Asian Journal of Mathematics & Statistics

Year: 2008 | Volume: 1 | Issue: 2 | Page No.: 69-79
DOI: 10.3923/ajms.2008.69.79

A Stochastic Analysis of the Effect of Sudden Increase in the Income of Individuals on the Economy

O. OSU Bright and C. Okoroafor Alfred

Abstract: In this study, we investigate the effect of sudden increase in the income of individuals on the economy. Such situation arises in countries where upward adjustment of salaries increases the income of individuals by a certain percentage. We show that accumulated wealth of individuals through capital investment follows the power law distribution. We quantify the effect of low and high propensity, respectively, of an individual to invest on the economy using an empirical illustration.

Keywords: stochastic multiplicative process, accumulated wealth (savings) and Power-law distribution

INTRODUCTION

It is well known that the distribution of many natural and artificial systems could be closely approximated by the power law distribution (Levy and Solomon, 1996).

This phenomenon was first noticed in 1897 by the Italian Economist V Pareto who discovered that the wealth w of individuals in the high income range is distributed according to the power-law distribution given by the following pdf:

Where:

Gibrat (1937) has shown that the probability distribution function can be approximated by the lognormal distribution:

Where:

Attempts have been made by several researchers to derive a single distribution that accommodates both income ranges (Aiyagari, 1994). Infact, the analysis of the personal income distribution is one of the important subjects in econophysics (Mategna and Stanley, 2000).

Since it can be easily deduced from the form of the pdf (1) that the distribution of the low-middle range has a thin tail and the entire distribution shifts along with mean income so that the changes on the distribution curve are driven by a stochastic multiplicative process which is the capital investment process that underlie the changes in the income owned by individuals, our first objective in this paper is to show that the wealth of an individual is paretocian (that is has power-law distribution).

Our next objective is to derive a wealth accumulation process (savings model) which is also paretocian.

Our main goal in this study is to propose and justify empirically, using illustrative examples from Nigerian, the effect of sudden increase in individual income on the economy using illustrative examples from the Nigerian case study.

Preliminary Results

Definition 1: If then the family of functions e > 0 is called a mollifier. Note that

Definition 2: Let φ_e be a mollifier, 1≥p<∞ and F∈ L^p (Rⁿ). then for each e > 0

Let φ_∈ be a Friedrichs` mollifier. If f ∈ L¹ (Rⁿ, loc) then the convolution exits for each x ∈ Rⁿ. Moreover

(ii) Supp (f * ρ_e) ≥ closed ∈ - neighborhood of supp f.

(iii) If and then as e → 0. Indeed

(iv) If k is a compact set of points of continuity of f then uniformly on k as ∈ > 0.

Definition 3: An Rⁿ -valued stochastic process indexed by an interval T of R, the real line is a map from T to L⁰ (Ω, f, Rⁿ).

Definition 4: A stochastic multiplicative process is a process in which the value of each element is multiplied by a random variable with each time step.

Definition 5: A probability distribution is a function ρ:Z → [0, 1] such that ρ)z)≥0 ∀z ε Z and

Suppose that X has the exponential density function

then y = e^x has the pareto distribution;

Consider y = e^x. Note that g (x) = e^x is strictly increasing and differentiable and that y = e^x if and only if x = h (y) = logy (Olkin et al., 1980).

Hence using the transformation

The probability density function of y gives

Suppose X has a uniform distribution on [0, 1] then has a Pareto distribution.

Since X has a uniform distribution

Then iff x = h(y) = y^-α which implies that is differentiable. Therefore, we have

Let (X, A, μ) be a measure space and let τ ∈ F (X, A, μ). Let h be a bounded A measurable function on X such that the set A = {x ∈ X:h (x) ≠ 0} is locally μ-null. Then ∫hd τ = 0 (Hewitt and Stromberg, 1960).

Consider a system consisting of a set of investors i = 1 …, N each owing a wealth w_i. Assume that the typical variations of w are characterized effectively by a multiplicative stochastic law:

With λ a stochastic variable with a finite support distribution of probability π(λ), does not generate a stationary PLD for W_i (t) instead, it gives rise to a time-dependent log-normal distribution. The effective transition probability distribution π(λ) is assumed not to depend on i or on the actual value of w_i. However if the shape of π(λ) varies in time during the process, our conclusion is not affected (Levy and Solomon, 1996).

Let us consider a discrete-time stochastic multiplicative process w_i (t), added with reset events in the following way: At each time step, n is reset with probability distribution P (w₀) if the reset event does not occur, w_i is multiplied by a stochastic positive factor λ with probability distribution π(λ). So that we have

between two consecutive reset events. w_i (t), then is a pure multiplicative process. When one of such events occurs, the multiplication sequence starts again.

To gain insight in the dynamics of Eq. 4, let us consider the simplest case where, w₀ (t) and λ (t) are constant for all t.

Since an arbitrary factor in the initial value of w₀ is irrelevant to its subsequent evolution, we take w₀ = 1 without loss of generality, (Manrubia and Zanette, 1999). Thus we have,

Now suppose the wealth w_i (t) follows geometric Brownian motion with killing rate where, the killing rate is μ, we denote the reset point by w_o (t) and write the dynamics of w_i (t) by:

Where, z (t) is a Brownian motion process and the inequality (7) means that w(o) is the reflective lower bound (Makoto and Wataru, 2002). Furthermore, suppose σ = 0 that is if the lower bound w(o) grows as fast as the drift of risky wealth, then:

This is because the risking capital income obeys a multiplicative process. Observe that the capital grows exponentially, that is,

or

Now we will be primarily concerned with the distribution of Eq. 5 which is essentially done in Levy and Solomon (1996). Here an alternative approach is considered which adopts the methods in Levy and Solomon (1996) with some novel modification.

