Algebraic Profit Maximization via Π'(Q)=0 vs. MR=MC
There are two equivalent algebraic approaches to solving a firm's profit maximization problem. The first is to differentiate the profit function, , and solve for the quantity where the derivative is zero (). The second approach involves finding marginal revenue (MR) and marginal cost (MC) by differentiating the revenue and cost functions respectively, and then solving for the quantity where MR = MC. Both methods will produce the identical profit-maximizing solution.
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Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
Ch.7 The firm and its customers - The Economy 2.0 Microeconomics @ CORE Econ
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Activity: Solving for the Profit-Maximizing Quantity (Q*) and Price (P*) Using Known Functions
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Figure 7.4b: Cheerios Profit Function Graph (Profit-Quantity Diagram)
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Algebraic Profit Maximization via Π'(Q)=0 vs. MR=MC
Profit Maximization for a Custom T-Shirt Business
A firm's profit (Π) is a function of the quantity (Q) it produces. The firm calculates the derivative of its profit function with respect to quantity, dΠ/dQ, at its current output level of 500 units and finds that the value is positive. Assuming the profit function is concave (meaning it has a single peak), what does this result imply about the firm's current production level?
Economic Rationale for the First-Order Condition
A firm's profit is maximized at the output level where the rate of change of its total revenue with respect to quantity is equal to the rate of change of its total cost with respect to quantity.
A firm's profit is depicted as a concave function of the quantity (Q) it produces, meaning the profit curve first rises to a peak and then falls. Three points are identified on this profit curve. Match each point's description with the correct mathematical statement about the first derivative of the profit function (dΠ/dQ) at that point.
Setting Up the Profit Maximization Problem
A company's profit (Π) as a function of the quantity (Q) it produces is given by the equation Π(Q) = -2Q² + 120Q - 500. To find the quantity that maximizes profit, the firm must first find the first derivative of the profit function with respect to quantity and set it equal to zero. The resulting equation, known as the first-order condition, is ____ = 0.
Comparing Profit Maximization Methods
A firm has an equation that expresses its profit (Π) solely as a function of the quantity (Q) it produces. To find the specific quantity that maximizes this profit, the firm must follow a set procedure. Arrange the following mathematical steps into the correct logical sequence.
A company's profit (Π) is described by a standard concave function of the quantity (Q) it produces, meaning the profit curve has a single peak. An analyst is tasked with finding the profit-maximizing output level. They correctly calculate the first derivative of the profit function with respect to quantity (dΠ/dQ). They then evaluate this derivative at two different output levels:
- At Q = 1,000 units, they find dΠ/dQ = +$15.
- At Q = 2,000 units, they find dΠ/dQ = -$10.
Based only on these two calculations, which of the following is the most logical conclusion about the profit-maximizing quantity, Q*?
Learn After
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