This page has been robot translated, sorry for typos if any. Original content here.

Model of economics - Vіtlіnsky V.V.

4.3.3. Imovіrnіsna model vplyvu official

{Smolyak S. A. Risk accounting when setting the discount rate // Economics and Mathematical Methods. - 1992. - T. 28. - Vol. 5-6. - S. 794.}

Problem statement. The process of gaining income in the form of functions is characterized by an intensive x ( t ). Todd dokhid, holders for finishing Maliy Interval hour ( t , t + dt ), x ( t ) dt . Please note that the income gain is x (0) = X per ear of function (at the moment t = 0). To falsify the dynamics of the whole indicator, pour in two groups of officials.

  • Fizichna znoshuvanіst basic concerns . It’s permissible that it’s fallowable in the main events for which the user is able to:

x ( t ) = X - bt . (4.22)

Vidomo, which object is fully operational, if you don’t see anything new. That is why in termination of exploitation (at the river T ) the max x ( T ) = 0. Sound

abo (4.23)

  • Riznogo genus vipadkovі officials, scho zumulyuyut Rizik vikoristannya ob'єkta. Yakbi ikh was not a blast, but it was unbearable, then for the standard norms, the discount is calculated according to the formula:


Respectfully three types of vipad officials, which can be added to the profitability of the case: vipad "zbos" in virobnitsvy; і з з м і серед серед серед серед серед серед серед катастроф серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед серед. Випадкові коливання цін, excuse and obedience to drink.

Shchob adequately visualize the officials in the normal discount R , almost two different versions of the expected form of the object.

The first, the Vicorist discount rate R , cannot be taken into consideration, and the integral discounts should not be taken into account (4.24).

For another, schochny on the “bezrizikovuyu” norm Rj , the officials are included in the model of the vipadkoy process of return of income intensively.

That is, the value of the norm for the discount R , as if I were the right ones to get rid of, thoroughly zrobiti such that offending variant rozrahunkіv odnakovo appreciated the effectiveness of the function of the object.

More information is presented to the income model from the list of the official bureaucrats.

Vipadkovі "zboї" at virobnitstvі. Don’t have a moment t the object was characterized by a real intensive income x ( t ). Todd, by extending the offensive small interval to the hour dt, anyway, there will be a “good” in the case of virtual w dt , otherwise the function is normal with normal 1 - w dt . As a matter of fact, it’s time to go on for the first time, (it’s probably a small amount, but not a little small), as well as to appreciate the added value x, seemingly, vipadkov.

Let’s say, since I’m turning around for the first time, I’m not going to change the time of the function.

"Disasters." The order given by the buyer (the hair attendant of the object) of the very same kind of dyalnosti zdіysnyuyut іnshі pіdprimєmtsі. It might be better to break a new efficient way (technology) of viral products from them, and we have a great deal to do with their products. As far as possible, the function of the first object is not even secured by income, but for the future, become an economic catastrophe. The situation is similar to that of the former, as long as the law is changed, the law is a bit political in the region.

There is no such disaster in the interval ( t , t + dt ) road kdt , de k is an intensive catastrophe, so do not lie down t . It is possible to assess such situations more expertly, due to the analysis of the scientific and technological progress in the case of a secondary diagnosis and the forecast of the economic situation.

The fragments of the quantity k і t are small, the “catastrophe” of the period of the discovery of the “evil” in the face of the least “immobilized”.

Kolivannya tsin podatkiv that obyag_v to a drink . By extending the period of functionality, the price is low for products, such as cooking, syringes, materials, complete virobi, as well as oaths of food and the rate of surcharge, you can pay. First and foremost, the official income intensity of income is also high. It is generally permissible, but at the time of evaluating the situation, the income was correctly marked, then the intensity was x ( t ), the group was enlarged, the rank and file were slightly matched, and I could be more dispersed. Of course, it is possible that at a small interval of a few hours x ( t ) change a little dispersion, and an intensive x ( t ) should not fall into the middle of such a clock at an alternate hour.

Looking at the price, I can let you know how to describe the model of the win-win vipadka process, so as to increase the profitability at a very close time t i t + dt

x ( t + dt ) = x ( t ) + s d w ( t ), (4.25)

de s - mean square viddilennaya vipadkovy colony income intensity x ( t ) per unit hour (the average square of such colony per hour dt is more valuable for all s2 dt ; w ( t ) is the highest percentage vipadkov.

Mathematically, I am able to achieve income at times the rate of official income in terms of maturity. Therefore, through V ( x ) is the value of the mathematically matched amount of integrated discounted income (beyond the standard Rj Don't worry about the cob moment the virobnosis function is “normal”, and the intensity of the income gained is x . Obviously, V (0) = 0. It is natural to see vipadoks, if x > 0. Respectfully, for V ( x ), the value is not significant, it’s a bit of a whisper; Then I can discount the time until t = 0.

Rosglyano maliy Interval hour (0, dt ). There are three possible situations.

  • З імовірністю w dt become Збій у віробництві. Vituni (vipadkova value) can be appreciated for the consolidation of the benefits, the value of x is discounted. Virobnitsvo normalize through vipadkovy promozhok hour t (vipadkova value), after which the object will go to the normal camp, such as vidpovіdaє mathematically combined with the integrated discounted income V ( x ). For the umpteenth time, the income should be up to one hour per month, due to the “zboyu”. Since it is possible to reach income up to the moment t = 0, then the value of V ( x ) should be multiplied by the available discounted factor .

