The elasticity answer key that split a department
Mr Abdulai teaches Economics to 48 S2 students at a public senior high school in Tamale. His end-of-term test included this item: 'The price of a bag of maize rises from GH¢400 to GH¢500 and quantity demanded at the market falls from 200 bags to 160 bags per week. Calculate the price elasticity of demand.' The marking scheme reads: (40 ÷ 200) ÷ (100 ÷ 400) = 20% ÷ 25% = 0.8, therefore demand is inelastic, 2 marks. Fifteen students used average (midpoint) values instead, obtained 1.0, and concluded 'unitary'. One student, Fati, wrote 0.8 but explained that '0.8 is less than one, so buyers do not respond to price at all'. The class average was 38 per cent. A parent has written to the headmaster demanding that 1.0 be accepted because 'another textbook uses averages'. The head of department, who is not an economist, tells Mr Abdulai to 'just add one mark to everybody so the complaint goes away'.
1. Show the full working for the initial-value and midpoint methods. Is the students' answer of 1.0 defensible, and how should Mr Abdulai rule on the parent's complaint?
Read the worked answer
Initial-value method (as in the key)
- %ΔQ = (160 − 200) ÷ 200 = −20%
- %ΔP = (500 − 400) ÷ 400 = +25%
- |Ed| = 20 ÷ 25 = 0.8 → inelastic
Midpoint (arc) method
- %ΔQ = 40 ÷ average of 200 and 160 (= 180) = 22.2%
- %ΔP = 100 ÷ average of 400 and 500 (= 450) = 22.2%
- |Ed| = 22.2 ÷ 22.2 = 1.0 → unitary
Both are arithmetically correct; they are different conventions, not different economics. The 1.0 answer is defensible only if the midpoint formula was taught and used in class for this topic.
Ruling
- Check the scheme of work, the class exercise book and the approved textbook to establish which method was taught.
- If midpoint was taught, the marking scheme is faulty: award full marks to students who showed correct midpoint working and correct interpretation, and correct the scheme for the future.
- If only the initial-value method was taught, do not award the mark - but state in writing that the item was poorly designed because it permits two valid conventions with different conclusions.
- Do not apply a blanket 'one mark for everybody'. That destroys the item's discrimination, rewards students who wrote nothing, and is unfair to the 15 students who used the taught method correctly.
Also note the deeper problem: with these numbers the classification flips between 'inelastic' and 'unitary', so the item cannot bear high-stakes weight whatever the ruling.
2. Diagnose Fati's misunderstanding and describe a short intervention that would move her from 'inelastic means no response' to a correct interpretation.
Read the worked answer
Diagnosis. Fati has collapsed a continuum into a binary. She has learned 'elastic = responds, inelastic = does not respond' and has no quantitative anchor for what a coefficient means. The error is conceptual, not computational - she calculated 0.8 correctly.
Intervention (10–12 minutes, small group of students with the same error)
- Translate the number into behaviour. 'A coefficient of 0.8 means that a 25% price rise cuts quantity demanded by about 20%. Did buyers respond? Yes - by 20%. Did they respond proportionally more than the price change? No.'
- Use a sharp contrast. Give a second case with Ed = 0.2: price up 10%, quantity down 2%. Ask: 'Is this zero response?' Then Ed = 3: price up 10%, quantity down 30%. Have her rank the three cases from least to most responsive and justify the ranking in words, not numbers only.
- Name the limiting cases. Perfectly inelastic (Ed = 0) is a vertical demand curve - e.g. insulin for a diabetic patient at any price. Perfectly elastic (Ed = ∞) is horizontal. Inelastic is the wide region between 0 and 1, not the point 0.
- Diagnostic exit question. 'A trader raises the price of sachet water by 10% and sales fall by 4%. Calculate Ed, classify it, and write one sentence a market trader would understand about her customers.'
- Follow-up. Re-test the concept in a different context (farm inputs, mobile data) within a week so she transfers rather than memorises.
3. Rewrite the test item and its marking scheme so that it is fair, unambiguous and tests interpretation as well as calculation.
Read the worked answer
Problems with the original item
- It does not specify the formula, so two conventions are legitimate.
- The numbers happen to sit exactly on the boundary, so the classification is unstable.
- It rewards a calculation only; the interpretation (what the coefficient means for revenue and for the trader) is untested.
- 'Calculate' with 2 marks gives no credit for method even when the final figure is wrong.
Rewritten item 'The price of a bag of maize rises from GH¢400 to GH¢440 and quantity demanded falls from 500 to 450 bags per week. (a) Using the initial-value formula Ed = (%ΔQ) ÷ (%ΔP), calculate the price elasticity of demand. Show your working. (b) State whether demand is elastic, inelastic or unitary, and explain in one sentence what this means for the total revenue of maize traders when they raise price. (c) Give one reason why demand for maize might be inelastic.'
Marking scheme
- (a) %ΔQ = −50/500 = −10%; %ΔP = 40/400 = +10%; Ed = −1.0 (3 marks: 1 for each percentage change, 1 for the ratio; award method marks even if arithmetic slips).
- (b) Unitary; total revenue unchanged because the percentage fall in quantity equals the percentage rise in price (2 marks: 1 classification, 1 revenue reasoning).
- (c) Any sound reason - staple food with few close substitutes, small share of budget, habit, no immediate alternative (1 mark).
Design principles applied: the formula is stated; numbers are away from the boundary (Ed = 1.0 is deliberate and unambiguous because the percentages are equal and exact); interpretation and economic reasoning carry their own marks; method marks protect students who make a slip.