Mathematics in Economics Lecture 1 Jiří Mazurek, Ph.D. Mathematics in Economics/PMAT Lecture 1 General information •Mathematics in Economics: 5 credits, 13 lectures and seminars. •Seminars are mandatory. •Materials can be found at Public files and Elearning. •Teachers: Jiří Mazurek •Exam: written, two parts (30+70 points). •Evaluation: A (100-91 points), B (90-81), C (80-71), D (70-66), E (65-60), F (59-0). • • • • Syllabus (short version) • 1. Real function of one real variable • 2. Introduction to differential calculus of one real variable 3. Course of a function of one real variable 4. Real function of two real variables • 5. Local and bounded extremes of a function of two variables 6. Indefinite integral of one real variable 7. Special substitutions in the indefinite integral 8. Definite integral of one real variable • 9. Applications of the definite integral 10. Infinite number series 11. Infinite function series 12. Introduction into ordinary differential equations 13. Linear differential equations Lecture 1 Literature •Basic: •CHIANG, C.C. Fundamental Methods of Mathematical Economics. New York: McGraw-Hill, Inc., 2000. ISBN 0-12-417890-1. • •Recommended: • •KLEIN, M. Mathematical Methods for Economics. Edinburgh: Pearson Education Limited, 2014. • •ASANO, A. An Introduction to Mathematics for Economics. Tokyo: Sophia University, 2012. Lecture 1 Youtube math courses •Bhagwan Singh Vishwakarma: •https://www.youtube.com/channel/UCBmdoqkBaIUh128ffq8XATw • •Nancy Pi: •https://www.youtube.com/channel/UCRGXV1QlxZ8aucmE45tRx8w • Lecture 1 A real function of one real variable •A function is a relation f between two sets X and Y such that each x from X is related to exactly one y from Y. •We write y = f(x). •Examples: • •The set X is called a domain, the set Y is called a range or a co-domain. •In economics the domain usually consists of non-negative real numbers. Lecture 1 A graph of a function •A graph of a function is the collection of all ordered pairs (x,y) displayed in a two-dimensial plane. •In a two dimensional plane it as a curve. • • Examples: •Linear function: a line, •Quadratic function: a parabola, •Reciprocal function: a hyperbola. • •Other functions are represented by more complex curves. • • • Lecture 1 A graph of a function: examples • • • • Lecture 1 graf08 Elementary function properties • • • • Lecture 1 • Domain and range, • Monotonicity (increasing, decreasing, non-increasing, non-decreasing, constant) • Extremes (local or global maximum or minimum), • Concavity and convexity, • Inflection points, • Bounded vs unbounded function, • Even functions and odd functions, • Peridiocity. Elementary functions • • • • Lecture 1 • Linear function: y = ax + b, • Quadratic function: • Polynomial function: • Linear reciprocal function: • Logarithmic function: In the logarithmic function, a is the so called a base. The decadic logarithm: a = 10, the natural logarithm: a = e = 2.718… For a larger than 1, the function is increasing, and for a smaller than 1 it is decreasing. Graphs of logarithmic functions • • • • Lecture 1 graf05 graf37 Elementary functions - continued • • • • Lecture 1 • Exponential function: • Goniometric functions: y = sinx, y = cosx, y = tgx, y = cotgx. • Cyclometric functions: y = arcsinx, y = arccosx, y = arctgx, y = arccotgx. Notes: exponential and logarithmic functions are inverse to each other. The same applies to goniometric and cyclometric functions. • Graphs of exponential functions • • • • Lecture 1 For a >1, the exponential function is increasing, for a < 1 is decreasing. Graphs of goniometric functions Lecture 1 graf33 graf34 graf35 graf36 Polynomials • • • • Lecture 1 In economics, various functions, such as demand or supply, are expressed by polynomials. Two main tasks when dealing with polynomials are transformation of a polyonomial into a product, and to find roots of a polynomial. Let denote a polynomial of a degree n. Polynomial roots are such values of x that . The equation above can be solved via known formulas or identities such as . Demand and supply function, equilibrium •The demand function expresses relationship between a price of a good (P) and a demanded quantity (Q) by customers. Usually, the demand function is denoted as Q= D(P) or QD , and it is assumed this function is decreasing. •The supply function expresses relationship between a price of a good (P) and a supplied quantity (Q) by sellers. Usually, the supply function is denoted as Q= S(P) or Qs , and it is assumed this function is increasing. •A point where demand is equal to supply, ant a market is cleared, is called an equilibrium. Lecture 1 Demand and supply function, equilibrium – cont. • • • • Lecture 1 Graf 31 Solved problem 1 •Find the domain of the function f: . • Solution: the expression under the square root sign must be non-negative, therefore we obtain: • • • •Hence, the domain is . Lecture 1 Solved problem 2 •Find the domain of the function f: . • Solution: the expression in the logarithm must be positive, therefore we obtain: • • We expand the term on the left hand side: • • •From the last inequality it follows that -5 and 5 are the roots that divide the x line into three intervals. By checking the sign in each interval we obtain the final solution: • • • • • • Lecture 1 Solved problem 3 •Find the domain of the function f: . • Solution: the expression in the arcsin is bounded by -1 from below and by 1 from above. Therefore, we obtain: • • • By dividing this inequality into two simple linear inequalities we obtain: • and • •Hence, we obtain the solution: • Lecture 1 Solved problem 4 •Let us asssume that the demand and the supply functions are given as follows: , . •Find the equilibrium. • •Solution: in the equilibrium both functions are equal: • • •Therefore, we obtain: PE = 6, and QE = 4. Draw both function! • •How will the situation change if there is a price floor P = 8? • • • Lecture 1 Solved problem 5 •Let us asssume that the demand and the supply functions are given as follows: , . •Find the equilibrium. • •Solution: in the equilibrium both functions are equal: • • • •Therefore, we obtain: PE = 3, and QE = 18. Draw both function! • • Lecture 1 Problems to solve •1. Find the domain of the following functions: • • • • • Lecture 1 Problems to solve – cont. •2. Draw a graph of the following functions: • • • • • • Lecture 1 Problems to solve – cont. •3. For the given functions of demand and supply find the equilibrium both geometrically and algebraically: • • • • • • Lecture 1 • • • • Lecture 1 Thank you for your attention!