<p>If (and that is a big if) you have a nonempty compact space of allocations and all utility functions are continuous, then one can prove in a highly nonconstructive way that a Pareto optimum exists.</p>
<p><strong>Theorem:</strong> Let <span class="math-container">$X$</span> be a nonempty compact space of allocation, <span class="math-container">$I$</span> be any nonempty set of agents, and for each <span class="math-container">$i\in I$</span> let <span class="math-container">$u_i:X\to\mathbb{R}$</span> be a continuous utility function. Then there exists some <span class="math-container">$x^*\in X$</span> such that there is no <span class="math-container">$x\in X$</span> with <span class="math-container">$u_i(x)\geq u_i(x^*)$</span> for all <span class="math-container">$i\in I$</span> and <span class="math-container">$u_i(x)&gt;u_i(x^*)$</span> for some <span class="math-container">$i\in I$</span>.</p>
<p><strong>Proof:</strong> Choose some well-ordering <span class="math-container">$&lt;$</span> of <span class="math-container">$I$</span> and construct recursively a transfinite sequence <span class="math-container">$(X_i)_{i\in I}$</span> such that <span class="math-container">$X_i$</span> is the set of maximizers of <span class="math-container">$u_i$</span> in <span class="math-container">$\bigcap_{j&lt;i}X_j$</span>. Clearly, each <span class="math-container">$X_i$</span> is nonempty and compact, provided <span class="math-container">$\bigcap_{j&lt;i}X_j$</span> is. The intersection of compact sets is always compact, and it is here nonempty by the finite intersection principle. Every element of <span class="math-container">$\bigcap_i X_i$</span> will do as <span class="math-container">$x^*$</span>. By transfinite induction, every agent has the same utility under every allocation in <span class="math-container">$\bigcap_i X_i$</span>. To see that these are Pareto optimal allocations, take such an allocation <span class="math-container">$x^*$</span> and let <span class="math-container">$x$</span> be any allocation under which some agent receives a different utility. Let <span class="math-container">$i$</span> be the first such agent. But then <span class="math-container">$x\in\bigcap_{j&lt;i}X_i$</span> by transfinite induction. Since <span class="math-container">$x^*$</span> maximizes <span class="math-container">$i$</span>'s  utility in this set, we must have <span class="math-container">$u_i(x)&lt;u_i(x^*)$</span>.
<span class="math-container">$~\Box$</span></p>
<p>Now, in the classical case of an exchange economy, aggregate feasibility is defined by an integral. The integral of a feasible allocation must equal the integral of the endowment allocation. Since one can always add to what a single agent receives without changing the integral, here, one usually takes the notion of Pareto optimality to be that in every feasible allocation in which a positive measure set of agents is better off, a positive measure set of agents must be worse off (or something similar).</p>

CC BY-SA 4.0
2023-02-26T14:51:36.883Z
Thanks! This works, but has the disadvantage that it is not symmetric with respect to the agents, due to the ordering (it is essentially a "serial dictatorship"). Do you know if there is a symmetric solution (such as maximizing the sum in the finite case)?
Erel Segal-Halevi
CC BY-SA 4.0
2023-02-26T18:20:10.890Z
I'm not sure how to define symmetric allocations in general,  but I will think about it.
Michael Greinecker
CC BY-SA 4.0
2023-02-25T20:47:44.497Z
2023-02-25T20:47:44.497Z
Michael Greinecker
<p>Consider an economy with some <span class="math-container">$n$</span> agents with continuous utility functions <span class="math-container">$u_1,\ldots,u_n$</span>. It is easy to prove that a Pareto-optimal allocation exists: define the <em>welfare</em> of an allocation <span class="math-container">$x$</span> as: <span class="math-container">$W(x) := \sum_{i=1}^n u_i(x)$</span>. This <span class="math-container">$W$</span> is continuous and the set of allocations is compact, so there exists an allocation maximizing <span class="math-container">$W$</span>. This allocation must be Pareto-optimal, since every Pareto-improvement would lead to a strictly larger welfare.</p>
<p>When the number of agents in the economy is infinite but countable, a similar proof can be used. Assume that the agents' utility functions are normalized such that all utilities are between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Define now <span class="math-container">$W(x) := \sum_{i=1}^{\infty} 2^{-i} \cdot u_i(x)$</span>. It is possible to prove that <span class="math-container">$W$</span> is continuous, so by the same reasoning as above, there is an allocation maximizing <span class="math-container">$W$</span>, and it is Pareto-optimal.</p>
<p>Now, suppose there is a continuum of agents. One could define a welfare function by e.g. <span class="math-container">$W(x) := \int_{i=0}^{\infty} 2^{-i} \cdot u_i(x) di$</span>, however, an allocation with maximum <span class="math-container">$W$</span> is not necessarily Pareto-optimal: there may exist a Pareto-improvement that gives a higher utility to a single agent, but this will not increase the welfare as this single agent has infinitesimal weight.</p>
<p>Is it possible to prove the existence of a Pareto-optimal allocation in this economy?</p>

