Closed-loop supply chain coordination through incentives with asymmetric information

A closed-loop supply chain seeks to enhance the consumers’ environmental consciousness to increase both the profits and the return of past-sold products. Even though, firms have misaligned interests for closing the loop: while all firms exploit consumers environmental consciousness to increase sales, only manufacturers use it for appropriating of returns’ residual value. Starting from a benchmark (no-incentive) scenario where a manufacturer (M) is the leader and a retailer (R) is the follower, we develop two incentive games through a profit-sharing contract to align firms’ motivations for closing the loop. In both incentive games, the incentive takes the form of a share of profits that M transfers to R. Our question is how the sharing fraction should be determined to make both players economically better-off. The first incentive game assumes that R has no-information on the sharing parameter, which is determined by M after R sets her strategies; thus the incentive has an endogenous nature. In the second incentive game the sharing parameter is common knowledge and both players know its values before the game starts, thus the incentive has an exogenous nature. We find that an endogenous incentive is never more economically and environmentally convenient than a no-incentive game. In contrast, an exogenous incentive can make both players economically better-off inside specific sharing parameter ranges. Nevertheless, when other forces (e.g., competition or legislation) impose the adoption of a profit-sharing contract, M should supply an endogenous incentive when the exogenous share is either too high or too low.

1. Later, we will refer to Endogenous incentive with the superscript I and to Exogenous incentive with the superscript II.

Authors and Affiliations

1. Department of Operations Management, ESSEC Business School, Avenue Bernard Hirsch, B.P. 105, 95021, Paris, France

   Pietro De Giovanni

Corresponding author

Correspondence to Pietro De Giovanni.

Appendix

Proof of Proposition 1

We need to establish the existence of bounded and continuously differentiable value functions \(V_M \left( G \right) ,V_R \left( G \right) \) such that there exists a unique solution G(t) to Eq. (1) and the HJB equations. The players’ HJBs in the benchmark scenario are:

Since the game is played à la Stackelberg and M is the leader, we first determine the necessary conditions of R as:

Substituting (27) into M’s HBJ equation, Eq. (25) becomes:

and therefore M’s necessary conditions are:

Using Eqs. (29), the pricing strategy is given by:

Replacing the players’ strategies inside Eqs. (26) and (28), we obtain:

We conjecture linear value functions, \(V_M =l_1 G+l_2 \) and \(V_R =l_3 G+l_4\) where \(l_1, l_2, l_3 \), and \(l_4 \) are the constant parameters to be identified. Substituting \(V_M \) and \(V_R \) and their derivatives inside Eqs. (31) and (32) it gives:

while we can identify the parameter values such as:

which are all strictly positive; thus, the result shows concave profit functions with respect to the decision variables and an unique equilibrium that maximizes players’ objective functions. \(\square \)

Proof of Proposition 2

We need to establish the existence of bounded and continuously differentiable value functions \(V_M^I \left( {G^{I}} \right) ,V_R^I \left( {G^{I}} \right) \) such that there exists a unique solution \(G^{I}(t)\) to Eq. (1) and the HJB equations. The players’ HJBs in Scenario I are:

As in the benchmark scenario, the game is played à la Stackelberg and M is the leader. First we substitute the wholesale price strategy of the benchmark scenario inside Eqs. (36) and (37), thus we determine the necessary conditions of R as:

Substituting (38) into M’s HBJ equation, Eq. (36) becomes:

and therefore M’s necessary conditions are:

Then, the pricing strategy becomes:

Replacing Eqs. (40) and (41) inside Eq. (37) and Eq. (39), we obtain:

We conjecture linear value functions, \(V_M^I =n_1 G^{I}+n_2 \) and \(V_R^I =n_3 G^{I}+n_4 \) where \(n_1, n_2, n_3 \), and \(n_4 \) are the constant parameters to be identified. Substituting \(V_M^I \) and \(V_R^I \) and their derivatives inside Eq. (42) and Eq. (43) it gives:

while we can identify the parameter values such as:

which are all strictly positive; thus, the result shows concave profit functions with respect to the decision variables and an unique equilibrium that maximizes players’ objective functions. \(\square \)

