Chapter Two
Budget Constraint
Where are We in the Course?
? We are working on the 1st of the 3
components of microeconomics,
Consumer behavior,production
theory,and market.
?There are three elements of
consumer behavior,budget
constraint,preference,and choices.
Consumption Choice Sets
?A consumption choice set is the
collection of all consumption choices
available to the consumer.
?What constrains consumption
choice?
–Budgetary,time and other
resource limitations.
Budget Constraints
?Q,When is a bundle (x1,…,x n)
affordable at prices p1,…,p n?
?A,When
p1x1 + … + p nxn ? m
where m is the consumer’s
(disposable) income.
Budget Constraints
?The consumer’s budget set is the set
of all affordable bundles;
B(p1,…,p n,m) =
{ (x1,…,x n) | x1 ? 0,…,x n ??0 and
p1x1 + … + p nxn ? m }
?The budget constraint is the upper
boundary of the budget set.
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Affordable
Just affordable
Not affordable
m /p2
Budget Set and Constraint for
Two Commoditiesx
2
x1
p1x1 + p2x2 = m is
x2 = -(p1/p2)x1 + m/p2
so slope is -p1/p2.
m /p1
Budget
Set
m /p2
Budget Constraints
?If n = 3 what do the budget constraint
and the budget set look like?
Budget Set for Three
Commodities
x2
x1
x3
m /p2
m /p1
m /p3
{ (x1,x2,x3) | x1 ? 0,x2 ? 0,x3 ?0 and
p1x1 + p2x2 + p3x3 ? m}
Budget Constraints
?For n = 2 and x1 on the horizontal
axis,the constraint’s slope is -p1/p2,
What does it mean?
?Increasing x1 by 1 must reduce x2 by
p1/p2.
x p
p
x m
p2
1
2
1
2
? ? ?
Budget Constraints
x2
x1
Opp,cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2,And
the opp,cost of an extra
unit of commodity 2 is
p2/p1 units foregone
of commodity 1,
-p2/p1
+1
Budget Sets & Constraints;
Income and Price Changes
?The budget constraint and budget
set depend upon prices and income,
What happens as prices or income
change?
Higher income gives more choice
Original
budget set
New affordable consumption
choices
x2
x1
Original and
new budget
constraints are
parallel (same
slope).
Budget Constraints - Income
Changes
?No original choice is lost and new
choices are added when income
increases,so higher income cannot
make a consumer worse off.
?An income decrease may (typically
will) make the consumer worse off.
Budget Constraints - Price
Changes
?What happens if just one price
decreases?
?Suppose p1 decreases.
How do the budget set and budget
constraint change as p1 decreases
from p1’ to p1”?
Original
budget set
x2
x1
m/p2
m/p1’ m/p1
”
New affordable choices
Budget constraint
pivots; slope flattens
from -p1’/p2 to
-p1”/p2
-p1’/p2
-p1”/p2
Budget Constraints - Price
Changes
?Reducing the price of one
commodity pivots the constraint
outward,No old choice is lost and
new choices are added,so reducing
one price cannot make the consumer
worse off.
Uniform Ad Valorem Sales Taxes
（ 从价税）
?An ad valorem sales tax levied at a
rate of 5% increases all prices by 5%,
from p to (1+0× 05)p = 1× 05p.
?An ad valorem sales tax levied at a
rate of t increases all prices by tp
from p to (1+t)p.
?A uniform sales tax is applied
uniformly to all commodities.
Uniform Ad Valorem Sales Taxes
?A uniform sales tax levied at rate t
changes the constraint from
p1x1 + p2x2 = m
to
(1+t)p1x1 + (1+t)p2x2 = m
i.e.
p1x1 + p2x2 = m/(1+t).
Uniform Ad Valorem Sales Taxesx
2
x1
m
t p( )1 2?
m
p2
m
t p( )1 1?
m
p1
Equivalent income loss
is
m m t t t m? ? ? ?1 1
The Food Stamp Program
?Food stamps are coupons that can
be legally exchanged only for food.
?How does a commodity-specific gift
such as a food stamp alter a family’s
budget constraint?
The Food Stamp ProgramG
F100
100
F + G = 100; before stamps.
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget set after 40 food
stamps issued.
140
The budget
set is enlarged.
40
The Food Stamp Program
?What if food stamps can be traded on
a black market for $0.50 each?
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget constraint after 40
food stamps issued.
