Marginal Revenue Curve From Demand Curve

Themarginal revenue curve from demand curve is a fundamental concept in microeconomics that explains how a firm’s additional revenue from selling one more unit interacts with the shape and slope of its demand curve. When a company faces a downward‑sloping demand curve—typical of monopolistic competition, oligopoly, or pure monopoly—the price it can charge must be reduced to sell additional units. This price‑adjustment mechanism creates a marginal revenue (MR) curve that lies below the demand curve, and understanding its derivation is essential for profit‑maximizing decisions. In this article we will explore the theoretical foundations, step‑by‑step derivation, graphical interpretation, and practical implications of the marginal revenue curve derived from a given demand curve.

Introduction to Demand and Revenue Concepts

A demand curve represents the relationship between the price (P) that consumers are willing to pay and the quantity (Q) they are prepared to purchase, holding other factors constant. For a monopolist, the demand curve is often expressed as an inverse function:

[P(Q)=a-bQ ]

where a and b are constants. Total revenue (TR) is the product of price and quantity:

[ TR(Q)=P(Q)\times Q = (a-bQ)Q = aQ-bQ^{2} ]

Marginal revenue is the additional revenue earned from selling one more unit of output. Mathematically, MR is the first derivative of the total revenue function with respect to quantity:

[ MR(Q)=\frac{dTR}{dQ}=a-2bQ ]

Notice that MR is a linear function with the same intercept as the demand curve but twice the slope. This linear relationship is the cornerstone of the marginal revenue curve from demand curve analysis.

Deriving the Marginal Revenue Curve from a Linear Demand Curve

Step‑by‑step Procedure

1. Specify the inverse demand function.
   Example: (P = 100 - 4Q).

2. Write total revenue as a function of quantity.
   (TR = P \times Q = (100 - 4Q)Q = 100Q - 4Q^{2}).

3. Differentiate TR with respect to Q to obtain MR.
   (MR = \frac{dTR}{dQ}=100 - 8Q).

4. Plot MR against Q. The resulting line intercepts the vertical axis at 100 and declines with a slope of –8, which is exactly twice the slope of the original demand curve (‑4).

5. Identify the intersection point where MR becomes zero; this quantity maximizes profit when combined with marginal cost.

General Formula

For any linear demand curve expressed as (P = a - bQ),

[ \boxed{MR = a - 2bQ} ]

Thus, the marginal revenue curve from demand curve is always a straight line with the same vertical intercept (a) but double the magnitude of the slope (‑2b). This simple transformation is the basis for many analytical results in monopoly theory.

Graphical Representation and Interpretation

Visual Layout

• Horizontal axis: Quantity (Q)
• Vertical axis: Price (P) and Revenue measures
• Demand curve: Downward‑sloping line starting at (0, a) and intersecting the quantity axis at (Q = \frac{a}{b}).
• Marginal revenue curve: Starts at the same price level (a) but falls twice as fast, crossing the horizontal axis at (Q = \frac{a}{2b}).

Economic Insight

Because MR lies below the demand curve, a monopolist cannot simply set price equal to marginal cost to achieve efficiency. Instead, the profit‑maximizing rule is:

[ \text{Produce where } MR = MC \text{ and set price from the demand curve at that quantity.} ]

The distance between the demand curve and MR curve at any output level reflects the price markup over marginal cost. The greater the divergence, the more market power the firm possesses.

Relationship Between MR, Demand, and Elasticity

The slope of the MR curve is directly linked to the price elasticity of demand (ε). At any point on the demand curve:

[ \varepsilon = \frac{dQ}{dP}\cdot\frac{P}{Q} ]

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When demand is elastic ((|\varepsilon|>1)), MR is positive; when demand is unit‑elastic ((|\varepsilon|=1)), MR equals zero; and when demand is inelastic ((|\varepsilon|<1)), MR becomes negative. This relationship explains why the MR curve intersects the horizontal axis at the quantity where demand transitions from elastic to inelastic.

