<p>We don't need to make any assumptions about the existence of a utility representation to show this. The law of diminishing returns is usually defined as the assumption that the preference relation <span class="math-container">$\succcurlyeq$</span> is <a href="https://en.wikipedia.org/wiki/Convex_preferences" rel="nofollow noreferrer">convex</a>. Convex preferences satisfy the following equation: <span class="math-container">$$y \succcurlyeq x, z \succcurlyeq x \implies \forall \theta \in [0, 1], \theta y + (1 - \theta) z \succcurlyeq x$$</span></p>
<p>Fix a bundle <span class="math-container">$x_0$</span> and consider its upper contour set <span class="math-container">$V(x_0)$</span>. Let <span class="math-container">$x_1, x_2$</span> be chosen such that <span class="math-container">$x_1 \succcurlyeq x_0$</span> and <span class="math-container">$x_2 \succcurlyeq x_0$</span>. Then by the definition of convexity above, <span class="math-container">$\forall \theta \in [0, 1], \theta x_1 + (1 - \theta) x_2 \succcurlyeq x_0$</span>, and so <span class="math-container">$\theta x_1 + (1 - \theta) x_2 \in V(x_0)$</span>.</p>
<p>Thus, we have shown <span class="math-container">$x_1, x_2 \in V(x_0) \implies \theta x_1 + (1 - \theta) x_2 \in V(x_0)$</span>, which is exactly the definition of a set being convex.</p>

CC BY-SA 4.0
2022-10-12T08:02:02.060Z
I think this is a neat way to define your way to it, but I think OP wants a proof of the following claim:

"That the assumption that marginal utilitity experiences diminishing returns is equivalent to the assumption that the preference relation, fixed at any arbitrary value, maps an indifference set that is convex."
Hemanth B
CC BY-SA 4.0
2022-10-12T07:40:10.347Z
2022-10-12T07:41:15.007Z
2022-10-12T07:41:15.007Z
Karthik Tadepalli
<p><strong>[Edited a la comments]</strong> <strong>[Proof attempt]</strong> We have a vector whose components are the real quantities of each good in a bundle of goods, <span class="math-container">$\vec{x_0} \in \mathbb{R}^n$</span>. We make standard neoclassical, <a href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#The_axioms" rel="nofollow noreferrer">von Neumann-Morgenstein assumptions</a> about the utility function, including that utility is a <a href="https://www.uv.es/%7Econtrera/Angles/Micro%20I.%20Lesson%204.pdf" rel="nofollow noreferrer">monotonic transformation</a> of preference relations and some appropriately continuous and differentiable function, <span class="math-container">$u(\vec{x}):\mathbb{R}^n \to \mathbb{R}$</span>. The indifference hypersurface<sup>1</sup> of <span class="math-container">$\vec{x_0}$</span> is defined as the set of bundles <span class="math-container">$V(\vec{x}_0):=\{\vec{x}:\vec{x}\succcurlyeq\vec{x}_0, \vec{x} \in \mathbb{R}^n\}$</span>, where a preference relation <span class="math-container">$\vec{x}\succcurlyeq\vec{x}_0 \implies u(\vec{x})\geq u(\vec{x}_0)$</span>. For any <em>rational</em> agent, given an arbitrary bundle <span class="math-container">$\vec{x_0}$</span>, the indifference set <span class="math-container">$V$</span> is a <a href="https://en.wikipedia.org/wiki/Convex_set" rel="nofollow noreferrer">convex set</a> as an implication of the assumption that <em>marginal</em> utility experiences <a href="https://en.wikipedia.org/wiki/Marginal_utility#Law_of_diminishing_marginal_utility" rel="nofollow noreferrer">diminishing marginal returns</a>.</p>
<p>Let <span class="math-container">$\vec{x_i} \in V(\vec{x_0})$</span>. Now, <span class="math-container">$\vec{x_i} \succcurlyeq \vec{x_0}$</span> and so <span class="math-container">$u(\vec{x_i}) \geq u(\vec{x_0}).$</span>
From diminishing marginal utility assumptions, utility is concave in quantity with respect to the origin, <span class="math-container">$\frac{\delta^2u}{\delta\textbf{x}^2} &lt; 0$</span> (or the hessian is negative-definite or <span class="math-container">$u''(\vec{x}) &lt; 0$</span>). As an implication of this, for any two arbitrary indices, <span class="math-container">$j,k \in \{1, \dots, |V|\}$</span>, and some <span class="math-container">$\alpha \in [0,1]$</span>, we have that the first utility weak inequality below is preserved:</p>
<p><span class="math-container">\begin{align}
u(\alpha \vec{x_j} + (1-\alpha)\vec{x_k}) &amp;\geq u(\vec{x_j} + 0\cdot \vec{x_k}) \\
&amp;\geq u( 0\cdot\vec{x_j} + \vec{x_k}) \\
&amp; \geq u(\vec{x_j}) \\
&amp; \geq u(\vec{x_0}) \\
\Big(\alpha \vec{x_j} + (1-\alpha)\vec{x_k}\Big) &amp;\succcurlyeq \vec{x_0}\\
\implies \Big(\alpha \vec{x_j} + (1-\alpha)\vec{x_k}\Big) &amp;\in V(\vec{x_0})\\
\implies V \text{ is a convex set.} \blacksquare
\end{align}</span></p>
<p>For any two arbitrary bundles in the preferred set <span class="math-container">$V$</span>, any affine combination of them is at least as utile as maintaining the quantities in one bundle and setting the other to zero. And by extension, such a combination bundle is at least as utile as the <span class="math-container">$\vec{x_0}$</span> bundle. If utility is concave in quantities, an utility-preserving indifference hypersurface of quantities is convex.</p>
<hr>
[1] <i>(1.8) The condition [<span class="math-container">$\vec{x}\succ\vec{x}_0 \implies u(\vec{x})&gt; u(\vec{x}_0)$</span>] implies that the set <span class="math-container">$u(\vec{x}) = constant$</span> is a hypersurface... called an indifference hypersurface. (Cassels, 1981, p. 1)</i>
<hr>
<p><strong>[My intuition]</strong> As a simplified 2D analog, imagine the quantity of one good as <span class="math-container">$x_0$</span>  and the quantity of another good that yields the same utility as <span class="math-container">$x_1 = V(x_0)$</span>. Here, it is easy to see that the plots of all combinations of <span class="math-container">$(x_0, x_1)$</span> is convex to the origin because of the law of diminishing marginal returns. Low but non-trivial quantities of either good yield the same utility as very high quantities of one good and very low quantities of the other because utility as a function of quantity is concave. Any new coordinate is an affine combination of other quantity coordinates on the curve (ie a point on the line connecting the two coordinates) and preserves the utility level <span class="math-container">$u(x_0, x_1)$</span>, ie remains on or inside the curve (in other words, <span class="math-container">$V$</span> is a convex set).</p>
<p><img src="https://i.sstatic.net/ai8SU.png" width="300">(<a href="https://economics.stackexchange.com/a/29584/42422">Source</a>)</p>
<p>We can generalize this idea up to vectors of quantities &quot;on either axis&quot;, ie bundles, rather than single-good scalars. Any quantity bundle that is an affine combination of indifferent quantity bundles preserves utility which means that such a combination bundle is also preferred or indifferent to <span class="math-container">$\vec{x_0}$</span>. So, any affine combination of indifferent bundles remains on the indifference hyperspace or inside the region bounded by it (ie convex set).</p>

