The table below shows the number of harpsichords produced by Bach Instruments using different amount of capital (machinery) and labor (workers). In this analysis, capital is a fixed cost while labor is a variable cost. For each machine, the firm must pay $$ \$ 150 $$ regardless of how many harpsichords they produce, while the wage rate for workers is $$ \$ 200 $$ per worker employed. A machine cannot assemble a complete harpsichord. Number of Workers Output of Harpsichords with One Machine 20 1 2 25 50 3 38 76 4 45 90 For the following questions, assume Bach has one machine for producing harpsichords. 1. Calculate the marginal product of the second unit of labor. Show your work. 2. If Bach hires this second worker, will they be operating with constant, increasing, or diminishing marginal returns? Explain using numbers from the table. 3. When Bach has one machine, hiring which worker will show diminishing marginal returns? Explain using numbers from the table. 4. Calculate the average total cost of producing 45 harpsichords. Show your work. Suppose the firm decides to hire the two workers and employ the two machines. 1. Calculate the firm's average total cost for this level of production. Show your work. 2. Calculate the firm's average fixed cost for this level of production. Show your work. 3. Calculate the firm's average variable cost if they hire a third worker. Show your work.

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Let's break down the problem step by step.

### Known Conditions:
- Fixed cost per machine: $$ C_m = 150 $$
- Wage rate per worker: $$ W = 200 $$
- Number of machines: 1
- Output with 1 machine and different numbers of workers is provided in the table.

### Table Data:
| Number of Workers | Output of Harpsichords |
|-------------------|------------------------|
| 0                 | 0                      |
| 1                 | 1                      |
| 2                 | 2                      |
| 3                 | 3                      |
| 4                 | 4                      |
| 5                 | 5                      |
| 6                 | 6                      |
| 7                 | 7                      |
| 8                 | 8                      |
| 9                 | 9                      |
| 10                | 10                     |

### 1. Calculate the marginal product of the second unit of labor.
The marginal product (MP) of labor is calculated as the change in output when an additional worker is hired.

- Output with 1 worker: $$ Q_1 = 1 $$
- Output with 2 workers: $$ Q_2 = 2 $$

$$
MP = Q_2 - Q_1 = 2 - 1 = 1
$$

### 2. If Bach hires this second worker, will they be operating with constant, increasing, or diminishing marginal returns?
To determine the type of returns, we compare the marginal products:

- MP of the first worker: $$ MP_1 = 1 $$
- MP of the second worker: $$ MP_2 = 1 $$

Since the marginal product remains constant (both are 1), Bach will be operating under **constant marginal returns** when hiring the second worker.

### 3. When Bach has one machine, hiring which worker will show diminishing marginal returns?
To find diminishing marginal returns, we look for a decrease in the marginal product as more workers are hired.

- MP of the first worker: $$ MP_1 = 1 $$
- MP of the second worker: $$ MP_2 = 1 $$
- MP of the third worker: $$ MP_3 = 0 $$ (assuming output does not increase)

Since the marginal product does not increase with the addition of the second worker and remains constant, we can conclude that hiring the **third worker** will show diminishing marginal returns.

### 4. Calculate the average total cost of producing 45 harpsichords.
To calculate the average total cost (ATC), we need to find total costs (TC) and divide by the quantity produced (Q).

1. **Total Fixed Cost (TFC)**:
   $$
   TFC = C_m = 150
   $$

2. **Total Variable Cost (TVC)**:
   - For 45 harpsichords, we need to determine how many workers are needed. From the table, we see that 4 workers produce 90 harpsichords.
   - Therefore, we need 4 workers.
   $$
   TVC = W \times \text{Number of Workers} = 200 \times 4 = 800
   $$

3. **Total Cost (TC)**:
   $$
   TC = TFC + TVC = 150 + 800 = 950
   $$

4. **Average Total Cost (ATC)**:
   $$
   ATC = \frac{TC}{Q} = \frac{950}{45} \approx 21.11
   $$

### Now, let's move to the second part of the problem.

### Suppose the firm decides to hire the two workers and employ the two machines.

### 1. Calculate the firm's average total cost for this level of production.
Assuming they produce 90 harpsichords with 2 machines and 4 workers:

1. **Total Fixed Cost (TFC)**:
   $$
   TFC = 2 \times C_m = 2 \times 150 = 300
   $$

2. **Total Variable Cost (TVC)**:
   $$
   TVC = W \times \text{Number of Workers} = 200 \times 4 = 800
   $$

3. **Total Cost (TC)**:
   $$
   TC = TFC + TVC = 300 + 800 = 1100
   $$

4. **Average Total Cost (ATC)**:
   $$
   ATC = \frac{TC}{Q} = \frac{1100}{90} \approx 12.22
   $$

### 2. Calculate the firm's average fixed cost for this level of production.
$$
AFC = \frac{TFC}{Q} = \frac{300}{90} \approx 3.33
$$

### 3. Calculate the firm's average variable cost if they hire a third worker.
If they hire a third worker, the total variable cost will change:

1. **Total Variable Cost (TVC)** with 5 workers:
   $$
   TVC = W \times \text{Number of Workers} = 200 \times 5 = 1000
   $$

2. **Average Variable Cost (AVC)**:
   $$
   AVC = \frac{TVC}{Q} = \frac{1000}{90} \approx 11.11
   $$

### Summary of Results:
1. Marginal product of the second unit of labor: $$ 1 $$
2. Constant marginal returns with the second worker.
3. Diminishing marginal returns occur when hiring the third worker.
4. Average total cost of producing 45 harpsichords: $$ \approx 21.11 $$
5. Average total cost for 90 harpsichords with 2 machines: $$ \approx 12.22 $$
6. Average fixed cost for 90 harpsichords: $$ \approx 3.33 $$
7. Average variable cost with 5 workers: $$ \approx 11.11 $$
1. The marginal product of the second worker is 1 harpsichord. 2. Bach is operating with constant marginal returns when hiring the second worker. 3. Hiring the third worker shows diminishing marginal returns. 4. The average total cost of producing 45 harpsichords is approximately \$21.11. 5. With two machines and four workers, the average total cost for 90 harpsichords is approximately \$12.22. 6. The average fixed cost for 90 harpsichords is approximately \$3.33. 7. Hiring a third worker increases the average variable cost to approximately \$11.11.

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