# Monopoly price

A monopoly price is set by a monopoly.[1][2] A monopoly occurs when a firm lacks any viable competition and is the sole producer of the industry's product.[1][2] Because a monopoly faces no competition, it has absolute market power and can set a price above the firm's marginal cost.[1][2] Since marginal cost is the increment in total cost required to produce an additional unit of the product, the firm can make a positive economic profit if it produces a greater quantity of the product and sells it at a lower price.[1]

The monopoly ensures a monopoly price exists when it establishes the quantity of the product.[1] As the sole supplier of the product within the market, its sales establish the entire industry's supply within the market, and the monopoly's production and sales decisions can establish a single price for the industry without any influence from competing firms.[1][2][3] The monopoly always considers the demand for its product as it considers what price is appropriate, such that it chooses a production supply and price combination that ensures a maximum economic profit,[1][2] which is determined by ensuring that the marginal cost (determined by the firm's technical limitations that form its cost structure) is the same as the marginal revenue (MR) (as determined by the impact a change in the price of the product will impact the quantity demanded) at the quantity it decides to sell.[1][2] The marginal revenue is solely determined by the demand for the product within the industry and is the change in revenue that will occur by lowering the price just enough to ensure a single additional unit is sold.[1][2] The marginal revenue is positive, but it is lower than its associated price because lowering the price will increase the demand for its product and increase the firm's sales revenue, and lower the price paid by those who are willing to buy the product at the higher price, which ensures a lower sales revenue on the product sales than those willing to pay the higher price.[1]

Marginal revenue can be calculated as ${\displaystyle MR=P+P'(Q)*Q}$, where ${\displaystyle 0>P'(Q)}$.[2][clarification needed]

Reduction in price increases the quantity demanded, but reduces payments by those who would be willing to pay a higher price: MR < P

Marginal cost (MC) relates to the firm's technical cost structure within production, and indicates the rise in total cost that must occur for an additional unit to be supplied to the market by the firm.[1] The marginal cost is higher than the average cost because of diminishing marginal product in the short run.[1] It can be calculated as ${\displaystyle MC=C'(Q)}$, where ${\displaystyle 0.[2][4][clarification needed]

Diminishing marginal product ensures the rise in cost from producing an additional item (marginal cost) is always greater than the average variable (controllable) cost at that level of production. Since some costs cannot be controlled in the short run, the variable (controllable) costs will always be lower than the total costs in the short run.

Samuelson[who?] indicates this point on the consumer demand curve is where the price is equal to one over one plus the reciprocal of the price elasticity of demand.[5] This rule does not apply to competitive firms, as they are price takers and do not have the market power to control either prices or industry-wide sales.[1]

Although the term markup is sometimes used in economics to refer to the difference between a monopoly price and the monopoly's MC,[6] it is frequently used in American accounting and finance to define the difference between the price of the product and its per unit accounting cost. Accepted neo-classical micro-economic theory indicates the American accounting and finance definition of markup, as it exists in most competitive markets, ensures an accounting profit that is just enough to solely compensate the equity owners of a competitive firm within a competitive market for the economic cost (opportunity cost) they must bear if they hold on to the firm's equity.[3] The economic cost of holding onto equity at its present value is the opportunity cost the investor must bear when giving up the interest earnings on debt of similar present value (they hold onto equity instead of the debt).[3] Economists would indicate that a markup rule on economic cost used by a monopoly to set a monopoly price that will maximize its profit is excessive markup that leads to inefficiencies within an economic system.[1][2][7][8]

## Mathematical derivation: how a monopoly sets the monopoly price

Mathematically, the general rule a monopoly uses to maximize monopoly profit can be derived through simple calculus. The basic equation for economic profit, in which the total economic cost varies directly with the quantity produced, can be expressed as

${\displaystyle \pi =P(Q)\times Q-C(Q)}$, where

• ${\displaystyle Q}$ = quantity sold,
• ${\displaystyle P(Q)}$ = inverse demand function; the price at which ${\displaystyle Q}$ can be sold given the existing demand
• ${\displaystyle C(Q)}$ = total cost of producing ${\displaystyle Q}$.
• ${\displaystyle \pi }$ = economic profit

