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Sagot :
Let's break down the problem and calculate the required values step by step, assuming we have the provided information.
### Step 1: Calculate Total Revenue (TR)
Total revenue for each price level is calculated using the formula:
[tex]\[ \text{Total Revenue} (TR) = \text{Price} \times \text{Quantity Demanded} \][/tex]
Using the given data:
- When [tex]\( \text{Price} = \$10 \)[/tex] and [tex]\( \text{Quantity Demanded} = 320,000 \)[/tex]:
[tex]\[ TR_1 = 10 \times 320,000 = 3,200,000 \][/tex]
- When [tex]\( \text{Price} = \$9 \)[/tex] and [tex]\( \text{Quantity Demanded} = 422,500 \)[/tex]:
[tex]\[ TR_2 = 9 \times 422,500 = 3,802,500 \][/tex]
- When [tex]\( \text{Price} = \$8 \)[/tex] and [tex]\( \text{Quantity Demanded} = 525,000 \)[/tex]:
[tex]\[ TR_3 = 8 \times 525,000 = 4,200,000 \][/tex]
- When [tex]\( \text{Price} = \$7 \)[/tex] and [tex]\( \text{Quantity Demanded} = 627,500 \)[/tex]:
[tex]\[ TR_4 = 7 \times 627,500 = 4,392,500 \][/tex]
- When [tex]\( \text{Price} = \$6 \)[/tex] and [tex]\( \text{Quantity Demanded} = 730,000 \)[/tex]:
[tex]\[ TR_5 = 6 \times 730,000 = 4,380,000 \][/tex]
So the total revenues (TR) are: [tex]\[ [3,200,000, 3,802,500, 4,200,000, 4,392,500, 4,380,000] \][/tex]
### Step 2: Calculate Marginal Revenue (MR)
Marginal revenue is the additional revenue that is generated by selling one more unit of a good or service. The formula is:
[tex]\[ \text{Marginal Revenue} (MR) = \frac{\Delta TR}{\Delta QD} \][/tex]
where [tex]\(\Delta TR\)[/tex] is the change in total revenue and [tex]\(\Delta QD\)[/tex] is the change in quantity demanded.
- From [tex]\( TR_1 \)[/tex] to [tex]\( TR_2 \)[/tex]:
[tex]\[ MR_1 = \frac{3,802,500 - 3,200,000}{422,500 - 320,000} = \frac{602,500}{102,500} = 5.878048780487805 \][/tex]
- From [tex]\( TR_2 \)[/tex] to [tex]\( TR_3 \)[/tex]:
[tex]\[ MR_2 = \frac{4,200,000 - 3,802,500}{525,000 - 422,500} = \frac{397,500}{102,500} = 3.8780487804878048 \][/tex]
- From [tex]\( TR_3 \)[/tex] to [tex]\( TR_4 \)[/tex]:
[tex]\[ MR_3 = \frac{4,392,500 - 4,200,000}{627,500 - 525,000} = \frac{192,500}{102,500} = 1.8780487804878048 \][/tex]
- From [tex]\( TR_4 \)[/tex] to [tex]\( TR_5 \)[/tex]:
[tex]\[ MR_4 = \frac{4,380,000 - 4,392,500}{730,000 - 627,500} = \frac{-12,500}{102,500} = -0.12195121951219512 \][/tex]
So the marginal revenues (MR) are: [tex]\[ [5.878048780487805, 3.8780487804878048, 1.8780487804878048, -0.12195121951219512] \][/tex]
### Step 3: Calculate Total Revenue Per Firm
Since the industry is composed of 100 identical firms, the total revenue for each firm is:
[tex]\[ \text{Total Revenue per Firm} = \frac{\text{Total Revenue}}{\text{Number of Firms}} \][/tex]
- For [tex]\( TR_1 \)[/tex]:
[tex]\[ \frac{3,200,000}{100} = 32,000 \][/tex]
- For [tex]\( TR_2 \)[/tex]:
[tex]\[ \frac{3,802,500}{100} = 38,025 \][/tex]
