Chapter Two
Budgetary and Other Constraints
on Choice
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
• A consumption bundle containing x1 units
of commodity 1, and x2 units of commodity
2 is denoted by the vector (x1, x2).
• Commodity prices are p1, p2
Budget Constraints
• Q: When is a consumption bundle
(x1, x2) affordable at given prices p1, p2?
• When
p1x1 + p2 x2  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, x2 ) | such that p1x1 + p2 x2 = m }.
• The budget constraint is the upper
boundary of the budget set
Budget Set and Constraint for
Two
Commodities
x
2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
m /p1
x1
Budget Set and Constraint for
Two
Commodities
x
2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
Just affordable
m /p1
x1
Budget Set and Constraint for
Two
Commodities
x
2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
Not affordable
Just affordable
m /p1
x1
Budget Set and Constraint for
Two
Commodities
x
2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
Not affordable
Just affordable
Affordable
m /p1
x1
Budget Set and Constraint for
Two
Commodities
x
2
m /p2
Budget constraint is
p1x1 + p2x2 = m.
the collection
of all affordable bundles.
Budget
Set
m /p1
x1
Budget Set and Constraint for
Two
Commodities
x
2
m /p2
p1x1 + p2x2 = m is
x2 = -(p1/p2)x1 + m/p2
so slope is -p1/p2.
Budget
Set
m /p1
x1
Budget Constraints
• For n = 2 and x1 on the horizontal axis,
the constraint’s slope is -p1/p2. What
does it mean?
p1
m
x2 =  x1

p2
p2
• Increasing x1 by 1 must reduce x2 by
p1/p2.
Budget Constraints
x2
Slope is -p1/p2
-p1/p2
+1
x1
Budget Constraints
x2
Opp. cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2.
-p1/p2
+1
x1
Budget Constraints
x2
Opp. cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2. And
the opp. cost of an extra
+1
unit of commodity 2 is
-p2/p1
p2/p1 units foregone
of commodity 1.
x1
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
x2
income m increases?
Original
budget set
x1
x2
Higher income gives more
choice
New affordable consumption
choices
Original and
new budget
constraints are
parallel (same
slope).
Original
budget set
x1
How do the budget set and
budget constraint change as
x2
income m decreases?
Original
budget set
x1
How do the budget set and
budget constraint change as
x2
income m decreases?
Consumption bundles
that are no longer
affordable.
New, smaller
budget set
Old and new
constraints
are parallel.
x1
Budget Constraints - Income
Changes
• Increases in income m shift the constraint
outward in a parallel manner, thereby
enlarging the budget set and improving
choice.
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
x2 decreases from p ’ to p ”?
1
1
m/p2
-p1’/p2
Original
budget set
m/p1’
m/p1
”
x1
How do the budget set and
budget constraint change as p1
x2 decreases from p ’ to p ”?
1
1
m/p2
New affordable choices
-p1’/p2
Original
budget set
m/p1’
m/p1
”
x1
How do the budget set and
budget constraint change as p1
x2 decreases from p ’ to p ”?
1
1
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
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 levied at a rate of
5% increases the price by 5%, from p to
(1+0 05)p = 1 05p.
• An ad valorem sales tax levied at a rate of
t increases the price 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
Taxes
x2
m
p2
p1x1 + p2x2 = m
m
p1
x1
Uniform Ad Valorem Sales
Taxes
x2
m
p2
m
(1  t ) p2
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)
m
(1  t ) p1
m
p1
x1
Uniform Ad Valorem Sales
Taxes
x2
m
p2
m
(1  t ) p2
Equivalent income loss
is
m
t
m
=
m
1 t 1 t
m
(1  t ) p1
m
p1
x1
Uniform Ad Valorem Taxes
x2
m
p2
m
(1  t ) p2
A uniform ad valorem
sales tax levied at rate t
is equivalent to an
income
t
tax levied at rate
.
1 t
m
(1  t ) p1
m
p1
x1
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.
• The budget constraint is then
F + G =100.
The Food Stamp Program
G
F + G = 100: before stamps.
100
100
F
The Food Stamp Program
G
F + G = 100: before stamps.
100
Budget set after 40 food
stamps issued.
The family’s budget
set is enlarged.
