The monthly per capita expenditure incurred by workers for an industrial centre during 1980 and 2005 on the following items are given below. The weights of these items are 75,10, 5, 6 and 4 respectively. Prepare a weighted index number for cost of living for 2005 with 1980 as the base.
To calculate the weighted index number for the cost of living for 2005 with 1980 as the base, we use the formula CPI = ΣWR / ΣW. First, we calculate the Price Relatives (R) for each item using the formula R = (P₁/P₀) × 100.
The table below shows the calculation:
| Items | Price in 1980 (P₀) | Price in 2005 (P₁) | Weight (W) | R = (P₁/P₀) × 100 | WR |
|---|---|---|---|---|---|
| Food | 100 | 200 | 75 | 200 | 15000 |
| Clothing | 20 | 25 | 10 | 125 | 1250 |
| Fuel & lighting | 15 | 20 | 5 | 133.33 | 666.65 |
| House rent | 30 | 40 | 6 | 133.33 | 799.98 |
| Misc | 35 | 65 | 4 | 185.71 | 742.84 |
| Total | ΣW = 100 | ΣWR = 18459.47 |
Using the totals from the table: CPI = ΣWR / ΣW CPI = 18459.47 / 100 CPI = 184.59
Thus, the weighted index number for the cost of living for 2005 is 184.59.
Explanation
This question requires the application of the weighted average of price relatives method to calculate the Consumer Price Index (CPI). The context provided in 'Example 4' explicitly outlines this method: calculating the price relative (R) for each item, multiplying it by the weight (W), summing these products (ΣWR), and dividing by the sum of weights (ΣW). The calculated index of 184.59 indicates that the cost of living has risen by approximately 84.59% from 1980 to 2005.
Solution Steps
Step 1: Calculate the Price Relative (R) for each item using the formula R = (Price in 2005 / Price in 1980) × 100.
Step 2: Multiply each Price Relative (R) by its respective Weight (W) to get the value of WR.
Step 3: Sum all the Weights to get ΣW (which equals 100) and sum all the WR values to get ΣWR.
Step 4: Divide ΣWR by ΣW to find the final index number (CPI = 18459.47 / 100 = 184.59).