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 € RoomRoom FoodFood 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 12 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( )  12

Similarities Scarcity Scarcity European lacks sheep, African lacks tobacco The student doesn’t have enough money Allocation and Reallocation 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 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.

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) AB 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) AB 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 xy xkxk ykyk x1x1 y1y1 Entertainment The reallocation of Entertainment Let denote x x = units of movie, y y = units of party The Problem: Given an initial allocation (x 1,y 1 ) for party and movie, find the set (x 2,y 2 ) that less decreases the current level of satisfaction. (x 1,y 1 ) U1 U0 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 xy x1x1 y1y1 (x 1,y 1 ) (x 2,y 2 ) The Strategy: To make infinitesimal reallocations x2x2 x1x1 y 2 y 1  x U 1 =  y U 1

Functions, Graphics and Derivatives Functions are represented by equations and graphics. Terminology fxfxfx f(x): f is a function of x. It is read “f of x” x x is an independent variable fx f is dependent of x Examples Straight lines  f(x) = ax + b Hyperbolic and parabolic functions  f(x) = x 

Some Functions and their Graphics: Straight Lines y = ax + b y x y = ½ x + 2 x0126 y22,535 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 y x 1 =  = -1 y x 2 =

Derivatives Terminology How a function f varies with respect to one variable: The derivative of f(x) with respect to x. f If f 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: d dx f(x) df(x) dx x  x f(x)

Calculating Derivatives f(x) = x n  x f(x) = nx n-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 right Assumptions about what is “right”  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 An Essay on the Nature and Significance of Economic Science.