33067 Words Feb 16th, 2012 133 Pages
Lecture Notes on Mathematics for Economists

Chien-Fu CHOU

September 2006

Contents
Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Static Economic Models and The Concept of Equilibrium Matrix Algebra Vector Space and Linear Transformation Determinant, Inverse Matrix, and Cramer’s rule Diﬀerential Calculus and Comparative Statics Comparative Statics – Economic applications Optimization Optimization–multivariate case Optimization with equality constraints and Nonlinear Programming General Equilibrium and Game Theory 1 5 10 16 25 36 44 61 74 89

1

1

Static Economic Models and The Concept of Equilibrium

Here we use three elementary examples to illustrate the general
In microeconomics, we even attempt to include every commodity (including money) in a general equilibrium model. Qd1 = D1 (P1 , . . . , Pn ; a) Qd2 = D2 (P1 , . . . , Pn ; a) . . . Qdn = Dn (P1 , . . . , Pn ; a) Qsn = Sn (P1 , . . . , Pn ; a) Qs2 = S2 (P1 , . . . , Pn ; a) Qs1 = S1 (P1 , . . . , Pn ; a) Qdn = Qsn Qd2 = Qs2 Qd1 = Qs1 Qdi : quantity demanded of commodity i Qsi : quantity supplied of commodity i Pi : market price of commodity i a: a factor that aﬀects the economy Di (P1 , . . . , Pn ; a): demand function of commodity i Si (P1 , . . . , Pn ; a): supply function of commodity i We have three variables and three equations for each commodity/market.

2 Substituting the demand and supply functions, we have D1 (P1 , . . . , Pn ; a) − S1 (P1 , . . . , Pn ; a) ≡ E1 (P1 , . . . , Pn ; a) = 0 D2 (P1 , . . . , Pn ; a) − S2 (P1 , . . . , Pn ; a) ≡ E2 (P1 , . . . , Pn ; a) = 0 . . . . . . Dn (P1 , . . . , Pn ; a) − Sn (P1 , . . . , Pn ; a) ≡ En (P1 , . . . , Pn ; a) = 0. For a given a, it is a simultaneous equation in (P1 , . . . , Pn ). There are n equations and n unknown. In principle, we can solve the simultaneous equation to ﬁnd the ∗ ∗ equilibrium prices (P1 , . . . , Pn ). A 2-market linear model: D1 = a0 + a1 P1 + a2 P2 , S1 = b0 + b1 P1 + b2 P2 , D2 = α0 + α1 P1 + α2 P2 , S2 = β0

## Related Documents

• ###### Linear Algebra Applied to Economics Essay

to Economics Input-Output Analysis Introduction: There are several non-mathematical subjects that linear algebra can be applicable too. Economics is a topic that linear algebra can be used to make a formal application, for example in Input-Output Analysis, econometrics, Game theory, and break-even point analysis. As a group we are going to be focusing on the Input-Output analysis, a type of analysis created for the purpose of describing and making predictions of complicated mathematical models…

Words: 1261 - Pages: 6
• ###### Essay Uses of Mathematics in Other Subjects

other. Without mathematics, subjects like are physics, chemistry, geography, economics etc. are incomplete. It possess a variety of applications in connection with other subjects as follows: 1) In geography, it is used in weather prediction, impact of atmospheric change on aquatic and non-aquatic life. 2) In history, it is used to find out the time period from which various artifacts, scriptures belong. 3) Study of economics is heavily based on it. It helps in calculating various index like inflation…

Words: 761 - Pages: 4
• ###### Essay International Student With Stress And International Students

Europe is much better. They are right because American education don’t teach necessary skill for student. The American workforce has some of the weakest mathematical and problem-solving skills in the developed world (the editorial board, 1). Mathematical and problem-solving skills are two important skills that students need in working after. Mathematical can help student to think more logically and calculate number. Also, students should have problem-solving skills to think and figure out problems in…

Words: 1440 - Pages: 6
• ###### Essay Kenneth J. Arrow : His Journey Into The Mathematical World

journey into the mathematical world. He started his journey to becoming a great mathematician by receiving his M.A. at the university of columbia. While doing his research here Kenneth meant the statistician economist Harold Hotelling. Kenneth was so influenced by the Harold’s work that he promptly switched over to the Economics Department for subsequent graduate work. Kenneth would then go on to obtain his Master Degree In 1941. Unfortunately while he was living in his mathematical world the world…

Words: 854 - Pages: 4
• ###### Essay Jan Tinbergen 's Legacy Of Ethics

already publishing models and explanations, along with the idea that mathematics and statistics were connected to economics. In 1965 he began working with the League of Nations. Finally this and every other one of his contributions led him to sharing the Nobel Prize. In 1969 he shared the prize with Ragnar Frisch, another contributor of similar studies. The two received it from the economic science category in honor of their involvement in econometrics. Even after receiving his prize, his life’s work…

Words: 2117 - Pages:
• ###### Essay The Theories Of Organizational Theory And Design

was respected by peers and community for my public service, and while not getting rich, was able to make a sustainable living and consider myself a functional member of society. Since the Prospect Theory is an economic theory of decision making, I can identify with it given the mathematical nature of the concept. The decision to pursue a degree in a field unrelated to my previous occupation meant that I, for the sake of expediency, would need to forsake 2 years of wages and benefits along with…

Words: 904 - Pages: 4
• ###### Essay Why Math Is Hard

numerous students every year who are unable to handle mathematical content. The primary goal, as a high school mathematics teacher, is to increase the student’s knowledge of algebra. However, many students are not ready for given information. Many students are lacking basic mathematical skills and are unable to do simple calculations, like long division. Nationally too many students are behind in mathematics. Every person no matter race, socio economics, or geographical location should be able to do…

Words: 1521 - Pages: 7
• ###### Science of Human Action Essay

universal. For example, the notion that humans engage in acts of choice implies that they have preferences, and this must be true for anyone who exhibits intentional behavior. The most common use of the term is in connection with the Austrian School of Economics, as established by economist Ludwig von Mises. Catallactics is the praxeological theory of the way the free market system reaches exchange ratios and prices. It aims to analyse all actions based on monetary calculation and trace the formation of…

Words: 5931 - Pages: 24
• ###### Essay about My Relationship With Finance

size and level of profit does not matter as poor decisions can still create loss. Making financial decisions is intriguing to me as I ultimately would like to apply my mathematical ability into a career in finance. The concept of investment is interesting to me as it involves calculated risks which depend on an array of mathematical equations to ensure that companies make well informed decisions on how they invest their money. When studying mathematics and finance at university I am keen to understand…

Words: 574 - Pages: 3
• ###### Essay American Student And International Student

is much better. They are right because American education don’t teach necessary skills for students. “The American workforce has some of the weakest mathematical and problem-solving skills in the developed world” (the editorial board, 1). Mathematical and problem-solving skills are two important skills that students need in working after. Mathematical can help students to think more logically and calculate the number. Also, students should have problem-solving skills to think and figure out problems…

Words: 1815 - Pages: 8