Budget Constraint,
Preferences and Utility
Varian: Intermediate Microeconomics, 8e, Chapters 2, 3 and 4
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Consumer Theory
Consumers choose the best bundles of goods they can afford.
• This is virtually the entire theory in a nutshell.
• But this theory has many surprising consequences.
Two parts to consumer theory
• “can afford” – budget constraint
• “best” – according to consumers’ preferences
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Consumer Theory (cont´d)
What do we want to do with the theory?
• Test it. See if it is adequate to describe consumer behavior.
• Predict how behavior changes as economic environment changes.
• Use observed behavior to estimate underlying values.
These values can be used for
• cost-benefit analysis,
• predicting impact of some policy.
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Budget Constraint
The first part of the lecture explains
• what is the budget constraint and
the budget line,
• how changes in income and prices
affect the budget line,
• how taxes, subsidies and rationing
affect the budget line.
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Consumption Bundle
For goods 1 and 2, the consumption bundle (x1, x2) shows how much
of each good is consumed.
Suppose that we can observe
• the prices of the two goods (p1, p2)
• and the amount of money the consumer has to spend m
(income).
The budget constraint can be written as p1x1 + p2x2 ≤ m.
The affordable consumption bundles are bundles that don’t cost more
than income.
The set of affordable consumption bundles is budget set of the
consumer.
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Two Goods
Theory works with more than two goods, but
can’t draw pictures.
We often think of good 2 (say) as a
composite good, representing money to
spend on other goods.
Budget constraint becomes p1x1 + x2 ≤ m.
Money spent on good 1 (p1x1) plus the money
spent on good 2 (x2) has to be less than or
equal to the available income (m).
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Budget Line
Budget line is p1x1 + p2x2 = m. It can be also written as
x2 =
m
p2
−
p1
p2
x1.
Slope of budget line = opportunity cost of good 1.
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Change in Income
Increasing m makes parallel shift out.
The vertical intercept increases and the slope remains the same.
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Change in One Price
Increasing p1 makes the budget line steeper.
The vertical intercept remains the same and the slope changes.
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Changes in More Variables
Multiplying all prices by t is just like dividing income by t:
tp1x1 + tp2x2 = m ⇐⇒ p1x1 + p2x2 =
m
t
.
Multiplying all prices and income by t doesn’t change budget line:
tp1x1 + tp2x2 = tm ⇐⇒ p1x1 + p2x2 = m.
A perfectly balanced inflation doesn’t change consumption
possibilities.
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Numeraire
We can arbitrarily assign one price or income a value of 1 and adjust
the other variables so as to describe the same budget set.
Budget line: p1x1 + p2x2 = m
The same budget line for p2 = 1:
p1
p2
x1 + x2 =
m
p2
.
The same budget line for m = 1:
p1
m
x1 +
p2
m
x2 = 1.
The price adjusted to 1 is called the numeraire price.
Useful when measuring relative prices; e.g. English pounds per dollar,
1987 dollars versus 1974 dollars, etc.
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Taxes
Three types of taxes:
• quantity tax – consumer pays amount t for each unit she
purchases.
→ Price of good 1 increases to p1 + t.
• value tax (or ad valorem tax) – consumer pays a proportion of
the price τ.
→ Price of good 1 increases to p1 + τp1 = (1 + τ)p1.
• lump-sum tax – amount of tax is independent of the consumer’s
choices.
→ The income of consumer decreases by the amount of the tax.
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Subsidies
Subsidies – opposite effect than the taxes
• quantity subsidy of s on good 1
→ Price price of good 1 decreases to p1 − s.
• ad valorem subsidy at a rate of σ on good 1
→ Price price of good 1 decreases to p1 − σp1 = (1 − σ)p1.
• lump-sum subsidy
→ The income increases by the amount of the subsidy.
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Rationing
Rationing – can’t consume more than a certain amount of some good.
Good 1 is rationed, no more than ¯x units of good 1 can be consumed
by any consumer.
