Intermediate Microeconomics, Economics 4011Lecture 4: Demand and Slutsky Equation6Demand 2010 W. W. Norton & Company, Inc.Properties of Demand FunctionsuComparative statics analysis of ordinary demand functions - the study of how ordinary demands x1*(p1,p2,y) and x2*(p1,p2,y) change as prices p1, p2 and income y change. 2010 W. W. Norton & Company, Inc.Own-Price ChangesuHow does x1*(p1,p2,y) change as p1 changes, holding p2 and y constant?uSuppose only p1 increases, from p1 to p1 and then to p1. 2010 W. W. Norton & Company, Inc. x1x2p1 = p1Fixed p2 and y.p1x1 + p2x2 = yOwn-Price Changes 2010 W. W. Norton & Company, Inc.Own-Price Changesx1x2p1= p1p1 = p1Fixed p2 and y.p1x1 + p2x2 = y 2010 W. W. Norton & Company, Inc.Own-Price Changesx1x2p1= p1p1=p1Fixed p2 and y.p1 = p1p1x1 + p2x2 = y 2010 W. W. Norton & Company, Inc.Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)Own-Price Changesp1 = p1Fixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)p1x1*(p1)p1x1*Own-Price ChangesFixed p2 and y.p1 = p1 2010 W. W. Norton & Company, Inc.x1*(p1)p1x1*(p1)p1p1 = p1x1*Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)p1x1*(p1)p1p1 = p1x1*Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)p1p1x1*Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)p1p1p1 = p1x1*Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)p1p1p1 = p1x1*Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)x1*(p1)p1p1p1x1*Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)x1*(p1)p1p1p1x1*Own-Price ChangesOrdinarydemand curvefor commodity 1Fixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)x1*(p1)p1p1p1x1*Own-Price ChangesOrdinarydemand curvefor commodity 1Fixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)p1x1*(p1)x1*(p1)x1*(p1)p1p1p1x1*Own-Price ChangesOrdinarydemand curvefor commodity 1p1 price offer curveFixed p2 and y. 2010 W. W. Norton & Company, Inc.Own-Price ChangesuThe curve containing all the utility-maximizing bundles traced out as p1 changes, with p2 and y constant, is the p1- price offer curve.uThe plot of the x1-coordinate of the p1- price offer curve against p1 is the ordinary demand curve for commodity 1. 2010 W. W. Norton & Company, Inc.Own-Price ChangesuWhat does a p1 price-offer curve look like for Cobb-Douglas preferences? 2010 W. W. Norton & Company, Inc.Own-Price ChangesuWhat does a p1 price-offer curve look like for Cobb-Douglas preferences?uTakeThen the ordinary demand functions for commodities 1 and 2 are 2010 W. W. Norton & Company, Inc.Own-Price ChangesandNotice that x2* does not vary with p1 so thep1 price offer curve is 2010 W. W. Norton & Company, Inc.Own-Price ChangesandNotice that x2* does not vary with p1 so thep1 price offer curve is flat 2010 W. W. Norton & Company, Inc.Own-Price ChangesandNotice that x2* does not vary with p1 so thep1 price offer curve is flat and the ordinarydemand curve for commodity 1 is a 2010 W. W. Norton & Company, Inc.Own-Price ChangesandNotice that x2* does not vary with p1 so thep1 price offer curve is flat and the ordinarydemand curve for commodity 1 is a rectangular hyperbola. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)Own-Price ChangesFixed p2 and y. 2010 W. W. Norton & Company, Inc.x1*(p1)x1*(p1)x1*(p1)p1x1*Own-Price ChangesOrdinarydemand curvefor commodity 1 isFixed p2 and y. 2010 W. W. Norton & Company, Inc.Own-Price ChangesuWhat does a p1 price-offer curve look like for a perfect-complements utility function? 2010 W. W. Norton & Company, Inc.Own-Price ChangesuWhat does a p1 price-offer curve look like for a perfect-complements utility function?Then the ordinary demand functionsfor commodities 1 and 2 are 2010 W. W. Norton & Company, Inc.Own-Price Changes 2010 W. W. Norton & Company, Inc.Own-Price ChangesWith p2 and y fixed, higher p1 causessmaller x1* and x2*. 