Finding the Profit-Maximizing Quantity Using the First-Order Condition (dΠ/dQ = 0)

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

Profit Maximization with Cost Function C(Q)=320+2Q+0.2Q^2 and Inverse Demand P=44−0.5Q

A firm faces an inverse demand curve given by P = 100 - 2Q and has a total cost function of C(Q) = 10Q + Q². Using the substitution method, which of the following equations correctly represents the firm's profit (Π) as a function of quantity (Q) alone?

An analyst for a company wants to build a model to see how profit changes as the quantity of goods produced changes. They have the company's cost function and the market's demand curve. Arrange the following steps in the correct logical sequence to derive an equation that expresses profit solely as a function of quantity (Q).

Deriving a Profit Function from Demand and Cost

Crafting a Profit Model for a New Product

A firm's profit (Π) is a function of the quantity (Q) it produces. This function can be derived from its total cost function, C(Q), and the market's inverse demand function, P(Q), using the formula Π(Q) = [P(Q) * Q] - C(Q). Match each set of cost and demand functions to the correct resulting profit function.

A firm's market is characterized by an inverse demand function of P = 200 - 4Q, and its total cost of production is given by C(Q) = 500 + 20Q. An analyst for the firm has determined that the profit function, expressed solely in terms of quantity (Q), is Π(Q) = -300 - 24Q.

A firm's profit as a function of quantity is given by Π(Q) = -2Q² + 80Q - 150. If the firm's total cost function is C(Q) = 20Q + 150, then the inverse demand function P(Q) faced by the firm must be ____.

The Strategic Value of a Single-Variable Profit Model

A company's total cost to produce a good is represented by the function C(Q) = 5Q² + 10Q + 50. The market demand for this good is given by the equation Q = 80 - 0.5P, where Q is the quantity demanded and P is the price. To analyze its pricing strategy, the company needs to express its profit (Π) solely as a function of the quantity produced (Q). Which of the following equations correctly represents this relationship?

Error Analysis in Profit Function Derivation

Finding the Profit-Maximizing Quantity Using the First-Order Condition (dΠ/dQ = 0)

Profit Maximization with Cost Function C(Q)=320+2Q+0.2Q^2 and Inverse Demand P=44−0.5Q

Profit Maximization with Cost Function C(Q) = 50 + 4Q + Q^2 and Inverse Demand P = 100 - 2Q

Profit Maximization for a Bicycle Manufacturer

A company's production costs are described by the total cost function C(Q) = 10Q + 50, and the market demand for its product is represented by the inverse demand function P = 70 - Q. To maximize its profit, what quantity (Q*) should the company produce, and what price (P*) should it charge?

Calculating Optimal Price and Quantity

A firm knows its total cost function and the market's inverse demand function. To find the specific price and quantity that will maximize its profit, it must follow a specific sequence of calculations. Arrange the following steps in the correct logical order.

A firm's production costs are described by the function C(Q) = 20 + 10Q + Q², and the market demand for its product is represented by the inverse demand function P = 70 - 2Q. An analyst correctly determines that the profit-maximizing quantity is Q* = 10. However, in their final step, they calculate the profit-maximizing price to be P* = $220. Which statement best describes the error made in calculating the price?

Evaluating a Suboptimal Production Decision

A firm's production process is described by the total cost function C(Q) = 100 + 4Q². The firm faces a linear inverse demand curve of the form P = A - 2Q, where 'A' is a positive constant representing the maximum possible price. If the firm sets its price at the profit-maximizing level of P* = $80, what must be the value of the constant 'A'?

Impact of a Demand Shift on a Firm's Pricing Strategy

A firm operates with a total cost function C(Q) = 5Q² + 10Q + 50 and faces an inverse demand function P = 100 - 5Q. A consultant advises the firm that to maximize profit, they should produce a quantity of 10 units.

For each firm described by a unique pair of cost and inverse demand functions, match it with its correct profit-maximizing output quantity (Q*).