Relationship between PED and the amount spent by consumers and revenue raised by firms

Allocation of Resources – Price Elasticity of Demand (PED)

Learning Objective

Explain how the price elasticity of demand determines the amount consumers spend on a good and the total revenue earned by firms, and evaluate the significance of PED for decision‑making by households, businesses, workers and governments.

1. Definition and Formula (percentage‑change method)

• Price Elasticity of Demand (PED): the percentage change in quantity demanded that results from a 1 % change in price.
• Formula

$$\text{PED}= \frac{\%\Delta Q_{d}}{\%\Delta P}$$

• For most normal goods the numerator and denominator have opposite signs, giving a negative value. When classifying elasticity we use the absolute value |PED| and ignore the sign.

2. Interpreting the Coefficient

Absolute value |PED|	Elasticity type (syllabus wording)	Implication
0	Perfectly inelastic	Quantity demanded does not respond to any price change.
< 1	Inelastic	Quantity changes by a smaller proportion than price.
= 1	Unit‑elastic (unitary)	Quantity changes by exactly the same proportion as price.
> 1 but < ∞	Elastic	Quantity changes by a larger proportion than price.
→ ∞	Perfectly elastic	Any price rise causes quantity demanded to fall to zero; a price fall leads to an infinite increase in quantity.

3. Determinants of PED (syllabus phrasing)

• Availability of close substitutes – the more close substitutes, the more elastic the demand.
• Proportion of income spent on the good – the larger the share of a consumer’s budget, the more elastic the demand.
• Nature of the good – luxuries are more elastic than necessities.
• Time‑period considered – demand is more elastic in the long run because consumers have more time to adjust their habits.

4. Calculating PED (percentage‑change method)

Use the formula in section 1. Remember to express the changes as percentages of the original (base‑year) values.

Example calculation

Price falls from £12 to £9 and quantity demanded rises from 80 to 120 units.

\[ \%\Delta P = \frac{9-12}{12}\times100 = -25\% \] \[ \%\Delta Q = \frac{120-80}{80}\times100 = 50\% \] \[ \text{PED}= \frac{50\%}{-25\%}= -2.0\;( \text{elastic, }|PED|=2) \]

5. Relationship Between PED and Consumer Expenditure

Consumer expenditure on a good is E = P \times Q.

• Elastic demand (|PED| > 1): a price fall produces a proportionally larger rise in Q, so E **increases**.
• Inelastic demand (|PED| < 1): a price fall produces a proportionally smaller rise in Q, so E **decreases**.
• Unit‑elastic demand (|PED| = 1): the percentage change in Q exactly offsets the price change, leaving E **unchanged**.
• Perfectly inelastic (|PED| = 0): expenditure changes in direct proportion to price (E rises when P rises).
• Perfectly elastic (|PED| → ∞): any price reduction causes Q to become arbitrarily large, so expenditure can increase dramatically.

6. Relationship Between PED and Firm Revenue

For a firm, total revenue (TR) is also TR = P \times Q. The effect of a price change depends on the elasticity of the product’s demand.

Elasticity	Effect of a Price Increase	Effect on TR	Effect of a Price Decrease	Effect on TR
Elastic (|PED| > 1)	Q falls proportionally more than P rises	TR falls	Q rises proportionally more than P falls	TR rises
Unit‑elastic (|PED| = 1)	Q falls proportionally the same as P rises	TR unchanged	Q rises proportionally the same as P falls	TR unchanged
Inelastic (|PED| < 1)	Q falls proportionally less than P rises	TR rises	Q rises proportionally less than P falls	TR falls
Perfectly inelastic (|PED| = 0)	Q unchanged	TR rises (price rise only)	Q unchanged	TR falls (price fall only)
Perfectly elastic (|PED| → ∞)	Any price rise drives Q to zero → TR falls to zero	TR falls	Any price cut makes Q infinite → TR can become very large	TR rises sharply

7. Graphical Illustration

8. Worked Examples

Example 1 – Price Decrease, Elastic Demand

Price falls from $10 to $8; quantity demanded rises from 100 to 150 units.

\[ \%\Delta P = \frac{8-10}{10}\times100 = -20\% \] \[ \%\Delta Q = \frac{150-100}{100}\times100 = 50\% \] \[ \text{PED}= \frac{50\%}{-20\%}= -2.5\;( |PED| = 2.5 > 1) \]

Because demand is elastic, total revenue rises:

\[ TR_{0}=10\times100 = \$1{,}000 \] \[ TR_{1}=8\times150 = \$1{,}200 \]

Revenue increases by $200, confirming the rule for elastic demand.

Example 2 – Price Increase, Inelastic Demand

Price rises from $5 to $6; quantity demanded falls from 200 to 180 units.

\[ \%\Delta P = \frac{6-5}{5}\times100 = 20\% \] \[ \%\Delta Q = \frac{180-200}{200}\times100 = -10\% \] \[ \text{PED}= \frac{-10\%}{20\%}= -0.5\;( |PED| = 0.5 < 1) \]

Since demand is inelastic, total revenue also rises:

\[ TR_{0}=5\times200 = \$1{,}000 \] \[ TR_{1}=6\times180 = \$1{,}080 \]

Revenue increases by $80 even though the price is higher, illustrating the inelastic case.

Example 3 – Perfectly Inelastic Demand

A life‑saving drug has a fixed quantity of 1 000 doses. Price rises from $50 to $60.

\[ \text{PED}=0\;( \text{perfectly inelastic}) \] \[ TR_{0}=50\times1{,}000 = \$50{,}000 \] \[ TR_{1}=60\times1{,}000 = \$60{,}000 \]

Revenue rises proportionally with price because quantity cannot change.

9. Significance of PED for Decision‑Making

10. Key Take‑aways

• |PED| > 1 (elastic): price cuts increase both consumer expenditure and firm revenue; price rises reduce them.
• |PED| < 1 (inelastic): price cuts reduce consumer expenditure and firm revenue; price rises increase them.
• |PED| = 1 (unit‑elastic): total expenditure and revenue are unchanged by price movements.
• |PED| = 0 (perfectly inelastic) and |PED| → ∞ (perfectly elastic) represent the two extreme cases required by the syllabus.
• Determinants – close substitutes, income share, nature of the good, and time‑period – explain why different markets exhibit different elasticities.
• Understanding PED equips households, businesses, workers and policymakers to make informed choices about pricing, taxation, wage setting and resource allocation.