EconomicsISC Class 11 · Index Numbers

Laspeyres, Paasche and Fisher Index Numbers (ISC Class 11 Economics)

Updated 17 October 2026 · Fact-checked

Weighted index numbers give each item importance in proportion to the quantity consumed. Laspeyres uses base-year quantities as weights: P01 = Sum P1Q0 / Sum P0Q0 x 100. Paasche uses current-year quantities: Sum P1Q1 / Sum P0Q1 x 100. Fisher's ideal index is the geometric mean of the two: square root of (Laspeyres x Paasche). WPI, CPI and IIP are important official indices.

Notice that Laspeyres leans high and Paasche leans low, so Fisher's geometric mean of the two balances both biases.

Illustration drawn for QuestionBuddy.

Understand Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP

Why weights? Items differ in importance (rice vs salt); weighted indices give each item importance according to its quantity or expenditure.

Laspeyres index. Weights: base-year quantities (Q0). P01(L) = Sum P1Q0 / Sum P0Q0 x 100. Merit: needs quantities of one year only, so comparisons are easy. Limit: tends to overstate price rise (it ignores substitution to cheaper goods).

Paasche index. Weights: current-year quantities (Q1). P01(P) = Sum P1Q1 / Sum P0Q1 x 100. Merit: reflects current consumption. Limit: needs new quantities every year, and tends to understate.

Fisher's ideal index. P01(F) = square root of [P01(L) x P01(P)]. Called ideal because it uses both base and current quantities, satisfies the time reversal and factor reversal tests, and balances the biases of L and P.

Official indices (meaning). *WPI (Wholesale Price Index):* measures changes in wholesale prices of goods. *CPI (Consumer Price Index):* measures changes in retail prices of a basket for consumers; used for cost of living and inflation; *CPI-IW* for industrial workers. *IIP (Index of Industrial Production):* a quantity index of output of industries. Base years are revised from time to time.

Problems in construction. Selecting items for the basket, choice of the base year (it should be normal), obtaining correct prices, choosing weights, choice of the average, and changes in quality and consumption patterns.

Uses and limitations. Measure inflation, adjust wages, deflate national income. Limitations: sampling errors, average may not match any one person's consumption, and quality changes are hard to adjust.

Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP: key facts

Laspeyres: P01(L) = Sum P1Q0 / Sum P0Q0 x 100 | Base-year quantities; Often overstates.
Paasche: P01(P) = Sum P1Q1 / Sum P0Q1 x 100 | Current-year quantities; Often understates.
Fisher: P01(F) = sqrt[P01(L) x P01(P)] | Geometric mean of L and P; Ideal index.

How to answer Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP questions

Build a table with the four products P0Q0, P1Q0, P0Q1, P1Q1.

1Compute P0Q0, P1Q0, P0Q1 and P1Q1 for each item.
2Total each column.
3Laspeyres = Sum P1Q0 / Sum P0Q0 x 100; Paasche = Sum P1Q1 / Sum P0Q1 x 100.
4Fisher = square root of their product; interpret.

Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Substitution

Suppose the price of apples rises sharply and a family buys fewer apples and more bananas.

Which index overstates the cost-of-living rise and why?

Try it first, then show the solutionSolution

Laspeyres keeps the base-year (larger) apple quantity as the weight.
So it gives too much weight to the dear item and overstates the rise.

Answer: Laspeyres overstates; Paasche, with the current quantities, understates.

Memory hook: L uses old weights, P uses new.

Tricks to remember Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP

Short memory aids. Say each one aloud once, then use it on a question the same day.

L = Last (base) quantities
Laspeyres weights are base-year quantities Q0.
Use it when: Laspeyres.
P = Present quantities
Paasche uses current quantities Q1.
Use it when: Paasche.
Fisher = between
Fisher lies between L and P.
Use it when: Check.

Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Weights · 1 mark · MCQBoard level

In Laspeyres' index the weights used are:
(a) current-year quantities (b) base-year quantities (c) average prices (d) no weights

Try it first, then show the solutionSolution

Base-year quantities Q0.

Answer: (b) base-year quantities

Example 2: Official indices · 3 marks · Short answerBoard level

Distinguish between WPI, CPI and IIP in one line each. (3 marks)

Try it first, then show the solutionSolution

WPI: wholesale prices of goods.
CPI: retail prices paid by consumers.
IIP: quantity index of industrial output.

Answer: WPI measures wholesale price changes; CPI measures retail price changes for consumers; IIP measures change in industrial output.

Example 3: Three indices · 8 marks · Long answerHarder board level

Item A: P0 = 10, Q0 = 20, P1 = 12, Q1 = 25. Item B: P0 = 20, Q0 = 10, P1 = 25, Q1 = 8. Item C: P0 = 5, Q0 = 40, P1 = 6, Q1 = 50. Find the Laspeyres, Paasche and Fisher price indices. (8 marks)

Try it first, then show the solutionSolution

P0Q0: 200, 200, 200; total 600. P1Q0: 240, 250, 240; total 730.
P1Q1: 300, 200, 300; total 800. P0Q1: 250, 160, 250; total 660.
Laspeyres = 730 / 600 x 100 = 121.67.
Paasche = 800 / 660 x 100 = 121.21.
Fisher = sqrt(121.67 x 121.21) = sqrt(14,747.5) = 121.44.

