The curse of low-valued recycling

Abstract

This paper discusses how to deal with low-valued recyclable wastes whose reprocessing itself does not pay financially. While such a recycling activity can potentially improve social welfare if the social costs associated with their disposal are sufficiently significant, governmental policies to promote recycling may lead to illegal disposal. Explicitly considering the monitoring cost in preventing firms from disposing of collected wastes illicitly, we show that the second-best policy for a low-valued recyclable is either one of the two following schemes: a deposit-refund scheme (DRS) that gives birth to a recycling market and an advanced-disposal fee that does not create a recycling market. However, in order to select the optimal policy scheme and implement it appropriately, a policymaker needs information available only in the recycling market. Thus, the structure of the second-best policy itself entails critical information issues in its implementation, which is in stark contrast to a DRS for a non-low-valued recyclable.

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Notes

  1. 1.

    Fullerton and Kinnaman (1995) and Palmer and Walls (1997) are seminal works in this literature. Fullerton and Wolverton (2005) show its effectiveness in a more general context.

  2. 2.

    On the other hand, illegal disposal by households has been examined by Fullerton and Kinnaman (1995) and Choe and Frazer (1999), for instance. Shinkuma (2003) and Calcott and Walls (2005) also carefully investigate households’ transactions of recyclable wastes.

  3. 3.

    In an earlier work that considers the firms’ illegal disposal and monitoring on them, Sullivan (1987) numerically analyzes three exogenously-given policy structures for controlling toxic wastes: laissez-faire, the subsidization of legal disposal, and the expansion of enforcement/monitoring efforts. More recently, D’Amato et al. (2015) consider a model where a waste management firm can choose illegal disposal options and examine the effects of a crime organization (a mafia) on the firm’s illegal waste business from both theoretical and empirical perspectives.

  4. 4.

    The environmental impacts of these disposal options would be even greater in the case of illegal dumping.

  5. 5.

    http://www.jcpra.or.jp/tabid/603/index.php (accessed February 6, 2019).

  6. 6.

    In one show, a plastic hunger that the then Minister of the Environment brought to the TV studio and claimed as an example of a successful plastic recycling program later turned out to be made almost entirely of virgin materials (broadcast on the NTV channel on Dec. 23, 2008).

  7. 7.

    In contrast to our focus on “downstream” cost such as monitoring cost, Acuff and Kaffine (2013) also find a case where an ADF should be favored over a DRS in the presence of “upstream” externalities as in the case of emissions of greenhouse gases. See also Palmer et al. (1997) and Kaffine (2014) for an empirical/simulation framework that compares an ADF with a DRS.

  8. 8.

    The inverse, an under-encouragement case, is also plausible.

  9. 9.

    All the results and implications of this paper can be readily extended to an economy with \(m\ge 1\) firms and \(n\ge 1\) households and also to a case where recyclers and producers of final products are separate entities (see also Footnotes 13 and 23).

  10. 10.

    When \(P_{r}<0\) (\(P_{r}>0\)), it is the household (the firm) which pays some positive amount in the recycling market. Note that \(P_{r}\) can be negative because the household may still be willing to pay to get rid of recyclable wastes in order to avoid the charge incurred for municipal waste disposal services.

  11. 11.

    We can also consider that \(C_{r} (\cdot )\) includes the cost incurred by some intermediate agents who conduct collection and sorting activities as long as all the markets where these intermediate agents trade are perfectly competitive.

  12. 12.

    This assumption is not essential for our results as long as \(C_{r}^{\prime }(0)\ge 0\).

  13. 13.

    We can show that our results hold even if we modify the model and separate the firm into the recycler and the final good producer as two independent entities, that is, the recycler maximizes the recycling profit (the second bracket), \(B(r^{d})-\left( P_{r}-s\right) r^{d}\), and the final good producer maximizes the production profit (the first bracket), \(\left( P_{x}-t\right) x^{s}-C_{x}(x^{s})\), respectively.

  14. 14.

