ECON5001
Commodity Spaces
Dr. Guy Mayraz
School of Economics
University of Sydney
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Outline
1 Introduction
2 Goods and bads
3 Indifference curves
4 Budget sets
5 Choice
6 MRS
7 Local optimality
8 Individual Demand
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Introduction
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Discrete-options and commodities
So far, we discussed choices between discrete options: apple
pie, chocolate cake, etc.
We can build a more interesting theory if we allow for:
1 Arbitrary quantities (one slice, two slices, 2020 slices)
2 Fractional quantities (0.742 slices, 2.994 slices, etc.)
We call such options commodities
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Discrete-options and commodities
So far, we discussed choices between discrete options: apple
pie, chocolate cake, etc.
We can build a more interesting theory if we allow for:
1 Arbitrary quantities (one slice, two slices, 2020 slices)
2 Fractional quantities (0.742 slices, 2.994 slices, etc.)
We call such options commodities
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Discrete-options and commodities
So far, we discussed choices between discrete options: apple
pie, chocolate cake, etc.
We can build a more interesting theory if we allow for:
1 Arbitrary quantities (one slice, two slices, 2020 slices)
2 Fractional quantities (0.742 slices, 2.994 slices, etc.)
We call such options commodities
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Commodity space
In the general case, we have n commodities: 1, 2,. . .n.
Preferences are defined over bundles, consisting of some
(possibly zero) quantity of each commodity.
A bundle (x1, x2, . . . , xn) consists of x1 units of commodity 1,
x2 units of commodity 2, etc., where xi ≥ 0.
The set of all bundles is called a commodity-space
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Commodity space
In the general case, we have n commodities: 1, 2,. . .n.
Preferences are defined over bundles, consisting of some
(possibly zero) quantity of each commodity.
A bundle (x1, x2, . . . , xn) consists of x1 units of commodity 1,
x2 units of commodity 2, etc., where xi ≥ 0.
The set of all bundles is called a commodity-space
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Commodity space
In the general case, we have n commodities: 1, 2,. . .n.
Preferences are defined over bundles, consisting of some
(possibly zero) quantity of each commodity.
A bundle (x1, x2, . . . , xn) consists of x1 units of commodity 1,
x2 units of commodity 2, etc., where xi ≥ 0.
The set of all bundles is called a commodity-space
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Commodity space
In the general case, we have n commodities: 1, 2,. . .n.
Preferences are defined over bundles, consisting of some
(possibly zero) quantity of each commodity.
A bundle (x1, x2, . . . , xn) consists of x1 units of commodity 1,
x2 units of commodity 2, etc., where xi ≥ 0.
The set of all bundles is called a commodity-space
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Two commodity case
We focus on the two commodity case:
• We need at least two commodities for there to be a
trade-off between different commodities
• There is relatively little additional insight from considering
trade-offs between three or more commodities
• Drawing a commodity space with three or more
commodities is next to impossible
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Graphical representation
x and y represent two possible consumption bundles
x1
x2
x = (3, 2)
y = (1, 5)
31
5
2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Goods and bads
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivation
It is technically convenient (and realistic) if consumers always
use up their entire budget
This works if consumers strictly prefer having more stuff
More specifically, there should always be at least one
commodity that consumers strictly prefer to have more of.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivation
It is technically convenient (and realistic) if consumers always
use up their entire budget
This works if consumers strictly prefer having more stuff
More specifically, there should always be at least one
commodity that consumers strictly prefer to have more of.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivation
It is technically convenient (and realistic) if consumers always
use up their entire budget
This works if consumers strictly prefer having more stuff
More specifically, there should always be at least one
commodity that consumers strictly prefer to have more of.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Goods
We say that commodity i is a good if the consumer prefers
(strictly or weakly) to have more of it
Examples
• chocolate
• vacation days
Note
We include the case where the consumer is indifferent about
some commodity
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Goods
We say that commodity i is a good if the consumer prefers
(strictly or weakly) to have more of it
Examples
• chocolate
• vacation days
Note
We include the case where the consumer is indifferent about
some commodity
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Goods
We say that commodity i is a good if the consumer prefers
(strictly or weakly) to have more of it
Examples
• chocolate
• vacation days
Note
We include the case where the consumer is indifferent about
some commodity
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Bads
We say that commodity i is a bad if the consumer strictly
prefers to have less of it:
Examples
• pollution
• years lost to disease
Reminder
If a consumer is indifferent about some commodity, it is
considered a good
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Bads
We say that commodity i is a bad if the consumer strictly
prefers to have less of it:
Examples
• pollution
• years lost to disease
Reminder
If a consumer is indifferent about some commodity, it is
considered a good
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Goods or bads
We assume that all commodities are either goods or bads
This rules out cases where there is a bliss point: an optimum
amount, beyond which utility is decreasing
Bliss point example
Drinking some wine may be nice, but too much can make you
sick, and can be dangerous.
