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ENVS 2010U (24)
Lecture

# 4 elasticity.doc

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School
Department
Environmental Science
Course
ENVS 2010U
Professor
Steve Hutchinson
Semester
Fall

Description
Principles of Microeconomics: Elasticity ELASTICITY Elasticity Definition: Elasticity measures the responsiveness of a change in a variable to the change in another variable => ε = %ΔQ/%ΔP Economists measure Elasticity as the ratio of percentages changes because a ratio of absolute change does not indicate the relative importance of a change (e.g., a \$1 increase in the price of a car would hardly affect car sales but a \$1 increase in the price of gasoline would significantly affect the sale of gasoline) Price Elasticity of Demand: Definition: Price Elasticity of Demand (η) is equal to the ratio of the percentage change in quantity demanded to the percentage change in price (responsible for that percentage change in quantity demanded) ε = %ΔQ /%DP e.g. If a 20% increase in the price of gasoline causes a 10% decrease in the quantity demanded of gasoline, then ε = -10%/+20% = -0.5 Note: Price Elasticity of Demand is usually defined for first year students as the absolute value of the ratio of the percentage changes of quantity demanded and price because the ratio is almost invariably negative (unless Demand is upward sloping, a rare and possibly non-existent case) and expression as an absolute value simplifies the relation between elasticity and changes in total revenue. I will not use the absolute value expression of elasticity because the sign has significance, particularly in relation to other types of elasticity. - 1 - Principles of Microeconomics: Elasticity Define: 1. Inelastic => %ΔQ /%DP< 1 i.e. the percentage response of Quantity is less than the percentage change in Price 2. Unit elastic => %ΔQ /%DP= 1 i.e. the percentage response of Quantity equals the percentage change in Price 3. Elastic => %ΔQ /%ΔP> 1 D i.e. the percentage response of Quantity is greater than the percentage change in Price Since Demand data in reality is in the form of Prices and Quantities, not percentage changes, we need to express elasticity in prices and quantities. Elasticity of Demand = %ΔQ/%ΔP = (100% * ΔQ/Q)/(100% * ΔP/P) = (ΔQ/Q)/(ΔP/P) = (ΔQ/ΔP) * P/Q) = 1/slope * P/Q (since (ΔQ/ΔP = 1/(ΔP/ΔQ) = 1/slope) We can express ΔQ and ΔP in terms of Po and P to 1et ε = (Q – Qo)/Q)/(P – Po)/P 1 1 What are the P and the Q that we divide by? Point Elasticity Point Elasticity calculates elasticity at a point, i.e., relative to the original Price and Quantity ε = (Q 1 Qo)/Qo)/(P –1Po)/Po - 2 - Principles of Microeconomics: Elasticity e.g. Suppose that the price of Ipods falls from \$250 to \$200 and the quantity sold of Ipods increases from 500,000 to 600,000. What is the point elasticity of demand? ε = (600,000 – 500,000)/500,000)/(200 – 250)/250 = - 1 or unit elasticity → Unit elasticity suggests that there is no change in Total Revenue with the change in price but Total Revenue actually falls from \$125 million to \$120 million Note also that point elasticity is not the same for the opposite direction, i.e., an increase in price from \$200 to \$250 with a decrease in quantity from 600,000 to 500,000. In this case, elasticity is –0.67, which suggests that the increase in price should increase Total Revenue as it does. . The reason for the failure of point elasticity to correctly predict the change in Total Revenue and for the difference between the elasticities for equal rises and falls in price is that point elasticity is only accurate for small changes relative to the original price and quantity because elasticity usually changes as P and Q change. It is really a calculus concept dQ/dP *P/Q Arc Elasticity Arc Elasticity calculates elasticity relative to the average price and quantity of the change. = ΔQ/ ΔP * (Average P)/(Average Q) or ΔQ/(Average Q)/ ΔP/(Average P) = (Q 1 Qo)/(P –1Po) * (Po + P )/21(Qo + Q )/2 1 e.g. The arc elasticity for a decrease in price from \$250 to \$200 causing an increase in quantity from \$500,000 to \$600,000 is the same as the arc elasticity of an increase in price from \$200 to \$250 with the opposite change in quantity = (600,000 – 500,000)/(200 – 250) * (250 + 200)/2/(500,000 + 600,000)/2 = -0.818 - 3 - Principles of Microeconomics: Elasticity Elasticity and Total Revenue Since movement along a Demand function implies an inverse relationship between Price and Quantity, what is the effect on Total Revenue of a change in Price or a change in Quantity? Should a firm lower price and thus increase quantity to increase Total Revenue or increase price and lower quantity to increase Total Revenue? The Elasticity of Demand gives us the answer. Suppose that a change in Price causes no change in Total Revenue (TR). ΔTR/ ΔP = 0 => Δ(P*Q)/ ΔP = 0 → ΔP/ΔP * Q + P * ΔQ/ΔP = 0 (Product rule) → P*ΔQ/ΔP = -Q → P/Q * ΔQ/ΔP = -1 → P/Q * ΔQ/ΔP= ε = 1 => Demand is unit elastic if a change in Price causes no change in Total Revenue. By similar calculations, we find that Δ (P*Q)/dP > 0 => P/Q * ΔQ/ΔP > -1 so that P/Q * ΔQ/ΔP= ε < 1 => Demand is inelastic if a change in Price causes an increase in Total Revenue Δ (P*Q)/ ΔP < 0 => P/Q * ΔQ/ΔP < -1 so that P/Q * ΔQ/ΔP= ε > 1 => Demand is elastic if a change in Price causes a decrease in Total Revenue Inverting this result (which we can do since elasticity is a monotonic function) means that for small changes in Price, 1. Inelastic => Total Revenue moves in the same direction as Price [Total Revenue increases (decreases) if Price increases (decreases) or Quantity decreases (increases)] - 4 - Principles of Microeconomics: Elasticity 2. Unit Elastic => Total Revenue does not change if Price changes →Total Revenue is at a maximum 3. Elastic => Total Revenue moves in the opposite direction as Price [Total Revenue increases (decreases) if Price decreases (increases) or Quantity increases (decreases)] An illustration of the relevance of price elasticity’s relevance for total revenue is the paradoxical result that farmers of wheat, an inelastic good, benefit more from harvest failure on an international scale than harvest success. e.g. Let us examine the behaviour of Elasticity for a Linear Demand function. Suppose that the Demand function is P = 6 – 0.5Q. We can compile the following table for elasticity at the price/quantity combination relative to the next price/quantity combination. Price Quantity Total Point Elasticity Point Elasticity Arc Elasticity Revenue %ΔQ/Δ%P ΔQ/ΔP*P/Q 6 0 0 5 2 10 100%/-20% = -5 2/-1*5/2 = -5 2/-1*4.5/3 = -3 4 4 16 50%/-25% = - 2 2/-1*4/4 = -2 2/-1*3.5/5 = -7/5 3 6 18 33%/-33% = -1 2/-1*3/6 = -1 2/-1*2.5/7 = -5/7 2 8 16 25%/-50% = -1/2 2/-1*2/8 = -1/2 2/-1*1.5/9 = -1/3 1 10 10 20%/-100%=-1/5 2/-1*1/10 = -1/5 2/-1*0.5/11=-1/11 0 12 0 - 5 - Principles of Microeconomics: Elasticity Demand: P = 6 - 0.5Q Price (\$s) 7 6 5 Elastic 4 3 Unit Elastic Inelastic 2 1 0 0 2 4 6 8 10 12 Quantity Note: A linear demand function has a constant slope but elasticity (-1/
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