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Model of economics - Vіtlіnsky V.V.

ROZDIL 4. APPLIED MATHEMATICAL MODELS OF FINANCIAL AND ECONOMIC PROCESSES

4.1. Organization of an advertising campaign

Nekhay deyaka firma (business, installation), repair, advertising, new goods, and chores. It is clear that the sale of Maybut is guilty of overriding Vitrati for the whole campaign. It is also clear that the reason for keeping Vitrati can be over the top, the smaller the frequency of potential purchasers, the more news will be informed. In the future, at the world’s best wishes for sale, you can even go for a little extra, I, right, at the moment, if you are full, and I don’t advertise, the goods will not be sensed.

The model of advertising campaigns is to run on such basic hypotheses. The value of dN / dt is the difference — the number of hours per hour, the number of people who have come to know about the product and the cost and the cost of buying it (the time is right before the start of the advertising campaign), N ( t ) is the time already. Get involved, but dN / dt is proportional to the number of buyers, I don’t know about the goods (services), because

a1 ( t ) ( N 0 - N ( t )),

de N 0 - ignorance of the potential plateau of potential buyers, a1 ( t )> 0 is characterized by the intensiveness of the advertising campaign (it is actually denoted by vitrates for advertising at a given time).

I must also allow myself to know about the product, so that I’ll extend it to the ripped information and middleware, and in the role of additional advertising “agents” of the company. Їхній extraordinary road value

a2 ( t ) ( N ( t ) ( N 0 - N ( t ))).

We’ll be more timid, we’ll have more agents. The value of a2 ( t )> 0 characterizes the stage of purchase of the purchases among themselves (there is a possibility of being installed).

The result is otrimaєmo rіvnyannya

(4.1)

It’s a1 ( t ) >> a2 ( t ) N ( t ), then h (4.1) is a model of the type of Malthus model (div. Section 2, clause 16), it’s also a1 ( t ) << a2 ( t ) N ( t ), - level of logistic curve (div. Div. 2, p. 18).

The analogy of the total number of zooms, because in the current model and models of the increase in the number of population of the population, the very idea of ​​“population” has been observed: the number of times increased in terms of the actual value N ( t ) (current value is () ) M izh її rivovnovimnim (population) and the border (purchase) th and more accurate values.

The analogy between the two processes will end, most likely at the present moment, the value of the zero will be displayed in full (for the whole, it is necessary that only one offending a1 ( t ), a2 become more visible). An important negative effect is often to be spent in advertising campaigns and guilty of national organizers on those who want to change the nature of the advertisements. Come with me because of the popularity of the product, you can fall into the highest value directly by the result of direct advertising, as well as mediated by advertisements.

Model (4.1) is not important, so it’s more expensive to zero at the last moment. Just take a look at model (4.1) in the vicinity of the point N ( t = 0) = N (0) = 0 ( t = 0 is the moment of the cob of the advertising campaign), we’re more involved, then N << N 0, a2 ( t ) N << a1 ( t ), then rivnyannya (4.1) typing in the look:

and yogo roz'vyazok -

(4.2)

I’m quite satisfied, naturally, I’m thinking of a cob, but t = 0.

З (4.2) it is really easy to bring in a couple of advertising strata and an ad side of the advertising campaign.

Apparently, through p, the value is a margin of a single sale, which would be like a bullion without advertising vitrates. It is acceptable for simplicity, but a leather buyer buys one more item. Koefіtsітnt a1 ( t ) for its zmіstom means kіlkіst equal-valued advertising dіy in the same hour (for example, distribution of the same posters). Through s, it is meaningfulness of the elementary act of advertisements. Todi Sumy Road:

(4.3)

and vitrati -

I’ve changed the vitrati for advertising for a reason pN 0> s , and if the advertisement is cheap, and the market is clean, then the winnings will be reached first and foremost (in days, I will pay for the advertisement, I will pay you the most in hours, which can be butchered more than in more detailed models). A vipad doesn’t get worse than an effective and expensive advertisement for a company that has the ears of a cob. Ale tse not privid, schob vіdmovitysya vіd advertisements. Fairly, viraz (4.3) it is removed on the first hand, pN 0> s is just less than the small N ( t ) value, since the functions P and S grow in the clock for the same laws. For information N ( t ) come in (4.1) warehouses to keep in mind, secure, to be advised by advertisements. That function N ( t ) can be the "fastest" function hour, lower than the formula (4.3). There is a non-linear effect at the constant value of N ( t ) for the constant pace of growth. You can help your financial failure to reach the nightly stage of the advertising campaign.

It is understandable that there is a confirmation of the private vipadka rivnyannya (4.1) with permanent factors a1, a2.

Vikonaєmo zamіnnykh:

Todi (4.1) be reduced to a logical rivnian

(4.4)

yake maє rozv'yazok

(4.5)

Here , otzhe, N (0) = 0;

З (4.4) it is clear that there are better functions I, from the very beginning, the function N ( t ) for thinking t > 0 can be more important, but not much more (for thinking, chi ) Maximum reach out if :

Tsey has a period of flowing, so as to be taken away in the same hour, the mother:

Віднімаючи від cob current

,

otrimaєmo:

Tobto riznitsya midzh by the cob and maximally accurate incomes can be buttes to finish significant. Summarized (integrated) economical effect of the campaign (necessary mental є, obviously, the vicariousness of ), be marked with all the excess, the characteristics of which are deducted from (4.4), (4.5) for the additional quadrature (integration).

Yak viplivaє s (4.4), much more often from the present moment the advertisement is old and permanent. Yes, if get close to , rivnyannya (4.4) can be written in the video:

(4.6)

Straight to the boundary value For t ® 0 (and the function N ( t ) ® N 0), we must have the exponential law. In one hour, there are only a few small new purchasers, and gaining a little extra for any kind of minds, you can’t overwhelm the extra-vitrats.

The analogous characteristics are deducted for the fourth year (4.1) and the third one is wider, and the wider the same for the description of the next technological innovations. Guidance on food analysis is low on the urgent tasks of micro-economic competition.