2010 W. W. Norton & Company, Inc.Own-Price ChangesWith p2 and y fixed, higher p1 causessmaller x1* and x2*.As 2010 W. W. Norton & Company, Inc.Own-Price ChangesWith p2 and y fixed, higher p1 causessmaller x1* and x2*.AsAs 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx1x2 2010 W. W. Norton & Company, Inc.p1x1*Fixed p2 and y.Own-Price Changesx1x2p1 p1 = p1 y/p2 2010 W. W. Norton & Company, Inc.p1x1*Fixed p2 and y.Own-Price Changesx1x2p1p1p1 = p1y/p2 2010 W. W. Norton & Company, Inc.p1x1*Fixed p2 and y.Own-Price Changesx1x2p1p1p1p1 = p1y/p2 2010 W. W. Norton & Company, Inc.p1x1*Ordinarydemand curvefor commodity 1 isFixed p2 and y.Own-Price Changesx1x2p1p1p1y/p2 2010 W. W. Norton & Company, Inc.Own-Price ChangesuWhat does a p1 price-offer curve look like for a perfect-substitutes utility function?Then the ordinary demand functionsfor commodities 1 and 2 are 2010 W. W. Norton & Company, Inc.Own-Price Changesand 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1Fixed p2 and y.p1 = p1 p2 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p1 = p1 p2 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p1 = p1 = p2 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p1 = p1 = p2 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p1 = p1 = p2 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p1 = p1 = p2p2 = p1 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p1p2 = p1 2010 W. W. Norton & Company, Inc.Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1p2 = p1p1p1 price offer curveOrdinarydemand curvefor commodity 1 2010 W. W. Norton & Company, Inc.Own-Price ChangesuUsually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?”uBut we could also ask the inverse question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?” 2010 W. W. Norton & Company, Inc.Own-Price Changesp1x1*p1Given p1, what quantity isdemanded of commodity 1? 2010 W. W. Norton & Company, Inc.Own-Price Changesp1x1*p1Given p1, what quantity isdemanded of commodity 1?Answer: x1 units.x1 2010 W. W. Norton & Company, Inc.Own-Price Changesp1x1*x1Given p1, what quantity isdemanded of commodity 1?Answer: x1 units.The inverse question is:Given x1 units are demanded, what is the price of commodity 1? 2010 W. W. Norton & Company, Inc.Own-Price Changesp1x1*p1x1Given p1, what quantity isdemanded of commodity 1?Answer: x1 units.The inverse question is:Given x1 units are demanded, what is the price of commodity 1? Answer: p1 2010 W. W. Norton & Company, Inc.Own-Price ChangesuTaking quantity demanded as given and then asking what must be price describes the inverse demand function of a commodity. 2010 W. W. Norton & Company, Inc.Own-Price ChangesA Cobb-Douglas example:is the ordinary demand function andis the inverse demand function. 2010 W. W. Norton & Company, Inc.Own-Price ChangesA perfect-complements example:is the ordinary demand function andis the inverse demand function. 2010 W. W. Norton & Company, Inc.Income ChangesuHow does the value of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant? 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y y 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y y 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curve 2010 W. W. Norton & Company, Inc.Income ChangesuA plot of quantity demanded against income is called an Engel curve. 