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### (a) Compute the values [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex].
1. Calculating [tex]\( P \)[/tex]:
- [tex]\( P \)[/tex] is the total product at 4 workers.
- Given the previous total product (16 kg for 3 workers) and the marginal product for adding the 4th worker (9 kg), we calculate:
[tex]\[ P = \text{Total Product for 3 workers} + \text{Marginal Product of 4th worker} = 16 + 9 = 25 \, \text{kg} \][/tex]
2. Calculating [tex]\( Q \)[/tex]:
- [tex]\( Q \)[/tex] is the marginal product for the 6th worker.
- Given the total product at 5 workers (32 kg) and at 6 workers (36 kg), we calculate:
[tex]\[ Q = \text{Total Product for 6 workers} - \text{Total Product for 5 workers} = 36 - 32 = 4 \, \text{kg} \][/tex]
3. Calculating [tex]\( R \)[/tex]:
- [tex]\( R \)[/tex] is the average product for 8 workers.
- Given the total product for 8 workers (34 kg), we calculate:
[tex]\[ R = \frac{\text{Total Product for 8 workers}}{\text{Number of workers}} = \frac{34}{8} = 4.25 \, \text{kg} \][/tex]
Therefore, [tex]\( P = 25 \, \text{kg} \)[/tex], [tex]\( Q = 4 \, \text{kg} \)[/tex], and [tex]\( R = 4.25 \, \text{kg} \)[/tex].
### (b) At what level(s) of employment does the company experience:
1. Increasing returns:
- Increasing returns occur when the marginal product of adding an additional worker is higher than the marginal product of the previous worker.
- The levels of employment with increasing returns are 2, 3, and 4 workers (since the marginal products 5, 8, and 9, respectively, are increasing).
2. Decreasing returns:
- Decreasing returns occur when the marginal product of adding an additional worker is lower than the marginal product of the previous worker, but still positive.
- The levels of employment with decreasing returns are 5 and 6 workers (since the marginal products 7 and 4, respectively, are positive but decreasing).
3. Constant returns:
- Constant returns occur when the marginal product remains the same as the previous marginal product.
- There are no levels of employment with constant returns in the given data.
4. Negative returns:
- Negative returns occur when the marginal product is negative.
- The level of employment with negative returns is 8 workers (since the marginal product is -2).
### (c) State the law exhibited in the table above.
The law exhibited in the table is the Law of Diminishing Marginal Returns. This law states that as additional units of a variable input (workers) are added to a fixed input, the additional output (marginal product) produced by each additional unit of the variable input will eventually decrease.
### (d) Explain why the firm should or should not employ a 9th worker.
The firm should not employ a 9th worker. This conclusion is based on the observation that at 8 workers, the marginal product is already negative (-2 kg). Hiring a 9th worker would likely decrease the total product further, leading to inefficiencies and reduced overall productivity.
### (a) Compute the values [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex].
1. Calculating [tex]\( P \)[/tex]:
- [tex]\( P \)[/tex] is the total product at 4 workers.
- Given the previous total product (16 kg for 3 workers) and the marginal product for adding the 4th worker (9 kg), we calculate:
[tex]\[ P = \text{Total Product for 3 workers} + \text{Marginal Product of 4th worker} = 16 + 9 = 25 \, \text{kg} \][/tex]
2. Calculating [tex]\( Q \)[/tex]:
- [tex]\( Q \)[/tex] is the marginal product for the 6th worker.
- Given the total product at 5 workers (32 kg) and at 6 workers (36 kg), we calculate:
[tex]\[ Q = \text{Total Product for 6 workers} - \text{Total Product for 5 workers} = 36 - 32 = 4 \, \text{kg} \][/tex]
3. Calculating [tex]\( R \)[/tex]:
- [tex]\( R \)[/tex] is the average product for 8 workers.
- Given the total product for 8 workers (34 kg), we calculate:
[tex]\[ R = \frac{\text{Total Product for 8 workers}}{\text{Number of workers}} = \frac{34}{8} = 4.25 \, \text{kg} \][/tex]
Therefore, [tex]\( P = 25 \, \text{kg} \)[/tex], [tex]\( Q = 4 \, \text{kg} \)[/tex], and [tex]\( R = 4.25 \, \text{kg} \)[/tex].
### (b) At what level(s) of employment does the company experience:
1. Increasing returns:
- Increasing returns occur when the marginal product of adding an additional worker is higher than the marginal product of the previous worker.
- The levels of employment with increasing returns are 2, 3, and 4 workers (since the marginal products 5, 8, and 9, respectively, are increasing).
2. Decreasing returns:
- Decreasing returns occur when the marginal product of adding an additional worker is lower than the marginal product of the previous worker, but still positive.
- The levels of employment with decreasing returns are 5 and 6 workers (since the marginal products 7 and 4, respectively, are positive but decreasing).
3. Constant returns:
- Constant returns occur when the marginal product remains the same as the previous marginal product.
- There are no levels of employment with constant returns in the given data.
4. Negative returns:
- Negative returns occur when the marginal product is negative.
- The level of employment with negative returns is 8 workers (since the marginal product is -2).
### (c) State the law exhibited in the table above.
The law exhibited in the table is the Law of Diminishing Marginal Returns. This law states that as additional units of a variable input (workers) are added to a fixed input, the additional output (marginal product) produced by each additional unit of the variable input will eventually decrease.
### (d) Explain why the firm should or should not employ a 9th worker.
The firm should not employ a 9th worker. This conclusion is based on the observation that at 8 workers, the marginal product is already negative (-2 kg). Hiring a 9th worker would likely decrease the total product further, leading to inefficiencies and reduced overall productivity.
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