A firm sells two goods (X and Y)that are related in consumption.The estimated inverse demand and cost functions are: $$\begin{array} { l } P _ { X } = 105 - 0.5 Q _ { X } - 0.75 Q _ { Y } \ P _ { Y } = 120 - Q _ { Y } - 0.5 Q _ { X } \ M C _ { X } = 10 + 0.25 Q _ { X } \ M C _ { Y } = 16 + 0.5 Q _ { Y } \end{array}$$ What are the profit-maximizing levels of output for the two goods? A) $Q _ { x }$ = 6, $Q _ { Y }$ = 3 B) $Q _ { x }$ = 40, $Q _ { Y }$ = 40 C) $Q _ { x }$ = 58, $Q _ { Y }$ = 30 D) $Q _ { x }$ = 240, $Q _ { Y }$ = 270 E)none of the above

Question

Quizplus · Accepted Answer

To find the profit maximizing levels of output for the two goods, we need to set up the profit function as follows:
$$ \pi = (105-0.5Q_X-0.75Q_Y-10-0.25Q_X)Q_X + (120-Q_Y-0.5Q_X-16-0.5Q_Y)Q_Y$$
Simplifying this expression, we get:
$$ \pi = 94Q_X + 103Q_Y - 0.75Q_X^2 - 1.25Q_XQ_Y - 0.5Q_Y^2$$
To find the profit-maximizing levels of output, we need to take partial derivatives of the profit function with respect to each variable, set them equal to zero, and solve for the variables. 
$$ \frac{\partial \pi}{\partial Q_X} = 94 - 1.5Q_X - 1.25Q_Y = 0$$
$$ \frac{\partial \pi}{\partial Q_Y} = 103 - 1.25Q_X - Q_Y = 0$$
Solving these two equations simultaneously, we get:
$$Q_X = 58$$
$$Q_Y = 30$$
Therefore, the answer is C.

A Firm Sells Two Goods (X and Y)that Are Related