CC BY-SA 4.0
2022-10-06T01:27:37.403Z
What's the "law of diminishing returns"?
Rick_Morty
CC BY-SA 4.0
2022-10-06T03:53:16.723Z
What does $u'(\vec x)$ mean? The derivative of $u$ with respect to the *vector* $\vec x$? Or some particular element of that vector? You should also doublecheck your definition of "diminishing marginal returns". It should be about $u''$ not $u'$.
Herr K.
CC BY-SA 4.0
2022-10-06T14:10:23.643Z
I think that your proof works, but you have to make it more precise. First, you have to specify the space in which you work, I guess $\mathbb{R^n}$,   and make more explicit your assumptions. In your proof you use a utility function, so you have a function $u$ of several variable, and therefore  you have to speak of partial derivatives (supposing that $u$ is differentiable as necessary). Then, I guess that you assume that the utility function is concave. You have to make explicit, in mathematical terms,  what do you mean by ‘law of  iminishing returns’, it should be about the hessian of $u$.
BakerStreet
CC BY-SA 4.0
2022-10-06T22:18:40.903Z
Thanks for the comments. Yes, I meant the hessian is negative definite, and $u'$ is a typo that's been fixed. Have edited to make it more precise.
Hemanth B
CC BY-SA 4.0
2022-10-05T23:50:47.757Z
2022-10-06T23:16:17.743Z
2022-10-06T23:16:17.743Z
Hemanth B
<p>To clarify notation in what follows, both <span class="math-container">$\vec{x}$</span> and <span class="math-container">$\vec{x}_0$</span> are <em>bundles of commodities</em>, that is, vectors in <span class="math-container">$\mathbb{R}^n$</span> such that the value in each coordinate represents the quantity of the commodity associated to such coordinate. Fixing an individual, one writes <span class="math-container">$\vec{x}\succ \vec{x}_0$</span> if the individual has a preference for <span class="math-container">$\vec{x}$</span> over <span class="math-container">$\vec{x}_0$</span>, and one writes <span class="math-container">$\vec{x}\succeq \vec{x}_0$</span> if <span class="math-container">$\vec{x}\succ \vec{x}_0$</span> or if the individual is indifferent between the two bundles.</p>
<hr />
<p>The following is an excerpt from Cassels' <em>Economics for Mathematicians</em>:</p>
<p><em>It is a standard assumption in economics that for any <span class="math-container">$x_0$</span> the set
<span class="math-container">$$V(\vec{x}_0):=\{\vec{x}:\vec{x}\succeq\vec{x}_0\}$$</span>
is convex. This may be regarded as a consequence of the &quot;law of diminishing returns&quot;.</em></p>
<hr />
<p>Why does the law of diminishing returns imply that the set <span class="math-container">$V(\vec{x}_0)$</span> is convex?</p>

