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

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Del curso dictado por National Research University Higher School of Economics

Mathematics for economists

6 calificaciones

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National Research University Higher School of Economics

Mathematics for economists

6 calificaciones

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.

De la lección

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

Conoce a los instructores

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

So, condition number one.

When we substitute into all these functions the coordinates, x coordinates,

all the point, we

get y value and that's true for all j from one to m inclusive.

Secondly, if we turn to the system,

our original system of equations,

when I'm taking jth equation and

substitute for y values,

the implicit functions,

I get zero, identically zero.

So, this is true for all x which belongs to the [inaudible] , B tilde.

Thirdly, all such derivatives can be found.

This phrase means that the formulas are derived but they're kind of

cumbersome and I'll show you using some examples that,

in each particular case,

such a derivative can be easily found,

more or less easily found.

Let's proceed with the examples from, first, in microeconomics.

Let us consider an IS-LM model

where we have two equilibrium equations or conditions.

There is an equilibrium in the goods market and an equilibrium in money market.

So, we have a system of two equations,

the outputs in an economy,

y, equals the consumption,

and this consumption function is the function of the output less of taxes imposed,

so this is a disposable income, plus the investment,

investment is a function of the interest rate,

and plus the government expenditures.

So, this equation defines equilibrium in the goods market.

The second equation concerns equilibrium in the money markets.

Money supply should equal demand on money,

which is a function of the output, y,

and the interest rate, r. The properties of the functions involved are known.

For instance, it is known that the derivative of the consumption function lies

between zero and one and it is called marginal propensity to consume.

As for the investment function,

its derivative is negative.

Demand and money, derivative of this function with respect to the output,

is positive and derivative with respect to the interest rate is negative.

Now, let's consider a comparative statics problem.

Let government expenditures be fixed and the money supply will be also

fixed and T taxes will change.

That means that dT increment in T, in taxes,

is not zero but

the increments in money supply and government expenditures are zero.

How will the change of taxes will affect the output and the interest rate?

So, we're interested in finding expressions for two derivatives:

the derivative of the output with respect to taxes

and derivative of the interest rate with respect to taxes,

so that's the question.

First of all, we need to check whether this third IFT is applicable.

In order to do that,

let us consider Jacobian of two equations.

Remember, the Jacobian is a matrix,

two by two matrix whose rows are made of derivatives.

So, we need to differentiate the first equation with respect to Y and

r which are endogenous variables and we fill in the first row of this Jacobian matrix.

Now, we proceed with the second equation and also we differentiate with

respect to Y and interest rate r and fill in the entries in the second row.

After that, we need to check that this square,

two by two matrix,

is a non-singular matrix,

so let us do it.

So, J, this is Jacobian, is a matrix.

First of all, let us

move all these three terms to the left and differentiate with respect to Y,

then I get one minus C prime.

After that, in order to fill in this missing entry in the first row,

I would like to remind you that this term was already moved to the left,

now we'll differentiate with respect to r. That's how we get minus I prime.

Now, here, we simply differentiate with respect

to the output and with respect to the interest rate.

Somewhere here, I will calculate the determinant of this matrix, it equals.

Okay, let's check the sign.

This difference within the brackets is positive,

this derivative is negative and

this derivative is also negative and this derivative is positive.

So, all in all, we have negative result which tells us that IFT

is applicable and the only thing is we need

to find out the expressions for these derivatives.

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