Elasticity and Expenditure

Definitions Elasticity = responsiveness of quantity demanded to price. Coefficient of elasticity = Percent change in quantity Percent change in price Percent change in Q = Change in Q / Q = Q 2 - Q 1 divided by Q 1 Percent change in P = Change in P / P = P 2 - P 1 divided by P 1

Relationship to expenditure % Δ (PQ) = % Δ P + % Δ Q If % Δ P = 3 percent and % Δ Q = -10 %, then % Δ (PQ) = - 7 % or revenue declines Coef of elas = ε = % Δ Q divided by % Δ P hence in the above example ε = 10 / 3 = 3.33 The coefficient of elasticity is greater than 1, so demand is ELASTIC

Problem: % Δ Q = - 10 %, and % Δ P = 4 % a. what is the coefficient of elasticity? b. is demand elastic or inelastic? c. what is the percent change in consumer expenditure? [See next slide for answers…but not before trying to solve the problem]

Solution: % Δ Q = - 10 %, and % Δ P = 4 % a.what is the coefficient of elasticity? ε = % Δ Q / % Δ P = 10 / 4 = 2.5 b. is demand elastic or inelastic? -- since ε > 1.0, demand is elastic. We should expect that consumer expenditure will decline with an increase in price. c. what is the percent change in consumer expenditure? Since % Δ PQ = % Δ P + % Δ Q, we have % Δ PQ = 4 % - 10 % = - 6 %

Problem: % Δ Q = + 10 %, and % Δ Revenue = 3 % a. what is the percent change in price? b. what is the coefficient of elasticity? c. is demand elastic or inelastic? d. what is the percent change in consumer expenditure? [See next slide for answers…but not before trying to solve the problem]

Solution: % Δ Q = + 10 %, and % Δ Revenue = + 3 % a. what is the percent change in price? Since % Δ PQ = % Δ P + % Δ Q 3 % = % Δ P + 10 % Price must have fallen by 7 percent b. what is the coefficient of elasticity? Since ε = % Δ Q / % Δ P = 10 % / 7 % = 1.43 c. is demand elastic or inelastic? Elastic demand – a reduction in price leads to an increase in revenue -- the coefficient of elasticity is greater than 1. d. what is the percent change in consumer expenditure? The same as the percent change in revenue, + 3 %

Problem: % Δ P = + 10 %, and % Δ Revenue = - 3 % a. what is the percent change in quantity? b. what is the coefficient of elasticity? c. is demand elastic or inelastic? d. what is the percent change in consumer expenditure? [See next slide for answers…but not before trying to solve the problem]

Solution: % Δ Q = + 10 %, and % Δ Revenue = - 3 % a. what is the percent change in price? Since % Δ PQ = % Δ P + % Δ Q - 3 % = % Δ P + 10 % Price must have fallen by 13 percent b. what is the coefficient of elasticity? Since ε = % Δ Q / % Δ P = 10 % / 13 % = c. is demand elastic or inelastic? Inelastic demand – a reduction in price leads to a decrease in revenue -- the coefficient of elasticity is less than 1. d. what is the percent change in consumer expenditure? The same as the percent change in revenue, - 3 %

Problem: % Δ P = + 10 %, and ε = 2.5 a. what is the percent change in quantity? b. is demand elastic or inelastic? c. what is the percent change in consumer expenditure? [See next slide for answers…but not before trying to solve the problem]

Solution: % Δ P = + 10 %, and ε = 2.5 a. what is the percent change in quantity? Since ε = 2.5 = % Δ Q / % Δ P = % Δ Q / 10 We have % Δ Q = - 25 % (when price goes up, Q down) b. is demand elastic or inelastic? Since ε = 2.5, demand is ELASTIC; we should expect consumer expenditure to decrease when price increases. c. what is the percent change in consumer expenditure? Since % Δ PQ = % Δ P + % Δ Q, we have % Δ PQ = % = - 15%

Problem: % Δ P = + 5 %, and ε = 1/4 a. what is the percent change in quantity? b. is demand elastic or inelastic? c. what is the percent change in consumer expenditure? [See next slide for answers…but not before trying to solve the problem]

Solution: % Δ P = + 5 %, and ε = ¼ a. what is the percent change in quantity? Since ε = ¼ = % Δ Q / % Δ P = % Δ Q / 5 We have % Δ Q = % (when price goes up, Q down) b. is demand elastic or inelastic? Since ε = ¼, demand is INELASTIC; we should expect consumer expenditure to increase when price increases. c. what is the percent change in consumer expenditure? Since % Δ PQ = % Δ P + % Δ Q, we have % Δ PQ = + 5 % % = %

Calculating elasticity from a table PriceQuantity% Δ P% Δ QElasticity

Calculating elasticity from a table PriceQuantity% Δ P% Δ QElasticity – 1 / 1 = 100 % 20 / 180 = 11.1 % (inelastic) – 2 / 2 = 50 % 20 / 160 = 12.5 % (inelastic) – 3 / 3 = 33 % 20 / 140 = 14.3 % (inelastic) – 4 / 4 = 25 % 20 / 120 = 16.7% (inelastic) – 5 / 5 = 20 % 20 / 100 = 20 % (unit elastic)

Elasticity and slope The table was derived from the demand equation: Q = P Note that for each dollar price goes up, Q goes down by 20 units. In mathematical symbolism, Δ Q / Δ P = - 20 This is close to the formula for the coefficient of elasticity, but not quite the same. What is the difference? [Pause for thought…]

Elasticity and slope Slope = Δ Q / Δ P = - 20 But coefficient of elasticity = Δ Q / Q divided by Δ P / P The expression for the coefficient of elastictity can be rearranged to get ε = Δ Q / Δ P times P / Q Which can be a handy computational rule for elasticity AT a point.

Problem: given the demand equation Q = 1500 – 25 P What is the coefficient of elasticity at: P = 10 ? First note that Q = 1500 – 250 = 1250, So that the coefficient of elasticity is: ε = Δ Q / Δ P times P / Q ε = 25 (10) / 1250 = 250 / 1250 ε = 0. 20

Problem: given the demand equation Q = 1500 – 25 P What is the coefficient of elasticity at: P = 50 ? First note that Q = 1500 – 1250 = 250, So that the coefficient of elasticity is: ε = Δ Q / Δ P times P / Q ε = 25 (50 / 250) = 1250 / 250 ε = 5.0