Specific Index Numbers and Applications

Consumer Price Index (CPI): Construction and Use

Definition

The Consumer Price Index (CPI), often referred to as the Cost of Living Index, is a measure designed to track the average change over time in the prices paid by a specific group of consumers for a fixed basket of consumer goods and services. It quantifies the relative change in the expenditure needed by households to purchase a representative selection of items they typically consume.

The CPI essentially compares the cost of a standard basket of goods and services in the current period to its cost in a designated base period. The base period cost is set to an index value of 100, and the current period index reflects the current cost relative to this base.

---

Purpose and Use

The CPI is a critically important economic indicator with widespread applications:

• Primary Measure of Consumer Inflation: CPI is the most commonly used gauge to measure the rate of inflation as experienced by average households. It tracks changes in the prices of goods and services purchased at the retail level, directly impacting consumers' purchasing power.
• Cost of Living Adjustments (COLA): CPI data is extensively used to adjust wages, salaries, pensions, social security benefits, and other forms of income to help individuals maintain their purchasing power in the face of rising prices. In India, the calculation of Dearness Allowance (DA) for government employees and pensioners is often directly linked to movements in specific CPI series, such as the CPI for Industrial Workers (CPI-IW).
• Guide for Economic Policy: Central banks, such as the Reserve Bank of India (RBI), closely monitor CPI inflation (specifically the CPI Combined) as a key indicator for formulating and implementing monetary policy, particularly regarding setting interest rates to control inflation. Governments use CPI data for fiscal policy planning and analysis.
• Deflating Consumer Expenditure: CPI is used to convert consumer expenditure data from nominal values (current prices) to real values (constant prices). This process of 'deflation' removes the effect of price changes, allowing analysts to measure changes in the actual volume or quantity of goods and services consumed.
• Inter-regional Comparisons: While the main CPI series measures change over time, specific studies or indices might be constructed to compare the cost of living or price levels between different cities or regions at a given point in time.
• Economic Analysis and Research: CPI provides essential data for economic researchers and analysts studying consumer behaviour, living standards, and price dynamics.

---

Construction (Conceptual Steps)

Constructing a reliable Consumer Price Index is a complex process involving several stages:

1. Define Target Population and Scope: The first step is to clearly define the specific group of consumers whose price experience the index aims to represent (e.g., all urban households, a particular income group, industrial workers in certain cities). The geographical coverage (national, state, specific cities) is also determined.
2. Select Base Period: A specific year or period is chosen as the reference base. The average price level during this period is set to an index value of 100. The base period should ideally be one of relative economic stability and should not be too far in the past to maintain relevance. In India, the current base year for the main CPI (Combined) series is 2012.
3. Select Basket of Goods and Services: This is a crucial step. Detailed Household Consumer Expenditure Surveys (CES) are conducted among the target population. These surveys collect data on what goods and services households buy and in what quantities or proportions they spend their money. Based on this survey data, a representative "basket" of goods and services is selected. This basket includes items that are commonly purchased and represent a significant portion of household expenditure. The composition of the basket is fixed between base year revisions.
4. Determine Weights: Weights are assigned to each item or group of items in the basket based on their relative importance in the average consumer's budget during the base period, as revealed by the CES data. The weight for an item reflects its share in the total expenditure on the basket. For a Laspeyres-type CPI, the weights ($w_i$) are typically based on the **base period value** of consumption for each item:

   $$w_i = p_{0i} \times q_{0i}$$

   ... (a)

   where $p_{0i}$ and $q_{0i}$ are the price and quantity of item $i$ in the base period. Alternatively, the weight can be expressed as the item's share in the total base period expenditure: $w_i = \frac{p_{0i} q_{0i}}{\sum p_{0i} q_{0i}}$.
5. Collect Price Data: Prices for all items in the selected basket are collected regularly (e.g., monthly, weekly) from a sample of retail outlets (shops, markets) in selected locations that are representative of the target population's purchasing points. Prices are collected for the base period ($p_{0i}$) and for each subsequent current period ($p_{1i}$).
6. Choose Formula and Compute Index: An appropriate index number formula is chosen to aggregate the price changes of the individual items into a single index number. Most official CPIs worldwide, including in India, use a formula that is conceptually equivalent to the **Laspeyres formula** (specifically, the Weighted Aggregate Method using base period quantities or the Weighted Average of Price Relatives using base period value weights). The reason for using a Laspeyres-type formula is that the consumption basket (and thus the implicit quantities $q_{0i}$ or explicit base-period expenditure weights $p_{0i}q_{0i}$) is typically kept fixed between base year revisions.
   • **Using Weighted Average of Price Relatives:** Calculate the price relative ($P_i = \frac{p_{1i}}{p_{0i}} \times 100$) for each item $i$. The CPI is then calculated as the weighted arithmetic mean of these price relatives, using base period expenditure shares as weights:

