Marginal Revenue Curve Twice as Steep
What is Marginal Revenue Curve Twice as Steep?
For a single-price monopolist with a straight-line demand curve, the marginal revenue curve has the same intercept but twice the slope, hitting the quantity axis at half the demand's intercept.
Because a single-price seller must lower price on every unit to sell one more, marginal revenue falls faster than price. With linear demand P = a − bQ, marginal revenue is MR = a − 2bQ: same vertical intercept (a), double the slope, so it reaches the horizontal axis at half the output where demand does. This 'twice as steep, half the quantity' rule is a fast graphing shortcut for monopoly and other price-maker diagrams and explains why MR lies below demand.
Marginal Revenue Curve Twice as Steep: a worked example
Take the linear demand curve P = 120 - 4Q. Total revenue is TR = 120Q - 4Q², so marginal revenue is MR = 120 - 8Q. Both curves start at $120 on the price axis, but demand reaches the quantity axis at Q = 30 while MR reaches it at Q = 15, exactly half. Now add a constant marginal cost of $40. Setting MR = MC gives 120 - 8Q = 40, so 8Q = 80 and Q = 10. Read the price off demand, not off MR: P = 120 - 4(10) = $80. Check that against discrete units. At Q = 9 the price is $84 and total revenue is $756; at Q = 10 the price is $80 and total revenue is $800. The tenth unit adds only $44, far below its $80 price, because the seller gave up $4 on each of the nine earlier units.
The mistake students make with marginal revenue curve twice as steep
Students hear 'twice as steep' and halve the wrong intercept, drawing MR from $60 when demand starts at $120. The phrase invites it, since doubling one number feels like it should halve another. Keep the two straight: MR shares demand's vertical intercept and halves its horizontal intercept. With P = 120 - 4Q, MR runs from $120 down to zero at Q = 15, never from $60. The second slip is reading the monopoly price off MR after solving MR = MC. The MR curve locates the quantity; the demand curve sets the price buyers will actually pay for it.
Marginal Revenue Curve Twice as Steep questions
Why is the marginal revenue curve twice as steep as the demand curve?
Marginal revenue falls twice as fast because a single-price seller must cut the price on every unit, not just on the extra one. Selling one more unit adds that unit's price but subtracts the price cut spread across all previous units. With demand P = a - bQ, total revenue is aQ - bQ², and the rate at which revenue changes as output rises is a - 2bQ. The b turns into 2b, which doubles the slope while leaving the intercept a untouched.
Does the twice-as-steep rule apply to a perfectly competitive firm?
Perfectly competitive firms face a horizontal demand curve at the market price, so the rule collapses to nothing useful. Doubling a slope of zero still gives zero, which is why price equals marginal revenue for a price taker. The shortcut matters only for price makers with downward-sloping demand: a monopoly, a monopolistically competitive firm, or the residual demand facing one cartel member. Any seller that must lower price to move more units sees MR sitting below demand.
Where does marginal revenue equal zero on a linear demand curve?
Marginal revenue hits zero at exactly half the quantity where demand meets the axis, and that output is where total revenue peaks. With P = 120 - 4Q, demand crosses at Q = 30, so MR crosses at Q = 15 and total revenue tops out at 15 × $60 = $900. Left of that point demand is elastic and MR is positive; right of it demand is inelastic and MR is negative, so a monopolist never chooses to operate there.
Formula / Example
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