Am from computer background and recently started to pick up my interest in learning economics basics for work related projects.
I do have some theoretical understanding of the very basic term , but not much. Lets say i have a inverse Demand equation Price $= \alpha -$ Quantity, and we have many firms each producing quantity $q$. so Quantity($Q$) = sum of q's of individual firms.
And also we have cost function given by $1.3bq^2 + 50$ (I know $50$ is the Fixed price). So total cost $TC =$ Variable cost ($1.3bq^2$) + Fixed cost of $50$.
So from this equation , I can calculate the Tc for producing each quantity , by just replacing $q$ with the quantity.
But how can I calculate the profit maximization for each company without knowing the Price per quantity ? To calculate Profit max i need both $TR$ and $TC$ term if am not wrong to apply logic of $MR=MC$.
$$ \begin{align} p(q)&=\alpha-q\\ c(q)&=50+1.3bq\\ r(q)&=p(q)\times q=\alpha q-q^2\\ \pi(q)&=r(q)-c(q)=(\alpha q-q^2)-(50+1.3bq)=(\alpha-1.3b)q-q^2-50\\ \max_q \pi(q)&=\max_q\{(\alpha-1.3b)q-q^2-50\} \\ \pi'(q)&=(\alpha-1.3b)-2q=0\quad\Longrightarrow \quad \boxed{ q_{\max}=\frac{2}{\alpha-1.3b}} \end{align} $$