - \theta )\alpha c$, let $F(A) = 0$, there are two roots of $F(A)$, one of the roots is $A_{1} = \frac{{Y + \sqrt {Y^{2} - 4N(1 - \theta )\alpha c} }}{2N}$ and $A_{2} = \frac{{Y - \sqrt {Y^{2} - 4N(1 - \theta )\alpha c} }}{2N}$.

For $A_{ 1}$, we assume $A_{1} \ge 1$, then we have

$$ \begin{aligned} \frac{{Y + \sqrt {Y^{2} - 4N(1 - \theta )\alpha c} }}{2N} \ge 1 & \Leftrightarrow \sqrt {Y^{2} - 4N(1 - \theta )\alpha c} \ge 2N - Y \\&\Leftrightarrow Y^{2} - 4N(1 - \theta )\alpha c \ge (2N - Y)^{2} \\ & \Leftrightarrow Y - N - (1 - \theta )\alpha c \ge 0. \\ \end{aligned} $$

As $s_{B}^{ * } = \left( {\frac{ - N + Y - (1 - \theta )\alpha c}{{2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon }}} \right)^{ + } \ge 0$ and $2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon \ge 0$, we can see $- N + Y - (1 - \theta )\alpha c \ge 0$. Thus, we can eliminate this root $A_{ 1}$.

For $A_{2} = \frac{{Y - \sqrt {Y^{2} - 4N(1 - \theta )\alpha c} }}{2N}$, we find when $0 \le A \le \frac{{Y - \sqrt {Y^{2} - 4N(1 - \theta )\alpha c} }}{2N}$, $F(A) \le 0$, then $s_{D}^{ * } \le 0$. This implies that the manufacturer will not overstate. □

1.4 Proof of Theorem 4

Recall $s_{B}^{ * } = \left( {\frac{ - N + Y - (1 - \theta )\alpha c}{{2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon }}} \right)^{ + }$ and $s_{D}^{ * } = \left( {\frac{{ - A^{2} N + AY - (1 - \theta )\alpha c}}{{2(1 - \theta )\eta + 2A^{2} \beta \varepsilon^{2} - 2A(1 - \theta )\alpha \varepsilon }}} \right)^{ + }$.

Suppose $s_{D}^{ * } \le s_{B}^{ * }$, then we have $\frac{{ - A^{2} N + AY - (1 - \theta )\alpha c}}{{2(1 - \theta )\eta + 2A^{2} \beta \varepsilon^{2} - 2A(1 - \theta )\alpha \varepsilon }} \le \frac{ - N + Y - (1 - \theta )\alpha c}{{2(1 - \theta )\eta + 2\beta \varepsilon^{2} - 2(1 - \theta )\alpha \varepsilon }}$

$$ \Leftrightarrow (1 - A^{2} )(1 - \theta )\eta N + (A^{2} - A)(1 - \theta )\alpha \varepsilon N - (1 - A)(1 - \theta )\eta Y $$

$$ (A - A^{2} )\beta \varepsilon^{2} Y - (1 - A^{2} )(1 - \theta )\beta \varepsilon^{2} \alpha c + (1 - A)(1 - \theta )^{2} \alpha^{2} \varepsilon c \le 0. $$

Dividing both sides by $(1 - A)$ and shifting the items, we can get when $0 \le A \le \frac{{(1 - \theta )\eta (Y - N){ + }(1 - \theta )\alpha \varepsilon c(\beta \varepsilon - (1 - \theta )\alpha )}}{{(1 - \theta )N(\eta - \alpha \varepsilon ) + \beta \varepsilon^{2} (Y - (1 - \theta )\alpha c)}}$, then, $s_{D}^{ * } \le s_{B}^{ * }$; otherwise, $s_{D}^{ * } > s_{B}^{ * }$.

