Why is a monopoly inefficient?

Monopolistic markets do not meet the criteria for the most important kind of social efficiency - allocative efficiency. If the market is allocatively efficient, firms will be producing at a point where price equals marginal cost. This means all consumers who value the good at more than it costs to produce the marginal unit of that good get to consume it.

In a perfectly competitive markets, firms' profit maximising level of production, where MC = MR, will be the same as the allocatively efficient point MC = AR. This is because in these markets, firms are price takers - the amount they produce has no effect on the price they get - and so MR=AR at all levels of output.

However, in a monopolistic market, consumers who do not buy their good from that firm cannot simply go elsewhere. Perhaps there are barriers to entry in the market preventing other firms from competing, or perhaps the firm produces a heterogenous good, so that even if other firms sell sandwiches too, this firm has a monopoly on a particular kind of sandwich that many people enjoy. This means that the AR curve is downward sloping - if the firm raises its price, some people will still want to buy - and so there will be a profit maximising equilibrium at an output level below the socially efficient level.

If the firm cut prices so that it was producing at the allocatively efficient level MC = AR, it would attract more customers that are willing to pay more than the marginal cost - but its profits would actually fall, because its existing customers would also be paying less. So profit maximising firms - absent government intervention - will produce at below the social optimum (allocatively efficient) quantity.