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Peer Pressure in Voluntary Environmental Programs: a Case of the Bag Rewards Program

Abstract

This paper studies stores that voluntarily reward consumers for changing their convenience-oriented, waste-increasing consumption behaviors. We focus on shopping bag rewards programs which encourage consumers to substitute reusable bags for plastic bags. Instead of assuming that consumers have environmental consciousness, we consider the peer pressure of participating in rewards programs among consumers as the driving force increasing the number of consumers behaving in environmentally friendly ways. The results show that rewards programs can be effective in reducing plastic bag users in the long term and can increase profits for the stores implementing them. Unlike former studies suggesting a plastic bag ban or levy to reduce plastic bag use, these findings suggest that the policymakers should support these long-term-effective bag rewards programs by educating consumers about the value of using reusable bags and offering subsidies for stores that implement bag rewards programs.

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Notes

  1. 1.

    Lrb represents the consumer choice to shop in store L using a reusable bag; Lpb represents the consumer choice to shop in store L using a plastic bag; Hrb represents the consumer choice to shop in store H using a reusable bag; Hpb represents the consumer choice to shop in store H using a plastic bag.

  2. 2.

    We find that \( 3\left({a}_H-{a}_L-\gamma \right)>0 \), \( \left(\overline{\theta}+\underline{\theta}+c\right)\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}+\underline{\theta}\right)\gamma >0 \) and \( \underline{\theta}\gamma >0 \)

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Acknowledgments

I would like to show my gratitude to Dr. Ana Espinola-Arredondo for providing insight and expertise that greatly assisted the research. I also thank the editor and the anonymous reviewers for helpful comments.

Author information

Correspondence to Jingze Jiang.

Electronic supplementary material

Below is the link to the electronic supplementary material.

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Appendices

Appendix 1

  1. 1)

    Consumers who are the type within \( \left[\underline{\theta},\;{\theta}_L^{t=1}\right) \) choose to shop in store L with a reusable bag, and the remainder of consumers shopping at store L, whose type is higher than \( {\theta}_L^{t=1}, \) use a plastic bag.

    Let \( {\tilde{\theta}}_L^{t=1} \) be the type of consumers shopping at store L in period one, \( {\theta}_L^{t=1} \) be the type of consumers indifferent between two stores, then

    The utility for consumer type θ t = 1 is

    $$ {U}_{rb}=\frac{y}{\theta_L^{t=1}}+{a}_L-\frac{p_L}{\theta_L^{t=1}}+\left(\frac{r_L^{t=1}}{\theta_L^{t=1}}-\gamma \right)={U}_{pb}=\frac{y}{\theta_L^{t=1}}+{a}_L-\frac{p_L}{\theta_L^{t=1}} $$
    (25)

    therefore

    $$ {U}_{rb}-{U}_{pb}=\frac{r_L^{t=1}}{\theta_L^{t=1}}-\gamma =0 $$
    (26)

    The utility for consumer type \( {\tilde{\theta}}_L^{t=1} \) gains from using a reusable bag is

    $$ \tilde{U_{rb}}=\frac{y}{{\tilde{\theta}}_L^{t=1}}+{a}_L-\frac{p_L}{{\tilde{\theta}}_L^{t=1}}+\left(\frac{r_L^{t=1}}{{\tilde{\theta}}_L^{t=1}}-\gamma \right) $$
    (27)

    The utility for consumer type \( {\tilde{\theta}}_L^{t=1} \) gains from using a plastic bag is

    $$ \tilde{U_{pb}}=\frac{y}{{\tilde{\theta}}_L^{t=1}}+{a}_L-\frac{p_L}{\theta_L^{t=1}} $$
    (28)

    therefore

    $$ \tilde{U_{rb}}-\tilde{U_{pb}}=\frac{r_L^{t=1}}{{\tilde{\theta}}_L^{t=1}}-\gamma $$
    (29)

    If \( {\tilde{\theta}}_L^{t=1}\le\ {\theta}_L^{t=1} \), then we have

    $$ \tilde{U_{rb}}-\tilde{U_{pb}}=\frac{r_L^{t=1}}{{\tilde{\theta}}_L^{t=1}}-\gamma \ge \frac{r_L^{t=1}}{\theta_L^{t=1}}-\gamma =0 $$
    (30)

    or

    $$ \tilde{U_{rb}}\ge \tilde{U_{pb}} $$
    (31)

    so the consumer type \( {\tilde{\theta}}_L^{t=1} \) has larger utility for using a reusable bag.

    If \( {\tilde{\theta}}_L^{t=1} > {\theta}_L^{t=1} \), then we have

    $$ \tilde{U_{rb}}-\tilde{U_{pb}}=\frac{r_L^{t=1}}{{\tilde{\theta}}_L^{t=1}}-\gamma <\frac{r_L^{t=1}}{\theta_L^{t=1}}-\gamma =0 $$
    (32)

    or

    $$ \tilde{U_{rb}}<\tilde{U_{pb}} $$
    (33)

    so the consumer type \( {\tilde{\theta}}_L^{t=1} \) has larger utility for using a plastic bag.

