Zar Island Gas Company Is the Sole Producer of Natural $$a x ^ { 2 } + b x + c = 0 \text { are } X = \frac { - b \pm \left( b ^ { 2 } - 4 a c \right) ^ { 5 } } { 2 a }$$

Essay

Zar Island Gas Company is the sole producer of natural gas in the remote island country of Zar.The company's operations are regulated by the State Energy Commission.The demand function for gas in Zar has been estimated as:
where Q is output (measured in units)and P is price (measured in dollars per unit).Zar Island's cost function is:
This total cost function does not include a "normal" return on the firm's invested capital of $4 million.
(a)In the absence of any government price regulation,determine Zar Island's optimal (i)output level,(ii)selling price,(iii)total profits,and (iv)rate of return on its asset base.
(b)The State Energy Commission has ordered the firm to charge a price which will provide it with no more than a 12 percent return on its total assets.Determine Zar Island's (i)output level,(ii)selling price,and (iii)total profits under this constraint.
Hint: The roots of the quadratic equation:
(a) In the absence of any government price regulation, determine Zar Island's optimal (i) output level, (ii) selling price, (iii) total profits, and (iv) rate of return on its asset base. The State Energy Commission has ordered the firm to charge a price which will provide it
(b) with no more than a 12 percent return on its total assets. Determine Zar Island's (1) output level, (ii) selling price, and (iii) total profits under this constraint. Hint: The roots of the quadratic equation:
$$a x ^ { 2 } + b x + c = 0 \text { are } X = \frac { - b \pm \left( b ^ { 2 } - 4 a c \right) ^ { 5 } } { 2 a }$$ $$a x ^ { 2 } + b x + c = 0 \text { are } X = \frac { - b \pm \left( b ^ { 2 } - 4 a c \right) ^ { 5 } } { 2 a }$$