Chapter 7ElasticityPrice Elasticity of DemandDistinguishing Values of eQ,PPrice Elasticity and Total ExpenditureSlide 6Slide 7Slide 8Income Elasticity of DemandCross-Price Elasticity of DemandRelationships Among ElasticitiesSlide 12Slide 13Slutsky Equation in ElasticitiesSlide 15Slide 16Slide 17HomogeneitySlide 19Cobb-Douglas ElasticitiesSlide 21Slide 22Slide 23Linear DemandSlide 25Slide 26Slide 27Slide 28Constant Elasticity FunctionsSlide 30Important Points to Note:Slide 32Slide 33Slide 34Chapter 7MARKET DEMAND AND ELASTICITYCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONElasticity•Suppose that a particular variable (B) depends on another variable (A)B = f(A…)•We define the elasticity of B with respect to A asBAABAABBABeAB// in change % in change %,–The elasticity shows how B responds (ceteris paribus) to a 1 percent change in APrice Elasticity of Demand•The most important elasticity is the price elasticity of demand–measures the change in quantity demanded caused by a change in the price of the goodQPPQPPQQPQePQ// in change % in change %,•eQ,P will generally be negative–except in cases of Giffen’s paradoxDistinguishing Values of eQ,PValue of eQ,P at a PointClassification of Elasticity at This PointeQ,P < -1ElasticeQ,P = -1Unit ElasticeQ,P > -1InelasticPrice Elasticity and Total Expenditure•Total expenditure on any good is equal tototal expenditure = PQ•Using elasticity, we can determine how total expenditure changes when the price of a good changesPrice Elasticity and Total Expenditure•Differentiating total expenditure with respect to P yieldsPQPQPPQ•Dividing both sides by Q, we getPQeQPPQQPPQ,/11Price Elasticity and Total Expenditure•Note that the sign of PQ/P depends on whether eQ,P is greater or less than -1–If eQ,P > -1, demand is inelastic and price and total expenditures move in the same direction–If eQ,P < -1, demand is elastic and price and total expenditures move in opposite directionsPQeQPPQQPPQ,/11Price Elasticity and Total ExpenditureResponses of PQDemand Price Increase Price DecreaseElastic Falls RisesUnit Elastic No Change No ChangeInelastic Rises FallsIncome Elasticity of Demand•The income elasticity of demand (eQ,I) measures the relationship between income changes and quantity changesQQQeQIIII in change % in change %,•Normal goods  eQ,I > 0–Luxury goods  eQ,I > 1•Inferior goods  eQ,I < 0Cross-Price Elasticity of Demand•The cross-price elasticity of demand (eQ,P’) measures the relationship between changes in the price of one good and and quantity changes in anotherQP'P'QP'QePQ in change % in change %',•Gross substitutes  eQ,P’ > 0•Gross complements  eQ,P’ < 0Relationships Among Elasticities•Suppose that there are only two goods (X and Y) so that the budget constraint is given byPXX + PYY = I•The individual’s demand functions areX = dX(PX,PY,I)Y = dY(PX,PY,I)Relationships Among Elasticities•Differentiation of the budget constraint with respect to I yields1IIYPXPYX •Multiplying each item by 11YYYPXXXPYXIIIIIIRelationships Among Elasticities•Since (PX · X)/I is the proportion of income spent on X and (PY · Y)/I is the proportion of income spent on Y,sXeX,I + sYeY,I = 1•For every good that has an income elasticity of demand less than 1, there must be goods that have income elasticities greater than 1Slutsky Equation in Elasticities•The Slutsky equation shows how an individual’s demand for a good responds to a change in priceIXXPXPXUXXconstant•Multiplying by PX /X yieldsXXXPXPPXXPPXXUXXXX1 constantISlutsky Equation in Elasticities•Multiplying the final term by I/I yieldsXXXPXPPXXPPXXUXXXXIII constantSlutsky Equation in Elasticities•A substitution elasticity shows how the compensated demand for X responds to proportional compensated price changes–it is the price elasticity of demand for movement along the compensated demand curveconstant, UXXSPXXPPXeXSlutsky Equation in Elasticities•Thus, the Slutsky relationship can be shown in elasticity formI,,, XXSPXPXeseeXX•It shows how the price elasticity of demand can be disaggregated into substitution and income components–Note that the relative size of the income component depends on the proportion of total expenditures devoted to the good (sX)Homogeneity•Remember that demand functions are homogeneous of degree zero in all prices and income•Euler’s theorem for homogenous functions shows that0II XXPPXPPXYYXHomogeneity•Dividing by X, we get0XXXPPXXPPXYYXII X•Using our definitions, this means that0I,,, XPXPXeeeYX•An equal percentage change in all prices and income will leave the quantity of X demanded unchangedCobb-Douglas Elasticities•The Cobb-Douglas utility function isU(X,Y) = XY•The demand functions for X and Y areXPXIYPYI•The elasticities can be calculated112 ,XXXXXXPXPPXPPXPPXeXIIICobb-Douglas Elasticities•Similar calculations show1I,Xe0YPXe,1 ,YPYe1I,Ye1YPYe,•Note thatIXPsXXIYPsYYCobb-Douglas Elasticities•Homogeneity can be shown for these elasticities•The elasticity version of the Slutsky equation can also be used0101 I,,, XPXPXeeeYXI,,, XXSPXPXeseeXX(1) ,SPXXe1 )-(1 ,SPXXeCobb-Douglas Elasticities•The price elasticity of demand for this compensated demand function is equal to (minus) the expenditure share of the other good •More generally )-(1 , XSPXseX where  is the elasticity of substitutionLinear DemandQ = a + bP + cI + dP’where:Q = quantity demandedP = price of the goodI = incomeP’ = price of other goodsa, b, c, d = various demand parametersLinear DemandQ = a + bP + cI + dP’•Assume that:Q/P = b  0 (no Giffen’s paradox)Q/I = c  0 (the good is a normal good)Q/P’ = d ⋛