,,,,,,,,,,,,,,Click to edit Master title style,Click to edit Master text styles,Second Level,Third Level,Fourth Level,Fifth Level,*,*,*,Chapter Six,Demand,需求函數(shù)的靜態(tài)比較分析,Chapter SixDemand,1,What Do We Do in This Chapter?,We conduct comparative statics analysis,of ordinary demand functions -- the study of how ordinary demands x,1,*(p,1,,p,2,,y) and x,2,*(p,1,,p,2,,y) change as prices p,1,, p,2,and income y change.,Theoretically, nothing new.,What Do We Do in This Chapter?,2,Own-Price Changes,How does x,1,*(p,1,,p,2,,y) change as p,1,changes, holding p,2,and y constant?,Suppose only p,1,increases, from p,1,’ to p,1,’’ and then to p,1,’’’.,Own-Price ChangesHow does x1*(,3,,,,x,1,*(p,1,’’’),x,1,*(p,1,’),x,1,*(p,1,’’),,,,,p,1,x,1,*(p,1,’),x,1,*(p,1,’’’),x,1,*(p,1,’’),p,1,’,p,1,’’,p,1,’’’,,,,,,,x,1,*,Own-Price Changes,Ordinarydemand curvefor commodity 1,Fixed p,2,and y.,x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x,4,,,,x,1,*(p,1,’’’),x,1,*(p,1,’),x,1,*(p,1,’’),,,,,p,1,x,1,*(p,1,’),x,1,*(p,1,’’’),x,1,*(p,1,’’),p,1,’,p,1,’’,p,1,’’’,,,,,,,x,1,*,Own-Price Changes,Ordinarydemand curvefor commodity 1,,p,1,price offer curve,Fixed p,2,and y.,x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x,5,Own-Price Changes,The curve containing all the utility-maximizing bundles traced out as p,1,changes, with p,2,and y constant, is the,p,1,- price offer curve,.,The plot of the x,1,-coordinate of the p,1,- price offer curve against p,1,is the,ordinary,demand curve for commodity 1.,Own-Price ChangesThe curve con,6,The Case of,Cobb-Douglas Utility Function,TakeThen the ordinary demand functions for commodities 1 and 2 are,The Case of Cobb-Douglas Utili,7,Own-Price Changes,and,Notice that x,2,* does not vary with p,1,so thep,1,price offer curve is,flat,and the ordinarydemand curve for commodity 1 is a,,rectangular hyperbola,.,Own-Price ChangesandNotice tha,8,x,1,*(p,1,’’’),x,1,*(p,1,’),x,1,*(p,1,’’),,,,Own-Price Changes,Fixed p,2,and y.,,,,x1*(p1’’’)x1*(p1’)x1*(p1’’)Own,9,x,1,*(p,1,’’’),x,1,*(p,1,’),x,1,*(p,1,’’),,,,p,1,,,,x,1,*,Own-Price Changes,Ordinarydemand curvefor commodity 1 is,Fixed p,2,and y.,,,,,,,,x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x,10,The Case of Perfect-Complements Utility Function,What does a p,1,price-offer curve look like for a perfect-complements utility function?,Then the ordinary demand functionsfor commodities 1 and 2 are,The Case of Perfect-Complement,11,The Case of Perfect-Complements Utility Function,The Case of Perfect-Complement,12,Own-Price Changes,With p,2,and y fixed, higher p,1,causessmaller x,1,* and x,2,*.,As,As,Own-Price ChangesWith p2 and y,13,p,1,x,1,*,Ordinarydemand curvefor commodity 1 is,Fixed p,2,and y.,Own-Price Changes,x,1,x,2,,,,p,1,’,p,1,’’,p,1,’’’,,,,,,,,,,,y/p,2,p1x1*Ordinarydemand curvefor,14,The Case of Perfect-Substitutes Utility Function,Then the ordinary demand functionsfor commodities 1 and 2 are,The Case of Perfect-Substitute,15,and,and,16,Fixed p,2,and y.,Own-Price Changes,x,2,x,1,p,1,x,1,*,Fixed p,2,and y.,p,1,’,p,2,= p,1,’’,,,,p,1,’’’,,,,,,,,,p,1,price