Elasticity of Demand: Mathematical Formulas, Indian Market Case Studies & Exam Numericals

Price elasticity of demand (Ep) measures the responsiveness of quantity demanded of a commodity to a change in its price. In microeconomics papers across Delhi University and central universities, examiners expect both calculus derivations (Point vs Arc Elasticity) and real-world empirical examples from the Indian market documented by MoSPI and NITI Aayog.

1. Mathematical Formulations: Point vs. Arc Elasticity

Point Price Elasticity (Calculus Formulation)

For a continuous demand function Q = f(P), point elasticity at a specific price-quantity point is:

Ep = (% Change in Q) / (% Change in P) = (dQ / dP) × (P / Q)

Arc Price Elasticity (Midpoint Formula)

When measuring elasticity over a discrete price interval between (P1, Q1) and (P2, Q2):

Earc = [(Q2 - Q1) / ((Q1 + Q2)/2)] ÷ [(P2 - P1) / ((P1 + P2)/2)] = [(Q2 - Q1)/(P2 - P1)] × [(P1 + P2)/(Q1 + Q2)]

2. Mathematical Proof: Marginal Revenue, Price, & Elasticity

Total Revenue TR is defined as TR = P × Q, where P = f(Q) is the inverse demand function.

Marginal Revenue MR is the derivative of Total Revenue with respect to Q:

MR = d(TR)/dQ = d(P × Q)/dQ = P + Q × (dP/dQ)
Factoring out P:
MR = P × [ 1 + (Q / P) × (dP / dQ) ]
Since Ep = (dQ / dP) × (P / Q), we have (Q / P) × (dP / dQ) = 1 / Ep:

MR = P × [ 1 - 1 / |Ep| ]

Implications of the MR Formula:

• If |Ep| > 1 (Elastic): 1 / |Ep| < 1 ⟹ MR > 0. Lowering price increases total revenue.
• If |Ep| = 1 (Unit Elastic): 1 / |Ep| = 1 ⟹ MR = 0. Total revenue is maximized.
• If |Ep| < 1 (Inelastic): 1 / |Ep| > 1 ⟹ MR < 0. Lowering price decreases total revenue.

3. Worked 10-Mark Exam Numericals

Solution Breakdown

1. Part (a): Point Elasticity at P = 10
   At P = 10, Q = 500 - 20(10) = 300. Derivative dQ/dP = -20.
   Ep = -20 × (10 / 300) = -200 / 300 = -0.67 (Inelastic, |Ep| < 1).
2. Part (b): Arc Elasticity between P1 = 10 (Q1 = 300) and P2 = 15 (Q2 = 200)
   Earc = [(200 - 300) / (15 - 10)] × [(10 + 15) / (300 + 200)] = [-100 / 5] × [25 / 500] = -20 × 0.05 = -1.0 (Unit Elastic).
3. Part (c): Total Revenue Maximizing Price
   Total Revenue TR = P × Q = P(500 - 20P) = 500P - 20P².
   dTR/dP = 500 - 40P = 0 ⟹ P_TR_max = 12.5.

4. Empirical Indian Market Case Studies

• Inelastic Demand (|Ep| < 1): Petrol & Essential Medicines
  Data published by the Petroleum Planning & Analysis Cell under the Ministry of Petroleum and Natural Gas shows retail fuel demand in urban India exhibits low short-run price elasticity (~ -0.2). Commuters have limited immediate public transit substitutes, making fuel consumption price-inelastic.
• Cross-Price Elasticity (Exy > 0): Delhi Metro vs DTC Buses
  When Delhi Metro fare revisions were implemented, commuter demand showed positive cross-elasticity between metro transit and municipal bus lines operated by DTC, illustrating classic substitute dynamics.

5. Related Micro & Quantitative Guides

For further practice on mathematical microeconomics and calculus techniques, check our guides on MME Survival & Distinction Guide and GDP Deflator vs CPI Inflation Analysis.