NOTE: This is a variation of the question I previously asked here, Economic price optimisation problem.
The problem is the same as the previous linked question, however I am now looking to find the prices set by the Firms if they undergo Cournot competition. The profit functions for Firm 1 and Firm 2 are, respectively:
$$\Pi_1(p_1,p_2)=\frac{25p_1}{(p_1-0.6p_2)^{1.1}}-0.0815[\frac{25}{(p_1-0.6p_2)^{1.1}}]^{1.9}$$ $$\Pi_2(p_1,p_2)=\frac{25p_2}{(p_2-0.6p_1)^{1.1}}-0.0645[\frac{25}{(p_2-0.6p_1)^{1.1}}]^{1.9}$$
In this case, the Firms now set their prices simultaneously, and I am looking to find the prices the Firms set in equilibrium.
I am just looking for some help in how I go about this question. I understand that I need to find optimal prices $p_1^*$, $p_2^*$ by maximising both profit functions.
$$max_{p_1}\Pi_1(p_1,p_2)$$ $$max_{p_2}\Pi_2(p_1,p_2)$$
This again has to be solved using a numerical method which I will be programming within MATLAB, just as I did with a previous question which can be seen here, Stackelberg Competition Matlab Problem. The numerical methods will find the maximum price in each case for the inputted price from the other firm. For instance, I can easily find $max_{p_1}\Pi_1(p_1,p_2)$ for each inputted value of $p_2$, and vice versa with the other maximisation problem.
I am then unsure how to pursue with this question. How do I go about finding the prices they both set in equilibrium in order to maximise their profits? I would also like to analyse the robustness of the profit functions, how they react to changes in parameters so any advice on that would also be appreciated.
Let $B_i(p_j)$ be the best response of firm $i$ to firm $j$'s strategy of choosing its price to be $p_j$, that is, for $i\ne j\in \{1,2\}$, $$ B_i(p_j) = \arg\max_{p_i} \Pi_i\left(p_1,p_2\right). $$ The Nash equilibrium $(p_1^*,p_2^*)$ obtains as the solution to the system of equations \begin{eqnarray} p_1^*&=&B_1(p_2^*)\\ p_2^*&=&B_2(p_1^*). \end{eqnarray}