Composite price indices. Basic formulas for calculating summary or general indices

“Index” translated from Latin is a pointer or indicator. In statistics, an index is an indicator of the relative change in a given level of the phenomenon under study compared to its other level, taken as the basis of comparison. As such a base, either the level for any past period of time (dynamic index) or the level of the same phenomenon in another territory (territorial index) can be used. Indices are an indispensable research tool in cases where it is necessary to compare two populations in time or space, the elements of which cannot be directly summarized.

In general, the index method is aimed at solving the following problems:

characterization of the general change in the level of a complex socio-economic phenomenon;

analysis of the influence of each factor on the change in the indexed value by eliminating the impact of other factors;

analysis of the influence of structural changes on changes in the indexed value.

In the further presentation of the index method, the following generally accepted notations will be used:

i - individual index;

I - composite index;

q - quantity;

1 - current period;
0 - base period.

The simplest indicator used in index analysis is individual index, which characterizes the change in time of economic values related to one object:

Price index,

where p 1 is the price of the product in the current period;

R 0 - price of goods in the base Period;

Change in the physical mass of goods sold in in kind measured by an individual index of the physical volume of sales:

A change in the value of trade turnover for a given product will be reflected in the value of the individual trade turnover index. To calculate it, the turnover of the current period (the product of the price and the quantity of goods sold) is compared with the turnover of the previous period:

This index can also be obtained as the product of an individual price index and an individual index of physical sales volume.

Individual indices are essentially relative indicators of dynamics or growth rates, and from data over several periods of time can be calculated in chain or basis forms.

Unlike individual indices, composite indices allow you to summarize indicators for several products. The initial form of a summary index is the aggregate form.

The aggregate form of the index makes it possible to find for a heterogeneous population such general indicator, in which you can combine all its elements. When analyzing price dynamics, it is unlawful to add up the individual prices of various goods, but summing up the turnover of these goods is quite acceptable. In the current period, such trade turnover is P goods will be:

If we compare trade turnover in the current period with its value in the base period, we get composite turnover index:

To illustrate this and subsequent indices, we will use the following conditional data (Table 10.1.):

Table 10.1 Prices and sales volumes of three products

Let's calculate the trade turnover index:

The calculated value of the index allows us to conclude that trade turnover in general for this product group in the current period compared to the base period increased by 8.9% /108.9% - 100.0%/. Note that the size of the product group and units of measurement of goods do not matter when calculating this and subsequent indices.

The value of the trade turnover index is formed under the influence of two factors - it is influenced by both changes in prices for goods and changes in their sales volumes. In order to evaluate changes in prices only (the indexed value), it is necessary to fix the number of goods sold (index weights) at some constant level. When studying the dynamics of such indicators as price and cost, the physical volume of sales is usually fixed at the level of the current period. In this way they get composite price index(according to Paasche's method):

For the example under consideration we get:

Thus, for this product group, prices in February increased by an average of 10.7% compared to January. When constructing this index, the price acts as the indexed value, and the quantity of goods sold acts as the weight.

Let's look at the summary price index in more detail. The numerator of this index contains the actual turnover of the current period. The denominator is a conditional value showing what trade turnover would be in the current period if prices remained at the basic level. Therefore, the ratio of these two categories reflects the changes in prices that have taken place.

The numerator and denominator of the composite price index can also be interpreted in another way. The numerator represents the amount of money actually paid by customers for goods in the current period. The denominator shows how much buyers would pay for the same goods if prices did not change. The difference between the numerator and the denominator will reflect the amount of savings (if the sign is “-”) or overexpenditure (“+”) of buyers in the region from price changes:

It should be noted that in statistical practice a summary price index is also used, constructed using the Laspeyres method, when weights or sales volumes are fixed at the level of the base rather than the current period:

The third index in the index system under consideration (including the price index calculated using the Pache method) is summary index of the physical volume of sales. It characterizes the change in the quantity of goods sold not in monetary terms, but in physical units measurements. Libra in in this case prices are fixed at a basic level:

In our case, the index will be:

The physical volume of sales (turnover) decreased by 1.6% (98.4%-100.0%). There is the following relationship between the calculated indices:

Or 1.107-0.984 = 1.089

Based on this relationship, using the values of two known indices, it is always possible to determine the unknown value of the third index.

