8.3. Solving consumer choice problem using Kuhn-Tucker conditions (part 2).

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Del curso dictado por National Research University Higher School of Economics

Mathematics for economists

11 calificaciones

National Research University Higher School of Economics

Mathematics for economists

11 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

Kunh-Tucker conditions. Homogeneous functions.

Week 8 of the Course is devoted to Kuhn-Tucker conditions and homogenous functions. This week students will find out how to use Kuhn-Tucker conditions for solving various economic problems.

8.1. Kuhn-Tucker conditions.7:28

8.2. Solving consumer choice problem using Kuhn-Tucker conditions (part 1).9:57

8.3. Solving consumer choice problem using Kuhn-Tucker conditions (part 2).11:15

8.4. Solving minimization costs problem using Kuhn-Tucker conditions (part 1).9:05

8.5. Solving minimization costs problem using Kuhn-Tucker conditions (part 2).7:51

8.6. Homogeneous functions.6:24

8.7. Homogeneous functions. Two propositions.5:24

8.8. Income-consumption curve.10:28

8.9. Euler's Theorem.12:21

Conoce a los instructores

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

So, first let's check that these point where x1

is consumed in positive amount but x2 is not consumed at all can be a corner solution.

So, that will be our corner solution point number one.

So, for that point we have x1* equals

I over P and x2* is 0.

So then, we can substitute,

since x1 is positive from the complimentary slackness condition,

we get that this should be 0,

x2 is also zero.

Then we get immediately λ* is 1 over P,

which is good, this is a positive number because price is positive, of course.

Then, we need to check whether this is fulfilled.

It's interesting, that provides us

the range of income values for which a consumer,

in his consumption, follows this pattern just spending money on buying good number one.

So, according to this inequality we have I over P,

minus one over P,

that's our multiplier, should be non-positive.

So, then we have a smaller income.

Finally, this point.

This point cannot be the maximization point,

clearly, but we need to check that still.

So, at this point x2* is I, x1* is zero,

then we get immediately

from these two values,

if we substitute that λ is zero, but that's impossible.

We've seen it when we begin solving the problem.

Remember, we were considering the case where the maximization point

lies below the budget line and that situation wasn't possible at all.

So, we have finished analyzing the first order conditions.

Now, we have found for the values of income greater than 1.

Well, we can still call it inner solution because this

is not a corner solution then, right?

For small values of income described here,

we have this particular corner solution.

How can we be sure that we've got the maximization points?

Once again, we referred to Weierstrass theorem.

Since the budget constraint is a closed bounded set.

So, Weierstrass theorem is applicable,

the utility function is continuous.

So, it takes the greatest value at some point here,

and we have found just one critical point or the Lagrangian.

So, that means that we have found exactly the optimal bundle or the consumer.

Now, we start another problem.

This will be a problem of minimization of costs.

So, we have a production function.

Let me write it here.

This is the output described by a letter Y.

This is our production function.

Now, we are minimizing costs.

This is a two factor production and that should be minimized. Subject two.

The minimal output level is Y,

so the constraint is written in the form of the value of the production function,

should exceed or at least be the same as y and x1,

x2, takes non-negative values.

We haven't analyzed yet the Kuhn-Tuckent conditions for minimization problems.

We may, but actually, as we know,

we can always replace a minimization problem with a maximization problem.

It's done by simply reversing the sign of the objective function.

So, if we take this condition on

minimization and replace by

minus a 100x1 minus x2, should be maximized.

Then, we get an equivalent problem and we can fill in,

we can form the Lagrangian function,

for this time for the maximization problem.

So, how will it look like?

L, Kuhn-Tucker is Lagrangian for maximization problem,

you start with objective function written there.

What now would take plus,

we take λ and what will be written here? Let's see.

For maximization problems, we subtract from

the bigger term the smaller term

that becomes like this,

and it's quite time to write

down the first order conditions using Kuhn-Tucker's conditions.

So, following their format. So, we have,

since no space left I will write the complementary slackness condition in the form of.

I have skipped NDCQ checking for the time being but I'll return to it later.

Now, λ takes on a non-negative values,

also I'm rewriting the constraint

and the signs of

the factors employed here.

Now, check NDCQ.

In order to check NDCQ,

we need to visualize then set the constraints set under which we maximize this function.

This is actually a separate problem because

we need to find out what will be the shape of the isoquant.

We get an isoquant when we rewrite inequality sign into equality sign.

Then, in the space of factors we have

a curve and this curve looks like that.

It can be done by,

this is single-variable calculus part of the problem,

but clearly you can solve for x2 as a function of

x1 and you get a P solve the hyperbola curve.

So, either we have a solution which belongs to

this isoquant in between the endpoints or we get some endpoint.

All in all, we have a gradient which is not 0,

and that means NDCQ holds no concerns about that.

Now, we proceed with

solving this system of inequalities, equalities.

It will require some time to do.

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