     $$CPI = \frac{\sum (P_i \times w_i)}{\sum w_i}$$

     ... (b)

     where $P_i = \frac{p_{1i}}{p_{0i}} \times 100$ and $w_i = p_{0i}q_{0i}$. Note that the $\times 100$ in $P_i$ means the numerator in (b) is $\sum \left( \frac{p_{1i}}{p_{0i}} \times 100 \right) (p_{0i}q_{0i}) = 100 \sum p_{1i}q_{0i}$, and the denominator is $\sum p_{0i}q_{0i}$. So, this becomes $\frac{100 \sum p_{1i}q_{0i}}{\sum p_{0i}q_{0i}}$.
   • **Using Weighted Aggregate Method (Laspeyres Type):** This method directly calculates the cost of the base period basket at current prices relative to its cost at base prices:

     $$CPI = \frac{\sum (p_{1i} \times q_{0i})}{\sum (p_{0i} \times q_{0i})} \times 100$$

     ... (c)

     where $q_{0i}$ are the base period quantities (derived from base period expenditure and price). As shown in previous sections, formula (b) with weights $w_i = p_{0i}q_{0i}$ is mathematically equivalent to formula (c). Formula (c) shows the concept of the cost of a fixed base period basket more directly.
   This fixed-weight approach simplifies data collection (only prices are needed regularly) but can lead to substitution bias over time as consumers switch to relatively cheaper goods, a limitation of the Laspeyres formula.
7. Index Calculation and Publication: The calculated index value for the current period is compiled and published, often alongside subgroup indices (e.g., food index, housing index) and commentary.
8. Base Year Revision: Periodically (e.g., every 5-10 years), the base year is revised, involving updated Consumer Expenditure Surveys to select a new basket of goods and services and calculate new weights to reflect current consumption patterns. This is necessary to keep the index relevant.

---

CPI Series in India

In India, different agencies compile CPI numbers for various target populations. The main series are:

• Consumer Price Index (Combined) [CPI(C)]: This is the headline CPI for India, covering both urban and rural populations. It uses a basket derived from the latest Household Consumption Expenditure Survey.
  • Compiled by: **National Statistical Office (NSO)**, Ministry of Statistics and Programme Implementation (MoSPI).
  • Current Base Year: **2012 = 100**.
  • Significance: This is the index primarily used by the Reserve Bank of India (RBI) for setting monetary policy targets (inflation targeting).
• Consumer Price Index for Industrial Workers (CPI-IW): This index specifically tracks price changes relevant to industrial workers living in selected urban centers.
  • Compiled by: **Labour Bureau**, Ministry of Labour and Employment.
  • Current Base Year: **2016 = 100**.
  • Significance: This index is predominantly used for determining the Dearness Allowance (DA) of government employees and workers in the organized sector.
• Consumer Price Index for Agricultural Labourers (CPI-AL): This index measures price changes relevant to agricultural labourers.
  • Compiled by: **Labour Bureau**, Ministry of Labour and Employment.
  • Current Base Year: **1986-87 = 100**.
  • Significance: Used for revising minimum wages for agricultural labour in various states.
• Consumer Price Index for Rural Labourers (CPI-RL): Similar to CPI-AL but includes non-agricultural rural labourers as well.
  • Compiled by: **Labour Bureau**, Ministry of Labour and Employment.
  • Current Base Year: **1986-87 = 100**.
  • Significance: Used for revising minimum wages for certain categories of rural labour.

It's important to note that the basket of goods and services, as well as the weights, differ significantly across these CPI series, reflecting the different consumption patterns of the target populations.

