Marginal Revenue (MR) Calculator

Derive the marginal revenue function from a linear demand curve defined by two price-quantity points. Visualize demand (AR) and MR curves, compute total revenue, average revenue, and marginal revenue at any quantity.

Use two points to define linear demand P = a - bQ (b ≥ 0). MR curve has same intercept, double slope. For perfect competition (constant price), b=0 → MR = P.
? Standard Demand: (0,100) & (100,0)
? Elastic scenario: (0,80) & (200,0)
? Inelastic region: (0,50) & (50,0)
? Premium product: (0,200) & (80,40)
? Perfect competition: (0,50) & (100,50)
Privacy-first: All calculations are performed locally in your browser. No data is transmitted or stored.

What is Marginal Revenue? Economic Foundation

Marginal Revenue (MR) is the additional revenue generated by selling one more unit of a good or service. In microeconomics, MR is crucial for profit maximization: a firm maximizes profit where MR = Marginal Cost (MC). For a firm facing a downward‑sloping demand curve (e.g., monopoly or monopolistic competition), MR lies below the demand curve because selling extra units requires lowering the price on all previous units.

For linear demand: P = a − b·Q   →   Total Revenue TR = P·Q = aQ − bQ²

Marginal revenue is the derivative: MR = d(TR)/dQ = a − 2b·Q

Thus, MR has the same vertical intercept (a) as demand but twice the slope (−2b). When b=0 (perfect competition), MR = a = constant price.

How to Use This Calculator

  • Step 1: Enter two points (Q₁,P₁) and (Q₂,P₂) that lie on the linear demand curve. Ensure Q₂ ≠ Q₁, non‑negative quantities and prices. For perfectly competitive markets, use two points with the same price.
  • Step 2: Specify the quantity (Q) at which you want to evaluate marginal revenue, total revenue, and average revenue.
  • Step 3: Click “Calculate & Draw Curves” to obtain the demand function, MR function, and a fully labelled graph with automatic axis scaling.
  • Step 4: Use example buttons to explore classic cases: standard linear demand, elastic/inelastic scenarios, premium product markets, or perfect competition.

Derivation & Intuition

Given two points (Q₁,P₁) and (Q₂,P₂) with Q₁ ≠ Q₂, the slope of the demand curve is b = (P₁ − P₂) / (Q₂ − Q₁) (positive if downward sloping, zero if constant price). The intercept is a = P₁ + b·Q₁. The marginal revenue function follows directly: MR(Q) = a − 2b·Q. The revenue‑maximizing quantity occurs when MR = 0, giving Q* = a/(2b) (if b>0). At that quantity, total revenue reaches its maximum value TR_max = a·Q* − b·(Q*)². The price elasticity of demand at any point can be expressed as ε = (P/Q)·(1/(−b)). In absolute terms, demand is elastic when |ε| > 1 (MR positive), inelastic when |ε| < 1 (MR negative), and unit elastic when MR = 0.

This calculator transforms raw market data into actionable managerial insights — from pricing decisions to output optimization.

Case Study: Monopoly Pricing Strategy

Application: Movie Theater Ticket Pricing

A local cinema estimates its demand curve using historical data: at $12 per ticket, 200 tickets sold (Q=200, P=12); at $8 per ticket, 400 tickets sold (Q=400, P=8). Using the calculator, the linear demand becomes P = 16 − 0.02Q, and MR = 16 − 0.04Q. The revenue‑maximizing quantity is Q = 400 (tickets) with price $8, maximum revenue $3,200. If the marginal cost is constant at $4, profit maximization sets MR = MC → 16 − 0.04Q = 4 → Q = 300, P = $10. This shows how marginal revenue guides optimal output beyond revenue maximization. Business students and managers can instantly test what-if scenarios.

Relationship Between MR, AR, and Elasticity

A fundamental identity: MR = AR × (1 − 1/|ε|), where ε is price elasticity of demand. For linear demand, this relationship holds exactly. Our calculator computes the elasticity at the selected quantity, helping you understand whether the firm is operating in the elastic (MR > 0) or inelastic (MR < 0) region. This is crucial for pricing: never set a price in the inelastic region if you aim to increase revenue.

Empirical Validation & Authoritative References

Source Key Concept Application
Varian, H. "Intermediate Microeconomics" Marginal revenue formula for linear demand Standard textbook derivation
Mankiw, N. G. "Principles of Economics" Monopoly revenue maximization MR = 0 → revenue peak
Brickley, Smith, Zimmerman "Managerial Economics" Pricing with MR and MC Real‑world decision framework

Frequently Asked Questions

This tool assumes a linear demand curve. For non‑linear functions (exponential, constant elasticity), MR would require calculus, but linear approximation is robust for many small ranges. For advanced modeling, consider our separate demand elasticity tool.

When MR is negative, selling additional units actually reduces total revenue. This occurs in the inelastic portion of the demand curve (|ε| < 1). A profit‑maximizing firm will never operate where MR < 0 if marginal cost is positive.

An upward‑sloping demand violates the law of demand. The calculator will display a warning and require corrected points (P₂ should be lower than P₁ when Q₂ > Q₁). For constant price (perfect competition) it is allowed.

The graph uses exact linear functions derived from your inputs, with dynamic scaling and axis ticks. It shows the demand curve, MR curve, and the evaluated quantity point with high precision.

For a firm, Average Revenue (AR) equals total revenue divided by quantity, which is simply the price per unit. Therefore, the demand curve is exactly the AR curve.
Academic references: Varian (2014), “Intermediate Microeconomics”; Pindyck & Rubinfeld (2018); and the foundational work by Cournot (1838) on monopoly revenue. Reviewed by GetZenQuery tech team, June 2025.