The index is a relative value that shows the dynamics of the process (phenomenon) under study both in time and in space, or reflects the level of implementation of the plan. In aggregate indices in the structural parts of a formula, arithmetic operations of addition, product, or the sum of the products of indicators of the phenomena or processes under study can be used.
The concept of individual considered objects
In order to come to the concept of an aggregate index, you need to first consider what an individual is.
The latter show the relationship of some elements of any combination, for example, the dynamics of the output of a particular product. In this case, the values are compared for two periods: base and reporting. The first of these is the one with which the comparison is made. Reporting is the current period. They are indicated by subscripts. Basic - 0, reporting - 1 (if several reporting periods, then subsequent natural numbers are used).
The quantity is usually designated q, the time spent on the production of a unit of production is t, the cost is z, and the price is p.
Based on this, not an aggregate volume index, but an individual, is expressed by the formula: i0 = q1 / q0. Similarly, with the substitution of the corresponding values, individual indices of prices, cost and time spent on manufacturing a unit of production are calculated.
The turnover index reflects the dynamics of the volume of this indicator in value terms: ipq = p1q1 / p0q0. Consequently, the dynamics of commodity circulation occurs when both the volume and the price of products sold change. Basically, indices are expressed in units. When multiplied by 100, a percentage transfer is possible.
Base and Chain Indexes
As already noted, to determine the considered objects must be present data for at least two periods.
If there is a certain amount of data for a certain series of levels or periods, either the initial level or that in relation to the previous period is taken as a base. For the first case, get indices with a constant base, called the base, for the second it is constantly changing, they are called chain.
The former are used to characterize the dynamics of some processes occurring over a long time period with respect to a certain point. If necessary, chain indexes are used to track current process changes.
The determination of the former through the latter and vice versa is carried out if, on the basis of them, a study of the same period occurs.
General indexes
They show the relation of such a totality of phenomena or processes consisting of elements that are directly incommensurable. These indices are called aggregate general indices. They are most prevalent in the economy.
So, for example, the aggregate turnover index is shown in Figure 2.
To construct any of the considered indices, three questions must be solved:
- determine the indexed value;
- with elements that are part of a process or phenomenon subject to indexing;
- choose weights.
The most typical representatives of these indices are the aggregate index of physical volume, prices, value.
Concept of weights
The latter are a value that is a measurer for indexed processes or phenomena. When calculating the index, the dynamics is in relation to the same value from the reporting period to the base, except for the labor productivity index for labor intensity.
Except where the value to be indexed is the entire product, the weights remain unchanged. The first is a qualitative indicator, and the second is quantitative in the indices of qualitative indicators, and vice versa, in such quantitative indicators. The composite indices are written so that the indexed value is in the first place, and the weight is in the last.
We choose a constant value - an example with a price index
The dynamics of prices is determined when taking as the considered number of sold goods for any one of the studied time periods. Regardless of which period is taken, the aggregate index formulas will look the same (shown in Figure 6). However, they cannot be considered identical. This is due to the fact that weights can be taken from either the reporting or the base period, which will give different values ββof the aggregate form of indices in the calculation.
If in their quality was taken the number of goods sold for the reporting period, the index reflects the price dynamics of the reporting period in comparison with the base for those goods for which sales were present in the reporting period and demonstrates the savings that could have taken place if prices for production declined.
Otherwise (weight is the number of goods sold in the base period), the index reflects the same dynamics, but only with respect to products sold in the base period, as well as the same savings. The latter is calculated as the difference between the numerator and denominator of the aggregate index. Thus, in this case, the variable to be indexed is the price, and the weight of the quantity of products sold in the reporting period is taken as a constant.
We choose a constant value - an example with a volume index
The aggregate index of the volume of goods turnover reflects the dynamics of the indicator embedded in the name when comparing the reporting period with the base one. For this, the prices of the base period are used as weights. With the help of this technique they get a true idea of ββthe change in the volume of trade.
The calculation of this index is carried out according to the formula shown in Figure 2.
The absolute dynamics is determined by the difference between the numerator and the denominator from the fraction of the calculation. Thus, weights for the base period are taken when constructing quantitative indices (number, physical volume), and for the reporting period - when constructing qualitative indices (cost, price, labor productivity ...).
Variable and constant weights for indices calculated over a number of periods
When calculating these values ββnot for one but for several periods, the weights can be the same for everyone (in this case we speak of indices with constant weights), or for each of them weights can be taken (with variable weights).
In this case, the classification of index types is carried out as follows:
- general basic price indices with constant weights;
- they are with variable weights;
- two of the same varieties for chain indexes.
They are obtained by comparing the prices of subsequent periods with the preceding each of them, weighted by the number of products taking place in the reporting period.
Base indices with constant weights are used if it is necessary to analyze the dynamics of prices, without taking it into account in the structure of manufactured goods. If accounting is carried out, then similar indices with variable weights are calculated.
If we study the dynamics of each period in comparison with the previous one, then we apply the corresponding chain indices.
The concept of the average index
Not always the information available makes it possible to determine aggregate indices. The averages in this case are calculated as arithmetic mean or harmonic mean values. Mathematically, they are equal to aggregate and coincide with them in meaning.
The value of the arithmetic mean index will be equal to the aggregate if the terms of the denominator of the latter are used as weights. This index in relation to the physical volume of production is determined by the formula: Ip = (Ξ£ lp x p0q0) / (Ξ£ p0q0).
The average harmonic index will be identical to the aggregate in the case if the terms of the numerator of the latter are used as weights. Calculation of the average harmonic index by the example of prime cost is shown in Figure 4.
It is calculated if in the reporting period individual values ββof prices and quantities of goods sold are not known, but the goods turnover for a given period, as well as individual price indices are known, and reporting weights are used to calculate the composite index.
The average value index is affected by the dynamics of the cost of production in each enterprise under consideration and changes in the contribution of the latter to the total volume of production. If the average values ββin the indices undergo dynamics due to the influence of indexed values, and the weights remain constant, then these indices are called constant composition. The change in the average indicators using indices is carried out after the breakdown of the entire population into expenditures by the characteristics that characterize structural shifts. In this case, calculations are performed for group averages.
Territorial indicators considered
They are used to compare indicators in space. If they are bilateral, then any territory can act as a comparable and basic.
Weights can also be used for any space. To avoid conflicting results, resort to the use of two methods. The first of them provides that the total weight of goods sold in two regions applies, which will be used in the formula of the territorial price index. Another method involves taking into account the ratio of weights in each of the compared spaces. First, the average price of goods is calculated over two territories, after which these data are substituted into the formula and the territorial index is calculated.
Other indices
- Aggregate cost index. With its help, the difference in costs per unit of production in the compared periods or planned indicator is demonstrated. Refers to quality indices, so weights are used in the reporting period.
- Labor productivity index . It represents the ratio of the time required to produce a unit of production for the two periods in question.
- Labor intensity index. It shows the dynamics of this indicator in the two periods under review. Also applies to quality indicators. For calculation, reporting weights are used.
Finally
Thus, when calculating aggregate indices in the numerators and denominators of calculation formulas, products, sums or sums of the products of some indicators of the studied processes or phenomena can be used. When calculating these coefficients, it is necessary to choose what will act as weights, for which it is necessary to determine: the calculated index is related to qualitative or quantitative. There are many similar coefficients that allow you to compare the analyzed indicators in the compared periods.