## Demand curves are basically CDFs

As my loyal reader knows, I’ve been trying to learn some econ, as I find it quite fascinating and continuous with several other interests. Anyway, last night I was on the phone to Jared, and mentioned a connection I’d noticed between a basic concept in microeconomics and one in statistics. I thought the connection was obvious, but apparently he hadn’t thought of it before, and suggested I blog about it.

#### Frequency distribution of values

Suppose I’m selling a bike on TradeMe (non-NZ readers: substitute “EBay”). Imagine I could mind-read everyone in New Zealand about their subjective value (the maximum price they would be willing to pay) for the bike. Lots of people don’t want or need the bike (they have better uses for their money), so I’d get lots of low answers. And a few would like it quite a lot, and they’ll have high values. Suppose the result is

$\textnormal{(everyone's values)} = \boldsymbol{v} = \{v_1, v_2, ..., v_n\}$

where $n$ is the population size. A histogram might look like this:

We could make a continuous approximation to this frequency distribution, using a density function $f(x)$, whose integral is $n$:

$\int_0^{\infty} f(x) \, dx = n$

The corresponding cumulative distribution gives the number with value less than $x$:

$\textnormal{Num}(\textnormal{value} < x) = F(x) = \int_0^x f(x') \, dx'$

All good. That’s just a CDF, applied to a measure normalised to the population size $n$, rather than 1 (as in the case of a probability distribution).

#### Demand Curve

Now, a demand curve is a function $D_q(p)$ which gives the quantity demanded (the number of bikes people would want to buy) as a function of price. For example, if $D_q(500)$ were 20, that means 20 people would want to buy the bike if its price were $500. (I’m assuming no single person would want to buy two or more). $D_q(p)$ has a negative gradient. If I were to set the price at$100, how many people would want to buy the bike? That’s the same thing as the number of people for whom $v$ is greater than 100:

$D_q(100) = \int_{100}^{\infty} f(x) \, dx = n - F(100)$.

The same argument works for all prices, not just \$100:

$D_q(p) = \int_{p}^{\infty} f(x) \, dx = n - F(p)$.

That is, the quantity demanded as a function of price is just the “complementary CDF” of people’s values. Since economists are odd, when they plot $D_q(p)$, they put $p$ on the y-axis, and call the plot a ‘demand curve’. After playing around with this idea a bit, I think I proved that a constant-elasticity demand curve corresponds to a Pareto distribution of values.

I hope this was interesting and/or useful.