This paper investigates the conditions under which a can find a set of prices and a set of products that are sustainable against competitive entry. By a natural monopoly we mean an industry whose cost function over some given set of products is such that no combination of several firms can produce an industry output vector as cheaply as it can be provided by a single supplier. A sustainable vector is a stationary equilibrium set of product quantities and prices which does not attract rivals into the industry. Even if a vector is not sustainable, a monopoly may still be able to protect itself from entry by changing its prices whenever and however necessary, in response to any entry that threatens at that moment. But, by definition, only a sustainable vector can prevent entry and yet remain stationary.
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