Editing Profit maximization (section)

==Marginal revenue – marginal cost perspective==
[[Image:profit max marginal small.png|right|thumb|250px|Profit maximization using the marginal revenue and marginal cost curves of a perfect competitor]]
[[File:Monopoly-surpluses.svg|thumb|upright=1.15|right|Price setting by a monopolist]]

An equivalent perspective relies on the relationship that, for each unit sold, [[marginal profit]] (<math>\text{M}\pi</math>) equals marginal revenue (<math>\text{MR}</math>) minus marginal cost (<math>\text{MC}</math>). Then, if marginal revenue is greater than marginal cost at some level of output, marginal profit is positive and thus a greater quantity should be produced, and if marginal revenue is less than marginal cost, marginal profit is negative and a lesser quantity should be produced. At the output level at which marginal revenue equals marginal cost, marginal profit is zero and this quantity is the one that maximizes profit.<ref name="lipsey"/> 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—where marginal cost equals marginal revenue—and where lower or higher output levels give lower profit levels.<ref name="lipsey">Lipsey (1975). pp. 245–47.</ref> In calculus terms, the requirement that the optimal output have higher profit than adjacent output levels is that:<ref name="lipsey"/>

:<math>\frac{\operatorname d^2 R}{{\operatorname dQ}^2} < \frac{\operatorname d^2 C}{{\operatorname dQ}^2}.</math>

The intersection of <math>\text{MR}</math> and <math>\text{MC}</math> is shown in the next diagram as point <math>\text{A}</math>. If the industry is perfectly competitive (as is assumed in the diagram), the firm faces a demand curve (<math>\text{D}</math>) that is identical to its marginal revenue curve (<math>\text{MR}</math>), and this is a horizontal line at a price determined by industry supply and demand. [[Average total cost]]s are represented by curve <math>\text{ATC}</math>. Total [[economic profit]] is represented by the area of the rectangle <math>\overline{\text{PABC}}</math>. The optimum quantity (<math>Q</math>) is the same as the optimum quantity in the first diagram.

If the firm is a [[monopolist]], the marginal revenue curve would have a negative slope as shown in the next graph, because it would be based on the downward-sloping market demand curve. The optimal output, shown in the graph as <math>Q_m</math>, is the level of output at which marginal cost equals marginal revenue. The price that induces that quantity of output is the height of the demand curve at that quantity (denoted <math>P_m</math>).

A generic derivation of the profit maximisation level of output is given by the following steps. Firstly, suppose a representative firm <math>i</math> has perfect information about its profit, given by:

:<math>\pi_i=\text{TR}_i-\text{TC}_i</math>

where <math>\text{TR}</math> denotes total revenue and <math>\text{TC}</math> denotes total costs. The above expression can be re-written as:

:<math>\pi_i=p_i\cdot q_i-c_i\cdot q_i</math>

where <math>p</math> denotes price (marginal revenue), <math>q</math> quantity, and <math>c</math> marginal cost. The firm maximises their profit with respect to quantity to yield the profit maximisation level of output:

:<math>\frac{\left(\partial\pi_i\right)}{\left(\partial q_i\right)}=p_i-c_i=0</math>

As such, the profit maximisation level of output is marginal revenue <math>p_i</math> equating to marginal cost <math>c_i</math>.

In an environment that is competitive but not perfectly so, more complicated profit maximization solutions involve the use of [[game theory]].

==Case in which maximizing revenue is equivalent==

In some cases a firm's demand and cost conditions are such that marginal profits are greater than zero for all levels of production up to a certain maximum.<ref name="samuelson">Samuelson, W and Marks, S (2003). p. 47.</ref> In this case marginal profit plunges to zero immediately after that maximum is reached; hence the <math>\text{M}\pi = 0</math> rule implies that output should be produced at the maximum level, which also happens to be the level that maximizes revenue.<ref name="samuelson" /> In other words, the profit-maximizing quantity and price can be determined by setting marginal revenue equal to zero, which occurs at the maximal level of output. 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 one more passenger on the flight are negligible until all the seats are filled. The airline would maximize profit by filling all the seats.

