Introduction and the Role of Mathematics in Economics

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Introduction and the Role of Mathematics in
Economics

• Is Economics a Science?
• Mathematics is for Describing Human Behavior in
  Economics

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Is Economics a Science?

• Etymology

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Adam Smith
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The XIX Century
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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 right
• Deontological
• Teleological

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Neville Keynes
Scope and Method of Political Economy
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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 right
• Deontological
• Teleological

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Lionel Robbins
1932. An Essay on the Nature and Significance of
Economic Science.
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).
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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

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The Role of Mathematics in Economics

• Mathematics is for Describing Human Behavior in
  Economics

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Case Behavior 1
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Case Behavior 2

• A student has 500 for her monthly expenditures

How to manage the budget?
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A Comparison
Clothes 3 ? 1 Save 40 Movie
2 ? 1 Save 15 Shamrock 3 ? 1
Save 25 Food 150 ? 140 Save
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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 marginally the levels of
  consumption of some goods.

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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.

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Decisions at the Margin and Mathematical Language
(Exchange)

• Exchange
• Scarcity ? Allocation ? Max. Satisfaction

U(B) U(A) (B - A)
U(B) - U(A) (B - A)
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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 for any
  point A?

is a tangent!
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Decisions at the Margin and Mathematical Language
(Exchange)

• is more than a tangent

U
A
The process of decreasing the horizontal distance
in order to find the right value of the tangent
is represented by a limit.
is the derivative of U
U
Decisions at the margin are represented by
derivatives
Calculus is the mathematical language of Economics
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Decisions at the Margin and Mathematical Language
(Reallocation)

• Scarcity ? (re)allocation ? max. satisfaction

• The reallocation of Entertainment

• Let denote
• x units of movie,
• y 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.

U1gtU0
U2gtU1
yk
xk
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Decisions at the Margin and Mathematical Language
(Reallocation)

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

• The Strategy
• To make infinitesimal reallocations

?xU1 ?yU1