A firm has demand function P = 108 5Q and the cost function C = 12Q +Q^2 . Find the price at which the profit is maximum. Also find the maximum profit.

Solution

Step 1: Define Profit Function Profit ($\pi$) is given by the difference between total revenue ($TR$) and total cost ($TC$). $$\pi =TR-TC$$ Total revenue is calculated as: $$TR=P\times Q=(108-5Q)\times Q=108Q-5Q^2$$ Given the cost function: $$C=-12Q+Q^2$$ Thus, the profit function becomes: $$\pi =(108Q-5Q^2)-(-12Q+Q^2)=108Q-5Q^2+12Q-Q^2=120Q-6Q^2$$

Step 2: Find the Critical Point for Maximum Profit To find the quantity ($Q$) that maximizes profit, we differentiate the profit function with respect to $Q$ and set it to zero: $$\frac{d\pi }{dQ}=120-12Q$$ Setting the derivative equal to zero: $$120-12Q=0⟹12Q=120⟹Q=10$$

Step 3: Verify the Maximum Profit To ensure this critical point is a maximum, we check the second derivative: $$\frac{d^2\pi }{d{Q}^2}=-12$$ Since the second derivative is negative, the profit function has a maximum at $Q=10$.

Step 4: Find the Optimal Price Substitute $Q=10$ into the demand function to find the price ($P$): $$P=108-5(10)=108-50=58$$

Step 5: Calculate Maximum Profit Substitute $Q=10$ into the profit function: $$\pi =120(10)-6(10{)}^2=1200-600=600$$

Final Answer

• Price at which profit is maximum: Rs. 58
• Maximum profit: Rs. 600