140
120
Budget constraint with
black market trading.
40
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget constraint after 40
food stamps issued.
140
120
Black market trading
makes the budget
set larger again.
40
Budget Constraints - Relative
Prices
?,Numeraire” means,unit of
account”.
?Suppose prices and income are
measured in dollars,Say p1=$2,
p2=$3,m = $12,Then the constraint
is
2x1 + 3x2 = 12.
Budget Constraints - Relative
Prices
?The constraint for p1=2,p2=3,m=12
2x1 + 3x2 = 12
is also 1.x1 + (3/2)x2 = 6,
the constraint for p1=1,p2=3/2,m=6,
Setting p1=1 makes commodity 1 the
numeraire and defines all prices
relative to p1; e.g,3/2 is the price of
commodity 2 relative to the price of
commodity 1,
Budget Constraints - Relative
Prices
?Any commodity can be chosen as
the numeraire without changing the
budget set or the budget constraint.
Budget Constraints - Relative Prices
?p1=2,p2=3 and p3=6 ?
?price of commodity 2 relative to
commodity 1 is 3/2,
?price of commodity 3 relative to
commodity 1 is 3.
?Relative prices are the rates of
exchange of commodities 2 and 3 for
units of commodity 1.
Shapes of Budget Constraints
?But what if prices are not constants?
?E.g,bulk buying discounts,or price
penalties for buying,too much”.
?Then constraints will be curved.
Shapes of Budget Constraints -
Quantity Discounts
?Suppose p2 is constant at $1 but that
p1=$2 for 0 ? x1 ? 20 and p1=$1 for
x1>20.
Shapes of Budget Constraints -
Quantity Discounts
?Suppose p2 is constant at $1 but that
p1=$2 for 0 ? x1 ? 20 and p1=$1 for
x1>20,Then the constraint’s slope is
- 2,for 0 ? x1 ? 20
-p1/p2 =
- 1,for x1 > 20
and the constraint is
{
Shapes of Budget Constraints
with a Quantity Discount
m = $100
50
100
20
Slope = - 2 / 1 = - 2
(p1=2,p2=1)
Slope = - 1/ 1 = - 1
(p1=1,p2=1)
80
x2
x1
Shapes of Budget Constraints
with a Quantity Discount
m = $100
50
100
20
Slope = - 2 / 1 = - 2
(p1=2,p2=1)
Slope = - 1/ 1 = - 1
(p1=1,p2=1)
80
x2
x1
Shapes of Budget Constraints
with a Quantity Discount
m = $100
50
100
20 80
x2
x1
Budget Set
Budget Constraint
Shapes of Budget Constraints
with a Quantity Penaltyx
2
x1
Budget Set
Budget
Constraint
Shapes of Budget Constraints -
One Price Negative
?Commodity 1 is stinky garbage,You
are paid $2 per unit to accept it; i.e.
p1 = - $2,p2 = $1,Income,other than
from accepting commodity 1,is m =
$10.
?Then the constraint is
- 2x1 + x2 = 10 or x2 = 2x1 + 10.
Shapes of Budget Constraints -
One Price Negative
10
Budget constraint’s slope is
-p1/p2 = -(-2)/1 = +2
x2
x1
x2 = 2x1 + 10
Shapes of Budget Constraints -
One Price Negative
10
x2
x1
Budget set is
all bundles for
which x1 ? 0,
x2 ? 0 and
x2 ? 2x1 + 10.
More General Choice Sets
?Choices are usually constrained by
more than a budget; e.g,time
constraints and other resources
constraints.
?A bundle is available only if it meets
every constraint.
More General Choice Sets
Food
Other Stuff
10
At least 10 units of food
must be eaten to survive
More General Choice Sets
Food
Other Stuff
10
Budget Set
Choice is also budget
constrained.
More General Choice Sets
Food
Other Stuff
10
Choice is further restricted by
a time constraint.
More General Choice Sets
So what is the choice set?
More General Choice Sets
Food
Other Stuff
10
More General Choice Sets
Food
Other Stuff
10
More General Choice Sets
Food
Other Stuff
10
The choice set is the
intersection of all of
the constraint sets.
Summary
1,The choice set is the set consists of all
possible choices of a consumer;
2,In the simplest case,the choice set is
typically bounded by a single budget
constraint whose slope is the negative
ratio of prices;
3,In general the choice set can be
bounded by multiple constraints.
