Unit 9: Index Numbers - Subjective Questions

Definition: An index number is a statistical measure that shows the relative change in the level of a variable or a group of related variables over time, place, or other characteristics. The base-period value is usually taken as .

Principal features:

• It measures relative rather than absolute change.
• It summarizes changes in several related items through a single figure.
• It requires a suitable base period for comparison.
• It is generally expressed as a percentage.
• It is a specialized type of average.

Uses:

• Measuring changes in the general price level and cost of living.
• Studying changes in production, trade, wages, and employment.
• Converting nominal values into real values through deflation.
• Revising wages, pensions, rents, and contractual payments.
• Assisting governments and businesses in planning and policy formulation.

Index numbers are needed because absolute figures alone do not clearly reveal the direction and magnitude of relative change.

Importance of index numbers:

• Measurement of inflation: Price indices indicate changes in the purchasing power of money.
• Cost-of-living comparisons: Consumer price indices measure changes in household expenses.
• Economic policy: Governments use indices when framing monetary, fiscal, wage, and trade policies.
• Business decisions: Firms use them for pricing, budgeting, forecasting, and market analysis.
• Deflation: A nominal series can be converted into a real series using an appropriate price index.
• Comparison: They facilitate comparisons across periods, regions, or sectors even when the original data use different units.
• Contract adjustment: Dearness allowances and escalation clauses may be linked to selected indices.

Thus, index numbers act as economic barometers, although their interpretation depends on the adequacy of the data, weights, base period, and construction method.

The construction of a reliable index number involves the following problems:

• Defining the objective: The purpose and population covered by the index must be clearly stated.
• Selecting a base period: The base should be normal, recent, and free from exceptional conditions such as war or severe recession.
• Selecting commodities: Items should be representative of the consumption, production, or trade being studied.
• Specifying quality and units: Each item must have a precise grade, quality, package size, and unit of measurement.
• Collecting prices: Reliable markets, outlets, quotations, and timing of price collection must be selected.
• Choosing weights: Weights should reflect the relative economic importance of different items.
• Selecting a formula: The choice among aggregative and price-relative methods affects the result.
• Handling changes: New goods, disappearing goods, quality changes, and altered consumption patterns require appropriate treatment.

A sound index therefore depends on both a suitable formula and careful statistical design.

Fixed-base index: Every period is compared directly with one common base period. If period is the base, the index for period is denoted by .

Chain-base index: Each period is first compared with the immediately preceding period. These link relatives are then chained together. If is the link relative, then

and

Differences:

• A fixed-base index uses one constant base, whereas a chain index uses a moving base.
• Fixed-base indices are easier to calculate and compare over time.
• Chain indices accommodate changes in commodities and weights more easily.
• A fixed base may become outdated, while chaining can suffer from chain drift.
• Chain indices require an uninterrupted series of link relatives.

The appropriate method depends on the stability of the market and the purpose of the index.

Under the simple aggregative method, prices of all selected commodities in the current period are added and compared with the sum of their base-period prices:

where and denote base-period and current-period prices.

Merits:

• It is simple to understand and calculate.
• It gives a single measure of the combined price change.
• It requires only price data.

Limitations:

• It assigns greater implicit importance to commodities with high numerical prices.
• It ignores the quantities consumed or economic importance of commodities.
• Its value changes when the units of measurement are changed.
• Prices expressed in different units are added, which may make the aggregate less meaningful.

Consequently, this method is suitable only when commodities are reasonably comparable and weighting is not essential.

First, the price relative of each commodity is calculated as

The individual relatives are then averaged.

Arithmetic mean form:

Geometric mean form:

Comparison:

• Both methods give every commodity equal explicit importance.
• The arithmetic mean is easier to compute but is more affected by extreme price relatives.
• The geometric mean is less affected by extreme values and treats proportionate increases and decreases more symmetrically.
• The geometric mean of price relatives satisfies the time-reversal test, whereas the arithmetic mean generally does not.
• Both methods ignore differences in expenditure or quantity unless weights are introduced.

The geometric mean is theoretically preferable when proportional changes are being averaged.

A weighted index number assigns different levels of importance to commodities according to their economic significance.