    For store L in period two, and store H in both periods, we can get the similar conclusion through the similar proof.

  2. 2)

    Proof for Lemma 1

    First of all, \( {\theta}^t \) and \( {\theta}_i^t \) should be in the interval \( \left[\underline{\theta}, \overline{\theta}\right] \), since \( \left[\underline{\theta}, \overline{\theta}\right] \) is the set of possible consumer types.

    Consumers with a type smaller than θ t shop at store L, and those with type larger than θ t shop at store H; therefore, consumers’ type θ L t indifferent between using a reusable bag and not using a reusable bag at store L must be no larger than θ t, and consumers’ type θ H t indifferent between using a reusable bag and not using a reusable bag at store L must be no smaller than θ t. Then we have

    $$ \underline{\theta}\le {\theta}_L^t\le {\theta}^t\le\ {\theta}_H^t\le \overline{\theta} $$
    (34)

    Moreover, Appendix 1) shows that in store i, consumers with type smaller than θ i t choose to use a reusable bag and those with a larger than θ i t choose to use a plastic bag. Therefore, θ i t cannot be equal to either \( \underline{\theta} \) or \( \overline{\theta} \) or θ t as to satisfy the existence of reusable bag users and plastic bag users. We conclude that

    $$ \underline{\theta}<{\theta}_L^t<{\theta}^t < {\theta}_H^t<\overline{\theta} $$
    (35)
  3. 3)

    Proof for the number of reusable bag users in period two in terms of the number of reusable bag users in period one.

    1. 3.1)

      Store L

      The number of reusable bag users in period two is

      $$ {\displaystyle {\int}_{\underline{\theta}}^{\theta_L^{t=2}}dF\left(\theta \right) = \frac{\theta_L^{t=2}-\underline{\theta}}{\left(\overline{\theta}-\underline{\theta}\right)}} $$
      (36)

      where

      $$ {\theta}_L^{t=2}=\frac{r_L^{t=2}+\lambda {\displaystyle {\int}_{\underline{\theta}}^{\theta_L^{t=1}}}dF\left(\theta \right)}{\gamma } $$
      (37)

      And optimal reward in period two is

      $$ {r}_L^{t=2}=\frac{\underline{\theta}\kern0.1em \gamma +{c}^b-\lambda {\displaystyle {\int}_{\underline{\theta}}^{\theta_L^{t=1}}}dF\left(\theta \right)}{2} $$
      (38)

      Then we plug 37 and r H t = 2 into 36, and we obtain

      $$ {\displaystyle {\int}_{\underline{\theta}}^{\theta_L^{t=2}}dF\left(\theta \right)=\left(1+\frac{\lambda }{2\left(\overline{\theta}-\underline{\theta}\right)\gamma}\right){\displaystyle {\int}_{\underline{\theta}}^{\theta_L^{t=1}}dF\left(\theta \right)}} $$
      (39)
    2. 3.2)

      Store H

      The number of reusable bag users in period two is

      $$ {\displaystyle {\int}_{\theta^{t=2}}^{\theta_H^{t=2}}dF\left(\theta \right) = \frac{\theta_H^{t=2}-{\theta}^{t=2}}{\left(\overline{\theta}-\underline{\theta}\right)}} $$
      (40)

      where

      $$ {\theta}_H^{t=2}=\frac{r_H^{t=2}+\lambda {\displaystyle {\int}_{\theta^{t=1}}^{\theta_H^{t=1}}}dF\left(\theta \right)}{\gamma } $$
      (41)

      And the optimal reward in period two is

      $$ {r}_H^{t=2}=\frac{c^b+\overline{\theta}\gamma -\lambda {\displaystyle {\int}_{\theta^{t=1}}^{\theta_H^{t=1}}}dF\left(\theta \right)}{2} $$
      (42)

      The consumer type indifferent between two stores in period two is

      $$ {\theta}^{t=2}=\frac{\left(\overline{\theta}+\underline{\theta}+c\right)\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}+\underline{\theta}\right)\gamma -\lambda {\displaystyle {\int}_{\theta^{t=1}}^{\theta_H^{t=1}}}dF\left(\theta \right)}{3\left({a}_H-{a}_L-\gamma \right)} $$
      (43)

      Then we plug 43, 42 and 41 into 40, and obtain

      $$ {\displaystyle {\int}_{\theta^{t=2}}^{\theta_H^{t=2}}dF\left(\theta \right)=\left(1+\frac{\lambda }{2\left(\overline{\theta}-\underline{\theta}\right)\gamma }+\frac{\lambda }{3{\left(\overline{\theta}-\underline{\theta}\right)}^2\left({a}_H-{a}_L-\gamma \right)}\right){\displaystyle {\int}_{\theta^{t=1}}^{\theta_H^{t=1}}dF\left(\theta \right)}} $$
      (44)
  4. 4)

    Proof for Corollary 1

    Proof. The optimal rewards offered by the store i at period one and two are \( {r}_L^{t=2}=\frac{\underline{\theta}\gamma +{c}^b-E\left({\varTheta}_{Lrb}^{t=1}\right)}{2} \) and \( {r}_L^{t=1}=\frac{\underline{\theta}\gamma +{c}^b}{2} \) respectively, hence \( {r}_L^{t=2}={r}_L^{t=1}-\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{2} \). Moreover, \( E\left({\varTheta}_{Lrb}^{t=1}\right) \) is the peer pressure in the period two. Therefore, given \( {r}_L^{t=1} \), the larger \( E\left({\varTheta}_{irb}^{t=1}\right) \) is, the smaller \( {r}_L^{t=2} \) is.

