Total Revenue and the Linear Demand Curve
What is Total Revenue and the Linear Demand Curve?
Along a straight-line demand curve, total revenue rises in the elastic upper half, peaks at the unit-elastic midpoint, and falls in the inelastic lower half.
On a linear demand curve elasticity is not constant: it is elastic at high prices/low quantities (upper left), unit elastic at the midpoint, and inelastic at low prices/high quantities (lower right). As price falls from the top, total revenue increases while demand is elastic, reaches its maximum exactly where demand is unit elastic (and marginal revenue equals zero), then decreases as demand becomes inelastic. This gives the familiar hump-shaped total-revenue curve and pinpoints the revenue-maximizing price.
Total Revenue and the Linear Demand Curve: a worked example
Take the linear demand curve P = 50 - Q, price in dollars and quantity in units. At P = $40 buyers take 10 units, so total revenue is 40 × 10 = $400. Drop the price to $25 and quantity rises to 25, giving 25 × 25 = $625. Drop it again to $10 and quantity reaches 40, giving 10 × 40 = $400, exactly where revenue started. Elasticity explains the round trip. A one dollar price change always moves quantity by exactly one unit here, so |E_d| = P ÷ (50 - P), which is 40 ÷ 10 = 4 at the first price, 25 ÷ 25 = 1 at the second and 10 ÷ 40 = 0.25 at the third. Revenue climbed $225 while demand was elastic and handed back the same $225 once demand turned inelastic. The peak sits at the midpoint quantity of 25.
The mistake students make with total revenue and the linear demand curve
The midpoint answers a revenue question, and students hand it in as the profit-maximizing quantity. Marginal revenue for P = 50 - Q is MR = 50 - 2Q, twice as steep as demand, and it reaches zero at Q = 25, which is the revenue peak. Give the same firm a marginal cost of $10 and profit maximization sets 50 - 2Q = 10, so Q = 20 at a price of $30. Revenue there is $600, below the $625 peak, because the firm gives up $25 of revenue to avoid the cost of five more units.
Total Revenue and the Linear Demand Curve questions
Where is total revenue maximized on a linear demand curve?
Total revenue peaks at the midpoint of a linear demand curve, halfway between the two intercepts. At that quantity price elasticity of demand equals exactly one and marginal revenue equals zero, which is the signature of a maximum. Above the midpoint, in the elastic region, cutting price still adds revenue. Below it, in the inelastic region, cutting price subtracts revenue. On a graph the midpoint sits at half the price intercept and half the quantity intercept, so the revenue-maximizing pair can be read straight off the axes.
Why does elasticity change along a straight-line demand curve?
Elasticity multiplies the responsiveness of quantity to price by the ratio of price to quantity, and only the first piece is constant on a straight line. Near the top of the curve price is high and quantity is tiny, so the ratio is huge and demand is very elastic. Near the bottom price is small and quantity is large, so the ratio shrinks and demand becomes inelastic. Elasticity therefore falls continuously as you move down the curve, which is why one slope never implies one elasticity.
What does the total revenue curve look like?
Plotting total revenue against quantity for linear demand traces a downward-opening hump. Revenue starts at zero when quantity is zero, rises through the elastic range, peaks at the midpoint quantity, then falls back to zero once price reaches zero. The shape follows straight from TR = P × Q with P sliding down as Q rises. The slope of that hump at any quantity is marginal revenue, which is why marginal revenue equals zero exactly at the top.
Formula / Example
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