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.

Monopoly: A pure monopoly exists when only one producer (seller) exists in a market. There are no competitors or rivals. However, the policies of a monopolist may be constrained by the indirect competition of all commodities for the consumers’ money and of reasonably adequate substitute goods and by the threat of potential competition if market entry is possible (Ferguson, 1975).

Duopoly: A simplification of oligopoly (few sellers’ market), where there are only two producers (sellers) selling an identical commodity (in our context). In such markets, strategic interaction among firms, hence rivalry, is clearly visible.

Cournot Duopoly: A specific type of duopoly model named after the economist who proposed it (Cournot, 2020.) where the two firms simultaneously choose their profit-maximising outputs. Unfortunately, little could be gathered from the existing literature regarding the research question. Cubic cost functions are discussed quite in detail in Vali (2014), and the analysis in the first part of Section 4 of the paper uses a similar cubic cost function for both market structures. Later on, the cost function in Vali (2014) was used to find and analyse the equilibrium outcomes of a firm in a perfectly competitive market and for firms in a monopolistically competitive market. After an extensive search for research and articles regarding the same, only one relevant piece of literature was found- where, once again, the authors focused more on the graphical (Nikutowski et al., 2013) part and other intuitive questions rather than the algebra. Moreover, nothing similar could be found in the case of Duopoly.

In light of the gaps observed during the literature review, the exercise to apply the cubic cost function along with the linear demand function in both forms of the market to find out their equilibrium outcomes and strike a comparison as far as possible was felt justified. Meanwhile, since the demand and cost function parameters in all the cases are arbitrary constants, it is meaningful to find out the signs and the range of values of the parameters and possible relationships between them, making the equilibrium outcomes comparable and feasible.

The methodology used is the universally applied optimisation (individual profit maximisation) technique. For the monopoly, the first order condition (F.O.C) (Sydsaeter et al., 2016) for optimisation is MR = MC to determine equilibrium output, and based on that output, equilibrium price and profit would be calculated by standard formulae. The second order condition (S.O.C) (Sydsaeter et al., 2016) for the same, i.e. this equilibrium should be on the rising portion of the MC curve, is also verified.

In Cournot Duopoly, the same optimisation technique is used in a simultaneous move game, where the two firms choose their profit-maximising output levels simultaneously. The equilibrium (Nash Equilibrium) occurs where the Best Response Function (BRF) (Osborne, 2009) of both firms has a meaningful intersection, that is, an intersection where the equilibrium output occurs on the rising portion of their MC Curves.

Finally, for comparability, the same linear demand function and the same cubic cost function are considered for firms in both the market structures considered here.

Ideally, the total duopoly profit should be less than the monopoly profit with identical demand and cost conditions (as has been done here), but in the present context with a cubic cost function, it will be really difficult to compare them.

Now, with the obtained results and the way they are obtained, it becomes quite clear why there is hardly any work towards this approach. Algebra is complicated, and it gradually gets more challenging to get a very clear picture of the meaningfulness of the results, especially when comparing them. In the paper, the relationship between the parameter in the demand function) and the parameter g (in the cost function) turned out to be crucial for comparability between these market structures; that is, the relationship must hold for positive profits in both markets. Finally, there is scope for further studies to explore similar analyses in other market structures using a cubic cost function.

1. Cournot, A. (2020). Researches into the Mathematical Principles of the Theory of Wealth |.
2. Retrieved December 7, 2024, from https://www.taylorfrancis.com/chapters/edit/10.4324/9781003051091-2/researches- mathematical-principles-theory-wealth-cournot
3. Eppstein et al, D. (n.d.). Sparse dynamic programming I: linear cost functions: Journal of the ACM: Vol 39, No 3. Retrieved December 7, 2024, from https://dl.acm.org/doi/abs/10.1145/146637.146650
4. Ferguson, C. (1975). Microeconomic theory | CiNii Research. Retrieved December 7, 2024, from https://cir.nii.ac.jp/crid/1130282270630389888
5. Koutsoyiannis, A. (1989). Modern microeconomics. MacMillan.
6. Nikutowski, O., Leis, V., & Von Weizsäcker, R. K. Frhr. (2013). Cubic Cost Functions and Major Market Structures. The Journal of Economic Education, 44(1), 91–91. https://doi.org/10.1080/00220485.2013.744630
7. Osborne, M. J. (2009). An Introduction to Game Theory. Oxford University Press. Sydsaeter, K., Hammond, P., Carvajal, A., & Strom, A. (2016). Essential Mathematics for Economic Analysis. Pearson Education.
8. Varian, Hal, R. (2014). Intermediate Microeconomics with Calculus: A Modern Approach: International Student Edition. W. W. Norton & Company.