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1、Chapter Six,Demand,Properties of Demand Functions,Comparative statics analysis (比较静态分析)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.,Structure,Own-price changes Price offer curve (价格提供曲线) Ordinary demand cu
2、rve Inverse demand curve (反需求函数) Income changes Income offer curve (收入提供曲线) Engel curve (恩格尔曲线) Cross-price effects,Own-Price Changes,How does x1*(p1,p2,y) change as p1 changes, holding p2 and y constant? Suppose only p1 increases, from p1 to p1 and then to p1.,x1,x2,p1 = p1,Fixed p2 and y.,p1x1 + p
3、2x2 = y,Own-Price Changes,Own-Price Changes,x1,x2,p1= p1,p1 = p1,Fixed p2 and y.,p1x1 + p2x2 = y,Own-Price Changes,x1,x2,p1= p1,p1= p1,Fixed p2 and y.,p1 = p1,p1x1 + p2x2 = y,p1 = p1,Own-Price Changes,Fixed p2 and y.,x1*(p1),Own-Price Changes,p1 = p1,Fixed p2 and y.,x1*(p1),p1,x1*(p1),p1,x1*,Own-Pri
4、ce Changes,Fixed p2 and y.,p1 = p1,x1*(p1),p1,x1*(p1),p1,p1 = p1,x1*,Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),p1,x1*(p1),p1,p1 = p1,x1*,Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),p1,p1,x1*,Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),p1,p1,p
5、1 = p1,x1*,Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),p1,p1,p1 = p1,x1*,Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),x1*(p1),p1,p1,p1,x1*,Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),x1*(p1),p1,p1,p1,x1
6、*,Own-Price Changes,Ordinary demand curve for commodity 1,Fixed p2 and y.,x1*(p1),x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),x1*(p1),p1,p1,p1,x1*,Own-Price Changes,Ordinary demand curve for commodity 1,Fixed p2 and y.,x1*(p1),x1*(p1),x1*(p1),p1,x1*(p1),x1*(p1),x1*(p1),p1,p1,p1,x1*,Own-Price Changes,Ordinary
7、 demand curve for commodity 1,p1 price offer curve,Fixed p2 and y.,Own-Price Changes,The curve containing all the utility-maximizing bundles traced out as p1 changes, with p2 and y constant, is the p1- price offer curve. The plot of the x1-coordinate of the p1- price offer curve against p1 is the or
8、dinary demand curve for commodity 1.,Own-Price Changes,What does a p1 price-offer curve look like for Cobb-Douglas preferences?,Own-Price Changes,What does a p1 price-offer curve look like for Cobb-Douglas preferences? Take Then the ordinary demand functions for commodities 1 and 2 are,Own-Price Cha
9、nges,and,Notice that x2* does not vary with p1 so the p1 price offer curve is,Own-Price Changes,and,Notice that x2* does not vary with p1 so the p1 price offer curve is flat,Own-Price Changes,and,Notice that x2* does not vary with p1 so the p1 price offer curve is flat and the ordinary demand curve
10、for commodity 1 is a,Own-Price Changes,and,Notice that x2* does not vary with p1 so the p1 price offer curve is flat and the ordinary demand curve for commodity 1 is a rectangular hyperbola.,x1*(p1),x1*(p1),x1*(p1),Own-Price Changes,Fixed p2 and y.,x1*(p1),x1*(p1),x1*(p1),p1,x1*,Own-Price Changes,Or
11、dinary demand curve for commodity 1 is,Fixed p2 and y.,Own-Price Changes,What does a p1 price-offer curve look like for a perfect-complements utility function?,Own-Price Changes,What does a p1 price-offer curve look like for a perfect-complements utility function?,Then the ordinary demand functions
12、for commodities 1 and 2 are,Own-Price Changes,Own-Price Changes,With p2 and y fixed, higher p1 causes smaller x1* and x2*.,Own-Price Changes,With p2 and y fixed, higher p1 causes smaller x1* and x2*.,As,Own-Price Changes,With p2 and y fixed, higher p1 causes smaller x1* and x2*.,As,As,Fixed p2 and y
13、.,Own-Price Changes,x1,x2,p1,x1*,Fixed p2 and y.,Own-Price Changes,x1,x2,p1,p1 = p1,y/p2,p1,x1*,Fixed p2 and y.,Own-Price Changes,x1,x2,p1,p1,p1 = p1,y/p2,p1,x1*,Fixed p2 and y.,Own-Price Changes,x1,x2,p1,p1,p1,p1 = p1,y/p2,p1,x1*,Ordinary demand curve for commodity 1 is,Fixed p2 and y.,Own-Price Ch