Concept of price index

Price index- this is an indicator in statistics that is used to calculate price dynamics in a certain time period.

Calculations are carried out in the following sequence:

1. Selection of objects for calculations using a representative sample (various sectors of the economy);

2. Selecting a system for weighing indicators;

3. Selecting a formula for calculating indices.

Types of price indices

Price indices are distinguished according to the basic objects for calculation. These include:

industrial price index;
agricultural price index;
transport tariff index;
index foreign trade;
capital investment index;
consumer index and indices - deflators.

Industrial price index shows the price level for goods and services that industrial enterprises (plants, factories, construction organizations, etc.) purchase for their production and technical purposes.

Agricultural Price Index shows the dynamics of food price fluctuations.

Transport Tariff Index includes prices for cargo transportation and transit payments (including transit of gas, oil and other resources).

Foreign trade price index shows the dynamics of prices for exported and imported goods. The price of goods that are produced for own consumption is not taken into account when calculating this index. For example, if one company produces the same product both for export and for the domestic market, then to calculate the foreign trade index, the price indicator of only that part of the product that was sold abroad is taken.

Deflator index- shows changes in one macroeconomic indicator (usually national accounts indicators) in the current period in relation to the base one.

Producer price indices indicate price dynamics in a certain sector of the economy. Unlike the industrial index, which tracks the dynamics of enterprise costs, the producer index tracks the dynamics of income from the sale of goods and services.

Each state creates a certain set of goods and services necessary to ensure a minimum standard of living. It is called consumer basket. An index that shows changes in the price of the consumer basket is called the consumer price index.

Consumer price index is an index display of the price of a typical market basket of domestic and imported consumer goods and services that are purchased in the country’s domestic market. When calculating it, the cost of a basket of goods and services of a fixed composition in the current and base periods is compared.

All price indices are used to track changes in prices and tariffs on the market, study its conditions, to calculate the standard of living and the impact of price dynamics on it. Also, all indices are used in the analysis of the macroenvironment and serve as the basis for calculating various indicators of the system of national accounts. These include gross external product (GDP), gross domestic product (GNP), national income and others. All these indicators are used to select and adjust the state's macroeconomic policy. As an inflation index, mainly two price indices are used: the consumer price index (CPI) and the GDP price index, that is, the GDP deflator (Defl).

Methods for calculating the price index

The methods and methods for calculating the price index are the same for all types of indices.

When calculating price indices, the actual index and the average price index are obtained. The actual index shows the absolute deviation of the price level, while the average price index takes into account specific gravity each product in a representative sample, adjusting not only the price level, but also its structure.

All price indices can be divided into individual and group.

An individual index takes into account only the change in price for one type of product:

p1 - prices of the reporting period;

p0 - prices of the base period;

The group price index takes into account the dynamics of the prices of all goods in the sample and is calculated as the sum of prices of the current period in relation to the sum of prices of the base period.

To calculate the price index in the economy, three methods are used:

Paasche index;
Laspeyres index;
Fisher index.

The Laspereys index shows how prices for products sold in the base period have changed. In other words, when calculating the index, we compare the cost of products that were sold in the previous period, but in the prices of the current period, in relation to the same number of goods, but in the prices of the previous period. Formula for calculating the Laspereys index:

p1 - prices of the reporting period;

p0 - prices of the base period;

q0 is the number of goods sold in the base period.

The Paasche price index displays how prices for products sold in the reporting period have changed, compared to the prices of the base period, by the number of goods sold in the reporting period.

q1 - the number of goods sold in the base period.

It is worth noting that, in Russian Federation Since 1991, the Laspeyres index has been used to calculate price indices. The Paasche index does not take into account the drop in demand for certain goods during periods of economic downturns and inflation, so its use becomes impractical.

The Paasche index somewhat underestimates the level of inflation, since it does not take into account assortment shifts in the current period relative to the base one. The Laspeyres index overestimates the inflation rate because it does not take into account the substitution effect of expensive goods with similar cheap goods. To eliminate these disagreements, it is proposed to use the I. Fisher index, which is calculated as the geometric mean value of the Laspeyres and Paasche indices:

But calculating the Fisher index is very labor-intensive. Therefore, in economic practice this index is calculated very rarely.

