This MCQ module is based on: Index Number Construction — Laspeyres, Paasche, Fisher
Index Number Construction — Laspeyres, Paasche, Fisher
This assessment will be based on: Index Number Construction — Laspeyres, Paasche, Fisher
Upload images, PDFs, or Word documents to include their content in assessment generation.
Index Numbers — Construction, Laspeyres, Paasche and Fisher Formulas
Imagine Rabi returning home from the bazaar. Tomatoes have become dearer, dal has become cheaper, soap has stayed the same and sugar has shot up. He rattles off twelve different price changes to his father; both end up confused. The industrial economy presents the same puzzle on a vastly bigger scale — coal output rising, steel falling, cement steady, sugar mills slowing down. Can a single figure summarise dozens or hundreds of changes at once? That is the job of an index number. Part 1 introduces the meaning of an index, walks through the simple aggregative method, the simple average of price relatives, and then the three weighted heavyweights — Laspeyres, Paasche and Fisher's ideal — using the worked NCERT examples step by step.
7.1 From Many Changes to One Number
You have already learned, in earlier chapters, how a single mean or median can summarise a mass of values for one variable. An index number stretches that idea to a group of related variables changing simultaneously. NCERT motivates the topic with three small cases: an industrial worker whose salary rose from Rs 1,000 in 1982 to Rs 12,000 today (has his standard of living really risen twelve times?); the BSE Sensex crossing 8,000 points and erasing Rs 1,53,690 crores of investor wealth when it dipped 600 points; and the government's claim that inflation will not accelerate even after a fuel price hike — a claim that only an inflation index can verify.
7.1.1 The Base and the Current Period
Conventionally, an index number is expressed as a percentage. The two situations being compared are called the base period? and the current period?. The base is the benchmark; its index is set, by convention, to 100. The current-period index is then expressed in proportion to that benchmark. An index number of 250 therefore means that the current value is two-and-a-half times the base value. An index of 138.5 means a 38.5 per cent rise; an index of 97.86 (we shall meet it later) means a 2.14 per cent fall.
7.2 Two Families of Construction Methods
NCERT lists two ways of building a price index from the underlying numbers — the aggregative method (add up the prices, then take a ratio) and the method of averaging relatives (compute a price relative for each item, then take the average). Each can be either unweighted (every item counted equally) or weighted (more important items counted more heavily). Combine the two axes and you get four possibilities:
7.3 Method 1 — Simple Aggregative Index
The first and easiest construction adds up the current-period prices, divides by the sum of base-period prices, and multiplies by 100.
Where P₀ is the base-period price of a commodity and P₁ is its current-period price. The subscript "01" reads "from period 0 to period 1".
7.3.1 Worked Example 1 — NCERT Table 7.1
| Commodity | Base price P₀ (Rs) | Current price P₁ (Rs) | Percentage change |
|---|---|---|---|
| A | 2 | 4 | +100% |
| B | 5 | 6 | +20% |
| C | 4 | 5 | +25% |
| D | 2 | 3 | +50% |
| Total | ΣP₀ = 13 | ΣP₁ = 18 | — |
Prices of the basket are said to have risen by 38.5 per cent. Notice that the four percentage changes (+100, +20, +25, +50) have been compressed into a single, communicable number.
7.4 Method 2 — Weighted Aggregative Index
The cure for the equality-of-items problem is to attach a weight to each commodity that reflects its importance in real consumption. The natural choice for weights is the quantity consumed — either in the base period (q₀) or in the current period (q₁). The weighted aggregative index has the form:
where the same set of quantities q appears in both numerator and denominator. The basket of goods is fixed; only the prices change. As the total expenditure on this fixed basket changes, the change must be due to price movement alone — this is the conceptual cleverness of the weighted index. Different choices of q give different formulas, of which three are crucial.
7.4.1 Laspeyres' Price Index
The Laspeyres index?, devised by the German economist Etienne Laspeyres in the 1870s, uses the base-period quantities q₀ as weights. It answers the question: "If the expenditure on the base-period basket was Rs 100, how much would it cost in the current period?"
7.4.2 Paasche's Price Index
The Paasche index?, named after another German economist, Hermann Paasche (1874), uses current-period quantities q₁ as weights. It answers: "If the current-period basket had been consumed in the base period at base prices, how much would Rs 100 of expenditure on it have grown to today?"
7.4.3 Fisher's Ideal Index
Both Laspeyres and Paasche have a flaw — Laspeyres tends to overstate the price rise (because base-period quantities ignore the way consumers shift away from items that have become dearer) while Paasche tends to understate it (because current-period quantities already reflect the shift away). The American economist Irving Fisher proposed an "ideal" formula that takes the geometric mean of the two, balancing the two biases against each other.