    2). The optimal rewards offered by the store H at period one and two are \( {r}_H^{t=2}=\frac{\overline{\theta}\gamma +{c}^b-E\left({\varTheta}_{Hrb}^{t=1}\right)}{2} \) and \( {r}_H^{t=1}=\frac{\overline{\theta}\gamma +{c}^b}{2} \) respectively, hence \( {r}_H^{t=2}={r}_H^{t=1}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{2} \). Moreover, \( E\left({\varTheta}_{Hrb}^{t=1}\right) \) is the peer pressure in the period two. Therefore, given \( {r}_H^{t=1} \), the larger \( E\left({\varTheta}_{Hrb}^{t=1}\right) \) is, the smaller \( {r}_H^{t=2} \) is.

  5. 5)

    Proof for Lemma 3

    1. i.

      The marginal profits with respect to reusable bag users at store L in period one

      $$ \mathrm{Baseline}:\kern7em \frac{\left(\overline{\theta}-2\underline{\theta}+c\right)\left({a}_H-{a}_L\right)}{3} $$
      (45)
      $$ \mathrm{Reward}:\kern6em \frac{\left(\overline{\theta}-2\underline{\theta}+c\right)\left({a}_H-{a}_L\right)}{3}+\frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{6} $$
      (46)

      The difference between 45 and 46 is that 46 has one more part, \( \frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{6} \). Moreover, \( \frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{6}=\frac{C^b-\overline{\theta}\gamma -\left(\overline{\theta}-\underline{\theta}\right)\gamma, }{6} \) , where \( -\left(\overline{\theta}-\underline{\theta}\right)\gamma <0. \) Since we show \( {\theta}_H^{t=1}=\frac{r_H^{t=1}}{\gamma }=\frac{c^b+\overline{\theta}\gamma }{2}<\overline{\theta} \) in lemma 2, \( {c}^b<\overline{\theta}\gamma \), or \( {c}^b-\overline{\theta}\gamma <0 \). Therefore \( \frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{6}<0 \), and we conclude that 45 is larger than 46, or the marginal profit with respect to reusable bag users at store L in period one is larger in the baseline.

    2. ii.

      The marginal profits with respect to plastic bag users at store L in period one

      $$ \mathrm{Baseline}:\kern7em \frac{\left(\overline{\theta}-2\underline{\theta}+c\right)\left({a}_H-{a}_L\right)}{3} $$
      (47)
      $$ \mathrm{Reward}:\kern6em \frac{\left(\overline{\theta}-2\underline{\theta}+c\right)\left({a}_H-{a}_L\right)}{3}-\frac{C^b+\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3} $$
      (48)

      The difference between 47 and 48 is that 48 has one more part, \( -\frac{C^b+\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3} \). Moreover, \( -\frac{C^b+\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3}=\frac{\underline{\theta}\gamma -{C}^b-\left(\overline{\theta}-\underline{\theta}\right)\gamma, }{6} \) , where \( -\left(\overline{\theta}-\underline{\theta}\right)\gamma <0. \) Since we show \( {\theta}_L^{t=1}=\frac{r_H^{t=1}}{\gamma }=\frac{c^b+\underline{\theta}\gamma }{2}>\underline{\theta} \) in lemma 2, \( {c}^b>\underline{\theta}\gamma \), or \( \underline{\theta}\gamma -{c}^b<0 \). Therefore \( -\frac{C^b+\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3}<0 \), and we conclude that 47 is larger than 48, or the marginal profit with respect to plastic bag users at store L in period one is larger in the baseline.

      In addition, we subtract 48 from 46 and get \( \frac{C^b-\underline{\theta}\gamma }{2}>0 \), so 46 is larger than 48. We conclude that, in the voluntary reward scenario, store L gains more profit from a reusable bag user than a plastic bag user at period one.

    3. iii.

      The marginal profits with respect to reusable bag users at store H in period one

      $$ \mathrm{Baseline}:\kern7em \frac{\left(2\overline{\theta}-\underline{\theta}-c\right)\left({a}_H-{a}_L\right)}{3} $$
      (49)
      $$ \mathrm{Reward}:\kern6em \frac{\left(2\overline{\theta}-\underline{\theta}-c\right)\left({a}_H-{a}_L\right)}{3}+\frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{3} $$
      (50)

      The difference between 49 and 50 is that 50 has one more part, \( \frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{3} \). We’ve proved \( \frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{6}<0 \) in the first part, hence \( \frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{3}<0 \). We conclude that 49 is larger than 50, or the marginal profit with respect to reusable bag users at store H in period one is larger in the baseline.

