Tornqvist index
Encyclopedia
In economics
the Törnqvist index is a price
or quantity index. Using price and quantity data, a Tornqvist index is a discrete approximation to a continuous Divisia index
. A Divisia index is a weighted sum of the growth rates of the various components, where the weights are the component's shares in total value. For a Törnqvist index, the growth rates are defined to be the difference in natural logarithms of successive observations of the components (i.e. their log-change) and the weights are equal to the mean of the factor shares of the components in the corresponding pair of periods (usually years).
The Törnqvist index represents an improvement over constant base-year weighted indexes, because as relative prices of inputs change, the Törnqvist index allows both quantities purchased of the inputs to vary and the weights used in summing the inputs to vary, reflecting the relative price changes. 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 data required are prices and quantities in two time periods (0 and t) for each of
goods which are indexed from 
to 
. Denote the price of item 
at time 0, the base period, by 
. Analogously let 
be the price of item 
at time t, 
be the quantity purchased of item 
at time 0, and 
be the quantity purchased of item 
at time t.
Then the Tornqvist index
at time t can be calculated this way:
The denominators in the exponent are the sums of total expenditure in each of the two periods. This is expressed more compactly in vector notation. Let
denote the vector of all prices at time 0 and analogously define vectors 
, 
, and 
. Then the above expression can be rewritten more compactly:
In this second expression it is easier to see that the overall exponent is the average share of expenditure on good
across the two periods. Often this is an average of two numbers which are very close to the same; e.g. food expenditures across a million households might be 20% of income in one period and 20.1% then next period. The Tornqvist index equally-weights the experiences in the two periods. Therefore it is a symmetric index.
In practice, Tornqvist indexes often computed using logarithms, as in the expression below which computes the same
as those above.
The Tornqvist index is a superlative index meaning it can approximate any smooth production
or cost function.
(1936), perhaps working with others at the Bank of Finland
.
Tornqvist indexes are used in a variety of official price and productivity statistics.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
the Törnqvist index is a price
Price index
A price index is a normalized average of prices for a given class of goods or services in a given region, during a given interval of time...
or quantity index. Using price and quantity data, a Tornqvist index is a discrete approximation to a continuous Divisia index
Divisia index
A Divisia index is a theoretical construct to create index number series for continuous-time data on prices and quantities of goods exchanged.It is designed to incorporate quantity and price changes over time from subcomponents which are measured in different units -- e.g...
. A Divisia index is a weighted sum of the growth rates of the various components, where the weights are the component's shares in total value. For a Törnqvist index, the growth rates are defined to be the difference in natural logarithms of successive observations of the components (i.e. their log-change) and the weights are equal to the mean of the factor shares of the components in the corresponding pair of periods (usually years).
The Törnqvist index represents an improvement over constant base-year weighted indexes, because as relative prices of inputs change, the Törnqvist index allows both quantities purchased of the inputs to vary and the weights used in summing the inputs to vary, reflecting the relative price changes. 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.
Theory
The Törnqvist or Törnqvist-Theil index is the geometric average of the price relative of the current to base period prices weighted by the arithmetic average of the value shares for the two periods. The price index at a specified time 0 is normalized usually to be 1 or 100.The data required are prices and quantities in two time periods (0 and t) for each of
goods which are indexed from to . Denote the price of item at time 0, the base period, by . Analogously let be the price of item at time t, be the quantity purchased of item at time 0, and be the quantity purchased of item at time t.Then the Tornqvist index
at time t can be calculated this way:The denominators in the exponent are the sums of total expenditure in each of the two periods. This is expressed more compactly in vector notation. Let
denote the vector of all prices at time 0 and analogously define vectors , , and . Then the above expression can be rewritten more compactly:In this second expression it is easier to see that the overall exponent is the average share of expenditure on good
across the two periods. Often this is an average of two numbers which are very close to the same; e.g. food expenditures across a million households might be 20% of income in one period and 20.1% then next period. The Tornqvist index equally-weights the experiences in the two periods. Therefore it is a symmetric index.In practice, Tornqvist indexes often computed using logarithms, as in the expression below which computes the same
as those above.The Tornqvist index is a superlative index meaning it can approximate any smooth production
Production function
In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...
or cost function.
History and Use
The Törnqvist index theory is attributed to Leo TörnqvistLeo Törnqvist
Leo Waldemar Törnqvist was one of the first professors of statistics in Finland. He taught at the University of Helsinki from 1950 to 1974, and developed techniques that are used in official price and productivity statistics....
(1936), perhaps working with others at the Bank of Finland
Bank of Finland
The Bank of Finland is the central bank of Finland. It is the fourth oldest central bank in the world.-History:The Bank of Finland was established on 1 March in 1812 in the city of Turku by Alexander I of Russia. In 1819 it was relocated to Helsinki...
.
Tornqvist indexes are used in a variety of official price and productivity statistics.