Mathematics in Economics

Mathematics is for Describing Human Behavior in Economics

Case Behavior 1:

Case Behavior 2:

A student has 500€ for her monthly expenditures

200

150

€

Room

Food

150

Others

Cleanliness

Clothes

Entertainment

Toothpaste, Soap, Dish cleaning, … 20€

1 T-shirt (20€), 1 trousers (20€), 1 pullover (20€) 60€

2*movie (30€), 3*Shamrock (40€) 70€

52€

VWL-Book

30€

Friend’s Happy B

How to manage the budget?

A Comparison

1 2

2

x

# Sheep

Clothes: 3 1 Save 40€

Movie: 2 1 Save 15€

Shamrock: 3 1 Save 25€

Food: 150€ 140€ Save 10€

=

P( ) =

P( ) P( )

1 2

Similarities ScarcityScarcity

European lacks sheep, African lacks tobacco The student doesn’t have enough money

Allocation and ReallocationAllocation and Reallocation Two sticks less for one sheep One sheep less for two sticks The student tries to reallocate the goods acquired with money

SatisfactionSatisfaction The European feels better with one sheep and two less sticks The African feels better with one sheep less but two sticks The student tries to preserve her level of satisfaction with small

changes. She reduces just marginally the levels of consumption of some

goods.

Decisions at the Margin and Satisfaction Exchange at the Margin

The European has many tobacco sticks and is willing to give two sticks.

The African has some sheep and is willing to give one sheep.

Reallocation at the Margin

The student is willing to reduce to some extent the consumption of some goods for other uses of money.

Satisfaction

The European and the African try to increase their levels of satisfaction with a marginal exchange.

The student tries to maintain her level of satisfaction with a marginal decrease of the consumption of some goods.

Decisions at the Margin and Mathematical Language (Exchange)

Exchange Scarcity Allocation Max. Satisfaction

U(Eur)

A B

U(A)

U(B)

Sheep

U(Goods) U(B) = U(A) + ƒ * (B - A)

U(B) - U(A) = ƒ * (B - A)

U(B) - U(A)(B - A)ƒ =

U(B) - U(A)

B - A

Decisions at the Margin and Mathematical Language (Exchange)

A more general (and formal) approach Given that U (the function of satisfaction) has a form (equation), what

is the marginal increase in the satisfaction at any given point of U?

I.e. what is the form of the function ƒ for any point A?

U(Eur)

A B

U(A)

U(B)

Sheep

U(Goods)

U(B) - U(A)(B - A)ƒ =

U(B) - U(A)

B - A

ƒ is a tangent!

Decisions at the Margin and Mathematical Language (Exchange)

ƒ is more than a tangent

U(B) - U(A)(B - A)ƒ =

A

U

The process of decreasing the horizontal distance in order to find the right value of the tangent is represented by a limit.

Lim B A

“Limit when B tends to A of ƒ”

U(B) - U(A)(B - A)ƒ = = U’

ƒ is the derivative of U

Decisions at the margin are represented by derivatives

Calculus is the mathematical language of Economics

Decisions at the Margin and Mathematical Language (Reallocation)

Scarcity (re)allocation max. satisfaction

xx

yy

xk

yk

x1

y1

The reallocation of EntertainmentEntertainment

Let denote xx = units of movie, yy = units of party

The Problem:Given an initial allocation (x1,y1) for party and movie, find the set (x2,y2) that less decreases the current level of satisfaction. (x1,y1)

U1U0

U1>U0

U2

U2>U1

Decisions at the Margin and Mathematical Language (Reallocation)

The Problem: To reallocate resources without changing the level of Satisfaction

U1

xx

yy

x1

y1(x1,y1)

(x2,y2)

The Strategy: To make infinitesimal reallocations

x2x1

y2

y1

xU1 = yU1

Functions, Graphics and Derivatives

Functions are represented by equations and graphics. Terminology

ff(xx): ff is a function of xx. It is read “ff of xx” xx is an independent variable ff is dependent of xx

Examples Straight lines

f(x) = ax + b Hyperbolic and parabolic functions

f(x) = x

Some Functions and their Graphics: Straight Linesy = ax + b

y

x

y = ½ x + 2

xx 0 1 2 6

yy 2 2,5 3 5

x=0

y=2

x=6

y=5

Hyperbolic Functions

f(x) = x

f(x)

x

0< <1

f(x)

x

>1

Hyperbolic Functions

f(x) = x f(x)

x

< 0

yx1

=

= -1

yx2

=

Derivatives

Terminology How a function f varies with respect to one variable:

The derivative of f(x) with respect to x.

If ff is a function of more than one variable f(x,y) then f may have two derivatives, one respect to x, and another with respect to y.

Notation The derivative of f(x) with respect to x:

ddxf(x) df(x)

dxxxf(x)

Calculating Derivatives

f(x) = xn

xf(x) = nxn-1

Is Economics a Science?

Physics: Descriptive science

Galileo: How and How much

Economics: Descriptive part Normative part

What ought to be: how things should be Assumptions about what is “rightright”

Deontological Teleological

The Difference between Economics and Management Economics

“The economist is not concerned with ends as such. He is concerned with the way in which the attainment of ends is limited. The ends may be noble or they may be base. They may be “material” or” immaterial” –if ends can be so described. But if the attainment of one set of ends involves the sacrifice of others, then it has an economic aspect” (Robbins, 1932, p. 25).

Management Ends:

Profit Share of consumers Prestige

Robbins, L. 1932. An Essay on the Nature and Significance of Economic Science.