Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

CEC 3304: Welfare Economics

Lecture Two: Consumer theory

Abdiaziz Ahmed
abdiazizahmedy@gmail.com

Department of Economics & Development Studies (DEDS)

University of Nairobi

February 2025

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Lecture Outline

Preferences and Indifference Curves

The Marginal Rate of Substitution

Utility Maximisation and Demand Functions

The Impact of Income and Price Changes

Elasticities of Demand

The Compensated Demand Curve

Welfare Measures

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Preferences and Utility

The theory of choice starts with rational preferences.

Generally, preferences are primitives in economics – you take these as given and
proceed from there.
The task of explaining why certain preferences exist in certain societies falls
largely under the domain of subjects such as anthropology or sociology.
However, to be able to create a model of choice that has some predictive power,
we do need to put some restrictions on preferences to rule out irrational behavior.
Just a few relatively sensible restrictions allow us to build a model of choice that
has great analytical power.
Varian sets out three restrictions on preferences: completeness, transitivity, and
non-satiation (’more is better’). These restrictions allow us to do something very
useful.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Ordinal versus Cardinal Utility

Preferences generally give us rankings among bundles rather than some absolute
measure of satisfaction derived from bundles.
You might prefer apple juice to orange juice but would have difficulty saying
exactly how much more satisfaction you derive from the former compared to the
latter.
Preferences therefore typically give us an ’ordinal’ ranking among bundles of
goods. Since utility is simply a representation of preferences, it is also an ordinal
measure.
This means that if your preferences can be represented by a utility function, then
a positive transformation of this function which preserves the ordering among
bundles is another function that is also a valid utility function.
In other words, there are many possible utility functions that can represent a
given set of preferences equally well.
However, there are some instances where we use cardinal utility and make
absolute comparisons among bundles. Money, for example, is a cardinal measure
– you know that 3000 KES is twice as good as 1500 KES. In general, though,
you should understand utility as an ordinal concept.
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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Indifference Curves

Definition: An indifference curve is the locus of different bundles of goods that yield
the same level of utility.
In other words, an indifference curve for a utility function u(x, y ) is given by:

u(x, y ) = k, where k is some constant.

As we vary k, we obtain an indifference map. These curves are analogous to
contour plots in geography, representing levels of utility instead of elevation.
Key Notes:
Indifference curves are derived from a utility function.
Different preferences lead to different shapes of indifference curves.
More utility means a higher (further from the origin) indifference curve.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Indifference Curves: Examples

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Indifference Curves: More Examples

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Indifference Curves: Examples

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Properties of Indifference Curves

Key restrictions and their implications:

1. Monotonicity: If an indifference curve is further from the origin than another,
any point on the former is preferred to any point on the latter. (Implied by
”more is better”).
2. Negative Slope: Indifference curves cannot slope upwards. (Again, ”more is
better”).
3. Thinness: Indifference curves cannot be thick. (Otherwise, the same bundle
would have different utilities).
4. No Crossing: Indifference curves cannot cross. (Implied by transitivity).
5. Completeness: Every bundle of goods lies on some indifference curve.
These properties ensure that preferences are well-behaved and consistent.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

The marginal rate of substitution

A further important property concerns the rate at which a consumer is willing to

substitute one good for another along an indifference curve.
The marginal rate of substitution (MRS) of a consumer between goods x and
y is the units of y the consumer is willing to substitute (i.e., willing to give up)
to obtain one more unit of x.
The slope of an indifference curve (with good y on the y -axis and good x on the
x-axis) is given by:

dy MUx
=−
dx u Constant MUy
The marginal rate of substitution is the absolute value of the slope:

MUx
MRSxy =
MUy

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Budget constraint

Once we have specified our model of preferences, we need to know the set of
goods that a consumer can afford to buy. This is captured by the budget
constraint.
Since consumers are generally taken to be price-takers (i.e. what an individual
consumer purchases does not affect the market price for any good), the budget
line is a straight line.
You should be aware that budget lines would no longer be a straight line if a
consumer buys different units at different prices.
This could happen if a consumer is a large buyer in a market or if the consumer
gets quantity discounts.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Utility maximisation
The consumer chooses the most preferred point in the budget set. If preferences
are such that indifference curves have the usual convex shape, the best point is
where an indifference curve is tangent to the budget line.
This is shown as point A the in Figure below:

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Utility maximisation
At A, the slope of the indifference curve coincides with the slope of the budget
constraint. So we have:

MUx Px
=− −
MUy Py
Multiplying both sides by −1, we can write this as the familiar condition:

Px
MRSxy =
Py
Let us derive this condition formally using a Lagrange multiplier approach. This
is the approach you are expected to use when faced with optimization problems
of this sort.
Note that the ‘more is better’ assumption ensures that a consumer spends all of
their income (if not, then the consumer could increase utility by buying more of
either good).
Therefore, the budget constraint is satisfied with equality. It follows that the
consumer maximizes u(x, y ) subject to the budget constraint:

Px x + Py y = M.
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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Set up the Lagrangian:

