Revenue curves and elasticity
| English | Français |
|---|---|
| marginal revenue/ˈmɑːdʒɪnl ˈrevənjuː/ | revenu marginal |
| average revenue/ˈævrɪdʒ ˈrevənjuː/ | revenu moyen |
A decision you can investigate
- A firm sells extra units by cutting the price of every unit. Extra sales bring receipts, but the price cut reduces receipts on units it could already sell.
- Marginal revenue 边际收益 is not automatically the new price.
Build the explanation
- Total revenue TR=P×Q. Average revenue 平均收益 AR=TR/Q equals price when every unit sells at the same price and Q is positive. Marginal revenue MR=ΔTR/ΔQ measures extra receipts per additional unit over a stated interval; in a smooth model it is the derivative of TR. With downward-sloping demand, a price cut on all units makes MR below price. In perfect competition the firm takes a fixed market price, so AR=MR=P for its relevant output range.
- On a demand curve, a price cut raises TR in an elastic region and lowers TR in an inelastic region; at a unit-elastic point of a smooth curve, TR is locally stationary. Distinguish point elasticity from finite-interval estimates. A revenue maximum is not necessarily a profit maximum because costs also change. Revenue diagrams need quantity on the horizontal axis and a clear choice of total receipts or per-unit receipts on the vertical axis.
Work through the evidence
- In a fictional smooth demand model, P=100−Q, so TR=100Q−Q², AR=100−Q and MR=100−2Q. At Q40, P60, TR2400 and point MR20. Point PED=(dQ/dP)×P/Q=−60/40=−1.5, elastic. At Q60, P40, TR2400 and point MR−20; point PED=−40/60≈−0.667, inelastic. TR reaches2500 at Q50, P50, with MR0 and point PED−1.
- Across a separate finite move Q40→41, prices60→59 and TR2400→2419. Interval MR=19, not the endpoint price59 or the derivative20 at Q40. Extra-unit receipts59 are offset by the price loss1 on forty earlier units, giving59−40=19. State whether a question asks for an interval calculation or a point value.
What is point MR at Q40 in the smooth model?
100−2×40=20; the finite one-unit change is a different convention.
What is interval MR from Q40 to41?
TR2419−TR2400=19 over one added unit.
With downward-sloping uniform-price demand, marginal revenue always equals the new unit price.
A price reduction also affects receipts from earlier units.
Test the limits
- The revenue/elasticity result concerns a price movement along unchanged demand, not a shift caused by advertising or income. Mixed prices, discounts and discrimination can make average receipts differ from one posted price. Point and midpoint elasticity are valid conventions for different tasks; do not mix a starting-point finite percentage with a derivative and call them identical.
- The linear demand curve is fictional and applies only over its stated positive-price range. Revenue maximization at Q50 does not show that selling more always helps: TR declines beyond that point. Cost, capacity, objectives and uncertainty determine the business decision. The diagram shows AR/MR per unit; TR belongs in a separate total-receipts curve.
What is point PED at the revenue maximum?
At P50/Q50 and slope dQ/dP=−1, PED=−1.
Apply and explain your answer
- Why is interval MR19 rather than the price59 in the one-unit expansion?
- Selling the extra unit adds59, but the lower price removes1 from each of forty earlier units; net receipts rise19.
Match the terms to their meanings.
Use each term for its stated economic relationship.
Use the terms precisely
- marginal revenue: Change in total receipts per additional unit over a stated interval, or its point derivative in a smooth model.
- average revenue: Total revenue divided by output; equal to price under uniform pricing.
In a fictional smooth demand model, P=100−Q, so TR=100Q−Q², AR=100−Q and MR=100−2Q. At Q40, P60, TR2400 and point MR20. Point PED=(dQ/dP)×P/Q=−60/40=−1.5, elastic. At Q60, P40, TR2400 and point MR−20; point PED=−40/60≈−0.667, inelastic. TR reaches2500 at Q50, P50, with MR0 and point PED−1. Across a separate finite move Q40→41, prices60→59 and TR2400→2419. Interval MR=19, not the endpoint price59 or the derivative20 at Q40. Extra-unit receipts59 are offset by the price loss1 on forty earlier units, giving59−40=19. State whether a question asks for an interval calculation or a point value.
The revenue/elasticity result concerns a price movement along unchanged demand, not a shift caused by advertising or income. Mixed prices, discounts and discrimination can make average receipts differ from one posted price. Point and midpoint elasticity are valid conventions for different tasks; do not mix a starting-point finite percentage with a derivative and call them identical. The linear demand curve is fictional and applies only over its stated positive-price range. Revenue maximization at Q50 does not show that selling more always helps: TR declines beyond that point. Cost, capacity, objectives and uncertainty determine the business decision. The diagram shows AR/MR per unit; TR belongs in a separate total-receipts curve.
Total revenue TR=P×Q. Average revenue AR=TR/Q equals price when every unit sells at the same price and Q is positive. Marginal revenue MR=ΔTR/ΔQ measures extra receipts per additional unit over a stated interval; in a smooth model it is the derivative of TR. With downward-sloping demand, a price cut on all units makes MR below price. In perfect competition the firm takes a fixed market price, so AR=MR=P for its relevant output range.