Chapter 6 Demand

```Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics
1
In Chapter 5, we talk about the optimal choice
under the budget constraint.
 Here, we consider how the changes of
exogenous variables affect decisions.


Recall
endogenous variables: quantities of goods x1, x2
exogenous variables: prices and income.
2
Consider the two-good case.
 Denote the ordinary demand functions as
x1*(p1,p2,m), x2*(p1,p2,m).

How does x1*(p1,p2,m) change as p1 changes,
holding p2 and m constant?
 Suppose only p1 increases, from p1’ to p1’’ and
then to p1’’’.

3
x2
Fixed p2 and m.
p1x1 + p2x2 = m
p1 = p1’
x1
4
x2
Fixed p2 and m.
p1x1 + p2x2 = m
p1 = p1’
p1= p1’’
x1
5
Fixed p2 and y.
x2
p1x1 + p2x2 = m
p1 = p1’
p1=
p1’’’
p1= p1’’
x1
6
Own-Price Changes
Fixed p2 and y.
p1 = p1’
x1*(p1’)
7
Own-Price Changes
p1
Fixed p2 and m.
p1 = p1’
p1’
x1*(p1’)
x1*
x1*(p1’)
8
Own-Price Changes
p1
Fixed p2 and m.
p1’’
p1’
x1*(p1’)
x1*
x1*(p1’’)
x1*(p1’)
x1*(p1’’)
9
Own-Price Changes
Fixed p2 and m.
p1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
10
Own-Price Changes
Fixed p2 and m.
p1
p1’’’
Ordinary
demand curve
for commodity 1
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
11
Own-Price Changes
Fixed p2 and m.
p1 price
offer
curve
p1
p1’’’
Ordinary
demand curve
for commodity 1
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
12


The curve containing all the utility-maximizing
bundles traced out as p1 changes, with p2 and y
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.
13

A good is called an ordinary good if the
quantity demanded of it always increases as its
own price decreases and vice versa (negatively
related), holding all other factors, such as prices,
income and preference constant.
Fixed p2 and y.
x2
p1

p1 price
offer
curve
Downward-sloping
demand curve
Good 1 is
ordinary
x1*
x1


If the quantity demanded of a good decreases as its
own-price decreases and vice versa (i.e. positively
related) holding all other factors constant, then the
good is called a Giffen Good.
Note: we need to hold other factors constant. Thus,
if the price change is also associated with change in
income or preference, then even if there’s a
positive relation between price and quantity, it is
not characterized as Giffen good.
Fixed p2 and y.
x2
p1
p1 price offer
curve
Demand curve has
a positively
sloped part

Good 1 is
Giffen
x1*
x1

What does a p1 price-offer curve look like for a
perfect-complements utility function?
U( x1 , x 2 )  minx1 , x 2.
m
x ( p1 , p2 , m)  x ( p1 , p2 , m) 
.
p1  p2
*
1
*
2
m
x ( p1 , p2 , m)  x ( p1 , p2 , m) 
.
p1  p2
*
1
*
2
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
As
m
p1  0,x  x  .
p2
As
p1  ,x  x  0.
*
1
*
2
*
1
*
2
x2
Fixed p2 and y.
x1
Perfect Complement
p1
Fixed p2 and y.
x2
y/p2
p1 = p1’
p1’
x2* 
m
p’1  p2
y
x 
p1 ' p2
*
1
x1* 
m
p1’ p2
x1
x1*
Perfect Complement
p1
Fixed p2 and y.
x2
y/p2
p1 = p1’’
p1’’
p1’
x2* 
m
*
x1 
p1" p2
m
p1" p2
x1* 
m
p1" p2
x1
x1*
Perfect Complement
Fixed p2 and y.
x2
p1 = p1’’’
y/p2
p1
p1’’’
p1’’
p1’
x2* 
m
p1 ' ' ' p2
m
x 
p1 ' ' ' p2
*
1
’’’
m
x 
p1' ' ' p2
*
1
x1
x1*
Perfect Complement
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
is
m
*
x1 
.
p1  p2
p1’’’
x2
p1’’
y/p2
x2* 
p1’
m
p1  p2
m
p2
x1* 
m
p1  p2
x1
x1*

What does a p1 price-offer curve look like for a
perfect-substitutes utility function?
U( x1 , x 2 )  x1  x 2 .
0,if p1  p2
x ( p1 , p2 , m)  
m / p1,if p1  p2
*
1
and
0,if p1  p2
x ( p1 , p2 , m)  
m / p2 ,if p1  p2 .
