6.6. Relaxing assumptions using second order conditions.

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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

Envelope theorems. Concavity and convexity.

Week 6 of the Course is devoted to envelope theorems, concavity and convexity of functions. This week students will understand how to interpret Lagrange multiplier and get to learn the criteria of convexity and concavity of functions in n-dimensional space.

6.1. Envelope Theorem.8:20

6.2. Envelope Theorem for constrained optimization.4:48

6.3. Examples of the Envelope Theorem application (part 1).7:13

6.4. Examples of the Envelope Theorem application (part 2).6:58

6.5. Interpretation of the Lagrangian multiplier.10:51

6.6. Relaxing assumptions using second order conditions.8:33

6.7. Concave and convex functions.7:57

6.8. Concave and convex functions in n-dimensional case.7:53

6.9. Inequality for concave function in n-dimensional space.9:45

6.10. Criteria of concavity and convexity of the function in n-dimensional space.5:35

6.11. Properties of concave functions.7:31

与讲师见面

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

A similar argument can

be expressed for the second envelope theorem which concerns constrained optimization.

So, we have an objective function

which we have to maximize subject to just one constraint.

Once again, alpha is a scalar parameter which belongs to sum interval I.

From the start, let's suppose that both functions,

f and g are twice continuously differentiable for all values of x and alpha.

So, x belongs to Rn space,

alpha belongs to I.

Let suppose that the Lagrangian function is formed according to the formula.

Now, let us write down the first-order conditions for the Lagrangian function.

Let's suppose that the solution was found in the form

of a critical point

and the corresponding lambda value, lambda star.

Earlier, we placed an assumption,

additional assumption on x star and lambda star also depends on alpha.

We said, let's suppose that x star,

lambda star belongs to C1 space of functions.

Actually, there is no need for such an assumption.

Instead, we can think that second order conditions are valid in this case,

and let us recall what it means.

These second order conditions are applied to

the definiteness of the Bordered Hessian metrics.

What will be Bordered Hessian metrics in this particular case?

Let it be H-tilde.

This is n plus one times n plus

one matrix made over border.

To fill in the entries in the first row,

we need to take g and differentiate with respect to the variables.

In the same fashion,

we fill in the entries in the first column,

s for the main block, a square block.

Will fill in these entries with the second order derivatives of

the Lagrangian function and I can write it using a shorthand,

that will be D squared L,

so all these variables are with respect to x.

Now, let's suppose that the second order condition for the maximization hold,

that means that partly, I mean that,

all I need to state at the moment,

the statement will concern the determinant of H-tilde alone.

We know that actually the condition,

sufficiency condition for the maximization problem will

include another leading principle minus all these H-tilde metrics,

but I'm not interested in.

So, let's suppose that the sign of the determinant of

H-tilde is the same as the sign of minus one to the nth power.

That means that this determinant is non-zero.

Now, can we apply implicit function theorem to the system of equations?

We can, the only thing we need to check will be

the non-singularity of the Jacobian matrix.

So, let's write down the Jacobian matrix for

this particular system of equations and compare with H-tilde,

that will produce the result.

So in order to form the Jacobian matrix,

I'll take the final equation where

we differentiate respect to lambda and I'll place it in the,

well, above the system,

so it will become equation number one.

You'll see why because it will resemble H-tilde.

So, J will look like that.

The l with a lambda being minus J.

So, if I differentiate respect to lambda,

g doesn't depend on lambda,

so I have simply zero.

Now, I take this function which is minus g,

and I will start differentiating with respect to x,

that's how I get minus g1 minus gn.

As for the rest of the entries,

we start differentiating with respect to x and we

get exactly Hessian, here.

So, if we compare two matrices H-tilde and J,

the only difference is that in geometrics,

the first row and first column have minuses in front of first-order derivatives of

g. But since we're interested only in determinant of J,

the determinant will not change if we multiply by negative one simultaneously,

the first row and the first column.

So, we get immediately H-tilde and the determinant of H-tilde,

as we know from second order conditions is a non-zero number.

That means that determinant of J is also a non-zero number,

that means the implicit function theorem can be applied.

The critical point or the Lagrangian function,

and also lambda star,

they are continuously differentiable.

So we can relax the initial assumption,

it will be more fulfilled automatically under

the assumption of the second order conditions which it should hold.

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