Answer :
To determine how many sweatshirts the firm should produce and the resulting total profit, we perform a detailed analysis using marginal analysis. We'll calculate the Total Revenue (TR), Marginal Cost (MC), and profit at each quantity level, and then determine the optimal quantity to maximize profit.
### Step 1: Define the Market Conditions
- Market Price (P): \[tex]$8 per sweatshirt ### Step 2: Define Quantity (Q) and Total Cost (TC) Table | Q (Quantity) | TC (Total Cost) | |--------------|-----------------| | 0 | 1 | | 1 | 5 | | 2 | 11 | | 3 | 19 | | 4 | 29 | | 5 | 42 | ### Step 3: Calculate Total Revenue (TR) Total Revenue (TR) is calculated as: \[ \text{TR} = \text{Price} \times \text{Quantity} \] | Q (Quantity) | TR (Total Revenue) | |--------------|--------------------| | 0 | \$[/tex]0 |
| 1 | \[tex]$8 | | 2 | \$[/tex]16 |
| 3 | \[tex]$24 | | 4 | \$[/tex]32 |
| 5 | \[tex]$40 | ### Step 4: Calculate Marginal Cost (MC) Marginal Cost is the additional cost of producing one more unit of output. It is calculated as the change in Total Cost for each additional unit produced. \[ \text{MC}_i = \text{TC}_i - \text{TC}_{i-1} \] | Q (Quantity) | TC (Total Cost) | MC (Marginal Cost) | |--------------|-----------------|--------------------| | 0 | 1 | 0 | | 1 | 5 | 4 | | 2 | 11 | 6 | | 3 | 19 | 8 | | 4 | 29 | 10 | | 5 | 42 | 13 | ### Step 5: Calculate Profit Profit is calculated as the difference between Total Revenue and Total Cost: \[ \text{Profit} = \text{TR} - \text{TC} \] | Q (Quantity) | TR (Total Revenue) | TC (Total Cost) | Profit | |--------------|--------------------|-----------------|---------------------| | 0 | 0 | 1 | \$[/tex] (0 - 1) = -1 |
| 1 | 8 | 5 | \[tex]$ (8 - 5) = 3 | | 2 | 16 | 11 | \$[/tex] (16 - 11) = 5 |
| 3 | 24 | 19 | \[tex]$ (24 - 19) = 5 | | 4 | 32 | 29 | \$[/tex] (32 - 29) = 3 |
| 5 | 40 | 42 | \[tex]$ (40 - 42) = -2 | ### Step 6: Determine Optimal Quantity and Maximum Profit To maximize profit, the firm should produce up to the quantity where Marginal Cost (MC) is less than or equal to the market price, as long as profit is increasing. - At \( Q = 2 \) and \( Q = 3 \), profit is maximized at \( \$[/tex]5 \).
- However, at [tex]\( Q = 3 \)[/tex], the Marginal Cost (\[tex]$8) equals the market price, making \( Q = 3 \) feasible. - The optimal quantity where profit remains the highest (even though it doesn't improve from \( Q = 2 \) to \( Q = 3 \)), within practical MC limits is \( Q = 2 \). So, the firm should produce 2 sweatshirts to maximize profit. ### Total Profit Calculation At \( Q = 2 \): - Total Revenue (\( \text{TR} \)) = \$[/tex]16
- Total Cost ([tex]\( \text{TC} \)[/tex]) = \[tex]$11 - Total Profit = \( \text{TR} - \text{TC} = 16 - 11 = 5 \) ### Conclusion - Optimal Number of Sweatshirts to Produce: 2 - Total Profit: \$[/tex]5
### Step 1: Define the Market Conditions
- Market Price (P): \[tex]$8 per sweatshirt ### Step 2: Define Quantity (Q) and Total Cost (TC) Table | Q (Quantity) | TC (Total Cost) | |--------------|-----------------| | 0 | 1 | | 1 | 5 | | 2 | 11 | | 3 | 19 | | 4 | 29 | | 5 | 42 | ### Step 3: Calculate Total Revenue (TR) Total Revenue (TR) is calculated as: \[ \text{TR} = \text{Price} \times \text{Quantity} \] | Q (Quantity) | TR (Total Revenue) | |--------------|--------------------| | 0 | \$[/tex]0 |
| 1 | \[tex]$8 | | 2 | \$[/tex]16 |
| 3 | \[tex]$24 | | 4 | \$[/tex]32 |
| 5 | \[tex]$40 | ### Step 4: Calculate Marginal Cost (MC) Marginal Cost is the additional cost of producing one more unit of output. It is calculated as the change in Total Cost for each additional unit produced. \[ \text{MC}_i = \text{TC}_i - \text{TC}_{i-1} \] | Q (Quantity) | TC (Total Cost) | MC (Marginal Cost) | |--------------|-----------------|--------------------| | 0 | 1 | 0 | | 1 | 5 | 4 | | 2 | 11 | 6 | | 3 | 19 | 8 | | 4 | 29 | 10 | | 5 | 42 | 13 | ### Step 5: Calculate Profit Profit is calculated as the difference between Total Revenue and Total Cost: \[ \text{Profit} = \text{TR} - \text{TC} \] | Q (Quantity) | TR (Total Revenue) | TC (Total Cost) | Profit | |--------------|--------------------|-----------------|---------------------| | 0 | 0 | 1 | \$[/tex] (0 - 1) = -1 |
| 1 | 8 | 5 | \[tex]$ (8 - 5) = 3 | | 2 | 16 | 11 | \$[/tex] (16 - 11) = 5 |
| 3 | 24 | 19 | \[tex]$ (24 - 19) = 5 | | 4 | 32 | 29 | \$[/tex] (32 - 29) = 3 |
| 5 | 40 | 42 | \[tex]$ (40 - 42) = -2 | ### Step 6: Determine Optimal Quantity and Maximum Profit To maximize profit, the firm should produce up to the quantity where Marginal Cost (MC) is less than or equal to the market price, as long as profit is increasing. - At \( Q = 2 \) and \( Q = 3 \), profit is maximized at \( \$[/tex]5 \).
- However, at [tex]\( Q = 3 \)[/tex], the Marginal Cost (\[tex]$8) equals the market price, making \( Q = 3 \) feasible. - The optimal quantity where profit remains the highest (even though it doesn't improve from \( Q = 2 \) to \( Q = 3 \)), within practical MC limits is \( Q = 2 \). So, the firm should produce 2 sweatshirts to maximize profit. ### Total Profit Calculation At \( Q = 2 \): - Total Revenue (\( \text{TR} \)) = \$[/tex]16
- Total Cost ([tex]\( \text{TC} \)[/tex]) = \[tex]$11 - Total Profit = \( \text{TR} - \text{TC} = 16 - 11 = 5 \) ### Conclusion - Optimal Number of Sweatshirts to Produce: 2 - Total Profit: \$[/tex]5
