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Take an agent with mean-variance utility over something that is uncertain: $$ U(x) = \mu_x^\theta - \sigma_x^\lambda $$

$A\in \{0,1\}$ happens if $U(x)>0$, and $x$ is a random variable

$$ A = \mathbf{1}\left(\mu_x^\theta - \sigma_x^\lambda >0\right) $$

Saha (1997) has shown that $\theta$ different from 1 corresponds to DARA or IARA preferences. If an agent isn't necessarily CARA, and $\mu = value - cost$, then the first term is $$ \mu_x^\theta = \left(E[value] - cost\right)^\theta $$ This assumes that cost isn't separable from value. In reality however, agents may not operate this way. The simplest example is a cash/credit constraint. You'd thus have $$ A = \mathbf{1}\left(\left(E[value] - cost\right)^\theta - \sigma_x^\lambda >0\right) s.t. cost < c $$

Alternatively, maybe cash expenditures are psychically painful. Like, even if I value something at \$5, the thought of not having that \$5 just bothers me. Maybe because of loss of option value, for example.

$$ A = \mathbf{1}\left(\left(E[value] - cost\right)^\theta - \sigma_x^\lambda - f(cost) >0\right) s.t. cost < c $$ where $f$ is weakly positive and monotonic.

My questions:

  1. Is there a specific term for this sort of separability in cost and value?
  2. Would it go away entirely if there were no credit constraint?
  3. And if I simply model $\left(value - cost\right)^\theta$ with a nonparametric $g(cost, value)$, would economists know what I am talking about if I justify it as represeting "non-classical behavior" or "deviations from the classical behavioral model?"
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