The summary index consists of two elements. Average indexes

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;
• foreign trade index;
• capital investment index;
• consumer index and indices - deflators.

Industrial price index shows the price level for goods and services purchased by industrial enterprises (factories, factories, construction organizations, etc.) 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, and the average price index takes into account the share of 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.

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

It is worth noting that in the 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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Composite trade turnover index (general trade turnover index):

Composite price index (general price index):

Consolidated index of physical volume of sales (general index of physical volume of sales):

General trade turnover index (for collective farm market No. 1)

Due to all factors, total trade turnover increased by 9%. Total trade turnover increased by ∆ T = 152-140 = 12 rub.

General price index (for collective farm market No. 1)

General index of physical production volume (for collective farm market No. 1)

Let us show the relationship between the indices I pq = I p I q = 1.08*1.01 = 1.09

To solve similar problems, select the number of rows and Object of analysis.

Example No. 1. The store's turnover in the reporting period amounted to 4,426 thousand rubles, prices increased by 20%. Determine the price index.
Solution. Here the price index is (100+20)/100 = 1.2

Example No. 2. How will the physical volume of sales of goods by retail enterprises change in the current period compared to the previous one, if turnover increased by 9.5% and prices increased by 2.1%.
Solution. Trade turnover index: I = iq*ip
I = (100+9.5)/100 = 1.095
ip = (100+2.1)/100 = 1.021
Where does iq = I/ip = 1.095/1.021 = 1.72 or 7.2%.

:

The numerator of this index reflects the production costs of the current period, and the denominator - the conditional value of costs while maintaining the cost at the basic level. The difference between the numerator and denominator shows the amount of savings (overspending) of the enterprise from changes in cost:

Index of physical volume of production:

Production cost index:

The index method also finds application in agricultural statistics: the index of gross crop yield (I rs) can be obtained through the yield index (I r) and the sown area index (I s).

Mean harmonic index

Summary price index in harmonic mean form:

Trade turnover for this product group increased by an average of 1.6%.

Summary index of physical volume of trade turnover in the form of harmonic average:

Due to changes in the price structure, the average price increased by 8%. If the structure of product sales across markets had not changed, the average price would have increased by 8%.

Index of structural changes

or I s.s. = I p.s. /I f.s. I s.s. = 1.073/1.08 =0.995

Due to changes in the sales structure, the average price decreased by 0.5%

“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 - summary 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 the product in the base Period;

The change in the physical mass of goods sold in physical terms is 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 a general indicator in which all its elements can be combined. 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, but in physical units of measurement. The scales in this case are prices that 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%). The following relationship exists 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.

Index; individual index; general (aggregate) index; chain indexes; basic indices; variable composition index; index of permanent (fixed) composition; general index of physical volume of production; general price index; general cost index; average price index

Indices are the most important type of generalizing statistical indicators. They are used to characterize the dynamics of phenomena, comparisons across different territories, for monitoring and developing plan targets. Along with averages, they represent one of the most common types of statistical indicators. The word “index” translated from Latin means pointer, indicator. In statistics, this term has a specific meaning. Index– this is a relative value that characterizes the change in complex social phenomena in time, space or in comparison with the plan.

The index is the result of comparing two quantities of the same name, so it is necessary to distinguish between the comparison value (the numerator of the index ratio) and the comparison base (the denominator). The choice of comparison base is determined by the purpose of the study; when studying dynamics, data from a previous period is used as a base; when monitoring the implementation of the plan - planned data; for territorial comparisons – data from another territory.

The comparison value is usually called the reporting period indicator, the comparison base is called the base period indicator. If the base level when calculating the index is taken as one, then the indices are calculated in the form of coefficients, and if the base level is taken as 100, then the index is calculated in the form of percentages. Based on the calculation, you can determine how many times the reported value is more or less than the base value or by what percentage it is more or less than the base value.

Statistics studies mainly complex economic phenomena that consist of directly incommensurable elements. Thus, if an electromechanical plant produces several types of products, then data on product output in physical terms cannot be summarized. In order to show the overall change in output for several types of products, indices are calculated. With their help, it is possible to give a generalized description of changes in costs, prices, and output for several types of products.