    In our model, the net benefit of proper recycling, B(r), is given exogenously, and B(r) can be interpreted in several different ways. For example, by letting \(\alpha \in (0,1]\) be the recycling content ratio that is exogenously determined by the firm’s recycling technology (thus, \(r/\alpha \) is the amount of recycled products), B(r) can be defined as

    $$\begin{aligned} B(r)=pr/\alpha -C(r/\alpha ), \end{aligned}$$

    where p is an exogenously given price of the completely recycled product and \(C(\cdot )\) is the cost of producing the recycled product including the reprocessing of the residual wastes. If the firm sells its recycled product in some market, p is simply the market price of the product or, a (hedonically) perfectly substitutable product made of virgin materials. If the firm uses the recycled material as an input for its own production, p is the price of a (hedonically) perfectly substitutable virgin input, v, provided that the production function of the good x is of the form \(x=f(r+v)\). In this case, \(\alpha =1\) in the above expression.

  15. 15.

    The case where \(B^{\prime }(0)>0\) is analyzed in Ino (2011).

  16. 16.

    Here, d includes not just the costs associated with the waste-processing services but also certain environmental damage costs while \(d_{h}\) consists of environmental damage costs of illegally-discarded wastes and, possibly, the clean-up costs if such activities are conducted. We suppose \(d<d_{h}\) since the marginal social cost of illegal disposal would typically be greater than that of legal disposal options, such as controlled landfills and proper incineration. This assumption is also adopted by Choe and Frazer (1999).

  17. 17.

    Following the tradition in the fields of the economics of crime and non-compliance (Becker 1968; Hayes 2000), \(\tau _{h}^{e}\) is defined by \(\tau _{h}^{e}=p_{h}\bar{f}_{h}\), where \(p_{h}\) is the probability of detection and \(\bar{f}_{h}\) is the largest available fine. Since the authorities must enhance \(p_{h}\) by controlling the stringency of their monitoring activities in order to heighten \(\tau _{h}^{e}\), the cost to monitor the household’s illegal disposal activities \(\Gamma _{h}(\tau _{h}^{e})\) is increasing in \(\tau _{h}^{e}\).

  18. 18.

    This is because, when \(\tau <\tau _{h}^{e}\), the authorities can save on the monitoring cost by reducing \(\tau _{h}^{e}\) and resetting \(\tau _{h}^{e}=\tau \), with other things being constant; and, when \(\tau >\tau _{h}^{e}\), all the illegal wastes from the household are turned into legal ones by reducing \(\tau \) and resetting \(\tau =\tau _{h}^{e}\), with other things being constant. We assume that the household chooses to dispose of its wastes legally if its private costs of two disposal options are equivalent, i.e., \(\tau =\tau _{h} ^{e}\).

  19. 19.

    See, among others, Fullerton and Kinnaman (1995), Palmer and Walls (1997), Fullerton and Wolverton (2005), and Ino (2011).

  20. 20.

    In the terms typically found in the environmental economics literature, \(r^{d}\) and \(r_{c}\) respectively correspond to the reported and actual levels of emission abatement while \(z_{f}\) is the level of the fabricated abatement (cheating) by the firm. Thus, our setting of the monitoring problem conforms to one widely used in the conventional literatures. See Harford (1978) and Lee (1984) among others.

  21. 21.

    As in Footnote 17, \(\tau _{f}^{e}\) is defined by \(\tau _{f} ^{e}=p_{f}\bar{f}_{f}\), where \(p_{f}\) is the probability of detection and \(\bar{f}_{f}\) is the largest available fine but for the firm’s illegal disposal in the case of \(\tau _{f}^{e}\). See also Sullivan (1987).

  22. 22.

    If it is so easy to trace the whole recycling processes that the monitoring cost is negligible, that is, if \(\Gamma _{f}^{\prime }\) is infinitesimally small, our model becomes close to the first-best situation (see Footnote 32).

  23. 23.

