Chapter Two
Budget Constraint
预算约束
Structure
?Describe budget constraint
–Algebra
–Graph
?Describe changes in budget
constraint
?Government programs and budget
constraints
?Non-linear budget lines
Consumption Sets
?A consumption set （ 消费集） is the
collection of all physically possible
consumption bundles （ 消费束） to
the consumer.
Consumption Bundle
?A consumption bundle containing x1 units
of commodity 1,x2 units of commodity 2
and so on up to xn units of commodity n
is denoted by the vector (x1,x2,…,x n).
Physical Constraints
?Non-negative:
Consumption set,
X={ (x1,…,x n) | x1 ? 0,…,x n ??0 }
?You only have 24 hours a day
?Subsistence need
?Etc.
Budget Constraint
?What constrains consumption
choice?
–Budgetary
–time
–other resource limitations.
Budget Constraints
?Commodity prices are p1,p2,…,p n.
?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 bundles that are only just
affordable form the consumer’s
budget constraint,This is the set
{ (x1,…,x n) | x1 ? 0,…,x n ??? and
p1x1 + … + p nxn = m }.
Budget Constraints
?The consumer’s budget set （ 预算集
） 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
m /p2
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.m /p2
m /p1
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
m /p2
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
Not affordable
m /p2
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
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Budget
Set
the collection
of all affordable bundles.
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
Meaning of the Slope
?
?Increasing x1 by 1 must reduce x2 by
p1/p2.
?Opportunity cost of consuming x1
?Or,the rate of exchange that market
allows.
x p
p
x m
p2
1
2
1
2
= ? ?
Budget Constraints
x2
x1
Slope is -p1/p2
+1
-p1/p2
Budget Constraints
x2
x1
+1
-p1/p2
Opp,cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2.
Budget Constraints
x2
x1
Opp,cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2,
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?
How do the budget set and budget
constraint change as income m
increases?
Original
budget set
x2
x1
Higher income gives more choice
Original
budget set
New affordable consumption
choices
x2
x1
Original and
new budget
constraints are
parallel (same
slope).
How do the budget set and budget
constraint change as income m
decreases?
Original
budget set
x2
x1
How do the budget set and budget
constraint change as income m
decreases?x2
x1
New,smaller
budget set
Consumption bundles
that are no longer
affordable.
Old and new
constraints
are parallel.
Budget Constraints - Income
Changes
?Increases in income m shift the
constraint outward in a parallel
manner,thereby enlarging the
budget set and improving choice.
?Decreases in income m shift the
constraint inward in a parallel
manner,thereby shrinking the
budget set and reducing choice.
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
”
-p1’/p2
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
-p1’/p2
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.
Budget Constraints - Price
Changes
?Similarly,increasing one price pivots
the constraint inwards,reduces
choice and may (typically will) make
the consumer worse off.
Uniform Ad Valorem Sales Taxes
? An ad valorem sales tax （ 从价营业税）
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
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
p2
m
p1
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)
m
t p( )1 1?
m
t p( )1 2?
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
Uniform Ad Valorem Sales Taxesx
2
x1
m
t p( )1 2?
m
p2
m
t p( )1 1?
m
p1
A uniform ad valorem
sales tax levied at rate t
is equivalent to an income
tax levied at rate
t
t1?,
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 Program
?Suppose m = $100,pF = $1 and the
price of,other goods” is pG = $1.
?“Other goods” is a composite good (
复合商品 )
–It simplifies the analysis to a 2-
good model.
?The budget constraint is then
F + G =100.
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
The Food Stamp Program
F + G = 100,before stamps.
Budget set after 40 food
stamps issued.
G
F100
100
14040
(F-40) + G = 100 for F?40
G=100 for F<40
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget set after 40 food
stamps issued.
140
The family’s budget
set is enlarged.
40
(F-40) + G = 100 for F?40
G=100 for F<40
The Food Stamp Program
?What if food stamps can be traded on
a black market for $0.50 each?
?F+G=100+0.5?(40-F) for F<40
?(F-40)+G=100 for F ? 40
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
?F+G=100+0.5?(40-F) for F<40
?(F-40)+G=100 for F ? 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” (记价物 ) means,unit of
account”.
? The budget line
p1x1 + p2x2 = m
? Can be equivalently expressed as
? Setting p2=1 makes commodity 2 the
numeraire and defines all prices relative to
p2;
1
12
22
p mxx
pp
?=
Budget Constraints - Relative
Prices
?Any commodity can be chosen as
the numeraire without changing the
budget set or the budget constraint.
?A relative price is the rates of
exchange of commodities 2 for units
of commodity 1.
Shapes of Budget Constraints
?Q,What makes a budget constraint a
straight line?
?A,A straight line has a constant
slope and the constraint is
p1x1 + … + p nxn = m
so if prices are constants then a
constraint is a straight line.
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{
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
Budget Constraints with a
Quantity Discount
?The constraint is
2x1+x2=m for 0 ? x1 ? 20
2?20+(x1-20)+x2=m for x1 > 20{
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.
