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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?

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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 $\bigcap_{j<i}X_j$. Clearly, each $X_i$ is nonempty and compact, provided $\bigcap_{j<i}X_j$ 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 $x\in\bigcap_{j<i}X_i$ by transfinite induction. Since $x^*$ maximizes $i$'s utility in this set, we must have $u_i(x)<u_i(x^*)$. $~\Box$

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).

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  • $\begingroup$ 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)? $\endgroup$ Feb 26, 2023 at 14:51
  • $\begingroup$ I'm not sure how to define symmetric allocations in general, but I will think about it. $\endgroup$ Feb 26, 2023 at 18:20

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