© 2010 W. W. Norton & Company, Inc. 6 Demand

© 2010 W. W. Norton & Company, Inc. 2 Properties of Demand Functions u 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.

© 2010 W. W. Norton & Company, Inc. 3 Own-Price Changes u How does x 1 *(p 1,p 2,y) change as p 1 changes, holding p 2 and y constant? u Suppose only p 1 increases, from p 1 ’ to p 1 ’’ and then to p 1 ’’’.

© 2010 W. W. Norton & Company, Inc. 4 x1x1 x2x2 p 1 = p 1 ’ Fixed p 2 and y. p 1 x 1 + p 2 x 2 = y Own-Price Changes

© 2010 W. W. Norton & Company, Inc. 5 Own-Price Changes x1x1 x2x2 p 1 = p 1 ’’ p 1 = p 1 ’ Fixed p 2 and y. p 1 x 1 + p 2 x 2 = y

© 2010 W. W. Norton & Company, Inc. 6 Own-Price Changes x1x1 x2x2 p 1 = p 1 ’’ p 1 = p 1 ’’’ Fixed p 2 and y. p 1 = p 1 ’ p 1 x 1 + p 2 x 2 = y

© 2010 W. W. Norton & Company, Inc. 7 Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 8 x 1 *(p 1 ’) Own-Price Changes p 1 = p 1 ’ Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 9 x 1 *(p 1 ’) p1p1 p1’p1’ x1*x1* Own-Price Changes Fixed p 2 and y. p 1 = p 1 ’

© 2010 W. W. Norton & Company, Inc. 10 x 1 *(p 1 ’) p1p1 p1’p1’ p 1 = p 1 ’’ x1*x1* Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 11 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) p1’p1’ p 1 = p 1 ’’ x1*x1* Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 12 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ x1*x1* Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 13 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 = p 1 ’’’ x1*x1* Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 14 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 = p 1 ’’’ x1*x1* Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 15 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 16 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 17 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 18 x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 p 1 price offer curve Fixed p 2 and y.

© 2010 W. W. Norton & Company, Inc. 19 Own-Price Changes u 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. u 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.

© 2010 W. W. Norton & Company, Inc. 20 Own-Price Changes u What does a p 1 price-offer curve look like for a perfect-complements utility function?

© 2010 W. W. Norton & Company, Inc. 21 Own-Price Changes u What does a p 1 price-offer curve look like for a perfect-complements utility function? Then the ordinary demand functions for commodities 1 and 2 are

© 2010 W. W. Norton & Company, Inc. 22 Own-Price Changes

© 2010 W. W. Norton & Company, Inc. 23 Own-Price Changes With p 2 and y fixed, higher p 1 causes smaller x 1 * and x 2 *.

© 2010 W. W. Norton & Company, Inc. 24 Own-Price Changes With p 2 and y fixed, higher p 1 causes smaller x 1 * and x 2 *. As

© 2010 W. W. Norton & Company, Inc. 25 Own-Price Changes With p 2 and y fixed, higher p 1 causes smaller x 1 * and x 2 *. As

© 2010 W. W. Norton & Company, Inc. 26 Fixed p 2 and y. Own-Price Changes x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 27 p1p1 x1*x1* Fixed p 2 and y. Own-Price Changes x1x1 x2x2 p1’p1’ ’ p 1 = p 1 ’ ’ ’ y/p 2

© 2010 W. W. Norton & Company, Inc. 28 p1p1 x1*x1* Fixed p 2 and y. Own-Price Changes x1x1 x2x2 p1’p1’ p 1 ’’ p 1 = p 1 ’’ ’’ y/p 2

© 2010 W. W. Norton & Company, Inc. 29 p1p1 x1*x1* Fixed p 2 and y. Own-Price Changes x1x1 x2x2 p1’p1’ p 1 ’’ p 1 ’’’ p 1 = p 1 ’’’ ’’’ y/p 2

© 2010 W. W. Norton & Company, Inc. 30 p1p1 x1*x1* Ordinary demand curve for commodity 1 is Fixed p 2 and y. Own-Price Changes x1x1 x2x2 p1’p1’ p 1 ’’ p 1 ’’’ y/p 2

© 2010 W. W. Norton & Company, Inc. 31 Own-Price Changes u What does a p 1 price-offer curve look like for a perfect-substitutes utility function? Then the ordinary demand functions for commodities 1 and 2 are

© 2010 W. W. Norton & Company, Inc. 32 Own-Price Changes and

© 2010 W. W. Norton & Company, Inc. 33 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 Fixed p 2 and y. p 1 = p 1 ’ < p 2 ’

© 2010 W. W. Norton & Company, Inc. 34 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 1 = p 1 ’ < p 2 ’ ’

© 2010 W. W. Norton & Company, Inc. 35 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 1 = p 1 ’’ = p 2

© 2010 W. W. Norton & Company, Inc. 36 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 1 = p 1 ’’ = p 2

