# Price competition; finding the equilibrium expression for price and profit

This question deals regarding the price competition between two firms developing products that directly compete on the market. Fundamentally basing on the game theory and the Nash equilibrium, the aim of this question is to find the correlation of how the price of a certain firm's product relate with the equilibrium price and profit of the other firm's product. The image below states the question:

Searching the relationship for a was pretty straightforward. Established a function between the net profit and P1, P2; derived the equation by P1 in order to find the equation for P1's value that maximizes the net profit of a firm.

This is what I got for P1:

Now the problem is b, as the expression I got is quite complex. Profit is the product between (Price of the product - Marginal costs of buying) and Quantity sold. As there are expressions for the price and quantity demanded, the expression for the equilibrium profit is structured using these.
And here's what I got:

I'm unsure regarding my way of approach, but this is what I learned. Are there any issues on the derived expressions, or are there any improvements needed or recalculations to be done? Please advise. Thanks.

• To my understanding, the price $p_2$ is endogenous, and is not a parameter. Consider using linear algebra for solving this type of questions (2 linear equations, two unknown). Oct 7 '21 at 6:28

## 1 Answer

I didn't work out the algebra, but you are on the right track. The thing to remember is that for the Nash equilibrium, each player chooses a strategy (in this case choosing price) and maximizes utility/profit assuming that the other player has chosen their optimum strategy.

Profit is $$$$\text{Profit of firm 1}=(p_1-c_1)q_1 = (p_1-c_1)\left(a + \frac{b}{2}p_2^* - p_1\right),$$$$ where $$p_2^*$$ is the other firms optimum price. Maximize this and solve for $$p_1^*$$ in terms of $$p_2^*$$, and similarly, for the corresponding maximization problem for firm 2: $$$$\text{Profit of firm 2}=(p_2-c_2)q_2 = (p_2-c_2)\left(a + \frac{b}{2}p_1^* - p_2\right),$$$$

You'll then need to solve the simultaneous equations for $$p_i^*$$ for the Nash equilibrium, that's the answer to (i). Then you can answer (ii), the profit as you stated (price - marginal cost)*qty.