The paper compares the equilibrium outcomes of Monopoly and Cournot Duopoly using the linear demand function and cubic cost function. Given the limited literature available on the subject, emphasis is placed on the signs, range of values, and interrelationships between the various parameters of the cost and demand functions to compare the equilibrium outcomes of both market structures. In both cases, using identical demand and cost functions is meaningful when the equilibrium outcomes are feasible and all the non-negativity conditions are satisfied.