54 



Generating Economic Cycles 



SWERBECK'S iNOCX NUMBERS (RWOUALi) 



Fourier Constitu£nt5 of e.73 w*o 7.38 years 



FiatTRE 15. Residuals of Sauerbeck's index numbers of general wholesale prices. 



Equation to the upper smooth curve: y = 3.8 sin (|.^L« + 9') + 



3.1 sin (y y-°« + 92'), origin at 1818. 



Middle smooth curve: y = 3.8 sin (| y"^« + 9')- 



Bottom smooth curve: y = 3.1 sin (5-^^< + 92'). 



exemplifies how the addition of Fourier terms gives an 

 increasingly accurate description of the general trend 

 of the Sauerbeck numbers. The compound curve in 

 Figure 14, I have regarded as the general trend of the 

 Sauerbeck numbers. If the ordinates of this compound 

 curve corresponding to the years between 1818 and 1913 

 are subtracted from the Sauerbeck index numbers 

 for those years, we obtain what I have called the Sauer- 

 beck residuals, which are listed in Table III and are 

 graphed in Figure 15. 



Table II, as we have seen, indicates that there is a 

 real cycle between 8.73 and 7.38 years in the Sauerbeck 

 index numbers. In Figure 15, three smooth curves have 

 been fitted to the Sauerbeck residuals, one of which is 

 made up of the two cycles of 8.73 and 7.38 years respec- 

 tively. This curve gives an excellent description of the 

 residuals. In the remaining curves the cycles of 8.73 



