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Model of economics - Vіtlіnsky V.V.


Maksimizatsiya a little way - the main criterion, on which are virobniks. Ale tse is not a single criterion. Maximization of the continuous flow is guilty of cooperation with a strategic forecast of development of the company.

8.1. Firm model

Nekhai virobnicha firm vipuskaє one product (chi bagato products, ale with a permanent structure). Natural food in natural speech form X - all products are of the same type (pure products and units).

Resource of Victoria: L - live praczya (at the average size of the population for the year of the year, for the year of the year); K - put together praxis (main virobnichi funds); M - subjects pratsі (vitrachene for rіk hot, energy, syrovin, materials, complete virobi skinny).

Kozhen from aggregated types of resources (prazya, funds, materials) ma ry singing kіlkіst іzraznovіv.

Let’s say that the vector-stovpchik of the oaths of vitamins of the winter species of resources through x = ( x 1, ..., xn ) ?. Todays technology companies have become aware of the virology functions, such as the rotation of the sound between the vitrates of the resources and the VIP:

X = F ( x ). (8.1)

There is a hypothesis to allow, that F ( x ) is continuously differentiated and neoclassical, before that there are many other matrices that have been marked.

For example, w = ( w 1, ..., wj , ..., wn ) is a row vector of price resources, and p is the price of products, then the skin vector vitrat x is available:

P ( x ) = pF ( x ) - wx . (8.2)

At (8.2), R = pX = pF ( x ) is the variability of the secondary insurance company, and C = wx is the vitality rate and the vitality of resources for the period.

If you don’t enter the last one, if you don’t have more resources, then the task is to get a maximum of a look at the look:


What is the task of non-linear programming with n minds that are not x ? 0, with the necessary minds розї rozv'yazannya є mind Kuhn — Tucker:


It’s possible that when you’re optimized, see all the resources, see x *> 0, then remember (8.4) to see the video:



tobto at the optimum point the variability of the boundary product given to the resource is guilty of the highest price.

This very (behind the form) ros'vyazok maє the task of maximum release for a given oath vitrat


All the task of non-linear programming with one linear interconnection and mental nevidnosti of meanings.

Stay up to date with the functions of Lagrange:

L ( x , l) = F ( x ) + l ( C - wx ),

Now maximizumo її for udov nevid'єmnostі zmіnnih.

For the whole one, the Kuhn-Tekker minds had popped up:


Yak bachimo, think (8.7) as a whole zbіgayutsya s (8.4), as a rule

The Vipusk application of the single-product company is set by the Cobb — Douglas virobiotic function:

Significantly the maximum issue, as a rule on the lease of funds and the payment of funds was seen 150 pennies. od., wart_ odd odinitsі fund_v wk = 5 pennies. od., wage rate w L = 10 pennies. od / people

How will the norm of substitution of one occupied by funds at the optimum point be boundary?

Razv'yazannya. Oskilki F (0, L ) = L ( K , 0) = 0, then to the optimal linking K *> 0, L *> 0, therefore, be able to (8.7) dial the look:


otherwise our vipadka:

Podilivshi perche rivnyannya on a friend, maєmo:


Having presented the thought of Viraz to the mind:


Razv'yazannya is possible geometrically. In fig. 8.1 image of the line (the linear postitarny vitrates for C = 50, 100, 150) and the image (line of the standard vitrines for X = 25.2; 37.8).

May be so takі rіvnyannya:

5 K + 10 L = C = const.

Izokvanti -

З The bones of the fast vitrates and the fasts of the vipus

Fig. 8.1. З The bones of the fast vitrates and the fasts of the vipus

At the optimum point, K * = 20, L * = 5 of the quanta X * = 37.8 and that of the C = 150, pass through the point, touch it, most of the time (8.8) are normal to these curves, given by the gradients Kolіnearnі.

The rate of replacement funds in optimal points:

tobto one pricyyuyuchy mozhe buti replacements dvoma odnitsami funds_v.

If the task model (8.3) is maximally expensive, you know the one optimal set of resources x *> 0 (look at the viaduct if all the resources enter before the set). The whole set of indicators is the value of vitrate: C * = wx *.

Rozv'yazuєmo task model firmi (8.6) to a maximum profit per task vitraty C *. As for F ( x ) - neoclassical, it’s optimal And why do you want to learn the яз line?

