Example:
Construct index numbers of price from the following data by applying:
1)Laspeyres method
2)Paasche method:
3)Bowley’s method
4) Fisher’s Ideal method
5) Marshall –Edgeworth method
1999 2000
Commodity Price Quantity Price Quantity
A 2 8 4 6
B 5 10 6 5
C 4 14 5 10
D 2 19 2 13
Solution:
Commodity | 1999 | 1999 | 2000 | 2000 | ||||
Price(Po) | Qty. (qo) | Price(P1) | Qty.(q1) | P1q0 | P0q0 | P1q1 | P0q1 | |
A | 2 | 8 | 4 | 6 | 32 | 16 | 24 | 12 |
B | 5 | 10 | 6 | 5 | 60 | 50 | 30 | 25 |
C | 4 | 14 | 5 | 10 | 70 | 56 | 50 | 40 |
D | 2 | 19 | 2 | 13 | 38 | 38 | 26 | 26 |
SUM | 200 | 160 | 130 | 103 | ||||
1) Laspeyres Method: po1: *100
= 200/160*100 = 125.
2) Paasche’s Method: P01: *100
= 130/103 *100
= 126.21
= 2.512/2 *100
=125.6
4) Fisher’s ideal Method: p01:
= 125.6
5) Marshall-Edge worth Method :
= 125.47
Example:
A food product company is contemplating the introduction of a revolutionary new product with new packaging, or replace the existing product at a much higher price (S1), or a moderate change in the composition of the existing product with a new packaging at a small increase in price (S2) or a small change in the composition of the existing product except the word “New” with a negligible increase in price (S3). The three possible states of nature or events are: (1) high increase in sales (N1) , (2) no change in sales (N2) and (iii) decrease in sales (N3) . The marketing department of the company worked out the payoffs in terms of yearly net profits for each of the strategies of the three events ( expected sales). This is represented in the following table:
Strategies N1 N2 N3
S1 700000 300000 150000
S2 500000 450000 0
S3 300000 300000 300000
Which strategy should the concerned executive choose on the basis of
(a) Laplace criterion
Solution:
The payoff matrix is rewritten as follows:
(D) Laplace Criterion:
Since we do not know the probability of the states of nature, assume that they are equal. For this example, we would assume that each state of nature has a one third probability of occurance.
Thus,
Expected return
S1 = (700000+300000+150000)/3 = 383333.33
S2 = (500000+450000+0)/3 = 316666.66
S3 = (300000+300000+300000)/3 = 300000
Since the largest expected return is from strategy S1, the executive must select strategy S1.
Compute the weighted aggregative price index numbers for with as base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number (4) Marshal Edgeworth Index Number.
Commodity | Prices | Quantities | ||
1980 | 1981 | 1980 | 1981 | |
A | 10 | 12 | 20 | 22 |
B | 8 | 8 | 16 | 18 |
C | 5 | 6 | 10 | 11 |
D | 4 | 4 | 7 | 8 |
Solution:
Commodity | Prices | Quantity | p1q0 | p0q0 | p1q1 | p0q1 | ||
1980 | 1981 | 1980 | 1981 | |||||
P0 | P1 | P0 | P1 | |||||
A | 10 | 12 | 20 | 22 | 240 | 200 | 264 | 220 |
B | 8 | 8 | 16 | 18 | 128 | 128 | 144 | 144 |
C | 5 | 6 | 10 | 11 | 60 | 50 | 66 | 55 |
D | 4 | 4 | 7 | 8 | 28 | 28 | 32 | 32 |
Laspeyre’s Index Number
Paashe’s Index Number
Fisher’s Ideal Index Number
Marshal Edgeworth Index Number
The Chain Index Number:
Example:
Compute the chain index number with 1995 prices as base from the following table giving the average wholesale prices of the commodities A,B and C for the year 1996.
Average wholesale price(in Rs.)
Commodity 1995 1996 1997 1998 1999
A 20 16 28 35 21
B 25 30 24 36 45
C 20 25 30 24 30
Solution:
Computation of Chain Indices
Relatives based on preceding year
Commodity | 1995 | 1996 | 1997 | 1998 | 1999 |
A | 100 | 16/20*100=80 | 28/16*100=175 | 35/28*100=125 | 21/35*100=60 |
B | 100 | 30/25*100=120 | 24/30*100=80 | 36/24*100=150 | 45/36*100=125 |
C | 100 | 25/20*100=125 | 30/25*100=120 | 24/30*100=80 | 30/24*100=125 |
Total of link Relatives | 300 | 325 | 375 | 355 | 310 |
Average of link relative | 100 | 108.33 | 125 | 118.33 | 103.33 |
Chain Index(1995=100) | 100 | (108.33*100)/100=108.33 | (125*108.33)*100=135.41 | (118.33*135.41)/100=160.23 | (103.33*160.23)/100=165.57 |
Exmple:
From the data below:
(A) What was the real average weekly wage for each year?
(B) In which year did the employees have the greatest buying power?
(C) What percentage increase in the weekly wages for the year 1998 is required to provide the same buying power that the employees enjoyed in the year in which they had the highest real wages?
solution:
( 1) Real average weekly wage can be obtained by the following formula:
Real wage: (monthly wage/ Price index )*100
CALCULATION OF REAL WAGES
Year | Weekly Take home pay | Consumer Price Index | Real Wages |
1993 | 109.50 | 112.8 | 109.5/112.8*100=97.07 |
1994 | 112.20 | 118.2 | 112.2/118.2*100=94.92 |
1995 | 116.40 | 127.4 | 116.4/127.4*100=91.37 |
1996 | 125.08 | 138.2 | 125.08/138.2*100=90.51 |
1997 | 135.40 | 143.5 | 135.4/143.5*100=94.36 |
1998 | 138.10 | 149.8 | 138.10/149.8*100=92.19 |
(b) Since real wage was maximum in the year 1993, the employees had the greatest buying power in the year.
(c) Percentage increase in the weekly wages for the year 1998 to provide the same buying power that the employees enjoyed in 1993.
Absolute difference = 97.07-92.19 = 6.88
Example:
The prices of 3 commodities for the 5 years are as follows:
Commodity | Prices (per kg) | ||||
2001 | 2002 | 2003 | 2004 | 2005 | |
Rice | 20 | 20 | 22 | 25 | 28 |
Sugar | 11 | 12 | 14 | 27 | 30 |
Tea | 178 | 176 | 174 | 180 | 180 |
Required:
Construct price index numbers using average of relatives’ method, taking 2001 as base year.
Solution:
Commodity | Prices (per kg) | ||||
2001 | 2002 | 2003 | 2004 | 2005 | |
Rice | |||||
Sugar | |||||
Tea | |||||
Total | 300 | 307.97 | 335.02 | 471.52 | 513.85 |
Mean(Index) | 100 | 102.66 | 111.67 | 157.17 | 171.28 |
(c) Weighted Index Numbers: This type of index can be further classified into two categories:
(i) Weighted Aggregative Index Numbers: In these index numbers, the quantities produced, sold or bought or consumed during the base year or current year are used as weights. These weights indicate the importance of the particular commodity. Some well-known weighted index numbers are given below:*
(1) Lespeyre’s Index: This index uses base year quantities as weights. For this reason, it is also known as ‘Base Year Weighted Index’:
Here W = Q_{o}
(2) Paasche’s Index: This index uses current years quantity as weights. For this reason, it is known as ‘Current Year Weighted Index’:
Here W = Q_{n}
(3) Fisher’s Ideal Index: This index number is the GM of the Lespeyre’s and Paasche’s index numbers. It is called ‘ideal’ because it satisfies two tests (Time Reversal and Factor Reversal Tests):
(4) Marshall-Edgeworth’s Index: This index number uses the average of the base year and current quantities as weights:
Example:
Commodities | 2001 | 2005 | ||
Price (Rs. / kg) | Qty. (kgs) | Price (Rs. / kg) | Qty. (kgs) | |
Rice | 20 | 100 | 28 | 160 |
Sugar | 11 | 18 | 30 | 37 |
Salt | 1 | 1 | 5 | 1.1 |
Milk | 18 | 57 | 32 | 149 |
Required:
Construct the following price index numbers using 2001 as base year:
(a) Lespeyre’s
(b) Paasche’s
(c) Fisher’s
(d) Marshall-Edgeworth’s
Solution:
2001 | 2005 | P_{o}Q_{o} | P_{n}Q_{o} | P_{n}Q_{n} | P_{o}Q_{n} | Q_{o}+Q_{n} | P_{o}(Q_{o}+Q_{n}) | P_{n}(Q_{o}+Q_{n}) | |||
P_{o } | Q_{o } | P_{n } | Q_{n } | ||||||||
Rice | 20 | 100 | 28 | 160 | 2000 | 2800 | 4480 | 3200 | 260 | 5200 | 7280 |
Sugar | 11 | 18 | 30 | 37 | 198 | 540 | 1110 | 407 | 55 | 605 | 1650 |
Salt | 1 | 1 | 5 | 1.1 | 1 | 5 | 5.5 | 1.1 | 2.1 | 2.1 | 10.5 |
Milk | 18 | 57 | 32 | 149 | 1026 | 1824 | 4768 | 2682 | 206 | 3708 | 6592 |
Total | 3225 | 5169 | 10363.5 | 6290.1 | 9515.1 | 15532.5 |
(a) Lespeyre’s:
(b) Paasche’s:
(c) Fisher’s:
(d) Marshall-Edgeworth’s:
(ii) Weighted Average of Relatives: The formula of weighted average of relatives is:
or
(Arithmetic Mean taken as average); where
(Geometric Mean taken as average)
The total value of the commodity is used as weights. If the base year value (P_{o}Q_{o}) is used as base, then the formula becomes:
or
If the current year value (P_{n}Q_{n}) is used as base, then the formula becomes:
Example
The index of prices for transport is 100 compared to a given base year. The index for housing is 160 (compared to the same base year). Calculate the composite index for transport and housing by using a weighting of 65 for transport and 35 for housing.
Solution:
Composite index = (Total of index * weighting) / Total weighting
Composite index = (100 * 65 + 160 * 35) / (65 + 35) = 12,100 / 100 = 121
Example
The following data are observed:
Beverage: Weight (120); Price index (210)
Clothing: Weight (100); Price index (200)
Food: Weight (80); Price index (190)
Then,
Composite index = (120 * 210 + 100 * 200 + 80 * 190) / (120 + 100 + 80) = 60,400 / 300 = 201.3
* W.A.I.N. is equal to