Index numbers
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Index numbers are used to show how economic variables change over time and to make comparisons between periods. There are several types of index numbers, each with advantages and disadvantages. Weighted index numbers, like the Laspeyre and Paasche indices, account for the relative importance or quantity of items in a basket. Index numbers facilitate comparisons over time but lose some information by representing complex data in a single figure. Care must be taken in choosing the appropriate index type and base year for the purpose of the analysis.
Index numbers
- 1. Index Numbers 1. Use of Index Numbers Uses: 1. To show how an economic variable (e.g. price) is changing over time. 2. To make comparisons. This is achieved by taking a typical year’s figure as base and expressing the figures for other years as a ratio or percentage of the base year figure. Advantages: 1. Facilitates comparisons. 2. Enables analysis of percentage or relative changes rather than absolute changes. Problems: 1. There are many types of index number. Which to use depends on circumstances. 2. Loss of information. One figure represents a mass of data. Note:As index numbers are used for comparisons it is conventional to work to 1 decimal place only. 2. Unweighted Index Numbers 2.1 Simple Index Number Pn Simple price index = × 100 where Pn is the price in year P0 n
- 2. And P0 is the price in the base year. Worked Example: The price of a Tyson 100 vacuum cleaner for each year from 1995-2000 is given below. Construct a simple index using 1997 as the base year. Year Price 1995 180 1996 185 1997 195 1998 199 1999 207 2000 210 The index for the base year is ALWAYS 100 Solution: Year Price Index 180 1995 180 × 100 = 92.3 195 185 1996 185 × 100 = 94.9 195 1997 195 100 199 1998 199 × 100 = 102.1 195 207 1999 207 × 100 = 106.2 195 210 2000 210 × 100 = 107.7 195
- 3. Examples: 1. The price of 500g of a brand of instant coffee for each year since 2000 is given below. Construct a simple index using 2000 as the base year. Yea Price Index r 2000 2.00 100 2001 2.20 2002 2.40 2003 2.50 2. Construct a simple index for the instant coffee using 2002 as the base year. Yea Price Index r 2000 2.00 2001 2.20 2002 2.40 100 2003 2.50
- 4. Usually we are concerned with more comprehensive changes than just one item. There is a need to combine price changes of several items to produce an aggregate index. 2.2 Simple Aggregate Index Considers a “basket” of items. Simple aggregate index = ∑P n × 100 ∑P 0 Worked Example: The table below shows the cost, in $’s, of running an average family car in 1998, 2000 and 2002. Find the simple aggregate index for 2000 and 2002 using 1998 as the base year. 1998 2000 2002 Road tax 100 110 115 Insurance 500 550 570 Fuel 800 830 950 Repairs/maintenance 200 220 230 Σ 1600 1710 1865 Solution: Year Index 1998 100 2000 1710 × 100 = 106.9 1600 2002 1865 × 100 = 116.6 1600
- 5. Examples: 1. The table below gives the prices, in £’s, of seats at the Aston Palladium Theatre. 1998 1999 2000 Front stalls 20 21 21 Rear stalls 16 17 18 Front circle 20 20 21 Rear circle 12 10 10 Box 25 28 30 Totals Find the simple aggregate index for 1999 and 2000 using 1998 as the base year. Year Index 1998 1999 2000
- 6. 2. The price of a typical English breakfast in 1980, 1990 and 2000 is given below: 1980(£) 1990(£) 2000(£) 1 egg 0.05 0.07 0.10 Rasher of bacon 0.30 0.35 0.40 1 sausage 0.20 0.15 0.20 100 gm baked 0.05 0.05 0.05 beans Totals Find the simple aggregate index for 1980 and 1990 using 2000 as the base year. Year Index 1980 1990 2000 Disadvantage: The value of the index does not depend on the number of items used or on the quantities chosen to quote the price in.
