Price index


A price index is a normalized average of price relatives for a given class of goods or services in a specific region over a defined time period. It is a statistic designed to measure how these price relatives, as a whole, differ between time periods or geographical locations, often expressed relative to a base period set at 100.
Price indices serve multiple purposes. Broad indices, like the Consumer price index, reflect the economy’s general price level or cost of living, while narrower ones, such as the Producer price index, assist producers with pricing and business planning. They can also guide investment decisions by tracking price trends.

Types of price indices

Some widely recognized price indices include:
The origins of price indices are debated, with no clear consensus on their inventor. The earliest reported research in this area came from Rice Vaughan, who in his 1675 book analyzed price level changes in England. Vaughan sought to distinguish inflation from precious metals imported by Spain from the New World from effects of currency debasement. By comparing labor statutes from his era to those under Edward III, he used wage levels as a proxy for a basket of goods, concluding prices had risen six- to eight-fold over a century. Though a pioneer, Vaughan did not actually compute an index.
In 1707, Englishman William Fleetwood developed perhaps the first true price index. Responding to an Oxford student facing loss of a fellowship due to a 15th-century income cap of five pounds, Fleetwood used historical price data to create an index of averaged price relatives. His work, published anonymously in Chronicon Preciosum, showed the value of five pounds had shifted significantly over 260 years.

Basic formula

Price indices measure relative price changes using price and quantity data for a set of goods or services. The total market value in period is: : where is the price and the quantity of item in period. If quantities remain constant across two periods, the price index simplifies to: :.
This ratio, weighted by quantities, compares prices between periods and. In practice, quantities vary, requiring more complex formulas.

Price index formulas

Over 100 formulas exist for calculating price indices, aggregating price and quantity data differently. They typically use expenditures or weighted averages of price relatives to track relative price changes. Categories include unilateral, bilateral, and unweighted indices, with modern applications favoring Laspeyres for simplicity and superlative indices like Fisher for accuracy in GDP and inflation metrics.

Unilateral indices

These indices use quantities from a single period—either the base or current —as fixed weights, meaning they do not adjust for changes in consumption patterns over time.

Laspeyres index

Developed in 1871 by Étienne Laspeyres, it uses base-period quantities:
It measures the cost of a fixed basket at new prices. This often overstates inflation because it does not account for consumers reacting to price changes by altering quantities purchased. For example, when applied to an individual consumer’s bundle, a Laspeyres index of 1 means the consumer can afford to buy the same bundle in the current period as consumed in the base period, assuming income hasn’t changed.

Paasche index

Introduced in 1874 by Hermann Paasche, it uses current-period quantities:
It understates inflation by assuming consumers instantly adjust to new quantities, ignoring that higher prices might reduce demand over time. For example, a Paasche index of 1 indicates the consumer could have consumed the same bundle in the base period as in the current period, given unchanged income.

Lowe index

Named after Joseph Lowe, this uses fixed quantity weights from an expenditure base period, typically earlier than both the base and current periods, where the principal modification is to draw quantity weights less frequently than every period:
Unlike Laspeyres or Paasche, which draw weights from indexed periods, Lowe indices inherit weights from surveys, often conducted every few years, while prices are tracked each period. For a consumer price index, these weights on various expenditures are typically derived from household budget surveys, which occur less often than price data collection. Used in most CPIs, it’s a "modified Laspeyres" where Laspeyres and Paasche are special cases if weights update every period. The Geary-Khamis method, used in the World Bank’s International Comparison Program, fixes prices while updating quantities.

Bilateral indices

These indices compare two periods or locations using prices and quantities from both, aiming to reduce bias from the single-period weighting of unilateral indices. They incorporate substitution effects by blending data symmetrically or averaging across periods, unlike unilateral indices that fix quantities and miss consumer adjustments.