With all their diversity, economic indices are divided into individual and general indices.

An index is called individual, characterizing changes in production volume, sales volume, level of labor productivity, etc. in relation to any one product. For example, the following data is available on the production of AC electric motors with a rotation axis height of 63-450 mm (thousand units) 1998 - 448; 1999 – 188. Let us determine the individual index of physical volume of production:

; , i.e. there was a decrease in production volume by 58%.

Individual indexes:

cost,

cost .

A general (aggregate) index is called, characterizing the general (average) change in production volume, sales volume, price levels, etc. in relation to a set of product series. For example, indices showing changes in the total volume of production of various types of products or changes in the price level of various types of goods in general. When calculating general indices, problems arise in comparing indicators for individual goods. The commensurability of individual indicators is achieved by weighing, the essence of which is that when calculating, one abstracts from the influence of changes in one of the aspects of the phenomenon being studied, taking it as a constant value. Thus, when calculating the index of the volume of products sold, prices will be constant values, and when calculating the price index, the quantity of products sold will be constant. That aspect of the phenomenon under study, the influence of which changes are abstracted from, taking it as unchangeable, is called index weights.

Volume indexes include indices of physical volume of production, number of workers, total consumption of materials. They measure the total, total volume of a particular phenomenon.

We will consider methods for constructing indices of volumetric indicators using the example of the index of physical volume of production. When calculating it, the task is to characterize the change in the volume of all products manufactured by an enterprise or group of enterprises.

Individual indices of the physical volume of products characterize the change in output for each type of product; their formula can be written as follows:

where and are the output of products of this type in the reporting and base periods, respectively.

In essence, these indices do not differ from relative values ​​and represent the ratio of the quantity of products of the reporting period to the quantity of products of the base period.

To obtain a generalized characteristic of the dynamics for the entire set of manufactured products, it is calculated aggregate (general) index of physical volume of production.

To ensure that the index reflects only the change in the indexed volume indicator, the weights in its numerator and denominator are fixed at the level of the same period. In this case, in order to show the change in production volume, it is necessary to eliminate the change in prices. This is achieved by the fact that the products of the reporting and base periods are calculated in the same (fixed) prices.

,

where is the indexed value;

– prices are comparable (basic).

Indices of quality indicators include price indices, product cost indices, average wage indices, labor productivity indices, and unit cost of materials indices. These indices characterize indicators that are of a calculated nature. They measure the intensity and effectiveness of a phenomenon and are either average or relative values.

Let's consider the calculation of the individual and general index of quality indicators using the price index as an example.

Individual price index characterizes the change in price for each type of product:

where and are the price of the reporting and base periods, respectively.

The general (aggregate) index of a quality indicator is tasked with measuring not only the relative change in the level, but also the absolute value of the economic effect that was obtained in the current period as a result of this change. In this case, the amount of savings customers save due to lower prices, or the amount of their additional expenses if prices have increased.

To obtain a general price index, it is necessary to construct it in such a way that only the influence of the factor of price changes is reflected, and the influence of changes in the number of goods sold is excluded. This is possible if the same number of goods sold is taken for both periods being compared. The quantity of goods sold should be taken in the current period, since only by purchasing this quantity can the consumer save as a result of lower prices or overspend as a result of their increase.

General price index:

– Paasche price index,

where is the indexed value;

The numerator of the index gives the total cost of goods sold in the current period at prices of the current period, and the denominator shows the cost of the same quantity of goods, but calculated at prices of the base period.

Savings (overexpenditure) from price changes: .

In statistics, other forms of presenting general price indices are used - Laspeyres and Fisher:

– Laspeyres price index,

– Fisher price index.

If, when calculating indices, there are three or more periods being compared, then the question arises about choosing a comparison base. Depending on the basis of comparison, chain and basic indices are distinguished.

Chained indexes obtained by comparing the indexed indicator of any period with the indicator of the period preceding it. Basic indices are calculated by comparing the indexed indicator of each period with the corresponding indicator of the period taken as the comparison base.

Chain aggregate index of physical volume of production:

; .

Consistently multiplying chain aggregate indices of the physical volume of production makes it possible to obtain a basic index

.

