Existence, uniqueness and global stability of Cournot equilibrium in oligopoly with cost externalities (Q2898760)

The authors study the following oligopoly model without product differentiation. There are \(n\) firms with the industry total output \(x =\sum_{k=1}^n x_k\), where \(x_k\) is the output of the \(k\)-th firm. It is assumed that the \(k\)-th firm output capacity is restricted by some finite limit \(L_k\), that is, \(0\leq x_k\leq L_k\). The profit \(\pi_k(x_k)\) of the \(k\)-th firm (if its output is \(x_k\)) is defined as \(\pi_k(x_k)=x_kf(x) - c_k(x_k, x-x_k)\), where \(f(x)\) is the income per output unit in the oligopoly and \(c_k(x_k, x-x_k)\) is the cost function of the \(k\)-th firm when \(x\) is the total oligopoly output and \(x_k\) is the output of the \(k\)-th firm. The model of the oligopoly described above is considered as the \(n\)-person noncooperative game \(\Gamma=(\{[0, L_k]\}_{1\leq k\leq n}, \{\pi_k\}_{1\leq k\leq n})\). It is proved in the paper that under some additional assumptions about the functions \(f\) and \(c_k\), the game \(\Gamma\) has a Nash equilibrium in pure strategies. Besides, the global stability condition for the equilibrium is derived using a firm's gradient output adjustment system.