From single-consumer-multi-units to multiple-consumers-single-unit When I took a course in consumer theory, the economy always had a single consumer, represented by a monotone positive utility function $u(x,y)$ and an income $I$. Given prices $p_x$ and $p_y$, it is possible to calculate the demands of the consumer for products $x$ and $y$.

Now, I deal with a different type of economy: there are many consumers, each of them wants only a single unit of each product. Each consumer is represented by three positive values: $u_x$ (utility for having $x$), $u_y$ (utility for having $y$) and $u_{xy} \geq \max(u_x,u_y)$ (utility for having both $x$ and $y$). Given prices $p_x$ and $p_y$, each consumer buys either $x$ or $y$ or both, whichever gives the highest net utility (utility of product/s minus price). So it is possible to calculate the aggregate demands for $x$ and $y$.

MY QUESTION IS: Is there a natural/standard way to convert between these two types of economies?

I.e, given a utility function $u(x,y)$ and income $I$ for a single consumer, is it possible to construct a set of consumers with different $u_x$, $u_y$  and $u_{xy}$, such that the demand curves in both economies are the same?
The cumulative distribution function (CDF) describes the probability that a random variable X with a given probability distribution will be found to have a value less than or equal to x. That is: $$F_X(x) = P(X \leq x) $$

In this context, think of $X$ as the valuation of a particular unit demander. The demands are distributed with a certain probability distribution. So the the CDF tells you the probability of  a randomly drawn demander of having a valuation at least as large as $x$. If everyone demands either one unit if the price $p \leq X$ and 0 otherwise, then this also gives a demand function:

$$Q_d(p) = P(X \geq p) = 1 - P(X \leq p) $$

because at each price $p$ everyone with a valuation at least as high as $p$ will buy their one unit. But the fraction of people with a valuation at least as high as $p$ is one minus the fraction of people with valuations $p$ or lower. 

Next, consider a single household with a demand function that just so happens to look like $1 - P(X \leq p) $. This household will have an identical demand for the good at all prices as the collective demand of the unit demanders above. 

I believe this generalizes to the more complex case you present in the question but I haven't worked out the details.