Crafting a Profit Model for a New Product
A company is launching a new smart widget. Market research indicates that the price (P) they can charge per widget is related to the weekly quantity (Q) sold by the inverse demand equation P = 200 - 0.5Q. The accounting department has determined that the total cost (C) to produce Q widgets per week is given by the cost function C(Q) = 5000 + 20Q. The management team wants a single equation that expresses the company's weekly profit (Π) solely as a function of the quantity (Q) produced and sold. Derive this profit function.
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Figure 7.4b: Cheerios Profit Function Graph (Profit-Quantity Diagram)
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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)