Isn't MC>P a better aim? Given the revenue you earn from each unit is more than cost in producing each unit?
closed as unclear what you're asking by Giskard, Bayesian, Herr K., dismalscience, emeryville Apr 26 at 21:04
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If $MC<P$, then the producer would want to produce one more unit of the good.
If $MC>P$, then she would want to produce one less unit of the good.
If $MC=P$, then she is maximizing her profits.
Your intuition is correct (assuming that you mean $MC < P$ rather than $MC > P$ as you wrote). For a given level of production, profits are indeed higher when $MC < P$ than when $MC = P$, as you'd expect from an increase in the price holding everything else constant.
So why do we say that a firm in a competitive market maximizes profit by setting $P = MC$? Because we're not looking at an increase in the price holding everything else constant, we're looking at a change in the quantity, holding price constant.
It may help to think first about the traditional model of monopoly, where a monopolist will maximize profit by choosing a quantity $Q$ that sets $MC = MR$, i.e. marginal cost = marginal revenue. Then, because they're a monopolist, they can sell that quantity $Q$ at a price $P > MR$ (and so we get $P > MR = MC$ or $P > MC$). This is because the monopolist has some control over their price: they can choose to go high price/low quantity or low price/high quantity. In the process of selling an additional unit, their revenue goes up ($MR$) by less than the price ($MR < P$), since they're moving towards the "low price/high quantity" end of the scale - they get paid the price $P$ by selling an additional unit, but they had to lower their price too and so make less on the other units they sell, making $MR < P$.
That takes us back to the competitive market. In the competitive market, each firm has no control over the price and so there's no price/quantity tradeoff - whatever quantity they pick, the price stays the same. So, in a competitive market, $P = MR$ by definition. So when they maximize their profit by choosing a quantity that sets $MR = MC$, since $P = MR$ we get $P = MR = MC$ or $P = MC$.
Intuitively, the reason the firm in the competitive market doesn't shoot for $P > MC$ is that the price is real darn stubborn. You can't coax it to go higher by picking a lower quantity (like you could if you were a monopoly). So it's not "I'd like to pick a price higher than my $MC$" because you can't pick a price. Instead, it's "I may as well continue producing more and more until my $MC$ meets the price". So, for a firm in a competitive market, profit is maximized at $P = MC$.