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 $$ 10 $$ 20 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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1. The output with one machine is given by:
   - 1 worker: $$10$$ harpsichords
   - 2 workers: $$25$$ harpsichords

The marginal product (MP) of the second unit of labor is the change in output when adding the second worker:
   $$
   \text{MP}_2 = 25 - 10 = 15.
   $$

2. For the first worker, the output is $$10$$ harpsichords. When the second worker is hired, output increases by $$15$$ (from $$10$$ to $$25$$). Since $$15$$ is greater than $$10$$, the second worker contributes more than the first worker did. This indicates increasing marginal returns at that stage.

3. Looking at the production with one machine:
   - With 2 workers, output is $$25$$ (marginal product from worker 2 is $$25-10=15$$).
   - With 3 workers, output is $$38$$ (marginal product from worker 3 is $$38-25=13$$).
   - With 4 workers, output is $$45$$ (marginal product from worker 4 is $$45-38=7$$).
   
   The marginal product falls from $$15$$ to $$13$$ when the third worker is added. Thus, hiring the third worker marks the beginning of diminishing marginal returns.

4. To calculate the average total cost (ATC) when producing $$45$$ harpsichords with one machine:
   - Fixed cost (machine): $$\$150$$
   - To produce $$45$$ harpsichords, the firm hires $$4$$ workers; labor cost is:
     $$
     4 \times 200 = \$800.
     $$
   - Total cost is:
     $$
     150 + 800 = \$950.
     $$
   - Average total cost is:
     $$
     \text{ATC} = \frac{950}{45} \approx 21.11.
     $$

Now suppose the firm hires $$2$$ workers and employs $$2$$ machines. The production with two machines is as follows:
   - With $$2$$ workers, output is $$50$$ harpsichords.

5. Calculate the firm's average total cost (ATC) at this level of production:
   - Fixed cost (2 machines): 
     $$
     2 \times 150 = \$300.
     $$
   - Labor cost (2 workers): 
     $$
     2 \times 200 = \$400.
     $$
   - Total cost:
     $$
     300 + 400 = \$700.
     $$
   - Average total cost:
     $$
     \text{ATC} = \frac{700}{50} = 14.
     $$

6. Calculate the firm's average fixed cost (AFC) for this production level:
   - Average fixed cost:
     $$
     \text{AFC} = \frac{300}{50} = 6.
     $$

7. Now, if the firm hires a third worker with two machines, the output becomes $$76$$ harpsichords. The additional cost here is the variable cost from labor:
   - Labor cost for $$3$$ workers:
     $$
     3 \times 200 = \$600.
     $$
   - Average variable cost (AVC) is:
     $$
     \text{AVC} = \frac{600}{76} \approx 7.89.
     $$

1. The marginal product of the second worker is 15 harpsichords. 2. Hiring the second worker results in increasing marginal returns because the output increases by 15 harpsichords. 3. Hiring the third worker shows diminishing marginal returns as the output increases by only 13 harpsichords. 4. The average total cost of producing 45 harpsichords is approximately \$21.11. 5. With two machines and two workers, the average total cost is \$14. 6. The average fixed cost is \$6. 7. Hiring a third worker with two machines results in an average variable cost of approximately \$7.89.

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