calculation of price, income and promotional elasticity of demand

Marketing Analysis – Elasticity

Objective: Calculate and interpret price, income and advertising (promotional) elasticity of demand, and use these measures to inform pricing, budgeting and product‑line decisions.

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1. What is Elasticity?

• Elasticity measures the responsiveness of quantity demanded (Q_d) to a change in another variable, holding all other factors constant (ceteris paribus).
• It is a core analytical tool in the Cambridge IGCSE/A‑Level syllabus (9609 – Section 8.1.1).

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2. Core Elasticities (Mid‑point / Arc Method)

Elasticity	Variable that Changes	Formula (mid‑point)
Price Elasticity of Demand (PED)	Price (P)	\[ E_{p}= \frac{\displaystyle\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\displaystyle\frac{P_{2}-P_{1}}{(P_{1}+P_{2})/2}} \]
Income Elasticity of Demand (YED)	Consumer Income (Y)	\[ E_{y}= \frac{\displaystyle\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\displaystyle\frac{Y_{2}-Y_{1}}{(Y_{1}+Y_{2})/2}} \]
Advertising (Promotional) Elasticity of Demand (AED)	Advertising Expenditure (A)	\[ E_{a}= \frac{\displaystyle\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\displaystyle\frac{A_{2}-A_{1}}{(A_{1}+A_{2})/2}} \]
Cross‑Price Elasticity of Demand (XED)	Price of a related good (Pr)	\[ E_{x}= \frac{\displaystyle\frac{Q_{A2}-Q_{A1}}{(Q_{A1}+Q_{A2})/2}}{\displaystyle\frac{P_{B2}-P_{B1}}{(P_{B1}+P_{B2})/2}} \]

Key Assumptions (Arc Method)

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3. Calculating Percentage (Arc) Changes

1. Record the initial value (subscript 1) and the new value (subscript 2) for the determinant and for Q_d.
2. Apply the midpoint formula for each variable: \[ \% \Delta X = \frac{X_{2}-X_{1}}{(X_{1}+X_{2})/2}\times 100 \] where \(X\) = P, Y, A or Q.
3. Insert the two percentage changes into the appropriate elasticity formula.

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4. Price Elasticity of Demand (PED)

Interpretation

• Elastic (|E_p| > 1): Quantity demanded changes proportionally more than price.
• Inelastic (|E_p| < 1): Quantity demanded changes proportionally less than price.
• Unitary (|E_p| = 1): Proportional change.

Business Implications

• Total‑Revenue Test:
  • If demand is elastic, a price cut increases total revenue; a price rise decreases it.
  • If demand is inelastic, a price rise increases total revenue; a price cut decreases it.
• Pricing Strategies (Cambridge syllabus link):
  • Highly elastic demand → consider penetration pricing or frequent price promotions.
  • Inelastic demand → firms may use price‑skimming, premium pricing, or price increases to maximise profit.
  • Unitary demand → total revenue is unchanged by price moves; focus may shift to cost control or product differentiation.

Worked Example – Discount Promotion

A sports‑wear brand is thinking about a 5 % discount on a sneaker.

	Before	After
Price (P)	£100	£95
Quantity demanded (Qd)	4,000 pairs	4,800 pairs

\[ \% \Delta P = \frac{95-100}{(100+95)/2}\times100 = -5.13\% \] \[ \% \Delta Q_{d} = \frac{4,800-4,000}{(4,000+4,800)/2}\times100 = 18.18\% \] \[ E_{p}= \frac{18.18\%}{-5.13\%}= -3.54 \] Because \(|-3.54|>1\), demand is elastic. The discount is likely to raise total revenue, supporting the promotion.

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5. Income Elasticity of Demand (YED)

Interpretation

• Normal good (E_y > 0): Demand rises when income rises.
• Inferior good (E_y < 0): Demand falls when income rises.
• Magnitude:
  • Luxury good: E_y > 1
  • Necessity: 0 < E_y < 1

Business Implications

• Forecast demand under economic growth (positive YED) or recession (negative YED).
• Guide product‑mix decisions – e.g., expand premium lines when YED > 1, or protect market share of inferior goods during downturns.