Wholesale Price Index (WPI): Construction and Use

Definition

The Wholesale Price Index (WPI) is a measure that tracks the average change over time in the prices of goods sold in bulk at the wholesale level. It captures price movements at the stage where goods are first transacted in bulk in the domestic market, typically at the level of manufacturers, producers, or large traders before they reach the retail stage.

A key characteristic of WPI is that its basket includes primarily **goods**, and it generally **excludes services**, unlike the CPI which includes both.

---

Purpose and Use

WPI serves various purposes, primarily related to monitoring price changes at the early stages of transactions:

• Measure of Wholesale or Producer Inflation: WPI tracks inflation at the producer or wholesale level. Changes in WPI can often serve as a leading indicator for potential future changes in consumer prices (CPI), though the pass-through is not always one-to-one or immediate.
• Economic Analysis and Policy: Government agencies, economists, and policymakers use WPI to analyse price movements in key sectors like manufacturing, agriculture (primary articles), and energy. It helps in understanding cost pressures faced by producers.
• Escalation Clauses in Contracts: WPI is sometimes used in commercial contracts (especially for supply or construction) to adjust prices or costs based on changes in the input costs faced by businesses.
• Deflating Value of Wholesale Trade: WPI can be used to convert the nominal value of wholesale trade or industrial production into real terms, allowing for the measurement of changes in the volume of goods traded or produced.
• Historical "Headline Inflation": Historically, before the CPI (Combined) gained prominence, WPI was the main measure of 'headline inflation' in India, widely tracked by media and analysts. While still important, its role as the primary measure for monetary policy has diminished compared to CPI(C).

---

Construction (Conceptual Steps)

The construction of a WPI series shares some similarities with CPI but differs in the basket and weighting:

1. Select Base Period: A base year is chosen as the reference point (Index = 100). The current base year for India's WPI is **2011-12 = 100**.
2. Select Basket of Commodities: A representative basket of commodities that are traded in bulk at the wholesale level is selected. This selection is based on data related to production, trade, and transaction values in the economy. The basket typically includes goods from agriculture, mining, and manufacturing sectors. Unlike CPI, services are generally excluded from the WPI basket.
3. Determine Weights: Weights ($w_i$) are assigned to each commodity or group of commodities based on their share in the total value of wholesale transactions or, more commonly in modern WPIs, based on their share in the total value of domestic production adjusted for foreign trade (Value of Output + Imports - Exports) during the base year. This reflects the relative importance of each commodity in the overall wholesale market.
4. Collect Price Data: Prices are collected regularly (e.g., weekly, monthly) for the selected commodities. The price points are typically at the first point of bulk sale, such as ex-factory prices, ex-mine prices, or wholesale market prices (mandi prices for agricultural goods). Prices are collected for the base period ($p_{0i}$) and for each subsequent current period ($p_{1i}$).
5. Choose Formula and Compute Index: WPI is typically calculated using a **Laspeyres-type formula**, similar to most CPIs. The Weighted Aggregate method is commonly used, either directly or implicitly through weighted price relatives.
   • **Using Weighted Aggregate Method (Laspeyres Type):** The index is calculated as the ratio of the total value of the base period quantities (implicit in weights) at current prices to their total value at base prices:

     $$WPI = \frac{\sum (p_{1i} \times q_{0i})}{\sum (p_{0i} \times q_{0i})} \times 100$$

     ... (d)

     where $q_{0i}$ are implicit base period quantities derived from the base year value weights.
   • **Using Weighted Average of Price Relatives:** Calculate price relatives ($P_i = \frac{p_{1i}}{p_{0i}} \times 100$) for each commodity. Compute the weighted average using base period value weights ($w_i = p_{0i}q_{0i}$):

     $$WPI = \frac{\sum (P_i \times w_i)}{\sum w_i}$$

     ... (e)

     As shown before, formula (e) with $w_i = p_{0i}q_{0i}$ is mathematically equivalent to formula (d).
6. Index Calculation and Publication: The calculated WPI is compiled and released regularly (e.g., monthly in India) along with indices for major groups and sub-groups.
7. Base Year Revision: Similar to CPI, the WPI base year and basket are revised periodically to reflect changes in the structure of the economy and the composition of wholesale trade.