Solution
• Think of xi as the quantity available for consumption
rather than quantity actually consumed
• The consumer is indifferent about having more wine than
he or she would ideally want to consume
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Utility over wine
With this interpretation, wine is a good:
Wine
Utility
ideal quantity
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Convention: all is good
By convention, we assume that all commodities are goods
• Every commodity is either an inherent good or the absence
of a bad
Examples
• Instead of modeling preferences over air pollution (a bad),
we model preferences over clean air (a good)
• Instead of modeling preferences over years in poor health
(a bad), we model preferences over years in good health (a
good)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Convention: all is good
By convention, we assume that all commodities are goods
• Every commodity is either an inherent good or the absence
of a bad
Examples
• Instead of modeling preferences over air pollution (a bad),
we model preferences over clean air (a good)
• Instead of modeling preferences over years in poor health
(a bad), we model preferences over years in good health (a
good)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Neutral goods
The concept of a good includes cases where utility is neither
increasing nor decreasing
This is called a neutral good.
As the wine example shows, a commodity may be a neutral
good only in some regions
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Neutral goods
The concept of a good includes cases where utility is neither
increasing nor decreasing
This is called a neutral good.
As the wine example shows, a commodity may be a neutral
good only in some regions
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Monotonicity
When there are two or more commodities, we further assume
that increasing the quantity of all commodities at the same
time, strictly increases utility.
• not all goods can be neutral goods
This further assumption is called monotonicity.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Monotonicity
When there are two or more commodities, we further assume
that increasing the quantity of all commodities at the same
time, strictly increases utility.
• not all goods can be neutral goods
This further assumption is called monotonicity.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Monotonicity
When there are two or more commodities, we further assume
that increasing the quantity of all commodities at the same
time, strictly increases utility.
• not all goods can be neutral goods
This further assumption is called monotonicity.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
This illustration shows 4 bundles. Some of preferences between
them depend on the specifics of the consumer’s preferences,
but there are some things that would always be true (given our
assumptions)
x1
x2
(2, 2)
(4, 4)(2, 4)
(4, 2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Commodity 1 is a good
Since Commodity 1 is a good (but maybe only a neutral good),
• (4, 2) (2, 2); and
• (4, 4) (2, 4)
x1
x2
(2, 2)
(4, 4)(2, 4)
(4, 2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Commodity 2 is a good
Since Commodity 2 is a good (but maybe only a neutral good),
• (2, 4) (2, 2); and
• (4, 4) (4, 2)
x1
x2
(2, 2)
(4, 4)(2, 4)
(4, 2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Preferences are monotonic
Since preferences are monotonic,
• (4, 4) (2, 2)
x1
x2
(2, 2)
(4, 4)(2, 4)
(4, 2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Conclusion
With these definitions, consumers will
• always use up their entire budget
• always prefer a larger budget to a smaller one
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves
For each bundle x, it is useful to consider the level set of u(x),
consisting of all bundles x′, such that u(x) = u(x′)
This level set is the indifference curve through x.
