To determine the Fisher’s Quantity Index based on the provided data, we first need to calculate the Laspeyres Quantity Index and the Paasche Quantity Index. Fisher's Quantity Index is the geometric mean of these two indices.
Let's start by organizing the given data:
Base Year Prices and Quantities:
- Item A: Price = 8, Quantity = 6
- Item B: Price = 10, Quantity = 5
- Item C: Price = 17, Quantity = 8
Current Year Prices and Quantities:
- Item A: Price = 12, Quantity = 5
- Item B: Price = 11, Quantity = 6
- Item C: Price = 8, Quantity = 5
### Step-by-Step Calculation
1. Laspeyres Quantity Index:
The Laspeyres Quantity Index (LQI) measures the change in quantities using the base year's prices as weights.
- Calculate the numerator:
[tex]\[ \sum (\text{Current Year Quantity} \times \text{Base Year Price}) \][/tex]
[tex]\[ = (5 \times 8) + (6 \times 10) + (5 \times 17) \][/tex]
[tex]\[ = 40 + 60 + 85 \][/tex]
[tex]\[ = 185 \][/tex]
- Calculate the denominator:
[tex]\[ \sum (\text{Base Year Quantity} \times \text{Base Year Price}) \][/tex]
[tex]\[ = (6 \times 8) + (5 \times 10) + (8 \times 17) \][/tex]
[tex]\[ = 48 + 50 + 136 \][/tex]
[tex]\[ = 234 \][/tex]
- Laspeyres Quantity Index:
[tex]\[ LQI = \frac{185}{234} = 0.7906 \][/tex]
2. Paasche Quantity Index:
The Paasche Quantity Index (PQI) measures the change in quantities using the current year's prices as weights.
- Calculate the numerator:
[tex]\[ \sum (\text{Current Year Quantity} \times \text{Current Year Price}) \][/tex]
[tex]\[ = (5 \times 12) + (6 \times 11) + (5 \times 8) \][/tex]
[tex]\[ = 60 + 66 + 40 \][/tex]
[tex]\[ = 166 \][/tex]
- Calculate the denominator:
[tex]\[ \sum (\text{Base Year Quantity} \times \text{Current Year Price}) \][/tex]
[tex]\[ = (6 \times 12) + (5 \times 11) + (8 \times 8) \][/tex]
[tex]\[ = 72 + 55 + 64 \][/tex]
[tex]\[ = 191 \][/tex]
- Paasche Quantity Index:
[tex]\[ PQI = \frac{166}{191} = 0.8691 \][/tex]
3. Fisher's Quantity Index:
Fisher’s Quantity Index (FQI) is the geometric mean of the Laspeyres and Paasche Quantity Indices.
[tex]\[ FQI = \sqrt{LQI \times PQI} \][/tex]
[tex]\[ = \sqrt{0.7906 \times 0.8691} \][/tex]
[tex]\[ = \sqrt{0.68707246} \][/tex]
[tex]\[ = 0.8289 \][/tex]
Therefore, the Fisher's Quantity Index is approximately 0.8289.
