Chapter 2 Budget Constraint

Report
Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics
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
Economists assume that consumers choose the
best bundle of goods they can afford.
This chapter first specifies in detail what
consumer can afford: the budget constraint or
the consumption possibility set.
 What is best for consumer, or the preference on
the possible consumption bundles, will be
discussed in the next chapter.

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

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, … , xn).
Commodity prices are p1, p2, … , pn.
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
Q: When is a consumption bundle (x1, … , xn)
affordable at prices p1, … , pn?
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
Q: When is a consumption bundle (x1, … , xn)
affordable at prices p1, … , pn?
A: When total expenditure is smaller than
income:
p1x1 + … + pnxn  m
where m is the consumer’s (disposable) income.
 That is, one’s total expenditure is smaller than
or equal to one’s income.

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
The bundles that are only just affordable by
the consumer is one’s budget constraint.
This is the set
{ (x1,…,xn) | x1  0, …, xn  and
p1x1 + … + pnxn = m }.
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

The consumer’s budget set is the set of all
affordable bundles;
B(p1, … , pn, m) =
{ (x1, … , xn) | x1  0, … , xn 0 and
p1x1 + … + pnxn  m }
The budget constraint is the upper boundary of
the budget set.
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x2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
m /p1
x1
8
x2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
m /p1
x1
9
x2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
Just affordable
m /p1
x1
10
x2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
Not affordable
Just affordable
m /p1
x1
11
x2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
Not affordable
Just affordable
Affordable
m /p1
x1
12
x2
m /p2
Budget constraint (budget line) is
p1x1 + p2x2 = m.
the collection
of all affordable bundles.
Budget
Set
m /p1
x1
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x2
m /p2
p1x1 + p2x2 = m is
x2 = -(p1/p2)x1 + m/p2
so slope is -p1/p2.
Budget
Set
m /p1
x1
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
For n = 2 and x1 on the horizontal axis, the
constraint’s slope is -p1/p2. What does it
mean?
x2 = 

p1
p2
x1 
m
p2
Note: For a linear line: y=mx+c, m is the
slope of the line, while c is the y-intercept.
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
For n = 2 and x1 on the horizontal axis, the
constraint’s slope is -p1/p2. What does it
mean?
x2 = 

p1
p2
x1 
m
p2
To hold income m constant, increasing x1 by
1 must reduce x2 by p1/p2 .
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x2
Slope is -p1/p2
-p1/p2
+1
x1
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x2
Opp. cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2.
-p1/p2
+1
x1
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x2
The opp. cost of an extra
unit of commodity 2 is
p2/p1 units foregone
of commodity 1.
+1
-p2/p1
x1
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
The budget constraint and budget set depend
upon prices and income. What happens as
prices or income change?
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x2
Original
budget set
x1
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x2
New affordable consumption
choices
Original and
new budget
constraints are
parallel (same
slope).
Original
budget set
x1
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x2
Original
budget set
x1
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x2
Consumption bundles
that are no longer
affordable.
New, smaller
budget set
Old and new
constraints
are parallel.
x1
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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.
 The slope –p1 / p2 does not change.

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

When income increases, NO original choice is
lost and new choices are added, so higher
income cannot make a consumer worse off.
When income decreases, the consumer may
(typically will) be worse off, as one can no
longer afford some of the bundles anymore.
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
What happens if just one price decreases?

Suppose p1 decreases.
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x2
m/p2
-p1’/p2
Original
budget set
m/p1’
m/p1”
x1
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x2
m/p2
New affordable choices
-p1’/p2
Original
budget set
m/p1’
m/p1”
x1
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x2
m/p2
New affordable choices
-p1’/p2
Original
budget set
Budget constraint
pivots; slope flattens
from -p1’/p2 to
-p1”/p2
-p ”/p
1
m/p1’
2
m/p1”
x1
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

