Index Numbers

An index number measures how much a group of related quantities — prices, output, wages — has changed relative to a base period, as a single percentage. This chapter covers the meaning and types of index numbers, the problems in constructing one, the simple and weighted formulae (Laspeyres, Paasche, Fisher and others), the tests an index should pass, base shifting and splicing, the cost of living index, and the uses of index numbers including deflating.

Meaning, types and the problems of construction

An index number is a specialised average of relative changes, expressed with the base period = 100. A price index tracks prices, a quantity (volume) index tracks quantities, and a value index tracks Σpq. Writing p₀, q₀ for base-period and p₁, q₁ for current-period prices and quantities, the simplest is the price relative (p₁/p₀) × 100 of a single item.

  • Purpose — what exactly the index should measure, and for whom.
  • Base period — a normal year, not too distant, free of booms, slumps and disasters.
  • Items — a representative basket, neither too few nor unmanageably many.
  • Prices — retail or wholesale, from reliable sources, for the same quality.
  • Weights — each item's importance, usually quantities or expenditure shares.
  • Average and formula — which average of relatives, and which weighted formula.

Simple and weighted formulae

The formulae (price index, × 100)
IndexFormulaWeights
Simple aggregativeΣp₁ ÷ Σp₀none
Simple average of relativesΣ(p₁/p₀) ÷ nnone
LaspeyresΣp₁q₀ ÷ Σp₀q₀base-year quantities
PaascheΣp₁q₁ ÷ Σp₀q₁current-year quantities
Fisher's ideal√(Laspeyres × Paasche)both
Marshall–EdgeworthΣp₁(q₀ + q₁) ÷ Σp₀(q₀ + q₁)sum of both quantities
Dorbish–Bowley(Laspeyres + Paasche) ÷ 2both
Kelly (fixed weight)Σp₁q ÷ Σp₀qfixed quantities q
Worked basket
Itemp₀q₀p₁q₁
A105126
B810108
C5465

Σp₁q₀ = 184, Σp₀q₀ = 150, Σp₁q₁ = 182, Σp₀q₁ = 149. Laspeyres = 184/150 × 100 ≈ 122.67; Paasche = 182/149 × 100 ≈ 122.15; Fisher = √(122.67 × 122.15) ≈ 122.41. Fisher's index always lies between the other two. Laspeyres tends to overstate a price rise because it ignores consumers shifting away from items that became dearer; Paasche tends to understate it.

Tests of adequacy

Which formula passes which test
TestRequirementSatisfied by
Time reversalP₀₁ × P₁₀ = 1Fisher, Marshall–Edgeworth, simple GM of relatives, Kelly
Factor reversalP₀₁ × Q₀₁ = Σp₁q₁ ÷ Σp₀q₀Fisher only
CircularP₀₁ × P₁₂ × P₂₀ = 1simple GM of relatives, simple aggregative, Kelly
🧠 Memory trick
Fisher is 'ideal' because it passes both reversal tests (time and factor). It does not pass the circular test.

Base shifting, splicing, chain index and cost of living

Base shifting: to move an index to a new base year, divide every value by the old index of the new base year and multiply by 100. If an index reads 125 in 2020 and 150 in 2022 (old base), on base 2020 the 2022 value becomes 150/125 × 100 = 120. Splicing joins an old series to a new one that starts where the old was discontinued: if the old series stood at 160 in 2020 and the new series (2020 = 100) reads 115 in 2022, the spliced value on the old base is 115 × 160/100 = 184. A chain index links each period to the one before (link relatives) and multiplies them up, which lets items and weights be revised every period.

The cost of living index (consumer price index) measures the change in what a defined class of people pays for its usual basket. It is built by the aggregate expenditure method, Σp₁q₀/Σp₀q₀ × 100 (Laspeyres), or the family budget method, ΣIW/ΣW with I = p₁/p₀ × 100 and weights W = p₀q₀ — the two give the same number. Its main use is deflating: real wage = money wage ÷ CPI × 100, so ₹12,000 at a CPI of 150 is worth ₹8,000 of base-year purchasing power. The purchasing power of money is the reciprocal of the price index.

⚠️ Exam trap
Base shifting divides by the old index of the new base year. Subtracting (150 − 125 = 25) is a tempting wrong answer.

Exam traps and a quick revision

  • Laspeyres weights by q₀, Paasche by q₁; Fisher is their geometric mean.
  • Fisher passes time and factor reversal, not the circular test.
  • New-base index = old index ÷ old index of the new base × 100.
  • Real wage = money wage ÷ CPI × 100.
  • Aggregate expenditure and family budget methods give the same CLI.