40
100 140
F
The Food Stamp Program
• What if food stamps can be traded on a
black market for $0.50 each?
The Food Stamp Program
G
F + G = 100: before stamps.
Budget constraint after 40
food stamps issued.
Budget constraint with
black market trading.
120
100
40
100 140
F
The Food Stamp Program
G
F + G = 100: before stamps.
Budget constraint after 40
food stamps issued.
Black market trading
makes the budget
set larger again.
120
100
40
100 140
F
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
• If prices and income are measured in
cents, then p1=200, p2=300, m=1200 and
the constraint is
200x1 + 300x2 = 1200,
the same as
2x1 + 3x2 = 12.
• Changing the numeraire changes neither
the budget constraint nor the budget set.
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
• Q: What makes a budget constraint a
straight line?
• A: 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
and the constraint is
{
Shapes of Budget Constraints
with a Quantity Discount
x2
100
m = $100
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
20
50
80
x1
Shapes of Budget Constraints
with a Quantity Discount
x2
100
m = $100
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
20
50
80
x1
Shapes of Budget Constraints
with a Quantity Discount
x2
m = $100
100
Budget Constraint
Budget Set
20
50
80
x1
Shapes of Budget Constraints
with a Quantity Penalty
x2
Budget
Constraint
Budget Set
x1
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
x2
x2 = 2x1 + 10
Budget constraint’s slope is
-p1/p2 = -(-2)/1 = +2
10
x1
Shapes of Budget Constraints One Price Negative
x2
Budget set is
all bundles for
which x1  0,
x2  0 and
x2  2x1 + 10.
10
x1
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
Other Stuff
At least 10 units of food
must be eaten to survive
10
Food
More General Choice Sets
Other Stuff
Choice is also budget
constrained.
Budget Set
10
Food
More General Choice Sets
Other Stuff
Choice is further restricted by
a time constraint.
10
Food
More General Choice Sets
So what is the choice set?
More General Choice Sets
Other Stuff
10
Food
More General Choice Sets
Other Stuff
10
Food
More General Choice Sets
Other Stuff
The choice set is the
intersection of all of
the constraint sets.
10
Food
Chapter Three
Preferences
Preference Relations
• Comparing two different consumption
bundles, x and y:
– strict preference: x is more preferred than is
y.
– weak preference: x is as at least as preferred
as is y.
– indifference: x is exactly as preferred as is y.
Preference Relations
• Those are ordinal relations; i.e. they state
only the order in which bundles are
preferred not the intensity of preference
Preference Relations
p
p
•
denotes strict preference;
x y means that bundle x is preferred
strictly to bundle y.
Preference Relations
p
p
•
denotes strict preference so
x y means that bundle x is preferred
strictly to bundle y.
 ~ denotes indifference; x ~ y means x and
y are equally preferred.
• f
denotes
weak
preference;
~
x f y means x is preferred at least as
~
much as is y.
Preference Relations
• x f y and y f x imply x ~ y.
p
~
~
• x f y and (not y f x) imply x
~
~
y.
Assumptions about Preference
Relations
• Completeness: For any two bundles x
and y it is always possible to make the
statement that either
x f y
~
or
y f x.
~
• Should compare any two
Assumptions about Preference
Relations
• Reflexivity: Any bundle x is always at
least as preferred as itself; i.e.
x
f x.
~
Assumptions about Preference
Relations
• Transitivity: If
x is at least as preferred as y, and
y is at least as preferred as z, then
x is at least as preferred as z; i.e.
x
f y and y f z
~
~
• otherwise cycles
x f z.
~
Indifference Curves
• Take a bundle x. The set of all bundles
equally preferred to x is the indifference
curve containing x
• Through every bundle x we can draw
exactly one indifference curve
Indifference Curves
x2
x’ ~ x’ ~ x”
x
x’
x”
x1
Weakly Better Set
WP(x), the set of
bundles weakly
preferred to x.
x2
x
Includes indifference
curve I(x).
I(x)
x1
Strictly Better Set
SP(x), the set of
bundles strictly
preferred to x,
does not include
indifference curve
I(x).
x2
x
I(x)
x1
Indifference Curves Cannot
Intersect
x2
I1
I2 From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z.
x
y
z
x1
Indifference Curves Cannot
Intersect
I1
I2 From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z. But y z, a
contradiction.
p
x2
x
y
z
x1
Slopes of Indifference Curves
• When more of a commodity is always
preferred, the commodity is a good.