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curve 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curvex1*yx1x1x1yyy 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curvex1*yx1x1x1yyyEngelcurve;good 1 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curvex2*yx2x2x2yyy 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curvex2*yx2x2x2yyyEngelcurve;good 2 2010 W. W. Norton & Company, Inc.Income ChangesFixed p1 and p2.y y yx1x1x1x2x2x2Incomeoffer curvex1*x2*yyx1x1x1x2x2x2yyyyyyEngelcurve;good 2Engelcurve;good 1 2010 W. W. Norton & Company, Inc.Income Changes and Cobb-Douglas PreferencesuAn example of computing the equations of Engel curves; the Cobb-Douglas case.uThe ordinary demand equations are 2010 W. W. Norton & Company, Inc.Income Changes and Cobb-Douglas PreferencesRearranged to isolate y, these are:Engel curve for good 1Engel curve for good 2 2010 W. W. Norton & Company, Inc.Income Changes and Cobb-Douglas Preferencesyyx1*x2*Engel curvefor good 1Engel curvefor good 2 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Complementary PreferencesuAnother example of computing the equations of Engel curves; the perfectly-complementary case.uThe ordinary demand equations are 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Complementary PreferencesRearranged to isolate y, these are:Engel curve for good 1Engel curve for good 2 2010 W. W. Norton & Company, Inc.Fixed p1 and p2.Income Changesx1x2 2010 W. W. Norton & Company, Inc.Income Changesx1x2y y yFixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changesx1x2y y yFixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changesx1x2y y yx1x1x2x2x2x1Fixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changesx1x2y y yx1x1x2x2x2x1x1*yyyyEngelcurve;good 1x1x1x1Fixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changesx1x2y y yx1x1x2x2x2x1x2*yx2x2x2yyyEngelcurve;good 2Fixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changesx1x2y y yx1x1x2x2x2x1x1*x2*yyx2x2x2yyyyyyEngelcurve;good 2Engelcurve;good 1x1x1x1Fixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changesx1*x2*yyx2x2x2yyyyyyx1x1x1Engelcurve;good 2Engelcurve;good 1Fixed p1 and p2. 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Substitutable PreferencesuAnother example of computing the equations of Engel curves; the perfectly-substitution case.uThe ordinary demand equations are 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Substitutable Preferences 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Substitutable PreferencesSuppose p1 p2. Then 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Substitutable PreferencesSuppose p1 p2. Thenand 2010 W. W. Norton & Company, Inc.Income Changes and Perfectly-Substitutable PreferencesSuppose p1 0.uThat is, the consumers MRS is the same anywhere on a straight line drawn from the origin.(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2)pppp 2010 W. W. Norton & Company, Inc.Income Effects - A Nonhomothetic ExampleuQuasilinear preferences are not homothetic.uFor example, 2010 W. W. Norton & Company, Inc.Quasi-linear Indifference Curvesx2x1Each curve is a vertically shifted copy of the others.Each curve intersectsboth axes. 2010 W. W. Norton & Company, Inc.Income Changes; Quasilinear Utilityx2x1x1 2010 W. W. Norton & Company, Inc.Income Changes; Quasilinear Utilityx2x1x1x1*yx1Engelcurveforgood 1 2010 W. W. Norton & Company, Inc.Income Changes; Quasilinear Utilityx2x1x1x2*yEngelcurveforgood 2 2010 W. W. Norton & Company, Inc.Income Changes; Quasilinear Utilityx2x1x1x1*x2*yyx1Engelcurveforgood 2Engelcurveforgood 1 2010 W. W. Norton & Company, Inc.Income EffectsuA good for which quantity demanded rises with income is called normal.uTherefore a normal goods Engel curve is positively sloped. 2010 W. W. Norton & Company, Inc.Income EffectsuA good for which quantity demanded falls as income increases is called income inferior.uTherefore an income inferior goods Engel curve is negatively sloped. 2010 W. W. Norton & Company, Inc.Income Changes; Goods1 & 2 Normalx1x1x1x2x2x2Incomeoffer curvex1*x2*yyx1x1x1x2x2x2yyyyyyEngelcurve;good 2Engelcurve;good 1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1Incomeoffer curve 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1x1*yEngel curvefor good 1 2010 W. W. Norton & Company, Inc.Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1x1*x2*yyEngel curvefor good 2Engel curvefor good 1 2010 W. W. Norton & Company, Inc.Ordinary GoodsuA good is called ordinary if the quantity demanded of it always increases as its own price decreases. 2010 W. W. Norton & Company, Inc.Ordinary GoodsFixed p2 and y.x1x2 2010 W. W. Norton & Company, Inc.Ordinary GoodsFixed p2 and y.x1x2p1 price offer curve 2010 W. W. Norton & Company, Inc.Ordinary GoodsFixed p2 and y.x1x2p1 price offer curvex1*Downward-sloping demand curve Good 1 isordinaryp1 2010 W. W. Norton & Company, Inc.Giffen GoodsuIf, for some values of its own price, the quantity demanded of a good rises as its own-price increases then the good is called Giffen. 2010 W. W. Norton & Company, Inc.Ordinary GoodsFixed p2 and y.x1x2 2010 W. W. Norton & Company, Inc.Ordinary GoodsFixed p2 and y.x1x2p1 price offer curve 2010 W. W. Norton & Company, Inc.Ordinary GoodsFixed p2 and y.x1x2p1 price offer curvex1*Demand curve has a positively sloped part Good 1 isGiffenp1 2010 W. W. Norton & Company, Inc.Cross-Price EffectsuIf an increase in p2increases demand for commodity 1 then commodity 1 is a gross substitute for commodity 2. reduces demand for commodity 1 then commodity 1 is a gross complement for commodity 2. 2010 W. W. Norton & Company, Inc.Cross-Price EffectsA perfect-complements example:soTherefore commodity 2 is a grosscomplement for commodity 1. 2010 W. W. Norton & Company, Inc.Cross-Price Effectsp1x1*p1p1p1Increase the price ofgood 2 from p2 to p2and 2010 W. W. Norton & Company, Inc.Cross-Price Effectsp1x1*p1p1p1Increase the price ofgood 2 from p2 to p2and the demand curvefor good 1 shifts inwards- good 2 is acomplement for good 1. 2010 W. W. Norton & Company, Inc.Cross-Price EffectsA Cobb- Douglas example:so 2010 W. W. Norton & Company, Inc.Cross-Price EffectsA Cobb- Douglas example:soTherefore commodity 1 is neither a grosscomplement nor a gross substitute forcommodity 2.8Slutsky EquationEffects of a Price ChangeWhat happens when a commoditys price decreases?Substitution effect: the commodity is relatively cheaper, so consumers substitute it for now relatively more expensive other commodities.Effects of a Price ChangeIncome effect: the consumers budget of $y can purchase more than before, as if the consumers income rose, with consequent income effects on quantities demanded.Effects of a Price Changex2x1Original choiceConsumers budget is $y.Effects of a Price Changex1Lower price for commodity 1pivots the constraint outwards.Consumers budget is $y.x2Effects of a Price Changex1Lower price for commodity 1pivots the constraint outwards.Consumers budget is $y.x2Now only $y are needed to buy the original bundle at the new prices, as if the consumers income has increased by $y - $y.Effects of a Price ChangeChanges to quantities demanded due to this extra income are the income effect of the price change.Effects of a Price ChangeSlutsky discovered that changes to demand from a price change are always the sum of a pure substitution effect and an income effect.Real Income ChangesSlutsky asserted that if, at the new prices,less income is needed to buy the original bundle then “real income” is increasedmore income is needed to buy the original bundle then “real income” is decreasedReal Income Changesx1x2Original budget constraint and choiceReal Income Changesx1x2Original budget constraint and choiceNew budget constraintReal Income Changesx1x2Original budget constraint and choiceNew budget constraint; real income has risenReal Income Changesx1x2Original budget constraint and choiceReal Income Changesx1x2Original budget constraint and choiceNew budget constraintReal Income Changesx1x2Original budget constraint and choiceNew budget constraint; real income has fallenPure Substitution EffectSlutsky isolated the change in demand due only to the change