- 7. 2.3 Simple Mean of Price Relatives 1 P Simple mean of price relatives = k ∑ Pn × 100 where k is the 0 number of items in the basket. Worked Example: P1 P2 P0 P1 P0 P2 P0 10 15 15/10 = 1.5 20 20/10 = 2 100 12 120/100 = 1.2 150 150/100 = 1.5 0 5 10 10/5 = 2 10 10/5 = 2 30 40 40/30 = 1.3 60 60/30 = 2 Σ P1/P0 = 6 Σ P2/P0 = 7.5 k = the number of items so in this case k = 4. Year Index 0 100 1 P 1 150 1 k ∑ P1 × 100 = 4 × 6 × 100 = 0 1 P 1 187.5 2 k ∑ P2 × 100 = 4 × 7.5 × 100 = 0
- 8. Example: The table below gives the prices in £’s of seats at the Aston Palladium Theatre for the 3 years 1998, 1999 and 2000. Find the Simple Mean of Price Relatives index for 1999 and 2000 using 1998 as the base year. 1998 1999 2000 P0 P1 P1/P0 P2 P2/P0 Front 20 21 21 stalls Rear stalls 16 17 18 Front 20 20 21 circle Rear circle 12 10 10 Box 25 28 30 Totals (Σ) k = _______ Year Index 1998 1999 2000 Disadvantages: Assumes all goods have equal importance although we do not spend equal amounts on all items.
- 9. 3. Weighted Index Numbers Introduction Disadvantage of unweighted index numbers: Prices are quoted in different units and give no indication of the relative importance of each item. Method of weighting: Multiply price by weight that will adjust the item’s size in proportion to its importance. Advantages: 1. Both the importance of the item and the unit in which the price is expressed are taken into consideration. 2. Weighted indices are directly comparable. 3.1 Laspeyre Price Index Base year weighted index. Laspeyre price index = ∑p q n 0 × 100 ∑p q 0 0 Worked Example: p0 q0 p0q0 p1 p1q0 p2 p2q0 10 1 10 × 1 = 10 15 15 × 1 = 15 20 20 × 1 = 20 100 2 100 × 2 = 200 120 120 × 2 = 240 15 150 × 2 = 300 0 5 10 5 × 10 = 50 10 10 × 10 = 100 10 10 × 10 = 100 30 5 30 × 5 = 150 40 40 × 5 = 200 60 60 × 5 = 300 Σ p0q0 = 410 Σ p1q0 = 555 Σ p2q0 = 720
- 10. Year Index 0 100 1 ∑pq 1 0 × 100 = 555 × 100 = 135.4 ∑p q 0 0 410 2 ∑p q 2 0 × 100 = 720 × 100 = ∑p q 0 0 410 175.6 Example: The table below gives the prices in £’s of seats at the Aston Palladium Theatre for 3 years. Find the Laspeyre Price Index for 1999 and 2000 using 1998 as the base year. 1998 1999 2000 p0 q0 p0q0 p1 p1q0 p2 p2q0 Front 20 120 21 21 Stalls Rear 16 150 17 18 Stalls Front 20 80 20 21 Circle Rear 10 100 12 14 Circle Box 25 20 28 30 Σ Σ Σ Year Index
- 11. 1998 1999 2000 3.2 Paasche Price Index Current year weighted index. Paasche price index = ∑p q n n × 100 ∑p q 0 n Worked Example: p0 p1 q1 p1q1 p 0q 1 p2 q2 p 2q 2 p0q2 10 15 2 15 × 2 = 30 10 × 2 = 20 20 4 20 × 4 = 80 10 × 4 = 40 100 120 4 120 × 4 = 480 100 × 4 = 400 150 6 150 × 6 = 900 100 × 6 = 600 5 10 5 10 × 5 = 50 5 × 5 = 25 10 1 10 × 10 = 100 5 × 10 = 50 0 30 40 2 40 × 20 = 800 30 × 20 = 600 60 1 60 × 10 = 600 30 × 10 = 300 0 0 Σp1q1= 1360 Σp0q1= 1045 Σp2q2= 1680 Σp0q2 = 990
- 12. Year Index 0 100 1 1360 ×100 = 130.1 1045 2 1680 ×100 = 169.7 990 Example: The table below gives the prices in £’s of seats at the Aston Palladium Theatre for 3 years. Find the Paasche Price Index for 1999 and 2000 using 1998 as the base year. p0 = price in 1998 , p1 = price in 1999, p2 = price in 2000 q1 = number of seats in 1999 , q2 = number of seats in 2000 p0 p1 q1 p1q1 p 0q 1 p2 q2 p2q2 p0q2 Front 20 21 120 21 130 Stalls Rear 16 17 160 18 160 Stalls Front 20 20 90 21 100 Circle Rear 12 10 90 10 80 Circle Box 25 28 22 30 24 Σ Year Index 1998 1999 2000
