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A monopolist maximizes profit where marginal cost equals marginal revenue. At this equilibrium point, the MC curve intersects the MR curve from below. Producing at this level balances additional costs against additional revenues, ensuring the firm captures maximum profitability given its unique market position as the sole producer.

A monopolist faces a downward-sloping marginal revenue curve because it must lower prices to sell additional units. Unlike firms in perfect competition, the monopolist cannot sell more output without reducing the price for all units sold. This steep downward slope reflects the trade-off between quantity and price inherent in the monopolist's market power.

If a monopolist produces below the equilibrium where MC equals MR, the firm misses potential profits. At lower production levels, the revenue from selling additional units exceeds their production cost, meaning the monopolist leaves money on the table by not expanding output to the profit-maximizing quantity.

Marginal cost is the additional cost of producing one more unit, while marginal revenue is the additional revenue gained from selling one more unit. As production increases, marginal cost typically rises due to diminishing productivity and higher variable costs like labor and raw materials. Marginal revenue, conversely, declines as the monopolist must lower prices to sell more.

Producing beyond the point where MC equals MR results in lower profits because the cost of making each additional unit exceeds the revenue it generates. At this stage, marginal cost surpasses marginal revenue, meaning each extra unit sold actually reduces total profit rather than increasing it.

Diminishing productivity causes a monopolist's marginal cost curve to slope upward. As production increases, each additional unit becomes more expensive to produce due to rising variable costs for labor and raw materials. This upward-sloping MC curve reflects the increasing expense of expanding output beyond efficient capacity levels.

A monopolist identifies the profit-maximizing output by locating where its marginal cost curve intersects its marginal revenue curve from below. At this intersection point, the firm produces the precise quantity that balances additional costs against additional revenues. This approach allows the monopolist to determine the optimal production level given unique market conditions as the sole producer.