Math 4. Math for Economists. Lecture 01. Introduction to the Course
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- Опубликовано: 5 авг 2013
- UCI Math 4: Math for Economists (Summer 2013)
Lec 01. Math for Economists
View the complete course: ocw.uci.edu/courses/math_4_mat...
Instructor: Jason Kronewetter, Ph.D.
License: Creative Commons CC-BY-SA
Terms of Use: ocw.uci.edu/info
More courses at ocw.uci.edu
Description: UCI Math 4 covers the following topics: linear algebra and multivariable differential calculus suitable for economic applications.
Recorded on August 5, 2013
Required attribution: Kronewetter, Jason. Math for Economists 4 (UCI OpenCourseWare: University of California, Irvine), ocw.uci.edu/courses/math_4_mat.... [Access date]. License: Creative Commons Attribution-ShareAlike 3.0 United States License. (creativecommons.org/licenses/b...)
As a note: the actual lecture starts at 20:00 (a bit earlier than that if you want to watch a clip of the Matrix movie)
The lesson theme of "Solving Systems of Linear Equations" begins at 37:05 and if you have the book "Mathematics for Economists" by Hoy, Livernois, et al. then it is Chapter 7.
How can I find the lectures of the earlier chapters (1-6)?
This playlist starts from chapter 7.
Allow me put the lecture outline in this thread since this time stamp is already pretty useful for future reference. :)
Lecture Outline:
1. Examples of math use / Building intuitions
1.1 Depreciation
1.2 Money multiplier and the geometric series
1.3 Graphing lines to build on intuition
2. Solving Systems of Linear Equations
2.1 Types of solutions to systems of linear equations: Unique, none or infinite
2.2 Ways to solve: Graphical, substitution and elimination method
2.3 Degenerate cases: Parallel lines and same line
3. Linear systems in n-variables
3.1 Just-identified equations: no. of equations = no. of variables.
(Note: Substitution and elimination method are appropriate)
3.2 Overdetermined equations: no. of equations > no. of variables
(Note: Substitution and elimination method are appropriate)
3.3 Underdetermined equations: no. of equations < no. of variables
(Note: Define a parameter to present the solution)
Thank you from the future! 🙏🏽
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00:00 Course information (topics, homework, quizzes, grading, schedule, textbook etc.)
06:00 Introduction: Why is there a maths for economics class (at UCI)? Who this course is for (it is especially for those who wish to go to graduate school)
10:50 Playing a clip from the movie, The Matrix --> "Downloading" information into your head
12:50 Dr Kronewetter's paedagogy - How to learn the content as quickly as possible
16:35 Dr Kronewetter's paedagogy - How to "download" math (into your head) --> Start with something very easy, and then build on it, increasing in difficulty/complexity.
**Simple method**: when you have a complete solution for a problem, take a clean piece of paper and until you can do it completely by yourself (without looking at the original solution), you aren't ready to move on
19:59 Introduction: Why do economists need math? Often, topics in economics have to be put into mathematical terms in order to understand it.
24:10 The money multiplier: The amount of money that gets created after you create $1. (e.g. if the federal reserve makes $1, you end up with $10 in the economy).