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Prices of various goods sold in retail trade, it is illegal to add up, but from an economic point of view it is permissible to sum up their turnover. If we compare trade turnover in the current period with its value in the base period, we obtain a composite trade turnover index:

Example 1. The following data is available on the sale of vegetable products at the city market:

Let's calculate the turnover index for example 1:

We find that trade turnover as a whole for the product group under consideration in the current period decreased by 0.9% (100-99.1) compared to the base period.

The value of this index is influenced by both changes in prices for goods and changes in their sales volumes. In order to evaluate changes in prices only (the indexed value), it is necessary to fix the number of goods sold (index weights) at some constant level. When studying the dynamics of such qualitative indicators as price, cost, labor productivity, the quantitative indicator is usually fixed at the current level. In this way, a summary price index is obtained:

The numerator of this index contains the actual turnover of the current period. The denominator is a conditional value showing what trade turnover would be in the current period if prices remained at the basic level. Therefore, the ratio of these two categories reflects price changes. A change in the quantity of products sold does not affect the value of the index.

Let's calculate the summary price index for example 1:

Consequently, prices for this product group in October compared to August decreased by 31.7%.

The numerator and denominator of the composite price index can be interpreted from the consumer's point of view. The numerator represents the amount of money actually paid by customers for goods purchased in the current period. The denominator shows how much buyers would pay for the same goods if prices did not change. The difference between the numerator and the denominator will reflect the amount of savings (if the sign is “-”) or overexpenditure (if the sign is “+”) of buyers from price changes:

The third index in this index system is the consolidated index of the physical volume of sales. It characterizes the change in the quantity of goods sold not in monetary, but in physical units of measurement:

The weights in this index are prices that are fixed at a basic level.

The index of physical sales volume in example 1 will be:

The physical volume of sales in October compared to August increased by 1.45 times, or by 51,500 rubles. (165,500 - 114,000).

There is the following relationship between the calculated indices:

IP × Iq = Ipq.

Using the relationship of indices, let's check the correctness of the calculations in example 1:

Ipq= IP × Iq= 0.683 × 1.452 = 0.991, or 99.1%.

Consequently, the decrease in turnover (by 0.9%) was due to an increase in the volume of products sold (by 45.2%) and a decrease in prices (by 68.3%), which in absolute terms amounted to -1000 rubles. (-52,500 + 51,500).

We examined the use of aggregate indices in the analysis of commodity price turnover and physical sales volume. When analyzing the results of production activities industrial enterprise the given summary indices are respectively called the product cost index, the wholesale price index and the physical volume index.

Chapter 10. Economic indices

An economic index is a relative value that characterizes the change in the phenomenon under study in time, in space, with some standard (planned, normative, previous level, etc.).

The individual index characterizes the change over time of individual elements of the population, individual price index, is calculated by the formula:

Where p i is the price in the current period, p 0 is the price in the base period.

For example, p i =30, p 0 =25

the price increased by 20% compared to the base level.

Individual index of physical sales volume:

Where q i is the quantity of goods sold in the current year, q 0 is the quantity of goods sold in the base year.

Individual turnover index:

A composite index is a relative indicator that characterizes the average change in a socio-economic phenomenon consisting of incommensurable indicators.

Composite turnover index calculated using the following formula:

Summary price index:

The quantity (weight) is fixed at a constant level. When studying the dynamics of such indicators as P-price, Z-cost, W-yield, the quantitative indicator is characterized by the current level.

Consolidated index of physical sales volume:

The weight is the price, which is fixed at the basic level.

There is the following relationship between the indices:

We examined the use of aggregate indices in the analysis of trade turnover and prices. When analyzing the results of production activities of an industrial enterprise, the above composite indices are respectively called the product cost index, the wholesale price index and the physical volume index of production.

Let's consider the use of the index method in the analysis of changes in production costs and production costs.

