L06

Demand

Model of choice parameters

Example 1: Cobb Douglass

Review

mpp ,, 21

),,( 211 mppx ),,( 212 mppx

21xxU

baxxxxU 2121 ),(

Perfect ComplementsmxxU ,p,p ),min( 2121

Perfect complements We know

Focus on one good (x1)

How the demand is affected by a change a) in “own” price b) in income

c) in price of other commodity

One variable at the time!

),,( ),,,( 21*221

*1 mppxmppx

Own-Price Changes

We focus on good 1

We hold p2 and m constant.

We change p1 The change represented by:- Price offer curve- Demand curve

),,( 21*1 mppx

Own-Price Change p1

x1

x2

Fix p2=1 and m=12. Vary p1=1, p1’=3, p1’’=4

(5,7)(3,3)(2.5,3)

p1 price offer curve

x1*

p1

Demand curvefor commodity 1

Own-Price Changes

The curve containing all the utility-maximizing bundles traced out as p1 changes, with p2 and m constant, is the p1- price offer curve.

The plot of optimal choice of x1

against p1 is the demand curve for commodity 1.

Ordinary and Giffen goods

x1*

p1

Ordinary and Giffen Goods

A good is called ordinary if the quantity always increases as the price decreases.

If, for some values of the price, the quantity demanded rises as the price increases, then the good is called Giffen.

Two examples

We find price offer and demand curve for Cobb-Douglas preferences

Perfect complements

In both cases we keep fixed

.),( 22

2121 xxxxU

),min(),( 2121 xxxxU

12,12 mp

Cobb-Douglass example

),,( 21*1 mppx

Data , variable12,1p , 222

21 mxxU

),,( 21*2 mppx

1p

Perfect Complements

Data , variable12,1p ),min( 221 mxxU 1p

Summary: Price offer curve

- Cobb-Douglas – flat line- Perfect Complements – optimal proportion line

Demand curve- Cobb-Douglas – downward slopping- Perfect Complements – downward slopping

Conclusion: both ordinary goods Preferences generating Giffen good?

Giffen Good

x1

x2 p1 price offer curve

x1*

Demand curve has a positively sloped part

Good 1 isGiffen

p1

Income Changes

We still focus on good 1

We hold p1 and p2 constant. We change m The change represented by:- Income offer curve- Engel curve

),,( 21*1 mppx

x1

x2

Fix p1=1, p2=1 Vary m=12, m’=6, m’’=4

(5,7)

(3,3)

(2,2)

income offer curve

x1*

mEngel curvefor commodity 1

Income Changes

Goods

A good for which quantity demanded rises with income is called normal.

(positive slope of Engel curve) A good for which quantity demanded

falls as income increases is called income inferior.

(negative slope of Engel curve)

Two examples

We find income offer and Engel curve for Cobb-Douglas preferences

Perfect complements

In both cases we assume

.),( 22

2121 xxxxU

),min(),( 2121 xxxxU

1,1 21 pp

Cobb-Douglass example

),,( 21*1 mppx

Data , variable1p1,p , 2122

21 xxU

),,( 21*2 mppx

m

Perfect Complements

Data , variable1p1,p ),min( 2121 xxU m

Summary: Income offer curve

- Cobb-Douglas – ray from origin- Perfect Complements – optimal proportion line

Engel curve- Cobb-Douglas – upward slopping- Perfect Complements – upward slopping

Conclusion: both normal goods

Preferences generating inferior good: textbook.

Cross-Price Effects

If 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.

Cobb Douglas example

Gross complements of substitutes?

Perfect Complements example

Gross complements