Damian owns a business that makes customized refrigerator magnets for other organizations to give away at job fairs and public events. His firm uses a Cobb-Douglas production function given by F(K, N) = zKN1 . Damian owns 54 machines that print the magnets and die-cut them into custom shapes, so his capital stock is K = 54. Labor (N) is measured as the number of workers. Each magnet sells at price p = 1. Our job is to determine the optimal labor input to maximize Damian's profit.

1. Suppose that z = 28, = 1 3 , and the market wage is w = 14 per hour.

(a) Write down the firm's profit maximization problem formally (mathematically).

(b) Solve the firm's problem to obtain the optimal labor choice, N .

(c) How many magnets (total output Y ) does Damian produce?

(d) How much profit () does Damian earn?

2. Suppose half of Damian's machines are offline for maintenance, so his capital stock decreases to K = 27. All else remains as in #1 above.

(a) Solve the firm's problem to obtain the optimal labor choice, N .

(b) How many magnets (total output Y ) does Damian produce?

(c) How much profit () does Damian earn?

(d) Explain why Damian changed his optimal choice for N compared to #1. (Hint: recall the optimal condition MPN = w.) How did the change affect the firm's output and profit? What would have been output and profit if the firm had maintained the same N you found in #1?

3. Suppose Damian attends a management training course where she learns techniques to improve organizational efficiencies in his business, and as a result his firm's total factor productivity increases to z = 56. All else remains as in #2 above.

(a) Solve the firm's problem to obtain the optimal labor choice, N .

(b) How many magnets (total output Y ) does Damian produce?

(c) How much profit () does Damian earn?

(d) Explain why Damian changed his optimal choice for N compared to #2. How did the change affect the firm's output and profit? What would have been output and profit if the firm had maintained the same N you found in #2?

4. Now suppose that the market wage rate increases to w = 16. All else remains as in #3 above.

(a) Solve the firm's problem to obtain the optimal labor choice, N .

(b) How many magnets (total output Y ) does Damian produce?

(c) How much profit () does Damian earn?

(d) Explain why Damian changed his optimal choice for N compared to #3. How did the change affect the firm's output and profit? What would have been output and profit if the firm had maintained the same N you found in #3?

5. Bonus: Suppose the price for magnets increases to p = 2. All else remains as in #4 above.

(a) Solve the firm's problem to obtain the optimal labor choice, N .

(b) How many magnets (total output Y ) does Damian produce?

(c) How much profit () does Damian earn?

(d) Explain why Damian changed his optimal choice for N compared to #4. How did the change affect the firm's output and profit? What would have been output and profit if the firm had maintained the same N you found in #4?