Unit 9: Index Numbers

I. Foundations and Governing Principle

An index number is a statistical measure that expresses the relative change in one variable, or a related group of variables, over time, place, or condition. The comparison period is measured against a base period, conventionally assigned the value 100.

• Relative measure: An index describes proportionate rather than absolute change. An index of 125 means a 25% increase over the base-period level.
• Base period: The reference period should be normal, recent, and free from exceptional events such as war, famine, or severe recession.
• Current period: The period whose level is compared with the base period.
• Dimensionless form: Because an index is usually a ratio multiplied by 100, unlike the original observations it has no physical unit.
• Principal types:
  • Price index: Measures changes in the prices of a basket of goods, such as a consumer price index.
  • Quantity index: Measures changes in physical quantities produced, sold, or consumed.
  • Value index: Measures changes in total monetary value, where value equals price multiplied by quantity.
• Composite character: A general index combines heterogeneous items—such as kilograms of rice, litres of fuel, and units of clothing—into one measure.
• Standard notation:
  • (p_0,p_1): Prices in the base and current periods.
  • (q_0,q_1): Quantities in the base and current periods.
  • (\Sigma): Summation across all included commodities.
  • (P{01},Q{01},V_{01}): Price, quantity, and value indices for period 1 relative to period 0.

II. Economic and Statistical Purpose — Why Index Numbers Are Required

A. Need for index numbers

Index numbers are needed because absolute data alone cannot conveniently reveal the overall direction and magnitude of changes involving many related variables.

• Measurement of price-level change: A price index condenses changes in numerous commodity prices into one figure. If an index rises from 100 to 118, the measured basket has become approximately 18% more expensive relative to the base period.
• Measurement of inflation and deflation: A sustained rise in a general price index indicates inflation, while a sustained fall indicates deflation. Policymakers use such movement when considering interest rates, taxation, and public expenditure.
• Cost-of-living comparison: Consumer price indices estimate changes in the expenditure needed to maintain a specified consumption pattern. They support revisions to wages, pensions, allowances, and dearness compensation.
• Purchasing-power assessment: The purchasing power of money moves inversely to the price level. An approximate purchasing-power index is:

Purchasing-power index = (100 / Price index) × 100

Here, the price index uses 100 as its base. If the price index is 125, purchasing power is (100/125 \times 100=80), indicating a 20% decline from the base level.

• Conversion of nominal values into real values: Monetary series can rise merely because prices increase. Deflation separates price change from real change:

Real value = (Nominal value / Relevant price index) × 100

If nominal income is 60,000 and the price index is 120, real income at base-period prices is (60,000/120 \times 100=50,000).

• Comparison across time and place: Production, trade, wages, and business activity measured in different years or regions become comparable after conversion to a common base.
• Economic policy and planning: Governments monitor wholesale prices, industrial production, agricultural output, and export-import indices to identify shortages, growth, or sectoral imbalance.
• Business decisions: Firms use indices to revise contracts, forecast demand, evaluate sales performance, and adjust inventories. An escalation clause may link a payment directly to a construction-cost index.
• Summary of complex data: Hundreds of observations can be represented by a single composite index, making broad trends easier to interpret.
• Limits of the need fulfilled: An index is an approximate indicator, not an exact measurement. Its usefulness depends on representative items, reliable quotations, suitable weights, and an appropriate base year.

III. Construction — From Individual Observations to a Composite Measure

A. Different methods of constructing index numbers

Construction methods differ mainly in whether commodities receive equal or unequal importance and whether prices or price relatives are aggregated.

• Preliminary decisions: Before selecting a formula, the investigator must define the objective, geographical area, population group, commodity basket, price quotations, base period, and system of weights.
• Fixed-base construction: Every period is compared directly with one constant base period. It supports easy long-term comparison but may become unrepresentative as consumption patterns change.
• Chain-base construction: Each period is compared with the immediately preceding period, and link relatives are chained together:

Chain index for period t = (Chain index for period t−1 × Link relative for t) / 100

The link relative compares period (t) with period (t-1). Chain indices accommodate changing baskets but may accumulate rounding errors.

1. Simple or unweighted methods

These methods treat all items as equally important, regardless of expenditure or quantity.

• Simple aggregative price index:

P₀₁ = (Σp₁ / Σp₀) × 100

(P_{01}) is the current-period price index; (\Sigma p_1) and (\Sigma p_0) are sums of current and base prices. The method is easy but adds prices quoted in potentially incomparable units and allows high-priced items to dominate.

• Simple average of price relatives: First calculate each commodity’s price relative:

R = (p₁ / p₀) × 100

Here, (R) is the individual price relative. The relatives are then averaged.