Recall the optimal retail price in the benchmark and the deferred payment are:

$$ \begin{aligned} r_{B}^{ * } (s) & = \frac{{(1 - \theta )(\alpha s + b) + \beta (\varepsilon s + w_{o} - \theta v)}}{2\beta (1 - \theta )} \\ & = \frac{((1 - \theta )\alpha + \beta \varepsilon )( - N + Y - (1 - \theta )\alpha c)}{{4\beta (1 - \theta )((1 - \theta )\eta + \beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon )}} + \frac{{(1 - \theta )b + \beta (w_{o} - \theta v)}}{2\beta (1 - \theta )} \\ \end{aligned} $$

$$ \begin{aligned} r_{D}^{ * } (s) & = \frac{{(1 - \theta )(\alpha s + b) + \beta (A(w_{o} + \varepsilon s) - \theta v)}}{2\beta (1 - \theta )} \\ & = \frac{{((1 - \theta )\alpha + A\beta \varepsilon )( - A^{2} N + AY - (1 - \theta )\alpha c)}}{{4\beta (1 - \theta )((1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon )}} + \frac{{(1 - \theta )b + \beta (Aw_{o} - \theta v)}}{2\beta (1 - \theta )}. \\ \end{aligned} $$

Compare $r_{B}^{*}$ and $r_{D}^{*}$, we can have

$$ \begin{aligned} & \frac{{((1 - \theta )\alpha + \beta \varepsilon )( - N + Y - (1 - \theta )\alpha c) + 2\beta w_{o} ((1 - \theta )\eta + \beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon )}}{{(1 - \theta )\eta + \beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon }} \\ & \ge \frac{{((1 - \theta )\alpha + A\beta \varepsilon )( - A^{2} N + AY - (1 - \theta )\alpha c) + 2A\beta w_{o} ((1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon ))}}{{(1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon )}}, \\ \end{aligned} $$

$r_{D}^{ * } \le r_{B}^{ * }$. We can rewrite the inequality to,

$$ \begin{aligned} & 4(A^{2} - 1)(1 - \theta )^{2} \eta \alpha \beta \varepsilon w_{o} + (1 - A)(1 - \theta )^{2} \eta \alpha Y + (1 - A^{2} )(1 - \theta )\eta \beta \varepsilon Y \\ & \quad+ (A - 1)(1 - \theta )^{2} \alpha \eta \beta \varepsilon c + 2(1 - A)(1 - \theta )^{2} \eta^{2} \beta w_{o} + (A^{2} - A)\beta \varepsilon^{2} (1 - \theta )\alpha Y \\ & \quad+ (1 - A^{2} )\beta \varepsilon^{2} (1 - \theta )^{2} \alpha^{2} c + (A - A^{2})\beta^{2} \varepsilon^{3} (1 - \theta )\alpha c \\ & \quad + 2(A^{2} - A)(1 - \theta )\eta \beta^{2} \varepsilon^{2} w_{o} + 4(A - A^{2} )(1 - \theta )^{2} \alpha^{2} \beta \varepsilon^{2} w_{o} \\ & \quad+ (A - 1)(1 - \theta )^{3} \alpha^{3} \varepsilon c + (A^{2} - A)(1 - \theta )\alpha \beta \varepsilon^{2} Y \ge 0. \\ \end{aligned} $$

We divide both sides by $(1 - A)$ and shift the items, we can get if

$$ 0 \le A \le \frac{{(1 - \theta )\eta Y((1 - \theta )\alpha + \beta \varepsilon ) - 2(1 - \theta )^{2} \eta \beta w_{o} (2\alpha \varepsilon - \eta ) + (1 - \theta )^{2} \alpha \varepsilon c(\alpha \beta \varepsilon - \beta \eta - (1 - \theta )\alpha^{2} )}}{{(1 - \theta )\beta \varepsilon Y(2\alpha \varepsilon - \eta ) + 2(1 - \theta )\beta \varepsilon w_{o} (\beta \varepsilon \eta - 2(1 - \theta )\alpha (\alpha \varepsilon - \eta )) - \beta \varepsilon^{2} (1 - \theta )\alpha c((1 - \theta )\alpha + \beta \varepsilon )}} , $$

then $r_{D}^{ * } \le r_{B}^{ * }$; otherwise, $r_{D}^{ * } > r_{B}^{ * }$. □

1.5 Proof of Theorem 5

Recall $s_{I}^{ * } { = }\left( {\frac{K - [K + ((1 - \theta )\alpha - \beta \varepsilon )(c - v)]\lambda }{L - [L - 2(1 - \theta )\eta ]\lambda }} \right)^{ + }$ and $s_{D}^{ * } = \left( {\frac{{ - A^{2} N + AY - (1 - \theta )\alpha c}}{{2(1 - \theta )\eta + 2A^{2} \beta \varepsilon^{2} - 2A(1 - \theta )\alpha \varepsilon }}} \right)^{ + }$.