    Similarly to the previous section, we can show that our results hold even if we consider the separate entities, that is, the recycler maximizes the recycling profit (the second bracket), \(B(r_{c})-(P_{r}-s)r^{d}-\tau _{f}^{e}z_{f}\), subject to \(r^{d} =r_{c}+z_{f}\) and the final good producer maximizes the production profit (the first bracket), \(\left( P_{x}-t\right) x^{s}-C_{x}(x^{s})\), respectively.

  24. 24.

    See also Caveat 1 for a related discussion and Footnote 14 for the specification of the benefit function B that is consistent with this interpretation.

  25. 25.

    The equilibrium outcomes in this case coincide with those in the always-compliant case of the previous section because \(r^{C}\) is also derived from \(B^{\prime }(r^{C})=C_{r}^{\prime }(r^{C})-(s+\tau )\) by (6).

  26. 26.

    This is also the case for \(r^{C}\) since \(B^{\prime }(r^{C})=C_{r}^{\prime }(r^{C})-(s+\tau )\) from (6).

  27. 27.

    Such a level of s exists because \(\tau _{f}^{e}>a\) in case (iii) and \(r^{C}\) (resp. \(-B^{\prime }(r^{C} )\)) goes to 0 (resp. \(-B^{\prime }(0)=a\)) as \(s+\tau \) approaches a.

  28. 28.

    If this policy change is implemented in case (iii), \(r_{c}^{*}\) is not affected because in case (iv), \(r_{c}^{*}\) satisfies \(C_{r}^{\prime }(r_{c}^{*})-(s+\tau )=B^{\prime }(r_{c}^{*})=-\tau _{f}^{e}\) when \(B^{\prime }(r^{C})=-\tau _{f}^{e}\). Thus, \(r_{c}^{*}\) is derived from the condition which is exactly the same as the one in case (iii).

  29. 29.

    We assume that the appropriate second-order conditions are met in the relevant ranges.

  30. 30.

    This monitoring rule for the optimality can be derived as follows. In order to obtain \(z_{f}^{*}=0\), the policies must be such that the equilibrium outcome is either in case (i) or in case (iv) of Proposition 1. Since \(z_{f}^{*}\) does not depend on t by Proposition 1, we can focus on the relationship between \(\tau _{f}^{e}\) and s under \(\tau =0\). For the equilibrium outcome to be in case (iv), we must have \(s>a\) and \(\tau _{f}^{e}\ge s-P_{r}^{C}(s,0)\), as Corollary 1 shows. To satisfy these two inequalities simultaneously, for \(s>a\), we need to have \(\tau _{f}^{e}=s-P_{r}^{C}(s,0)\) since, if \(\tau _{f}>s-P_{r}^{C}(s,0)\), the government can save on the monitoring cost. For the outcome to be in case (i), Proposition 1 indicates that \(s\le a\) and \(\tau _{f}^{e}\ge s\) must hold. Thus, if \(0<s\le a\), the monitoring level should be \(\tau _{f} ^{e}=s\) in order to save on the monitoring cost. On the other hand, if \(s\le 0\), \(\tau _{f}^{e}=0\) is sufficient to yield \(\tau _{f}^{e}\ge s\).

  31. 31.

    For \(s>a\), \(P_{r}^{C}(s,0)=C_{r}^{\prime }(r^{C})\) goes to zero when \(s\rightarrow a\) because of the fact that \(r^{C}=0\) if and only if \(s\le a\), as was argued in Sect. 2.3.

  32. 32.

    When the cost of monitoring on the firm is infinitesimally small, i.e., \(\Gamma _{f}^{\prime }\) is almost zero, \(s^{*}=d-A\) is close to d because \(A=-B^{\prime \prime }\Gamma _{f}^{\prime }\). Therefore, our second-best DRS approches the first-best DRS (i.e., \(t=s=d\)).

  33. 33.

    Corollary 1 confirms that these levels of the subsidy and expected penalty create a legal recycling market where all the transacted wastes are recycled properly.

  34. 34.

    See also Footnote 32 for a consistent interpretation with the potentially non-compliant firm model.

  35. 35.

