12.5 Profit Maximization for a Competitive Firm
When we defined the revenue function for a competitive firm, we said that its inverse demand function was simply $p(q) = p$, where $p$ is the market price that the firm takes as given.
Therefore the competitive firm’s profit maximization problem is
\[\overbrace{\pi(q)}^\text{PROFIT} = \overbrace{p \times q}^\text{REVENUE} - \overbrace{c(q)}^\text{COST}\]its marginal profit is
\[\pi^\prime(q) = \overbrace{p}^{MR} - \overbrace{c^\prime(q)}^{MC}\]and so its optimal quantity is found by setting $p = MC$. Note that this is really just a special case of $MR = MC$, since in this case the firm’s marginal revenue from each unit sold is just $p$. The diagram below shows the total and unit revenues and costs for the case where $p = 12$:
As you can see, the profit-maximizing point occurs at $q = 24$, at which point $p = MC = 12$. As with the case of a firm facing a downward-sloping inverse demand curve, the profit may be represented as the area $(AR - AC) \times q$.
Since a competitive firm’s optimal choice depends on the price, a critical question to ask is how the quantity the firm supplies responds to a change in the market price. This is the subject of our next chapter: the supply curve of the competitive firm.