Question 5 (a)

Laspeyres, Paasche and Fisher price indices from base prices and expenditures, and the ideal index

9 marks

Time to write: About 18 minutes
Topic: Index numbers

Question

Question 5 (a) · 9 marks

In a survey related to inflation, a national agency collected the following data for five commodities:

Item	Price in Base Year	Expenditure in Base Year	Expenditure in Current Year	Quantity in Current Year
A	15	1500	1530	90
B	18	1260	1520	80
C	12	960	990	90
D	10	1100	1200	100
E	20	1000	1080	60

Find out the price index by:

Laspeyres's Method,
The method of Paasche,
Fisher's Index Method.
Of the three methods above, which one is the ideal index, and why?

Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

18 minfor 9 marks

Read from the question

three methodsThe question fixes the number: three methods. Give that many, each under its own heading.
Find…Which…?Two asks in one question: answer both. Where it asks why, give reasons, not a description.
Find outFormula first, then the working, then the answer with its unit (₹, %, units).

Write it in this order

Read the data; note exactly what is asked 1 min
1 Base quantities and current prices from the expenditures 2 marks 4 min
2 Laspeyres' index 2 marks 3 min
3 Paasche's index 2 marks 3 min
4 Fisher's index 1½ marks 3 min
5 The ideal index and the reasons 1½ marks 3 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

12345

Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Finding the missing prices and quantities. Expenditure is price × quantity. The base-year quantity therefore comes from base-year expenditure ÷ base-year price, q₀ = (p₀q₀) / p₀, and the current-year price from current-year expenditure ÷ current-year quantity, p₁ = (p₁q₁) / q₁. For item A: q₀ = 1,500 / 15 = 100 and p₁ = 1,530 / 90 = 17. The formulae used are those of Laspeyres (base-year quantities as weights), Paasche (current-year quantities as weights) and Fisher (the geometric mean of the two); they are the three methods the syllabus prescribes.

Working table

Item	p₀	q₀ = p₀q₀ ÷ p₀	p₁ = p₁q₁ ÷ q₁	q₁	p₀q₀	p₁q₀	p₀q₁	p₁q₁
A	15	100	17	90	1,500	1,700	1,350	1,530
B	18	70	19	80	1,260	1,330	1,440	1,520
C	12	80	11	90	960	880	1,080	990
D	10	110	12	100	1,100	1,320	1,000	1,200
E	20	50	18	60	1,000	900	1,200	1,080
Total					5,820	6,130	6,070	6,320

(i) Laspeyres' price index

P₀₁ (L) = Σp₁q₀ / Σp₀q₀ × 100 = 6,130 / 5,820 × 100 = 105.33.

The base-year basket would cost 5.33% more at current prices.

(ii) Paasche's price index

P₀₁ (P) = Σp₁q₁ / Σp₀q₁ × 100 = 6,320 / 6,070 × 100 = 104.12.

The current-year basket costs 4.12% more than it would have at base-year prices.

(iii) Fisher's price index

P₀₁ (F) = √[L × P] = √[(6,130 / 5,820) × (6,320 / 6,070)] × 100 = √(105.33 × 104.12) = 104.72.

Prices have risen by about 4.72% on Fisher's measure, which lies between the other two, as a geometric mean of them must.

(iv) The ideal index

Fisher's index is the ideal index. The reasons:

It satisfies the time reversal test: the index of the current year on the base year and the index of the base year on the current year multiply to one, so the formula gives consistent results whichever way in time it is worked. Laspeyres' and Paasche's indices do not satisfy this test.
It satisfies the factor reversal test: Fisher's price index multiplied by Fisher's quantity index equals the value index, Σp₁q₁ / Σp₀q₀. Because it meets both tests, it is called the ideal index.
It uses the quantities of both years. Laspeyres' index keeps the old basket and tends to overstate a price rise, since buyers shift away from items that became dearer; Paasche's index uses the new basket and tends to understate it. Taking their geometric mean balances the two biases, as the figures here show (105.33 and 104.12 against 104.72).

(The DU guideline minutes keep the tests of adequacy of index numbers to theory only, so they are stated here and not worked numerically.)

Where the 9 marks sit

#	Component	Marks
1	Base quantities and current prices from the expenditures	2
2	Laspeyres' index	2
3	Paasche's index	2
4	Fisher's index	1.5
5	The ideal index and the reasons	1.5
	Total	9

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