Closing the loop in a duopolistic circular economy model

We analyze recycling as firm's optimal strategy in a duopolistic market, under the cases of a linear economy (no recycling technology), semi-circular economy (one firm recycles), and a circular economy (both firms recycle). Furthermore, our analysis will be separated into two cases: when only one firm recycles, then it can either choose to use the recycled material in its production or sell part (or all) of it to the other firm; similarly, if both firms can recycle, they can either use the recycled material they produced, or sell all of it or just part of it to the other firm. We characterize the resulting equilibria showing under which circumstances the economy is linear, semi-circular, or when it "closes the loop", being fully circular (both firms recycle under alternative organizational structures).