This below is a Nash equilibrium problem, I'm stuck in the math part. I solved the first part but I'm confused on the second one. I believe there is a mistake on denominator and it should be $4\alpha^2-\beta^2$ (I wrote down problem as it is). The equilibrium I found is $p_1=\frac{A(2\alpha+\beta)+2\alpha^2c_1+\alpha\beta c_2}{4\alpha^2 - \beta^2}$ and $p_2=\frac{A(2\alpha+\beta)+2\alpha^2c_2+\alpha\beta c_1}{4\alpha^2 - \beta^2}$. I need some clues on how to answer part (b). Thanks!
Consider two firms, 1 and 2, competing in a differentiated market. Each firm faces demand for their product given by $ q_i = A - \alpha p_i + \beta p_j$, where $ \alpha > \beta $. Each firm has a constant marginal cost of $c_1$ and $c_2$ respectively, with $c_1<c_2<\frac{A(2\alpha+\beta)+\alpha\beta c_1}{2\alpha^2 - \beta^2}$. Assume there are no fixed costs.
(a) Find the Cournot Nash equilibrium for each firm? Which has the higher price.
(b) Explain why your answer in (a) no longer holds if $c_1<\frac{A(2\alpha+\beta)+\alpha\beta c_1}{2\alpha^2 - \beta^2}<c_2$ ?
I'm not checking the computations, please use Wolfram Alpha for this. But to answer your question and give you the hint you asked: in part (b) the marginal cost of firm 2 is higher than in part (a) right? What happens with $q_{2}$ if you just use the solution of part (a) for part (b)? It is a quantity so we must have $q_2\ge 0$. If you get that $q_2<0$ then clearly you do not have an answer for part (b). In this case the economics logic seems that the high cost first will produce zero and the low cost firm will produce the monopoly quantity (make sure this is a Nash eq).