Demand
PROPORTIES 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?
Suppose only p1 increases, from p1 =
1 to p1 = 2 and then to p1= 3.
OWN-PRICE CHANGES
x2 Fixed p2 and M
p1x1 + p2x2 = M
p1 = 1
x1
OWN-PRICE CHANGES
Fixed p2 and M
x2
p1x1 + p2x2 = M
p1 = 1
p1= 2
x1
OWN-PRICE CHANGES
Fixed p2 and M
x2
p1x1 + p2x2 = M
p1 = 1
p1= 3
p1= 2
x1
p1
Own-Price Changes
Fixed p2 and M
x2
p1 = 1
P1= 1
x1*(p1=1) x1*
x1*(p1=1)
x1
p1
Own-Price Changes
Fixed p2 and M
x2
p1 = 2
P1=1
x1*(p1=1) x1*
x1*(p1=2) x1*(p1=1) x1
p1
Own-Price Changes
Fixed p2 and M
x2
p1 = 2
P1=2
P1=1
x1*(p1=2) x1*(p1=1) x1*
x1*(p1=2) x1*(p1=1) x1
p1
Own-Price Changes
Fixed p2 and M
x2 P1=3
p1 = 3
P1=2
P1=1
x1*(p1=3) x1*(p1=2) x1*(p1=1) x1*
x1*(p1=3) x1*(p1=2) x1*(p1=1) x1
p1
Ordinary
Own-Price Changes demand curve
Fixed p2 and M for product 1
x2 P1=3
P1=2
P1=1
x1*(p1=3) x1*(p1=2) x1*(p1=1) x1*
x1*(p1=3) x1*(p1=2) x1*(p1=1) x1
p1
Ordinary
Own-Price Changes demand curve
Fixed p2 and M for product 1
x2 P1=3
P1=2
P1=1
P1 price
offer
x1*(p1=3) x1*(p1=2) x1*(p1=1) x1*
curve
x1*(p1=3) x1*(p1=2) x1*(p1=1) x1
OWN-PRICE CHANGES
The curve containing all the utility-
maximizing bundles traced out (in x1,
x2 space) as p1 changes, with p2 and
M constant, is the price offer curve.
The plot of the x1co-ordinates of the
price offer curve against p1 is the
ordinary demand curve for product 1.
[Q = F(P) or X = F(P)]
Own-Price Changes
Takingquantity demanded as given
and then asking what must be price
describes the inverse demand
function of a good. [P = F(Q) or P =
F(X)]
OWN-PRICE CHANGES
A Cobb-Douglas example:
is the ordinary demand function and
is the inverse demand function.
OWN-PRICE CHANGES
A perfect complements example:
is the ordinary demand function and
is the inverse demand function.
OWN-PRICE CHANGES
What does a p1 price-offer curve look
like for Cobb-Douglas preferences?
Take
Then the ordinary demand functions
for goods 1 and 2 are
OWN-PRICE CHANGES
and
Notice that x2* does not vary with p1 so the
price offer curve is flat and the ordinary
demand curve for product 1 is a
rectangular hyperbola.
Own-Price Changes
Fixed p2 and M
x2
X2*=bM/
(a+b)p2
X1*=aM/ x1
(a+b)p1
p1
Own-Price Changes
Fixed p2 and M
x2
X2*=bM/
(a+b)p2 x 1*
X1*=aM/ x1
(a+b)p1
p1
Ordinary
Own-Price Changes demand curve
Fixed p2 and M for product 1
x2 is
X1*=aM/(a+b)p1
X2*=bM/
(a+b)p2
x 1*
X1*=aM/
x1
(a+b)p1
OWN-PRICE CHANGES
PERFECT COMPLEMENTS
What does a p1 price-offer curve look
like for a perfect-complements utility
function?
The ordinary demand functions
for products 1 and 2 are:
OWN-PRICE CHANGES
PERFECT COMPLEMENTS
With p2 and M fixed, higher p1 causes
smaller x1* and x2*.
