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Lecture 5: Elasticity of Demand and Supply Text: Chapter 5 ( pages 101-105) Responsiveness of the Quantity Demanded to Price Changes Earlier, we indicated that, ceteris paribus, the quantity of a product demanded will vary inversely to the price of that product. That is, the direction of change in quantity demanded following a price change is clear. What is not known is the extent by which quantity demanded will respond to a price change. To measure the responsiveness of the quantity demanded to change in price, we use a measure called PRICE ELASTICITY OF DEMAND. Own Price Elasticity of Demand The price elasticity of demand is a measure of the responsiveness of quantity demanded to a price change Own Price Elasticity of Demand: The percentage change in the quantity demanded relative to a percentage change in its own price. Q P Q P ED Q P P Q For a smooth (differentiable) demand curve, the price elasticity of demand is given by Q P ED P Q Using Price Elasticity of Demand Elasticity is a pure ratio independent of units. Since price and quantity demanded generally move in opposite direction, the sign of the elasticity coefficient is generally negative. Interpretation: If ED = - 2.72: A one percent increase in price results in a 2.72% decrease in quantity demanded Own Price Elasticity of Demand (ED) A numerical example Quantity Q2-Q1 Price P2-P1 ED 1 125 2 1 100 -25 (1/-25)X(125/1) = - 0.04x125 = - 5 4 2 50 -50 (2/-50)x(100/2) = - 0.04x50 = - 2 5 1 10 -40 (1/-40)x(50/4) = - 0.025x12.5 = - 0.3 Price Elasticity of Demand … Classifications of Own Price Elasticity of Demand: Inelastic demand ( |ED| < 1 ): a change in price brings about a relatively smaller change in quantity demanded (ex. gasoline). Total Revenue = P×Q rises as a result of a price increase Unitary elastic demand ( |ED| = 1 ): a change in price brings about an equivalent change in quantity demanded. TR= P×Q remains the same as a result of a price increase Elastic demand ( |ED| > 1 ): a change in price brings about a relatively larger change in quantity demanded (ex. expensive wine). TR = P×Q falls as a result of a price increase Total Revenue and ED Price Inelastic Demand Price Elastic Demand | ED | < 1 | ED| > 1 P P Q Q TR TR Price Elasticity along Linear Demand Curves Linear Demand Curve: P Q = a – bP a/b Price elasticity of this demand Ed = (∂Q/ ∂P)(P/Q) = − b(P/Q) a/2b M Any downward sloping demand curve has a corresponding inverse demand curve. 0 Q a/2 a Inverse linear Demand Curve: P = a/b – (1/b)Q At P= a/b, Ed = − ∞; at P = 0, Ed = 0; at P= a/2b, Ed = −1 In the region of the demand curve to the left of the mid-point M, demand is elastic, that is − ∞ ≤ Ed < – 1 In the region to the right of the mid-point M, demand is inelastic, – 1 < Ed ≤ 0 Constant Elasticity Demand Curve Another commonly used demand curve is the constant elasticity demand curve, given by Q = aP-b For this demand curve, the price elasticity of demand is Ed = (∂Q/ ∂P)(P/Q) = − b Thus, the price elasticity of demand is always the same (-b) on every point of this demand curve. However, such a demand curve could be elastic, or inelastic, or unit-elastic (depending on the value of b) . Own Price Elasticity of Demand (ED) Calculating Own Price Elasticity of Demand from a Demand Function: Q P Using calculus: Ed P Q Given a demand function: Qb = 100 – 30 Pb – 20 Pc + .005I, where, Qb = Quantity demanded of beer in billion 6-packs, Pb = Price of beer per 6- pack ($5), Pc = Price of a pack of chips ($1), and I = Annual household income ($25,000). Qb = 100 – 30*(5) – 20*(1) + .005*(25000) = 55 Taking partial derivative of the demand function with respect to price and substituting values for P and Q we get: Q P 5 Ed ( 30 ) 2.7272 P Q 55 Price Elasticity of Demand for some Commodities in the US Product Own Price Elasticity (Demand) Turkey - 1.56 Margarine - 0.84 Beef - 0.64 Cheese - 0.46 Potatoes - 0.30 Bread - 0.15 Cross Price Elasticity of Demand Shows the percentage change in the quantity demanded of good Y in response to a change in the price of good X. EDYX = % Change in QDY / % change in PX Algebraically: Qy Px Qy Px EdYX Qy Px Px Qy Read as the cross-price elasticity of demand for commodity Y with respect to commodity X. Units of Y demanded Price of X EDYX___________ 60 $10 40 $12 (-20/2)x(10/60) = - 1.66 Cross Price Elasticity of Demand (Edyx) Calculating Cross