<mathematical-economics>
Why does the law of diminishing returns imply that the set of commodity bundles preferable to $\vec{x}_0$ is convex?
Sam
# Why does the law of diminishing returns imply that the set of commodity bundles preferable to $\vec{x}_0$ is convex?

(Asked by: username_0 on 2022-10-01)

To clarify notation in what follows, both $\vec{x}$ and $\vec{x}\_0$ are *bundles of commodities*, that is, vectors in $\mathbb{R}^n$ such that the value in each coordinate represents the quantity of the commodity associated to such coordinate. Fixing an individual, one writes $\vec{x}\succ \vec{x}\_0$ if the individual has a preference for $\vec{x}$ over $\vec{x}\_0$, and one writes $\vec{x}\succeq \vec{x}\_0$ if $\vec{x}\succ \vec{x}\_0$ or if the individual is indifferent between the two bundles.

---

The following is an excerpt from Cassels' *Economics for Mathematicians*:

*It is a standard assumption in economics that for any $x\_0$ the set
$$V(\vec{x}\_0):=\{\vec{x}:\vec{x}\succeq\vec{x}\_0\}$$
is convex. This may be regarded as a consequence of the "law of diminishing returns".*

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Why does the law of diminishing returns imply that the set $V(\vec{x}\_0)$ is convex?

**Tags:** mathematical-economics


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## 2 answers

### Answer by username_2 on 2022-10-05. Score: 3 | Accepted answer

**[Edited a la comments]** **[Proof attempt]** We have a vector whose components are the real quantities of each good in a bundle of goods, $\vec{x\_0} \in \mathbb{R}^n$. We make standard neoclassical, [von Neumann-Morgenstein assumptions](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#The_axioms) about the utility function, including that utility is a [monotonic transformation](https://www.uv.es/%7Econtrera/Angles/Micro%20I.%20Lesson%204.pdf) of preference relations and some appropriately continuous and differentiable function, $u(\vec{x}):\mathbb{R}^n \to \mathbb{R}$. The indifference hypersurface1 of $\vec{x\_0}$ is defined as the set of bundles $V(\vec{x}\_0):=\{\vec{x}:\vec{x}\succcurlyeq\vec{x}\_0, \vec{x} \in \mathbb{R}^n\}$, where a preference relation $\vec{x}\succcurlyeq\vec{x}\_0 \implies u(\vec{x})\geq u(\vec{x}\_0)$. For any *rational* agent, given an arbitrary bundle $\vec{x\_0}$, the indifference set $V$ is a [convex set](https://en.wikipedia.org/wiki/Convex_set) as an implication of the assumption that *marginal* utility experiences [diminishing marginal returns](https://en.wikipedia.org/wiki/Marginal_utility#Law_of_diminishing_marginal_utility).

Let $\vec{x\_i} \in V(\vec{x\_0})$. Now, $\vec{x\_i} \succcurlyeq \vec{x\_0}$ and so $u(\vec{x\_i}) \geq u(\vec{x\_0}).$
From diminishing marginal utility assumptions, utility is concave in quantity with respect to the origin, $\frac{\delta^2u}{\delta\textbf{x}^2} < 0$ (or the hessian is negative-definite or $u''(\vec{x}) < 0$). As an implication of this, for any two arbitrary indices, $j,k \in \{1, \dots, |V|\}$, and some $\alpha \in [0,1]$, we have that the first utility weak inequality below is preserved:

\begin{align}
u(\alpha \vec{x\_j} + (1-\alpha)\vec{x\_k}) &\geq u(\vec{x\_j} + 0\cdot \vec{x\_k}) \\
&\geq u( 0\cdot\vec{x\_j} + \vec{x\_k}) \\
& \geq u(\vec{x\_j}) \\
& \geq u(\vec{x\_0}) \\
\Big(\alpha \vec{x\_j} + (1-\alpha)\vec{x\_k}\Big) &\succcurlyeq \vec{x\_0}\\
\implies \Big(\alpha \vec{x\_j} + (1-\alpha)\vec{x\_k}\Big) &\in V(\vec{x\_0})\\
\implies V \text{ is a convex set.} \blacksquare
\end{align}

