Hey there, Apple
Apple Inc. is a company that makes smartphones and a variety of other products.
Along with making smartphones, it makes the operating system for smartphones as well. Along with making the operating system for the smartphones it makes, Apple operates the App Store - a place where people who use Apple’s software on Apple’s phones can download apps made by other developers.
The Three Faces of Apple
People think of Apple as one company, but it is more useful to think of it as three companies with similar interests. The first company is Apple the Phone Maker. The second company is Apple the OS maker. The third company is Apple the App Store Owner.

Tim Cook - the CEO of Apple. (All three of them)
Apple the Phone Maker is in a commodity business. Pieces of hardware are not hard to come by. A large number of Chinese and Korean companies are at Apple’s throats making really cheap hardware that can be used as a phone. Samsung, Oppo, Vivo and Xiaomi compete with Apple in this and they don’t face major barriers to entry while doing it.
Apple the OS maker is in a duopoly. There are two major operating systems for smartphones - iOS and Android. iOS is owned (and used) exclusively by Apple, but Android is open source and used by almost anybody who wants to sell smartphones.
Apple the OS maker sells products that are complements to Apple the Phone Maker. If you buy a phone, you use the OS too. If the OS cost a lot of money, you wouldn’t buy the phone. So, Apple the Phone Maker and Apple the OS Maker decide to not charge for the OS separately. Whatever you pay for the iPhone ends up paying for the OS too.
So, according to the laws of technology, Apple should make the complement of the iPhone cheaper. And they do that. When you buy the iPhone you get iOS free. (Or rather you end up paying for both at the same time).
The incentives for Apple in this (both the Apples) are to help the consumer. It wants to price its phone optimally so that lots of people can buy it, and it wants to give the OS for no extra charge so that people buy its phone.
Apple the App Store Owner is different. To begin with, it doesn’t sell anything itself. The App Store is a marketplace. They connect sellers and buyers. If you want to download an app, you have to do it via the App Store. If a developer wants to sell an app, they have to do it via the app store. Its not possible to download iOS apps on iPhones without going through Apple.
Imagine there was a road between developers and the users of iPhones. the App Store is that road. Apple is the toll-booth owner. It decides who comes in, who goes out, and how much money they pay.
The App Store does do lots of useful things. It ensures that unscrupulous entities don’t sell to customers. If you had an app which for example tracked a user’s location without asking for their consent, Apple would not let it on the road (App Store) on the way to the customers.
But as history and economic theory suggests, when a company gets a monopoly position it begins to abuse it. Apple can charge 10% of revenues from developers if it wanted to. It could charge 20% if it wanted to, the developers have no choice if they want their app sold to iPhone users.
Apple does do that. It in fact does worse. It charges 30% of a company’s In App Purchases, requires that developers pay Apple $99 and prevents developers from suggesting that users buy from any other source other than In App Purchases.
That is damaging to the developer ecosystem, because it reduces the profits for developers with no benefit to them. Apple doesn’t make the 30% because the App Store is better than all the other competitors, it makes the 30% because it prohibits its users from getting apps from anywhere else - destroying the competition.
Hey There, Apple
All of Apple the App Store Owner’s problems came into public attention when a new email app - Hey launched its iOS app.

Hey’s app. Source: ZDNet
When Hey launched its app last week, they had a problem. Apple told them that they wouldn’t allow them to have a subscription unless they used Apple’s payment tools, and the 30% cut Apple takes.
Apple did initially accept the Hey app, but then decided that it would not accept a bug fix because it violated their rules.
This got Basecamp - the creator of Hey - angry and Basecamp CEO David Heinemeier Hansson showed it on Twitter.

After a few days of negotiations back and forth (and claims of unfairness because Apple does allow Spotify and Amazon to use their own payment tools, but not small companies like Basecamp), Apple did allow Hey’s bug fix on the App Store, and the issue was solved.
United States v. Microsoft Corp. 2.0?
The saga was enough for House Antitrust Comittee Chairman David Cicilline to call it “highway robbery”, and for several developers to contact tech blogger Ben Thompson to complain that their updates were not allowed in the App store.

This builds to a large extent a good case for antitrust for Apple under the Sherman Act. Apple owns a monopoly (Apple the App Store Owner) on the market for iPhone apps, and has abused their market power by “restraining trade”. There is an ongoing lawsuit in the State of California where several developers charged Apple under the Sherman Act. But what remains to be seen is how these claims will hold up in court.
Appendix
Does Apple even have a monopoly?
The best counterargument to my assertion that Apple does have a monopoly which has restrained consumer welfare is that Apple is not a monopoly to begin with. In the US it has 55% of the market, in Canada it has 50% of the market, in China it has only 15% of the market, and in India it has only 3% of the market.
That is true. But we have to remember that Apple as a phone company is actually three separate companies. Apple the Phone Maker is not a monopoly and Apple the OS Maker is also not a monopoly. But Apple the App Store Owner is a monopoly because its relevant market is people who have iPhones. And Apple the company ensures that the only place where consumers can buy apps for their iPhones is Apple’s App store.
And so, Apple does have a monopoly in selling to iPhone users, but not in the entire smartphone market.