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Can someone tell my how i can figure out which type of utility function the following maximization problem has. It is for an overlapping generations model.

$$\underset{x'}{\max} x'\big(E_t (P_{t+1}+δ_{t+1})-(1+r^f)P_t\big) - \tfrac{\gamma}{2} x'Ωx $$
Where: $x'=$ portfolio of shares

$E_t (P_{t+1}+δ_{t+1}) =$ expected future payoff

$\gamma =$ agent i risk aversion

$Ω=$ covariance matrix

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  • $\begingroup$ Could you tidy up the notation? Lower script is _{what you want to have as lower index}. E.g.: P_{t+1} $\endgroup$
    – Giskard
    May 24, 2015 at 21:31

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I would call this a mean-variance utility function. The agent likes higher mean values, which is the first term, but trades that off against higher variance, which is the second term.

If the random variable of interest is normally distributed with mean $P_{t+1} + \delta_{t+1} - (1+r^f)P_t$ and covariance matrix $\Omega$, and if the agent has constant absolute risk aversion utility, such as $u(w) = 1-e^{-\gamma w}$, then maximizing that utility is equivalent to maximizing mean-variance utility. Here are some details I found with a quick google search.

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  • $\begingroup$ Hi Nick, thank you very much for you response. I still have one question though, more specifically would you consider this a quadratic, exponential or isoelastic utility function? As far as i remember mean-variance utility can be derived as either of the beforementioned? For instance you can derive CAPM which uses mean-variance utility, in either form isoelastic, quadratic, or expontential? $\endgroup$
    – nvanlaer
    Jun 7, 2015 at 10:32
  • $\begingroup$ Just to clarify wouldn't this be log utility function where $q{z}(R)=R-γ/2R^2$ ? $\endgroup$
    – nvanlaer
    Jun 7, 2015 at 13:01
  • $\begingroup$ @nvanlaer The specific function you have above is a quadratic function. When risk is normally distributed, then maximizing an exponential utility function (defined over outcomes, i.e. payoffs), is equivalent to maximizing a quadratic function (defined over level of exposure to the risk asset) But, for example, it won't be true that maximizing expected quadratic utility (again, defined over outcomes) or isoelastic utility will have the same property. I'm not sure if that answers your question because I'm not sure I understand your question. I don't know how to interpret your second comment. $\endgroup$
    – NickJ
    Jun 7, 2015 at 17:05

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