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so if we assume r is something like 5%, i is close to 3%, and initial year expenses of $24k, your starting endowment should be

a_0 = (24000*1.05)/(.05-.03) = 25200/.02 = $1.26M


edit: note: this assumes a tax-free world, so r should be after-tax return and e_0 should be the amount of pre-tax income one would need to net living expenses in year 0. adjusting the numbers above for an effective tax rate of 30% gives


a_0 = (25200/.7)/(.05*.7-.03) = 36000/.005 = $7.2M

which is a big difference.

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Is the second formula correct?
Wouldn't it be this?
a_0 = (25200/.7)/(.05-.03)*.7 = 36000/.014 = $2.57M

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depends ... i think of inflation as an after-tax return, in which case mine is correct ... if the inflation rate used needed to be adjusted for taxes, yours would be correct.

this does point out how sensitive this model is to your assumptions, though.