Maximizing Profit: Calculating Optimal Production
To maximize profit, a firm should produce at the level where Marginal Cost (MC) equals Marginal Revenue (MR). The profit is maximized when this condition holds, provided the price is greater than the average variable cost (which is implied here as we seek maximization).
1. Define Cost and Revenue Functions
- Cost Function: $ C(q) = 5q^2 $
- Revenue Function: $ R(q) = \text{Price} \times q = 50q $
2. Calculate Marginal Cost (MC)
Marginal Cost is the derivative of the Cost Function with respect to quantity ($q$): $ MC = \frac{dC}{dq} = \frac{d}{dq}(5q^2) = 10q $
3. Calculate Marginal Revenue (MR)
Marginal Revenue is the derivative of the Revenue Function with respect to quantity ($q$): $ MR = \frac{dR}{dq} = \frac{d}{dq}(50q) = 50 $
4. Find Production Quantity for Maximum Profit
Set MC equal to MR to find the profit-maximizing quantity ($q$): $ MC = MR $ $ 10q = 50 $ $ q = \frac{50}{10} $ $ q = 5 $
The second-order condition for profit maximization requires the second derivative of the profit function to be negative. Profit $ \Pi(q) = R(q) - C(q) = 50q - 5q^2 $. $ \frac{d\Pi}{dq} = 50 - 10q $ $ \frac{d^2\Pi}{dq^2} = -10 $ Since $ -10 < 0 $, the profit is indeed maximized at $ q=5 $.
Therefore, the firm should produce 5 units to maximize profit.
Important Questions from Maxima & Minima
Which of the following statements is false about convex minimization problem?
For what value of 'x' will the function y = x2 - 4x have the maximum or minimum value?
For a right-angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have a maximum area of the triangle, the angle between the hypotenuse and the side is
The optimum value of the function f(x) = x2 – 4x + 2 is
As \(\rm x\) varies from \(\rm −1\ to \ +3\), which one of the following describes the behaviour of the function \(\rm f(x) = x^3 – 3x^2 + 1\)?