7.5 Worked Example 2 — Computing Laspeyres and Paasche
| Commodity | P₀ (Rs) | q₀ | P₁ (Rs) | q₁ |
|---|---|---|---|---|
| A | 2 | 10 | 4 | 5 |
| B | 5 | 12 | 6 | 10 |
| C | 4 | 20 | 5 | 15 |
| D | 2 | 15 | 3 | 10 |
7.5.1 Building the Sums Step by Step
The four key sums are best computed in a working table. The numerator of Laspeyres is ΣP₁q₀; the denominator is ΣP₀q₀. The numerator of Paasche is ΣP₁q₁; the denominator is ΣP₀q₁.
| Commodity | P₀q₀ | P₁q₀ | P₀q₁ | P₁q₁ |
|---|---|---|---|---|
| A | 2×10 = 20 | 4×10 = 40 | 2×5 = 10 | 4×5 = 20 |
| B | 5×12 = 60 | 6×12 = 72 | 5×10 = 50 | 6×10 = 60 |
| C | 4×20 = 80 | 5×20 = 100 | 4×15 = 60 | 5×15 = 75 |
| D | 2×15 = 30 | 3×15 = 45 | 2×10 = 20 | 3×10 = 30 |
| Total | 190 | 257 | 140 | 185 |
7.5.2 Laspeyres — Using Base Quantities
Reading: holding the base-period basket fixed, the price level has risen by 35.3 per cent from period 0 to period 1. If a household used to spend Rs 100 on its 1990 basket, the same basket will cost Rs 135.3 today.
7.5.3 Paasche — Using Current Quantities
Using the current basket as weights, the price rise is 32.1 per cent. Notice that Paasche (132.1) is a little smaller than Laspeyres (135.3) — exactly the substitution-bias gap predicted by theory. Quantities of A and B fell sharply (10→5 and 12→10) precisely because their prices doubled or rose; current weights downplay these expensive items, lowering the index.
7.5.4 Fisher's Ideal — The Geometric Mean
Fisher's index lands between the two, at 33.7 per cent rise — a balanced compromise between the over-statement of Laspeyres and the under-statement of Paasche.
7.6 Method 3 — Simple Average of Price Relatives
Instead of adding up rupees and rupees of different commodities (an apples-and-oranges objection), the second family of methods first standardises each price by computing a price relative — the ratio of current price to base price, multiplied by 100 — and then averages the relatives across commodities.
The simple (unweighted) average of price relatives is an arithmetic mean over n commodities:
7.6.1 NCERT Example — Simple Average on the Same Four Commodities
Reusing the data of Table 7.1, the price relatives are 200%, 120%, 125% and 150% for A, B, C and D respectively. The simple average is:
The unweighted average reports a 49 per cent price rise — much higher than the simple aggregative figure (38.5%). The reason is that the relatives method gives the cheap-but-doubled commodity A as much voice as the expensive-but-only-mildly-rising commodity B. NCERT rounds the figure to 149.
7.7 Method 4 — Weighted Average of Price Relatives
The weighted version of the relatives method takes a weighted arithmetic mean, where each weight Wi reflects the share of expenditure on commodity i (usually in the base period). The formula is:
Weights may be expressed as percentages (in which case ΣW = 100), or as actual rupee value-shares of expenditure. NCERT prefers base-period weights because (a) computing weights every year is impractical and (b) using current weights makes successive indices strictly non-comparable, since the basket itself keeps changing.
7.7.1 NCERT Example 3 — Weighted Price Relatives
| Commodity | Weight (W) % | Base price (Rs) | Current price (Rs) | Price relative R = (P₁/P₀)×100 | WR |
|---|---|---|---|---|---|
| A | 40 | 2 | 4 | 200 | 40×200 = 8000 |
| B | 30 | 5 | 6 | 120 | 30×120 = 3600 |
| C | 20 | 4 | 5 | 125 | 20×125 = 2500 |
| D | 10 | 2 | 3 | 150 | 10×150 = 1500 |
| Total | 100 | — | — | — | 15600 |
The weighted price index is 156, signalling a 56 per cent rise. It is sharply higher than the unweighted figure (148.75), because the most-doubled item, A, also carries the largest weight (40 per cent) — its outsized rise pulls the average up. The contrast between unweighted and weighted figures is precisely the reason policy makers insist on weighting.
Interchange the current-period values with the base-period values in the data given in Example 2 (Table 7.2). Calculate Laspeyres' and Paasche's indices for the reversed data. What difference do you observe from the original calculation?
7.8 Side-by-Side — The Three Weighted Heavyweights
Laspeyres P₀₁L
- Weight: base-period quantity q₀
- Asks: "How much would the 1990 basket cost today?"
- Bias: tends to overstate price rise (no substitution allowed)
- Used by: Indian CPI, US CPI, most national statistics offices
- Easy to update: q₀ is fixed once at base year
Paasche P₀₁P
- Weight: current-period quantity q₁
- Asks: "What did today's basket cost in 1990?"
- Bias: tends to understate price rise (substitution already embedded)
- Used by: theoretical work, GDP deflator (Paasche-style)
- Hard to update: q₁ must be re-collected every period
Fisher's Ideal P₀₁F — The Geometric Mean
- Definition: √(L × P), the geometric mean of Laspeyres and Paasche.