    The following holds all the more if the first-best level of the refund, i.e., \(s=d\), is provided, since it is true even when a refund is provided on a reduced second-best scale, i.e., \(s=d-A\).

  36. 36.

    In this case, the monitoring cost on the firm’s illegal activities is just too significant, compared to the social cost potentially saved by proper recycling.

  37. 37.

    This can be easily checked under the specification illustrated in Figs. 2 and 3. The optimal subsidy level, \(s^{*}\), includes \(A=b\gamma \), and the threshold value of the social cost, \(\bar{d} =a+b\gamma +2\sqrt{a(b+c)\gamma }\), contains the information only available in a recycling market, i.e., a, b and c. More generally, in finding the actual level of \(s^{M}\), the value of \(A(s^{M})\) depends on \(B^{\prime \prime } \), which is the information that can be estimated from the demand elasticity in a legal recycling market. Furthermore, as is in the proof of Proposition 2, we must compare two locally maximized welfare levels, \(W^{*}(d,s^{M})\) and \(W^{*}(d,0)\), in order to obtain \(\bar{d}\). Thus, \(\bar{d}\) also consists of the information available only in the legal recycling market.

  38. 38.

    Since \(a=0\), very small positive levels of subsidy and monitoring efforts immediately lead to the minimal but legal recycling activities (Corollary 1). The required market information A(0) in creating the recycling market (at the point X=Y in the figure) can be obtained once the smallest possible recycling market emerges and the demand elasticity is estimated (around a realized price, which is close to 0 since \(P_{r}=C_{r}^{\prime }(r)\rightarrow 0\) when \(r\rightarrow 0\)).

  39. 39.

    In Fig. 1, this corresponds with the case where we have “Yes” for C1 because \(s>min[\tau _{f}^{e},a]=0\) and “No” for C2 because \(\tau _{f}^{e}<a\).

  40. 40.

    See Proposition 2 in Ino (2011).

  41. 41.

    See Lemma 2 in Ino (2011), where the critical information that is related to the policy switch is the level of \(\hat{s}\), which can be obtained through a simple market-based criterion: the subsidy level, s, exceeds the price in the recycling market if and only if \(s>\hat{s}.\)

  42. 42.

    It is modified as \(t^{*}=d+P_{x}/\eta _{x}\), where \(\eta _{x}<0\) represents the price elasticities of the final good’s demand, i.e., as is usual Pigouvian tax, the optimal tax should be reduced by considering monopoly power (Misiolek 1980).

  43. 43.

    In the absense of the monitoring problem, Ino (2007) investigates the optimal disposal fee when both the product and recycling markets are oligopolistic. David and Sinclair-Desgagne (2005, 2010) investigate the optimal environmental policies under imperfect competition both in the product and abatement markets in a more general context.

  44. 44.

    Some reader might think that if we introduce a fixed penalty that does not depend on the size of violation, most of our results would disappear. But the problem is not so simple, because if a policymaker increases the level of the subsidy to encourage proper recycling, it simultaneously increases the benefit of illegal disposal as has been discussed in this paper. Hence, the necessary level of the expected fixed penalty to prevent the firm from switching from legal to illegal recycling activities also increases. Whenever monitoring efforts have to be expended to enhance the expected fixed penalty, a similar trade-off to a linear case arises between the encouragement of recycling and the expansion of the monitoring cost.

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Acknowledgements

The authors thank useful comments by an anonymous referee of this journal, Jiro Akita, Alain Ayong le Kama, Paul Calcott, Etienne Farvaque, Ida Ferrara, Lionel Ragot, Hiroaki Sakamoto, Akihisa Shibata, Takayoshi Shinkuma, Hide-Fumi Yokoo, the participants of workshops at the Victoria University of Wellington, Kyoto University, Kwansei Gakuin University, the 15th Annual Meeting of the Association for Public Economic Theory at the University of Washington (Seattle), the 21st Annual Conference of the European Association of Environmental and Resource Economists at the University of Helsinki, the 15th Journées Louis-André Gérald-Varet (International Conference in Public Economics) in Aix-en-Provence, and the 26th Annual Conference of the Canadian Resource and Environmental Economists in Banff, Alberta. The usual disclaimer applies. This work was supported by JSPS KAKENHI (23730260 and 25285087).

