thumb|right|Linear marginal revenue (MR) and average revenue (AR) curves for a firm that is not in [[perfect competition]]
Marginal revenue (or marginal benefit) is a central concept in microeconomics that describes the additional total revenue generated by increasing product sales by 1 unit. Marginal revenue is the increase in revenue from the sale of one additional unit of product, i.e., the revenue from the sale of the last unit of product. It can be positive or negative. Marginal revenue is an important concept in vendor analysis. To derive the value of marginal revenue, it is required to examine the difference between the aggregate benefits a firm received from the quantity of a good and service produced last period and the current period with one extra unit increase in the rate of production. Marginal revenue is a fundamental tool for economic decision-making within a firm's setting, together with marginal cost to be considered.
In a perfectly competitive market, the incremental revenue generated by selling an additional unit of a good is equal to the price the firm is able to charge the buyer of the good. This is because a firm in a competitive market will always get the same price for every unit it sells regardless of the number of units the firm sells since the firm's sales can never impact the industry's price.
Definition
Marginal revenue is equal to the ratio of the change in revenue for some change in quantity sold to that change in quantity sold. This can be formulated as:
<math>MR = \frac{\Delta TR}{\Delta Q}</math>
This can also be represented as a derivative when the change in quantity sold becomes arbitrarily small. Define the revenue function to be
:<math>R(Q)=P(Q)\cdot Q ,</math>
where Q is output and P(Q) is the inverse demand function of customers. By the product rule, marginal revenue is then given by
:<math>R'(Q)=P(Q) + P'(Q)\cdot Q,</math>
where the prime sign indicates a derivative. For a firm facing perfect competition, price does not change with quantity sold so marginal revenue is equal to price. For a monopoly, the price decreases with quantity sold so marginal revenue is less than price for positive <math>Q</math> (see Example 1).
<math> R(Q)=(Q)\cdot (200 - Q) </math>
<math> R(Q)=200Q - Q^2 </math>
Then, by first order derivation, marginal revenue would be expressed as
<math> MR = R'(Q)=200- 2Q</math>
Therefore, if Q = 40,
MR = 200 − 2(40) = $120
Marginal revenue curve
thumb|Marginal revenue under perfect competitionthumb|Marginal revenue under monopolyThe marginal revenue curve is affected by the same factors as the demand curve – changes in income, changes in the prices of complements and substitutes, changes in populations, etc. These factors can cause the MR curve to shift and rotate. Marginal revenue curve differs under perfect competition and imperfect competition (monopoly).
Under perfect competition, there are multiple firms present in the market. Changes in the supply level of a single firm does not have an impact on the price in the market. Firms follow the price determined by market equilibrium of supply and demand and are price takers. The marginal revenue curve is a horizontal line at the market price, implying perfectly elastic demand and is equal to the demand curve.
Under monopoly, one firm is a sole seller in the market with a differentiated product. The marginal revenue for a monopolist is the private gain of selling an additional unit of output. The marginal revenue curve is downward sloping and below the demand curve and the additional gain from increasing the quantity sold is lower than the chosen market price. Under monopoly, the price of all units lowers each time a firm increases its output sold, this causes the firm to face a diminishing marginal revenue.
Marginal revenue curve and marginal cost curve
A company will stop producing a product/service when marginal revenue (money the company earns from each additional sale) equals marginal cost (the cost the company costs to produce an additional unit). Therefore, a company is making money when MR is greater than marginal cost (MC). And when MC = MR, it is called profit maximization. After this point; the company can no longer make a profit. Therefore, it is in their interest to stop production.
Relationship between marginal revenue and elasticity
The relationship between marginal revenue and the elasticity of demand by the firm's customers can be derived as follows:
:<math>R=P(Q)\cdot Q,</math>
:Taking the first order derivative of total revenue:
:<math>\left ( \frac{dR}{dQ} \right )= \left ( \frac{dQ}{dQ} \right )\cdot P + \left ( \frac{dP}{dQ} \right ) \cdot Q</math>
:<math>MR = dR/dQ = P + \frac{dP}{dQ} \cdot Q = P + \left(\frac{dP}{dQ} \frac{Q}{P}\right) \cdot P = P \cdot \left(1 + \frac{1}{e} \right),</math>
where R is total revenue, P(Q) is the inverse of the demand function, and e < 0 is the price elasticity of demand written as <math>e = \left(\frac{dQ}{dP}\frac{P}{Q}\right)</math>.
When a monopolist firm is facing an Inelastic demand curve (e<1), it implies that a percentage change in quantity is less than the percentage change in price. By increasing quantity sold, the firm is forced to accept a reduction of price for all the current and previous production units,
Increases in consumer's responsiveness to small changes in prices leads represents an elastic demand curve (e>1), resulting in a positive marginal revenue (MR) under monopoly competition. This signifies that a percentage change in quantity outweighs the percentage change in price. Firms in the imperfect competition market that lower prices by a small portion benefit from a large percentage increase in quantity sold and this generates greater marginal revenue. With that, a rational firm will recognize the value of price effects under an elastic demand function for its products and would avoid increasing prices as the quantity (demand) lost would be amplified due to the elastic demand curve.
Law of increasing marginal returns
In contrast to the law of diminishing marginal returns, in a knowledge-dependent economy, as knowledge and technological inputs increase, the output increases and the producer's returns tend to increase. This is an example of increasing marginal revenue; suppose a company produces toy airplanes. After some production, the company spends $10 in materials and labor to build the 1st toy airplane. The 1st toy airplane sells for $15, which means the profit on that toy is $5. Now, suppose that the 2nd toy airplane also costs $10, but this time it can be sold for $17. The profit on the 2nd toy airplane is $12 greater than the profit on the 1st toy airplane.
Marginal revenue and markup pricing
Profit maximization requires that a firm produces where marginal revenue equals marginal costs. Firm managers are unlikely to have complete information concerning their marginal revenue function or their marginal costs. However, the profit maximization conditions can be expressed in a “more easily applicable form”:
:MR = MC,
:MR = P(1 + 1/e),
:MC = P(1 + 1/e),
:MC = P + P/e,
:(P − MC)/ P = −1/e.
Markup is the difference between price and marginal cost. The formula states that markup as a percentage of price equals the negative (and hence the absolute value) of the inverse of the elasticity of demand. The Lerner index is a measure of market power — the ability of a firm to charge a price that exceeds marginal cost. The index varies from zero (when demand is infinitely elastic (a perfectly competitive market) to 1 (when demand has an elasticity of −1). The closer the index value is to 1, the greater is the difference between price and marginal cost. The Lerner index increases as demand becomes less elastic.