All, The following equations generate business cycles purely from the falling profit rate. The end result is satisfyingly catastrophic. You can plug them into a spreadsheet (details below) or I could send mine, currently Excel 5, as a UUencoded file which most mail programmes will extract automatically. This is a 'toy' model designed to show that you can generate business cycle behaviour *purely* on the basis of the falling rate of profit, without recourse to any special behavioural assumptions. In particular, unlike Goodwin's (extremely interesting) models it does not rely on any assumptions about employment or the wage. The maths does not therefore hold workers responsible for the crisis. These endogenous cycles illustrate Marx's thesis that the contradictions of capital are generated by the self-movement of capital. Definition of terms: ____________________ F=Fixed capital/capital stock S=profit or surplus value (the same) I=investment r=profit rate (S/F) Equations _________ 1) F(0)=200 2) S(t)=50 3) I(0)=2 4) r(t)=S(t)/F(t)*100 5) deltaI(t)=S(t)*[r(t)-r(threshold)]*Accelerator 6) I(t)=I(t-1)+deltaI(t-1) 7) F(t)=F(t-1)+I(t) Parameters __________ threshold=5 Accelerator=1/500 The spreadsheet: ________________ F S t I r delta(I) A4=200; B3=50; C3=0; D3=2; E3=100*B3/A3; F3=B3*(E3-5)/500 A4=A3+D3;B4=50; C4=C3+1;D4=D3+F3;E4=100*B4/A4; F4=B4*(E4-5)/500 I may have messed up the description so if anyone tries and it does not work please contact me and I will attempt to post a bug fix. The parameters 'threshold' and 'accelerator' can be adjusted for varying results and graphs. A further refinement is to increase S in line with population growth, say by 0.01 per period. A particularly interesting graph is an XY-plot of I versus F. It turns out, roughly, that the bigger the accelerator the shorter the wavelength. Chaotic behaviour sets in for small Accelerators or thresholds. The cycles are divergent and unstable. If any 'theory' can be said to attach to this, it would be the following story: capitalists set a target profit rate (threshold). If the observed profit rate is higher than this, they dedicate an increased proportion of profits S to investment. Profits higher than the threshold lead to rising investment which raises the capital stock and eventually leads to a falling profit rate. Profits lower than the threshhold lead to falling investment and, eventually, disinvestment, raising the profit rate again. Just to stress that I do not claim or believe this is the actual mechanism of the business cycle. I just wanted to show that on the basis of very simple mechanisms and in particular the falling rate of profit, cyclic behaviour can be demonstrated without recourse to anything other than the endogenous effects of the accumulation process. A request: can anyone produce an equation (5) which results in stable cycles? I couldn't. Alan Freeman