*
2
p1
Perfect Substitutes
Fixed p2 and y.
x2
p1 = p1’ < p2
p1’
m
x 
p1 '
*
1
x*2  0
m x1
x 
p1
*
1
x1*
Perfect Substitutes
p1
Fixed p2 and y.
x2
p1 = p1’’ = p2
p1’
x1*
x1
Perfect Substitutes
p1
Fixed p2 and y.
x2
p1 = p1’’ = p2
p1’
x1*
x1
p1
Perfect Substitutes
Fixed p2 and y.
x2
p1 = p1’’ = p2
p2 = p1’’
m
x 
p2
*
2






*
x 2  0 

x1* 
x*1  0
p1’


m
0 x 
p2
*
1
m
p2
x1
x1*
Perfect Substitutes
Fixed p2 and y.
p1
p1’’’
x2
p2 = p1’’
m
x 
p2
*
2
x*1  0
p1’
x*1  0
x1
x1*
Perfect Substitutes
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
p1’’’
x2
x1* 
p2 = p1’’
m
p2
p1 price
offer
curve
m
p1
p1’


m
0 x 
p2
*
1
x1
x1*



Usually we ask “Given the price for commodity 1
what is the quantity demanded of commodity 1?”
But we could also ask the inverse question “At
what price for commodity 1 would a given quantity
of commodity 1 be demanded?”
Taking quantity demanded as given and then
asking what must be price describes the inverse
demand function of a commodity.
p1
Given p1’, what quantity is
demanded of commodity 1?
p1’
x1’
x1*
p1
Given p1’, what quantity is
demanded of commodity 1?
p1’
x1’
The inverse question is:
Given x1’ units are
demanded, what is the
price of
commodity 1?
x1*

If an increase in p2, holding p1 constant,
 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.


Substitutes: if the price is higher for one good, you
turn to buy other goods to give you satisfaction.
E.g. drinks: bubble tea vs fruit juice;
entertainment: movie vs magazine.
Complements: things tend to be used together so
when one of the prices increases, the demand for
the other will decrease.
E.g.: camera and memory cards
38
A perfect-complements example:
m
x 
p1  p2
*
1
so
x
m

 0.
2
 p2
 p1  p2 
*
1
Therefore commodity 2 is a gross complement
for commodity 1.
p1
p1’’’
Increase the price of
good 2 from p2’ to p2’’
and
p1’’
p1’
m
p2 '
x1*
p1
Increase the price of
good 2 from p2’ to p2’’
and the demand curve
for good 1 shifts inwards
-- good 1 is a
complement for good 2.
p1’’’
p1’’
p1’
m
p2 ' '
x1*


For perfect substitutes, how will the quantity
demanded x1 change when p2 increases?
Note: it only changes the point where x1 starts
to be positive.
42
p1
p1’’’
p1’’
p1’
Generally, an increase the
price of good 2 from p2’
to p2’’ and the demand
curve for good 1 shifts
outwards
-- good 1 is a
substitute for good 2.
x1*

How does the value of x1*(p1,p2,m) change as m
changes, holding both p1 and p2 constant?
Income Changes
Fixed p1 and p2.
m’ < m’’ < m’’’
Income Changes
Fixed p1 and p2.
m’ < m’’ < m’’’
x2’’’
x2’’
x2’
x1’
x1’’’
x1’’
Income Changes
Fixed p1 and p2.
m’ < m’’ < m’’’
Income
offer curve
x2’’’
x2’’
x2’
x1’
x1’’’
x1’’

A plot of income against quantity demanded is
called an Engel curve.
Income Changes
Fixed p1 and p2.
m’ < m’’ < m’’’
Income
offer curve
Engel
curve;
good 2
y
m’’’
m’’
m’
y
x2’’’
x2’
x2*
x2’’
x2’’’
m’’’
x2’’
( a  b) p
m  (a  b2) px2*
2 *
m b
x2
b
x2’
x1’
x1’’’
x1’’
Engel
curve;
good 1
m’’
m’
x1’
x1’’’
x1’’
x1*
p1
p1’’’
When income increases, the
curve shifts outward for each give
price, if the good is a normal
good.
p1’’
p1’
x1*
A good for which quantity demanded rises with
income is called normal.
 Therefore a normal good’s Engel curve is
positively sloped.
 Generally, most goods are normal goods.

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.
 E.g.: low-quality goods.

Income Changes; Goods
y
1 & 2 Normal
Engel
curve;
good 2
y’’’
y’’
y’
Income
offer curve
y
x2’’’
x2’
x2*
x2’’
x2’’’
y’’’
x2’’
Engel
curve;
good 1
y’’
x2’
y’
x1’
x1’’’
x1’’
x1’
x1’’’
x1’’
x1*
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
Income
offer curve
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
y
Engel curve
for good 1
x1
x1*

Another example of computing the equations
of Engel curves; the perfectly-complementary
case.
U( x1 , x 2 )  minx1 , x 2.

