6.1. Envelope Theorem.

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來自 National Research University Higher School of Economics 的課程

Mathematics for economists

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National Research University Higher School of Economics

Mathematics for economists

8 個評分

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.

從本節課中

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

與講師見面

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

We proceed with the envelope theorems.

There are at least two theorems which are called envelope,

and one is about unconstrained optimization,

another one is about constrained and we start with the unconstrained firstly.

We have an objective function,

a function of many variables but also it depends on some scalar parameter alpha.

So we are considering a problem of unconstrained either maximization or minimization,

and alpha as I said is scalar parameter so it belongs to some interval.

As for the function f,

this function is continuously

differentiable in all the variables.

Now, let's suppose we

have found the solution of this optimization problem.

To be more concrete,

let's suppose the function has been maximized.

So then, the maximum point depends on alpha.

Now we'll also suggest that this vector function of

alpha is still continuous differentiable on the interval I.

Later on we'll find out how to guarantee this condition.

For the time being, let's suppose they found maximum point

is continuously differentiable and now we'll substitute x star into the function itself,

having a function of alpha alone and we'll call it value function.

So by definition I'll be using phi letter,

this value function is defined exactly like this.

Now the question is,

how to find derivative of this value function with respect to parameter.

The envelope theorem is

exactly the theoretical statement which simplifies the answer to this question.

We form it as a theorem.

I will need just one formula.

So under condition that a function is continuously differentiable,

that x star is also continuous differentiable,

this derivative can be formed by partial differentiation or

the objective function with respect to alpha and later on we

substitute into the argument of the function the maximum point.

We'll provide the proof.

The proof is quite simple.

Actually, to find this derivative means

just to apply the chain rule to the composite function.

This is our composite function, so then becomes.

Don't forget that x star is the maximum point and that

means that all first order derivatives

of the objective function turns zero at this point.

So what's left?

All this terms under the summation symbol disappear and we get just one term

where we substitute x star point.

Now, there are a lot of

applications to the microeconomic theory of this particular theorem.

I will just provide one.

Let's suppose we have a perfectly competitive firm

whose cost function is c(y) function which is continuously differentiable,

and we are considering a profit maximization problem where p is

the price of the production per unit and this is the revenue,

y is the output.

So the firm maximizes its profits.

Maximization takes place with respect to the output.

Now the question is how to find

the derivative of the profit function with respect to the price.

Without knowing the envelope theorem,

the problem is solved through some steps.

Firstly, you need to find the supply function of the firm by

differentiation of the pie function and having the first or the conditions.

After that, actually you have to apply implicit function theorem in

order to find y as the function of price p and later,

you have to substitute into the pie function or

the supply function and only then

you are able to differentiate with respect to the price.

But the application or the envelope theorem makes

it much easier because all we need to do,

we need to take this formula and we need to simply differentiate with respect to

the parameter which is p.

Now this pie star function which is called in microeconomic theory a profit function,

is exactly the value function.

All we need to do, we need to take

the formula and differentiate with respect to p then we get the answer.

It's y star the supply which is

positive and that means that with the growth of the price,

the profit function which is the maximum profit

attainable at a given price is also positive.

The derivative is positive so it's growing.

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