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Marginal revenue

Marginal revenue (MR) is the additional revenue a firm receives from selling one more unit of output, calculated as the change in total revenue divided by the change in quantity sold, or in continuous terms, the derivative of the total revenue function with respect to quantity.^[1]^[2] In economics, it serves as a critical concept for understanding firm behavior in various market structures, guiding decisions on output levels to maximize profits. Under perfect competition, where firms are price takers, marginal revenue equals the market price because selling an additional unit does not affect the price of the product.^[3] In contrast, in monopoly or other imperfectly competitive markets, the downward-sloping demand curve means that selling more units requires lowering the price on all units sold, resulting in marginal revenue being less than the price.^[3] This distinction highlights how market power influences revenue dynamics and pricing strategies. Firms across market structures use marginal revenue in profit maximization by producing up to the point where marginal revenue equals marginal cost (MR = MC), as this equates the additional revenue and cost of the last unit produced.^[4] Beyond production decisions, marginal revenue analysis extends to resource allocation, such as in labor markets where marginal revenue product determines optimal input use.^[5]

Fundamentals

Definition

Marginal revenue (MR) is defined as the additional revenue generated by selling one more unit of a good or service. It is formally expressed as the change in total revenue (TR) divided by the change in quantity sold (ΔQ), or MR = ΔTR / ΔQ.^[1] Total revenue represents the overall income from sales and is calculated as the product of the price per unit (P) and the quantity sold (Q), so TR = P × Q.^[5] Marginal revenue captures the incremental effect of increasing output by one unit, reflecting how additional sales impact overall earnings, often accounting for potential price adjustments needed to sell more units.^[6] In contrast, average revenue (AR) measures the revenue per unit sold and is computed as AR = TR / Q, which typically equals the price P in standard market conditions. This distinction highlights that while average revenue provides an overall average, marginal revenue focuses on the specific contribution of the next unit to total revenue.^[7] The concept of marginal revenue emerged in neoclassical economics, particularly through the work of Alfred Marshall in his Principles of Economics (1890), where it formed a key part of analyzing firm behavior and resource allocation.^[8]

Calculation

Marginal revenue can be calculated in discrete or continuous terms, depending on whether output changes are treated as finite increments or infinitesimal variations. In the discrete case, marginal revenue for the nth unit, denoted MR_n, is computed as the change in total revenue divided by the change in quantity: MR_n = (TR_n - TR_{n-1}) / (Q_n - Q_{n-1}), where TR represents total revenue and Q is quantity sold. This approach approximates the additional revenue from selling one more unit, assuming quantity changes by one unit.^[9] For continuous analysis, marginal revenue is the derivative of total revenue with respect to quantity: MR = dTR / dQ. When total revenue is expressed as TR = P(Q) \cdot Q, where P is the price as a function of quantity, the chain rule yields MR = P + Q (dP / dQ).^[10] This formula highlights that under a downward-sloping demand curve (where dP / dQ < 0), marginal revenue is less than the price, as the second term is negative.^[10] The calculation assumes either constant price per unit, in which case MR equals P for all units since additional sales do not affect price, or a downward-sloping demand curve, where selling more units requires lowering the price on all units, reducing MR below P.^[11] To illustrate, consider a linear demand curve P = 100 - Q. Total revenue is TR = Q(100 - Q) = 100Q - Q^2. The marginal revenue function is then MR = 100 - 2Q.^[10] The following table computes discrete values for the first few units:

Q	P	TR	MR
0	100	0	-
1	99	99	99
2	98	196	97
3	97	291	95
4	96	384	93
5	95	475	91

Here, MR decreases by 2 for each additional unit, matching the continuous formula at integer points.^[12] Edge cases include the initial output level where Q=0, at which MR equals the price P (as no prior revenue exists), and the point where total revenue is maximized, where MR=0 since further increases in quantity yield no additional revenue.^[11] This maximization occurs where the slope of the total revenue curve flattens.^[10]

Graphical and Analytical Representation

Marginal Revenue Curve

The marginal revenue (MR) curve represents the additional revenue gained from selling one more unit of output, derived from the total revenue function. For a linear demand curve expressed as P = a - bQ, where P is price, Q is quantity, a is the vertical intercept, and b > 0 is the slope, total revenue is TR = PQ = aQ - bQ^2. Differentiating with respect to Q yields the MR curve: MR = \frac{dTR}{dQ} = a - 2bQ.^[10]^[13] Thus, the MR curve has the same vertical intercept as the demand curve but twice the negative slope, making it steeper. Graphically, the MR curve for a linear demand is downward-sloping and lies below the demand curve for all Q > 0, reflecting that to sell additional units, price must fall on all prior units, reducing revenue from those sales.^[14] It intersects the horizontal axis at Q = \frac{a}{2b}, the quantity where total revenue is maximized, as MR = 0 at that point.^[15] For non-linear demand curves, the MR curve can take varied shapes, including discontinuities—such as in cases of kinked demand where the slope changes abruptly, leading to a vertical gap in MR—or become negative in regions where demand is inelastic.^[16] A key property is that MR equals price (P) when demand is perfectly elastic (horizontal), but MR is less than P otherwise, due to the revenue loss from lowering price on inframarginal units.^[17] The MR curve's intersection with the marginal cost curve determines the profit-maximizing output.^[18]

