Why do markets that are dominated by a single seller have a tendency to maximize profits at lower levels of output than their competitive counterpart? Why is the competitive market equilibrium price lower than the monopolist price?
The monopolist will restrict output to be able to charge a higher price.
It is true that if the monopolist reduced the price, it could sell more units, but the benefit of selling more units is more than offset by the fact that the monopolist would have to charge a lower price on all units that were previously sold at a higher price.
Profits are maximised when marginal revenue (MR) equals marginal cost (MC). Producing in a region where marginal revenue is lower than marginal cost lowers profits.
In the case of perfect competition, the firm can sell any quantity at the market price, so marginal revenue equals the price (MR = P). It will choose to sell a quantity where marginal revenue equals marginal cost (MR = MC). From these two equations we have MR = MC = P.
For the monopolist, marginal revenue is lower than price (MR < P). This is the crucial point. When the monopolist sells an extra unit of output, it earns price, BUT it also loses some money as it has to lower the price on all units previously sold at a higher price. This is why the monopolists marginal revenue at every quantity on the chart below is lower than the price. This is why on the chart below the marginal revenue curve lies below the demand curve.
The monopolist will choose to sell quantity Q* corresponding to Point A, where MR = MC < P. This quantity is lower than the quantity were price equals marginal revenue (where the D curve intersects the MC curve above), which is the quantity chosen in perfect competition. So the quantity sold by the monopolist is lower than the quantity sold in perfect competition.
This is why a monopoly is undesirable from the consumer's perspective. Look at the chart above to the right of Point B. There are people (represented by the D curve) who are willing to pay a higher price for an extra unit of output than the marginal cost. Still, the monopolist will not produce that extra unit.