Deterministic approach to bubbles description: goals of the bubble Organizer

George G. Dimitriadi (george-d@yandex.ru) The Institute for System Analysis of the Russian Academy of Sciences

Introduction. In the paper [2] the main approaches to mathematical description of bubbles in literature are described.

Papers [1-8] devoted to development of the approach, offered by S. V. Dubovsky [9-10] and entitled scenario or deterministic one.

Let's state its main ideas.

The Organizer creates a bubble. The Organizer can be a natural person or an organization. So terms "firm", "organization", "company" etc. are regarded as synonyms of the term "the Organizer".

To consider as more situations as possible and not to be restricted by some concrete organizational scheme a bubble is defined as the following finance scheme.

Definition. During some period of time the bubble Organizer sells it commitments according to which it engages to pay some concrete sum to their bearer in the future. It's supposed that the Organizer fulfills all its commitments up to the moment called "the bubble burst".

Without any loss of generality let's suppose that the Organizer commitments are bonds without coupon ("zero" bonds). It's obvious that any set of commitments having known cash flows in the future (bills of exchange, bonds and shares, saving certificates [11] and other financial instruments including derivatives in some cases) can be reduced to set of bonds without coupon. So this consideration is common enough.

Simplifying assumptions.

1. Let's suppose that the Organizer fulfills its commitments to its creditors using only the revenue of bubble securities cells that is not using funds beyond the bubble.

Such assumption is natural enough from the point of view what is usually understood as a bubble; so investigation of this case is of the most interest.

2. All bonds without coupon are with the equal time to run.

This assumption strongly restricts the field of situations being considered.

Notations. Let's assume that the bubble begins at the moment of time t = 0, V(t) is the bubble Organizer revenue, g(t) is a total face-value of the bubble securities sold exactly at the moment t, 9 > 0 is a fixed period of time, after which the moment of commitments fulfillment takes place and counted off from the moment of their sale (i.e. the period to run of the bonds), cg e [0;1] is a price (noted in parts of the face-

value) which is the real price of the security sells at the moment t.

Main equations describing the dynamics of the model look like:

dG=&g (t ' <9 , v (0)=0,

dt lg(0-g(t-9),t>e' K' '

V K(t)g(t),t<e , = 0, dt \cg(t)g(t)-g(t-e),t>e w '

dw &0,t<e

— = [r n , W(0) = 0.

dt [[1 - Cg (t)] g(t-e), t >e

(compare with [2])

Using these formulae one can calculate functions G (t), V (t) and W (t) at any moment using known function g (t).

In writing these formulae it isn't supposed constancy of price. It (as other quantities) generally speaking depends on time: cg = cg (t).

Besides that, in the model a demand function on the bubble Organizer bonds is introduced (it is described in details in [4]). Then

d- = Cg (t) g (t) - g (t -e)

g (0) = 0, t e [-e, 0) is a dependence of the bubble Organizer revenue on time.

g (t) = ga® (Cg (t ))f (t)

The function g (t) is expanded into production of 2 functions: the demand function 0(cg), which depends on the bubble Organizer commitments price, and the given growth function f (t) (i.e. a priori scenario of the bubble investments inflow depending on time). g0 is a constant of proportionality for units chosen.

The model, which is analogous to the model in question, is described in [1-4]. Evolution of the model. As an evolution of the model one more variable (which is a control variable) is introduced in [5-8]:

^ e [0,1] is a part of investments made into advertising campaign of the bubble (part of the current revenue cgg (t) of the Organizer securities cells).

Using of this variable allows to take into account expenses of the Organizer to advertise its commitments and to consider additional investors inflow caused by that.

In case when we take into account advertising campaign of the Organizer commitments the main equations describing the dynamics of the model look like:

dG = 'g(t),t <e , G(0) = 0, dt *g(t)-g(t-e),t>e w '

dv =&&cg (t) *(t) g (tXt <e = 0

dt [cg (t)s(t)g(t) - g(t -e), t >e '

dw &0,t<e

dW = [r i , W (0) = 0.

dt [[i - cg (t)] g(t-e), t >e

The second equation is changed in comparison with the previous version for the model without advertising campaign.

In the model when we take into account advertising campaign g(t) = goers^(cg) f (t) accordingly when t > 0: the function g(t) is expanded into production of 3 functions: the function of effectiveness of investments in advertising campaign eYS, which depends on part of investments in advertising, the demand func-

tion 0(cg), which depends on the bubble Organizer commitments price, and the given growth function f (t) (i.e. a priori scenario of the bubble investments inflow depending on time). g0 is a constant.

