Effective Demand seeks to identify the end of a business expansion in the business cycle. This post will present a method to determine the coefficients in the effective demand equation to determine the effective demand limit upon the business cycle.
Defining the Top of the Business Cycle
In order to determine an equation to describe the top of a business cycle, we first need to identify the tops of a business cycle. I measure the business cycle by multiplying capacity utilization by (1  unemployment rate). I call this composite measure T in the graph below. This measure gives a view into how capital and labor is being utilized throughout the business cycle.
The graph shows that the utilization of capital and labor rises and falls throughout the business cycle. I have connected the tops of the cycle with a yellow line. This yellow line will represent how the top potential of the business cycle has changed through time. The tops of business cycles are connected with a straight line. Whether the yellow line should dip below or hover above a straight line is not explored at this stage of the research. So the tops are connected with a straight line.
The effective demand equation will be determined by regressing variables against the yellow line. The idea is to develop an equation that will describe the behavior between T and effective demand at the top of the past business cycles since 1967.
Independent Variables to determine Effective Demand
In order to determine an equation for Effective Demand, we need to have independent variables to put into our regression. The dependent variable in the regression will be the yellow line connecting the tops of the "T" business cycle above.
Here is a list of the independent variables to be regressed. (All data is quarterly begining in 1967 because capacity utilization data began in 1967 according to data source, FRED.)
 Labor Share Index of National Income. (data) The labor share index as given by the US Bureau of Labor Statistics. Labor share will represent the final consumption power of society apart from the income of profits, which are left over after paying out labor income.
 Headline Inflation. (data) The CPI of all items. Inflation represents how prices cut into demand. A higher inflation will lower potential demand.
 Looseness or Tightness of the Federal Funds rate. A loose Fed rate will raise potential demand. This variable will be determined using the NGDP effective demand rule for monetary policy. This variable depends upon the coefficients used for the other independent variables, so there will be a series of iterations of the regression which change the other coefficients. The iterations will arrive at stable coefficients for all independent variables. (shown later) (Note: Core inflation is factored into the NGDP effective demand rule. See this post for its comparison to Headline inflation.)
 Looseness or Tightness of Longterm Interest rates. (data) The yearoveryear change of the difference between the 10year Treasury (constant maturity) minus the Fed rate. If the 10year falls in relation to the Fed rate, then demand potential is raised.
 Population Growth rate. (data) The faster the population grows, the more demand potential would grow too.
 Yearly Change of Real GDP. (data from US Bureau of Economic Analysis, table 1.1.6) As real GDP grows, demand potential should fall. More products and services, all other variables equal, should lower demand potential resulting in less capital and labor utilized in businesses.
 Government Expenditures as a percentage of Real GDP. (data from US Bureau of Economic Analysis, table 1.1.6) The more the government spends, the more demand potential there should be.
 Net Exports as a percentage of Real GDP. (data from US Bureau of Economic Analysis, table 1.1.6) The more net exports are in relation to total production, the more demand potential there should be.
 Private Investment as a percentage of Real GDP. (data from US Bureau of Economic Analysis, table 1.1.6) The more private investment there is in relation to total production, the more demand potential there should be.
Iterations of Regressions to Determine the Coefficients
These independent variables are put into a regression. Estimated coefficients are used to start the iterations of the regression due to the variable (EDrule  Fed rate) needing a start. The starting estimated coefficients do not affect the ultimate result.
Effective Demand limit; Top potential of business cycle (yellow line in first graph) = 0.80*LSI  1*CPIall + 1*(EDrule  Fed rate)  1*(10year  Fed rate) + 1*popgrowth  1*realGDPgrowth + 10*G/rGDP + 10*NX/rGDP + 10*I/rGDP
No residual factor is used in order to constrict the result to the variables chosen.
The initial estimation of this equation against the top demand potential (yellow line above) looks like this.
The first iteration of the regression gives these results.