    4. iv.

      The marginal profits with respect to plastic bag users at store H in period one

      $$ \mathrm{Baseline}:\kern7em \frac{\left(2\overline{\theta}-\underline{\theta}-c\right)\left({a}_H-{a}_L\right)}{3} $$
      (51)
      $$ \mathrm{Reward}:\kern6em \frac{\left(2\overline{\theta}-\underline{\theta}-c\right)\left({a}_H-{a}_L\right)}{3}-\frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{6} $$
      (52)

      The difference between 51 and 52 is that 52 has one more part \( -\frac{C^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma }{3} \). We’ve proved \( -\frac{C^b+\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3} \) in the first part, hence \( -\frac{C^b+\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{6}<0 \). We conclude that 51 is larger than 52, or the marginal profit with respect to plastic bag users at store H in period one is larger in the baseline.

      In addition, we subtract 52 from 50 and get \( \frac{C^b-\overline{\theta}\gamma }{2}<0 \), so 52 is larger than 50. We conclude that, in voluntary reward scenario, store H gains more profit from a plastic bag user than a reusable bag user at period one.

  6. 6)

    Proof for Lemma 4

    The type of consumers indifferent between to store is θ baseline, and we plug in the optimal prices and rewards,

    $$ {\theta}^{baseline}=\frac{\overline{\theta}+\underline{\theta}+c}{3} $$
    (53)

    Therefore the store L’s market share in the baseline is

    $$ \frac{\overline{\theta}-2\underline{\theta}+c}{3} $$
    (54)

    The store H’s market share in the baseline is

    $$ \frac{2\overline{\theta}-\underline{\theta}-c}{3} $$
    (55)

    The type of consumers indifferent between to store is θ t = 1, and we plug in the optimal prices and rewards,

    $$ {\theta}^{t=1}=\frac{\overline{\theta}+\underline{\theta}+c}{3}+\frac{\gamma c-{c}^b}{3\left({a}_H-{a}_L-\gamma \right)} $$
    (56)

    Therefore the store L’s market share in the baseline is

    $$ \frac{\overline{\theta}-2\underline{\theta}+c}{3}+\frac{\gamma c-{c}^b}{3\left({a}_H-{a}_L-\gamma \right)} $$
    (57)

    The store H’s market share in the baseline is

    $$ \frac{2\overline{\theta}-\underline{\theta}-c}{3}-+\frac{\gamma c-{c}^b}{3\left({a}_H-{a}_L-\gamma \right)} $$
    (58)

    if  > c b, 56 is larger than 53 and 57 is smaller than 54, hence the store L takes share away from store H in the voluntary reward scenario, compared with the baseline; if  < c b, 56 is smaller than 53 and 57 is larger than 54; hence, store H takes shares away from store L in voluntary reward scenario, compared with the baseline; if  = c b, 55 is equal to 57 and 56 is equal to 58; hence, the market shares do not change in the voluntary reward scenario compared with the baseline.

  7. 7)

    Proof for Corollary 2

    The profits in period one in the voluntary reward scenario are

    $$ {\pi}_L^{t=1, reward}=\frac{{\left[9{A}_{LH}\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}-2\underline{\theta}\right)\gamma \right]}^2}{9\left(\overline{\theta}-\underline{\theta}\right)\left({a}_H-{a}_L-\gamma \right)}+\frac{{\left({c}^b-\underline{\theta}\gamma \right)}^2}{4\left(\overline{\theta}-\underline{\theta}\right)\gamma } $$
    (59)
    $$ {\pi}_H^{t=1, reward}=\frac{{\left[9{A}_{HL}\left({a}_H-{a}_L\right)+{C}^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma \right]}^2}{9\left(\overline{\theta}-\underline{\theta}\right)\left({a}_H-{a}_L-\gamma \right)}+\frac{{\left(\overline{\theta}\gamma -{c}^b\right)}^2}{4\left(\overline{\theta}-\underline{\theta}\right)\gamma } $$
    (60)

    The profits in period two in the voluntary reward scenario are

    $$ {\pi}_L^{t=2, reward}=\frac{{\left[9{A}_{LH}\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}-2\underline{\theta}\right)\gamma -E\left({\varTheta}_{Hrb}^{t=1}\right)\right]}^2}{9\left(\overline{\theta}-\underline{\theta}\right)\left({a}_H-{a}_L-\gamma \right)}+\frac{{\left({c}^b+\lambda {\displaystyle {\int}_{\underline{\theta}}^{\theta_L^{t=1}}}dF\left(\theta \right)-\underline{\theta}\gamma \right)}^2}{4\left(\overline{\theta}-\underline{\theta}\right)\gamma } $$
    (61)
    $$ {\pi}_H^{t=2, reward}=\frac{{\left[9{A}_{HL}\left({a}_H-{a}_L\right)+{C}^b-\left(2\overline{\theta}-\underline{\theta}\right)\gamma +E\left({\varTheta}_{Hrb}^{t=1}\right)\right]}^2}{9\left(\overline{\theta}-\underline{\theta}\right)\left({a}_H-{a}_L-\gamma \right)}+\frac{{\left(\overline{\theta}\gamma -{c}^b-E\left({\varTheta}_{Hrb}^{t=1}\right)\right)}^2}{4\left(\overline{\theta}-\underline{\theta}\right)\gamma } $$
    (62)

    We compare 59 with 61, 60 with 62, and we find that the differences are in period two the optimal profits introduce the peer pressure E(Θ irb t = 1 ), i=L,H.