Price Elasticity of Demand from a Demand Function: Qy Px Using calculus: Edyx Px Qy Given a demand function: Qb = 100 – 30 Pb – 20 Pc + .005I, where, Qb = Quantity demanded of beer in billion 6-packs, Pb = Price of beer per 6- pack ($5), Pc = Price of a pack of chips ($1), and I = Annual household income ($25,000). Qb = 100 – 30*(5) – 20*(1) + .005*(25000) = 55 Taking partial derivative of the demand function for beer with respect to price of chips and substituting values for Pc and Q we get: Qb Pc 1 Edbc ( 20 ) 0.3636 Pc Qb 55 Classification of Cross-price elasticity of Demand Interpretation: If Edyx = - 0.36: A one percent increase in price of chips results in a 0.36% decrease in quantity demanded of beer Classification: If (Edyx > 0): implies that as the price of good X increases, the quantity demanded of Good Y also increases. Thus, Y and X are substitutes in consumption (ex. chicken and pork). If (Edyx < 0): implies that as the price of good X increases, the quantity demanded of Good Y decreases. Thus Y & X are Complements in consumption (ex. bear and chips). If (Edyx = 0): implies that the price of good X has no effect on quantity demanded of Good Y. Thus, Y & X are Independent in consumption (ex. bread and coke) Income Elasticity of Demand (EI) Shows the percentage change in the quantity demanded of good Y in response to a percentage change in Income. EI = % Change in QY / % change in I Algebraically: Qy I Qy I EI Qy I I Qy Units of Y demanded Income EI 100 $1200 150 $1600 (50/400)x(1200/100) = 1.5 Income Elasticity of Demand (EI) Calculating Income Elasticity of Demand from a Demand Function: Qy I Using calculus: EI I Qy Given a demand function: Qb = 100 – 30 Pb – 20 Pc + .005I, where, Qb = Quantity demanded of beer in billion 6-packs, Pb = Price of beer per 6-pack ($5), Pc = Price of a pack of chips ($1), and I = Annual household income ($25,000). Qb = 100 – 30*(5) – 20*(1) + .005*(25000) = 55 Taking partial derivative of the demand function with respect to income and substituting values for Q and I we get: Qb I 25000 EI (0.005 ) 2.2727 I Qb 55 Income Elasticity of Demand (EI) Interpretation: If EI = 2.27: A one percent increase income results in a 2.27% increase in quantity demanded of beer Classification: If EI > 0, then the good is considered a normal good (ex. beef). If EI < 0, then the good is considered an inferior good (ex. ramen noodles) High income elasticity of demand for luxury goods Low income elasticity of demand for necessary goods Income Elasticity of Demand for Some Commodities in the US Product Income Elasticity of Demand Total food 0.31 Food away from home 0.53 Fresh fruits 0.25 Fish 0.20 Beef 0.10 Pork 0.04 Managerial decisions and elasticities You are a marketing manager and your costs have increased (energy, salaries) reducing your net revenues. You think about increasing your price but need to know the effect on sales, total and net revenues. You are a supermarket manager and you want to offer a 10% price cut in margarine this week. You want to know how much more margarine you need to have to satisfy your clients. Example of price elasticity use Your supermarket is selling 1000 containers of margarine a week at $ 1.50 each. You know that the own price elasticity for margarine is –0.8. If you decide to reduce the price by 10%, how many more margarine containers would you be selling that week? Example.. Since E= Q/Q / P/P = -0.8, and P/P=-0.10 Q/Q= -0.8 *- 0.10 Q=-0.8*-0.10 * 1000 = 80 more margarine containers to be sold What would be the total revenue gain? Revenue without price reduction= 1000*1.50 = 1500 New revenue= 1080 *1.35 = 1458 Was it a good decision? Forecasting price using price elasticity of demand Example using Price Elasticity of Demand Elasticity of Demand for Beef: Ed = − 0.67 Beef disappeared in quarter 3 of 2009 = 6.64 bill. Lbs Beef disappeared in quarter 4 of 2009 = 6.45 bill. Lbs % change in Eq. Quant. Dem. = ∆Q/Q = (6.45-6.64)/6.64 = − 0.029 Ed = (∆Q/Q ) /(∆P/P ) − 0.67 = − 0.029 /(∆P/P ) ∆P/P = − 0.029/ − 0.67 = 0.043 If price of beef in quarter 3 of 2010 is 277 cents/lb, then forecasted beef price for quarter 4 of 2010 is P* = 277 + 0.043×277 = 277+11.98 = 288.98 cents/lb. Equilibrium Displacement Model The concept if elasticity is also useful for forecasting changes in prices and quantities resulting from supply and demand curve shifts. Supply of pork decreases due to a tougher regulation