<p>In <a href="https://economics.stackexchange.com/a/57222/23231">this answer</a>, I was pointed to the utility function
<span class="math-container">$$
 U(c, y) = c - \frac{c^2}{2} + y
$$</span>
with corresponding demand function
<span class="math-container">$$
 c(p) = 1 -p
$$</span>
which is inelastic for <span class="math-container">$p &lt; 1/2$</span>. Something that escaped my attention then was the following: if I have understood correctly, the implication is that demand is most inelastic when <span class="math-container">$p \to 0$</span> (where consumption is maximised) and most elastic when <span class="math-container">$p \to 1$</span> (where consumption vanishes).</p>
<p>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 <span class="math-container">$p \to 0$</span> (since consumption is large here), but becomes increasingly inelastic as <span class="math-container">$p \to 1$</span> (since the agent becomes increasingly unwilling to forego consumption of the necessary good). So something like
<span class="math-container">$$
 \epsilon_p = -1 + p
$$</span>
which would imply the demand function
<span class="math-container">$$
 c(w, p) = f(w) \frac{e^p}{p} \ ,
$$</span>
where <span class="math-container">$f$</span> is an increasing function of <span class="math-container">$w$</span>, the available income. In fact, applying similar reasoning to income elasticity, and demanding
<span class="math-container">$$
 \epsilon_w = w
$$</span>
we arrive at
<span class="math-container">$$
 c(w, p) = e^w \frac{e^p}{p}\ . 
$$</span>
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.</p>

CC BY-SA 4.0
2024-06-25T14:39:28.630Z
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
CC BY-SA 4.0
2024-06-30T15:26:38.873Z
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.
mark leeds
CC BY-SA 4.0
2024-07-08T08:35:01.047Z
@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?
Anthony
CC BY-SA 4.0
2024-07-08T08:40:58.737Z
@Amit Also, would you agree with my assessment of the quadratic utility function?
Anthony
CC BY-SA 4.0
2024-07-08T11:40:06.627Z
@Anthony Please see this functional form of utility, if you are looking for necessities vs other goods: https://x.com/amit_k_goyal/status/1717156615315427769
Amit
<microeconomics><reference-request><utility><consumer-theory><elasticity>
Implications of assuming quadratic utility; the search for an alternative
Anthony
# Implications of assuming quadratic utility; the search for an alternative

(Asked by: username_0 on 2024-06-25)

In [this answer](https://economics.stackexchange.com/a/57222/23231), 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.

**Tags:** microeconomics, reference-request, utility, consumer-theory, elasticity

> **Comments:**
> **username_1**: 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.  
> **username_2**: 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.  
> **username_0**: @username_1 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?  
> **username_0**: @username_1 Also, would you agree with my assessment of the quadratic utility function?  
> **username_1**: @username_0 Please see this functional form of utility, if you are looking for necessities vs other goods: https://x.com/amit_k_goyal/status/1717156615315427769  

---

## 0 answers

