# Price Discrimination Enables New Products and Services to Exist

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.

# Can We Develop Less Wasteful Price Discrimination Techniques?

Request: discuss any literature on and/or speculate wildly about ways to make price discrimination more efficient.

One component of pd is often arbitrary time wasting. Could that be equally effective but more socially useful? Instead of clipping coupons, complete some mechanical Turk task requested by a charity.

Or watch khan academy videos for coupons might effectively achieve the same useful discrimination without actually wasting time (might require some irrationality).

Basically, what’s the total friction cost of current pd schemes? Is that waste totally unavoidable?

On the literature in general, everything I know about price discrimination I learned from Alex Tabarrok (standard disclaimer!), so probably the best I can do is point you to his graduate IO syllabus, which contains a nice list of price discrimination articles.

Fmb is right that price discrimination often imposes costs. In addition to the costs that it imposes on consumers, sometimes producers bear extra costs to be able to price discriminate. The Cowen and Tabarrok principles text discusses the example of HP printers, which are built to force you to buy HP ink, facilitating a cheap-printer, expensive ink price discrimination strategy. The strategy works because HP has a patent on the printer head, and it builds the head into the ink cartridge rather than into the printer. It would be socially efficient for it to be built into the printer, but we instead dispose of perfectly good HP printer heads every time we swap out our ink cartridges.

How about if instead of the disposable printer head strategy, HP relied on cryptography in the printer and the cartridge to authenticate genuine HP ink and reject non-HP ink? It would be imperfect, and hackers would surely find a way to break the system, but it might work well enough. After all, this is how video game console makers prevent a lot (but not all) video game piracy. Sure, you can buy a chip to mod your XBox, but most people just pay the high price for games. Similarly, maybe most people would just suck it up and pay a high price for ink.

One way we could bear fewer costs as consumers is if we were willing to part with more privacy. Governments tax you on the basis of your income (sort of like price discrimination), which you report to them. Universities charge you on the basis of your and your parents incomes, which you report to them. These are pretty substantial invasions of privacy, yet we tolerate them, and many people even seem to think that they are “fair.” This model could easily be extended to other industries, such as groceries. When you apply for your discount card at the grocery store, simply bring last year’s tax return to become eligible for bigger discounts. No more clipping coupons! I’m sure this suggestion (which I don’t necessarily support!) will stir the ire of privacy advocates, who would be quick to point out that loss of privacy is a cost. So it’s really just a tradeoff of one kind of cost for another. However, to the extent that your tax return more directly correlates with what firms want to know in the first place, it could be a more efficient kind of price discrimination even if we value our privacy relatively highly.

More generally, there are lots of ways that we can trade off one kind of cost versus another to effectively signal that we deserve the lower price. For instance, to the extent that we signal with dull time wasting, we can switch to shorter, more unpleasant expenditures of time. Tim Lee complains about spending ten minutes on the phone with Comcast lying to them to get a better price on broadband service. Having to lie, he says, imposes a psychological cost. But it seems better to me than spending 30 minutes on the phone not lying. Tim concludes that “Comcast’s price discrimination strategy is only socially efficient if we assume the aggravation consumers feel from haggling with Comcast isn’t important,” but this is false! It can be, and probably is, socially efficient despite this aggravation. In addition, cable companies probably also rely on information other than willingness to have an unpleasant phone call: they know what neighborhood you live in, your credit score, your service history, and whether you have an HDTV. These do not impose very many costs.

Ideally, you would want pricing to be based on something that is costless for low-value types and so costly for high-value types that they do not even attempt to get the low price. If, in equilibrium, there is no posing as a low-value type, then this form of price discrimination imposes zero social costs. If we accept that in many cases low and high willingness-to-pay are simply manifestations of low and high income, and that say, extreme obesity, is negatively correlated with income, then we should observe some companies offering “fat discounts” to some customers. I suspect that this would get politicized and hashed out in court, if it is already not illegal. And there is obviously something unappealing about creating an incentive, however small, to be so fat.

Fmb makes a good point: it would be great if proof of low willingness-to-pay could be based on public goods production. For this to work, the public good would need to be something that can be produced effectively by low-value types and is preferable to just working, earning more money, and spending the extra money on the good. No American is going to collect litter for an hour to save \$2 on groceries. Unfortunately, I think this fact puts a ceiling on the gains that we can really expect from better price discrimination techniques. If low-value types were that much better at producing public goods than they were at producing market goods, they would probably be compensated for that value some other way already. If not, they are better off just producing market goods and spending their extra income at the higher price. That’s one reason that wasteful modes of price discrimination stick around.