We provide necessary and sufficient conditions for the existence of Cournot equilibrium for a class of demand functions when firms are identical and produce under constant returns to scale. We focus on inelastic and convex demand yielding non-concave profit functions. We show that equilibrium is unique, stable and yields well-behaved comparative statics. The model can yield huge welfare losses, non-downward sloping Laffer curves and tax reforms that are never Pareto improving. In the Appendix we generalize our results on existence and uniqueness to more general inverse demand functions and non-identical firms.