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0 
#N/A 
#N/A 
#N/A 
#N/A 
#N/A 
lsi 
0.724 
0.050 
14.574 
0.000 
0.626 
0.822 
cpiall 
1.056 
0.069 
15.302 
0.000 
1.192 
0.920 
edff 
86.338 
5.589 
15.449 
0.000 
75.312 
97.364 
10ff 
0.656 
0.088 
7.456 
0.000 
0.829 
0.482 
G 
32.153 
11.453 
2.808 
0.006 
9.558 
54.749 
r growth 
0.483 
0.082 
5.869 
0.000 
0.645 
0.320 
I 
25.174 
20.518 
1.227 
0.221 
15.307 
65.654 
nx 
34.331 
19.060 
1.801 
0.073 
3.274 
71.935 
pop 
2.080 
0.969 
2.146 
0.033 
3.992 
0.168 
According to the Pvalues, Investment (I) and Net Exports (NX) are not significant enough to describe the effective demand limit by having values over 5%. The highest T stats point to labor share, headline inflation, monetary effects and real GDP growth as the most significant variables.
After 6 iterations, the results look like this.

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0 
#N/A 
#N/A 
#N/A 
#N/A 
#N/A 
lsi 
0.69316 
0.04859 
14.26542 
1.41E31 
0.597294 
0.789026 
cpiall 
1.13543 
0.071273 
15.9306 
1.79E36 
1.27605 
0.99481 
edff 
87.10909 
5.554219 
15.68341 
9.45E36 
76.15095 
98.06723 
10ff 
0.44419 
0.083452 
5.32273 
2.95E07 
0.60884 
0.27955 
G 
43.9263 
11.15365 
3.93829 
0.000116 
21.92082 
65.93178 
r growth 
0.19435 
0.079358 
2.44901 
0.015262 
0.35092 
0.03778 
I 
29.42429 
20.27896 
1.450976 
0.148489 
10.5849 
69.43348 
nx 
47.67543 
18.85586 
2.528415 
0.012298 
10.47395 
84.87692 
pop 
2.97525 
0.96935 
3.06932 
0.00247 
4.88772 
1.06278 
Now the only independent variable with a Pvalue too high for significance is Investment (I). Investment will be dropped from the regression. Then the regression iterations will be run over again.
Here is how the coefficients change over the 6 iterations giving two examples. All coefficients trend toward a stable value.
Second Run of Regression Iterations without Investment
After 6 iterations of the regression without investment, the results look like this.

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0 
#N/A 
#N/A 
#N/A 
#N/A 
#N/A 
lsi 
0.759053 
0.018787 
40.40267 
1.01E93 
0.721988 
0.796118 
cpiall 
1.13232 
0.071133 
15.9182 
1.68E36 
1.27265 
0.99198 
edff 
88.26595 
5.387935 
16.38215 
7.43E38 
77.63626 
98.89565 
10ff 
0.46773 
0.081525 
5.73728 
3.87E08 
0.62857 
0.30689 
G 
29.49403 
5.457594 
5.404218 
1.99E07 
18.72691 
40.26115 
r growth 
0.12814 
0.06538 
1.95997 
0.051502 
0.25713 
0.000844 
nx 
27.20015 
12.30226 
2.210989 
0.028262 
2.929399 
51.4709 
pop 
2.77093 
0.955917 
2.89871 
0.0042 
4.65683 
0.88503 
The variable pushing the limits of significance for a Pvalue is real GDP growth. The regression iterations will be run again without real GDP growth.
Third Run of Regression Iterations without Real GDP Growth
After 6 iterations of the regression without real GDP growth, the results look like this.

