# Conditional distributions in model with continuum of agents

Many economic models consider a continuum of agents, $$i \in [0,1]$$. Suppose these agents have characteristics $$(x_i, y_i)$$, which are independently distributed. Are all possible values of $$(x_i,y_i) \in \mathbb{R} \times \mathbb{R}$$ held by some agent $$i$$? If I pin down a specific value $$\bar{x}$$, is there still a continuum of individuals with $$x_i = \bar{x}$$ and holding all possible values of $$y_i$$? Some models appear to rely on this being the case (see e.g. the 'island-economy' proof in Appendix B.1 of http://www.jonathanheathcote.com/hsv_taxation_final.pdf - it seems like on each 'island' there is a continuum of agents with different values for $$\epsilon_i$$).

Is there a more mathematically formal way of setting this up which could clarify it? Does my confusion have something to do with the agent space being one-dimensional, whereas the characteristic space is two-dimensional?

Is the case different if each agent has characteristics $$(x_i, y_i(j))$$, where $$y_i(j)$$ are a set of values for each $$j \in [0,1]$$? In this sense, the characteristic vector exists in an infinite-dimensional space. In the previous example, the characteristic vectors have values in $$\mathbb{R}^2$$, which has the same cardinality as the agent space $$[0,1]$$. But in this example, the characteristic space $$\mathbb{R} \times \mathbb{R}^{[0,1]}$$ has greater cardinality than the agent space. I'm not sure if this is relevant though.

Thanks!

• Seems like several of these would depend on the parameter space of the model used. Are you asking about the specific model you linked or models in general? (In the latter case the answer is "it depends".) Sep 9, 2020 at 5:28
• "If I pin down a specific value $\bar{x}$, is there still a continuum of individuals with $x_i = \bar{x}$ and holding all possible values of $y_i$?" Seems like this is an immediate consequence of the independence of $x_i$ and $y_i$. Sep 9, 2020 at 5:29
• "where $y_i(j)$ are a set of values for each $j \in [0,1]$" What is this undefined $j$ here? Sep 9, 2020 at 5:31
• One can make all these arguments rigorous using what is an "often skipped-over measure-theoretic set-up which formalises this" but this set up is quite subtle. See here for the appropriate set-up and here for results that show what exactly you have to assume in terms of the space of agents. Sep 9, 2020 at 7:41
• @John "...can it still exist when the characteristic space is ...$\mathbb{R}^{[0,1]}$?"---According to Theorem 5 of Podczeck, it's enough that the original factor spaces in the product are Polish with atomless measures. So perhaps $\mathbb{R}^{[0,1]}$ is too much to ask for but, e.g. $C[0,1]$ or the Skorohod space $D[0,1]$ with an atom-less measure would be covered by the construction. Sep 10, 2020 at 11:20