Introduction and the Role of Mathematics in Economics

1. Introduction and the Role of Mathematics in Economics Is Economics a Science? Mathematics is for Describing Human Behavior in Economics

2. Is Economics a Science? • Etymology:

3. Adam Smith

4. The XIX Century

5. 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

6. Neville Keynes Scope and Method of Political Economy

7. 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

8. 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).

9. 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

10. The Role of Mathematics in Economics • Mathematics is for Describing Human Behavior in Economics

11. Case Behavior 1:

12. € 200 150 150 Room Food 52€ 30€ Others VWL-Book Friend’s Happy B Toothpaste, Soap, Dish cleaning, … 20€ Cleanliness Clothes 1 T-shirt (20€), 1 trousers (20€), 1 pullover (20€) 60€ Entertainment 2*movie (30€), 3*Shamrock (40€) 70€ Case Behavior 2: A student has 500€ for her monthly expenditures How to manage the budget?

13.  P( )  P( ) 1 2 x P( ) = = 2 1 2 # Sheep A Comparison Clothes: 3  1 Save 40€ Movie: 2  1 Save 15€ Shamrock: 3  1 Save 25€ Food: 150€  140€ Save 10€

14. Similarities • Scarcity • European lacks sheep, African lacks tobacco • The student doesn’t 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.

15. 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.

16. U(Eur) U(Goods) U(B) U(B) - U(A) U(A) B - A U(B) - U(A) ƒ = A B Sheep (B - A) Decisions at the Margin and Mathematical Language (Exchange) • Exchange • Scarcity  Allocation  Max. Satisfaction U(B) = U(A) + ƒ * (B - A) U(B) - U(A) = ƒ * (B - A)

17. U(Eur) U(Goods) U(B) U(B) - U(A) U(A) B - A U(B) - U(A) ƒ = A B Sheep (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? ƒ is a tangent!

18. Lim U(B) - U(A) U(B) - U(A) ƒ = ƒ = B  A (B - A) (B - A) “Limit when B tends to A of ƒ” Decisions at the Margin and Mathematical Language (Exchange) • ƒ is more than a tangent U The process of decreasing the horizontal distance in order to find the right value of the tangent is represented by a limit. A ƒ is the derivative of U = U’ Decisions at the margin are represented by derivatives Calculus is the mathematical language of Economics

19. y (x1,y1) U2 y1 U1 U0 x1 x 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. U1>U0 U2>U1 yk xk

20. y (x2,y2) y2 (x1,y1) y1 U1 y1 x1 x x1 x2 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