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]

A monopolist will set a price and production quantity where MC=MR, such that MR is always below the monopoly price set. A competitive firm's MR is the price it gets for its product, and will have Price=MC.

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:

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

Note that the price elasticity of demand (and its reciprocal) is negative, ${\displaystyle {\epsilon }<0}$, so a more intuitive formula, using the absolute value of the elasticity, is

${\displaystyle P={\frac {|\epsilon |}{|\epsilon |-1}}MC}$.

This shows clearly that the profit-maximizing price is set at a point where "demand is elastic", namely, the price elasticity of demand must be greater than unity in absolute terms, in order for the profit-maximizing price to be positive.

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]

In the case of price elasticity of demand, it also called Lerner index. The formula can be expressed: ${\displaystyle L={\frac {P-MC}{P}}}$, ${\displaystyle P}$ means monopoly price set by firms ${\displaystyle MC}$ means the marginal cost of production

The Lerner index measures the level of market power and monopoly power that a firm owned.The higher Lerner index indicated the more monopoly power allows a company have chance to establish prices that are higher than their marginal costs and then lead a higher monopoly price. In conclusion, a monopoly price is established by a monopolistic firm while they have no rivals in the market and feasible to raise price further above their marginal cost. In order to ensure a maximum economic return, the monopoly price is established at the point where marginal revenue equals marginal cost based on the firm's evaluation of the demand for its product. The Lerner index can be used to measured the degree of monopoly power and monopoly price.

In addition, monopoly price will prevent new business from entering the market and restrict innovation. A monopoly would not like to invest more on research and development or innovation due to it already has a captive market. Then the lack of innovation may block market competition and limit the industry’s growth potential in long run. The monopoly’s entrance restrictions also make it difficult for new businesses to enter the market, which reduces the scope for innovation and new ideas.

In sum up, monopoly pricing generally has negative consequences on consumers and the overall economy, resulting in higher costs, lower quantity desired, inefficiencies and a lack of innovation.

A market structure is defined by three factors which are barriers to entry, number of firms in the market, and product substitutability.

Below is the market structure for a monopoly:

Unlike perfect competition where firms can freely enter and exit the market, it is not the case for monopolistic competition. For a monopoly to exist, there must be high barriers to entry for new firms. Barriers to entry must be strong enough to discourage potential competitors from entering. However, if the number of firms in the market for a specific good or service increases, the perceived value of firms in the market will decrease. Therefore, the likelihood for firms to exit the market is higher, leaving one firm to monopolise the market.

The difference between the products or services of a perfect competition and one in a monopoly is if the products or services are differentiated. Thereby, the products or services sold in the monopolistic market are not perfect substitutes for one another.[12]

Market power is the firm's ability to affect terms and conditions of exchange. [13] A monopoly possesses a substantial amount of market power, however, it is not unlimited. A monopoly is a price maker, not a price taker, meaning that a monopoly has the power to set the market price. [14] The firm in monopoly is the market as it sets its price based on their circumstances of what best suits them.

To easily identify a monopoly, it would have one or more of these five characteristics