Elasticity

Lessons

Senior 4 · Economics

Elasticity

4 lessons

Lesson 1 of 4

Price elasticity of demand: measurement and categories

35 min

The same price rise, two different results

Suppose the price of salt and the price of a bus ticket to Musanze both rise by 20%. Households will hardly change how much salt they buy: there is no substitute, and it costs very little. Many travellers, though, will take a shared car or a motorcycle, or simply travel less. The law of demand tells us that quantity falls in both cases; it does not tell us by how much. Elasticity measures that "how much". It decides whether a firm earns more or less when it raises its price, how much revenue a tax will bring in, and whether farmers gain or lose from a good harvest.

Definition

Elasticity: the degree to which one variable responds to a change in another, measured as a ratio of percentage changes.

The four you will use: price elasticity of demand (PED), income elasticity of demand (YED), cross elasticity of demand (XED) and price elasticity of supply (PES).

Price elasticity of demand: the responsiveness of quantity demanded to a change in the good's own price, ceteris paribus:

\text{PED} = \frac{\%\ \text{change in quantity demanded}}{\%\ \text{change in price}}

1. Three ways to calculate PED

(a) The percentage (simple) method uses the original price $P_{1}$ and quantity $Q_{1}$ as the base:

\text{PED} = \frac{\Delta Q}{Q_1} \div \frac{\Delta P}{P_1} = \frac{\Delta Q}{\Delta P} \times \frac{P_1}{Q_1}

(b) The arc (midpoint) method uses the averages, so it gives the same answer whether the price rises or falls:

\text{PED} = \frac{Q_2 - Q_1}{\frac{1}{2}(Q_1 + Q_2)} \div \frac{P_2 - P_1}{\frac{1}{2}(P_1 + P_2)}

(c) The point method measures elasticity at a single point. For a straight-line demand curve $Q = a - b P$ , the change in $Q$ for each franc is $- b$ , so

\text{PED} = -b \times \frac{P}{Q}

Because price and quantity demanded move in opposite directions, PED is negative. Economists usually quote its absolute value, $|\text{PED}|$ , and you should say so when you do.

Worked example 1: why the method matters

In Unit 2's bean market, when the price rises from Frw 800 to Frw 1,000 per kg, the quantity demanded falls from 16 kg to 11 kg a week.

Simple method, price rising (base 800 and 16):

\frac{-5}{16} \div \frac{200}{800} = -0.3125 \div 0.25 = -1.25

Simple method, price falling from 1,000 to 800 (base 1,000 and 11):

\frac{5}{11} \div \frac{-200}{1\,000} \approx 0.4545 \div (-0.2) \approx -2.27

Arc method (either direction):

\frac{-5}{13.5} \div \frac{200}{900} \approx -0.370 \div 0.222 \approx -1.67

Conclusion: the simple method gives different answers for the same stretch of the curve, depending on the direction of the change. The arc method avoids this. All three say demand is elastic here ( $|\text{PED}| \gt 1$ ).

Worked example 2: point elasticity along one straight line

Tomato demand is $Q_d = 1\,600 - 2P$ , so $b = 2$ .

Price (Frw)	Quantity (kg)	$\text{PED} = -2 \times \frac{P}{Q}$	Category
300	1,000	$-2 \times \frac{300}{1\,000} = -0.6$	Inelastic
400	800	$-2 \times \frac{400}{800} = -1$	Unit elastic
600	400	$-2 \times \frac{600}{400} = -3$	Elastic

The slope is the same everywhere on a straight line, yet the elasticity changes, because the ratio $\frac{P}{Q}$ changes. High up the curve a franc is a small share of a high price but the quantity is small, so each change is a big percentage of it.

Warning

Slope is not elasticity. A steep curve looks inelastic, but elasticity also depends on where you are on the curve and on the units of the axes. Only the vertical line (PED = 0), the horizontal line (PED = ∞) and the unit-elastic curve have one elasticity all along. Calculate; don't guess from the picture.

2. The five categories of PED

Category	Value	Meaning	Example
Perfectly inelastic	$\|\text{PED}\| = 0$	Quantity does not change at all	A life-saving drug with no substitute (in the short run)
Inelastic	$0 \lt \|\text{PED}\| \lt 1$	Quantity changes by a smaller percentage than price	Salt, fuel, cigarettes
Unit elastic	$\|\text{PED}\| = 1$	Quantity changes by the same percentage as price	The midpoint of a straight-line demand curve
Elastic	$\|\text{PED}\| \gt 1$	Quantity changes by a larger percentage than price	One brand of soap; a particular bus company's tickets
Perfectly elastic	$\|\text{PED}\| = \infty$	Any rise in price loses every buyer	One farmer's tomatoes in a market full of identical tomatoes

Definition

Total revenue (TR): the amount a seller receives, which is the total spending of buyers: $\text{TR} = P \times Q$ .

