Properties of Demand Functions. Chapter Six. Own-Price Changes Fixed p 2 and y. Own-Price Changes. Demand

Transcription

1 Properties of Demand Functions Chapter Six Demand Comparative statics analsis of ordinar demand functions -- the stud of how ordinar demands (,p 2,) and (,p 2,) change as prices, p 2 and income change. How does (,p 2,) change as changes, holding p 2 and constant? Suppose onl increases, from to and then to. Fixed p 2 and. + p 2 1

2 The curve containing all the utilitmaximizing bundles traced out as changes, with p 2 and constant, is the - price offer curve. The plot of the -coordinate of the - price offer curve against is the ordinar demand curve for commodit 1. What does a price-offer curve look like for Cobb-Douglas preferences? Take Ux ( 1, x2) a x b 2. Then the ordinar demand functions for commodities 1 and 2 are and a ( p1,, ) a+ b p1 b x2 ( p1,, ). a+ b p Notice that does not var with so the price offer curve is flat and the ordinar demand curve for commodit 1 is a rectangular hperbola. 2 2

3 What does a price-offer curve look like for a perfect-complements utilit function? Ux ( 1, x2) min, x2. Then the ordinar demand functions for commodities 1 and 2 are { } ( p1,, ) x2( p1,, ). p1+ With p 2 and fixed, higher causes smaller and. As p1 0, x2. As p1, x2 0. Fixed p 2 and. /p 2 x2 p1+ p1+ Ordinar demand curve for commodit 1 is. p1+ p 2 3

4 perfect-substitutes utilit What does a price-offer curve look like for a perfect-substitutes utilit function? Ux ( 1, x2) + x2. Then the ordinar demand functions for commodities 1 and 2 are 0, if p1 > p ( p1, 2, ) / p1, ifp1 < and 0, if p1 < p x2 ( p1, 2, ) /, if p1 >. Fixed p 2 and. Fixed p 2 and. < p 2 p 2 p 2 0 p1 x 1 p1 x

5 p 2 Fixed p 2 and. price offer curve p 2 Ordinar demand curve for commodit 1 p1 0 Inverse demand function Usuall we ask Given the price for commodit 1 what is the quantit demanded of commodit 1? But we could also ask the inverse question At what price for commodit 1 would a given quantit of commodit 1 be demanded? Given, what quantit is demanded of commodit 1? Answer: units. The inverse question is: Given units are demanded, what is the price of x commodit 1? 1 Answer: Taking quantit demanded as given and then asking what must be price describes the inverse demand function of a commodit. 5

6 A Cobb-Douglas example: a p1 is the ordinar demand function and a p1 x 1 is the inverse demand function. A perfect-complements example: p1+ is the ordinar demand function and p1 p 2 is the inverse demand function. Income Changes How does the value of (,p 2,) change as changes, holding both and p 2 constant? 6

7 Income Changes A plot of quantit demanded against income is called an Engel curve. Income Changes and Cobb- Douglas Preferences An example of computing the equations of Engel curves; the Cobb-Douglas case. Ux ( 1, x2) a x b 2. The ordinar demand equations are a b ; x2. p1 Income Changes and Cobb- Douglas Preferences a b ; x2. p1 Rearranged to isolate, these are: p 1 Engel curve for good 1 a p 2 Engel curve for good 2 b 7

8 Income Changes and Cobb- Douglas Preferences p 1 a Engel curve for good 1 x 1 p 2 b Engel curve for good 2 Income Changes and Perfectl-Complementar Preferences Another example of computing the equations of Engel curves; the perfectlcomplementar case. Ux (, x ) min x, x The ordinar demand equations are x2. p1 + { } Income Changes and Perfectl-Complementar Preferences x2. p1 + Rearranged to isolate, these are: ( p1+ ) Engel curve for good 1 ( p1+ ) x2 Engel curve for good 2 Income Changes Fixed and p 2. < < Engel curve; good 2 Engel curve; good 1 x x 1 1 8

9 Income Changes Fixed and p 2. ( p1+ ) x 2 ( p1 + ) x 1 Engel curve; good 2 Engel curve; good 1 Income Changes and Perfectl-Substitutable Preferences Another example of computing the equations of Engel curves; the perfectlsubstitution case. Ux ( 1, x2) + x2. The ordinar demand equations are Income Changes and Perfectl- Substitutable Preferences 0, if p1 > p ( p1, 2, ) / p1, ifp1 < 0, if p1 < p x2 ( p1, 2, ) /, if p1 >. Suppose < p 2. Then and x2 0 p1 p 1 and 0. Income Changes and Perfectl- Substitutable Preferences 0. 0 Engel curve for good 1 Engel curve for good 2 9

10 Income Changes In ever 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. Homotheticit A consumer s preferences are homothetic if and onl if (, ) ( 1, 2 ) (k,k ) (k 1,k 2 ) for ever k > 0. That is, the consumer s MRS is the same anwhere on a straight line drawn from the origin. Income Effects -- A Nonhomothetic Example Quasilinear preferences are not homothetic. For example, Ux ( 1, x2) f( ) + x2. Ux ( 1, x2) + x2. Quasi-linear Indifference Curves Each curve is a verticall shifted cop of the others. Each curve intersects both axes. 10

11 Income Changes; Quasilinear Utilit x ~ 1 x ~ 1 Engel curve for good 2 Engel curve for good 1 Income Effects A good for which quantit demanded rises with income is called normal. Therefore a normal good s Engel curve is positivel sloped. Income Effects A good for which quantit demanded falls as income increases is called income inferior. Therefore an income inferior good s Engel curve is negativel sloped. 11

12 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior Engel curve for good 2 Engel curve for good 1 Ordinar Goods A good is called ordinar if the quantit demanded of it alwas increases as its own price decreases. Fixed p 2 and. Ordinar Goods price offer curve Downward-sloping demand curve Good 1 is ordinar Giffen Goods If, for some values of its own price, the quantit demanded of a good rises as its own-price increases then the good is called Giffen. 12

13 Fixed p 2 and. Ordinar Goods p1 price offer curve Demand curve has a positivel sloped part Good 1 is Giffen Cross-Price Effects If an increase in p 2 increases demand for commodit 1 then commodit 1 is a gross substitute for commodit 2. reduces demand for commodit 1 then commodit 1 is a gross complement for commodit 2. Cross-Price Effects A perfect-complements example: p1 + so p1 2 < 0. + ( ) Therefore commodit 2 is a gross complement for commodit 1. Cross-Price Effects p 2 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. 13

14 Cross-Price Effects A Cobb- Douglas example: b x2 so x2 0. p1 Therefore commodit 1 is neither a gross complement nor a gross substitute for commodit 2. Warning of the concept When there are more than 2 goods, ma be a substitutes for x3 but x3 ma be a complement for. Net substitutes and net complement are more precise. 14