    Therefore, using optimal profits at period two to do comparative statics, one can show that,

    $$ \frac{d{\pi}_L^{t=2, reward}}{dE\left({\varTheta}_{Lrb}^{t=1}\right)}>0;\ \frac{d{\pi}_L^{t=2, reward}}{dE\left({\varTheta}_{Hrb}^{t=1}\right)}<0 $$
    (63)
    $$ \frac{d{\pi}_H^{t=2, reward}}{dE\left({\varTheta}_{Hrb}^{t=1}\right)}>0;\ \frac{d{\pi}_H^{t=2, reward}}{dE\left({\varTheta}_{Lrb}^{t=1}\right)}=0 $$
    (64)

Appendix 2

  1. 1).

    Optimal solutions in period two under full participation only at store H consumer partition

    Period 2:

    Equilibrium Consumer Partition:

    The θ t = 2 who is indifferent between two stores at period two are obtained from setting v + a L θ t = 2 − p L  = v + (a H  − γ)θ t = 2 + r H t = 2  + E(Θ Hrb t = 1 ) − p H . Therefore, θ t = 2 is solved as

    $$ {\theta}^{t=2}=\frac{p_H-{p}_L-{r}_H^{t=2}}{a_H-{a}_L-\gamma }-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{a_H-{a}_L-\gamma } $$
    (65)

    A consumer who is indifferent as to whether to use or not use a reusable bag at store L in period 2, is characterized by θ L t = 2 satisfying:

    $$ \begin{array}{c}\hfill {\mathcal{U}}_{Lrb}^{t=2}={\mathcal{U}}_{Lpb}^{t=2}\kern1.25em \iff \hfill \\ {}\hfill v+\left({a}_L-\gamma \right){\theta}_L^{t=2}+\left({r}_L^{t=2}+E\left({\varTheta}_{Hrb}^{t=1}\right)\right)-{p}_L=v+{a}_L{\theta}_L^{t=2}-{p}_L\hfill \end{array} $$
    (66)

    Therefore, solving for θ L t = 2 yields

    $$ {\theta}_L^{t=2}=\frac{r_L^{t=2}}{\gamma }+\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{\gamma } $$
    (67)

    Profit functions:

    $$ {\pi}_L=\left({p}_L-{a}_Lc-{r}_L\right){{\displaystyle \int}}_{\varTheta_{Lrb}}dF\left(\theta \right) + \left({p}_L-{a}_Lc-{c}^b\right){{\displaystyle \int}}_{\varTheta_{Lpb}}dF\left(\theta \right) $$
    (68)
    $$ {\pi}_H=\left({p}_H-{a}_Hc-{r}_H\right){{\displaystyle \int}}_{\varTheta_{Hrb}}dF\left(\theta \right) $$
    (69)

    The store i’s profit at second stage in period two is denoted as π i t = 2 , where i= L,H, given store’s rewards in period two r i t = 2 and the set of reusable bag user formed in period two Θ i,rb t = 2 . We can get the optimal grocery prices p L t = 2 * , p H t = 2 * in terms of r i t = 2 and Θ i,rb t = 1 by plugging the characterized θ L t = 2 , θ H t = 2 and θ t = 2 and , differentiating profit functions with respect to prices, then setting equal to zero. At the first stage in period two, stores choose the optimal rewards. Substituting p L t = 2 * and p H t = 2 * to Eqs. (68) and (69), the profit functions for either store are in terms of rewards. By differentiating the profit function with respect to rewards, one can obtain the interior optimal rewards r L t = 2 *  and r H t = 2 * in the second period. Therefore, the equilibrium prices and rewards in period two {p L t = 2 * , p H t = 2 * , r L t = 2 * , r H t = 2 * } are given by

    $$ {p}_L^{t=2*}={A}_{LH}{a}_H-{A}_{LL}{a}_L+\frac{2{c}^b-\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{3} $$
    (70)
    $$ {p}_H^{t=2*}={A}_{HH}{a}_H-{A}_{HL}{a}_L+\frac{5{c}^b-\left(\overline{\theta}-\underline{\theta}\right)\gamma }{6}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{6} $$
    (71)
    $$ {r}_L^{t=2*}=\frac{\underline{\theta}\gamma +{c}^b}{2}-\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{2}<{\theta}^{t=2}\gamma $$
    (72)
    $$ {r}_H^{t=2*}=\frac{c^b+\overline{\theta}\gamma }{2}\ge \overline{\theta}\gamma $$
    (73)
  2. 2).

