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Price Elasticity of Demand Overheads How much would your roommate pay to watch a live fight? How does Showtime decide how much to charge for a live fight? What about Hank and Son’s Concrete? How much should they charge per square foot? Can ISU raise parking revenue by raising parking fees? Or will the increase in price drive demand down so far that revenue falls? All of these pricing issues revolve around the issue of how responsive the quantity demanded is to price. Elasticity is a measure of how responsive one variable is to changes in another variable? The Law of Demand The law of demand states that when the price of a good rises, and everything else remains the same, the quantity of the good demanded will fall. The real issue is how far it will fall. The demand function is given by Q D h(P, ZD ) QD = quantity demanded P = price of the good ZD = other factors that affect demand The inverse demand function is given by P h 1 D (Q , ZD ) P g(Q , ZD ) D To obtain the inverse demand function we just solve the demand function for P as a function of Q Examples QD = 20 - 2P 2P + QD = 20 2P = 20 - QD P = 10 - 1/2 QD Slope = - 1/2 Examples QD = 60 - 3P 3P + QD = 60 3P = 60 - QD P = 20 - 1/3 QD Slope = - 1/3 One measure of responsiveness is slope For demand Q D h(P, ZD ) The slope of a demand curve is given by the change in Q divided by the change in P ΔQ D slope ΔP For inverse demand P g(Q , ZD ) D The slope of an inverse demand curve is given by the change in P divided by the change in Q ΔP slope ΔQ D Examples QD = 60 - 3P Slope = - 3 P = 20 - 1/3 QD Slope = - 1/3 Examples QD = 20 - 2P Slope = - 2 P = 10 - 1/2 QD Slope = - 1/2 We can also find slope from tabular data Q P 0 10 Q 2 9 P 4 8 6 7 8 6 10 5 slope ΔQ D 2 2 ΔP 1 Demand for Handballs Q P 0 10 1 9.5 2 9 3 8.5 4 8 5 7.5 6 7 7 6.5 8 6 9 5.5 10 5 11 4.5 12 4 13 3.5 14 3 15 2.5 16 2 17 1.5 18 1 19 0.5 20 0 Q P 0 10 Demand for Handballs 1 9.5 2 9 11 3 8.5 Price 4 8 10 5 7.5 9 6 7 7 6.5 8 8 6 7 9 5.5 6 10 5 5 11 12 4.5 4 P 4 13 3.5 14 3 3 15 2.5 2 16 2 1 17 1.5 18 1 0 19 0.5 0 2 4 6 8 10 12 14 16 18 20 22 20 0 Quantity Q P 0 10 Demand for Handballs 1 9.5 Q 2 9 3 P 11 8.5 Price 4 8 10 Q = 2 - 4 = -2 5 7.5 9 6 7 7 6.5 8 P = 9 - 8 = 1 8 6 7 9 5.5 6 10 5 11 4.5 5 12 4 4 13 3.5 14 3 3 slope ΔP 1 15 2.5 16 2 2 ΔQ D 2 1 17 1.5 18 1 0 19 0.5 0 2 4 6 8 10 12 14 16 18 20 22 20 0 Quantity Problems with slope as a measure of responsiveness Slope depends on the units of measurement The same slope can be associated with very different percentage changes Examples QD = 200 - 2P 2P + QD = 200 2P = 200 - QD P = 100 - 1/2 QD slope ΔP 1 ΔQ D 2 Q P Consider data on racquets 0 100 1 99.5 Let P change from 95 to 96 2 99 3 98.5 4 98 P = 96 - 95 = 1 5 97.5 6 97 Q = 8 - 10 = -2 7 96.5 Q 8 9 10 96 95.5 95 P 11 94.5 12 94 13 93.5 14 93 A $1.00 price change when P = $95.00 is tiny Graphically for racquets Demand for Racquets 102 Price 100 98 96 Slope = - 1/2 94 