I want to estimate a demand function, and for convenience, I would suppose it has constant price and income elasticities: $\varepsilon$ and $\eta$, let's say. That is, demand $x_i(p, w)$ would be something of the form
$x_i = Ap_i^{\varepsilon}w^{\eta}$
where $A$ is just a constant (possibly including other prices $p_j$). What are possible utility functions that would generate such demand?
I know a similar question has been asked here, with a good general answer on the condition to be satisfied by a utility function for its corresponding demand function to have a constant price-elasticity; what I'm looking for is examples of such utility functions where the elascity could be a fixed parameter of unknown value. I know that a Cobb-Douglas utility function would lead to constant (unit) elascitiy, but I would like it to be a parameter I can estimate, and not a known constant value.