Key takeaways

  • An index number is a specialised average of relative changes with base = 100.
  • Laspeyres = Σp₁q₀/Σp₀q₀ × 100; Paasche = Σp₁q₁/Σp₀q₁ × 100; Fisher = √(L × P).
  • Fisher's ideal index passes the time-reversal and factor-reversal tests.
  • Base shifting divides by the new base's old index; splicing rescales the new series by the old series' value at the link year.
  • Real wage = money wage ÷ CPI × 100; the CPI is built by the aggregate expenditure or family budget method.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. Laspeyres' price index uses as weights the:

    1. current-year quantities
    2. base-year quantities
    3. average of base and current quantities
    4. base-year prices
    Show answer

    Answer: B — base-year quantities

    Laspeyres = Σp₁q₀/Σp₀q₀ × 100, weighting by q₀. Current-year quantities are Paasche's weights.

  2. Paasche's price index is:

    1. Σp₁ ÷ Σp₀ × 100
    2. Σp₁q₁ ÷ Σp₀q₁ × 100
    3. Σp₁q₀ ÷ Σp₀q₀ × 100
    4. Σq₁p₀ ÷ Σq₀p₀ × 100
    Show answer

    Answer: B — Σp₁q₁ ÷ Σp₀q₁ × 100

    Paasche weights both sums by current quantities q₁. Σp₁q₀ ÷ Σp₀q₀ is Laspeyres, Σp₁ ÷ Σp₀ the simple aggregative index, and Σq₁p₀ ÷ Σq₀p₀ a Laspeyres quantity index.

  3. Which index number satisfies both the time-reversal test and the factor-reversal test?

    1. Laspeyres' index
    2. simple aggregative index
    3. Paasche's index
    4. Fisher's ideal index
    Show answer

    Answer: D — Fisher's ideal index

    Fisher's √(L × P) is the only standard formula that passes both reversal tests, which is why it is called ideal. Laspeyres and Paasche fail both.

  4. For a basket, Σp₁q₀ = 184 and Σp₀q₀ = 150. Laspeyres' price index is about:

    1. 81.52
    2. 122.67
    3. 122.15
    4. 134
    Show answer

    Answer: B — 122.67

    Laspeyres = 184/150 × 100 ≈ 122.67. Inverting the ratio gives 81.52; 122.15 is the Paasche index of the same basket, 182/149 × 100.

  5. Laspeyres' index is 144 and Paasche's index is 100 for the same data. Fisher's ideal index is:

    1. 122
    2. 144
    3. 244
    4. 120
    Show answer

    Answer: D — 120

    Fisher = √(144 × 100) = √14400 = 120. 122 is the arithmetic mean (Dorbish–Bowley), always at least as large as the geometric mean.

  6. An index (base 2018 = 100) reads 125 in 2020 and 150 in 2022. With 2020 as the new base, the 2022 index is:

    1. 187.5
    2. 25
    3. 120
    4. 125
    Show answer

    Answer: C — 120

    New index = 150/125 × 100 = 120. Subtracting gives 25, and multiplying instead of dividing gives 187.5.

  7. A worker's money wage is ₹12,000 and the consumer price index is 150. His real wage in base-year rupees is:

    1. ₹11,850
    2. ₹12,150
    3. ₹8,000
    4. ₹18,000
    Show answer

    Answer: C — ₹8,000

    Real wage = money wage ÷ CPI × 100 = 12000/150 × 100 = ₹8,000. Prices are 1.5 times the base, so the wage buys two-thirds as much.

  8. The prices of three items in the base year are ₹20, ₹30 and ₹50, and in the current year ₹25, ₹33 and ₹62. The simple aggregative price index is:

    1. 120
    2. 100
    3. 83.33
    4. 125
    Show answer

    Answer: A — 120

    Σp₁/Σp₀ × 100 = 120/100 × 100 = 120. Each item's own relative (125, 110, 124) is not needed for this formula.

  9. An old index series stood at 160 in 2020, when it was discontinued. A new series with 2020 = 100 reads 115 in 2022. Spliced to the old series, the 2022 value is:

    1. 184
    2. 275
    3. 139.13
    4. 71.88
    Show answer

    Answer: A — 184

    Splicing rescales the new series so that 2020 equals 160: 115 × 160/100 = 184. Adding 160 + 115 gives the meaningless 275.

  10. Which of the following satisfies the circular test?

    1. Laspeyres' index
    2. Paasche's index
    3. the simple geometric mean of price relatives
    4. Fisher's ideal index
    Show answer

    Answer: C — the simple geometric mean of price relatives

    The circular test P₀₁ × P₁₂ × P₂₀ = 1 is met by the simple geometric mean of relatives (and by fixed-weight aggregative indices), because the intermediate prices cancel. Fisher passes both reversal tests but not this one.