Product Design for Recycling and Recycling Industry

Abstract

This paper treats the analysis of a recycling industry in an economy where the quality of product design is incorporated. We assume three types of market failures caused by a monopoly power, environmental damage by wastes, and cost of the product design for recycling. In order to solve these market failures, we consider three policy instruments and characterize these policies in this present economy. Finally, a comparative statics analysis is carried out with respect to the consumption good supply, production design quality, and recycling ratio in response to changes in the recycled material price, landfilling cost of wastes, and degree of environmental damage by wastes.

Appendix

The first-order conditions for maximizing the social welfare are

$$ \begin{aligned}&\beta -\delta y^{*}-F\left(q^{*}\right)-G\left(q^{*},\alpha^{*}\right)-\left({c}^w+{c}^d\right)\left(1-\alpha^{*}\right)+\alpha^{*}{p}^r=0,\\ &\quad-y{F}_q-y{G}_q=0,\\ &\quad -y{G}_{\alpha }+y{c}^w+y{c}^d+y{p}^r=0.\end{aligned} $$

The Jacobian matrix of (y, q, α) of these equations is

$$ \begin{aligned}&\left[\arraycolsep6pt\begin{array}{ccc} -\delta & -{F}_q-{G}_q & -{G}_{\alpha }+{c}^w+{c}^d+{p}^r \\ -{F}_q-{G}_q & -y{F}_{qq}-y{G}_{qq} & -y{G}_{\alpha q} \\ -{G}_{\alpha }+{c}^w+{c}^d+{p}^r & -y{G}_{\alpha q} & -{G}_{\alpha \alpha} \end{array}\right]\\ &\quad =\left[\begin{array}{ccc} -\delta & 0 & 0 \\ 0 & -y{F}_{qq}-y{G}_{qq} & -y{G}_{\alpha q} \\ 0 & -y{G}_{\alpha q} & -{G}_{\alpha \alpha} \end{array}\right]\end{aligned} $$

Assume that G _αα > 0, Gqq > 0, Fqq > 0, and the convexity of G(q, α) w.r.t. (q, α).

Then, in this Jacobian matrix, all principal minors of orders of 1 and 3 are negative and all second-order principal minors are positive. Thus, the second-order optimal condition is satisfied.