Mathematical Economics

Main textbook

• Chiang A.C., Wainwright K., Fundamental Methods of

Mathematical Economics, McGraw-Hill/Irwin, Boston,
Mass., (4th edition) 2005.
 The Theory of Consumer Choice

• The Budget Constraint

• The Budget Line Changes (Increasing Income,
Increasing Price)
• Consumer Preferences
• Assumptions about Preferences
• Indifference Curves: Normal Good, Perfect
Substitutes, Perfect Complements, Bads,
Neutrals
• The Marginal Rate of Substitution
 Consumers choose the best bundle of
goods they can afford
• How to describe what a consumer can afford?

• What does mean the best bundle?

• The consumer theory uses the concepts of a

budget constraint and a preference map to
analyse consumer choices.
 The budget constraint – the two-good case

• It represents the combination of goods that

consumer can purchase given current prices
and income.
• x1,x2 , xi - consumer’s
0, i 1, 2
consumption bundle (the objects of consumer
choice)

• p1,p2 , pi 0, i 1, 2 - market prices

of the two goods
 The budget constraint – the two-good case

• The budget constraint of the consumer (the amount of

money spent on the two goods is no more than the total
amount the consumer has to spend)

p1x1 p2x2 I

• I 0 - consumer’s income (the amount of money the

consumer has to spend)
• p1x1 - the amount of money the consumer is spending on
good 1
• p2x2 - the amount of money the consumer is spending on
good 2
 Graphical representation of the budget set and the budget line

• The set of affordable consumption bundles at

given prices and income is called the budget
set of the consumer.
The Budget Line
 The Budget Line Changes

• Increasing (decreasing) income – an increase (decrease) in

income causes a parallel shift outward (inward) of the
budget line (a lump-sum tax; a value tax)
 The Budget Line Changes

• Increasing price – if good 1

becomes more expensive,
the budget line becomes
steeper.
• Increasing the price of good
1 makes the budget line
steeper; increasing the price
of good 2 makes the budget
line flatter.
• A quantity tax
A value tax (ad valorem tax)
A quantity subsidy
Ad valorem subsidy
Exercise 1 Consumer Preferences
Consumer Preferences
Ps (x,y) X Xx

y relation of strict preference

I (x,y) X Xx ~

y relation of indifference
P (x,y) X Xxy relation of weak
preference
~
Assumptions about Preferences
Assumptions about Preferences
Assumptions about Preferences
Assumptions about Preferences
The relations of strict preference, weak preference and
indifference are not independent concepts!
Exercise 2
Exercise 3
 Indifference Curves

• The set of all consumption bundles that are

indifferent to each other is called an
indifference curve.

• Points yielding different utility levels are each

associated with distinct indifference curves.
Indifference curves are
Indifference curve for normal goods
 Perfect substitutes

• Two goods are perfect

substitutes if the consumer is
willing to substitute one good
for the other at a constant rate.
• The simplest case of perfect
substitutes occurs when the
consumer is willing to substitute
the goods on a one-to-one basis.
• The indifference curves has a constant slope since the consumer
is willing to trade at a fixed ratio.
 Perfect complements

• Perfect complements are

goods that are always
consumed together in
fixed proportions.

• L-shaped indifference
curves.

Bads: a bad is a commodity that consumer doesn’t like

Neutrals: a good is a neutral good if the consumer
doesn’t care about it one way or the other
The Marginal Rate of Substitution (MRS)
• The marginal rate of substitution measures the slope of the
indifference curve.

The Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (MRS)
• The MRS is different at each point along the
indifference curve for normal goods.

• The marginal rate of substitution between

perfect substitutes is constant.
 The Utility Function,
• Examples of Utility Functions: Normal Good,
Perfect Substitutes, Perfect Complements,
• The Quasilinear and Homothetic Utility
Functions,
• The Marginal Utility and The Marginal Rate of
Substitution,
• The Optimal Choice,
• The Utility Maximization Problem,
• The Lagrange Method
 The Utility Function

• A utility is a measure of the relative satisfaction

from consumption of various goods.

• A utility function is a way of assigning a

number to every possible consumption bundle
such that more-preferred bundles get assigned
larger numbers then less-preferred bundles.
 The Utility Function

• The numerical magnitudes of utility levels have no intrinsic meaning

– the only property of a utility assignment that is important is how it
orders the bundles of goods.

• The magnitude of the utility function is only important insofar as it

ranks the different consumption bundles.

• Ordinal utility - consumer assigns a higher utility to the chosen

bundle than to the rejected. Ordinal utility captures only ranking and
not strength of preferences.

• Cardinal utility theories attach a significance to the magnitude of

utility. The size of the utility difference between two bundles of goods
is supposed to have some sort of significance.
 Existence of a Utility Function

• Suppose preferences are complete, reflexive,

transitive, continuous, and strongly monotonic.
• Then there exists a continuous utility function

u: 2

which represents those preferences.

 The Utility Function

• A utility function is a function u assigning a real

number to each consumption bundle so that for
a pair of bundles x and y:
Examples of Utility Functions Exercise
1
 The Quasilinear Utility Function

• The quasilinear (partly linear) utility function is

linear in one argument.
• For example the utility function linear in good 2
is the following:

u x1,x2 v x1 x2
The Quasilinear Utility Function
• Specific examples of quasilinear utility would
be:

u x1, x2 x1 x 2 or

u x1, x2 ln x1 x2
The Homothetic Utility Function
The Homothetic Utility Function
 The Homothetic Utility Function

• Slopes of indifference curves are constant along

a ray through the origin.
• Assuming that preferences can be represented
by a homothetic function is equivalent to
assuming that they can be represented by a
function that is homogenous of degree 1
because a utility function is unique up to a
positive monotonic transformation.
Exercise 2
 The Marginal Utility

The Marginal Rate of Substitution

• Suppose that we increase the amount of good i;
how does the consumer have to change their
consumption of good j in order to keep utility
constant?

The Marginal Rate of Substitution

 The Optimal Choice

• Consumers choose the most preferred bundle from their budget sets.
• The optimal choice of consumer is that bundle in the consumer’s budget set
that lies on the highest indifference curve.
The Optimal Choice
The Optimal Choice
 The Optimal Choice
• Utility functions

• Budget line
The Optimal Choice
 The Utility Maximization

• The problem of utility maximization can be written as:

• Consumers seek to maximize utility subject to their budget constraint.

• The consumption levels which solve the utility maximization

problem are the Marshallian demand functions.
 The Lagrange Method

• The method starts by defining an auxiliary

function known as the Lagrangean:

• The new variable l is called a Lagrange

multiplier since it is multiplied by constraint.
The Lagrange Method