Editing Reputation tokenomics (section)

=== Fundamental results ===
The basic results from which all other applications can be derived are summarized in the following theorems, which give the [[present value]] of a single REP token when it is minted.

'''Theorem 1.'''  (''Infinite Life Tokens'') <math display="block">R(t)=\int_{-\infty}^{t} m*f' (s)ds=R_0+\int_{0}^t m*f' (s)ds</math> ''where <math>R_0=R(0)</math>. The reputational salary of a single token is therefore given by the income stream ''<math display="block">f_0^1 (t)=\int_0^t\frac{f'(s)}{R(s)}  ds.</math>The present value at time <math>t_0=0</math> when a single token is minted in a DAO is <math display="block">PVf_0^1=\int_0^\infty e^{-rt} \frac{d}{dt} f_0^1 (t)dt=\int_0^\infty e^{-rt}\frac{f'(t)}{R(t)}dt.</math> <br>
[[Proof of PV formula for Infinite LIfe REP|Proof.]] <br><br>
'''Theorem 2.''' (''Constant fees'') ''''' '''''

''Assume the DAO is in the market position of earning fees with a constant rate <math>f_0'</math> and the lifetime of a token is infinite, <math>L=\infty</math>.''

''Then the reputational salary of your single REP token is <math display="block">f_0^1 (t)=\frac{1}{m} ln\frac{R_0+mtf_0'}{R_0}</math>''

''and the present value is <math display="block">PVf_0^1=\int_0^\infty \frac{f_0'}{R_0+mtf_0'} e^{-rt} dt=\frac{1}{m} exp\biggl({\frac{rR_0}{mf_0'}}\biggr) \int_{\frac{rR_0}{mf_0'}}^\infty \frac {e^{-s}}{s} ds </math>'' <br>
[[Proof.]] <br>
 


'''Theorem 3.'''  (''Finite Life Tokens'')

''Assume the REP tokens have finite lifetime <math>L<\infty</math>''.'' Then the total number of active REP tokens at any time <math>t</math> is <math display="block">R(t)=\int_{t-L}^t m*f' (s)ds.</math>''Then, assuming a single token was minted at time <math>t_0=0</math>  the fees it earns is given by <math display="block">f_0^1 (t)=\int_0^{min\{t,L\}}\frac{f' (s)}{R(s)}ds.</math>''(Constant Fees)''

''Now assume the rate of fees <math>f_0'</math> is constant. At any time <math>t</math> after the DAO reaches token number equilibrium <math>t>L</math>, there will always be <math>R(t)=R_{\infty}=mf_0' L</math> tokens in the system. The income stream of a single token is then <math display="block">f_0^1 (t)= \begin{cases} \frac{t}{mL} & \text{if }t<L \\ \frac{1}{m} & \text{if }t\geq L \end{cases}</math>and the present value of 1 token at time <math>t_0=0</math> when it is minted is <math display="block">PVf_0^1=\frac{1}{m} \frac{1-e^{-rL}}{rL}.</math>'''''('''''Exponential Fees'')

''Now assume a DAO has exponentially growing fees <math>f'(t)=f_0' e^{ct}</math>'' ''and lifetime <math>L<\infty</math>. After the DAO has been running <math>L</math> units of time, the number of active tokens will grow at a proportional exponential rate. The income stream of a single token is then <math display="block">f_0^1 (t)=\frac{c}{m(1-e^{-cL} )}  min\{t,L\}</math>''

''with present value <math display="block">PVf_0^1=\frac{1}{m} \biggl(\frac{1-e^{-rL}}{rL}\biggr) \biggl(\frac{cL}{1-e^{-cL}}\biggr) </math>''

[[Proof.]]