Two firms, firm 1 and 2 , are competing in prices in two differentiated product markets. The demand for respective firms products are given by the following demand functions; $$ q_1(p_1, p_2)=a-p_1+bp_2,\\ q_2(p_1,p_2)= a-p_2+bp_1$$
where $p_1$ and $p_2$ are the prices of the products,$q_1$ and $q_2$ are the quantities produced by the respective firms, and $b>0$ reflects the extent to which firm i’s product is a substitute for firm j’s product. As for the production cost,there is no fixed cost, and both firm have a constant marginal cost of $c <a$. We assume that $b<2$ and also that the firms choose their prices simultaneously. What are the prices at the Nash equiliburium?
This is a Bertrand duopoly situation. As the game is symmetric, there exists a Nash equilibrium in which both players use the same strategy. Now each firm is a profit maximizer. So we have:
$$\pi_{i} = (p_{i} - c)q_{i}(p_{i}, p_{-i})$$
Player $i$ wishes to maximize its profit $\pi_{i}$ and can only vary its price. This leads to the first order condition:
$$\frac{\partial \pi_{i}}{\partial p_{i}} = 0$$
So:
$$a - 2p_{i} - bp_{-i} + c = 0$$
In a symmetric Nash equilibrium, $p_{i}^{*} = p_{-i}^{*}$. So you substitute this into the first order condition and solve for $p_{i}$. I will leave the algebra to you.