◄UNIT 3: INTEGRALSUNIT 5: INTRODUCTION TO PROBABILITY►

UNIT 4: INDEX NUMBERS AND APPLICATIONS

  • UNIT 4: INDEX NUMBERS AND APPLICATIONS

    Key unit Competence: Apply index numbers in solving financial related 
    problems, interpreting a value index, and drawing 

    appropriate decisions.

    Introductory activity

    An industrial worker was earning a salary of 100,000Frw in the year 2000. 

    Today, he earns 1,200,000Frw. 

    i. Can his standard of living be said to have risen 12 times during this 

    period? 

    ii. Which statistical measure used to detect changes in a variable (rising 

    or decreasing)

    iii. By how much should his salary be raised so that he is as well off as 

    before?

    iv. Describe the methods can be used detect the rise of the price 

    4.1. Introduction to index numbers

    4.1.1 Meaning, types and characteristics of index number

    Learning activity 4.1.1

    If a businessman wants to measure changes in the price level for a specific 
    group of consumers in his region of business.
    a) Which statistical measure he can use to detect changes in a variable 
    or group of variables in his business
    b) Help him how he can calculate consumer price index for industrial 
    workers, city workers and agricultural workers.

    1. Meaning of index number

    An index number is a statistical measure used to detect changes in a variable 
    or group of variables. In addition, a single ratio (or percentage) that measures 
    the combined change of several variables between two different times, places, 
    or situations is referred to as an index number. Index numbers are expressed 

    as percentages.

    The relative change in price, quantity, or value in comparison to a base period. 
    An index number is used to track changes in raw material prices, employee and 

    customer numbers, annual income and profits.

    A simple index is one that is used to measure the relative change in only 
    one variable, such as wages per hour in manufacturing. A composite index 
    is a number that can be used to measure changes in the value of a group of 
    variables such as commodity prices, volume of production in different sectors 

    of an industry, production of various agricultural crops and cost of living.

    The index number measures the average change in a group of related variables 
    over two different situations, such as commodity prices, volume of production 
    in various sectors of an industry, production of various agricultural crops, and 
    cost of living. Index numbers measure the changing value of a variable over 
    time in relation to its value at some fixed point in time, the base period, when it 

    is given the value of 100

    2. Characteristics of index numbers 

    Index numbers have the following important characteristics:

    – Index numbers are of the type of average that measures the relative 
    changes in the level of a particular phenomenon over time. It is a special 
    type of average that can be used to compare two or more series made up of 

    different types of items or expressed in different units.

    – Index numbers are expressed as percentages to show the level of relative 

    change.

    – Index numbers are used to calculate relative changes. They assess the 
    relative change in the value of a variable or a group of related variables 

    over time or between locations.

    – Index numbers can also be used to quantify changes that are not directly 
    measurable. For example, the cost of living, price level, or business activity 
    in a country are not directly measurable, but relative changes in these 
    activities can be studied by measuring changes in the values of variables/

    factors that affect these activities.

    3. Types of index numbers 

    a) Consumer price index 

    A consumer price index (CPI) tracks price changes in a basket of consumer 
    goods and services purchased by households. CPI measures changes in the price 
    level for a specific group of consumers in a given region. CPI can be calculated 
    for industrial workers, city workers and agricultural workers. Consumer price 

    index is given by:

    e) Uses of index numbers

    The price indices such as CPI and PPI used to:
    • Measure the cost of living;
    • Measure of inflationary and deflationary tendencies in the economy,
    • Means of adjusting income payments: nominal interest rates, wage 
    determination, taxes and other allowances 

    The quantity/production Indices used to 

    • Used in national accounts to assess the performance of the economy
    • Good indicator of the economic progress taking place in different 

    sectors, regions and countries to facilitate comparisons

    Application activity 4.1.1

    1. Discuss why index numbers are called as economic barometer

    2. Explain the characteristics of index number.

    4.1.2. Construction of indices

    Learning activity 4.1.2

    For the given data of prices in certain country, answer the questions that 

    follows:

    a) How do you describe these prices over the given years?
    b) Describe the methods can be used to detect changes in these variable?
    c) Find simple aggregative index for the year 1999 over the year 1998 
    d) Find the simple aggregative index for the year 2000 over the year 