==Maximizing profits in the real world==
{{unreferenced section|date=November 2022}}
In the real world, it is not easy to achieve profit maximization. The company must accurately know the marginal income and the [[marginal cost]] of the last commodity sold because of [[marginal revenue|MR]].

The [[price elasticity]] of demand for goods depends on the response of other companies. When it is the only company raising prices, demand will be elastic. If one family raises prices and others follow, demand may be inelastic.

Companies can seek to maximize profits through estimation. When the price increase leads to a small decline in demand, the company can increase the price as much as possible before the demand becomes elastic. Generally, it is difficult to change the impact of the price according to the demand, because the demand may occur due to many other factors besides the price. 

The company may also have other goals and considerations. For example, companies may choose to earn less than the maximum profit in pursuit of higher [[market share]]. Because price increases maximize profits in the short term, they will attract more companies to enter the market.

Many companies try to minimize costs by shifting production to foreign locations with [[cheap labor]] (e.g. [[Nike, Inc.]]). However, moving the production line to a foreign location may cause unnecessary transportation costs. Close market locations for producing and selling products can improve demand optimization, but when the production cost is much higher, it is not a good choice.

===Tools===
;[[Profit analysis]]: Habitually recording and analyzing the business costs of all products/services sold. There are many miscellaneous items in the cost including labor, materials, transportation, advertising, storage, etc. related to any goods or services sold, which become expenses.
;[[Business intelligence]] tools: may be needed to integrate all financial information to record expense reports so that the business can clearly understand all costs related to operations and their accuracy.
;Planning and actual execution: when implementing a "what if" solution to help in sales and operation planning process, familiarity with the company's operations, including the supply chain, inventory management and sales process is useful. Constraints are required to prevent corporate plans from becoming unfeasible.

==Changes in total costs and profit maximization==
A firm maximizes profit by operating where marginal revenue equals marginal cost. This is stipulated under neoclassical theory, in which a firm maximizes profit in order to determine a level of output and inputs, which provides the price equals marginal cost condition.<ref name="Desai, M 2017">Desai, M (2017).</ref>{{Full citation needed|date=November 2022}} In the short run, a change in fixed costs has no effect on the profit maximizing output or price.<ref>Samuelson, W and Marks, S (2003). p. 52.</ref> The firm merely treats short term fixed costs as sunk costs and continues to operate as before.<ref>Landsburg, S (2002).</ref> This can be confirmed graphically. Using the diagram illustrating the total cost–total revenue perspective, the firm maximizes profit at the point where the slopes of the total cost line and total revenue line are equal.<ref name="samuelson"/> An increase in fixed cost would cause the total cost curve to shift up rigidly by the amount of the change.<ref name="samuelson"/> There would be no effect on the total revenue curve or the shape of the total cost curve. Consequently, the profit maximizing output would remain the same. This point can also be illustrated using the diagram for the marginal revenue–marginal cost perspective. A change in fixed cost would have no effect on the position or shape of these curves.<ref name="samuelson"/> In simple terms, although profit is related to total cost, <math>\text{Profit} = \text{TR}-\text{TC}</math>, the enterprise can maximize profit by producing to the maximum profit (the maximum value of <math>\text{TR}-\text{TC}</math>) to maximize profit. But when the total cost increases, it does not mean maximizing profit Will change, because the increase in total cost does not necessarily change the marginal cost. If the marginal cost remains the same, the enterprise can still produce to the unit of (<math>\text{MR}=\text{MC}=\text{Price}</math>) to maximize profit. In the long run, a firm will theoretically have zero expected profits under the competitive equilibrium. The market should adjust to clear any profits if there is perfect competition. In situations where there are non-zero profits, we should expect to see either some form of long run disequilibrium or non-competitive conditions, such as barriers to entry, where there is not perfect competition between firms.<ref name="Desai, M 2017"/>{{Full citation needed|date=November 2022}}