    Optimal solutions in period two under Full participation only at store L consumer partition

    Period 2:

    Equilibrium Consumer Partition:

    The θ t = 2 who is indifferent between two stores at period two are obtained from setting v + (a L  − γ)θ t = 2 + (r L t = 2  + E(Θ Lrb t = 1 )) − p L  = v + (a H  − γ)θ t = 2 + (r H t = 2  + E(Θ Hrb t = 1 )) − p H . Therefore, θ t = 2 is solved as:

    $$ {\theta}^{t=2}=\frac{p_H-{p}_L-{r}_H^{t=2}+{r}_L^{t=2}}{a_H-{a}_L}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{a_H-{a}_L}+\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{a_H-{a}_L} $$
    (74)

    A consumer who is indifferent as to whether to use or not use a reusable bag at store H in period 2, is characterized by θ H t = 2 satisfying:

    $$ \begin{array}{c}\hfill {\mathcal{U}}_{Hrb}^{t=2}={\mathcal{U}}_{Hpb}^{t=2}\kern1.25em \iff \hfill \\ {}\hfill v+\left({a}_H-\gamma \right){\theta}_H^{t=2}+\left({r}_H^{t=2}+E\left({\varTheta}_{Hrb}^{t=1}\right)\right)-{p}_H=v+{a}_H{\theta}_H^{t=2}-{p}_H\hfill \end{array} $$
    (75)

    Therefore, solving for θ L t = 2 yields

    $$ {\theta}_H^{t=2}=\frac{r_H^{t=2}}{\gamma }+\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{\gamma } $$
    (76)

    Profit functions:

    $$ {\pi}_L=\left({p}_L-{a}_Lc-{r}_i\right){{\displaystyle \int}}_{\varTheta_{Lrb}}dF\left(\theta \right) $$
    (77)
    $$ {\pi}_H=\left({p}_H-{a}_Hc-{r}_H\right){{\displaystyle \int}}_{\varTheta_{Hrb}}dF\left(\theta \right) + \left({p}_H-{a}_Hc-{c}^b\right){{\displaystyle \int}}_{\varTheta_{Hpb}}dF\left(\theta \right) $$
    (78)

    The store i’s profit at second stage in period two is denoted as π i t = 2 , where i= L,H, given store’s rewards in period two r i t = 2 and the set of reusable bag user formed in period two Θ i,rb t = 2 . We can get the optimal grocery prices p L t = 2 * , p H t = 2 * in terms of r i t = 2 and Θ i,rb t = 1 by plugging the characterized θ L t = 2 , θ H t = 2 and θ t = 2 and , differentiating profit functions with respect to prices, then setting equal to zero. At the first stage in period two, stores choose the optimal rewards. Substituting p L t = 2 * and p H t = 2 * to Eq. (10), the profit functions for either store are in terms of rewards. By differentiating the profit function with respect to rewards, one can obtain the interior optimal rewards r L t = 2 *  and r H t = 2 * in the second period. Therefore, the equilibrium prices and rewards in period two {p L t = 2 * , p H t = 2 * , r L t = 2 * , r H t = 2 * } are given by

    $$ {p}_L^{t=2*}={A}_{LH}{a}_H-{A}_{LL}{a}_L+\frac{2{c}^b-\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{3}+\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{3} $$
    (79)
    $$ {p}_H^{t=2*}={A}_{HH}{a}_H-{A}_{HL}{a}_L+\frac{5{c}^b-\left(\overline{\theta}-\underline{\theta}\right)\gamma }{6}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{6}+\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{6} $$
    (80)
    $$ {r}_L^{t=2*}=\frac{\underline{\theta}\kern0.1em \gamma +{c}^b}{2}-\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{2} $$
    (81)
    $$ {r}_H^{t=2*}=\frac{c^b+\overline{\theta}\gamma }{2}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{2} $$
    (82)
  3. 3).

    Optimal solutions in period two under full participation at both stores consumer partition

    Period 2:

    Equilibrium Consumer Partition:

    The θ t = 2 who is indifferent between two stores at period two are obtained from setting v + (a L  − γ)θ t = 2 + (r L t = 2  + E(Θ Lrb t = 1 )) − p L  = v + (a H  − γ)θ t = 2 + (r H t = 2  + E(Θ Hrb t = 1 )) − p H . Therefore, θ t = 2 is solved as:

    $$ {\theta}^{t=2}=\frac{p_H-{p}_L-{r}_H^{t=2}+{r}_L^{t=2}}{a_H-{a}_L}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{a_H-{a}_L}+\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{a_H-{a}_L} $$
    (83)

