It is difficult in a natural monopoly market for the firm to achieve both efficiency and zero economic profit simultaneously, even with regulation.
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Natural Monopoly

A market condition where a single firm can supply a good or service to an entire market at a lower cost than two or more firms, due to economies of scale.

Economic Profit

The difference between a firm's total revenues and its total costs, including both explicit and implicit costs.

Regulation

Rules or directives made and maintained by an authority to regulate behavior or achieve an outcome.