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What is the equation for an inflation adjusted annuity held in perpetuity?
Let P denote the amount of the investment, R the rate of return and I the rate of inflation.  For simplicity, assume that the payment p is made annually right after the return has been earned.  Thus, at the end if the year, the investment P has increased to P*(1+R) and p is returned as the annuity payment. If I = 0, the entire return can be paid out as the payment, and thus p = P*R. That is, at the end of the year, when the dust settles after the return P*R has been collected and paid out as the annuity payment, P is again available at the beginning of the next year to earn return at rate R. We have P*(1+R) - p = P If I > 0, then at the end of the year, after the dust settles, we cannot afford to have only P available as the investment for next year. Next year's payment must be p*(1+I) and so we need a larger investment since the rate of return is fixed.  How much larger?  Well, if the investment at the beginning of next year is P*(1+I), it will earn exactly enough additional money to pay out the increased payment for next year, and have enough left over to help towards future increases in payments. (Note  that we are assuming that R > I. If R < I, a perpetuity cannot be created.)  Thus, suppose that we choose p such that P*(1+R) - p = P*(1+I)   Multiplying this equation by (1+I), we have [P(1+I)]*(1+R) - [p*(1+I)] = P*(1+I)^2 In words, at the start of next year, the investment is P*(1+I) and the return less the increased payout of p*(1+I) leaves an investment of P*(1+I)^2 for the following year.  Each year, the payment and the amount to be invested for the following year increase by a factor of (1+I). Solving P*(1+R) - p = P*(1+I) for p, we get  p = P*(R-I)  as the initial perpetuity payment and the payment increases by a factor (1+I) each year.  The initial investment is P and it also increases by a factor of (1+I) each year. In later years, the investment is P*(1+I)^n at the start of the year, the payment is p*(1+I)^n and the amount invested for the next year is P*(1+I)^{n+1}. This is the same result as obtained by the OP but written in terms that I can understand, that is, without the financial jargon about discount rates, gradients, PV, FV and the like.