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I want to model an Expected Utility Function for risk aversion but my problem is uncertainty in itself.

I want a function(a special case) $$f(x, y) =\left\{\begin{matrix} h(bx^{1-C}+ay^{1-C}),C\neq 1\\ h(b\ln x+a\ln y),C=1\end{matrix}\right.$$ where $h(x)$ is an arbitrary smooth function.

C is the Elasticity of Substitution between x and y. A and B are the Weights.

When C is 0 a special case is Von Neumann Morgenstern, Linear Utility.

When C is 1 Cobb-Douglas.

When C is 2 we have Harmonic Mean.

When C approaches infinity Maximax.

When C approaches negative infinity Minimax.

C=0, C infinite, C negatively infinite are easy because you do not need to know the current value. Only the weights matter.

I need a function for $C\in(0,\infty)$ even when C does not approach infinity. I particulary like a situation where C is 2. But anything higher than 1 does the job.

Such that I can decide even if I am uncertain of my current position and I only have information on what is ahead of me and I am deciding.

$f(x,y)=xy$

$MRS(x,y)=-\dfrac{y}{x}$

The Marginal rate of substitution depends on the entirety of x and y.

I only know what would be added to x and not x itself, what would be added to y, the probabilities of that(which I need somehow to model to), and the weights a and b.

For von Neumann the MRS is a simple Ratio of a and b. For Minimax and Maximax only the worst and best need to be considered. Maximizing the best and worst case scenario.

Which is a function that solves my problem?

Edit

Actually I figured it out after some reading that I can just consider a and b probabilities. Since Relative Frequency might just as well be considered a Weight.

But what if x and y have opposing signs; I get a big problem.

Another problem is that the function I want(I asked) is a multivariate function.

The choices are at least in part mutually exclusive.

I am in need of a single variable function that does the same job.

A function monotonous to an harmonic mean which does not degenerate to linear when considering one variable

XY is a cobb douglas X is linear ln(x) is a function that models risk aversion

maxln(x) + ln(y) = maxXY They have the same elasticity of substitution, supplementarity, and the same MRS.

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