- Why "ideal": satisfies the time-reversal test (index 0→1 = 1/index 1→0) and the factor-reversal test (price index × quantity index = value ratio).
- Bias: averages the over- and under-statements; close to the underlying truth.
- Drawback: requires both q₀ and q₁, so demands more data; rarely used by national agencies for routine inflation reporting.
If Paasche is theoretically just as legitimate as Laspeyres, why does the Reserve Bank of India and the Ministry of Statistics overwhelmingly prefer Laspeyres for the official CPI and WPI series?
For the NCERT Example 2 data, Laspeyres reports 135.3, Paasche 132.1 and Fisher 133.7. Discuss in pairs why all three are simultaneously "correct" yet give different numbers, and which one a policy-maker should quote.
7.9 A Worked Case-Based Question
📋 Case-Based Question — Building the Mini-CPI for a Hostel Mess
7.10 Assertion–Reason Questions
Choose: (A) Both A and R are true and R is the correct explanation of A. (B) Both A and R are true but R is not the correct explanation of A. (C) A is true, R is false. (D) A is false, R is true.
Continue to Part 2 — Some Important Indian Index Numbers (CPI, WPI, IIP, Sensex, HDI), Issues in Construction, NCERT Exercises and Summary.
Frequently Asked Questions — Index Numbers — Construction, Laspeyres, Paasche and Fisher Formulas
What is an index number in NCERT Class 11 Statistics Chapter 7?
An index number is a statistical device that measures relative changes in a variable or a group of related variables across time, geographic locations or other characteristics, with a chosen reference period (base year) set to 100. NCERT Class 11 Statistics Chapter 7 Part 1 explains that index numbers help compare changes in prices, quantities, wages or production over time. Common examples include the Consumer Price Index (CPI), Wholesale Price Index (WPI), Index of Industrial Production (IIP) and Sensex. They are called economic barometers because they summarise complex movements into a single number that can be tracked easily.
What is the formula for Laspeyres price index in Class 11 Statistics?
The Laspeyres price index uses the base-year quantities as weights and is calculated as P01(L) = (Σp1·q0 / Σp0·q0) × 100, where p1 and p0 are current-year and base-year prices respectively, and q0 is the base-year quantity. NCERT Class 11 Statistics Chapter 7 Part 1 explains that Laspeyres is the most widely used formula because it requires only base-year quantity data, which is fixed over time. However, it overstates inflation because it ignores how consumers substitute away from goods whose prices rise — the substitution bias.
What is the difference between Laspeyres and Paasche price index?
Laspeyres uses base-year quantities (q0) as weights, giving P01(L) = (Σp1·q0/Σp0·q0)×100, while Paasche uses current-year quantities (q1), giving P01(P) = (Σp1·q1/Σp0·q1)×100. NCERT Class 11 Statistics Chapter 7 Part 1 notes that Laspeyres tends to overstate price increases (upward bias) because it ignores substitution toward cheaper goods, while Paasche tends to understate them (downward bias) because it gives less weight to goods whose prices have risen most. Fisher's ideal index averages the two using a geometric mean to balance these biases and satisfies more tests of a good index number.
Why is Fisher's index called the ideal index in NCERT Class 11 Statistics?
Fisher's index is called the ideal index in NCERT Class 11 Statistics Chapter 7 Part 1 because it is the geometric mean of Laspeyres and Paasche indexes — P01(F) = √(L × P) — and balances out the upward bias of Laspeyres and the downward bias of Paasche. It also satisfies two important tests of a good index number: the time reversal test (P01 × P10 = 1) and the factor reversal test (price index × quantity index = value index). Despite these theoretical advantages, Fisher's index requires both base-year and current-year quantity data, making it harder to compute regularly than Laspeyres.
What is the difference between simple and weighted index numbers?
Simple index numbers treat all items as equally important and are calculated either by simple aggregative method P01 = (Σp1/Σp0)×100 or by simple average of price relatives. Weighted index numbers assign different weights to different items based on their economic importance — typically quantities consumed, expenditure shares or production volumes. NCERT Class 11 Statistics Chapter 7 Part 1 stresses that weighted indexes (Laspeyres, Paasche, Fisher) are far more reliable because they reflect real economic significance: a price rise in rice matters more to a household than a price rise in saffron, so weights are essential for meaningful CPI or WPI calculations.
How do you choose a base year for an index number in NCERT Class 11?
NCERT Class 11 Statistics Chapter 7 Part 1 lists four rules for selecting a base year: (1) it should be a normal year, free from war, famine, natural disaster or unusual economic events; (2) it should not be too distant from the current year, so the basket of goods and economic structure remains broadly comparable; (3) reliable and complete data must be available for the base year; and (4) it should be reviewed and updated periodically — India's WPI base year was revised from 2004-05 to 2011-12, for example. Older base years lose relevance as consumption patterns and technology change.