Given `P_(01)(M -E) =120, sum p_1q_1=300 ,sump_0q_0= 120 sum p_0q_1 = 320 ` Find `p_(01)(L)`

Answer

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Understand the given information
We are given:
- \( P_{01}(M - E) = 120 \) (Marshall Edgeworth Price Index)
- \( \sum P_1 Q_1 = 300 \)
- \( \sum P_0 Q_0 = 120 \)
- \( \sum P_0 Q_1 = 320 \)

### Step 2: Write down the formula for the Marshall Edgeworth Price Index
The formula for the Marshall Edgeworth Price Index \( P_{01}(M - E) \) is given by:
\[ P_{01}(M - E) = \frac{\sum P_1 Q_0 + \sum P_1 Q_1}{\sum P_0 Q_0 + \sum P_0 Q_1} \times 100 \]

### Step 3: Substitute the known values into the formula
We know \( P_{01}(M - E) = 120 \), so we can set up the equation:
\[ 120 = \frac{\sum P_1 Q_0 + 300}{120 + 320} \times 100 \]

### Step 4: Simplify the denominator
Calculate the denominator:
\[ \sum P_0 Q_0 + \sum P_0 Q_1 = 120 + 320 = 440 \]

### Step 5: Rewrite the equation with the simplified denominator
Now, substituting back into the equation:
\[ 120 = \frac{\sum P_1 Q_0 + 300}{440} \times 100 \]

### Step 6: Eliminate the fraction by multiplying both sides by 440
\[ 120 \times 440 = \sum P_1 Q_0 + 300 \times 100 \]
\[ 52800 = \sum P_1 Q_0 + 300 \]

### Step 7: Solve for \( \sum P_1 Q_0 \)
Now, isolate \( \sum P_1 Q_0 \):
\[ \sum P_1 Q_0 = 52800 - 300 \]
\[ \sum P_1 Q_0 = 52500 \]

### Step 8: Find \( P_{01}(L) \) using the Lapsar's Price Index formula
The formula for Lapsar's Price Index \( P_{01}(L) \) is:
\[ P_{01}(L) = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100 \]

### Step 9: Substitute the known values into Lapsar's formula
Substituting the values we have:
\[ P_{01}(L) = \frac{52500}{120} \times 100 \]

### Step 10: Calculate \( P_{01}(L) \)
\[ P_{01}(L) = \frac{52500}{120} \times 100 = 43750 \]

### Final Answer
Thus, the value of \( P_{01}(L) \) is \( 43750 \).

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