Sunday, October 30, 2011

Marginal utility revisited

Thinking further into my last post, I think perhaps a better way to look at marginal utility is that, rather than assuming marginal returns go as 1/(wealth) (log function), asume instead that marginal returns go as a  gaussian function that is truncated by a natural lower bound.  Most things that we like are things that you cannot have less than zero of: I cannot have less than zero cookies, less than zero shelter, less than zero bicycles, etc.  If this means that the utility returns to wealth are right-hand tail of a Gaussian curve, it looks just like the 1/(wealth) function that we know from Econ 101.

Notice that the right hand side of this graph:
Normal curve
Is shaped a lot like the right hand side of this graph:

Similarly, the integral (total utility) of the gaussian function looks like this:

Which has a right hand side that looks an awful lot like the log function, or the integral of 1/x.

So, what this means to me is that perhaps for each good we choose some value to consume, constrained by the natural bound, and receive some marginal utility that results from a truncated gaussian function.  For something like making the pile of bills go away, or getting the debt collectors off of our back, we might only see the left hand side of the curve, constrained by the natural upper bound of "they've stopped bothering me."  This would make sense that the marginal utility would increase, since paying off the first bill doesn't make much of a difference, but paying off the last one at the upper bound does.  

For more "normal" goods, like furniture or jewelry, or whatever else, we are constrained by the natural lower bound of zero and then have decreasing returns (the right hand side of the gaussian curve) above that point.

This sort of curve, in aggregate, could give all sorts of overall utility functions, and could be used to explain all sorts of "irrational" behavior in terms of utility maximizing rational actors. 

I really like this concept of a gaussian underlying marginal utility function constrained by natural bounds, and I think I'll keep exploring it.

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