中级微观-第六次课件.ppt

Chapter SixDemand需求需求Properties of Demand FunctionsuComparative statics analysis （（比较比较静态分析）静态分析）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.StructureuOwn-price changes–Price offer curve （（价格提供曲线）价格提供曲线）–Ordinary demand curve–Inverse demand curve （（反需求函数）反需求函数）uIncome changes–Income offer curve （（收入提供曲线）收入提供曲线）–Engel curve （（恩格尔曲线）恩格尔曲线）uCross-price effects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’’’. x1x2p1 = p1’Fixed p2 and y.p1x1 + p2x2 = yOwn-Price ChangesOwn-Price Changesx1x2p1= p1’’p1 = p1’Fixed p2 and y.p1x1 + p2x2 = yOwn-Price Changesx1x2p1= p1’’p1=p1’’’Fixed p2 and y.p1 = p1’p1x1 + p2x2 = yx1*(p1’)Own-Price Changesp1 = p1’Fixed p2 and y.x1*(p1’)p1x1*(p1’)p1’x1*Own-Price ChangesFixed p2 and y.p1 = p1’x1*(p1’)x1*(p1’’)p1x1*(p1’)p1’p1 = p1’’x1*Own-Price ChangesFixed p2 and y.x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’)p1’p1’’x1*Own-Price ChangesFixed p2 and y.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’)p1’p1’’p1 = p1’’’x1*Own-Price ChangesFixed p2 and y.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’’)x1*(p1’’)p1’p1’’p1’’’x1*Own-Price ChangesOrdinarydemand curvefor commodity 1Fixed p2 and y.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’’)x1*(p1’’)p1’p1’’p1’’’x1*Own-Price ChangesOrdinarydemand curvefor commodity 1p1 price offer curveFixed p2 and y.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.Own-Price ChangesuWhat does a p1 price-offer curve look like for Cobb-Douglas preferences?uThen the ordinary demand functions for commodities 1 and 2 areTake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.x1*(p1’’’)x1*(p1’)x1*(p1’’)Own-Price ChangesFixed p2 and y.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*Own-Price ChangesOrdinarydemand curvefor commodity 1 isFixed p2 and y.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Own-Price ChangesWith p2 and y fixed, higher p1 causessmaller x1* and x2*.AsAsFixed p2 and y.Own-Price Changesx1x2p1x1*Fixed p2 and y.Own-Price Changesx1x2p1’ ’p1 = p1’ ’ ’y/p2p1x1*Fixed p2 and y.Own-Price Changesx1x2p1’p1’’p1 = p1’’’’’’’’y/p2p1x1*Fixed p2 and y.Own-Price Changesx1x2p1’p1’’p1’’’p1 = p1’’’’’’’’’’’’y/p2p1x1*Ordinarydemand curvefor commodity 1 isFixed p2 and y.Own-Price Changesx1x2p1’p1’’p1’’’y/p2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Own-Price ChangesandFixed p2 and y.Own-Price Changesx2x1Fixed p2 and y.p1 = p1’ < p2’Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p1 = p1’ < p2’’Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p1 = p1’’ = p2Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p1 = p1’’ = p2Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p1 = p1’’ = p2’’Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p1 = p1’’ = p2p2 = p1’’Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p1’’’p2 = p1’’p1 = p1”’ > p2Fixed p2 and y.Own-Price Changesx2x1p1x1*Fixed p2 and y.p1’p2 = p1’’p1’’’p1 price offer curveOrdinarydemand curvefor commodity 1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?”Own-Price Changesp1x1*p1’Given p1’, what quantity isdemanded of commodity 1?Own-Price Changesp1x1*p1’Given p1’, what quantity isdemanded of commodity 1?Answer: x1’ units.x1’Own-Price Changesp1x1*x1’Given 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?Own-Price Changesp1x1*p1’x1’Given 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’Own-Price ChangesuTaking quantity demanded as given and then asking what must be price describes the inverse demand function of a commodity.Own-Price ChangesA Cobb-Douglas example:is the ordinary demand function andis the inverse demand function.Own-Price ChangesA perfect-complements example:is the ordinary demand function andis the inverse demand function.Meaning of the Inverse Demand FunctionuAt optimal choiceuOruIf taking good 2 as money on other goods, then p2=1 and p1=MRS.uThis is the marginal willingness to pay.Income ChangesuHow does the value of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant?Income ChangesFixed p1 and p2.y’ < y’’ < y’’’Income ChangesFixed p1 and p2.y’ < y’’ < y’’’Income ChangesFixed p1 and p2.y’ < y’’ < y’’’x1’’’x1’’x1’x2’’’x2’’x2’Income ChangesFixed p1 and p2.y’ < y’’ < y’’’x1’’’x1’’x1’x2’’’x2’’x2’Incomeoffer curveIncome ChangesuA plot of quantity demanded against income is called an Engel curve.Income ChangesFixed p1 and p2.y’ < y’’ < y’’’x1’’’x1’’x1’x2’’’x2’’x2’Incomeoffer curveIncome ChangesFixed p1 and p2.y’ < y’’ < y’’’x1’’’x1’’x1’x2’’’x2’’x2’Incomeoffer curvex1*yx1’’’x1’’x1’y’y’’y’’’Engelcurve;good 1Income ChangesFixed p1 and p2.y’ < y’’ < y’’’x1’’’x1’’x1’x2’’’x2’’x2’Incomeoffer curvex2*yx2’’’x2’’x2’y’y’’y’’’Engelcurve;good 2Income ChangesFixed p1 and p2.y’ < y’’ < y’’’x1’’’x1’’x1’x2’’’x2’’x2’Incomeoffer curvex1*x2*yyx1’’’x1’’x1’x2’’’x2’’x2’y’y’’y’’’y’y’’y’’’Engelcurve;good 2Engelcurve;good 1Income Changes and Cobb-Douglas PreferencesuAn example of computing the equations of Engel curves; the Cobb-Douglas case.uThe ordinary demand equations areIncome Changes and Cobb-Douglas PreferencesRearranged to isolate y, these are:Engel curve for good 1Engel curve for good 2Income Changes and Cobb-Douglas Preferencesyyx1*x2*Engel curvefor good 1Engel curvefor good 2Income Changes and Perfectly-Complementary PreferencesuAnother example of computing the equations of Engel curves; the perfectly-complementary case.uThe ordinary demand equations areIncome Changes and Perfectly-Complementary PreferencesRearranged to isolate y, these are:Engel curve for good 1Engel curve for good 2Fixed p1 and p2.Income Changesx1x2Income Changesx1x2y’ < y’’ < y’’’Fixed p1 and p2.Income Changesx1x2y’ < y’’ < y’’’Fixed p1 and p2.Income Changesx1x2y’ < y’’ < y’’’x1’’x1’x2’’’x2’’x2’x1’’’Fixed p1 and p2.Income Changesx1x2y’ < y’’ < y’’’x1’’x1’x2’’’x2’’x2’x1’’’x1*yy’y’’y’’’Engelcurve;good 1x1’’’x1’’x1’Fixed p1 and p2.Income Changesx1x2y’ < y’’ < y’’’x1’’x1’x2’’’x2’’x2’x1’’’x2*yx2’’’x2’’x2’y’y’’y’’’Engelcurve;good 2Fixed p1 and p2.Income Changesx1x2y’ < y’’ < y’’’x1’’x1’x2’’’x2’’x2’x1’’’x1*x2*yyx2’’’x2’’x2’y’y’’y’’’y’y’’y’’’Engelcurve;good 2Engelcurve;good 1x1’’’x1’’x1’Fixed p1 and p2.Income Changesx1*x2*yyx2’’’x2’’x2’y’y’’y’’’y’y’’y’’’x1’’’x1’’x1’Engelcurve;good 2Engelcurve;good 1Fixed p1 and p2.Income Changes and Perfectly-Substitutable PreferencesuAnother example of computing the equations of Engel curves; the perfectly-substitution case.uThe ordinary demand equations areIncome Changes and Perfectly-Substitutable PreferencesSuppose p1 < p2. ThenandandIncome Changes and Perfectly-Substitutable Preferencesyyx1*x2*0Engel curvefor good 1Engel curvefor good 2Income ChangesuIn every example so far the Engel curves have all been straight lines?Q: Is this true in general?uA: No. Engel curves are straight lines if the consumer’s preferences are homothetic.Homotheticity （位似偏好）uA consumer’s preferences are homothetic if and only iffor every k > 0.uThat is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.ÛÛ(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2)ppppIncome Effects -- A Nonhomothetic ExampleuQuasilinear preferences are not homothetic.For example,uOptimal interior consumption:Quasi-linear Indifference Curvesx2x1Each curve is a vertically shifted copy of the others.Each curve intersectsboth axes.Income Changes; Quasilinear Utilityx2x1x1~Income Changes; Quasilinear Utilityx2x1x1~x1*yx1~Engelcurveforgood 1Income Changes; Quasilinear Utilityx2x1x1~x2*yEngelcurveforgood 2Income Changes; Quasilinear Utilityx2x1x1~x1*x2*yyx1~Engelcurveforgood 2Engelcurveforgood 1Income EffectsuA good for which quantity demanded rises with income is called normal （（正常品）正常品）.uTherefore a normal good’s Engel curve is positively sloped.Income EffectsuA good for which quantity demanded falls as income increases is called income inferior （（劣质品）劣质品）.uTherefore an income inferior good’s Engel curve is negatively sloped.Income Changes; Goods1 & 2 Normalx1’’’x1’’x1’x2’’’x2’’x2’Incomeoffer curvex1*x2*yyx1’’’x1’’x1’x2’’’x2’’x2’y’y’’y’’’y’y’’y’’’Engelcurve;good 2Engelcurve;good 1Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1Incomeoffer curveIncome Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1x1*yEngel curvefor good 1Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferiorx2x1x1*x2*yyEngel curvefor good 2Engel curvefor good 1Ordinary Goods （一般商品）uA good is called ordinary if the quantity demanded of it always increases as its own price decreases.Ordinary GoodsFixed p2 and y.x1x2p1 price offer curveOrdinary GoodsFixed p2 and y.x1x2p1 price offer curvex1*Downward-sloping demand curve Good 1 isordinaryÛÛp1Giffen Goods （吉芬商品）uIf, 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.Ordinary GoodsFixed p2 and y.x1x2p1 price offer curveOrdinary GoodsFixed p2 and y.x1x2p1 price offer curvex1*Demand curve has a positively sloped part Good 1 isGiffenÛÛp1Cross-Price EffectsuIf an increase in p2–increases 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.Cross-Price EffectsA perfect-complements example:soTherefore commodity 2 is a grosscomplement for commodity 1.Cross-Price Effectsp1x1*p1’p1’’p1’’’’Increase the price ofgood 2 from p2’ to p2’’and Cross-Price Effectsp1x1*p1’p1’’p1’’’’’Increase the price ofgood 2 from p2’ to p2’’and the demand curvefor good 1 shifts inwards-- good 2 is acomplement for good 1. Cross-Price EffectsA Cobb- Douglas example:soTherefore commodity 1 is neither a grosscomplement nor a gross substitute forcommodity 2.。