Introduction Mathematics in Economics-1

8/2/2019 Introduction Mathematics in Economics-1

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Introduction and the

Role of Mathematics inEconomics

Is Economics a Science?

Mathematics is for DescribingHuman Behavior in Economics


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Etymology:


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Adam Smith


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The XIX Century


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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 right

Deontological

Teleological


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Neville KeynesScope and Method of Political Economy


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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 right

Deontological

Teleological


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Lionel Robbins1932. An Essay on the Nature and Significance of Economic Science.

The economist is not concerned with ends as such. He is concernedwith 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 immaterialif ends can be so described. But if the attainment of one set of endsinvolves the sacrifice of others, then it has an economic aspect(Robbins, 1932, p. 25).


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The Difference betweenEconomics and Management

Economics

The economist is not concerned with ends as such. He isconcerned with the way in which the attainment of ends islimited. The ends may be noble or they may be base. They maybe material or immaterial if ends can be so described. But ifthe 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


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The Role of Mathematics inEconomics

Mathematics is for Describing HumanBehavior in Economics


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CaseBehavior 1:


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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

Friends

Happy B

How to manage the budget?


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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
http://www.tsl.state.tx.us/ld/projects/trc/2006/manual/illos/sheep.jpg


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Similarities

Scarcity European lacks sheep, African lacks tobacco The student doesnt have enough money

Allocation and Reallocation

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

Satisfaction 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 marginallythe levels of consumption of some

goods.


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Decisions at the Margin andSatisfaction

Exchange at the Margin

The European has manytobacco sticks and is willing to give twosticks.

The African has somesheep and is willing to give onesheep.

Reallocation at the Margin

The student is willing to reduce to some extentthe consumption of some goods forother uses of money.

Satisfaction

The European and the African try to increase their levels of satisfaction with amarginalexchange.

The student tries to maintain her level of satisfaction with a marginal decreaseof theconsumption of some goods.


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Decisions at the Margin andMathematical Language (Exchange)

Exchange Scarcity Allocation Max. Satisfaction

U(Eur)

A B

U(A)

U(B)

Sheep

U(Goods)U(B) = U(A) + * (B - A)

U(B) - U(A) = * (B - A)U(B) - U(A)

(B - A) =

U(B) - U(A)

B - A


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A more general (and formal) approach

Given that U (the function of satisfaction) has a form (equation), whatis the marginal increase in the satisfaction at any given point of U?

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

U(Eur)

A B

U(A)

U(B)

Sheep

U(Goods)

U(B) - U(A)

(B - A) =

U(B) - U(A)

B - A

is a tangent!


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is more than a tangent

U(B) - U(A)

(B - A) =

A

U

The process of decreasing thehorizontal distance in order tofind the right value of the tangentis represented by a limit.

LimB A

Limit when B tends to A of

U(B) - U(A)

(B - A) = = U

is the derivative of U

Decisions at the margin arerepresented by derivatives

Calculus is the mathematicallanguage of Economics


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Decisions at the Margin andMathematical Language (Reallocation)

Scarcity (re)allocation max. satisfaction

x

y

xk

yk

x1

y1

The reallocation of Entertainment

Let denotex= units of movie,y= units of party

The Problem:Given an initial allocation(x

1,y

1) for party and movie,

find the set (x2,y2) that lessdecreases the current levelof satisfaction. (x1,y1)

U1U0

U1>U0

U2

U2>U1


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Decisions at the Margin andMathematical Language (Reallocation)

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

U1

x

y

x1

y1(x1,y1)

(x2,y2)

The Strategy:

To make infinitesimal reallocations

x2 x1

y2

y1

xU1 = yU1

Introduction Mathematics in Economics-1

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