Proof of Proposition 3

Hereby we follow the same procedure of the previous proof with the difference that the sharing parameter is exogenous. We need to establish the existence of bounded and continuously differentiable value functions \(V_M^{II} \left( {G^{II}} \right) ,V_R^{II} \left( {G^{II}} \right) \) such that there exists a unique solution \(G^{II}(t)\) to Eq. (1) and the HJB equations. The players’ HJBs in Scenario II are:

As in the benchmark scenario, the game is played à la Stackelberg and M is the leader. First the wholesale price in the benchmark scenario inside Eqs. (47) and (48), thus we determine the necessary conditions of R as:

Substituting (49) into M’s HBJ equations Eq. (48) becomes:

and therefore M’s necessary condition is:

Replacing the players’ strategies inside Eqs. (48) and (50), we obtain:

We conjecture linear value functions, \(V_M^{II} =m_1 G^{II}+m_2 \) and \(V_R^{II} =m_3 G^{II}+m_4 \)where \(m_1, m_2, m_3 \), and \(m_4 \) are the constant parameters to be identified. Substituting \(V_M^{II} \) and \(V_R^{II} \) and their derivatives inside Eqs. (52) and (53), it gives:

while we can identify the parameter values such as:

which are all strictly positive; thus, the result shows concave profit functions with respect to the decision variables and an unique equilibrium that maximizes players’ objective functions. \(\square \)

Proof of Proposition 9

Compute the difference \(V_M -V_M^I \) to show that \(V_M -V_M^I =\frac{c_L s\left[ {\left( {b^{2}\mu _M +a^{2}\mu _R} \right) \left( {\delta +\rho } \right) +a^{2}\mu _R \rho } \right] \Omega }{128\beta \delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}\) with

To proof that \(V_M -V_M^I >0\), we solve the third degree polynomial \(\Omega \) with respect to \(\theta +\beta s\Delta \). Among the three solutions, only one is feasible, that is \(\theta +\beta s\Delta =\frac{\beta \left( {3c_L s+2c_R h} \right) }{2}\). Therefore, the result \(V_M -V_M^I >0\) always holds because it meets the assumption \(\psi _1 \ge \beta c_L s\); further, computing the difference \(V_M -V_M^{II} \), we can demonstrate that:

1. (a)

   when \(\phi ^{II}=0,\quad V_M -V_M^{II} =\frac{\psi _1 \left[ {\mu _M b^{2}\left( {\rho +\delta } \right) \left( {\psi _1^3 -\psi _2^3} \right) +\mu _R a^{2}\left( {2\rho +\delta } \right) \psi _1 \left( {\psi _1^2 +\psi _2^2} \right) } \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\);

2. (b)

   when \(\phi ^{II}=1,\quad V_M -V_M^{II} =\frac{\psi _1^4 \left[ {\left( {\rho +\delta } \right) \left( {\mu _M b^{2}+\mu _R a^{2}} \right) +\rho \mu _R a^{2}} \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\);

3. (c)

   when

with \(\psi _2 =\psi _1 -2c_L s\beta \ge 0\), where define the interval inside which \(V_M -V_M^{II} \le 0\) holds.

Finally, compute the difference \(V_M^I -V_M^{II} \), we demonstrate that:

1. (a)

   when \(\phi ^{II}=0,V_M^I -V_M^{II} =\frac{\left( {V_M^I -V_M^{II}} \right) _{\left| {\phi ^{II}=1} \right. } -\psi _1 \psi _2^2 \left[ {a^{2}\rho \mu _R \psi _1 +\left( {\rho +\delta } \right) \left( {a^{2}\mu _R \psi _1 +b^{2}\mu _M \psi _2} \right) } \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\);

2. (b)

   when \(\phi ^{II}=1,V_M^I -V_M^{II} =\frac{\left( {\psi _1 -c_L s\beta } \right) ^{4}\left[ {\left( {\rho +\delta } \right) \left( {\mu _M b^{2}+\mu _R a^{2}} \right) +\rho \mu _R a^{2}} \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\);