Need for weights:

• Commodities do not have equal importance in a consumer budget or in national production.
• A small change in the price of a heavily consumed item may matter more than a large change in a rarely purchased item.
• Weights make the index more representative of actual economic behavior.

Quantity weights: Quantities such as or are used with prices. Examples include Laspeyres' index, which uses , and Paasche's index, which uses .

Value weights: Expenditure or value shares such as may be used to weight price relatives. A weighted average of price relatives is

where represents the importance of item .

Weights should be representative, reliable, and periodically revised so that the index does not become obsolete.

Laspeyres' price index uses base-period quantities as weights. The cost of purchasing the base-period basket at current prices is compared with its cost at base prices:

The numerator is the hypothetical current cost of the old basket, while the denominator is its actual base-period cost.

Advantages:

• Only base-period quantity data are required.
• The same basket facilitates comparisons across several periods.
• It is comparatively simple and economical to construct.

Disadvantages:

• The weights may become outdated.
• It ignores substitution toward relatively cheaper commodities.
• It often overstates the rise in the cost of maintaining utility because consumers may alter their purchases.
• It generally fails the time-reversal and factor-reversal tests.

Laspeyres' formula is widely used where current quantity information is difficult or expensive to obtain.

Paasche's price index uses current-period quantities as weights. It compares the current cost of the current basket with the hypothetical base-period cost of that same basket:

Here, is actual current expenditure, while values the current basket at base prices.

Advantages:

• Current-period quantities reflect recent consumption or production patterns.
• The index incorporates substitutions and newly important commodities.
• It does not rely on an old fixed basket.

Disadvantages:

• Current quantity data must be collected for every period, making it costly.
• Comparisons across periods use different baskets.
• It may understate increases in the cost of living because the current basket can reflect substitution toward cheaper goods.
• It generally fails the time-reversal and factor-reversal tests.

Paasche's method is appropriate when reliable current-period quantity data are available.

The formulas are

and

Comparison:

• Laspeyres uses base-period quantities ; Paasche uses current-period quantities .
• Laspeyres requires less frequent quantity surveys; Paasche requires current quantity information.
• Laspeyres maintains a fixed basket; Paasche uses a changing basket.
• Laspeyres commonly has an upward substitution bias; Paasche commonly has a downward substitution bias.

They differ because relative prices and quantities usually change together. Consumers tend to reduce purchases of goods whose relative prices rise and increase purchases of relatively cheaper goods. Thus, the base-period and current-period weighting systems give different importance to individual price changes. If quantities remain unchanged or all prices change in the same proportion, the two indices coincide.

Fisher's price index is the geometric mean of Laspeyres' and Paasche's price indices:

In ratio form,

If the result is required with base , the ratio is multiplied by .

Reasons for calling it ideal:

• It uses both base-period and current-period quantities.
• It balances the upward tendency of Laspeyres' index and the downward tendency of Paasche's index.
• It satisfies the time-reversal test.
• It satisfies the factor-reversal test when the corresponding Fisher quantity index is used.
• It treats the two periods symmetrically.

Its main limitations are greater computational and data requirements, since prices and quantities for both periods are needed.

Marshall-Edgeworth index: It uses the sum of base and current quantities as weights:

Dorbish-Bowley index: It is the arithmetic mean of Laspeyres' and Paasche's indices:

Walsh index: It uses the geometric mean of quantities as weights:

Kelly index: It uses a fixed set of representative quantity weights :

These methods attempt to improve representativeness by using combined, average, or fixed representative weights. Their suitability depends on the available data and purpose of the index.

Under the weighted average of price relatives method, each price relative

is assigned a weight . The index is

If base-period value weights are selected, then . Substituting gives

After cancellation of in the numerator,

This is precisely Laspeyres' price index. Thus, a weighted arithmetic mean of price relatives using base-period expenditure weights is algebraically equivalent to the Laspeyres aggregative index.

A consumer price index, or cost-of-living index, measures the average change in retail prices paid by a specified class of consumers for a representative basket of goods and services.

Aggregate expenditure method:

It compares the current cost of the base-period basket with its base-period cost.

Family budget method: Price relatives are weighted by base-period expenditure:

where and .

