7.5. Constrained optimization with inequality constraints.

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Из курса от партнера National Research University Higher School of Economics

Mathematics for economists

1 оценка

National Research University Higher School of Economics

Mathematics for economists

1 оценка

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.

Из урока

Global extrema. Constrained optimization with inequality constraints.

Week 7 of the Course is devoted to identification of global extrema and constrained optimization with inequality constraints. This week students will grasp the concept of binding constraints and complementary slackness conditions.

7.1. How to identify global extrema? (part 1)11:30

7.2. How to identify global extrema? (part 2)10:14

7.3. How to identify global extrema? (part 3)12:31

7.4. How to identify global extrema? (part 4)10:03

7.5. Constrained optimization with inequality constraints.8:47

7.6. Binding constraints. Complementary slackness conditions.5:40

7.7. Constrained optimization problem with inequalities for n-dimensional space.7:14

7.8. Constrained optimization problem with inequalities. Theorem.6:48

7.9. Example of solving constrained optimization problem with inequalities (part 1).9:11

7.10. Example of solving constrained optimization problem with inequalities (part 2).10:27

7.11. Example of solving constrained optimization problem with inequalities (part 3).3:53

Познакомьтесь с преподавателями

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

So, the change in the maximum value

of the value function f_star will be roughly equal to.

Now, these two derivatives can be found from the Envelope theorem.

So, in order to find this derivative,

we'll take the original function.

Let me remind you,

the original function whose

differentiate with respect to a and substitute x_star, which was.

And the same for another derivative.

So then, it becomes, and accordingly.

So, then we're going back and we get,

and that concludes our solution.

Now, it's the proper time to get an introduction to

the optimization theory with the constraints which are inequalities.

The thing is that if we consider micro-economic problems,

the majority of the problems is all about inequality constraints.

Even for utility maximization problem,

the budget constraint should be written as inequality in the form of,

utility should be maximized subject to constraint,

the price of x times the prize of y,

no graves on the income and also no negativity constraints.

In almost all problems in micro,

we'll have to count for non-negativity of the economic variables.

So, that tells us that we need to consider

how to solve problems at least with one inequality, here we have three.

So, let's consider a problem where there is a function of

two variables and we would like to

maximize it's subject to just one inequality constraint.

Both functions, f and g,

are considered continuously differentiable,

and we start with drawing.

So, we draw the coordinate plane and that will be

the graphical representation over the equation when we equate g and b.

So, below this graph,

the set of (x,y) coordinates satisfy inequality here,

g(x,y) is strictly less than b.

Above the curve, g is strictly greater than b.

As for the objective function f(x,y),

we'll be considering the point where that level curve of f touches this constraint.

This is a very similar picture to the one which

was depicted earlier when we started considering optimization with equality constraints.

The thing is that since we have inequality here,

we should consider not only points which belong to the curve

but also the points which belong to the set under the curve,

and that makes more cases to consider.

Clearly, we have the old case.

Here, if the level curves provides the growth,

so their depiction shows the growth of f in the direction of North-East.

Then, here, the gradient of f points exactly in this direction,

North-East and as for the gradient of g function,

it points in the same direction.

Now, two gradients points in the same direction,

so they are related.

Gradient f is lambda gradient g,

where g is necessarily a non-negative parameter,

and we form Lagrangian function according to a concrete formula.

So, we follow the rule.

Start with objective function as before,

choose plus sign, after that goes lambda,

and within the square brackets,

we fill in with the difference

where we take the constraints and for our maximization problem,

we subtract from the bigger value the smaller value.

That's how we get b minus g(x,y).

Now, if we set the system of first-order conditions,

then we get immediately a similar equation relating gradient f and gradient g,

because the dl over dx is df over dx,

minus lambda dg over dx equals zero,

and the same for dl over dy.

Now, how to consider additional case when we need to reconsider the picture,

maybe a function f has a maximum within this set.

So, if it does,

then these are the level curves

of function f and it takes the greatest swell here at this point.

So here, this point being x_star, y_star,

we have df over dx equals zero,

df over dy equals zero.

What about lambda then?

Lambda then should be zero also because actually,

we're considering an unconstrained maximization problem here.

Both cases, when m(x) of f is taken at some point

within the set g less than b or on the boundary or the set at at this point,

it can be combined in one statement.

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