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Maybe I am just confusing myself, but what does charging a different price at every unit have to do with an output level being allocatively efficient? I understand what price discrimination is, I just do not understand that if only the last unit of output is where the marginal cost equals demand then it must mean it is allocativally efficient. I mean not every unit is sold at this price level, right?

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If the monopolist is able to engage in perfect price discrimination every consumer will be charged their marginal willingness to pay. This means that the $q$th item will be sold at price $P(q)$ (where $P(.)$ is the inverse demand function). Here, effectively all the consumer surplus is being captured by the producer.

The profit maximising monopolist will keep on selling items (at prices tracing out the inverse demand function) as long as the additional income $P(q)$ is larger than its marginal cost of producing an additional output $MC(q)$. As such, the last output will equate demand and supply, which means that the output level will be allocative efficient.

Of course, all consumer and producer surplus will go to the monopolist.

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