# Implications of assuming quadratic utility; the search for an alternative

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.

• I don't think it is clear that consumption is elastic when price, $P\rightarrow 0$, when law of demand holds. In fact, when $P$ is close to to $0$ and consumption level $(Q)$ is high, then the percentage change in quantity $\frac{\Delta Q}{Q}$ is small, and percentage change in price is $\frac{\Delta P}{P}$ is relatively high, so that the elasticity $\frac{\Delta Q}{Q}/\frac{\Delta P}{P}$ is unambiguously small.
– Amit
Commented Jun 25 at 14:39
• I don't follow all the things you were saying ( background is not economics ) but Herb Simon derived certainty equivalence in the 50's which basically allows for the use of the expected value operator in dynamic programming-optimization. Note that certainty equivalence only holds under quadratic utility so that may be why that utility formulation is so popular. Commented Jun 30 at 15:26
• @Amit thanks, you make a good point. What would be a reasonable functional form for the price-elasticity $\epsilon_p$ that captures the intuition that the good is necessary? Commented Jul 8 at 8:35
• @Amit Also, would you agree with my assessment of the quadratic utility function? Commented Jul 8 at 8:40
• @Anthony Please see this functional form of utility, if you are looking for necessities vs other goods: x.com/amit_k_goyal/status/1717156615315427769
– Amit
Commented Jul 8 at 11:40