• Arithmetic-mean form:

P₀₁ = ΣR / n

In this formula, (n) is the number of commodities.

• Geometric-mean form:

P₀₁ = (R₁R₂...Rₙ)^(1/n)

(R_1,\ldots,R_n) are individual price relatives. The geometric mean is less affected by extreme relatives and is consistent with multiplicative changes.

2. Weighted methods

Weighted indices assign greater influence to commodities of greater economic importance.

• Laspeyres price index: Uses base-period quantities as weights:

Pᴸ₀₁ = (Σp₁q₀ / Σp₀q₀) × 100

(P^L_{01}) is the Laspeyres index. It is economical because only base quantities are required, but it may overstate price increases when consumers substitute away from goods that become expensive.

• Paasche price index: Uses current-period quantities as weights:

Pᴾ₀₁ = (Σp₁q₁ / Σp₀q₁) × 100

(P^P_{01}) is the Paasche index. It reflects the current consumption pattern but requires fresh quantity data and may understate inflation because substitution has already occurred.

• Fisher’s ideal price index: Takes the geometric mean of Laspeyres and Paasche indices:

Pᶠ₀₁ = √(Pᴸ₀₁ × Pᴾ₀₁)

(P^F_{01}) is Fisher’s index. It balances base- and current-period weighting and satisfies important consistency tests.

• Marshall–Edgeworth price index: Uses the sum of base and current quantities as weights:

Pᴹᴱ₀₁ = [Σp₁(q₀ + q₁) / Σp₀(q₀ + q₁)] × 100

(P^{ME}_{01}) is the Marshall–Edgeworth index. It incorporates both periods but requires quantity information for each.

• Weighted average of price relatives:

P₀₁ = ΣwR / Σw

Here, (w) is the assigned weight and (R) is the price relative. Expenditure shares or base-period values commonly provide the weights.

• Worked comparison: For two goods, let (p_0=(10,20)), (p_1=(12,25)), (q_0=(5,4)), and (q_1=(4,5)). Then:

Laspeyres = [(12×5 + 25×4) / (10×5 + 20×4)] × 100 = 123.08
Paasche   = [(12×4 + 25×5) / (10×4 + 20×5)] × 100 = 123.57
Fisher    = √(123.08 × 123.57) = 123.32

Thus, the combined price level increased by approximately 23.32% under Fisher’s method.

IV. Consistency Criteria — Evaluating Formula Quality

A. Tests for judging the soundness of an index number

A sound index formula should give logically consistent results when periods are reversed, prices and quantities are related, or comparisons are linked.

• Unit or commensurability test: The index should not change merely because the unit of quotation changes—for example, from kilograms to grams. Price-relative and properly weighted aggregative methods generally avoid arbitrary unit effects; the simple aggregative method may fail because it directly adds quoted prices.

• Time-reversal test: Reversing the base and current periods should produce the reciprocal result:

P₀₁ × P₁₀ = 1

This form assumes indices are expressed as ratios; when expressed with base 100, the corresponding product is (10,000). (P_{10}) compares period 0 with period 1. Fisher’s index satisfies this test, whereas Laspeyres and Paasche individually do not.

• Factor-reversal test: The product of the price and quantity indices should equal the value index:

P₀₁ × Q₀₁ = V₀₁
V₀₁ = Σp₁q₁ / Σp₀q₀

All indices are expressed as ratios here. (Q{01}) measures quantity change, while (V{01}) measures change in total expenditure or value. Fisher’s price and quantity indices satisfy this test.

• Circular test: Chained comparisons should return consistently to the starting point:

P₀₁ × P₁₂ × P₂₀ = 1

The symbols represent index ratios between periods 0, 1, and 2. The test is especially relevant to chain-base indices. Simple aggregative indices and geometric means of price relatives satisfy it under consistent coverage; Fisher’s index does not generally satisfy it.

• Adequacy beyond mathematical tests: Formula consistency alone cannot guarantee a useful index.

  • Representative coverage: The basket should reflect the target group; a worker-consumer index must emphasize goods and services purchased by workers.
  • Appropriate weights: Weights should measure economic importance and be revised when expenditure patterns change substantially.
  • Comparable quality: A price increase caused by improved size, durability, or specification should not automatically be treated as pure inflation.
  • Reliable observations: Prices should come from comparable markets, outlets, dates, and transaction levels.
  • Transparent interpretation: An index of 140 means a 40% rise from its own base, not necessarily a 40% rise from another index with a different basket or base year.

• Overall judgement: Fisher’s formula is called “ideal” because it satisfies both the time-reversal and factor-reversal tests. Nevertheless, the best practical index is the one whose data, weights, coverage, and construction most closely match its stated purpose.