- For [tex]\( TR_3 \)[/tex]:
[tex]\[ \frac{4,200,000}{100} = 42,000 \][/tex]
- For [tex]\( TR_4 \)[/tex]:
[tex]\[ \frac{4,392,500}{100} = 43,925 \][/tex]
- For [tex]\( TR_5 \)[/tex]:
[tex]\[ \frac{4,380,000}{100} = 43,800 \][/tex]
So the total revenue per firm are: [tex]\[ [32,000, 38,025, 42,000, 43,925, 43,800] \][/tex]
### Step 4: Calculate Marginal Revenue Per Firm
Similarly, the marginal revenue per firm is:
[tex]\[ \text{Marginal Revenue per Firm} = \frac{\text{Marginal Revenue}}{\text{Number of Firms}} \][/tex]
- For [tex]\( MR_1 \)[/tex]:
[tex]\[ \frac{5.878048780487805}{100} = 0.058780487804878045 \][/tex]
- For [tex]\( MR_2 \)[/tex]:
[tex]\[ \frac{3.8780487804878048}{100} = 0.03878048780487805 \][/tex]
- For [tex]\( MR_3 \)[/tex]:
[tex]\[ \frac{1.8780487804878048}{100} = 0.018780487804878048 \][/tex]
- For [tex]\( MR_4 \)[/tex]:
[tex]\[ \frac{-0.12195121951219512}{100} = -0.0012195121951219512 \][/tex]
So the marginal revenue per firm are: [tex]\[ [0.058780487804878045, 0.03878048780487805, 0.018780487804878048, -0.0012195121951219512] \][/tex]
### Final Result
For each of the 100 firms in this purely competitive industry:
- Total Revenues (TR) are: [tex]\[32,000, 38,025, 42,000, 43,925, 43,800\][/tex]
- Marginal Revenues (MR) are: [tex]\[0.058780487804878045, 0.03878048780487805, 0.018780487804878048, -0.0012195121951219512\][/tex]
### Step 1: Calculate Total Revenue (TR)
Total revenue for each price level is calculated using the formula:
[tex]\[ \text{Total Revenue} (TR) = \text{Price} \times \text{Quantity Demanded} \][/tex]
Using the given data:
- When [tex]\( \text{Price} = \$10 \)[/tex] and [tex]\( \text{Quantity Demanded} = 320,000 \)[/tex]:
[tex]\[ TR_1 = 10 \times 320,000 = 3,200,000 \][/tex]
- When [tex]\( \text{Price} = \$9 \)[/tex] and [tex]\( \text{Quantity Demanded} = 422,500 \)[/tex]:
[tex]\[ TR_2 = 9 \times 422,500 = 3,802,500 \][/tex]
- When [tex]\( \text{Price} = \$8 \)[/tex] and [tex]\( \text{Quantity Demanded} = 525,000 \)[/tex]:
[tex]\[ TR_3 = 8 \times 525,000 = 4,200,000 \][/tex]
- When [tex]\( \text{Price} = \$7 \)[/tex] and [tex]\( \text{Quantity Demanded} = 627,500 \)[/tex]:
[tex]\[ TR_4 = 7 \times 627,500 = 4,392,500 \][/tex]
- When [tex]\( \text{Price} = \$6 \)[/tex] and [tex]\( \text{Quantity Demanded} = 730,000 \)[/tex]:
[tex]\[ TR_5 = 6 \times 730,000 = 4,380,000 \][/tex]
So the total revenues (TR) are: [tex]\[ [3,200,000, 3,802,500, 4,200,000, 4,392,500, 4,380,000] \][/tex]
### Step 2: Calculate Marginal Revenue (MR)
Marginal revenue is the additional revenue that is generated by selling one more unit of a good or service. The formula is:
[tex]\[ \text{Marginal Revenue} (MR) = \frac{\Delta TR}{\Delta QD} \][/tex]
where [tex]\(\Delta TR\)[/tex] is the change in total revenue and [tex]\(\Delta QD\)[/tex] is the change in quantity demanded.
