Suppose the demand for a chemical is given by Q=100-3P

In a perfectly competitive market, the equilibrium is defined by P = MC. To get P, we need to manipulate the demand function to isolate price. We get P = (100 - Q) / 3. We can now solve when marginal cost is 6. This yields 6 * 3 = 100 - Q or Q = 100 - 18 = 82. How about when marginal cost is 8? We can use the same formula to get 8 * 3 = 100 - Q or Q = 100 - 24 = 76. So there is a difference in the quantity produced at the two levels of marginal cost.

To get marginal revenue we can use a little trick. Marginal revenue is just twice the slope of the demand curve. If Q = 100 - 3P, MR = 100-6P. Finally we get to marginal profit. Marginal profit is marginal revenue minus marginal cost, which makes some sense since a firm's profits are whatever it bring in minus whatever it has to pay to produce the good. In both cases, the marginal revenue curve is the same but the marginal cost curves are different. Therefore the market and socially optimal marginal profit amounts are different! At the competitive equilibrium MP = 100 - 6 * 6 - 6 = 58. At the socially optimal point, 100 - 6 * 8 -8 = 44. Firms are making more profit than society would prefer. This is because there is an externality.