Two firms, $1$ and $2$, producing one product, with a quantity of $Q_1$ and $Q_2$ respectively.
Price of the product $= P = a - b Q$.
Profit function for each firm, $i=1,2$ given by $ \pi_i = Q_i(P-C_i)$.
Question is: Express the stationary point $Q_i^*$ of each firm's profit function, given an arbitrary production of the other firm $Q_j$.
How do I find the stationary point for the profit function? I know if it was for revenue I would make $MR=MC$ and solve for $0$, but in this case how do I? Obviously you would substitute $P$ into $ \pi $ as a start, but apart from that I am lost, I have not seen this with the profit function before. Thanks. Sorry if I have not provided enough work.
Total demand is $Q=Q_1+Q_2$, so
$$ P = a-b(Q_1+Q_2). $$
Hence the profit of firm $i$ is given by
$$ \pi_i(Q_i) = Q_i(a-b(Q_i+Q_j)-C_i). $$
The first order condition for the problem of maximizing profits by choosing $Q_i$ is given by $\pi_i'(Q_i)=0$, which implies (using the product rule)
$$ (a-b(Q_i+Q_j)-C_i) + Q_i(-b) = 0.$$
Solving for $Q_i$ yields
$$Q_i = \frac{a-C_{i}-bQ_{j}}{2b} .$$