Static Aspect of Heterogeneous Competition in Duopoly

The study gives a comparison between price policy and quantity policy in duopoly producing differentiated goods with different production costs and indicates which is more beneficial. Further, it is investigated that in a non-linear duopoly with differentiated goods and two different policies, firms may earn more profit if they choose a quantity policy in a stable economy when the marginal production cost of both the firms is the same. If the production cost of both firms is different, then the price policy is better only when the firm is efficient.


Introduction
There are mainly two types of market systems, one is Monopoly and the other is Perfect Competition and they are completely contrary to each other in character [1] [2]. Cournot proposed a new market system named Oligopoly [3]. Oligopoly is the transitional case of monopoly and perfect competition [4] [5]. Duopoly is the sub-case of oligopoly. A duopoly is a market system that is controlled by two firms. There are two classical models in the theory of duopoly. One is of Cournot and the other is of Bertrand [6]. Both mentioned models firms were generating the same kind of goods and adopting homogeneous strategies. Cournot model firms use quantity strategies while Bertrand model firms use price strategies [7] [8]. Numerous researchers have established mathematical models to deal with diverse situations of duopoly [9] [10]. In most established studies, re-How to cite this paper: Kapoor, B. (2022) Static Aspect of Heterogeneous Competition in Duopoly. Open Journal of Applied Sciences, 12, 455-468. searchers assumed that in a duopoly, both firms produce homogeneous goods and adopt the same kind of strategy either quantity strategy or price strategy [11] [12]. But in real-life situations, firms may produce different kinds of products which provide customers a variety of goods to choose from them and it provides opportunities for firms to improve their products because of competition. Also, it is not necessary that both firms adopt homogeneous strategies [13]. Instead, in order to get maximum benefit and avail market situations, firms use different strategies. Various studies have been conducted to give a realistic outlook to the classical model [14] [15]. Many theories are established on the idea of product differentiation [16] [17] as well. Some researchers established the relation between relative profit objective and the advertising competition model [18] [19].
But, there is not abundant work on the non-linear duopoly model with the heterogeneous cost of production and heterogeneous strategies. In this paper, the strategic behavior in a heterogeneous nonlinear differentiated duopoly is observed. For this, a duopoly model is examined after introducing nonlinearity through heterogeneous strategies and differentiated goods while the cost of production of both firms is also heterogeneous. The main objective of the paper is to analyze the static effect of non-linearity on the behavior of the competitors.
Non-linearity in the duopoly model is due to various reasons like heterogeneity in production cost, heterogeneity in strategy, and heterogeneity in competition, which means that different firms choose different strategies. This paper is systematized like this: In Section 2, there is an outline of the general demand function in dissimilar duopoly; Section 3 describes the differentiated duopoly with linear demand function; Section 4 explains the duopoly with non-linear demand with heterogeneous stratagems as well as heterogeneous manufacturing cost and a conclusion is given in Section 5.

Duopoly with General Demand Function
As in a duopoly, two firms compete against each other. So, let there be two firms firm 1 and firm 2. Let us suppose that firm 1 produces 1 x quantity of a product with price 1 x p and firm 2 produces 2 x quantity of product with price 2 x p .
The motive of both firms is to earn maximum profit. Let us consider the case when both players follow heterogeneous strategy. In that case, one of the players follows quantity strategy taking price variable of other as given and other follows price strategy taking quantity variable of other firm as given. , , , , Thus, profit in CB competition of both the firms is a function of two variables, where quantity is variable for the first firm and price for the second firm, as both the firms want to earn maximum profit. For that take partial derivatives of these profit functions w.r. p give CB equilibrium [20]. Similarly, in BC competition, firm 1 is the price setter and firm 2 is quantity setter. Value of 1 x is substituted in Equations (2.3) and (2.4) respectively as follows: As discussed above, take partial derivatives of these profit functions w.r.t. As Also, by using Equation ( Now from Equations (3.7) and (3.8), it is clear that If we consider the case that production cost of both firms is positive and same i.e. homogeneous production cost. Take 1 c and 2 c both zero, and this is typical postulation of linear models. Solving reaction functions given in Equations  Above observations clearly indicate that in both CB and BC competition, out of two firms producing differentiated goods and following heterogeneous strategy, the quantity setter firm produces more output, have fewer prices and enjoys more profits.

Duopoly Model with Non-Linear Demand
Here we take an assumption that demand is iso-elastic [22]. Then inverse nonlinear demand function is given by In CB competition with non-linear demand function, reaction functions will be obtained as in case of linear demand functions. We will get CB reaction functions for firm 1 and firm 2 as Similarly, we get ( ) Dividing (4.7) by (4.8) Also from (4.8) In the similar way Taking ratio of output, prices and profits of both the firms, we get ( ) ( ) and 1 x x From (4.9) and (4.10), it is clear that = . This can be interpreted that whichever firm produces more output sells at lower prices.

Bertrand Cournot Competitions
In BC competition, firm 1 is the price setter and the second is quantity setter.
Following the above stated procedure, the reaction functions are The reaction functions given in above equations are defined in ( ) 1 2 , x p x space. BC equilibrium will be converted in quantity space ( ) This means output of firm 1 is more than that of firm 2, and its prices are lesser above the critical line 1 1 α θ = + . But in the region below the critical line, firm 2 produces more output than the firm 1 and set lower prices.
When repeating the above procedure, we reach the conclusion that efficient firm produces more output and sells at lower price. Here efficient means production cost is less.

Conclusions
In a duopoly market system when both firms choose heterogeneous strategies i.e. one of the firms chooses quantity strategies and the other chooses price strategies, it is observed that when production costs are homogeneous, then quantity setter firms produce more output, face lower prices, and make larger profits than price setter firm. So, the quantity strategy is more profitable. When strategies and production costs are heterogeneous, the efficiency of the firms is the determining factor to decide which firm will produce more and sell at a lower price.
This paper studies the static aspects of the non-linear heterogeneous duopoly model. There is a future scope for investigating the dynamical aspects with numerical simulation.