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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 like that 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.

The cubic cost function (which would be identical for each firm throughout this paper for the sake of comparison)

C(q) – Total cost, q – output, and f, g, h are constants

The fixed cost component is avoided for simplicity since it does not affect the equilibrium quantity and, hence, the equilibrium price.

To start with, we must find out about the restrictions on the constants f, g and h, so that the marginal cost (MC) and the average cost (AC) are ‘U’ shaped and the relevant portions of these curves lie in the feasible region; that is their minimum values are positive (non-negative at least)

Since, in this case C(0) = 0, we must  have C(1)  = f + g + h > 0  …(1.2)

Hence, to obtain the MC function, we differentiate both sides of (1.1) w.r.t. q

Which is a quadratic function and would be strictly convex for f > 0

For the minimum point of the MC curve, we must have the following;

The second-order condition for strict convexity of the MC curve is automatically satisfied, since;

Now, for the value of the MC at its minimum to be strictly positive, we must have the following;

Now, the average cost (AC) function is;

Which is also a quadratic function in q and would be strictly convex for f > 0

For the minimum point of the AC curve, we must have;

Also, for the value of AC at this minimum point to be strictly greater than zero, we must have;

Monopoly

The inverse demand function (linear) faced by the monopolists is given by;

Where p – market price and q – output.

The cost function is the same as in (1.1)

For the MC curve to intersect the MR curve twice in the non-negative quadrant, we must have

a < h

For the profit function to be strictly concave, we must have;

Which is strictly greater than zero, given the above conditions. To simplify calculations,

Now, substituting the values from (2.7) into (2.4), we have;

 which suggests an output level when

the MC curve is falling. To have the equilibrium output on the rising portion of the MC curve, we must have the equilibrium monopoly output (q^m ) as;

Hence, the equilibrium monopoly price (p^m ) will be;

Now, to calculate the monopoly profit, substitute the value of (q^m ) from (2.10) into the profit function in (2.3) and using the simplifications of (2.7), we have;

q – total output, which is the sum of individual outputs q₁ and q₂

We also assume that both firms have identical cost functions (the same as considered for the monopoly firm), which is;

Where the MR would be different from the previous case, but the MC would be the same. Here, we also consider a simultaneous move Cournot game in which both firms decide upon their equilibrium quantities simultaneously.

Now, to find out the best response function (BRF) of each firm, we have to set;

which is obvious since k < 0 and the term in the third bracket is strictly positive

Now, the duopoly market price p^d would be;

(substituting the value of q^d and after a bit of algebraic manipulations)

Hence, for the duopoly profit to be strictly positive, we must have;

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.

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