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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.

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 [

・ Cost Function

A1: There are two identical firms competing in the market. They produce homogeneous products and the cost function is:

・ Demand Curve

A2: Suppose that price and demand satisfy a linear relationship:

・ 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.

Let

Then we derive three critical prices to determine the Nash equilibrium price interval.

・ Zero Profit Price

Let

It requires that:

・ Maximum Profit Price

Let

・ Identical Profit Price

Let

Lemma 1. If

Proof.

Lemma 2. If

Proof. Let

Lemma 3.

Proof.

Lemma 4. If

Proof.

Lemma 5. If

Proof.

Theorem. A Bertrand game satisfying assumptions A1 through A3 has Nash equilibria

Proof. First of all, we claim that

3:

price below

Proposition: In a Bertrand game satisfying assumptions A1 through A3 with strictly positive profits, the price strategies of the two firms to earn maximum profits are 1)

Proof. It follows from the theorem and lemma 5.

This work was supported by the Ministry of Science and Technology of the People’s Republic of China under National Science and Technology Supporting Project 2015BAG10B00 “Research and Demonstration of Electric Vehicle Time Sharing Rental Pattern and Supporting Technologies in Mountainous City”.

Yongjian Pu,Lei Zhu, (2016) A Pure Strategy Nash Equilibrium Bertrand Game with Strictly Positive Profits. Open Journal of Social Sciences,04,54-56. doi: 10.4236/jss.2016.43009