on manure treatment and disposal Demand for pork increases due to an increase in the price of beef If we know the elasticities of demand and supply, we can calculate the changes in price and quantity demanded (or supplied) by incorporating elasticities into a model called an equilibrium displacement model. Equilibrium Displacement Model… Demand Equation Qd = (price, income, tastes and preferences, expectations, and prices of other goods) Demand changes (shifts) if any right hand side factor other than price of the commodity changes Supply Equation Qs = (price, input prices, prices of other goods, expectations, technological change, number of producers) Supply changes (shifts) if any right hand side factor other than price of the commodity changes Equilibrium Displacement Model… Equilibrium: Price and quantity are determined by the intersection of the demand and supply curves, where Qd = Qs (i.e., quantity demanded = quantity supplied) Equilibrium changes (gets displaced) if demand and/or supply changes because of changes in any right hand side demand and/or supply factor other than the price of the commodity. Suppose that an increase in the production cost will shift the supply curve to the left. As a result, new equilibrium price will be higher and equilibrium quantity will be lower. Equilibrium Displacement Model… We can formalize this concept and write the demand equation as %∆Qd = Ed×(%∆P) + Sd Where, Ed is the elasticity of demand Sd represents any exogenous demand shift – the percentage change in quantity demanded due to a change in the value of any right hand side variable other than own price Similarly, we can write the supply equation as %∆Qs = Es×(%∆P) + Ss Where, Es is the elasticity of supply Where, Ss represents any exogenous demand shift – the percentage change in quantity supplied due to a change in the value of any right hand side variable other than own price Equilibrium Displacement Model… In equilibrium the percentage change in quantity demanded must equal the percentage changed in quantity supplied, i.e., %∆Qd = %∆Qs Ed×(%∆P) + Sd = Es×(%∆P) + Ss Es×(%∆P) − Ed×(%∆P) = Sd − Ss [Es − Ed]×(%∆P) = Sd − Ss %∆P = [Sd − Ss]/[Es − Ed] Thus, once we know the values of percentage change in demand and/or supply because of an exogenous shock, we can easily calculate the percentage change in price Equilibrium Displacement Model… %∆P = [Sd − Ss]/[Es − Ed] Note that the denominator [Es − Ed] is always positive, because Es is positive and Ed is negative. If Sd >0 and Ss =0, then %∆P >0 If Sd =0 and Ss >0, then %∆P <0 If Sd >0 and Ss >0 and Sd > Ss , then %∆P >0 If Sd >0 and Ss >0 and Sd < Ss , then %∆P <0 Once we calculate the percentage change in price %∆P, we can substitute that value into the demand or supply equation to calculate the percentage change in quantity demanded or supplied %∆Qd = Ed×(%∆P) + Sd = Ed× [Sd − Ss]/[Es − Ed] + Sd Equilibrium Displacement Model… General steps in solving an equilibrium displacement model Step 1: Determine the values of percentage change in demand (Sd) and supply (Ss) Step 2: Specify the % changes in quantity demanded and supplied as %∆Qd = Ed×(%∆P) + Sd %∆Qs = Es×(%∆P) + Ss Step 3: Set %∆Qd = %∆Qs and solve for %∆P Step 4: Plug the calculated value for %∆P into the %∆Qd or %∆Qs equation to calculate the percentage change in quantity. Equilibrium Displacement Model… Example: Impact of manure regulation in the pork market Elasticity of pork demand, Ed = −1.96 Elasticity of pork supply, Es = 2.15 Because of a newly introduced tighter manure regulation, pork supply falls by 4.3%, i.e., Ss= − 4.3% There is no change in pork demand, i.e., Sd = 0 Suppose that the current price for pork is $100/cwt and the quantity bought and sold is 1,000 cwt per day. What would be the new equilibrium price and quantity? Equilibrium Displacement Model… % change in pork price and quantity due to the supply shift %∆P = [Sd − Ss]/[Es − Ed] = [0 − (− 4.3)]/[2.15 − (− 1.95)] = 4.3/4.10 = 1.05% Thus the pork price increases by 1.05% So, the new equilibrium price would be P* = (1+ %∆P)*P = (1+0.0105)*50 = 50.525/cwt. %∆Qd = Ed× [Sd − Ss]/[Es − Ed] + Sd = (− 1.95) × [0 − (− 4.3)]/[2.15 − (− 1.95)] + 0 = (− 1.95) × 4.3/4.10 = (− 1.95) × 1.05 = 2.05% Thus the pork price decreases by 2.05% So, the new equilibrium quantity would be, Q* = (1+ %∆Q)*Q = (1- 0.0205)*1000 = 979.5 cwt.

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