    A consumer who is indifferent as to whether to use or not use a reusable bag at store H in period 2, is characterized by θ H t = 2 satisfying:

    $$ \begin{array}{c}\hfill {\mathcal{U}}_{Hrb}^{t=2}={\mathcal{U}}_{Hpb}^{t=2}\kern0.75em \iff \hfill \\ {}\hfill v+\left({a}_H-\gamma \right){\theta}_H^{t=2}+\left({r}_H^{t=2}+E\left({\varTheta}_{Hrb}^{t=1}\right)\right)-{p}_H=v+{a}_H{\theta}_H^{t=2}-{p}_H\hfill \end{array} $$
    (84)

    Therefore, solving for θ L t = 2 yields

    $$ {\theta}_H^{t=2}=\frac{r_H^{t=2}}{\gamma }+\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{\gamma } $$
    (85)

    Profit functions:

    $$ {\pi}_L=\left({p}_L-{a}_Lc-{r}_i\right){{\displaystyle \int}}_{\varTheta_{Lrb}}dF\left(\theta \right) $$
    (86)
    $$ {\pi}_H=\left({p}_H-{a}_Hc-{r}_H\right){{\displaystyle \int}}_{\varTheta_{Hrb}}dF\left(\theta \right) $$
    (87)

    The store i’s profit at second stage in period two is denoted as π i t = 2 , where i= L,H, given store’s rewards in period two r i t = 2 and the set of reusable bag user formed in period two Θ i,rb t = 2 . We can get the optimal grocery prices p L t = 2 * , p H t = 2 * in terms of r i t = 2 and Θ i,rb t = 1 by plugging the characterized θ L t = 2 , θ H t = 2 and θ t = 2 and , differentiating profit functions with respect to prices, then setting equal to zero. At the first stage in period two, stores choose the optimal rewards. Substituting p L t = 2 * and p H t = 2 * to Eq. (10), the profit functions for either store are in terms of rewards. By differentiating the profit function with respect to rewards, one can obtain the interior optimal rewards r L t = 2 *  and r H t = 2 * in the second period. Therefore, the equilibrium prices and rewards in period two {p L t = 2 * , p H t = 2 * , r L t = 2 * , r H t = 2 * } are given by

    $$ {p}_L^{t=2*}={A}_{LH}{a}_H-{A}_{LL}{a}_L+\frac{2{c}^b-\left(\overline{\theta}-2\underline{\theta}\right)\gamma }{3}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{3}+\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{3} $$
    (88)
    $$ {p}_H^{t=2*}={A}_{HH}{a}_H-{A}_{HL}{a}_L+\frac{5{c}^b-\left(\overline{\theta}-\underline{\theta}\right)\gamma }{6}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{6}+\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{6} $$
    (89)
    $$ {r}_L^{t=2*}=\frac{\underline{\theta}\kern0.1em \gamma +{c}^b}{2}-\frac{E\left({\varTheta}_{Lrb}^{t=1}\right)}{2} $$
    (90)
    $$ {r}_H^{t=2*}=\frac{c^b+\overline{\theta \kern0.1em }\gamma }{2}-\frac{E\left({\varTheta}_{Hrb}^{t=1}\right)}{2} $$
    (91)

Appendix 3

We extended two-period model to infinite-horizon model in order to consider the impact of the rewards program on the reusable bag adoption in the steady state. Based on the Eqs. 36, 37 and 38, we can get the evolution function of the dynamic system for the number of plastic bag users at store i, i= L, H, as:

$$ \varDelta {{\displaystyle \int}}_{\varTheta_{irb}^t}dF\left(\theta \right)={{\displaystyle \int}}_{\varTheta_{irb}^{t+1}}dF\left(\theta \right)-{{\displaystyle \int}}_{\varTheta_{irb}^t}dF\left(\theta \right)={\alpha}_i+{\beta}_i{{\displaystyle \int}}_{\varTheta_{irb}^t}dF\left(\theta \right)+{\eta}_i{r}_i^t,\kern1.25em where\ i=L,H $$
(92)

Where \( {\alpha}_L=-\underline{\theta}/\left(\overline{\theta}-\underline{\theta}\right) \), \( {\beta}_L=\left(\lambda -\left(\overline{\theta}-\underline{\theta}\right)\gamma \right)/\left(\gamma \left(\overline{\theta}-\underline{\theta}\right)\right) \) and \( {\eta}_L=1/\gamma \left(\overline{\theta}-\underline{\theta}\right) \) for store L; \( {\alpha}_H=\gamma /\left(\gamma \left(\overline{\theta}-\underline{\theta}\right)\right) \), \( {\beta}_H=\left(3\left({a}_H-{a}_L-\gamma \right)\left(\lambda -\gamma \left(\overline{\theta}-\underline{\theta}\right)\right)+\lambda \gamma \right)/\left(3\lambda \left(\overline{\theta}-\underline{\theta}\right)\left({a}_H-{a}_L-\gamma \right)\right) \) and \( {\eta}_H=\left(\left(\overline{\theta}+\underline{\theta}+C\right)\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}+\underline{\theta}\right)\gamma \right)/\left(3\lambda \left(\overline{\theta}-\underline{\theta}\right)\left({a}_H-{a}_L-\gamma \right)\right) \) for store H.

