relationship between price elasticity of demand and total expenditure on a product

1. Demand and Supply – Foundations (AS‑Level)

1.1 Demand

Demand curve: shows the quantity of a good that consumers are willing and able to buy at each price, ceteris paribus. It is derived from the consumer’s utility‑maximising behaviour (marginal utility per dollar is equalised across all goods).

Determinants of demand (shift the whole curve):

• Consumer income
• Tastes and preferences
• Expectations of future price or income
• Number of buyers
• Prices of related goods (substitutes and complements)

Own price causes a movement along the curve, not a shift.

1.2 Supply

Supply curve: shows the quantity of a good that producers are willing and able to sell at each price, ceteris paribus. It reflects the profit‑maximising rule where marginal cost (MC) equals price (P).

Determinants of supply (shift the whole curve):

• Input (factor) prices
• Technology
• Number of sellers
• Expectations of future price
• Taxes, subsidies and regulations

1.3 Price Elasticity of Supply (ES)

Definition: the responsiveness of quantity supplied to a change in price.

Formulae

Type	Formula
Point elasticity of supply	\(E_{s}= \dfrac{dQ}{dP}\times\dfrac{P}{Q}\)
Arc (mid‑point) elasticity of supply	\(E_{s}^{\text{arc}}= \dfrac{\Delta Q}{\Delta P}\times\dfrac{\bar P}{\bar Q}\) where \(\bar P=\dfrac{P_{1}+P_{2}}{2}\) and \(\bar Q=\dfrac{Q_{1}+Q_{2}}{2}\)

Determinants of ES

• Time period (short‑run vs. long‑run)
• Availability of spare capacity
• Flexibility of inputs (e.g., labour, raw materials)
• Mobility of factors of production

In the long run most supply curves are more elastic because firms can adjust plant size and entry/exit is possible.

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2. Elasticities of Demand (AS‑Level)

2.1 Definitions & Formulae

Elasticity	Point formula	Arc (mid‑point) formula
Price elasticity of demand (PED)	\(E_{p}= \dfrac{dQ}{dP}\times\dfrac{P}{Q}\)	\(E_{p}^{\text{arc}}= \dfrac{\Delta Q}{\Delta P}\times\dfrac{\bar P}{\bar Q}\)
Income elasticity of demand (YED)	\(E_{y}= \dfrac{dQ}{dY}\times\dfrac{Y}{Q}\)	\(E_{y}^{\text{arc}}= \dfrac{\Delta Q}{\Delta Y}\times\dfrac{\bar Y}{\bar Q}\)
Cross‑price elasticity of demand (XED)	\(E_{x}= \dfrac{dQ^{A}}{dP^{B}}\times\dfrac{P^{B}}{Q^{A}}\)	\(E_{x}^{\text{arc}}= \dfrac{\Delta Q^{A}}{\Delta P^{B}}\times\dfrac{\bar P^{B}}{\bar Q^{A}}\)

When to use arc elasticity: when the price change is large (generally >5 %) or when only two discrete data points are available. Point elasticity is appropriate for infinitesimally small changes and for calculus‑based analysis.

2.2 Sign Conventions

• PED is negative (law of demand). In practice we refer to the absolute value \(|E_{p}|\).
• YED is positive for normal goods, negative for inferior goods.
• XED is positive for substitutes, negative for complements, zero for unrelated goods.

2.3 Interpreting the Magnitude of PED

• Elastic demand \(|E_{p}|>1\) – quantity changes proportionally more than price.
• Inelastic demand \(|E_{p}|<1\) – quantity changes proportionally less than price.
• Unitary elastic \(|E_{p}|=1\) – proportional change in quantity equals the proportional change in price.
• Perfectly elastic \(|E_{p}|=\infty\) – horizontal demand curve; any price rise eliminates demand.
• Perfectly inelastic \(|E_{p}|=0\) – vertical demand curve; quantity demanded does not change when price changes.

2.4 Variation of PED Along a Linear Demand Curve

For a straight‑line demand function \(Q = a - bP\):

• Upper (high‑price) segment: \(|E_{p}|>1\) (elastic).
• Mid‑point: \(|E_{p}|=1\) (unitary).
• Lower (low‑price) segment: \(|E_{p}|<1\) (inelastic).

Remember the exam tip: “elasticity falls as you move down a linear demand curve”.

2.5 Quick Reference – Real‑World Examples

Elasticity Category	Typical Example
Very elastic (|E| ≫ 1)	Luxury watches, designer clothing
Elastic (|E| > 1)	Restaurant meals, cinema tickets
Unitary (|E| = 1)	Many agricultural products over a short range
Inelastic (|E| < 1)	Petrol, basic utilities (water, electricity)
Very inelastic (|E| ≈ 0)	Life‑saving medicines, essential insulin

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3. Relationship Between Price Elasticity of Demand and Total Expenditure (Total Revenue) (AS‑Level)

3.1 Definition of Total Expenditure (TE)

\[ \text{TE}=P \times Q \]

TE is the amount of money consumers spend on a good. Whether TE rises or falls after a price change depends on the price elasticity of demand.

