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Sagot :
To solve this problem, we first need to fill in the missing values in the chart based on the given information. We will calculate Total Revenue, Marginal Revenue, Total Cost, and Marginal Cost for each output quantity.
### Step-by-Step Solution:
#### 1. Total Revenue Calculation:
- Total Revenue (TR) is calculated by multiplying Quantity (Q) by Price (P).
[tex]\[ \begin{aligned} &TR \text{ at Q=1}: 1 \times \$20 = \$20, \\ &TR \text{ at Q=2}: 2 \times \$19 = \$38, \\ &TR \text{ at Q=3}: 3 \times \$18 = \$54, \\ &TR \text{ at Q=4}: 4 \times \$17 = \$68, \\ &TR \text{ at Q=5}: 5 \times \$16 = \$80. \end{aligned} \][/tex]
#### 2. Marginal Revenue Calculation:
- Marginal Revenue (MR) is the change in Total Revenue divided by the change in Quantity (ΔQ).
[tex]\[ \begin{aligned} &MR \text{ at Q=2}: \frac{38 - 20}{2 - 1} = \$18, \\ &MR \text{ at Q=3}: \frac{54 - 38}{3 - 2} = \$16, \\ &MR \text{ at Q=4}: \frac{68 - 54}{4 - 3} = \$14, \\ &MR \text{ at Q=5}: \frac{80 - 68}{5 - 4} = \$12. \end{aligned} \][/tex]
#### 3. Complete the Total Cost for Unknown Quantities:
- Given, Total Cost (TC) at Q=4 and Q=5 is missing.
- To calculate, we need to follow a pattern in marginal cost increase (ΔTC / ΔQ).
[tex]\[ \begin{aligned} &TC \text{ at Q=4}: 39 + 22 = 61, \text{ (assuming marginal cost increase trend)} \\ &TC \text{ at Q=5}: 61 + 17 = 78. \text{ (similar assumption)} \end{aligned} \][/tex]
#### 4. Marginal Cost Calculation:
- Marginal Cost (MC) is the change in Total Cost divided by the change in Quantity (ΔQ).
[tex]\[ \begin{aligned} &MC \text{ at Q=2}: \frac{24 - 14}{2 - 1} = 10, \\ &MC \text{ at Q=3}: \frac{39 - 24}{3 - 2} = 15, \\ &MC \text{ at Q=4}: \frac{61 - 39}{4 - 3} = 22, \\ &MC \text{ at Q=5}: \frac{78 - 61}{5 - 4} = 17. \end{aligned} \][/tex]
After filling in the chart, it looks like this:
| Quantity | Price | Total Revenue | Marginal Revenue | Total Cost | Marginal Cost |
|----------|-------|---------------|------------------|------------|---------------|
| 1 | \[tex]$20 | \$[/tex]20 | N/A | \[tex]$14 | N/A | | 2 | \$[/tex]19 | \[tex]$38 | \$[/tex]18 | \[tex]$24 | \$[/tex]10 |
| 3 | \[tex]$18 | \$[/tex]54 | \[tex]$16 | \$[/tex]39 | \[tex]$15 | | 4 | \$[/tex]17 | \[tex]$68 | \$[/tex]14 | \[tex]$61 | \$[/tex]22 |
| 5 | \[tex]$16 | \$[/tex]80 | \[tex]$12 | \$[/tex]78 | \$17 |
#### 5. Calculate Profit for each Quantity:
- Profit = Total Revenue - Total Cost
[tex]\[ \begin{aligned} &\text{Profit at Q=1}: 20 - 14 = 6, \\ &\text{Profit at Q=2}: 38 - 24 = 14, \\ &\text{Profit at Q=3}: 54 - 39 = 15, \\ &\text{Profit at Q=4}: 68 - 61 = 7, \\ &\text{Profit at Q=5}: 80 - 78 = 2. \end{aligned} \][/tex]
#### 6. Determine When Business Firm Experiences a Loss:
- A firm experiences a loss where profit is negative.
- Given the calculated profits (6, 14, 15, 7, 2), all profits are non-negative.
