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Let's jump into things quickly: what is the profit maximization condition? The answer is:
\begin{equation}\label{MR=MC}MR=MC\end{equation}
That's all. This is the most important conclusion in economics. This is the reason why we need to learn marginal / average / total * concepts before. This is also the reason why we need to learn consumer theory, firm theory, cost, revenue, profit, etc. All of these background are necessary in order to derive this equation. \\
This equation also makes sense in real life. If marginal revenue is greater than marginal cost, that means you still have room to earn some money. Then why not keep investing? If marginal revenue is less than marginal cost, that means you are losing money. Then you should reduce your investment to stop losing money, in this way, you can get more profit (for example, your profit is from -10 to -5).
\section{Profit Maximization in a Perfectly Competitive Market}
The most important charactistic of a perfectly competitive market is that firm is a \textit{price taker}. Once the price is given, a single firm cannot change it. Thus, for a firm in a perfectly competitive market, it always sells the product according to the same price. Suppose a perfect competitive firm is selling a football game ticket. The price is \$50. This number (\$50) is fixed in a perfectly competitive market. Thus, if a firm sells one more additional unit of product (here one more ticket), its marginal revenue is just the fixed price (\$50). In this way, we have the following equation in a perfectly competitive market:
\begin{equation}\label{MR=P}MR=P\end{equation}
Remember, \eqref{MR=P} only holds in perfectly competitive market because price is only fixed in this type of market.\\
Combining \eqref{MR=MC} and \eqref{MR=P}, we then get the profit maximization condition in the perfectly competitive market:
\begin{equation}\label{MRMCP}MR=MC=P\end{equation}
The difference between \eqref{MR=MC} and \eqref{MRMCP} is that \eqref{MR=MC} holds in any type of market, while \eqref{MRMCP} only holds in perfectly competitive market.LaTeX Warning: Reference `MR=P' on page 3 undefined on input line 46
LaTeX Warning: Reference `MR=MC' on page 3 undefined on input line 48.
LaTeX Warning: Reference `MR=P' on page 3 undefined on input line 48.
LaTeX Warning: Reference `MR=MC' on page 3 undefined on input line 51.
LaTeX Warning: Reference `MRMCP' on page 3 undefined on input line 51.
LaTeX Warning: Reference `MR=MC' on page 3 undefined on input line 51.
LaTeX Warning: Reference `MRMCP' on page 3 undefined on input line 51.What happened?