The goal of this article is to consider and formulate different variants of the bubble Organizer goals, their quality discussion and comparison and obtaining results about goals' comparison in their extreme forms.

Goals of the bubble Organizer. In this model behavior of the Organizer is not restricted. In the real life the Organizer, when it creates a bubble, always purposes some aims. Usually this goal, stated explicitly or implicitly, is a goal of obtaining benefit and increasing of personal wealth. This statement can mean direct obtaining of money and at the same time indirect obtaining, for example, by using funds got from the bubble to cover current expenses. After the bubble burst the Organizer supposes to use some method to escape from fulfillment of its commitments or to use other funds to fulfill them. Other variants of actions are possible (for example, intermediate ones).

It's assumed that the Organizer creates new or uses existing trade procedure [11], which allows to create external debts of the same type (for examples, by selling of securities of definite type) during long period of time. Let's formulate two possible goals of the bubble Organizer.

Goal 1a. The goal of the bubble Organizer is to maximize its own benefit obtained in the bubble.

Such aim is purposed by small companies, well-known persons etc. However, Organizers, who publicly manifests the goal of type 1 a, can obtain a very good deal of money.

Other possible goal looks like:

Goal 2a. The goal of the bubble Organizer is to exactly cover all current expenses (budget deficit) at the expense of the bubble.

Goals of type 2a usually corresponds to large organizations or state structures having authorities to implement state budget and to manage public debts.

For example, a large company, who purposes the aim of type 2a, thereby (due to predictability of its behavior to external observers and existence of a liquid market of its commitments) diminishes for itself expenses of external market borrowings and obtains other indirect benefits. At the same time a part of funds obtained can be used to cover budget deficit.

Another example is an authorized state structure whose goal may be long-term issue of public debt commitments. This goal may be caused by two reasons: covering of (part of) state budget deficit and / or creating of the liquid market of public securities for increasing of the liquidity of the market as whole and increasing its stability.

Let's rewrite these goals within the frame of the bubble description model in question:

Goal 1. The goal of the bubble Organizer is to maximize its own revenue V(T) at the moment of the bubble burst T. Let's assume after this moment the Organizer doesn't fulfill its commitments.

Goal 2. The goal of the bubble Organizer is to maximize the bubble lifetime, i.e. quantity T, which is a moment of the bubble burst. Let's assume after this moment the Organizer doesn't fulfill its commitments.

The goal 1 is close enough to the goal 1 a. At the same time the goals 2 and 2a strongly differ. The Organizer, who purposes the aim 2, wishes to cover current needs at the expense of the bubble. Firstly, it isn't always possible: there can be more needs than the total revenue of the bubble by the same moment of time. Secondly, when formulating this goal it isn't clear what happens to the revenue of the bubble which remains after covering all current needs. The goals 1 and 2 are formulated clearer.

Starting from examples abovementioned and meaning of the goals one can see that both goals 1 and 2 formulated are different, interrelated and alternative: funds, obtained early or at the current moment, can be used to cover the current budget deficit or to finance the Organizer commitments which are in time to be fulfilled; the last variant allows to prolong the lifetime of the scheme of obtaining of external borrowings. Often organizations try to combine both these goals reaching both only partially. Below both goals 1 and 2 will be regarded separately as extreme cases of the Organizer behavior.

Conditions of the bubble finish. Let's consider two alternative conditions of the bubble finish.

a) The condition — = 0: the bubble exists (the Organizer fulfills its commit-

dt

ments) as long as it's true that cg(t)(1 - s(t))g(t) > g(t - e).

b) The condition V = 0 : to fulfill its commitments the Organizer can use not only the revenue of the current moment cg (t )(1 - s(t)) g (t), but one obtained before too. In

this case the condition of the bubble finish can be written as V = 0 .

The conditions of the bubble finish a) and b) correspond to the goals 1 and 2.

As a condition of bubble finish one can choose any intermediate variants between two ones described, for example, such one: to fulfill its commitments the Organizer can use not only the revenue of the current moment, but a fixed part of the revenue obtained before (for example, not more than a half: < 50%).

Let's mark that in papers [1-8] the goal 1 is used always when the author describes models and formulates tasks of optimal behavior of the Organizer.

The following assertion takes place:

Assertion. The assertion about goals comparison. The bubble for the Organizer, whose goal is the goal 1, exists not longer than the bubble for the Organizer, whose goal is the goal 2, on equal other conditions

or equivalently:

the bubble, whose finish condition is the d- = 0, exists not longer than the bubble, whose finish condition is the V = 0 , on equal other conditions.