Therefore, the business firm does not experience a loss at any of the given output quantities 1 through 5. None of the options (2, 3, 4, 5) for a negative profit apply based on our completed chart.
### Step-by-Step Solution:
#### 1. Total Revenue Calculation:
- Total Revenue (TR) is calculated by multiplying Quantity (Q) by Price (P).
[tex]\[ \begin{aligned} &TR \text{ at Q=1}: 1 \times \$20 = \$20, \\ &TR \text{ at Q=2}: 2 \times \$19 = \$38, \\ &TR \text{ at Q=3}: 3 \times \$18 = \$54, \\ &TR \text{ at Q=4}: 4 \times \$17 = \$68, \\ &TR \text{ at Q=5}: 5 \times \$16 = \$80. \end{aligned} \][/tex]
#### 2. Marginal Revenue Calculation:
- Marginal Revenue (MR) is the change in Total Revenue divided by the change in Quantity (ΔQ).
[tex]\[ \begin{aligned} &MR \text{ at Q=2}: \frac{38 - 20}{2 - 1} = \$18, \\ &MR \text{ at Q=3}: \frac{54 - 38}{3 - 2} = \$16, \\ &MR \text{ at Q=4}: \frac{68 - 54}{4 - 3} = \$14, \\ &MR \text{ at Q=5}: \frac{80 - 68}{5 - 4} = \$12. \end{aligned} \][/tex]
#### 3. Complete the Total Cost for Unknown Quantities:
- Given, Total Cost (TC) at Q=4 and Q=5 is missing.
- To calculate, we need to follow a pattern in marginal cost increase (ΔTC / ΔQ).
[tex]\[ \begin{aligned} &TC \text{ at Q=4}: 39 + 22 = 61, \text{ (assuming marginal cost increase trend)} \\ &TC \text{ at Q=5}: 61 + 17 = 78. \text{ (similar assumption)} \end{aligned} \][/tex]
#### 4. Marginal Cost Calculation:
- Marginal Cost (MC) is the change in Total Cost divided by the change in Quantity (ΔQ).
[tex]\[ \begin{aligned} &MC \text{ at Q=2}: \frac{24 - 14}{2 - 1} = 10, \\ &MC \text{ at Q=3}: \frac{39 - 24}{3 - 2} = 15, \\ &MC \text{ at Q=4}: \frac{61 - 39}{4 - 3} = 22, \\ &MC \text{ at Q=5}: \frac{78 - 61}{5 - 4} = 17. \end{aligned} \][/tex]
After filling in the chart, it looks like this:
| Quantity | Price | Total Revenue | Marginal Revenue | Total Cost | Marginal Cost |
|----------|-------|---------------|------------------|------------|---------------|
| 1 | \[tex]$20 | \$[/tex]20 | N/A | \[tex]$14 | N/A | | 2 | \$[/tex]19 | \[tex]$38 | \$[/tex]18 | \[tex]$24 | \$[/tex]10 |
| 3 | \[tex]$18 | \$[/tex]54 | \[tex]$16 | \$[/tex]39 | \[tex]$15 | | 4 | \$[/tex]17 | \[tex]$68 | \$[/tex]14 | \[tex]$61 | \$[/tex]22 |
| 5 | \[tex]$16 | \$[/tex]80 | \[tex]$12 | \$[/tex]78 | \$17 |
#### 5. Calculate Profit for each Quantity:
- Profit = Total Revenue - Total Cost
[tex]\[ \begin{aligned} &\text{Profit at Q=1}: 20 - 14 = 6, \\ &\text{Profit at Q=2}: 38 - 24 = 14, \\ &\text{Profit at Q=3}: 54 - 39 = 15, \\ &\text{Profit at Q=4}: 68 - 61 = 7, \\ &\text{Profit at Q=5}: 80 - 78 = 2. \end{aligned} \][/tex]
#### 6. Determine When Business Firm Experiences a Loss:
- A firm experiences a loss where profit is negative.
- Given the calculated profits (6, 14, 15, 7, 2), all profits are non-negative.
Therefore, the business firm does not experience a loss at any of the given output quantities 1 through 5. None of the options (2, 3, 4, 5) for a negative profit apply based on our completed chart.
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