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# Chapter 20

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• 1. Introductory Mathematics & Statistics Chapter 20 Index Numbers Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-1
• 2. Learning Objectives • Interpret and use a range of index numbers commonly used in the Australian business sector • Define an index number and explain its use • Perform calculations involving simple, composite and weighted index numbers • Understand the basic structure of the consumer price index (CPI) and perform calculations involving its use • Understand other indexes used in the Australian business sector Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-2
• 3. 20.1 Introduction • An index number is a statistical value that measures the change in a variable with respect to time • Two variables that are often considered in this analysis are price and quantity • With the aid of index numbers, the average price of several articles in one year may be compared with the average price of the same quantity of the same articles in a number of different years • There are several sources of ‘official’ statistics that contain index numbers for quantities such as food prices, clothing prices, housing, wages and so on Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-3
• 4. 20.2 Simple index numbers • We will examine index numbers that are constructed from a single item only • Such indexes are called simple index numbers • Current period = the period for which you wish to find the index number • Base period = the period with which you wish to compare prices in the current period • The choice of the base period should be considered very carefully • The choice itself often depends on economic factors 1. It should be a ‘normal’ period with respect to the relevant index 2. It should not be chosen too far in the past Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-4
• 5. 20.2 Simple index numbers (cont…) • The notation we shall use is: – pn = the price of an item in the current period – po = the price of an item in the base period • Price relative – The price relative of an item is the ratio of the price of the item in the current period to the price of the same item in the base period – The formal definition is: pn Price relative = po Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-5
• 6. 20.2 Simple index numbers (cont…) • Simple price index – The price relative provides a ratio that indicates the change in price of an item from one period to another – A more common method of expressing this change is to use a simple price index – The formal definition is: Simple price index = price relative × 100 = pn × 100 po Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-6
• 7. 20.2 Simple index numbers (cont…) • The simple price index finds the percentage change in the price of an item from one period to another – If the simple price index is more than 100, subtract 100 from the simple price index. The result is the percentage increase in price from the base period to the current period – If the simple price index is less than 100, subtract the simple price index from 100. The result is the percentage by which the item cost less in the base period than it does in the current period Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-7
• 8. 20.3 Composite index numbers • A composite index number is constructed from changes in a number of different items • Simple aggregate index – the simple aggregate index has appeal because its nature is simplistic and it is easy to find – The formal definition is: Simple aggregate index = ∑p ∑p n × 100 o Where Σpn = the sum of the prices in the current period Σpo = the sum of the prices in the base period Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-8
• 9. 20.3 Composite index numbers (cont…) • Simple aggregate index (cont…) – Even though the simple aggregate index is easy to calculate, it has serious disadvantages: 1. An item with a relatively large price can dominate the index 2. If prices are quoted for different quantities, the simple aggregate index will yield a different answer 3. It does not take into account the quantity of each item sold – Disadvantage 2 is perhaps the worst feature of this index, since it makes it possible, to a certain extent, to manipulate the value of the index Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-9
• 10. 20.3 Composite index numbers (cont…) • Averages of relative prices – This index also does not take into account the quantity of each item sold, but it is still a vast improvement on the simple aggregate index – The formal definition is: Average of relative prices = ∑  pn   × 100  p   o  k where k = the number of items pn = the price of an item in the current period po = the price of an item in the base period Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-10
• 11. 20.4 Weighted index numbers • The use of a weighted index number or weighted index allows greater importance to be attached to some items • Information other than simply the change in price over time can then be used, and can include such factors as quantity sold or quantity consumed for each item • Laspeyres index – The Laspeyres index is also known as the average of weighted relative prices – In this case, the weights used are the quantities of each item bought in the base period Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-11
• 12. 20.4 Weighted index numbers (cont…) – The formula is: ∑p q Laspeyres index = ∑p q n o × 100 o o Where: qo = the quantity bought (or sold) in the base period pn = price in current period po = price in base period – Thus, the Laspeyres index measures the relative change in the cost of purchasing these items in the quantities specified in the base period Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-12
