In this answer, I was pointed to the utility function $$ U(c, y) = c - \frac{c^2}{2} + y $$ with corresponding demand function $$ c(p) = 1 -p $$ which is inelastic for $p < 1/2$. Something that escaped my attention then was the following: if I have understood correctly, the implication is that demand is most inelastic when $p \to 0$ (where consumption is maximised) and most elastic when $p \to 1$ (where consumption vanishes).
This is counter-intuitive to me. I was envisioning a demand function for a necessary good which captures the intuition that consumption is elastic when $p \to 0$ (since consumption is large here), but becomes increasingly inelastic as $p \to 1$ (since the agent becomes increasingly unwilling to forego consumption of the necessary good). So something like $$ \epsilon_p = -1 + p $$ which would imply the demand function $$ c(w, p) = f(w) \frac{e^p}{p} \ , $$ where $f$ is an increasing function of $w$, the available income. In fact, applying similar reasoning to income elasticity, and demanding $$ \epsilon_w = w $$ we arrive at $$ c(w, p) = e^w \frac{e^p}{p}\ . $$ I lack the experience to answer the following: is this a reasonable demand function, and does it show up anywhere in the literature? If so, how does one derive the associated utility function? I have browsed some related posts on this site but they all begin with the indirect utility function, which I don't have here. Finally, if there are established utility functions which yield demand functions with the desired properties, I would be grateful for a reference.