3.2 The “PED ↔ TE” Rules

Elasticity Category	Effect of a Price rise	Effect of a Price fall
Elastic \(|E_{p}|>1\)	Quantity falls **more** than price rises → TE falls	Quantity rises **more** than price falls → TE rises
Inelastic \(|E_{p}|<1\)	Quantity falls **less** than price rises → TE rises	Quantity rises **less** than price falls → TE falls
Unitary \(|E_{p}|=1\)	Percentage change in Q exactly offsets price change → TE unchanged	Same as above → TE unchanged

3.3 Limitations of the PED‑TE Rule

• Assumes all other factors (income, tastes, prices of related goods) remain constant (ceteris paribus).
• Ignores income effects that may arise when a price change also changes real purchasing power.
• Applies only to short‑run movements unless the elasticity used already reflects long‑run behaviour.

3.4 Numerical Example (Arc Elasticity)

Price of a video game falls from £60 to £54 (a 10 % fall). Quantity demanded rises from 1 000 to 1 300 units.

• Arc PED: \(\displaystyle E_{p}^{\text{arc}}= \frac{\Delta Q}{\Delta P}\times\frac{\bar P}{\bar Q} =\frac{300}{-6}\times\frac{57}{1150}\approx -2.6\) (elastic).
• Initial TE = £60 × 1 000 = £60 000.
• New TE = £54 × 1 300 = £70 200.
• Because demand is elastic, TE **increases** after the price fall – exactly as the rule predicts.

3.5 Graphical Illustration

• Elastic segment: after a price rise the TE rectangle is smaller.
• Inelastic segment: after a price rise the TE rectangle is larger.
• For perfectly elastic (horizontal) demand, TE moves in the same direction as price; for perfectly inelastic (vertical) demand, TE also moves in the same direction as price.

3.6 Using the Mid‑point (Arc) Elasticity to Predict TE

When only two price‑quantity points are given, calculate the arc PED with the midpoint formula and then apply the table in 3.2. The midpoint method ensures the elasticity is the same whichever direction the price change is measured.

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4. Link to A‑Level Extensions (7.1 & 7.2, 2.5)

4.1 Utility, Marginal Utility & Indifference Curves

• Utility is the satisfaction derived from consuming a good. Marginal utility (MU) is the extra satisfaction from one more unit.
• The consumer’s equilibrium condition:
  \(\displaystyle \frac{MU_{x}}{P_{x}} = \frac{MU_{y}}{P_{y}}\) (equal marginal utility per pound).
• Indifference curves and budget lines illustrate how changes in price affect the optimal consumption bundle – the graphical basis for deriving demand curves.

4.2 Consumer & Producer Surplus

• Consumer surplus (CS): area between the demand curve and the price line up to the quantity purchased.
• Producer surplus (PS): area between the supply curve and the price line up to the quantity sold.
• Elasticity influences the size of CS and PS: a more elastic demand leads to a larger loss of CS when price rises, and vice‑versa.

4.3 Joint and Derived Demand

• Joint demand: two goods are used together (e.g., cars and petrol). A price change in one affects the demand for the other – captured by cross‑price elasticity.
• Derived demand: demand for a factor of production is derived from the demand for the final product (e.g., steel for cars).

4.4 Preview of Market Failure & Government Intervention (A‑Level 8)

Understanding elasticity is essential when evaluating taxes, subsidies, price controls, and externalities because the welfare impact depends on how quantity responds to price changes.

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5. Summary for Examination

5.1 AS‑Level Checklist

1. Define and draw demand and supply curves; list all determinants.
2. State and apply the formulas for point and arc PED, YED, XED.
3. Explain sign conventions and interpret the magnitude of each elasticity.
4. Describe how PED varies along a linear demand curve.
5. Define price elasticity of supply, give its formulae, and list its determinants.
6. Apply the “PED ↔ TE” rules, noting the limitations.
7. Calculate arc PED from two price‑quantity points and predict the effect on total expenditure.
8. Be able to sketch:
   • Linear demand curve with elastic, unitary and inelastic sections.
   • TE rectangles before and after a price change for both elastic and inelastic cases.
   • Horizontal (perfectly elastic) and vertical (perfectly inelastic) demand curves.
   • Supply curve and a brief diagram showing a more elastic supply in the long run.
9. Practice short calculations: given %ΔP and %ΔQ, compute PED, classify elasticity, and state the expected change in TE.

5.2 A‑Level Extension Checklist (for 7.1, 7.2 & 2.5)

1. Explain marginal utility and the equal‑MU‑per‑dollar rule.
2. Draw and interpret indifference curves and budget constraints.
3. Define consumer and producer surplus; illustrate how a price change shifts the CS and PS areas.
4. Discuss how elasticity affects the welfare impact of taxes, subsidies and price controls.
5. Link cross‑price elasticity to joint demand and derived demand concepts.
6. Recognise that the concepts covered here form the foundation for later topics on market failure and government intervention.