Answer: Laspeyres 121.67; Paasche 121.21; Fisher 121.44 (between the two).

Example 4: Cost of living · 5 marks · Long answerHarder board level

A family spent Rs 6,000 a month at base prices on its base-year basket. At current prices the same basket costs Rs 7,500; the current basket costs Rs 7,920 at current prices and Rs 6,400 at base prices. Find the Laspeyres and Paasche indices and the extra monthly income needed to maintain the base-year standard of living. (5 marks)

Try it first, then show the solutionSolution

Laspeyres = 7,500 / 6,000 x 100 = 125.
Paasche = 7,920 / 6,400 x 100 = 123.75.
Extra income to buy the base basket = 7,500 - 6,000 = Rs 1,500 a month (25% more).
Paasche is lower because the family has shifted to cheaper goods.

Answer: Laspeyres 125; Paasche 123.75; Rs 1,500 more is needed to buy the base-year basket.

Example 5: Evaluate the index · 4 marks · Short answerHOTS / competency

The government is deciding whether to link pensions to a Laspeyres or a Fisher index. Using the figures L = 125, P = 123.75 and F = sqrt(125 x 123.75), find F and discuss which is fairer to pensioners and to the budget. (4 marks)

Try it first, then show the solutionSolution

F = sqrt(15,468.75) = 124.37.
Laspeyres is the highest (favours pensioners, burdens the budget); Paasche is lowest.
Fisher lies between and removes bias, so it is the fairest compromise.

Answer: F = 124.37; Fisher is a balanced choice between L = 125 and P = 123.75.

Where students lose marks in Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP

The slips examiners see most often, and what to write instead.

Mistake: Swapping the quantities
L and P confusion.
Wrong working
Laspeyres with Q1
Right working
Laspeyres with Q0
Do this: L = Last/base quantities.
Usually costs: 2-3 marks.
Mistake: Arithmetic mean for Fisher
Averaging mistake.
Wrong working
(121.67 + 121.21) / 2
Right working
sqrt(121.67 x 121.21)
Do this: Geometric mean.
Usually costs: 2 marks.
Mistake: Forgetting x100
Index is a percentage.
Wrong working
730 / 600 = 1.217
Right working
121.7
Do this: Multiply by 100.
Usually costs: 1 mark.

Exam tips

Always draw the four-column table.
Check that Fisher lies between L and P.
Write the formulas before substituting.

Practice questions on Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Fisher's index is the:
(a) arithmetic mean of L and P (b) geometric mean of L and P (c) median of L and P (d) harmonic mean of price
Hint
Square root of the product.
Answer
(b) geometric mean of L and P
Q2 · 4 marks · Short answerBoard level
State two merits and two limitations of Laspeyres' index. (4 marks)
Hint
One year's quantities; substitution bias.
Answer
Merits: requires quantities of the base year only; easy comparison between years. Limitations: overstates price rise because it ignores shifts to cheaper goods; the base-year weights become out of date.
Q3 · 7 marks · Long answerHarder board level
Item X: P0 = 8, Q0 = 30, P1 = 10, Q1 = 25. Item Y: P0 = 15, Q0 = 20, P1 = 18, Q1 = 25. Item Z: P0 = 4, Q0 = 50, P1 = 5, Q1 = 40. Find the Laspeyres, Paasche and Fisher indices. (7 marks)
Hint
P0Q0: 240, 300, 200.
Answer
P0Q0 = 240 + 300 + 200 = 740. P1Q0 = 300 + 360 + 250 = 910. L = 910 / 740 x 100 = 122.97. P1Q1 = 250 + 450 + 200 = 900. P0Q1 = 200 + 375 + 160 = 735. P = 900 / 735 x 100 = 122.45. F = sqrt(122.97 x 122.45) = 122.71.
Q4 · 4 marks · Short answerHarder board level
The Laspeyres index of a city is 140 and the Paasche index is 126. Find the Fisher index, the implied substitution effect and say what a Rs 40,000 monthly salary (base-year standard) must become to keep the same standard using Laspeyres. (4 marks)
Hint
sqrt(140 x 126).
Answer
F = sqrt(17,640) = 132.82. L is 14 points above P, indicating consumers shifted to cheaper goods. Salary needed = 40,000 x 140 / 100 = Rs 56,000.
Q5 · 3 marks · Short answerHOTS / competency
A newspaper reports that the CPI rose 6% but a retired couple that spends mostly on medicines and rent says their costs rose 11%. Explain the difference in terms of weights and basket, and suggest a better index for them. (3 marks)
Hint
Different baskets.
Answer
The CPI uses the average household basket; the couple's basket has a larger share of medicines and rent, which rose faster, so their personal inflation is higher. A group-specific index with their weights would suit them better.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Laspeyres weights?Show answer
Base-year quantities.
2.Paasche weights?Show answer
Current-year quantities.
3.Fisher is?Show answer
Geometric mean of L and P.
4.IIP measures?Show answer
Industrial output.
5.CPI measures?Show answer
Retail prices for consumers.

Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP: frequently asked questions

Which is bigger usually?

Laspeyres is usually higher.

Is Fisher always between L and P?

Yes.

Why revise base years?

To keep weights relevant.

Who publishes CPI?

Government statistical authorities; check the current source.