3.8. Implicit Function Theorem for the system of implicit functions. Jacobian matrix.

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来自 National Research University Higher School of Economics 的课程

Mathematics for economists

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National Research University Higher School of Economics

Mathematics for economists

1 个评分

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.

从本节课中

Implicit Function Theorems and their applications.

Week 3 of the Course is devoted to implicit function theorems. In this week three different implicit function theorems are explained. This week students will grasp how to apply IFT concept to solve different problems.

3.1. Implicit functions.7:05

3.2. Implicit Function Theorem.5:33

3.3. Applications of the Implicit Function Theorem (part 1).10:33

3.4. Applications of the Implicit Function Theorem (part 2).7:01

3.5. Gradient is perpendicular to a level curve of a function.11:41

3.6. Implicit function theorem for the function of many variables.6:20

3.7. Example of application of the IFT for the function of many variables.8:45

3.8. Implicit Function Theorem for the system of implicit functions. Jacobian matrix.10:53

3.9. Example of application IFT for the system of implicit functions (part 1).8:41

3.10. Example of application IFT for the system of implicit functions (part 2).7:05

3.11. Example of application in microeconomics.6:26

3.12. Cramer's rule.2:45

与讲师见面

Kirill Bukin
Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics

Previously, we dealt with one equation,

defining just one implicit function.

So, we either considered a function which had x and y two variables,

and we were looking for a function y as a function of x,

or in the latter case,

we dealt with one equation,

which defined a function y.

This time a function of many variables,

but quite often in economics,

we have to deal with the systems of equations.

We're looking for several implicit functions or

sometimes we can call it vector valued functions,

but I will try to abstain from this term.

So, let's consider m functions,

they are functions of x and y.

So, the arguments for all of

these functions are written in the form of,

I'm using x letters and also y letters.

So, this is equation number one

and I continue until I reach the final equation.

So, the question is whether it's possible to find m functions denoted by y letters,

all of them being the function of x alone.

But firstly I need to provide a definition of

a very important and quite useful concept

in the calculus of several variables,

which is called Jacobian or Jacobian

matrix of a system of functions.

To begin with, we have written within the braces,

the system, each of

them depends on n variables and within the system we have m functions.

Now, let us introduce a matrix,

each row of which consists of a gradient of all these functions,

starting from the first.

So, in order to fill in the entries in this matrix,

we take the very first function, f1,

and we differentiate with respect to x and fill in the entries.

So, it becomes.

Now, in order to fill in the second row,

we need to take the second function, f2,

and differentiate it with respect to these variables,

that's how we get

and we continue until we reach the nth function at the very bottom.

So, here I put dots indicating that there are more rows until we reach the nth row.

That's how we get this matrix,

which has the name a Jacobian matrix or simply Jacobian for short,

and this is m by n matrix.

As I said earlier,

it's made of the gradients of all these functions drawn from the system.

Now, going back to this implicit function theorem number three this time.

Let's suppose that all these functions are defined in some ball,

so they're continuously differentiable, there is a ball.

Ball centered at point x0, y0,

it belongs to the n plus m dimensional vector space,

R n plus one,

n plus m. We are considering a system,

let me unite all these m equations with

a brace and I'll write a number for this system of equations.

We assume that this particular point satisfies all m equations being after substitution.

Now, we need to formulate a condition under which

the system of implicit functions exists.

This condition is formulated in terms of

the non-singularity of the Jacobian matrix.

This Jacobian matrix is based on

these m functions but we differentiate only with respect to y variables,

that's how we get a square m by m matrix.

So, the condition which we need to check in order to be

sure that m implicit function will exist,

takes the form the determinant of the Jacobian matrix,

J, is not zero at a given point.

Where J is a matrix, as I said,

a square matrix where we differentiate respectively,

all these functions with respect to y variables.

So, we have here,

I'll fill in the final row.

So, once again let me repeat,

we deal with a system of equations based on a system of functions.

All these functions are continuous and differentiable on some ball.

The ball has a center at a given point,

and this particular point satisfies all these equations.

Now, we check at a given point

the non-singularity of the Jacobian matrix which is a square matrix.

We have information, this is a non-zero number.

Now, the claim of the theorem is as follows,

there exists a ball,

B tilda, in n-dimensional space centered at x0.

There exist a system of implicit functions.

I will be using a shorthand notations in the vector form to make it shorter.

This theorem claims that there exists a ball, B tilda,

in n-dimensional space centered at exactly x0 point.

There exists a system of implicit functions from y1 through y nth,

and the following conditions hold.

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