---

WPI Series in India

• Compiled by: The **Office of the Economic Adviser**, Department for Promotion of Industry and Internal Trade (DPIIT), Ministry of Commerce and Industry.
• Current Base Year: **2011-12 = 100**.
• The WPI basket in India is broadly categorized into three major groups with specific weights (based on 2011-12):
  1. **Primary Articles:** Includes Food Articles (cereals, vegetables, fruits, milk, eggs, meat, fish, etc.) and Non-Food Articles (oilseeds, fibres, minerals, crude petroleum, etc.). Weight: ~22.6%.
  2. **Fuel & Power:** Includes Coal, Mineral Oils (petrol, diesel, LPG, naphtha, etc.), and Electricity. Weight: ~13.2%.
  3. **Manufactured Products:** This is the largest group, covering a wide range of goods from textiles, chemicals, plastics, metals, machinery, food products (manufactured), etc. Weight: ~64.2%.
• The weekly WPI was discontinued, and it is now released on a monthly basis.

---

Key Differences between WPI and CPI (in the Indian Context)

While both WPI and CPI are price indices measuring inflation, they differ significantly in their scope, coverage, and purpose:

Feature	Wholesale Price Index (WPI)	Consumer Price Index (CPI)
**Price Level Tracked**	Prices at the wholesale or producer level (bulk sales, ex-factory, mandi prices).	Prices at the retail level (prices paid by the final consumer).
**Basket Composition**	Covers only **Goods** (Primary Articles, Fuel & Power, Manufactured Products). Generally **excludes services**.	Covers both **Goods and Services** (Food & Beverages, Housing, Clothing, Footwear, Fuel & Light, Health, Transport, Education, Recreation, etc.).
**Target Audience / Level**	Reflects price changes relevant to producers, manufacturers, and wholesalers.	Reflects price changes relevant to households and final consumers.
**Weighting Basis**	Based on value of domestic production (adjusted for imports/exports) and value of wholesale trade during the base year.	Based on household consumption expenditure patterns (share of spending on different items) as determined by Consumer Expenditure Surveys during the base period.
**Formula Type**	Typically Laspeyres-type (Fixed base year weights based on quantities/values).	Typically Laspeyres-type (Fixed base year weights based on quantities/values).
**Significance / Main Use in India (Current)**	Used for tracking producer/input inflation; analytical purposes; contract escalation. Historically used as headline inflation.	Used for tracking consumer inflation; primary index for RBI's monetary policy inflation targeting; used for DA adjustments and wage negotiations.
**Compiling Authority in India**	Office of the Economic Adviser (DPIIT), Ministry of Commerce & Industry.	NSO, MoSPI (CPI Combined); Labour Bureau, MoLE (CPI-IW, AL, RL).

The different coverage and purpose mean that WPI and CPI can sometimes show divergent inflation rates, reflecting price pressures at different stages of the economy.

Index Numbers of Industrial Production

Definition

The Index of Industrial Production (IIP) is a quantitative measure that reflects the change in the **volume of production** of a selected basket of industrial products over a period of time, relative to a base period. It is a volume index, meaning it measures changes in the physical quantity or output, not price changes.

The IIP provides a single composite figure that indicates the general level of industrial activity and growth within an economy. An increase in the IIP signifies an overall growth in the volume of goods produced by the industrial sector, while a decrease indicates contraction.

---

Purpose and Use

The IIP is a key short-term economic indicator used for various analytical and policy purposes:

• Indicator of Industrial Growth: IIP is the primary measure used to assess the performance and growth trends of the industrial sector, covering mining, manufacturing, and electricity generation.
• Economic Analysis and Forecasting: It is a vital input for economic planners, analysts, and researchers to monitor short-term trends in industrial output. It contributes to quarterly and annual estimates of Gross Domestic Product (GDP), specifically the GDP originating from the industry sector.
• Short-Term Monitoring: As IIP data is typically released monthly, it provides a much more frequent update on industrial activity compared to the quarterly GDP figures. This allows for timely monitoring and assessment of the state of the industrial sector.
• Policy Formulation: Governments and central banks use IIP data to understand the momentum in industrial production and inform policy decisions related to industry, infrastructure, and overall economic management.
• Business Decision Making: Businesses can use IIP data, particularly for their specific sector, to understand market demand, production trends, and make decisions related to production planning, inventory management, and investment.

