Profit function in the context of Firm

⭐ Core Definition: Profit function

In economics, profit maximization is the short run or long run process by which a firm may determine the price, input and output levels that will lead to the highest possible total profit (or just profit in short). In neoclassical economics, which is currently the mainstream approach to microeconomics, the firm is assumed to be a "rational agent" (whether operating in a perfectly competitive market or otherwise) which wants to maximize its total profit, which is the difference between its total revenue and its total cost.

Measuring the total cost and total revenue is often impractical, as the firms do not have the necessary reliable information to determine costs at all levels of production. Instead, they take more practical approach by examining how small changes in production influence revenues and costs. When a firm produces an extra unit of product, the additional revenue gained from selling it is called the marginal revenue ( ${\text{MR}}$ ), and the additional cost to produce that unit is called the marginal cost ( ${\text{MC}}$ ). When the level of output is such that the marginal revenue is equal to the marginal cost ( ${\text{MR}}={\text{MC}}$ ), then the firm's total profit is said to be maximized. If the marginal revenue is greater than the marginal cost ( ${\text{MR}}>{\text{MC}}$ ), then its total profit is not maximized, because the firm can produce additional units to earn additional profit. In other words, in this case, it is in the "rational" interest of the firm to increase its output level until its total profit is maximized. On the other hand, if the marginal revenue is less than the marginal cost ( ${\text{MR}}<{\text{MC}}$ ), then too its total profit is not maximized, because producing one unit less will reduce total cost more than total revenue gained, thus giving the firm more total profit. In this case, a "rational" firm has an incentive to reduce its output level until its total profit is maximized.

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Profit function in the context of Objective function

In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy.

In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss.

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⭐ Core Definition: Profit function

Profit function in the context of Objective function