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Model of economics - Vіtlіnsky V.V.


In the beginning, I’ve got an hour of vivchenennaya food. I’ll have to understand how to establish, in some kind of oath, I’m in good condition of a comrade and servant for the given price and until

7.1. Overcome with the same function of corinosity.

Rishennya with the right to buy a singing set of goods, mathematically it is possible to give such a point at the space of goods. Nehai n - how many are the number of different goods, de x = ( x 1, ..., xn )? - vector-stovpchik of the living goods (oaths), who added them to the singing term (for example, with a stretch of rock) for the assignment of prices, may be the singing oath of income for the whole period.

Simple goods - with a large number of different sets of goods x non-specified coordinates.

C = { x : x ? 0}.

At the theory of living, there is a concession, which means that leather is on my way to the wide range of goods:

X I { x : x ? 0}.

This means that for skin betting x I X , y I Y, there’s one thing with three things:

x y - nabir x є addictive, lower y ;

x y - nabіr x ш mensch is adaptable, not y ;

x ~ y - for hitting offense, dial the equivalent.

Vydoshennya overpower may be such a power:

1) is x y x z then x z (transitivity);

2) is x y then x y (non-insignificance: a larger set, wait for a lesser number).

There is such a theorem:

Debreu's theorem . As a rule, the number of Xs is sound without dir, and the indisputable overstatement is uninterrupted, then the function is corrupted and weak (weak hypotheses). Relocations are available at the form of an indicator of overdrive, as such function corinth u ( x ), w x x y is hoarseє u ( x )> u ( y ), while x ~ y is hoarseє u ( x ) = u ( y ).

For dermal spozhivacha takane sub-indicator є bagatovarіantnim. For example, u ( x ) is the corinosity function, then cu ( x ), ln u ( x ) is also an overdrive indicator.

Recognized functionalities of permissiveness to replace the desired performance with the sound functionalities of the following numbers: more, less, more.

At the theory of living, some hypotheses and invasions are allowed, which are the functions of corinosity and power:

1) - zі zrostannyam blessings cinnamon zrostaє;

2) - a small increase in the benefit for the first on-the-go co-working day and night;

3) - for the benefit of living; the goodness of growth; for the growth of corporeality;

4) - if уже a great oath of good is blowing, you shouldn’t bring it away from the growth to the growth of corinosity.

Umova 3, speak up to the widest possible interpretation - as a matrix of other pohidnyh (Hessian matrix)

і є vіd'єmany signposted.

Boundary goods

showing, on skilki zrostaє cinnamon, as a rule, there’s a little product zrostaє in a small amount.

I call the surface baiyuzhity hypersurface rozmіrnіstyu ( n - 1), on a basic corinth post_yna:

abo maє diferentsiyovanu form:


Umova (7.1) means that it is dotical to the superficial canal and perpendicular to the gradients of corinas.

Tse znachaє (I can see right now) you can replace one product with a singing product.

Don't bother in (7.1) dxi = 0 for i = 3, ..., n , so that the best match is:



tobto is the boundary norm for replacing the first product with another product, another for the boundary corinosity of the first one and other product.

Norma zamіshchennya show ск, skіlki need to have one other product, shchob zamіniti maliy oath of the first product, a kind of vibuv.

The budget number is called the number of quietly set of goods, as if you can give me some good, mayuchi dohid obedience M :

de p = ( p 1, ..., pn ) is a row vector of values.

Live behavior model

At the theory of living, get involved (working the hypothesis), then, please, wait for Prague to maximize your cinnamon, and є ине ине ине, й ри ри є є є - - - - - - -


The task of intelligent extremum is reduced to the value of the crazy crazy extremum of the Lagrange function:

L ( x ) = u ( x ) - l ( px - M ).

Neobkhіdnі wash local extremum:


. (7.5)

Tseisno viznacha a point to a maximum, more matrix u - I’m much marked. З (7.5) bachimo, schozhach for foksovannogo income so pick up x *, schoch at the point of view of the border corinosity dorіvnyє v_dnoshennyu price:

Yakshcho rozv'yazati (7.4), (7.5) is clearly x *, we wither away the function of living well :