Properties of Demand Functions

• Comparative statics analysis of ordinary demand functions -- the study of how ordinary demands x1*(p1,p2,m) and x2*(p1,p2,m) change as prices p1, p2 and income m change.

• Own-Price changes:

How does x1*(p1,p2,m) change as p1 changes, holding p2 and m constant?

x2

x1x1*(p1’’’) x1*(p1’)

x1*(p1’’)

p1

x1*(p1’)x1*(p1’’’)

x1*(p1’’)

p1’

p1’’

p1’’’

x1*

Own-Price Changes Ordinarydemand curvefor commodity 1Fixed p2 and m.

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 the x1-coordinate of the p1- price offer curve against p1 is the ordinary demand curve for commodity 1.

Own-Price Changes: Cobb-Douglas

• Take

• Then the ordinary demand functions for commodities 1 and 2 are

• What does the demand curve look like?

• What does a p1 price-offer curve look like?

U x x x xa b( , ) .1 2 1 2

121

*1 ),,(

pm

baa

mppx

221

*2 ),,(

pm

bab

mppx

x1*(p1’’’) x1*(p1’)

x1*(p1’’)

x2

x1

p1

x1*

Own-Price Changes Ordinarydemand curvefor commodity 1 is

Fixed p2 and m.

2

*2

)( pbabm

x

1

*1 )( pba

amx

1

*1 )( pba

amx

Own-Price Changes: Perfect complements

• Take

• Then the ordinary demand functionsfor commodities 1 and 2 are

• What does the demand curve look like?

• What is the p1 price-offer curve?

U x x x x( , ) min , .1 2 1 2

p1

x1*

Ordinarydemand curvefor commodity 1 is

Fixed p2 and m.

21

*2

ppm

x

21

*1 pp

mx

.21

*1 pp

mx

Own-Price Changes

x1

x2

p1’

p1’’

p1’’’

2pm

m/p2

Own-Price Changes:Perfect Substitutes

U x x x x( , ) .1 2 1 2 • Take

• Then the ordinary demand functionsfor commodities 1 and 2 are

• What does the demand curve look like?

• What is the p1 price-offer curve?

Own-Price Changes

x2

x1

p1

x1*

Fixed p2 and m.

p1’

p2 = p1’’

p1’’’

1

*1 p

mx

2

*10

pm

x

2pm

p1 price offer curve

Ordinarydemand curvefor commodity 1

Income Changes• How does the value of x1*(p1,p2,m) change

as m changes, holding both p1 and p2 constant?

• A plot of quantity demanded against income is called an Engel curve.

• A plot of bundles chosen as we vary income is called the income-offer curve.

• Draw these two curves for Perfect complements, Cobb-Douglas, and Perfect Substitutes.

Income Changes

• In every example so far the income offer curves have all been straight lines for the origin?Q: Is this true in general?

• A: No. Income offer curves are straight lines only if the consumer’s preferences are homothetic.

Homotheticity

• A consumer’s preferences are homothetic if and only if

for every k > 0.

• That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.

(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2)

Why the connection?• Take a budget and choice (x1*,x2*). Double the budget

so new budget is 2m. • Notice if (x1,x2) were in the old budget, (2 x1, 2 x2) is

in the new budget. • Homotheticity implies that if (x1*,x2*)> (x1,x2) then

(2x1*,2x2*)>(2x1,2x2), thus new choice is (2x1*,2x2*).

• Notice if you draw a line from origin to (x1*,x2*) it goes through (2x1*,2x2*).

• Since these are the choices, the MRS must be the same at both points to be tangent to the ICs.

Homogeneous Utility Functions

• U(k x1,k x2)= g(k) u(x1,x2) and g(k)>0.

• This implies homotheticity!

• U(x1,x2)>U(y1,y2) =>

g(k) U(x1,x2)> g(k) U(y1,y2) =>

U(k x1,k x2)>U(k y1,k y2)

• Does Cobb-Douglas, Perfect Substitutes, Perfect Complements satisfy this?

Income Effects -- A Nonhomothetic Example

• Quasilinear preferences are not homothetic.

• For example,

• U(1,0)>U(0,3/4) but when k=4, we have U(4,0)<U(0,3)!

U x x f x x( , ) ( ) .1 2 1 2

U x x x x( , ) .1 2 1 2

Income Effects• A good for which quantity demanded rises with income is

called normal.

• Therefore a normal good’s Engel curve is positively sloped.

• A good for which quantity demanded falls as income increases is called income inferior.

• Therefore an income inferior good’s Engel curve is negatively sloped.

• A luxury good is where the proportion of income spent on it goes up with income.

• The opposite of a luxury good is a necessary good.

• Is a necessary good always inferior?

• Is a inferior good always necessary?

• Can we have a necessary good w/o a luxury good? Vice-versa?

• Can we have an inferior good w/o a luxury good? Vice-versa?

Ordinary and Giffen goods.

• A good is called ordinary if the quantity demanded of it always increases as its own price decreases.

• 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.

• Example of a Giffen good – U(x1,x2)=-e-x1-e-x2

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.

• What happens with Cobb-Douglas preferences?