Optimal design of deposit–refund systems considering allocation of unredeemed deposits

Abstract

In a deposit–refund system, not all deposits are refunded; this yields unrefunded deposits that are called unredeemed deposits. If the system is mandatory, it is difficult to decide who should hold these—suppliers, recyclers or government. However, from the perspective of economics, examining the allocation of the unredeemed deposits is not enough. We investigate the optimal design of deposit–refund systems considering the allocation of unredeemed deposits in detail from the economics aspect. The conclusions can be summarized as follows. The refund should be equal to the sum of the following three: (1) the suppliers’ marginal net revenue from collecting and treating used deposit–refund goods, (2) the marginal negative externality, and (3) the deposit multiplied by the share of the unredeemed deposits that the government and the recycler collect from the supplier. The weight for the sum of (2) and (3) is the marginal product for the supplier multiplied by that for the recycler.

Appendix: When the return rate is 100%

The Jacobian is u _xx  − v _yy (<0). From the implicit function theorem, the following identities can be set:

$$ \begin{aligned}&Y-T+ry^{S}(p+d,r,Y-T,w)-(p+d)x^{D}(p+d,r,Y-T,w)-z^{D}(p+d,r,Y-T,w)=0,\\& x^D(p+d,r,Y-T,w)-y^S(p+d,r,Y-T,w)=0,\\& u_{x}(x^{D}(p+d,r,Y-T,w))-\lambda_{1}(p+d,r,Y-T,w)(p+d)+\lambda_2(p+d,r,Y-T,w)=0,\end{aligned} $$

(29)

$$ -v_{y}(y^{S}(p+d,r,Y-T,w))+\lambda_{1}(p+d,r,Y-T,w)r-\lambda_2(p+d,r,Y-T,w)=0, $$

(30)

and

$$ 1-\lambda_{1}(p+d,r,Y-T,w)=0. $$

(31)

From comparative statics, the characteristics of x ^D and y ^S with respect to p + d, r and Y − T is as follows:

$$ \begin{aligned}\frac{\partial x^D}{\partial (p+d)}&=\frac{1}{u_{xx}-v_{yy}}({<}0),\quad \frac{\partial y^S}{\partial (p+d)}=\frac{1}{u_{xx}-v_{yy}}({<}0),\\ \frac{\partial x^D}{\partial r}&=\frac{-1}{u_{xx}-v_{yy}}({>}0),\quad \frac{\partial y^S}{\partial r}=\frac{-1}{u_{xx}-v_{yy}}({>}0),\\ \frac{\partial x^D}{\partial (Y-T)}&=0,\quad\hbox{and}\quad\frac{\partial y^S}{\partial (Y-T)}=0.\\ \end{aligned} $$

The consumer comparative statics of λ₂ with respect to Y − T is 0. Therefore, in the same way as the body, x ^D and y ^S are not the functions of Y − T. Equation (14) becomes as follows:

$$ \begin{aligned}Y&=I+[\{p+(1-\alpha)d\}f(X^D(p,\alpha,\beta,d,\hat{p}))-\hat{p}X^D(p,\alpha,\beta,d,\hat{p})] \\&\quad+\{s+\tau-(1-\alpha)r\}y^S(p+d,r,w)-C^y(y^S(p+d,r,w))\\&\quad+\hat{p}g(y^D(\hat{p},\tau))-\tau y^D (\hat{p},\tau),\\ \end{aligned} $$

which results in $Y(I,p,\alpha,d,\beta,\hat{p},s,r,\tau,w).$ Equation (15) becomes as follows:

$$ \begin{aligned}V&\equiv u(x^D(p+d,r,w))-v(y^S(p+d,r,w)))\\&\quad+z^D(p+d,r,Y-T,w)-N(x^D(p+d,r,w)-y^S(p+d,r,w)).\\\end{aligned} $$

Equations (16) and (17) are the same as those in the body:

$$ \begin{aligned} \max_{\alpha,\beta,d,r,s,T}V&\\\hbox{subject to}\quad&\alpha(dx^S-ry^S)+T=sy^S \\ \end{aligned} $$

Even though x = y, unredeemed deposits are generated when d is larger than r. In the same way as the body, the indirect utility function becomes as follows:

$$ \begin{aligned}V&\equiv u(x^D(p+d,r,w))-v(y^S(p+d,r,w))+I+(p+d)[f(X^D(p,\alpha,\beta,d,\hat{p}))-x^D(p+d, r,w)]\\&\quad-\hat{p}[X^D(p,\alpha,\beta,d,\hat{p})-g(y^D(\hat{p},\tau))]\\&\quad+\tau [y^S(p+d,r,w)-y^D(\hat{p},\tau)]-C^y(y^S(p+d,r,w))-N(x^D(p+d,r,w)-y^S(p+d,r,w)).\\ \end{aligned} $$

(32)

