The problem asks us to calculate Laspeyres and Paasche indices for the cost of living in different African cities (Dakar, Abidjan, and Bamako) using the provided price (P) and quantity (Q) data for logement (housing), nourriture (food), and électricité (electricity). We need to calculate these indices with different base cities and then determine if the Laspeyres index is circular.

1. Problem Description

The problem asks us to calculate Laspeyres and Paasche indices for the cost of living in different African cities (Dakar, Abidjan, and Bamako) using the provided price (P) and quantity (Q) data for logement (housing), nourriture (food), and électricité (electricity). We need to calculate these indices with different base cities and then determine if the Laspeyres index is circular.

2. Solution Steps

First, let's define the formulas for the Laspeyres and Paasche indices:

Laspeyres Index:

$L = \frac{\sum P_t Q_0}{\sum P_0 Q_0}$

Paasche Index:

$P = \frac{\sum P_t Q_t}{\sum P_0 Q_t}$

where $P_t$ is the price in the current period, $Q_t$ is the quantity in the current period, $P_0$ is the price in the base period, and $Q_0$ is the quantity in the base period.

1.a. Dakar (t), Abidjan (0)

Laspeyres:

$L = \frac{(2*2 + 4*2 + 2*1)}{(3*2 + 5*2 + 9*1)} = \frac{4 + 8 + 2}{6 + 10 + 9} = \frac{14}{25} = 0.56$

Paasche:

$P = \frac{(2*4 + 4*2 + 2*1)}{(3*4 + 5*2 + 9*1)} = \frac{8 + 8 + 2}{12 + 10 + 9} = \frac{18}{31} = 0.5806$

1.b. Abidjan (t), Dakar (0)

Laspeyres:

$L = \frac{(3*4 + 5*2 + 9*1)}{(2*4 + 4*2 + 2*1)} = \frac{12 + 10 + 9}{8 + 8 + 2} = \frac{31}{18} = 1.7222$

Paasche:

$P = \frac{(3*2 + 5*2 + 9*1)}{(2*2 + 4*2 + 2*1)} = \frac{6 + 10 + 9}{4 + 8 + 2} = \frac{25}{14} = 1.7857$

2.a. Abidjan (t), Bamako (0)

Laspeyres:

$L = \frac{(3*3 + 5*4 + 9*2)}{(3*3 + 3*4 + 1*2)} = \frac{9 + 20 + 18}{9 + 12 + 2} = \frac{47}{23} = 2.0435$

Paasche:

$P = \frac{(3*2 + 5*2 + 9*1)}{(3*2 + 3*2 + 1*1)} = \frac{6 + 10 + 9}{6 + 6 + 1} = \frac{25}{13} = 1.9231$

2.b. Bamako (t), Dakar (0)

Laspeyres:

$L = \frac{(3*4 + 3*2 + 1*1)}{(2*4 + 4*2 + 2*1)} = \frac{12 + 6 + 1}{8 + 8 + 2} = \frac{19}{18} = 1.0556$

Paasche:

$P = \frac{(3*3 + 3*4 + 1*2)}{(2*3 + 4*2 + 2*1)} = \frac{9 + 12 + 2}{6 + 8 + 2} = \frac{23}{16} = 1.4375$

3. Circularity of Laspeyres Index

To check for circularity, we need to see if the Laspeyres index from city A to city B, multiplied by the Laspeyres index from city B to city C, equals the Laspeyres index from city A to city C. Let L(A, B) denote the Laspeyres index of A with B as the base. We need to check if L(Dakar, Abidjan) * L(Abidjan, Bamako) = L(Dakar, Bamako).

We have L(Dakar, Abidjan) = 0.56 (from 1a). Then L(Abidjan, Dakar) = 1/0.56 = 1.7857 (Using reciprocal property).

We have L(Abidjan, Bamako) = 2.0435 (from 2a). Then L(Bamako, Abidjan) = 1/2.0435 = 0.4893

We have L(Bamako, Dakar). To Calculate L(Dakar, Bamako), we compute:

L(Dakar, Bamako) = $\frac{(2*3 + 4*4 + 2*2)}{(3*3 + 3*4 + 1*2)} = \frac{6 + 16 + 4}{9 + 12 + 2} = \frac{26}{23} = 1.1304$

So, we require: L(Dakar, Abidjan) * L(Abidjan, Bamako) = L(Dakar, Bamako) i.e., $0.56 * 2.0435 \approx 1.1304$ which is true.

Alternatively,

L(Abidjan, Dakar) = 1.7222 (from 1b).

L(Bamako, Abidjan) can be calculated as 1/2.0435 = 0.4893 (Using reciprocal property)

Now, L(Bamako, Dakar) = $\frac{(3*4 + 3*2 + 1*1)}{(2*4 + 4*2 + 2*1)} = \frac{19}{18} = 1.0556$

We require L(Abidjan, Dakar) * L(Bamako, Abidjan) = L(Bamako, Dakar) i.e. $1.7222 * 0.4893 \approx 1.0556$, which is true.

To be thorough, we can calculate L(Dakar, Bamako) directly as follows:

$L(Dakar, Bamako) = \frac{(2*3 + 4*4 + 2*2)}{(3*3 + 3*4 + 1*2)} = \frac{6 + 16 + 4}{9 + 12 + 2} = \frac{26}{23} \approx 1.1304$

Also, calculate L(Abidjan, Dakar) as 31/18 = 1.7222 (as found in part 1b) and L(Abidjan, Bamako) as 47/23 = 2.0435 (as found in part 2a).

Then, we calculate L(Dakar, Bamako) from the indices calculated earlier:

$L(Dakar, Bamako) = \frac{L(Abidjan, Bamako)}{L(Abidjan, Dakar)} = \frac{47/23}{31/18} = \frac{47*18}{23*31} = \frac{846}{713} \approx 1.1865$. This value is different.

We conclude that the Laspeyres index is *not* circular.

3. Final Answer

1.a. Dakar (t), Abidjan (0): Laspeyres = 0.56, Paasche = 0.5806

1.b. Abidjan (t), Dakar (0): Laspeyres = 1.7222, Paasche = 1.7857

2.a. Abidjan (t), Bamako (0): Laspeyres = 2.0435, Paasche = 1.9231

2.b. Bamako (t), Dakar (0): Laspeyres = 1.0556, Paasche = 1.4375

3. The Laspeyres index is not circular.