<p>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: $$F_X(x) = P(X \leq x) $$</p>

<p>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:</p>

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

<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. </p>

<p>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. </p>

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

CC BY-SA 3.0
2015-03-09T12:46:29.140Z
2015-03-09T12:46:29.140Z
BKay
<p>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 $y$.</p>

<p>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 $y$.</p>

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

<p>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?</p>

CC BY-SA 3.0
2015-03-07T21:28:28.950Z
Not clear what the three values mean. Are you assuming separability of preferences?
FooBar
CC BY-SA 3.0
2015-03-08T06:47:11.707Z
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
CC BY-SA 3.0
2015-03-08T06:49:18.397Z
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
CC BY-SA 3.0
2015-03-08T19:05:27.333Z
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
CC BY-SA 3.0
2015-03-09T09:19:53.587Z
@BKay this sounds like the answer I am looking for... can you please elaborate?
Erel Segal-Halevi
<utility><supply-and-demand><consumer-theory>
From single-consumer-multi-units to multiple-consumers-single-unit
Erel Segal-Halevi
# From single-consumer-multi-units to multiple-consumers-single-unit

(Asked by: username_0 on 2015-03-07)

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?

**Tags:** utility, supply-and-demand, consumer-theory

> **Comments:**
> **username_1**: Not clear what the three values mean. Are you assuming separability of preferences?  
> **username_0**: 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.  
> **username_0**: 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.  
> **username_2**: 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.  
> **username_0**: @username_2 this sounds like the answer I am looking for... can you please elaborate?  

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## 1 answers

### Answer by username_2 on 2015-03-09. Score: 1 | Accepted answer

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.


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