Relationship with Demand Curve

In microeconomics, the marginal revenue (MR) curve is derived from the firm's demand curve and lies below it for downward-sloping demand, reflecting the revenue implications of increasing output.^[19] Geometrically, when a firm sells an additional unit, it gains revenue equal to the price of that unit (a rectangular area under the demand curve at the current quantity), but it must lower the price on all previously sold units, incurring a revenue loss (another rectangular area representing the price reduction times the prior quantity).^[19] For linear demand curves, this net effect positions the MR curve below the demand curve, with its slope twice as steep, as the revenue gain is a rectangle while the offsetting loss expands with quantity.^[10] Analytically, total revenue is given by TR = P(Q) \times Q, where P(Q) is the inverse demand function. Differentiating yields the marginal revenue:

MR = \frac{dTR}{dQ} = P(Q) + Q \frac{dP}{dQ}.

Since the demand curve slopes downward, \frac{dP}{dQ} < 0, making the second term negative and thus MR < P for Q > 0.^[10] This divergence arises because the additional revenue from the extra unit is partially offset by the revenue forfeited from lower prices on inframarginal units.^[19] A special case occurs with perfectly elastic demand, as in perfect competition, where the demand curve is horizontal at the market price. Here, increasing output does not require a price reduction, so \frac{dP}{dQ} = 0, and the MR curve coincides exactly with the demand curve.^[19] Illustrations of this relationship often depict the areas under the curves: the revenue gain as a thin rectangle along the demand curve at the margin, subtracted by a larger rectangle of loss to the left, resulting in the MR curve tracing a path below the demand for non-constant elasticity scenarios.^[19]

Applications in Market Structures

In Perfect Competition

In a perfectly competitive market, individual firms are price-takers, meaning they have no influence over the market price and must accept the prevailing price P determined by industry supply and demand.^[20] This structure arises because there are numerous buyers and sellers, homogeneous products, and free entry and exit, ensuring no single firm can affect the price by changing its output.^[21] Consequently, the demand curve facing each firm is perfectly elastic, represented as a horizontal line at the market price P. Under these conditions, marginal revenue (MR) equals the price (P) and average revenue (AR) for all levels of output Q, so MR = P = AR. The marginal revenue curve thus coincides with the demand curve, forming a horizontal line at P, reflecting that each additional unit sold brings in exactly P in revenue without lowering the price for subsequent units.^[20] This equality implies that total revenue (TR) is a linear function of quantity, given by TR = P \times Q, as revenue increases proportionally with output without any discounting effect from price reductions.^[22] For instance, in agricultural markets like wheat production, a small farmer can sell any amount of output at the market price without impacting it, due to the vast number of producers and the commodity nature of the product.^[21] In contrast to monopolistic markets, where firms face downward-sloping demand and thus MR < P, perfect competition maintains constant marginal revenue aligned with price.

In Monopoly and Imperfect Competition

In a monopoly, a single firm is the sole seller in the market and thus faces the entire market demand curve, which is downward-sloping.^[23] To increase sales, the monopolist must lower the price, reducing revenue from all previous units sold, which causes marginal revenue to decline more rapidly than the demand curve (price).^[4] For a linear demand curve of the form P = a - bQ, where P is price, a is the vertical intercept, b is the slope, and Q is quantity, the corresponding marginal revenue function is MR = a - 2bQ.^[23] This structure implies that marginal revenue is always less than price for output levels above zero, as the price reduction applies to the entire quantity sold.^[24] A classic example is a natural monopoly in utilities, such as electricity distribution, where the firm derives its marginal revenue directly from the market demand curve due to barriers to entry and economies of scale.^[25] In imperfectly competitive markets like oligopolies and monopolistic competition, firms also face downward-sloping demand curves, leading to marginal revenue below price for similar reasons.^[4] However, in oligopoly models such as the kinked demand curve proposed by Sweezy, rivals are assumed to match price decreases but ignore increases, resulting in a kink in the demand curve and a corresponding discontinuity (vertical gap) in the marginal revenue curve. This discontinuity creates a range of marginal costs over which the profit-maximizing output remains stable, contributing to price rigidity observed in concentrated industries.^[26]