offer,curve,Ordinarydemand curvefor commodity 1,Fixed p2 and y.Own-Price Chang,17,Own-Price Changes,Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?”,But we could also ask the,inverse,question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?”,Own-Price ChangesUsually we as,18,Own-Price Changes,Taking quantity demanded as given and then asking what must be price describes the,inverse demand function,of a commodity.,Own-Price ChangesTaking quanti,19,Inverse Demand Function,A Cobb-Douglas example:,is the ordinary demand function and,is the inverse demand function.,Inverse Demand FunctionA Cobb-,20,Income Changes,How does the value of x,1,*(p,1,,p,2,,y) change as y changes, holding both p,1,and p,2,constant?,Income ChangesHow does the val,21,Income Changes,Fixed p,1,and p,2,.,y’ < y’’ < y’’’,,,,,,,x,1,’’’,x,1,’’,x,1,’,x,2,’’’,x,2,’’,x,2,’,,,,Incomeoffer curve,Income ChangesFixed p1 and p2.,22,The Engel,C,urve,A plot of quantity demanded against income is called an,Engel curve,.,The Engel CurveA plot of quant,23,Income Changes,Fixed p,1,and p,2,.,y’ < y’’ < y’’’,,,,,,,x,1,’’’,x,1,’’,x,1,’,x,2,’’’,x,2,’’,x,2,’,,,,Incomeoffer curve,x,1,*,x,2,*,y,y,,,,x,1,’’’,x,1,’’,x,1,’,,,,x,2,’’’,x,2,’’,x,2,’,,,,,,,y’,y’’,y’’’,y’,y’’,y’’’,,,,,,,Engelcurve;,good 2,Engelcurve;,good 1,Income ChangesFixed p1 and p2.,24,Income Changes and Cobb-Douglas Preferences,An example of computing the equations of Engel curves; the Cobb-Douglas case.,,The ordinary demand equations are,Income Changes and Cobb-Dougla,25,Income Changes and Cobb-Douglas Preferences,Rearranged to isolate y, these are:,Engel curve for good 1,Engel curve for good 2,Income Changes and Cobb-Dougla,26,Income Changes and Cobb-Douglas Preferences,y,y,x,1,*,x,2,*,Engel curvefor good 1,Engel curvefor good 2,Income Changes and Cobb-Dougla,27,Income Changes and Perfectly-Complementary Preferences,The ordinary demand equations are,Income Changes and Perfectly-C,28,Income Changes and Perfectly-Complementary Preferences,Rearranged to isolate y, these are:,Engel curve for good 1,Engel curve for good 2,Income Changes and Perfectly-C,29,Income Changes,x,1,*,x,2,*,y,y,x,2,’’’,x,2,’’,x,2,’,y’,y’’,y’’’,y’,y’’,y’’’,x,1,’’’,x,1,’’,x,1,’,,,,Engelcurve;,good 2,Engelcurve;,good 1,Fixed p,1,and p,2,.,,,,Income Changesx1*x2*yyx2’’’x2’,30,Income Changes and Perfectly-Substitutable Preferences,Another example of computing the equations of Engel curves; the perfectly-substitution case.,,The ordinary demand equations are,Income Changes and Perfectly-S,31,Income Changes and Perfectly-Substitutable Preferences,Income Changes and Perfectly-S,32,Income Changes and Perfectly-Substitutable Preferences,Suppose p,1,< p,2,. Then,and,and,,Income Changes and Perfectly-S,33,Income Changes and Perfectly-Substitutable Preferences,y,y,x,1,*,x,2,*,0,Engel curvefor good 1,Engel curvefor good 2,Income Changes and Perfectly-S,34,Income Changes,In every example so far the Engel curves have all been straight lines?Q: Is this true in general?,A: No. Engel curves are straight lines if the consumer’s preferences