<p>For a formal treatment (if you aim at estimating some demand data), I recommend <a href="https://www.aeaweb.org/aer/top20/70.3.312-326.pdf" rel="nofollow noreferrer">Deaton and Muellbauer (1980)</a>'s <strong>almost ideal demand system</strong>. Also see the <a href="https://en.wikipedia.org/wiki/Almost_ideal_demand_system" rel="nofollow noreferrer">Wikipedia page</a>. To make the expenditure share on some good <span class="math-container">$i$</span> (<span class="math-container">$w_i$</span>) decrease in wealth <span class="math-container">$m$</span>, simply choose <span class="math-container">$\beta_i &lt; 0$</span>.</p>
<p>However, according to your references, you might be looking for some simple examples. I constructed a simple example as the following:
<span class="math-container">$$u(x,y) = \sqrt{x} + \sqrt{y+a}, \; a &gt; 0$$</span>
subject to
<span class="math-container">$$px + y = m$$</span>
The interior demand is
<span class="math-container">$$x(p,m) = \frac{m+a}{p+p^2}, y(p,m) = \frac{p^2m - ap}{p+p^2}$$</span>
which exists when <span class="math-container">$mp \geq a$</span>. Although both goods are normal, the share of expenditure on <span class="math-container">$x$</span> decreases in <span class="math-container">$m$</span>, i.e.,
<span class="math-container">$$w_x = \frac{px(p,m)}{m} = \frac{p+a\frac{p}{m}}{p+p^2}$$</span></p>
<p>On the other hand, if <span class="math-container">$mp &lt; a$</span>, we have a corner solution with
<span class="math-container">$$x(p,m) = \frac{m}{p}, y(p,m) = 0$$</span>
In this case, <span class="math-container">$w_x \equiv 1$</span> as <span class="math-container">$m$</span> increases. Fix <span class="math-container">$p$</span>, increasing <span class="math-container">$m$</span> from <span class="math-container">$0$</span> to <span class="math-container">$\infty$</span>, we see a continuous decrease of <span class="math-container">$w_x$</span>.</p>
<p>Finally, it is easy to verify that the utility function exhibits diminishing marginal utility in both goods.</p>

CC BY-SA 4.0
2024-07-29T03:48:47.703Z
thanks. This example looks perfect as neither of the goods are inferior and diminishing MU for both. Thanks! I am aware of AIDS but somehow forgot to explore that, so thanks for that as well. Will check that also now.
Dayne
CC BY-SA 4.0
2024-07-27T03:06:13.943Z
2024-07-27T03:06:13.943Z
DiZ
<p>My question is somewhat related to utility functions with inferior goods (some answers <a href="https://economics.stackexchange.com/q/3381/24967">here</a>).</p>
<p>Suppose there are two goods, <span class="math-container">$a, b$</span>. I am looking for a utility function such that it satisfies the usual condition of positive but diminishing marginal utilities. However, I would want that the share of income spent on good <span class="math-container">$a$</span> is decreasing with income. For example, spending share on food may decline with income even though it may not be an inferior good.</p>
<p>Clearly, if good <span class="math-container">$a$</span> is an inferior good, the above requirement is satisfied (see proof below). But as the linked answers mention, such utility functions become rapidly intractable. I am hoping that since my requirement is less stringent, it may be possible to come up with somewhat easier-to-deal with utility functions.</p>
<p>Basically, let <span class="math-container">$a^*(p_a,p_b,m), b^*(p_a,p_b,m)$</span> be the demand functions derived from <span class="math-container">$U(a,b)$</span>. What I want is:</p>
<p><span class="math-container">$$\begin{align} 
\frac{\partial}{\partial m}\left(\frac{p_a a^*}{m}\right) &amp;&lt; 0 \\
\implies \frac{p_a}{m}\frac{\partial a^*}{\partial m}-p_a a^*\frac{1}{m^2} &amp;&lt;0 \\
\implies \frac{\partial a^*}{\partial m}&amp;&lt;\frac{m}{a^*}
\end{align}$$</span></p>
<p>Clearly, if the LHS is non-positive (inferior or zero income effect good) the inequality is satisfied. My requirement is therefore less stringent.</p>
<p>I also found a function for zero income effect <a href="https://economics.stackexchange.com/q/5933/24967">here</a>. This is so far my best alternative but I do not like the fact that second good is constant marginal utility (I would prefer diminishing). Given that what I have in mind is quite realistic for some goods, there should certainly be literature on this.</p>
<p>Please do share the references if your answer is motivated by some paper/book.</p>