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.
Similarly, increasing one price pivots the
constraint inwards (consider a price change
from p1” to p1’), reduces choice and may
(typically will) make the consumer worse off.
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Quantity/per-unit tax: price increases from p to
p+t.
 Quantity/per-unit subsidy: price decreases from
p to p-s.
 Ad valorem/value tax: price increases from p to
(1+t)p
 Ad valorem/value subsidy: price decreases from
p to (1-s)p

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
A uniform sales tax levied at rate t on all goods
changes the constraint from
p1x1 + p2x2 = m
to
(1+t)p1x1 + (1+t)p2x2 = m
i.e.
p1x1 + p2x2 = m/(1+t).
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x2
m
p2
m
( 1  t ) p2
Equivalent income loss
is
m
m
1 t
=
t
1 t
m
m
( 1  t ) p1
p1
m
x1
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x2
m
p2
m
( 1  t ) p2
A uniform ad valorem
sales tax levied at rate t
is equivalent to an income
tax levied at rate t
.
1 t
m
m
( 1  t ) p1
p1
x1
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Lump-sum tax: government tax a fixed sum of
money, T, regardless of individual’s behavior.
 This is equivalent to a decrease in income by T,
implying an inward parallel shift of budget line.


Similarly, lump-sum subsidy S implies an
outward parallel shift of budget line
corresponding to an amount S.
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


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?
Here we assume one of the two goods is food.
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

Suppose m = $100, pF = $1 and the price of
“other goods” is pG = $1.
The budget constraint is then
F + G =100.
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G
F + G = 100: before stamps.
100
100
F
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G
F + G = 100: before stamps.
100
Budget set after $40 food
stamps issued.
40
100 140
F
40
G
F + G = 100: before stamps.
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Budget set after $40 food
stamps issued.
The family’s budget
set is enlarged.
40
100 140
F
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

How does the unit of account affect the budget
constraints and budget set?
Suppose prices and income are measured in
dollars. Say p1=$2, p2=$3, m = $12. Then the
constraint is
2x1 + 3x2 = 12.
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

If prices and income are measured in cents,
then p1=200, p2=300, m=1200 and the
constraint is
200x1 + 300x2 = 1200,
upon simplification, it is the same as
2x1 + 3x2 = 12.
Changing the unit of account changes neither
the budget constraint nor the budget set.
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


The constraint for p1=2, p2=3, m=12
2x1 + 3x2 = 12
is also 1x1 + (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.
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


Multiplying all prices and income by any constant
k does not change the budget constraint.
kp1x1 +kp2x2 =km
Any commodity can be chosen as the numeraire
(by taking k=1/pi) without changing the budget set
or the budget constraint.
It is also clear from the graph that, only the ratios
p1/p2, m/p1 and m/p2 are relevant to the budget
line and budget set.
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

Q: What makes a budget constraint a straight
line?
A: A straight line has a constant slope and the
constraint is
p1x1 + … + pnxn = m
so if prices are constants then a constraint is a
straight line.
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


But what if prices are not constants?
E.g. bulk buying discounts, or price penalties (or
tax) for buying “too much”.
Then constraints will be curved or have kinks.
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
Suppose p2 is constant at $1 but that p1=$2 for
0  x1  20 and p1=$1 for x1>20.
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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

{

Also assume m=100.
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x2
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
100
m = $100
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
20
50
80
x1
50
x2
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
100
m = $100
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
20
50
80
x1
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x2
m = $100
100
Budget Constraint
Budget Set
20
50
80
x1
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x2
Budget
Constraint
Budget Set
x1
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The budget set describes what consumption
bundles are affordable to the consumers.
 The budget constraint is typically described by
p1 x1 + p2 x2 = m, which is a straight line when
prices are constant.
 When income increases, budget set shifts
outward, enlarging the budget set.
 When prices increases, the slope of budget line
changes, and the it shrinks the budget set.

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

The next chapter will introduce preference,
which describes the ordering of what a
consumer likes among the consumption
bundles.
Then we can combine both preference and
budget constraint to analyze consumer’s choice.
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