Total revenue follows elasticity

A price change has two opposing effects.

Total revenue is price times quantity. A price increase raises revenue per unit but reduces units sold. With inelastic demand, the percentage quantity loss is smaller, so total revenue rises. With elastic demand, the quantity loss is larger, so total revenue falls. At unit elasticity, the effects offset.

The rule can be remembered as a direction test. With elastic demand, price and total revenue move in opposite directions. With inelastic demand, they move in the same direction. At unit elasticity, a small price change leaves total revenue unchanged. This follows from the competing price and quantity effects rather than from a separate formula.

This total-revenue test applies to movement along a fixed demand curve. If demand itself shifts, price and revenue can change for a different reason. A monopolist with nonnegative marginal cost will not choose an output on the inelastic portion of demand because reducing output there raises revenue and lowers cost.

That monopoly result uses two gains from reducing output on the inelastic portion. A small price increase raises total revenue because demand is inelastic, and producing fewer units lowers total cost when marginal cost is nonnegative. Profit must rise, so the original output could not have been profit maximizing. The firm chooses on the elastic portion or, in a limiting case, at unit elasticity when marginal cost is zero.

Midpoint elasticity and revenue

Price falls from $10 to $8 while quantity rises from 40 to 60. Midpoint quantity change is 20/50=40%. Price change magnitude is 2/9=22.2%. Elasticity is about 1.8, so demand is elastic. Revenue rises from $400 to $480, as the total-revenue rule predicts.

Notice that the price cut is 20 percent if $10 is used as the original base, while the price increase back to $10 is 25 percent if $8 is used as the original base. The midpoint denominator of $9 avoids that asymmetry. An item that explicitly asks for ordinary percentage change may use an initial base. An item asking for two-point elasticity usually expects midpoint.

Total revenue is not profit. A price cut can raise revenue while reducing profit if additional units have sufficiently high cost. The elasticity rule identifies the revenue effect only. Profit requires comparing the revenue change with the cost change, usually through marginal revenue and marginal cost.

Revenue direction without exact elasticity

A theater knows demand for weekday tickets is elastic. It lowers price while remaining on the same demand curve. Attendance rises by a larger percentage than price falls, so total ticket revenue rises. Whether economic profit rises cannot be determined without the theater’s cost response.

If a problem gives two price-quantity pairs, calculate revenue at both points as a direct audit. A classification and the actual P× Q figures should tell the same story. If not, recheck the percentage bases, direction, or arithmetic.

When the stem asks what a seller should do to increase revenue, first identify the elasticity on the relevant portion of demand. A seller facing elastic demand can raise revenue by lowering price. A seller facing inelastic demand can raise revenue by increasing price. This is a conditional prediction, not a universal pricing recommendation. It assumes movement along the stated demand curve and answers only the revenue question. A complete profit decision must also account for how the resulting output change affects total cost.