Under perfect competition, $MR=MC=P$, but $P$ is also the point where the supply and demand curves intersect. Why is it that those will always correspond to the same point? Or is the idea just that if it wasn't, the market would adjust such that it became the same point and that it will be able to do so under the assumptions of perfect competition? I understand why each condition is true in order for profit to be maximized, but I'm struggling with relating them to each other.
In a perfectly competitive market, the sellers and the buyers are price takers, that is, they believe that no matter what they do, they cannot individually affect the price of the commodity (so $MR=P$ for firms always, and $MC=P$ for consumers always). Thus, they take the price as given and choose the quantity to maximise their profit. Similarly, the consumers take the market price as given and choose the quantity to consume given the price.
When calculating the equilibrium, we calculate the optimal quantity supplied and quantity demanded for every possible price and not just one price. So, for every possible $P$, we solve $MC(Q)=P$ for $Q$ and call the solution $S(P)$. Supply is thus a function and not a specific number: if price is 1, then supply 2 units, if price is 1.5 then supply 4 units, if price is 2, then supply 10 units etc. Same for demand $D(P)$.
After finding the supply function and demand function, to find the market equilibrium, we find out that price which makes the quantity supplied equal to the quantity demanded.
So it is not the case that coincidentally the price where $MR=MC=P$ is also the price where $S=D$. What we are actually doing is, we are finding the quantity supplied (or the quantity where $MC(Q) = P$) and quantity demanded ($MU(Q) = P$) for every possible $P$ and then finding that $P$ where $S(P)=D(P)$.
In perfect competition, firms are price takers, so a firm producing one more unit won’t affect the price. The only thing that would change revenue-wise is that the firm would sell that “marginal” extra unit for its market price $P$, so $MR_i = P$.
A firm would only produce that “marginal” extra unit if its “marginal” profit from it is not negative. The minimum market price which would lead a firm to produce that “marginal” $Q_i$-th unit is such that $ MR_i(Q_i) = MC_i(Q_i) $, but we know that $MR_i(Q_i) = P$, so firm $i$’s supply curve is given by $P = MC_i(Q_i)$.
The supply curve for the industry is defined for each price level $P$ by adding the corresponding supplies from each firm $Q_i$, as $Q^s = Q^s(P)$ where $Q^s = Q_1 + \dots + Q_N$.
We derive the individual demand from each consumer $Q^j = Q^j(P)$ from the consumer’s utility maximization problem.
Similarly, the aggregate demand for the good is defined for each price level $P$ by adding the corresponding demands from each consumer $Q^j$, as $Q^d = Q^d(P)$ where $Q^d = Q^1 + \dots + Q^M$.
The equilibrium (actual) market price $P$ can be found from the point where supply $=$ demand, i.e., $Q^d(P) = Q^s(P)$
We plug this $P$ into each firm’s supply curve $P = MC_i(Q_i)$ to find each firm’s supply $Q_i$.
We can either sum all of the $Q_i$’s or plug that same $P$ into the industry’s supply curve to find the aggregate supply $Q^s$.
Similarly, we plug this $P$ into each consumer’s demand curve $P = P(Q^j)$ to find each consumer’s demand/consumption $Q^j$.
From the equilibrium condition $Q^d(P) = Q^s(P)$, we know the aggregate demand has to be equal to the aggregate supply from #8.