Profit maximization

Profit maximization

Overview
In economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, profit maximization is the (short run) process by which a firm determines the price
Price
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...

 and output level that returns the greatest profit
Profit (economics)
In economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...

. There are several approaches to this problem. The total revenue–total cost method relies on the fact that profit equals revenue minus cost, and the marginal revenue
Marginal revenue
In microeconomics, marginal revenue is the extra revenue that an additional unit of product will bring. It is the additional income from selling one more unit of a good; sometimes equal to price...

marginal cost
Marginal cost
In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. That is, it is the cost of producing one more unit of a good...

 method is based on the fact that total profit in a perfectly competitive market reaches its maximum point where marginal revenue equals marginal cost.

Any costs incurred by a firm
Corporation
A corporation is created under the laws of a state as a separate legal entity that has privileges and liabilities that are distinct from those of its members. There are many different forms of corporations, most of which are used to conduct business. Early corporations were established by charter...

 may be classed into two groups: fixed costs and variable costs.
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Encyclopedia
In economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, profit maximization is the (short run) process by which a firm determines the price
Price
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...

 and output level that returns the greatest profit
Profit (economics)
In economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...

. There are several approaches to this problem. The total revenue–total cost method relies on the fact that profit equals revenue minus cost, and the marginal revenue
Marginal revenue
In microeconomics, marginal revenue is the extra revenue that an additional unit of product will bring. It is the additional income from selling one more unit of a good; sometimes equal to price...

marginal cost
Marginal cost
In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. That is, it is the cost of producing one more unit of a good...

 method is based on the fact that total profit in a perfectly competitive market reaches its maximum point where marginal revenue equals marginal cost.

Basic definitions


Any costs incurred by a firm
Corporation
A corporation is created under the laws of a state as a separate legal entity that has privileges and liabilities that are distinct from those of its members. There are many different forms of corporations, most of which are used to conduct business. Early corporations were established by charter...

 may be classed into two groups: fixed costs and variable costs. Fixed costs are incurred by the business at any level of output, including zero output. These may include equipment maintenance, rent, wages, and general upkeep. Variable costs change with the level of output, increasing as more product is generated. Materials consumed during production often have the largest impact on this category. Fixed cost and variable cost, combined, equal total cost
Total cost
In economics, and cost accounting, total cost describes the total economic cost of production and is made up of variable costs, which vary according to the quantity of a good produced and include inputs such as labor and raw materials, plus fixed costs, which are independent of the quantity of a...

.

Revenue
Revenue
In business, revenue is income that a company receives from its normal business activities, usually from the sale of goods and services to customers. In many countries, such as the United Kingdom, revenue is referred to as turnover....

 is the amount of money that a company receives from its normal business activities, usually from the sale of goods and services (as opposed to monies from security sales such as equity shares or debt issuances).

Marginal
Marginal
Marginal may refer to:* For marginal constituency in politics, see “Marginal seat”* For the manga, see Marginal * For marginal model in hierarchical linear modeling, see “Marginal model”...

 cost and revenue, depending on whether the calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

 approach is taken or not, are defined as either the change in cost or revenue as each additional unit is produced, or the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of cost or revenue with respect to quantity output. It may also be defined as the addition to total cost or revenue as output increase by a single unit. For instance, taking the first definition, if it costs a firm 400 USD to produce 5 units and 480 USD to produce 6, the marginal cost of the sixth unit is approximately 80 dollars, although this is more accurately stated as the marginal cost of the 5.5th unit due to linear interpolation
Linear interpolation
Linear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation, which can also be used as a verb .-Linear interpolation between two known points:...

. Calculus is capable of providing more accurate answers if regression
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...

 equations can be provided.

Total revenue - total cost method



To obtain the profit maximising output quantity, we start by recognizing that profit is equal to total revenue (TR) minus total cost (TC). Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph. Finding the profit-maximizing output is as simple as finding the output at which profit reaches its maximum. That is represented by output Q in the diagram.

There are two graphical ways of determining that Q is optimal. First, the profit curve is at its maximum at this point (A). Secondly, at the point (B) the tangent on the total cost curve (TC) is parallel to the total revenue curve (TR), meaning that the surplus of revenue net of costs (B,C) is at its greatest. Because total revenue minus total costs is equal to profit, the line segment C,B is equal in length to the line segment A,Q.

Computing the price at which to sell the product requires knowledge of the firm's demand
Supply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...

 curve. The price at which quantity demanded equals profit-maximizing output is the optimum price to sell the product
Product (business)
In general, the product is defined as a "thing produced by labor or effort" or the "result of an act or a process", and stems from the verb produce, from the Latin prōdūce ' lead or bring forth'. Since 1575, the word "product" has referred to anything produced...

.

Marginal revenue-marginal cost method



An alternative argument says that for each unit sold, marginal profit (Mπ) equals marginal revenue (MR) minus marginal cost (MC). Then, if marginal revenue is greater than marginal cost, marginal profit is positive, and if marginal revenue is less than marginal cost, marginal profit is negative. When marginal revenue equals marginal cost, marginal profit is zero. Since total profit increases when marginal profit is positive and total profit decreases when marginal profit is negative, it must reach a maximum where marginal profit is zero - or where marginal cost equals marginal revenue. If there are two points where this occurs, maximum profit is achieved where the producer has collected positive profit up until the intersection of MR and MC (where zero profit is collected), but would not continue to after, as opposed to vice versa, which represents a profit minimum. In calculus terms, the correct intersection of MC and MR will occur when:

The intersection of MR and MC is shown in the next diagram as point A. If the industry is perfectly competitive (as is assumed in the diagram), the firm faces a demand curve (D) that is identical to its Marginal revenue curve (MR), and this is a horizontal line at a price determined by industry supply and demand. Average total costs are represented by curve ATC. Total economic profit are represented by area P,A,B,C. The optimum quantity (Q) is the same as the optimum quantity (Q) in the first diagram.