Proof. For the first bubble the Organizer's goal is to maximize its own revenue V (T) at the moment of the bubble burst T. It means that the bubbles bursts when the

derivative dV(t) changes sign to minus first time1. Hence, dV(t) > 0 when t e [0, T].

dt dt

So, V (0) = 0 and dV(t) > 0. Hence, V (t ) > 0 when t e [0, T ] [12], and the second bubble

dt

necessarily exists when t e [0, T ]. Assertion is proved.

The formula of dependence of the bubble Organizer revenue on time. As an

illustration let's derive common formula of dependence of the bubble Organizer revenue V (t) on time for two different cases. It allows to find the bubble lifetime on different conditions of its finish.

The bubble in case when there are no alternative sources of income. The main equation of changes of the Organizer revenue looks like :

^=(t ) -g (t

Here it's assumed that V(0) = 0, g(t) > 0 when t > 0, g(t) = 0 when t e [-0,0). Let's integrate the main equation and than one can obtain:

V(t) = j[cg(£)g(£) - g(Ç-8)]

This formula considers both cases cg = const and cg = cg(t).

The bubble in case when there are alternative sources of income. Let's r is an effective annual interest rate of alternative investments (for simplicity let's assume r = const). The Organizer can invest funds obtained at this rate. Let's assume that one can invest an arbitrary sum at this rate and one can return his deposit in an arbitrary time period obtaining increment of exactly that period of time. Continuous interest is considered.

The main equation of changes of the Organizer revenue now looks like:

dVdT = cg (t) - g (t -9) + rV (t).

When writing this formula it is assumed that total positive revenue is invested at invest rate r and returned from the deposit in a quality, which is necessarily to exactly fulfill the Organizer commitments, and only at those moments of time when

cgg (t) - g (t -9) < 0.

Let's mark that for the organization, whose goal is the goal 1, it is not supposed "accumulation" of the bubble revenue by the firm for following fulfillment of its commitments; so, existence of alternative investments doesn't influence on the bubble. Therefore, further the case of the Organizer, who can use the revenue to pay to off bonds in future moments of time. Thus it can be the Organizer, whose goal is the goal 2 or the goal which is intermediate between the goals 1 and 2.

1 It's supposed that the functions f (t) (a priori scenario of the bubble investments inflow depending on time) and g (t) are monotone increasing.

2 Here only the model without taking into account advertising campaign is used for simplicity. It's easy to consider such campaign but it unnecessarily overloads the formulae.

Let's decide the differential equation obtained to find V (t). Let's f (t) = cgg (t) - g (t -e). So:

^ = f (t) + rV (t). at

The decision of the homogeneous linear first-order differential equation

dV(t) = rV (t) looks like: dt '

V(t) = Dert, where D is an arbitrary constant.

To get the decision of the heterogeneous equation let's use the method of variation of constants. Let's D = D(t) and substitute V(t) = D(t)ert into the equation in question:

D'(t)ert + rertD(t) = f (t) + rertD(t), D'(t) = f (t)e r ,

t

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D(t) = J f (£)e~+ C, where C is an arbitrary constant. So:

0

9

V(t) = ert I j f (£)e~rZdZ + C < 0 =

But V (0) = 0 ^ C = 0. As a result one has got the formula for V (t) in case when there are alternative sources of income:

V(t) = ert j[c, (Z)g(£) - g(Z - 0)] e"

Conclusions. For the approach to bubbles description [1-8], offered by S. V. Dubovsky [9-10], in the article different possible goals of the bubble Organizer are described. Extreme alternative goals of the bubble Organizer are singled out; an assertion about their comparison is deduced. Moreover, common formulae for the Organizer revenue depending on time are obtained for the model in question without advertising campaign and on an additional consumption: when the Organizer is able to invest funds obtained in bank (any other profitable active), i.e. when the Organizer has alternative sources of income.

It's rigorously devoted that the bubble for the Organizer, who uses (at least part) of funds obtained before to fulfill its commitments, exists not less time than the bubble for the Organizer, who immediately takes out funds obtained in the bubble, for example, to finance current budget deficit.

So, the partial answer to the following question is obtained: if the Organizer "keeps" (at least partially) funds obtained before to fulfill its future commitments and invest them for the period of this "waiting", how should temporal value of money be considered and how can one compare the bubble lifetime and the bubble Organizer revenue in this case in comparison with case when there is no alternative investments? However more detailed and general investigation of this case and of results caused by the formulae obtained is an affair of the future.

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