Implementing the United Nations sustainable development Goals to supply chains with behavioral consumers

Abstract

We identify that the United Nations (UN) Sustainable Development Goals (SDGs) of 7, 9, 12 pertaining to industry investment, innovation, affordability, clean product, and responsible consumption/production are relevant for the firms operating in closed-loop supply chain (CLSC) structures. We show how these goals can be implemented into a CLSC in which downstream manufacturer serves different types of consumers and engages in vertical relations with a supplier in the upstream. While the supplier invests in a green component, the manufacturer produces the final green product. The manufacturer diversifies its customer base and meet their demands through contracts and a variety of pricing options. Specifically, at the end of the supply chain there are three different customer groups: “contract customers”, “green-conscious customers”, and “wholesale market customers”. While the firms execute sustainability Goals of 7 and 9, the behavioral consumers who are the contract and green-conscious customers are altruistic, consume responsibly, and hence fulfill the Goal 12. We show that green consciousness and altruism play an important role on firm profitability and production. Furthermore, the SDGs proposed by the UN have welfare-improving implications. In particular, when the SDGs are implemented, all parties (firms and consumers) are better off compared to the benchmark case under which no SDGs are applied.

Appendix

Proof of Proposition 1:

Assuming that all three goals (of 7,9, and 12) are executed the optimality conditions are as follows.

The first order necessary condition with respect to optimal output in market T1 is

$\frac{\partial E\Pi ^{M}}{\partial q_{1}}=p_{1}-2(c_{M}-d_{I})q_{1}-w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})=0$. The second order condition satisfies $\frac{\partial ^2\Pi ^{M}}{\partial q_{1}^2}=-2(c_{M}-d_{I})<0$ because $c_{M}-d_{I}>0$ by assumption. This implies

$$\begin{aligned} q_{1}(w,Ep_{3})=\frac{p_{1}-w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})}{2(c_{M}-d_{I})} \end{aligned}$$

(32)

The expected profit maximizing level of production in market T2 satisfies

$\frac{\partial E\Pi ^{M}}{\partial q_{2}}=a_{2}-2b_{2}q_{2}+c_{2}eI-2(c_{M}-d_{I})q_{2}-w=0$. The second order condition satisfies $\frac{\partial ^2\Pi ^{M}}{\partial q_{2}^2}=-2(b_{2}+c_{M}-d_{I})<0$ because $c_{M}-d_{I}>0$ and $b_{2}>0$ by assumption. This implies

$$\begin{aligned} q_{2}(w,I)=\frac{a_{2}+c_{2}eI-w}{2(c_{M}-d_{I})+2b_{2}} \end{aligned}$$

(33)

In the wholesale market T3, the optimal output satisfies

$\frac{\partial E\Pi ^{M}}{\partial q_{3}}=Ep_{3}-2(c_{M}-d_{I})q_{3}-w=0$. The second order condition is $\frac{\partial ^2\Pi ^{M}}{\partial q_{3}^2}=-2(c_{M}-d_{I})<0$ because $c_{M}-d_{I}>0$ by assumption. This implies

$$\begin{aligned} q_{3}(w,Ep_{3})=\frac{Ep_{3}-w}{2(c_{M}-d_{I})} \end{aligned}$$

(34)

The outputs in (32)–(34) are the best response functions of M for a given level of wholesale price w and the investment I chosen by S to satisfy the Goal 9. Observe that the Goal 7 leads to higher outputs in the respective customer groups.

In the upstream layer of the industry, given the production strategies of M, S will maximize its profit function in (11) to choose its investment level (Goal 9) as well as its wholesale price, which will be affected by the SGs in the CLSC:

$$\begin{aligned} \underset{w,\,I}{max}\Pi ^{S}=(q_{1}(w)+q_{2}(w,\,I)+q_{3}(w))(w-(f_{S}-f_{I}))-D(I) \end{aligned}$$

The first order necessary condition with respect to wholesale price is

$\frac{\partial \Pi ^{S}}{\partial w}=(w-(f_{S}-f_{I}))[-2/2(c_{M}-d_{I})-1/(2(c_{M}-d_{I})+2b_{2})]+(q_{1}(w)+q_{2}(w,\,I)+q_{3}(w))=0$. The second order condition is $\frac{\partial ^2\Pi ^{S}}{\partial w^2}=-[6(c_{M}-d_{I})+4b_{2}]/[4(c_{M}-d_{I})(c_{M}-d_{I}+b_{2})]<0$ because $c_{M}-d_{I}>0$ by assumption. Here the total output produced by the S or the M in all markets is

$$\begin{aligned} \sum _{i=1}^{3}q_{i}(w,I)=\frac{(c_{M}-d_{I}+b_{2})[p_{1}+Ep_{3}-2w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+(c_{M}-d_{I})[a_{2}+c_{2}eI-w]}{2(c_{M}-d_{I})(c_{M}-d_{I}+b_{2})}\nonumber \\ \end{aligned}$$