Key Relationships

With Marginal Cost

The interaction between marginal revenue (MR) and marginal cost (MC) forms the basis for short-run production decisions by firms across market structures. The profit-maximizing output level is determined at the point where the MR curve intersects the MC curve, as this equality ensures that the additional revenue from the last unit produced exactly matches the additional cost.^[27] At quantities where MR exceeds MC, firms should expand production because each extra unit adds more to total revenue than to total cost, increasing overall profit.^[28] Conversely, if MR is less than MC, firms should contract output to eliminate units that contribute negatively to profit.^[29] This MR = MC rule operates under short-run assumptions, where at least one factor of production is fixed, resulting in an upward-sloping MC curve as output increases.^[30] The upward slope of the MC curve arises from diminishing marginal returns to variable inputs, which eventually raise the cost of producing each additional unit.^[31] Graphically, the MR and MC curves are plotted on a graph with quantity on the horizontal axis and dollars per unit on the vertical axis; the intersection point identifies the optimal output quantity, denoted as Q*.^[32] In this diagram, total profit is represented as a rectangle formed by the vertical distance between the price (or MR) and the average total cost at Q*, multiplied by Q* itself, highlighting the area of economic surplus when the intersection lies above average costs.^[28] A key short-run decision tied to this relationship is the shutdown rule: firms should halt production if the market price falls below the minimum average variable cost (min AVC), as continuing operations would fail to cover variable expenses and exacerbate losses beyond fixed costs.^[33] In perfect competition, where MR equals price, this condition simplifies to P < min AVC, prompting temporary shutdown while fixed costs are still incurred.^[34] For illustration, consider a competitive firm with an MC curve intersecting a horizontal MR line at Q* = 200 units and a price of $15; if price drops below the MC-AVC minimum at $10, the firm shuts down, avoiding further variable cost outlays.^[35]

With Price Elasticity of Demand

The relationship between marginal revenue (MR) and the price elasticity of demand (\varepsilon) provides a key insight into how changes in quantity demanded affect revenue, where \varepsilon = \frac{dQ}{dP} \cdot \frac{P}{Q} < 0 measures the responsiveness of quantity to price changes.^[36] To derive the formula, begin with total revenue TR = P(Q) \cdot Q, where P is price as a function of quantity Q. Marginal revenue is the derivative:

MR = \frac{dTR}{dQ} = P + Q \cdot \frac{dP}{dQ}.

The price elasticity \varepsilon relates to the slope via \varepsilon = \frac{dQ}{dP} \cdot \frac{P}{Q}, so \frac{dP}{dQ} = \frac{P}{Q \varepsilon}. Substituting yields:

MR = P + Q \cdot \left( \frac{P}{Q \varepsilon} \right) = P \left(1 + \frac{1}{\varepsilon}\right).

This equation links MR directly to price and elasticity, with \varepsilon < 0 ensuring the term $1 + \frac{1}{\varepsilon} is less than 1.^[37] The sign and magnitude of MR depend on the absolute value of elasticity |\varepsilon|: MR > 0 when |\varepsilon| > 1 (elastic demand, where quantity responds strongly to price drops, increasing revenue); MR = 0 when |\varepsilon| = 1 (unit elastic demand, where total revenue is maximized); and MR < 0 when |\varepsilon| < 1 (inelastic demand, where quantity responds weakly, reducing revenue).^[36] This interpretation highlights that firms operating in elastic segments can expand output profitably, while inelastic segments may require caution to avoid revenue loss. In pricing decisions, the MR-\varepsilon relationship guides firms toward elastic demand segments for revenue growth, as positive MR allows sales increases without proportional revenue decline, whereas inelastic segments signal potential for higher markups but risk overproduction.^[37]^[36]

With Marginal Benefit

Marginal benefit (MB) refers to the additional satisfaction or utility that a consumer derives from consuming one more unit of a good or service, and it is typically represented by the demand curve, where the price a consumer is willing to pay reflects this incremental value.^[38] In consumer equilibrium, marginal benefit equals the price (MB = P), as individuals purchase units up to the point where the utility gained no longer exceeds the cost.^[14] This concept parallels marginal revenue (MR) on the producer side, particularly in competitive markets where MR also equals price (MR = P), allowing firms to sell additional units without affecting the market price.^[39] In such equilibria, producers set output where MR = marginal cost (MC), mirroring the consumer's MB = P condition and ensuring market clearing.^[38] However, in imperfect competition, MR falls below P, diverging from the direct equivalence seen in perfect competition. In welfare economics, efficient resource allocation occurs when marginal benefit equals marginal cost across the economy (MB = MC), maximizing total surplus by balancing consumer valuation with production costs.^[38] Marginal revenue influences this on the producer side by guiding output decisions toward the MC intersection, contributing to overall efficiency when markets are competitive, though distortions like monopoly can prevent MB = MC alignment.^[14] A key distinction lies in their nature: marginal revenue measures the objective, monetary addition to a firm's total revenue from selling one more unit, calculated as the change in revenue divided by the change in quantity.^[39] In contrast, marginal benefit captures the subjective value or utility to the buyer, which diminishes with additional consumption due to the law of diminishing marginal utility, without a direct monetary formula comparable to MR.^[38]