The total revenue test: if a price rise raises TR, demand is inelastic; if it lowers TR, demand is elastic; if TR does not change, demand is unit elastic.

3. Elasticity and total revenue

If demand is…	A price rise makes TR…	A price cut makes TR…
Elastic	Fall	Rise
Unit elastic	Stay the same	Stay the same
Inelastic	Rise	Fall

Worked example 3: the total revenue test on the tomato market

Inelastic stretch: raise the price from Frw 300 to Frw 350.

\text{TR}: 300 \times 1\,000 = 300\,000 \;\rightarrow\; 350 \times 900 = 315\,000

TR rises by Frw 15,000, so demand is inelastic here (point PED at Frw 300 is $- 0.6$ ).

Elastic stretch: raise the price from Frw 500 to Frw 550.

\text{TR}: 500 \times 600 = 300\,000 \;\rightarrow\; 550 \times 500 = 275\,000

TR falls by Frw 25,000, so demand is elastic here. A seller who wants more revenue should raise the price only where demand is inelastic.

Tip

"Inelastic: price and revenue go the same way." In the inelastic range, price up means revenue up, and price down means revenue down. In the elastic range they go opposite ways.

Important

Discuss. A bus company on the Kigali–Musanze route raises its fares by 10% and its ticket revenue falls. What does this tell you about the demand for its tickets, and why might the demand for bus travel as a whole on that route be less elastic than the demand for one company's tickets?

Jacob Clifford, "Elasticity of Demand". He asks why petrol stations rarely hold sales; watch for the total revenue test and check it against Worked example 3.

✦

Practice

Define price elasticity of demand and state its formula.
When the price of a loaf of bread rises from Frw 1,000 to Frw 1,100, sales fall from 500 to 470 a day. Calculate PED by the simple method and state the category.
For the same change, calculate PED by the arc method, to 2 decimal places.
Demand is $Q_{d} = 600 - 3 P$ . Find the point PED at $P = 50$ and at $P = 150$ .
At what price is demand $Q_{d} = 600 - 3 P$ unit elastic? What is TR there?
Draw demand curves for perfectly elastic and perfectly inelastic demand, and give an example of each.
A cooperative cuts the price of its milk from Frw 400 to Frw 350 a litre and sales rise from 2,000 to 2,400 litres a day. Use the total revenue test to decide whether demand is elastic or inelastic.
"A steep demand curve is always inelastic." Discuss.

Show answersHide answers

The responsiveness of quantity demanded to a change in the good's own price: PED = % change in quantity demanded ÷ % change in price.
$\frac{-30}{500} \div \frac{100}{1\,000} = -0.06 \div 0.1 = -0.6$ : inelastic.
$\frac{-30}{485} \div \frac{100}{1\,050} \approx -0.0619 \div 0.0952 \approx -0.65$ .
At 50: $Q = 450$ , PED $= -3 \times \frac{50}{450} \approx -0.33$ (inelastic). At 150: $Q = 150$ , PED $= -3 \times \frac{150}{150} = -3$ (elastic).
Unit elastic at the midpoint: $3 \times \frac{P}{600 - 3P} = 1 \Rightarrow 3P = 600 - 3P \Rightarrow P = 100$ , $Q = 300$ , TR $= 100 \times 300 = \text{Frw } 30\,000$ .
Perfectly elastic: horizontal (one farmer's produce in a competitive market). Perfectly inelastic: vertical (an essential drug with no substitute, in the short run).
TR before $= 400 \times 2\,000 = 800\,000$ ; after $= 350 \times 2\,400 = 840\,000$ . A price cut raised TR, so demand is elastic.
Not always: elasticity depends on the ratio of price to quantity as well as the slope, and on the units on the axes. On a straight line it varies from elastic to inelastic. Steepness is only a rough guide when comparing curves drawn on the same axes, through the same point.

Quick check

Key points

PED = % change in quantity demanded ÷ % change in price; it is negative, and usually quoted as an absolute value.
Calculate it by the simple, arc (midpoint) or point method; the arc method gives the same answer in both directions; for $Q = a - b P$ , point PED $= -b \frac{P}{Q}$ .
Categories: perfectly inelastic (0), inelastic (below 1), unit (1), elastic (above 1), perfectly elastic (∞).
Total revenue test: inelastic, price and TR move together; elastic, they move in opposite directions; TR is greatest where |PED| = 1.

Sources and credits

Video: Elasticity of Demand - Micro Topic 2.3, Jacob Clifford, embedded from YouTube.

Written by Ganzaa: the text, the elasticity diagrams (the straight-line demand and total-revenue curves drawn to scale from $Q = 1\,600 - 2P$ ), the tables, the worked examples and the practice items. All prices and quantities are illustrative.

1The same price rise, two different results

21. Three ways to calculate PED

32. The five categories of PED

43. Elasticity and total revenue

The same price rise, two different results

1. Three ways to calculate PED

2. The five categories of PED

3. Elasticity and total revenue