Construction steps:

• Define the consumer group and geographical area.
• Conduct a family expenditure survey.
• Select representative commodities and services.
• Assign expenditure weights.
• Collect retail prices regularly.
• Combine group indices into the overall index.

With consistent base-period expenditure weights, the two methods yield the same result.

The unit test, also called the commensurability test, requires that an index number should not change merely because the units used to quote commodities are changed. For example, quoting a commodity per kilogram instead of per gram should not alter the measured price movement.

Application:

• The simple aggregative index generally fails the test because changing a unit changes the numerical magnitude of the corresponding prices and therefore changes and .
• The price-relative method generally satisfies the test because

where is the common conversion factor.

• Properly constructed weighted aggregative indices also satisfy the test when price and quantity units are adjusted consistently, because a change in the price unit is offset by the inverse change in quantity.

The test ensures that the index is not an arbitrary consequence of measurement units.

The time-reversal test requires consistency when the base and current periods are interchanged. In ratio form, a satisfactory index should obey

When indices are expressed with base , the condition is

For Fisher's index in ratio form,

On reversing time,

Multiplying these expressions causes every numerator to cancel with a corresponding denominator, giving

Therefore, Fisher's ideal index satisfies the time-reversal test. Laspeyres' and Paasche's indices individually generally fail it.

The factor-reversal test requires that the product of the price index and quantity index should reproduce the value index. In ratio form,

Fisher's price and quantity indices are

and

Multiplying and cancelling common terms gives

Thus, Fisher's index satisfies the factor-reversal test. If all indices have base , the equivalent relation is

This property shows consistency between changes in prices, quantities, and total value.

The circular test requires consistency in comparisons involving three or more periods. For periods , , and , the condition in ratio form is

Equivalently,

For indices expressed with base , suitable divisions by must be made when chaining.

Significance:

• It ensures that a direct comparison between two periods agrees with the result obtained through intermediate periods.
• It is particularly relevant to chain-base index numbers.
• It prevents the final index from depending on the chosen comparison path.

Limitations:

• Most weighted aggregative indices, including Fisher's index, do not generally satisfy the circular test.
• Passing the test does not by itself guarantee representative commodities, reliable data, or appropriate weights.
• Changes in quality, products, and consumption patterns can make strict circularity difficult to achieve.

The circular test is therefore one criterion of consistency, not a complete measure of quality.

Unit test: Checks whether the index is unaffected by a change in units of measurement. Price-relative methods and consistently weighted indices generally satisfy it.

Time-reversal test: Requires that reversing the two periods should produce the reciprocal index:

Fisher's index satisfies this test.

Factor-reversal test: Requires the price and quantity indices to reproduce the value change:

Fisher's index satisfies this test.

Circular test: Requires consistency through a sequence of periods:

It is especially relevant for chain indices.

Comparison: The unit test concerns measurement invariance; time reversal concerns symmetry between two periods; factor reversal links price, quantity, and value changes; and circularity concerns multiperiod consistency. No single test examines every practical issue, so theoretical tests must be supplemented by checks on data quality, coverage, weights, and relevance.

A sound price index may be constructed through the following procedure:

1. Specify the objective: Define what change is to be measured and for which population, region, and period.
2. Choose a base period: Select a normal and reasonably recent period, and assign it the value .
3. Select representative items: Include important goods and services purchased by the target population.
4. Fix specifications: Clearly state quality, brand, size, unit, market, and type of price for each item.
5. Collect reliable data: Obtain comparable prices from representative outlets at consistent intervals.
6. Choose weights: Use quantity or expenditure weights reflecting the relative importance of items.
7. Select a formula: Choose an appropriate method such as Laspeyres, Paasche, Fisher, or a weighted average of price relatives.
8. Calculate group indices: Construct subgroup indices before combining them into an overall index.
9. Test the index: Examine unit consistency, time reversal, factor reversal, and circularity where relevant.
10. Review and revise: Update the basket, weights, outlets, and base period to address quality changes and changing consumption patterns.

A theoretically elegant formula cannot compensate for unrepresentative items or poor data; statistical relevance and formula consistency are both essential.