^{1}

^{2}

The Cobb-Douglas function not only leads to a long-term relationship between the rate of output change and the interest rate, but also analyzes why they fluctuate in the short-term. This paper first divides the fluctuation cycle of the interest rate in the statistical data of the past 45 years by using the mathematical phase diagram method, and draws the phase diagram of the rate of output change on the interest rate according to the cycle equation of output. From this phase diagram, we explain the reason that the phase difference between the interest rate and the rate of output change in the fluctuation. Then, according to the optimal relation between L and K in the Cobb-Douglas function, we further derive the employment equation and its relation to the real interest rate and the rate of real output change, and verify the theoretical speculation with statistical data. Finally, it is concluded that the business cycle is a kind of endogenous production phenomenon.

Neoclassic theories believe that the surplus of output and unemployment would be cleaned out during competition and equilibrium is the normality of the economic system, therefore the fluctuation of macroeconomic variables is generated by external factors. The “Real Business Cycle Model” (RBC) arisen during 1980s considers the total factor productivity or the random perturbations of technology determine fluctuation of other variables [

Hayek said: “the incorporation of cyclical phenomena into the system of economic equilibrium theory, which they are on apparent contradiction.” [

From the Cobb-Douglas function

In a competitive market, assuming that the marginal cost of using K is determined by the market interest rate r, the optimal allocation condition for K in production is

differential on both sides of the equation, so:

according to the basic equation,

so Equation (3) can be rewritten as:

When r and

In

This is the business cycle equation. The change of

By

As shown in

At point B or D,

According to the phase diagram

Based on the phase diagram

This is caused by the collection of data has longer interval time than the real fluctuation. Take semi-annual data to redraw this diagram, it is the dotted curve rather than the solid curve during 1983-1985. Therefore, 1976-1983 and 1983-1986 are two different cycles. In the same way, 2003-2009 and 2009-2012 are also two cycles in 2003-2012.

According to annual statistical data and the dividing rule showed in

higher than 2015, so we guess 2016 is also in the cycle since 2012.

According to Equation (6), the change rate of output

Point a and c on

As shown in

In

The foundation equation

Based on algebraic rules, no matter what are the original state of

Apparently, the condition of

Equation (8)

since

among them,

In

The statistical data Y and r are the nominal values with money when calculate

When

Since

Since

As

Above statistical data show that the change rate of employment

According to Equation (6) assume

The structure of Equation (13) is similar to that of

into

As

According to Equation (13), the reason of short-term fluctuation of _{r} is small, then the fluctuating paths of

The phase diagram

In order to discuss the relationship between the unemployment rate

Based on Equation (15) we can convert the phase diagram

The

While the above analysis explains the reasons for the cycle in the unemployment rate, there is one fundamental problem that remains unsolved: We cannot determine the value of

Friedman considered although the unemployment state would be affected by “market imperfections, stochastic variability in demands and supplies, the cost of gathering information about job vacancies and labor availabilities, the costs of mobility, and so

on” [

² Production function in the market system:

² A marginal condition:

²

² The cycle equation about the output:

² The cycle equation about the employment:

² Traditional macroeconomics cannot logically explain contradictions of economic problems in long-term and short-term. Classical theory seems to be handy in explaining relationships between variables in long-term, but it is difficult to understand the phenomenon in short-term. Keynesian theory, while able to explain some phenomenon in short-term, but there will be ridiculous inference in long-term. From the model in this paper and “A kind of neither Keynesian nor neoclassical model (1): the fundamental equation” [

² The cycle is affected by many factors, but as long as the marginal product of the economic system is not zero, there is the business cycle even without these external stochastic factors. Since r and

² Due to the limited data sources and the heavy workload of processing data, this paper is limited to the verification of annual data. It is not known whether these periodic equations also apply to quarterly or monthly data.

Zhan, M.A. and Zhan, Z. (2016) A Kind of Neither Keyne- sian Nor Neoclassical Model (2): The Busi- ness Cycle. Open Access Library Journal, 3: e3215. http://dx.doi.org/10.4236/oalib.1103215