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.

Appendix

1.1 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).$\square$

1.2 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

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 $r^{*}(s^{*})=0$.$\square$