• If x′ is on the indifference curve through x, then x ∼ x′
Because of monotonicity,
• points below it have lower utility
• points above it have higher utility
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves
For each bundle x, it is useful to consider the level set of u(x),
consisting of all bundles x′, such that u(x) = u(x′)
This level set is the indifference curve through x.
• If x′ is on the indifference curve through x, then x ∼ x′
Because of monotonicity,
• points below it have lower utility
• points above it have higher utility
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves
For each bundle x, it is useful to consider the level set of u(x),
consisting of all bundles x′, such that u(x) = u(x′)
This level set is the indifference curve through x.
• If x′ is on the indifference curve through x, then x ∼ x′
Because of monotonicity,
• points below it have lower utility
• points above it have higher utility
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Graphical representation
Note that better and worse are in terms of the consumer’s
preferences; a ‘better’ bundle may make the consumer worse
(more heroin for a heroin addict)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
Problem
Suppose u(x1, x2) = x1x2
What is the indifference curve through (2, 3)?
Solution
It is the set of bundles such that
u(x) = x1x2 = u(2, 3) = 2 · 3 = 6
Thus,
x2 =
6
x1
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
Problem
Suppose u(x1, x2) = x1x2
What is the indifference curve through (2, 3)?
Solution
It is the set of bundles such that
u(x) = x1x2 = u(2, 3) = 2 · 3 = 6
Thus,
x2 =
6
x1
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves cannot cross
Since X ∼ Z and Z ∼ Y , we have X ∼ Y . Therefore X and
Y belong to the same indifference curve.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves cannot slope up
An indifference curve that slopes upwards would include
bundles x and x′ such that x > x′ (for both goods).
But by monotonicity, x x′, so the two bundles are not on the
same indifferent curve
For the same reason, indifference curves have zero width
Conclusion
Indifference curve have zero width, never cross, and weakly
slope down
• indifference curve can be horizontal or vertical if one of the
two goods is a neutral good
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves cannot slope up
An indifference curve that slopes upwards would include
bundles x and x′ such that x > x′ (for both goods).
But by monotonicity, x x′, so the two bundles are not on the
same indifferent curve
For the same reason, indifference curves have zero width
Conclusion
Indifference curve have zero width, never cross, and weakly
slope down
• indifference curve can be horizontal or vertical if one of the
two goods is a neutral good
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference curves cannot slope up
An indifference curve that slopes upwards would include
bundles x and x′ such that x > x′ (for both goods).
But by monotonicity, x x′, so the two bundles are not on the
same indifferent curve
For the same reason, indifference curves have zero width
Conclusion
Indifference curve have zero width, never cross, and weakly
slope down
• indifference curve can be horizontal or vertical if one of the
two goods is a neutral good
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Several indifference curves
We often draw several indifference curves. Bundles in higher
indifference curves have strictly higher utilities, but only the
order of utilities is significant.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget sets
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget sets
Choice sets in the market are typically defined by an income m
and prices p1 and p2.
The bundles for which p1x1 + p2x2 ≤ m constitute the budget
set.
Points inside the budget set are affordable (strictly affordable if
strictly inside the budget set).
Points outside the budget set are unaffordable
The bundles for which p1x1 + p2x2 = m constitute the budget
line.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget sets
Choice sets in the market are typically defined by an income m
and prices p1 and p2.
The bundles for which p1x1 + p2x2 ≤ m constitute the budget
set.
Points inside the budget set are affordable (strictly affordable if
strictly inside the budget set).
Points outside the budget set are unaffordable
The bundles for which p1x1 + p2x2 = m constitute the budget
line.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget sets
Choice sets in the market are typically defined by an income m
and prices p1 and p2.