Aggregate indices of quality indicators are always indices with variable weights, since the weights of the reporting period are always used in their calculation. Therefore, the chain method of calculating basic indices is not acceptable for them.

The aggregate method of calculating general indices is the main one, but not the only one in statistics. In some cases, due to the lack of certain data, it is impossible to calculate using the aggregate index formula. This may occur if there is no data on the absolute value of the indexed value, i.e. the value of the indicator characterizing that side of the phenomenon, the change of which is being studied (for example, when calculating the index of physical volume of production, there is no data on the volume of production as a whole). In this case, average indices are used.

18. Indexes – these are relative indicators that characterize average measurements in time, space in comparison with the plan or standard of individual or complex social phenomena, the elements of which cannot be directly summed up.

For convenience of working with indexes, we will use the following notation:

g 1 and g 0 are the physical volume (quantity) of products produced or sold in the reporting (g 1) and base (g 0) periods, respectively;

p 1 and p 0 – unit price;

p 1 g 1 and p 0 g 0 – cost (turnover) of manufactured or sold products;

z 1 and z 0 – cost of manufactured products.

There are indices of volumetric (quantitative) and qualitative indicators.

To the indices of volumetric indicators include indices of physical volume of production, gross harvest, etc.

To the indexes of quality indicators include indices of prices, costs, labor productivity, etc.

Depending on the coverage of population units, indices are divided into individual and general.

Individual indices– this is the ratio of the level of an indicator in the current (reporting) period to the same indicator in the base period (i).

General indices are used to compare directly incommensurable heterogeneous phenomena.

Aggregate indices consist of two elements: the indexed value and the weight attribute.

The indexed value is an indicator of change that is reflected by the index.

A weight attribute (commensurate) is an indicator that allows you to move from incommensurable elements to commensurable ones.

In statistics, there is a rule for constructing aggregate indices, according to which the weights in the indices of volume indicators are taken at the level of the base period, and the weights in the indices of quality indicators are taken at the level of the reporting period.

Aggregate index of physical volume of products (turnover)

Aggregate price index

Aggregate index of the cost of manufactured or sold products (turnover)

The relationship between these indices I pg = I p ·I g

- aggregate index of the cost of manufactured products

- aggregate index of the physical volume of manufactured products

- aggregate index of production costs Relationship between these indices zg = I z ·I g

To correctly compile a general index, the following requirements must be taken into account:

1) the numerator and denominator of the general index will always contain the sum of the products of the indexed value and the indicator taken as the weight of the index;

2) the choice of index weights is determined by the economic content of the phenomenon being studied. When indexing quality indicators, weighing is carried out according to reporting scales; when indexing volumetric (quantitative) indicators, weighing is carried out using basic scales;

3) when indexing two indicators, such as trade turnover - pq; production costs - zq, etc.

The general index is constructed as a relative value of dynamics: in the numerator – the reporting period – p 1 × q; in the denominator the base – p 0 × q 0 (compared period);

4) when compiling a system of interrelated indices, first establish relationships between the initial indicators, then move on to a system of interrelated indices.

For example:

pq = p × q; Jpq = Jр × Jq.

Let's look at the construction of the aggregate form of the index using an example.

The prices and quantities of goods sold in the market of the city are known.

Table 6.1

Determine the change in prices and quantities of goods in general for all goods in the reporting period compared to the base one.

Individual indices for individual types of vegetables are calculated as follows: for potatoes, the number of sales was - , i.e. the quantity of potatoes sold increased by 1.2 times or by 20% = 120 – 100. for potatoes 8.0: 6.0 = 1.333, thus the price increased by 1.333 times or by 33% = 133 – 100.

So, we need to construct general indices of prices and quantities of goods sold - J p; Jq.

According to the above rule, the price index is equal to

We take the quantity of goods sold as the weight, but since the indexed value is a qualitative indicator, we take the weight in the reporting period.

Thus, prices for all three goods increased by 69.2% = 169.2 – 100. This is in relative terms, but in absolute terms they increased by 103,500 rubles. = 253,000 – 149,500.

