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KackToodles
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Joined: August 28th, 2005, 10:46 pm

exponential versus power-law utility functions

August 29th, 2005, 6:59 pm

For simple consumption-investment dynamic portfolio optimization problems in discrete time,most models seem to use power law (or CRRA) utility functions. Why not use exponential(CARA) utilities? What are the limitations of the exponential utility that drive people to the power law?Onyee
 
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Rez
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exponential versus power-law utility functions

August 29th, 2005, 9:24 pm

Since typically it is the logarithm that is modelled as a stochastic process, the power utility offers tractability: suppose that the utility is given by U_t = S_t^a and s_t = log(S_t) is a stoch process, say a random walk with S_0=1. Then we can write S_t = exp(s_t), and U_t = exp(a s_t), both having the same form. Since it is the MRS that we normally use, we can differentiate the utility, which will maintain its exponential form: M_t = U'_t/U'_0 = exp(a s_t)In addition, such a MRS is `equivalent' to Girsanov's theorem and the Esscher transform, providing a link between discrete and continuous time models.I don't think that exponential utility is that tractable, too many nested exponentials.Kyriakos
 
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KackToodles
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exponential versus power-law utility functions

August 29th, 2005, 10:37 pm

I agree that, if one wants to take the continuous time limit, the power-law utility is more tractible.However, it seems to me that in many cases the exponential utility is equally (or even more) tractibleif one wants to work in discrete time -- especially if one wants to assume cash flows are normally distributed(e.g., no limited liability). For instance, in a 2 period model, the exponential utility is equivalentto mean-variance optimization. Since tractibility is not an issue under such assumptions, I'm mainly worried there might be an economic disadvantage of assuming exponentialutilities.
 
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Alan
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exponential versus power-law utility functions

August 30th, 2005, 1:03 am

Power utility (of wealth) is invariant under scaling the wealth bya multiplicative constant. Exponential utility will introducea characteristic wealth level. If you're modeling the market asa whole, and you want your theory to work 100 years from now,clearly you need the first case. If there *is* a characteristiclevel in your problem, then the exponential might be better.regards,