14、anges,x1,x2,p1,p1,p1,y/p2,Own-Price Changes,What does a p1 price-offer curve look like for a perfect-substitutes utility function?,Then the ordinary demand functions for commodities 1 and 2 are,Own-Price Changes,and,Fixed p2 and y.,Own-Price Changes,x2,x1,Fixed p2 and y.,p1 = p1 p2,Fixed p2 and y.,O
15、wn-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p1 = p1 p2,Fixed p2 and y.,Own-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p1 = p1 = p2,Fixed p2 and y.,Own-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p1 = p1 = p2,Fixed p2 and y.,Own-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p1 = p1 = p2,Fix
16、ed p2 and y.,Own-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p1 = p1 = p2,p2 = p1,Fixed p2 and y.,Own-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p1,p2 = p1,Fixed p2 and y.,Own-Price Changes,x2,x1,p1,x1*,Fixed p2 and y.,p1,p2 = p1,p1,p1 price offer curve,Ordinary demand curve for commodity 1,Own
17、-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 Changes,p1,x1*,p1,Given p1, what quantity is demanded
18、of commodity 1?,Own-Price Changes,p1,x1*,p1,Given p1, what quantity is demanded of commodity 1? Answer: x1 units.,x1,Own-Price Changes,p1,x1*,x1,Given p1, what quantity is demanded of commodity 1? Answer: x1 units.,The inverse question is: Given x1 units are demanded, what is the price of commodity
19、1?,Own-Price Changes,p1,x1*,p1,x1,Given p1, what quantity is demanded 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 Changes,Taking quantity demanded as given and then asking what must be price describes
20、the inverse demand function of a commodity.,Own-Price Changes,A Cobb-Douglas example:,is the ordinary demand function and,is the inverse demand function.,Own-Price Changes,A perfect-complements example:,is the ordinary demand function and,is the inverse demand function.,Income Changes,How does the v
21、alue of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant?,Income Changes,Fixed p1 and p2.,y y y,Income Changes,Fixed p1 and p2.,y y y,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income offer curve,Income Changes,A p
22、lot of quantity demanded against income is called an Engel curve.,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income offer curve,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income offer curve,x1*,y,x1,x1,x1,y,y,y,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income
23、offer curve,x1*,y,x1,x1,x1,y,y,y,Engel curve; good 1,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income offer curve,x2*,y,x2,x2,x2,y,y,y,Income Changes,Fixed p1 and p2.,y y y,x1,x1,x1,x2,x2,x2,Income offer curve,x2*,y,x2,x2,x2,y,y,y,Engel curve; good 2,Income Changes,Fixed p1 and p2.,y y
24、 y,x1,x1,x1,x2,x2,x2,Income offer curve,x1*,x2*,y,y,x1,x1,x1,x2,x2,x2,y,y,y,y,y,y,Engel curve; good 2,Engel curve; good 1,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 Cob
25、b-Douglas Preferences,Rearranged to isolate y, these are:,Engel curve for good 1,Engel curve for good 2,Income Changes and Cobb-Douglas Preferences,y,y,x1*,x2*,Engel curve for good 1,Engel curve for good 2,Income Changes and Perfectly-Complementary Preferences,Another example of computing the equati
26、ons of Engel curves; the perfectly-complementary case. The ordinary demand equations are,Income Changes and Perfectly-Complementary Preferences,Rearranged to isolate y, these are:,Engel curve for good 1,Engel curve for good 2,Fixed p1 and p2.,Income Changes,x1,x2,Income Changes,x1,x2,y y y,Fixed p1
27、and p2.,Income Changes,x1,x2,y y y,Fixed p1 and p2.,Income Changes,x1,x2,y y y,x1,x1,x2,x2,x2,x1,Fixed p1 and p2.,Income Changes,x1,x2,y y y,x1,x1,x2,x2,x2,x1,x1*,y,y,y,y,Engel curve; good 1,x1,x1,x1,Fixed p1 and p2.,Income Changes,x1,x2,y y y,x1,x1,x2,x2,x2,x1,x2*,y,x2,x2,x2,y,y,y,Engel curve; good
28、 2,Fixed p1 and p2.,Income Changes,x1,x2,y y y,x1,x1,x2,x2,x2,x1,x1*,x2*,y,y,x2,x2,x2,y,y,y,y,y,y,Engel curve; good 2,Engel curve; good 1,x1,x1,x1,Fixed p1 and p2.,Income Changes,x1*,x2*,y,y,x2,x2,x2,y,y,y,y,y,y,x1,x1,x1,Engel curve; good 2,Engel curve; good 1,Fixed p1 and p2.,Income Changes and Per