• 13. 20.4 Weighted index numbers (cont…) • Paasche index – The Paasche index uses the consumption in the current period – It measures the change in the cost of purchasing items, in terms of quantities relating to the current period – The formal definition of the Paasche index is: ∑p q Paasche index = ∑p q n n × 100 o n Where: pn = the price in the current period po = the price in the base period qn = the quantity bought (or sold) in the current period Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-13
• 14. 20.4 Weighted index numbers (cont…) • Comparison of the Laspeyres and Paasche indexes – The Laspeyres index measures the ratio of expenditures on base year quantities in the current year to expenditures on those quantities in the base year – The Paasche index measures the ratio of expenditures on current year quantities in the current year to expenditures on those quantities in the base year – Since the Laspeyres index uses base period weights, it may overestimate the rise in the cost of living (because people may have reduced their consumption of items that have become proportionately dearer than others) Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-14
• 15. 20.4 Weighted index numbers (cont…) • Comparison of the Laspeyres and Paasche indexes (cont…) – Since the Paasche index uses current period weights, it may underestimate the rise in the cost of living – The Laspeyres index is usually larger than the Paasche index – With the Paasche index it is difficult to make year-to-year comparisons, since every year a new set of weights is used – The Paasche index requires that a new set of weights be obtained each year, and this information can be expensive to obtain – Because of the last 2 points above, the Laspeyres index is the one most commonly used Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-15
• 16. 20.4 Weighted index numbers (cont…) • Fisher’s ideal index – Fisher’s ideal index is the geometric mean of the Laspeyres and Paasche indexes – Although it has little use in practice, it does demonstrate the many different types of index that can be used – The formal definition is: Fisher' s index = ( Laspeyres index )( Paasche index ) ∑p q ∑p q ∑p q ∑p q n o n n o o = o n × 100 Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-16
• 17. 20.5 The Consumer Price Index (CPI) • The measure most commonly used in Australia as a general indicator of the rate of price change for consumer goods and services is the consumer price index • The Australian CPI assumes the purchase of a constant ‘basket’ of goods and services and measures price changes in that basket alone • The description of the CPI commonly adopted by users is in terms of its perceived uses; hence there are frequent references to the CPI as – a measure of inflation – a measure of changes in purchasing power, or – a measure of changes in the cost of living Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-17
• 18. 20.5 The Consumer Price Index (CPI) (cont…) • • • The CPI has been designed as a general measure of price inflation for the household sector. The CPI is simply a measure of the changes in the cost of a basket, as the prices of items in it change From the September quarter 2005 onwards, the total basket has been divided into the following 11 major commodity groups: – food – alcohol and tobacco – clothing and footwear – housing – household contents and services – health – transportation – communication – research – education – financial and insurance services Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-18
• 19. 20.5 The Consumer Price Index (CPI) (cont…) • Historical details of the CPI – Retail prices of food, other groceries and average rentals of houses have been collected by the ABS for the years extending back to 1901 – From its inception in 1960, the CPI covered the six state capital cities. In 1964 the geographical coverage of the CPI was extended to include Canberra. From the June quarter in 1982 geographic coverage was further extended to include Darwin – Index numbers at the ‘Group’ and ‘All groups’ levels are published for each capital city and for the weighted average of eight capital cities Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-19
• 20. 20.5 The Consumer Price Index (CPI) (cont…) • The conceptual basis for measuring price changes – The CPI is a quarterly measure of the change in average retail price levels – In measuring price changes, the CPI aims to measure only pure price changes – The CPI is a measure of changes in transaction prices, the prices actually paid by consumers for the goods and services they buy – It is not concerned with nominal, recommended or list prices – The CPI measures price change over time and does not provide comparisons between relative price levels at a particular date Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-20
• 21. 20.5 The Consumer Price Index (CPI) (cont…) • The index population – Because the spending patterns of various groups in the population differ somewhat, the pattern of one large group, fairly homogeneous in its spending habits, is chosen for the purpose of calculating the CPI – The CPI population group is, in concept, metropolitan employee households – For this purpose, employee households are defined as those households that obtain the major part of their household income from wages and salaries Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-21
