- Determine the profit maximizing output for a firm exhibiting the following:
Demand Schedule: P = 4000 – 33Q
TC = 2Q3 – 3Q2 + 400Q + 5000
The Correct Answer and Explanation is :
To determine the profit-maximizing output, we need to find the quantity (Q) at which the firm’s marginal revenue (MR) equals marginal cost (MC). Here’s the step-by-step process:
Step 1: Determine the Revenue Function
The revenue (R) is given by price (P) times quantity (Q):
[
R = P \times Q
]
From the demand schedule ( P = 4000 – 33Q ), we substitute it into the revenue formula:
[
R = (4000 – 33Q) \times Q = 4000Q – 33Q^2
]
Step 2: Determine Marginal Revenue (MR)
Marginal revenue is the derivative of the revenue function with respect to Q:
[
MR = \frac{dR}{dQ} = 4000 – 66Q
]
Step 3: Determine the Marginal Cost (MC)
The total cost (TC) function is given as:
[
TC = 2Q^3 – 3Q^2 + 400Q + 5000
]
Marginal cost is the derivative of the total cost function with respect to Q:
[
MC = \frac{dTC}{dQ} = 6Q^2 – 6Q + 400
]
Step 4: Set MR Equal to MC
To find the profit-maximizing output, set MR equal to MC:
[
4000 – 66Q = 6Q^2 – 6Q + 400
]
Rearrange the equation:
[
6Q^2 + 60Q – 3600 = 0
]
Step 5: Solve the Quadratic Equation
Solve the quadratic equation using the quadratic formula:
[
Q = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
Where ( a = 6 ), ( b = 60 ), and ( c = -3600 ):
[
Q = \frac{-60 \pm \sqrt{60^2 – 4(6)(-3600)}}{2(6)}
]
[
Q = \frac{-60 \pm \sqrt{3600 + 86400}}{12}
]
[
Q = \frac{-60 \pm \sqrt{90000}}{12}
]
[
Q = \frac{-60 \pm 300}{12}
]
Thus, the two possible solutions for Q are:
[
Q = \frac{240}{12} = 20 \quad \text{or} \quad Q = \frac{-360}{12} = -30
]
Since quantity cannot be negative, the profit-maximizing output is ( Q = 20 ).
Step 6: Confirm the Profit-Maximizing Output
At ( Q = 20 ), we can check that MR equals MC to confirm:
[
MR = 4000 – 66(20) = 4000 – 1320 = 2680
]
[
MC = 6(20)^2 – 6(20) + 400 = 6(400) – 120 + 400 = 2400 – 120 + 400 = 2680
]
Since MR equals MC at ( Q = 20 ), this is the profit-maximizing output.
Conclusion
The profit-maximizing output for the firm is 20 units. At this output, the firm’s marginal revenue equals its marginal cost, ensuring maximum profit.