Worked Example – Coffee Shop

	Before	After
Average weekly disposable income (Y)	£400	£440
Weekly cups sold (Qd)	1,200	1,380

\[ \% \Delta Y = \frac{440-400}{(400+440)/2}\times100 = 9.76\% \] \[ \% \Delta Q_{d} = \frac{1,380-1,200}{(1,200+1,380)/2}\times100 = 15.38\% \] \[ E_{y}= \frac{15.38\%}{9.76\%}= 1.58 \] Positive and > 1 → coffee behaves as a luxury good for this market, suggesting the shop could introduce premium blends.

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6. Advertising (Promotional) Elasticity of Demand (AED)

Interpretation

• Positive AED: Advertising increases demand.
• Negative AED: Advertising reduces demand (e.g., negative publicity).
• The larger the absolute value, the more effective the advertising spend.

Business Implications

• Estimate the optimal advertising budget by comparing the marginal revenue from additional sales with the marginal cost of extra advertising.
• Remember the time lag between ad spend and sales response – an exam‑level requirement.

Worked Example – TV Campaign

Month 1: £20,000 ad spend, 5,000 units sold.
Month 2: £30,000 ad spend, 6,300 units sold.

\[ \% \Delta A = \frac{30,000-20,000}{(20,000+30,000)/2}\times100 = 40.00\% \] \[ \% \Delta Q_{d} = \frac{6,300-5,000}{(5,000+6,300)/2}\times100 = 20.00\% \] \[ E_{a}= \frac{20.00\%}{40.00\%}= 0.50 \] A coefficient of 0.5 means a 1 % rise in ad spend yields a 0.5 % rise in sales – modest but positive. The firm must weigh this against the extra cost and the expected lag before sales materialise.

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7. Cross‑Price Elasticity of Demand (XED) – Brief Understanding + Worked Example

Interpretation

• Positive XED: The two goods are substitutes (e.g., tea vs. coffee).
• Negative XED: The two goods are complements (e.g., printers vs. ink cartridges).
• The larger the absolute value, the stronger the relationship.

Worked Example – Coffee vs. Tea

When the price of tea rises from £2.00 to £2.40 per cup, coffee sales increase from 1,800 to 2,100 cups per week.

	Before	After
Price of tea (Ptea)	£2.00	£2.40
Coffee sales (Qcoffee)	1,800	2,100

\[ \% \Delta P_{tea}= \frac{2.40-2.00}{(2.00+2.40)/2}\times100 = 18.18\% \] \[ \% \Delta Q_{coffee}= \frac{2,100-1,800}{(1,800+2,100)/2}\times100 = 16.67\% \] \[ E_{x}= \frac{16.67\%}{18.18\%}= 0.92 \] The positive value indicates coffee and tea are substitutes, but |E_x| < 1 shows the substitution effect is relatively weak.

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8. Short‑Run vs. Long‑Run Elasticities

• Short‑run: Consumers have limited ability to change habits, find substitutes or adjust income allocations, so elasticity is usually lower.
• Long‑run: More time allows for behavioural change, product redesign, or entry of new substitutes; therefore, elasticities (especially PED and YED) tend to be higher.
• Examiners often ask you to state whether the calculated elasticity is more likely a short‑run or long‑run figure and why.

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9. Limitations of Elasticity Analysis

• Ceteris paribus assumption: In reality other factors (taste, competitor actions, seasonality) may change simultaneously.
• Data quality & time‑frame: Elasticities differ between short‑run and long‑run; outdated or inaccurate data give misleading results.
• Aggregate vs. individual demand: Market‑wide elasticities may not apply to a specific segment or niche.
• Linear (arc) approximation: The midpoint method assumes a constant elasticity over the interval; large changes can distort the estimate.

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10. Quick Reference Checklist for Exam Questions

1. Identify the required elasticity (PED, YED, AED, or XED).
2. Record initial (subscript 1) and new (subscript 2) values for the determinant and for Q_d.
3. Calculate % changes using the midpoint (arc) formula.
4. Insert the percentages into the correct elasticity formula.
5. Interpret the sign and magnitude:
   • Elastic vs. inelastic (|E| > 1 or < 1)
   • Normal vs. inferior (YED) or substitute vs. complement (XED)
   • Positive vs. negative (AED)
6. State the relevant business implication (pricing, revenue, advertising budget, product‑mix).
7. Note any assumptions or limitations (ceteris paribus, short‑run/long‑run, time lag, arc‑method constancy).

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