are,homothetic,.,Income ChangesIn every example,35,Homotheticity,A consumer’s preferences are,homothetic,if and only iffor every k > 0.,That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.,Û,(x,1,,x,2,) (y,1,,y,2,) (kx,1,,kx,2,) (ky,1,,ky,2,),p,p,HomotheticityA consumer’s pref,36,Income Effects -- A Nonhomothetic Example,Quasilinear preferences are not homothetic.,For example,,Income Effects -- A Nonhomothe,37,,,,,Income Changes; Quasilinear Utility,x,2,x,1,x,1,~,x,1,*,x,2,*,y,y,x,1,~,Engelcurve,forgood 2,Engelcurve,forgood 1,Income Changes; Quasilinear Ut,38,Income Effects,A good for which quantity demanded rises with income is called,normal,.,Therefore a normal good’s Engel curve is positively sloped.,Income EffectsA good for which,39,Income Effects,A good for which quantity demanded falls as income increases is called,income inferior,.,Therefore an income inferior good’s Engel curve is negatively sloped.,Income EffectsA good for which,40,Income Changes; Goods1 & 2 Normal,,,,,,,x,1,’’’,x,1,’’,x,1,’,x,2,’’’,x,2,’’,x,2,’,,,,Incomeoffer curve,x,1,*,x,2,*,y,y,,,,x,1,’’’,x,1,’’,x,1,’,,,,x,2,’’’,x,2,’’,x,2,’,,,,,,,y’,y’’,y’’’,y’,y’’,y’’’,,,,,,,Engelcurve;,good 2,Engelcurve;,good 1,Income Changes; Goods1 & 2 No,41,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,,,,,,,,,x,2,x,1,x,1,*,x,2,*,y,y,,Engel curvefor good 2,Engel curvefor good 1,,,Income Changes; Good 2 Is Norm,42,Ordinary Goods,A good is called,ordinary,if the quantity demanded of it always increases as its own price decreases.,Ordinary GoodsA good is called,43,Ordinary Goods,Fixed p,2,and y.,x,1,x,2,,,,,,,p,1,price offer curve,x,1,*,,,,,Downward-sloping  demand curve,Good 1 is,ordinary,Û,p,1,Ordinary GoodsFixed p2 and y.x,44,Giffen Goods,If, 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,.,Giffen GoodsIf, for some value,45,Ordinary Goods,Fixed p,2,and y.,x,1,x,2,,,,,,,Ordinary GoodsFixed p2 and y.x,46,,Ordinary Goods,Fixed p,2,and y.,x,1,x,2,,,,,,p,1,price offer,curve,x,1,*,Demand curve has a positively,sloped part,Good 1 is,Giffen,Û,p,1,,,,,,Ordinary GoodsFixed p2 and y.x,47,Cross-Price Effects,If an increase in p,2,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 EffectsIf an incre,48,Cross-Price Effects,A perfect-complements example:,so,Therefore commodity 2 is a grosscomplement for commodity 1.,Cross-Price EffectsA perfect-c,49,Cross-Price Effects,p,1,x,1,*,p,1,’,p,1,’’,p,1,’’’,,,,,,,,,’’,Increase the price ofgood 2 from p,2,’ to p,2,’’and the demand curve,for good 1 shifts inwards-- good 2 is acomplement for good 1.,Cross-Price Effectsp1x1*p1’p1’,50,Cross-Price Effects,A Cobb- Douglas example:,so,Therefore commodity 1 is neither a grosscomplement nor a gross substitute forcommodity 2.,Cross-Price EffectsA Cobb- Dou,51,Summary,Own Price Effect,Price Offer curve;,Ordinary demand curve;,Inverse demand function;,Ordinary goods vs. Giffen Goods.,Income Effect,Income offer curve;,Engle curve;,Normal goods vs. income inferior goods;,Homothetic preferences.,Cross Price Effect,Gross substitutes;,Gross complements.,SummaryOwn Price Effect,52,