The firm's primary objective of profit maximization can be formally expressed as a constrained optimization problem. The task is to choose the price (P) and quantity (Q) that maximize the profit function, Π(P, Q), while being subject to the constraint imposed by the demand curve, which is defined by the equation P = f(Q).

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The demand curve embodies the trade-off a firm must navigate between price and quantity. A firm's desire for high prices and high quantities to maximize profit is constrained by the fact that raising the price will reduce the number of consumers willing to buy. Therefore, the demand curve acts as the firm's feasible frontier, outlining all viable combinations of price (P) and quantity (Q). A profit-maximizing firm will always select a point on this frontier because, for any given quantity, the highest possible price—and thus the highest profit—is found on the demand curve itself. [1] This makes the curve the central constraint in the firm's optimization problem. [1]

The Demand Curve as the Firm's Feasibility Frontier and Price-Quantity Trade-off

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CORE Econ - The Economy 2.0: Microeconomics

To achieve the highest possible profit, a firm must select a price and quantity from within its feasible set, which is defined by the area on or below the demand curve. The profit-maximizing combination occurs at the point on the demand curve that is tangent to the highest achievable isoprofit curve. At this optimal point, the slope of the demand curve (the firm's marginal rate of transformation) equals the slope of the isoprofit curve (the firm's marginal rate of substitution).

Profit Maximization at the Tangency of the Demand Curve and an Isoprofit Curve

The firm's profit maximization problem is a type of constrained choice problem with a structure analogous to a consumer's utility maximization problem. In both scenarios, the decision is depicted graphically. The objective, whether it is profit for the firm or utility for the consumer, is represented by indifference curves, while the feasible set of outcomes is determined by a constraint. The optimal solution for both problems is located at the tangency point between an indifference curve and the constraint.

Price-Quantity Profit Maximization as a Constrained Choice Problem Analogous to Consumer Choice

In a firm's price and quantity selection model, the feasible set includes all possible combinations of price and quantity that the firm can achieve, as determined by market demand. [1] Graphically, this set is the area enclosed by the demand curve and the price and quantity axes. [1] While any point in this set is attainable, a profit-maximizing firm will select a point on the edge of this set, specifically on the demand curve itself. [1]

Feasible Set in a Firm's Price-Quantity Model

Firm's Profit Maximization as a Constrained Optimization Problem

For a price-taking firm, such as a bakery in a competitive market, the feasible frontier is a horizontal line at the established market price (P*). The firm cannot set a price above this level, as it would lose all its customers to competitors. Consequently, the firm's feasible set of possible price and quantity combinations is the entire area at or below this horizontal frontier line.

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