Q.Explain why, for a monopolist, marginal revenue is always less than average revenue at every level of output beyond the first unit.

A monopolist cannot sell more without lowering price on everything it already sells, and this single fact is what pulls MR below AR.

Under perfect competition a firm can sell as much as it likes without affecting price at all, which is why its AR and MR coincide. A monopolist has no such luxury, because it is the entire market and therefore faces the market's own downward-sloping demand curve directly: to sell a larger quantity, it must lower the price, and — since it cannot practically charge different buyers different prices for identical units bought under ordinary market conditions — that lower price applies to every unit sold, not merely to the additional unit that induced the price cut in the first place.

Consider what happens when a monopolist decides to sell one more unit. Two things happen simultaneously: it gains revenue equal to the new, lower price on that additional unit, but it also loses revenue on every unit it was already selling, because all of those units must now also be sold at the new, lower price rather than the higher price that prevailed before. Marginal revenue — the net addition to total revenue from selling the one extra unit — is therefore the price received on the new unit minus the revenue given up on all the previously sold units, and this net figure is necessarily smaller than the price (average revenue) at which the extra unit itself sold. As output expands further and price keeps falling along the downward-sloping demand curve, this gap between price and marginal revenue does not close; marginal revenue remains below average revenue throughout, and can even turn negative at high enough output, if the loss on earlier units outweighs the gain from the additional one. …

A common slip is assuming AR and MR are equal under monopoly as they are under perfect competition — under monopoly they are never equal beyond the …