2
$\begingroup$

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?

Improve this question
$\endgroup$
5
  • $\begingroup$ Not clear what the three values mean. Are you assuming separability of preferences? $\endgroup$ – FooBar Mar 7 '15 at 21:28
  • $\begingroup$ 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. $\endgroup$ – Erel Segal-Halevi Mar 8 '15 at 6:47
  • $\begingroup$ 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. $\endgroup$ – Erel Segal-Halevi Mar 8 '15 at 6:49
  • $\begingroup$ 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. $\endgroup$ – BKay Mar 8 '15 at 19:05
  • $\begingroup$ @BKay this sounds like the answer I am looking for... can you please elaborate? $\endgroup$ – Erel Segal-Halevi Mar 9 '15 at 9:19
1
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.