# 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?

• Not clear what the three values mean. Are you assuming separability of preferences? – FooBar Mar 7 '15 at 21:28
• No. I mean that, if the consumer has only $x$ then his utility is $u_x$, if the consumer has only $y$ then his utility is $u_y$, and if the consumer has both $x$ and $y$ then his utility is $u_{xy}$. This includes the family of separable preferences (if $u_{xy}=u_x+u_y$) but also the family of unit-demand preferences (if $u_{xy}=\max(u_x,u_y)$) and also other families of preferences. – Erel Segal-Halevi Mar 8 '15 at 6:47
• But I am assuming monotonicity, if this matters. I.e. the utility from having both $x$ and $y$ is at least the utility from having only one of them. – Erel Segal-Halevi Mar 8 '15 at 6:49
• Unit demand + a CDF of the distribution of reserve prices is analogous a traditional representative household demand function with a functional form of the CDF. – BKay Mar 8 '15 at 19:05
• @BKay this sounds like the answer I am looking for... can you please elaborate? – Erel Segal-Halevi Mar 9 '15 at 9:19

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.