Let $G = [(1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon ]$,$J = [ - A^{2} N + AY - (1 - \theta )\alpha c]$ and $Z = [K + ((1 - \theta )\alpha - \beta \varepsilon )(c - v)]$.

We find that if $\frac{2KG - JL}{2ZG - J[L - 2(1 - \theta )\eta ]} \le \lambda \le 1$, then $s_{D}^{ * } \ge s_{I}^{ * }$; otherwise, $s_{D}^{ * } < s_{I}^{ * }$.

For the retail price in the inspection,

$$ r_{I}^{ * } = \frac{{(1 - \theta )(\alpha s + b) + \beta (w_{o} + \varepsilon s - \theta v)}}{2\beta (1 - \theta )} $$

$$\begin{aligned} &= \frac{((1 - \theta )\alpha + \beta \varepsilon )K - ((1 - \theta )\alpha + \beta \varepsilon )[K + ((1 - \theta )\alpha - \beta \varepsilon )(c - v)]\lambda }{{4\beta (1 - \theta )((1 - \theta )\eta { + }\beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon ) - 4\beta (1 - \theta )[\beta \varepsilon^{2} - (1 - \theta )\varepsilon \alpha ]\lambda }} \\&\quad + \frac{{(1 - \theta )b + \beta (w_{o} - \theta v)}}{2\beta (1 - \theta )} \end{aligned}$$

Recall the expression of the optimal sale price in the deferred payment at Proof of Theorem 4, when $r_{D}^{ * } \ge r_{I}^{ * }$, we have,

$$ \begin{aligned} & \frac{{((1 - \theta )\alpha + \beta \varepsilon )K + 2\beta w_{o} ((1 - \theta )\eta { + }\beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon ) - [((1 - \theta )\alpha + \beta \varepsilon )[K + ((1 - \theta )\alpha - \beta \varepsilon )(c - v)] - 2\beta w_{o} \varepsilon ((1 - \theta )\alpha - \beta \varepsilon )]\lambda }}{{2((1 - \theta )\eta { + }\beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon ) - 2[\beta \varepsilon^{2} - (1 - \theta )\varepsilon \alpha ]\lambda }} \\ & \le \frac{{((1 - \theta )\alpha + A\beta \varepsilon )( - A^{2} N + AY - (1 - \theta )\alpha c) + 2A\beta w_{o} ((1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon )}}{{2((1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon )}} \\ \end{aligned} . $$

In order to simplify calculation, let $X = ((1 - \theta )\alpha + \beta \varepsilon )K + 2\beta w_{o} ((1 - \theta )\eta { + }\beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon )$,

$$ \begin{aligned} Q & = [((1 - \theta )\alpha + \beta \varepsilon )[K + ((1 - \theta )\alpha - \beta \varepsilon )(c - v)] - 2\beta w_{o} \varepsilon ((1 - \theta )\alpha - \beta \varepsilon )]\,\,{\text{and}} \\ H & = ((1 - \theta )\alpha + A\beta \varepsilon )( - A^{2} N + AY - (1 - \theta )\alpha c) + 2A\beta w_{o} ((1 - \theta )\eta \\&\quad + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon ) \end{aligned} , $$

the inequality can be simplified as,

$$ \frac{X - Q\lambda }{{2((1 - \theta )\eta { + }\beta \varepsilon^{2} - (1 - \theta )\alpha \varepsilon ) - 2[\beta \varepsilon^{2} - (1 - \theta )\varepsilon \alpha ]\lambda }} \le \frac{H}{{2((1 - \theta )\eta + A^{2} \beta \varepsilon^{2} - A(1 - \theta )\alpha \varepsilon )}} , $$

Thus, if $0 \le \lambda \le \frac{{(1 - \theta )\eta (H - X) + \beta \varepsilon^{2} (H - A^{2} X) + (1 - \theta )\alpha \varepsilon (AX - H)}}{{-(1 - \theta )\eta Q + \beta \varepsilon^{2} (H - A^{2} Q) + (1 - \theta )\alpha \varepsilon (AQ - H)}}$, $r_{D}^{ * } \ge r_{I}^{ * }$; otherwise, $r_{D}^{ * } < r_{I}^{ * }$.(Q.E.D.)