<mathematical-economics><theory><pareto-efficiency>
Pareto optimal allocations with uncountably many agents
Erel Segal-Halevi
# Pareto optimal allocations with uncountably many agents

(Asked by: username_0 on 2023-02-25)

Consider an economy with some $n$ agents with continuous utility functions $u\_1,\ldots,u\_n$. It is easy to prove that a Pareto-optimal allocation exists: define the *welfare* of an allocation $x$ as: $W(x) := \sum\_{i=1}^n u\_i(x)$. This $W$ is continuous and the set of allocations is compact, so there exists an allocation maximizing $W$. This allocation must be Pareto-optimal, since every Pareto-improvement would lead to a strictly larger welfare.

When the number of agents in the economy is infinite but countable, a similar proof can be used. Assume that the agents' utility functions are normalized such that all utilities are between $0$ and $1$. Define now $W(x) := \sum\_{i=1}^{\infty} 2^{-i} \cdot u\_i(x)$. It is possible to prove that $W$ is continuous, so by the same reasoning as above, there is an allocation maximizing $W$, and it is Pareto-optimal.

Now, suppose there is a continuum of agents. One could define a welfare function by e.g. $W(x) := \int\_{i=0}^{\infty} 2^{-i} \cdot u\_i(x) di$, however, an allocation with maximum $W$ is not necessarily Pareto-optimal: there may exist a Pareto-improvement that gives a higher utility to a single agent, but this will not increase the welfare as this single agent has infinitesimal weight.

Is it possible to prove the existence of a Pareto-optimal allocation in this economy?

**Tags:** mathematical-economics, theory, pareto-efficiency


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## 1 answers

### Answer by username_1 on 2023-02-25. Score: 3 | Accepted answer

If (and that is a big if) you have a nonempty compact space of allocations and all utility functions are continuous, then one can prove in a highly nonconstructive way that a Pareto optimum exists.

**Theorem:** Let $X$ be a nonempty compact space of allocation, $I$ be any nonempty set of agents, and for each $i\in I$ let $u\_i:X\to\mathbb{R}$ be a continuous utility function. Then there exists some $x^\*\in X$ such that there is no $x\in X$ with $u\_i(x)\geq u\_i(x^\*)$ for all $i\in I$ and $u\_i(x)>u\_i(x^\*)$ for some $i\in I$.

**Proof:** Choose some well-ordering $<$ of $I$ and construct recursively a transfinite sequence $(X\_i)\_{i\in I}$ such that $X\_i$ is the set of maximizers of $u\_i$ in $\bigcap\_{j. Clearly, each $X\_i$ is nonempty and compact, provided $\bigcap\_{j is. The intersection of compact sets is always compact, and it is here nonempty by the finite intersection principle. Every element of $\bigcap\_i X\_i$ will do as $x^\*$. By transfinite induction, every agent has the same utility under every allocation in $\bigcap\_i X\_i$. To see that these are Pareto optimal allocations, take such an allocation $x^\*$ and let $x$ be any allocation under which some agent receives a different utility. Let $i$ be the first such agent. But then $x\in\bigcap\_{j by transfinite induction. Since $x^\*$ maximizes $i$'s utility in this set, we must have $u\_i(x).
$~\Box$

Now, in the classical case of an exchange economy, aggregate feasibility is defined by an integral. The integral of a feasible allocation must equal the integral of the endowment allocation. Since one can always add to what a single agent receives without changing the integral, here, one usually takes the notion of Pareto optimality to be that in every feasible allocation in which a positive measure set of agents is better off, a positive measure set of agents must be worse off (or something similar).

> **Comments:**
> **username_0**: Thanks! This works, but has the disadvantage that it is not symmetric with respect to the agents, due to the ordering (it is essentially a "serial dictatorship"). Do you know if there is a symmetric solution (such as maximizing the sum in the finite case)?  
> **username_1**: I'm not sure how to define symmetric allocations in general,  but I will think about it.  

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