Answer :
To compute the Marshall-Edgeworth's and Dorbish-Bowley's Price Index numbers from the given data, we need to follow several steps. We'll start by organizing the given data and then proceed with the calculations.
### Step 1: Organize the Data
We are given the prices and quantities in the base year and the current year.
| Item | Price (Base Year) P_0 | Price (Current Year) P_1 | Quantity (Base Year) Q_0 | Quantity (Current Year) Q_1 |
|------|--------------------|-----------------------|----------------------|-------------------------|
| A | 6 | 10 | 50 | 56 |
| B | 2 | 2 | 100 | 120 |
| C | 4 | 6 | 60 | 60 |
| D | 10 | 12 | 30 | 24 |
| E | 8 | 12 | 40 | 36 |
### Step 2: Calculate the Summations Needed
#### Summation 1: [tex]\(\sum (P_1 \cdot Q_0)\)[/tex]
This is the total value of quantities in the base year evaluated at current year prices.
[tex]\[ \sum (P_1 \cdot Q_0) = (10 \cdot 50) + (2 \cdot 100) + (6 \cdot 60) + (12 \cdot 30) + (12 \cdot 40) = 500 + 200 + 360 + 360 + 480 = 1900 \][/tex]
#### Summation 2: [tex]\(\sum (P_1 \cdot Q_1)\)[/tex]
This is the total value of quantities in the current year evaluated at current year prices.
[tex]\[ \sum (P_1 \cdot Q_1) = (10 \cdot 56) + (2 \cdot 120) + (6 \cdot 60) + (12 \cdot 24) + (12 \cdot 36) = 560 + 240 + 360 + 288 + 432 = 1880 \][/tex]
#### Summation 3: [tex]\(\sum (P_0 \cdot Q_1)\)[/tex]
This is the total value of quantities in the current year evaluated at base year prices.
[tex]\[ \sum (P_0 \cdot Q_1) = (6 \cdot 56) + (2 \cdot 120) + (4 \cdot 60) + (10 \cdot 24) + (8 \cdot 36) = 336 + 240 + 240 + 240 + 288 = 1344 \][/tex]
#### Summation 4: [tex]\(\sum (P_0 \cdot Q_0)\)[/tex]
This is the total value of quantities in the base year evaluated at base year prices.
[tex]\[ \sum (P_0 \cdot Q_0) = (6 \cdot 50) + (2 \cdot 100) + (4 \cdot 60) + (10 \cdot 30) + (8 \cdot 40) = 300 + 200 + 240 + 300 + 320 = 1360 \][/tex]
### Step 3: Compute the Price Indices
#### Marshall-Edgeworth's Price Index
The formula for Marshall-Edgeworth's Price Index is:
[tex]\[ \text{ME Price Index} = \left( \frac{\sum (P_1 \cdot Q_0) + \sum (P_1 \cdot Q_1)}{\sum (P_0 \cdot Q_0) + \sum (P_0 \cdot Q_1)} \right) \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{ME Price Index} = \left( \frac{1900 + 1880}{1360 + 1344} \right) \times 100 = \left( \frac{3780}{2704} \right) \times 100 = 139.79289940828403 \][/tex]
#### Dorbish-Bowley's Price Index
The formula for Dorbish-Bowley's Price Index is:
[tex]\[ \text{DB Price Index} = \left( \frac{\sum (P_1 \cdot Q_0) + \sum (P_1 \cdot Q_1)}{2 \times \sum (P_0 \cdot Q_0)} \right) \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{DB Price Index} = \left( \frac{1900 + 1880}{2 \times 1360} \right) \times 100 = \left( \frac{3780}{2720} \right) \times 100 = 138.97058823529412 \][/tex]
### Step 4: Final Answer
- Marshall-Edgeworth's Price Index: [tex]\(139.79\)[/tex]
- Dorbish-Bowley's Price Index: [tex]\(138.97\)[/tex]
These indices provide us with a measure of how much prices have changed from the base year to the current year, taking different aspects of quantities into account.