The Corollary 1 indicates the evolution function of money rewards at store i, i= L, H, as:

$$ \varDelta {r}_i^t={r}_i^{t+1}-{r}_i^t=-\frac{\lambda {{\displaystyle \int}}_{\varTheta_{irb}^t}dF\left(\theta \right)}{2},\kern1.25em where\kern.5em i=L,H $$
(93)

We obtain the phase lines by setting \( \varDelta {\displaystyle {\int}_{\varTheta_{irb}^t}dF}\left(\theta \right) \) and Δr t equal to zero:

$$ {{\displaystyle \int}}_{\Theta_{Lrb}}dF\left(\theta \right)=\frac{1}{\boldsymbol{\gamma} \left(\overline{\boldsymbol{\theta}}-\underline {\boldsymbol{\theta}}\right)-\boldsymbol{\lambda}}\ {r}_L+\frac{\underline {\boldsymbol{\theta}}\gamma }{\boldsymbol{\lambda} -\boldsymbol{\gamma} \left(\overline{\boldsymbol{\theta}}-\underline {\boldsymbol{\theta}}\right)} $$
(94)
$$ {{\displaystyle \int}}_{\varTheta_{Hrb}}dF\left(\theta \right)=\frac{1}{\left(\boldsymbol{\gamma} \left(\overline{\boldsymbol{\theta}}-\underline {\boldsymbol{\theta}}\right)-\boldsymbol{\lambda} \right)}\ {r}_H+\frac{\left(\overline{\theta}+\underline{\theta}+c\right)\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}+\underline{\theta}\right)\gamma }{3\left({a}_H-{a}_L-\gamma \right)\left(\boldsymbol{\lambda} -\boldsymbol{\gamma} \left(\overline{\boldsymbol{\theta}}-\underline {\boldsymbol{\theta}}\right)\right)} $$
(95)
$$ {{\displaystyle \int}}_{\varTheta_{irb}}dF\left(\theta \right)=0 $$
(96)

Since \( \underline{\theta}\gamma \), \( 3\left({a}_H-{a}_L-\gamma \right) \), and \( \left(\overline{\theta}+\underline{\theta}+c\right)\left({a}_H-{a}_L\right)-{C}^b-\left(\overline{\theta}+\underline{\theta}\right)\gamma \) are all larger than zero, the sign of the slope and y-intercept in Eqs. (94) and (95) is determined by the sign of \( \gamma \left(\overline{\theta}-\underline{\theta}\right)-\lambda \). However, the slope and y-intercept have opposite sign. Therefore, the phase lines in the (\( {r}_i \), \( {\int}_{\Theta_{irb}}dF\left(\theta \right) \)) panel under two different scenarios are discussed as following:

  • Opportunity Cost Dominance

    If the opportunity cost of using reusable bags, \( \gamma \), is so larger that we have \( \gamma \left(\overline{\theta}-\underline{\theta}\right)-\lambda >0 \), then we will have the slopes for Eqn. (94) and (95) are positive and the y-intercepts are negative. Considering the restriction of \( {r}_i\ge 0 \) and \( 0\le {\int}_{\Theta_{irb}}dF\left(\theta \right)\le {\int}_{\Theta_{ib}}dF\left(\theta \right) \), we developed the phase diagram for scenario in Fig. (5). In addition, \( \frac{\partial \varDelta {\displaystyle {\int}_{\varTheta_{irb}^t}dF\left(\theta \right)}}{\partial {\displaystyle {\int}_{\varTheta_{irb}^t}dF\left(\theta \right)}} \) and \( \frac{\partial \varDelta {f}_i^t}{r_i}<0 \) help to decide the arrows of motion in the phase diagram.

  • Peer Pressure Dominance

    If the imputed value, \( \lambda \), is so larger that we have \( \gamma \left(\overline{\theta}-\underline{\theta}\right)-\lambda <0 \), then we will have the slopes for Eqs. (94) and (95) are negative and the y-intercepts are positive. Considering the restriction of \( {r}_i\ge 0 \) and \( 0\le {\int}_{\Theta_{irb}}dF\left(\theta \right)\le {\int}_{\Theta_{ib}}dF\left(\theta \right) \), we developed the phase diagram for scenario in Fig. (6). In addition, \( \frac{\partial \varDelta {\displaystyle {\int}_{\varTheta_{irb}^t}dF\left(\theta \right)}}{\partial {\displaystyle {\int}_{\varTheta_{irb}^t}dF\left(\theta \right)}}>0 \) and \( \frac{\partial \varDelta {r}_i^t}{r_i}<0 \) help to decide the arrows of motion in the phase diagram.

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Jiang, J. Peer Pressure in Voluntary Environmental Programs: a Case of the Bag Rewards Program. J Ind Compet Trade 16, 155–190 (2016). https://doi.org/10.1007/s10842-015-0208-6

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Keywords

  • Voluntary rewards program
  • Peer pressure
  • Plastic shopping bag
  • Reusable bag
  • Duopoly market

JEL Classification

  • L13
  • D21
  • Q53