The ordinary demand equations are
m
x x 
.
p1  p2
*
1
*
2
m
x x 
.
p1  p2
*
1
*
2
Rearranged to isolate y, these are:
m  ( p1  p2 ) x
*
1
m  ( p1  p2 ) x
*
2
Engel curve for good 1
Engel curve for good 2
Income Changes
Fixed p1 and p2.
x2
x1
Income Changes
Fixed p1 and p2.
x2
y’ < y’’ < y’’’
x2’’’
x2’’
x2’
x1’
x1’’’
x1’’
x1
Income Changes
Fixed p1 and p2.
x2
Engel
curve;
good 2
y
y’’’
y’’
y’ < y’’ < y’’’
y’
y
x2’’’
x2’
x2*
x2’’
x2’’’
y’’’
x2’’
Engel
curve;
good 1
y’’
x2’
y’
x1’
x1’’’
x1’’
x1
x1’’’
x1’
x1’’
x1*
Income Changes
Fixed p1 and p2.
*
y  (p1  p2 )x 2
Engel
curve;
good 2
y
y’’’
y’’
y’
y
x2’’’
x2’
x2*
x2’’
y’’’
*
y  (p1  p2 )x1
Engel
curve;
good 1
y’’
y’
x1’’’
x1’
x1’’
x1*

Another example of computing the equations
of Engel curves; the perfectly-substitution case.
U( x1 , x 2 )  x1  x 2 .

The ordinary demand equations are
0, if p1  p2
x ( p1 , p2 , m)  
m / p1, if p1  p2
*
1
0, if p1  p2
x ( p1 , p2 , m)  
m / p2 , if p1  p2 .
*
2
Suppose p1 < p2
y
y
*
x 2  0.
*
y  p1x1
x1*
Engel curve
for good 1
0
Engel curve
for good 2
It is the opposite when p1 > p2.
x2*

Quasi-linear preferences
U( x1 , x 2 )  f ( x1 )  x 2 .

For example,
U( x1 , x 2 )  x1  x 2 .
Income Changes; Quasilinear
Utility
x
2
~
x1
x1
Income Changes; Quasilinear
y
Utility
x
2
Engel
curve
for
good 2
x2*
Engel
curve
for
good 1
y
~
x1
x1
~
x1
x1*
Income Changes; Good 2 Is Normal,
Good 1 Becomes yIncome Inferior
x2
Engel curve
for good 2
y
x2*
Engel curve
for good 1
x1
x1*



If income and prices all change by the same
proportion k, how does the consumer demand
change? (e.g. same rate of inflation for prices and
income)
Recall that prices and income only affects the
budget constraint: p1 x1 + p2 x2 = m.
Now it becomes: kp1 x1 + kp2 x2 = km.
Which clearly gets back to the original one.
Thus, xi (kp1 , kp2 , km) = xi (p1 , p2 , m)
We sometimes call this no money illusion.
70


How are the demand curves and Engel curves
look like for a Cobb-Douglas utility?
Recall the Cobb-Douglas utility function:
a b
U( x1 , x 2 )  x1 x 2 .

On solving using the MRS=p1/ p2 and the
budget constraint, you will have:
71
Then the ordinary demand functions for
commodities 1 and 2 are
a
m
x ( p1 , p2 , m) 

a  b p1
*
1
b
m
x ( p1 , p2 , m) 
 .
a  b p2
*
2



Own-Price effect: inversely related to its own
price.
Cross-Price effect: no cross price effect
Income Effect: Engle curve is a straight line
passing through the origin.
73
Own-Price Changes
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
is
am
x 
(a  b) p1
*
1
x2* 
bm
( a  b) p2
x1*
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
am
bm
*
x 
; x2 
.
(a  b) p1
( a  b ) p2
*
1
Rearranged to isolate m, these are:
(a  b) p1 *
m
x1
a
( a  b) p2 *
m
x2
b
Engel curve for good 1
Engel curve for good 2
m
m
(a  b) p1 *
m
x1
a
Engel curve
for good 1
x1*
( a  b ) p2 *
m
x2
b
x2*
Engel curve
for good 2


Demand Elasticity measures the percentage
change of demand as a result of one percent
change in exogenous variable.
Own price elasticity of demand:
dxi / xi
pi dxi
 

dpi / pi xi dpi
i
i
77

Cross Price Elasticity of demand:
p j dxi
dxi / xi
i 

dp j / p j xi dp j
j

Income Elasticity of demand:
dxi / xi m dxi
 

dm / m xi dm
m
i
78

 ii  0
Ordinary Good:
Giffen Good:   0
i
i

Gross Substitutes:   0
Gross Complements:   0
j
i
j
i

Normal Good:
Inferior Good: 
 im  0
m
i
0
79
The consumer demand function for a good
generally depends on prices of all goods and
income.
 Ordinary: demand decreases with own price
Giffen: demand increases with own price
 Substitute: demand increases with other price
Complement: demand decreases with other price
 Normal good: demand increases with income
Inferior good: demand decreases with income

80
```