Testing last year's naive intuition on consumption utility with a lifetime utility model

Last year in this post (My Own Private Idaho Of Utility - A Case Study In Spending Control) --before I had figured out the idea of multi-period lifetime consumption utility modeling -- I had made a naive stab at creating my own single period consumption utility formula just for fun.  It was ugly and un-informed and more or less tongue-in-cheek. It looked like this: U(Ct) = 7e-11*x^3 -6.85e-06*x^2+.1396x-565.56 ...for reasons I can't recall but could if I re-read my own post.  The basic idea was that I wanted a hump-shaped function because I figured that spending 35k has utility and spending 36k has slightly higher utility and spending 37k has even higher but diminishing utility, BUT spending 100k might ruin you and might be wasteful to boot. The implication for me at the time was that consumption utility (if perpetuated over time) would not really rise as more is spent, it would go down at some critical point.  The typical formulas for consumption utility, however, like CRRA or log utility have monotonically rising and diminishing utility behavior so I made up a new formula (above) to get the hump.   I wanted:


a single-period amateur utility calc hack like this (using the formula above from last year)

not the formal, academically-loved CRRA version like this

Except I had no real idea what I was doing or why.  Now that I have a lifecycle model (actually two of them if we include my AWOL simulator) with an estimate of discounted utility of lifetime consumption (note: this is deterministic since I am working with my spreadsheet model rather than my missing simulator) I wanted to see if my single period "intuition" from last year still stands up to the lifetime version.  The lifetime version uses the standard CRRA approach in the second chart above but it also sums utility over the multiple periods of a lifetime where there is a chance that spending, if it over-consumes a nest egg, will snap to available income if wealth runs out before death.  This is where periodic utility during that "wealth depletion time" will take a big hit putting the sum of total lifetime utility at risk.  The spreadsheet deterministic approach was described here (Supplement to "optimizing spend rates in a WDT model").

Running both (single period "hump" math and the multi-period lifetime model I'll call DULC: discounted utility of lifetime consumption) using a panoply of generally kinda similar assumptions (but not listed...note: no annuities in there) , and then overlaying one on the other we get something like this:

Conclusion?

On the one hand it's all a bit off and one could endlessly debate assumptions and scale issues (where I am cheating just a bit, I think). I also think the framing of start age is off by a couple of years. I'm also guessing that the optimum is pretty sensitive to the particular parameter-set used.  I'll try to re-run some of this when (or if) I get my simulator software back.

On the other hand, it looks like I was not a total moron last year (maybe half-moron as some have pointed out) since the intuition seems to have held up in at least the shape of the utility curve.  In the end I guess I was naively and hackishly trying to do something last year with a single-period formula that, even though it looks like it was playing a similar game, makes more sense using traditional math in a multi-period life-cycle model.  I'll end by saying that the differences between, and the insights from, thinking in multiple periods vs single seems to be a recurring lesson in the retirement version of my financial self-education.