Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Certainly! Let's find the Laspeyre's, Paasche's, and Dorbish-Bowley's price index numbers for the given data step by step.
### Data Analysis
We have the following data for items A, B, C, and D:
#### 2004:
- Price (Rs.): [tex]\( P_0 \)[/tex]
- Quantity: [tex]\( Q_0 \)[/tex]
| Item | Price (P_0) | Quantity (Q_0) |
|------|-------------|----------------|
| A | 10 | 5 |
| B | 15 | 8 |
| C | 6 | 3 |
| D | 3 | 4 |
#### 2008:
- Price (Rs.): [tex]\( P_1 \)[/tex]
- Quantity: [tex]\( Q_1 \)[/tex]
| Item | Price (P_1) | Quantity (Q_1) |
|------|-------------|----------------|
| A | 12 | 4 |
| B | 18 | 7 |
| C | 4 | 5 |
| D | 3 | 5 |
### Laspeyre's Price Index
Laspeyre's Price Index is calculated using the formula:
[tex]\[ \text{Laspeyres Index} = \left( \frac{\sum (P_1 \times Q_0)}{\sum (P_0 \times Q_0)} \right) \times 100 \][/tex]
1. Numerator [tex]\( \sum (P_1 \times Q_0) \)[/tex]:
- [tex]\( (12 \times 5) + (18 \times 8) + (4 \times 3) + (3 \times 4) = 60 + 144 + 12 + 12 = 228 \)[/tex]
2. Denominator [tex]\( \sum (P_0 \times Q_0) \)[/tex]:
- [tex]\( (10 \times 5) + (15 \times 8) + (6 \times 3) + (3 \times 4) = 50 + 120 + 18 + 12 = 200 \)[/tex]
3. Laspeyres Index:
- [tex]\[ \text{Laspeyres Index} = \left( \frac{228}{200} \right) \times 100 = 114 \% \][/tex]
### Paasche's Price Index
Paasche's Price Index is calculated using the formula:
[tex]\[ \text{Paasche Index} = \left( \frac{\sum (P_1 \times Q_1)}{\sum (P_0 \times Q_1)} \right) \times 100 \][/tex]
1. Numerator [tex]\( \sum (P_1 \times Q_1) \)[/tex]:
- [tex]\( (12 \times 4) + (18 \times 7) + (4 \times 5) + (3 \times 5) = 48 + 126 + 20 + 15 = 209 \)[/tex]
2. Denominator [tex]\( \sum (P_0 \times Q_1) \)[/tex]:
- [tex]\( (10 \times 4) + (15 \times 7) + (6 \times 5) + (3 \times 5) = 40 + 105 + 30 + 15 = 190 \)[/tex]
3. Paasche Index:
- [tex]\[ \text{Paasche Index} = \left( \frac{209}{190} \right) \times 100 = 110 \% \][/tex]
### Dorbish-Bowley's Price Index
Dorbish-Bowley's Price Index is calculated as the simple average of Laspeyres and Paasche indices:
[tex]\[ \text{Dorbish-Bowley Index} = \frac{\text{Laspeyres Index} + \text{Paasche Index}}{2} \][/tex]
- Dorbish-Bowley Index:
- [tex]\[ \text{Dorbish-Bowley Index} = \frac{114 + 110}{2} = 112 \% \][/tex]
### Summary
The price index numbers based on the given data are:
- Laspeyre's Price Index: [tex]\( 114 \% \)[/tex]
- Paasche's Price Index: [tex]\( 110 \% \)[/tex]
- Dorbish-Bowley's Price Index: [tex]\( 112 \% \)[/tex]
### Data Analysis
We have the following data for items A, B, C, and D:
#### 2004:
- Price (Rs.): [tex]\( P_0 \)[/tex]
- Quantity: [tex]\( Q_0 \)[/tex]
| Item | Price (P_0) | Quantity (Q_0) |
|------|-------------|----------------|
| A | 10 | 5 |
| B | 15 | 8 |
| C | 6 | 3 |
| D | 3 | 4 |
#### 2008:
- Price (Rs.): [tex]\( P_1 \)[/tex]
- Quantity: [tex]\( Q_1 \)[/tex]
| Item | Price (P_1) | Quantity (Q_1) |
|------|-------------|----------------|
| A | 12 | 4 |
| B | 18 | 7 |
| C | 4 | 5 |
| D | 3 | 5 |
### Laspeyre's Price Index
Laspeyre's Price Index is calculated using the formula:
[tex]\[ \text{Laspeyres Index} = \left( \frac{\sum (P_1 \times Q_0)}{\sum (P_0 \times Q_0)} \right) \times 100 \][/tex]
1. Numerator [tex]\( \sum (P_1 \times Q_0) \)[/tex]:
- [tex]\( (12 \times 5) + (18 \times 8) + (4 \times 3) + (3 \times 4) = 60 + 144 + 12 + 12 = 228 \)[/tex]
2. Denominator [tex]\( \sum (P_0 \times Q_0) \)[/tex]:
- [tex]\( (10 \times 5) + (15 \times 8) + (6 \times 3) + (3 \times 4) = 50 + 120 + 18 + 12 = 200 \)[/tex]
3. Laspeyres Index:
- [tex]\[ \text{Laspeyres Index} = \left( \frac{228}{200} \right) \times 100 = 114 \% \][/tex]
### Paasche's Price Index
Paasche's Price Index is calculated using the formula:
[tex]\[ \text{Paasche Index} = \left( \frac{\sum (P_1 \times Q_1)}{\sum (P_0 \times Q_1)} \right) \times 100 \][/tex]
1. Numerator [tex]\( \sum (P_1 \times Q_1) \)[/tex]:
- [tex]\( (12 \times 4) + (18 \times 7) + (4 \times 5) + (3 \times 5) = 48 + 126 + 20 + 15 = 209 \)[/tex]
2. Denominator [tex]\( \sum (P_0 \times Q_1) \)[/tex]:
- [tex]\( (10 \times 4) + (15 \times 7) + (6 \times 5) + (3 \times 5) = 40 + 105 + 30 + 15 = 190 \)[/tex]
3. Paasche Index:
- [tex]\[ \text{Paasche Index} = \left( \frac{209}{190} \right) \times 100 = 110 \% \][/tex]
### Dorbish-Bowley's Price Index
Dorbish-Bowley's Price Index is calculated as the simple average of Laspeyres and Paasche indices:
[tex]\[ \text{Dorbish-Bowley Index} = \frac{\text{Laspeyres Index} + \text{Paasche Index}}{2} \][/tex]
- Dorbish-Bowley Index:
- [tex]\[ \text{Dorbish-Bowley Index} = \frac{114 + 110}{2} = 112 \% \][/tex]
### Summary
The price index numbers based on the given data are:
- Laspeyre's Price Index: [tex]\( 114 \% \)[/tex]
- Paasche's Price Index: [tex]\( 110 \% \)[/tex]
- Dorbish-Bowley's Price Index: [tex]\( 112 \% \)[/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
