The Mathematical Equivalence of Profit Maximization Rules
A firm's profit is the difference between its total revenue and its total cost. One rule for finding the profit-maximizing level of output is to find the quantity where the slope of the profit function is zero. Another rule is to find the quantity where the additional revenue from selling one more unit is exactly equal to the additional cost of producing that one more unit. Using the basic definitions of profit, marginal revenue, and marginal cost in the context of calculus, explain why these two rules are mathematically identical and must therefore yield the same result.
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Demonstrating Equivalent Profit Maximization Methods
Verifying a Profit Maximization Recommendation
A firm's total revenue function is R(Q) = 15Q - Q², and its total cost function is C(Q) = 2Q. Which of the following equations, when solved for Q, will yield the profit-maximizing output level?
A firm is trying to find its profit-maximizing output level. It correctly calculates that at an output of Q=50, the slope of its total revenue curve is $7 and the slope of its total cost curve is also $7. Based on this information, the firm can conclude that the slope of its profit function at Q=50 is zero.
The Mathematical Equivalence of Profit Maximization Rules
A firm's total revenue is given by the function R(Q) and its total cost by the function C(Q). The firm's profit is given by Π(Q) = R(Q) - C(Q). To find the profit-maximizing level of output, a manager can use two equivalent approaches: setting the derivative of the profit function to zero (Π'(Q) = 0) or setting marginal revenue equal to marginal cost (MR = MC). Match each mathematical expression with its correct economic term.
Applying Equivalent Profit Maximization Methods
Reconciling Profit Maximization Approaches
Diagnosing an Error in Profit Maximization