THE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics Business Statstcs 1 MTU 07203 INDEX NUMBERS
Topic content 1) Define index numbers 2) Explain the uses of index numbers 3) Explain the problems in index number construction 4) Compute the unweighted index numbers 5) Define Laspeyres index, Paasche index and Fisher s ideal index 6) Compute the Weighted Index numbers (Laspeyres, Paasche index and Fisher s Ideal index)
Index Number Defniton An index number is, in part, a rato of a measure taken during one tme frame to that same measure taken during another tme frame. Or An Index number is a number that expresses the relatve change in price, quantty, or value compared to a base period.
Example of Index Number Examples of specific indexes are a. Employment cost index b. Price index for construction c. Index of manufacturing capacity d. Producer price index e. Consumer price index f. Dow Jones industrial average g. Index of output etc.
Characteristics of Index numbers The Index Number a. Usually expressed in terms of a base of 100 b. Has a base period c. Is a ratio often multiplied by 100 d. Expressed as a percentage that serve as an alternative to compare raw numbers.
Uses of Index Number Index numbers are descriptve measures that are used to Compare phenomena from one tme period to another (comparison between two diferent periods). Relate informaton about stock markets, infaton, sales, exports and imports, agriculture etc.
Problems in Index Number Constructon i. Collection of data The availability and comparability of data to get the correct data is very difficult as we know that the primary data which are always the appropriate ones is costly and time consuming ii. Selection of Base time Period In the calculation of index number we have two important periods; a. The base period where the reference is to be made b. The current period period whose data compared to those of base year
iii. The choice of the suitable formula There are several methods in which the index number can be derived. Different method gives different index numbers. The choice among the different formulae should depend on the particular use to be made on it iv. Selection of component commodity For the single item or commodity it is easier to get a price index, but if the index is of general purpose i.e. to compare the cost of link, here the selection of item to be included should be properly made considering the place, habit and time.
The equation for computing a simple index number is as follows where; X 0 is the quantity, price, or cost in the base year X t is the quantity, price, or cost in the year of interest I t is the index number for the year of interest
Price Relative (PR) Price Relative is the ratio of the price of the single commodity in a given period called current period to its price in another period called base period. where, P t is the price for current period P o is the price for base period
Example
Example 2 The following table shows the average price of one liter of petrol in Dar es Salaam from 2005 to 2010 Year 2005 2006 2007 2008 2009 2010 Tshs 450 560 750 900 1000 1100 Using the 2006-2007 as the base year, find the price relatives of one liter of petrol compounding to all the given years
Solution The arithmetic mean of the price for the year 2006 2007, Year Tshs Price relative 2005 450 2006 560 2007 750 2008 900 2009 1000 2010 1100
Simple Average of Relatve Price Index
Example 3: The price for five commodities for two years, 2000 and 2010 and their consumed quantities by a certain family is given by the following table below Item 2000 2010 Price Po (Tshs) Eggs(dozen) 185 298 Milk (liter) 88 198 mangoes (500g) 146 175 Orange juice (355ml) 158 170 Tea bags (500g) 440 475 Price Pt (Tshs) Calculate the simple average of relative price index use 2000 as a base period
Solution: Item 2000 2010 Simple Index Price Po (Tshs) Price Pt (Tshs) Eggs(dozen) 185 298 Milk (liter) 88 198 mangoes (500g) 146 175 Orange juice (355ml) 158 170 Tea bags (500g) 440 475 Total P t P 0 100 746. 5
Simple Aggregate Index (SAI)
Characteristcs of Simple Aggregate Index Simple Aggregate Index i. Is the sum the prices (rather than the indexes) for the two periods ii. Determine the index based on the totals. iii. Is constructed by collecting a number of similar items.
Example 4 The average price in Tshs ( 000 ) for stone, cement, sand and building block for the two years 2005 and 2010 are given in table below Item 2005 2010 1ton Stone 87 160 1ton cement 56 125 1ton building block 98 200 1 ton sand 67 135 Calculate the simple aggregate price index for the year 2010 using 2005 as a base.
Soluton Item P 0 (2005) P n (2010) 1 ton Stone 87 160 1 ton cement 56 125 1 ton building block 98 200 1 ton sand 67 35
Disadvantage of the Simple Aggregate Index The method do not put in considerations the unity used to give the weight of the difference commodities It assumes equal qualities have been used on each commodity
Exercises 1. Statistics Canada results show that the number of farms in Canada dropped from 276 548 in 1996, to an estimated 246 923 in 2001. What is the index for the number of farms in 2001 based on the number in 1996? 2. Suppose the whole sales price of maize, wheat flour and rice per bag varies as here below Commodity 2000 2010 Price in Tshs per bag Maize 2500 3500 Wheat flour 3000 4500 Rice 3500 5000 Calculate the simple average of relative price index use 2000 as base year.
