Marginal revenue includes the price-cut effect

For linear demand, MR has twice the slope.

Suppose a seller can sell three units at $10 or four at $9. Revenue rises from $30 to $36, so MR of the fourth unit is $6, below its $9 price. The seller gains $9 from the new unit but loses $1 on each of the previous three. This is why MR lies below demand.

For a single-price seller, lowering price applies to all units sold. The marginal revenue of the fourth unit can be decomposed:

MR = $9 from the new unit – (3 earlier units × $1 lost per unit) = $6.

The price-cut effect is absent for a competitive firm because it takes price as fixed.

For inverse linear demand P=a-bQ, total revenue is TR=aQ-bQ^2, so marginal revenue is MR=a-2bQ. MR shares the price intercept and has twice the negative slope. Do not double the intercept or shift MR horizontally by guesswork.

Choose Q_m where MR meets MC. Then move vertically to demand for P_m. Moving to demand first or setting demand equal to MC produces the wrong monopoly output. Profit is (P_m-ATC)Q_m.

Setting demand equal to MC finds the allocatively efficient quantity, not the monopoly quantity. Because MR lies below demand, MR meets MC at a smaller output. The monopolist then uses demand to learn the highest single price buyers will pay for that output.

The two-step rule

MR equals MC at 30 units and $12. Demand shows consumers will pay $20 for 30. ATC is $15. The monopolist charges $20 and earns (20-15)30=$150. It does not charge the $12 marginal revenue.

The $12 is the revenue added by the marginal unit after accounting for price reductions, not a posted price. Charging $12 would correspond to a larger quantity on demand and would invalidate the selected MR-MC point. Always keep a vertical guide from Q_m to demand.

If fixed cost rises, ATC rises and profit falls, but MR and MC remain unchanged, so output and price remain the same. A per-unit tax raises MC, reducing quantity and raising price. The type of cost determines which step changes.

The monopolist may earn profit, break even, or lose money. Monopoly power does not guarantee profit because demand may be weak or cost high. The firm still chooses the loss-minimizing quantity with MR=MC and applies the relevant shutdown rule.

For a linear inverse demand curve, MR shares the price intercept and crosses the quantity axis at half demand’s intercept. This is equivalent to twice the negative slope. Draw both from the equation rather than estimating their spacing by eye.

Total revenue is maximized where MR is zero. The profit maximum normally occurs earlier when MC is positive, because the firm stops where MR has fallen only to MC. Choosing the largest total revenue ignores the resource cost of additional output.