Budget Constraint
预算约束
Structure
?Describe budget constraint
–Algebra
–Graph
?Describe changes in budget
constraint
?Government programs and budget
constraints
?Non-linear budget lines
Consumption Sets
?A consumption set （ 消费集） is the
collection of all physically possible
consumption bundles （ 消费束） to
the consumer.
Consumption Bundle
?A consumption bundle containing x1 units
of commodity 1,x2 units of commodity 2
and so on up to xn units of commodity n
is denoted by the vector (x1,x2,…,x n).
Physical Constraints
?Non-negative:
Consumption set,
X={ (x1,…,x n) | x1 ? 0,…,x n ??0 }
?You only have 24 hours a day
?Subsistence need
?Etc.
Budget Constraint
?What constrains consumption
choice?
–Budgetary
–time
–other resource limitations.
Budget Constraints
?Commodity prices are p1,p2,…,p n.
?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 bundles that are only just
affordable form the consumer’s
budget constraint,This is the set
{ (x1,…,x n) | x1 ? 0,…,x n ??? and
p1x1 + … + p nxn = m }.
Budget Constraints
?The consumer’s budget set （ 预算集
） 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
m /p2
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.m /p2
m /p1
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
m /p2
Budget Set and Constraint for
Two Commoditiesx
2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
Not affordable
m /p2
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
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Budget
Set
the collection
of all affordable bundles.
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
Meaning of the Slope
?
?Increasing x1 by 1 must reduce x2 by
p1/p2.
?Opportunity cost of consuming x1
?Or,the rate of exchange that market
allows.
x p
p
x m
p2
1
2
1
2
= ? ?
Budget Constraints
x2
x1
Slope is -p1/p2
+1
-p1/p2
Budget Constraints
x2
x1
+1
-p1/p2
Opp,cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2.
Budget Constraints
x2
x1
Opp,cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2,
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?
How do the budget set and budget
constraint change as income m
increases?
Original
budget set
x2
x1
Higher income gives more choice
Original
budget set
New affordable consumption
choices
x2
x1
Original and
new budget
constraints are
parallel (same
slope).
How do the budget set and budget
constraint change as income m
decreases?
Original
budget set
x2
x1
How do the budget set and budget
constraint change as income m
decreases?x2
x1
New,smaller
budget set
Consumption bundles
that are no longer
affordable.
Old and new
constraints
are parallel.
Budget Constraints - Income
Changes
?Increases in income m shift the
constraint outward in a parallel
manner,thereby enlarging the
budget set and improving choice.
?Decreases in income m shift the
constraint inward in a parallel
manner,thereby shrinking the
budget set and reducing choice.
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
”
-p1’/p2
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
-p1’/p2
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.
Budget Constraints - Price
Changes
?Similarly,increasing one price pivots
the constraint inwards,reduces
choice and may (typically will) make
the consumer worse off.
Uniform Ad Valorem Sales Taxes
? An ad valorem sales tax （ 从价营业税）
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
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
p2
m
p1
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)
m
t p( )1 1?
m
t p( )1 2?
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
Uniform Ad Valorem Sales Taxesx
2
x1
m
t p( )1 2?
m
p2
m
t p( )1 1?
m
p1
A uniform ad valorem
sales tax levied at rate t
is equivalent to an income
tax levied at rate
t
t1?,
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 Program
?Suppose m = $100,pF = $1 and the
price of,other goods” is pG = $1.
?“Other goods” is a composite good (
复合商品 )
–It simplifies the analysis to a 2-
good model.
?The budget constraint is then
F + G =100.
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
The Food Stamp Program
F + G = 100,before stamps.
Budget set after 40 food
stamps issued.
G
F100
100
14040
(F-40) + G = 100 for F?40
G=100 for F<40
The Food Stamp ProgramG
F100
100
F + G = 100,before stamps.
Budget set after 40 food
stamps issued.
140
The family’s budget
set is enlarged.
40
(F-40) + G = 100 for F?40
G=100 for F<40
The Food Stamp Program
?What if food stamps can be traded on
a black market for $0.50 each?
?F+G=100+0.5?(40-F) for F<40
?(F-40)+G=100 for F ? 40
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
?F+G=100+0.5?(40-F) for F<40
?(F-40)+G=100 for F ? 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” (记价物 ) means,unit of
account”.
? The budget line
p1x1 + p2x2 = m
? Can be equivalently expressed as
? Setting p2=1 makes commodity 2 the
numeraire and defines all prices relative to
p2;
1
12
22
p mxx
pp
?=
Budget Constraints - Relative
Prices
?Any commodity can be chosen as
the numeraire without changing the
budget set or the budget constraint.
?A relative price is the rates of
exchange of commodities 2 for units
of commodity 1.
Shapes of Budget Constraints
?Q,What makes a budget constraint a
straight line?
?A,A straight line has a constant
slope and the constraint is
p1x1 + … + p nxn = m
so if prices are constants then a
constraint is a straight line.
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{
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
Budget Constraints with a
Quantity Discount
?The constraint is
2x1+x2=m for 0 ? x1 ? 20
2?20+(x1-20)+x2=m for x1 > 20{
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.