---

Construction (Conceptual Steps)

Constructing an Index of Industrial Production involves steps similar to price indices, but focusing on quantities and using value/value-added for weighting:

1. Define Scope and Coverage: Identify the industrial sector to be covered (e.g., all registered factories, specific industries). This involves selecting representative establishments.
2. Select Base Period: Choose a reference year or period for which the index will be set to 100. This period's industrial structure and output levels serve as the benchmark. In India, the current base year for IIP is **2011-12 = 100**.
3. Select Basket of Industrial Products: Identify a representative list of specific industrial products whose production volumes will be tracked. This selection is based on their significance in terms of value of output or contribution to the industrial sector's value added during the base period.
4. Determine Weights: Assign weights ($w_i$) to each selected product or group of products based on their relative importance in the total value of output or, more commonly, their share in the **value added** of the industrial sector during the base period. The weight of a product reflects its contribution to the overall industrial activity.
5. Collect Production Data: Regularly collect the physical production quantity data ($q_{0i}$ for the base period, $q_{1i}$ for the current period) for all selected items from the reporting industrial units. Data collection frequency is typically monthly.
6. Choose Formula and Compute Index: The IIP is conventionally calculated using a **Laspeyres-type quantity index formula**. This uses the weights (based on base period values/value added) applied to the quantity relatives or directly in an aggregate form.
   • **Using Weighted Average of Quantity Relatives:** Calculate the quantity relative ($Q_i = \frac{q_{1i}}{q_{0i}} \times 100$) for each item $i$. The IIP is then calculated as the weighted arithmetic mean of these quantity relatives, using the base period value/value-added weights ($w_i$):

     $$IIP = \frac{\sum (Q_i \times w_i)}{\sum w_i}$$

     ... (a)

     where $Q_i = \frac{q_{1i}}{q_{0i}} \times 100$ and $w_i$ is the weight based on base period value added/output share. Similar to CPI, the $\times 100$ is usually managed such that the final index is a percentage.
   • **Using Weighted Aggregate Method (Laspeyres Type):** This method uses the base period value weights $w_i = p_{0i}q_{0i}$ (where $p_{0i}$ is the base period price, representing value) and quantities. It is equivalent to formula (a) if the weights $w_i$ are the base period value weights $p_{0i}q_{0i}$:

     $$IIP = \frac{\sum (q_{1i} \times p_{0i})}{\sum (q_{0i} \times p_{0i})} \times 100$$

     ... (b)

     Formula (b) shows the concept of valuing the current and base period quantities using fixed base period prices (implicitly captured by the value-based weights in formula a).
7. Index Calculation and Publication: The calculated IIP for the current period is compiled and released, typically monthly, along with indices for major industrial sectors and use-based categories.
8. Base Year Revision: The base year, the basket of products, and the weighting structure are revised periodically to reflect changes in the industrial structure and production patterns of the economy.

---

IIP in India

• Compiled and released on a monthly basis by the **National Statistical Office (NSO)**, under the Ministry of Statistics and Programme Implementation (MoSPI).
• Current Base Year: **2011-12 = 100**. The previous base was 2004-05.
• The IIP in India is classified in two ways:
  1. **Sectoral Classification:** Based on the National Industrial Classification (NIC), covering three main sectors:

     **Mining**

     **Manufacturing** (Largest share)

     **Electricity**

  2. **Use-Based Classification:** Categorizing products based on their end-use:

     **Primary Goods**

     **Capital Goods**

     **Intermediate Goods**

     **Infrastructure/Construction Goods**

     **Consumer Durables**

     **Consumer Non-durables**

• The 'eight core industries' (Coal, Crude Oil, Natural Gas, Refinery Products, Fertilisers, Steel, Cement, Electricity) have a significant weight (around 40%) in the overall IIP.

IIP is a crucial indicator for monitoring the pulse of the industrial sector and is widely used for economic forecasting and policy analysis.

Limitations of Index Numbers

While index numbers are invaluable statistical tools for summarizing complex changes over time or space and facilitating comparisons, they are not without limitations. These limitations arise from various aspects of their construction and interpretation. Users should be aware of these drawbacks to avoid misinterpreting or overstating the conclusions drawn from index numbers.