    1998

    The construction of index numbers considers the definition of the purpose of 
    the index, selection of the base period, selection of commodities, obtaining price 
    quotations, choice of an average, selection of weights and then selection of a 
    suitable formula. Price index numbers are used to explain the various methods 
    of index number construction. Price index number construction methods can 

    be divided into three broad categories, as shown below:

    Un-weighted index 

    Weights are not assigned to the various items used in the calculation of the 
    un-weighted index number. The following are two un-weighted price index 

    numbers:

    i. Simple Aggregate Method
    This method assumes that different items and their prices are quoted in the 
    same units. All of the items are given equal weight. The following is the formula 

    for a simple aggregative price index:

    From the following data compute price index number for the year 2014 taking 

    2013 as the base year using simple aggregative method:

    This price index of 140 indicates that the aggregate of the prices of the given 
    group of commodities increased by 40% between 2013 and 2014. This price 
    index number, calculated using a simple aggregative method, is only of limited 

    utility. The following are the reasons: 

    a) This method disregards the relative importance of the various 

    commodities used in the calculation. 

    b) The various items must be expressed in the same unit. In practice, the 

    various items may be expressed in different units.

    c) The index number obtained by this method is untrustworthy because it 
    is influenced by the unit in which the prices of various commodities are 

    quoted.

    ii. Simple Average of Price Relatives Method

    This method is superior to the previous one because it is unaffected by the 
    unit in which the prices of various commodities are quoted. Because the price 
    relatives are pure numbers, they are independent of the original units in which 
    they are quoted. The price index number is defined using price relatives as 

    follows:

    Example1: 

    Using the data of example 1 the index number using price relative method can 

    be calculated as follows:

    As a result, the price in 2014 is 112.5% higher than in 2013. The index number, 
    which is based on a simple average of price relatives, is unaffected by the units 
    in which the commodities’ prices are quoted. However, this method, like the 
    simple aggregative method, gives equal weight to all items, ignoring their 

    relative importance in the group.

    Weighted Index Number

    All items or commodities are given rational weights in a weighted index number. 
    These weights indicate the relative importance of the items used in the index 
    calculation. In most cases, the quantity of usage is the most accurate indicator 

    of importance.

    i. Weighted Aggregative Price Indices

    Weights are assigned to each item in the basket in various ways in weighted 
    aggregative price indices, and the weighted aggregates are also used in various 
    ways to calculate an index. In most cases, the price index number is calculated 
    using the quantity of usage. The two most important methods for calculating 
    weighted price indices are Laspeyre’s price index and Paasche’s price index. 
    Laspeyre’s price index number is a weighted aggregative price index number 

    that uses the quantity from the base year as the weights. It is provided by:

    In the given example 2 (above), Paasche’s price index number can be calculated 

    as follows:

    Weights in a weighted price relative index can be determined by the proportion 
    or percentage of total expenditure on them during the base or current period. 
    The base period weight is generally preferred over the current period weight. 

    It is due to the inconvenience of calculating the weight every year.

    Example 3: From the following data compute an index number by using 

    weighted average of price relative method:

    Application activity 4.1.2

    Compute Laspeyre’s and Paasche’s index numbers for 2000 from the following 

    data

    4.2 End unit assessment

    1. Calculate cost of living index number using Family Budget method from 

    the following data (Price in dollars).

    i. Calculate the Laspeyre’s index

    ii. Paasche’s index 

    3. Collect data from the local vegetable market over a week for, at 
    least 10 items. Try to construct the daily price index for the week. 
    What problems do you encounter in applying both methods for the 

    construction of a price index?

    ◄UNIT 3: INTEGRALSUNIT 5: INTRODUCTION TO PROBABILITY►