A firm sells two goods (X and Y)that are related in consumption.The estimated inverse demand and cost functions are: $$\begin{array} { l } P _ { X } = 105 - 0.5 Q _ { X } - 0.75 Q _ { Y } \ P _ { Y } = 120 - Q _ { Y } - 0.5 Q _ { X } \ M C _ { X } = 10 + 0.25 Q _ { X } \ M C _ { Y } = 16 + 0.5 Q _ { Y } \end{array}$$ Good X and Y are: A)complements B)substitutes C)independent D)normal E)inferior

Question

Quizplus · Accepted Answer

The goods X and Y are substitutes because the coefficient of Qx in the inverse demand function of Y and the coefficient of Qy in the inverse demand function of X are both negative, indicating that an increase in the price of one good will increase the demand for the other good. To find the best choice, we need to calculate the profit-maximizing quantity of X and Y. Using the profit function, we get:

π = (105-0.5Qx-0.75Qy-Qy)*(Qx) - (10+0.25Qx)*(Qx) + (120-Qy-0.5Qx)*(Qy) - (16+0.5Qy)*(Qy)
π = -0.25Qx^2-0.75QxQy+104Qx-0.75Qy^2+104Qy-480

To find the first-order conditions, we take the derivative of the profit function with respect to Qx and Qy and set them equal to zero:

∂π/∂Qx = -0.5Qx-0.75Qy+104+0.5(∂Qy/∂Qx)(Qy) + 0.25(∂Qy/∂Qx)Qx - 0.5(∂Qy/∂Qx)(Qy) - 0.25Qx = 0
-0.75Qx - 1.5Qy + 104 = 0
-0.5Qy-0.5Qx+120+0.5(∂Qx/∂Qy)(Qx) - 0.5(∂Qx/∂Qy)(Qy) - 0.5(∂Qx/∂Qy)(Qx) - 0.25Qy = 0
-0.5Qx - 0.75Qy + 60 = 0

Solving these two equations simultaneously, we get:
Qx = 60
Qy = 40

Therefore, the best choice is to produce and sell 60 units of X and 40 units of Y.

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