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Taxing Consumption Greater than ¯x1
Taxed only consumption of good 1 in excess of ¯x1, the budget line
becomes steeper right of ¯x1
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The Food Stamp Program
Before 1979 was an ad valorem subsidy on food
• paid a certain amount of money to get food stamps which were
worth more than they cost
• some rationing component — could only buy a maximum
amount of food stamps
After 1979 got a straight lump-sum grant of food coupons. Not the
same as a pure lump-sum grant since could only spend the coupons
on food.
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Summary
• The budget set consists of bundles of goods
that the consumer can afford at given prices
and income. Typically assume only 2 goods –
one of the goods might be composite good.
• The budget line can be written as
p1x1 + p2x2 = m.
• Increasing income shifts the budget line
outward. Increasing price of one good changes
the slope of the budget line.
• Taxes, subsidies, and rationing change the
position and slope of the budget line.
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Preferences
The second part of the lecture explains
• what are consumer’s preferences,
• what properties have well-behaved
preferences,
• what is marginal rate of
substitution.
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Preferences - Introduction
Economic model of consumer behavior – people
choose the best things they can afford
• up to now, we clarified “can afford”
• next, we deal with “best things”
Several observations about optimal choice from
movements of budget lines
• perfectly balanced inflation doesn’t change
anybody’s optimal choice
• after a rise of income, the same choices are
available – consumer must be at least as well of
as before
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Preferences
Preferences are relationships between bundles.
• If a consumer chooses bundle (x1, x2) when (y1, y2) is available,
then it is natural to say that bundle (x1, x2) is preferred to
(y1, y2) by this consumer.
• Preferences have to do with the entire bundle of goods, not with
individual goods.
Notation
• (x1, x2) (y1, y2) means the x-bundle is strictly preferred to the
y-bundle.
• (x1, x2) ∼ (y1, y2) means that the x-bundle is regarded as
indifferent to the y-bundle.
• (x1, x2) (y1, y2) means the x-bundle is at least as good as
(or weakly preferred) the y-bundle.
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Assumptions about Preferences
Assumptions about “consistency” of consumers’ preferences:
• Completeness — any two bundles can be compared:
(x1, x2) (y1, y2), or (x1, x2) (y1, y2), or both
• Reflexivity — any bundle is at least as good as itself:
(x1, x2) (x1, x2)
• Transitivity — if the bundle X is at least as good as Y and Y
at least as good as Z, then X is at least as good as Z:
If (x1, x2) (y1, y2), and (y1, y2) (z1, z2), then
(x1, x2) (z1, z2)
Transitivity necessary for theory of optimal choice. Otherwise, there
could be a set of bundles for which there is no best choice.
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Indifference Curves
Weakly preferred set are all consumption bundles that are weakly
preffered to a bundle (x1, x2).
Indifference curve is formed by all consumption bundles for which
the consumer is indifferent to (x1, x2) – like contour lines on a map.
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Indifference Curves (cont’d)
Note that indifference curves describing two distinct levels of
preference cannot cross.
Proof — we know that X ∼ Z and Z ∼ Y . Transitivity implies that
X ∼ Y . This contradicts the assumption that X Y .
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Examples: Perfect Substitutes
Perfect substitutes have constant rate of trade-off between the two
goods; constant slope of the indifference curve (not necessarily −1).
E.g. red pencils and blue pencils; pints and quarts.
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Examples: Perfect Complements
Perfect complements are consumed in fixed proportion (not
necessarily 1:1).
E.g. right shoes and left shoes; coffee and cream.
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Examples: Bad Good
A bad is a commodity that the consumer doesn’t like.
Suppose consumer is doesn’t like anchovies and likes pepperoni.
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Examples: Neutral Good
Consumer doesn’t care about the neutral good.
Suppose consumer is neutral about anchovies and likes pepperoni.
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Examples: Satiation Point
Satiation or bliss point is the most preferred bundle ( ¯x1, ¯x2)
• When consumer has too much of good, it becomes a bad –
reducing consumption of the good makes consumer better off.