Example using simplistic model of $1: Fractional Reserve Bank keeps 10% of deposits, so they can only lend 90%. If the first loan is $1 and goes into the bank from the federal reserve, and because the bank keeps 10% of $1 (i.e. 10c), there is only 90c available to loan out to people and it goes out into the economy. Eventually some or all of it will go back into the bank, so the next loan will be 90c of that 90c (i.e. 0.9*0.9), and so on
28:24 The above pattern explained above (i.e. for the total loans) can be described by a geometric series
30:00 Formula for the money multiplier (1/f where f is the fractional reserve rate)
31:10 Goals for this class --> 1a) Survey of linear algebra, 1b) Survey of multivariable calculus, 2a) Mathematical maturity (not being intimidated by notation), 2b) Mathematical abstraction (thinking about things abstractly), 2c) Mathetmical generalisation, 2d) Proof-writing
3) Intuition (the mind stores details in pictures)
37:06 Solving systems of linear equations
38:40 Linear equation: can have any number of variables (n), but variables can only have powers of 1, and the coefficients can be any real number. Equations have an equal (=) sign
40:50 Levels of variables
1 variable e.g. x=2, or y=3 (a line)
2 variables e.g. 2x+y=1 (generalised: ax+by=c, where a,b & c are coefficents)
3 variables e.g. 2x+3y-z=6 (the third variable, z is projected from flat 2D) --> plane
44:37 Use of subscript in instances where there is more than 3 variables. Coefficients associated with these variables can also be labelled with indices
48:00 Examples of a non-linear equation: root(x)+y=3, 1/x-y=7, x^2+y^2=4
49:19 y=root(x) --> Law of Diminishing Returns
50:30 y=1/x (focusing on the positive x axis only) --> Indifference curve
52:30 "System" means there are multiple linear equations
Possibilities: 1) Unique solution, 2) No solution, 3) Infinitely many solutions
54:10 Two variable system of linear equations --> applications include: supply & demand, equilibrium
1) Unique solution: 1 point which satisfies both equations
2) No solution: Parallel lines (do not intersect)
3) Infinitely many solutions: 2 overlapping lines (the same line)
57:40 Imagining the third dimension for the above scenarios
1) Unique solution: for two intersecting planes, the point where they meet now becomes a line in the 3rd dimension
2) No solution: 2 planes stacked on each other - they still do not intersect
3) Infinitely many solutions: the same plane
58:35 Examples of two variable systems of linear equations
01:02:24 Solving systems of linear equations
01:02:34 1) Graphical method (not effective for more complex situations)
01:04:20 2) Substitution method
01:07:20 3) Elimination method
01:10:56 Degenerate cases
01:10:58 2) Parallel lines - coefficents of x and y in both lines are multiples of each other. The same slope means they are parallel line (unless they are the exact same line). Because the constant number is not a multiple of both lines, we know they are parallel lines (rather than the same line). Algebraically, you get a nonsense answer when trying to solve this system. There are no solutions.
01:13:55 3) Same line - both lines have a multiple of each other that produces the exact same line. When you solve this algebraically, you get 0=0. Thus, there are infinitely many solutions.
01:17:00 Multivariable systems of linear equations. Two things to consider: 1) the number of equations, 2) the number of variables
01:17:48 Example using 3 variables and 3 equations
01:22:24 Degenerate case: Overdetermined system - situation where there are too many solutions (more equations than variables). Example using 3 equations and 2 variables
01:27:18 It's not always the case that overdetermined systems have no solution, but usually there isn't (at least in this course)
01:28:16 Underdetermined system - situation where there are not enough solutions (fewer equations than variables). Example using 1 equations and 2 variables. Every point on the line is a solution, so there are infinitely many solutions to this system.
01:32:04 (For the undetermined system, from previous example) we can present the solution using a parameter, t. The equations can be rewritten in terms of t, where t is any real number
01:34:28 Example of underdetermined system where there are 3 variables but only 2 equations. There are two planes which are intersecting and the goal is to find the line where they intersect (remember the point where they intersect is actually a line in the 3rd dimension). Parameterisation is used - 1 variable is set to equal t, and the equations can then be written in terms of t
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He sure is good :) However, mathematical economics is different from mathematics for economics!
@@jqn8361 elaborate plz
I know it's different
But how exactly?
@@yydd4954 Well, mathematical economics could be defined as the condensation of economic theory and economic ideas in mathematical models, rather than in, say, essays. An example of mathematical economics is how the money multiplier is modeled with a geometric series. The money multiplier can be and has been discussed in economics essays and books, but it is only once the concept is mathematically modeled that we can talk about the multiplier being part of mathematical economics. That is, economic concepts can be expressed in different ways and mathematical economics is one of them. So mathematical economics is just a way of “doing” economics
Mathematics for economics relates to the set of mathematical techniques commonly used by economists within mathematical economics (e.g. Lagrange multipliers)
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@@harshvardhansingh9057 Jason Kronewetter. Better late than never lol
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@@arkamukherjee4027 on the website, the link is in the video description :)
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At 37:22 he starts teaching from the book. Section 7.1, Solving Systems of Linear Equations.
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Yeah please include the titles of the topics discussed in each lecture!! Big help! Thanks :-)
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FYI- money multiplier is a fallacy. Banks don't lend reserves :)
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5 years man
You might be able to understand an economic phenomenon without putting into mathematical terms, but you'll have immense difficulty COMMUNICATING such understanding to others. This is one principle reason why simplifying the understanding to mathematical equations is so popular. Understanding is deep. Math is relatively shallow in comparison.
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Syllabus material starts @ 19:51
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