Introductory topics

Mathematical Economics

Vilen Lipatov

Fall 2014

Goals

I To understand the basic mathematical concepts of economicanalysis.

I To grasp tools and ideas used in further Economics andFinance courses.

I To highlight the importance of microeconomic methods.

Course outline

I Part 1.

I Introductory topics.I Functions of one variable. Properties of functions.I Differentiation. Derivatives. Single-variable optimization.I Integration. Interest rates and present values.I Functions of many variables. Multivariable optimization.

Constrained optimization.I Matrix and vector algebra. Linear programming.

Course outline

I Part 2.

I Consumer theory. Choice under uncertainty.I Equilibrium and Welfare.I Producer theory. Market structure.

Reading

I Sydsaeter, K., Hamond, P., 2008. Essential Mathematics forEconomic Analysis, 3rd Edition, Prentice Hall.

I Varian, H., 2006. Intermediate Microeconomics: A ModernApproach, 7th Edition, W.W. Norton & Co.

Grading and Contact

I Take-home exam.

I 180 examination points.

I Lipatov@em.uni-frankfurt.de

Today’s session outline

I Mathematics in social sciences.

I The use of symbols.

I Real numbers.

I Some aspects of logic.

I Mathematical proofs.

I Set theory.

Reading: Sydsaeter and Hammond, chapter 1-3

Mathematical analysis (calculus)

I Differential and integral calculus and their extensions.

I Developed at the end of the XVIIth century by Newton andLeibnitz.

I Calculus is now employed in many areas of economics. It isused to study

I influence of relative price changes on demand;I influence of oil price changes on the production process;I the consequence of population growth for the economy;I the extent to which a tax on energy use might reduce CO2

emissions.

Scientific method in social sciences

I Elements/steps:

1. Qualitative and quantitative observations of phenomena, eitherdirectly or by means of experiments.

2. Numerical and statistical processing of the observed data.3. Constructing theoretical models that describe the observed

phenomena and explain the relationships between them.4. Using these theoretical models to derive predictions.5. Correcting and improving models to get better predictions.

I In other words, the crucial processes are

1. Observation2. Modeling3. Verification

Models and reality

I Models are necessarily a simplification of reality.

I Aim at pinning down the most important factors determiningcertain phenomenon.

I Often one of many explanations of reality.

I Select models by checking which one gives the bestdescription of reality (that may be very difficult!).

Symbols in mathematics I

I Mathematics is a language, so it has its own symbols.

I The unique symbols are necessary to convey a messageunambiguously.

I Logical constants: 3,√

2, π, [0, 1].I Variables: usually denoted by Latin or Greek letters, such as

x .I Domain of variation: the objects that a variable is meant to

represent.

Symbols in mathematics II

I Equality:x2 − 16 = (x + 4)(x − 4)

I The above equality is called an identity, since it is valididentically for any x .

I If we want to be precise, we use the symbol 5 ≡ 5 to denotethe identity.

I Other equalities:I Formulas:

A = πr2

I Equations:x2 + x − 12 = 0

Unlike an identity, this equality is only true for x = 3 andx = −4. These values are called the solutions.

Example 1

I A farmer has 1000 meters of fence wire with which to enclosea rectangle. If one side of a rectangle is x (measured inmeters), find the area enclosed when x is chosen to be 150,250, 350, and for general x .

I Which value of x gives the greatest possible area?

Example 2

I Consider a simple macroeconomic model:

Y = C + I ,

C = a + bY ,

where Y is GDP, C is consumption, and I is investment.

I I , a, b are positive numerical constants (e.g.I = 100, a = 500, b = 0.8)

I Use the two equations to obtain Y in terms of parametersI , a, b.

Integers

I The everyday numbers we use for counting (1,2,3,..) arecalled natural numbers.

I If we add or multiply two natural numbers, we get a naturalnumber.

I Subtraction and division suggest the need for havingI number zero,I negative numbers,I fractions.

I 0,±1,±2,±3, ... are called integers.

I They can be represented on a number line.

Real numbers

I Real numbers can be rational and irrational.

I Rational numbers are those that can be written in the formp/q, where both p and q are integers.

I Examples:

1

2,

11

70,

125

7,−11

10, 0 =

0

1,−1.26 = −126

100.

I Numbers such as√

2 that cannot be represented in the formp/q are called irrational numbers.

I Real numbers can also be represented on a number line.

The decimal system

I The decimal system uses 10 as a base number.

I Every natural number can be written using symbols

0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

called digits.

I Decimal point allows us to express rational numbers otherthan integers, e.g.

3.7526 = 3 · 100 + 7/101 + 5/102 + 2/103 + 6/104.

I This is a finite decimal fraction

Infinite decimal fraction

I Infinite decimal fraction has an infinite number of digits,e.g. 100/3 = 33.3333...

I If a decimal fraction is a rational number, it is alwaysperiodic, e.g. 11/70 = 0.15714285714285...

I Real number can be defined as an arbitrary (finite or infinite)decimal fraction.

I When applied to the real numbers, the four basic arithmeticoperations always result in a real number.

I The only exception is division by zero, which is not definedfor any real number.

Inequalities

I Inequality is a relation that holds between two numbers whenthey may not be equal in value.

I It establishes order between real numbers.

I When x ≥ y , we say that x is greater or equal to y .

I When x > y , we say that x is (strictly) greater than y .I Example:

I Show that if a ≥ 0 and b ≥ 0, then

√ab ≤ a + b

2.

Intervals

I If a and b are two numbers on the real line, then the numbersthat lie between a and b are called interval.

