INTMIC9.jpg Chapter 6 Demand Properties of Demand Functions uComparative 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. Own-Price Changes uHow 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’’’. x1 x2 p1 = p1’ Fixed p2 and y. p1x1 + p2x2 = 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 Own-Price Changes Fixed p2 and y. 06-07.png x1*(p1’) Own-Price Changes p1 = p1’ Fixed p2 and y. 06-07.png x1*(p1’) p1 x1*(p1’) p1’ x1* Own-Price Changes Fixed p2 and y. p1 = p1’ 06-07.png x1*(p1’) p1 x1*(p1’) p1’ p1 = p1’’ x1* Own-Price Changes Fixed p2 and y. 06-07.png x1*(p1’) x1*(p1’’) p1 x1*(p1’) p1’ p1 = p1’’ x1* Own-Price Changes Fixed p2 and y. 06-07.png x1*(p1’) x1*(p1’’) p1 x1*(p1’) x1*(p1’’) p1’ p1’’ x1* Own-Price Changes Fixed p2 and y. 06-07.png x1*(p1’) x1*(p1’’) p1 x1*(p1’) x1*(p1’’) p1’ p1’’ p1 = p1’’’ x1* Own-Price Changes Fixed p2 and y. 06-07.png x1*(p1’’’) x1*(p1’) x1*(p1’’) p1 x1*(p1’) x1*(p1’’) p1’ p1’’ p1 = p1’’’ x1* Own-Price Changes Fixed p2 and y. 06-07.png x1*(p1’’’) x1*(p1’) x1*(p1’’) p1 x1*(p1’) x1*(p1’’’) x1*(p1’’) p1’ p1’’ p1’’’ x1* Own-Price Changes Fixed p2 and y. 06-07.png 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. 06-07.png 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. 06-07.png 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 p1 price offer curve Fixed p2 and y. 06-07.png Own-Price Changes uThe 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 Changes uWhat does a p1 price-offer curve look like for Cobb-Douglas preferences? Own-Price Changes uWhat does a p1 price-offer curve look like for Cobb-Douglas preferences? uTake Then the ordinary demand functions for commodities 1 and 2 are Own-Price Changes 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 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 Ordinary demand curve for commodity 1 is Fixed p2 and y. Own-Price Changes uWhat does a p1 price-offer curve look like for a perfect-complements utility function? Own-Price Changes uWhat does a p1 price-offer curve look like for a perfect-complements utility function? Then the ordinary demand functions 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. 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 Changes x1 x2 p1’ p1’’ p1’’’ y/p2 Own-Price Changes uWhat 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. 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 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 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-Price Changes uUsually 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 Changes p1 x1* p1’ Given p1’, what quantity is demanded 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 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 uTaking quantity demanded as given and then asking what must be price describes 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 uHow does the value of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant? Income Changes Fixed p1 and p2. y’ < y’’ < y’’’ 06-07.png Income Changes Fixed p1 and p2. y’ < y’’ < y’’’ 06-07.png Income Changes Fixed p1 and p2. y’ < y’’ < y’’’ x1’’’ x1’’ x1’ x2’’’ x2’’ x2’ 06-07.png Income Changes Fixed p1 and p2. y’ < y’’ < y’’’ x1’’’ x1’’ x1’ x2’’’ x2’’ x2’ Income offer curve 06-07.png Income Changes uA plot 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 06-07.png 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’’’ 06-07.png 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’’’ Engel curve; good 1 06-07.png 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’’’ 06-07.png 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 06-07.png Income Changes Fixed p1 and p2. y’ < y’’ < 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 06-07.png Income Changes and Cobb-Douglas Preferences uAn example of computing the equations of Engel curves; the Cobb-Douglas case. u uThe ordinary demand equations are 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-Douglas Preferences y y x1* x2* Engel curve for good 1 Engel curve for good 2 Income Changes and Perfectly-Complementary Preferences uAnother example of computing the equations of Engel curves; the perfectly-complementary case. u uThe 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 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 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 Perfectly-Substitutable Preferences uAnother example of computing the equations of Engel curves; the perfectly-substitution case. u uThe ordinary demand equations are Income Changes and Perfectly-Substitutable Preferences Income Changes and Perfectly-Substitutable Preferences Suppose p1 < p2. Then Income Changes 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 uIn 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 uA consumer’s preferences are homothetic if and only if for every k > 0. uThat is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin. Û (x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2) p p Income Effects -- A Nonhomothetic Example uQuasilinear preferences are not homothetic. uFor example, Quasi-linear Indifference Curves x2 x1 Each curve is a vertically shifted copy of the others. Each curve intersects both axes. Income Changes; Quasilinear Utility x2 x1 x1 ~ Income Changes; Quasilinear Utility x2 x1 x1 ~ x1* y x1 ~ Engel curve for good 1 Income Changes; Quasilinear Utility x2 x1 x1 ~ x2* y Engel curve for good 2 Income Changes; Quasilinear Utility x2 x1 x1 ~ x1* x2* y y x1 ~ Engel curve for good 2 Engel curve for good 1 Income Effects uA good for which quantity demanded rises with income is called normal. uTherefore a normal good’s Engel curve is positively sloped. Income Effects uA good for which quantity demanded falls as income increases is called income inferior. uTherefore an income inferior good’s 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 06-07.png 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, 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 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 Goods uA 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 Good 1 is ordinary p1 Giffen Goods 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 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* Demand curve has a positively sloped part Good 1 is Giffen p1 Cross-Price Effects uIf 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 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.