In This Article

  1. Introduction
  2. The Demand Curve
  3. Revenue Concepts
  4. Elasticity and Revenue
  5. Revenue Optimization
  6. Conclusion

Introduction

Demand and revenue analysis is fundamental to managerial decision-making. Understanding how consumers respond to prices and how this translates into revenue helps managers make optimal pricing and output decisions.

This unit examines the relationship between demand, price, and revenue. We explore how the demand curve shape affects revenue and how elasticity determines whether price increases or decreases will maximize revenue.

The Demand Curve

Law of Demand

The Law of Demand states that, ceteris paribus (all else equal), as the price of a good increases, the quantity demanded decreases, and vice versa.

Demand Function:

Qd = f(P, Y, Ps, Pc, T, E, N)

Where: P = Price, Y = Income, Ps = Price of substitutes, Pc = Price of complements, T = Tastes, E = Expectations, N = Number of buyers

Linear Demand Function

A common simplified form is the linear demand function:

Q = a - bP

Or in inverse form: P = (a/b) - (1/b)Q

Where: a = intercept, b = slope coefficient

Shifts vs. Movements

  • Movement along the curve: Caused by price changes of the good itself
  • Shift of the curve: Caused by changes in other factors (income, preferences, etc.)

Revenue Concepts

Total Revenue (TR)

TR = P × Q

For linear demand P = a - bQ:

TR = (a - bQ) × Q = aQ - bQ²

Average Revenue (AR)

AR = TR / Q = P

Average revenue equals price (the demand curve)

Marginal Revenue (MR)

Marginal revenue is the additional revenue from selling one more unit.

MR = ΔTR / ΔQ = dTR / dQ

For linear demand P = a - bQ:

MR = a - 2bQ

MR has twice the slope of the demand curve

Key Insight: For a downward-sloping demand curve, MR < P because to sell one more unit, the firm must lower the price on all units sold.

Example: Revenue Calculations

Given demand: P = 100 - 2Q

  • TR = P × Q = (100 - 2Q)Q = 100Q - 2Q²
  • MR = dTR/dQ = 100 - 4Q
  • At Q = 10: P = 80, TR = 800, MR = 60
  • At Q = 20: P = 60, TR = 1200, MR = 20
  • At Q = 25: P = 50, TR = 1250, MR = 0 (max TR)

Elasticity and Revenue

Price Elasticity of Demand

Ep = (% Change in Q) / (% Change in P)

Ep = (dQ/dP) × (P/Q)

Elasticity Categories

ElasticityValueMeaningRevenue Effect of Price Increase
Elastic|E| > 1Quantity very responsive to priceTR decreases
Unitary|E| = 1Proportional responseTR unchanged
Inelastic|E| < 1Quantity not very responsiveTR increases

Relationship Between MR and Elasticity

MR = P(1 + 1/E)

When demand is elastic (E < -1): MR > 0

When demand is unitary (E = -1): MR = 0

When demand is inelastic (E > -1): MR < 0

Revenue Optimization

Revenue Maximization

Total revenue is maximized where MR = 0 (at the midpoint of a linear demand curve, where elasticity = -1).

Example: Finding Maximum Revenue

Given: P = 100 - 2Q, so MR = 100 - 4Q

Set MR = 0: 100 - 4Q = 0

Q* = 25, P* = 50

Maximum TR = 25 × 50 = ₹1,250

Profit Maximization

Note that revenue maximization ≠ profit maximization. Profit is maximized where MR = MC, not where MR = 0.

Conclusion

Key Takeaways

  • The demand curve shows the inverse relationship between price and quantity
  • Total Revenue = Price × Quantity
  • Marginal Revenue is below price for a downward-sloping demand curve
  • For linear demand, MR has twice the slope of the demand curve
  • Elasticity determines how price changes affect revenue
  • Revenue is maximized where MR = 0 (elasticity = -1)
  • Profit is maximized where MR = MC, not where revenue is maximized