L = u(x, y ) + λ (M − Px x − Py y ) .
The first-order conditions for a constrained maximum are:
∂L ∂u
= − λPx = 0
∂x ∂x

∂L ∂u
= − λPy = 0
∂y ∂y
∂L
= M − Px x − Py y = 0
∂λ
From the first two conditions, we get:
∂u
Px ∂x
= ∂u
= MRSxy
Py ∂y

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Demand functions
The maximization exercise above gives us the demand for goods x and y at given
prices and income. As we vary the price of good x, we can trace out the demand curve
for good x. Below we compute demand functions in a specific example. A consumer
has the following Cobb-Douglas utility function:

u(x, y ) = x α y β
where α, β > 0. The price of x is normalized to 1 and the price of y is p. The
consumer’s income is m. Let us derive the demand functions for x and y . The
consumer’s problem is as follows:

max x α y β subject to x + py ≤ m.
x,y

Using the Lagrange multipliers method, we get:

MUx Px
= .
MUy Py

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Demand functions

Using this, we get:

αx α−1 y β 1
= .
βx α y β−1 p
Simplifying:
αy 1
= .
βx p
Using this in the budget constraint and solving, we get the demand functions:

αm βm
x(p, m) = and y (p, m) = .
α+β p(α + β)

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The impact of income and price changes

Now that we have derived demand curves, we can try to understand various
properties of demand by varying income and prices.
Depending on the values of the income elasticity of demand, goods can be
broadly categorized as
1. Normal goods: a consumer buys more of these when income
increases.
2. Inferior goods: a consumer buys less of these when income increases.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

The income–consumption curve

The income–consumption curve of a consumer traces out the path of optimal

bundles as income varies (keeping all prices constant).
Using this exercise, we also plot the relationship between quantity demanded and
income directly.
The curve that shows this relationship is called the Engel curve.
The slope of the income–consumption curve indicates the sign of income
elasticity of demand.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Price changes

It is very important to understand fully the decomposition of the total price

effect into income and substitution effects.
This decomposition is, of course, a purely artificial thought experiment.
But this thought experiment is extremely useful in understanding how the
demand for different goods responds to a change in price at different levels of
income and given different opportunities to substitute out of a good.
You should understand how these effects (and, therefore, the total price effect)
differ across normal and inferior goods, and understand how the effect known as
Giffen’s paradox can arise.
The idea of income and substitution effects can help us understand the design of
an optimal tax scheme

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Price changes

To isolate the substitution effect, we must change the price of x, but also take
away income so that the consumer is on the original indifference curve.
In other words, we must keep the utility at u0 . In Figure below, the dashed
budget line is the one after the compensating reduction in income. The point B
is the optimal point on this compensated budget line. The movement from the
original point A to B shows the substitution effect.
Finally, you should also study the impact on demand for a good of changes in
prices of some other good, and how this effect differs depending on whether the
other good is a substitute or a complement.
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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Price elasticity of demand

This is the percentage change in quantity demanded of a good in response to a given

percentage change in price of the good.

dQ/Q P dQ
ε= = .
dP/P Q dP
Note that ε < 0 since demand is typically downward-sloping. Demand is said to be
elastic if ε < −1, unit elastic if ε = −1, and inelastic if ε > −1.
Your textbook outlines a variety of uses of this concept, which you should read
carefully. You should know how to calculate demand elasticity at different points on a
demand curve, and how the elasticity varies along a linear demand curve.
Price elasticity of demand is the most common measure of elasticity and often referred
to as just elasticity of demand.
Other than price elasticity, we can define income elasticity and cross-price elasticity.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Income elasticity of demand

Denoting income by M, income elasticity of demand is given by:

P dM
εM =
.
M dP
This is positive for normal goods, and negative for inferior goods. When εM exceeds 1,
we call the good a luxury good. Necessities like food have income elasticities much
lower than 1.
Cross-price elasticity
Let us consider the elasticity of demand for good i with respect to the price of good j.
The cross-price elasticity of demand for good i is given by:

Pj dQi
.εij =
Qi dPj
This is negative for complements and positive for substitutes.

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The compensated demand curve

We derived the demand function for a good above.
To derive the demand function for good x, we vary the price of good x but hold
constant the prices of other goods and income.
Of course, as the price changes so that the optimal choice changes, the utility of
the consumer at the optimal point also changes.
This is the usual demand curve, and is also known as the Marshallian demand
curve or the uncompensated demand curve.
Indeed, if we simply mention a demand curve without putting a qualifier before
it, it refers to the Marshallian or uncompensated demand curve.
A compensated, or Hicksian, demand curve can be derived as follows. Suppose
as the price of a good changes, we keep utility constant while allowing income to
vary.
In other words, if the price of x, say, falls (so that the new optimal bundle of the
consumer would be associated with a higher level of utility if income is left
unchanged), we take away enough income to leave the consumer at the original
level of utility.
It is clear that this process eliminates the income effect and simply captures the
substitution effect.
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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

The compensated demand curve

Below, we list some properties of compensated demand curves.