Budget Constraint
Where are We in the Course?
? We are working on the 1st of the 3
components of microeconomics,
Consumer behavior,production
theory,and market.
?There are three elements of
consumer behavior,budget
constraint,preference,and choices.
Consumption Choice Sets
?A consumption choice set is the
collection of all consumption choices
available to the consumer.
?What constrains consumption
choice?
–Budgetary,time and other
resource limitations.
Budget Constraints
?Q,When is a bundle (x1,…,x n)
affordable at prices p1,…,p n?
?A,When
p1x1 + … + p nxn ? m
where m is the consumer’s
(disposable) income.
Budget Constraints
?The consumer’s budget set is the set
of all affordable bundles;
B(p1,…,p n,m) =
{ (x1,…,x n) | x1 ? 0,…,x n ??0 and
p1x1 + … + p nxn ? m }
?The budget constraint is the upper
boundary of the budget set.
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Affordable
Just affordable
Not affordable
m /p2
Budget Set and Constraint for
Two Commoditiesx
2
x1
p1x1 + p2x2 = m is
x2 = -(p1/p2)x1 + m/p2
so slope is -p1/p2.
m /p1
Budget
Set
m /p2
Budget Constraints
?If n = 3 what do the budget constraint
and the budget set look like?
Budget Set for Three
Commodities
x2
x1
x3
m /p2
m /p1
m /p3
{ (x1,x2,x3) | x1 ? 0,x2 ? 0,x3 ?0 and
p1x1 + p2x2 + p3x3 ? m}
Budget Constraints
?For n = 2 and x1 on the horizontal
axis,the constraint’s slope is -p1/p2,
What does it mean?
?Increasing x1 by 1 must reduce x2 by
p1/p2.
x p
p
x m
p2
1
2
1
2
? ? ?
Budget Constraints
x2
x1
Opp,cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2,And
the opp,cost of an extra
unit of commodity 2 is
p2/p1 units foregone
of commodity 1,
-p2/p1
+1
Budget Sets & Constraints;
Income and Price Changes
?The budget constraint and budget
set depend upon prices and income,
What happens as prices or income
change?
Higher income gives more choice
Original
budget set
New affordable consumption
choices
x2
x1
Original and
new budget
constraints are
parallel (same
slope).
Budget Constraints - Income
Changes
?No original choice is lost and new
choices are added when income
increases,so higher income cannot
make a consumer worse off.
?An income decrease may (typically
will) make the consumer worse off.
Budget Constraints - Price
Changes
?What happens if just one price
decreases?
?Suppose p1 decreases.
How do the budget set and budget
constraint change as p1 decreases
from p1’ to p1”?
Original
budget set
x2
x1
m/p2
m/p1’ m/p1
”
New affordable choices
Budget constraint
pivots; slope flattens
from -p1’/p2 to
-p1”/p2
-p1’/p2
-p1”/p2
Budget Constraints - Price
Changes
?Reducing the price of one
commodity pivots the constraint
outward,No old choice is lost and
new choices are added,so reducing
one price cannot make the consumer
worse off.
Uniform Ad Valorem Sales Taxes
（ 从价税）
?An ad valorem sales tax levied at a
rate of 5% increases all prices by 5%,
from p to (1+0× 05)p = 1× 05p.
?An ad valorem sales tax levied at a
rate of t increases all prices by tp
from p to (1+t)p.
?A uniform sales tax is applied
uniformly to all commodities.
Uniform Ad Valorem Sales Taxes
?A uniform sales tax levied at rate t
changes the constraint from
p1x1 + p2x2 = m
to
(1+t)p1x1 + (1+t)p2x2 = m
i.e.
p1x1 + p2x2 = m/(1+t).
Uniform Ad Valorem Sales Taxesx
2
x1
m
t p( )1 2?
m
p2
m
t p( )1 1?
m
p1
Equivalent income loss
is
m m t t t m? ? ? ?1 1
The Food Stamp Program
?Food stamps are coupons that can
be legally exchanged only for food.
?How does a commodity-specific gift
such as a food stamp alter a family’s
budget constraint?
The Food Stamp ProgramG
F100
100
F + G = 100; before stamps.
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget set after 40 food
stamps issued.
140
The budget
set is enlarged.
40
The Food Stamp Program
?What if food stamps can be traded on
a black market for $0.50 each?
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget constraint after 40
food stamps issued.
