The fraction of income saved by the consumers: an mathematical "output" of a utility function.

Predescu, Iuliana ; Toader, Stela ; Predescu, Antoniu 等

1. INTRODUCTION

In microeconomics, it is of great importance to turn into numbers and formulae the (economic) behavior of the average consumer--in our paper, of the producer, in its quality as a taxpayer (Maddala & Miller, 1989). Our starting point is based on a rather simple observation, that is the fact one will save some of his incomes if the remainder of his incomes are sufficient for consumption, and, as a result of his own thinking about the utility of saving (Romer, 1996). This research will lead to a more structured and analytic vision of the relation between the saving phenomenon and utility.

2. QUANTIFYING THE SAVING RATE THROUGH THE UTILITY OF CONSUMPTION

Every individual is presumed to have some particular desires of his own concerning what he wants; in order to measure the amplitude of the desires of the tax payer, concerning consumption, it is used, especially in this mathematical model of human economic behavior, the [theta] factor, factor which has the following properties: the size of this coefficient diminishes every time the consumers will change his mind about his consumption desires and decisions, and will decrease with an increment which is as large as the number of those changes in 'consumption strategy'. This [theta] factor, called the factor of constant relative risk aversion is lower as the individual is more willing to decide, more frequently, how much that person consumes in a period of their life and how much in another period (Romer, 1996).

In the mathematical modeling of this psycho-socioeconomic mechanism, the factor of relative risk aversion is constant, and therefore, the utility function which uses this factor is called constant relative risk aversion (CRRA) utility.

In principle, the utility function with constant relative risk aversion has the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

where C is the input.

The utility function counts on constant relative risk aversion, using two different inputs, for two periods:

[U.sub.t] [C.sup.1-[theta].sub.1t] / 1 - [theta] + 1 / 1+ [rho] x [C.sup.1-[theta].sub.2t+1] / 1 - [theta], where [rho] > -1. (2)

[rho] is the discount rate, respectively the rate whose value determines the position of the individual with respect to the values granted to present and future input related to the balance position.

3. THE LAGRANGE MULTIPLIER

The multiplier Lagrange ([lambda]) is used to express mathematically the utility function and with its help we build the Lagrange of our utility function. We have the function objectively and, respectively, its related restriction (r = real interest rate):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Young people share their income from work between the two "accounts", namely the period between consumption and savings. The amount of the savings noted with [S.sub.t], noting, on the other hand, the proportion of income that an individual saves--in the first period--with s(r), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

(1 + p)6 + (1 + rt+1) 6 We say that we have a regulator of the macroeconomic policy, which is the real interest rate (Schiller, 2003). It is indeed a regulator, not the only one, but through it, it is no less important than it is in fact, and--obviously--in particular from the perspective of balance that any individual achieves (in one way or another), for themselves and their own efforts, between personal consumption and savings (hence, investments).

4. MAXIMIZATION OF THE SAVING RATE-THE CASE OF LOGARITHMIC UTILITY

We calculate the condition of order I for the maximization of s(r), depending on [r.sub.t]--by applying the conditions on the numerator only:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The value of the numerator--and of s(r)--increases if the value of 6 is higher than 1 and decreases if the value of [theta] is less than 1. As regards the situation in which [theta] = 1, to observe its influence on the value of s(r) (i.e., the rate of saving for young individuals), we analyze the behavior of that economic entity r--the real interest rate.

An increase in the value of r presents both an income effect and substitution effect, i.e. the substitution effect is the phenomenon of increasing savings rate, when the individual wishes a higher consumption in the second period of their life than the first period (Mishkin, 2004).

The effect of income is characterized by the decrease of the savings rate--the individual relying on that, whereas the (real) interest rate has grown, they can consume "nearly" (we say "nearly", because that level of consumption in the first period will be equal to the level of consumption in the next period namely, the concept of the concerned individual; levels will be, however, of similar size) as much as in the first period of their life and in the second.

Therefore, when [theta] < 1, the behavior of the individual is one in which the substitution effect prevails, while, conversely, the same behavior will be characterized by a predominance of the effect of income--a situation which, mathematically, is defined as one in which [theta] > 1. When [theta] = 1, the two effects, the substitution and income are in balance, more specifically, the saving rate of young individuals is independent from r. This case is called the logarithmic utility (Romer, 1996).

This case is one where we have something to say, especially in regard to taxation (i.e. rate income tax): as in many other studies, we see that saving is a more or less flexible phenomenon--i.e. the savings rate is not (very) sensitive to the rate of return on saving. Thus, fiscal measures to encourage savings through reducing the tax on interest income will not increase the rate of savings, but will have two main results, more or less undesirable:

On the one hand, the holders of bank deposits will record higher revenue--by paying a lower tax on such income.

On the other hand, revenues available to the government are reduced by reducing the tax rate on income from interest, which may, alone or in conjunction with others, generate (a) budget deficit (Von Mises, 1996). What happens, however, where [theta] = 1? Like we said, in this case savings--even savings rate, i.e. the proportion of (total) saved income--is practically independent of the real interest rate.

For [theta] = 1, the savings rate is calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The condition for [rho] is [rho] > -1; which means that, if [theta] = 1, s([r.sub.t+l]) it cannot have a value higher than the one it would have if [rho] = -1; therefore,

s([r.sub.t+1]) = 1 / 2 + (-1) = 1 / 2 - 1 = 1/1 = 1. (8)

This means that the entire income would be saved [??] [C.sub.1t] = 0.

If [rho] = 0, the value of the savings rate is

s([r.sub.t+1]) = 1 / 2 + 0 = 1 / 2. (9)

5. CONCLUSIONS

Like any other model, the model we have presented hereby uses certain simplifications which, as in any effective model, do not impair the outcome: the fact the saving decisions taken by any individual consumer (and, of course, tax payer) have a strong connection with his psychological constitution and social habits.

This demonstration illustrates that when, for an individual, the consumption of the first period of their life has equal value as consumption in the second period, the individual will save in the first period, as we said--half of their income.

The individual considers that, because the (real) interest rate has increased, they will be able to consume in the second period of their life "nearly" as much as in the first period of their life (Lipsey & Chrystal, 1999); this is, considering its beginnings, a psychological principle, not an economic one, but with firm economic consequences.

6. REFERENCES

Lipsey, R. G.& Chrystal, K. A. (1999). Economia Pozitiva, Economica Publishing House, ISBN 973-590-088-2, Bucharest (Romania)

Maddala, G.S.& Miller, E. (1989). Microeconomics: Theory and Applications, The McGraw-Hill Companies, Inc., ISBN 0-07-039415-6, U.S.A.

Mishkin, F.S. (2004). The Economics of Money, Banking and Financial Markets (7th edition--The Addison-Wesley series in economics), Pearson Addison-Wesley, ISBN 0-32112235-6, U.S.A.

Romer, D. (1996). Advanced Macroeconomics, The McGraw-Hill Companies, Inc., ISBN 0-07-053667-8, U.S.A.

Schiller, B.R. (2003). The Economy Today (9th edition), The McGraw-Hill Companies, Inc., INTERNATIONAL EDITION ISBN 0-07-115114-1, New York

Von Mises, L. (1996), Human Action: A Treatise on Economics (Fourth revised edition), Fox&Wilkes, ISBN 0-930073-185, San Francisco, CA (U.S.A.)