# Pareto optimal allocations with uncountably many agents

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?

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). $$~\Box$$