Equilibrium with substitute goods I am attempting to solve the following problem
The demand functions for two substitute goods, the production cost of which equals $c_1$ and $c_2$,
are $q_1 = a_1 + b_{11}p_1 + b_{12}p_2$ and $q_2 = a_2 + b_{21}p_1 + b_{22}p_2$.
How can I find equilibrium under simultaneous and sequential interaction in a price oligopoly
(the values of the coefficients $a_1,a_2,b_{11} ...$ can be selected by yourself.
Although, if I'm not mistaken, the sign matter)
I need help in understanding the difference between simultanious and sequential interaction solution
Let's start by writing down the profit functions:
$$
\pi_1 = p_1 q_1 - c_1 q_1 = (p_1 - c_1) q_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2)
$$
and similarlily:
$$
\pi_2 = (p_2 - c_2)(a_2 + b_{21} p_1 +b_{22} p_2)
$$
In a simultaneous equilibrium, both firms simultaneously maximize their profits taking the strategies of the players as given.
Doing this will give you a best response of player 1 that will be a function of $p_2$: $p_1 = p_1^\ast(p_2)$. Similarly, the best response of player 2 will be a function of $p_1$: $p_2 = p_2^\ast(p_1)$.
In a Nash equilibrium, you are looking for a solution for both equations simultaneously:
$$
p_1 = p_1^\ast(p_2),\\
p_2 = p_2^\ast(p_1).
$$
In a sequential equilibrium on the other hand, you assume that one firm, say 1, chooses $p_1$ first and next firm 2 decides in $q_2$, after being informed about the value of $q_1$ chosen by firm 1.  Firm 1, however, is smart enough, such that it can predict the optimal action that firm 2 is going to take in stage 2.
As such, we solve the game by backward induction, first solving for firm 2, giving $p_2 = p_2^\ast(p_1)$ and then we substitute this into the profit function of firm 1.
$$
\pi_1 = (p_1 - c_1)(a_1 + b_{11} p_1+b_{12} p_2^\ast(p_1))
$$
This substitution conforms to the fact that player 1 knows that by choosing $p_1$ it will influence the price of firm 2. So firm 1 takes the future behaviour of firm 2 into account when choosing its optimal level of $q_1$.
Maximizing $\pi_1$ with respect to $p_1$ will give an optimal value $p_1^\ast$ (which will no longer be a function of $p_2$).
Then, to compute the equilibrium, you substitute this value into the best response for firm 2:
$$
p_2 = p_2^\ast(p_1^\ast).
$$