in relative prices by asking “What is the change in demand when the consumers income is adjusted so that, at the new prices, she can only just buy the original bundle?”Pure Substitution Effect Onlyx2x1x2x1Pure Substitution Effect Onlyx2x1x2x1Pure Substitution Effect Onlyx2x1x2x1Pure Substitution Effect Onlyx2x1x2x2x1x1Pure Substitution Effect Onlyx2x1x2x2x1x1Pure Substitution Effect Onlyx2x1x2x2x1x1Lower p1 makes good 1 relativelycheaper and causes a substitutionfrom good 2 to good 1.Pure Substitution Effect Onlyx2x1x2x2x1x1Lower p1 makes good 1 relativelycheaper and causes a substitutionfrom good 2 to good 1. (x1,x2) (x1,x2) is the pure substitution effect.And Now The Income Effectx2x1x2x2x1x1(x1,x2)And Now The Income Effectx2x1x2x2x1x1(x1,x2)The income effect is (x1,x2) (x1,x2).The Overall Change in Demandx2x1x2x2x1x1(x1,x2)The change to demand due to lower p1 is the sum of the income and substitution effects, (x1,x2) (x1,x2).calculating pivot incomeLet m be the amount of income that will keep original bundle affordableSince (x1, x2) is affordable at both (p1, p2, m) and (p1, p2, m), we have m= p1x1 + p2 x2 and m= p1 x1 + p2 x2subtracting the second equation from the first givesm-m = x1 (p1 p1)The change in money income necessary to make the old bundle affordable at the new prices is just the original amount of consumption of good 1 times the change in pricesSlutskys Effects for Normal GoodsMost goods are normal (i.e. demand increases with income).The substitution and income effects reinforce each other when a normal goods own price changes.Slutskys Effects for Normal Goodsx2x1x2x2x1x1(x1,x2)Good 1 is normal becausehigher income increasesdemand x2x1x2x2x1x1(x1,x2)Good 1 is normal becausehigher income increasesdemand, so the income and substitution effects reinforce each other. Slutskys Effects for Normal GoodsSlutskys Effects for Normal GoodsSince both the substitution and income effects increase demand when own-price falls, a normal goods ordinary demand curve slopes down.The Law of Downward-Sloping Demand therefore always applies to normal goods.Slutskys Effects for Income-Inferior GoodsSome goods are income-inferior (i.e. demand is reduced by higher income).The substitution and income effects oppose each other when an income-inferior goods own price changes.Slutskys Effects for Income-Inferior Goodsx2x1x2x1Slutskys Effects for Income-Inferior Goodsx2x1x2x1Slutskys Effects for Income-Inferior Goodsx2x1x2x1Slutskys Effects for Income-Inferior Goodsx2x1x2x2x1x1Slutskys Effects for Income-Inferior Goodsx2x1x2x2x1x1The pure substitution effect is as fora normal good. But, .Slutskys Effects for Income-Inferior Goodsx2x1x2x2x1x1(x1,x2)The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. Slutskys Effects for Income-Inferior Goodsx2x1x2x2x1x1(x1,x2)The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. Good 1 is income-inferior because an increase to income causes demand to fall. Slutskys Effects for Income-Inferior Goodsx2x1x2x2x1x1(x1,x2)The overall changes to demand arethe sums of the substitution and income effects.Giffen GoodsIn rare cases of extreme income-inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to fall as own-price rises.Such goods are Giffen goods.Slutskys Effects for Giffen Goodsx2x1x2x1A decrease in p1 causes quantity demanded of good 1 to fall.Slutskys Effects for Giffen Goodsx2x1x2x1x1x2A decrease in p1 causes quantity demanded of good 1 to fall.Slutskys Effects for Giffen Goodsx2x1x2x2x1x1x1x2Substitution effectIncome effectA decrease in p1 causes quantity demanded of good 1 to fall.Slutskys Effects for Giffen GoodsSlutskys decomposition of the effect of a price change into a pure substitution effect and an income effect thus explains why the Law of Downward-Sloping Demand is violated for extremely income-inferior goods.