• E.g. amount of chocolate cake and ice cream per week
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Examples: Discrete Good
Discrete good is only available in integer amounts.
• Indiference “curves” – sets of discrete points; weakly preferred
set – line segments.
• Important if consumer chooses only few units of the good per
time period (e.g. cars).
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Well-Behaved Preferences
Monotonicity – more is better (we have only goods, not bads) =⇒
indifference curves have negative slope (see Figure 3.9):
If (y1, y2) has at least as much of both goods as (x1, x2) and more of
one, then (y1, y2) (x1, x2).
Convexity – averages are preferred to extremes =⇒ slope gets
flatter as you move further to right (see Figure 3.10):
If (x1, x2) ∼ (y1, y2), then (tx1 + (1 − t)y1, tx2 + (1 − t)y2) (x1, x2)
for all 0 ≤ t ≤ 1
• non convex preferences – olives and ice cream
• strict convexity – If the bundles (x1, x2) ∼ (y1, y2), then
(tx1 + (1 − t)y1, tx2 + (1 − t)y2) (x1, x2) for all 0 ≤ t ≤ 1
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Marginal Rate of Substitution
Marginal rate of substitution (MRS) is the slope of the
indifference curve: MRS = ∆x2/∆x1 = dx2/dx1.
Sign problem — natural sign is negative, since indifference curves will
generally have negative slope.
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Marginal Rate of Substitution (cont’d)
MRS measures how the consumer is willing to trade off consumption
of good 1 for consumption of good 2 (see Figure 3.12).
For strictly convex preferences, the indifference curves exhibit
diminishing marginal rate of sustitution
Other interpretation: marginal willingness to pay – how much of
good 2 is one willing to pay for a extra consumption of good 1.
If good 2 is a composite good, the willingness-to-pay interpretation is
very natural.
Not the same as how much you have to pay.
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Example: Slope of the Indifference Curve
1) Calculate the slope of the indifference curve x2 = 4/x1 at the point
(x1, x2) = (2, 2).
Slope of the indifference curve = MRS =
dx2
dx1
=
−4
x2
1
= −1.
2) Calculate the slope of the indifference curve x2 = 10 − 6
√
x1 at the
point (x1, x2) = (4, 5).
Slope of the indifference curve = MRS =
dx2
dx1
=
−3
√
x1
=
−3
2
.
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Summary
• Economists assume that a consumer can rank
consumption bundles. The ranking describes
the consumer’s preferences.
• The preferences are assumed to be complete,
reflexive and transitive.
• Well-behaved preferences are monotonic and
convex.
• MRS measures the slope of the indifference
curve. MRS can be interpreted as how much of
good 2 is one willing to pay for an extra
consumption of good 1.
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Utility
The third part of the lecture
explains
• what is utility,
• what is a utility function,
• what is a monotonic
tranformation of a utility
function,
• how can we use utility
function to calculate MRS.
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Utility
Two ways of viewing utility:
Old way - measures how “satisfied” you are
(cardinal utility)
• not operational
• many other problems
New way - summarizes preferences, only the
ordering of bundles counts (ordinal utility)
• operational
• gives a complete theory of demand
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Ordinal Utility
A utility function assigns a number to each bundle of goods so that
more preferred bundles get higher numbers.
If (x1, x2) (y1, y2), then u(x1, x2) > u(y1, y2).
Three ways to assign utility that represent the same preferences:
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Utility Function is Not Unique
A positive monotonic transformation f (u) is any increasing
function.
Examples: f (u) = 3u, f (u) = u + 3, f (u) = u3
.
If u(x1, x2) is a utility function that represents some preferences,
then f (u(x1, x2)) represents the same preferences.
Why? Because u(x1, x2) > u(y1, y2) only if f (u(x1, x2)) > f (u(y1, y2)).
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Constructing Utility Functions
Mechanically using the indifference curves.