For any two arbitrary bundles in the preferred set $V$, any affine combination of them is at least as utile as maintaining the quantities in one bundle and setting the other to zero. And by extension, such a combination bundle is at least as utile as the $\vec{x\_0}$ bundle. If utility is concave in quantities, an utility-preserving indifference hypersurface of quantities is convex.

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[1] *(1.8) The condition [$\vec{x}\succ\vec{x}\_0 \implies u(\vec{x})> u(\vec{x}\_0)$] implies that the set $u(\vec{x}) = constant$ is a hypersurface... called an indifference hypersurface. (Cassels, 1981, p. 1)*

---

**[My intuition]** As a simplified 2D analog, imagine the quantity of one good as $x\_0$ and the quantity of another good that yields the same utility as $x\_1 = V(x\_0)$. Here, it is easy to see that the plots of all combinations of $(x\_0, x\_1)$ is convex to the origin because of the law of diminishing marginal returns. Low but non-trivial quantities of either good yield the same utility as very high quantities of one good and very low quantities of the other because utility as a function of quantity is concave. Any new coordinate is an affine combination of other quantity coordinates on the curve (ie a point on the line connecting the two coordinates) and preserves the utility level $u(x\_0, x\_1)$, ie remains on or inside the curve (in other words, $V$ is a convex set).

![](https://i.sstatic.net/ai8SU.png)([Source](https://economics.stackexchange.com/a/29584/42422))

We can generalize this idea up to vectors of quantities "on either axis", ie bundles, rather than single-good scalars. Any quantity bundle that is an affine combination of indifferent quantity bundles preserves utility which means that such a combination bundle is also preferred or indifferent to $\vec{x\_0}$. So, any affine combination of indifferent bundles remains on the indifference hyperspace or inside the region bounded by it (ie convex set).

> **Comments:**
> **username_3**: What's the "law of diminishing returns"?  
> **username_4**: What does $u'(\vec x)$ mean? The derivative of $u$ with respect to the *vector* $\vec x$? Or some particular element of that vector? You should also doublecheck your definition of "diminishing marginal returns". It should be about $u''$ not $u'$.  
> **username_5**: I think that your proof works, but you have to make it more precise. First, you have to specify the space in which you work, I guess $\mathbb{R^n}$,   and make more explicit your assumptions. In your proof you use a utility function, so you have a function $u$ of several variable, and therefore  you have to speak of partial derivatives (supposing that $u$ is differentiable as necessary). Then, I guess that you assume that the utility function is concave. You have to make explicit, in mathematical terms,  what do you mean by ‘law of  iminishing returns’, it should be about the hessian of $u$.  
> **username_2**: Thanks for the comments. Yes, I meant the hessian is negative definite, and $u'$ is a typo that's been fixed. Have edited to make it more precise.  

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### Answer by username_1 on 2022-10-12. Score: 0

We don't need to make any assumptions about the existence of a utility representation to show this. The law of diminishing returns is usually defined as the assumption that the preference relation $\succcurlyeq$ is [convex](https://en.wikipedia.org/wiki/Convex_preferences). Convex preferences satisfy the following equation: $$y \succcurlyeq x, z \succcurlyeq x \implies \forall \theta \in [0, 1], \theta y + (1 - \theta) z \succcurlyeq x$$

Fix a bundle $x\_0$ and consider its upper contour set $V(x\_0)$. Let $x\_1, x\_2$ be chosen such that $x\_1 \succcurlyeq x\_0$ and $x\_2 \succcurlyeq x\_0$. Then by the definition of convexity above, $\forall \theta \in [0, 1], \theta x\_1 + (1 - \theta) x\_2 \succcurlyeq x\_0$, and so $\theta x\_1 + (1 - \theta) x\_2 \in V(x\_0)$.

Thus, we have shown $x\_1, x\_2 \in V(x\_0) \implies \theta x\_1 + (1 - \theta) x\_2 \in V(x\_0)$, which is exactly the definition of a set being convex.

> **Comments:**
> **username_2**: I think this is a neat way to define your way to it, but I think OP wants a proof of the following claim:  "That the assumption that marginal utilitity experiences diminishing returns is equivalent to the assumption that the preference relation, fixed at any arbitrary value, maps an indifference set that is convex."  

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