So H is negative definite; [[PI].sup.C] is jointly concave with respect to [p.sup.C.sub.i], [p.sup.C.sub.3-i], [p.sup.C.sub.j], and [p.sup.C.sub.3-j]; thus, the optimal retailing and repurchase prices can be obtained by the first-order conditions as follows:
By combining (11), (12), (13), and (14) we can obtain the optimal retail prices [p.sup.*.sub.i] and [p.sup.*.sub.j], and optimal repurchase prices [p.sup.*.sub.Ri] and [p.sup.*.sub.Rj].
By assumptions 0 [less than or equal to] [gamma] < [beta] < 1 and 0 [less than or equal to] s < [beta] < 1, we can obtain that d[p.sup.CC.sub.j]/d[gamma] > 0; thus, the repurchase prices [p.sup.C.sub.Ri] and [p.sup.C.sub.Rj] are strictly increasing with respect to the recycling substitution effect.
By combining (22), (23), (24), and (25), we can obtain the optimal retail prices [p.sub.i] and [p.sub.j] and optimal repurchase prices [P.sub.Ri] and [p.sub.Rj].
And it follows from (19) that the repurchase prices [p.sub.Ri] and [p.sub.Rj] are decreasing with respect to the wholesale price w, while increasing with respect to the buy-back payment b.
Then we derive the optimal retail price, the optimal repurchase price, and the optimal profits of the manufacturer and the retailers.