(35)

Then the equilibrium wholesale price as a function of the “green investment” is

$$\begin{aligned} w(I)=\frac{(2b_{2}+3(c_{M}-d_{I}))(f_{S}-f_{I})+((c_{M}-d_{I})+b_{2})[p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+(c_{M}-d_{I})(a_{2}+c_{2}eI)}{2(2b_{2}+3(c_{M}-d_{I}))}\nonumber \\ \end{aligned}$$

(36)

Observe that the wholesale price is increasing in the expected price ($Ep_{3}$) charged to the wholesale market customers T3, and increasing in greenness level of the product (eI), defined in expression (4).

The profit maximizing level of green investment satisfies

$\frac{\partial \Pi ^{S}}{\partial I}=(w-(f_{S}-f_{I}))\frac{\partial q_{2}}{\partial I}-h_{1}I=0$

The second order condition satisfies $\frac{\partial ^2\Pi ^{S}}{\partial I ^2}=-h_{1}<0$ because $h_{1}>0$ by assumption. This implies

$$\begin{aligned} I(w)=\frac{(w-(f_{S}-f_{I}))c_{2}e}{2h_{1}(c_{M}-d_{I}+b_{2})} \end{aligned}$$

(37)

Solving (36) and (37) together we obtain Stackelberg equilibrium wholesale price and investment as a function of model parameters.

The expression in (36) becomes

$$\begin{aligned} w=\frac{2h_{1}K_{0}[K_{1}f+K_{0}(p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1}))+ca_{2}]-cfc_{2}^{2}e^{2}}{4h_{1}K_{1}K_{0}-cc_{2}^{2}e^{2}} \end{aligned}$$

(38)

where $c=(c_{M}-d_{I})$, $f=(f_{S}-f_{I})$, $K_{0}=c+b_{2}$, and $K_{1}=3c+2b_{2}$.

Note that the equilibrium wholesale price w charged to the manufacturer increases in retail prices. $\square$

Proof of Lemma 1:

This case boils down to involve no investment, $I=0$, because the Goal 9 is not pursued. Then, demand, return, and cost functions reach to their reduced forms as such the following model parameters become zero: $\gamma =c_{2}=e=f_{I}=d_{I}=0$. Furthermore, $h_{1}=0$ which is implied by zero investment cost so that $D(I)=0$.

Inserting these parameter values into the equilibrium conditions in the proof of Proposition 1 we obtain that

$$\begin{aligned} q_{1}(w)=\frac{p_{1}-w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})}{2c_{M}} \end{aligned}$$

The production for the second market is

$$\begin{aligned} q_{2}(w)=\frac{a_{2}-w}{2(c_{M}+b_{2})} \end{aligned}$$

In market $T_{3}$, the optimal output is

$$\begin{aligned} q_{3}(w)=\frac{Ep_{3}-w}{2c_{M}}. \end{aligned}$$

The total output produced for all markets is

$$\begin{aligned} {\displaystyle \sum _{i=1}^{3}}q_{i}(w)=\frac{(c_{M}+b_{2})[p_{1}+Ep_{3}-2w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+(c_{M})[a_{2}-w]}{2(c_{M})(c_{M}+b_{2})} \end{aligned}$$

(39)

When S does not invest its objective function becomes

$$\begin{aligned} \underset{w}{max}\Pi ^{S}=(q_{1}(w)+q_{2}(w)+q_{3}(w))(w-f_{S}) \end{aligned}$$

The first order necessary condition with respect to the wholesale price is

$\frac{\partial \Pi ^{S}}{\partial w}=(w-f_{S})[-2/2c_{M}-1/2(c_{M}+b_{2}]+(q_{1}(w)+q_{2}(w)+q_{3}(w))=0$

which implies that

$$\begin{aligned} w=\frac{(2b_{2}+3c_{M})f_{S}+(c_{M}+b_{2})[p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+c_{M}a_{2}}{2(2b_{2}+3c_{M})} \end{aligned}$$

Hence, the result. $\square$

Proof of Proposition 2:

When the investment (Goal 9) provides benefit only to returns and demand so that the Goal 12 is satisfied, all model parameters will be positive but $f_{I}=d_{I}=0$, which are the cost improvement parameters. Inserting these parameters into the equilibrium outcomes in the proof of Proposition 1 we obtain that in the T1 market

$$\begin{aligned} q_{1}(w,Ep_{3})=\frac{p_{1}-w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})}{2c_{M}} \end{aligned}$$