© 2010 W. W. Norton & Company, Inc. 37 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 1 = p 1 ’’ = p 2 ’’

© 2010 W. W. Norton & Company, Inc. 38 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 1 = p 1 ’’ = p 2 p 2 = p 1 ’’

© 2010 W. W. Norton & Company, Inc. 39 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 1 ’’’ p 2 = p 1 ’’

© 2010 W. W. Norton & Company, Inc. 40 Fixed p 2 and y. Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and y. p1’p1’ p 2 = p 1 ’’ p 1 ’’’ p 1 price offer curve Ordinary demand curve for commodity 1

© 2010 W. W. Norton & Company, Inc. 41 Own-Price Changes u Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?” u But 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. 42 Own-Price Changes p1p1 x1*x1* p1’p1’ Given p 1 ’, what quantity is demanded of commodity 1?

© 2010 W. W. Norton & Company, Inc. 43 Own-Price Changes p1p1 x1*x1* p1’p1’ Given p 1 ’, what quantity is demanded of commodity 1? Answer: x 1 ’ units. x1’x1’

© 2010 W. W. Norton & Company, Inc. 44 Own-Price Changes p1p1 x1*x1* x1’x1’ Given p 1 ’, what quantity is demanded of commodity 1? Answer: x 1 ’ units. The inverse question is: Given x 1 ’ units are demanded, what is the price of commodity 1?

© 2010 W. W. Norton & Company, Inc. 45 Own-Price Changes p1p1 x1*x1* p1’p1’ x1’x1’ Given p 1 ’, what quantity is demanded of commodity 1? Answer: x 1 ’ units. The inverse question is: Given x 1 ’ units are demanded, what is the price of commodity 1? Answer: p 1 ’

© 2010 W. W. Norton & Company, Inc. 46 Own-Price Changes u Taking 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. 47 Own-Price Changes A Cobb-Douglas example: is the ordinary demand function and is the inverse demand function.

© 2010 W. W. Norton & Company, Inc. 48 Own-Price Changes A perfect-complements example: is the ordinary demand function and is the inverse demand function.

© 2010 W. W. Norton & Company, Inc. 49 Income Changes u How does the value of x 1 *(p 1,p 2,y) change as y changes, holding both p 1 and p 2 constant?

© 2010 W. W. Norton & Company, Inc. 50 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’

© 2010 W. W. Norton & Company, Inc. 51 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’

© 2010 W. W. Norton & Company, Inc. 52 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’

© 2010 W. W. Norton & Company, Inc. 53 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve

© 2010 W. W. Norton & Company, Inc. 54 Income Changes u A plot of quantity demanded against income is called an Engel curve.

© 2010 W. W. Norton & Company, Inc. 55 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve

© 2010 W. W. Norton & Company, Inc. 56 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x1*x1* y x 1 ’’’ x 1 ’’ x1’x1’ y’ y’’ y’’’

© 2010 W. W. Norton & Company, Inc. 57 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x1*x1* y x 1 ’’’ x 1 ’’ x1’x1’ y’ y’’ y’’’ Engel curve; good 1

© 2010 W. W. Norton & Company, Inc. 58 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x2*x2* y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’

© 2010 W. W. Norton & Company, Inc. 59 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x2*x2* y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ Engel curve; good 2

© 2010 W. W. Norton & Company, Inc. 60 Income Changes Fixed p 1 and p 2. y’ < y’’ < y’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x1*x1* x2*x2* y y x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ Engel curve; good 2 Engel curve; good 1

© 2010 W. W. Norton & Company, Inc. 61 Income Changes and Cobb- Douglas Preferences u An example of computing the equations of Engel curves; the Cobb- Douglas case. u The ordinary demand equations are

© 2010 W. W. Norton & Company, Inc. 62 Income Changes and Cobb- Douglas Preferences Rearranged to isolate y, these are: Engel curve for good 1 Engel curve for good 2

© 2010 W. W. Norton & Company, Inc. 63 Income Changes and Cobb- Douglas Preferences y y x1*x1* x2*x2* Engel curve for good 1 Engel curve for good 2

© 2010 W. W. Norton & Company, Inc. 64 Income Changes and Perfectly- Complementary Preferences u Another example of computing the equations of Engel curves; the perfectly-complementary case. u The ordinary demand equations are

© 2010 W. W. Norton & Company, Inc. 65 Income Changes and Perfectly- Complementary Preferences Rearranged to isolate y, these are: Engel curve for good 1 Engel curve for good 2

