A firm must choose total employee hours and robot hours k knowing that the hourly wage is $20, hourly robot rental rate is $10,

A firm must choose total employee hours ℓ and robot hours k knowing that the hourly wage is $20, hourly robot rental rate is $10, and total quantity pro- duced will be q = f (k, ℓ). The robot must at all times be supervised by the employee. Each unit of output can be sold at a price $p, which the firm takes as given. 

a) (2 points) Formally write the firm's profit maximisation problem. 

b) (2 points) How would you confirm this problem has a maximum? 

c) (4 points) How would you find optimal employee and robot hours? 

(Someone should be able to do so by following your answer exactly.) The optimum is the critical point (k∗, ℓ∗, μ∗), where μ∗ > 0. For the rest of the problem, assume the firm is at the optimum and justify your answers. 

d) (2 points) How do employee hours compare to robot hours? 

e) (4 points) By about how much do maximised profits change in the following scenarios, all else equal? 

• The firm can leave the robot unattended for one hour. 

• The robot rental rate increases to $11/hour. f) 

(4 points) Compute the derivative that describes.