The bundles for which p1x1 + p2x2 ≤ m constitute the budget
set.
Points inside the budget set are affordable (strictly affordable if
strictly inside the budget set).
Points outside the budget set are unaffordable
The bundles for which p1x1 + p2x2 = m constitute the budget
line.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget sets
Choice sets in the market are typically defined by an income m
and prices p1 and p2.
The bundles for which p1x1 + p2x2 ≤ m constitute the budget
set.
Points inside the budget set are affordable (strictly affordable if
strictly inside the budget set).
Points outside the budget set are unaffordable
The bundles for which p1x1 + p2x2 = m constitute the budget
line.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget sets
Choice sets in the market are typically defined by an income m
and prices p1 and p2.
The bundles for which p1x1 + p2x2 ≤ m constitute the budget
set.
Points inside the budget set are affordable (strictly affordable if
strictly inside the budget set).
Points outside the budget set are unaffordable
The bundles for which p1x1 + p2x2 = m constitute the budget
line.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Graphical representation
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Any optimal choice is on the budget line
If x = (x1, x2) is strictly affordable, p1x1 + p2x2 < m.
But then the consumer can also afford a bundle that has more
of both goods
• Since preferences are monotonic, the consumer would
strictly prefer this bundle
Conclusion
When looking for the optimal choice, we can restrict attention
to bundles that are just affordable (on the budget line)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Any optimal choice is on the budget line
If x = (x1, x2) is strictly affordable, p1x1 + p2x2 < m.
But then the consumer can also afford a bundle that has more
of both goods
• Since preferences are monotonic, the consumer would
strictly prefer this bundle
Conclusion
When looking for the optimal choice, we can restrict attention
to bundles that are just affordable (on the budget line)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Revealed preferences implications
Suppose the consumer chooses x∗, and that y is strictly
affordable, then x∗ y.
(if y is just affordable, we cannot be sure whether x∗ y or
x∗ ∼ y)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Revealed preferences implications
If the consumer chooses (x1, x2), we can conclude that
(x1, x2) (y1, y2).
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Revealed preferences implications
If the consumer chooses (y1, y2) from the second budget set,
we conclude that (y1, y2) (z1, z2), and hence that
(x1, x2) (z1, z2).
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Opportunity cost
(The absolute value of) the slope of the budget line is the
market exchange rate between the two goods.
Budget line:
p1x1 + p2x2 = m
Slope:
−p1
p2
Market exchange rate
• One unit of good 1 can be traded for p1/p2 units of good
2.
• One unit of good 2 can be traded for p2/p1 units of good
1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Opportunity cost
(The absolute value of) the slope of the budget line is the
market exchange rate between the two goods.
Budget line:
p1x1 + p2x2 = m
Slope:
−p1
p2
Market exchange rate
• One unit of good 1 can be traded for p1/p2 units of good
2.
• One unit of good 2 can be traded for p2/p1 units of good
1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget line slope
The market makes it possible to trade p1/p2 units of good 2 for
one unit of good 1 (or vice versa)
x1
x2
m/p1
m/p2
1
p1/p2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Opportunity cost
From the point of view of the consumer, this is the opportunity
cost of good 1 in terms of good 2.
If the consumer wants one extra unit of good 1, the consumer
has to give up p1/p2 units of good 2.
Similarly, if the consumer wants one extra unit of good 2, the
consumer has to give up p2/p1 units of good 1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Opportunity cost
From the point of view of the consumer, this is the opportunity
cost of good 1 in terms of good 2.
If the consumer wants one extra unit of good 1, the consumer
has to give up p1/p2 units of good 2.
Similarly, if the consumer wants one extra unit of good 2, the
consumer has to give up p2/p1 units of good 1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Non-standard budget sets
In some special cases there are additional restrictions.
Examples
• Only a certain number of units is available
• Discounts for purchasing large quantities
• Discounts limited by quantity
• Tax for higher quantities
• Vouchers that can only be used on certain goods
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Non-standard budget sets
In some special cases there are additional restrictions.