Equations (19–22) become as follows:

$$ x^{D}(p+d,r,w)=f(X^D(p,\alpha,\beta,d,\hat{p})), $$

(33)

$$ y^D(\hat{p},\tau)=y^S(p+d,r,w), $$

(34)

$$ X^D(p,\alpha,\beta,d,\hat{p})=g(y^D(\hat{p},\tau)),\quad\hbox{and} $$

(35)

$$ C^y(y^S(p+d,r,w))+z^{D}(p+d,r,Y(I,p,\alpha,\beta,d,\hat{p},s,r,\tau,w)-T,w)=I, $$

(36)

and Walras law holds in the same way as the body. Equation (34) can be seen as the definitional equation of τ. Therefore, we consider the equilibrium as (33) and (35). We substitute (34) into V. As a result, the indirect utility function is as follows:

$$ \begin{aligned}V&=u(x^D(p+d,r,w))-v(y^S(p+d,r,w))+I+(p+d)[f(X^D(p,\alpha,\beta,d,\hat{p}))-x^D(p+d,r,w)]\\&\quad-\hat{p}[X^D(p,\alpha,\beta,d,\hat{p})-g(y^S(p+d,r,w))]-C^y(y^S(p+d,r,w))\\&\quad-N(x^D(p+d,r,w)-y^S(p+d,r,w)),\\ \end{aligned} $$

which is the objective function. In (33), (35), and the objective function, s does not appear, therefore, we delete s from the control variables for the government.

Then, the following equations hold, using (29), (30), (31) and (12):

$$ \begin{aligned} \frac{\partial V}{\partial \alpha}&=(\alpha+\beta)df_X \frac{\partial X^D}{\partial \alpha},\\ \frac{\partial V}{\partial \beta}&=(\alpha+\beta)df_X \frac{\partial X^D}{\partial \beta},\\ \frac{\partial V}{\partial d}&=[-\lambda_2-N_{w}]\frac{\partial x^D}{\partial (p+d)}+[-r+\lambda_2+\hat{p}g_y+N_{w}-C^{y'}]\frac{\partial y^S}{\partial (p+d)}\\&\quad+(\alpha+\beta)df_X \frac{\partial X^D}{\partial d}+[f(X^D)-x^D],\quad\hbox{and}\quad\\ \frac{\partial V}{\partial r}&=(-\lambda_2-N_{w})\frac{\partial x^D}{\partial r}+[-r+\lambda_2+\hat{p}g_y+N_{w}-C^{y'}]\frac{\partial y^S}{\partial r}.\\ \end{aligned} $$

Therefore,

$$ \begin{aligned}&\max_{\alpha,\beta,d,r}V\\&\hbox{subject to}\quad x^D(p+d,r,w)=f(X^D(p,\alpha,\beta,d,\hat{p})),\quad\hbox{and}\\& X^D(p,\alpha,\beta,d,\hat{p})=g(y^S(p+d,r,w)).\\ \end{aligned} $$

The Lagrangian becomes as follows:

$$ \begin{aligned} L&=V+\lambda_3[f(X^D(p,\alpha,\beta,d,\hat{p}))-x^D(p+d,r,w)]\\&\quad+\lambda_4[g(y^S(p+d,r,w))-X^D(p,\alpha,\beta,d,\hat{p})].\\ \end{aligned} $$

From this Lagrangian, using the result of the comparative statics above, the first-order conditions of the government behavior are as follows:

$$ \begin{aligned}\frac{\partial L}{\partial \alpha}&=[\{(\alpha+\beta)d+\lambda_3\} f_X-\lambda_4]\frac{\partial X^D}{\partial \alpha}=0,\\ \frac{\partial L}{\partial \beta}&= [\{(\alpha+\beta)d+\lambda_3\} f_X-\lambda_4]\frac{\partial X^D}{\partial \beta}=0,\\ \frac{\partial L}{\partial d}&=(-\lambda_2-N_{w}-\lambda_3)\frac{\partial x^D}{\partial (p+d)}+[-r+\lambda_2+\hat{p}g_y+N_{w}-C^{y'}+\lambda_4 g_y]\frac{\partial y^S}{\partial (p+d)}\\&\quad+[\{(\alpha+\beta)d+\lambda_3\}f_X-\lambda_4] \frac{\partial X^D}{\partial d}+[f(X^D)-x^D]=0,\\ \frac{\partial L}{\partial r}&=(-\lambda_2-N_{w}-\lambda_3)\frac{\partial x^D}{\partial r}+[-r+\lambda_2+\hat{p}g_y+N_{w}-C^{y'}+\lambda_4 g_y]\frac{\partial y^S}{\partial r}=0,\\ \frac{\partial L}{\partial \lambda_3}&=f(X^D)-x^D=0,\quad\hbox{and}\quad\\\frac{\partial L}{\partial \lambda_4}&=g(y^S)-X^D=0.\\ \end{aligned} $$

Therefore,

$$ \begin{aligned}&\{(\alpha+\beta)d+\lambda_3\}f_X =\lambda_4,\quad{\text{and}} \\&-\lambda_3-r+(\hat{p}+\lambda_4) g_y-C^{y'}=0. \\ \end{aligned} $$

These results are almost the same as those in the body, except λ₂ + λ₃ =  −N _w . λ₂ is the marginal utility of the waste and λ₃ is the marginal utility of the excess supply of the deposit–refund goods.