If the firm is operating in a non-competitive market, minor changes would have to be made to the diagrams. For example, the Marginal Revenue would have a negative gradient, due to the overall market demand curve. In a non-competitive environment, more complicated profit maximization solutions involve the use of game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

.

Maximizing revenue method


In some cases a firm's demand and cost conditions are such that marginal profits are greater than zero for all levels of production. In this case the Mπ = 0 rule has to be modified and the firm should maximize revenue. In other words the profit maximizing quantity and price can be determined by setting marginal revenue equal to zero. Marginal revenue equals zero when the Total revenue curve has reached its maximum value. An example would be a scheduled airline flight. The marginal costs of flying the route are negligible. The airline would maximize profits by filling all the seats. The airline would determine the conditions by maximizing revenues.

Changes in total costs and profit maximization


A firm maximizes profit by operating where marginal revenue equal marginal costs. A change in fixed costs has no effect on the profit maximizing output or price. The firm merely treats short term fixed costs as sunk costs and continues to operate as before. This can be confirmed graphically. Using the diagram illustrating the total cost total revenue method the firm maximizes profits at the point where the slope of the total cost line and total revenue line are equal. A change in total cost would cause the total cost curve to shift up by the amount of the change. There would be no effect on the total revenue curve or the shape of the total cost curve. Consequently, the profit maximizing point would remain the same. This point can also be illustrated using the diagram for the marginal revenue marginal cost method. A change in fixed cost would have no effect on the position or shape of these curves.

Markup pricing


In addition to using methods to determine a firm's optimal level of output, a firm can also set price to maximize profit. Profit maximization requires that a firm produce where marginal revenue equals marginal costs. Firm managers are unlikely to have complete information concerning their marginal revenue function or their marginal costs. Fortunately the profit maximization conditions can be expressed in a "more easily applicable" form or rule of thumb. The first step is to rewrite the expression for marginal revenue as MR = ∆TR/∆Q =(P∆Q+Q∆P)/∆Q=P+Q∆P/∆Q. The marginal revenue from an "incremental unit of quantity" has two parts: first, the revenue the firm gains from selling the additional units or P∆Q. The additional units are called the marginal units. Producing one extra unit and selling it a P brings in revenue of P. Second, "the revenue the firm loses on the units it could have sold at the higher price." These lost units are called the infra-marginal units. That is selling the extra unit results in a small drop in price which reduces the revenue for all units sold. Q(∆P/∆Q) Thus MR = P + Q(∆P/∆Q) = P +P (Q/P)((∆P/∆Q) = P + P(1/(PED) Then setting MR = MC MC = P + P(1/(PED) P - MC/P = - 1/PED P = MC/1 + (1/PED). The optimal markup rule is:
/P = 1/ -Ep

P = (Ep/(1 + Ep)) MC


Where MC equals marginal costs and Ep equals price elasticity of demand for the firm (not the market PED). Ep is a negative number. Therefore, -Ep is a positive number.

In English the rule is that the size of the markup is inversely related to the price elasticity of demand for a good.

The optimal markup rule also implies that a non-competitive firm will produce on the elastic region of its market demand curve. Marginal costs is positive. The term 1/ -Ep would be positive only if the PED is between -1 and -∝ that is if demand is elastic.

MPL, MRPL and profit maximization


The general rule is that firm maximizes profit by producing that quantity of output where marginal revenue equals marginal costs. The profit maximization issue can also be approached from the input side. That is, what is the profit maximizing usage of the variable input? To maximize profits the firm should increase usage "up to the point where the input's marginal revenue product equals its marginal costs". So mathematically the profit maximizing rule is MRPL = MCL
The marginal revenue product is the change in total revenue per unit change in the variable input assume labor. That is MRPL = ∆TR/∆L. MRPL is the product of marginal revenue and the marginal product of labor or MRPL = MR x MPL.

See also

  • Business organization
  • Corporation
    Corporation
    A corporation is created under the laws of a state as a separate legal entity that has privileges and liabilities that are distinct from those of its members. There are many different forms of corporations, most of which are used to conduct business. Early corporations were established by charter...

  • Market forms
  • Microeconomics
    Microeconomics
    Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

  • Pricing
    Pricing
    Pricing is the process of determining what a company will receive in exchange for its products. Pricing factors are manufacturing cost, market place, competition, market condition, and quality of product. Pricing is also a key variable in microeconomic price allocation theory. Pricing is a...

  • Production, costs, and pricing
    Production, costs, and pricing
    The following outline is provided as an overview of and topical guide to industrial organization:Industrial organization – describes the behavior of firms in the marketplace with regard to production, pricing, employment and other decisions...

  • Rational choice theory
    Rational choice theory
    Rational choice theory, also known as choice theory or rational action theory, is a framework for understanding and often formally modeling social and economic behavior. It is the main theoretical paradigm in the currently-dominant school of microeconomics...

  • Supply and demand
    Supply and demand
    Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...

  • Marginal revenue
    Marginal revenue
    In microeconomics, marginal revenue is the extra revenue that an additional unit of product will bring. It is the additional income from selling one more unit of a good; sometimes equal to price...

  • Total revenue
    Total revenue
    Total revenue is the total receipts of a firm from the sale of any given quantity of a product.It can be calculated as the selling price of the firm's product times the quantity sold, i.e...


External links

  • Profit Maximization in Perfect Competition by Fiona Maclachlan, Wolfram Demonstrations Project
    Wolfram Demonstrations Project
    The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...

    .