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0 
#N/A 
#N/A 
#N/A 
#N/A 
#N/A 
lsi 
0.758384 
0.018787 
40.3675 
5.85E94 
0.721321 
0.795447 
cpiall 
1.09859 
0.068206 
16.1069 
4.08E37 
1.23315 
0.96404 
edff 
87.68955 
5.366844 
16.33913 
8.52E38 
77.10184 
98.27726 
10ff 
0.39175 
0.071212 
5.50122 
1.24E07 
0.53224 
0.25127 
G 
28.0139 
5.426011 
5.162891 
6.22E07 
17.30947 
38.71834 
nx 
26.34758 
12.29211 
2.143454 
0.033376 
2.097702 
50.59745 
pop 
2.85751 
0.95646 
2.98759 
0.003191 
4.74442 
0.97061 
The variable with less significance for a Pvalue when compared to the other variables is Net Exports/rGDP. The regression iterations will be run again without Net Exports/rGDP.
Third Run of Regression Iterations without Net Exports/rGDP
After 6 iterations of the regression without Net Exports/rGDP, the results look like this.

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0 
#N/A 
#N/A 
#N/A 
#N/A 
#N/A 
lsi 
0.729557 
0.012576 
58.01381 
1.7E121 
0.704748 
0.754365 
cpiall 
1.02953 
0.059659 
17.2568 
1.54E40 
1.14722 
0.91184 
edff 
86.61256 
5.30586 
16.32394 
8.11E38 
76.14552 
97.07959 
10ff 
0.37225 
0.070746 
5.26183 
3.88E07 
0.51182 
0.23269 
G 
33.8623 
4.607057 
7.350093 
5.99E12 
24.77381 
42.95078 
pop 
2.03106 
0.866964 
2.34273 
0.020195 
3.74135 
0.32077 
The variable with less significance for a Pvalue when compared to the other variables is population growth. The regression iterations will be run again without population growth.
Fourth Run of Regression Iterations without Population Growth
After 6 iterations of the regression without population growth, the results look like this.

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0 
#N/A 
#N/A 
#N/A 
#N/A 
#N/A 
lsi 
0.713623 
0.009986 
71.45959 
3.2E138 
0.693923 
0.733323 
cpiall 
1.01453 
0.059451 
17.0648 
4.71E40 
1.1318 
0.89725 
edff 
85.47596 
5.282628 
16.18057 
1.85E37 
75.05511 
95.8968 
10ff 
0.37906 
0.070897 
5.3467 
2.58E07 
0.51892 
0.23921 
G 
32.05953 
4.617768 
6.942646 
6.07E11 
22.95023 
41.16883 
The Pvalues all show high significance.
Evaluating the Final Equation for Effective Demand
Using the coefficients in the last regression run, the effective demand equation looks like this.
Effective Demand limit; Top potential of business cycle (yellow line in first graph) = 0.714*LSI  1.015*CPIall + 0.855*(EDrule  Fed rate)  0.379*(10year  Fed rate) + 0.32*G/rGDP
and plots like this.
In this above graph, the effective demand limit (orange line) is currently rising. Towards the end of 2014, there was concern about the economy. The stock market had a correction. Then the price of oil started to fall and the 10year longterm rates started to fall. The combination of those two made effective demand start to rise. The economy and the stock markets have been rising upon this resuscitation of effective demand.
The resulting Effective Demand rule rate is now compared to the Fed rate. When the Fed rate (red line) is below the blue line, the Fed rate is looser than the effective demand limit would prescribe.
Concluding thoughts
The resulting equation could be used as a guide to assess when the end of the business cycle might occur. Still, the economic environment must be evaluated because the effective demand limit may just slow down utilization of capital and labor instead of lead it into recession. The research into effective demand continues.
The effective demand blog here has spent almost 2 years using labor share as the only variable for effective demand. The Pvalue of the above regression results show that labor share has the highest significance for the effective demand limit. So the principles developed over the last two years are not lost. They have paved the way to a better understanding of the current equation.
Real GDP growth may still have a significant influence. If I add real GDP growth with a coefficient of 0.40, the plot matches up better to the peaks, for example in the late 1960's, 1980, 1984 and 2005. This addition also shows how close T and effective demand came together before the stock market correction in the 3rd quarter of 2014. I want to put the graph here for reference.