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Mathematical Economics dr Wioletta Nowak

Lecture 1

Syllabus

Mathematical Theory of Demand

Utility Maximization Problem

Expenditure Minimization Problem

Mathematical Theory of Production

Profit Maximization Problem

Cost Minimization Problem

General Equilibrium Theory

Neoclassical Growth Models

Models of Endogenous Growth Theory

Dynamic Optimization

Syllabus


• Budget Constraint

• Consumer Preferences

• Utility Function

• Utility Maximization Problem

• Optimal Choice

• Properties of Demand Function

• Indirect Utility Function and its Properties

• Roy’s Identity

Syllabus


• Expenditure Minimization Problem

• Expenditure Function and its Properties

• Shephard's Lemma

• Properties of Hicksian Demand Function

• The Compensated Law of Demand

• Relationship between Utility Maximization and Expenditure Minimization Problem

Syllabus


• Production Functions and Their Properties

• Perfectly Competitive Firms

• Profit Function and Profit Maximization

Problem

• Properties of Input Demand and Output

Supply

Syllabus


• Cost Minimization Problem

• Definition and Properties of Conditional Factor Demand and Cost Function

• Profit Maximization with Cost Function

• Long and Short Run Equilibrium

• Total Costs, Average Costs, Marginal Costs, Long-run Costs, Short-run Costs, Cost Curves, Long-run and Short-run Cost Curves

Syllabus


Monopoly

Oligopoly

• Cournot Equilibrium

• Quantity Leadership – Slackelberg Model

Syllabus

General Equilibrium Theory

• Exchange

• Market Equilibrium

Syllabus

Neoclassical Growth Model

• The Solow Growth Model

• Introduction to Dynamic Optimization

• The Ramsey-Cass-Koopmans Growth Model

Models of Endogenous Growth Theory

Convergence to the Balance Growth Path

Recommended Reading

• Chiang A.C., Wainwright K., Fundamental Methods of Mathematical Economics, McGraw-Hill/Irwin, Boston, Mass., (4th edition) 2005.

• Chiang A.C., Elements of Dynamic Optimization, Waveland Press, 1992.

• Romer D., Advanced Macroeconomics, McGraw-Hill, 1996.

• Varian H.R., Intermediate Microeconomics, A Modern Approach, W.W. Norton & Company, New York, London, 1996.

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.

• - consumer’s

consumption bundle (the objects of consumer

choice)

• - market prices

of the two goods

2,1i,0x,x,x i21

2,1i,0p,p,p i21

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)

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

consumer has to spend)

• - the amount of money the consumer is spending on good 1

• - the amount of money the consumer is spending on good 2

Ixpxp 2211

11xp

22xp

0I

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

preferenceweakofrelationyxXX)y,x(P

ofrelationy~xXX)y,x(I

preferencestrictofrelationyxXX)y,x(P

~

s

ceindifferen

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 MRS is different at each point along the

indifference curve for normal goods.

• The marginal rate of substitution between

perfect substitutes is constant.


Lecture 2

• 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 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

which represents those preferences.

2:u


• 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:

2121, xxvxxu

The Quasilinear Utility Function

• Specific examples of quasilinear utility would

be:

or

2121, xxxxu

2121 ln, xxxxu

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

• 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


Lecture 3-4

• Properties of the Demand Function: the Marginal Demand, the Price, Income and Cross Price Elasticity of Demand,

• Classification of Goods: Normal Goods, Inferior Goods, Ordinary Goods, Giffen Goods, Perfect Substitutes, Perfect Complements,

• The Total Change in Demand: The Substitution Effect and the Income Effect,

• Comparative Statics: Income Offer Curve, Price Offer Curves and Engel Curves

• The Indirect Utility Function: Definition and Properties,

• The Roy's Identity,

The Demand Function

• The value of that solves the utility maximization problem

such that

is the consumer’s demanded bundle.

• It expresses how much of each good the consumer desires at a given level of prices and income.

21,

,max21

xxuxx

Ixpxp 2211

x~

The Demand Function

• The function that relates p and I to the

demanded bundle is called consumer’s demand

function:

23:

xIpIp ~,,:

IppIpp ,,,,, 212211

Elasticities of Demand

• Elasticity is the ratio of the per cent change in

one variable to the per cent change in other

variable (is a measure of sensitivity of one

variable to another).

Price elasticity of demand

Cross price elasticity of demand

Income elasticity of demand

Elasticities of Demand

• Price elasticity of demand is defined as the measure of responsiveness in the quantity demanded for a good as a result of change in price of the same good. It is a measure of how consumers react to a change in price of a given good (a measure of the sensitivity of quantity demanded to changes in price).

• Cross price elasticity of demand measures the responsiveness of the quantity demanded of a good to a change in the price of another good.

• Income elasticity of demand measures the responsiveness of the quantity demanded of a good to the change in the income of the consumer.

The Classification of Goods – An Ordinary Versus a

Giffen Good

The Classification of Goods – Perfect Substitutes

Versus Perfect Complements

The Classification of Goods – A normal Versus an

Inferior Good

• Normal goods: if a good is normal, then the demand for

it increases when income increases, and decreases when

income decreases. The quantity demanded always

changes in the same way as income changes.

• If income elasticity of demand of a commodity is less

than 1, it is a necessity good. If the elasticity of demand

is greater than 1, it is a luxury or a superior good.


Inferior Good

• Interior goods: An inferior good is one for which the

demand decreases when income increases.


Inferior Good

The Total Change in Demand: The Substitution

Effect and the Income Effect

• When the price of good decreases, there will be two effects on consumption.

• The change in relative prices makes consumer want to consume more of the cheaper good. The increase in purchasing power due to the lower price may increase or decrease consumption, depending on whether the good is a normal good or an inferior good.

• The change in demand due to the change in relative prices is called the substitution effect; the change due to the change in purchasing power is called the income effect.

The Total Change in Demand: The Substitution

Effect and the Income Effect

The Income Offer Curve (Income Expansion Path)

and the Price Offer Curve

• The income offer curve depicts how consumption changes with income.

• The price offer curve represents the bundles that would be demanded at

different prices for a given good.

The Engle Curve

• The Engle curve is a graph of the demand for a one of the

goods as the function of income, with all prices being held

constant.

The Indirect Utility Function

Example

Properties of the Indirect Utility Function

Roy’s Identity


Lecture 5-6

• The Expenditure Minimization Problem,

• Properties of the Hicksian Demand Function,

• The Expenditure Function and its Properties,

• The Shephard's Lemma,

• Relationship between the Utility Maximization

and the Expenditure Minimization Problem,

• The Slutsky Equation

The Expenditure Minimization Problem

• Instead of maximizing utility given a budget

constraint we can consider the dual problem of

minimizing the expenditure necessary to

obtain a given utility level:

The Hicksian demand function

• The solution to this problem is the optimal

consumption bundle as function of p and u i.e.

• It is the expenditure-minimizing bundle

necessary to achieve utility level u at prices p.

),( upf


• The Hicksian demand function is sometimes

called compensated demand function.

• This terminology comes from viewing the

demand function as being constructed by

varying prices and income so as to keep the

consumer at fixed level of utility.

• Thus, the income changes are arranged to

compensate for the price changes.


• The Hicksian demand functions are not

directly observable since they depend on

utility, which is not directly observable.

• The Marshallian demand functions expressed

as function of prices and income are

observable.

Example 1

Properties of the Hicksian Demand Function

The Expenditure Function

Example 2

Properties of the expenditure function

Relationship between the Utility Maximization and the

Expenditure Minimization Problem

The Slutsky Equation

The Hicks Decomposition of a Demand Change

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