Cournot duopoly Cournot model in game theory In 1838,
Antoine Augustin Cournot published a book titled "Researches Into the Mathematical Principles of the Theory of Wealth" in which he introduced and developed this model for the first time. As an imperfect competition model, Cournot duopoly (also known as Cournot competition), in which two firms with identical cost functions compete with homogenous products in a static context, is also known as
Cournot competition. The Cournot model, shows that two firms assume each other's output and treat this as a fixed amount, and produce in their own firm according to this. The Cournot duopoly model relies on the following assumptions: • Each firm chooses a quantity to produce independently • All firms make this choice simultaneously • The cost structures of the firms are public information In this model, two companies, each of which chooses its own quantity of output, compete against each other while facing constant marginal and average costs. The market price is determined by the sum of the output of two companies. P(Q)=a-bQ is the equation for the market demand function. • Market with two firms with constant marginal cost • Inverse market demand for a homogeneous good: • Where is the sum of both firms' production levels: • Firms choose their quantity simultaneously (static game) • Firms maximize their profit: \begin{aligned} \Pi_1(q_1,q_2) &= \left(P(q_1 + q_2) - c_1\right)*q_1\,, \\ \Pi_2(q_1,q_2) &= \left(P(q_1 + q_2) - c_2\right)*q_2 \end{aligned} The general process for obtaining a Nash equilibrium of a game using the
best response functions is followed in order to discover a Nash equilibrium of Cournot's model for a specific cost function and demand function. A Nash Equilibrium of the Cournot model is a such that For a given q_2^* solves: \begin{aligned} \operatorname{MAX}_{q1} \Pi_1(q_1, q_2^*) &= (P(q_1 + q_2^*) - c_1)q_1\,, \\ \operatorname{MAX}_{q2} \Pi_2(q_1^*, q_2) &= (P(q_1^* + q_2) - c_1)q_2 \end{aligned} Given the other firm's optimal quantity, each firm maximizes its profit over the residual inverse demand. In equilibrium, no firm can increase profits by changing its output level. The two first order conditions equal to zero are the
best response. Cournot's duopoly marked the beginning of the study of oligopolies, and specifically duopolies, as well as the expansion of the research of market structures, which had previously focussed on the extremes of perfect competition and monopoly. In the Cournot duopoly model, firms choose the quantity of output they produce simultaneously, taking into consideration the quantity produced by their competitor. Each firm's profit depends on the total output produced by both firms, and the market price is determined by the sum of their outputs. The goal of each firm is to maximize its profit given the output produced by the other firm. This process continues until both firms reach a Nash equilibrium, where neither firm has an incentive to change its output level given the output of the other firm.
Bertrand duopoly Bertrand model in game theory The
Bertrand competition was developed by a French mathematician called
Joseph Louis François Bertrand after investigating the claims of the Cournot model in "Researches into the mathematical principles of the theory of wealth, 1838". Bertrand took issue with this. In this market structure, each firm could only choose whole amounts and each firm receives zero payoffs when the aggregate demand exceeds the size of the amount that they share with each other. The market demand function is Q(P)=a-bP. The Bertrand model has similar assumptions to the Cournot model: • Two firms • Homogeneous products • Both firms know the market demand curve • However, unlike the Cournot model, it assumes that firms have the same marginal cost (MC). It also assumes that the MC is constant. The Bertrand model, in which, in a
game of two firms, competes in price instead of output. Each one of them will assume that the other will not change prices in response to its price cuts. When both firms use this logic, they will reach a
Nash equilibrium. • Consider price competition among two firms () selling homogeneous good • Downward sloping market demand , with • Constant, symmetric marginal cost • Static game: firms set prices simultaneously • Rationing rule of demand: • lowest priced firm wins all demand at its price • if prices are tied, each firm gets half of market demand at this price • Firm s profits: \Pi_i = (p_i-c)D_i(p_i, p_j) Let be the monopoly price, p^m = \operatorname{argmax}_p(p-c)D(p) • Firm s best response is: R_i(p_j) = \begin{cases} p^m, & \text{if } p_j > p^m \\ p_j - c, & \text{if } c For rival prices above cost, each firm has incentive to undercut rival to get the whole demand. If rival prices below cost, firms make losses when it attracts demand; firm better off charging at cost level. Nash equilibrium is .
Bertrand paradox Under static price competition with homogenous products and constant, symmetric marginal cost, firms price at the level of marginal cost and make no
economic profits. In contrast to the Cournot model, the Bertrand duopoly model assumes that firms compete on price rather than quantity. Each firm sets its price simultaneously, anticipating that the other firm will not change its price in response. When both firms use this logic, they will reach a Nash equilibrium, where neither firm has an incentive to change its price given the price set by the other firm. In this model, firms tend to price their products at the level of their marginal cost, resulting in zero economic profits, a phenomenon known as the
Bertrand paradox. == Characteristics of duopoly ==