EEGenreith-18

18 Analysis of the Basic Equations

The simplified basic system

(18.1 a)

and (18.1 b)

allows at hand of the structure already some analytical statements. We consider the following a closed economy, i.e. the foreign terms are zero or close to zero. So obviously, GDP growth stops ifand capital growth at . Further on, GDP growth stops when at least one of the three right hand side factors is zero:

or or (18.2).

The case K=0 is trivial, the case, however, surprising at first glance. It means that the flow of credit dries up, if the return on investment is zero. From the last factor, we can now estimate the time, when the GDP growth stops non trivially:

(18.3).

With the realistic choisethis results inyears. After this time GDP in the medium term falls ever further. An effect which now shows up in developed economies like the U.S. or the EU as well. It can be mitigated, however, due to the effect of population growth since then the additive termcan feed continuing growth:

(18.1 c)

and from that rules:

(18.4 a).

On the capital side, it looks a little better. There does the savings rate obviously the same task as population growth in GDP:

(18.4 b)

In the following short analysis, we will distinguish between direct capital driven growth (cdg) and the indirect growth (idg) that is driven from the coefficient of population growthand the savings rate. Let us now consider the three factors on the right hand side more precisely. The nominal interest rateis generally a with time decreasing positive factor, but the value of total capital coefficienthowever continues to rise roughly linearly with time.

It remains the key factor, which lies between -1 and 1 and is negative until, after then it gets positive. So we can write abbreviated, whereis a number between 0 and 1 :

and (18.4 c)

The importance of these two equations is that when they rule, indirect growth stops. If one wants to achieve indirect growth, it must apply the reverse:

for idg-GDP growth

for idg-Capital growth

for idg-GDP growth

for idg-Capital growth

for idg-GDP growth

for idg-Capital growth

(18.4 d)

Let us now consider the situations of different time lines. The situation foris relatively trivial, as a further GDP growth is possible by a positive population growth and/or a positive savings rate. For the period before the turning pointthe capital coefficienthas values between about 1/3 and 3, while the nominal interest rates slowly fall from initially more than 10% against almost 0 at the end of the period. So we divide in the two time periods prior to and after reaching a coefficient of capital. This was achieved in the FRG in 1967. If we now insert typical values for the coefficients for the three phases, then the following rough estimates hold:

”Economic miracle”-years ³⁶

% for idg-growth

or more generally (18.5 a)

normal growth years

% for idg-growth

or more generally (18.5 b)

years of crisis

% for idg-growth

or more generally (18.5 c)

Population growth is a phenomenon that can be controlled very badly through policy measures. As one can see a stable population was important in the beginning, and the actually slightly declining population of the German could never hurt much GDP growth. After the onset of the crisis years, however, only a significant population growth could avoid decline in the GDP in the medium-term. However, the savings rateis quite suggestible. Thus we see that in the so-called “economic miracle years”, the savings rate had to be possibly positive ³⁷ to allow for growth. In the later years of growth, also a slightly negative savings rate can afford growth. Since the years of crisis a positive savings rate then is even required to allow the indirect growth of capital to continue. Finally, we take a look at the complete system of equations:

(18.1 d)

(18.1 e)

The foreign contributionsandcan therefore be used at any time to compensate for deficits. Because when considering the case, thentakes over the role of andtakes over the role ofin the prior consideration. Due to the complex mutual dependencies of all parameters, the possibilities of correction in practice are unfortunately not so simple.
After this introductory treatment of the structure of our system of differential equations, we now come to the necessary integration. With numerical ³⁸methods this is not a significant problem, however, there will be lost inevitably important analytical relationships. This can be avoided by the analytic integration. Analytical solutions are so important because they only allow universally valid quantitative results. The analytical integration even easier systems of differential equations are often not trivial. Already our simplified model (18.1) is not integrable in closed form. In the alternative, but can be integrated piecewise to gain an impression of the analytical behavior of the functions. Thus we may regardas piecewise constant, where the integration runs only over a limited number of years.

As a powerful tool to scientists today are good software packages for the manipulation of algebraic equations avaiable. Such programs ³⁹ can save days on computing to try the "crack" of the integrals. It shows out that, depending on whether the expression generated

(18.6)

is positive, negative or zero, different solutions must be discussed within their respective local scope:

(negative) (18.7 a and b)

(null) (18.8 a and b)

(positive) (18.9 a and b)

Thefunction has the dimension of a frequency and is crucial for the kind of growth. The growth rateis zero foror. Therefore, we need to make at this point a further case distinction:

For (18.10 a and b)

For (18.11 a and b)

This can be written with the specific growth factors

and (18.12 a and b)

in the following specific forms

for (18.13 a and b)

and

for (18.14 a and b).

The area aroundis the upper turning point of the GDP development. At beginning of the national economy, the "good times",is negative, and after the reversal point it runs into the positive range (“bad times” of crisis). The results are: