Difficulty: Intermediate | Prerequisites: basic algebra, familiarity with linear demand curves
Price elasticity of demand sits at the intersection of demand analysis and revenue theory in intermediate microeconomics. You need it whenever a firm is deciding whether raising its price will increase or decrease total revenue, which is one of the most practical questions in the entire course. Before tackling this, you should be comfortable with linear demand functions and basic calculus (derivatives). If you can plug a price into a demand equation and compute total revenue, you are ready.
For a linear demand curve Q = 300 − 5P, you can compute quantity demanded at any price, calculate point price elasticity using ε = (dQ/dP) × (P/Q), and find total revenue as P × Q. Elasticity tells you whether a price increase raises or lowers revenue: inelastic demand (|ε| < 1) means revenue rises with price, elastic demand (|ε| > 1) means it falls, and unit elastic (|ε| = 1) is the revenue-maximising sweet spot.
Demand function (linear)
An equation expressing quantity demanded as a function of price, taking the form Q = a − bP, where a is the intercept (quantity demanded when price is zero) and b is the slope (how much Q drops per unit increase in P).
In simple terms, it is the rule that tells you how many units people will buy at any given price.
Price elasticity of demand (ε)
A unit-free measure of how responsive quantity demanded is to a change in price, defined as ε = (dQ/dP) × (P/Q). For a downward-sloping demand curve, ε is always negative.
Think of it as the answer to: "If I raise price by 1%, by what percentage does quantity demanded fall?"
Elastic demand
|ε| > 1. Quantity demanded is highly responsive to price. A price increase causes total revenue to fall because the percentage drop in quantity outweighs the percentage gain in price.
In simple terms, buyers are price-sensitive here, so hiking the price backfires on revenue.
Inelastic demand
|ε| < 1. Quantity demanded is relatively unresponsive to price. A price increase causes total revenue to rise because the percentage gain in price outweighs the percentage drop in quantity.
In simple terms, buyers are not very sensitive to price, so the firm can charge more and still come out ahead.
Unit elastic demand
|ε| = 1. The percentage change in quantity exactly offsets the percentage change in price. Total revenue is at its maximum at this point.
In simple terms, this is the price where revenue peaks.
Total revenue (TR)
TR = P × Q. The total amount of money a firm receives from selling Q units at price P.
Given Q = 300 − 5P, substitute the price to find quantity
At P = 10: Q = 300 − 5(10) = 250
At P = 20: Q = 300 − 5(20) = 200
At P = 30: Q = 300 − 5(30) = 150
At P = 50: Q = 300 − 5(50) = 50
The slope of the demand curve here is dQ/dP = −5, which stays constant along the entire curve
The formula is ε = (dQ/dP) × (P/Q)
For a linear demand curve, the slope dQ/dP is constant (here, −5), but elasticity changes at every point because P/Q changes
Worked examples from Q = 300 − 5P:
At P = 10, Q = 250: ε = (−5)(10/250) = −0.2 (inelastic)
At P = 20, Q = 200: ε = (−5)(20/200) = −0.5 (inelastic)
At P = 30, Q = 150: ε = (−5)(30/150) = −1.0 (unit elastic)
At P = 50, Q = 50: ε = (−5)(50/50) = −5.0 (elastic)
Notice that elasticity moves from inelastic to elastic as you slide up the demand curve (higher P, lower Q)
Total revenue = P × Q
At P = 10: TR = 10 × 250 = 2,500
At P = 20: TR = 20 × 200 = 4,000
At P = 30: TR = 30 × 150 = 4,500 (maximum, where |ε| = 1)
At P = 50: TR = 50 × 50 = 2,500
When demand is inelastic (|ε| < 1), raising price increases TR
When demand is elastic (|ε| > 1), raising price decreases TR
TR is maximised at unit elasticity (|ε| = 1), here at P = 30
Q = a - bP\varepsilon = \frac{dQ}{dP} \cdot \frac{P}{Q}TR = P \times QFor Q = 300 − 5P specifically, the revenue-maximising price is found by setting |ε| = 1, which gives P = 30 and TR = 4,500.
This is exactly the calculation a concert promoter runs when deciding ticket prices: set the price too high and empty seats cost you more in lost volume than you gain per ticket. Airlines use the same logic in reverse, segmenting passengers by elasticity (business travellers are inelastic, leisure travellers are elastic) and pricing each group differently to maximise total revenue across the plane.
Students often think that because the slope of a linear demand curve is constant, elasticity must be constant too. It is not. Elasticity = slope × (P/Q), and the ratio P/Q changes at every price point.
Students sometimes confuse the sign of elasticity with its economic meaning. Elasticity of demand is negative (price goes up, quantity goes down), but we compare |ε| to 1 when deciding elastic vs inelastic.
A common error is thinking that the revenue-maximising price is always the highest price. In fact, revenue peaks at unit elasticity, which is typically in the middle of the demand curve, not at the top.
Students occasionally compute TR by multiplying price by the slope rather than by the quantity demanded. TR = P × Q, always.
⚠️ You will almost certainly be asked to fill in a table of Q, ε, and TR for various prices along a demand curve. Practise until this is automatic.
⚠️ Exam questions often ask: "If the firm raises its price from $20 to $30, what happens to total revenue?" The answer depends on whether demand is elastic or inelastic at the starting price.
⚠️ Be ready to identify the revenue-maximising price (where |ε| = 1) and explain why raising price beyond that point lowers TR.
True or False: Elasticity is the same at every point on a linear demand curve.
False. The slope is constant, but P/Q changes, so elasticity changes.
Fill in the blank: When |ε| < 1, demand is ______ and a price increase will ______ total revenue.
Inelastic; increase.
True or False: Total revenue is maximised where demand is perfectly inelastic.
False. TR is maximised where |ε| = 1 (unit elastic).
If Q = 300 − 5P, what is quantity demanded at P = 40?
Q = 300 − 200 = 100.
True or False: The slope dQ/dP for Q = 300 − 5P is −5.
True.
Q: Given Q = 300 − 5P, calculate the price elasticity of demand at P = 20. Is demand elastic or inelastic at this price?
A: At P = 20, Q = 200. ε = (−5)(20/200) = −0.5. Since |ε| = 0.5 < 1, demand is inelastic.
Q: A firm faces Q = 300 − 5P. If it currently charges $20 and considers raising its price to $30, should it expect total revenue to rise or fall? Explain.
A: At P = 20, demand is inelastic (|ε| = 0.5), so revenue rises with a price increase. TR goes from 4,000 to 4,500. Revenue rises.
Q: At what price is total revenue maximised for Q = 300 − 5P? What is total revenue at that price?
A: Set |ε| = 1: (5)(P/Q) = 1, so 5P = Q = 300 − 5P, giving 10P = 300, P = 30. TR = 30 × 150 = 4,500.
Q: Explain why total revenue falls when a firm raises its price from $30 to $50 along Q = 300 − 5P.
A: At P = 30, |ε| = 1 (unit elastic). At any price above 30, demand is elastic (|ε| > 1), so the percentage drop in quantity exceeds the percentage gain from the higher price. TR falls from 4,500 to 2,500.
Elasticity feeds directly into monopoly pricing (a monopolist chooses to produce only on the elastic portion of its demand curve, because operating on the inelastic portion means it could raise price and increase both revenue and profit). It also connects to consumer surplus and deadweight loss: how much surplus a tax or price change destroys depends on how elastic demand is. In perfect competition (covered in question 3 of this homework), firms are price-takers facing perfectly elastic demand at the market price.
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