==Markup pricing==

In addition to using methods to determine a firm's optimal level of output, a firm that is not perfectly competitive can equivalently set price to maximize profit (since setting price along a given demand curve involves picking a preferred point on that curve, which is equivalent to picking a preferred quantity to produce and sell). The profit maximization conditions can be expressed in a "more easily applicable" form or rule of thumb than the above perspectives use.<ref name="pindyck">Pindyck, R and Rubinfeld, D (2001) p. 333.</ref>{{Full citation needed|date=November 2022}} The first step is to rewrite the expression for marginal revenue as

<math>
   \begin{align}
   \text{MR} =  & \frac{\Delta \text{TR}}{\Delta Q} \\ 
  = & \frac{P\Delta Q+Q \Delta P}{\Delta Q} \\ 
   = & P+\frac{Q \Delta P}{\Delta Q} \\
\end{align}
</math>

, where <math>P</math> and <math>Q</math> refer to the midpoints between the old and new values of price and quantity respectively.<ref name="pindyck"/> The marginal revenue from an incremental unit of output has two parts: first, the revenue the firm gains from selling the additional units or, giving the term <math>P\Delta Q</math>. The additional units are called the marginal units.<ref name="besanko">Besanko, D. and Beautigam, R, (2001) p. 408.</ref>{{Full citation needed|date=November 2022}} Producing one extra unit and selling it at price <math>P</math> brings in revenue of <math>P</math>. Moreover, one must consider "the revenue the firm loses on the units it could have sold at the higher price"<ref name="besanko"/>—that is, if the price of all units had not been pulled down by the effort to sell more units. These units that have lost revenue are called the infra-marginal units.<ref name="besanko"/> That is, selling the extra unit results in a small drop in price which reduces the revenue for all units sold by the amount <math>Q \cdot \left( \frac{\Delta P}{\Delta Q}\right)</math>. Thus, <math>\text{MR} = P + Q \cdot \frac{\Delta P}{\Delta Q} = P + P \cdot \frac{Q}{P} \cdot \frac{\Delta P}{\Delta Q} = P + \frac{P}{\text{PED}}</math>, where <math>\text{PED}</math> is the [[price elasticity of demand]] characterizing the demand curve of the firms' customers, which is negative. Then setting <math>\text{MC} = \text{MR}</math> gives <math>\text{MC} = P + \frac{P}{\text{PED}}</math> so <math>\frac{P - \text{MC}}{P} = \frac{-1}{\text{PED}}</math> and <math>P = \frac{MC}{1 + \left(\frac{1}{\text{PED}}\right)}</math>. Thus, the optimal markup rule is:

:<math>\frac{\left( P - \text{MC} \right)}{P} = \frac{1}{\left( -\text{PED}\right)}</math>

:or equivalently

:<math>P = \frac{\text{PED}}{1 + \text{PED}} \cdot \text{MC}</math>.<ref name="samuelson103">Samuelson, W and Marks, S (2003). p. 103–05.</ref><ref>Pindyck, R and Rubinfeld, D (2001) p. 341.</ref>{{Full citation needed|date=November 2022}}

In other words, the rule is that the size of the markup of price over the marginal cost is inversely related to the absolute value of the price elasticity of demand for the good.<ref name="samuelson103"/>

The optimal markup rule also implies that a non-competitive firm will produce on the elastic region of its market demand curve. Marginal cost is positive. The term <math>\frac{PED}{1+\text{PED}}</math> would be positive so <math>P>0</math> only if <math>\text{PED}</math> is between <math>-1</math> and <math>-\infty</math> (that is, if demand is elastic at that level of output).<ref>Besanko and Braeutigam (2005) p. 419.</ref>{{Full citation needed|date=November 2022}} The intuition behind this result is that, if demand is inelastic at some value <math>Q_1</math> then a decrease in <math>Q</math> would increase <math>P</math> more than proportionately, thereby increasing revenue <math>P \cdot Q</math>; since lower <math>Q</math> would also lead to lower total cost, profit would go up due to the combination of increased revenue and decreased cost. Thus, <math>Q_1</math> does not give the highest possible profit.