Marshall-Edgeworth index

Credited to Alfred Marshall and Francis Ysidro Edgeworth, it averages quantities:
It uses a simple arithmetic mean of base and current quantities, making it symmetric and intuitive. However, its use can be problematic when comparing entities of vastly different scales.

Superlative indices

Introduced by W. Erwin Diewert in 1976, superlative indices are a subset of bilateral indices defined by their ability to exactly match flexible economic functions with second-order accuracy, unlike the Marshall-Edgeworth index, which uses a basic arithmetic average lacking such precision. They adjust for substitution symmetrically, making them preferred for inflation and GDP measurement over simpler bilateral or unilateral indices.
Fisher index
Named for Irving Fisher, it’s the geometric mean of Laspeyres and Paasche:
It balances Laspeyres’ base-period bias and Paasche’s current-period bias, offering greater accuracy than Marshall-Edgeworth’s arithmetic approach. It requires data from both periods, unlike unilateral indices, and in chaining, it multiplies geometric means of consecutive period-to-period indices.
Törnqvist index
A geometric mean weighted by average value shares:
It weights price relatives by economic importance, providing precision over Marshall-Edgeworth’s simpler averaging, but it’s data-intensive, needing detailed expenditure data.
Walsh index
Uses geometric quantity averages:
It reduces bias from period-specific weighting with geometric averaging, outperforming Marshall-Edgeworth’s arithmetic mean in theoretical alignment, though it shares superlative data demands.

Unweighted indices

These compare prices of single goods between periods without quantity or expenditure weights, often as building blocks for indices like Laspeyres or Paasche within broader measures like CPI or PPI. For example, a Carli index of bread prices might feed into a Laspeyres index for a food category. They are called "elementary" because they’re applied at lower aggregation levels, assuming prices alone capture consistent quality and economic importance—a simplification that fails if quality changes or substitutes shift demand, unlike weighted indices that adjust via quantity or expenditure data.

Carli index

From Gian Rinaldo Carli, an arithmetic mean of price relatives over a set of goods :
Simple and intuitive, it overweights large price increases, causing upward bias. Used in part in the British retail price index, it can record inflation even when prices are stable overall because it averages price ratios directly—e.g., a 100% increase and a 50% decrease yield 1.25, not 1.

Dutot index

By Nicolas Dutot, a ratio of average prices:
Easy to compute, it’s sensitive to price scale and assumes equal item importance.

Jevons index

By W.S. Jevons, a geometric mean:
It’s the unweighted geometric mean of price relatives. It was used in an early Financial Times index, but it was inadequate for that purpose because if any price falls to zero, the index drops to zero. That is an extreme case; in general, the formula will understate the total cost of a basket of goods unless their prices all change at the same rate. Also, as the index is unweighted, large price changes in selected constituents can transmit to the index to an extent not representing their importance in the average portfolio.

Harmonic mean indices

Related unweighted indices include the harmonic mean of price relatives :
and the ratio of harmonic means:
These dampen large price drops, offering stability but less economic grounding than weighted indices.

CSWD index

Named for Carruthers, Sellwood, Ward, and Dalén, a geometric mean of Carli and harmonic indices:
In 1922 Fisher wrote that this and the Jevons were the two best unweighted indexes based on Fisher’s test approach to index number theory, balancing Carli’s bias with harmonic stability, though it lacks economic weighting.

Geometric mean index

Weighted by base-period expenditure shares:
A geometric mean of price relatives, it weights by economic importance, offering stability over arithmetic means like Laspeyres, but it’s fixed to base-period behavior.

Dynamic Equilibrium Price Index (DEPI)

The Dynamic Equilibrium Price Index is a theoretical concept and a type of price index that measures the ex ante intertemporal cost of living by incorporating both consumer prices and asset prices. The index is constructed by allocating weights to the Consumer Price Index and asset prices through a parameter determined by time preference and the length of the planning horizon for living costs. When this parameter equals 1, the DEPI reduces to the CPI. In other words, DEPI is a universal price index, with the CPI being a special case of it.
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