92 90 88 0 2 4 6 8 10 12 14 16 18 Large % change in Q Quantity Small % change in P Graphically for hand balls Demand for Handballs 11 Price 10 P=7-6=1 9 8 7 6 Q = 6 - 8 = -2 5 P 4 3 Slope = - 1/2 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 Quantity Large % change in Q Large % change in P So slope is not such a good measure of responsiveness Instead of slope we use percentage changes The ratio of the percentage change in one variable to the percentage change in another variable is called elasticity The Own Price Elasticity of Demand is given by ΔQ D D D %ΔQ Q εD %ΔP ΔP P There are a number of ways to compute percentage changes Initial point method for computing The Own Price Elasticity of Demand Price Elasticity of Demand (Initial Point Method) P Q Q 6 8 ΔQ D P 5.5 9 5 10 %ΔQ D QD εD 4.5 11 %ΔP ΔP 4 12 P (8 10) εD 8 (8 10) 6 (6 5) 8 (6 5) 6 2 6 12 1.5 8 1 8 Final point method for computing The Own Price Elasticity of Demand Price Elasticity of Demand (Final Point Method) P Q Q 6 8 ΔQ D P 5.5 9 5 10 %ΔQ D QD εD 4.5 11 %ΔP ΔP 4 12 P (8 10) εD 10 (8 10) 5 (6 5) 10 (6 5) 5 2 5 10 1.0 10 1 10 The answer is very different depending on the choice of the base point So we usually use The midpoint method for computing The Own Price Elasticity of Demand Elasticity of Demand Using the Mid-Point ΔQ D εD %ΔQ D Q D %ΔP ΔP P ΔQ D Q1 0 or Q0 1 Q Q For QD we use the midpoint of the Q’s 1 Q D (Q1 Q0 ) 2 Similarly for prices ΔP P1 0 or P0 1 P P For P we use the midpoint of the P’s 1 P ( P1 P0 ) 2 ΔQ D D D εD %ΔQ Q %ΔP ΔP P (Q1 0 ) Q 1 (Q1 0 ) Q ( Q1 Q0 ) 2 (Q1 Q0 ) ε (P1 0 ) P (P1 0 ) P 1 ( P1 P0 ) ( P1 P0 ) 2 Price Elasticity of Demand (Mid-Point Method) Q P ΔQ D 8 6 9 5.5 %ΔQ D QD 10 5 εD 11 4.5 %ΔP ΔP 12 4 P (Q1 Q0 ) (Q1 Q0 ) (Q1 0 ) (P1 P0 ) Q (P1 P0 ) (P1 0 ) (Q1 Q0 ) P (P1 P0 ) (8 10) (6 5) (6 5) (8 10) 2) ( (11) 22 11 (1) (18) 18 9 Classification of the elasticity of demand Inelastic demand When the numerical value of the elasticity of demand is between 0 and -1.0, we say that demand is inelastic. %ΔQ D %ΔP < 1 %ΔQ D < %ΔP Classification of the elasticity of demand Elastic demand When the numerical value of the elasticity of demand is less than -1.0, we say that demand is elastic. %ΔQ D %ΔP > 1 %ΔQ D > %ΔP Classification of the elasticity of demand Unitary elastic demand When the numerical value of the elasticity of demand is equal to -1.0, we say that demand is unitary elastic. %ΔQ D 1 %ΔP %ΔQ D %ΔP Classification of the elasticity of demand Perfectly elastic - D = - horizontal Perfectly inelastic - D = 0 vertical Elasticity of demand with linear demand Consider a linear inverse demand function P A D BQ The slope is (-B) for all values of P and Q For example, P 12 0.5Q D The slope is -0.5 = - 1/2 P Q Demand for Diskettes 12 0 11.5 1 13 11 2 Price 12 11 10.5 3 10 10 4 9 9.5 5 8 7 9 6 6 8.5 7 5 P Q 4 8 8 3 7.5 9 2 1 7 10 0 6.5 11 0 2 4 6 8 10 12 14 16 18 20 22 6 12 Quantity 5.5 13 P Q 5 14 ΔQ D 4.5 15 2.0 4 16 ΔP 3.5 17 3 18 The slope