Törnqvist index¶

In economics, the Törnqvist index is a price or quantity index.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12583
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomains: Index Numbers, Price Statistics → Economics & Finance

Core Idea¶

Törnqvist index is treated here as the recurring social_sciences_humanities_arts identity summarized by this source-grounded definition: In economics, the Törnqvist index is a price or quantity index.

In economics, the Törnqvist index is a price or quantity index. In practice, Törnqvist index values are calculated for consecutive periods, then these are strung together, or "chained". Thus, the core calculation does not refer to a single base year.

There is research on which method to use based partly on whether price changes or quantity changes are more volatile. A Divisia index is a theoretical construct, a continuous-time weighted sum of the growth rates of the various components, where the weights are the component's shares in total value. The Törnqvist index is a superlative index, meaning it can approximate any smooth production or cost function. "Smooth" here means that small changes in relative prices for a good will be associated with small changes in the quantity of it used.

For Törnqvist index, the abstraction is narrower than the article's general subject matter: a positive case must preserve In economics, the Törnqvist index is a price or quantity index. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in social_sciences_humanities_arts, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The data used are prices and quantities in two time-periods, (t-1) and (t), for each of n goods which are indexed by i.
Constitutive relation — In practice, Törnqvist indexes are often computed using an equation that results from taking logs of both sides, as in the expression below which computes the same P_t as those above.
Operating condition — Törnqvist indexes are close to the figures given by the Fisher index.
Recognition evidence — For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive.
Admissible variation — If we denote the price of item i at time t-1 by p_{i,t-1} , and, analogously, we define q_{i,t} to be the quantity purchased of item i at time t, then, the Törnqvist price index P_t at time t can be calculated as follows.
Characteristic consequence — If one needs matched quantity and price indexes they can be calculated directly from these equations, but it is more common to compute a price index by dividing total expenditure each period by the quantity index so the resulting indexes multiply out to total expenditure.
Failure boundary — The price index for some period is usually normalized to be 1 or 100, and that period is called "base period.".

What It Is Not¶

Not the whole field of social_sciences_humanities_arts. The node requires the specific identity stated by In economics, the Törnqvist index is a price or quantity index.
Not an over-broad reading. The Fisher index is sometimes preferred in practice because it handles zero-quantities without special exceptions, whereas in the equations above a quantity of zero would make the Törnqvist index calculation break down.
Not an over-broad reading. Quantity indexes are used in computing aggregate indexes for physical "capital" summarizing equipment and structures of different types into one time series.
Not an over-broad reading. This approach, called the indirect way of calculating a Törnqvist index, generates numbers that are not exactly the same as a direct calculation.
Not automatically Index (Economics). Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Törnqvist index applies literally inside social_sciences_humanities_arts wherever the source-defined carrier and relation can be established. Its documented habitats include:

Theory. The Törnqvist index is a superlative index, meaning it can approximate any smooth production or cost function. "Smooth" here means that small changes in relative prices for a good will be associated with small changes in the quantity of it used.
Computation. For multifactor productivity calculations, the indirect method is used.
Computation. The data used are prices and quantities in two time-periods, (t-1) and (t), for each of n goods which are indexed by i.
Computation. In practice, Törnqvist indexes are often computed using an equation that results from taking logs of both sides, as in the expression below which computes the same P_t as those above.
Computation. Quantity indexes are used in computing aggregate indexes for physical "capital" summarizing equipment and structures of different types into one time series.
Computation. There is research on which method to use based partly on whether price changes or quantity changes are more volatile.

Outside social_sciences_humanities_arts, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Measurement or should be marked as analogy.

Clarity¶

A clear use of Törnqvist index names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In economics, the Törnqvist index is a price or quantity index. The strongest recognition evidence in the frozen account is: For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The Fisher index is sometimes preferred in practice because it handles zero-quantities without special exceptions, whereas in the equations above a quantity of zero would make the Törnqvist index calculation break down. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Törnqvist index compresses multiple social_sciences_humanities_arts details into a stable diagnostic relation. The source shows both the central mechanism—in practice, Törnqvist indexes are often computed using an equation that results from taking logs of both sides, as in the expression below which computes the same P_t as those above.—and the practical consequence—if one needs matched quantity and price indexes they can be calculated directly from these equations, but it is more common to compute a price index by dividing total expenditure each period by the quantity index so the resulting indexes multiply out to total expenditure. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the social_sciences_humanities_arts entities to which the claim applies.
State the relation. Use the source-grounded identity: In economics, the Törnqvist index is a price or quantity index.
Check operation and conditions. Törnqvist indexes are close to the figures given by the Fisher index.
Demand recognition evidence. For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive.
Test variation. Change an implementation or setting while preserving if we denote the price of item i at time t-1 by p_{i,t-1} , and, analogously, we define q_{i,t} to be the quantity purchased of item i at time t, then, the Törnqvist price index P_t at time t can be calculated as follows.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Measurement.

Knowledge Transfer¶

Within the home domain. Knowledge about Törnqvist index transfers literally when a new case preserves the same carrier type, relation, and recognition test. The Törnqvist index is a superlative index, meaning it can approximate any smooth production or cost function. "Smooth" here means that small changes in relative prices for a good will be associated with small changes in the quantity of it used. For multifactor productivity calculations, the indirect method is used.

