Market demand facing a monopolist is Qd=-5P+20. If the monopoly practices perfect price discrimination, what is the profit-maximizing level of output when MC=$2?
To find the profit-maximizing level of output for a monopolist practicing perfect price discrimination, we need to equate the marginal cost (MC) with the marginal revenue (MR) and solve for the corresponding quantity.
In perfect price discrimination, the monopolist charges each consumer the maximum they are willing to pay, resulting in no consumer surplus. Therefore, the price (P) will be equal to the marginal revenue.
The marginal revenue can be found by taking the derivative of the total revenue function with respect to quantity. In this case, since the demand function is given as Qd = -5P + 20, the total revenue (TR) can be calculated by multiplying the price (P) by the quantity (Qd).
TR = P * Qd = P * (-5P + 20)
Next, we differentiate the total revenue function with respect to quantity to find the marginal revenue.
MR = d(TR)/dQd = d(P * (-5P + 20))/dQd
To find MR, we differentiate each term in the product using the product rule:
MR = (dP/dQd) * (-5P + 20) + P * (d(-5P + 20)/dQd)
Since P is a constant with respect to Qd, its derivative is zero:
MR = (dP/dQd) * (-5P + 20)
Since P = MR, we can set the marginal cost (MC) equal to the marginal revenue (MR):
MC = MR
Substituting the previously derived expression for MR, we get:
2 = (dP/dQd) * (-5P + 20)
Solving for dP/dQd:
2 = -5P * (dP/dQd) + 20 * (dP/dQd)
Rearranging the equation:
-5P * (dP/dQd) + 20 * (dP/dQd) = 2
Combining like terms:
(dP/dQd) * (-5P + 20) = 2
Now we have an equation that relates the price P and the derivative of price with respect to quantity (dP/dQd).
To find the profit-maximizing level of output, we need to solve this equation for P. However, we do not have enough information to solve for P given the MC in this scenario. We would need additional information about the price function to determine the profit-maximizing level of output.