Supply and Demand Curves under Perfect Competition

Question

I am real learning microeconomics with a bit more math under my belt, but could not understand why the MR=P in competitive markets. This was my attempt at reasoning Given, $R= P*q$
For a linear demand curve $q=a-bp$
$p= a-q/b$
$R= aq-q^2/b$
$MR= dR/dq$
$MR=a-2q/b$
Which doesn't seem equal to price
I have two questions
a) How did I go wrong ?
b) How do we go from this to the MC curve being the supply curve mathematically ?

Answer 1

First, there is an error in your last equation: You forgot to divide $2q$ by $b$. But that doesn't help here, because the problem is that you are mixing things up. Your equation for $p$ describes the demand curve of the market, but in a competitive market individual firms are by assumption price takers, so they treat the price $p$ as a constant. Then, $R=pq$ and therefore $MR=p$. (What you calculated is the $MR$ curve of a monopolist instead.)

Given this, and remembering that the profit maximizing $q$ is the one where $MR=MC$, you can substitute $p$ for $MR$ and get that the supplied quantity $q$ is the one where $p=MC$.