Individual cost index characterizes the change in the cost of a particular type of product in the current period compared to the base period:

To determine the overall change in the cost level of several types of products manufactured by the enterprise, a composite cost index is calculated. In this case, the cost is weighted by production volume individual species products of the current period:

The numerator of this index reflects the production costs of the current period, and the denominator is the conditional value of costs while maintaining the cost at the basic level. The difference between the numerator and the denominator shows the amount of savings an enterprise can make from reducing costs:

Composite index of physical volume of production, weighted by cost. has the following form:

The third indicator in this index system is composite production cost index:

All three indices are interconnected:

Another area of application of index method - analysis changes in labor productivity. In this case, two approaches to calculating indices are possible. The first approach is based on taking into account the amount of products produced per unit of time (w).

With such calculations, it is necessary to solve a number of methodological problems - which indicator of production to use, how to evaluate the products of service sector workers, etc.

In the second approach, labor productivity is determined by the cost of working time per unit of production (t). In practice, these calculations are also fraught with certain difficulties, since it is not always possible to assess the contribution of a particular employee to the production of a particular product.

The amount of products produced per unit of time (in physical terms) and the time spent per unit of production are interrelated:

For example, if an employee spends 15 minutes on each product. (t=0.25 h), then per hour its production will be 4 products. Note that production can be measured not only in physical terms, but in value terms (pq).

Individual labor productivity indices, based on these indicators, have the following form:

;

where T is the total time spent on producing a given product in man-hours, man-days or man-months (in the latter case it corresponds to the total number of employees).

Labor intensity is an inverse indicator, so a decrease in labor intensity in the current period compared to the base period indicates an increase in labor productivity.

Having data on labor intensity various types products and their production volumes can be calculated composite labor productivity index (by labor intensity):

The denominator of this index reflects the actual total time spent on the production of all products in the current period (T 1). The numerator is a conditional value showing what the time spent on producing this product would be if labor intensity did not change.

The labor productivity index by labor intensity is related to the index costs of working time (labor) and with index physical volume of products, weighted by labor intensity:

When calculating composite index of labor productivity in value terms (by output) it is necessary to weigh the quantity of products produced for each period at some prices accepted as comparable. Prices of the current, base or any other period or average prices can be comparable. The index in this version is calculated using the formula:

The first part of this formula represents the average output in the reporting period, the second part - in the base period.

Multiplying the labor productivity index for output by the working time cost index leads to index of physical volume of production, weighted by price:

Summary indices in arithmetic mean and harmonic mean forms. In some cases, in practice, instead of indexes in aggregate form, it is more convenient to use arithmetic averages and harmonic averages. Any composite index can be represented as a weighted average of the individual indices. However, in this case, the form of the average must be chosen in such a way that the resulting average index is identical to the original aggregate index.

Suppose we have data on the cost of products sold in the current period (p 1 q 1) and individual price indices obtained, for example, as a result of sample observation. Then in the denominator of the summary price index you can use the following replacement:

Thus, the summary price index will be expressed in the form of the harmonic mean of the individual indices:

Indices of constant and variable composition. All indices discussed above were calculated for several goods sold in one place, or types of products produced at one enterprise. Let us now consider the case when one product is sold in several places or a type of product is produced at a number of enterprises.

If only one type of product is sold, it is quite legitimate to calculate it average price in each period. Variable Composition Index represents the ratio of the two obtained average values:

This index characterizes not only changes in individual prices at points of sale, but also changes in the sales structure of retail or wholesale trade, markets, cities, regions. To assess the impact of this factor, it is calculated structural change index:

The last one in this system is the one discussed above. fixed composition price index, which does not take into account structure changes:

There is the following relationship between these indices:

This discrepancy is explained by the influence of changes in the structure of sales of goods by region: in June there were more high price sold twice as much goods, but in July the situation changed fundamentally (in this conditional example, for clarity, the numbers were selected in such a way that this difference in the sales structure was obvious).

Let's calculate the index of structural changes:

Or 89.1%.

The first part of this expression allows us to answer the question of what the average price would be in July if prices in each region remained at the same June level. the second part reflects the actual average price in June. In general, based on the obtained index value, we can conclude that due to structural changes, prices decreased by 10.9%.

The calculated fixed composition price index is 1.098, or 109.8%. This leads to the conclusion: if the structure of sales of product A by region had not changed, the average price would have increased by 9.8%. However, the influence on the average price of the first factor turned out to be stronger, which is reflected in the following relationship:

1,098*0,891=0,978.

Indices of structural changes, variable and fixed compositions are constructed similarly to analyze changes in cost, yield, etc.