The problem provides data on prices and quantities of three raw materials (A, B, C) in two locations (Lieu 0 and Lieu 1). We need to calculate the Laspeyres and Paasche indices for both price and quantity, comparing Lieu 1 to Lieu 0 and Lieu 0 to Lieu 1. Finally, we need to calculate the Fisher index for price and quantity comparing Lieu 1 to Lieu 0.

1. Problem Description

The problem provides data on prices and quantities of three raw materials (A, B, C) in two locations (Lieu 0 and Lieu 1). We need to calculate the Laspeyres and Paasche indices for both price and quantity, comparing Lieu 1 to Lieu 0 and Lieu 0 to Lieu

1. Finally, we need to calculate the Fisher index for price and quantity comparing Lieu 1 to Lieu

0.

2. Solution Steps

First, let's define the notation:

- $P_{0i}$: Price of product $i$ in Lieu 0

- $Q_{0i}$: Quantity of product $i$ in Lieu 0

- $P_{1i}$: Price of product $i$ in Lieu 1

- $Q_{1i}$: Quantity of product $i$ in Lieu 1

The data is:

- A: $P_{0A} = 3.80$, $Q_{0A} = 300$, $P_{1A} = 4.05$, $Q_{1A} = 200$

- B: $P_{0B} = 4.75$, $Q_{0B} = 300$, $P_{1B} = 4.50$, $Q_{1B} = 400$

- C: $P_{0C} = 5.50$, $Q_{0C} = 400$, $P_{1C} = 5.50$, $Q_{1C} = 400$

1. Calculate Laspeyres and Paasche indices comparing Lieu 1 to Lieu 0:

Laspeyres Price Index (Lieu 1 vs Lieu 0):

$$L_P = \frac{\sum P_{1i} Q_{0i}}{\sum P_{0i} Q_{0i}}$$

$L_P = \frac{(4.05 * 300) + (4.50 * 300) + (5.50 * 400)}{(3.80 * 300) + (4.75 * 300) + (5.50 * 400)} = \frac{1215 + 1350 + 2200}{1140 + 1425 + 2200} = \frac{4765}{4765} = 1$

Laspeyres Quantity Index (Lieu 1 vs Lieu 0):

$$L_Q = \frac{\sum P_{0i} Q_{1i}}{\sum P_{0i} Q_{0i}}$$

$L_Q = \frac{(3.80 * 200) + (4.75 * 400) + (5.50 * 400)}{(3.80 * 300) + (4.75 * 300) + (5.50 * 400)} = \frac{760 + 1900 + 2200}{1140 + 1425 + 2200} = \frac{4860}{4765} \approx 1.020$

Paasche Price Index (Lieu 1 vs Lieu 0):

$$P_P = \frac{\sum P_{1i} Q_{1i}}{\sum P_{0i} Q_{1i}}$$

$P_P = \frac{(4.05 * 200) + (4.50 * 400) + (5.50 * 400)}{(3.80 * 200) + (4.75 * 400) + (5.50 * 400)} = \frac{810 + 1800 + 2200}{760 + 1900 + 2200} = \frac{4810}{4860} \approx 0.990$

Paasche Quantity Index (Lieu 1 vs Lieu 0):

$$P_Q = \frac{\sum P_{1i} Q_{1i}}{\sum P_{1i} Q_{0i}}$$

$P_Q = \frac{(4.05 * 200) + (4.50 * 400) + (5.50 * 400)}{(4.05 * 300) + (4.50 * 300) + (5.50 * 400)} = \frac{810 + 1800 + 2200}{1215 + 1350 + 2200} = \frac{4810}{4765} \approx 1.010$

2. Calculate Laspeyres and Paasche indices comparing Lieu 0 to Lieu 1:

Laspeyres Price Index (Lieu 0 vs Lieu 1):

$$L_P = \frac{\sum P_{0i} Q_{1i}}{\sum P_{1i} Q_{1i}}$$

$L_P = \frac{(3.80 * 200) + (4.75 * 400) + (5.50 * 400)}{(4.05 * 200) + (4.50 * 400) + (5.50 * 400)} = \frac{760 + 1900 + 2200}{810 + 1800 + 2200} = \frac{4860}{4810} \approx 1.010$

Laspeyres Quantity Index (Lieu 0 vs Lieu 1):

$$L_Q = \frac{\sum P_{1i} Q_{0i}}{\sum P_{1i} Q_{1i}}$$

$L_Q = \frac{(4.05 * 300) + (4.50 * 300) + (5.50 * 400)}{(4.05 * 200) + (4.50 * 400) + (5.50 * 400)} = \frac{1215 + 1350 + 2200}{810 + 1800 + 2200} = \frac{4765}{4810} \approx 0.991$

Paasche Price Index (Lieu 0 vs Lieu 1):

$$P_P = \frac{\sum P_{0i} Q_{0i}}{\sum P_{1i} Q_{0i}}$$

$P_P = \frac{(3.80 * 300) + (4.75 * 300) + (5.50 * 400)}{(4.05 * 300) + (4.50 * 300) + (5.50 * 400)} = \frac{1140 + 1425 + 2200}{1215 + 1350 + 2200} = \frac{4765}{4765} = 1$

Paasche Quantity Index (Lieu 0 vs Lieu 1):

$$P_Q = \frac{\sum P_{0i} Q_{0i}}{\sum P_{0i} Q_{1i}}$$

$P_Q = \frac{(3.80 * 300) + (4.75 * 300) + (5.50 * 400)}{(3.80 * 200) + (4.75 * 400) + (5.50 * 400)} = \frac{1140 + 1425 + 2200}{760 + 1900 + 2200} = \frac{4765}{4860} \approx 0.980$

3. Calculate Fisher index comparing Lieu 1 to Lieu 0:

Fisher Price Index (Lieu 1 vs Lieu 0):

$$F_P = \sqrt{L_P * P_P} = \sqrt{1 * 0.990} = \sqrt{0.990} \approx 0.995$$

Fisher Quantity Index (Lieu 1 vs Lieu 0):

$$F_Q = \sqrt{L_Q * P_Q} = \sqrt{1.020 * 1.010} = \sqrt{1.0302} \approx 1.015$$

3. Final Answer

1. Laspeyres Price Index (Lieu 1 vs Lieu 0): 1

Laspeyres Quantity Index (Lieu 1 vs Lieu 0): 1.020

Paasche Price Index (Lieu 1 vs Lieu 0): 0.990

Paasche Quantity Index (Lieu 1 vs Lieu 0): 1.010

2. Laspeyres Price Index (Lieu 0 vs Lieu 1): 1.010

Laspeyres Quantity Index (Lieu 0 vs Lieu 1): 0.991

Paasche Price Index (Lieu 0 vs Lieu 1): 1

Paasche Quantity Index (Lieu 0 vs Lieu 1): 0.980

3. Fisher Price Index (Lieu 1 vs Lieu 0): 0.995

Fisher Quantity Index (Lieu 1 vs Lieu 0): 1.015