is constant but the P 12 Q 0 elasticity of demand will vary 11.5 11 1 2 ΔQ D 10.5 3 10 4 %ΔQ D QD εD 9.5 5 %ΔP ΔP 9 6 8.5 7 P P Q 8 8 ΔQ (P1 P0 ) 7.5 9 7 10 ΔP (Q1 Q0 ) 6.5 11 6 12 (8 10) (8 7) 5.5 13 (8 7) (8 10) 5 14 4.5 15 4 16 2) ( (15) 30 5 3.5 17 (1) (18) 18 3 3 18 The slope is constant but the P 12 Q 0 elasticity of demand will vary 11.5 11 1 2 ΔQ D 10.5 3 10 4 %ΔQ D QD 9.5 5 εD 9 6 %ΔP ΔP 8.5 7 P 8 8 ΔQ (P1 P0 ) 7.5 9 7 10 ΔP (Q1 Q0 ) 6.5 11 6 12 (14 16) (5 4) 5.5 13 (5 ) (14 16) P 4 Q 5 14 4.5 15 4 16 2) ( (9 ) 18 3 3.5 17 (1) (30) 30 5 3 18 The slope is constant but the elasticity of demand will vary A linear demand curve becomes more inelastic as we lower price and increase quantity P smaller εD %ΔQ D ΔQ (P1 P0 ) %ΔP ΔP (Q1 Q0 ) Q larger The elasticity gets closer to zero The slope is constant but the elasticity of demand will vary Q P Elasticity Expenditure 0 12 0 2 11 -23.0000 22 4 10 -7.0000 40 6 9 -3.8000 54 8 8 -2.4286 64 10 7 -1.6667 70 12 6 -1.1818 72 14 5 -0.8462 70 16 4 -0.6000 64 18 3 -0.4118 54 20 2 -0.2632 40 22 1 -0.1429 22 24 0 -0.0435 0 The slope is constant but the elasticity of demand will vary Q P Elasticity Expenditure 0 12 0 2 11 -23.0000 22 4 10 -7.0000 40 6 9 -3.8000 54 8 8 -2.4286 64 10 7 -1.6667 70 12 6 -1.1818 72 14 5 -0.8462 70 16 4 -0.6000 64 18 3 -0.4118 54 20 2 -0.2632 40 22 1 -0.1429 22 24 0 -0.0435 0 Note We do not say that demand is elastic or inelastic ….. We say that demand is elastic or inelastic at a given point Example ΔQ D D D %ΔQ Q εD %ΔP ΔP P ΔQ (P1 P0 ) ΔP (Q1 Q0 ) Constant with linear demand The Own Price Elasticity of Demand and Total Expenditure on an Item How do changes in an items price affect expenditure on the item? If I lower the price of a product, will the increased sales make up for the lower price per unit? Expenditure for the consumer is equal to revenue for the firm Revenue = R = price x quantity = PQ Expenditure = E = price x quantity = PQ Modeling changes in price and quantity P = change in price Q = change in quantity The Law of Demand says that as P increases Q will decrease P Q So P = initial price P = change in price P + P = final price Q = initial quantity Q = change in quantity Q + Q = final quantity So Initial Revenue = PQ P + P = final price Q + Q = final quantity Final Revenue = (P + P) (Q + Q) = P Q + P Q + P Q + P Q Now find the change in revenue R = final revenue - initial revenue = P Q + P Q + P Q + P Q - P Q = P Q + P Q + P Q %R = R / R = R / P Q ΔR ΔP Q P ΔQ ΔP ΔQ PQ PQ We can rewrite this expression as follows ΔR ΔP Q P ΔQ ΔP ΔQ PQ PQ PQ PQ ΔP ΔQ ΔP ΔQ %ΔR P Q PQ %ΔR %ΔP %ΔQ Classification of the elasticity of demand Inelastic demand D %ΔQ %ΔP < 1 %ΔQ D < %ΔP + - %ΔR %ΔP %ΔQ % Q and % P are of opposite sign so %R has the same sign as %P Classification of the elasticity of demand Inelastic demand D %ΔQ %ΔP < 1 %ΔQ D < %ΔP - + %ΔR %ΔP %ΔQ % Q and % P are of opposite sign so %R has the same sign as %P Lower price lower revenue Higher price higher