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Appendix

Appendix

Proof of Proposition 1

Proof

First, we show that when \(s+\tau \le \min [\tau _{f},a]\), \(r^{*}=0\). Suppose to the contrary that \(r^{*}>0\). Then, \(\mu =C_{r}^{\prime }(r^{*} )-(s+\tau )\) by (2) and (10). \(r^{*}>0\) implies \(r_{c}^{*}>0\) or \(z_{f}^{*}>0\). If \(r_{c}^{*}>0\), \((s+\tau )+B^{\prime }(r^{*})=C_{r}^{\prime }(r^{*})\) by (11). If \(z_{f}^{*}>0\), \((s+\tau )-\tau _{f}^{e}=C_{r}^{\prime }(r^{*})\) by (12). In both cases, the left hand side is negative by the assumption \(s+\tau \le \min [\tau _{f}^{e},a]\). The right hand side is strictly positive by \(r^{*}>0\), which is a contradiction. Thus, we obtain the result (i).

Next, we show that, when \(s+\tau >\min [\tau _{f}^{e},a]\), \(r^{*}>0\). This is because \(B^{\prime }(r_{c}^{*}),-\tau _{f}^{e}\le P_{r}-s\le C_{r}^{\prime }(r^{*})-(s+\tau )\) from the first-order conditions concerning the recycling market. Thus, if \(r^{*}=0\), we must have \(a\ge s+\tau \) and \(\tau _{f} ^{e}\ge s+\tau \), which contradicts \(s+\tau >\min [\tau _{f}^{e},a]\). Consequently, in cases (ii)–(iv), we must have \(r^{*}>0\).

Now, consider case (ii) where \(s+\tau >\min [\tau _{f}^{e},a]\) and \(\tau _{f} ^{e}\le a\), that is, \(-a\le -\tau _{f}^{e}\) and \(\tau _{f}^{e}<s+\tau \). If \(z_{f}^{*}=0\), \(r^{*}>0\) implies \(r_{c}^{*}>0\). Therefore, by (11), \(\mu =B^{\prime }(r^{*})<B^{\prime }(0)=-a\le -\tau _{f}^{e}\), where the inequalities stems form the assumptions, \(r^{*}>0\) and \(-a\le -\tau _{f}^{e}\) of case (ii). This contradicts to (12). Thus, \(z_{f}^{*}>0\) must hold in this case. Then, \(\mu =-\tau _{f}^{e}\) by (12). If \(r_{c}^{*}>0\), we have \(B^{\prime }(r_{c}^{*} )=\mu =-\tau _{f}^{e}\) by (11) but \(B^{\prime }(r_{c}^{*} )<B^{\prime }(0)=-a\le -\tau _{f}^{e}\) by the assumptions, which is a contradiction. On the other side, if \(r_{c}^{*}=0\), the condition (11) becomes \(-a\le \mu =-\tau _{f}^{e}\) and is satisfied for case (ii). Further, since \(r^{*}>0\), \(C_{r}^{\prime }(r^{*})=(s+\tau )-\tau _{f}^{e} \) must hold by (2) and (10). Since the right-hand side is strictly positive by the assumption \(\tau _{f}^{e}<s+\tau \), solving this yields \(r^{*}>0\) uniquely. Thus, we obtain the result (ii) in the Proposition.

Finally, we consider cases (iii) and (iv) where \(s+\tau >\min [\tau _{f}^{e},a]\) and \(a<\tau _{f}^{e}\), that is, \(-a>-\tau _{f}^{e}\) and \(a<s+\tau \). If \(r_{c}^{*}=0\), \(r^{*}>0\) implies \(z_{f}^{*}>0\). Therefore, (11) and (12) implies \(-a\le \mu =-\tau _{f}^{e}\), which contradicts the assumption, \(a<\tau _{f}^{e}\). Thus, it must be true that \(r_{c}^{*}>0\) in this case. Then, \(\mu =B^{\prime }(r_{c}^{*})\) by (11).

