Demand - Carnegie Mellon University · Price Offer and Demand Curves Price Offer Curve: Demand...
Transcript of Demand - Carnegie Mellon University · Price Offer and Demand Curves Price Offer Curve: Demand...
Demand
Consumer Demand
Consumer’s demand functions:
( )mppxx ,, 2111 =
( )mppxx ,, 2122 =
Changes in Income Given Prices
Normal Goods
Both goods 1 and 2 are normal
2x
1x
X
Y 0,, 211
>∂
∂
mmppx
0,, 212
>∂
∂
mmppx
An Inferior Good
Good 1 is normal.Good 2 is inferior:
2x
1x
X Y0
,, 212<
∂∂
mmppx
Income Offer and Engel Curves
m
1x1x
2x Income Offer Curve Engel Curve
Cobb-Douglas
Demand function for good 1:
Demand function for good 2:
2x
1x
*2x
*1x
11
pmcx =
( )2
2 1pmcx −=
Cobb Douglas
Income Offer Curve: Engel Curve:
2x
1x
m
1x
cp1
Perfect Substitutes
2x
1x
1−
2
1
pp−
*1x
Demand function for good 1:
if
if
if
21 pp <11 / pmx =
( )11 /,0 pmx =
01 =x 21 pp >
21 pp =
Perfect Substitutes (with )
2x
1x
m
1x
Income Offer Curve:
21 pp <
Engel Curve:
1p
Perfect Complements
Optimal choice:
Budget line:
Demand function for goods 1 and 2:
*1x
*2x
1x
2x12 xx =
12 xx =
mxpxp =+ 2211
2121
ppmxx+
==
Perfect Complements
Income Offer Curve: Engel Curve:2x
1x 1x
m
21 pp +
Homothetic Preferences
Consumer’s preferences only depend on the ratio of the two goods:
If Then, for
Example: Cobb-Douglas, Perfect substitutes, Perfect Complements.Properties: straight income offer curve and Engel curve.
),(~),( 2121 yyxx
),(~),( 2121 tytyxtxt0>t
Luxury Good
m
1x
Necessary Good
m
1x
Changes in Prices
Fix income and price of one good and change price of the other.
Ordinary Goods
Price of good 1 decreases.Demand for good 1 increases.
2x
1x
YX
Giffen Goods
Price of good 1 decreases.Demand for good 1 decreases.
2x
1x
X
Y
Price Offer and Demand Curves
Price Offer Curve: Demand Curve:
2x
1x
1p
1x
Perfect Complements
Price Offer Curve: Demand Curve:2x
1x
1p
1x
Substitutes
Good 1 is a substitute for good 2 when:
( ) 0,,2
211 >∂
∂p
mppx
Complements
Good 1 is a complement to good 2:
( ) 0,,2
211 <∂
∂p
mppx
Inverse Demand Function
Consider a demand function
The inverse demand function is
Cobb-Douglas example:
( )mppxx ,, 2111 =
( )111 xpp =
11
pmcx =
11
xmcp =
Inverse Demand Curve
Inverse Demand Curve
1p
1x
Optimal choice:
Suppose:(composite good)Rearrange:
12 =p
MRSpp −=
2
1
MRSp −=1