- From [tex]\( TR_1 \)[/tex] to [tex]\( TR_2 \)[/tex]:
[tex]\[ MR_1 = \frac{3,802,500 - 3,200,000}{422,500 - 320,000} = \frac{602,500}{102,500} = 5.878048780487805 \][/tex]
- From [tex]\( TR_2 \)[/tex] to [tex]\( TR_3 \)[/tex]:
[tex]\[ MR_2 = \frac{4,200,000 - 3,802,500}{525,000 - 422,500} = \frac{397,500}{102,500} = 3.8780487804878048 \][/tex]
- From [tex]\( TR_3 \)[/tex] to [tex]\( TR_4 \)[/tex]:
[tex]\[ MR_3 = \frac{4,392,500 - 4,200,000}{627,500 - 525,000} = \frac{192,500}{102,500} = 1.8780487804878048 \][/tex]
- From [tex]\( TR_4 \)[/tex] to [tex]\( TR_5 \)[/tex]:
[tex]\[ MR_4 = \frac{4,380,000 - 4,392,500}{730,000 - 627,500} = \frac{-12,500}{102,500} = -0.12195121951219512 \][/tex]
So the marginal revenues (MR) are: [tex]\[ [5.878048780487805, 3.8780487804878048, 1.8780487804878048, -0.12195121951219512] \][/tex]
### Step 3: Calculate Total Revenue Per Firm
Since the industry is composed of 100 identical firms, the total revenue for each firm is:
[tex]\[ \text{Total Revenue per Firm} = \frac{\text{Total Revenue}}{\text{Number of Firms}} \][/tex]
- For [tex]\( TR_1 \)[/tex]:
[tex]\[ \frac{3,200,000}{100} = 32,000 \][/tex]
- For [tex]\( TR_2 \)[/tex]:
[tex]\[ \frac{3,802,500}{100} = 38,025 \][/tex]
- For [tex]\( TR_3 \)[/tex]:
[tex]\[ \frac{4,200,000}{100} = 42,000 \][/tex]
- For [tex]\( TR_4 \)[/tex]:
[tex]\[ \frac{4,392,500}{100} = 43,925 \][/tex]
- For [tex]\( TR_5 \)[/tex]:
[tex]\[ \frac{4,380,000}{100} = 43,800 \][/tex]
So the total revenue per firm are: [tex]\[ [32,000, 38,025, 42,000, 43,925, 43,800] \][/tex]
### Step 4: Calculate Marginal Revenue Per Firm
Similarly, the marginal revenue per firm is:
[tex]\[ \text{Marginal Revenue per Firm} = \frac{\text{Marginal Revenue}}{\text{Number of Firms}} \][/tex]
- For [tex]\( MR_1 \)[/tex]:
[tex]\[ \frac{5.878048780487805}{100} = 0.058780487804878045 \][/tex]
- For [tex]\( MR_2 \)[/tex]:
[tex]\[ \frac{3.8780487804878048}{100} = 0.03878048780487805 \][/tex]
- For [tex]\( MR_3 \)[/tex]:
[tex]\[ \frac{1.8780487804878048}{100} = 0.018780487804878048 \][/tex]
- For [tex]\( MR_4 \)[/tex]:
[tex]\[ \frac{-0.12195121951219512}{100} = -0.0012195121951219512 \][/tex]
So the marginal revenue per firm are: [tex]\[ [0.058780487804878045, 0.03878048780487805, 0.018780487804878048, -0.0012195121951219512] \][/tex]
### Final Result
For each of the 100 firms in this purely competitive industry:
- Total Revenues (TR) are: [tex]\[32,000, 38,025, 42,000, 43,925, 43,800\][/tex]
- Marginal Revenues (MR) are: [tex]\[0.058780487804878045, 0.03878048780487805, 0.018780487804878048, -0.0012195121951219512\][/tex]
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