revenue Classification of the elasticity of demand Elastic demand %ΔQ D %ΔP > 1 %ΔQ D > %ΔP + - %ΔR %ΔP %ΔQ % Q and % P are of opposite sign so %R has the opposite sign as %P Higher price lower revenue Lower price higher revenue Classification of the elasticity of demand Unitary elastic demand %ΔQ D 1 %ΔQ D %ΔP %ΔP + - %ΔR %ΔP %ΔQ % Q and % P are of opposite sign so their effects will cancel out and %R = 0. Tabular data Elastic Q P Elasticity Revenue Price falls 0 12 0 2 11 -23.0000 22 4 10 -7.0000 40 6 9 -3.8000 54 8 8 -2.4286 64 10 7 -1.6667 70 Inelastic 12 6 -1.1818 72 Price falls 14 5 -0.8462 70 16 4 -0.6000 64 18 3 -0.4118 54 20 2 -0.2632 40 Revenue falls 22 1 -0.1429 22 24 0 -0.0435 0 Graphical analysis Q P Elasticity Revenue Demand for Diskettes 0 12 0 Demand 2 11 -23.0000 22 13 4 10 -7.0000 40 Price 12 P0, Q0 6 9 -3.8000 54 11 8 8 -2.4286 64 10 P1, Q1 10 7 -1.6667 70 9 B 12 6 -1.1818 72 8 7 14 5 -0.8462 70 6 16 4 -0.6000 64 5 18 3 -0.4118 54 4 3 C A 20 2 -0.2632 40 2 22 1 -0.1429 22 1 24 0 -0.0435 0 0 0 2 4 6 8 10 12 14 16 18 20 22 Quantity Lose B, gain A, revenue rises Graphical analysis Q P Elasticity Revenue Demand for Diskettes 0 12 0 Demand 2 11 -23.0000 22 13 4 10 -7.0000 40 Price 12 P0, Q0 6 9 -3.8000 54 11 8 8 -2.4286 64 10 P1, Q1 10 9 7 -1.6667 70 8 12 6 -1.1818 72 7 14 5 -0.8462 70 6 16 4 -0.6000 64 5 18 3 -0.4118 54 4 3 2 A 20 22 2 1 -0.2632 40 -0.1429 22 1 B 24 0 -0.0435 0 0 0 2 4 6 8 10 12 14 16 18 20 22 Quantity Lose A, gain B, revenue falls Factors affecting the elasticity of demand Availability of substitutes Importance of item in the buyer’s budget Availability of substitutes The easier it is to substitute for a good, the more elastic the demand With many substitutes, individuals will move away from a good whose price increases Examples of goods with “easy “substitution Gasoline at different stores Soft drinks Detergent Airline tickets Local telephone service Narrow definition of product The more narrowly we define an item, the more elastic the demand With a narrow definition, there will lots of substitutes Examples of narrowly defined goods Lemon-lime drinks Corn at a specific farmer’s market Vanilla ice cream Food Transportation “Necessities” tend to have inelastic demand Necessities tend to have few substitutes Examples of necessities Salt Insulin Food Trips to Hawaii Sailboats Demand is more elastic in the long-run There is more time to adjust in the long run Examples of short and long run elasticity Postal rates Gasoline Sweeteners Factors affecting the elasticity of demand Importance of item in the buyer’s budget The more of their total budget consumers spend on an item, the more elastic the demand for the good The elasticity is larger because the item has a large budget impact “Big ticket” items and elasticity Housing Big summer vacations Table salt College tuition The End

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posted: | 7/20/2012 |

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