# Monopoly Pricing

A monopolist should set its price such that the difference between the price and marginal cost as a percentage of price equals the inverse of the elasticity of demand of its product.

The profit-maximizing output and price of a monopolist occur at output level at which its marginal revenue is equal to its marginal cost. Marginal revenue is the incremental revenue from each additional unit of sales and marginal cost is the incremental cost of the additional unit.

Since the demand curve in case of a monopoly slopes downward (unlike perfect competition in which it is a horizontal line), increase in sales is possible only when the monopolist reduces its price. But reduction in price reduces the revenue from all units. The total reduction equals quantity (Q) multiplied by per-unit change in price (∆P/∆Q).

It follow that the marginal revenue of a monopolist is given by the following equation:

$$\text{MR}\ =\ \text{P}\ +\ \text{Q}\times\frac{\Delta \text{P}}{\Delta \text{Q}}$$

MR is marginal revenue, P is price, Q is quantity and ∆P/∆Q is the reduction in price necessary to increase quantity by ∆Q.

## Monopoly Price and Elasticity of Demand

Let’s multiply and divide the right-hand side of the above expression by P:

$$\text{MR}\ =\ \text{P}\ +\text{P}\times\ \frac{\text{Q}}{\text{P}}\times\frac{\Delta \text{P}}{\Delta \text{Q}}$$

(Price) elasticity of demand is defined as the responsiveness of quantity demanded to change in price. It equals percentage change in quantity divided by percentage change in price. It can be calculated using the following equation:

$$\text{E} _ \text{d}=\frac{\text{P}}{\text{Q}}\times\frac{\Delta \text{Q}}{\Delta \text{P}}$$

Taking inverse of the above equation, we get

$$\frac{\text{1}}{\text{E} _ \text{d}}=\frac{\text{Q}}{\text{P}}\times\frac{\Delta \text{P}}{\Delta \text{Q}}$$

Substituting the value of Q/P×∆P/∆Q in the expression for marginal revenue, we get

$$\text{MR}\ =\ \text{P}\ +\text{P}\times\ \frac{\text{1}}{\text{E} _ \text{d}}$$

It means that marginal revenue of a monopolist equals price P plus the price divided by elasticity of demand. Since elasticity of demand is negative in most cases, the second expression on the right-hand side is negative which means that marginal revenue is less than price P. Secondly, when elasticity of demand is low, the second expression has high absolute value, and the marginal revenue decreases more steeply and vice versa.

## Monopoly Profit-Maximizing Price

Since monopoly profit-maximization occurs when MR = MC, we can write the following expression:

$$\text{MR}\ =\ \text{P}\ +\text{P}\times\ \frac{\text{1}}{\text{E} _ \text{d}}=\text{MC}$$

Just a bit of mathematical rearrangement:

$$\text{P}-\text{MC}=\text{P}\times\ -\frac{\text{1}}{\text{E} _ \text{d}}$$

$$\frac{\text{P}-\text{MC}}{\text{P}}=-\frac{\text{1}}{\text{E} _ \text{d}}$$

But (P – MC)/P is the markup over marginal cost as a percentage of price. It is also called Lerner index. The equation above has important implications for monopoly pricing. It says that a monopolist facing low elasticity of demand can charge a higher mark-up i.e. set a higher price and vice versa.

We can write the above expression such that P is the independent variable:

$$\text{P}\ +\text{P}\times\ \frac{\text{1}}{\text{E} _ \text{d}}=\text{MC}$$

$$\text{P}\ \times\left(\text{1}\ +\ \frac{\text{1}}{\text{E} _ \text{d}}\right)=\text{MC}$$

$$\text{P}\ =\frac{\text{MC}}{\left(\text{1}\ +\ \frac{\text{1}}{\text{E} _ \text{d}}\right)}$$

The above formula can be used directly to determine a monopoly’s profit-maximizing price. Since marginal cost and elasticity of demand changes as we move along a demand curve, the above equation can’t be used directly to find the profit-maximizing output.