Monopoly Profit Maximization Explained

To find the profit-maximizing output for a pure monopoly firm, we need to determine the quantity (Q) where Marginal Revenue (MR) equals Marginal Cost (MC).

Deriving Marginal Revenue (MR)

First, we calculate the Total Revenue (TR) using the inverse demand function, $P = 500 - 5 Q$.

$ TR = P \times Q $

$ TR = (500 - 5 Q) \times Q $

$ TR = 500Q - 5Q^2 $

Marginal Revenue (MR) is the derivative of TR with respect to Q:

$ MR = \frac{d(TR)}{dQ} = 500 - 10 Q $

Deriving Marginal Cost (MC)

The Total Cost (TC) function is given as $TC = 50 + 20 Q + Q^2$.

Marginal Cost (MC) is the derivative of TC with respect to Q:

$ MC = \frac{d(TC)}{dQ} = 20 + 2 Q $

Calculating Profit-Maximizing Output

Profit maximization occurs where $MR = MC$. Set the derived MR and MC functions equal to each other:

$ 500 - 10 Q = 20 + 2 Q $

Now, solve for Q:

• Subtract $2Q$ from both sides: $500 - 12 Q = 20$
• Subtract $500$ from both sides: $-12 Q = 20 - 500$
• Simplify: $-12 Q = -480$
• Divide by $-12$: $Q = \frac{-480}{-12}$
• Result: $Q = 40$

The profit-maximizing output is 40 units.