Editing Executive governance (section)

==== Exponential fees ====
Now we assume, as above, the fees follow an exponential rate [[Validation Pool#Vote|<math display="inline">f'(t):=f_0'e^{ct}</math>]]

This gives[[Validation Pool#Vote|<math display="block">R(t)=R_0+\int_{0}^t m*f' (s)ds=R_0+mf_0'\frac{e^{ct}-1}{c} </math>]]Using Equation 11 we get[[Validation Pool#Vote|<math display="block">P_0^1(t)  =exp\Biggl( p_3m\int _0^t \frac{f'(t)}{R(t)}\Biggr)=\Biggl(1+\frac{mf_0'}{R_0}\frac{e^{ct}-1}{c} \Biggr)^{p_3}</math>]]The single policing token gives the reputational salary[[Validation Pool#Vote|<math display="block">f_0^{1,P}(t)=\int_0^t f'(s)\frac{P_0^1(s)}{R(s)}ds=\frac{1}{mp_3}\Biggl(\biggl(1+\frac{mf_0'}{R_0}\frac{e^{ct}-1}{c}\biggr)^{p_3}-1\Biggr).</math>]]

Notice policing makes a qualitative difference under exponential fees, since a passive token had salary [[Validation Pool#Vote|<math display="inline">f_0^1(t) \sim \frac {1}{m}ct  </math>]] asymptotically linear, whereas policing gives [[Validation Pool#Vote|<math display="inline">f_0^{1,P}(t) \sim e^{cp_3t}</math>]] asymptotically exponential salary.

This salary’s present value is<math display="block">PVf_0^{1,P}=\int_0^\infty e^{-rt} \frac{d}{dt} f_0^{1,P} (t)dt</math>which is not expressible using elementary functions, but is given by the hypergeometric function <math>2F_1</math>. Notice <math display="inline">PVf_0^{1,P}</math> is finite when <math>cp_3<r</math> and infinite otherwise. Therefore, we see that when the policing ratio <math>p_3</math> and the exponential growth rate <math>c</math> overcome the interest rate <math>r</math> we can expect explosive returns, which is a threat to the stability of the DAO since it is likely a hype cycle would form, generating unreasonable expectations of future earnings. 

Even though exponentially growing fees gives an exponentially growing salary, notice that the <math>R_0</math> original tokens, even if they all participate reliably in policing will be diluted in relative power according to <math display="block">\frac{R_0P_0^1(t)}{R(t)}=\frac{R_0\Biggl(1+\frac{mf_0'}{R_0}\frac{e^{ct}-1}{c} \Biggr)^{p_3}}{R_0+mf_0'\frac{e^{ct}-1}{c}}=\Biggl(1+\frac{mf_0'}{R_0}\frac{e^{ct}-1}{c} \Biggr)^{p_3-1}\rightarrow0</math>

as <math>t\rightarrow \infty</math> so the DAO will predictably decentralize, even if the founders continually police the newcomers. The only way founders can maintain majority power is by bring the majority of new fees to the DAO by continually performing the majority of productive work.