Furthermore, suppose that \(z_{f}^{*}>0\) in cases (iii) and (iv). Then, \(B^{\prime }(r_{c}^{*})=-\tau _{f}^{e}\) by (11) and (12), and solving this yields \(r_{c}^{*}>0\) uniquely where the strict positivity holds by the assumption \(B^{\prime }(0)=-a>-\tau _{f}^{e}\). Also, by (2), (10) and (12), \(C_{r}^{\prime }(r^{*})=(s+\tau )-\tau _{f}^{e}\) must hold for \(r^{*}>0\). Then, \(z_{f}^{*}=r^{*}-r_{c}^{*}>0\) holds if and only if \(\tau _{f}^{e}<-B^{\prime }(r^{C})\). In order to show this, suppose \(\tau _{f}^{e}\lessgtr -B^{\prime }(r^{C})\). Then, \(B^{\prime }(r_{c}^{*})=-\tau _{f}^{e}\gtrless B^{\prime }(r^{C})\) and \(C_{r}^{\prime }(r^{*})=(s+\tau )-\tau _{f}^{e}\gtrless (s+\tau )+B^{\prime }(r^{C})=C_{r}^{\prime }(r^{C})\), where the last equality is obtained from (6). These inequalities implies that \(r_{c}^{*}\lessgtr r^{C}\) by \(B^{\prime \prime }<0\) and \(r^{*}\gtrless r^{C}\) by \(C_{r}^{\prime \prime }>0\), respectively. Thus, \(r^{*}>r_{c}^{*}\) if and only if \(\tau _{f}^{e}<-B^{\prime }(r^{C})\). We obtain the result (iii).

Alternatively, suppose that \(z_{f}^{*}=0\) in cases (iii) and (iv). Then, by (2), (11) and (10), \(r^{*}=r_{c}^{*}>0\) implies \(B^{\prime }(r^{*})=C_{r}^{\prime }(r^{*})-(s+\tau )\). Solving this yields \(r^{*}=r_{c}^{*}>0\), where the strict positivity holds because \(r^{*}=r^{C}\) by (6) and \(r^{C}>0\) by the assumption \(a<s+\tau \). Since \(\mu =B^{\prime }(r_{c}^{*})\), (12) becomes \(-\tau _{f}^{e}\le B^{\prime }(r_{c}^{*})\). Therefore, in this case, (12) satisfies if and only if \(\tau _{f}^{e}\ge -B^{\prime }(r^{C})\) by \(r_{c}^{*}=r^{C}\). Thus, we obtain the result (iv).\(\square \)

Proof of Proposition 2

Proof

(i) The first-order condition for the welfare maximization problem of (13) with respect to t is

$$\begin{aligned} 0=\frac{\partial W^{*}(t,s)}{\partial t}=U^{\prime }\frac{\partial x^{*} }{\partial t}-C_x^{\prime }\frac{\partial x^{*}}{\partial t}-d\frac{\partial x^{*}}{\partial t}=0. \end{aligned}$$
(15)

Plugging the first-order conditions, (1) and (3), obtained for the final good market, into (15), and solving the equation with respect to t, we obtain \(t^{*}=d\).

(ii-1) Suppose \(r^{*}=r_{c}^{*}>0\) is optimal. Then, by Corollary 1(i), we must have \(s>a\) and thus, \(\hat{\tau }_{f} (s)=s-P_{r}^{C}(s,0)>0\) by (14), where the positivity is due to the fact that \(\hat{\tau }_{f}(s)=s-P_{r}^{C}(s,0)\) increases in \(s>a\) as can be derived in (16) below and \(\hat{\tau }_{f}(a)=a>0\). Substituting \(\hat{\tau }_{f}=s-P_{r}^{C}\) into \(W^{*}\), the first-order condition for (13) with respect to s is

