Need recommendations for a function with certain properties

Question

Not sure if this might be a strange way of posing a question, but anyway:

I need a function $f(v,p,a)$ with the following properties:

• it is defined for $v \in [-\infty,\infty]$
• it is defined for $p \in [0,1]$
• $f$ equals 0 at $p=0$ and $v$ at $p=1$
• it is either concave or convex (in $f$) depending on the value of $a$, and a straight line for one value of $a$. not sure if $a$ is defined on the real line or strictly positive

the exact form doesn't matter so long as it fits those characteristics -- the parameter $a$ is imaginary inasmuch as it depends on the functional form.

for context, I am dabbling in economics and trying to find a way to reconcile Klibanoff's smooth ambiguity model with mean-sd utility, given objective probability measures for the probability of a mean model and a conditional heteroskedasticity model.

Answer 1

If I read you right (assuming you mean concavity/convexity in $v$), I think this should do the job

$f : \mathbb{R} \times [0,1] \times \mathbb{R} \times \mathbb{R}_{++} \rightarrow \mathbb{R}$ with $f(v,p,a,c) = (1-p)p(1-a)[c^v] + pv$

$c$ is an additional positive parameter you can chose the value of.

You can check that, as required

• $f$ is defined for any real value of $v$
• $p=0$ implies $f=0$ and $p=1$ implies $f = v$
• The convexity/concavity in $v$ of the function depend on the value of $a$ with the function being a straight line for $a = 1$.

I encoded the function on desmos at https://www.desmos.com/calculator/v8gniwvz6e. There you can play with the parameters and see how the shape of the function changes.