So, from one side,

and from Іншого -



ale to

Through the development of tasks (8.3) ний line, then

From now on, the task for the maximum offset is max ний single isolation x * > 0, then the third task is the maximum release for the tasks of vitrates С * = wx * , and therefore the same isolation, as the first one (div. Fig. 8.1):

Geometrically, the point is dotik izokost and izokvant for the highest value of vitrates C is the sign of the pre-term hat development of the company X ( C ), so I will show it, like a prostate (recession), let it go, but it’s easy to grow. Oskilki qya fallow is monotonous, then monotonous function of vitrate С = С ( Х ) is reversed.

Oskilki X ( C ) is the maximum allowance for the task of Vitrat C , then Vitrati C ( X ), as a whole, the maximum allowance of X , is minimal, and the optimal oath is for the start of the call and be recognized for thinking of the maximum allowance:

. (8.9)

Prior to zero pohidnu

bachimo, at the optimum point of the border vitrati dorіvnuyut prіnі vipusku:

In addition, a maximum of a profit is reached for

(bo )

Rosglyano n spivvidoshen (8.5):

The number of conditions may be different in relation to x at the optimum point x *, as it were de

Tse meansє, what is guilty of buti vidimnim vіd scratch Hessian virobnic functions (ale H vіd'єmany signatures, ), Todi



In order to set the function of the feed (on the resources), see the help for the additional model of behavior of the company. Functions on resources can also be known experimentally for the additional method of mathematical statistics for the latest vibrant data.

Function of the offer -


It’s okay to demonstrate Slutsky’s reaction, I’ll show the reaction to the winter price, the analogous description of the reaction to the winter price and the resource.

For your understanding of the p , w price of the virobnik’s behavior, you can be assigned to such a match (a maximum ( n + 1) match):


Nekhai teper price vipusu zmіnilasya chi zmіnilasya price іn resources, otherwise they are those.

1. The reaction of the virobnik to the winter price.

Differential (8.11) for p :

but in matrix designation:

de - row vector - stovpchik vector, abo


Rivnyannya (8.12) is a virobnik reaction (winter meal) (winter meal for resources) to winter season river .

2. The response of the virobnik to the winter price of resources.

Nekhai zmilinilsya price k- th resource wk , Todd differentiation єmo rivnyannya (8.11) for wk :


I mean

then n ( n + 1) рівняння (8.13) in the matrix form can be written with the following rank (the reaction of the virobnik to the winter price):


3. The response of the virobnik to an immediate winter price and the resource. Podnannya (8.12) and (8.14) yes, basically matrix theory of theory:


I’ll show the reaction of the virobnik to an instant winter price and VIP resource.

Rozv'yazuyuchi (8.15) v_noznno zmіini vipusku I will eat on resources , let’s say:


Having complained about the rule of block matrix matrices, mamo:

Pіdstavlyayuchi stayed viraz u (8.16), we’ll remove the system rivnyan firmi vidnozhno zmіn vipusku i poit on resources:

. (8.17)

First of all the system (8.17) I will show how I can change the start-up price for the production of the company. Oskilka Hessian matrix H is much more marked, then H –1 is also



tobto zrostannyam price _ vipu oath vipu production рост zrostaє.

So rank ,


Ale (for neoclassical function boundary products are optional), that is obov'yazkovo deyaki tobro zrostannya price _ vipusku bring to zrostannya poit on deyakі resources.

Resource l- th species is called malotsіnnim , yakshcho s (8.17) it is visible (the friend is the third group of the Rivnyan), but at the flashed viglyad_ -


That is why the price for the production of zoom-in vision is lower (see below) for the view, the maximum price for the whole view of the resources should be reduced to the speed of the optimal release. Zokrem, the best price for a low resource is hidden for the best way.

Pіdstavlyayuchi (8.20) at (8.19),

to that whip, tobto zrostannya price for a deyy kind of resources to zoom in a quick way.

Згідно з (8.17)

the matrix - vіd' визmuch marked, otzhe, to wait for the price of a deyy resource wait to bring down to a decline on one, the same, crooked drink є down.

Oskilki matrix - symmetric then


so that you can fill the winter price for the l - th resource for winter food for the j- th resource and the winter price for j- resource for winter money for the l- th resource for the same.

Vitrati j- th and l- th kind of resources є vzamozamіnyuvany (vzakomodopovnyuchimi), as skinny.