$$\begin{aligned} 0=\frac{\partial W^{*}(t,s)}{\partial s}&=B^{\prime }\frac{\partial r^{*}}{\partial s}-C_{r}^{\prime }\frac{\partial r^{*}}{\partial s}+d\frac{\partial r^{*}}{\partial s}-\Gamma _{f}^{\prime }\left[ \frac{\partial (s-P_{r}^{C})}{\partial s}\right] \nonumber \\&=\frac{1}{C_{r}^{\prime \prime }-B^{\prime \prime }}(B^{\prime }-C_{r}^{\prime }+d)-\Gamma _{f}^{\prime }\frac{-B^{\prime \prime }}{C_{r}^{\prime \prime }-B^{\prime \prime }}, \end{aligned}$$
(16)

where the comparative static results, \(\partial r^{C}/\partial s=1/(C_{r} ^{\prime \prime }-B^{\prime \prime })>0\) and \(\partial P_{r}^{C}/\partial s=C_{r}^{\prime \prime }/(C_{r}^{\prime \prime }-B^{\prime \prime })>0\) are derived from (6) and \(\partial r^{C}/\partial s=\partial r^{*}/\partial s\) by Proposition 1(iv). By the supposition of \(r^{*}=r_{c}^{*}>0\), (2) implies \(C_{r}^{\prime }=P_{r}\), and (10) and (11) implies \(B^{\prime }=P_{r}-s\). Plugging these first-order conditions for the recycling market into (16) and solving the equation with respect to s, we obtain \(s^{*}=d+B^{\prime \prime }\Gamma _{f}^{\prime }\).

(ii-2) Suppose \(r^{*}=r_{c}^{*}=z_{f}^{*}=0\) is optimal. By Proposition 1(i), we must have \(s\le a\). If \(0<s\le a\), \(\hat{\tau }_{f}(s)=s\) by (14). Thus, \(W^{*}(t,s)\) is decreasing in s since it depends on s only through \(\Gamma _{f}(\hat{\tau }_{f} (s))=\Gamma _{f}(s)\) when \(r^{*}=0\). If \(s\le 0\), \(\hat{\tau }_{f}(s)=0\) by (14). Thus, \(W^{*}(d,s)\) is constant in s when \(r^{*}=0\) and \(\Gamma _{f}(\hat{\tau }_{f}(s))=\Gamma _{f}(0)\). Therefore, any non-positive subsidy can be optimal, i.e., \(s^{*}\le 0\), and \(\tau _{f}^{*}=0\).

(iii) Suppose that the tax is set at \(t=d\) according to (i) and the policymaker monitors the firm according to \(\hat{\tau }_{f}(s)\) defined in (14). From (ii-1) and (ii-2), in order to identify which policy set is globally optimal, we must compare two locally maximized welfare levels, \(W^{*}(d,s^{M})\) and \(W^{*}(d,0)\). In the former, \(\hat{\tau }_{f} =s^{M}-P_{r}^{C}(s^{M},0)\) if \(s^{M}>a\) is satisfied, which we will be check later, and, in the latter, \(\hat{\tau }_{f}=0\) by definition.

Choose the value of \(d^{\prime }\) that satisfies \(d^{\prime }=a+A(a)\), where \(A(a)=-B{^{\prime \prime }}(r^{*}(a))\Gamma _{f}^{\prime }(\hat{\tau } _{f}(a))=-B{^{\prime \prime }}(0)\Gamma _{f}^{\prime }(a)\) since \(r^{*}(a)=0\) under the monitoring rule of \(\hat{\tau }_{f}(a)=a\). Note that \(s^{M}=a\) when \(d=d^{\prime }\) by the definition of \(d^{\prime }\). Furthermore, \(s^{M}\gtrless a\) if only if \(d\gtrless d^{\prime }\). This is because if \(s^{M}\gtrless a\), \(d\gtrless d^{\prime }\) must hold since

$$\begin{aligned} d=s^{M}+A(s^{M})\gtreqless s^{M}-B{^{\prime \prime }}(r^{*}(s^{M}))\Gamma _{f}^{\prime }(a)\gtreqless s^{M}-B{^{\prime \prime }}(0)\Gamma _{f}^{\prime }(a)\gtrless d^{\prime }, \end{aligned}$$

