Long-run Equilibrium of the monopolist:

In the long-run, the monopolist is endowed with the time to expand her plant or to use her existing plant at any level which maximises her profit. Unlike perfect competition, it is not necessary for the monopolist to reach a optimal scale (i.e., to build up her plant until she reaches the minimum point of long run average cost curve). But we can say for sure that a monopolist will most probably continue to earn supernormal profits even in the long-run (given that entry is barred) and she will not stay in business if she makes losses in the long-run. However, the size of the plant and degree of its utilisation crucially depend upon the market demand conditions.

In the following three figures, we depict three situations. In the first Figure, we show that the market size does not permit the monopolist to expand to the minimum point of LAC. In this case not only her plant is of suboptimal size but also is underutilised. The optimum use of the existing plant is at 'a' and the minimum point of long-run average cost is given by 'b'. Since the firm utilises the capacity 'c', there is excess capacity.  

In the 2  figure we show that the monopolist, in order to maximise profit, must build a plant size greater than the optimal size (to the right of minimum point of LAC) and will over utilise it. This happens when the market size is unduly large. Thus, the plant that maximises the monopolist's profit leads to higher costs  for two reasons: first, because it is larger than the optimal size and second, because it is over utilised. This is often the case with public utility companies operating at national level.

Finally, in the 3rd figure we show the case in which the market size is just large enough to permit the monopolist to build the optimal plant and to use it at full capacity.

Thus, there is no certainty that in the long-run the monopolist will reach optimal plant size as in the case of perfect competition.
Whether a monopolist stays in business in the long-run will depend on the long run average cost curve. She will exit in the long run unless all costs can be covered. Long run equilibrium for the monopolist requires that

LRMC = SRMC = MR

so that profit is maximised and P LRAC, and full opportunity cost is covered. But if the monopolist is making profit, such a project provides an incentive for the new firms to enter the industry. If entry occurs, then the equilibrium position will change, and since there are more than one firm, there will be no longer a monopoly market structure. Thus, in the long-run, for a profitable monopoly to survive, there must be barriers to entry. Sometimes these barriers are created at the time the monopoly is established. In other cases, the monopoly is created through threats and coercion. If a monopolist has a cost advantage over its rivals, then it can indulge in 'pre-emptive price' cutting to deter rivals from entering the market. This is explained in the following figure.

In the absence of any potential rival, the monopolist charges price PM and produces XM to maximize profit. However, if there is a rival with average cost curve ACr threatening to enter, the monopolist lowers its price to P* and produces X*. At price P* the rival cannot cover it's cost and hence does not enter. Thus, P* is the 'pre-emptive price'.

Generalising, we can say that a monopolist maximises profit in the long-run by producing that output for which LRMC (long-run marginal cost) equals marginal revenue and short-run marginal cost. The optimal plant is the one whose short-run average total cost curve is tangent to the long-run average cost curve at the point corresponding to long-run equilibrium output.