Foundationsmicroeconomics
Revenue: Total, Average, and Marginal
Overview
Revenue is the other half of profit - every profit decision weighs revenue against cost. This lesson defines total, average, and marginal revenue, shows why a price-taking firm faces constant MR equal to price while a firm with market power faces MR below price, and locates the output where total revenue is maximized.
Learning objectives
- Define total, average, and marginal revenue and compute them.
- Explain why AR always equals price.
- Show why MR = P for price takers but MR < P with downward-sloping demand.
- Find the output where total revenue is maximized.
Three Revenue Measures
Total revenue (TR) is price times quantity: TR = P·Q. Average revenue (AR = TR/Q) is revenue per unit. Marginal revenue (MR = ΔTR/ΔQ) is the extra revenue from selling one more unit. Profit analysis pairs each with its cost twin - TR with TC, MR with MC - so fluency with all three is non-negotiable for what follows.
AR = TR/Q = P·Q/Q = P, identically, for any firm. Average revenue carries no independent information - but it earns its keep as the firm's demand curve: the AR curve is the demand curve the firm faces.
The Price Taker's Revenue: MR = P
A perfectly competitive firm sells any quantity at the market price - its demand curve is horizontal. Selling one more unit adds exactly P to revenue, so MR = P at every output, and TR = P·Q is a straight ray from the origin. For the price taker, the revenue side of profit maximization is trivially simple: every unit is worth P.
Horizontal demand makes price, average revenue, and marginal revenue one and the same number. This equality is why the competitive firm's rule 'produce where P = MC' is just the general MR = MC rule in disguise.
Market Power: Why MR Falls Below Price
A firm facing downward-sloping demand must cut price to sell more - and the cut applies to all units, not just the marginal one. Selling the 11th unit at 10 adds 1 on each of the 10 inframarginal units: MR = 9 - 10 = -9 price. In general MR < P whenever demand slopes down, and the gap widens as output grows.
MR has two parts: the quantity effect (+P from the new unit) and the price effect (−ΔP × Q on all previous units). For price takers the price effect is zero; with market power it is negative and growing - the heart of monopoly pricing.
The MR Curve Lies Below Demand
With linear demand P = a - bQ, marginal revenue is MR = a - 2bQ: same intercept, twice the slope, lying everywhere below the demand (AR) curve except at zero output. This doubled-slope rule is an exam staple - memorize it for linear demand, and understand the price-effect logic behind it for everything else.
Where Total Revenue Peaks
TR rises while MR is positive, peaks where MR = 0, and falls where MR turns negative - the marginal-average logic once more. With linear demand, the TR peak sits at the quantity where elasticity equals one (unit-elastic demand): cutting price raises revenue while demand is elastic, lowers it once demand turns inelastic. Revenue maximization and profit maximization are different goals - the next lesson shows why.
Revenue Curves at a Glance
Price taker: TR a straight line, AR and MR horizontal and identical at P. Firm with market power: TR hump-shaped, AR the downward-sloping demand curve, MR below AR falling twice as fast (linear case). Sketching these from memory, for both market structures, is the gateway skill for monopoly and competition analysis.
Key points
- TR = P·Q; AR = TR/Q = P always; MR = ΔTR/ΔQ.
- The AR curve is the firm's demand curve.
- Price taker: MR = P at all outputs; TR is a straight ray from the origin.
- With downward-sloping demand, MR < P because price cuts apply to inframarginal units.
- Linear demand P = a - bQ gives MR = a - 2bQ: same intercept, twice the slope.
- TR is maximized where MR = 0 (unit-elastic demand); negative MR means cutting price loses revenue.
Common misconceptions
✓ Truth: Only for price takers. With market power, MR equals the new unit's price minus the revenue lost on all previous units from the required price cut - which is why MR lies below price and can even be negative.
✓ Truth: Revenue maximization ignores costs - it is the right goal only if someone else pays the costs. Profit maximization (MR = MC) generally picks a smaller output than revenue maximization (MR = 0).
✓ Truth: AR = TR/Q = P holds by definition for every firm in every market structure. What differs across structures is marginal revenue, not average.
Practice questions
Q1. A firm sells 10 units at $10 each, then cuts price to $9 to sell 11 units. Marginal revenue of the 11th unit is:
- $9
- $10
- -$1
- $99
TR rises from 99, so MR = -9 quantity effect is outweighed by the 1 lost on each of the 10 previous units) - MR far below price.
Q2. With linear demand P = 20 - 2Q, the marginal revenue curve is:
- MR = 20 - 2Q
- MR = 20 - 4Q
- MR = 10 - 2Q
- MR = 20 - Q
For P = a - bQ, MR = a - 2bQ: same vertical intercept, twice the slope. Here MR = 20 - 4Q, hitting zero at Q = 5 where total revenue peaks.
Q3. For a perfectly competitive firm, which equality holds at every output?
- MR < P
- P = AR = MR
- MR = 0
- AR < MR
Horizontal demand means each extra unit sells at the market price with no price effect, so price, average revenue, and marginal revenue coincide at every quantity.
Q4. Total revenue is maximized at the output where:
- MR = MC.
- MR = 0.
- P = MC.
- AR is maximized.
TR rises while MR > 0 and falls while MR < 0, so its peak is where MR = 0 (unit-elastic demand). Profit maximization instead requires MR = MC - a stricter, cost-aware condition.
Click a question to reveal the answer and explanation.
Frequently asked questions
Can marginal revenue really be negative?
Yes. If demand is inelastic, the price cut needed to sell more destroys more revenue on existing units than the new unit adds. No profit-maximizing firm operates there - cutting output would raise both revenue and (by saving costs) profit.
Why does the price effect grow with output?
The price effect is ΔP × Q: each extra unit forces a price cut applied to an ever larger base of inframarginal units. At high output the base is huge, so even a tiny required cut makes MR deeply negative.
How does elasticity connect to the TR peak?
While demand is elastic, a price cut raises quantity proportionally more, so TR rises (MR > 0). When demand turns inelastic, quantity responds weakly and TR falls (MR < 0). The peak sits at unit elasticity - the boundary between the two.
Does a competitive firm ever face MR below price?
No - as long as it truly cannot influence the market price. If a 'competitive' firm is large enough that its extra output moves the price, it has market power by definition and the MR < P analysis applies.