1. A compensated demand curve always slopes downward.
2. For a normal good, the compensated demand curve is less elastic
compared to the uncompensated demand curve.
3. For an inferior good, the compensated demand curve is more elastic
compared to the uncompensated demand curve.
You should understand that all three properties result from the fact that only the
substitution effect matters for the change in compensated demand when price
changes.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

The compensated demand curve

We calculate compensated demand curves in the following example. Suppose
u(x, y ) = x 1/2 y 1/2 . Income is M. The compensated demand curves for x and y are
calculated as follows.
To do this, we must first calculate the Marshallian demand curves. These are given by
(you should do the detailed calculations to show this):

M M
x= and y= .
2px 2py
The optimised value of utility is:

M
V = √ .
2 px py
Holding utility constant at V implies adjusting M to the value M ∗ so that:
√
M ∗ = 2V px py .

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The compensated demand curve

This is the value of income, compensated to keep utility constant at the level given by
the original choices of x and y . It follows that the compensated demand functions are:

M∗
…
py
xc = =V
2px px
and:

M∗ px
…
yc = =V .
2py py
Note that the Marshallian demand for x does not depend on py , but the Hicksian or
compensated demand does. This is because changes in py require income adjustments,
which generate an income effect on the demand for x.

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Welfare measures: ∆CS

When drawing demand curves, we typically draw the inverse demand curve (price
on vertical axis, quantity on horizontal axis).
In such a diagram, the consumer surplus (CS) is the area under the (inverse)
demand curve and above the market price up to the quantity purchased at the
market price.
This is the most widely-used measure of welfare. We can measure the welfare
effect of a price rise by calculating the change in CS (denoted by ∆CS).
Much of our discussion of policy will be based on this measure. Any part of ∆CS
that does not get translated into revenue or profits is a deadweight loss. The
extent of deadweight loss generated by any policy is a measure of inefficiency
associated with that policy.
However, ∆CS is not an exact measure because of the presence of an income
effect. Ideally, we would use the compensated demand curve to calculate the
welfare change. CV and EV give us two such measures. You should use these
measures to understand the design of ideal policies, but when measuring welfare
change in practice, use ∆CS.

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Other Welfare measures: (CV and EV)

Compensating Variation (CV) is the amount of money that must be given to a

consumer to offset the harm from a price increase, i.e., to keep the consumer on
the original indifference curve before the price increase.
Equivalent Variation (EV) is the amount of money that must be taken away from
a consumer to cause as much harm as the price increase. In this case, we keep
the price at its original level (before the rise) but take away income to keep the
consumer on the indifference curve reached after the price rise.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Comparing the three measures

Consider welfare changes from a price rise. For a normal good, we have
CV > ∆CS > EV , and for an inferior good we have CV < ∆CS < EV . The measures
would coincide for preferences that exhibit no income effect [Quasilinear preferences].
The example that follows shows an application of these concepts.
1/2 1/2
Suppose that a consumer has the utility function u(x1 , x2 ) = x1 x2 . He originally
faces prices (1,1) and has income 100. Then the price of good x1 increases to 2. Let
us calculate the compensating and equivalent variations.
Suppose income is M and the prices are p1 and p2 . You should work out that the
demand functions are:

M M
x1 = and x2 = .
2p1 2p2
Therefore, utility is:

M
u ∗ (p1 , p2 , M) = √ .
2 p1 p2

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Comparing the three measures

√
At the initial prices, u ∗ = M/2. Once the price of good x1 increases, u ∗∗ = M/(2 2).
CV is the extra income that restores utility to the original level. Therefore, it is given
by:

M + CV M
√ = .
2 2 2
Solving:
√
CV = ( 2 − 1)M.
Using the value M = 100, this is 41.42.
EV is the variation in income equivalent to the price change. This is given by:

M M − EV
√ = .
2 2 2
Solving:
√
( 2 − 1)M
EV = √ .
2
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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Comparing the three measures

Using M = 100, this is 29.29. The consumer surplus (CS) is given by:
Z p1
CS = x1 (p) dp
p2

where the Marshallian demand for x1 * is:

100 50
=x1 =
2p p
We compute consumer surplus before and after the price change:
1. CS before price increase (p1 = 1):
Z ∞
50
CSbefore = dp
1 p
2. CS after price increase (p2 = 2):
Z ∞
50
CSafter = dp
2 p

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Comparing the three measures

Thus, the change in consumer surplus is:

Z 2
50
∆CS = dp.
1 p
Solving the definite integral:
Å ã
2
∆CS = 50 ln
1

∆CS = 50 ln 2
Approximating:

∆CS ≈ 50 × 0.693 = 34.66.

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Preferences and Indifference Curves The Marginal Rate of Substitution Utility Maximisation and Demand Functions The Impact of

Practice exercise

Suppose Ann has the utility function:

U = X 0.1 Y 0.9 .
She has an income of 100 and Px = 1 and Py = 1. Calculate the compensating
variation, equivalent variation, and consumer surplus change when the price of X
increases to 2.

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