Implications for Monopolistic Pricing

1. Higher markup – The steeper the MR curve relative to the demand curve, the larger the markup over marginal cost.
2. Output restriction – A monopolist restricts production to the point where MR = MC, which is always at a lower quantity than the competitive equilibrium.
3. Deadweight loss – The gap between the competitive quantity and the monopoly quantity creates inefficiency, known as deadweight loss, which is a key justification for antitrust regulation.

Frequently Asked Questions (FAQ)

How does the marginal revenue curve change if the demand curve is non‑linear?

For non‑linear demand (e., (P = aQ^{-b})), the MR function must be derived by first expressing TR = P(Q)·Q, then differentiating. g.The resulting MR curve may be curvilinear, but the principle that MR lies below demand still holds.

Can MR ever intersect the demand curve?

In a linear demand setting, MR intersects the demand curve only at the vertical axis (Q = 0). For more complex demand functions, intersections can occur, but they are generally not economically meaningful for profit maximization.

Why is MR always below the demand curve for a price‑setting firm?

Because to sell an additional unit, the firm must lower the price for all units sold, not just the marginal one. This price reduction reduces revenue on existing units, causing the extra revenue from the new unit to be smaller than the price of that unit—hence MR < P.

What happens to MR if the firm faces a perfectly elastic demand curve?

If demand is perfectly elastic (horizontal), the firm is a price taker. In this case, MR coincides with the demand curve, and the firm’s pricing decision is irrelevant; MR equals the market price.

Practical Example: A Numerical IllustrationSuppose a monopolist faces the demand curve (P = 200 - 5

Suppose a monopolist facesthe demand curve (P = 200 - 5Q). Total revenue is

[TR(Q)=P\cdot Q = (200-5Q)Q = 200Q-5Q^{2}. ]

Marginal revenue follows from differentiating TR:

[ MR(Q)=\frac{dTR}{dQ}=200-10Q. ]

Assume the firm’s marginal cost is constant at (MC = 20). Profit maximization requires (MR = MC):

[ 200-10Q = 20 ;\Longrightarrow; Q^{*}=18. ]

The corresponding price is obtained from the demand curve:

[ P^{*}=200-5(18)=200-90=110. ]

Profit calculation (ignoring fixed costs for simplicity):

[ \pi = (P^{}-MC)Q^{}= (110-20)\times 18 = 90\times 18 = 1{,}620. ]

Competitive benchmark: In a perfectly competitive market, price equals marginal cost, so

[ P_{c}=MC=20 \quad\Rightarrow\quad 200-5Q_{c}=20 ;\Longrightarrow; Q_{c}=36. ]

The monopoly restricts output to 18 units, whereas competition would supply 36 units. The resulting deadweight loss (DWL) is the area of the triangle between the demand and marginal‑cost curves from (Q^{*}) to (Q_{c}):

[DWL = \frac{1}{2},(Q_{c}-Q^{}),(P^{}-MC) = \frac{1}{2},(36-18),(110-20) = \frac{1}{2}\times 18 \times 90 = 810. ]

This numerical illustration reinforces the theoretical insights discussed earlier: a steeper MR curve (relative to demand) yields a higher markup, output is reduced below the competitive level, and the gap creates a measurable efficiency loss that antitrust policy seeks to mitigate.

Conclusion
The marginal revenue curve is a important tool for understanding how price‑setting firms translate market power into pricing and output decisions. Its position beneath the demand curve reflects the revenue‑reducing effect of lowering price on all prior units, while its slope encodes the elasticity of demand. By equating MR with marginal cost, a monopolist determines the profit‑maximizing quantity and price, inevitably producing less than the socially optimal output and generating deadweight loss. Quantitative examples, such as the one above, make these abstract relationships concrete and highlight why regulators monitor markets where MR lies significantly below demand. When all is said and done, grasping the MR‑demand elasticity nexus equips policymakers, managers, and students to evaluate the welfare implications of market power and to design interventions that promote competitive efficiency.