The probability distribution of the wealth process given by Eq. 5 is approximated by power law distribution (Pareto distribution)

Let f (w, t) and f (w, t+1) be the wealth distribution at time t and t+1, respectively. Then, because the wealth of the i^th investor changes from w_i (t) at time t to λw_i (t) at time t+1 that is (t), the wealth distribution at t+1 (Levy, 2001) is given by:

Then

Let x = Inw and μ = Inλ, then the corresponding probability distributions p and π become in the new variable:

In terms of the f, x, ρ, μ and by definition 2, Eq. 12 becomes

At each time step where no reset occurs we can write at t = 0

and by definition 1, Eq. 15 becomes for

Since x and μ are independent random variables, suppose that both follow exponential distribution such that

and

(Manrubia and Zanette, 1999),

then

Putting we have

We can also see that T= ∞ for n = 0.

Translating back to the original variables we get using Eq. 13a

The wealth accumulation process which we shall call saving, in the sequel, is that part of disposable income not spent on current consumption. It is amount of money set aside for future use.

In recent years, attempt had been made to model the distribution of savings. Kaldor model in (1960) is an attempt to make the saving-income ratio a variable in the growth process. It is based on the classical saving function which implies that saving equals the ratio of profits to national income ie

Where, s, p, y, stand for savings, profit and income (Jhingan, 2003).

The object of this part is to present a savings model and show that it has a power-law distribution.

We shall make the following assumptions;

Given these assumptions, let S_i (t) and S_i (t+1) be the of ith individual at time t and t+1, respectively where, t = 0, 1, 2, 3,….Let us now introduce the aggregate savings as a sum of additive random and multiplicative terms:

(Aoyama et al., 2000), where, λ(t) and R(t) are assumed independent random variables.

Furthermore, by A3, we add that

Where, γ and β are the marginal propensity to consume and marginal propensity to save, respectively so that Eq. 23 becomes:

or

We shall now show

Theorem 2
Let, then the distribution of S_i (t+1) is closely approximated by the Pareto (Power Law) distribution.

Proof
Let λ (t) be the total growth of saving i such that Eq. 26 holds. If the growth rates are independently and identically distributed random variables with density function f (λ) and given that the average normalized size must stay constant, that is , then Eq. 26 expressed in term of the cumulative distribution function of S_i(t), G (s, t) gives (Ioannides and Overman, 2000).

Where, all values of λ such that the savings at t+1 is equal, is are accounted for and (as in lemma 3). For the limiting stationary savings distribution we have ,(Levy, 2001), so that Eq. 27 becomes:

This equals

(using Eq. 11, 12 and 13a) and gives:

Using Eq. 19.Translating back to original variable S = e^x, Eq. 29 becomes:

Equation 30 compared Eq. 1 gives:

Equation 31 is the slope of the saving line which reflects the rate of change in saving relative to a change in disposable income.

Here, we attempt to fit the expected return rate of investments made by investors base on the knowledge of state of income to Eq. 30. For this purpose, we define the lower bound w₀ as:

Where:

We convert the vertical axis to the accumulated probability P (≥w₀), the probability that a given investor has wealth equal to or greater than w₀ and the horizontal axis to w_is.

In terms of the normalized wealth, the probability that a given individual has wealth W. is given by simple rescaling:

Where,

Since Eq. 30 is paretosian, we employ the maximum likelihood method and estimate our parameter as follows; for W≥W_m, we conclude that

Using Eq. 31 and 37 we have

Present analysis implies (and the experimental avalable data collected from the FMS owerri confirm) that if the individual income distribution is fitted by Eq. 30, then the savings of an individuals increase, as β→1 (Fig. 1,2). This is due to A₃, where high amount spent on investment can increase the rate of return. On the other hand, as β→1, the savings decreases. With β = 1, the saving increases without bound. For β < 0, the individual is fairly alright due to the implementation of salary adjustment policy. But this behaviour affects the economic indicator as individuals tend to relax their investment activity and so, the per Capita Gross National Product (GNP) in such nation will be very low (Wolken and Glocker, 1988).

Equation 30 is paretosian and Pareto distribution is not yet recognized by many programming languages. In actuarial field, pareto distribution is widely used to estimate portfolio costs. As a matter of fact. It can be quite demanding to get data from this particular probability distribution.

One can easily generate a random sample from Pareto distribution by making use of transformation given in lemmas 1 and 2. Figure 3 confirms the different values of β, given the simulated available data generated by lemmas 1 and 2.

Individuals can use their income either for consumption or (financial investment). Essentially, economy can grow only if the individuals are able to find ways to produce a little extra over an extended period of time and produce enough to exceed their immediate need so that they can consume less than the labour income they earn.

The ability to invest the surplus to finance future consumption is crucial to growth. Many of the poor nations of the world have the people (labour), but they have been unable to accumulate savings in quantities sufficient enough to purchase tools and machinery. Unless money is available, no new machinery, tools, or factories will be produced and as the already existing capital goods begin to wear out, they will not be replaced.

Therefore, with proper implementation of the government policy on salary adjustment and proper investment, the economy grows.

REFERENCES