For an allowance, right now, there’s a need for “obstruction”, a secondary law and an exponential law for middle values ​​q, mathematically maturing a discountable factor can be given to a viewer


I’m more important, I’ll need to increase my expenses in the process of assuring that I’ve got a good reason, and I’ll have to change z in one hour, I know what is mathematically discounted, I have to say one:


  • Selling Interval (0, dt ) to the hour of the future kdt to be economical catastrophe. At such a time, the lawsuit is closed, and the integral discount of income from the offensive function of the zero value.
  • At the interval (0, dt ) from the international 1 - (w + k ) dt , the function is normal. Todd per hour dt income becomes x ( t ) dt , due to the fact that intensity is gained by the value of bdt for the round of the physical old keynote, and also in the case of (4.25) for the most time ( s ) abo) Moreover, at the time of hour dt , an object is characterized by an intensive income x - bdt + s d w ( t ). For the benefit of the integral discounting (up to the moment 0, not d ( t )), the income, the right price

Urakhovuchi ymіvіrnіst skin and oslozhnutіh situation and that camp, at a certain interruption of the object for them, you can write down for mathematical calculation of integrated income in the form of discounted income function:

With Urahuvannyam (4.25), (4.27) the main reason is that, to the smallest magnitude of the order, which rises for the first one, you can replace this:


It is permissible that the functionality of V is smooth, and for thinking x 0 of a friend it’s always V '' ( x ) існує і є uninterrupted. Todd, the main part of the Taylor series and the number of columns, the value of d w ( t ) is less than the value of the mathematical method and dispersion , know:

Zvidsy maєmo rivnyannya



d = Rj + k + (1 - q ) w. (4.30)

Line function

V 0 ( x ) = ( x - C w) / (s - b / s2) (4.31)

є one із розв'язків tsogo rіvnyannya. Zagalny rozv'yazok (4.29) write down as a sum V 0 ( x ) і rozv'yazok v_dpovіdnogo odnoridnogo rivnyannya


Obviously, scho (4.32) has two linearly independent distributions - е l х і е m х . Here

. (4.33)

The values ​​of l, m are the root of the characteristic characteristic curve.

Otzhe, the ignorant rose (4.29)

V ( x ) = V 0 ( x ) + Ce l x + C 0 e m x .

B (4.33) is bachimo, m> 0> l. Ozhe, if C 0? 0, then the function V ( x ), except x ® +?, Exponentially grows to +? anyhow exponentially spadatime to - ?. Even with a determinate vipad, bachimo, the function V ( x ) is no bigger than x ( t ). It is much less than Todi, if C 0 = 0.

Obviously, with the brain, V (0) = 0 mamo C = - V 0 ( x ). Dali is writable

V ( x ) = x / d - ( C w / d + b / d2) [1 - e l x ]. (4.34)

On the other hand, mathematically matched integrally discounted income as a function of the moment, as long as I relied on it, I could take into account that 4.3, I could take it easy to take into account that 4.3.


Aggregation in vipadkovy officials in one indicator. Zauzimo, since w = k = s = 0 i Rj = R , then formula (4.34) is shifted to the formula (4.24). Obviously, the real dynamics of revenue is not to wait for a linear model (4.22).

Todd zruchnіche obmeshitsya prediction of the dynamics of the average value of income, aggregated in one indicator, I’m waking up information about the flow of VIP officials.

In more and more countries, there is a lot of bureaucratic and non-significant risks, according to the day, established by the standard norms of discount (basically, just before).

For this reason, listed below the formulas (4.24) and (4.34) with the norms, the discount R i Rj yielded the same results, but rather satisfied


We will take such a designation:

With w / X = g, d (s T / C ) 2 = n , d T = a, RT = r. (4.36)

Legal notice (4.35) is possible to pay at the Wagley:


From now on, why do you want to know the invisible rate of discount, would you like to know how much you want to know? (Razv'yazok Rivnyannya (4.37)) i, nareshti, calculate R :


What does it mean, what is the norm of the discount R z urahuvannyam rizik vidrіznya vid? incisive kofіtsієntom . The value of this factor is, as bachimo, to lie more than a, g and n .

It is possible to absorb the analysis of the skin cut-out from the official officials.

Formulate this important importance of the garment with the model encouraged: the discount rate R , which is recognized in a clear way, cannot be paid at the sight of the sum of the non-recessionary warehouse Rj that is the kind of allowance, the same is the price for the free, the good I do not have to lie down Rj і vrahovuє rizik.

The smart butt rozrahunkіv normi discount to urakhuvannyam dignitary riziku . For one unit vimіryuvannya hour vіzmemo one rіk. Dearly, why do you want to go to the middle one once in a row: , and the hour on liquidation due to the need to expose the exponential law to average values ​​q = 0.04 (approximately two tizhnyam). It is permissible, however, to have vitality, to associate it with “evil”, to proportionate vitrates at a single hour, which is why it is more expensive to “lose” more than 50% of your income, and you need to save 50% of your income. For a change of mind iz (4.26) and (4.27) we know:

The emergence of new technologies, the improvement of how to tidy up rozglyaduvan virobnitsvo zbitkovim, you can respect the small malovym. Let’s let it go, but also the “revolution” in technology, two times a week for the second, you can take k = 0.03.
We write the statement for the parameters, which should be entered before the statement (4.37):

For the first time, R is less expensive than Rf і T and that is .

Porvnyuvannya value R and Rj pіdtverdzhuє vnovnovok about non-additiveness and non-multiplіkativnost in the pouring of officials to the rate of discount.