How does a monopoly firm perfectly price discriminate?

Perfect price discrimination is also called as first degree price discrimination. Under this the monopolist charges each consumer according to their willingness to pay. The monopolist in this case exhausts the entire consumer surplus by charging each consumer the exact price that they would be willing to pay for the good. Usually the monopolist set its level of output at the point where the marginal revenue is equal to the marginal cost and the price is set at a level corresponding to this point in the average revenue curve. But if the monopolist is to charge the consumers, their willingness to pay, it should equate the marginal revenue with the price and set its level of output where MR = MC = price. This means that, at this level, the firm is not only willing to sell higher output but also the output sold is allocatively efficient. But the firm will charge the consumer the maximum price for this that the consumer is willing to pay, and thus exhaust the entire consumer surplus and thus convert the consumer surplus into profits.