Examples
• Only a certain number of units is available
• Discounts for purchasing large quantities
• Discounts limited by quantity
• Tax for higher quantities
• Vouchers that can only be used on certain goods
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
Only x¯1 units are available for purchase
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
Purchases above x¯1 are taxed, increasing the effective price of
additional units.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
The US food stamps program used to subsidise food up to
some limit (A) and now provides vouchers than can only be
spent on food.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Choice
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Optimal choice
Since we assume preferences are monotonic, the optimal choice
is on the budget line
This still leaves us with infinitely many candidates.
Identifying the (globally) optimal bundle can be hard, so it is
often easier to first identify the bundles that are locally optimal
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Optimal choice
Since we assume preferences are monotonic, the optimal choice
is on the budget line
This still leaves us with infinitely many candidates.
Identifying the (globally) optimal bundle can be hard, so it is
often easier to first identify the bundles that are locally optimal
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Optimal choice
Since we assume preferences are monotonic, the optimal choice
is on the budget line
This still leaves us with infinitely many candidates.
Identifying the (globally) optimal bundle can be hard, so it is
often easier to first identify the bundles that are locally optimal
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Local optimality
The market makes it possible to trade p1/p2 units of good 2 for
every unit of good 1 (or vice versa)
x1
x2
m/p1
m/p2
1
p1/p2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Example
x∗ is not optimal, since the consumer can trade x∗ for x, and
u(x) > u(x∗).
indifference curve
through x∗
x1
x2
m/p1
m/p2
1
p1/p2
x∗
x
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Local optimality (cont.)
Starting from a point x∗, we want to determine whether such
local trades increase utility.
In order to do so, we need to know the consumer’s preferences
around x∗ even when the indifference curve is non-linear.
We do that using the marginal rate of substitution at x∗, which
is the local slope of the indifference curve.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Local optimality (cont.)
Starting from a point x∗, we want to determine whether such
local trades increase utility.
In order to do so, we need to know the consumer’s preferences
around x∗ even when the indifference curve is non-linear.
We do that using the marginal rate of substitution at x∗, which
is the local slope of the indifference curve.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Local optimality (cont.)
Starting from a point x∗, we want to determine whether such
local trades increase utility.
In order to do so, we need to know the consumer’s preferences
around x∗ even when the indifference curve is non-linear.
We do that using the marginal rate of substitution at x∗, which
is the local slope of the indifference curve.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
MRS
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Marginal rate of substitution (MRS)
The marginal rate of substitution at a point (x∗1, x∗2) (more
fully, marginal rate of substitution of good 2 for good 1) is the
slope of the indifference curve at (x∗1, x∗2)
Since the indifference curve slopes down, the MRS is negative
It’s absolute value is the the number of units of good 2 that the
consumer would (locally) trade for one extra unit of good 1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivating example
Consider a linear indifference curve:
ax1 + bx2 = c
c/b
c/a
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivating example (cont.)
Since this is an indifference curve, the consumer is indifferent
between any two bundles on this line.
Since the slope is −a/b, the consumer is indifferent between a
bundle (x1, x2) and any bundle (x1 + ∆x1, x2 − (a/b)∆x1) for
any positive or negative number ∆x1.
The consumer would thus strictly prefer
• selling a unit of x1 for any more than a/b units of x2
• buying a unit of x1 for any less than a/b units of x2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivating example (cont.)
Since this is an indifference curve, the consumer is indifferent
between any two bundles on this line.
Since the slope is −a/b, the consumer is indifferent between a
bundle (x1, x2) and any bundle (x1 + ∆x1, x2 − (a/b)∆x1) for
any positive or negative number ∆x1.
The consumer would thus strictly prefer
• selling a unit of x1 for any more than a/b units of x2
• buying a unit of x1 for any less than a/b units of x2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Motivating example (cont.)