3. The average price in Tshs for rice, maize and sugar for the two years 2000 and 2011 are given in table Item 2000 2011 1kg rice 650 1200 1kg maize 400 800 1 kg sugar 700 1000 Calculate the simple aggregate price index for the year 2011 using 2000 as a base.
Weighted Index Number There two main types of weighted aggregate indexes we will discuss in this course i. Laspeyre s ii. Paasche s The two methods of computing a weighted price index they differ only in the period used for weighting.
Laspeyre s index The Laspeyres method uses base-period weights that is, the original prices and quanttes of the items bought to fnd the percent change over a period of tme in either price or quantty consumed depending on the problem.
Laspeyre s Price Index Laspeyre s Quantty Index
Example 5 The price for six commodities for two years, 2000 and 2010 and their consumed quantities by a certain family is given by the following table below. Use 2000 as a base year to calculate Laspeyre s Price Index and Laspeyre s Quantity Index Item 2000 2010 Price Po (Tshs) Quantity (q o ) Price Pt (Tshs) Brown bread (loaf) 77 50 198 55 Eggs(dozen) 185 26 298 20 Milk (liter) 88 102 198 130 mangoes (500g) 146 30 175 40 Orange juice (355ml) 158 40 170 41 Tea bags (500g) 440 12 475 12 Quantity (q n )
Soluton Item 2000 2010 p o q o p t q o p o q t Price P o (Tshs) Quantity (q o ) price P t (Tshs) Quantity (q t ) Brown bread (loaf) 77 50 198 55 3850 9900 4235 Eggs(dozen) 185 26 298 20 4810 7748 3700 Milk (liter) 88 102 198 130 8976 20196 11440 mangoes (500g) 146 30 175 40 4380 5250 5840 Orange juice (355ml) 158 40 170 41 6320 6800 6478 Tea bags (500g) 440 12 475 12 5280 5700 5280 Total po q o 33616 pt q o 55594 P q 0 36973 t
Laspeyre s Price Index LPI LPI P q t P o 55594 33616 q o o 100 100 LPI 165.4 Based on this analysis we conclude that the price of this group of items has increased 65.4 percent in the ten year period.
Laspeyre s Quantity Index LQI q q t o p p o 0 100 LQI 36973 33616 100 LQI 110 Based on this analysis we conclude that the quantity of this group of items has increased 10 percent in the ten year period
Paasche s Index The Paasche s method uses current - year weights for the denominator of the weighted index. Paasche s Price Index This is the type of weighted aggregate index number considering the quantity of the current year s period as weights. It is defined as;
Paasche s Quantty Index
Example 6: Use example 5 Item 2000 2010 Price p o Quantity Price p t (Tshs) (q o ) (Tshs) Quantity q n p o q t P t q t q o p t Brown bread (loaf) 77 50 198 55 4235 1089 0 9900 Eggs(dozen) 185 26 298 20 3700 5960 7748 Milk (liter) 88 102 198 130 1144 0 2574 0 2019 6 Mangoes (500g) 146 30 175 40 5840 7000 5250 Orange juice 158 40 170 41 6478 6970 6800 (355ml) Tea bags (500g) 440 12 475 12 5280 5700 5700 p o q 36973 t p q 62260 t t q o P t 55590
Paasche s Price Index PPI PPI P q P 100 Paasche s Quantty Index t o q t t 62250 36973 PPI 168.4 PQI PQI q q t o p p 185500 162500 PQI 112 t t 100 100 100
The Paasche index is more reflective of the current situation. It should be noted that the Laspeyres index is more widely used. The Consumer Price Index, the most widely reported index, is an example of a Laspeyres index.
Advantage and disadvantages of Laspeyres and Paasche s Laspeyres Advantages Disadvantages Requires quantity data from only the base period. This allows a more meaningful comparison over time. The changes in the index can be attributed to changes in the price. Does not reflect changes in buying patterns over time. Also, it may overweight goods whose prices increase. Paasche s Advantages Disadvantages Because it uses quantities from the current period, it reflects current buying habits. Disadvantages It requires quantity data for each year, which may be difficult to obtain. Because different quantities are used each year, it is impossible to attribute changes in the index to changes in price alone. It tends to overweight the goods whose prices
Advantage of Weighted over the Simple Aggregate In weighted index the weight of each of the items is considered.
Fisher s index It is the geometric mean of the Laspeyres and Paasche indexes. It is determined by taking the kth root of the product of k positive numbers Fisher s index combines the best features of both Laspeyres and Paasche. That is, it balances the effects of the two indexes.