Activity: Solving for the Profit-Maximizing Quantity (Q*) and Price (P*) Using Known Functions

Conceptual Interpretation of the First-Order Condition as a Tangency Condition

Figure 7.4b: Cheerios Profit Function Graph (Profit-Quantity Diagram)

Deriving the Price Markup-Demand Elasticity Relationship from the First-Order Condition

Profit Maximization at the Intersection of Marginal Revenue and Marginal Cost Curves

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*?