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Sagot :
To find the change in the Herfindahl-Hirschman Index (HHI) following the merger of firm B and firm D, we can follow these steps:
### Step 1: Determine Firm D's Market Share
Given the market shares:
- Firm A: 40%
- Firm B: 30%
- Firm C: 18%
The total market must add up to 100%, so the market share of Firm D is calculated as:
[tex]\[ \text{Firm D} = 100\% - (40\% + 30\% + 18\%) = 100\% - 88\% = 12\% \][/tex]
### Step 2: Calculate the Initial HHI
The HHI is calculated by squaring the market share of each firm (as a percentage) and summing these squares. Let's convert the percentages to fractions and compute the initial HHI:
[tex]\[ \text{Firm A's share} = 0.40 \][/tex]
[tex]\[ \text{Firm B's share} = 0.30 \][/tex]
[tex]\[ \text{Firm C's share} = 0.18 \][/tex]
[tex]\[ \text{Firm D's share} = 0.12 \][/tex]
The initial HHI is:
[tex]\[ \text{Initial HHI} = (0.40^2 + 0.30^2 + 0.18^2 + 0.12^2) \times 10000 \][/tex]
Calculating each term:
[tex]\[ 0.40^2 = 0.16, \quad 0.30^2 = 0.09, \quad 0.18^2 = 0.0324, \quad 0.12^2 = 0.0144 \][/tex]
Adding these results:
[tex]\[ \text{Initial HHI} = (0.16 + 0.09 + 0.0324 + 0.0144) \times 10000 = 0.2968 \times 10000 = 2968 \][/tex]
### Step 3: Calculate the HHI Post-Merger
After the merger, the combined market share of firm B and firm D is:
[tex]\[ \text{Firm BD's share} = 30\% + 12\% = 42\% = 0.42 \text{ as a fraction} \][/tex]
Now, recalculate the HHI with firms A, merged firm BD, and C:
[tex]\[ \text{Firm A's share} = 0.40 \][/tex]
[tex]\[ \text{Merged Firm BD's share} = 0.42 \][/tex]
[tex]\[ \text{Firm C's share} = 0.18 \][/tex]
The new HHI is:
[tex]\[ \text{New HHI} = (0.40^2 + 0.42^2 + 0.18^2) \times 10000 \][/tex]
Calculating each term:
[tex]\[ 0.40^2 = 0.16, \quad 0.42^2 = 0.1764, \quad 0.18^2 = 0.0324 \][/tex]
Adding these results:
[tex]\[ \text{New HHI} = (0.16 + 0.1764 + 0.0324) \times 10000 = 0.3688 \times 10000 = 3688 \][/tex]
### Step 4: Calculate the Change in HHI
Finally, the change in HHI due to the merger is:
[tex]\[ \text{Change in HHI} = \text{New HHI} - \text{Initial HHI} = 3688 - 2968 = 720 \][/tex]
Thus, the change in HHI is 720. Therefore, the correct choice is:
[tex]\[ \boxed{720} \][/tex]
### Step 1: Determine Firm D's Market Share
Given the market shares:
- Firm A: 40%
- Firm B: 30%
- Firm C: 18%
The total market must add up to 100%, so the market share of Firm D is calculated as:
[tex]\[ \text{Firm D} = 100\% - (40\% + 30\% + 18\%) = 100\% - 88\% = 12\% \][/tex]
### Step 2: Calculate the Initial HHI
The HHI is calculated by squaring the market share of each firm (as a percentage) and summing these squares. Let's convert the percentages to fractions and compute the initial HHI:
[tex]\[ \text{Firm A's share} = 0.40 \][/tex]
[tex]\[ \text{Firm B's share} = 0.30 \][/tex]
[tex]\[ \text{Firm C's share} = 0.18 \][/tex]
[tex]\[ \text{Firm D's share} = 0.12 \][/tex]
The initial HHI is:
[tex]\[ \text{Initial HHI} = (0.40^2 + 0.30^2 + 0.18^2 + 0.12^2) \times 10000 \][/tex]
Calculating each term:
[tex]\[ 0.40^2 = 0.16, \quad 0.30^2 = 0.09, \quad 0.18^2 = 0.0324, \quad 0.12^2 = 0.0144 \][/tex]
Adding these results:
[tex]\[ \text{Initial HHI} = (0.16 + 0.09 + 0.0324 + 0.0144) \times 10000 = 0.2968 \times 10000 = 2968 \][/tex]
### Step 3: Calculate the HHI Post-Merger
After the merger, the combined market share of firm B and firm D is:
[tex]\[ \text{Firm BD's share} = 30\% + 12\% = 42\% = 0.42 \text{ as a fraction} \][/tex]
Now, recalculate the HHI with firms A, merged firm BD, and C:
[tex]\[ \text{Firm A's share} = 0.40 \][/tex]
[tex]\[ \text{Merged Firm BD's share} = 0.42 \][/tex]
[tex]\[ \text{Firm C's share} = 0.18 \][/tex]
The new HHI is:
[tex]\[ \text{New HHI} = (0.40^2 + 0.42^2 + 0.18^2) \times 10000 \][/tex]
Calculating each term:
[tex]\[ 0.40^2 = 0.16, \quad 0.42^2 = 0.1764, \quad 0.18^2 = 0.0324 \][/tex]
Adding these results:
[tex]\[ \text{New HHI} = (0.16 + 0.1764 + 0.0324) \times 10000 = 0.3688 \times 10000 = 3688 \][/tex]
### Step 4: Calculate the Change in HHI
Finally, the change in HHI due to the merger is:
[tex]\[ \text{Change in HHI} = \text{New HHI} - \text{Initial HHI} = 3688 - 2968 = 720 \][/tex]
Thus, the change in HHI is 720. Therefore, the correct choice is:
[tex]\[ \boxed{720} \][/tex]
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