© 2010 W. W. Norton & Company, Inc. 66 Fixed p 1 and p 2. Income Changes x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 67 Income Changes x1x1 x2x2 y’ < y’’ < y’’’ Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 68 Income Changes x1x1 x2x2 y’ < y’’ < y’’’ Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 69 Income Changes x1x1 x2x2 y’ < y’’ < y’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ x 1 ’’’ Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 70 Income Changes x1x1 x2x2 y’ < y’’ < y’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ x 1 ’’’ x1*x1* y y’ y’’ y’’’ Engel curve; good 1 x 1 ’’’ x 1 ’’ x1’x1’ Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 71 Income Changes x1x1 x2x2 y’ < y’’ < y’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ x 1 ’’’ x2*x2* y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ Engel curve; good 2 Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 72 Income Changes x1x1 x2x2 y’ < y’’ < y’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ x 1 ’’’ x1*x1* x2*x2* y y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ Engel curve; good 2 Engel curve; good 1 x 1 ’’’ x 1 ’’ x1’x1’ Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 73 Income Changes x1*x1* x2*x2* y y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ x 1 ’’’ x 1 ’’ x1’x1’ Engel curve; good 2 Engel curve; good 1 Fixed p 1 and p 2.

© 2010 W. W. Norton & Company, Inc. 74 Income Changes and Perfectly- Substitutable Preferences u Another example of computing the equations of Engel curves; the perfectly-substitution case. u The ordinary demand equations are

© 2010 W. W. Norton & Company, Inc. 75 Income Changes and Perfectly- Substitutable Preferences

© 2010 W. W. Norton & Company, Inc. 76 Income Changes and Perfectly- Substitutable Preferences Suppose p 1 < p 2. Then

© 2010 W. W. Norton & Company, Inc. 77 Income Changes and Perfectly- Substitutable Preferences Suppose p 1 < p 2. Thenand

© 2010 W. W. Norton & Company, Inc. 78 Income Changes and Perfectly- Substitutable Preferences Suppose p 1 < p 2. Thenand

© 2010 W. W. Norton & Company, Inc. 79 Income Changes and Perfectly- Substitutable Preferences yy x1*x1*x2*x2* 0 Engel curve for good 1 Engel curve for good 2

© 2010 W. W. Norton & Company, Inc. 80 Income Effects u A good for which quantity demanded rises with income is called normal. u Therefore a normal good’s Engel curve is positively sloped.

© 2010 W. W. Norton & Company, Inc. 81 Income Effects u A good for which quantity demanded falls as income increases is called income inferior. u Therefore an income inferior good’s Engel curve is negatively sloped.

© 2010 W. W. Norton & Company, Inc. 82 Income Changes; Goods 1 & 2 Normal x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x1*x1* x2*x2* y y x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ Engel curve; good 2 Engel curve; good 1

© 2010 W. W. Norton & Company, Inc. 83 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 84 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 85 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 86 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 87 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 88 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1 Income offer curve

© 2010 W. W. Norton & Company, Inc. 89 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1 x1*x1* y Engel curve for good 1

© 2010 W. W. Norton & Company, Inc. 90 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2x2 x1x1 x1*x1* x2*x2* y y Engel curve for good 2 Engel curve for good 1

© 2010 W. W. Norton & Company, Inc. 91 Ordinary Goods u A good is called ordinary if the quantity demanded of it always increases as its own price decreases.

© 2010 W. W. Norton & Company, Inc. 92 Ordinary Goods Fixed p 2 and y. x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 93 Ordinary Goods Fixed p 2 and y. x1x1 x2x2 p 1 price offer curve

© 2010 W. W. Norton & Company, Inc. 94 Ordinary Goods Fixed p 2 and y. x1x1 x2x2 p 1 price offer curve x1*x1* Downward-sloping demand curve Good 1 is ordinary  p1p1

© 2010 W. W. Norton & Company, Inc. 95 Giffen Goods u 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.

© 2010 W. W. Norton & Company, Inc. 96 Ordinary Goods Fixed p 2 and y. x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 97 Ordinary Goods Fixed p 2 and y. x1x1 x2x2 p 1 price offer curve

© 2010 W. W. Norton & Company, Inc. 98 Ordinary Goods Fixed p 2 and y. x1x1 x2x2 p 1 price offer curve x1*x1* Demand curve has a positively sloped part Good 1 is Giffen  p1p1

© 2010 W. W. Norton & Company, Inc. 99 Cross-Price Effects u 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.

© 2010 W. W. Norton & Company, Inc. 100 Cross-Price Effects A perfect-complements example: so Therefore commodity 2 is a gross complement for commodity 1.

© 2010 W. W. Norton & Company, Inc. 101 Cross-Price Effects p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ ’ Increase the price of good 2 from p 2 ’ to p 2 ’’ and

© 2010 W. W. Norton & Company, Inc. 102 Cross-Price Effects p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ ’’ Increase the price of good 2 from p 2 ’ to p 2 ’’ and the demand curve for good 1 shifts inwards -- good 2 is a complement for good 1.

© 2010 W. W. Norton & Company, Inc. 103 Cross-Price Effects A Cobb- Douglas example: so

© 2010 W. W. Norton & Company, Inc. 104 Cross-Price Effects A Cobb- Douglas example: so Therefore commodity 1 is neither a gross complement nor a gross substitute for commodity 2.