Assuming identical firms, a linear market demand function and a single factor of production, labour, we analyse the existence and stability of a homogeneous Cournot duopoly facing imperfect competition in both product and factor markets. We establish the possibility of antisymmetric equilibria separated by a unique symmetric equilibrium. Under certain assumptions on the wage function we are able to demonstrate local stability of the unique symmetric equilibrium, and its global stability under the assumption of a convex reaction function.