I If a < b, there are four possible types of intervals:I an open interval (a, b) consists of all x satisfying a < x < b;I a closed interval [a, b] consists of all x satisfying a ≤ x ≤ b;I a half-open interval (a, b] consists of all x satisfying a < x ≤ b;I a half-open interval [a, b) consists of all x satisfying a ≤ x < b.

I All four have the same length, b − a.I The intervals can also be unbounded:

I (− inf, b) consists of all x satisfying x < b;I [a, inf) consists of all x satisfying x ≥ b.

Absolute value

I The positive distance between a real number x and zero iscalled the absolute value of x and denoted by |x |.

I From this definition, it follows that

|x | =

{x , if x ≥ 0;−x , if x < 0.

I Examples:I |13| = 13I | − 5| = 5I |0| = 0

I Note:I |x | < a means that −a < x < aI |x | ≤ a means that −a ≤ x ≤ a

Examples

I Consider y = |x − 2|I Compute y for x = −3, x = 0, and x = 4.I Rewrite y using the bracket notation.

I Consider y = |3x − 2|I Find all x that y ≤ 5.I Check if this inequality holds for

x = −3, x = 0, x = 7/3, x = 10.

I Find a possible solution for the equation

x + 2 =√

4− x .

Propositions

I Assertions that are either true or false are called statements,or propositions.

I “All cats are mammals” is a true proposition.I “All mammals are cats” is a false proposition.I “Please enter your password” is neither true nor false,

therefore it is not a proposition.I “Some blobs are flobs” lacks a precise meaning, therefore it is

not a proposition.

I Some propositions are true under certain conditions, and falseotherwise. These are called open propositions.

I x2 − 1 = 0 is true for x = 1 and x = −1, but false for anyother value of x .

Implications

I Implication arrows help to keep track of each step in a chainof logical reasoning.

I Suppose P and Q are two propositions such that whenever Pis true, then Q is necessarily true. In this case, we usuallywrite

P ⇒ Q

I Examples:I x > 2⇒ x2 > 4I xy = 0⇒ x = 0 or y = 0I x is a square ⇒ x is a rectangle

Equivalence

I Sometimes, it may be possible to draw the logical conclusionin the opposite direction as well:

Q ⇒ P

I When both implications P ⇒ Q and Q ⇒ P hold, we canwrite them as a single logical equivalence:

Q ⇔ P

Necessary and sufficient conditions

I P is a sufficient condition for Q means P ⇒ QI Example: A sufficient condition to write an x on the board is

to write an x on the board with a green marker.

I P is a necessary condition for Q means Q ⇒ PI Example: A necessary condition to get a job is to show up at

the interview.

I P is a necessary and sufficient condition for Q meansP ⇔ Q

I Example: A necessary and sufficient condition to win a WorldCup is to win its final.

Examples

I Find all x such that

(2x − 1)2 − 3x2 = 2

(1

2− 4x

)I Find all x such that

x + 2 =√

4− x

Theorems and Proofs

I Most important results in mathematics are called theorems.

I Every theorem can be formulated as an implication

P ⇒ Q

I P is a proposition or a series of propositions called premise(s)(what we know)

I Q is a proposition or a series of propositions calledconclusion(s) (what we want to know)

I Sometimes we prove P ⇒ Q starting P and workingsuccessively towards Q. This is a direct proof.

I Sometimes we prove not Q ⇒ not P, i.e. supposing that Q isfalse and working to show that P must also be false. This iscalled proof by contradiction.

Sets

I Sets are used to group objects based on common properties.I For example, companies with more than 100 employees, OECD

countries, stocks in the S&P index.

I Each object in a set is called an element or a member of thisset.

I The simplest way to specify a set is to list its elements, e.g.S = {a, b, c}.

I Two sets are equal if every element of one set is also theelement of the other set.

I The order of elements does not matter for equality; repeatedelements do not change the set

I For example, sets A = {1, 1, 2, 3} and B = {3, 2, 1} are equal.

Infinite Sets

I Some sets can be infinite, so their elements cannot be listed.

I We define such sets by general specification

S = {typical member : defining properties}

I For example, a budget set

B = {(x , y) : px + qy ≤ m, x ≥ 0, y ≥ 0}

Set membership

I If x is an element of a set S , we write x ∈ S .I For example, a ∈ {a, b, c}, 2 ∈ N, 6 ∈ {x : 0 ≤ x < 10}

I If x is not an element of a set S , we write x /∈ S .I For example, d /∈ {a, b, c}, −3.75 /∈ N, 10 /∈ {x : 0 ≤ x < 10}

I A set that contains no element is called an empty set. It isdenoted by ∅

Subsets

I If every member of set A is also member of set B, then we saythat A is a subset of B and write A ⊆ B.

I A = {1, 2} and B = {1, 2, 3} implies A ⊆ B.I A = {x , y , z} and B = {x , y , z} implies A ⊆ B (and also

B ⊆ A).I N ⊆ R

I If A is a subset of B, but A 6= B, we call it a proper subsetand write A ⊂ B.

Basic Set Operations

I Union, A ∪ B - all the elements that belong to at least one ofthe sets A and B.

I Intersection, A ∩ B - all the elements that belong to both Aand B.

I Difference, A \ B - all the elements that belong to A but notto B.

I Let A = {1, 2, 3, 4, 5} and B = {3, 6}. FindI A ∪ B,I A ∩ B,I A \ B,I B \ A

Euler-Venn Diagrams

I When considering relationships between several sets, it ishelpful to represent each set by a region in a plane.

I All elements of a certain set are contained within some closedregion of a plane.

I Such representations are called Euler-Venn diagrams.

I draw an Euler-Venn diagram