As
As
OWN-PRICE CHANGES
PERFECT COMPLEMENTS
x2 Fixed p2 and M
x1
OWN-PRICE CHANGES
p1
PERFECT COMPLEMENTS
Fixed p2 and M
x2
p1 = p11
M/p2
p11
x 1*
x1
OWN-PRICE CHANGES p1
PERFECT COMPLEMENTS
Fixed p2 and M
x2
p1 = p12
M/p2 p12
p11
x 1*
2
x1
OWN-PRICE CHANGES
p1
PERFECT COMPLEMENTS
Fixed p2 and M. p13
x2
p1 = p13
M/p2 p12
p11
x 1*
x1
p1
OWN-PRICE CHANGES Ordinary
PERFECT COMPLEMENTS demand curve
Fixed p2 and M p13 for product 1
x2 is
M/p2 p12
p11
x 1*
x1
OWN-PRICE CHANGES
PERFECT SUBSITUTES
What does a price-offer curve look
like for a perfect-substitutes utility
function?
Then the ordinary demand functions
for products 1 and 2 are
OWN-PRICE CHANGES
PERFECT SUBSTITUTES
and
OWN-PRICE CHANGES
PERFECT SUBSTITUTES
Homework: Draw the diagrams.
INCOME CHANGES
How does the value of x1*(p1,p2,M)
change as M changes, holding both
p1 and p2 constant?
INCOME CHANGES
A plot of quantity demanded against
income is called an Engel curve.
INCOME CHANGES
Fixed p1 and p2
M1 (y1, y2) (tx1, tx2) > (ty1, ty2)
for all positive t.
HOMOTHETICITY
Linear expansion path through the
origin
X2
X1
Income Effects
A product for which quantity
demanded rises with income is
called normal.
Therefore a normal product’s Engel
curve is positively sloped.
Income Effects
A product for which quantity
demanded falls as income increases
is called inferior.
Therefore an inferior product’s Engel
curve is negatively sloped.
Income Changes: Products 1 & 2
M Engel
Normal curve;
M3 good 2
M2
M1
Income
offer curve M x21 x23 x2*
x22
x23 M3
x22 Engel
M2
x21 curve;
M1
good 1
x11 x13 x11 x13 x1*
x12 x12
As income
changes… Engel curves
M
x2 product 2
x2*
M Inferior x1/M0
x1 x1*
Product 2 Is Normal, Product 1 Becomes Inferior
ORDINARY PRODUCTS
A product is called ordinary if the
quantity demanded always increases
as its own price decreases.
ORDINARY PRODUCTS
Fixed p2 and M. Downward sloping
x2 p1 demand curve
p1 price
offer Product 1 is
curve ordinary
x 1*
x1
GIFFEN PRODUCTS
If,for some values of its own price,
the quantity demanded of a product
rises as its own price increases then
the product is called a Giffen
product.
GIFFEN PRODUCTS
Fixed p2 and M Demand curve has
x2 p1 a positively
p1 price offer sloped part
curve
Product 1 is
Giffen
x 1*
x1
CROSS PRICE EFFECTS
Ifan increase in p2
– increases demand for product 1 then
product 1 is a gross substitute for
product 2.
– reduces demand for product 1 then
product 1 is a gross complement for
product 2.
CROSS PRICE EFFECTS
A perfect complements example:
so
Therefore product 2 is a gross
complement for product 1
CROSS PRICE EFFECTS
p1
Increase the price of
p13 product 2 from p21 to p22
and
p12
p11
x 1*
CROSS PRICE EFFECTS
p1 Increase the price of
product 2 from p21 to p22
p13 and the demand curve
for product 1 shifts inwards
p12 -- product 2 is a
complement for product 1.
p11
x 1*
CROSS PRICE EFFECTS
A Cobb- Douglas example:
so
CROSS PRICE EFFECTS
A Cobb- Douglas example:
so
Therefore product 1 is neither a gross
complement nor a gross substitute for
product 2.
SUMMARY I
Changes in income
Cobb Douglas
Perfect Substitutes
Perfect Complements
SUMMARY II
Cross Price Changes
Cobb Douglas
Perfect Substitutes
(Be Careful!)
Perfect Complements