3. (c)

   when ,

   $$\begin{aligned}&V_M^I -V_M^{II} \\&=\frac{\left( {V_M^I -V_M^{II}} \right) _{\left| {\phi ^{II}=1} \right. } -\psi _1 \left( {1-\phi ^{II}} \right) \left( {\psi _2 +\phi ^{II}\psi _1} \right) ^{2}\left[ {\left( {2\rho +\delta } \right) a^{2}\mu _R \psi _1 \left( {1-\phi ^{II}} \right) +\left( {\delta +\rho } \right) b^{2}\mu _M \left( {\psi _2 +\psi _1 \phi ^{II}} \right) } \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}\le 0, \end{aligned}$$

   where define the interval inside which \(V_M^I -V_M^{II} \le 0\) holds. \(\square \)

Proof of Proposition 10

Compute the difference \(V_R -V_R^I \) to show that \(V_R -V_R^I =\frac{c_L s\left[ {\left( {b^{2}\mu _M +4a^{2}\mu _R} \right) \left( {\delta +\rho } \right) +b^{2}\mu _M \rho } \right] \Omega }{512\beta \delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\) because \(\Omega >0\). The sign of the difference \(V_R -V_R^{II} \) depends on \(\phi ^{II}\). Fixing \(\psi _3 =4a^{2}\mu _R \left( {\delta +\rho } \right) +b^{2}\mu _M \left( {\delta +2\rho } \right) >0\), the following cases can be displayed:

1. (1)

   when \(\phi ^{II}=0, \quad V_R -V_R^{II} =\frac{\psi _3 \psi _1^4 -\psi _2^3 \left[ {4a^{2}\mu _R \psi _1 \left( {\rho +\delta } \right) +b^{2}\mu _M \psi _2 \left( {\delta +2\rho } \right) } \right] }{512\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\), as \(\psi _1 >\psi _2 \);

2. (2)

   when \(\phi ^{II}=1, V_R -V_R^{II} =\frac{\psi _3 \psi _1^4 -\left( {\psi _2 +\psi _1} \right) ^{4}b^{2}\mu _M \left( {\delta +2\rho } \right) }{512\beta \delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}<0\). Assume that \(\delta =\rho =\mu _M =\mu _R =a=b=1\); substituting for \(\psi _2 =\psi _1 -2c_L s\beta \), the numerator becomes \(11\psi _1^4 -48\left( {\psi _1 -c_L s\beta } \right) ^{4}\le 0\). Solving this polynomial with respect to \(\psi _1\) gives four solutions: the first two are not feasible, while the second two are feasible and negative. Therefore, \(V_R -V_R^{II} <0\) always holds.

Finally, cases (1) and (2) allow one to verify that \(V_R -V_R^{II} \left\{ {\begin{array}{ll} \ge 0&{}\quad \phi ^{II}\in \left[ {0,\bar{{\phi }}^{II}} \right] \\ <0&{}\quad \phi ^{II}\in \left( {\bar{{\phi }}^{II},1} \right] \\ \end{array}} \right. \).

The sign of the difference \(V_R^I -V_R^{II} \) also depends on \(\phi ^{II}\) and it results:

According to \(\phi ^{II}\), three cases may be displayed:

1. (1)

   when \(\phi ^{II}=0, V_R^I -V_R^{II} =\frac{\psi _3 \left( {\psi _1 -c_L s\beta } \right) ^{4}-\psi _2^3 \left[ {4a^{2}\mu _R \psi _1 \left( {\rho +\delta } \right) +b^{2}\mu _M \psi _2 \left( {\delta +2\rho } \right) } \right] }{516\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\, \psi _1 -c_L s\beta >\psi _2 \);

2. (2)

   when \(\phi ^{II}=1, V_R^I -V_R^{II} =\frac{\psi _3 \left( {\psi _1 -c_L s\beta } \right) ^{4}-\left( {\psi _2 +\psi _1 } \right) ^{4}b^{2}\mu _M \left( {\delta +2\rho } \right) }{516\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}<0\). Solving the numerator with respect to \(\psi _1 \) gives four solutions: two are not feasible, two are feasible and negative, \(V_R^I -V_R^{II} <0\) always holds;

3. (3)

   Cases (1) and (2) allow one to verify that \(V_R^I -V_R^{II} \left\{ {\begin{array}{ll} \ge 0&{}\quad \phi ^{II}\in \left[ {0,\bar{{\bar{{\phi }}}}^{II}} \right] \\ <0&{}\quad \phi ^{II}\in \left( {\bar{{\bar{{\phi }}}}^{II},1} \right] \\ \end{array}} \right. \). \(\square \)

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