which is true because \(s^{M}>a\) (resp. \(s^{M}<a\)) implies that \(\hat{\tau } _{f}(s^{M})=s^{M}-P_{r}^{C}(s^{M},0)>a\) (resp. \(\hat{\tau }_{f}(s^{M} )=\min [s^{M},0]<a\)) and \(r^{*}(s^{M})>0\) (resp. \(r^{*}(s^{M})=0\)). Note that the equality is included in the first and the second inequalities above because \(\Gamma _{f}^{\prime \prime }\ge 0\) and \(r^{*}(s^{M})\) can be 0.

Consider the case where \(d>d^{\prime }\). Then, \(s^{M}>a\) holds. In this case, \(W^{*}(d,s^{M})\) under \(\hat{\tau }_{f}=s^{M}-P_{r}^{C}(s^{M},0)\) needs to be compared to \(W^{*}(d,0)\) (see Fig. 4). For \(s>a\) (including \(s=s^{M}\)), invoking the envelope theorem, we have \(\partial W^{*}(d,s)/\partial d=r^{*}(s)-x^{*}(d)\) since \(r^{*}(s)>0\). From the envelope theorem and the fact that \(r^{*}(0)=0\), we also have \(\partial W^{*}(d,0)/\partial d=-x^{*}(d)\). Thus, (i) \(W^{*}(d,s^{M})-W^{*}(d,0)\) is strictly increasing in \(d>d^{\prime }\). Now, take an arbitrary subsidy level \(s^{\prime }\) such that \(s^{\prime }>a\). Then, \(W^{*} (d,s^{M})\ge W^{*}(d,s^{\prime })\) since \(s=s^{M}\) gives the local maximum in this range. Moreover, \(\partial (W^{*}(d,s^{\prime })-W^{*}(d,0))/\partial d=r^{*}(s^{\prime })>0\), where \(r^{*}(s^{\prime })\) is constant for any d. Hence, if d is sufficiently large, (ii) there exists \(d>d{^{\prime }}\) such that \(W^{*}(d,s^{M})-W^{*}(d,0)\ge W^{*}(d,s^{\prime })-W^{*}(d,0)>0\). Furthermore, (iii) \(W^{*}(d,s^{M} )-W^{*}(d,0)\le 0\) holds if \(d=d^{\prime }\) (with equality if and only if \(a=0\)) since \(s^{M}=a\) when \(d=d^{\prime }\) and \(W^{*}(d,a)\le W^{*}(d,0)\) (note that \(W^{*}(d,s)\) is decreasing in s when \(0<s\le a\) as is shown in the proof of (ii-2)). From (i)–(iii), there exists a value of \(\bar{d}\ge d^{\prime }\) such that \(W^{*}(d,s^{M})-W^{*}(d,0)>0\) if and only if \(d>\bar{d}\), where \(\bar{d}=d^{\prime }\) if and only if \(a=0\).

Fig. 4
figure4

Welfare comparison in the case \(d\ge d^{\prime }\) (\(s^{M}>a\)): the left panel is the case where \(W^{*}(d,s^{M})<W^{*}(d,0)\), and the right panel is the case where \(W^{*}(d,s^{M})>W^{*}(d,0)\)

Finally, consider the case where \(d\le d^{\prime }\). Then, \(W^{*}(d,s)\) is decreasing in s when \(s>a\) since \(s^{M}\le a\). Therefore, under the optimal policy set, \(s^{*}\le 0\) and \(r^{*}(s^{*})=0\).\(\square \)

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Ino, H., Matsueda, N. The curse of low-valued recycling. J Regul Econ 55, 282–306 (2019). https://doi.org/10.1007/s11149-019-09378-9

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Keywords

  • Hazardous and household solid waste
  • Illegal waste disposal
  • Enforcement
  • Compliance
  • Government-induced recycling
  • Optimal taxation

JEL Classification

  • H21
  • Q21
  • Q28