The expected profit maximizing level of production in market T2 satisfies

$$\begin{aligned} q_{2}(w,I)=\frac{a_{2}+c_{2}eI-w}{2(c_{M}+b_{2})} \end{aligned}$$

In the wholesale market T3, the optimal output satisfies

$$\begin{aligned} q_{3}(w,Ep_{3})=\frac{Ep_{3}-w}{2c_{M}} \end{aligned}$$

In the upstream layer of the industry, the equilibrium wholesale price as a function of the “green investment” is

$$\begin{aligned} w(I)=\frac{(2b_{2}+3c_{M})f_{S}+(c_{M}+b_{2})[p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+c_{M}(a_{2}+c_{2}eI)}{2(2b_{2}+3c_{M})} \end{aligned}$$

The profit maximizing level of innovation satisfies

$\frac{\partial \Pi ^{S}}{\partial I}=(w-f_{S})\frac{\partial q_{2}}{\partial I}-h_{1}I=0$

which implies

$$\begin{aligned} I(w)=\frac{(w-f_{S})c_{2}e}{2h_{1}(c_{M}+b_{2})} \end{aligned}$$

Inserting this investment function into the wholesale price results in

$$\begin{aligned} w=\frac{2h_{1}K_{2}[K_{3}f_{S}+K_{2}(p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1}))+ca_{2}]-c_{M}f_{S}c_{2}^{2}e^{2}}{4h_{1}K_{2}K_{3}-c_{M}c_{2}^{2}e^{2}} \end{aligned}$$

where $K_{2}=c_{M}+b_{2}$, and $K_{3}=3c_{M}+2b_{2}$. $\square$

Proof of Proposition 3:

M will maximize its expected profit function:

$$\begin{aligned} E\Pi ^{M}(q_{1},q_{2},q_{3})&=q_{1}p_{1}+q_{2}p_{2}(q_{2})+q_{3}Ep_{3}(Q)-C(q_{1},q_{2},q_{3})\\&\quad +(r_{1}(q_{1})+r_{2}(I))(Ep_{3}-p_{1}-\epsilon _{1}) \end{aligned}$$

The first order necessary condition with respect to output in market T1 is

$\frac{\partial E\Pi ^{M}}{\partial q_{1}}=p_{1}-2(c_{M}-d_{I})q_{1}-w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})=0$

This condition implies

$$\begin{aligned} q_{1}(w,Ep_{3})=\frac{p_{1}-w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})}{2(c_{M}-d_{I})} \end{aligned}$$

The expected profit maximizing level of production in market T2 satisfies

$\frac{\partial E\Pi ^{M}}{\partial q_{2}}=a_{2}-2b_{2}q_{2}-2(c_{M}-d_{I})q_{2}-w=0$

which implies

$$\begin{aligned} q_{2}(w)=\frac{a_{2}-w}{2(c_{M}-d_{I})+2b_{2}} \end{aligned}$$

In the wholesale market T3, the optimal output satisfies

$\frac{\partial E\Pi ^{M}}{\partial q_{3}}=Ep_{3}-2(c_{M}-d_{I})q_{3}-w=0$

which implies

$$\begin{aligned} q_{3}(w,Ep_{3})=\frac{Ep_{3}-w}{2(c_{M}-d_{I})} \end{aligned}$$

Given the production strategies of M, S will maximize its profit function:

$$\begin{aligned} \underset{w,\,I}{max}\Pi ^{S}=(q_{1}(w)+q_{2}(w)+q_{3}(w))(w-(f_{S}-f_{I}))-D(I) \end{aligned}$$

The first order necessary condition with respect to wholesale price is

$\frac{\partial \Pi ^{S}}{\partial w}=(w-(f_{S}-f_{I}))[-2/2(c_{M}-d_{I})-1/(2(c_{M}-d_{I})+2b_{2})]+(q_{1}(w)+q_{2}(w,\,I)+q_{3}(w))=0$

where the total output in all markets is

$$\begin{aligned} {\displaystyle \sum _{i=1}^{3}}q_{i}(w)=\frac{(c_{M}-d_{I}+b_{2})[p_{1}+Ep_{3}-2w+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+(c_{M}-d_{I})[a_{2}-w]}{2(c_{M}-d_{I})(c_{M}-d_{I}+b_{2})}. \end{aligned}$$

Then the equilibrium wholesale price as a function of the “green investment” is

$$\begin{aligned} w=\frac{(2b_{2}+3(c_{M}-d_{I}))(f_{S}-f_{I})+((c_{M}-d_{I})+b_{2})[p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+(c_{M}-d_{I})(a_{2})}{2(2b_{2}+3(c_{M}-d_{I}))} \end{aligned}$$

Observe that the wholesale price is increasing in the expected price ($Ep_{3}$) charged to the wholesale market customers.