Since this is an indifference curve, the consumer is indifferent
between any two bundles on this line.
Since the slope is −a/b, the consumer is indifferent between a
bundle (x1, x2) and any bundle (x1 + ∆x1, x2 − (a/b)∆x1) for
any positive or negative number ∆x1.
The consumer would thus strictly prefer
• selling a unit of x1 for any more than a/b units of x2
• buying a unit of x1 for any less than a/b units of x2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
What if the indifference curve is not linear?
As long as the utility function is differentiable, we can locally
approximate the indifference curve by a line.
The linear approximation for the change in utility is
du = ∂u
∂x1
dx1 +
∂u
∂x2
dx2
On the indifference curve, utility is constant, so du = 0
Conclusion
The local slope of the indifference curve at (x∗1, x∗2) is:
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
What if the indifference curve is not linear?
As long as the utility function is differentiable, we can locally
approximate the indifference curve by a line.
The linear approximation for the change in utility is
du = ∂u
∂x1
dx1 +
∂u
∂x2
dx2
On the indifference curve, utility is constant, so du = 0
Conclusion
The local slope of the indifference curve at (x∗1, x∗2) is:
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
What if the indifference curve is not linear?
As long as the utility function is differentiable, we can locally
approximate the indifference curve by a line.
The linear approximation for the change in utility is
du = ∂u
∂x1
dx1 +
∂u
∂x2
dx2
On the indifference curve, utility is constant, so du = 0
Conclusion
The local slope of the indifference curve at (x∗1, x∗2) is:
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Linear case
To see that this works, consider the linear indifference curve we
considered before:
ax1 + bx2 = c
which has a (constant) slope of −a/b.
Computing the MRS
∂u
∂x1
= a
∂u
∂x2
= b
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
= −a
b
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Linear case
To see that this works, consider the linear indifference curve we
considered before:
ax1 + bx2 = c
which has a (constant) slope of −a/b.
Computing the MRS
∂u
∂x1
= a
∂u
∂x2
= b
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
= −a
b
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Non-linear example
u = x1x2
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
= −x2
x1
Interpretation
The consumer prefers a balanced quantity of both goods:
• If x1 is close to zero, the consumer would exchange a very
large quantity of x2 for one more unit of x1
• If x2 is close to zero, the consumer would exchange a very
large quantity of x1 for one more unit of x2
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Non-linear example
u = x1x2
MRS2→1(x∗1, x∗2) = −
∂u/∂x1
∂u/∂x2
∣∣∣∣
(x∗1,x∗2)
= −x2
x1
Interpretation
The consumer prefers a balanced quantity of both goods:
• If x1 is close to zero, the consumer would exchange a very
large quantity of x2 for one more unit of x1
• If x2 is close to zero, the consumer would exchange a very
large quantity of x1 for one more unit of x2
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
The MRS does not depend on the utility-function
For example, suppose we replace u = x1x2 with
v = ln u = ln x1 + ln x2. Then,
∂v
∂x1
= 1
x1
∂v
∂x2
= 1
x2
MRS2→1(x∗1, x∗2) = −
∂v/∂x1
∂v/∂x2
∣∣∣∣
(x∗1,x∗2)
= −1/x11/x2 = −
x2
x1
The same result as when computing the MRS using u.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Explanation for math nerds (not for exam)
If v represents the same preferences, then v = f(u) for some
strictly positive function f .
Therefore,
∂v
∂x1
= f ′(u(x∗1, x∗2))
∂u
∂x1
∂v
∂x2
= f ′(u(x∗1, x∗2))
∂u
∂x2
And so,
−∂v/∂x1
∂v/∂x2
= −∂u/∂x1
∂u/∂x2
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
The sign of the MRS
The MRS is defined with a minus sign, because the slope of the
indifference curves is negative.