29、fectly-Substitutable Preferences,Another example of computing the equations of Engel curves; the perfectly-substitution case. The ordinary demand equations are,Income Changes and Perfectly-Substitutable Preferences,Income Changes and Perfectly-Substitutable Preferences,Suppose p1 p2. Then,Income Cha
30、nges and Perfectly-Substitutable Preferences,Suppose p1 p2. Then,and,Income Changes and Perfectly-Substitutable Preferences,Suppose p1 p2. Then,and,and,Income Changes and Perfectly-Substitutable Preferences,y,y,x1*,x2*,0,Engel curve for good 1,Engel curve for good 2,Income Changes,In every example s
31、o far the Engel curves have all been straight lines? Q: Is this true in general? A: No. Engel curves are straight lines if the consumers preferences are homothetic.,Homotheticity (位似偏好),A consumers preferences are homothetic if and only if for every k 0. That is, the consumers MRS is the same anywhe
32、re on a straight line drawn from the origin.,(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2),p,p,Income Effects - A Nonhomothetic Example,Quasilinear preferences are not homothetic. For example, Optimal interior consumption:,Quasi-linear Indifference Curves,x2,x1,Each curve is a vertically shifted copy of the o
33、thers.,Each curve intersects both axes.,Income Changes; Quasilinear Utility,x2,x1,Income Changes; Quasilinear Utility,x2,x1,x1*,y,x1,Engel curve for good 1,Income Changes; Quasilinear Utility,x2,x1,x2*,y,Engel curve for good 2,Income Changes; Quasilinear Utility,x2,x1,x1*,x2*,y,y,x1,Engel curve for
34、good 2,Engel curve for good 1,Income Effects,A good for which quantity demanded rises with income is called normal (正常品). Therefore a normal goods Engel curve is positively sloped.,Income Effects,A good for which quantity demanded falls as income increases is called income inferior (劣质品). Therefore
35、an income inferior goods Engel curve is negatively sloped.,Income Changes; Goods 1 & 2 Normal,x1,x1,x1,x2,x2,x2,Income offer curve,x1*,x2*,y,y,x1,x1,x1,x2,x2,x2,y,y,y,y,y,y,Engel curve; good 2,Engel curve; good 1,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,Income Changes;
36、Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,Income Changes; Good 2 Is Normal
37、, Good 1 Becomes Income Inferior,x2,x1,Income offer curve,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,x1*,y,Engel curve for good 1,Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior,x2,x1,x1*,x2*,y,y,Engel curve for good 2,Engel curve for good 1,Ordinary Good
38、s (一般商品),A good is called ordinary if the quantity demanded of it always increases as its own price decreases.,Ordinary Goods,Fixed p2 and y.,x1,x2,Ordinary Goods,Fixed p2 and y.,x1,x2,p1 price offer curve,Ordinary Goods,Fixed p2 and y.,x1,x2,p1 price offer curve,x1*,Downward-sloping demand curve,Go
39、od 1 is ordinary,p1,Giffen Goods (吉芬商品),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.,Ordinary Goods,Fixed p2 and y.,x1,x2,Ordinary Goods,Fixed p2 and y.,x1,x2,p1 price offer curve,Ordinary Goods,Fixed p2 and y.
40、,x1,x2,p1 price offer curve,x1*,Demand curve has a positively sloped part,Good 1 is Giffen,p1,Cross-Price Effects,If 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 co
41、mmodity 2.,Cross-Price Effects,A perfect-complements example:,so,Therefore commodity 2 is a gross complement for commodity 1.,Cross-Price Effects,p1,x1*,p1,p1,p1,Increase the price of good 2 from p2 to p2 and,Cross-Price Effects,p1,x1*,p1,p1,p1,Increase the price of good 2 from p2 to p2 and the demand curve for good 1 shifts inwards - good 2 is a complement for good 1.,Cross-Price Effects,A Cobb- Douglas example:,so,Cross-Price Effects,A Cobb- Douglas example:,so,Therefore commodity 1 is neither a gross complement nor a gross substitute for commodity 2.,
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