Abstract
This paper explores the existence of pure strategy Nash equilibrium of a Bertrand game with strictly positive profits. We show that when fixed cost is small enough, there always exists pure strategy Nash equilibrium with strictly positive profits if firms have quadratic cost functions and linear demand curve.
Highlights
The Bertrand paradox indicates that zero profits are earned if two identical firms produce homogeneous products in a duopoly market
There has been some work discussing the existence of mixed-strategy Nash equilibrium of a Bertrand game with positive profits [1], [2]
[1] assumed that when several firms set the same lowest price, the profit of each firm is the monopoly profit divided by the number of the firms setting the same lowest price
Summary
The Bertrand paradox indicates that zero profits are earned if two identical firms produce homogeneous products in a duopoly market. There has been some work discussing the existence of mixed-strategy Nash equilibrium of a Bertrand game with positive profits [1], [2]. Both [1] and [2] adopted impractical assumptions. Cost Function A1: There are two identical firms competing in the market They produce homogeneous products and the cost function is: C= kq2 2 + sq + e. Market Share A3: Since the two firms produce homogeneous products, any one setting a lower price will own the entire market. If the two firms set the same price, they split the demand evenly
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