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Examples: Utility to Indifference Curves
Easy — just plot all points where the utility is constant
Utility function u(x1, x2) = x1x2;
Indifference curves: k = x1x2 ⇐⇒ x2 = k
x1
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Examples: Indifference Curves to Utility
More difficult - given the preferences, what combination of goods
describes the consumer’s choices.
Perfect substitutes
• All that matters is total number of pencils, so u(x1, x2) = x1 + x2
does the trick.
• Can use any monotonic transformation of this as well, such as
ln(x1 + x2).
Perfect complements
• What matters is the minimum of the left and right shoes you
have, so u(x1, x2) = min{x1, x2} works.
• In general, if it not 1:1, the utility function is
u(x1, x2) = min{ax1, bx2}, where a and b are positive numbers.
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Examples: Indifference Curves to Utility (cont’d)
Quasilinear preferences
• Indifference curves are vertically parallel (see Figure 4.4). Not
particularly realistic, but easy to work with.
• Utility function has form u(x1, x2) = v(x1) + x2
• Specific examples: u(x1, x2) =
√
x1 + x2 or u(x1, x2) = ln x1 + x2
Cobb-Douglas preferences
• Simplest algebraic expression that generates well-behaved
preferences.
• Utility function has form u(x1, x2) = xb
1 xc
2 (See Figure 4.5).
• Convenient to take transformation f (u) = u
1
b+c and write
x
b
b+c
1 x
c
b+c
2 or xa
1 x1−a
2 , where a = b/(b + c).
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Marginal Utility
Marginal utility (MU) is extra utility from some extra consumption
of one of the goods, holding the other good fixed.
A partial derivative – this just means that you look at the derivative
of u(x1, x2) keeping x2 fixed — treating it like a constant.
Examples:
• if u(x1, x2) = x1 + x2, then MU1 = ∂u/∂x1 = 1
• if u(x1, x2) = xa
1 x1−a
2 , then MU1 = ∂u/∂x1 = axa−1
1 x1−a
2
Note that marginal utility depends on which utility function you
choose to represent preferences.
• If you multiply utility 2x, you multiply marginal utility 2x =⇒
it is not an operational concept.
• However, MU is closely related to MRS, which is an operational
concept.
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Relationship between MU and MRS
An indifference curve u(x1, x2) = k, where k is a constant.
We want to measure slope of indifference curve, the MRS.
So consider a change (∆x1, ∆x2) that keeps utility constant. Then,
MU1∆x1 + MU2∆x2 = 0
∂u
∂x1
∆x1 +
∂u
∂x2
∆x2 = 0.
Hence,
∆x2
∆x1
= −
MU1
MU2
.
So we can compute MRS from knowing the utility function.
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Example: Utility for Commuting
Question: Take a bus or take a car to work?
Each way of transport represents bundle of
different characteristics: Let x1 be the time of
taking a car, y1 be the time of taking a bus.
Let x2 be cost of car, etc.
Suppose utility function takes linear form
U(x1, ..., xn) = β1x1 + ... + βnxn.
We can observe a number of choices and use
statistical tech- niques to estimate the
parameters βi that best describe choices.
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Example: Utility for Commuting (con’t)
Domenich a McFadden (1975) report a utility function
U(TE, TT, C) = −0.147TW − 0.0411TT − 2.24C,
where
TW = total walking time to and from bus or car in minutes
TT = total time of trip in minutes
C = total cost of trip in dollars.
Once we have the utility function we can do many things with it:
• Calculate the marginal rate of substitution between two
characteristics. How much money would the average consumer
give up in order to get a shorter travel time?
• Forecast consumer response to proposed changes.
• Estimate whether proposed change is worthwhile in a benefit-cost
sense.
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Summary
• A utility function is a way to represent a
preference ordering. The numbers
assigned to different utility levels have no
intrinsic meaning.
• Any monotonic transformation of a utility
function will represent the same
preferences.
• The marginal rate of substitution is equal
to MRS = ∆x2/∆x1 = −MU1/MU2.
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