The economic effect, or otherwise the amount of money saved or overspent due to price changes, is calculated according to the general price index and is equal to the difference between the numerator and denominator of the index: Σр 1 q 1 – Σp 0 q 0 ; therefore, due to a price increase of 69.8%, the population spent an additional 103,500 rubles in the reporting period. for the purchase of these goods.

Let's determine the general index of physical volume

since physical volume is a quantitative indicator, the weights are taken in the base period.

Consequently, not only prices increased, but also the number of vegetables sold increased by 20.5% = 120.5 - 100, which in absolute terms is: 25,500 rubles. = 149,500 – 124,000.

If the absolute value, i.e. the difference between the numerator and denominator is positive, then the sales effect is received by the seller. If the absolute value is minus, then the buyer receives the amount of savings.

Now let's see what the seller received from the sale of these goods. according to the third rule for constructing general indices, when two factors influence simultaneously, i.e. on the dynamics of trade turnover.

Consequently, trade turnover will increase by 2.04 times, and in absolute terms this amounted to 129,000 rubles.

So, we traced the impact of each factor separately in relative and absolute terms on the price and quantity of vegetables sold, and also identified the influence of two factors at once.

Now let's see how the general indices are interrelated. In mathematics, p × q = pq; exactly like this in the indexes

J pq =J p × J q,

according to our example: 1.692 × 1.205 = 2.046.

Therefore, the indexes are compiled correctly.

Any aggregate index can be represented as a weighted value of individual indices

Let's substitute it into the general price index

then we get the harmonic mean weighted index

from here q 1 = iq ×q 0, substitute into the aggregate form of the general index of physical volume

We obtained a weighted average index. These are the purposes for which an individual index is used, i.e. expands the capabilities of the aggregate form of the index.

The use of the original form of the aggregate index or the harmonic mean, weighted average index depends on the initial data available to the researcher.

19 Depending on the methodology for calculating individual and consolidated indices, they are distinguished arithmetic averages And average harmonic indices. In other words, the overall index constructed from the individual index takes the form of an arithmetic mean or harmonic index, i.e., it can be converted into an arithmetic mean and a harmonic mean.

The idea of ​​constructing a composite index in the form of an average value from individual (group) indices is quite understandable: after all, the composite index is a general measure characterizing the average change in the indexed indicator, and, of course, its value should depend on the values ​​of individual indices. And the criterion for the correct construction of a composite index in the form of an average value (average index) is its identity with the aggregate index.

The transformation of the aggregate index into the average of the individual (group) indexes is carried out as follows: either in the numerator or in the denominator of the aggregate index, the indexed indicator is replaced by its expression through the corresponding individual index. If such a replacement is made in the numerator, then the aggregate index will be transformed into the arithmetic average, but if in the denominator, then into the harmonic average of the individual indices.

For example, the individual physical volume index is known IQ y = K1/value q0 and the cost of products of each type in the base period (d0 p0). The initial basis for constructing the average of individual indices is the composite index of physical volume:

(aggregate form of the Laspeyres index).

From the available data, only the denominator of the formula can be obtained by direct summation. The numerator can be obtained by multiplying the cost of a particular type of product of the base period by an individual index:

Then the formula for the summary index will take the form:

i.e., we obtain the arithmetic average index of physical volume, where the weights are the cost of individual types of products in the base period.

Let us assume that information is available on the dynamics of the volume of output of each type of product (r^) and the cost of each type of product in the reporting period (p1q1). To determine the overall change in the enterprise's output in this case, it is convenient to use Paasche's formula:

The numerator of the formula can be obtained by summing the quantities q1P1, and the denominator - by dividing the actual cost of each type of product by the corresponding individual index of the physical volume of production, i.e. by dividing: p1q1/on IQ, Then:

Thus, we obtain the formula for the average weighted harmonic index of the physical volume.

The application of one or another formula for the physical volume index (aggregate, arithmetic mean and harmonic mean) depends on the information available. You also need to keep in mind that the aggregate index can be transformed and calculated as the average of the individual indices only if the list of types of products or goods (their assortment) coincides in the reporting and base periods, i.e. when the aggregate index is built according to comparable circle units (aggregate indices of quality indicators and aggregate indices of volume indicators, subject to a comparable assortment).