Mathematics / Mathematics for Economists. Andrzej Malawski & Beata Ciałowicz and Marta Kornafel

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1 Mathematics / Mathematics for Economists by Andrzej Malawski & Beata Ciałowicz and Marta Kornafel Department of Mathematics, CUE General description: 1-semester course = lecture + classes (60 h = 30 lectures + 30 classes, a week: 2 +2) Assessment: - of the course: final exam (written) - of problem sessions (classes): credit in each semester Bibliography/course materials L.D.Hoffmann, G.L. Bradley, Calculus for Business, Economics, and the Social and Life Sciences, McGraw-Hill, M.L.Bittinger, J.A.Beecher, College Algebra, Addison-Wesley, Additional readings delivered by lecturers.

2 What is mathematics? - traditionally: Introductory remarks the formal, exact science dealing with quantitative and spatial aspects (properties) of the real world phenomena and processes - nowadays: ex.: - arithmetic, geometry the collection of theories of mathematical structures calculus - differential structures (smooth transformations) algebra - algebraic systems (groups, vector spaces, Boolean algebras,...) topology - continuous structures (theory of limits,...) What is economics? (loosely speaking) - the empirical, social science studying the economic life of societies, i.e. real economies based on economic phenomena and/or processes such as production, consumption, exchange, etc. on their micro- and macro level - multiparadigmatic discipline.

3 Why do economists need mathematics? - all economic phenomena and processes are measureable (ex.: costs, expenditures, taxes, GDP, interest rates, inflation, growth, etc ), - natural sciences (physics, astronomy, ) are based on mathematics and economists follow their example Course Content: 1. Fundamentals (12h = 6 + 6) - logic, operations on sets, - relations, functions, - applications to economics: o commodity/price space R l, o preference relation, utility function 2. Calculus (36 h = ) - metric space, - limits and continuity, - differentiation: o derivatives, higher-order derivatives, o investigation of function, o practical optimization problems: applications to business and economics - antidifferentiation: o antiderivatives, o integration by substitution, o integration by parts - integration: o definite integral,

4 o area and integration, o applications to economics - functions of several variables: o partial derivatives, o relative maxima and minima, o Lagrange multipliers, o the method of least squares - differential equations (?): o elementary differential equations, o economic growth models, o tatonnement process. 3. Linear Algebra (12 h = 6 + 6) - matrices, - determinants, - simultaneous equations, - input-output model, - algebraic structures: o group, field, o vector space

5 How to study mathematics for economists?

6 - No problem! (entrance qualifications..)

7 Antoine Augustin Cournot ( ), father of m.e., openminded man, lawyer, PhD in mechanics and astronomy, professor of analysis and mechanics in Lyon, rector of the Academy of Grenoble Researches into the Mathematical Principles of the Theory of Wealth (1838) - this book opened a new chapter in the history of economics because of the systematic analysis of profit maximization by firms based on differential calculus The law of demand: the quantity demanded D is a function of market price p D = F(p) Postulates: the demand curve is monotonically decreasing, continuous, and twice differentiable. The demand curve is used to derive the consumers expenditure as a function of price: R = pf(p) This represents also the revenue of the seller. Remark: under the above postulates it has at least one maximum (R = 0, for p = 0 and for p sufficiently great ) These maxima can be found by the conditions: and R (p) = F(p) + pf (p) = 0 R (p) =... < 0 The farther analysis: monopoly, competition, imperfections, taxation,... in the same mathematical apparatus

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9 Test (example) 1. Define determinant of a matrix det A and solve the system of linear equations 2x y + z = 2 x + y z = 3 4x + y z = 9 2. Determine where the function f(x) = 1 + 2x + 18/x is convex and concave. Find the inflection points. 3. Draw the domain D R 2 of the function f(x,y) = ln (2x + y) and compute the partial derivatives of the first order. 4. Define the Riemannian integral and find the area of the region R bounded by the curve y = x 3 and the line y = 2x. 5. Suppose the total revenue in dollars from the sale of q units of a certain commodity is R(q) = -2q q 128. (a) At what level of sales is the average revenue per unit equal to the marginal revenue? (b) Verify that the average revenue is increasing if the level of sales is less than the level in part (a) and decreasing in the opposite case. 6. Interpret vector space R l as a commodity-price space.