### Step 1: Organize the Data
We are given the prices and quantities in the base year and the current year.
| Item | Price (Base Year) P_0 | Price (Current Year) P_1 | Quantity (Base Year) Q_0 | Quantity (Current Year) Q_1 |
|------|--------------------|-----------------------|----------------------|-------------------------|
| A | 6 | 10 | 50 | 56 |
| B | 2 | 2 | 100 | 120 |
| C | 4 | 6 | 60 | 60 |
| D | 10 | 12 | 30 | 24 |
| E | 8 | 12 | 40 | 36 |
### Step 2: Calculate the Summations Needed
#### Summation 1: [tex]\(\sum (P_1 \cdot Q_0)\)[/tex]
This is the total value of quantities in the base year evaluated at current year prices.
[tex]\[ \sum (P_1 \cdot Q_0) = (10 \cdot 50) + (2 \cdot 100) + (6 \cdot 60) + (12 \cdot 30) + (12 \cdot 40) = 500 + 200 + 360 + 360 + 480 = 1900 \][/tex]
#### Summation 2: [tex]\(\sum (P_1 \cdot Q_1)\)[/tex]
This is the total value of quantities in the current year evaluated at current year prices.
[tex]\[ \sum (P_1 \cdot Q_1) = (10 \cdot 56) + (2 \cdot 120) + (6 \cdot 60) + (12 \cdot 24) + (12 \cdot 36) = 560 + 240 + 360 + 288 + 432 = 1880 \][/tex]
#### Summation 3: [tex]\(\sum (P_0 \cdot Q_1)\)[/tex]
This is the total value of quantities in the current year evaluated at base year prices.
[tex]\[ \sum (P_0 \cdot Q_1) = (6 \cdot 56) + (2 \cdot 120) + (4 \cdot 60) + (10 \cdot 24) + (8 \cdot 36) = 336 + 240 + 240 + 240 + 288 = 1344 \][/tex]
#### Summation 4: [tex]\(\sum (P_0 \cdot Q_0)\)[/tex]
This is the total value of quantities in the base year evaluated at base year prices.
[tex]\[ \sum (P_0 \cdot Q_0) = (6 \cdot 50) + (2 \cdot 100) + (4 \cdot 60) + (10 \cdot 30) + (8 \cdot 40) = 300 + 200 + 240 + 300 + 320 = 1360 \][/tex]
### Step 3: Compute the Price Indices
#### Marshall-Edgeworth's Price Index
The formula for Marshall-Edgeworth's Price Index is:
[tex]\[ \text{ME Price Index} = \left( \frac{\sum (P_1 \cdot Q_0) + \sum (P_1 \cdot Q_1)}{\sum (P_0 \cdot Q_0) + \sum (P_0 \cdot Q_1)} \right) \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{ME Price Index} = \left( \frac{1900 + 1880}{1360 + 1344} \right) \times 100 = \left( \frac{3780}{2704} \right) \times 100 = 139.79289940828403 \][/tex]
#### Dorbish-Bowley's Price Index
The formula for Dorbish-Bowley's Price Index is:
[tex]\[ \text{DB Price Index} = \left( \frac{\sum (P_1 \cdot Q_0) + \sum (P_1 \cdot Q_1)}{2 \times \sum (P_0 \cdot Q_0)} \right) \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{DB Price Index} = \left( \frac{1900 + 1880}{2 \times 1360} \right) \times 100 = \left( \frac{3780}{2720} \right) \times 100 = 138.97058823529412 \][/tex]
### Step 4: Final Answer
- Marshall-Edgeworth's Price Index: [tex]\(139.79\)[/tex]
- Dorbish-Bowley's Price Index: [tex]\(138.97\)[/tex]
These indices provide us with a measure of how much prices have changed from the base year to the current year, taking different aspects of quantities into account.