Beyond the home domain. No canonical parent is asserted for Törnqvist index. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Usually, that share doesn't change much; e.g. food expenditures across a million households might be 20% of income in one period and 20.1% the next period. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In economics, the Törnqvist index is a price or quantity index; recognition evidence → For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive

Applied / In Practice¶

For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Theory; invariant → In economics, the Törnqvist index is a price or quantity index; boundary → the case exits the class when the Fisher index is sometimes preferred in practice because it handles zero-quantities without special exceptions, whereas in the equations above a quantity of zero would make the Törnqvist index calculation break down

Structural Tensions¶

T1 — Stable identity versus admissible variation. The Fisher index is sometimes preferred in practice because it handles zero-quantities without special exceptions, whereas in the equations above a quantity of zero would make the Törnqvist index calculation break down. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Quantity indexes are used in computing aggregate indexes for physical "capital" summarizing equipment and structures of different types into one time series. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. This approach, called the indirect way of calculating a Törnqvist index, generates numbers that are not exactly the same as a direct calculation. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Thus, the core calculation does not refer to a single base year. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The data used are prices and quantities in two time-periods, (t-1) and (t), for each of n goods which are indexed by i. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Törnqvist index literally, co-instantiate Measurement, or only resemble it?

T6 — Autonomy versus reduction. In practice, Törnqvist indexes are often computed using an equation that results from taking logs of both sides, as in the expression below which computes the same P_t as those above. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Törnqvist index distinguish that the broader parent Measurement leaves together?

Structural–Framed Character¶

Törnqvist index is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In economics, the Törnqvist index is a price or quantity index. Its framed side is the social_sciences_humanities_arts vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Törnqvist indexes are close to the figures given by the Fisher index. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Measurement. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In economics, the Törnqvist index is a price or quantity index. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The data used are prices and quantities in two time-periods, (t-1) and (t), for each of n goods which are indexed by i. In practice, Törnqvist indexes are often computed using an equation that results from taking logs of both sides, as in the expression below which computes the same Pt as those above. It further constrains recognition and variation through: Törnqvist indexes are close to the figures given by the Fisher index. For example, a Törnqvist index summarizing labor input may weigh the growth rate of the hours of each group of workers by the share of labor compensation they receive.

What is domain-bound. social sciences humanities arts supplies the operative entities, technical vocabulary, warrants, and exceptions that make Törnqvist index literal. Its documented scope includes the condition that The Törnqvist index is a superlative index, meaning it can approximate any smooth production or cost function. "Smooth" here means that small changes in relative prices for a good will be associated with small changes in the quantity of it used. Another bounded application condition is that For multifactor productivity calculations, the indirect method is used. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—If we denote the price of item i at time t-1 by p{i,t-1} , and, analogously, we define q{i,t} to be the quantity purchased of item i at time t, then, the Törnqvist price index Pt at time t can be calculated as follows.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry typically is a kind of Aggregation.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Törnqvist index. The reviewed identity is: In economics, the Törnqvist index is a price or quantity index. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Törnqvist index Domain-specific

Parents (1) — more general patterns this builds on

Törnqvist index is a kind of, typical Aggregation Prime

The Törnqvist index is a value-share-weighted geometric mean collapsing many price or quantity ratios into one chained index number.

Hierarchy path (1) — routes to 1 parentless root

Törnqvist index → Aggregation → Micro Macro Linkage

Neighborhood in Abstraction Space¶

Törnqvist index sits in a moderately populated region (54^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Financial Indices & Trading Indicators (15 abstractions)

Nearest neighbors

Volume Index — 0.87
Merton's portfolio problem — 0.87
PRIX index — 0.86
Value at risk — 0.86
Substitution bias — 0.85

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Measurement. The parent omits the specialist differentia. Tell: Can the case establish In economics, the Törnqvist index is a price or quantity index?
Index (Economics). A normalized statistic comparing the level or change of a specified economic aggregate across periods, places, or populations relative to a declared base using explicit items, weights, and an aggregation formula. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Price Level. A normalized aggregate of prices for a defined economic domain and period whose ratios track the common monetary-price component used to measure inflation, purchasing-power change, and real-versus-nominal magnitudes. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Divisia Index. A continuous-time economic index whose logarithmic growth rate is a current value-share-weighted sum of component quantity or price growth rates, with discrete data ordinarily implemented through superlative approximations such as Törnqvist or Fisher indexes. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Törnqvist index remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside social_sciences_humanities_arts lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Measurement?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/T%C3%B6rnqvist_index (revision 1345461368).
Preserved source candidate: http://stats.oecd.org/glossary/detail.asp?ID=2711
Preserved source candidate: http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers
Preserved source candidate: https://web.archive.org/web/20131224111339/http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers
Preserved source candidate: http://www.imf.org/external/np/sta/tegppi/gloss.pdf
Preserved source candidate: https://www.jstor.org/stable/view/1924361
Preserved source candidate: http://econpapers.repec.org/RePEc:eee:ecolet:v:76:y:2002:i:2:p:251-258
Preserved source candidate: http://www.bls.gov/mfp/mprover.htm
Preserved source candidate: https://www.bls.gov/opub/hom/msp/home.htm

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.