There are two settings where we derive equilibrium price and quantity. The first involves a price taking (i.e. perfectly competitive) industry, and the second involves a monopoly. Let's consider each setting.
A. Finding Equilibrium in a Perfectly Competitive Industry:
1. Set Demand equal to Marginal Cost, and then solve for Q*:
2. Plug Q* into either the Demand or Supply equation, and solve for P*:
B. Finding Equilibrium in a Monopolistic Industry:
2. Plug Q* into the Demand equation (not MC), and solve for P*:
Note that both industries face the same Market Demand and MC curves. However, the equilibrium price and quantities which result from each industry are not the same. Comparing the two outcomes, we find that perfect competition leads to lower prices and greater output. This makes good, intuitive sense since we expect that greater competition should keep prices down.