Critiquing the Derivation of an Optimal Choice Condition
A producer's net benefit from producing a quantity (Q) of a good is described by the objective function B(Q) = 120Q - 3Q². A student attempts to find the quantity that maximizes this benefit. Their work is shown below:
- Start with the objective function: B(Q) = 120Q - 3Q²
- To find the maximum, set the function equal to zero: 120Q - 3Q² = 0
- Solve for Q: 3Q(40 - Q) = 0, so Q = 0 or Q = 40.
- Conclusion: The optimal quantity is 40.
Critique the student's method. In your response, identify any specific errors in their reasoning or calculations. Explain why their approach is incorrect for finding the maximum benefit. Finally, provide the correct procedure to derive the first-order condition and determine the optimal quantity.
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