If you have a demand function Q1= 20 + 3/4 p2 - p1 and Q2= 20 + 3/4 p1 - p2, how does one go about solving for the Nash equilibrium if you know there are no costs?
I've tried solving for Q and then taking the derivative; however, I'm left with the extra price variable of the other firm's price.
Thank you for the help!
Take the given demand functions, written as a system of linear equation, and invert them. Write $p_1$ and $p_2$ as a function of total firm output. That is, solve for $p_1$ and $p_2$ as a function of $Q_1=q_1^a + q_1^b$ and $Q_2=q_2^a + q_2^b$.
Next, maximize the profit function of firm A over $q_1^a$ and $q_2^a$ given the output of firm B, $q_1^b$ and $q_2^b$. Do the same for B given A. This will give you four first-order conditions, with $q_1^a, q_2^a, q_1^b,$ and $q_2^b$ as the four unknowns that you will solve for.
EDIT: As a hint, you should begin by inverting the system of demand equations $$ \begin{bmatrix} Q_1 \\ Q_2 \end{bmatrix} = \begin{bmatrix} -1 & 3/4 \\ 3/4 & -1 \end{bmatrix} \begin{bmatrix} p_1 \\ p_2 \end{bmatrix} + \begin{bmatrix} 20 \\ 20 \end{bmatrix} $$ to get $$ \begin{bmatrix} p_1 \\ p_2 \end{bmatrix} = \begin{bmatrix} -16/7 & -12/7 \\ -12/7 & -16/7 \end{bmatrix} \begin{bmatrix} Q_1 \\ Q_2 \end{bmatrix} + \begin{bmatrix} 80 \\ 80 \end{bmatrix}. $$ Now, the prices are written in terms of quantity only. You can then proceed to find the first-order conditions for each firm, $a$ and $b$. The maximization problem for firm $a$, written in matrix notation, is given below: $$ \max_{\vec q_a}\vec q_a^\intercal P(\vec q_a + \vec q_b) = \vec q_a^\intercal (A(\vec q_a + \vec q_b) + b). $$ With both sets of first-order conditions, you will end up with 4 equations and 4 unknowns that will have a unique solution.