140
120
Budget constraint with
black market trading.
40
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget constraint after 40
food stamps issued.
140
120
Black market trading
makes the budget
set larger again.
40
Budget Constraints - Relative
Prices
?,Numeraire” means,unit of
account”.
?Suppose prices and income are
measured in dollars,Say p1=$2,
p2=$3,m = $12,Then the constraint
is
2x1 + 3x2 = 12.
Budget Constraints - Relative
Prices
?The constraint for p1=2,p2=3,m=12
2x1 + 3x2 = 12
is also 1.x1 + (3/2)x2 = 6,
the constraint for p1=1,p2=3/2,m=6,
Setting p1=1 makes commodity 1 the
numeraire and defines all prices
relative to p1; e.g,3/2 is the price of
commodity 2 relative to the price of
commodity 1,
Budget Constraints - Relative
Prices
?Any commodity can be chosen as
the numeraire without changing the
budget set or the budget constraint.
Budget Constraints - Relative Prices
?p1=2,p2=3 and p3=6 ?
?price of commodity 2 relative to
commodity 1 is 3/2,
?price of commodity 3 relative to
commodity 1 is 3.
?Relative prices are the rates of
exchange of commodities 2 and 3 for
units of commodity 1.
Shapes of Budget Constraints
?But what if prices are not constants?
?E.g,bulk buying discounts,or price
penalties for buying,too much”.
?Then constraints will be curved.
Shapes of Budget Constraints -
Quantity Discounts
?Suppose p2 is constant at $1 but that
p1=$2 for 0 ? x1 ? 20 and p1=$1 for
x1>20.
Shapes of Budget Constraints -
Quantity Discounts
?Suppose p2 is constant at $1 but that
p1=$2 for 0 ? x1 ? 20 and p1=$1 for
x1>20,Then the constraint’s slope is
- 2,for 0 ? x1 ? 20
-p1/p2 =
- 1,for x1 > 20
and the constraint is
{
Shapes of Budget Constraints
with a Quantity Discount
m = $100
50
100
20
Slope = - 2 / 1 = - 2
(p1=2,p2=1)
Slope = - 1/ 1 = - 1
(p1=1,p2=1)
80
x2
x1
Shapes of Budget Constraints
with a Quantity Discount
m = $100
50
100
20
Slope = - 2 / 1 = - 2
(p1=2,p2=1)
Slope = - 1/ 1 = - 1
(p1=1,p2=1)
80
x2
x1
Shapes of Budget Constraints
with a Quantity Discount
m = $100
50
100
20 80
x2
x1
Budget Set
Budget Constraint
Shapes of Budget Constraints
with a Quantity Penaltyx
2
x1
Budget Set
Budget
Constraint
Shapes of Budget Constraints -
One Price Negative
?Commodity 1 is stinky garbage,You
are paid $2 per unit to accept it; i.e.
p1 = - $2,p2 = $1,Income,other than
from accepting commodity 1,is m =
$10.
?Then the constraint is
- 2x1 + x2 = 10 or x2 = 2x1 + 10.
Shapes of Budget Constraints -
One Price Negative
10
Budget constraint’s slope is
-p1/p2 = -(-2)/1 = +2
x2
x1
x2 = 2x1 + 10
Shapes of Budget Constraints -
One Price Negative
10
x2
x1
Budget set is
all bundles for
which x1 ? 0,
x2 ? 0 and
x2 ? 2x1 + 10.
More General Choice Sets
?Choices are usually constrained by
more than a budget; e.g,time
constraints and other resources
constraints.
?A bundle is available only if it meets
every constraint.
More General Choice Sets
Food
Other Stuff
10
At least 10 units of food
must be eaten to survive
More General Choice Sets
Food
Other Stuff
10
Budget Set
Choice is also budget
constrained.
More General Choice Sets
Food
Other Stuff
10
Choice is further restricted by
a time constraint.
More General Choice Sets
So what is the choice set?
More General Choice Sets
Food
Other Stuff
10
More General Choice Sets
Food
Other Stuff
10
More General Choice Sets
Food
Other Stuff
10
The choice set is the
intersection of all of
the constraint sets.
Summary
1,The choice set is the set consists of all
possible choices of a consumer;
2,In the simplest case,the choice set is
typically bounded by a single budget
constraint whose slope is the negative
ratio of prices;
3,In general the choice set can be
bounded by multiple constraints.
