Demand. Consumer Demand Consumer’s demand functions:

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

m

mppx

0,, 212

m

mppx

An Inferior Good

Good 1 is normal.Good 2 is inferior:

2x

1x

X Y0

,, 212

m

mppx

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

p

mcx

2

2 1p

mcx

Cobb Douglas

Income Offer Curve: Engel Curve:

2x

1x

m

1x

c

p1

Perfect Substitutes

2x

1x

1

2

1

p

p

*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

pp

mxx

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 tytyxtxt0t

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

p

mcx

11

x

mcp

Inverse Demand Curve

Inverse Demand Curve

1p

1x

Optimal choice:

Suppose:

(composite good)

Rearrange:

12 p

MRSp

p

2

1

MRSp 1