# Indexes

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## Indexes

in statistics, relative quantities that quantitatively characterize the overall dynamics (or, less often, a change in space) of a heterogeneous aggregate. For example, the index

means that the overall level of all retail prices in state trade in the USSR in 1964 was 0.76 (or 76 percent) of the 1950 price (in other words, taken in aggregate the average price declined by a factor of 0.24, or 24 percent, between 1950 and 1964).

An aggregate is heterogeneous with respect to some characteristic if the total magnitude of this characteristic throughout the entire aggregate cannot be calculated by direct summation of the values of the individual units (for example, the physical quantity of a product consisting of materially different physical units or parts) or if such summation, although formally possible, leads to a result that has no economic meaning (for example, the sum of the prices of materially different commodities taken only with respect to one unit of physical measurement). The four elements of any index are the indexed quantities, the type (form) of the index, the weight of the index, and the time period in the calculation. Indexed quantities may be price indexes, physical (in-kind) production volume indexes, indexes of labor productivity, or other types. Two types (forms) of indexes are found: aggregate and average indexes; the latter include arithmetic indexes, geometric (median) indexes, and modal indexes. There are simple (unweighted) and weighted indexes, and among the latter are indexes with constant (invariant) weights and indexes with variable weights (which are reviewed over time if necessary). Depending on the time base of calculation, there are base indexes (indexes with a constant base that is invariant in time) and chain indexes (indexes in which the numerical values of the indexed quantity at given “current” time are compared with their values in the preceding period or in which the base is variable). In the general case, the product of the chain indexes should give the corresponding base index. For example,

Indexes can be calculated not only for an entire heterogeneous aggregate (overall or “total” indexes) but also for any characteristic part thereof or for any significant group of units (group indexes, or subindexes): for example, the overall index of the wholesale prices of all commodities in general, the group indexes of prices of food products and of prices of nonfood products or of manufactured and agricultural products, or the price index of textiles or the prices of leather goods. The ordinary relative quantity of a change in a characteristic of some commodity (such as the relative change

of the cost *z* of a commodity *I* in a given three-year period) is not an index, although in practice it is usually called by analogy an “individual index” (of production costs).

The most difficult problem in constructing an index is selecting the weights and making the most accurate possible calculation of the weight of each group, and sometimes of each unit, belonging to the indexed population. A system of such weights should represent a model of the structure of the socioeconomic phenomenon whose dynamics are given numerical expression in the index. Thus, the weight of a price index should reflect the commodity composition of trade turnover (retail and wholesale), and the physical quantities of commodities and services that are found in the “market basket” of goods and services should be the weights of the budget index. In physical volume indexes, the role of weighting the physical qualities of commodities is played by fixed prices, as a result of which it becomes possible to “commensurate” and bring together all parts of a heterogeneous physical aggregate. Out of this possibility arises the frequently seen but inaccurate interpretation of any index weights as “commensuration coefficients” and “reduction factors” of the parts of a heterogeneous aggregate.

Rudimentary prototype indexes were first used more than two centuries ago. For example, in 1738, Dutot (France) compared the sums of the prices of a set of units of some commodities and published their ratio (Σ *p*_{1}/Σ *p*_{0}; the simple aggregate index); in 1764, G. Carli (Italy) calculated the primitive unweighted arithmetic index of the change in the prices of three commodities (bread, wine, olive oil) over a quarter of a millennium (between 1500 and 1750). In 1798, independently of Carli, G. Shuckburgh (Great Britain) began to calculate in the same manner the wholesale price index of ten goods; and in 1812, A. Young (Great Britain) introduced weights (from one to five for various commodities) into this index. However, only 50 years later, with the devaluation of silver and the consequent overall rise in world prices, especially in the 1860’s, did systematic calculation and publication of wholesale price indexes start in Great Britain. Two of the oldest published indexes are those of *The Economist* and of A. Sauerbeck. The index of the journal *The Economist* has used the formula *Σ ^{pl}/n* since 1869, first for 22 commodities and then after 1920 for 44 commodities; this is the oldest of the currently existing indexes. The index of the journal

*Statist*is a continuation of Sauerbeck’s (begun in 1886) and covers 36 commodities according to the same formula as

*The Economist’s*index. In the United States the price index was first calculated in 1881, for 1824–80. The foundations of the modern theory of price indexes were laid by the works of W. Jevons (Great Britain, 1863 and 1865), E. Laspeyres (1871), and H. Paasche (Germany, 1874). In Russia the first wholesale price indexes were published in a series of yearbooks, the

*Svod tovarnykh tsen*(for 45 commodities over the period 1890–1915, according to the formula of the unweighted arithmetic mean).

World War I and its aftermath brought tremendous price shifts to the world market and the economies of individual countries. Many new, previously unknown indexes were needed to study and measure these shifts, including the retail price index, the “cost of living” index (first in Great Britain in 1918, then in the United States in 1919), indexes of the physical volume of economic events (which eliminated the factor of continually changing prices), the purchasing-power index of currency units (in connection with the collapse of the world gold monometallism system and the attempts to replace currencies with the “purchasing power parities” of currencies), and various indexes for studying market conditions. Therefore, the last 50 years (since 1918) has become a new stage in the history of indexes, heralded by the unprecedented development of the index method of statistical science and by the expansion of the practice of indexing. In the USSR the calculation of the subsistence minimum for workers, which in 1922 was converted into the calculation of the budget index, started as early as 1918; in the period 1919–21 the calculation and publication of the indexes of the Market Institute were started; and the publication of wholesale price indexes of Gosplan (State Planning Commission) commenced in August 1922. In the planned economy of the USSR (and after World War II in other socialist states as well), the creation and regular calculation of a multitude of new indexes were needed, especially the planning assignment indexes and the degree of plan fulfillment indexes. The 1920’s, and subsequently the decade 1956–65, were years of especially intensive development of the theory of the Soviet indexing method as a powerful cognitive tool of contemporary Soviet statistics.

### REFERENCES

Nemchinov, V. S. “Sel’skokhoziaistvennaia statistika s osnovami obshchei teorii.”*Izbr. proizv*., vol. 2. Moscow, 1967. Chapter 19.

Suslov, I. P.

*Obschaia teoriia statistiki*. Moscow, 1970.

*Statisticheskii slovar’*. Moscow, 1965. (See the articles concerning indexes.)

*Uch. zap. po statistike AN SSSR*, 1955, vol. 1; 1959, vol. 5; 1963, vol. 7.

Iugenburg, S. M.

*Indeksnyi metod v sovetskoi statistike*. Moscow, 1958.

Peregudov, V. N.

*Teoreticheskie voprosy indeksnogo analiza*. Moscow, 1960.

Kazinets, L. S.

*Teoriia indeksov (Osnovnye voprosy)*. Moscow, 1963.

Ianovskii, A. S. “Russkie indeksy.” In I. Fisher,

*Poslroenie indeksov*. Moscow, 1928. Appendix 6, pp. 391–438. (Translated from English.)

Fisher, I. “Etapy istorii indeksov.”

*Ibid*., appendix 4, pp. 378–81.

Chetverikov, N. S.

*Statisticheskie i stokhasticheskie issledovaniia*. Moscow, 1963. Pages 13–56.

F. D. LIVSHITS