A common sentiment that I encounter in the tech policy world is a visceral opposition to price discrimination. This is odd to me, because as an economist, I know that price discrimination often leads to more efficient outcomes. One particular element of this added efficiency is that when fixed costs are present, price discrimination allows products and services to be profitable that would not be profitable under standard pricing. This means that if we were to ban price discrimination, we would not get these products at all.

The tech world is filled with lots of smart people who understand math, so for this post, I am going to try to make the case with algebra and a wee bit of calculus. If you can follow along, great; if not, I’ll get you in some other post.

Let’s start with a basic, normalized linear demand function:

$$Q=1-P$$

$Q$ is quantity and $P$ is price. The results of this exercise will translate easily to *any* linear demand function, and they will apply broadly to all demand functions, so why not make it easy on ourselves?

Let’s assume that firms have a fixed cost $F$ and a marginal cost $C$. Firms' total costs are:

$$F+QC$$

Total revenue for the firm is just price times quantity, so it is equal to $QP$. If we are concerned that not even one firm might serve this market, then it is useful to look at the monopoly case, where market $P$ and $Q$ are equal to firm $P$ and $Q$. In this context, we can substitute $1-Q$ for $P$, and therefore, total revenue is equal to $Q(1-Q)$.

Total profit is simply total revenue minus total costs. Therefore:

$$\pi = Q(1-Q) - F - QC$$

What prices and quantities maximize profit? To calculate this, we can take a partial derivative of profit with respect to $Q$ and set it to zero. This condition will hold where profit is maximized.

$$\dfrac{\partial\pi}{\partial Q} = 0 = (1 - Q) - Q - C = 1 - 2Q - C$$

Solving for $Q$,

$$Q = \dfrac{(1-C)}{2}$$

Plugging this expression for $Q$ into the demand function lets us solve for $P$:

$$P = 1 - \dfrac{(1-C)}{2}$$

$$P = \dfrac{(1+C)}{2}$$ This is the profit-maximizing $P$ and $Q$ for a monopolist in this market in terms of $C$. Note that $F$ drops out. The profit-maximizing values don’t depend on $F$.

What *does* depend on $F$, however, is whether the firm is earning enough at these values of $P$ and $Q$ to stay in business. In particular, profit needs to be zero or positive for the firm not to shut down.

$$\pi = Q(1-Q) - F - QC \geq 0$$

Substituting $\frac{(1-C)}{2}$ for $Q$:

$$\frac{(1-C)}{2}(1-\frac{(1-C)}{2}) - F - \frac{(1-C)}{2}C \geq 0$$

Gathering terms:

$$\frac{(1-C)}{2}(1 - \frac{(1-C)}{2} - C) \geq F$$

Simplifying:

$$\dfrac{(1-C)}{2}\dfrac{(1-C)}{2} \geq F$$

$$\dfrac{(1-C)^2}{4} \geq F$$

So without price discrimination, this market will be served if and only if $F \leq \frac{(1-C)^2}{4}$. If we want to plug in some numbers, assume that $C=0$; in this case the market will be served only if $F \leq 0.25$.

Want to try it with price discrimination now?

With marginal cost equal to $C$, a monopolist would produce $1-C$ units. Assuming that the monopolist is able to charge each consumer the maximum they are willing to pay, then profit can be expressed like this:

$$\pi = \int^{1-C}_0 (1 - Q - C) dQ - F$$

Computing the integral:

$$\pi = [Q - \dfrac{Q^2}{2} - CQ]_0^{1-C} - F$$

This is equal to:

$$\pi = (1 - C) - \dfrac{(1 - C)^2}{2} - C(1 - C) - F$$

$$\pi = \dfrac{(1 - C)^2}{2} - F$$

Since profits must be non-negative for the firm to stay in business:

$$\dfrac{(1 - C)^2}{2} - F \geq 0$$ or

$$\dfrac{(1 - C)^2}{2} \geq F$$

So this market, with perfect price discrimination, will be served if $F \leq \frac{(1 - C)^2}{2}$. This means that the fixed cost can be twice as high (with linear demand) and the product or service will still be provided. If we want to plug in $C = 0$, then the market will be served as long as $F \leq 0.5$.

Why does this matter in the tech world? Because a lot of tech products and services have very high fixed costs. Building out wired and wireless broadband networks, for instance, is extremely costly. Marginal costs are often relatively low.

If we want to reap the benefits of new and innovative tech products, we must be prepared to accept price discrimination at least some of the time. There are products that are viable with price discrimination that are not viable without it—and if we ban price discrimination like some people thoughtlessly advocate, we won’t get them.