This is done by equating the derivative of ${\displaystyle \pi }$ with respect to ${\displaystyle Q}$ to 0. The profit of a firm is given by total revenue (price times quantity sold) minus total cost:

${\displaystyle P'(Q)Q+P-C'(Q)=0}$, where

• ${\displaystyle Q}$ = quantity sold,
• ${\displaystyle P'(Q)}$ = the partial derivative of the inverse demand function, and the price at which ${\displaystyle Q}$ can be sold given the existing demand
• ${\displaystyle C'(Q)}$ = marginal cost, or the partial derivative of the total cost of producing ${\displaystyle Q}$

which yields

${\displaystyle P'(Q)*Q+P=C'(Q)}$
where marginal revenue equals marginal cost. This is usually called the first order conditions for a profit maximum.[2]

According to Samuelson,

${\displaystyle P(P'(Q/P)+1)=MC}$

By definition, ${\displaystyle P'(Q/P)}$ is the reciprocal of the price elasticity of demand (or ${\displaystyle 1/\epsilon }$), or

${\displaystyle P(1+1/{\epsilon })=MC}$

This gives the markup rule:

${\displaystyle P={\frac {\epsilon }{\epsilon +1}}MC}$

or, letting ${\displaystyle \eta }$ be the reciprocal of the price elasticity of demand,

${\displaystyle P={\frac {1}{1+\eta }}MC}$

Thus the monopolistic firm chooses the quantity at which the demand price satisfies this rule. Since ${\displaystyle \eta <0}$ for a price setting firm, it means that a firm with market power will charge a price above marginal cost and earn a monopoly rent. On the other hand, a competitive firm by definition faces a perfectly elastic demand, ${\displaystyle \eta =0}$, which means that it sets price equal to marginal cost.

The rule also implies that, absent menu costs, a monopolistic firm will never choose a point on the inelastic portion of its demand curve. For an equilibrium to exist in a monopoly or in an oligopoly market, the price elasticity of demand must be less than negative one (${\displaystyle {\frac {1}{\eta }}<-1}$), for marginal revenue to be positive.[4] The mathematical profit maximization conditions ("first order conditions") ensure the price elasticity of demand must be less than negative one,[2][7] since no rational firm that attempts to maximize its profit would incur additional cost (a positive marginal cost) in order to reduce revenue (when MR < 0).[1]

## References

1. Roger LeRoy Miller, Intermediate Microeconomics Theory Issues Applications, Third Edition, New York: McGraw-Hill, Inc, 1982.
2. Tirole, Jean, "The Theory of Industrial Organization", Cambridge, Massachusetts: The MIT Press, 1988.
3. ^ a b c John Black, "Oxford Dictionary of Economics", New York: Oxford University Press, 2003.
4. ^ a b Henderson, James M., and Richard E. Quandt, "Micro Economic Theory, A Mathematical Approach. 3rd Edition", New York: McGraw-Hill Book Company, 1980. Glenview, Illinois: Scott, Foresmand and Company, 1988. Usually, in many textbooks, economic cost, here presented by ${\displaystyle C(Q)}$, is divided into two categories; labor costs and capital costs: ${\displaystyle C(Q)=Lw+KR}$, where
• ${\displaystyle L}$ = labor hired,
• ${\displaystyle w}$ = wage rate,
• ${\displaystyle K}$ = total amount of capital financed by both debt and equity,
• ${\displaystyle R}$ = cost of capital, including both interest expense and the minimum required rate of return on equity
• ${\displaystyle Q}$ = ${\displaystyle Q(L,K)}$ = a function of the quantity of labor and capital employed in production
5. ^ Samuelson; Marks (2003). p.104
6. ^ Nicholson, Walter and Christopher Snyder, Microeconomic Theory: Basic Principles and Extensions, Mason, OH: Thomson/South-Western, 2008.
7. ^ a b Henderson, James M., and Richard E. Quandt, "Micro Economic Theory, A Mathematical Approach. 3rd Edition", New York: McGraw-Hill Book Company, 1980. Glenview, Illinois: Scott, Foresmand and Company, 1988.
8. ^ Bradley R. chiller, "Essentials of Economics", New York: McGraw-Hill, Inc., 1991.