• 22. 20.5 The Consumer Price Index (CPI) (cont…) • Details of the 15th series CPI – Since 1960, when the CPI was first compiled, the ABS has maintained a program of periodic reviews of the CPI – The main objective of these reviews is to update item weights, but they also provide an opportunity to reassess the scope and coverage of the index – The latest (15th series) review has resulted in three main outcomes: 1. updating the weighting pattern for the CPI 2. incorporating a price index for financial services into the CPI 3. introducing an Australian hedonic price index for computers Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-22
• 23. 20.5 The Consumer Price Index (CPI) (cont…) • Collecting prices – This involves collecting prices from many sources, including supermarkets, department stores, footwear stores, restaurants, motor vehicle dealers and service stations, dental surgeries, etc. – In total, around 100 000 separate price quotations are collected each quarter – Prices of the goods and services included in the CPI are generally collected quarterly – The prices used in the CPI are those that any member of the public would have to pay on the pricing day to purchase the specified good or service – Any sales or excise taxes that the consumer must pay when purchasing specific items are included in the CPI price Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-23
• 24. 20.5 The Consumer Price Index (CPI) (cont…) • Periodic revision of the CPI – The CPI is periodically revised in order to ensure it continues to reflect current conditions – CPI revisions have usually been carried out at approximately 5-yearly intervals • Changes in quality – it is necessary to ensure that identical or equivalent items are priced in successive time periods – This involves evaluating changes in the quality of goods and services Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-24
• 25. 20.5 The Consumer Price Index (CPI) (cont…) • Long-term linked series – A single series of index numbers has been constructed by linking together selected retail price index series – The index numbers are expressed on a reference base 1945 = 100 • International CPIs – A comparison of the CPIs for a number of countries, including Australia, as measured in September quarters – The base year is 2000 – During the 8-year period up to 2008–09, of the countries listed, Japan had the smallest increase in CPI (0.4%), while Indonesia had the largest (111.0%) Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-25
• 26. 20.6 Using the CPI • There are a number of situations where the CPI is used to make adjustments to prices charged and payments made • Examples include changes in rent, pension payments and child support payments • Such adjustments are often made by the relevant government agency, but if members of the community wish to use the CPI for such purposes, it is their responsibility to ensure that this index is suitable Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-26
• 27. 20.7 Index numbers as a measure of deflation • One of the uses for price indexes is to measure the changes in the purchasing power of the dollar • This is known as deflation • In order to eliminate the effect of inflation and obtain a clear picture of the ‘real’ change, the values must be deflated • For example, to deflate an actual salary and express it in terms of ‘real’ salary (of the base year), use: ' Real' salary = actual salary in current year × CPI in base year CPI in current year Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-27
• 28. 20.8 Other types of Australian indexes • Producer price indexes – Several producer price indexes (PPIs) are produced and published – PPIs can be constructed as either output measures or input measures – Output indexes measure change in the prices of goods and/or services sold by a defined sector of the economy – Input indexes measure changes in the prices of goods and/or services purchased by a particular economic sector Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-28
• 29. 20.8 Other types of Australian indexes (cont…) • International trade price indexes – The international trade price indexes are intended to broadly measure changes in the prices of goods imported into Australia (the export price index, EPL) – As the prices used in the indexes are expressed in Australian currency, changes in the relative value of the Australian dollar and overseas currencies can have a direct impact Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-29
• 30. 20.8 Other types of Australian indexes (cont…) • The All-Ordinaries index – The All-Ordinaries index (also know as the ‘All-Ords’) is an Australian Stock Exchange (ASX) measure of the share-price movements of the largest 500 Australian companies – The market capitalisation of these companies totals about 95% of the value of shares listed on the exchange • The trade-weighted index – The trade-weighted index (TWI) provides an indication of movements in the average value of the Australian dollar against the currencies of our main trading partners – The method of calculating the TWI has been basically unchanged since 1974, but new weights were announced by the Reserve Bank of Australia as from 1 October 2008 Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-30
• 31. Summary • We have interpreted and used a range of index numbers commonly used in the Australian business sector • We defined an index number and explained its use • We performed calculations involving simple, composite and weighted index numbers • We understood the basic structure of the consumer price index (CPI) and performed calculations involving its use • We understood other indexes used in the Australian business sector Copyright © 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 20-31