Markup Pricing

Markup pricing refers to the strategy where firms set prices as a markup over their marginal cost, a practice particularly prevalent in markets with imperfect competition such as monopolies. This approach is directly informed by the marginal revenue concept, as profit maximization occurs where marginal revenue equals marginal cost (MR = MC). In a monopoly, the relationship between marginal revenue and price is given by the formula MR = P \left(1 + \frac{1}{\epsilon_d}\right), where P is the price and \epsilon_d is the price elasticity of demand (a negative value). Setting MR equal to MC yields the optimal pricing rule: P = \frac{MC}{1 + \frac{1}{\epsilon_d}}, indicating that the markup over marginal cost decreases as demand becomes more elastic (more negative \epsilon_d).^[40] The Lerner Index provides a standardized measure of this markup and the degree of market power, defined as L = \frac{P - MC}{P} = -\frac{1}{\epsilon_d}. This index ranges from 0 in perfect competition, where price equals marginal cost and no markup is possible, to higher values in monopolies with inelastic demand, reflecting greater pricing discretion. For instance, if \epsilon_d = -2, then L = 0.5, implying the price is twice the marginal cost. The index thus quantifies how far a firm can deviate from competitive pricing based on consumers' sensitivity to price changes.^[41] In practice, firms in imperfectly competitive markets estimate the perceived elasticity of demand to determine the appropriate markup over marginal cost, often using market research or historical data to avoid over- or under-pricing. This elasticity-based markup allows firms to balance revenue gains from higher prices against potential volume losses. A prominent example is the pharmaceutical industry, where patented drugs face highly inelastic demand due to patients' urgent medical needs and limited substitutes, enabling substantial markups that can exceed 100% over production costs despite relatively low marginal costs per unit. Such pricing sustains innovation incentives but raises concerns about access and affordability.^[42]

Role in Profit Maximization

In microeconomics, a firm's profit \pi is defined as total revenue (TR) minus total cost (TC), expressed as \pi = TR - TC. To maximize profit, the firm adjusts its output quantity Q such that the first-order condition holds: the derivative \frac{d\pi}{dQ} = MR - MC = 0, where marginal revenue (MR) equals marginal cost (MC). This condition implies that the additional revenue from selling one more unit exactly offsets the additional cost of producing it, ensuring no further gains from expanding or contracting output.^[43] In the short run, where some inputs are fixed, a profit-maximizing firm produces the output level where MR = MC, provided this yields positive economic profit or at least covers variable costs to minimize losses. If MR > MC at all feasible outputs, the firm produces at capacity; if MR < MC everywhere, it shuts down temporarily. In the long run, with all inputs variable, firms enter the market if economic profits are positive (attracting new competitors until MR = MC aligns with zero economic profit) or exit if losses persist, driving the industry toward long-run equilibrium where average total cost equals price. Diminishing returns to variable inputs contribute to the upward-sloping MC curve in both periods, influencing these decisions.^[43]^[44] The second-order condition confirms that the MR = MC point is a maximum rather than a minimum or inflection: the slope of the MC curve must exceed the slope of the MR curve at the intersection, or \frac{dMC}{dQ} > \frac{dMR}{dQ}. This is typically satisfied when MC is upward-sloping, as is common due to increasing marginal costs at higher output levels, ensuring that deviations from the optimal quantity reduce profit. In perfect competition, where MR is horizontal (equal to price), the condition simplifies to MC having a positive slope at the equilibrium.^[29] Recent extensions in behavioral economics critique the standard model's assumption of perfect rationality in estimating MR for profit maximization, incorporating bounded rationality where firms use heuristics or misspecified beliefs about demand and costs. For instance, models show firms may assume fixed prices or constant unit costs, leading to suboptimal output choices that undetectably deviate from the rational MR = MC rule in equilibrium, with implications for pricing and efficiency. Post-2020 studies further highlight how limited information and computational constraints cause monopolists to approximate marginal profit via finite differences or gradient adjustments based on past data, resulting in dynamic instabilities like chaos or bifurcations rather than static optima. These critiques suggest bounded rationality can bias MR estimates downward in uncertain environments, reducing realized profits compared to rational benchmarks.^[45]^[46]

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