A Los Angeles firm uses a single input to produce a recreational commodity according to a production function f(x) = 4√x, where x is the number of units of input. The commodity sells for$100 per unit. The input costs $50 per unit.

(a) Write down a function that states the firm’s profit as a function of the amount of input. 

(b) What is the profit-maximizing amount of input ______________ of output ___________ How much profits does it make when it maximizes profits ______?

(c) Suppose that the firm is taxed $20 per unit of its output and the price of its input is subsidized by $10. What is its new input level___________ . What is its new output level ___________ .How much profit does it make now? _________ . (A good way to solve this is to write an expression for the firm’s profit as a function of its input and solve for the profit-maximizing amount of input.)

(d) Suppose that instead of these taxes and subsidies, the firm is taxed at 50% of its profits. Write down its after-tax profits as a function of the amount of input. ______________. What is the profit-maximizing amount of output? _________. How much profit does it make after taxes _______.