Overview
The lecture explains elasticity of demand, how to calculate it using derivatives, and interprets the meaning of inelastic demand in relation to price changes and revenue.
Elasticity of Demand Basics
- Elasticity measures how sensitive demand is to changes in the product's price.
- If price increases and demand drops significantly, demand is elastic; if not, it is inelastic.
- The elasticity equation: Elasticity = -x × (derivative of demand function / original demand function).
Calculating the Derivative of the Demand Function
- To find elasticity, calculate the derivative of the demand function using the quotient rule.
- Quotient rule: (Denominator × derivative of numerator) - (Numerator × derivative of denominator), all over (denominator squared).
- Simplify the numerator and denominator by distributing and combining like terms.
Building the Elasticity Formula
- Substitute the negative x and the derivative of demand into the elasticity formula.
- The derivative goes in the numerator, the original demand function in the denominator.
- Simplify by factoring out common factors in the denominator and numerator.
- Cancel out and simplify further to get the final elasticity formula.
Example Calculation
- To find elasticity at x = 4, substitute 4 for every x in the formula.
- Calculate and simplify numerically to get elasticity as a decimal.
- If the result is less than 1, demand is inelastic at that price.
Interpretation of Inelastic Demand
- Inelastic demand means consumers aren’t very sensitive to price increases.
- Raising the price when demand is inelastic generally increases total revenue.
- If elasticity < 1, a price rise does not significantly reduce quantity demanded.
Key Terms & Definitions
- Elasticity of Demand — A measure of how much quantity demanded responds to a change in price.
- Inelastic Demand — Demand with elasticity less than 1; quantity demanded changes little with price changes.
- Quotient Rule — A calculus rule for finding the derivative of a fraction.
Action Items / Next Steps
- Practice calculating elasticity for various demand functions and values of x.
- Review properties of elastic vs. inelastic demand for exam preparation.