The profit maximizing level of innovation satisfies

$\frac{\partial \Pi ^{S}}{\partial I}=-h_{1}I<0$

which implies $I=0$.

No investment by the supplier will imply that the CLSC will experience no cost reductions. Therefore, the cost coefficients will be $d_{I}=f_{I}=0$. Consequently the result in the proposition holds. $\square$

Proof of Proposition 4:

When the industry innovation (i.e., investment) only impacts returns and demand (Proposition 2), so that the production costs are intact, the following parameters will be zero: $f_{I}=d_{I}=0$. Clearly $c_{2},\,h_{1},\,e\,>0$ which are demand and investment function coefficients.

At $f_{I}=d_{I}=0$ the wholesale price in (21) gets

$$\begin{aligned} w_{P2}(I)=\frac{(2b_{2}+3c_{M})f_{S}+(c_{M}+b_{2})[p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+c_{M}(a_{2}+c_{2}eI_{P2})}{2(2b_{2}+3c_{M})} \end{aligned}$$

When SDGs are not implemented we know from Lemma 1 that

$$\begin{aligned} w_{L1}=\frac{(2b_{2}+3c_{M})f_{S}+(c_{M}+b_{2})[p_{1}+Ep_{3}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})]+c_{M}a_{2}}{2(2b_{2}+3c_{M})} \end{aligned}$$

Clearly $w_{P2}>w_{L1}$ because $I_{P2}>0$ in Proposition 2, and $c_{2},\,e>0$, and $p_{1}$ is fixed at the contract, and $Ep_{3}$ is the same market shock in both cases.

As the wholesale price is higher in the upstream, downstream prices will also be higher because output prices increase in input prices. Alternatively, when SDGs are not applied, the outputs satisfy, as in Lemma 1,

$$\begin{aligned} q_{1,L1}(w)=\frac{p_{1}-w_{L1}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})}{2c_{M}} \end{aligned}$$

The production for the second market is

$$\begin{aligned} q_{2,L1}(w)=\frac{a_{2}-w_{L1}}{2(c_{M}+b_{2})} \end{aligned}$$

In the third market the optimal output is

$$\begin{aligned} q_{3,L1}(w)=\frac{Ep_{3}-w_{L1}}{2c_{M}}. \end{aligned}$$

When the investment is strategic so that it benefits the demand and returns in market $T_2$, we obtain

$$\begin{aligned} q_{1,P2}(w)= & {} \frac{p_{1}-w_{P2}+\alpha (Ep_{3}-p_{1}-\epsilon _{1})}{2c_{M}} \\ q_{2,P2}(w,I)= & {} \frac{a_{2}+c_{2}eI_{P2}-w_{P2}}{2(c_{M}+b_{2})} \\ q_{3,P2}(w)= & {} \frac{Ep_{3}-w_{P2}}{2c_{M}} \end{aligned}$$

Because $w_{P2}(I)>w_{L1}$ and $I_{P2}>0$ we obtain that

$q_{1,L1}>q_{1,P2}$ and $q_{3,L1}>q_{3,P2}$. Therefore, the M produces less in markets T1 and T3. As M is a price taker in market T3 and already set the price in market T1 through contract, prices $p_{1}$ and $p_{3}$ will be intact.

However, the M will produce more and charge a higher price when it invests because this shifts demand upward. Therefore, $q_{2,P2}>q_{2,L1}$ and $p_{2,P2}>p_{2,L1}$. Alternatively, as wholesale price increases, that is $w_{P2}>w_{L1}$, the retail price goes up $p_{2,P2}>p_{2,L1}$ too.

An alternative proof to show that $q_{2,P2}>q_{2,L1}$ is as follows:

It holds that$\underset{I_{P2}\rightarrow 0}{lim}\,q_{2,P2}=q_{2,L1}$ and $\partial q_{2,P2}/\partial I_{P2}>0$, that is the output with SDGs is increasing in investment and the lower bound of output with SDGs approaches the output without SDGs, therefore $q_{2,P2}>q_{2,L1}$ must hold.

The result with respect to returns directly follows from the ranking of outputs in T1 market (higher output higher returns) and investment ranking in T2 market (higher investment lower returns).

Finally, using the limit argument $\underset{I_{P2}\rightarrow 0}{lim}\,\Pi _{P2}^{S}=\Pi _{L1}^{S}$ furthermore $\Pi _{P2}^{S}$ is concave in investment $I_{P2}$ as it gets its maximum in Proposition 2. Therefore, an incremental investment by S under the SDGs would lead its profit to go up. Similar arguments also hold for M. $\square$