CC BY-SA 4.0
2024-07-27T13:01:47.020Z
Also see https://economics.stackexchange.com/a/54591/11824
Amit
CC BY-SA 4.0
2024-07-29T03:46:58.927Z
@Amit: I saw this earlier. As I have mentioned in my question, I was hoping for a utility function that has diminishing marginal utilities in both goods. In any case the answer below gives a nice example.
Dayne
CC BY-SA 4.0
2024-07-29T06:52:02.707Z
Diminishing marginal utility is not a problem. It can always be achieved by taking an appropriate monotonic transformation. For example: $u_1(x_1,x_2;y) = x_1^{\frac{1}{1+y}}x_2^{\frac{y}{1+y}}$ and $u_2(x_1,x_2)=\sqrt{2\sqrt{x_1}+x_2}$ satisfy diminishing marginal utility.
Amit
CC BY-SA 4.0
2024-07-29T09:13:08.813Z
@Amit: is y income in your first example. Ad for second yes but makes it even less tractible I think. I like the first example except that it is unusual to include income..but nice example nevertheless. Thanks!
Dayne
<microeconomics><utility><income-effect>
Utility function with two goods with declining spending share on one good
Dayne
# Utility function with two goods with declining spending share on one good

(Asked by: username_0 on 2024-07-26)

My question is somewhat related to utility functions with inferior goods (some answers [here](https://economics.stackexchange.com/q/3381/24967)).

Suppose there are two goods, $a, b$. I am looking for a utility function such that it satisfies the usual condition of positive but diminishing marginal utilities. However, I would want that the share of income spent on good $a$ is decreasing with income. For example, spending share on food may decline with income even though it may not be an inferior good.

Clearly, if good $a$ is an inferior good, the above requirement is satisfied (see proof below). But as the linked answers mention, such utility functions become rapidly intractable. I am hoping that since my requirement is less stringent, it may be possible to come up with somewhat easier-to-deal with utility functions.

Basically, let $a^\*(p\_a,p\_b,m), b^\*(p\_a,p\_b,m)$ be the demand functions derived from $U(a,b)$. What I want is:

$$\begin{align}
\frac{\partial}{\partial m}\left(\frac{p\_a a^\*}{m}\right) &< 0 \\
\implies \frac{p\_a}{m}\frac{\partial a^\*}{\partial m}-p\_a a^\*\frac{1}{m^2} &<0 \\
\implies \frac{\partial a^\*}{\partial m}&<\frac{m}{a^\*}
\end{align}$$

Clearly, if the LHS is non-positive (inferior or zero income effect good) the inequality is satisfied. My requirement is therefore less stringent.

I also found a function for zero income effect [here](https://economics.stackexchange.com/q/5933/24967). This is so far my best alternative but I do not like the fact that second good is constant marginal utility (I would prefer diminishing). Given that what I have in mind is quite realistic for some goods, there should certainly be literature on this.

Please do share the references if your answer is motivated by some paper/book.

**Tags:** microeconomics, utility, income-effect
> **Comments:**
> **username_1**: Also see https://economics.stackexchange.com/a/54591/11824  
> **username_0**: @Amit: I saw this earlier. As I have mentioned in my question, I was hoping for a utility function that has diminishing marginal utilities in both goods. In any case the answer below gives a nice example.  
> **username_1**: Diminishing marginal utility is not a problem. It can always be achieved by taking an appropriate monotonic transformation. For example: $u_1(x_1,x_2;y) = x_1^{\frac{1}{1+y}}x_2^{\frac{y}{1+y}}$ and $u_2(x_1,x_2)=\sqrt{2\sqrt{x_1}+x_2}$ satisfy diminishing marginal utility.  
> **username_0**: @Amit: is y income in your first example. Ad for second yes but makes it even less tractible I think. I like the first example except that it is unusual to include income..but nice example nevertheless. Thanks!  

---

## 1 answers

### Answer by username_2 on 2024-07-27. Score: 2 | Accepted answer

For a formal treatment (if you aim at estimating some demand data), I recommend [Deaton and Muellbauer (1980)](https://www.aeaweb.org/aer/top20/70.3.312-326.pdf)'s **almost ideal demand system**. Also see the [Wikipedia page](https://en.wikipedia.org/wiki/Almost_ideal_demand_system). To make the expenditure share on some good $i$ ($w\_i$) decrease in wealth $m$, simply choose $\beta\_i < 0$.

However, according to your references, you might be looking for some simple examples. I constructed a simple example as the following:
$$u(x,y) = \sqrt{x} + \sqrt{y+a}, \; a > 0$$
subject to
$$px + y = m$$
The interior demand is
$$x(p,m) = \frac{m+a}{p+p^2}, y(p,m) = \frac{p^2m - ap}{p+p^2}$$
which exists when $mp \geq a$. Although both goods are normal, the share of expenditure on $x$ decreases in $m$, i.e.,
$$w\_x = \frac{px(p,m)}{m} = \frac{p+a\frac{p}{m}}{p+p^2}$$

On the other hand, if $mp < a$, we have a corner solution with
$$x(p,m) = \frac{m}{p}, y(p,m) = 0$$
In this case, $w\_x \equiv 1$ as $m$ increases. Fix $p$, increasing $m$ from $0$ to $\infty$, we see a continuous decrease of $w\_x$.

Finally, it is easy to verify that the utility function exhibits diminishing marginal utility in both goods.

> **Comments:**
> **username_0**: thanks. This example looks perfect as neither of the goods are inferior and diminishing MU for both. Thanks! I am aware of AIDS but somehow forgot to explore that, so thanks for that as well. Will check that also now.  

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