- 13. Laspeyre V Paasche 1. Compares cost of buying 1. Compares cost of buying base year quantities at current year quantities current year prices with base at current year prices with year prices. base year prices. 2. Assumes that if prices had 2. Assumes that if prices had risen would still purchase same risen would have bought same quantities as in base year. Quantities as in current year. (Overestimates inflation) (Underestimates inflation) 3. Easy to calculate as weights 3. Difficult, expensive and time fixed. consuming to keep re- calculating weights. 4. Same basket of goods so 4. Difficult to make comparisons as different years are directly changes in index reflect both comparable. changes in price and weights. 3.3 Quantity Indices. Measure changes in quantity rather than price. For example the Index of Industrial Production. Laspeyre quantity index = ∑q n po × 100 ∑q 0 p0 Paasche quantity index = ∑q n pn × 100 ∑q 0 pn
- 14. Example 1: Laspeyre Quantity Index p0 q0 p0q0 q1 p0q1 q2 p 0q 2 10 1 1 2 100 2 3 5 5 10 15 10 30 5 10 20 Year Index 0 1 2 Example 2: Paasche Quantity Index q0 p1 q1 p1q1 p1q0 p2 q2 p2q2 p2q0 0 15 2 20 4 2 120 4 150 6 5 10 5 10 10 30 40 20 60 10
- 15. Year Index 0 1 2 4. Use of Index Numbers 4.1 Constructing Indices Choice of index: depends on the purpose of constructing the index. Selecting items: must be representative, Must be unambiguous and their absolute values ascertainable. Choice of weights: changing weights does not appear to have much effect on an index. Choice of base year: should be “normal”, Should not be too distant. 4.2 Deflators Can be used to produce comparisons in real rather than monetary terms. I.e. eliminates the effects of inflation. Worked Example: Calculate the profit in 1996 terms for the following data. Year Actual Profits Index of Profit in 1996 terms (£000,000’s) inflation (£000,000’s) 1996 12 120 12 1997 15 125 120 15 × = 14.4 125 1998 20 140 120 20 × = 17.1 140
- 16. Example: Calculate the profit in 2001 terms for the following data: .Year Actual Profits Index Profit in 2001 (£000,000’s) of terms (£000,000’s) Inflation 1999 44 115 2000 46 120 2001 50 129 2002 54 131 4.3 Linking Series Together The base year needs to be updated every few years as: • The index becomes out of date • The weights become out of date. Year 1980=100 1985=100 1987=100 1983 103.3 1984 106.7 1985 110.7 100 1986 102.1 1987 105.8 Completed table: Year 1980 = 100 1985 = 100 1987 = 100 1983 103.3 100 100 103.3 × = 93.3 93.3 × = 88.2 110.7 105.8 1984 106.7 100 100 106.7 × = 96.4 96.4 × = 91.1 110.7 105.8 1985 110.7 100 100 100 × = 94.5 105.8 1986 110.7 102.1 100 102.1 × = 113.0 102.1 × = 96.5 100 105.8 1987 110.7 105.8 100 105.8 × = 117.1 100
- 17. Example: Complete the following table: Year 1990 = 100 1996 = 100 2000 = 100 1994 104.4 1995 107.8 1996 111.8 100 1997 103.2 1998 106.9 1999 108.8 2000 110
- 18. End of Topic Test. 1. The base year has an index of 2. A simple index is where item(s) is/are monitored. 3. The Laspeyre index is an example of a weighted index. 4. The Paasche index is an example of a weighted index. 5. The weights used in the RPI are derived from the expenditure survey. 6. The RPI is updated each 7. The Laspeyre index is easier to calculate than the Paasche index. TRUE/FALSE. 8. You cannot directly compare years with the Paasche index. TRUE/FALSE. 9. It is not possible to have an index below 100. TRUE/FALSE. 10. The RPI in 1989 was 115.2 and by 1992 it was 138.5. This represents a rise in the price of goods by: (a) 15% (b) 20.2% (c) 23.3% (d) £11.8 11. The turnover by a company in 1997 was £54.5m and in 1999 it was £75m. If the RPI has increased from 101.9 in 1997 to 115.2 by 1999, the real change in turnover has been: (a) £20.5m (b) £16m (c) £11.8m 12. If the price index of bananas was 100 in 1993 and that for apples was 110, then the price of a kg of bananas was: (a) Less than for apples (b) More than for apples (c) Impossible to compare Uploaded By Iftikhar Changazi