The key limitations inherent in the construction and use of index numbers include:

1. Representativeness of the Basket: The basket of goods, services, or industrial products included in an index is necessarily a sample of the vast universe of items being measured (e.g., consumer goods, wholesale goods, industrial output). If the selected sample (the basket) is not truly representative of the consumption patterns, production structure, or trade composition it aims to capture, the resulting index may not accurately reflect the overall changes in the target variable.

2. Weighting Issues: Assigning appropriate weights that accurately reflect the relative importance of different items is challenging.

a) Choice of Weights: Determining what constitutes 'importance' (e.g., expenditure share, production value, value added) and how to measure it precisely can be complex.

b) Outdated Weights (Laspeyres Bias): Indices using fixed base period weights (like Laspeyres-type CPI and WPI, and IIP) do not account for changes in consumption, production, or trade patterns that occur over time. Consumers substitute towards relatively cheaper goods, producers shift production, and new products emerge. Using outdated weights from the base period tends to overstate price increases (Laspeyres price index's upward bias) or understate quantity increases.

c) Changing Weights (Paasche Issues): Indices using current period weights (like Paasche) reflect current patterns but mean that the basket of goods changes every period. This makes direct period-to-period comparisons less meaningful and requires extensive data collection for every period.

3. Formula Choice and Bias: Different index number formulas (Laspeyres, Paasche, Fisher, Marshall-Edgeworth, etc.) use different averaging and weighting schemes, which can lead to different index values even with the same data. As noted, Laspeyres and Paasche have known biases (upward and downward, respectively, for price indices). While Fisher's index is considered theoretically superior for satisfying test properties, it is complex to calculate regularly. There is no single formula that is universally 'best' for all purposes and free from any theoretical issues.

4. Quality Changes: Accounting for changes in the quality of goods and services over time is one of the most difficult challenges in constructing price indices. If the price of an item increases because its quality has improved (e.g., a car includes new safety features), the index should ideally attribute only part of the price increase to 'inflation'. However, isolating the pure price change from the value of the quality change is complex and often involves imputation methods or hedonic regression, which are themselves subject to limitations.

5. Introduction of New Products and Disappearance of Old Products: Index baskets are typically fixed between base year revisions. New goods and services constantly enter the market (e.g., smartphones, streaming services), and old ones become obsolete or disappear. Incorporating new products into a fixed-base index is challenging as they were not available in the base period. Failing to include new, potentially cheaper or higher-quality options, or failing to remove obsolete, potentially more expensive ones, can reduce the index's relevance and introduce bias.

6. Sampling and Non-Sampling Errors: Data (prices, quantities) for index calculation is collected from a sample of sources (retail outlets, factories, markets). This sampling process is subject to sampling errors. Additionally, non-sampling errors can occur during data collection, processing, or estimation (e.g., errors in reporting prices, incorrect units, data entry mistakes).

7. Averaging Effect: An index number represents an average change for the entire basket and the entire target population or sector. It does not necessarily reflect the price change or production change experienced by any specific individual consumer, household, firm, or sub-sector, whose consumption/production patterns and specific items of interest may differ significantly from the average represented by the index basket and its weights.

8. Comparability Issues: Comparing index numbers across different countries, different time periods (if the base year or methodology has changed), or different types of indices (e.g., WPI vs. CPI) can be misleading if the differences in basket composition, weights, base years, coverage, and methodology are not carefully considered and adjusted for.

9. Purpose Specificity: An index number is constructed for a specific purpose and target audience. Using an index designed for one purpose (e.g., WPI measuring producer input costs) for a different purpose (e.g., measuring the cost of living for consumers) is inappropriate and can lead to incorrect conclusions.

10. **Subjectivity in Base Selection:** While the base period is ideally "normal" and recent, the choice can still involve some subjective judgment. The chosen base period affects the visual representation and magnitude of index values in subsequent periods, although the percentage change between any two non-base periods (in a fixed-base index) should be consistent regardless of the base year choice.

Despite these limitations, index numbers are indispensable tools in economics, business, and government for summarizing complex data, tracking macroeconomic trends, facilitating comparisons, and informing policy and decisions. Awareness of their limitations is essential for appropriate interpretation and use.

---