• If every commodity is a good then
indifference curves are negatively sloped.
Slopes of Indifference Curves
Good 2
Two goods
a negatively sloped
indifference curve.
Good 1
Perfect Substitutes
• If a consumer is willing to trade
commodities 1 and 2 at a constant rate -the commodities are perfect substitutes
For Jean only the total amount
of salmon and trout matters
x2
15 I2
8
I1
IC Slopes are constant at -1.
Bundles in I2 all have a total
of 15 units and are strictly
preferred to all bundles in
I1, which have a total of
only 8 units in them.
x1
8
15
For Jill only the nutritional content
of salmon and trout matters
Salmon is twice more nutritious.
s
IC Slopes are constant at -1/2.
I2
8
4
Still perfect substitutes
I1
8
16
t
Perfect Complements
• If commodities 1 and 2 are always
consumed in fixed proportion then they are
perfect complements
• only the number of bundles of the two
commodities determines the ordering of
bundles.
Perfect Complements
x2
45o
9
Each of (5,5), (5,9)
and (9,5) contains
5 pairs so each is
equally preferred.
5
5
9
x1
Perfect Complements
x2
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs,
each is less
I2 preferred than the
bundle (9,9) which
I1 contains 9 pairs.
45o
9
5
5
9
x1
Perfect complements
• Goods that are consumed in fixed, not
only one-to-one proportion, are also
perfect complements
Preferences Exhibiting Satiation
• A bundle strictly preferred to any other is a
satiation point.
Preferences Exhibiting Satiation
x2
Better
Satiation
point
x1
Indifference Curves Exhibiting
Satiation
x2
Better
Satiation
(bliss)
point
x1
Well-Behaved Preferences
• A preference relation is “well-behaved” if it
is monotonic and convex.
• Monotonicity: More of any commodity is
always preferred (i.e. no satiation and
every commodity is a good).
Well-Behaved Preferences
• Convexity: Mixtures of bundles are (at
least weakly) preferred to the bundles
themselves.
• 50-50 mixture of the bundles x and y is
z = (0.5)x + (0.5)y.
Well-Behaved Preferences -Convexity.
x
x2
x+y is strictly preferred
z=
2 to both x and y.
x2+y2
2
y
y2
x1
x1+y
1
2
y1
Well-Behaved Preferences -Convexity.
x
x2
z =(tx1+(1-t)y1, tx2+(1-t)y2)
is preferred to x and y
for all 0 < t < 1.
y
y2
x1
y1
Well-Behaved Preferences -Convexity.
x
x2
y2
x1
Preferences are strictly convex
when all mixtures z
are strictly
z
preferred to their
component
bundles x and y.
y
y1
Well-Behaved Preferences -Weak Convexity.
Preferences are
weakly convex if at
least one mixture z
is equally preferred
to a component
bundle.
x’
z’
x
z
y
y’
Non-Convex Preferences
x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
Counterexample: coffee and olives
More Non-Convex Preferences
x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
Slopes of Indifference Curves
• The slope of an indifference curve is its
marginal rate-of-substitution (MRS).
Marginal Rate of Substitution
x2
MRS at x’ is the slope of the
indifference curve at x’
x’
x1
Marginal Rate of Substitution
x2
MRS at x’ is
dx2/dx1 at x’
D x2
x’
Dx1
x1
Marginal Rate of Substitution
x2
dx2 x’
dx1
dx2 = MRS  dx1 so, at x’,
MRS is the rate at which the
consumer is only just willing
to exchange commodity 2
for a small amount of
commodity 1
x1
Good 2
a negatively sloped
indifference curve
MRS is negative
Good 1
Diminishing Marginal Rate of
Substitution
Good 2
MRS = - 5
MRS always increases with x1
(becomes less negative) if and
only if preferences are strictly
convex
MRS = - 0.5
Good 1
Intuition
• If you have a lot of good 2 you are willing
to sacrifice a lot of it to get some amount
of good 1.
MRS & Ind. Curve Properties
x2
MRS is not always increasing as x1
increases
nonconvex
preferences.
MRS = - 1
MRS
= - 0.5
MRS = - 2
x1