The number of units of good 2 that the consumer would trade
for a unit of good 1 corresponds to the absolute value of the
MRS
People sometimes write the MRS itself in absolute value
(without the minus sign).
You can use whichever convention you prefer, as long as you
don’t get confused. In exams, I will accept both.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Kink points
Some preferences have kink points, in which two lines of
different slopes meet. The MRS is not defined at these kink
points
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Local optimality
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
MRS
With the exception of kink points, the consumer’s indifference
curve can be locally approximated by a line that has MRS2→1
as its slope.
x1
x2
1
|MRS2→1|
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Indifference about trades at the MRS
The consumer is indifferent about trades at the MRS exchange
rate.
x1
x2
1
|MRS2→1|
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Preference for trades at better rates
The consumer strictly prefers getting more of good 1 at less
than the MRS
x1
x2
1
|MRS2→1|
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Preference for trades at better rates
The consumer also strictly prefers getting more of good 2 at
more than the MRS
x1
x2
1
|MRS2→1|
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget line is flatter
If the budget line at x∗ is flatter, the consumer can get more of
good 1 at better than the MRS, so x∗ is not optimal.
x1
x2
1
|MRS2→1|
m/p2
x∗
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Budget line is steeper
If the budget line at x∗ is steeper, the consumer can get more
of good 2 at better than the MRS, so x∗ is not optimal.
x1
x2
1
|MRS2→1| x∗
m/p1
m/p2
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Tangency
If x∗ is optimal, the budget line must be tangent to the
indifference curve (|MRS2→1| = p1/p2).
x1
x2
1
MRS2→1
m/p2
m/p1
x∗
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Tangency at internal solution
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Exception 1: x1 = 0
If x1 = 0, the budget line can be steeper (though not flatter).
x1
x2
x∗
m/p1
m/p2
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Exception 2: x2 = 0
If x2 = 0, the budget line can be flatter (though not steeper).
x1
x2
x∗
m/p1
m/p2
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Exception 3: changing budget line slope
If the slope of the budget changes from flatter to steeper, x∗
can be optimal if the MRS is between the slopes the two
budget line segments.
x1
x2
x∗
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Marginal dollar interpretation
An equivalent perspective considers the impact of a marginal
dollar.
Spending a marginal dollar on good i raises utility by
MUi
pi
= 1
pi
∂u
∂xi
Optimality
1 If MU2/p2 > MU1/p1 and x1 > 0, the consumer can
increase utility by switching a dollar from good 1 to good
2.
2 If MU1/p1 > MU2/p2 and x2 > 0, the consumer can
increase utility by switching a dollar from good 2 to good
1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Marginal dollar interpretation
An equivalent perspective considers the impact of a marginal
dollar.
Spending a marginal dollar on good i raises utility by
MUi
pi
= 1
pi
∂u
∂xi
Optimality
1 If MU2/p2 > MU1/p1 and x1 > 0, the consumer can
increase utility by switching a dollar from good 1 to good
2.
2 If MU1/p1 > MU2/p2 and x2 > 0, the consumer can
increase utility by switching a dollar from good 2 to good
1.
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Individual Demand
ECON5001
Commodity
Spaces
Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Demand
The (individual) demand is the optimal consumption bundle
seen as a function of prices and income:
x1(p1, p2,m)
and
x2(p1, p2,m)
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Demand as a function of one variable
We are often interested in demand as a function of one
variable: p1, p2, or m.
For example, the demand for good 1 as a function of its price
p1 at some point (p∗1, p∗2,m∗) is
x1(p1, p∗2,m∗)
and
x2(p1, p∗2,m∗)
ECON5001
Commodity
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Introduction
Goods and
bads
Indifference
curves
Budget sets
Choice
MRS
Local
optimality
Individual
Demand
Demand curves
The demand curve plots demand for a good as a function of its
own price.
The Engel curve plots demand for a good as a function of
income
We will see plenty of examples in the next section